Home / Math and Physics Files / Math / Math Book Downloads

Communication Systems 4Ed Haykin 2001

PDF · 838 pages · 25.6 MB

A commercial textbook by Simon Haykin of McMaster University, not Phil's own work, kept among downloaded math books. The preface outlines chapters on random processes, CW modulation, pulse modulation, baseband and passband data transmission, signal-space analysis, spread spectrum, multiuser radio, information theory and error-control coding. It also includes MATLAB computer experiments.

AI-written summary; may contain errors.

Extracted text (machine-read; may contain errors)

COMMUNICRTION ~ SYSTEMS ---I f""'T'"1 :::::!i: ~ Ha~kin 4thEd4thEI1ition SimonHaijkin COMMUNICATION SYSTEMS 4THEDITION COMMUNICATION SYSTEMS 4THEDITION SinumHaykin McMaster University JOHNWILEY&SONS,INC. NewYorkili!Chichester IIIWeinheirn iiiBrisbane I!iSingapore I!iIToronto Editor Marketing Manager Associate Production Director SeniorProduction Editor CoverDesigner Illustration Coordinator Illustration StudioBillZobrist Katherine Hepburn LucilleBuonocore Monique Calello Madelyn Lesure GeneAiello Wellington Studios CoverPhotoNASAJPhoto Researchers, Inc. Thisbookwassetin10/12TimesRomanbyUG1GGSInformation Services,Inc.andprintedandbound byHamilton PrintingCompany. ThecoverwasprintedbyPhoenixColorCorporation. Thisbookisprintedonacid-ftee paper.8 Thepaperinthisbookwasmanufactured byamillwhoseforestmanagement programs includesustained yieldharvesting ofitstimberlands. Sustained yieldharvesting principles ensurethatthenumbers oftrees cuteachyeardoesnotexceedtheamountofnewgrowth. Copyright ©2001,JohnWiley&Sons,Inc.AllrightsReserved. Nopartofthispublication maybereproduced, storedinaretrieval systemortransmitted inanyformorbyanymeans,electronic, mechanical, photocopying, recording, scanning orotherwise, exceptaspermitted underSections107or1089ofthe1976UnitedStates Copyright Act,without eitherthepriorwrittenpermission ofthePublisher, or authorization throughpayment oftheappropriate per-copy feetotheCopyright Clearance Center,222Rosewood Drive,Danvers, MA01923,(508)750-8400, fax (508)750-4470. Requests tothePublisher forpermission shouldbeaddressed tothe Permissions Department, JohnWiley&Sons,Inc.,605ThirdAvenue,NewYork,NY 10158-0012, (212)850-6011, fax(212)850-6008, E-Mail:[email protected]. Toorderbooksorforcustomer servicecall1-800-CALL-WlLEY (225-5945). LibraryofCongress Cataloging-in-Publication Data Haykin,Simon Communication system'1SimonHaykin.-4th ed. p.em. ISBN0-471-17869-1 (cloth:alk.paper) 1.Telecommunication. 2.Signaltheory(Telecommunication) I.Title. TK5101 .H372000 621.382-dc21 PrintedintheUnitedStatesofAmerica 109 8 7 6 5 4 3 299-042977 InlovingmemoryofVera Electrical engineering education hasundergone someradicalchangesduringthepastcou pleofdecadesandcontinues todoso.Amodernundergraduate program inelectrical engineering includesthefollowing twointroductory courses: 1>0SignalsandSystems, whichprovides abalanced andintegrated treatment ofcontin uous-time anddiscrete-time formsofsignalsandsystems.TheFouriertransform (in itsdifferent forms),Laplacetransform, andz-transform aretreatedindetail.Typi cally,thecoursealsoincludes anelementary treatment ofcommunication systems. ..PrDbability andRandDm PrDcesses, whichdevelops anintuitivegraspofdiscreteand CDntinuous randomvariables andthenintroduces thenDtionofarandomprocess anditscharacteristics. Typically, thesetwointroductory coursesleadtoasenior-level CDurseoncommunicatiDn systems. Thefourtheditionofthisbookhasbeenwrittenwiththisbackground andprimary objectiv,e inmind.Simplyput,thebookprovides amoderntreatment Dfcommunication systemsatalevelsuitableforaone-DrtwD-semester seniorundergraduate course.The emphasis isonthestatistical underpinnings ofcommunication theorywithapplications. Thematerial ispresented inalogicalmanner, anditisillustrated withexamples, withtheoverallaimbeingthatofhelpingthestudentdevelopanintuitive graspofthe theoryunderdiscussion. ExceptfortheBackground andPreviewchapter, eachchapter endswithnumerous problems designed notonlytohelpthestudentstesttheirunderstand ingofthematerialcoveredinthechapterbutalsotochallenge themtoextendthismaterial. Everychapterincludesnotesandreferences thatprovidesuggestions forfurtherreading. Sectionsorsubsections thatcanbebypassed withoutlossofcontinuity areidentified with afootnote. Adistinctive featureofthebookistheinclusiDn ofeightcomputer experiments using MATLAB. Thissetofexperiments prpvides thebasisofa"Software Laboratory", with eachexperiment beingdesigned toex/;endthematerial coveredinthepertinent chapter. Mostimportant, theexperiments expld>ittheuniquecapabilities ofMATLAB inaninstruc tivemanner.TheMATLAB codesforalltheseexperiments areavailable ontheWileyWeb site:http://www.wiley.com!collegelhaykinl. TheBackground andPreviewchapterpresents introductory andmotivational ma terial,pavingthewayfordetailedtreatment DfthemanyfacetsDfcommunication systems inthesubsequent 10chapters. Thematerial inthesechapters isDrganized asfollows: ..Chapter 1develops adetailedtreatment DfrandDm, orstDchastic, processes, with particular emphasis ontheirpartialcharacterizatiDn (i.e.,second-order statistics). In effect,thediscussion isrestricted towide-sense stationary processes. Thecorrelation vii viiiPREFACE properties andpowerspectraofrandomprocesses aredescribed indetail.Gaussian processes andnarrowband noisefeatureprominently inthestudyofcommunication systems, hencetheirtreatment inthelatterpartofthechapter.Thistreatment nat urallyleadstotheconsideration oftheRayleigh andRiciandistributions thatarise inacommunications environment. i>"Chapter2presents anintegrated treatment ofcontinuous-wave (CW)modulation (i.e.,analogcommunications) andtheirdifferent types,asoutlined here: (i)Amplitude modulation, whichitselfcanassumeoneofthefollowing forms(de pendingonhowthespectralcharacteristics ofthemodulated wavearespecified): I>Fullamplitude modulation PDoublesideband-suppressed carriermodulation I>Quadrature amplitude modulation I>Singlesideband modulation I>Vestigial sideband modulation (ii)Anglemodulation, whichitselfcanassumeoneoftwointerrelated forms: '"Phasemodulation I>Frequency modulation Thetime-domain andspectralcharacteristics ofthesemodulated waves,methods for theirgeneration anddetection, andtheeffectsofchannelnoiseontheirperformances arediscussed. fl>Chapter 3coverspulsemodulation anddiscusses theprocesses ofsampling, quan tization, andcodingthatarefundamental tothedigitaltransmission ofanalogsig nals.Thischapter may beviewedasthetransition fromanalogtodigitalcommu nications. Specifically, thefollowing typesofpulsemodulation arediscussed: (i)Analogpulsemodulation, whereonlytimeisrepresented indiscreteform;it embodies thefollowing specialforms: ~Pulseamplitude modulation ...Pulsewidth(duration) modulation ~Pulepositionmodulation Thecharacteristics ofpulseamplitude modulation arediscussed indetail,asitis basictoallformsofpulsemodulation, betheyoftheanalogordigitaltype. (ii)Digitalpulsemodulation, inwhichbothtime and signalamplitude arerepre sentedindiscreteform;itembodies thefollowing specialforms: '"Pulse-code modulation i>Deltamodulation Differential pulse-code modulation Indeltamodulation, thesampling rateisincreased farinexcessofthatusedinpulse codemodulation soastosimplify implementation ofthesystem.Incontrast, in differential pulse-code modulation, thesampling rateisreducedthroughtheuseof apredictor thatexploitsthecorrelation properties oftheinformation-bearing signal. (iii)MPEG/audio codingstandard, whichincludesapsychoacoustic modelasakey elementinthedesignoftheencoder. Ii>Chapter4coversbaseband pulsetransmission, whichdealswiththetransmission of pulse-amplitude modulated signalsintheirbaseband form.Twoimportant issuesare discussed: theeffectsofchannelnoiseandlimitedchannelbandwidth ontheperfor manceofadigitalcommunication system.Assuming thatthechannelnoiseisadditive PREFACE ix andwhite,thiseffectisminimized byusingamatched filter,whichisbasictothe designofcommunication receivers. Asforlimitedchannelbandwidth, itmanifests itselfintheformofaphenomenon knownasintersymbol interference. Tocombat thedegrading effectsofthissignal-dependent interference, wemayuseeitherapulse shapingfilterorcorrelative encoder/decoder; bothoftheseapproaches arediscussed. Thechapterincludesadiscussion ofdigitalsubscriber linesfordirectcommunication between asubscriber andanInternetserviceprovider. Thisisfollowed byaderiva tionoftheoptimum linearreceiverforcombatting thecombined effectsofchannel noiseandintersymbol interference, which,inturn,leadstoanintroductory treatment ofadaptive equalization. I>Chapter5discusses signal-space analysisforanadditivewhiteGaussian noisechan nel.Inparticular, thefoundations forthegeometric representation ofsignalswith finiteenergyareestablished. Thecorrelation receiverisderived,anditsequivalence withthematched filterreceiverisdemonstrated. Thechapterfinisheswithadiscus sionoftheprobability oferroranditsapproximate calculation. I>Chapter6discusses passband datatransmission, whereasinusoidal carrierwaveis employed tofacilitate thetransmission ofthedigitallymodulated waveoveraband passchannel. Thischapterbuildsonthegeometric interpretation ofsignalspresented inChapterS.Inparticular, theeffectofchannelnoiseontheperformance ofdigital communication systemsisevaluated, usingthefollowing modulation techniques: (i)Phase-shift keying,whichisthedigitalcounterpart tophasemodulation with thephaseofthecarrierwavetakingononeofaprescribed setofdiscretevalues. (ii)Hybridamplitude/phase modulation schemes including quadrature-amplitude modulation (QAM),andcarrierless amplitude/phase modulation (CAP). (iii)Frequency-shift keying,whichisthedigitalcounterpart offrequency modulation withthefrequency ofthecarrierwavetakingononeofaprescribed setofdiscrete values. (iv)Genericmultichannel modulation, followed bydiscretemultitone, theuseof whichhasbeenstandardized inasymmetric digitalsubscriber lines. Inadigitalcommunication system,timingiseverything, whichmeansthatthere ceivermustbesynchronized tothetransmitter. Inthiscontext, wespeakofthe receiverbeingcoherent ornoncoherent. Inacoherent receiver, provisions aremade fortherecovery ofboththecarrierphaseandsymboltiming.Inanoncoherent receiverthecarrierphaseisignoredandprovision isonly'made forsymboltiming. Suchastrategyisdictatedbythefactthatthecarrierphasemayberandom, making phaserecovery acostlyproposition. Synchronization techniques arediscussed inthe latterpartofthechapter,withparticular emphasis ondiscrete-time signalprocessing. I:>Chapter7introduces spread-spectrum modulation. Unliketraditional formsofmod ulationdiscussed inearlierchapters, channelbandwidth ispurposely sacrificed in spread-spectrum modulation forthesakeofsecurityorprotection againstinterfering signals.Thedirect-sequence andfrequency-hop formsofspread-spectrum modula tionarediscussed. I>Chapter 8dealswithmultiuser radiocommunications, whereamultitude ofusers haveaccesstoacommon radiochannel. Thistypeofcommunication channeliswell represented insatelliteandwirelesscommunication systems, bothofwhicharedis cussed.Thechapterincludesapresentation oflinkbudgetanalysis, emphasizing the relatedantennaandpropagation concepts, andnoisecalculations. I»Chapter9develops thefundamental limitsininformation theory,whichareembod iedinShannon's theorems fordatacompaction, datacompression, anddatatrans- xPREFACE mission. Thesetheorems provideupperboundsontheperformance ofinformation sourcesandcommunication channels. Twoconcepts, basictoformulation ofthe theorems, are(1)theentropyofasource(whosedefinition isanalogous tothatof entropyinthermodynamics), and(2)channelcapacity. !l>Chapter10dealswitherror-control coding,whichencompasses techniques forthe encoding anddecoding ofdigitaldatastreamsfortheirreliabletransmission over noisychannels. Fourtypesoferror-control codingarediscussed: (i)Linearblockcodes,whicharecompletely described bysetsoflinearlyindepen dentcodewords,eachofwhichconsistsofmessagebitsandparity-check bits. Theparity-check bitsareincluded forthepurposeoferrorcontrol. (ii)Cycliccodes,whichformasubclassoflinearblockcodes. (iii)Convolutional codes,whichinvolveoperating onthemessagesequence contin uouslyinaserialmanner. (iv)Turbocodes,whichprovideanovelmethodofconstructing goodcodesthat approach Shannon's channelcapacityinaphysically realizable manner. Methods forthegeneration ofthesecodesandtheirdecoding arediscussed. Thebookalsoincludes supplementary material inthefonnofsixappendices as follows: i'>Appendix 1reviewsprobability theory. I>cAppendix 2,ontherepresentation ofsignalsandsystems,reviewstheFouriertrans formanditsproperties, thevariousdefinitions ofbandwidth, theHilberttransform, andthelow-pass equivalents ofnarrowband signalsandsystems. f»Appendix 3presentsanintroductorytreatment oftheBesselfunctionanditsmodified form.Besselfunctions ariseinthestudyoffrequency modulation, noncoherent de tectionofsignalsinnoise,andsymboltimingsynchronization. ~Appendix 4introduces theconiluent hypergeometric function, theneedforwhich arisesintheenvelope detection ofamplitude-modulated signalsinnoise. /»-Appendix 5provides anintroduction tocryptography, which isbasictosecure communications. \>-Appendix 6includes12usefultablesofvariouskinds. Asmentioned previously, theprimarypurposeofthisbookistoprovideamodern treatment ofcommunication systemssuitableforuseinaone-ortwo-semester under graduate courseattheseniorlevel.Themakecup ofthematerialforthecourseisnaturally determined bythebackground ofthestudents andtheinterestsoftheteachersinvolved. Thematerial coveredinthebookisbothbroadanddeepenoughtosatisfyavarietyof backgrounds andinterests, therebyproviding considerable flexibility inthechoiceof coursematerial. Asanaidtotheteacherofthecourse,adetailedsolutions manualforall theproblems inthebookisavailable fromthepublisher. IAcknowledgments Iwishtoexpressmydeepgratitude toDr.Gregory].Pottie(University ofCalifornia, Los Angeles), Dr.SantoshVenkatesh (University ofPennsylvania), Dr.StephenG.Wilson(Uni versityofVirginia), Dr.GordonStuber(Georgia InstituteofTechnology), Dr.Venugopal Veeraralli (CornellUniversity), andDr.Granville E.Ott(University ofTexasatAustin) PREFACE xi forcriticalreviewsofanearlierversionofthemanuscript andformakingnumerous sug gestionsthathavehelped m~_shape thebookintoitspresentform.Thetreatment ofthe effectofnoiseonenvelope detection presented inChapter2isbasedoncoursenotesmade available tomebyDr.SantoshVenkatesh, forwhichIamgrateful. IamgratefultoDr. GordonStiiberforgivingpermission toreproduce Figure6.32. Iamindebted toDr.MichaelMoher(Communications Research Centre,Ottawa) forreadingfivechapters ofanearlierversionofthemanuscript andformakingmany constructive comments onturbocodes.Iamequallyindebted toDr.Brendan Frey(Uni versityofWaterloo, Ontario) forhisinvaluable helpinrefiningthematerial onturbo codes,comments onlow-density parity-check codes,forproviding thesoftware toplot Fig.9.18,andgivingmethepermission toreproduce Figures10.27and10.33.Iamgrate fultoDr.DavidConn(McMaster University, Ontario) forhiscriticalreadingoftheBack groundandPreviewChapterandformakingsuggestions onhowtoimprove thepresen tationofthematerialtherein. IalsowishtothankDr.Jean-Jacque Werner(LucentTechnologies, Holmdel), Dr. JamesMazo(LucentTechnologies, MurrayHill),Dr.AndrewViterbi(Qualcom, SanDi ego),Dr.Radford Neal(University ofToronto,Ontario), Dr.Yitzhak(Irwin)Kaler(Tech nion,Israel),Dr.WalterChen(Motorola), Dr.JohnCioffi(Stanford University), Dr.Jon Mark(University ofWaterloo, Ontario), andDr.RobertDony(University ofGuelph, Ontario); Ithankthemallfortheirhelpfulcomments onselectedsectionsinthebook. Corrections andsuggestions forimprovements tothebookmadebyDr.DonaldWunsch II(University ofMissouri) arealsoappreciated. Iamgratefultomygraduate studentMathiniSellathurai (McMaster University) for performing thecomputer experiments inthebook,andHughPasika(McMaster Univer sity)formanyusefulcomments ontheBackground andPreviewChapterandfordoing thecomputations onsomegraphical plotsinthebook.Proofreading ofthepageproofs byMathiniSellathurai andAlpeshPatelismuchappreciated. Iamparticularly gratefultomyeditoratWiley,BillZobrist,forhisstrongsupport andhelpthroughout thewritingofthebook.Iamindebted toMonique Calello,Senior Production EditoratWiley,forhertirelesseffortinoverseeing theproduction ofthebook initsvariousstages.IthankKatherine Hepburn foradvertising andmarketing thebook. IthankKarenTongishforhercarefulcopyediting ofthemanuscript, KatrinaAveryfor hercarefulproofreading ofthepageproofs,andKristenMausforcomposing theindex ofthebook. Lastbutbynomeansleast,asalways,IamgratefultomyTechnical Coordinator, LolaBrooks,forhertirelesseffortintypingthemanuscript ofthebook.Ialsowishto recordmygratitude toBrigitteMaier,Assistant Librarian, andReginaBendig,Reference Librarian, atMcMaster University, forhelpingmeonnumerous occasions intracingref erencesforthebibliography. SimonHaykin Ancaster, Ontario January, 2000 IBACKGROUND ANDPREVIEW 1.TheCommunication Process 1 2.PrimaryCommunication Resources ~ 3.SourcesofInformation 3 4.Communication Networks 10 ~5.Communication Channels 15 6.Modulation Process 19 7.AnalogandDigitalTypesofCommunication 21 8.Shannon's Information Capacity Theorem 23 9.ADigitalCommunication Problem 24 10.Historical Notes26 NotesandReferences 291 ICHAPTER1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14Random Processes Introduction 31 Mathematical Definition ofaRandom Process 32 Stationary Processes 33 Mean,Correlation, andCovariance Functions 35 ErgodicProcesses 41 Transmission ofaRandom ProcessThrough aLinearTime-Invariant Filter PowerSpectralDensity 44 Gaussian Process 54 Noise58 Narrowband Noise64 Representation ofNarrowband NoiseinTermsofIn-phase andQuadrature Components 64 Representation ofNarrowband NoiseinTermsofEnvelope andPhase Components 67 SineWavePlusNarrowband Noise69 Computer Experiments: Flat-Fading Channel 7131 42 xiii xivCONTENTS 1.15Summary andDiscussion 75 NotesandReferences 77 Problems 78 ICHAPTER2Continuous-Wave Modulation 2.1Introduction 88 2.2Amplitude Modulation 90 2.3LinearModulation Schemes 93 2.4Frequency Translation 103 2.5Frequency-Division Multiplexing 105 2.6AngleModulation 107 2.7Frequency Modulation 109 2.8Nonlinear EffectsinFMSystems 126 2.9Superheterodyne Receiver 128 2.10NoiseinCWModulation Systems 130 2.11NoiseinLinearReceivers usingCoherent Detection 132 2.12NoiseinAMReceivers usingEnvelope Detection 135 2.13NoiseinFMReceivers 142 2.14Computer Experiments: Phase-locked Loop157 2.15Summary andDiscussion 162 NotesandReferences 165 Problems 16688 ICHAPTER3 3.1 3.2 3.3 3.4 3.53.6 3.7 3.8 3.93.103.11 3.123.13 3.14 3.15PulseModulation 183 Introduction 183 Sampling Process 184 Pulse-Amplitude Modulation 188 OtherFormsofPulseModulation 191 Bandwidth-Noise Trade-off 193 Quantization Process 193 Pulse-Code Modulation 201 NoiseConsiderations inPCMSystems 209 Time-Division Multiplexing 211 DigitalMultiplexers 214 VIrtues,Limitations, andModifications ofPCM217 DeltaModulation 218 LinearPrediction 223 Differential Pulse-Code Modulation 227 Adaptive Differential Pulse-Code Modulation 229 CONTENTS xv 3.16Computer Experiment: Adaptive DeltaModulation 232 3.17MPEGAudioCodingStandard 234 3.18Summary andDiscussion 236 NotesandReferences. 238 Problems 239 ICHAPTER4Baseband PulseTransmission 247 4.1 4.2 4.3 4.4 4.54.6 4.74.84.9 4.10 4.11 4.12 ICHAPTER5Introduction 247 Matched Filter248 ErrorRateDue.toNoise253 Intersymbol Interference 259 Nyquist's Criterion forDistortionless Baseband BinaryTransmission Correlative-Level Coding 267 Baseband M-aryPAMTransmission 275 e DigitalSubscriber Lines277 Optimum LinearReceiver 282 Adaptive Equalization 287 Computer Experiments: EyePatterns 293 Summary andDiscussion 296 NotesandReferences 297 Problems 300 Signal-Space A1Jalysis261 309 5.1Introduction 309 5.2Geometric Representation ofSignals 311 5.3Conversion oftheContinuous AWGNChannelintoaVectorChannel 318 5.4Likelihood Functions 322 5.5Coherent Detection ofSignalsinNoise:Maximum Likelihood Decoding 322 5.6Correlation Receiver 326 5.7Probability ofErrpr328 5.8Summary andDiscussion 337 NotesandReferences 337 Problems 338 ICHAPTER 6Passband DigitalTransmissiOn 344 6.1 6.2 6.3Introduction 344 Passband Transmission Model Coherent Phase-Shift Keying348 349 xviCONTENTS 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12. 6.136.14 6.15 6.16HybridAmplitude/Phase Modulation Schemes 368 Coherent Frequency-Shift Keying 380 Detection ofSignalswithUnknown Phase403 Noncoherent Orthogonal Modulation 407 Noncoherent BinaryFrequency-Shift Keying 413 Differential Phase-Shift Keying 414 Comparison ofDigitalModulation Schemes UsingaSingleCarrier Voiceband Modems 420 Multichannel Modulation 431 Discrete Multitone 440 Synchronization 448 Computer Experiments: CarrierRecovery andSymbolTiming 458 Summary andDiscussion 464 NotesandReferences 465 Problems 468417 ICHAPTER7Spread-Spectru1n Modulation 479 7.1Introduction 479 7.2Pseudo-Noise Sequences 480 7.3ANotionofSpreadSpectrum 488 7.4Direct-Sequence SpreadSpectrum withCoherent BinaryPhase-Shift Keying 490 7.5Signal-Space Dimensionality andProcessing Gain493 7.6Probability ofError497 7.7Frequency-Hop SpreadSpectrum 499 7.8Computer Experiments: Maximal-Length andGoldCodes 505 7.9Summary andDiscussion 508 NotesandReferences 509 Problems 509 ICHAPTER 8Multiuser RadioC01n1nunications 512 547 5508.1 8.2 8.38.48.58.6 8.7 8.88.9Introduction 512 Multiple-Access Techniques 513 SatelliteCommunications 514 RadioLinkAnalysis 517 Wireless Communications 529 Statistical Characterization ofMultipath Channels 535 BinarySignaling overaRayleigh FadingChannel 542 TDMAandCDMAWireless Communication Systems SourceCodingofSpeechforWireless Communications CONTENTS xvii 8.10Adaptive Antenna ArraysforWireless Communications 553 8.11Summary andDiscussion 559 NotesandReferences 560 Problems 562 !CHAPTER 9 9.1 9.2 9.3 904 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 ICHAPTER10Fundamental Li1ffitsinInfonnation Theory Introduction 567 Uncertainty, Information, andEntropy 568 Source-Coding Theorem 574 DataCompaction 575 Discrete Memoryless Channels 581 MutualInformation 584 Channel Capacity 587 Channel-Coding Theorem 589 Differential EntropyandMutualInformation forContinuous Ensembles Information Capacity Theorem 597 Implications oftheInformation Capacity Theorem 601 Information Capacity ofColoredNoiseChannel 607 RateDistortion Theory 611 DataCompression 614 Summary andDiscussion 616 NotesandReferences 617 Problems 618 Error-Control Coding567 593 626 10.1Introduction 626 10.2Discrete-Memoryless Channels 629 10.3LinearBlockCodes632 lOACyclicCodes641 10.5Convolutional Codes 654 10.6Maximum Likelihood Decoding ofConvolutional Codes660 10.7Trellis-Coded Modulation 668 10.8TurboCodes674 10.9Computer Experiment: TurboDecoding 682 10.10Low-Density Parity-Check Codes683 10.11Irregular Codes691 10.12Summary andDiscussion 693 NotesandReferences 694 Problems 696 xviii CONTENTS APPENDIX 1 APPENDIX 2 APPENDIX 3 APPENDIX 4 APPENDIX 5 APPENDIX 6 GLOSSARY BIBLIOGRAPHY INDEXProbability Theory 703 Representation ofSignalsandSystems 715 BesselFunctions 735 Confluent Hypergeometric Functions 740 Cryptography 742 Tables 761 771 777 792 BACKGROUND ANDPREVIEW Thebackground andpreview material presented hereinsetsthestageforastatistical treatment ofcommunication systemsinsubsequent chapters. Inparticular, wedescribe the following: ~Thecommunication process. ~Primarycommunication resources, namely,transmitted powerandchannelbandwidth. ~Sourcesofinformation. ~Thetwoprimarytypesofswitching: circuitswitching andpacketswitching. ~Communication channels forthetransportation ofinformation-bearing signalsfromthe transmitter tothereceiver. ~Themodulation process,whichisbasictocommunication systems. ~Analoganddigitaltypesofcommunication systems. ~Shannon's information capacitytheorem. ~Adigitalcommunications problem. Thechapterconcludes withsomehistorical notes,asasourceofmotivation forthe reader. ITheCommunication Process Today,communication entersourdailylivesinsomanydifferent waysthatitisveryeasy tooverlook themultitude ofitsfacets.Thetelephones atourhands,theradiosandtele visionsinourlivingrooms,thecomputer terminals withaccessteitheInternetinouroffices andhomes,andournewspapers areallcapableofproviding rapidcommunications from everycorneroftheglobe.Communication provides thesensesforshipsonthehighseas, aircraftinflight,androcketsandsatellites inspace.Communication throughawireless telephone keepsacardriverintouchwiththeofficeorhomemilesaway.Communication keepsaweatherforecaster informed ofconditions measured byamultitude ofsensors. Indeed,thelistofapplications involving theuseofcommunication inonewayoranother isalmostendless. 1 2 IlaBACKGROUND A<"IDPREVIEW Inthemostfundamental sense,communication involvesimplicitly thetransmission ofinformation fromonepointtoanotherthroughasuccession ofprocesses, asdescribed here: 1.Thegeneration ofamessagesignal:voice,music,picture,orcomputer data. 2.Thedescription ofthatmessagesignalwithacertainmeasure ofprecision, byaset ofsymbols: electrical, aural,orvisual. 3.Theencoding ofthesesymbols inaformthatissuitablefortransmission overa physicalmediumofinterest. 4.Thetransmission oftheencoded symbolstothedesireddestination. 5.Thedecoding andreproduction oftheoriginalsymbols. 6.There-creation oftheoriginalmessagesignal,withadefinable degradation inqual ity;thedegradation iscausedbyimperfections inthesystem. Thereare,ofcourse,manyotherformsofcommunication thatdonotdirectlyinvolve thehumanmindinrealtime.Forexample, incomputer communications involving com munication betweentwoormorecomputers, humandecisions mayenteronlyinsetting uptheprograms orcommands forthecomputer, orinmonitoring theresults. Irrespective oftheformofconununication processbeingconsidered, therearethree basicelements toeveryconununication system,namely,transmitter, channel,andreceiver, asdepicted inFigure1.Thetransmitter islocatedatonepointinspace,thereceiveris locatedatsomeotherpointseparatefromthetransmitter, andthechannelisthephysical mediumthatC01Ulects them.Thepurposeofthetransmitter istoconvertthemessagesignal produced bythesourceofinformation intoaformsuitablefortransmission overthe channel. However, asthetransmitted signalpropagates alongthechannel, itisdistorted duetochannelimperfections. Moreover, noiseandinterfering signals(originating from othersources) areaddedtothechanneloutput,withtheresultthatthereceivedsignalis acorrupted versionofthetransmitted signal.Thereceiverhasthetaskofoperating on thereceivedsignalsoastoreconstruct arecognizable formoftheoriginalmessagesignal forauser. Therearetwobasicmodesofcommunication: 1.Broadcasting, whichinvolvestheuseofasinglepowerful transmitter andnumerous receivers thatarerelatively inexpensive tobuild.Hereinformation-bearing signals flowonlyinonedirection. 2.Point-tocpoint communication, inwhichthecommunication processtakesplaceover alinkbetween asingletransmitter andareceiver. Inthiscase,thereisusuallya bidirectional flowofinformation-bearing signals,whichrequirestheuseofatrans mitterandreceiverateachendofthelink. Communication System FIGURE 1Elements ofacommunication system. So....cesofltiformation 3 Thebroadcasting modeofcommunication isexemplified byradioandtelevision, andthe ubiquitous telephone provides themeansforoneformofpoint-to-point communication. Another example ofpoint-to-point communication isthelinkbetween anEarthstation andarobotnavigating thesurfaceofadistantplanet. Allthesedifferent communication systemsaswellasothersnotmentioned hereshare acommon feature:Theunderlying communication processineachandeveryone ofthem isstatistical innature.Indeed,itisforthisimportant reasonthatmuchofthisbookis devotedtothestatistical underpinnings ofcommunication systems.Insodoing,wedevelop anexposition ofthefundamental issuesinvolved inthestudyofdifferent communication methodologies andtherebyprovideanaturalforumfortheircomparative evaluations. P,,"il1'Ui.rv Comntu.nication Resou.rces Inacommunication system,twoprimaryresources areemployed: transmitted powerand channelbandwidth. Thetransmitted poweristheaveragepowerofthetransmitted signal. Thechannelbandwidth isdefinedasthebandoffrequencies allocated forthetransmission ofthemessage signal.Ageneralsystemdesignobjective istousethesetworesources as efficiently aspossible. Inmostcommunication channels, oneresource maybeconsidered moreimportant thantheother.Wemaytherefore classifycommunication channels as powerlimitedorbandlimited.Forexample, thetelephone circuitisatypicalband-limited channel, whereas aspacecommunication linkorsatellitechannel istypically power limited. Whenthespectrum ofamessagesignal extendsdowntozeroorlowfrequencies, we definethebandwidth ofthesignalasthatupperfrequency abovewhichthespectralcontent ofthesignalisnegligible andtherefore unnecessary fortransmitting information. For example, theaveragevoicespectrum extendswelTbeyond10kHz,thoughmostofthe averagepowerisconcentrated intherangeof100to600Hz,andabandfrom300to 3100Hzgivesgoodarticulation. Accordingly, wefindthattelephone circuitsthatrespond welltothislatterrangeoffrequencies givequitesatisfactory commercial telephone service. Another important pointthatwehavetokeepinmindistheunavoidable presence ofnoiseinacommunication systeIIJ,.Noisereferstounwanted wavesthattendtodisturb thetransmission andprocessing ofmessage signalsinacommunication system.The sourcesofnoisemaybeinternalorexternaltothesystem. Aquantitative waytoaccountfortheeffectofnoiseistointroduce signal-to-noise ratio(SNR)asasystemparameter. Forexample, wemaydefinetheSNRatthereceiver inputastheratiooftheaveragesignalpowertotheaveragenoisepower,bothbeing measured atthesamepoint.Thecustomary practice istoexpresstheSNRindecibels (dBs),definedas10timesthelogarithm (tobase10)ofthepowerratio.Forexample, signal-to-noise ratiosof10,100,and1,000correspond to10,20,and30dBs,respectively. ISourcesofInforntation Thetelecommunications environment isdominated byfourimportant sourcesofinfor mation:speech,music,pictures, andcomputer data.Asourceofinformation maybe characterized intermsofthesignalthatcarriestheinformation. Asignalisdefinedasa single-valued function oftimethatplaystheroleoftheindependent variable; atevery instantoftime,thefunction hasauniquevalue.Thesignalcanbeone-dimensional, asin thecaseofspeech,music,orcomputer data;two-dimensional, asinthecaseofpictures; 4 I!JBACKGROUND ANDPREVIEW three-dimensional, asinthecaseofvideodata;andfour-dimensional, asinthecaseof volumedataovertime.Inthesequel,weelaborate ondifferent sourcesofinformation. (i)Speechistheprimary methodofhumancommunication. Specifically, thespeech communication processinvolves thetransfer ofinformation fromaspeakertoa listener,whichtakesplaceinthreesuccessive stages: '"Production. Anintended messageinthe speaker's mindisrepresented byaspeech signalthatconsistsofsounds(i.e.,pressure waves)generated insidethevocaltract andwhosearrangement isgoverned bytherulesoflanguage. It>Propagation. Thesoundwavespropagate throughtheairataspeedof300mis, reaching thelistener's ears. I>-Perception. Theincoming soundsaredeciphered bythelistenerintoareceived message, therebycompleting thechainofeventsthatculminate inthetransferof information fromthespeakertothelistener. Thespeech-production processmaybeviewedasaformoffiltering,inwhichasound sourceexcitesavocaltractfilter.Thevocaltractconsistsofatubeofnonuniform cross-sectional area,beginning attheglottis(i.e.,theopening between thevocal cords)andendingatthelip.Asthesoundpropagates alongthevocaltract,the spectrum (i.e.,frequency content) isshapedbythefrequency selectivity ofthevocal tract;thiseffectissomewhat similartotheresonance phenomenon observed inorgan pipes.Theimportant pointtonotehereisthatthepowerspectrum (i.e.,the distri butionoflong-term averagepowerversusfrequency) ofspeechapproaches zerofor zerofrequency andreachesapeakintheneighborhood ofafewhundred hertz.To putmattersintoproperperspective, however, wehavetokeepinmindthatthe hearingmechanism isverysensitive tofrequency. Moreover, thetypeofcommuni cationsystembeingconsidered hasanimportant bearingonthebandoffrequencies considered tobe"essential" forthecommunication process.Forexample, asmen tionedpreviously, abandwidth of300to3100Hzisconsidered adequate forcom mercialtelephonic communication. (tl)Thesecondsourceofinformation, music,originates frominstruments suchasthe piano,violin,andflute.Thenotemadebyamusicalinstrument maylastforashort timeintervalasinthepressing ofakeyonapiano,oritmaybesustained foralong timeintervalasintheexample ofafluteplayerholdingaprolonged note.Typically, musichastwostructures: amelodicstructure consisting ofatimesequence ofsounds, andaharmonic structure consisting ofasetofsimultaneous sounds.Likeaspeech signal,amusicalsignalisbipolar.However, amusicalsignaldiffersfromaspeech signalinthatitsspectrum occupies amuchwiderbandoffrequencies thatmayextend uptoabout15kHz.Accordingly, musicalsignalsdemandamuchwiderchannel bandwidth thanspeechsignalsfortheirtransmission. (iii)Thethirdsourceofinformation, pictures, reliesonthehumanvisualsystemforits perception. Thepicturecanbedynamic, asintelevision, orstatic,asinfacsimile. Takingthecaseoftelevision first,thepicturesinmotionareconverted intoelectrical signalstofacilitate theirtransport fromthetransmitter tothereceiver. Todoso, eachcomplete pictureissequentially scanned. Thescanning processiscarriedoutin aTVcamera.Inablack-and-white TV;thecameracontains opticsdesigned tofocus animageonaphotocathode consisting ofalargenumberofphotosensitive elements. Thechargepatternsogenerated onthephotosensitive surfaceisscannedbyanelec tronbeam,therebyproducing anoutputcurrentthatvariestemporally withtheway inwhichthebrightness oftheoriginalpicturevariesspatially fromonepointto another. Theresulting outputcurrentiscalledavideosignal.Thetypeofscanning SourcesofInformation 5 usedintelevision isaformofspatialsampling calledrasterscanning, whichconverts atwo-dimensional imageintensity intoaone-dimensional waveform; itissomewhat analogous tothemannerinwhichwereadaprintedpaperinthatthescanning is performed fromlefttorightonaline-by-line basis.InNorthAmerican analogtele vision,apictureisdividedinto525lines,whichconstitute aframe.Eachframeis decomposed intotwointerlaced fields,eachofwhichconsistsof262.5lines.For convenience ofpresentation, wewillrefertothetwofieldsasIandII.Thescanning procedure isillustrated inFigure2.ThelinesoffieldIaredepicted assolidlines,and thoseoffieldIIaredepicted asdashedlines.Thestartandendofeachfieldarealso included inthefigure.FieldIisscannedfirst.Thescanning spotoftheTVcamera moveswithconstant velocityacrosseachlineofthefieldfromlefttoright,andthe imageintensity atthecenterofthespotismeasured; thescanning spotitselfispartly responsible forlocalspatialaveraging oftheimage.Whentheendofaparticular lineisreached, thescanning spotquicklyfliesback(inahorizontal direction) tothe startofthenextlinedowninthefield.Thisflybackiscalledthehorizontal retrace. Thescanning processdescribed hereiscontinued untilthewholefieldhasbeenac countedfor.Whenthiscondition isreached, thescanning spotmovesquickly(ina verticaldirection) fromtheendoffieldItothestartoffieldII.Thissecondflyback iscalledtheverticalretrace.FieldIIistreatedinthesamefashionasfieldI.Thetime takenforeachfieldtobescannedis1/60s.Correspondingly, thetimetakenfora frameoracomplete picturetobescannedis1/30s.With525linesinaframe,the line-scanning frequency equals15.75kHz.Thus,byflashing30stillpicturesper secondonthedisplaytubeoftheTVreceiver, thehumaneyeperceives themtobe movingpictures. Thiseffectisduetoaphenomenon knownasthepersistence of vision.Duringthehorizontal- andvertical-retrace intervals, thepicturetubeismade inoperative bymeansofblanking pulsesthataregenerated atthetransmitter. More over,synchronization between thevariousscanning operations atbothtransmitter andreceiverisaccomplished bymeansofspecialpulsesthataretransmitted during theblanking periods;thus,thesynchronizing pulsesdonotshowonthereproduced picture.Thereproduction qualityofaTVpictureislimitedbytwobasicfactors: 1.Thenumberoflinesavailable inarasterscan,whichlimitsresolution ofthe pictureintheverticaldirection. 2.Thechannelbandwidth available fortransmitting thevideosignal,whichlimits resolution ofthepictureinthehorizontal direction. StartoffieldI /StartoffieldII --->------------~--------~ ----~-------------- --------------------- EndoffieldI? FIGURE 2Interlaced rasterscan. 6'"BACKGROUND A>"IDPREVIEW Foreachdirection, resolution isexpressed intermsofthemaximum numberoflines alternating betweenblackandwhitethatcanberesolved intheTVimagealongthe pertinent direction byahumanobserver. IntheNTSC(National Television System Committee) system,whichistheNorthAmerican standard, theparameter values usedresultinavideobandwidth of4.2MHz,whichextendsdowntozerofrequency. Thisbandwidth isordersofmagnitude largerthanthatofaspeechsignal.Notealso thatwhereasaspeechsignalisbipolar,avideo(television) signalisinherently positive (i.e.,unipolar). IncolorTV,theperception ofcolorisbasedonthethreetypesofcolorrecep tors(cones)inthehumaneye:red,green,andblue,whosewavelengths are570nm, 535nm,and445nm,respectively. Thesethreecolorsarereferredtoasprimary colorsbecauseanyothercolorfoundinnaturecanbeapproximated byanadditive mixtureofthem.Thisphysicalrealityisindeedthebasisforthetransmission ofcolor incommercial TVbroadcasting. Thethreeprimary colorsarerepresented bythe videosignalsmR(t),mdt),andmB(t),respectively. Toconserve bandwidth andpro duceapicturethatcanbeviewedonaconventional black-and-white (monochrome) television receiver, thetransmission ofthesethreeprimarycolorsisaccomplished by observing thattheycanbeuniquely represented byanythreesignalsthatareinde pendent linearcombinations ofmR(t),mG(t),andmB(t).Thethreesignalsareas follows: i>-Aluminance signal,mL(t),whichproduces ablack-and-white versionofthecolor picturewhenitisreceivedonaconventional monochrome television receiver. ~Apairofsignals,mr(t)andmQ(t),calledthechrominance signals,whichindicate thewaythecolorofthepicturedepartsfromshadesofgray. Theluminance signalmdt)isassigned theentire4.2MHzbandwidth. Owingto certainproperties ofhumanvision,testsshowthatifthenominalbandwidths ofthe chrominance signalsmr(t)andmQ(t)are1.6MHzand0.6MHz,respectively, sat isfactory colorreproduction ispossible. Turning nexttoafacsimile (fax)machine, thepurpose ofthismachine isto transmit stillpictures overacommunication channel(mostnotably, atelephone channel). Suchamachine provides ahighlypopularfacilityforthetransmission of handwritten orprintedtextfromonepointtoanother;transmitting textbyfacsimile istreatedsimplyliketransmitting apicture.Thebasicprinciple employed forsignal generation inafacsimile machine istoscananoriginaldocument (picture) anduse animagesensortoconvertthelighttoanelectrical signal. (iv)Finally,personal computers (PCs)havebecomeanintegralpartofourdailylives. Weusethemforelectronic mail,exchange ofsoftware, andsharingofresources. The texttransmitted byaPCisusuallyencoded usingtheAmerican Standard Codefor Information Interchange (ASCII), whichisthefirstcodedeveloped specifically for computer communications. Eachcharacter inASCIIisrepresented bysevendatabits constituting auniquebinarypatternmadeupofOsandis;bitisacronym forbinary digit.Thusatotalof27=128different characters canberepresented inASCII.The characters arevariouslowercase anduppercase letters,numbers, specialcontrolsym bols,andpunctuation symbols commonly usedsuchas@,$,and%.Someofthe special"control" symbols, suchasBS(backspace) andCR(carriage return),areused tocontroltheprintingofcharacters onapage.Othersymbols, suchasENQ(enquiry) andETB(endoftransmission block),areusedforcommunication purposes. (Acom pletelistingofASCIIcharacters isgiveninTableA6.1.)Thesevendatabitsare orderedstartingwiththemostsignificant bithdowntotheleastsignificant bitb}) SourcesofInformation 7 Idle High Low StartI bit:Databits IParityIStop :bit:bit FIGURE 3Thebitformatforsendingasynchronous serialdatausedintheRS-232 standard. asillustrated inFigure3.Attheendofthedatabits,anextrabitb8isappended for thepurposeoferrordetection. Thiserror-detection bitiscalledaparitybit.Ase quenceofeightbitsisreferredtoasabyte,oranoctet.Theparitybitissetinsuch awaythatthetotalnumberofisineachbyteisoddforoddparityandevenfor evenparity.Suppose, forexample, thecommunicators agreetouseevenparity;then theparitybitwillbea°whenthenumberofisinthedatabitisevenanda 1when itisodd.Hence,ifasinglebitinabyteisreceived inerrorandtherebyviolatesthe evenparityrule,itcanbedetected andthencorrected throughretransmission. Per sonalcomputers areoftenconnected viatheirRS(recommended standard)-232 ports. WhenASCIIdata(infact,allcharacter dataYare transmitted throughtheseports,a startbit,setto0,andoneormorestopbits,setto1,asshowninFigure3,areadded toprovidecharacter framing. Whenthetransmission isidle,alongseriesofisis sentsoastokeepthecircuitconnection alive.InFigure3, symbols°and1are designated as"low"and"high,"respectively. Theyarealsosometimes referredto as"space"and"mark," respectively; thelatterterminology comesfromthedaysof telegraphy. Thetextprepared onaPCisusuallystoredandthentransmitted overa communication channel(e.g.,atelephone channel) withasinglecharacter beingsent atatime.Thisformofdatatransmission iscalledasynchronous transmission, as opposed tosynchronous transmission inwhichawholesequence ofencoded char actersissentoverthechannelinonelongtransmission. Encoded characters produced byamixtureofasynchronous andsynchronous terminals arecombined bymeans ofdatamultiplexers. Themultiplexed streamofdatasoformedisthenappliedtoa devicecalledamodem(modulator-demodulator) forthepurpose oftransmission overthechannel. Insummary, computer-generated dataandtelevision signalsarebothwide bandsignals,inthattheirpowercontentoccupies awiderangeoffrequencies. An otherimportant characteristic ofdatacommunication betweenpersonal computers isburstiness, whichmeansthatinformation isusuallytransmitted fromoneterminal toanotherinburstswithsilentperiodsbetweenbursts.Indeed,datatrafficinvolving computers inoneformoranother tendstobeofaburstynature.Thisistobe contrasted withtrafficinadigitaltransmission network duetovoiceorinteractive video,which,relatively speaking, iscontinuous. Another wayinwhichweusethecomputer istodownload compressed forms oftext,audio,andvideodatafromaserviceprovider ataremotelocation. Data compression provides apractical meansfortheefficientstorageandtransmission of thesekindsofdata.Adatacompression systemconsistsofanencoderandadecoder, wherethecOrbpression ofanincoming datastreamanditsreconstruction areper formed,respectiyely. Basically, therearetwoformsofdatacompression: 1.Losslesscompression operates byremoving theredundant information contained inthedataofinterest.Thecompression issaidtobelosslessbecauseitiscom- 8'"BACKGROUND ANDPIIEVIEW pletelyreversible inthattheoriginaldatacanbereconstructed exactly.Lossless compression isalsoreferredtoasdatacompaction. 2.Lossycompression involves thelossofinformation inacontrolled manner; the compression maytherefore notbecompletely reversible. Lossycompression is, however, capableofachieving acompression ratiohigherthanthatattainable withlosslessmethods. Fordigitaltext,losslesscompression isrequired. Inthiscontext, wemention the Lempel-Ziv algorithm, whichisintrinsically adaptive andcapableofencoding groupsofsourcesymbols thatoccurfrequently. Itachieves acompression ofap proximately 55percentonordinary Englishtext,which,looselyspeaking, corre spondstothecompression thatwouldbeachieved byencoding pairsofletters.The Lempel-Ziv algorithm isaformofentropiccoding,orsourcecoding,whichisdis cussedinChapter9. Inmanyotherapplications, lossycompression isusuallythepreferred approach asitsusecansubstantially reducethedatasizewithout significantly alteringthe perceptual qualityofanimageoraudiosignal.Forsuchapplications, thisformof datacompression isacceptable, andinhigh-throughput data-transmission applica tionssuchastheInternet, itisanecessity. Butinsomeotherapplications suchasa clinicalsetting,thequalityofamedicalimage(e.g.,digitalx-rayradiograph) must notbedegraded onreconstruction. Fordigitalaudioandvideoapplications involving storageortransmission to beviableintoday'smarketplace, weneedstandard compression algorithms that enabletheinteroperability ofequipment produced bydifferent manufacturers. Inthis context,wemention threeprominent standard compression algorithms thatcaterto different needs: II>ThefPEGimagecodingstandard2isdesigned tocompress full-color orgrayscale imagesofnatural,real-world scenesbyexploiting knownlimitations ofthehuman visualsystem;JPEGstandsforJointPhotographic ExpertsGroup.Attheinputto theencoder,pictureelements, orpixels,aregrouped into8X8blocks,whichare appliedtoarelativeoftheFouriertransform knownasthediscretecosinetrans form(DCT)3.TheDCTdecomposes eachblockofpixelsintoasetof64coeffi cientsthatcloselysatisfytworelatedobjectives: 1.Thecoefficients shouldbeasuncorrelated aspossible. 2.Theenergyoftheinputsignalshouldbepackedintothesmallestnumberof coefficients possible. Thenextoperation intheencoder isthatofquantization, whereeachofthe64 DCTcoefficients isroundedoff.InJPEG,quantization isperformed inconjunction withaquantization tablesupplied bytheuserasaninputtotheencoder. Each elementofthetableisanintegerfrom1to255thatspecifies thestepsizeofthe DCTcoefficients, which,inturn,permitstherepresentation ofeachquantized DCTcoefficient byan8-bitcodeword.Basically, thepurposeofquantization is todiscardinformation thatisnotperceptually discernible. Quantization isamany to-onemapping andtherefore theprincipal sourceoflossiness intheencoder. The finaloperation intheencoder isthatofHuffman coding,whichisaformof entropic (source)codingalsodiscussed inChapter 9.Huffman codingachieves additional datacompression inalosslessmannerbyencoding thequantized DCT coefficients inaccordance withtheirstatistical characteristics. Atthedecoder, data reconstruction isperformed throughasequence ofoperations thataretheinverse SourcesofInformation 9 ofthoseintheencoder, namely,Huffman decoding, dequantization inaccordance withthequantization table,andfinallytheinverseDCT. TheMPEG-1Ivideo codingstandard4isdesigned primarily tocompress videosig nalsat30framespersecond(fps)intobitstreamsrunningattherateof1.5 megabits persecond(Mb/s);MPEGstandsforMotionPhotographic Experts Group.TheMPEG-l videocodingstandard achievesthisdesigngoalbyexploiting fourbasicformsofredundancy inherently presentinvideodata: 1.Interframe (temporal) redundancy. 2.Interpixel redundancy withinaframe. 3.Psychovisual redundancy. 4.Entropic codingredundancy. Itistheexploitation ofinterframe redundancy thatdistinguishes MPEG-l from JPEG.Inprinciple, neighboring framesintypicalvideosequences arehighlycor related.Themeaning ofthishighcorrelation isthat,inanaveragesense,avideo signaldoesnotchangerapidlyfromoneframetothenext,andasaresult,the difference between adjacent frameshasavariance (i.e.,averagepower)thatis muchsmaUerthanthevariance ofthevideosignalitself.Accordingly, theinter frameredundancy canbesignificantly reducedtoproduce amoreefficiently com pressedvideosignal.Thisreduction isachieved throughtheuseofprediction to estimateeachframefromitsneighbors; theresulting prediction erroristransmitted formotionestimation andcompensation. Theprediction isnonlinear byvirtueof thenatureoftheproblem. AswithJPEG,theinterpixel redundancy isreduced throughthecombined useoftheDCT,quantization, andlosslessentropiccoding. Thenetresultisthatfull-motion, videobecomes a1.5Mb/sstreamofcomputer datathatcanbestoredoncompact discsorintegrated withtextsandgraphics. Mostimportant, thefull-motion videoandassociated audiocanbedelivered over existingcomputer andtelecommunication networks, which,inturn,makesitpos sibletofulfilltheneedforvideo-on-demand ontheInternet. l>TheMPEG-1Iaudio codingstandards isbasedonperceptual coding,whichisa waveform-preserving process;thatis,theamplitude-time waveform ofthedecoded audiosignalcloselyapproximates thatoftheoriginalaudiosignal.Inbasicterms, theencoding processencompasses fourdistinctoperations: 1.Time-frequency mapping, whereby theinputaudiosignalisdecomposed into multiple subbands. 2.Psychoacoustic modeling, whichsimultaneously operates ontheinputaudio signaltocompute certainthresholds usingknownrulesfromthepsychoacous ticbehavior ofthehumanauditory system. 3.Quantization andcoding,which,inconjunction withthepsychoacoustic model,worksontheoutputofthetime-frequency mappersoastomaintain thenoiseresulting fromquantization processataninaudible level. 4.Frame-packing, whichisusedtoformatthequantized audiosamplesintoa decodable bitstream. Thepsychoacoustic modelbuildsonaperceptual phenomenon knownasauditory masking. Specifically, thehumaneardoesnotperceive quantization noiseina givenfrequency bandiftheaveragenoisepowerliesbelowthemasking threshold (i.e.,thethreshold ofjustnoticeable distortion). Foragivenfrequency bandof interest, themasking threshold varieswithfrequency acrossthatband.Themin- 10..BACKGROUND ANDPREVIEW imummaskingthreshold istheonethatisemployed inthepsychoacoustic model onaband-by-band basis.Forexample, thenetresultofusingtheMPEG-1 stan dardonthetwoaudiochannels ofastereoprogram isthateachdigitized audio signal,cominginattherateof768kilobitspersecond(kb/s),iscompressed toa rateaslowas16kb/s.(Theincoming datarateof768kb/scorresponds toa sampling rateof48kHz,witheachsamplebeingrepresented bya16-bitcode word.)ThustheMPEG-1/audio codingstandard issuitable forthestorageof audiosignalsininexpensive mediaortheirtransmission overch,annels withlimited bandwidth, whileatthesametimemaintaining perceptual quality. ICom:munication Networks6 Acommunication network (orsimplynetwork), illustrated inFigure4,consists ofan interconnection ofanumberofroutersmadeupofintelligent processors (e.g.,microproc essors).Theprimary purpose oftheseprocessors istoroutedatathrough thenetwork, hencethename.Eachrouterhasoneormorehostsattached toit;hostsaredevicesthat communicate withoneanother. Thenetwork isdesigned toserveasasharedresource for movingdataexchanged between hostsinanefficientmannerandtoprovideaframework tosupportnewapplications andservices. Thetelephone network isanexample ofacommunication network inwhichcircuit switching isusedtoprovideadedicated communication path,orcircuit,between two hosts.Thecircuitconsistsofaconnected sequence oflinksfromsourcetodestination. For example, thelinksmayconsistoftimeslotsforwhichacommon channelisavailable for accessbyamultitude ofusers.Thecircuit,onceinplace,remainsuninterrupted forthe duration oftransmission. Circuitswitching isusuallycontrolled byacentralized hierar chicalcontrolmechanism withknowledge ofthenetwork's organization. Toestablish a circuit-switched connection, anavailable paththrough thenetwork isseizedandthen dedicated totheexclusive useofthetwohostswishingtocommunicate. Inparticular, a call-request signalmustpropagate allthewaytothedestination andbeacknowledged beforetransmission canbegin.Then,thenetwork iseffectively transparent totheusers. Thismeansthatduringtheconnection time,thebandwidth andresources allocated tothe circuitareessentially "owned" bythetwohostsuntilthecircuitisdisconnected. Thecircuit Boundary ofsubnet FIGURE 4Communication network. Communication Networks 11 thusrepresents anefficientuseofresources onlytotheextentthattheallocated bandwidth isproperly used.Although thetelephone network isusedtotransmit data,voiceconstitutes thebulkofthenetwork's traffic.'Indeed, circuitswitching iswellsuitedtothetransmission ofvoicesignals,sincevoicegivesrisetoastreamtrafficandvoiceconversations tendto beoflongduration (about2minutesontheaverage) compared tothetimerequired for settingupthecircuit(about0.1to0.5seconds). Moreover, inmostvoiceconversations, thereisinformation flowforarelatively largepercentage oftheconnection time,which makescircuitswitching allthemoresuitableforvoiceconversations. Incircuitswitching, acommunication linkissharedbetween thedifferent sessions usingthatlinkonafixedallocation basis.Inpacketswitching, ontheotherhand,the sharingisdoneonademand basis,soithasanadvantage overcircuitswitching inthat whenalinkhastraffictosend,thelinkmaybemorefullyutilized. Thenetwork principle ofpacketswitching is"storeandforward." Specifically, ina packet-switched network, anymessage larger.thanaspecified sizeissubdivided priorto transmission intosegments notexceeding thespecified size.Thesegments arecommonly referredtoaspackets.Theoriginalmessage isreassembled atthedestination onapacket by-packet basis.Thenetwork maybeviewedasadistributed poolofnetwork resources (i.e.,channelbandwidth, buffers,andswitching processors) whosecapacity isshareddy namically byacommunity ofcompeting hostswishingtocommunicate. Incontrast, ina circuit-switched network, resources arededicated toapairofhostsfortheentireperiod theyareinsession.Accordingly, packetswitching isfarbettersuitedtoacomputer communication environment inwhichburstsofdataareexchanged between hostsonan occasional basis.TheuseofP<icketswitching, however, requiresthatcarefulcontrolbe exercised onuserdemands; otherwise, thenetwork maybeseriously abused. Thedesignofadatanetwork (i.e.,anetwork inwhichthehostsareallmadeupof computers andterminals) mayproceedinanorderlywaybylookingatthenetwork in termsofalayeredarchitecture, regarded asahierarchy ofnestedlayers.Layerreferstoa processordeviceinsideacomputer system,designed toperform aspecificfunction. Nat urally,thedesigners ofalayerwillbeintimately familiarwithitsinternaldetailsand operation. Atthesystemlevel,however, auserviewsthelayermerelyasa"blackbox" thatisdescribed intermsofinputs,outputs, andthefunctional relationbetween outputs andinputs.Inalayeredarchitecture, eachlayerregardsthenextlowerlayerasoneor moreblackboxeswithsomegivenfunctional specification tobeusedbythegivenhigher layer.Thus,thehighlycomplex communication problem indatanetworks isresolved as amanageable setofwell-defined interlocking functions. Itisthislineofreasoning thathas ledtothedevelopment oftheopensystemsinterconnection (OSIfreference modelbya subcommittee oftheInternational Organization forStandardization. Thetermopenrefers totheabilityofanytwosystemsconforming tothereference modelanditsassociated standards tointerconnect. IntheOSIreference model,thecommunications andrelated-connection functions areorganized asaseriesoflayers,orlevels,withwell-defined interfaces, andwitheach layerbuiltonitspredecessor. Inparticular, eachlayerperforms arelatedsubsetofprimitive functions, anditreliesonthenextlowerlayertoperform additional primitive functions. Moreover, eachlayerofferscertainservicestothenexthigherlayerandshieldsthelatter fromtheimplementation detailsofthoseservices. Between eachpairoflayers,thereisan interface. Itistheinterface thatdefinestheservicesofferedbythelowerlayertotheupper layer. TheOSImodeliscomposed ofsevenlayers,asillustrated inFigure5;thisfigurealso includes adescription ofthefunctions oftheindividual layersofthemodel.Layerkon systemA,say,communicates withlayerkonsomeothersystemBinaccordance witha ..... l'" Layer 6 5 4End-userX SystemALayer3 protocol PhysicallinkLayer7protocol Layer6protocol Layer5protocol Layer4protocol SubnetnodeLayer3 protocol PhysicallinkEnd-user Y Function Provision ofaccesstotheOSlenvironment forend-users. Transformation oftheinputdatatoprovideservicesselected bytheapplication layer;anexampleofdatatransformation isencryption toprovidesecurity. Provision ofthecontrolstructure forcommunication betweentwocooperating users,andtheorderlymanagement ofthedialogue betweenthem. End~to~end (i.e.,source-ta-destination) controlofthe messages exchanged betweenusers, Routingofpacketsthroughthenetworkandflawcontrol designedtoguarantee goodperformance overacommunication linkfoundbytheroutingprocedure. Errorcontrolforthereliabletransferofinformation acrossthechannel. Transmission ofrawbitsofdataoveraphysicalchannel; thislayerdealswiththemechanical, electrical, functional, andprocedural requirements toaccessthechannel. SystemB FIGURE 5OSImodel;theacronym DLCinthemiddleofthefigurestandsfordatalinkcontrol. Comm....icationNetworks 13 setofrulesandconventions, collectively constituting thelayerkprotocol, wherek=1, 2,...7.(Thetermprotocol hasbeenborrowed fromcommon usage,describing conven tionalsocialbehavior betweenhumanbeings.)Theentitiesthatcomprise thecorrespond inglayersondifferent systemsarereferred.toaspeerprocesses. Inotherwords,commu nication isachieved byhavingthepeerprocesses intwodifferent systemscommunicate viaaprotocol, withtheprotocol itselfbeingdefinedbyasetofrulesofprocedure. Physical communication between peerprocesses existsonlyatlayer1.Ontheotherhand,layers2 through 7areinvirtualcommunication withtheirdistantpeers.However, eachofthese sixlayerscanexchange dataandcontrolinformation withitsneighboring layers(below andabove)throughlayer-to"layer interfaces. InFigure5,physicalcommunication isshown bysolidlinesandvirtualcommunication bydashedlines. Ourprimaryinterestinthisbookisinthephysical layerofthe05Imodel. INTERNET Anydiscussion ofcomputer networks naturally leads'totheInternet. IntheInternetpar adigm,theunderlying network technology isdecoupled fromtheapplications athandby adopting an abstractdefinition ofnetwork service.Inmorespecificterms,wemaysaythe following: ,..Theapplications arecarriedoutindependently ofthetechnology employed tocon structthenetwork. ~Bythesametoken,thenetwork technology iscapableofevolving withoutaffecting theapplications. TheInternetarchitecture, depicted inFigure6,hasthreefunctional blocks:hosts, subnets, androuters.Thehostsconstitute nodesofthenetwork, wheredataoriginate or wheretheyaredelivered. Theroutersconstitute intermediate nodesthatareusedtocross subnetboundaries. Withinasubnet,allthehostsbelonging tothatsubnetexchange data directly;see,forexample, subnets1and3inFigure6. Likeothercomputer networks, theInternethasalayeredsetofprotocols. Inpartic ular,theexchange ofdatabetween thehostsandroutersisaccomplished bymeansofthe Internetprotocol (IP),asillustrated inFigure7.TheIPisauniversal protocol thatresides inthenetwork layer(i.e.,layer3ofthe05Ireference model).Itissimple,definingan addressing planwithabuilt-incapability totransport dataintheformofpacketsfrom nodetonode.Incrossing asubnetwork boundary, theroutersmakethedecisions asto howthepacketsaddressed foraspecified destination shouldberouted.Thisisdoneon thebasisofroutingtablesthataredeveloped through theuseofcustomprotocols for Hosts Hosts FIGURE 6Aninterconnected network ofsubnets. 14 1>1BACKGROUND &'IDPREVIEW AP TCP/UDP IPAP TCP/UDP IPAP TCP/UDP IPAP TCP/UDP IP AP,Application protocol UDP,Userdatagram protocol Tep:Transmission controlprotocol IP:Internetprotocol FIGURE 7Illustrating thenetwork architecture oftheInternet. exchanging pertinent information withotherrouters.Thenetresultofusingthelayered setofprotocols isthedeliveryofbesteffortservice.Thatis,theInternetofferstodeliver eachpacketofdata,buttherearenoguarantees onthetransittimeexperienced indelivery orevenwhetherthepacketswillbedelivered totheintended recipient. Ii!BROADBAND NETWORKS Withtheever-increasing demand fornewservices(e.g.,videoondemand, multimedia communications) andtheavailability ofkeyenabling technologies (e.g.,opticalfibers, digitalswitches), thetelephone network isevolving intoanall-purpose broadband network knownasthebroadband integrated servicesdigitalnetwork (B-ISDN). Theunderlying technology thatmakesB-ISDN possible isauser-network interface protocol calledthe asynchronous transfermode(ATM).ATMisahigh-bandwidth, low-delay, packet-like technique usedforswitching andmultiplexing; itisindependent ofthephysicalmeansof transport. Thelow-delay featureofthetechnique isneededtosupportreal-time services suchasvoice.Thehigh-bandwidth featureisrequiredtohandlevideoondemand. Simply put,ATMisbothatechnology thatishiddenfromtheusersandaconnection-oriented servicethatisvisibletotheusers. Asthenameimplies,ATMisnotsynchronous (i.e.,tiedtoamasterclock).Itallows forthetransport ofdigitalinformation intheformofsmall,fixed-size packetscalledcells. ThekeyfeatureofATMtonotehereisthattheconnection-oriented servicepreserves call sequencing, whichmeansthatnoreassembly ofcellsisneededpriortopresenting thetraffic streamtothedestination host.Thedeployment ofacell-switching technology inB-ISDN isagiganticbreakwiththetraditional useofcircuitswitching inthetelephone network. TheprimarypurposeofATMistoallocatenetwork resources (i.e.,bandwidth, buf fers,andprocessing horsepower) efficiently soastoguarantee theexpected qualityof service(QoS)foreachconnection. QoSismeasured intermsofthreeparameters: l>Celllossratio,definedastheratioofthenumberofcellslostintransport acrossthe network tothetotalnumberofcellspumped intothenetwork. I»Celldelay,definedasthetimetakenforacellofaparticular connection totransit acrossthenetwork. ~Celldelayvariation, definedasthedispersion orjitteraboutthemeancelldelay. QualityofserviceofferedinB-ISDNistobecontrasted withbesteffortserviceofferedby theInternet. Communication CJu.nnels 15 TABLE1Hierarchy ofSONET datarates Level" OC-1 OC-3 OC-9 OC-12 OC-18 OC-24OC-36 OC-48DataRate(Mb/s) 51.84 155.52 466.56 622.08 933.12 1,244.161,866.24 2,488.32 aoestandsforopticalcarrierlevel. Aftertheirgeneration, theATMcellsarestructured fortransport acrossthenetwork. TheceJlsinB-ISDNareplacedonanopticaltransmission systemcaJledthesynchronous opticalnetwork (SONET);8 opticalfibersarediscussed inthenextsection.SONET uses time-division multiplexing, whereby theentirebandwidth ofanopticalfiberisdevotedto different incoming datastreamsonatime-shared basis,hencetheneedforasynchronous operation. SONET iscontrolled byamasterclockwithanaccuracy ofabout1partin 109•ThusbitsofdataaresentonaSONETlineatextremely preciseintervals, controJled bythemasterclock.Nevertheless, SONET permitstheirregular timearrivalsofATM cells. ThebasicSONETframeisablockof810bytesputoutevery125/.LSforanoveraJi datarateof51.84Mb/s.Having8000frameseverysecondexactlymatches thesampling rateof8kHz,whichisthestandard sampling rateforthedigitaltransmission ofvoice signalsacrossthetelephone network. Thebasicdataratesof51.84Mb/saresynchronously byte-interleaved togenerate ahierarchy ofdatarates,assummarized inTable1. ICommunication Channels Thetransmission ofinformation acrossacommunication network isaccomplished inthe physical layerbymeansofacommunication channel. Depending onthemodeoftrans missionused,wemaydistinguish twobasicgroupsofcommunication channels: channels basedonguidedpropagation andthosebasedonfreepropagation. Thefirstgroupincludes telephone channels, coaxialcables,andopticalfibers.Thesecondgroupincludeswireless broadcast channels, mobileradiochannels, andsatellitechannels. Thesesixchannels are described inwhatfollows. (i)Asmentioned earlier,atypicaltelephone network usescircuitswitching toestablish anend-to-end communication linkonatemporary basis.Theprimary purposeof thenetwork istoensurethatthetelephone transmission between aspeakeratone endofthelinkandalistenerattheotherendisanacceptable substitute for face-to-face conversation. Inthisformofcommunication, themessage sourceisthe soundproduced bythespeaker's voice,andtheultimate destination isthelistener's ear.Thetelephone channel, however, supports onlythetransmission ofelectrical signals.Accordingly, appropriate transducers areusedatthetransmitting andre ceivingendsofthesystem.SpecificaJly, amicrophone isplacednearthespeaker's 16 "BACKGROUNDAND PREVIEW mouthtoconvertsoundwavesintoanelectrical signal,andtheelectrical signalis converted backintoacoustic formbymeansofamoving-coil receiverplacednear thelistener's ear.Present-day designsofthesetwotransducers havebeenperfected soastorespondwelltofrequencies rangingfrom20to8000Hz;moreover, apair ofthemcanbecompactly packaged insideasingletelephone setthatiseasytospeak intoorlistenfrom.Thetelephone channelisabandwidth-limited channel. There striction onbandwidth arisesfromtherequirement ofsharingthechannelamonga multitude ofusersatanyone time.Apractical solution tothetelephonic commu nication problem musttherefore minimize thechannelbandwidth requirement, sub jecttoasatisfactory transmission ofhumanvoice.Tomeetthisrequirement, the transducers andchannelspecifications mustconform tostandards basedonsubjec tiveteststhatareperformed ontheintelligibility, orarticulation, oftelephone signals byrepresentative maleandfemalespeakers. Aspeechsignal(maleorfemale)ises sentially limitedtoabandfrom300to3100Hzinthesensethatfrequencies outside thisbanddonotcontribute muchtoarticulation efficiency. Thisfrequency bandmay therefore beviewedasaroughguideline forthepassband ofatelephone channel thatprovides asatisfactory service,asillustrated inFigure8foratypicaltollcon nection.Figure8ashowstheinsertion lossofthechannelplottedversusfrequency; insertion loss(indB)isdefinedas1010glO(PoIPrJ,wherePListhepowerdelivered toaloadfromasourceviathechannelandPoisthepowerdelivered tothesame loadwhenitisconnected directlytothesource.Figure8bshowsthecorresponding plotoftheenvelope (group)delay(inmilliseconds) versusfrequency; envelope delay isdefinedasthenegative ofthederivative ofthephaseresponse withrespecttothe angularfrequency w211"[.TheplotsofFigure8clearlyillustrate thedispersive natureofthetelephone channel. Thetelephone channelisbuiltusingtwistedpairsforsignaltransmission. A twistedpairconsistsoftwosolidcopperconductors, eachofwhichisencasedina polyvinylchloride (PVC)sheath.Typically, eachpairhasatwistrateof2to12twists perfoot,andacharacteristic impedance of90to110ohms.Twistedpairsareusually madeupintocables,witheachcableconsisting ofmanypairsincloseproximity to 20~--_--_---~~-_--~ 4 (bi2 3 Frequency (kHz)4.---.------,------,-----;--,--,---- 4 2 Frequency (kHz) (a)15 FIGUIlE 8Characteristics ofl}picaltelephone connection: (a)Insertion loss.(b)Envelope delay. (Adapted fromBellamy, 1991.) Comm....kationCJumnels 17 eachother.Twisted pairsarenaturally susceptible toelectromagnetic interference (EMI),theeffectsofwhicharemitigated throughtwistingthewires. (ii)Acoaxialcableconsistsofaninnerconductor andanouterconductor, separated by adielectric insulating materiaLThe innerconductor ismadeofacopperwireencased insidethedielectric material. Asfortheouterconductor, itismadeofcopper,tinned copper,orcopper-coated steel.Typically, acoaxialcablehasacharacteristic imped anceof50or75ohms.Compared toatwisted-pair cable,acoaxialcableoffersa greaterdegreeofimmunity toEMI.Moreover, becauseoftheirmuchhigherband width,coaxialcablescansupportthetransmission ofdigitaldataatmuchhigherbit ratesthantwistedpairs.Ratesupto20Mb/sarefeasibleusingcoaxialcables,with 10Mb/sbeingthestandard. Whereas theuseofatwistedpairhasbeenconfined mainlytopoint-to-point service,acoaxialcablecanoperateasamultiple-access medium byusinghigh impedance taps.Acommon application ofcoaxialcablesisasthetransmission me diumforlocalareanetworks inanofficeenvironment. Another common application ofcoaxialcablesisincable-television systems, alsoknownascommunity-antenna television (CATV)systems. Inthisapplication coaxialcablesareusedtodistribute television, audio,anddatasignalsfromthehead endtothesubscribers, Theheadendisthecentraloriginating unitoftheCATV system,whereallsignals·arecarriedandprocessed. (iii)Anopticalfiberisadielectric waveguidethattransports lightsignalsfromoneplace toanotherjustasatwisted-wire pairoracoaxialcabletransports electrical signals. Itconsistsofacentralcorewithinwhichthepropagating electromagnetic fieldis confined andwhichissurrounded byacladding layer,whichisitselfsurrounded by athinprotective jacket.9Thecoreandcladding arebothmadeofpuresilicaglass, whereas thejacketismadeofplastic.Opticalfibershaveuniquecharacteristics that makethemhighlyattractive asatransmission medium. Inparticular, theyofferthe following uniquecharacteristics: i'-Enormous potential bandwidth, resulting fromtheuseofopticalcarrierfrequen ciesaround2X10'4Hz;withsuchahighcarrierfrequency andabandwidth roughlyequalto10percentofthecarrierfrequency, thetheoretical bandwidth of alightwave systemisaround2X1013Hz,whichisverylargeindeed. Lowtransmission losses,aslowasO.ldB/km. l>Immunity toelectromagnetic interference, whichisaninherent characteristic of anopticalfiberviewedasadielectric waveguide. ""Smallsizeandweight,characterized byadiameter nogreaterthanthatofahuman hair. i>-Ruggedness andflexibility, exemplified byveryhightensilestrengths andthepos- sibilityofbeingbentortwistedwithoutdamage. Last,butbynomeansleast,opticalfibersofferthepotential forlow-cost linecom munications sincetheyarefabricated fromsand,which,unlikethecopperusedin metallicconductors, isnotascarceresource. Theuniqueproperties ofopticalfibers havefuelledphenomenal advances inlightwave systemstechnology, whichhave,in turn,revolutionized long-distance communications andcontinue todoso. (iv)Wirelessbroadcast channels supportthetransmission ofradioandtelevision signals. Theinformation-bearing signal,representing speech,music,orpictures, ismodulated ontoacarrierfrequency thatidentifies thetransmitting station;modulation isde scribedinthenextsection.Thetransmission originates fromanantennathatactsas thetransition ormatching unitbetween thesourceofthemodulated signaland 18'"BACKGROUND ANDPREVIEW electromagnetic wavesinfreespace.Theobjective indesigning theantenna isto excitethewavesintherequired direction ordirections, asefficiently aspossible. Typically, thetransmitting antenna ismounted onatowertoprovideanunob structed viewofthesurrounding area,asfarafieldaspossible. Byvirtueofthe phenomenon ofdiffraction, whichisafundamental property ofwavemotion,radio wavesarebentaroundtheearth'ssurface.Propagation beyondthelineofsightis therebymadepossible, albeitwithsomewhat greaterlossthanisincurred infree space. Atthereceiving end,anantennaisusedtopickuptheradiated waves,estab lishingacommunication linktothetransmitter. Mostradioreceivers areofthe superheterodyne type.Thistechnique consistsofdown-converting thereceivedsignal tosomeconvenient frequency band,calledtheintermediate frequency (IF)band,and thenrecovering theoriginalinformation-bearing signalbymeansofanappropriate detector. (v)Amobileradiochannelextendsthecapability ofthepublictelecommunications net workbyintroducing mobility intothenetwork byvirtueofitsabilitytobroadcast. Thetermmobileradioisusuallymeanttoencompass terrestrial situations wherea radiotransmitter orreceiver iscapableofbeingmoved,regardless ofwhether it actually movesornot.Themajorpropagation effectsencountered intheuseofa mobileradioinbuilt-upareasareduetothefactthattheantennaofthemobileunit mayliewellbelowthesurrounding buildings. Simplyput,thereisno"Iine-of-sight" pathforcommunication; rather,radiopropagation takesplacemainlybywayof scattering fromthesurfacesofthesurrounding buildings andbydiffraction overand/ oraroundthem.Theendresultisthatenergyreachesthereceiving antennaviamore thanonepath.Inamobileradioenvironment, wethusspeakofamultipath phe nomenon inthatthevariousincoming radiowavesreachtheirdestination from different directions andwithdifferent timedelays.Indeed,theremaybeamultitude ofpropagation pathswithdifferent electrical lengths,andtheircontributions tothe receivedsignalcouldcombine inavarietyofways.Consequently, thereceivedsignal strength varieswithlocation inaverycomplicated fashion,andsoamobileradio channelmaybeviewedasalineartime-varying channelthatisstatistical innature. (vi)Finally,asatellitechanneladdsanotherinvaluable dimension tothepublictelecom munications network byproviding broad-area coverage inbothacontinental and anintercontinental sense.Moreover, accesstoremoteareasnotcoveredby conven tionalcableorfibercommunications isalsoadistinctfeatureofsatellites.Inalmost allsatellitecommunication systems, thesatellites areplacedingeostationary orbit. Fortheorbittobegeostationary, ithastosatisfytworequirements. First,theorbit isgeosynchronous, whichrequiresthesatellite tobeatanaltitudeof22,300miles; ageosynchronous satelliteorbitstheEarthin24hours(i.e.,thesatelliteissynchro nouswiththeEarth'srotation). Second,thesatelliteisplacedinorbitdirectlyabove theequatoronaneastward heading(i.e.,ithaszeroinclination). ViewedfromEarth, asatelliteingeostationary orbitappears tobestationary inthesky.Consequently, anEarthstationdoesnothavetotrackthesatellite; rather,itmerelyhastopointits antennaalongafixeddirection, pointing towardthesatellite.Bysodoing,thesystem designissimplified considerably. Communications satellites ingeostationary orbit offerthefollowing uniquesystemcapabilities: l>Broad-area coverage. I>Reliable transmission links. l>Widetransmission bandwidths. Modulation Process 19 Intermsofservices, satellites canprovidefixedpoint-to-point linksextending over longdistances andintoremoteareas,communication tomobileplatforms (e.g.,air craft,ships),orbroadcast capabilities. Indeed,communications satellitesplayakey roleinthenotionofthewholeworldbeingviewedasa"globalvillage." Inatypical satellitecommunication system,amessagesignalistransmitted fromanEarthstation viaanuplinktoasatellite, amplified inatransponder (i.e.,electronic circuitry) on boardthesatellite,andthenretransmitted fromthesatelliteviaadownlink toanother Earthstation.Withthe satellite positioned ingeostationary orbit,itisalwaysvisible toalltheEarthstationslocatedinsidethesatelliteantenna's coverage zonesonthe Earth'ssurface.Ineffect,thesatelliteactsasapowerful repeaterinthesky.Themost popularfrequency bandforsatellitecommunications is6GHzfortheuplinkand 4GHzforthedownlink. Theuseofthisfrequency bandoffersthefollowing attributes: s>Relatively inexpensive microwave equipment. Lowattenuation duetorainfall;rainfallisaprimaryatmospheric causeofsignal loss. l>-Insignificant skybackground noise;theskybackground noise(duetorandom noiseemissions fromgalactic, solar,andterrestrial sources)reachesitslowestlevel between 1and10GHz. Inthe6/4-GHz band,atypicalsatelliteisassigned a500MHzbandwidth thatis dividedamong12transponders onboardthesatellite. Eachtransponder, usingap proximately 36MHzofthe satellite bandwidth, corresponds toaspecificradiochan nel.Asingletransponder cancarryatleastonecolortelevision signal,1200voice circuits,ordigitaldataatarateof50Mbfs. Tosummarize, acommunication channeliscentraltotheoperation ofacom munication system.Itsproperties determine boththeinformation-carrying capacity ofthesystemandthequalityofserviceofferedbythesystem.Wemayclassifycom munication channels indifferent ways: Ii>Achannelmaybelinearornonlinear; awirelessradiochannelislinear,whereas asatellitechannelisusually(butnotalways)nonlinear. i>'Achannelmaybetimeinvariant ortimevarying;anopticalfiberistimeinvariant, whereas amobileradiochannelistypically timevarying. ..Achannelmaybebandwidth limitedorpowerlimited(i.e.,limitedintheavailable transmitted power);atelephone channelisbandwidth limited,whereas anoptical fiberlinkandasatellitechannelarebothpowerlimited. Nowthatwehavesomeunderstanding ofsourcesofinformation andcom munication channels, wemayreturntotheblockdiagram ofacommunication sys temshowninFigure1. IModulation Process Thepurposeofacommunication systemistodeliveramessagesignalfromaninformation sourceinrecognizable formtoauserdestination, withthesourceandtheuserbeing physically separated fromeachother.Todothis,thetransmitter modifies themessage signalintoaformsuitablefortransmission overthechannel. Thismodification isachieved bymeansofaprocessknownasmodulation, whichinvolves varyingsomeparameter of acarrierwaveinaccordance withthemessage signal.Thereceiverre-creates theoriginal 20 ,,;BACKGROUND Al'lDPREVIEW messagesignalfromadegraded versionofthetransmitted signalafterpropagation through thechannel. Thisre-creation isaccomplished byusingaprocessknownasdemodulation, whichisthereverseofthemodulation processusedinthetransmitter. However, owing totheunavoidable presence ofnoiseanddistortion inthereceivedsignal,wefindthatthe receivercannotre-create theoriginalmessage signalexactly.Theresulting degradation in overallsystemperformance isinfluenced bythetypeofmodulation schemeused.Specifi cally,wefindthatsomemodulation schemes arelesssensitive totheeffectsofnoiseand distortion thanothers. Wemayclassifythemodulation processintocontinuous-wave modulation andpulse modulation. Incontinuous-wave (CW)modulation, asinusoidal waveisusedasthecar rier.Whentheamplitude ofthecarrierisvariedinaccordance withthemessage signal, wehaveamplitude modulation (AM),andwhentheangleofthecarrierisvaried,wehave anglemodulation. ThelatterformofCWmodulation maybefurthersubdivided into frequency modulation (FM)andphasemodulation (PM),inwhichtheinstantaneous fre quencyandphaseofthecarrier,respectively, arevariedinaccordance withthemessage signal. Inpulsemodulation, ontheotherhand,thecarrierconsistsofaperiodic sequence ofrectangular pulses.Pulsemodulation canitselfbeofananalogordigitaltype.Inanalog pulsemodulation, theamplitude, duration, orposition ofapulseisvariedinaccordance withsamplevaluesofthemessage signal.Insuchacase,wespeakofpulse-amplitude modulation (PAM),pulse-duration modulation (PDM),andpulse-position modulation (PPM). Thestandard digitalformofpulsemodulation isknownaspulse-code modulation (PCM)thathasnoCWcounterpart. PCMstartsoutessentially asPAM,butwithan important modification: Theamplitude ofeachmodulated pulse(i.e.,sampleoftheoriginal message signal)isquantized orroundedofftothenearestvalueinaprescribed setof discreteamplitude levelsandthencodedintoacorresponding sequence ofbinarysymbols. Thebinarysymbolsaand1arethemselves represented bypulsesignalsthataresuitably shapedfortransmission overthechannel. Inanyevent,asaresultofthequantization process,someinformation isalwayslostandtheoriginalmessagesignalcannottherefore bereconstructed exactly.However, provided thatthenumberofquantizing (discrete am plitude)levelsislargeenough,thedistortion produced bythequantization processisnot discernible tothehumanearinthecaseofaspeechsignalorthehumaneyeinthecaseof atwo-dimensional image.Amongallthedifferent modulation schemes, pulse-code mod ulationhasemerged asthepreferred methodofmodulation forthetransmission ofanalog message signalsforthefollowing reasons: 1J>Robustness innoisyenvironments byregenerating thetransmitted signalatregular intervals. />Flexibleoperation. ""Integration ofdiversesourcesofinformation intoacommon format. />Securityofinformation initstransmission fromsourcetodestination. Inintroducing theideaofmodulation, westresseditsimportance asaprocessthat ensuresthetransmission ofamessage signaloveraprescribed channel. Thereisanother important benefit,namely,multiplexing, thatresultsfromtheuseofmodulation. Multi plexingistheprocessofcombining severalmessage signalsfortheirsimultaneous trans missionoverthesamechannel. Threecommonly usedmethods ofmultiplexing areas follows: ~Frequency-division multiplexing (FDM),inwhichCWmodulation isusedtotrans lateeachmessage signaltoresideinaspecificfrequency slotinsidethepassband of AnalogandDigitalTypesofCmmnunication 21 thech,annelbyassigning itadistinctcarrierfrequency; atthereceiver, abankof filtersisusedtoseparate thedifferent modulated signalsandpreparethemindivid uallyfordemodulation. l>Time-division multiplexing (TDM),inwhichpulsemodulation isusedtoposition samplesofthedifferent message signals innonoverlapping timeslots. '"Code-division multiplexing (CDM),inwhicheachmessagesignalisidentified bya distinctive code. InFDMthemessagesignals overlapwitheachotheratthechannelinput;hencethesystem maysufferfromcrosstalk (i.e.,interaction betweenmessagesignals)ifthechannelisnon linear.InTDMthemessage signalsusethefullpassband ofthechannel, butonatime sharedbasis.InCDMthemessage signalsarepermitted tooverlapinbothtimeand frequency acrossthechannel. Mention shouldalsobemadeofwavelength-division multiplexing (WDM), which isspecialtoopticalfibers.InWDM,wavelength isusedasanewdegreeoffreedom by concurrently operating distinctportions ofthewavelength spectrum (i.e.,distinctcolors) thatareaccessible withintheopticalfiber.However, recognizing thereciprocal relation shipthatexistsbetween thewavelength andfrequency ofanelectromagnetic wave,we maysaythatWDMisaformofFDM. IAnalogandDigitalTypesofCommunication Typically, inthedesignofacommunication systemtheinformation source,communica tionchannel, andinformation sink(enduser)areallspecified. Thechallenge istodesign thetransmitter andthereceiverwiththefollowing guidelines inmind: l>Encode/modulate themessagesignalgenerated bythesourceofinformation, transmit itoverthechannel,andproducean"estimate" ofitatthereceiveroutputthatsatisfies therequirements oftheenduser. ~Doallofthisatanaffordable cost. Wehavetheoptionofusingadigitaloranalogcommunication system. Consider firstthecaseofadigitalcommunication systemrepresented bytheblock diagramofFigure9,therationale forwhichisrootedininformation theory.Thefunctional blocksofthetransmitter andthereceiver, startingfromthefarendofthechannel, are pairedasfollows: l>Sourceencoder-decoder. I>Channelencoder-decoder. 1>0Modulator-demodulator. Thesourceencoderremoves redundant information fromthemessage signalandisre sponsible fortheefficientuseofthechannel. Theresulting sequence ofsymbols iscalled thesourcecodeword.Thedatastreamisprocessed nextbythechannelencoder, which produces anewsequence ofsymbolscalledthechannelcodeword.Thechannelcodeword islongerthanthesourcecodewordbyvirtueofthecontrolled redundancy builtintoits construction. Finally,themodulator represents eachsymbolofthechannelcodewordby acorresponding analogsymbol,appropriately selectedfromafinitesetofpossibleanalog symbols. Thesequence ofanalogsymbolsproduced bythemodulator iscalledawaveform, whichissuitablefortransmission overthechannel. Atthereceiver, thechanneloutput (received signal)isprocessed inreverseordertothatinthetransmitter, therebyrecon- 22 BACKGROUND &"IDPREVIEW Transmitterr------ -----j I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Il J WaveformReceived signal FIGURE 9Blockdiagramofdigitalcommunication system. structing arecognizable versionoftheoriginalmessagesignal.Thereconstructed message signalisfinallydelivered totheuserofinformation atthedestination. Fromthisdescription itisapparent thatthedesignofadigitalcommunication systemisrathercomplex in conceptual termsbuteasytobuild.Moreover, thesystemisrobust,offeringgreatertol eranceofphysical effects(e.g.,temperature variations, aging,mechanical vibrations) than itsanalogcounterpart. Incontrast, thedesignofananalogcommunication systemissimpleinconceptual termsbutdifficulttobuildbecauseofstringent requirements onlinearity andsystemad justment. Forexample, voicecommunication requires nonlinear distortion products at least40dBbelowthewantedmessage signal.Insignal-processing terms,thetransmitter consistsofamodulator andthereceiverconsistsofademodulator, thedetailsofwhich aredetermined bythetypeofCWmodulation used. Theconceptual simplicity ofanalogcommunications isduetothefactthatanalog modulati2-n_techniques, exemplified bytheirwideuseinradioandtelevision, makerela tivelysuperficial changestothemessagesignalinordertoprepareitfortransmission over thechannel. Morespecifically, thereisnosignificant effortmadebythesystemdesigner totailorthewaveform ofthetransmitted signaltosuitthechannelatanydeeperlevel. Ontheotherhand,digitalcommunication theoryendeavors tofindafinitesetofwave formsthatarecloselymatched tothecharacteristics ofthechannelandwhicharetherefore moretolerantofchannelimpairments. Insodoing,reliablecommunication isestablished overthechannel. Intheselection ofgoodwaveforms fordigitalcommunication overa noisychannel, thedesignisinfluenced solelybythechannelcharacteristics. However, once theappropriate setofwaveforms fortransmission overthechannelhasbeenselected, the sourceinformation canbeencoded intothechannelwaveforms, andtheefficienttrans- Shannon's Information Capacity Theorem 23 missionofinformation fromthesourcetotheuseristherebyensured. Insummary, the useofdigitalcommunications provides thecapability forinformation transmission thatis bothefficientandreliable. Fromthisdiscussion, itisapparent thattheuseofdigitalcommunications requires aconsiderable amountofelectroniC circuitry, butnowadays electronics areinexpensive, duetotheever-increasing availability ofvery-large-scale integrated (VLSI)circuitsinthe fOrInofsiliconchips.Thusalthough cost considerations usedtobeafactorinselecting analogcommunications overdigitalcommunications inthepast,thatisnolongerthecase. Despitethetrendtowardtheever-increasing useofdigitalcommunications, astrong casecanbemadeforthestudyofanalogcommunications fortwoimportant reasons: 1.Aslongaswehearandseeanalogcommunications aroundusviaradioandtelevi sion,weneedtounderstand howthesecommunications systemswork.Moreover, thestudyofanalogmodulation motivates otherdigitalmodulation schemes. 2.Analogdevicesandcircuitshaveanaturalaffinityforoperating atveryhighspeeds andtheyconsume verylittlepowercompared totheirdigitalcounterparts. Accord ingly,theimplementation ofveryhigh-speed orverylow-power communication sys temsdictatestheuseofananalogapproach. IShannon's Information Capacity Theorem Thegoalofacommunication systemdesigner istoconfigure asystemthattransports a message signalfromasourceofinterestacrossanoisychanneltoauserattheotherend ofthechannelwiththefollowing objective: Themessagesignalisdelivered totheuserbothefficiently andreliably,subjectto certaindesignconstraints: allowable transmitpower,available channelbandwidth, andaffordable costofbuildingthesystem. Inthecaseofadigitalcommunication system,reliability iscommonly expressed interms ofbiterrorrate(BER)orprobability ofbiterrormeasured atthereceiveroutput.Clearly, thesmallertheBER,themorereliablethecommunication systemis.Aquestion thatcomes tomindinthiscontextiswhether itispossible todesignacommunication systemthat operates withzeroBEReventhroughthechannelisnoisy.Inanidealsetting,theanswer tothisquestion isanemphatic yes.Theanswerisembodied inoneofShannon's celebrated theorems,"o whichiscalledtheinformation capacity theorem. LetBdenotethechannelbandwidth, andletSNRdenotethereceivedsignal-to-noise ratio.Theinformation capacitytheoremstatesthatideallythesetwoparameters arerelated as C=Blog2(1+SNR)b/s (1) whereCistheinformation capacity ofthechannel. Theinformation capacity isdefined asthemaximum rateatwhichinformation canbetransmitted acrossthechannelwithout error;itismeasured inbitspersecond(b/s).Foraprescribed channelbandwidth Band received SNR,theinformation capacity theorem tellsusthatamessage signalcanbe transmitted through thesystemwithouterrorevenwhenthechannelisnoisy,provided thattheactualsignaling rateRinbitspersecond,atwhichdataaretransmitted through thechannel, islessthantheinformation capacity C. 24 I!IBACKGROUND ANDPREVIEW Unfortunately, Shannon's information capacity theorem doesnottellushowtode signthesystem.Nevertheless, fromadesignpointofview,thetheorem isveryvaluable forthefollowing reasons: 1.Theinformation capacity theorem provides abaundonwhatrateofdatatransmis sionistheoretically attainable forprescribed valuesofchannelbandwidth Band received SNR.Onthisbasis,wemayusetheratio RTJ=-C asameasure oftheefficiency ofthedigitalcommunication systemunderstudy.The closer TJistounity,themoreefficientthesystemis. 2.Equation (1)provides abasisforthetrade-off between channelbandwidth Band received SNR.Inparticular, foraprescribed signaling rateR,wemayreducethe required SNRbyincreasing thechannelbandwidth B,hencethemotivation forusing awidebandmodulated scheme(e.g.,pulse-code modulation) forimproved noise performance. 3.Equation (1)provides anidealized framework forcomparing thenoiseperformance ofonemodulation schemeagainstanother. Communication Problem Whenwespeakofadigitalcommunication systemhavingalowbiterrorrate,say,the implication isthatonlyasmallfractioninalongstreamofbinarysymbols isdecoded in errorbythereceiver. Theissueofthereceiverdetermining whether abinarysymbolsent overanoisychannelisdecodedinerrorornotisoffundamental importance tothedesign ofdigitalcommunication systems.Itistherefore appropriate brieflytodiscussthisbasic issuesoastomotivate thestudyofcommunication systems. Suppose wehavearandombinarysignal,m(t),consisting ofsymbols 1and0that areequallylikely.Symbol1isrepresented byaconstant level+1,andsymbol0isrep resented byaconstant level-1,eachofwhichlastsforaduration T.Suchasignalmay represent theoutputofadigitalcomputer orthedigitized versionofaspeechsignal.To facilitate thetransmission ofthissignaloveracommunication channel, weemployasimple modulation schemeknownasphase-shift keying.Specifically, theinformation bearing signalm(t)ismultiplied byasinusoidal carrierwaveAccos(2'1Tfct), whereAcisthecarrier amplitude, fcisthecarrierfrequency, andtistime.FigurelOashowsablockdiagram of thetransmitter, theoutputofwhichisdefinedby forsymbol1 forsymbol0(2) where0:$t:$T.Thecarrierfrequency fcisamultiple oflIT. Thechannelisassumed tobedistortionless butnoisy,asdepicted inFigurelOb.The received signalx(t)isthusdefinedby x(t)=s(t)+w(t) (3) wherew(t)istheadditivechannelnoise. Thereceiverconsistsofacorrelator followed byadecision-making device,asde pictedinFigure10c.Thecorrelatar multiplies thereceivedsignalx(t) byalocallygenerated MessagE!~ Transmitted signalmet)t signalset) Carrierwave Accos(2'1rfct) (a) CorrelatorADig'talComm..ffwatlon Problem 25 Transmitted~ Channeloutput sigpalset)t+ (received signal)xlt) Noise wet) (b) Received signalxlt)Say1ifYr>0 Otherwise, say0 (c) FIGURE10Elements ofadigitalcommunication system.(ajBlockdiagram oftransmitter. (b)Blockdiagramofchannel. (c)Blockdiagramofreceiver. carrierCOS(27Tlct) andthenintegrates theproductoverthesymbolinterval0:5t:5T, producing theoutput Yrfx(t)COS(27Tfct) dt (4) Substituting Equations (2)and(3)into(4)andinvoking theassumption thatthecarrier frequencyIeisamultiple ofliT,weobtain(afterthesimplification ofterms) forsymbol1 (5) forsymbol0 whereWristhecontribution ofthecorrelator outputduetothechannelnoisew(t).To reconstruct theoriginalbinarysignalm(t),thecorrelator outputYriscompared againsta threshold ofzerovoltsbythedecision-making device,theoperation ofwhichisbasedon thefollowing rule: IfthecorreiatoroutputYTisgreaterthanzero,thereceiveroutputssymbol1; otherwise, itoutputssymbolO. Withthisbackground, wemaynowdiscusslraise somebasicissues.First,fromFou rieranalysiswefindthatthetime-bandwidth productofapulsesignalisconstant. This meansthatthebandwidth ofarectangular pulseofduration Tisinversely proportional toT.Thetransmitted signalinFigurelOaconsistsoftheproductofthisrectangular signal andthesinusoidal carrierAcCOS(27Tlct). Themultiplication ofasignalbyasinusoid has theeffectofshiftingtheFouriertransform ofthesignaltotherightbyIeandtotheleft byanequalamount, exceptforthescalingfactorof1/2.Itfollowstherefore thatthe bandwidth ofthetransmitted signalm(t),andtherefore therequired channelbandwidth, isinversely proportional tothereciprocal ofthesymbolduration T.Fortheproblem at hand,thereciprocal ofTisalsothesignaling rateofthesysteminb/s. 26..BACKGROUND ANDPREVIEW Thereare,however, someotherissuesthatrequiretheoretical considerations: 1.Whatisthejustification forthereceiverstructure ofFigure1Dc? 2.Thenoisecontribution WTisthevalueofarandomvariableWproduced bysampling acertainrealization w(t)ofthechannelnoiseattimet=Tinaccordance with Equations (3)and(4).Howdowerelatethestatistics oftherandomvariableWto thestatistical characteristics ofthechannelnoise? 3.ThereceiverofFigure10cmakesoccasional errorsduetotherandomnatureofthe correIatoroutput.Thatis,thereceiverdecidesinfavorofsymbol0giventhatsymbol 1wasactuallytransmitted, andviceversa.Whatistheprobability ofdecisionerrors? Moreover, therearesomeimportant practical issuesthatneedattention: 1.Channelbandwidth isahighlyvaluable resource. Howdowechooseamodulation schemethatconserves bandwidth inacost-effective manner? 2.Thebinarysignalm(t)mayincluderedundant symbolsintroduced intoitthrough theuseofchannelencoding soastoprovideprotection againstchannelnoise.How dowedesignthechannelencoderinthetransmitter andthechanneldecoderinthe receiversoastocomeveryclosetoShannon's information capacity theorem ina physically realizable manner? 3.Thelocallygenerated carrierinthereceiverofFigure10cisphysically separatefrom thecarriersourceusedformodulation inthetransmitter. Howdowesynchronize thereceivertothetransmitter withrespecttoboththecarrierphaseandsymbol timingsoastojustifytheuseofEquation (4)asthebasisofdecision-making inthe reconstruction oftheoriginalbinarysignal? Thetheoretical andpractical issuesraisedhereinthecontextofthesimpledigital communication systemofFigure10areaddressed inthefollowing chapters ofthebook. IHistorical Notes!! Apreviewofcommunications wouldbeincomplete withoutahistory of thesubject.In thisfinalsectionofthisintroductory chapterwepresentsomehistorical notesoncom munications; eachparagraph focusesonsomeimportant andrelatedevents.Itishoped thatthismaterialwillprovideasenseofinspiration andmotivation forthereader. In1837,thetelegraph wasperfected bySamuelMorse,apainter.Withthewords "WhathathGodwrought," transmitted byMorse'selectrictelegraph betweenWashing ton,D.C.,andBaltimore, Maryland, in1844,acompletely revolutionary meansofreal time,long-distance communications wastriggered. Thetelegraph istheforerunner ofdig italcommunications inthattheMorsecodeisavariable-length ternarycodeusingan alphabet offoursymbols: adot,adash,aletterspace,andawordspace;shortsequences represent frequent letters,whereaslongsequences represent infrequent letters.Thistype ofsignaling isidealformanualkeying.Subsequently, EmileBaudotdeveloped afixed lengthbinarycodefortelegraphy in1875.InBaudot's telegraphic code,well-suited for usewithteletypewriters, eachcodewordconsistsoffiveequal-length codeelements, and eachelementisassignedoneoftwopossiblestates:amarkoraspace(i.e.,symbol1or0 intoday'sterminology). In1864,JamesClerkMaxwell formulated theelectromagnetic theoryoflightand predicted theexistence ofradiowaves;theunderlying setofequations bearshisname.The Historical Notes 27 existence ofradiowaveswasestablished exjJerimentally byHeinrich Hertzin1887.In 1894,OliverLodgedemonstrated wirelesscommunication overarelatively shortdistance (150yards).Then,onDecember 12,1901,Guglielmo Marconi receivedaradiosignalat SignalHillinNewfoundland; theradiosignalhadoriginated inCornwall, England, 1700 milesawayacrosstheAtlantic. Thewaywastherebyopenedtowardatremendous broad eningofthescopeofcommunications. In1906,Reginald Fessenden, aself-educated aca demic,madehistorybyconducting thefirstradiobroadcast. In1875,thetelephone wasinvented byAlexander Graham Bell,ateacherofthe deaf.Thetelephone madereal-time transmission ofspeechbyelectrical encoding and replication ofsoundapractical reality.Thefirstversionofthetelephone wascrudeand weak,enabling peopletotalkovershortdistances only.Whentelephone servicewasonly afewyearsold,interestdeveloped inautomating it.Notably, in1897,A.B.Strowger, an undertaker fromKansasCity,Missouri, devisedtheautomatic step-by-step switchthat bearshisname;ofalltheelectromechanical switches devisedovertheyears,theStrowger switchwasthemostpopularandwidelyused. In1904,JohnAmbrose Fleminginvented thevacuum-tube diode,whichpavedthe wayfortheinvention ofthevacuum-tube triodebyLeedeForestin1906.Thediscovery ofthetriodewasinstrumental inthedevelopment oftranscontinental telephony in1913 andsignaledthedawnofwirelessvoicecommunications. Indeed,untiltheinvention and perfection ofthetransistor, thetriodewasthesupreme deviceforthedesignofelectronic amplifiers. In1918,EdwinH.Armstrong invented thesuperheterodyne radioreceiver; tothis day,almostallradioreceivers areofthistype.In1933,Armstrong demonstrated another revolutionary concept, namely,amodulation schemethathecalledfrequency modulation (FM);Armstrong's papermakingthecaseforFMradiowaspublished in1936. Thefirstall-electronic television systemwasdemonstrated byPhiloT.Farnsworth in1928,andthenbyVladimirK.Zworykin in1929.By1939,theBritishBroadcasting Corporation (BBC)wasbroadcasting television onacommercial basis. In1928,HarryNyquistpublished aclassicpaperonthetheoryofsignaltransmission intelegraphy. Inparticular, Nyquistdeveloped criteriaforthecorrectreception oftele graphsignalstransmitted overdispersive channelsintheabsenceofnoise.MuchofNy quist'searlyworkwasappliedlatertothetransmission ofdigitaldataoverdispersive channels. In1937,AlecReevesinvented pulse-code modulation (PCM)forthedigitalencoding ofspeechsignals.Thetechnique wasdeveloped duringWorldWarIItoenabletheen cryption ofspeechsignals;indeed,afull-scale, 24-channel systemwasusedinthefieldby theUnitedStatesmilitaryattheendofthewar.However, PCMhadtoawaitthediscovery ofthetransistor andthesubsequent development oflarge-scale integration ofcircuitsfor itscommercial exploitation. In1943,D.O.Northdevisedthematched filterfortheoptimum detection ofa knownsignalinadditivewhitenoise.Asimilarresultwasobtained in1946independently byJ.H.VanVleckandD.Middleton, whocoinedthetermmatched filter. In1947,thegeometric representation ofsignalswasdeveloped byV.A.Kotel'nikov inadoctoral dissertation presented beforetheAcademic CounciloftheMolotov Energy Institute inMoscow. Thismethodwassubsequently broughttofullfruitionbyJohnM. Wozencraft andIrwinM.Jacobsinalandmark textbook published in1965. In1948,thetheoretical foundations ofdigitalcommunications werelaidbyClaude Shannoninapaperentitled"AMathematical TheoryofCommunication." Shannon's paperwasreceivedwithimmediate andenthusiastic acclaim.Itwasperhapsthisresponse 28"BACKGROUND AN/)PREVIEW thatemboldened Shannon toamendthetitleofhispaperto"TheMathematical Theory ofCommunication" whenitwasreprinted ayearlaterinabookco-authored withWarren Weaver.Itisnoteworthy thatpriortothepublication ofShannon's 1948classicpaper,it wasbelievedthatincreasing therateofinformation transmission overachannelwould increase theprobability oferror;thecommunication theorycommunity wastakenby surprisewhenShannon provedthatthiswasnottrue,provided thatthetransmission rate wasbelowthechannelcapacity. Shannon's 1948paperwasfollowed bysomesignificant advances incodingtheory,whichincludethefollowing: ,.Development ofthefirstnontrivial error-correcting codesbyM.J.E.Golayin1949 andRichardW.Hamming in1950. 1'-Development ofturbocodesbyC.Berrou,A.Glavieux, andP.Thitimajshima in 1993;turbocodesprovidenear-optimum error-correcting codinganddecoding per formance intheShannon sense. Thetransistor wasinvented in1948byWalterH.Brattain, JohnBardeen, andWil liamShockley atBellLaboratories. Thefirstsiliconintegrated circuit(IC)wasproduced byRobertNoycein1958.Theselandmark innovations insolid-state devicesandintegrated circuitsledtothedevelopment ofvery-large-scale integrated (VLSI)circuitsandsingle chipmicroprocessors, andwiththemthenatureofsignalprocessing andthetelecommu nications industry changed forever. Theinvention ofthetransistor in1948spurredtheapplication ofelectronics to switching anddigitalcommunications. Themotivation wastoimprovereliability, increase capacity, andreducecost.Thefirstcallthrough astored-program systemwasplacedin March1958atBellLaboratories, andthefirstcommercial telephone servicewithdigital switching beganinMorris,Illinois,inJune 1960. ThefirstT-lcarriersystemtransmission wasinstalled in1962byBellLaboratories. Duringtheperiod1943to1946,thefirstelectronic digitalcomputer, calledthe ENIAC,wasbuiltattheMooreSchoolofElectrical Engineering oftheUniversity ofPenn sylvania underthetechnical direction ofJ.PresperEckert,Jr.,andJohnW.Mauchly. However, JohnvonNeumann's contributions wereamongtheearliestandmostfunda mentaltothetheory,design,andapplication ofdigitalcomputers, whichgobacktothe firstdraftofareportwrittenin1945.Computers andterminals startedcommunicating witheachotheroverlongdistances intheearly1950s.Thelinksusedwereinitiallyvoice gradetelephone channels operating atlowspeeds (300 to1200b/s).Variousfactorshave contributed toadramatic increaseindatatransmission rates;notableamongthemarethe ideaofadaptive equalization, pioneered byRobertLuckyin1965,andefficientmodula tiontechniques, pioneered byG.Ungerboeck in1982.Another ideawidelyemployed in computer communications isthatofautomatic repeat-request (ARQ).TheARQmethod wasoriginally devisedbyH.C.A.vanDuurenduringWorldWarIIandpublished in 1946.Itwasusedtoimprove radio-telephony fortelextransmission overlongdistances. From1950to1970,variousstudiesweremadeoncomputer networks. However, themostsignificant ofthemintermsofimpactoncomputer communications wasthe Advanced Research ProjectAgencyNetwork (ARPANET), firstputintoservicein1971. Thedevelopment ofARPANET wassponsored bytheAdvanced Research ProjectsAgency oftheU.S.Department ofDefense. Thepioneering workinpacketswitching wasdoneon ARPANET. In1985,ARPANET wasrenamed theInternet. Theturningpointintheevo lutionoftheInternet occurred in1990whenTimBerners-Lee proposed ahypermedia software interface totheInternet, whichhenamedtheWorldWideWeb.12Thereupon, in NotesandReferences 29 thespaceofonlyabouttwoyears,theWebwentfromnonexistence toworldwide popu larity,culminating initscommercialization in1994.Howdoweexplaintheexplosive growthoftheInternet? Wemayanswerthisquestion byofferingthesereasons:13 II>BeforetheWebexploded intoexistence, theingredients foritscreation werealready inplace.Inparticular, thankstoVLSI,personal computers (PCs)hadalreadybecome ubiquitous inhomesthroughout theworld,andtheywereincreasingly equipped with modems forinterconnectiviry totheoutsideworld. II>Forabouttwodecades, theInternet hadgrownsteadily(albeitwithinaconfined community ofusers),reaching acriticalthreshold ofuser-value basedelectronic mail andfiletransfer. ~Standards fordocument description andtransfer, hypertext markup language (HTML), andhypertext transferprotocol (HTTP)hadbeenadopted. Thus,everything neededforcreating theWebwasalreadyinplaceexceptfortwocritical ingredients: asimpleuserinterface andabrilliantserviceconcept. In1955,JohnR.Pierceproposed theuseofsatellites forcommunications. This proposal waspreceded, however, byanearlierpaperbyArthurC.Clarkthatwaspub lishedin1945,alsoproposing theideaofusinganEarth-Drbiting satelliteasarelaypDint fDrcommunication between twDEarthstatiDns. In1957,theSovietUnionlaunched Sput nikI,whichtransmitted telemetry signalsfor21days.ThiswasfDllDwed shDrtlybythe launching DfExplorer IbytheUnitedStatesin1958,whichtransmitted telemetry signals fDraboutfivemonths. AmajDrexperimental stepincommunicatiDns satellitetechnology wastakenwiththelaunching DfTelstarIfromCapeCanaveral DnJuly10,1962.The TelstarsatellitewasbuiltbyBellLabDratories, whichhadacquired cDnsiderable knDwl edgefrompioneering wDrkbyPierce.Thesatellitewascapable DfrelayingTVprograms acrosstheAtlantic; thiswasmadepDssible onlythrough theuserofmaserreceivers and largeantennas. Theuseofopticalmeans(e.g.,smokeandfiresignals)forthetransmissiDn DfinfDr matiDndatesbacktoprehistoric times.HDwever, nDmajorbreakthrough inDpticalCDm munications Wasmadeuntil1966,whenK.C.KaDandG.A.HDckham DfStandard TelephDne Laboratories, U.K.,propDsed theuseDfacladglassfiberasadielectric wave guide.Thelaser(anacrDnym fDrlightamplificatiDn bystimulated emission ofradiation) hadbeeninvented anddevelDped in1959and1960.KaoandHDckham pDinted Dutthat (1)theattenuatiDn inanopticalfiberwasduetDimpurities intheglass,and(2)theintrinsic lDss,determined byRayleigh scattering, isverylDw.Indeed,theypredicted thatalossof 20dB/kmshouldbeattainable. Thisremarkable prediction, madeatatimewhenthe pDwerlDssinaglassfiberwasabout1000dB/km,wastDbedemDnstrated later.Nowa days,transmissiDn lDssesaslDwas0.1dB/kmareachievable. Thespectacular advances inmicroelectrDnics, digitalcDmputers, andlightwave sys temsthatwehavewitnessed tDdate,andthatwillcontinue intothefuture,areallre spDnsible fDrdramatic changes inthetelecommunications environment; manyDfthese changesarealreadyinplace,andmorechangeswilleVDlveastimegoeson. INOTES ANDREFERENCES 1.ForessaysonanearlyaccountofcDmmunications andotherrelateddisciplines (e.g.,elec tronics,computers, radar,radioastronomy, satellites), seeOverhage (1962);inparticular, seethechapteron"Communications" by1.V.Berkner,pp.35-50. 30"BACKGROUND ANDPREVIEW 2.The]PEGimagecodingstandard isdiscussed inthepapersbyWallace(1991);seealsothe articlebyT.A.Ramstad inthehandbook editedbyMadisetti andWilliams (1998). 3.Thediscretecosinetransform (DCT)anditsinverseforablockof8X8sourceimage samplesarerespectively definedby F(u,v)=~C(U)C(v{io,to f(x,y)cose2X:61 )U1T)cos((2Y:61 )V1T)] f(x,y)=~[~o~oC(u)C(v)F(u, v)cose2X:61 )U1T)cose2Y:61 )V1T)] where C(U),C(v)={~forU=0andv=6 otherwise Forafulltreatment oftheDCT,seeRaoandYip(1990). 4.TheMPEG-1 videocodingstandard isdiscussed inthepaperbyGall(1991);seealsothe articlebyA.M.Tekalpinthehandbook editedbyMadisetti andWilliams (1998),which discusses thefollow-up versionsoftheMPEGvideocodingstandard. 5.TheMPEG-1 audiocodingstandard isdiscussed inthepapersbyBrandenburg andStoll (1994)andPan(1993);seealsothearticlebyP.Nollinthehandbook editedbyMadisetti andWilliams (1998),whichalsodiscusses thefollow-up versions oftheMPEGaudio codingstandard.Inparticular, thewidespread useofthemorecurrentstandard, MPEG-3 audio,isresulting inalevelofpiracythatmaydwarftheearlierproblems of"bootleg" cassettetapes. 6.Foradetaileddiscussion ofcommunication networks, seeTanenbaum (1996). 7.TheOS1reference modelwasdeveloped byasubcommittee oftheInternational Organi zationforStandardization (ISO)in1977.Foradiscussion oftheprinciples involved in arrivingatthesevenlayersoftheOS1modelandadescription ofthelayersthemselves, seeTanenbaum (1996). 8.SONETwasoriginally proposed byTelcordia Technologies Inc.(thenknownasBellcore) andstandardized bytheAmerican National Standards Institute (ANSI).Later,CCITT approved aSONETstandard andissuedasetofparallelrecommendations calledsynchro nousdigitalhierarchy (SDH).Thedifferences between SONETandSDHareofaminor nature. 9.Forathorough andpreciseanalysisofthepropagation oflightwavesinanopticalfiber, weneedtotreatitasadielectric waveguide anduseMaxwell's equations tocarryoutthe analysis; suchananalysis ishighlymathematical innature.Forareadable accountofthe analysis, seeChapter3ofGreen,]r. (1993). 10.Forasemitechnical overview ofShannon's theorems oninformation theorypresented ina highlyreadable fashion,seethebookentitledSiliconDreamsbyLucky(1989). 11.Forareadable accountofthehistoryofcommunications, seeLebow(1995). 12.Forahistorical accountofthedevelopment oftheInternet, seeLeineretal.(1997). 13.Foraninsightful essayonnewtelecommunications servicesandhowsocietyreactstotheir development, seeLucky(1997).ThispaperpointstoMetcalf's law,according towhichit seemsasifanynewtelecommunications servicemusttakealongtimeforittobuildto universal acceptance. LuckycitestheWorldWideWebasastartling counterexample to Metcalf's lawandgivesthereasonswhy. RANDOM PROCESSES Thischapterpresentsanintroductory treatment ofstationary randomprocesses with emphasis onsecond-order statistics. Inparticular, itdiscusses thefollowing issues: ..Thenotionofarandomprocess. ..Therequirement thathastobesatisfiedforarandomprocesstobestationary . ..Thepartialdescription ofarandomprocessintermsofitsmean,correlation, and covariance functions. ~Theconditions thathavetobesatisfiedforastationary randomprocesstobeergodic,a property thatenablesustosubstitute timeaverages forensemble averages. ..Whathappenstoastationary randomprocesswhenitistransmitted throughalinear time-invariant filter? ..Thefrequency-domain description ofarandomprocessintermsofpowerspectraldensity. ..Thecharacteristics ofanimportant typeofrandomprocessknownasaGaussian process. ..Sourcesofnoiseandtheirnarrowband form. ..Rayleigh andRiciandistributions, whichrepresent twospecialprobability distributions thatariseinthestudyofcommunication systems. I1.1Introduction Theideaofamathematical modelusedtodescribeaphysicalphenomenon iswellestab lishedinthephysicalsciencesandengineering. Inthiscontext,wemaydistinguish two classesofmathematical models:deterministic andstochastic. Amodelissaidtobedeter ministicifthereisnouncertainty aboutitstime-dependent behavior atanyinstantoftime. However, inmanyreal-world problems theuseofadeterministic modelisinappropriate becausethephysicalphenomenon ofinterestinvolvestoomanyunknown factors.Nev ertheless, itmaybepossibletoconsider amodeldescribed inprobabilistic termsinthat wespeakoftheprobability ofafuturevaluelyingbetweentwospecified limits.Insucha case,themodelissaidtobestochastic orrandom. Abriefreviewofprobability theoryis presented inAppendix 1. Consider, forexample, aradiocommunication system.Thereceivedsignalinsuch asystemusuallyconsistsofaninformation-bearing signalcomponent, arandominterfer encecomponent, andchannelnoise.Theinformation-bearing signalcomponent mayrep resent,forexample, avoicesignalthat,typically, consistsofrandomly spacedburstsof energyofrandomduration. Theinterference component mayrepresent spurious electro magnetic wavesproduced byothercommunication systemsoperating inthevicinityofthe 31 32 CHAPTER 1.,RANDOM PROCESSES radioreceiver. Amajorsourceofchannelnoiseisthermalnoise,whichiscausedbythe randommotionoftheelectrons inconductors anddevicesatthefrontendofthereceiver. Wethusfindthatthereceived signalisrandominnature.Although itisnotpossibleto predicttheexactvalueofthesignalinadvance, itispossibletodescribethesignalinterms ofstatistical parameters suchasaveragepowerandpowerspectraldensity,asdiscussed inthischapter. 1.2Mathematical Definition ofaRandom Process Inlightoftheseintroductory remarks, itisapparent thatrandom processes havetwo properties. First,theyarefunctions oftime.Second,theyarerandom inthesensethat beforeconducting anexperiment, itisnotpossibletoexactlydefinethewaveforms that willbeobserved inthefuture. Indescribing arandom experiment itisconvenient tothinkintermsofasample space.Specifically, eachoutcome oftheexperiment isassociated withasamplepoint.The totalityofsamplepointscorresponding totheaggregate ofallpossible outcomes ofthe experiment iscalledthesamplespace.Eachsamplepointofthesamplespaceisafunction oftime.Thesamplespaceorensemble composed offunctions oftimeiscalledarandom orstochastic process.1Asanintegralpartofthisnorion,weassumetheexistence ofa probability distribution definedoveranappropriate classofsetsinthesamplespace,so thatwemayspeakwithconfidence oftheprobability ofvariousevents. Consider, then,arandomexperiment specified bytheoutcomes sfromsomesample spaceS,bytheeventsdefinedonthesamplespaceS,andbytheprobabilities ofthese Outcomeofthe firsttrialof theexperiment Outcomeofthe secondtrialof theexperiment Outcomeofthe nthtrialof +Ttheexperiment t~ FIGlJRE 1.1Anensemble ofsamplefunctions. 1.3Statumary Processes 33 events.Suppose thatweassigntoeachsamplepointsafunction oftimeinaccordance withtherule: X(t,s),-T:5t:5T (1.1) where2Tisthetotalobservation interval. Forafixedsamplepointsi'thegraphofthe function X(t,si)versustimetiscalledarealization orsamplefunction oftherandom process.Tosimplifythenotation, wedenotethissamplefunction as xi(t)=X(t,si) (1.2) Figure1.1illustrates asetofsamplefunctions {xi(t)Ij1,2,...,n}.Fromthisfigure, wenotethatforafixedtimetkinsidetheobservation interval, thesetofnumbers {Xr(tk)'X2(tk),...,Xn(tk)}={X(tbsl),X(tbS2),...,X(tk,s,,)} constitutes arandom variable. Thuswehaveanindexedensemble (family)ofrandom variables {X(t,s)},whichiscalledarandomprocess.Tosimplifythenotation, thecustom arypracticeistosuppress thesandsimplyuseX(t)todenotearandomprocess.Wemay nowformally definearandomprocessX(t)asanensemble oftimefunctions together with aprobability rulethatassignsaprobability toanymeaningful eventassociated withan observation ofoneofthesamplefunctions oftherandom process.Moreover, wemay distinguish'between arandomvariableandarandomprocessasfollows: Forarandom variable, theoutcome ofarandom experiment ismapped intoa number. ~Forarandomprocess,theoutcome ofarandomexperiment ismappedintoawave formthatisafunction oftime. I1.3Stationary Processes Indealingwithrandomprocesses encountered intherealworld,weoftenfindthatthe statistical characterization ofaprocessisindependent ofthetimeatwhichobservation of theprocessisinitiated. Thatis,ifsuchaprocessisdividedintoanumberoftimeintervals, thevarioussectionsoftheprocessexhibitessentially thesamestatistical properties. Such aprocessissaidtobestationary. Otherwise, itissaidtobenonstationary. Generally speaking, astationary processarisesfromastablephysicalphenomenon thathasevolved intoasteady-state modeofbehavior, whereas anonstationary processarisesfroman unstable phenomenon. Tobemoreprecise,consider arandomprocessX(t)thatisinitiatedatt=-00.Let X(tl),X(t2),...,X(tk)denotetherandomvariables obtained byobserving therandom processX(t)attimest1>t2,•••,tk,respectively. Thejointdistribution function ofthisset ofrandomvariables isFX(t,)"",X(tk)(Xl,...,Xk)'Supposenextweshiftalltheobservation timesbyafixedamount T,therebyobtaining anewsetofrandomvariables X(t,+T), X(t2+T),•••,X(tk+T).Thejointdistribution function ofthislattersetofrandom variables isFX(tl+T),...,X(t,+T)(X1>...,Xk).TherandomprocessX(t)issaidtobestationary inthestrictsenseorstrictlystationary ifthefollowing condition holds: FX(t,+T),...,X(t,+T)(X1>"" Xk)=FX(t,),...,Xit,)(X1>"" Xk) (1.3) foralltimeshiftsT,allk,andallpossiblechoicesofobservation timestl,.••,tk'Inother words,arandom processX(t),initiated attimet=-00,isstrictlystationary ifthejoint distribution ofanysetofrandom variables obtained byobserving therandomprocessX(t) isinvariant withrespecttothelocationoftheorigint=O.Notethatthefinite-dimensional 34 CHAPTER 1'"RANDOM PROCESSES distributions inEquation (1.3)dependontherelativetimeseparation between random variables butnotontheirabsolute time.Thatis,therandomprocesshastheSameprob abilisticbehavior throughalltime. Similarly, wemaysaythattworandomprocesses X(t)andY(t)arejointlystrictly stationary ifthejointfinite-dimensional distributions ofthetwosetsofrandomvariables X(t1),•••,X(tk)andY(ti),...,Y(tj)areinvariant withrespecttotheorigint=0forall kandjandallchoicesofobservation timest"...,tkandti,...,ti. Returning toEquation (1.3),wemaydistinguish twosituations ofspecialinterest: 1.Fork=1,wehave foralltandT (1.4) Thatis,thefirst-order distribution functionofastationary randomprocessisinde pendentoftime. 2.Fork=2andT=-t"wehave forallt1andt2(1.5) Thatis,thesecond-order distribution functionofastationary randomprocessde pendsonlyonthetimedifference betweentheobservation timesandnotonthe particular timesatwhichtherandomprocessisobserved. Thesetwoproperties haveprofound implications forthestatistical parameterization ofa stationary randomprocess;thisissueisdiscussed inSection1.4. ~EXAMPLE 1.1 Consider Figure1.2,depicting threespatialwindows locatedattimest1>tz,t,.Wewishto evaluate theprobability ofobtaining asamplefunction x(t)ofarandomprocessX(t)that passesthroughthissetofwindows, thatis,theprobability ofthejointevent A[ai<X(ti)$bi},i=1,2,3 Intermsofthejointdistribution function, thisprobability equals PIA)=FX1t,),X(t2),X(t3)(b1> b"b3)-FX(t,),XI0.),xlt3)(a" a"a3) SupposenowtherandomprocessX(t)isknowntobestrictlystationary. Animplication ofstrictstationarity isthattheprobability ofthesetofsamplefunctions ohhisprocesspassing throughthewindows ofFigure1.3aisequaltotheprobability ofthesetofsamplefunctions passingthroughthecorresponding tiDle-shifted windows ofFigure1.3b.Note,however, that itisnotnecessary thatthesetwosetsconsistoftheSamesamplefunctions. ~ /"--" Aposible Ia3 "sample I function I Ta2 FIGURE 1.2Illustrating theprobability ofajointevent. 1.4Mean,Correlation, alUiCovariance Functkms 35 '2 II b213 Ta2 (a) Ib1L3 al a3 t2+'T" t1+1" b2 t3+T Ta2 (b) FIGURE1.3Illustrating theconceptofstationarity inExample 1.1. 1.4Mean,Correlation, andCovariance Functions Consider astrictlystationary randomprocessX(t).WedefinethemeanoftheprocessX(t) astheexpectation oftherandomvariableobtained byobserving theprocessatsometime t,asshownby f.kX(t)=E[X(t)] =rooX!x(II(X)dx(1.6) where!X(I)(X)isthefirst-order probability densityfunction oftheprocess.FromEquation (1.4)wededucethatforastrictlystationary randomprocess, !X(t)(x)isindependent of timet.Consequently, themeanofastrictlystationary processisaconstant, asshownby f.kX(t)=f.kxforallt (1.7) Wedefinetheautocorrelation functionoftheprocessX(t)astheexpectation oftheproduct oftworandomvariables, X(t,)andX(2),obtained byobserving theprocessXU)attimes t1andt2,respectively. Specifically, wewrite (1.8) where !X(tj),X(t2)(Xh X2)isthesecond-order probability densityfunction oftheprocess. FromEquation (1.5),wededucethatforastrictlystationary random process, !X(lj),X(t2)(Xh X2)depends onlyonthedifference between theobservation timest1andt2• (1.10)36 CHAPTER 1.,RANDOM PROCESSES This,inturn,impliesthattheautocorrelation functionofastrictlystationary process depends onlyonthetimedifference t2-t "asshownby Rx(t"t2)=RX(t2-t,)forallt,andt2 (1.9) Similarly, theautocovariance function ofastrictlystationary processX(t)iswritten'as Cx(tj,t2)E[(X(t1) -JLX)(X(t2) -/Lx)] =RX(t2-t,)JL5.c Equation (1.10)showsthat,liketheautocorrelation function, theautocovariance function ofastrictlystationary processX(t)depends onlyonthetimedifference t2-t1.This equation alsoshowsthatifweknowthemeanandautocorrelation function oftheprocess, wecanuniquely determine theautocovariance function. Themeanandautocorrelation function aretherefore sufficient todescribethefirsttwomoments oftheprocess. However, twoimportant pointsshouldbecarefully noted: 1.Themeanandautocorrelation function onlyprovideapartialdescription ofthe distribution ofarandomprocessX(t). 2.Theconditions ofEquations (1.7)and(1.9),involving themeanandautocorrelation function, respectively, arenotsufficient toguarantee thattherandomprocessX(t) isstrictlystationary. Nevertheless, practical considerations oftendictatethatwesimplylimitourselves toa partialdescription oftheprocessgivenbythemeanandautocorrelation function. The classofrandomprocesses thatsatisfyEquations (1.7)and(1.9)hasbeengivenvarious names,suchassecond-order stationary, wide-sense stationary, orweaklystationary pro cesses.Henceforth, weshallsimplyrefertothemasstationary processes.2 Astationary processisnotnecessarily strictlystationary becauseEquations (1.7)and (1.9)obviously donotimplytheinvariance ofthejoint(k-dimensional) distribution of Equation (1.3)withrespecttothetimeshiftTforallk.Ontheotherhand,astrictly stationary processdoesnotnecessarily satisfyEquations (1.7)and(1.9)asthefirst-and second-order moments maynotexist.Clearly,however, theclassofstrictlystationary processes withfinitesecond-order moments formsasubclassoftheclassofallstationary processes. !!!PROPERTIES OFTHEAUTOCORRELATION FUNCTION Forconvenience ofnotation, weredefinetheautocorrelation function ofastationary pro cessX(t)as RX(T) E[X(t+T)X(t)] forallt (1.11) Thisautocorrelation function hasseveralimportant properties: 1.Themean-square valueoftheprocessmaybeobtained fromRx(T)simplybyputting T=0inEquation (1.11),asshownby Rx(O)=E[X2(t)] (1.12) 2.Theautocorrelation function Rx(T)isanevenfunction ofT,thatis, RX(T)=Rx(-T) (1.13) Thisproperty followsdirectlyfromthedefining equation (1.11).Accordingly, we mayalsodefinetheautocorrelation function RX(T)as RX(T)=E[X(t)X(t T)] 1.4Mean,Correl..tion, andCrwariance Functions 37 RX(T) Slowlyfluctuating randomprocess o FIGURE 1.4Illustrating theautocorrelation functions ofslowlyandrapidlyfluctuating random processes. 3.Theautocorrelation functionRX(T)hasitsmaximum magnitude atT=0,thatis, (1.14) Toprovethisproperty, consider thenonnegative quantity E[(X{t+T)±X(tW] 2:0 Expanding termsandtakingtheirindividual expectations, wereadilyfindthat E[X2(t+T)]±2E[X(t+T)X(t)]+E[X2{t)]2:0 which,inlightofEquations (1.11)and(1.12),reducesto 2Rx(0)±2Rx(T)2:0 Equivalently, wemaywrite -Rx(O) SRx(T)sRx{O) fromwhichEquation (1.14)followsdirectly. Thephysical significance oftheautocorrelation function Rx{T)isthatitprovides a meansofdescribing theinterdependence oftworandomvariables obtained byobserving arandom processX(t)attimes Tsecondsapart.Itistherefore apparent thatthemore rapidlytherandomprocessX(t)changeswithtime,themorerapidlywilltheautocorre lationfunction RX(T)decrease fromitsmaximum Rx(O)asTincreases, asillustrated in Figure1.4.Thisdecrease maybecharacterized byadecorrelation timeTo,suchthatfor T>TO,themagnitude oftheautocorrelation functionRx{T)remainsbelowsomeprescribed value.Wemaythusdefinethedecorrelation timeTOofastationary processX{t)ofzero meanasthetimetakenforthemagnitude oftheautocorrelation function Rx(T)todecrease to1percent,say,ofitsmaximum valueRx{O). l1>EXAMPLE 1.2Sinusoidal WavewithRandom Phase Consider asinusoidal signalwithrandomphase,definedby X(t)=Acos(2Trfct+0) (1.15) whereAandfoareconstants and0isarandomvariable Cthatisuniformly distributed over theinterval[-Tr,Tr],thatis, feU})={2~' 0,elsewhere(1.16) 38 CHAPTER 1'"RANDOM PROCESSES A' 2 FIGURE1.5Autocorrelation function ofasinewavewithrandomphase. Thismeansthattherandomvariable®isequallylikelytohaveanyvalue0intheinterval [-'1r,'IT].Eachvalueof®corresponds toasampleinthesamplespaceoftherandomprocess X(t). TheprocessX(t)definedbyEquations (1.15)and(1.16)mayrepresent alocallygen eratedcarrierinthereceiverofacommunication system,whichisusedindemodulation of thereceived signal.Inparticular, therandomvariable®denotesthephasedifference between thislocallygenerated carrierandthesinusoidal carrierwaveusedtomodulate themessage signalinthetransmitter. Theautocorrelation function ofX(t)is Rx(7')=E[X(t+7')X(t)] =E[A2cos(2'lTD+2'1rfc7'+®)cos(2'lTD+e)] A2 A2 =2E[cos(4'lTD +2'lTfc7'+2®)]+2E[cos(2'Trfc7')] A2Jrr1 A2 = --2cos(4'lT];t+2'lT];7'+20)dO+-cos(2'lTf,7')2orr'IT 2 Thefirstrermintegrates tozero,andsoweget A2 Rx(7')=2cos(2'1r];7') (1.17) whichisplottedinFigure1.5.Weseetherefore thattheautocorrelation function ofasinu soidalwavewithrandomphaseisanothersinusoidatthesamefrequency inthe"7'domain" ratherthantheoriginaltimedomain. -<ll ~EXAMPLE 1.3Random BinaryWave Figure1.6showsthesamplefunctionx(t)ofaprocessX(t)consisting ofarandomsequence ofbinarysymbols 1andO.Thefollowing assumptions aremade: 1.Thesymbols 1and0arerepresented bypulsesofamplitude+Aand-Avolts,respec tively,andduration Tseconds. 2.Thepulsesarenotsynchronized, sothestartingtimetdofthefirstcomplete pulsefor positivetimeisequallylikelytolieanywhere between zeroandTseconds. Thatis,td isthesamplevalueofauniformly distributed randomvariableTd,withitsprobability densityfunction definedby 0::0;td::0;T elsewhere 3.Duringanytimeinterval(nI)T<t-t"<nT,wherenisaninteger,thepresence ofa 1ora 0isdetermined bytossingafaircoin;specifically, iftheoutcome isheads, 1.4Mean,Correlation, andCovariance Functions 39 x(t) +A,....--r- - a ---J:L-A ~td~ FIGURE 1.6Samplefunction ofrandombinarywave. wehavea 1andiftheoutcome istails,wehaveaO.Thesetwosymbolsarethusequally likely,andthepresence ofa 1or0inanyone intervalisindependent ofallother intervals. Sincetheamplitude levels-Aand+Aoccurwithequalprobability, i[followsimme diatelythatE[X{t)]=0forallt,and[hemeanoftheprocessistherefore zero. Tofindtheautocorrelation function Rx{t.,til,wehavetoevaluateE[X{tk)X{til], where X{tk)and,X(ti)arerandomvariables obtained byobserving therandomprocessX(t)attimes tkandti,respectively. Consider firstthecasewhenItktiI>T.Underthiscondition therandomvariables X{tk)andX(ti)occurindifferent pulseintervals andaretherefore independent. Wethushave E[X(tk)X{til] =E[X(tk)]E[X(t i)]=0, Consider nextthecasewhenItk-tjI<T,withtk=0andtj<tk'Insuchasituation weobservefromFigure1.6thattherandomvariables X(tk)andX(ti)oceminthesamepulse intervalifandonlyifthedelaytdsatisfiesthecondition td<T-1tk-tjI.Wethusobtain theconditional expectation: td<T-Itk-tjI elsewhere Averaging thisresultoverallpossiblevaluesoftd,weget Itk-t,j) T ' Bysimilarreasoning foranyothervalueoft.,weconclude [hattheautocorrelation function ofarandombinarywave,represented bythesamplefunction showninFigure1.6,isonlya function ofthetimedifference 7"=tk-ti,asshownby {2(17"1)Rx(7")=A1 -T' 0, TbisresultisplottedinFigure1.7.17"1<T 17"1;0:T(U8) 40 CHAPTER 1 "RANDOM PROCESSES A' FIGURE1.7Autocorrelation function ofrandombinarywave. 11IICROSS-CORRELATION FUNCTIONS Consider nextthemoregeneralcaseoftworandomprocesses X(t)andY(t)withauto correlation functions Rx(t,u)andRy(t,u),respectively. Thetwocross-correlation func tionsofX(t)andY(t)aredefinedby andRxy(t,u)=E[X(t)Y(u)] Ryx(t,u)=E[Y(t)X(u)](1.19) (1.20) wheretandudenotetwovaluesoftimeatwhichtheprocesses areobserved. Inthiscase, thecorrelation properties ofthetworandomprocesses X(t)andY(t)maybedisplayed conveniently inmatrixformasfollows: R(t,u)=[Rx(t,u) Ryx(t,u) whichiscalledthecorrelation matrixoftherandomprocesses X(t)andY(t).Iftherandom processes X(t)andY(t)areeachstationary and,inaddition, theyarejointlystationary, thenthecorrelation matrixcanbewrittenas R(r)=[Rx(r) Ryx(r)(1.21) wherer=t-u. Thecross-correlation function isnotgenerally anevenfunction ofraswastruefor theautocorrelation function, nordoesithaveamaximum attheorigin.However, itdoes obeyacertainsymmetry relationship asfollows(seeProblem 1.9): Rxy(r) =Ryx(-r) (1.22) ..ExAMPLE 1.4Quadrature-Modulated Processes Consider apairofquadrature-modulated processes X,(t)andX2(t)ma[arerelatedtoasta tionaryprocessXU)asfollows: X,(t)=X(t)cos(2",fot+8) X2(t)=X(t)sin{2Trfot+8) 1.5Ergodic PrO£esses 41 whereI,isacarrierfrequency, andtherandomvariable Elisuniformly distributed overthe interval[0,27T].Moreover, Elisindependent ofX(t).Onecross-correlation functionofXl(t) andXl(t)isgivenby RutT)=E[Xl(t)Xl(t -T)] =E[X(t)X(t -T)cos(27Tfct+El)sin(27Tfct -27TfcT+Ell] =E[X(t)X(t T)]E[cos(27TfJ +El)Sin(27Tfct -27TfcT+El)] =fRx(T)E[sin(47TfJ -27TfcT+2El)-sin(27TfcT)] =-fRx(T) sin(27TI,T)(1.23) where,inthelastline,wehavemadeuseoftheuniformdistribution oftherandomvariable Elrepresenting phase.NorethatatT=0,thefactorsin(27TfcT) iszeroandtherefore RutO)=E[Xl(t)Xl(t)] =0 Thisshowsthattherandomvariables obtained bysimultaneously observing thequadrature modulated processes Xl(t)andXl(t)atsomefixedvalueoftimetareorthogonal toeach other. "Ill I1.5Ergodic Processes Theexpectations orensemble averages ofarandomprocessX(t)areaverages "acrossthe process." Forexample, themeanofarandom processX(t)atsomefixedtimetkisthe expectation oftherandomvariableX(tk)thatdescribes allpossible values ofthesample functions oftheprocessobserved attimet=tk'Naturally, wemayalsodefinelong-term sampleaverages, ortimeaverages thatareaverages "alongtheprocess." Wearetherefore interested inrelatingensemble averages totimeaverages, fortimeaverages represent a practical meansavailable tousfortheestimation ofensemble averages ofarandompro cess.Thekeyquestion, ofcourse,is:Whencanwesubstitute timeaverages forensemble averages? Toexplorethisissue,consider thesamplefunction x(t)ofastationary process X(t),withtheobservation intervaldefinedas-T:s;t:s;T.TheDCvalueofx(t)isdefined bythetimeaverage 1IT/Lx(T)==2T-Tx(t)dt (1.24) Clearly, thetimeaverage/Lx(T)isarandomvariable, asitsvaluedepends ontheobser vationintervalandwhichparticular samplefunction oftherandomprocessX(t)ispicked foruseinEquation (1.24).SincetheprocessX(t)isassumed tobestationary, themeanof thetimeaverage/Lx(T)isgivenby(afterinterchanging theoperation ofexpectation and integration): 1ITE[/Lx(T)] =2T-TE[x(t)]dt 1IT=2T-T/Lxdt =/Lx(1.25) (1.26)42 CHAPTER 1IIIRANDOM PROCESSES whereiLxisthemeanoftheprocessX(t).Accordingly, thetimeaverageiLxlT)represents anunbiased estimateoftheensemble-averaged meaniLx.WesaythattheprocessX(t)is ergodicinthemeaniftwoconditions aresatisfied: I>Thetimeaverage iLx(T)approaches theensemble average iLxinthelimitasthe observation intervalTapproaches infinity;thatis, limiLx(T)=iLxT_oo I>Thevariance ofiLjT),treatedasarandomvariable, approaches zerointhelimitas theobservation intervalTapproaches infinity;thatis, limvar[iLx(T)] =0T_oo Theothertimeaverageofparticular interestistheautocorrelation functionRx(r,T) definedintermsofthesamplefunction x(t)observed overtheinterval-T:=;t:=;T. Following Equation (1.24),wemayformally definethetime-averaged autocorrelation function ofasamplefunction x(t)asfollows: 1ITRx(r,T)=2T_TX(t+r)x(t)dt Thissecondtime-average shouldalsobeviewedasarandomvariablewithameanand variance ofitsown.InamaImersimilartoergodicity ofthemean,wesaythattheprocess x(t)isergodicintheautocorrelation function ifthefollowing twolimitingconditions are satisfied: limRx(r,T)=Rx(r)T_oo limvar[Rx(r, T)]=0 T_= Wecould,ofcourse,gooninasimilarwaytodefineergodicity inthemostgeneral sensebyconsidering higher-order statistics oftheprocessX(t).Inpractice, however, er godicityinthemeanandergodicity intheautocorrelation function, asdescribed here,are often(butnotalways)considered tobeadequate. NotealsothattheuseofEquations (1.24)and(1.26)tocompute thetimeaverages iLx(T)andRx(t,T)requiresthattheprocess X(t)bestationary. Inotherwords,forarandomprocesstobeergodic,ithastobesta tionary;however, theconverse isnotnecessarily true. 1.6Transmission ofaRandom Process Through aLinearTIme-Invariant Filter SupposethatarandomprocessX(t)isappliedasinputtoalineartime-invariant filterof impulse response hit),producing anewrandomprocessY(t)atthefilteroutput,asin Figure1.8.Ingeneral,itisdifficulttodescribetheprobability distribution oftheoutput randomprocessY(t),evenwhentheprobability distribution oftheinputrandomprocess X(t)iscompletely specified for-00<t<00. Inthissection,wedetermine thetime-domain formoftheinput-output relations of thefilterfordefiningthemeanandautocorrelation functions oftheoutputrandomprocess Y(t)intermsofthoseoftheinputX(t),assuming thatX(t)isastationary process. 1.6TraflSmission ofaRandom ProcessThrough aLinearTi....,·Invariant Filter 43 X(t) Y(r) FIGURE 1.8Transmission ofarandomprocessthrough alineartime-invariant filter. Thetransmission ofaprocessthroughalineartime-invariant filterisgoverned by theconvolution integral; forareviewofthisoperation, seeAppendix 2. Fortheproblem athand,wemaythusexpresstheoutputrandomprocess Y(t)intermsoftheinputrandom processX(t)as whereT1istheintegration variable. Hence,themeanofY(t)is f.Ly(t)=E[Y(t)] =E[r~h(T1)X(t-T1)dT1](1.27) Provided thattheexpectation E[X(t)]isfiniteforalltandthesystemisstable,wemay interchange the orderofexpectation andintegration inEquation (1.27)andsowrite f.Ly(t)=r~hh)E[X(t T1)]dT1 =f:~h(T1)f.LX(t -T1)dT1(1.28) (1.29)WhentheinputrandomprocessX(t)isstationary, themeanf.Lx(t)isaconstant f.Lx,so thatwemaysimplifyEquation (1.28)asfollows: f.Ly=f.Lxf:~h(T1)dT1 =f.LxH(O) whereH(O)isthezero-frequency (DC)response ofthesystem.Equation (1.29)statesthat themeanoftherandomprocessY(t)produced attheoutputofalineartime-invariant systeminresponse toX(t)actingastheinputprocessisequaltothemeanofX(t)multiplied bytheDCresponse ofthesystem,whichisintuitively satisfying. Consider nexttheautocorrelation function oftheoutputrandomprocessY(t).By definition, wehave Ry(t,u)=E[Y(t)Y(u)] wheretandudenotetwovaluesofthetimeatwhichtheoutputprocessisobserved. We maytherefore usetheconvolution integraltowrite (1.30) 44 CHAPTER 1RANDOM PROCESSES Hereagain,provided thatthemean-square valueE[X2(t)Jisfiniteforalltandthesystem isstable,wemayinterchange theorderoftheexpectation andtheintegrations withrespect to1'1and1'2inEquation (1.30),obtaining Ry(t,u)=f~dTth(Tt)f~dT2h(T2)E[X(t -Tj)X(U-T2)J =J:~dTth(Tt) J:~dT2h(T2)Rx(t -Tt,U-1'2)(1.31) WhentheinputX(t)isastationary process,theautocorrelation function ofX(t)isonlya function ofthedifference betweentheobservation timest-TtandU-1'2'Thus,putting l'=t-uinEquation (1.31),wemaywrite (1.32) Oncombining thisresultwiththatinvolving themean }J-y,weseethatiftheinputtoa stablelineartime-invariant filterisastationary process,thentheoutputofthefilterisalso astationary process. SinceRy(O)=E[y2(t)J,itfollowsthatthemean-square valueoftheoutputrandom processY(t)isobtained byputting l'=0inEquation (1.32).Wethusgettheresult (1.33) whichisaconstant. I1.7PowerSpectral Density Thusfarwehaveconsidered thecharacterization ofstationary processes inlinearsystems inthetimedomain. Weturnnexttothecharacterization ofrandomprocesses inlinear systemsbyusingfrequency-domain ideas.Inparticular, wewishtoderivethefrequency domainequivalent totheresultofEquation (1.33)definingthemean-square valueofthe filteroutput. Bydefinition, theimpulseresponse ofalineartime-invariant filterisequaltothe inverseFouriertransform ofthefrequency response ofthesystem;areviewoftheFourier transform ispresented inAppendix 2.UsingH(!)todenotethefrequency response ofthe system,wemaythuswrite Substituting thisexpression forh(Tt)intoEquation (1.33),weget E[y2(t)J=f~f=[f~H(!)exp(j27rfTtl df}(T2)Rx(T2-Tt)dTtdT2 =f=dfH(!)f~dT2h(Tz)r~Rx(T2-Tt)exp(j27rf Tt)dTt(1.34) (1.35) 1.7PowerSpectral Density 45 Inthelastintegralontheright-hand sideofEquation (1.35),defineanewvariable ThenwemayrewriteEquation (1.35)intheform E[y2(t)] =r~dfR(f)r~dT2h(T2)exp(j27Tf T2)r~Rx(T)exp(-j27TfT) dT(1.36) However, themiddleintegralontheright-hand sideinEquation (1.36)issimplyH*(f), thecomplex conjugate ofthefrequency response ofthefilter,andsowemaysimplifythis equation as (1.37) whereIH(f)[isthemagnitude response ofthefilter.WemayfurthersimplifyEquation (1.37)byrecogniiing thatthelastintegralissimplytheFouriertransform oftheauto correlation functionRx(T)oftheinputrandomprocessX(t).Thisprompts ustointroduce thedefinition ofanewparameter (1.38) ThefunctionSx(f)iscalledthepowerspectraldensity,orpowerspectrum, ofthestation aryprocessX(t).Thussubstituting Equation (1.38)into(1.37),weobtainthedesired relation: (1.39) Equation (1.39)statesthatthemean-square valueoftheoutputofastablelineartime invariant filterinresponse toastationary processisequaltotheintegraloverallfrequen ciesofthepowerspectraldensityoftheinputprocessmultiplied bythesquaredmagnitude responseofthefilter.Thisisthedesiredfrequency-domain equivalent tothetime-domain relationofEquation (1.33). Toinvestigate thephysical significance ofthepowerspectraldensity,supposethat therandomprocessX(t)ispassedthroughanidealnarrowband filterwithamagnitude response centered aboutthefrequency fe>asshowninFigure1.9;thatis, [H(f)[={1, 0,[f::':fe[<~Jif [f::':fe'>~Jif(1.40) IH(fll ---------~~-------- FIGURE1.9Magnitude response ofidealnarrowband filter. 46 CHAPTER 1..RANDOM PROCESSES whereI1fisthebandwidth ofthefilter.ThenfromEquation (1.39)wefindthatifthe filterbandwidth !:J.fissufficiently smallcompared tothemidband frequency f,andSx(f) isacontinuous function, themean-square valueofthefilteroutputisapproximately (1.41) (1.42) (1.43) (1.44)Thefilter,however, passesonlythosefrequency components oftheinputrandomprocess X(t)thatlieinsideanarrowfrequency bandofwidthI1fcentered aboutthefrequency :!:.fe.ThusSx(fe)represents thefrequency densityoftheaveragepowerintherandom processX(t),evaluated atthefrequency f=fe.Thedimensions ofthepowerspectral densityaretherefore inwattsperHertz(WIHz). !IIIPROPERTIES OFTHEPOWER SPECTRAL DENSI'IY ThepowerspectraldensitySx(f)andtheautocorrelation function Rx(T)ofastationary processX(t)formaFourier-transform pairwithTandfasthevariables ofinterest,as shownbythepairofrelations Sx(f)=rooRX(T)exp(-j211'fT) dT Rx(T)=rooSx(f)exp(j211'fT) df Equations (1.42)and(1.43)arebasicrelations inthetheoryofspectralanalysisofrandom processes, andtogether theyconstitute whatareusuallycalledtheEinstein- Wiener Khintchine relations.3 TheEinstein- Wiener-Khintchine relations showthatifeithertheautocorrelation function orpowerspectraldensityofarandomprocessisknown,theothercanbefound exactly.Butthese functions displaydifferent aspectsofthecorrelation information about theprocess.Itiscommonly accepted thatforpractical purposes, thepowerspectraldensity isthemoreuseful"parameter." Wenowwishtousethispairofrelations toderivesomegeneralproperties ofthe powerspectraldensityofastationary process. Property 1 Thezero-frequency valueofthepowerspectraldensityofastationary processequalsthe totalareaunderthegraphoftheautocorrelation function; thatis, Sx(O)==rooRx(T)dT Thisproperty followsdirectlyfromEquation (1.42)byputtingf=O. Property 2 Themean-square valueofastationary processequalsthetotalareaunderthegraphof thepowerspectraldensity;thatis, (1.45) Thisproperty followsdirectlyfromEquation (1.43)byputting T=0andnotingthat Rx(O)=E[X2(t)]. 1.7PowerSpectral Density 47 Property 3 Thepowerspectraldensityofastationary processisalwaysnonnegative; thatis, forallf (1.46) Thisproperty isanimmediate consequence ofthefactthat,inEquation (1.41),the mean-square valueE[y2(t)]mustalwaysbenonnegative. Property 4 Thepowerspectraldensityofareal-valued randomprocessisanevenfunctionoffre quency;thatis, (1.47) Thisproperty isreadilyobtained bysubstituting -fforfinEquation (1.42): Sx(-f) =rooRx(T)exp(j2'TTfT) dT Next,substituting -TforT,andrecognizing thatRx(-T)=Rx(T),weget whichisthedesiredresult. Property 5 Thepowerspectraldensity,appropriately normalized, hastheproperties usuallyassociated withaprobability densityfunction. Thenormalization wehaveinmindhereiswithrespecttothetotalareaunderthe graphofthepowerspectraldensity(i.e.,themean-square valueoftheprocess). Consider thenthefunction (1.48) InlightofProperties 2and3,wenotethatpx(f)2:0forallf.Moreover, thetotalarea underthefunction Px(f)isunity.Hence,thenormalized form ofthepowerspectralden sity,asdefinedinEquation (1.48),behavessimilartoaprobability densityfunction. ~ExAMPLE 1.5Sinusoidal WavewithRandom Phase(continued) Consider therandomprocessX(t)=Acos(271"M+0),where®isauniformly distributed randomvariableovertheinterval[-71",71"].Theautocorrelation function ofthisrandompro cessisgivenbyEquation (1.17),whichisreproduced hereforconvenience: A2 Tcos(271"f/r) 48 CHAPTER 1illRANDOM PROCESSES Sx(jJ (1.49) ITI<T ITI~T---~_-t,Lc ------L----t,-!-,----f FIGURE1.10Powerspectraldensityofsinewavewithrandomphase;8(j)denotesthedelta function atf=O. Let8(f)denotethedeltafunction atf=0;forthedefinition ofthedeltafunction andits properties, seeAppendix 2.TakingtheFonriertransform ofbothsidesoftherelationdefining Rx(T),wefindthatthepowerspectraldensityofthesinusoidal processX(t)is A2 "4[8(f-fc)+8(f+f,)] whichconsistsofapairofdeltafunctions weighted bythefactorAz/4andlocatedat±f"as illustrated inFignre1.10.Wenotethatthetotalareaunderadeltafunction isone.Hence, thetotalareaunderSx(f)isequaltoA2/2,asexpected. <\!l fl>-EXAJI'IPLE 1.6Random BinaryWave(continued) Consider againarandom binarywaveconsisting ofasequence ofIsandOsrepresented by thevalues+Aand-A,respectively. InExample 1.3weshowedthattheautocorrelation function ofthisrandomprocesshasatriangular waveform, asshownby {z(ITI)RX(T)=A1 -T' 0, Thepowerspectraldensityoftheprocessistherefore Sx(f)(AZ(1 -1;1)exp(-j211fT) dT UsingtheFonriertransform ofatriangular function (seeTableA6.3),weobtain Sx(f)=AZTsincZ(fT) (1.50) (1.52)whichisplottedinFigure1.11.Hereagainweseethatthepowerspectraldensityisnonneg ativeforallfandthatitisanevenfunction off.NotingthatRx(O)=AZandusingProperty 2,wefindthatthetotalareaunderSx(f),ortheaveragepoweroftherandom binarywave described here,isAZ,whichisintuitively satisfying. <\!l TheresultofEquation (1.50)maybegeneralized asfollows.Wenotethattheenergy spectraldensity(i.e.,thesquaredmagnitude oftheFouriertransform) ofarectangulat pulseg(t)ofamplitude AanddurationTisgivenby 'fog(f)=A2T2sinc2(fT) (1.51) Wemaytherefore rewriteEquation (1.50)intermsof'fog(f)simplyas Sx(f)='fog(f) T 1.7PowerSpectral Density 49 FIGURE1.11Powerspectraldensityofrandombinarywave. Equation (1.52)statesthatforarandom binarywaveinwhichbinarysymbols1and0 arerepresented bypulsesg(t)and-g(t),respectively, thepowerspectral densitySx(f)is equaltotheenergyspectral density<tbg(f)ofthesymbolshapingpulseg(t),dividedbythe symbolduration T. ~EXAMPLE 1.7MixingofaRandom Process withaSinusoidal Process Asituation thatoftenarisesinpracticeisthatofmixing(i.e.,multiplication) ofastationary processX(t)withasinusoidal wavecos(2'lIfct+e),wherethephase@isarandomvariable thatisuniformly distributed overtheinterval[0,21T].Theadditionoftherandomphase@ inthismannermerelyrecognizes thefactthatthetimeoriginisarbitrarily chosenwhen X(t)andCOS(21Tfct+@)comefromphysically independent sources,asisusuallythecase.We areinterested indetermining thepowerspectraldensityoftherandomprocessY(t),defined by Y(t)=X(t)COS(21Tfct+@) (1.53) Usingthedefinition ofautocorrelation function ofastationary processandnotingthatthe randomvariableeisindependent ofX(t),wefindthattheautocorrelation functionofY(t)is givenby Ry(,.)E[Y(t+,.)Y(t)] =E[X(t+,.)COS(21Tfct+21Tfc"+@)X(t)COS(21Tfct+e)] =E[X(t+,.)X(t)]E[cos(21Tfct +21Tf,"+@)cos(21Tfct+e)] =!Rx(,.)E[cos(21Tfc") +COS(41Tfct+21Tf<"+2@)] =!Rx(,.)cos(21Tfc")(1.54) BecausethepowerspectraldensityistheFouriertransform oftheautocorrelation function, wefindthatthepowerspectraldensities oftherandomprocesses X(t)andY(t)arerelatedas follows: (1.55) According toEquation (1.55),thepowerspectraldensityoftherandomprocessY(t)defined inEquation (1.53)isobtained asfollows:WeshiftthegivenpowerspectraldensitySx(f)of randomprocessX(t)totherightbyf"shiftittotheleftbyf"addthetwoshiftedpower spectra,anddividetheresultby4. """i 50 CHAPTER I..RANDOM PROCESSES RELATION AMONG THEPOWER SPECTRAL DENSITIES OFTHEINPUT ANDOUTPUT RANDOM PROCESSES LetSx(f)denotethepowerspectraldensityoftheoutputrandomprocessY(t)obtained bypassingtherandomprocessX(t)through alinearfilteroffrequency response R(f). Then,recognizing bydefinition thatthepowerspectraldensityofarandom processis equaltotheFouriertransform ofitsautocorrelation function andusingEquation (1.32), weobtain Sy(f)=rooRy(T)exp(-j2'TTfT) dT =rooroorooh(Tl)h(T2)R x(T-Tl+T2)exp(-j2'TTf T)dT1dT2dT(1.56) LetT-Tl+T2=TO,or,equivalently, T=TO+Tl-T2'Thenbymakingthissubstitution inEquation (1.56),wefindthatSx(f)maybeexpressed astheproductofthreeterms:the frequency responseR(f)ofthefilter,thecomplex conjugate ofR(f),andthepowerspec traldensitySx(f)oftheinputrandomprocessX(t).WemaythussimplifyEquation (1.56) as Sy(f)=R(f)H*(f)Sx(f) (1.57) Finally,since'R(f) 12=R(f)R*(f),wefindthattherelationship amongthepowerspectral densities oftheinputandoutputrandomprocesses isexpressed inthefrequency domain bywriting (1.58) Equation (1.58)statesthatthepowerspectraldensityoftheoutputprocessY(t)equals thepowerspectraldensityoftheinputprocessX(t)multiplied bythesquaredmagnitude responseofthefilter.Byusingthisrelation, wecantherefore determine theeffectofpassing arandomprocessthrough astable,linear,time-invariant, filter.Incomputational terms, Equation (1.58)isusuallyeasiertohandlethanitstime-domain counterpart ofEquation (1.32),involving theautocorrelation function. IIIRELATION AMONG THEPOWER SPECTRAL DENSny ANDTHEMAGNITUDE SPECTRUM OFASAMPLE FUNCTION WenowwishtorelatethepowerspectraldensitySx(f)directlytothespectralproperties ofasamplefunction x(t)ofastationary processX(t)thatisergodic. Forthesample function x(t)tobeFouriertransformable, however, itmustbeabsolutely integrable; that IS rooIx(t) 1dt<00 (1.59) (1.60)Thiscondition canneverbesatisfied byanystationary samplefunction x(t)ofinfinite duration. InordertousetheFouriertransform technique, weconsider atruncated segment ofx(t),definedovertheobservation interval-T:5t:5T,say.Thus,usingX(f,T)to denotetheFouriertransform ofthetruncated samplefunction sodefined,wemaywrite X(f,T)=fTx(t)exp(-j27Tft) dt (1.61) (1.62) (1.63)1.7PowerSpectral DeNSity 51 Assuming thattheprocessx(t)isalsoergodic,wemayevaluatetheautocorrelation functionRxtr)ofX(t)usingthetime-average formula(seeSection1.5) 1JTRx(r)==~~2T-Tx(t+r)x(t)dt Itiscustomary toviewthesamplefunction x(t)asapowersignal(i.e.,asignalwithfinite averagepower).Hence,wemayformulate thefollowing Fourier-transform pair: 1JT 12T-Tx(t+r)x(t)dt¢2TIX(f,TJ!2 Theparameter ontheleft-hand sideisatime-averaged autocorrelation function. Thepa rameterontheright-hand sideiscalledtheperiodogram, whosedimensions arethesame asthoseofthepowerspectraldensity.Thisterminology isamisnomer, however, sincethe periodogram isafunction offrequency, notperiod.Nevertheless, ithaswideusage.The quantity wasfirstusedbystatisticians tolookforperiodicities suchasseasonal trendsin data. UsingtheformulafortheinverseFouriertransform intheFourier-transform pairof Equation (1.62),wemayexpressthetime-averaged autocorrelation function ofthesample function x(t)intermsoftheperiodogram as 1JT J~12T-Tx(t+r)x(t)dt=-002TIX(f,TJ!2exp(j2TrfT) df Hence,substituting Equation (1.63)into(1.61),weget Rx(r)=~~roo2~IX(f,TJ!2exp(j2Trfr) df (1.64) (1.65) (1.67)Forafixedvalueofthefrequencyf,theperiodogram isarandomvariableinthat itsvaluevariesinarandommannerfromonesamplefunction oftherandomprocessto another. Thus,foragivensamplefunction x(t),theperiodogram doesnotconverge inany statistical sensetoalimitingvalueasTtendstoinfinity.Assuch,itwouldbeincorrect to interchange theorderoftheintegration andlimitingoperations inEquation (1.64).Sup pose,however, thatwetaketheexpectation ofbothsidesofEquation (1.64)overthe ensemble ofallsamplefunctions oftherandomprocessandrecognize thatforanergodic processtheautocorrelation function Rx(r)isunchanged bysuchanoperation. Then,since eachsamplefunction ofanergodicprocesseventually takesonnearlyallthemodesof behavior ofeachothersamplefunction, wemaythuswrite Rx(r)==~~r~2~E[IX(f, TJ!2Jexp(j2Trfr) df Nowwemayinterchange theorderoftheintegration andlimitingoperations andsoobtain Rx(r)=roo{~~2~E[IX(f, TJ!2J}exp(j2Trfr) df (1.66) Hence,comparing Equations (1.66)and(1.43),weobtainthedesiredrelationbetween thepowerspectraldensitySx(f)ofanergodicprocessandthesquaredmagnitude spectrum IX(f,T) 12ofatruncated samplefunction oftheprocess: Sx(f)=~~2~E[IX(f, TJ!2J ==~~2~E[lfTx(t)exp(-j2Trft) df] 52 CHAPTER IIIIRANDOM PROCESSES Itisimportant tonotethatinEquation (1.67)itisnotpossibletoletT~00beforetaking theexpectation. Equation (1.67)provides themathematical basisforestimating4thepower spectraldensityofanergodicrandomprocess,givenasamplefunction x(t)oftheprocess observed overtheinterval[-T,Tj. CROSS-SPECTRAL DENSITIES Justasthepowerspectraldensityprovides ameasure ofthefrequency distribution ofa singlerandomprocess,cross-spectral densities provideameasure ofthefrequency inter relationship betweentworandomprocesses. Inparticular, letX(t)andY(t)betwojointly stationary processes withtheircross-correlation functions denotedbyRxy(T)andRyx(T). Wethendefinethecross-spectral densities SXy(f)andSyx(f)ofthispairofrandompro cessestobetheFouriertransforms oftheirrespective cross-correlation functions, as shownby andSxy(f)=r~Rxy(T)exp(-j2TTfT) dT Syx(f)=r~Ryx(T)exp(-j2TTfT) dT(1.68) (1.69) Thecross-correlation functions andcross-spectral densities thusformFourier-transform pairs.Accordingly, usingtheformula forinverseFouriertransformation wemayalso write andRXy(T) r~SXy(f)exp(j2TTfT) df RYX(T) =r~Syx(f)exp(j2TTfT) df(1.70) (1.71) Thecross-spectral densities SXy(f)andSyx(f)arenotnecessarily realfunctions of thefrequencyf.However, substituting therelationship RXy(T) =RYX(-T) intoEquation (1.68)andthenusingEquation (1.69)wefindthatSXy(f)andSyx(f)are relatedby ~EXAMPLE 1.8Sxy(f)=Syx(-f)=S~x(f) (1.72) Supposethattherandomprocesses X(t}andY(t)havezeromean,andtheyareindividually stationary. Considerthesumrandomprocess Z(t)=X(t)+Y(t) Theproblemistodetermine thepowerspectraldensityofZ(t}. 1.7PowerSpectral Density 53 Theautocorrelation function ofZ(t)isgivenby Rz(t,u)=E[Z(t)Z(u)] =E[(X(t)+Y(t))(X(u)+Y(u))J E[X(t)X(u)] +E[X(t)Y(u)] +E[Y(t)X(u)] +E[Y(t)Y(u)] =Rx(t,u)+Rxy(t,u)+Ryx(t,u)+Ry(t,u) Defining T=tu,wemaytherefore write RZ(T)=RX(T)+RXy(T)+Ryx(T)+Ry(T) (1.73) whentherandomprocesses X(t)andY(t)arealsojointlystationary. Accordingly, takingthe Fouriertransform ofbothsidesofEquation (1.73),weget (1.74) Wethusseethatthecross-spectral densities SXy(f)andSyx(f)represent thespectralcompo nentsthatmustbeaddedtotheindividual powerspectraldensities ofapairofcorrelated randomprocesses inordertoobtainthepowerspectraldensityoftheirsum. Whenthestationary processes X(t)andY(t)areuncorrelated, thecross-spectral densities SXy(f)andSyx(f)arezero,andsoEquation (1.74)reducesasfollows: (1.75) (1.76)Wemaygeneralize thislatterresultbystatingthatwhenthereisamultiplicity ofzero-mean stationary processes thatareuncorrelated witheachother,thepowerspectraldensityoftheir sumisequaltothesumoftheirindividual powerspectraldensities. -<l:I Ii>EXAMPLE 1.9 Consider nexttheproblem ofpassingtwojointlystationary processes throughapairofsep arate,stable,linear,time-invariant filters,asshowninFigure1.12.Inparticular, supposethat therandomprocessX(t)istheinputtothefilterofimpulseresponse hI(t)andthattherandom processY(t)istheinputtothefilterofimpulseresponse hz(t).LetV(t)andZ(t)denotethe randomprocesses attherespective filteroutputs. Thecross-correlation function ofV(t)and Z(t)istherefore Ryz(t,u)=E[V(t)Z(u)] =E[rooh,(T,)X(t-T,)dTlroohz(TZ)Y(uTZ)dTz] =roorooh,(T,)hz(Tz)E[X(t -T,)Y(U-Tz)]dTldTz whereRxy(t,u)isthecross-correlation function ofX(t)andY(t).Becausetheinputrandom processes arejointlystationary (byhypothesis), wemaysetT=t-uandsorewriteEquation (1.76)asfollows: RvZ(T)=r~r~hl!Tl)hz(Tz)RxY(T T,+TZ)dTldTz (1.77) X(tJ-&V(tJ Y<tJ-&Z(tl FIGURE1.12Apairofseparate lineartime-invariant filters. 54 CHAPTER 1..RANDOM PROCESSES TakingtheFouriertransform ofbothsidesofEquation (1.77)andusingaprocedure similartothatwhichledtothedevelopment ofEquation (1.39),wefinallyget (1.78) (1.79) (1.80)whereH,(f)andH2(f)arethefrequency responses oftherespective filtersinFigure1.12,and H;(f)isthecomplexconjugate ofH2(f).Thisisthedesiredrelationship betweenthecross spectraldensityoftheoutputprocesses and thatoftheinputprocesses. <ill I1.8Gaussian Process Thematerial wehavepresented onrandomprocesses uptothispointinthediscussion hasbeenofafairlygeneralnature.Inthissection,weconsider animportant familyof randomprocesses knownasGaussian processes.5 LetussupposethatweobservearandomprocessX(t)foranintervalthatstartsat timet=0andlastsuntilt=T.Suppose alsothatweweighttherandomprocessX(t)by somefunction g(t)andthenintegrate theproductg(t)X(t)overthisobservation interval, therebyobtaining arandomvariableYdefinedby Y=rg(t)X(t)dt WerefertoYasalinearfunctional ofX(t).Thedistinction between afunction anda functional shouldbecarefully noted.Forexample, thesumY=2:;:la)C, wheretheaiare Constants andtheXiarerandomvariables, isalinearfunction oftheXi;foreachobserved setofvaluesfortherandomvariables Xi'wehaveacorresponding valuefortherandom variable Y.Ontheotherhand,inEquation (1.79)thevalueoftherandom variable Y depends onthecourseoftheargument function g(t)X(t)overtheentireobservation in tervalfrom0toT.Thusafunctional isaquantity thatdepends ontheentirecourseof oneormorefunctions ratherthanonanumberofdiscretevariables. Inotherwords,the domainofafunctional isasetorspaceofadmissible functions ratherthanaregionofa coordinate space. IfinEquation (1.79)theweighting function g(t)issuchthatthemean-square value oftherandomvariable Yisfinite,andiftherandomvariable YisaGaussian-distributed randomvariableforeveryg(t)inthisclassoffunctions, thentheprocessX(t)issaidtobe aGaussian process.Inotherwords,theprocessX(t)isaGaussian processifeverylinear functional ofX(t)isaGaussian randomvariable. Wesaythattherandom variable YhasaGaussian distribution ifitsprobability densityfunction hastheform 1[(Y-/LYf]fv(y)=Y21Tuyexp-20{ where/LYisthemeanand0{isthevariance oftherandomvariable Y.Aplotofthis probability densityfunction isgiveninFigure1.13forthespecialcasewhentheGaussian randomvariable Yisnormalized tohaveamean/LYofzeroandavariance0{ofone,as shownby Suchanormalized Gaussian distribution iscommonly writtenas}flO,1). 1.8Gaussian Process 55 0.6 0.4 -3 FIGURE 1.13Normalized Gaussian distribution. AGaussian processhastwomainvirtues.First,theGaussian processhasmany properties thatmakeanalyticresultspossible; wewilldiscusstheseproperties laterinthe section.Second,therandomprocesses produced byphysical phenomena areoftensuch thataGaussian modelisappropriate. Furthermore, theuseofaGaussian modeltodescribe thephysicalphenomena isusuallyconfirmed byexperiments. Thusthefrequent occurrence ofphysicalphenomena forwhichaGaussian modelisappropriate, together with theease withwhichaGaussian processishandledmathematically, maketheGaussian processvery important inthestudyofcommunication systems. CENTRAL LIMITTHEOREM Thecentrallimittheorem provides themathematical justification forusingaGaussian processasamodelforalargenumberofdifferent physical phenomena inwhichthe observed randomvariable, ataparticular instantoftime,istheresultofalargenumber ofindividual randomevents.Toformulate thisimportant theorem, letX;,i=1,2,..., N,beasetofrandomvariables thatsatisfiesthefollowing requirements: 1.TheXiarestatistically independent. 2.TheXihavethesameprobability distribution withmean/Lxandvarianceai. TheXisodescribed aresaidtoconstitute asetofindependently andidentically distributed (i.i.d.)randomvariables. Lettheserandomvariables benormalized asfollows: sothatwehave1Y,= -(Xi-/Lx), CTxi=1,2,..., N andElYi]=0 varlYi]=1 Definetherandomvariable 56 CHAPTER 1..RANDOM PROCESSES Thecentrallimittheoremstatesthattheprobability distribution ofVNapproaches anor malizedGaussian distribution .N'(O,1)inthelimitasthenumberofrandomvariablesN approaches infinity. Itisimportant torealize,however, thatthecentrallimittheorem givesonlythe "limiting" formoftheprobability distribution ofthenormalized randomvariableVNas Napproaches infinity.WhenNisfinite,itissometimes foundthattheGaussian limitgives arelatively poorapproximation fortheactualprobability distribution ofVNeventhough Nmaybequitelarge. !illPROPERTIES OFAGAUSSIAN PROCESS AGaussian processhassomeusefulproperties thataredescribed inthesequeL Property 1 IfaGaussian processX(t)isappliedtoastablelinearfilter,thentherandomprocessY(t) developed attheoutputofthefilterisalsoGaussian. (1.81) O:5t<ooThisproperty isreadilyderivedbyusingthedefinition ofaGaussian processbased onEquation (1.79).Consider thesituation depicted inFigure1.8,wherewehavealinear time-invariant filterofimpulseresponse h(t),withtherandomprocessX(t)asinputand therandomprocessY(t)asoutput.WeassumethatX(t)isaGaussian process.Therandom processes Y(t)andX(t)arerelatedbytheconvolution integral Y(t)rh(t-r)X(r)dr, Weassumethattheimpulseresponse h(t)issuchthatthemean-square valueoftheoutput randomprocessY(t)isfiniteforalltintherange0:5t<00forwhichY(t)isdefined.To demonstrate thattheoutputprocessY(t)isGaussian, wemustshowthatanylinearfunc tionalofitisaGaussian randomvariable. Thatis,ifwedefinetherandomvariable z=rgy(t)rh(tr)X(r)drdt (1.82) thenZmustbeaGaussian randomvariableforeveryfunction gy(t),suchthatthemean squarevalueofZisfinite.Interchanging theorderofintegration inEquation (1.82),we get (1.83) where g(r)=rgy(t)h(t-r)dr (1.84) SinceX(t)isaGaussian processbyhypothesis, itfollowsfromEquation (1.83)thatZ mustbeaGaussian randomvariable. WehavethusshownthatiftheinputX(t)toalinear filterisaGaussian process,thentheoutputY(t)isalsoaGaussian process.Note,however, 1.8Gaussian Process 57 thatalthough ourproofwascarriedoutassuming atime-invariant linearfilter,thisprop ertyistrueforanyarbitrary stablelinearsystem. Property 2 Consider thesetofrandomvariables orsamplesX(tj),X(t2),...,X(tn),obtained by observing arandomprocessX(t)attimestbt2,•••,twIftheprocessX(t)isGaussian, thenthissetofrandomvariables isjointlyGaussial1 foranyn,withtheirn-foldjoint probability densityfunctionbeingcompletely determined byspecifying thesetofmeans /-LXlt,)=E[X(ti)], andthesetofcovariance functionsi=1,2,...,n k,i=1,2,...,n Letthen-by-1vectorXdenotethesetofrandomvariables X(tt),...,X(t.)derivedfrom theGaussian processX(t)bysampling itattimest"...,tn'LetxdenoteavalueofX. According toProperty 2,therandomvectorXhasamultivariate Gaussian distribution definedinmatrixformas !X(tl),...,X(t,)(X" •••,X2)=(21T)~2a1l2 exp(-~(x-...YI-t(x-....)) wherethesuperscript Tdenotestransposition and(1.85) ....=meanvector =[/-L"/-L2,•••,/-L.F I=covariance matrix ={Cx(tk)ti)}k,i-t I-t=inverseofcovariance matrix a=determinant ofcovariance matrixI Property 2isfrequently usedasthedefinition ofaGaussian process.However, this definition ismoredifficulttousethanthatbasedonEquation (1.79)forevaluating the effectsoffilteringonaGaussian process. WemayextendProperty 2totwo(ormore)randomprocesses asfollows.Consider thecomposite setofrandomvariables X(tt),X(t2),•••,X(tn),Y(Ut),Y(U2),'..,Y(um) obtained byobserving arandomprocessX(t)attimes(ti, i=1,2,...,n),andasecond randomprocessY(t)attimes{Uk'k=1,2,...,m}.Wesaythattheprocesses X(t)and Y(t)arejointlyGaussian ifthiscomposite setofrandomvariables isjointlyGaussian for anynandm.Notethatinaddition tothemeanandcorrelation functions oftherandom processes X(t)andY(t)individually, wemustalsoknowthecross-covariance function E[(X(ti)-/-LX(til)(Y(Uk) -/-LY(ukl)]=Rxy(ti,Uk)-/-LX(til/-LY(Ukl foranypairofobservation instants(t"Uk)'Thisadditional knowledge isembodied inthe cross-correlation function, Rxy(t"Uk),ofthetwoprocesses X(t)andY(t). Property 3 IfaGaussian processisstationary, thentheprocessisalsostrictlystationary. ThisfollowsdirectlyfromProperty 2. 58 CHAPTER 1ISRANDOM PROCESSES Property 4 Iftherandomvariables X(t1),X(tz],...,X(t,J,obtained bysampling aGaussian process X(t)attimest1,tb•••,tn,areuncorrelated, thatis, i"*k thentheserandomvariables arestatistically independent. Theuncorrelatedness ofX(t1),•••,X(tn)meansthatthecovariance matrixIisa diagonal matrixasshownby lui0] 1=01... ° u~ where a}=E[(X(ti)-E[X(ti)]n i=1,2,...,n Underthiscondition, themultivariate Gaussian distribution ofEquation (1.85)simplifies to n !x(x)=II!X(Xi)i=l I whereXi=X(ti)and 1((Xi-J-Lxi)!X,(Xi)=,~exp- 2 2 V2'1TUi Ui Inwords,iftheGaussian randomvariables X(t,),...,X(tn)areuncorrelated, thenthey arestatistically independent, which,inturn,meansthatthejointprobability densityfunc tionofthissetofrandomvariables canbeexpressed astheproductoftheprobability densityfunctions oftheindividual randomvariables intheset. I1.9Noise Thetermnoiseisusedcustomarily todesignate unwanted signalsthattendtodisturbthe transmission andprocessing ofsignalsincommunication systemsandoverwhichwehave incomplete control.Inpractice, wefindthattherearemanypotential sourcesofnoisein acommunication system.Thesourcesofnoisemaybeexternal tothesystem(e.g.,at mospheric noise,galacticnoise,man-made noise),orinternaltothesystem.Thesecond category includesanimportant typeofnoisethatarisesfromspontaneous fluctuations of currentorvoltageinelectrical circuits.6Thistypeofnoiserepresents abasiclimitation on thetransmission ordetection ofsignalsincommunication systemsinvolving theuseof electronic devices.Thetwomostcommon examples ofspontaneous fluctuations inelec tricalcircuitsareshotnoiseandthermalnoise,whicharedescribed inthesequel. Ii!SHOTNOISE Shotnoisearisesinelectronic devicessuchasdiodesandtransistors becauseofthediscrete natureofcurrentflowinthesedevices.Forexample, inaphotodetector circuitacurrent (1.86)1.9Noise 59 pulseisgenerated everytimeanelectronisemittedbythecathodeduetoincidentlight fromasourceofconstant intensity. Theelectrons arenaturally emittedatrandomtimes denotedbyTbwhere-00<k<00.Itisassumed thattherandomemissions ofelectrons havebeengoingonforalongtime.Thus,thetotalcurrentflowingthroughthephoto detectormaybemodeled asaninfinitesumofcurrentpulses,asshownby X(t)=2:h(t-Tk) k~-~ whereh(t-Tk)isthecurrentpulsegenerated attimeTk.TheprocessX(t)definedby Equation (1.86)isastationary processcalledshotnoise. Thenumberofelectrons, N(t),emittedinthetimeinterval[0,t]constitutes adiscrete stochastic process,thevalueofwhichincreases byoneeachtimeanelectronisemitted. Figure1.14showsasamplefunction ofsuchaprocess.Letthemeanvalueofthenumber ofelectrons, v,emittedbetweentimestandt+tobedefinedby E[v]=Ato (1.87) Theparameter Aisaconstant calledtherateoftheprocess.Thetotalnumberofelectrons emittedintheinterval [t,t+toJ,thatis, v=N(t+to)-N(t) followsaPoissondistribution withameanvalueequaltoAto.Inparticular, theprobability thatkelectrons areemittedintheinterval [t,t+to]isdefinedby (Ato)kP(v=k)=--e-oI.tok=0,1,.. . (1.88)k! Unfortunately, adetailedstatistical characterization oftheshot-noise processX(t) definedinEquation (1.86)isadifficultmathematical task.Herewesimplyquotetheresults pertaining tothefirsttwomoments oftheprocess: I>-ThemeanofX(t)is /-Lx=Ar~h(t)dt whereAistherateoftheprocessandh(t)isthewaveform ofacurrentpulse.(1.89) N(t) 6 4r I I,...---, I I I I .....-< I I I I I I I_II I I I I I I I II I I I r---1 I I I I I I I II I I I I ,...-I I I I I I I I I I I I I I I I I o FIGURE1.14Samplefunction ofaPoissoncounting process. 60 CHAPTER 110RA1'lDOM PROCESSES Il>Theautocovariance function ofX(t)is CX(7)=Ar~h(t)h(t+7)dt (1.90) ThissecondresultisknownasCampbell's theorem. Forthespecialcaseofawaveform h(t)consisting ofarectangular pulseofamplitude Aandduration T,themeanoftheshot-noise processX(t)isAAT,anditsautocovariance functionis whichhasatriangular formsimilartothatshowninFigure1.7. IIITHERMAL NOISE Thermalnoiseisthenamegiventotheelectrical noisearisingfromtherandommotionof electrons inaconductor. Themean-square valueofthethermalnoisevoltageVTNap pearingacrosstheterminals ofaresistor,measured inabandwidth of!1fHertz,is,forall practical purposes, givenby (1.91) (1.92)wherekisBoltzmann's constant equalto1.38X10-23joulesperdegreeKelvin,Tisthe absolute temperature indegreesKelvin,andRistheresistance inohms.Wemaythus modelanoisyresistorbytheThevenin equivalent circuitconsisting ofanoisevoltage generator ofmean-square valueE[V}Nlinserieswithanoiseless resistor, asinFigure LISa.Alternatively, wemayusetheNortonequivalent circuitconsisting ofanoisecurrent generator inparallelwithanoiseless conductance, as inFigure1.1Sb.Themean-square valueofthenoisecurrentgenerator is E[I}Nl=;2E[VhJl =4kTG!1famps2 whereG=1/Ristheconductance. Itisalsoofinteresttonotethatbecausethenumber ofelectrons inaresistorisverylargeandtheirrandommotions insidetheresistorare statistically independent ofeachother,thecentrallimittheorem indicates thatthermal noiseisGaussian distributed withzeromean. R (a) (b)G FIGURE1.15Modelsofanoisyresistor.(a)Thevenin equivalent circuit.(b)Nortonequivalent circuit. (1.93)1.9Noise 61 Noisecalculations involvethetransferofpower,andsowefindthattheuseofthe maximum-power transfertheorem isapplicable tosuchcalculations. Thistheorem states thatthemaximum possiblepoweristransferred fromasourceofinternalresistance Rto aloadofresistance R[whenR[=R.Underthismatched condition, thepowerproduced bythesourceisdividedequallybetweentheinternalresistance ofthesourceandtheload resistance, andthepowerdelivered totheloadisreferredtoastheavailable power.Ap plyingthemaximum-power transfertheoremtotheThevenin equivalent circuitofFigure 1.15aortheNortonequivalent circuitofFigure1.15b,wefindthatanoisyresistorpro ducesanavailable noisepowerequaltokT6./watts. iilWHITE NOISE Thenoiseanalysisofcommunication systemsiscustomarily basedonanidealized form ofnoisecalledwhitenoise,thepowerspectraldensityofwhichisindependent ofthe operating frequency. Theadjective whiteisusedinthesensethatwhitelightcontains equal amounts ofallfrequencies withinthevisiblebandofelectromagnetic radiation. Weexpress thepowerspectraldensityofwhitenoise,withasamplefunction denotedbywIt),as NoSw(f)=2 whichisillustrated inFigure1.16a.Thedimensions ofNoareinwattsperHertz.The parameter Noisusuallyreferenced totheinputstageofthereceiverofacommunication system. It maybeexpressed as No=kTe (1.94) wherekisBoltzmann's constant andTeistheequivalent noisetemperature ofthereceiver.7 Theequivalent noisetemperature ofasystemisdefinedasthetemperature atwhicha noisyresistorhastobemaintained suchthat,byconnecting theresistortotheinputofa noiseless versionofthesystem,itproduces thesameavailable noisepowerattheoutput ofthesystemasthatproduced byallthesourcesofnoiseintheactualsystem.Theim portantfeatureoftheequivalent noisetemperature isthatitdependsonlyontheparam etersofthesystem. Sincetheautocorrelation function istheinverseFouriertransform ofthepower spectraldensity,itfollowsthatforwhitenoise Sw(f}(1.95) ------..JoL-----t (a)o (b) FIGURE 1.16Characteristics ofwhitenoL.e.(a)Powerspectral density.(b)Autocorrelation function. 62 CHAPTER 1"RANDOM PROCESSES Thatis,theautocorrelation function ofwhitenoiseconsistsofadeltafunction weighted bythefactorNo/2andoccurring at7"=0,asinFigure1.16b.WenotethatRw(7")iszero forT*"O.Accordingly, anytwodifferent samples ofwhitenoise,nomatterhowclosely together intimetheyaretaken,areuncorrelated. Ifthewhitenoisew(t)isalsoGaussian, thenthetwosamples arestatistically independent. Inasense,whiteGaussian noiserep resentstheultimate in"randomness." Strictlyspeaking, whitenoisehasinfiniteaveragepowerand,assuch,itisnotphys icallyrealizable. Nevertheless, whitenoisehassimplemathematical properties exemplified byEquations (1.93)and(1.95),whichmakeitusefulinstatistical systemanalysis. Theutilityofawhitenoiseprocessisparalleltothatofanimpulsefunction ordelta function intheanalysisoflinearsystems. Justaswemayobservetheeffectofanimpulse onlyafterithasbeenpassedthrough asystemwithafinitebandwidth, soitiswithwhite noisewhoseeffectisobserved onlyafterpassingthrough asimilarsystem.Wemaystate, therefore, thataslongasthebandwidth ofanoiseprocessattheinputofasystemis appreciably largerthanthatofthesystemitself,thenwemaymodelthenoiseprocessas whitenoise. ExAMPLE 1.10IdealLow-Pass Filtered "WhiteNoise SupposethatawhiteGaussian noisew(t)ofzeromeanandpowerspectraldensityNo/2is appliedtoanideal[ow-passfilterofbandwidth Bandpassband magnitude responseofone. Thepowerspectraldensityofthenoisen(t) appearing atthefilteroutputistherefore (see Figure1.17) -B<f<B IfI>B(1.96) (1.97)Theautocorrelation functionofn(t)istheinverseFourierrransform ofthepowerspectral densityshowninFigure1.17a: fBN RNlT)=-Biexp(j2r.fT) df =NoBsinc(2BT) Thisautocorrelation functionisplottedinFigure1.17b.WeseethatRN(7")hasitsmaximum valueofNoBattheorigin,anditpassesthroughzeroatT=:tk/2B,wherek=1,2,3,.... No 2 ------"=B----:---B-'----- f {al (bJ FIGURE1.17Characteristics oflow-passfilteredwhitenoise.(a)Powerspectraldensity.(b)Auto correlation function. 1.9Noise 63 Sincetheinputnoisewit)isGaussian (byhypothesis), itfollowsthattheband-limited noisenit)atthefilteroutputisalsoGaussian. Supposenowthatn(t)issampledattherateof 2Btimespersecond.FromFigure1.17b,weseethattheresulting noisesamplesareuncor relatedand,beingGaussian, theyarestatistically independent. Accordingly, thejointproba bilitydensityfunction ofasetofnoisesamplesobtained inthiswayisequaltotheproduct oftheindividual probability densityfunctions. Notethateachsuchnoisesamplehasamean ofzeroandvarianceofNoB. ... l'>ExAMPLE 1.11Correlation of\Vhite NoisewithaSinusoidal Wave Consider thesamplefunction w'(t)=J?rrw(t)COS(27Tj,t) dt (1.98) whichistheoutputofacorrelator withwhiteGaussian noisew(t)andsinusoidal wave V2ffCOS(27Tfct) asinputs;thescalingfactorV2ffisincluded heretomakethesinusoidal waveinputhaveunitenergyovertheinterval0:0::t:0::T.(Thisproblem wasencountered in theBackground andPreviewchapterbutwasnotelaborated onatthattime.)Withthe noise wit)havingzeromean,itimmediately followsthatthecorrelator outputw'(t)haszeromean, too.Thevariance ofthecorreiatoroutputisdefinedby rT=E[~foTfoTw(t,)COS(27Tj,tj)w(t 2)COS(27Tj,t2) dtjdt2] 2ITfT=Too E[W(t,)W(t2)]COS(27Tfct j)COS(27Tfct2) dtjdt2 =~rrRw(tht2)COS(21Tjij) COS(21Tjh) dtjdt2 whereRw(tJ,t2)istheautocorrelation function ofthewhitenoisew(t).ButfromEquation (1.95): whereNo/2isthepowerspectraldensityofthewhitenoisew(t).Accordingly, wemaysimplify theexpression forthevariancerTas rT=No..?:.ITITo(tjtJCOS(21Tj,t ,)COS(21Tfct2) dt,dt22Too Wenowinvokethesiftingproperty ofthedeltafunction, namely, roog(t)o(t)dt=g(O) whereg(t)isacontinuous functionoftime,assuming thevalueg(O)attimet=O.Hence,we mayfurthersimplifyrTas N2ITrT=-f'T0cos2(27Tj,t)dt No 2(1.99) whereitisassumed thatthefrequencyj,ofthesinusoidal waveinputisanintegermultiple ofthereciprocal ofT. ... 64 CHAPTER I'"RANDOM PROCESSES n(t) r=t------1 "- / / w W FIGURE LIS(a)Powerspectraldensityofnarrowband noise.(b)Samplefunction ofnarrow bandnoise. I1.10Narrowband Noise Thereceiverofacommunication systemusuallyincludessomeprovision forpreprocessing thereceived signal.Thepreprocessing maytaketheformofanarrowband filterwhose bandwidth isjustlargeenoughtopassthemodulated component ofthereceivedsignal essentially undistorted butnotsolargeastoadmitexcessive noisethroughthereceiver. Thenoiseprocessappearing attheoutputofsuchafilteriscallednarrowband noise.With thespectralcomponents ofnarrowband noiseconcentrated aboutsomemidband fre quency±fcasinFigure1.18a,wefindthatasamplefunctionn(t) ofsuchaprocessappears somewhat similartoasinewaveoffrequency fe,whichundulates slowlyinbothamplitude andphase,asillustrated inFigure1.18b. Toanalyzetheeffectsofnarrowband noiseontheperformance ofacommunication system,weneed·amathematical representation ofit.Depending ontheapplication of interest,therearetwospecificrepresentations ofnarrowband noise: 1.Thenarrowband noiseisdefinedintermsofapairofcomponents calledthein-phase andquadrature components. 2.Thenarrowband noiseisdefinedintermsoftwoothercomponents calledtheen- velopeandphase. Thesetworepresentations aredescribed inwhatfollows.Fornowitsufficestosaythat giventhein-phaseandquadrature components, wemaydetermine theenvelope andphase components, andviceversa.Moreover, intheir own individual ways,thetworepresen tationsarenotonlybasictothenoiseanalysisofcommunication systemsbutalsotothe characterization ofnarrowband noiseitself. 1.11Representation ofNarrowband Noise inTermsofIn-Phase andQuadratu.re Components Consider anarrowband noisen(t)ofbandwidth 2Bcenteredonfrequency fe,asillustrated inFigure1.18.Inlightofthetheoryofband-pass signalsandsystemspresented inAp pendix2,wemayrepresent n(t)inthecanonical (standard) form: n(t)=nI(t)cos(271'fctJ -ndt)sin(271'fJ) (1.100) 1.11In-Phase andQuadrature Ctmtponents 65 wherenj(t)iscalledthein-phase component ofn(t),andnQ(t)iscalledthequadrature component ofn(t).Bothnj(t)andnQ(t)arelow-pass signals.Exceptforthemidband frequency fe,thesetwocomponents arefullyrepresentative ofthenarrowband noisenit). Giventhenarrowband noisenit),wemayextractitsin-phase andquadrature com ponentsusingtheschemeshowninFigure1.19a.Itisassumedthatthetwolow-pass filters usedinthisschemeareideal,eachhavingabandwidth equaltoB(i.e.,one-halftheband widthofthenarrowband noisen(t)).TheschemeofFigure1.19afollowsfromtherep resentation ofEquation (1.100).Wemay,ofcourse,usethisequation directlytogenerate thenarrowband noisenit),givenitsin-phase andquadrature components, asshownin Figure1.19b.TheschemesofFigures1.19aand1.19bmaythusbeviewedasnarrowband noiseanalyzer andsynthesizer, respectively. Thein-phase andquadrature components ofanarrowband noisehaveimportant properties thataresummarized here: 1.Thein-phasecomponent n,(t)andquadrature component nQ(t)ofnarrowband noise n(t)havezeromean. 2.Ifthenarrowband noisen(t)isGaussian, thenitsin-phase component n,(t)and quadrature component nQ(t)arejointlyGaussian. 3.Ifthenarrowband noisen(t)isstationary, thenitsin-phase component nj(t)and quadrature component nQ(t)arejointlystationary. 4.Boththein-phasecomponent nj(t)andquadrature component nQ(t)havethesame powerspectraldensity,whichisrelatedtothepowerspectraldensitySN(f)ofthe narrowband noisen(t)as (1.102) -B:5f:5B otherwiseSNI(f)=SNQ(f)={SON,(f-fe)+SN(f+fel,-B:5f:5B(1.101) otherwise whereitisassumedthatSN(f)occupies thefrequency intervalfc-B:5IfI:5fe+B, andfe> B. 5.Thein-phasecomponent nj(t)andquadrature component nQ(t)havethesamevari anceasthenarrowband noisen(t). 6.Thecross-spectral densityofthein-phase andquadrature components ofnarrow bandnoisen(t)ispurelyimaginary, asshownby SNjNQ(f) =-SNQNI(f) ={j[SN(f+fe)-SN(f-fc)], 0, nit) n,(t) n(t) 2cos(2TrfcP n(t) nQW nQ(t) -2sin(2rrfct) sin(2rrfct) (al (b) FIGURE1.19(a)Extraction ofin-phase andquadrature components ofanarrowband process. (b)Generation ofanarrowband processfromitsin-phase andquadrature components. 66 CHAPTER 1IIIRANDOM PROCESSES 7.Ifthenarrowband noisen(t)isGaussian anditspowerspectral densitySN(t)issym metricaboutthemid-band frequencytothenthein-phase component nr(t)and quadrature component nQ(t)arestatistically independent. Forfurtherdiscussions oftheseproperties, thereaderisreferred toProblems 1.28and 1.29. ill>ExAMPLE 1.12IdealBand-Pass Filtered \\'biteNoise Consider awhiteGaussian noiseofzeromeanandpowerspectraldensityNo/2,whichis passedthroughanidealband-pass filterofpassband magnitude response equaltoone,mid bandfrequency!" andbandwidth 2B.Thepowerspectraldensitycharacteristic ofthefiltered noisen(t)willtherefore beasshowninFigure1.20a.Theproblem istodetermine theauto correlation functions ofnit)anditsin-phaseandquadrature components. Theautocorrelation function ofnit)istheinverseFouriertransform ofthepowerspec traldensitycharacteristic showninFigure1.20a: J-UBN (o+BN RN(T)=-Ie-Biexp(j27rfT) df+JI,-Biexp(j21T!T) d! =NoBsinc(2BT)[exp(-j21T!cT) +exp(j21T!cT)] (1.103) =2NoBsinc(lBT) COS(21ThT) whichisplottedinFigure1.20b. Thespectraldensitycharacteristic ofFigure1.20aissymmetric about±!,.Therefore, wefindthatthecorresponding spectraldensitycharacteristic ofthein-phasenoisecomponent __-I------J,----.JL..- ..1.-- -I---+---.J!-_1No 2 ~---r------- ------- (a) ---'---'---'---1-BaB (b) (c) FIGURE1.20Characteristics ofidealband-pass filteredwhitenoise.(a)Powerspectraldensity. (b)Autocorrelation function. (e)Powerspectraldensityofin-phase andquadrature components. 1.12Envelope atulPhase Components 67 nr(t)orthequadrature noisecomponent nQ(t)isasshowninFigure1.21c.Theautocorrelation functionofnr(t)ornQ(t)istherefore (seeExample 1.10): RN/r)=RNQ(r) 2NoBsinc(2Br) (1.104) -<ll 1.12Representation ofNarrowband Noise inTermsofEn-velope andPhaseComponents InSection1.11weconsidered therepresentation ofanarrowband noisen(t)intermsof itsin-phase andquadrature components. Wemayalsorepresent thenoisen(t)intermsof itsenvelope andphasecomponents asfollows: where andn(t)=r(t)COS[21T.fct +"'(t)] r(t)=[nf(t)+nt(tWl2(1.105) (1.106) "'(t)=tan-1[ndt)] (1.107) nr(t) Thefunctionr(t)iscalledtheenvelope ofn(t),andthefunction if!(t)iscalledthephaseof n(t). Theenvelope r(t)andphase"'(t)arebothsamplefunctions oflow-pass random processes. Asillustrated inFigure1.18b,thetimeintervalbetweentwosuccessive peaks oftheenvelope r(t)isapproximately liB,where2Bisthebandwidth ofthenarrowband noisen(t). Theprobability distributions ofr(t)andif!(t)maybeobtained fromthoseofn,(t) andnQ(t)asfollows.LetN,andNQdenotetherandomvariables obtained byobserving (atsomefixedtime)therandomprocesses represented bythesamplefunctions n,(t)and nQ(t),respectively. WenotethatN1andNQareindependent Gaussian randomvariables ofzeromeanandvariancecr,andsowemayexpresstheirjointprobability densityfunc tionby 1(n;+nt)fNj.NQ(n" nQl=21Tcrexp-~ (1.108) Accordingly, theprobability ofthejointeventthatN,liesbetween n1andn,+dn1and thatNQliesbetween nQandnQ+dnQ(i.e.,thepairofrandomvariablesN,andNQlies jointlyinsidetheshadedareaofFigure1.21a)isgivenby fN1.NQ(n" nQldn,dnQ=2:crexp(-nf2~nt)dn1dnQ (1.109) Definethetransformation (seeFigure1.21a) n,=rcosif! (1.110) nQ=rsin'" (1.111) Inalimitingsense,wemayequatethetwoincremental areasshownshadedinFigures 1.21aand1.21bandthuswrite (1.112) 68CHAPTER 1'"RANDOM PROCESSES (al (b] FIGURE 1.21Illustrating thecoordinate systemforrepresentation ofnarrowband noise:(a)in termsofin-phase andquadrature components, and(b)intermsofenvelope andphase. Now,letRandqrdenotetherandomvariables obtained byobserving (atsometimet)the randomprocesses represented bytheenvelope r(t)andphaseifi(t},respectively. Then, substituting Equations (1.110}-(1.112) into(1.109),wefindthattheprobability ofthe randomvariables RandqrlyingjointlyinsidetheshadedareaofFigure1.21bisequalto r(r2 ) --exp--drdifi 27TCT221? Thatis,thejointprobability densityfunction ofRandqris r(r2 )!R,'I'(r,ifi}=2TTl?exp-21? (1.113) Thisprobabilitydensityfunction isindependent oftheangleifi,whichmeansthatthe randomvariables Randqrarestatistically independent. Wemaythusexpress!R,'I'(r,ifi) astheproductof!R(r}and1",(ifi}.Inparticular, therandomvariableqrrepresenting phase isuniformly distributed insidetherange0to2TT,asshownby O:s;ifi:S;2TT elsewhere(1.114) Thisleavestheprobability densityfunction oftherandomvariableRas r2:0 elsewhere(1.115) whereI?isthevariance oftheoriginalnarrowband noisen(t}.Arandomvariablehaving theprobability densityfunctionofEquation (1.115)issaidtobeRayleigh distributed.8 Forconvenience ofgraphical presentation, let rv= IT(1.116) (1.117) 1.13SineWavePlusNarrowband Noise 69 0.8 FIGURE1.22Normalized Rayleigh distribution. ThenwemayrewritetheRayleigh distribution ofEquation (1.115)inthenormalized form v;?0 elsewhere(1.118) Equation (1.118)isplottedinFigure1.22.Thepeakvalueofthedistribution Iv(v)occurs atv=1andisequalto0.607.Notealsothat,unliketheGaussian distribution, theRayleigh distribution iszerofornegative valuesofv.Thisisbecausetheenveloper(t)canassume onlynonnegative values. I1.13SineWavePlusNarrowband Noise Suppose nextthatweaddthesinusoidal waveACOS(2'T1fet) tothenarrowband noisenit), whereAandIearebothconstants. Weassumethatthefrequency ofthesinusoidal wave isthesameasthenominal carrierfrequency ofthenoise.Asamplefunction ofthesinu soidalwaveplusnoiseisthenexpressed by x(t)=Acos(2-rrfct) +nit) (1.119) Representing thenarrowband noisenit)intermsofitsin-phase andquadrature compo nents,wemaywrite wherex(t)=nj(t)cos(2-rrfct) -nQ(t)sin(2-rrfct) nj(t)=A+nr(t)(1.120) (1.121) (1.122)Weassumethatnit)isGaussian withzeromeanandvariancecr.Accordingly, wemay statethefollowing: 1.Bothn;(t)andndt)areGaussian andstatistically independent. 2.ThemeanofnHt)isAandthatofnQ(t)iszero. 3.Thevariance ofbothnl(t)andnQ(t)iscr. Wemaytherefore expressthejointprobability densityfunction oftherandomvariables N;andNQ,corresponding ton;(t)andnQ(t),asfollows: 1[(n'-A)2+n2]INi,NQ(n;, nd=2-rru2exp- r2cr Q 70CHAPl'ER 1IIIRANDOM PROCESSES Letr(t)denotetheenvelope ofx(t)andI/J(t)denoteitsphase.FromEquation (1.120), wethusfindthat andr(t)=([nl(t}F+n~(tWI2 I/J(t)=tan-1[ndt)]n;(t)(1.123) (1.124) (1.125)Following aprocedure similartothatdescribed inSection1.12forthederivation ofthe Rayleigh distribution, wefindthatthejointprobability densityfunction oftherandom variables Rand'1',corresponding tor(t)andI/J(t)forsomefixedtimet,isgivenby r(r2+A2 -2ArcosI/J)!R.",(r, I/J)=271'trexp- 2tr Weseethatinthiscase,however, wecannotexpressthejointprobability densityfunction !R,'I!(r, I/J)asaproduct!R(r)!'I!(I/J). Thisisbecausewenowhaveaterminvolving thevalues ofbothrandom variables multiplied together asrcosI/J.Hence,Rand'I'aredependent randomvariables fornonzerovaluesoftheamplitude Aofthesinusoidal wavecomponent. Weareinterested, inparticular, intheprobability densityfunction ofR.Todetermine thisprobability densityfunction, weintegrate Equation (1.125)overallpossiblevaluesof I/Jobtaining themarginal density 12" !R(r)=0!R,'I!(r, I/J)dI/J r(r2+A2 )12 "(Ar)=271'trexp-~ 0exptrcosI/JdI/J(1.126) Theintegralintheright-hand sideofEquation (1.126)canbeidentified intermsofthe defining integralforthemodified Besselfunctionofthefirstkindofzeroorder(seeAp pendix3);thatis, 1rh Io(x)=271'Joexp(xcosI/J)dI/J Thus,lettingx=Ar/tr,wemayrewriteEquation (1.126)inthecompact form: r(r2+A2)(Ar)A(r)=trexp-~ 10(J'2(1.127) (1.128) ThisrelationiscalledtheRiciandistribution.9 AswiththeRayleigh distribution, thegraphical presentation oftheRiciandistribu tionissimplified byputting rv= (J' Aa= (J'(1.129) (1.130) (1.131) 1.14C.....puteTExperiments: F1at-Fadi ..gcha....el71 8 (1.132)FIGURE1.23Nonnalized Riciandistribution. ThenwemayexpresstheRiciandistribution ofEquation (1.128)inthenormalized form (t?+a2 )fv(v)=vexp---2- 10(av) whichisplottedinFigure1.23forthevalues0,1,2,3,5,oftheparameter a.Basedon thesecurves,wemaymakethefollowing observations: 1.Whenaiszero,theRiciandistribution reducestothe~Rayleigh distribution. 2.Theenvelope distribution isapproximately Gaussian inthevicinityofv=awhena islarge,thatis,whenthesine-wave amplitude Aislargecompared withu,thesquare rootoftheaveragepowerofthenoisen{t). 1.14Computer Experiments: Flat-Fading Channel Inthissectionweusecomputer simulations tostudyamultipath channelcharacterized by Rayleigh fading,examples ofwhichariseinwirelesscommunications andlong-range radio transmission viatheionosphere. Fadingoccursbecauseofinterference between different versions ofthetransmitted signal,whichreachthereceiveratcorrespondingly different times.Thenetresultisthatthereceived signalcanvarywidelyinbothamplitude and phase.Undercertainconditions, thestatistical time-varying natureofthereceivedsignal's envelope iscloselydescribed byaRayleigh distribution asdemonstrated herein. Figure1.24presentsamodelofamultipath channel.Itconsistsofalargecollection ofscatterers randomly positioned inspace,whereby asingleincidentbeamisconverted intoacorrespondingly largenumberofscattered beamsatthereceiving antenna. The transmitted signalissetequaltoACOS(27Tfct). Itisassumed thatallthescattered beams travelatthesamemeanvelocity. However, theydifferfromeachotherinamplitude and phasebyvirtueofdifferences inpathlossandpathdelay.Thusthekthscattered beamis ,givenbyAkCOS(27Tfct+Elk),wheretheamplitude AkandphaseElkarerandomvariables thatvaryslowlywithtime.Moreover, theElkareallindependent ofoneanotherand 72 CHAPTER 1'"RANDOM PROCESSES Incident beam Transmitting antennaRandom mediumScattered beams Receiving antenna FIGURE1.24Modelofamultipath channel. uniformly distributed insidetheinterval[0,21T].Thetypeoffadingexhibited bythemul tipathchanneldescribed hereinisreferredtoas"flatfading"becausethespectralchar acteristics ofthetransmitted signalarecompletely preserved atthechanneloutput.How ever,thestrengthofthechanneloutputchangeswithtimeduetorandomfluctuations in thegainofthechannelcausedbythemultipath phenomenon. Summing thecontributions ofallthescatterers, assumed tobeNinnumber,wemay expresstherandomprocessrepresenting thereceivedsignalas N X(t)=2:AkCOS(21Tfct+Ok) k~l whichmayberewriten intheequivalent form X(t)=XICOS(21Tfct) -XQsin(21T{ct) wheretheXIandXQarerespectively definedby N XI=2:AkcosOk k~l and N XQ=2:AksinOk k~l(1.33) (1.134) (1.135) (1.136) Forconvenience ofpresentation andwithoutlossofgenerality, wemayassumethatAk liesintheclosedinterval[-1,1]forallk. Experiment 1.Gaussian Distributions Fromthecentrallimittheorem wenotethatasthenumberofscatterers, N,approaches infinity,bothXIandXQshouldapproach Gaussian randomvariables. Totestthevalidity ofthisstatement, theprobability distributions ofthein-phase component XIandquad raturecomponent XQarecomputed forN=10,100,1000,and 10,000. Totestthe validityofthecentrallimittheorem, weneedameasure ofthegoodness-of-fit thattests theequivalence ofthemeasured probability distribution ofthesampled dataforvarying 1.14Computer Experiments: Flat-Fading Channel 73 (1.137) f31Ntothetheoretical Gaussian distribution. Onewayofperforming suchatestistouse centralmoments ofadistribution (uptoorder4)todefinethefollowing twoparameters: J-L~ JJJ and (1.138)f3-J-L4 2 - J-L~ whereJ-Lz,J-L3'andJ-L4arethesecond,third,andfourthcentralmoments, respectively. The parameters f3,andf32togetherprovideameasure oftheskewness ofthedistribution under test.Thecloserthevaluesf31andf32forthemeasured distribution aretothecorresponding onesforthetheoretical distribution, thebetteristhegoodness-of-fit. ForaGaussian ran domvariableXofmeanJ-Lxandvarianceoiwehave J-L2=oi J-L3=0 J-L4=3<T3c whichyield /31=0 and f3z=3 Table1.1presents thevaluesoff3,andf32computed forboththein-phase component XI andquadrature component XQforvaryingN.Comparing thesevalueswiththecorre sponding onesforaGaussian distribution, weclearlyseethatasthenumberofscatterers, N,increases thedistributions ofbothXIandXQdoapproach azero-mean Gaussian distribution inaccordance withthecentrallimittheorem. ITABLE1.1 ~Valuesforin-phase andquadrature components (a)Measured Distribution NumberofScatterers, N 10 100 1000 10,000 In-phase component XI f31 0.2443 0.0255 0.0065 0.0003 f32 2.1567 2.8759 2.8587 3.0075 Quadrature component XQ f31 0.0874 0.0017 0.0004 0.0000 f32 1.9621 2.7109 3.1663 3.0135 (b)Theoretical Distribution: Gaussian f31=0 f32=3 74 CHAPTER 1!!lRANDOM PROCESSES Experiment 2.Rayleigh Distribution InEquation (1.134)therandomprocessX(t)isexpressed intermsofitsin-phase and quadrature components. Equivalently, wemayexpressX(t)intermsofitsenvelope and phaseas where andX(t)=RCOS(27Tfct+'1') R=YX;+X~(1.139) (1.140) (1.141) Notethatintheexperiments considered herethein-phase component XI'quadrature component XQ,envelope R,andphase'I'areallindependent oftime. IfXIandXQapproach Gaussian randomvariables forincreasing N,thenfromthe theorypresented inSection1.12wenotethattheenvelope Rwillapproach aRayleigh distribution, andthephase'I'willapproach auniform distribution. InFigure1.25we presenttheactualprobability densityfunction ofrfordatagenerated forthecaseof N=10,000,with100histograms and100ensemble averages beingcomputed. Thisfigure alsoincludes thetheoretical curve.Thereiscloseagreement between thesetwocurves, substantiating theassertion thattheenvelope Rofthereceivedsignalapproaches aRay leighdistribution. Figure1.26illustrates theeffectofRayleigh fadingonthewavefonn ofthereceived signalx(t),asamplefunction ofX(t),forthecaseofasinusoidal transmitted signalwith unitamplitude (i.e.,A=1)andfrequency fc=1MHz.Specifically, thetransmitted signal andthecorresponding receivedsignalareshowninpartsaandbofFigure1.26,respec tively.Comparing thesetwowaveforms, weseethattransmission throughthemultipath 0.7,-------r-,----,--,-------r-,----,--,-------r-, FIGURE 1.25Probability densityfunction oftheenvelope ofrandomprocessX(t):comparing theoryandexperiment. 1.15SummaryandDiscussion 75 4.55 X1063.54 1.50.5 ~-c .-E0 ~«0.5 1 00.5 22.53 TIme(s) (b) FIGURE1.26EffectofRayleigh fadingonasinusoidal wave.(a)Inputsinusoidal wave. (b)Waveform oftheresulting signal.-g '"~«0.5 channelofFigure1.24hasresultedinareceivedsignalwhoseamplitude andphasecom ponentsvaryrandomly withtime,asexpected. I1.15Summary andDiscussion Muchofthematerial presented inthischapterhasdealtwiththecharacterization ofa particular classofrandomprocesses knowntobestationary andergodic.Theimplication of(wide-sense) stationarity isthatwemaydevelopapartialdescription ofarandom processintermsoftwoensemble-averaged parameters: (1)ameanthatisindependent of time,and(2)anautocorrelation functionthatdependsonlyonthedifference betweenthe timesatwhichtwoobservations oftheprocessaremade.lOErgodicity enablesustouse timeaverages as"estimates" oftheseparameters. Thetimeaverages arecomputed using asamplefunction (i.e.,singlerealization) oftherandomprocess. Anotherimportant parameter ofarandomprocessisthepowerspectraldensity.The autocorrelation function andthepowerspectraldensityconstitute aFourier-transform pair.Theformulas thatdefinethepowerspectraldensityintermsoftheautocorrelation function andviceversaareknownastheEinstein-Wiener-Khintchine relations. InTable1.2wepresentagraphical summary oftheautocorrelation functions and powerspectraldensities ofimportant randomprocesses. Alltheprocesses described inthis tableareassumed tohavezeromeanandunitvariance. Thistableshouldgivethereader afeelingfor(1)theinterplay between theautocorrelation function andpowerspectral densityofarandomprocess,and(2)theroleoflinearfilteringinshapingtheautocorre lationfunction or,equivalently, thepowerspectraldensityofawhitenoiseprocess. ThelatterpartofthecrapterdealtwithanoiseprocessthatisGaussian andnar rowband, whichisthekindoffilterednoiseencountered atthefrontendofanidealized formofcommunication receiver. Gaussianity meansthattherandomvariableobtained by 76 CHAPTER 1"RANDOM PROCESSES TABLE1.2Graphical summary ofautocorrelation functions andpowerspectral densities ofrandom processes ofzeromeanandunitvariance TypeofProcess,X(t) Autocorrelation Function, Rx(T) PowerSpectralDensity, Sx(f) Sinusoidal processofunit frequency andrandom phase -1.0 1.0 f Random binarywaveofunit symbol-duration -4 -2 -2 1.0 2.0 RClow-pass filteredwhite noise -4 -0.5 0.5 f,,,, -1.0Ideallow-pass filteredwhite noise -4 -2 -0.5 0.5f Idealhand-pass filtered0.5 whitenoise ,--, ,-,- -4-~-4 -1.0 1.0f RLC-filtered whitenoise NotesandReferences 77 observing theoutputofthefilteratsomefixedtimehasaGaussian distribution. The narrowband natureofthenoisemeansthatitmayberepresented intermsofanin-phase andaquadrature component. Thesetwocomponents arebothlow-pass, Gaussian pro cesses,eachwithzeromeanandavariance equaltothatoftheoriginalnarrowband noise. Alternatively, aGaussian narrowband noisemayberepresented intermsofaRayleigh distributed envelope andauniformly distributed phase.Eachoftheserepresentations has itsownspecificareaofapplication, asshowninsubsequent chapters ofthebook. LNOTES ANDREFERENCES 1.Forarigorous treatment ofrandomprocesses, seetheclassicbooksofDoob(1953),Loeve (1963),andCramerandLeadbetter (1967). 2.Thereisanotherimportant classofrandomprocesses commonly encountered inpractice, themeanandautocorrelation function ofwhichexhibitperiodicity, asin /Lx(t,+T)=/Lx(t,) Rx(t,+T,t2+T)=Rx(t"t2) forallt,andt2•ArandomprocessX(t)satisfying thispairofconditions issaidtobe cyclostationary (inthewidesense).Modeling theprocessX(t)ascydostationary addsa newdimension, namely,periodTtothepartialdescription oftheprocess.Examples of cyc1ostationary processes includeatelevision signal obtained byraster-scanning arandom videofield,andamodulated processobtained byvaryingtheamplitude, phase,orfre quencyofasinusoidal carrier.Fordetaileddiscussion ofcyc1ostationary processes, see Franks(1969),pp.204-214, andthepaperbyGardner andFranks(1975). 3.Traditionally, Equations (1.42)and(1.43)havebeenreferredtointheliterature asthe Wiener-Khintchine relations inrecognition ofpioneering workdonebyNorbertWiener andA.I.Khintchine; fortheiroriginalpapers,seeWiener(1930)andKhintchine (1934). Adiscovery ofaforgotten paperbyAlbertEinsteinontime-series analysis(delivered atthe SwissPhysicalSociety's February 1914meetinginBasel)revealsthatEinsteinhaddiscussed theautocorrelation function anditsrelationship tothespectralcontentofatimeseries manyyearsbeforeWienerandKhintchine. AnEnglishtranslation ofEinstein's paperis reproduced intheIEEEASSPMagazine, vol.4,October1987.Thisparticular issuealso contains articlesbyW.A.Gardner andA.M.Yaglom, whichelaborate onEinstein's originalwork. 4.Forfurtherdetailsofpowerspectrum estimation, seeBlackman andTukey(1958),Box andJenkins(1976),Marple(1987),andKay(1988). 5.TheGaussian distribution andassociated Gaussian processarenamedafterthegreatmath ematician C.F.Gauss.Atage18,Gaussinvented themethodofleastsquaresforfinding thebestvalueofasequence ofmeasurements ofsomequantity. Gausslaterusedthemethod ofleastsquaresinfittingorbitsofplanetstodatameasurements, aprocedure thatwas published in1809inhisbookentitledTheoryofMotionoftheHeavenly Bodies.Incon nectionwiththeerrorofobservation, hedeveloped theGaussian distribution. Thisdistri butionisalsoknownasthenormaldistribution. Partlyforhistorical reasons,mathemati cianscommonly usethetermnormal,whileengineers andphysicists commonly usethe termGaussian. 6.Foradetailedtreatment ofelectrical noise,seeVanderZiel(1970)andthecollection of paperseditedbyGupta(1977). Anintroductory treatment ofshotnoiseispresented inHelstrom (1990).Foramorede tailedtreatment, seethepaperbyYue,Luganani, andRice(1978). 78 CHAPTER 1"RANDOM PROCESSES Thermal noisewasfirststudiedexperimentally byJ.B.Johnson in1928,andforthisreason itissometimes referred toastheJohnson noise.Johnson's experiments wereconfirmed theoretically byNyquist(1928). 7.Thenoisiness ofareceivermayalsobemeasured intermsoftheso-called noisefigure.The relationship betweenthenoisefigureandtheequivalent noisetemperature isdeveloped in Chapter 8. 8.TheRayleigh distribution isnamedaftertheEnglishphysicist J.W.Strutt,LordRayleigh. 9.TheRiciandistribution isnamedinhonorofStephenO.Ricefortheoriginalcontribution reported inapairofpaperspublished in1944and1945,whicharereproduced inWax (1954). 10.Thestatistical characterization ofcommunication systemspresented inthisbookiscon finedtothefirsttwomoments, meanandautocorrelation function (equivalently, autoco variance function) ofthepertinent randomprocess.However, whenarandomprocessis transmitted throughanonlinear system,valuable information iscontained inhigher-order moments oftheresulting outputprocess.Theparameters usedtocharacterize higher-order moments inthetimedomainarecalledcumulants, andtheirmultidimensional Fourier transforms arecalledpolyspectra. Foradiscussion ofhigher-order cumulants andpolys pectraandtheirestimation, seethepaperbyNikiasandRaghuveer (1987). IPROBLEMS Stationarity andErgodieity 1.1Consider arandomprocessX(t)definedby X(t)=sin(Z7Tfct) inwhichthefrequency Ieisarandomvariable uniformly distributed overtheinterval [0,Wl.ShowthatX(t)isnonstationary. Hint:Examine specificsamplefunctions ofthe randomprocessX(t)forthefrequencyI=W/4,W/Z,andW,say. 1.2Consider thesinusoidal process X(t)=Acos{27Tfct) wherethefrequency Ieisconstant andtheamplitude Aisuniformly distributed: {I,0:5a:51Ma)= .0,otherwise Determine whetherornotthisprocessisstrictlystationary. 1.3ArandomprocessX{t)isdefinedby X{t)=Acos(27Tfct) whereAisaGaussian~distributed randomvariableofzeromeanandvariancea7..This randomprocessisappliedtoanidealintegrator, producing theoutput Y(t)=J:X(T)dT (a)Determine theprobability densityfunction oftheoutputY{t)ataparticular timetk' (b)Determine whetherornotY{t)isstationary. (c)Determine whetherornotY{t)isergodic. 1.4LetXandYbestatistically independent Gaussian-distributed randomvariables, eachwith zeromeanandunitvariance. DefinetheGaussian process Z(t)=Xcos(Z7Tt)+Ysin{Z7Tt) Problems 79 (a)Determine thejointprobability densityfunction oftherandomvariablesZ(t,)and Z(t2)obtained byobserving Zit)attimest,andt2,respectively. (b)IstheprocessZ(t)stationary? Why? Correlation andSpectral Density Functions 1.5Provethefollowing twoproperties oftheautocorrelation functionRx('T)ofarandom processX(t): (a)IfX(t)contains aDCcomponent equaltoA,thenRx('T)willcontainaconstant component equaltoA2• (b)IfX(t)contains asinusoidal component, thenRx('T)willalsocontainasinusoidal component ofthesamefrequency. 1.6Thesquarewavex(t)ofFigureP1.6ofconstant amplitude A,periodTo,anddelaytd, represents thesamplefunction ofarandomprocessX(t).Thedelayisrandom, described bytheprobability densityfunction 1 1-2To:,;:td:,;:2To otherwise (a)Determine theprobability densityfunctionoftherandomvariableX(tk)obtained by observing therandomprocessX(t)attimetk. (b)Determine themeanandautocorrelation function ofX(t)usingensemble-averaging. (c)Determine themeanandautocorrelation function ofX(t)usingtime-averaging. (d)Establish whetherornotX(t)isstationary. Inwhatsenseisitergodic? FIGUREP1.6 H<T11)T '1.7Abinarywaveconsistsofarandomsequence ofsymbols1and0,similartothatdescribed inExample 1.3,withonebasicdifference: symbol1isnowrepresented byapulseof amplitude Avoltsandsymbol0isrepresented byzerovolts.Allotherparameters arethe sameasbefore.ShowthatforthisnewrandombinarywaveX(t): (a)Theautocorrelation function is {A2A2(-+- 14 4 Rx('T)=A2 4 ' (b)Thepowerspeetraldensityis A2A2TSx(f)=""4fJ(f)+4sincVT) Whatisthepercentage powercontained intheDCcomponent ofthebinarywave? 80 CHAPTER IIIIRANDOM PROCESSES 1.8ArandomprocessY(t)consistsofaDCcomponent ofV3fivolts,aperiodiccomponent g(t),andarandomcomponent X(t).Theautocorrelation function ofY(t)isshownin FigureP1.8. (a)Whatistheaveragepoweroftheperiodiccomponent g(t)? (b)Whatistheaveragepoweroftherandomcomponent X(t)? RyeT) (volts)2 -5T -4T -3T -2T -T 0 T FIGUREP1.82T 3T 4T 5T 1.9Consider apairofstationary processes X(t)andY(t).Showthatthecross-correlations RXy(T)andRyx(T)oftheseprocesses havethefollowing properties: (a)RXy(T)=RYX(-T) (b)IRXy(T)1:5HRx(O)+Ry(O)] whereRx(T)andRy(T)aretheautocorrelation functions ofX(t)andY(t),respectively. 1.10Consider twolinearfiltersconnected incascadeasinFigurePLiO.LetX(t)beastationary processwithautocorrelation function Rx(T).Therandomprocessappearing atthefirst filteroutputisV(t)andthatatthesecondfilteroutputisY(t). (a),Findtheautocorrelation functionofY(t). (b)Findthecross-correlation function RVy(T)ofV(t)andY(t). ~xCt)~Y(t) FIGUREPI.to 1.11Astationary processX(t)isappliedtoalineartime-invariant filterofimpulseresponse h(t),producing anoutputY(t). (a)Showthatthecross-correlation functionRYX(T)oftheoutputY(t)andtheinputX(t) isequaltotheimpulseresponse h(1")convolved withtheautocorrelation function RX(T)oftheinput,asshownby RYX(T)=r~h(u)Rx(T -u)du Showthatthesecondcross-cortelation function Rxy(T)equals RXy(T)=r~h(-u)Rx(T -u)du (b)Findthecross-spectral densities Syx(f)andSxy(f). Problems 81 (c)Assuming thatX(t)isawhitenoiseprocesswithzeromeanandpowerspectraldensity No/2,showthat NoRyx(r)=Th(r) Comment onthepractical significance ofthisresult. 1.12ThepowerspectraldensityofarandomprocessX(t)isshowninFigurePl.12.Itconsists ofadeltafunction atf=0andatriangular comppnent. (a)Determine andsketchtheautocorrelation functionRx(r)ofX(t). (h)WhatistheDCpowercontained inX(t)? (clWhatistheACpowercontained inX(t)? (d)Whatsampling rateswillgiveuncorrelated samplesofX(t)?Arethesamplesstatis ticallyindependent? 6(/) 1.0 ---:fo~----!;-----~fo-f FIGUREPI.12 1.13Apairofnoiseprocesses n1(t)andn2(t)arerelatedby n2(t)=n/(t)cos(27rfet+0)n/(t)sin(27rfct+0) wherefeisaconstant, and0isthevalueofarandomvariable Elwhoseprobability density function isdefinedby fa(O)={2~' 0,O:s;o:s;27r otherwise Thenoiseprocessnl(t)isstationary anditspowerspectraldensityisasshowninFigure Pl.l3.Findandplotthecorresponding powerspectraldensityofn2(t). a ---:!-:----~---~~-f FIGUREPI.n 1.14Arandomtelegraph signalX(t),characterized bytheautocorrelation function Rx(r)=exp(-2vlrl) 82 CHAPTER 1IIIRANDOM PROCESSES wherevisaconstant, isappliedtothelow-pass RCfilterofFigurePl.14.Determine the powerspectraldensityandautocorrelation function oftherandomprocessatthefilter output. FIGUREPl.14 1.15Arunningintegrator isdefinedby wherex(t)istheinput,y(t)istheoutput,andTistheintegration period.Bothx(t)and y(t)aresamplefunctions ofstationary processes X(t)andY(t),respectively. Showthat thepowerspectraldensityoftheintegrator outputisrelatedtothatoftheintegrator input as 1.16Azero-mean stationary processX(t)isappliedtoalinearfilterwhoseimpulseresponse isdefinedbyatruncated exponential: h(t)={ae-at , 0,o:$;t:$;T otherwise ShowthatthepowerspectraldensityofthefilteroutputY(t)isdefinedby ciSy(f)=a2+4~P(1- 2exp(-aT) cos(2nfT)+exp(-2aT))Sx(f) whereSx(f)isthepowerspectraldensityofthefilterinput. 1.17Theoutputofanoscillator isdescribed by X(t)=ACOS(27Tft -e) whereAisaconstant, andfandeareindependent randomvariables. Theprobability densityfunction ofeisdefinedby f..(O){21 7T' 0,otherwise FindthepowerspectraldensityofX(t)intermsoftheprobability densityfunctionofthe frequencyf.Whathappens tothispowerspectraldensitywhenthefrequencyfassumes aconstant value? yz:O y<OProblems 83 Gaussian ],'rocesses 1.18Astationary, Gaussian processX(t)haszeromeanandpowerspectraldensitySx(f). Determine theprobability densityfunction ofarandomvariableobtained byobserving theprocessX(t)atsometimetk' 1.19AGaussian processX(t)ofzeromeanandvarianceaiispassedthrough afull-wave rectifier, whichisdescribed bytheinput-output relationofFigurePl.19.Showthatthe probability densityfunctionoftherandomvariableY(tk),obtained byobserving theran domprocessY(t)attherectifieroutputattimetbisasfollows: {t1(y2)-exp---fy(,,)(y) = 'Tr"x 2ai' 0, Y Y=-X Y=X ------"'oC-----X FIGUREP1.19 1.20LetX(t)beazero-mean, stationary, Gaussian processwithautocorrelation function Rx(1").Thisprocessisappliedtoasquare-law device,whichisdefinedbytheinput-output relation Y(t)=X2(t) whereY(t)istheoutput. (a)ShowthatthemeanofY(t)isRx(O). (b)Showthattheautocovariance function ofY(t)is2Rk(1"). 1.21Astationary, Gaussian processX(t)withmean/Lxandvarianceaiispassedthroughtwo linearfilterswithimpulseresponses hi(t)andh2(t),yieldingprocesses Y(t)andZ(t),as showninFigureP1.21. (a)Determine thejointprobability densityfunction oftherandomvariables Y(t,)and Z(t2). (b)Whatconditions arenecessary andsufficient toensurethatY(t,)andZ(t2)arestatis ticallyindependent? X(dyet) Z(t) FIGUREP1.21 84 CHAPTER 1..~l'I/DOM PROCESSES 1.22Astationary, Gaussian processX(t)withzeromeanandpowerspectraldensitySx(f)is appliedtoalinearfilterwhoseimpulseresponse h(t)isshowninFigureP1.22.Asample YistakenoftherandomprocessatthefilteroutputattimeT. (a)Determine themeanandvariance ofY. (b)Whatistheprobability densityfunction ofY? J,----L, T FIGUREP1.22 Noise 1.23Consider awhiteGaussian noiseprocessofzeromeanandpowerspectraldensityNol2 thatisappliedtotheinputofthehigh-pass RLfiltershowninFigureP1.23. (a)Findtheautocorrelation function andpowerspectraldensityoftherandomprocess attheoutputofthefilter. (b)Whatarethemeanandvariance ofthisoutput? FIGUREP1.23 1.24Awhitenoisew(t)ofpowerspectraldensityNol2isappliedtoaButterworth low-pass filterofordern,whosemagnitude response isdefinedby 1IH(f)I=[1+(flfo)2"]'/2 (a)Determine thenoiseequivalent bandwidth forthislow-pass filter.(SeeAppendix 2 forthedefinition ofnoiseequivalent bandwidth.) (b)Whatisthelimitingvalueofthenoiseequivalent bandwidth asnapproaches infinity? 1.25Theshot-noise processX(t)definedbyEquation (1.86)isstationary. Why? 1.26WhiteGaussian noiseofzeromeanandpowerspectraldensityNo/2isappliedtothe filteringschemeshowninFigureP1.26a.Thefrequency responses ofthesetwofiltersare showninFigureP1.26b.Thenoiseatthelow-pass filteroutputisdenotedbyn(t). (a)Findthepowerspectraldensityandtheautocorrelation function ofn(t). (b)Findthemeanandvariance ofn(t). Problems 85 (c)WhatistherateatwhichnIt)canbesampledsothattheresulting samplesareessen tiallyuncorrelated? While noise (a)Oulpul net)1.0 (b)l(n, La J2:L f FIGUREPl.26 1.27LetX(t)beastationary processwithzeromean,autocorrelation function Rx(r),and powerspectraldensitySx(f).Wearerequired tofindalinearfilterwithimpulseresponse h(t),suchthatthefilteroutputhasthesamestatistical characteristics asX(t)whenthe inputiswhitenoiseofpowerspectraldensityNoll. (a)Determine thecondition whichtheimpulseresponse h(t)mustsatisfytoachievethis requirement. (b)Whatisthecorresponding condition onthefrequency response H(f)ofthefilter? Narrowband Noise 1.28InthenoiseanalyzerofFigure1.19a,thelow-pass filtersareidealwithabandwidth equal toone-half thatofthenarrowband noisenIt)appliedtotheinput.Usingthisscheme, derivethefollowing results: (a)Equation (1.101),definingthepowerspectraldensities ofthein-phase noisecom ponentn,(t)andquadrature noisecomponent nQ(t)intermsofthepowerspectral densityofn(t). (b)Equation (1.102),definingthecross-spectral densities ofn,(t)andndt). 1.29Assumethatthenarrowband noisenIt)isGaussian anditspowerspectraldensitySN(f) issymmetric aboutthemidband frequency tcoShowthatthein-phase andquadrature components ofn(t)arestatistically independent. 1.30Thepowerspectraldensityofanarrowband noisen(t)isasshowninFigureP1.30.The carrierfrequency is5Hz. (a)Findthepowerspectraldensities ofthein-phase andquadrature components ofn(t). (b)Findtheircross-spectral densities. La ___ J-----'----l..__.....I..__....J.-..!..._l-__ f(Hz) a FIGUREPl.30 86 CHAPTER 1 "RANDOM PROCESSES 1.31Consider aGaussian noisenit)withzeromeanandrhepowerspectraldensitySN(f) showninFigurePl.31. (a)Findtheprobability densityfunction oftheenvelope ofn(t). (b)Whatarethemeanandvariance ofthisenvelope? -_--L--L--L __ ~--+__-'-----_!_--f -feaL~j FIGUREPI.31 Computer Experiments 1.32Inthiscomputer experiment westudythestatistical characterization ofarandomprocess X(t)definedby X(t)=Acos(l7ffct+El)+W(t) wherethephaseElofthesinusoidal component isauniformly distributed randomvariable overtheinterval[-7f,7f],andW(t)isawhiteGaussian noisecomponent ofzeromean andpowerspectraldensityNoll.Thetwocomponents ofX(t)arestatistically indepen dent;hencetheautocorrelation function ofX(t)is A2N RX(T)=Tcos(27ffcT)+20Il(T) Thisequation showsthatfor[T[>atheautocorrelation function RX(T)hasthesame sinusoidal waveform asthesignalcomponent ofX(t). Thepurposeofthiscomputer experiment istoperform thecomputation ofRX(T) usingtwodifferent methods: (a)Ensemble averaging. Generate M=50randomly pickedrealizations oftheprocess X(t).Hencecompute theproductx(t+T)X(t)forsomefixedtimet,wherex(t)isa realization ofX(t).Repeatthecomputation ofx(t+T)X(t)fortheMrealizations of X(t),andtherebycompute theaverageofthesecomputations overM.Repeatthis sequence ofcomputations fordifferent valuesofT. (b)Timeaveraging. Compute thetime-averaged autocorrelation function 1ITRx(T,T)=2T_TX(t+T)X(t)dt wherex(t)isaparticular realization ofX(t),andITisthetotalobservation intervaL Forthiscomputation, usetheFourier-transform pair: Problems 87 where IXT(f) 12/2Tistheperiodogram oftheprocessX(t).Specifically, compute the Fouriertransform XT(f)ofthetime-windowed function XT(t)={X(t), -T:St:ST 0,otherwise Hencecompute theinverseFouriertransform of1Xy{f)12/2T. Compare theresultsofyourcomputation ofRx(T)usingthesetwoapproaches. 1.33Inthiscomputer experiment wecontinue thestudyofthemultipathchanneldescribed in Section1.14.Specifically, considerthesituation wherethe received signalincludesaline of-sightcomponent, asshownby N X(t)=LAkcos(2Trfct+Elk)+acos(27rfct) k=l whereacos(27rfct) isthedirectlyreceivedcomponent. Following thematerialpresented inSection1.14,compute theenvelope ofX(t)forN=10,000,anda=0,1,2,3,5. Compare yourresultswiththeRiciandistribution studiedinSection1.13. CONTINUOUS-WAVE MODULATION Inthischapterwestudycontinuous-wave modulation, whichisbasictotheoperation of analogcommunication systems. Thechapterisdividedintotworelatedparts.Inthefirst partwestudythetime-domain andfrequency-domain descriptions oftwobasicfamilies of continuous-wave modulation: ' ~Amplitude modulation, inwhichtheamplitude ofasinusoidal carrierisvariedin accordance withanincoming message signal. ~Anglemodulation, inwhichtheinstantaneous frequency orphaseofthesinusoidal carrier isvariedinaccordance withthemessage signal. Thesecondpartofthechapterfocusesontheeffectsofchannel noiseontheperformance ofthereceivers pertaining tothesemodulation schemes. Advantages anddisadvantages ofthedifferent methods ofcontinuous-wave modulation arehighlighted inlightofthematerial presented herein. I2.1Introductwn Thepurposeofacommunication systemistotransmitinformation-bearing signalsthrough acommunication channelseparating thetransmitter fromthereceiver. Information bearingsignalsarealsoreferredtoasbaseband signals.Thetermbaseband isusedto designate thebandoffrequencies representing theoriginalsignalasdelivered byasource ofinformation. Theproperuseofthecommunication channelrequiresashiftoftherange ofbaseband frequencies intootherfrequency rangessuitablefortransmission, andacor responding shiftbacktotheoriginalfrequency rangeafterreception. Forexample, aradio systemmustoperatewithfrequencies of30kHzandupward,whereasthebaseband signal usuallycontainsfrequencies intheaudiofrequency range,andsosomeformoffrequency bandshiftingmustbeusedforthesystemtooperatesatisfactorily. Ashiftoftherangeof frequencies inasignalisaccomplished byusingmodulation, whichisdefinedastheprocess bywhichsomecharacteristic ofacarrierisvariedinaccordance withamodulating wave (signal).Acommon formofthecarrierisasinusoidal wave,inwhichcase wespeakofa continuous-wave modulation" process.Thebaseband signalisreferredtoasthemodulat ingwave,andtheresultofthemodulation processisreferredtoasthemodulated wave. Modulation isperformed atthetransmitting endofthecommunication system.Atthe receiving endofthesystem,weusuallyrequiretheoriginalbaseband signaltoberestored. Thisisaccomplished byusingaprocessknownasdemodulation, whichisthereverseof themodulation process. Inbasicsignal-processing terms,wethusfindthatthetransmitter ofananalogcom munication systemconsi·stsofamodulator andthereceiverconsistsofademodulator, as 88 Message signal Sinusoidal carrierwaveModulated wave2.1Introductifm 89 (b) <aJ FIGURE2.1Components ofacontinuous-wave modulation system:(a)translllitter, and(b) receiver. depicted inFigure2.1.Inaddition tothesignalreceivedfromthetransmitter, thereceiver inputincludeschannelnoise.Thedegradation inreceiverperformance duetochannelnoise isdetermined bythetypeofmodulation used. Inthischapterwestudytwofamiliesofcontinuous-wave (CW)modulation systems, namely,amplitude modulation andanglemodulation. Inamplitude modulation, theam plitudeofthesinusoidal carrierwaveisvariedinaccordance withthebaseband signal.In anglemodulation, theangleofthesinusoidal carrierwaveisvariedinaccordance withthe baseband signal.Figure2.2displays th~waveforms ofamplitude-modulated andangle modulated signalsforthecaseofsinusoidal modulation. Parts(a)and(b)ofthefigure showthesinusoidal carrierandmodulating waves,respectively. Parts(c)and(d)showthe ~""IlfIII"1\f\f\1\ f\"""" VVVVVVVVVVVV VVVVV (a) (c) (d) Time~ FIGURE2.2Illustrating AMandFMsignalsproduced byasingletone.(alCarrierwave.(b) Sinusoidal modulating signal.(c)Amplitude-modulated signal.(d)Frequency-modulated signal. 90 CHAPTER 2..CONTINUOUS-WAVE MODULATION corresponding amplitude-modulated andfrequency-modulated waves,respectively; fre quencymodulation isaformofanglemodulation. Thisfigureclearlyillustrates thebasic differences between amplitude modulation andanglemodulation, whicharediscussedin whatfollows. I2.2Amplitude Modulation Consider asinusoidal carrierwavec(t)definedby (2.1) whereAeisthecarrieramplitude andIeisthecarrierfrequency. Tosimplifytheexposition withoutaffecting theresultsobtained andconclusions reached, wehaveassumed thatthe phaseofthecarrierwaveiszeroinEquation (2.1).Letm(t)denotethebaseband signal thatcarriesthespecification ofthemessage. Thesourceofcarrierwavecit)isphysically independent ofthesourceresponsible forgenerating m(t).Amplitude modulation (AM)is definedasaprocessinwhichtheamplitude ofthecarrierwavecrt}isvariedaboutamean value,linearlywiththebaseband signalm(t}.Anamplitude-modulated (AM)wavemay thusbedescribed, initsmostgeneralform,asafunction oftimeasfollows: (2.2) wherekaisaconstant calledtheamplitude sensitivity ofthemodulator responsible forthe generation ofthemodulated signals(t).Typically, thecarrieramplitude Aeandthemessage signalm(t)aremeasured involts,inwhichcasekaismeasured involt-i. Figure2.3ashowsabaseband signalm(t),andFigures2.3band2.3cshowthecor responding AMwavesit)fortwovaluesofamplitude sensitivity kaandacarrieramplitude Ae=1volt.Weobservethattheenvelope ofs(t)hasessentially thesameshapeasthe baseband signalm(t)provided thattworequirements aresatisfied: 1.Theamplitude ofkam(t)isalwayslessthanunity,thatis, Ikam(t)I<1forallt (2.3) Thiscondition isillustrated inFigure2.3b;itensuresthatthefunction 1+kam(t) isalwayspositive, andsinceanenvelope isapositivefunction, wemayexpressthe envelope oftheAMwavesit)ofEquation (2.2)asAe[l+kam(t)].Whentheam plitudesensitivity kaofthemodulator islargeenoughtomakeIkam(t)I>1forany t,thecarrierwavebecomes overmodulated, resulting incarrierphasereversals when everthefactor1+kam(t)cr(jsseszero.Themodulated wavethenexhibitsenvelope distortion, asinFigure2.3c.Itistherefore apparent thatbyavoiding overmodula tion,aone-to-one relationship ismaintained between theenvelope oftheAMwave andthemodulating waveforallvaluesoftime-ausefulfeature,asweshallseelater on.Theabsolute maximum valueofk.m(t)multiplied by100isreferredtoasthe percentage modulation. 2.Thecarrierfrequency Ieismuchgreaterthanthehighestfrequency component Wof themessage signalm(t),thatis fc»W (2.4) WecallWthemessagebandwidth. Ifthecondition ofEquation (2.4)isnotsatisfied, anenvelope cannotbevisualized (andtherefore detected) satisfactorily. 2.2Amplitude Modulation 91 (a) s(t} of-\--I-t-Hr-f-++-\-f--JH-'--- -1 (b)+1- -1 ,/,/,/ (c) FIGURE2.3Illustrating theamplitude modulation process. (a)Baseband signalmit).(b)AM waveforIk.m(t)I<1forallt.(e)AMwaveforIk.m(t)I>1forsomet. FromEquation (2.2),wefindthattheFouriertransform oftheAMwaves(t)isgiven by S(f)A k A2c[8(1-Ic)+8(1+fc)]+~[M(I-fc)+M(I+Ic)](2.5) Supposethatthebaseband signalm(t)isband-limited totheinterval- W:s;I:s;W,asin Figure2Aa.Theshapeofthespectrum showninthisfigureisintended forthepurposeof illustration only.WefindfromEquation (2.5)thatthespectrum S(f)oftheAMwaveis asshowninFigure2AbforthecasewhenIc>W.Thisspectrum consistsoftwodelta functions weighted bythefactorAi2andoccurring at~Ic>andtwoversions ofthe baseband spectrum translated infrequency by~Icandscaledinamplitude bykaA)2. Fromthespectrum ofFigure2Ab,wenotethefollowing: 1.Asaresultofthemodulation process,thespectrum ofthemessage signalmit)for negative frequencies extending from- Wto0becomes completely visibleforpositive (i.e.,measurable) frequencies, provided thatthecarrierfrequency satisfiesthecon ditionIc>W;hereinliestheimportance oftheideaof"negative" frequencies. 2.Forpositivefrequencies, theportionofthespectrum ofanAMwavelyingabovethe carrierfrequency Icisreferred toastheuppersideband, whereas thesymmetric portionbelowIcisreferredtoasthelowersideband. Fornegative frequencies, the uppersideband isrepresented bytheportionofthespectrum below-Icandthe lowersideband bytheportionabove-fc.Thecondition Ic>Wensuresthat thesidebands donotoverlap. 92 CHAPTER 2IIICONTINUOUS-WAVE MODUlATION M(f) M(O) ---w'---'------'--- f (a)S(fl (bl.....-----f FIGURE 2.4(a)Spectrum ofbaseband signal.(b)Spectrum ofAMwave. 3.Forpositivefrequencies, thehighestfrequency component oftheAMwaveequals fc+W,andthelowestfrequency component equalsIe-W.Thedifference between thesetwofrequencies definesthetransmission bandwidth ByforanAMwave,which isexactlytwicethemessagebandwidth W,thatis, By=2W (2.6) Ill!VIRTUES ANDLIMITATIONS OFAMPLITUDE MODULATION Amplitude modulation istheoldestmethodofperforming modulation. Itsgreatestvirtue isthesimplicity ofimplementation: ~Inthetransmitter, amplitude modulation isaccomplished usinganonlinear device. Forexample, intheswitching modulator discussed inProblem 2.3,thecombined sumofthemessage signalandcarrierwaveisappliedtoadiode,withthecarrier amplitude beinglargeenoughtoswingacrossthecharacteristic curveofthediode. Fourieranalysisofthevoltagedeveloped acrossaresistiveloadrevealsthegeneration ofanAMcomponent, whichmaybeextracted bymeansofaband-pass filter. 1>0Inthereceiver, amplitude demodulation isalsoaccomplished usinganonlinear de vice.Forexample, wemayuseasimpleandyethighlyeffectivecircuitknownasthe envelope detector, whichisdiscussed inProblem 2.5.Thecircuitconsistsofadiode connected inserieswiththeparallelcombination ofacapacitor andloadresistor. Someversionofthiscircuitisfoundinmostcommercial AMradioreceivers. Pro videdthatthecarrierfrequency ishighenoughandthepercentage modulation isless than100percent,thedemodulator outputdeveloped acrosstheloadresistorisnearly thesameastheenvelope oftheincoming AMwave,hencethename"envelope detector." Recall,however, thattransmitted powerandchannelbandwidth areourtwoprimary communication resources, andtheyshouldbeusedefficiently. Inthiscontext,wefindthat thestandard formofamplitude modulation definedinEquation (2.2)suffersfromtwO majorlimitations: 1.Amplitude modulation iswastefulofpower.Thecarrierwavecrt)iscompletely independent oftheinformation-bearing signalmit).Thetransmission ofthecarrier wavetherefore represents awasteofpower,whichmeansthatinamplitude modu lationonlyafractionofthetotaltransmitted powerisactuallyaffectedbymit). 2.3LinearMotLUation ScJ.emes 93 2.Amplitude modulation iswastefulofbandwidth. Theupperandlowersidebands of anAMwaveareuniquely relatedtoeachotherbyvirtueoftheirsymmetry about thecarrierfrequency; hence,giventhemagnitude andphasespectraofeitherside band,wecanuniquely determine theother.Thismeansthatinsofarasthetransmis sionofinformation isconcerned, onlyonesideband isnecessary, andthecommu nicationchanneltherefore needstoprovideonlythesamebandwidth asthebaseband signal.Inlightofthisobservation, amplitude modulation iswasteful ofbandwidth asitrequiresatransmission bandwidth equaltotwicethemessage bandwidth. Toovercome theselimitations, wemustmakecertainmodifications: suppress the carrierandmodifythesidebands oftheAMwave.Thesemodifications naturally resultin increased systemcomplexity. Ineffect,wetradesystemcomplexity forimproved useof communication resources. Thebasisofthistrade-off islinearmodulation, whichisdis cussedinthenextsection.Inastrictsense,fullamplitude modulation doesnotqualifyas linearmodulation becauseofthepresence ofthecarrierwave. l2.3LinearModulation Schemes Initsmostgeneralform,linearmodulation isdefinedby (2.7) whereSI(t)isthein~phase component ofthemodulated wavesit),andsQ(t)isitsquad raturecomponent. Equation (2.7)isrecognized asthecanonical representation ofanar rowband signal,whichisdiscussed indetailinAppendix 2.Inlinearmodulation, both SI{t)andsQ(t)arelow-pass signalsthatarelinearlyrelatedtothemessage signalm(t). Indeed,depending onhowthesetwocomponents ofsit)aredefined,wemayidentify threetypesoflinearmodulation involving asinglemessage signal: 1.Doublesideband-suppressed carrier(DSB-SC) modulation, whereonlytheupperand lowersidebands aretransmitted. 2.Singlesideband (SSB)modulation, whereonlyonesideband (thelowersideband or theuppersideband) istransmitted. 3.Vestigial sideband (VSB)modulation, whereonlyavestige(i.e.,trace)ofoneofthe sidebands andacorrespondingly modified versionoftheothersideband are transmitted. Table2.1presents asummary ofthedefinitions ofthesethreespecialformsoflinear modulation. Therearetwoimportant pointstonotefromTable2.1: 1.Thein-phase component SI{t)issolelydependent onthemessage signalmit). 2.Thequadrature component sQ(t)isafilteredversionofmit).Thespectralmodifi cationofthemodulated wavesit)issolelyduetosQ{t). Tobemorespecific,theroleofthequadrature component (ifpresent)ismerelytointerfere withthein-phase component, soastoreduceoreliminate powerinoneofthesidebands ofthemodulated signalsit),depending onhowthequadrature component isdefined. 94 CHAPTER 2mCONTINUOUS-WAVE MODULATION ITABLE2.1Different frmnsoflinearmodulation In-Phase Quadrature Component Component TypeofModulation s,(t) sQ(t) Comments DSB-SC m{t) 0 m(t)=messagesignal SSB:' (a)Uppersideband !m{t) 1m{t)m(t)=Hilberttransfonn ofm(t) transmitted (b)Lowersideband 1m{t)-1m{t) transmitted VSB: (a)Vestigeoflowersideband 1m(t) !m'(t)!m'('j;"_,ofcl"fil.,oftransmitted frequency response HQ(f) (b)Vestigeofupper 1m(t) -!m'(t) duerom{t). sideband transmitted Forthedefinition ofHQ(f), seeEq.(2.16) 'Forthemathematical description ofsinglesidebandmodulation, seeProblem2.16. Ii!DOUBLE SIDEBAND-SUPPRESSED CARRIER (DSB-SC) MODUlATION Thisformoflinearmodulation isgenerated byusingaproductmodulator thatsimply multiplies themessage signalm(t)bythecarrierwaveAcCOS{27Tfct), asillustrated inFigure 2.5a.Specifically, wewrite s{t)=Acm(t)COS(27Tfct) m(t)(2.8) Baseband signalm(t)DSB-SC modulated wave .(,)=Acm(l)cos(21f/J) Or------------'~----r-- Carrier Accos(2'1rfctl (a) .(1) Phasereversals ort--+-+-+-t-I----'\---i'lidY-+-++P-{h} (c) FIGURE2.5(a)Blockdiagramofproductmodulator. (b)Baseband signal.(c)DSB-SC modu latedwave. 2.3LinearModulation Schemes 95 M(f) ,11(0) --L------.J-----'--f-w0W (a)S(j)liAcM(O) ~----- -----~ L-:~-J 0L:~-J (b)f FIGURE2.6(a)Spectrum ofbaseband signal.(b)Spectrum ofDSB-SC modulated wave. Figure2.Scshowsthemodulated signals(t)forthearbitrary messagewaveform ofFigure 2.Sb.Themodulated signals(t)undergoes aphasereversalwhenever themessage signal m(t)crosseszero.Consequently, theenvelope ofaDSB-SCmodulated signalisdifferent fromthemessage signal;thisisunlikethecaseofanAMwavethathasapercentage modulation lessthan100percent. FromEquation (2.8),theFouriertransform ofs(t)isobtained as S(f)=~AJM(f-fc)+M(f+fell (2.9) Forthecasewhenthebaseband signalm(t)islimitedtotheinterval-W:s;f:s;W,asin Figure2.6a,wethusfindthatthespectrum S(f)oftheDSB-SC waves(t)isasillustrated inFigure2.6b.Exceptforachangeinscalefactor,themodulation processsimplytranslates thespectrum ofthebaseband signalby±fc'Ofcourse,thetransmission bandwidth re quiredbyDSB-SCmodulation isthesameasthatforamplitude modulation, namely,2W. IIICOHERENT DETECTION Thebaseband signalm(t)canbeuniquely recovered fromaDSB-SC waves(t)byfirst multiplying s(t)withalocallygenerated sinusoidal waveandthenlow-pass filteringthe product, asinFigure2.7.Itisassumed thatthelocaloscillator signalisexactlycoherent orsynchronized, inbothfrequency andphase,withthecarrierwavec(t)usedintheprod uctmodulator togenerate s(t).Thismethodofdemodulation isknownascoherent detec tionorsynchronous demodulation. Itisinstructive toderivecoherent detection asaspecialcaseofthemoregeneral demodulation processusingalocaloscillator signalofthesamefrequency butarbitrary phasedifference ,p,measured withrespecttothecarrierwavec(t).Thus,denoting thelocal s(t) FIGURE2.7Coherent detector fordemodulating DSB-SC modulated wave. (2.10)96 CHAPTER 2Ii!CONTINUOUS-WAVE MODULATION V(f) __ J-~_~ __ f--L_--'-_---''-----~_-':- __f FIGURE2.8Illustrating thespectrum ofaproduct modulator outputwithaDSB-SC modulated waveasinput. oscillator signalbyA~COS(27Tfet+<jJ),andusingEquation (2.8)fortheDSB-SCwavesit), wefindthattheproductmodulator outputinFigure2.7is v(t)=A~COS(27Tfct+<jJ)s(t) =AcA~COS(27Tfct) COS(27Tfct+<jJ)m(t) 1 1 =2"AcA~COS(47Tfct+<jJ)m(t)+2"AcA~cos<jJm(t) ThefirstterminEquation (2.10)represents aDSB-SCmodulated signalwithacarrier frequency 2fnwhereasthesecondtermisproportional tothebaseband signalmit).This isfurtherillustrated bythespectrum V(f)showninFigure.2.8,whereitisassumed that thebaseband signalmit)islimitedtotheinterval-W:5:f:5:W.Itistherefore apparent thatthefirstterminEquation (2.10)isremoved bythelow-pass filterinFigure2.7, provided thatthecut-offfrequency ofthisfilterisgreaterthanWbutlessthan2fe-W. Thisrequirement issatisfied bychoosing fc>W.Atthefilteroutputwethenobtaina signalgivenby (2.11) Thedemodulated signalVolt)istherefore proportional tomit)whenthephaseerror'" isaconstant. Theamplitude ofthisdemodulated signalismaximum when <jJ=0,and itisminimum (zero)when <jJ=:!:.7T/2.Thezerodemodulated signal,whichoccursfor <jJ=:!:.7T/2,represents thequadrature nulleffectofthecoherent detector. Thusthephase error<jJinthelocaloscillator causesthedetectoroutputtobeattenuated byafactorequal tocos<jJ.Aslongasthephaseerror <jJisconstant, thedetector provides anundistorted versionoftheoriginalbaseband signalmit).Inpractice, however, weusuallyfindthatthe phaseerror<jJvariesrandomly withtime,duetorandomvariations inthecommunication channel. Theresultisthatatthedetectoroutput,themultiplying factorcos<jJalsovaries randomly withtime,whichisobviously undesirable. Therefore, provision mustbemade inthesystemtomaintain thelocaloscillator inthereceiverinperfectsynchronism, inboth frequency andphase,withthecarrierwaveusedtogenerate theDSB-SCmodulated signal inthetransmitter. Theresulting systemcomplexity isthepricethatmustbepaidfor suppressing thecarrierwavetosavetransmitter power. COSTAS RECEIVER Onemethodofobtaining apractical synchronous receiversystem,suitablefordemodu latingDSB-SC waves,istousetheCostasreceiver2showninFigure2.9.Thisreceiver 2.3Li_rModulati ....Schemes 97 I-channel DSB-SCsignal A,cos(2"'I,t)m(t)r---"""'-~ Demodulated signal Q-channel FIGURE2.9Costasreceiver. consistsoftwocoherent detectors supplied withthesameinputsignal,namely,theincom ingDSB-SCwaveA,cos(2'1Tlct)m(t), butwithindividual localoscillator signalsthatare inphasequadrature withrespecttoeachother.Thefrequency ofthelocaloscillator is adjusted tobethesameasthecarrierfrequencyI"whichisassumed knownapriori.The detector intheupperpathisreferredtoasthein-phase coherent detector orI-channel, andthatinthelowerpathisreferred toasthequadrature-phase coherent detector or Q-channel. Thesetwodetectors arecoupledtogether toformanegativefeedback system designed insuchawayastomaintain thelocaloscillator synchronous withthecarrier wave. Tounderstand theoperation ofthisreceiver, supposethatthelocaloscillator signal isofthesamephaseasthecarrierwaveAccos(2'1Tlet) usedtogenerate theincoming DSB-SC wave.Undertheseconditions, wefindthattheI-channel outputcontains the desireddemodulated signalmIt),whereas theQ-channel outputiszeroduetothequad raturenulleffectoftheQ-channel. Supposenextthatthelocaloscillator phasedriftsfrom itspropervaluebyasmallangle,pradians.TheI-channel outputwillremainessentially unchanged, buttherewillnowbesomesignalappearing attheQ-channel output,which isproportional tosin,p=,pforsmall,p.ThisQ-channel outputwillhavethesamepolarity astheI-channel outputforonedirection oflocaloscillator phasedriftandopposite po larityfortheopposite direction oflocaloscillator phasedrift.Thus,bycombining the I-andQ-channel outputsinaphasediscriminator (whichconsistsofamultiplier followed byalow-pass filter),asspowninFigure2.9,aDCcontrolsignalisobtained thatauto matically correctsforlocalphaseerrorsinthevoltage-controlled oscillator. Itisapparent thatphasecontrolintheCostasreceiverceaseswithmodulation and thatphase-lock hastobereestablished withthereappearance ofmodulation. Thisisnot aseriousproblem whenreceiving voicetransmission, becausethelock-upprocessnormally occurssorapidlythatnodistortion isperceptible. Il!IQUADRATURE-CARRIER MULTIPLEXING Thequadrature nulleffectofthecoherent detector mayalsobeputtogooduseinthe construction oftheso-called quadrature-carrier multiplexing orquadrature-amplitude 98 CHAPTER 2..CONTINUOIJS-WAVE MODIJLATION Message signalml{t} Message signal1n2(t)Multipje_xed signals(t) Multiplexed signals(t) (a) FIGIJRE 2.10(b) Quadrature-carrier multiplexing system.(a)Transmitter. (b)Receiver. modulation (QAM).ThisschemeenablestwoDSB-SCmodulated waves(resulting from theapplication oftwophysically independent messagesignals)tooccupythesamechannel bandwidth, andyetitallowsfortheseparation ofthetwomessagesignalsatthereceiver output.Itistherefore abandwidth-conservation scheme. Ablockdiagram ofthequadrature-carrier multiplexing systemisshowninFigure 2.10.Thetransmitter partofthesystem,showninFigure2.10a,involves theuseoftwo separate productmodulators thataresupplied withtwocarrierwavesofthesamefre quencybutdiffering inphaseby-90degrees.Thetransmitted signalsIt)consistsofthe sumofthesetwoproductmodulator outputs, asshownby (2.12) wherem,(t)andm2(t)denote the twodifferent message signals appliedtotheproduct modulators. ThussIt)occupies achannelbandwidth of2Wcentered atthecarrierfre quencyfe'whereWisthemessage bandwidth ofm,(t)orm2(t).According toEquation (2.12),wemayviewAem,(t)asthein-phase component ofthemultiplexed band-pass signals(t)and-Aem2(t) asitsquadrature component. ThereceiverpartofthesystemisshowninFigure2.l0b.Themultiplexed signals(t) isappliedsimultaneously totwoseparate coherent detectors thataresupplied withtwo localcarriersofthesamefrequency butdiffering inphaseby-90degrees.Theoutputof thetopdetectorisAem,(t), whereastheoutputofthebottomdetectorisAcm2(t). Forthe systemtooperatesatisfactorily, itisimportant tomaintain thecorrectphaseandfrequency relationships betweenthelocaloscillators usedinthetransmitter andreceiverpartsofthe system. Tomaintain thissynchronization, wemaysendapilotsignaloutsidethepassband ofthemodulated signal.Inthismethod,thepilotsignaltypically consistsofalow-power sinusoidal tonewhosefrequency andphasearerelatedtothecarrierwavecrt);atthe receiver, thepilotsignalisextracted bymeansofasuitablytunedcircuitandthentrans latedtothecorrectfrequency foruseinthecoherent detector. IIISINGLE-SIDEBAND MODUlATION Insingle-sideband modulation, onlytheupperorlowersideband istransmitted. Wemay generate suchamodulated wavebyusingthefrequency-discrimination methodthatcon sistsoftwostages: 2.3LlHearModulatron Schemes 99 Ii>Thefirststageisaproductmodulator, whichgenerates aDSB-SCmodulated wave. '"Thesecondstageisaband-pass filter,whichisdesigned topassoneofthesidebands ofthismodulated waveandsuppress theother. Fromapractical viewpoint themostsevererequirement ofSSBgeneration usingthefre quencydiscrimination methodarisesfromtheunwanted sideband. Thenearestfrequency component oftheunwanted sideband isseparated fromthedesiredsideband bytwicethe lowestfrequency component ofthemessage (modulating) signal.Theimplication hereis thatforthegeneration ofanSSBmodulated signaltobepossible, themessagespectrum musthaveanenergygapcentered attheorigin,asillustrated inFigurel.lla.Thisrequire mentisnaturally satisfiedbyvoicesignals,whoseenergygapisabout600Hzwide(i.e., itextendsfrom-300to+300Hz).Thus,assuming thattheuppersideband isretained, thespectrum oftheSSBmodulated signalisasshowninFigure1.IIb. Indesigning theband-pass filterusedinthefrequency-discriminator forgenerating aSSB-modulated wave,wemustmeetthethreebasicrequirements: Il>Thedesiredsideband liesinsidethepassband ofthefilter. ""Theunwanted sideband liesinsidethestopband ofthefilter. ..Thefilter'stransition band,whichseparates thepassband fromthestopband, istwice thelowestfrequency component ofthemessagesignal. Thiskindoffrequency discrimination usuallyrequirestheuseofhighlyselective filters, whichcanonlyberealizedinpracticebymeansofcrystalresonators. Todemodulate aSSBmodulated signals(t),wemayuseacoherent detector, which multiplies s(t)byalocallygenerated carrierandthenlow-pass filterstheproduct. This methodofdemodulation assumesperfectsynchronism between theoscillator intheco herentdetectorandtheoscillator usedtosupplythecarrierwaveinthetransmitter. This requirement isusuallymetinoneoftwoways: ~Alow-power pilotcarrieristransmitted inadditiontotheselectedsideband. Il>Ahighlystableoscillator, tunedtothesamefrequency asthecarrierfrequency, is usedinthereceiver. Inthelattermethod, itisinevitable thattherewouldbesomephaseerrorcPinthelocal oscillator outputwithrespecttothecarrierwaveusedtogenerate theincoming SSBmod ulatedwave.Theeffectofthisphaseerroristointroduce aphasedistortion inthede modulated signal,whereeachfrequency component oftheoriginalmessagesignalunder goesaconstant phaseshiftcP.Thisphasedistortion istolerable invoicecommunications, !M{jll ----;.'b"-------''--O~<------'''''- f " -fafa fb --->1I<'-Energygap (a)IS{j)1 o (bl FIGURE2.11(a)Spectrum ofamessage signalm(t)withanenergygapofwidth2f.centered ontheorigin.(b)Spectrum ofcorresponding SSBsignalcontaining theuppersideband. 100 CHAPTER 2IIICONTINUOUS-WAVE MODUlATION becausethehumanearisrelatively insensitive tophasedistortion. Inparticular, thepres enceofphasedistortion givesrisetoaDonaldDuckvoiceeffect.Inthetransmission of musicandvideosignals,ontheotherhand,thepresence ofthisformofwaveform distor tionisutterlyunacceptable. !lilVESTIGIAL SIDEBAND MODUlATION Investigialsideband (VSB)modulation, oneofthesidebands ispartially suppressed and avestigeoftheothersideband istransmitted tocompensate forthatsuppression. Apopular method forgenerating aVSB-modulated waveisto'usethefrequency discrimination method. First,wegenerate aDSB-SC modulated waveandthenpassitthrough aband. passfilter,asshowninFigure2.12;itisthespecialdesignoftheband-pass filterthat distinguishes VSBmodulation fromSSBmodulation. Assuming thatavestigeofthelower sideband istransmitted, thefrequency responseH(f)oftheband-pass filterta"estheform showninFigure2.13.Tosimplify matters, onlytheresponse forpositivefrequencies is shownhere.Thisfrequency response isnormalized, sothatatthecarrierfrequencyhwe haveIH(fe)I=1/2.Theimportant featuretonotefromFigure2.13isthatthecutoff portionofthefrequency response aroundthecarrierfrequency feexhibitsoddsymmetry. Thatis,insidethetransition intervalfe-fv:=;IfI:=;fe+fvthefollowing twoconditions aresatisfied: 1.Thesumofthevalues9fthemagnitude response IH(f)Iatanytwofrequencies equallydisplaced aboveandbelowfeisunity. 2.Thephaseresponse arg(H(f)) islinear.Thatis,H(f)satisfiesthecondition H(f-fe)+H(f+fel=1for-W:=;f:=;W (2.13) Notealsothatoutsidethefrequency bandofinterest(i.e.,Ifl>fe+W),thefrequency response H(f)mayhaveanarbitrary specification. Accordingly, thetransmission band widthofVSBmodulation is BT=W+fv (2.14) whereWisthemessage bandwidth, andfvisthewidthofthevestigial sideband. According toTable2.1,theVSBmodulated waveisdescribed inthetimedomainas s(t)=~Aem(t)cos(27rfet) :!:~Acm'(t)sin(27rfet) (2.15) wheretheplussigncorresponds tothetransmission ofavestigeoftheuppersideband, andtheminussigncorresponds tothetransmission ofavestigeofthelowersideband. The signalm'(t)inthequadrature component ofs(t)isobtained bypassingthemessagesignal Message signalm{t) Accos(27Tfctl carrierwaveVSB modulated wave FIGURE2.12Filtering schemeforthegeneration ofVSBmodulated wave. 2.3LinearModulation Schemes 101 lH(fJl 1.0 0.5 oL----""-------'---,--'----fe--'+-W-----'''''''---f FIGURE2.13Magnitude response ofVSBfilter;onlythepositive-frequency portionisshown. mit)throughafilterwhosefrequency responseHdf)satisfiesthefollowing requirement (seeProblem 2.20): Hdf)=j[H(f-tel-H(f+fe)]for- W:sf:sW (2.16) Figure2.14displaysaplotofthefrequency response HQ(f),scaledbyl/j.Theroleofthe quadrature component determined byHQ(f)istointerfere withthein-phase component inEquation (2.15)soastopartiallyreducepowerinoneofthesidebands ofthemodulated waves(t)andretainsimplyavestigeoftheothersideband, asdesired. ItisofinteresttonotethatSSBmodulation maybeviewedasaspecialcaseof VSBmodulation. Specifically, whenthevestigialsideband isreducedtozero(i.e.,weset fv=0),themodulated wavesit)ofEquation (2.15)takesthelimitingformofasingle sideband modulated wave. iiiTELEVISION SIGNALS Adiscussion ofyestigialsideband modulation wouldbeincomplete withoutamention of itsroleincommercial television (TV)broadcasting. Theexactdetailsofthemodulation formatusedtotransmit thevideosignalcharacterizing aTVsystemareinfluenced bytwo factors: 1.Thevideosignalexhibitsalargebandwidth andsignificant low-frequency content, whichsuggesttheuseofvestigialsideband modulation. 2.Thecircuitry usedfordemodulation inthereceivershouldbesimpleandtherefore inexpensive; this suggests theuseofenvelope detection, whichrequirestheaddition ofacarriertotheVSB-modulated wave. ---..L.--\----'i---f FIGURE2.14Frequency response ofafilterforproducing thequadrature component ofthe VSBmodulated wave. 102 CHAPTER 2illCONTINUOUS-WAVE MODUlATION Withregardtopoint1,however, itshouldbestressedthatalthough thereisindeed abasicdesiretoconserve bandwidth, incommercial TVbroadcasting thetransmitted signalisnotquiteVSBmodulated. Thereasonisthatatthetransmitter thepowerlevels arehigh,withtheresultthatitwouldbeexpensive torigidlycontrolthefilteringofside bands.Instead,aVSBfilterisinsertedineachreceiver, wherethepowerlevelsarelow. Theoverallperformance isthesameasconventional vestigial-sideband modulation, except forsomewastedpowerandbandwidth. Theseremarksareillustrated inFigure2.15.In particular, Figure2.I5ashowstheidealized spectrum ofatransmitted TVsignal.The uppersideband, 25percentofthelowersideband, andthepicturecarrieraretransmitted. Thefrequency response oftheVSBfilterusedtodotherequired spectrum shapinginthe receiverisshowninFigure2.15b. Thechannelbandwidth usedforTVbroadcasting inNorthAmerica is6MHz,as indicated inFigure2.15b.Thischannelbandwidth notonlyaccommodates thebandwidth requirement of theVSBmodulated videosignalbutalsoprovides fortheaccompanying soundsignalthatmodulates acarrierofitsown.Thevaluespresented onthefrequency axisinFigures2.I5aand2.15bpertaintoaspecificTVchannel. According tothisfigure, thepicturecarrierfrequency isat55.25MHz,andthesoundcarrierfrequency isat59.75 MHz.Note,however, thattheinformation contentoftheTVsignalliesinabaseband spectrum extending from1.25MHzbelowthepicturecarrierto4.5MHzaboveit. Withregardtopoint2,theuseofenvelope detection (appliedtoaVSBmodulated ~1.0 ~ l" ".~ ~0.5 z4~~;r-- 4.5MHz------>-II+-0.25MHz 4~0.75MHZ r Picture Sound carrier carrier, 1 1 ;1!(MHz)54 56 58 60 (al Picture Sound carrier carrier o'---\\--5--'4'-----CJ'--5...L6---5...L8-----'6.L0--!(MHz) IChannelbandwidth~I-<---- 6MHz (b) -FIGURE 2.15(a)Idealized magnitude spectrum ofatransmitted TVsignal.(b)Magnitude re sponseofVSBshapingfilterinthereceiver. 2.4FrequeflC)/ Translation 103 (2.17)wavepluscarrier)produces waveform distortion inthevideosignalrecovered atthede tectoroutput.Thedistortion isproduced bythequadrature component oftheVSBmod ulatedwave;thisissueisdiscussed next. Theuseofthetime-domain description giveninEquation (2.15)enablesthedeter mination ofthewaveform distortion causedbytheenvelope detect~r. Specifically, adding thecarriercomponent Aecos(27rlet) totheVSB-modulated waveofEquation (2.15),the latterbeingscaledbyafactork.,modifies themodulated signalappliedtotheenvelope detectorinputas sit)=Ae[1+~kam(t)]cos(27rfct) :!:~kaAem'(t) sin(27rlet) (2.18)[1]2}'/2 2"kam'(t) [1]{[~kam'(t) ]2}'/2 =Ae1+2"kam(t) 1+-1- 1+2"kam(t)wheretheconstantkadetermines thepercentage modulation. Theenvelope detectorout put,denotedbya(t),istherefore a(t)=Ac{[1+~kam(tT+ Equation (2.18)indicates thatthedistortion iscontributed bym'(t),whichisresponsible forthequadrature component oftheincoming VSB-modulated signal.Thisdistortion can bereducedbyusingtwomethods: ~Reducing thepercentage modulation toreducetheamplitude sensitivity ka' l>Increasing thewidthofthevestigialsideband toreducem'(t). Bothmethods areinfactusedinpractice. Incommercial TVbroadcasting, thewidthof thevestigialsideband (whichisabout0.75MHz,orone-sixth ofafullsideband) isdeter minedtokeepthedistortion duetom'(t)withintolerable limitswhenthepercentage modulation isnearly100. I2.4Frequency Translation Thebasicoperation involved insingle-sideband modulation isinfactaformoffrequency translation, whichiswhysingle-sideband modulation issometimes referredtoasfrequency changing, mixing,orheterodyning. Thisoperation isclearlyillustrated inthespectrum of thesignalshowninFigure2.11bcompared tothatoftheoriginalmessagesignalinFigure 2.11a.Specifically, weseethatamessagespectrum occupying thebandfromlatoIbfor positivefrequencies inFigure2.11aisshiftedupwardbyanamountequaltothecarrier frequency IeinFigure2.11b,andthemessagespectrum fornegative frequencies istrans lateddownward inasymmetric fashion. Theideaoffrequency translation described hereinmaybegeneralized asfollows. Suppose thatwehaveamodulated waves,(t)whosespectrum iscentered onacarrier frequencyI"andtherequirement istotranslate itupwardinfrequency suchthatitscarrier frequency ischanged from1,toanewvalue[,.Thisrequirement maybeaccomplished usingthemixershowninFigure2.16.Specifically, themixerisadevicethatconsistsofa productmodulator followed byaband-pass filter. 104 CHAPTER 2IIICONTINUOUS-WAVE MODULATION Modulated wave'1(tl withcarrierfrequency 11Modulated waveS2(t) withcarrierfrequencyfz A,cos(2"t,ll FIGURE2.16Blockdiagramofmixer. Toexplaintheactionofthemixer,consider thesituation depicted inFigure2.17, where,forthepurposeofillustration, itisassumed thatthemixerinputSl(t)isanAM signalwithcarrierfrequencyIIandbandwidth 2W.Part(a)ofFigure2.17displaysthe AMspectrum Sl(f)assuming that1,>W.Part(b)ofthefiguredisplaysthespectrum S'(f)oftheresulting signals'(t)attheproductmodulator output. Thesignals'(t)maybeviewedasthesumoftwomodulated components: onecom ponentrepresented bytheshadedspectrum inFigure2.17b,andtheothercomponent represented bytheunshaded spectrum inthisfigure.Depending onwhethertheincoming carrierfrequencyIIistranslated upwardordownward, wemayidentifytwodifferent simations, asdescribed here: Upconversion. Inthiscasethetranslated carrierfrequency 12isgreaterthanthe incoming carrierfrequencyI"andtherequired localoscillator frequencyftisthere foredefinedby or12=/l+ft ft=/2-/1 ~',"'! ~t L:~j0 L:~j (al (bl FIGURE2.17(a)Spectrum ofmodulated signal5,(t)atthemixerinput.(b)Spectrum ofthe corresponding signal5'(t)attheoutputoftheproductmodulator inthemixer. 2.5Frequency-Division Multiplexing 105 Theunshaded partofthespectrum inFigure2.17bdefinesthewantedmodulated signalS2(t),andtheshadedpartofthisspectrum definestheimagesignalassociated withS2(t).Forobviousreasons,themixerinthiscaseisreferredtoasafrequency upconverter. Downconversion. Inthissecondcasethetranslated carrierfrequencyhissmaller thantheincoming carrierfrequency fhandtherequired oscillator frequency fiis therefore definedby or fi=f,-f2 Thepicturewehavethistimeisthereverseofthatpertaining toupconversion. In particular, theshadedpartofthespectrum inFigure2.17bdefinesthewantedmod ulatedsignal S2(t),andtheunshaded partofthisspectrum definestheassociated imagesignal.Themixerisnowreferredtoasafrequency-down converter. Notethat inthiscasethetranslated carrierfrequency f2hastobelargerthanW(i.e.,onehalf ofthebandwidth ofthemodulated signal)toavoidsideband overlap. Thepurpose oftheband-pass filterinthemixerofFigure2.16istopassthewanted modulated signalS2(t)andeliminate theassociated imagesignal.Thisobjective isachieved byaligningthemidband frequency ofthefilterwiththetranslated carrierfrequency f2and assigning itabandwidth equaltothatoftheincoming modulated signals,(t). Itisimportant tonotethatmixingisalinearoperation. Accordingly, therelationof thesidebands oftheincoming modulated wavetothecarrieriscompletely preserved at themixeroutput. I2.5Frequency-Division Multiplexing Anotherimportant signalprocessing operation ismultiplexing, whereby anumberofin dependent signalscanbecombined intoacomposite signalsuitablefortransmission over acommon channel. Voicefrequencies transmitted overtelephone systems,forexample, rangefrom300to3100Hz.Totransmit anumberofthesesignalsoverthesamechannel, thesignalsmustbekeptapartsothattheydonotinterferewitheachother,andthusthey canbeseparated atthereceiving end.Thisisaccomplished byseparating thesignalseither infrequency orintime.Thetechnique ofseparating thesignalsinfrequency isreferred to asfrequency-division multiplexing (FDM),whereasthetechnique ofseparating thesignals intimeiscalledtime-division multiplexing (TDM).Inthissection,wediscussFDMsys tems,andTDMsystemsarediscussed inChapter3. AblockdiagramofanFDMsystemisshowninFigure2.18.Theincoming message signalsareassumed tobeofthelow-pass type,buttheirspectradonotnecessarily have nonzerovaluesallthewaydowntozerofrequency. Following eachsignalinput,wehave shownalow-pass filter,whichisdesigned toremovehigh-frequency components thatdo notcontribute significantly tosignalrepresentation butarecapableofdisturbing other message signalsthatsharethecommon channel. Theselow-pass filtersmaybeomitted onlyiftheinputsignalsaresufficiently bandlimitedinitially.Thefilteredsignalsareapplied 106 CHAPTER 2illCONTINLJOLJS-WAVE MODLJLATION Message Low~pass inputs filters ModulatorsBand-pass filtersBand-pass filters DemodulatorsLow~pass Message filters outputs N N Transmitter Receiver FIGLJRE2.18BlockdiagramofFDMsystem. tomodulators thatshiftthefrequency rangesofthesignalssoastooccupymutually exclusive frequency interv'als. Thenecessary carrierfrequencies neededtoperform these frequency translations areobtained fromacarriersupply.Forthemodulation, wemay useanyoneofthemethods described inprevious sectionsofthischapter. However, the mostwidelyusedmethodofmodulation infrequency-division multiplexing issingleside bandmodulation, which,inthecaseofvoicesignals,requires abandwidth thatisap proximately equaltothatoftheoriginalvoicesignal.Inpractice, eachvoiceinputisusually assigned abandwidth of4kHz.Theband-pass filtersfollowing themodulators areused torestrictthebandofeachmodulated wavetoitsprescribed range.Theresulting band passfilteroutputsarenextcombined inparalleltoformtheinputtothecommon channel. Atthereceiving terminal, abankofband-pass filters,withtheirinputsconnected inpar allel,isusedtoseparate themessage signalsonafrequency-occupancy basis.Finally,the originalmessage signalsarerecovered byindividual demodulators. NotethattheFDM systemshowninFigure2.18operates inonlyonedirection. Toprovidefortwo-way transmission, asintelephony, forexample, wehavetocompletely duplicate themulti plexingfacilities, withthecomponents connected inreverseorderandwiththesignal wavesproceeding fromrighttoleft. ~ExAMPLE 2.1 Thepractical implementation ofanFDMsystemusuallyinvolvesmanystepsofmodulation and demodulation, asillustrated inFigure2.19.Thefirstmultiplexing stepcombines 12voice inputsintoabasicgroup,whichisformedbyhavingthenthinputmodulate acarrierat frequencyte=60+4nkHz,wheren=1,2,...,12.Thelowersidebands arethenselected byband-pass filteringandcombined toformagroupof12lowersidebands (oneforeach voiceinput).Thusthebasicgroupoccupiesthefrequency band60to108kHz.Thenextstep intheFDMhierarchy involvesthecombination offivebasicgroupsintoasupergroup. Thi' isaccomplished byusingthenthgrouptomodulate acarrieroffrequencyte=372+48n kHz,wheren=1,2,...,5.Hereagainthelowersidebands areselectedbyfilteringandthen 60 Basicgroupof12 voiceinputs12 11 10 9 8 7 6 5 4 3 2 14kHz ~}Carrierfrequencies (inkHz) ofvoiceinputs \ 108 104100 96 92 88 ~: 76 72 68 642.6AngleModulation 107 Carrierfrequencies 108#kHZ (inkHz)°tOUijPS 5~~~kHz 564 4456 516 3408 468 2360 420 1312 Supergroup of5groups Voiceband FIGURE2.19Illustrating themodulation stepsinanFDMsystem. combined toformasupergroup occupying theband312to552kHz.Thusasupergroup is designedtoaccommodate 60independent voiceinputs.Thereasonforforming thesupergroup inthismanneristhateconomical filtersoftherequiredcharacteris[ics areavailableonlyover alimitedfrequency range.Inasimilarmanner,supergroups arecombined intomastergroups, andmastergroups arecombined intoverylargegroups. -<IiI Intheprevious sectionsofthischapter, weinvestigated theeffectofslowlyvaryingthe amplitude ofasinusoidal carrierwaveinaccordance withthebaseband (information carrying) signal.Thereisanotherwayofmodulating asinusoidal carrierwave,namely, anglemodulation inwhichtheangleofthecarrierwaveisvariedaccording tothebaseband signal.Inthismethodofmodulation, theamplitude ofthecarrierwaveismaintained constam. Animportant featureofanglemodulation isthatitcanprovidebetterdiscrim inationagainstnoiseandinterference thanamplitude modulation. Aswillbeshownlater inSection2.7,however, thisimprovemem inperformance isachieved attheexpenseof increased transmission bandwidth; thatis,anglemodulation provides uswithapractical meansofexchanging channelbandwidth forimproved noiseperformance. Suchatrade offisnotpossiblewithamplitude modulation, regardless ofitsform. !!!BASIC DEFINITIONS Letl1i(t)denotetheangleofamodulated sinusoidal carrier,assumed tobeafunction of themessage signal.Weexpresstheresulting angle-modulated waveas s(t)=Accos[l1i(t)] (2.19) lOS CHAPTER 2IIICONTINUOUS-WAVE MODULATION whereAeisthecarrieramplitude. Acomplete oscillation occurswhenever lIi{t)changes by21Tradians.IflIi{t)increases monotonically withtime,theaveragefrequency inHertz, overanintervalfromttot+I1t,isgivenby I()=1I;(t+I1t)-lIi{t) Mt 21TI1t(2.20) Wemaythusdefinetheinstantaneous frequency oftheangle-modulated signals{t)as follows: {;{t)=limIu,{t) It.t_O =lim[Oi{t+I1t)-lIi{t)] "'~O 21TI1t 1dOi{t) =21T--;It(2.21) Thus,according toEquation (2.19),wemayinterpret theangle-modulated signal s(t)asarotatingphasoroflengthAeandanglelIi{t).Theangularvelocityofsuchaphasor isdlli{t)ldt measured inradianspersecond,inaccordance withEquation (2.2l).Inthe simplecaseofanunmodulated carrier,theanglelIi{t)is andthecorresponding phasorrotates withaconstant angularvelocityequalto21Tlc'The constant 4>eisthevalueofO;(t)att=O. ThereareaninfinitenumberofwaysinwhichtheangleOi(t)maybevariedinsome mannerwiththemessage (baseband) signal.However, weshallconsider onlytwocom monlyusedmethods, phasemodulation andfrequency modulation, definedasfollows: 1.Phasemodulation (PM)isthatformofanglemodulation inwhichtheangleO;(t)is variedlinearlywiththemessage signalm(t),asshownby (2.22) Theterm21Tlctrepresents theangleoftheunmodulated carrier;andtheconstantkp represents thephasesensitivity ofthemodulator, expressed inradianspervolton theassumption thatm(t)isavoltagewaveform. Forconvenience, wehaveassumed inEquation (2.22)thattheangleoftheunmodulated carrieriszeroatt=O.The phase-modulated signals(t)isthusdescribed inthetimedomainby (2.23) 2.Frequency modulation (FM)isthatformofanglemodulation inwhichtheinstan taneous frequency {;(t)isvariedlinearlywiththemessage signalm(t),asshownby (2.24) ThetermIerepresents thefrequency oftheunmodulated carrier,andtheconstant kfrepresents thefrequency sensitivity ofthemodulator, expressed inHertzper volt 2.7Frequency Modulation 109 Mod,lating wave (a)Modulating wave (b) FIGURE2.20Illustrating therelationship between frequency modulation andphasemodulation. (a)Scheme forgenerating anFMwavebyusingaphasemodulator. (b)Scheme forgenerating a PMwavebyusingafrequency modulator. ontheassumption thatm(t)isavoltagewaveform. Integrating Equation (2.24)with respecttotimeandmultiplying theresultby21T,weget Oi(t)=21Tfet+21TkfJ:m(T)dT (2.25) where,forconvenience, wehaveassumed thattheangleoftheunmodulated carrier waveiszeroatt=O.Thefrequency-modulated signalistherefore described inthe timedomainby (2.26) Aconsequence ofallowing theangleOi(t)tobecomedependent onthemessagesignal m(t)asinEquation (2.22)oronitsintegralasinEquation (2.25)isthatthezerocrossings ofaPMsignalorFMsignalnolongerhaveaperfectregularity intheirspacing; zero crossings refertotheinstantsoftimeatwhichawaveform changesfromanegative toa positivevalueorviceversa.Thisisoneimportant featurethatdistinguishes bothPMand FMsignalsfromanAMsignal.Another important difference isthattheenvelope ofaPM orFMsignalisconstant (equaltothecarrieramplitude), whereas theenvelope ofanAM signalisdependent onthemessage signal. Comparing Equation (2.23)with(2.26)revealsthatanFMsignalmayberegarded asaPMsignalinwhichthemodulating waveisf~m(T)dTinplaceofm(t).Thismeans thatanFMsignalcanbegenerated byfirstintegrating m(t)andthenusingtheresultas theinputtoaphasemodulator, asinFigure2.20a.Conversely, aPMsignalcanbegen eratedbyfirstdifferentiating m(t)andthenusingtheresultastheinputtoafrequency modulator, asinFigure2.20b.Wemaythusdeducealltheproperties ofPMsignalsfrom thoseofFMsignalsandviceversa.Henceforth, weconcentrate ourattention onFM signals. l2.7Frequency Modulation TheFMsignals(t)definedbyEquation (2.26)isanonlinear function ofthemodulating signalm(t),whichmakesfrequency modulation anonlinear modulation process.Conse quently, unlikeamplitude modulation, thespectrum ofanFMsignalisnotrelatedina simplemannertothatofthemodulating signal;rather,itsanalysisismuchmoredifficult thanthatofanAMsignal. 110 CHAPTER 2"CONTINUOUS-WAVE MODULATION HowthencanwetacklethespectralanalysisofanFMsignal?Weproposetoprovide anempirical answertothisimportant question byproceeding inthefollowing manner: ~Weconsiderthesimplestcasepossible, namely,thatofasingle-tone modulation that produces anarrowband FMsignal. ~Wenextconsiderthemoregeneralcasealsoinvolving asingle-tone modulation, but thistimetheFMsignaliswideband. Wecould,ofcourse,goonandconsiderthemoreelaborate caseofamultitone FMsignal. However, weproposenottodoso,becauseourimmediate objective istoestablish an empirical relationship betweenthetransmission bandwidth ofanFMsignalandthemes sagebandwidth. Asweshallsubsequently see,thetwo-stage spectralanalysisdescribed hereprovides uswithenoughinsighttoproposeasolutiontotheproblem. Consider thenasinusoidal modulating signaldefinedby Theinstantaneous frequency oftheresulting FMsignalequals tit)=fe+kfAmcos(2'T1fm t) =fe+!:1fcos(2'T1fm t) where(2.27) (2.28) (2.29) Thequantity I1fiscalledthefrequency deviation, representing themaximum departure oftheinstantaneous frequency oftheFMsignalfromthecarrierfrequencyt.Afunda mentalcharacteristic ofanFMsignalisthatthefrequency deviation I1fisproportional totheamplitude ofthemodulating signalandisindependent ofthemodulation frequency. UsingEquation (2.28),theangleei(t)oftheFMsignalisobtained as ei(t)=2'TTJ:t(T)dT =2'TTfct+~:sin(2'TTfmt)(2.30) Theratioofthefrequency deviation!:1f tothemodulation frequency fmiscommonly called themodulation indexoftheFMsignal.Wedenoteitbyf3,andsowrite andf3=!:1f fm(2.31) (2.32) FromEquation (2.32)weseethat,inaphysicalsense,theparameter f3represents thephase deviation oftheFMsignal,thatis,themaximum departure oftheangleei(t)fromthe angle21ff,toftheunmodulated carrier;hence,f3ismeasured inradians. TheFMsignalitselfisgivenby (2.33) 2.7Frequency Modulafion III Depending onthevalueofthemodulation index13,wemaydistinguish twocasesof frequency modulation: PNarrowband FM,forwhich13issmallcompared tooneradian. ~WidebandFM,forwhich13islargecompared tooneradian. Thesetwocasesareconsidered next,inthatorder. IIINARROWBAND FREQUENCY MODULATION Consider Equation (2.33),whichdefinesanFMsignalresulting fromtheuseofasinusoidal modulating signal.Expanding thisrelation, weget s(t)=Accos(2nfct) cos[f3sin(21Tfmt)] -Acsin(21Tfct) sin[f3sin(21Tfmt)] (2.34) Assuming thatthe modulation index13issmallcompared tooneradian,wemayusethe following approximations: and sin[f3sin(21Tfmt)] '"13sin(21Tfmt) Hence,Equation (2.34)simplifies to (2.35) Equation (2.35)definestheapproximate formofanarrowband FMsignalproduced bya sinusoidal modulating signalAmCOS(21Tfmt). Fromthisrepresentation wededucethemod ulatorshowninblockdiagramforminFigure2.21.Thismodulator involvessplittingthe carrierwaveAcCOS(21Tfct) intotwopaths.Onepathisdirect;theotherpathcontains a -90degreephase-shifting network andaproductmodulator, thecombination ofwhich generates aDSB-SCmodulated signal.Thedifference betweenthesetwosignalsproduces anarrowband FMsignal,butwithsomedistortion. Ideally,anFMsignalhasaconstant envelope and,forthecaseofasinusoidal mod ulatingsignaloffrequency fm'theanglee,(t)isalsosinusoidal withthesamefrequency. Narrowband FMwaveI I I II I I 1-Ei--4-+-- Carrierwavei IAccos(21T!ct) I I ~-------------- I"--------------------1 I I I i I IModulating wave Narrowband phasemodulator FIGURE2.21Blockdiagramofamethodforgenerating anarrowband PMsignal. 112 CHAPTER 2OJCONTINUOUS-WAVE MODUlATION Butthemodulated signalproduced bythenarrowband modulator ofFigure2.21differs fromthisidealcondition intwofundamental respects: 1.Theenvelope contains aresidualamplitude modulation and,therefore, varieswith time. 1.Forasinusoidal modulating wave,theangleIJi(t)containsharmonic distortion in theformofthird-andhigher-order harmonics ofthemodulation frequencytn' However, byrestricting themodulation indexto{3:s:0.3radians,theeffectsofresidUal AMandharmonic PMarelimitedtonegligible levels. Returning toEquation (2.35),wemayexpanditasfollows: 1sit)=AcCOS(271fct)+2"{3Aclcos[21T(fc +fm)t]-COS[21T(fc -fm)tJ) (2.36) Thisexpression issomewhat similartothecorresponding onedefining anAMsignal, whichisasfollows: wheref.Listhemodulation factoroftheAMsignal.Comparing Equations (2.36)and (2.37),weseethatinthecaseofsinusoidal modulation, thebasicdifference between an AMsignalandanarrowband FMsignalisthatthealgebraic signofthelowersidefre quencyinthenarrowband FMisreversed. Thus,anarrowband FMsignalrequiresessen tiallythesametransmission bandwidth (i.e.,2fm)astheAMsignal. Wemayrepresent thenarrowband FMsignalwithaphasordiagram asshownin Figure2.22a,wherewehaveusedthecarrierphasorasreference. Weseethattheresultant CarrierResultant CalSumofside~ frequency phasors,,,, "0(m ,,,, C. ,Sumof_______ :..:a:..:"..:'o:...' ~'- *~side-frequency // phasors / / / ~ 1m (bl FIGURE2.22Aphasorcomparison ofnarrowband FMandAi\'!wavesforsinusoidal modula tion.(a)Narrowband FMwave.(b)AMwave. (2.38)2.7Frequency Modulation 113 ofthetwoside-frequency phasorsisalwaysatrightanglestothecarrierphasor.Theeffect ofthisistoproduce aresultant phasorrepresenting thenarrowband FMsignalthatis approximately ofthesameamplitude asthecarrierphasor,butoutofphasewithrespect toit.Thisphasordiagramshouldbecontrasted withthatofFigure2.22b,representing anAMsignaLInthislattercase weseethattheresultant phasorrepresenting theAM signalhasanamplitude thatisdifferent fromthatofthecarrierphasorbutalwaysinphase withit. C iliIWIDEBAND FREQUENCY MODULATION Wenextwishtodetermine thespectrum ofthesingle-tone FMsignalofEquation (2.33) foranarbitrary valueofthemodulation indexf3.Ingeneral,anFMsignalproduced bya sinusoidal modulating signal,asinEquation (2.33),isinitselfnonperiodic unlessthe carrierfrequency Ieisanintegralmultiple ofthemodulation frequency 1m.However, we maysimplifymattersbyusingthecomplex representation ofband-pass signalsdescribed inAppendix 2.Specifically, weassumethatthecarrierfrequency Ieislargeenough(com paredtothebandwidth oftheFMsignal)tojustifyrewriting thisequation intheform sIt)=Re[Acexp(j27Tlct +jf3sin(27Tlmt))] =Re[s(t)exp(j27Tlct)] wheres(t)isthecomplex envelope oftheFMsignals(t),definedby s(t)=Aeexp[jf3sin(27T{mt)] (2.39) Thus,unliketheoriginalFMsignals(t),thecomplex envelope s(t)isaperiodicfunction oftimewithafundamental frequency equaltothemodulation frequency 1m'Wemay therefore expandsIt)intheformofacomplex Fourierseriesasfollows: wherethecomplex Fouriercoefficient c"isdefinedby J"/2/m Cn=1m s(t)eXp(-j27Tnl mt)dt -1/2/m J1I2fm =ImAc exp[jf3sin(27Tlmt) -j27Tnlmt]dt -1I2f", Defineanewvariable: x=27Tlmt Hence,wemayrewriteEquation (2.41)inthenewform Cn=2AcJr.exp[j(f3sinx-nx)]dx 7T-r.(2.40) (2.41) (2.42) (2.43) Theintegralontheright-hand sideofEquation (2.43),exceptforascalingfactor,is recognized asthenthorderBesselfunctionofthefirstkind3andargument f3.Thisfunction iscommonly denotedbythesymbolIn(f3),asshownby 1Jr.In(f3)=27T-r.exp[j(f3sinx-nx)]dx (2.44) 114 CHAPTER 2"CONTINUOUS-WAVE MODULATION Accordingly, wemayreduceEquation (2.43)to Cn=Acfn(f3) (2,45) Substituting Equation (2,45)in(2,40),weget,intermsoftheBesselfunction 1,,(f3),the following expansion forthecomplex envelope oftheFMsignal: s(t)=AeLIn(f3)exp(j27rnfm t) Next,substituting Equation (2.46)in(2.38),weget(2,46) (2,47) Interchanging theorderofsummation andevaluation oftherealpartintheright-hand sideofEquation (2.47),wefinallyget ~ s(t)=Ae2:1"(f3)cos[27r(fc+nfm)t] (2.4S) ThisisthedesiredformfortheFourierseriesrepresentation ofthesingle-tone FMsignal s(t)foranarbitrary valueoff3.Thediscretespectrum ofs(t)isobtained bytakingthe Fouriertransforms ofbothsidesofEquation (2.48);wethushave A-S(f)= ;JLIn(f3)[S(f -fe-nfm)+o(f+fe+nfm)] (2,49) InFigure2.23wehaveplotredtheBesselfunctionln(f3)versusthemodulation index f3fordifferent positiveintegervaluesofn.Wecandevelopfurtherinsightintothebehavior 0.6 0.4 0.2 -0.2 -0.4 FIGURE2.23PlotsofBesselfunctions ofthefirstIdndforvaryingorder. 2.7Frequency Modulation 115 oftheBesselfunction !n(f3)bymakinguseofthefollowing properties (seeAppendix 3for moredetails): 1.!n(f3)=(-I)n!_n(f3) foralln,bothpositiveandnegative 2.Forsmallvaluesofthemodulation indexf3,wehave(2.50) !o(f3)=1 !1(f3)=~ !n(f3)=0,nJ(2.51) 3. L!~(f3)=1 (2.52) (2.53) (2.54)Thus,usingEquations (2.49)-(2.52) andthecurvesofFigure2.23,wemaymake hefollowing observations: 1.Thespectrum ofanFMsignalcontains acarriercomponent andaninfinitesetof sidefrequencies locatedsymmetrically oneithersideofthecarrieratfrequency sep arationsof1m,21m,31m,....Inthisrespect,theresultisunlikethatwhichprevails inanAMsystem,sinceinanAMsystemasinusoidal modulating signalgivesrise toonlyonepairofsidefrequencies. 2.Forthespecialcaseoff3smallcompared withunity,onlytheBesselcoefficients !o(f3) and!1(f3)havesignificant values,sothatthePMsignaliseffectively composed ofa carrierandasinglepairofsidefrequencies atIe::t:1m.Thissituation corresponds to thespecialcaseofnarrowband FMthatwasconsidered earlier. 3.Theamplitude ofthecarriercomponent varieswithf3according to!o(f3).Thatis, unlikeanAMsignal,theamplitude ofthecarriercomponent ofanFMsignalis dependent onthemodulation indexf3.Thephysicalexplanation forthisproperty is thattheenvelope ofanFMsignalisconstant, sothattheaveragepowerofsucha signaldeveloped acrossaI-ohmresistorisalsoconstant, asshownby P=.!.A2 2' Whenthecarrierismodulated togenerate theFMsignal,thepowerintheside frequencies mayappearonlyattheexpenseofthepoweroriginally inthecarrier, therebymakingtheamplitude ofthecarriercomponent dependent onf3.Notethat theaveragepowerofanFMsignalmayalsobedetermined fromEquation (2.48), obtaining P=.!.A~i!~(f3)2 n~-~ Substituting Equation (2.52)into(2.54),theexpression fortheaveragepowerP reducestoEquation (2.53),andsoitshould. •EXAMPLE 2.2 Inthisexample, wewishtoinvestigate thewaysinwhichvariations intheamplitude and frequency ofasinusoidal modulating signalaffectthespectrum oftheFMsignal.Consider 116 CHAPTER 2IIICONTINUOUS-WAVE MODULATION 1.0 _:-:- --"'--'--L....L.....L.-.L.1Il f il=1.0 (a) 1.0 _.,..,-- --'-L-'--'--'-'-'-i-''-- f il=2.0 (b) 1.0 i3--=-5.-0~--"'---'----IL--'---'---1IL-'_f,L,. --'-"'---'--..j-'--,...Lr-'-f,-"'m--'--- f h-----2t:.f (c) FIGURE2.24Discreteamplitude spectraofanPMsignal,normalized withrespecttothecarrier amplitude, forthecaseofsinusoidal modulation offixedfrequency andvaryingamplitude. Only thespectraforpositivefrequencies areshown. firstthecasewhenthefrequency ofthemodulating signalisfixed,butitsamplitude isvaried, producing acorresponding variation inthefrequency deviation /!J.f.Thus,keepingthemod ulationfrequency fmfixed,wefindthattheamplitude spectrum oftheresulting PMsignalis asshownplottedinFigure2.24forf3=1,2,and5.Inthisdiagramwehavenormalized the spectrum withrespecttotheunmodulated carrieramplitude. Consider nextthecasewhentheamplitude ofthemodulating signalisfixed;thatis,tho frequency deviation /!J.fismaintained constant, andthemodulation frequency fmisvaried. Inthiscasewefindthattheamplitude spectrum oftheresulting PMsignalisasshown plottedinFigure2.25forf3=1,2,and5.Weseethatwhen/!J.fisfixedandf3isincreased, wehaveanincreasing numberofspectrallinescrowding intothefixedfrequency interval f,/!J.f<If1<f,+/!J.f·Thatis,whenf3approaches infinity,thebandwidth oftheFMwave 2.7Frequency Modulation 117 1.0 -------"'"---L-----'------'------'-------'--------"---f/l=1.0 (a) 1.0 -- ..A-_L-----L_--'-------l_---'---_l..----l._..A- f /l=2.0 (b) 1.0 ---------'--"L.L...L...J-'-""--'-'--'--..!rL.L-'----'---'---"'--J. f /l~5.0 L- fe I 2~f-------+I (0) FIGURE2.25Discrete amplitude spectraofanFMsignal,normalized withrespecttothecarrier amplitude, forthecaseofsinusoidal modulation ofvaryingfrequency andfixedamplitude. Only thespectraforpositivefrequencies areshown. approaches thelimitingvalueof2!!.f,whichisanimportant pointtokeepinmindforlater discussion. ..qj iliITRANSMISSION BANDWIDTH OFFMSIGNALS Intheory,anFMsignalcontainsaninfinitenumberofsidefrequencies sothattheband widthrequiredtotransmitsuchasignalissimilarly infiniteinextent.Inpractice, however, wefindthattheFMsignaliseffectively limitedtoafinitenumberofsignificant side frequencies compatible withaspecified amountofdistortion. Wemaytherefore specify aneffective bandwidth required forthetransmission ofanFMsignal.Consider firstthe (2.55)lIS CHAPTER 2;1lCONTINUOUS-WAVE MODULATION caseofanFMsignalgenerated byasingle-tone modulating waveoffrequency fm.Insuch anFMsignal,thesidefrequencies thatareseparated fromthecarrierfrequency fcbyan amountgreaterthanthefrequency deviation !!.fdecrease rapidlytowardzero,sothatthe bandwidth alwaysexceedsthetotalfrequency excursion, butnevertheless islimited.Spe cifically, forlargevaluesofthemodulation indexf3,thebandwidth approaches, andis onlyslightlygreaterthan,thetotalfrequency excursion 2!!.finaccordance withthesitu ationshowninFigure2.25.Ontheotherhand,forsmallvaluesofthemodulation index f3,thespectrum of theFMsignaliseffectively limitedtothecarrierfrequency fcandone pairofsidefrequencies atfc:!:fm,sothatthebandwidth approaches 2fm.Wemaythus defineanapproximate ruleforthetransmission bandwidth ofanFMsignalgenerated by asingle-tone modulating signaloffrequency fmasfollows: By=2!!.f+2fm=2!!.f(1+~) Thisempirical relationisknownasCarson's rule" Foranalternative assessment ofthebandwidth requirement of anFMsigna~we mayuseadefinition basedonretaining themaximum numberofsignificant sidefrequen cieswhoseamplitudes areallgreaterthansomeselectedvalue.Aconvenient choicefor thisvalueis1percentoftheunmodulated carrieramplitude. Wemaythusdefinethe transmission bandwidth ofanFMwaveastheseparation between thetwofrequencies beyondwhichnoneofthesidefrequencies isgreaterthan1percentofthecarrieramplitude obtained whenthemodulation isremoved. Thatis,wedefinethetransmission bandwidth as2nmaJm,wherefmisthemodulation frequency andnmaxisthelargestvalueofthe integernthatsatisfiestherequirement IJn(f3)I>0.01.Thevalueofnmaxvarieswiththe modulation indexf3andcanbedetermined readilyfromtabulated valuesoftheBessel function In(f3).Table2.2showsthetotalnumberofsignificant sidefrequencies (including boththeupperandlowersidefrequencies) fordifferent valuesoff3,calculated onthe1 percentbasisexplained herein.Thetransmission bandwidth Bycalculated usingthispro cedurecanbepresented intheformofauniversal curvebynormalizing itwithrespectto thefrequency deviation !!.fandthenplottingitversusf3.ThiscurveisshowninFigure 2.26,whichisdrawnasabestfitthroughthesetofpointsobtained byusingTable2.2. InFigure2.26wenotethatasthemodulation indexf3isincreased, thebandwidth occupied TABLE2.2Numberofsignifreant side frequencies ofawideband FMsignalforvarying modulation index Modulation Index 13 0.1 0.3 0.5 1.0 2.0 5.0 10.0 20.0 30.0NumberofSignificant SideFrequencies 2nmilx 2 4 4 6 8 16 28 50 70 2.7Frequency Modulation 119 40 2------------------------------------- f3 FIGURE2.26Universal curveforevaluating the1percentbandwidth ofanFMwave. bythesignificant sidefrequencies dropstowardthatoverwhichthecarrierfrequency actuallydeviates. Thismeansthatsmallvaluesofthemodulation index{3arerelatively moreextravagant intransmission bandwidth thanarethelargervaluesof{3. Consider nextthemoregeneralcaseofanarbitrary modulating signalmit)withits highestfrequency component denoted byW.Thebandwidth required totransmit an FMsignalgenerated bythismodulating signalisestimated byusingaworst-case tone modulation analysis. Specifically, wefirstdetermine theso-called deviation ratioD,defined astheratioofthefrequency deviation11/,whichcorresponds tothemaximum possible amplitude ofthemodulation signalmit),tothehighestmodulation frequency W;these conditions represent theextremecasespossible. Thedeviation ratioDplaysthesamerole fornonsinusoidal modulation thatthemodulation index{3playsforthecaseofsinusoidal modulation. Then,replacing {3byDandreplacing/mwithW,wemayuseCarson's rule givenbyEquation (2.55)ortheuniversal curveofFigure2.26toobtainavalueforthe transmission bandwidth oftheFMsignal.Fromapractical viewpoint, Carson's rulesome whatunderestimates thebandwidth requirement ofanFMsystem,whereas usingtheuni versalcurveofFigure2.26yieldsasomewhat conservative result.Thus,thechoiceofa transmission bandwidth thatliesbetweentheboundsprovided bythesetworulesofthumb isacceptable formostpractical purposes. Ii>-EXAMPLE 2.3 InNorthAmerica, themaximum valueoffrequency deviation 11[isfixedat75kHzfor commercial FMbroadcasting byradio.Ifwetakethemodulation frequency W=15kHz, whichistypicallythe"maximum" audiofrequency ofinterestinFMtransmission, wefind thatthecorresponding valueofthedeviation ratiois D=75=5 15 UsingCarson'sruleofEquation (2.55),replacing f3byD,andreplacing [mbyW,theap proximate valueofthetransmission bandwidth oftheFMsignalisobtained as BT=2(75+15)=180kHz 120 CHAPTER 2..CONTINUOUS-WAVE MODUlATION Ontheotherhand,useofthecurveofFigure2.26givesrhetransmission bandwidth ofthe FMsignaltobe By=3.2!:J.f=3.2X75=240kHz Inpractice,abandwidth of200kHzisallocated toeachFMtransmitter. Onthisbasis, Carson'sruleunderestimates thetransmission bandwidth by10percent,whereastheuniversal curveofFigure2.26overestimates itby20percent. ... IIiIIGENERATION OFFMSIGNALS Thereareessentially twobasicmethods ofgenerating frequency-modulated signals, namely,directFMandindirectFM.Inthedirectmethodthecarrierfrequency isdirectly variedinaccordance withtheinputbaseband signal,whichisreadilyaccomplished using avoltage-controlled oscillator. Intheindirectmethod, themodulating signalisfirstused toproduceanarrowband FMsignal,andfrequency multiplication isnextusedtoincrease thefrequency deviation tothedesiredlevel.Theindirectmethodisthepreferred choice forfrequency modulation whenthestabilityofcarrierfrequency isofmajorconcernasin commercial radiobroadcasting, asdescribed next. IndirectFM' Asimplified blockdiagramofanindirectFMsystemisshowninFigure2.27.The message (baseband) signalm(t)isfirstintegrated andthenusedtophase-modulate a crystal-controlled oscillator; theuseofcrystalcontrolprovides frequency stability. To minimize thedistortion inherent inthephasemodulator, themaximum phasedeviation ormodulation indexf3iskeptsmall,therebyresulting inanarrowband FMsignal;forthe implementation ofthenarrow-band phasemodulator, wemayusethearrangement de scribedinFigure2.21.Thenarrowband FMsignalisnextmultiplied infrequency bymeans ofafrequency multiplier soastoproduce thedesiredwidebandFMsignal. Afrequency multiplier consistsofanonlinear devicefollowed byaband-pass filter, asshowninFigure2.28.Theimplication 0,£thenonlinear devicebeingmemoryless isthat ithasnoenergy-storage elements. Theinput-output relationofsuchadevicemaybe expressed inthegeneralform (2.56) whereahaz,•..,anarecoefficients determined bytheoperating pointofthedevice,and nisthehighestorderofnonlinearity. Inotherwords,thememoryless nonlinear deviceis annthpower-law device.Theinputs(t)isanFMsignaldefinedby sit)=Accos[27rfct+27rkf1:m(T)dT] Baseband signal m(tjFMsignal s{t) FIGURE2.27Blockdiagramoftheindirectmethodofgenerating awideband FMsignal. 2.7Frequency Modulation 121 FMsignals(t) withcarrierfrequency Ieandmodulation index13Memoryless nonlinear devicevet) Band-pass filter withmidband frequency nfr:FMsignal,'(t)with carrierfrequency njt andmodulation indexnf3 FIGURE2.28Blockiliagramoffrequency multiplier, whoseinstantaneous frequency is (2.57) (2.58)Themid-band frequency oftheband-pass filterinFigure2.28issetequaltonl"whereIe isthecarrierfrequency oftheincoming FMsignals(t).Moreover, theband-pass filteris designed tohaveabandwidth equaltontimesthetransmission bandwidth ofsit).In Section2.8dealingwithnonlinear effectsinFMsystems, wedescribe thespectralcontri butionsofsuchnonlinear termsasthesecond-andthird-order termsintheinput-output relationofEquation (2.56).Fornowitsufficestosaythatafterband-pass filteringofthe nonlinear device'soutputv(t),wehaveanewFMsignaldefinedby s'(t)=A~cos[21Tnfct+21Tnkff:miT)dT] whoseinstantaneous frequency is If(t)=nle+nkfm(t) (2.59) Thus,comparing Equation (2.59)with(2.57),weseethatthenonlinear processing circuit ofFigure2.28actsasafrequency multiplier. Thefrequency multiplication ratioisdeter minedbythehighestpowernintheinput-output relationofEquation (2.56),character izingthememoryless nonlinear device. !!!lDEMODUlAnON OFFMSIGNALS Frequency demodulation istheprocessthatenablesustorecovertheoriginalmodulating signalfromafrequency-modulated signal.Theobjective istoproduce atransfercharac teristicthatistheinverseofthatofthefrequency modulator, whichcanberealizeddirectly orindirectly. Herewedescribe adirectmethodoffrequency demodulation involving the useofapopulardeviceknownasafrequency discriminator, whoseinstantaneous output amplitude isdirectlyproportional totheinstantaneous frequency oftheinputFMsignal. InSection2.14,wedescribe anindirectmethodoffrequency demodulation thatuses anotherpopulardeviceknownasaphase-locked loop. Basically, thefrequency discriminator consistsofaslopecircuitfollowed byanen velopedetector.Anidealslopecircuitischaracterized byafrequency response thatis purelyimaginary, varyinglinearlywithfrequency insideaprescribed frequency interval. Consider thefrequency response depicted inFigure2.29a,whichisdefinedby (BT) BT BTj21TaI-Ie+2' Ie-2:=;I:=;Ie+2 HI(!)=j21Ta(1+Ie-~T), -Ie-~T:=;I:=;-Ie+~T (2.60) 0, elsewhere 122 CHAPTER 2illCONTINUOUS-WAVE MODUlATION ------:'----:-B-:'-T---- 1 2BT-!c+2 ----r'---r----~----...J--1..._::_-1 (a) (bl Slope=-2,," _t._BT-f,+1!I.r:""2 c2--,----.------+------'---'--1o BT'+1!I.!c-T Je2 (e) FIGURE2.29(a)Frequency response ofidealslopecircuit.(blFrequency response oftheslope circuit's complex low-pass equivalent. (c)Frequency response oftheidealslopecircuitcomple mentary tothatofpart(al. (2.61) 1>0whereaisaconstant. Wewishtoevaluate theresponse ofthisslopecircuit,denoted by S,(t),whichisproduced byanFMsignals(t)ofcarrierfrequency Ieandtransmission bandwidth BT•Itisassumed thatthespectrum ofs(t)isessentially zerooutsidethefre quencyintervalIe-By/2:5III:5Ie+BT/2. Wemaysimplifytheanalysisofthefrequency discriminator byinvoking theiso morphism between areal-valued band-pass filterandacorresponding complex-valued low-pass filter.Thisisomorphism isdiscussed inAppendix 2.According tothematerial presented inthatappendix, wemayreplacetheband-pass filterwithfrequency response H,(f)withanequivalent low-pass filterwithfrequency responsefI,(f)bydoingtwo things: 1.WeshiftfI,(f)totherightbyIe'whereIeisthemidband frequency oftheband passfilter;thisoperation alignsthetranslated frequency response oftheequivalent low-pass filterwiththatoftheband-pass filter. 2.WesetfI,(1-Ie)equalto2Htlf)forI>O. Thusfortheproblemathandweget fI,(1-Ie)=2H,(f), Hence,usingEquations (2.60)and(2.61),weget fI,(f)={i47Ta(1+~T), 0,BTBT} --:51:5-2 2 elsewhere(2.62) whichisplottedinFigure2.29b. 2.7Frequettey Modulation 123 Theincoming FMsignals(t)isdefinedbyEquation (2.26),whichisreproduced here forconvenience: s(t)=A,cos[21T'lct+21T'ktJ:m(7')d7'] Giventhatthecarrierfrequency Ieishighcompared tothetransmission bandwidth ofthe FMsignals(t),thecomplex envelope ofs(t)is s(t)=Acexp[j21T'k t1:m(7')d7'] (2.63) LetS,(t)denote the complex envelope oftheresponse oftheslopecircuitdefinedby Figure2.29aduetos(t).Then,following thematerial presented inAppendix 2,wemay expresstheFouriertransform ofs,(t)asfollows: 5,(f)=1H,(f)5(f) ={j21T'a(1+~Y)5(f), 0,By By--:0;/:0;-2 2 elsewhere(2.64) (2.65) (2.66)where5(f)istheFouriertransform ofs(t).FromFourieranalysisweknowthatmultipli cationoftheFouriertransform ofasignalbyj21T'1isequivalent todifferentiating the signalinthetimedomain; seeitem8ofTableA6.2.Hence,fromEquation (2.64)we deduce S,(t)=a[d~~)+j1T'BYS(t)] Substituting Equation (2.63)into(2.65),weget S,(t)=j1TB7"lAc[ 1+~;m(t)]exP[j21T'ktJ:m(r)d7'] Thedesiredresponse oftheslopecircuitistherefore S,(t)=Re[s,(t)exp(j21Tfct)] =1T'BJ<lAc[1+~k;m(t)]cos[21Tlct+21T'kfJ:m(7')d7'+I](2.67) ThesignalS,(t)isahybrid-modulated signalinwhichbothamplitude andfrequency of thecarrierwavevarywiththemessagesignalm(t).However, provided thatwechoose I¥:m(t)1<1forallt thenwemayuseanenvelope detectortorecovertheamplitude variations andthus,except forabiasterm,obtaintheoriginalmessagesignal.Theresulting envelope-detector output istherefore Is,(t)'=1TB7"lAc[ 1+¥:m(t)] (2.68) Thebiasterm1T'B7"lAcintheright-hand sideofEquation (2.68)isproportional to theslopeaofthetransferfunction oftheslopecircuit.Thissuggeststhatthebiasmaybe 124 CHAPTER 2'"CONTINUOUS-WAVE MODULATION FMwaveBaseband signal FIGURE2.30Blockdiagram offrequency discriminator. removed bysubtracting fromtheenvelope-detector output151(t)Itheoutputofasecond envelope detectorpreceded bythecomplementary slopecircuitwithafrequency response H2(f)asdescribed inFigure2.29c.Thatis,thetwoslopecircuitsarerelatedby (2.69) (2.70) (2.71)LetS2(t)denotetheresponse ofthecomplementary slopecircuitproduced bytheincoming FMsignals(t).Then,following aprocedure similartothatjustdescribed, wemaywrite I52(t)I=7TB7i1Ac[ 1 -¥;m(t)] where52(t)isthecomplex envelope ofthesignalS2(t).Thedifference between thetwo envelopes inEquations (2.68)and(2.70)is so(t)=I51(t)I- I52(t)I =47TkrtAcm(t) whichisascaledversionoftheoriginalmessagesignalm(t)andfreefrombias. Wemaythusmodeltheidealfrequency discriminator asapairofslopecircuitswith theircomplex transferfunctions relatedbyEquation (2.69),followed byenvelope detectors andfinallyasummer, asinFigure2.30.Thisschemeiscalledabalanced frequency discriminator. Ill!FMSTEREO MVLTIPLEXING6 Stereomultiplexing isaformoffrequency-division multiplexing (FDM)designed totrans mittwoseparate signalsviathesamecarrier.ItiswidelyusedinFMradiobroadcasting tosendtwodifferent elements ofaprogram (e.g.,twodifferent sectionsofanorchestra, avocalistandanaccompanist) soastogiveaspatialdimension toitsperception bya listeneratthereceiving end. Thespecification ofstandards forFMstereotransmission isinfluenced bytwo factors: 1.Thetransmission hastooperatewithintheallocated FMbroadcast channels. 2.Ithastobecompatible withmonophonic radioreceivers. Thefirstrequirement setsthepermissible frequency parameters, including frequency de viation.Thesecondrequirement constrains thewayinwhichthetransmitted signalis configured . .Figure2.31ashowstheblockdiagram ofthemultiplexing systemusedinanFM stereotransmitter. Letm/(t)andmr(t)denotethesignalspickedupbyleft-hand and 2.7FrequencyModulatwn 125 Matrixer m,(rlo-_......;~ K (a) Matrixer m(t)o--...t-..,.,+ 2m,(t) (b) FIGURE2.31(a)Multiplexer intransnritter ofFMstereo.(b)Demultiplexer inreceiverofFM stereo. right-hand microphones atthetransmitting endofthesystem.Theyareappliedtoa simplematrixer thatgenerates thesumsignal,mitt)+mr(t),andthedifference signal, mitt)-m,(t).Thesumsignalisleftunprocessed initsbaseband form;itisavailable for monophonic reception. Thedifference signalanda38-kHzsubcarrier (derivedfroma19 kHzcrystaloscillator byfrequency doubling) areappliedtoaproductmodulator, thereby producing aDSB-SCmodulated wave.Inaddition tothesumsignalandthisDSB-SC modulated wave,themultiplexed signalm(t)alsoincludes a19-kHzpilottoprovidea reference forthecoherent detection ofthedifference signalatthestereoreceiver.Thusthe multiplexed signalisdescribed by m(t)=[mitt)+mr(t)]+[mi-mr(t)]COS(41Tfct)+KCOS(27Tlct) (2.72) whereIe=19kHz,andKistheamplitude ofthepilottone.Themultiplexed signalm(t) thenfrequency-modulates themaincarriertoproducethetransmitted signal.Thepilotis allottedbetween 8and10percentofthepeakfrequency deviation; theamplitude Kin Equation (2.72)ischosentosatisfythisrequirement. Atastereoreceiver, themultiplexed signalm(t)isrecovered byfrequency demodu latingtheincoming FMwave.Thenm(t)isappliedtothedemultiplexing systemshown 126 CHAPTER 2"CONTINUOUS-WAVE MODULATION inFigure2.31h.Theindividual components ofthemultiplexed signalm(t)areseparated bytheuseofthreeappropriate filters.Therecovered pilot(usinganarrowband filtertuned to19kHz)isfrequency doubledtoproducethedesired38-kHzsubcarrier. Theavailability ofthissubcarrier enablesthecoherent detection oftheDSB-SCmodulated wave,thereby recovering thedifference signal,ml(t)-mr(t).Thebaseband low-pass filterinthetop pathofFigure2.31bisdesigned topassthesumsignal,ml(t)+mr(t).Finally,thesimple matrixer reconstructs theleft-hand signalml(t)andright-hand signalm,(t),exceptfor scalingfactors,andappliesthemtotheirrespective speakers. I2.8Nonlinear EffectsinPMSystems Inthepreceding twosections, westudiedfrequency modulation theoryandmethods for itsgeneration anddemodulation. Wecomplete thediscussion offrequency'modulation by considering nonlinear effectsinFMsystems. Nonlinearities, inoneformoranother, arepresentinallelectrical networks. There aretwobasicformsofnonlinearity toconsider: 1.Thenonlinearity issaidtobestrongwhenitisintroduced intentionally andina controlled marmerfor'somespecificapplication. Examples ofstrongnonlinearity includesquare-law modulators, hard-limiters, andfrequency multipliers. 2.Thenonlinearity issaidtobeweakwhenalinearperformance isdesired,butnon linearities ofaparasitic natureariseduetoimperfections. Theeffectofsuchweak nonlinearities istolimittheusefulsignallevelsinasystemandtherebybecome an important designconsideration. Inthissectionweexamine theeffectsofweaknonlinearities onfrequency modulation. Consider acommunications channel, thetransfercharacteristic ofwhichisdefined bythenonlinear input-output relation (2.73) whereVi(t)andvolt)aretheinputandoutputsignals,respectively, andaI,al,anda3are constants; Equation (2.73)isatruncated versionofEquation (2.56)usedinthecontext offrequency multiplication. Thechanneldescribed inEquation (2.73)issaidtobeme moryless inthattheoutputsignalvolt)isaninstantaneous function oftheinputsignal v;(t)(i.e.,thereisnoenergystorageinvolved inthedescription). Wewishtodetermine the effectoftransmitting afrequency-modulated wavethroughsuchachannel.TheFMsignal isdefinedby Vi(t)=Aecos[2'lTfct+«;b(t)] where Forthisinputsignal,theuseofEquation (2.73)yields vo(t)=alAecos[2'lTfet+«;b(t)]+alA~cos1[2'lTfct+«;b,(t)] +a3A:cos3[2'lTfct+«;b(t)](2.74) (2.75)2.8Nonlinear EffectsinFMSystems 127 Expanding thesquared andcubedcosinetermsinEquation (2.74)andthencollecting common terms,weget volt)=~a2A~+(alA,+%a3A~)cos[27rI,t+<p(t)] I+2:a2A~cos[47rlet+2<p(t)] I+4a3A~cos[67rlet+3<p(t)] ThusthechanneloutputconsistsofaDCcomponent andthreefrequency-modulated signalswithcarrierfrequencies Ie'21"and31e;thelattercomponents arecontributed by thelinear,second-order, andthird-order termsofEquation (2.73),respectively. ToextractthedesiredFMsignalfromthechanneloutputvolt),thatis,theparticular component withcarrierfrequency,I"itisnecessary toseparate theFMsignalwiththis carrierfrequency fromtheonewiththeclosestcarrierfrequency, 21e.Let!:;.Idenotethe frequency deviation oftheincoming FMsignalVi(t),andWdenotethehighestfrequency component ofthemessage signalmit).Then,applying Carson's ruleandnotingthatthe frequency deviation aboutthesecondharmonic ofthecarrierfrequency isdoubled, we findthatthenecessary condition forseparating thedesiredFMsignalwiththecarrier frequency Iefromthatwiththecarrierfrequency 21eis 21e-(2!:;.1+W)>Ie+!:;.I+W or Ie>3!:;.1+2W (2.76) (2.77)Thus,byusingaband-pass filterofmidband frequency Ieandbandwidth 2!:;.1+2W,the channeloutputisreducedto v~(t)=(alAe+%a3A~)cos[27rlet+<p(t)] Weseetherefore thattheonlyeffectofpassinganFMsignalthrough achannelwith amplitude nonlinearities, followed byappropriate filtering, issimplytomodifyitsampli tude.Thatis,unlikeamplitude modulation, frequency modulation isnotaffectedbydis tortionproduced bytransmission through achannelwithamplitude nonlinearities. Itis forthisreasonthatwefindfrequency modulation usedinmicrowave radiosystems: It permitstheuseofhighlynonlinear amplifiers andpowertransmitters, whichareparticu larlyimportant toproducing amaximum poweroutputatradiofrequencies. AnFMsystemisextremely sensitive tophase nonlinearities, however, aswewould intuitively expect.Acommon typeofphasenonlinearity thatisencountered inmicrowave radiosystemsisknownasAM-to-PM conversion. Thisistheresultofthephasecharac teristicofrepeaters oramplifiers usedinthesystembeingdependent ontheinstantaneous amplitude oftheinputsignal.Inpractice, AM-to-PM conversion ischaracterized bya constantK,whichismeasured indegreesperdBandmaybeinterpreted asthepeakphase changeattheoutputforaI-dBchangeinenvelope attheinput.WhenanFMwaveis transmitted throughamicrowave radiolink,itpicksupspurious amplitude variations due tonoiseandinterference duringthecourseoftransmission, andwhensuchanFMwave ispassedthrougharepeaterwithAM-to-PM conversion, theoutputwillcontainunwanted 128 CHAPTER:2" CONTINUOUS-WAVE MODUlATION phasemodulation andresultant distortion. Itistherefore important tokeeptheAM-to_ PMconversion atalowlevel.Forexample, foragoodmicrowave repeater, theAM-to_ PMconversion constant Kislessthan2degreesperdB. I2.9Superheterodyne Receiver? Inabroadcasting system,irrespective ofwhetheritisbasedonamplitude modulation or frequency modulation, thereceivernotonlyhasthetaskofdemodulating theincoming modulated signal,butitisalsorequiredtoperformsomeothersystemfunctions: l-Carrier-frequency tuning,thepurpose ofwhichistoselectthedesiredsignal(i.e., desiredradioorTVstation). ""Filtering, whichisrequired toseparate thedesiredsignalfromothermodulated sig nalsthatmaybepickedupalongthe.way. Ii>Amplification, whichisintended tocompensate forthelossofsignalpowerincurred inthecourseoftransmission. Thesuperheterodyne receiver, orsuperhet asitisofrenreferredto,isaspecialtypeof receiverthatfulfillsallthreefunctions, particularly thefirsttwo,inanelegantandpractical fashion.Specifically, itovercomes thedifficulty ofhavingtobuildatunablehighlyselective andvariablefilter.Indeed,practically allradioandTVreceivers nowbeingmadeareof thesuperheterodyne type. Basically, thereceiverconsistsofaradio-frequency (RF)section,amixerandlocal oscillator, anintennediate-frequency (IF)section,demodulator, andpoweramplifier. Typ icalfrequency parameters ofcommercial AMandFMradioreceivers arelistedinTable 2.3.Figure2.32showstheblockdiagram ofasuperheterodyne receiverforamplitude modulation usinganenvelope detectorfordemodulation. Theincoming amplitude-modulated waveispickedupbythereceiving antelU1aand amplified intheRFsectionthatistunedtothecarrierfrequency oftheincoming wave. Thecombination ofmixerandlocaloscillator (ofadjustable frequency) provides ahet erodyning function, whereby theincoming signalisconverted toapredetermined fixed intermediate frequency, usuallylowerthantheincoming carrierfrequency. Thisfrequency translation isachieved withoutdisturbing therelationofthesidebands tothecarrier;see Section2.4.Theresultoftheheterodyning istoproduceanintermediate-frequency carrier definedby (2.78) where fLOisthefrequency ofthelocaloscillator andfRFisthecarrierfrequency ofthe incoming RFsignal.Wereferto!IFastheintermediate frequency (IF),becausethesignal TABLE2.3Typicalfrequency parameters ofAMandFM radioreceivers RFcarrierrange Midband frequency ofIFsection IFbandwidthAMRadio 0.535-1.605 MHz 0.455MHz 10kHzFMRadio 88-108MHz 10.7MHz 200kHz 2.9Superheterodyne Receiver 129 FIGURE2.32Basicelements ofanAMradioreceiverofthesuperheterodyne type. isneitherattheoriginalinputfrequency nOratthefinalbaseband frequency. Themixer localoscillator combination issometimes referredtoasthefirstdetector, inwhichcase thedemodulator iscalledtheseconddetector. TheIFsectionconsistsofoneormOrestagesoftunedamplification, withaband widthcorresponding tothatrequiredfortheparticular typeofmodulation thatthereceiver isintended tohandle.TheIFsectionprovides mostoftheamplification andselectivity in thereceiver.TheoutputoftheIFsectionisappliedtoademodulator, thepurposeofwhich istorecoverthebaseband signal.Ifcoherent detection isused,thenacoherent signal sourcemustbeprovided inthereceiver. Thefinaloperation inthereceiveristhepower amplification oftherecovered messagesignal. Inasuperheterodyne receiverthemixerwilldevelopanintermediate frequency out putwhentheinputsignalfrequency isgreaterorlessthanthelocaloscillator frequency byanamountequaltotheintermediate frequency. Thatis,therearetwoinputfrequencies, namely,IlLO±IIFI,whichwillresultinIIFatthemixeroutput.Thisintroduces the possibility ofsimultaneous reception oftwosignalsdiffering infrequency bytwicethe intermediate frequency. Forexample, areceivertunedto0.65MHzandhavinganIFof 0.455MHzissubjecttoanimageinterference at1.56MHz;indeed,anyreceiverwiththis valueofIF,whentunedtoanystation,issubjecttoimageinterference atafrequency of 0.910MHzhigherthanthedesiredstation.Sincethefunction ofthemixeristoproduce thedifference between twoappliedfrequencies, itisincapable ofdistinguishing between thedesiredsignalanditsimageinthatitproduces anIFoutputfromeitheroneofthem. Theonlypractical cureforimageinterference istoemployhighlyselective stagesinthe RFsection(i.e.,betweentheantennaandthemixer)inordertofavorthedesiredsignal anddiscriminate againsttheundesired orimagesignal.Theeffectiveness ofsuppressing unwanted imagesignalsincreases asthenumberofselective stagesintheRFsectionin creases,andastheratioofintermediate tosignalfrequency increases. Thebasicdifference betweenA.t\1andFMsuperheterodyne receivers liesintheuse ofanFMdemodulator suchaslimiter-frequency discriminator. InanFMsystem,the message information istransmitted byvariations oftheinstantaneous frequency ofa sinusoidal carrierwave,anditsamplitude ismaintained constant. Therefore, anyvaria tionsofthecarrieramplitude atthereceiverinputmustresultfromnoiseorinterference. Anamplitude limiter,following theIFsection,isusedtoremoveamplitude variations byclipping themodulated waveattheIFsectionoutputalmosttothezeroaxis.The resulting rectangular waveisrounded offbyaband-pass filterthatsuppresses har monicsofthecarrierfrequency. Thusthefilteroutputisagainsinusoidal, withanampli tudethatispractically independent ofthecarrieramplitude atthereceiverinput(see Problem 2.42). 130 CHAPTER2 OJCONTINUOUS-WAVE MODULATION I2.10NoiseinCWModulation Systems Uptothispointinourdiscussion wehavefocusedattention onthecharacterization of continuous-wave (CW)modulation techniques, entirelyfromadeterministic perspective. Intheremainder ofthechapter,westudytheeffectsofchannelnoiseonthereception of CWmodulated signalsandtherebydevelopadeeperunderstanding ofthebehavior of analogcommunications. Toundertake suchastudywefollowthecustomary practicebyformulating two models: 1.Channel model,whichassumesacommunication channelthatisdistortionless but perturbed byadditivewhiteGaussian noise(AWGN). 2.Receiver model,whichassumesareceiverconsisting ofanidealband-pass filterfol lowedbyanidealdemodulator appropriate fortheapplication athand;theband passfilterisusedtominimize theeffectofchannelnoise. Thesesimplifying assumptions aremadeinordertoobtainabasicunderstanding ofthe wayinwhichnoiseaffectstheperformance ofthereceiver. Moreover, theyprovidea framework forthecomparison ofdifferent CWmodulation-demodulation schemes. Figure2.33showsthenoisyreceivermodelthatcombines theabovetwoassump tions.Inthisfigure,s(t)denotestheincoming modulated signalandw(t)denotesthe channelnoise.Thereceivedsignalistherefore madeupofthesumofs(t)andw(t);thisis thesignalthatthereceiverhastoworkon.Theband-pass filterinthemodelofFigure 2.33represents thecombined filteringactionofthetunedamplifiers usedintheactual receiverforthepurposeofsignalamplification priortodemodulation. Thebandwidth of thisband-pass filterisjustwideenoughtopassthemodulated signals(t)withoutdistor tion.Asforthedemodulator inthemodelofFigure2.33,itsdetailsnaturally dependon thetypeofmodulation used. III!SIGNAL-TO-NOISE RAnos: BASICDEFINITIONS Letthepowerspectraldensityofthenoisewit)bedenotedbyNo/2,definedforboth positiveandnegativefrequencies; thatis,Noistheaveragenoisepowerperunitbandwidth measured atthefrontendofthereceiver. Wealsoassumethattheband-pass filterinthe receivermodelofFigure2.33isideal,havingabandwidth equaltothetransmission band widthByofthemodulated signals(t)andamidband frequency equaltothecarrierfre quencyfe.Thelatterassumption isjustifiedfordoublesideband-suppressed carrier(DSB SC)modulation, fullamplitude modulation (AM),andfrequency modulation (FM);the casesofsinglesideband (SSB)modulation andvestigialsideband (VSB)modulation require specialconsiderations. Takingthemidband frequency oftheband-pass filtertobethe sameasthecarrierfrequency fe'wemaymodelthepowerspectraldensitySNit)ofthe noisenit),resulting fromthepassageofthewhitenoisew(t)throughthefilter,asshown + Modulated signal----i;.{ s(t) Noise wet) FIGURE2.33Receiver model.Output signal 2.10NoiseinCWModulation Syst....... 131 inFigure2.34.Typically, thecarrierfrequency fcislargecompared tothetransmission bandwidth BT•Wemaytherefore treatthefilterednoisenit)asanarrowband noiserep resented inthecanonical form (2.79) wherenr(t)isthein-phase noisecomponent andnQ(t)isthequadrature noisecomponent, bothmeasured withrespecttothecarrierwaveAcCOS(27Tfct). Thefilteredsignalx(t)avail ablefordemodulation isdefinedby x(t)=sit)+n(t) (2.80) Thedetailsofsit)dependonthetypeofmodulation used.Inanyevent,theaveragenoise poweratthedemodulator inputisequaltothetotalareaunderthecurveofthepower spectraldensitySN(f).FromFigure2.34wereadilyseethatthisaveragenoisepoweris equaltoNoBT.Giventheformatofs(t),wemayalsodetermine theaveragesignalpower atthedemodulator input.Withthedemodulated signals(t) andthefilterednoise(n(t) appearing additively atthedemodulator inputinaccordance withEquation (2.80),we maygoontodefineaninputsignal-to-noise ratio,(SNR)1>astheratiooftheaverage powerofthemodulated signals(t)totheaveragepowerofthefilterednoisen(t). Amoreusefulmeasure ofnoiseperformance, however, istheoutputsignal-to-noise ratio,(SNR)o, definedastheratiooftheaveragepowerofthedemodulated messagesignal totheaveragepowerofthenoise,bothmeasured atthereceiveroutput.Theoutputsignal to-noise ratioprovides anintuitive measure fordescribing thefidelitywithwhichthede modulation processinthereceiverrecovers themessagesignalfromthemodulated signal inthepresence ofadditive noise.Forsuchacriterion tobewelldefined,therecovered message signalandthecorruptive noisecomponent mustappearadditively atthe demod ulatoroutput.Thiscondition isperfectly validinthecaseofareceiverusingcoherent detection. Ontheotherhand,whenthereceiverusesenvelope detection asinfullAMor frequency discrimination asinFM,wehavetoassumethattheaveragepowerofthefiltered noisen(t)isrelatively lowtojustifytheuseofoutputsignal-to-noise ratioasameasure of receiverperformance. Theoutputsignal-to-noise ratiodepends, amongotherfactors,onthetypeofmod ulationusedinthetransmitter andthetypeofdemodulation usedinthereceiver. Thusit isinformative tocompare theoutputsignal-to-noise ratiosfordifferent modulation demodulation systems. However, forthiscomparison tobeofmeaningful value,itmust bemadeonanequalbasisasdescribed here: I>Themodulated signalsit)transmitted byeachsystemhasthesameaveragepower. I>Thechannelnoisewit)hasthesameaveragepowermeasured inthemessage band widthw. No2rBT-'"j---- -----~--------- ---'--.,1---1'-----'-----'-----"--'---1-Ie 0 Ie FIGURE2.34Idealized characteristic ofband-pass filterednoise. 132 CHAPTER 2IIICONTINUOUS-WAVE MODUlATION Output (2.81)FIGURE2.35Thebaseband transmission model,assuming amessagesignalofbandwidth \V, usedforcalculating thechannelsignal-to-noise ratio. Accordingly, asaframeofreference wedefinethechannelsignal-to-noise ratio,(SNRlc, astheratiooftheaveragepowerofthemodulated signaltotheaveragepowerofchannel noiseinthemessage bandWidth, bothmeasured atthereceiverinput.Thisdefinition is illustrated inFigure2.35. Forthepurposeofcomparing different continuous-wave (CW)modulation systems, wenormalize thereceiverperformance bydividingtheoutputsignal-to-noise ratiobythe channelsignal-to-noise ratio.Wethusdefineafigureofmeritforthereceiverasfollows: F· f . (SNR)o Igureament=(SNRlc Clearly,thehigherthevalueofthefigureofmerit,thebetterwillthenoiseperformance ofthereceiverbe.Thefigure.ofmeritmayequalone,belessthanone,orbegreaterthan one,depending onthetypeofmodulation used,whichwillbecomeapparent fromthe discussion thatfollows. 2.11NoiseinLinearReceivers UsingCoherent Detection FromSections 2.2and2.3werecallthatthedemodulation ofanamplitude-modulated wavedependsonwhetherthecarrierissuppressed ornot.Whenthecarrierissuppressed weusuallyrequiretheuseofcoherent detection, inwhichcasethereceiverislinear.On theotherhand,whentheamplitude modulation includes transmission ofthecarrier,de modulation isaccomplished simplybyusinganenvelope detector, inwhichcasethere ceiverisnonlinear. Inthissectionwestudytheeffectofnoiseontheperformance ofa linearreceiver. Themoredifficultcaseofanonlinear receiverisdeferred toSection2.12. Consider thecaseofDSB-SCmodulation Figure2.36showsthemodelofaDSB-SC receiverusingacoherent detector. Theuseofcoherent detection requiresmultiplication ofthefilteredsignalx(t)byalocallygenerated sinusoidal wavecos(2'Trf"t) andthenlow passfilteringtheproduct. Tosimplify theanalysis, weassumethattheamplitude ofthe locallygenerated sinusoidal waveisunity.Forthisdemodulation schemetooperatesat isfactorily, however, itisnecessary thatthelocaloscillator besynchronized bothinphase andinfrequency withtheoscillator generating thecarrierwaveinthetransmitter. We assumethatthissynchronization hasbeenachieved. TheDSB-SCcomponent ofthefilteredsignalx(t)isexpressed as (2.82) whereAccos(2'Trfct) isthesinusoidal carrierwaveandm(t)isthemessage signal.Inthe expression fors(t)inEquation (2.82)wehaveincluded asystem-dependent scalingfactor C,thepurposeofwhichistoensurethatthesignalcomponent s(t)ismeasured inthesarne unitsastheadditivenoisecomponent n(t).Weassumethatm(t)isthesamplefunction of 2.11NoiseinLinearReceivers UsingCoherent Detection 133 DSB-SC signal,{t)y(t) FIGliRE2.36ModelofDSB-SC receiverusingcoherent detection. (2.83) (2.85)astationary processofzeromean,whosepowerspectraldensitySM(f)islimitedtoa maximum frequency W;thatis,Wisthemessagebandwidth. TheaveragepowerPofthe messagesignalisthetotalareaunderthecurveofpowerspectraldensity,asshownby P=fwSM(f)d[ Thecarrierwaveisstatistically independent ofthemessage signal.Toemphasize this independence, thecarriershouldincludearandomphasethatisuniformly distributed over 27Tradians.Inthedefiningequation fors(t)thisrandomphaseanglehasbeenomittedfor convenience ofpresentation. UsingtheresultofExample 1.7ofChapter1onamodulated random process, wemayexpresstheaveragepoweroftheDSB-SC modulated signal component s(t)asC2A~PI2.WithanoisespectraldensityofNo12,theaveragenoisepower inthemessage bandwidth WisequaltoWNo.Thechannelsignal-to-noise ratioofthe DSB·SCmodulation systemistherefore C2A2P (SNRlc,DsB =2~o (2.84) wheretheconstant C2inthenumerator ensuresthatthisratioisdimensionless. Next,wewishtodetermine theoutputsignal-to-noise ratioofthesystem.Usingthe narrowband representation ofthefilterednoisen(t),thetotalsignalatthecoherent detec torinputmaybeexpressed as x(t)=s(t)+n(t) =CAeCOs(27Tfct)m(t) +nI(t)COS(27T[,t) ndt)sin(27Tfct) wheren,(t)andnQ(t)arethein-phase andquadrature components ofn(t)withrespectto thecarrier.Theoutputoftheproduct-modulator component ofthecoherent detectoris therefore v(t)=x(t)COS(27Tfct) =!CA.m(t)+!nI(t) +HCAcm(t)+n,(t)]COS(47T[ct) -!ndt)sin(47Tht) Thelow-pass filterinthecoherent detector inFigure2.36removes thehigh-frequency components ofv(t),yieldingthereceiveroutput y(t)=!CA,m(t)+!nI(t) (2.86) Equation (2.86)indicates thefollowing: 1.Themessagesignalm(t)andin-phase noisecomponent ndt)ofthefilterednoisen(t) appearadditively atthereceiveroutput. (2.87) (2.88)134 CHAPTER 2.,CONTINUOUS-WAVE MODUlATION 2.Thequadrature component nQ(t)ofthenoisen(t)iscompletely rejectedbytheco herentdetector. Thesetworesultsareindependent oftheinputsignal-to-noise ratio.Thus,coherent detec tiondistinguishes itselffromotherdemodulation techniques inanimportant property: The outputmessage component isunmutilated andthenoisecomponent alwaysappearsad ditivelywiththemessage, irrespective oftheinputsignal-to-noise ratio. Themessage signalcomponent atthereceiveroutputisCAcm(t)/2.Therefore, the averagepowerofthiscomponent maybeexpressed asC2A;P/4,wherePistheaverage poweroftheoriginalmessage signalm(t)andCisthesystem-dependent scalingfactor referredtoearlier. InthecaseofDSB-SC modulation, theband-pass filterinFigure2.36hasaband widthByequalto2Winordertoaccommodate theupperandlowersidebands ofthe modulated signals(t).Itfollowstherefore thattheaveragepowerofthefilterednoisen(t) is2WNo.Fromthediscussion ofnarrowband noisepresented inSection1.11,weknow thattheaveragepowerofthe(low-pass) in-phase noisecomponent nj(t)isthesameas thatofthe(band-pass) filterednoisen(t).SincefromEquation (2.86).thenoisecomponent atthereceiveroutputisnj(t)/2,itfollowsthattheaveragepowerofthenoiseatthereceiver outputis (~)22WNo=~WNo Theoutputsignal-to-noise foraDSB-SCreceiverusingcoherent detection istherefore C2A2P/4 (SNR)o, DSJl.-SC=~o/2 C2A;P 2WNo UsingEquations (2.84)and(2.87),weobtainthefigureofmerit (SNRlol _1 (SNRlc DSB-SC NotethatthefactorC2iscommon toboththeoutputandchannelsignal-to-noise ratios, andtherefore cancelsoutinevaluating thefigureofmerit. Following throughthenoiseanalysisofacoherent detector forSSB,wefindthat, despitethefundamental differences between itandthecoherent detectorforDSB-SCmod ulation,thefigureofmeritisexactlythesameforbothofthem;seeProblem 2.49. Theimportant conclusions to.bedrawnfromthediscussions presented inthissection andProblem 2.49aretwo-fold: 1.Forthesameaveragetransmitted ormodulated signalpowerandthesameaverage noisepowerinthemessage bandwidth, acoherent SSBreceiverwillhaveexactlythe sameoutputsignal-to-noise ratioasacoherent DSB-SCreceiver. 2.Inbothcases,thenoiseperformance ofthereceiverisexactlythesameasthatob tainedbysimplytransmitting themessagesignalinthe presence ofthesamechannel noise.Theonlyeffectofthemodulation processistotranslate themessage signalto adifferent frequency bandtofacilitate itstransmission overaband-pass channel. Simplyput,neitherDSB-SCmodulation norSSBmodulation offersthemeansforatrade offbetween improved noiseperformance andincreased channelbandwidth. Thisisase riousproblem whenhighqualityofreception isarequirement. 2.12NoiseinAMReceivers UsingEnveu.pe DefectUm 135 ~ NoiseinAMRecei~ers Using En~elope Detection Thenextnoiseanalysisweperform isforanamplitude modulation (AM)systemusingan envelope detectorinthereceiver, asshowninthemodelofFigure2.37.InafullAMsignal, bothsidebands andthecarrierwavearetransmitted, asshownby (2.89) (2.90) (2.91) (2.92)whereA,cos(2'T1'f,t) isthecarrierwave,m(t)isthemessage signal,andkaisaconstant thatdetermines thepercentage modulation. Intheexpression fortheamplitude-modulated signalcomponent s(t)giveninEquation (2.89),weseenoneedfortheuseofascaling factor,becauseitisreasonable toassumethatthecarrieramplitude Achasthesameunits astheadditivenoisecomponent. Theaveragepowerofthecarriercomponent intheAMsignals(t)isA~/2.The averagepoweroftheinformation-bearing component Ack"m(t) cos(2'T1'fct) isA~k;P/2, wherePistheaveragepowerofthemessagesignalm(t).Theaveragepowerofthefull AMsignals(t)istherefore equaltoA~(1+k;P)/2.AsfortheDSB-SCsystem,theaverage powerofnoiseinthemessage bandwidth isWNo•Thechannelsignal-to-noise ratiofor AMistherefore (SNR)=A~(1+k;P) C,AM 2WNo Toevaluate theoutputsignal-to-noise ratio,wefirstrepresent thefilterednoisen(t) intermsofitsin-phase andquadrature components. Wemaytherefore definethefiltered signalx(t)appliedtotheenvelope detectorinthereceivermodelofFigure2.37asfollows: x(t)=s(t)+n(t) =[A,+A)am(t)+nl(t)]cos(2'T1'f,t) -nQ(t)sin(2'T1'f,t) Itisinformative torepresent thecomponents thatcomprise thesignalx(t)bymeansof phasors, asinFigure2.38a.Fromthisphasordiagram, thereceiveroutputisreadilyob tainedas y(t)=envelope ofx(t) =(lA,+A,kam(t)+nl(tW+n~(t)}ll2 Thesignaly(t)definestheoutputofanidealenvelope detector. Thephaseofx(t)isofno interesttous,becauseanidealenvelope detectoristotallyinsensitive tovariations inthe phaseofx(t). Theexpression definingy(t)issomewhat complex andneedstobesimplified insome mannertopermitthederivation ofinsightful results.Specifically, wewouldliketoap proximate theoutputy(t)asthesumofamessagetermplusatermduetonoise.Ingeneral, thisisquitedifficulttoachieve.However, whentheaveragecarrierpowerislargecom- AMsignal sit) FIGURE2.37ModelofAMreceiver. r(t)136 CHAPTER 2..CONTINUOUS-WAVE MODUlATION --I nQ<t) : I A,[l+kam(t)] n,(t) (a) ___======~Re~S:UI:ta:nt=y=(t:) ====::~~~::r::-A,[l+kam(t)] ;>-)-- 'If(t) (b) FIGURE 2.38(a)Phasordiagram forAMwaveplusnarrowband noiseforthecaseofhighcar rier-to-noise ratio.(b)Phasordiagram forAMwaveplusnarrowband noiseforthecaseoflow carrier~to-noise ratio. paredwiththeaveragenoisepower,sothatthereceiverisoperating satisfactorily, then thesignaltermAc[l+kam(t)]willbelargecompared withthenoisetermsnJ(t)andnQit), atleastmostofthetime.Thenwemayapproximate theoutputy(t)as(seeProblem 2.51): (2.93) (2.94) (2.95)Thepresence oftheDCorconstant termA,intheenvelope detectoroutputy(t)of Equation (2.93)isduetodemodulation ofthetransmitted carrierwave.Wemayignore thisterm,however, becauseitbearsnorelationwhatsoever tothemessagesignalm(t).In anycase,itmayberemoved simplybymeansofablocking capacitor. Thusifweneglect theDCtermAcinEquation (2.93),wefindthattheremainder has,exceptforscaling factors,aformsimilartotheoutputofaDSB-SCreceiverusingcoherent detection. Ac cordingly, theoutputsignal-to-noise ratioofanAMreceiverusinganenvelope detector isapproximately A2k2p(SNR) '"_,_a_ O,AM2WNo Equation (2.94)is,however, validonlyifthefollowing twoconditions aresatisfied: 1.Theaveragenoisepowerissmallcompared totheaveragecarrierpoweratthe envelope detectorinput. 2.Theamplitude sensitivity kaisadjusted forapercentage modulation lessthanor equalto100percent. UsingEquations (2.90)and(2.94),weobtainthefollowing figureofmeritforamplitude modulation: (SNR)O/ k;P (SNR)c AM'"1+k;P Thus,whereas thefigureofmeritofaDSB-SCreceiverorthatofanSSBreceiverusing coherent detection isalwaysunity,thecorresponding figureofmeritofanAMreceiver usingenvelope detection isalwayslessthanunity.Inotherwords,thenoiseperformance ofafullAMreceiver isalwaysinferiortothatofaDSB-SC receiver. Thisisduetothe 2.12NoiseinA,MReceivers UsingEnvelope Detection 137 wastageoftransmitter power,whichresultsfromtransmitting thecarrierasacomponent oftheAMwave. IVEXAMPLE 2.4Single-Tone Modulation Consider rhespecialcaseofasinusoidal waveoffrequency fmandamplitude Amasthe modulating wave,asshownby m(t)=Amcos(2'Trfmt) Thecorresponding AMwaveis s(t)=A,[1+JLcos(2'Trfmt)] cos(2'Trfct) whereJL=k.,Amisthemodulation factor.Theaveragepowerofthemodulating wavemit)is (assuming aloadresistorof1ohm) P=!A;;, Therefore, usingEquation (2.95),weget (2.96)(SNR)ol (SNR)c AM.!FA22am 1+1FA'2am JLl =2+JL2 WhenJL=1,whichcorresponds to100percemmodulation, wegetafigureofmeritequal to1/3.Thismeansthat,otherfactorsbeingequal,anAMsystem(usingenvelope detection) musttransmit threetimesasmuchaveragepowerasasuppressed-carrier system(usingco herentdetection) toachievethesamequalityofnoiseperformance. 4ll IITHRESHOLD EFFECT Whenthecarrier-to-noise ratioissmallcompared withunity,thenoisetermdominates andtheperformance oftheenvelope detectorchangescompletely fromthatjustdescribed. Inthiscaseitismoreconvenient torepresent thenarrowband noisen(t)intermsofits envelope r(t)andphaseif!(t),asshownby n(t)=r(t)cos[2'lT.fct+if!(t)] (2.97) Thecorresponding phasordiagram forthedetectorinputx(t)=sit)+n(t)isshownin Figure2.38b,wherewehaveusedthenoiseenvelope asreference, becauseitisnowthe dominant term.Tothenoisephasorr(t)wehaveaddedaphasorrepresenting thesignal termAc[1+kam(t)],withtheanglebetween thembeingequaltothephaseif!(t)ofthe noisen(t).InFigure2.38bitisassumed thatthecarrier-to-noise ratioissolowthatthe carrieramplitude Acissmallcompared withthenoiseenvelope r(t),atleastmostofthe time.Thenwemayneglectthequadrature component ofthesignalwithrespecttothe noise,andthusfindfromFigure2.38bthattheenvelope detectoroutputis y(t)=r(t)+Accos[if!(t)]+AJ<am(t) cos[if!(t)] (2.98) Thisrelationrevealsthatwhenthecarrier-to-noise ratioislow,thedetector outputhas nocomponent strictlyproportional tothemessage signalm(t).Thelasttermoftheex pression definingy(t)contains themessagesignalm(t)multiplied bynoiseintheformof 138 CHAPTER 2"CONTINUOUS-WAVE MODULATION cos[lfJ(t)]. FromSection1.11werecallthatthephaselfJ(t)ofthenarrowband noisen(t)is uniformly distributed over271'radians.Itfollowstherefore thatwehaveacomplete loss ofinformation inthatthedetectoroutputdoesnotcontainthemessage signalm(t)atall. Thelossofamessageinanenvelope detectorthatoperates atalowcarrier-to-noise ratio isreferredtoasthethreshold effect.Bythreshold wemeanavalueofthecarrier-to-noise ratiobelowwhichthenoiseperformance ofadetector deteriorates muchmorerapidly thanproportionately tothecarrier-to-noise ratio.Itisimportant torecognize thatevery nonlinear detector(e.g.,envelope detector) exhibitsathreshold effe(.;:.Ontheotherhand, suchaneffectdoesnotariseinacoherent detector. Arigorous mathematical analysisofthethreshold effectforthegeneralcaseofan AMwaveisbeyondthescopeofthisbook.Inthenextsubsection wesimplifymattersby considering thecaseofanunmodulated carrier.Despitethissimplification, wecanstill developagreatdealofinsightintothethreshold effectexperienced inanenvelope detector. General Formulafor (SNR)oinEnvelope Detection" Consider anenvelope detector whoseinputsignalisdefinedby x(t)=Accos(271'[j)+n(t) (2.99) (2.100) otherwiseforI [ -[eI:sWwhereAccos(271'[,t) istheunmodulated carrierandn(t)isthesamplefunction ofband limited,zero-mean, whiteGaussian noiseN(t).ThepowerspectraldensityofN(t)is {No SN(t)=02 Representing thenarrowband noisen(t)intermsofitsin-phase component nI(t)and quadrature component nQ(t),wemayexpressthenoisysignalatthedetectorinputas (2.101) Thenoisecomponents nI(t)andnQ(t)arezero-mean, jointlyGaussian, mutually indepen· dentlow-pass random processes withidentical powerspectral densities (seeEquation 1.101): forI[I:sW otherwise(2.102) Fortheproblem athand,theinputsignalconsistsofanunmodulated carrierwith averagepowerequaltoA~/2.Theaveragenoisepoweratthedetectorinputis (2.103) (2.104)~=2WNo Thecarrier-to-noiser ratioistherefore definedby A~/2P=-2erN A~ 4WNo Wemaythinkofpasaninputsignal-to-noise ratiofortheproblem described herein. '"Areaderwhoisnotinterested inthemathematical detailsofhownoiseaffectstheenvelope detection ofan AMsignalmayskipthematerial uptoEq.(2.124)andreadthetwulimitingcasesoftheformulainthatequation. 2.12NoiseinAMReceOve1"S UsingEnvelope Detection 139 However, determination oftheoutputsignal-to-noise ratioisamoredifficultun dertaking becausetheenvelope detectoroutput (2.105) isanonlinear combination ofsignalandnoiseterms.Withnoclear-cut separation between signalandnoiseatthedetectoroutputy(t),howthendoweisolatethecontribution of thesignals(t)toy(t)fromthecontribution duetothenoisen(t)?Toresolvethisissue,we adoptaheuristic approach basedonsignalaveraging: Specifically, weintroduce thefol lowingtwodefinitions: 1.Themeanoutputsignal, SO)isthedifference betweentheexpectation ofy(t)inthe combined presence ofsignalandnoiseandtheexpectation ofy(t)inthepresence of noisealone,asshownby So=E[y(t)]-E[Yo(t)] wherey(t)isitselfdefinedbyEquation (2.105)andyo(t)isdefinedby yo(t)=v'n1(t)+n~(t)(2.106) (2.107) 2.Themeanoutputnoisepoweristhedifference betweenthemean-square valueofthe detectoroutputy(t)andthesquareofthemeanvalueofy(t),asshownby var[y(t)] =E[y2(t)]-(E[y(t)])2 Onthisbasis,wedefinetheoutputsignal-to-noise ratioas 52 (SNR)o =varb(t)](2.108) (2.109) FromSection1.12,werecallthattheenvelope detectoroutputduetonoisealoneis Rayleigh distributed; thatis {-4-exp(-4),Idy)=~N 2UN Theexpectation ofyo(t)isthereforey2:0 otherwise(2.110) (2.111)E[Yo(t)] =r~yfyJy)dy f~y2(y2)=-exp---dyou~ 2uKr Fromthedefinition ofthegammafunction forrealpositivevaluesoftheargument x,we have f(x)=rz,,--lexp(-z)dz Wemaytherefore rewriteEquation (2.111)as E[Yo(t)]=Y2UNrG) =ftUN(2.112) (2.113) 140 CHAPTER:2" CONTINUOUS-WAVE MODUlATION (2.114)fory2':0 otherwisewherewehaveusedthevalue[(3/2)=V1T12.Tocalculate rhemeansignal Soatthe detector output,wealsoneedtheexpecration ofy(t).Duetothecombined presence of signalandnoise,werecallfromSection1.13thaty(t)isRiciandistributed, asshownby {y(y2+A~)(A~Y) ~exp ---~ I -fy{y)=~~ 2(T~ 0(T~ whereIo{')isthemodified Besselfunction ofthefirstkindofzeroorder(seeAppendix 3). Hence, {~y2(y2+A2)(Ay)E[y(t)]=Jo(T~exp-2(T~ CIo~dy (2.115) Putring Acy/~=uandrecognizing thatp=A~/2(T~, wemayrecastthisexpectation in theform (2.116) E[y(t)]=(2:r12exp(-p)ru2exp( -:;)Io{U) du TheintegralinEquation (2.116)canbewritteninaconciseformbyusingconfluent hypergeometric functions; seeAppendix 4.Inparticular, usingtheintegralrepresentation (2.118)rum -1exp{-bV)Io{u) du=r~:~~\lFl(~;1;4~2)) (2.117) withm=3,[(mI2) =V1T12andb2=1/4p,wemayexpresstheexpectation ofy{t)in termsoftheconfluent hypergeometric function IF1{3/2;1;p) as E[y{t)] =~(TNeXP(-p)("FIG;l;P)) Wemayfurthersimplifymattersbyusingthefollowing identity: exp{-u)(lFl{a;{:l;u)) =lFl({:l-a;{:l;-u) andsofinallyexpresstheexpectation ofy{t)intheconciseform E[y(t)] =~(TN(lFl(-i;l;-P))(2.119) (2.120) ThususingEquations (2.113)and(2.120)inEquation (2.106)yieldsthemeanoutput signalas (2.121) whosedependence onthestandard deviation (TNofthenoisenit)istestimony tothe intermingling ofsignalandnoiseatthedetectoroutput. Following asimilarprocedure, wemayexpressthemean-square valueofthedetector outputy(t)as 2 {~y3(y2+A~)(AcY)E[y(t)]=Jo(T~exp-2~Io(T~dy =2(T~hFl{-1;1;-p))(2.122) 2.12NaiseinAMReceivers UsingEnvelope Detedion 141 HenceusingEquations (2.120)and(2.122)inEquation (2.108)yieldsthemeanoutput noisepoweras var[y(t)] =2tri.z(lFl(-1;1;-P) -~(lFl(-~;l;-P)r) (2.123) Finally,usingEquations (2.121)and(2.123)inEquation (2.109)yieldstheoutput signal-to-noise ratiofortheenvelope detection problem athandas (2.124) Equation (2.124)isthegeneralformulafortheoutputsignal-to-noise ofanenvelope detector whoseinputconsistsofanunrnodulated carrierandband-limited, whiteGaussian noise.Twolimitingcasesofthisgeneralformula areofparticular interest: 1.Largecarrier-to-noise ratio.Forlargep,wemayusethefollowing asymptotic for mula(seeAppendix 4) IF,(-~;l;-P) =2j¥;. forp~00 (2.125) Moreover, thefollowing identity IF,(-l;l;-p) =1+p (2.126) holdsexactlyforallp.Accordingly, theuseofEquations (2.125)and(2.126)in Equation (2.124)yieldsthefollowing approximate formulafortheoutputsignal-to noiseratio: (SNR)o=pforp~00 (2.127) wherewehaveignoredcontributions duetop1l2andpOinthenumerator ofEquation (2.124)asbeingsubdominant compared topforlargep.Equation (2.127)shows thatforlargecarrier-to-noise ptheenvelope detectorbehaveslikeacoherent detector, inthattheoutputsignal-to-noise ratioisproportional totheinputsignal-to-noise ratio. 2.Smallcarrier-to-noise ratio.Forsmallp,wehave(seeAppendix 4) forp~0 (2.128) (2.129)Hence,usingthisasymptotic formula, wemayapproximate theoutputsignal-to noiseratioforsmallpas 'TTp2 (SNR)o=16-4'TT =0.91p2forp~0 where,inthedenominator, wehaveignoredcontributions duetopandp2asbeing subdominant compared topOforsmallp.Equation (2.129)showsthatforasmall carrier-to-noise ratio,theoutputsignal-to-noise ratiooftheenvelope detector ispro portional tothesquaredinputsignal-to-noise ratio. 142 CHAPTER 2'"CONTINUOUS-WAVE MODULATION 100.0 Carrier-to-noise ratio,p FIGURE2.39Outputsignal-to-noise ratioofanenvelope detector forvaryingcarrier-to-noise ratio. Theconclusions drawnfromthetwolimitingcasesconsidered hereinarethatan envelope detector favorsstrongsignalsandpenalizes weaksignals.Thephenomenon of weaksignalsbeingpenalized bythedetector isreferredtoasweaksignalsuppression, whichisamanifestation ofthethreshold effect. UsingtheformulaofEquation (2.124),inFigure2.39wehaveplottedtheoutput signal-to-noise ratio(SNR)ooftheenvelope detector versusthecarrier-to-noise ratiop usingtabulated valuesofconfluent hypergeometric functions. Thisfigurealsoincludesthe twoasymptotes forlargepandsmallp.FromFigure2.39weseethattheoutputsignal, to-noiseratiodeviatesfromalinearbehavior aroundacarrier-to-noise ratioof10dB(i.e., p=10). I2.13NoiseinFMReceivers Finally,weturnourattention tothenoiseanalysisofafrequency modulation (FM)system, forwhichweusethereceivermodelshowninFigure2.40.Asbefore,thenoisew(t)is modeled aswhiteGaussian noiseofzeromeanandpowerspectral densityNo/2.The FM signalsit) Noise wet) FIGURE2.40ModelofanFMreceiver.Oulput signal 2.13NoiseinFMReceivers 143 received FMsignals(t)hasacarrierfrequency Ieandtransmission bandwidth BT,such thatonlyanegligible amountofpowerliesoutsidethefrequency bandIe±Br/2for positivefrequencies, andsimilarly fornegative frequencies. AsintheAMcase,theband-pass filterhasamidband frequency fcandbandwidth BTandtherefore passestheFMsignalessentially without distortion. Ordinarily, BTis smallcompared withthemidband frequency fc,sothatwemayusethenarrowband representation forn(t),thefilteredversionofchannelnoisew(t),intermsofitsin-phase andquadrature components. InanFMsystem,themessagesignalistransmitted byvariations oftheinstantaneous frequency ofasinusoidal carrierwave,anditsamplitude ismaintained constant. Therefore, anyvariations ofthecarrieramplitude atthereceiverinputmustresultfromnoiseor interference. Theamplitude limiter,following theband-pass filterinthereceivermodelof Figure2.40,isusedtoremoveamplitude variations byclipping themodulated waveat thefilteroutputalmosttothezeroaxis.Theresulting rectangular waveisrounded offby anotherband-pass filterthatisanintegralpartofthelimiter,therebysuppressing har monicsofthecarrierfrequency. Thus,thefilteroutputisagainsinusoidal, withanam plitudethatispractically independent ofthecarrieramplitude atthereceiverinput. Thediscriminator inthemodelofFigure2.40consistsoftwocomponents: 1.Aslopenetwork ordifferentiator withapurelyimaginary frequency response that varieslinearlywithfrequency. Itproduces ahybrid-modulated waveinwhichboth amplitude andfrequency varyinaccordance withthemessage signal. 2.Anenvelope detectorthatrecovers theamplitude variation andthus reproduces the message signal. Theslopenetwork andenvelope detector areusuallyimplemented asintegralpartsofa singlephysical unit. Thepostdetection filter,labeled"baseband low-pass filter"inFigure2.40,hasa bandwidth thatisjustlargeenoughtoaccommodate thehighestfrequency component of themessage signal.Thisfilterremoves theout-of-band components ofthenoiseatthe discriminator outputandtherebykeepstheeffectoftheoutputnoisetoaminimum. Thefilterednoisen(t)attheband-pass filteroutputinFigure2.40isdefinedinterms ofitsin-phase andquadrature components by n(t)=nr(t)cos(271"fct) -ndt)sin(271fct) Equivalently, wemayexpressn(t)intermsofitsenvelope andphaseas n(t)=r(t)COS[(271fct)+l/F(t)] wheretheenvelope is r(t)=[nr(t)+nb(t)],12 andthephaseis(2.130) (2.131) (2.132) l/F(t)=tan-1[nQ(t)] nr(t) Theenvelope r(t)isRayleigh distributed, andthephasel/F(t)isuniformly distributed over 271radians(seeSection1.12). Theincoming FMsignals(t)isdefinedby (2.133) (2.134)144 CHAPTER 2..CONTINUOUS-WAVE MODULATION whereAeisthecarrieramplitude, Ieisthecarrierfrequency, kfisthefrequency sensitivity andm(t)isthemessagesignal.Notethat,aswiththestandard AM,inFMthereisn~ needtointroduce ascalingfactorinthedefinition ofthemodulated signals(t),sinceitis reasonable toassumethatitsamplitude Achasthesameunitsastheadditivenoise COill_ ponentn(t).Toproceed, wedefine </>(t)=2Tfkf!:m('T)d'T Wemaythusexpresss(t}inthesimpleform s(t)=Aecos[2TfIJ+</>(t)] Thenoisysignalattheband-pass filteroutputistherefore x(t)=s(t)+n(t) =Accos[2Tffct+</>(t)]+r(t)cos[2Tffct+l/J(t)](2.135) (2.136) (2.137)Itisinformative torepresent x(t)bymeansofaphasordiagram, asinFigure2.41.Inthis diagramwehaveusedthesignaltermasreference. Thephaseott)oftheresultant phasor representing x(t)isobtained directlyfromFigure2.41as ott)=</>(t)+tan-1{ r(t)sin[l/J(t)-</>(t)]} Ac+r(t)cos[l/J(t)-</>(t)] Theenvelope ofx(t)isofnointeresttous,becauseanyenvelope variations attheband passfilteroutputareremoved bythelimiter. Ourmotivation istodetermine theerrorintheinstantaneous frequency ofthecarrier wavecausedbythepresence ofthefilterednoisen(t).Withthediscriminator assumed ideal,itsoutputisproportional toO'(t)f2Tfwhere O'(t)isthederivative ofott)withrespect totime.Inviewofthecomplexity oftheexpression definingott),however, weneedto makecertainsimplifying approximations, sothatouranalysismayyieldusefulresults. Weassumethatthecarrier-to-noise ratiomeasured atthediscriminator inputislarge compared withunity.LetRdenotetherandomvariableobtained byobserving (atsome £Xedtime)theenvelope processwithsamplefunctionr(t)[duetothenoisen(t)].Then,at leastmostofthetime,therandomvariableRissmallcompared withthecarrieramplitude Anandsotheexpression forthephaseott)simplifies considerably asfollows: r(t). ott)=</>(t)+~sm[l/J(t) </>(t)] or,usingtheexpression for</>(t)giveninEquation (2.134), {' r(t). ott)=2TfkfJom('T)d'T+~sm[l/J(t)-</>(t)](2.138) (2.139) ~)---- FIGURE2.41Phasordiagram forFMwaveplusnarrowband noiseforthecaseofhighcarrier to-noise ratio. 2.13NoiseinFMRecewers 145 Thediscriminator outputistherefore v(t)=1dO(t) 2'Trdt =kfm(t)+nAt) wherethenoisetermnd(t)isdefinedby(2.140) (2.142) (2.143) (2.144)1 dnd(t)=2'TrAcdt{r(t)sin[!{I(t)-q,(t)]} (2.141) Wethusseethatprovided thecarrier-to-noise ratioishigh,thediscriminator outputv(t) consistsoftheoriginalmessage signalm(t)multiplied bytheconstant factorkf,plusan additivenoisecomponent nd(t).Accordingly, wemayusetheoutputsignal-to-noise ratio aspreviously definedtoassessthequalityofperformance oftheFMreceiver. Beforedoing this,however, itisinstructive toseeifwecansimplifytheexpression definingthenoise nd(t). FromthephasordiagramofFigure2.41,wenotethattheeffectofvariations inthe phasel/1(t)ofthenarrowband noiseappearreferredtothesignaltermq,(t).Weknowthat thephase!{I(t)isuniformly distributed over2'Trradians.Itwouldtherefore betempting to assumethatthephasedifference !{I(t)-q,(t)isalsouniformly distributed over2'Trradians. Ifsuchanassumption weretrue,thenthenoisetid(t)atthediscriminator outputwould beindependent ofthemodulating signalandwoulddependonlyonthecharacteristics of thecarrierandnarrowband noise.Theoretical considerations showthatthisassumption isjustifiedprovided thatthecarrier-to-noise ratioishigh.9ThenwemaysimplifyEquation (2.141)as: nd(t)=2~Ac~{r(t)sin[l/1(t)]} However, fromthedefiningequations forr(t)and!{I(t),wenotethatthequadrature com ponentng(t)ofthefilterednoisen(t)is ng(t)=r(t)sin[l/1(t)] Therefore, wemayrewriteEquation (2.142)as nd(t)=_1_dnQ(t) 2'TrAcdt Thismeansthattheadditivenoisend(t)appearing atthediscriminator outputisdeter minedeffectively bythecarrieramplitude Acandthequadrature component nQ(t)ofthe narrowband noisen(t). Theoutputsignal-to-noise ratioisdefinedastheratiooftheaverageoutputsignal powertotheaverageoutputnoisepower.FromEquation (2.140),weseethatthemessage component inthediscriminator output,andtherefore thelow-pass filteroutput,iskfm(t). Hence,theaverageoutputsignalpowerisequaltok}P,wherePistheaveragepowerof themessage signalm(t). Todetermine theaverageoutputnoisepower,wenotethatthenoisend(t)atthe discriminator outputisproportional tothetimederivative ofthequadrature noisecom ponentnQ(t).Sincethedifferentiation ofafunction withrespecttotimecorresponds to multiplication ofitsFouriertransform byj2'Trf,itfollowsthatwemayobtainthenoise processnd(t)bypassingndt)through alinearfilterwithafrequency response equalto j2'Trfif 2'TrAcAc (2.145)146 CHAPTER 2"CONTINUOUS-WAVE MODULATION ThismeansthatthepowerspectraldensitySNJf)ofthenoisena(t)isrelatedtothepOWer spectraldensitySNQ(f)ofthequadrature noisecomponent nQ(t)asfollows: FSNJf)=A~SNQ(f) Withtheband-pass filterinthereceivermodelofFigure2.40havinganidealfre quencyresponse characterized bybandwidth BTandmidband frequency foitfollowsthat thenarrowband noisen(t)willhaveapowerspectraldensitycharacteristic thatissimilarly shaped.Thismeansthatthequadrature component ng(t)ofthenarrowband noisen(t) willhavetheideallow-pass characteristic showninFigure2.42a.Thecorresponding power spectraldensityofthenoisena(t)isshowninFigure2.42b;thatis, IfI~BT 2 otherwise(2.146) InthereceivermodelofFigure2.40,thediscriminator outputisfollowed byalow-pass filterwithabandwidth equaltothemessagebandwidth W.ForwidebandFM,weusually findthatWissmallerthanBT/2,whereBTisthetransmission bandwidth oftheFM signal.This meanschattheout-of-band components ofnoisena(t)willberejected.there fore,thepowerspectraldensitySNJf)ofthenoiseno(t)appearing atthereceiveroutput isdefinedby ifI~W otherwise(2.147) (2.148)asshowninFigure2.42c.Theaverageoutputnoisepowerisdetermined byintegrating thepowerspectraldensitySNJf)from- WtoW.Wethusgetthefollowing result: w Averagepowerofoutputnoise=~~Lwf2df c 2NoW3 3A~ BT 2---'BT'----~LO----B...LT--f 2 2 (a)---'~~----'''-'...-<:.-----'--f BT 2" (b) (c) FIGURE 2....2NoiseanalysisofFMreceiver. (a)Powerspectraldensityofquadrature compo nentnQ(t)ofnarrowband noisen(t).(b)Powerspectral densityofnoisen.l(t)atthediscriminator output.(c)Powerspectraldensityofnoiseno(t)atthereceiver output. (2.150)2.13NoiseinFMReceivers 147 Notethattheaverageoutputnoisepowerisinversely proportional totheaveragecarrier power A~/2.Accordingly, inanFMsystem,increasing thecarrierpowerhasanoise quietingeffect. Earlierwedetermined theaverageoutputsignalpoweraskJP.Therefore, provided thecarrier-to· noiseratioishigh,wemaydividethis average outputsignalpowerbythe averageoutputnoisepowerofEquation (2.148)toobtaintheoutputsignal-to-noise ratio 3A~kJP (SNR)o,FM =2N oW3 (2.149) Theaveragepowerinthemodulated signalsit)isA~I2,andtheaveragenoisepowerin themessage bandwidth isWNo•Thusthechannelsignal-to-noise ratiois A2 (SNR)c,FM =2W~o Dividing theoutputsignal-to-noise ratiobythechannelsignal-to-noise ratio,wegetthe following figureofmeritforfrequency modulation: (SNR)ol _3kJP (2.151) (SNRlc FM-W2 FromSection2.7werecallthatthefrequency deviation Ilfisproportional tothe frequency sensitivity kfofthemodulator. Also,bydefinition, thedeviation ratioDisequal tothefrequency deviation Ilfdividedbythemessage bandwidth W.Inotherwords,the deviation ratioDisproportional totheratiokfPlI2/W.Itfollowstherefore fromEquation (2.151)thatthefigureofmeritofawidebandFMsystemisaquadratic function ofthe deviation ratio.Now,inwideband FM,thetransmission bandwidth BTisapproximately proportional tothedeviation ratioD.Accordingly, wemaystatethatwhenthecarrier to-noiseratioishigh,anincreaseinthetransmission bandwidth BTprovidesacorrespond ingquadratic increaseintheoutputsignal-to-noise ratioorfigureofmeritoftheFM system.Theimportant pointtonotefromthisstatement isthat,unlikeamplitude modu lation,theuseoffrequency modulation doesprovideapractical mechanism fortheex changeofincreased transmission bandwidth forimproved noiseperformance. if>ExAMPLE 2.5Single-Tone Modulation Consider thecaseofasinusoidal waveoffrequency fmasthemodulating signal,andassume apeakfrequency deviationaf.Themodulated FMsignalisthusdefinedby s(t)Accos[27rfct+~~Sin(2T1fmt)] Therefore, wemaywrite r' af27rkfJrm(1")d1"= -sin(27rfmt)o fm Differentiating bothsideswithrespecttotimeandsolvingform(t),weget ilf m(t)=k fcos(27rfmt) Hence,theaveragepowerofthemessagesignalm(t),developed acrossaI-ohmload,is p=(ilf)2 2k} 148 CHAPTER 2IJlCONTINUOUS-WAVE MODUlATION Substituting thistesultintothefotmulafortheoutputsignal-to-noise ratiogiveninEquation (2.149),weget (SNR).=3A~(Lif)2 O,FM4NoW3 3A~fP =4NoW wheref3LiflWisthemodulation index.UsingEquation (2.151)toevaluatethecorrespond. ingfigureofm.erit,weget (2.152) Itisimportant tonotethatthemodulation indexf3=t>flWisdetermined bythebandwidth Wafthepostdetection low-pass filterandisnotrelatedtothesinusoidal messagefrequency f~,exceptinsofarasthisfilterisusuallychosensoastopassthespectrum ofthedesired message; thisismerelyamatterofconsistent design.Foraspecified systembandwidth W,the sinusoidal messagefrequency fmmaylieanywhere between0andWandwouldyieldthesame outputsignal-to-noise ratio. Itisofparticular interesttocompare thenoiseperformance ofAMandFMsystems: Aninsightful wayofmakingthiscomparison istoconsider thefiguresofmeritofthetwo systemsbasedonasinusoidal modulating signal.ForanAMsystemoperating withasinu soidalmodulating signaland100percentmodulation, wehave(fromExample 2.4): (SNR)ol (SNRlc A.'-'l1 3 Comparing thisfigureofmeritwiththecorresponding resultdescribed inEquation (2.1521 foranFMsystem,weseethattheuseoffrequency modulation offersthepossibility ofim· provednoiseperformance overamplitude modulation when ~f32>t thatis, V2 f3>"""3=0,471 Wemaytherefore consider f3=0.5asdefiningroughlythetransition between narrowband FMandwideband FM.Thisstatement, basedonnoiseconsiderations, furtherconfirms asimilarobservation thatwasmadeinSection2.7whenconsidering thebandwidth of ~-~ ~ !illCAPTURE EFFECT Theinherent abilityofanFMsystemtominimize theeffectsofunwanted signals(e,g" noise,asjustdiscussed) alsoappliestointerference produced byanotherfrequenct modulated signalwhosefrequency contentisclosetothecarrierfrequency ofthedesired FMwave.However, interference suppression inanFMreceiverworkswellonlywhenthe interference isweakerthanthedesiredFMinpnt.Whentheinterference isthestronger oneofthetwo,thereceiverlocksontothestronger signalandtherebysnppresses the 2.13N.meinFMReceivers 149 desiredFMinput.Whentheyareofnearlyequalstrength, thereceiverfluctuates backand forthbetween them.Thisphenomenon isknownasthecaptureeffect,whichdescribes anotherdistinctive characteristic offrequency modulation. ~FMTHREsHoLD EFFECT TheformulaofEquation (2.149),definingtheoutputsignal-to-noise ratioofanFMre ceiver,isvalidonlyifthecarrier-to-noise ratio,measured atthediscriminator input,is highcompared withunity.Itisfoundexperimentally thatastheinputnoisepoweris increased sothatthecarrier-to-noise ratioisdecreased, theFMreceiver breaks.Atfirst, individual clicksareheardinthereceiveroutput,andasthecarrier-to-noise ratiodecreases stillfurther,theclicksrapidlymergeintoacrackling orsputtering sound.Nearthebreak ingpoint,Equation (2.149)beginstofailbypredicting valuesofoutputsignal-to-noise ratiolargerthantheactualones.Thisphenomenon isknownasthethreshold effect.10The threshold isdefinedastheminimum carrier-to-noise ratioyieldinganFMimprovement thatisnotsignificantly deteriorated fromthevaluepredicted bytheusualsignal-to-noise formulaassuming asmallnoisepower. Foraqualitative discussion oftheFMthreshold effect,consider firstthecasewhen thereisanosignalpresent,sothatthecarrierwaveisunmodulated. Thenthecomposite signalatthefrequency discriminator inputis (2.153) wherenI(t)andnQ{t)arethein-phase andquadrature components ofthenarrowband noisenit)withrespecttothecarrierwave.Thephasordiagram ofFigure2.43displays thephaserelations between thevariouscomponents ofx(t)inEquation (2.153).Asthe amplitudes andphasesofnI(t)andnQ!t)changewithtimeinarandommanner, thepoint PI[thetipofthephasorrepresenting x(t)]wanders aroundthepointP2(thetipofthe phasorrepresenting thecarrier).Whenthecarrier-to-noise ratioislarge,nI{t)andnQ(t) areusuallymuchsmallerthanthecarrieramplitude Anandsothewandering pointPIin Figure2.43spendsmostofitstimenearpointP2•Thustheangle8(t)isapproximately nQ(t)/A ctowithinamultiple of217.Whenthecarrier-to-noise ratioislow,ontheother hand,thewandering pointPIoccasionally sweepsaroundtheoriginand8(t)increases or decreases by217radians.Figure2.44illustrates howinaroughwaytheexcursions in8(t), depicted inFigure2.44a,produce impulselike components in8'(t)=d8/dt.Thediscrim inatoroutputv(t)isequalto8'(t)1217. Theseimpulselike components havedifferent heights depending onhowclosethewandering pointPIcomestotheorigin0,butallhaveareas nearlyequalto±217radians, asillustrated inFigure2.44b.Whenthesignalshownin Figure2.44bispassedthroughthepostdetection low-pass filter,corresponding butwider impulselike components areexcitedinthereceiveroutputandareheardasclicks.The clicksareproduced onlywhen8(t)changesby±217radians. O....",==---------r=-------------;M<---'T---; ... FIGURE2.43Phasordiagram interpretation ofEquation (2.1;3). 150 CHAPTER 2'"CONTINUOUS-WAVE MODULATION 4" 2" (2.154/o1V'tN''<f-\::f--r------'----;--.-----c-+------- -2" (bl FIG\JRE2.44Illustrating impulselike components inO'(t)=dO(t)/dtproduced bychanges of21r inO(t):(a)and(b)aregraphsofott)andO'(t),respectively. Fromthephasordiagram ofFigure2.43,wemaydeducetheconditions required fo! clickstooccur.Apositive-going clickoccurswhentheenveloper(t)andphaset/J(t)ofthe narrowband noisen(t)satisfythefollowing conditions: r(t)>A, l/J(t)<7T:5l/J(t)+dl/J(t) dl/J(t)>0 dt Theseconditions ensurethatthephasee(t)oftheresultant phasorx(t)changes by271 radiansinthetimeincrement dt,duringwhichthephaseofthenarrowband noiseincreases byincremental amountdt/J(t).Similarly, theconditions foranegative-going clicktooccur areasfollows: r(t)>Ac t/J(t)>-7T>l/J(t)+dl/J(t) dt/J(t)<0 dt Theseconditions ensurethate(t)changesby-27Tradiansduringthetimeincrement dt. Thecarrier-to-noise ratioisdefinedby A: p=2BTNo Aspisdecreased, theaveragenumberofclicksperunittimeincreases. Whenthisnumber becomes appreciably large,threshold issaidtooccur. 2.13NoiseinFMReceivers 151 Theoutputsignal-to-noise ratioiscalculated asfollows: 1.Theoutputsignalistakenasthereceiveroutputmeasured intheabsenceofnoise. Theaverageoutputsignalpoweriscalculated assuming asinusoidal modulation that produces afrequency deviationAtequaltoB~2,sothatthecarrierswingsback andforthacrosstheentireinputfrequency band. 2.Theaverageoutputnoisepoweriscalculated whenthereisnosignalpresent;that is,thecarrierisunmodulated, withnorestriction imposed onthevalueofthecarrier to-noiseratiop. Onthisheuristic basis,theoryll yieldsCurveIofFigure2.45presenting aplotofthe outputsignal-to-noise ratioversusthecarrier-to-noise ratiowhentheratioBT/2Wisequal to5.Thiscurveshowsthattheoutputsignal-to-noise ratiodeviatesappreciably froma linearfunction ofthecarrier-to-noise ratiopwhenpislessthanabout10dB.CurveIIof Figure2.45showstheeffectofmodulation ontheoutputsignal-to-noise ratiowhenthe modulating signal(assumed sinusoidal) andthenoisearepresentatthesametime.The averageoutputsignalpowerpertaining tocurveIImaybetakentobeeffectively thesame asforcurve1.Theaverageoutputnoisepower,however, isstrongly dependent onthe presence ofthemodulating signal,whichaccounts forthenoticeable deviation ofcurveII fromcurve1.Inparticular, wefindthataspdecreases frominfinity,theoutputsignal-to- 42 40 38 36(BT)3 34(SNR)o~\~~/ B]'-=5 32 2W / /30 / / /28 / / /26 24 22 20 6..18 o16 14 12 10"---'--_"----'-_'------'-_'------'-_-'----'--o 4 8 Carrier-to-noise ratio10loglOp,dB FIGURE2.45Dependence ofoutputsignal-to-noise ratiooninputcarrier-to-noise ratioforFM reciever. IncurveI,theaverageoutputnoisepoweriscalculated assuming anunmodulated car rier.IncurveII,theaverageoutputnoisepoweriscalculated assuming asinusoidally modulated carrier.BothcurvesIandIIarecalculated fromtheory. (2.155)152 CHAPTER 2"CONTINUOUS-WAVE MODUlATION noisedeviatesappreciably fromalinearfunction ofpwhenpisabout11dB.Alsowhen thesignalispresent,theresulting modulation ofthecarriertendstoincreasetheaverage numberofclickspersecond.Experimentally, itisfoundthatoccasional clicksareheard inthereceiveroutputatacarrier-to-noise ratioofabout13dB,whichappearstobeonly slightlyhigherthanwhattheoryindicates. Alsoitisofinteresttonotethattheincreasein theaveragenumberofclickspersecondtendstocausetheoutputsignal-to-noise ratioto falloffsomewhat moresharplyjustbelowthethreshold levelinthepresence of modulation. Fromtheforegoing discussion wemayconclude thatthreshold effectsinFMreceivers maybeavoidedinmostpractical casesofinterestifthecarrier-to-noise ratiopisequalto orgreaterthan20or,equivalently, 13dB.ThususingEquation (2.154)wefindthatthe lossofmessageatthediscriminator outputisnegligibleif ~~202BTNo or,equivalently, iftheaveragetransmitted power A~12satisfiesthecondition A2 ;~20BTNo Tousethisformula, wemayproceedasfollows: 1.Foraspecified modulation index{3andmessagebandwidth W,wedetermine the transmission bandwidth oftheFMwave,BT,usingtheuniversal curveofFigure2.26 orCarson's rule. 2.Foraspecified averagenoisepowerperunitbandwidth, No,weuseEquation (2.155) todetermine theminimum valueoftheaveragetransmitted power A~/2thatisnec essarytooperateabovethreshold. IIIFMTHRESHOLD REDUCTION Incommunication systemsusingfrequency modulation, thereisparticular interestinre' ducingthenoisethreshold inanFMreceiversoastosatisfactorily operatethereceiver withtheminimum signalpowerpossible. Threshold reduction inFMreceivers maybe achieved byusinganFMdemodulator withnegativefeedback12(commonly referred toas anFMFBdemodulator), orbyusingaphase-locked loopdemodulator. Suchdevicesare referredtoasextended-threshold demodulators, theideaofwhichisillustrated inFigure 2.46.Thethreshold extension showninthisfigureismeasured withrespecttothestandard frequency discriminator (i.e.,onewithoutfeedback). TheblockdiagramofanFMFBdemodulator13isshowninFigure2.47.Weseethat thelocaloscillator oftheconventional FMreceiverhasbeenreplaced byavoltage controlled oscillator (VCO)whoseinstantaneous outputfrequency iscontrolled bythe demodulated signaLInordertounderstand theoperation ofthisreceiver, supposeforthe moment thattheVCOisremoved fromthecircuitandthefeedback loopisleftopen: Assumethatawideband FMsignalisappliedtothereceiverinput,andasecondFM signal,fromthesamesourcebutwhosemodulation indexisafractionsmaller,isapplied totheVCOterminalofthemixer.Theoutputofthemixerwouldconsistofthedifference frequency component, becausethesumfrequency component isremoved bytheband-pasS filter.Thefrequency deviation ofthemixeroutputwouldbesmall,although thefrequency deviation ofbothinputFMwavesislarge,sincethedifference betweentheirinstantaneouS deviations issmalLHence,themodulation indiceswouldsubtractandtheresultingFM waveatthemixetoutputwouldhaveasmallermodulation index.TheFMwavewith 2.13NoiseinFMReceiwen 153 Extended thresholdThreshold Carrier-to-noise ratio,dB FIGlJRE2.46FMthreshold extension. reducedmodulation indexmaybepassedthrough aband-pass filter,whosebandwidth needonlybeafractionofthatrequired foreitherwideband FM,andthenfrequency demodulated. Itisnowapparent thatthesecondwideband FMsignalappliedtothemixer maybeobtained byfeedingtheoutputofthefrequency discriminator backtotheYeo. Itwillnowbearguedthatthesignal-to-noise ratioofanFMFBreceiveristhesame asthatofaconventional FMreceiverwiththesameinputsignalandnoisepowerifthe carrier-to-noise ratioissufficiently large.Assumeforthemoment thatthereisnofeed backaroundthedemodulator. Inthecombined presence ofanunmodulated carrier AcCOS(271fct) andnarrowband noise nit)=nr(t)COS(271fct) -nQ(t)sin(27Tfct) thephaseofthecomposite signalx(t)atthelimiter-discriminator inputisapproximately equaltonQ(t)/Ac> assuming thatthecarrier-to-noise ratioishigh.Theenvelope ofx(t)is ofnointeresttous,becausethelimiterremoves allvariations intheenvelope. Thusthe composite signalatthefrequency discriminator inputconsistsofasmallindexphase modulated wavewiththemodulation derivedfromthecomponent nQ(t)ofnoisethatis inphasequadrature withthecarrier.Whenfeedback isapplied,theyeOgenerates a frequency-modulated signalthatreducesthephase-modulation indexofthewaveinthe band-pass filteroutput,thatis,thequadrature component nQ(t)ofnoise.Thusweseethat aslongasthecarrier-to-noise ratioissufficiently large,theFMFBreceiverdoesnotrespond tothein-phase noisecomponent nrlt),butthatitwoulddemodulate thequadrature noise component nQlt)inexactlythesamefashionasitwoulddemodulate signalmodulation. Received FM wave FIGlJRE2.47FMdemodulator withnegative feedback.Output signal 154 CHAPTER 2IIICONTINUOUS-WAVE MODULATION Signalandquadrature noisearereducedinthesameproportion bytheappliedfeedback, withtheresultthatthebaseband signal-to-noise ratioisindependent offeedback. Forlarge carrier-to-noise ratios,thebaseband signal-to-noise ratioofanFMFBreceiveristhenthe sameasthatofaconventional FMreceiver. ThereasonthatanFMFBreceiver isabletoextendthethreshold isthat,unlikea conventional FMreceiver, itusesaveryimportant pieceofaprioriinformation, namely, thateventhoughthecarrierfrequency oftheincoming FMwavewillusuallyhavelarge frequency deviations, itsrateofchangewillbeatthebaseband rate.AnFMFBdemodu_ latorisessentially atracking filterthatcantrackonlytheslowlyvaryingfrequency ofa wideband FMsignal,andconsequently itresponds onlytoanarrowband ofnoisecentered abouttheinstantaneous carrierfrequency. Thebandwidth ofnoisetowhichtheFMFB receiverresponds isprecisely thebandofnoisethattheVCOtracks.Theendresultisthat anFMFBreceiver iscapableofrealizing athreshold extension ontheorderof5-7dB, whichrepresents asignificant improvement inthedesignofminimum powerFMsystems. LiketheFMFBdemodulator, thephase-locked loop(discussed laterinSection2.14) isalsoatracking filterand,assuch,thenoisebandwidth towhichitresponds isprecisely thebandofnoisetrackedbythe VCO. Indeed,thephase-locked loopdemodulator offers athreshold extension capability witharelatively simplecircuit.Unfortunately, theamount ofthreshold extension isnotpredictable byanyexistingtheory,anditdependsonsignal parameters. Roughly speaking, improvement byafew(ontheorderof2to3)decibels is achieved intypicalapplications, whichisnotasgoodasanFMFBdemodulator. I!OPRE-EMPHASIS ANDDE-EMPHASIS INFM Equation (2.147)showsthatthepowerspectraldensityofthenoiseattheoutputofan FMreceiverhasasquare-law dependence ontheoperating frequency; thisisillustrated in Figure2.48a.InFigure2.48b,wehaveincluded thepowerspectraldensityofatypical messagesource;audioandvideosignalstypically havespectraofthisform.Inparticular, weseethatthepowerspectraldensityofthemessageusuallyfallsoffappreciably athigher frequencies. Ontheotherhand,thepowerspectraldensityoftheoutputnoiseincreases rapidlywithfrequency. Thusaroundf=±W,therelativespectraldensityofthemessage isquitelow,whereasthatoftheoutputnoiseisquitehighincomparison. Clearly, the message isnotusingthefrequency bandallottedtoitinanefficientmanner.Itmayappear thatonewayofimproving the noiseperformance ofthesystemistoslightlyreducethe bandwidth ofthepostdetection low-pass filtersoastorejectalargeamountofnoisepower whilelosingonlyasmallamountofmessagepower.Suchanapproach, however, isusually notsatisfactory becausethedistortion ofthemessage causedbythereducedfilterband width,eventhoughslight,maynotbetolerable. Forexample, inthecaseofmusic,we findthatalthough thehigh-frequency notescontribute onlyaverysmallfractionofthe totalpower,nonetheless, theycontribute agreatdealfromanestheticviewpoint. . Amoresatisfactory approach totheefficientuseoftheallowedfrequency bandis basedonrheuseofpre-emphasis inthetransmitter andde-emphasis inthereceiver, as ~-w wt -w -----"-w--- t FIGURE2.48(a)PowerspectraldensityofnoiseatFMreceiveroutput.(b)Powerspectralden sityofatypicalmessagesignal. 2.13NoiseinFMReceivers 155 met)Message plusnoise NOise w(t) FIGURE2.49Useofpre-emphasis andde-emphasis inanFMsystem. (2.156)illustrated inFigure2.49.Inthismethod, weartificially emphasize thehigh-frequency components ofthemessage signalpriortomodulation inthetransmitter, andtherefore beforethenoiseisintroduced inthereceiver.Ineffect,thelow-frequency andhigh frequency portions ofthepowerspectraldensityofthemessage areequalized insucha waythatthemessagefullyoccupies thefrequency bandallottedtoit.Then,atthediscrim inatoroutputinthereceiver, weperform theinverseoperation byde-emphasizing the high-frequency components, soastorestoretheoriginalsignal-power distribution ofthe message.Insuchaprocess,thehigh-frequency components ofthenoiseatthediscriminator outputarealsoreduced, therebyeffectively increasing theoutputsignal-to-noise ratioof thesystem.Suchapre-emphasis andde-emphasis processiswidelyusedincommercial FMradiotransmission andreception. Inordertoproduce anundistorted versionoftheoriginalmessageatthereceiver output,thepre-emphasis filterinthetransmitter andthede-emphasis filterinthereceiver mustideallyhavefrequency responses thataretheinverseofeachother.Thatis,ifHpe(f) designates thefrequency response ofthepre-emphasis filter,thenthefrequency response Hde(f)ofthede-emphasis filtermustideallybe(ignoring transmission delay) 1 Hde(f)=Hpe(f), (2.157)Ifl:s;BT 2 otherwiseThischoiceoffrequency responses makestheaveragemessagepoweratthereceiveroutput independent ofthepre-emphasis andde-emphasis procedure. Fromourprevious noiseanalysisinFMsystems, assuming ahighcarrier-to-noise ratio,thepowerspectraldensityofthenoisend(t)atthediscriminator outputisgivenby Equation (2.146).Themodified powerspectraldensityofthenoiseatthede-emphasis filteroutputistherefore {NoF I12IHde(fWSN)f) =A~Hde(f) , 0, (2.158)Recognizing, asbefore,thatthepostdetection low-pass filterhasabandwidth Wthatis, ingeneral,lessthanBTI2,wefindthattheaveragepowerofthemodified noiseatthe receiveroutputisasfollows: (Average outputnoise) _NofW21 12d. . -""'2fHde(f)fpowerwithde-emphaSIS Ac-W (2.159)Because theaveragemessage poweratthereceiveroutputisideallyunaffected bythe combined pre-emphasis andde-emphasis procedure, it follows thattheimprovement in outputsignal-to-noise ratioproduced bytheuseofpre-emphasis inthetransmitter and de-emphasis inthereceiverisdefinedby I=averageoutputnoisepowerwithoutpre-emphasis andde-emphasis averageoutputnoisepowerwithpre-emphasis andde-emphasis 156 CHAPTER 2!:llCONTINUOUS-WAVE MODULn'ION Earlierweshowedthattheaverage outputnoisepowerwithout pre-emphasis andde emphasis isequalto(2NoW3/3A;);seeEquation (2.148). Therefore, aftercancellation of common terms,wemayexpresstheimprovement factorIas 2W3 I=---=-----~- 3(wF1Hd,(fW df(2.160) Itmustbeemphasized thatthisimprovement factorassumes theuseofahighcarrier-to_ noiseratio:\tthediscriminator inputinthereceiver. IS>EXAMPLE 2.6 Asimplepre-emphasis filterthatemphasizes highfrequencies andiscommonly usedinpractice isdefinedbythefrequency response Hpc(f)=1+* whichiscloselyrealizedbytheRC-amplifier network showninFigure2.50a,provided that R«rand21TfCr«1insidethefrequency bandofinterest.TheamplifierinFigure2.S0a isintended tomakeupfortheattenuation introduced bytheRCnetworkatlowfrequencies. Thefrequency parameter fois1/(21TCr}. Thecorresponding de-emphasis filterinthereceiverisdefinedbythefrequency response whichcanberealiz<;dusingthesimpleRCnetworkofFigure2.50b. Theimprovement inoutputsignal-to-noise ratiooftheFMreceiver,resulting fromthe combined useofthepre-emphasis andde-emphasis filtersofFigure2.50,istherefore I2W' Jwf2df 3-w1+(flfo)2 (Wlfo)3 3[(Wlfo) tan1(Wlfo)](2.161) Incommercial FMbroadcasting, wetypically havefo=2.1kHz,andwemayreason. ablyassumeW=15kHz.ThissetofvaluesyieldsI=22,whichcorresponds toanimprove mentof13dBintheoutputsignal-to-noise ratioofthereceiver. Theoutputsignal-to-noise c (a)ROutput signalr In~ot Si~al (bJ FIGURE 2.50(a)Pre-emphasis filter.(b)De-emphasis filter. 2.14Computer Experiments: Phase-Locked Loop 157 ratioof.anFMreceiverwithoutpre"emphasis andde-emphasis istypically40-50dB.Wesee, therefore, thatbyusingthesimplepre-emphasis andde-emphasis filtersshowninFigure2.50, wecanrealizeasignificant improvement inthenoiseperformance ofthereceiver. -<ll Theuseofthe simple linearpre-emphasis andde-emphasis filtersjustdescribed isan example ofhowtheperformance ofanFMsystemmaybeimproved byusingthediffer encesbetween characteristics ofsignalsandnoiseinthesystem.Thesesimplefiltersalso findapplication inaudiotape-recording. Specifically, nonlinear pre-emphasis andde emphasis techniques havebeenappliedsuccessfully totaperecording. Thesetechniques'4 (knownasDolby-A, Dolby-B, andDBXsystems) useacombination offilteringanddy namicrangecompression toreducetheeffectsofnoise,particularly whenthesignallevel islow. l2.14 Computer Experiments: Phase-Locked Loop Theexperimental studypresented inthissectionfocusesontheuseofaphase-locked loop forthedemodulation ofafrequency modulated signal.Beforeproceeding withtheexper iments,however, wefirstpresentabriefexposition ofphase-locked looptheory. Basically, thephase-locked loopconsistsofthreemajorcomponents: amultiplier, a loopfilter,andavoltage-controlled oscillator (VeO)connected together intheformofa feedback system,asshowninFigure2.51.Theveoisasinusoidal generator whose frequency isdetermined byavoltageappliedtoitfromanexternalsource.Ineffect,any frequency modulator mayserveasaYeo.Weassumethatinitiallywehaveadjusted the veosothatwhenthecontrolvoltageiszero,twoconditions aresatisfied: 1.Thefrequency oftheveoisprecisely setattheunmodulated carrierfrequency fe. 2.Theveooutputhasa90degreephase-shift withrespecttotheunmodulated carrier wave. Supposethenthattheinputsignalappliedtothephase-locked loopisanFMsignaldefined by s(t)=Aesin[2?Tfct+<PI(t)] whereAcisthecarrieramplitude. Withamodulating signalm(t),theangle<PI(t)isrelated tom(t)bytheintegral FMwave Output ~ oW FIGURE2.51Phase-locked loop. 158 CHAPTER 2tilCONTINUOUS-WAVE MODULATION wherekfisthefrequency sensitivity ofthefrequency modulator. Lettheveooutputin thephase-locked loopbedefinedby r(t)=A"COS[27T[,t+'!>2(t)] whereA"istheamplitude. Withacontrolvoltagev(t)appliedtotheveoinput,theangle '!>2(t)isrelatedtov(t)bytheintegral <P2(t)=27Tk"J:V(T)dT (2.162) wherek"isthefrequency sensitivity oftheveo,measured inHertzpervolt.Theobject ofthephase-locked loopistogenerate aveooutputr(t)thathasthesamephaseangle (exceptforthefixeddifference of90degrees)astheinputFMsignals(t).Thetime-varying phaseangle<P1(t)characterizing s(t)maybeduetomodulation byamessage signalm(t), inwhichcasewewishtorecover <P1(t)andtherebyproduce anestimate ofm(t).Inother applications ofthephase-locked loop,thetime-varying phaseangle<P1(t)oftheincoming signals(t)maybeanunwanted phaseshiftcausedbyfluctuations inthecommunication channel; inthislattercase,wewishtotrack<Pl(t)soastoproduce asignalwiththesame phaseangleforthepurposeofcoherent detection (synchronous demodulation). MODEL OFTHEPHASE-LOCKED Loopl; Todevelopanunderstanding ofthephase-locked loop,itisdesirable tohaveamodelof theloop.Westartbydeveloping anonlinear model,whichissubsequently linearized to simplifytheanalysis. According toFigure2.51,theincoming FMsignals(t)andtheveo outputr(t)areappliedtothemultiplier, producing twocomponents: 1.Ahigh-frequency component, represented bythedouble-frequency term kmAfi"sin[47T[ct+<P1(t)+<P2(t)] 2.Alow-frequency component represented bythedifference-frequency term kmAfi"sin[<p1(t) -<P2(t)] wherekmisthemultiplier gain,measured invole1• Theloopfilterinthephase-locked loopisalow-pass filter,anditsresponse tothehigh ,frequency component willbenegligible. Theveoalsocontributes totheattenuation of thiscomponent. Therefore, discarding thehigh-frequency component (i.e.,thedouble frequency term),theinputtotheloopfilterisreducedto e(t)=kmAfi"sin[<p)t)] where<p,(t)isthephaseerrordefinedby <Pe(t)=<P1(t)-cP2(t) =<Pl(t)-27Tk"J:V(T)dT(2.163) (2.1641 Theloopfilteroperates ontheerrore(t)toproduce anoutputv(t)definedbytheconvo lutionintegral: v(t)=rooe(T)h(t T)dT (2.165) 2.14'Computer Experiments: PluIse-LockedLoop 159 whereh(t)istheimpulseresponse oftheloopfilter.UsingEquations (2.164)and(2.165) torelate<Pe(t)and<Pl(t),weobtainthefollowing nonlinear integro-differential equation asthedescriptor ofthedynamic behavior ofthephase-locked loop: d<Pe(t) dt7")d7" (2.166) whereKoisasloop-gain parameter definedby (2.167) Theamplitudes AcandAvarebothmeasured involts,themultiplier gainkminvole'and thefrequency sensitivity kvinHertzpervolt.Hence,itfollowsfromEquation (2.167)that Kohasthedimensions offrequency. Equation (2.166)suggeststhemodelshowninFigure 2.52foraphase-locked loop.Inthismodelwehavealsoincluded therelationship between v(t)ande(t)asrepresented byEquations (2.163)and(2.165).Weseethatthemodelof Figure2.52resembles theactualblockdiagramofFigure2.51.Themultiplier attheinput ofthephase-locked loopisreplaced byasubtracter andasinusoidal nonlinearity, andtheveobyanintegrator. Thesinusoidal nonlinearity inthemodelofFigure2.52complicates thetaskofan alyzingthebehavior ofthephase-locked loop.Itwouldbehelpfultolinearize thismodel tosimplifytheanalysisandyetgiveagoodapproximate description oftheloop'sbehavior incertainmodesofoperation. Whenthe'phaseerror<Pe(t)iszero,thephase-locked loop issaidtobeinphase-lock. When<Pe(t)isatalltimessmallcompared withoneradian,we mayusetheapproximation sin[<p.(t)]'"<Pe(t) whichisaccurate towithin4percentfor<p,(t)lessthan0.5radians.Inthiscase,theloop issaidtobenearphase-lock, andthesinusoidal nonlinearity ofFigure2.52maybedis regarded. Underthiscondition, v(t)isapproximately equaltom(t),exceptforthescaling factorkflkv' Thecomplexity ofthephase-locked loopisdetermined bythefrequency response H(f)oftheloopfilter.Thesimplest formofaphase-locked loopisobtained when H(f)=1;thatis,thereisnoloopfilter,andthe'resulting phase-locked loopisreferredto asafirst-order phase-locked loop.Amajorlirnitation ofafirst-order phase-locked loopis thattheloopgainparameter Kocontrols boththeloopbandwidth aswellasthehold-in frequency rangeoftheloop;thehold-infrequency rangereferstotherangeoffrequencies v(t) FIGURE2.52Nonlinear modelofthephase-locked loop. 160 CllAPTER 2'"CONTINUOUS-WAVE MODULATION forwhichtheloopremainsphase-locked totheinputsignal.Wemayovercome thislim itationbyusingaloopfilterwiththefrequency response a H(f)=1+Ii (2.168) whereaisaconstant. Thenwiththisloopfilterinplaceandthephase-locked loopoper atinginitslinearmode,wefindfromEquation (2.166)thatthephase-locked loopbehaves asasecond-order feedback system,asshownbythestandard frequency response cI>e(f)_ Uflfn)2 cI>1(f) 1+2Wflfn)+Uflfn)2(2.169) wherecI>e(f)andcI>t(f)aretheFouriertransforms ofcPe(t)andcP1(t),respectively. The systemisparameterized bythenaturalfrequency, fn,anddamping factor,{,whichare respectively definedby (2.170) and (2.171) Thesecond-order phase-locked loopsodescribed isthesubjectofthecomputer experi mentspresented next. Experiment 1:Acquisition Mode Whenaphase-locked loopisusedforcoherent detection (synchronous demodulation), the loopmustfirstlockontotheinputsignalandthenfollowthevariations ofitsphaseangle withtime.Theprocessofbringing aloopintophase-lock iscalledacquisition, andthe ensuingprocessoffollowing angularvariations intheinputsignaliscalledtracking. In theacquisition modeandquitepossiblythetrackingmode,thephaseerrorcPe(t)between theinputsignals(t)andtheVCOoutputr{t)willcertainly belarge,therebymandating theuseofthenonlinear modelofFigure2.52.However, anonlinear analysisoftheac quisitionprocessbasedonthislattermodelisbeyondthescopeofthisbook.Inthis experiment, weusecomputer simulations tostudytheacquisition processandthereby developinsightintosomeofitsfeatures. Consider asecond-order phase-locked loopusingtheloopfilterofEquation (2.168) andhavingthefollowing parameters: 1Naturalfrequencyin=271'Hz Damping factor{=0.3,0.707,1.0 Toaccommodate variation in{,thefilterparameter aisvariedinaccordance withthe formula ina=-2? whichfollowsfromEquations (2.170)and(2.171).Figure2.53presentsthevariation in thephaseerrorcP.(t)withtimeforeachofthethreespecified valuesofdamping factor~, 2.14Computer Experiments: Phase-Locked Loop 161 0.6 0.5 ~0.4cm '0 ::!0.3 <:: ~0.2 ~ "0.1~m £ D- O -0.1 -0.2 0 25 Timet,seconds FIGURE2.53Variation ofthephaseerrorforthreedifferent valuesofdamping factor. assuming afrequency srepof0.125Hz.Theseresultsshowthatthedamping factor ?=0.707givesrhebesrcompromise betweenafasrresponse timeandanunderdamped oscillatory behavior. Experiment 2:Phase-Plane Portrait Aphase-plane portraitisafamilyoftrajectories, witheachtrajecrory representing asingle solurionofEquation (2.166).Forthesecondexperiment weplotthephase-plane portrait ofasecond-order phase-locked loopforthecaseofsinusoidal modulation. Thesystem parameters oftheloopareasfollows: . K50Loop-gam parameter 0=2-rrHz 50Loop-natural frequency fn=•f1Hz2v2 Sinusoidal modulation frequency 1m=:~Hz2-rrv2-rr Figure2.54presentsthephase-plane portraitofthephase-locked loopadjusted for criticaldamping, wherethetrajectories (frequency errorversusphaseerror)areplotted fordifferent startingpoints.Fromthisportraitwemakethefollowing observations: 1.Forasinusoidal nonlinearity, thephase-plane portraitisitselfperiodicwithperiod 2-rrinthephaseerrorcfJ"butitisaperiodic indcfJJdt. 2.Foraninitialfrequency error .!dcfJe Kdt withanabsolute valuelessthanorequalto1,thephase-locked loopisassuredof attaining astable(equilibrium) pointat(0,0)or(0,2-rr);themultiplicity ofequilib riumpointsisamanifestation ofperiodicity ofthephase-plane portrait. 162 CHAPTER 2IICONTINUOUS-WAVE MODUlATION 3 2 -2 -3 -4 -5L----::__ ---::-----'~""'::-__--:-__--=-_____"'__::_""=--- Phaseerror,radians FIGURE2.54Phase-plane portraitforcriticaldamping andsinusoidal modulation. 3.Foraninitialfrequency error .!.dcPe Kdt withanabsolutevalueequalto2,wehaveasaddlepointat(0,1T)wheretheslightest perturbation appliedtothephase-locked loopcausesittoshifttotheequilibrium point(0,0)or(0,21T). I2.15Summory andDiscussion Inthischapterwestudiedtheprinciples ofcontinuous-wave (CW)modulation. Thisan alogformofmodulation usesasinusoidal carrierwhoseamplitude orangleisvariedin accordance withamessagesignal.Wemaythusdistinguish twofamiliesofCWmodula tion:amplitude modulation andanglemodulation. IIIAMPLITUDE MODULATION Amplitude modulation mayitselfbeclassified intofourtypes,depending onthespectral contentofthemodulated signal.Thefourtypesofamplitude modulation andtheirprac ticalmeritsareasfollows: 1.Fullamplitude modulation (AM),inwhichtheupperandlowersidebands aretrans mittedinfull,accompanied bythecarrierwave. Accordingly, demodulation ofanAMsignalisaccomplished rathersimplyinthereceiver byusinganenvelope detector, forexample.ItisforthisreasonwefindthatfullAMis commonly usedincommercial AMradiobroadcasting, whichinvolvesasinglepowerful transmitter andnumerous receivers thatarerelatively inexpensive tobuild. 2.15SummaryandDiscussWn 163 2.Doublesideband-suppressed carrier(DSB-SC) modulation, inwhichonlytheupper andlowersidebands aretransmitted. Thesuppression ofthecarrierwavemeansthatDSB-SCmodulation requiresmuchless powerthanfullAMtotransmit thesamemessagesignal;thisadvantage ofDSB-SCmod ulationoverfullAMis,however, attainedattheexpenseofincreased receivercomplexity. DSBcSCmodulation istherefore wellsuitedforpoint-to-point communication involving onetransmitter andonereceiver; inthisformofcommunication, transmitted powerisat apremium andtheuseofacomplex receiveristherefore justifiable. 3.Singlesideband (SSB)modulation, inwhichonlytheuppersideband orlowersideband istransmitted. SSBmodulation istheoptimum formofCWmodulation inthesensethatitrequiresthe minimum transmitted powerandtheminimum channelbandwidth forconveying ames sagesignalfromonepointtoanother. However, itsuseislimitedtomessagesignalswith anenergygapcentered onzerofrequency. 4.Vestigial sideband modulation, inwhichalmostallofonesideband andavestigeof theothersideband aretransmitted inaprescribed complementary fashion. VSBmodulation requiresachannelbandwidth thatisbetweenthatrequired forSSBand DSB-SCsystems,andthesavinginbandwidth canbesignificant ifmodulating signalswith largebandwidths arebeinghandled, asinthecaseoftelevision signalsandhigh-speed data. DSB-SC, SSB,andVSBareexamples oflinearmodulation, whereas, strictlyspeaking, full AMisnonlinear. However, thedeviation offullAMfromlinearity isofamildsort. Accordingly, allfourformsofamplitude modulation lendthemselves readilytospectral analysisusingtheFouriertransform. ANGLE MODUlATION Anglemodulationinay beclassified intofrequency modulation (FM)andphasemodula tion(PM).InFM,theinstantaneous frequency ofasinusoidal carrierisvariedinpropor tiontothemessagesignal.InPM,ontheotherhand,itisthephaseofthecarrierthatis variedinproportion tothemessagesignal.Theinstantaneous frequency isdefinedasthe derivative ofthephasewithrespecttotime,exceptforthescalingfactor1/(21T).Accord ingly,FMandPMarecloselyrelatedtoeachother;ifweknowtheproperties oftheone, wecandetermine thoseoftheother.Forthisreason,andbecauseFMiscommonly used inbroadcasting, muchofthematerial onanglemodulation inthechapterwasdevoted toFM. Unlikeamplitude modulation, FMisanonlinear modulation process.Accordingly, spectralanalysisofFMismoredifficultthanforAM.Nevertheless, bystudying single toneFM,wewereabletodevelopagreatdealofinsightintothespectralproperties of FM.Inparticular, wederivedanempirical ruleknownasCarson's ruleforanapproximate evaluation ofthetransmission bandwidth BTofFM.According tothisrule,BTiscontrolled byasingleparameter: themodulation index{3forsinusoidal FM,orthedeviation ratio Dfornonsinusoidal FM. IIINOISE ANALYSIS Weconclude thechapteronCWmodulation systemsbypresenting acomparison oftheir noiseperformances. Forthiscomparison, weassumethatthemodulation isproduced by 164 CHAPTER 2!IICONTINUOUS-WAVE MODUlATION 70 60 50 '""Co ~ ill400g ~.. .~30 lso 20 10 10 20 30IV ill II 40 50 Channelsignal-to-noise ratio,dB FIGURE2.55Comparison ofthenoiseperformance ofvariousCWmodulation systems. CurveI: FullAM,JL=1.CurveII:DSB-SC, SSB.CurveIII:FM,{3=2.CurveIV:FM,{3=5.(Curves IIIandIVincludeB-dBpre-emphasis, de-emphasis improvement.) asinusoidal wave.Forthecomparison tobemeaningful, wealsoassumethatthemodu lationsystemsoperate with exactlythesamechannelsignal-to-noise ratio.Wemaythus plottheoutputsignal-to-noise ratioversusthechannelsignal-to-noise ratioasinFigure 2.55forthefollowing modulation schemes: ~FullAMwith100percentmodulation I>-Coherent DSB-SC, SSB I>-FMwithf3=2andf3=5 Figure2.55alsoincludestheAMandFMthreshold effects.Inmakingthecomparison, it isinformative tokeepinmindthe transmission bandwidth requirement ofthemodulation systeminquestion.Inthisregard,weuseanormalized transmission bandwidth definedby B=BT nW whereBTisthetransmission bandwidth ofthemodulated signal,andWisthemessage bandwidth. Table2.4presentsthevaluesofBnforthedifferentCWmodulation schemes. FromFigure2.55andTable2.4wemakethefollowing observations: ~AmongthefamilyofAMsystems, SSBmodulation isoptimum withregardtonoise performance aswellasbandwidth conservation . .,.TheuseofFMimproves noiseperformance butattheexpenseofanexcessive trans' missionbandwidth. ThisassumesthattheFMsystemoperates abovethreshold for thenoiseimprovement toberealized. NotesandReferences 165 TABLE204ValuesofBnfor variousCWmodulation schemes PM AM,DSB-SC 2SSB 132 813=5 16 Onanimportant pointtoconclude thediscussion onCWmodulation, onlyfrequency modulation offersthecapability totradeofftransmission bandwidth forimproved noise performance. Thetrade-off followsasquarelaw,whichisthebestthatwecandowith CWmodulation (i.e.,analogcommunications). InChapter 3wedescribe pulse-code mod ulation, whichisbasictothetransmission ofanaloginformation-bearing signalsbya digitalcommunication system,andwhichcanindeeddomuchbetter. INOTES ANDREFERENCES 1.Itappearsthatthetermscontinuous waveandheterodyning werefirstusedbyReginald Fessenden intheearly1900s. 2.TheCostasreceiverisnamedinhonorofitsinventor; seethepaperbyCostas(1956). 3.Besselfunctions playanimportant roleinthestudyofbothanaloganddigitalcommuni cationsystems.Theycanbeoftheso-called firstkindorsecondkind.Appendix 3discusses mathematical detailsandproperties ofbothkindsofBesselfunctions. AtableofBessel functions ofthefirstkindispresented inTableA6.5. 4.Carson's ruleforthebandwidth ofFMsignalsisnamedinhonorofitsoriginator; Carson andFry(1937)wroteoneoftheearlyclassicpapersonfrequency modulation theory. 5.Theindirectmethodofgenerating awideband FMwavewasfirstproposed byArmstrong (1936).Armstrong wasalsothefirsttorecognize thenoise-robustness properties offre quencymodulation. 6.Stereomultiplexing usuallyinvolves theuseoffrequency modulation forradiotransmis sion.However, itmayalsobetransmitted usingamplitude modulation asdiscussed in Problem 2.14;formoredetails,seethepaperbyMennie(1978). 7.Fordetaileddescription ofthesuperheterodyne receiver, seetheRadioEngineering Hand bookeditedbyHenney(1958,pp.19-34-19-41). 8.Thequalitative studyofthreshold inenvelope detection presented herefollowsDowning (1964,p.71). 9.Forajustification ofthecriticalassumption onwhichthesimplification presented inEqua tion(2.142)rests,seeRice(1963). 10.Foradetailed discussion ofthethreshold effectinFMreceivers, seethepaperbyRice (1963)andthebookbySchwartz, Bennett, andStein(1966,pp.129-163). 11.Figure2.45isadapted fromRice(1963).Thevalidityofthetheoretical curveIIinthis figurehasbeenconfirmed experimentally; seeSchwartz, Bennett, andStein(1966,p.153). Forsomeearlierexperimental workonthethreshold phenomenon inFM,seethepaper byCrosby(1937). 12.Theideaofusingfeedback aroundanFMdemodulator wasoriginally proposed byChaffee (1939). 166 CHAPTER 2'"CON'I'lNUOUS-WAVE MODULATION 13.Thetreatment oftheFMFBdemodulator presented inSection2.13isbasedonthepaper byEnloe(1962);seealsoRoberrs(1977,pp.166-181). 14.Foradetaileddiscussion ofDolbysystemsmentioned inthelatterpartofSection2.13,see Stremler (1990,pp.732-734). 15.Forafulltreatment ofthenonlinear analysisofaphase-locked loop,seeGardner (1979) Lindsey(1972),andViterbi(1966). ' IPROBLEMS Amplitude Modulation 2.1Suppose thatnonlinear devicesareavailable forwhichtheoutputcurrentioandinput voltage Viarerelatedby whereajanda3areconstants. Explainhowthesedevicesmaybeusedtoprovide: (a)a productmodulator and(b)anamplitude modulator. 2.2FigureP2.2showsthecircuitdiagramofasquare-law modulator. Thesignalappliedto thenonlinear deviceisrelatively weak,suchthatitcanberepresented byasquarelaw: whereajanda2areconstants, V,(t)istheinputvoltage,andv,(t)istheoutputvoltage. Theinputvoltageisdefinedby wherem(t)isamessagesignalandA,cos(2'7tf,t) isthecarrierwave. (a)Evaluate theoutputvoltagev,(t). (b)Specifythefrequency response thatthetunedcircuitinFigureP2.2mustsatisfyin ordertogenerate anAMsignalwithfcasthecarrierfrequency. (c)Whatistheamplitude sensitivity ofthisAMsignal? Tuned tQf~ FIGUREP2.2 2.3FigureP2.3ashowsthecircuitdiagramofaswitching modulator. Assumethatthecarrier wavec(t)appliedtothediodeislargeinamplitude, sothatthediodeactslikeanideal switch:itpresents zeroimpedance whenforward biased(i.e.,cit)>0).Wemaythus Problems 167 approximate thetransfer characteristic ofthediode-load resistorcombination byapiece wise-linear characteristic definedas(seeFigureP2.3b) c(t)>0 c(t)<0 Thatis,theloadvoltageV2(t)variesperiodically betweenthevaluesv,(t)andzeroata rateequaltothecarrierfrequency fe.Hence,wemaywrite 1.2(t)=[Accos(2'1TfJ)+m(t)]gYo(t) wheregyo(t)isaperiodicpulsetraindefinedby 1 2 ~(_1)"-' gyo(t)=-2+-~-2-1cos[2'17fet(2n -1)] '1T'n=ln (a)FindtheAMwavecomponent contained intheoutputvoltageV2(t). (b)Specifytheunwanted components inV2(t)thatneedtoberemoved byaband-pass filterofsuitabledesign. (a)-""0'----------- VI (b) 2.4Consider theAMsignalFIGUREP2.3 produced byasinusoidal modulating signaloffrequency fm.Asswnethatthemodulation factorisf.L=2,andthecarrierfrequency j;ismuchgreaterthanfm-TheAMsignals(t) isappliedtoanidealenvelope detector, producing theoutputv(t). (a)Determine theFourierseriesrepresentation ofv(t). (b)Whatistheratioofsecond-harmonic amplitude tofundamental amplitude inv(t)? 2.5FigureP2.5showsthecircuitdiagram ofanenvelope detector. Itconsistssimplyofa diodeandresistor-capacitor (RC)filter.Onapositivehalf-cycle oftheinputsignal,the diodeisforward-biased andthecapacitor Cchargesuprapidlytothepeakvalueofthe inputsignal.WhentheinputsignalfaUsbelowthisvalue,thediodebecomes reverse biasedandthecapacitor Cdischarges slowlythroughtheloadresistorRioThedischarging processcontinues untilthenextpositivehalf-cycle. Thereafter, thecharging-discharging routineiscontinued. (a)Specifythecondition thatmustbesatisfiedbythecapacitor Cforittochargerapidly andtherebyfollowtheinputvoltageuptothepositivepeakwhenthediodeis conducting. (b)Specifythecondition whichtheloadresistorRzmustsatisfysothatthecapacitor C discharges slowlybetweenpositivepeaksofthecarrierwave,butnotsolongthatthe 168 CHAPTER 2OJC01',TINUOUS-WAVE MODULATION capacitor voltagewillnotdischarge atthemaximum rateofchangeofthemodulating wave. c AMwave s(t)1 R[Output ~J FIGUREP2.5 2.6Consider asquare-law detector, usinganonlinear devicewhosetransfercharacteristic is definedby wherea,anda2areconstants, v,(t)istheinput,andV2(t)istheoutput.Theinputconsists oftheAMwave v,(t)=Acll+kam(t)]COS(21Tfct) (a)Evaluate theoutputV2(t). (b)Findtheconditions forwhichthemessagesignalm(t)mayberecovered fromV2(t). 2.7TheAMsignal sit)=A,[l+kam(t)]COS(21Tfct) isappliedtothesystemshowninFigureP2.7.Assuming thatIkam(t)I<1foralltand themessage signalmit)islimitedtotheinterval- W:5f:5Wandthatthecarrier frequency fe>2Wshowthatm(t)canbeobtained fromthesquare-rooter outputVj(t), s(t) FIGUREP2.7 2.8Consider amessage signalm(t)withthespectrum showninFigureP2.8.Themessage bandwidth W=1kHz.Thissignalisappliedtoaproductmodulator, together witha carrierwaveAeCOS(21Tfet), producing theDSB-SCmodulated signals(t).Themodulated signalisnextappliedtoacoherent detector. Assuming perfectsynchronism betWeen the carrierwavesinthemodulator anddetector, determine thespectrum ofthedetector outputwhen:(a)thecarrierfrequency fe=1.25kHzand(b)thecarrierfrequency fe=0.75kHz.Whatisthelowestcarrierfrequency forwhicheachcomponent ofthe modulated signals(t)isuniquely determined bym(t)? Problems 169 M(f) ---..L-~...1---f FIGUREP2.8 2.9FigureP2.9showsthecircuitdiagramofabalanceimodulator. Theinputappliedtothe topAMmodulator ism(t),whereas thatappliedtothelowerAMmodulator is-m(t); thesetwomodulators havethesameamplitude sensitivity. Showthattheoutputs(t)of thebalanced modulator consistsofaDSB-SCmodulated signal. Sl{t) m(t) A,eDS(21lf,tl+ Ls{t) Accas(21T!ct) -met)"'2(tl FIGUREP2.9 2.10ADSB-SCmodulated signalisdemodulated byapplying ittoacoherent detector. (a)Evaluate theeffectofafrequency errorlifinthelocalcarrierfrequency ofthede tector,measured withrespecttothecarrierfrequency oftheincoming DSB-SCsignal. (b)Forthecaseofasinusoidal modulating wave,showthatbecauseofthisfrequency error,thedemodulated signalexhibits beatsattheerrorfrequency. Illustrate your answerwithasketchofthisdemodulated signal. 2.11Consider theDSB-SCsignal s(t)=Accos(21rfct)m(t) whereA,cos(27rfct) isthecarrierwaveandm(t)isthemessagesignal.Thismodulated signalisappliedtoasquare-law devicecharacterized by y(t)=s2(t) Theoutputy(t)isnextappliedtoanarrowband filterwithapassband magnitude response ofone,midband frequency 2foandbandwidth ilf.Assumethatilfissmallenoughto treatthespectrum ofy(t)asessentially constant insidethepassband ofthefilter. (a)Determine thespectrum ofthesquare-law deviceoutputy(t). (b)Showthatthefilteroutputv(t)isapproximately sinusoidal, givenby A2 v(t)=2'Eilfcos(47rfc t) whereEistheenergyofthemessagesignalm(t). 170 CHAPTER 2"CONTINUOUS-WAVE MODULATION 2.12Consider thequadrature-carrier multiplex systemofFigure2.10.Themultiplexed signal s(t)produced atthetransmitter outputinFigure2.10aisappliedtoacommunication channeloffrequency response H(f).Theoutputofthischannelis,inturn,appliedtothe receiverinputinFigure2.10b.Provethatthecondition H(fe+f)=W(f,-f), isnecessary forrecovery ofthemessage signalsm,(t)andm2(t)atthereceiveroutputs. feisthecarrierfrequency, andWisthemessage bandwidth. Hint:Evaluate thespectr~ ofthetworeceiveroutputs. 2.13Suppose thatinthereceiverofthequadrature-carrier multiplex systemofFigure2.10b thelocalcarrieravailable fordemodulation hasaphaseerror1>withrespecttothecarrier sourceusedinthetransmitter. Assuming adistortionless communication channelbetween transmitter andreceiver, showthatthisphaseerrorwillcausecross-talk toarisebetween thetwodemodulated signalsatthereceiveroutputs. Bycross-talk wemeanthatapottion ofonemessagesignalappearsatthereceiveroutputhelonging totheothermessagesignal, andviceversa. 2.14Aparticular versionofAMstereousesquadrature multiplexing. Specifically, thecarrier A,cos(2'1rfet) isusedtomodulate thesumsignal m,(t)=Vo+mAt)+mAt) whereVoisaDCoffsetincluded forthepurposeoftransmitting thecarriercomponent, mAt)istheleft-hand audiosignal,andm,(t)istheright-hand audiosignal.Thequadrature carrierAcsin(21TfJ) isusedtomodulate thedifference signal m2(t)=mAt)-m,(t) (a)Showthatanenvelope detector mayheusedtorecoverthesumm,(t)+m,(t)from thequadrature-multiplexed signal.Howwouldyouminimize thesignaldistortion produced bytheenvelope detector? (b)Showthatacoherent detectorcanrecoverthedifference me(t)-m,(t). (c)HowarethedesiredmAt)andmAt)finallyobtained? 2.15Usingthemessagesignal 1 m(t)=1+t' determine andsketchthemodulated wavesforthefollowing methods ofmodulation: (a)Amplitude modulation with50percentmodulation. (b)Doublesideband-suppressed carriermodulation. (c)Singlesideband modulation withonlytheuppersideband transmitted. (d)Singlesideband modulation withonlythelowersideband transmitted. 2.16TheHilberttransform ofaFouriertransformable signalm(t),denotedbym(t),isdefined by(seeAppendix 2) ,1foom(T)m(t)=---dT '1r-00t-T Inthefrequency domain, wehave M(f)=-jsgn(f)M(f) wherem(t)~M(f),m(t)~M(f),andsgn(f)isthesignumfunction. Problems 171 Usingthedefinition oftheHilberttransform, showthatasingle-sideband modu latedsignalresulting fromthemessagesignalm(t)andcarriercos(2'1Tf,t) ofunitamplitude isgivenby(seeTable2.1) s(t)=Im(t)cos(2'1TM) ::'::im(t)sin(2'1T.fct) wheretheminussigncorresponds tothetransmission oftheuppersideband andtheplus signcorresponds tothetransmission ofthelowersiCleband. 2.17Thelocaloscillator usedforthedemodulation ofanSSBsignals(t)hasafrequency error lifmeasured withrespecttothecarrierfrequency feusedtogenerate s(t).Otherwise, thereisperfectsynchronism betweenthisoscillator inthereceiverandtheoscillator sup plyingthecarrierwaveinthetransmitter. Evaluate thedemodulated signalforthefol lowingtwosituations: (a)TheSSBsignals(t)consistsoftheuppersideband only. (b)TheSSBsignals(t)consistsofthelowersideband only. 2.18FigureP2.18showstheblockdiagramofWeaver's methodforgenerating SSBmodulated waves.Themessage (modulating) signalm(t)islimitedtothebandfa:5IfI:5fb'The auxiliary carrierappliedtothefirstpairofproductmodulators hasafrequency fo,which liesatthecenterofthisband,asshownby fo=fa+fb 2 Thelow-pass filtersinthein-phase andquadrature channels areidentical, eachwitha cutofffrequency equalto(1&fa)/2.Thecarrierappliedtothesecondpairofproduct modulators hasafrequency f,thatisgreaterthan(fb-fa)/2.Sketchthespectraatthe variouspointsinthemodulator ofFigureP2.18,andhenceshowthat: (a)Forthelowersideband, thecontributions ofthein-phase andquadrature channels areofopposite polarity, andbyaddingthematthemodulator output,thelower sideband issuppressed. (b)Fortheuppersideband, thecontributions ofthein-phase andquadrature channels areofthesamepolarity, andbyaddingthem,theuppersideband istransmitted. (c)Howwouldyoumodifythemodulator ofFigureP2.18sothatonlythelowerside bandistransmitted? In-phasechannel m(,)SSB wave sin(2,,!0" sin(2,,!,') Quadrature channel FIGUREP2.18 172 CHAPTER:1 IIICONTINUOUS-WAVE MODUlATION 2.19Thespectrum ofavoicesignalm(t)iszerooutsidetheinterval fa:5IfI:s/b.Toensure communication privacy,thissignalisappliedtoascrambler thatconsistsofthefollowing cascadeofcomponents: productmodulator, high-pass filter,secondproductmodulator andlow-pass filter.Thecarrierwaveappliedtothefirstproductmodulator hasafre: quencyequaltofc,whereasthatappliedtothesecondproductmodulator hasafrequency equaltofb+fe;bothofthemhaveunitamplitude. Thehigh-pass andlow-pass filters havethesamecutofffrequency atfe.Assumethatfe>Ji,. (a)Deriveanexpression forthescrambler outputs(t),andsketchitsspectrum. (b)Showthattheoriginalvoicesignalm(t)mayberecovered froms(t)byusingan unscrambler thatisidentical totheunitjustdescribed. 2.20Inthisproblem wederiveEquation (2.16)thatdefinesthefrequency response HQ(f)of thefilterthatoperates onthemessage signalm(t)toproducem'(t)forVSBmodulation. Thesignalm'(t),exceptforascalingfactor,constitutes thequadrature component of5(t). Todothederivation, weapplys(t)tothecoherent detector ofFigureP2.20soasto recoverascaledversionoftheoriginalmessagesignalm(t). (a)StartingwiththeblockdiagramofFigure2.12forthegeneration ofaVSBmodulated wave,determine theFouriertransform V(f)oftheproductmodulator outputu(t)in FigureP2.20intermsoftheFouriertransform ofthemessage signalm(t)andthe frequency response H(f)oftheband-pass filterinFigure2.12. (b)Hence,byevaluating theFouriertransform ofthelow-pass filteroutputinFigure P2.20,determine theconditions thatmustbesatisfiedbyHQ(f)intermsofH(f)to assureperfectrecovery oftheoriginalmessagesignalm(t),exceptforascalingfactor. VSBmodulated signals(t) FIGURE P:1.:10Outputsignal representing ascaled versionofthemessage signalm(t) 2.21Thesingle-tone modulating signalm(t)=Amcos(271"fmt} isusedtogenerate theVSBsignal 1 1s(t)=:2aA.,A,cos[271"(fe +fm}t]+"2A.,AA1-a)cos[271"(fc -fm)t] whereaisaconstant, lessthanunity,representing theattenuation oftheupperside frequency. (a)Findthequadrature component oftheVSBsignals(t). (b)TheVSBsignal,plusthecarrierAecos(271"f,t), ispassedthroughanenvelope detector. Determine thedistortion produced bythequadrature component. (c)Whatisthevalueofconstant aforwhichthisdistortion reachesitsworstpossible condition? 2.22Inthisproblem westudytheideaofmixinginasuperheterodyne receiver.Tobespecific, consider theblockdiagram ofthemixershowninFigureP2.22thatconsistsofaproduct modulator withalocaloscillator ofvariable frequencyft,followed byaband-pass filter. TheinputsignalisanAMwaveofbandwidth 10kHzandcarrierfrequency thatrna)'lie anywhere intherangeof0.535to1.605MHz;theseparameters aretypicalofAMradio broadcasting. Itisrequired totranslate thissignaltoafrequency bandcenteredatafixed Problems 173 intermediate frequency (IF)of0.455MHz.Findtherangeoftuningthatmustbeprovided inthelocaloscillator toachievethisrequirement. s(t) CDS(brI,l) FIGUREP2.22 2.23FigureP2.23showstheblockdiagram ofaheterodyne spectrum analyzer. Itconsistsof avariable-frequency oscillator, multiplier, band-pass filter,androotmeansquare(RMS) meter.Theoscillator hasanamplitude Aandoperates overtherangefatofa+W,where faisthemidband frequency ofthefilterandWisthesignalbandwidth. Assumethat fa=2W,thefilterbandwidth li.fissmallcompared withfaandthatthepassband mag nituderesponse ofthefilterisone.Detennine thevalueoftheRMSmeteroutputfora low-pass inputsignalg(t). Input signalge,) FIGUREP2.23Output signal AngleModulation 2.24SketchthePMandFMwavesproduced bythesawtooth waveshowninFigureP2.24. FIGUREP2.24 2.25Inafrequency-modulated radar,theinstantaneous frequency ofthetransmitted carrier isvariedasinFigureP2.25,whichisobtained byusingatriangular modulating signal. Theinstantaneous frequency ofthereceivedechosignalisshowndashedinFigureP2.25, whereTistheround-trip delaytime.Thetransmitted andreceivedechosignalsareapplied toamixer,andthedifference frequency component isretained. Assuming thatfoT«1, determine thenumberofbeatcyclesatthemixeroutput,averaged overonesecond,in 174 CHAPTER 2illCONTINUOUS-WAVE MODUlATION termsofthepeakdeviation tJ.fofthecarrierfrequency, thedelay 7",andtherepetition frequency faofthetransmitted signal. 1;(') Transmitted signal I I I FIGUREP2.25 2.26Theinstantaneous frequency ofasinewaveisequaltof,tJ.fforItiST12,andfcfor Itl>T12.Determine thespectrum ofthisfrequency-modulated wave.Hint:Divideup thetimeintervalofinterestintothreeregions:-00<t< -T12,-TI2stsT12,and TI2<t<00. 2.27Single-sideband modulation maybeviewedasahybridformofamplitude modulation andfrequency modulation. Evaluate theenvelope andinstantaneous frequency ofanSSB waveforthefollowing twocases: (a)Whenonlytheuppersideband istransmitted. (b)Whenonlythelowersideband istransmitted. 2.28Consider anarrowband FMsignalapproximately definedby s(t)=Accos(27rfct) -Mesin(27rfct) sin(27rfmt) (a)Determine theenvelope ofthismodulated signal.Whatistheratioofthemaximum totheminimum valueofthisenvelope? Plotthisratioversus(3,assuming thatf3is restricted totheinterval0Sf3S0.3. (b)Determine theaveragepowerofthenarrowband FMsignal,expressed asapercentage oftheaveragepoweroftheunmodulated carrierwave.Plotthisresultversusf3, assuming thatf3isrestricted totheinterval0Sf3S0.3. (c)Byexpanding theangletJi(t)ofthenarrow-band FMsignals(t)intheformofapower series,andrestricting themodulation indexf3toamaximum valueof0.3radians, showthat Whatisthepowerratioofthirdharmonic tofundamental component for(3=0.3? 2.29Thesinusoidal modulating wave m(t)=Amcos(27rfmt) isappliedtoaphasemodulator withphasesensitivity kp•Theunmodulated carrierwave hasfrequency f~andamplitude A,. (a)Determine thespectrum oftheresulting phase-modulated signal,assuming thatthe maximum phasedeviation f31,=kpAmdoesnotexceed0.3radians. (b)Construct aphasordiagram forthismodulated signal,andcompare itwiththat01 thecorresponding narrowband FMsignal. 2.30Suppose thatthephase-modulated signalofProblem 2.29hasanarbitrary valueforthe maximum phasedeviation f3p.Thismodulated signalisappliedtoanidealband-pass filter withmidband frequency fcandapassband extending fromfe1.5fmtofc+1.5/",. Problems 175 Determine theenvelope, phase,andinstantaneous frequency ofthemodulated signalat thefilteroutputasfunctions oftime. 2.31Acarrierwaveisfrequency-modulated usingasinusoidal signaloffrequency 1mand amplitude Am. (a)Determine thevaluesofthemodulation index{3forwhichthecarriercomponent of theFMsignalisreducedtozero.Forthiscalculation youmayusethevaluesofJo({3) giveninTableA6.5. (b)Inacertainexperiment conducted with1m=1kHzandincreasing Am(startingfromovolts),itisfoundthatthecarriercomponent oftheFMsignalisreducedtozero forthefirsttimewhenAm=2volts.Whatisthefrequency sensitivity ofthemodu lator?WhatisthevalueofAmforwhichthecarriercomponents isreducedtozero forthesecondtime? 2.32AnFMsignalwithmodulation index{3=1istransmitted throughanidealband-pass filterwithmidband frequency Ieandbandwidth 51m'whereIeisthecarrierfrequency and 1misthefrequency ofthesinusoidal modulating wave.Determine themagnitude spectrum ofthefilteroutput. 2.33Acarrierwaveoffrequency 100MHzisfrequency-modulated byasinusoidal waveof amplitude 20voltsandfrequency 100kHz.Thefrequency sensitivity ofthemodulator is25kHzpervolt. (a)Determine theapproximate bandwidth oftheFMsignal,usingCarson's rule. (b)Determine thebandwidth bytransmitting onlythosesidefrequencies whoseampli tudesexceed1percentoftheunmodulated carrieramplitude. Usetheuniversal curve ofFigure2.26forthiscalculation. (c)Repeatyourcalculations, assuming thatthe amplitude ofthemodulating signalis doubled. (d)Repeatyourcalculations, assuming thatthemodulation frequency isdoubled. 2.34Consider awideband PMsignalproduced byasinusoidal modulating wave Amcos(27rlmt), usingamodulator withaphasesensitivity equaltokpradianspervolt. (a)Showthatifthemaximum phasedeviation ofthePMsignalislargecompared with oneradian,thebandwidth ofthePMsignalvarieslinearlywiththemodulation fre quency1m. (b)Compare thischaracteristic ofawideband PMsignalwiththatofawideband FM signaL 2.35FigureP2.35showstheblockdiagramofareal-time spectrum analyzer working onthe principle offrequency modulation. Thegivensignalg(t) andafrequency-modulated signal s(t)areappliedtoamultiplier andtheoutputg(t)s(t) isfedintoafilterofimpulseresponse h(t).Thes(t)andh(t)arelinearFMsignalswhoseinstantaneous frequencies varylinearly withtimeatopposite rates,asshownby s(t)=cos(27rfct -7rkf) h(t)=cos(27rI,t+7rkf) wherekisaconstant. Showthattheenvelope ofthefilteroutputisproportional tothe magnitude spectrum oftheinputsignalg(t)withktplayingtheroleoffrequency I.Hint: Usethecomplex notations described inAppendix 2fortheanalysisofband-pass signals andband-pass filters. g(t)~outPut s(t) FIGUREP2.35 176 CHAPTER 2IIICONTINlJOlJS-WAVE MODlJLA'nON 2.36AnFMsignalwithafrequency deviation of10kHzatamodulation frequency of5kB~ isappliedtotwofrequency multipliers connected incascade.Thefirstmultiplier doubles the.fr~quency andthesecondmultiplier triplesthefreque~cy. Determine thefrequency devlaoon andthemodulatIOn mdexoftheFMsignalobtamed atthesecondmultiplier output.Whatisthefrequency separation oftheadjacent sidefrequencies ofthisFM signal? 2.37FigureP2.37showstheblockdiagram ofawideband frequency modulator usingthe indirectmethod.Thismodulator isusedtotransmit audiosignalscontaining frequencies intherangeof100Hzto15kHz.Thenarrowband phasemodulator issupplied witha carrieroffrequency f1=0.1MHzbyacrystal-controlled oscillator. Asecondcrystal. controlled oscillator suppliesasinusoidal waveoffrequency 9.5MHztothemixer.The systemspecifications areasfollows: Carrierfrequency atthetransmitter output,fe=100MHz Minimum frequency deviation, I1f=75kHz Maximum modulation indexinthephasemodulator =0.3radians (a)Calculate thefrequency multiplication ratiosn1andn2(preceding andfollowing the mixer),whichwillsatisfythesespecifications. (b)Specifythevaluesofthecarrierfrequency andfrequency deviation atthevarious pointsinthemodulator ofFigureP2.37. Baseband signal FIGlJREP2.37FM signal 2.38FigureP2.38showsthefrequency-determining networkofavoltage-eontrolled oscillator. Frequency modulation isproduced byapplying themodulating signalAmsin(21Tfmt) plus abiasVbtoapairofvaractor diodesconnected acrosstheparallelcombination ofa 200-~inductor and100-pFcapacitor. Thecapacitor ofeachvaractor diodeisrelated tothevoltageV(involts)appliedacrossitselectrodes by C100V-1/2pF Theunmodulated frequency ofoscillation is1MHz.TheVCOoutputisappliedtoa frequency multiplier toproduceanFMsignalwithacarrierfrequency of64MHzanda modulation indexof5.Determine (a)themagnitude ofthebiasvoltageVband(b)the amplitude Amofthemodulating wave,giventhatfm=10kHz. J 200 I'-H 1100 pF F1GlJREP2.38 Problems 177 2.39TheFMsignal s(t)=Accos[l'1Tfct+l'1TkfJ:miT)dT] isappliedtothesystemshowninFigurePl.39consisting ofahigh-pass RCfilterandan envelope detector. Assumethat(a)theresistance Rissmallcompared withthereactance ofthecapacitor Cforallsignificant frequency components ofsit),and(b)theenvelope detector doesnotloadthefilter.Determine theresulting signalattheenvelope detector output,assuming thatkfIm(t)I<feforallt. FMwave ,<t)REnvelope detectorOutput signal FIGUREP2.39 2.40Consider thefrequency demodulation schemeshowninFigureP2.40inwhichtheincom ingFMsignals(t)ispassedthroughadelaylinethatproduces aphase-shift of'1T/2radians atthecarrierfrequency !C.Thedelay-line outputissubtracted fromtheincoming FM signal,andtheresulting composite signalisthenenvelope-detected. Thisdemodulator findsapplication indemodulating microwave FMsignals.Assuming that s(t)=Accos[2'1Tfct+f3sin(2'1Tf",t)] analyzetheoperation ofthisdemodulator whenthemodulation indexf3islessthanunity andthedelayTproduced bythedelaylineissufficiently smalltojustifymakingthe approximations cos(2'1TfmT) =1 and FM wave s(t) FIGUREP2.40Output signal 2.41Figure1.41showstheblockdiagramofazero-crossing detectorfordemodulating anFM signaLItconsistsofalimiter,apulsegenerator forproducing ashortpulseateachzero crossingoftheinput,andalow-pass filterforextracting themodulating wave. (a)Showthattheinstantaneous frequency oftheinputFMsignalisproportional tothe numberofzerocrossings inthetimeintervalt-(T,ll)tot+(T,/2),dividedbyT,. Assumethatthemodulating signalisessentially constant duringthistimeintervaL 178 CHAPTER 2.,CONTINUOUS-WAVE MODUlATION (b)Illustrate theoperation ofthisdemodulator, usingthesawtooth waveofFigureP2.24 asthemodulating wave. FM wave FIGUREP2.41Output signa! 2.42SupposethatthereceivedsignalinanFMsystemcontains someresidualamplitude mod. ulationofpositiveamplitude a(t),asshownby s(t)=a(t)COS[217M+¢(t)] wherefeisthecarrierfrequency. Thephase¢(t)isrelatedtothemodulating signalmit) by ¢(t)217kf1:m(T)dT wherekfisaconstant. Assumethatthesignals(t)isrestricted toafrequency bandof widthBT,centered atfe'whereByisthetransmission bandwidth oftheFMsignalinthe absenceofamplitude modulation, andthattheamplitude modulation isslowlyvarying compared with¢(t).Showthattheoutputofanidealfrequency discriminator produced bys(t)isproportional toa(t)m(t). Hint:Usethecomplex notation described inAppendix 2torepresent themodulated waves(t). 2.43(a)Letthemodulated waves(t)inProblem 2.42beappliedtoahardlimiter,whose outputz(t)isdefinedby z(t)=sgn[s(t)] ={+1, -1,s(t)>0 s(t)<0 Showthatthelimiteroutputmaybeexpressed intheformofaFourierseriesas follows: 4m(-1)"z(t)= -2:--COS[217M(2n +1)+(2n+1)¢(t)] 17n~O2n+1 (b)Supposethatthelimiteroutputisappliedtoaband-pass filterwithapassband mag nituderesponse ofoneandbandwidth Bycentered aboutthecarrierfrequency In whereByisthetransmission bandwidth oftheFMsignalintheabsenceofamplitude modulation. Assuming thatfeismuchgreaterthanBy,showthattheresulting filter outputequals 4y(t)= -COS[217fet+¢It)] 17 Bycomparing thisoutputwiththeoriginalmodulated signals(t)definedinProblem 2.42,comment onthepractical usefulness oftheresult. 2.44(a)Consider anFMsignalofcarrierfrequency fe'whichisproduced byamodulating signalm(t).AssumethatfeislargeenoughtojustifytreatingthisFMsignalasa narrowband signal.Findanapproximate expression foritsHilberttransform. (b)Forthespecialcaseofasinusoidal modulating wavem(t)=A=cos(2rrf,,,t), findthe exactexpression fortheHilberttransform oftheresulting FMsignal.Forthiscase, whatistheerrorintheapproximation usedinpart(a)1 Problems 179 2.45Thesinglesideband versionofanglemodulation isdefinedby s(t)=exp[-cf,(t)]cos[2'lTIet+"'(t)] wherecf,(t)istheHilberttransform ofthephasefunction "'(t),andIeisthecarriet ftequency" (a)Showthatthespectrum ofthemodulated signals(t)contains nofrequency compo- nentsintheinterval-Ie<I<Ie>andisofinfiniteextent. ' (b)Giventhatthephasefunction "'(t)=f3sin(2'lTImt) whetef3isthemodulation indexandIrnisthemodulation frequency, detivethe corresponding expression forthemodulated waves(t). Note:ForProblems 2.44and2.45,youneedtorefertoAppendix 2foratreatment of theHilberttransform. NoiseinC\VModulation Systems 2.46ADSB-SCmodulated signalistransmitted ovetanoisychannel, withthepowerspectral densityofthenoisebeingasshowninFigureP2.46.Themessagebandwidth is4kHz andthecarrierfrequency is200kHz.Assuming thattheaveragepowerofthemodulated waveis10watts,determine theoutputsignal-to-noise ratioofthereceiver. SN(j) W/Hz --~,-------::------;~--f(kHz) FIGUREP2.46 2.47Evaluate theautocorrelation functions andcross-correlation functions ofthein-phaseand quadrature components ofthenarrowband noiseatthecoherent detector inputfor (a)theDSB-SCsystem,(b)anSSBsystemusingthelowersideband, and(c)anSSBsystem usingtheuppersideband. 2.48Inareceiverusingcoherent detection, thesinusoidal wavegenerated bythelocaloscillator suffersfromaphaseerrorott)withrespecttothecarrierwavecos(2'lTIet). Assuming that ott)isasamplefunction ofazero-mean Gaussian processofvariance ~,andthatmost ofthetimethemaximum valueofIi(t)issmallcompared withunity,findthemean-square errorofthereceiveroutputforDSB-SCmodulation. Themean-square errorisdefinedas theexpected valueofthesquareddifference betweenthereceiveroutputandthemessage signalcomponent ofthereceiveroutput. 2.49Following aprocedure similartothatdescribed inSection2.11 fortheDSB-SCreceiver, extendthisnoiseanalysistoaSSBreceiver.Specifically, evaluatethefollowing: (a)Theoutputsignal-to-noise ratio. (b)Thechannelsignal-to-noise ratio. Hence,showthatthefigureofmeritfortheSSBreceiverisexactlythesameasthatfor theDSB-SCreceiver.NotethatunliketheDSB-SCreceiver,themidband frequency ofthe spectraldensityfunction ofthenarrowband-filtered noiseatthefrontendoftheSSB 180 CHAPTER 2..CONTINUOUS-WAVE MODULATION receiverisoffsetfromthecarrierfrequency fcbyanamountequaltoW/2,whereWis themessagebandwidth. 2.50Letamessage signalm(t)betransmitted usingsingle-sideband modulation. ThePower spectraldensityofm(t)is IfI,,;W otherwise whereaandWareconstants. WhiteGaussian noiseofzeromeanandpowerspectral densityNo/2isaddedtotheSSBmodulated waveatthereceiverinput.Findanexpression fortheoutputsignal-to-noise ratioofthereceiver. 2.51Consider theoutputofanenvelope detectordefinedbyEquation (2.92),whichisrepro. ducedhereforconvenience y(t)=([Ac+A»am(t)+n,(t)f+nt(t))'12 (a)Assumethattheprobability oftheevent InQ(t)I>eAc11+kam(t)I isequaltoorlessthan8bwheree«1.Whatistheprobability thattheeffectof thequadrawre component nQ(t)isnegligible? (b)Supposethatkaisadjustedrelativetothemessagesignalm(t)suchthattheprobability oftheevent AAl+kam(t)]+n,(t)<0 isequalto8,.Whatistheprobability thattheapproximation y(t)=AAl+kam(t)]+n,(t) isvalid? (c)Comment onthesignificance oftheresultinpart(b)forthecasewhen0,and8,are bothsmallcompared withunity. 2.52Anunmodulated carrierofamplitude Acandfrequency fcandband-limited whitenoise aresummed andthenpassedthroughanidealenvelope detector. Assumethenoisespec traldensitytobeofheightNo/2andbandwidth 2W,centered aboutthecarrierfrequency fc.Determine theoutputsignal-to-noise ratioforthecasewhenthecarrier-to-noise ratio ishigh. 2.53LetRdenotetherandomvariableobrained byobserving theoutputofanenvelope de tectoratsomefixedtime.Intuitively, theenvelope detector isexpected tobeoperating wellintothethreshold regioniftheprobability thattherandomvariableRexceedsthe carrieramplitude Acis0.5.Ontheotherhand,ifthissameprobability isonly0.01,the envelope detector isexpected toberelatively freeoflossofmessage andthethreshold effect. (a)Assuming thatthenarrowband noiseatthedetectorinputiswhite,zero-mean, Gaus sianwithspectraldensityNo/2andthemessagebandwidth isW,showthattheprob abilityoftheeventR;=:Acis P(R;=:AJ=exp(-p) wherepisthecarrier-to-noise ratio: A2 p=4~o (b)Usingtheformulaforthisprobability, calculate thecarrier-to-noise ratiowhen(1)the envelope detectorisexpected tobewellintothethreshold region,and(2)itisexpecred tobeoperating satisfactorily. Problems 181 2.54Consider aphasemodulation (PM)system,withthemodulated wavedefinedby s(t)=AcCOS[27Tfct+kpm(t)] wherekpisaconstant andm(t)isthemessagesignal.Theadditivenoisen(t)atthephase detectorinputis n(t)=n,(t)COS(27Tf,t) -nQ(t)sin(27Tf,t) Assuming thatthecarrier-to-noise ratioatthedetectorinputishighcompared withunity, determine (a)theoutputsignal-to-noise ratioand(b)thefigureofmeritofthesystem. Compare yourresultswiththeFMsystemforthecaseofsinusoidal modulation. 2.55AnFDMsystemusessingle-si~eband modulation tocombine 12independent voicesignals andthenusesfrequency modulation totransmit thecomposite baseband signal.Each voicesignalhasanaveragepowerPandoccupies thefrequency band0.3to3.4kHz;the systemallocates itabandwidth of4kHz.Foreachvoicesignal,onlythelowersideband istransmitted. Thesubcarrier wavesusedforthefirststageofmodulation aredefinedby 0:5k:511 Thereceived signalconsistsofthetransmitted FMsignalpluswhiteGaussian noiseof zeromeanandpowerspectraldensityNo/2. (a)Sketchthepowerspectraldensityofthesignalproduced atthefrequency discrimi natoroutput,showing boththesignalandnoisecomponents. (b)Findtherelationship between thesubcarrier amplitudes Aksothatthemodulated voicesignalshaveequalsignal-to-noise ratios. 2.56Inthediscussion onFMthreshold effectpresented inSection2.13,wedescribed the conditions forpositive-going andnegative-going clicksintermsoftheenvelope r(t)and phaseI/J(t)ofthenarrowband noisen(t).Reformulate theseconditions intermsofthein phasecomponent nl(t)andquadrature component nQ(t)ofn(t). 2.57Byusingthepre-emphasis filtershowninFigure2.50aandwithavoicesignalasthe modulating wave,anFMtransmitter produces asignalthatisessentially frequency modulated bytheloweraudiofrequencies andphase-modulated bythehigheraudio frequencies. Explainthereasonsforthisphenomenon. 2.58Supposethatthetransferfunctions ofthepre-emphasis andde-emphasis filtersofanFM systemarescaledasfollows: and Thescalingfactorkistobechosensothattheaveragepoweroftheemphasized message signalisthesameasthatoftheoriginalmessagesignalm(t). (a)Findthevalueofkthatsatisfiesthisrequirement forthecasewhenthepowerspectral densityofthemessagesignalm(t)is elsewhere (b)Whatisthecorresponding valueoftheimprovement factorIproduced byusingthis pairofpre-emphasis andde-emphasis filters?Compare thisratiowiththatobtained inExample 2.6. The improvement factorIisdefinedbyEquation (2.160). 182 CHAPrER 2..CONTINUOUS·WAVE MODUlATION 2.59Aphasemodulation (PM)systemusesapairofpre-emphasis andde-emphasis filters definedbythetransferfunctions and 1 Hd,(f)=1+(jflfo) Showthattheimprovement inoutputsignal-to-noise ratioproduced byusingthispairof filtersis I=Wlfo tanI(Wlfo} whereWisthemessage bandwidth. Evaluate thisimprovement forthecasewhen W=15kHzandfo=2.1kHz,andcompare yourresultwiththecorresponding value foranFMsystem. Computer Experiments 2.60Inthisexperiment westudythebehavior oftheenvelope detectorshowninFigureP2.5 forthefollowing specifications: Sourceresistance, R,=75n Loadresistance, R1=10kfi Capacitance, C=O.OlJ.LF Thediodehasaresistance of25nwhenitisforward-biased andinfiniteresistance when reverse-biased. Compute thewaveform oftheenvelope detectoroutput,assuming aninputsiuu' soidalAMwavewith50percentmodulation. Themodulation frequency is1kHz,and thecarrierfrequency is20kHz. 2.61Inthisexperiment wecontinue thestudyofthephase-locked loopconsidered inSection 2.14: (a)Compute variations intheinstantaneous frequency ofthevoltage-controlled oscillator intheloopforthefollowing loopparameters: Loop-gain parameter, Ko=~~Hz 1Naturalfrequency, fn=21THz Damping factor,{=0.707 Perform thecomputations forthefollowing valuesoffrequency step:1:;.f=0.125, 7 2 0.5,12'"3Hz. (b)Fortheparameters ofthephase-locked loopasspecified inExperiment 2inSection 2.14,compute howvariations intherelativefrequency deviation 1:;.f•f"Jf~affect thepeakphaseerrorofthephase-locked loop. PULSE MODULATION Thischapter, representing thetransition fromanalogtodigitalcommunications, covers thefollowing topics: ~Sampling, whichisbasictoallformsofpulsemodulation. ~pulse-amplitude modulation, whichisthesimplestformofpulsemodulation. ~Quantization, which,whencombined withsampling, permitstherepresentation ofan analogsignalindiscreteforminbothamplitude andtime. ~pulse-code modulation, whichisthestandard methodforthetransmission ofananalog messagesignalbydigitalmeans. ~Time-division multiplexing, whichprovides forthetimesharingofacommon channelby apluralityofusersbymeansofpulsemodulation. ~Digitalmultiplexers, whichcombine manyslowbitstreamsintoasinglefasterstream. ~Otherformsofdigitalpulsemodulation, namely,deltamodulation anddifferential pulse codemodulation. ~Linearprediction, whichisbasictotheencoding ofanalogmessagesignalsatreducedbit ratesasindifferential pulse-code modulation. Ii'Adaptive formsofdifferential pulse-code modulation anddeltamodulation. ~TheMPEG-lIaudio codingstandard, whichisatransparent, perceptually lossless compression system. L3.1Introduction Incontinuous-wave (CW)modulation, whichwestudiedinChapter2,someparameter ofasinusoidal carrierwaveisvariedcontinuously inaccordance withthemessagesignal. Thisisindirectcontrasttopulsemodulation, whichwestudyinthepresentchapter.In pulsemodulation, someparameter ofapulsetrainisvariedinaccordance withthemessage signal.Wemaydistinguish twofamiliesofpulsemodulation: analogpulsemodulation anddigitalpulsemodulation. Inanalogpulsemodulation, aperiodicpulsetrainisused asthecarrierwave,andsomecharacteristic featureofeachpulse(e.g.,amplitude, duration, orposition) isvariedinacontinuous mannerinaccordance withthecorresponding sample valueofthemessagesignal.Thusinanalogpulsemodulation, information istransmitted basically inanalogform,butthetransmission takesplaceatdiscretetimes.Indigitalpulse modulation, ontheotherhand,themessagesignalisrepresented inaformthatisdiscrete inbothtimeandamplitude, therebypermitting itstransmission indigitalformasase quenceofcodedpulses;thisform ofsignaltransmission hasnoCWcounterpart. 183 184 CHAPTER 3"PULSE MODUlATION Theuseofcodedpulsesforthetransmission ofanaloginformation-bearing ~ignals represents abasicingredient intheapplication ofdigitalcommunications. Thischapter maytherefore beviewedasatransition fromanalogtodigitalcommunications inOur studyoftheprinciples ofcommunication systems. Webeginthediscussion bydescribing thesampling process,whichisbasictoallpulsemodulation systems, whether theyare analogordigital. I3.2Sampling Process Thesampling processisusuallydescribed inthetimedomain. Assuch,itisanoperation thatisbasictodigitalsignalprocessing anddigitalcommunications. Through useofthe sampling process,ananalogsignalisconverted intoacorresponding sequence ofsamples thatareusuallyspaceduniformly intime.Clearly,forsuchaprocedure tohavepractical utility,itisnecessary thatwechoosethesampling rateproperly, sothatthesequence of samplesuniquely definestheoriginalanalogsignal.Thisistheessenceofthesampling theorem, whichisderivedinwhatfollows. Consider anarbitrary signalg(t)offiniteenergy,whichisspecified foralltime.A segment ofthesignalg(t)isshowninFigure3.1a.Supposethatwesamplethesignalg(t) instantaneously andatauniformrate,onceeveryT,seconds. Consequently, weobtainan infinitesequence ofsamplesspacedT,secondsapartanddenoted by{g(nT,)}, wheren takesonallpossible integervalues.WerefertoT,asthesampling period,andtoits reciprocalf,=liT,asthesampling rate.Thisidealformofsampling iscalledinstantaneous sampling. Letga(t)denotethesignalobtained byindividually weighting theelements ofape riodicsequence ofdeltafunctions spacedT,secondsapartbythesequence ofnumbers {g(nT,)}, asshownby(seeFigure3.1h) g5(t)=2:g(nT,)ottnT,) (3.1) WerefertogB(t)astheidealsampled signal.Thetermo(t-nT,)represents adeltafunction positioned attimet=nT,.Fromthedefinition ofthedeltafunction, werecallthatsuch anidealized functionhasunitarea;seeAppendix 2.Wemaytherefore viewthemultiplying factorg(nT,)inEquation (3.1)asa"mass"assigned tothedeltafunction ott-nT,).A deltafunction weighted inthismanneriscloselyapproximated byarectangular pulseof get) (a) (b) FIGURE 3.1Thesampling process.(a)Analogsignal.(b)Instantaneously sampled versionofthe analogsignal. 3.2Sampling Process 185 duration litandamplitude g(nTsJ/,dt; thesmallerwemakeL1tthebetterwillbethe approximation. UsingthetableofFourier-transform pairs,wemaywrite(seethelastitemofTable A6.3) gs(t)~Is2:G(f-mls) (3.2) whereG(f)istheFouriertransform oftheoriginalsignalg(t),andIsisthesampling rate. Equation (3.2)statesthattheprocessofuniformly sampling acontinuous-time signalof finiteenergyresultsinaperiodicspectrum withaperiodequaltothesampling rate. Anotherusefulexpression fortheFouriertransform oftheidealsampledsignalga(t) maybeobtained bytakingtheFouriertransform ofbothsidesofEquation (3.1)andnoting thattheFouriertransform ofthedeltafunction 8(t-nTJisequaltoexp(-j27rnfT,). Let Gs(f)denotetheFouriertransform ofgs(t).Wemaytherefore write Gs(f)=2:g(nT,)exp(-j27T"nfT,) (3.3) Thisrelationiscalledthediscrete-time Fouriertransform. Itmaybeviewedasacomplex Fourierseriesrepresentation oftheperiodicfrequency function Ga(f),withthesequence ofsamples{g(nT,))definingthecoefficients oftheexpansion. Therelations, asderivedhere,applytoanycontinuous-time signalg(t)offinite energyandinfiniteduration. Suppose, however, thatthesignalg(t)isstrictlyband-limited, withnofrequency components higherthanWHertz.Thatis,theFouriertransform G(f) ofthesignalg(t)hastheproperty thatG(f)iszeroforIfI2:W,asillustrated inFigure 3.2a;theshapeofthespectrum showninthisfigureisintended forthepurposeofillus trationonly.Suppose alsothatwechoosethesampling periodT,=1/2W.Thenthe corresponding spectrum Ga(f)ofthesampled signalgs(t)isasshowninFigure3.2b. PuttingT,=1/2WinEquation (3.3)yields (3.4) FromEquation (3.2),wereadilyseethattheFouriertransform ofg.(t)mayalsobe expressed as Ga(f)=IsG(f)+Is2:G(f-mls) (3.5) m*D G(/) G8(j) G(O) I I-IV0IV -2/, -I,-IV IVI, 21, (a) (b) FIGURE3.2(a)Spectrum ofastrictlyband-limited signalg(t). (b)Spectrum ofthesampled versionofget)forasampling periodT,=I/2W. 186 CHAPTER 3,.PULSE MODUlATION Hence,underthefollowing twoconditions: 1.G(f)=0forIfI~W 2.f,=2W wefindfromEquation (3.5)that -W<f<W (3.6) Substituting Equation (3.4)into(3.6)1wemayalsowrite 1~(n) (jTrnf)G(f)=2Wn~~g2Wexp-W'-W<f<W (3.7) (3.8) (3.9)Therefore, ifthesamplevaluesg(n/2W)ofasignalg(t)arespecified foralln,thenthe Fouriertransform G(f)ofthesignalisuniquely determined byusingthediscrete-time Fouriertransform ofEquation (3.7).Becauseg(t)isrelatedtoG(f)bytheinverseFourier transform, itfollowsthatthesignalg(t)isitselfuniquely determined bythesamplevalues g(n/2W)for-00<n<00.Inotherwords,thesequence {g(n/2W)}hasalltheinformation contained ing(t). Consider nexttheproblem ofreconstructing thesignalg(t)fromthesequence of samplevalues{g(n/2W)).Substituting Equation (3.7)intheformulafortheinverseFourier transform definingg(t)intermsofG(f),weget g(t)=r~G(f)exp(j2Trft) df =fw2~n~~g(2;')exp(-j~f) exp(j2Trft) df Interchanging theorderofsummation andintegration: g(t)=n~~gU~)2~fweXP[i2Trf(t-2;')]df TheintegralterminEquation (3.8)isreadilyevaluated, yieldingthefinalresult ()~(n)sin(2TrWt -nTr)gt=LJg- n~-~2W(2TrWt-nTr) n~~g(2;')sinc(2Wt -n), -00<t<00 Equation (3.9)provides aninterpolation formulaforreconstructing theoriginalsignalg(t) fromthesequence ofsamplevalues{g(nl2W)j, withthesincfunction sinc(2Wt) playing theroleofaninterpolation function. Eachsampleismultiplied byadelayedversionof theinterpolation function, andalltheresulting waveforms areaddedtoobtaing(t). Wemaynowstatethesampling theorem forstrictlyband-limited signalsoffinite energyintwoequivalent parts,whichapplytothetransmitter andreceiverofapulse· modulation system,respectively: 1.Aband-limited signaloffiniteenergy,whichhasnofrequency components higher thanWHertz,iscompletely described byspecifying thevaluesofthesignalatinstants oftimeseparated by1/2Wseconds. 2.Aband-limited signaloffiniteenergy,whichhasnofrequency components higher thanWHertz,maybecompletely recovered fromaknowledge ofitssamplestaken attherateof2Wsamplespersecond. 3.2Sampling Process 187 G(I) --<C----'----""'--- \/ \/ \ V V \/\ /\ \ /\ /\ \ _..-..:.__----:J..<.:.__--'--J..<.:./__'...:....J..<.:./__'...:....J..<.:.__"""'"""':--__-"~_I I, (b) FIGURE3.3(aJSpectrum ofasignal.(bJSpectrum ofanundersampled versionofthesignal exhibiting thealiasingphenomenon. Thesampling rateof2Wsamplespersecond,forasignalbandwidth ofWHertz,iscalled theNyquistrate;itsreciprocal 1/2W(measured inseconds) iscalledtheNyquistinterval. Thederivation ofthesampling theorem, asdescribed herein,isbasedontheas sumption thatthesignalg(t)isstrictlybandlimited.Inpractice,however, aninformation bearingsignalisnotstrictlybandlimited,withtheresultthatsomedegreeofundersam pIingisencountered. Consequently, somealiasingisproduced bythesampling process. Aliasingreferstothephenomenon ofahigh-frequency component inthespectrum ofthe signalseemingly takingontheidentityofalowerfrequency inthespectrum ofitssampled version,asillustrated inFigure3.3.Thealiasedspectrum, shownbythesolidcurvein Figure3.3b,pertainstoan"undersampled" versionofthemessagesignalrepresented by thespectrum ofFigure3.3a. Tocombattheeffectsofaliasinginpractice, wemayusetwocorrective measures, asdescribed here: 1.Priortosampling, alow-pass anti-aliasing filterisusedtoattenuate thosehigh frequency components ofthesignalthatarenotessential totheinformation being conveyed bythesignal. 2.ThefilteredsignalissampledatarateslightlyhigherthantheNyquistrate. Theuseofasampling ratehigherthantheNyquistratealsohasthebeneficial effectof easingthedesignofthereconstruction filterusedtorecovertheoriginalsignalfromits sampled version.Consider theexample ofamessagesignalthathasbeenanti-alias (low pass)filtered,resulting inthespectrum showninFigure3Aa.Thecorresponding spectrum oftheinstantaneously sampled versionofthesignalisshowninFigure3Ab,assuming a sampling ratehigherthantheNyquistrate.According toFigure3Ab,wereadilyseethat thedesignofthereconstruction filtermaybespecified asfollows(seeFigure3Ac): ~Thereconstruction filterislow-pass withapassband extending from- WtoW, whichisitselfdetermined bytheanti-aliasing filter. ~Thefilterhasatransition bandextending (forpositivefrequencies) fromWtof,-W, whereIsisthesampling rate. 188 CHAPTER 3Il:IPULSE MODULATION G(f) --_W--"'-------J..----""l-V-- f --"I \ /\ / \ / \ I \ I '\ / '- -I,-W -fs-I,+W/"I \ I \ / \ I \ I \ I \ / '- -W 0 W (b) Magnitude/"I \ I \ / \ / \ I \ I \ / '- I,-W I,I,+W --"-cc:--L-----L----'--~--f-f,,+w -W 0 (c) FIGURE3.4(a)Anti-alias filteredspectrum ofaninformation-bearing signal.(b)Spectrum of instantaneously sampled versionofthesignal,assuming theuseofasampling rategreaterthan theNyquistrate.(c)Magnitude response ofreconstruction filter. Thefactthatthereconstruction filterhasawell-defined transition bandmeansthatitis physically realizable. I3.3Pulse-Amplitude Modulation Nowthatweunderstand the essenceofthesampling process,wearereadytoformally definepulse-amplitude modulation, whichisthesimplestandmostbasicformofanalog pulsemodulation. Inpulse-amplitude modulation (PAM),theamplitudes ofregularly spacedpulsesarevariedinproportion tothecorresponding samplevaluesofacontinuous message signal;thepulsescanbeofarectangular formorsomeotherappropriate shape. Pulse-amplitude modulation asdefinedhereissomewhat similartonaturalsampling, wherethemessagesignalismultiplied byaperiodictrainofrectangular pulses.However, innaturalsampling thetopofeachmodulated rectangular pulsevarieswiththemessage signal,whereasinPAMitismaintained flat;naturalsampling isexplored furtherinProb lem3.2. Thewaveform ofaPAMsignalisillustrated inFigure3.5.Thedashedcurveinthis figuredepictsthewaveform ofamessage signalm(t),andthesequence ofamplitude- 3.3Pulse-AmplitudeModUlatlon 189 FIGURE3.5Flat-top samples, representing ananalogsignal. modulated rectangular pulsesshownassolidlinesrepresents thecorresponding PAMsig nals(t).Therearetwooperations involved inthegeneration ofthePAMsignal: 1.Instantaneous sampling ofthemessage signalm(t)everyTsseconds, wherethesam plingrateIs=liT,ischoseninaccordance withthesampling theorem. 2.Lengthening theduration ofeachsamplesoobtained tosomeconstant valueT. Indigitalcircuittechnology, thesetwooperations arejointlyreferredtoas"sample and hold."Oneimportant reasonforintentionally lengthening theduration ofeachsampleis toavoidtheuseofanexcessive channelbandwidth, sincebandwidth isinversely propor tionaltopulseduration. However, carehastobeexercised inhowlongwemakethe sampledurationT,asthefollowing analysisreveals. Lets(t)denotethesequence offlat-toppulsesgenerated inthemannerdescribed in Figure3.5.WemayexpressthePAMsignalas s(t)=2:m(nTs)h(t -nT,) (3.10) whereTsisthesampling periodandm(nTslisthesamplevalueofm(t)obtained attime t=nTs.Theh(t)isastandard rectangular pulseofunitamplitude andduration T,defined asfollows(seeFigure3.6a): {I, h(t)=~, 0,0<t<T t=0,t=T otherwise(3.11) Bydefinition, theinstantaneously sampled versionofm(t)isgivenby m8(t)=2:m(nTs)8(t-nTsl (3.12) where8(tnTJisatime-shifted deltafunction. Therefore, convolving m8(t)withthe pulseh(t),weget m8(t)*h(t)=r~m.(T)h(t -T)dT =r~n~~m(nT,)8(T-nTslh(t-T)dT =im(nTJr8(T-nTs)h(t-T)dT n=-oc -:):>(3.13) 190 CHAPTER 3.,PULSEMODULATION h(t) oT (a) IH(f)1 T arg[H(J)] 3T (b) FIGURE 3.6(a)Rectangular pulseh(t).(b)Spectrum H(fl,madeupofthemagnitude IH(fll, andphasearg[H(fl]. Usingthesiftingproperty ofthedeltafun<.tion (seeAppendix 2),wethusobtain 00 m8(t)*h(t)=2:m(nT,)h(t -nT,) (3.14) FromEquations (3.10)and(3.14)itfollowsthatthePAMsignals(t}ismathematically equivalent totheconvolution ofm8(t),theinstantaneously sampled versionofm(t),and thepulseh(t),asshownby s(t)=m8(t)*h(t) (3.15) TakingtheFouriertransform ofbothsidesofEquation (3.15)andrecognizing that theconvolution oftwotimefunctions istransformed intothemultiplication oftheirreo spectiveFouriertransforms, weget S(f)=M8(f)H(f) (3.16) whereS(f)=F[s(t)],M8(f)=F[m8(t)], andH(f)=F[h(t)].Adapting Equation (3.2)to theproblem athand,wenotethattheFouriertransform M8(f)isrelatedtotheFourier transform M(f)oftheoriginalmessage signalm(t)asfollows: 00 M8(f)=f,2:M(f-kf,) (3.17) k~-oo wheref,isthesampling rate.Therefore, substitution ofEquation (3.17)into(3.16)yields S(f)=f,2:M(f-kf,)H(f) k~-oo(3.18) GivenaPAMsignals(t)whoseFouriertransform S(f)isasdefinedinEquation (3.18),howdowerecovertheoriginalmessage signalm(t)?Asafirststepinthisrecon' 3.4OtherFonttSoJPurseModulation 191 PAMsign.I ,(I)Message sign.1m(t) FIGURE3.7Systemforrecovering message signalmit)fromPAMsignalsit). struction, wemaypasssit)throughalow-pass filterwhosefrequency response isdefined inFigure3Ac;hereitisassumed thatthemessage islimitedtobandwidth Wandthe sampling ratef,islargerthantheNyquistrate2W.Then,fromEquation (3.18)wefind thatthespectrum of theresulting filteroutputisequaltoM(f)H(f). Thisoutputisequiv alenttopassingtheoriginalmessage signalmit)throughanotherlow-pass filteroffre quencyresponseH(f). FromEquation (3.11)wenotethattheFouriertransform oftherectangular pulse h(t)isgivenby H(f)=Tsinc(fT) exp(-j7rfT) (3.19) whichisplottedinFigure3.6b.Weseetherefore thatbyusingflat-topsamplestogenerate aPAMsignal,wehaveintroduced amplitude distortion aswellasadelayofT12.This effectisrathersimilartothevariation intransmission withfrequency thatiscausedbythe finitesizeofthescanning aperture intelevision. Accordingly, thedistortion causedbythe useofpulse-amplitude modulation totransmit ananaloginformation-bearing signalis referredtoastheapertureeffect. Thisdistortion maybecorrected byconnecting anequalizer incascadewiththelow passreconstruction filter,asshowninFigure3.7.Theequalizer hastheeffectofdecreasing thein-bandlossofthereconstruction filterasthefrequency increases insuchamanneras tocompensate fortheaperture effect.Ideally,themagnitude response oftheequalizer is givenby 1 IH(f)I1=~ Tsinc(fT) sin(7rfT)(3.20) Theamountofequalization neededinpractice isusuallysmall.Indeed,foradutycycle TIT,:s;0.1,theamplitude distortion islessthan0.5percent,inwhichcasetheneedfor equalization maybeomittedaltogether. Thetransmission ofaPAMsignalimposesratherstringent requirements onthemag nitudeandphaseresponses ofthechannel, becauseoftherelatively shortduration ofthe transmitted pulses.Furthermore, thenoiseperformance ofaPAMsystemcanneverbe betterthanbaseband-signal transmission. Accordingly, wefindthatfortransmission over longdistances, PAMwouldbeusedonlyasameansofmessage processing fortime divisionmultiplexing, fromwhichconversion tosomeotherformofpulsemodulation is subsequently made;time-division multiplexing isdiscussed inSection3.9.' L3.4OtherFormsofPulseModulation Inapulsemodulation systemwemayusetheincreased bandwidth consumed bythepulses toimprovethenoiseperformance ofthesystem.Thiscanbeachieved byrepresenting the samplevaluesofthemessagesignalsbysomeproperty ofthepulseotherthanamplitude: i>-Pulse-duration modulation (PDM),alsoreferred toaspulse-width modulation, wheresamplesofthemessagesignalareusedtovarytheduration oftheindividual pulsesinthecarrier. 192 CHAPTER 3.,PULSEMODUlATION l>Pulse-position modulation (PPM),wheretheposition ofapulserelativetoitsUn. modulated timeofoccurrence isvariedinaccordance withthemessagesignal. Thesetwootherformsofpulsemodulation areillustrated inFigure3.8forthecaseofa sinusoidal modulating wave. InPDM,longpulsesexpendconsiderable powerwhilebearingnoadditional infor. mation.Ifthisunusedpowerissubtracted fromPDMsothatonlytimetransitions are preserved, weobtainPPM.Accordingly, PPMisamoreefficientformofpulsemodulation thanPDM. SinceinaPPMsystemthetransmitted information iscontained intherelativepo sitionsofthemodulated pulses,thepresence ofadditivenoiseaffectstheperformance of suchasystembyfalsifying thetimeatwhichthemodulated pulsesarejudgedtoOCCUr. Immunity tonoisecanbeestablished bymakingthepulsebuildupsorapidlythatthe timeintervalduringwhichnoisecanexertanyperturbation isveryshort.Indeed,additive noisewouldhavenoeffectonthepulsepositions ifthereceived pulseswereperfectly rectangular, becausethepresence ofnoiseintroduces onlyverticalperturbations. However thereception ofperfectly rectangular pulseswouldrequireaninfinitechannelbandwidth' whichisofcourseimpractical. Thuswithafinitechannelbandwidth inpractice, wefind thatthereceivedpulseshaveafiniterisetime,sotheperformance ofthePPMreceiver is affectedbynoise,whichistobeexpected. (a) (b) (c) Time-g,. (d) FIGURE3.8Illustrating twodifferent formsofpulse-time modulation forthecaseofasinusoi· dalmodulating wave.(a)Modulating wave.(b)Pulsecarrier.(e)POMwave.(d)PPMwave. 3.6QuanthatitmProcess 193 AsinaCWmodulation system,thenoiseperformance ofaPPMsystemmaybe described intermsoftheoutputsignal-TO-noise ratio(SNR).Also,tofindthenoiseim provement produced byPPMoverbaseband transmission ofamessagesignal,wemayuse thefigureofmeritdefinedastheoutputsignal-to-noise ratioofthePPMsystemdivided bythechannelsignal-to-noise ratio;seeSection2.10.Assuming thattheaveragepowerof thechannelnoiseissmallcompared tothepeakpulsepower,thefigureofmeritofthe PPMsystemisproportional tothesquareofthetransmission bandwidth Bynormalized withrespecttothemessage bandwidth W.When,however, theinputsignal-to-noise ratio dropsbelowacriticalvalue,thesystemsuffersalossofthewantedmessage signalatthe receiveroutput.Thatis,aPPMsystemsuffersfromathreshold effectofitsown. L3.5Bandwidth-Noise Trade-Off Inthecontextofnoiseperformance, aPPMsystemistheoptimum formofanalogpulse modulation. ThenoiseanalysisofaPPMsystemrevealsthatpulse-position modulation (PPM)andfrequency modulation (FM)systemsexhibitasimilarnoiseperformance, as summarized here.' 1.Bothsystemshaveafigureofmeritproportional tothesquareofthetransmission bandwidth normalized withrespeL"ttothe message bandwidth. 2.Bothsystemsexhibitathreshold effectasthesignal-to-noise ratioisreduced. Thepractical implication ofpoint1isthat,intermsofatrade-off ofincreased transmission bandwidth forimproved noiseperformance, thebestthatwecandowithcontinuous-wave (CW)modulation andanalogpulsemodulation systemsistofollowasquarelaw.Aques tionthatarisesatthispointinthediscussion is:Canweproduce atrade-off betrerthana squarelaw?Theanswerisanemphatic yes,anddigitalpulsemodulation isthewaytodo it.Theuseofsuchamethodisaradicaldeparture fromCWmodulation. Specifically, inabasicformofdigitalpulsemodulation knownaspulse-code mod ulation(PCM)/ amessage signalisrepresented indiscreteforminbothtimeandampli tude.Thisformofsignalrepresentation permitsthetransmission ofthemessage signalas asequence ofcodedbinarypulses.Givensuchasequence, theeffectofchannelnoiseat thereceiveroutputcanbereducedtoanegligible levelsimplybymakingtheaveragepower ofthetransmitted binaryPCMwavelargeenoughcompared totheaveragepowerofthe noise. Twofundamental processes areinvolved inthegeneration ofabinaryPCMwave: sampling andquantization. Thesampling processtakescareofthediscrete-time represen tationofthemessage signal;foritsproperapplication, wehavetofollowthesampling theorem described inSection3.2.Thequantization processtakescareofthediscrete amplitude representation ofthemessage signal;quantization isanewprocess,thedetails ofwhicharedescribed inthenextsection.Fornowitsufficestosaythatthecombined useofsampling andquantization permitsthetransmission ofamessage signalincoded form.This,inturn,makesitpossibletorealizeanexponential lawforthebandwidth noisetrade-off, whichisalsodemonstrated inthenextseL"tion. L3.6Quantization Process3 Acontinuous signal,suchasvoice,hasacontinuous rangeofamplitudes andtherefore its sampleshaveacontinuous amplitude range.Inotherwords,withinthefiniteamplitude 194 CHAPTER 3!llPULSE MODUlATION Continuous sample mDiscrete samplev ~ W FIGURE3.9Description ofamemoryless quantizer. rangeofthesignal,wefindaninfinitenumberofamplitude levels.Itisnotnecessary in facttotransmittheexactamplitudes ofthesamples. Anyhumansense(theearortheeye), asultimatereceiver,candetectonlyfiniteintensity differences. Thismeansthattheoriginal continuous signalmaybeapproximated byasignalconstructed ofdiscreteamplitudes selectedonaminimum errorbasisfromanavailable set.Theexistence ofafinitenumber ofdiscreteamplitude levelsisabasiccondition ofpulse-code modulation. Clearly,ifWe assignthediscreteamplitude levelswithsufficiently closespacing,wemaymaketheap proximated signalpractically indistinguishable fromtheoriginalcontinuous signaL Amplitude quantization isdefinedastheprocessoftransforming thesampleampli. tudem(nT,)ofamessagesignalm(t)attimet=nT,intoadiscreteamplitude v(nT,)taken fromafinitesetofpossible amplitudes. Weassumethatthequantization process is memoryless andinstantaneous, whichmeansthatthetransformation attimet=nT,is notaffected byearlierorlatersamplesofthemessagesignal.Thissimpleformofscalar quantization, thoughnotoptimum, iscommonly usedinpractice. Whendealingwithamemoryless quantizer, wemaysimplifythenotation bydrop pingthetimeindex.Wemaythususethesymbolminplaceofm(nT,),asindicated inthe blockdiagram ofaquantizer showninFigure3.9a.Then,asshowninFigure.3.9b,the signalamplitude misspecified bytheindexkifitliesinsidethepartition cell k=1,2,...,L (3.21) whereListhetotalnumberofamplitude levelsusedinthequantizer. Thediscreteampli tudesmbk=1,2,...,L,atthequantizer inputarecalleddecision levelsordecision thresholds. Atthequantizer output,theindexkistransformed intoanamplitude Vkthat represents allamplitudes ofthecellffik;thediscreteamplitudes Vk,k=1,2,...,L,are calledrepresentation levelsorreconstruction levels,andthespacingbetweentwoadjacent representation levelsiscalledaquantum orstep-size. Thus,thequantizer outputvequals Vkiftheinputsignalsamplembelongstotheinterval ffik.Themapping (seeFigure3.9a) v=g(m) (3.22) isthequantizer characteristic, whichisastaircase function bydefinition. Quantizers canbeofauniformornonuniform type.Inauniform quantizer, the representation levelsareuniformly spaced;otherwise, thequantizer isnonuniform. Inthis section,weconsider onlyuniform quantizers; nonuniform quantizers areconsidered in Section3.7.Thequantizer characteristic canalsobeofmidtread ormidrisetype.Figure 3.10ashowstheinput-output characteristic ofauniformquantizer ofthemidtread type, whichissocalledbecausetheoriginliesinthemiddleofatreadofthestaircaselike graph. Figure3.10bshowsthecorresponding input-output characteristic ofauniformquantizer ofthemidrisetype,inwhichtheoriginliesinthemiddleofarisingpartofthestaircaseiike graph.Notethatboththemidtread andmidrisetypesofuniformquantizers illustrated in Figure3.10aresymmetric abouttheorigin. Output level 4 --'----'---.-of-'--'---L- Input4level -4 (0)3.6Q....nti%mitmProcess 195 Output level 4 -'----'----+---'-----'- 11~~~i -4 (b) FIGURE3.10Twotypesofquantization: (a)midtread and(b)midrise. II!QUANTIZATION NOISE Theuseofquantization introduces anerrordefinedasthedifference betweentheinput signalmandtheoutputsignalv.Theerroriscalledquantization noise.Figure3.11illus tratesatypicalvariation ofthequantization noiseasafunction oftime,assuming theuse ofauniformquantizer ofthemidtread type. Letthequantizer inputmbethesamplevalueofazero-mean randomvariableM. (Iftheinputhasanonzeromean,wecanalwaysremoveitbysubtracting themeanfrom theinputandthenaddingitbackafterquantization.) Aquantizer g(.)mapstheinput 1Inputwave 2Quantized output Time~ FIGURE3.11Illustration ofthequantization process. (Adapted fromBennett, 1948,with permission ofAT&T.) 196 CHAPTER 3IIIPULSEMODUlAUON randomvariable Mofcontinuous amplitude intoadiscreterandomvariable V;their respective samplevaluesmandvarerelatedbyEquation (3.22).Letthequantization error bedenotedbytherandomvariableQofsamplevalueq.Wemaythuswrite or,correspondingly,q=m-v (3.23) Q=M-V (3.24) WiththeinputMhavingzeromean,andthequantizer assumed tobesymmetric asin Figure3.10,itfollowsthatthequantizer outputVandtherefore thequantization errOr Q,willalsohavezeromean.Thusforapartialstatistical characterization ofthequantizer intermsofoutputsignal-to-(quantization) noiseratio,weneedonlyfindthemean-square valueofthequantization errorQ. Consider thenaninputmofcontinuous amplitude intherange(-mmax, mmax). Assuming auniformquantizer ofthemidrisetypeillustrated inFigure3.1Db,wefindthat thestep-sizeofthequantizer isgivenby (3.25) whereListhetotalnumberofrepresentation levels.Forauniformquantizer, thequan tizationerrorQwillhaveitssamplevaluesbounded by-M2$q$M2.Ifthestep-size 8issufficiently small(i.e.,thenumberofrepresentation levelsLissufficiently large),itis reasonable toassumethatthequantization errorQisauniformly distributed random variable, andtheinterfering effectofthequantization noiseonthequantizer inputissimilar tothatofthermalnoise.Wemaythusexpresstheprobability densityfunction ofthe quantization errorQasfollows: 8 8-2"<q$2" otherwise(3.26) Forthistobetrue,however, wemustensurethattheincoming signaldoesnotoverload thequantizer. Then,withthemeanofthequantization errorbeingzero,itsvarianceut isthesameasthemean-square value: Substituting Equation (3.26)into(3.27),weget 1J!>J207=- q2dq Q8-!>J2 ~ 12(3.27) (3.28) Typically, theL-arynumberk,denoting thekthrepresentation levelofthequanti.zet, istransmitted tothereceiverinbinaryform.LetRdenotethenumberofbitspersample usedintheconstruction ofthebinarycode.Wemaythenwrite L=~ ~~ 3.6Quantizatitm Process 197 or,equivalently, R=log2L Hence,substituting Equation (3.29)into(3.25),wegetthestepsize ThustheuseofEquation (3.31)in(3.28)yields(3.30) (3.31) (3.32) LetPdenotetheaveragepowerofthemessagesignalm(t).Wemaythenexpresstheoutput signal-to-noise ratioofauniformquantizer as (3.33) Equation (3.33)showsthattheoutputsignal-to-noise ratioofthequantizer increases ex ponentially withincreasing nwnberofbitspersample,R.Recognizing thatanincreasein Rrequires aproportionate increase inthechannel(transmission) bandwidth BT,wethus seethattheuseofabinarycodefortherepresentation ofamessage signal(asinpulse codemodulation) provides amoreefficientmethodthaneitherfrequency modulation (FM) orpulse-position modulation (PPM)forthetrade-off ofincreased channelbandwidth for improved noiseperformance. Inmakingthisstatement, wepresume thattheFMandPPM systemsarelimitedbyreceivernoise,whereas thebinary-coded modulation systemislim itedbyquantization noise.WehavemoretosayonthelatterissueinSection3.8. ~ExAMPLE 3.1Sinusoidal Modulating Signal Considerthespecialcaseofafull-loadsinusoidal modulating signalofamplitude Am'which utilizesalltherepresentation levelsprovided. Theaveragesignalpoweris(assuming aload of1ohm) p=A~ 2 Thetotalrangeofthequantizer inputis2Am,becausethemodulating signalswingsbetween -AmandAm-Wemaytherefore setm=x=Am'inwhichcasetheuseofEquation (3.32) yieldstheaveragepower(variance) ofthequantization noiseas a1=tMn2-2R Thustheoutputsignal-to-noise ratioofauniformquantizer, forafull-loadtesttone,is A~2 32R(SNR)o =A~2-2R/3 =2:(2) Expressing thesignal-to-noise ratioindecibels,weget 10loglO(SNR)o =1.8+6R(3.34) (3.35) 198 CHAPTER. 3"PULSE MODULATION TABLE3.1Signal-to-(quantization) noiseratio forvaryingnumberofrepresentation levels forsinusoidal modulation NumberofRepresentation Levels,L 32 64 128 256Numberof Bitsper Sample,R 5 6 7 8Signal-to- Noise Ratio(dB) 31.8 37.8 43.8 49.8 ForvariousvaluesofLandR,thecorresponding valuesofsignal-to-noise ratioareasgiven inTable3.1.FromTable3.1wecanmakeaquickestimate ofrhenumberofbirspersample required foradesiredoutputsignal-to-noise ratio,assuming sinusoidal modulation. .... Thusfarinthissectionwehavefocusedonhowtocharacterize memoryless scalar quantizers andassesstheirperformance. Insodoing,however, weavoidedtheoptimum designofquantizers, thatis,theissueofselecting therepresentation levelsandpartition cellssoastominimize theaveragequantization powerforaprescribed numberofrepre· sentation levels.Unfortunately, thisoptimization problem doesnotlenditselftoaclosed· formsolutionbecauseofthehighlynonlinear natureofthequantization process.Rather, wehaveeffectivealgorithms forfindingtheoptimum designinaniterativemanner.Awell· knownalgorithm thatdeserves tobementioned inthiscontextistheLloyd-Max quantizer, whichisdiscussed next. IIICONDITIONS FOROPTIMALITY OFSCALAR QUANTIZERS Indesigning ascalarquantizer thechallenge ishowtoselecttherepresentation levelsand surrounding partition cellssoastominimize theaveragequantization powerforafixed numberofrepresentation levels. Tostatetheproblem inmathematical terms,consider amessagesignalm(t)drawn fromastationary processM(t).Let-A:5,m:5,Adenotethedynamic rangeofm(t),which ispartitioned intoasetofLcells,asdepicted inFigure3.12.Theboundaries ofthe partition cellsaredefinedbyasetofrealnumbers mllm2'•••,mL+lthat'satisfy the fonowing threeconditions: m,=-A mL+l=A mk:5,mk+1fork1,2,...,L Thekthpartition cellisdefinedby :Jk:mk<m:5,mk+lfork=1,2,..., L ~I~__,-- I ml=-Am2 m3 mL-1 mL mL +1=+A /- 2A-------4'1(3.36) FIGURE3.12Illustrating thepartitioning ofthedynamic range-Asm:5,Aofamessage signalm(t)intoasetofLcells. 3.6Quanti::;atw..Process 199 Lettherepresentation levels(i.e.,ql\antization values)bedenotedbyVbk=1,2,...,L. Then,assuming thatd(m,Vk)denotesadistortion measure forusingVktorepresent all thosevaluesoftheinputmthatlieinsidethepartition cell!Jbthegoalistofindthetwo sets,{Vk}t~,and{!Jk}t~"thatminimize theaveragedistortion D=kt,LE§kd(m,vk)fM(m) dm (3.37) wherefM(m)istheprobability densityfunction oftherandomvariableMwithsample valuem. Acommonly useddistortion measure is (3.38) inwhichcasewespeakofthemean-square distortion. Inanyevent,theoptimization problem statedhereinisnonlinear, defyinganexplicit,closed-form solution. Togetaround thisdifficulty, weresorttoanalgorithmic approach forsolvingtheproblem inaniterative manner. Structurally speaking, thequantizer consistsoftwocomponents withinterrelated designparameters: I>Anencodercharacterized bythesetofpartition cells (!Jk}t~,;itislocatedinthe transmitter. l>Adecodercharacterized bythesetofrepresentation levels (Vk}t~,;itislocatedinthe receiver. Accordingly, wemayidentifytwocritically important conditions thatprovidethemath ematical basisforallalgorithmic solutions totheoptimum quantization problem. One condition assumes thatwearegivenadecoderandtheproblem istofindtheoptimum encoderinthetransmitter. Theothercondition assumesthatwearegivenanencoderand theproblem istofindtheoptimum decoderinthereceiver. Henceforth, thesetwocondi tionsarereferredtoascondition Iandcondition IT,respectively. Condition 1. oftheEncoder foraGivenDecoder Theavailability ofadecodermeansthatwehaveacertaincodebook inmind.Letthe codebookbedefinedby '€:(Vk}t~, (3.39) Giventhecodebook'€,theproblem istofindthesetofpartition cells {!JkJt~,thatmini mizestheaveragedistortion D.Thatis,wewishtofindtheencoderdefinedbythenon linearmapping k=1,2,...,L (3.40) suchthatwehave D=fAd(m,g(m))fM{m) dM;::kt,L",[~~~d(m,Vk)]fM(m) dm(3.41) Forthelowerboundspecified inEquation (3.41)tobeattained, werequirethatthenon linearmapping ofEquation (3.40)besatisfiedonlyifthecondition d(m,Vk):;;dim,Vi)holdsforallji=k (3.42) (3.43)200 CHAPTIlR;} illPULSEMODULATION Thenecessary condition described inEquation (3.42)foroptimality oftheencoderfora specified codebook ~isrecognized asthenearestneighbor condition. Inwords,thenearest neighbor condition requiresthatthepartition celljkshouldembodyallthosevaluesof theinputmthatareclosertoVkthananyotherelementofthecodebook C.Thisoptimality condition isindeedintuitively satisfying. Condition II.Optimality oftheDecoder foraGivenEncoder Consider nextthereversesituation tothatdescribed undercondition I,whichmaybe statedasfollows:Optimize thecodebook ~={Vk}t~lforthedecoder, giventhattheset ofpartition cells{jk}t~lcharacterizing theencoderisfixed.Thecriterion foroptimization istheaverage(mean-square) distortion: D=±r(m-Vk)2fM(m) dm k=lJmE.'Jk Theprobability densityfunction fM(m)isclearlyindependent ofthecodebook ~.Hence, differentiating Dwithrespecttotherepresentation levelVbwereadilyobtain (3.44) Setting aDlaVkequaltozeroandthensolvingforVbweobtaintheoptimum value (3.45) Thedenominator inEquation (3.45)isjusttheprobability, Pk,thattherandomvariable Mwithsamplevaluemliesinthepartition celljk,asshownby Pk=P(mk<M$mk+1) =J.fM(m)dm fflE:Jk(3.46) Accordingly, wemayinterpret theoptimality condition ofEquation (3.45)aschoosing therepresentation levelVktoequaltheconditional meanoftherandomvariableM,given thatMliesinthepartition celljk'Wecanthusformally statethecondition foroptimality ofthedecoderforagivenencoderasfollows: (3.47) whereEistheexpectation operator. Equation (3.47)isalsointuitively satisfying. Notethatthenearestneighbor condition (condition I)foroptimality oftheencoder foragivendecoderwasprovedforagenericaveragedistortion. However, theconditional meanrequirement (condition II)foroptimality ofthedecoderforagivenencoder was provedforthespecialcaseofamean-square distortion. Inanyevent,thesetwoconditions arenecessary foroptimality ofascalarquantizer. Basically, thealgorithm fordesigning thequantizer consistsofalternately optimizing theencoderinaccordance withcondition I,thenoptimizing thedecoder inaccordance withcondition II,andcontinuing intbis 3.7Pulse-Code Modulation 201 manneruntiltheaveragedistortion Dreachesaminimum. Anoptimum quantizer designed inthismanneriscalledaLloyd-Max quantizer.4 Pulse-Code Modulation Withthesampling andquantization processes atourdisposal, wearenowreadytode scribepulse-code modulation, which,asmentioned previously, isthemostbasicformof digitalpulsemodulation. Inpulse-code modulation {PCMj,amessagesignalisrepresented byasequenceofcodedpulses,whichisaccomplished byrepresenting thesignalindiscrete forminbothtimeandamplitude. Thebasicoperations performed inthetransmitter ofa PCMsystemaresampling, quantizing, andencoding, asshowninFigure3.13a;thelow passfilterpriortosampling isincluded topreventaliasingofthemessage signal.The quantizing andencoding operations areusuallyperformed inthesamecircuit,whichis calledananalog-to-digital converter. Thebasicoperations inthereceiverareregeneration ofimpaired signals,decoding, andreconstruaion ofthetrainofquantized samples, as showninFigure3.13c.Regeneration alsooccursatintermediate pointsalongthetrans missionpathasnecessary, asindicated inFigure3.13b.Whentime-division multiplexing isused,itbecomes necessary tosynchronize thereceivertothetransmitter fortheoverall systemtooperatesatisfactorily, asdiscussed inSection3.9.Inwhatfollows,wedescribe thevariousoperations thatconstitute abasicPCMsystem. SAMPLING Theincoming messagesignalissampled withatrainofnarrowrectangular pulsessoas tocloselyapproximate theinstantaneous sampling process.Toensureperfectreconstruc tionofthemessagesignalatthereceiver, thesampling ratemustbegreaterthantwicethe highestfrequency component Wofthemessage signalinaccordance withthesampling theorem. Inpractice, alow-pass anti-aliasing filterisusedatthefrontendofthesampler toexcludefrequencies greaterthanWbeforesampling. Thustheapplication ofsampling Sourceof continuous timemessage signal (a)TransmitterPCMsignal appliedto channelinput Distorted PCM signalproduced atchanneloutputRegenerated PCMsignal appliedtothe receiver (b)Transmission path Final channel output (c)Receiver FIGVRE 3.13Thebasicelements ofapeMsystem. 202 CHAPTER 3..PVLSE MODVLATION permitsthereduction ofthecontinuously varyingmessagesignal(ofsomefiniteduration) toalimitednumberofdiscretevaluespersecond. !iiiQUANTIZATION Thesampled versionofthemessage signalisthenquantized, therebyproviding anew representation ofthesignalthatisdiscreteinbothtimeandamplitude. Thequantization processmayfollowauniform lawasdescribed inSection3.6.Intelephonic communica_ tion,however, itispreferable touseavariableseparation betweentherepresentation levels. Forexample, therangeofvoltagescoveredbyvoicesignals,fromthepeaksofloudtalk totheweakpassages ofweaktalk,isontheorderof1000to1.Byusinganonuniform quantizer withthefeaturethatthestep-size increases astheseparation fromtheoriginof theinput-output amplitude characteristic isincreased, thelargeendstepsofthequantizer cantakecareofpossibleexcursions ofthevoicesignalintothelargeamplitude ranges thatoccurrelatively infrequently. Inotherwords,theweakpassages, whichneedmore protection, arefavoredattheexpenseoftheloudpassages. Inthisway,anearlyuniform percentage precision isachieved throughout thegreaterpartoftheamplitude rangeofthe inputsignal,withtheresultthatfewerstepsareneededthanwouldbethecaseifauniform quantizer wereused. Theuseofanonuniform quantizer isequivalent topassingthebaseband signal throughacompressor andthenapplying thecompressed signaltoauniformquantizer. A particular formofcompression lawthatisusedinpracticeistheso-called fJ.,-law,'which isdefinedby Ivl=log!l+fJ.,lml) log(l+fJ.,)(3.48) wheremandvarethenormalized inputandoutputvoltages, andfJ.,isapositiveconstant. InFigure3.14a,wehaveplottedtheJL-lawforthreedifferent valuesoffJ.,.Thecaseof uniform quantization corresponds tofJ.,=O.ForagivenvalueoffJ."thereciprocal slope 1.0 0.2 0.4 0.6 0.8 Normalized input,Im\LO0 0.2 0.4 0.6 0.8 Normalized input,1m!o-"" ~.t0.6 ]!0.4 z0.8LO,--..,---,--------,---,---=_ (al (bl FIGURE3.14Compression laws.(a)wlaw.(b)A-law. 3.7Pulse-Code Modulation 203 (3.49)ofthecompression curve,whichdefinesthequantum steps,isgivenbythederivative of ImIwithrespecttoIvI;thatis, dlml=10g(1+JL)(1II)dlvl JL +JLm (3.50)1 O$;Iml$;-A 1-:s;Iml:S;1AWeseetherefore thattheJL-lawisneitherstrictlylinearnotstrictlylogarithmic, butitis approximately linearatlowinputlevelscorresponding toJLImI«1,andapproximately logarithmic athighinputlevelscorresponding toJLIml»1. Another compression lawthatisusedinpractice istheso-calledA-lawdefinedby {I~I~~A'Ivl--1+10g(Alml) 1+logA (3.51)1 O$;Iml$;A 1-:s;Iml:s;1AwhichisplottedinFigure3.14bforvaryingA.Thecaseofuniform quantization corre spondstoA=1.Thereciprocal slopeofthissecondcompression curveisgivenby thederivative ofImIwithrespecttoIvI,asshownby(depending onthevalueassigned tothenormalized inputImI) {I+logA dimIA' dlvl=(1+A)lml, Torestorethesignalsamplestotheircorrectrelativelevel,wemust,ofcourse,use adeviceinthereceiverwithacharacteristic complementary tothecompressor. Sucha deviceiscalledanexpander. Ideally,thecompression andexpansion lawsareexactly inversesothat,exceptfortheeffectofquantization, theexpander outputisequaltothe compressor input.Thecombination ofacompressor andanexpander iscalleda compander. ForboththeJL-lawandA-law,thedynamic rangecapability ofthecompander im proveswithincreasing JLandA,respectively. TheSNRforlow-level signalsincreases at theexpense oftheSNRforhigh-level signals.Toaccommodate thesetwoconflicting requirements (i.e.,areasonable SNRforbothlow-andhigh-level signals), acompromise isusuallymadeinchoosing thevalueofparamenter JLfortheJL-lawandparameter Afor theA-law.Thetypicalvaluesusedinpracticeare:JL=255andA=87.6. ItisalsoofinteresttonotethatinactualpeMsystems, thecompanding circuitry doesnotproduce anexactreplicaofthenonlinear compression curvesshowninFigure 3.14.Rather,itprovides apiecewise linearapproximation tothedesiredcurve.Byusing alargeenoughnumberoflinearsegments, theapproximation canapproach thetruecom pression curveveryclosely.Thisformofapproximation isillustrated inExample 3.2. ENCODING Incombining theprocesses ofsampling andquantization, thespecification ofacontinuous message (baseband) signalbecomes limitedtoadiscretesetofvalues,butnotintheform bestsuitedtotransmission overatelephone lineorradiopath.Toexploittheadvantages ofsampling andquantizing forthepurposeofmakingthetransmitted signalmorerobust tonoise,interference andotherchannelimpairments, werequiretheuseofanencoding 204 CHAPTER 3"PULSE MODULATION TABLE3.2Binarynumbersystem forR=4bits/sample OrdinalNumberofLevelNumberExpressed asBinary Representation Level SumofPowersof2 Number 0 0000 1 2° 0001 2 2' 0010 3 2'+2° 0011 4 220100 5 2'+2° 0101 6 22+2' 0110 7 22+2'+2° 0111 8 231000 9 23+2° 1001 10 23+2' 1010 11 23+2'+2° 1011 12 23+221100 13 23+22+2° 1101 14 23+22+2' 1110 15 23+22+2'+2° 1111 processtotranslate thediscretesetofsamplevaluestoamoreappropriate formofsignal. Anyplanforrepresenting each ofthisdiscretesetofvaluesasaparticular arrangement of discreteeventsiscalledacode.Oneofthediscreteeventsinacodeiscalledacodeelement orsymbol. Forexample, thepresence orabsenceofapulseisasymbol.Aparticular arrangement ofsymbols usedinacodetorepresent asinglevalueofthediscretesetis calledacodewordorcharacter. Inabinarycode,eachsymbolmaybeeitheroftwodistinctvaluesorkinds,suchas thepresence orabsenceofapulse.Thetwosymbols ofabinarycodearecustomarily denotedas0and1.Ina'ternarycode,eachsymbolmaybeoneofthreedistinctvaluesor kinds,andsoonforothercodes.However, themaximum advantage overtheeffectsof noiseinatransmission medium isobtained byusingabinarycode,becauseabinary symbolwithstands arelatively highlevelofnoiseandiseasytoregenerate. Supposethat, inabinarycode,eachcodewordconsistsofRbits:bitisanacronym forbinarydigit; thusRdenotesthenumberofbitspersample.Then,usingsuchacode,wemayrepresent atotalof2Rdistinctnumbers. Forexample, asamplequantized intooneof256levels mayberepresented byan8-bitcodeword. Thereareseveralwaysofestablishing aone-to-one correspondence between repre· sentation levelsandcodewords.Aconvenient methodistoexpresstheordinalnumber oftherepresentation levelasabinarynumber. Inthebinarynumbersystem,eachdigit hasaplace-value thatisapowerof2,asillustrated inTable3.2forthecaseoffourbits persample(i.e.,R=4). LineCodes Anyofseverallinecodescanbeusedfortheelectrical representation ofabinary datastream.Figure3.15displaysthewaveforms offiveimportant linecodesfortheex' ampledatastream01101001. Figure3.16displaystheirindividual powerspectra(for 3.7Pulse-Code ilIad.ration 205 Binarydata0 o o (a) A Ol--I----II--+-+-------f--- -A (b) A of"---'--J.....L--'- __.L....J'-- ....L--'-__ (c) (d)-AoI-....L--l-r--T--L..L------,---,--A A OH--+--+-+------II--+--+-++-----I-- -A Time~ (e) FIGURE3.15Linecodesfortheelectrical representations ofbinarydata.(a)Unipolar NRZ signaling. (b)PolarNRZsignaling. (c)Unipolar RZsignaling. (d)BipolarRZsignaling. (e)Split-phase orManchester code. positivefrequencies) forrandomly generated binarydata,assuming that(1)symbols0and 1areequiprobable, (2)theaveragepowerisnormalized tounity,and(3)thefrequencyf isnormalized withrespecttothebitrate11Tb•(Fortheformulas usedtoplotthepower spectraofFigure3.16,thereaderisreferredtoProblem3.11.)Thefivelinecodesillustrated inFigure3.15aredescribed here: 1.Unipolar nonreturn-to-zero (NRZ)signaling Inthislinecode,symbol1isrepresented bytransmitting apulseofamplitude Aforthe duration ofthesymbol,andsymbol0isrepresented byswitching offthepulse,asinFigure 3.15a.Thislinecodeisalsoreferredtoason-offsignaling. Disadvantages ofon-offsig nalingarethewasteofpowerduetothetransmitted DClevelandthefactthatthepower spectrum ofthetransmitted signaldoesnotapproach zeroatzerofrequency. 2.Polarnonreturn-to-zero (NRZ)signaling Inthissecondlinecode,symbols 1and0arerepresented bytransmitting pulsesofampli tudes+Aand-A,respectively, asillustrated inFigure3.15b.Thislinecodeisrelatively easytogenerate butitsdisadvantage isthatthepowerspectrum of thesignalislargenear zerofrequency. 206 CHAPTER 3IIIPULSE MODUIATION Deltafunction ofweight112 Normalized frequency (alNormalized frequency (bJ Deltafunction ofweight1 Deltafunction ofweight0.1 Normalized frequency (cJ "".~ -80.5.~ ~ I IJl0.5 I Normalized frequency (d) Normalized frequency (d FiGURE3.16Powerspectraofnnecodes:(a)Unipolar NRZsignal.(b)PolarNRZsignal. (c)Unipolar RZsignal.(d)BipolarRZsigrml.(e)Manchester-encoded signal.Thefrequency is normalized ,,~threspecttothebitrateliTh,andtheaveragepowerisnormalized tounity. 3.7Pulse-Code Modulation 207 3.Unipolar return-to-zero (RZ)signaling Inthisotherlinecode,symbol1isrepresented byarectangular pulseofamplitude Aand half-symbol width,andsymhol°isrepresented bytransmitting nopulse,asillustrated in Figure3.15c.Anattractive featureofthislinecodeisthepresence ofdeltafunctions at f=0,±1/Tbinthepowerspectrum ofthetransmitted signal,whichcanbeusedforbit timingrecovery atthereceiver. However, itsdisadvantage isthatitrequires 3dBmore powerthanpolarreturn-to-zero signaling forthesameprobability ofsymbolerror;this issueisaddressed inChapter4underProblem 4.10. 4.Bipolarreturn-t(j-zero (BRZ)signaling Thislinecodeusesthreeamplitude levelsasindicated inFigure3.15d.Specifically, positive andnegative pulsesofequalamplitude (i.e.,+Aand-A)areusedalternately forsymbol 1,witheachpulsehavingahalf-symbol width;nopulseisalwaysusedforsymbol0.A usefulproperty oftheBRZsignaling isthatthepowerspectrum ofthetransmitted signal hasnoDCcomponent andrelatively insignificant low-frequency components forthecase whensymbols 1and°occurwithequalprobability. Thislinecodeisalsocalledalternate markinversion (AMI)signaling. 5.Split-phase (Manchester code) Inthismethodofsignaling, illustrated inFigure3.15e,symbol1isrepresented byapositive pulseofamplitude Afollowed byanegativepulseofamplitude-A,withbothpulsesbeing half-symbol wide.Forsymbol0,thepolarities ofthesetwopulsesarereversed. TheMan chestercodesuppresses theDCcomponent andhasrelatively insignificant low-frequency components, regardless ofthesignalstatistics. Thisproperty isessential insome applications. Differential Encoding Thismethodisusedtoencodeinformation intermsofsignaltransitions. Inpartic ular,atransition isusedtodesignate symbol°intheincoming binarydatastream,while notransition isusedtodesignate symbol1,asillustrated inFigure3.17.InFigure3.17b weshowthedifferentially encoded datastreamfortheexample dataspecified inFigure 3.17a.TheoriginalbinarydatastreamusedhereisthesameasthatusedinFigure3.15. Thewaveform ofthedifferentially encoded dataisshowninFigure3.17c,assuming the useofunipolar nometurn-to-zero signaling. FromFigure3.17itisapparent thatadiffer entiallyencoded signalmaybeinverted withoutaffecting itsinterpretation. Theoriginal binaryinformation isrecovered simplybycomparing thepolarityofadjacent binarysym bolstoestablish whetherornotatransition hasoccurred. Notethatdifferential encoding requirestheuseofareference bitbeforeinitiating theencoding process.InFigure3.17, symbol1isusedasthereference bit. (a)Originalbinarydata (b)Differentially encodeddatao o0 0oo0 o (c)Waveform Reference bit0f-----...l------''-- ......----- Time ----.!'o- FIGURE3.17(a)Original binarydata.(b)Differentially encoded data,assuming reference bit1. (c)Waveform ofdifferentially encoded datausingunipolar NRZsignaling. 208 CHAPTER 3,.PULSE MODUlATION Regenerated PCMwaVe FIGURE 3.18Blockdiagramofregenerative repeater. I!!IREGENERATION Themostimportant featureofPCMsystemsliesintheabilitytocontroltheeffectsof distortion andnoiseproduced bytransmitting aPCMsignalthrough achannel. This capability isaccomplished byreconstructing thePCMsignalbymeansofachainobe generative repeaters locatedatsufficiently closespacingalongthetransmission route.As illustrated inFigure3.18,threebasicfunctions areperformed byaregenerative repeater: equalization, timing,anddecision making.Theequalizer shapesthereceived pulsessoas tocompensate fortheeffectsofamplitude andphasedistortions produced bythenonideal transmission characteristics ofthechannel. Thetimingcircuitry provides aperiodic pulse train,derivedfromthereceivedpulses,forsampling theequalized pulsesattheinstantsof timewherethesignal-to-noise ratioisamaximum. Eachsamplesoextracted iscompared toapredetermined threshold inthedecision-making device.Ineachbitinterval,adecision isthenmadewhetherthereceivedsymbolisa 1ora 0onthebasisofwhetherthethreshold isexceeded ornot.Ifthethreshold isexceeded, acleannewpulserepresenting symbol 1istransmitted tothenextrepeater. Otherwise, anothercleannewpulserepresenting symbol0istransmitted. Inthisway,theaccumulation ofdistortion andnoiseinarepeater spaniscompletely removed, provided thatthedisturbance isnottoolargetocausean errorinthedecision-making process. Ideally,exceptfordelay,theregenerated signalis exactlythesameasthesignaloriginally transmitted. Inpractice, however, theregenerated signaldepartsfromtheoriginalsignalfortwomainreasons: 1.Theunavoidable presence ofchannelnoiseandinterference causestherepeater to makewrongdecisions occasionally, therebyintroducing biterrorsintotheregener atedsignal. 2.Ifthespacingbetween received pulsesdeviatesfromitsassigned value,ajitteris introduced intotheregenerated pulseposition, therebycausingdistortion. JiilDECODING Thefirstoperation inthereceiveristoregenerate (i.e.,reshapeandcleanup)thereceived pulsesonelasttime.Thesecleanpulsesarethenregrouped intocodewordsanddecoded (i.e.,mappedback)intoaqnantized PAMsignal.Thedecoding processinvolvesgenerating apulsetheamplitude ofwhichisthelinearsumofallthepulsesinthecodeword,with eachpulsebeingweighted byitsplacevalue(2°,2\22,•••,2R-1)inthecode,whereRis thenumberofbitspersample. IIIFILTERING Thefinaloperation inthereceiveristorecoverthemessagesignalbypassingthedecoder outputthrough alow-pass reconstruction filterwhosecutofffrequency isequaltothe message bandwidth W.Assuming thatthetransmission pathiserrorfree,therecovered 3.8NoiseConsiderations inPCMSystems 209 signalincludes nonoisewiththeexception oftheinitialdistortion introduced bythe quantization process. ~NoiseConsiderations inPCMSystems Theperformance ofarCMsystemisinfluenced bytwomajorsourcesofnoise: 1.Channel noise,whichisintroduced anywhere between thetransmitter outputand thereceiverinput.Channel noiseisalwayspresent,oncetheequipment isswitched on. 2.Quantization noise,whichisintroduced inthetransmitter andiscarriedalltheway alongtothereceiver output.Unlikechannel noise,quantization noiseissignal dependent inthesensethatitdisappears whenthemessage signalisswitched off. Naturally, thesetwosourcesofnoiseappearsimultaneously oncetherCMsystemisin operation. However, thetraditional practiceistoconsider themseparately, sothatwemay developinsightintotheirindividual effectsonthesystemperformance. Themaineffectofchannelnoiseistointroduce biterrorsintothereceived signal. InthecaseofabinaryrCMsystem,thepresence ofabiterrorcausessymbol1tobe mistaken forsymbol0,orviceversa.Clearly,themorefrequently biterrorsoccur,the moredissimilar thereceiveroutputbecomes compared totheoriginalmessagesignal.The fidelityofinformation transmission byrCMinthepresence ofchannelnoisemaybe measured intermsoftheaverageprobability ofsymbolerror,whichisdefinedasthe probability thatthereconstructed symbolatthereceiveroutputdiffersfromthetransmit tedbinarysymbol,ontheaverage. Theaverageprobability ofsymbolerror,alsoreferred toasthebiterrorrate(BER),assumesthatallthebitsintheoriginalbinarywaveareof equalimportance. When,however, thereismoreinterestinreconstructuring theanalog waveform oftheoriginalmessagesignal,different symbolerrorsmayneedtobeweighted differently; forexample, anerrorinthemostsignificant bitinacodeword(representing aquantized sampleofthemessage signal)ismoreharmful thananerrorintheleast significant bit. Tooptimize systemperformance inthepresence ofchannelnoise,weneedtomini mizetheaverageprobability ofsymbolerror.Forthisevaluation, itiscustomary tomodel thechannelnoiseasadditive, white,andGaussian. Theeffectofchannelnoisecanbe madepractically negligible byensuring theuseofanadequate signalenergy-to-noise den sityratiothroughtheprovision ofshort-enough spacingbetweentheregenerative repeaters intherCMsystem.Insuchasituation, theperformance oftherCMsystemisessentially limitedbyquantization noiseactingalone. Fromthediscussion ofquantization noisepresented inSection3.6,werecognize that quantization noiseisessentially underthedesigner's control.Itcanbemadenegligibly smallthroughtheuseofanadequate numberofrepresentation levelsinthequantizer and theselection ofacompanding strategymatched tothecharacteristics ofthetypeofmessage signalbeingtransmitted. WethusfindthattheuseofrCMoffersthepossibility ofbuilding acommunication systemthatisruggedwithrespecttochannelnoiseonascalethatis beyondthecapability ofanyCWmodulation oranalogpulsemodulation system. iiiERROR THRESHOLD Theunderlying theoryofbiterrorratecalculation inarCMsystemisdeferred until Chapter4.Forthepresent,itsufficestosaythattheaverageprobability ofsymbolerror inabinaryencodedrCMreceiverduetoadditivewhiteGaussian noisedependssolelyon 210 CHAPTER 3IIIPULSE MODUlATION EblNo,whichisdefinedastheratioofthetransmitted signalenergyperbit,Eb,tothe noisespectral density,No.NotethattheratioEblNoisdimensionless eventhoughthe quantities EbandNohavedifferent physicalmeaning. InTable3.3wepresentasummary ofthisdependence forthecaseofabinaryPCMsystemusingpolarnonreturn-to-zero signaling. Theresultspresented inthelastcolumnofthetableassumeabitrateof105b/s. FromTable3.3itisclearthatthereisanerrorthreshold (atabout11dB).For EblNobelowtheerrorthreshold thereceiverperformance involvessignificant numbers of errors,andaboveittheeffectofchannelnoiseispractically negligible. Inotherwords provided thattheratioEblNoexceedstheerrorthreshold, channelnoisehasvirtually n~ effectonthereceiverperformance, whichisprecisely thegoalofPCM.When,however EblNodropsbelowtheerrorthreshoLd, thereisasharpincreaseintherateatwhicherror: occurinthereceiver. Becausedecisionerrorsresultintheconstruction ofincorrect code words,wefindthatwhentheerrorsarefrequent, thereconstructed messageatthereceiver outputbearslittleresemblance totheoriginalmessage. Comparing thefigureof11dBfortheerrorthreshold inaPCMsystemusingpolar NRZsignaling withthe60-70dBrequired forhigh-quality transmission ofspeechusing amplitude modulation, weseethatPCMrequiresmuchlesspower,eventhoughtheav eragenoisepowerinthePCMsystemisincreased bytheR-foldincreaseinbandwidth, whereRisthenumberofbitsinacodeword(i.e.,bitspersample). Inmosttransmission systems, theeffectsofnoiseanddistortion fromtheindividual linksaccumulate. Foragivenqualityofoveralltransmission, thelongerthephysical sep arationbetweenthetransmitter andthereceiver, themoreseverearetherequirements on eachlinkinthesystem.InaPCMsystem,however, becausethesignalcanberegenerated asoftenasnecessary, theeffectsofamplitude, phase,andnonlinear distortions inonelink (ifnottoosevere)havepractically noeffectontheregenerated inputsignaltothenext link.Wehavealsoseenthattheeffectofchannelnoisecanbemadepractically negligible byusingaratioEblNoabovethreshold. Forallpractical purposes, then,thetransmission requirements foraPCMlinkarealmostindependent ofthephysical lengthofthecom munication channel. Another important characteristic ofaPCMsystemisitsruggedness tointerference, causedbystrayimpulses orcross-talk. Thecombined presence ofchannelnoiseandin terference causestheerrorthreshold necessary forsatisfactory operation ofthePCMsys temtoincrease. 1£anadequate marginovertheerrorthreshold isprovided inthefirst place,however, thesystemcanwithstand thepresence ofrelatively largeamounts ofin terference. Inotherwords,aPCMsystemisrobusttochannelnoiseandinterference. TABLE3.3Influence ofEb/NOonthe probability oferror ForaBitRateoflOSb/s, Probability of ThisIsAboutOne EbtNo ErrorP, ErrorEvery 4.3dB 10-210-3second 8.4 10-410-1second 10.6 10-610seconds 12.0 10-820minutes 13.0 10-101day 14.0 10-123months 3.9Time-Division Multiplexing 211 ~Time-Division Multiplexing Thesampling theorem provides thebasisfortransmitting theinformationcontained ina band-limited message signalm(t)asasequence ofsamplesofm(t)takenuniformly ata ratethatisusuallyslightlyhigherthantheNyquist rate.Animportant featureofthe sampling processisaconservation oftime.Thatis,thetransmission ofthemessagesamples engagesthecommunication channelforonlyafractionofthesampling intervalona periodicbasis,andinthiswaysomeofthetimeintervalbetweenadjacentsamplesiscleared forusebyotherindependent messagesourcesonatime-shared basis.Wetherebyobtain atime-division multiplex (TDM)system,whichenablesthejointutilization ofacommon communication channelbyaplurality ofindependent message sourceswithoutmutual interference amongthem. TheconceptofTDMisiIIusttated bytheblockdiagramshowninFigure3.19.Each inputmessage signalisfirstrestricted inbandwidth byalow-pass anti-aliasing filterto removethefrequencies thatarenonessential toanadequate signalrepresentation. The low-pass filteroutputsarethenapplied toacommutator, whichisusuallyimplemented usingelecttonic switching circuitry. Thefunction ofthecommutator istwofold: (1)totake anarrowsampleofeachoftheNinputmessages atarateIsthatisslightlyhigherthan 2W,whereWisthecutofffrequency oftheanti-aliasing filter,and(2)tosequentially interleave theseNsamplesinsidethesampling intervalT,.Indeed,thislatterfunction is theessenceofthetime-division multiplexing operation. Following thecommutation pro cess,themultiplexed signalisappliedtoapulsemodulator, thepurposeofwhichisto transform themultiplexed signalintoaformsuitablefortransmission overthecommon channel. Itisclearthattheuseoftime-division multiplexing introduces abandwidth ex pansionfactorN,becausetheschememustsqueezeNsamplesderivedfromNindependent message sourcesintoatimeslotequaltoonesampling interval. Atthereceiving endof thesystem,thereceived signalisappliedtoapulsedemodulator, whichperforms the reverseoperation ofthepulsemodulator. Thenarrowsamplesproduced atthepulsede modulator outputaredisttibuted totheappropriate low-pass reconstruction filtersby meansofadecommutator, whichoperates insynchronism withthecommutator inthe transmitter. Thissynchronization isessential forasatisfactory operation ofthesystem. Thewaythissynchronization isimplemented dependsnaturally onthemethodofpulse modulation usedtotransmit themultiplexed sequence ofsamples. TheTDMsystemishighlysensitive todispersion inthecommon channel, thaIis,to variations ofamplitude withfrequency orlackofproportionality ofphasewithfrequency. Accordingly, accurate equalization ofbothmagnitude andphaseresponses ofthechannel isnecessary 10ensureasatisfactory operation ofthesystem;thisissueisdiscussed in Lowwpass M(anti-aliasing) essage filters inputs 1~~ /2~ I:r---l';'/N~ Co~;';ator ClockpulsesSynchronized ClockpulsesLow-pass (reconsruction) filters Message outputs _,~1 ~~~2 \~/~.'/ . Dec~-;'-;ator LPF ~ FIGURE3.19BlockdiagramofTDMsystem. 212 CHAPTER 3IIIPULSE MOIlUlATION Chapter 4.However, unlikeFDM,toafirst-order approximation TDMisimmune to nonlinearities inthechannelasasourceofcross-talk. Thereasonforthisbehavior isthat different message signalsarenotsimultaneously appliedtothechannel. I!!lSYNCHRONIZATION Inapplications usingPCM,itisnaturaltomultiplex different messages sourcesbytime division, whereby eachsourcekeepsitsindividuality throughout thejourneyfromthe transmitter tothereceiver. Thisindividuality accounts forthecomparative easewithwhich message sourcesmaybedropped orreinserted inatime-division multiplex system.Asthe numberofindependent message sourcesisincreased, thetimeintervalthatmaybeallotted toeachsourcehastobereduced, sinceallofthemmustbeaccommodated intoatime intervalequaltothereciprocal ofthesampling rate.This,inturn,meansthattheallowable duration ofacodewordrepresenting asinglesampleisreduced. However, pulsestendto becomemoredifficulttogenerate andtotransmit astheirduration isreduced. Further more,ifthepulsesbecometooshort,impairments inthetransmission medium beginto interfere withtheproperoperation ofthesystem.Accordingly, inpractice, itisnecessary torestrictthenumberofindependent message sourcesthatcanbeincluded withinatime divisiongroup. Inanyevent,foraPCMsystemwithtime-division multiplexing tooperatesatisfac torily,itisnecessary thatthetimingoperations atthereceiver, exceptforthetimelostin transmission andregenerative repeating, followcloselythecorresponding operations at thetransmitter. Inageneralway,thisamounts torequiring alocalclockatthereceiver tokeepthesametimeasadistantstandard clockatthetransmitter, exceptthatthelocal clockissomewhat slowerbyanamountcorresponding tothetimerequired totransport themessage signalsfromthetransmitter tothereceiver. Onepossible procedure t.osyn chronize thetransmitter andreceiverclocksistosetasideacodeelementorpulseatthe endofaframe(consisting ofacodewordderivedfromeachoftheindependent message sourcesinsuccession) andtotransmit thispulseeveryotherframeonly.Insuchacase, thereceiverincludes acircuitthatwouldsearchforthepatternof1sandOsalternating at halftheframerate,andtherebyestablish synchronization between thetransmitter and receiver. Whenthetransmission pathisinterrupted, itishighlyunlikelythattransmitter and receiverclockswillcontinue toindicatethesametimeforlong.Accordingly, incarrying outasynchronization process, wemustsetupanorderlyprocedure fordetecring the synchronizing pulse.Theprocedure consistsofobserving thecodeelements onebyone untilthesynchronizing pulseisdetected. Thatis,afterobserving aparticular codeelement longenoughtoestablish theabsenceofthesynchronizing pulse,thereceiverclockisset backbyonecodeelementandthenextcodeelementisobserved. Thissearching process isrepeated untilthesynchronizing pulseisdetected. Clearly,thetimerequired forsyn chronization depends ontheepochatwhichpropertransmission isre-established. EXAMPLE 3.2TheTlSystem Inthisexample, wedescribetheimportant characteristics ofapeMsystemknownastheT1 system,6which carries 24voicechannelsoverseparatepairsofwireswithregenerative re peatersspacedatapproximately 2-kmintervals. TheT1carriersystemisbasictotheNorth American DigitalSwitching Hierarchy described inSection3.10. . Avoicesignal(maleorfemale)isessentially limitedtoabandfrom300to3100Hz,0 thatfrequencies outsidethisbanddonotcontribute muchtoarticulation efficiency. Indeed, 3.9Time-Divisi_ Multiplexing 213 telephone circuitsthatrespond tothisrangeoffrequencies givequitesatisfactory service. Accordingly, itiscustomary topassthevoicesignalthroughalow-pass filterwithacutoff frequency ofabout3.1kHzpriortosampling. Hence,withW=3.1kHz,thenominal value oftheNyquistrateis6.2kHz.Thefilteredvoicesignalisusuallysampled ataslightlyhigher rate,namely,8kHz,whichisthestandard sampling rateintelephone systems. Forcompanding, theT1systemusesapiecewise-linear characteristic (consisting of 15linearsegments) toapproximate thelogarithmic JL-IawofEquation (3.48)withtheconstant JL=255.Thisapproximation isconstructed insu<:hawaythatthesegmentendpointslieon thecompression curvecomputed fromEquation (3.48),andtheirprojections ontothevertical axisarespaceduniformly. Table3.4givestheprojections ofthesegmentendpointsontothe horizontal axisandthestep-sizes oftheindividual segments. Thetableisnormalized to8159, sothatallvaluesarerepresented asintegernumbers. Segment 0oftheapproximation isa colinearsegment, passingthroughtheorigin;itcontains atotalof30uniformdecisionlevels. Linearsegments la,Za,...,7alieabovethehorizontal axis,whereas linearsegments 1b, 2b,...,7bliebelowthehorizontal axis;eachofthese14segments contains 16uniform decision levels.Forcolinear segment O.thedecision levelsatthequantizer inputare±1, ±3,,±31,andthecorresponding representation levelsatthequantizer outputare0, ±1,,±15.Forlinearsegments 1aand1b,thedecisionlevelsatthequantizer inputare ±31,±35, ,±95,andthecorresponding representation levelsatthequantizer outputare ±16, ±17, ,±31,andsoonfortheotherlinearsegments. Thereareatotalof31+(14X16)=255representation levelsassociated withthe IS-segment companding characteristic described above.Toaccommodate thisnumberofrep resentation levels,eachofthe24voicechannels usesabinarycodewithan8-bitword.The firstbitindicates whethertheinputvoicesampleispositiveornegative; thisbitisa 1ifpositive anda 0ifnegative. Thenextthreebitsofthecodewordidentifytheparticular segmentinside whichtheamplitude oftheinputvoicesamplelies,andthelastfourbitsidentifytheactual representation levelinsidethatsegment. Withasampling rateof8kHz,eachframeofthemultiplexed signaloccupies aperiod of125JLS.Inparticular, itconsistsoftwenty-four 8-bitwords,plusasinglebitthatisadded attheendoftheframeforthepurposeofsynchronization. Hence,eachframeconsistsofa totalof(24X8)+1=193bits.Correspondingly, theduration ofeachbitequals0.647JLS, andtheresulting transmission rateis1.544megabits persecond(Mb/s). Inaddition tothevoicesignal,atelephone systemmustalsopassspecialsupervisory signalstothefarend.Thissignaling information isneededtotransmit dialpulses,aswellas ITABLE3.4The15-segment companding characteristic (J1=255) Projections ofSegmentEndPoints LinearSegment Number Step-Size ontotheHorizontal Axis 0 2 ±31 1a,lb 4 ±95 2a,2b 8 ±223 3a,3b 16 ±479 4a,4b 32 ±991 5a,5b 64 ±2015 6a,6b 128 ±4063 7a,7b 256 ±8159 214 CHAPTER 3!!lPULSEMODULATION telephone off-hooklon-hook signals.IntheTlsystem,thisrequirement isaccomplished as follows.Everysixthframe,theleastsignificant (thatis,theeighth)bitofeachvoicechannel isdeletedandasignaling bitisinsertedinitsplace,therebyyieldinganaverage7i-bitoperation foreachvoiceinput.Thesequence ofsignaling bitsisthustransmitted atarateequalto sampling rateof8kHzdividedbysix,thatis,1.333kb/s.Thissignalingrateappliestoeach ofthe24inputchannels. ... I3.10DigitalMultiplexers InSection3.9weintroduced theideaoftime-division multiplexing whereby agroupof analogsignals(e.g.,voicesignals)aresampled sequentially intimeatacommon sampling rateandthenmultiplexed fortransmission overacommon line.Inthissectionweconsider themultiplexing ofdigitalsignalsatdifferent bitrates.Thisenablesustocombine several digitalsignals,suchascomputer outputs, digitized voicesignals,digitized facsimile, andtelevision signals,intoasingledatastream(ataconsiderably higherbitratethan anyoftheinputs).Figure3.20showsaconceptual diagram ofthedigitalmultiplexing demultiplexing operation. Themultiplexing ofdigitalsignalsisaccomplished byusingabit-by-bit interleaving procedure withaselectorswitchthatsequentially takesabitfromeachincoming lineand thenappliesittothehigh-speed common line.Atthereceiving endofthesystemtheoutput ofthiscommon lineisseparated outintoitslow-speed individual components andthen delivered totheirrespective destinations. Digitalmultiplexers arecategorized intotwomajorgroups.Onegroupofmultiplex ersisusedtotakerelatively lowbit-ratedatastreamsoriginating fromdigitalcomputers andmultiplex themforTDMtransmission overthepublicswitched telephone net work.Theimplementation ofthisfirstgroupofmultiplexers requirestheuseofmodems (modulators-demodulators), whicharediscussed inChapter 6. Thesecondgroupofdigitalmultiplexers formspartofthedatatransmission service provided bytelecommunication carrierssuchasAT&T.Inparticular, thesemultiplexers constitute adigitalhierarchy thattime-division multiplexes low-rate bitstreamsintomuch higher-rate bitstreams. Thedetailsofthebitratesthatareaccommodated inthehierarchy varyfromonecountrytoanother. However, aworldwide featureofthehierarchy isthat itstartsat64kb/s,whichcorresponds tothestandard PCMrepresentation ofavoice signal.Anincoming bitstreamatthisrate,irrespective ofitsorigin,iscalledadigitalsig/tal zero(DSO).IntheUnitedStates,Canada, andJapan7thehierarchy followstheNorth American digitalTDMhierarchy asdescribed here: Il>Thefirst-level hierarchy combines twenty-four DSObitstreamstoobtainadigital signalone(DS1)at1.544Mb/s,whichiscarriedontheTlsystemdescribed in N DatasourcesN Destinations FIGURE 3.20Conceptual diagramofmultiplexing-demultiplexing. 3.10DigiudMulfipkxes 215 Example 3.2.ThesebitstreamsarecaIledtheprimaryrateinthedigitalhierarchy, becauseitisthelowestbitratethatexistsoutsideadigitalswitch.Thedigitalswitch isadeviceconsisting ofmemoryandlogic,thefunctionofwhichismerelytheswitch ingofdigitalsignals,hencethename. II>Thesecond-level multiplexer combines fourDS1bitstreamstoobtainadigitalsignal two(DS2)at6.312Mb/s. il>Thethird-level multiplexer combines sevenDS2bitstreamstoobtainadigitalsignal three(DS3)at44.736Mb/s. II>Thefourth-level multiplexer combines sixDS3bitstreamstoobtainadigitalsignal four(DS4)at274.176 Mb/s. II>Thefifth-level multiplexer, thefinaloneinthehierarchy, combines twoDS4bit streamstoobtainadigitalsignalfive(DS5)at560.160 Mb/s. Notethatthebitrateofadigitalsignalproduced byanyone ofthesemultiplexers is slightlyhigherthantheprescribed multipleoftheincoming bitratebecauseofbitstuffing builtintothedesignofeachmultiplexer; bitstuffingisdiscussed inthesequel. Moreover, itisimportant torecognize thatthefunctions ofadigitaltransmission facilityismerelytocarryabitstreamwithoutinterpreting whatthebitsthemselves mean. However, thedigitalswitches atthetwoendsofthefacilitydohaveacommon under standing ofhowtointerpret thebitswithinthestream,suchaswhetherthebitsrepresent voiceordata,framingformat,signaling format,andsoon. Therearesomebasicproblems involved inthedesignofadigitalmultiplexer, irre spectiveofitsgrouping: 1.Digitalsignalscannotbedirectlyinterleaved intoaformatthataIlowsfortheireven tualseparation unlesstheirbitratesarelockedtoacommon clock.Rather,provision hastobemadeforsynchronization oftheincoming digitalsignals,sothattheycan beproperly interleaved. 2.Themultiplexed signalmustincludesomeformofframingsothatitsindividual components canbeidentified atthereceiver. 3.Themultiplexer hastohandlesmaIlvariations inthebitratesoftheincoming digital signals.Forexample, a1000-km coaxialcablecarrying 3X108pulsespersecond wiIlhaveaboutonemillionpulsesintransit,witheachpulseoccupying aboutone meterofthecable.A0.01percentvariation inthepropagation delay,produced by a1°Fdecrease intemperature, willresultin100fewerpulsesinthecable.Clearly, thesepulsesmustbeabsorbed bythemultiplexer. Totailortherequirements ofsynchronization andrateadjustment toaccommodate smaIl variations intheinputdatarates,wemayuseatechnique knownasbitstuffing.Theidea hereistohavetheoutgoing bitrateofthemultiplexer slightlyhigherthanthesumofthe maximum expected bitratesoftheinputchannels bystuffinginadditional non-informa tioncarryingpulses.AUincoming digitalsignalsarestuffedwithanumberofbitssufficient toraiseeachoftheirbitratestoequalthatofa10caIlygenerated clock.Toaccomplish bit stuffing, eachincoming digitalsignalorbitstreamisfedintoanelasticstoreatthemul tiplexer. Theelasticstoreisadevicethatstoresabitstreaminsuchamannerthatthe streammaybereadoutataratedifferent fromtherateatwhichitisreadin.Atthe demultiplexer, thestuffedbitsmustobviously beremoved fromthemultiplexed signal. Thisrequiresamethodthatcanbeusedtoidentifythestuffedbits.Toillustrate onesuch method, andalsoshowonemethodofproviding framesynchronization, wedescribethe signalformatoftheAT&TM12multiplexer, whichisdesigned tocombine fourDS1bit 216 CHAPTER 3!>PULSE MODUlATION streamsintooneDS2bitstream.Thisisthesecondlevelofthedigitalhierarchy discussed earlier. Il>ExAMPLE 3.3SignalFormatoftheAT&TM12Multiplexer Figure3.21 illustrates thesignalformatoftheM12multiplexer. Eachframeissubdivided intofoursubframes. Thefirstsubframe (firstlineinFigure3.21)istransmitted, thenthe second,thethird,andthefourth,inthatorder. Bit-by-bit interleaving oftheincoming fourDSIbitstreamsisusedtoaccumulate a totalof48bits,12fromeachinput.Acontrolbitistheninsertedbythemultiplexer. Each framecontainsatotalof24controlbits,separated bysequences of48databits.Threetypes ofcontrolbitsareusedintheM12multiplexer toprovidesynchronization andframeindi cation,andtoidentifywhichofthefourinputsignalshasbeenstuffed.Thesecontrolbitsare labeledF,M,andCinFigure3.21.Theirfunctions areasfollows: 1.TheF-control bits,rwopersubframe, constitUte themainframingpulses.Thesubscripts ontheF-control bitsdenotetheactualbit(Oor1)transmitted. Thusthemainframing sequence isFoF,FOF,FOF,FoF, or01010101. 2.TheM-control bits,onepersubframe, formsecondary framingpulsestoidentifythe foursubframes. Hereagainthesubscripts ontheM-control bitsdenotetheactualbit (Oor1)transmitted. Thusthesecondary framingsequence isMoM,M,M,or0111. 3.TheC-eontrol bits,threepersubframe, arestuffingindicators. Inparticular, C[refers toinputchannelI,CnreferstoinputchannelII,andsoforth.Forexample, therhree C-eontrol bitsinthefirstsubframe following M ointhefirstsubframe arestuffingin dicatorsforthefirstDSIbitstream.Theinsertion ofastuffedbitinthisDSIbitstream isindicated bysettingallthreeC-control bitsto1.Toindicatenostuffing,allthreeare settoO.IfthethreeC-control bitsindicatestuffing,thestuffedbitislocatedinthe positionofthefirstinformation bitassociated withthefirstDSIbitstreamthatfollows theF,-controlbitinthesamesubframe. Inasimilarway,thesecond,third,andfourth DSIbitstreamsmaybestuffed,asrequired. Byusingmajority logicdecoding in[he receiver,asingleerrorinanyofthethreeC-eontrol bitscanbedetected. Thisformof decoding meanssimplythatthemajority oftheC-control bitsdetermine whetheran all-oneorall-zerosequence wastransmitted. ThusthreeIsorcombinations oftwoIs anda 0indicatethatastuffedbitispresentintheinformation sequence, following the controlbitF,inthepertinent subframe. Ontheotherhand,threeOsorcombinations oftwOOsanda 1indicatethatnostuffingisused. Thedemultiplexer atthereceiving M12unitfirstsearches forthemainframing sequence FoF,FoF,FoF,FoF" Thisestablishes identityforthefourinputDSIbitstreamsandalsoforthe M-andC-eonrrol bits.FromtheMoM,M,M,sequence, thecorrectframingoftheC-control bitsisverified.Finally,thefourDSIbitstreamsareproperly demultiplexed anddestuffed. Thesignalformatdescribed abovehastwosafeguards: 1.Itispossible, although uulikely, thatwithjusttheFoF,FoF,FoF,FoF, sequence, oneof theincoming DSIbitstreamsmaycontainasinlilarsequence. Thiscouldthencause MO[48]Cl[48]Fa[48]Cl[48]Cl[48]Fl[48] Ml[48]Cll[48]Fa[48]Cll[48]Cll[48]Fl[48] Ml[48]Clll[48]'Fa[48]Clll[48]Clll[48]Fl[48] Ml[48]ClY[48]Fo[48]ClY[48]ClY[48]Fl[48] tttttt Subframe First Frame Second Third FrameStuffed markers stuffiog markers stuffing stuffing markers bits indicators indicators indicators FIGURE 3.21SignalfonnatofAT&TM]2multiplexer. 3.11Virtues, Limitations, andModifications ofPCM217 thereceivertolockontothewrongsequence. ThepresenceoftheMoM,M,M, sequence providesverification ofthegenuineFoF,FoF,FoF,FoF, sequence, therebyensuringthat thefourD51bitstreamsareproperlydemultiplexed. 2.Thesingle-error correction capability builtintotheC-control bitsensuresthatthefour DS1bitstreamsareproperlydestuffed. ThecapacityoftheM12multiplexer toaccommodate smallvariations intheinputdata ratescanbecalculated fromtheformatofFigure3.21.IneachMframe,definedastheinterval containing onecycleofMaM,M,M, bits,onebitcanbestuffedintoeachoffourinputD51 bitstreams.Eachsuchsignalhas12x 6 x 4 =288positions ineachMframe.Also,theTl bitstreamhasabitrateequalto1.544Mb/s.Hence,eachinputcanbeincremented by 1.544X103X2~8=5.4kb/s Thisresultismuchlargerthantheexpected changeinthebitrateoftheincoming D51"bit stream.Itfollowstherefore thattheuseofonlyonestuffedbitperinputchannelineachframe issufficient toaccommodate expectedvariations intheinputsignalrate. Thelocalclockthatdetermines theoutgoingbitratealsodetermines thenominalstuffing rateS,definedastheaveragenumberofbitsstuffedperchannelinanyframe.The M12multiplexer isdesigned forS=1/3.Accordingly, thenominalbitrateoftheD52bit streamis 49288 1.544X4X48X288-5=6.312Mb/s ThisalsoensuresthatthenominalDS2clockfrequency isamultipleof8kHz(thenominal sampling rateofavoicesignal),whichisadesirable feature. <II 3.11Virtues, LinJitations, andModifications ofPCM Inagenericsense,pulse-code modulation (PCM)hasemerged asthemostfavoredmod ulationschemeforthetransmission ofanaloginformation-bearing signalssuchasvoice andvideosignals.Theadvantages ofPCMmayaUbetracedtotheuseofcodedpulses forthedigitalrepresentation ofanalogsignals,afeaturetbatdistinguishes itfromallother analogmetbods ofmodulation. Wemaysummarize theimportant advantages ofPCMas follows: 1.Robustness tochannelnoiseandinterference. 2.Efficient regeneration ofthecodedsignalalongthetransmission path. 3.Efficient exchange ofincreased channel bandwidth forimproved signal-to-noise ra tio,obeyinganexponential law. 4.Auniform formatforthetransmission ofdifferent kindsofbaseband signals,hence tbeirintegration withotherformsofdigitaldatainacommon network. 5.Comparative easewithwhichmessage sourcesmaybedropped orreinserted ina time-division multiplex system. 6.Securecommunication through theuseofspecialmodulation schemesorencryption; theencryption anddecryption ofdataarediscussed inAppendix 5. Theseadvantages, however, areattained atthecostofincreased systemcomplexity and increased channelbandwidth. Thesetwoissuesareconsidered inthesequelinturn. 218 CHAPTER 3"PULSE MODULATION Although theuseofPCMinvolves manycomplex operations, todaytheycanallbe implemented inacost-effective fashionusingcommercially available and/orcustom-made very-large-scale integrated (VLSI)chips.Inotherwords,therequisite devicetechnology fortheimplementation ofaPCMsystemisalreadyinplace.Moreover, withcontinuing improvements inVLSItechnology, wearelikelytoseeanever-expanding useofPCMfor thedigitaltransmission ofanalogsignals. If,however, thesimplicity ofimplementation isanecessary requirement, thenWe mayusedelta~odula~ion a~analternative topulse-code m~dulation. Indeltamodulation, thebaseband slgnallsmtentlOnally "o,:ersampled" toperm~ttheuseofasl~plequantizing strategyforconstructmg theencoded signal;deltamodulatIOn ISdiscussed mSectIOn3.12 Turning nexttotheissueofbandwidth, wedorecognize thattheincreased band: widthrequirement ofPCMmayhavebeenareasonforjustifiable concern inthepast. Today,however, itisofnorealconcernfortwodifferent reasons. First,theincreasing availability ofwideband communication channels meansthatbandwidth isnolongera systemconstraint inthetraditional wayitusedtobe.Liberation fromthebandWidth constraint hasbeenmadepossible bythedeployment ofcommunication satellites for broadcasting andtheever-increasing useoffiberopticsfornetworking; adiscussion of thesecommunication channelconcepts waspresented intheBackground andPreview chapter. Thesecondreasonisthatthrough theuseofsophisticated datacompression tecb niques,itisindeedpossibletoremovetheredundancy inherently presentinaPCMsignal andtherebyreducethebitrateofthetransmitted datawithout seriousdegradation in systemperformance. Ineffect,increased processing complexity (andtherefore increased costofimplementation) istradedoffforareducedbitrateandtherefore reducedband widthrequirement. Amajormotivation forbit-ratereduction isforsecurecommunication overradiochannels thatareinherently oflowcapacity. I3.12DeltaModulation IndeltamodulationS (DM),anincoming message signalisoversampled (i.e.,atarate muchhigherthantheNyquistrate)topurposely increasethecorrelation betweenadjacent samplesofthesignal.Thisisdonetopermittheuseofasimplequantizing strategy for constructing theencoded signal. Initsbasicform,DMprovides astaircase approximation totheoversampled version ofthemessagesignal,asillustrated inFigure3.11a.Thedifference between theinputand theapprOXimation isquantized intoonlytwolevels,namely,:til,corresponding topositive andnegative differences. Thusiftheapproximation fallsbelowthesignalatanysampling epoch,itisincreased byIl.Ifontheotherhand,theapproximation liesabovethesigna~ itisdiminished byd..Provided thatthesignaldoesnotchangetoorapidlyfromsample tosample,wefindthatthestaircase approximation remainswithin:tilofrheinputsignal. Letm(t)denotetheinput(message) signal,andmq(t)denoteitsstaircase approxi mation.Forconvenience ofpresentation, weadoptthefollowing notation thatiscom' monlyusedinthedigitalsignalprocessing literatnre: m[n]=m(nTJ, n=0,:t1,:t2,... whereTsisthesampling periodandm(nTJisasampleofthesignalm(t)takenattime t=nT"andlikewise forthesamples ofothercontinuous-time signals.Wemaythen 3.12DeltaModulation 219 m(t) Staircase approximation mq(t) (oj Binary ~~~;~C~ator 0 0 1 0 1 outputo1000 0 0 0 (b) FIGURE3.22Illustration ofdeltamodulation. formalize thebasicprinciples ofdeltamodulation inthefollowing setofdiscrete-time relations: ern]=m[n]-mq[n-~.] (3.52) eq=8sgn(e[n]) (3.53) mq[n]=mq[n-1]+eq[n] (3.54) whereern]isanerrorsignalrepresenting thedifference between thepresentsamplem[n] oftheinputsignalandthelatestapproximation mq[n-1]toit,eq[n]isthequantized versionofern]'andsgn(')isthesignumfunction. Finally,thequantizer outputmq[n]is codedtoproducetheDMsignal. Figure3.22aillustrates thewayinwhichthestaircase approximation mq(t)follows variations intheinputsignalm(t)inaccordance withEquations (3.52)-(3.54), andFigure 3.22bdisplays thecorresponding binarysequence atthedeltamodulator output.Itis apparent thatinadeltamodulation systemtherateofinformation transmission issimply equaltothesampling rateis=l/Ts. Theprincipal virtueofdeltamodulation isitssimplicity. Itmaybegenerated by applying thesampledversionoftheincoming messagesignaltoamodulator thatinvolves acomparator, quantizer, andaccumulator interconnected asshowninFigure3.23a.The blocklabeledZ-linsidetheaccumulator represents aunitdelay,thatis,adelayequalto onesampling period.(Thevariableziscommonly usedinthez-transform, whichisbasic totheanalysisofdiscrete-time signalsandsystems.) Detailsofthemodulator followdi rectlyfromEquations (3.52)-(3.54). Thecomparator computes thedifference betweenits twoinputs.Thequantizer consistsofahardlimiterwithaninput-output relationthatis ascaledversionofthesignumfunction. Thequantizer outputisthenappliedtoanaccu mulator, producing theresult n mq[n]=8Lsgn(e[i]) i=l n =Leq[i] i=l(3.55) 220 CHAPTIlR 3"PULSE MODULATION Sampled messagesignal mIn] mq[n-l],- I I I I II I I : mqln] I ~ J Accumulator (a)DM wave Sampled channel outputReconstructed messagesignal (b) FIGURE3.23DMsystem.(a)Transmitter. (b)Receiver. whichisobtained bysolving Equations (3.53)and(3.54)formq[n].Thus,atthesampling instantnT"theaccwnulator increments theapproximation byastep~inapositiveor negative direction, depending onthealgebraic signoftheerrorsampleern].Iftheinput samplem[n]isgreaterthanthemostrecentapproximation mq[n],apositiveincrement +~isappliedtotheapproximation. If,ontheotherhand,theinputsampleissmaller,a negative increment -~isappliedtotheapproximation. Inthisway,theaccwnulator does thebestitcantotracktheinputsamplesbyonestep(ofamplitude +~or-Matatime. InthereceivershowninFigure3.23b,thestaircase approximation mq(t)isreconstructed bypassingthesequence ofpositiveandnegative pulses,produced atthedecoderoutput, throughanaccumulator inamannersimilartothatusedinthetransmitter. Theout-of bandquantization noiseinthehigh-frequency staircase waveform mq(t)isrejectedby passingitthroughalow-pass filter,asinFigure3.23b,withabandwidth equaltothe originalmessagebandwidth. Deltamodulation issubjectto.twotypesofquantization error:slopeoverload dis tortionandgranular noise.Wevvilldiscussthecaseofslopeoverload distortion first. WeobservethatEquation (3.54)isthedigitalequivalent ofintegration inthesense thatitrepresents theaccumulation ofpositiveandnegative increments ofmagnitude ~. Also,denoting thequantization errorbyq[n],asshownby mq[n]=m[n]+q[n] weobservefromEquation (3.52)thattheinputtothequantizer is ern]=m[n]-m[n-1]-q[n-1](3.56) (3.57) 3.12DeltaModulation 221 Granular noise Staircase approximation mq(t) FIGURE3.24Illustration ofthetwodifferent formsofquantization errorindeltamodulation. Thusexceptforthequantization errorq[n-1],thequantizer inputisafirstbackward difference oftheinputsignal,whichmaybeviewedasadigitalapproximation tothe derivative oftheinputsignalor,equivalently, astheinverseofthedigitalintegration process.Ifweconsider themaximum slopeoftheoriginalinputwaveform mit),itis clearthatinorderforthesequence ofsamples {mq[n]}toincrease asfastastheinput sequence ofsamples {m[n]}inaregionofmaximum slopeofmit),werequirethatthe condition ~2:maxldm(t)I T, dt(3.58) besatisfied. Otherwise, wefindthatthestep-sizeaistoosmallforthestaircase approxi mationmqlt)tofollowasteepsegnIent oftheinputwaveform mit),withtheresultthat mq(t)fallsbehindmit),asillustrated inFigure3.24.Thiscondition iscalledslopeoverload, andtheresulting quantization erroriscalledslope-overload distortion (noise).Notethat sincethemaximum slopeofthestaircase approximation mq(t)isfixedbythestepsizeA, increases anddecreases inmqlt)tendtooccuralongstraightlines.Forthisreason,adelta modulator usingafixedstepsizeisoftenreferredtoasalineardeltamodulator. Incontrasttoslope-overload distortion, granular noiseoccurswhenthestepsizeA istoolargerelativetothelocalslopecharacteristics oftheinputwaveform mit),thereby causingthestaircase approximation mq(t)tohuntaroundarelatively flatsegmentofthe inputwaveform; thisphenomenon isalsoillustrated inFigure3.24.Granular noiseis analogous toquantization noiseinareMsystem. Wethusseethatthereisaneedtohavealargestep-size toaccommodate awide dynamic range,whereas asmallstepsizeisrequired fortheaccurate representation of relatively low-level signals.Itistherefore clearthatthechoiceoftheoptimum stepsize thatminimizes themean-square valueofthequantization errorinalineardeltamodulator willbetheresultofacompromise between slope-overload distortion andgranular noise. Tosatisfysucharequirement, weneedtomakethedeltamodulator "adaptive," inthe sensethatthestepsizeismadetovaryinaccordance withtheinputsignal;thisissueis discussed furtherinacomputer experiment presented inSection3.16. IIIDELTA-SIGMA MODUlATION Asmentioned earlier,thequantizer inputintheconventional formofdeltamodulation maybeviewedasanapproximation tothederivative oftheincoming messagesignal.This behavior leadstoadrawback ofdeltamodulation inthattransmission disturbances such asnoiseresultinanaccumulative errorinthedemodulated signal.Thisdrawback canbe 222 CHAPTElt 3"PULSE MODUlATION overcome byintegrating themessagesignalpriortodeltamodulation. Theuseofintegra. tioninthemannerdescribed herehasalsothefollowing beneficial effects: 1>-Thelow-frequency contentoftheinputsignalispre-emphasized. ~Correlation between adjacent samplesofthedeltamodulator inputisincreased whichtendstoimproveoverallsystemperformance byreducing thevariance ofth~ errorsignalatthequantizer input. l'-Designofthereceiverissimplified. Adeltamodulation schemethatincorporates integration atitsinputiscalleddelta-sigma modulation (D-~M).9 Tobemoreprecise,however, itshouldbecalledsigma-delta /nod_ ulation,becausetheintegration isinfactperformed beforethedeltamodulation. Never. theless,theformerterminology istheonecommonly usedintheliterature. Figure3.25ashowstheblockdiagram ofadelta-sigma modulation system.Inthis diagram, themessagesignalmit)isdefinedinitscontinuous-time form,whichmeansthat thepulsemodulator nowconsistsofahard-limiter followed byamultiplier; thelatter component isalsofedfromanexternalpulsegenerator (clock)toproduceai-bitencoded signal.Theuseofintegration atthetransmitter inputclearlyrequiresaninversesignal emphasis, namely,differentiation, atthereceiver.Theneedforthisdifferentiation is,how ever,eliminated becauseofitscancellation byintegration intheconventional DMreceiver. Message signal m(tl Message signal mIt)Pulsemodulator I I I I _______________ 1 Integrator 2 Transmitter (al Transmitter (b)Receiver ReceiverEstimateof message signal Estimateof message signal FIGURE3.25Twoequivalent versions ofdelta-sigma modulation system. (3.59)3.13LinearPredktion 223 Thusthereceiverofadelta-sigma modulation systemconsistssimplyofalow-pass filter, asindicated inFigure3.25a. Moreover, wenotethatintegration isbasically alinearoperation. Accordingly, we maysimplify thedesignofthetransmitter bycombining thetwointegrators 1and2of Figure3.25aintoasingleintegrator placedafterthecomparator, asshowninFigure3.25b. Thislatterformofthedelta-sigma modulation systemisnotonlysimplerthanthatof Figure3.25a,butitalsoprovides aninteresting interpretation ofdelta-sigma modulation asa"smoothed" versionofI-bitpulse-code modulation: Thetermsmoothness refersto thefactthatthecomparator outputisintegrated priortoquantization, andtheterml-bit merelyrestatesthatthequantizer consistsofahard-limiter withonlytworepresentation levels. Indeltamodulation, simplicity ofimplementations ofboththetransmitter andre ceiverisattained byusingasampling ratefarinexcessofthatneededforpulse-code modulation. Thepricepaidforthisbenefitisacorresponding increaseinthetransmission andtherefore channelbandwidth. Thereare,however, applications wherechannelband widthisatapremium, inwhichcasewehavetheopposite requirement tothatindelta modulation. Specifically, wemaywishtotradeincreased systemcomplexity forareduced channelbandwidth. Asignal-processing operation basictotheattainment ofthislatter designobjective isprediction, thelinearformofwhichisdiscussed next. l3,13LinearPrediction Consider afinite-duration impulseresponse (FIR)discrete-time filterconfigured asinFig ure3.26,whichinvolves theuseofthreefunctional blocks: 1.Setofpunit-delay elements, eachofwhichisrepresented byZ-l. 2.Setofmultipliers involving thefiltercoefficients w"W2,•••,wP' 3.Setof"adders" usedtosumthescaledversions ofthedelayedinputsx[n-1], x[n-2],...,x[n-p]toproduce theoutputx[n].Thefilteroutputx[n]ormore precisely, thelinearprediction oftheinput,isthusdefinedbytheconvolution sum p i[n]=2:wkx[n-k] k~l wherep,thenumberofunit-delay elements, iscalledtheprediction order. TheactualsampleattimenT,isx[n].Theprediaion error,denotedbyern],isdefined asthedifference between x[n]andtheprediction i[n],asshownby Prediction ,In]Input x[.]ern]=x[n]i[nl FIGURE3.26Blockdiagramofalinearprediction filteroforderp.(3.60) 224 CHAPTER 3IIIPuLSE MODULATION Thedesignobjective istochoosethefiltercoefficients W"W2,•••,Wpsoastominimize anindexofperformance,], definedasthemean-square error: ] =E[~[n]] (3.61) Substituting Equations (3.59)and(3.60)into(3.61)andthenexpanding terms,wemay reformulate theindexofperformance as p p p ] =E[x2[n]]-22:wkE[x[n]x[n -k]]+2:2:WjwkE[x[n -J1x[n-k]](3.62) k=l ;=1k=l Weassumethattheinputsignalx(t)isthesamplefunction ofastationary process X(t)ofzeromean;thatis,E[x[n]]iszeroforalln.Define uJ.:=variance ofasampleoftheprocessX(t)attimenTs =E[x2[n]]-(E[x[n]W =E[x2[n]] Rx(kT s)=autocorrelation oftheprocessX(t)foralagofkT, =Rx[k] =E[x[n]x[n -k]] Accordingly, wemayrewriteEquation (3.62)inthesimplified form p p p ] =(fi:-22:WkRX[k]+2:2:WjwkRX[k- i] k=l ;=1k=l{3.631 Hencedifferentiating theindexofperformance] withrespecttothefiltercoefficient Wk, settingtheresultequaltozero,andthenrearranging terms,weobtain p 2:wjRx[k-J1=Rx[k]=Rx[-k], j=lk=1,2,...,P (3.64l Rx[p-1]1 Rx[p-2] Rx[O]Theoptimality equations (3.64)arecalledtheWiener-Ropf equations forlinearprediction. Wefinditconvenient toreformulate theWiener-Hopf equations (3.64)inmatrix form.Let wo=p-by-1optimum coefficient vector =[w"W2,"" wpV rx=p-by-1autocorrelation vector =[Rx[1],Rx[2],...,Rx[P]f Rx=p-by-pautocorrelation matrix [Rx[O] Rx[1] Rx[1] Rx[O] =RxlP'-11Rx[P:-21 Wemaythussimplifythesetofequations (3.64)as (3.65) 3.13LinearPrediction 225 Weassumethattheautocorrelation matrixRxisnonsingular, sothatitsinverseexists. WemaythensolveEquation (3.65)forthecoefficient vector Wobymultiplying bothsides ofthisequation bytheinversematrixRx\obtaining theoptimum solution (3.66) Notethatalltheelements onthemaindiagonal oftheautocorrelation matrixRx areequaltoRx[O]=O{,andtheelements onanyotherdiagonal paralleltothemain diagonal arealsoequal.Asquarematrixhavingthisproperty issaidtobeToeplitz, which isadirectconsequence oftheassumption thattheinputsignalx(t)isdrawnfromasta tionaryprocess.Thepractical significance oftheToeplitz property isthatthecorrelation matrixRxisuniquely definedbythesetofautocorrelation valuesRx[O],Rx[l],..., Rx[p-1].Theautocorrelation vectorrxisdefinedbythesetofautocorrelation values Rx[1],Rx[2],...,Rx[p].Itfollowstherefore thatthepfiltercoefficients ofthelinear optimum predictor areuniquely definedbythevariance0{=Rx[O]andpvaluesofthe autocorrelation function oftheprocessK(t)forlagsofT"2T"...,pTs• Theminimum mean-square valueoftheprediction errorisobtained bysubstituting Equation (3.64)into(3.63),whichyields(aftersimplification) (3.67) Thequadratic termrIRx"rx isalwayspositive. Accordingly, themean-square errorJminof theoptimum linearpredictor definedbyEquation (3.67)isalwayslessthanthevariance akoftheinputsamplethatisbeingpredicted. IIILINEAR ADAPTIVE PREDICTION TheuseofEquation (3.66)forcalculating theweightvectorofalinearpredictor requires knowledge oftheautocorrelation function Rx[k]oftheinputsequence {x[n]}forlags k=0,1,...,p,wherepistheprediction order.Whatifknowledge ofRx[k]forvarying kisnotavailable? Inthesesituations, whichoccurfrequently inpractice, wemayresort totheuseofanadaptive predictor. Thepredictor isadaptiveinthefollowing sense: I>-Computation ofthetapweights Wk>k=1,2,...,p,proceeds ina"recursive" manner, startingfromsomearbitrary initialvaluesofthetapweights. ~Thealgorithm usedtoadjustthetapweights(fromoneiteration tothenext)is"self- designed," operating solelyonthebasisofavailable data. Theaimofthealgorithm istofindtheminimum pointofthebowl-shaped errorsurface thatdescribes thedependence ofthecostfunctionJonthetapweights. Itistherefore intuitively reasonable thatsuccessive adjustments tothetap-weights ofthepredictor be madeinthedirection ofthesteepestdescentoftheerrorsurface,thatis,inadirection opposite tothegradient vectorwhoseelements aredefinedby (3.68) k=1,2,...,PJJgk=--, JWk Thisisindeedtheideabehindthemethodofsteepestdescent.Letwk[n]denotethevalue ofthekthtap-weight atiteration n.Thentheupdated valueofthisweightatiteration n+1isdefinedby 1wk[n+1]=wk[n]-2"jLgk, k=1,2,...,P (3.69) 226 CHAPTER 3IIPULSE MODUlATION (3.70/ k=1,2,...,Pp =-2E[x[n]x[n -k]]+22:w;E[x[n-nx[n-k]], ;=1where /.Lisastep-sizeparameter thatcontrolsthespeedofadaptation, andthefactor1/2 isincluded forconvenience ofpresentation. Differentiating thecostfunction] ofEquation (3.63)withrespecttoWk,wereadilyfindthat p gk=-2Rx[k]+22:w;Rx[k-n i=l Thisformulaforgkcoulddowithfurthersimplification, whichisachieved byusingin. stantaneous valuesasestimates oftheautocorrelation functions Rx[k]andRx[k-n'That is,weignoretheexpectation operatorsinEquation (3.70)tofacilitate theadaptive process onastep-by-step basis.Wemaythusexpressthecorresponding estimateofgkatiteration nas p gk[n]=-2x[n]x[n -k]+22:w;[n]x[n -l1x[n-k], j=1k=1,2,...,P(3.71) Notethatforaninputx[n]drawnfromastationary processthegradientgk isadetermin. isticquantity, whereastheestimategk[n]isthesamplevalueofarandomvariable. Inanyevent,substituting Equation (3.71)into(3.69)andfactoring thecommon termx[n-k],wemaywrite Wk[n+1]=wk[n]+/.LX[n-k](X[n]-;tw;[n]x[n -II) =wk[n]+/.LX[n-k]e[n], k=1,2,...,p whereern]istheprediction errordefinedas(3.72/ p ern]=x[n]-2:w;[n]x[n -j] ;=1(3.73) InEquations (3.72)and(3.73),wehaveusedWkasanestimate ofthekthtap-weightto distinguish itfromtheactualvalueWk.Notealsothatx[n]playstheroleofa"desired response" forcomputing therecursive adjustments applied tothetap-weights ofthe predictor. Equations (3.72)and(3.73)constitute thepopularleast-mean-square (LMS)alga' rithmforlinearadaptive prediction, theoperation ofwhichisdepicted inFigure3.27.The reasonforpopularity ofthisadaptive filteringalgorithm isthesimplicity ofitsimplemen· tation.Inparticular, thecomputational complexity ofthealgorithm, measured interms ofthenumberofadditions andmultiplications, islinearintheprediction orderp. Input Prediction xW ;W FIGURE3.27Blockdiagram illustrating thelinearadaptive prediction process. 3.14Differential Pulse-Code Modulation 227 TheLMSalgorithm isastochastic adaptive filteringalgorithm, stochastic inthesense that,startingfromtheinitialcondition definedby(Wk[O]M:-I, itseekstofindtheminimum pointoftheerrorsurfacebyfollowing azig-zagpath.Moreover, itneverfindsthismini mumpointexactly.Rather,itexecutes arandommotionaroundtheminimum pointof theerrorsurface,oncesteady-state conditions areestablished. Withthismaterial onlinearprediction athand,wearereadytodiscusspractical improvements ontheperformance ofpulse-code modulation. ~ Differential Pulse-Code Modulation WhenavoiceOrvideosignalissampled atarateslightlyhigherthantheNyquistrateas usuallydoneinpulse-code modulation, theresulting sampled signalisfoundtoexhibita highdegreeofcorrelation between adjacent samples. Themeaning ofthishighcorrelation isthat,inanaveragesense,thesignaldoesnotchangerapidlyfromonesampletothe next,andasaresult,thedifference betweenadjacent sampleshasavariancethatissmaller thanthevariance ofthesignalitself.Whenthesehighlycorrelated samplesareencoded, asinthestandard PCMsystem,theresulting encoded signalcontains redundant infor mation.Thismeansthatsymbols thatarenotabsolutely essential tothetransmission of information aregenerated asaresultoftheencoding process.Byremoving thisredundancy beforeencoding, weobtainamoreefficientcodedsignal,whichisthebasicideabehind differential pulse-code modulation. Nowifweknowthepastbehavior ofasignaluptoacertainpointintime,wemay useprediction tomakeanestimate ofafuturevalueofthesignalasdescribed inSection 3.13.Suppose thenabaseband signalm(t)issampledattheratej,=lIT,toproduce the sequence {m[n]}whosesamplesareT,secondsapart.Thefactthatitispossibletopredict futurevaluesofthesignalm(t)provides motivation forthedifferential quantization scheme showninFigure3.28a.Inthisscheme,theinputsignaltothequantizer isdefinedby ern]=m[n]m[n] (3.74) whichisthedifference betweentheunquantized inputsamplem[n]andaprediction ofit, denotedbym[n].Thispredicted valueisproduced byusingalinearprediction filterwhose input,aswewillsee,consistsofaquantized versionoftheinputsamplem[n].Thediffer encesignalern]istheprediction error,sinceitistheamountbywhichtheprediction filter failstopredicttheinputexactly.Byencoding thequantizer output,asinFigure3.28a,we obtainavariantofPCMknownasdifferential pulse-code modulation'O(DPCM). Thequantizer outputmaybeexpressed as eq[n]=ern]+q[n] (3.75) whereq[n]isthequantization error.According toFigure3.28a,thequantizer outputeq[n] isaddcdtothepredicted valuem[n]toproduce theprediction-filter input mq[n]=m[n]+eq[n] Substituting Equation (3.75)into(3.76),weget mq[n]=m[n]+ern]+q[n](3.76) (3.77) However, fromEquation (3.74)weobservethatthesumtermm[n]+ern]isequaltothe inputsamplem[n].Therefore, wemaysimplifyEquation (3.77)as mq[n]=m[n]+q[n] (3.78) 228 CHAPTER 3"PULSEMODULATION Sampled input m[n]DPCM wave Input(a) I)------~--',.. Output (b) FIGURE 3.28DPCMsystem.(a)Transmitter. (b)Receiver. whichrepresents aquantized versionoftheinputsamplemin].Thatis,irrespective ofthe properties oftheprediction filter,thequantized samplemq[n]attheprediction filterinput differsfromtheoriginalinputsamplemin]bythequantization errorq[n].Accordingly, if theprediction isgood,thevariance oftheprediction errore[n]willbesmallerthanthe variance ofm[n],sothataquantizer withagivennumberoflevelscanbeadjusted to produceaquantization errorwithasmallervariance thanwouldbepossibleiftheinput samplemin]werequantized directlyasinastandard PCMsystem. Thereceiverforreconstructing thequantized versionoftheinputisshowninFigule 3.28b.Itconsistsofadecodertoreconstruct thequantized errorsignal.Thequantized versionoftheoriginalinputisreconstructed from thedecoderoutputusingthesame prediction filterusedinthetransmitter ofFigure3.28a.Intheabsenceofchannelnoise, wefindthattheencoded signalatthereceiverinputisidentical totheencoded signalat thetransmitter output.Accordingly, thecorresponding receiveroutputisequaltornq[n], whichdiffersfromtheoriginalinputmin]onlybythequantization errorq[n]incUlredas aresultofquantizing theprediction errorern]. Fromtheforegoing analysisweobservethat,inanoise-free environment, thepre dictionfiltersinthetransmitter andreceiveroperateonthesamesequence ofsamples, mq[n].Itiswiththispurposeinmindthatafeedback pathisaddedtothequantizer inthe transmitter, asshowninFigure3.28a. Differential pulse-code modulation includes deltamodulation asaspecialcase.In particular, comparing theDPCMsystemofFigure3.28withtheDMsystemofFigure 3.23,weseethattheyarebasically similar,exceptfortwoimportant differences: theuse ofaone-bit(two-level) quantizer inthedeltamodulator andthereplacement ofthepre dictionfilterbyasingledelayelement(i.e.,zeroprediction order).Simplyput,DMisthe 1-bitversionofDPCM.Notethatunlikeastandard PCMsystem,thetransmitters ofboth theDPCMandDMinvolvetheuseoffeedback. DPCM,likeDM,issubjecttoslope-overload distortion whenever theinputsignal changestoorapidlyfortheprediction filtertotrackit.Also,likePCM,DPCMsuffers fromquantization noise. 3.15Adaptive Differential Pulse-Code Modulation 229 PROCESSING GAIN Theoutputsignal-to-noise ratiooftheDPCMsystemshown IIIFigure3.28is,by definition, 2 (SNR)o =aMat(3.79) where~isthevariance oftheoriginalinputsamplem[n],assumed tobeofzeromean, anda~isthevariance ofthequantization errorq[n].WemayrewriteEquation (3.79)as theproductoftwofactorsasfollows: (SNR)o= (3.80) (3.81)wherea'icisthevariance oftheprediction error.Thefactor(SNR)Q isthesignal-to quantization noiseratio,whichisdefinedby 2 (SNR) =aE Qa~ TheotherfactorGpistheprocessing gainproduced bythedifferential quantization scheme;itisdefinedby G=~\1 Pa'ic(3.82) Thequantity Gp,whengreaterthanunity,represents againinsignal-to-noise ratiothat isduetothedifferential quantization schemeofFigure3.28.Now,foragivenbaseband (message) signal,thevariance ~isfixed,sothatGpismaximized byminimizing the variancea'icoftheprediction errorern].Accordingly, ourobjective shouldbetodesign theprediction filtersoastominimize a'ic. Inthecaseofvoicesignals,itisfoundthattheoptimum signal-to-quantization noise advantage ofDPCMoverstandard PCMisintheneighborhood of4to11dB.Thegreatest improvement occursingoingfromnoprediction tofirst-order prediction, withsomead ditionalgainresulting fromincreasing theorderoftheprediction filterupto4or5,after whichlittleadditional gainisobtained. Since6dBofquantization noiseisequivalent to 1bitpersamplebyvirtueofEquation (3.35),theadvantage ofDPCMmayalsobeex pressedintermsofbitrate.Foraconstant signal-to-quantization noiseratio,andassuming asampling rateof8kHz,theuseofDPCMmayprovideasavingofabout8to16kb/s (i.e.,1to2bitspersample)compared tothestandard PCM. 3.15Adaptive Differential Pulse-Code Modulation TheuseofPCMforspeechcodingatthestandard rateof64kb/sdemands ahighchannel bandwidth foritstransmission. Incertainapplications, however, suchassecuretransmis sionoverradiochannels thatareinherently oflowcapacity, channelbandwidth isata premium. Inapplications ofthiskind,thereisadefiniteneedforspeechcodingatlowbit rates,whilemaintaining acceptable fidelityorqualityofreproduction. 230 CHAPTER 3"PULSE MODUlATION Forcodingspeechatlowbitrates,awaveform coderofprescribed configuration i optimized byexploiting bothstatistical characterization ofspeechwaveforms andPTOP~ ertiesofhearing.Inparticular, thedesignphilosophy hastwoaimsinmind: 1.Toremoveredundancies fromthespeechsignalasfaraspossible. 2.Toassigntheavailable bitstocodethenonredundant partsofthespeechsignalina perceptually efficientmanner. Aswestrivetoreducethebitratefrom64kb/s(usedinstandard PCM)to32,16,8,and 4kb/s,the.sc~emesusedforredundancy r~movalandbitassignment becomeincreasingly moresophistICated. Asaruleofthumb,illthe64to8kb/srange,thecomputational complexity (measured intermsofmultiply-add operations) required tocodespeechin creasesbyanorderofmagnitude whenthebitrateishalved,forapproximately equal speechquality. Inthissection,wedescribeadaptive differential pulse-code modulation (ADPCMj,ll whichpermitsthecodingofspeechat32kb/sthrough thecombined useofadaptive quantization andadaptive prediction; thenumberofeightbitspersamplerequiredinthe standard PCMistherebyreducedtofour.Thetermadaptive usedhereinmeansbeing responsive tochanging levelandspectrum of theinputspeechsignal.Thevariation of performance withspeakers andspeechmaterial, together withvariations insignallevel inherentinthespeechcommunication process,makethecombined useofadaptive quan tizationandadaptive prediction necessary toachievebestperformance overawiderange ofspeakers andspeaking situations. Adaptive quantization referstoaquantizer thatoperates withatime-varying step. sizeMn].Atanygivensampling instantidentified bytheindexn,theadaptive quantizel isassumed tohaveauniformtransfercharacteristic. Thestep-size A[n]isvariedsoasto matchthevarianceairoftheinputsamplem[n].Inparticular, wewrite (3.83) where4>isaconstant, anduM[n]isanestimateofthestandard deviation uM[n](i.e.,square rootofthevarianceair).Foranonstationary input,uAdn]istimevarying.Theproblem ofadaptive quantization according toEquation (3.83)is,therefore, oneofcomputing the estimate uM[n]continuously. Theimplementation ofEquation (3.83)mayproceedinoneoftwoways: 1.Adaptive quantization withforward estimation (AQF),inwhichunquantized sam· piesoftheinputsignalareusedtoderiveforwardestimates ofuM[n]. 2.Adaptive quantization withbackward estimation (AQB),inwhichsamplesofthe quantizer outputareusedtoderivebackward estimates ofuM[n]. TheAQFschemerequirestheuseofabuffertostoreunquantized samplesoftheinput speechsignalneededforthelearningperiod.Italsorequirestheexplicittransmission of levelinformation (typically, about5to6bitsperstep-size sample)toaremotedecoder, therebyburdening thesystemwithadditional sideinformation thathastobetransmitted tothereceiver. Moreover, aprocessing delay(ontheorderof16msforspeech)inthe encoding operation resultsfromtheuseofAQF,whichisunacceptable insomeapplica tions.Theproblems ofleveltransmission, buffering, anddelayintrinsic toAQFareall avoidedinAQB.Inthelatterscheme,therecenthistoryofthequantizer outputisusedto extractinformation forthecomputation ofthestepsizeA[n].Inpractice, AQBistherefore usuallypreferred overAQF. . Figure3.29showstheblockdiagram ofanadaptive quantizer withbackward esti mation.Itrepresents anonlinear feedback system;hence,itisnotobviousthatthesysteID 3.15Adaptive Differentiol Pulse-Code Modulation 231 Input m[n] Transmitter ReceiverOutput FIGURE3.29Adaptive quantization "ithbackward estimation (AQB). willbestable.However, ifthequantizer inputm[n]isbounded, thenthebackward estimate UM[n]andthecorresponding stepsize.i[n]areaswell;undersuchacondition, thesystem isindeedstable. Theuseofadaptive prediction inADPCM isjustifiedbecausespeech signals are inherently nonstationary, aphenomenon thatmanifests itselfinthefactthattheautocor relationfunction andpowerspectraldensityofspeechsignalsaretime-varying functions oftheirrespective arguments. Thisimpliesthatthedesignofpredictors forsuchinputs shouldlikewisebetimevarying,thatis,adaptive. Aswithadaptive quantization, thereare twoschemesforperforming adaptive prediction: 1.Adaptive prediction withforwardestimation (APF),inwhichunquantized samples oftheinputsignalareusedtoderiveestimates ofthepredictor coefficients. 2.Adaptive prediction withbackward estimation (APB),inwhichsamplesofthequan tizeroutputandtheprediction errorareusedtoderiveestimates ofthepredictor coefficients. However, APFsuffersfromthesameintrinsicdisadvantages (sideinformation, buffering, anddelay)asAQF.Thesedisadvantages areeliminated byusingtheAPBschemeshown inFigure3.30,wheretheboxlabeled"logicforadaptive prediction" represents theal gorithmforupdating thepredictor coefficients. Inthelatterscheme,theoptimum predictor coefficients areestimated onthebasisofquantized andtransmitted data;theycantherefore beupdatepasfrequently asdesired,say,fromsampletosample.Accordingly, APBisthe preferred methodofprediction forADPCM. TheLMSalgorithm forthepredictor, described inSection3.13,andanadaptive schemeforthequantizer, basedonEquation (3.83),havebeencombined inasynchronous Input m[n] Prediction min]Tochannel FIGURE3.30Adaptive prediction withbackward estimation (APB). 232 CHAPTER 3.,PULSE MODULATION fashionforthedesignofboththeencoderanddecoder. Theperformance ofthiscombi_ nationissoimpressive at32kblsthatADPCM isnowaccepted internationally asastan_ dardcodingtechnique forvoicesignals,alongwith64kblsusingstandard PCM. 3.16Computer Experiment: Adaptive DeltaModulation AsimpleformofAQBistobefoundinthemodific~ti~n oflinear ~eltamodulation (LDM) toformadaptIVe deltamodulatIOn (ADM).Thepnnclple underlymg allADMalgorithms istwo-fold: 1.Ifsuccessive errorsareofopposite polarity, thenthedeltamodulator isoperating in itsgranular mode;inthiscase,itmaybeadvantageous toreducethestepsize. 2.If,however, successive errorsareofthesamepolarity, thenthedeltamodulator IS operating initsslope-overload mode;inthissecondcase,thestepsizeshouldbe increased. (3.84) ifA[n-1]<AminifA[n-1]2:AminThusbyvaryingthestep-size inaccordance withthisprinciple, thedeltamodulator is enabledtocopewithchangesintheinputsignal. Figure3.31showstheblockdiagram ofanADMbasedonincreasing ordecreasing thestepsizebyafactorof50percentateachiteration oftheadaptive process. Theal. gorithmforadaptation ofthestepsizeisdefinedby12 {IAln-1]1 A[n]=mq[n] (mq[n]+0.5mq[n-1]) Amin whereA[n]isthestepsizeatiteration (timestep)nofthealgorithm, andmAn]isthei-bit quantizer outputthatequals±1. Inthisexperiment weuseasinusoidal inputsignaltodemonstrate thereconstruction performance oftheADMalgorithm basedonEquation (3.84),andcompare ittotne performance ofacorresponding lineardeltamodulator (LDM).Detailsoftheexperiment areasfollows: Inputsignal: m(t)=Asin(27T[mt) whereamplitude A=10,frequency [m=[,/100,and!,=sampling frequency. Lineardeltamodulation (LDM): StepsizeA[n]=1foralln Adaptive deltamodulation (ADM): Theresultsoftheexperiment areplottedinFigure3.32.PartaofFigure3.32isfot LDM,andpartbofthefigureisforADM.Fromthewaveforms presented here,wemaY makethefollowing observations: ~ADMtrackschangesinthesinusoidal inputsignalmuchbetterthanLDM.'!his improvement intheperformance ofADMisduetoadaptation ofthestepsizeIII Sampled messagesignal mud Sampled channeloutput3.16Computer Experiment: Adaptive DeltaModulation 233 1-.....--------- .....-,..mq[n] (a) Reconstructed)------<1'--;;... message signal (b) FIGURE3.31Adaptive deltamodulation system:(a)Transmitter. (b)Receiver. successive iterations ofthealgorithm. Inparticular, thereducedstepsizeoftheADM resultsinsmallerquantization errorsneartheextremities oftheinputsignalthanthe LDM.However, bothmodulation schemesproducecomparable quantization errors inregionsoftheinputsignalwheretheslopeismoderately high. i>Theimproved tracking performance oftheADMresultsinanoutputsignalwitha muchlowerbitrate,ontheaverage,thantheLDM. 15 15 10Approximation tomodulator input10 Approximation tomodulator input -20:::--;,::--""-c-_~~_~_~~_~_~_o102030405060708090100 Numberofiterations (a)-20l-~~~_~_~_~~_~_~_~_ o102030405060708090100 Numberofiterations (bl FIGURE3.32Waveforms resulting fromthecomputer experiment ondeltamodulation: (a)Lineardeltamodulation. (b)Adaptive deltamodulation. 234 CHAPTER 3,.PULSE MODUUCION I3.17MPEGAudio CodingStandard Speech(voice)andaudiosignalsaresimilarinthat,inbothcases,thequalityofacoding. schemeisbasedontheproperties ofhumanauditory perception. InthecaseofSpeech signals,wehaveefficientcodingschemes (e.g.,ADPCM) because a speechproduction modelisavailable. Unfortunately, nothingsimilarexistsforaudiosignals. Inthis section, werevisittheMPEG-llaudio codingstandard brieflydescribed inthe Background andPreviewchapter;MPEGstandsforMotionPictureExpertsGroup,and thesuffix1isintended tomeanitisthefirstinaseriesofseveralstandardsY LikeADPCM theMPEG-l/audio codingstandard isalossycompression system,butitdiffers fro~ ADPCM inanimportant practical respect:TheMPEG-l standard iscapableofachieVing transparent, perceptually losslesscompression ofstereophonic audiosignalsathigh sampling rates.Inparticular, subjective listening testsperformed bytheMPEGlaudio committee, underverydifficultlistening conditions, haveshownthatevenwitha6-to.! compression ratio,thecodedandoriginalaudiosignalsareperceptually indistinguishable. TheMPEG-l/audio codingstandard achieves thisremarkable performance byex ploitingtwopsychoacoustic characteristics ofthehumanauditory system: 1.Criticalbands. Theinnerear14oftheauditory systemrepresents thepowerspectraofincoming signals onanonlinear scaleintheformoflimitedfrequency bandscalledthecriticalbands.The audiblefrequency band,extending upto20kHz,iscoveredby25criticalbands,whose individual bandwidths increasewithfrequency. Looselyspeaking, theauditorysystemmay bemodeled asaband-pass filterbank,consisting of25overlapping band-pass filterswith bandwidths lessthan100Hzforthelowestaudiblefrequencies andupto5kHzforthe highestaudiblefrequencies. 2.Auditory masking. Auditory masking ornoisemasking isafrequency-domain phenomenon thatariseswhen alow-level signal(themaskee)andahigh-level signal(themasker)occursimultaneously andarecloseenoughtoeachotherinfrequency.Ifthelow-level signalliesbelowamasking threshold, itismadeinaudible (i.e.,masked) bythestrongersignal.Theauditory-masking phenomenon ismostpronounced whenbothsignalslieinthesamecriticalband,andless effectivewhentheylieinneighboring bands. Figure3.33illustrates thedefinition ofmaskingthreshold andrelatedparameters for apairofadjacentfrequency bands;itisassumed thatthemasker(i.e.,thehigh-level signal) liesinside.thedark-shaded criticalband.Thelow-level signalslyinginsidethisdarkarea andbelowthemaskingthreshold aremaskedbythestrongersignal.FromFigure3.33we seethatthemaskingthreshold varieswithfrequency acrossthecriticalband.Accordingly, wemaydefineaminimum masking threshold foracriticalband,belowwhichalllow, levelsignalsthatlieinsidethatbandaremadeinaudible bythestrongersignal.Thepower. difference, expressed indecibels, betweenthemaskerandtheminimum maskingthreshold, istermedthesignal-to-mask ratio(SMR).Figure3.33alsoincludes thesignal-to-noise ratio(SNR)foranR-bitquantizer. Thedifference betweenSMRandSNRisthenoise-to, maskratio(NMR)foranR-bitquantizer asshownby NMR=SMR-SNR (3.85) whereallthreetermsareexpressed indBs.Withinacriticalband,thequantization noise isinaudible aslongastheNMRforthepertinent quantizer isnegative. 3.17MPEGAudio CodingStatUlard 235 Quantization noiselevelMinimum -~--masking thresholdHigh-Ievei----------------- signal (masker) 1---:~[ maskratio Signal-to-JnoiseratioL- ---;;.. Criticalband Neighboring Frequency criticalband FIGURE3.:nIllustrating thedefinitions ofmasking threshold andrelatedparameters. Thehigh levelsignal(masker) liesinsidethedarker-shaded criticalhand,hencethemasking ismore effective inthisbandthanintheneighboring bandshowninlightershading. (Adapted fromNoll (1998)withpermission oftheCRCPress.) Withthisbackground onthepsychoacoustics oftheauditory system,wearenow readytodescribetheoperation oftheMPEG-l/audio codingstandard. Figure3.34shows thebasicblockdiagrams oftheencoderanddecoder. Theencoderconsistsoffourfunc tionalunits:time-to-frequency mapping network, psychoacoustic model,quantizer and coder,andframe-packing unit.Thedecoderconsistsofthreefunctional units:frame unpacking unit,frequency-sample reconstruction network, andfrequency-to-time map pingnetwork. Thepsychoacoustic modelisthusonlynecessary intheencoder. Startingwithadescription oftheencoderfirst,thefunction ofthetime-to-frequency mapping network istodecompose theinputaudiosignalintomultiplesubbands forcod ing.Themapping isperformed inthreelayers,labeledI,II,andIII,whichareofincreasing complexity, delay,andsubjective perceptual performance. Thealgorithm iIilayerIusesa Digital(PCM)-_--i;> audiosignalTime-to frequency mapping networkEncoded bitstream (a) Encoded bitstreamFrequency to-time mapping networkDigital(PCM) audiosignal (b) FIGURE 3.34MPEG/Audio codingsystem.(a)Transmitter. (b)Receiver. 236 CHAPTER 3..PULSE MODULATION band-pass filterbankthatdividestheaudiosignalinto32constant-width subbands; thi filterbankisalsofoundinlayersIIandill.Inlightofourprevious remarksonthenon~ uniformly spacedcriticalbands,thedesignofthisfilterbankisacompromise between computational efficiency andperceptual performance. Thealgorithm inlayerIIisasimple enhancement oflayerI;itimproves thecompression performance bycodingthedatain largergroups.Finally,thelayerillalgorithm ismuchmorerefinedinthatitisdesigned toachievefrequency resolutions closertothepartitions between thecriticalbands. Thepsychoacoustic modelisthekeycomponent intheencoder. Itsfunction isto analyzethespectralcontentoftheinputaudiosignalandtherebycompute thesignal-to. maskratioforeachsubband ineachofthethreelayers.Thisinformation is,inturn,used bythequantizer-coder todecidehowtoapportion theavailable numberofbitsforthe quantization ofthesubband signals.Thisdynamic allocation ofbitsisperformed soasto minimize theaudibility ofthequantization noise.Finally,theframe-packing unitassembles thequantized audiosamplesintoadecodable bitstream. Thedecodersimplyreversesthesignal-processing operations performed intheen coder,converting thereceivedstreamofencoded bitsintoatime-domain audiosignal. Tosumup,theMPEG-l/audio codingstandard represents thestateoftheartinthe codingofaudiosignals.LayerIachievesacompression ratioof4atanapproximate stereo bitrateof384kb/sfortransparent qualityofperformance. Thecorresponding compres sionratiosforlayersIIandIIIare8and12atapproximate stereobitratesof192kb/s and128kb/s,respectively. Thesubjective qualityoftheMPEG-1/audio codingstandard isequivalent tocompact discquality(16-bitPCM)formanytypesofmusic;thecompact disc(CD)istoday'sdefactostandard ofdigitalaudiorepresentation. I3.18Summary andDiscussion Inthischapterweintroduced twofundamental andcomplementary processes: I>-Sampling, whichoperates inthetimedomain; thesampling processisthelinkbe· tweenananalogwaveform anditsdiscrete-time representation. I>Quantization, whichoperates intheamplitude domain; thequantization processis thelinkbetween ananalogwaveform anditsdiscrete-amplitude representation. Thesampling processbuildsonthesamplingtheorem, whichstatesthatastrictlyband limitedsignalwithnofrequency components higherthanWHzisrepresented uniquely byasequence ofsamplestakenatauniformrateequaltoorgreaterthantheNyquistrate of2Wsamplespersecond.Thequantization processexploitsthefactthatanyhuman sense,asultimatereceiver, canonlydetectfiniteintensity differences. Thesampling processisbasictotheoperation ofallpulsemodulation systems, which maybeclassified intoanalogpulsemodulation anddigitalpulsemodulation. Thedistin guishing featurebetweenthemisthatanalogpulsemodulation systemsmaintain acontin uousamplitude representation of themessage signal,whereas digitalpulsemodulation systemsalsoemployquantization toprovidearepresentation ofthemessagesignalthatis discreteinbothtimeandamplitude. Analogpulsemodulation resultsfromvaryingsomeparameter ofthetransmitted pulses,suchasamplitude, duration, orposition, inwhichcasewespeakofpulse-amplitude modulation (PAM),pulse-duration modulation (PDM),orpulse-position modulation (PPM),respectively. Intime-division multiplexing (TDM)ofseveralchannels, signalpro cessingusuallybeginswithPAM.TousePDMorPPMinsuchanapplication, wehave toensurethatfull-scale modulation willnotcauseapulsefromonemessagesignaltoenter atimeslotbelonging toanothermessagesignal.Thisrestriction resultsinawasteful use oftimespaceintelephone systemsthatarecharacterized byhighpeakfactors,whichis onereasonfornotusingPDMorPPMintelephony. Also,despitethefactthatPPMis moreefficientthanPDM,theybothfallshortoftheidealsystemforexchanging trans missionbandwidth forimproved noiseperformance. Digitalpulsemodulation systemstransmit analogmessage signalsasasequence of codedpulses,whichismadepossiblethroughthecombined useofsampling andquanti zation.Pulse-code modulation isanimportant formofdigitalpulsemodulation thatis endowed withsomeuniquesystemadvantages, which,inturn,havemadeitthestandard methodofmodulation forthetransmission ofsuchanalogsignalsasvoiceandvideo signals.Theadvantages ofpulse-code modulation includerobustness tonoiseandinter ference,efficientregeneration ofthecodedpulsesalongthetransmission path,andauni formformatfordifferent kindsofbaseband signals. Indeed,itisbecauseofthislistofadvantages uniquetopulse-code modulation that ithasbecomethemethodofchoicefortheconstruction ofpublicswitched telephone networks (PSTNs). Inthiscontext, thereadershouldcarefully notethatthetelephone channelviewedfromthePSTNtoanInternetserviceprovider, forexample, isnonlinear duetotheuseofcompanding and,mostimportantly, itisentirelydigital.Thisobservation hasasignificant impactonthedesignofhigh-speed modems forcommunication between acomputer userandserver,asdiscussed inChapter6. Deltamodulation anddifferential pulse-code modulation aretwootherusefulforms ofdigitalpulsemodulation. Theprincipal advantage ofdeltamodulation isthesimplicity ofitscircuitry. Incontrast, differential pulse-code modulation employs increased circuit complexity toreducechannelbandwidth. Theimprovement isachieved byusingtheidea ofprediction toremoveredundant symbols fromanincoming datastream.Afurther improvement intheoperation ofdifferential pulse-code modulation canbemadethrough theuseofadaptivity toaccountforstatistical variations intheinputdata.Bysodoing, bandwidth requirement isreduced significantly without seriousdegradation insystem performance. Unlikeadaptive differential pulse-code modulation, theMPEGaudiocodingstan dardachievesthecompression ofstereophonic audiosignalsinatransparent, perceptually losslessmanner. Thisimpressive performance isrealizedbyexploiting certainpsycho acousticproperties oftheauditory system. Atthispointinthediscussion, itisinformative totakeacriticallookatthedifferent formsofpulsemodulation thatwehavedescribed inthischapter. Inastrictsense,the termpulsemodulation isamisnomer inthatallofitsdifferent forms,bethey analog or digital,areinfactsourcecodingtechniques. Wesaythisforthesimplereasonthata message signalremainsabaseband signalafterundergoing allthechangesinvolved ina pulsemodulation process.Thebaseband natureofapulse-modulated signalisexemplified bythefactthat,irrespective ofitsexactdescription, itcanbetransmitted overabaseband channelofadequate bandwidth. Indeed,thematerial presented inthenextchapteris devotedtothebaseband transmission ofdatarepresented byasequence ofpulses. Itisalsoimportant torecognize thatpulsemodulation techniques arelossyinthe sensethatsomeinformation islostasaresultofthesignalrepresentation thattheyperform. Forexample, inpulse-amplitude modulation, thecustomary practice istouselow-pass anti-alias filteringpriortosampling; insodoing,information islostbyvirtueofthefact thathigh-frequency components considered tobeunessential areremoved bythefilter. Thelossynatureofpulsemodulation ismostvividlyseeninpulse-code modulation that ischaracterized bythegeneration ofquantization noise(i.e.,distortion); thetransmitted 238 CHAPTER 3..PULSEMODULATION sequence ofencoded pulsesdoesnothavetheinfiniteprecision neededtorepresent Can. tinuoussamples exactly. Nevertheless, thelossofinformation incurred bytheuseofa pulse-modulation processisunderthedesigner's controlinthatitcanbemadesmall enoughforittobe nondiscernible bytheenduser. Thematerial presented inthischapteronpulsemodulation hasbeenfromasignal processing perspective. Wewillrevisitpulse-code modulation inChapter 9,whichisde. votedtoinformation-theoretic considerations ofcommunication systems.InsodoingWe willdevelopdeeperinsightintoitsoperation asasourcecodingtechnique. INOTES ANDREFERENCES 1.Theclassicbookonpulsemodulation isBlack(1953).Amoredetailedtreatment ofthe subjectispresented inthebookbyRowe(1965).ForthenoiseanalysisofaPPMsystem seethethirdeditionofthebookbyHaykin(1994). ' 2.Pulse-code modulation wasinvented byReevesin1937.Forahistorical accountofthis invention, seethepaperbyReeves(1975).ThebookbyJayantandNoll(1984)presents detailed treatment ofpulse-code modulation, differential pulse-code modulation, deha modulation, andtheirvariants. ThebookeditedbyJayant(1976)provides acolleetionof earlypaperswrittenonwaveform quantization andcoding. 3.Foradetaileddiscussion ofquantization noiseinPCMsystems, seethepaperbyBennett (1948)andalsothebookbyRowe(1965,pp.311-321). 4.Thetwonecessary conditions ofEquations (3.42)and(3.47)foroptimality ofascalar quantizer werereported independently byLloyd(1957)andMax(1960),hencethename "Lloyd-Max quantizer." Thederivation ofthesetwooptimality conditions presented in thischapterfollowsthebookbyGershoandGray(1992). 5.TheJL-lawusedforsignalcompression isdescribed inSmith(1957).The JL-Iawisusedin theUnitedStates,Canada, andJapan.InEurope,theA-lawisusedforsignalcompression; thiscompression lawisdescribed inCattermode (1969,pp.133-140). Foradiscussianof theJL-lawandA-law,seealsothepaperbyKaneko(1970). 6.Foradescription oftheoriginalversionoftheT1-carrier system,seethepaperbyFultz andPenick(1965).Thedescription giveninExample 3.2isbasedonanupdated version ofthissystem;seeHenning andPan(1972). 7.TheNorthAmerica/Japan standards fordigitalmultiplexers Wereoriginally adopted by AT&T.Another setofstandards hasbeenadoptedbyCCITTfortherestofthewarld. TheCCITTdigitalhierarchy issimilartothatdescribed inSection3.10,exceptfarcertain changesinthespecifications ofthenumberofchannelinputstothefivedigitalmultiplexers andtheirindividual bitrates.FordetailsoftheCCITTdigitalhierarchy, seeCouch(1997). 8.Fortheoriginalpapersondeltamodulation, seeSchouten, Dejager, andGreefkes (1952) andDejager(1952).Forareviewpaperondeltamodulation, seethepaperbySchindler (1970). 9.Delta-sigma modulation isdescribed inthebookbyJayantandNoll(1984,pp.399-400); seealsothepaperbyInose,Yasuda,andMurakami (1962). 10.Differential pulse-code modulation wasinvented byCutler;theinvention isdescribed ina patentissuedin1952.Foracomparison ofthenoiseperformances ofPCMandDPcM, seethepaperbyJayant(1974);seealsoRabinerandSchafer(1978,Chapter5). 11.Foradiscussion ofadaptivedifferential pulse-code modulation, seeJayantandNoll(1984). Problems 239 12.Theadaptive deltamodulation algorithm (ADM)ofEquation (3.84)isthecorrected ver sionofanalgorithm presented inSklar(1988,p.641).Sklar'salgorithm wasadaptedfrom anearlierpaperbySongetal.(1971),whereanoptimum ADMsystemisderived; the highlynonlinear equations characterizing theoptimum systemareapproximated inthe latterpaperbypiecewise-linear equations forthepurposeofimplementation. 13.TheMPEG-1/audio codingstandard isdescribed inthepapersbyBrandenburg andStoll (1994),Pan(1993),andthearticlebyPeterNollinthehandbook onDigitalSignalPro cessingeditedbyMadisetti andWilliams (1998);thelatterarticlealsodiscusses thefollow upstandards toMPEG-1. Inparticular, theMPEG-2 offersstereophonic audiocodingat sampling rateslowerthanMPEG-1. 14.Theear,theorganofhearing, responds toincoming acoustical waves.Ithasthreemain parts,withtheirfunctions assummarized here: ..Theouterearaidsinthecollection ofsounds. ..Themiddleearprovidesanacousticimpedance matchbetweentheairandthecochleafluids, therebyconveying thevibrations ofthetympanic membrane (eardrum) duetotheincoming soundstotheinnerearinanefficientmanner. ..Theinnerearconvertsthemechanical vibrations fromthemiddleeartoanelectrochemical orneuralsignalfortransmission tothebrainforprocessing. LPROBLEMS Sampling Process 3.1Anarrowband signalhasabandwidth of10kHzcentered onacarrierfrequency of 100kHz.Itisproposed torepresent thissignalindiscrete-time formbysampling itsin phaseandquadrature components individually. Whatistheminimum sampling ratethat canbeusedforthisrepresentation? Justifyyouranswer.Howwouldyoureconstruct the originalnarrowband signalfromthesampled versions ofitsin-phase andquadrature components? 3.2Innaturalsampling, ananalogsignalg(t)ismultiplied byaperiodictrainofrectangular pulsescrt).Giventhatthepulserepetition frequency ofthisperiodic trainisJ,andthe duration ofeachrectangular pulseisT(withI,T«1),dothefollowing: (a)Findthespectrum ofthesignals(t)thatresultsfromtheuseofnaturalsampling; you mayassumethattimet=0corresponds tothemidpoint ofarectangular pulsein crt). (b)Showthattheoriginalsignalm(t)mayberecovered exactlyfromitsnaturally sampled version,provided thattheconditions embodied inthesampling theorem aresatisfied. 3.3SpecifytheNyquistrateandtheNyquistintervalforeachofthefollowing signals: (a)g(t)=sinc(200t) (b)g(t)=sinc2(200t) (c)g(t)=sinc(200t) +sinc2(200t) 3.4(a)Plotthespectrum ofaPAMwaveproduced bythemodulating signal m(t)=Amcos(2'11"I",t) assuming amodulation frequency 1m=0.25Hz,sampling periodT,=1s,andpulse duration T=0.45s. (b)Usinganidealreconstruction filter,plotthespectrum ofthefilteroutput.Compare thisresultwiththeoutputthatwouldbeobtained iftherewerenoaperture effect. IfI<BT IfI>BT240 CHAPTER 3..PULSE MODUlATION Pulse-Amplitude Modulation 3.5FigureP3.5showstheidealized spectrum ofamessage signalm(t).Thesignalissampled atarateequalto1kHzusingflat-toppulses,witheachpulsebeingofunitamplitude andduration 0.1ms.Determine andsketchthespectrum oftheresulting PAMsignal. ___~~A'-.o__,,,,n,ci---4"'"O-o--f<Hz) FIGUREP3.5 3.6Inthisproblem, weevaluate theequalization neededfortheaperture effectinaPAM system.Theoperating frequency f=fj2,whichcorresponds tothehighestfrequency component ofthemessage signalforasampling rateequaltotheNyquist rate.Plot I/sinc(O.5TIT,) versusTIT"andhencefindtheequalization neededwhenTIT,=0.1. 3.7Consider aPAMwavetransmitted through achannelwithwhiteGaussian noiseand minimum bandwidth BT=lI2T"whereT,isthesampling period.Thenoiseisofwo meanandpowerspectraldensityNo/2.ThePAMsignalusesastandard pulseg(t)with itsFouriertransform definedby G(f)={2~T' 0, Byconsidering afull-load sinusoidal modulating wave,showthatPAMandbaseband· signaltransmission haveequalsignal-to-noise ratiosforthesameaveragetransmitted power. 3.8Twenty-four voicesignalsaresampled uniformly andthentime-division multiplexed. The sampling operation usesflat-topsampleswith1J.I1lduration. Themultiplexing operation includes provision forsynchronization byaddinganextrapulseofsufficient amplitude andalso1p.sduration. Thehighestfrequency component ofeachvoicesignalis3.4kHz, (a)Assuming asampling rateof8kHz,calculate thespacingbetween successive pulses ofthemultiplexed signal. (b)Repeatyourcalculation assuming theuseofNyquistratesampling. 3.9Twelvedifferent messagesignals,eachwithabandwidth of10kHz,aretobemultiplexed andtransmitted. Determine theminimum bandwidth required foreachmethodifthe multiplexing/modulation methodusedis (a)FDM,SSB. (b)TDM,PAM. 3.10APAMtelemetry systeminvolves themultiplexing offourinputsignals:Silt),i=1,2,3, 4.TwoofthesignalsSl(t)andS2(t)havebandwidths of80Hzeach,whereastheremaining twosignalsS3(t)andS4(t)havebandwidths of1kHzeach.ThesignalsS3(t)andS4(t)ace eachsampledattherateof2400samplespersecond.Thissampling rateisdividedby2R (i.e.,anintegerpowerof2)toderivethesampling rateforSl(t)andS2(t). (a)Findthemaximum valueofR. (b)·UsingthevalueofRfoundinpart(a),designamultiplexing systemthatfirstmulti· plexesSl(t)andS2{t)intoanewsequence, ss(t),andthenmultiplexes S3(t),S4(t),and ss(t). Problems 241 LineCodes 3.11Inthisproblem wederivetheformulas usedtocompute thepowerspectraofFigure3.16 forthefivelinecodesdescribed inSection3.7.Inthecaseofeachlinecode,thebit duration isTbandthepulseamplitude Aisconditioned tonormalize theaveragepower ofthelinecodetounityasindicated inFigure3.16.Assumethatthedatastreamis randomly generated, andsymbols0and1areequallylikely. Derivethepowerspectraldensities oftheselinecodesassummarized here: (a)Unipolar nonreturn-to-zero signals: A'Tb., (1 )S(f)=-4-slnc-(fTb)1+T ba(f) (b)Polarnonreturn-to-zero signals: S(f)=A'Tbsinc2(fTb) (c)Unipolar return-to-zero signals: S(f)=A2 Tbsinc2(fTb)[1+..!..ia(f-!!...)J16 2 Tbn--ro Tb (d)Bipolarreturn-to-zero signals: S(f)=A:Tbsinc2(f~b) sin2(7TfTb) (e)Manchester-encoded signals: S(f)=A2Tbsinc2(f~b) sinl(7T~Tb) Hence,confinnthespectralplotsdisplayed inFigure3.16. 3.12Suppose arandombinarydatastream(withequiprobable symbols) isdifferentially en codedandthentransmitted usingoneofthefivelinecodesdescribed inProblem 3.11. Howisthepowerspectraldensityofthetransmitted dataaffectedbytheuseofdifferential encoding? Justifyyouranswer. 3.13Arandomly generated datastreamconsistsofequiprobable binarysymbols0and1.Itis encodedintoapolarnonreturn-to-zero wavefonn witheachbinarysymbolbeingdefined asfollows: s(t)={cos(;:), 0,TbTb-2<t:s2 otherwise (a)Sketchthewaveform sogenerated, assuming thatthedatastreamis00101110. (b)Deriveanexpression forthepowerspectraldensityofthissignal,andsketchit.Hint: useEquation (1.52). (c)Compare thepowerspectraldensityofthisrandomwaveform withthatdefinedin part(b)ofProblem3.11. 3.14Giventhedatastream1110010100, sketchthetransmitted sequence ofpulsesforeach ofthefollowing linecodes: (a)Unipolar nonreturn-to-zero (b)Polarnonreturn-to-zero (c)Unipolar return-to-zero (d)Bipolarreturn-to-zero (e)Manchester code 242 CHAPTER 3"PULSE MODULATION 3.15Suppose thebinarydatastreamconsidered inProblem 3.14isdifferentially encoded and thentransmitted usingoneofthefivelinecodesconsidered therein.Sketcheachofth transmitted datastreams, assuming theuseofsymbol1forthereference bit.Howisth: resultaffectedifsymbolaisusedforthereference bit? Pulse-Code Modulation 3.16Aspeechsignalhasatotalduration of10s.Itissampledattherateof8kHzandthen encoded. Thesignal-to-(quantization) noiseratioisrequired tobe40dB.Calculate the minimum storagecapacity neededtoaccommodate thisdigitized speechsignal. 3.17Consider auniform quantizer characterized bytheinput-output relationillustrated in Figure3.10a.AssumethataGaussian-distributed randomvariablewithzeromeanand unitvariance isappliedtothisquantizer input. (a)Whatistheprobability thatthe amplitude oftheinputliesoutsidetherange-4to +4? (b)Usingtheresultofpart(a),showthattheoutputsignal-to-noise ratioofthequantizer isgivenby (SNR)o=6R-7.2dB whereRisthenumberofbitspersample.Specifically, youmayassumethatthe quantizer inputextendsfrom-4to+4.Compare theresultofpart(b)withthar obtained inExample 3.1. 3.18APCMsystemusesauniformquantizer followed bya7-bitbinaryencoder. Thebitrate ofthesystemisequalto50X106b/s. (a)Whatisthemaximum message bandwidth forwhichthesystemoperates satisfactorily? (b)Determine theoutputsignal-to-(quantization) noiseratiowhenafull-load sinusoidal modulating waveoffrequency 1MHzisappliedtotheinput. 3.19Showthat,withanonuniform quantizer, themean-square valueofthequantization error isapproximately equalto(1/12)~i ~?iP;,where.6.,istheithstepsizeandPiistheproba· bilitythattheinputsignalamplitude lieswithintheithinterval.Assumethatthestepsize .6.iissmallcompared withtheexcursion oftheinputsignal. 3.20(a)Asinusoidal signal,withanamplitude of3.25volts,isappliedtoauniformquantizer ofthemidtread typewhoseoutputtakesonthevalues0,±1,±2,±3volts.Skerch thewaveform oftheresulting quantizer outputforonecomplete cycleoftheinput. (b)Repeatthisevaluation forthecasewhenthequantizer isofthemidrisetypewhose outputtakesonthevalues=0.5,±1.5, ±2.5, ±3.5volts. 3.21Thesignal m(t)= 6sin(21Tt) volts istransmitted usinga4-bitbinaryPCMsystem.Thequantizer isoftbemidrisetype,with astepsizeof1volt.Sketchtberesulting PCMwaveforonecomplete cycleoftheinput. Assumeasampling rateoffoursamplespersecond,withsamplestakenatt=±1/8, ±3/8,±5/8,...,seconds. 3.22FigureP3.22showsaPCMsignalinwhichthe amplitude levelsof+1voltand-1volt areusedtorepresent binarysymbols 1and0,respectively. Tbecodewordusedconsists ofthreebits.Findthesampled versionofananalogsignalfromwhichthisPCMsignal isderived. problems 243 +1 Q---'------'--+++-+--L-t-'--'-1'--'--f-f-t--L--'---+--'-- -I FIGllREP3.22 3.23Consider achainof(n-1)regenerative repeaters, withatotalofnsequential decisions madeonabinaryPCMwave,including thefinaldecisionmadeatthereceiver. Assume thatanybinarysymboltransmitted throughthesystemhasanindependent probability PIofbeinginvertedbyanyrepeater. LetPnrepresent theprobability thatabinarysymbol isinerroraftertransmission throughthecomplete system. (a)Showthat Pn=HI-(1-2p,)"] (b)IfPIisverysmallandnisnottoolarge,whatisthecorresponding valueofPn? 3.24Discussthebasicissuesinvolved inthedesignofaregenerative repeater forpulse-code modulation. DeltaModulation 3.25Consider atestsignalm(t)definedbyahyperbolic tangentfunction: m(t)=Atanh(f3t) whereAandf3areconstants. Determine theminimum stepsizeLlfordeltamodulation ofthissignal,whichisrequired toavoidslopeoverload. 3.26Consider asinewaveoffrequency fmandamplitude Am'whichisappliedtoadelta modulator ofstepsizeLl.Showthatslope-overload distortion willoccurif LlA>-- m27rfmT, whereT,isthesampling period.Whatisthemaximum powerthatmaybettansmitted withoutslope-overload distortion? 3.27Alineardeltamodulator isdesigned tooperateonspeechsignalslimitedto3.4kHz.The specifications ofthemodulator areasfollows: ~Sampling rate=10fNyqu~" wherefNyqu'"istheNyquistrateofthespeechsignal. ~StepsizeLl=100mY. Themodulator istestedwithaI-kHzsinusoidal signal.Determine themaximum ampli tudeofthistestsignalrequired toavoidslopeoverload. 3.28Inthisproblem, wederiveanempirical formulafortheaveragesignal-to-(quantization) noiseratioofaDMsystemwithasinusoidal signalofamplitude Aandfrequency fmas thetestsignal.Assumethatthepowerspectraldensityofthegranular noisegeneratedby thesystemisgoverned bytheformula Lll SN!j)=6j, wherej,isthesampling rateandLlisthestepsize.(Notethatthisformulaisbasically thesameasthatforthepowerspectraldensityofquantization noiseinaPCMsystem 244 CHAPTER 3"'PULSE MODUlATION with!JJ2 forPCMbeingreplaced byAforDM.)TheDMsystemisdesigned tohandl analogmessagesignalslimitedtobandwidth W. e (a)Showthattheaveragequantization noisepowerproduced bythesystemis N=41T2A2f~ W 3f; whereitisassumed thatthestepsizeAhasbeenchoseninaccordance withthe formulausedinProblem 3.27soastoavoidslopeoverload. (b)Hencedetermine thesignal-to-(quantization) noiseratiooftheDMsystemfora sinusoidal input. 3.29Consider aDMsystemdesigned toaccommodate analogmessagesignalslimitedtoband. widthW=5kHz.Asinusoidal testsignalofamplitude A=1voltandfrequency fm=1kHzisappliedtothesystem.Thesampling rateofthesystemis50kHz. (a)Calculate thestepsizeArequired tominimize slopeoverload. (b)Calculate thesignal-to-(quantization) noiseratioofthesystemforthespecified sinusoidal testsignal. Forthesecalculations, usetheformulas derivedinProblems 3.27and3.28. 3.30Consider alow-pass signalwithabandwidth of3kHz.Alineardeltamodulation system, withstepsizeA=O.lV,isusedtoprocessthissignalatasampling ratetentimesthe Nyquistrate. (a)Evaluate themaximum amplitude ofatestsinusoidal signaloffrequency 1kHz, whichcanbeprocessed bythesystemwithoutslope-overload distortion. (b)Forthespecifications giveninpart(a),evaluatetheoutputsignal-to-noise ratiounder (i)prefiltered, and(ii)postfiltered conditions. LinearPrediction 3.31Aone-step linearpredictor operates onthesampled versionofasinusoidal signal.The sampling rate isequalto10fowherefaisthefrequency ofthesinusoid. Thepredictor hasasinglecoefficient denotedbyW1' (a)Determine theoptimum valueofw,required tominimize theprediction error variance. (b)Determine theminimum valueoftheprediction errorvariance. 3.32Astationary processX(t)hasthefollowing valuesforitsautocorrelation function: Rx(O)=1 Rx(l)=0.8 Rx(2)=0.6 Rx(3)=0.4 (a)Calculate thecoefficients ofanoptimum linearpredictor involving theuseofthree unit-delays. (b)Calculate thevariance oftheresulting prediction error. 3.33Repeatthecalculations ofProblem 3.32,butthistimeusealinearpredictor withtwO unit-delays. Compare theperformance ofthissecondoptimum linearpredictor withthat considered inProblem 3.32. Differential Pulse-Code Modulation 3.34ADPCMsystemusesalinearpredictor withasingletap.Thenormalized autocorrelation function oftheinputsignalforalagofonesampling intervalis0.75.Thepredictor IS Problems 245 designed tominimize theprediction errorvariance. Determine theprocessing gainattained bytheuseofthispredictor. 3.35Calculate theimprovement inprocessing gainofaDPCMsystemusingtheoptimized three-tap linearpredictor ofProblem 3.32overthatoftheoptimized two-taplinearpre dictorofProblem 3.33. Forthiscalculation, usetheautocorrelation function valuesof theinputsignalspecified inProblem 3.32. 3.36Inthisproblem, wecompare theperformance ofaDPCMsystemwiththatofanordinary PCMsystemusingcompanding. Forasufficiently largenumberofrepresentation levels,thesignal-to-{ quantization) noiseratioofPCMsystems,ingeneral,isdefinedby 1010g,0{SNR)o =a+6ndB where2"isthenumberofrepresentation levels.Foracompanded PCMsystemusingthe wlaw,theconstant aisitselfdefinedby a=4.77-2010glO{log(1+p,))dB ForaDPCMsystem,ontheotherhand,theconstant aliesintherange- 3<a<15 dBs.Theformulas quotedhereinapplytotelephone-quality speechsignals. Compare theperformance oftheDPCMsystemagainstthatofthep,-companded PCMsystemwithp,=255foreachofthefollowing scenarios: (a)Theimprovement in{SNR)orealizedbyDPCMovercompanded PCMforthesame numberofbitspersample. (b)Thereduction inthenumberofbitspersamplerequired byDPCM,compared tothe companded PCMforthesame(SNR)o. 3.37IntheDPCMsystemdepicted inFigureP3.37,showthatintheabsenceofchannelnoise, thetransmitting andreceiving prediction filtersoperateonslightlydifferent inputsignals. + ~i~lt--"..,...-----;;..( I Transmitter FIGUREP3.37I)------'t--;;>- Output + Receiver Computer Experiments 3.38Asinusoidal signaloffrequency fa=104/27THzissampledattherateof8kHzandthen appliedtoasample-and-hold circuittoproduceaflat-topped PAMsignals(t)withpulse durationT=500JLS. (a)Compute thewaveform ofthePAMsignals{t). (b)Compute1S(f)I,denoting themagnitude spectrum ofthePAMsignals(t). (c)Compute theenvelope ofIS(f)I.Henceconfirmthatthefrequency atwhichthis envelope goesthroughzeroforthefirsttimeisequalto(liT)=20kHz. 3.39Inthisproblem, weusecomputer simulation tocompare theperformance ofacompanded PCMsystemusingthep,-lawagainstthatofthecorresponding systemusingauniform k=1,2,...,P246 CHAPTER 3111PULSE MODULA1:10N quantizer. Thesimulation istobeperformed forasinusoidal inputsignalofvaryin amplitude. g (a)Usingthep,-lawdescribed inTable3.4,plottheoutputsignal-to-noise ratioas function oftheinputsignalcto-noise ratio,bothratiosbeingexpressed indecibels. a (b)Compare theresultsofyourcomputation inpart(a)withauniformquantizer havin 256representation levels. g 3.40Inthisexperiment westudythelinearadaptive prediction ofasignalx[n]governed by thefollowing recursion: x[n]=0.8x[n 1]-O.lx[n-2]+O.lv[n] wherev[n]isdrawnfromadiscrete-time whitenoiseprocessofzeromeanandunit variance. (Aprocessgenerated inthismannerisreferredtoasanautoregressive process ofordertwo.)Specifically, theadaptive prediction isperformed usingthenormalized LMs algorithm definedby p x[n]=Lwk[n]x[n -k] k~lern]=x[n]-x[n] wk[n+1]=wk[n]+ (pp,)x[n-k]e[n],Lx2[n-k] k=l wherepistheprediction orderandp,isthenormalized step-sizeparameter. Theimportant pointtonotehereisthatp,isdimensionless andstabilityofthealgorithm isassuredby choosing itinaccordance withtheformula Thealgorithm isinitiated bysetting Wk[O]=0forallk Thelearningcurveofthealgorithm isdefinedasaplotofthemean-square error versusthenumberofiterations nforspecified parameter values,whichisobtained by averaging theplotofe'[n]versusnoveralargenumberofdifferent realizations ofthe algorithm. (a)Plotthelearningcurvesfortheadaptive prediction ofx[n]forafixedprediction order p=5andthreedifferent valuesofstep-size parameter: p,=0.0075,0.05, and0.5. (b)Whatobservations canyoumakefromthelearningcurvesofpart(a)? BASEBAND PULSE TRANSMISSION Thischapterdiscusses thetransmission ofdigitaldataoverabaseband channel, with emphasis onthefollowing topics: ~Thematched filter,whichistheoptimum systemfordetecting aknownsignalinadditive whiteGaussian noise. ~Calculation ofthebiterrorrateduetothepresenceofchannelnoise. ~Intersymbol interference, whichariseswhenthechannelisdispersive asiscommonly the caseinpractice. ~Nyquist's criterion fordistortionless baseband datatransmission. ~Correlative-level codingorpartial-response signaling forcombatting theeffectsof intersymbol interference. ~Digitalsubscriber lines. ~Equalization ofadispersive baseband channel. ~Theeyepatternfordisplaying thecombined effectsofintersymbol interference and channelnoiseindatatransmission. I4.1Introduction InChapter3wedescribed techniques forconverting ananaloginformation-bearing signal intodigitalform.Thereisanotherwayinwhichdigitaldatacanariseinpractice: The datamayrepresent theoutputofasourceofinformation thatisinherently discretein nature(e.g.,adigitalcomputer). Inthischapterwestudythetransmission ofdigitaldata (ofwhatever origin)overabaseband channel.1Datatransmission overaband-pass channel usingmodulation iscoveredinChapter6. Digitaldatahaveabroadspectrum withasignificant low-frequency content.Base bandtransmission ofdigitaldatatherefore requirestheuseofalow-pass channelwitha bandwidth largeenoughtoaccommodate theessential frequency contentofthedata stream.Typically, however, thechannelisdispersive inthatitsfrequency response deviates fromthatofanideallow-pass filter.Theresultofdatatransmission oversuchachannel isthateachreceivedpulseisaffectedsomewhat byadjacentpulses,therebygivingriseto acommon formofinterference calledintersymbol interference (lSI).Intersymbol interfer enceisamajorsourceofbiterrorsinthereconstructed datastreamatthereceiveroutput. Tocorrectforit,controlhastobeexercised overthepulseshapeintheoverallsystem. Thusmuchofthematerialcoveredinthischapterisdevotedtopulseshapinginoneform oranother. 247 248 CHAPTER 4..BASEBAND PULSE TRANSMISSION Another sourceofbiterrorsinabaseband datatransmission systemistheubiquitous channel noise.Naturally, noiseandlSIariseinthesystemsimultaneously. However, to understand howtheyaffecttheperformance ofthesystem,wefirstconsider themsepa_ rately;lateroninthechapter, westudytheircombined effects. Wethusbeginthechapterbydescribing afundamental resultincommunication theory,whichdealswiththedetection ofapulsesignalofknownwaveform thatisim mersedinadditive whitenoise.Thedevicefortheoptimum detection ofsuchapulse involves theuseofalinear-time-invariant filterknownasamatched filter/whichisso calledbecauseitsimpulseresponse ismatched tothepulsesignal. I4.2Matched Filter Abasicproblemthatoftenarisesinthestudyofcommunication systemsisthatofdetecting apulsetransmitted overachannelthatiscorrupted bychannelnoise(i.e.,additive noise atthefrontendofthereceiver). Forthepurposeofthediscussion presented inthissection, weassumethatthemajorsourceofsystemlimitation isthechannelnoise. Consider thenthereceivermodelshowninFigure4.1,involving alineartime-invar iantfilterofimpulseresponse h(t).Thefilterinputx(t)consistsofapulsesignalg(t} corrupted byadditivechannelnoisew(t),asshownby x(t)=g(t)+w(t),O:st:sT (4.1) whereTisanarbitrary observation interval. Thepulsesignalg(t)mayreptesent abinary symbol1or0inadigitalcommunication system.Thew(t)isthesamplefunction ofa whitenoiseprocessofzeromeanandpowerspectraldensityNo/2.Itisassumed thatthe receiverhasknowledge ofthewaveform ofthepulsesignalg(t).Thesourceofuncertainty liesinthenoisew(t).Thefunction ofthereceiveristodetectthepulsesignalg(t)inan optimum manner, giventhereceived signalx(t).Tosatisfythisrequirement, wehaveto optimize thedesignofthefiltersoastominimize theeffectsofnoiseatthefilteroutput insomestatistical sense,andtherebyenhance thedetection ofthepulsesignalg(t). Sincethefilterislinear,theresulting outputy(t)maybeexpressed as y(t)=galt)+n(t) (4.2) wheregalt)andn(t)areproduced bythesignalandnoisecomponenrs oftheinputx(t}, respectively. Asimplewayofdescribing therequirement thattheoutputsignalcomponent galt)beconsiderably greaterthantheoutputnoisecomponent n(t)istohavethefilter maketheinstantaneous powerintheoutputsignalgalt),measured attimet=T,aslarge aspossiblecompared withtheaveragepoweroftheoutputnoisen(t).Thisisequivalent tomaximizing thepeakpulsesignal-to-noise ratio,definedas (4.3) Signal g(l)Lineartime invariantfilterof impulseresponse h{t)y(,)~ Sampleat timet=T Whitenoise W(I) FIGURE4.1Linearreceiver. 4.2Matched Filter 249 (4.4)whereIgo(T) 12istheinstantaneous powerintheoutputsignal,Eisthestatistical expec tationoperator, andE[n2(t)]isameasureoftheaverageoutputnoisepower.Therequire mentistospecifytheimpulseresponse hit)ofthefiltersuchthattheoutputsignal-to noiseratioinEquation (4.3)ismaximized. LetG(f)denotetheFouriertransform oftheknownsignalg(t),andH(f)denote thefrequency response ofthefilter.ThentheFouriertransform oftheoutputsignalgo(t) isequaltoH(f)G(f), andgalt)isitselfgivenbytheinverseFouriertransform galt)=fooH(f)G(f) exp(j27T'ft) df Hence,whenthefilteroutputissampled attimet=T,wehave(intheabsenceofchannel noise) (4.5) Igo(TW =IfooH(f)G(f) exp(j27T'fT) dfl2 Consider nexttheeffectonthefilteroutputduetothenoisew(t)actingalone.The powerspectraldensitySN(f)oftheoutputnoisen(t)isequaltothepowerspectraldensity oftheinputnoisewit)timesthesquaredmagnitude response IH(fW(seeSection1.7). Sincew(t)iswhitewithconstant powerspectraldensityNo/2,itfollowsthat (4.6) Theaveragepoweroftheoutputnoisen(t)istherefore E[n2(t)]=fooSN(f)df =~of~IH(fWdf(4.7) (4.8) 71=Thussubstituting Equations (4.5)and(4.7)into(4.3),wemayrewritetheexpression forthepeakpulsesignal-to-noise ratioas If~H(f)G(f) exp(j27T'fT) dfr ~ofooIH(fWdf Ourproblem istofind,foragivenG(f),theparticular formofthefrequency response H(f)ofthefilterthatmakes 71amaximum. Tofindthesolution tothisoptimization problem, weapplyamathematical resultknownasSchwarz's inequality tothenumerator ofEquation (4.8). Aderivation ofSchwarz's inequality isgiveninChapter5.Fornowitsufficestosay thatifwehavetwocomplex functions (f>t(x)and(h(x)intherealvariablex,satisfying theconditions and 250 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION thenwemaywrite (rooq,,(X)q,2(X) dxl2~roo1q,,(XW dxroo1q,2(XW dx (4.91 Theequality in(4.9)holdsif,andonlyif,wehave (4.]0) wherekisanarbitrary constant, andtheasteriskdenotescomplex conjugation. Returning totheproblem athand,wereadilyseethatbyinvoking Schwarz's in. equality (4.9),andsettingq,l(X)=H(f)andq,2(X)=G(f)exp(j'TrfT), thenumerator in Equation (4.8)mayberewritten as IrooH(f)G(f) exp(j2'TrfT) dfl2~rooIH(fWdfrooIG(fWdf(4.11) UsingthisrelationinEquation (4.8),wemayredefinethepeakpulsesignal-to-noise ratio as (4.12) Theright-hand sideofthisrelationdoesnotdependonthefrequency response H(fIof thefilterbutonlyonthesignal energy andthenoisepowerspectraldensity.Consequently, thepeakpulsesignal-to-noise ratio1)willbeamaximum whenH(f)ischosensothatthe equality holds;thatis, 2fOO 1)max=No .00IG(f)12df (4.13) Correspondingly, H(f)assumes itsoptimum valuedenoted byHopt(f).Tofindthisopti mumvalueweuseEquation (4.10),which,forthesituation athand,yields Hopt(f)=kG*(f)exp(-j2'TrfT) (4.14) whereG*(f)isthecomplex conjugate oftheFouriertransform oftheinputsignalg(t), andkisascalingfactorofappropriate dimensions. Thisrelationstatesthat,exceptfot thefactorkexp(-j2'TrfT),thefrequency response oftheoptimum filteristhesameasthe complex conjugate oftheFouriertransform oftheinputsignal. Equation (4.14)specifiestheoptimum filterinthefrequency domain.Tocharacterize itinthetimedomain, wetaketheinverseFouriertransform ofHop,(f)inEquation (4,14) toobtaintheimpulseresponse oftheoptimum filteras hop,(t)=krooG*(f)exp[-j2'Trf(T -t)]df (4.15) Sinceforarealsignalg(t)wehaveG*(f)=G(-f),wemayrewriteEquation (4.15)as hop,(t)=krooG(-f)exp[-j2'Trf(T -t)]df =krooG(f)expU2'Trf(T-t)]df (4.161 =kg(T-t) Equation (4.16)showsthattheimpulse response oftheoptimum filter,exceptfOIt~e scalingfactork,isatime-reversed anddelayedversionoftheinputsignalg(t);thatis,It (4.17)4.2Matched Filter 251 is"matched" totheinputsignal.Alineartime-invariant filterdefinedinthiswayiscalled amatched filter.Notethatinderiving thematched filtertheonlyassumption wehave madeabouttheinputnoisew(t)isthatitisstationary andwhitewithzeromeanand powerspectraldensityNo/2.Inotherwords,noassumption wasmadeonthestatistics of thechannelnoisew(t). I!IlPROPERTIES OFMATCHED FILTERS Wenotethatafilter,whichismatched toa pulsesignalg(t) ofdurationT,ischaracterized byanimpulseresponse thatisatime-reversed anddelayedversionoftheinputg(t),as shownby hop,(t)=kg(Tt) Inotherwords,theimpulseresponse hopt(t)isuniquely defined,exceptforthedelayTand ..thescalingfactork,bythewaveform ofthepulsesignalg(t)towhichthefilterismatched. Inthefrequency domain,thematched filterischaracterized byafrequency response that is,exceptforadelayfactor,thecomplex conjugate oftheFouriertransform oftheinput g(t),asshownby HOf"(f)=kG*(f)exp(-j2'l1fT) Themostimportant resultinthecalculation oftheperformance ofsignalprocessing sys temsusingmatched filtersisperhapsthefollowing: Thepeakpulsesignal-to-noise ratioofamatched filterdepends onlyontheratioofthe signalenergytothepowerspectraldensityofthewhitenoiseatthefilterinput. Todemonstrate thisproperty, considerafiltermatched toaknownsignalg(t). TheFourier transform oftheresulting matched filteroutputgo(t)is Go(f)=Hopt(f)G(f) =kG*(f)G(f) exp(-j2nfT) =k1G(f) 12exp(-j27TfT) UsingEquation (4.17)intheformulafortheinverseFouriertransform, wefindthatthe matched filteroutputattimet=Tis ga(T)=rooGo(f)exp(j27TfT) df =kr~[G(fWdf According toRayleigh's energytheorem, theintegralofthesquaredmagnitude spectrum ofapulsesignalwithrespecttofrequency isequaltothesignalenergyE: E=roog2(t)dt =r~ 1G(f) 12df Hence (4.18) (4.19)Substituting Equation (4.14)into(4.7),wefindthattheaverageoutputnoisepoweris E[n2(t)]=P~orooIG(fWdf =PNoEI2 252 CHAPTER 4tllBASEBAND PULSE TRANSMISSION whereagainwehavemadeuseofRayleigh's energytheorem. Therefore, thepeakpul signal-to-noise ratiohasthemaximum value Se (kE)2 2E 7)max=(PNoEI2)=No (4.20l FromEquation (4.20)weseethatdependence onthewaveform oftheinputg(t)hasbee completely removed bythematched filter.Accordingly, inevaluating theabilityofn matched-filter receivertocombatadditivewhitenoise,wefindthatallsignalsthathava thesameenergyareequallyeffective. NotethatthesignalenergyEisinjoulesandth: noisespectraldensityNo/2isinwattsperHertz,sothattheratio2EINoisdimensionless. however, thetwoquantities havedifferent physicalmeaning. WerefertoEINoasthesignal energy-to-noise spectraldensityratio. ~ExAMPLE 4.1Matched FilterforRectangular Pulse Consider asignalg(t)intheformofarectangular pulseofamplitude Aandduration T,as showninFigure4.2a.Inthisexample, theimpulseresponse h(t)ofthematched filterbas exactlythesamewaveform asthesignalitself.Theoutputsignalgo(t)ofthematched filter produced inresponse totheinputsignalg(t}hasatriangular waveform, asshowninFigure 4.2b. Themaximum valueoftheoutputsignalgo(t)isequaltokA2T,whichistheenergyof theinputsignalg(t)scaledbythefactork;thismaximum valueoccursatt=T,asindicated inFigure4.2b. T (c) FIGURE 4.2(a)Rectangular pulse.(b)Matched filteroutput.(e)Integrator output. 4.3ErrorRateDuetoNoise 253 Integrator~~ ~mPleal timet=TRectangular-1 pulseL- _ FIGURE4.3Integrate-and-dump circuit. Forthespecialcaseofarectangular pulse,thematchedfiltermaybeimplemented using acircuitknownastheintegrate-and-dump circuit,ablockdiagramofwhichisshownin Figure4.3.Theintegrator computes theareaundertherectangular pulse,andtheresulting outputisthensampledattimet=T,whereTisthedurationofthepulse.Immediately after t=T,theintegrator isrestoredtoitsinitialcondition; hencethenameofthecircuit.Figure 4.2cshowstheoutputwaveform oftheintegrate-and-dump circuitfortherectangular pulse ofFigure4.2a.Weseethatfor0:5t:5T,theoutputofthiscircuithasthesamewaveform asthatappearing attheoutputofthematchedfilter;thedifference inthenotations usedto describetheirpeakvaluesisofnopracticalsignificance. <l!l 1J.3ErrorRateDuetoNoise InSection3.8wepresented aqualitative discussion oftheeffectofchannelnoiseonthe performance ofabinaryPCMsystem.Nowthatweareequipped withthematched filter astheoptimum detector ofaknownpulseinadditivewhitenoise,wearereadytoderive aformula fortheerrorrateinsuchasystemduetonoise. Toproceed withtheanalysis, consider abinaryPCMsystembasedonpolarnon return-to-zero (NRZ)signaling.Inthisformofsignaling, symbols 1and0arerepresented bypositiveandnegative rectangular pulsesofequalamplitude andequalduration. The channelnoiseismodeled asadditivewhiteGaussian noisew(t)ofzeromeanandpower spectral densityNo/2;theGaussian assumption isneededforlatercalculations. Inthe signaling interval0os;tos;Tb,thereceived signalisthuswrittenasfollows: (4.21){+A+w(t), symbol1wassentx(t)--A+w(t), symbol0wassent whereTbisthebitduration, andAisthetransmitted pulseamplitude. Itisassumed that thereceiverhasacquired knowledge ofthestartingandendingtimesofeachtransmitted pulse;inotherwords,thereceiver haspriorknowledge ofthepulseshape,butnotits polarity. Giventhenoisysignalx(t),thereceiverisrequired tomakeadecision ineach signaling intervalastowhether thetransmitted symbolisa 1oraO. Thestructure ofthereceiverusedtoperform thisdecision-making processisshown inFigure4.4.Itconsists ofamatched filterfollowed byasampler, andthenfinallya PCMwave set)Say1ify>,l Say0ify<,l WnileGaussian noisewet)Tnresnold ,l FIGURE4.4Receiverforbaseband transmission ofbinary-encoded PCMwaveusingpolarNRZ signaling. 254 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION decisiondevice.Thefilterismatched toarectangular pulseofamplitude Aandduratio Tb,exploiting thebit-timing information available tothereceiver. Theresulting matche~ filteroutputissampledattheendofeachsignaling interval.Thepresence ofchannelnoise w(t)addsrandomness tothematched filteroutput. Letydenote the samplevalueobtained attheendofasignaling interval.TheSaInple valueyiscompared toapresetthreshold Ainthedecision device.Ifthethreshold 18 exceeded, thereceivermakesadecisioninfavorofsymbol1;ifnot,adecision ismadein favorofsymbol O.Weadopttheconvention thatwhenthesamplevalueyisexactlyequal tothethreshold A,thereceiverjustmakesaguessastowhichsymbolwastransmitted. suchadecisionisthesameasthatobtained byflippingafaircoin,theoutcome ofwhkh willnotaltertheaverageprobability oferror. Therearetwopossiblekindsoferrortobeconsidered: 1.Symbol1ischosenwhena 0wasactuallytransmitted; werefertothiserrorasan errorofthefirstkind. 2.Symbol0ischosenwhena 1wasactuallytransmitted; werefertothiserrorasan errorofthesecondkind. Todetermine theaverageprobability oferror,weconsider thesetwosituations separately. Suppose thatsymbol0wassent.Then,according toEquation (4.21),thereceived signalis ~(t)=-A+w(t), (4.221 Correspondingly, thematched filteroutput,sampledattimet=Tb,isgivenby(inlight ofExample 4.1withkATbsetequaltounityforconvenience ofpresentation) (b Y=Jox(t)dt 1fTb=-A+-Tw(t)dt b0(4.23) (4.24)whichrepresents thesamplevalueofarandomvariable Y.Byvirtueofthefactthatthe noisew(t)iswhiteandGaussian, wemaycharacterize therandomvariableYasfollows: ~TherandomvariableYisGaussian distributed withameanof-A. ~Thevariance oftherandomvariableYis crt=E[(Y+A)2] 1[(b(1'b] =Tt,EJoJow(t)w(u) dtdu 1rTb(b =T~Jo JoE[w(t)w(u)] dtdu 1rTb(b =Tt,Jo JoRw(t,u)dtdu whereRw(t,u)istheautocorrelation function ofthewhitenoisew(t).Sincew(t)iswhite withapowerspectraldensityNo/2,wehave NoRw(t,u)="28(t-u) (4.251 4.3ErrorRateDuetoNoise 255 where8(t-u}isatime-shifted deltafunction. Hence,substituting Equation (4.25)into (4.24)yields 1{Tb(bN lT~=nJo Joi8(t-u)dtdu No 2Tb(4.26) (4.27)wherewehaveusedthesiftingproperty ofthedeltafunction andthefactthatitsareais unity.Theconditional probability densityfunction oftherandomvariable Y,giventhat symbol0wassent,istherefore 1 ( (y+A)2) fy(yIO)=~exp No/T b ,Thisfunction isplottedinFigure4.5(a}.LetPIOdenotetheconditional probability oferror, giventhatsymbol0wassent.Thisprobability isdefinedbytheshadedareaunderthe curveoffy(yIO}fromthethreshold Atoinfinity,whichcorresponds totherangeofvalues assumed byyforadecision infavorofsymbol1.Intheabsenceofnoise,thematched filteroutputysampledattimet=Tbisequalto-A.Whennoiseispresent,yoccasionally assumes avaluegreaterthanA,inwhichcaseanerrorismade.Theprobability ofthis error,conditional onsendingsymbol0,isdefinedby PlO=pry>AIsymbol0wassent) =rfy(yI0)dy (4.28) 1Jm((y+Af)d =V7TNo/Tb'exp-No/Tby Atthispointinthediscussion wedigressbrieflyandintroduce thedefinition ofthe so-called complementary errorfunction:3 2Jmerfc(u}=v;: uexp(-~}dz (4.29) (4.30)whichiscloselyrelatedtotheGaussian distribution. For largepositivevaluesofu,we havethefollowing upperboundonthecomplementaty errorfunction: f()exp(-u2) ercu<vrru (al (bl FIGURE4.5NoiseanalysisofPCMsystem.(a)Probability densityfunction ofrandomvariableY atmatched filteroutputwhen0istransmitted. (b)Probability densityfunction ofYwhenIis transmitted. 256 CUAPTER4 BASEBAl"lD PULSE TRANSMISSION (4.31) (4.33) (4.34)Toreformulate theconditional probability oferrorPtointermsofthecompleme . taryerrorfunction, wefirstdefineanewvariable n y+Az=---YNo/T b Accordingly, wemayrewriteEquation (4.28)inthecompact form 1f~PtO=- exp(-r) dzy";(A+A)IVNorrb =&erfc(~N:/~J Assumenextthatsymbol1wastransmitted. ThistimetheGaussian randomvariable Yrepresented bythesamplevalueyofthematched filteroutputhasamean+Aand variance No/2Tb•Notethat,compared tothesituation whensymbol0wassent,themean oftherandomvariable Yhaschanged, butitsvariance isexactlythesameasbefore.The conditional probability densityfunction ofY,giventhatsymbol1wassent,istherefore 1 ( (y-Af) fy(y11)=Y7TNo/T bexpNo/Tb(4.32) whichisplottedinFigure4.5b.LetPOldenotetheconditional probability oferror,given thatsymbol1wassent.Thisprobability isdefinedbytheshadedareaunderthecurveof fy(y11)extending from-00tothethreshold A,whichcorresponds totherangeofvalues assumed byyforadecision infavorofsymbol O.Intheabsenceofnoise,thematched filteroutputysampledattimet=Tbisequalto+A.Whennoiseispresent,yoccasionally assumes avaluelessthanA,andanerroristhenmade.Theprobability ofthiserror, conditional onsendingsymbol1,isdefinedby POl=P(y<AIsymbol1wassent) =ffy(yI1)dy _-001fAexp(-(yA)l)dy -V7TNo/Tb-00 No/Tb Toexpress POIintermsofthecomplementary errorfunction, this time wedefineanew variable A-yz=---YNo/T b Accordingly, wemayreformulate Equation (4.33)inthecompact form 1fooPal=- exp(-r) dzy";(A-A)IVNuIT b .!.erfc(~)2YNo/T b Havingdetermined theconditional probabilities oferror,PIOandPo"ournexttask istoderivetheformulafortheaverageprobability ofsymbolerror,denotedbyP,.Here wenotethatthesetwopossiblekindsoferroraremutually exclusive eventsinthatifthe receiver, ataparticular sampling instant,choosessymbol1,thensymbol0isexcludell (4.35) (4.36)4.3ErrorRateDuetoNoise 257 fromappearing, andviceversa.LetPoandPIdenote the aprioriprobabilities oftrans mittingsymbols 0and1,respectively. Hence,theaverageprobability ofsymbolerrorPe inthereceiverisgivenby Pe=POPlO+P,POI =f12erfc(~) +fJJcerfc(~)2VNo/T b2VNo/T b FromEquation (4.35)weseethatPeisinfactafunction ofthethreshold A,which immediately suggeststheneed for formulating anoptimum threshold thatminimizes Pe. Forthisoptimization weuseLeibniz's rule. Consider theintegral fblU) f(z,u)dz a(u) Leibniz's rulestatesthatthederivative ofthisintegralwithrespecttouis dd(bIU)f(z,u)d7;=[(b(u),u)dbd(u)_f(a(u),u)d(da(u)+(blu)8f(7;,u)d7; U)a(u) u u)oIU)au Fortheproblem athand,wenotefromthedefinition ofthecomplementary errorfunction inEquation (4.29)that 2f(7;,u)=y:;;:exp(-Z2) a(u)=u b(u)=00 Theapplication ofLeibniz's ruletothecomplementary errorfunction thusyields d 1-erfc(u)=--exp(-u2)du y:;;: Hence,differentiating Equation (4.35)withrespecttoAbymakinguseoftheformulain Equation (4.36),thensettingtheresultequaltozeroandsimplifying terms,weobtainthe optimum threshold as Ao~=4~~b10g(~) Forthespecialcasewhensymbols1and0areequiprobable, wehave 1 PI=Po=2: inwhichcaseEquation (4.37)reducesto(4.37) Aopt=0 Thisresultisintuitively satisfying asitstatesthat,forthetransmission ofequiprobable binarysymbols, weshouldchoosethethreshold atthemidpoint betweenthepulseheights -Aand+Arepresenting thetwosymbols0and1.Notethatforthisspecialcasewealso have POI=PlO 258 CHAPTER 4iiiBASEBAND PUI.SE TllANS1\USSION Achannelforwhichtheconditional probabilities oferrorPOIandPlOareequalissaidt bebinarysymmetric. Correspondingly, theaverageprobability ofsymbolerrorinEquatio~ (4.35)reducesto P=.!erfc(_A_) e2VNolTb(4.38) Nowthetransmitted signalenergyperbitisdefinedby Eh=A2Tb (4.39) Accordingly, wemayfinallyformulate theaverageprobability ofsymbolerrorforthe receiverinFigure4.4as (4.40) (4.41)whichshowsthattheaverageprobability ofsymbolerrorinabinarysymmetric channel depends solelyonEJNo,theratioofthetransmitted signalenergyperbittothenoise spectraldensity. UsingtheupperboundofEquation (4.30)onthecomplementary errorfunction, we maycorrespondingly boundtheaverageprobability ofsymbolerrorforthePCMreceiver as Pexp(-EblNo) e<.~lV7rE biNo ThePCMreceiverofFigure4.4therefore exhibitsanexponential improvement inthe averageprobability ofsymbolerrorwithincreaseinEJNo. Thisimportant resultisfurtherillustrated inFigure4.6wheretheaverageprobability ofsymbolerrorPeisplottedversusthedimensionless ratioEJNo.Inparticular, wesee thatPedecreases veryrapidlyastheratioEJNoisincreased, sothateventually avery "smallincrease" intransmitted signalenergywillmakethereception ofbinarypulses almosterrorfree,asdiscussed previously inSection3.8.Note,however, thatinpractical termstheincreaseinsignalenergyhastobeviewedinthecontextofthebias;forexample, 10-2 10-4 .... ~10-6 b .~ ~10-8 d: 10-10 10-12 15 FIGURE 4.6Probability oferrorinapeMreceiver. 4.4IntersymbollnterferelUJe 259 a3-dBincreaseinEJNoismucheasiertoimplement whenEbhasasmallvaluethanwhen itsvalueisordersofmagnitude larger. ~lntersymbol Interference Thenextsourceofbiterrorsinabaseband-pulse transmission systemthatwewishto studyisintersymbol interference (lSI),whichariseswhenthecommunication channelis dispersive. Firstofall,however, weneedtoaddressakeyquestion: Givenapulseshape ofinterest,howdoweuseittotransmit datainM-aryform?Theanswerliesintheuse ofdiscretepulsemodulation, inwhichtheamplitude, duration, orpositionofthetrans mittedpulsesisvariedinadiscretemannerinaccordance withthegivendatastream. However, forthebaseband transmission ofdigitaldata,theuseofdiscretepulse-amplitude modulation (PAM)isoneofthemostefficientschemesintermsofpowerandbandwidth utilization. Accordingly, weconfineourattention todiscretePAMsystems.Webeginthe studybyfirstconsidering thecaseofbinarydata;laterinthechapter, weconsider the moregeneralcaseofM-arydata. Consider thenabaseband binaryPAMsystem,agenericformofwhichisshownin Figure4.7.Theincoming binarysequence {bk}consistsofsymbols 1and0,eachofdu rationTb•Thepulse-amplitude modulator modifies thisbinarysequence intoanewse quenceofshortpulses(approximating aunitimpulse), whoseamplitude akisrepresented inthepolarform ak={+1-1ifsymbolbkis1 ifsymbolbkis0(4.42) Thesequence ofshortpulsessoproduced isappliedtoatransmitfilterofimpulseresponse g(t),producing thetransmitted signal s(t)=La"g(t-kTb) (4.43) k Thesignals(t)ismodified asaresultoftransmission throughthechannelofimpulse response h(t).Inaddition, thechanneladdsrandomnoisetothesignalatthereceiver input.Thenoisysignalx(t)isthenpassedthroughareceivefilterofimpulseresponse crt). Theresulting filteroutputy(t)issampled synchronously withthetransmitter, withthe sampling instantsbeingdetermined byaclockortimingsignalthatisusuallyextracted fromthereceivefilteroutput.Finally,thesequence ofsamplesthusobtained isusedto reconstruct theoriginaldatasequence bymeansofadecision device.Specifically, the amplitude ofeachsampleiscompared toathreshold A.Ifthethreshold Aisexceeded, a decisionismadeinfavorofsymbol1.Ifthethreshold Aisnotexceeded, adecisionismade infavorofsymbol O.Ifthesampleamplitude equalsthethreshold exactly,theflipofa Input binary data I,,)Say1ifyeti)>A Say0ifyeti)<A I---- Transmitter ---~'>t-I~' --Channel------,.I""'------Receiver-----~ FIGURE4.7Baseband binarydatatransmission system. 260 CHAPrER 4IIIBASEBA,"ID PULSETR,\NSMISSION faircoinwilldetermine whichsymbolwastransmitted (i.e.,thereceiversimplymakesa randomguess). Thereceivefilteroutputiswrittenas y(t)=p.,2:a"p(t-kTb)+nIt) k(4.44) wherep.,isascalingfactor,andthepulsep(t)istobedefined.Tobeprecise,anarbitrary timedelaytoshouldbeincluded intheargument ofthepulsepIt-kTb)inEquation (4.44) torepresent theeffectoftransmission delaythroughthesystem.Tosimplifytheexposition, wehaveputthisdelayequaltozeroinEquation (4.44)withoutlossofgenerality. Thescaledpulsep.,p(t)isobtained byadoubleconvolution involving theimpulse response g(t)ofthetransmit filter,theimpulseresponse hIt)ofthechannel, andtheimpulse responsecrt)ofthereceivefilter,asshownby p.,p(t)=g(t)*h(t)*c(t) (4.45) wherethestardenotesconvolution. WeassumethatthepulsepIt)isnormalized bysetting prO)=1 (4.46) whichjustifiestheuseofp.,asascalingfactortoaccountforamplitude changesincurred inthecourseofsignaltransmission throughthesystem. Sinceconvolution inthetimedomainistransformed intomultiplication inthefre. quencydomain, wemayusetheFouriertransform tochangeEquation (4.45)intothe equivalent form p.,P(f)=G(f)H(f)C(f) (4.47) whereP(f),G(f),H(f),andC(f)aretheFouriertransforms ofp(t),g(t),h(t),andc(t), respectively. Finally,thetermnIt)inEquation (4.44)isthenoiseproduced attheoutputofthe receivefilterduetothechannelnoisew(t).Itiscustomary tomodelw(t)asawhiteGaus· siannoiseofzeromean. Thereceivefilteroutputy(t)issampled attimeti=iTb(withitakingoninteger values),yielding[inlightofEquation (4.46)] (4.48) =p.,ai+p.,La"p[(i-k)Tb]+n(ti) k~-~ k*i InEquation (4.48),thefirsttermp.,airepresents thecontribution oftheithtransmitted bit. Thesecondtermrepresents theresidualeffectofallothertransmitted bitsonthedecoding oftheithbit;thisresidual effectduetotheoccurrence ofpulsesbeforeandafterthe sampling instanttiiscalledintersymbol interference (lSI).Thelasttermn(ti)represents the noisesampleattime'i' IntheabsenceofbothlSIandnoise,weobservefromEquation (4.48)that y(til=p.,ai whichshowsthat,undertheseidealconditions, theithtransmitted bitisdecodedcorrectly. Theunavoidable presence oflSIandnoiseinthesystem,however, introduces errorsinthe decisiondeviceatthereceiveroutput.Therefore, inthedesignofthetransmit andreceive filters,theobjective istominimize theeffectsofnoiseandlSIandtherebydeliverthedigitll datatotheirdestination withthesmallesterrorratepossible. 4.5Distortionless Basebmul BinaryTransmission 261 Whenthesignal-to-noise ratioishigh,asisthecaseinatelephone system,forex ample,theoperation ofthesystemislargelylimitedbylSIratherthannoise;inother words,wemayignoren(t;l.Inthenextcoupleofsections, weassumethatthiscondition holdssothatwemayfocusourattention onlSIandthetechniques foritscontrol.In particular, theissuewewishtoconsider istodetermine thepulsewavefonn p(t)forwhich thelSIiscompletely eliminated. ~.5Nyquist's Criterion forDistortionless .Baseband Binary·Transmission Typically, thefrequency response ofthechannelandthetransmitted pulseshapearespec ified,andtheproblem istodetermine thefrequency responses ofthetransmit andreceive filterssoastoreconstruct theoriginalbinarydatasequence {bk}'Thereceiverdoesthisby extracting andthendecoding thecorresponding sequence ofcoefficients, {ak},fromthe outputy(t).Theextraction involvessampling theoutputy(t)attimet=iTb•Thedecoding requiresthattheweighted pulsecontribution akP(iT b-kTb)fork=ibefreefromlSI duetotheoverlapping tailsofallotherweighted pulsecontributions represented byk"*i. This,inturn,requiresthatwecontroltheoverallpulsep(t),asshownby i=k i"*k(4.49) wherep(O)=1,bynormalization. Ifp(t)satisfiestheconditions ofEquation (4.49),the receiveroutputy(ti)giveninEquation (4.48)simplifies to(ignoring thenoiseterm) y(t,)=JlAiforalli whichimplieszerointersymbol interference. Hence,thetwoconditions ofEquation (4.49) ensureperfectreception intheabsenceofnoise. Fromadesignpointofview,itisinformative totransform theconditions ofEquation (4.49)intothefrequency domain. Consider thenthesequence ofsamples{p(nTb)},where n=0,±1,±2,•...Fromthediscussion presented inChapter3onthesampling process, werecallthatsampling inthetimedomainproduces periodicity inthefrequency domain. Inparticular, wemaywrite (4.50) whereRb=IITbisthebitrateinbitspersecond(b/s);Pa(f)istheFouriertransform of aninfiniteperiodicsequence ofdeltafunctions ofperiodTb,whoseindividual areasare weighted bytherespective samplevaluesofp(t).Thatis,Pa(f)isgivenby (4.51) (4.52)Lettheintegerm=i -k.Then,i=kcorresponds tom=0,andlikewisei"*kcorresponds tom"*0.Accordingly, imposing theconditions ofEquation (4.49)onthesamplevalues ofp(t)intheintegralofEquation (4.51),weget Pa(f)=roop(O)B(t)exp(-j271ft) dt =p(O) 262 CHAPTER 4..BASEBAND PULSETRANSMISSION wherewehavemadeuseofthesiftingproperty ofthedeltafunction. SincefromEqualio (4.46)wehavep(O)=1,itfollowsfromEquations (4.50)and(4.52)thatthecondition forzerointersymbol interference issatisfiedif n (4.53) WemaynowstatetheNyquistcriterion' fardistartianless baseband transmission in theabsenceofnoise:Thefrequency functianP(f)eliminates intersymbal interference for samplestakenatintervalsTbprovided thatitsatisfiesEquation (4.53).NotethatP(f) referstotheoverallsystem,incorporating thetransmit filter,thechannel, andthereceive filterinaccordance withEquation (4.47). IlilIDEAL NYQUIST CHANNEL Thesimplestwayofsatisfying Equation (4.53)istospecifythefrequency function P(f)to beintheformofarectangular function, asshownby -W<f< W IfI>W(4.54) whererlXt(f)standsforarectangular functionofunitamplitude andunitsupportcentered onf=0,andtheoverallsystembandwidth Wisdefinedby (4.55) (4.56)According tothesolution described byEquations (4.54)and(4.55),nofrequencies of absolute valueexceeding halfthebitrateareneeded.Hence,fromFourier-transform pair 2ofTableA6.3wefindthatasignalwaveform thatproduces zerointersymbol interference isdefinedbythesincfunction: ()_sin(27TWt) pt-27TWt =sinc(2Wt) ThespecialvalueofthebitrateRb=2WiscalledtheNyquistrate,andWisirseH calledtheNyquistbandwidth. Correspondingly, theidealbaseband pulsetransmission, systemdescribed byEquation (4.54)inthefrequency domainor,equivalendy, Equation (4.56)inthetimedomain, iscalledtheidealNyquistchannel. Figures4.8aand4.8bshowplotsofP(f)andp(t),respectively. InFigure4.8a,the normalized formofthefrequency function P(f)isplottedforpositiveandnegative fre quencies. InFigure4.8b,wehavealsoincluded thesignaling intervals andthecorrespond ingcentered sampling instants. Thefunction p(t)canberegarded astheimpulseresponse ofanideallow-pass filterwithpassband magnitude response 1/2Wandbandwidth VI. Thefunctionp(t)hasitspeakvalueattheoriginandgoesthroughzeroatintegermultiples ofthebitdurationTb•Itisapparent thatifthereceivedwaveform y(t)issampled atthe 4.5Distartirmless Baseband BinaryTransmission 263 p{tJ 1.0 0.5 t ttt t t t----=-1--=r---=-+--------,:-I---:+---+,.--+.,.-- I-3 3T" 1.02WP(jJ Sampling instants -1 of W Signaling intervals w W FIGURE4.8(a)Idealmagnitude response. (b)Idealbasicpulseshape. instantsoftimet=0,±Tb,±2Tb,"',thenthepulsesdefinedbyiLP(t-iTb)with arbitrary amplitude iLandindexi=0,±1,±2,"', willnotinterfere witheachother. Thiscondition isillustrated inFigure4.9forthebinarysequence 1011010. Although theuseoftheidealNyquistchanneldoesindeedachieveeconomyinband widthinthatitsolvestheproblem ofzerointersymbol interference withtheminimum Binarysequence 1 0 1 1 0 1 0 1.0 0.5 ~ .""0.0Q. E« -0.5 -1.0 -4-2o 4 Time6810 12 FIGURE4.9Aseriesofsinepulsescorresponding tothesequence 1011010. (4.57)264 CHAPTER 4!!lBASEBAND PULSE TR&'iSMISSION bandwidth possible, therearetwopractical difficulties thatmakeitanundesirable objec tIveforsystemdesign: 1.Itrequiresthatthemagnitude characteristic ofP(f)beflatfrom- WtoW,andZer elsewhere. Thisisphysically unrealizable becauseoftheabrupttransitions att~ bandedges±W. 2.Thefunction pit)decreases as1/1tIforlargeItI,resulting inaslowrateofdecay Thisisalsocausedbythediscontinuity ofP(f)at±W.Accordingly, thereisprac: ticallynomarginoferrorinsampling timesinthereceiver. Toevaluate theeffectofthistimingerror,consider thesampleofy(t)att==At where!i.tisthetimingerror.Tosimplifytheexposition, wemayputthecorrectsarnplin' timetiequaltozero.Intheabsenceofnoise,wethushave(fromEquation (4.48)) g y(!i.t)=/L2:akP(!i.t-kTb) k "sin[2'1TW(!i.t -kTb)] =/L~ak2'1TW(!i.t -kTb) Since2WTb=1,bydefinition, wemayrewriteEquation (4.57)as (A )=a.(2W A)+/Lsin(2'1TW !i.t)"(-l)kak yut/L0smcut 7T ~(2W!i.t_k) (4.58) k"O Thefirsttermontheright-hand sideofEquation (4.58)definesthedesiredsymbol,whereas theremaining seriesrepresents theintersymbol interference causedbythetimingerrorAt insampling theoutputy(t).Unfortunately, itispossibleforthisseriestodiverge,thereby causingerroneous decisions inthereceiver. l!!lRAIsED COSINE SPECTRUM Wemayovercome thepractical difficulties encountered withtheidealNyquistchannel by extending thebandwidth fromtheminimum valueW=Rb/2toanadjustable valuebe tweenWand2W.Wenowspecifytheoverallfrequency response P(f)tosatisfyacon ditionmoreelaborate thanthatfortheidealNyquistchannel;speCifically, weretainthree termsofEquation (4.53)andrestrictthefrequency bandofinterestto[-W,Wl,asshown by 1P(f)+P(f-2W)+P(f+2W)=2W' -W~f~ W (4.591 Wemaydeviseseveralband-limited functions thatsatisfyEquation (4.59).Aparticular formofP(f)thatembodies manydesirable featuresisprovided byaraisedcosinespectrum. Thisfrequency response consistsofaflatportionandarolloffportionthathasasinusoidal form,asfollows: 1 O~IfI<Ii2W' P(f)=1 { . ['1T(IfI-W)]}fl~IfI<2W-f,(4.60] 4W1 -sm2W-2fl' 0, Ifl~2W-fl 4.5Distortionless Baseband Bi....ryTransmission 265 Thefrequency parameter 11andbandwidth Warerelatedby ex=1 _11 W(4.61) Theparameter exiscalledtherollofffactor;itindicates theexcessbandwidth overthe idealsolution, W.Specifically, thetransmission bandwidth BTisdefinedby BT=2W-/l =W(1+ex) Thefrequency response P(f),normalized bymultiplying itby2W,isplottedinFigure 4.10aforthreevaluesofex,namely,0,0.5,and1.Weseethatforex=0.5or1,the 2WP{j) f -2 3-1_la 1 3 IV-22 2 2 (a) pIt) 1.0 0.5 (b) FIGURE4.10Responses fordifferent rollofffactors.(a)Frequencyresponse. (b)Timeresponse. 266 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION functionP(f)cutsoffgradually ascompared withtheidealNyquistchannel(i.e.,Q'"0) andistherefore easiertoimplement inpractice. AlsothefunctionP(f)exhibitsoddsYl!i. merrywithrespecttotheNyquistbandwidth W,makingitpossibletosatisfytheconditio ofEquation (4.59). II Thetimeresponse p(t)istheinverseFouriertransform ofthefrequency response P(f).Hence,usingtheP{f)definedinEquation (4.60),weobtaintheresult(seeProblem 4.13) (COS(27TCIWt)) p(t)=(sinc(2Wt)) 1 _16a'W2t2 (4.62) whichisplottedinFigure4.10bforQ=0,0.5,and1. Thetimeresponse p(t)consistsoftheproductoftwofactors:thefactorsinc(2WI) characterizing theidealNyquistchannelandasecondfactorthatdecreases as1/1tI'for largeItI.Thefirstfactorensureszerocrossings ofp(t)atthedesiredsampling instantsof timet=iTwithianinteger(positive andnegative). Thesecondfactorreducesthetails ofthepulseconsiderably belowthatobtained fromtheidealNyquistchannel, sothatthe transmission ofbinarywaves using suchpulsesisrelatively insensitive tosampling time errors.Infact,forQ=1wehavethemostgradualrolloffinthattheamplitudes ofthe oscillatory tailsofp(t)aresmallest. Thustheamountofintersymbol interference resulting fromtimingerrordecreases astherollofffactoraisincreased fromzerotounity. Thespecialcasewithr:x=1(i.e.,f1=0)isknownasthefull-cosine rallaffcharac. teristic,forwhichthefrequency response ofEquation (4.60)simplifies to P(f)={4~[1+cos(;~)J. 0, Correspondingly, thetimeresponse p(t)simplifies to ( )_sinc(4Wt) p t-1 -16W2r Thistimeresponse exhibitstwointeresting properties:0<IfI<2W Ifi~2W(4.63) (4.641 1.Att=±Til2=±1/4W,wehavep(t)=0.5;thatis,thepulsewidthmeasured at halfamplitude isexactlyequaltothebitdurationTb• 2.Therearezerocrossings att=±3Tb/2,±5Tb/2,...inaddition totheusualzero crossings atthesampling timest=±Tb,±2Tb,···. Thesetwoproperties areextremely usefulinextracting atimingsignalfromthereceived signalforthepurposeofsynchronization. However, thepricepaidforthisdesirable prop· ertyistheuseofachannelbandwidth doublethatrequired fortheidealNyquist channel corresponding toQ=O. !l>EXAMPLE 4.2Bandwidth Requirement oftheTISystem InExample 3.2ofChapter3,wedescribed thesignalformatfortheTlcarriersystemthad! usedtomultiplex 24independent voiceinputs,basedonan8-bitPCMword.Itwasshown 4.6Correlative-Level Coding 267 thatthebitdurationoftheresultingtime-division multiplexed signal(including aframingbit) is Tb=0.647/.LS Assuming theuseofanidealNyquistchannel,itfollowsthattheminimum transmission bandwidth ByoftheT1systemis(fora=0) By=W=~=772kHz2Tb However, amorerealisticvalueforthenecessary transmission bandwidth isobtained byusing afull-cosine rolloffcharacteristic witha=1.Inthiscase,wefindthat 1BT=W(1+a)=2W=T;,=1.544MHz L4.6Correlative-Level Coding Thusfarwehavetreatedintersymbol interference asanundesirable phenomenon that produces adegradation insystemperformance. Indeed,itsverynameconnotes anuisance effect.Nevertheless, byaddingintersymbol interference tothetransmitted signalinacon trolledmanner, itispossible toachieveasignaling rateequaltotheNyquistrateof2W symbols persecondinachannelofbandwidth WHertz.Suchschemes arecaUedcorrel ative-level codingorpartial-response signaling schemes.5Thedesignoftheseschemes is basedonthefollowing premise: Sinceintersymbol interference introduced intothetrans mittedsignalisknown,itseffectcanbeinterpreted atthereceiverinadeterministic way. Thuscorrelative-level codingmayberegarded asapractical methodofachieving the theoretical maximum signaling rateof2Wsymbolspersecondinabandwidth ofWHertz, aspostulated byNyquist, usingrealizable andperturbation-tolerant filters. !illDlJOBINARY SIGNALING Thebasicideaofcorrelative-level codingwillnowbeillustrated byconsidering thespecific example ofduobinary signaling, where"duo"impliesdoubling ofthetransmission ca pacityofastraightbinarysystem.Thisparticular formofcorrelative-level codingisalso calledclassIpartialresponse. Consider abinaryinputsequence {bkJconsisting ofuncorrelated binary symbols 1 and0,eachhavingdurationTb•Asbefore,thissequence isappliedtoapulse-amplitude modulator producing atwo-level sequence ofshortpulses(approximating aunitimpulse), whoseamplitude akisdefinedby ak={+1-1ifsymbolbkis1 ifsymbolbkis0(4.65) Whenthissequence isappliedtoaduobinary encoder, itisconverted intoathree leveloutput,namely,-2,0,and+2.Toproduce thistransformation, wemayusethe schemeshowninFigure4.11.Thetwo-level sequence {ak}isfirstpassedthroughasimple filterinvolving asingledelayelementandsummer. Foreveryunitimpulseappliedtothe 268 CHAPTER 4"BASEBAND PULSE TRANSMIS~ION 1---------------- Input I + II two-level-H..,...-----';-{ I sequence I {ak} I I I I I Tb I 1 I ~---------_-_--_I FilterHM) FIGURE4.11Duobinary signaling scheme. inputofthisfilter,wegettwounitimpulses spacedTbsecondsapartatthefilteroutput, Wemaytherefore expresstheduobinary coderoutputCkasthesumofthepresentinput pulseakanditsprevious valueak-hasshownby (4.66) Oneoftheeffectsofthetransformation described byEquation (4.66)istochangethe inputsequence {ak}ofuncorrelated two-level pulsesintoasequence [Ck}ofcorrelated three levelpulses.Thiscorrelation between theadjacent pulsesmaybeviewedasintroducing intersyrnbol interference intothetransmitted signalinanartificialmanner.However, the intersyrnbol interference sointroduced isunderthedesigner's control,whichisthebasis ofcorrelative coding. Anidealdelayelement, producing adelayofTbseconds, hasthefrequency response exp(-j27TfTb),sothatthefrequency response ofthesimpledelay-line filterinFigure4.11 is1+exp(-j27rfTb).Hence,theoverallfrequency response ofthisfilterconnected in cascadewithanidealNyquistchannelis Hr(f)=HNyqu,,,(fHl+exp(-j27rfTb)] =HNyqui,,(fHexp(j7rfTb) +exp(-j7rfTb)] exp(-j7rfT bl(4.67) =2HNyqui,,(f) cos(7rfTb)exp(-j7rfT b) wherethesubscript IinHdf)indicates thepertinent classofpartialresponse. Foranideal Nyquistchannelofbandwidth W=1/2Th,wehave(ignoring thescalingfactorTb) {1, HNyqui,,(f)=0,Ifl0;1/2T b otherwise(4.68) Thustheoverallfrequency response oftheduobinarysignaling schemehastheformofa half-cycle cosinefunction, asshownby IfI0;1/2Tb otherwise(4.69) forwhichthemagnitude response andphaseresponse areasshowninFigures4.12aand 4.12b,respectively. Anadvantage ofthisfrequency response isthatitcanbeeasilyap proximated, inpractice, byvirtueofthefactthatthereiscontinuity atthebandedges. FromthefirstlineinEquation (4.67)andthedefinition ofHNyqui,,(f) inEquation (4.68),wefindthattheimpulseresponse corresponding tothefrequency responseHMl 4.6Correlative-Level Coding 269 arg[HIU)] 2.0 --L----'c-------':--f (a)--~------C""k------'-::"'-f (b) (4.70)FIGURE4.12Frequency response oftheduobinary conversion filter.(a)Magnitude response. (b)Phaseresponse. consistsoftwosine(Nyquist) pulsesthataretime-displaced byTbsecondswithrespectto eachother,asshownby(exceptforascalingfactor) hI(t)=sin(mITb)+_si_n[,--'1T...;.(_t _-_T...;.b=)/,-T=,,] '1TtlTb '1T(t-Tb)ITb sin(mlTb)sin(mlTb) '1TtlTb'1T(t-Tb)ITbnsin('1TtITb) '1Tt(Tb-t) Theimpulseresponse hI(t)isplottedinFigure4.13,whereweseethatithasonlytwo distinguishable valuesatthesampling instants. TheformofhI(t)shownhereexplainswhy wealsorefertothistypeofcorrelative codingaspartial-response signaling. Theresponse toaninputpulseisspreadovermorethanonesignaling interval; statedinanotherway, theresponse inanysignaling intervalis"partial." NotealsothatthetailsofhI(t)decayas 11ItI2,whichisafasterrateofdecaythanthe11ItIencountered intheidealNyquist channel. Theoriginaltwo-level sequence {ak}maybedetected fromtheduobinary-coded sequence {Ck}byinvoking theuseofEquation (4.66).Specifically, letakrepresent the estimate oftheoriginalpulseakasconceived bythereceiverattimet=ktb•Then,sub tractingtheprevious estimateak-lfrom Ck>weget (4.71) Itisapparent thatifCkisreceived withouterrorandifalsotheprevious estimateak-lat timet=(k-1)Tbcorresponds toacorrectdecision, thenthecurrentestimate akwillbe h" ~-2T~Tb 0 Tb2T~3Tb 4T;;' FIGURE4.13Impulse response oftheduobinary conversion filter. 270 CHAPTER 4"'BASEBAND PULSE TRANSMISSION correcttoo.Thetechnique ofusingastoredestimate oftheprevious symboliscalled decisionfeedback. Weobservethatthedetection procedure justdescribed isessentially aninverseofthe operation ofthesimpledelay-line filteratthetransmitter. However, amajordrawback of thisdetection procedure isthatonceerrorsaremade,theytendtopropagate throughthe outputbecauseadecisiononthecurrentinputakdependsonthecorrectness ofthedecision madeontheprevious inputak-l' Apractical meansofavoiding theerror-propagation phenomenon istouseprecoding beforetheduobinary coding,asshowninFigure4.14.Theprecoding operation performed onthebinarydatasequence{bdconverts itintoanotherbinarysequence{dkldefinedby dk=bkEBdk-1 (4.72) wherethesymbolEBdenotesmodulo-two addition ofthebinarydigitsbkanddk-1•This addition isequivalent toatwo-input EXCLUSIVE ORoperation, whichisperformed as follows: d k={symbOl 1ifeithersymbolbkorsymboldk-1(butnotboth)is1 symbol0otherwise (4.73) Theprecoded binarysequence {dkJisappliedtoapulse-amplitude modulator, producing acorresponding two-level sequence ofshortpulses{akl,whereak=:!:1asbefore.This sequence ofshortpulsesisnextappliedtotheduobinarycoder,therebyproducing the sequence {cdthatisrelatedto{adasfollows: (4.74) Notethatunlikethelinearoperation ofduobinarycoding,theprecoding described by Equation (4.72)isanonlinear operation. Thecombined useofEquations (4.72)and(4.74)yields {oifdatasymbolbkis1 Ck= (4.75):!:2ifdatasymbolbkis0 whichisillustrated inExample 4.3.FromEquation (4.75)wededucethefollowing decision rulefordetecting theoriginalbinarysequence {bdfrom{ckl: IfICkI<1,saysymbolbkis1(4.76)IfICkI>1,saysymbolbkis0 I I I I I{d'_I' II : Tb IL ~ Precoder FIGURE4.14Aprecoded duobinary scheme; detailsoftheduobinarycoderaregiveninFigUJe 4.1I. 4.6CorreJatwe-Level Coding 271 Threshold =1 FIGURE4.15Detector forrecovering originalbinarysequence fromtheprecoded duobinary coderoutput. WhenICkI=1,thereceiver simplymakesarandom guessinfavorofsymbol1orO. According tothisdecision rule,thedetector consistsofarectifier, theoutputofwhichis compared inadecision devicetoathreshold of1.Ablockdiagram ofthedetector is showninFigure4.15.Ausefulfeatureofthisdetector isthatnoknowledge ofanyinput sampleotherthanthepresentoneisrequired. Hence,errorpropagation cannotoccurin thedetectorofFigure4.15. ~ExAMPLE 4.3Duobinary CodingwithPrecoding Considerthebinarydatasequence 0010110. Toproceedwiththeprecoding ofthissequence, whichinvolvesfeedingtheprecoder outputbacktothe input, weaddanextrabittothe precoderoutput.Thisextrabitischosenarbitrarily tobe1.Hence,usingEquation (4.73),we findthatthesequence [d.}attheprecoderoutputisasshowninrow2ofTable4.1.Thepolar representation oftheprecoded sequence {dkJisshowninrow3ofTable4.1.Finally,using Equation (4.74),wefindthattheduobinary coderoutputhastheamplitude levelsgivenin row4ofTable4.1. Todetecttheoriginalbinarysequence, weapplythedecisionruleofEquation (4.76), andsoobtainthebinarysequence giveninrow5ofTable4.1.Thislatterresultshowsthat, intheabsenceofnoise,theoriginalbinarysequence isdetectedcorrectly. <!Ill illMODIFIED DVOBINARY SIGNALING Intheduobinary signaling technique thefrequency responseH(f),andconsequently the powerspectraldensityofthetransmitted pulse,isnonzeroattheorigin.Thisisconsidered tobeanundesirable featureinsomeapplications, sincemanycommunications channels cannottransmit aDCcomponent. Wemaycorrectforthisdeficiency byusingtheclass IVpartialresponse ormodified duobinary technique, whichinvolves acorrelation span oftwobinarydigits.Thisspecialformofcorrelation isachieved bysubtracting amplitude modulated pulsesspaced2Tbsecondsapart,asindicated intheblockdiagram ofFigure ITABLE4.1Illustrating Example 4.3onduobinarycoding Binarysequence lb.} 0010 1 1 0 Precoded sequence{d.l 1 1 1 0 0 1 0 0 Two-level sequence{a.} +1 +1 +1 -1-1+1-1-1 Duobinary coderoutputIc.l +2+2 0-2 0 0 -2 Binarysequenceobtained by 0 0 10 1 1 0 applyingdecisionruleofEq.(4.76) 272 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION Input binary,-------------.., sequence 1Modulo-2 adder : Ib,lI I{d,l , Output f~three-levl!l :Sampleatsequence It""-kTbirk} I I I I I 2Tb :L I Modifiedduobinary conversion filterHr..(f) FIGURE4.16Modified duobinary signaling scheme. 4.16.Theprecoder involvesadelayof2Tbseconds.Theoutputofthemodified duobinary conversion filterisrelatedtotheinputtwo-level sequence {aklatthepulse-amplitude mod ulatoroutputasfollows: (4.77/ Here,again,wefindthatathree-level signalisgenerated. Withak=:!:1,wefindthatCI takesononeofthreevalues:+2,0,and-2. Theoverallfrequency response ofthedelay-line filterconnected incascadewithan idealNyquistchannel, asinFigure4.16,isgivenby (4.78/HlV(f)=HNyqui,,(f)[l-exp(-j41TIT b)] =2jHNyquis,(f)sin(21TITb) exp(-j21TIT b) wherethesubscript IVinH]v(f)indicates thepertinent classofpartialresponse and HNyqui,,(f) isasdefinedinEquation (4.68).Wetherefore haveanoverallfrequency re sponseintheformofahalf-cycle sinefunction, asshownby I11:$1/2Th elsewhere(4.79) Thecorresponding magnitude response andphaseresponse ofthemodified duobinary coderareshowninFigures4.17aand4.17b,respectively. Ausefulfeatureofthemodified duobinarycoderisthefactthatitsoutputhasnoDCcomponent. Notealsothatthis 2.0 1f 1r--------1 1 0 1 -2Tb-4Tb 4Tb2Tb2 (a) (b) FIGURE4.17Frequency response ofthemodilied duobinary conversion filter.(a)Magnitude response. (b)Phaseresponse. (4.80) ifeithersymbolbkorsymboldk-2(butnotboth)is1(4.81) otherwise (4.82)4.6Correlative-Level Coding 273 secondformofcorrelative-level codingexhibitsthesamecontinuity atthebandedgesas induobinarysignaling. FromthefirstlineofEquation (4.78)andthedefinition ofHNyqUist(f) inEquation (4.68),wefindthattheimpulseresponse ofthemodified duobinary coderconsistsoftwo sinc(Nyquist) pulsesthataretime-displaced by2Tbsecondswithrespecttoeachother,as shownby(exceptforascalingfactor) hry(t)=sin(7T't/T b)_sin[7T'(t -=-2Tb)/Tb)] 7T't/Tb 7T'(t-2Tb)/Tb sin(7T't/Tb)sin(7T't/Tb) 7T't/Tb7T'(t-2Tb)/Tb 2Ttsin(7T't/Tb) 7T't(2Tb-t) Thisimpulseresponse isplottedinFigure4.18,whichshowsthatithasthreedistinguish ablelevelsatthesampling instants. Notealsothat,aswithduobinarysignaling, thetails ofhIV(t)forthemodified duobinary signaling decayas1/1t12• Toeliminate thepossibility oferrorpropagation inthemodified duobinarysystem, weuseaprecoding procedure similartothatusedfortheduobinary case.Specifically, priortothegeneration ofthemodified duobinarysignal,amodulo-two logicaladdition isusedonsignals2Tbsecondsapart,asshownby(seethefrontendofFigure4.16) dk=bkEBdk-2 ={symbOl 1 symbol0 where{bkJistheincoming binarydatasequence and[dk}isthesequence attheprecoder output.Theprecoded sequence [dk}thusproduced isthenappliedtoapulse-amplitude modulator andthentothemodified duobinaryconversion filter. InFigure4.16,theoutputdigitCkequals-2,0,or+2,assuming thatthepulse amplitude modulator usesapolarrepresentation fortheprecoded sequence {dkJ.Alsowe findthatthedetected digitbkatthereceiveroutputmaybeextracted from Ckbydisre gardingthepolarityofCk'Specifically, wemayformulate thefollowing decisionrule: If1Ck1>1,saysymbolbkis1 IfICk1<1,saysymbolbkis0 1.0 FIGURE4.18Impulseresponse ofthemodified duobinary conversion filter. 274 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION WhenICkI=1,thereceivermakesarandomguessinfavorofsymbol1orO.Aswiththe duobinary signaling, wemaynotethefollowing: l>Intheabsenceofchannelnoise,thedetectedbinarysequence[bk}isexactlythesam astheoriginalbinarysequence (bk}atthetransmitter input. e I>TheuseofEquation (4.81)requirestheaddition oftwoextrabitstotheprecoded sequence [ad.Thecomposition ofthedecodedsequence {bk}usingEquation (4.82) isinvariant totheselection madeforthesetwobits. !llGENERALIZED FORMOFCORRELATIVE-LEVEL CODING (PARTIAL-RESPONSE SIGNALING) Theduobinary andmodified duobinarytechniques havecorrelation spansof1binarydigit and2binarydigits,respectively. Itisastraightforward mattertogeneralize thesetwo techniques tootherschemes, whichareknowncollectively ascorrelative-level codingor partial-response signaling schemes. Thisgeneralization isshowninFigure4.19,where HNyqui,,(f) isdefinedinEquation (4.68).Itinvolves theuseofatapped-delay-line filter withtap-weights Wo,w"...,WN-l'Specifically, different classesofpartial-response sig. Input two·level--...."..,..---l sequence [ak} FIGURE4.I9Generalized correlative codingscheme. 4.7BasebandM-ary PAMTra.......issWn 275 TABLE4.2Different classesofpartial-response signaling schemes referring toFigure4.19 TypeofClass N Wo W, W2 W3 W4 Comments I 2 11 Duobinary coding II 3 12 1 III 3 21-1 IV 310-1 Modified duobinary coding V 5-1 0 2 0-1 nalingschemesmaybeachieved byusingaweighted linearcombination ofNidealNyquist (sinc)pulses,asshownby N-i ( )h(t)=2:WnsincTt-n n=O b(4.83) Anappropriate choiceofthetap-weights inEquation (4.83)resultsinavarietyofspectral shapesdesigned tosuitindividual applications. Table4.2presentsthespecificdetailsof fivedifferent classesofpartial-response signaling schemes. Forexample, intheduobinary case(classIpartialresponse), wehave Wo=+1 Wi=+1 andWn=0forn;:,2.Inthemodified duobinarycase(classIVpartialresponse), wehave Wo=+1 Wi=0 W2=-1 andWn=0forn;:,3. Theusefulcharacteristics ofpartial-response signaling schemesmaynowbesum marizedasfollows: ~Binarydatatransmission overaphysicalbaseband channelcanbeaccomplished at arateclosetotheNyquist rate,usingrealizable filterswithgradual cutoff characteristics. ~Different spectralshapescanbeproduced, appropriate fortheapplication athand. However, thesedesirable characteristics areachieved ataprice:Alargersignal-to-noise ratioisrequired toyieldthesameaverageprobability ofsymbolerrorinthepresence of noiseasinthecorresponding binaryPAMsystemsbecauseofanincreaseinthenumber ofsignallevelsused. ~.7Baseband M-aryPAMTransmission Inthebaseband binaryPAMsystemofFigure4.7,thepulse-amplitude modulator pro ducesbinarypulses,thatis,pulseswithoneoftwopossibleamplitude levels.Ontheother 276 CHAPTER 4IIIBASEBAND PULSE TRA<"ISMISSION hand,inabaseband M-aryPAMsystem,thepulse-amplitude modulator produces oneof Mpossibleamplitude levelswithM>2.Thisformofpulsemodulation isillustrated in Figure4.20aforthecaseofaquaternary (M=4)systemandthebinarydatasequence 0010110111. Thewaveform showninFigure4.20aisbasedontheelectrical representa. tionforeachofthefourpossibledibits(pairsofbits)giveninFigure4.20b.Notethatthis representation isGrayencoded, whichmeansthatanydibitinthequaternary alphabet differsfromanadjacent dibitinasinglebitposition. InanM-arysystem,theinformation sourceemitsasequence ofsymbolsfroman alphabet thatconsistsofMsymbols. Eachamplitude levelatthepulse-amplitude modu. latoroutputcorresponds toadistinctsymbol,sothatthereareMdistinctamplitude leve~ tobetransmitted. Consider thenanM-aryPAMsystemwithasignalalphabet thatcon. tainsMequallylikelyandstatistically independent symbols, withthesymbolduration denotedbyTseconds. WerefertolITasthesignaling rateofthesystem,whichisexpressed insymbolspersecond,orbauds.Itisinformative torelatethesignaling rateofthissystem tothatofanequivalent binaryPAMsystemforwhichthevalueofMis2andthesuccessive binarysymbols1and0areequallylikelyandstatistically independent, withtheduration ofeithersymboldenotedbyTbseconds. Undertheconditions described here,thebinary PAMsystemproduces information attherateofIITbbitsperseconds. Wealsoobserve thatinthecaseofaquaternary PAMsystem,forexample, thefourpossiblesymbols may beidentified withthedibits00,01,10,and11.Wethusseethateachsymbolrepresents 2bitsofinformation, and1baudisequalto2bitspersecond.Wemaygeneralize this resultbystatingthatinanM-aryPAMsystem,1baudisequaltolo~Mbitspersecond, andthesymbolduration ToftheM-aryPAMsystemisrelatedtothebitduration Tbof theequivalent binaryPAMsystemas T=Tblog2M (4.84) Therefore, inagivenchannelbandwidth, wefindthatbyusinganM-aryPAMsystem, weareabletotransmit information ataratethatislog2Mfasterthanthecorresponding binaryPAMsystem.However, torealizethesameaverageprobability ofsymbolerror,an M-aryPAMsystemrequiresmoretransmitted power.Specifically, wefindthatforMmuch largerthan2andanaverageprobability ofsymbolerrorsmallcompared to1,thetrans, +3 +1Binarya a data '"":ec. ~-1a a Dibit Amplitude 00 -3 01 -1 11 +1 10 +3 -3 w W FIGURE4.20Outputofaquaternary system.(a)Waveform. (h)Representation ofthe4possible dibits,basedonGrayencoding. 4.8DigitalSubscriber Lines 277 mitted power mustbeincreased bythefactorM2/logzM,compared toabinaryPAM system. Inabaseband M-arysystem,firstofall,thesequence ofsymbols emittedbythe information sourceisconverted intoanM-levelPAMpulsetrainbyapulse-amplitude modulator atthetransmitter input.Next,aswiththebinaryPAMsystem,thispulsetrain isshapedbyatransmitfilterandthentransmitted overthecommunication channel,which corrupts thesignalwaveform withbothnoiseanddistortion. Thereceivedsignalispassed throughareceivefilterandthensampledatanappropriate rateinsynchronism withthe transmitter. Eachsampleiscompared withpresetthreshold values(alsocalledslicinglev els),andadecisionismadeastowhichsymbolwastransmitted. Wetherefore findthat thedesignsofthepulse-amplitude modulator andthedecision-making deviceinanM-ary PAMaremorecomplex thanthoseinabinaryPAMsystem.Intersymbol interference, noise,andimperfect synchronization causeerrorstoappearatthereceiveroutput.The transmit andreceivefiltersaredesigned tominimize theseerrors.Procedures used forthe designofthesefiltersaresimilartothosediscussed inSections4.5and4.6forbaseband binaryPAMsystems. L4.8DigittllSubscriber Lines Atthispointinourstudyofbaseband datatransmission itisratherappropriate thatwe digressfromtheoretical aspectsofthestudyandconsider afast-growing application: dig italsubscriber lines.6Adigitalsubscriber line(DSL)operates overalocalloop(lessthan 1.5Ian)thatprovides adirectconnection betweenauserterminal (e.g.,computer) anda telephone company's centraloffice(CO),asillustrated inFigure4.21.Through theCO,a DSLuserisconnected toabroadband backbone datanetwork, whichisbasedontech nologiessuchastheasynchronous transfermode(ATM)andInternetprotocol (IP);these technologies andrelatednetwork resources (i.e.,opticalfibers,SONET) arediscussed in theBackground andPreviewchapter.Accordingly, theinformation-bearing signaliskept inthedigitaldomainallthewayfromtheuserterminal toanInternetserviceprovider, withthesignalbeingswitched orroutedatregularintervalsinthecourseofitstransmission throughthedatanetwork. Intheinterestofaninexpensive implementation, digitalsubscriber linesusetwisted pairsconfigured toprovideahighdata-rate, fullduplex,digitaltransmission capability. (Twisted pairsarealsousedforordinary telephonic communication, asdiscussed inthe Background andPreviewchapter.) Toachievefull-duplex, two-wire transmission, wemay useoneoftwopossiblemodesofoperation: 1.Timecompression (TC)multiplexing, wheredatatransmission inthetwoopposite directions onthecommon lineareseparated intime.Specifically, blocksofbits ofdataaresentinburstsineachdirection onanalternate basis,asillustrated in User Central office --+-Upstream Downstream FIGURE4.21Blockdiagram depicting theoperational environment ofdigitalsubscriber lines. 278 CHAPTER 4'"BASEBAND PULSE TRANSMISSION (a) (b) FIGURE 4.22Full-duplex operation using(altimecompression multiplexing, and (b1echo-cancellation. Figure4.220.ToaccoWltforpropagation timeacrosstheline,aguardtimeisinserted betweenindividual burstsofdata.Accordingly, thelinerateisslightlygreaterthan twicethedatarate. 2.Echo-cancellation mode,whichsupports thesimultaneous flowofdataalongthe common lineinbothdirections. Forthisformoftransmission tobefeasible, each transceiver (transmitter/receiver) includesahybridfortwopurposes: theseparation ofthetransmitted signalfromthereceivedsignalandthetwo-to-four-wire conver sion,asshowninFigure4.22b.Thehybrid,ormoreprecisely, thehybridtrans former,isbasically abridgecircuitwiththreeports(terminal pairs),asdepicred in Figure4.23.Ifthebridgeisnotperfecrly balanced, thetransmitter portofthehybrid -----';> Transmitter~ ...- Receiver ----::l-IIr:criberLOP FIGURE 4.23Simplified circuitofhybridtransformer. Forthebridgetobebalanced, the reference impedance Z",shouldequalthelineimpedance z,. 4.8DigitalSubscriber Lines 279 becomes coupledtothereceiverport,therebygivingrisetoanechoduetoleakage ofthenear-end (local)transmitted signaltothenear-end (local)receiver. Tocancel theunwanted echo,eachtransceiver includes anechocanceller, asshowninFigure 4.22b.Sincedatacanflowthrough thelinesimultaneously inbothdirections, the linerateisthesameasthedatarate. Fromthisdiscussion, itisapparent thattheecho-cancellation schemeoffersamuch betterdata-transmission performance thanthetimecOIl}pression multiplexing scheme,but attheexpenseofincreased complexity. However, byimplementing theentiretransceiver inasingleverylarge-scale integrated (VLSI)chip,thecostismadeaffordable despitethe increased complexity. InNorthAmerica, theecho-cancellation schemehasbeenadopted asthebasisfordesigning thetransceivers. Anadaptive implementation oftheechocan cellerisdiscussed inProblem 4.31. Inaddition toecho,thereareotherimpairments ofthetransmission mediumthat needtobeconsidered. Thetwodominant impairments areintersymbol interference and crosstalk, whicharediscussed inwhatfollowsinthatorder. Toafirst-order approximation, thesquaredmagnitude response ofatwistedpairis givenby whereexp(-aV!) (4.85) (4.86) a=ki 10 InEquation (4.85),thefrequencyfismeasured inkHz,kisaphysical constant ofthe twistedpair,10isareference length(e.g.,kilometers), and1istheactuallengthofthe twistedpair.Equation (4.85)pointstoamajorimpairment intheuseofatwistedpairfor baseband datatransmission: thegradualfalloffinthefrequency response, which,inturn, givesrisetointersymbol interference. Turningnexttocrosstalk, theprimarycauseforitsoccurrence isthecapacitive cou plingthatexistsbetween adjacent twistedpairsinacable.Typically, thenearestfiveto seventwistedpairsinthecablecausemostofthecrosstalk. Inanyevent,twokindsof crosstalk canbeobserved inareceiverofinterest: 1.Near-end crosstalk (NEXT), whichisgenerated bytransmitters locatedatthesame endofthecableasthereceiver, asillustrated inFigure4.24a. Disturbing end Transmitter Receh/f:r Disturbed end'Cablecontaining a bundleoftwistedpaJrs NEXT CalTransmitterDisturbing endCablecontaining a bundleoftwistedpairs (blDisturbed endReceiver FIGURE4.24(u)Near-end crosstalk (NEXT).(b)Far-endcrosstalk (FEXT). 280 CHAPTER 4IIIBASEBAND PULSE TRANSMISSION Transmitted signal ofinterestOutput Interfering signalresponsible for generating near-end crosstalk (NEXT) FIGURE 4.25Modeloftwisted-pair channel. 2.Far-endcrosstalk (FEXT), whichisgenerated bytransmitters locatedfurtheraway fromthereceiver, asillustrated inFigure4.24b. FEXTnaturally suffersthesamelinelossasthesignal,whereas NEXTdoesnot.Accord ingly,intheecho-cancellation schemeofFigure4.22bwheresignalstravelinbothdirec tionsinthecable,NEXTwillbemuchstronger thanFEXT.Henceforth, weignorethe effectofFEXT. Indeed,near-end crosstalk andintersymbol interference arethetwomostimportant factorsindetermining theperformance ofadigitalsubscriber loop.Figure4.25showsthe modelofatwisted-pair channeldominated bythesetwoimpairments. Sincealltwisted pairsareusuallytransmitting similarsignals,wemaymodeltheNEXTasasignalwith thesamepowerspectral densityasthetransmitted signalpassingthrough acrosstalk frequency response HNEXT(f), whichisapproximated by HNEXT(f) ={3f3/2 (4.87) where{3isaconstant ofthecable.Theinteresting pointtonotefromFigure4.25isthat boththetransmitted signalandtheinterfering signalhavethesamepowerspectraldensity; theydifferfromeachothermerelyintheirassociated frequency responses, asshownin Equations (4.85)and(4.87),respectively. Whenthemodeldescribed hereinisusedfor simulation study,thetransmitted signalisrepresented byarandomdatasequence, while theinterference isrepresented byaGaussian noisesequence. ~LINECODES FORDIGITAL SUBSCRIBER LINES Nowthatwehaveidentified themajortransmission impairments, wemaydescribe the desirable featuresthespectrum ofatransmitted signalshouldexhibit: 1.Thepowerspectraldensityofthetransmitted signalshouldbezeroatzerofrequency, sincenoDCtransmission throughahybridtransformer ispossible. 2.Thepowerspectraldensityofthetransmitted signalshouldbelowathighfrequencies forthefollowing reasons: I>Transmission attenuation inatwistedpairismostsevereathighfrequencies. I>Crosstalk betweenadjacent twistedpairsincreases dramatically athighfrequencies becauseofincreased capacitive coupling. Inthisregard,recallthattheimpedance ofacapacitor isinversely proportional tofrequency. Tosatisfythesedesirable properties, wehavetobecarefulinchoosing thelinecode thatmapstheincoming streamofdatabitsintoelectrical pulsesfortransmission on theline.Variouspossibilities, eachwithitsownadvantages anddisadvantages, exist 4.8Digit..lSubsc.-iber Lines 281 forsuchachoice.Thelistofpotential candidates forlinecodesincludes the following: I>Manchester code,whichissimpleandhaszeroDCcomponent. Itsdisadvantage is theoccupation ofalargespectrum, whichmakesitvulnerable tonear-end crosstalk andintersymbol interference. (TheManchester codewasdiscussed inSection3.7.) I>Modified duobinary code,whichhaszeroDC,ismoderately spectrally efficient, andcausesminimal intersymbol interference.-However, simulation studiesofthe crosstalk performance ofthemodified duobinarycodehaveshownthatitsim munitytonear-end crosstalk andintersymbol interference isabout2to3dB poorerthanthatofblockcodesonworst-case subscriber lines.(Themodified duobinary codewasdiscussed inSection4.6.) I>-Bipolarcode,inwhichsuccessive Isarerepresented alternately bypositiveand negative butequallevels,andsymbol0isrepresented byazerolevel.Bipolar signaling haszeroDC.Computer simulations haveshownthatitsnear-end cross talkandintersymbol interference performance isslightlyinferiortothemodified duobinarycodeonalldigitalsubscriber loops.(Thebipolarcode,alsoknownas thealternate markinversion (AMI)codes,wasdiscussed inSection3.7.) I>2B1Qcode,whichstandsfortwobinarydigitsencodedintoonequaternary sym bol.Thiscodeisablockcoderepresenting afour-level PAMsignal,asillustrated inFigure4.20.Assuming thatsymbols1and0areequiprobable, the2BIQcode haszeroDContheaverage.Moreover, amongallthelinecodesconsidered herein, itoffersthegreatestbaudreduction, andthebestperformance withrespectto near-end crosstalk andintersymbol interference. Itisbecauseofthedesirable properties ofthe2BIQcodecompared totheManchester code,modified duobinary code,thebipolarcode,andotherlinecodesnotmentioned here/ thatthe2BIQcodehasbeenadopted astheNorthAmerican standard fordigitalsub scriberloops. Usingthe2BIQasthelinecodeandVLSIimplementation ofatransceiver that incorporates adaptive equalizers andechocancellers, itispossibletoachieveabiterror rateof10-7operating fullduplexat160kb/sonthevastmajority oftwisted-pair sub scriberlines.Abiterrorrateof10-7with12dBnoisemargin,when1percentworst-case NEXTispresent,isanaccepted performance criterion fordigitalsubscriber lines.Noise marginistheamountofreceivernoise(including uncancelled echo)thatcanbetolerated withoutexceeding the10-7errorrate. IlilAsYMMETRIC DIGITAL SUBSCRIBER LINES Another important typeofDSListheasymmetric digitalsubscriber line(ADSL),whichis alocaltransmission systemdesigned tosimultaneously supportthreeservicesonasingle twisted-wire pair: 1.Datatransmission downstream (towardthesubscriber) atbitratesofupto9Mb/s. 2.Datatransmission upstream (awayfromthesubscriber) atbitratesofupto1Mb/s. 3.Plainoldtelephone service(POTS). Thedownstream andupstream bitratesdependonthelengthofthetwistedpairusedto dothetransmission. TheDSLissaidtobe"asymmetric" becausethedownstream bitrate ismuchhigherthantheupstream bitrate.Analogvoiceistransmitted atbaseband fre quencies andcombined withthepassband transmissions ofdownstream andupstream 282 CHAPTER 4OJBASEBAND PULSE TRANSMISSION Transmit power ~ POTSGuard bandband (alFrequencyDigital subscriber line (b)Telephone FIGURE4.26(a)Illustrating thedifferent bandallocations foranFDM-based ADSLsystem. (b)Blockdiagramofsplitterperfonning thefunction ofamultiplexer ordemultiplexer. Note:both filtersinthesplitterarebidirectional filters. datausingfrequency-division multiplexing (FDM).Asillustrated inFigure4.26a,theup streamdatatransmission isplacedinafrequency banddifferent fromthedownstream datatransmission toavoidcrosstalk. Moreover, aguardbandisinsertedbetweenthePOTS bandandtheupstream transmission band.Thecoexistence ofADSLandPOTSsignalson thelocalloopismadepossiblethroughtheuseofapairofsplitters; onesplinerisplaced attheCOendofthelocalloopandtheotheroneisplacedattheuserend.Infunctional terms,asplitterisdividedintotwobidirectional filters,asshowninFigure4.26b: ~Alow-pass filterforthebaseband transmission orextraction ofvoicesignals. 1;0-Ahigh-pass filterforthepassband transmission orextraction ofADSLdata. Ineffect,thesplitterperforms theroleofafrequency-division multiplexer ordemultiplexer, depending onthedirection ofsignaltransmission. Themotivation formakingDSLasymmetric istoaccommodate "video-on-demand." Insuchapplications, asubscriber needsahigh-throughput channeltodownload high bandwidth videodatafromacentralofficeondemand. Inthereversedirection, amuch lowerthroughput channelisadequate tosendorderinformation aswellasreal-time con trolcommands. Forexample, anADSLforInternetproviding downstream transmission attheDS1rateof1.544Mb/sandupstream transmission ofabout160kb/swouldmeet therequirements ofthisapplication. Theapproximate 10:1asymmetry ratiorealizedby suchasystemprevents theflowofacknowledgment packetsintheIPfrombecoming a bottleneck tothefasterdirection ofdatatransmission. Itisverydifficult totransmit dataoveratwistedpairattheDS1rateandhigher, anddoingsorequirestheuseofsophisticated modulation techniques. Thetreatment of thissubjectisdeferredtoChapter6. I4.9Optimum LinearReceiver Resuming ourstudyofthebaseband datatransmission systemdepicted inFigure4.7,we havethusfartreatedthefollowing twochannelconditions separately: '"Channelnoiseactingalone,whichledtoformulation ofthematched filterreceiver. '"Intersymbol interference actingalone,whichledtoformulation ofthepulse-shaping transmit filtersoastorealizetheNyquistchannel. 4.9Optimum LinearReceiver 283 Inareal-lifesituation, however, channelnoiseandintersymbol interference acttogether, affecting thebehavior ofadatatransmission systeminacombined manner. Inthissection, weformulate thebasisfordesigning alinearreceiveroptimized forthegeneralcaseofa linearchannelthatisbothdispersive andnoisy. Inoneapproach rothedesignofalinearreceiver, thereceiverisviewedasazero forcingequalizer followed byadecision-making device.Theobjective ofthisformof equalization istohavethe"inrersymbol interference forcedtozero"atalltheinstants t=kTatwhichthechanneloutputissampled, exceptfork=0wherethesymbolof interestisassumed tooccur.Underthiscondition, symbol-to-symbol detection isassured tobeoptimalinaccordance withtheNyquist criterion, provided thatthechannelnoise w(t)iszero. Thezero-forcing equalizer isrelatively easytodesignbecauseitignorestheeffectof thechannelnoisew(t).Aseriousconsequence ofthisoversight, however, isthatitleads tooverallperformance degradation duetonoiseenhancement, aphenomenon thatisan inherent featureofzero-forcing equalization; seeProblem 4.32.Amorerefinedapproach forthereceiverdesignistousethemean-square errorcriterion, whichprovides abalanced solutiontotheproblem ofreducing theeffectsofbothchannelnoiseandintersymbol interference. Indeed,foraprescribed computational complexity, anequalizer designed on thislatterbasisalwaysperforms aswellas,andoftenbetterthan,itszero-forcing coun terpart.Henceforth, weconcentrate onthemean-square errorcriterion forreceiverdesign. Referring backtothebaseband binarydatatransmission systemofFigure4.7,the receivefiltercharacterized bytheimpulseresponse c(t)produces thefollowing response duetothechanneloutputx(t): y(t)=r~C(7)X(t-7)d7 (4.88) (4.89)Thechanneloutputx(t)isitselfdefinedby x(t)=2:akq(t-kTb)+w(t) k whereakisthesymboltransmitted attimet=kTbandw(t)isthechannelnoise.Thetime function q(t)istheconvolution oftwoimpulseresponses: g(t)pertaining tothepulse shapingtransmit filter,andh(t)pertaining tothechannel. Substituting Equation (4.89) into(4.88)andsampling theresulting outputy(t)attimet=iTb,wemaywrite y(iTb)=gj+nj wheregiisthesignalcomponent definedby andniisthenoisecomponent definedby(4.90) (4.91) (4.92) (4.93)Thecondition forperfectoperation ofthereceiveristohavey(iTb)=a"whereaiisthe transmitted symbol.Deviation fromthiscondition resultsintheerrorsignal ei=y(iTbl-aj =gi+ni-ai 284 CHAPTER 4iiiBASEBAND PULSE TRANSMISSION Accordingly, wemayformally definethemean-square erroras 1J=2"E[etJ (4.94) (4.96)whereEisthestatistical expectation operator, andthefactor1/2isintroduced forCon_ venience ofpresentation. Substituting Equation (4.93)irito(4.94)andthenexpanding terms,weget111J=2"Elm+2"E[ntJ+2"E[atJ+E[(;iniJ-E[n,aiJ-E[(;ia;](4.95) Wenowevaluatethesixexpectation termsinthisequation intheordertheyappearhere: 1.Inastationary environment themean-square termE[(;tJisindependent oftheinstant oftimet=iTbatwhichthereceivefilteroutputissampled. Hence,wemaysimplify theexpression ofthistermbywriting Elm=~~E[a/akJroorooC(Tl)C(T2)q(lT b-T,)q(kT b-T2)dTldT2 Assuming that,first,thebinarysymbolsak=±1asinEquation (4.42)and,second, thetransmitted symbolsarestatistically independent, thatis, {IforI=k E[a,akJ=0otherwise wemayfurtherreducetheexpression forthemean-square termE[(;tJto E[m=roorooRq(Tj,T2)c(T,)C(T2) dT,dT2 where(4.97) (4.99) (4.101)Rq(ThT2)=2:q(kTb-T,)q(kT b-T2) (4.98) k ThefactorRq(Tj,T2)isthetemporal autocorrelation function ofthesequence (q(kTb)).Stationarity ofthissequence meansthat(seeSection1.5) Rq(ThT2)=Rq(T2-T,)=Rq(Tl-T2) 2.Themean-square termE[ntJduetochannelnoiseisgivenby(usingEquation (4.92)) E[ntJ=roorooc(T,)c(T2)E[w(iT b-T,)w(iT b-T2)JdT,dT2 =J:ooJ:ooC(T,)C(T2)R w(T2-T,)dT,dT2 whereRw(T2-T,)istheensemble-averaged autocorrelation function ofthechannel noisew(t).Withw(t)assumed tobewhitewithpowerspectraldensityNo/2,wehave Rw(T2-T,)=~o8(T2-T,) (4.1001 Hence,theexpression forE[ntJsimplifies to 2NoJOOJ=E[n']="2_oo_ooC(T,)C(T2) 8(T2-T,)dT,dT2 4.9Optimum LinearReceiver 285 3.Themean-square termE[ar]duetothetransmitted symbolaiisunitybyvirtueof Equation (4.96);thatis, E[ar]=1foralli (4.102) 4.Theexpectation ofthecross-product term!;injiszerofortworeasons:first!;jandnj areindependent and,second,thechannelnoisew(t),andtherefore n;,haszeromean; thatis, foralli (4.103) 5.Forsimilarreasons,theexpectation ofthecross-product termniaiisalsozero;that is, E[njai]=0foralli (4.104) 6.Finally,theexpectation ofthecross-product term!;jajisgivenby(usingEquation (4.91)) (4.105) Byvirrueofthestatistical independence ofthetransmitted symbols described in Equation (4.96),thisexpectation reducesto (4.106) (4.108) (4.109)Thussubstituting Equations (4.97),(4.101)to(4.104)and(4.106)into(4.95),we mayexpressthemean-square errorJforthebinarydatatransmission systemof Figure4.7as 1 1J~J~(N) J- J=2+2_~ -00Rq(t-T)+2°5(t-T)C(t)C(T)dtdT--00c(t)q(-t) dt (4.107) Forconvenience ofpresentation, wehavemadethefollowing changesinvariables: TlandT2inthefirstintegralarereplaced bytandT,respectively, andTisreplaced withtinthesecondintegral.Notealsothatthisexpression forthemean-square error Jisinactualfactnormalized withrespecttothevarianceofthetransmitted symbols akbyvirtueoftheassumption madeinEquation (4.96). WiththeformulaofEquation (4.107)forthemean-square errorJathand,we arenowreadytospecifythedesignofthereceivefilterinFigure4.7.Differentiating Equation (4.107)withrespecttotheimpulseresponse cit)ofthereceivefilter,and thensettingtheresultequaltozero,weget roo(Rq(t-T)+~o5(tT))C(T)dT=q(-t) Equation (4.108)istheformulaforfindingtheimpulseresponse crt)oftheequalizer optimized inthemean-square errorsense.Anequalizer sodesigned isreferredtoas theminimum-mean squareerror(mmse)equalizer. TakingtheFouriertransform ofbothsidesofEquation (4.108),weobtain (Sq(f)+~o)C(f)=Q*(f) 286 CHAPTER 4'"BASEBAND PULSE TR&-.SMISSION wherec(t)~C(f),q(t)~Q(f),andRq~Sq(f).SolvingEquation (4.109)forC(f), weget ' C(f)=Q*(f) Sq(f)+~o(4.110) InProblem 4.33itisshownthatthepowerspectraldensityofthesequence !q(kTb)l canbeexpressed as (4.111) whichmeansthatthefrequency response C(f)oftheoptimum linearreceiveris periodicwithperiod11Th,Equation (4.110)suggeststheinterpretation oftheopti. mumlinearreceiverasthecascadeconnection oftwobasiccomponents:s ~Amatched filterwhoseimpulseresponse isq(-t),whereq(t)=g(t)*h(t). I>-Atransversal (tapped-delay-line) equalizer whosefrequency response istheinverse oftheperiodicfunction Sq(j')+(No/2). Toimplement Equation (4.110)exactlyweneedanequalizer ofinfinitelength.Inpractice, wemayapproximate theoptimum solution byusinganequalizer withafinitesetof coefficients {CkH'~-N' providedNislargeenough.Thusthereceivertakestheformshown inFigure4.27.NotethattheblocklabeledZ-linFigure4.27introduces adelayequalto Th,whichmeansthatthetapspacingoftheequalizer isexactlythesameasthebitduration Th•Anequalizer soconfigured issaidtobesynchronous withthetransmitter. PRACTICAL CONSIDERATIONS ThemmsereceiverofFigure4.27workswellinthelaboratory, wherewehaveaccess10 thesystemtobeequalized, inwhichcasewemaydetermine atransversal equalizer char acterized bythesetofcoefficients {cklr~-N' whichprovides anadequate approximation tothefrequency response C(f)ofEquation (4.110).Inareal-lifetelecommunications environment, however, thechannel isusuallytimevarying. Forexample, inapublic Rel:eived signal .l(t)Matched filterTransversal eqlJalizerr--------------------------------------------------, I I I I I I I I eNI I I I I :,.-l-----'----------'-----'------'---~------''----_4: I L I ~__________________ _ --J FIGURE4.27Optimum linearreceiverconsisting ofthecascadeconnection ofmatchedfilter andtransversal equalizer. 4.10Adaptive Equalization 287 switched telephone network, wefindthattwofactorscontribute tothedistribution of pulsedistortion ondifferent linkconnections: r>-Differences inthetransmission characteristics oftheindividual linksthatmaybe switched together. r>-Differences inthenumberoflinksinaconnection. Theresultisthatthetelephone channelisrandominthesenseofbeingoneofanensemble ofpossiblephysical realizations. Consequently, theuseofafixedpairofmatched filter andequalizer designed onthebasisofaveragechannelcharacteristics maynotadequately reducetheeffectsofintersymbol interference andchannelnoise.Torealizethefulltrans missioncapability ofthetelephone channel, weneedanadaptive receiver9thatprovides fortheadaptive implementation ofboththematched filterandtheequalizer inacombined manner. Thereceiver isadaptive inthesensethattheequalizer coefficients areadjusted automatically inaccordance withabuilt-inalgorithm. Another pointofinterestisthatitmaybedesirable tohavethetapsoftheequalizer spacedbyanamountcloserthanthesymbolperiod;typically, thespacingbetweenadjacent tapsissetequaltoT12.Theresulting structure isknownasafractionally spacedequalizer (FSE).AnFSEhasthecapability ofcompensating fordelaydistortion muchmoreeffec tivelythanaconventional synchronous equalizer. Anotheradvantage oftheFSEisthefact thatdatatransmission maybeginwithanarbitrary sampling phase.However, mathemat icalanalysisoftheFSEismorecomplicated thanforasynchronous equalizer andwill therefore notbepursuedhere.to L4.10Adaptive Equalization Inthissectionwedevelopasimpleandyeteffectivealgorithm fortheadaptive equalization ofalinearchannelofunknown characteristics. Figure4.28showsthestructure ofan adaptive synchronous equalizer, whichincorporates thematched filteringaction.Theal gorithmusedtoadjusttheequalizer coefficients assumestheavailability ofadesiredre sponse.One'sfirstreaction totheavailability ofareplicaofthetransmitted signalis:If suchasignalisavailable atthereceiver, whydoweneedadaptive equalization? Toanswer thisquestion, wefirstnotethatatypicaltelephone channelchangeslittleduringanaverage datacall.Accordingly, priortodatatransmission, theequalizer isadjustedundertheguid- Output y[n] Errorsignal'- "'e[;;;:n]_--{ 1: + Desired response d[n] FIGURE4.28Blockdiagram ofadaptive equalizer. 288 CHAPTER 4.,BASEJlA,'IO PULSETRANSMISSION anceofatrainingsequence transmitted throughthechannel. Asynchronized versionof thistrainingsequence isgenerated atthereceiver, where(afteratimeshiftequaltoth transmission delaythrough thechannel) itisappliedtotheequalizer asthedesired le~ sponse.Atrainingsequence commonly usedinpracticeisthepseudonoise (PN)sequence whichconsistsofadeterministic periodic sequence withnoise-like characteristics. Tw' identical PNsequence generators areused,oneatthetransmitter andtheotherattk receiver. Whenthetrainingprocessiscompleted, thePNsequence generator isswitched off,andtheadaptive equalizer isreadyfornormaldatatransmission. Detailed description ofPNsequence generators ispresented inChapter7. I!i!LEAST-MEAN-SQUARE ALGORITHM (REVISITED) Tosimplifynotational matters,welet x[n]=x(nT) y[n]=y(nT) Then,theoutputy[n]ofthetapped-delay-line equalizer inresponse totheinputsequence {x[n]}isdefinedbythediscreteconvolution sum(seeFigure4.28) N y[n]=2:wkx[n-k] k~O(4.112) whereWkistheweightatthekthtap,andN+1isthetotalnumberoftaps.Thetap weightsconstitute theadaptive filtercoefficients. Weassumethattheinputsequence [x[nJl hasfiniteenergy.Wehaveusedanotation fortheequalizer weightsinFigure4.28thatIS different fromthecorresponding notationinFigure4.27toemphasize thefactthatthe equalizer inFigure4.28alsoincorporates matched filtering. Theadaptation maybeachieved byobserving theerrorbetween thedesiredpulse shapeandtheactualpulseshapeatthefilteroutput,measured atthesampling instants, andthenusingthiserrortoestimate thedirection inwhichthetap-weights ofthefilter shouldbechanged soastoapproach anoptimum setofvalues.Fortheadaptation, we mayuseacriterion basedonminimizing thepeakdistortion, definedastheworst-case intersymbol interference attheoutputoftheequalizer. Thedevelopment ofanadaptive equalizer usingsuchacriterion buildsonthezero-forcing conceptdescribed brieflyin Section4.9.However, theequalizer isoptimum onlywhenthepeakdistortion atitsinput islessthan100percent(i.e.,theintersymbol interference isnottoosevere).Abetter approach istouseamean-square errorcriterion, whichismoregeneralinapplication; alsoanadaptive equalizer basedonthemean-square errorcriterion appears tobeless sensitive totimingperturbations thanonebasedonthepeakdistortion criterion. Accord ingly,inwhatfollowsweusethemean-square errorcriteriontoderivetheadaptive equal izationalgorithm. Leta[n]denotethedesiredresponse definedasthepolarrepresentation ofthenth transmitted binarysymbol.Letern]denotetheerrorsignaldefinedasthedifference be· tweenthedesiredresponse a[n]andtheactualresponse y[n]oftheequalizer, asshownby ern]=a[n]-y[n] (4.113J Intheleast-mean-square (LMS)algorithmllforadaptiveequalization, theerrorsignalern] actuates theadjustments appliedtotheindividual tapweightsoftheequalizer asthe algorithm proceeds fromoneiteration tothenext.Aderivation oftheLMSalgorithm for 4.10Adaptive Equalization 289 adaptive prediction waspresented inSection3.13.Recasting Equation (3.72)intoitsmost generalform,wemaystatetheformulafortheLMSalgorithm inwordsasfollows: (U~1~::t:al~e) =(o~IZt:~:e_) + (Step-size) .(I~~~~i:~g~~l) (~rror) (4.114) 'ghP . h P parameter kthtap- signalweit weigt weight LetJLdenotethestep-size parameter. FromFigure4.28weseethattheinputsignalapplied tothekthtap-weight attimestepnisx[n-k].Hence,usingwk(n)astheoldvalueof thekthtap-weight attimestepn,theupdatedvalueofthistap-weight attimestepn+1 is,inlightofEquation (4.114),definedby wk[n+1]=wk[n]+JLX[n-k]e[n], k=0,1,..., N (4.115) where N ern]=a[n]-2:wk[n]x[n -k] k~O(4.116) Thesetwoequations constitute theLMSalgorithm foradaptiveequalization. Notethat thelengthoftheadaptive equalizer inFigure4.28isnottobeconfused withthelengthof theequalizer inFigure4.27. Wemaysimplifytheformulation oftheLMSalgorithm usingmatrixnotation. Let the(N+1)-by-1vectorx[n]denotethetap-inputs oftheequalizer: x[n]=[x[n],...,x[n-N+1],x[n-NJV (4.117) wherethesuperscript Tdenotes matrixtransposition. Correspondingly, letthe (N+1)-by-1vectorw[n]denotethetap-weights oftheequalizer: w[n]=[worn],wl[n],•..,wN[n]f (4.118) Wemaythenusematrixnotation torecasttheconvolution sumofEquation (4.112)in thecompact form y[n]=xT[n]w[n] (4.119) wherexT[n]w[n] isreferredtoastheinnerproductofthevectorsx[n]andw[n].Wemay nowsummarize theLMSalgorithm foradaptive equalization asfollows: 1.Initialize thealgorithm bysettingw[l]=0(i.e.,setallthetap-weights oftheequalizer tozeroatn=1,whichcorresponds totimet=T). 2.Forn=1,2,...,compute y[n]=xT[n]w[n] ern]=a[n]-y[n] w[n+1]=w[n]+JLe[n]x[n] whereJListhestep-size parameter. 3.Continue theiterative computation untiltheequalizer reachesa"steadystate,"by whichwemeanthattheactualmean-square erroroftheequalizer essentially reaches aconstant value. TheLMSalgorithm isanexample ofafeedback system,asillustrated intheblock diagram ofFigure4.29,whichpertainstothekthfiltercoefficient. Itistherefore possible 290 CHAPTER 4'"BASEJIA,'1D PULSE TRANSMISSION FIGURE 4.29Signal-flow graphrepresentation oftheLMSalgorithm involving thekthtap weight. forthealgorithm todiverge(i.e.,fortheadaptive equalizer tobecomeunstable). Unfor tunately, theconvergence behavior oftheLMSalgorithm isdifficulttoanalyze.Neverthe. less,provided thatthestep-size parameter f.J,isassigned asmallvalue,wefindthataftera largenumberofiterations thebehavior oftheLMSalgorithm isroughlysimilartothatof thesteepest-descent algorithm, whichusestheactualgradientratherthananoisyestimate forthecomputation ofthetap-weights. (Thesteepest-descent algorithm wasdiscussed in Section3.13.) IIOPERATION OFTHEEQUALIZER Therearetwomodesofoperation foranadaptive equalizer, namely,thetraining mode anddecision-directed mode,asshowninFigure4.30.Duringthetrainingmode,asex plainedpreviously, aknownPNsequence istransmitted andasynchronized versionofit isgenerated inthereceiver, where(afteratimeshiftequaltothetransmission delay)itis appliedtotheadaptive equalizer asthedesiredresponse; thetap-weights oftheequalizer aretherebyadjusted inaccordance withtheLMSalgorithm. Whenthetrainingprocessiscompleted, theadaptive equalizer isswitched toits secondmodeofoperation: thedecision-directed mode.Inthismodeofoperation, theerror signalisdefinedby ern]=a[n]-y[n] (4.120) FIGURE 4.30Illustrating thetwooperating modesofanadaptive equalizer: Forthetraining mode,theswitchisinposition I;andforthetracking mode,itismovedtoposition 2. 4.10Adaptive Equalization 291 wherey[n]istheequalizer outputattimet=nT,andd[n]isthefinal(notnecessarily) correctestimate ofthetransmitted symbola[n].Now,innormaloperation thedecisions madebythereceiverarecorrectwithhighprobability. Thismeansthattheerrorestimates arecorrectmostofthetime,therebypermitting theadaptive equalizer tooperatesatisfac torily.Furthermore, anadaptive equalizer operating inadecision-directed modeisableto trackrelatively slowvariations inchannelcharacteristics. Itturnsoutthatthelargerthestep-sizeparameter JL,thefasterthetrackingcapability oftheadaptive equalizer. However, alargestep-size parameter JLmayresultinanunac ceptably highexcessmean-square error,definedasthatpartofthemean-square valueof theerrorsignalinexcessoftheminimum attainable valueJmin(whichresultswhenthe tap-weights areattheiroptimum settings). Wetherefore findthatinpracticethechoiceof asuitablevalueforthestep-size parameter JLinvolvesmakingacompromise betweenfast tracking andreducing theexcessmean-square error. IIIDECISION-FEEDBACK EQUALIZATION Todevelopfurtherinsightintoadaptive equalization, consider abaseband channelwith impulseresponse denotedinitssampled formbythesequence {h[n]}whereh[n]=h(nT). Theresponse ofthischanneltoaninputsequence {x[n]},intheabsenceofnoise,isgiven bythediscreteconvolution sum y[n]=Lh[k]x[n-k] k =h[O]x[n]+Lh[k]x[n-k]+Lh[k]x[n-k] k<o k>O(4.121) ThefirsttermofEquation (4.121)represents thedesireddatasymbol.Thesecondtermis duetotheprecursors ofthechannelimpulseresponse thatoccurbeforethemainsample h[O]associated withthedesireddatasymbol.Thethirdtermisduetotheposteursors of thechannelimpulseresponse thatoccurafterthemainsampleh[O].Theprecursors and postcursors ofachannelimpluseresponse areillustrated inFigure4.31.Theideaofde cision-feedback equalization12istousedatadecisions madeonthebasisofprecursors of thechannelimpulseresponse totakecareofthepostcursors; fortheideatowork,however, thedecisions wouldobviously havetobecorrect.Provided thatthiscondition issatisfied, h[O] Precursorso Postcursors FIGURE4.31Impulse response ofadiscrete-time channel, depicting theprecursors and postcursors. 292 CHAPTER 4!IIBASERAND PlJLSE TRANSMISSION Feedback sectionEstimate oftransmitted symbol.an FIGlJRE 4.32Blockdiagram ofdecision-feedback equalizer. adecision-feedback equalizer isabletoprovideanimprovement overtheperformance of thetapped-delay-line equalizer. Adecision-feedback equalizer (DFE)consistsofafeedforward section,afeedback section,andadecisiondeviceconnected togetherasshowninFigure4.32.Thefeedforward sectionconsistsofatapped-delay-line filterwhosetapsarespacedatthereciprocal ofthe signaling rate.Thedatasequence tobeequalized isappliedtothissection.Thefeedback sectionconsistsofanothertapped-delay-line filterwhosetapsarealsospacedatthereCIp rocalofthesignaling rate.Theinputappliedtothefeedback sectionconsistsofthedeci sionsmadeonpreviously detected symbols oftheinputsequence. Thefunction ofrhe feedback sectionistosubtractoutthatportionoftheintersymbol interference produced bypreviously detectedsymbolsfromtheestimates offuturesamples. Notethattheinclusion ofthedecisiondeviceinthefeedback loopmakestheequal izerintrinsically nonlinear andtherefore moredifficulttoanalyzethananordinary tapped delay-line equalizer. Nevertheless, themean-square errorcriterion canbeusedtoobtaina mathematically tractable optimization ofadecision-feedback equalizer. Indeed,theLMS algorithm canbeusedtojointlyadaptboththefeedforward tap-weights andthefeedback tap-weights basedonacommon errorsignal;seeProblem 4.37. Onthebasisofextensive comparative evaluations ofalinearequalizer anddecision feedback equalizer reported intheliterature,'3 wemayreportthatwhenthefrequency response ofalinearchannelischaracterized bysevereamplitude distortion orrelatively sharpamplitude cutoff,thedecision-feedback equalizer offersasignificant improvement inperformance overalinearequalizer foranequalnumberoftaps.Itispresupposed here thatthefeedback decisions intheDFEareallcorrect.Foranexample ofsharpanlplitude cutoff,seethefrequency response ofatelephone channeldepicted inFigure8intheBack groundandPreviewchapter. Unlikealinearequalizer, adecision-feedback equalizer suffersfromerrorpropaga tion.However, despitethefactthattheDFEisafeedback system,errorpropagation will notpersistindefinitely. Rather,decisionerrorstendtooccurinbursts.Tojustifythiskind ofbehavior, weofferthefollowing intuitive reasoning:'4 '"'LetLdenote the numberoftapsinthefeedback sectionofaDFE.Afterasequence ofLconsecutive correctdecisions, alldecisionerrorsinthefeedback sectionwillbe flushedout.Thispointstoanerrorpropagation offiniteduration. I»Whenadecisionerrorismade,theprobability ofthenextdecisionbeingerroneoUS tooisclearlynoworsethan1/2. ~LetKdenotetheduration oferrorpropagation, thatis,thenumberofsymbo~ neededtomakeLconsecutive correctdecisions. ThentheaverageerrorrateIS (KI2)Po,whereKI2istheaveragenumberoferrorsproduced byasingledecision error,andPoistheprobability oferrorgiventhatthepastLdecisions areallcorrect. 4.11Computer Experiments: EyePatterns 293 ~Inafair-coin tossingexperiment, theaveragenumberofcointosses,K,neededto getLsuccessive heads(representing noerrors)turnsouttobe2(2L-1). Itfollowstherefore thattheeffectoferrorpropagation inadecision-feedback equalizer is toincreasetheaverageerrorratebyafactorapproximately equal to2\compared tothe probability ofmakingthefirsterror.Forexample, forL=3theaverageerrorrateis increased bylessthananorderofmagnitude duetoerrorpropagation. ~11Computer Experinreffts: EyePatterns Inprevious sectionsofthischapterwehavediscussed varioustechniques fordealingwith theeffectsofchannelnoiseandintersymbol interference ontheperformance ofabaseband pulse-transmission system.Inthefinalanalysis, whatreallymattersishowtoevaluate the combined effectoftheseimpairments onoverallsystemperformance inanoperational environment. Anexperimental toolforsuchanevaluation inaninsightful manneristhe so-called eyepattern,whichisdefinedasthesynchronized superposition ofallpossible realizations ofthesignalofinterest(e.g.,receivedsignal,receiveroutput)viewedwithina particular signaling interval.Theeyepatternderivesitsnamefromthefactthatitresembles thehumaneyeforbinarywaves.Theinteriorregionoftheeyepatterniscalledtheeye opening. Aneyepatternprovides agreatdealofusefulinformation abouttheperformance of adatatransmission system,asdescribed inFigure4.33.Specifically, wemakethefollowing statements: l>Thewidthoftheeyeopeningdefinesthetimeintervaloverwhichthereceivedsignal canbesampled withouterrorfromintersymbol interference; itisapparent thatthe preferred timeforsampling istheinstantoftimeatwhichtheeyeisopenthewidest. I>Thesensitivity ofthesystemtotimingerrorsisdetermined bytherateofclosureof theeyeasthesampling timeisvaried. ~Theheightoftheeyeopening, ataspecified sampling time,definesthenoisemargin ofthesystem. Best sampling time I I I Margin---- overnoise Timeintervaloverwhich thereceivedsignalcan besampled FIGURE4.33Interpretation oftheeyepattern. 294 CHAPTER 4'"BASEBA-l\ID PULSETRANSMISSION Whentheeffectofintersymbol interference issevere,tracesfromtheupperportionoftne eyepatterncrosstracesfromthelowerportion, withtheresultthattheeyeiscompletely closed.Insuchasituation, itisimpossible toavoiderrorsduetothecombined presence ofintersymbol interference andnoiseinthesystem. InthecaseofanM-arysystem,theeyepatterncontains(M-1)eyeopenings stacked upvertically oneontheother,whereMisthenumberofdiscreteamplitude levelsusedto construct thetransmitted signal.Inastrictlylinearsystemwithtrulyrandomdata,allthese eyeopenings wouldbeidentical. Inthenexttwoexperiments, weusecomputer simulations tostudytheeyepatterns foraquaternary (M=4)baseband PAMtransmission systemundernoiseless, noisy,and band-limited conditions. Theeffectofchannelnonlinearity oneyepatterns isdiscussedin Problem 4.38. Experiment 1:EffectofChannel Noise Figure4.34ashowstheeyediagram ofthesystemunderidealized conditions: nochannel noiseandnobandwidth limitation. Thesourcesymbols usedarerandomly generated on acomputer, withraisedcosinepulse-shaping. Thesystemparameters usedforthegener ationoftheeyediagram areasfollows: Nyquist bandwidth W=0.5Hz,rollofffactor a=0.5,andsymboldurationT=TblogzM=2Tb•Theopenings inFigure4.34are perfect,indicating reliableoperation ofthesystem.NotethatthisfigurehasM - 1=3 openings. Figures4.34band4.34cshowtheeyediagrams forthesystem,butthistimewith channelnoisecorrupting thereceived signal.Thesetwofiguresweresimulated forsignal to-noise ratioSNR=20dBand10dB,respectively, withtheSNRbeingmeasured arthe channeloutput.WhenSNR=20dBtheeffectofchannelnoiseishardlydiscernible in Figure4.34b,butwhenSNR=10dBtheopenings oftheeyediagram inFigure4.34care barelyvisible. Experiment 2:EffectofBandwidth Limitation Figures4.35aand4.35bshowtheeyediagrams forthequaternary systemusingthesame parameters asbefore,butthistimeunderabandwidth-limited condition andanoiseless channel. Specifically, thechannelisnowmodeled byalow-pass Butterworth filter,whose squaredmagnitude response isdefinedby IH(fW =1+(}'fo)2N whereNistheorderofthefilter,andfoisits3-dBcutofffrequency. Forthecomputer experiment described inFigure4.35a,thefollowing valuesareused: N=25andfo=0.975Hz Thebandwidth required bythePAMtrasmission systemiscomputed tobe BT=W(l+a)=0.75Hz Although thechannelbandwidth (i.e.,cutofffrequency) isgreaterthanabsolutely neces sary,itseffectonthepassband isobserved asadecrease inthesizeoftheeyeopenings compared tothoseinFigure4.34a.Insteadofthedistinctvaluesattimet=1 s(asshown inFigure4.34a),nowthereisablurredregion. (a) 0.5 w '"~ C-O E« -0.5 -1 0.20.40.60.8 1.21.41.6 1.8 (bl (e)0.5 -0.5 -1 0.5 -0.5 -1Normalized timetfTb 0.20.40.60.811.21.41.61.8 Normalized timetIT/) 0.20.40.60.8 1.21.41.6 1.8 2 Normalized timet/Tb FIGURE4.34(a)Eyediagram fornoiseless quaternary system.(b)Eyediagram forquaternary systemwithSNR=20dB.(e)Eyediagram forquaternary systemwithSNR=10dB. 295 296 CHAPTER 4mBASEBAND PULSE TRAl\lSMISSION (al 0.5 ~-0 Ea ~« -0.5 -1 a0.20.40.6 0.8 11.2 1.4 1.6 1.8 Normalized timetlTt, (b) 0.5 ~ ~a.a E« -0.5 -1 o0.20.40.6 0.8 11.21.41.6 1.8 Normalized timetlTb FIGURE4.35(a)Eyediagram fornoiseless band-limited quaternary system:cutofffrequency f0=0.975Hz.(b)Eyediagram fornoiseless band-limited quaternary system:cutofffrequency fo0.5Hz. InFigure4.35bthechannelbandwidth isreducedfurtherbymodeling thechannel asalow-pass Butterworth filterwithN=25andfo=0.5Hz.Theeffectofreduced channelbandwidth istofurtherreducetheextenttowhichtheeyesareopen. I4.12Summary andDiscussion Inthischapter,westudiedtheeffectsofchannelnoiseandintersymbol interference onthe performance ofbaseband-pulse transmission systems.Intersymbol interference (lSI)isdif ferentfromnoiseinthatitisasignal-dependent formofinterference thatarisesbecause ofdeviations inthefrequency response ofachannelfromtheideallow-pass filter(Nyquist channel); itdisappears whenthetransmitted signalisswitched off.Theresultofthese deviations isthatthereceivedpulsecorresponding toaparticular datasymbolisaffected Notesam!References 297 bythetailendsofthepulsesrepresenting theprevious symbolsandthefrontendsofthe pulsesrepresenting thesubsequent symbols. Depending onthereceived signal-to-noise ratio,wemaydistinguish threedifferent situations thatcanariseinbaseband-pulse transmission systemsforchannels withfixed characteristics: 1.TheeffectoflSIisnegligible incomparison tothatofchannelnoise. Theproperprocedure inthiscaseistouseamatched filter,whichistheoptimum linear time-invariant filterformaximizing thepeakpulsesignal-to-noise ratio. 2.Thereceivedsignal-to-noise ratioishighenoughtoignoretheeffectofchannelnoise. Inthiscase,weneedtoguardagainsttheeffectsoflSIonthereconstruction ofthetrans mitteddataatthereceiver.Inparticular, controlmustbeexercised overtheshapeofthe receivedpulse.Thisdesignobjective canbeachieved inoneoftwodifferent ways: p..Usingaraisedcosinespectrum fortheoverallfrequency response ofthebaseband pulsetransmission system. !l>Usingcorrelative-level codingorpartial-response signaling thataddslSItothetrans mittedsignalinacontrolled manner. 3.ThelSIandnoisearebothsignificant. Foramathematically tractable solution tothismoredifficultsituation, wemayusethe mean-square errorcriterion. Theresulting optimum linearreceiveriscalledtheminimum mean-square error(mmse)receiver. Itconsistsofthecascadeconnection ofamatched filter andlineartransversal (tapped-delay-line) equalizer. When,however, thechannelisrandominthesenseofbeingoneofanensemble of possiblephysical realizations, whichisfrequently thecaseinatelecommunications envi ronment, theuseoffixedfilterdesignsbasedonaveragechannelcharacteristics maynot beadequate.Insituations ofthiskind,thepreferred approach istouseanadaptive equal izer,thepurposeofwhichistocompensate for variations inthefrequency response ofthe channelautomatically duringthecourseofdatatransmission. Thecombined useofa tapped-delay-line filterandtheleast-mean-square (LMS)algorithm foradjusting thetap weightsprovides thebasisofasimpleandyethighlyeffective methodforimplementing theadaptive equalizer. Suchadeviceiscapableofdealingwiththecombined effectsofISI andreceivernoiseinanonstationary environment. Itspractical valueliesinthefactthat almosteverymodem(modulator-demodulator) incommercial usetodayforthetransmis sionofdigitaldataoveravoice-grade telephone channelusesanadaptive equalizer asan integralpart. Another important application ofadaptive filteringisinthedesignofechocancellers thatconstitute acriticalcomponent oftransceivers fordigitalsubscriber lines.Typically, adigitalsubscriber lineusesatwistedpairasthetransmission medium, theverysameone usedinordinary telephone channels. However, unliketelephone channels, digitalsub scriberlinesaredesigned toprovideahighdata-rate digitaltransmission capability be tweenadigitalnetwork andsubscriber plants,withadatarateof64kb/sandup. LNOTESANDREFERENCES 1.Theclassicbooksonbaseband-pulse transmission areLucky,Salz,andWeldon(1968)and Sunde(1969).Fordetailedtreatment ofdifferentaspectsofthesubject,seeGitlin,Hayes, andWeinstein (1992).Proakis(1995),andBenedetto, Biglieri,andCastellani (1987). 298 CHAPTER 4IIIBASEBA1'lD PULSE TRANSMISSION 2.Thecharacterization ofamatched filterwasfirstderivedbyNorthinaclassified repa (RCALaboratories ReportPTR-6C, June1943),whichwaspublished 20yearslater-sn thepaperbyNorth(1963).Asimilarresultwasobtained independently byVanV1eck'a:~ Middleton, whocoinedthetermmatched filter:seethepaperbyVanVleckandMiddlet (1946).Forreviewmaterialonthematched filteranditsproperties, seethepapers bYTu~n (1960,1976). In 3.Theerrorfunction denotedbyerf(u),isdefinedinanumberofdifferent waysintheliter_ ature.Weshallusethefollowing definition: 2(U erf(u)=y;.Joexp(-z2) dz Theerrorfunction hastwousefulproperties: (i)erf(-u) =-erf(u) Thisisknownasthesymmetry relation. (ii)Asuapproaches infinity,erf(u)approaches unity;thatis, 2100 •,exp(-z2) dz=1 V'iTa Thecomplementary errorfunction isdefinedby 2fOOerfc(u)=y;. uexp(-z2) dz whichisrelatedtotheerrorfunction asfollows: erfc(u)=1 -erf(u) TableA6.6givesvaluesoftheerrorfunction erf(u)foruintherange0to3.3. Forlargepositivevaluesofu,wehavetwosimpleboundsonerfc(u),onelowerand theotherupper,asshownby exp(-u2 )(11 )rf()exp(-u2)v;.u 2u2<e cu<v;.u Thecomplementary errorfunction provides thebasisforacompact formulation of theprobability ofsymbolerror,asexplained inSection4.3.Another function thatisalso commonly usedintheliterature forthispurpose istheQ-function. Consider astandardized Gaussian randomvariableXofzero·meananduuitvariance. Theprobability thatan observed valueoftherandomvariableXwillbegreaterthanvisgivenbytheQ-func~on: Q(v)=vbrexp(-~)dx TheQ-function definestheareaunderthestandardized Gaussian tail.TheQ-function is relatedtothecomplementary errorfunction as Q(v)=~erfc(0) Conversely, puttingu=v/V2,wehave erfc(u)=2Q(V2u) 4.Thecriterion described inEquation (4.49)orEquation (4.53)wasfirstformulated by Nyquistinthestudyoftelegraph transmission theory;the1928paperbyNyquist isa classic.Intheliterature, thiscriterion isreferredtoasNyquist's firstcriterion. Inhis192~ paper,Nyquistdescribed anothermethod, referredtointheliterature asNyquist's setOn NotesamiRefere>rees 299 criterion. Thesecondmethodmakesuseoftheinstantsoftransition betweenunlikesym bolsinthereceivedsignalratherthancenteredsamples. Adiscussion ofthefirstandsecond criteriaispresented inBennett(1970,pp.78-92)andinthepaperbyGibbyandSmith (1965).Athirdcriterion attributed toNyquist isdiscussed inSunde(1969);seealsothe papersbyPasupathy (1974)andSayarandPasupathy (1987). 5.Correlative-level codingandpartial-response signaling aresynonymous; bothtermsare usedintheliterature. Theideaofcorrelative codingwasoriginated byLender(1963). Lender's workwasgeneralized forbinarydatatransmission byKretzmer (1966).Forfur therdetailsoncorrelative codingtechniques, seethebookbyGitlin,Hayes,andWeinstein (1992);seealsothepapersbyPasupathy (1977),KabalandPasupathy (1975),andSousa andPasupathy (1983). 6.Thematerialondigitalsubscriber linespresented inSection4.8isbasedonthetwopapers byLinandTzeng(1988),andLechleider (1989),andthebooksbyStarr,Cioffi,and Silverman (1999)andChen(1998). 7.Foradiscussion oflinecodesfor digital subscriber loops,seeGitlinetal.(1992). 8.InEricson(1971)itisshownthatforevery"reasonable" performance criterion, theopti mumreceivercanberealizedasamatched filterfollowed byatapped-delay-line equalizer, asshowninFigure4.27.Inaddition tothemean-square errorcriterion considered in Section4.9,reasonable performance criteriaofinterestincludethefollowing: (i)Minimization oftheprobability ofsymbolerror. (ii)Zero-forcing equalization (toreducetheintersymbol interference tozero),followed byminimization oftheprobability ofsymbolerrorsubjecttothisconstraint. (iii)Minimization ofsignal-to-noise ratioatthesampling instants. Criterion (i)isthemostnaturalapproach totheoptimization ofalinearreceiver; this approach, pursuedinAaronandTufts(1966),is,unfortunately, complicated. Criterion (ii),duetoLuckyetal.(1968),isamuchsimplerapproach. Criterion (iii)isduetoGeorge (1965). 9.Adaptive equalization oftelephone channels waspioneered byLucky(1965,1966).Since thattime,numerous adaptive equalization schemes havebeenpublished intheliterature, whichprovideequalization forspecificsynchronous data-transmission systems.Forreview papersonadaptive equalization, seeProakis(1975)andQureshi(1982,1985).Adaptive equalization isalsodiscussed indetailinthebooksbyGitlin,Hayes,andWeinstein (1992, Chapter 8)andProakis(1995,Chapter6). 10.Itappearsthatearlyworkonfractionally spacedequalizers wasinitiated byBrady(1970). Othercontributions tothesubjectincludesubsequent workbyUngerboeck (1976)and GitlinandWeinstein (1981).Adetaileddiscussion offractionally spacedequalizers isalso presented inGitlinetal.(1992). 11.TheLMSalgorithm wasoriginated byWidrow andHoff,Jr.(1960).Foradetailedcon vergence analysisoftheLMSalgorithm, seeHaykin(1996,Chapter9),andWidrow and Stearns(1985,Chapter6). 12.Decision-feedback equalization wasfirstdescribed inareportbyAustin(1967).Theop timization ofthedecision-feedback equalizer forminimum mean-square errorwasfirst accomplished byMonsen(1971).Areadable accountofdecision-feedback equalization is presented inthebookbyGitlin,Hayes,andWeinstein (1992,pp.500-510). Tomlinson (1971)andHarashima andMiyakawa (1972)describeadeviceforelim inatingerrorpropagation inadecision-feedback equalizer. Thedevice,knownastheTom linson-Harashima precoder, appearsinthetransmitter asapreprocessor tothemodulator. Thebasicideaofthisprecoder istomovethefeedback sectioninthedecisionfeedback equalizer tothetransmitter whereitisimpossible tomakedecisionerrors.However, this 300 CHAPTER 4..BASEBA<'ilD PULSE TRANSMISSION 13. 14. IPROBLEMSmodification mayresultina.significa';lt increaseintransmit power;moduloarithmetic is usedtoovercome mostofthIspowerIncrease. Forperformance comparison between linearequalizers anddecision-feedback equalize seeGitlinetal.(1992)andProakis(1995). rs, Theintuitive discussion onerrorpropagation indecision-feedback equalizers presented· Section4.10followsGitlinetaJ.(1992). In Forarigorous evaluation oftheprobability ofsymbolerrorP,inadecision-feedback equalizer witherrorpropagation, seeDuttweiler etaJ.(1974).Inthispaperitisshown thatintheworst-case intersymbol interference, P,ismultiplied byafactorof2Lrelativ totheprobability oferrortltatresultsintheabsenceofdecisionerrorsathighsigual-to~ noiseratios,whereListhenumberoftapsinthefeedback section.Theresultderivedby Duttweiler etal.provides tlteoretical justification fortheintuitive arguments presentedin Section4.10. Matched Filters 4.1Consider tltesignalsit)showninFigureP4.1. (a)Determine theimpulseresponse ofafiltermatched tothissignalandsketchitasa function oftime. (b)Plotthematched filteroutputasafunction oftime. (c)Whatisthepeakvalueoftheoutput? s(t) A "2 T T ;> FIGUREP4.1 4.2FigureP4.2ashowsapairofpulsestltatareorthogonal toeachotherovertheinterval [0,Tj.Inthisproblem weinvestigate theuseofthispulse-pair tostudyatwo-dimensional matched filter. (a)Determine thematched filtersforthepulsesSI(t)andS2(t)considered individually; forSI(t)tltefilteristhesameasthatconsidered inProblem 4.1. (b)Formatwo-dimensional matched filterbyconnecting tltetwomatched filtersofPart (a)inparallel,asshowninFigureP4.2b.Hence,demonstrate thefollowing: (i)Whenthepulses,It)isappliedtothistwo-dimensional filter,theresponse ofrhe lowermatched filteriszero. (ii)WhentltepulseS2(t)isappliedtothetwo-dimensional filter,tlteresponse ofme uppermatched filteriszero. Generalize theresultsofyourinvestigation. Problems 301 ,,(I)'2(r) 1-A 2- 3TT44" T TInput-0T T "2 "2 AA -2-2 (aJ FIGUREP4.2 4.3Consider arectangular pulsedefinedby(b)Output1 Output2 {A,g(t)=0,D:st:sT otherwise Itisproposed toapproximate thematched filterforg(t)byanideallow-pass filterof bandwidth B;maximization ofthepeakpulsesignal-to-noise ratioistheprimary objective. (a)Determine theoptimum valueofBforwhichtheideallow-pass filterprovides the bestapproximation tothematched filter. (b)Byhowmanydecibelsistheideallow-pass filterworseoffthanthematched filter? 4.4Inthisproblem weexploreanothermethodfortheapproximate realization ofamatched filter,thistimeusingthesimpleresistance-capacitance (RC)low-pass filtershowninFig ureP4.4.Thefrequency resonseofthisfilteris 1 H(f)=1+jf/fo wherefo=1I21TRC. Theinputsignalg(t)isarectangular pulseofamplitude Aand durationT.Therequirement istooptimize theselection ofthe3-dBcutofffrequencyf0 ofthefiltersothatthepeakpulsesignal-to-noise ratioatthefilteroutputismaximized. Withthisobjective inmind,showthattheoptimum valueoffoisD.2/T,forwhichthe lossinsignal-to-noise ratiocompared tothematched filterisabout1dB. FIGUREP4.4 Probability ofErrorCalculation 4.5Theformulafortheoptimum threshold inthereceiverofFigure4.4is,ingeneral,given byEquation (4.37).Discuss, ingraphical terms,howthisoptimum choiceaffectsthe 302 CHAPTER 4illBASEBAND PuLSE TRANSMISSION contributions ofthetwotermsinEquation (4.35)fortheaverageprobability ofsYmbol errorP,byconsidering thefollowing twocases: (a)Po>PI (b)P,<Po wherePoandP,aretheaprioriprobabilities ofsymbolsaand1,respectively. 4.6InabinaryPCMsystem,symbolsaand1haveaprioriprobabilities PoandP"respec_ tively.Theconditional probability densityfunction oftherandomvariableY(withsampl valuey)obtained bysampling thematched filteroutputinthereceiverofFigure4.4atthe e?d.ofasignaling interval, giventh~~symbolaw~~transm.itted, isdenotedby!Y(yIOl~ Sumlarly, Jy(y11)denotestheconditIOnal probabIlity denSityfunctIon ofY,giventhat symbol1wastransmitted. LetAdenotethethreshold usedinthereceiver, sothatifthe samplevalueyexceeds A,thereceiverdecidesinfavorofsymbol1;otherwise, itdecides infavorofsymbol O.Showthattheoptimum threshold A."p"forwhichtheaverageprob abilityoferrorisaminimum, isgivenbythesolutionof fy(Aoptll)=f!2. fy(Aopt10)P, 4.7AbinaryPCMsystemusingpolarNRZsignaling operates justabovetheerrorthreshold withanaverageprobability oferrorequalto10-6•Suppose thatthesignaling rate~ doubled. Findthenewvalueoftheaverageprobability oferror.YoumayuseTableA6.6 toevaluatethecomplementary errorfunction. 4.8Acontinuous-time sigrialissampled andthentransmitted asaPCMsignal.Therandom variableattheinputofthedecision deviceinthereceiverhasavariance of0.01volts'. (a)Assuming theuseofpolarNRZsignaling, determine thepulseamplitude thatmust betransmitted fortheaverageerrorratenottoexceed1bitinlOSbits. (b)Iftheaddedpresence ofinterference causestheerrorratetoincreaseto1bitin106 bits,whatisthevariance oftheinterference? 4.9AbinaryPCMwaveusesunipolar NRZsignaling totransmit symbols1and0;symbol 1isrepresented byarectangular pulseofamplitude Aandduration Tb'Thechannelnoise ismodeled asadditive, whiteandGaussian, withzeromeanandpowerspectraldensity No/2.Assuming thatsymbols 1andaoccurwithequalprobability, findanexpression fortheaverageprobability oferroratthereceiveroutput,usingamatched filterasde scribedinSection4.3. 4.10RepeatProblem4.9forthecaseofunipolar return-to-zero signaling, inwhichcasesymbol 1isrepresented byapulseofamplitude AanddurationTbl2andsymbolaisrepresented bytransmitting nopulse. Henceshowthatthisunipolar typeofsignaling requirestwicetheaveragepower ofunipolar nonreturn-to-zero (i.e.,on-off)signaling forthesameaverageprobability of symbolerror. 4.11Inthisproblem, werevisitthePCMreceiverofFigure4.4,butthistimeweconsidet the useofbipolarnonreturn-to-zero signaling, inwhichcasethetransmitted signals(t)is definedby Binarysymbol1:s(t)=±Afora<t:5T Binarysymbol0:s(t)0,a<t:5T Determine theaverageprobability ofsymbolerrorP,forthisreceiverassuming thatthe binarysymbolsaand1areequiprobable. RaisedCosineSpectrum 4.12Thenonreturn-to-zero pulseofFigureP4.12maybeviewedasaverycrudeformofa Nyquistpulse.Compare thespectralcharacteristics ofthesetwopulses. Problems 303 p(f) 1.0 TOT-22 FIGUREP4.12 4.13Determine theinverseFouriertransform ofthefrequency functionP(t)definedinEqua tion(4.60). 4.14Ananalogsignalissampled, quantized, andencodedintoabinaryPCMwave.Thespec ifications ofthePCMsystemincludethefollowing: Sampling rate=8kHz Number ofrepresentation levels=64 ThePCMwaveistransmitted overabaseband channelusingdiscretepulse-amplitude modulation. Determine theminimum bandwidth requiredfortransmitting thePCMwave ifeachpulseisallowed totakeonthefollowing numberofamplitude levels:2,4,or8. 4.15Consider abaseband binaryPAMsystemthatisdesigned tohavearaised-cosine spectrum P(f).Theresulting pulsepit)isdefinedinEquation (4.62).Howwouldthispulsebe modified ifthesystemwasdesigned tohavealinearphaseresponse? 4.16Acomputer putsoutbinarydataattherateof56kb/s.Thecomputer outputistransmitted usingabaseband binaryPAMsystemthatisdesigned tohavearaised-cosine spectrum. Determine thetransmission bandwidth required foreachofthefollowing rollofffactors: a=0.25,0.5,0.75,1.0. 4.17RepeatProblem 4.16,giventhateachsetofthreesuccessive binarydigitsinthecomputer outputarecodedintooneofeightpossibleamplitude levels,andtheresulting signalis transmitted usinganeight-level PAMsystemdesigned tohavearaised-cosine spectrum. 4.18Ananalogsignalissampled, quantized, andencodedintoabinaryPCMwave.Thenum berofrepresentation levelsusedis128.Asynchronizing pulseisaddedattheendofeach codewordrepresenting asampleoftheanalogsignal.Theresulting PCMwaveistrans mittedoverachannelofbandwidth 12kHzusingaquaternary PAMsystemwithraised cosinespectrum. Therollofffactorisunity. (a)Findtherate(b/s)atwhichinformation istransmitted throughthechannel. (b)Findtherateatwhichtheanalogsignalissampled. Whatisthemaximum possible valueforthehighestfrequency component oftheanalogsignal? 4.19AbinaryPAMwaveistobetransmitted overabaseband channelwithanabsolute max imumbandwidth of75kHz.Thebitduration is10JLS.Findaraised-cosine spectrum thatsatisfiestheserequirements. Correlative-Level Coding 4.20Theduobinary,ternary,andbipolarsignaling techniques haveonecommon feature:They allemploythreeamplitude levels.Inwhatwaydoestheduobinarytechnique differfrom theothertwo? 4.21Thebinarydatastream001101001 isappliedtotheinputofaduobinary system. (a)Construct theduobinary coderoutputandcorresponding receiveroutput,withouta precoder. 304 CHAPTER 4..BASEIlk'lD PULSE TRANSMISSION (b)Supposethatowingtoerrorduringtransmission, thelevelatthereceiverinPUtp ducedbytheseconddigitisreducedtozero.Construct thenewreceiveroutput, roo 4.22RepeatProblem 4.21,assuming theuseofaprecoder inthetransmitter. 4.23TheschemeshowninFigureP4.23maybeviewedasadifferential encoder(consisti ofthemodulo-2 adderandthe1-unitdelayelement) connected incascadewithaspec~ formofcorrelative coder(consisting ofthe1-unitdelayelementandsummer). Asingl delayelementisshowninFigureP4.23sinceitiscommon toboththedifferential encod: andthecorrelative coder.Inthisdifferential encoder, atransition isrepresented bysYmboloandnotransition bysymbol1. (a)Findthefrequency response andimpulseresponse ofthecorrelative coderpartofth schemeshowninFigureP4.23. e (b)Showthatthisschememaybeusedtoconverttheon-offrepresentation ofabinary sequence (appliedtotheinput)intothebipolarrepresentation ofthesequence atthe output.Youmayillustrate thisconversion byconsidering thesequence 010001101. For.descriptions ofon-off,bipolar,anddifferential encoding ofbinarysequences, see Semon3.7. Modulo-2 adder Bipolarrepresentation ofbinarysequence FIGURE P4.23 4.24Consider arandombinarywavex(t)inwhichthe1sandOsoccurwithequalprobability, thesymbols inadjacent timeslotsarestatistically independent, andsymbol1isrepre· sentedbyAvoltsandsymbol0byzerovolts.Thison-offbinarywaveisappliedtothe circuitofFigureP4.23. (a)UsingtheresultofProblem 4.23,showthat-thepowerspectraldensityofthebipolar wavey(t)appearing attheoutputofthecircuitequals Sx(f)=Tt.A2sin2(1TfTb)sinc2(fTb) (b)Plotthepowerspectraldensities oftheon-offandbipolarbinarywaves,andcompare them. 4.25Thebinarydatastream011100101 isappliedtotheinputofamodified duobinarysystern. (a)Construct themodified duobinary coderoutputandcorresponding receiver outpU~ withoutaprecoder. (b)Supposethatduetoerrorduringtransmission, thelevelproduced bythethirddigit isreducedtozero.Construct thenewreceiveroutput. 4.26RepeatProblem 4.25assuming theuseofaprecoder inthetransmitter. M-aryPAMSystems 4.27Consider abaseband M·arysystemusingMdiscreteamplitude levels.Thereceivermo?el isasshowninFigureP4.27,theoperation ofwhichisgoverned bythefolloWIng assumptions: Problems 305 (a)Thesignalcomponent inthereceived wave is mIt)=~ansinc(~-n) wherelITisthesignaling rateinbauds. (b)Theamplitude levelsarean=±N2,±3N2,... ,±(M-1)N2ifMiseven,and an=0,±A,...,±(M-l}A/2ifMisodd. (e)TheMlevelsareequiprobable, andthesymbolstransmitted inadjacemtimeslotsare statistically independent. (d)Thechannelnoisew(t)iswhiteandGaussian withzeromeanandpowerspectral densityNo/2. (e)Thelow-pass filterisidealwithbandwidth B=l/2T. (f)Thethreshold levelsusedinthedecisiondeviceare0,±A,...,±(M-2)A/2ifM iseven,and±A/2,±3N2,...,±(M-2)A/2ifMisodd. Theaverageprobability ofsymbolerrorinthissystemisdefinedby P=(1-~)erfc(~) e M 2V2IT where ITisthestandard deviation ofthenoiseattheinputofthedecisiondevice.Dem onstrate the validity ofthisgeneralformula bydetermining P,forthefollowing three cases:M=2,3,4. met) FIGUREP4.27Output 4.28Suppose thatinabaseband M-aryPAMsystemwithMequallylikelyamplitude levels, asdescribed inProblem 4.27,theaverageprobability ofsymbolerrorPeislessthan10-6 soastomaketheoccurrence ofdecoding errorsnegligible. Showthattheminimum value ofreceivedsignal-to-noise ratioinsuchasystemisapproximarely givenby (SNR)min =7.8(M2-1) DigitalSubscriber Lines 4.29Theamplitude distribution ofcross-talk inadigitalsubscriber linemaybemodeled as Gaussian. Justifythevalidityofsuchamodel.Hint:Typically, acablecontains many twistedpairs. 4.30(a)Derivetheformulaforthepowerspectraldensityofatransmitted signalusingthe 2B1Qlinecode. (b)Plotthepowerspectrum ofthefollowing linecodes: I>-Manchesrer code I>Modified duobinary code II>Bipolarrerum-ro-zero code II>2B1Qcode Hencecompare therelativemeritsoftheselinecodesfortheirsuitability inadigital subscriber loop. 306 CHAPTER 4IIBASEBAND PULSE 'I'RA,'iSMISSION 4.31Inthisproblem weusetheLMSalgorithm toformulate anadaptive echocanceller1 useinadigitalsubscriber line.Thebasicprinciple ofadaptive echocancellation isOr synthesize areplicaoftheechoandsubtract itfromthereturned signalinanadaptito manner,asillustrated inFigureP4.31.Thesynthesized echo,denotedbyf[n],isgenera:~ bypassingthetransmitted signalthroughanadaptive filterthatideallymatchesthe!ran ferfunction oftheechopath.Thereturned signal,consisting ofthesumofactualech8 r[n]andthereceivedsignalx[n],maybeviewedasthedesiredresponse fortheadapti0 filteringprocess. ve UsingtheLMSalgorithm, formulate theequations thatdefinetheoperation ofth adaptive echocanceller inFigureN.31. e Received signal,xln]---i..,.-----, FIGUREP4.31 Equalization 4.32FigureP4.32showsthecascadeconnection ofalinearchannelandasynchronous tapped. delay-line equalizer. Theimpulseresponse ofthechannelisdenotedbycit),andthat01 theequalizer isdenotedbyh(t).Theh(t)isdefinedby N h(t)=L.Wk8(t-kT) k~-N whereTisthespacingbetween adjacent tapsoftheequalizer, andtheWkareitstap weights(coefficients). Theimpulseresponse ofthecascaded systemofFigureP4.32is denotedbyp(t).Thep(t)issampled uniformly attheratelIT.Toeliminate intersymbol interference, werequirethattheNyquistcriterion fordistortionless transmission besat isfied,asshownby {I,p(nT)=0,n=0 n*0 (a)Byimposing thiscondition, showthatthe(2N+1)tap-weights oftheresulting zero forcingequalizer satisfythefollowing setof(2N+1)simultaneous equations: N{In=0 k~NWkc,,-k=0:n*:tl,:t2,...,:tN where Cn=c(nT).Hence,showthatthezero-forcing equalizer isaninversefilterin thatitstransferfunction isequaltothereciprocal ofthetransferfunction ofthe channel. (b)Ashortcoming ofthezero-forcing equalizer isnoiseenhancement thatcanresultio poorperformance inthepresence ofchannelnoise.Toexplorethisphenomenon. consider alow-pass channelwithanotchattheNyquistfrequency, thatis,H(f)IS zeroatf=112T.Assuming thatthechannelnoiseisadditiveandwhite,showthaI thepowerspectraldensityofthenoiseattheequalizer outputapproaches infinityal f=1I2T. Problems 307 Evenifthechannelhas nonotchinitsfrequency response, thepowerspectral densityofthenoiseattheequalizer outputcanassumehighvalues.Justifythevalidity ofthisgeneralstatement. Transmitted signal FIGUREP4.32Output 4.33Consider Equation (4.108),whichdefinestheimpulseresponse ofaminimum mean squareerrorreceiver. (a)JustifythevalidityofEquation (4.109)thatistheFourier-transformed versionof Equation (4.108). (b)ThepowerspectraldensitySq{f)inEquation (4.109)istheFouriertransform ofthe autocorrelation Rq{T"T2)ofthetimefunctionq(t).TheRq{T"T2)isdefinedbyEqua tion(4.98).StartingwithEquation (4.98),derivetheformulaofEquation (4.111). 4.34Someradiosystemssufferfrommultipath distortion, whichiscausedbytheexistence of morethanonepropagation pathbetween thetransmitter andthereceiver. Consider a channeltheoutputofwhich,inresponse toasignalsIt),isdefinedby{intheabsenceof noise} x(t)=als(t-toll+a2s(t-t02) wherealanda2areconstant, andtalandt02represent transmission delays.Itisproposed tousethethree-tap delay-line-filter ofFigureP4.34toequalizethemultipath distortion produced bythischannel. (a)Evaluate thettansferfunctionofthechannel. (b)Evaluate theparameters oftherapped-delay-line filterintermsofa"a2,to"andt02, assuming thata2«alandt02>tOI' Input signal Output signal FIGUREP4.34 4.35Letthesequence [x(nT)Jdenotetheinputappliedtoatapped-delay-line equalizer. Show thatintersyrnbol interference iseliminated completely bytheequalizer provided thatits frequency response satisfiesthecondition whereTisthesymbolduration. Asthenumberoftapsintheequalizer approaches infinity,thefrequency response oftheequalizer becomes aFourierserieswithrealcoefficients andcantherefore approx- 308 CHAPTER 4'"BASEBAND PULSE TRANSMISSION imateanyfunction intheinterval(-1/2T,1/2T).Demonstrate thisproperty ofthe equalizer. 4.36Thestep-size parameter J.Lplaysacriticalroleintheoperation oftheLMSalgorithm. In thiscontext, discussthefollowing twoissues: (a)Stability. IfJ.Lexceedsacertaincriticalvalue,thealgorithm diverges (Le.,thesystem becomes unstable). (b)Memory. Thereciprocal ofJ.Lmaybeviewedasameasureofthealgorithm's memory. AswemakeJ.Lsmaller,moreofthepastsamplesoftheinputsignalinfluence operatio~ ofthealgorithm. 4.37LetthevectorswI1)[n]andwI2)[n]denotethetap-weights ofthefeed-forward andfeed. backsectionsofthedecision-feedback equalizer inFigure4.32.Formulate theLMSal. gorithmforadjusting thetap-weights ofthisequalizer. Computer Experiments 4.38InSection4.11westudiedtheeyediagram ofaquaternary (M=4)PAMbaseband transmission systemunderbothnoisyandband-limited conditions. Inthatexperimen~ thechannelwasassumed linear.Inastrictlylinearsystemwithtrulyrandomdata,allthe eyeopenings wouldbeidentical. Inpractice, however, itisoftenpossibletodiscernasym metriesintheeyepatrern, whicharecausedbynonlinearities inthecommunication channel. Inthisexperiment, westudytheeffectofanonlinear channelontheopenings of aneyepatrern.Specifically, werepeatthecomputer experiment pertaining tothenoiseless eyepatternofFigure4.34aforM=4,butthistimeassumethatthechannelisnonlinear withthefollowing input-output relation: x(t)=s(t)+as2(t) wheres(t)isthechannelinputandx(t)isthechanneloutput,andaisaconstant. (a)Dotheexperiment fora=0,0.05,0.1, 0.2. (b)Hence,discusshowvaryingaaffectstheshapeoftheeyepattern. 4.39Inthisexperiment westudytherootraised-cosine pulseduetoChennakeshu andSaulnier (1993).Thispulse,denotedbyp(t),hasthefollowing properties: Il>Thepulsep(t)issymmetric intime,thatis,p(-t)=p(t). I>ThesquaredFouriertransform ofp(t),namely,p2(f),satisfiestheraisedcosinespectrum ofEquation (4.60),buttheFouriertransform P(f)itselfdoesnot. ~Thepulsep(t)satisfiestheorthogonality constraint: roop(t)p(t nT)dt=0,n=::!:1,:!:2,... whereTisthesymbolperiod. (alCompute thebaseband waveform ofthebinarydatastream101100forrollofffaetor a=0.3. (b)Compare thewaveform computed inpart(a)withthatobtained usingtheordinary raised-cosine spectrum. SIGNAL-SPACE ANALYSIS Thischapter discusses somebasicissuesthatpertaintothetransmission ofsignalsoveran additivewhiteGaussian noise(AWGN)channel. Specifically, itaddresses thefollowing topics: ~Geometric representation ofsignalswithfiniteenergy,whichprovides amathematically elegantandhighlyinsightful toolforthestudyofdatatransmission. ~Maximum likelihood procedure forthedetectionofasignalinAWGNchannel. ~Derivation ofthecorrelation receiverthatisequivalent to thematched filterreceiver discussed intheprevious chapter. ~Probability ofsymbolerrorandtheunionboundforitsapproximate calculation. Thematerial presented hereinnaturally leadstothestudyofpassband datatransmission coveredinChapter6. I5.1Introduction Consider themostbasicformofadigitalcommunication systemdepicted inFigure5.1. Amessage sourceemitsonesymboleveryTseconds, withthesymbols belonging toan alphabet ofMsymbolsdenotedbym"m2,...,mM'Consider, forexample, theremote connection oftwodigitalcomputers, withonecomputer actingasaninformation source thatcalculates digitaloutputsbasedonobservations andinputsfedintoit.Theresulting computer outputisexpressed asasequence ofOsandIs,whicharetransmitted toasecond computer overacommunication channel. Inthiscase,thealphabet consistssimplyoftwo binarysymbols: 0and1.Asecondexample isthatofaquaternary PCMencoderwithan alphabet consisting offourpossiblesymbols: 00, 01,10,and11.Inanyevent,theapriori probabilities PhP2'...,PMspecifythemessagesourceoutput.Intheabsenceofprior information, itiscustomary toassumethattheMsymbolsofthealphabet areequally likely.Thenwemayexpresstheprobability thatsymbolmiisemittedbythesourceas Pi=P(mi} 1f .MorI=1,2,...,M(5.1) Thetransmitter takesthemessagesourceoutputmiandcodesitintoadistinctsignalSi(t) suitablefortransmission overthechannel. ThesignalSi(t}occupies thefullduration T allottedtosymbolmi'Mostimportant, Si(t)isareal-valued energysignal(i.e.,asignal withfiniteenergy),asshownby T Ei=fasf(t)dt,i=1,2,..., M (5.2) 309 310 CHAPTER 5..SIGNAL-SPACE ANALYSIS m=estimateofmj FIGlJRE 5.1Blockdiagram ofagenericdigitalcommunication system. Thechannelisassumedtohavetwocharacteristics: 1.Thechannelislinear,withabandwidth thatiswideenoughtoaccommodate the transmission ofsignalSilt)withnegligible ornodistortion. 2.Thechannelnoise,w(t),isthesamplefunctionofazero-mean whiteGaussian noise process.Thereasonsforthissecondassumption arethatitmakesreceivercalcula_ tionstractable, anditisareasonable description ofthetypeofnoisepresentinmany practical communication systems. Werefertosuchachannelasanadditive whiteGaussian noise(AWGN) channel. Ac cordingly, wemayexpressthereceived signalx(t)as x(t)=silt)+w(t),{O:S;toS;T i=1,2,..., M(5.3) andthusmodelthechannelasinFigure5.2. Thereceiverhasthetaskofobserving thereceived signalx(t)foraduration ofT secondsandmakingabestestimate ofthetransmitted signalsilt)or,equivalently, the symbolmi'However, owingtothepresence ofchannelnoise,thisdecision-making process isstatistical innature,withtheresultthatthereceiverwillmakeoccasional errors.The requirement istherefore todesignthereceiversoastominimize theaverageprobability ofsymbolerror,definedas M Pe=LPiP(m*m,Imi) i=l(5.4) wherem,isthetransmitted symbol,mistheestimate produced bythereceiver, and P(m*m,1m,)istheconditional errorprobability giventhattheithsymbolwassent.The resulting receiverissaidtobeoptimum intheminimum probability oferrorsense. Thismodelprovides abasisforthedesignoftheoptimum receiver, forwhichwe willusegeometric representation oftheknownsetoftransmitted signals,(silt)).This method, discussed inSection5.2,provides agreatdealofinsight,withconsiderable sim plification ofdetail. Transmitted Received signal signalT WhiteGaussian noise lV(,) FIGlJRE 5.2Additive whiteGaussian noise(AWGN) modelofachannel. 5.2Geometric Representation afSig....ls311 ~Geometric Representation ofSigHals Theessenceofgeometric representation ofsignals 1istorepresent anysetofMenergy signals{Si(t))aslinearcombinations ofNorthonormal basisfunctions, whereN,;;M. Thatistosay,givenasetofreal-valued energysignalss,(t),S2(t),.•.,SM(t),eachof durationTseconds, wewrite N Si(t)=LSi;<Pj(t), ;=1{O';;t,;;T i=1,2,..., M(5.5) (5.6)wherethecoefficients oftheexpansion aredefinedby({i=1,2,...,MSij=J,Si(t)<P;(t) dt, . _o J-1,2,..., N Thereal-valued basisfunctions <p!(t),<P2(t),•••,<PN(t)areorthonormal, bywhichwe mean (5.7) ( {Iiii=jJo<Pi(t)<Pj(t) dt=fJij=°ifi*"j wherefJijistheKronecker delta.Thefirstcondition ofEquation (5.7)statesthateachbasis function isnormalized tohaveunitenergy.Thesecondcondition statesthatthebasis functions <Pl(t),<P2(t),..•,<PN(t)areorthogonal withrespecttoeachotheroverthein terval°,;;t,;;T. Thesetofcoefficients (Sij}j:::!maynaturally beviewedasanN-dimensional vector, denotedbySi'Theimportant pointtonotehereisthatthevector Sibearsaone-to-one relationship withthetransmitted signalSi(t): 110GiventheNelements ofthevectors Si(i.e.,Sibsa,•..,SiN)operating asinput,we mayusetheschemeshowninFigure5.3atogenerate thesignalSi(t),whichfollows Sil ~,(t) silt)Si2 si(t) ~2(t) SiN ~N(t) (b) (0) FIGURE 5.3(a)Synthesizerfor generating thesignals,(t).(b)Analyzer forgenerating thesetof signalvectors{s,}. 312 CHAPTER 5IiSIGNAL-SPACE ANALYSIS directlyfromEquation (5.5).ItconsistsofabankofNmultipliers, witheachmul. tiplierhavingitsownbasisfunction, followed byasummer. Thisschememaybe viewedasasynthesizer. i>Conversely, giventhesignals Si(t),i=1,2,...,M,operating asinput,wemayUse theschemeshowninFigure5.3btocalculate thecoefficients SiloSi2,•••,SiNwhich followsdirectlyfromEquation (5.6).ThissecondschemeconsistsofabankofN product-integrators orcorrelators withacommon input,andwitheachoneofthem supplied withitsownbasisfunction. TheschemeofFigure5.3bmaybeviewedas ananalyzer. Accordingly, wemaystatethateachsignalintheset{SiU))iscompletely determined bythevectorofitscoefficients [SillSi2 Si=:' SiNi=1,2,..., M (5.8) (5.9) i=1,2,..., MThevector Siiscalledasignalvector.Furthermore, ifweconceptually extendourconven· tionalnotionoftwo-andthree-dimensional Euclidean spacestoanN-dimensional Eu clideanspace,wemayvisualize thesetofsignalvectors lSiIi=1,2,...,M}asdefining acorresponding setofMpointsinanN-dimensional Euclidean space,withNmutually perpendicular axeslabeled4>"4>2,..•,4>N'ThisN-dimensional Euclidean spaceiscalled thesignalspace. Theideaofvisualizing asetofenergysignalsgeometrically, asjustdescribed, isof profound importance. Itprovides themathematical basisforthegeometric representation ofenergysignals,therebypavingthewayforthenoiseanalysisofdigitalcommunication systemsinaconceptually satisfying manner. Thisformof representation isillustrated in Figure5.4forthecaseofatwo-dimensional signalspacewiththreesignals,thatis,N=2 andM=3. InanN-dimensional Euclidean space,wemaydefinelengthsofvectorsandangles between vectors.Itiscustomary todenotethelength(alsocalledtheabsolute valueor norm)ofasignalvector SibythesymbolIISiII.Thesquared-length ofanysignalvectors, isdefinedtobetheinnerproductordotproductofSiwithitself,asshownby IISi112=STSi N =Ls;, ;=1 where Sijisthejthelementofs;,andthesuperscript Tdenotesmatrixtransposition. Thereisaninteresting relationship between theenergycontentofasignalandits representation asavector.Bydefinition, theenergyofasignalsilt)ofduration Tseconds is Ei=rsf(t)dt Therefore, substituting Equation (5.5)into(5.10),weget(5.10) 5.2Geometric Representation ofSignals 313 -3 FIGURE5.4Illustrating thegeometric representation ofsignalsforthecasewhen N=2andM=3. . Interchanging theorderofsummation andintegration, andthenrearranging terms,weget (5.11) Butsincethe'hit)formanorthonormal set,inaccordance withthetwoconditions of Equation (5.7),wefindthatEquation (5.11)reducessimplyto N Ei=2:st j=l (5.12) ThusEquations (5.9)and(5.12)showthattheenergyofasignalsilt)isequaltothe squaredlengthofthesignalvectorsilt)representing it. Inthecaseofapairofsignalssilt)andSk(t),represented bythesignalvectors Siand Sbrespectively, wemayalsoshowthat (5.13) Equation (5.13)statesthattheinnerproductofthesignalssilt)andskit)overtheinterval [0,T],usingtheirtime-domain representations, isequaltotheinnerproduct oftheir respective vectorrepresentations SiandSk'Notethattheinnerproductofsilt)andskit)is invariant tothechoiceofbasisfunctions (1>j(t)}~l inthatitonlydepends onthecompo nentsofthesignalsSilt)andSk(t)projected ontoeachofthebasisfunctions. 314 CHAPTER 5I:lSIGNAL-SPACE ANALYSIS (5.14)Yetanother usefulrelation involving thevectorrepresentations ofthesignals5.(t) andsk(t)isdescribed by , N IISi-SkII2=L(Sij-Skj)2 ;=1 =f(Si(t)-sk(t)fdt whereIISi-skIIistheEuclidean distance, dik,between thepointsrepresented bythe signalvectors SiandSk. Tocomplete thegeometric representation ofenergysignals,weneedtohavearep_ resentation fortheangleeiksubtended between twosignalvectors SiandSk'Bydefinition, thecosineoftheangleeikisequaltotheinnerproductofthesetwovectorsdividedbythe productoftheirindividual norms,asshownby coseik=IISiII IISkII(5.15) Thetwovectors SiandSkarethusorthogonal orperpendicular toeachotheriftheirinner product STSkiszero,inwhichcaseeik=90degrees;thiscondition isintuitively satisfying. ~EXAMPLE 5.1Schwan Inequality Consider anypairofenergysignals 5,(t)and52(t).TheSchwarzinequality statesthat (5.16) Theequalityholdsifandonlyif52(t)=cs,(t),wherecisanyconstant. Toprovethisimpor[ant inequality, let5,(t)and52(t)beexpressed intermsofthepal! oforthonormal basisfunctions "'1(t)and"'2(t)asfollows: 5dt)=511"',(t)+512"'2(t) 52(t)=52,,,,,{t)+522"',{t) where",,(t)and"'2(t)satisfy[heorthonormality conditions over[heentiretimeinterval (-00,(0): f~ {1fori=i __"'i(t)"'i{t)dt =(iij=a ~ otherwise Onthisbasis,wemayrepresent thesignals5,{t)and52(t)bythefollowing respective pairof vectors,asillus[rated inFigure5.5: (5.17)cose=IISIII IIS2II roo5,(t)52{t)dts,=[511] 5'2 S2=[::] FromFigure5.5wereadilyseethatangle0sub[endedbecweenthevectors 51and52is STS2 5.2Geometric Representation ofSignals 315 --=-I'''''-----~------'---- <1>1 FIGURE5.5Vectorrepresentations ofsignalsSl(t)andS2(t),providing thebackground picture forprovingtheSchwarzinequality. wherewehavemadeuseofEquations (5.15),(5.13)and(5.9).Recognizing that1cos81:s1, theSchwarzinequality ofEquation (5.16)immediately followsfromEquation (5.17).More over,fromthefirstlineofEquation (5.17)wenotethat 1cos8'=1ifandonlyifS2=cs" thatis,S2(t)=cs,(t),wherecisanarbitrary constant. TheproofoftheSchwarz inequality, aspresented here,appliestoreal-valued signals. Itmaybereadilyextended tocomplex-valued signals,inwhichcaseEquation (5.16)isrefor mulatedas (5.18) wheretheequalityholdsifandonlyifS2(t)=cs,(t),wherecisaconstant; seeProblem 5.9. Itisthecomplex formoftheSchwarz inequality thatwasusedinChapter4toderivethe matched filter. ... l1liGRAM-SCHMIDT ORTHOGONALIZATION PROCEDURE Havingdemonstrated theelegance ofthegeometric representation ofenergysignals,how dowejustifyitinmathematical terms?TheanswerliesintheGram-Schmidt orthogon alization procedure, forwhichweneedacomplete orthonormal setofbasisfunctions. To proceedwiththeformulation ofthisprocedure, supposewehaveasetofMenergysignals denotedbys,(t),S2(t),.••,SM(t).StartingwithSl(t)chosenfromthissetarbitrarily, the firstbasisfunction isdefinedby <P1(t)=5,(t) ~ whereE,istheenergyofthesignal51(t).Then,clearly,wehave 5,(t)=~<P1(t) ==S11<P,(t) wherethecoefficient 511=~and<p,(t)hasunitenergy,asrequired. Next,usingthesignal52(t),wedefinethecoefficient 52'as 521=faT52(t)<P1(t)dt Wemaythusintroduce anewintermediate function(5.19) (5.20) (5.21) (5.22) 316 CHAPTER 5"SIGNAL-SPACE ANALYSIS whichisorthogonal tocP1(t)overtheinterval0:0;t:o;TbyvirtueofEquation (5.21)and thefactthatthebasisfunction cP1(t)hasunitenergy.Now,wearereadytodefineth secondbasisfunction as e (5.23) Substituting Equation (5.22)into(5.23)andsimplifying, wegetthedesiredresult (5.24) whereE2istheenergyofthesignalS2(t).ItisclearfromEquation (5.23)thatrcPi(t)dt=1 andfromEquation (5.24)that Thatistosay,cP1(t)andcP2(t)formanorthonormal pair,asrequired. Continuing inthisfashion,wemayingeneraldefine i-I gilt)=silt)-LSijcPj(t) j=l wherethecoefficients Sijarethemselves definedby(5.25) j=1,2,...,i-1 (5.26) Equation (5.22)isaspecialcaseofEquation (5.25)withi=2.Notealsothatfori=1, thefunction gilt)reducestosilt). Giventhegilt),wemaynowdefinethesetofbasisfunctions cPi(t)=~g~i(=t)=rgf(t)dt'i=1,2,..., N (5.27) whichformanorthonormal set.Thedimension Nislessthanorequaltothenumberof givensignals,M,depending ononeoftwopossibilities: I>-ThesignalsSl(t),S2(t),..•,SM(t)formalinearlyindependent set,inwhichcase N=M. Il<-ThesignalsSI(t),S2(t),...,SM(t)arenotlinearlyindependent, inwhichcaseN<M, andtheintermediate function gilt)iszerofori>N. 5.2Geometric Represenmtion ofSignals 317 TABLE5.1Amplitude Levelsofthe2B1QCode SymbolSignal Amplitude -3 -1 +1 +3Gray code 00 01 11 10 Notethattheconventional Fourierseriesexpansion ofaperiodicsignalisanexample ofaparticular expansion ofthetype described herein.Also,therepresentation ofaband limitedsignalintermsofitssamplestakenattheNyquist ratemaybeviewedasanother sampleofaparticular expansion ofthistype.However, twoimportant distinctions should bemade: 1.Theformofthebasisfunctions <Pl(t),<P2(t),•••,<PN(t)hasnotbeenspecified. That istosay,unliketheFourierseriesexpansion ofaperiodic signalorthesampled representation ofaband-limited signal,wehavenotrestricted theGram-Schmidt orthogonalization procedure tobeintermsofsinusoidal functions orsincfunctions oftime. 2.Theexpansion ofthesignalSi(t)intermsofafinitenumberoftermsisnot'an approximation wherein onlythefirstNtermsaresignificant butratheranexact expression whereNandonlyNtermsaresignificant. ~EXAMPLE 5.22B12Code The2B1Qcodewasdescribed inChapter4astheNorthAmerican linecodefordigital subscriber lines.Itrepresents aquaternary PAMsignalasshownintheGray-encoded alphabet ofTable5.1.Thefourpossiblesignals,S,(t),S2(t), S3(t), andS4(t),areamplitude-scaled versions ofaNyquistpulse.Eachsignalrepresents adibit.Wewishtofindthevectorrepresentation ofthe2B1Qcode. Thisexample issimpleenoughforustosolveitbyinspection. Letq,,(t)denotethe Nyquistpulse,normalized tohaveunitenergy.Theq,,(t)sodefinedistheonlybasisfunction forthevectorrepresentation ofthe2BIQcode.Accordingly, thesignal-space representation ofthiscodeisasshowninFigure5.6.Itconsistsoffoursignalvectorss,'S2, S3,andS4,which arelocatedontheq,raxisinasymmetric mannerabouttheorigin.Inthisexample, wethus haveM=4andN=1. Wemaygeneralize theresultdepictedinFigure5.6forthe2BIQcodeasfollows.The signal-space diagramofanM-arypulse-amplitude modulated signal,ingeneral,isone dimensional withMsignalpointsuniformly positioned ontheonlyaxisofthediagram. <II SI•S2.i.S3 S4 ,}, ,{,•"'12------1 FIGURE5.6Signal-space representation ofthe2BIQcode. ~,/-- 318 CHAPTER 5"SIGNAL-SPACE ANALYSIS 5.3Conversion oftheContinuous AWGNChannel intoaVectorChannel Suppose thattheinputtothebankofNproductintegrators orcorrelators inFigure5.3b isnotthetransmitted signalsilt)butratherthereceivedsignalx{t)definedinaccordance withtheidealized AWGNchannelofFigure5.2.Thatistosay, {o:$t:$T x(t)=silt)+w{t), {5.28}i=1,2,..., M wherew{t)isasamplefunction ofawhiteGaussian noiseprocessW(t)ofzeromeanand powerspectraldensityNo/2.Correspondingly, wefindthattheoutputofcorrelator j,say isthesamplevalueofarandomvariableX;,asshownby , x;=fx(t)tp;(t)dt (5.29)=Si;+wi'j=1,2,..., N Thefirstcomponent, Si;'isadeterministic quantity contributed bythetransmitted signal silt);itisdefinedby (5.30) Thesecondcomponent, wi'isthesamplevalueofarandomvariableW;thatarisesbecause ofthepresence ofthechannelnoisew(t);itisdefinedby w;=fw{t)cP;{t)dt (5.31) Consider nextanewrandomprocessX'{t)whosesamplefunction x'(t)isrelatedto thereceivedsignalx(t)asfollows: N x'(t)=x(t)-2:x;cP;{t) j=1(5.32) Substituting Equations (5.28)and(5.29)into(5.32),andthenusingtheexpansion of Equation (5.5),weget N x'(t)=silt)+w{t)-2:(Si;+w;)cP;{t) N j=1 =w{t)-2:w;cP;(tj (5.33) 1=1=w'(t) Thesamplefunctionx'(t)therefore dependssolelyonthechannelnoisew(t).Onthebasis ofEquations (5.32)and(5.33),wemaythusexpressthereceivedsignalas N x{t)=2:x;cP;(t)+x'{t) ;~1 (5.34) =2:x;cP;{t)+w'{t) j=1 Accordingly, wemayvieww'(t)asasortofremainder termthatmustbeincluded onthe righttopreservetheequalityinEquation (5.34).Itisinformative tocontrasttheexpansIOn 5.3Conversion ofAWGNChannel intoVectorChannel 319 ofthereceived signalx(t)giveninEquation (5.34)withthecorresponding expansion of thetransmitted signalsilt)giveninEqufltion (5.5).Thelatterexpansion isentirelydeter ministic, whereasthatofEquation (5.34)jsrandom(stochastic), whichistobeexpected. Ill!STATISTICAL CHARACTERlZATION OFTHECORREIATOR OUTPUTS Wenowwishtodevelopastatistical characterization ofthesetofNcorrelator outputs. LetX(t)denotetherandom process, asamplefunction ofwhichisrepresented bythe received signalx(t).Correspondingly, letXidenotetherandom variable whosesample valueisrepresented bythecorrelator outputXj,j=1,2,...,N.According totheAWGN modelofFigure5.2,therandomprocessX(t)isaGaussian process. Itfollowstherefore thatXjisaGaussian randomvariable forallj(seeProperty 1ofaGaussian process, Section1.8).Hence,Xjischaracterized completely byitsmeanandvariance, whichare determined next. LetWjdenotetherandomvariablerepresented bythesamplevalueWjproduced by thejthcorrelator inresponse tothewhiteGaussian noisecomponent wIt).Therandom variable Wihaszeromean,becausethenoiseprocessWit)represented bywIt)inthe AWGNmodelofFigure5.2haszeromeanbydefinition. Consequently, themeanofXi depends onlyonSii,asshownby /Lx;=E[Xi] =E[Sij+Wj] =Sii+E[Wj] =Sij(5.35) (5.36)Tofindthevariance ofXi'wenotethat 01;=var[Xi] =E[(Xi-Sij)2] =E[WTJ wherethelastlinefoHowsfromEquation (5.29)withXjandWjreplaced byXiandWj, respectively. According toEquation (5.31),therandomvariable Wiisdefinedby Wj=fW(t)'Mt)dt Wemaytherefore expandEquation (5.36)asfollows: 01;=E[fW(t)<Pi(t)dtfW(U)<Pj(U)dU] =E[I:f<Pi(t)<PAU)W(t)W(U)dtdu] Interchanging theorderofintegration andexpectation: O"~=((<pi(t)<pj(u)E[W(t)W(u)]dtdu 1JoJo =fr<Pj(t)<pi(u)Rw(t, u)dtdu(5.37) (5.38) 320 CHAPTER 5iiiSIGNAL-SPACE ANALYSIS where,Rw(t,u)istheautocorrelation functi~n of~enoiseprocessW(t).Sincethisnoise ISstatlOnary, Rw(t,u)dependsonlyonthetimedIfference t-u.Furthermore, sincet~ noiseWIt)iswhite w~thaconstant powerspectraldensityNo/2,wemayexpressRw(t,u) asfollows[seeEquation (1.95)]: NoRw(t,u)='28(t-u) (5.39) (5.40)Therefore, substituting Equation (5.39)into(5.38),andthenusingthesiftingproperty of thedeltafunction8(t),weget NJTJTu3c=----.Q c/>i(t)c/>i(u) 8(t-u)dtdu, 2 0 0 NJT=----.Q cP2(t)dt20 1 SincethecPi(t)haveunitenergy,bydefinition, wefinallygetthesimpleresult 2Noux;='2 forallj (5.41) Thisimportant resultshowsthatallthecorrelator outputsdenoted byXiwithj=1, 2,...,N,haveavariance equaltothepowerspectraldensityNo/2ofthenoiseprocess W(t). Moreover, sincethecPi(t)formanorthogonal set,wefindthattheXiaremutually uncorrelated, asshownby cov[XiX,j =E[(Xi-/Lx)(X k-/Lx,)] =E[(Xi-s'i)(Xk-Sik)] =E[WiW k] =E[rW(t)cPi(t)dtrW(u)cPk(u)du] =rrcP;(t)cPk(u)Rw(t, u)dtdu (5.42) NfTfT =Too cPi(t)cPk(U) 8(t-u)dtdu NoJT ='20cPi(t)cPk(t)dt =0,j*-k SincetheXiareGaussian randomvariables, Equation (5.42)impliesthattheyarealso statistically independent (seeProperty 4ofaGaussian Process,Section1.8). DefinethevectorofNrandomvariables (5.43) whoseelements areindependent Gaussian randomvariableswithmeanvaluesequaltoSq andvariances equaltoNo/2.Sincetheelements ofthevectorXarestatistically indepen' 5.3Conversion ofAWGNChannel intoVectorChannel 321 (5.44) i=1,2,..., Mdent,wemayexpresstheconditional probability densityfunction ofthevectorX,given thatthesignalSilt)orcorrespondingly thesymbolmiwastransmitted, astheproductof theconditional probability densityfunctions ofitsindividual elements asshownby N fx(xlmi)=ITfX(xilmi),j=l 1 wherethevectorxandscalarXjaresamplevaluesoftherandomvectorXandrandom variableXj'respectively. Thevectorxiscalledtheobservation vector;correspondingly, Xiiscalledanobservable element.AnychannelthatsatisfiesEquation (5.44)iscalleda memorylesschannel. SinceeachXiisaGaussian randomvariablewithmeanSijandvariance No/2,we have i=1,2,,N i=1,2,,M(5.45) Therefore, substituting Equation (5.45)into(5.44)yields fx(xlm j)=(7TNo)-NI2eXP[-N1f,(Xi-SiifJ, i=1,2,...,M(5.46) 01=1 Itisnowclearthattheelements oftherandomvectorXcompletely characterizethe summation term2.iXlPj(t), whosesamplevalueisrepresented bythefirstterminEquation (5.34).However, thereremainsthenoisetermw'(t)inthisequation, whichdependsonly onthechannelnoisew(t).SincethenoiseprocessW(t)represented byw(t)isGaussian withzeromean,itfollowsthatthenoiseprocessW'(t)represented bythesamplefunction w'(t)isalsoazero-mean Gaussian process. Finally,wenotethatanyrandomvariable W'(tk),say,derivedfromthenoiseprocessW'(t)bysampling itattimetbisinfact statistically independent ofthesetofrandomvariables {Xi};thatistosay(seeProblem 5.10), (5.47){i=1,2,..., N o:5tk:5T Sinceanyrandomvariablebasedontheremainder noiseprocessW'(t)isindependent of thesetofrandomvariables {Xi}aswellasthesetoftransmitted signals(silt)},Equation (5.47)statesthattherandomvariable W'(tk)isirrelevant tothedecision astowhich particular signalwasactuallytransmitted. Inotherwords,thecorrelator outputsdeter minedbythereceivedsignalx(t)aretheonlydatathatareusefulforthedecision-making processand,hence,represent sufficient statistics fortheproblem athand.Bydefinition, sufficient statistics summarize thewhole of therelevant information supplied byanob servation vector. Wemaynowsummarize theresultspresented inthissectionbyformulating the theoremofirrelevance: Insofarassignaldetection inadditivewhiteGaussian noiseisconcerned, onlythe projections ofthenoiseontothebasisfunctions ofthesignalset{Si(t)}i'!,affects thesufficient statistics ofthedetection problem; theremainder ofthenoiseis irrelevant. Asacorollary tothistheorem, wemaystatethattheAWGNchannelofFigure5.2is equivalent toanN-dimensional vectorchanneldescribed bytheobservation vector x=Si+W,i=1,2,..., M (5.48) 322 CHAPTER 5IiSIGNAL-SPACE ANALYSIS wherethedimension Nisthenumberofbasisfunctions involved informulating thesignal vector Si'Theindividual components ofthesignalvector Siandnoisevector Waledefined byEquations (5.6)and(5.31),respectively. Thetheorem ofirrelevance anditscorollary areindeedbasictotheunderstanding ofthesignaldetection problem asdescribed next. I5.4LikeliJwod Functions Theconditional probability densityfunctions fx{xImil,i=1,2,...,M,arethevery characterization ofanAWGNchannel.Theirderivation leadstoafunctional dependence ontheobservation vectorx,giventhetransmitted messagesymbolmi'However, atthe receiverwehavetheexactopposite situation: Wearegiventheobservation vectorxand therequirement istoestimatethemessagesymbolmithatisresponsible forgenerating x. Toemphasize thislatterviewpoint, weintroduce theideaofalikelihood function, denoted byL{mi)anddefinedby L(mi)=fx(xlmi)' i=1,2,..., M (5.49) Itisimportant however torecognize thatalthough theL(mi)andfx{xImilhaveexactly thesamemathematical form,theirindividual meanings aredifferent. Inpractice, wefinditmoreconvenient toworkwiththelog-likelihood function, denotedbyl(mi)anddefinedby l(mi)=logL{mi), i=1,2,..., M (5.50) Thelog-likelihood function bearsaone-to-one relationship tothelikelihood function for tworeasons: 1.Bydefinition, aprobability densityfunction isalwaysnonnegative. ItfollowstheIe forethatthelikelihood function islikewiseanonnegative quantity. 2.Thelogarithmic function isamonotonically increasing function ofitsargument. TheuseofEquation (S.46)in(S.SO)yieldsthelog-likelihood functions foranAWGN channelas 1N l{mi)= -Noi~(Xi-5ii)2,i=1,2,...,M (5.51) wherewehaveignoredtheconstant term-(N/2)10g(7TNo)asitbearsnorelationwhat soevertothemessagesymbolmi'Notethatthe5ii'j=1,2,...,N,aletheelements of thesignalvector Sirepresenting themessage symbolmi.With Equation (S.Sl)atour disposal, wealenowreadytoaddIessthebasicreceiverdesignproblem. 5.5Coherent Detection ofSignalsinNoise: Maximum Likelihood Decoding Supposethatineachtimeslotofduration Tseconds, oneoftheMpossiblesignalsS1(t), 52(t),...,SM(t)istransmitred withequalprobability, 11M.Forgeometric signalrepresen tation,thesignal5i(t),i=1,2,...,M,isappliedtoabankofcorrelators, withacommon inputandsuppliedwithanappropriate setofNorthonormal basisfunctions. Theresulting correlator outputsdefinethe5ignalvector Si'Sinceknowledge ofthesignalvector Siisas goodasknowing thetransmitred signal5i(t)itself,andviceversa,wemayrepresent St(t) byapointinaEuclidean spaceofdimension N,;;M.Werefertothispointasthetrans- 5.5Maxi........Likelihood Decodi..g323 mittedsignalpointormessagepoint.Thesetofmessage pointscorresponding totheset oftransmitted signals {Si(t)}~l iscalledasignalconstellation. However, therepresentation ofthereceived signalx(t)iscomplicated bythepresence ofadditive noisew(t).Wenotethatwhenthereceived signalx(t)isappliedtothebank ofNcorrelators, thecorrelator outputsdefinetheobservation vectorx.FromEquation (5.48),thevector Xdiffersfromthesignalvectors;bythenoisevectorw whoseorientation iscompletely random. Thenoisevectorwiscompletely characterized bythenoisew(t); theconverse ofthisstatement, however, isnottrue.Thenoisevectorwrepresents that portionofthenoisew(t)thatwillinterfere withthedetection process; theremaining portionofthisnoise,denoted byw'(t),istunedoutbythebankofcorrelators. Now,basedontheobservation vectorx,wemayrepresent thereceived signalx(t) byapointinthesameEuclidean spaceusedtorepresent thetransmitted signal.Werefer tothissecondpointasthereceivedsignalpoint.Thereceived signalpointwanders about themessage pointinacompletely randomfashion,inthesensethatitmaylieanywhere insideaGaussian-distributed "cloud" centered onthemessage point.Thisisillustrated in Figure5.7aforthecaseofathree-dimensional signalspace.Foraparticular realization ofthenoisevectorw(i.e.,aparticular pointinsidetherandomcloudofFigure5.7a),the relationship between theobservation vectorxandthesignalvectors;isasillustrated in Figure5.7b. Wearenowreadytostatethesignaldetection problem: Giventheobservation vectorx,performamappingfrom xtoanestimatelizofthe transmitted symbol,m"inawaythatwouldminimize theprobability oferrorin thedecision-making process. (5.52)Suppose that,giventheobservation vectorX,wemakethedecisionliz=mi'The probability oferrorinthisdecision, whichwedenotebyPe(m;1x),issimply PAmilx)=P(m;notsentIx) =1 -P(misentix) (5.53)forallk*iThedecision-making criterion istominimize theprobability oferrorinmapping each givenobservation vectorxintoadecision. OnthebasisofEquation (5.52),wemaythere forestatetheoptimum decisionrule: Setliz=miif P(m;sentIx)~P(mksent'x) Noise Received vector signalpoint w ~==------------<Pl (a) (b) FIGURE5.7Illustrating theeffectofnoiseperturbation, depicted in(a),onthelocation ofthe received signalpoint,depicted in(b). (5.54) (5.55) (5.56)324 CHAPTER 5 "SIGNAL-SPACE ANALYSIS wherek=1,2,...,M.Thisdecision ruleisreferredtoasthemaximum aposterio. probability (MAP)rule. 1) Thecondition ofEquation (5.53)maybeexpressed moreexplicitly intermsofthe aprioriprobabilities ofthetransmitted signalsandintermsofthelikelihood functions UsingBayes'ruleinEquation (5.53),andforthemoment ignoring possible tiesinth~ decision-making process,wemayrestatetheMAPruleasfollows: Setm=miif PJX(xlmk).. . [(ISmaXImum fork=1xx) wherePkistheaprioriprobability oftransmitting symbolmb[xixImk)istheconditional probability densityfunction oftherandomobservation vectorXgiventhetransmission ofsymbolmband[x(x)istheunconditional probability densityfunction ofX.InEquation (5.54)wemaynotethefollowing: ~Thedenominator term[xix)isindependent ofthetransmitted symbol. '"Theaprioriprobability Pk=PiwhenallthesOUlcesymbols aretransmitted with equalprobability. ~Theconditional probability densityfunction[x(xImk)bearsaone-to-one relatIOn shiptothelog-likelihood function I(mk)' Accordingly, wemayrestate.thedecisionruleofEquation (5.54)intermsofI(mk)simply asfollows: Setm=miif l(mk)ismaximum fork=i Thisdecision ruleisreferredtoasthemaximum likelihood rule,andthedeviceforits implementation iscorrespondingly referredtoasthemaximum likelihood decoder. Ac cordingtoEquation (5.55),amaximum likelihood decodercomputes thelog-likelihood functions asmetricsforalltheMpossible message symbols, compares them,andthen decidesinfavorofthemaximum. Thusthemaximum likelihood decoderdiffersfromthe maximum aposteriori decoderinthatitassumesequallylikelymessage symbols. Itisusefultohaveagraphical interpretation ofthemaximum likelihood decision rule.Let2denotetheN-dimensional spaceofallpossibleobservation vectorsx.Werefer tothisspaceastheobservation space.Becausewehaveassumed thatthedecision rule mustsaym=mi,wherei=1,2,...,M,thetotal0bservation space2iscorrespondingly partitioned intoM-decision regions,denoted by2"22,•••,2M,Accordingly, wemay restatethedecisionruleofEquation (5.55)asfollows: Observation vectorxliesinregion2,if l(mk)ismaximum fork=i Asidefromtheboundaries between thedecisionregions2"22,•••,2,\1:>itisclearthat thissetofregionscoverstheentirespaceofpossibleobservation vectorsx.Weadoptthe convention thatalltiesareresolvedatrandom; thatis,thereceiversimplymakesaguess. Specifically, iftheobservation vectorxfallsontheboundary between anytwodeciSIOn regions,2,and2bsay,thechoicebetweenthetwopossibledecisionsmmiandfll=In. isresolved aprioribytheflipofafaircoin.Clearly,theoutcome ofsuchaneventdoes notaffe~theultimate valueoftheprobability oferrorsince,onthisboundary, thecon ditionofEquation (5.53)issatisfiedwiththeequalitysign. 5.5Maxi.....mLikelilwodDecoding 325 Themaximum likelihood decisionruleofEquation (5.55)oritsgeometric counter partdescribed inEquation (5.56)isofagenerickind,withthechannelnoisew(t}being additiveastheonlyrestriction imposedonit.Wenextspecialize thisruleforthecasewhen w(t)isbothwhiteandGaussian. Fromthelog-likelihood function definedinEquation (5.51)foranAWGNchannel wenotethatl(mk}attainsitsmaximum valuewhenthesummation term N L(x;-Ski}' ;=1 isminimized bythechoicek=i.Accordingly, wemayformulate themaximum likelihood decisionruleforanAWGNchannelas Observation vectorxliesinregionZiif NL(x;-sk;fisminimum fork=i ;=1(5.57) (5.58) (5.59) (5.60)Next,wenotefromourearlierdiscussion that(seeEquation (5.14)forcomparison) NL(x;-Sk;)2=IIx -SkII2 ;-1 whereIIx -SkIIistheEuclidean distancebetweenthereceivedsignalpointandmessage point,represented bythevectorsxandSk>respectively. Accordingly, wemayrestatethe decisionruleofEquation (5.57)asfollows: Observation vectorxliesinregionZiif theEuclidean distance IIx -SkIIisminimum fork=i Equation (5.59)statesthatthemaximum likelihood decisionruleissimplytochoosethe messagepointclosesttothereceivedsignalpoint,whichisintuitively satisfying. Inpractice, theneedforsquarers inthedecisionruleofEquation (5.59)isavoided byrecognizing that N N N N L(Xj-Ski=Lxf-2LX;Skj+LS~j ;=1 i=1 ;=1 1'=1 Thefirstsummation termofthisexpansion isindependent oftheindexkandmaytherefore beignored. Thesecondsummation termistheinnerproductoftheobservation vectorx andsignalvector Sk.Thethirdsummation termistheenergyofthetransmitted signal Sk(t).Accordingly, wemayformulate adecisionruleequivalent tothatofEquation (5.59) asfollows: Observation vectorxliesinregionZiif N 1 ~XjSk;-2Ekismaximum fork=i whereEkistheenergyofthetransmitted signal Sk(t): N Ek=LsZ; ;=1(5.61) (5.62) FromEquation (5.61)wededucethat,foranAWGN channel, thedecision regions ateregionsoftheN-dimensional observation spaceZ,bounded bylinear[(N-1) dimensional hyperplane] boundaries. Figure5.8showstheexampleofdecisionregionsfor 326 CHAPTER 5l!lSIGNAL-SPACE ANALYSIS Decision boundary Region ZI ---------<0--- "'1 Decision boundary FIGURE 5.8Illustrating thepartitioning oftheobservation spaceintodecision regionsforthe casewhenN=2andM=4;itisassumed thattheMtransmitted symbols areequallylikely. M=4signalsandN=2dimensions, assuming thatthesignalsaretransmitted withequal energy,E,andequalprobabiliry. I5.6Correlation Recei17er Fromthematerialpresented intheprevious sections, wefindthatforanAWGNchannel andforthecasewhenthetransmitted signalsS,(t),S2,•••,SM(t)areequallylikely,the optimum receiverconsistsoftwosubsystems, whicharedetailedinFigure5.9andde scribedhere: 1.ThedetectorpartofthereceiverisshowninFigure5.9a.ItconsistsofabankofM product-integrators orcorrelators, supplied withacorresponding setofcoherent reference signalsororthonormal basisfunctions cPl(t),cP2(t),•••,cPN(t)tharare generated locally.Thisbankofcorrelators operates onthereceived signalx(t), os;ts;T,toproducetheobservation vectorx. 2.Thesecondpartofthereceiver, namely,thesignaltransmission decoderisshownin Figure5.9b.Itisimplemented intheformofamaximum-likelihood decoderthat operates ontheobservation vectorxtoproduceanestimate,m,ofthetransmitted symbolm"i=1,2,...,M,inawaythatwouldminimize theaverageprobability ofsymbolerror.Inaccordance withEquation (5.61),theNelements oftheobser· vationvectorxarefirstmultiplied bythecorresponding Nelements ofeachofthe Msignalvectorss"S2,•••,SM,andtheresulting products aresuccessively sUIJl!Ded inaccumulators toformthecorresponding setofinnerproducts {XTSkIk=1,2,.." M}.Next,theinnerproducts arecorrected forthefactthatthetransmitted signal energiesmayb~unequal. Finally,thelargestintheresulting setofnumbers isselected; andanappropriate decisiononthetransmitted messageismade. Theoptimum receiverofFigure5.9iscommonly referred toasacorrelation receiver. 5.6Correlation Receiver 327 x, <I>,(t) xlI)x,Observation vector• <1>,(1) xN <l>N(t) (a) Select larges!Estimate in (b) FIGURE5.9(u)Detector ordemodulator. (b)Signaltransmission decoder. (5.63)l!llEQUIVALENCE OFCORRElATION ANDMATCHED FILTER RECEIVERS ThedetectorshowninFigure5.9ainvolvesasetofcorreiators.Alternatively, wemayuse acorresponding setofmatched filterstobuildthedetector; thematched filterandits properties wereconsidered inSection4.2.Todemonstrate theequivalence ofacorrelator andamatched filter,consider alineartime-invariant filterwithimpulseresponse hilt}. Withthereceived signalx(t)usedasthefilterinput,theresulting filteroutput,Yj(t),is definedbytheconvolution integral: Yj(t)=r~x(T)hi(t-T)dT 328 CHAPTER 5"SIGNAL-SPACE ANALYSIS Received signal x(t) Matched filtersSample att=TObservation vector x FIGURE5.10Detector partofmatchedfilterreceiver;thesignaltransnUssion decoderisas showninFig.5.9b. Fromthedefinition ofamatched filterpresented inSect!()n4.2,werecallthattheimpulse response hj(t)ofalineartime-invariant filtermatched toaninputsignal<Pj(t)isatime reversedanddelayedversionoftheinput<Pi(t).Supposethatweset hj(t)=<pj(T-t) (5.64) Thentheresulting filteroutputis Yj(t)=r~x(7')<pj(T -t+7')d7' Sampling thisoutputattimet=T,weget Yi(T)=r~x(7')<pj(7')d7'(5.65) Since,bydefinition, <Pj(t)iszerooutsidetheinterval0oS:toS:T,wefindthatYi(T)isin actualfactthejthcorrelator output Xjproduced bythereceivedsignalx(t)inFigure5.9a, asshownby (5.66) Accordingly, thedetectorpartoftheoptimum receivermayalsobeimplemented usinga bankofmatched filters,asshowninFigure5.10.Itisimportant tonote,however, that theoutputofeachcorreIatorinFigure5.9aisequivalent totheoutputofacorresponding matched filterinFigure5.10onlywhenthatoutputissampledattimet=T. I5.7Probability ofError Tocomplete thestatistical characterization ofthecorrelation receiverdepicted inFig~e 5.9,weneedtoevaluate itsnoiseperformance. Todoso,supposethattheobservatIon spaceZispartitioned, inaccordance withthemaximum likelihood decisionrule,intO,a setifMregions[Z;}fi,.Suppose alsothatsymbolm,(or,equivalently, signalvector5,)IS (5.67) (5.68)5.7Probability ofError 329 transmitted, andanobservation vectorxisreceived. Thenanerroroccurswhenever the received signalpointrepresented byxdoesnotfallinsideregionZ,associated withthe message pointrepresented bys,.Averaging overallpossible transmitted symbols, we readilyseethattheaverageprobability ofsymbolerror,Peis M Pe=2:PiP(xdoesnotlieinZiIm,sent) i=l 1M - =M~PixdoesnotlieinZ,Imisent) M =1~~PixliesinZiImisent) wherewehaveusedstandard notation todenotetheprobability ofaneventandthe conditional probability ofanevent.SincexisthesamplevalueofrandomvectorX,we mayrewriteEquation (5.67)intermsofthelikelihood function (whenmiissent)as follows: 1'MfPe=1 -M~z,fx(xlmi) dx ForanN-dimensional observation vector,theintegralinEquation (5.68)islikewise N-dimensional. IIIINVARIANCE OFTHEPROBABILITY OFERROR TO ROTATION ANDTRANSLATION Thewayinwhichtheobservation spaceZispartitioned intothesetofregionsZl,Z2,..., ZM,inthemaximum likelihood detection ofasignalinadditivewhiteGaussian noise,is uniquely definedbythesignalconstellation understudy.Accordingly, changesintheori entation ofthesignalconstellation withrespecttoboththecoordinate axesandoriginof thesignalspacedonotaffecttheprobability ofsymbolerrorPedefinedinEquation (5.68). Thisresultisaconsequence oftwofacts: 1.Inmaximum likelihood detection, theprobability ofsymbolerrorPedependssolely ontherelativeEuclidean distances betweenthemessagepointsintheconstellation. 2.TheadditivewhiteGaussian noiseisspherically symmetric inalldirections inthe signalspace. Consider firsttheinvariance ofPewithrespl;cttorotation. Theeffectofarotation appliedtoallthemessage pointsinaconstellation isequivalent tomultiplying the N-dimensional signalvectOr SibyanN-by-Northonormal matrixdenotedbyQforalli. ThematrixQsatisfiesthecondition (5.69) whereIistheidentitymatrixwhosediagonal elements areallunityanditsoff-diagonal elements areallzero.Notethataccording toEquation (5.69),theinverseofareal-valued orthonormal matrixisequaltoitstransposed form.Thusthesignalvector Siisreplaced byitsrotatedversion Si,rotate=QShi=1,2,..., M (5.70) Correspondingly, theN-by-1noisevectorwisreplaced byitsrotatedversion Wrotate=Qw (5.71) 330 CHAPTER 5I1lSIGNAL-SPACE ANALYSIS However, thestatistical characteristics ofthenoisevectorareunaffected bythisrotar forthefollowing reasons: IOn ""FromChapter 1werecallthatalinearcombination ofGaussian randomvariabl isalsoGaussian. SincethenoisevectorwisGaussian, byassumption, itfollowsthes therotatednoisevectorWrotateisalsoGaussian. at ~Sincethenoisevectorwhaszeromean,therotated~oise vectorWrotatealsohaszero mean,asshownby E[wrota,,]=E[Qw] =QE[w] =0(5.72) (5.73)""Thecovariance matrixofthenoisevectorwisequalto(No/2)1,whereNollisthe powerspectraldensityoftheAWGNw(t);thatis, E[ww'l)=NoI 2 Hence,thecovariance matrixoftherotatednoisevectorwweateis E[wrotateW;;'tate] =E[QW(QW)T] =E[QwwTQ""] =QE[WW7)QT =NoQQT 2 =NoI 2(5.74) whereinthelasttwolineswehavemadeuseofEquations (5.73)and(5.69). Inlightoftheseobservations, wemayexpresstheobservation vectorfortherotated signalconstellation as Xrotate=QSi+W,i=1,2,..., M (5.75) FromEquation (5.59)weknowthatthedecision ruleformaximum likelihood detection isbasedontheEuclidean distancefromtheobservation vector Xwtatetotherotatedsignal vector Si.wta"=QSi'Comparing Equation (5.75)toEquation (5.48),wereadilyseethat IIx"ot.te-Si,n'''''''II=IIx -SiIIforalli (5.76) Wemaytherefore formally statetheprincipleofrotational invariance asfollows: Ifasignalconstellation isrotatedbyanorthonormal transformation, thatis, i=1,2,..., M whereQisanorthonormal matrix,thentheprobability ofsymbolerrorPeincurred inmaximum likelihood signaldetection overanAWGNchanneliscompletely unchanged. Weillustrate thisprinciple withanexample. Thesignalconstellation showninFigur: 5.11bisthesameasthatofFigure5.1la,exceptthatithasbeenrotatedthrough4) degrees. Although thesetwoconstellations doindeedlookdifferent, theprinciple ofroo tationalinvariance tellsusimmediately thatthePeisthesameforbothofthem. 5.7PTobabilityofETTOT 331 T--_a---, I I I I -----=a;}:-----=-o+---:[-;ao-- 4>, I I 0------.-a4>, ..fi" / / / / / / -..fi""-,,-0 "-"-"-"- -..fi""-"-"-"- ""- /..fi"4>, / / / / / (a] (b) FIGURE 5.11Apairofsignalconstellations forillustrating theprinciple ofrotational invariance. Consider nexttheissueofinvariance totranslation. Suppose allthemessagepoints inasignalconstellation aretranslated byaconstant vectoramounta,asshownby 5i.translate =5i-a,i=1,2,...,M (5.77) Theobservation vectoriscorrespondingly translated bythesamevectoramount, asshown by Xnanslate=X -a (5.78) FromEquations (5.77)and(5.78)weseethatthetranslate aiscommon toboththe translated signalvector Siandtranslated observation vectorx.Wetherefore immediately deducethat IIX"on<la,e -Si,"amla" II=IIx -SiIIforalli (5.79) andthusformulate theprincipleoftranslational invariance asfollows: Ifasignalconstellation istranslated biaconstantvectoramount,thentheprob abilityofsymbolerrorPeincurredinmaximum likelihood signaldetection overan AWGNchanneliscompletely unchanged. Asanexample, consider thetwosignalconstellations showninFigure5.12,which pertaintoapairofdifferent 4-levelPAMsignals.Theconstellation ofFigure5.12bisthe sameasthatofFigure5.12a,exceptforatranslation of3a12totherightalongthe <PI-axis.Theprinciple oftranslational invariance saysthatthePeisthesameforbothof theseconstellations. IiMINIMUM ENERGY SIGNALS Ausefulapplication oftheprinciple oftranslational invariance isinthetranslation ofa givensignalconstellation insuchawaythattheaverageenergyisminimized. Toexplore 4>,oj• •4>, -30/2 -,,120,,12 3,,12 2" 3" (a) (b) FIGURE 5.12Apairofsignalconstellations forillustrating theprinciple oftranslational invariance. 332 CHAPTER 5'"SIGNAL-SPACE ANALYSIS thisissue,consider asetofsymbolsm"m2,...,mMrepresented bythesignalvectorss S2,•••,SM,respectively. Theaverageenergyofthissignalconstellation translated b),h. avectoramountaIS M 't:tran,late =LIISi-aII2Pi i=1(5.80) wherePiistheprobability thatsymbolmiisemittedbythesourceofinformation. The squaredEuclidean distance between Siandaisexpanded as IISf-aII2==IISfII2 -2aTSi+IIaII2 Wemaytherefore rewriteEquation (5.80)intheexpanded form M M M 't:tr""lme =LIISiII2Pi-2LaTsiPi+IIa112LPi i=l ;=1 i=1 ='t:-2aTE[s]+IIaII2 where 't:istheaverageenergyoftheoriginalsignalconstellation, and M E[s]=LSiPi i=1(5.81) (5.82) Differentiating Equation (5.81)withrespecttothevectoraandthensettingtheresult equaltozero,wereadilyfindthattheminimizing translate is amin=E[s] Theminimum averageenergyofthesignalconstellation translated inthiswayis(5.83) ~translate.min =~IIamin112(5.84) Wemaynowstatetheprocedure forfindingtheminimum energytranslate: Givenasignalconstellation {Si}J';;"thecorresponding signalconstellation withmin imumaverageenergyisobtained bysubtracting fromeachsignalvector Siinthe givenconstellation anamountequaltotheconstant vectorE[s],whereE[s]isde finedbyEquation (5.82). Recalling thattheenergy(orpower)neededforsignaltransmission isaprimaryresource, theminimum energytranslate provides aprincipled methodfortranslating asignalcon stellation ofinterestsoastominimize theenergyrequirement. Forexample, theconstel lationofFigure5.12ahasminimum averageenergy,whereas thatofFigure5.12bdoes not. i'SUNION BOUND ONTHEPROBABILITY OFERROR2 ForAWGNchannels, theformulation oftheaverageprobability ofsymbolerror,P"is conceptually straightforward. WesimplywritePeinintegralformbysubstituting Equation (5.46)intoEquation (5.68).Unfortunately, however, numerical computation oftheinte gralisimpractical, exceptinafewsimple(butimportant) cases.Toovercome thiscom putational difficulty, wemayresorttotheuseofbounds, whichareusuallyadequate to predictthesignal-to-noise ratio(withinadecibelorso)required tomaintain aprescribed errorrate.Theapproximation totheintegraldefiningPeismadebysimplifying theintegral orsimplifying theregionofintegration. Inthesequel,weusethelatterprocedure tode velopasimpleyetusefulupperboundcalledtheunionboundasanapproximation tothe 5.7P..obabiHtyofE ....o..333 (5.85) i=1,2,...,Maverageprobability ofsymbolerrorforasetofMequallylikelysignals(symbols) inan AWGNchannel. LetAibwith(i,k)=1,2,...,M,denote the eventthattheobservation vectorxis closertothesignalvectorSk<thantoSi'whenthesymbolmi(vectorSi)issent.Thecon ditionalprobability ofsymbol.errorwhensymbolmiissent,PAmi),isequaltotheprob abilityoftheunionofevents,Ail,Ail,•••,Ai,i-bAi,i+b...,Ai,M'Fromprobability theoryweknowthattheprobability ofafiniteunionofeventsisoverbounded bythesum oftheprobabilities oftheconstituent events.Wemaytherefore write M Pe(mi) "s;2:P(Aik), k-1 k# Thisrelationship isillustrated inFigure5.13forthecaseofM=4.InFigure5.13a,we showthefourmessagepointsandassociated decisionregions,withthepoint S1assumed torepresent thetransmitted symbol.InFigure5.13b,weshowthethreeconstituent signal spacedescriptions where,ineachcase,thetransmitted message points,andoneother messagepointareretained. According toFigure5.13atheconditional probability ofsym bolerror,Pe(mi),isequaltotheprobability thattheobservation vectorxliesintheshaded regionofthetwo-dimensional signal-space diagram. Clearly,thisprobability islessthan thesumoftheprobabilities ofthethreeindividual eventsthatxliesintheshadedregions ofthethreeconstituent signalspacesdepicted inFigure5.13b. Itisimportant tonotethat,ingeneral,theprobability P(Aik)isdifferent fromthe probability P(m=mk'mi)'Thelatteristheprobability thattheobservation vectorxis (a) (b) FIGURE5.13Illustrating theunionbound.(a)Constellation offourmessage points.(b)Three constellations withacommon message pointandoneothermessage pointretained fromtheorigi nalconstellation. 334 CHAPTER 5..SIGNAL-SPACE ANALYSIS closertothesignalvectorSkthaneveryother,whenSi(ormilissent.Ontheotherhand theprobability P(Aik)depends ononlytwosignalvectors, SiandSk·Toemphasize tho' difference, werewriteEquation (5.85)byadopting P2(s;,Sk)inplaceofP(Aik).WethIs. ~wnte M Pe(mi):;;2:P2(Si,Sk), k=l k*ii=1,2,..., M(5.86) (5.87)Theprobability P2(Si,Sk)iscalledthepairwiseerrorprobability inthatifadatatransmis_ sionsystemusesonlyapairofsignals, SiandSk>thenP2(Si,Sk)istheprobability ofthe receivermistaking SkforSi' Consider thenasimplified digitalcommunication systemthatinvolvestheuseoftwo equallylikelymessages represented bythevectors SiandSk'SincewhiteGaussian noiseis identically distributed alonganysetoforthogonal axes,wemaytemporarily choosethe firstaxisinsuchasetasonethatpassesthroughthepoints SiandSk;forthreeexamples seeFigure5.13b.Thecorresponding decision boundary isrepresented bythebiseetortha; isperpendicular tothelinejoiningthepoints SiandSk'Accordingly, whenthesymbolm (vector Si)issent,andiftheobservation vectorxliesonthesideofthebisectorwheres; lies,anerrorismade.Theprobability ofthiseventisgivenby P2(Si,Sk)=P(xisclosertoSkthanSi'when Siissent) =f~_1exp(-z?)dv d~/2\1~No No wheredikistheEuclidean distance between SiandSk;thatis, dik=IISi-SkII Fromthedefinition ofthecomplementary errorfunction, wehave(5.88) 2I.- erfc(u)=Vir uexp(-~)dz Thus,intermsofthisfunction, withzsetequaltovlYFJ:."wefindthatEquation (5.87) takesonthecompact form i=1,2,..., M1 (dk) P2(s;,Sk)=:2erfc2~ Substituting Equation (5.89)intoEquation (5.86),weget 1~(dik)Pe(mi):;;-2L.erfc.;':" k~l2vNo k#(5.89) (5.90) Theprobability ofsymbolerror,averaged overalltheMsymbols, istherefore overbounded asfollows: M Pe=2:PiPe(mi) i=l wherePiistheprobability oftransmitting symbolmi'(5.91) 5.7ProbabilityofErmr 335 Therearetwospecialformsof'Equation (5.91)thatwewilJfindusefulinChapter6 onpassband datatransmission: 1.Supposethatthesignalconstellation iscircularly symmetric abouttheorigin.Then theconditional probability oferrorPe(mi)isthesameforalli,inwhichcaseEquation (5.91)reducesto (5.92) 2.Definetheminimum distanceofasignalconstellation, dm;n,asthesmallestEuclidean distancebetweenanytwotransmitted signalpointsintheconstellation, asshownby dmin=mindikk#foralliandk (5.93) Then,recognizing thatthecomplementary errorfunction erfc(u)isamonotonically de creasingfunction ofitsargument u,wemaywrite erfc(2~) ~erfcC~) foralliandk (5.94) Wemaytherefore, ingeneral,simplifytheboundontheaverageprobability ofsymbol errorinEquation (5.91)as P(M-1)f (dmin)~---erc-- e2 2~ Thecomplementary errorfunction isitselfbounded as3 £(dmin)1(d;"in)eriC2~ ~v'1Texp-4No(5.95) (5.96) Accordingly, wemayfurthersimplifytheunionboundonPegiveninEquation (5.95)as P(M-1)(d;;'in)e~---- exp--- 2v'1T 4No(5.97) Equation (5.97)showsthatforaprescribed AWGNchannel, theaverageprobability of symbolerrorPedecreases exponentially asthesquaredminimum distance,d;;'in' IIIBITVERSUS SYMBOL ERROR PROBABILITIES Thusfar,theonlyfigureofmeritwehaveusedtoassessthenoiseperformance ofadigital passband transmission systemhasbeentheaverageprobability ofsymbolerror.Thisfigure ofmeritisthenaturalchoicewhenmessages oflengthm=log2Maretransmitted, such asalphanumeric symbols. However, whentherequirement istotransmit binarydatasuch asdigitalcomputer data,itisoftenmoremeaningful touseanotherfigureofmeritcalled thebiterrorrate(BER).Although, ingeneral,therearenouniquerelationships between thesetwofiguresofmerit,itisfortunate thatsuchrelationships canbederivedfortwo casesofpractical interest,asdiscussed next. 336 CHAPIER 5IIISIGNAL-SPACE ANALYSIS Case1 Inthefirstcase,weassumethatitispossibletoperformthemapping frombinary toM-arysymbolsinsuchawaythatthetwobinaryM-tuples corresponding toanypair ofadjacent symbolsintheM-arymodulation schemedifferinonlyonebitposition. This mapping constraint issatisfiedbyusingaGraycode.Whentheprobability ofsymbol errorPeisacceptably small,wefindthattheprobability ofmistaking onesymbolfor eitheroneofthetwo"nearest" symbols ismuchgreaterthananyotherkindofsymbol error.Moreover, givenasymbolerror,themostprobable numberofbiterrorsisone, subjecttotheaforementioned mapping constraint. Sincetherearelog2Mbitspersymbo~ itfollowsthattheaverageprobability ofsymbolerrorisrelatedtothebiterrorrateas follows: (,ag2M ) Pe=Pi~'lithbitisinerror} log2M :s2:PUthbitisinerror) i=l =log2M .(BER) Wealsonotethat Pe<'::PUthbitisinerror)=BER Itfollowstherefore thatthebiterrorrateisbounded asfollows:(5.97) (5.98) (5.99) Case2 LetM=2K,whereKisaninteger.Weassumethatallsymbolerrorsareequally likelyandoccurwithprobability Pe M-1 wherePeistheaverageprobability ofsymbolerror.Whatistheprobability thattheith bitinasymbolisinerror?Well,thereare2K-1casesofsymbolerrorinwhichthis particular bitischanged, andthereare2K-1casesinwhichitisnotchanged. Hence,the biterrorrateis or,equivalently,(2K-l ) BER=2K_1Pe (M/2)BER=M_1Pe(5.100) (5.101) NotethatforlargeM,thebiterrorrateapproaches thelimitingvalueofPe12.Thesame ideadescribed herealsoshowsthatbiterrorsarenotindependent, sincewehave P(ithandjthbitsareinerror)=2;K~21Peif'(BER)2 NotesandReferences 337 ~Summary andDiscussion Theprimarygoalofthematerial presented inthischapteristheformulation ofasystematic procedure fortheanalysisanddesignofadigitalcommunication receiverinthepresence ofadditivewhiteGaussian noise(AWGN).Theprocedure, knownasmaximum likelihood detection, decideswhichparticular transmitted symbolisthemostlikelycauseofthenoisy signalobserved atthechannel output.Theapproach thatledtotheformulation ofthe maximum likelihood detector (receiver) iscalledsignal-space analysis. Thebasicideaofthe approach istorepresent eachmember ofasetoftransmitted signalsbyanN-dimensional vector,whereNisthenumberoforthonormal basisfunctions neededforauniquegeo metricrepresentation ofthetransmitted signals.Thesetofsignalvectorssoformeddefines asignalconstellation inanN-dimensional signalspace. Foragivensignalconstellation, the(average) probability ofsymbolerrorP,incurred inmaximum likelihood signaldetection overanAWGNchannelisinvariant torotation ofthesignalconstellation aswellasitstranslation. However, exceptforafewsimple(but important) cases,thenumerical calculation ofP,isanimpractical proposition. Toover comethisdifficulty, thecustomary practice istoresorttotheuseofboundsthatlend themselves tocomputation inastraightforward manner. Inthiscontext, wedescribed the unionboundthatfollowsdirectlyfromthesignal-space diagram. Theunionbound isbased onanintuitively satisfying idea:Theprobability ofsymbolerrorPeisdominated bythe nearestneighbors tothetransmitted signal.Theresultsobtained usingtheunion bound areusuallyfairlyaccurate whenthesignal-to-noise ratioishigh. Withthematerial onsignal-space analysisandrelatedissuesonhand,wearewell equipped tostudypassband datatransmission systems, whichwedoinChapter6. INOTES ANDREFERENCES 1.Thegeometric representation ofsignalswasfirstdeveloped byKotel'nikov in1947:V.A. Kote!'nikov, TheTheoryofOptimum NoiseImmunity (DoverPublications, 1960),which isatranslation oftheoriginaldoctoral dissertation presented inJanuary1947beforethe Academic CounciloftheMolotov EnergyInstituteinMoscow. Inparticular, seePartIIof thebook.Thismethodwassubsequently broughttofullerfruitionintheclassicbookby Wozencraft andJacobs(1965).Signal-space analysisisalsodiscussed inCioffi(1998), Anderson (1999),andProakis(1995). 2.InSection5.7,wederivedtheunionboundontheaverageprobability ofsymbolerror;the classicreference forthisboundisWozencraft andJacobs(1965).Forthederivation oftighter bounds,seeViterbiandOmura(1979,pp.58-59). 3.InChapter4,weusedthefollowing upperboundonthecomplementary errorfunction exp(-u2)eric(u)<------:;==-- V7rU Forlargepositiveu,asecondboundonthecomplementary errorfunction isobtained by omitting themultiplying factorlIuintheaboveupperbound,asshownby £()exp(-u2 )encu<V7T ItisthissecondupperboundthatisusedinEquation (5.97). 338 CHAPTER 5'"SIGNAL-SPACE ANALYSIS IPROBLEMS Representation ofSignals 5.1InSection3.7wedescribed linecodesforpulse-code modulation. Referring tothematerial presented therein,formulate thesignalconstellations forrhefollowing linecodes: (a)Unipolar nonretum-to-zero code (b)Polarnonrerurn-to-zero code (c)Unipolar retum-to-zero code (d)Manchester code 5.2An8-levelPAMsignalisdefinedby Silt)=Airect(~-0 whereAi=±1, ±3,±5,±7.Formulate thesignalconstellation of(Si(t))~~I' 5.3FigureP5.3displaysthewaveforms offoursignalsS,(t),S2(t),S3(t),ands.(t). (a)UsingtheGram-Schmidt orthogonalization procedure, findanorthonormal basisfor thissetofsignals. (b)Construct thecorresponding signal-space diagram. S3(t)tiL 1 - tOTT "3 FIGUREP5.3 5.4(a)UsingtheGram-Schmidt orthogonalization procedure, findasetoforrhonormal basis functions torepresent thethreesignalss,it),S2(t),andS3(t)showninFigureP5.4. (b)Expresseachofthesesignalsintermsofthesetofbasisfunctions foundinpartla). S,(t) ',(t) '3(t) 4 4- 4 3 3- 3 2 2- 2 1 1 - 2 1 0 0 023 3 23-1 -1 -1 -2 -2 -2 -3 -3 -3 -4 -4 -4 -5I- FIGUREP5.4 5.5Anorthogonal setofsignalsischaracterized bytheproperty thattheinnerproductof anypairofsignalsinthesetiszero.FigureP5.5showsapairofsignalsS,(t)anddt) thatsatisfythiscondition. Construct thesignalconstellation forS,(t)andS2(t). o -1T/2 TProblems 339 ~L-----T-- FIGUREP5.5 5.6Asourceofinformation emitsasetofsymbolsdenotedby{mi}~"Twocandidate mod ulationschemes, namely,pulse-duration modulation (PDM)andpulse-position modula tion(PPM),areconsidered fortheelectrical representation ofthissetofsymbols.InPDM, theithsymbolisrepresented byapulseofunitamplitude andduration (i/M)T.Onthe otherhand,inPPM,theithsymbolisrepresented byashortpulseofunitamplitude and fixedduration, whichistransmitted attimet=(jIM)T.ShowthatPPMistheonlyone ofthetwothatcanproduceanorthogonal setofsignalsovertheintervala:s;t:s;T. 5.7Asetof2Mbiorthogonal signalsisobtained fromasetofMorthogonal signalsby augmenting itwiththenegativeofeachsignalintheset. (a)Theextension oforthogonal tobiorthogonal signalsleavesthedimensionality ofthe signalspaceunchanged. Why? (b)Construct thesignalconstellation forthebiorthogonal signalscorresponding tothe pairoforthogonal signalsshowninFigureP5.5. 5.8(a)Apairofsignalssilt)andSkIt)haveacommon duration T.Showthattheinner productofthispairofsignalsisgivenbyrSi(t)Sk(t)dt =STSk where SiandSkarethevectorrepresentations ofsilt)andSkIt),respectively. (b)Asafollowup topart(a),showthatr(Si(t)-sk(t)j2dt =IISi-Sk[12 5.9Consider apairofcomplex-valued signalss,(t)andS2(t)thatarerespectively represented by S,(t)=all<P,(t)+a'2<P2(t), -c;o<t<c;o S2(t)=a2,<p,(t)+a22<P2(t), -c;o<t<c;o wherethebasisfunctions <p,(t)and<P2(t)arebothrealvalued,butthecoefficients all' au,a2banda22arecomplex valued.Provethecomplex formoftheSchwarzinequality; IrooS,(t)S;(t)df :s;roo[s,(t)12dt[ IS2(tl!'dt wheretheasteriskdenotescomplex conjugation. Whenisthisrelationsatisfiedwiththe equalitysign? Random Processes 5.10Consider arandomprocessX(t)expanded intheform N X(t)=2:X/<p/(t)+W'(t), j=1O:s;t:s;T x(t)=Sk(t)+w(t),340 CHAPTER 5I!!SIGNAL-SPACE ANALYSIS whereW'(t)isaremainder noiseterm.The{eMt));!., formanorthonormal setOverthe inrerval0:=;t:=;T,andtheXjaredefinedby Xj=fX(t)<Mt)dt LetW'(tk)denotearandomvariableobtained byobserving W'(t)attimet=tk.Show that {i=1,2,..., N o:=;tk:=;T 5.11Consider theoptimwn detection ofthesinusoidal signal s(t)=sin(8;t), 0:=;t:=;T inadditivewhiteGaussian noise. (a)Determine thecorrelator outputassuming anoiseless input. (b)Determine thecorresponding matched filteroutput,assuming thatthefilterincludes adelayTtomakeitcausal. (c)Henceshowthatthesetwooutputsarethesameonlyattimeinstantt=T. Probability ofError 5.12FigureP5.12showsapairofsignalss,(t)andS2(t)thatareorthogonal toeachotherover theobservation interval0:=;t:=;3T.Thereceivedsignalisdefinedby O:=;t:=;3T k=1,2 wherew(t)iswhiteGaussian noiseofzeromeanandpowerspectraldensityNoll. (a)Designareceiverthatdecidesinfavorofsignalss,(t)orS2(t),assuming thatthese twosignalsareequiprobable. (b)Calculate theaverageprobability ofsymbolerrorincurred bythisreceiver 101 EINo=4,whereEisthesignalenergy. 92(1) -1Tok--+-T---I-3T--S-T-+--3--'T--"222 -1 FIGUREP5.12 5.13IntheManchester code,binarysymbol1isrepresented bythedoubletpulses(t)shown inFigureP5.13,andbinarysymbol0isrepresented bythenegative ofthispulse.Derive theformulafortheprobability oferrorincurred bythemaximum likelihood detection procedure appliedtothisformofsignaling overanAWGNchannel. Problems 341 sltl o -1T/2 T FIGUREP5.13 5.14IntheBayestest,appliedtoabinaryhypothesis testingproblemwherewehavetochoose oneoftwopossiblehypotheses HoandHi>weminimize theriskRdefinedby R=CoopoP(say HoIHoistrue) +C,OPOP(say H,IHoistrue) +C"p,P(say H,IH,istrue) +COJP,P(say HolH,istrue) ThetermsCoo,C,o,Cll,andCo,denotethecostsassignedtothefourpossibleoutcomes oftheexperiment: Thefirstsubscriptindicates thehypothesis chosen,andthesecondthe hypothesis thatistrue.AssumethatC'O>CooandCo,>Cll•ThePoandp,denotethe aprioriprobabilities ofhypotheses HoandHi>respectively. (a)Giventheobservation vectorx,showthatthepartitioning oftheobservation space soastominimize theriskRleadstothelikelihood ratiotest: sayHoifA(x)<A sayH,ifA(x)>A whereA(x)isthelikelihood ratio A(x)=!x(xIH,) fx(xIHo) andAisthethreshold ofthetestdefinedby A=Po(ClO-Coo) p,(COl-Cll) (h)WhatarethecostvaluesforwhichtheBayes'criterion reducestotheminimum probability oferrorcriterion? Principles ofRotational andTranslational Invariance 5.15Continuing withthefourlinecodesconsidered inProblem 5.1,identifythelinecodes thathaveminimum averageenergyandthosethatdonot.Compare youranswerswith theobservations madeontheselinecodesinSection3.7. 5.16Consider thetwoconstellations showninFigure5.11.Determine theorthonormal matrix Qthattransforms theconstellation showninFigure5.11aintotheoneshowninFigure 5.11b. 342 CHAPTER 5..SIGNAL-SPACE ANALYSIS 5.17(a)Thetwosignalconstellations showninFigureP5.17exhibitthesameaveragePtoh- abilityofsymbolerror.Justifythevalidityofthisstatement. (b)Whichofthesetwoconstellations hasminimum averageenergy?JustifyyouransWer. Youmayassumethatthesymbols perraining tothemessagepointsdisplayed inFigUrP5.17areequallylikely. e <1>2<1>2 -I2a /""" f-_3---, /, I I /" /"I I /, I<1>,/ "<1>,al0a 0"- //2-12aI I""I I"- // .---~--4 "/ --I2a"-'w/ (a) (b) FIGUREP5.17 5.18Simplex(transorthogonal) signalsareequallylikelyhighly-correlated signalswiththe mostnegative correlation thatcanbeachieved withasetofMorthogonal signals.That is,thecorrelation coefficient between anypairofsignalsinthesetisdefinedby {1fori=j Pij=~M _ 1 fori1=j Onemethodofconstructing simplexsignalsistostartwithasetofMorthogonal signals, eachwithenergyE,andthenapplytheminimum energytranslate. Consider asetofthreeequaJJylikelysymbolswhosesignalconstellation consiSISof theverricesofanequilateral triangle. Showthatthesethreesymbolsconstitute asimplex code. BoundsonProbability ofError 5.19Inthisproblem weexploretheapproximations totheprobability ofanerror,P"forthe pairofantipodal signalsshowninFigureP5.19inthepresence ofadditivewhiteGaussian noiseofpowerspectraldensityNo/2.TheexactformulaforPeis Pe=ierfc(~) (ThisformulaisderivedinSection6.3) (a)Usingthetwoupperboundsforthecomplementary errorfunction giveninNOle3, derivethecorresponding approximations toPc. (b)Compare theapproximations derivedinparr(a)forPctotheexactformulafor EblNo=9.Fortheexactcalculation ofP"youmayuseTableA6.6ontheerrOl function. :..fib------<>-----co-+:----<O>----- <1>, I FIGUREP5.19 Problems 343 5.20Consider thespecialcaseofasignalconstellation thathasasymmetric geometry with respecttotheorigin.AssumethattheMmessagepointsoftheconstellation, pertaining tosymbolsm"m2'...,mM,areequallylikely.Usingtheupperboundonthecomple mentary errorfunction giveninEquation (5.94),showthattheaverageprobability of symbolerrorfortheconstellation isbounded as Mmin( •(dfk)) PeoS--exp-mill_--2v'1T i,k4No #k wheredikistheEuclidean distancebetweenmessagepointsiandk,andMministhenumber oftransmitted signalsthatattaintheminimum Euclidean distanceforeachmi' PASSBAND DATA TRANSMISSION Thischapterbuildsonthematerial developed inChapter 5onsignal-space analysis.It discusses thesubjectofdigitaldatatransmission overaband-pass channelthatcanbe linearornonlinear. Aswithanalogcommunications, thismodeofdatatransmission relies ontheuseofasinusoidal carrierwavemodulated bythedatastream. Specifically, thefollowing topicsarecovered: ~Different methodsofdigitalmodulation, namely,phase-shift keying,quadrature amplitude modulation, andfrequency-shift keying,andtheirindividual variants. ~Coherent detectionofmodulated signalsinadditivewhiteGaussian noise,whichrequires thereceivertobesynchronized tothetransmitter withrespecttobothcarrierphaseand bittiming. ~Noncoherent detectionofmodulated signalsinadditivewhiteGaussian noise, disregarding phaseinformation inthereceivedsignal. ~Modems forthetransmission andreception ofdigitaldataoverthepublicswitcbed telephone network. ~Sophisticated modulation techniques, namely,carrierless amplitude/phase modulation and discretemultitone, fordatatransmission overawidebandchannelwithmediumtosevere intersymbol interference. ~Techniques forsynchronizing thereceivertothetransmitter. I6.1Introduction Inbaseband pulsetransmission, whichwestudiedinChapter4,adatastreamrepresented intheformofadiscretepulse-amplitude modulated (PAM)signalistransmitted directly overalow-pass channel.Indigitalpassband transmission, ontheotherhand,theincoming datastreamismodulated ontoacarrier(usuallysinusoidal) withfixedfrequency limits imposed byaband-pass channelofinterest;passband datatransmission isstudiedinthis chapter. Thecommunication channelusedforpassband datatransmission maybeamicro waveradiolink,asatellitechannel, orthelike.Yetotherapplications ofpassband data transmission areinthedesignofpassband linecodesforuseondigitalsubscriber looPS andorthogonal frequency-division multiplexing techniques forbroadcasting. Inanyeveo~ themodulation processmakingthetransmission possibleinvolvesswitching (keying)~ amplitude, frequency, orphaseofasinusoidal carrierinsomefashioninaccordance WI theincoming data.Thustherearethreebasicsignaling schemes, andtheyareknownas 344 6.1Introduction 345 Binary data 0 0 0 0l~I\I\{JI~f\D ~f\D VV VV oIVVVV Tb---+1 (al I If\f\~f\f\ f\nAA~f\nf\Df\f\~Du o~v1JvVVvvv1vvvvVlJ1VVi (b) O~{\Df\f\{\f\f\f\f\f\f\f\IVVVVVVVVVvVV (c) FIGURE 6.1Illustrative waveforms forthethreebasicfonnsofsignaling binaryinfonnation. (a) Amplitude-shift keying.(b)Phase-shift keying.(c)Frequency-shift keyingwithcontinuous phase. amplitude-shift keying(ASK),frequency-shift keying(FSK),andphase-shift keying(PSK). Theymaybeviewedasspecialcasesofamplitude modulation, frequency modulation, and phasemodulation, respectively. Figure6.1illustrates thesethreemethods ofmodulation forthecaseofasource supplying binarydata.Thefollowing pointsarenoteworthy fromFigure6.1: &>Although incontinuous-wave modulation itisusuallydifficulttodistinguish between phase-modulated andfrequency-modulated signalsbymerelylookingattheirwave forms,thisisnottrueforPSKandFSKsignals. ~UnlikeASKsignals,bothPSKandFSKsignalshaveaconstant envelope. Thislatterproperty makesPSKandFSKsignalsimpervious toamplitude nonlinearities, commonly encountered inmicrowave radioandsatellitechannels.Itisforthisreason,in practice, wefindthatPSKandFSKsignalsarepreferred toASKsignalsforpassband data transmission overnonlinear channels. IIIHIERARCHY OFDIGITAL MODULATION TECHNIQUESl Digitalmodulation techniques maybeclassified intocoherent andnoncoherenttechniques, depending onwhetherthereceiverisequipped withaphase-recovery circuitornot.The phase-recovery circuitensuresthattheoscillator supplying thelocallygenerated carrier waveinthereceiverissynchronized (inbothfrequency andphase)totheoscillator sup plyingthecarrierwaveusedtooriginally modulate theincoming datastreaminthe transmitter. Asdiscussed inChapter4,inanM-arysignaling scheme,wemaysendanyoneof MpossiblesignalsSl(t),S2(t),•••,SM(t),duringeachsignaling intervalofdurationT.For 346 CHAPTER 6l!lPASSBAND DATATRANSMISSION almostallapplications, thenumberofpossiblesignalsM=ln,wherenisaninteger.The symbolduration T=nTb,whereTbisthebitduration. Inpassband datatransmissio thesesignalsaregenerated bychanging theamplitude, phase,orfrequency ofasinusoidi carrier i~Mdiscretesteps.ThuswehaveM-a~ASK,M-a?PSK:andM-a~FSKdigit~1 modulation schemes. Another wayofgeneratmg M-arysIgnals IStocombme different methods ofmodulation intoahybridform.Forexample, wemaycombine discretechange inboththeamplitude andphaseofacarriertoproduce M-aryamplitude-phase keyinS (APK).Aspecialformofthishybridmodulation isM-aryquadrature-amplitude mol ulation(QAM), whichhassomeattractive properties. M-aryASKisaspecialcaseof M-aryQAM. M-arysignaling schemesarepreferred overbinarysignaling schemesfortransmitting digitalinformation overband-pass channels whentherequirement istoconserve band. widthattheexpenseofincreased power.Inpractice, werarelyfindacommunication channelthathastheexactbandwidth required fortransmitting theoutputofaninfor. mationsourcebymeansofbinarysignaling schemes. Thuswhenthebandwidth ofthe channelislessthantherequiredvalue,wemayuseM-arysignaling schemesformaximum efficiency. Toillustrate thebandwidth-conservation capability ofM-arysignaling schemes considerthetransmission ofinformation consisting ofabinarysequence withbitduratio~ Tb•Ifweweretotransmit thisinformation bymeansofbinaryPSK,forexample, We wouldrequireabandwidth thatisinversely proportional toTb•However,ifwetakeblocks ofnbitsanduseanM-aryPSKschemewithM=Inandsymbolduration T=nTbthe bandwidth requiredisproportional toIlnTb•ThisshowsthattheuseofM-aryPSKenables areduction intransmission bandwidth bythefactorn=log2MoverbinaryPSK. M-aryPSKandM-aryQAMareexamples oflinearmodulation. However, theydiffer fromeachotherinoneimportant respect:AnM-aryPSKsignalhasaconstant envelope, whereas anM-aryQAMsignalinvolves changesinthecarrieramplitude. Accordingly, M-aryPSKcanbeusedtotransmit digitaldataoveranonlinear band-pass channel, whereasM-aryQAMrequirestheuseofalinearchannel. M-aryPSK,M-aryQAM,andM-aryFSKarecommonly usedincoherent systems. Amplitude-shift keyingandfrequency-shift keyinglendthemselves naturally touseinnon· coherent systemswhenever itisimpractical tomaintain carrierphasesynchronization. But inthecaseofphase-shift keying,wecannothave"noncoherent PSK"becausetheterm noncoherent meansdoingwithout carrierphaseinformation. Instead, weemploya "pseudo PSK"technique knownasdifferential phase-shift keying(DPSK), which(ina loosesense)maybeviewedasthenoncoherent formofPSK.Inpractice, M-aryFSKand M-aryDPSKarethecommonly usedformsofdigitalmodulation innoncoherent systems. IIIPROBABILITY OFERROR Ama.jorgoalofpassband datatransmission systemsistheoptimum designofthereceiver soastominimize theaverageprobability ofsymbolerrorinthepresence ofadditivewhite Gaussian noise(AWGN).Withthisgoalinmind,muchofthematerialpresented int~S chapterbuildsonthesignal-space analysis toolspresented inChapter5.Specifically, 111 thestudyofeachsystemwebeginwiththeformulation ofasignalconstellation andthe construction ofdecisionregionsinaccordance withmaximum likelihood. signaldetec??D overanAWGNchannel. Theseformulations setthestageforevaluating theprobabilitY ofsymbolerrorPe•Depending onthemethodofdigitalmodulation understudy,the evaluation ofP,proceeds in oneoftwoways: i>-Inthecaseofcertainsimplemethods suchascoherent binaryPSKandcoherent binaryFSK,exactformulas arederivedforP,. (6.1)6.1Introduction 347 10>Inthecaseofmoreelaborate methods suchascoherent M-aryPSKandcoherent M-aryFSK,weresorttotheuseoftheunionboundforderiving anapproximate formulaforP,. illJPOWER SPECTRA Tofullyappreciate thepractical virtuesofdifferent methods ofdigitalmodulation, we alsoneedtostudythepowerspectraoftheresulting modulated signals.Thislatterissue isparticularly important intwocontexts: occupancy ofthechannelbandwidth andco channelinterference inmultiplexed systems. Givenamodulated signalsit),wemaydescribeitintermsofitsin-phase andquad raturecomponents as sit)=SI(t)cos(2'11fcT) -sQ(t)sin(2'11fct) =Re[s(t)exp(j2'11fct)] whereRe[·]istherealpartoftheexpression contained insidethesquarebrackets. Wealso have (6.2) and exp(j27TfcT) =COS(27Tfct)+jsin(27Tfct) (6.3) Thesignals(t)isthecomplex envelope (i.e.,baseband version)ofthemodulated (band pass)signals(t).Thecomponents SI(t)andsdt)andtherefore s(t)arealllow-pass signals. Theyareuniquely definedintermsoftheband-pass signals(t)andthecarrierfrequency fc,provided thatthehalf-bandwidth ofs(t)islessthanthecarrierfrequency fc. LetSB(f)denotethepowerspectraldensityofthecomplex envelope s(t).Werefer toSB(f)asthebaseband powerspectraldensity.Thepowerspectraldensity,Ss(f),ofthe originalband-pass signals(t)isafrequency-shifted versionofSB(f),exceptforascaling factor,asshownby (6.4) Itistherefore sufficient toevaluate thebaseband powerspectraldensitySB(f).Sinces(t) isalow-pass signal,thecalculation ofSn(f)shouldbesimplerthanthecalculation ofSs(f). (SeeExample 1.7.) I!llBANDWIDTH EFFICIENCY Throughout thisbookwehaveemphasized thatchannelbandwidth andtransmitted power constitute twoprimary"communication resources," theefficientutilization ofwhichpro videsthemotivation forthesearchforspectrally efficientschemes. Theprimaryobjective ofspectrally efficientmodulation istomaximize thebandwidth efficiency definedasthe ratioofthedatarateinbitspersecondtotheeffectively utilizedchannelbandwidth. A secondary objective istoachievethisbandwidth efficiency ataminimum practical expen ditureofaveragesignalpoweror,equivalently, inachannelperturbed byadditivewhite Gaussian noise,aminimum practical expenditure ofaveragesignal-to-noise ratio. WiththedataratedenotedbyRbandtheeffectively usedchannelbandwidth byB, wemayexpressthebandwidth efficiency, p,as p=?bits/slHz (6.5) 348 CHAPTER 6111PASSBAND DATATBANSMISSION RecallfromChapter4thatbandwidth efficiency istheproductoftwoindependent factor. oneduetothepossibleuseofmultilevel encoding andtheotherduetospectralshapin;' Inmultilevel encoding, information transmission throughthechanneliscarriedoutonth. basisofblocksofbitsratherthansinglebits.Withefficientspectralshaping, bandwid~ requirement onthechannelisreducedbytheuseofpulse-shaping filtersthatsmooth OUt thesharptransitions inthetransmitted waveform. Thesetwofactorsaretherefore impor_ tantintheirownindividual waysindetermining thebandwidth efficiency ofapassband datatransmission systemofinterest. I6.2Passband Transmission Model Inafunctional sense,wemaymodelapassband datatransmission systemasshownin Figure6.2.First,thereisassumed toexistamessage sourcethatemitsonesymbolevery Tseconds, withthesymbolsbelonging toanalphabet ofMsymbols, whichwedenoteby milm2,...,mM'Theaprioriprobabilities P(mt),P(m2),...,P(mM)specifythemessage sourceoutput.WhentheMsymbolsofthealphabet areequallylikely,wewrite Pt=P(mi) 1 Mforalli(6.6) TheM-aryoutputofthemessagesourceispresented toasignaltransmission encoder, producing acorresponding vector SimadeupofNrealelements, onesuchsetforeach0/ theMsymbolsofthesourcealphabet; thedimension NislessthanorequaltoM.With thevector Siasinput,themodulator thenconstructs adistinctsignal Si(t)ofdurationT secondsastherepresentation ofthesymbolmigenerated bythemessagesource.Thesignal Si(t)isnecessarily anenergysignal,asshownby Ei=fsf(t)dt,i=1,2,..., M 16.7) Notethats,(t)isrealvalued.Onesuchsignalistransmitted everyTseconds. Theparticular signalchosenfortransmission depends insomefashionontheincoming message and possibly onthesignalstransmitted inpreceding timeslots.Withasinusoidal carrier,the featurethatisusedbythemodulator todistinguish onesignalfromanotherisastepchange intheamplitude, frequency, orphaseofthecarrier.(Sometimes, ahybridformofmod ulationthatcombines changesinbothamplitude andphaseoramplitude andfrequency isused.) carrierwave tran~~~:~ion Sj encoder I I I I I ~ TransmitterI I I I I I 1----- 1 Receiver FIGURE6.2Functional modelofpassband datatransmission system. 6.3Coherent Phnse-Shift Keying 349 Returning tothefunctional modelofFigure6.2,thebandpass conununication chan nel,coupling thetransmitter tothereceiver, isassumed tohavetwocharacteristics: 1.Thechannelislinear,withabandwidth thatiswideenoughtoaccommodate the transmission ofthemodulated signalsilt)withnegligible ornodistortion. 2.Thechannelnoisewit)isthesamplefunction ofawhiteGaussian noiseprocessof zeromeanandpowerspectraldensitYNoll. Theassumptions madehereinarebasically thesameasthoseinvokedinChapter5dealing withsignal-space analysis. Thereceiver,whichconsistsofadetectorfollowed byasignaltransmission decoder, performs twofunctions: 1.Itreversestheoperations performed inthetransmitter. 2.Itminimizes theeffectofchannelnoiseontheestimatemcomputed forthetrans mittedsymbol mi' ~.3Coherent Phase-Shift Keying Withthebackground materialonthecoherent detection ofsignalsinadditivewhiteGaus siannoisethatwaspresented inChapter 5atourdisposal, wearenowreadytostudy specificpassband datatransmission systems. Inthissectionwefocusoncoherent phase shiftkeying(PSK)byconsidering binaryPSK,QPSKanditsvariants, andfinishupwith M-aryPSK. IIIBINARY PHASE-SHIFT KEYING Inacoherent binaryPSKsystem,thepairofsignalsslit)andS2(t)usedtorepresent binary symbols1and0,respectively, isdefinedby (2E;,slit)={r;:cos(27rfct) (6.8) S2(t)=f!fcos(27rfct+7r)= -f!fcos(27rfct) (6.9) where0:s;:t:s;:Tb,andEbisthetransmitted signalenergyperbit.Toensurethateach transmitted bitcontains anintegralnumberofcyclesofthecarrierwave,thecarrierfre quencyfcischosenequaltonJTbforsomefixedintegerncoApairofsinusoidal waves thatdifferonlyinarelativephase-shift of180degrees,asdefinedinEqautions (6.8)and (6.9),arereferredtoasantipodal signals. Fromthispairofequations itisclearthat,inthecaseofbinaryPSK,thereisonly onebasisfunction ofunitenergy,namely, <P1(t)=Ifcos(27rfct), 0:s;:t<Tb (6.10) Thenwemayexpressthetransmitted signalsslit)andS2(t)intermsof<PI(t)asfollows: slit)=\!E;,<Pl(t), 0:s;:t<Tb (6.11) and (6.12) (6.13) (6.14)350 CHAPTER 6'"PASSBAND DATATRANSMISSION Decision boundary 1 RegIonIRegion ,L.22 I 2, I --.fEbI-.fEb I. I. Message 01 Message point I point 2 1 1 I FIGUBE6.3Signal-space diagram forcoherent binaryPSKsystem.Thewaveforms depicting the transmitted signalss,(t)ands,,(t),displayed intheinserts,assumeno=2. Acoherent binaryPSKsystemistherefore characterized byhavingasignalspace thatisone-dimensional (i.e.,N=1),withasignalconstellation consisting oftwomessage points(i.e.,M=2).Thecoordinates ofthemessagepointsare fTb Sl1=0Sl(t)<Pl(t) dt =+YE;; and S21=foTb S2(t)<Pl(t) dt =-YE;; Themessagepointcorresponding toSl(t)islocatedatS11=+YE;;,andthemessagepoint corresponding toS2(t)islocatedatS21=-YE;;.Figure6.3displaysthesignal-space di agramforbinaryPSK.Thisfigurealsoincludestwoinserts,showing example waveforms ofantipodal signalsrepresenting Sl(t)andS2(t).Notethattheconstellation ofFigure6.3 hasminimum averageenergy. ErrorProbability ofBinaryPSK Torealizearuleformakingadecision infavorofsymbol1orsymbol0,weapply Equation (5.59)ofChapter5.Specifically, wepartition thesignalspaceofFigure6.3into tworegions: ..Thesetofpointsclosesttomessagepoint1at+YE;;. I>Thesetofpointsclosesttomessagepoint2at-VE;;. 6.3Coherent Phase-Shift Keying 351 Thisisaccomplished byconstructing themidpoint ofthelinejoiningthesetwomessage points,andthenmarking offtheappropriate decisionregions.InFigure6.3thesedecision regionsaremarked21and2z,according tothemessage pointaroundwhichtheyare constructed. Thedecisionruleisnowsimplytodecidethatsignalslit)(i.e.,binarysymbol1)was transmitted ifthereceived signalpointfallsinregion21>anddecidethatsignalsz(t)(i.e., binarysy'mbol0)wastransmitted ifthereceived signalpointfallsinregion2z.Twokinds oferroneous decisions may,however, be'made. Signalsz(t)istransmitted, butthenoiseis suchthatthereceived signalpointfallsinsideregion21andsothereceiverdecidesinfavor ofsignalSlit).Alternatively, signalslit)istransmitted, butthenoiseissuchthatthere ceivedsignalpointfallsinsideregion2zandsothereceiverdecidesinfavorofsignalsz(t). Tocalculate theprobability ofmakinganerrorofthefirstkind,wenotefromFigure 6.3thatthedecision regionassociated withsymbol1orsignalSIlt)isdescribed by 21:0<Xl<00 wheretheobservable elementXlisrelatedtothereceived signalx(t)by (b Xl=Jox(t)q,l(t)dt (6.15) (6.16)Theconditional probability densityfunction ofrandomvariableXl'giventhatsymbol0 [i.e.,signalsz(t)]wastransmitted, isdefinedby fxl(XIIO)=V~Noexp[-~o(Xl-SZ1)ZJ =V~Noexp[-~o(Xl+~)zJ Theconditional probability ofthereceiverdeciding infavorofsymbol1,giventhatsymbol owastransmitted, istherefore (6.17) (6.18)Putting 1z=•'"(Xl+~)vNo andchanging thevariable ofintegration fromXltoZ,wemayrewriteEquation (6.17)in thecompact form =~erfe(~) whereeric(·)isthecomplementary errorfunction.(6.19) (6.20)352 CHAPTER 6"lPASSBAND DATATRANSMISSION Consider nextanerrorofthesecondkind.WenotethatthesignalspaceofFigur6.3issymmetric withrespecttotheorigin.Itfollowstherefore thatPOI'theconditio~ probability ofthereceiverdeciding infavorofsymbol0,giventhatsymbol1wastran_ mitted,alsohasthesamevalueasinEquation (6.19). s Thus,averaging theconditional errorprobabilities PIOandPOhwefindthatth averageproba?ility o(symbolerroror,equivalently, thebiterrorrateforcoherent bina~ PSKIS(assummg eqUlprobable symbols) . P,=&erfc(~) Asweincrease thetransmitted signalenergyperbit,Eb,foraspecified noisespectral densityNo,themessagepointscorresponding tosymbols1and0movefurtherapart,and theaverageprobability oferrorPeiscorrespondingly reducedinaccordance withEquation (6.20),whichisintuitively satisfying. Generation andDetection ofCoherent BinaryPSKSignals Togenerate abinaryPSKsignal,weseefromEquations (6.8)-(6.10) thatwehave torepresent theinputbinarysequence inpolarformwithsymbols1and0represented by constant amplitude levelsof+VB;;and-VB;;,respectively. Thissignaltransmission en. codingisperformed byapolarnonreturn-to-zero (NRZ)levelencoder. Theresulting bi narywaveandasinusoidal carrier"'1(t),whosefrequencyt=(njTb)forsomefixed integer no>areappliedtoaproductmodulator, asinFigure6.4a.Thecarrierandthe timingpulsesusedtogenerate thebinarywaveareusuallyextracted fromacornmon masterclock.ThedesiredPSKwaveisobtained atthemodulator output. Todetecttheoriginalbinarysequence of1sandOs,weapplythenoisyPSKsignal x(t)(atthechanneloutput)toacorrelator, whichis.alsosupplied withalocallygenerated coherent reference signal<PI(t),asinFigure6,4b.Thecorrelator output, Xhiscompared withathreshold ofzerovolts.IfXl>0,thereceiverdecidesinfavorofsymbol1.Onthe Binary data sequence (0)Binary PSK signal ,it) x(t)rChoose1ifxl>01Chooseaifx1<0 (b) FIGURE6.4Blockdiagrams for(a)binaryPSKtransmitter and(b)coherent binaryPSK receiver. (6.22)6.3Coherent PMse-Shlft Keying 353 otherhand,ifX,<0,itdecidesinfavorofsymbol O.IfX,isexactly zero, thereceiver makesarandomguessinfavorof0or1. PowerSpectraofBinaryPSKSignals Fromthemodulator ofFigure6.4a,weseethatthecomplex envelope ofabinary PSKwaveconsistsofanin-phase component only.Furthermore, depending onwhether wehavesymbol1orsymbol0atthemodulator inputduringthesignaling intervalo:5t:5Tb,wefindthatthisin-phase component equals+g(t)or-g(t),respectively, whereg(t)isthesymbolshapingfunction definedby g(t)={J3!f, 0:5t:5Tb (6.21) 0, otherwise Weassumethattheinputbinarywaveisrandom, withsymbols1and0equallylikelyand thesymbolstransmitted duringthedifferent timeslotsbeingstatistically independent. In Example 1.6ofChapter1itisshownthatthepowerspectraldensityofarandombinary wavesodescribed isequaltotheenergyspectraldensityofthesymbolshapingfunction dividedbythesymbolduration. TheenergyspectraldensityofaFouriertransformable signalg(t)isdefinedasthesquaredmagnitude ofthesignal'sFouriertransform. Hence, thebaseband powerspectraldensityofabinaryPSKsignalequals S(f)=2Ebsin2 (1TTd) B (1TTbf)2 =2Ebsinc2(Td) Thispowerspectrum fallsoffastheinversesquareoffrequency, asshowninFigure6.5. Figure6.5alsoincludesaplotofthebaseband powerspectraldensityofabinary FSKsignal,detailsofwhicharepresented inSection6.5.Comparison ofthesetwospectra isdeferredtothatsection. oDeltafunction (parlofFSKspectrum) Q5 1.0 I~ Normalized frequency,jTb2.0 FIGURE6.5PowerspectraofbinaryP8KandFSKsignals. 354 CHAYI'ER 6illPASSBAND DATATRANSMISSION !!!QVADRIPHASE-SHIFT KEYING Theprovision ofreliableperformance, exemplified byaverylowprobability oferror. oneimportant goalinthedesignofadigitalcommunication system.Another imPOrt~ IS goalistheefficientutilization ofchannelbandwidth. Inthissubsection, westudyaban: width-conserving modulation schemeknownascoherent quadriphase-shift keying,which isanexample ofquadrature-carrier multiplexing. ~quadripha~e-shift ~eyin?(QPSK),aswithbi~aryPSK,information carriedbythe transmItted signal IScontamed mthephase.Inparticular, thephaseofthecarriertakes ononeoffourequallyspacedvalues,suchas'TT'/4,3'TT'/4,5'TT'/4,and7'TT'/4.Forthissetof valueswemaydefinethetransmitted signalas s;(t)={!¥cos[2'TT'fct+(2i-1)~J, 0, elsewhere(6.23) wherei=1,2,3,4;Eisthetransmitted signalenergypersymbol,andTisrhesymbol duration. Thecarrierfrequency fcequalsnJTforsomefixedintegerncoEachpossible valueofthephasecorresponds toauniquedibit.Thus,forexample, wemaychoosethe foregoing setofphasevaluestorepresent theGray-encoded setofdibits:10,00,01,and 11,whereonlyasinglebitischangedfromonedibittothenext. Signal-Space Diagram. ofQPSK Usingawell-known trigonometric identity, wemayuseEquation (6.23)toredefine thetransmitted signals;(t)fortheinterval°:5t:5Tintheequivalent form: s;(t)=!¥cos[(2i-1)~]cos(2'TT'fct) -!¥sin[(2i-1)~]sin(2'TT'fct) (6.24) wherei=1,2,3,4.Basedonthisrepresentation, wecanmakethefollowing observations: \l>Therearetwoorthonormal basisfunctions, cP,(t)andcP2(t),contained intheexpan. sionofsift).Specifically, cP,(t)andcP2(t)aredefinedbyapairofquadrature carriers: cP,(t)=j$,cos(2'TT'fct), 0:5t:5T (6.25) cP2(t)=j$,sin(2'TT'fct), 0:5t:5T (6.26) TABLE6.1Signal-space characterization ofQPSK Gray-encoded InputDibitPhaseof QPSKSignal (radians)Coordinates of Message Points Sit Sn 10 00 01 11'TT'/4 3'TT'/4 5'TT'/4 7'TT'/4+VEfi -vEfi -vEfi +VEfi-vEfi -vEfi +VEfi +VEfi 6.3Coherent Phase-Shift Keying 355 Decision boundary-----'--<PI fIGURE6.6Signal-space diagramofcoherent QPSKsystem. I;>Therearefourmessagepoints,andtheassociated signalvectorsaredefinedby Sl= [VBCOS((2i-1)~)j, -VBSin((2i-1)~)i=1,2,3,4 (6.27) Theelements ofthesignalvectors,namely, SilandSil'havetheirvaluessummarized inTable6.1.Thefirsttwocolumns ofthistablegivetheassociated dibitandphase oftheQPSKsignal. Accordingly, aQPSKsignalhasatwo-dimensional signalconstellation (i.e.,N=2)and fourmessagepoints(i.e.,M=4)whosephaseanglesincreaseinacounterclockwise di rection,asillustrated inFigure6.6.AswithbinaryPSK,theQPSKsignalhasminimum averageenergy. II>ExAMPLE 6.1 Figure6.7illustrates thesequences andwaveforms involved inthegeneration ofaQPSK signal.Theinputbinarysequence 01101000 isshowninFigure6.7a.Thissequence isdivided intotwoothersequences, consisting ofodd-andeven-numbered bitsoftheinputsequence. Theserwosequences areshowninthetoplinesofFigures6.7band6.7c.Thewaveforms representing therwocomponents oftheQPSKsignal,namely,SilcP,(t)andSi2cP2(t),arealso showninFigures6.7band6.7c,respectively. Theserwowaveforms mayindividually be viewedasexamples ofabinaryPSKsignal.Addingthem,wegettheQPSKwaveform shown inFigure6.7d. Todefinethedecisionruleforthedetection ofthetransmitted datasequence, wepar titionthesignalspaceintofourregions,inaccordance withEquation (5.59)ofChapter5. Theindividual regionsaredefinedbythesetofpointsclosesttothemessagepointrepresented bysignalvectorss"S2'S3,andS4-Thisisreadilyaccomplished byconstructing theperpen dicularbisectors ofthesquareformedbyjoiningthefourmessagepointsandthenmarking 356 CHAPTER 6..PASSBAND DATATRANSJlUSSION Input binary sequenceo 1~ Dibit011 D '-------y-----J Dibit1D (al1 0~ Dibit10o 0 '-------y-----J Dibit00 Odd-numbered sequence 0 Polarityofcoefficient sil- + +o /\/\r'\/\/\/\11/\/\~V~v ~v\.Tt/V ~t (bl Even-numbered sequence Polarityofcoefficient si2 +o o o L\L\L\/\/\L\/$\VVV~~V~t (c) f\f\~f\f\f\nf\f\ s(tl7V\JVV V VV$\ t (d) FIGURE6.7(a)Inputbinarysequence. (b)Odd-numbered bitsofinputsequence andassociated binaryPSKwave.(c)Even-numbered bitsofinputsequence andassociated binaryPSKwave. (d)QPSKwaveform definedas5(t)=5il"'1(t)+5'2"'2(t). offtheappropriate regions.Wethusfindtbatthedecisionregionsarequadrants whosevertices coincidewiththeorigin.Theseregionsaremarked2"22,23,and24,inFigure6.6,accordlJ1g tothemessagepointaroundwhichtheyareconstructed. <II x(t)=Silt)+wit),ErrorProbability ofQPSK Inacoherent QPSKsystem,thereceived signalx(t)isdefinedby {O:=;t:=;T i=1,2,3,4(6.28) wherewIt)isthesamplefunction ofawhiteGaussian noiseprocessofzeromeanand powerspectral densityNo/2.Correspondingly, theobservation vectorxhastwoelements, X,andX2,definedby X,=fX(t)<Pl(t) dt =vBCOS[(2i-1)~]+W, =+@+W,-{i(6.29) (6.30)6.3Coherent Phase-Shift Keying 357 and X2=rX(t)cP2(t) dt =-vBsin[(2i-1)~J+W2 ==+=~+W2 Thustheobservable elements XlandX2aresamplevaluesofindependent Gaussian random variables withmeanvaluesequalto~v1ffiand=+=v1ffi,respectively, andwithacornmon variance equaltoNo/2. Thedecisionruleisnowsimplytodecidethat51(t)wastransmitted ifthereceived signalpointassociated withtheobservation vectorxfallsinsideregion21>decidethat S2(t)wastransmitted ifthereceived signalpointfallsinsideregion22,andsoon.An erroneous decisionwillbemadeif,forexample, signalS4(t)istransmitted butthenoise w(t)issuchthatthereceivedsignalpointfallsoutsideregion24, Tocalculate theaverageprobability ofsymbolerror,wenotefromEquation (6.24) thatacoherent QPSKsystemisinfactequivalent totwocoherent binaryPSKsystems working inparallelandusingtwocarriersthatareinphasequadrature; thisismerelya statement ofthequadrature-carrier multiplexing property ofcoherentQPSK.Thein-phase channeloutputXlandthequadrature channeloutputX2(i.e.,thetwoelements ofthe observation vectorx)maybeviewedastheindividual outputsofthetwocoherent binary PSKsystems.Thus,according toEquations (6.29)and(6.30),thesetwobinaryPSKsys temsmaybecharacterized asfollows: l>ThesignalenergyperbitisEI2. g..ThenoisespectraldensityisNol2. Hence,usingEquation (6.20)fortheaverageprobability ofbiterrorofacoherent binary PSKsystem,wemaynowstatethattheaverageprobability ofbiterrorineachchannelof thecoherent QPSKsystemis 1(fEii)P'=:2erfc{N;; =~erfc(~)(6.31) Another important pointtonoteisthatthebiterrorsinthein-phase andquadrature channels ofthecoherent QPSKsystemarestatistically independent. Thein-phase channel makesadecisionononeofthetwobitsconstituting asymbol(dibit)oftheQPSKsignal, andthequadrature channeltakescareoftheotherbit.Accordingly, theaverageprobability ofacorrectdecision resulting fromthecombined actionofthetwochannels working together is (6.32) (6.33)358 CHAPTER 6IIIPASSBAND DATA'TRANSMISSION Theaverageprobability ofsymbolerrorforcoherent QPSKistherefore Pe=1 -Pc =erfc(fE)_1erfc2(fE) ~2iia 4 ~2iia Intheregionwhere(EI2Na)»1,wemayignorethequadratic termontheright-hand sideofEquation (6.33),soweapproximate theformula fortheaverageprobability of symbolerrorforcoherent QPSKas Pe=erfc(~) (6.34l TheformulaofEquation (6.34)mayalsobederivedinanotherinsightful way,using thesignal-space diagram ofFigure6.6.Sincethefourmessage pointsofthisdiagram are circularly symmetric withrespecttotheorigin,wemayapplyEquation (5.92),reproduced hereintheform foralli (6.35) Consider, forexample, message pointml(corresponding todibit10)chosenasthetraos mittedmessage point.Themessage pointsm2andm4(corresponding todibits00and11) aretheclosesttom"FromFigure6.6wereadilyfindthatm,isequidistant fromm,and m4inaEuclidean sense,asshownby d12=dt4=v'2E Assuming thatEINoislargeenoughtoignorethecontribution ofthemostdistantmessage pointm3(corresponding todibit01)relativetomt,wefindthattheuseofEquation (6.35) yieldsanapproximate expression forPethatisthesameasEquation (6.34).Notethatin mistaking eitherm2orm4formt>asinglebiterrorismade;ontheotherhand,inmistaking m3formt>twobiterrorsaremade.ForahighenoughEINa,thelikelihood ofbothbits ofasymbolbeinginerrorismuchlessthanasinglebit,whichisafurtherjustification forignoring m3incalculating Pewhenm,issent. InaQPSKsystem,wenotethatsincetherearetwobitspersymbol,thetransmitted signalenergypersymbolistwicethesignalenergypetbit,asshownby (6.361 Thusexpressing theaverageprobability ofsymbolerrorintermsoftheratioEblNo,we maywrite Pe=erfc(Hi)(6.37) WithGrayencoding usedfortheincoming symbols, wefindfromEquations (6.31) and(6.36)thatthebiterrorrateofQPSKisexactly BER=~erfc(Hi) (6.38) Wemaytherefore statethatacoherent QPSKsystemachievesthesameaverageprobabilitY ofbiterrorasacoherent binaryPSKsystemforthesamebitrateandthesameEblNo, butusesonlyhalfthechannelbandwidth. Statedinadifferent way,forthesameEblt!0 andtherefore thesameaverageprobability ofbiterror,acoherent QPSKsystem transI11l~ information attwicethebitrateofacoherent binaryPSKsystemforthesamechanne 6.3Coherent Plwse-Shift Keying 359 bandwidth. Foraprescribed performance, QPSKuseschannelbandwidth betterthanbi naryPSK,whichexplainsthepreferred useofQPSKoverbinaryPSKinpractice. Generation andDetectUm ofCoherent QPSKSignals Consider nextthegeneration anddetection ofQPSKsignals.Figure6.8ashowsa blockdiagram ofatypicalQPSKtransmitter. Theincoming binarydatasequence isfirst transformed intopolarformbyanonreturn-to-zero levelencoder. Thus,symbols1and0 arerepresented by+v'"E;;and-v'"E;;,respectively. Thisbinarywaveisnextdividedby meansofademultiplexer intotwoseparate binarywavesconsisting oftheodd-andeven numbered inputbits.Thesetwobinarywavesaredenotedbya,(t)anda2(t).Wenotethat inanysignaling interval, theamplitudes ofa,(t)anda2(t)equalSitandSa,respectively, depending ontheparticular dibitthatisbeingtransmitted. Thetwobinarywavesa,(t) anda2(t)areusedtomodulate apairofquadrature carriersororthonormal basisfunctions: <p,(t)equaltov2ftcos(27rfct) and<P2(t)equaltov2ftsin(27rfct). Theresultisapairof r---------i;;>o( x)---..., Binary data sequencePolarnonreturn to-zerolevel encoder + '----.:.--...;;..( x)-__ ...J <p,{t)~{2iisin(2'1rj,t) {a} Thteshold ~aQPSK signal s(t) Received signal x{t)Estimate af transmitted binary sequence Threshold=0 Quadrature channel (b) FIGURE6.8Blockdiagrams of(a)QPSKtransmitter and(b)coherent QPSKreceiver. 360 CHAPTER 6IIIPASSBAND DATATBANSMISSION binaryPSKsignals,whichmaybedetectedindependently duetotheorthogonality of</J(I) and<P2(t).Finally,thetwobinaryPSKsignalsareaddedtoproducethedesiredQPS\( signal. .TheQPSKreceiver co~sistsofapairofcorrelato~s withacommon inputandsupplied Withalocallygenerated patrofcoherent reference signals <PI(t)and<P2(t},asinFigur 6.8b.Thecorrelator outputs XIandX2,produced inresponse tothereceivedsignalX(I)e areeachcompared withathreshold ofzero.IfXl>0,adecision ismadeinfavorof symbol1forthein-phase channeloutput,butifXl<0,adecision ismadeinfavorof symbol0.Similarly, ifX2>0,adecisionismadeinfavorofsymbol1forthequadrature channeloutput,butifX2<0,adecisionismadeinfavorofsymbol0.Finally,thesetwo binarysequences atthein-phase andquadrature channeloutputsarecombined inamul. tiplexertoreproduce theoriginalbinarysequence atthetransmitter inputwiththemini. mumprobability ofsymbolerrorinanAWGNchannel. PowerSpectraofQPSKSignols Assumethatthebinarywaveatthemodulator inputisrandom, withsymbols1and°beingequallylikely,andwiththesymbolstransmitted duringadjacent timeslotsbeing statistically independent. Wemakethefollowing observations pertaining tothein·phase andquadrature components ofaQPSKsignal: (6.39) otherwise1.Depending onthedibitsentduringthesignaling interval-TboStoSTb,thein·phase component equals+g(t)or-g(t),andsimilarly forthequadrature component. The g(t)denotesthesymbolshapingfunction, definedby g(t)={ft, 0, Hence,thein-phase andquadrature components haveacommon powerspectral density,namely,Esinc2(Tf). ~ ~z0.1 a 0.5 0.75 Normalized frequencY.fTb1.0 FIGURE6.9PowerspectraofQPSKandMSKsignals. (6.40)6.3Coherent Phnse-Shift Keying 361 2.Thein-phaseandquadrature components arestatistically independent. Accordingly, thebaseband powerspectraldensityoftheQPSKsignalequalsthesumoftheindi vidualpowerspectraldensities ofthein-phase andquadrature components, sowe maywrite SB(f) 2Esinc2(Tf) =4Ebsinc2(2Tbf) Figure6.9plotsSB(f),normalized withrespectto4Eb,versusthenormalized fre quencyfTb•Thisfigurealsoincludes aplotofthebaseband powerspectraldensityofa certainformofbinaryFSKcalledminimum shiftkeying,theevaluation ofwhichispre sentedinSection6.5.Comparison ofthesetwospectraisdeferredtothatsection. !illOFFSETQPSK ThesignalspacediagramofFigure6.10aembodies allthepossiblephasetransitions that canariseinthegeneration ofaQPSKsignal.Morespecifically, examining theQPSK waveform illustrated inFigure6.7forExample 6.1,wemaymakethefollowing observations: 1.Thecarrierphasechangesby:t180degreeswhenever boththein-phase andquad raturecomponents oftheQPSKsignalchangessign.Anexample ofthissituation is illustrated inFigure6.7whentheinputbinarysequence switches fromdibit01to dibit10. 2.Thecarrierphasechanges by:t90degreeswhenever thein-phase orquadrature component changessign.Anexample ofthissecondsituation isillustrated inFigure 6.7whentheinputbinarysequence switches fromdibit101:0dibit00,duringwhich thein-phase component changes sign,whereas thequadrature component is unchanged. 3.Thecarrierphaseisunchanged whenneitherthein-phase component northequad raturecomponent changessign.Thislastsituation isillustrated inFigure6.7when dibit10istransmitted intwosuccessive symbolintervals. Situation 1and,toamuchlesserextent,situation 2canbeofaparticular concernwhen theQPSKsignalisfilteredduringthecourseoftransmission, priortodetection. Specifi cally,the180-and90-degree shiftsincarrierphasecanresultinchangesinthecarrier amplitude,(i.e., envelope oftheQPSKsignal),therebycausingadditional symbolerrorson detection. (0)t/J, r'----"'1 I I I !t/J: I0 I I I f--~.-oE--- (bl FIGURE6.10Possible pathsforswitching between themessage pointsin(a)QPSKand (b)offsetQPSK. 362 CHAPTER 6"PASSBAND DATATRANSMISSION Theextentofamplitude fluctuations exhibited byQPSKsignalsmaybereducedb usingoffsetQPSK.2InthisvariantofQPSK,thebitstreamresponsible forgenerating thY quadrature component isdelayed(i.e.,offset)byhalfasymbolintervalwithrespecttothe bitstreamresponsible forgenerating thein-phase component. Specifically, thetwoba/ functions ofoffsetQPSKaredefinedby IS <PI(t)=ftCOS(2'11'fct), 0oStoST (6.41) g T3T <P2(t)={fsin(2'11'fct), '2oStoS"2 (6.42) Accordingly, unlikeQPSK,thephasetransitions likelytooccurinoffsetQPSKareconfined to±90degrees,asindicated inthesignalspacediagram ofFigure6.10b.However, :'::90 degreephasetransitions inoffsetQPSKoccurtwiceasfrequently butwithhalftheintensity encountered inQPSK.Since,inaddition to±90-degree phasetransitions, ±ISO-degree phasetransitions alsooccurinQPSK,wefindthatamplitude fluctuations inoffsetQPSK duetofilteringhaveasmalleramplitude thaninthecaseofQPSK. DespitethedelayTI2appliedtothebasisfunction <P2(t)inEquation (6.42)compared tothatinEquation (6.26),theoffsetQPSKhasexactlythesameprobability ofsymbol errorinanAWGNchannelasQPSK.Theequivalence innoiseperformance betweenthese phase-shift keyingschemesassumestheuseofcoherent detection. Thereasonfortheequiv. alenceisthatthestatistical independence ofthein-phase andquadrature components appliestobothQPSKandoffsetQPSK.Wemaytherefore saythattheerrorprobability inthein-phase orquadrature channelofacoherent offsetQPSKreceiveds stillequalto (1/2)erfc(YEI2N o).HencetheformulaofEquation (6.34)appliesequallywelltotheoffset QPSK. !lil'IT/4-SHIFTED QPSK Anordinary QPSKsignalmayresideineitheroneofthetwocommonly usedconstellations showninFigures6.11aand6.11b,whichareshiftedby'11'/4radianswithrespectto eacb other.InanothervariantofQPSKknownas'I1'14-shifted QpSK,3thecarrierphaseused forthetransmission ofsuccessive symbols (i.e.,dibits)isalternately pickedfromoneof thetwoQPSKconstellations inFigure6.11andthentheother.Itfollowstherefore thata 'I1'/4-shifted QPSKsignalmayresideinanyoneofeightpossiblephasestates,asindicated (a) (b) FIGVRE6.11Twocommonly usedsignalconstellations forQPSK;thearrowsindicate thepaths alongwhichtheQPSKmodulator canchangeitsstate. (6.43)6.3Colu.rent Phase-Shift Keying 363 FIGURE6.12Eightpossible phasestatesforthe1'/4-shifted QPSKmodulator. inFigure6.12.Thefourdashedlinesemanating fromeachpossible message pointin Figure6.12definethephasetransitions thatarefeasiblein1T/4-shifted QPSK. Table6.2summarizes apossiblesetofrelationships between thephasetransitions inthisnewdigitalmodulation schemeandtheincoming Gray-encoded dibits.Forexample, ifthemodulator isinoneofthephasestatesportrayed inFigure6.llb,thenonreceiving thedibit00itshiftsintoaphasestateportrayed inFigure6.llabyrotatingthrough 1T/4 radiansinacounterclockwise direction. Attractive featuresofthe1T/4-shifted QPSKschemeincludethefollowing: ~Thephasetransitions fromonesymboltothenextarerestricted to±1T/4and±31T/4 radians,whichistobecontrasted withthe±1T/2and±1Tphase transitions inQPSK. Consequently, envelope variations of1T14-shifted QPSKsignalsduetofilteringare significantly reduced, compared tothoseinQPSK. ~UnlikeoffsetQPSKsignals,1T/4-shifted QPSKsignalscanbenoncoherently detected, therebyconsiderably simplifying thereceiverdesign.Moreover, likeQPSKsignals, 1T/4-shifted QPSKcanbedifferently encoded, inwhichcaseweshouldreallyspeak of1T14-shifted DQPSK. Thegeneration of1T14-shifted DQPSKsymbols, represented bythesymbolpair(I,Q), isdescribed bythefollowing pairofrelationships (seeProblem 6.13): h=COS(Ok-l+/!l.Ok) =cosOk Qk=sin(Ok-l+/!l.0k) =sinOk TABLE6.2Correspondence betweeninput dibitandphasechangefor'fT/4-shlfted DQPSK(6.44) Gray-Encoded InputDibit 00 01 11 10PhaseChange, .dO(radians) 1'/4 31'/4 -31'/4 -1'/4 {6.451364 CHAPTER 6illPASSBAND DATATBANSMISSION ITABLE6.37'l14-shi.fted DQPSK resultsforExample 6.2 Phase8k-1 PhaseChangell8kTransmitted Phase8, Stepk(radians) InputDibit (radians) (radians) 1 1T14 00 1T14 1T12 2 1T12 10 -1T14 1T14 3 1T14 10 -1T14 0 4 0 01 31T14 31T14 whereIh-listheabsolute phaseangleofsymbolk1,and!llhisthedifferentially encoded phasechangedefinedinaccordance withTable6.2. ~ExAMPLE 6.2 Continuing withtheinputbinarysequence ofExample 6.1,namely,01101000, supposethat thephaseangle60=1T14intheconstellation ofFigure6.11bisassigoedastheinitialphase stateofthe1T14-shifted DQPSKmodulator. Then,arranging theinputbinarysequence asa sequence ofdibitsandfollowing theconvention ofTable6.2,wegettheresultspresented in Table6.3fortheexampleathand. ... Detection of7r/4~Shifted DQPSK Signals Havingfamiliarized ourselves withthegeneration of1T/4-shifted DQPSK signals,we goontoconsider theirdifferential detection. Giventhenoisychannel outputx(t),the receiverfirstcomputes theprojections ofx{t)ontothebasisfunctions <Pl(t)and <P2{t).The resulting outputs, denoted byIandQ,respectively, areappliedtoadifferential detector thatconsistsofthefollowing components, asindicated inFigure6.13: II>Arctangent computer forextracting thephaseangleI}ofthechanneloutput(received signal). ~Phase-difference computer fordetermining thechangeinthephaseI}occurring over onesymbolinterval. ~Modulo-21T correction logicforcorrecting errorsduetothepossibility ofphaseangles wrapping aroundtherealaxis. Elaborating furtheronthelatterpoint,let!ll}kdenotethecomputed phasedifference be tween I}kandI}k-lrepresenting thephaseanglesofthechanneloutputforsymbolskand k-1,respectively. Thenthemodulo-21T correction logicoperates asfollows: IF!ll}k<-180degreesTHEN !ll}k=!l(Jk+360degrees IF!l(Jk>180degreesTHEN !l(Jk=!l(Jk-360degrees In-phase component, I Quadrature component, QArctan(QIl) computerPhase-difference computer Modulo-21T correction logicEstimateat transmitted data sequence FIGURE6.13Blockdiagramofthe1T14-shifted DQPSKdetector. 6.3Coherent Phase-Shift Keying 365 Imaginary Symbolk -----t-:lk102:=------- Real Symbolk-1 FIGURE6.14Illustrating thepossibility ofphaseangleswrapping aroundthepositiverealaxis. Toillustrate theneedforthisphasecorrection, consider thesituation depicted inFigure 6.14,where0k-l=350degreesandOk=60degrees,bothphaseanglesbeingmeasured inacounterclockwise direction. Fromthisfigurewereadilyseethatthephasechange flOk, measured inacounterclockwise direction, is70degrees.However, withoutcorrection the phasechange flOkiscomputed as60degrees-350degrees =-290degrees.Applying thefitstlineofEquation (6.45),themodulo-2'lTcorrection logiccompensates forthewrap aroundthepositiverealaxis,yieldingthecorrected result flOk=-290degrees+360degrees=70degrees Thetangenttypedifferential detectorofFigure6.13forthedemodulation of'IT/4 shiftedDQPSKsignalsisrelatively simpletoimplement. Itoffersasatisfactory perfor manceinaRayleigh fadingchannelasinastaticmultipath environment. However, when themultipath environment istimevaryingasexperienced inacommercial digitalwireless communication system,computer simulation resultsappeartoshowthatthereceiverper formance degrades veryrapidly.4 1mM-ARyPSK (6.46) i=1,2,..., MQPSKisaspecialcaseofM-aryPSK,wherethephaseofthecarriertakesononeofM possiblevalues,namely, Oi=2(i-l)'lT/M,wherei=1,2,...,M.Accordingly, during eachsignaling intervalofduration T,oneoftheMpossiblesignals f2E( 2'lT ) s;(t)={rcos2'lTfct+M(i-1), issent,whereEisthesignalenergypersymbol.Thecarrierfrequency fc=n)Tforsome fixedinteger nco EachSi(t)maybeexpanded intermsofthesametwobasisfunctions <PI(t)and<P2(t) definedinEquations (6.25)and(6.26),respectively. Thesignalconstellation ofM-aryPSK istherefore two-dimensional. TheMmessagepointsareequallyspacedonacircleofradius vBandcenterattheorigin,asillustrated inFigure6.15a,forthecaseofoctaphase shift-keying (i.e.,M=8). FromFigllIe6.15awenotethatthesignal-space diagramiscircularly symmetric. We maytherefore applyEquation (5.92),basedontheunionbound,todevelopanapproxi mateformulafortheaverageprobability ofsymbolerrorforM-aryPSK.Supposethatthe 366 CHAPTER 6"PASSIlM'D DATATRANSMISSION (a) ~, (b) FIGURE6.15(a)Signal-space diagram foroctaphase-shift keying(i.e.,M=8).Thedecision boundaries areshownasdashedlines.(b)Signal-space diagramillustrating theapplication ofthe unionboundforoctaphase-shift keying. 6.3Coherent Phase-Shift Keying 367 transmitted signalcorresponds tothemessage pointm"whosecoordinates alongthe <prand<P2-axesare+YEand0,respectively. SupposethattheratioEINoislargeenough toconsiderthenearesttwomessagepoints,oneoneithersideofm"aspotential candidates forbeingmistaken formiduetochannelnoise.ThisisilJusrrated inFigure6.15bforthe caseofM=8.TheEuclidean distanceofeachofthesetwopointsfrommiis(forM=8) d12=dlB~2YEsin(~) Hence,theuseofEquation (5.92).ofChapter5yieldstheaverageprobability ofsymbol errorforcoherent M-aryPSKas (6,47) whereitisassumed thatM""4.Theapproximation becomes extremely tight,forfixed M,asEINoisincreased. ForM=4,Equation (6,47)reducestothesameformgivenin Equation (6.34)forQPSK. . PowerSpectraofM-aryPSKSignals Thesymbolduration ofM-aryPSKisdefinedby T=Tblog2M (6,48) whereTbisthebitduration. Proceeding inamannersimilartothatdescribed foraQPSK signal,wemayshowthatthebaseband powerspectraldensityofanM-aryPSKsignalis givenby SB(f)=2Esinc2(Tf) =2Eblo~Msinc2(Tdlog2M)(6,49) InFigure6.16,weshowthenormalized powerspectraldensitySB(f)12E bplottedversus thenormalized frequency fTbforthreedifferent valuesofM,namely,M=2,4,8. 11.0r---- ~ 1z0.5 Normalized frequency1.0 FIGURE6.16PowerspectraofM-aryPSKsignalsforM=2,4,8. 368 CHAPTER 6"PASSBAND DATATRANSMISSION TABLE6.4Bandwidth effu:iency of M-aryPSKsignals M 2 4 8 163264 p(bits/slHz) 0.511.522.53 I'llBANDWIDTH EFFICIENCY OFM-ARYPSKSIGNALS ThepowerspectraofM-aryPSKsignalspossessamainlobebounded bywell-defined spectralnulls(i.e.,frequencies atwhichthepowerspectraldensityiszero).Accordingly thespectralwidthofthemainlobeprovides asimpleandpopularmeasurefortheband: widthofM-aryPSKsignals.Thisdefinition isreferredto<tSthenull-to-null bandwidth Withthenull-to-null bandwidth encompassing themainlobeofthepowerspectrum of~ M-arysignal,wefindthatitcontains mostofthesignalpower.Thisisreadilyseenby lookingatthepowerspectralplotsofFigure6.16. Forthepassband basisfunctions definedinEquations (6.25)and(6.26),thechannel bandwidth required topassM-aryPSKsignals(moreprecisely, themainspectrallobeof M-arysignals)isgivenby B=3.T(6.50) (6.51)whereTisthesymbolduration. ButthesymboldurationTisrelatedtothebitduration TbbyEquation (6.48).Moreover, thebitrateRb=I/Tb.Hence,wemayredefinetbe channelbandwidth ofEquation (6.50)intermsofthebitrateRbas B=2Rb logzM Basedonthisformula, thebandwidth efficiency ofM-aryPSKsignalsisgivenby RbP=J3 logzM 2(6.52) Table6.4givesthevaluesofpcalculated fromEquation (6.52)forvaryingM. InlightofEquation (6.47)andTable6.4,wemakethefollowing observation intbe contextofM-aryPSK:Asthenumberofstates,M,isincreased, thebandwidth efficiency isimproved attheexpenseoferrorperformance. Toensurethatthereisnodegradation inerrorperformance, wehavetoincreaseEblNotocompensate fortheincreaseinM. 6.4HybridAmplitude/Phase Modulation Schem.es InanM-aryPSKsystem,thein-phase andquadrature components ofthemodulated si~l areinterrelated in suchawaythattheenvelope isconstrained toremainconstant. ~s constraint manifests itselfinacircularconstellation forthemessagepoints.However,if 6.4HybridA>nplituJeIPJw.se Modulation Schemes 369 thisconstraint isremoved, andthein-phase andquadrature components aretherebyper mittedtobeindependent, wegetanewmodulation schemecalledM-aryquadrature am plitudemodulation (QAM).ThislatterJ;llodulation schemeishybridinnatureinthatthe carrierexperiences amplitude aswellasphasemodulation. Thepassband basisfunctions inM-aryQAMmaynotbeperiodic foranarbittary choiceofthecarrierfrequency fcwithrespecttothesymbolratelIT.Ordinarily, this aperiodicity isofnorealconcern. Byreformulating theexpression forthetransmitted signalinacertainway,itispossibletoeliminate thetimevariation ofthebasisfunctions onsuccessive symboltransmissions, andthere1:>Y simplifyimplementation ofthetransmit ter.Inparticular, thetransmitter ismadetoappear"carrierless," whilefullyretaining the essenceofthehybridized amplitude andphasemodulation process.Thisisindeedtheidea behindthecarrierless amplitude/phase modulation (CAP). Despitethedifferences betweenQAMandCAPintheirimplementation details,they haveexactlythesamesignalconstellations. Accordingly, theyarefundamentally equivalent inperformance foraprescribed receivercomplexity. InwhatfollowswefirstdiscussQAM andthenCAP. IIM-ARY QUADRATURE AMPLITUDE MODUlATION InChapters 4and5,westudiedM-arypulseamplitude modulation (PAM),whichisone dimensional. M-aryQAMisatwo-dimensional generalization ofM-aryPAMinthatits formulation involves twoorthogonal passband basisfunctions, asshownby <PI{t)=ftCOS{27Tfct), 0:5t:5T (6.53) <P2(t)=ftsin(27Tfct), 0:5t:5T (6.54) LettheithmessagepointSiinthe(<Ph<P2)planebedenotedby(aidmiJ2,bidmin/2),where dmmistheminimum distance between anytwomessagepointsinthe constellation, aiand biareintegers, andi1,2,...,M.Let(dmj2)=VE.;,whereEoistheenergyofthe signalwiththe'lowest amplitude. Thetransmitted M-aryQAMsignalforsymbolk,say, isthendefinedby ~Eo ~Eo. 0:5t:5TSk(t)-Ta"COS(27Tfct) --Tbksm{27Tfct), k (6.55) =0,±1,±2,... ThesignalSk(t)consistsoftwophase-quadrature carrierswitheachonebeingmodulated byasetofdiscreteamplitudes, hencethenamequadrature amplitude modulation. Depending onthenumberofpossible symbols M,wemaydistinguish twodistinct QAMconstellations: squareconstellations forwhichthenumberofbitspersymbolis even,andcrossconstellations forwhichthenumberofbitspersymbolisodd.Thesetwo casesareconsidered inthesequelinthatorder. QAMSquareConstellations Withanevennumberofbitspersymbol,wemaywrite L=VM (6.56) whereLisapositiveinteger.Underthiscondition, anM-aryQAMsquareconstellation canalwaysbeviewedastheCartesian productofaone-dimensional L-aryPAMconstel- 370 CHAPTER 6illPASSBM"ID DATATUANSMISSION lationwithitself.Bydefinition, theCartesian product oftwosetsofcoordinates (rept• sentingapairofone-dimensional constellations) ismadeupofthesetofallpossibf orderedpairsofcoordinates withthefirstcoordinate ineachsuchpairtakenfromthefirs~ setinvolved intheproduct andthesecondcoordinate takenfromthesecondsetinthe product. InthecaseofaQAMsquareconstellation, theorderedpairsofcoordinates natutally formasquarematrix,asshownby {ai'hi}= [(-L+1,L-1) (-L+1,L-3) (-L+1,-L+1)(-L+3,L1) (-L+3,L-3) (-L+3,-L+1)(L-1,L-1)] (L-1,L-3) (L-1,-L+1) (6..17) ~ExAMPLE 6.3 Consider a16-QAM whosesignalconstellation isdepictedinFigure6.17a.Theencoding of themessagepointsshowninthisfigureisasfollows: I>Twoofthefourbits,namely,theleft-most twobits,specifythequadrant inthe(<Ph<P2) planeinwhichamessagepointlies.Thus,startingfromthefirstquadrant andpro ceedingcounterclockwise, thefourquadrants arerepresented bythedibits11,10,00, and01. I>Theremaining twohitsareusedtorepresent oneofthefourpossihlesymbolslying withineachquadrant ofthe(<Ph<pz)-plane. Notethattheencoding ofthefourquadrants andalsotheencoding ofthesymbolsineach quadrant followtheGraycodingrule. <P2 ·.3d/2•· 1011 1001 1110 1111 • •d/2·•I 1010 1000 1100 1101 I I <PI.•1•• ¢l -3d12 -d/201dl2 3d12I••-dl2··1 0001 0000 0100 0110I (b) ·.-3d12· · 0011 0010 0101 0111 (a) FIGURE 6.17(a)Signal-space diagramofM-atyQAMforM=16;themessagepointsineach quadrant areidentified withGray-encoded quadbits. (b)Signal-space diagramofthecorrespond ing4-PAMsignal. 6.4HybridAtnplitudelPhase Modulation Sc"hetnes 371 Fortheexampleathand,wehaveL=4.Thusthesquareconstellation ofFigure6.17a istheCartesian productofthe4-PAMconstellation showninFigure6.17bwithitself.More over,thematrixofEquation (6.57)hasthevalue [(-3,3) _(-3,1) {ai'bi}-(-3,~1) (-3,-3)(-1,3) (-1,1) (-1,-1) (-1,-3)(1,3) (1,1) (1,-1) (1,-3)(3,3)](3,1) (3,-1) (3,-3) Tocalculate theprobability ofsymbolerrorforM-aryQAM,weexploittheproperty thataQAMsquareconstellation canbefactored intotheproductofthecorresponding PAMconstellation withits.elf.Wemaythusproceedasfollows: 1.Theprobability ofcorrectdetection forM-aryQAMmaybewrittenas (6.58) whereP;istheprobability ofsymbolerrorforthecorresponding L-aryPAMwith L=\IM. 2.Theprobability ofsymbolerrorP;isdefinedby (6.59) (NotethatL=\1MintheM-aryQAMcorresponds toMinrheM-aryPAMcon sideredinProblem 4.27.) 3.Theprobability ofsymbolerrorforM-aryQAMisgivenby Pe=1-Pc =1(1-P;)2 ""2P;(6.60) whereitisassumed thatP;issmallenoughcompared tounitytojustifyignoring the quadratic term. Hence,usingEquations (6.58)and(6.59)inEquation (6.60),wefindthattheprobability ofsymbolerrorforM-aryQAMisapproximately givenby (6.61) Thetransmitted energyinM-aryQAMisvariable inrhatitsinstantaneous value dependsontheparticular symboltransmitted. Itistherefore morelogicaltoexpressPein termsoftheaveragevalueofthetransmitted energyratherthanEo.Assuming thattheL amplitude levelsofthein-phase orquadrature component areequallylikely,wehave (6.62) 372 CHAPTER 6Ii!PASSBAND DATATRANSMISSION wherethemultiplying factorof2outsidethesquarebrackets accounts fortheequalco tributions madebythein-phase andquadrature components. Thelimitsofthesummati n andthemultiplying factorof2insidethesquarebrackets takeaccountofthesymme:n natureofthepertinent amplitude levelsaroundzero.Summing theseriesinEquatio~ (6.62),weget E=2(L2 -I)Ea av 3 2(M-I)Ea 3 Accordingly, wemayrewriteEquation (6.61)intermsofEavas Pe=2(1-~)erfc(2(M3~a;)NJ(6.63) (6.64) whichisthedesiredresult. ThecaseofM=4isofspecialinterest.Thesignalconstellation forthisvalueofM isthesameasthatforQPSK.Indeed,puttingM=4inEquation (6.64)andnotingthat forthisspecialcaseE.vequalsE,whereEistheenergypersymbol, wefindthatthe resulting formulafortheprobabilityofsymbolerrorbecomes identical tothatinEquation (6.34),andsoitshould. QAMCrossConstellation TogenerateanM-ary QA1v1signalwithanoddnumberofbitspersymbol,werequire theuseofacrossconstellation. Asillustrated inFigure6.18,wemayconstruct sucha signalconstellation withnbitspersymbolbyproceeding asfollows: I>-StartwithaQA1v1squareconstellation withn-lbitspersymbol. lr>ExtendeachsideoftheQA1v1squareconstellation byadding2n-3symbols. il>Ignorethecornersintheextension. Theinnersquarerepresents 2n-1symbols. Thefoursideextensions add4X2n3=2n-1 symbols. Thetotalnumberofsymbols inthecrossconstellation istherefore 2,,-1+2n-" whichequals2"andtherefore represents nbitspersymbolasdesired. UnlikeQAMsquareconstellation, itisnotpossible toexpressaQA1v1crosscon stellation astheproductofa PA1v1constellation withitself.Theabsenceofsuchafactor- FIGURE 6.18Illustrating howasquareQAMconstellation canbeexpanded toformaQAM cross-constellation. (6.65)6.4HybridAmplitudelPhase Modulation Schemes 373 izationcomplicates thedetermination oftheprobability ofsymbolerrorPeincurred inthe useofM-aryQAMcharacterized byacrossconstellation. Wetherefore simplystatethe formulaforPewithoutproof,asshownhere Pe=2(1 -vk)erfc(fti)forhighEolNo whichagreeswiththeformula ofEquation (6.61)forasquareconstellation, exceptfor theinclusion ofanextra0.5bitperdimension intheconstellation.5Notealsothatitis notpossibletoperfectly GraycodeaQAMcrossconstellation. IIICARRIERLESS A!\lPLlTUDEIPHASE MODUlATION Thepassband basisfunctions ofEquations (6.53)and(6.54)assumetheuseofarectan gularpulseforthepulse-shaping function. Forreasonsthatwillbecomeapparent, we redefinethetransmitted M-aryQAMsignalofEquation (6.55)intermsofageneralpulse shapingfunction g(t)as °~t~T k=0,±1,±2,... (6.66) Itisassumed thatcarrierfrequencythasanarbitrary valuewithrespecttothesymbol ratelIT.OnthebasisofEquation (6.66),wemayexpressthetransmitted M-aryQAM signals(t)foraninfinitesuccession ofsymbolsas s(t)=2:sdt) k=-ec =2:[akg(t-kT)COS(21Tfct) -bkg(t-kT)sin(21Ttt)] k~-~(6.67) Thisequation showsthatforanarbitraryt,thepassband functions g(t-kT)COS(21Ttt) andg(t-kT)sin(21Ttt) areaperiodic inthattheyvaryfromonesymboltoanother. Howcanweeliminate thetimevariations ofthesepassband basisfunctions from symboltosymbol?Toanswerthisquestion, wefinditconvenient tochangeourformalism fromrealtocomplex notation. Specifically, werewriteEquation (6.67)intheequivalent form s(t)=ReL~oo (ak+jbk)g(t-kT)eXP(j21Ttt)} =ReL~oo Akg(t-kT)eXP(j21Tfct)} whereAkisacomplex numberdefinedby Ak=ak+jbk(6.68) (6.69) andRe[·}denotestherealpartofthecomplex quantityenclosed insidethebraces.Clearly, Equation (6.68)isunchanged bymultiplying thesummand inthisequation byunityex- (6.70) (6.73)374 CHAPTER 6IIIPASSBAND DATATRANSMISSION pressedastheproductofthecomplex exponential exp(-;2'iTfckT)anditscomplex jugateexp(i2'iTfckT). WemaythusrewriteEquation (6.68)inthenewform COn· s(t)=ReL~~ Akg(t-kT)exp(i2'iTfct) exp(-;2'iTfckT) eXP(i2'iTfckT)} =ReL~~ Akexp(i2'iTfckT)g(t -kT)exp(i2'iTfc(t -kT))} Define Ak=Akexp(i2'iTfckT) (6.71) g+(t)=g(t)exp(i2'iTfct) (6.72) ThescalarAkissimplyarotatedversionofthecomplex representation ofthecoordinate ofthekthtransmitted symbolinthe(<Ph<P2)-plane. Beforepresenting aninterpretatio~ ofthecomplex-valued signalg+(t),weassumethatthepulse-shaping function g(t)isa low-pass signalwhosehighestfrequency component issmallerthanthecarrierfrequency fc.Thenfollowing thematerialpresented inAppendix 2,werecognize g+(t)astheanalytic signal,orpre-envelope, representation oftheband-pass signalg(t)cos(2'iTfct).Tobemore specific,weexpandg+(t)as g+(t)=g(t)cos(2'iTfct)+;g(t)sin(2'iTfct) =p(t)+;p(t) wherep(t)andP(t)aredefinedby andp(t)=g(t)cos(2'iTJ:t) p(t)=g(t)sin(2'iTfct)(6.74) (6.75) (6.76)Wemaythensaythatthequadrature (imaginary) component p(t)oftheanalytic signal g+(t)istheHilberttransform ofthein-phase (real)component p(t).Notethatwhereasthe pulse-shaping function g(t)isabaseband function, thein-phase andquadrature compo nentsofthecorresponding analyticsignalg+(t)arebothpassband functions. Henceforth, g(t)isreferred toasthebaseband pulse,andp(t)andp(t)arereferredtoaspassband In phaseandpassband quadrature pulses,respectively. Withthedefinitions ofAkandg+(t)athand,wearereadytofinallyredefinethe transmitted signalofEquation (6.70)simplyas s(t)=ReL~~ Akg+(tkT)} Threeimportant observations arenoteworthy fromthisnewformulation oftherransmit tedsignal: Ii>Thetransmitted signals(t)appearstobecarrierless. I>Sincetheformulation ofs(t)inEquation (6.76)isindeedequivalent tothatofthe M-aryQAMsignalpresented inEquation (6.67),thenewformulation ofs(t)fully retainsthehybridized amplitude andphasemodulation characterizing theorigmal M-aryQAMsignaL ,.Thetransmitted signals(t)represents asymbol-time-invariant realization ofthishy bridmodulation process. 6.4HybridAmplitudelPhase Modulation Schemes 375 Foraprescribed carrierfrequencyfc,thesequence ofrotations described inEquation (6.71)isknown.Hence,thereceiverneedonlydetectJibinwhichcase wemaycompute thecorresponding valueofAkbyapplying thereverserotations asdescribed here: k=0,±1,±2,... (6.77)Inpractice, however, therotations areignoredbecausetheydonothaveanybearingon operation orperformance ofthehybridmodulation system;application oftherotations isinfactnecessary onlywhenitsequivalence toQAMisanissueofinterest.Accordingly, wemayignoreEquation (6.71)andredefinethetransmitted signalofEquation (6.76) simplyas s(t)=ReL~ro A~+(t-kT)} =ReL~ro (ak+jbk)(p(t kT)+jp(t-kT))} =L[akP(t-kT)-bkfi(t-kT)] k~-ro where,asmentioned previously, p(t)istheHilberttransform ofp(t).Forobviousreasons, thetransmitted signalofEquation (6.77)isreferredtoascarrierlessamplitude/phase mod ulation(CAP).6 Properties oftlu?Passband ,,.-phase andQuadrature Pulses Fromthedefinitions ofthepassband in-phase andquadrature pulsesgiveninEqua tions(6.74)and(6.75),wededucethefollowing properties: Property 1 Thepassband in-phase pulsep(t)andquadrature pulsep(t)areevenandoddfunctions of timet,respectively, giventhatthebaseband pulseg(t)isanevenfunctionoftimet. Thisproperty followsdirectlyfromEquations (6.74)and(6.75). Property 2 Thepassband pulsesp(t)andp(t)formanorthogonal setovertheentireinterval(-00,00) asshownby froP(t)p(t)dt=0 (6.78) Thereasonfornotrestricting theintegration intervalinEquation (6.78)toasymbolperiod Tisthatthepulsesp(t)andp(t)arebandwidth-efficient forhigh-performing CAPsystems. ToproveProperty 2,wetransform Equation (6.78)intothefrequency domainbyusing theFouriertransform towrite f~P(t)p(t)dt=r:P(f)P'(f) df (6.79) wherep(t)~P(f)andp(t)~p(f).TheasteriskinP'(f)denotescomplex conjugation. (Thefrequency function p(f)shouldnotbeviewedastheHilberttransform ofP(f); p(f)=-jsgn(f)P(f)376 CHAPTER 6.,PASS)lA,'1D DATATRANSMISSION rather,following theusualterminology inFourieranalysis, p(f)issimplytheFou. transform ofp(t).)TheFouriertransform p(f)isrelatedtotheFouriertransform p(f)t~er (seeAppendix 2) y (6.80) wheresgn(f)isthesignumfunction. TheFouriertransforms P(f)andp(f)havetheSam magnitude spectrum, buttheirphasespectradifferby+90degreesfornegativefrequenc' e and-90degreesforpositivefrequencies. Wemaytherefore rewriteEquation (6.79)~es frop(t)P(t)dt=jfroP(f)P*(f)sgn(f)df =jfroIP(f)12sgn(f)df(6.81) Recognizing thatthemagnitude responseIP(f)1isanevenfunction offrequencyfand thesignumfunction sgn(f)isanoddfunction offrequencyf,theintegralofEquation (6.81)iszero,provingProperty 2. Next,letthepassband pulsesp(t)andp(t)bepassedthroughalineartime-invariant channelofimpulseresponse h(t),yieldingthefollowing passband outputs: andu(t)=p(t)*h(t) u(t)=p(t)*h(t)(6.821 (6.83) wherethesymbol*denotesconvolution. Wemaythenformulate thethirdproperty. Property 3 Thepassband pulsesu(t)andu(t),definedinEquations (6.82)and(6.83),formaHilbert transform pairandaretherefore orthogonal overtheentireinterval(-00,00)foranyh(t). Itisthisimportant property thatmakesitpossible fortheCAPreceivertoseparate the transmitted realandimaginary symbols, akandbk>giventhechanneloutput.Toprove Property 3,weagainusetheFouriertransform (inamannersimilartoEquations (6.79) and(6.81))towrite frou(t)u(t)dt=froU(f)U*(f) df =fro(P(f)H(f))(-j sgn(f)P(f)H(f))*df =frojIP(f)12IH(f)12sgn(f)df =0 ~EXAMPLE 6.4Bandwidth- Efficient Spectral Shaping TheCAPsignalofEquation (6.77)usesmultilevel encoding viathecomplex scalarAlfor spectralefficiency. Wemayfurtherimprovethebandwidth efficiency ofCAPbyusingaspec trallyefficientformulation ofthebaseband pulseg(t).Fortheselection ofg(t),wemaydraw upontheraised-cosine familyofpulse-shaping functions discussedinChapter4. 6.4HybndAmplitude/Phase Modulation Schemes 377 3 2 -2],-----~---cc--~--7T",.,-----~--~--~ 0.8 0.6 ~0.4 ~0.2 «0 -0.2 -0.4L-__ ---'--_~_-'--- __-'-- _'_____ ___'___ _____.J -3 FIGURE6.19Thebaseband pulseg(t)forrollofffactora=0.2. (6.84)Consider, forexample, abaseband raised-cosine shapingfilterwitharollofffactor a=0.2(i.e.,excessbandwidth of20percent). Theformulaforthebaseband pulseg(t)is' .cos(?Tat) g(t)=(smc(t))1 _4tl-f- wheretimetisnormalized withrespecttothesymboldurationT.Henceputtinga=0.2in Equation (6.84),wegettheplotofFigure6.19.UsingEquations (6.74)and(6.75),wemay compute thepassband in-phase pulsepit)andquadrature pulsePit)plottedinFignres6.20a and6.20bforanormalized carrierfrequency fcT=0.5(1+a)~0.6.Thewaveforms of Figure6.20showthatpit)andPit)areevenandoddfunction oftimet,respectively, in accordance withProperty 1andtheyareorthogonal overtheinterval(-00,00)inaccordance withProperty 2. ... 1 0.8 m0.6 "00.4"c.0.2E«0 -0.2 -0.4-3 -2 -1 0 2 (al 0.5m"0 "C.0 E« -0.5 -1-3 -2 -1 0 3 Normalized time,tIT (bl FIGURE 6.20(a)In-phase pulsepit),and(b)quadrature pulsepit)forrollofffactora=0.2. 'Equation (6.84)isobtained byusingthenormalized timetinplaceof2.WtinEq.(4.62.)ofChapter4.Recall [hattheNyquistbandwidth Wisequalto0.5timesthesymbol(bit)rate.Notealsothatthepit)inChapter4is nottobeconfused withthepit)inthischapter. 378 CHAPTER 6PASSBAND DATATRA1'lSMISSION Input bit streamTransmitted signal sit) FIGURE6.21Blockdiagram ofCAPtransmitter. BasicStructureoftheCAPSystem Figure6.21showsafunctional blockdiagram oftheCAPtransmitter, whichbuilds onthematerial embodied inEquations (6.74), (6.75), and(6.77).Thetransmitter consists ofamultilevel encoderandapairofpassband filters.Themultilevel encoderpartItions theincoming serialdatastreamintosuccessive blocksofnbitseach;theseblocksare,in turn,mappedintomultilevel symbols akandbk,wherekreferstothekthsymbolperiod. Thepassband in-phase andquadrature filtersprocessthesymbolstreams{adand(bk!in parallel,respectively. Theimpulseresponses ofthesetwofilters,namely, p{t)andPit),are designed inaccordance withEquations (6.74)and(6.75),forachosenbaseband pulse g(t).Theresulting outputsofthesetwofiltersaresubtracted toproduce thetransmitted CAPsignals(t)inaccordance withEquation (6.77). Thetransmitted signals(t)propagates through achannelcharacterized byimpulse response h(t)andadditivenoisew{t).Theresulting channeloutputisdefinedby x(t)=s{t)*h(t)+w(t) =2:[akP(t-kT)*h(t)-b"p{t-kT)*h(t)]+w(t) (6.85) k~;;~ 2:[aku(t-kT) k~-~bku(t-kT)]+w{t) whereu{t)andu(t)aredefinedbyEquations (6.82)and(6.83),respectively. Givenx{t)as theinputsignal,thefunction ofthereceiveristorecoverthetransmitted symbols akand bkinanoptimum fashionandonasymbol-by-symbol basis. Thereceiverconsistsofatwo-dimensional optimum receiver, thedetailsofwhich dependonthechannelimpairments. Specifically, wemaymention twodifferent situations: II>Additive whiteGaussian noiseistheonlyimpairment. Inthisidealized situation, the optimum CAPreceiverconsistsofatwo-dimensional matched filter,asindicated in Received signal x(t)Estimate ofthe.origin'll bitstream FIGURE6.22Blockdiagram ofCAPreceiver usingatwo-dimcnsional matched filterforthe caseofanidealwhiteGaussian noisechannel. 6.4HybridAmplitudelPlu.se Modulation Schemes 379 Figure6.22.Theoptimum in-phasefilterconsistsofafiltermatchedtothepassband in-phase pulsepit).Theoptimum quadrature filterconsistsofafiltermatched tothe passband quadrature pulsep(t). !>Intersymbol interference andadditivewhiteGaussian noisearethechannelimpair ments.Inthismorerealisticsituation inpractical terms,thestructure oftheoptimum CAPreceiverfollowsfromtheoptimum linearreceivertheorypresented inSection 4.9.Inparticular, theCAPreceiverconsistsofatwo-dimensional matched filter, followed byapairofidentical equalizers andapairofsynchronous samplers, as showninFigure6.23.Theequalizers (implemented intapped-delay-line form)com pensatefordispersion inthechannel. FromProperty 3,wededucethattheoutputofthein-phase matched filterduetothe passband quadrature pulseiszero,andviceversa.Accordingly, inbothofthesesituations, thetransmitted symbols akandbkcanbedetected separately bythetwo-dimensional optimum CAPreceiver. DigitalImplementation oftheCAPReceiver Thereceiverstructure ofFigure6.23canbefurthersimplified byfirstreformulating itintheformshowninFigure6.24a,wherewehaveintroduced ananalog-to-digital (AID)converter atthereceiverinputtofacilitate theuseofdigitalsignalprocessing. Next, werecognize thatthematched filtersandequalizers inFigure6.24aarealllinearsystems. Accordingly, thesamplesatpointsAandBinFigure6.24aarelinearcombinations ofthe inputsamplesx[n]=x(nT.),whereT.isthesampling period.Itfollowstherefore thatwe mayreplacethecombination ofmatched filterandequalizer ineachreceiverpathinFigure 6.24abyasinglefinite-duration impulseresponse (FIR)filteroperating atthesampling rateliT.,asshowninFigure6.24b.Withacommon input,itisnaturalforthetwoFIR filtersinFigure6.24btoshareacommon setofunit-delay elements butdifferent setsof coefficients oftheirown.Notealsothatthereceiverstructure ofFigure6.24bisthemirror imageofthetransmitter structure inFigure6.21. Onelastcomment isinorder.Inpractice, thetwoFIRfiltersinthereceiverofFigure 6.24baremadeadaptive soastoaccommodate operation oftheCAPreceiverinanun knownenvironment. Withadaptive equalization (filtering) asthemethodofchoice,we havetwooptionsinlightofthematerialpresented inChapter4:linearequalization and decisionfeedback equalization (DFE).Whenthefrequency response ofthechannelisap proximately flat,theuseoflinearequalization isadequate forthetaskathand.However, whenthefrequency response ofthechannelandlorthenoisepowerspectrum arenot approximately flat,performance oftheCAPreceivercanbeimproved significantly bythe useofDFE. Received signal x(t)Estimate oftheoriginal bitstream FIGURE6.23BlockdiagramofCAPreceiverusingapairofoptimum linearreceivers forthe caseofanoisy,dispersive channel. 380 CHAPTER 6"PASSBAND DATATRANSMISSION Sampling rate=las Ca)DecoderEstimate oftheoniln~ bitstrl?arn Sampling rate=lITsDecoderEstimate oftheoriginal bitstream (b) FIGURE6.24Digital imple~entation oftheCAPreceiver. (a)Reformulation ofthereceiverin Figure6.23usinganNOconverter andassociated circuitry. (b)Replacement ofthematched fil ter/equalizer pairswithequivalent (digitally implemented) FIRfilters. AnApplication ofCAP Animportant application ofCAP(using32or64constellation points)isinthe passband transmission ofdigitaldataovertwisted-pair wiringoflengthslessthan100m, asinlocalareanetworks (LANs)suitableforpremises' distribution systems. (Allmodern LANstandards limitthelengthofthewiringtoamaximum of100m;theCAPsystems canactuallyoperateoverlongerloops,ifsorequired.) Thedataratesmayrangefrom51 upto155Mb/s,withtheusablechannelbandwidth beingstrictlylimitedto30MHz. Thetwomajorimpairments fortransceivers providing duplexoperation over twisted-pairs cablesarepropagation lossandnear-end crosstalk (NEXT); dIeseimpair· mentswerediscussed inSection4.8,whichdealtwidIdigitalsubscriber lines. IndIeenvironment described herein,apractical issueishowtoadaptdIereceiverto widevariations intwisted-pair cables.TocatertodIisrequirement, adaptive filtersare usedtoimplement theoptimum in-phase andquadrature filtersindIereceiver, asremarked earlier.Thewidertherangeoftwisted-pair cablesdIathavetobeaccommodated, dIemore complex theseadaptive filtersmustbe. I6.5Coherent Frequency·Shift Keying M-aryPSKandM-aryQAMshareacommon property: BodIareexamples oflinearmod· ulation.IndIissectionwestudyanonlinear medIodofpassband datatransmission, namely,coherent frequency-shift keying(FSK).Webeginthestudybyconsidering the simplecaseofbinaryFSK. 6.5CoJwrenf Frequency-Shill Keying 381 IIBINARYFSK InabinaryFSKsystem,symbols1and0aredi~tinguished fromeachotherbytransmitting oneoftwosinusoidal wavesthatdifferinfrequency byafixedamount.Atypicalpairof sinusoidal wavesisdescribed by {~silt)=Fr;:cos(27rJit), 0,o:st:sTb elsewhere(6.86) wherei=1,2,andEbisthetransmitted signalenergyperbit;thetransmitted frequency is (6.87) forsomefixedintegerncandi=1,2nc+i Ji=-----y;- Thussymbol1isrepresented byS,(t),andsymbol0byS2(t).TheFSKsignal described hereisknownasSunde'sFSK.Itisacontinuous-phase signalinthesensethatphase continuity isalwaysmaintained, including theinter-bitswitching times.Thisformofdig italmodulation isanexample ofcontinuous-phase frequency-shift keying(CPFSK), on whichwehavemoretosaylateroninthesection. FromEquations (6.86)and(6.87),weobservedirectlythatthesignalsS,(t)andS2(t) areorthogonal, butnotnormalized tohaveunitenergy.Wetherefore deducethatthemost usefulformforthesetoforthonormal basisfunctions is eMt)={Ifcos(27rJit), 0,o:st:sTb elsewhere(6.88) wherei=1,2.Correspondingly, thecoefficient Si;fori=1,2,and;=1,2isdefinedby i=; i"*;(b Si;=JoSj(t)<P;(t) dt (b~ (2 =Jo~-tcos(27rJit){f;'cos(27rjjt) dt ={VB;" 0,(6.89) Thus,unlikecoherent binaryPSK,acoherent binaryFSKsystemischaracterized byhaving asignalspacethatistwo-dimensional (i.e.,N=2)withtwomessagepoints(i.e.,M=2), asshowninFigure6.25.Thetwomessagepointsaredefinedbythe _[VB;,]S,-0 (6.90) and (6.91) withtheEuclidean distance between themequaltoV'iE;,.Figure6.25alsoincludes a coupleofinserts,whichshowwaveforms representative ofsignalsS,(t)andS2(t). 382 CHAPTER 6'IIIPASSBAND DATATRANSMISSION / / / / //V24/7 Region ZI FIGURE 6.25Signal-space diagram forbinaryFSKsystem.Thediagram alsoincludes two insertsshowing example waveforms ofthetwomodulated signals51(t)and52(t). ErrorProbability ofBinaryFSK Theobservation vectorxhastwoelements XlandX2thataredefinedby,respectively, (6.92) and (6.93) wherex(t)isthereceivedsignal,theform ofwhichdependsonwhichsymbolwastrans mitted.Giventhatsymbol1wastransmitted, x(t)equalsslit)+w(t),wherew(t)isthe samplefunctionofawhiteGaussian noiseprocessofzeromeanandpowerspectraldenslt}' Noll.H,ontheotherhand,symbol0wastransmitted, x(t)equalssz(t)+w(t). . Now,applying thedecision ruleofEquation (5.59),wefindthattheobservanon spaceispartitioned intotwodecision regions,labeledZlandZ2inFigure6.25.Th~ decisionboundary, separating regionZlfromregionZ2istheperpendicular bisectora 6.5Colwrent Frequency-Shift Keying 383 thelinejoiningthetwomessage points.Thereceiverdecidesinfavorofsymbol1ifthe received signalpointrepresented bytheobservation vectorxfallsinside region 2 "This occurswhenX,>X2'If,ontheotherhand,wehaveX,<X2,thereceived signalpoint fallsinsideregion22,andthereceiverdecidesinfavorofsymbol0.Onthedecision boundary, wehaveX,=X2,inwhichcasethereceivermakesarandomguessinfavorof symbol1or0. DefineanewGaussian random variableYwhosesamplevalueyisequaltothe difference between X,andX2;thatis, (6.94) ThemeanvalueoftherandomvariableYdepends onwhichbinarysymbolwastrans mitted.Giventhatsymbol1wastransmitted, theGaussian randomvariables X,andX2, whosesamplevaluesaredenotedbyX,andX2,havemeanvaluesequaltoVE;,andzero, respectively. Correspondingly, theconditional meanoftherandomvariable Y,giventhat symbol1wastransmitted, is E[YI1] =E[X,11]-E[X211] =+VE;,(6.95) Ontheotherhand,giventhatsymbol°wastransmitted, therandomvariables X,andX2 havemeanvaluesequaltozeroandVE;"respectively. Correspondingly, theconditional meanoftherandomvariable Y,giventhatsymbol°wastransmitted, is E[YIO]=E[X,IO]-E[X210] =-VE;,(6.96) (6.97) (6.98)Thevariance oftherandomvariableYisindependent ofwhichbinarysymbolwastrans mitted.Sincetherandomvariables X,andX2arestatistically independent, eachwitha variance equaltoNo/2,itfollowsthat var[¥]=var[X,]+var[X 2] =No Supposeweknowthatsymbol°wastransmitted. Theconditional probability density function oftherandomvariableYisthengivenby 1[(y+VE;,)2] fy(yIO)=Y21TNoexp 2No Sincethecondition X,>Xl>orequivalently, y>0,corresponds tothereceivermakinga decision infavorofsymbol1,wededucethattheconditional probability oferror,given thatsymbol°wastransmitted, is PutPlO=P(y>°Isymbol°wassent} =r[y(yIO)dy '=Y2~Norexp[ y+VE;,~=z(6.99) (6.100) 384 CHAPTER 6IIIPASSBAND DATATRANSMISSION Then,changing thevariableofintegration fromytoZ,wemayrewriteEquation (6.99) asfollows: =~erfc(#rf)(6.101) Similarly, wemayshowthePOlltheconditional probability oferrorgiventhatSYIllbol1 wastransmitted, hasthesamevalueasinEquation (6.101).Accordingly, averagingp andPOllwefindthattheaverageprobability ofbiterroror,equivalently, thebiterrorra~ forcoherent binaryFSKis(assuming equiprobable symbols) Pe=~erfc(#rf) (6.102) Comparing Equations (6.20)and(6.102),weseethat,inacoherent binaryFSK system,wehavetodoublethebitenergy-to-noise densityratio,EhlNo,tomaintain the samebiterrorrateasinacoherent binaryPSKsystem.Thisresultisinperfectaccordwith thesignal-space diagrams ofFigures6.3and6.25,whereweseethatinabinaryPSK systemtheEuclidean distancebetweenthetwomessagepointsisequalto2YE;;,whereas inabinaryFSKsystemthecorresponding distance isv'2E;;.Foraprescribed Eb,the minimum distance dmininbinaryPSKisthereforevItimesthatinbinaryFSK.Recall fromChapter 5thattheprobability oferrordecreases exponentially asd;;'in,hencethe difference betweentheformulas ofEquations (6.20)and(6.102). Generatron andDetectUm ofCoherent Bi_ryFSKSignals TogenerateabinaryFSKsignal,wemayusetheschemeshowninFigure6.26a.The incoming binarydatasequence isfirstappliedtoanon-offlevelencoder, attheoutputof whichsymbol1isrepresented byaconstant amplitude ofvB;,voltsandsymbol0is represented byzerovolts.ByusinganinverterinthelowerchannelinFigure6.26a,wein effectmakesurethatwhenwehavesymbol1attheinput,theoscillator withfrequency 11intheupperchannelisswitched onwhiletheoscillator withfrequency 12inthelower channelisswitched off,withtheresultthatfrequency 11istransmitted. Conversely, when wehavesymbol0attheinput,theoscillator intheupperchannelisswitched offandthe oscillator inthelowerchannelisswitched on,withtheresultthatfrequency 12istrans' mitted.Thetwofrequencies 11and12arechosentoequaldifferent integermultiples ofthe bitrate11Th,asinEquation (6.87). Inthetransmitter ofFigure6.26a,weassumethatthetwooscillators aresynchro nized,sothattheiroutputssatisfytherequirements ofthetwoorthonormal basisfunctions <P1(t)and<P2(t),asinEquation (6.88).Alternatively, wemayuseasinglekeyed(voltage controlled) oscillator. Ineithercase,thefrequency ofthemodulated waveisshiftedwith acontinuous phase,inaccordance ,withtheinputbinarywave. Todetecttheoriginalbinarysequence giventhenoisyreceivedsignalx(t),wemay usethereceivershowninFigure6.26b.Itconsistsoftwocorrelators withacommon input, whicharesupplied withlocallygenerated coherent reference signals <P1(t)and<P2(t).'The correlator outputsarethensubtracted, onefromtheother,andtheresulting difference,Y, iscompared withathreshold ofzerovolts.Ify>0,thereceiverdecidesinfavorof1.On theotherhand,ify<0,itdecidesinfavorofO.Ifyisexactlyzero,thereceiverruakesa randomguessinfavorof1orO. 6.5Coherent Frequency-Shift Keying 385 Binary data sequenceOn-off level encoder (a)+ +Binary FSK signal s(t) Choose1ify>0 Choose0ify<0 (6.103) (6.104)(b) FIGURE6.26Blockdiagrams for(a)binaryFSKtransmitter and(b)coherent binaryFSK receiver. PowerSpectraofBi_ryFSKSignals Consider thecaseofSunde'sFSK,forwhichthetwotransmitted frequencies 11and hdifferbyanamountequaltothebitrate11Tb,andtheirarithmetic meanequalsthe nominal carrierfrequency fc;phasecontinuity isalwaysmaintained, including inter-bit switching times.WemayexpressthisspecialbinaryFSKsignalasfollows: [2E;,( 71't)s(t)={"T;:cos271'fct±T b' andusingawell-known trigonometric identity,weget s(t)=Rfcos(±;:)cos(271'fct) -RfSin(±;:)sin(271'fct) =Rfcos(;:)cos(271'fct)+Rfsin(;:)sin(27rfct) InthelastlineofEquation (6.104),theplussigncorresponds totransmitting symbol0, andtheminussigncorresponds totransmitting symbol1.Asbefore,weassumethatthe symbols1and0intherandombinarywaveatthemodulator inputareequallylikely,and thatthesymbols transmitted inadjacent timeslotsarestatistically independent. Then, 386 CHAPTER 6 "PASSBAND DATATRANSMISSION basedontherepresentation ofEquation (6.104),wemaymakethefollowing observati pertaining tothein-phase andquadrature components ofabinaryFSKsignalwithc°fJI; tinuousphase: on.. 1.Thein-phase component iscompletely independent ofthe'inputbinaryWaveI equalsv'2Eb/Tbcos(7Tt/Tb)forallvaluesoftimet.Thepowerspectraldensity' ~ thiscomponent therefore consistsoftwodeltafunctions, weighted bythefactO Eb/2Tb,andoccurring at1=±1/2Tb• Or 2.Thequadrature component isdirectlyrelatedtotheinputbinarywave.Duringthe signaling interval0,,;t,,;Tb,itequals-g(t)whenwehavesymbol1,andtg(f) whenwehavesymbolO.Thesymbolshapingfunction g(t)isdefinedby {~.(7Tt)g(t)=vT;smTb' 0,0,,;t,,;Tb elsewhere(6.105) Theenergyspectraldensityofthissymbolshapingfunction equals (6.106) Thepowerspectraldensityofthequadrature component equals'Pg(f)/Tb•Itisalso apparent thatthein-phase andquadrature components ofthebinaryFSKsignalarein dependent ofeachother.Accordingly, thebaseband powerspectraldensityofSunde'sFSK signalequalsthesumofthepowerspectraldensities ofthesetwocomponents, asshown by (6.107) Substituting Equation (6.107)inEquation (6.4),wefindthatthepowerspectrum of thebinaryFSKsignalcontains twodiscrete frequency components locatedat(Ictl/2Tbl =1,and(Ic-112Tb)=12'withtheiraveragepowersaddinguptoone-halfthetotal powerofthebinaryFSKsignal.Thepresence ofthesetwodiscretefrequency components provides ameansofsynchronizing thereceiverwiththetransmitter. Notealsothatthebaseband powerspectraldensityofabinaryFSKsignalwith continuous phaseultimately fallsoffastheinversefourthpoweroffrequency. Thisis readilyestablished bytakingthelimitinEquation (6.107)asIapproaches infinity.If, however, theFSKsignalexhibitsphasediscontinuity attheinter-bit switching instants (thisariseswhenthetwooscillators applying frequenciesIiand12operateindependently ofeachother),thepowerspectraldensityultimately fallsoffastheinversesquareof frequency; seeProblem 6.23.Accordingly, anFSKsignalwithcontinuous phasedoesnot produce asmuchinterference outsidethesignalbandofinterestasanFSKsignalwith discontinuous phase. InFigure6.5,wehaveplottedthebaseband powerspectraofEquations (6.22)and (6.107).(Tosimplifymatters,wehaveonlyplottedtheresultsforpositivefrequency.) In bothcases,SB(f)isshownnormalized withrespectto2Eb,andthefrequency isnormalized withrespecttothebitrateRb=lITb•Thedifference inthefalloffratesofthesespectra canbeexplained onthebasisofthepulseshapeg(t).Thesmoother thepulse,thefaster thedropofspectraltailstozero.Thus,sincebinaryFSK(withcontinuous phase)hasa smoother pulseshape,ithaslowersidelobes thanbinaryPSK. (6.110)6.5Coherent Frequency·Shift Keying 387 mMINIMUM SHIFT KEYING Inthecoherent detection ofbinaryFSKsignal,thephaseinformation contained inthe received signalisnotfullyexploited, otherthantoprovideforsynchronization ofthe receivertothetransmitter. Wenowshowthatbyproperuseofthephasewhenperforming detection, itispossibletoimprovethenoiseperformance ofthereceiversignificantly. This improvement is,however, achieved attheexpenseofincreased receivercomplexity. Consider acontinuous-phase frequency-shift keying(CPFSK) signal,whichisdefined fortheinterval0:5t:5Tbasfollows: {RfCOS[27T/,t+ 0(0)] forsymbol1 sit)= (6.108)~yT;;COS[27T/2t+0(0)] forsymbol0 whereEbisthetransmitted signalenergyperbit,andTbisthebitduration. Thephase 0(0),denoting thevalueofthephaseattimet=0,sumsupthepasthistoryofthe modulation processuptotimet=o.Thefrequencies 1,andAaresentinresponse to binarysymbols1and0appearing atthemodulator input,respectively. Another usefulwayofrepresenting theCPFSKsignals(t)istoexpressitinthe conventional formofanangle-modulated signalasfollows: ~sit)=yT;;COS[27T.fct+O(t)] (6.109) whereO(t)isthephaseofs(t).WhenthephaseO(t)isacontinuous function oftime,we findthatthemodulated signalsit)itselfisalsocontinuous atalltimes,including theinter bitswitching times.ThephaseO(t)ofaCPFSKsignalincreases ordecreases linearlywith timeduringeachbitduration ofTbseconds, asshownby 7ThO(t)=0(0)±T bt, wheretheplussigncorresponds tosendingsymbol1,andtheminussigncorresponds to sendingsymbol0;theparameter histobedefined.Substituting Equation (6.110)into (6.109),andthencomparing theangleofthecosinefunctionwiththatofEquation (6.108), wededucethefollowing pairofrelations: h .fc+2T b=1, h .fc-2T b=12 SolvingEquations (6.111)and(6.112)for.fcandh,wethusget 1 .fc=2:(/,+12) and(6.111) (6.112) (6.113) (6.114) Thenominalcarrierfrequency fcistherefore thearithmetic meanofthefrequencies 1,and 120Thedifference between thefrequencies 1,andA,normalized withrespecttothebit rateliT,,,definesthedimensionless parameter h,whichisreferred toasthedeviation ratio. (6.llS)388 CHAPTER 6II!PASSBM'D DATATRANSMISSION PhaseTrellis FromEquation (6.110)wefindthatattimet=Tb, {7f'hforsymbol1 O(Tb)-0(0)=hfbi-7f' orsym00 Thatistosay,thesendingofsymbol1increases thephaseofaCPFSKsignals(t)by7Th radians,whereasthesendingofsymbol0reducesitbyanequalamount. Thevariation ofphaseO(t)withtimetfollowsapathconsisting ofasequence of straightlines,theslopesofwhichrepresent frequency changes. Figure6.27depictspossible pathsstartingfromtimet=O.AplotlikethatshowninFigure6.27iscalledaphasetree. Thetreemakesclearthetransitions ofphaseacrossintervalboundaries oftheincoming sequence ofdatabits.Moreover, itisevidentfromFigure6.27thatthephaseofaCPFSK signalisanoddorevenmultipleof7f'hradiansatoddorevenmultiples ofthebitduration Tb,respectively. Thephasetreedescribed inFigure6.27isamanifestation ofphasecontinuity, whicQ isaninherent characteristic ofaCPFSKsignal.Toappreciate thenotionofphaseconti nuity,letusgobackforamoment toSunde'sFSK,whichisaCPFSKschemeaspreviously described. Inthiscase,thedeviation ratiohisexactlyunity.Hence,according toFigure 6.27thephasechangeoveronebitintervalis:±:7f'radians.But,achangeof+7f'radiansis exactlythesameasachangeof-7f'radians,modulo 27f'.Itfollowstherefore thatinthe caseofSunde'sFSKthereisnomemory; thatis,knowing whichparticular changeoccurred intheprevious bitintervalprovides nohelpinthecurrentbitintervaL Incontrast, wehaveacompletely different situation whenthedeviation ratiohis' assigned thespecialvalueof1/2.Wenowfindthatthephasecantakeononlythetwo values:±:7f'/2atoddmultiples ofTb,andonlythetwovalues0and7f'atevenmultiples of Tb,asinFigure6.28.Thissecondgraphiscalledaphasetrellis,sincea"trellis"isatreelih structure withremerging branches. EachpathfromlefttorightthroughthetrellisofFigure 6.28corresponds toaspecificbinarysequence input.Forexample, thepathshownin boldfaceinFigure6.28corresponds tothebinarysequence 1101000 with0(0)=O.Hence forth,weassumethath=1/2. 5--c"1E---'-'l*---'*---*--*----<>: I FIGURE 6.27Phasetree. 6.5Coherent Frequency-Shift Keying 389 ~ I ~-1r12 FIGURE6.28Phasetrellis;boldfaced pathrepresents thesequence 1101000. Withh=1/2,wefindfromEquation (6.114)thatthefrequency deviation (i.e.,the difference between thetwosignaling frequencies [,and[2)equalshalfthebitrate.Thisis theminimum frequency spacingthatallowsthetwoFSKsignalsrepresenting symbols1 and0,asinEquation (6.108),tobecoherently orthogonal inthesen;ethattheydonot interfere withoneanotherintheprocessofdetection.ItisforthisreasonthataCPFSK signalwithadeviation ratioofonehalfiscommonly referredtoasminimum shiftkeying (MSK)? Signal-Space Diagram ofMSK UsingaweU-known trigonometric identityinEquation (6.109),wemayexpressthe CPFSKsignals(t)intermsofitsin-phase andquadrature components asfoUows: s(t)=~cos[O(t)] cos(27rfct) -~sin[O(t)] sin(27rfct) (6.116) Consider firstthein-phase component V2EblTbcos[lI(t)]. Withthedeviation ratio h=1/2,wehavefromEquation (6.110)that 7r lI(t)=11(0)±2T bt, (6.117) (6.118)wheretheplussigncorresponds tosymbol1andtheminussigncorresponds tosymbol o.AsimilarresultholdsforlI(t)intheinterval-Tb:5t:50,exceptthatthealgebraic signisnotnecessarily thesameinbothintervals. Sincethephase11(0)is0or7r,depending onthepasthistoryofthemodulation process,wefindthat,intheinterval-Tb:5t:5Tb, thepolarity ofcos[O(t)] depends onlyon0(0),regardless ofthesequence ofIsandOs transmitted beforeoraftert=o.Thus,forthistimeinterval,thein-phase component sIlt) consistsofahalf-cycle cosinepulsedefinedasfollows: g§ sAt)=-JT:cos[lI(t)] ~ (7r)=-JT:cos[II(O)] cos2T bt =±~cos(~t),-JT: 2Tb wheretheplussigncorresponds to11(0)=0andtheminussigncorresponds to11(0)=7r. Inasimilarway,wemayshowthat,intheinterval0:5t:52Tb,thequadrature component 390 CHAPTER 6"PASSBAl'iD DATATRANSMISSION sQ(t)consistsofahalf-cycle sinepulse,whosepolaritydepends onlyonII(Td,assh ~ ~ sdt)=~sin[lI(t)] =~sin[II(T b)]sin(2;bt) (6.1191 =:!:~sin(2;bt),O:=;t:=;2Tb wheretheplussigncorresponds toII(Tb)=Trl2andtheminussigncorresponds to II(Tb)=-T112. Fromtheforegoing discussion weseethatsincethephasestates11(0)and/I(Tb)can eachassumeoneoftwopossiblevalues,anyoneoffourpossibilities canarise,asdescribed here: I>Thephase11(0)=0andII(Tb)=Tr12,corresponding tothetransmission ofsymboll. I>Thephase11(0)=TrandII(Tb)=Tr12,corresponding tothetransmission ofsymbolO. '"Thephase11(0)=TrandII(Tb)=-Tr12(or,equivalently, 3Tr/2modulo 27T),corre- sponding tothetransmission ofsymbol1. I>Thephase11(0)=0andII(Tb)=-TrI2,corresponding tothetransmission of symbolO. This,inturn,meansthattheMSKsignalitselfmayassumeanyoneoffourpossibleforms, depending onthevaluesof11(0)andII(Tb). Fromtheexpansion ofEquation (6.116),wededucethattheorthonormal basis functions 1>,(t)and1>2(t)forMSKaredefinedbyapairofsinusoidally modulated quad· raturecarriers: 1>,(t)=itcos(2;bt)cos(2Trfct), O:=;t:=;Tb (6.120) 1>2(t)=itsin(2;bt)sin(2Trfct), 0:=;t:=;Tb (6.121) Correspondingly, wemayexpresstheMSKsignalintheexpanded form sIt)=S,1>,(t)+S21>2(t), 0:=;t:=;Tb (6.122) wherethecoefficients 5,and52arerelatedtothephasestates11(0)andII(Tb),respectively. Toevaluate 5"weintegrate theproducts(t)1>l(t)betweenthelimits-TbandTb,asshown by 5,=r:,s(t)1>,(t)dt '" (6.123) =VB;,cos[II(O)], -Tb:=;t:=;Tb Similarly, toevaluate 52weintegrate theproductS(t)1>2(t) between thelimits0and2Tb' asshownby 2Tb 52=faS(t)1>2(t)dt =-VB;,sin[II(Tb)],(6.124) 6.5Coherent Frequency-Shift Keying 391 Messagepointm3:Symbol0 [8(0)=",8(Tb)=,,12] Region Z,Region Zl Messagepointml:Symbol0 [8(0)=0,MTb)=-,,12]------, :,;E;,: I I I FIGURE6.29Signal-space diagramforMSKsystem. NotethatinEquations (6.123)and(6.124): ...Bothintegrals areevaluated foratimeintervalequaltotwicethebitduration. !>Boththelowerandupperlimitsoftheproductintegration usedtoevaluate the coefficient 51areshiftedbythebitdurationTbwithrespecttothoseusedtoevaluate thecoefficient 52' I»Thetimeinterval0::st::sTb,forwhichthephasestates0(0)andO(Tb)aredefined, iscommon tobothintegrals. Accordingly, thesignalconstellation foranMSKsignalistwo-dimensional (i.e., N=2),withfourpossiblemessage points(i.e.,M=4),asillustrated inFigure6.29. Thecoordinates ofthemessage pointsareasfollowsinacounterclockwise direction: (+VB;;,+VB;;),(-VB;;,+VB;;),(-VB;;,-VB;;),and(+VB;;,-VB;;).Thepossible valuesof0(0)and9(T{,),corresponding tothesefourmessagepoints,arealsoincluded in Figure6.29.Thesignal-space diagramofMSKisthussimilartothatofQPSKinthatboth ofthemhavefourmessagepoints.However, theydifferinasubtlewaythatshouldbe carefully noted:InQPSKthetransmitted symbolisrepresented byanyone ofthefour messagepoints,whereasinMSKoneoftwomessagepointsisusedtorepresent thetrans mittedsymbolatanyonetime,depending onthevalueof0(0). Table6.5presentsasummary ofthevaluesof0(0)andO(Tb),aswellasthecorre sponding valuesof51and52thatarecalculated forthetimeintervals -Tb:::st::sTband o:::st::s2Tb,respectively, Thefirstcolumnofthistableindicates whethersymbol1or symbol0wassentintheinterval0::st::sTb•Notethatthecoordinates ofthemessage points, 51and52,haveopposite signswhensymbol1issentinthisinterval, butthesame signwhensymbol0issent.Accordingly, foragiveninputdatasequence, wemayusethe (6.125)392 CHAPTER 6IIIPASSBAND DATA TRANS~lISSION ITABLE6.5Signal-space characterization ofMSK Coordinates PhaseStates ofMessage Transmitted (radians) Points BinarySymbol, 0:0;t:o;Tba(O) a(Tb) S1 S2 0 0-7r/2+vB;;+vB;; 1 7r-7r/2-vB;;+vB;; 0 7r+7r/2-vB;;-vB;; 1 0+7r/2+vB;;-vB;; entriesofTable6.5toderive,onabit-by-bit basis,thetwosequences ofcoefficients re quiredtoscale<p,(t)and<Pz(t),andtherebydetermine theMSKsignal5(t). ~ExAMPLE 6.5 Figure6.30showsthesequences andwaveforms involvedinthegeneration ofanMSKsignal forthebinarysequence pOlOOO. Theinputbinarysequence isshowninFigure6.30a.The twomodulation frequences are:f,=5/4Tbandf2=3/4Tb•Assuming that,attimet=0the phase8(0)iszero,thesequence ofphasestatesisasshowninFigure6.30,modulo2".The polarities ofthetwosequences offactorsusedtoscalethetimefunctions <I>,(t)and<1>2(1)are showninthetoplinesofFigures6.30band6.30c.Notethatthesetwosequences areoffset relativetoeachotherbyanintervalequaltothebitduration Tb•Thewaveforms ofthe resultingtwocomponents ofsit),namely, 5,<1>,(1)andS2<1>2(t),arealsoshowninFigures6.30b and6.30c.Addingthesetwomodulated waveforms, wegetthedesiredMSKsignals(t)shown inFigure6.30d. ... ErrorProbability ofMSK InthecaseofanAWGNchannel, thereceived signalisgivenby x(t)=5(t)+w(t) where5(t)isthetransmitted MSKsignal,andw(t)isthesamplefunction ofawhiteGaus siannoiseprocessofzeromeanandpowerspectral densityNo/2.Todecidewhether symbol! orsymbol0wastransmitted intheinterval0:0;t:o;Tb,say,wehavetoestablish aprocedure fortheuseofx(t)todetectthephasestates1:1(0)and8(Tb).Fortheoptimum detection of1:1(0),wefirstdetermine theprojection ofthereceived signalx(t)ontothe reference signal<p,(t)overtheinterval-Tbst:o;Tb,obtaining x,=r:bx(t)<p,(t) dt =5,+w"-TbStsTb where5,isasdefinedbyEquation (6.123)andw,isthesamplevalueofaGaussian random variableofzeromeanandvariance No/2.Fromthesignal-space diagram ofFigure6.29, weobservethatifx,>0,thereceiverchoosestheestimate 1:1(0)=O.Ontheotherhand, ifx,<0,itchoosestheestimate0(0)='IT. 6.5Coherent Frequency-Shift Keying 393 Inputbinarysequence I Trmescale 0o (a)oo0 I 9(kTb) Polarityof8,o +" (b)"o + 9(kTb) PolarityofSz 8(t)(e) \AA;A;tv\TVVV (d) FIGURE6.30(a)Inputbinarysequence. (b)Waveform ofscaledtimefunction S,<PI(t). (c)Waveform ofscaledtimefunction S2<P2(t).(d)Waveform oftheMSKsignals(t)obtained by addings,<p,(t) andS2<P2(t)onabit-by-bit basis. Similarly, fortheoptimum detection offJ(Tb),wedetermine theprojection ofthe receivedsignalx(t)ontothesecondreference signalq,2(t)overtheinterval0:5t:52Tb, obtaining (2Tb X2=Jox(t)q,2(t)dt =S2+W2,(6.126) where S2isasdefinedbyEquation (6.124)andW2isthesamplevalueofanotherindepen dentGaussian randomvariableofzeromeanandvariance No/2.Referring againtothe signalspacediagram ofFigure6.29,weobservethatifX2>0,thereceiverchoosesthe estimateO(Tb)=-11l2.If,ontheotherhand,X2<0,itchoosestheestimateO(Tb)=11/2. Toreconstruct theoriginalbinarysequence, weinterleavetheabovetwosetsofphase decisions, asdescribed next(seeTaoJe6.5): I>Ifwehavetheestimates 0(0)=0andO(Tb)=-11/2,oralternatively ifwehavethe estimates 0(0)='ITandO(Tb)='lT12,thereceivermakesadecisioninfavorofsym bolO. 394 CHAPTER 6"PASSBAND DATATRANSMISSION ~Ifwehave,theestimates .iJ(O)=7Tand9(Tb)=-7T/2,oralternatively ifwehaveth estimates 8(0)=0and8(Tb)=7T/2,thereceivermakesadecisioninfavorofsye boll. In· Thusreferring tothesignal-space diagram ofFigure6.29,weseethatthedecisio madebythereceiverisbetweenthemessagepointsm1andm3forsymbol1,orbetwen themessagepointsm2andm4forsymbol O.Thecorresponding decisions whether erO)e,n oor7Tandwhether 8(Tb)is-7TI2,or+7T(2(i.e.,thebitdecisio~s) aremadealternately ~ theI-andQ-channels oftherece1ver, witheachchannellookmg atthemputsignalfo 2Tbseconds. Thesignalfromotherbitsdoesnotinterfere withthereceiver's decisionfor agivenbitineitherchannel. ThereceivermakesanerrorwhentheI-channel assignsth: wrongvalueto8(0)ortheQ-channel assignsthewrongvalueto8(Tb)·Accordingly, Usmg thestatistical characterizations oftheproduct-integrator outputs X,andX2ofthesetwo channels, definedbyEquations (6.125)and(6.126),respectively, wereadilyfindthatthe biterrorrateforcoherent MSKisgivenby Pe=~erfc(ffi) (6.127) whichisexactlythesameasthatforbinaryPSKandQPSK.Itisimportant tonote however, thatthisgoodperformance istheresultofthedetection oftheMSKsignal bein~ performed inthereceiveronthebasisofobservations over2Tbseconds. Generation andDetection ofMSKSignals Consider nextthegeneration anddemodulation ofMSK.Figure6.31ashowsthe blockdiagramofatypicalMSKtransmitter. Theadvantage ofthismethodofgenerating MSKsignalsisthatthesignalcoherence anddeviation ratioarelargelyunaffected by variations intheinputdatarate.Twoinputsinusoidal waves,oneoffrequency fc=nJ4Tb forsomefixedintegernoandtheotheroffrequency 1/4Tb,arefirstappliedtoaproduct modulator. Thisproduces twophase-coherent sinusoidal wavesatfrequenciesIIandA, whicharerelatedtothecarrierfrequency fcandthebitratelITbbyEquations (6.111) and(6.112)forh=1/2.Thesetwosinusoidal wavesareseparated fromeachotherby twonarrowband filters,onecentered atAandtheotherat12'Theresulting filteroutputs arenextlinearlycombined toproducethepairofquadrature carriersororthonormal basis functions <P1(t)and<P2(t).Finally,<P1(t)and<P2(t)aremultiplied withtwobinarywaves al(t)anda2(t),bothofwhichhaveabitrateequalto1I2Tb•Thesetwobinarywavesare extracted fromtheincoming binarysequence inthemannerdescribed inExample 6.5. Figure6.31bshowstheblockdiagramofatypicalMSKreceiver.Thereceived signal x(t)iscorrelated withlocallygenerated replicasofthecoherent reference signals <P1(t)and <P2(t).Notethatinbothcasestheintegration intervalis2Tbseconds, andthattheinte· grationinthequadrature channelisdelayedbyTbsecondswithrespecttothatinthein· phasechannel.Theresulting in-phase andquadrature channelcorreiatoroutputs, XIand X2,areeachcompared withathreshold ofzero,andestimates ofthephase8(0)ande(Tb) arederivedinthemannerdescribed previously. Finally,thesephasedecisions areinter' leavedsoastoreconstruct theoriginalinputbinarysequence withtheminimum average probability ofsymbolerrorinanAWGNchannel. PowerSpectraofMSKSignols AswiththebinaryFSKsignal,weassumethattheinputbinarywaveisrandom. withsymbols1and0equallylikely,andthesymbolstransmitted duringdifferenttune slotsbeingstatistically independent. Inthiscase,wemakethefollowing observations: 6.5Coherent Frequency-Shift Keying 395 COS(2~J Narrowband filters (a) Threshold =aOutput binary sequenceLogic circuit for interleaving phase decisionsPhase estimate D~~~f~~nf-_IJ ,;.(0,;.)-JX,In-phasechannel ~------; Input x(t) Quadrature channel r2T ifbdtoX2Decision devicePhase estimate Ii(Tb) 4>,(t) Threshold =D (b) FIGlJRE6.31Blockdiagrams for(a)MSKtransmitter and(b)coherent MSKreceiver. 1.Depending onthevalueofphasestate0(0),thein-phasecomponent equals+g(t)or -g(t),where {P£:,(1Tt)g(t)={r;:cos2Tb' 0,-Tb:$t:$T b otherwise(6.128) (6.129)Theenergyspectraldensityofthissymbol-shaping function is ([)=32EbTb[COS(21TTd)]2 I/Jg 1T216T~P-1 Hence,thepowerspectraldensityofthein-phase component equalsI/Jg(f)/2T b• 2.Depending onthevalueofthephasestateO(Tb),thequadrature component equals +g(t)or-g(t),wherewenowhave {P£:,.(1Tt) g(t)=rT;sm2Tb' 0,0:$t:$2Tb otherwise(6.130) (6.131)396 CHAPTER 6"PASSBAND DATATllANSMISSION Theenergyspectraldensityofthissecondsymbol-shaping function isalsogivenby Equation (6.129).Hence,thein-phase andquadrature components havetheSa powerspectraldensity. me 3.Thein-phase andquadrature component oftheMSKsignalarealsostatisticall independent. Hence,thebaseband powerspectraldensityoftheMSKsignalis.,;vY ~ .~ SB(f)=2[~~)] =32Eb[COS(271"Tbf)]2 71"216TiP-1 Thebaseband powerspectrum ofEquation (6.131)isplottedinFigure6.9,where thepowerspectrum isnormalized withrespectto4Ebandthefrequencyfisnormalized withrespecttothebitrateliTb.Thisfigurealsoincludesthecorresponding plotofEqua. tion(6.40)fortheQPSKsignal,whichwasconsidered earlier.Forf»1IT&,thebaseband powerspectraldensityoftheMSKsignalfallsoffastheinversefourthpoweroffrequency, whereas inthecaseoftheQPSKsignalitfallsoffastheinversesquareoffrequency. Accordingly, MSKdoesnotproduce asmuchinterference outsidethesignalbandofin. terestasQPSK.Thisisadesirable characteristic ofMSK,especially whenthedigitalcom. munication systemoperates withabandwidth limitation. Il!lGALJSSIAN-FILTERED MSK Fromthedetailedstudyofminimum shiftkeying(MSK)justpresented, wemaysummarize thedesirable properties oftheMSKsignalasfollows: Il>-Constant envelope I>Relatively narrowbandwidth I>Coherent detection performance equivalent tothatofQPSK However, theout-of-band spectralcharacteristics ofMSKsignals,asgoodastheyare,still donotsatisfythestringent requirements ofcertainapplications suchaswirelesscommu· nications. Toillustrate thislimitation, wefindfromEquation (6.131)thatatfTb=0.5, thebaseband powerspectraldensityoftheMSKsignaldropsbyonly10loglo9=9.54 dBbelowitsmidband value.Hence,whentheMSKsignalisassigned atransmission bandwidth of1ITb,theadjacent channelinterference ofawirelesscommunication system usingMSKisnotlowenoughtosatisfythepractical requirements ofsuchamultiuser communications environment. Recognizing thattheMSKsignalcanbegenerated bydirectfrequency modulation ofavoltage-controlled oscillator, wemayovercome thisseriouslimitation ofMSKby modifying itspowerspectrum intoacompact form,whilemaintaining theconstant envelope property oftheMSKsignaLThismodification canbeachieved throughtheuse ofapremodulation low-pass filter,hereafter referredtoasabaseband pulse-shaping filter. Desirably, thepulse-shaping filtershouldsatisfythefollowing properties: 1.Frequency response withnarrowbandwidth andsharpcutoffcharacteristics. 2.Impulseresponse withrelatively lowovershoot. 6.5Coherent Frequency-Shift Keying 397 3.Evolution ofaphasetrelliswherethecarrierphaseofthemodulated signalassumes thetwovalues±71'/2atoddmultiples ofTbandthetwovalues0and71'ateven multiples ofTbasinMSK. Condition 1isneededtosuppress thehigh-frequency components ofthetransmitted signal. Condition 2avoidsexcessive deviations intheinstantaneous frequency oftheFMsignal. Finally,condition 3ensuresthatthemodified FMsignalcanbecoherently detectedinthe samewayastheMSKsignal,oritcanbenoncoherently detectedasasimplebinaryFSK signal. Thesedesirable properties canbeachieved bypassinganonreturn-to-zero (NRZ) binarydatastreamthroughabaseband pulse-shaping filterwhoseimpulseresponse (and likewiseitsfrequency response) isdefinedbyaGaussian function. Theresulting method ofbinaryfrequency modulation isnaturally referredtoasGaussian-filtered MSKorjust GMSK.8 LetWdenotethe3dBbaseband bandwidth ofthepulse-shaping filter.Wemaythen definethetransferfunction H(f)andimpulseresponse h(t)ofthepulse-shaping filteras follows,respectively: and h-~ (271'222)(t)-.jI;;giWexp-10g2Wt(6.132) (6.133) Theresponse ofthisGaussian filtertoarectangular pulseofunitamplitude and durationTb(centered ontheorigin)isgivenby(seeProblem 6.28) (6.134) whichmaybeexpressed asthedifference betweentwocomplementary errorfunctions, as shownby g(t)=.!.[erfc(7r[2WTb(~-.!.))-erfc(7r[2WTb(~+.!.))](6.135)2vr;;g2 Tb2 vr;;g2 Tb2 Thepulseresponse g(t}constitutes thefrequency shapingpulseoftheGMSKmodulator, withthedimensionless time-bandwidth productWTbplayingtheroleofadesign parameter. Thefrequency-shaping pulseg(t),asdefinedinEquation (6.135),isnoncausal inthat itisnonzerofort<-Tb/2,wheret=-Tb/2isthetimeatwhichtheinputrectangular pulse(symmetrically positioned aroundtheorigin)isappliedtotheGaussian filter.Fora causalresponse, g(t}mustbetruncated andshiftedintime.Figure6.32presentsplotsof g(t),whichhasbeentruncated att=±2.5Tbandthenshiftedintimeby2.5Tb•Theplots shownhereareforWTb=0.2,0.25,and0.3.NotethatasWTbisreduced,thetimespread ofthefrequency-shaping pulseiscorrespondingly increased. 0.2'""g0.4 .tea.E0.3..398 CHAPTER 6'lPASSBAND DATA'TRANSMISSION 0.91~~~~~~~-~---r====::::;l 0.8 0.7 0.6 0.5 ""/'",I011,;'//" -0~--~-- - -0.2'--~-~_~_~_~_~_~_~_L-_o0.5 1.522.533.544.5 Normalized time,t/Tb FIGURE6.32Frequency-shaping pulseg(t)ofEquation (6.135)shiftedintimeby2.5T,and truncated at:±:2.5T bforvaryingtime-bandwidth productWT,. Figure6.33showsthemachine-computed powerspectraofMSKsignals(e:Kpressed indecibels) versusthenormalized frequency difference (f-fc)Tb,wherefcisthemid bandfrequency andTbisthebitduration.9TheresultsplottedinFigure6.33arefor varyingvaluesofthetime-bandwidth productWTb•Fromthisfigurewemaymakethe following observations: I>Thecurveforthelimitingcondition WTb=00corresponds tothecaseofordinary MSK. t»WhenWTbislessthanunity,increasingly moreofthetransmit powerisconcentrated insidethepassband oftheGMSKsignal. Anundesirable featureofGMSKisthattheprocessing ofNRZbinarydatabya Gaussian filtergenerates amodulating signalthatisnolongerconfined toasinglebit intervalasinordinary MSK,whichisreadilyapparent fromFigure6.32.Statedinanother way,thetailsoftheGaussian impulseresponse ofthepulse-shaping filtercausethemod ulatingsignaltospreadouttoadjacent symbolintervals. Thenetresultisthegeneration ofintersymbol interference, theextentof which increases withdecreasing WTb•Inlight ofthisobservation andtheobservation wemadeonthebasisofFigure6.33onthepower spectraofGMSKsignals,wemaysaythatthechoiceofthetime-bandwidth productWT, offersatrade-off betweenspectralcompactness andperformance loss. Toexploretheissueofperformance degradation, consider theprobability oferror PeofGMSKusingcoherent detection inthepresence ofadditivewhiteGaussian noise. Recognizing thatGMSKisaspecialkindofbinaryfrequency modulation, wemayexpress Pebytheempirical formula Pe=~erfc(~) (6.136) where,asbefore,EbisthesignalenergyperbitandNo/2isthenoisespectraldensity.The factoraisaconstant whosevalue depends onthetime-bandwidth productWTb'CoJll paringtheformula ofEquation (6.136)forGMSKwiththatofEquation (6.127)for 6.5Coherent Frequency.Shift Keying 399 -----MSK - - - - WTb=O,2--- WTb=0,25 - - - WTb=0.3 , \...-...\',, I \",...., , , \I ,,I,I \I ,t II \I I' I, " :1!-20 i£:s-40-'" .~ ~ i-60 ~(f -80 -100 -1200,0 0,5 1.0 1.5 2,0 2.5 Normalized frequency,If-f,ITb FIGURE6.33PowerspectraofMSKandGMSKsignalsforvaryingtime-bandwidth product. (Reproduced withpermission fromDr.GordonStuber,GeorgiaTech.) ordinary MSK,wemayview10l0g lO(al2),expressedIndecibels, asameasure ofperfor mancedegradation ofGMSK(withprescribed WTb)compared toordinary MSK.Figure 6.34showsthemachine-computed valueof1010g1o(al2)versusWTb•Forordinary MSK wehaveWTb=00,inwhichcaseEquation (6.136)witha=2assumesexactlythesame formasEquation (6.127)andthereisnodegradation inperformance, whichisconfirmed byFigure6.34.ForGMSKwithWTb=0.3wefindfromFigure6.34thatthereisa 3,--,---,,------,----,-----, FIGURE6.34Theoretical EhlNodegradation ofGMSKforvaryingtime-bandwidth product, (TakenfromMurataandHirade,1981,withpermission oftheIEEE.) 400 CHAPTER 6!illPASSBAND DATATRANSMISSION degradation inperformance ofabout0.46dB,whichcorresponds to(ai/l)=0.9.Thi degradation inperformance isasmallpricetopayforthehighlydesirable spectral cOIU~ paetness oftheGMSKsignal. li>EXAMPLE 6.6GMSKforGSMWireless Communications Animportant application ofGMSKisinastandardized wirelesscommunication systemwidel known a~GSM~w~ichisatime-division multiple-access systemthati.sdiscussed inChapte~ 8.Forthisapplication, thetime-bandwIdth productWTbofGMSK ISstandardized at0.3 whichprovides thebestcompromise between increased bandwidth occupancy andresistan~ toco-channel interference. Ninety-nine percentoftheradiofrequency (RF)powerofGMSK signalssospecified isconfined toabandwidth of250kHz,whichmeansthat,forallpractical purposes, thesidelobes arevirtually zerooutsidethisfrequency band. Theavailable spectrum isdividedinto200kHz-wide subchannels. Eachsubchannel is assignedtoaGSMsystemtransmitting dataat271kb/s.Figure6.35depictsthepOwerSpec. trumofasubchannel inrelationtoitstwoadjacent subchannels; thisplotisthepassband versionofthebaseband powerspectrum ofFigure6.33corresponding toWTb;0.3.From Figure6.35wemaymakethefollowing intportant observation: TheRFpowerspectrum of thesubchannel shownshadedisdownbyanamountlargerthan40dBatthecarrierfre. quencies ofbothadjacent subchannels, whichmeansthattheeffectofco"channel interference ispractically negligible. "C iiiM-ARyFSK Consider nexttheM-aryversionofFSK,forwhichthetransmitted signals.aredefinedby Si(t)=Rcos[¥(nc+i)tJ.0::£t::£T (6.137/ ----;.-Frequency, kHz FIGURE6.35Powerspectrum ofGMSKsignalforGSMwirelesscommunications. 6.5Co'.erent Frequency-Shift Keying 401 wherei=1,2,...,M,andthecarrierfrequency fc=nJ2Tforsomefixedintegernco Thetransmitted symbols areofequalduration TandhaveequalenergyE.Sincethe individual signalfrequencies areseparated by1/2THz,thesignalsinEquation (6.137)are orthogonal; thatis i=1=j (6.138) Thisproperty ofM-aryFSKsuggeststhatwemayusethetransmitted signalssilt) themselves, exceptforenergynormalization, asacomplete orthonormal setofbasisfunc tions,asshownby 1 .0"';t",;T <Pi(t)=vBSi(t), i=1,2,..., M(6.139) (6.140)Accordingly, theM-aryFSKisdescribed byanMcdimensional signal-space diagram. Forcoherent M-aryFSK,theoptimum receiverconsistsofabankofMcorrelators ormatched filters,withthe<Pi(t)ofEquation (6.139)providing thepertinent reference signals.Atthesampling timest=kT,thereceivermakesdecisions basedonthelargest matched filteroutputinaccordance withthemaximum likelihood decoding rule.Anexact formulafortheprobability ofsymbolerrorishowever difficulttoderiveforacoherent M-aryFSKsystem.Nevertheless, wemayusetheunionboundofEquation (5.96)of Chapter5toplaceanupperboundontheaverageprobability ofsymbolerrorforM-ary FSK.Specifically, notingthattheminimum distancedmininM-aryFSKisv'lE,theuseof Equation (5.96)yields(assuming equiprobable symbols) Pc",;~(M1)erfc()&) ForfixedM,thisboundbecomes increasingly tightasEINoisincreased. Indeed,itbecomes agoodapproximation toPcforvaluesofPc",;10-3•Moreover, forM=2(i.e.,binary FSK),theboundofEquation (6.140)becomes anequality. PowerSpectraofM-aryFSKSigfU"s ThespectralanalysisofM-aryFSKsignals'°ismuchmorecomplicated thanthatof M-aryPSKsignals.Acaseofparticular interestoccurswhenthefrequencies assigned to themultilevels makethefrequency spacinguniformandthefrequency deviation k=0.5. Thatis,theMsignalfrequencies areseparated bylI2T,whereTisthesymbolduration. Fork=0.5,thebaseband powerspectraldensityofM-aryFSKsignalsisplottedinFigure 6.36forM=2,4,8. Bandwidth Efficiency ofM-aryFSKSignals Whentheorthogonal signalsofanM-aryFSKsignalaredetected coherently, the adjacent signalsneedonlybeseparated fromeachotherbyafrequency difference lI2Tso astomaintain orthogonality. Hence,wemaydefinethechannelbandwidth required to transmit M-aryFSKsignalsas M B=2T(6.141) 402 CHAPTER 6'"PASSBAND DATATRANSMISSION 1.0 o 1.5 2.0 Normalized frequency, fTh2.5 3.0 FIGURE 6.36PowerspectraofM-aryPSKsignalsforM=2,4,8. Formultilevels withfrequency assignments thatmakethefrequency spacinguniformand equalto1/2T,thebandwidth BofEquation (6.141)contains alargefractionofthesignal power.Thisisreadilyconfirmed bylookingatthebaseband powerspectralplotsshown inFigure6.36.FromEquation (6.48)werecallthatthesymbolperiodTisequalto Tblog2M.Hence,usingRb=lITb,wemayredefinethechannelbandwidth BforM-ary FSKsignalsas B=Rr,M 2log2M Thebandwidth efficiency ofM-arysignalsistherefore RbP=J3 2log2M M(6.142) (6.143) Table6.6givesthevaluesofpcalculated fromEquation (6.143)forvaryingM. Comparing Tables6.4and6.6,weseethatincreasing thenumberoflevelsMtends toincreasethebandwidth efficiency ofM-aryPSKsignals,butitalsotendstodecrease the bandwidth efficiency ofM-aryFSKsignals.Inotherwords,M-aryPSKsignalsarespec trallyefficient, whereasM-aryFSKsignalsarespectrally inefficient. ITABLE6.6 FSKsignalsBandwidth efficiency ofM-ary M 2 p(bits/slHz)48 0.7516 0.532 0.312564 0.1875 (6.145)6.6Detection ofSignalswithUnknown Phose 403 I~T·~__Detection ofSignalswith ~Phase Uptothispointinourdiscussion, wehaveassumed thatthereceiverisperfectly synchro nizedtothetransmitter, andtheonlychannelimpairment isnoise.Inpractice, however, itisoftenfoundthatinaddition totheuncertainty duetochannelnoise,thereisalso uncertainty duetotherandonmess ofcertainsignalparameters. Theusualcauseofthis uncertainty isdistortion inthetransmission medium. Perhapsthemostcommon random signalparameter isthecarrierphase,whichisespecially truefornarrowband signals.For example, transmission overamultiplicity ofpathsofdifferent andvariable lengths,or rapidlyvaryingdelaysinthepropagating mediumfromtransmitter toreceiver, maycause thephaseofthereceived signaltochangeinawaythatthereceivercannotfollow.Syn chronization withthephaseofthetransmitted carriermaythenbetoocostly,andthe designer maysimplychoosetodisregard thephaseinformation inthereceived signalat theexpenseofsomedegradation innoiseperformance. Adigitalcommunication receiver withnoprovision madeforcarrierphaserecovery issaidtobenoncoherent. i!§OPTIMUM QUADRATIC RECEIVER Consider abinarydigitalcommunication systemipwhichthetransmitted signalis flE O:s;t:s;T Si(t)=-TCOS(27Tfit,)' (6.144)i=1,2 whereEisthesignalenergy,Tistheduration ofthesignaling interval, andthecarrier frequency fiforsymboliisanintegralmultiple of112T.Thesystemisassumed tobe noncoherent, inwhichcasethereceived signalforanAWGNchannelmaybewrittenin theform flE O:s;t:s;T x(t)=-TCOS(27Tfit+ Ii)+wit), . _ t-1,2, whereIiistheunknown carrierphase,andwit)isthesamplefunction ofawhiteGaussian noiseprocessofzeromeanandpowerspectraldensityNo/2.Inareal-lifesituation itis realistictoassumecomplete lackofpriorinformation aboutIiandtotreatitasarandom variablewithuniformdistribution: [..(Ii)={21 7T' 0,-7T<Ii:s;7T otherwise(6.146) Thebinarydetection problem tobesolvedmaynowbestatedasfollows: Giventhereceivedsignalx(t)andconfronted withtheunknown carrierphasefi, designanoptimum receiverfordetecting symbolSirepresented bythesignalcom ponentVEI2TCOS(27Tj,t+fi)thatiscontained inx(t). Proceeding inamannersimilartothatdescribed inSections5.3-5.6, wemayformulate theconditional likelihood function ofsymbolSi,giventhecarrierphaseIi,as L(Si(Ii))=exp(Jbrx(t)COS(27Tfit+Ii)dt) (6.147) 404 CHAPTER 6tlIPASSBAND DATATRANSMISSION Toproceedfurther,wehavetoremovedependence ofL(si(O))onphase0,whichisachievd byintegrating itoverallpossiblevaluesof0.Wemaythuswrite e L(si)=f"L(si(O))!..(O)do =Lrf"exp(jl;fx(t)cos(2'lrj;t +0)dt)do(6.148) Notethatthedependence on0wasremoved byintegrating thelikelihood function and notthelog-likelihood function. Usingawell-known trigonometric formula, wemayexpandcos(2'lrj;t+0)as Correspondingly, wemayrewritetheintegralintheexponent ofEquation (6.148)as faTx(t)COS(271f;t+0)dt=cos0faTx(t)COS(271f;t) dtsin0faTx(t)sin(217'j;t) dt(6.149) Define (T )2]'/2fax(t)sin(2'lrj;t) dt li=[(fx(t)~os(2'lrj;t) dt)2+ _(fx(t)sin(2'lrj;t) dt) f3i=tan1----'.rT------ fax(t)cos(2'lrj;t) dt Hence,wemaygoonestepfurtherandsimplifyEquation (6.149)to faTx(t)cos(2'lrj;t+0)dt=li(COS0cosf3i-sin0sinf3i) =licos(O+f3i)(6.150) (6.151) (6.152) (6.153)Accordingly, usingEquation (6.152)inEquation (6.148),weobtain L(si)=2~f"exp(jl;licos(O+f3,))dO 1f,,+f3i((E) =2'lr-"+f3iexp...jNJlicos0dO =2~f"exp(jl;licos0)dO FromAppendix 3onBesselfunctions, weimmediately recognize theintegralofEqua tion(6.153)asthemodified Besselfunctionofzeroorder: (6.154) (6.155)6.6DetectUmofSignalswithUnk......,.. Phnse 405 Hence,wemayexpressthelikelihood functionforthesignaldetection problemdescribed hereininthecompact form L(Si)=Io(JlrIi) Thebinaryhypothesis test(i.e.,thehypothesis thatsignalS,(t)orsignalS2(t)was transmitted) cannowbewrittenas (6.156) wherehypothesis H,andHzcorrespond tosignalss,(t)andS2(t),respectively. FromAp pendix3wenotethatthemodified BesselfunctionI(')isamonotonically increasing func tionofitsargument. Hencethehypothesis testcanbecarriedoutintermsofeither Io(VE/NoTl i)orsimplyIi'Forconvenience ofimplementation, however, thehypothesis testiscarriedoutintermsofftinsteadofIi,asshownby (6.157) AreceiverbasedonEquation (6.157)isknownasthequadratic receiver. Inlightofthe definition ofIigiveninEquation (6.150),thereceiverstructure forcomputing Iiisasshown inFigure6.37a.Notethatthetestdescribed inEquation (6.157)isindependent ofthe symbolenergyE.Itisforthisreasonthatthishypothesis testissaidtobeuniformly most powerful withrespecttothesymbolenergyE. l1liTwoEQUIVALENT FORMS OFTHEQUADRATIC RECEIVER Wenextderivetwoequivalent formsofthequadrature receivershowninFigure6.37a. Thefirstformisobtained easilybyreplacing eachcorrelator inFigure6.37awithacor responding equivalent matched filter.Wethusobtainthealternative formofquadrature receivershowninFigure6.37b.Inonebranchofthisreceiver, wehaveafiltermatched tothesignalCOS(21Tj;t), andintheotherbranchwehaveafiltermatched tosin(277'j;t), bothofwhicharedefinedforthetimeinterval0:s;t:s;T.Thefilteroutputsaresampled attimet=T,squared, andthenaddedtogether. Toobtainthesecondequivalent formofthequadrature receiver, supposewehave afilterthatismatched tos(t)=COS(21Tj;t+9)for0:s;t:s;T.Theenvelope ofthematched filteroutputisobviously unaffected bythevalueofphase9.Therefore, forconvenience, we maysimplychooseamatchedfilterwithimpulseresponsecos[21Tj;(T -t)],corresponding to()=O.Theoutputofsuchafilterinresponse tothereceivedsignalx(t)isgivenby yet)=rX(T)COS[27Tfi(T -t+T)]d7 (6.158) =COS[27Tfi(T -Il]fX(7)cos(277'fi7) d7-Sin[27Tfi(T -I)]rX(T)sin(27TfiT) dT Theenvelope ofthematched filteroutputisproportional tothesquarerootofthesumof thesquaresoftheintegrals inEquation (6.158).Theenvelope, evaluated attimet=T,is therefore {[T J2[T JZ}'/2Ii=faX(7)COS(21Tj;7) d7+faX(T)Sin(21Tj;7) d7 (6.159) 406 CHAPTER 6"PASSRAND DATATRANSMISSION x{t)Square rooterOutput Ii FIGURE6.37Noncoherent receivers. (a)Quadrature receiverusingcorrelators. (b)Quadrature receiverusingmatchedfilters.(c)Noncoherent matchedfilter. Butthisisjusttheoutputofthequadrature receiver. Therefore, theoutput(attimeT)of afiltermatched tothesignalCOS(2TTf,t+e),ofarbitrary phasee,followed byanenvelope detectoristhesameasthecorresponding outputofthequadrature receiverofFigure6.37a. Thisformofreceiver isshowninFigure6.37c.Thecombination ofmatched filterand envelope detectorshowninFigure6.37ciscalledanoncoherent matched filter. Theneedforanenvelope detectorfollowing thematched filterinFigure6.37cmay alsobejustifiedintuitively asfollows.Theoutputofafiltermatched toarectangular RF wavereachesapositivepeakatthesampling instantt=T.If,however, thephaseofthe filterisnotmatched tothatofthesignal,thepeakmayoccuratatimedifferent fromthe sampling instant.Inactualfact,ifthephasesdifferby180degrees,wegetanegativepeak atthesampling instant.Figure6.38illustrates thematched filteroutputforthetwolimiting conditions:e=0ande=180degrees.Toavoidpoorsampling thatarisesintheabsence ofpriorinformation aboutthephasee,itisreasonable toretainonlytheenvelope ofthe matched filteroutput,sinceitiscompletely independent ofthephasemismatch e. 6.7Nancoherent Orthogonal Modulation 407 2T f<'---T .1 (a) l~------------~-_ 2T (6.160)(b) FIGURE6.38Outputofmatched filterforarectangular RFwave:(a)()=0,and(b)()=180 degrees. I6.7Noncoherent Orthogonal Modulation Withthenoncoherent receiverstructures ofFigure6.37atourdisposal, wemaynow proceedtostudythenoiseperformance ofnoncoherent orthogonal modulation thatin cludestwononcoherent receivers asspecialcases:noncoherent binaryfrequency-shift key inganddifferential phase-shift keying. Consider abinarysignaling schemethatinvolvestheuseoftwoorthogonal signals .,(t)and'2(t),whichhaveequalenergy.Duringtheinterval0:s;t:s;T,oneofthesetwo signalsissentoveranimperfect channelthatshiftsthecarrierphasebyanunknown amount. Letg,{t)andg2(t)denotethephase-shifted versionsof.,(t)andS2(t),respectively. Itisassumed thatthesignalsg,(t)andg2(t)remainorthogonal andhavethesameenergy E,regardless oftheunknown carrierphase.Werefertosuchasignaling schemeasnon coherent orthogonal modulation. Depending onhowwedefinetheorthogonal pairof signalss,(t)and'2(t),noncoherent binaryFSKandDPSKmaybetreatedasspecialcases ofthismodulation scheme. Thechannelalsointroduces anadditivewhiteGaussian noisew(t)ofzeromeanand powerspectraldensityNo/2.Wemaythusexpressthereceivedsignalx(t)as {gl(t)+w(t), s,(t)sent,0:s;t:s;Tx(t)= g2(t)+w(t), S2(t)sent,0:s;t:s;T Therequirement istousex(t)todiscriminate between s,(t)andS2(t),regardless ofthe carrierphase. 408 CHAPTER 6"PASSBAND D1I:£ATBANS1IUSSION Forthispurpose, weemploythereceivershowninFigure6.39a.ThereceiverCons' ofapairoffiltersmatched tothetransmitted signalsSl(t)andS2(t).Becausethecar:~ts phaseisunknown, thereceiverreliesonamplitude astheonlypossiblediscriminant. ~~ cordingly, thematched filteroutputsareenvelope detected, sampled, andthencompared witheachother.IftheupperpathinFigure6.39ahasanoutputamplitude 11greaterth theoutputamp!itude l~oft~elo,:"erpath,thereceivermakesadecisioninfav~rofSl(~ Iftheconverse IStrue,ItdeCides Infavorofslit).Whentheyareequa~thedeCisionma bemadebyflippingafaircoin.Inanyevent,adecision erroroccurswhenthematche~ filterthatrejectsthesignalcomponent ofthereceivedsignalx(t)hasalargeroutpUtam. plitude(duetonoisealone)thanthematched filterthatpassesit. Fromthediscussion presented inSection6.6,wenotethatanoncoherent matched filter(constituting theupperorlowerpathinthereceiverofFigure6.39a)maybeviewed asbeingequivalent toaquadrature receiver. Thequadrature receiveritselfhastwochan. nels.Oneversionofthequadrature receiverisshowninFigure6.39b.Intheupper channe~ calledthein.phasechannel, thereceived signalx(t)iscorrelated withthefunction 1/J;lt) whichrepresents ascaledversionofthetransmitted signal Sl(t)orS2(t)withzerocarrie; phase.Inthelowerchannel, calledthequadrature channel, ontheotherhand,x(t)is xU) (a)If/,>/,. choosesl(t). If/,</" chooses2(t). xl,) (b) FIGURE6.39(a)Generalized binaryreceiver fornoncoherent orthogonal modulation. (b)Quad raturereceiver equivalent toeitheroneofthetwomatched filtersinpart(a);theindexi='1,2. 6.7Noncoherent Orthogonal Modulotu- 409 correlated withanotherfunction (J;,(t),whichrepresents theversionof!/Ji(t)thatresults fromshiftingthecarrierphaseby-90degrees.Naturally, !/Ji(t)and(J;i(t)areorthogonal toeachother. Thesignal(J;i(t)isinfacttheHilberttransform of!/Ji(t);theHilberttransform is discussed inAppendix 2.Toillustrate thenatureofthisrelationship, let !/Ji(t)=mit)cos(2nf;t) (6.161) wheremit)isaband-limited message signal.Typically, thecarrierfrequency [;isgreater thanthehighestfrequency component ofmit).Then(inamannersimilartothediscussion onCAPpresented inSection6.4)theHilberttransform of!/Ji(t)isdefinedby Since(J;i(t)=mit)sin(27T[;t) (6.162) cos(27T[;t-~)=sin(27T[;t) weseethat(J;i(t)isindeedobtained from!/Ji(t)byshiftingthecarrierCOS(27T[;t) by-90 degrees.Animportant property ofHilberttransformation isthatasignalanditsHilbert transform areorthogonal toeachother.Thus,!/Ji(t)and(J;i(t)areorthogonal toeachother, asalreadystated. Theaverageprobability oferro~forthenoncoherent receiverofFigure6.39aisgiven bythesimpleformula p=1.exp(-~)e2 2No whereEisthesignalenergypersymbol,andNo/2isthenoisespectraldensity.(6.163) !i!lDERIVATION OFEQUATION (6.163) .... Toderivetheformula ofEquation (6.163),wemakeuseoftheequivalence depicted in Figure6.39.Inparticular, weobservethatsincethecarrierphaseisunknown, noiseatthe outputofeachmatched filterinFigure6.39ahastwodegreesoffreedom, namely,in-phase andquadrature. Accordingly, thenoncoherent receiverofFigure6.39ahasatotaloffour noisyparameters thatareconditionally independent giventhephasee,andalsoidentically distributed. Thesefournoisyparameters havesamplevaluesdenotedbyXI1,XQhXI2'and XQ2;thefirsttwoaccountfordegreesoffreedom associated withtheupperpathofFigure 6.39a,andthelattertwoaccountfordegreesoffreedom associated withthelowerpath. ThereceiverofFigure6.39ahasasymmetric structure. Hence,theprobability of choosing S2(t),giventhatS,(t)wastransmitted, isthesameastheprobability ofchoosing S,(t),giventhatS2(t)wastransmitted. Thismeansthattheaverageprobability oferror maybeobtained bytransmitting S,{t)andcalculating theprobability ofchoosing S2(t),or viceversa,assuming thatS,(t)andS2(t)areequiprobable. Suppose thatsignals,(t)istransmitted fortheinterval0:st:sT.Anerroroccursif thechannelnoisewit)issuchthattheoutput12ofthelowerpathinFigure6.39aisgreater thantheoutput1,oftheupperpath.Thenthereceivermakesadecision infavorofS2(t) *Readers whoarenotinterested inrheformalderivation ofEq.(6.163)mayarrhispoinrwishtomoveonto thetreatment ofnoncoherent binaryfrequency-shift keying(inSection6.7)anddifferential phase-shift keying (inSection6.8)asspecialcasesofnoncoherent orthogonal modulation, without lossofcontinuity. 410 CHAPTER 6IIIPASSBAND DATATR&"ISMISSION XI, (noise) (a) (h) (6.165) (6.167)FIGURE6.40Geometric interpretations ofthetwopathoutputs I]and12inthegeneralized non coherent receiver. ratherthanSI(t).Tocalculate theprobability oferrorsomade,wemusthavetheproba bilitydensityfunction oftherandomvariableL2(represented bysamplevalue12),Since thefilterinthelowerpathismatched toS2(t),andS2(t)isorthogonal tothetransmitted signalSI(t),itfollowsthattheoutputofthismatched filterisduetonoisealone.LetXI2 andXQ2denotethein-phase andquadrature components ofthematched filteroutputin thelowerpathofFigure6.39a.Then,fromtheequivalent structure depicted inFigure 6.39b,weseethat(fori=2) 12=VXJ2+xtz (6.164) Figure6.40ashowsageometric interpretation ofthisrelation. Thechannelnoisew(t)is bothwhite(withpowerspectraldensityNo/2)andGaussian (withzeromean).Corre spondingly, wefindthattherandomvariables XI2andXQ2(represented bysamplevalues XnandxQz)arebothGaussian-distributed withzeromeanandvarianceNol2,giventhe phasee.Hence,wemaywrite 1(XlZ)fXI2(XI2)=v;:N;;exp-No and fxQ2(xQ2)=kexp(-~02) (6.166) Next,weuseawell-known resultinprobability theory,namely,thefactthattheenvelope ofaGaussian processisRayleigh-distributed andindependent ofthephasee(seeSection 1.12).Specifically, forthesituation athand,wemaystatethattherandomvariableL2 [whosesamplevalue12isrelatedtoXI2andXQ2byEquation (6.164)]hasthefollowing probability densityfunction: {21Z(Ii)-exp--,h2(12)=No No 0, elsewhere Figure6.41showsaplotofthisprobability densityfunction. Theconditional probability that12>I"giventhesamplevalueI"isdefinedbytheshadedareainFigure6.41.Hence, wehave (6.168) (6.169)6.7Noncoherent Onoogonal Modulatitm 411 I, FIGURE6.41Calculation oftheconditional probability that12>II>given1" Substituting Equation (6.167)intoEquation (6.168)andintegrating, weget P(l2>11111)=exp( -~J Consider nexttheoutputamplitudeI"pertaining totheupperpathinFigure6.39a.Since thefilterinthispathismatched toS1(t),anditisassumed thatS1(t)istransmitted, it followsthat11isduetosignalplusnoise.LetXnandXQldenotethecomponents atthe outputofthematched filter(intheupperpathofFigure6.39a)thatareinphaseandin quadrature withrespecttothereceivedsignal,respectively. Thenfromtheequivalent struc turedepicted inFigure6.39b,weseethat(fori=1) 11=v'XYl+X~l (6.170) Figure6.40bpresents ageometric interpretation ofthisrelation. SinceaFourier transformable signalanditsHilberttransform formanorthogonal pair,itfollowsthatXIl isduetosignalphisnoise,whereas XQ1isduetonoisealone.Thismeansthat(1)the randomvariableXnrepresented bythesamplevalueXl1isGaussian distributed withmean \IEandvariance No/2,whereEisthesignalenergypersymbol,and(2)therandom variableXQ1represented bythesamplevalueXQ1isGaussian distributed withzeromean andvariance No/2.Hence,wemayexpresstheprobability densityfunctions ofthesetwo independent randomvariables asfollows: (6.171) (6.172)and _ 1(xb)!XQ1(XQ1)-V7iNi,exp-No Sincethetworandomvariables XIlandXQ,areindependent, their jointprobability den sityfunction issimplytheproductoftheprobability densityfunctions giveninEquations (6.171)and(6.172). Tofindtheaverageprobability oferror,wehavetoaveragetheconditional proba bilityoferrorgiveninEquation (6.169)overallpossible valuesofI,.Naturally, this calculation requiresknowledge oftheprobability densityfunction ofrandomvariables L, represented bysamplevalueI,.Thestandard methodisnowtocombine Equations (6.171) and(6.172)tofindtheprobability densityfunction ofL1duetosignalplusnoise.However, thisleadstorathercomplicated calculations involving theuseofBesselfunctions. This (6.175) (6.177)412 CHAPTEIl 6'"PASSBAND DATATRANSMISSION analyticdifficulty maybecircumvented bythefollowing approach. GivenXIlandx erroroccurswhen,inFigure6.39a,thelowerpath'soutputamplitude 12duetonoise~l~an exceeds 11duetosignal plus noise;fromEquation (6.170)wehave ne Ii=XiI+Xbl (6.173) Theprobability ofsuchanoccurrence isobtained bysubstituting Equation (6.173)into Equation (6.169),asshownby (XiI+xb1)P(errorlxlh XQl)=exp No (6.174) Thisisnowaconditional probability oferror,conditional ontheoutputofthematched filterintheupperpathtakingonvaluesXnandXQ1•Thisconditional probability mul. tipliedbythejointprobability densityfunction ofXnandXQ1isthentheerror-density give~XIland.XQ1'SinceXIlandXQ1are~t~tist.ic~lly indepen~ent, theirjointprobabili~ denSItyfunctIOn equalstheproductoftheumdlvldual probabIlIty denSItyfunctions. The resulting error-density isacomplicated expression inXIlandXQl'However, theaverage probability oferror,whichistheissueofinterest,maybeobtained inarelatively simple manner.WefirstuseEquations (6.171), (6.172), and(6.174)toevaluatethedesirederror. densityas P(errorlxIl' XQl)!XI1(XIl)!XQ1(XQ1) _ 1 {,12 2 -'1T'N oexp-No[Xll+XQl+(Xll Completing thesquareintheexponent ofEquation (6.175),wemayrewritetheexponent exceptfor-llNoas 2 2 .IT'2 2 ( vE)2 ,EXll+XQl+(xn-VE)+XQl=2XIl-2+2XQl+2"(6.176) Next,wesubstitute Equation (6.176)intoEquation (6.175)andintegrate theerror-density overallXllandXQI.Wethusevaluate theaverageprobability oferroras Pe=J:ooJ:ooP(errorlxIl' XQl)!XI1(Xll)!XQ,(XQI) dXIldXQl ='1T'~Oexp(-2~Jrooexp[ -~o(XIl-~r]dXn .rooexp( -2~~1) dXQl Wenowusethefollowing twoidentities: foo[2(v'E)2] ~ -00exp-NoXIl-2dxI1=~2 and(6.178) rooexp( -2~~,) dXQl=fF (6.179) TheidentityofEquation (6.178)isobtained byconsidering aGaussian-distributed variable withmeanVEfiandvariance No/4,andrecognizing thatthetotalareaunderthec~ve ofarandomvariable's probability densityfunction equalsunity;theidentityofEquatIOn 6.8Noncoherent BinaryFrequency-Shift Keying 413 (6.179)followsasaspecialcaseofEquation (6.178).Thus,inlightofthesetwoidentities, Equation (6.177)simplifies asfollows: p=.!exp(-~) e22No whichisthedesiredresultpresented previously asEquation (6.163). Withthisformulaatourdisposal, wearereadytoconsider noncoherent binaryFSK andDPSKasspecialcases,whichwedointhenexttwosections, respectively.ll 6.8Noncoherent Binary Frequency-Shift Keying IIIthebinaryFSKcase,thetransmitted signalisdefinedby silt)={ftcos(27rfit), 0::=;t::=;Tb (6.180) 0, elsewhere wherethecarrierfrequency fiequalsoneoftwopossiblevalues,I,and12;toensurethat thesignalsrepresenting thesetwofrequencies areorthogonal, wechoosefi=n;lT&,where niisaninteger.Thetransmission offrequency 11represents symbol1,andthetransmission offrequency 12represents symbol O.Forthenoncoherent detection ofthisfrequency modulated wave,thereceiverconsistsofapairofmatched filtersfollowed byenvelope detectors, asinFigure6.42.Thefilterintheupperpathofthereceiver ismatched to cos(27r/,t), andthefilterinthelowerpathismatched tocos(211'12t), andinbothcases o::=;t::=;Tb•Theresulting envelope detector outputsaresampled att=Tb,and their valuesarecompared. Theenvelope samplesoftheupperandlowerpathsinFigure6.42 areshownasI,and12,respectively. Then,ifI,>12,thereceiverdecidesinfavorofsymbol 1,andif11<12,itdecidesinfavorofsymbols O.If11=12,thereceiversimplymakesa guessinfavorofsymbol1orO. Thenoncoherent binaryFSKdescribed hereinisaspecialcaseofnoncoherent or thogonal modulation withT=TbandE=Eb,whereTbisthebitduration andEbisthe If11>[2' choose1. III,<I,. chooseO. I,Comparison deviceampleat met=Tb ampleat met=TbS Filter timatchedtoEnvelope r--'" CDS(27T/,tJf-detectorf--o a-;;'t~Tb ....;;.. Filter matched toEnvelopeL.;.-eos(27T/2t)40- --0detectorSo:s;t"5,.Ttix(1) FIGURE6.42Noncoherent receiverforthedetection ofhinaryFSKsignals. (6.181)414 CHAPTER 6"PASSRAND DATATRA-1\ISMISSION signalenergyperbit.Hence,using(Equation (6.163),wefindthatthebiterrorrate~ noncoherent binaryFSKis Or 1(Eb)Pe="2exp-2No TheformulaofEquation (6.181)isderivedasaspecialcaseofnoncoherent orthogonal modulation. InProblem 6.31weaddressthesameissueusingadirectapproach that invokestheapplication ofRayleigh andRiciandistributions; thesedistributions pertain respectively totherandomvariables L2andL,whosesamplevaluesaredefinedbyEqua_ tions(6.164)and(6.170),respectively. l6.9Differential Phase-Shift Keying Asremarked earlierinSection6.1,wemayviewdifferential phase-shift keying(DPSK)as thenoncoherent versionofPSK.Iteliminates theneedforacoherent reference signalat thereceiverbycombining twobasicoperations atthetransmitter: (1)differential encodillg oftheinputbinarywaveand(2)phase-shift keying-hence, thename,differential phase shiftkeying(DPSK). Ineffect,tosendsymbol0,wephaseadvance thecurrentsignal waveform by180degrees,andtosendsymbol1weleavethephaseofthecurrentsignal waveform unchanged. Thereceiverisequipped withastoragecapability, sothatitcan measure therelativephasedifference between thewaveforms received duringtwosucces sivebitintervals. Provided thattheunknown phaseecontained inthereceivedwavevaries slowly(thatis,slowenoughforittobeconsidered essentially constant overtwobitintet vals),thephasedifference between waveforms receivedintwosuccessive bitintervalswill beindependent ofe. . DPSKisanother example ofnoncoherent orthogonal modulation, whenitis considered overtwobitintervals. Suppose thetransmitted DPSKsignalequals VEbl2TbCOS(27Tfct) for0::st::sTb,whereTbisthebitduration andEbisthesignal energyperbit.LetS,(t)denotethetransmitted DPSKsignalfor0::st::s2Tbforthecase whenwehavebinarysymbol1atthetransmitter inputforthesecondpartofthisinterva~ namely,Tb::st::S2Tb•Thetransmission ofsymbol1leavesthecarrierphaseunchanged overtheinterval0::st::s2Tb,andsowedefineS,(t)as (6.182) LetS2(t)denotethetransmitted DPSKsignalfor0::st::s2Tbforthecasewhenwehave binarysymbol0atthetransmitter inputforTb::st::s2Tb•Thetransmission of0advances thecarrierphaseby180degrees,andsowedefineS2(t)as (6.183l 6.9Differential Phase-Shift Keying 415 WereadilyseefromEquations (6.182)and(6.183)thatS,(t)andS2(t)areindeedorthog onaloverthetwo-bitinterval0$t$2Tb•Inotherwords,DPSKisaspecialcaseof noncoherent orthogonal modulation withT=2TbandE=2Eb•Hence,usingEquation (6.163),wefindthatthebiterrorrateforDPSKisgivenby Pe=~exp( -~) (6.184) whichprovides againof3dBovernoncoherent FSKforthesameEb/No• Generation andDetection ofDPSK Thenextissuetobeconsidered'is'the generation ofDPSKsignals.Thedifferential encoding processatthetransmitter inputstartswithanarbitrary firstbit,servingasref erence.Let{dkldenote the differentially encoded sequence withthisaddedteference bit. Wenowintroduce thefollowing definitions inthegeneration ofthissequence: '"Iftheincoming binarysymbolbkis1,leavethesymboldkunchanged withrespect totheprevious bit. ~Iftheincoming binarysymbolbkis0,changethesymboldkwithrespecttothe previous bit. Thedifferentially encoded sequence{dklthusgenerated isusedtophase-shift acarrier withphaseangles0and7Tradiansrepresenting symbols 1and0,respectively. The diffetential-phase encoding processisillustrated inTable6.7.Notethatdkisthecomple mentofthemodulo-2 sumofbkanddk-1• Theblockdiagram ofaDPSKtransmitter isshowninFigure6.43a.Itconsists, in part,ofalogicnetwork andaone-bitdelayelementinterconnected soastoconvertthe tawbinarysequence{bklintoadifferentially encoded sequence [dkl.Thissequence is amplitude-level encodedandthenusedtomodulate acarrierwaveoffrequency !C,thereby producing thedesiredDPSKsignal. Suppose next,indifferentially coherent detection ofbinaryDPSK,thecarrierphase isunknown. Then, inlightofthereceiverbeingequipped withanin-phase andaquad raturechannel, wehaveasignalspacediagram wherethereceived signalpointsare (Acos£J,Asin£J)and(-Acos£J,-Asin£J),with£Jdenoting theunknown phaseandA denoting theamplitude. Thisgeometry ofpossiblesignalsisillustrated inFigure6.44.The receivermeasures thecoordinates (Xlo'xQo)attimet=Tband(XI"XQ,)attimet=2Tb• Theissuetoberesolved iswhether thesetwopointsmaptothesamesignalpointor different ones.Recognizing thatthetwovectors XoandXl>withendpoints(Xlo'xQ(»)and (XI"XQ,)arepointedroughlyinthesamedirection iftheirinnerproductispositive, we mayformulate thehypothesis testasfollows: Istheinnerproductx5x,positiveornegative? ITABLE6.7Illustrating thegeneration ofDPSKsignal (bk) 1 0 0 1 0 0 1 [dk-,} 1 1 0 1 1 0 1 Differentially encoded 1 0 1 1 0 1 1 sequence [dk} Transmitted phase 0 0 7T0 0 7T0 0 0 (radians) 416 CHAPTER 6O!!PASSBA-l\ID DATATRANSMISSION Input binary sequence (bk)DPSK signal In-phase channel x(t)Saylify>O Sayoify<O Quadrature channel (b) FIGURE 6.43Blockdiagrams of(u)DPSKtransmitter and(b)DPSKreceiver. Accordingly, wemaywrite say1 XloXI,+XQoXQ, "'"0 say0(6.185) Wenownotethefollowing identity; XloXI,+xQoxQ,=~[(Xlo+XIl-(Xlo-XI,)2+(xQo+XQ,)2-(xQo-xQ.)'J Hencesubstituting thisidentityintoEquation (6.185)andmultiplying borhsidesofthe testby4,wegettheequivalent test: say1 (Xlo+X1,)2+(XQo+XQl-(Xlo-XI,)2-(XQo-XrY"'"0 say0(6.186) Thedecision-making processmaytherefore bethoughtofastestingwherherthepoint (Xlo'xQo)iscloserto(XI"XQ,)oritsimage(-XI"-XQ'). 6.10Comparison ofDigitalModulation Schemes 417 Asine -Acose I' I I I FIGVRE6.44Signal-space diagramofreceivedDPSKsignal. Thus,theoptimum receiver12fordifferentially coherent detection ofbinaryDPSKis asshowninFigure6.43b,whichfollowsdirectlyfromEquation (6.185).Thisimplemen tationmerelyrequiresthatsamplevaluesbestored,therebyavoiding theneedforfancy delaylinesthatmaybeneededotherwise. Theequivalent receiverimplementation that testssquaredelements asinEquation (6.186)ismorecomplicated, butitsusemakesthe analysiseasiertohandleinthatthetwosignalstobeconsidered areorthogonal overthe interval(0,2Tb);hence,thenoncoherent orthogonal demodulation analysisapplies. 6.10Comparison ofDigitalModulation Schemes UsingaSingleCarrier PROBABILITY OFERROR InTable6.8wehavesummarized theexpressions forthebiterrorrate(BER)forcoherent binaryPSK,conventional coherent binaryFSKwithone-bitdecoding, DPSK,noncoherent binaryFSK,coherent QPSK,andcoherent MSK,whenoperating overanAWGNchannel. InFigure6.45wehaveusedtheexpressions summarized inTable6.8toplottheBERas afunction ofthesignalenergyperbit-to-noise spectraldensityratio,EblNo• TABLE6.8Sunmwry offormulas forthebiterrorrateofdifferent digitalmodulation schemes Signaling Scheme (a)Coherent binaryPSK} Coherent QPSK Coherent MSK (b)Coherent binary FSK (c)DPSK (d)Noncoherent binaryFSKBitErrorRate !erfc(YEbIN o) erfc(YEbI2No) exp(-EbIN o} exp(-Eb/2No) 418 CHAPTER 6OJPASSBfu'\lD DATATRANSMISSION 0.5....--...,----r--.,---,---,---------,-----, 10-1f---+--'''''-t---''l*--'k----t---f----j 10-2f---+--t-~--t.,___\_-r\_~--'\_+-----1 (a)CoherentbinaryPSK} (b)Coherent QPSK (c)Coherent MSK g10-3f---+--t----t---f---1rlr---1rl-r----1.. iii lO-4f---+--t----t---f----t-\-++--\----\I 1O-5_'::-5-----=2L.5=---O~---=2~.5=------=5~.0:------=7~.5=------L.,.u---,-"12.5 ~dBNo' FIGURE6.45Comparison ofthenoiseperformance ofdifferent PSKandFSKschemes, Basedontheperforrp.ance curvesshowninFigure6.45,thesummary offormulas giveninTable6,8,andthedefiningequations forthepertinent modulation formats, we canmakethefollowing statements: 1.Thebiterrorratesforallthesystemsdecrease monotonically withincreasing values ofEb/No;thedefiningcurveshaveasimilarshapeintheformofawaterfall. 2.ForanyvalueofEb/No,coherent binaryPSK,QPSK,andMSKproduceasmallet biterrorratethananyoftheothermodulation schemes, 3.Coherent binaryPSKandDPSKrequireanEb/Nothatis3dBlessthanthecorre sponding valuesforconventional coherent binaryFSKandnoncoherent binaryFSK, respectively, torealizethesamebiterrorrate, 4.AthighvaluesofEb/No,DPSKandnoncoherent binaryFSKperformalmostaswell (towithinabout1dB)ascoherent binaryPSKandconventional coherent binaJY FSK,respectively, forthesamebitrateandsignalenergyperbit, 5.Incoherent QPSK,twoorthogonal carriersV2ftcos(27Tf;t) andV2iTsin(211'fct) areused,wherethecarrierfrequencytisanintegermultiple ofthesymbolrate 6.10Comparison ofDigitalModulafion Schemes 419 lIT,withtheresultthattwoindependent bitstreamscanbetransmitted simulta neouslyandsubsequently detectedinthereceiver. 6.Inthecaseofcoherent MSK,therearetwoorthogonal carriers, namely, V21TbCOS(27Tfct) andV21T bsin(27Tfct), whicharemodulated bythetwoantipodal symbolshapingpulsescos(7Tt/2T b)andsin(7TtI2T b),respectively, over2Tbintervals, whereTbisthebitduration. Correspondingly, thereceiverusesacoherent phase decoding processovertwosuccessive bitintervals torecovertheoriginalbitstream. 7.TheMSKschemediffersfromitscounterpart, theQPSK,inthatitsreceiverhas memory. Inparticular, theMSKreceivermakesdecisions basedonobservations over twosuccessive bitintervals. Thus,although thetransmitted signalhasabinaryfor matrepresented bythetransmission oftwodistinctfrequencies, thepresenceofmem oryinthereceivermakesitassumeatwo-dimensional signalspacediagram. There arefourmessagepoints,depending onwhichbinarysymbol(0or1)wassentand thepastphasehistoryoftheFSKsignal. f;llBANDWIDTH EFFICIENCY OFM-ARY DIGITAL MODULATION TECHNIQUES InTable6.9,wehavesummarized typicalvaluesofpower-bandwidth requirements for coherent binaryandM-aryPSKschemes, assuming anaverageprobability ofsymbolerror equalto10-4andthesystemsoperating inidentical noiseenvironments. Thistableshows that,amongthefamilyofM-aryPSKsignals,QPSK(corresponding toM=4)offersthe besttrade-off betweenpowerandbandwidth requirements. Forthisreason,wefindthat QPSKiswidelyusedinpractice. ForM>8,powerrequirements becomeexcessive; ac cordingly, M-aryPSKschemeswithM>8arenotaswidelyusedinpractice. Also,co herentM-aryPSKschemesrequireconsiderably morecomplex equipment thancoherent binaryPSKschemesforsignalgeneration ordetection, especially whenM>8.(Coherent 8-PSKisusedindigitalsatellitecommunications.) Basically, M-aryPSKandM-aryQAMhavesimilarspectralandbandwidth char acteristics. ForM>4,however, thetwoschemeshavedifferent signalconstellations. For M-aryPSKthesignalconstellation iscircular,whereasforM-aryQAMitisrectangular. Moreover, acomparison ofthesetwoconstellations revealsthatthedistancebetweenthe messagepointsofM-aryPSKissmallerthanthedistancebetweenthemessagepointsof M-aryQAM,forafixedpeaktransmitted power.Thisbasicdifference betweenthetwo schemes isillustrated inFigure6.46forM=16.Accordingly, inanAWGNchannel, M-aryQAMoutperforms thecorresponding M-aryPSKinerrorperformance forM>4. TABLE6.9Comparison ofpower-bandwidth requirementsfor M-aryPSKwithbinary PSK.Probability ofsymbolerror=10--4 ValueofM 4 8 16 32(Bandwidth) M~cy (Bandwidth )B;nary 0.5 0.333 0.25 0.2(Average power)M_acy (Average power)Binary 0.34dB 3.91dB 8.52dB 13.52dB FromSharunugan (1979.p.424). 420 CHAPTER 6IIIPASSBAND DATATRANSMISSION <P2 .-/0- ,.~ /•" II• f \ II, <P,,°\I"...-/.... (a)<P2 ,.-/.~·'"/ \ I•.·.\ I \<P, \·.0.! \ f \ /•~.·/"- (b) FIGURE6.46Signalconstellations for(a)M-aryPSKand(b)corresponding M-aryQAM,for M=16. However, thesuperiorperformance ofM-aryQAMcanberealizedonlyifthechannel IS freeofnonlinearities. AsforM-aryFSK,wefindthatforafixedprobability oferror,increasing Mresults inareducedpowerrequirement. However, thisreduction intransmitted powerisachieved atthecostofincreased channelbandwidth. Inotherwords,M-aryFSKbehavesinan opposite mannertothatofM-aryPSK.Wewillrevisitthisissueinaninformation theoretical contextinChapter9,andtherebydevelopfurtherinsightintothecontrasting behaviors ofM-aryPSKandM-aryFSK. 16.11Voiceband Modems The"modem," acontraction ofthetermmodulator-demodulator, isaconversion device thatfacilitates thetransmission andreception ofdataoverthepublicswitched telephone network (PSTN).13 Thedataofinterestmaybedigitalsignalsgenerated bycomputers or serviceproviders. Insuchanapplication, themodulator portionofthemodernconverts theincoming digitalsignalintoastandard formsuitablefortransmission overatelephone channelinthePSTN.Thedemodulator portionofthemodemreceivesthechanneloutput andreconverts itintotheoriginaldigitalsignalformat.Inyetanotherapplication, namely, faxmodems, ormoreprecisely moderns withfacsimile capability, thedatamayrepresent text,graphics, pictures, orcombinations thereof.Inthislatterapplication, thedocument ofinterestiscodedintoaseriesofcompressed pictureelements (pixels),whicharethen transmitted overthetelephone channelbymodulating theirvaluesaccording toaprede finedmodulation standard. Whenthefaxmodemisinareceiving modeofoperation, the demodulator portionofthemodemoperates onthereceived analogsignalanddecom pressesthecorresponding binarydatarepresentation ofthedemodulated signalintoanear oractualduplicate oftheoriginaltransmitted image.Inwhatfollows,wefocusouratten tiononmodems thatprovidecommunication betweenauserandanInternetServicePro' vider(ISP)overthePSTN. Traditionally, thePSTNhasbeenviewedasananalognetwork. Inreality,however, thePSTNaswepresently knowithasbecomeanalmostentirelydigitalnetwork. Inmost cases,theonlypartofthePSTNthathasremained analog(andwilllikelyremainsofor manyyearstocome)isthelocalloop,whichrepresents therelatively shortconnection fromahometothecentraloffice.Thus,depending onhowthePSTNisused,wemaY identifytwodistinctclassesofmodemconfigurations, symmetric andasymmetric, asde scribednext. 6.11VowebamlModems 421 iiiSYMMETRIC MODEM CONFIGURATIONS Thesimplestapproach tothedesignofmodems istotreattheentirePSTNasalinear analognetwork, asindicated inFigure6,47a.(RecallfromChapter3thatthePSTNis almostentirelydigitalduetotheuseofpulse-code modulation (PCM)forthetransmission ofvoicesignals.)Insuchasetting,analog-to-digital anddigital-to-analog conversions are neededwhenever themodems sendsignalstoandreceivesignalsfromthePSTN.The modemconfiguration depictedinFigure6,47aexhibits-"symmetry" inthatbothmodems areidentical andthedataratedownstream (fromtheISPtotheuser)isexactlythesame asthedatarateupstream (fromtheusertotheISP). Thesymmetric modemconfiguration ofFigure6,47aembodies alargenumberof modemtypes,rangingindataratefrom300blsto36,600bls,assummarized inTable A6.7onaselection ofstandard modems. Thedesignofmodems beganwithfrequency shifrkeying,whichcateredtorelatively lowdatarates.Asthedemandfordatatransmis sionovertelephone channels increased, increasingly moresophisticated modulation tech niqueswereemployed tobetterusetheinformation capacityofthetelephone channel. Consider, forexample, thepopular V.32modemstandard that.has thefollowing characteristics: Carrierfrequency =1,800Hz Modulation rate=2,400bauds Datarate=9,600bls Thesignaling datarateof9,600blsassumesahighsignal-to-noise ratio.TheV.32standard specifiestwoalternative modulation schemes: Nonredundant coding.Underthisscheme,theincoming datastreamisdividedinto quadbits (i.e.,groupsoffoursuccessive bits)andthentransmitted overthetelephone channelas16-QAM. Ineachquadbit, themostsignificant inputdibitundergoes phasemodulation, whereas theleastsignificant inputdibitundergoes amplitude modulation. Discussing thephasemodulation first,practical considerations favor theuseofdifferential phasemodulation forthereceiverneedonlybeconcerned with thedetection ofphasecharges.Thismatteristakencareofbyusingadifferential encoder, whichconsistsofaread-only memory andacoupleofdelayunits,as showninFigure6,48a.LetQ"nQ2,n denotethecurrentvalueofthemostsignificant UpstreamPublicDownstream User's----;.. ~ Server's analogswitchedanalog Analog telephone Analog modemnetworkmodem (a) UpstreamPublicDownstream User's----;.. ~ Server's analogswitcheddigitaltelephone Digitalmodem Analog networkmodem (b) FIGURE6.47(u)Environmental overview ofsymmetric modemconfiguration: theupstream and downstream dataratesareequal.(b)Environmental overview of"asymmetric" modemconfigura tion:dataratedownstream ishigherthanupstream. 422 CHAPTER 6.,PASSBAND DATATRANSMISSION an'}16-QAM outputbn16-point signal-space mapper,------------: 12,n Most{Q2,n-+-Read-only 1 significant Imemory III,n inputdibitQ"n+- I'-.,r--~ 112,n-1 I I Ii ...1Least {Q4,n significant inputdibitQ3,n-----------~ Differential encoder (T=symbolperiod) (a) 32-polnt signal-space mapperLeast{Q4,n significant inputdlbitQ3,n------------ ~ Sig~~i~~nt{Q2,n-f-~e~d~o:: -- - --II:::',:'-c-o-n-VO-Iu-tl-on-a'i ~::: inputdibit QI,1f+- memory encoder Yo,ll I '-----,---~I'__ __ -----J :12,11_1 I I I 1------...1 Differential encoder (T=symbolperiod)a;}32-QAM b noutput (b) FIGURE6.48Blockdiagrams ofV.32modem.(a)Nonredundant coding.(b)Trelliscoding, inputdibit,andlet11,n-112,,,-1denotetheprevious valueofthecorresponding dlbit outputbytheencoder. Then,inresponse tothedibitsQ1,,,Q2,,, and1,.,,-tl2,I.-l' the differential encoderproduces thedibit1"n1z,n>which,inturn,inducesaphasechange inthemodulated signal.Thisphasechange,measured inthecounterclockwise di rection,isgoverned bytheGraycodingschemeofTable6.10.Notethatthepbase changeisdetermined entirelybytheinputdibitQ1,,,Q2,n' Insofarasthedifferential phasemodulation isconcerned, thereisoneothermatterthatneedstobeaddressed: acodeforidentifying thefourquadrants ofthetwo-dimensional signalspace.Tbis secondmatterisresolved byadopting theGraycodingschemeincluded inFigure 6.49. Turningnexttotheamplitude modulation, acodehastobespecified fortbe fourpossiblevalueswhichtheleastsignificant inputdibit,denotedbyQ],,,Q4,,,, can assumein,say,thefirstquadrant. Thismatteristakencareofbyadopting theGray codeforthefoursignalpointsinthefirstquadrant shownlightlyshadedinFigure 6.49. Thefinalissuethatneedstoberesolved isthe90°rotational inzlariance, which ismandated bytheuseofdifferential encoding. Thisformofinvariance meansthat theoverallM-aryQAMconstellation looksexactlythesamewhenitisrotated 6.11Voiceba..dModems 423 TABLE6.10Phasechanges induced bydifferential encoding intheV.32modemduetovarying inputdibits Currentinputdibit Phasechange Q"n Q2.' (degrees) 0 0 90 0 1 0 1 0 180 1 1 270 throughanintegermultiple of90degrees,regardless ofwhether itiscodedorun coded;thenthereceivercancorrectly decodethetransmitted messagesequence when thelocaloscillator phasediffersfromthecarrierphasebyanintegermultiple of90 degrees.Thisfinalrequirement issatisfied byfillingintheGraycodesforthesignal pointsintheremaining threequadrants inthemannershowninFigure6.49.Dashed arrowsareincluded inFigure6.49toillustrate the90°rotational invariance. Puttingallofthesematterstogether forthecombined amplitude andphase modulation, wegetthel6-QAM constellation shownpreviously inFigure6.l7a, whichisreproduced hereasFigure6.50a.Correspondingly, theencoding system consistsofadifferential encoderfollowed byal6-point signal-space mapper, as showninFigure6.48a.TheV.32modemsoconfigured issaidtobenonredundant because, with16constellation points,thetransmitted 4-bitcodewordhasnoredun dantbits. FIGURE6.49Illustrating theGrayencoding ofthefourquadrants anddibitsineachquadrant fortheV.32modem. Thedashedarrowsillustrate the90'rotational invariance. 424 CHAPTER 6l!!PASSBAND DATATRANSMISSION B •-3@ 0010 01011010 1000 -3 -1 [email protected] 0001 0000. 1011 . 0011D@3 1001 1110 1100 •0100•1111 ·0110 •0111H,,:ln~ 01~00@00~101 01~1O 10010 10101 10011 10100• •2. • 00000 01111 00010.. 24" OOll1®01~01 00~11O 01~11 OOlOO 10000 10111 10001 10111• •-2. • 01110 00~001®01100 11100 11011•-4 • (al (b) FIGURE 6.50(a)Signalconstdlation ofV.32modemusingnomedundant coding.(h)Signal constellation ofV.32modemusingtrelliscoding. Asanillustrative example ofhowthisparticular V.32modemoperates, letthecur rentgroupoffourinputbitsbe1001andthedibitpreviously outputbythemodembe 11.Forthisexample, wethushave Ql,nQ2,n =10 Q3,nQ4,n =01 I1,n-1I2,n-l =11 Theninlightofthecodingschemeforthefourquadrants specified inFigure6.49,the previous outputdibit11meansthatthemodulator waspreviously residinginthefirst quadrant. Becausethecorresponding inputdibitis10,itfollowsfromTable6.10thatthe modulator experiences aphasechangeof1800inthecounterclockwise direction, thereby switching itsoperation intothethirdquadrant identified bythedibit00.Finally,withthe currentvalueoftheleastsignificant dibitQ3,nQ4,n being01,themodulator outputsa QAMsignalwhosecoordinates arean=-3(alongthe4>l·axis)andbn=-1(alongthe 4>raxis). Thisoutputcorresponds tothecodeword0001. Whenthesignal-to-noise ratioisnothighenough,theV.32modemswitchestoits QPSKmode,operating atthereducedrateof4,800b/s.Inthislattermodeofoperation, thefourstatesofthemodemaresignified bythepointslabeledA,B,C,andDinFigure 6.50a. TrellisCoding Trelliscodingisaforward-error correction schemewherecodingandmodulaci°haretreatedasacombined entityratherthanastwoseparate operations. Figure6,48 6.11Voiceband Modems 425 showstheencoding systemoftheV.32modemwithtrelliscoding.Theincoming data streamisdividedintoquadbits, butunlikethecaseofnonredundant coding,theyare transmitted overthechannelasa32-QAM signal. Asindicated inFigure6ASb,thetrellisencoderinvolves theuseofaconvolu tionalencoder, whichoperates ontheoutputofthedifferential encoder. (Convolutional encoders arediscussed inChapter10.)However, thechoiceofconvolutional encoding is restricted intheV.32modemtoaccommodate theuseofdifferential encoding (i.e.,90 degreesrotational invariance). Indeed,thisrequirement cannotbesatisfied byalinear convolutional encoder. Rather,theconvolutional encodermustbenonlinear;'4 seeProb lem10.30. Thedata-encoding processintheV.32modemwithtrelliscodingproceeds in three stages: 1.Thedifferential encoderinFigure6ASb,inresponse tothecurrentinputdibit Q"nQ2,n andtheprevious differentially encodeddibitI"n-,I2,n-1> produces thedibit I "nI2,no 2.Thedifferentially encodedcurrentdibitI ".!".isinputtotheconvolutional encoder inFigure6048b,whichproduces athree-bit output.Oneofthesebitsisaparity checkbit,denotedbyYO,n'ThevalueofYO,ndependsontheothertwobits,Y "nand Y2""produced bytheconvolutional encoder. 3.ThebitsYO,n,YI,nandY2,nproduced bytheconvolutional encoder, togetherwith theleastsignificant inputdibitQ3,nQ4,n areappliedtothesignal-space mapperin Figure6ASb,whichselectsoneofthestatesinthe32-point constellation shownin Figure6.50basthemodemoutput. Theparity-check bitYO•nprovides amodemwithtrelliscodingbetterimmunity to channelimpairments thanaV.32modemwithnonredundant coding,anadvantage that isgainedwithoutanincreaseinbandwidth requirements. Inquantitative terms,trellis codingprovides aneffective codinggainof4dBcompared to16-QAM. Codinggain expresses howmuchmoresignalenergyperdatabitisneededbytheuncoded modemfor thesamelevelofnoiseperformance. However, forthisadvantage oftrelliscodingtoberealizedinpractice, thesignal to-noise ratiomustbehighenough. Otherwise, theV.32modemisswitched toits QPSKmodeofoperation, whichissignified bythefourstateslabeledA,B,C,andD inFigure6.50b.Inthislattermodeofoperation, thedatarateofthemodemisreduced to4,800b/s. !!IIAsYMMETRIC MODEM CONFIGURATIONS ForamoreefficientuseofthePSTN,weshouldtreatitaswhatitreallyis:analmost entirelydigitalnetwork thatisnonlinear. Inparticular, sincetheISPisdigitallyimple mented,theneedforanalog-to-digital conversion attheISPmodemiseliminated. This meansthatthecommunication between theISPandthePSTNcanbeentirelydigital,as portrayed inFigure6047b.However, theuser'smodemhastoremainanalogbecausethe localloopisanalog.This,inturn,requirestheuseofanalog-to-digital anddigital-to-analog conversions eachtimetheuser'smodemsendssignalstoandreceivessignalsfromthe PSTN.Themodemconfiguration depicted inFigure6047bis"asymmetric" inthatitis possibleforthedownstream signaling dataratetobemuchhigherthantheupstream signaling datarate,asexplained next. 426 CHAPTER 6IIIPASSBAND DATATRANSMISSION Asmentioned earlier,adigitalPS1NisbasedontheuseofPCMforthetransmissi ofvoicesignals.Features ofthesystemrelevanttothepresentdiscussion areasfollo~~ (seeChapter3): I>Datasignaling rateof64kb/s,whichismadeupofasampling rateof8kHzand therepresentation ofeachvoicesamplebyan8-bitcodeword. I>Fifteen-segment companding law(e.g.,alogaritlunic wlawwithJ.L=255)forCom_ pressingthevoicesignalatthetransmitter andexpanding itatthereceiver. Fromthediscussion onPCMpresented inChapter3wealsorecallthatquantization only affectsanalog-to-digital conversion butnotdigital-to-analog conversion. Theseobserva_ tionshaveaprofound impactontheoptimum strategyforthedesignofasymmetric modems. Supposethereisnoanalog-to-digital conversion betweenadigitalmodemattheISP andthedigitalportionofthePSTN,andthedigitallyconnected transmitter ofthemodem isdesigned toproperly usethenonuniformly spaced256(discrete) threshold levelsofthe digitalPS1N.Then,sincedigital-to-analog conversion iscompletely unaffected byquan tizationnoise,itfollowsthattheinformation transmitted bytheISP'sdigitalmodem reachestheuser'sanalogmodemwithnolosswhatsoever. Onthebasisofthesearguments, intheory,itshouldbepossibletotransmit datafromtheISPtotheuseratarateequalto the64kb/sdatarateofthedigitalPS1N.Butsystemlimitations inherent tothePSlN reducetheattainable dataratedownto56kb/s,asexplained inthesequel. DigitalModem Fromthedescription ofaPCMvoiceband channelpresented inChapter3,wefind thatthedesignofthedigitalmodemisconstrained bythreefactorsnotunderourcontrol. Thedesignconstraints are: 1.Asampling ratef,=8kHz. 2.AsetofM=256allowable threshold levelsbuiltintotheconstruction ofthecom pressor(i.e.,transmitter portionofthecompander). 3.Abaseband (antialiasing) filterofabout3.5kHzbandwidth, builtintothefrontend ofthePCMtransmitter. Inlightoftheseconstraints, wemaynowstatethefundamental philosophy underlying the designofthedigitalmodemasfollows: Designasignals(t)atthedigitalmodem's inputsuchthateachofitssamplestaken attheratef,=8kHzmatchesoneoftheM=256threshold levelsofthecom pressor,andthetransmitted signalsatisfiesNyquist's criterionforzerointersymbol interference. (Nyquist's criterion forzerointersymbol interference wasdiscussed inChapter4.) OneRealization oftkeDigitalModem Asolutiontothissignaldesignproblem ismadeparticularly difficultbythefactl~l thePCMtransmitfilterhasabandwidth ofabout3.5kHzandnot4kHz(halfthesamphng ratef,).Theimmediate implication ofthisconstraint isthatinsteadofthedesiredseto~ 8,000samples,wecanonlygenerate 2X3,500=7,000independent sampleseverysecon inaccordance withNyquist's criterion forzerointersymbol interference. Howthendowe 6.11Voiceband Modems 427 • • I~~~:~. • • • • I • • • • • •• •Time -'6",_-'-"·,_-' -C- 1,--<'I""'O-'''2'~O -'·3,0-'4·,0----<'5 ....,0--<'6,>-0-,..7'O->c-I-'IO-,,--<'2 ....,,-'-03,~1-'·4,,-'5·,1---<'6 ....,1--<'7,....,--'--'·1,2- , (N+I)T,=IOOO}', FIGURE 6.51GroupofNuniformly spacedsamples, repeating every(N+l)T.seconds, fit7,000independent samplespersecondwithintheprescribed framework of8,000sam plespersecond? Toanswerthisfundamental question, wemakeuseoftherecurrent nonuniform equivalent formofthesampling theorem. Tobemorespecific,consider thesituation de pictedinFigure6.51,wherethesamplesaredividedintogroups,witheachgroupcon tainingNuniformly spacedsamples, andthegroupshavingarecurrent periodof(N+1)T, seconds, where7',=1/fs.Theillustration presented inFigure6.51isfortheproblem at hand:7',=125J1!5andN=7.Thesampling instantsinthenonuniform distribution of Figure6.51arewrittenas tk,l=tk+(N+1)/7', =(k-1)7',+(N+1)/7'"k=1,2,..., N I=0,±1,±2,..,(6.187) Thestageisnowsetforustodefinetheband-limited signals(t)asfollowsY N s(t)=2:2:S(tk,l)o/dt -(N+1)/7',) l~-= k~l(6.188) (6.189) fork=1,2,...,7wheretheinterpolation function o/k(t)isitselfdefinedby )Nsin(N'ITl)T(t-tq)), (t-tk (+, o/k(t)=SInC(N+1)7',D,(7r ) q*ksm(N+1)7',(tk-tq) Computing Equation (6.189)forN=7,weobtainthesevenstandard pulsesplottedin Figure6.52,wheretimeisnormalized withrespecttothesampling period7',.Thesepulses exhibitthefollowing properties: ' I>-Eachstandard pulseisnormalized sothatwehave o/{~)=o/k(k-1)=1 Note,however, thatthepeakofthekthpulsedoesnotoccurattimetk=(k-l)T" I>Fork=1,2,...,7thepulseo/k(tiT.)goesthroughzeroattimest*(k1)7',modulo (N+1),exceptatthosetimesthatarecongruent tot=(-1)modulo(N+1). Accordingly, thesignaling schemeforthedigitalmodemconsistsofarecurrent non uniformpulseamplitude modulation scheme.Theamplitudes ofsevenuniformly spaced samplesineachgroupofeightsamplesaredetermined bytheincoming datastreamand inconformity tothethreshold levelsofthecompressor inthePCMtransmitter. Ineffect, thesesevensamplesaretheindependent samplesthatareresponsible forcarrying the 10428 CHAPTER 6..PASSBAND DATATRANSMISSION J30j3 ~~Ff0Ej o 5 10 0 5 10·,:bEE ~J+39 o 5 10 0 5 10 ~:RAB"'-----'-- ~~E-A o 5 10 0 5 -J:P+=1 o 5 10 FIGLJRE6.52Adigitalmodem'swavefonns ofthestandardpulseslfik(t),k=1,2,...,7. incoming datastreamacrossthePSTNevery1,000p.,s.Moreover, theydeliverthedatato thereceiverwithzerointersymbol interference. Theremaining "eighth" samplesarecom· pletelydetermined bytheindependent samplesandknownbeforehand tothesystem;they donotcarryinformation andaretherefore discarded atthereceiver. Thusthedigital modemiscapableoftransmitting digitaldataacrossthePSTNalmosterrorless atarate equalto56kbls,whichiscalculated asfollows: 7X1,000Xlog2256=56,000bls Onelastcomment isinorder.Thestandard pulsesrfJ,,(t)canbeconstructed soasto decayataratefasterthanlIt.Todoso,wesimplyreplacethesinefunctioninEquation (6.189)byaNyquistpulsewitharolloHinamannersimilartothatdescribed inChap ter4. Another Re~H%ation oftheDigitalModem Thekindofdigitalmodemjustdescribed isbidirectional, assuming thatbothends ofthedatalinkareanalog.However, asimplersolution tothedigitalmodem ~esi~ problemensueswhenoneendofthelinkisdigitalandasymmetric dataratesarepossible. Consider whathappens whenadatasequence consisting ofoctets(i.e.,8-bitcode words)arrivesatthePSTN.Theretheywillbetreatedasoctetsrepresenting speechen codedinaccordance withthep.,-IaworA-law,depending onthepartoftheworldwhere 6.11Voiceband Modems 429 thePSTNislocated.Consequently, theD/Aconverter, whichdrivestheanalogmodem, produces acontinuous-time signaldefinedby s(t)=2:a(ck)g(t-kTsl (6.190) k where Ckisthekthoctetinthedatasequence, a(Ck)istherepresentation levelspecified by thepertinent companding law,T"isthesampling interval(equalto125f.Ls),andg(t)isan interpolation function bandlimited toafrequency below1/2T",orabout4kHz,tosatisfy thereconstruction partofthesampling theorem; seeSection3.2. Inthenormaloperation ofthePSTN,thesignals(t)represents areconstructed speech signal.However, inthecaseofinputdata,s(t)appearslikenoise.Inanyevent,froma communication theoretical perspective, thesignals(t)inEquation (6.190)maybeviewed asapulse-amplitude modulated signal.Hereinliesthetheoretical basisforthedesignof thedigitalmodem.Specifically, thedesignisbasedonasignalconstellation asinananalog modem, exceptthattheconstellation isconstructed fromone-dimensional PCMsymbols ratherthantwo-dimensional QAMsymbols. Ordinarily, thedatarateachievable byadigitalmodemislimitedtoabout56kbls becauseofthefollowing factors: 1.Theinnedevels ofthecompander inthePSTNareverycloselyspaced,asshownin Table3.4;hencetheyaresusceptible toresidualintersymbol interference andnoise following themodem's equalizer. 2.Leastsignificant bits(LSBs)arerobbedfromthedatastreamforvariouspurposes internaltothePSTN;this"bit-robbing" canbeasmuchas(butusuallylessthan) 8kb/sandalwaysinaperiodic pattern. AnalogModem Unlikethedigitalmodem, thenoiseperformance oftheanalogmodemislimited essentially byquantization noiseinthef.L-laworA-lawgoverning theoperation ofthe PCMcompander. Typically, thesignal-to-noise ratioonagoodPCMvoiceband channel isontheorderof34to38dB.Theotherchannelimpairment thatlimitstheoperation of theanalogmodemistheeffectofbandlimiting imposed bytheantialiasing andinterpo lationfilters,which,asalreadymentioned, istypically about3.5kHz. Asophisticated choicefortheanalogmodemisthestandard V.34modem, which operates atratesextending upto33.6kb/s.Thefundamental designphilosophy ofthis modemembodies fivedistinctive features. 17 1.960-QAM super-constellation. Thesignalconstellation issaidtobeasuper-ornested-constellation inthatitconsistsof fourconstellations: theQAMconstellation showninFigure6.53with240messagepoints, anditsrotatedversionsthrough90,180,and270degrees. 2.Adaptive bandwidth. Thetransmitter probesthechannelbysendingasetoftones,whichpermitsmeasurement ofthesignal-to-noise ratioatthechanneloutputasafunction offrequency. Themodem istherebyenabledtoselecttheappropriate carrierfrequency andbandwidth according to theprobingresultsandavailable symbolrates. 3.Adaptive bitrates. Duringthetrainingofthereceiver, thebitrateisselectedaccording tothereceiver's esti mateofthemaximum bitrate,whichthemodemcansupportatbiterrorratesaslowas 10-6to10-5• 430 CHAPTER 6IIPASSBAND DATA'I'RAt"lSMISSION -39 -35 -31-27 -23 -19-15-11-7-3 2723 3119 3515113) 2529 21 1713 17 2125 29 3337 f.!II I I I(lin)I I I IIT-, 236224216 212 218228·•0•· ·r- 234205185 173 164162170 181 197 220 f- •·•·•· · ·•·226193165146133123121125 137 154179207·•·•· ·•· ·•· ·r- 229189 156 13111096873392100117140172208 f-·· ·0•· · · · ···•·2011601269879 64 5854627190112141180221··•···•···0··· ·r- 222177135102775541353137486591118 155 198 f-· ·0•···•·• •···•·203158 119 846039241715203049721011381822301· ···•·• •····•·•··r- 194148108755028136 4 8 21386393127171219 f-··••·•· ··• ••· · ··• 238186 142 1036943229 1 0 5 16325685122163213 l-••···· ·•·•· · · ·•· ··1901441~67.34525113271.,836598.81~4166217(Re)• •·•·•·• • •·f- -199 152 1138052331912 10 1426426697134 174 225 l-···•· ····• •· · · · ·• 2101671289467473427 23 29405781111147187237- f-•···· ·• ••· · · · ·• •· 232183149115896853464451617899132 168 209- l-•···•·• •c.•·• ••• • 214175 139 116 958274 70 76 86104 129 157 195 235- f-··•· ····•·•· ·0·- 205176 150 130114107105109 120 136161 191 227 l-···•·•·•···•· -215184169153145143151159178202231·• •· · · · ·•0·f- -233211200192188196204223 l-··· · ·•··- 239 C-o- I I I I I I37 2933 2125 17 13 -7-3 -15 -19-11 -27 -31 -35-23 -39 -35 -31-27 -23 -19-15-11-7-3 913172125 29 3337 FIGURE6.53Quarter-supereonstellation ofV.34modemwith240signa!points.ThefuJI,uper constellation isobtained bycombining therotatedversions ofthesepointsby0,90,180,and270 degrees. (TakenfromForneyetaJ.,1996,withpermission oftheIEEE.) 4.Trelliscoding. Thiserror-control codingtechnique isusedtoprovideaneffective codinggainofabout 3.6dB;thereisanoptional morepowerful trelliscodewithaneffective codinggainof about4.7dB. 5.Decision feedback equalization. Tomakefulluseoftheavailable telephone channelbandwidth, including frequencies near thebandedgeswheretherecanbeattenuation asmuchas10to20dB,adecisionfeedback equalizer (DFE)isused.(TheDFEisdiscussed inChapter4.)However, itisnotastraight forwardmattertocombine codingwithDFEbecausedecisionfeedback requiresimmediate decisions, whereascodinginherently involvesdecoding delay.Theovercome thisproblell1, thefeedback sectionoftheDFEismovedtothetransmitter, whichismadeposslb!e throughtheuseoftheTomlinson-Harashima precoding. (Thisformofequalization VIa precoding isdiscussed brieflyinNote12ofChapter4.) 6.12MulticJmnnel Modulation 431 V.90Modem TheV.90modemstandard embodies digiralandanalogmodems. Thedigitalmodem attheISPendisbasedonthesecondrealization described earlier;itsendsdatadownstream attherateof56kb/s.Theanalogmodemattheuser'sendisaV.34modemstandard, transmitting dataupstream attherateof33.6kb/s.Thesetwohighlydifferentratesconfirm theasymmetric natureoftheV.90modem. Theoutstanding featureoftheV.90modem, namely,thedownstream datarateof 56kb/s,makesitsuitableforuseontheInternetfordownloading graphics inintensive Webpages,audio,andvideoatnear-ISDN speeds. 13::.-1_2_M_"_I_tic_h_"_ff_ff_e_I_M_o_d_"_'_"_t_Wn_ Theasymmetric digitalsubscriber line(ADSL), described inSection4.8,isadatatrans missionsystemcapableofrealizing megabitratesoverexistingtwisted-pair telephone lines. Specifically, ADSLrunsatadownstream datarateupto9Mb/sandanupstream data rateupto1Mb/s.Thesedatasignaling ratesfittheaccessrequirements oftheInternet perfectly. (Asmentioned inSection4.8,theupstream bitrateshouldbeabout10percent ofthedownstream bitrateforefficientoperation oftheInternetprotocol.) Thechallenge indesigning ADSListodevelopalinecodethatexploitstheinformation capacity ofthe channelasfullyaspossible. Thecarrierless amplitude phasemodulation (CAP),discussed inSection6.4,provides oneapproach forsolvingthisdifficultpassband datatransmission problem. Another approach istouseanequallyelegantmodulation technique calleddis cretemultitone. Thislatterapproach isaformofmultichannel modulation'8thatallows themodulator characteristics tobeafunction ofmeasured channelcharacteristics. Itis fittingthatwebeginthediscussion bydescribing multichannel modulation, whichwedo inthissection,followed bydiscretemultitone inthenextsection. Thebasicideaofmultichannel modulation isrootedinacommonly usedengineering principle: divideandconquer. According tothisprinciple, adifficultproblem issolvedby dividingitintoanumberofsimplerproblems, andthencombining thesolutions tothose simpleproblems. Inthecontextofourpresentdiscussion, thedifficultproblem isthatof datatransmission overawidebandchannelwithsevereintersymbol interference, andthe simplerproblems areexemplified bydatatransmission overAWGNchannels. Wemay thussummarize theessenceofmultichannel modulation asfollows: Datatransmission overadifficultchannelistransformed throughtheuseofad vancedsignalprocessing techniques intotheparalleltransmission ofthegivendata streamoveralargenumberofsubchannels, suchthateachsubchannel maybe viewedeffectively asanAWGNchannel. Naturally, theoveralldatarateisthesumoftheindividual dataratesoverthesuhchannels operating inparallel. IICAPACITY OFAWGN CHANNEL FromtheBackground andPreviewmaterial presented intheopeningchapter, werecall that,according toShannon's information capacitytheorem, thecapacity ofanAWGN channel(thatisfreefromintersymhol interference) isdefinedby C=Blog2(1+SNR)b/s (6.191) 432 CHAPTER 6..PASSBAND DATATRANSMISSION whereBisthechannelbandwidth,. a~dSNRdenotesthesignal-to-noise ratiomeasureda thechanneloutput.AproofofthIsl1llportant theorem ISformally presented inChat 9.FornowitsufficestosaythatforagivenSNR,wecantransmit dataoveranAW~~ channelofbandwidth Batthemaximum rateofCbitspersecondwitharbitrarily smII probability oferror,provided thatweemployanencoding systemofsufficiently hi~ complexity. Equivalently, wemayexpressthecapacity Cinbitspertransmission orchan. neluseas 1C=2:log2(1+SNR) bitsltransmission(6.192) Inpractice, weusuallyfindthataphysically realizable encoding systemmusttransmit dataatarateRlessthanthemaximum possiblerateCforittobereliable.ForanimpJe. mentable systemoperating atlowenoughprobability ofsymbolerror,wethusneedto introduce asignal-to-noise ratiogaporjustgap,denotedbyr.Thegapisafunction of thepermissible probability ofsymbolerrorPcandtheencoding systemofinterest.Itpro videsameasure ofthe"efficiency" ofanencoding systemwithrespecttotheidealtrans missionsystemofEquation (6.192).WithCdenoting thecapacity oftheidealencodmg systemandRdenoting thecapacityofthecorresponding implementable encoding system thegapisdefinedby , 22C-1r=22R_1 SNR =22R-1(6.193) Equivalently, wemaywrite 1 ( SNR)R=2:log21+r bits/transmission (6.194) ForencodedPAMorQAMoperating atPc=10-6,forexample, thegaprisconstantat 8.8dB.Through theuseofcodes(e.g.,trelliscodesdiscussed inChapter10),thegapr maybereducedtoaslowas1dB. LetPdenotethetransmitted signalpower,andrrdenotethechannelnoisevariance measured overthebandwidth B.Thesignal-to-noise ratioistherefore SNR=!..rr where rr=NoB Wemaythusfinallydefinetheattainable datarateas R=110g2(1+r~) bits/transmission (6.195) Withthisformulaathand,wearereadytodescribemultichannel modulation inquanti tativeterms. iiiCONTINuous-TIME CHANNEL PARTITIONING Consider alinearwideband channel(e.g.,twistedpair)withanarbitrary frequency ct" sponseH(f).Letthesquaredmagnitude responseIH(f)Ibeapproximated byastajrca5e 6.12M..ltidumnelModul ..tion433 IH(j)I ---:----------------f FIGVRE6.54Staircase approximation ofanarbitrary magnitude responseIH(f)I;onlypositive frequency portionoftheresponse isshown. function asillustrated inFigure6.54,withI1fdenoting thewidthofeachstep.Inthelimit, asthefrequency increment I1fapproaches zero,thestaircase approximation ofthechannel approaches theactualH(!).Alongeachstepoftheapproximation, thechannelmaybe assumed tooperateasanAWGNchannelfreefromintersymbol interference. Theproblem oftransmitting asinglewideband signalistherebytransformed intothetransmission ofa setofnarrowband orthogonal signals.Eachnarrowband orthogonal signal,withitsown carrier,isgenerated usingaspectrally efficientmodulation technique suchasM-aryQAM, withadditivewhiteGaussian noisebeingessentially theonlyprimarysourceoftransmis sionimpairment. This,inturn,meansthatdatatransmission overeachsubchannel of bandwidth I1fcanbeoptimized byinvoking Shannon's information capacity theorem, withtheoptimization ofeachsubchannel beingperformed independently ofalltheothers. Thus,inpractical signal-processing terms,theneedforcomplicated equalization ofawide bandchannelisreplacedbytheneedformultiplexing anddemultiplexing thetransmission oftheincoming datastreamoveralargenumberofnarrowband subchannels thatare contiguous anddisjoint. Although theresulting complexity ofamulticarrier systemis indeedhighforalargenumberofsubchannels, implementation oftheentiresystemcan beaccomplished inacost-effective mannerthroughtheuseofVLSItechnology. Figure6.55showsablockdiagram ofthemultichannel datatransmission systemin itsmostbasicform.Thesystemisconfigured hereusingquadrature-amplitude modulation whosechoiceisjustifiedbyvirtueofitsspectralefficiency. Theincoming binarydatastream isfirstappliedtoademultiplexer (notshowninthefigure),therebyproducing asetofN substreams.Eachsubstreamrepresents asequence oftwo-element subsymbols, which,for thesymbolinterval0:s:t:s:T,isdenotedby n=1,2,...,N where anandbnareelementvaluesalongthetwocoordinates ofsubchannel n. Correspondingly, thepassband basisfunctions ofthequadrature-amplitude modu latorsaredefinedbythefunction pairs ("'(t)COS(21Tfnt), "'(t)sin(21Tfnt)], n=1,2,...,N (6.196) wherethecarrierfrequencyfnofthenthmodulator isanintegermultipleofthesymbol ratelIT,asshownby n fn=T'n=1,2,..., N 434 CHAPTER 6..PASSBAND DATATRANSMISSION Symbols ModulatorsMaximum likelihOOd detectors Transmitter Receiver FIGURE 6.55Blockdiagram ofmultichannel datatransmission system. (6.197) -oo<t<ooandthelow-pass function ¢(t)isthesinefunction: ¢(t)=ftSinc(~), Thepassband basisfunctions definedherehavethefollowing desirable properties (see Problem 6.41fortheirproofs): Property 1 Foreachn,thetwoqU4drature-modulated sincfunctions formanorthogonal pairas shownby fro(¢(t)cos(21l'fnt))(¢(t) sin(27Tfnt))dt=0foralln (6.198) Thisorthogonal relationship provides thebasisforformulating thesignalconstellation fot eachoftheNmodulators intheformofasquaredlattice. 6.12Multichannel Modulation 435 Property 2 Recognizing that exp(j2'Trf"t) =cos(2'Trlnt) +jsin(2'Trf"t) wemaycompletely redefinethepassband basisfunctions inthecomplex form {~cP(t)eXP(j2'Trlnt)},n=1,2,..., N (6.199) wherethefactor11V2hasbeenintroduced toensurethatthescaledfunction q,(t)/V2 hasunitenergy.Hence,thesepassband basisfunctions formanorthonormal set,asshown by wheretheasteriskdenotescomplex conjugation.k=n k"*n(6.200) Equation (6.200)provides themathematical basisfor ensuring thattheNmodulator demodulator pairsoperateindependently ofeachother. Property 3 Thesetofchannel-output functions (h(t)*q,(t)]remainsorthogonal foralinearchannel witharbitrary impulseresponse hit),where*denotesconvolution. Thechannelisthuspartitioned intoasetofindependent subchannels operating incontin uoustime. Figure6.55alsoincludes thestructure ofthereceiver. ItconsistsofabankofN coherent detectors, withthechanneloutputbeingsimultaneously appliedtothedetector inputs.Eachdetector issupplied withalocallygenerated pairofquadrature modulated sinefunctions operating insynchrony withthepairofpassband basisfunction appliedto thecorresponding modulator inthetransmitter. Eachsubchannel mayhavesomeresidualintersymbol interference (151).However, asthenumberofsubchannels Napproaches infinity,the151disappears. Thus,fora sufficiently largeN,thebankofcoherent detectors inFigure6.55operates asmaximum likelihood detectors, independently ofeachotherandonasubsymbol-by-subsymbol basis. Todefinethedetectoroutputsinresponse totheinputsubsymbols, wefinditcon venienttousecomplexnotation. LetAndenotethesubsymbol appliedtothenthmodulator duringthesymbolinterval0:;;t:;;T: An=an+jbn> Thecorresponding detectoroutputis Yn=HnAn+w,.,n=1,2,...,N n=1,2,...,N(6.201) (6.202) whereHnisthecomplex-valued frequency response ofthechannelevaluated atthesub channelcarrierfrequencyI=In: Hn=H(f,,), n=1,2,..., N (6.203) TheWnisacomplex-valued randomvariableduetothechannelnoisew(t);therealand imaginary partsofWnhavezeromeanandvariance No/2.Withknowledge ofthemea- (6.204) (6.205) (6.206) (6.207)436 CHAPTER 6!ilPASSBAND DATATRANSMISSION suredfrequency response H(f)available, wemaytherefore useEquation (6.202)tocom. puteamaximum likelihood estimate ofthetransmitted subsymbol An.TheestimatesA A2,•••,ANsoobtained arefinallymultiplexed toproducethecorresponding estimateof theoriginalbinarydatatransmitted duringtheinterval0:5t:5T. Tosummarize, forasufficiently largeN,wemayimplement thereceiverasanop. timummaximum likelihood detector, operating asNsubsymbol-by-subsymbol detectors Thereasonwhyitispossibletobuildamaximum likelihood receiverinsuchasimpleWa• isthefactthatthepassbandbasisfunctions constitute anorthonormal set,andtheir0:' thogonality ismaintained foranychannelimpulseresponse h(t). GEOMETRIC SIGNAL-TO-NOISE RATIO Inthemultichannel transmission systemofFigure6.55,eachsubchannel ischaracterized byaSNRofitsown.Itwouldbehighlydesirable toderiveasinglemeasure forthe performance oftheentiresystemofFigure6.55. Tosimplifythederivation ofsuchameasure, weassumethatallofthesubchannels inFigure6.55arerepresented byone-dimensional constellations. Thenthechannelca. pacityoftheentiresysteminbitspertransmission isgivenby 1NR=-LRNn=l n 1N (p) =2NLlog21+r\ .n=l Un1 N (P)=2Nlog2I11+rn2n=l Un =~IOg2[D(1+r~~)rm Let(SNR)ovemli denotetheoverallsignal-to-noise ratiooftheentiresystem.Wemaythen expressRinbitspertransmission as R-1I(1(SNR)oYCnlI)-2:0g2+r Comparing Equations (6.205)with(6.204),wemaythuswrite (N( P)11N)(SNR)oYecall =rD1+r;~-j Assuming thatPnlrtT~ishighenoughtoignorethetwounitytermsinEquation (6.206), wemayapproximate theoverallSNRas (SNR)=D(:~)liN Wemaythuscharacterize theoverallsystembyaSNRthatisthegeometric meanofthe SNRsoftheindividual subchannels. Thegeometric SNRofEquation (6.207)canbeimproved considerably bydisrri~ utingtheavailable transmit poweramongtheNsubchannels onanonuniform basis.This objective isattainedthroughtheuseofloadingasdiscussed next. 6.12MultkhannelMod..lation 437 IJjjLOADING OFTHEMULTICHANNEL TRANSMISSION SYSTEM Equation (6.204)forthebitrateoftheentiremultichannel systemignorestheeffectofthe channelonsystemperformance. Toaccountforthiseffect,define n=1,2,..., N (6.208) Thenassuming thatthenumberofsubchannels Nislargeenough,wemayassumethat gnisconstant overtheentirebandwidth lifassigned tosubchannel nforalln.Insucha case,wemaymodifythesecondlineofEquation (6.204)fortheoverallSNRofthesystem as 1N ( ;,z»R=2N2:logz1+gf; n=l Un(6.209) (6.210)Theg;;andfareusuallyfixed.Thenoisevarianceu;;islifNoforalln,wherelifisthe bandwidth ofeachsubchannel andNo/2isthenoisepowerspectral density.Wemay therefore optimize theoverallbitrateRthroughaproperallocation ofthetotaltransmit poweramongthevariouschannels. However, forthisoptimization tobeofpractical value, wemustmaintain thetotaltransmit poweratsomeconstant valueP,say,asshownby N 2:Pn=P=constant n=l Theoptimization wetherefore havetodealwithisaconstrained optimization problem, whichmaybestatedasfollows: Maximize thebitrateRfortheentiremultichannel transmission systemthrough anoptimalsharingofthetotaltransmitpowerPbetweentheNsubchannels, subject totheconstraint thatPismaintained constant. Tosolvethisoptimization problem, wefirstusethemethodofLagrange multipliers19to setupanobjective functionthatincorporates theconstraint ofEquation (6.210),asshown by (6.211)1N ( g;,z>n) ( N )J=2N~1logz1+fu;;+i\P-~1Pn 1N ( ;,z»(N)=2Nlogze~1log.1+~u~+i\P -~1Pn wherei\istheLagrange multiplier. Hence,differentiating JwithrespecttoPnothensetting theresultequaltozeroandfinallyrearranging terms,weget 1 2Nlogze -----=:---:;-=i\ Pfu;; n+-zgn(6.212) (6.213) forn=1,2,..., NThisresultindicates thatthe solution toourconstrained optimization problem istohave Pfu;;=K n+zgn 438 CHAPTER 6IIIPASSBAND DATATRANSMISSION whereKisaprescribed constant underthedesigner's control.Thatis,thesumofth transmit powerandthenoisevariance (power)scaledbytheratioflg~mustbemaintain ~ constant foreachsubchannel. Theprocessofallocating thetransmit powerPtotheie_ dividualsubchannels soastomaximize thebitrateoftheentiremultichannel transmiss: systemiscalledloading. n IIIWATER-FILLING INTERPRETATION OFTIlEOPTIMIZATION PROBLEM Insolvingtheconstrained optimization problem justdescribed, twoconditions mustb satisfied, namely,Equations (6.210)and(6.213).Theoptimum solutionsodefinedhas~ interesting interpretation asillustrated inFigure6.56forN=6,assuming thatthegapr isconstant overallthesubchannels. Tosimplifytheillustration inFigure6.56WehaveSet lT~=NoD.f=1,thatis,theaveragenoisepowerisunityforallNsubchannels. Referring tothisfigure,wemaynowmakethefollowing observations: ""ThesumofpowerPnallocated tochannelnandthescalednoisepowerfig;;satisfies theconstraint ofEquation (6.213)forfour ofthesubchannels foraprescribed value oftheconstant K. ~Thesumofpowerallocations tothesefoursubchannels consumes alltheavailable transmit power,maintained attheconstant valueP. ...Theremaining twosubchannels havebeeneliminated fromconsideration because theywouldeachrequirenegativepowertosatisfyEquation (6.213)fortheprescribed valueoftheconstant K;thiscondition isclearlyunacceptable. Theinterpretation illustrated inFigure6.56prompts ustorefertotheoptimum solution ofEquation (6.213), subjecttotheconstraint ofEquation (6.210), asthewater-filling solution. Thisterminology followsfromanalogyofouroptimization problemwithafixed amountofwater(standing fortransmit power)beingpouredintoacontainer witha numberofconnected regions,eachhavingadifferent depth(standing fornoisepower). Thewaterdistributes itselfinsuchawaythataconstant waterlevelisattainedacrossthe wholecontainer. Wehavemoretosayonthewater-filling interpretation ofinformation capacity inChapter9. Returning tothetaskofhowtoallocatethefixedtransmit powerPamongthe varioussubchannels ofamultichannel transmission systemsoastooptimize thebitrate FIGURE6.56Water-filling interpretation oftheloadingproblem. 6.12MultichannelModulation 439 oftheentiresystem,wemayproceedasfollows.Letthetotaltransmit powerbefixedat theconstant valuePasinEquation (6.210).LetKdenotetheconstant valueprescribed forthesumPn+r(T~/g~forallnasinEquation (6.213).Wemaythenusethispairof equations tosetupthefollowing systemofsimultaneous equations: PI+P2+...PN=P PI-K =-r~/gi P2-K =-r~/g~ =-r~/gF.r(6.214) wherewehaveatotalof(N+1)unknowns and(N+1)equations tosolveforthem.We mayrewritethissetofsimultaneous equations inmatrixformas 1 1 10PI P 100-1P2-r(T2/gi 01 0 -1-r(T2/g~ (6.215) PN 0 0 1-1K -r~/g~ Premultiplying bothsidesofEquation (6.215)bytheinverseofthe(N+l)-by-(N+1) matrixontheleft-hand sideoftheequation, weobtainsolutions fortheunknowns P1, P2,•••,PN,andK.WeshouldalwaysfindthatKispositive, butitispossible forsome ofthePstobenegative. Thenegative Psarediscarded aspowercannotbenegative. to-EXAMPLE 6.7 Consideralinearchannelwhosesquaredmagnitude response IH(f) 12hasthepiecewise-linear formshowninFigure6.57.Tosimplifytheexample, wesetthegapr=1andthenoise variancerr=1.Inthesituationsodescribed, theapplication ofEquation (6.214)yields P,+P,=P P,-K=-1 P2-K=-11/ IH(fll' 1.0 -----',----------',-------:-----:----'---1-1,-1, 0I,I, FIGURE6.57Squaredmagnitude responseforExample6.7. 440 CHAPTER 6 "PASSBAND DATATRANSMISSION 12 10 8 10 4 2 1o1 2 Indexofsubchannel' n FIGURE6.58Water-filling profileforExample 6.7. wherethetotalttansmit powerPisnormalized withrespecttothenoisevariance. Solving thesethreesimultaneous equations forP"P2,andK,weget P,=Hp-1+7) P2=Hp+1-7) K=i(p+1+7) Since0<I<1,itfollowsthatP,>0,butitispossibleforP2tobenegative. Thislatter condition canariseif 1<_1_ P+1 ButthenP,exceedstheprescribed valueoftransmitpowerP.Itfollowstherefore thatinthis example theonlyacceptable solution istohavel/(P+1)<I<1.Supposethenwehave P=10andI=0.1,forwhichthesolutionis K=10.5 P,=9.5 P2=0.5 Thecorresponding water-filling pictureisportrayed inFigure6.58. I6.13Discrete Multitone Thematerialpresented inSection6.12provides aninsightful introduction tothenotiOD ofmultichannel modulation. Inparticular, thecontinuous-time channel partitionin~ ~. ducedbythepassband basisfunctions ofEquation (6.196)orequivalently (6.199)exhibJlS 6.13Discrete Multittme 441 ahighlydesirable property: Orthogonality ofthebasisfunctions (andtherefore thechannel partitioning) ispreserved despitetheirconvolution withtheimpulseresponse ofthechan nel.However, thesystemhastwoshortcomings: 1.Thepassband basisfunctions useasincfunction thatisnonzeroforaninfinitetime interval, whereaspractical considerations favorafiniteobservation interval. 2.Forafinitenumberofsubchannels, N,thesystemissuboptimal; optimality ofthe systemisassuredonlywhenNapproaches infinity. Wemayovercome theseshortcomings byusingdiscretemultitane (DMT),thebasic ideaofwhichistotransform awidebandchannelintoasetofNsubchannels operating inparallel.WhatmakesDMTdistinctive isthefactthatthetransformation isperformed indiscretetimeaswellasdiscretefrequency. Consequently, thetransmitter input-output behavior oftheentirecommunicatioll_ systemadmitsalinearmatrixrepresentation, which lendsitselftoimplementation usingthediscreteFouriertransform. Toexplorethisnewapproach, wefirstrecognize thatinarealisticsituation the channelhasitsnonzero impulseresponse, h(t),essentially confined toafiniteinterval [0,Tb].So,letthesequence ho,hI'...,hvdenotethebaseband equivalent impulseresponse ofthechannelsampledattherateliT"with Tb=(1+p)T, (6.216) Thesampling rateliT,ischosentobegreaterthantwicethehigherfrequency component ofinterestinaccordance withthesampling theorem. Tocontinue withthediscrete-time description ofthesystem,lets[n]=s(nT,)denoteasampleofthetransmitted symbols(t), turn]=w(nT,)denoteasampleofthechannelnoisew(t),andx[n]=x(nT,)denotethe corresponding sampleofthechanneloutput(received signal).Thechannelperforms linear convolution ontheincoming symbolsequence {s[n]}oflengthN,producing achannel outputsequence {x[n]}oflengthN+P.Extension ofthechanneloutputsequence byP samplescompared tothechannelinputsequence isduetotheintersymbol interference produced bythechannel. Toovercome theeffectofintersymbol interference, wecreateacyclically extended guardintervalwhereby eachsymbolsequence ispreceded byaperiodic extension ofthe sequence itself.Specifically, thelastvsamplesofthesymbolsequence arerepeated atthe beginning ofthesequence beingtransmitted, asshownby s[-k]=s[N-k]fork=1,2,...,P (6.217) Thiscondition iscalledacyclicprefix.Theexcessbandwidth factorduetotheinclusion ofthecyclicprefixisthereforevlN,whereNisthenumberoftransmitted samplesafter theguardinterval. Withthecyclicprefixinplace,thematrixdescription ofthechanneltakestheform x[N-l] x(N-2] x[N-v-l] x[N-v-2] x[O]=It:f:I•••:f:'T,'f:...II":~~~:'l+I}~~~~:'l] hv0 0 0 0 ho'"hV-1s[N-v-2] w[N-v-2] :::::::: : hIh2h3hv00·..ho s[O] wlO](6.218) Equivalently, wemaydescribethediscrete-time representation ofthechannelinthecom pactmatrixform x=Hs+w (6.219) 442 CHAPTER 6OJPASSBAND DATATRANSMISSION w FIGURE6.59Discrete-time representation ofmultichannel datatransmission system. wherethetransmitted symbolvectors,thechannelnoisevectorw,andthereceivedsignal vectorxareallN-by-lvectorswhicharerespectively definedby s=[s[N-1],s[N-2],,slOW w=[w[N-1],w[N-2],,w[oW(6.220) (6.221) and o oox=[x[N-1],x[N-2],...,x[OW (6.222) Wemaythusdepictthediscrete-time representation ofthechannelasinFigure6.59.The N-by-NchannelmatrixHisdefinedby hoh,hz ohoh, H=0 0 0 hv0 0o o(6.223) hvo o Fromthisdefinition, wereadilyseethatthematrixHhasthefollowing structural com position: Everyrowofthematrixisobtained byapplying aright-shift totheprevious row byoneposition, withtheaddedprovisothattherightmost elementoftheprevious row spillsoverintheshiftingprocesstobe"circulated" backtotheleftmost elementofthe newrow.Accordingly, thematrixHisreferredtoasacirculant matrix. Beforeproceeding further,itisbefitting thatwebrieflyreviewthediscreteFourier transform anditsroleinthespectraldecomposition ofthecirculant matrixH. l;DISCRETE FOURIER TRANSFORM Consider theN-by-lvectorxofEquation (6.222).ThediscreteFouriertransform (DFT) ofthevectorxisdefinedbytheN-by-lvector x=[X[N-1],X[N-2],...,X[OW (6.224) where (6.226)(6.225) k=0,1,..., N - 1 n=0,1,..., N - 11N-I(2)X[k]=.""2:x[n]exp-j-!!.kn,vNn~1 N Theexponential termexp{-j2TrknIN) isreferredtoasthekerneloftheDFT.Correspond ingly,theinversediscreteFouriertransform (IDFT)oftheN-by-lvectorXisdefinedby1N-I(2)x[n]=.""2:X[k]expj-!!.kn,vNk~O N 6.13Discrete Mulfitone 443 Although Equations (6.225)and(6.226)appeartobesimilar,theyhavedifferent inter pretations. Giventhesignalvectorx,Equation (6.225)provides aspectralrepresentation ofthesignalcomputed atasetofdiscretefrequencies: h=kIN,whicharenormalized withrespecttothesampling rate.Giventhetransformed vectorX,Equation (6.226)re coverstheoriginalsignalvectorx.Wemaytherefore viewEquation (6.225)astheanalysis equation andEquation (6.226)asthesynthesis equation. Animportant property ofacirculant matrix,exemplified bythechannelmatrixH ofEquation (6.223),isthatitpermitsspectraldecomposition asshownby H=QTAQ (6.227) wherethesuperscript TdenotesHermitian transposition (i.e.,thecombination ofcomplex conjugation andordinary matrixtransposition). Descriptions ofthematricesQandAare presented inthesequelinthatorder. ThematrixQisasquarematrixdefinedintermsofthekerneloftheN-pointDFT asfollows: exp(-j~(N-1)) 11Q=y'N(2rrexp-jN(N (2rrexp-jN(Nl)(N l)(N1))exp(-j~2(N-1)) 2))exp(-j~2(N2)) (.2rr)exp-IN2 1exp(-j~(N-1)) exp(-j~(N-2)) exp(-j~) 1 (6.228) (6.229) (k,I)=0,1,..., N - 1Fromthisdefinition, wereadilyseethatthekithelementoftheN-by-Nmatrix,Q,starting fromthebottomrightatk=0and1=0andcounting upstep-by-step, is 1(27T)qkl=y'Nexp-jNkl, ThematrixQisanorthonormal matrixorunitarymatrixinthatitsatisfiesthecondition QTQ=I (6.230) whereIistheidentitymatrix.Thatis,theinversematrixofQisequaltotheHermitian transpose ofQ. ThematrixAisadiagonal matrixthatcontains theNdiscreteFouriertransform valuesofthesequence ho,hb•••,hvcharacterizing thechannel. Denoting thesetransform valuesbyAN-i,...,A1)Ao,wemayexpressAas [A~'0 AN-2A= : 0 0(6.231) (TheAsherearenottobeconfused withtheLagrange multipliers inSection6.12.) TheDFThasestablished itselfasoneoftheprincipal toolsofdigitalsignalprocessing byvirtueofitsefficientcomputation usingthefastFouriertransform (FFT)algorithm.20 444 CHAPTIlR 6IIPASSRAND DATATRANSMISSION Specifically, theFFTalgorithm requiresontheorderofNlog2Noperations ratherth theN2operations fordirectcomputation oftheDFT.Forefficientimplementation of~n FFTalgorithm, weshouldchoosetheblocklengthNanintegerpoweroftwo.Thecoe putational savingsobtained byusingtheFFTalgorithm aremadepossiblebyexploitt thespecialstructure oftheDFTdefinedinEquation (6.225).Moreover, thesesavinng becomemoresubstantial asweincreasethedatalengthN. gs IiliFREQUENCY-DOMAIN DESCRIPTION OFTHECHANNEL Withthisbriefdescription oftheDFTonhand,wearereadytoresumeourdiscussion of discretemultitone. First,wedefine s=Qts (6.232) whereSisthefrequency-domain vectorrepresentation ofthetransmitter input.Eachele mentoftheN-by-1vectorSmaybeviewedasacomplex-valued pointinatwo-dimen_ sionalQAMsignalconstellation. Giventhechanneloutputvector X,wedefineitscorre sponding frequency-domain representation as X=Qx (6.233) UsingEquations (6.227),(6.232)and(6.233),wemayrewriteEquation (6.219)inthe equivalent form Hence,usingtherelationofEquation (6.230),wesimplyget X=AS+W where W=Qw Inexpanded form,Equation (6.235)readsas(6.234) (6.235) (6.236) k=0,1,..., N - 1 (6.237) wherethesetoffrequency-domain values{AkW=-O'isknownforaprescribed channeL Forachannelwithadditivewhitenoise,Equation (6.237)impliesthatthereceiver iscomposed ofasetofindependent processors operating inparallel.WiththeAkallknown, wemaythususetheblockoffrequency-domain values {Xk}~dtocompute estimates of thecorresponding transmitted blockoffrequency domain-values {SkH",,::-O'. IIDFf-BASED DMTSYSTEM Equations (6.235), (6.225), (6.226),and(6.237)providethemathematical basisforthe implementation ofDMTusingtheDFT.Figure6.60illustrates theblockdiagram ofthe systemderivedfromtheseequations andtheirpractical implications. Thetransmitter consistsofthefollowing functional blocks: P>Demultiplexer, whichconverts theincoming serialdatastreamintoparallelforlD. I>Constellation encoder, whichmapstheparalleldataintoNI2multibitsubchannels witheachsubchannel beingrepresented byaQAMsignalconstellation. Bitallocat~on amongthesubchannels isalsoperformed hereinaccordance withaloading algorithm. 6.13Discrete Multitane 445 Binary dala input TransmitterInverse discrete Fourier transformerDiscrete Fourier transformerEstimateofthe originalbinary datainput Receiver Parallel-to serial converter and guard-interval providerSerial-to parallel converter and guard-interval remover FIGURE6.60Blockdiagramofthediscrete-multitone (DMT)data-transmission system. ~InversediscreteFouriertransformer (IDFT),whichtransforms thefrequency-domain paralleldataattheconstellation encoderoutputintoparalleltime-domain data.For efficientimplementation oftheIDFTusingthefastFouriertransform (FFT)algo rithm,weneedtochooseN=2kwherekisapositiveinteger. po.Parallel-to-serial converter, whichconverts theparalleltime-domain dataintoserial form.Guardintervals stuffedwithcyclicprefixesareinsertedintotheserialdataon aperiodicbasisbeforeconversion intoanalogform. •Digital-to-analog converter (DAC),whichconvertsthedigitaldataintoanalogform readyfortransmission overthechannel. Typically, theDACincludesatransmit filter.Accordingly, thetimefunctionh(t)should beredefined asthecombined impulseresponse ofthecascadeconnection ofthetransmit filterandthechannel. Thereceiverperforms theinverseoperations ofthetransmitter, asdescribed here: '"Analog-to-digital converter (ADC),whichconverts theanalogchanneloutputinto digitalform. po.Serial-to-parallel converter, whichconvertstheresulting bitstreamintoparallelform. Beforethisconversion takesplace,theguardintervals (cyclicprefixes) areremoved. ~DiscreteFouriertransformer (DFT),whichtransforms thetime-domain paralleldata intofrequency-domain paralleldata;aswiththeIDFT,theFFTalgorithm isusedto implement theDFT. 446 CHAPTER 6OJPASSBAND DATATRANSMISSION II>Decoder, whichusestheDFToutputtocompute estimates oftheoriginalmulti-b' subchannel datasuppliedtothetransmitter. It '"Multiplexer, whichcombines theestimates socomputed toproduceareconstructi ofthetransmitted serialdatastream. on I!liAPPLICATIONS OFDMT Animportant application ofDMTisinthetransmission ofdataovertwo-way channels Indeed,DMThasbeenstandardized foruseonasymmetric digitalsubscriber lines(ADSls) usingtwistedpairs.TheADSLwasdescribed inChapter4.Forexample, DMTprovides forthetransmission ofdatadownstream (i.e.,fromanInternetserviceprovider toasub scriber)attheDS1rateof1.544Mb/sandthesimultaneous transmission ofdataupstream (i.e.,fromthe subscriber totheInternetserviceprovider) at160kb/s.Thiskindofdata transmission capability iswellsuitedforhandling data-intensive applications suchas video-an-demand. DMTisalsoacoretechnology inimplementing theasymmetric very-high-rate digital subscriber lines21(VDSL),whichdiffersfromallotherDSLtransmission techniques be causeofitsabilitytodeliverextremely highdatarates.Forexample, VDSLcanprovide dataratesof13to26Mb/sdownstream and2to3MBisupstream overtwistedpairsthat emanate fromanopticalnetwork unitandconnect tothe subscriber overdistances less thanabout1km.ThesehighdataratesallowthedeliveryofdigitalTV,super-fast Web surfingandfiletransfer, andvirtualofficesathome. TheuseofDMTforADSLandVDSLprovides anumberofadvantages: '"Theabilitytomaximize thetransmitted bitrate,whichisprovided bytailoringthe distribution ofinformation-bearing signalsacrossthechannelaccording tochannel attenuation andnoiseconditions. I>Adaptivity tochanging lineconditions, whichisrealizedbyvirtueofthefactthat thechannelispartitioned intoanumberofsubchannels. fl>Reduced sensitivity toimpulsenoise,whichisachieved byspreading itsenergyover themanysubchannels ofthereceiver. Asthenameimplies,impulsenoiseischar acterized bylong,quietintervals followed bynarrowpulsesofrandomly varying amplitude. InanADSLorVDSLenvironment, impulsenoisearisesduetoswitching transients coupledtowirepairsinthecentralofficeandtovariouselectrical devices ontheuser'spremises. IIICOMPARISON OFDIGITAL SUBSCRIBER LINESANDVOICEBAND MODEMS InSection6.11wediscussed voiceband modemsthatarealreadyclosetooperating attheir theoretical limitsof33.6kb/supstream and56kb/sdownstream. Inthissectionwehave discussed theapplication ofDMTtoVDSLsthatcanoperateatdataratesofabout2to 3Mb/supstream and13to26Mb/sdownstream. Thesetwovastlydifferent setsofup stream/downstream dataratespromptthefollowing question: HowisitpossibleforVDSL tooperateatratesaboutthreeordersofmagnitude fasterthanvoiceband modems o~er thesametwistedpairs(i.e.,phonelines)?Thereasonforthisvastdifference inoperatlllg dataratesbetweenvoiceband modemsandVDSLsisnotthetwistedpairs;rather,itisthe digitalswitches builtintoapublicswitched telephone network thatpreventthetran~p.o~ ofbroadband datatosubscribers (users)viavoiceband modems. Simplyput,the,dlgl~ switches treatdigitaldatainthesamewayasvoicesignalsforwhichtheyareprunarY designed. 6.13DiscreteM ..ltitotw 447 User's environment1--------------1 1 1 1 1I'--_----...J 1 1 1 1 1 1I ...J (a) User'senvironment,---------------------1 1 1 1 1 1 1 1 1 1 1 1 L J eb) FIGURE6.61(a)Voiceband modemenvironment. (b)xDSL(digitalsubscriber line)environ ment,wherexstandsforHas)'Dlmetric" or"veryhigh-rate." Figure6.61highlights theoperational environments ofvoiceband modems and xDSLs,wherexstandsforAinADSLandVinVDSL.InthemodelofFigure6.61a pertaining toavoiceband modem,wehavearelatively longtransmission pathbetweenan Internetserviceprovider (ISP)andasubscriber. Mostimportantly, thetransmission path traverses throughanarrowband publicswitched telephone network (PSTN),whichlimits theavailable channelbandwidth toabout3.5kHz.Incontrast, inthemodelofFigure 6.61bpertaining toxDSL,thetransmission pathaccommodates thetransport ofbroad banddatabetween theISPandsubscriber viaabroadband integrated servicesdigital network andarelatively shortlocalloopconsisting ofatwistedpair.Thesystempermits thecoexistence ofPOTSandxDSLsignalsonthesamelocalloop,whichismadepossible throughtheuseofapairofsplitters, asindicated inFigure6.61b;splitters, consisting of bidirectional low-pass andhigh-pass filters,arediscussed inSection4.8. IIIORTHOGONAL FREQUENCY DIVISION MULTIPLEXING22 Discretemultitone isoneparticular discreteformofmultichannel modulation. Another closelyrelatedformofthismethodofmodulation isorthogonal frequency-division mul tiplexing (OFDM) thatdiffersfromDMTinareasofapplication andsomeaspectsofits design. OFDMisusedfordatatransmission overradiobroadcast channels andwireless communication channels. Thisdomainofapplication requiressomechanges tothedesign 448 CHAPTER 6IIPASSBAND DATATRANSMISSION oftheOFDMsystem.UnlikeDMTthatusesloadingforbitallocation, OFDM Use fixednumberofbitspersubchannel. Thisrestriction ismadenecessary bythefacttha:a broadcast channelinvolvesone-way transmission, andinawirelesscommunications e~ vironment thechannelisvaryingtoorapidly.Accordingly, inbothcasesitisnotfeasib~ forthetransmitter toknowthechannelandhowto"load"it. e Thus,theblockdiagramofFigure6.60appliesequallytoOFDMexceptforthefact thatthesignalconstellation encoderdoesnotincludealoadingalgorithm forbitallocation Inaddition, twootherchangeshavetobemadetothedesignofthesystem: . ;>Inthetransmitter, anupconverter isincluded afterthedigital-to-analog converteno translate thetransmitted frequency, therebyfacilitating thepropagation ofthetrans. mittedsignaloveraradiochannel. ~Inthereceiver,adownconverter isincluded beforetheanalog-to-digital converter to undothefrequency translation thatwasperformed bytheupconverter inthe transmitter. Applications ofOFDMincludethefollowing: 1.Wirelesscommunications. OFDM,combined withcodingandinterleaving, provides aneffectivetechnique tocombat multipath fadingthatisacharacteristic featureofwirelesscommunication channels. 2.Digitalaudiobroadcasting. OFDMhasbeenadoptedasthestandard fordigitalaudiobroadcasting inEurope.Here againthesysteminvolvesthecombined useofcodingandinterleaving. (Error-control codingandrelatedissuesarediscussed inChapter10.) I6.14Synchroniz.ation Thecoherent reception ofadigitally modulated signal,irrespective ofitsform,requires thatthereceiverbesynchronous tothetransmitter. Wesaythattwosequences ofevents (representing atransmitter andareceiver) aresynchronous relativetoeachotherwhenthe eventsinonesequence andthecorresponding eventsintheotheroccursimultaneously. Theprocessofmakingasituation synchronous, andmaintaining itinthiscondition, is calledsynchronization.23 Fromthediscussion presented ontheoperation ofdigitalmodulation techniques, we recognize theneedfortwobasicmodesofsynchronization: 1.Whencoherent detection isused,knowledge ofboththefrequency andphaseofthe carrierisnecessary. Theestimation ofcarrierphaseandfrequency iscalledcarrier recovery orcarriersynchronization. 2.Toperform demodulation, thereceiverhastoknowtheinstantsoftimeatwhich themodulation canchangeitsstate.Thatis,ithastoknowthestartingandfinishing timesoftheindividual symbols, sothatitmaydetermine whentosampleandwhen toquenchtheproduct-integrators. Theestimation ofthesetimesiscalledclockreo coveryorsymbolsynchronization. Thesetwomodesofsynchronization canbecoincident witheachother,ortheycanoc.cur sequentially oneaftertheother.Naturally, inanoncoherent system,carriersynchroJllZ3' tionisofnoconcern. 6.14Synchronization 449 Synchronization canbeimplemented inoneoftwofundamentally different ways: 1.Data-aided synchronization. Indata-aided synchronization systems, apreamble istransmitted alongwiththedata bearingsignalinatime-multiplexed manneronaperiodic basis.Thepreamble contains information aboutthecarrierandsymboltiming,whichisextracted byappropriate pro cessingofthechalU1eloutputatthereceiver.Suchanapproach iscommonly usedindigital satelliteandwirelesscommunications, wherethemotivation istominimize thetimere quiredtosynchronize thereceivertothetransmitter. Itslimitations aretwo-fold: (1)re duceddata-throughput efficiency thatisincurred byassigning acertainportionofeach transmitted frametothepreamble, and(2)reducedpowerefficiency byallocating acertain fractionofthetransmitted powertothetransmission ofthepreamble. 2.Nondata-aided synchronization. Inthissecondapproach, theuseofapreamble isavoided, andthereceiverhasthetaskof establishing synchronization byextracting thenecessary information fromthemodulated signaLBoththroughput andpowerefficiency aretherebyimproved but attheexpenseof anincreaseinthetimetakentoestablish synchronization. Inanyevent,synchronization isbasically astatistical parameter estimation problem. Aprincipled approach forsolvingsuchaproblem ismaximum likelihood estimation (see Section5.5),whichproceeds byfirstformulating alog-likelihood functionoftheparameter ofinterestgiventhereceivedsignaLThisformulation isrelatively straightforward bytreat ingthechalU1elnoiseasaGaussian process.Mostimportant, itrequiresnopriorinfor mationaboutthemodulated signaL Inthissectionweconfineourattention tonondata-aided formsofcarrierandtiming synchronization systems.Inthiscontext,wemayidentifytwoapproaches forsolvingthe synchronization problem, givenamodulated signalwithsuppressed carriertoconserve power: 1.Classical approach. Anessential buildingblockintheclassicalapproach tosynchronization isthephase-locked loop.(Thephase-locked loopwasdiscussed inChapter2.)Specifically, forcarrierrecovery thereceiverrequirestheuseofasuppressed-carrier trackingloopforproviding acoherent secondary carrier(subcarrier) reference. Forexample, wemayuseavariantoftheCostas looportheMthpowerloopforM-aryPSK.Thestandard Costasloopfordoublesideband suppressed carrier(DSB-SC) modulation wasdiscussed inChapter2.AsfortheMthpower loop,itconsistsofthecascadeconnection ofanMthpower-law device,band-pass filter, phase-locked loop,andfrequency dividerbyM.Theobjective hereistoexploittheac quisition andttacking properties ofthephaseclocked loop.Forfurtherdiscussion ofthe Mthpowerloop,thereaderisreferredtoProblem 6.47. 2.Algorithmic (modern) approach. Inthemodernapproach, thesolutiontomaximum likelihood estimation isformulated in algorithmic formusingdiscrete-time signalprocessing. Specifically, implementation ofthe synchronizer isbuiltonanalgorithm thatprovides anestimate ofcarrierphaseorsymbol timingonaniteration-by-iteration basis.Theprocessing isperformed inthebaseband domaintopavethewayfortheuseofdiscrete-time (digital)signalprocessing. Inthissectionwedescribe thealgorithmic approach tosynchronization forM-aryPSK systemsforbothcarrierrecovery andsymbol-timing recovery. 450 CHAPTER 6..PASSRAND DATATRANSMISSION Theapproach takenintheexposition issequential inthattimingrecovery isp formedbeforephaserecovery. Thereasonforsodoingisthatifweknowthegroupdetr. incurred bytransmission throughthechannel, thenonesamplepersymbolatthematc~~ filteroutputinthereceiverissufficient forestimating theunknown carrierphase.Mar over,thecomputational complexity ofthereceiverisminimized byusingsynchronizatiof' algorithms thatoperateatthesymbolrate1fT. n DECISION-DIRECTED RECURSIVE ALGORfIHM FORPHASE RECOVERY Asremarked earlier,thefirstimportant stepinsolvingthesynchronization problem ist formulate thelog-likelihood function forthecarrierphase&,giventheGaussian noise~ contaminated received signal.Let1(&)denotethislog-likelihood function, whichservesas theobjective function forestimating &.Thenextstepistodetermine thederivative of1(0) withrespectto&.Thefinalstepistoformulate arecursive (iterative) algorithm forcom. putingamaximum likelihood estimate oftheunknown &inastep-by-step manner. Evaluation ofiH(0)/0fJ" O:=;t:=;TLetSk(t)denotethetransmitted signalforsymbolk=0,1,..., M -1: f2Esdt)={Tcos(27rtt+O'k), whereEisthesymbolenergy,Tisthesymbol period, and 27r 27r O'k=0,M '...,(M-1)M Equivalently, wemaywrite f2ESk(t)={Tcos(27rtt+O'k)g(t)(6.238) (6.239) (6.240) (6.241) (6.242)whereg(t)istheshapingpulse,namely,arectangular pulseofunitamplitude andduration T.Let'Tcdenotethecarrier(phase)delay,and'Tgdenotetheenvelope (group)delay,both ofwhichareintroduced bythechannel. Bydefinition, 'Tcaffectsthecarrierand'Tgaffects th,eenvelope. Thenthereceived signalis f2Ex(t)={rcos(27rt(t -'Tel+O'k)g(t-'Tg)+w(t) f2E=VTcos(27rfJ+&+O'k)g(t-'Tg)+w(t) wherew(t)isthechannel noiseand&isdefinedas-27rfc'Tctobeconsistent withthe notation inSection6.6.Boththecarrierphase&andgroupdelay 'Tgareunknown. HoW' ever,itisassumed thattheyremainconstant overtheobservation interval0:=;t~Toor through thetransmission ofLo=TofTsymbols. Equivalently, wemaywrite(usingrin placeof'Tgtosimplifymatters) f2E x(t)=VTcos(27rfct+&+O'k)+wit), 'Areaderwhoisnotinterested intheformalderivation ofa/(e)la8mayomitthissubsection andmoveoneath' nextsubsectionwithout lossofcontinuity. 6.14Synchronizafion 451 Atthereceiverthebasisfunctions aredefinedby cPi(t)=~COS(21Tfct), fl. cP2(t)={fsm(21Tfct),(6.243) (6.244) Hereitisassumed thatthereceiverhasperfectknowledge ofthecarrierfrequency fc; otherwise, acarrierfrequency offsethastobeincluded, whichcomplicates theanalysis. Accordingly, wemayrepresent thereceived signalx(t)bythevector whereX(T)=[X1(T)] X2(T) fT+T Xi(T)=TX(t)cPi(t) dt,i=1,2(6.245) (6.246) Inacorresponding fashion,wemayexpressthesignalcomponent ofx(t)bythevector whereakisthetransmitted symboland IT!Tf2E si(ak,e,T)= T{TCOS(21Tfct+e+C<k)cPi(t)dtfori=1,2 Assuming thatfcisanintegermultiple ofthesymbolrateliT,wehave si(abe,T)=\IEcos(e+C<k) sZ(abe,T)=-\IEsin(e+C<k) Wemaythuswrite wherewisthenoisevector w=[::] with(6.247) (6.248) (6.249) (6.250) (6.251) (6.252) Wif+TW(t)cPi(t) dt,i=1,2 (6.253) (6.254)TheWiisthesamplevalueofaGaussian randomvariable Wofzeromeanandvariance No/2,whereNo/2isthe(two-sided) powerspectraldensityofthechannelnoisew(t). Theconditional probability densityfunction oftherandomvectorX,giventhetrans missionofsymbolakandtheoccurrence ofcarrierphaseeandgroupdelayT,is fx(xlab e,T)=1T~Oexp(-~oIIXdT)-slabe,T)liZ) 452 CHAPTER 6II!PASSBAND DATATRANSMISSION IIs(ab0,'1') 112)Forak=0thereceivedsignalx(t)equalsthechannelnoisew(t),so tx(xlak =0)=-Nlexp(-~ IIXk(T) 112) 1r0No Hencewemaydefinethelikelihood function forM-aryPSKatthereceiveras L(0)=tx(xlab 0,'1') ab,'1'tx(xlak =0) =exp(~oXnT)s(ab 0,'1')-~o(6.255) (6.256) InM-aryPSK, (6.259)IIs(ab0,'1')II=constant asthemessagepointslieonacircleofradiusv'E.Hence,ignoring thesecondterminthe exponent, wemaysimplifythelikelihood function as L(ab0,'1')=exp(~oXnT)s(ab 0,'1')) (6.257) Assuming thatwetransmit asequence ofLostatistically independent symbols, namely, a=lao,at>.•.,aLa-IV (6.258) theresulting likelihood function is L(a,0,'1')=In'eXP(N2 Xk(T)s(ab 0,'1')) k~O 0 Thelog-likelihood function istherefore I(a,0,'1')=logL(a,0,'1') 2Lo-I =N2:XnT)s(ab 0,'1') ok=O(6.260) (6.261) k=0,1,...,La-1FromEquations (6.249)and(6.250)wededuce Sk(0)=stab0,'1') =v'E[cos(ak+0)] -sin(ak+0), whereakisanestimate oftheactualOIkproduced atthedetectoroutputforthesymbol ak.Correspondingly, wemayexpressthematched filteroutputas x.[_XI,k] X2,k Hence,usingthisdefinition andEquation (6.261)inEquation (6.260),weget 2v'ELo-I 1(0)=~N2:[X"kcos(ak+0)+X2,ksin(ak+0)] ak~O 2v'ELo-I (6.262)=~N2:[(XI,kcosak+X2,ksinak)cos0 ak~a (X"ksinak-X2,kcosak)sin0] 6.14Synchrrmi:z;ution 453 Differentiating lie)withrespecttoe,weobtain al(e)2YELo-l--=--NL[(Xl,kcosak+X2,ksinak)sineae 0k~O (6.263) +(Xl,ksinakX2,kcosak)cose] WemaysimplifyEquation (6.263)byintroducing thefollowing notations: Xk=Xl,k+jX2,k (6.264) and ak=ejak =cosctk+jsinctk(6.265) (6.266)whereXkisthecomplex envelope (i.e.,baseband value)ofthematched filteroutputdue tothekthtransmitted symbol,andakisasymbolindicator inthemessage constellation oftheM-aryPSK.Wemaythuswrite Re[dtxk] =Re[(cosak-jsinak)(xl,k+jX2,k)] =Xl,kcosak+X2,ksinak Im[dtXk]=Im[(cosak-jsinak)(xl,k+jX2,k)] = -Xl,ksinak+X2,kcosak WemayalsonotefromEuler'sformula: e-jO=cose-jsine Accordingly, wemayrewriteEquation (6.263)inthecompact form: al(e)2YEL~l * .'" * .'"- = L..{(Re[akxk])(Im[e- 1J)+(Im[akxk])(Re[e- 1J))aeNo k~O 2YELo-l •. =--LIm[aZxke-l"j No k~O whereakisanestimate ofak>andtheasteriskdenotescomplex conjugation.(6.267) (6.268) (6.269) (6.270)RECURSIVE ALGORITHM FORMAxIMUM LIKELIHOOD ESTIMATION OFTHECARRIER PHASE WiththeformulaofEquation (6.269)forthederivative ofthelog-likelihood functionlie) withrespecttothecarrierphaseeathand,wearenowreadytoformulate analgorithm thatseekstomaximize lie).Wewouldliketoperform themaximization inaniterative fashionsothatthereceiverisenabledtorespondtothereceivedsignalonasymbol-by symbolbasis.Tothatend,wemaybuildonthefollowing algorithmic ideaborrowed from adaptive filtering (seethediscussions ontheLMSalgorithm presented inChapters 3 and4): (UPdated) (Old) (Step-size) (Error) estimate =estimate+parameter signal wheretheerrorsignal,ortheadjustment signaltobemoreprecise,isdefinedasthe instantaneous valueofthegradient ofthelog-likelihood functionl(e)withrespecttoe. Notethattheparameter adjustment appliedtotheoldestimate inEquation (6.270)is 454 CHAPTER 6illPASSBAND DATATRANSMISSION Complexenvelopeof matchedfilteroutput attimet=nT: I I III I I l 1 Loopfiller", FIGURE6.62Recursive Costasloop. positiveastheobjective hereistoperformgradient ascent.FromEquation (6.269)we readilyseethattheerrorsignal(i.e.,theinstantaneous valueofal(e)/aeduetothetrans missionofasinglesymbol)isgivenby ern]=Im[a:x"e-iB] (6.271) wherethescalingfactor2YE/Noisaccounted forinwhatfollows.Also,wehaveusedn inplaceofktodenoteatimesteporiteration ofthealgorithm. Accordingly, weuse Equation (6.270)towrite ern+1]=ern]+ye[n] (6.272) whereern]istheoldestimate ofthecarrierphasee,ern+1]istheupdatedestimateof e,andyisthestep-sizeparameter; thescalingfactor2YE/Noisabsorbed iny. Equations (6.271)and(6.272)definetherecursive algorithm forphaserecovery. This algorithm isimplemented usingthesystemshowninFigure6.62,whichmaybeviewed asarecursive generalization oftheCostasloop.Wemaytherefore refertoitastherecursIVe Costasloopforphasesynchronization. Thefollowing pointsshouldbenotedinFigure6.62: I>-Thedetector suppliesanestimate ofthetransmitted symbolamgiventhematched filteroutput. 1>0Thelook-uptablesuppliesthevalueofexp(-je[n]) =cose[nJ-sinern]foran inputern]. 1>0Theoutputoftheerrorgenerator istheerrorsignale[n]_ I>Theblocklabeled Z-lisaunit-delay elementwiththedelayequaltothesymbol periodT. Therecursive CostasloopofFigure6.62usesafirst-order digitalfilter.Toimprove thetrackingperformance ofthissynchronization systemwemayuseasecond-order digital filter.Figure6.63showsanexample ofasecond-order digitalfiltermadeupofacascade oftwofirst-order sections, withpasanadjustable loopparameter. Animportant property ofasecond-order filterusedintheCostasloopforphaserecovery isthatitwilleventually lockontotheincoming carrierwithnostaticerror,provided thatthefrequency errOl betweenthereceiverandtransmitter isinitiallysmall. 6.14Sym;h......i_ti.... 455 p Input sequence FIGURE6.63Second-order digitalfilter.Output sequence IIINONDATA-MDED RECURSIVE ALGORITHM FORSYMBOL TIMING Fortimingsynchronization theonlyassumption madeisthatthereceiverhasknowledge ofthecarrierfrequency /C.Therequirement istodevelopanalgorithm forrecursive esti mationofthegroupdelay Tincurred inthecourseoftransmitting themodulated signal throughthechannel. LetL(ak)e,T)denotethelikelihood function ofT,whichisalsoafunction oftrans mittedsymbolakandcarrierphasee.Thelikelihood function isdefinedbyEquation (6.257).Toproceedfurtherwemustremovethedependencies ofL(ak)e,T)onthetrans mitteddatasequence {ak}andcarrierphasee,asdescribed next. Toremovethedependence oneweaveragethelikelihood functionL(ak)e,T),but notitslogarithm, overallpossiblevaluesofeinsidetherange[0,217].Assuming thateis uniformly distributed insidethisrange,whichisusuallyjustifiable, wemaywrite (2~ Lav(ak)T)=)0L(ak)e,T)f,,(e)de 1{2rr(2 ) =217)0expNoxJ:(T)s(ak) e,T)de Theexponent inL(ak)e,T)isexpressed by(seeProblem 6.49) 2T 2YE <_-eNoXk(T)s(ak) e,T)=NoRe[akxk(T)e'] 2YE=NoRe[Iakxk(T)! exp(j(arg[Xk(T)] -arg[ak]-e))](6.273) 2YE =Nolakxk(T)Icos(arg[xk(T)] arg[ak] Ii) Hence, 1{2~(2YE )Lav(ak)T)=217)0expNolakxdT)Icos(arg[xk(T)] -arg[ak]-e)de 1J2~-a,g[£k(TIJ+a,g[a,] (2YE ) = exp--lakxdT)1 cos('f')d'f'27T-arglxk( ..)j+arglak] No where,inthelastline,wehavemadethesubstitution(6.274) (jl=arg[xdT)] -arg[ad-a Wenowinvokethedefinition ofthemodified Besselfunction ofzeroorder,asshownby (seeAppendix 3) (6.275) 456 CHAPTER 6I!1lPASSBAND DATATRANSMISSION Hence,wemayexpresstheaveragelikelihood function L.v(ak, 'T)as (6.276) wherexk(T)isthecomplex envelope ofthematched filteroutputinthereceiverduetolh kthtransmitted symbol ak'ForM-aryPSK,wehave e lakl=1 Hence,Equation (6.276)reducestoforallk (6.277) Wethusseethataveraging thelikelihood function overthecarrierphase()hasalsore moveddependence onthetransmitted symbol akforM-aryPSK. Finally,takingaccountofthetransmission ofLoindependent symbols ao,"1>..., a'r"wemayexpresstheoveralllikelihood function ofTas Lo-l Lav(T)=ITL.v(ak>T) k=O (6.278) Nowwecantakethelogarithm ofLav(T)toobtainthelog-likelihood function ofTas (6.279) Toproceedfurther,weneedtoapproximate lav(T).Tothatendwefirstnotethatthe modified Besselfunction Io(x)maybeexpanded inapowerseriesas(seeAppendix 3) _Gxrm Io(x)=~o(m!f Forsmallvaluesofxwemaythusapproximate Io(x)as forsmallx4Wemayfurthersimplifymattersbyusingtheapproximation logIo(x)=log(1+~2) x2 Fortheproblem athand,smallxcorresponds tosmallsignal-to-noise ratio.Underthis condition, wemayapproximate Equation (6.279)as (6.280) (6.281) (6.282)6.14Synchroni;z;ation 457 where,asmentioned earlier,Xk(T)isthecomplex envelope ofthematched filteroutput duetothekthtransmitted symbol. Differentiating Iav('T)withrespecttothegroupdelay'T,weobtain alav('T)_E'~1a1_( ) 12--;;;--N~1:-20a:;:xk'T 2E"0-1 =N22:Re[xZ('T)Xk('T)]ok~O wherex;('T)isthecomplex conjugate ofXk('T)andXk('T)isitsderivative withrespectto'T. Accordingly, wemaydefinetheerrorsignalfortimingrecovery as(accounting forthe scalingfactor2E1Ni;inwhatfollows) ern]=Re[x~('T)x~('T)] wherewehaveusedninplaceofktobeconsistent withthenotationinFigure6.62.Let Tndenotetheestimate oftheunknown delay'Tattimet=nT.Then,introducing the definitions and X~('T)=x'(nT+Tn) wemayreformulate theerrorsignale(n)as ern]=Re[x*(nT+Tn)x'(nT+Tn)] Calculation oftheerrorsignalern]requirestheuseoftwofilters: 1.Thecomplex matched filterforgeneratingxn('T). 2.Thederivative matched filterforgenerating x~('T). Thereceiver isalreadyequipped withthefirstfilter.Thesecondoneisnew.Inpractice, theadditional computational complexity duetothederivative matched filterisobjection able.Wemaydispensewiththeneedforitbyusingafinitedifference toapproximate the derivativex~('T)as x'(nT+Tn)=~[x(nT+f+Tn+112)-x(nT-f+Tn-1I2)1(6.283) whereTn±1/2arethetimingestimates computed atnT±T12.Itisdesirable tomakeone furthermodification toaccountforthefactthattimingestimates areupdatedatmultiples ofthesymbolperiodTandtheonlyavailable quantities areTn.Consequently, wereplace 1'''+112byTn(whichrepresents thelatestestimateof'T)andreplaceTn-112byTn-l(whichis theestimateof'Tbeforethelastone).WemaythusrewriteEquation (6.283)as x'(nT+Tn)=~[x(nT+f+Tn)-x(nTf+Tn-I)] (6.284) andsofinallyredefinetheerrorsignalas ern]=Re{x*(nT+Tn{x(nT+f+Tn)-x(nT-f+Tn-I)]}(6.285) wherethescalingfactorliTisalsoaccounted forinwhatfollows. Wearenowreadytoformulate therecursive algorithm fortimingrecovery: e[n+1]=ern]+ye[n] (6.286) Sampleat t=nT+T n (6.287)458 CHAPTER 6.,PASSBAND DATATRA,"SMISSION Complexenvelope of matchedfilteroutput----;>--0 attImet: x(t) FIGVRE6.64Nondata-aided early-late delaysynchronizer. where')' isthestep-size parameter inwhich 2EIN~andliTareabsorbed, andtheerror signalern]isdefinedbyEquation (6.285).Thec[n]isarealnumberemployed asthecontrol forthefrequency ofanosdllator, referredtoasanumber-controlled oscillator (NCO). Theschemeforimplementing thetimingrecovery algorithm ofEquations (6.285)and (6.286)isshowninFigure6.64.Thisschemeisanalogous tothecontinuous-time version oftheearly-late gatesynchronizer widelyusedfortimingrecovery. Itisthusreferredto asanondata-aided early-late delay(NDA-ELD) synchronizer. Ateveryiteration, itworks onthreesuccessive samples ofthematched filteroutput,namely,x(nT+f+T,,), x(nT+Tn)andx(nT+f-Tn-I).Thefirstsampleisearlyandthelastoneislate, bothwithrespecttothemiddleone. Notethatwecouldhavesimplified thederivations presented inthissectionbyusing theband-pass tocomplex low-pass transformation described inAppendix 2.Wedidnot dosomerelyforthesakeofsimplifying theunderstanding ofthematerial presented here. 6.15Computer Experiments: Carrier Recovery andSymbolTiming Inthissectionweillustrate theoperations oftherecursive Costasloopandnondata-aided early-late delaysynchronizer byconsidering acoherent QPSKsystemwiththefollowing specifications: (i)Channel response: raisedcosine(Nyquist) withrolloHfactora=0.5. (ii)Loopfilter:first-order digitalfilterwithitstransferfunction definedby 1 H(z)=z~(1-')'A) where')'isthestep-size parameter andAisaparameter tobedefined. (iii)Loopbandwidth, BL=2%ofthesymbolrateliT;thatis,BLT=0.02. 6.15Computer Experiments 459 , Experiment 1:CarrierPhaseRecovery Inordertoinvestigate thephase-acquisition behavior oftherecursive Costasloop,weneed tohavetheso-called S-curveofthephase-error generator. Thisisdefinedastheexpectation oftheadjustment signalern],conditioned onafixedvalueofthephaseerror Ip=()-e where()istheactualvalueofthecarrierphaseandeisitsestimate. Thatis, S(Ip)=E[e[n]lip] (6.288) Experimentally, S(rp)ismeasured byopeningtherecursive CostasloopofFigure6.62and measuring theaverageoftheadjustment signalern],asindicated inFigure6.65. Theimplementation procedure consistsofthefollowing steps.First,thecomplex envelope ofthereceivedsignalisgenerated, whichisgivenby (6.289) whereak=0,7r/2,7r,37r/4;TcisthecarrierdelayandTgisthegroupdelay;andiii(t)is thecomplex-valued channelnoise.Theoverallchannelresponse g(t)isgivenbytheNy quistpulse(seeSection4.5) ( ) _sin(7rtIT). cos(7ratlT) gt-(7rtIT) 1 -4a2t21T2(6.290) wherea=0.5.Aspointedoutearlier,weassumethatthesymboltiming(i.e.,group delayTg)isknown,andtheproblem istoestimate thecarrierphase()=-27rfcT c'The effectof()istoshiftanelementofthesignalconstellation inthemannerindicated inFigure 6.66. Usingtheexperimental procedure described inFigure6.65,theS-curveoftheQPSK systemmaynowbemeasured. Figure6.67ashowstheidealS-curve,assuming aninfinitely largesignal-to-noise ratio.Thiscurvedisplays discontinuities atrp=±m7r/4, where m=0,1,3,..., because ofambiguity encountered inthedetection ofthetransmitted exp(-jih S(<f» FIGURE6.65Schemeformeasuring theS-curveforcarrierphaserecovery. 460 CHAPTER 6'"PASSBAND DATA'TRANSMISSION Decision boundary "".0X//\ / \ I J I I / / / Decision boundary •Regularstatesoftransmitted QPSK oDisplaced stateofreceivedQPSK FIGURE6.66Illustrating theeffectofcarrierphase0onastateoftheQPSKsignal. symbolak.Thepresence ofchannelnoisetendstoroundoff thediscontinuities, asshown intheexperimentally measured S-curveofFigure6.67b.Theresultspresented inFigure 6.67bwereobtained forEINo=10dB.Recallthatthein-phase andquadrature compo nentsofthenarrowband noisehaveanidentical Gaussian distribution withzeromean andthesamevariance astheoriginalnarrowband noise;thesetwocomponents define w(t). -0.5 Co-1'--_-l__--"'"-'--'----'-- __-'---__L__-l__--"-__~__ -4-3-2-1 0 1 3 4 (fI,radians0.5 0.3r---,---,------,r--...,..---,--~~-_r--.___-_. 0.2 0.1 9-0 "'"-0.1 -0.2 -0.3 -0·~'---_-"3,-----_..,2----'----':0---'-1 ~--:---~3---4L----:5 rp,radians FIGURE6.67Performance ofrecursive Costasloop.(a)S-curvefor(EINo)=00.(b)S-curvefor (EINo)=lOdB. 6.15Computer Experinumts 461 ~",g0.3 ~ "lid0.4 il: 0.5 0.6·1--r=0.51--r=O.1....... ,-."." . (6.291)0.7O:---;CIO=-----:2-=-O---:3C:O,---------'c40:-----:5c:-O--6C:0,--------=-70:-----:a'=0--g:':0,-------,JI 00 Normalized time,tiT FIGURE6.68Effectsofvaryingthestep-size parameter onconvergence behavior oftherecur siveCostasloop. Whensteady-state conditions havebeenestablished, theestimated phase8willfluc tuatearoundthetruevalue8.Theextentofthesefluctuations depends onthestep-size parameter 'Yandthereceivedsignal-to-noise ratio: (i)Figure6.68plotsthephaseerrorcpversusthenormalized timetiTfortwodiHerent valuesofstep-size parameter 'Y,namely,0.1and0.5,andfixedEINo=20dB.This figureclearlyshowsthatthesmallerwemake'Ythesmallerthesteady-state fluctu ationsinthephaseerrorcpwillbe.However, thisimprovement isattained atthe expenseofaslowerrateofconvergence ofthealgorithm. Thenumberofiterations neededbythealgorithm toreachsteady-state isapproximately givenby L__1_ 0-2BLT Thenormalized bandwidth BLTisitselfapproximately givenby BLT='YA 4(6.292) whereAistheslopeoftheS-curvemeasured attheongm.For'Y=0.1,and BLT=0.02,Equation (6.291)yieldsLo=25iterations, whichcheckswiththesolid curveplottedinFigure6.68.Moreover, fromEquations (6.291)and(6.292)wesee thatLoisinversely proportional to'Y,whichagaincheckswiththeresultspresented inFigure6.68. (ii)Figure6.69plotsthephaseerrorcpversusthenormalized timetiTforthreedifferent valuesofEINo,namely,S, 10,and30dB,andfixed'Y=0.08.Wenowseethatthe largerwemakethesignal-to-noise ratio,thesmallerthesteady-state fluctuations in thephaseerrorcpwillbe.Moreover, therateofconvergence ofthealgorithm also improves withincreased signal-to-noise ratio,whichisintuitively satisfying. 462 CHAPTER 6..PASSBAND DATATRANSMISSION 1.2r---~---'-------'---~--~---, 0.2-_.EINo=5dB '-'-EINo=10dB --EIN,=30dB (6.293)-0.2oL----~20---4~0--~60---80~--1~0-0---..J120 Normalized time,tiT FIGURE6.69Convergence behavior oftherecursive CoslasloopforvaryingElNo. Figure6.70plotsthevariance ofthephaseerror(averaged over100trialsofthe experiment) versusE/No(measured indecibels) forBLT=0.02and'Y=0.08.Thisfigure alsoincludesaplotofthemodified Cramer-Rao bounddefinedby24 1 MCRB(e) =2Lo(E/No) Thisboundisamodification oftheordinary Cramer-Rao bound,whichisalowerbound onthevariance ofanyunbiased estimator. Themodification tothisboundismadeto overcome computational difficulties encountered inpractical synchroni2ation problems. 10-2,r----~---c---____,---~---_, ---e--Tracking---MCRB QPSK.BLT= 0.02 10-5 5:---~1""0----:-15=----2"'0'----""2::5'-----::30 EINo'dB FIGURE6.70Comparison ofthemeasured tracking-error variance oftherecursive Costasloop againsttheoryforvaryingE/No• 6.15Computer Experiments 463 Complexenvelope t=nT+;:/1 ofmatched filter---i!>--O outputattimet: xl') S(8) FIGUR£ 6.71Scheme formeasuring the.'i-curvefortherecursive early-late-delay synchronizer. Inanyevent,theexperimental andtheoretical resultspresented inFigure6.70areinvery closeagreement for(EINo)~10dB. Experiment 2:SymbolTimingRecovery Tomeasure theS-curveforthenondata-aided early-late delaysynchronizer forsymbol timingrecovery, wemayusetheexperimental set-upshownionFigure6.71,wherethe15 inS(15)referstothetimingoffset.TheS-curvesomeasured isplottedinFigure6.72for EINo=10dBandEINo=00. Figure6.73plotsthenormalized valueoftheexperimentally measured symboltiming errorversusEINofortwodifferent valuesofstep-size parameter y,namely,T/20and T/200.Thisfigurealsoincludestheoretical plotsofthecorresponding modified Cramer RaoboundofEquation (6.293)adaptedforsymbol-timing error.Fromtheresultspre sentedhere,weobservethatasthestep-sizeparameter yisreduced, thenormalized timing -0.4 -0.6 -0.8L-~L--7=--7=-~c--':c-~=----':-=-~=----:--cc--=0.5 0.4 0.3 0.2 0.1 00.10.20.3 0.4 0.5 Normalized timingoffset8fT FIGURE6.72S-curveofNDA-ELDsynchronizer measured undernoiseless andnoisy conditions. 464 CHAPTER 6IIIPASSBAND DATATRANSMISSION 10-9 0::---"5=----=1-=0---:'1-=5---:'2""0 --=-25=---::30=-""3=:5=-~4:-:0"--4:-:5"---:50 EINo•dB FIGURE 6.73Comparison oftracking-error variance oftheNDA-ELD synchronizer against theoryforvaryingE/Noandtwostep-size parameters. errorisreducedandtherangeofEINoforwhichthemodified Cramer-Rao boundholds (albeitinanapproximate fashion)isenlarged. I6.16Summary andDiscussion Withthebasicbackground theory onoptimum receivers ofChapter5atourdisposal,in thischapterwederivedformulas for,orboundson,thebiterrorrateforsomeimportant digitalmodulation techniques inanAWGNchannel: 1.Phase-shift keying(PSK),represented by ~Coherent binaryphase-shift keying(BPSK). II>Coherent quadriphase shiftkeying(QPSK)anditsvariants, namely,theoffset QPSKand1T14-shifted QPSK. l>Coherent M-aryPSK,whichincludesBPSKandQPSKasspecialcaseswithM=2 andM=4,respectively. Coherent M-aryPSKisusedindigitalsatellite communications. l>Differential phase-shift keying(DPSK),whichmaybeviewedasthepseudo-no n coherent formofPSK. 2.Coherent M-aryquadrature amplitude modulation (QAM),whichisahybridform ofmodulation thatcombines amplitude andphase-shift keying.ForM=4itincludes QPSKasaspecialcase.M-aryQAMisbasictotheconstruction ofhigh-speeD voiceband modems. 3.Frequency-shift keying(FSK),represented by !>Coherent binaryfrequency-shift keying. l>Coherent formsofminimum shiftkeying(MSK)andGaussian minimum shift keying(GMSK); GMSKisbasictotheconstruction ofGSMwireless communications. NotesandReferences 465 ~Coherent M-aryFSK. ""Noncoherent binaryFSK. Inthischapterwealsostudiedrwoalternative techniques forpassband datatrans mission: carrierless amplitude/phase modulation (CAP)anddiscretemultitone (DMT).In thecaseofanAWGNchannel, theperformance ofCAPandDMTareequivalent because theDMTmaybeviewedasalinearreversible transformation ofasingle-carrier modulated signal.However, theyperform quitedifferently inapractical settingthatdeviatesfrom thisidealized model.25DMThasbeenstandardized foruseonasymmetric digitalsub scriberlines(ADSLs) usingrwistedpairs.CAP,withtheuseofdecisionfeedback equali zation,provides anotherapproach forsolvingtheADSLproblem. CAPisalsousedfor datatransmission inlocalareanerworks forpremises' distribution systems. DMTisaformofmultichannel modulation, andsoisorthogonal frequency-division multiplexing (OFDM). Thebasicdifference berweenDMTandOFDMisthatDMTper mitstheuseofloadingtooptimize information capacity, whereas OFDMdoesnot.This difference arisesbecauseoftheirdifferent domains ofapplication. DMTappliestorwo wirechannels suchasADSLs, whereas OFDMappliestobroadcasting andwireless channels. Irrespective ofthedigitalmodulation systemofinterest,synchronization ofthere ceivertothetransmitter isessential totheoperation ofthesystem.Symboltimingrecovery isrequired whether thereceiveriscoherent ornot.Ifthereceiveriscoherent, wealso requireprovision forcarrierrecovery. Inthelatterpartofthechapter, wediscussed non data-aided synchronizers tocatertothesetworequirements withemphasis onM-ary phase-shift keyingsignalsinwhichthecarrierissuppressed. Thepresentation focusedon iterative synchronization techniques thatarenaturally suitedfortheuseofdigitalsignal processing. INOTESANDREFERENCES 1.Foranearlytutorialpaperreviewing different digital modulation techniques (ASK,FSK, andPSK)basedouageometric viewpoint, seeArthursandDym,(1962).Seealsothe following listofbooks: lI>Auderson (1998,Chapter3) >Benedetto andBiglieri(1999,Chapters4and5) Il-LeeandMessersehmitt (1994,PartII) !'-Proakis(1995,Chapter5) ",.Sklar(1988,Chapter3) I>ViterbiandOmura(1979,pp.47-127) 2.ForanearlypaperonrheoffsetQPSK,seeGitlinandHo(1975). 3.The1T/4-shifted QPSKwasfirstdescribed intheopenliterature inAkaiwaandNegata (1987). 4.Chennakeshu andSauliner(1993)usecomputer simulations tostudytheperformance of 1T14-shifted QPSKinadigitalwirelesscommunications environment. Thepulse-shaping signalusedinthegeneration ofthe1T14-shifted QPSKsignalisbasedonthesquareroot raisedcosinespectrum (seeProblem4.38).Inthislatterpaper,itisshownthartheperfor manceof1T/4-shifted QPSKmaydegraderapidlyinsuchanenvironment. Thedifferential detectorofFigure6.13followsChennakeshu andSauliner(1993). 466 CHAPTER 6"PASSBAND DATATRANSMISSION 5.Foraderivation ofEquation (6.65),seeCioffi(1998). 6.Thederivation ofEquation (6.76)wasfirstreported ina1975internalBellLaborat. memorandum authored byWe~er..Alittlela~er,Falconer (1975)issued ~notherBell~~~ oratones memorandum, mwhichItwaspomtedoutthesymbolrotatIOn isnotteU neededifwedidnotwanttobecompatible withexistingQAMorotiJerbandpass sign~y therebysimplifying themathematical representation ofCAPsignals(andtherefore th~, implementation), asshowninEquation (6.77).However, theterminology "CAP"Waselr coineduntil1987whencarrierless amplitude/phase modulation wasreplaced byCAP~t thestandards representative, GarrySmitiJ,ofBellLaboratories. Thefirstdetaileddiscussiy ofCAPinthecontextofdigitalsubscriber lineswaspresented inatwo-part repOtt ~n Werner(1992,1993).In'aseparate reportbyChen,Im,andWerner(1992),thefeasibili~ ofCAPforuseondigitalsubscriber lineswasstudied;seealsothebookbyChen(1998) pp.461-473. Theapplication ofCAPtolocalareanetworks, involving theuseoftwisted pairsforlengtiJslessthan100m,isdiscussed inthepaperbyImandWerner(1995)'the maximum lengthof100misspecified byastandard forthewiringofpremises. ' Thedigitalimplementation ofabaseband equalizer similartotheCAPreceivetof Figure6.24isdiscussed inMuellerandWerner(1982). 7.TheMSKsignalwasfirstdescribed inDoelzandHeald(1961).Foratutorialreviewof MSKandcomparison withQPSK,seePasupathy (1979).SincetiJefrequency spacingis onlyhalfasmuchastheconventional spacingofliTbthatisusedinthecoherent detecoon ofbinaryFSKsignals,thissignaling schemeisalsoreferredtoasfastFSK;seedeBuda (1972). 8.Forearlydiscussions ofGaussian MSK,seeMurotaandHirade(1981)andIshizuka and Hirade(1980). 9.Theanalytical specification ofthepowerspectraldensityofdigitalFMisdifficulttohan dle,exceptforthecaseofarectangular shapedmodulating pulse.ThepaperbyGarrison (1975)presents aprocedure basedontheselection ofanappropriate duration-limited! level-quantized approximation forthemodulating pulse.Theequations developed thelein areparticularly suitableformachine computation ofthepowerspectraofdigitalFMsig· nals;seetiJebookbyStuber(1996). 10.AderailedanalysisofthespectraofM-aryFSKforanarbitrary valueoffrequency deviation ispresented inthepaperbyAnderson andSalz(1965).TheresultsshownplottedinFigure 6.36represent aspecialcaseofaformuladerivedinthatpaperforafrequency deviation ofk=0.5. 11.Thestandard methodofderivingthebiterrorratefornoncoherent binaryFSK,presented inMcDonough andWhalen(1995)andthatfordifferential phase-shift keyingpresented inArthursandDym(1962),involves tiJeuseoftheRiciandistribution. Thisdistribution ariseswhentheenvelope ofasinewaveplusadditiveGaussian noiseisofinterest; see Chapter1foradiscussion oftheRiciandistribution. Thederivations presented inSection 6.6avoidthecomplications encountered inthestandard method. 12.Theoptimum receiverfordifferential phase-shift keyingisdiscussed inSimonandDivsalar (1992). 13.Foratechnical discussion ofvariouskindsofmodems, withemphasis ontheiroperational characteristics, seethebooksofLewart(1988)andHold(1997). 14.Inatwo-part paperbyWei(1984),differential encoding isappliedtoconvolutional chan nelcoding.Severaleight-state convolutional encoders aredescribed therein,whichresuh incodesthataretransparent tosignalelementrotations. Inparticular, inpartIIof,the paper,Weidescribes designrulesandprocedures fora90-degree rotationally invanant convolutional codethathasbeenadoptedforuseintheV.32modemwithtrelliscoding. NotesatulReferences 467 15.Nonuniform sampling ofband-limited signalsisdiscussed inthepaperbyYen(1956).The mainresultsderivedinthatpaperarecontained infourgeneralized theorems. Equation (6.188)isbasedonTheorem IIIofYen'spaper. InthepaperbyKalet,Maw,andSaltzberg (1993),thisparticular theorem dueto Yenisusedtoformulate thefundamental philosophy underlying thedesignofthebidirec tionaldigitalmodem;seealsothepatentbyAyanoglu etal.(1995). 16.Foradiscussion ofthesecondrealization ofadigitalmodem, seethearticlebyHumblet andTroulis(1996). 17.Foradetaileddescription oftheV.34high-speed modemstandard, seeForneyetal.(1996). ThetrelliscodesusedintheV.34modemareduetoWei(1984,1987). 18.Theideaofmultichannel modulation maybetracedtotheearlyworkofChang(1966), Saltzberg (1967),andWeinstein andEbert(1971).Amathematical treatment oftheopti malityofmultitone modulation foralinearchannelwithsevereintersymbol interference ispresented inKalet(1989).However, itwastheworkdonebyCioffiandco-workers that ledtothestandardization ofdiscretemultitone (DMT)forasymmetric digitalsubscriher lines;fordetails,seeRuizetal.(1992),ChowandCioffi(1995),Section7.2ofthebook byStarretal.(1999),andChapter11ofCioffi(1998).Problem 6.44isadaptedfromCioffi (1998). 19.ThemethodofLagrange multipliers fordetermining theextremevaluesofthefunction y=f(x) subjecttotheconstraint cp(x)=0 followsfromthefollowing theorem: Anecessary andsufficient condition foranextremum ofacontinuously differentiable function[(x)isthatitsdifferential withrespecttoxvan ishesatthecritical(i.e.,maximum andminimum) pointsofthefunction. Accordingly, at thecriricalpointsof!(x)wehave ~dx=0 (1)ax Moreover, since",(x)=0,itsdifferential alsovanishes asshownby acpdx=0 (2) ax Hencemultiplying (2)bysomeparameter Aandthenaddingtheresultto(1),weget (a[+Aacp)dx=0 ax ax Sincedxisanindependent increment, weimmediately deducethat aax([(x)+Acp(x))=0 Thisequation isamathematical statement ofthemethodofLagrange multipliers. The parameter AiscalledtheLagrange multiplier. Thematerial presented inthisnotefollows Sokolnikoff andRedheffer (1966,pp.341-344). 20.InthediscreteFouriertransform (DIT),boththeinputandtheoutputconsistofsequences ofnumbers definedatuniformly spacedpointsintimeandfrequency, respectively. This featuremakestheDITideallysuitedfornumerical computation usingthefastFourier transform (FFT)algorithm. FITalgorithms areefficientbecausetheyuseagreatlyreduced numberofarithmetic operations compared tothebrute-force computation oftheDFT. Basically, anFITalgorithm attainsitscomputational efficiency byfollowing a"divideand conquer" strategy, whereby theoriginalDFTcomputation isdecomposed successively into smallerDFTcomputations. ForthecaseofanN-pointDFTandN=2L,theFFTalgorithm 468 CHAPTER 6..PASSBAND DATATRANSMISSION requiresL=log2Nstagesofcomputation, witheachstageofthecomputation involv' complex multiplications andadditions oforderN.Fordetaileddiscussion oftheFFrI1Irgorithrn, seeOppenheim andSchafer(1989,Chapter9). a. 21.Anoverview ofvery-high-rate digitalsubscriber lines(VDSL)ispresented inthepapeb Cioffietal.(1999);thispaperalsoincludes acomparative discussion onVDSLs:X Voiceband moderns. n 22.Fordiscussion ofOFDManditsapplications, seeCasasandLeung(1991),LeFlocheta!. (1989),andZouandWu(1995).FortutonalnotesonOFDMandanextensive listf references, seeCiminiandLi(1999). 0 23.Fordetailed descriptions ofphaserecovery andsymbol-timing recovery usingclassical synchronization systems, seeStiffler(1971),Lindsey(1972),andLindseyandSimon(1973 Chapters 2and9). ' Foramodemtreatment ofsynchronization systemswithemphasis ontheuseof discrete-time signalprocessing algorithms, seeMengali andD'Andrea (1997),Meyr, MoenecJaey, andFechte](1998). . 24.Equation (6.293)onthemodified Cramer-Rao boundforphaserecovery isderivedin Mengali andD'Andrea (1997). 25.Saltzberg (1998)discusses howtheperformances ofCAPandDMTareaffectedbychannel impairments andsystemimperfections inthecontextofADSLapplication. Theimpair_ ments/imperfections considered thereinincludeimpulsenoise,narrowband interference (e.g.,RFingressfromanover-the-air AMradiotransmission), timingjittercausedby imperfect synchronization, andsystemnonlinearities. IPROBLEMS Amplitude-Shift Keying 6.1Intheon-offkeyingversionofVASKsystem,symbol1isrepresented bytransmitting a sinusoidal carrierofamplitude 2Eb/Tb,whereEbisthesignalenergyperbitandTbis thebitduration. Symbol0isrepresented byswitching offthecarrier.Assumethatsymbols 1and0occurwithequalprobability. ForanAWGNchannel, determine theaverageprobability oferrorforthisASK systemunderthefollowing scenarios: (a)Coherent reception. (b)Noncoherent reception, operating withalargevalueofbitenergy-to-noise spectral densityratioEb/No• Note:Whenxislarge,themodified Besselfunction ofthefirstkindofzeroordermaybe approximated asfollows(seeAppendix 3): I( )_exp(x) oX-~ Phase-Shift Keying 6.2APSKsignalisappliedtoacorrelator supplied withaphasereference thatlieswithin'P radiansoftheexactcarrierphase.Determine theeffectofthephaseerror<pontheaverage probability oferrorofthesystem. 6.3Consider aphase-locked loopconsisting ofamultiplier, loopfilter,andvoltage-conttolled oscillator (VCO).Letthesignalappliedtothemultiplier inputbeaPSKsignaldefinedby sit)=A,COS[217"kt+kpm(t)] Problems 469 wherekpisthephasesensitivity, andthedatasignalm(t)takesonthevalue+1forbinary symbol1and-1forbinarysymbolo.TheVCOoutputis r(t)AcSin[21Tfct+19(t)] (a)Evaluate theloopfilteroutput,assuming thatthisfilterremovesonlymodulated com ponentswithcarrierfrequency 2fc. (b)Showthatthisoutputisproportional tothedatasignalmit)whentheloopisphase locked,thatis,19(t)=O. 6.4Thesignalcomponent ofacoherent PSKsystemisdefinedby sit)Acksin(l1rfct) ::'::AcY1=k2 COS(21Tfct) where0:5t:5Tb,andtheplussigncorresponds tosymbol1andtheminussigncorre spondstosymbolO.Thefirsttermrepresents acarriercomponent includedforthepurpose ofsynchronizing thereceivertothe transmitter. (a)Drawasignal-space diagram fortheschemedescribed here;whatobservations can youmakeaboutthisdiagtam? (b)Showthat,inthepresence ofadditivewhiteGaussian noiseofzeromeanandpower speettaldensityNo/2,theaverageprobability oferroris Pe=~erfc(~(1-P)) where 12Eb=l:AcT b (c)Suppose that10percentofthetransmitted signalpowerisallocated tothecatrier component. Determine theE/,1Norequired torealizeaprobability oferrorequalto 10-4• (d)Compare thisvalueofEblNowiththatrequired foraconventional PSKsystemwith thesameprobability oferror. 6.5(a)Giventheinputbinarysequence 1100100010, sketchthewaveforms ofthein-phase andquadrature components ofamodulated waveobtained byusingtheQPSKbased onthesignalsetofFigure6.6. (b)SketchtheQPSKwaveform itselffortheinputbinarysequence specified inpart(a). 6.6LetPelandPeQdenotetheprobabilities ofsymbolerrorfotthein-phase andquadrature channels ofanarrowband digitalcommunication system.Showthattheaverageproba bilityofsymbolerrorfortheoverallsystemisgivenby Pc=Per+PeQ-PerPeQ 6.7Equation (6.47)isanapproximate formulafortheaverageprobability ofsymbolerror forcoherent M-aryPSK.Thisformulawasderivedusingtheunionboundinlightofthe signal-space diagramofFigure6.1Sb.Giventhatmessagepointmlwastransmitted, show thattheapproximate formulaofEquation (6.47)maybederiveddirectlyfromFigure 6.15b. 6.8FindthepowerspectraldensityofanoffsetQPSKsignalproduced byarandombinary sequence inwhichsymbols1and0(represented by::'::1)areequallylikely,andthesymbols indifferent timeslotsarestatistically independent andidentically distributed. 6.9Vestigial sideband modulation (VSB),discussed inChapter2,offersanothermodulation methodforpassband datatransmission. (a)Inparticular, adigitalVSBtransmission systemmaybeviewedasatime-varying one dimensional systemoperating atarateof21Tdimensions persecond,whereTisthe symbolperiod.Justifythevalidityofthisstatement. (b)ShowthatdigitalVSBisindeedequivalent inperformance totheoffsetQPSK. 470 CHAPTER 6Ii!PASSB&"ID DATATRANSMISSION 6.10Thebinarydatastream01101000 isappliedtoa1T/4-shifted DQPSKmodulator tht' initiallyinthestate(<Pl=v'E,<P2=0)inFigure6.11a.Usingtherelationships bet:IS inputdibitsandcarrier-phase shiftssummarized inTable6.2,determine thephaseSt:ten occupied bythemodulator inresponse tothespecified datastream. es 6.11Justasinanordinary QPSKmodulator, theoutputofa1T/4-shifted DQPSKmodulat maybeexpressed intermsofitsin-phase andquadrature components asfollows: Or s(t)=sr(t)COS(21Tfct) -sQ(t)sin(21Tfct) Formulate thein-phase component s,(t)andquadrature component Sg(t)ofthe71"/4 shiftedDQPSKsignal.Hence,outlineaschemeforthegeneration of1T/4-shifted DQPSK signals. 6.12Aninteresting property of1T/4-shifted DQPSKsignalsisthattheycanbedemodulated usinganFMdiscriminator. Demonstrate thevalidityofthisproperty. TheFMdiscrimi_ natorisdiscussed inChapter2. 6.13Let:iJhdenotethedifferentially encoded phaseinthe1T/4-shifted DQPSK. ThesYmbol pairs(I,Q)generated by thisschememaybedefinedas h=h-lCOS(~Ok) -Qk-l sin(~Ok) Qk=h-lsin(~Okl+Qk-l COS(~Ok) wherehandQkarethein-phase andquadrature components corresponding tothekth symbol.Showthatthispairofrelations canbeexpressed simplyas h=cosOk Qk=sinOk whereOkistheabsolute phaseangleforthekthsymbol. Quadrature-Amplitude Modulation 6.14Figure6.53showsa240-QAM signalconstellation, whichmaybeviewedasanextended formofQAMcrossconstellation. (a)IdentifytheportionofFigure6.53thatisaQAMsquareconstellation. (b)Buildonpart(a)toidentify theportionofFigure6.53thatisaQAMcross constellation. (c)Hence,identifytheportionofFigure6.53thatisanextension toQAMcross constellation. 6.15Determine thetransmission bandwidth reduction andaveragesignalenergyof256-QAM, compared to64-QAM. 6.16Twopassband datatransmission systemsaretobecompared. Onesystemuses16-PSK, andtheotheruses16-QAM. Bothsystemsarerequired toproduceanaverageprobability ofsymbolerrorequalto10-3.Compare thesignal-to-noise ratiorequirements ofthese twosystems. Carrierless AmplitudelPhase Modulation (CAP) 6.17Thetwo-dimensional CAPandM-aryQAMschemesarecloselyrelated.Dothefollowing: (a)GivenaQAMsystem,withaprescribed numberofamplitude levels,derivetheequiV alentCAPsystem. (b)Performthereverseofpart(a). 6.18ShowthatthepowerspectraldensityofaCAPsignalwithatotalofLamplitude levels isdefinedby S(f)=1:IP(f)12 Problems 471 where IP(f)Iisthemagnitude spectrum ofthepassband in-phase pulsep(t);theO"~isthe variance ofthecomplex symbolsAi=ai+jbi,whichisdefinedby 1L O"~= -2:(al+bl) Li~' 6.19Youaregiventhebaseband raised-eosine spectrum G(f)pertaining toacertainrolloff factora.Describe afrequency-domain procedure forevaluating thepassband in-phase pulsep(t)andquadrature pulsep(t)thatcharacterize thecorresponding CAPsignal. Frequency.Shift Keying 6.20Thesignalvectorss,andS2areusedtorepresent binarysymbols1and0,respectively, in acoherent binaryFSKsystem.Thereceiverdecidesinfavorofsymbol1when XTS,>XTS2 where XTSiistheinnerproductoftheobservation vectorxandthesignalvector Si,where i=1,2.Showthatthisdecisionruleisequivalent tothecondition x,>X2,wherex,and X2aretheelements oftheobservation vectorx.Assumethatthesignalvectorss,and52 haveequalenergy. 6.21AnFSKsystemtransmits binarydataattherateof2.5X106bitspersecond.Duringthe courseoftransmission, whiteGaussian noiseofzeromeanandpowerspectraldensity 10-20WIHzisaddedtothesignal.Intheabsenceofnoise,theamplitude ofthereceived sinusoidal wavefordigit1or0is1 mV.Determine theaverageprobability ofsymbol errorforthefollowing systemconfigurations: (a)Coherent binaryFSK (b)Coherent MSK (c)Noncoherent binaryFSK 6.22(a)Inacoherent FSKsystem,thesignalss,(t)andS2(t)representing symbols 1and0, respectively, aredefinedby s,(t),S2(t)=Accos[2'lT(fc:!:a;)tJ, 0StsTb Assuming thatfc>af,showthatthecorrelation coefficient ofthesignalss,(t)andS2(t) isapproximately givenby fT'S,(t)S2(t) dt p=0T =sinc(2afT b) fobsilt)dt (b)Whatistheminimum valueoffrequency shifrafforwhichthesignalsSI(tlandS2(t) areorthogonal? (c)Whatisthevalueofafthatminimizes theaverageprobability ofsymbolerror? (d)Forthevalueof!:>.fobtained inpart(c),determine theincreaseinEblNorequired so thatthiscoherent FSKsystemhasthesamenoiseperformance asacoherent binary PSKsystem. 6.23AbinaryFSKsignalwithdiscontinuous phaseisdefinedby forsymbol1 forsymbol0 472 CHAPTER 6"PASSBAND DATATRANSMISSION whereEbisthesignalenergyperbit,Tbisthebitduration, andli,andli2aresaI valuesofuniformly distributed randomvariables overtheinterval0to2'17.Ineffec~~ twooscillators supplying thetransmitted frequencies j;±;::"f/2operateindependent! eachother.Assumethatj;»;::"f. Yof (a)Evaluate thepowerspectraldensityoftheFSKsignal. (b)Showthatforfrequencies farremoved fromthecarrierfrequency j;,thepowersp traldensityfallsoffastheinversesquareoffrequency. ec- 6.24Setupablockdiagram forthegeneration ofSunde'sFSKsignals(t)withcontinuo phasebyusingtherepresentation giveninEquation (6.104),whichisreproduced here~s s(t)=%!'cos(;:)cos(2'17ht) :;:%!'sin(;:)sin(2'17j;t) 6.25Discussthesimilarities between MSKandoffsetQPSK,andthefeaturesthatdistinguisb them. 6.26Therearetwowaysofdetecting anMSKsignal.Onewayistouseacoherent receivelto takefullaccountofthephaseinformation contentoftheMSKsignal.Another Wayisto useanoncoherent receiveranddisregard thephaseinformation. Thesecondmethodoffers theadvantage ofsimplicity ofimplementation, attheexpenseofadegraded noiseper formance. Byhowmanydecibelsdowehavetoincreasethebitenergy-to-noise density ratioEb/Nointhesecondcasesoastorealizeanaverageprobability ofsymbolerrOl equalto10-5inbothcases? 6.27(a)Sketchthewaveforms ofthein-phase andquadrature components oftheMSKsignal inresponse totheinputbinarysequence 1100100010. (b)SketchtheMSKwaveform itselfforthebinarysequence specified inpart(a). 6.28Anonreturn-to-zero darastream(ofamplitude levels±1)ispassedthroughalow-pass filterwhoseimpulseresponse isdefinedbytheGaussian function yr;r ('172~)h(t)= -exp--a a2 whereaisadesignparameter definedintermsofthefilter's3-dBbandwidth by a=fJ;;giJ:..VTw (a)Showthatthetransferfunction ofthefilterisdefinedby H(f)=exp(-a2f2) Hencedemonstrate thatthe3-dBbandwidth ofthefilterisindeedequaltoW.You mayuseTableA6.3onFourier-transform pairs. (b)Showthattheresponse ofthefiltertoarectangular pulseofunitamplitude and duration Tcentered ontheoriginisdefinedbyEquation (6.135). 6.29Plotthewaveform ofaGMSKmodulator produced inresponse tothebinarysequence 1101000, assuming theuseofagain-bandwidth productWTb=0.3.Compare yourresult withthatofExample 6.5. 6.30Summarize thesimilarities anddifferences between thestandard MSKandGaussian filteredMSKsignals. Noncoherent Receivers 6.31InSection6.8wederivedtheformulaforthebiterrorrateofnoncoherent binaryFS~ asaspecialcaseofnoncoherent otthogonal modulation. Inthisproblem werevisittbis Problems 473 issue.Asbefore,weassumethatbinarysymbol1represented bysignal S1(t)istransmitted. According tothematerialpresented inSection6.8,wenotethefollowing: ""TherandomvariableL2represented bythesamplevalue/2ofEquation (6.164)isRayleigh distributed. ~TherandomvariableL1represented bythesamplevalue/1ofEquation (6.170)isRician- distributed. TheRayleigh andRiciandistributions arediscussed inChapter LUsingtheprobability distributions definedinthatchapter,derivetheformulaofEquation (6.181)fortheBER ofnoncoherent binaryFSK. 6.32FigureP6.32ashowsanoncoherent receiverusingamatched filterforthedetection ofa sinusoidal signalofknownfrequency butrandomphase,inthepresence ofadditivewhite Gaussian noise.Analternative implementation ofthisreceiverisitsmechanization inthe frequency domainasaspectrum analyzer receiver, asinFigureP6.32b,wherethecor relatorcomputes thefinitetimeautocorrelation function Rx('1")definedby Rx('1")=r-T x(t)x(t+'1")dt, Showthatthesquare-law envelope detectoroutputsampledattimet=TinFigureP6.32a istwicethespectraloutputoftheFouriertransformer sampled atfrequencyf=fcin FigureP6.32b. (a)~OUIPUI Sampleal txT FIGUREP6.32(b)OUlpul sampled al t=t, 6.33Thebinarysequence 1100100010 isappliedtotheDPSKtransmitter ofFigure6.43a. (a)Sketchtheresulting waveform atthetransmitter output. (blApplying thiswaveform totheDPSKreceiverofFigure6.43b,showthat,inthe absenceofnoise,theoriginalbinarysequence isreconstructed atthereceiveroutput. 6.34Differential M-aryPSKistheM-aryextension ofbinaryDPSK.Thepresentphaseangle Onofthemodulator atsymboltimenisdetermined recursively bytherelation On=On-l+(~)mn> modulo 211" where 0n-1istheprevious phaseangleandmnE{O,1,..•, M -1}isthepresentmod ulatorinput.Theprobability ofsymbolerrorforthisM-arymodulation schemeisap proximately givenby M;e;4 whereitisassumed thatEINoislarge. (alDetermine thefactorbywhichthetransmitted energypersymbolwouldhavetobe increased forthedifferential M-aryPSKtoattainthesameprobability ofsymbol errorascoherent M-aryPSKforM;e;4. (b)ForM=4,byhowmanydecibels isdifferential QPSKpoorerinperformance than coherent QPSK? 474 CHAPTER 6..PASSBAND DATATRANSMISSION Comparison ofDigitalModulation Schemes UsingaSingleCarrier 6.35Binarydataaretransmitted overamicrowave linkattherateof106b/s,andthepo spectraldensityofthenoiseatthereceiverinputis10-10W1Hz.Findtheaveragecarw,er powerrequired tomaintain anaverageprobability oferrorPe:$10-4for(a)coherner binaryPSK,and(b)DPSK. ent 6.36ThevaluesofEblNorequired torealizeanaverageprobability ofsymbolerrorP,~10-' usingcoherent binaryPSKandcoherent FSK(conventional) systemsareequalto7.2a d 13.5,respectively. Usingtheapproximation n 1erfc(u)=•Iexp(-u2)v7rU determine theseparation inthevaluesofEblNoforPe=10-4,using (a)Coherent binaryPSKandDPSK. (b)Coherent binaryPSKandQPSK. (c)Coherent binaryFSK(conventional) andnoncoherent binaryFSK. (d)Coherent binaryFSK(conventional) andcoherent MSK. 6.37InSection6.10wecompared thenoiseperformances ofcoherent binaryPSK,coherent binaryFSK,QPSK,MSK,DPSK,andnoncoherent FSKbyusingthebiterrorrateasthe basisofcomparison. Inthisproblemwetakeadifferent viewpoint andusetheaverage probability ofsymbolerror,Pe>todothecomparison. PlotPeversusEblNoforeachof theseschemesandcomment onyourresults. 6.38Thenoiseequivalent bandwidth ofabandpass signalisdefinedasthevalueofbandwidth thatsatisfiestherelation 2BS(fc)=PI2 where2Bisthenoiseequivalent bandwidth centered aroundthemidband frequencyt, S(fc)isthemaximum valueofthepowerspectraldensityofthesignalatf=fc,andPis theaveragepowerofthesignal.Showthatthenoiseequivalent bandwidths ofbinary PSK,QPSK,andMSKareasfollows: TypeofModulation BinaryPSK QPSK MSKNoiseBandwidth/Bit Rate 1.0 0.5 0.62 Note:Youmayusethedefiniteintegrals inTableA6.10.Adiscussion ofnoiseequivalent bandwidth ispresented inAppendix 2. Voiceband Modems 6.39(a)Refertothedifferential encoderusedinFigure6.48a.Table6.10definesthephase changesinducedintheV.32modembyvaryinginputdibits.Expandthistableby including thecorresponding previous andcurrentvaluesofthedifferential encoder's output.NotethatforeveryinputdibitQ1.•Q2,n)therearefourpossiblevaluesforthe differentially encoded dibit11,.12,.andlikewiseforitsprevious value11,0-112,.-1' (b)ThecurrentquadbitappliedtotheV.32modemwithnonredundant codingis0001. Theprevious outputofthemodemis01.Findthecodewordoutputproduced bytbe modemanditscoordinates. 1.1Problems 475 6.40TheV.32modemstandard withnonredundant codingusesarectangular 16-QAM con stellation. Themodelspecifications areasfollows: Carrierfrequency =1,800Hz Symbolrate=2,400bauds Datarate=9,600bls Calculate (a)theaveragesignal-to-noise ratio,and(b)theaverageprobability ofsymbol errorforthismodem, assuming thatE.JNo=20dB. Multichannel LineCodes 6.41Consider thepassband basisfunctions definedinEquation (6.196),wheret/>(t)isitself definedbyEquation (6.197).Demonstrate thevalidityofProperties 1,2,and3ofthese passband basisfunctions mentioned onpages434and435. 6.42Thewater-filling solutionfortheloadingproblem isdefinedbyEquation (6.213)subject totheconstraint ofEquation (6.210).Usingthispairofrelations, formulate arecursive algorithm forcomputing theallocation ofthetransmitpowerPamongtheNsubchannels. Thealgorithm shouldstartwith(a)aninitialtotalorsumnoise-to-signal ratio NSR(i)=0foriterationi=0,and(b)thesubchanne1s sortedintermsofthosewiththe smallestpowerallocation tothelargest. 6.43Thesquaredmagnitude response ofalinearchannel, denotedby1H(f) 12,isshownin FigureP6.43.Assumethatthegapr=1andthenoisevariance ~=1forallsubchannels. (a)Derivetheformulas fortheoptimum powersP"P2,andP3allocated tothethree subchannels offrequency bands(0,W,),(W"W2),and(W"W). (b)Giventhatthetotaltransmit power P=10,II=2/3and12=1/3,calculate the corresponding valuesofPhP2,andP3• IH(f)I, I ..-_~LL__~I- L.<L.-_....., I I I_....L_---L __l-_-+__ L-_..L-~___L_f -w-w, -W, 0 W, W, W FIGUREP6.43 6.44Inthisproblemweexploretheuseofsingularvaluedecomposition (SVD)asanalternative tothediscreteFouriertransform forvectorcoding.Thisapproach avoidstheneedfora cyclicprefix,withthechannelmatrixbeingformulated as rhohIh2hv0 oheh,hv-1hvH=: o0 0 heh, 476 CHAPTER 6IIIPASSBAND DATATRANSMISSION wherethesequence ho,h10•••,hvdenotesthesampled impulseresponse ofthechaI TheSVDofthematrixHisdefinedby -nne. HU[A:ON,v]vt wher.eUisanN-by-NunitarymatrixandVisan(N+v)-by-(N+v)unitarymatt'. ~~ ~ uutI vvt=I whereIistheidentitymatrixandthesuperscripttdenotesHermitian transposition, Th AisanN-by-Ndiagonal matrixwithsingularvaluesAmn=1,2,...,N.The0.e anN-by-vmatrixofzeros. N"IS (a)Usingthisdecomposition, showthattheNsubchannels resulting fromtheuseof vectorcodingaremathematically described by Xn=""An+Wn TheXnisanelementofthematrixproductutx,wherexisthereceivedsignal(channel output)vector.TheAnisthenthsymbolan+jbnandWnisarandomvariabledue tochannelnoise. (b)Showthatthesignal-to-noise ratioforvectorcodingasdescribed hereinisgivenby (SNR)"etoc coding=r(it(1+(S~R)n))'/I N+V)r whereN*isthenumberofchannels foreachofwhichtheallocated transmit power isnonnegative, (SNR)nisthesignal-to-noise ratioofsubchannel n,andrisapre scribedgap. (c)AstheblocklengthNapproaches infinity,thesingularvaluesapproach themagni tudesofthechannelFouriertransform. Usingthisresult,comment ontherelationship between vectorcodinganddiscretemultitone. 6.45Compare theperformance ofDMTandCAPwithrespecttothefollowing channel impairments: (a)Impulsenoise. (b)Narrowband interference. Assumethat(1)theDMThasalargenumberofsubehannels, and(2)theCAPsysremis uncoded anditsreceiverusesapairofadaptive filtersforimplementation. 6.46Orthogonal frequency-division multiplexing maybeviewedasageneralization ofM-ary FSK.Validate therationale ofthisstatement. Synchronization 6.47FigureP6.47showstheblockdiagram ofacontinuous-time Mthpowerloopforphase recovery inanM-aryPSKreceiver. (a)ShowthattheoutputoftheMthpower-law devicecontains atoneoffrequencyMj;, whereiistheoriginalcarrier. (b)Theoscillator inthephase-locked loopissettoafrequency equaltoMi.Justifythis choice. (c)TheMthpowerloopsuffersfromaphaseambiguity problem inthatitexhibits~ phaseambiguities intheinterval[0,27T].Explainhowthisproblem arisesintheMl powerloop.Howwouldyouovercome theproblem? Problems 477 Phase-locked loop Received M-aryPSK signalMlh power-law deviceBand-pass filter Frequency divider byMLow-pass filter Voltage controlled oscillator Mreference signals FIGlJREP6.47 6.48(a)~ntherecursive algorithm ofEguation (6.272)forphaserecovery, theoldestimateo[n]andtheupdatedestimate0[n+I]ofthecarrierphase0arebothmeasured in radians.Discusstheunitsinwhichtheerrorsignale[n]andstep-sizeparameter "yare measured. (b)Intherecursive algorithm ofEquation (6.286)forsymboltimingrecovery, thecontrol signalsern]ande[n+I]arebothdimensionless. Discusstheunitsinwhichtheerror signale[n]andstep-size parameter "yaremeasured. 6.49Usingthedefinitions ofEquations (6.264)and(6.265)forXkandabrespectively, show thattheexponent inthelikelihood functionL(ab0,r)canbeexpressed asinEquation (6.273). 6.50InSection6.14westudiedanon-data-aided schemeforcarrierphaserecovery, basedon thelog-likelihood function ofEquation (6.260).Inthisproblem weexploretheuseof thisequation fordata-aid carrierphaserecovery. (a)Consider areceiverdesigned foralinearmodulation system.Giventhatthereceiver hasknowledge ofapreamble oflengthLo,showthatthemaximum likelihood esti mateofthecarrierphaseisdefinedby {LO-' } iJ=arg'&::0atx(k) wherethepreamble{aklt«o' isaknownsequence ofcomplex symbols, and (x(k)lt«o' isthecomplex envelope ofthecorresponding receivedsignal. (b)Usingtheresultderivedinpart(a),construct ablockdiagram forthemaximum likelihood phaseestimator. 478 CHAPTER 6"PASSBAND DATATUANSl\lISSION Computer Experiment 6.51Thepurposeofthiscomputer experiment istocompare theeffectofadispersive chanI onthewaveforms generated bythefollowing passband modulation techniques: ne (a)Binaryphase-shift keying(BPSK) (b)Quadriphase-shift keying(QPSK) (c)Minimum shiftkeying(MSK) (d)Gaussian MSKwithtime-bandwidth productWTb=0.3 Thechannelconsistsofaband-pass Butterworth filteroforder2N=10and3-dBband_ width2Bcentered onthemidband ftequencyt.Thelow-pass equivalent ofthecharmI hasthesquaredmagnitude response e I 12 _ 1 H(f)-1+(f/B)2N Thechannelbandwidth isvariablesoastoillustrate itseffectonthefilteredmodulated wave. Assuming theuseofacoherent receiver, plotthewaveforms ofthemodulated signalsunder(a),(b),(c)and(d)forthefollowing channelbandwidths: (i)2B=12kHz (ii)2B=16kHz (iii)2B=20kHz (iv)2B=24kHz (v)2B=30kHz Comment onyourresults. Hint.Toperformthecomputations neededforthisexperiment, itisadvisable toperform thecomputations inbaseband byperforming theband-pass tolow-pass transformation described inAppendix 2. SPREAD-SPECTRLTM MODULATION Thischapterintroduces amodulation technique calledspread"spectrum modulation, whichisradically different fromthemodulation techniques thatarecovered inpreceding chapters. Inspread"spectrum modulation, channel bandwidth andtransmit powerare sacrificed forthesakeofsecurecommunications. Specifically, wecoverthefollowing topics: ~Spreading sequences intheformofpseudo"noise sequences, theirproperties, andmethods ofgeneration. ~Thebasicnotionofspread"spectrum modulation. ~Thetwocommonly usedtypesofspread"spectrum modulation: directsequence and frequency hopping. Thematerial presented inthischapterisbasictowireless communications usingcode" divisionmultiple access,whichiscovered inChapter 8. Amajorissueofconcerninthestudyofdigitalcommunications asconsidered inChapters 4,5,and6isthatofproviding fortheefficientuseofbandwidth andpower.Notwith standing theimportance ofthesetwoprimarycommunication resources, therearesitua tionswhereitisnecessary tosacrificethisefficiency inordertomeetcertainotherdesign objectives. Forexample, thesystemmayberequired toprovideaformofsecurecom munication inahostileenvironment suchthatthetransmirted signalisnoteasilydetected orrecognized byunwanted listeners. Thisrequirement iscateredtobyaclassofsignaling techniques knowncollectively asspread-spectrum modulation. Theprimaryadvantage ofaspread-spectrum communication systemisitsabilityto rejectinterference whetheritbetheunintentional interference byanotherusersimulta neouslyartempting totransmit throughthechannel, ortheintentional interference bya hostiletransmirter artempting tojamthetransmission. Thedefinition ofspread-spectrum modulation! maybestatedintwoparts: 1.Spreadspectrum isameansoftransmission inwhichthedatasequence occupies a bandwidth inexcessoftheminimum bandwidth necessary tosendit. 2.Thespectrum spreading isaccomplished beforetransmission throughtheuseofa codethatisindependent ofthedatasequence. Thesamecodeisusedinthereceiver 479 480 CHAPTER 7"SPREAD-SPECTRUM MODULATION {operating insynchronism withthetransmitter} todespread thereceivedsignalso thattheoriginaldatasequence mayberecovered. Although standard modulation techniques suchasfrequency modulation andpulse-code modulation dosatisfypart1ofthisdefinition, theyarenotspread-spectrum techniques becausetheydonotsatisfypart2ofthedefinition. Spread-spectrum modulation wasoriginally developed formilitary applications whereresistance tojamming {interference} isofmajorconcern. However, there arecivilia~ applications thatalsobenefitfromtheuniquecharacteristics ofspread-spectrum modu lation.Forexample, itcanbeusedtoprovidemultipath rejection inaground-based mobile radioenvironment. Yetanotherapplication isinmultiple-access communications inwhich anumberofindependent usersarerequiredtoshareacommon channelwithoutanex ternalsynchronizing mechanism; here,forexample, wemaymentionaground-based radio environment involving mobilevehiclesthatmustcommunicate withacentralstation.Mote issaidaboutthislatterapplication inChapter8. Inthischapter,wediscussprinciples ofspread-spectrum modulation, withemphasis ondirect-sequence andfrequency-hopping techniques. Inadirect-sequence spread spectrum technique, twostagesofmodulation areused.First,theincoming datasequence isusedtomodulate awidebandcode.Thiscodetransforms thenarrowband datasequence intoanoiselike wideband signal.Theresulting widebandsignalundergoes asecondmod ulationusingaphase-shift keyingtechnique. Inafrequency-hop spread-spectrum tech nique,ontheotherhand,thespectrum ofadata-modulated carrieriswidenedbychanging thecarrierfrequency inapseudo-random manner.Fortheiroperation, bothofthesetech niquesrelyontheavailability ofanoiselike spreading codecalledapseudo-random or pseudo-noise sequence. Sincesuchasequence isbasictotheoperation ofspread-spectrllll1 modulation, itislogicalthatwebeginourstudybydescribing thegeneration andprop ertiesofpseudo-noise sequences. I7.2Pseudo-Noise Sequences Apseudo-noise (PN)sequence isaperiodic binarysequence withanoiselike wavefotm thatisusuallygenerated bymeansofafeedback shiftregister,ageneralblockdiagtam of whichisshowninFigure7.1.Afeedback shiftregisterconsistsofanordinary shiftregister madeupofmflip-flops {two-state memory stages}andalogiccircuitthatareintercon nectedtoformamultiloop feedback circuit.Theflip-flops intheshiftregisterareregulated byasingletimingclock.Ateachpulse(tick)oftheclock,thestateofeachflip-flopis shiftedtothenextonedowntheline.Witheachclockpulsethelogiccircuitcomputes a Output sequence Clock ..... ..... ---l FIGURE7.1Feedback shiftregister. (7.1)7.2Pse,"",-Noise Sequences 481 Booleanfunction ofthestatesoftheflip-flops. Theresultisthellfedbackastheinputto thefirstflip-flop, therebypreventing theshiftregisterfromemptying. ThePNsequence so generated isdetermined bythelengthmoftheshiftregister,itsinitialstate,andthefeed backlogic. Letsj(k)denotethestateofthejthflip-flopafterthekthclockpulse;thisstatemay berepresented bysymbol0or1.Thestateoftheshiftregisterafterthekthclockpulseis thendefinedbytheset{s,(k),s2(k),...,sm(k)),wherek:2:O.Fortheinitialstate,kis zero.Fromthedefinition ofashiftregister,wehave {k:2:0 lsjSm whereso(k)istheinputappliedrothefirstflip-flopafterthekthclockpulse.According to theconfiguration described inFigure7.1,so(k)isaBoolean function oftheindividual statess,(k),s2(k),...,sm(k).Foraspecified lengthm,thisBoolean function uniquely determines thesubsequent sequence ofstatesandtherefore thePNsequence produced at theoutputofthefinalflip-flopintheshiftregister.Withatotalnumberofmflip-flops, thenumberofpossiblestatesoftheshiftregisterisatmost2m•Itfollowstherefore thatthe PNsequence generated byafeedbac): shiftregistermusteventually becomeperiodicwith aperiodofatmost2m• Afeedback shiftregisterissaidtobelinearwhenthefeedback logicconsistsentirely ofmodulo-2 adders.Insuchacase,thezerostate(e.g.,thestateforwhichalltheflip-flops areinstate0)isnotpermitted. Wesaysobecauseforazerostate,theinputso(k)produced bythefeedback logicwouldbe0,theshiftregisterwouldthencontinue toremaininthe zerostate,andtheoutputwouldtherefore consistentirelyofOs.Consequently, theperiod ofaPNsequence produced byalinearfeedback shiftregisterwithmflip-flops cannot exceed2m-1.Whentheperiodisexactly2m-1,thePNsequence iscalledamaximal length-sequence orsimplym-sequence. ~ExAMPLE 7.1 Consider thelinearfeedback shiftregistershowninFigure7.2,involving threeflip-flops. The inputSoappliedtothefirstflip-flopisequaltothemodulo-2 sumof5,and53'Itisassumed thattheinitialstateoftheshiftregisteris100(readingthecontentsofthethreeflip-flops from lefttoright).Then,thesuccession ofstateswillbeasfollows: 100,110,111,011,101,010,001,100,.... Modulo-2 adder Output sequence Clock--_*- -<l>-- ....J FIGURE7.2Maximal-length sequence generator form=3. 482 CHAPTER 7..SPREAD-SPECTRUM MODULATION Theoutputsequence(thelastpositionofeachstateoftheshiftregister)istherefore 00111010... whichrepeatsitselfwithperiod23-1=7. Notethatthechoiceof100astheinitialstateisarbitrary. Anyoftheothersixpermis_ siblestatescouldserveequallywellasaninitialstate.Theresultingoutputsequence would thensimplyexperience acyclicshift. ... PROPERTIES OFMAxIMAL-LENGTH SEQUENCES2 Maximal-length sequences havemanyoftheproperties possessed byatrulyrandom binarysequence. Arandombinarysequence isasequence inwhichthepresence ofbinary symbol1or0isequallyprobable. Someproperties ofmaximal-length sequences areas follows: 1.Ine'achperiodofamaximal-length sequence, thenumberofisisalwaysonemorethan thenumberofOs.Thisproperty iscalledthebalanceproperty. 2.AmongtherunsofisandofOsineachperiodofamaximal-length sequence, one halftheruns.ofeachkindareoflengthone,one-fourth areoflengthtwo,one-eighth are oflengththree,andsoonaslongasthesefractions represent meaningful numbers of runs.Thisproperty iscalledtherunproperty. Bya"run"wemeanasubsequence ofidentical symbols (isorOs)withinoneperiodofthesequence. Thelengthofthis subsequence isthelengthoftherun.Foramaximal-length sequence generated bya linearfeedback shiftregisteroflengthm,thetotalnumberofrunsis(N+1)12,where N=2m-1. 3.Theautocorrelation function ofamaximal-length sequence isperiodic andbinary valued.Thisproperty iscalledthecorrelation property. Theperiodofamaximum-length sequence isdefinedby N=2m-1 (7.2) wheremisthelengthoftheshiftregister.Letbinarysymbols 0and1ofthesequence be denotedbythelevels-1and+1,respectively. Letc(t)denotetheresulting waveform of themaximal-length sequence, asillustrated inFigure7.3aforN=7.Theperiodofthe waveform c(t)is(basedOnterminology usedinsubsequent sections) (7.3) whereTcistheduration assigned tosymbol1or0inthemaximal-length sequence. By definition, theautocorrelation function ofaperiodic signalc(t)of period Tbis 1fT;/2 RAr)=-y,. c(t)c(t-r)dt b-Tb12(7.4) 7.2Pseudo-Noise Sequences 483 Binarysequence 0 0 1 1 1 0 1 0 0 1 1 1 0 1 +1 -1 (0) 1.0 (b) seCt) \ \ I \ I \ ------:2-::1..u..J.::..o-1d..J..LJ..LJ..LJ..LJ..LJ..h-...:1-'--'-.l::..~-- f Tc Tc Tc (c) FIGURE 7.3(a)Waveform ofmaximal-length sequence forlengthm=3orperiodN=7. (b)Autocorrelation function. (e)Powerspectral density.Allthreepartsrefertotheoutputofthe feedback shiftregisterofFigure7.2. (7.5) fortheremainder oftheperiodwherethelagTliesintheinterval(-Tb12,TbI2);Equation (7.4)isaspecialcaseofEquation (1.26).Applying thisformulatoamaximal-length sequence represented bye(t),weget __{l__-lNN;c1 H,Rc(T) N' ThisresultisplottedinFigure7.3bforthecaseofm=3orN=7. 484 CHAPTER 7IIISPREAD-SPECTRUM MODUlATION FromFouriertransform theoryweknowthatperiodicity inthetimedomainistrans_ formedintouniform sampling inthefrequency domain. Thisinterplay between thetime andfrequency domains isborneoutbythepowerspectraldensityofthemaximal-length wavec(t).Specifically, takingtheFouriertransform ofEquation (7.5),wegetthesampled spectrum 1 1 +N=.(n)(n)SA!)=N215(f)+~2Lsm~N15f-NT c n*O(7.6) whichisplottedinFigure7.3cform=3orN=7. Comparing theresultsofFigure7.3foramaximal-length sequence withthecorre sponding resultsshowninFigure1.11forarandombinarysequence, wemaymakethe following observations: I>"Foraperiodofthemaximal-length sequence, theautocorrelation functionRJr}is somewhat similartothatofarandombinarywave. ~Thewaveforms ofbothsequences havethesameenvelope, sinc2(fT),fortheirpower spectraldensities. Thefundamental difference between themisthatwhereastheran dombinarysequence hasacontinuous spectraldensitycharacteristic, thecorrespond ingcharacteristic ofamaximal-length sequence consistsofdeltafunctions spaced lINTcHzapart. Astheshift-register lengthm,orequivalently, theperiodNofthemaximal-length sequence isincreased, themaximal-length sequence becomes increasingly similartotherandom binarysequence. Indeed,inthelimit,thetwosequences becomeidentical whenNismade infinitely large.However, thepricepaidformakingNlargeisanincreasing storagerequire ment,whichimposesapractical limitonhowlargeNcanactuallybemade. !IiCHOOSING AMAxIMAL-LENGTH SEQUENCE Nowthatweunderstand theproperties ofamaximal-length sequence andthefactthat wecangenerate itusingalinearfeedback shiftregister,thekeyquestion thatweneedto addressis:Howdowefindthefeedback logicforadesiredperiodN?Theanswertothis ITABLE7.1Maximal-length sequencesofshiftdegister lengths2-8 Shift-Register Length,m Feedback Taps 2" [2,1] 3" [3,1] 4 [4,1] 5* [5,2],[5,4,3,2], [5,4,2,1] 6 [6,1],[6,5,2,1],[6,5,3,2] 7" [7,1],[7,3],[7,3,2,1],[7,4,3,2], [7,6,4,2], [7,6,3,1],[7,6,5,2], [7,6,5,4,2,1],[7,5,4,3,2,1] 8 [8,4,3,2],[8,6,5,3],[8,6,5,2], [8,5,3,1],[8,6,5,1],[8,7,6,1], [8,7,6,5,2, 1],[8,6,4,3,2,1] 7.2Pseudo-Noise Sequences 485 question istobefoundinthetheoryoferror-control codes,whichiscoveredinChapter 10.Thetaskoffindingtherequired feedback logicismadeparticularly easyforusby virtueoftheextensive tablesofthenecessary feedback connections forvarying shift registerlengthsthathavebeencompiled intheliterature. InTable7.1,wepresentthesets ofmaximal (feedback) tapspertaining toshift-register lengthsm=2,3,...,8.3Note thatas-mincreases, thenumberofalternative schemes (codes)isenlarged. Also,forevery setoffeedback connections showninthistable,thereisan"image" setthatgenerates an identical maximal-length code,reversed intimesequence. Theparticular setsidentified withanasteriskinTable7.1correspond toMersenne primelengthsequences, forwhichtheperiodNisaprimenumber. !>-ExAMPLE 7.2 Consider amaximal-length sequence requiring theuseofalinearfeedback-shift registerof lengthm=5.Forfeedback taps,weselecttheset[5,2]fromTable7.1.Thecorresponding configuration ofthecodegenerator isshowninFigure7Aa.Assuming thattheinitialstateis 10000,theevolution ofoneperiodofthemaximal-length sequence generated bythisscheme isshowninTable7.2a,whereweseethatthegenerator returnstotheinitial10000after31 iterations; thatis,theperiodis31,whichagreeswiththevalueobtainedfromEquation (7.2). Supposenextweselectanothersetoffeedback tapsfromTable7.1,namely,[5,4,2,1]. Thecorresponding codegenerator isthusasshowninFigure7Ab.Fortheinitialstate10000, wenowfindthattheevolution ofthemaximal-length sequence isasshowninTable7.2b. Hereagain,thegenerator returnstotheinitialstate10000after31iterations, andsoitshould. Butthemaximal-length sequence generated isdifferentfromthatshowninTable7.2a. Clearly,thecodegenerator ofFigure7Aahasanadvantage overthatofFigure7Ab, asitrequiresfewerfeedback connections. .;I Modulo-2 adder Output sequence Clock--.....---....--4--- .....-----J (aJ Output sequence Clock--.....---....--......--- ......--- (b) FIGURE7.4Twodifferentconfigurations offeedback shiftregisteroflengthm=5.(a)Feed backconnections [5,2].(b)Feedback connections [5,4,2,1]. 486 CHAPTER 7IiSPREAJ)-SPECTRUMMODUlATION TABLE7.2"Evolution ofthemaximal- lengthsequence generated bythefeedback- shiftregisterofFig.7.4a FeedbackStateofShiftRegister Output Symbol 1000 0 Symbol 0 010 0 0 0 1 1 010 0 0 0 01010 0 1 10101 0 1 1 1 010 1 1 11101 0 0 011 1 0 1 1 1 011 1 0 1 1101 1 1 0 0 1101 1 0 0 0 110 1 0 0 0 0 1 1 0 1 10..001 1 1 11000 1 1 1 1100 0 1 1 1 110 0 1 11 1 1 1 0 0 01 1 1 1 1 0 0 0 11 1 1 1 10011 1 1 1 10 0 1 1 0 0 1 1 00 1 1 10110 0 0 01011 0 0 00101 1 1 1 0 0 10 1 0 0 10 0 1 0 0 00100 1 0 00010 0 0 000 0 1 0 1 100 0 0 1 Code:0000101011101100011111001101001 7.2PSeruW-Noise Sequences 487 TABLE7.2bEvolution ofthemaximal- lengthsequence generated bythefeedback- shiftregisterofFig.7.4b StateofShiftRegister Feedback Output Symbol 10 0 0 0 Symbol 1 11 0 0 0 0 0 0 11 0 0 0 1 1 0 1 10 0 0 0101 1 0 1 1 0 1 0 1 1 0 0 1 0 10 1 0 0 01 0 1 0 1 1 0 0 10 1 0 0 1 0 0 1 0 0 0 01 0 0 1 0 00 0 10 0 1 10 0 0 1 0 0 0 1 0 00 1 1 101,0 0 0 1 11 0 10 0 1 1 1101 0 1 111 1 0 1 1 11 1 1 1 0 0 0 11 1 1 1 1 1 01 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 0 001 1 0 1 1 1 0 0 1 1 0 1 11 0 0 1 1 1 11 1 0 0 1 0 0 1 1 10 0 0 001 1 1 0 0 0 0 0 1 1 1 0 000 0 1 1 1 10000 1 Code:0000110101001000101111101100111 488 CIIAYl'ER 7OllSPREAll-SPECTRUM MODUL.4.TION I7.3ANotionofSpreadSpectrum Animportant attribute ofspread-spectrum modulation isthatitcanprovideprotection againstexternally generated interfering (jamming) signalswithfinitepower.Thejamming signalmayconsistofafairlypowerful broadband noiseormultitone waveform thatis directedatthereceiverforthepurposeofdisrupting communications. Protection against jamming waveforms isprovided bypurposely makingtheinformation-bearing signaloc cupyabandwidth farinexcessoftheminimum bandwidth necessary totransmit it.This hastheeffectofmakingthetransmitted signalassumeanoiselike appearance soasto blendintothebackground. Thetransmitted signalisthus enabled topropagate through thechannelundetected byanyonewhomaybelistening. Wemaytherefore thinkofspread spectrum asamethodof"camouflaging" theinformation-bearing signal. Onemethodofwidening thebandwidth ofaninformation-bearing (data)sequence involvestheuseofmodulation. Let(bkldenoteabinarydatasequence, and(Ck)denotea pseudo-noise (PN)sequence. Letthewaveforms b(t)andc(t)denotetheirrespective polar nonreturn-to-zero representations intermsoftwolevelsequalinamplitude andopposite inpolarity, namely,::tl.Wewillrefertob(t)astheinformation-bearing (data)signal,and toc(t)asthePNsignal.Thedesiredmodulation isachieved byapplying thedatasignal b(t)andthePNsignalcrt)toaproductmodulator ormultiplier, asinFigure7.5a.We knowfromFouriertransformtheory thatmultiplication oftwosignalsproduces asignal whosespectrum equalstheconvolution ofthespectraofthetwocomponent signals.Thus, ifthemessagesignalb(t)isnarrowband andthePNsignalcrt)iswideband, theproduct (modulated) signalm(t)willhaveaspectrum thatisnearlythesameasthewideband PN signal.Inotherwords,inthecontextofourpresentapplication, thePNsequence pedorrns theroleofaspreading code. Bymultiplying theinformation-bearing signalb(t)bythePNsignalc(t),eachinfor mationbitis"chopped" upintoanumberofsmalltimeincrements, asillustrated inthe waveforms ofFigure7.6.Thesesmalltimeincrements arecommonly referredtoaschips. Forbaseband transmission, theproductsignalm(t)represents thetransmitted signal. Wemaythusexpressthetransmitted signalas m(t)=c(t)b(t) (7.7)·"rm ", c(t) (a)m"'i'· i(t) (b) ret)Saylifv>O SayOifv<0 (c) FIGURE7.5Idealized modelofbaseband spread-spectrum system.(a)Transmitter. (b)Channel, (e)Receiver. 7.3ANotionofSpreadSpectrum 489 (a)Datasignalb(r) +1 Of---+_--+-+-t--+_---j--+-- -1 (b)Spreading codec(r) +1 of---+----+--+~--+_--+-+-- -1 (c)Productsignalm(t) FIGURE7.6Illustrating thewaveforms inthetransmitter ofFigure7.Sa. Thereceivedsignalr(t)consistsofthetransmitted signalm(t)plusanadditive interference denotedbyi(t),asshowninthechannelmodelofFiguIe7.Sb.Hence, r(t)=m(t)+i(t) =c(t)b(t)+i(t)(7.8) (7.9)Torecovertheoriginalmessage signalb(t),thereceived signalr(t)isappliedtoa demodulator thatconsistsofamultiplier followed byanintegrator, andadecisiondevice, asinFigure7.Sc.Themultiplier issupplied withalocallygenerated PNsequence thatis anexactreplicaofthatusedinthetransmitter. Moreover, weassumethatthereceiver operates inperfectsynchronism withthetransmitter, whichmeansthatthePNsequence inthereceiverislinedupexactlywiththatinthetransmitter. Themultiplier outputinthe receiveristherefore givenby z(t)=c(t)r(t) =c2(t)b(t)+c(t)i(t) Equation (7.9)showsthatthedatasignalb(t)ismultiplied twicebythePNsignalc(t), whereas theunwanted signali(t)ismultiplied onlyonce.ThePNsignalc(t)alternates betweenthelevels-1and+1,andthealternation isdestroyed whenitissquared; hence, c2(t)=1forallt (7.10) Accordingly, wemaysimplifyEquation (7.9)as z(t)=b(t)+c(t)i(t) (7.11) 490 CUAPTER 7'"SPREAD-SPECTRUM MODULtHION WethusseefromEquation (7.11)thatthedatasignalb(t)isreproduced atthemultiplier outputinthereceiver, exceptfortheeffectoftheinterference represented bytheadditive termc(t)i(t).Multiplication oftheinterference i(t)bythelocallygenerated PNsignalc(t) meansthatthespreading codewillaffecttheinterference justasitdidtheoriginalsignal atthetransmitter. Wenowobservethatthedatacomponent b(t)isnarrowband, whereas thespurious component c(t)i(t)iswideband. Hence,byapplying themultiplier outputto abaseband (low-pass) filterwithabandwidth justlargeenoughtoaccommodate the recovery ofthedatasignalb(t),mostofthepowerinthespurious component c(t)i(t)is filteredout.Theeffectoftheinterference i(t)isthussignificantly reducedatthereceiver output. InthereceivershowninFigure7.5c,thelow-pass filteringactionisacruallyper formedbytheintegraror rhatevaluares theareaunderthesignalproduced atthemultiplier output.Theintegrarion iscarriedoutforthebitinterval0<;t<;Tb,providing thesample valuev.Finally,adecision ismadebythereceiver:Ifvisgreaterthanthethreshold of zero,thereceiversaysthatbinarysymbol1oftheoriginaldatasequence wassentinthe interval0<;t<;Tb,andifvislessthanzero,thereceiversaysthatsymbol0wassent;if visexactlyzerothereceivermakesarandomguessinfavorof1orO. Insummary, theuseofaspreading code(withpseudo-random properties) inthe transmitter produces awidebandtransmitted signalthatappearsnoiselike toareceiver thathasnoknowledge ofthespreading code.Fromthediscussion presented inSection 7.2,werecallthat(foraprescribed datarate)thelongerwemaketheperiodof thespread ingcode,thecloserwillthetransmitted signalbetoatrulyrandombinarywave,andthe harderitistodetect.Naturally, thepricewehavetopayfortheimproved protection againstinterference isincreased transmission bandwidth, systemcomplexity, andprocess ingdelay.However, whenourprimary concern isthesecurityoftransmission, theseare notunreasonable,costs topay. 7.4Direct-Sequence SpreadSpectrum withCoherent BinaryPhase-Shift Keying Thespread-spectrum technique described intheprevious sectionisreferredtoasdirect sequence spreadspectrum. Thediscussion presented therewasinthecontextofbaseband transmission. Toprovidefortheuseofthistechnique inpassband transmission overa satellitechannel, forexample, wemayincorporate coherent binaryphase-shift keying (PSK)intothetransmitter andreceiver, asshowninFigure7.7.Thetransmitrer ofFigure 7.7afirstconverts theincoming binarydatasequence {bklintoapolarNRZwaveform b(t),whichisfollowed bytwostagesofmodulation. Thefirststageconsistsofaproduct modulator ormultiplier withthedatasignalb(t)(representing adatasequence) andthe PNsignalc(t)(representing thePNsequence) asinputs.Thesecondstageconsistsofa binaryPSKmodulator. Thetransmitted signalx(t)isthusadirect-sequence spreadbiliary phase-shift-keyed (DS/BPSK) signal.Thephasemodulation Ott)ofx(t)hasoneoftwo values,0and'TT,depending onthepolarities ofthemessage signalb(t)andPNsignalcrt) attimetinaccordance withthetruthtableofTable7.3. Figure7.8illustrates thewaveforms forthesecondstageofmodulation. Partofthe modulated waveform showninFigure7.6cisreproduced inFigure7.8a;thewaveform shownherecorresponds tooneperiodofthePNsequence. Figure7.8bshowsthewave formofasinusoidal carrier,andFigure7.8cshowstheDS/BPSK waveform thatresults fromthesecondstageofmodulation. 7.4Direct-Sequence SpreadSpectrum 491 Binary d_atasequence lbk)xC,) Coherent detectorCal Received signa! y('1Say1ifv>0 SayOifv<O Cbl FIGURE7.7Direct-sequence spreadcoherent phase-shift keying.(a)Tra.nsmitter. (b)Receiver. Thereceiver, showninFigure7.7b,consistsoftwostagesofdemodulation. Inthe firststage,thereceived signaly(t)andalocallygenerated carrierareappliedtoaproduct modulator followed byalow-pass filterwhosebandwidth isequaltothatoftheoriginal message signalm(t).Thisstageofthedemodulation processreversesthephase-shiftkeying appliedtothetransmitted signal.Thesecondstageofdemodulation performs spectrum despreading bymultiplying thelow-pass filteroutputbyalocallygenerated replicaofthe PNsignalc(t),followed byintegration overabitinterval0:5t:5Tb,andfinallydecision makinginthemannerdescribed inSection7.3. IIIMODEL FORANALYSIS Inthenormalformofthetransmitter, showninFigure7.7a,thespectrum spreading is performed priortophasemodulation. Forthepurpose ofanalysis, however, wefindit moreconvenient tointerchange theorderoftheseoperations, asshowninthemodelof ITABLE7.3Truthtableforphasemodulation O(t),radians PolarityofData Sequence b(t)atTimet +- PolarityofPN + 0 1T sequence crt)attimet 1T 0 (7.12)492 CHAPTER 7roSPREAD-SPECTRUM MODULATION (a) ~~~nnn~nnnnnnnnnn -A,lJVlJVVVVVV\f\f\fVV (b) (2E; ""~ ~Ti; A~AAAMAAAAA nMA b -A,VVVVVVVVWVVV (e) FIGURE7.8(a)ProductsignalmIt)=c(t)b(t).(b)Sinusoidal carrier.(e)DSIBPSK signal. Figure7.9.Wearepermitted todothisbecausethespectrum spreading andthebinary phase-shift keyingarebothlinearoperations; likewise forthephasedemodulation and spectrum despreading. Butfortheinterchange ofoperations tobefeasible,itisimportant tosynchronize theincoming datasequence andthePNsequehce. ThemodelofFigure7.9 alsoincludesrepresentations ofthechannelandthereceiver. Inthismodel,itisassumed thattheinterference j(t)limitsperformance, sothattheeffectofchannelnoisemaybe ignored. Accordingly, thechanneloutputisgivenby y(t)=x(t)+j(t) =c(t)s(t)+j(t) Estimate ofb(t)----~Transmitter ----"'i Data signal b(t)I Channel fo'l,----- Receiver---- I I Iy(t) FIGURE7.9Modelofdirect-sequence spreadbinaryPSKsystem. 7.5Signal-Space Dimensi<mality andProcessing Gain 493 wheres(t)isthebinaryPSKsignal,andc(t)isthePNsignal.Inthechannelmodelincluded inFigure7.9,theinterfering signalisdenotedbyj(t).Thisnotation ischosenpurposely tobedifferent fromthatusedfortheinterference inFigure7.5b.Thechannelmodelin Figure7.9ispassband inspectralcontent,whereasthatinFigure7.5bisinbaseband form. Inthereceiver, thereceivedsignaly(t)isfirstmultiplied bythePNsignalcit)yielding anoutputthatequalsthecoherent detectorinputu(t).Thus, u(t)=c(t)y(t) =c2(t)s(t)+c(t)j(t) =sit)+c(t)j(t)(7.13) InthelastlineofEquation (7.13),wehavenotedthat,bydesign,thePNsignalcit)satisfies theproperty described inEquation (7.10),reproduced hereforconvenience: forallt Equation (7.13)showsthatthecoherent detectorinputu(t)consistsofabinaryPSKsignal s(t)embedded inadditivecode-modulated interference denotedbyc(t)j(t).Themodulated natureofthelattercomponent forcestheinterference signal(jammer) tospreaditsspec trumsuchthatthedetection ofinformation bitsatthereceiveroutputisafforded increased reliability. SYNCHRONIZATION Foritsproperoperation, aspread-spectrum communication systemrequiresthatthelocally generated PNsequence usedinthereceivertodespread thereceivedsignalbesynchronized tothePNsequence usedtospreadthetransmitted signalinthetransmitter." Asolution tothesynchronization problem consistsoftwoparts:acquisition andtracking. Inacqui sition,orcoarsesynchronization, thetwoPNcodesarealignedtowithinafractionofthe chipinasshortatimeaspossible. Oncetheincoming PNcodehasbeenacquired, tracking, orfinesynchronization, takesplace.Typically, PNacquisition proceeds intwosteps.First, thereceived signalismultiplied byalocallygenerated PNcodetoproduce ameasure of correlation between itandthePNcodeusedinthetransmitter. Next,anappropriate decision-rule andsearchstrategyisusedtoprocessthemeasure ofcorrelation soobtained todetermine whetherthetwocodesareinsynchronism andwhattodoiftheyarenot.As fortracking,itisaccomplished usingphase-lock techniques verysimilartothoseusedfor thelocalgeneration ofcoherent carrierreferences. Theprincipal difference betweenthem liesinthewayinwhichphasediscrimination isimplemented. 7.5Signal-Space Dimensi01udity andProcessing Gain Havingdeveloped aconceptual understanding ofspread-spectrum modulation anda methodforitsimplementation, wearereadytoundertake adetailedmathematical analysis ofthetechnique. Theapproach wehaveinmindisbasedonthesignal-space theoretic ideasofChapter5.Inparticular, wedevelopsignal-space representations ofthetransmit tedsignalandtheinterfering signal(jammer). 494 CHAPTER 7 "SPREAD-SPECTRUM MODULATION Inthiscontext,consider thesetoforthonormal basisfunctions: cPk(t)={~COS(27rfJ)' 0, ~k(t}={~sin(27rf,t), 0, k=0,1,..., N~1otherwise otherwise(7.14) (7.15) whereTcisthechipduration, andNisthenumberofchipsperbit.Accordingly, wemay describethetransmitted signalx(t)fortheintervalofaninformation bitasfollows: (7.16) whereEbisthesignalenergyperbit;theplussigncorresponds toinformation bit1,and theminussigncorresponds toinformation bitO.Thecodesequence (co,Cl>•••,CN-I} denotesthePNsequence, with Ck=:!:1.Thetransmitted signalx(t)istherefore N dimensional inthatitrequires aminimum ofNorthonormal functions forits representation. Consider nexttherepresentation oftheinterfering signal(jammer), j(t).Ideally,the jammerlikestoplaceallofitsavailable energyinexactlythesameN-dimensional signal spaceasthettansmitted signalx(t);otherwise, partofitsenergygoestowaste.However, thebestthatthejammercanhopetoknowisthetransmitted signalbandwidth. Moreover, thereisnowaythatthejammercanhaveknowledge ofthesignalphase.Accordingly, we may-represent thejammerbythegeneralform whereN-l N-l j(t)=LikcPk(t)+L7k~k(t), k~o k~O(7.17) andrTb jk=Joj(t)cPdt)dt,.k=0,1,..., N-1 k=0,1,..., N - 1(7.1S) (7.19) Thustheinterference j(t)is2N-dimensional; thatis,ithastwicethenumberofdimensions required forrepresenting thetransmitted DSIBPSK signalx(t).Intermsoftherepresen- 7.5Signal-Space Dimensionality andProcessing Gain 495 tationgiveninEquation (7.17),wemayexpresstheaveragepoweroftheinterference j(t) asfollows: 1IT.J=T;;0?(t)dt 1N-l 1N-l_ =-Lj~+-Lj~Tbk-OTbk-O(7.20) Moreover, duetolackofknowledge ofsignalphase,thebeststrategyajammercanapply istoplaceequalenergyinthecosineandsinecoordinates definedinEquations (7.18)and (7.19);hence,wemaysafelyassume Correspondingly, wemaysimplifyEquation (7.20)as 2N-lJ=-Lj~Tbk-O(7.21) (7.22) Ouraimistotietheseresultstogether byfindingthesignal-to-noise ratiosmeasured at theinputandoutputoftheDSIBPSK receiverinFigure7.9.Tothatend,weuseEquation (7.13)toexpressthecoherent detectoroutputas f2ITb V=~T;; 0u(t)cos{2'Trfct) dt (7.23) wherethecomponents VsandVcjareduetothedespread binaryPSKsignal,s(t),andthe spreadinterference, c(t)j(t),respectively. Thesetwocomponents aredefinedasfollows: andfIIT.v,=~T;; 0s(t)cos(2'Trfct) dt fIITb Vcj=~T;; 0c(t)j(t)cos(2'Trfct) dt(7.24) (7.25) Consider firstthecomponent Vsduetothesignal.Thedespread binaryPSKsignal s(t)equals ~s(t)=i:~I;;"cos(2'Trfct), (7.26) wheretheplussigncorresponds toinformation bit1,andtheminussigncorresponds to information bitO.Hence,assuming thatthecarrierfrequency Ieisanintegermultiple of 11Thwehave Vs=±~ (7.27) 496 CHAPTER 7"SPREAD-SPECTRUM MODULATION Consider nextthecomponent vc;duetointerference. Expressing thePNsignalc(t) intheexplicitformofasequence, (co,C1>...,CN-tl,wemayrewriteEquation (7.25)in thecorresponding form (7.28) UsingEquation (7.14)forcPk(t),andthenEquation (7.18)forthecoefficient;k, wemay redefinevc;as (7.29) Wenextapproximate thePNsequence asanindependent andidentically distributed (i.i.d.) binarysequence. Weemphasize theimplication ofthisapproximation byrecasting Equa tion(7.29)intheform (7.30) whereVc;andCkarerandom variables withsamplevalues Vc;andCk,respectively. In Equation (7.30),thejammerisassumed tobefixed.WiththeCktreatedasU.d.random variables, wefindthattheprobability oftheeventCk=±1is P(Ck=1)=P(Ck=-1)=f (7.31) Accordingly, themeanoftherandomvariableV;jiszerosince,forfixedk,wehave E[Ckik!;k]=;kP(Ck=1)-;kP(Ck=-1) 1·1·=Ilk-Ilk =0(7.32) (7.33)Forafixedvectorj,representing thesetofcoefficients ;0';1>...,;N-1>thevariance ofV,; isgivenby 1N-l var[Vc;liJ = -2:ifNk~O SincethespreadfactorN=Tb/T"wemayuseEquation (7.22)toexpressthisvariancein termsoftheaverageinterference powerJas (7.34) ThustherandomvariableVc;haszeromeanandvarianceJTJ2. FromEquation (7.27),wenotethatthesignalcomponent atthecoherent detector output(duringeachbitinterval) equals±~,whereEt,isthesignalenergyperbit.Hence, thepeakinstantaneous powerofthesignalcomponent isEboAccordingly, wemaydefine (7.35)7.6Probability ojErroT 497 theoutputsignal-to-noise ratioastheinstantaneous peakpowerE"dividedbythevariance oftheequivalent noisecomponent inEquation (7.34).Wethuswrite 2Eb(SNR)o =]T e TheaveragesignalpoweratthereceiverinputequalsE,jTb•Wethusdefineaninputsignal to-noiseratioas (SNRh =EblTb ](7.36) Hence,eliminating Ebl]betweenEquations (7.35)and(7.36),wemayexpresstheoutput signal-to-noise ratiointermsoftheinputsignal-to-noise ratioas 2Tb(SNR)o =T(SNRh (7.37) (7.38)Itiscustomary practicetoexpresssignal-to-noise ratiosindecibels.Tothatend,weintro duceatermcalledtheprocessing gain(PG),whichisdefinedasthegaininSNRobtained bytheuseofspreadspectrum. Specifically, wewrite PG=T!? Tc whichrepresents thegainachieved byprocessing aspread-spectrum signaloveranun spreadsignal.WemaythuswriteEquation (7.37)intheequivalent form: (7.39) The3-dBtermontheright-hand sideofEquation (7.39)accounts forthegaininSNR thatisobtained throughtheuseofcoherent detection (whichpresumes exactknowledge ofthesignalphasebythereceiver). ThisgaininSNRhasnothingtodowiththeuseof spreadspectrum. Rather,itisthelastterm,10loglo(PG), thataccounts fortheprocessing gain.Notethatboththeprocessing gainPGandthespreadfactorN(i.e.,PNsequence length)equaltheratioT,jT,.Thus,thelongerwemakethePNsequence (or,correspond ingly,thesmallerrhechiptimeTeis),thelargerwilltheprocessing gainbe. ~_~Probability ofError Letthecoherent detector ourputvinthedirect-sequence spreadBPSKsystemofFigure 7.9denotethesamplevalueofarandomvariableV.Lettheequivalent noisecomponent vcjproduced byexternal interference denotethesamplevalueofarandomvariable V,j' Then,fromEquarions (7.23)and(7.27)wededucethat V=:tV£;;+V,j (7.40) whereEbisthetransmitted signalenergyperbit.Theplussignreferstosendingsymbol (information bit)1,andtheminussignreferstosendingsymbolO.Thedecisionruleused bythecoherent detector ofFigure7.9istodeclarethatthereceivedbitinaninterval(0, 7[,)is1ifthedetector outputexceedsathreshold ofzero,andthatitis0ifthedetector outputislessthanrhethreshold; ifthedetectoroutputisexactlyzero,thereceivermakes arandomguessinfavorof1orO.Withbothinformation bitsassumed equallylikely,we 498 CHAPTER 7"SPREAD-SPECTRUM MODUlATION findthat(because ofthesymmetric natureoftheproblem) theaverageprobability oferrOr Peisthesameastheconditional probability of(say)thereceivermakingadecisionin favorofsymbol1,giventhatsymbol0wassent.Thatis, Pe=P(V>0Isymbol0wassent) =P(Vcj>VE;;)(7.41) Naturally, theprobability oferrorPedependsontherandomvariable VcjdefinedbyEqua_ tion(7.30).According tothisdefinition, VcjisthesumofNidentically distributed random variables. Hence,fromthecentrallimittheorem, wededucethatforlargeN,therandom variableVcjassumes aGaussian distribution. Indeed,thespreadfactororPNsequence lengthNistypically largeinthedirect-sequence spread-spectrum systemsencountered in practice, underwhichcondition theapplication ofthecentrallimittheorem isjustified. Earlierweevaluated themeanandvariance ofVcj;seeEquations (7.32)and(7.34). Wemaytherefore statethattheequivalent noisecomponent Vcjcontained inthecoherent detectoroutputmaybeapproximated asaGaussian randomvariablewithzeromeanand varianceJTJ2,whereJistheaverageinterference powerandTcisthechipduration. With thisapproximation athand,wemaythenproceedtocalculate theprobability oftheevent Vcj>VE;;,andthusexpresstheaverageprobability oferrorinaccordance withEquation (7.41)as (7.42) Thissimpleformula, whichinvokestheGaussian assumption, isappropriate forDSIBPSK binarysystemswithlargespreadfactorN. IIIANTIJAM CHARACTERISTICS Itisinformative tocompare Equation (7.42)withtheformulafortheaverageprobabiliry oferrorforacoherent binaryPSKsystemreproduced hereforconvenience ofpresentation [seeEquation (6.20)] Pe=ierfc(Hi)(7.43) Basedonthiscomparison, weseethatinsofarasthecalculation ofbiterrorrateinadirect sequence spreadbinaryPSKsystemisconcerned, theinterference maybetreatedas widebandnoiseofpowerspectraldensityNo/2,definedby NoJT, 2 2(7.44) Thisrelationissimplyarestatement ofanearlierresultgiveninEquation (7.34). SincethesignalenergyperbitEb=PTb,wherePistheaveragesignalpowerandTb isthebitduration, wemayexpressthesignalenergyperbit-to-noise spectraldensityratio as ~=(~)(7)(7.45) (7.46)7.7Frequency-Hop SpreadSpectrum 499 Usingthedefinition ofEquation (7.38)fortheprocessing gainPGwemayreformulate thisresultas LPG PEblNo TheratioJlPistermedthejamming margin.Accordingly, thejamming marginandthe processing gain,bothexpressed indecibels, arerelatedby (Jamming margin)dB=(Procesing gain)dB-1010glo(Eb) (7.47) . Nomin where(EbINo)m;n istheminimum valueneededtosupportaprescribed averageprobability oferror. !!'>EXAMPLE 7.3 Aspread-spectrum communication systemhasthefollowing parameters: Information bitduration, Tb=4.095ms PNchipduration, Tc=1p.s Hence,usingEquation (7.38)wefindthattheprocessing gainis PG=4095 Correspondingly, therequired periodofthePNsequence isN=4095,andtheshift-register lengthism12. Forasatisfactory reception, wemayassumethattheaverageprobability oferror isnottoexceed10-5•Fromtheformula foracoherent binaryPSKreceiver, wefindthat EblNo=10yieldsanaverageprobability oferrorequalto0.387X10-5•Hence,usingthis valueforEblNo,andthevaluecalculated fortheprocessing gain,wefindfromEquation (7.47) thatthejamming marginis (Jamming margin)dB =10loglo4095-10loglo(10) =36.110 =26.1dB Thatis,information bitsatthereceiveroutputcanbedetected reliablyevenwhenthenoise orinterference atthereceiverinputisupto409.5tinlesthereceived signalpower.Clearly, thisisapowerful advantage againstinterference (jamming), whichisrealizedthroug\l the cleveruseofspread,spectrum modulation. 41 L7.7Frequency-Hop SpreadSpectrum- Inthetypeofspread-spectrum systemsdiscussed inSection7.4,theuseofaPNsequence tomodulate aphase-shift-keyed signalachievesinstantaneous spreading ofthetransmisc sianbandwidth. Theabilityofsuchasystemtocombattheeffectsofjammersisdetermined bytheprocessing gainofthesystem,whichisafunction ofthePNsequence period.The processing gaincanbemadelargerbyemploying aPNsequence withnarrowchipdura tion,which,inturn,permitsagreatertransmission bandwidth andmorechipsperbit. However, thecapabilities ofphysical devicesusedtogenerate thePNspread-spectrum signalsimposeapractical limitontheattainable processing gain.Indeed,itmayturnout thattheprocessing gainsoattained isstillnotlargeenoughtoovercome theeffectsof 500 CHAPTER 7OJSPREAD-SPECTRUM MODULATION somejammers ofconcern, inwhichcasewehavetoresorttoothermethods. Onesuch alternative methodistoforcethejammertocoverawiderspectrum byrandomly hOPPing thedata-modulated carrierfromonefrequency tothenext.Ineffect,thespectrum ofthe transmitted signalisspreadsequentially ratherthaninstantaneously; theterm"sequen_ tially"referstothepseudo-random-ordered sequence offrequency hops. Thetypeofspreadspectrum inwhichthecarrierhopsrandomly fromonefrequency toanotheriscalledfrequency-hop (FH)spreadspectrum. Acommon modulation format forFHsystemsisthatofM-aryfrequency-shift keying(MFSK). Thecombination ofthese twotechniques isreferredtosimplyasFHlMFSK. (Adescription ofM-aryFSKispresented inChapter6.) Sincefrequency hopping doesnotcovertheentirespreadspectrum instantaneously, weareledtoconsider therateatwhichthehopsoccur.Inthiscontext,wemayidentify twobasic(technology-independent) characterizations offrequency hopping: 1.Slow-frequency hopping, inwhichthesymbolrateR,oftheMFSKsignalisaninteger multiple ofthehoprateRh•Thatis,severalsymbols aretransmitted oneachfre quencyhop. 2.Fast-frequency hopping, inwhichthehoprateRhisanintegermultipleoftheMFSK symbolrateR,.Thatis,thecarrierfrequency willchangeorhopseveraltimesduring thetransmission ofonesymbol. Obviously, slow-frequency hopping andfast-frequency hopping aretheconverse ofone another. Inthefollowing, thesetwocharacterizations offrequency hoppingareconsidered inturn. !>!ISWW-FREQUENCY HOPPING Figure7.10ashowstheblockdiagram ofanFHlMFSK transmitter, whichinvolves fre quencymodulation followed bymixing.First,theincoming binarydataareappliedtoan M-aryFSKmodulator. Theresulting modulated waveandtheoutputfromadigitalfre quencysynthesizer arethenappliedtoamixerthatconsistsofamultiplier followed bya band-pass filter.Thefilterisdesigned toselectthesumfrequency component resulting fromthemultiplication processasthetransmitted signal.Inparticular, successive k-bit segments ofaPNsequence drivethefrequency synthesizer, whichenablesthecarrierfre quencytohopover2kdistinctvalues.Onasinglehop,thebandwidth ofthetransmitted signalisthesameasthatresulting fromtheuseofaconventional MFSKwithanalphabet ofM=2Korthogonal signals.However, foracomplete rangeof2kfrequency hops,the transmitted FHlMFSK signaloccupies amuchlargerbandwidth. Indeed,withpresent-day technology, FHbandwidths ontheorderofseveralGHzareattainable, whichisanorder ofmagnitude largerthanthatachievable withdirect-sequence spreadspectra.Animpli cationoftheselargeFHbandwidths isthatcoherent detection ispossibleonlywithineach hop,becausefrequency synthesizers areunabletomaintain phasecoherence oversucces sivehops.Accordingly, mostfrequency-hop spread-spectrum communication systemsuse noncoherent M-arymodulation schemes. Inthereceiverdepicted inFigure7.10b,thefrequency hopping isfirstremoved by mixing(down-converting) thereceivedsignalwiththeoutputofalocalfrequency synthe sizerthatissynchronously controlled inthesamemannerasthatinthetransmitter. The resulting outputisthenband-pass filtered,andsubsequently processed byanoncoherent M-aryFSKdetector. Toimplement thisM-arydetector, wemayuseabankofMnonCD herentmatched filters,eachofwhich ismatched tooneoftheMFSKtones.(Noncoherent matched filtersaredescribed inChapter6.)Anestimateoftheoriginalsymboltransmitted isobtained byselecting thelargestfilteroutput. 7.7Frequency-Hop SpreadSpectrum 501 Mixer Binary data (0) MixerFH/MFSK signal Received signalEstimate ofbinary data (7.49)(b) FIGURE7.10Frequency-hop spreadM-aryfrequency-shift keying.(a)Transmitter. (b)Receiver. Anindividual FHlMFSK toneofshortestduration isreferredtoasachip;thister minology shouldnotbeconfused withthatusedinSection7.4describing DSIBPSK. The chiprate,Re,foranFH/MFSK systemisdefinedby Re=max(R h,Rs) (7.48) whereRhisthehoprate,andRsisthesymbolrate. AslowFH/MFSK signalischaracterized byhavingmultiplesymbolstransmitted per hop.Hence,eachsymbolofaslowFHlMFSK signalisachip.Correspondingly, inaslow FH/MFSK system,thebitrateRboftheincoming binarydata,thesymbolrateR,ofthe MFSKsignal,thechiprateRoandthehoprateRharerelatedby RbRc=Rs=K;;=:Rh whereK=log2M. Ateachhop,theMFSKtonesareseparated infrequency byanintegermultiple of thechiprateRe=R"ensuring theirorthogonality. Theimplication ofthiscondition is thatanytransmitted symbolwillnotproduceanycrosstalk intheotherM - 1noncoherent matched filtersconstituting theMFSKdetectorofthereceiverinFigure7.10b.By"cross talk"wemeanthespillover fromonefilteroutputintoanadjacent one.Theresulting performance oftheslowFH/MFSK systemisthesameasthatforthenoncoherent detection 502 CHAPTER 7'"SPIlEAD-SPECTRUM MODULATION ofconventional (unhopped) MFSKsignalsinadditive whiteGaussian noise.Thusthe interfering (jamming) signalhasaneffectontheFHlMFSK receiver, intermsofaverage probability ofsymbolerror,equivalent tothatofadditive whiteGaussian noiseOna conventional noncoherent M-aryFSKreceiverexperiencing nointerference. Onthebasis ofthisequivalence, wemayuseEquation (6.140)forapproximate evaluation oftheprob abilityofsymbolerrorintheFHlMFSK system. Assuming thatthejammer decidestospreaditsaveragepower] overtheentire frequency-hopped spectrum, thejammer's effectisequivalent toanAWGNwithpOWer spectraldensityNoll,whereNo=]/WeandWeistheFHbandwidth. Thespread-spectrum systemisthuscharacterized bythesymbolenergy-to-noise spectraldensityratio: EPI] NoWjR,(7.50) (7.51)wheretheratioPI]isthereciprocal ofthejamming margin.Theotherratiointhedenom inatorofEquation (7.50)istheprocessing gainoftheslowFHlMFSK system,whichis definedby PG=We R, =lk Thatis,theprocessing gain(expressed indecibels) isequalto10loglolk""3k,wherek isthelengthofthePNsegment employed toselectafrequency hop. ThisresultassumesthatthejammerspreadsitspowerovertheentireFHspectrum. However, ifthejammerdecidestoconcentrate onjustafewofthehoppedfrequencies, thentheprocessing gainrealizedbythereceiverwouldbelessthan3kdecibels. !!?EXAMPLE 7.4 Figure7.11aillustrates thevariation ofthefrequency ofaslowFH/MFSK signalwithtime foronecomplete periodofthePNsequence. TheperiodofthePNsequence is241=15. TheFH/MFSK signalhasthefollowing parameters: NumberofbitsperMFSKsymbol NumberofMFSKtones LengthofPNsegmentperhop Totalnumberoffrequency hopsK=2 M2K=4 k=3 2k8 Inthisexample, thecarrierishoppedtoanewfrequency aftertransmitting twosymbolsor equivalently, fourinformation bits.Figure7.11aalsoincludestheinputbinarydata,andthe PNsequencecontrolling theselectionofFHcarrierfrequency. Itisnoteworthy thatalthough thereareeightdistinctfrequencies available forhopping, onlythreeofthemareutilizedby thePNsequence. Figure7.11bshowsthevariation ofthedehopped frequency withtime.Thisvariation isrecognized tobethesameasthatofaconventional MFSKsignalproduced bythegiven inputdata. <41 IilIFAST-FREQUENCY HOPPING AfastFHlMFSK systemdiffersfromaslowFHlMFSK systeminthattherearemultiple hopsperM-arysymbol.Hence,inafastFHlMFSK system,eachhopisachip.Ingeneral, 7.7Frequency-Hap SpreadSpect....... 503 FHcarrier Frequency We Inputbinarydata01 1 110 0 0 10 0 1 111 0 1 0 PNsequenceI001 110 011 001 001 (a) R,{ Time~ (b) FIGURE 7.11Illustrating slow·frequency hopping. (a)Frequency variation foronecomplete periodofthePNsequence. (b)Variation ofthedehopped frequency withtime. fast-frequency hopping isusedtodefeatasmartjammer's tacticthatinvolvestwofunc tions:measurements ofthespectralcontentofthetransmitted signal,andretuning ofthe interfering signaltothatportionofthefrequency band.Clearly,toovercome thejammer, thetransmitted signalmustbehopped toanewcarrierfrequency beforethejammeris abletocomplete theprocessing ofthesetwofunctions. Fordatarecovery atthereceiver, noncoherent detection isused.However, thede tectionprocedure isquitedifferent fromthatusedinaslowFHlMFSK receiver.Inpartic ular,twoprocedures maybeconsidered: 1.ForeachFHlMFSK symbol,separate decisions aremadeontheKfrequency-hop chipsreceived, andasimplerulebasedonmajority voteisusedtomakeanestimate ofthedehopped MFSKsymbol. 2.ForeachFHlMFSK symbol,likelihood functions arecomputed asfunctions ofthe totalsignalreceivedoverKchips,andthelargestoneisselected. 504 CHAPTER 7..SPREAD-SPECIRUM MODUlATION Areceiverbasedonthesecondprocedure isoptimum inthesensethatitminimizes the averageprobability ofsymbolerrorforagivenEhiNo• ~EXAMPLE 7.5 Figure7.12aillustrates thevariation ofthetransmitted frequency ofafastFHlMFSK Signal withtime.Thesignalhasthefollowing parameters: Number ofbitsperMFSKsymbol Number ofMFSKtones LengthofPNsegment perhop Totalnumberoffrequency hopsK=2 M=2K=4 k=3 2'=8 Inthisexample, eachMFSKsymbolhasthesamenumberofbitsandchips;thatis,thechip rateRcisthesameasthebitrateRb•Aftereachchip,thecarrierfrequency ofthetransmitted MFSKsignalishoppedtoadifferent value,exceptforfewoccasions whenthek-chipsegment ofthePNsequence repeatsitself. 1\1III fH carrier11IIIII Time-;:.- MfSK symbol Inputbinarydataf..?'::-iJ 1110001001111010 PNsequence 001110011001001001110011001001001110011001001001110011001001 (a) Time----?'" (h) FIGURE7.12Illustrating fast-frequency hopping. (a)Variation ofthetransmitter frequency with. time.(b)Variation ofthedehopped frequency withtime. 7.8Cmnputer Experiments: Maximal-Length andGoldCOMS 505 Figure7.12bdepictsthetimevariationofthefrequency ofthedehopped MFSKsignal, whichisthesameasthatinExample 7.4. 4il 7.8C01nputer Experim-ents: Maxi1nal-Length andGoldCodes Code-division multiplexing (CDM)provides analternative tothetraditional methods of frequency-division multiplexing (FDM)andtime-division multiplexing (TDM).Itdoesnot requirethebandwidth allocation ofFDM(discussed inChapter2)northetimesynchro nizationneededinTDM(discussed inChapter3).Rather,usersofacornmon channelare permitted accesstothechannelthrough theassignment ofa"spreading code"toeach individual userundertheumbrella ofspread-spectrum modulation. The purpose ofthis computer experiment istostudyacertainclassofspreading codesforCDMsystemsthat provideasatisfactory performance. InanidealCDMsystem,thecross-eorrelation between anytwousersofthesystem iszero.Forthisidealcondition toberealized, werequirethatthecross-correlation function between thespreading codesassignedtoanytwousersofthesystembezeroforallcyclic shifts.Unfortunately, ordinary PNsequences donotsatisfythisrequirement becauseof theirrelatively poorcross-correlation properties. Asaremedyforthisshortcoming ofordinary PNsequences, wemayuseaspecial classofPNsequences calledGoldsequences (codes),' thegeneration ofwhichisembodied inthefollowing theorem: Letg,(X)andg2(X)beapreferred pairofprimitive polynomials ofdegreenwhose corresponding shiftregistersgeneratemaximal-length sequences ofperiod2n-1 andwhosecross-correlation functionhasamagnitude lessthanorequalto or2(n+1)12+1fornodd (7.52) 2(n+2)12+1fornevenandn*-Omod4 (7.53) Thentheshiftregistercorresponding totheproductpolynomial gl(X).g2(X)will generate 2"+1differentsequences, witheachsequence havingaperiodof2n-1, andthecross-correlation betweenanypairofsuchsequences satisfying thepreced ingcondition. Hereafter, thistheorem isreferredtoasGold'stheorem. Tounderstand Gold'stheorem, weneedtodefinewhatwemeanbyaprimitive polynomial. Consider apolynomial g(X)definedoverabinaryfield(i.e.,afinitesetoftwo elements, 0and1,whichisgoverned bytherulesofbinaryarithmetic). Thepolynomial g(X)issaidtobeanirreducible polynomial ifitcannotbefactored usinganypolynomials fromthebinaryfield.Anirreducible polynomial g(X)ofdegreemissaidtobeaprimitive polynomial ifthesmallest integermforwhichthepolynomial g(X)dividesthefactor X"+1isn=2~-1.Furtherdiscussion ofthistopicisdeferredtoChapter8;inparticular, seeExample 8.3. 506 CHAPTER 7"SPREAD-SPECTRUM MODUlATION Experiment 1.Correlation Properties ofPNSequences Consider apairofshiftregistersforgenerating twoPNsequences ofperiod27-1=:127 Onefeedback shiftregisterhasthefeedback taps[7,1]andtheotheronehasthefeedback taps[7,6,5,4]. Bothsequences havethesameautocorrelation function showninFigure 7.134,whichfollowsreadilyfromthedefinition presented inEquation (7.5). However, thecalculation ofthecross-correlation function betweenPNsequences is amoredifficultproposition, particularly forlargen.Toperformthiscalculation, wereSOrt totheuseofcomputer simulation forvarying cyclicshift 'Tinsidetheinterval 0<'T:;;2n-1.Theresultsofthiscomputation arepresented inFigure7.13b.Thisfigure confirms thepoorcross-correlation property ofPNsequences compared totheirautocor. relationfunction. Themagnitude ofthecross-correlation function exceeds40. 120 _100.;:. '"§BO j §60 ~§40 ~ <t 20 o1'---------,-..JI...----------4 40-100 -50 o Delay'T (a)50 100 -30 -40 -100 -50 o Delay T50 100 (b) FIGURE7.13(a)Autocorrelation functionRe(T),and(b)cross-correlation functionR,2(T)ofthe twoPNsequences [7,1]and[7,6,5,4]. 7.8Ctnnputer Experiments: Maximal-Length andGoldCodes 507 Gold sequence Clock FIGURE7.14Generator foraGoldsequence ofperiod2'- 1=127. Experiment 2.Correlation Properties ofGoldSequences Forournextexperiment, weconsider Goldsequences withperiod27-1=127.To generate suchasequence forn=7weneedapreferred pairofPNsequences thatsatisfy Equation (7.52)(nodd),asshownby 2(n+lj/2+1=24+1=17 Thisrequirement issatisfiedbythePNsequences withfeedback taps[7,4]anq[7,6,5,4]. TheGold-sequence generator isshowninFigure7.14thatinvolvesthemodulo-2 addition ofthesetwosequences. According toGold'stheorem, thereareatotalof 2"+1=27+1=129 sequences thatsatisfyEquation (7.52). The cross-correlation between anypairofsuch sequences isshowninFigure7.15,whichisindeedinfullaccordwithGold'stheorem. In particular, themagnitude ofthecross-correlation islessthanorequalto17. 20,---~--~---~---~---~~ -15 -20~-_-cl~00----~5-'-0---~0 ---':'50::----:-1~OO::----.J Delay'T FIGURE7.15Cross-correlation function R12(T)ofapairofGoldsequences basedonthetwo PNsequences [7,4]and[7,6,5,4]. 508 CHAPTER 7'"SPREAD-SPECTRUI\I MODUlATION I7.9Sutnmary andDiscussion Direct-sequence M-aryphaseshiftkeying«DS/MPSK) andfrequency-hop M-aryfrequenc shift-keying (FHlMFSK) represent twoprincipal categories ofspread-spectrum comrn:' nications. Bothofthemrelyontheuseofapseudo-noise (PN)sequence, whichisapplied differently inthetwocategories. InaDS/MPSK system,thePNsequence makesthetransmitted signalassumea noiselike appearance byspreading itsspectrum overabroadrangeoffrequencies simul_ taneously. Forthephase-shift keying,wemayusebinaryPSK(i.e.,M=2)withasingle carrier.Alternatively, wemayuseQPSK(i.e.,M=4),inwhichcasethedataaretrans_ mittedusingapairofcarriersinphasequadrature. (BothPSKandQPSKarediscussedin Section6.3.)Theusualmotivation forusingQPSKistoprovideforimproved bandwidth efficiency. Inaspread-spectrum system,bandwidth efficiency isusuallynotofprimecon cern.Rather,theuseofQPSKismotivated bythefactthatitislesssensitive tosometypes ofinterference (jamming). InanFHlMFSK system,thePNsequence makesthecarrierhopoveranumberof frequencies inapseudo-random manner, withtheresultthatthespectrum ofthetrans mittedsignalisspreadinasequential manner. Naturally, thedirect-sequence andfrequency-hop spectrum-spreading techniques maybeemployed inasinglesystem.Theresulting systemisreferredtoashybridDSIFH spread-spectrum system.Thereasonforseekingahybridapproach isthatadvantages of boththedirect-sequence andfrequency-hop spectrum-spreading techniques arerealizedin thesamesystem. Adiscussion ofspread-spectrum communications wouldbeincomplete withoutsome reference tojammerwaveforms. Thejammers encountered inpracticeincludethefollow ingtypes: 1.Thebarragenoisejammer, whichconsistsofband-limited whiteGaussian noiseof highaveragepower.Thebarragenoisejammerisabrute-force jammerthatdoesnot exploitanyknowledge oftheantijamcommunication systemexceptforitsspread bandwidth. 2.Thepartial-band noisejammer, whichconsistsofnoisewhosetotalpowerisevenly spreadoversomefrequency bandthatisasubsetofthetotalspreadbandwidth. Owingtothesmallerbandwidth, thepartial-band noisejammeriseasiertogenerate thanthebarragenoisejammer. 3.Thepulsednoisejammer, whichinvolvestransmitting widebandnoiseofpower forafractionpofthetime,andnothingfortheremaining fraction1 -Pofthetime. Theaveragenoisepowerequals]. 4.Thesingle-tone jammer, whichconsistsofasinusoidal wavewhosefrequency lies insidethespreadbandwidth; assuch,itistheeasiestofalljamming signalsto generate. 5.Themultitone jammer,whichisthetoneequivalent ofthepartial-band noisejammer. Inaddition tothesefive,manyotherkindsofjamming waveforms occurinpractice. In anyevent,thereisnosinglejamming waveform thatisworstforallspread-spectrum Problems 509 systems, andthereisnosinglespread-spectrum systemthatisbestagainstallpossible jamming waveforms. ~OTES ANDREFERENCES 1.Thedefinition ofspread-spectrum modulation presented intheIntroduction isadaptedfrom Pickholtz, Schilling, andMilstein (1982).Thispaperpresentsatutorialreviewofthetheory ofspread-spectrum communications. Forintroductory papersonthesubject,seeViterbi(1979),andCookandMarsh (1983).Forbooksonthesubject,seeDixon(1984),Holmes(1982),ZiemerandPeterson (1985,pp.327-649), CooperandMcGillem (1986,pp.269--411), andSimon,Omura, Scholtz,andLevitt(1985,Volumes I,II,andill).Thethree-volume bookbySimonetal.is themostexhaustive treatment ofspread-spectrum communications available intheopen literature. Thedevelopment ofspread-spectrum communications datesbacktoaboutthe mid-1950s. Forahistorical accountofthesetechniques, seeScholtz(1982).Thislatterpaper ttacestheoriginsofspread-spectrum communications backtothe1920s.Muchofthehis toricalmaterial presented inthispaperisreproduced inChapter2,Volume I,ofthebook bySimonetal. ThebookeditedbyTantaratana andAhmed(1998)includes introductory andad vancedpapersonwireless applications ofspread-spectrum modulation. Thepapersare grouped intothefollowing categories: spread-spectrum technology, cellulatmobilesystems, satellitecommunications, wireless localareanetworks, andglobalpositioning systems (GPS). 2.Forfurtherdetailsonmaximal-length sequences, seeGolomb (1964,pp.1-32),Simon, Omura,Scholtz,andLevitt(1985,pp.283-295), andPeterson andWeldon(1972).The lastreference includesanextensive listofpolynomials forgenetating maximal-length se quences; seealsoDixon(1984).Foratutorialpaperonpseudo-noise sequences, seeSarwate andPursley(1980). 3.Table7.1isextracted fromthebookbyDixon(1984,pp.81-83),wherefeedback connec tionsofmaximal-length sequences aretabulated forshift-register lengthmextending up to89. 4.Fordetaileddiscussion ofthesynchronization probleminspread-spectrum communications, seeZiemerandPeterson (1985,Chapters 9and10)andSimonetal.(1985,Volumeill). 5.TheoriginalpapersonGoldsequences areGold(1967,1968).Adetaileddiscussion ofGold sequences ispresented inHolmes(1982). lPROBLEMS Pseudo-Noise Sequences 7.1Apseudo-noise (PN)sequence isgenerated usingafeedback shiftregisteroflength m=4.Thechiprateis107chipspersecond.Findthefollowing parameters: (a)PNsequence length. (h)Chipduration ofthePNsequence. (c)PNsequence period. 510 CHAl'TER 7IIISPREAD-SPECTRUM MODULATION 7.2FigureP7.2showsafour-stage feedback shiftregister.Theinitialstateoftheregisteris 1000.Findtheoutputsequence oftheshiftregister. Modulo-2 adder Output sequence FIGUREP7.2 7.3Forthefeedback shiftregistergiveninProblem 7.2,demonstrate thebalanceproperty andrunproperty ofaPNsequence. Also,calculate andplottheautocorrelation function ofthePNsequence produced bythisshiftregister. 7.4Referring toTable7.1,developthemaximal-length codesforthethreefeedback config urations[6,1],[6,5,2,1], and[6,5,3,2],whoseperiodisN=63. 7.5FigureP7.5showsthemqdular multitap versionofthelinearfeedback shift-register showninFigure7Ab.Demonstrate thatthePNsequence generated bythisschemeis exactlythesameasthatdescribed inTable7.2b. ~.I,~o~"sequence Clock FIGUREP7.5 DirectSequencelPhase-Shift KeyingSystem 7.6ShowthatthetruthtablegiveninTable7.3canbeconstructed bycombining thefollow ingtwosteps: (a)Themessagesignalb(t)andPNsignalc(t)areaddedmodulo-2. (b)Symbols 0and1atthemodulo-2adderoutputarerepresented byphaseshiftsof0 and180degrees,respectively. 7.7Asingle-tone jammer j(t)=Vi]cos(271fct+(J) isappliedtoaDS/BPSK system.TheN-dimensional transmitted signalx(t)isdescribed byEquation (7.16).Findthe2Ncoordinates ofthejammeritt). 7.8Theprocessing gainofaspread-spectrum systemmaybeexpressed astheratioofthe spreadbandwidth ofthetransmitted signaltothedespread bandwidth ofthereceived signal.Justifythisstatement fortheDS/BPSK system. Problems 511 7.9Adirect-sequence spreadbinaryphase-shift keyingsystemusesafeedback shiftregister oflength19forthegeneration ofthePNsequence. Calculate theprocessing gainofthe system. 7.10InaDS/BPSK system,thefeedback shiftregisterusedtogenerate thePNsequence has lengthm=19.Thesystemisrequired tohaveanaverageprobability ofsymbolerror duetoexternally generated interfering signalsthatdoesnotexceed10-5•Calculate the following systemparameters indecibels: (a)Processing gain. (b)Antijammargin. 7.11InSection7.5,wepresented ananalysisonthesignal-space dimensionality andprocessing gainofadirectsequence spread-spectrum systemusingbinaryphase-shift keying.Extend theanalysispresented thereintothecaseofsuchasystemusingquadriphase-shift keying. Frequency-Hop SpreadSpectrum 7.12AslowFHlMFSK systemhasthefollowing parameters: Number ofbitsperMFSKsymbol=4 Number ofMFSKsymbolsperhop=5 Calculate theprocessing gainofthesystem. 7.13AfastFHlMFSK systemhasthefollowing parameters: NumberofbitsperMFSKsymbol =4 NumberofhopsperMFSKsymbol=4 Calculate theprocessing gainofthesystem. Computer Experiments 7.14Consider twoPNsequences ofperiodN=63.Onesequence hasthefeedback taps[6,1] andtheothersequence hasthefeedback taps[6,5,2,1],whicharepickedinaccordance withTable7.1. (a)Compute theautocorrelation function ofthesetwosequences, andtheircross correlation function. (b)Compare thecross-correlation function computed inpart(a)withthecross correlation function betweenthesequence [6,52,1] anditsmirrorimage[6,5,4,1]. Comment onyourresults. 7.15(a)Compute thepartialcross-correlation function ofaPNsequence withfeedback taps [5,2]anditsimagesequence definedbythefeedback taps[5,3]. (b)Repeatthecomputation forthePNsequence withfeedback taps[5,2]andthePN sequence withfeedback taps[5,4,2,1]. (c)Repeatthecomputation forthePNsequence withfeedback taps[5,4,3,2]andthe PNsequence withfeedback taps[5,4,2,1]. Thefeedback taps[5,2],[5,4,3,2],and[5,4,2,1]arepossibletapsforamaxirnal lengthsequence ofperiod31,inaccordance withTable7.1. MULTIUSER RADIO COMMUNICATIONS Asitsnameimplies, multiuser communications referstothesimultaneous useofa communication channel byanumber ofusers.Inthischapter, wediscussmultiuser communication systemsthatrelyonradiopropagation forlinkingthereceivers tothe transmitters. . Inparticular, wefocusonthefollowing topics: ~Multiple-access techniques, whicharebasictomultiuser communication systems. ~Satellitecommunications, offeringglobalcoverage. ~Radiolinkanalysis, highlighting therolesoftransmitting andreceiving antennas andfree- spacepropagation. ~Wireless communications withemphasis onmobility andthemultipath phenomenon. ~Speechcodingforwirelesscommunications. ~Adaptive antennas forwirelesscommunications. L8.1Introduction Muchofthematerial oncommunication theorypresented inearlierchapters hasbeen basedonaparticular idealization ofthecommunication channel,namely,achannelmodel limitedinbandwidth andcorrupted byadditivewhiteGaussian noise(AWGN).Theclas sicalcommunication theorysodeveloped ismathematically elegant,providing asound introduction totheever-expanding fieldofcommunication systems.Anexampleofaphys icalchannelthatiswellrepresented bysuchamodelisthesatellitecommunications chan nel.Itistherefore befitting thatthefirsttypeofmultiuser communications discussed in thischapterissatellitecommunications. Asatellitecommunication systemingeostationary orbitreliesonline-of-sight radio propagation fortheoperation ofitsuplinkfromanearthterminaltothetransponder and thedownlink fromthetransponder toanotherearthterminal. Thusthediscussion of satellitecommunications naturally leadstotheanalysisofradiopropagation infreespace, linkingareceiving anteunatoatransmitting antenna. Theuseofsatellitecommunications offersglobalcoverage. Theothermultiuser com munication systemstudiedinthischapter,namely,wirelesscommunications, offersmo bilitywhich,inconjunction withexistingtelephone networks andsatellitecommunication systems,permitsamobileunittocommunicate withanyone,anywhere intheworld.An othercharacteristic featureofwirelesscommunication systemsisthattheyaretetherless 512 8.2Multiple-Access Techniques 513 (i.e.,totalfreedom oflocation ispermitted), hencetheinterestintheiruseforlocalarea networks (i.e.,datanetworks confined tobuildings uptoafewkilometers insize)dueto significant advantages overconventional cabling:elimination ofwiringandrewiring, flex ibilityofcreatingnewcommunication services, andmobility ofusers. Theradiopropagation channel characterizing wireless communications deviates fromtheidealized AWGNchannelmodelduetothepresence ofmultipath, whichisa non-Gaussian formofsignal-dependent phenomenon thatarisesbecauseofreflections of thetransmitted signalfromfixedandmovingobjects.Thepresence ofmultipath raises practical difficulties intheuseofaradiopropagation channelandcomplicates itsmath ematical analysis. Simplyput,multipath isaphysicalphenomenon thatisintrinsictothe operation ofindoorandoutdoorformsofwirelesscommunications. Beforeproceeding todiscussspecificaspectsofsatellitecommunications andwireless communications, however, itisappropriate thatwebeginthediscussion bydescribing multiple-access techniques, whichenabledifferent userstosimultaneously (ornearlyso) accessacommon channel. L8.2Multiple-Access Techniques Multiple accessisatechnique whereby manysubscribers orlocalstationscansharethe useofacommunication channelatthesametimeornearlyso,despitethefactthattheir individual transmissions mayoriginate fromwidelydifferent locations. Statedinanother way,amultiple-access technique permitsthecommunication resources ofthechannelto besharedbyalargenumberofusersseekingtocommunicate witheachother. Therearesubtledifferences betweenmultipleaccessandmultiplexing thatshouldbe noted: I>Multiple accessreferstotheremotesharingofacommunication channelsuchasa satelliteorradiochannelbyusersinhighlydispersed locations. Ontheotherhand, multiplexing referstothesharingofachannelsuchasatelephone channelbyusers confined toalocalsite. I>Inamultiplexed system,userrequirements areordinarily fixed.Incontrast, ina multiple-access systemuserrequirements canchangedynamically withtime,inwhich caseprovisions arenecessary fordynamic channelallocation. Forobviousreasonsitisdesirable thatinamultiple-access systemthesharingof resources ofthechannelbeaccomplished withoutcausingseriousinterference between usersofthesystem.Inthiscontext,wemayidentifyfourbasictypesofmultipleaccess: 1.Frequency-division multipleaccess(FDMA). Inthistechnique, disjointsubbands offrequencies areallocated tothedifferent userson acontinuous-time basis.Inordertoreduceinterference betweenusersallocated adjacent channelbands,guardbandsareusedtoactasbufferzones,asillustrated inFigure8.la. Theseguardbandsarenecessary becauseoftheimpossibility ofachieving idealfiltering forseparating thedifferent users. 2.Time-division multipleaccess(TDMA). Inthissecondtechnique, eachuserisallocated thefullspectraloccupancy ofthechannel, butonlyforashortduration oftimecalledatimeslot.AsshowninFigure8.lb,buffer zonesintheformofguardtimesareinsertedbetweentheassigned timeslots.Thisisdone 514 CHAPTER 8..MULTIUSER RADIO COMMUNICATIONS Time (alTime (blTime (Cl FIGURE 8.1Illustrating theideasbehindmultiple-access techniques. (a)Frequency-division multiple access.(b)Time"division multiple access.(c)Frequency-hop multiple access. toreduceinterference between usersbyallowing fortimeuncertainty thatarisesdueto systemimperfections, especially insynchronization schemes. 3.Code-division multipleaccess(CDMA). InFDMA,theresources ofthechannelaresharedbydividingthemalongthefrequency coordinate intodisjointfrequency bands,asillustrated inFigure8.1a.InTDMA, the resources aresharedbydividing themalongthetimecoordinate intodisjointtimeslots, asillustrated inFigure8.1b.InFigure8.1c,weillustrate anothertechnique forsharingthe chamlelresources byusingahybridcombination ofFDMAandTDMA,whichrepresents aspecificformofcode-division multiple access(CDMA). Forexample, frequency hopping maybeemployed toensurethatduringeachsuccessive timeslot,thefrequency bands assignedtotheusersarereordered inanessentially randommanner.Tobespecific,during timeslot1,user1occupies frequency band1,user2occupies frequency band2,user3 occupies frequency band3,andsoon.Duringtimeslot2,user1hopstofrequency band 3,user2hopstofrequency band1,user3hopstofrequency band2,andsoon.Suchan arrangement hastheappearance oftheusersplayingagameofmusicalchairs.Animpor tantadvantage ofCDMAoverbothFDMAandTDMAisthatitcanprovideforsecure communications. InthetypeofCDMAillustrated inFigure8.1c,thefrequency hopping mechanism canbeimplemented throughtheuseofapseudo-noise (PN)sequence. 4.Space-division multipleaccess(SDMA). Inthismultiple-access technique, resource allocation isachieved byexploiting thespatial separation oftheindividual users.Inparticular, multibeam antennas areusedtoseparate radiosignalsbypointing themalongdifferent directions. Thus,different usersareenabled toaccessthechannelsimultaneously onthesamefrequency orinthesametimeslot. Thesemultiple-access techniques shareacommon feature:allocating thecommunication resources ofthechannelthrough theuseofdisjointedness (ororthogonality inaloose sense)intime,frequency, orspace. Withthisbackground materialathand,wearenowreadytodiscusssomeimportant multiuser communication systems. I8.3Satellite Communications Inageostationary satellitecommunication system,' amessagesignalistransmitted from anearthstationviaanuplinktoasatellite, amplified inatransponder (i.e.,electronic 8.3Satellite Communications 515 Earth transmitting station--- Earth •NorthPole____ ~~nk --- ------~ ___.L:------;atellite-&----- (ingeostationary orbit)_---- Downlink Earth receiving station FIGURE8.2Satellite communications system. circuitry} onboardthesatellite,andthenretransmitted fromthesatelliteviaadownlink toanotherearthstation,asillustrated inFigure8.2.Themostpopularfrequency bandfor satellitecommunications is6GHz(e-band) fortheuplinkand4GHzforthedownlink. Theuseofthisfrequency bandoffersthefollowing advantages: I>Relatively inexpensive microwave equipment. I>Lowattenuation duetorainfall;rainfallistheprimaryatmospheric causeofsignal degradation. 100Insignificant skybackground noise;theskybackground noise(duetorandomnoise emissions fromgalactic,solar,andterrestrial sources)reachesitslowestlevelbetween 1and10GHz. However, radiointerference limitstheapplications ofcommunication satellites operating inthe6/4GHzband,becausethetransmission frequencies ofthisbandcoincidewiththose usedforterrestrial microwave systems.Thisproblem iseliminated inthemorepowerful "second-generation" communication satellites thatoperateinthe14/12GHzband(Le., Ku-band); moreover, theuseofthesehigherfrequencies makesitpossibletobuildsmaller andtherefore lessexpensive antennas. TheblockdiagramofFigure8.3showsthebasiccomponents ofasingletransponder channelofatypicalcommunication satellite.Specifically, thereceiving antennaoutputof theuplinkisappliedtothecascadeconnection ofthefollowing components: I>Band-pass filter,designed toseparatethereceivedsignalfromamongthedifferent radiochannels. •Low-noise amplifier. Uplink signals Receiving antenna FIGURE 8.3Blockdiagram oftransponder.Downlink signals ~ 516 CHAPTER 8!"MULTIUSER RADIO COMMUNICATIONS ~Frequency down-converter, thepUIpose ofwhichistoconvertthereceivedradi frequency (RF)signaltothedesireddownlink frequency. 0 I>Traveling-wave tubeamplifier, whichprovides highgainoverawidebandoffre_ quencies. Inatraveling-wave tube(TWT),anelectromagnetic signaltravelsalonga helix(i.e.,aspring-shaped coilofwire),whileelectrons inahigh-voltage beamtravel through thehelixataspeedclosetothatofthesignalwave;thenetresultisthe transferofpowerfromtheelectrons tothewave,whichgrowsrapidlyasthesignal wavetravelsdownthehelix. Thechannelconfiguration showninFigure8.3usesasinglefrequency translation. Other channelconfigurations dothefrequency conversion fromtheuplinktothedownlink fre quencyintwostages:down-conversion toanintermediate frequency, followed byampli_ fication,andthenup-conversion tothedesiredtransmit frequency. Propagation timedelaybecomes particularly pronounced inasatellitechannelbe causeofthelargedistances involved. Specifically, speechsignalssentbysatelliteincura transmission delayofapproximately 270ms.Hence,forspeechsignals,anyimpedance mismatch atthereceiving endofasatellitelinkresultsinanechoofthespeaker's voice whichisheardbackatthetransmitting endafteraround-trip delayofapproximately 540 ms.Wemayovercome thisproblem byusinganechocanceller, whichisadevicethat subtracts anestimateoftheechofromthereturnpath;elimination oftheechoisperformed bymeansofaspecialfilterthatadaptsitselftothechanging channelcharacteristics. Thesatellitechannel is,closelyrepresented byanadditive whiteGaussian noise (AWGN)model,whichappliestoboththeuplinkanddownlink portions ofthesatellite communication system.Accordingly, muchofthematerial presented inChapter 6on passband systemsforthetransmission ofdata,withparticular reference tophase-shift keyingandfrequency-shift keyingtechniques, isdirectly applicable todigitalsatellite communications. Asatellitetransponder differsfromaconventional microwave line-of-sight repeater inthatmanyearthstationscanaccessthesatellitefromwidelydifferent locations onearth atthesametimeornearlyso.Thiscapability ismadepossiblebyusingoneofthemultiple accesstechniques discussed inSection8.2.Inthiscontextwemayofferthefollowing observations: II>Inasatellitechannel, nonlinearity ofthetransponder istheprimary causeofinter ferencebetween users.Tocontainthisseriousproblem, thetraveling-wave tnbeam plifierinthetransponder ispUIposely operated belowcapacity. Consequently, we findthatinanFDMAsystemthepowerefficiency ofthesystemisreducedbecause ofthenecessary powerbackoffofthetraveling-wave tubeamplifier. I>InaTDMAsystem,theusersaccessthesatellitetransponder oneatatime.Accord ingly,thesatellitetransponder isnowabletooperateclosetofullpowerefficiency bypermitting thetraveling-wave tubeamplifier torunintosaturation. This,intum, meansthatTDMAusesthetransponder moreefficiently thanFDMA,henceitswide useintheimplementation ofdigitalsatellitecommunication systems. !I>SDMAoperates byexploiting thespatiallocations ofearthstations, whichis achieved bymeansofonboard switching. Specifically, thetransponder isequipped withmultiple antennas, withtheproperantennabeambeingselectedforradiorrans missiontotheparticular earthstationdemanding useofthetransponder. Inaddition tomultiple access,anothercapability ofasatellitechannelisthatof broadcasting withemphasis onbroadareacoverage. Herewemention broadcasting sat ellites,whicharecharacterized bytheirhighpowertransmission toinexpensive receivers. 8.4Had...LinkAtudysls 517 Thischaracteristic isexploited intheuseofdirectbroadcast satellites (DBS),designed for homereception oftelevision servicesonaverywidescale.Bycomparison withthelarge earthstationsusedforsatellitecommunications, theearthstationsforDBSareverysimple andtherefore inexpensive. l8.4RadioLinkAnalysis Animportant issuethatarisesinthedesignofsatellitecommunication systemsisthatof linkbudgetanalysis.2Asitsnameimplies,alinkbudget,ormoreprecisely "linkpower budget," isthetotalingofallthegainsandlossesincurred inoperating acommunication link.Inparticular, thebalancesheetconstituting thelinkbudgetprovides adetailedac counting ofthreebroadlydefineditems: 1.Apportionment oftheresources available tothetransmitter andthereceiver. 2.Sourcesresponsible forthelossofsignalpower. 3.Sourcesofnoise. Puttingalltheseitemstogether intothelinkbudget,weendupwithanestimation pro cedureforevaluating theperformance ofaradiolink,whichcould betheuplinkordown linkofasatellitecommunication system.Needlesstosay,theessenceofthecommunication linkanalysispresented inthissectionalsoappliestootherradiolinksthatrelyonlineof sightfortheiroperation. Itisforthisreasonthetreatment ofradiolinkanalysispresented inthissectionisofagenericnature.Thesectionfinisheswithanillustrative example on thebudgetanalysisofadownlink ofadigitalsatellitecommunication system. Fromthematerialpresented inChapter6welearnedthattheperformance ofadigital communication system,inthepresence ofchannelnoisemodeled asadditivewhiteGaus siannoise,isdefinedbyaformulahavingtheshapeofa"waterfall" curveasshownin Figure8.4.Thisfigureportrays theprobability ofsymbolerror,P"plottedversusthebit Probability oferror I I II --------+I I I Linkmargins~ (EbINo)required EblNo,dB FIGURE8.4'Waterfall" curverelatingtheprobability oferrortotheEb/Noratio. 518 CHAPTER 8"MULTIUSER RADIO COMl"IUNICATIONS energy-to-noise spectraldensityratio,EJNo.Onceamodulation schemehasbeenchosen thefirstdesigntaskistospecifytwoparticular valuesofEJNoasdescribed here: ' 1.Required EFfNo. Suppose forexample, theprescribed probability ofsymbolerrorisPe=10-3•Usingthe waterfall curveofFigure8.4pertaining tothemodulation schemeofinterest,theEblN required torealizetheprescribed Peisdetermined. Let(EbINo),eq denotethevalueof EblNoobtained fromthiscalculation. Theprescribed Peandthecalculated (EbINo)"de fineapointonthewaterfall curveofFigure8.4,whichisdesignated asoperating PO~t1. 2.Received EbINO' Toassurereliableoperation ofthecommunication link,thelinkbudgetincludesasafety measure calledthelinkmargin.Thelinkmarginprovides protection againstchangeand theunexpected. Thusthe(EbINo)actuallyreceived bythesystemissomewhat largerthan (EbINo),eq' Let(EbINo)«c denotetheactualorreceivedEblNo,whichdefinesasecondpoint onthewaterfall curveofFigure8.4,designated asoperating point2.ThePecorresponding tooperating point2isshownas10-5inFigure8.4merelyforthepurposeofillustration. Inanyevent,introducing thelinkmargindenoted byM,wemaywrite (Eb)_M(Eb) Noree Noreq(8.1) Equivalently, expressing thetwoEblNovaluesofinterestindecibels, wemaydefinethe linkmarginas M(dB)=(::)(dB)-(::)(dB) oree 0req(8.2) Clearly,thelargerwemakethelinkmarginM,themorereliableisthecommunication link.However, theincreased reliability ofthelinkisattained atthecostofahigher EbiNo. FREE-SPACE PROPAGATION MODEL Thenextstepinformulating thelinkbudgetistocalculate thereceived signalpower. Naturally, thiscalculation accounts forallthegainsandlossesincurred inthetransmission andreception ofthecarrier. Inaradiocommunication system,thepropagation ofthemodulated signalisaccom plishedbymeansofatransmitting antenna, thefunction ofwhichistwofold: ..Toconverttheelectrical modulated signalintoanelectromagnetic field.Inthisca pacity,thetransmitting antenna actsasan"impedance-transforming" transducer, matching theimpedance oftheantennatothatoffreespace. II>Toradiatetheelectromagnetic energyindesireddirections. Atthereceiver, wehaveareceiving antennawhosefunction istheopposite ofthatofthe transmitting antenna: Itconverts theelectromagnetic fieldintoanelectrical signalfrom whichthemodulated signalisextracted.In addition, thereceiving antennamayberequired tosuppress radiation originating fromdirections whereitisnotwanted. Typically, thereceiverislocatedinthefarfieldofthetransmitting antenna, inwhich case,forallpractical purposes, wemayviewthetransmitting antennaasafictitious vol umelessemitterorpointsource.Acomplete description ofthefarfieldofthepointsource requires knowledge oftheelectromagnetic fieldasafunction ofbothtimeandspace. (8.3)8.4RadioLinkAnalysis 519 However, insofaraslinkcalculations areconcerned, suchacomplete knowledge isnot necessary. Rather,itissufficient tomerelyspecifythevariation ofthepowerdensityfor theantenna. Bydefinition, thePoynting vectororpowerdensityistherateofenergyflowperunit area;ithasthedimensions ofwattspersquaremeter.Thetreatment ofthetransmitting antenna asapointsourcegreatlysimplifies mattersinthatthepowerdensityofapoint sourcehasonlyaradialcomponent; thatis,theradiated energystreamsfromthesource alongradiallines. Itisusefultohavea"reference" antenna againstwhichtheperformance ofthe transmitting andreceiving antennas canbecompared. Thecustomary practiceistoassume thatthereference antennaisanisotropic source,definedasanomnidirectional (i.e.,com pletelynondireetional) antenna thatradiatesuniformly inalldirections. Anisotropic sourceishypothetical because,inreality,allradioantennas havesomedirectivity, however small.Nonetheless, thenotionofanisotropic sourceisuseful,especially forgaincompar isonpurposes. Consider thenanisotropic sourceradiating atotalpowerdenoted byP"measured inwatts.Theradiated powerpassesuniformly through asphereofsurfacearea41Td2, wheredisthedistance (inmeters)fromthesource.Hence,thepowerdensity,denotedby p(d),atanypointonthesurfaceofthesphereisgivenby Pt 2p(d)=41Td2watts/m Equation (8.3)statesthatthepowerdensityvariesinversely asthesquareofthedistance fromapointsource.Thisstatement isthefamiliarinverse-square lawthatgovernsthe propagation ofelectromagnetic wavesinfreespace. Multiplying thepowerdensityp(d)bythesquareofthedistancedatwhichitis measured, wegetaquantity calledradiation intensity denoted by<1>.Wemaythuswrite (8.4) Whereas thepowerdensityp(d)ismeasured inwattspersquaremeter,theradiation intensity <1>ismeasured inwattsperunitsolidangle(wattspersteradian). Inthecaseofatypicaltransmitting orreceiving radioantenna, theradiation intensity isafunction ofthespherical coordinates ()andcPdefinedinFigure8.5.Thus,ingeneral, J'iOl!I__~~Elementofarea r--~---1--f7':7i ,2sinOdOd¢> FIGURE8.5Illustrating thespherical coordinates ofapointsource. (8.6)520 CHAPI'ER 8..MULTIUSER RADIO COMMUNICATIONS wemayexpresstheradiation intensity as<1>(11,<!»,andsospeakofaradiation-intensity pattern.Thepowerradiated insideaninfinitesimal solidangledOisgivenby<1>(11,<!»dO. where(referring toFigure8.5) , dO=sinIIdlld<!>steradians (8.5) Thetotalpowerradiated istherefore P=J<1>(11,<!»dOwatts whichisamathematical statement ofthepowertheorem. Inwords,thepowertheorem statesthatiftheradiation-intensity pattern <1>(11,<!»isknownforallvaluesofanglepair (II,<!»,thenthetotalpowerradiated isgivenbytheintegralof<1>(11,<!»overasolidangle of47Tsteradians. Theaveragepowerradiatedperunitsolidangleis watts/steradianPav=41 7TJ<1>(11,<!»dn P 47T(8.7) (8.8)whichrepresents theradiation intensity thatisproduced byanisotropic sourceradiating thesametotalpowerP. Directive Gain,Directivity, andPowerGain3 Nowtheabilityofanantennatoconcentrate theradiatedpowerinagivendirection asinthecaseofthetransmitting antennaor,conversely, toeffectively absorbtheincident powerfromthatdirection asinthecaseofthereceiving antenna, isspecified intermsof itsdirectivegainordirectivity. Foradirection specified bytheanglepair(II,<!»,thedirective gainofanantenna, denotedbyg(lI,<!»isdefinedastheratiooftheradiation intensity in thatdirection totheaverageradiated power,asshownby g(lI,<!»=<1>~:v<!» =<1>(11,<!» P/47T Thedirectivity ofanantenna, denotedbyD,isdefinedastheratioofthemaximum radiation intensity fromtheantenna totheradiation intensity fromanisotropic source. Thatis,thedirectivity Disthemaximum valueofthedirective gaing(lI,<!».Thus,whereas thedirective gainoftheantennaisafunction oftheanglepair(II,<!»,thedirectivity Dis aconstant thathasbeenmaximized foraparticular direction. Thedefinition ofdirectivity isbasedontheshapeoftheradiation-intensity pattern <1>(11,<!»;assuch,itdoesnotinvolvetheeffectofantennaimperfections duetodissipation lossandimpedance mismatch. Aquantity calledpowergaindoesinvolvetheradiation efficiency oftheantenna. Specifically, thepowergainofanantenna, denoted byG,is definedastheratioofthemaximum radiation intensity fromtheantennatotheradiation intensity fromalosslessisotropic source,undertheconstraint thatthesameinputpower isappliedtobothantennas. Specifically, using7j"di"iontodenotetheradiation efficiency factoroftheantenna, wemayrelatethepowergainGtothedirectivity Das G=7JradiationD(8.9) Thus,thepowergainofanantennaoveralosslessisotropic sourceequalsthedirectivity iftheantennais100percentefficient(i.e.,7jmdiarion=1),butitislessthanthedirectivity 8.4RadioUnkAnalysis 521 ifanylossesarepresentintheantenna (i.e.,1Jmliation<1).Henceforth, weassumethatthe antennais100percentefficientandtherefore referonlytothepowergainoftheantenna. Theconceptofpowergain,whichisbasedonthetransmitted power-pattern shape, canbeextended toareceiving antenna byvirtueofthereciprocity principle. Anantenna issaidtobereciprocalifthetransmission medium islinear,passiveandisotropic. Fora givenantennastructure, thepowergainsoftransmitting andreceiving antennas arethen identical. Thepowergainofanantenna istheresultofconcentrating thepowerdensityina restricted regionsmallerthan417"steradians, asillustrated inFigure8.6.Inlightofthe pictureportrayed inthisfigure,wemayintroduce thefollowing twoparameters: 1.Effective radiatedpowerreferenced toanisotropic source(EIRP);theEIRPisdefined astheproductofthetransmitted power,P"andthepowergainofthetransmitting antenna, G"asshownby EIRP=PtGtwatts (8.10) (8.11)2.Antenna beamwidth, representing a"planar" measure oftheantenna's solidangle ofview;thebeamwidth, indegreesorradians, isdefinedastheanglethatsubtends thetwopointsonthemainlobe ofthefield-power patternatwhichthepeakfield powerisreducedby3dBs.Thehigherthepowergainoftheantenna, thenarrower istheantennabeamwidth. Another matterofinterestdiscernible fromFigure8.6isthesidelobes ofthefield-power pattern. Unfortunately, everyphysical antenna hassidelobes, whichareresponsible for absorbing unwanted interfering radiations. Effective Aperture Atermthathasaspecial significanc~ forareceiving antennaistheeffectiveaperture oftheantenna, whichisdefinedastheratioofthepoweravailable attheantennaterminals tothepowerperunitareaoftheappropriately polarized incidentelectromagnetic wave. Theeffective aperture, denotedbyA,isdefinedintermsoftheantenna's powergainGas A=~G 417" whereAisthewavelength ofthecarrier.Thewavelength Aandfrequencyfarereciprocally relatedas cA=-f wherecisthespeedoflight(approximately equalto3X108mis).(8.12) Transmitting If?~-~ antennaPointofpeak outputpower SidelobesPointswheretheoutputpower is3dBsbelowitspeakvatue FIGURE 8.6Illustrating theconcentration ofpowerdensityofatransmilling antenna insidea regionsmallerthan417"steradians. (8.13) (8.15)522 CHAPTER 8elMULTIUSER RADIO COMMUNICATIONS Thetermeffective aperture hasparticular significance inthecontextofreflector antennas andelectromagnetic hornsthatarecharacterized byawell-defined aperture. For theseantennas, theratiooftheantenna's effective aperture toitsphysical aperture isa directmeasureoftheantenna's aperture efficiency, 'TI,p'rture,inradiating powertoadesired direction orabsorbing powerfromthatdirection. Nominal valuesfortheefficiency 'TIape'tureofreflector antennas lieintherangeof45to75percent. FriisFree-Space Equation Withthisintroductory materialonantennas athand,wearenowreadytoformulate thebasicpropagation equation foraradiocommunication linleConsider atransmitting antenna withanEIRPdefinedinEquation (8.10).Invoking theinverse-square law ofEquation (8.3),wemayexpressthepowerdensityofthetransmitting antenna as EIRP/417d2,wheredisthedistancebetweenthereceiving andtransmitting antennas. The powerPrabsorbed bythereceiving antennaistheproductofthispowerdensityandthe antenna's effectiveareadenotedbyA"asshownby PT=(~:;)AT P,G,AT =417d2watts According tothereciprocity principle, wemayuseEquation (8.11)toexpresstheeffective areaofthereceiving antennaas ,1.2 AT=417GT whereGTisthepowergainofthereceiving antenna. Substituting thisformulaforATinto Equation (8.13),wemayexpressthereceivedsignalpowerintheequivalent form PT=P'G,GT(4~d)2 (8.14) Equation (8.14)iscalledtheFriisfree-space equation: Thepathloss,PL,representing signal"attenuation" indecibelsacrosstheentire communication link,isdefinedasthedifference (indecibels) betweenthetransmitted signal powerP,andreceivedsignalpowerProasshownby PL=1010glo(t) (417d)2=-1010glO(G,G r)+1010gloT Theminussignassociated withthefirstterminEquation (8.15)signifiesthefactthatthis termrepresents a"gain."Thesecondterm,duetothecollection ofterms(417dJAf, iscalled thefree-space loss,denotedbyLfree'pace'Notethatincreasing thedistance dseparating thereceiving antennafromthetransmitting antennacausesthefree-space losstoincrease, which,intum,compelsustooperatetheradiocommunication linkatlowerfrequencies soastomaintain thepathlossatamanageable level. TheFriisfree-space equation enablesustocalculate thepathlossPLforspecified valuesofpowergainsG,andGnthecarrierwavelength A,anddistanced.Tocomplete (8.16)8.4RadioLinkAnalysis 523 thebudgetlinkanalysis, weneedtocalculate theaveragenoisepowerinthereceived signal,whichisconsidered next. NOISE FIGURE Toperformnoiseanalysisatthereceiverofacommunication system,weneedaconvenient measure ofthenoiseperformance ofalineartwo-port device.Onesuchmeasure isfur nishedbytheso-called noisefigure.Consider alineartwoeport deviceconnected toasignal sourceofinternalimpedance Z(f)=R(f)+jX(f)attheinput,asinFigure8.7.Thenoise voltagev(t)represents thethermalnoiseassociated withtheinternalresistance R(f)ofthe source.Theoutputnoiseofthedeviceismadeupoftwocontributions, oneduetothe sourceandtheotherduetothedeviceitself.Wedefinetheavailable outputnoisepower inabandofwidthillcenteredatfrequencyIasthemaximum averagenoisepowerin thisband,obtainable attheoutputofthedevice.Themaximum noisepowerthatthetwo portdevicecandelivertoanexternal loadisobtained whentheloadimpedance isthe complex conjugate oftheoutputimpedance ofthedevice,thatis,whentheresistance is matched andthereactance istunedout.Wedefinethenoisefigureofthetwo-port device astheratioofthetotalavailable outputnoisepower(duetothedeviceandthesource) perunitbandwidth totheportionthereofduesolelytothesource. Letthespectraldensityofthetotalavailable noisepowerofthedeviceoutputbe SNO(f),andthespectraldensityoftheavailable noisepowerduetothesourceatthedevice inputbeSNS(f).AlsoletG(f)denotetheavailable powergainofthetwo-port device, definedastheratiooftheavailable signalpowerattheoutputofthedevicetotheavailable signalpowerofthesourcewhenthesignalisasinusoidal waveoffrequencyf.Thenwe mayexpressthenoisefigureFofthedeviceas SNo(f) F=G(f)SNS(f) Hthedevicewerenoisefree,SNO(f)=G(f)SNS(f), andthenoisefigurewouldthenbe unity.Inaphysical device,however, SNO(f)islargerthanG(f)SNS(f), sothatthenoise figureisalwayslargerthanunity.Thenoisefigureiscommonly expressed indecibels,that is,as1010gloF. Thenoisefiguremayalsobeexpressed inanalternative form.LetPs(f)denotethe available signalpowerfromthesource,whichisthemaximum averagesignalpowerthat canbeobtained. Forthecaseofasourceproviding asingle-frequency signalcomponent l' r----02' Linear two-port device 1----02 FIGURE 8.7Lineartwo-port device. (8.17)524 CHAPTER 8'"MULl'IUSER RADIO COMMUNICATIONS withopen-circuit voltageVacos(2'lTft), theavailable signalpowerisobtained whenthe loadconnected tothesourceis Z*(f)=R(f)jX(f) wheretheasteriskdenotescomplex conjugation. Underthiscondition, wefindthat Ps(f)=[2~°f)r R(f) Vf> 4R(f) Theavailable signalpowerattheoutputofthedeviceistherefore Po(f)=G(f)P,s(f) (8.18) (8.19)Then,multiplying boththenumerator anddenominator oftheright-hand sideofEquation (8.16)byPs(f)Ii(f),weobtain F=PS(f)SNO(f)lif G(f)PS(f)SNS(f) lif PsU)SNO(f)lif PO(f)SNS(f)lif Ps(f) Po(f) where (8.20) (8.21) WerefertoPs(f)astheavailable signal-to-noise ratioofthesourceandtoPo(f)asthe available signal-to-noise ratioatthedeviceoutput,bothmeasured inanarrowbandof widthlifcentered atf.Sincethenoisefigureisalwaysgreaterthanunity,itfollowsfrom Equation (8.19)thatthesignal-to-noise ratioalwaysdecreases withamplification, which isasignificant result. ThenoisefigureFisafunction oftheoperating frequencyf;itistherefore referred toasthespotnoisefigure.Incontrast, wemaydefineanaveragenoisefigureFaofatwo portdeviceastheratioofthetotalnoisepoweratthedeviceoutputtotheoutputnoise powerduesolelytothesource.Thatis, r~SNO(f)df Fo=------- r~G(f)SNS(f) df(8.22) Itisapparent thatinthecaseofthermalnoiseintheinputcircuitwithR(f)constant and constant gainthroughout afixedbandwithzerogainatotherfrequencies, thespotnoise figureFandtheaveragenoisefigureFaareidentical. Equivalent NoiseTemperature Adisadvantage ofthenoisefigureFisthatwhenitisusedtocompare low-noise devices,thevaluesobtained areallclosetounity,whichmakesthecomparison rather 8.4RadioLinkAnalysis 525 22'l' , Rin=Rs Lineartwo-port device:--NoisefigureF ?--;r.. ,...,... 14KI'R/J.fR Available noisepower N1=kTlJ.fAvailable noisepower N2=GN1+Nd FIGURE8.8Lineartwo-port devicematched totheinternal resistance ofasourceconnected to theinput. difficult. Insuchcases,itispreferable tousetheequivalent noisetemperature. Consider a lineartwo-port devicewhoseinputresistance ismatched totheinternalresistance ofthe sourceasshowninFigure8.8.Inthisdiagram, wehavealsoincluded thenoisevoltage generator associated withtheinternalresistance Rsofthesource.Themean-square value ofthisnoisevoltageis4kTR s1:J.j,wherekisBoltzmann's constant. Hence;theavailable noisepoweratthedeviceinputis N1=kT1:J.j (8.23) LetNddenotethenoisepowercontributed bythetwo-port devicetothetotalavailable outputnoisepowerNz.WedefineNdas (8.24) (8.25)whereGistheavailable powergainofthedeviceandT,isitsequivalent noisetemperature. Thenitfollowsthatthetotalontputnoisepoweris Nz=GN!+Nd =Gk(T+Te)1:J.j Thenoisefigureofthedeviceistherefore (seetheoutputportofFigure8.8) F=Nz NzNd Solvingfortheequivalent noisetemperature:(8.26) T,=T(F1) (8.27) ThenoisefigureFismeasured undermatched inputconditions, andwiththenoisesource attemperature T.Byconvention thetemperature Tistakenas"roomtemperature," namely290K,whereKstandsfor"degreeKelvin." Cascade Connection ofTwo-Port Networks Itisoftennecessary toevaluatethenoisefigureofacascadeconnection oftwo-port networks whoseindividual noisefiguresareknown.Consider Figure8.9,consisting ofa 526 CHAPTER 8OJMULTIUSER RADIO COMMUNICATIONS (F,-l)N, (F,-l)N, Available --:...,.. Available powergain=G1 powergain=02 N,Noisefigure=F1 F1G1N1Noisefigure=F2 F1G1N1G2+ -(F,l)N,G, FIGURE 8.9Acascadeoftwonoisytwo-portnetworks. pairoftwo-port networks ofnoisefiguresF1andF2andpowergainsGIandG2,connected incascade.Itisassumed thatthedevicesarematched, andthatthenoisefigureF2ofthe secondnetwork isdefinedassuming aninputnoisepowerN1• Attheinputofthefirstnetwork, wehaveanoisepowerNIcontributed bythesource plusanequivalent noisepower(FI-l)NIcontributed bythenetwork itself.Theoutpu~ noisepowerfromthefirstnetwork istherefore FINIGI.Addedtothisnoisepoweratthe inputofthesecondnetwork, wehavetheequivalent extrapower(F2-l)NIcontributed bythesecondnetwork itself.Theoutputnoisepowerfromthissecondnetwork istherefore equaltoFIGINIG2+(F2-1)NIG2•Wemayconsider thenoisefigureFastheratioof theactualoutputnoisepowertotheoutputnoisepowerassuming thenetworks tobe noiseless. Wemaytherefore expresstheoverallnoisefigureofthecascadeconnection of Figure8.9as F==FIGIN1G2+(F2-1)N1G2 N1G1G2 =F+F 2-1 1GI(8.28) Theresultmaybereadilyextended tothecascadeconnection ofanynumberoftwo-port networks, asshownby F-lF-lF-l F=F+_2__+_3__+_4__._+... 1G1G1G2GIG2G3(8.29) whereFhF2,F3,•••aretheindividual noisefigures,andGbG2,G3,•••aretheavailable powergains,respectively. Equation (8.29)showsthatifthefirststageofthecascade connection inFigure8.9hasahighgain,theoverallnoisefigureFisdominated bythe noisefigureofthefirststage. Correspondingly, wemayexpresstheoverallequivalent noisetemperature ofthe cascadeconnection ofanynumberofnoisytwo-port networks asfollows: (8.30)T2T, T4T=T1+ - +--+---+... e G1GIG2G1G2G3 whereThT2,T3, •••aretheequivalent noisetemperatures oftheindividual networks, andGhG2,G3,•••aretheavailable powergains,respectively. Equation (8.30)isknown astheFriisformula. HereagainwenotethatifthegainG1ofthefirststageishigh,the equivalent noisetemperature Teisdominated bythatofthefirststage. ~ExAMPLE 8.1NoiseTemperature ofEarth-Terminal Receiver Figure8.10showsatypicalearth-terminal receiver,consisting ofalow-noise radio-frequency (RF)amplifier (LNA),frequency down-converter (mixer),andintermediate frequency (IF) 8.4RadioLinkA....lysis527 ~a~:ltrom ~ite Receiving antenna FIGURE8.10Blockdiagramofearthterminal receiver.Output amplifier. Theequivalent noisetemperatures ofthesecomponents, including thereceiving antenna, are Tant",""=50K TRF=50K T=oo"=500K TIF=1000K Theavailable powergainsofthetwoamplifiers are GRF=200=23dB G1F=1000=30dB Tocalculate [heequivalent noisetemperature ofthereceiver, weuseEquation (8.30), obtaining T T +TRP+Tmixer+TIP e=antennaGRF O500+1000 =50+5+200 =107.5K ~EXAMPLE 8.2Downlink BudgetAnalysis ofaDigitalSatellite Communication System Inadigitalsatel!i[ecommunication system,oneofthekeyelements intheoveralldesignand analysisofthesystemisthedownlink powerbudget,whichisusuallymorecriticalthanthe uplinkpowerbudgetbecauseofthepractical constraints imposed ondownlink powerand satelliteantennasize.Theexample presented hereaddresses asampledownlink budgetanal ysis,assuming thatanyrequired uplinkpower(withinlimits)isavailable forsatisfactory operation ofthesystem. Thecriticalparameter tobecalculated istheratioofreceivedcarrierpower-to-noise spectraldensity,denotedbyCINo•According to[heFriisfree-space equation (8.14),theav eragepowerreceivedattheearthterminal totheaveragepowerP,transmitted bythesatellite is (8.31)528 CHAPTER 8 "MULTIUSER RADIO COMMUNICATIONS where,inthisexample, G,isthepowergainofthesatelliteantenna, G,isthepowergaino£ thereceiving earth-terminal antenna, Aisthecarrierwavelength forthedownlink, anddis thedistance between thesatelliteandtheearthterminal. Giventhattheequivalent noise temperamre ofthesystemisT.,wemayuseEquation (1.94)ofChapter1toexpressthenoise spectraldensityNoaskT"wherekisBoltzmann's constant. Moreover, fromEquation (8.10) wenotethatF,G,isequaltotheErR!'ofthesatellite. Hence,dividingF,byNo,WeI!lay expresstheCINoratioforthedownlink as (G) (G')(A)21 - =(EIR!')",ellito - --- Nodownlink Teearthtermina.l47Tdk Foragivensatellitesystem,thefree-space loss(4'1TdlA)2 isaconstant. Viewing thesystemfrom theearthterminal, weseefromEquation (8.31)thatthe(GINo)ratioisproportional to G,IT,.TheratioG,IT,maytherefore beusedtoassessthe"quality" ofanearthterminal;it isusuallyshortened totheGITratio,whichisreferred toasthefigureofmeritofthereceiving earthterminal. Thus,rewriting theformula(8.31)forthe(GINo)ratiomeasured indecibels wemayexpressitasthesumofgainsandlossesasitemized here: ' 1.(EIR!'),,"I1i'" measured indBW,wheredBWdenotesdecibelsreferenced to1watt,that is,0dBW. 2.(GIT)e=h '"mi,,,l>measured indBIK,whereKreferstodegreeKelvin. 3.L,='pace,denoting thefree-space loss10log,o(4'1TdJA)2 indB. 4.-10log,ok,representing thegainindBWIK-HzduetodivisionbytheBoltzmann constantk=1.38X10-23joulelK. Table8.1presentsthevaluesofthesefourtermsforthedownlink ofatypicaldomestic digital satellitecommunication system,basedonthefollowing: 1.Thetransponder isoperated atitsmaximum outputpower(i.e.,nopowerbackoffis employed), yieldinganErR!'of46.5dBW. 2.Thereceiving earthterminal usesa2m-dishantennawithapowergainG=45dB,and thereceiverisconfigured asinExample 8.1withequivalent temperamre T=107.5K. Hence GT45-1010glO107.5 =45-20.3 =24.7dBIK 3.Thefree-space lossis Lfr'Npa= =92.4+20log,of+2010glOddB TABLE8.1Downlink power budgetforExample 8.2(8.32) Variable EIRP GITratio Free-space loss Boltzmann constant ONoValue +46.5dBW +24.7dBIK -206dB +228.6dBWIK-Hz 93.8dB-Hz (8.33)8.5Wireless Communications 529 wherethedownlink carrierfrequencyfisinGHzandthedisrancedbetweenthesatellite andtheearthterminal isinkilometers. Forageostationary satellite,thedistancebetween thesatelliteandanearthterminal liesintherangeof36,000to41,000km.Thus choosing d40,000kmandassumingf=12GHz,theuseofEquation (8.32)yields L,m.,p",=92.4+20log,o12+2010glO40,000 92.4+21.6+92.0 =206dB 4.WiththeBoltzmann constantk=1.39X10-23joulelK,itscontribution totheCINa ratiois -1010glOk=1010glO1.38X10-23 =228.6dBWIK-Hz Totaling thegainsandlosses,wethusget (NC)=93.8dB-Hz odownlink The"received" downlink valueofthe(CINo)ratiomayalsobeexpressed intermsof the"required" valueofthebitenergy-to-noise spectraldensityratio,(EbINo),"'l dB,atthe receiving earthterminal as(seeEquation (8.2)) (~JdoWn];nk =(~L+10log,oM+1010glORdB where1010glOMisthelinkmarginindecibels, andRisthedatarateinb/s.Thelinkmargin allowsforexcessrainlossesinpropagation andotherpowerdegradations. Typically, thelink marginisselectedas4dBforC-band,6dBforKu-hand, andhigherforthehigherK-band frequencies becauseofthehigherrainlosses.Foroperation attheKu-band frequency of 12GHz,wechoosealinkmarginof6dB.Thus,usingthevalueCINo=93.8dB-Hzcalculated fromthelinkbudget,thelinkmargin10log,aM=6dB,andassuming (EliNo),,"=12.5dB, theuseofEquation (8.33)yields 10log1OR=93.812.5-6 =75.3 Hence, R=33.9Mb/s Assuming theuseofcoherent 8-PSKforthetransmission ofdigitaldataviathesatellite, andsubstituting (EbINoJ 12.5dBinEquation (6.47)ofChapter6,wefindthattheprob- abilityofsymbolerrorPe=0.6X10-3• Tosummarize, thedigitalsatellitecommunication systemanalyzed inthisexampleper-' mits,undertheworstoperating conditions, datatransmission onthedownlink atarate R=33.9Mb/sandwithaprobability ofsymbolerrorP,=0.6X10-3,assuming theuseof 8-phasePSK. ~ l8.5Wireless Communications Inthissectionwestudythesecondtypeofmultiuser radiocommunication system,namely, wirelesscommunications, whichissynonymous withmobileradio.Thetermmobileradio isusuallymeanttoencompass indoororoutdoorformsofwirelesscommunications where 530 CHAPTER 8"MULTIUSER RADIO COMMUNICATIONS aradiotransmitter orreceiveris~apableofbeingmoved,regardless of.whether itactually movesornot.DuetothestochastIc natureofthemobileradiochannel, itscharacterization mandates theuseofpractical measurements andstatistical analysis. Theaimofsuchan evaluation istoquantify twofactorsofprimaryconcern: 1.Mediansignalstrength, whichenablesustopredictthemimmum powerneededto radiatefromthetransmitter soastoprovideanacceptable qualityofcoverage over apredetermined servicearea. 2.Signalvariability, whichcharacterizes thefadingnatureofthechannel. Ourspecificinterestinwirelesscommunications isinthecontextofcellularradioS thathastheinherentcapability ofbuildingmobility intothetelephone network. Withsuch acapability, ausercanmovefreelywithinaserviceareaandsimultaneously communicate withanytelephone subscriber intheworld.Anidealized modelofthecellularradiosystem, illustrated inFigure8.11,consistsofanarrayofhexagonal cellswithabasestationlocated atthecenterofeachcell;atypicalcellhasaradiusof1to12miles.Thefunction ofthe basestationsistoactasaninterface between mobilesubscribers andthecellularradio system.Thebasestationsarethemselves connected toaswitching centerbydedicated wirelines. Themobileswitching centerhastwoimportant roles.First,itactsastheinterface between thecellularradiosystemandthepublicswitched telephone network. Second,it performs overallsupervision an.dcontrolofthemobilecommunications. Itperfonns the latterfunction bymonitoring thesignal-to-noise ratioofacallinprogress, asmeasured atthebasestationincommunication withthemobilesubscriber involved inthecall.When theSNRfallsbelowaprescribed threshold, whichhappens whenthemobilesubscriber leavesitscellorwhentheradiochannelfades,itisswitched toanotherbasestation.This switching process,canedahandover orhandof(,isdesigned tomoveamobilesubscriber fromonebasestationtoanotherduringacallinatransparentfashion,thatis,without interruption ofservice. Thecellularconceptreliesontwoessential features, asdescribed here: 1.Frequency reuse.Thetermfrequency reusereferstotheuseofradiochannels onthe samecarrierfrequency tocoverdifferent areas,whicharephysically separated from Cell Basestation FIGURE8.11Idealized modelofcellularradio. 8.5Wireless Com.....nications 531 eachothersufficiently toensurethatco-channel interference isnotobjectionable. Thus,insteadofcovering anentirelocalareafromasingletransmitter withhigh poweratahighelevation, frequency reusemakesitpossibletoachievetwocom monsense objectives: keepthetransmitted powerfromeachbasestationtoamini mum,andposition theantennas ofthebasestationsjusthighenoughtoprovidefor theareacoverage oftherespective cells. 2.Cellsplitting. Whenthedemandforserviceexceedsthenumberofchannels allocated toaparticular cell,cellsplitting isusedtohandletheadditional growthintraffic withinthatparticular cell.Specifically, cellsplittinginvolvesarevisionofcellbound aries,sothaithelocalareaformerly regarded asasinglecellcannowcontaina numberofsmallercellsandusethechannelcomplements ofthesenewcells.Thenew cells,whichhaveasmallerradiusthantheoriginalcells,arecalledmierocells. The transmitter powerandtheantennaheightofthenewbasestationsarecorrespond inglyreduced, andthesamesetoffrequencies arereusedinaccordance withanew plan. Forahexagonal modelofthecellularradiosystem,wemayexploitthebasicprop ertiesofhexagonal cellulargeometry tolayoutaradiochannel assignment planthat determines whichchannelsetshouldbeassignedtowhichcell.Webeginwithtwointegers iandj(i~j),calledshiftparameters, whicharepredetermined insomemanner. Wenote thatwithahexagonal cellulargeometry therearesix"chains" ofhexagons thatemanate fromeachhexagon andthatextendindifferent directions. Thus,startingwithanycellas areference, wefindthenearestco-channel cellsbyproceeding asfollows: ~Moveicellsalonganychainofhexagons, turncounterclockwise 60degrees,and movejcellsalongthechainthatliesonthisnewdirection. Thejthcellssolocated andthereference cellconstitute thesetofco-channel cells. Thisprocedure isrepeated foradifferent reference cell,untilallthecellsinthesystemare covered. Figure8.12illustrates theapplication ofthisprocedure forasinglereference cell andtheexample ofi=2andj=2. InNorthAmerica, thebandofradiofrequencies assigned tothecellularsystemis 800-900 MHz.Thesubband 824-849 MHzisusedtoreceivesignalsfromthemobile units,andthesubband 869-894 MHzisusedtotransmit signalstothemobileunits.The FIGURE8.12Illustrating thedetermination ofco-channel cells. 532 CHAPTER 8i'"MULTIUSER RADIO COMMUNICATIONS useoftheserelatively highfrequencies hasthebeneficial featureofproviding agoodpor tablecoverage bypenetrating buildings. InEuropeandelsewhere, thebase-mobile and mobile-base subbands arereversed. IIIPROPAGATION EFFECTS6 Themajorpropagation problems encountered intheuseofcellularradioinbuilt-upareas areduetothefactthattheantenna ofamobileunitmayliewellbelowthesurrounding buildings. Simplyput,thereisno"line-of-sight" pathtothebasestation.Instead,radio propagation takesplacemainlybywayofscattering homthesurfacesofthesurrounding buildings andbydiffraction overand/oraroundthem,asillustrated inFigure8.13.The important pointtonotefromFigure8.13isthatenergyreachesthereceiving antennavia morethanonepath.Accordingly, wespeakofamultipath phenomenon inthatthevariolis incoming radiowavesreachtheirdestination fromdifferent directions andwithdifferent timedelays. Tounderstand thenatureofthemultipath phenomenon, consider firsta"static" multipath environment involving astationary receiverandatransmitted signalthatcon sistsofanarrowband signal(e.g.,unmodulated sinusoidal carrier).Letitbeassumedthat twoattenuated versions ofthetransmitted signalarrivesequentially atthereceiver. The effectofthedifferential timedelayistointroduce arelativephaseshiftbetween thetwo components ofthereceived signal.Wemaythenidentifyoneoftwoextreme ca.sesthat canarise: !'>Therelativephaseshihiszero,inwhichcasethetwocomponents addconstructively, asillustrated inFigure8.14a. Ii'Therelativephaseshiftis180degrees, inwhichcasethetwocomponent addde- structively, asillustrated inFigure8.14b. Wemayalsousephasorstodemonstrate theconstructive anddestructive effectsofmul tipath,asshowninFigures8.15aand8.15b,respectively. Notethatinthestaticmultipath environment described herein,theamplitude ofthereceivedsignaldoesnotvarywithtime. Consider nexta"dynamic" multipath environment inwhichthereceiverisinmotion andtwoversions ofthetransmitted narrowband signalreachthereceiverviapathsof Direction toelevated basestation Obstructed IinlHlf-sight path" "---------- FIGURE8.13Illustrating themechanism ofradiopropagation inurbanareas.(FromParsons, 1992,withpennission.) Direct-path signal~ Reflected signal'.i\.f\v Composite signalM ---'----------> Time (0)8.5Wireless C........unicatirms 533 Direct-path signal~ Reflected signal'VV' ~ Time (b) FIGURE8.14(a)Constructive and(b)destruetive formsofthemultipath phenomenon forsinu soidalsignals. different lengths.Duetomotionofthereceiver, thereisacontinuous changeinthelength ofeachpropagation path.Hence,therelativephaseshiftbetween thetwocomponents of thereceived signalisafunction ofspatiallocation ofthereceiver. Ai>thereceivermoves, wenowfindthatthereceived amplitude (envelope) isnolongerconstant aswasthecase inastaticenvironment; rather,itvarieswithdistance, asillustrated inFigure8.16.Atthe topofthisfigure,wehavealsoincluded thephasorrelationships forthetwocomponents ofthereceived signalatvariouslocations ofthereceiver. Figure8.16showsthatthereis constructive addition atsomelocations, andalmostcomplete cancellation atsomeother locations. Thisphenomenon isreferredtoassignalfading. Inamobileradioenvironment encountered inpractice, theremayofcoursebea multitude ofpropagation pathswithdifferent lengths,andtheircontributions tothere- ~",jl".. representing representing direct-transmission reflected signal signal Phasor representing composite signal (0),,=,j~~'"~ representing representing direct-transmission reflected signal signal Phasor trepresenting composite signal (b) FIGURE8.15Phasorrepresentations of(a)constructive and(b)destructive formsofmultipath. 534 CHAPTER 8 "MULTIUSER RADIO CO,V1MUNICATIONS nI I I :1\ 11I 1\ <1t\I j1\II1\II1\II I II II II~ I I"C I I I~ I I ""I II IIQ. I I..... E....I II I«....,I II II..I II, I II" ,II I , ,II IIFadingI II,IIIIenvelope.. I)I ....I.... ......-'" Distance FIGURE8.16Illustrating howtheenvelope fadesastwoincoming signalscombine withdiffer entphases.(FromParsons, 1992,withpennission.) ceivedsignalcouldcombine inavaIietyofways.Thenetresultisthattheenvelope ofthe receivedsignalvaIieswithlocationinacomplicated fashion,asshownbytheexperimental recordofreceivedsignalenvelope inanurbanateathatispresented inFigure8.17.This figurecleaIlydisplaysthefadingnatureofthereceivedsignal.Thereceivedsignalenvelope inFigure8.17ismeasured indBm.TheunitdBmisdefinedas10loglo(PIP o),withP denoting thepowerbeingmeasured andPo=1milliwatt. InthecaseofFigure8.17,Pis theinstantaneous powerinthereceivedsignalenvelope. Signalfadingisessentially aspatialphenomenon thatmanifests itselfinthetime domainasthereceivermoves.Thesevariations canberelatedtothemotionofthereceiver asfollows.Tobespecific,consider thesituation illustrated inFigure8.18,wherethere ceiverisassumed tobemovingalongthelineAA'withaconstant velocityv.Itisalso -60 E ~ ~~~l ~·in ~ .~-90 '" -100 0 10 15 20 Distance inmeters FIGURE8.17Experimental recordofreceived signalenvelopeinanurbanarea.(FromParsons, 1992,withpennission.) 8.6Smtistical Characteri:z;ation ofMultipath Channels 535 s Direction ofmotion FIGliRE8.18Illustrating thecalculation ofDopplershift. assumed thatthereceivedsignalisduetoaradiowavefromascatterer labeledS.LettH denotethetimetakenforthereceivertomovefrompointAtoA'.Usingthenotation described inFigure8.18,theincremental changeinthepathlengthoftheradiowaveis deduced tobe !J.l=dcosa =-vtHcosa(8.34) whereaisthespatialanglebetween theincoming radiowaveandthedirection ofmotion ofthereceiver. Correspondingly, thechangeinthephaseangleofthereceived signalat pointAIwithrespecttothatatpointAisgivenby t1,p=27T!J.l 27TV!J.t =---A- cosa(8.35) whereAistheradiowavelength. Theapparent changeinfrequency, ortheDoppler-shift, istherefore v=_~!J.,p 27T!J.t v Acos'"(8.36) TheDoppler-shift vispositive(resulting inanincreasein&equency) whentheradiowaves arrivefromaheadofthemobileunit,anditisnegativewhentheradiowavesarrivefrom behindthemobileunit. 8.6Statistical Characteriz.ation ofMultipath Channels Thenarrowband characterization ofthemultipath environment described inSection8.5 isappropriate formobileradiotransmissions wherethesignalbandwidth isverysmall 536 CHAPTER 8..MULTIUSER RADIO COMMUNICATIONS compared tothereciprocal ofthespreadinpropagation pathdelays.Multipath insuch anenvironment resultsintwoeffects:rapidfadingofthereceivedsignalenvelope anda spreadinDoppler shiftsinthereceived spectrum. Real-life signalsradiated inamobile radioenvironment may,however, occupyabandwidth wideenoughtorequiremorede tailedconsiderations oftheeffectsofmultipathpropagation onthereceivedsignal.Inthis section,wepresentastatistical characterization ofamobileradiochannel.* Consider amobileradiochannelwithmultiple propagation paths.Inaccordance withthecomplex notation described inAppendix 2,wemayexpressthetransmitted band. passsignalas s(t)=Re[s(t}exp(j27Tfet)] (8.37) wheres(t}isthecomplex (low-pass) envelope ofs(t},andfeisanominalcarrierfrequency. Sincethechannel istimevaryingduetomultipath effects,theimpulseresponse ofthe channelisdelaydependent andtherefore atime-varying function. Lettheimpulseresponse ofthechannelbeexpressed as h(T;t)=Re[h(T;t}exp(j27Tfct)] (8.38) whereh(T;t}isthe(low-pass) complex ~pulseresponse ofthechannel, andTisadelay variable. Thecomplex impulseresponse h(T;t)iscalledtheinputdelay-spread functionof thechannel. The(low-pass) complex envelope ofthechanneloutputisdefinedbythe convolution integral (8.39) wherethescalingfactor ~istheresultofusingcomplex notation. Ingelllira~he behavior ofamobileradiochannelcanbedescribed onlyinstatistical terms.Foranalyticpurposes, thedelay-spread functionh(T;t}maythusbemodeled asa zero-mean complex-valued Gaussian process.Then,atanytimettheenvelope Ih(T;t)Iis Rayleigh distributed, andthechannelisreferredtoasaRayleigh fadingchannel.When, however, themobileradioenvironment includesfzxedscatterers, wearenolongerjustified inusingazero-mean modeltodescribetheinputdelay-spread function h(T;t).Insucha case,itismoreappropriate touseaRiciandistribution todescribetheenvelopeIh(r,t}I, andthechannelisreferred toasaRicianfadingchannel. TheRayleigh andRiciandistri butionsforareal-valued randomprocesswereconsidered inChapter 1.Inthediscussion presented inthischapter,weconsider onlyaRayleigh fadingchannel. Thetime-varying transferfunction ofthechannelisdefinedastheFouriertransform oftheinputdelay-spread functionh(T;t)withrespecttothedelayvariable T,asshownby H(f;t}=rooh(T;t}exp(-;27TfT} dT (8.40) wherefdenotesthefrequency variable. Thetime-varying transferfunctionH(f;t}maybe viewedasafrequency transmission characteristic ofthechannel. *Readerswhoarenotinterestedinthemathematical detailspertaining tothestatistical characterization offading multipath channels, mayskipthematerial presented inthissection,exceptforthesubsection ontheclassification' ofmultipath channels attheendofthesection. 8.6Statistical Characteri%atUm ofMultipath Channels 537 Forastatistical characterization ofthechannel, wemakethefollowing assumptions: l>Theinputdelay-spread function h(7';t)isazero-mean, complex-valued Gaussian process.Ourinterestisconfined toshort-term fading;itistherefore reasonable to assumethath(7';t)isalsostationary. BecauseFouriertransformation islinear,the time-varying transferfunction H(f;t)hassimilarstatistics. !l>Thechannelisanuncorrelated scattering channel, whichmeansthatcontributions fromscatterers withdifferent propagation delaysareuncorrelated. Consider thentheautocorrelation function oftheinputdelay-spread functionh(7';t). Since h(7';t)iscomplex valued,weusethefollowing definition fortheautocorrelatiori function: (8.41) whereEisthestatistical expectation operator, theasteriskdenotescomplex conjugation, 7']and7'zarethepropagation delaysoftherwopathsinvolved inthecalculation, andt1 andtzarethetimesatwhichtheoutputsofthetwopathsareobserved. Invoking station arityinthetimevariable tanduncorrelated scattering inthetime-delay variable 7',wemay reformulate theautocorrelation function ofh(7';t)as R;;(7'b7'2;llt) =E[h*{Tr;t)hh;t +Ilt)] =r;;(7'1;1lt) 8(7'1-7'2)(8.42) whereIltisthedifference berweentheobservation times,and8(7'1-7'2)isadeltafunction. Using 7'inplaceof7'"theremaining function inEquation (8.42)isredefined as r;;(7';llt) =E[h(r,t)h*(7';t +Ilt)] (8.43) Thefunction r;;(7';llt)iscalledthemultipath autocorrelation profileofthechannel. Consider nextastatistical characterization ofthechannelintermsofthecomplex valued,time-varying transfer function H(f;t).Following aformulation similartothatde scribedinEquation (8.41),theautocorrelation function ofH(f;t)isdefinedby (8.44) whereIIand!zrepresent rwofrequencies inthespectrum ofatransmitted signal.The autocorrelation function RA(f"t,;!z,tz)provides astatistical measure oftheextent towhichthesignalisdistorted bytransmission through thechannel. FromEquations (8.40), (8.41), and(8.44)wefindthattheautocorrelation functions RA(f"t,;!z,tz)and R;;(7'1,t,;7'2,tz)arerelatedbyaformofrwo-dimensional Fouriertransformation asfollows: Invoking stationarity inthetimedomain, wemayreformulate Equation (8.44)as (8.46) Thisdefinition suggeststhattheautocorrelation functionRA(f"fz;llt) maybemeasured bypairsofspacedtonestocarryoutcross-correlation measurements ontheresulting ;38 CHAPTER 8..MULTIUSER RADIO COMMUNICATIONS channeloutputs.Suchameasurement presumes stationarity inthetimedomain.Ifweals assumestationarity inthefrequency domain, wemaygoonestepfurtherandwrite 0 RFi(f,f+!1f;Jit)=rn(!1f;Jit) =E[H*(f;t)H(f +!1f;t+!1t)](8A7) Thisspecialized formoftheautocorrelation function ofH(f;t)isinfacttheFouriertrans formofthemultipath autocorrelation profilerj;(r,!1t)withrespecttothedelay-time vari ableT,asshownby rFf(!1f;!1t) =frorj;(r,!1t)exp(-j211'T!1f)dT (8A8) Thefunction rFi!1f;!1t) iscalledthespaced-frequency spaced-time correlation functionof thechannel. Finally,weintroduce afunction S(T;v)thatformsaFourier-transform pairwiththe multipath autocorrelation profilerj;(T;!1t)withrespecttothevariable!1t,asshownby andS(T;V)=frorj;(T;!1t)exp(-j211'V !1t)d(!1t) (8.49) (8.50) ~T;!1t) =froS(T;V)exp(j211'v!1t) dv Thefunction S(T;V)mayalsobedefinedintermsofrn(!1f;!i.t) byapplying aformofdouble Fouriertransformation: aFouriertransform withrespecttothetimevariable!1tandan inverseFouriertransform withrespecttothefrequency variable!1f.Thatistosay, S(T;V)=frofrorfI(!1f;!1t) exp(-j211'V !1t)exp(j21rT!1f) d(!1t)d(!1f)(8.51) Figure8.19displaysthefunctional relationships between rj;(T;!1t),rfI(!1f;!i.t), andS(r,v) intermsoftheFouriertransform anditsinverse. Spaced-frequency Spaced-time Correlation function rii(Aj;At)Multipath autocorrelation profile r;;(T;,at}FM[']----...~ F;'[.]Scattering function S(r;v) F7'[·l:Fouriertransform withrespecttodelay 1" Fi}H:InverseFouriertransform withrespecttofrequency increment8/ FiltH:Fouriertransform withrespecttotimeincrement tJ.t F;l[-];InverseFouriertransform withrespecttoDopplershiftp FIGURE 8.19Functional relationships between themultipath autocorrelation profiler;;(r~t),the spaced-frequency spaced-time correlation function rn(ilf;ilt), andthescattering function S(r,v). 8.6Statistkal Characteri:zat'r>n ofMult'path Channels 539 Thefunction S(7";1')iscalledthescattering function ofthechannel. Foraphysical interpretation ofit,consider thetransmission ofasingletoneoffrequency f'(relativeto thecarrier).Thecomplex envelope oftheresulting filteroutputis so(t)=exp(j27Tf't)H.(f';t) Theautocorrelation function ofso(t)is E[s~(t)So(t +dt)]=exp(j21Tf' dt)E[H*(f';t)H(f';t +dt)] =exp(j21Tf' dt)rtf(O;dt)(8.52) (8.53) where,inthelastline,wehavemadeuseofEquation (8.47).Puttingdf=0inEquation (8.48),andthenusingEquation (8.50),wemaywrite rH(O;dt) =roor';(7";dt) d7" =roo[rooS(7";1')d7"]exp(j21T1' dt)d1' Hence,wemayviewtheintegral(8.54) asthepowerspectraldensityofthechanneloutputrelativetothefrequency f'ofthe transmitted tone,andwiththeDoppler shiftvactingasthefrequency variable. General izingthisresult,wemaystatethatthescattering function S(7";v)provides astatistical measUre oftheoutputpowerofthechannel, expressed asafunction ofthetimedelay7" andtheDopplershift1'. !!ilDELAY SPREAD ANDDOPPLER SPREAD Puttingdt=0inEquation (8.43),wemaywrite P';(7")=r';(7";O) =E[Ih(7";tW](8.55) Thefunction P,;(7")describes theintensity (averaged overthefadingfluctuations) ofthe scattering processatpropagation delay7".Accordingly, P';(7")iscalledthedelaypower spectrum orthemultipath intensity profileofthechannel.Thedelaypowerspectrum may alsobedefinedintermsofthescattering function S(7";v)byaveraging itoverallDoppler shifts.Specifically, puttingdt=0inEquation (8.50)andthenusingthefirstlineofEqua tion(8.55),wemaywrite (8.56) Figure8.20showsanexample ofadelaypowerspectrum thatdepictsatypicalplot ofthepowerspectraldensityversusexcessdelay;theexcessdelayismeasured withrespect tothetime delay fortheshortestechopath.Note,asinFigure8.17,thepowerismeasured indBm.The"threshold level"included inFigure8.20definesthepowerlevelbelowwhich thereceiverfailstooperatesatisfactorily. 540 CHAPTER 8IIIMULTIUSER RADIOCOMMUNICATIONs Excessdelay FIGURE 8.20Example ofapower-delay profileforamobileradiochannel. (FromParsons, 1992,,,1thpermission.) Twostatistical moments ofPh(T)ofinterestaretheaveragedelay, Tavoandthedelay spread, crT'Theaveragedelayisdefinedasthefirstcentralmoment(i.e.,themean)ofPi,(r), asshownbyrTPh(T)dTrPi,(T)dT(8.57) Thedelayspreadisdefinedasthesquarerootofthesecondcentralmoment ofPi,(T),as shownby ( )V2 =r(T-TavfPh(T) dT crT JOO oPi,(T)dT(8.58) Thereciprocal ofthedelayspread crTisameasure ofthecoherence bandwidth ofthe channel, whichisdenotedbyBe. Consider nexttheissueofrelatingtheDopplereffectstotimevariations ofthechan nel.Forthispurpose, wefirstset!1f=0,whichcorresponds tothetransmission ofasingle tone(ofsomeappropriate frequency) overthechannel.Thespaced-frequency spaced-time correlation functionofthechannelthenreducestorfl(O;!1t). Hence,evaluating theFourier transform ofthisfunction withrespecttothetimevariable !1t,wemaywrite (8.59) ThefunctionSfl(p)definesthepowerspectrum ofthechanneloutputexpressed asafunc tionoftheDopplershiftP;itistherefore calledtheDoppler spectrum ofthechannel. The 8.6Sfafisti<:al CJumu>fe,.izafi",. ofMulfipafh Cha....els541 Doppler spectrum mayalsobedefinedintermsofthescattering function byaveraging it overallpossiblepropagation delays,asshownby (8.60) TheDoppler shiftvmayassumepositiveandnegativevalueswithequallikelihood. The meanDopplershiftistherefore zero.ThesquarerootofthesecondmomentoftheDoppler spectrum isthusdefinedby (8.61) Theparameter Upprovides ameasureofthewidthoftheDopplerspectrum; itistherefore calledtheDoppler spreadofthechannel. Thereciprocal oftheDoppler spreadiscalled thecoherence timeofthechannel, whichisdenotedby7'e' Another usefulparameter thatisoftenusedinmeasurements isthefaderateofthe channel. ForaRayleigh fadingchannel, theaveragefaderateisrelatedtotheDoppler spreadu.as Ie=1.475uvcrossings persecond (8.62) Asthenameimplies,thefaderateprovides ameasure oftherapidityoffadingofthe channel. Sometypicalvaluesencountered inamobileradioenvironment areasfollows: ~Thedelayspread, Unamounts toabout20IJ-S. ~TheDopplerspread, Up>duetothemotionofavehiclemayextendupto40-80Hz. iiiClASSIFICATION OFMULTIPATH CHANNELS Theparticular formoffadingexperienced byamultipath channeldependsonwhetherthe channelcharacterization isviewedinthefrequency domainorthetimedomain. Whenthechannelisviewedinthefrequency domain, theparameter ofconcern is thechannel's coherence bandwidth, Be>whichisameasureofthetransmission bandwidth forwhichsignaldistortion acrossthechannelbecomesnoticeable. Amultipath channelis saidtobefrequency selectiveifthecoherence bandwidth ofthechannelissmallcompared tothebandwidth ofthetransmitted signal.Insuchasituation, thechannelhasafiltering effectinthattwosinusoidal components, withafrequency separation greaterthanthe channel's coherence bandwidth, aretreateddifferently. If,however, thecoherence band widthofthechamielislargecompared tothemessagebandwidth, thefadingissaidtobe frequency nonselective, orfrequency flat. Whenthechannelisviewedinthetimedomain, theparameter ofconcernisthe coherence time,7'e,whichprovides ameasureofthetransmitted signalduration forwhich distortion acrossthechannelbecomes noticeable. Thefadingissaidtobetimeselectiveif thecoherence timeofthechannelissmallcompared totheduration ofthereceivedsignal (i.e.,thetimeforwhichthesignalisinflight).Fordigitaltransmission, thereceivedsignal's duration istakenasthesymbolduration plusthechannel's delayspread.If,however, the channel's coherence timeislargecompared tothereceivedsignalduration, thefadingis 542 CHAPTER S "MULTIUSER RADIO COMMUNICATIONS Bandwidth oI III I I Time-flat INon-flatinboth :timeandfrequency I I II_________ L _ I I I Flat-flat :Frequency-flat I II I I Timeduration FIGURE8.21Illustrating thefourclassesofmultipath channels: 'Tc=coherence time, Be=coherence band"idth. saidtobetimenonselective, ortimeflat,inthesensethatthechannelappearstothe transmitted signalastimeinvariant. Inlightofthisdiscussion, wemayclassifymultipath channels asfollows: Flat-flatchannel,whichisflatinbothfrequency andtime. ,.Frequency-flat channel, whichisflatinfrequency only. >Time-flat channel,whichisflatintimeonly. Nonflatchannel, whichisflatneitherinfrequency norintime;suchachannelis sometimes referredtoasadoublydispersive channel. Theclassification ofmultipath channels, basedonthisapproach, isshowninFigure8.21. Theforbidden area,shownshadedinthisfigure,followsfromtheinverserelationship that existsbetweenbandwidth andtimeduration. 8.7BinarySignaling overa Rayleigh FadingChannel InChapter6,wedetermined theaverageprobability ofsymbolerrorforthetransmission ofbinarydataoverachannelcorrupted byadditivewhiteGaussian noise.Inamobile radioenvironment, wehaveanadditional effecttoconsider, namely,thefluctuations in theamplitude andphaseofthereceived signalduetomultipath effects.Tobespecific, consider thetransmission ofbinarydataoveraRayleigh fadingchannel, forwhichthe (low-pass) complex envelope ofthereceivedsignalismodeled asfollows: x(t)=IXexp(-j.p):5(t) +w(t) (8.63) where:5(t)isthecomplex envelope ofthetransmitted (band-pass) signal, IXisaRayleigh distributed randomvariabledes<;:ribing theattenuation intransmission, ¢isauniformly 8.7BinarySignaling overaRayleigh FadingClumnel 543 distributed randomvariabledescribing thephase-shift intransmission, andw(t)isacom plex-valued whiteGaussian noiseprocess. Itisassumed thatthechannel isflatinboth timeandfrequency, sothatwecanestimate thephase-shift <f>fromthereceived signal withouterror.Supposethenthatcoherent binaryphase-shift keyingisusedtodothedata transmission. Underthecondition thataisfixedorconstant overabitinterval, wemay adaptEquation (6.20)ofChapter 6forthesituation athandbyexpressing theaverage probability ofsymbolerror(i.e.,biterrorrate)duetotheadditivewhiteGaussian noise actingaloneasfollows: Pe('Y) ~erfc(yY) (8.64) (8.65)where 'Yisanattenuated versionofthetransmitted signalenergyperbit-to-noise spectral densityratioEb/No,asshownby elEb 'Y=No Now,insofarasamobileradiochannelisconcerned, wemayviewPe('Y)asaconditional probability giventhataisfixed.Thus,toevaluatetheaverageprobability ofsymbolerror inthecombined presence offadingandnoise,wemustaveragePet'Y)overallpossible valuesof'Y,asshownby Pe=rPe('Y)f('Y)d'Y (8.66) (8.68)wheref('Y)istheprobability densityfunction of'Y.FromEquation (8.65)wenotethat'Y depends onthesquaredvalueofa.SinceaisRayleigh distributed, wefindthat'Yhasa chi-square distribution withtwodegreesoffreedom.7Inparticular, wemayexpressthe probability densityfunctionof'Yas f('Y)=1-exp(-..r.), 'Y~0 (8.67) 'Yo 'Yo Theterm'Yoisthemeanvalueofthe received signalenergyperbit-to-noise spectraldensity ratio,whichisdefinedby . 'Yo=E['Y] =EbE[trJ No whereE[el]isthemean-square valueoftheRayleigh-distributed random variable a. Substituting Equations (8.64)and(8.67)into(8.66),andcarrying outtheintegration, we getthefinalresult Pe=I(1-J1:0'YJ (8.69) Equation (8.69)definesthebiterrorrateforcoherent binaryphase-shift keying(PSK) overaflat-flatRayleigh fadingchannel. Following asimilarapproach, wemayderivethe corresponding biterrorratesforcoherent binaryfrequency-shift keying(FSK),binary differential phase-shift keying(DPSK),andnoncoherent binaryFSK.Theresultsofthese evaluations aresummarized inTable8.2.InFigure8.22,wehaveusedtheexactformulas ofTable8.2toplotthebiterrorrateversus 'Yoexpressed indecibels. Forthesakeof comparison, wehavealsoincluded inFigure8.22plotsforthebiterrorratesofcoherent binaryPSKandnoncoherent binaryFSKforanonfading channel. WeseethatRayleigh 544 CHAPTER 8iiiMULTIUSER RADIO COMMUNICATIONS TABLE8.2Biterrorratesforbinarysignaling overaflat-flat Rayleigh fadingchannel Approximate Formula ExactFormula forthe fortheBitErrorRate, TypeofSignaling BitErrorRatePe Assuming Large'Yo Coherent binaryPSK 1(1~)1 2 1 +'Yo4'Yo Coherent binaryFSK 1(1~)1 2 2 +'Yo2'Yo BinaryDPSK 1 1 2(1+'Yo) 2'Yo Noncoherent binaryFSK 1 2+'Yo 'Yo fadingresultsinaseveredegradation inthenoiseperformance ofadigitalpassbandtrans missionsystem,thedegradation beingmeasured intensofdecibelsofadditional mean signal-to-noise ratiocompared toanonfading channelforthesamebiterrorrate.Inpar ticular,forlarge'Yowemayderivetheapproximate formulas giveninthelastcolumnof Table8.2,according towhichtheasymptotic decreaseinthebiterrorratewiththeaverage signalenergyperbit-to-noise spectraldensityratio'Yofollowsaninverselaw.Thisbehavior isdramatically different fromthecaseofanonfading channel, forwhichtheasymptotic decrease inthebiterrorratewith"Yofollowsanexponential law. Thepractical implication ofthisdifference isthatinamobileradioenvironment, we havetoprovidealargeincrease inmeansignal-to-noise ratio(relative toanonfading environment), soastoensureabiterrorratethatislowenoughforpractical use.Tomeet sucharequirement, wehavetoincreasethetransmitted power,antennasize,andsoon, whichcanbecostlyintermsofimplementation. Alternatively, wemayutilizespecialmod· ulationandreception techniques thatarelessvulnerable tofadingeffects.Amongthese techniques, thebestknownandmostwidelyusedarethemultiple-receiver combining techniques referredtocollectively asdiversity, abriefdiscussion ofwhichispresenred next. !i1lDIVERSITY TECHNIQUES Diversity maybeviewedasaformofredundancy. Inparticular, ifseveralreplicasofthe messagesignalcanbetransmitted simultaneously overindependentlyfading channels, then thereisa·goodlikelihood thatatleastoneofthereceived signalswillnotbeseverely degraded byfading.Thereareseveralmethods formakingsuchaprovision. Inthecontext ofourpresentdiscussion, thefollowing diversity techniques areofparticular interest: P>Frequency diversity II>Time(signal-repetition) diversity ~Spacediversity Infrequency diversity, themessage signalistransmitted usingseveralcarriersthat arespacedsufficiently apartformeachothertoprovideindependently fadingversionsof 8.7BinarySignaling rn>eraRayleigh FadingChannel 545 100,----,.----,.----,.----,.----,---,-----, FIGURE8.22Performance ofbinarysignaling schemes overaRayleigh fadingchannel, shown ascontinuous curves;thedashedcurvespertaintoanonfading channel. thesignal.Thismaybeaccomplished bychoosing afrequency spacingequaltoorlarger thanthecoherence bandwidth ofthechannel. Intimediversity, thesamemessage signalistransmitted indifferent timeslots,with thespacingbetweensuccessive timeslotsbeingequaltoorgreaterthanthecoherence time ofthechannel. Timediversity maybelikenedtotheuseofarepetition codeforerror controlcoding.(Error-control codingisdiscussed inChapter 10.) Inspacediversity, multiple transmitting orreceiving antennas (orboth)areused, withthespacingbetweenadjacent antennas beingchosensoastoassuretheindependence offadingevents;thismaybesatisfied byspacingtheadjacent antennas byatleastseven timestheradiowavelength. GiventhatbyoneofthesemeanswecreateLindependently fadingchannels, we maythenusealineardiversity combining structure involving Lseparate receivers, as depicted inFigure8.23.Thesystemisdesigned tocompensate onlyforshort-term effects ofafadingchannel. Moreover, itisassumed thatnoise-free estimates ofthechannelat tenuation factors[aelandthechannel phase-shifts [<Pelareavailable. Then,thelinear combiner achieves optimum performance forbinarydatatransmission (discussed herefor 546 CHAPTER 8IIIMULTIUSER RADIO COMMUNICATIONS Say1ifRe[J,(t)],Re[6 0(t)1 Say0ifRe[,1,(t)]<Re[Jolt)] FIGURE8.23Blockdiagram illustrating thespacediversity technique. 40 35 30 20 15 10~ "-l\~"-1\\\'',".....--FSK(noncoherent) - '\ ---DPSK-'-PSK(coherent) - \\,\\\'\."1"'-\1'\"~.,""""" \"~\\\."',""",,"\1\\\1\, :'........ 1\'."\\\\\\""""I'\...., \\.~\\" ...,:'........ \"\ \, I\." \.\ 1\\\\"- L=1 \ \ ..... 1\\\\\\ \ \.\\.\ \ L=4\.\\\~~L=21* "'t\ \\\~\..lr' \\.'\\2 25 10-2 10-3 10-510-4 10-6 5 25 Yo-dB FIGURE8.24Performance ofbinarysignaling schemes withdiversity. (FromProws, 1995, withpermission ofMcGraw-Hill.) 8.8mMAandCDMAWireless Cummunkation Systems 547 (8.70) k=0,1thepurposeofillustration) byproceeding asfollows:Theoutputofthekthmatched filter intheethreceiver, V'k(t),ismultiplied bya,exp(j<p,) thatrepresents thecomplex conju gateofthefthchannelgain,wheref=1,2,...,L,andk=0,1.Thus,thelinearcombiner resultsintwooutputcomplex envelopes definedby L Vk(t)=2:a,exp(i<Pe)V,k(t), e=1 according towhichatexp(Ne)playstheroleofaweighting factor.Oneoutputcomplex envelope vo(t)corresponds tothetransmission ofsymbol0,andtheotherV,(t)corresponds tothetransmission ofsymbol1.Therealpartsofvo(t)andv,(t)arethenusedinthe decision-making process.Thesituation described hereappliestobinaryFSK.Inthecase ofbinaryPSK,onlyasinglematched filterisneeded,inwhichcasethelinearcombiner produces asingleoutputcomplex envelope. Hereagain,however, therealpartofthe combiner outputisusedinthedecision-making process. Inthelinearcombiner described herein,the"instantaneous" outputsignal-to-noise ratio(SNR)isthesumoftheinstantaneous SNRsontheindividual diversity branches (channels). Thisoptimum formofalinearcombiner istherefore referredtoasamaximal ratiocombiner; seeProblem 8.17. Figure8.24showsthenoiseperformance ofcoherent binaryPSK,binaryDPSK,and noncoherent binaryFSKforL=2,4independently fadingchannels. Forthesakeof comparison, wehavealsoincluded inthisfigurethecorresponding graphsforafading channelwithnodiversity (i.e.,L=1).Figure8.24clearlyillustrates theeffectiveness of diversity asameansofmitigating theshort-term effectsofRayleigh fading. 8.8TDMAandCDMAWireless Communication Systems8 Inwireless communications, aswithordinary telephony, auserwouldliketotalkand listensimultaneously. Tocatertothisnaturaldesire,someformofduplexing isrequired. Onewayinwhichthisrequirement canbesatisfied istoprovidetwofrequency bands, onefortheforward linkfromthebasestationtoamobileandtheotherforthereverse linkfromthemobiletothebasestation.Aspointedoutearlier,inNorthAmerica the subband 869-894 MHzisusedfortheforward link,andthesubband 824-849 MHzis usedforthereverselink.Thisformofduplexing iscalledfrequency divisionduplexing (FDD).Indeed,FDDisanintegralpartofthetwowidelyusedwirelesscommunication systemssummarized inTable8.3.' Thefirstofthesesystems, namely, GSM,usesTDMA.FromSection8.2werecall thatinaTDMAsystemeachsubscriber ispermitted toaccesstheradiochannelduringa setofpredetermined timeslots,duringwhichtimethatparticular subscriber willhavefull useofthechannel. Consequently, dataaretransmitted overthechannelinbursts,asshown intheframestructure ofFigure8.25.ThebasicframeofGSMiscomposed ofeight 577}LSslots.TheI-bitflagpreceding eachdataburstof57bitsisusedtoidentifywhether thedatabitsaredigitized speechorsomeotherinformation-bearing signal.The3tailbits, alllogicalzeros,areusedinconvolutional decoding ofthechannel-encoded databits. (Convolutional codesarediscussed inChapter 10.)The26-bittrainingsequence inthe middleofthetimeslotisusedforchannelequalization. Finally,theguardtime,occupying 8.25bits,isincluded attheendofeachslottopreventdataburstsreceived atthebase 548 CHAPTER 8"MULTIUSER RADIO COMMUNICATIONS TABLE8.3Summary oftwowidelyused wirelesscommunication systems Item GSM' IS-95t Number ofduplex 125 20 channels Channel bandwidth 200 1,250 (kHz) Typeofmultiple access TDMA CDMA Accessusersper 8 20to35 channel Modulation type Datarate(kb/s) Frameperiod(ms)GMSK 270.833 4.615BPSKlQPSK 9.6or14.4 20Comments ATDMAsystemisdeterministic inthatthe numberofaccessusersperchannelisdefined bythenumberofavailable timeslots.On theotherhand,aCDMAsystemis interference-limited inthatithasasoftlimit onthenumberofaccessusersperchannel. InCDMA,dataaremodulated asBPSK,but thespreading isQPSK ForCDMA,theframeperiodequalsthatof thespeechcodec(coder/decoder) 'GSMstandsforGlobalSystemforMobileCommunications; originally, itwasintroduced asanacronymfor GroupedetravailSpecialepourlesservicesMobiles. tISstandsforInterimStandard. stationfrommobilesfromoverlapping witheachother;thisisachieved bytransmitting nosignalatallduringtheguardtime.Witheachslotconsisting of156.25bits,ofwhich 40.25bitsareoverhead (ignoring the2flagbits),theframeefficiency ofGSMis (40.25) 01 -156.25 X100=74.24Yo Thesecondwireless communication system,15-95,summarized inTable8.3uses CDMA.FromSection8.2werecallthatinCDMA, eachsubscriber isassigned adistinct spreading code(PNsequence), therebypermitting thesubscriber fullaccesstothechannel allofthetime.Consequently, inaCDMAsystemwehaveanewformofinterference calledmultiple-access interference (MAl),whicharisesbecauseofdeviation ofmespread ingcodesfromperfectortiIogonality. Arelatedphenomenon thatneedsattention isthe near-farproblem, whichoccursifmereceived signalsfromthemobileunitsdonothave equalpoweratthebasestation.Insuchasituation, mestrongest received signalfroma mobileusercaptures medemodulation processatthebasestationtothedetriment ofthe T:Tail(bits) F:Flag(bit) Train:Trainingintervalforequalizer Guard:Guardtimeinterval f*---- Timeslot=156.25bits=577jJ.s~ FIGURE 8.25Framestructure oftheGSMwirelesscommunication system. 8.8IDMAandCDMAWireless C..........nication Systems 549 otherusers,Toovercome thenear-farproblem, itiscustomary tousepowercontrolat thebasestation,whereby thebasestationmaintains controloverthepowerlevelofthe transmitted signalfromeverymobilebeingservedbythatbasestation.Theuseofpower controlisparticularly important inCDMAsystemsforanotherreason.Agoalofmultiple accesssystemsistomaximize systemcapacity, whichisdefinedasthelargestpossible numberofusersthatcanbereliablyservedbythesystem,givenprescribed resources. Clearly,systemcapacityiscompromised ifeachmobileisfreetoraiseitstransmitted power levelregardless ofotherusers,sincethatincreaseintransmitted powerwill,inturn,raise thelevelofmultiple-access interference inthesystem.Tomaximize systemcapacity, itis therefore essential thateachmobile's transmitter beunderthecontroloftheservingbase stationsothatthesignal-to-interference ratioismaintained attheminimum acceptable levelneededforreliableservice. IIIRAKERECEIVER Adiscussion ofwirelesscommunications usingCDMAwouldbeincomplete withouta description oftheRAKEreceiver.9TheRAKEreceiverwasoriginally developed inthe 19505asa"diversity" receiverdesigned expressly toequalizetheeffectofmultipath. First, andforemost, itisrecognized thatusefulinformation aboutthetransmitted signaliscon tainedinthemultipath component ofthereceivedsignal.Thus,takingtheviewpoint that multipath maybeapproximated asalinearcombination ofdifferently delayedechoes,the RAKEreceiverseekstocombattheeffectofmultipath byusingacorrelation methodto detecttheechosignalsindividually andthenaddingthemalgebraically. Inthisway,in tersymbol interference duetomultipath isdealtwithbyreinserting different delaysinto thedetectedechoessothattheyperformaconstructive ratherthandestructive role. Figure8.26showsthebasicideabehindtheRAKEreceiver. Thereceiverconsistsof anumberofcorrelators connected inparallelandoperating inasynchronous fashion. Eachcorrelator hastwoinputs:(1)adelayedversionofthereceivedsignaland(2)areplica ofthepseudo-noise (PN)sequence usedasthespreading codetogenerate thespread- Phase andgain adjustors Reference PN_o\---__-+__ ~I---+--___ll---+-------J sequence Received .....-.....;;.j signal FIGURE8.26Blockdiagram ofthcRAKEreceiver. 550 CHAPTER 8"MULTIUSER RADIO COMMUNICATIONS spectrum modulated signalatthetransmitter. Ineffect,thePNsequence actsasa"refer_ encesignal."Letthenominal bandwidth ofthePNsequence bedenotedasW;liT whereTcisthechipduration. Fromthediscussion ofspread-spectrum modulation pr;: sentedinChapter 7,werecallthattheautocorrelation function ofaPNsequence hasa singlepeakofwidth1/W,anditdisappears towardzeroelsewhere insideoneperiodof thePNsequence (i.e.,onesymbolperiod).Thusweneedonlymakethebandwidth Wof thePNsequence sufficiently largeto"identify" thesignificant echoesinthereceivedsignal. Tobesurethatthecorrelator outputsalladdconstructively, twootheroperations are performed inthereceiverbythefunctional blockslabeled"phaseandgainadjustors": 1.Anappropriate delayisintroduced intoeachcorrelator outputsothatthephase anglesofthecorrelator outputsareinagreement witheachother. 2.Thecorrelator outputsareweighted sothatthecorrelators responding tostrong pathsinthemultipath environment havetheircontributions accentuated, whilethe correlators notsynchronizing withanysignificant patharecorrespondingly suppressed. Theweighting coefficients, lXk>arecomputed inaccordance withthemaximal ratiocom biningprinciple:'o Thesignal-to-noise ratioofaweighted sum,whereeachelementofthesumconsists ofasignalplusadditivenoiseoffixedpower,ismaximized whentheamplitude weighting isperformed inproportion torhepertinent signalstrength. Thelinearcombiner outputis M y(t)=2:lXkZk(t) k~'(8.71) whereZk(t)isthephase-compensated outputofthekthcorrelator, andMisthenumber ofcorrelators in/thereceiver. Provided weuseenoughcorrelators inthereceivertospan aregionofdelayssufficiently widetoencompass allthesignificant echoesthatarelikely tooccurinthemultipath environment, theoutputy(t)behavesessentially asthoughthere wasasinglepropagation pathbetweenthetransmitter andreceiverratherthanaseriesof multiple pathsspreadintime. Tosimplifythepresentation, thereceiverofFigure8.26assumes theuseofbinary phase-shift keyinginperforming spread-spectrum modulation atthetransmitter. Thusthe finaloperation performed inFigure8.26isthatofintegrating thelinearcombiner output y(t)overthebitintervalTbandthendetermining whether binarysymbol1or0was transmitted inthatbitinterval. TheRAKEreceiverderivesitsnamefromthefactthatthebankofparallelcorrelators hasanappearance similartothefingersofarake.Becausespreadspectrum modulation isbasictotheoperation ofCDMAwirelesscommunications, itisnaturalfortheRAKE receiver tobecentraltothedesignofthereceiverusedinthistypeofmultiuser radio communication." 8.9SourceCodingofSpeechfor Wireless Cmnmunications Fortheefficientuseofchannelbandwidth, digitalwirelesscommunication systems, be theyoftheTDMAorCDMAtype,relyontheuseofspeechcodingtoremovealmostall 8.9SourceCodingofSpeechforWireless Communications 551 ofthenaturalredundancy inspeech,whilemaintaining ahigh-quality speechondecoding. Thecommon approach istousesourcecoding,which,inoneformoranother, exploits thelinearpredictive coding(LPC)ofspeech. Inthissection,wedescribetwodifferent techniques forspeechcoding:multi-pulse excitedLPCandcode-excited LPC,versionsofwhichareusedinGSMand15-95,respec tively.Ourtreatment ofbothofthesespeechcodingtechniques isinconceptual terms.12 !ilMULTI-PULSE EXCITED LPC Thisformofspeechcodingexploitstheprincipleofanalysisbysynthesis, whichmeans thattheencoderincludes areplicaofthedecoderinitsdesign.Specifically, theencoder consistsofthreemainpartsasindicated inFigure8.27a: 1.Synthesis filterforthepredictive modeling ofspeech.Itmayconsistofanall-pole filter(i.e.,afilterwhosetransferfunctionhaspolesonly),whichisdesigned tomodel theshort-term spectralenvelope ofspeech;thetermshort-term referstothefactthat thefilterparameters arecomputed onthebasisofpredicting thepresentsampleofthe speechsigna!usingeighttosixteenprevious samples. Thesynthesis filtermayalso includealong-term predictor formodeling thefinestructure ofthespeechspectrum; insuchacase,thelong-term predictor isconnected incascadewiththeshort-term predictor. Inanyevent,thefunction ofthesynthesis filteristoproduceasynthetic versionoftheoriginalspeechthatisofhighquality. 2.Excitation generator forproducing theexcitation appliedtothesynthesis filter.The excitation consistsofadefinitenumberofpulsesevery5to15ms.Theamplitudes andpositions oftheindividual pulsesareadjustable. 3.Errorminimization foroptimizing theperceptually weighted errorbetweentheorig inalspeechandsynthesized speech.Theaimofthisminimization istooptimize the amplitudes andpositions ofthepulsesusedintheexcitation. Typically, amean squareerrorcriterion isusedfortheminimization. Thus,asshowninFigure8.27a,thethreepartsoftheencoderformaclosed-loop optimization procedure, whichpermitstheencodertooperateatabitratebelow16 kb/s,whilemaintaining high-quality speech. Theencoding procedure itselfhastwomainsteps: Il>Thefreeparameters ofthesynthesis filterarecomputed usingtheactualspeech samplesasinput.Thiscomputation isperformed outsidetheoptimization loopover Input speech (aJReceived signal (b)Synthetic speech FIGURE8.27Multi-pulse excitedlinearpredictive codec.(a)Encoder. (b)Decoder whoseinput (thereceived signal)consists ofquantized filterparameters andquantized excitation asproduced bytheencoder. 552 CHAPTER 8IIMULTIUSER RADIO COMMUNICATIONS aperiodof10to30ms,duringwhichthespeechsignalistreatedaspseudo_ stationary. I>Theoptimum excitation forthesynthesis filteriscomputed byminimizing theper- ceptually weighted errorwiththeloopclosedasinFigure8.27a. Thusthespeechsamplesaredividedintoframes(10to30mslong)forcomputing the filterparameters, andeachframeisdividedfurtherintosubframes (5to15ms)foroPti mizingtheexcitation. Thequantized filterparameters andquantized excitation constitute thetransmitted signal. Notethatbyfirstpermitting thefilterparameters tovaryfromoneframetothenext andthenpermitting theexcitation tovaryfromonesubframe tothenext,theencoderi; enabledtotrackthenonstationary behavior ofspeech,albeitonabatch-by-batch basis. Thedecoder, locatedinthereceiver, consistssimplyoftwoparts:excitation generator andsynthesis filter,asshowninFigure8.27b.Thesetwopartsareidentical tothecorre sponding onesintheencoder. Thefunction ofthedecoderistousethereceivedsignalto produce asynthetic versionoftheoriginalspeechsignal.Thisisachieved bypassingthe decoded excitation throughthesynthesis filterwhoseparameters aresetequaltothosein theencoder. Toreducethecomputational complexity ofthecodec(i.e.,contraction ofcoder! decoder), theintervals between theindividual pulsesintheexcitation areconstrained to assumeacommon value.Theresulting analysis-by-synthesis codecissaidtohavearegular pulseexcitation. iiiJCODE-ExcITED LPC Figure8.28showstheblockdiagram ofthecode-excited LPC,commonly referredtoas CELP.Thedistinguishing featureofCELPistheuseofapredetermined codebook of stochastic (zero-mean whiteGaussian) vectorsasthesourceofexcitation forthesynthesis filter.Thesynthesis filteritselfconsistsoftwoall-polefiltersconnected incascade,oneof whichperforms short-term prediction andtheotherperforms long-term prediction. Aswiththemulti-pulse excitedLPC,thefreeparameters ofthesynthesis filterare computed first,usingtheactualspeechsamplesasinput.Next,thechoiceofaparticular vector(code)storedintheexcitation codebook andthegainfactorGinFigure8.28is optimized byminimizing theaveragepoweroftheperceptually weighted errorbetween Code#1 Code#2-----1 1 11-----'tSynthetic speechInpul speech Code#N Excitation codebook ofsizeN FIGURE8.28Encoder ofthecode-excited linearpredictive codec(CELP): thetransmitted sig· nalconsists oftheaddressofthecodeselected fromthecodebook, quantized G,andquantized filterparameters. 8.10Adoptive Antenna ArraysfurWireless Communicatiuns 553 theoriginalspeechandsynthesized speech(i.e.,outputofthesynthesis filter).Theaddress ofthestochastic vectorselectedfromthecodebook andthecorresponding quantized gain factor,together withthequantized filterparameters, constitute thetransmitted signal. Anidentical copyofthecodebook ismadeavailable tothedecoder, andlikewisefor thesynthesis filter.Hence,giventhereceivedsignal,thedecoderisenabledtoparameterize itsownsynthesis filteranddetermine theappropriate excitation forthesynthesis filter, therebyproducing asynthetic versionoftheoriginalspeechsignal. CELPiscapableofproducing good-quality speechatbitratesbelow8kb/s.How ever,itscomputational complexity isintensive because oftheexhaustive searchofthe excitation codebook. Inparticular, theweighted synthesized speechintheencoderhasto becomputc;d foralltheentriesinthecodebook andthencompared withtheweighted originalspeech.Nevertheless, real-time implementation ofCELPcodecshasbeenmade possiblebyvirtueofadvances indigitalsignalprocessing andVLSItechnology. 8.10Adaptive Antenna Arraysfor W'ireless Communicationsl3 Thegoalofwirelesscommunications istoallowasmanyusersaspossibletocommunicate reliablywithoutregardtolocationandmobility. Fromthediscussion presented inSections 8.5and8.6,wefindthatthisgoalisseriously impeded bythreemajorchannelimpairments: 1.Multipath cancauseseverefadingduetophasecancellation betweendifferent prop agationpaths.Fadingleadstoareduction inavailable signalpowerandtherefore a degraded noiseperformance. 2.Delayspreadresultsfromdifferences inpropagation delaysamongthemultipleprop agationpaths.Whenthedelayspreadexceedsabout10percentofthesymboldu ration,theintersymbol interference experienced bythereceivedsignalreachesasig nificantlevel,therebycausingareduction intheattainable datarate. 3.Co-channel interference arisesincellularsystemswheretheavailable frequency chan nelsaredividedintodifferent sets,witheachsetbeingassigned toaspecificcelland withseveralcellsinthesystemusingthesamesetoffrequencies. Co-channel inter ferencelimitsthesystemcapacity (i.e.,thelargestpossiblenumberofusersthatcan bereliablyservedbythesystem). Typically, cellularsystemsuse1200sectorization ateachbasestation,andonly oneuseraccesses asectorofabasestationatagivenfrequency. Wemaycombatthe effectsofmultipath fadingandco-channel interference atthebasestationbyusingthree identical butseparate antenna arrays,oneforeachsectionofthebasestation.The compensation ofdelayspreadisconsidered laterinthesection.Figure8.29showsthe blockdiagram ofanarraysignalprocessor, whereitisassumed thatthereareNusers whosesignalsarereceivedataparticular sectorofthebasestation,andthearrayforthat sectorconsistsofMidentical antennaelements. Aparticular useristreatedastheoneof interest, andtheremaining N-1usersgiverisetoco-channel interference. Inaddition to theco-channel interference, eachcomponent ofthearraysignalprocessor's inputiscor nfptedbyadditivewhiteGaussian noise(AWGN).Theanalysispresented hereinisfor baseband signals,which,ingeneral,arecomplex valued.This,inturn,meansthatboth thechannelandarraysignalprocessor requirecomplex characterizations oftheirown. Thestructure depicted inFigure8.29isdrawnforoneoutputpertaining totheuserof 554 CHAPTER 8"MULTWSER RADIOCOMMUNICATIONS Channel matrix CUserof interest m,(£)o----~ Interfering1m,(£) users mli(t)o-----i~L- -----.J Multipath channel"M(t) Sourcesof AWGNArray signal processor ReceiverArray processor output yet) FIGURE 8.29Blockdiagramofarraysignalprocessor thatinvolves Mantenna elements, and thatisbeingdrivenbyamultipath channel. interest.Thearraysignalprocessor isduplicated forusersatotherfrequencies atthebase station. Themultipathchannelis~haracterized bythechannelmatrix,whichisdenotedby C.ThematrixChasdimensions M-by-N andmaytherefore beexpanded intoNcolumn vectors,asshownby (8.72) whereeachcolumnvectorisofdimension M. Giventheconfiguration described inFigure8.29,thegoalistodesignalineararray signalprocessor forthereceiver, whichsatisfiestworequirements: 1.Theco-channel interference produced bytheN-linterfering usersiscancelled. 2.Theoutputsignal-to-noise ratio(SNR)fortheuserofinterestismaximized. Hereafter, thesetworequirements arereferredtoasdesignrequirements 1and2. Toproceedwiththisdesigntask,itisassumed thatthemultipath channelisdescribed byflatRayleigh fading.Then,inlightofthematerialpresented inSection8.7,wefindthat theuseofdiversity permitsthetreatment ofthecolumnvectors CbC2,•••,CNaslinearly independent, whichisjustifiedprovided thatthespacingbetween antennaelements ofthe arrayislargeenough(e.g.,seventimesthewavelength) forindependent fad.ing.Tosimplify thepresentation, wesupposethatuser1istheuserofinterestandtheremaining N-1 usersareresponsible forco-channel interference, asindicated inFigure8.29.Thekey designissueishowtofindtheweightvectordenotedbyw,whichcharacterizes thearray signalprocessor. Tothatend,wemayproceedasfollows: 1.WechoosetheM-dimensional weightvectorwtobeorthogonal tothevectors C2,•••,CN,whichareassociated withtheinterfering users.Thischoicefulfillsdesign requirement 1(i.e.,cancellation ofco-channel interference). 2.Tosatisfydesignrequirement 2(i.e.,maximization oftheSNR),wewillbrieflydi gressfromtheissueathandtointroduce thenotionofasubspace. Givenavector 8.10Adoptive A..te......ArraysforWireless Comm-.katWns 555 space,orjustspace,formedbyasetoflinearlyindependent vectors,asubspace of thespaceisasubsetthatsatisfiestwoconditions: '4 (i)Ifweaddanytwovectors ZlandZ2inthesubspace, theirsumZlandZ2isstill inthesubspace. (ii)Ifwemultiply anyvector Zinthesubspace byanyscalara,themultipleazis stillinthesubspace. Returning totheissueofhowtomaximize theoutputSNRforuser1,wefirst construct asubspace denotedby'lV,whosedimension isequaltothedifference be tweenthenumberofantennaelements andthenumberofinterfering users,thatis, M -(N-1)=M-N+1.Next,weprojectthecomplex conjugate ofthechannel vector Cl(pertaining touser1)ontothesubspace 'lV.Theprojection socomputed definestheweightvectorw. II>EXAMPLE 8.3 Toillustrate thetwo-step subspace methodfordetermining theweightvectorw,considerthe simpleexampleofasysteminvolving twouserscharacterized bythechannelvectorsc,and C2,andanantennaarrayconsisting ofthreeelements; thatis,N=2andM=3.Then,for thisexample, thesubspace Wistwo-dimensional, asshownby M-N+l=3-2+1=2 Withuser1viewedastheuserofinterestanduser2viewedastheinterferer, wemayconstruct thesignal-space diagramshowninFigure8.30.Thesubspace W,shownshadedinthisligure, isorthogonal tochannelvector C2'Theweightvectorwofthearraysignalprocessor isde termined bytheprojection ofthecomplex-conjugated channelvectorofuser1,thatis,c~, ontothesubspace W,asdepictedinFigure8.30. "'ll Theimportant conclusion drawnfromthisdiscussion isthatalinearreceiverusing optimum combining withMantennaelements andinvolving N-1interfering usershas thesameperformance asalinearreceiverwithM - N+1antennaelements without interference, independent ofthemultipath environment. Forthisequivalence toberealized, ·2(Interferer) .~ (Userofinterest) FIGURE8.30Signal-space diagramforExample 8.3,involving auserofinterest,asingleinter ferer,andanantennaarrayof3elements. Thesubspace W,shownshaded,istwo-dimensional in thisexample. 556 CHAPTER 8 "MULTIUSER RAmo COMMUNICATIONS weofcourserequirethatM>N -1.Provided thatthiscondition issatisfied, thereceiver cancelstheco-channel interference withadiversity improvement equaltoM-N+1 whichrepresents anN-foldincreaseinsystemcapacity. ' Thedesignofanarraysignalprocessor inaccordance withthetwo-step subspace procedure described hereinisofthezero-forcing kind.Wesaysobecause, givenMantenna elements, thearrayhasenoughdegreesoffreedom toforcetheoutputduetotheN-1 interfering usersrepresented bythelinearlyindependent channelvectors ~,...,CMto zerosolongasMisgreaterthanN -1.Notealsothatthisprocedure includes N~1 (i.e.,asingleuserwithnointerfering users)asaspecialcase.Inthiscase, the channel matrixconsistsofvector C1>whichliesinthesubspace'lV,andthezero-forcing solution wequals c~. Theanalysispresented thusfarhasbeenentirelyofaspatialkind,whichignoresthe effectofdelayspread.Whatifthedelayspreadissignificant compared tothesymbol duration andcannottherefore beignored? Recognizing thatdelayspreadisresponsible forintersymbol interference, wemay,inlightofthematerial presented inChapter4on theequalization ofatelephone channel, incorporate alinearequalizer ineachantenna branchofthearraytocompensate fordelayspread.Theresulting arraysignalprocessor takestheformshowninFigure8.31,whichcombines temporal andspatialprocessing. Ele~entI-';l--......~ Antenna arrayFIRfilters FIGURE8.31Baseband space-time processor. Theblockslabeledz-1areunit-delay elements witheachdelaybeingequaltothesymbolperiod.Thefiltercoefficients arecomplexvalued.The FIRfiltersareallassumed tobeoflength1. 8.10AJU,pti"e A..tennaArr..ysforWireless Co..........ic..tums557 Spatialprocessing isprovided bytheantennaarray,andthetemporal processing ispro videdbyabankoffinite-duration impulseresponse (FIR)filters.Forobviousreasons,this structure iscalledaspace-time processor.15 IIIADAPTIVE ANTENNA AImAv Thesubspace designprocedure forthearraysignalprocessor inFigure8.29assumesthat thechannelimpairments arestationary, andthatwehaveknowledge ofthechannelmatrix C.Inreality,however, multipath fading,delayspread,andco-channel interference areall nonstationary intheirownindividual ways.Also,thechannelcharacterization maybe unknown. Todealwiththesepractical issues,weneedtomakethereceiving arraysignal processor inFigure8.29adaptive. Bearinginmindthescopeofthisbook,weconfinethe discussion toadaptive spatialprocessing, assuming thatthedelayspreadisnegligible. We furtherassumethatthemultipath fadingphenomenon isslowenoughtojustifytheleast mean-square (LMS)algorithm toperformtheadaptation. Figure8.32showsthestructure ofanadaptive antennaarray,wheretheoutputof eachantennaelementismultiplied byanadjustable (controllable) weight,andthenthe weighted elemental outputsofthearrayaresummed toproducethearrayoutputsignal. Theadaptive antennaarraydoesnotrequireknowledge ofthedirection ofarrivalofthe desiredsignaloriginating fromauserofinterestaslongasthesystemissupplied witha reference signal,whichiscorrelated withthedesiredsignal.Theoutputsignalofthearray issubtracted fromthereference signaltogenerate anerrorsignal,whichisusedtoapply theappropriate adjustments totheelemental weightsofthearray.Inthisway,afeedback systemtocontroltheelemental weightsisbuiltintotheoperation oftheantennaarray, therebymakingitadaptive tochangesintheenvironment. Notethattheblockdiagram ofFigure8.32isdrawnforbaseband processing, hencethecomplex conjugation ofthe elemental weights.Inapractical system,aquadrature hybridisusedforeachantenna elementofthearraytosplitthecomplex-valued receivedsignalateachelementintotwo components: onerealandtheotherimaginary. Theuseofahybridhasbeenomittedin Figure8.32tosimplifythediagram. Arra~of M antenna elementsArrayoutput yen] Reference signal den] FIGURE8.32Blockdiagram ofadaptive antenna array. (8.74)558 CHAPTER 8illMULTIUSER RADIO COMMUNI(:ATIONS Tooptimize theperformance oftheadaptive antennaarray,itiscustomary toUse themean-square error J=E[ie[n]l2] (8.73) asthecostfunction tobeminimized. Theern]istheerrorsignalattimet=nT,whereT isthesymbolperiodandnisanintegerservingasdiscretetime.Minimization oftheCOst functionJsuppresses theinterfering signalsandenhances thedesiredsignalinthearray output.However, theLMSalgorithm minimizes theinstantaneous valueofthecostfunc tionJand,through successive iterations, itstrivestoreachtheminimum mean-square error(MMSE) (i.e.,optimum solutionfortheelemental weights).Inlightofthediscussion presented inChapter4ontemporal equalizers, whichcarriesovertothespatialdomain wemaysaythatanadaptive antenna arraybasedontheminimum mean-square erro; criterion ishighlylikelytoprovideabettersolutionthanonebasedonthezero-forcing criterion embodied inthetwo-step subspace method. Letxk[n]denotetheoutputofthekthelementinthearrayatdiscretetimen,andlet wk[n]denotethecorresponding valueoftheweightconnected tothiselement.TheOutput signalofthearray(consisting ofMantennaelements) istherefore M y[n]=LwHn]xk[n] k~l whereWan]xk[n] istheinnerproductofthecomplex-valued quantities wk[n]andxkln]. Denoting thereference signalasd[n],wemayevaluatetheerrorsignalas ern]=d[n]-y[n] (8.75) Hence,theadjustment appliedtothekthelemental weightis .6.wk[n]=JLi?*[n]xk[n], k=1,2,..., M (8.76) where/Listhestep-sizeparameter, andtheupdatedvalueofthisweightis wk[n+1]=wk[n]+.6.wk[n], k=1,2,..., M (8.77) Equations (8.74)-(8.77), inthatorder,constitute thecomplex LMSalgorithm, whichin cludestheLMSalgorithm forrealsignals(studiedinChapters 3and4)asaspecialcase. Thealgorithm isinitiated bysettingWk[O]=0forallk.Thederivation ofthecomplex LMSalgorithlfl isposedasProblem 8.19. Theadvantages ofanadaptive antennaarrayusingthecomplex LMSalgorithm are three-fold: II>Simplicity ofimplementation. Lineargrowthincomplexity withthenumberofantennaelements. l'-Robustperformance withrespecttodisturbances. However, thesystemsuffersfromthefollowing drawbacks: I>Slowrateofconvergence, whichistypically tentimesthenumberofweights. This limitstheuseofthecomplex LMSalgorithm toaslow-fading environment, forwhich theDoppler spreadissmallcompared tothereciprocal oftheduration oftheobser· vationinterval. I>Sensitivity oftheconvergence behavior tovariations inthereference signalandco' channelinterference powers. Theselimitations ofthecomplex LMSalgorithm canbeovercome byusinganal· gorithmknownasdirectmatrixinversion (DMI),whichfollowsdirectlyfromtheWiener filterdiscussed inChapter 4;seeProblem 8.21.UnliketheLMSalgorithm, theDMIaJ,. S.HSummary andDiscussion 559 gorithmoperates inthebatchmodeinthatthecomputation oftheelemental weightsis basedonabatchofKsnapshots. ThebatchsizeKischosenasacompromise between twoconflicting requirements: ~ThesizeKshouldbesmallenoughforthebatchofsnapshots usedinthecomputation tobejustifiably treatedaspseudo-stationary. I>ThesizeKshouldbelargeenoughforthecomputed valuesoftheelemental weights toapproach theMMSEsolution. TheDMIalgorithm istheoptimum combining technique forarrayantennas currently deployed inmanybasestationstoday.TheDMIalgorithm maybereformulated forre cursivecomputation,'6 ifsodesired. Whentheteletraffic ishigh,thebasestationsareordinarily configured asmicrocells, whicharesmallcellssuchasanofficefloororastationdeployed alongahighway with directional antennas. Insuchaconfiguration, therearemanyinexpensive basestationsin closeproximity toeachother.Theuseofadaptive antennaarraysprovides themeansfor analternative configuration wheretherearefewer(butmoreexpensive) basestationsand furtherapartfromeachotherthaninthecorresponding microcellular system. L8.11Summary andDiscussion Inthischapter,wediscussed twoimportant typesofmultiuser communications: satellite communications andwireless communications. Satellite communication systemsoffer globalcoverage, whereaswirelesscommunication systemsoffermobility. Theglobalcov erageandmobility offeredbythesetwocommunication systemshaveprofoundly trans formedthewaywecommunicate, bothlocallyandglobally. Although satellitecommunication andwirelesscommunication systemsfunction in entirelydifferent ways,bothrelyonradiopropagation tolinkthereceivertothetrans mitter. In satellitecommunications, wehaveanuplinkfromanearthterminal tothe satellitetransponder andadownlink fromthesatellitetoanotherearthterminal. The satelliteoperates likearepeater inthesky.Moreover, withthesatellitepositioned ina geostationary orbit,theuplinkanddownlink operateasline-of-sight pathsoffixedlengths. Accordingly, thesatellitecommunication channel, encompassing bothoftheselinks,is closelymodeled asanadditivewhiteGaussian noise(AWGN)channel. Thewirelesscommunication systemalsohastwolinksofitsown:anuplink,or reverselink,forthemobile-to-base stationtransmission, andadownlink, orforwardlink, forthebasestation-to-mobile transmission. Thebasestationisfixed,beinglocatedatthe centerorontheedgeofacoverage region;itconsistsofradiochannels, andtransmitter, andreceiverantennas mounted onatower.Threemajorsourcesofdegradation inwireless communications, discussed inthechapter,areco-channel interference, fading,anddelay spread;thelattertwoarebyproductsofmultipath. Acommon characteristic ofthese channelimpairments isthattheyareallsignal-dependent phenomena. Unliketheubiqui touschannelnoise,thedegrading effectsofinterference andmultipath cannottherefore becombatted bysimplyincreasing thetransmitted signalpower.Rather,bothinterference andmultipath requiretheuseofspecialized techniques, tailor-made totheirpatticular physicalcharacteristics. Thesespecialized techniques includediversity, adaptive arrayan tennas,andtheRAKEreceiver. Weclosethediscussion withremarkscontrasting wirelesscommunications towired communications. FromChapter3werecallthatamajorsourceofconcerninwiredcom- 560 CHAPTER 8IIIMULTIUSER RADIO COMMUNICATIONS munication systemsisnoise;thesesystemshavesufficient channel bandwidth topermit theuseofpulse-code modulation (PCM)asthestandard methodforconverting speech intoa64kb/sstream, which provides thebasicdataforanalmostnoise-free performance. Inwirelesscommunications, ontheotherhand,channelbandwidth isaprecious resource, theconservation ofwhichnecessitates theuseofspectrally efficientspeechcodingtech niquestoproduce toll-quality digitized speechatratesthatareasmallfractionofthePCM rate.Unfortunately, the waveform codersexemplified byadaptive differential pulse-code modulation, disClissed inChapter3,donotsatisfythisstringent requirement. Thepreferred approach istousethespectrally efficient source-coding techniques: multi-pulse excited linearpredictive coding(LPC)oritsregular-pulse excitedvariant,andcode-excited LPC (CELP); thesesourcecodingtechniques produce bitratesbelow16kb/sbyremoving almostallofthenaturalredundancy inspeech,whilemaintaining high-quality speech, albeitofasynthetic kind.Toprovideprotection againstnoise,channel codingisused whereby redundant bitsareinsertedintothetransmitted datastreaminacontrolled man ner.Theuseofchannelcodingalsohelpsinotherways:Itextendstherangeoflow-power handsets aswellasbatterylife.Channel codingisdiscussed inChapter 10. INOTES ANDREFERENCES 1.Fordetailedtreatment ofsatellitecommunications andrelatedissues,seethefollowing books:Sklar(1988),PrattandBostian(1986),Wu(1984),Bhargava etal.(1981),and Spilker,Jr.(1977).Thefirst,third,fourth,andfifthbooksemphasize theuseofsatellites fordigitalcommunications. ThebookbyPrattandBostianpresentsabroadtreatment of satellitecommut:lications, emphasizing suchdiversetopicsasradio-wave propagation, an tennas,orbitalmechanics, signalprocessing, andradioelectronics. 2.Linkbudgetanalysisisdiscussed inthebooksbySklar(1988)andAnderson (1999);for satellitecommunications, itisdiscussedinBhargava etal.(1981). 3.Forthefundamentals ofantennas, seethebookbyKraus(1950)andChapter11ofthe bookbyJordanandBalmain(1973). 4.Thefree-space equation (Equation 8.14)isnamedinhonorofFriis(1946).Fortheorigin oftheFriisformulaofEquation (8.30),seeFriis(1944). 5.Foranoriginaltreatment ofcellularradio,seethepaperbyMacDonald (1979). 6.Foracomprehensive treatment ofthemobileradiopropagation channel, seethebookby Parsons(1992).Thisbookpresentsthefundamentals ofVHFandUHFpropagation, prop agationoverirregular terrainandinbuilt-upareas,andastatistical characterization ofthe mobileradiochannel. Thestatistical characterization ofamobileradiochannelisalso discussed inProakis(1995).Thisbookprovides areadable accountoftheeffectoffading ontheerrorperformance ofRayleigh fadingchannels andagooddiscussion ofdiversity techniques. Forafulltreatment ofthesubject,seeChapters 9-11bySteininthebook editedbySchwartz, Bennett,andStein(1966). 7.Thechi-square distribution isaspecialcaseofthegammadistribution. Theprobability densityfunction ofagamma-distributed randomvariableXhastwoparameters: a:>a andA>0;itisdefinedby O<x<oo NotesandRefenmces 561 wheref(a)isthegammafunction, whichisitselfdefinedby Thegammafunction hasthefollowing properties: f(1/2)=y:;;: f(a+1)=af(a),a>0 Byletting,\=1/2anda=k12,wherekisapositiveinteger,wegetthechi-square distri butionwith2kdegreesoffreedom, asshownby 8.Forasurveyarticleontheevolution ofwirelesscommunications, seeOliphant (1999).For booksonthefundamentals ofwireless communication systems, seeSteeleandHanzo (1999),StUber(1996)andRappaport (1996). Foradetaileddescription ofGSM,seeChapter 8ofthebookbySteeleandHanzo (1999).Foradetaileddescription oftheIS-95system,seethehandbook byLeeandMiller (1998). 9.TheclassicpaperontheRAKEreceiverisduetoPriceandGreen(1958). 10.Fortheoriginalpaperonhowtomaximize thesignal-to-noise ratiorealizable fromthe sumofseveralnoisysignals,seetheclassicpaperbyBrennan (1955). 11.Theapplication oftheRAKEreceiverinCDMAwirelesscommunication systemsisdis cussedindetailinthebookbyViterbi(1995). 12.Theideaofmulti-pulse excitation forspeechcodingisduetoAtalandRemde(1982). Code-excited linearprediction (CELP)ofspeechwasfirstintroduced byAtalandSchroeder (1984).Foradetailedmathematical discussion ofmulti-pulse excited,regular-pulse ex cited,andcode-excited typesofspeechcoding,particularly astheyrelatetowirelesscom munications, seeChapter3inthebookeditedbySteeleandHanzo(1999). 13.Inthewirelesscommunications literature, adaptive antennaarraysareoftenreferredtoas smartantennas. Foranoverview ofthevariousissuesinvolvedintheuseofadaptive antenna arraysforwireless communications, seethearticlebyWinters (1998)andthe coursenotesbyWinters(1999).Thetwo-step subspace procedure for designing thearray signalprocessor inFigure8.29isbasedonmaterialpresented inWinters(1999).Thebook byRappaport (1999)presents acollection ofpapersonadaptive antenna arrays,which aregrouped intoalgorithms, architectures, hardware applications, channelmodels,and performance evaluation. 14.Theideaofsubspace isrootedinmatrixalgebra.Foradiscussion ofthisidea,seeStrang (1980)andStewart(1973).Foradiscussion ofsubspace decomposition inthecontextof statistical signalprocessing, seeScharf(1991). 15.Fortutorialdiscussions ofspace-time processing forwirelesscommunications, seethear ticlesbyPaulrajandNg(1998),PaulrajandPapadias (1997),andKohno(1998). 16.Recursive implementation oftheDMIalgorithm leadstoanewalgorithm commonly re ferredtoastherecursive leastsquares(RLS)algorithm; foraderivation oftheRLSalgo rithmanditsvariants, seeHaykin(1996). = 1dBw =4GHz562 CHAPTER 8IIIMULTIUSER RADIO COMMUNICATIONS IPROBLEMS Free-Space Propagation 8.1Aradiolinkusesapairof2mdishantennas withanefficiency of60percenteach,as transmitting andreceiving antennas. Otherspecifications ofthelinkare: Transmitted power Carrierfrequency Distance ofthereceiver fromthetransmitter =150m Calculate (a)thefree-space loss,(b)thepowergainofeachantenna, and(c)thereceived powerindBW. 8.2RepeatProblem 8.1foracarrierfrequency of12GHz. 8.3Equation (8.14)isoneformulation oftheFriisfree-space equation. Showthatthisequa tioncanalsobeformulated inthefollowing equivalent forms: ( )P,AA aP,=A2d2 (b)P=P,AtG,,47Td2 wherePtisthetransmitted power,Atistheeffective areaofthetransmitting antenna, A isthecarrierwavelength, disthedistanceofthereceiverfromthetransmitter, G,isthe powergainofthereceiving antenna,A,istheeffectiveareaofthereceiving antenna,and P,isthereceivedpower. Discussthesituations thatfavortheuseofoneoftheseequations overtheother. 8,4Fromthemathematical definition ofthefree-space loss weseethatitisdependent onthecarrierwavelength Aorfrequencyf.Howcanthis dependence onwavelength orfrequency bejustifiedinphysicalterms? 8.5Inasatellitecommunication system,thecarrierfrequency usedontheuplinkisalways higherthanthecarrierfrequency usedonthedownlink. Justifytherationale forthis choice. 8.6Acontinuous-wave (CW)beacontransmitter islocatedonasatelliteingeostationary orbit.Thebeacon's 12GHzoutputismonitored byanearthstationpositioned 40,000 kmfromthesatellite. Thesatellitetransmitting antenna isa1mdishwithanaperture efficiency of70percent,andtheearthstationreceiving antenna isa10mdishwithan aperture efficiency of55percent.Calculate thereceived power,giventhatthebeacon's outputpoweris100mW. NoiseFigure 8.7Consider a75-ilresistormaintained at"roomtemperature" of290K.Assuming aband' widthof1MHz,calculate thefoliowing: (a)Theroot-mean-square (RMS)valueofthevoltageappearing acrosstheterminals of thisresistorduetothermalnoise. (b)Themaximum available noisepowerdelivered toamatched load. Problems 563 8.8Inthisproblem, we revisit Example 8.1basedonthereceiverconfiguration ofFigure 8.10.Suppose thatalossywaveguide isinsertedbetween thereceiving antennaandthe low-noise amplifier. Thewaveguide lossis1dB,anditsphysical temperature is290K. Recalculate theeffectivenoisetemperature ofthereceiver. 8.9Consider thereceiverofFigureP8.9,whichconsistsofalossywaveguide, low-noise RF amplifier, frequency down-converter (mixer),andIFamplifier. Thefigureincludes the noisefiguresandpowergainsofthesefourcomponents. Theantennatemperature is50K. (a)Calculate theequivalent noisetemperature foreachofthefourcomponents inFigure P8.9,assuming aroomtemperature T=290K. (b)Calculate theeffective noisetemperature ofthewholereceiver. F=1.7 G=10F=3 G=5F=5 G=5,000Output 14GHz -81dBW/m2 1.9dBIKFIGlJREP8.9 BudgetLinkCalculations 8.10Inthisproblem weaddresstheuplinkpowerbudgetofthedigitalsatellitecommunication systemconsidered inExample 8.2.Theparameters ofthelinkareasfollows: Carrierfrequency PowerdensityattheTWT amplifier insaturation Satellitefigureofmerit,G/T Distance ofthesatellitefromthe transmitting earthterminal =40,000km (a)Assuming nopowerbackoffoftheTWT,calculate theCINaratioatthesatellite. (b)Giventhatthedatarateintheuplinkisthesameasthatcalculated forthedownlink inExample 8.2,calculate theprobability ofsymbolerrorincurred intheuplinkal lowingforalinkmarginof6dB.Compare yourresultwiththatinExample 8.2. 8.11Thedownlink ClNaratioinadirectbroadcast satellite(DBS)systemisestimated tobe 85dB-Hz.Thespecifications ofthelinkare: SatelliteEIRP =57dBW Downlink carrierfrequency =12.5GHz Datarate =10Mb/s Required ErJNaatthereceiving earthterminal =10dB Distance ofthesatellitefromthereceiving earthterminal=41,000km Calculate theminimum diameter ofthedishantennaneededtoprovideasatisfactory TV reception, assuming thatthedishhasanefficiency of55percentanditislocatedalongside thehoinewherethetemperature is310K.Forthiscalculation, assumethattheoperation oftheDBSsystemisessentially downlink-limited. 564 CHAPTER 8IIIMULTIUSER RADIO COMMUNICATIONS Wireless Communications 8.12Bothwirelesscommunications andsatellitecommunications relyonradiopropagation fortheiroperations. Summarize (a)thesimilarities ofthesetwomultiuser communication systems, and(b)themajordifferences thatdistinguish themfromeachother. 8.13Inwirelesscommunication systems, thecarrierfrequency ontheuplink(reverselink)is smallerthanthecarrierfrequency onthedownlink (forward link).Justifytherationale forthischoice. 8.14FigureP8.14depictsthedirect(line-of-sight) andindirect(reflected) pathsofaradiolink operating overaplaneearth.Theheightsofthetransmitting antennaatthebasestation andthereceiving antenna ofamobileunitarehbandhm,respectively. Assumethe following: ~Thereflection coefficient ofthegroundis-l. ~Thedistancedbetweenthetwoantennas islargeenoughtomakethephasedifference '" betweenthereflected anddirectpathssmallcompared to1raillan,sothatwemayset sin</>=</>. Hence,showthatthereceived powerP,isgivenbytheapproximation whereP,isthetransmitted power,andGbandGmarethepowergainsofthetransmitting baseandmobileantennas, respectively. Compare thisresultwiththeFriisfree-space equation. FIGUREP8.14 8.15Thetwo-path modeldefinedbytheimpulseresponse h(t)=al8(t-7"1)+azexp(-je) 8(t-7"z) isfrequently usedintheanalytictreatment ofwirelesscommunication systems.Themodel parameters arethedelaytimes 7"1and7"z,theuniformly distributed phasee,andthereal coefficients alandaz. (a)Determine (i)thetransferfunction ofthemodel,and(ii)itspower-delay profile. (b)Showthatthemodelexhibitsfrequency-selective fadingduetovariations intheco efficients alandaz. 8.16IntheRAKEreceiverillustrated inFigure8.26,eachcorrelator issynchronized byin sertingtherightdelayintothereceived signal. (a)Showthat,intheory,thesameresultisobtained byinserting therightdelayintothe reference signal(i.e.,pseudo-noise sequence). (b)Inpractice, thepreferred methodistousetheprocedure described inFigure8.26. Whatreasoncanyousuggestforthispreference? Problems 565 8.17Inthisproblem westudythemaximal-ratio combining diversity scheme.Toproceed, consider asetofnoisysignals[Xj(t)}):" whereXj(t)isdefinedby Xj(t)=Sj(t)+nAt),i=1,2,..., N Assumethefollowing: I»Thesignalcomponents Sj(t)arelocallycoherent, thatis, i=1,2,..., N wheretheZjarepositiverealnumbers, andm(t)denotesamessagesignalwithunit power. I>Thenoisecomponents n;(t)havezeromean,andtheyarestatistically independent, thatis, fork=i otherwise Theoutputofthelinearcombiner isdefinedby N x(t)=L,,;xAt) j=l wheretheparameters ex;aretobedetermined. (a)Showthattheoutputsignal-to-noise ratiois (b)Set(SNR)o=(~ajz;r N 2:afar i=J Uj=a,ai v.=!i. Ju; andreformulate theexpression for(SNR)o. Hence,applying theSchwarz inequality tothisreformulation, showthat N (i)(SNR)o :5L(SNR); j=1 where(SNR)j=zj/ar. (ii)Theoptimum valuesofthecombiner's coefficients aredefinedby Zj Ctj=aJ inwhichcasetheSchwarz inequality issatisfied withtheequality sign. TheSchwarz inequality isdiscussed inSection5.2. Adaptive Antenna Arrays 8.18Consider thearraysignalprocessor ofFigure8.29wherethereareonlytwousers(N= 2)andthearrayconsistsoftwoelements (M=2).Construct thesubspace Wforthis problem. Hence,usingasignal-space diagram, illustrate thecomputation oftheweight characterizing thearraysignalprocessor. 566 CHAPTER 8"MULTIUSER RADIO COM1UUNICATIONS 8.19Inthisproblem wederivethecomplex LMSalgorithm. Referring toFigure8.32and startingwiththeinstantaneous costfunction 1]=-le[n]12 2 wheree[n]istheerrorsignalandMisthenumberofantennaelements, dothefollowing: (a)Determine thederivative ofthecostfunction] withrespecttothekthelemental weight wk[n]. (b)Usingtheinstantaneous derivative a]lawkl.n], denoted byV![k],determine thead justment.lwk[n]madetothekthelemental weightinaccordance withtherule Ilwk[n] =-JLV][k] (c)Verifythecomposition ofthecomplex LMSalgorithm described inEquations (8.75) to(8.77). Notethatwk[n]iscomplex valued,andyouneedtoconsider itsrealandimaginary partsseparately. 8.20Apractical limitation ofanadaptive antennaarrayusingtheLMSalgorithm isthedy namicrangeoverwhichthearraycanoperate.Thislimitation isduetothefactthatthe speedofresponse oftheweightsintheLMSalgorithm isproportional totheaverage signalpoweratthearrayinput. (a)Justifytheassertion thatthedynamic rangeofaveragesignalpoweratthearrayinput isproportional toRblfm;"whereRbisthedatarateinblsandfm""isthemaximum faderateinHz. (b)Assuming aproportionality factorof0.2,bywhichtheratioRblf=xisscaled,cal culatethedynamic rangeofanadaptive antennaarrayusingtheLMSalgorithm for Rb=32kb/sandfm.x=70Hz.Comment onyourresult.(Theproportionality factor of0.2isareasonable choiceforsystemsusingPSK.) 8.21Inthisproblem we derive thedirectmatrixinversion algorithm foradjusting theweights ofanadaptive antenna array.Todoso,werevisitthederivation oftheWienerfilter presented inChapter3. (a)Showthat Rxw=rxd whereItisanestimate ofthecorrelation matrixoftheinputvectorx[k]: 1K Rx=K~,x[k]r[k] andrXdisanestimate ofthecross-correlation vectorbetweenx[k]andthereference signald[k]: 1K Cxd=K~,x[k]d'[k] Thesuperscript HintheformulaforItdenotesHermitian transportation (i.e.,trans position andcomplex conjugation), sox[k]r[k] denotestheouterproductofx[k] withitself.Thesummations forbothRxandCxdareperformed overatotalofK snapshots, witheachsnapshot beingrepresented bythepair{x[k],d[k]}. (b)Usingtheformulas ofpart(a),describeanalgorithm forcomputing theweightvector w,givenadatasetconsisting ofKsnapshots. Hencedemonstrate thatthecomplexity ofthisalgorithm growsasM3withthesizeoftheweightvectorwdenotedbyM. FUNDAMENTAL LIMITS ININFORMATION THEORY Shannon's landmark paperoninformation theoryin1948,anditsrefinements byother researchers, wereindirectresponse totheneedofelectrical engineers todesign communication systemsthatarebothefficientandreliable.Efficient communication from asourcetoauserdestination isattainedthroughsourcecoding. Reliable communication overanoisychannelisattainedthrougherror-control coding.Thischapteraddresses these important issuesassummarized here: ~Entropyasthebasicmeasureofinformation. ~Sourcecodingtheoremanddatacompaction algorithms. ~Mutualinformation anditsrelationtothecapacityofacommunication channelfor information transmission. ~Channelcodingtheorem asthebasisforreliablecommunication. ~Information capacitytheoremasthebasisforatradeoffbetweenchannelbandwidth and signal-to-noise ratio. ~Rate-distortion theoryforsourcecodingwithafidelitycriterion. I9.1Introduction Asmentioned intheBackgroWld andPreviewchapterandreiterated alongtheway,the purposeofacommunication systemistocarryinformation-bearing baseband signalsfrom oneplacetoanotheroveracommunication channel. Inpreceding chapters ofthebook, wehavedescribed avarietyofmodulation schemesforaccomplishing thisobjective. But whatdowemeanbytheterminformation? Toaddressthisissue,weneedtoinvoke information theory.'Thisbroadlybasedmathematical discipline hasmadefundamental contributions, notonlytocommunications, butalsotocomputer science,statistical phys ics,statistical inference, andprobability andstatistics. Inthecontextofcommunications, information theorydealswithmathematical mod elingandanalysisofacommWlication systemratherthanwithphysicalsourcesandphys icalchannels. Inparticular, itprovides answerstotwofundamental questions (among others): I»Whatistheirreducible complexity belowwhichasignalcannotbecompressed? I>Whatistheultimate transmission rateforreliablecommunication overanoisy channel? 567 568 CHAPTER 9..FUNDAMENTAL LIMITS ININFORMATION THEORY Theanswerstothesequestions lieintheentropyofasourceandthecapacityofachanne~ respectively. Entropyisdefinedintermsoftheprobabilistic behavior ofasourceofinfor_ mation;itissonamedindeference totheparalleluseofthisconceptinthermodynamics . .Capacity isdefinedastheintrinsicabilityofachanneltoconveyinformation; itisnaturally relatedtothenoisecharacteristics ofthechannel. Aremarkable resultthatemergesfrom information theoryisthatiftheentropyofthesourceislessthanthecapacity ofthe channel, thenerror-free communication overthechannelcanbeachieved. Itistherefore befitting thatwebeginourstudyofinformation theorybydiscussing therelationships amonguncertainty, information, andentropy. I9.2Uncertainty, lnformatitm, andEntropy Supposethataprobabilistic experiment involvestheobservation oftheoutputemittedbv adiscretesourceduringeveryunitoftime(signaling interval). Thesourceoutputismod eledasadiscreterandomvariable, S,whichtakesonsymbolsfromafixedfinitealphabet withprobabilities(9.1) k=0,1,...,K-1 (9.2) Ofcourse,thissetofprobabilities mustsatisfythecondition (9.3) (9.4)Weassumethatthesymbols emittedbythesourceduringsuccessive signaling intervals arestatistically independent. Asourcehavingtheproperties justdescribed iscalledadis cretememorylesssource,memoryless inthesensethatthesymbolemittedatanytimeis independent ofprevious choices. Canwefindameasure ofhowmuchinformation isproduced bysuchasource?To answerthisquestion, wenotethattheideaofinformation iscloselyrelatedtothatof uncertainty orsurprise, asdescribed next. Consider theeventS=Sk>describing theemission ofsymbol Skbythesourcewith probability Pk'asdefinedinEquation (9.2).Clearly,iftheprobability Pk=1andPi=0 foralli*k,thenthereisno"surprise," andtherefore no"information," whensymbol Skisemitted,becauseweknowwhatthemessagefromthesourcemustbe.If,ontheother hand,thesourcesymbolsoccurwithdifferent probabilities, andtheprobability Pkislow, thenthereismoresurprise, andtherefore information, whensymbol Skisemittedbythe sourcethanwhensymbol s"i*k,withhigherprobability isemitted. Thus,thewords uncertainty, surprise, andinformation areallrelated.BeforetheeventS=Skoccurs,there isanamountofuncertainty. WhentheeventS=Skoccursthereisanamountofsurprise. Aftertheoccurrence oftheeventS=Sk>thereisgainintheamountofinformation, the essenceofwhichmaybeviewedastheresolution ofuncertainty. Moreover, theamount ofinformation isrelatedtotheinverseoftheprobability ofoccurrence. Wedefinetheamountofinformation gainedafterobserving theeventS=Sk>which occurswithprobability Pk>asthelogarithmic function2 I(sk)=10g(~) 9.2Uncerlainty, InfonnatUm, andEntropy 569 Thisdefinition exhibitsthefollowing important properties thatareintuitively satisfying: 1. forPk=1 (9.5) 2.Obviously, ifweareabsolutely certainoftheoutcome ofanevent,evenbeforeit occurs,thereisnoinformation gained. (9.6) 3.Thatistosay,theoccurrence ofaneventS=Skeitherprovides someornoinfor mation,butneverbringsaboutalossofinformation. forPk<Pi (9.7) Thatis,thelessprobable aneventis,themoreinformation wegainwhenitoccurs. 4.I(SkSt)=I(Sk)+I(Si)ifSkandSiarestatistically independent. Thebaseofthelogarithm inEquation (9.4)isquitearbitrary. Nevertheless, itisthe standard practicetodaytousealogarithm tobase2.Theresulting unitofinformation is calledthebit(acontraction ofbinarydigit).Wethuswrite I(Sk)=logz(p~) =-logzPkfork=0,1,...,K-1(9.8) WhenPk=1/2,wehaveI(Sk)=1bit.Hence,onebitistheamountofinformation that wegainwhenoneoftwopossibleandequallylikely(i.e.,equiprobable) eventsoccurs. Notethattheinformation I(Sk)ispositive, sincethelogarithm ofanumberlessthanone, suchasaprobability, isnegative. Theamountofinformation I(Sk)produced bythesourceduringanarbitrary signaling intervaldepends onthesymbol Skemittedbythesourceatthattime.Indeed,I(Sk)isa discreterandomvariablethattakesonthevaluesI(so),I(sIl,...,I(SK-I)withprobabilities Po,PI>...,PK-Irespectively. ThemeanofI(Sk)overthesourcealphabetgisgivenby (9.9) Theimportant quantityHW)iscalledtheentropy' ofadiscretememoryless sourcewith sourcealphabetg.Itisameasure oftheaverageinformation contentpersourcesymbol. NotethattheentropyHW)depends onlyontheprobabilities ofthesymbols intheal phabetgofthesource.ThusthesymbolginHW)isnotanargument ofafunction but ratheralabelforasource. 570 CHAPTER 9.,FUNDAMENTAL LIMITS ININFORMATION THEORY !iiiSOMEPROPERTIES OFENTROPY Consider adiscretememoryless sourcewhosemathematical modelisdefinedbyEquations (9.1)and(9.2).TheentropyH(9')ofsuchasourceisbounded asfollows: o:5H(9'):5log2K (9.10) whereKistheradix(number ofsymbols) ofthealphabet g'ofthesource.Furthermore wemaymaketwostatements: ' 1.H(9')='0,ifandonlyiftheprobability Pk=1forsomek,andtheremaining probabilities inthesetareallzero;thislowerboundonentropycorresponds toflO uncertainty . 2.H(9')=log2K,ifandonlyifPk=11Kforallk(i.e.,allthesymbols inthealphabet g'areequiprobable); thisupperboundonentropy corresponds tomaximum uncertainty . Toprovetheseproperties ofH(g'),weproceedasfollows. First,sinceeachproba bilityPkislessthanorequaltounity,itfollowsthateachtermPklog2(1IPk) inEquation (9.9)isalwaysnonnegative, andsoH(9')2:O.Next,wenotethattheproductterm Pklog2(1IPk) iszeroif,andonlyif,Pk=0or1.Wetherefore deducethatH(g')=0if, andonlyif,Pk=0or1,that i~,P"=1forsomekandalltherestarezero. Thiscompletes theproofsofthelowerboundinEquation (9.10)andstatement (1). ToprovetheupperboundinEquation (9.10)andstatement (2),wemakeuseofa property ofthenaturallogarithm: logx:5x-I, x2:0 (9.11) Thisinequality canbereadilyverifiedbyplottingthefunctions logxand(x-1)versus x,asshowninFigure9.1.Hereweseethattheliney=x1alwaysliesabovethecurve y=logx.Theequalityholdsonlyatthepointx=1,wherethelineistangential tothe curve. FIGURE9.1Graphsofthefunctions x1andlogxversusx. 9.2Vncertainty, Informafion, atulEntropy 571 Toproceed withtheproof,consider fustanytwoprobability distributions {Po,Pi>...,PK-l}and{qo,q"...,qK-donthealphabet 9'=(so,51>...,sK-dofa discretememoryless source.Then,changing tothenaturallogarithm, wemaywrite K-l (qk) 1K-l (qk)LPklog2-=-I2LPklog k~O Pkog k~O Pk Hence,usingtheinequality ofEquation (9.11),weget ~1Pklog2(Cf.!!.) .,;::_1_~lPk(qk_1) k~O Pklog2k~OPk 1K-l .,;::-I2L(qk-Pk)og k~O 1(K-lK-l) .,;::--LqkLPk=0log2k~O k~O Wethushavethefundamental inequality ~1PkIOg2(Cf.!!.) .,;::0 k~O Pk wheretheequalityholdsonlyifqk=Pkforallk. Suppose wenextput(9.12) (9.13) k=0,1,..., K - 11 qk=Fe' whichcorresponds toanalphabet 9'withequiprobable symbols. Theentropyofadiscrete memoryless sourcewithsuchacharacterization equals (9.14)K-l(1)Lqklog2-=lo~K k~O qk Also,theuseofEquation (9.13)inEquation (9.12)yields K-l(1)LPklog2-.,;::log2K k~O Pk Equivalently, theentropyofadiscretememoryless sourcewithanarbitrary probability distribution forthesymbolsofitsalphabet 9'isbounded as H(9'):Slog2K ThusH(9')isalwayslessthanorequaltolog2K.Theequalityholdsonlyifthesymbols inthealphabet 9'areequiprobable, asinEquation (9.13).Thiscompletes theproofof Equation (9.10)andstatements (1)and(2). ExAMPLE 9.1EntropyofBinaryMemoryless Source Toillustrate theproperties ofH(9'),weconsider abinarysourceforwhichsymbol0occurs withprobability Poandsymbol1withprobability p,=1 -Po.Weassumethatthesource ismemoryless sothatsuccessive symbolsemittedbythesourcearestatistically independent. (9.15)572 CHAPTER 910FUNDAMENTAL LIMITS ININI'ORMATION THEORY 1.0------- 0.8 ~0.6 0.4 0.2 000.2 0.4 0.50.6 0.8 Symbolprobability. Po FIGURE9.2Entropyfunction 'Jf(Po). Theentropyofsuchasourceequals H(g)=-Polog2Po-Pilog2Pi -Polog2Po(1-Po)log2(1-Po)bits fromwhichweobservethefollowing: 1.WhenPo=0,theentropy H(9') 0;thisfollowsfromthefactthatxlogx--->0as x--->O. 2.WhenPo=1,theentropyH(.'f)=O. 3.TheentropyH(&')attainsitsmaximum value,Hmax=1bit,whenPi=Po=112,that is,symbols1and0areequallyprobable. Thefunction ofPogivenontheright-hand sideofEquation (9.15)isfrequently en countered ininformation-theoretic problems. Itistherefore customary toassignaspecial symboltothisfunction. Specifically, wedefine 'Jf(Po)=-Polog2Po-(1-Po)log2(1-Po) (9.16) Wereferto'Jf(Po)astheentropyfunction. Thedistinction betweenEquation (9.15)andEqua tion(9.16)shouldbecarefully noted.TheH(9')ofEquation (9.15)givestheentropyof adiscretememoryless sourcewithsourcealphabet g.The'Jf(Po)ofEquation (9.16),on theotherhand,isafunctionofthepriorprobability Podefinedontheinterval[0,1].Accord ingly,wemayplottheentropyfunction 'Jf(Po)versusPo,definedontheinterval[0,1],as inFigure9.2.ThecurveinFigure9.2highlights theobservations madeunderpoints1,2, and3. -<l !l!lExTENSION OFADISCRETE MEMORYLESS SOURCE Indiscussing information-theoretic concepts, weoftenfinditusefultoconsider blocks ratherthanindividual symbols, witheachblockconsisting ofnsuccessive sourcesymbols. Wemayvieweachsuchblockasbeingproduced byanextended sourcewithasource alphabet gmthathasKndistinctblocks,whereKisthenumber ofdistinctsymbols inthe sourcealphabet g'oftheoriginal source.Inthecaseofadiscretememoryless source,the sourcesymbols arestatistically independent. Hence,theprobability ofasourcesymbolin gnisequaltotheproduct oftheprobabilities ofthensourcesymbols ing'constituting theparticular sourcesymboling'n.Wemaythusintuitively expectthatH(g'"),theentropy 9.2Uncertainty, Information, andEntropy 573 oftheextended source,isequaltontimesH(g'),theentropyoftheoriginal source.That is,wemaywrite H(:J")=nH(g') (9.17) ~EXAMPLE 9.2EntropyofExtended Source Consider adiscretememoryless sourcewithsourcealphabet [f=(so,51>sz)withrespective probabilities Po=i PI=i P2=! Hence,theuseofEquation (9.9)yieldstheentropyofthesourceas HW)=Polog2(~)+P,log2(f,)+P210gZ(f,) 1 1 1 ="410g2(4)+"4logz(4) +2:logz(2) 3b'=:2Its Consider nextthesecond-order extension ofthesource.Withthesourcealphabet [f consisting ofthreesymbols, itfollowsthatthesourcealphabet [f2oftheextended sourcehas ninesymbols. ThefirstrowofTable9.1presentstheninesymbolsof[f2,denotedas<To, <T"..•,<T8'Thesecondrowofthetablepresentsthecomposition oftheseninesymbolsin termsofthecorresponding sequences ofsourcesymbols so,5"and52,takentwoatatime. Theprobabilities oftheninesourcesymbolsoftheextended sourcearepresented inthelast rowofthetable.Accordingly, theuseofEquation (9.9)yieldstheentropyoftheextended sourceas 8 1 H([fz)=~P(<Ti)logzP(<Ti) 1 1 1 1 =16IOg2(16)+16IOg2(16)+glogz(8)+16log2(16) 1 1 1 1 1+16logz(16)+gIOg2(8)+gIOg2(8)+glog2(8)+4Iog2(4) =3bits WethusseethatH([f2)=2HW)inaccordance withEquation (9.17). TABLE9.1Alphabet particulars ofsecol1d-order extel1siol1 ofadiscretememoryless source , ;;:1 '81 '8..L 16, 16, '8<Tz , 16, 16Symbols of[f2 <To <T, Corresponding sequences 5050 50S, ofsymbolsof[f Probabiliry P(<Ti), i=0,1,...,8 574 CHAPTER 9IIIFUNDAMENTAL LIMnS ININFORMATION THEORY I9.3Source-Coding TheorenJ Animportant problem incommunications istheefficientrepresentation ofdatagenerated byadiscretesource.Theprocessbywhichthisrepresentation isaccomplished iscalled sourceencoding. Thedevicethatperforms therepresentation iscalledasourceencoder. Forthesourceencodertobeefficient, werequireknowledge ofthestatistics oftheSOurce. Inparticular, ifsomesourcesymbols areknowntobemoreprobable thanothers,then wemayexploitthisfeatureinthegeneration ofasourcecodebyassigning shortcode wordstofrequent sourcesymbols, andlongcodewordstoraresourcesymbols. Werefer tosuchasourcecodeasavariable-length code.TheMorsecodeisanexample ofavariable_ lengthcode.IntheMorsecode,thelettersofthealphabet andnumerals areencoded into streamsofmarksandspaces,denoted asdots"."anddashes"-",respectively. Inthe Englishlanguage, theletterEoccursmorefrequently thantheletterQ,forexample, so theMorsecodeencodesEintoasingledot".",theshortestcodewordinthecode,and itencodesQinto"-- .-",thelongestcodewordinthecode. Ourprimaryinterestisinthedevelopment ofanefficientsourceencoderthatsatisfies twofunctional requirements: 1.Thecodewordsproduced bytheencoderareinbinaryform. 2.Thesourcecodeisuniquely decodable, sothattheoriginalsourcesequence canbe reconstructed perfectly fromtheencoded binarysequence. Consider thentheschemeshowninFigure9.3,whichdepictsadiscretememoryless sourcewhoseoutput Skisconverted bythesourceencoder intoablockofOsand1s, denotedbybk•Weassumethatthesourcehasanalphabet withKdifferent symbols, and thatthekthsymbol Skoccurswithprobability Pk>k=0,1,...,K1.Letthebinary codewordassignedtosymbol SkbytheencoderhavelengthIk>measured inbits.Wedefine theaveragecode-word length,I,ofthesourceencoderas K-l I=2:Pklk (9.18) k~O Inphysicalterms,theparameterIrepresents theaveragenumberofbitspersources)lmbol usedinthesourceencoding process. LetLm;ndenotetheminimum possible valueofr. Wethendefinethecodingefficiency ofthesourceencoderas Lrnin 11=-=-L(9.19) WithI2:Lm;",weclearlyhave11s;1.Thesourceencoderissaidtobeefficientwhen17 approaches unity. Buthowistheminimum valueLm;ndetermined? Theanswertothisfundamental question isembodied inShannon's firsttheorem: thesource-coding theorem,4 whichmay bestatedasfollows: GivenadiscretememorylesssourceofentropyH(:fl,theaveragecode-word length Iforanydistortionless sourceencoding schemeisbounded as I2:H(9') (9.20) b"Binary ----;...sequence FIGURE9.3Sourceencoding. 9.4DataCompaction 575 (Aproofofthistheorem foraparticular classofsourcecodesispresented inthenext section.) According tothesource-coding theorem, theentropyH(Ef)represents afunda mentallimitontheaveragenumberofbitspersourcesymbolnecessary torepresent a discretememoryless sourceinthatitcanbemadeassmallas,butnosmallerthan,the entropyH(Ef).ThuswithLmin=H(Ef),wemayrewritetheefficiency ofasourceencoder intermsoftheentropyH(Ef)as 19.4DataCompactionH(Ef) T/=----=-L(9.21) Acommon characteristic ofsignalsgenerated byphysical sourcesisthat,intheirnatural form,theycontainasignificant amountofinformation thatisredundant, thetransmission ofwhichistherefore wasteful ofprimary communication resources. Forefficientsignal transmission, theredundant information shouldberemoved fromthesignalpriortotrans mission. Thisoperation, withnolossofinformation, isordinarily performed onasignal indigitalform,inwhichcasewerefertoitasdatacompaction orlosslessdatacompression. Thecoderesulting fromsuchanoperation provides arepresentation ofthesourceoutput thatisnotonlyefficientintermsoftheaveragenumberofbitspersymbolbutalsoexact inthesensethattheoriginaldatacanbereconstructed withnolossofinformation. The entropyofthesourceestablishes thefundamentallirnit ontheremovalofredundancy from thedata.Basically, datacompaction isachieved byassigning shortdescriptions tothemost frequent outcomes ofthesourceoutputandlongerdescriptions tothelessfrequent ones. Inthissection,wediscusssomesource-coding schemes fordatacompaction. We beginthediscussion bydescribing atypeofsourcecodeknownasaprefixcode,whichis notonlydecodable butalsooffersthepossibility ofrealizing anaveragecode-word length thatcanbemadearbitrarily closetothesourceentropy. IIIPREFIX CODING Consider adiscrete memoryless sourceofalphabet {so,SIo"" SK-rlandstatIstics {Po,Pb...,PK-tl.Forasourcecoderepresenting theoutputofthissourcetobeof practical use,thecodehastobeuniquely decodable. Thisrestriction ensuresthatforeach finitesequence ofsymbolsemittedbythesource,thecorresponding sequence ofcodewords isdifferent fromthesequence ofcodewordscorresponding toanyothersourcesequence. Wearespecifically interested inaspecialclassofcodessatisfying arestriction known astheprefixcondition. Todefinetheprefixcondition, letthecodewordassigned to sourcesymbol Skbedenoted by(mkj,mk2,•••,mkJ,wheretheindividual elements mk j,••• ,mknareOsand1s,andnisthecode-word length.Theinitialpartofthecode wordisrepresented bytheelements mk "...,.1Jlkiforsomei,;:;n.Anysequence madeup oftheinitialpartofthecodewordiscalledaprefixofthecodeword.Aprefixcodeis definedasacodeinwhichnocodewordistheprefixofanyothercodeword. Toillustrate themeaning ofaprefixcode,consider thethreesourcecodesdescribed inTable9.2.CodeIisnotaprefixcodesincethebit0,thecodewordforso,isaprefix of00,thecodewordforS2'Likewise, thebit1,thecodewordfors"isaprefixof11,the codewordforS3'Similarly, wemayshowthatcodeIIIisnotaprefixcode,butcodeIIis. Todecodeasequence ofcodewordsgenerated fromaprefixsourcecode,thesource decodersimplystartsatthebeginning ofthesequence anddecodesonecodewordata time.Specifically, itsetsupwhatisequivalent toadecision tree,whichisagraphical 576 CHAPTER 9IlFUNDAMENTAL LIMITS ININFORMATION THEORY ITABLE9.2Illustrating thedefinition ofaprefIXcode SourceSymbol Probability ofOccurrence CodeI CodeII Code!II- 50 0.5 0 0 0 5, 0.25 1 10 01 52 0.125 00 110 011 53 0.125 11 111 0111 portrayal ofthecodewordsintheparticular sourcecode.Forexample, Figure9.4depicts thedecisiontreecorresponding tocodeIIinTable9.2.Thetreehasaninitial5tateand fourterminal 5tate5corresponding tosourcesymbols 50'5"52'and53'Thedecoderalways startsattheinitialstate.Thefirstreceived bitmovesthedecodertotheterminal stateSo ifitis0,orelsetoaseconddecisionpointifitis1.Inthelattercase,thesecondbitmoves thedecoderonestepfurtherdownthetree,eithertoterminal state5,ifitis0,orelseto athirddecisionpointifitis1,andsoon.Onceeachterminal stateemitsitssymbol,the decoderisresettoitsinitialstate.Notealsothateachbitinthereceivedencodedsequence isexamined onlyonce.Forexample, theencoded sequence 1011111000 ...isreadily decoded asthesourcesequence 5,53525050' •••Thereaderisinvitedtocarryoutthis decoding. Aprefixcodehastheimpprtant property thatitisalway5uniquely decodable. But theconverse isnotnecessarily true.Forexample, codeIIIinTable9.2doesnotsatisfythe prefixcondition, yetitisuniquely decodable sincethebit0indicates thebeginning ofeach codewordinthecode. Moreover, ifaprefixcodehasbeenconstructed foradiscretememoryless source withsourcealphabet (so,5"•••,5K-')andsourcestatistics (Po,p"...,PK-1)andthe codewordforsymbol 5khaslengthlhok=0,1,...,K-1,thenthecode-word lengths ofthecodealwayssatisfyacertaininequality knownastheKraft-McMillan Inequality,' asshownby K-lLr1k:O;1 k~O(9.22) Initial state FIGURE 9.4Decision treeforcodeIIofTable9.2. 9.4Da'aC.....pac'hm 577 wherethefactor2referstotheradix(number ofsymbols) inthebinaryalphabet.Itis important tonote,however, thattheKraft-McMillan inequality doesnottellusthata sourcecodeisaprefixcode.Rather,itismerelyacondition onthecode-word lengthsof thecodeandnotonthecodewordsthemselves. Forexample, referring tothethreecodes listedinTable9.2,wenotethefollowing: i>CodeIviolatestheKraft-McMillan inequality; itcannottherefore beaprefixcode. ..TheKraft-McMillan inequality issatisfiedbybothcodesIIandIII;butonlycodeII isaprefixcode. Prefixcodesaredistinguished fromotheruniquely decodable codesbythefactthat theendofacodewordisalwaysrecognizable. Hence,thedecoding ofaprefixcanbe accomplished assoonasthebinarysequence representing asourcesymbolisfullyreceived. Forthisreason,prefixcodesarealsoreferredtoasinstantaneous codes. Givenadiscretememoryless sourceofentropyH(:f'),aprefixcodecanbeconstructed withanaveragecode-word lengthI,whichisbounded asfollows: HW)osI<HUt)+1 (9.23) Theleft-hand boundofEquation (9.23)issatisfiedwithequalityunderthecondition that symbol Skisemittedbythesourcewithprobability (9.24) (9.25) (9.26)wherelkisthelengthofthecodewordassigned tosourcesymbol Sk.Wethenhave K":-l K-l 2.:2-1•=2.:Pk=1 k~O k~O Underthiscondition, theKraft-McMillan inequality ofEquation (9.22)tellsusthatwe canconstruct aprefixcode,suchthatthelengthofthecodewordassigned tosourcesymbol Skis-10g2Pk' Forsuchacode,theaveragecode-word lengthis _K-llk L=2.:21k k~O andthecorresponding entropyofthesourceis K-l(1 )HUt)=t;o21.log2(21.) K-lh k~O21• Hence,inthisspecial(rathermeretricious) case,wefindfromEquations (9.25)and(9.26) thattheprefixcodeismatched tothesourceinthatI=H(;t). Buthowdowematchtheprefixcodetoanarbitrary discretememoryless source? Theanswertothisproblem liesintheuseofanextended code.LetIndenotetheaverage code-word lengthoftheextended prefixcode.Forauniquely decodable code,Inisthe smallestpossible. FromEquation (9.23),wededucethat (9.27) Substituting Equation (9.17)foranextended sourceintoEquation (9.27),weget nHW)osIn<nH(;t)+1 578 CHAPTER 9..FUNDAMENTAL LIMITS ININ.'ORMATION THEORY or,equivalently, H(9')oSIn<H(9')+~n n(9.28) Inthelimit,asnapproaches infinity, thelowerandupperboundsinEquation (9.28) converge, asshownby 1-lim-Ln=H(9') n~=n(9.29) Wemaytherefore statethatbymakingtheordernofanextended prefixsauce encoderlargeenough,wecanmakethecodefaithfully represent thediscretememoryless source9'ascloselyasdesired.Inotherwords,theaveragecode-word lengthofanextended prefixcodecanbemadeassmallastheentropyofthesourceprovided theextended code hasahighenoughorder,inaccordance withthesource-coding theorem. However, the pricewehavetopayfordecreasing theaveragecode-word lengthisincreased decoding complexity, whichisbroughtaboutbythehighorderoftheextended prefixcode. HUFFMAJ." CODING Wenextdescribeanimportant classofprefixcodesknownasHuffman codes.Thebasic ideabehindHuffman coding6ist,oassigntoeachsymbolofanalphabet asequence ofbits roughlyequalinlengthtotheamountofinformation conveyed bythesymbolinquestion. Theendresultisasourcecodewhoseaveragecode-word lengthapproaches thefunda mentallimitsetbytheentropyofadiscretememoryless source,namely,H(9').Theessence ofthealgorithm usedtosynthesize theHuffman codeistoreplacetheprescribed setof sourcestatistics ofadiscretememoryless sourcewithasimplerone.Thisreduction process iscontinued inastep-by-step manneruntilweareleftwithafinalsetofonlytwosource statistics (symbols), forwhich(0,1)isanoptimalcode.Startingfromthistrivialcode,we thenworkbackward andtherebyconstruct theHuffman codeforthegivensource. Specifically, theHuffman encoding algorithm proceeds asfollows: 1.Thesourcesymbols arelistedinorderofdecreasing probability. Thetwosource symbolsoflowestprobability areassigned a 0anda1.Thispartofthestepisreferred toasasplittingstage. 2.Thesetwosourcesymbols areregarded asbeingcombined intoanewsourcesymbol withprobability equaltothesumofthetwooriginalprobabilities. (Thelistofsource symbols, andtherefore sourcestatistics, istherebyreducedinsizebyone.)Theprob abilityofthenewsymbolisplacedinthelistinaccordance withitsvalue. 3.Theprocedure isrepeated untilweareleftwithafinallistofsourcestatistics (sym bols)ofonlytwoforwhicha 0anda 1areassigned. Thecodeforeach(original) sourcesymbolisfoundbyworking backward andtracingthe sequence ofOsand1sassigned tothatsymbolaswellasitssuccessors. ~ExAMPLE 9.3Huffman Tree Thefivesymbolsofthealphabetofadiscretememorylesssourceand their probabilities are showninthetwoleftmostcolumnsofFigure9.5a.Following throughtheHuffman algorithm, wereachtheendofthecomputation infoursteps,resultingintheHuffman treeshownin Figure9.5a.ThecodewordsoftheHuffman c.odeforthesourcearetabulated inFigure9.5b. 9.4DataCompaction 579 Symbol So S,SlageI SlageII SlageIII SlageIV Symbol Probability Codeword OAOA~f03=F0'JsD 0.4 00 sl 0.2 10s, 0.2 11 O'~O' 0.0 OA0.1 010 S3o 1 S4 0.1 0110.2 0.2 0.2 o 1(b) 0.1 0.2 10.1 (9.30)(a) FIGlJRE9.5(a)ExampleoftheHuffman encoding algorithm. (b)Sourcecode. Theaveragecode-word lengrhistherefore L=0.4(2)+0.2(2)+0.2(2)+0.1(3)+0.1(3) =2.2 Theentropyofthespecifieddiscrerememoryless sourceiscalculared asfollows[see Equation (9.9)]: H(:f)=0.4log2(..l.)+0.2IOfu(..l.)+0.2IOg2(..l.2)0.4 0.2 O. +0.llog2(0\)+0.llog2(0\) =0.52877+0.46439+0.46439+0.33219+0.33219 =2.12193bits Fortheexampleathand,wemaymakelwoobservations: 1.Theaveragecode-word lengthLexceedstheentropyH(:f)byonly3.67percent. 2.Theaveragecode-word lengthLdoesindeedsatisfyEquation (9.23). ""l Itisnoteworthy thattheHuffman encoding process(i.e.,theHuffman tree)isnot unique.Inparticular, wemaycitetwovariations intheprocessthatareresponsible for thenonuniqueness oftheHuffman code.First,ateachsplitting stageintheconstruction ofaHuffman code,thereisarbitrariness inthewaya 0anda 1areassigned tothelast twosourcesymbols. Whichever waytheassignments aremade,however, theresulting differences aretrivial.Second,ambiguity ariseswhentheprobability ofacombined symbol (obtained byaddingthelasttwoprobabilities pertinent toaparticular step)isfoundto equalanotherprobability inthelist.Wemayproceed byplacingtheprobability ofthe newsymbolashighaspossible, asinExample 9.3.Alternatively, wemayplaceitaslow aspossible. (Itispresumed thatwhichever waytheplacement ismade,highorlow,itis consistently adheredtothroughout theencoding process.) Butthistime,noticeable differ encesariseinthatthecodewordsintheresulting sourcecodecanhavedifferent lengths. Nevertheless, theaveragecode-word lengthremainsthesame. Asameasure ofthevariability incode-word lengthsofasourcecode,wedefinethe variance oftheaveragecode-word lengthIovertheensemble ofsourcesymbols as K-l a2=2:Pk(lk-If k-O wherePo,Ph...,PK-larethesourcestatistics, andlkisthelengthofthecodeword assigned tosourcesymbol Sk'Itisusuallyfoundthatwhenacombined symbolismoved 580 CHAPTER 9I!lFUNDAMENTAL LIMITS ININFORMATION THEORY ashighaspossible, theresulting Huffman codehasasignificantly smallervariancec?than whenitismovedaslowaspossible. Onthisbasis,itisreasonable tochoosetheformer Huffman codeoverthelatter. 1>1LEMPEL-ZIV CODING Adrawback oftheHuffman codeisthatitrequiresknowledge ofaprobabilistic model ofthesource;unfortunately, inpractice, sourcestatistics arenotalwaysknownaPriori. Moreover, inmodeling textwefindthatstoragerequirements preventtheHuffman code fromcapturing thehigher-order relationships between wordsandphrases, therebycom promising theefficiency ofthecode.Toovercome thesepractical limitations, wemayuse theLempel-Ziv algorithm,7whichisintrinsically adaptive andsimplertoimplement than Huffman coding. Basically, encoding intheLempel-Ziv algorithm isaccomplished byparsingthe source data streamintosegments thataretheshortestsubsequences notencountered pre viously.Toillustrate thissimpleyetelegantidea,consider theexample ofaninputbinary sequence specified asfollows: 000101110010100101 ... Itisassumed thatthebinarysymbols 0and1arealreadystoredinthatorderinthecode book.Wethuswrite Subsequences stored: Datatobeparsed:0,1 000101110010100101 ... Theencoding processbeginsattheleft.Withsymbols 0and1alreadystored,theshortest subsequence ofthedatastreamencountered forthefirsttimeandnotseenbeforeis00; sowewrite Subsequences stored: Datatobeparsed:0,1,00 0101110010100101 ... Thesecondshortestsubsequence notseenbeforeis01;accordingly, wegoontowrite Subsequences stored: Datatobeparsed:0,1,00,01 01110010100101 ... Thenextshortestsubsequence notencountered previously is011;hence,wewrite Subsequences stored: Datatobeparsed:0,1,00,01,011 10010100101 ... Wecontinue inthemannerdescribed hereuntilthegivendatastreamhasbeencompletely parsed.Thus,fortheexample athand,wegetthecodebookofbinarysubsequences shown inthesecondrowofFigure9.6. Numerical positions: 1 2 3 4 5 6 7 8 9 Subsequences: 0 1 0001011 10010100101 Numerical representations: 11124221416162 Binaryencodedblocks: 0010001110010100100011001101 FIGURE 9.6Illustrating theencodingprocessperformed bytheLempel-Ziv algorithm onthe binarysequence 000I0III0010100101.... 9.5 Disc~te Memoryless Channels 581 Thefirstrowshowninthisfiguremerelyinc!icates thenumerical positions ofthe individual subsequences inthecodebook.Wenowrecognize thatthefirstsubsequence of thedatastream,00,ismadeupoftheconcatenation ofthefirstcodebookentry,0,with itself;itistherefore represented bythenumber11.Thesecondsubsequence ofthedata stream,01,consistsofthefirstcodebookentry,0,concatenated withthesecondcode bookentry,1;itistherefore represented bythenumber12.Theremaining subsequences aretreatedinasimilarfashion. Thecomplete setofnumerical representations forthe varioussubsequences inthecodebookisshowninthethirdrowofFigure9.6.Asafurther example illustrating thecomposition ofthisrow,wenotethatthesubsequence 010consists oftheconcatenation ofthesubsequence 01inposition4andsymbol°inposition1;hence, thenumerical representation 41.ThelastrowshowninFigure9.6isthebinaryencoded representation ofthedifferent subsequences ofthedatastream. Thelastsymbolofeachsubsequence inthecodebook(i.e.,thesecondrowofFigure 9.6)isaninnovation symbol,whichissocalledinrecognition ofthefactthatitsappendage toaparticular subsequence distinguishes itfromallprevious subsequences storedinthe codebook.Correspondingly, thelastbitofeachuniform blockofbitsinthebinaryen codedrepresentation ofthedatastream(i.e.,thefourthrowinFigure9.6)represents the innovation symbolfortheparticular subsequence underconsideration. Theremaining bits providetheequivalent binaryrepresentation ofthe"pointer" totherootsubsequence that matchestheoneinquestion exceptfortheinnovation symbol. Thedecoderisjustassimpleastheencoder. Specifically, itusesthepointertoidentify therootsubsequence andthenappendstheinnovation symbol.Consider, forexample, the binaryencoded block1101inposition 9.Thelastbit,1,istheinnovation symbol.The remaining bits,110,pointtotherootsubsequence 10inposition6.Hence,theblock1101 isdecodedinto101,whichiscorrect. Fromtheexample described here,wenotethat,incontrasttoHuffman coding,the Lempel-Ziv algorithm usesfixed-length codestorepresent avariablenumberofsource symbols; thisfeaturemakestheLempel-Ziv codesuitableforsynchronous transmission. Inpractice, fixedblocksof12bitslongareused,whichimpliesacodebookof4096 entries. Foralongtime,Huffman codingwasunchallenged asthealgorithm ofchoicefor datacompaction. However, theLempel-Ziv algorithm hastakenoveralmostcompletely fromtheHuffman algorithm. TheLempel-Ziv algorithm isnowthestandard algorithm forfilecompression. Whenitisappliedtoordinary Englishtext,theLempel-Ziv algorithm achieves acompaction ofapproximately 55percent.Thisistobecontrasted withacom pactionofapproximately 43percentachieved withHuffman coding.Thereasonforthis behavior isthat,asmentioned previously, Huffman codingdoesnottakeadvantage ofthe intercharacter redundancies ofthelanguage. Ontheotherhand,theLempel-Ziv algorithm isabletodothebestpossiblecompaction oftext(withincertainlimits)byworking effec tivelyathigherlevels. I9.5Discrete Menwryless Channels Uptothispointinthechapter,wehavebeenpreoccupied withdiscretememoryless sources responsible forinformation generation. Wenextconsider theissueofinformation trans mission, withparticular emphasis onreliability. Westartthediscussion byconsidering a discretememoryless channel, thecounterpart ofadiscretememoryless source. Adiscretememoryless channelisastatistical modelwithaninputXandanoutput ythatisanoisyversionofX;bothXandYarerandomvariables. Everyunitoftime,the 582 CHAPTER 9'"FUNDAMENTAL LIMITS ININFORl\1ATION THEORY FIGURE 9.7Discrete memoryless channel. channelacceptsaninputsymbolXselectedfromanalphabet 2eand,inresponse, itemits anoutputsymbolYfromanalphabet qy.Thechannelissaidtobe"discrete" whenboth ofthealphabets 2eandqyhavefinitesizes.Itissaidtobe"memoryless" whenthecurrent outputsymboldependsonlyonthecurrentinputsymbolandnotanyoftheprevious ones. Figure9.7depictsaviewofadiscretememoryless channel.Thechannelisdescribed intermsofaninputalphabet anoutputalphabet, qy={Yo,Yh•..,YK-'}, andasetoftransition probabilities(9.31) (9.32) foralljandk (9.33) Naturally, wehave foralljandk (9.34) Also,theinputalphabet 2eandoutputalphabet qyneednothavethesamesize.For example, inchannelcoding,thesizeKoftheoutputalphabet qymaybelargerthanthe sizeJoftheinputalphabet 2e;thus,K;;:;"J.Ontheotherhand,wemayhaveasituation inwhichthechannelemitsthesamesymbolwheneitheroneoftwoinputsymbolsissent, inwhichcasewehaveKSJ. Aconvenient wayofdescribing adiscretememoryless channelistoarrangethe varioustransition probabilities ofthechannelintheformofamatrixasfollows: [p(Yol·o)p(Y,lxo)p(,<-,I·o) ] p(YoIx,) p(Yllx,) P(YK-llxl)(9.35) P=p(Yoixl-,)p(Y,lxj-1) P(YK-~Ixl-') TheJ-by-KmatrixPiscalledthechannelmatrix,ortransition matrix.NotethateachroW ofthechannelmatrixPcorresponds toafixedchannelinput,whereaseachcolumnofthe matrixcorresponds toafixedchanneloutput.Notealsothatafundamental property of thechannelmatrixP,asdefinedhere,isthatthesumoftheelements alonganyrowofthe matrixisalwaysequaltoone;thatis, K-'Lp(Yklx;) =1 k~Oforallj (9.36) 9.5Discrete Memoryless Channels 583 Suppose nowthattheinputstoadiscretememoryless channelareselectedaccording totheprobability distribution {p(xj),j=0,1,...,J1}.Inotherwords,theeventthat thechalU1elinputX=Xjoccurswithprobability forj=0,1,...,J-1 (9.37) Havingspecified therandomvariable Xdenoting thechannelinput,wemaynowspecify thesecondrandomvariable Ydenoting thechanneloutput.Thejointprobability distri butionoftherandomvariables XandYisgivenby p(Xj,Yk)=P(X=Xj,Y=Yk) =pry=YkIX=Xj)P(X=Xj) =P(YkIXj)p(Xj)(9.38) Themarginal probability distribution oftheoutputrandom variable Yisobtained by averaging outthedependence ofp(Xj,Yk)onXj'asshownby P(Yk)=P(Y=Yk) J-1 =2:P(Y=YklX=Xj)P(X=Xj) j=o J-l =2:P(YkIXi)P(xi) fork=0,1,...,K-1 j=O(9.39) Theprobabilities P(xi)forj=0,1,...,J-1,areknownastheaprioriprobabilities ofthevariousinputsymbols. Equation (9.39)statesthatifwearegiventheinputapri oriprobabilities P(xi)andthechalU1elmatrix[i.e.,thematrixoftransition probabilities P(YkIXi)]'thenwemaycalculate theprobabilities ofthevariousoutputsymbols, theP(Yk)' IPExAMPLE 9.4BinarySymmetric Channel Thebinarysymmetric channelisofgreattheoretical interestandpracticalimportance. Itisa specialcaseofthediscretememorylesschannelwith]=K=2.Thechannelhastwoinput symbols (xo=0,Xl=1)andtwooutputsymbols (Yo=0,Yl1).Thechannelissymmetric becausetheprobability ofreceiving a1ifa0issentisthesameastheprobability ofreceiving a 0ifa 1issent.Thisconditional probability oferrorisdenotedbyp.Thetransition prob abilitydiagramofabinarysymmetric channelisasshowninFigure9.8. ~ I-p Xo=0ec----~----7'yo =0 FIGURE9.8Transition probability diagramofbinarysymmetric channel. (9.40)584 CHAPTER 9IIIFUNDAMENTAL LIMITS ININFORMATION THEORY Itisofinteresttorelatethetransition probability diagramofFigure9.8totheCon_ ditionalprobabilities oferrorPlOandPOlthatweredetermined fortherCMreceiverin Section3.3.Forthecasewhenthebinarysymbols0and1areequiprobable, weshowed thattheoptimized valuesofthesetwoerrorprobabilities areequal.Indeed,recalling the following definitions (usingtheterminology ofFigure9.8): PlO=P(y=llx=0) and POl=P(y=0Ix=1) weimmediately seethatfortherCMreceiverofFigure3.4: PlO=POl=P I9.6Mutuallnformation GiventhatwethinkofthechanneloutputY(selected fromalphabet <J.li)asanoisyversion ofthechannelinputX(selected fromalphabet iJf),andthattheentropyH(iJf)isameasure oftheprioruncertainty aboutX,howcanwemeasure theuncertainty aboutXafter observing Y?Toanswerthisquestion, weextendtheideasdeveloped inSection9.2by definingtheconditional entropyofXselectedfromalphabet iJf,giventhatY=Yk'Spe cifically,wewrite J-l [1 ]H(iJflY=Yk)=LP(XjIYk)log2-(-1-) i-O PXiYk Thisquantity isitselfarandom variable thattakesonthevalues H(iJfIY=yo),••.,H(iJfIY=YK-,)withprobabilities p(Yo),•••,P(YK-l), respectively. ThemeanofentropyH(iJfIY=Yk)overtheoutputalphabet <J.liistherefore givenby K-l H(iJfI<J.li)=LH(iJf1Y=Yk)P(Yk) k~O (9.41) where,inthelastline,wehavemadeuseoftherelation (9.42) ThequantityH(iJf 1<J.li)iscalledaconditional entropy.Itrepresents theamountofuncer taintyremaining aboutthechannelinputafterthechanneloutputhasbeenobserved. SincetheentropyH(iJf)represents ouruncertainty aboutthechannelinputbefore observing thechanneloutput,andtheconditional entropyH(iJfI<J.li)represents ouruncer taintyaboutthechannelinputafterobserving thechanneloutput,itfollowsthatthe difference H(iJf)-H(iJfI<J.li)mustrepresent ouruncertainty aboutthechannelinputthatis resolved byobserving thechanneloutput.Thisimportant quantity iscalledthemutual 9.6M..'....llnformation 585 information ofthechannel. Denoting themutualinformation by1(2e;q]/),wemaythus write Similarly, wemaywriteI(gf;DY)=H(gf)-H(gfI'\lJ) I('\lJ;gf)=H('\lJ)-H(q]/Igf)(9.43) (9.44) whereH(q]/)istheentropyofthechanneloutputandH(q]/Igf)istheconditional entropy ofthechanneloutputgiventhechannelinput. Il1lPROPERTIES OFMUTUAL INFORMATION Themutualinforamtion I(gf;'\lJ)hasthefollowing important properties. Property 1 Themutualinformation ofachannelissymmetric; thatis (9.45) wherethemutualinformation I(gf;q]/)isameasure oftheuncertainty aboutthechannel inputthatisresolved byobserving thechannel output,andthemutualinformation I(q]/;gf)isameasureoftheuncertainty aboutthechanneloutputthatisresolvedbysending thechannelinput. Toprovethisproperty, wefirstusetheformulaforentropyandthenuseEquations (9.36)and(9.38),inthatorder,toexpressH(gf)as (9.46) Hence,substituting Equations (9.41)and(9.46)intoEquation (9.43)andthencombining terms,weobtain (9.47) FromBayes'ruleforconditional probabilities, wehave[seeEquations (9.38)and(9.42)] P(X;IYk)=p(Yklx;) p(x;) P(Yk)(9.48) 586 CHAPTER 9iiiFUNDAMENTAL LIMITS ININFORMATION THEORY Hence,substituting Equation (9.48)intoEquation (9.47)andinterchanging theorderof summation, wemaywrite whichisthedesiredresult. Property 2 Themutualinformation isalwaysnonnegative; thatis I(?£;0J)2:0 Toprovethisproperty, wefirstnotefromEquation (9.42)that (I)-p(Xj,Yk) PXjYk-P(Yk)(9.49) (9.50) (9.51) (9.52)Hence,substituting Equation (9.51)intoEquation (9.47),wemayexpressthemutual information ofthechannelas J-lK-l (p(Xj,Yk)) I(?£;0J)=LLp(xj,Yk)log2p()p() ,~ok~O x,Yk Next,adirectapplication ofthefundamental inequality [definedbyEquation (9.12)]yields thedesiredresult I(?£;0J)2:0 withequalityif,andonlyif, p(Xj,Yk)=p(Xj)P(Yk) foralljandk (9.53) Property 2statesthatwecannotloseinformation, ontheaverage,byobserving the outputofachannel. Moreover, themutualinformation iszeroif,andonlyif,theinput andoutputsymbolsofthechannelarestatistically independent, asinEquation (9.53). Property 3 Themutualinformation ofachannelisrelatedtothejointentropyofthechannelinput andchanneloutputby I(?£;0J)=H(?£)+H(0J)-H(?£,0J) (9.54) (9.55)wherethejointentropyH(?£,0J)isdefinedby J-lK-[ (1 ) H(?£,0J)=LLp(Xj,Yk)log2-(--) j~Ok~O PXj'Yk ToproveEquation (9.54),wefirstrewritethedefinition forthejointentropy H(?£,0J)as (9.56) (9.57)9.7Channel Capacity 587 H(X,')}) FIGURE9.9Illustrating therelations amongvariouschannel entropies, Thefirstdoublesummation termontheright-hand sideofEquation (9.56)isrecognized asthenegative ofthemutualinformation ofthechannel, 1(:1£;'Y),previously givenin Equation (9,52).Asforthesecondsummation term,wemanipulate itasfollows: J-lK-l[1]J-l[1] K-lLLP(Xj,Yk)log2p()p()=Llog2-P()Lp(Xj,Yk) ,~ok~O x,Yk ,~o Xj k~O K-J[1] J-l +t:ologzP(Yk)~P(xi'Yk) 1-1 [1 ]=LP(xi)log,-(-) ,~o Pxi K-l [1 ]+LP(Yk)log2P-() k~o Yk H(:I£)+H('Y) Accordingly, usingEquations (9,52)and(9.57)inEquation (9.56),wegettheresult H(:I£,'Y)-1(:1£;'Y)+H(:I£)+H('Y) (9.58) Rearranging termsinthisequation, wegettheresultgiveninEquation (9.54),thereby confirming Property 3. Weconclude ourdiscussion ofthemutualinformation ofachannelbyproviding a diagramatic interpretation ofEquations (9.43), (9.44), and(9.54).Theinterpretation is giveninFigure9.9.TheentropyofchannelinputXisrepresented bythecircleontheleft, TheentropyofchanneloutputYisrepresented bythecircleontheright.Themutual information ofthechannelisrepresented bytheoverlapbetweenthesetwocircles. I9.7Channel Capacity Consider adiscretememoryless channelwithinputalphabet :1£,outputalphabet 'Y,and transition probabilities P(YkIXi)'wherej=0,1,. , .,J-1andk=0,1,...,K-1.The mutualinformation ofthechannelisdefinedbythefirstlineofEquation (9.49),whichis reproduced hereforconvenience: 588 CHAPTER 9..FUNDAMENTAL LIMITS ININFORMATION THEORY Herewenotethat[seeEquation (9.38)] P(x;,Yk)=P(YkIx;)P(x;) Also,fromEquation (9.39),wehave J-l P(Yk)=L.P(YkIx;)p(x;) ;=0 Fromthesethreeequations weseethatitisnecessary forustoknowtheinputprobability distribution [P(x;)Ii=0,1,...,J-1)sothatwemaycalculate themutualinformation 1(1£;qiJ).Themutualinformation ofachanneltherefore dependsnotonlyonthechannel butalsoonthewayinwhichthechannelisused. Theinputprobability distribution {p(x;))isobviously independent ofthechannel. Wecanthenmaximize themutualinformation 1(1£;qiJ)ofthechannelwithrespectto [p(x;)).Hence,wedefinethechannel capacityofadiscretememoryless channelasthe maximum mutualinformation 1(1£;qiJ)inanysingleuseofthechannel (i.e.,signaling interval), wherethemaximization isoverallpossibleinputprobability distributions [P(x;)} on1£.Thechannelcapacity iscommonly denotedbyC.Wethuswrite C=max1(1£;qiJ) (P(xi))(9.59) Thechannelcapacity Cismeasured inbitsperchanneluse,orbitspertransmission. Notethatthechannelcapacity Cisafunction onlyofthe transition probabilities P(YkIx;),whichdefinethechannel. Thecalculation ofCinvolves maximization ofthe mutualinformation 1(1£;qiJ)overJvariables [i.e.,theinputprobabilities p(xo),..•,P(XJ-I)] subjecttotwoconstraints: and J-12:p(x;)=1 ;=0 Ingeneral,thevariational problem offindingthechannelcapacityCisachallenging task. ~ExAMPLE 9.5BinarySymmetric Channel (Revisited) Consider againthebinarysymmetric channel,whichisdescribed bythetransition probability diagramofFigure9.8.Thisdiagramisuniquely definedbytheconditionalprobability oferror p. TheentropyH(X)ismaximized whenthechannelinputprobability p(xo)=p(x,)= 1/2,whereXoandx,areeach0or1.Themutualinformation I(X;'!II)issimilarly maximized, sothatwemaywrite FromFigure9.8,wehave p(YoIx,)=p(YIIxo)P and p(YoIxu)=P(YI!X,) =1P 9.8Channel-Coding TIzeorem 589 1.0 0.5 Transition probability p FIGURE9.10Variation ofchannelcapacityofabinarysymmetric channelwithtransition proba bilityp. Therefore, substituting thesechannel transltlon probabilities intoEquation (9.49)with J=K=2,andthensettingtheinputprobability p(xo)=P(X,)inaccordance withEquation (9.59),wefindthatthecapacityofthebinarysymmetric channelis C=1+Plog,P+(1-p)10g,(1-p) (9.60) Usingthedefinition oftheentropyfunction giveninEquation (9.16),wemayreduceEquation (9.60)to C=1 -H(p) Thechannelcapacity Cvarieswiththeprobability oferror(transirion probabiliry) pin aconvexmannerasshowninFigure9.10,whichissymmetric aboutp=1/2.Comparing the curveinthisfigurewiththatinFigure9.2,wemaymakethefollowing observations: 1.Whenthechannelisnoise{ree,permitting ustosetp=0,thechannelcapaciryCattains itsmaximum valueofonebitperchanneluse,whichisexactlytheinformation ineach channelinput.Atthisvalueofp,theentropyfunctionH(p)attainsitsminimum value ofzero. 2.Whenthecondirional probability oferrorp=1/2duetonoise,thechannelcapacityC attainsitsminimum valueofzero,whereastheentropyfunctionH(p)attainsitsmax imumvalueofunity;insuchacasethechannelissaidtobeuseless. <Ill I9.8Channel-Coding Theorem Theinevitable presence ofnoiseinachannel causesdiscrepancies (errors) between the outputandinputdatasequences ofadigitalcommunication system.Forarelatively noisy channel (e.g.,wireless communication channel), theprobability oferrormayreachavalue ashighas10-\whichmeansthat(ontheaverage) only9outof10transmitted bitsare received correctly. Formanyapplications, thislevelofreliability isunacceptable. Indeed, aprobability oferrorequalto10-6orevenlowerisoftenanecessary requirement. To achievesuchahighlevelofperformance, weresorttotheuseofchannel coding. Thedesigngoalofchannel codingistoincrease theresistance ofadigitalcommu nication systemtochannel noise.Specifically, channelcodingconsists ofmapping the incoming datasequence intoachannel inputsequence, andinversemapping thechannel outputsequence intoanoutputdatasequence insuchawaythattheoveralleffectof 590 CHAPTER 9'"FUNDAMENTAL LIMITS ININFORMATION THEORY channelnoiseonthesystemisminimized. Thefirstmapping operation isperformed inth transmitter byachannelencoder, whereas theinversemapping operation isperformed ~ thereceiverbyachanneldecoder, asshownintheblockdiagramofFigure9.11;tosimplify theexposition, wehavenotincluded sourceencoding (beforechannelencoding) andSource decoding (afterchanneldecoding) inFigure9.11. Thechannelencoderandchannel decoder inFigure9.11arebothunderthede signer'scontrolandshouldbedesigned tooptimize theoverallreliability ofthecommu_ nication system.Theapproach takenistointroduce redundancy inthechannelencoder soastoreconstruct theoriginalsourcesequence asaccurately aspossible. Thus,inarather loosesense,wemayviewchannelcodingasthedualofsourcecodinginthattheformer introduces controlled redundancy toimprove reliability, whereasthelatterreducesredun dancytoimprove efficiency. ThesubjectofchannelcodingistreatedindetailinChapter 10.Forthepurposeof ourpresentdiscussion, itsufficestoconfineourattention toblockcodes.Inthisclassof codes,themessagesequence issubdivided intosequential blockseachkbitslong,andeach k-bitblockismapped intoann-bitblock,wheren>k.Thenumberofredundant bits addedbytheencodertoeachtransmitted blockisn-kbits.Theratiokiniscalledthe coderate.Usingrtodenotethecoderate,wemaythuswrite kr=n where,ofcourse,rislessthanunity.Foraprescribed k,thecoderater(andtherefore the system's codingefficiency) approaches zeroastheblocklengthnapproaches infinity. Theaccurate reconstruction oftheoriginalsourcesequence atthedestination re quiresthattheaverageprobability ofsymbolerrorbearbitrarily low.Thisraisesthe following important question: Doesthereexistachannelcodingschemesuchthatthe probability thatamessage bitwillbeinerrorislessthananypositivenumber E(i.e.,as smallaswewantit),andyetthechannelcodingschemeisefficientinthatthecoderate neednotbetoosmall?Theanswertothisfundamental question isanemphatic "yes." Indeed,theanswertothequestion isprovided byShannon's secondtheorem intermsof thechannelcapacity C,asdescribed inwhatfollows. Upuntilthispoint,timehasnot playedanimportant roleinourdiscussion ofchannelcapacity. Supposethenthediscrete memoryless sourceinFigure9.11hasthesourcealphabet :JandentropyH(:J)bitsper sourcesymbol.Weassumethatthesourceemitssymbols onceeveryTsseconds. Hence, theaverageinformation rateofthesourceisH(:J)/T,bitspersecond.Thedecoderdelivers decoded symbols tothedestination fromthesourcealphabet :Jandatthesamesource rateofonesymboleveryTsseconds. Thediscretememoryless channelhasachannelca pacityequaltoCbitsperuseofthechannel. Weassumethatthechanneliscapableof beingusedonceeveryTcseconds. Hence,thechannelcapacity perunittimeisaT,bits persecond,whichrepresents themaximum rateofinformation transferoverthechannel. WearenowreadytostateShannon's secondtheorem, knownasthechannelcoding theorem. Noise FIGURE 9.11Blockdiagramofdigitalcommunication system. (9.61)9.8Channel-CodingTlu!Orem 591 Specifically, thechannelcodingtheorems foradiscretememoryless channelisstated intwopartsasfollows. (i)Letadiscretememoryless sourcewithanalphabet ~haveentropyHW)andproduce symbols onceeveryT,seconds. Letadiscretememoryless channelhavecapacity C andbeusedonceeveryT,seconds. Then,if H(~)<C T,-Tc thereexistsacodingschemeforwhichthesourceoutputcanbetransmitted overthe channel andbereconstructed withanarbitrarily smallprobability oferror.The parameter CITeiscalledthecriticalrate.WhenEquation (9.61)issatisfied with theequalitysign,thesystemissaidtobesignaling atthecriticalrate. (ii)Conversely, if H(~)>..f T,T, itisnotpossibletotransmit information overthechannelandreconstruct itwithan arbitrarily smallprobability oferror. Thechannelcodingtheorem isthesinglemostimportant resultofinformation the ory.Thetheorem specifies thechannelcapacity Casafundamental limitontherateat whichthetransmission ofreliableerror-free messages cantakeplaceoveradiscrete memorylesschannel. However, itisimportant tonotethefollowing: l>-Thechannelcodingtheorem doesnotshowushowtoconstruct agoodcode.Rather, thetheorem shouldbeviewedasanexistence proofinthesensethatittellsusthat ifthecondition ofEquation (9.61)issatisfied, thengoodcodesdoexist.(Laterin Chapter10wedescribeseveralgoodcodesfordiscrete memoryless channels.) I>Thetheorem doesnothaveapreciseresultfortheprobability ofsymbolerrorafter decoding thechanneloutput.Rather,ittellsusthattheprobability ofsymbolerror tendstozeroasthelengthofthecodeincreases, againprovided thatthecondition ofEquation (9.61)issatisfied. Notealsothatpowerandbandwidth constraints werehiddeninthediscussion presented here.Nevertheless, thesetwosystemconstraints doactuallyshowupinthechannelmatrix Pofthediscretememoryless channel. Thisobservation isreadilyconfirmed bylinkingthe resultsofExample 9.5onthebinarysymmetric channelwiththenoiseanalysisforthe PCMreceiverpresented inSection5.3. iiiAPPLICATION OFTHECHANNEL CODING THEOREM TOBINARY SYMMETRIC CHANNELS Consider adiscretememoryless sourcethatemitsequallylikelybinarysymbols (Osand Is)onceeveryT,seconds.Withthesourceentropyequaltoonebitpersourcesymbol(see Example 9.1),theinformation rateofthesourceis(lIT,)bitspersecond.Thesource sequence isappliedtoachannelencoderwithcoderater.Thechannelencoderproduces asymbolonceeveryTcseconds. Hence,theencoded symboltransmission rateis(lITe) symbolspersecond.Thechannelencoderengagesabinarysymmetric channelonceevery Teseconds. Hence,thechannelcapacity perunittimeis(CITe)bitspersecond,whereC 592 CHAPTER 9IIIFUNDAMENTAL LIMITS ININFORMATION THEORY (9.62)isdetermined bytheprescribed channeltransition probability pinaccordance withEqua tion(9.60).Accordingly, thechannelcodingtheorem [part(i)]impliesthatif 1 C-<-Ts-Tc (9.63)theprobability oferrorcanbemadearbitrarily lowbytheuseofasuitablechannel encoding scheme.ButtheratioTJTsequalsthecoderateofthechannelencoder: Tcr=Ts Hence,wemayrestatethecondition ofEquation (9.62)simplyas r:=;C (9.64) Thatis,forr:=;C,thereexistsacode(withcoderatelessthanorequaltoC)capableof achieving anarbitrarily lowprobability oferror. !>ExAMPLE 9.6Repetition Code Inthisexample,wepresentagraphical interpretation ofthechannelcodingtheorem. Wealso bringoutasurprising aspectofthetheorembytakingalookatasimplecodingscheme. 1.0 10-1 10-2 ..' ~10-3 '5 ~ :010-4 .22 Q. ~10-5 ~ 10-6 10-7Limitingvalue E.=10~8IChannel :capacityc I 10-8~"""-~--..LL,-----0.01 0.1 1.0 Coderate,r FIGURE 9.12Illustrating significance ofthechannelcodingtheorem. 9.9Differential Entropy amIMutualInftwmatWnfor Continuous Ensembles 593 TABLE9.3Averageprobability oferror forrepetition code CodeRate,r =1In 1 J 1< 1 7 1"1ITAverageProbability ofError,P, 10-2 3X10-' 10-6 4X10-7 10-' 5X10-10 Consider firstabinarysymmetric channelwithtransition probability p=10-2•Forthis valueofp,wefindfromEquation (9.60)thatthechannelcapacityC=0.9192.Hence,from thechannelcodingtheorem, wemaystatethatforanyE>0andr:s0.9192,thereexistsa codeoflargeenouighlengthnandcoderater,andanappropriate decoding algorithm, such thatwhenthecodedbitstreamissentoverthegivenchannel, theaverageprobability of channeldecoding errorislessthanE.Thisresultisdepicted inFigure9.12forthelimiting valueE=10-'. Toputthesignificance ofthisresultinperspective, considernextasimplecodingscheme thatinvolvestheuseofarepetition code,inwhicheachbitofthemessageisrepeated several times.Leteachbit(0or1)berepeated ntimes,wheren=2m+1isanoddinteger.For example, forn=3,wetransmit 0and1as000and111,respectively. Intuitively, itwould seemlogicaltouseamajority rulefordecoding, whichoperates asfollows:Ifinablockofn received bits(representing onebitofthemessage), thenumberofOsexceedsthenumberof 1s,thedecoderdecidesinfavorofao.Otherwise, itdecidesinfavorofa1.Hence,anerror occurswhenm+1ormorebitsoutofn=2m+1bitsarereceivedincorrectly. Becauseofthe assumedsymmetric natureofthechannel,theaverageprobability oferrorP,isindependent of theaprioriprobabilities of0and1.Accordingly, wefindthatP,isgivenby(seeProblem9.24) P,=i~t+1(7)pi(l-p)"-i (9.65) wherepisthetransition probability ofthechannel. Table9.3givestheaverageprobability oferrorP,forarepetition code,whichiscal culatedbyusingEquation (9.65)fordifferentvaluesofthecoderater.Thevaluesgivenhere assumetheuseofabinarysymmetric channelwithtransition probability p10-2•The improvement inreliability displayed inTable9.3isachieved atthecostofdecreasing code rate.Theresultsofthistablearealsoshownplottedasthecurvelabeled"repetition code"in Figure9.12.Thiscurveillustrates theexchange ofcoderateformessagereliability, whichis acharacteristic ofrepetition codes. Thisexample highlights theunexpected resultpresented tousbythechannelcoding theorem. Theresultisthatitisnotnecessary tohavethecoderaterapproach zero(asin thecaseofrepetition codes)soastoachievemoreandmorereliableoperation ofthecom munication linleThetheorem merelyrequiresthatthecoderatebelessthanthechannel capacity C. <!ll 9.9Differential Entropy andMutual Information forContinuous Ensembles Thesourcesandchannels considered inourdiscussion ofinformation-theoretic concepts thusfarhaveinvolved ensembles ofrandom variables thatarediscreteinamplitude. In 594 CHAPTER 9mFUNDAMENTAL LIMITS ININFORMATION THEORY thissection,weextendsomeoftheseconcepts tocontinuous randomvariables andrandom vectors.Themotivation fordoingsoistopavethewayforthedescription ofanother fundamental limitininformation theory,whichwetakeupinSection9.10. Consider acontinuous randomvariableXwiththeprobability densityfunction fx(x).Byanalogywiththeentropyofadiscreterandomvariable, weintroduce thefol lowingdefinition: (9.66) Werefertoh(X)asthedifferential entropyofXtodistinguish itfromtheordinary or absolute entropy. Wedosoinrecognition ofthefactthatalthough h(X)isausefulmath ematical quantity toknow,itisnotinanysenseameasure oftherandomness ofX. Nevertheless, wejustifytheuseofEquation (9.66)inwhatfollows.Webeginbyviewing thecontinuous randomvariableXasthelimitingform ofadiscreterandomvariablethat assumesthevalueXk=k~x,wherek=0,±1, ±2,...,and~approaches zero.By definition, thecontinuous randomvariableXassumesavalueintheinterval[XbXk+/lx] withprobability fX(Xk)/lx.Hence,permitting ~xtoapproach zero,theordinary entropy ofthecontinuous randomvariableXmaybewritteninthelimitasfollows: H(X)l~~ok~OOfX(Xk)/lxIOg2(fx(X~) ~J =lx~o[k~OOfX(Xk)IOg2(fx~xJ ~X-log2/lxk~oofX(Xk)~x] =roofx(X)IOg2(fx~x)) dx-l~~olog2~roofx(X)dx =h(X)-limlogz~x ,6,x--+o(9.67) where,inthelastline,wehavemadeuseofEquation (9.66)andthefactthatthetotal areaunderthecurveoftheprobability densityfunction fx(x)isunity.Inthelimitas /lxapproaches zero,-logz/lxapproaches infinity.Thismeansthattheentropyofacon tinuousrandomvariableisinfinitely large.Intuitively, wewouldexpectthistobetrue, becauseacontinuous random variable mayassumeavalueanywhere intheinterval (-00,00)andtheuncertainty associated withthevariableisontheorderofinfinity.We avoidtheproblem associated withthetermlogz/lxbyadopting h(X)asadifferential entropy, withtheterm-log2 ~xservingasreference. Moreover, sincetheinformation transmitted overachannelisactuallythedifference betweentwoentropytermsthathave acommon reference, theinformation willbethesameasthedifference betweenthecor responding differential entropyterms.Wearetherefore perfectly justifiedinusingtheterm h(X),definedinEquation (9.66),asthedifferential entropyofthecontinuous random variableX. Whenwehaveacontinuous randomvectorXconsisting ofnrandomvariables Xl> Xz,...,Xmwedefinethedifferential entropyofXasthen-foldintegral h(X)=roofx(X)logz[fx~x,] dx wherefxix)isthejointprobability densityfunction ofX.(9.68) 9.9DijJerentlal Entropy andMutualInformation forContinuous Ensembles 595 I>ExAMPLE 9.7Uniform Distribution Consider arandomvariableXuniformly distributed overtheinterval(0,a).Theprobability densityfunctionofXis fx(x)={~' 0,O<x<a otherwise (9.69) (9.70)Applying Equation (9.66)tothisdistribution, weget La1 h(X)= -log(a)dxoa =loga Notethatloga<0fora<1.Thusthisexampleshowsthat,unlikeadiscreterandomvariable, thedifferential entropyofacontinuous randomvariablecanbenegative. "'1l I>ExAMPLE 9.8Gaussian Distribution Consider anarbitrary pairofrandomvariablesXandY,whoseprobability densityfunctions arerespectively denotedbyfy(x)andfx(x)wherexismerelyadummyvariable. Adapting thefundamental inequality ofEquation (9.12)tothesituation athand,wemaywrite" [fy(x)IOg2(j:i:;) dx,;0 or,equivalently,-r~fy(x)log2fy(x)dx,;-r~fy(x)log2fx(x)dx (9.71) Thequantityontheleft-hand sideofEquation (9.71)isthedifferential entropyoftherandom variable Y;hence, h(Y)::=:-r~fy(x)log2fx(x)dx SupposenowtherandomvariablesXandYaredescribed asfollows: I>TherandomvariablesXandYhavethesamemeanILandthesamevariancecr. I>-TherandomvariableXisGaussian distributed asshownby(9.72) (9.73) fx(x)=vbuexp( -(x;)Lf) Hence,substituting Equation (9.73)intoEquation (9.72),andchanging thebaseofthelog arithmfrom2toe=2.7183,weget J~((X-1L)2 •r-o)h(Y),;-log,e_~fy(x)------zT-log(v27TU)dx (9.74) Wenowrecognize thefollowing properties oftherandomvariableY(giventhatitsmeanis ILanditsvariance iscr): [fy(x) dx=1 r~(x-ILffy(x) dx=cr 596 CHAPTER 9IIFUNDAMENTAL LIMITS ININFORMATION THEORY Wemaytherefore simplifyEquation (9.74)as (9.75) Thequantityontheright-hand sideofEquation (9.75)isinfactthedifferential entropyof theGaussian randomvariableX: h(X)=tlog2(27re~) Finally,combining Equations (9.75)and(9.76),wemaywrite(9.76) h(Y):Sh(X),{X:Gaussian randomvariable Y:anotherrandomvariable(9.77) whereequalityholdsif,andonlyif,Y=X. Wemaynowsummarizethe resultsofthisimportant exampleasfWOentropicproperties ofaGaussian randomvariable: 1.Forafinitevariance tI2,theGaussian randomvariablehasthelargestdifferential en tropyattainable byanyrandomvariable. 2.TheentropyofaGaussian randomvariable X isuniquely determined bythevariance ofX(i.e.,itisindependent ofthemeanofX). Indeed,itisbecauseofProperty 1thattheGaussian channelmodelissowidelyusedasa conservative modelinthestudyofdigitalcommunication systems. <Ill IlIIMUTUAL INFORMATION Consider nextapairofcontinuous random variables XandY.Byanalogy withEqua tion(9.47),wedefinethemutualinformation between therandom variables XandYas follows: fOOfoo [fx(X1y)]fiX;Y)=-00 -00fx,Y(x,y) log2fx(x)dxdy (9.78) wherefx,Y(x,y)isthejointprobability densityfunction ofXandY,andfx(xIy)isthe conditional probability densityfunction ofX,giventhatY=y.Also,byanalogy with Equations (9.45), (9.50), (9.43), and(9.44)wefindthatthemutualinformation fiX;Y) hasthefoHowing properties: 1.f(X;Y)=flY;X) 2.fiX;Y)2:0 3.fiX;Y)=h(X)-h(XIY) =h(Y)-h(YIX)(9.79) (9.80) (9.81) Theparameter h(X)isthedifferential entropy ofX;likewise forh(Y).Theparameter h(XIY)istheconditional differential entropyofX,givenY;itisdefinedbythedouble integral (seeEquation (9.41)) (9.82) 9.10Information Capacity Theorem 597 Theparameter h(YIX)istheconditional differential entropyofY,givenX;itisdefined inamannersimilartoh(XIV). L9.10Information Capacity Theorem Inthissection,weusetheideaofmutualinformation toformulate theinformation capacity theorem forband-limited, power-limited Gaussian channels. Tobespecific,consider a zero-mean stationary processX(t)thatisband-limited toBhertz.LetXbk=1,2,..., K,denotethecontinuous randomvariables obtained byuniformsampling oftheprocess X(t)attheNyquistrateof2Bsamplespersecond.Thesesamplesaretransmitted inT secondsoveranoisychannel, alsoband-limited toBhertz.Hence,thenumberofsamples, K,isgivenby K=2BT (9.83) WerefertoXkasasampleofthetransmitted signal.Thechanneloutputisperturbed byadditive whiteGaussian noise(AWGN)ofzeromeanandpowerspectraldensity No/2.Thenoiseisband-limited toBhertz.Letthecontinuous randomvariables Yb k=1,2,...,Kdenotesamplesofthereceivedsignal,asshownby k=1,2,...,K (9.84) ThenoisesampleNkisGaussian withzeromeanandvariance givenby (9.85) WeassumethatthesamplesYbk=1,2,..., Karestatistically independent. Achannelforwhichthenoiseandthereceivedsignalareasdescribed inEquations (9.84)and(9.85)iscalledadiscrete-time, memoryless Gaussian channel.Itismodeled as inFigure9.13.Tomakemeaningful statements aboutthechannel, however, wehaveto assignacosttoeachchannelinput.Typically, thetransmitter ispowerlimited;itisthere forereasonable todefinethecostas E[Xi]=P,k=1,2,..., K (9.86) wherePistheaveragetransmitted power.Thepower-limited Gaussian channeldescribed hereinisofnotonlytheoretical butalsopractical importance inthatitmodelsmany communication channels, including line-of-sight radioandsatellitelinks. Theinformation capacity ofthechannelisdefinedasthemaximum ofthemutual information betweenthechannelinputXkandthechanneloutputYkoveralldistributions ontheinputXkthatsatisfythepowerconstraint ofEquation (9.86).LetI(Xk;Yk)denote XkrYk Nk FIGURE9.13Modelofdiscrete-time, memoryless Gaussian channel. 598 CHAPTER 9Ii!FUNDAMENTAL LIMITS ININFORMATION THEORY themutualinformation betweenXkandYk•Wemaythendefinetheinformation capacity ofthechannelas C=max{I(X k;Yk):E[XtJ=P} fxk.(X)(9.87) wherethemaximization isperformed withrespectto!Xk(X),theprobability densityfunc tionofXk• Themutualinformation I(Xk;Yk)canbeexpressed inoneofthetwoequivalent formsshowninEquation (9.81).Forthepurposeathand,weusethesecondlineofthis equation andsowrite (9.88) SinceXkandNkareindependent randomvariables, andtheirsumequalsY,,,asinEqua tion(9.84),wefindthattheconditional differential entropyofYbgivenXbisequalto thedifferential entropyofNk(seeProblem 9.28): (9.89) Hence,wemayrewriteEquation (9.88)as (9.90) Sinceh(Nk)isindependent ofthedistribution ofXbmaximizing I(Xk;Yk)inaccor dancewithEquation (9.87)requiresmaximizing h(Yk),thedifferential entropyofsample Ykofthereceived signal.Forh(Yk)tobemaximum, YkhastobeaGaussian random variable(seeExample 9.8).Thatis,thesamplesofthereceivedsignalrepresent anoiselike process.Next,weobservethatsinceNkisGaussian byassumption, thesampleXkofthe transmitted signalmustbeGaussian too.Wemaytherefore statethatthemaximization specified inEquation (9.87)isattainedbychoosing thesamplesofthetransmitted signal fromanoiselike process ofaveragepowerP.Correspondingly, wemayreformulate Equa tion(9.87)as E[X~J=P (9.91) wherethemutualinformation I(Xk;Yk)isdefinedinaccordance withEquation (9.90). Fortheevaluation oftheinformation capacity C,weproceedinthreestages: 1.Thevariance ofsampleYkofthereceived signalequalsP+u2•Hence,theuseof Equation (9.76)yieldsthedifferential entropyofYkas h(Yk)=!log2[21Te(P + (2)J (9.92) 2.Thevariance ofthenoisesampleNkequalsu2•Hence,theuseofEquation (9.76) yieldsthedifferential entropyofNkas h(Nk)=!log2(21Teu2) (9.93) 3.Substituting Equations (9.92)and(9.93)intoEquation (9.90)andrecognizing the definition ofinformation capacitygiveninEquation (9.91),wegetthedesiredresult: C=~log2(1+:2)bitspertransmission (9.94) WiththechannelusedKtimesforthetransmission ofKsamplesoftheprocessX(t) inTseconds, wefindthattheinformation capacityperunittimeis(KIT)timestheresult 9.10Injtw1tUJtion Capacity 'Theorem 599 giveninEquation (9.94).ThenumberKequalsIBT,asinEquation (9.83).Accordingly, wemayexpresstheinformation capacity intheequivalent form: C=Blog2(1+~)bitspersecond (9.95)NoB wherewehaveusedEquation (9.85)forthenoisevariance u2• BasedontheformulaofEquation (9.95),wemaynowstateShannon's third(and mostfamous)theorem, theinformation capacitytheorem;'o asfollows: Theinformation capacity ofacontinuous channelofbandwidth Bhertz,perturbed by additivewhiteGaussian noiseofpowerspectraldensityNollandlimitedinbandwidth toB,isgivenby C=Blog2(1+~)bitspersecondNoB wherePistheaveragetransmitted power. Theinformation capacity theorem isoneofthemostremarkable resultsofinfor mationtheoryfor,inasingleformula, ithighlights mostvividlytheinterplay amongthree keysystemparameters: channelbandwidth, averagetransmitted power(or, equivalently, averagereceived signalpower),andnoisepowerspectraldensityatthechanneloutput. Thedependence ofinformation capacity Conchannelbandwidth Bislinear,whereasits dependence onsignal-to-noise ratioPINoBislogarithmic. Accordingly, itiseasiertoin creasetheinformation capacityofacommunication channelbyexpanding itsbandwidth thanincreasing thetransmitted powerforaprescribed noisevariance. Thetheorem impliesthat,forgivenaveragetransmitted powerPandchannelband widthB,wecantransmit information attherateofCbitspersecond,asdefinedin Equation (9.95),witharbitrarily smallprobability oferrorbyemploying sufficiently com plexencoding systems.Itisnotpossibletotransmit ataratehigherthanCbitspersecond byanyencoding systemwithoutadefiniteprobability oferror.Hence,thechannelcapacity theorem definesthefundamental limitontherateoferror-free transmission forapower limited,band-limited Gaussian channel.Toapproach thislimit,however, thetransmitted signalmusthavestatistical properties approximating thoseofwhiteGaussian noise. illSPHERE PACKINGll Toprovideaplausible argument supporting theinformation capacity theorem, suppose thatweuseanencoding schemethatyieldsKcodewords,oneforeachsampleofthe transmitted signal.Letndenotethelength(i.e.,thenumberofbits)ofeachcodeword.It ispresumed thatthecodingschemeisdesigned toproduceanacceptably lowprobability ofsymbolerror.Furthermore, thecodewordssatisfythepowerconstraint; thatis,the averagepowercontained inthetransmission ofeachcodewordwithnbitsisnP,where Pistheaveragepowerperbit. Supposethatanycodewordinthecodeistransmitted. Thereceivedvectorofnbits isGaussian distributed withmeanequaltothetransmitted codewordandvariance equal tonu2,whereu2isthenoisevariance. Withhighprobability, thereceivedvectorliesinside asphereofradiusV;U;Z,centered onthetransmitted codeword.Thissphereisitself contained inalargersphereofradiusVn(P+(2),wheren(P+(2)istheaveragepower ofthereceivedvector. 600 CHAPTER 9 "FUNDAMENTAL LIMITS ININFORMATION THEORY FIGURE9.14Thesphere-packing problem. (9.96) =2(nl2)logz(1+PI'?)Wemaythusvisualize thepictureportrayed inFigure9.14.Witheverything inside asmallsphereofradiusvnc;zassigned tothecodewordonwhichitiscentered, itis reasonable tosaythatwhenthisparticular codewordistransmitted, theprobability that thereceivedvectorwilllieinsidethecorrect"decoding" sphereishigh.Thekeyquestion is:Howmanydecoding spherescanbepackedinsidethelargersphereofreceivedvectors? Inotherwords,howmanycodewordscanweinfactchoose?Toanswerthisquestion, wefirstrecognize thatthevolumeofann-dimensional sphereofradiusrmaybewritten asAnrn,whereAnisascalingfactor.Wemaytherefore makethefollowing statements: I;>ThevolumeofthesphereofreceivedvectorsisAn[n(P+lT2)]nl2. ~Thevolumeofthedecoding sphereisAn(nlT2r12• Accordingly, itfollowsthatthemaximum numberofnonintersecting decoding spheres thatcanbepackedinsidethesphereofpossiblereceivedvectorsis An[n(P+lT2)]nl2=(1+~)nl2 An(nlT2)n/2 lT2 Takingthelogarithm ofthisresulttobase2,wereadilyseethatthemaximum numberof bitspertransmission foralowprobability oferrorisindeedasdefinedpreviously in Equation (9.94). .,.EXAMPLE 9.9Reconfiguration ofConstellation forReduced Power Toillustrate theideaofspherepacking, consider the64-QAM squareconstellation ofFigure 9.15a.Thefiguredepictstwo-dimensional nonintersecting decoding spher~scentered onthe messagepointsintheconstellation. Intryingtopackthedecoding spheresastightlyaspossible whilemaintaining thesameEuclidean distance between themessage pointsasbefore,we obtainthealternative constellation showninFigure9.15b.Withacommon Euclidean dis tancebetween themessage points,thetwoconstellations ofFigure9.15produce approxi matelythesamebiterrorrate,assUllling theuseofahighenoughsignal-to-noise ratioover anAWGNchannel; see,forexample, Equation (5.95).However, comparing thesetwocon stellations, wefindthatthesumofsquaredEuclidean distances fromthemessagepointsco theorigininFigure9.15bissmallerthanthatinFigure9.15a.Itfollowstherefore thatthe tightlypackedconstellation ofFigure9.15bhasanadvantage overthesquareconstellation 109.11Implications oftheInformation Capacity Theorem 601 10 -1~1·'--0---_":::5---"---c-----'-------,J (a)-1~1'--0---_.l.5----0L----l5------.JI0 (b) FIGURE9.15(a)Square64-QAM constellation. (b)Themosttightlycoupledalternative tothat ofparta. ofFigure9.15a:asmallertransmitted averagesignalenergypersymbolforthesamebiterror rateonanAWGNchannel. <Ill 9.11Implications oftheInformation Capacity Theorem Nowthatwehaveanintuitivefeelfortheinformation capacity theorem, wemaygoon todiscussitsimplications inthecontextofaGaussian chalU1elthatislimitedinboth powerandbandwidth. Forthediscussion tobeuseful,however, weneedanidealframe workagainstwhichtheperformance ofapractical communication systemcanbeassessed. Tothisend,weintroduce thenotionofanidealsystemdefinedasonethattransmits data atabitrateRbequaltotheinformation capacity C.Wemaythenexpresstheaverage transmitted poweras (9.97) whereEbisthetransmitted energyperbit.Accordingly, theidealsystemisdefinedbythe equation C ( EbC)-=logz1+--B NoB(9.98) (9.99)Equivalently, wemaydefinethesignalenergy-per-bit tonoisepowerspectraldensityratio Eb/NointermsoftheratioC/Bfortheidealsystemas Eb2C/B-1 No C/B Aplotofbandwidth efficiency Rb/BversusEb/Noiscalledthebandwidth-efficiency dia gram.Agenericformofthisdiagram isdisplayed inFigure9.16,wherethecurvelabeled (9.100) (9.101)602 CHAPTER 9..FUNDAMENTAL LIMITS ININFORMATION THEORY "capacity boundary" corresponds totheidealsystemforwhichRb=C.BasedonFigure 9.16,wecanmakethefollowing observations: 1.Forinfinitebandwidth, theratioEblNoapproaches thelimitingvalue (~t-;~(~) =log2=0.693 ThisvalueiscalledtheShannon limitforanAWGNchannel, assuming acoderate ofzero.Expressed indecibels, itequals-1.6dB.The corresponding limitingvalue ofthechannelcapacity isobtained bylettingthechannelbandwidth BinEquation (9.95)approach infinity;wethusfindthat P Nolog2e whereeisthebaseofthenaturallogarithm. 2.Thecapacityboundary, definedbythecurveforthecriticalbitrateRb=C,separates combinations ofsystemparameters thathavethepotential forsupporting error-free transmission (Rb<C)fromthoseforwhicherror-free transmission isnotpossible (Rb>C).ThelatterregionisshownshadedinFigure9.16. 3.Thediagram highlights potential trade-offs amongEblNo,RbIB,andprobability of symbolerrorPe•Inparticular, wemayviewmovement oftheoperating pointalong 30 20 10 0.1Regionforwhich Rb<C FIGURE 9.16Bandwidth-efficiency diagram. 9.11Implications oftheInformation Capacity Theorem 603 ahorizontal lineastradingPeversusEb/NoforafixedRb/B.Ontheotherhand,we mayviewmovement oftheoperating pointalongaverticallineastradingPeversus Rb/BforafixedEb/No• ""EXAMPLE 9.10M-aryPCM Inthisexample, we lookatanM-aryPCMsysteminlightofthechannelcapacity theorem undertheassumption thatthesystemoperates abovetheerrorthreshold. Thatis,theaverage probability oferrorduetochannelnoiseisnegligible. WeassumethattheM-aryPCMsystemusesacodewordconsisting ofncodeelements, eachhavingoneofMpossible discreteamplitude levels;hencethename"M-ary." From Chapter 3werecallthatforaPCMsystemtooperateabovetheerrorthreshold, theremust beprovision foranoisemarginthatissufficiently largetomaintain anegligible errorratedue tochannelnoise.This,inturn,meanstheremustbeacertainseparation between theseM discreteamplitude levels.Callthisseparation ku,wherekisaconstant andu2=NoBisthe noisevariance measured inachannelbandwidth B.Thenumberofamplitude levelsMis usuallyanintegerpowerof2.Theaveragetransmitted powerwillbeleastiftheamplitude rangeissymmetrical aboutzero.Thenthediscreteamplitude levels,normalized withrespect totheseparation ku,willhavethevalues±l/2,+3/2,...,±(M-1)/2.Weassumethat theseMdifferent amplitude levelsareequallylikely.Accordingly, wefindthattheaverage transmitted powerisgivenby (9.102) Suppose thattheM-aryPCMsystemdescribed hereinisusedtotransmit amessage signalwithitshighestfrequency component equaltoWhertz.Thesignalissampled atthe Nyquist rateof2Wsamplespersecond.Weassumethatthesystemusesaquantizer ofthe midrisetype,withLequallylikelyrepresentation levels.Hence,theprobability ofoccurrence ofanyoneoftheLrepresentation levelsis1IL.Correspondingly, theamountofinformation carriedbyasinglesampleofthesignalislog2Lbits.Withamaximum sampling rateof2W samplespersecond,themaximum rateofinformation transmission ofthePCMsystem,mea suredinbitspersecond,isgivenby Rb=2Wlog2Lbitspersecond (9.103) SincethePCMsystemusesacodewordconsisting ofncodeelements, eachhavingoneofM possible discreteamplitude values,wehaveM"different possible codewords.Foraunique encoding process,werequire L=M" (9.104) Clearly,therateofinformation transmission inthesystemisunaffected bytheuseof anencoding process. Wemaytherefore eliminate Lbetween Equations (9.103)and(9.104) toobtain Rb=2Wnlog2Mbitspersecond (9.105) Equation (9.102)definestheaveragetransmitted powerrequired tomaintain anM-ary PCMsystemoperating abovetheerrorthreshold. Hence,solvingthisequation forthenumber ofdiscreteamplitude levels,M,weget (12P)'12 M=1+PNoB(9.106) 604 CHAPTER 9'"FUNDAl"IENTAL LIMITS ININFORMATION THEORY wherecr2=NoBisthevariance ofthechannelnoisemeasured inabandwidth B.Therefore subsrituting Equation (9.106)intoEquarion (9.105),weobtain ' (12P)Rb=Wnlogz1+PNoB (9.107) Thechannelbandwidth Brequired totransmit arectangular pulseofduration 112nW(rep_ resenting acodeelementinthecodeword)isgivenby(seeChapter3) B=KnW where Kisaconstant withavaluelyingbetween1and2.Usingtheminimum possiblevalue K=1,wefindthatthechannelbandwidth B=nW.Wemaythus rewrite Equation (9.107) as (12P)Rb=Blog21+PNoB (9.108) Theidealsystemisdescribed byShannon's channelcapacitytheorem, giveninEquation (9.95). Hence,comparing Equation (9.108)withEquation (9.95),weseethattheyareidentical if theaveragetransmitted powerinthePCMsystemisincreased bythefactorP1l2,compared withtheidealsystem.Perhapsthemostinteresting pointtonoteaboutEquation (9.108)is thattheformoftheequation isright:Powerandbandwidth inaPCMsystemareexchanged onalogarithmic basis,andtheinformation capacityCisproportional tothechannelband widthB. <$ !>'>ExAMPLE 9.11M-aryPSKand M-aryFSK Inthisexample, wecompare thebandwidth-power exchange capabilities ofM-aryPSKand M-aryFSKsignalsinlightofShannon's information capacity theorem. Consider firstaco herentM-aryPSKsystemthatemploys anonorthogonal setofMphase-shifted signalsforthe transmission ofbinarydata.Eachsignalinthesetrepresents asymbolwithlog2Mbits.Using thedefinition ofnull-to-null bandwidth, wemayexpressthebandwidth efficiency ofM-ary PSKasfollows[seeEquation (6.51)]: InFigure9.17a,weshowtheoperating pointsfordifferent numbers ofphaselevelsM=2, 4,8,16,32,64. Eachpointcorresponds toanaverageprobability ofsymbolerrorP,=10-5• Inthefigurewehavealsoincluded thecapacity boundary fortheidealsystem.Weobserve fromFigure9.17athatasMisincreased, thebandwidth efficiency isimproved, buttbevalue ofE;,INorequired forerror-free transmission movesawayfromtheShannon limit. Consider nextacoherent M-aryFSKsystemthatusesanorthogonal setofMfrequency shiftedsignalsforthetransmission ofbinarydata,withtheseparation betweenadjacentsignal frequencies setatlI2T,whereTisthesymbolperiod.AswiththeM-aryPSK,eachsignalin thesetrepresents asymbolwithlogzMbits.Thebandwidth efficiency ofM-aryFSKisas follows[seeEquation (6.143)]: Rb=210gzM BM InFigure9.17b,weshowtheoperating pointsfordifferent numbers offrequency levelsM= 2,4,8,16,32,64foranaverageprobability ofsymbolerrorP,=10-5•Inthefigure,we havealsoincluded thecapacity boundary fortheidealsystem.Weseethatincreasing Min (orthogonal) M-aryFSKhastheopposite effecttothatin(nonorthogonal) M-aryPSK.In particular, asMisincreased, whichisequivalent toincreased bandwidth requirement, the operating pointmovesclosertotheShannon limit. 41 30 209.11Implications oftheInformation Capacity Theorem 605 30 20 -6ifl'"10 I I 0.5M=21 0.4 0.3 0.2 0.1 (a)~,dB~I~10 5 4 M=2 IIII I /12 18 24 30 36 /M=8EbN;'dB ]0.5JM=16 0.4JM=32 0,M=64 ~0.3 0.2 0.1 (b) FIGURE 9.17(a)Comparison ofM-aryPSKagainsttheidealsystemforPo105andincreas- ingM.(b)Comparison ofM-aryFSKagainsttheidealsystemforPe=10-5andincreasing M. Ii>-ExAMPLE 9.12Capacity ofBinary-Input AWGNChannel Inthisexample, weinvestigate thecapacityofanAWGNchannelusingencoded binaryan tipodalsignaling (i.e.,levels-1and+1forbinarysymbols 0and1,respectively). Inpartic ular,weaddresstheissueofdetermining theminimum achievable biterrorrateasafunction ofEJNoforvaryingcoderater.Itisassumed thatthebinarysymbols 0and1are equiprobable. Lettherandomvariables XandYdenotethechannelinputandchanneloutput,re spectively; Xisadiscretevariable, whereas Yisacontinuous variable. Inlightofthesecond lineofEquation (9.81),wemayexpressthemutualinformation between thechannelinput andchanneloutputas I(X;Y)=h(Y)-h{YIX) Thesecondterm,h(YIX),istheconditional differential entropyofthechannelOutputY,given thechannelinputX.ByvirtueofEquations (9.89)and(9.93),thistermisjusttheentropyof aGaussian distribution. Hence,using,rtodenotethevarianceofthechannelnoise,wemay write Next,thefirstterm,h(Y),isthedifferential entropyofthechanneloutputY.Withtheuseof binaryantipodal signaling, theprobability densityfunction ofY,givenX=x,isamixture oftwoGaussian distributions withcornmon variance 0'2andmeanvalues-1and+1,as shownby I1[exP(-(Yi+1)2120'2) exp(-(Yi 1)212<r')] fY(Yix)=2:VhO' +-vr--mT(9.109) 606 CHAPTER 9IIFUNDAMENTAL LIMITS ININFORMATION THEORY Hence,wemaydetermine thedifferential entropyofYusingtheformula h(Y)=-roofAYiIx)log2[fy(Yil x)]dYi wherefY(Yilx)isdefinedbyEquation (9.109).Fromtheformulas ofh(YIX)andh(Y),itis clearthatthemutualinformation issolelyafunction ofthenoisevariance (I2.UsingM(ul) todenotethisfunctional dependence, wemaythuswrite I(X;Y)=M(ul) Unfortunately, thereisnoclosedformulathatwecanderiveforM((I2)becauseofthedifficulty ofdetermining h(Y).Nevertheless, thedifferential entropyh(Y)canbewellapproximated usingMonteCarlointegration, whichisstraightforward toprogram onadigitalcomputer; seeProblem 9.36. Because symbols 0and1areequiprobable, itfollowsthatthechannel capacity Cis equaltothemutualinformation between XandY.Hence,forerror-free datatransmission overtheAWGNchannel, thecoderatermustsatisfythecondition r<M(ul) (9.110) Arobustmeasure oftheratioEJNois EbP P No=Nor=2ulr wherePistheaveragetransmitted power,andNol2isthetwo-sided powerspectraldensity ofthechannelnoise.Without lossofgenerality, wemaysetP=1.Wemaythenexpressthe noisevariance as (9.111) ~ 2Bbr Substituting Equation (9.111)into(9.110)andrearranging terms,wegetthedesiredrelation: Eb1 No=2rM-1(r)(9.112) 0.5 -1-0.5 EbINo'dB (b)-1.510-1 c:::=::::=-~==::::=:=====~r:: r;:1 co10-2 ~ ~ 1510-3 E E'c :'E10-4 1/16 1/8 1/5 1/4 1/3112 Coderater,bits/transmission (a)34,-----~--~-~~~-~-_____n -2L-__ ~__ ~__ ~~~_~ __-.J 1/32-1 FIGURE 9.18Binaryantipodal signaling overanAWGNchannel. (a)Minimum Eb/Noversus thecoderater.(b)Minimum biterrorrate(BER)versusEblNoforvaryingcoderater. 9.12l...frwmatw..Capacity ofColored NoiseClumnel 607 whereM-1(r)istheinverseofthemutualinformation betweenthechannelinputandoutput, expressed asafunctionofthecoderater. UsingtheMonteCarlomethodtoestimatethedifferential entropyh(Y)andtherefore M-1(r),theplotsofFigure9.18arecomputedY Figure9.18aplotstheminimum EJNoversus thecoderaterforerror-free communication. Figure9.18bplotstheminimum achievable bit errorrateversusEb/Nowiththecoderaterasarunningparameter. FromFigure9.18we maydrawthefollowing conclusions: I>-Foruncodedbinarysignaling (i.e.,r=1),aninfiniteEb/Noisrequiredforerror-free communication, whichagreeswithwhatweknowaboutuncodeddatatransmission overanAWGNchannel. ..Theminimum EJNodecreases withdecreasing coderater,whichisintuitively satis fying.Forexample, forr1/2,theminimum valueofEb/Noisslightlylessthan0.2 dB. I>Asrapproaches zero,theminimum Eb/Noapproaches thelimitingvalueof-1.6dB, whichagreeswiththeShannonlimitderivedearlier;seeEquation (9.100). <il!I 9.12Information Capacity ofColored NoiseChannel13 Theinformation capacity theorem asformulated inEquation (9.95)appliestoaband limitedwhitenoisechannel. Inthissection,weextendShannon's information capacity theorem tothemoregeneralcaseofanonwhite, orcolored,noisechannel. Tobespecific, consider thechannel modelshowninFigure9.19awherethetransfer function ofthe channelisdenotedbyH(f).Thechannelnoisen(t),whichappearsadditively atthechannel output,ismodeled asthesamplefunction ofastationary Gaussian processofzeromean andpowerspectraldensitySN(f).Therequirement istwofold: 1.Findtheinputensemble, described bythepowerspectraldensitySx(f),thatmaxi mizesthemutualinformation betweenthechanneloutputy(t)andthechannelinput x(t),subjecttotheconstraint thattheaveragepowerofx(t)isfixedataconstant valueP. 2.Hence,determine theoptimum information capacity ofthechannel. Thisproblem isaconstrained optimization problem. Tosolveit,weproceedasfollows: I>Because thechannelislinear,wemayreplacethemodelofFigure9.19awiththe equivalent modelshowninFigure9.19b.Fromtheviewpoint ofthespectralchar acteristics ofthesignalplusnoisemeasured atthechanneloutput,thetwomodels ofFigure9.19areequivalent, provided thatthepowerspectraldensityofthenoise Input--il>-~~ Output M~t yoo Colorednoise net) (a)X(t)~outPutT '-------J yet) Modified colorednoise n'(t) (b) FIGURE9.19(a)Modelofband-limited, power-limited noisychannel. (h)Equivalent modelof thechannel. 608 CHAPTER 9iiiFUNDAMENTAL LIMITS ININFORMATION THEORY n'(t)inFigure9.19bisdefinedintermsofthepowerspectraldensityofthenoise n(t)inFigure9.19aas (9.111) whereIH(f)Iisthemagnitude response ofthechannel. !>-Tosimplifytheanalysis, weusethe"principle ofdivideandconquer" inamanner similartothatdescribed inSection6.12.Specifically, thechannelisdividedintoa largenumberofadjoining frequency slots,asillustrated inFigure9.20.Thesmaller wemaketheincremental frequency intervalJ1fofeachsubchannel, thebetteristhis approximation. ThenetresultofthesetwopointsisthattheoriginalmodelofFigure9.19aisreplaced by theparallelcombination ofafinitenumberofsubchannels, N,eachofwhichiscorrupted essentially by"band-limited whiteGaussian noise." Thekthsubchannel intheapproximation tothemodelofFigure9.19bisdescribed by k=1,2,..., N (9.14) Theaveragepowerofthesignalcomponent Xk(t)is k=1,2,...,N (9.115) whereSx(/k)isthepowerspectraldensityoftheinputsignalevaluated atthefrequency f=fk'Thevariance ofthenoisecomponent nk(t)is k=1,2,...,N (9.116) whereSN(/k)andIH(fk)Iarethenoisespectraldensityandthechannel's magnitude re sponseevaluated atthefrequencyfbrespectively. Theinformation capacity ofthekth subchannel is IH(j)Ik=1,2,..., N Staircase approximation ..((9.117) --.1. 0---------------1 FIGURE 9.20Staircase approximation ofanarbitrary magnitude responseIH(j)I;onlypositive frequency portionoftheresponse isshown. 9.12Information Capacity ofColored NoiseeJu.nnel 609 wherethefactor1/2accounts forthefactthat/:ifappliestobothpositiveandnegative frequencies. AlltheNsubchannels areindependent ofoneanother. Hencethetotalca pacityoftheoverallchannelisapproximately givenbythesummation N C=2:Ck k=l1N(P)=-22:/:iflog21+-1 k~l U'k(9.118) Theproblem wehavetoaddressistomaximize theoverallinformation capacity Csubject totheconstraint: N 2:Pk=P=constant k~l(9.119) Theusualprocedure tosolveaconstrained optimization problem istousethemethodof Lagrange multipliers; seeNote19inChapter 6.Toproceedwiththisoptimization, we firstdefineanobjective function thatincorporates boththeinformation capacity Cand theconstraint [i.e.,Equations (9.118)and(9.119)], asshownby J=i~,aflog2(1+:1)+A(P-~1Pk) (9.120) whereAistheLagrange multiplier. Next,differentiating theobjective functionJwith respecttoPkandsettingtheresultequaltozero,weobtain aflog2e ----'-------'''''-=- -A=0 Pk+U'~ Tosatisfythisoptimizing solution, weimposethefollowing requirement: fork=1,2,..., N (9.121) whereKisaconstant thatisthesameforallk.TheconstantKischosentosatisfythe averagepowerconstraint. Inserting thedefiningvaluesofEquations (9.115)and(9.116)intheoptimizing con ditionofEquation (9.121),simplifying, andrearranging terms,weget k=1,2,..., N (9.122) Let:IFAdenotethefrequency rangeforwhichtheconstantKsatisfiesthecondition :>SN(f) K-IH(fW Then,astheincremental frequency intervalafisallowedtoapproach zeroandthenumber ofsubchannels Ngoestoinfinity,wemayuseEquation (9.122)toformally statethatthe powerspectraldensityoftheinputensemble thatachieves theoptimum information ca pacityisanonnegative quantity definedby forfE:IFA otherwise(9.123) 610 CHAPTER 9"FUNDA1"1ENTAL LIMITS ININFORMATION 'THEORY Sincetheaveragepowerofarandomprocessisthetotalareaunderthecurveofthepower spectraldensityoftheprocess, wemayexpresstheaveragepowerofthechannelinput x(t)as P=J(K-SN(f))df fE7YAIH(fW(9.124) Foraprescribed Pandspecified SN(f)andH(!),theconstant KisthesolutiontoEquation (9.124). Theonlythingthatremainsforustodoistofindtheoptimum information capacity. Substituting theoptimizing solution ofEquation (9.121)intoEquation (9.118)andthen usingthedefiningvaluesofEquations (9.115)and(9.116),weobtain C"".!."£~flog2(KIH(fkW) 2k~l SN(fd Whentheincremental frequency intervalJ1fisallowedtoapproach zero,thisequation takesthelimitingform: 1J~(IH(fW)c=2:-00log2KSN(!)df (9.125) wheretheconstant KischosenasthesolutiontoEquation (9.124)foraprescribed input signalpowerP. I!'lWATER-FILLING INTERPRETATION OFTHEINFORMATION CAPACITY THEOREM Equations (9.123)and(9.124)suggestthepictureportrayed inFigure9.21.Specifically, wemakethefollowing observations: ..Theappropriate inputpowerspectraldensitySx(f)isdescribed asthebottomregions ofthefunction SN(f)/1H(!) 12thatliebelowtheconstant levelK,whichareshown shaded. £>TheinputpowerPisdefinedbythetotalareaoftheseshadedregions. Thespectral domainpictureportrayed hereiscalledthewater-filling (pouring) inter pretation inthesensethattheprocessbywhichtheinputpowerisdistributed across ---------0'----------/ FIGURE9.2IWater-filling interpretation ofinformation-capacity theorem foracolorednoisy channel. B B 0,,;Ie2";III,,;Ie+"2 otherwise9.13RateDistortion Theory 611 thefunction SN(f)1 IH(f) 12isidentical tothewayinwhichwaterdistributes itselfina vessel. Consider nowtheidealized caseofaband-limited signalinadditivewhiteGaussian noiseofpowerspectraldensityN(f)=No/2.Thetransferfunction H(f)isthatofanideal band-pass filterdefinedby H(f)={1' 0, whereIeisthemidband frequency andBisthechannelbandwidth. Forthisspecialcase, Equations (9.124)and(9.125)reduceto,respectively, P2B(K_~o) and Hence,eliminating Kbetweenthesetwoequations, wegetthestandard formofShannon's capacity theorem, definedbyEquation (9.95). I'l>ExAMPLE 9.13Capacity ofNEXT-Dominated Channel Fromthediscussion presented inSection4.8,werecallthatamajorchannelimpairment in digitalsubscriber linesisnear-end crosstalk (NEXT). Thepowerspectraldensityofthiscross talkmaybetakenas (9.126) cwhereSx(f)isthepowerspectraldensityofthetransmitted signalandHNEXT(f) isthetransfer function thatcouplesadjacent twistedpairs.Theonlyconstraint wehavetosatisfyinthis example isthatrhepowerspectraldensityfunction Sx(f)benonnegative forallf.Substituting Equation (9.126)into(9.123), wereadilyfindthatthiscondition issatisfied bysolvingfor Kas K=(1+IH"'EXT(fW)s (f)IH(fW x Finally,usingthisresultinEquation (9.125),wefindthatthecapacity oftheNEXT-dominated digitalsubscriber channelisgivenby 1] (IH(fW)2'O<Alog21+IHNExr(fW df where;!IiAistheserofpositiveandnegative frequencies forwhichSx(f)>O. I9.13HateDistortion Theory InSection9.3weintroduced thesourcecodingtheorem foradiscretememoryless source, according towhichtheaveragecode-word lengthmustbeatleastaslargeasthesource entropyforperfectcoding(i.e.,perfectrepresentation ofthesource).However, inmany practical situations thereare constraints thatforcethecodingtobeimperfect, thereby 612 CHAPTER 9'"FUNDAMENTAL LIMITS ININFORMATION THEORY resulting inunavoidable distortion. Forexample, constraints imposed byacommunication channelmayplaceanupperlimitonthepermissible coderateandtherefore averagecode_ wordlengthassigned totheinformation source.Asanotherexample, theinformation sourcemayhaveacontinuous amplitude asinthecaseofspeech,andtherequirement is toquantize theamplitude ofeachsamplegenerated bythesourcetopermititsrepresen_ tationbyacodewordoffinitelengthasinpulse-code modulation. Insuchcases,the problem isreferredtoassourcecodingwithafidelitycriterion, andthebranchofinfor mationtheorythatdealswithitiscalledratedistortion theory.I' Ratedistortion theory findsapplications intwotypesofsituations: I>Sourcecodingwherethepermitted codingalphabet cannotexactlyrepresent the information source,inwhichcaseweareforcedtodolossydatacompressiOll. ~Information transmission atarategreaterthanchannelcapacity. Accordingly, ratedistortion theorymaybeviewedasanaturalextension ofShannon's codingtheorems. !IllRATEDISTORTION FUNCTION Consider adiscretememoryless sourcedefinedbyanM-aryalphabet X:{XiIi=1,2,..., M},whichconsistsofasetofstatistically independent symbolstogether withtheassociated symbolprobabilities [PiIi=1,2,...,M}.LetRbetheaveragecoderateinbitspercode word.Therepresentation codewordsaretakenfromanotheralphabet Y:{YjIi=1,2,..., N}.Thesourcecodingtheorem statesthatthissecondalphabet provides aperfectrepre sentation ofthesourceprovided thatR>H,whereHisthesourceentropy. Butifweare forcedtohaveR<H,thenthereisunavoidable distortion andtherefore lossof information. LetP(Xi,Yj)denotethejointprobability ofoccurrence ofsourcesymbolXiandrep resentation symbolYi'Fromprobability theory,wehave (9.127) wherep(YjIXi)isatransition probability. Letd(x"y;ldenoteameasureofthecostincurred inrepresenting thesourcesymbolXibythesymbolYi;thequantity d(x"Yi)isreferredto asasingle-letter distortion measure. Thestatistical averageofd(xi,Yj)overallpossible sourcesymbols andrepresentation symbols isgivenby M N d=2,2,P(xi)p(YjIXi)d(Xi'Yj) i=lj='l(9.128) Notethattheaveragedistortiondisanonnegative continuous function ofthetransition probabilities P(YiIXi)thataredetermined bythesourceencoder-decoder pair. Aconditional prob~bility assignment P(YiIXi)issaidtobeD-admissible ifandonly iftheaveragedistortion dislessthanorequaltosomeacceptable valueD.Thesetofall D-admissible conditional probability assignments isdenotedby Foreachsetoftransition probabilities, wehaveamutualinformation M N (P(Ylx))I(X;Y)=~~P(Xi)P(Yilxi) logp(Yj),(9.129) (9.130) 9.13RateDistortion Theory 613 Yj FIGURE9.22Summary ofratcdistortion theory. Aratedistortion functionR(D)isdefinedasthesmallestcodingratepossibleforwhich theaveragedistortion isguaranteed nottoexceedD.LetPDdenotethesettowhichthe conditional probability P(YiIXi)belongsforaprescribed D.Then,forafixedDwewrite'S R(D)=minI(X;Y) p(YjIXj)EPD subjecttotheconstraint(9.131) N 2:p(Yilxi)=1 1=1fori=1,2,...,M (9.132) Theratedistortion function R(D)ismeasured inunitsofbitsifthebase-2logarithm is usedinEquation (9.130).Intuitively, weexpectthedistortion Dtodecrease astherate distortion function R(D)isincreased. Wemaysayconversely thattolerating alargedis tortionDpermitstheuseofasmallerrateforcodingandlortransmission ofinformation. Figure9.22summarizes themainparameters ofratedistortion theory.Inparticular, giventhesourcesymbols [Xi}andtheirprobabilities {Pi}andgivenadefinition ofthesingle letterdistortion measure d(xi'Yi)'thecalculation oftheratedistortion function R(D) involves findingtheconditional probability assignment P(YiIXi)subjecttocertaincon straintsimposed onP(YiIXi)'Thisisavariational problem, thesolution ofwhichisun fortunately notstraightforward ingeneral. ~EXAMPLE 9.14Gaussian Source Consider adiscrete-time, memoryless Gaussian sourCewithzeromeanandvariancecr.Letx denotethevalueofasamplegenerated bysuchasource.Letydenoteaquantized versionof xthatpermitsafiniterepresentation ofit.Thesquarederrordistortion d(x,y)(x-y)" provides adistortion measurethatiswidelyusedforcontinuous alphabets. Theratedistortion function fortheGaussian sourcewithsquarederrordistortion, asdescribed herein,isgiven by (9.133) Inthiscase,weseethatR(D)-->00asD-->0,andR(D)=0forD=cr. 614 CHAPTER 9OJFUNDAMENTAL LIMITS ININFORMATION THEORY 345 6 Sourceindex:i FIGURE 9.23Reversewater-filling pictureforasetofparallelGaussian processes. ~EXAMPLE 9.I 5SetofParallel Gaussian Sources Consider nextasetofNindependent Gaussian randomvariables[X,j)'i"whereXihaszero meanandvariancecTf.Usingthedistortion measure N d=2:(Xi-xy i"'"1 andbuildingontheresultofExample 9.14,wemayexpresstheratedistortion functionfor thesetofparallelGaussian sourcesdescribed hereas whereDiisirselfdefinedbyN1(cTf)R(D)=2:-2log--": 1=1 Dz(9.134) Di={~ifA<O"i ifA2':O"t(9.135) andtheconstantAischosensoastosatisfythecondition N 2:Di=D i=l(9.136) Equations (9.135)and(9.136)maybeinterpreted asakindof"warer-filling inreverse," as illustrated inFigure9.23.Firsr,wechooseaconstant Aandonlythesubsetofrandomvariables whosevariances exceedtheconstant A.Nobitsareusedtodescribetheremaining subsetof randomvariables whosevatiances arelessthantheconstant A. <li! I9.14DataCompression Ratedistortion theorynaturally leadsustoconsider theideaofdatacompression that involves apurposeful orunavoidable reduction intheinformation contentofdatafroma continuous ordiscretesource.Specifically, wemaythinkofadatacompressor, orsignal compressor, asadevicethatsupplies acodewiththeleastnumber ofsymbols forthe representation ofthesourceoutput,subjecttoapermissible oracceptable distortion. The datacompressor thusretainstheessential information contentofthesourceoutputby blurring finedetailsinadeliberate butcontrolled manner. Accordingly, datacompression (9.137)9.14DataCmnpression 615 isalossyoperation inthesensethatthesourceentropy isreduced (i.e.,information is lost),irrespective ofthetypeofsourcebeingconsidered. Inthecaseofadiscretesource,thereasonforusingdatacompression istoencode thesourceoutputataratesmallerthanthesourceentropy. Bysodoing,thesourcecoding theorem isviolated, whichmeansthatexactreproduction oftheoriginaldataisnolonger possible. Inthecaseofacontinuous source,theentropyisinfinite,andtherefore asignal compression codemustalwaysbeusedtoencodethesourceoutputatafiniterate.Con sequently, itisimpossible todigitallyencodeananalogsignalwithafinitenumberofbits withoutproducing somedistortion. Thisstatement isinperfectaccordwiththeideaof pulse-code modulation, whichwasstudiedinChapter3.Thereitwasshownthatquan tization, whichisbasictotheanalog-to-digital conversion processinpulse-code modula tion,alwaysintroduces distortion (knownasquantization noise)intothetransmitted sig nal.Aquantizer maytherefore beviewedasasignalcompressor. Theuniformandnonuniform quantizers considered inChapter3aresaidtobescalar quantizers inthesensethattheydealwithsamplesoftheanalogsignal(i.e.,continuous sourceoutput)oneatatime.Eachsampleisconverted intoaquantized value,withthe conversion beingindependent fromsampletosample.Ascalarquantizer isarathersimple signalcompressor, whichmakesitattractive forpractical use.Yetitcanprovideasur prisingly goodperformance; thisisespecially soifnonuniform quantization isused. Thereisanotherclassofquantizers knownasvectorquantizers thatuseblocksof consecutive samplesofthesourceoutputtoformvectors,eachofwhichistreatedasa singleentity.Theessential operation inavectorquantizer isthequantization ofarandom vector16byencoding itasabinarycodeword.Thevectorisencodedbycomparing itwith acodebook consisting ofasetofstoredreference vectorsknownascodevectorsorpat terns.Eachpatterninthecodebookisusedtorepresent inputvectorsthatareidentified bytheencodertobesimilartotheparticular pattern,subjecttothemaximization ofan appropriate fidelitycriterion. Theencoding processinavectorquantizer maythusbe viewedasapatternmatching operation. LetNbethenumberofcodevectorsinthecodebook, kbethedimension ofeach vector(i.e.,thenumberofsamplesineachpattern), andrbethecodedtransmission rate inbitspersample.Thesethreeparameters arerelatedasfollows: log2Nr=-k- Then,assuming thatthesizeofthecodebookissufficiently large,thesignal-to-quanti zationnoiseratio(SNR)forthevectorquantizer isgivenby (log2N)1010glo(SNR) =6-k-+CkdB (9.138) whereCkisaconstant (expressed indB)thatdependsonthedimensions k.According to Equation (9.138),theSNRforavectorquantizer increases approximately attherateof 6/kdBforeachdoubling ofthecodebooksize.Equivalently, wemaystatethattheSNR increases by6dBperunitincreaseinrate(bitspersample)asinthestandard PCMusing auniformscalarquantizer. Theadvantage ofthevectorquantizer overthescalarquantizer isthatitsconstant termCkhasahighervalue,becausethevectorquantizer optimally exploitsthecorrelations amongthesamplesconstituting avector.Specifically, theconstant Ckincreases withthedimension k,approaching theultimate rate-distortion limitfora 616 CHAPTER 9FUNDAMENTAL LIMITS ININFORMATION THEORY givensourceofinformation. However, theimprovement inSNRisattained atthecostof increased encoding complexity, whichgrowsexponentially withthedimension kfora specified rater.Unfortunately, thisisthemainobstacletothewideuseofvectorquanti zationinpractice. Nevertheless, incertainapplications, theissueofcomputational com plexityismitigated byexploiting thecapability ofVLSItechnology toconcentrate ahighly complex signalprocessor onasiliconchip.Forexample, thatisprecisely whatisdonein theuseofcode-excited linearpredictive (CELP)modeling ofspeechinwirelesscommu_ nication systemsoftheCDMAtype,namely,theIS-95system.Fromthedescription of CELPpresented inSection8.9,itisclearthattheCELPmodeling ofspeechisanexample ofvectorquantization. I9.15Summary andDiscussion Inthischapterweestablished fourfundamental limitsondifferent aspectsofacommu_ nicationsystem.Thelimitsareembodied inthesourcecodingtheorem, thechannelcoding theorem, theinformation capacity theorem, andtheratedistortion function. Thesourcecodingtheorem, Shannon's firsttheorem, provides themathematical tool forassessing datacompaction, thatis,losslesscompression ofdatagenerated byadiscrete memoryless source.Thetheorem tellsusthatwecanmaketheaveragenumberofbinary codeelements (bits)persourcesymbolassmallas,butnosmallerthan,theentropyofthe sourcemeasured inbits.Theentropyofasourceisafunction oftheprobabilities ofthe sourcesymbols thatconstitute thealphabet ofthesource.Sinceentropyisameasureof uncertainty, theentropyismaximum whentheassociated probability distribution gener atesmaximum uncertainty. Thechannelcodingtheorem, Shannon's secondtheorem, isboththemostsurprising andthesinglemostimportant resultofinformation theory.Forabinarysymmetric chan nel,thechannelcodingtheorem tellsusthatforanycoderaterlessthanorequaltothe channelcapacity C,codesdoexistsuchthattheaverageprobability oferrorisassmallas wewantit.Abinarysymmetric channelisthesimplest formofadiscrelememoryless channel.Itissymmetric becausetheprobability ofreceiving a 1ifa 0issentisthesame astheprobability ofreceiving a 0ifa 1issent.Thisprobability, theprobability thatan errorwilloccur,istermedatransition probability. Thetransition probability pisdeter minednotonlybytheadditivenoiseatthechanneloutputbutalsobythekindofreceiver used.Thevalueofpuniquely definesthechannelcapacity C. Shannon's thirdremarkable theorem, theinformation capacitytheorem, tellsusthat thereisamaximum totherateatwhichanycommunication systemcanoperatereliably (i.e.,freeoferrors)whenthesystemisconstrained inpower.Thismaximum rateiscalled theinformation capacity, measured inbitspersecond.Whenthesystemoperates atarate greaterthantheinformation capacity, itiscondemned toahighprobability oferror, regardless ofthechoiceofsignalsetusedfortransmission orthereceiverusedforpro cessingthereceived signal. Finally,theratedistortion function provides themathematical toolforsignalcom pression (i.e.,solvingtheproblem ofsourcecodingwithafidelitycriterion): Therate distortion function canbeappliedtoadiscreteaswellascontinuous memoryless source. Whentheoutputofasourceofinformation iscompressed ina lossless manner,the resulting datastreamusuallycontains redundant bits.Theseredundant bitscanberemoved byusingalosslessalgorithm suchasHuffman codingortheLempel-Ziv algorithm for datacompaction. Wemaythusspeakofdatacompression followed bydatacompaction astwoconstituents ofthedissection ofsourcecoding,whichissocalledbecauseitrefers NotesandReferences 617 exclusively tothesourcesofinformation. Insomesourcecodingapplications, wehavea thirdconstituent, namely,dataencryption, whichfollowsdatacompaction. Thepurpose ofdataencryption istodisguisethedata(bit)streaminsuchawaythatithasnomeaning toanunauthorized receiver. Somebasicaspectsofcryptography, whichencompasses both encryption anddecryption, followquitenaturally frominformation theory,asdiscussed inAppendix 5.Otherissuesrelatingtocryptography arealsodiscussed inthatappendix. Onelastcomment isinorder.Shannon's information theory,aspresented inthis chapter, hasbeenentirelyinthecontextofmemorylesssourcesandchannels. Thetheory canbeextended todealwithsourcesandchannels withmemory, inwhichcaseasymbol ofinterestdepends onpreceding symbols; however, thelevelofexposition neededtodo thisisbeyondthescopeofthisbookY INOTES ANDREFERENCES 1.According toLucky(1989),thefirstmentionoftheterminformation theorybyShannon occursina1945memorandum entitled"AMathematical TheoryofCryptography." Itis rathercuriousthatthetermwasneverusedintheclassic1948paperbyShannon, which laiddownthefoundations ofinformation theory.Foranintroductory treatment ofinfor mationtheory,seeChapter2ofLucky(1989)andthepaperbyWyner(1981);seealso thebooksofAdamek(1991),Hamming (1980),andAbramson (1963).Formoreadvanced treatments ofthesubject,seethebooksofCoverandThomas (1991),Blahut(1987), McEliece (1977),andGallager (1968).Foracollection ofpapersonthedevelopment of information theory(including the1948classicpaperbyShannon), seeSlepian(1974).For acollection ofthepaperspublished byShannon, seeSloaneandWyner(1993). 2.Theuseofalogarithmic measure ofinformation wasfirstsuggested byHartley(1928); however, Hartleyusedlogarithms tobase10. 3.Instatistical physics,theentropyofaphysicalsystemisdefinedby(Reif,1967,p.147) Ef=klogn wherekisBoltzmann's constant,nisthemlmberofstatesaccessible tothesystem,and logdenotesthenaturallogarithm. Thisentropyhasthedimensions ofenergybecauseits definition involvestheconstant k.Inparticular, itprovides aquantitative measureofthe degreeofrandomness ofthesystem.Comparing theentropyofstatistical physicswiththat ofinformation theory,weseethattheyhaveasimilarform.Foradetaileddiscussion of therelationbetweenthem,seePierce(1961,pp.184-207) andBrillouin (1962). 4.Fortheoriginalproofofthesourcecodingtheorem, seeShannon (1948).Ageneralproof ofthesourcecodingtheorem isalsogiveninthefollowing books:ViterbiandOmura (1979,pp.13-19),McEliece (1977,Chapter3),andGallager (1968,pp.38-55).The sourcecodingtheorem isalsoreferredtointheliterature asthenoiseless codingtheorem, noiseless inthesensethatitestablishes thecondition forerror-free encoding tobepossible. 5.ForproofoftheKraft-McMillan inequality, seeCoverandThomas(1991,pp.82-84), Blahut(1990,pp.298-299), andMcEliece (1977,pp.239-240). ForaproofofEquation (9.23),seeCoverandThomas (1991),pp.87-88),Blahut(1990,pp.300-301), and McEliece (1977,pp.241-242). 6.TheHuffman codeisnamedafteritsinventor: D.A.Huffman (1952).Forareadable accountofHuffman codinganditsuseindatacompaction, seeAdamek (1991). 7.TheoriginalpapersontheLempel-Ziv algorithm areZivandLempel(1977,1978).For readable descriptions oftheLempel-Ziv algorithm, seeLucky(1989,pp.118-122), Blahut 618 CHAPTER 9'"FUNDAMENTAL LIMITS ININFORMATION THEORY (1990,pp.314-319), andGitlin,Hayes,andWeinstein (1992,pp.120-122). Fotthe application oftheLempel-Ziv algorithm tothecompaction ofEnglishtext,seeLucky (1989,pp.122':'128) andthepaperbyWelch(1984);seealsothereviewpaperbyWeiss andShremp(1993). 8.Thechannelcodingtheorem isalsoknownasthenoisycodingtheorem. Theoriginalproof ofthetiIeorem isgiveninShannon (1948).Aproofofthetheorem isalsopresented in Hamming (1980,Chapters 9and10)insufficient detailsothatageneralappreciation of relevantresultsisdeveloped. ThesecondpartofthetiIeorem isreferredtointiIeliterature astheconverse tothecodingtheorem. AproofoftiIistheorem ispresented inthefollowing references: ViterbiandOmura(1979,pp.28-34)andGallager (1968,pp.76-82). 9.Thequantity [jy(x)IOg2(j:i:;) dx ontheleft-hand sideofEquation (9.70)iscalledrelativeentropyortiIeKullback-Leibler divergence betweentheprobability densityfunctions fx(x)andjy(x);seeKullback (1968). 10.Shannon's information capacitytiIeoremisalsoreferredtointiIeliterature astheShannon Hartleylawinrecognition ofearly"YorkbyHartleyoninformation transmission (Hartley, 1928).Inparticular, Hartleyshowedthattheamountofinformation thatcanbetrans mittedoveragivenchannelisproportional totheproduct of thechannelbandwidth and thetimeofoperation. 11.Alucidexposition ofspherepackingispresented inCoverandThomas (1991,pp.242 243);seealsoWozencraft andJacobs(1965,pp.323-341). 12.PartsaandbofFigure9.18followthecorresponding partsofFigure6.2inthebookby Frey(1998). 13.Forarigorous treatment oftiIeinformation capacity ofacolorednoisychannel, seeGal lager(1968).Theideaofreplacing thechannelmodelofFigure9.19awiththatofFigure 9.19bisdiscussed inGitlin,Hayes,andWeinstein (1992). 14.Foracomplete treatment ofratedistortion theory,seethebookbyBerger(1971);this subjectisalsotreatedinsomewhat lessdetailinCoverandThomas (1991),McEliece (1977),andGallager (1968). 15.Forthederivation ofEquation (9.131),seeCoverandThomas(1991,p. 345).Analgorithm forcomputation oftheratedistortion function R(D)definedinEquation (9.131)isde scribedinBlahut(1987,pp.220-221) andCoverandThomas (1991,pp.364-367). 16.Fortheearlypapersonvectorquantization, seeGersho(1979)andLinde,Buzo,andGray (1980).Foratutorialreviewofvectorquantization, seeGray(1984).Equation (9.138), definingtheSNRforavectorquantizer, isdiscussed inGershoandCuperman (1983).For acomplete treatment ofvectorquantization, seethebookbyGershoandGray(1992). 17.Fordetaileddiscussion ofdiscretechannels withmemory, seeGallager (1968,pp.97-112) andAsh(1965,pp.211-229). IPROBLEMS Entropy 9.1Letpdenotetheprobability ofsomeevent.Plottheamountofinformation gainedbythe occurrence ofthiseventfor0~p~1. k=1,2,...,nProblems 619 9.2Asourceemitsoneoffourpossiblesymbolsduringeachsignaling interval.Thesymbols occurwiththeprobabilities: Po=0.4 P,=0.3 pz=0.2 P3=0.1 Findtheamountofinformation gainedbyobserving thesourceemitting eachofthese symbols. 9.3Asourceemitsoneoffoursymbols So,s,'sz,andS3withprobabilities 1/3,1/6, 1/4, and 1/4,respectively. Thesuccessive symbolsemittedbythesourcearestatistically indepen dent.Calculate theentropyofthesource. 9.4LetXrepresent theoutcome ofasinglerollofafairdie.WhatistheentropyofX? 9.5ThesamplefunctionofaGaussian processofzeromeanandunitvariance isuniformly sampledandthenappliedtoauniformquantizer havingtheinput-output amplitude char acteristic showninFigureP9.5.Calculate theentropyofthequantizer output. Output ----...:r----::+---:-----Input FIGUREP9.5 9.6Consider adiscretememoryless sourcewithsourcealphabet :J'={so,S"•••,SK-1}and sourcestatistics {Po,p"...,PK-I}'Thenthextension ofthissourceisanotherdiscrete memoryless sourcewithsourcealphabet gm=lao,a"...,aM-I},whereM=Kn.Let P(ai)denotetheprobability ofai' (a)Showthat M-I 2:P(ai)=1 i=O whichistobeexpected. (b)Showthat M-1 (1 )2:P(ai)logz---:-=H(:J'), 1=0 P'k wherePi,istheprobability ofsymbol Si"andHW)istheentropyoftheoriginal source. (c)Hence,showthat M-1 1 H(gm)=~P(ai)logzP(ai) =nHW) 620 CHAPIER 9illFUNDAMENTAL LIMITS ININFORMATION THEORY 9.7Consider adiscretememoryless sourcewithsourcealphabet g'={so,s"S2}andsource statistics {a.?,0.15,0.15}. (a)Calculate theentropyofthesource. (b)Calculate theentropyofthesecond-order extension ofthesource. 9.8Itmaycomeasasurprise, butthenumberofbitsneededtostoretextismuchlessthan thatrequired tostoreitsspokenequivalent. Canyouexplainthereasonforit? DataCompaction 9.9Consider adiscrete memorylesssourcewhosealphabet consists ofKequiprobable symbols. (a)Explainwhytheuseofafixed-length codefortherepresentation ofsuchasourceis aboutasefficientasanycodecanbe. (b)Whatconditions havetobesatisfied byKandthecode-word lengthforthecoding efficiency tobe100percent? 9.10Consider thefourcodeslistedbelow: Symbol Code1Code11 Code111 CodeIV SO a a a 00 51 10 01 01 01 S2 110 001 all 10 5, 1110 0010 110 110 54 1111 0011 111 111 (a)Twoofthesefourcodesareprefixcodes.Identifythem,andconstruct theirindividual decisiontrees. (b)ApplytheKraft-McMillan inequality tocodesI,II,III,andIV.Discussyourresults inlightofthoseobtained inpart(a). 9.11Consider asequence ofletersoftheEnglishalphabet withtheirprobabilities ofoccurrence asgivenhere: Letter Probabilitya 0.10.1I 0.2m 0.1n 0.1a 0.2P 0.1y 0.1 Compute twodifferent Huffman codesforthisalphabet. Inonecase,moveacombined symbolinthecodingprocedure ashighaspossible, andinthesecondcase,moveitas lowaspossible. Hence,foreachofthetwocodes,findtheaveragecode-word lengthand thevariance oftheaveragecode-word lengthovertheensemble ofletters. 9.12Adiscretememoryless sourcehasanalphabet ofsevensymbols whoseprobabilities of occurrence areasdescribed here: Symbol ProbabilitySo 0.25SI 0.25 0.125 0.125 0.125S5 0.0625S6 0.0625 Compute theHuffman codeforthissource,movinga"combined" symbolashighas possible. Explainwhythecomputed sourcecodehasanefficiency of100percent. 9.13Consider adiscretememoryless sourcewithalphabet {so,s"S2}andstatistics {a.?,0.15, 0.15}foritsoutput. (a)ApplytheHuffman algorithm tothissource.Hence,showthattheaveragecode wordlengthoftheHuffman codeequals1.3bits/symbol. Problems 621 (b)Letthesourcebeextended toordertwo.ApplytheHuffman algorithm totheresulting extended source,andshowthattheaveragecode-word lengthofthenewcodeequals 1.1975bits/symbol. (c)Compare theaveragecode-word lengthcalculated inpart(b)withtheentropyofthe originalsource. 9.14FigureP9.14showsaHuffman tree.WhatisthecodewordforeachofthesymbolsA, B,C,D,E,F,andGrepresented bythisHuffman tree?Whataretheirindividual code wordlengths? 3/8A 3/16 B C3/16 a a1/8a a G FIGUREP9.14 9.15Acomputer executes fourinstructions thataredesignated bythecodewords (00,01,10, 11).Assuming thattheinstructions areusedindependently withprobabilities (1/2,118,1/8,1/4),calculate thepercentage bywhichthenumberofbitsusedforthe instructions maybereducedbytheuseofanoptimum sourcecode.Construct aHuffman codetorealizethereduction. 9.16Consider thefollowing binarysequence 11101001100010110100 ... UsetheLempel-Ziv algorithm toencodethissequence. Assumethatthebinarysymbolsoand1arealreadyinthecodebook. BinarySymmetric Channel 9.17Consider thetransition probability diagram ofabinarysymmetric channelshownin Figure9.8.Theinputbinarysymbols 0and1occurwithequalprobability. Findthe probabilities ofthebinarysymbols0and1appearing atthechanneloutput. 9.18Repeatthecalculation inProblem 9.17,assuming thattheinputbinarysymbols0and1 occurwithprobabilities 1/4and3/4,respectively. Mutuallnfonnation andChannel Capacity 9.19Consider abinarysymmetric challllelcharacterized bythetransition probability p.Plot themutualinformation ofthechallllelasafunction ofPhtheaprioriprobability of symbol1atthechannelinput;doyourcalculations forthetransition probability p=0, 0.1,0.2,0.3, 0.5. 622 CUAPTER 9"FUNDAMENTAL LIMITS ININI'ORMATION ThEORY 9.20Figure9.10depictsthevariation ofthechannelcapacityofabinarysymmetric charmI withthetransition probability p.UsetheresultsofProblem9.19toexplainthisvariatioen. 9.21Consider thebinarysymmetric channeldescribed inFigure9.8.LetPodenotetheprob. abilityofsendingbinarysymbolXo=0,andletP,=1 -Podenotetheprobability f sendingbinarysymbolx,1.Letpdenotethetransition probability ofthechannel. 0 (a)Showthatthemutualinformation betweenthechannelinputandchanneloutputis givenby [('if;'Y)=~(z)-~(p) where H(z)=zlog2G)+(1-z)log2(;~z) z=PoP+(1-Po)(1-p) and H(p)=plog2G)+(1-p)log2(1~p) (b)ShowthatthevalueofPothatmaximizes I('if;'Y)isequalto112. (c)Hence,showthatthechannelcapacityequals C=1 -H(p) 9.22Twobinarysymmetric channels areconnected incascade,asshowninFigureP9.22.Find theoverallchannelcapacity ofthecascaded connection, assuming thatbothchannels havethesametransition probability diagramshowninFigure9.8. Intpul FIGUREP9.22Oulpul 9.23Thebinaryerasurechannelhastwoinputsandthreeoutputsasdescribed inFigureP9.23. Theinputsarelabeled0and1,andtheoutputsarelabeled0,1,ande.Afractionaof theincoming bitsareerasedbythechannel.Findthecapacityofthechannel. I-a00<:;:::----....;..-----0<) 0 I-a FIGUREP9.23 Problems 623 9.24Consider adigitalcommunication systemthatusesarepetition codeforthechannel encoding/decoding. Inparticular, eachtransmission isrepeated ntimes,wheren=2m+ 1isanoddinteger.Thedecoderoperates asfollows.Ifinablockofnreceived bits,the numberofOsexceedsthenumberof1s,thedecoderdecidesinfavorofaO.Otherwise, itdecidesinfavorofa1.Anerroroccurswhenm+1ormoretransmissions outofn= 2m+1areincorrect. Assumeabinarysymmetric channel. (a)Forn=3,showthattheaverageprobability oferrorisgivenby Pe=3p2(1p)+p3 wherepisthetransition probability ofthechannel. (b)Forn=5,showthattheaverageprobability oferrorisgivenby P,=10P3(1_p)2+5p4(1p)+p5 (c)Hence,forthegeneralcase,deducethattheaverageprobability oferrorisgivenby Pe=±(~)pi(1 p)"-i i=m+l t Differential Entropy 9.25LetX"Xl>'..,Xndenotetheelements ofaGaussian vectorX.TheXiareindependent withmean f.L;andvariance a},i=1,2,...,n.Showthatthedifferential entropyofthe vectorXequals nh(X)=2:log2[21Te(aiai ...a~)'h'] Whatdoesh(X)reducetoifthevariances areequal? 9.26Acontinuous random variableXisconstrained toapeakmagnitude M;thatis, -M<X<M. (a)Showthatthedifferential entropyofXismaximum whenitisuniformly distributed, asshownby {112M, fx(x)=0,-M<x::5M otherwise (b)Showthatthemaximum differential entropyofXislog22M. 9.27Provetheptoperties giveninEquations (9.79)to(9.81)forthemutualinformation [(X;Y). 9.28Consider thecontinuous randomvariableYdefinedby Y=X+N whereXandNarestatistically independent. Showthattheconditional differential en tropyofY,givenX,equals h(YIX) =h(N) whereh(N)isthedifferential entropyofN. Information Capacity 9.29Avoice-grade channelofthetelephone network hasabandwidth of3.4kHz. (a)Calculate theinformation capacityofthetelephone channelforasignal-to-noise ratio of30dB. (b)Calculate theminimum signal-to-noise ratiorequired tosupportinformation trans missionthroughthetelephone channelattherateof9,600b/s. 624 CHAPTER 9..FUNDAMENTAL LIl\lITS ININFORMATION THEORY 9.30Alphanumeric dataareenteredintoacomputer fromaremoteterminal throughaVoice_ gradetelephone channel. Thechannelhasabandwidth of3.4kHzandoutputsignal-to_ noiseratioof20dB.Theterminal hasatotalof128symbols. AssumethatthesYmbol areequiprobable andthesuccessive transmissions arestatistically independent. s (a)Calculate theinformation capacity ofthechannel. (h)Calculate themaximum symbolrateforwhicherror-free transmission overthechan_ nelispossible. 9.31Ablack-and-white television picturemaybeviewedasconsisting ofapproximately 3 x105elements, eachofwhichmayoccupyoneof10distinctbrightness levelswith equalprobability. Assumethat(1)therateoftransmission is30pictureframespersecond and(2)thesignal-to-noise ratiois30dB. ' Usingtheinformation capacity theorem, calculate theminimum bandwidth re quiredtosupportthetransmission oftheresulting videosignal. Note:Asamatterofinterest,commercial television transmissions actuallyemployaband widthof4.2MHz,whichfitsintoanallocated bandwidth of6MHz. 9.32Inthisproblem, wecontinue withExample 9.9.SupposethatthetightlypackedCOnstel lationofFigure9.15bisscaledupwardsothatthetransmitted signalenergypersymbol ismaintained atthesameaveragevalueasthatconsumed bythe64-QAM squarecon stellation ofFigure9.15a.Construct thenewconstellation thatresultsfromthisscaling. Howdoesthebiterrorrateofthisnewconstellation compare withthatofFigure9.15a? Justifyyouranswer. 9.33Thesquaredmagnitude response ofatwisted-pair channelcanbemodeled as IH(fW=exp(-aYf) Theconstant aisdefinedby kl a=I;; wherekisaconstant depending onwiregauge,10isareference linelength,andIisthe actuallengthofthetwistedpairunderstudy.Thesquaredmagnitude response ofthe coupling responsible forNEXThastheform IHNEdf) 12={3f3/2 where{3isaconstant thatdependsonthetypeofcableused. Formulate theexpression fortheinformation capacity oftheNEXT-dominated channeldescribed here. DataCompression 9.34Equation (9.138)forthesignal-to-noise ratio(SNR)ofavectorquantizer includesthe SNRformulaofEquation (3.33)forstandard pulse-code modulation asaspecialcasefor whiclJk=1.Justifythevalidityofthisinclusion. 9.35Allpractical datacompression anddatatransmission schemesliebetweentwolimitsset bytheratedistortion function andthechannelcapacitytheorem. Bothofthesetheorems involvethenotionofmutualinformation, butindifferent ways.Elaborate ontheissues raisedbythesetwostatements. Computer Experiment 9.36Inthisproblem, we revisit Example 9.12,whichdealswithcodedbinaryantipodal sig nalingoveranadditivewhiteGaussian noise(AWGN) channel. StartingwithEquation (9.112)andtheunderlying theory,developasoftware packageforcomputing themini mumEb/Norequired foragivenbiterrorrate,whereEbisthesignalenergyperbit,and Problems 625 AreaA ----f-- Shadedarea =fg(Y)dy =pA wherepisthefractionof randomly chosenpointsthat lieunderthecurveofg(y). FIGUREP9.36 No/2isthenoisespectraldensity.Hence,compute theresultsplotted in partsaandbof Figure9.18. Asmentioned inExample 9.12,thecomputation ofthemutualinformation between thechannelinputandchanneloutputiswellapproximated usingMonteCarlointegra tion.Toexplainhowthismethodworks,consider afunction g(y)thatisdifficultto sample randomly, whichisindeedthecasefortheproblem athand.(Forourproblem, thefunc tiong(y)represents thecomplicated integrand intheformulaforthedifferential entropy ofthechanneloutput.)Forthecomputation, proceedasfollows: '"FindanareaAthatincludestheregionofinterestandthatiseasilysampled. l>ChooseNpoints,uniformly randomly insidetheareaA. ThentheMonteCarlointegration theorem statesthattheintegralofthefunction g(y) withrespecttoyisapproximately equaltotheareaAmultiplied bythefractionofpoints thatresidebelowthecurveofg,asillustrated inFigureP9.36.Theaccuracy oftheap proximation improves withincreasing N. ERROR-CONTROL CODING Thischapteristhenaturalsequeltothepreceding chapteronShannon's information theory.Inparticular, inthischapterwepresenterror-control codingtechniques that provide different waysofimplementing Shannon's channel-coding theorem. Eacherror controlcodingtechnique involves theuseofachannelencoderinthetransmitter anda decoding algorithm inthereceiver. Theerror-control codingtechniques described hereinincludethefollowing important classesofcodes: ~Linearblockcodes. ~Cycliccodes. ~Convolutional codes. ~Compound codesexe111plifiedbyturbocodesandlow-density parity-check codes,and theirirregular variants. I10.1Introductron Thetaskfacingthedesignerofadigitalcommunication systemisthatofproviding acost effectivefacilityfortransmitting information fromoneendofthesystematarateanda levelofreliability andqualitythatareacceptable toauserattheotherend.Thetwokey systemparameters available tothedesigner aretransmitted signalpowerandchannel bandwidth. Thesetwoparameters, togetherwiththepowerspectraldensityofreceiver noise,determine thesignalenergyperbit-to-noise powerspectraldensityratioEbiNo•In Chapter6,weshowedthatthisratiouniquely determines thebiterrorrateforaparticular modulation scheme.Practical considerations usuallyplacealimitonthevaluethatwecan assigntoEbiNo•Accordingly, inpractice,weoftenarriveatamodulation schemeandfind thatitisnotpossibletoprovideacceptable dataquality(i.e.,lowenougherrorperfor mance).ForafixedEblNo,theonlypractical optionavailable forchanging dataquality fromproblematic toacceptable istouseerror-control coding. Anotherpractical motivation fortheuseofcodingistoreducetherequired EblNo forafixedbiterrorrate.Thisreduction inEblNomay,inturn,beexploited toreducethe requiredtransmitted powerorreducethehardware costsbyrequiring asmallerantenna sizeinthecaseofradiocommunications. Errorcontrol'fordataintegritymaybeexercised bymeansofforwarderrorcor rection(FEe).FigurelD.lashowsthemodelofadigitalcommunication systemusingsuch anapproach. Thediscretesourcegenerates information intheformofbinarysymbols. Thechannelencoderinthetransmitter acceptsmessagebitsandaddsredundancy accord ingtoaprescribed rule,therebyproducing encodeddataatahigherbitrate.Thechannel 626 10.1lntroo.reti<no 627 Noise (al Encoder/modulator Noise (blDetector/decoder FIGURE 10.1Simplified modelsofdigitalcommunication system.(a)Codingandmodulation performed separately. (b)Codingandmodulation combined. decoderinthereceiverexploitstheredundancy todecidewhichmessagebitswereactually transmitted. Thecombined goalofthechannelencoderanddecoderistominimize the effectofchannelnoise.Thatis,thenumberoferrorsbetween thechannelencoderinput (derived fromthesource)andthechannel decoder output(delivered totheuser)is minimized. Forafixedmodulation scheme,theaddition ofredundancy inthecodedmessages impliestheneedforincreased transmission bandwidth. Moreover, theuseoferror-control codingaddscomplexity tothesystem,especially fortheimplementation ofdecoding op erations inthereceiver. Thus,thedesigntrade-offs intheuseoferror-control codingto achieveacceptable errorperformance includeconsiderations ofbandwidth andsystem complexity. Therearemanydifferent error-correcting codes(withrootsindiversemathematical disciplines) thatwecanuse.Historically, thesecodeshavebeenclassified intoblockcodes andconvolutional codes.Thedistinguishing featureforthisparticular classification isthe presence orabsenceofmemory intheencoders forthetwocodes. Togenerate an(n,k)blockcode,thechannelencoderacceptsinformation insuc cessivek-bitblocks;foreachblock,itaddsn-kredundant bitsthatarealgebraically relatedtothekmessage bits,therebyproducing anoverallencoded blockofnbits,where n>k.Then-bitblockiscalledacodeword,andniscalledtheblocklengthofthecode. Thechannelencoderproduces bitsattherateRo=(nlk)R"whereRsisthebitrateofthe information source.Thedimensionless ratior=kiniscalledthecoderate,where o<r<1.ThebitrateRo,comingoutoftheencoder, iscalledthechanneldatarate. Thus,thecoderateisadimensionless ratio,whereasthedatarateproduced bythesource andthechanneldataratearebothmeasured inbitspersecond. Inaconvolutional code,theencoding operation maybeviewedasthediscrete timeconvolution oftheinputsequence withtheimpulseresponse oftheencoder. The duration oftheimpulse response equalsthememory oftheencoder. Accordingly, the encoder foraconvolutional code operates ontheincoming message sequence, using 628 CHAPTER 10"ERROR-CONTROL CODING a"slidingwindow" equalinduration toitsownmemory. This,inturn,meansthatin aconvolutional code,unlikeablockcode,thechannelencoderacceptsmessagebitsasa continuous sequence andtherebygenerates acontinuous sequence ofencoded bitsat ahigherrate. InthemodeldepictedinFigure10.la,theoperations ofchannelcodingandmodu lationareperformed separately inthetransmitter; likewisefortheoperations ofdetection anddecoding inthereceiver. When,however, bandwidth efficiency isofmajorconcern themosteffective methodofimplementing forwarderror-control correctiDn codingist~ combine itwithmodulation asasinglefunction, asshowninFigure10.lb.Insuchan approach, codingisredefined asaprocessofimposing certainpatternsonthetransmitted signal. Ill!AUTOMATIC-REpEAT REQUEST Feed-forward errorcorrection (FEe)reliesonthecontrolled useofredundancy inthe transmitted codewordforboththedetection andcorrection oferrorsincurred during thecDurseoftransmissiDn Dveranoisychannel. Irrespective ofwhetherthedecoding of thereceived cDdewordissuccessful, nDfurtherprocessing isperfDrmed atthereceiver. AccDrdingly, channelcDdingtechniques suitable fDrFECrequireDnlyaone-way linkbe tweenthetransmitter andreceiver. ThereisanDtherapprDach knDwnasautDmatic-repeat request(ARQ)2forsDlving theerror-cDntrDl problem. Theunderlying philDsDphy DfARQisquitedifferent frDmthat DfFEC.Specifically, ARQusesredundancy merelyfDrthepurpDse DferrDrdetectiDn. Upon thedetectiDn Dfanerrorinatransmitted cDdewDrd,thereceiverrequestsarepeattrans mission DfthecDrrupted cDdewDrd,whichnecessitates theuseDfareturnpath(i.e.,a feedback channel). Assuch,ARQcanbeusedDnlyDnhalf-duplex Drfull-duplex links.In ahalf-duplex link,datatransmissiDn overthelinkcanbemadeineitherdirectiDn butnot simultaneDusly. OntheDtherhand,inafull-duplex link,itispDssiblefordatatransmission toproceedDverthelinkinbothdirectiDns simultaneously. Ahalf-duplex linkusesthesimplestARQschemeknDwnasthestDp-and-wait strat egy.Inthisapproach, ablDckofmessagebitsisencDdedintDacDdewDrdandtransmitted overthechannel.Thetransmitter thenStDpSandwaitsfDrfeedback frDmthereceiver.The feedback signalcanbeacknowledgment DfacorrectreceiptDfthecDdewDrdorarequest fDrtransmissiDn DfthecodewDrdbecause DfanerrorinitsdecDding. Inthelattercase, thetransmitter resendsthecDdewordinquestiDn befDremovingontothenextblDckof messagebits. Theidlingprobleminstop-and-wait ARQresultsinreduceddatathrDughput, which isalleviated inanDthertypeDfARQknownascontinuDus ARQwithpullback. ThissecDnd strategyusesafull-duplex link,therebypermitting thereceivertosendafeedback signal whilethetransmitter isengagedinsendingcDdewDrdsDverthefDrwardchannel. Specif ically,thetransmitter cDntinues tosendasuccessiDn DfcDdewordsuntilitreceivesa requestfromthereceiver(onthefeedback channel) fDraretransmissiDn. AtthatpDint, thetransmitter stops,pullsbacktDtheparticular cDdewDrdthatwasnDtdecodedcDrrectly bythereceiver, andretransmits thecDmplete sequence DfcDdewordsstartingwiththe corrupted Dne. InarefinedversiDnDfcDntinuDus ARQknDwnasthecontinuDus ARQwithselective repeat,datathrDughDut isimproved furtherbyDnlyretransmitting thecDdewDrdthat. wasreceivedwithdetectederrors.InDtherwords,theneedfDrretransmitting thesuccess fullyreceivedcodewDrdsfDllDwing thecorrupted cDdewordiseliminated. 10.2Discrete-Memoryless CJuz..nels629 ThethreetypesofARQdescribed hereoffertrade-offs oftheirownbetween the needforahalf-duplex orfull-duplex linkandtherequirement forefficientuseofcom munication resources. Inanyevent,theyallrelyontwopremises: l>Errordetection, whichmakesthedesignofthedecoderrelatively simple. I!>Noiseless feedback channel, whichisnotasevererestriction becausetherateof information flowoverthefeedback channelistypically quitelow. Forthesereasons,ARQisWidelyusedincomputer-communication systems. Nevertheless, thefactthatFECrequiresonlyone-way linksforitsoperation makes theFECmuchwiderinapplication thanARQ.Moreover, theincreased decoding com plexityofFECduetothecombined needforerrordetection andcorrection isnolonger apressing practical issuebecausethedecoderusuallylendsitselftomicroprocessor or VLSIimplementation inacost-effective manner. L10.2Discrete-Memoryless Clulnnels Returning tothemodelofFigure10.la,thewaveform channelissaidtobememoryless ifthedetector outputinagivenintervaldepends onlyonthesignaltransmitted inthat interval, andnotonanyprevious transmission. Underthiscondition, wemaymodelthe combination ofthemodulator, thewaveform channel, andthedetector asadiscrete memoryless channel. Suchachanneliscompletely described bythesetoftransition prob abilitiesp(j'i),whereidenotesamodulator inputsymbol,jdenotesademodulator output symbol,andp(jIi)denotestheprobability ofreceiving symbolj,giventhatsymboliwas sent.(Discrete memoryless channels weredescribed previously atsomelengthinSection 9.5.) Thesimplest discretememoryless channelresultsfromtheuseofbinaryinputand binaryoutputsymbols. Whenbinarycodingisused,themodulator hasonlythebinary symbols 0and1asinputs.Likewise, thedecoderhasonlybinaryinputsifbinaryquan tizationofthedemodulator outputisused,thatis,aharddecisionismadeonthedemod ulatoroutputastowhichsymbolwasactuallytransmitted. Inthissituation, wehavea binarysymmetric channel(BSC)withatransition probability diagramasshowninFigure 10.2.Thebinarysymmetric channel, assuming achannelnoisemodeled asadditivewhite Gaussian noise(AWGN)channel, iscompletely described bythetransition probability p. Themajority ofcodeddigitalcommunication systemsemploybinarycodingwithhard decision decoding, duetothesimplicity ofimplementation offeredbysuchanapproach. Hard-decision decoders, oralgebraic decoders, takeadvantage ofthespecialalgebraic Symbol1«:------; .....----::'P Symbol1 Symbol0_-----; .....--......:,., Symbol0 I-p FIGURE10.2Transition probability diagram ofbinarysymmetric channel. 630 CHAPTER 10"ERROR-CONTROL CODING x)---~ (a) Output ------=t---,------Input (b)Symbol1transmitted qCl~§==~~2~b2 -.IE b, Symbol2transmitted -d~~~==:s~~b6--.IE v ~ b, (e) FIGURE 10.3BinaryinputQ-ary"output discretememoryless channel. (a)Receiver forbinary phase-shift keying.(b)Transfer characteristic ofmultilevel quantizcr. (e)Channel transition prob abilitydiagram. Parts(b)and(e)areillustrated foreightlevelsofquantization. structure thatisbuiltintothedesignofchannelcodestomakethedecoding relatively easy toperform. Theuseofharddecisions priortodecoding causesanirreversible lossofinformation inthereceiver. Toreducethisloss,soft-decision codingisused.Thisisachieved byin cludingamultilevel quantizer atthedemodulator output,asillustrated inFigurelO.3a forthecaseofbinaryPSKsignals.Theinput-output characteristic ofthequantizer is showninFigurelO.3b.Themodulator hasonlythebinarysymbols 0and1asinputs,but thedemodulator outputnowhasanalphabet withQsymbols. Assuming theuseofthe quantizer asdescribed inFigurelO.3b,wehaveQ=8.Suchachanneliscalledabinary inputQ-aryoutputdiscretememorylesschannel. Thecorresponding channeltransition probability diagram isshowninFigurelO.3c.Theformofthisdistribution, andconse quentlythedecoderperformance, dependsonthelocation oftherepresentation levelsof thequantizer, which,inturn,depends onthesignallevelandnoisevariance. Accordingly, thedemodulator mustincorporate automatic gaincontrolifaneffective multilevel quan tizeristoberealized. Moreover, theuseofsoftdecisions complicates theimplementation ofthedecoder. Nevertheless, soft-decision decoding offerssignificant improvement inper formance overhard-decision decoding bytakingaprobabilistic ratherthananalgebraic approach. Itisforthisreasonthatsoft-decision decoders arealsoreferredtoasprobabi listicdecoders. illCHANNEL CODING THEOREM REVISI'IED InChapter 9,weestablished theconcept ofchannelcapacity, which,foradiscrete memoryless channel, represents themaximum amountofinformation transmitted per 10.2Discrete-Memoryless Channels 631 channeluse.Thechannelcodingtheoremstatesthatifadiscretememoryless channelhas capacity Candasourcegenerates information ataratelessthanC,thenthereexistsa codingtechnique suchthattheoutputofthesourcemaybetransmitted overthechannel withanarbitrarily lowprobability ofsymbolerror.Forthespecialcaseofabinarysym metricchannel,thetheoremtellsusthatifthecoderaterislessthanthechannelcapacity C,thenitispossibletofindacodethatachieveserror-free transmission overthechannel. Conversely, itisnotpossibletofindsuchacodeifthecoderaterisgreaterthanthe channelcapacity C. Thechannelcodingtheoremthusspecifiesthechannelcapacity Casafundamental limitontherateatwhichthetransmission ofreliable(error-free) messages cantakeplace overadiscretememoryless channel. Theissuethatmattersisnotthesignal-to-noise ratio, solongasitislargeenough,buthowthechannelinputisencoded. Themostunsatisfactory featureofthechannelcodingtheorem, however, isitsnon constructive nature.Thetheorem assertstheexistence ofgoodcodesbutdoesnottellus howtofindthem.Bygoodcodeswemeanfamiliesofchannelcodesthatarecapableof providing reliabletransmission ofinformation (i.e.,atarbitrarily smallprobability ofsym bolerror)overanoisychannelofinterestatbitratesuptoamaximum valuelessthan thecapacityofthatchannel.Theerror-control codingtechniques described inthischapter providedifferent methods ofdesigning goodcodes. !!iNOTATION Thecodesdescribed inthischapterarebinarycodes,forwhichthealphabet consistsonly ofsymbols0and1.Insuchacode,theencoding anddecoding functions involvethebinary arithmetic operations ofmodulo-2 addition andmultiplication performed oncodewords inthecode. Throughout thischapter,weuseanordinary plussign(+)todenotemodul0-2 ad dition.Theuseofthisterminology willnotleadtoconfusion becausethewholechapter reliesonbinaryarithmetic. Insodoing,weavoidtheuseofaspecialsymbolEEl,aswedid inpreceding chapters. Thus,according tothenotation usedinthischapter,therulesfor modulo-2 addition areasfollows: 0+0=0 1+0=1 o+1=1 1+1=0 Because1+1=0,itfollowsthat1=-1.Hence,inbinaryarithmetic, subtraction isthe sameasaddition. Therulesformodul0-2 multiplication areasfollows: Ox0=0 1X0=0 Ox1=0 1X1=1 Division istrivialinthatwehave 1-;-1=1 0-;-1=0 632 CHAPTER 10"ERROR-CONTROL CODING anddivisionby0isnotpermitted. Modulo-2 addition istheEXCLUSIVE-OR operation inlogic,andmodulo-2 multiplication istheANDoperation. I10.3LinearBlockCodes Acodeissaidtobelinearifanytwocodewordsinthecodecanbeaddedinmodulo-2 arithmetic toproduce athirdcodewordinthecode.Consider thenan(n,k)linearblock code,inwhichkbitsofthencodebitsarealwaysidentical tothemessage sequence tobe transmitted. Thenkbitsintheremaining portionarecomputed fromthemessagebits inaccordance withaprescribed encoding rulethatdetermines themathematical structure ofthecode.Accordingly, thesen-kbitsarereferredtoasgeneralized paritycheckbits orsimplyparitybits.Blockcodesinwhichthemessage bitsaretransmitted inunaltered formarecalledsystematic codes.Forapplications requiring botherrordetection anderror correction, theuseofsystematic blockcodessimplifies implementation ofthedecoder. Letmo,mb•..,mk-lconstitute ablockofkarbitrary message bits.Thuswehave 2kdistinctmessage blocks.Letthissequence ofmessage bitsbeappliedtoalinearblock encoder, producing ann-bitcodewordwhoseelements aredenoted byco,C1o'••,cn-1• Letbo,b10•••,bn-k-1denotethe(n-k)paritybitsinthecodeword.Forthecodeto possessasystematic structure, acodewordisdividedintotwoparts,oneofwhichis occupied bythemessage bitsandtheotherbytheparitybits.Clearly,wehavetheoption ofsendingthemessagebitsofa'codewordbeforetheparitybits,orviceversa.Theformer optionisillustrated inFigurelOA,anditsuseisassumed inthesequel. According totherepresentation ofFigurelOA,the(n-k)left-most bitsofacode wordareidentical tothecorresponding parity bits,andthekright-most bitsofthecode wordareidentical tothecorresponding messagebits.Wemaytherefore write {bi' i=0,1,...,n-k-1 Ci=mi+k-m i=n-k,n-k.+1,...,n-1 (10.1) The(n-k)paritybitsarelinearsumsofthekmessage bits,asshownbythegeneralized relation bi=POimO+P1iml+...+Pk-l,imk-l wherethecoefficients aredefinedasfollows:(10.2) ifbidepends onmj otherwise(10.3) Thecoefficients Pijarechoseninsuchawaythattherowsofthegenerator matrixare linearlyindependent andtheparityequations areunique. ThesystemofEquations (10.1)and(10.2)definesthemathematical structure ofthe (n,k)linearblockcode.Thissystemofequations mayberewritten inacompact form Paritybits Messagebits FIGURE10.4Structure ofsystematic codeword. 10.3LinearBlockCodes 633 usingmatrixnotation. Toproceedwiththisreformulation, wedefinethe1-by-kmessage vector,orinformation vector,m,the1-by-(n-k)parityvectorb,andthe1-by-ncode vectorcasfollows: m=[mo,m",mk-,] b=lbo,b",bn-k-1] c=[co,c",Cn-1t(lOA) (10.5) (10.6) Notethatallthreevectorsarerowvectors.Theuseofrowvectorsisadoptedinthischapter forthesakeofbeingconsistent withthenotation commonly usedinthecodingliterature. Wemaythusrewritethesetofsimultaneous equations defining theparitybitsinthe compact matrixform: Pk-l,lPOt P11b=mP wherePisthek-by-(n-k)coefficient matrixdefinedby PO,n-k-l ] Pl,n-k-l Pk-l~-k-l(10.7) (10.8) wherePi;is0or1. Fromthedefinitions giveninEquations (1004)-(10.6),weseethatcmaybeexpressed asapartitioned rowvectorintermsofthevectorsmandbasfollows: c=[bom] (10.9) Hence,substituting Equation (10.7)intoEquation (10.9)andfactoring outthecommon messagevectorm,weget (10.10) whereIkisthek-by-kidentitymatrix: (10.11) Definethek-by-ngenerator matrix (10.12) Thegenerator matrixGofEquation (10.12)issaidtobeinthecanonical forminthatits krowsarelinearlyindependent; thatis,itisnotpossibletoexpressanyrowofthematrix Gasalinearcombination oftheremaining rows.Usingthedefinition ofthegenerator matrixG,wemaysimplifyEquation (10.10)as c=mG (10.13) Thefullsetofcodewords,referredtosimplyasthecode,isgenerated inaccordance withEquation (10.13)bylettingthemessage vectorillrangethrough thesetofall2k binaryk-tuples(l-by-kvectors). Moreover, thesumofanytwocodewordsisanother 634 CHAYI'ER 10IIIERROR-CONTROL CODING codeword.Thisbasicproperty oflinearblockcodesiscalledclosure.Toproveitsvalidity consider apairofcodevectors Ciandc;corresponding toapairofmessagevectorsmiand m;,respectively. UsingEquation (10.13)wemayexpressthesumofCiandc;as c,+c;=m,G+m;G =(m,+rn;)G Themodulo-2 sumofmiandrn;represents anewmessagevector.Correspondingly, the modulo-2 sumofc,andc;represents anewcodevector. Thereisanotherwayofexpressing therelationship between themessage bitsand parity-check bitsofalinearblockcode.LetHdenotean(n-k)-by-nmatrix,definedas H=[I,,-k:pI] (10.14) wherepTisan(n-k)-by-kmatrix,representing thetranspose ofthecoefficient matrixP, andI,,-kisthe(n-k)-by-(n -k)identitymatrix.Accordingly, wemayperformthe following multiplication ofpartitioned matrices: HGT=[I,,-k:PI][~] =pT+pT wherewehaveusedthefactthatmultiplication ofarectangular matrixbyanidentity matrixofcompatible dimensions leavesthematrixunchanged. Inmodulo-l arithmetic, wehavepT+pT=0,where0denotesan(n-k)-by-knullmatrix(i.e.,amatrixthathas zerosforallofitselements). Hence, (10.15) (10.16)=0Equivalently, wehaveGHT=0,where0isanewnullmatrix.Postmultiplying bothsides ofEquation (10.13)byHT,thetranspose ofH,andthenusingEquation (10.15),weget cHT=mGHT ThematrixHiscalledtheparity-check matrixofthecode,andthesetofequations spec ifiedbyEquation (10.16)arecalledparity-check equations. Thegenerator equation (10.13)andtheparity-check detectorequation (10.16)are basictothedescription andoperation ofalinearblockcode.Thesetwoequations are depicted intheformofblockdiagrams inFigure10.Saand10.Sb,respectively. Messagevector mCodelIector c (a) Codevector cNullvector o (b) FIGURE 10.5Blockdffigramrepresentations ofthegenerator equation (10.13)andtheparity checkequation (10.16). 10.3LinearBlockCodes 635 ~EXAMPLE 10.1Repetition Codes Repetition codesrepresent thesimplesttypeoflinearblockcodes.Inparticular, asinglemes sagebitisencodedintoablockofnidenticalbits,producing an(n,1)blockcode.Sucha codeallowsprovision foravariableamountofredundancy. Thereareonlytwocodewords inthecode:anall-zerocodewordandanall-onecodeword. Consider, forexample, thecaseofarepetition codewithk=1andn=5.Inthiscase, wehavefourparitybitsthatarethesameasthemessagebit.Hence,theidentitymatrixI.= 1,andthecoefficient matrixPconsistsofa1-by-4vectorthathas1forallofitselements. Correspondingly, thegenerator matrixequalsarowvectorofallIs,asshownby G=[11 11:1] Thetranspose ofthecoefficient matrixP,namely,matrixpT,consistsofa4-by-1vectorthat has1forallofitselements. TheidentitymatrixIn-.consistsofa4-by-4matrix.Hence,the parity-check matrixequals [1 0 0 0 ~1]o100'1H= :o0 10:1 o0 01:1 Sincethemessagevectorconsistsofasinglebinarysymbol,0or1,itfollowsfromEquation (10.13)thatthereareonlytwocodewords:00000and11111inthe(5,1)repetition code, asexpected. NotealsothatHGT=0,modulo-2, inaccordance withEquation (10.15). <1/l !illSYNDROME: DEFINITION ANDPROPERTIES Thegenerator matrixGisusedintheencoding operation atthetransmitter. Ontheother hand,theparity-eheck matrixHisusedinthedecoding operation atthereceiver.Inthe contextofthelatteroperation, letrdenotetheI-by-nreceivedvectorthatresultsfrom sendingthecodevectorcoveranoisychannel. Weexpressthevectorrasthesumofthe originalcodevectorcandavectore,asshownby r=c+e (10.17) Thevectoreiscalledtheerrorvectororerrorpattern.Theithelementofeequals0ifthe corresponding elementofristhesameasthatofc.Ontheotherhand,theithelementof eequals1ifthecorresponding elementofrisdifferent fromthatofc,inwhichcasean errorissaidtohaveoccurred intheithlocation. Thatis,fori=1,2,...,n,wehave e.={1,0ifanerrorhasoccurred intheith location otherwise(10.18) Thereceiverhasthetaskofdecoding thecodevectorcfromthereceived vectorr. Thealgorithm commonly usedtoperform thisdecoding operation startswiththecom putation ofa1-by-(n-k)vectorcalledtheerror-syndrome vectororsimplythesyn drome.3Theimportance ofthesyndrome liesinthefactthatitdepends onlyuponthe errorpattern. Givena1-by-nreceived vectorr,thecorresponding syndrome isformally definedas (10.19) Accordingly, thesyndrome hasthefollowing important properties. 636 CHAPTER 10IIIERROR-CONTROL CODING Property 1 Thesyndrome dependsonlyontheerrorpattern,andnotonthetransmitted codeword. Toprovethisproperty, wefirstuseEquations (10.17)and(10.19),andthenEquation (10.16)toobtain s=(c+e)HT =cHT+eHT =eHT(10.20) (10.22)Hence,theparity-check matrixHofacodepermitsustocompute thesyndrome s,which depends onlyupontheerrorpatterne. Property 2 Allerrorpatternsthatdifferbyacodewordhavethesamesyndrome. Forkmessage bits,thereare2kdistinctcodevectorsdenoted asCi,i=0,1,..., 2k-1.Correspondingly, foranyerrorpatterne,wedefinethe2kdistinctvectors eias ei=e+c;,i=0,1,...,2k-1 (10.21) Thesetofvectorslei'i=0,1,...,2k-IJsodefinediscalledacosetofthecode.In otherwords,acosethasexactly2kelements thatdifferatmostbyacodevector.Thus, an(n,k)linearblock code has2n-kpossible cosets.Inanyevent,multiplying bothsides ofEquation (10.21)bythematrixHT,weget eiHT=eHT+C;HT =eHT whichisindependent oftheindexi.Accordingly, wemaystatethateachcosetofthecode ischaracterized byauniquesyndrome. WemayputProperties 1and2inperspective byexpanding Equation (10.20). Spe cifically, withthematrixHhavingthesystematic formgiveninEquation (10.14),where thematrixPisitselfdefinedbyEquation (10.8),wefindfromEquation (10.20)thatthe (n-k)elements ofthesyndrome sarelinearcombinations ofthenelements oftheerror patterne,asshownby So=eo+en-kPOO+en-k+1PI0+ + en-1Pk-l,a Sl=el+en-kPOl+en-k+lPll+ + en-lPk-l,l(10.23) Sn-k-l=en-k-l+en-kPO,n-k-l +...+en-1Pk-l,n-k-l Thissetof(n-k)linearequations clearlyshowsthatthesyndrome contains information abouttheerrorpatternandmaytherefore beusedforerrordetection, However, itshould benotedthatthesetofequations isunderdetermined inthatwehavemoreunknowns thanequations. Accordingly, thereisnouniquesolution fortheerrorpattern. Rather, thereare2nerrorpatterns thatsatisfyEquation (10.23)andtherefore resultinthesame syndrome, inaccordance withProperty 2andEquation (10.22),Inparticular, with2"-1 possible syndrome vectors,theinformation contained inthesyndrome sabouttheerror patterneisnotenoughforthedecodertocompute theexactvalueofthetransmitted code vector.Nevertheless, knowledge ofthesyndrome sreducesthesearchforthetrueerror 10.3LifUlarBlockCodes 637 patternefromrto2n-kpossibilities. Giventhesepossibilities, thedecoderhasthetask ofmakingthebestselection fromthecosetscorresponding tos. IIIMINIMUM DISTANCE CONSIDERATIONS Consider apairofcodevectorsc,andCzthathavethesamenumberofelements. The Hamming distanced(cl>cz)betweensuchapairofcodevectorsisdefinedasthenumber oflocations inwhichtheirrespective elements differ. TheHamming weightw(c)ofacodevectorcisdefinedasthenumberofnonzero elements inthecodevector.Equivalently, wemaystatethattheHamming weightofa codevectoristhedistancebetweenthecodevectorandtheall-zerocodevector. Theminimum distancedminofalinearblockcodeisdefinedasthesmallestHamming distancebetween anypairofcodevectorsinthecode.Thatis,theminimum distanceis thesameasthesmallestHamming weightofthedifference between anypairofcode vectors.Fromtheclosureproperty oflinearblockcodes,thesum(ordifference) oftwo codevectorsisanothercodevector.Accordingly, wemaystatethattheminimum distance ofalinearblockcodeisthesmallestHamming weightofthenonzerocodevectorsinthe code. Theminimum distance dxmnisrelatedtothestructure oftheparity-check matrixH ofthecodeinafundamental way.FromEquation (10.16)weknowthatalinearblock codeisdefinedbythesetofallcodevectorsforwhichcHT=0,whereHTisthetranspose oftheparity-check matrixH.LetthematrixHbeexpressed intermsofitscolumns as follows: H=[hi>hz,...,hnJ (10.24) Then,for acodevectorctosatisfythecondition cHT=0,thevectorcmusthaveIsin suchpositions thatthecorresponding rowsofHTsumtothezerovectorO.However, by definition, thenumberofIsinacodevectoristheHamming weightofthecodevector. Moreover, thesmallestHamming weightofthenonzerocodevectorsinalinearblock codeequalstheminimum distanceofthecode.Hence,theminimum distanceofalinear blockcodeisdefinedbytheminimum numberofrowsofthematrixHTwhosesumis equaltothezerovector. Theminimum distanceofalinearblockcode,dmin,isanimportant parameter ofthe code.Specifically, itdetermines theerror-correcting capability ofthecode.Suppose an (n,k)linearblockcodeisrequiredtodetectandcorrectallerrorpatterns(overabinary symmetric channel), andwhoseHamming weightislessthanorequaltot.Thatis,ifa codevector Ciinthecodeistransmitted andthereceivedvectorisr=Cj+e,werequire thatthedecoderoutputC=Cj,whenever theerrorpatternehasaHamming weight w(e):5t.Weassumethatthe2kcodevectorsinthecodearetransmitted withequal probability. Thebeststrategyforthedecoderthenistopickthecodevectorclosesttothe receivedvectorr,thatis,theoneforwhichtheHamming distance d(c"r)isthesmallest. Withsuchastrategy, thedecoderwillbeabletodetectandcorrect aJ.lerrorpatternsof Hamming weightw(e):5t,provided thattheminimum distanceofthecodeisequaltoor greaterthan2t+1.Wemaydemonstrate thevalidityofthisrequirement byadopting a geometric interpretation oftheproblem.illparticular, theI-by-ncodevectorsandthe I-by-nreceivedvectorarerepresented aspointsinann-dimensional space.Supposethat weconstruct twospheres,eachofradiust,aroundthepointsthatrepresent codevectors c,andCj'Letthesetwospheresbedisjoint,asdepictedinFigure10.6a.Forthiscondition tobesatisfied, werequirethatd(c"c;)~2t+1.Ifthenthecodevector Cjistransmitted andtheHamming distance d(c"r):5t,itisclearthatthedecoderwillpickCiasitisthe 638 CHAPTER 10"ERROR-CONTROL CODING (al (b) FIGURE 10.6(a)Hamming distance d(c"Cj)2:2t+1.(b)Hamming distance d(c"c)<2t. Thereceived vectorisdenotedbyr. codevectorclosesttothereceived vectorr.If,ontheotherhand,theHamming distance d(e;,Cj)~2t,thetwospheresaround CiandCjintersect, asdepicted inFigure10.6b.Here weseethatifCiistransmitted, thereexistsareceived vectorrsuchthattheHamming distance d(ci'r)~t,andyetrisasclosetoCjasitistoCi'Clearly, thereisnowthe possibility ofthedecoderpickingthevector Cj,whichiswrong.Wethusconclude thatan (n,k)linearblockcodehasthepowertocorrectallerrorpatternsofweighttorlessif, andonlyif, d(ci'Cj);::::2t+1forallCiandCj Bydefinition, however, thesmallest distance between anypairofcodevectorsinacode istheminimum distance ofthecode,dmin•Wemaytherefore statethatan(n,k)linear blockcodeofminimum distance'd mincancorrectuptoterrorsif,andonlyif, (10.25) whereLJdenotesthelargestintegerlessthanorequaltotheenclosed quantity. Equation (10.25)givestheerror-correcting capability ofalinearblockcodeaquantitative meaning. IiSYNDROME DECODING Wearenowreadytodescribe asyndrome-based decoding schemeforlinearblockcodes. Letc"C2'...,C2kdenotethe2kcodevectorsofan(n,k)linearblockcode.Letrdenote thereceived vector,whichmayhaveoneofrpossiblevalues.Thereceiverhasthetask ofpartitioning the2npossiblereceivedvectorsinto2kdisjointsubsets'2ll"'2ll2,...,'2ll2kin suchawaythattheithsubset'2lljcorresponds tocodevectorCifor1~i~2k•Thereceived vectorrisdecoded intoCjifitisintheithsubset.Forthedecoding tobecorrect,rmust beinthesubsetthatbelongstothecodevectorCithatwasactuallysent. The2ksubsetsdescribed hereinconstitute astandard arrayofthelinearblockcode. Toconsttuct it,wemayexploitthelinearstructure ofthecodebyproceeding asfollows: 1.The2kcodevectorsareplacedinarowwiththeall-zerocodevectorC,astheleft mostelement. 2.Anerrorpatterne,ispickedandplacedunderc"andasecondrowisformedby addinge2toeachoftheremaining codevectorsinthefirstrow;itisimportant that theerrorpatternchosenasthefirstelementinarownothavepreviously appeared inthestandard array. 3.Step2isrepeated untilallthepossibleerrorpatterns havebeenaccounted for. Figure10.7illustrates thestructure ofthestandard arraysoconstructed. The2kcolumns ofthisarrayrepresent thedisjoint subsets '2ll"'2ll2,•••,'2ll2,.Ther-krowsofthearray 10.3LinearBlockCodes 639 .,=0.,., Ci .z' ·2&2+82 &3+82 Ci+82C2k+82.,C2+83 C3+83 Ci+83 &:/+B3 .j C2+8] C3+Bj Ci+8] C2k+Bj B2n-kC2+82,,-1::C3+e;tl-k Ci+ez.'!-k C2k+82,,-1< FIGURE10.7Standard arrayforan(n,k)blockcode. represent thecosetsofthecode,andtheirfirstelements eb...,e2n-karecalledcoset leaders. Foragivenchannel, theprobability ofdecoding errorisminimized whenthemost likelyerrorpatterns (i.e.,thosewiththelargestprobability ofoccurrence) arechosenas thecosetleaders.Inthecaseofabinarysymmetric channel, thesmallertheHamming weightofanerrorpatternthemorelikelyitistooccur.Accordingly, thestandard array shouldbeconstructed witheachcosetleaderhavingtheminimum Hamming weightinits coset. Wemaynowdescribe adecoding procedure foralinearblockcode: 1.Forthereceived vectorr,compute thesyndrome s =rHT• 2.Withinthecosetcharacterized bythesyndrome s,identifythecosetleader(i.e.,the errorpatternwiththelargestprobability ofoccurrence); calliteo. 3.Compute thecodevector c=r+eo asthedecoded versionofthereceived vectorr. Thisprocedure iscalledsyndrome decoding. ~ExAMPLE 10.2Hamming Codes4 Consider afamilyof(n,k)linearblockcodesthathavethefollowing parameters:(10.26) Blocklength: Nwnherofmessagebits: Nwnberofparitybits:k=2m-m-1 nk=m wherem2:3.Thesearetheso-calledHamming codes. Consider, forexample, the(7,4)Hamming codewithn=7andk=4,corresponding tom=3.Thegenerator matrixofthecodemusthaveastructurethatconforms toEquation (10.12).Thefollowing matrixrepresents anappropriate generator matrixforthe(7,4)Ham mingcode: [1 1 0 ~1 0 0 0]o11·0 1 0 0G= : 1 11:00 1 0 101:0001'---,---'.'-,----' P I k 640 CHAPTER 10IIIERROR-CONTROL CODING TABLE10.1Code'Wordsofa(7,4)Hamming code Message Weightof Message Weightof Word CodeWord CodeWord Word CodeWord CodeWord 0000 0000000 0 1000 1101000 3 0001 1010001 3 1001 0111001 4 0010 1110010 4 1010 0011010 3 0011 0100011 03 10111001011 4 0100 0110100 3 1100 1011100 4 0101 1100101 4 1101 0001101 3 0110 1000110 3 1110 0101110 4 0111 0010111 4 111 11111111 7 Thecorresponding parity-check matrixisgivenby [1 00:1 0 1 H=010~11 0 001·011'-,--' .~-v-- .... L.-. pT Withk=4,thereare2'=16distinctmessagewords,whicharelistedinTable 10.1.Foragivenmessageword,thecorresponding codewordisobtained byusingEqua tion(10.13).Thus,theapplication ofthisequation resultsinthe16codewordslistedin Table10.1. InTable10.1,wehavealsolistedtheHamming weightsoftheindividual codewords inthe(7,4)Hamming code.SincethesmallestoftheHamming weightsforthenonzerocode wordsis3,itfollowsthattheminimum distanceofthecodeis3.Indeed,Hamming codes havetheproperty thattheminimum distancedmm=3,independent ofthevalueassignedto thenumberofparitybitsm. Toillustrate therelationbetweentheminimum distancedminandthestructure ofthe parity-check matrixH,considerthecodeword0110100. Inthematrixmultiplication defined byEquation (10.16),thenonzeroelements ofthiscodeword"sift"outthesecond,third,and fifthcolumnsofthematrixHyielding Wemayperformsimilarcalculations fortheremaining 14nonzerocodewords.Wethusfind thatthesmallestnumberofcolumns inHthatsumstozerois3,confirming theearlierstate mentthatdmm=3. Animportant property ofHamming codesisthattheysatisfythecondition ofEquation (10.25)withtheequalitysign,assuming thatt=1.thismeansthatHamming codesate single-error correcting binaryperfectcodes. Assuming single-error patterns, wemayformulate thesevencosetleaderslistedinthe right-hand columnofTable10.2.Thecorresponding 23syndromes, listedintheleft-hand column,arecalculated inaccordance withEquation (10.20).Thezerosyndrome signifies nO transmission errors. Suppose, forexample, thecodevector[1110010] issent,andthereceivedvectoris 10.4CyclicCodes 641 TABLE10.2Decoding tableforthe(7,4) Hamming codedefined inTable10.1 Sy"drome 000 1 0 0 010 001 110 011 111 101ErrorPattern 0000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 [1~00010] withanerrorinthethirdbit.UsingEquation (10.19),thesyndrome iscalculated tobe 1 0 0 010 001 s=[1100010] 1 10 01 1 111 10 1 =[001] FromTable10.2thecorresponding cosetleader(i.e.,errorpatternwiththehighestprobability ofoccurrence) isfoundtobe[0010000], indicating correctly thatthethirdbitofthereceived vectoriserroneous. Thus,addingthiserrorpatterntothereceivedvector,inaccordance with Equation (10.26),yieldsthecorrectcodevectoractuallysent. <il Ill!DUALCODE Givenalinearblockcode,wemaydefineitsdualasfollows. Takingthetranspose ofboth sides.ofEquation (10.15), wehave whereHTisthetranspose oftheparity-check matrixofthecode,and0isanewzero matrix.Thisequation suggests thatevery(n,k)linearblockcodewithgenerator matrix Gandparity-check matrixHhasadualcodewithparameters (n,n-k),generator matrix Handparity-check matrixG. I10.4CyclicCodes Cycliccodesformasubclass oflinearblockcodes.Indeed,manyoftheimportant linear blockcodesdiscovered todateareeithercycliccodesorcloselyrelatedtocycliccodes.An (10.28)642 CHAPTER 10IIERROR-CONTROL CODING advantage ofcycliccodesovermostothertypesofcodesisthattheyareeasytoencode. Furthermore, cycliccodespossessawell-defined mathematical structure, whichhasledto thedevelopment ofveryefficientdecoding schemesforthem. Abinarycodeissaidtobeacycliccodeifitexhibitstwofundamental properties: 1.Linearity property: Thesumofanytwocodewordsinthecodeisalsoacodeword. 2.Cyclicproperty: Anycyclicshiftofacodewordinthecodeisalsoacodeword. Property 1restatesthefactthatacycliccodeisalinearblockcode(i.e.,itcanbedescribed asaparity-check code):TorestateProperty 2inmathematical terms,letthen-tuple (co,Ch•••,Cn-l)denoteacodewordofan(n,k)linear block code.Thecodeisacyclic codeifthen-tuples (en-1'Co,...,Cn-2), (Cn-2'Cn-b••.,Cn-3), areallcodewordsinthecode. Todevelopthealgebraic properties ofcycliccodes,weusetheelements co,CI,•••, Cn-lofacodewordtodefinethecodepolynomial c(X)=Co+cIX+C2X2+...+cn_Ixn-1 (10.27) whereXisanindeterminate. Naturally, forbinarycodes,thecoefficients are15andOs. EachpowerofXinthepolynomial c(X)represents aone-bitshiftintime.Hence,multi plication ofthepolynomial c(X)byXmaybeviewedasashifttotheright.Thekey question is:Howdowemakesuchashiftcyclic?Theanswertothisquestion isaddressed next. Letthecodepolynomial c(X)bemultiplied byXi,yielding Xic(X)=Xi(co+c,X+...+Cn_i_Ixn-i-1 +Cn_ixn-i +...+cn_Ixn-l) =COXi+CIXi+1+...+Cn_i_Ixn-1 +Cn_ixn +...+cn_1xn+i-1 =Cn_ixn+...+cn_Ixn+i-1 +COXi+CIXi+1 +...+Cn_i_Ixn-1 where,inthelastline,wehavemerelyrearranged terms.Recognizing, forexample, that Cn-i+Cn-i=0inmodulo-2 addition, wemaymanipulate thefirstitermsofEquation (10.28)asfollows: XiC(X)=Cn-i+...+Cn_IXi-1+COXi+C,Xi+1+...+Cn_i_Ixn-1 +Cn_i(xn+1)+...+cn_IXi-l(xn +1) Next,weintroduce thefollowing definitions: c(i)(X)=Cn-i+...+Cn_IXi-1+eaXi+CIXi+1 +...+Cn_i_Ixn-1 q(X)=Cn-i+Cn-i+IX+...+Cn_IXi-1(10.29) (10.30) (10.31) 10.4CyclicCodes 643 Accordingly, Equation (10.29)isreformulated inthecompact form Xic(X)=q(x)(xn+1)+di)(X) (10.32) Thepolynomial c(i)(X)isrecognized asthecodepolynomial ofthecodeword(Cn-i' , cn-"Co,C"••.,Cn-i-i)obtained byapplying icyclicshiftstothecodeword(co,c", Cn-i-"Cn-i'...,cn-ll.Moreover, fromEquation (10.32)wereadilyseethatdi)(X)isthe remainder thatresultsfromdividing Xic(X)by(xn+1).Wemaythusformally statethe cyclicproperty inpolynomial notation asfollows:Ifc(X)isacodepolynomial, thenthe polynomial (10.33) isalsoacodepolynomial foranycyclicshifti;thetermmodistheabbreviation formodulo. Thespecialformofpolynomial multiplication described inEquation (10.33)isreferredto asmultiplication moduloxn+1.Ineffect,themultiplication issubjecttotheconstraint xn=1,theapplication ofwhichrestoresthepolynomial Xic(X)toordern1forall i<n.(Notethatinmodulo-2 arithmetic, xn+1hasthesamevalueasXn-1.) IlilGENERATOR POLYNOMIAL Thepolynomial xn+1anditsfactorsplayamajorroleinthegeneration ofcycliccodes. Letg(X)beapolynomial ofdegreen-kthatisafactorofxn+1;assuch,g(X)isthe polynomial ofleastdegreeinthecode.Ingeneral,g(X)maybeexpanded asfollows: n-k-] g(X)=1+LgiXi+xn-k i=1(10.34) wherethecoefficient giisequalto0or1.According tothisexpansion, thepolynomial g(X)hastwotermswithcoefficient 1separated byn-k-1terms.Thepolynomial g(X) iscalledthegenerator polynomial ofacycliccode.Acycliccodeisuniquely determined bythegenerator polynomial g(X)inthateachcodepolynomial inthecodecanbeex pressedintheformofapolynomial productasfollows: c(X)=a(X)g(X) (10.35) wherea(X)isapolynomial inXwithdegreek-1.Thec(X)soformedsatisfiesthe condition ofEquation (10.33)sinceg(X)isafactorofxn+1. Supposewearegiventhegenerator polynomial g(X)andtherequirement istoencode themessagesequence (mo,m"...,mk-i)intoan(n,k)systematic cycliccode.Thatis, themessage bitsaretransmitted inunaltered form,asshownbythefollowing structure foracodeword(seeFigurelOA): (bo,bi,•••,bn-k-"'-----------,- n~kparitybitsrna,mb...,mk-l)'---y--------' kmessage bits Letthemessage polynomial bedefinedby m(X)=mo+miX+...+mk_iXk-i andlet(10.36) (10.37) 644 CHAPTER 10IIIERROR-CONTROL CODING According toEquation (10.1),wewantthecodepolynomial tobeintheform c(X)=b(X)+xn-km(X) Hence,theuseofEquations (10.35)and(10.38)yields a(X)g(X) =b(X)+xn-km(X) Equivalently, inlightofmodulo-2 addition, wemaywrite Xn-km(X) (b(X) g(X) =aX)+g(X)(10.38) (10.39) Equation (10.39)statesthatthepolynomial b(X)istheremainder leftoverafterdiViding xn-km(X) byg(X). Wemaynowsummarize thestepsinvolved intheencoding procedure foran(n,k) cycliccodeassuredofasystematic structure. Specifically, weproceedasfollows: 1.Multiply themessagepolynomial m(X)byxn-k. 2.Dividexn-km(X) bythegenerator polynomial g(X),obtaining theremainder b(X). 3.Addb(X)toxn-km(X), obtaining thecodepolynomial c(X). !IiPARITY-CHECK POLYNOMiAL An(n,k)cycliccodeisuniquely specified byitsgenerator polynomial g(X)oforder (n-k).Suchacodeisalsouniquely specified byanotherpolynomial ofdegreek,which iscalledtheparity-check polynomial, definedby k-l h(X)=1+LhiX'+Xk i=J(lOAD) wherethecoefficients hiare0or1.Theparity-check polynomial h(X)hasaformsimilar tothegenerator polynomial inthattherearetwotermswithcoefficient 1,butseparated byk-1terms. Thegenerator polynomial g(X)isequivalent tothegenerator matrixGasadescrip tionofthecode.Correspondingly, theparity-check polynomial, denoted byh(X),isan equivalent representation oftheparity-check matrixH.Wethusfindthatthematrixre lationHGT=0presented inEquation (10.15)forlinearblockcodescorresponds tothe relationship g(X)h(X) mod(Xn+1)=0 (lOA1) Accordingly, wemaystatethatthegenerator polynomial g(X)andtheparity-check poly nomialh(X)arefactorsofthepolynomial X"+1,asshownby g(X)h(X) =X"+1 (lDA2) Thisproperty provides thebasisforselecting thegenerator orparity-check polynomial of acycliccode.Inparticu~r, wemaystatethatifg(X)isapolynomial ofdegree(n-k) anditisalsoafactorofxn+1,theng(X)isthegenerator polynomial ofan(n,k)cyclic code.Equivalently, wemaystatethatifh(X)isapolynomial ofdegreekanditisalsoa factorofxn+1,thenh(X)istheparity-check polynomial ofan(n,k)cycliccode. Afinalcomment isinorder.Anyfactorofxn+1withdegree(n-k),thenumber ofparitybits,canbeusedasagenerator polynomial. Forlargevaluesofn,thepolynomialxn+1mayhavemanyfactorsofdegreen-k.Someofthesepolynomial factorsgenerate 10.4CyclicCodes 645 goodcycliccodes,whereassomeofthemgenerate badcycliccodes.Theissueofhowto selectgenerator polynomials thatproduce goodcycliccodesisverydifficulttoresolve. Indeed,codingtheorists haveexpended mucheffortinthesearchforgoodcycliccodes. IEGENERATOR ANDPARlIT-CHECK MATRICES Giventhegenerator polynomial g(X)ofan(n,k)cycliccode,wemayconstruct thegen eratormatrixGofthecodebynotingthatthekpolynomials g(X),Xg(X),...,Xk-1g(X) spanthecode.Hence,then-tuplescorresponding tothesepolynomials maybeusedas rowsofthek-by-ngenerator matrixG. However, theconstruction of theparity-check matrixHofthecycliccodefromthe parity-check polynomial h(X)requires specialattention, asdescribed here.Multiplying Equation (10.42)bya(x)andthenusingEquation (10.35),weobtain c(X)h(X) =a(X)+Xna(X) (10,43) Thepolynomials c(X)andh(X)arethemselves definedbyEquations (10.27)and(10.40), respectively, whichmeansthattheirproductontheleft-hand sideofEquation (10.43) contains termswithpowersextending upton+k-1.Ontheotherhand,thepolynomial a(X)hasdegreek-1orless,theimplication ofwhichisthatthepowersofXk,Xk+\..., xn-ldonotappearinthepolynomial ontheright-hand sideofEquation (10.43).Thus, settingthecoefficients ofxk,Xk-1,•••,Xn-1intheexpansion oftheproductpolynomial c(X)h(X) equaltozero,weobtainthefollowing setofn-kequations: j+k 2:Cihk+i-i=0 i=jforOsjsn-k-l (10.44) Comparing Equation (10.44)withthecorresponding relationofEqu'!tion (10.16),wemay makethefollowing important observation: Thecoefficients oftheparity-check polynomial h(X)involved inthepolynomial multiplication described inEquation (10.44)arearranged inreversed orderwithrespecttothecoefficients oftheparity-check matrixHinvolved in formingtheinnerproductofvectorsdescribed inEquation (10.16).Thisobservation sug geststhatwedefinethereciprocal oftheparity-check polynomial asfollows: k-l =1+2:hk_Xi+Xk i=l(10.45) whichisalsoafactorofxn+1.Then-tuplespertaining tothe(n-k)polynomials Xkh(X-1),Xk+1h(X-1),•••,Xn-1h(X-1)maynowbeusedinrowsofthe(n-k)-by-n parity-check matrixH. Ingeneral,thegenerator matrixGandtheparity-check matrixHconstructed inthe mannerdescribed herearenotintheirsystematic forms.Theycanbeputintotheirsys tematicformsbyperforming simpleoperations ontheirrespective rows,asillustrated in Example 10.3. ~ENCODER FORCYCLIC CODES Earlierweshowedthattheencoding procedure foran(n,k)cycliccodeinsystematic form involves threesteps:(1)multiplication ofthemessagepolynomial m(X)byxn-k,(2)di- 646 CHAPTER 10..ERROR-CONTROL CODING Flip-flop Modulo-2 adderCode WIJ'd Message bits0----1-;0.0" FIGURE10.8Encoder foran(n,k)cycliccode. visionofXn-km(X) bythegenerator polynomial g(X)toobtaintheremainder b(X),and (3)addition ofb(X)toXn-km(X) toformthedesiredcodepolynomial. Thesethreesteps canbeimplemented bymeansoftheencodershowninFigure10.8,consisting ofalinear feedback shiftregisterwith(n-k)stages. TheboxesinFigure10.8represent flip-flops, orunit-delay elements. Theflip-flopis adevicethatresidesinoneoftwopossible statesdenoted by0and1.Anextemal clock (notshowninFigure10.8)controls theoperation ofalltheflip-flops. Everytimetheclock ticks,thecontents oftheflip-flops (initially settothestate0)areshiftedoutinthedirection ofthearrows.Inaddition totheflip-flops, theencoderofFigure10.8includes asecond setoflogicelements, namely,adders,whichcompute themodulo-2 sumsoftheirrespective inputs.Finally,themultipliers multiply theirrespective inputsbytheassociated coeffi cients.Inparticular, ifthecoefficient gi=1,themultiplier isjustadirect"connection." If,ontheotherhand,thecoefficient gi=0,themultiplier is"noconnection." Theoperation oftheencodershowninFigure10.8proceeds asfollows: 1.Thegateisswitched on.Hence,thekmessage bitsareshiftedintothechannel. Ali soonasthekmessage bitshaveenteredtheshiftregister,theresulting (n-k)bits intheregisterformtheparitybits[recallthattheparitybitsarethesameasthe coefficients oftheremainder b(X)]. 2.Thegateisswitched off,therebybreaking thefeedback connections. 3.Thecontents oftheshiftregisterarereadoutintothechannel. iilCALCULATION OFTHESYNDROME Suppose thecodeword(co,c"...,Cn-l)istransmitted overanoisychannel, resulting in thereceived word(Yo,r"_._,rn-l)'FromSection10.3,werecallthatthefirststepinthe decoding ofalinearblockcodeistocalculate thesyndrome forthereceived word.Hthe syndrome iszero,therearenotransmission errorsinthereceived word.If,ontheother hand,thesyndrome isnonzero, thereceivedwordcontains transmission errorsthatrequire correction. Inthecaseofacycliccodeinsystematic form,thesyndrome canbecalculated easily. Letthereceivedwordberepresented byapolynomial ofdegreen-1orless,asshown by r(X)=ro+r,X+...+rn_lxn-l (10.46) 10.4CyclicCOtks 647 Letq(X)denotethequotient ands(X)denotetheremainder, whicharetheresultsof dividingr(X)bythegenerator polynomialg(X). Wemaytherefore expressr(X)asfollows: r(X)=q(X)g(X)+s(X) (10.47) Theremainder s(X)isapolynomial ofdegreen-k-1orless,whichistheresultof interest.Itiscalledthesyndrome polynomial becauseitscoefficients makeupthe(n-k) by-lsyndrome s. Figure10.9showsasyndrome calculator thatisidentical totheencoderofFigure 10.8exceptforthefactthatthereceivedbitsarefedintothe(n-k)stagesofthefeedback shiftregisterfromtheleft.Assoonasallthereceivedbitshavebeenshiftedintotheshift register,itscontents definethesyndrome s. Thesyndrome polynomial s(X)hasthefollowing usefulproperties thatfollowfrom thedefinition giveninEquation (10.47). 1.Thesyndrome ofareceivedwordpolynomial isalsothesyndrome ofthecorresponding errorpolynomial. Giventhatacycliccodewithpolynomial c(X)issentoveranoisychannel, thereceived wordpolynomial isdefinedby r(X)=c(X)+e(X) wheree(X)istheerrorpolynomial. Equivalently, wemaywrite e(X)=r(X)+c(X) Hence,substituting Equations (10.35)and(10.47)into(10.49),weget e(X)=u(X)g(X)+s(X)(10.48) (10.49) (10.50) wherethequotient isu(X)=a(X)+q(X).Equation (10.50)showsthats(X)isalsothe syndrome oftheerrorpolynomial e(X).Theimplication ofthisproperty isthatwhenthe syndrome polynomial s(X)isnonzero, thepresence oftransmission errors inthereceived wordisdetected. 2.Lets(X)bethesyndrome ofareceivedwordpolynomial r(X).Then,thesyndrome of Xr(X),acyclicshiftofr(X),isXs(X). Applying acyclicshifttobothsidesofEquation (10.47),weget Xr(X)=Xq(X)g(X)+Xs(X) (10.51) Flip-flop Modulo-2 adder FIGlJRE 10.9Syndrome calculator for(n,k)cycliccode. 648 CHAPTER 10"ERROR-CONTROL CODING fromwhichwereadilyseethatXs(X)istheremainder ofthedivisionofXr(X)byg(X). Hence,thesyndrome ofXr(X)isXs(X)asstated.Wemaygeneralize thisresultbystating thatifsIX)isthesyndrome ofr(X),thenX's(X)isthesyndrome ofX'r(X). 3.Thesyndrome polynomial sIX)isidentical totheerrorpolynomial e(X),assuming that theerrorsareconfined tothe(n-k)parity-check bitsofthereceived wordpolynomial r(X). . Theassumption madehereisanotherwayofsayingthatthedegreeoftheerrorpolynomial e(X)islessthanorequalto(n-k-1).Sincethegenerator polynomial g(X)isofdegree (n-k),bydefinition, itfollowsthatEquation (10.50)canonlybesatisfiedifthequotient u(X)iszero.Inotherwords,theerrorpolynomial e(X)andthesyndrome polynomial sIX) areoneandthesame.Theimplication ofProperty 3isthat,undertheaforementioned conditions, errorcorrection canbeaccomplished simplybyaddingthesyndrome poly nomials(X)tothereceived wordpolynomial r(X). il>EXAMPLE 10.3Hamming CodesRevisited Toillustratetheissuesrelatingtothepolynomial representation ofcycliccodes,weconsider thegeneration ofa(7,4)cycliccode.Withtheblocklengthn=7,westartbyfactorizing X7+1intothreeirreducible polynomials: X7+1=c(I+X)(I+X2+X3)(1+X+X3) Byan"irreducible polynomial" wemeanapolynomial thatcannotbefactoredusingonly polynomials withcoefficients fromthebinaryfield.Anirreducible polynomial ofdegree missaidtobeprimitive ifthesmallestpositiveintegernforwhichthepolynomial divides xn+1isn=2m-1.Fortheexampleathand,thetwopolynomials (1+X2+X3)and (1+X+X3)areprimitive. Letustake g(X)=1+X+X3 asthegenerator polynomial, whosedegreeequalsthenumberofparitybits.Thismeansthat theparity-check polynomial isgivenby h(X)=(1+X)(1+X2+X3) =1+X+X2+X4 whosedegreeequalsthenumberofmessagebitsk=4. Next,weillustrate theprocedure fortheconstruction ofacodewordbyusingthis generator polynomial toencodethemessagesequence1001.Thecorresponding messagepoly nomialisgivenby m(X)=1+X3 Hence,multiplying m(X)byxn-k=X3,weget Xn-km(X) =Xl+X. Thesecondstepistodividexn-km(X) byg(X),thedetailsofwhich(fortheexampleathand) aregivenbelow: Xl+X Xl+X+l)X· X·+X3 +X4+Xl +X2+ X2+X 10.4CyclicCodes 649 Notethatinthislongdivisionwehavetreatedsubtraction thesameasaddition, sinceweare operating inmodulo-2 arithmetic. Wemaythuswrite X3+X·=X+X3+X+X2 1+X+X31+X+X3 Thatis,thequotienta(X)andremainder b(X)areasfollows,respectively: a(X)=X+X3 b(X)=X+X2 Hence,fromEquation (10.38)wefindthatthedesiredcodepolynomial is c(X)=b(X)+Xn-km(X) X+X2+X3+X6 Thecodewordistherefore 0111001. Thefourright-most bits,1001,arethespecifiedmessage bits.Thethreeleft-most bits,011,aretheparity-check bits.Thecodewordthusgenerated is exactlythesameasthecorresponding oneshowninTable10.1fora(7,4)Hamming code. Wemaygeneralize thisresultbystatingthatanycycliccodegenerated byaprimitive polynomial isaHamming codeofminimum distance3. Wenextshowthatthegenerator polynomial g(X)andtheparity-check polynomial h(X)uniquely specifythegenerator matrixGandtheparity-<:heck matrixH,respectively. Toconstruct the4-by-7generator matrixG,westartwithfourpolynomials represented byg(X)andthreecyclic-shifted versionsofit,asshownby g(X)=1+X+X3 Xg(X) =X+X2+X4 X2g(X)=Xl+X3+XS X3g(X)=X3+X4+X6 Thepolynomials g(X),Xg(X),Xlg(X),andX3g(X)represent codepolynomials inthe(7,4) Hamming code.Ifthecoefficients ofthesepolynomials areusedastheelements oftherows ofa4-by-7matrix,wegetthefollowing generator matrix: G'=[~~~~~~~]o0 1 1 0 1 0 000 1101 Clearly,thegenerator matrixG'soconstructed isnotinsystematic form.Wecanputitinto asystematic formbyaddingthefirstrowtothethirdrow,andaddingthesumofthefirst tworowstothefourthrow.Thesemanipulations resultinthedesiredgenerator matrix: [1 1 0 1 0 0 0]o1 1 0 1 0 0G=1 1100 1 0 1 0 1000 1 whichisexactlythesameasthatinExample 10.2. Wenextshowhowtoconstruct the3-by-7parity-check matrixHfromtheparity-check polynomial h(X).Todothis,wefirsttakethereciprocal ofh(X),namely,X4h(X-1).Forthe problem athand,weformthreepolynomials represented byX4h(X-1)andtwoshiftedver sionsofit,asshownby X4h(X-1)=1+X"+X3+X4 X'h(X-') =X+X3+X4+X' X6h(X-1)=X2+X4+X5+X6 650 CHAPTER 10"EnROn-CONTIlOL CODING Flip-flop Modulo-2 adder Message bits0---- ......._/Code word FIGURE10.10Encodcrfor the(7,4)cycliccodegenerated byg(X)=1+X+X'. Usingthecoefficients ofthesethreepolynomials astheelements oftherowsofthe3-by-7 parity-check matrix,weget 11]1 0 1 1H'=[~~~~~~] 001 0 1 1 HereagainweseethatthematrixH'isnotinsystematic form.Toputitintoasystematic form,weadd[hethirdrowtothefirstrowtoobtain H=[~~~~~ o0101 whichisexactlythesameasthatofExample 10.2. Figure10.10showstheencoderforthe(7,4)cyclicHamming codegenerated bythe polynomial g(X)=1+X+X3•Toillustrate theoperation ofthisencoder, consider the message sequence (1001).Thecontents oftheshiftregisteraremodified bytheincoming message bitsasinTable10.3.Afterfourshifts,thecontents oftheshiftregister,andtherefore theparitybits,are(011).Accordingly, appending theseparitybitstothemessagebits(1001), wegetthecodeword(0111001); thisresultisexa~'tlythesameasthatdetermined earlierin theexample. Figure10.11showsthecorresponding syndrome calculator forthe(7,4)Hamming code.Letthetransmitted codewordbe(0111001) andthereceivedwordbe(0110001); that is,themiddlebitisinerror.Asthereceived bitsarefedintotheshiftregister, initiallysetto zero,itscon£ents aremodified asinTablelOA.A[theendoftheseventhshift,thesyndrome isidentified fromthecontents oftheshiftregisteras11O.Sincethesyndrome isnonzero, the received wordisinerror.Moreover, fromTable10.2,weseethattheerrorpatterncorre sponding tothissyndrome is0001000. Thisindicates thattheerrorisinthemiddlebitofthe received word,whichisindeedthecase. ... TABLE10.3Contents ofthe shiftregisterintheencoder ofFigure10.10formessage sequence (1001) 1 o o 11 2 3 4Shift Input RegisterContents o0 0(initialstate) 1 1 0 o1 1 111 011 10.4CyclkCodes 651 ,1.~Received 0.~)0 )0bits Modulo-2 Flip-flop adder FIGURE 10.11Syndrome calculator forthe(7,4)cycliccodegenerated bythepolynomial g(X)=1+X+X'. ~EXAMPLE 10.4Maximal-Length Codes Foranypositive integerm2"3,thereexistsamaximal-length codewiththefollowing parameters: Blocklength: Number ofmessage bits: Minimum distance:n=2m k=m Maximal-length codesaregenerated bypolynomials oftheform 1+X"g(X)=~ (10.52) whereh(X)isanyprimitive polynomial ofdegreem.Earlierwestatedthatanycycliccode generated byaprimitive polynomial isaHamming codeofminimum distance3(seeExample 10.3).Itfollowstherefore thatmaximal-length codesarethedualofHamming codes. Thepolynomial h(X)definesthefeedback connections oftheencoder. Thegenerator polynomial g(X)definesoneperiodofthemaximal-length code,assuming thattheencoderis intheinitialstate00...01.Toillustrate thesepoints,consider theexample ofa(7,3) maximal-length code,whichisthedualofthe(7,4)Hamming codedescribed inExample 10.3.Thus,choosing h(X)1+X+X' wefindthatthegenerator polynomial ofthe(7,3)maximal-length codeis g(X)1+X+X'+X4 TABLE10.4Contents ofthesyndrome calculator inFigure10.11forthe received word0110001 Shift InputBit ContentsofShiftRegister o0 0(initialstate) 1 1 100 2 0 010 3 0 001 4 0 110 5 1 111 6 1 001 7 0 1 1 0 652 CHAPTER 10illERROR-CONTROL CODING Modulo-2 adderFlip-flop FIGURE10.12Encoderforthe(7,3)maximal-length code;theinitialstateoftheencoderis showninthefigure. Figure10.12showstheencoderforthe(7,3)maximal-length code,thefeedbackconnections ofwhichareexactlythesameasthoseshowninFigure8.2inChapter8.Theperiodofthe codeisn=7.Thus,assuming thattheencoderisintheinitialstate001,asindicated inFigure 10.12,wefindtheoutputsequence isdescribed by 100 1 1101 0 0 '----y------' iriitial g(X)=1+X+X2+X' state Thisresultmaybereadilyvalidated bycyclingthroughtheencoderofFigure10.12. Notethatifweweretochoosetheotherprimitivepolynomial heX)=1+X2+X' forthe(7,3)maximal-length code,wewouldsimplygetthe"image"ofthecodedescribed above,andtheoutputsequence wouldbe"reversed" intime. ~ IIIOTHER CYCLIC CODES Weconclude thediscussion ofcycliccodesbypresenting thecharacteristics ofthreeother important classesofcycliccodes. CyclicRedundancy CheckCodes Cycliccodesareextremely well-suited forerrordetection. Wemakethisstatement fortworeasons. First,theycanbedesigned todetectmanycombinations oflikelyerrors. Second,theimplementation ofbothencoding anderror-detecting circuitsispractical. Itis forthesereasonsthatmanyoftheerror-detecting codesusedinpracticeareofthecyclic codetype.Acycliccodeusedforerror-detection isreferredtoascyclicredundancy check (GRG)code. WedefineanerrorburstoflengthBinann-bitreceived wordasacontiguous sequence ofBbitsinwhichthefirstandlastbirsoranynumberofintermediate bitsare received inerror.Binary(n,k)CRCcodesarecapableofdetecting thefollowing error patterns: 1.Allerrorburstsoflengthn-korless. 2.Afraction oferrorburstsoflengthequalton-k+1;thefracrion equals 1 -2-ln-k-1i• 3.Afraction oferrorburstsoflengthgreaterrhann-k+1;thefracrion equals 12-(n-k-1i. 4.Allcombinations ofdmin-1(orfewer)errors. 5.Allerrorpatterns withanoddnumberoferrorsifthegenerator polynomial g(X) forthecodehasanevennumberofnonzero coefficients. 10.4CyclicCodes 653 ITABLE10.5CRCcodes Code CRC-12code CRC-16code(USA) CRC-lTIJ codeGenerator Polynomial, g(X) 1+X+X2+X3+XU+X12 1+X2+X15+X'6 1+XS+X12+X'6n-k 12 1616 Table10.5presentsthegenerator polynomials ofthreeCRCcodesthathavebecome international standards. Allthreecodescontain1+Xasaprimefactor.TheCRC-12 codeisusedfor6-bitcharacters, andtheothertwocodesareusedfor8-bitcharacters. CRCcodesprovideapowerful methodoferrordetection foruseinautomatic-repeat request(ARQ)strategies discussed inSection10.1,anddigitalsubscriber linesdiscussed inChapter4. Bose-Clwudhf-lri-Hocqfienghem (BCH)Codes5 Oneofthemostimportant andpowerful classesoflinear-block codesareBCHcodes, whicharecycliccodeswithawidevarietyofparameters. Themostcommon binaryBCH codes,knownasprimitive BCHcodes,arecharacterized foranypositiveintegersm(equal toorgreaterthan3)andt[lessthan(2m-1)/2]bythefollowing parameters: Blocklength: n=2m-1 Number ofmessage bits:k2:n-mt Minimum distance: dm;n2:2t+1 EachBCHcodeisat-errorcorreaing codeinthatitcandetectandcorrectuptotrandom errorspercodeword.TheHamming single-error correcting codescanbedescribed as BCHcodes.TheBCHcodesoffer'flexibility inthechoiceofcodeparameters, namely, blocklengthandcoderate.Furthermore, forblocklengthsofafewhundred bitsorless, theBCHcodesareamongthebestknowncodesofthesameblocklengthandcoderate. Adetailed treatment oftheconstruction ofBCHcodesisbeyondthescopeof ourpresentdiscussion. Toprovideafeelfortheircapability, wepresentinTable10.6,the codeparameters andgenerator polynomials forbinaryblockBCHcodesoflengthupto 25-1.Forexample, supposewewishtoconstruct thegenerator polynomial for(15,7) ITABLE10.6BinaryBCRcodesoflengthupto25-1 nk Generator Polynomial 741 1011 15111 10011 1572 111010001 155 3 10100110111 31261 100101 31212 11101101001 31163 1000111110101 111 31115 101100010011 011 010101 316711001 011 011 110101000100111 Notation;n blocklength k=numberofmessagebirs t=maximum numberofdetectable errors Thehigh-order coefficients ofthegeneratorpolynomial g(X)areattheleft, 654 CHAPIER 10IIIERROR-CONTROL CODING BCHcode.FromTable10.6wehave(111010001)forthecoefficients ofthegenerator polynomial; hence,wewrite g(X)=X8+X7+X6+X4+1 Heed-Solmnon Codes6 TheReed-Solomon codesareanimportant subclassofnonbinary BCHcodes;they areoftenabbreviated asRScodes.Theencoder foranRScodediffersfromabinary encoderinthatitoperates onmultiple bitsratherthanindividual bits.Specifically, anRS (n,k)codeisusedtoencodem-bitsymbolsintoblocksconsisting ofn=2m-1symbols, thatis,m(2m-1)bits,wherem;;,:1.Thus,theencoding algorithm expands ablockofk symbolstonsymbols byaddingn-kredundant symbols. WhenmisanintegerpOwer oftwo,them-bitsymbols arecalledbytes.Apopularvalueofmis8;indeed,8-bitRS codesareextremely powerful. At-error-correcting RScodehasthefollowing parameters: Blocklength: n=2m-1symbols Message size: ksymbols Parity-check size: n-k=2tsymbols Minimum distance: dmiD=2t+1symbols TheblocklengthoftheRScodeisonelessthanthesizeofacodesymbol,andtheminimum distanceisonegreaterthanthenumberofparity-check symbols. TheRScodesmakehighly efficientuseofredundancy, andblocklengthsandsy~bolsizescanbeadjusted readilyto _accommodate awiderangeofmessagesizes.Moreover, theRScodesprovideawiderange ofcoderatesthatcanbechosentooptimize performance. Finally,efficientdecoding tech niquesareavailable forusewithRScodes,whichisonemorereasonfortheirwideap plication (e.g.,compact discdigitalaudiosystems). I10.5Convolutional Codes7 (10.53) bits/symbolr=n(L+M)Inblockcoding,theencoderacceptsak-bitmessage blockandgenerates ann-bitcode word.Thus,codewordsareproduced onablock-by-block basis.Clearly,provision must bemadeintheencodertobufferanentiremessageblockbeforegenerating theassociated codeword.Thereareapplications, however, wherethemessagebitscomeinseriallyrather thaninlargeblocks,inwhichcasetheuseofabuffermaybeundesirable. Insuchsitua tions,theuseofconvolutional codingmaybethepreferred method.Aconvolutional coder generates redundant bitsbyusingmodulo-2 convolutions, hencethename. Theencoderofabinaryconvolutional codewithratelIn,measured inbitsper symbol,maybeviewedasafinite-state machine thatconsistsofanM-stageshiftregister withprescribed connections tonmodulo-2 adders,andamultiplexer thatserializes the outputsoftheadders.AnL-bitmessage sequence produces acodedoutputsequence of lengthn(L+M)bits.Thecoderateistherefore givenby L Typically, wehaveL»M.Hence,thecoderatesimplifies to 1r=~nbits/symbol (10.54) 10.5Cotttloluti<mal Codes 655 Theconstraint lengthofaconvolutional code,expressed intermsofmessagebits,isdefined asthenumberofshiftsoverwhichasinglemessagebitcaninfluence theencoderoutput. InanencoderwithanM-stageshiftregister,thememoryoftheencoderequalsMmessage bits,andK=M+1shiftsarerequired foramessage bittoentertheshiftregisterand finallycomeout.Hence,theconstraint lengthoftheencoderisK. Figure10.13ashowsaconvolutional encoderwithn=2andK=3.Hence,the coderateofthisencoderis1/2.TheencoderofFigure10.13aoperates ontheincoming messagesequence, onebitatatime. Wemaygenerate abinaryconvolutional codewithratekinbyusingkseparateshift registers withprescribed connections tonmodulo-2 adders,aninputmultiplexer and Input Flip-flop Path2 (a) Flip-flop Input Modulo-2 adder (b) FIGURE10.13(a)Constraint length-3, rate-tconvolutional encoder.(b)Constraint length-2, rate-~convolutional encoder. 656 CHAPTER 10'"EitROR-CoNTROL CODING anoutputmultiplexer. Anexample ofsuchanencoder isshowninFigure10.13b wherek=2,n=3,andthetwoshiftregisters haveK=2each.Thecoderateis2/3.k thissecondexarn]:Jle, theencoderprocesses theincoming message sequence twobitsata time. Theconvolutional codesgenerated bytheencoders ofFigure10.13arenonsystem<ltic codes.Unlikeblockcoding,theuseofnonsystematic codesisordinarily preferred Over systematic codesinconvolutional coding. Eachpathconnecting theoutputtotheinputofaconvolutional encoder maybe characterized intermsofitsimpulseresponse, definedastheresponse ofthatpathtoa symbol1appliedtoitsinput,witheachflip-flop intheencodersetinitiallyinthezero state.Equivalently, wemaycharacterize eachpathintermsofagenerator polynomial, definedastheunit-delay transform oftheimpulseresponse. Tobespecific,letthegenerator sequence (ggl,g~l,gyJ,,gl:})denotetheimpulseresponse oftheithpath,wherethe coefficients gg),g~l,gy),,gl:}equal0or1.Correspondingly, thegenerator polynomial oftheithpathisdefinedby (10.55) whereDdenotestheunit-delay variable. Thecomplete convolutional encoderisdescribed bythesetofgenerator polynomials {g(l)(D),g(2)(D),...,g(n)(D)}.Traditionally, different variables areusedforthedescription ofconvolutional andcycliccodes,withDbeing commonly usedforconvolutional ~odesandXforcycliccodes. ~ExAMPLE 10.5 Considertheconvolutional encoderofFigure10.13a,whichhastwopathsnumbered 1and 2forconvenience ofreference. Theimpulseresponseofpath1(i.e.,upperpath)is(1,1,1). Hence,thecorresponding generator polynomial isgivenby glll(D)=1+D+D2 Theimpulseresponseofpath2(i.e.,lowerpath)is(1,0,1).Hence,thecorresponding gen eratorpolynomial isgivenby Forthemessagesequence(10011),say,wehavethepolynomial representation m(D)= 1+D3+D4 AswithFouriertransformation, convolution inthetimedomainistransformed intomulti plicationintheD-domain. Hence,theoutputpolynomial ofpath1isgivenby d'1(D)=glll(D)m(D) =(1+D+D2)(1+D3+D4) =1+D+D2+D3+D6 Fromthisweimmediately deducethattheoutputsequenceofpath1is(1111001). Similarly, theoutputpolynomial ofpath2isgivenby d21(D)=g21(D)m(D) =(1+D2)(1+D3+D4) =1+D2+D3+D4+D5+D6 10.5Cunvolutional Codes 657 Theoutputsequenceofpath2istherefore (1011111). Finally,multiplexing thetwooutput sequences ofpaths1and2,wegettheencodedsequence c=(11,10,11,11,01, 01,11) NotethatthemessagesequenceoflengthL=5bitsproduces anencodedsequenceoflength n(L+K1)=14bits.Notealsothatfattheshiftregistettobetestotedtoitszeroinitial state,aterminating sequence ofK1=2zelOSisappended tothelastinputbitofthe messagesequence. Theterminating sequenceofK-1ZetaSiscalledthetailofthemessage. <1!l !IllCODETREE,TRELLIS, ANDSTATE DIAGRAM Traditionally, thestructural properties ofaconvolutional encoderareportrayed ingraph icalformbyusinganyoneofthreeequivalent diagrams: codetree,trellis,andstatedia gram.Wewillusetheconvolutional encoderofFigure10.13aasarunningexample to illustrate theinsightsthateachoneofthesethreediagrams canprovide. Webeginthediscussion withthecodetreeofFigure10.14.Eachbranchofthetree represents aninputsymbol,withthecorresponding pairofoutputbinarysymbols indi catedonthebranch.Theconvention usedtodistinguish theinputbinarysymbols 0and 1isasfollows. Aninput0specifies theupperbranchofabifurcation, whereas input1 specifies thelowerbranch.Aspecificpathinthetreeistracedfromlefttorightinaccor dancewiththeinput(message) sequence. Thecorresponding codedsymbols onthe branches ofthatpathconstitute theinput(message) sequence. Consider, forexample, the messagesequence (10011)appliedtotheinputoftheencoderofFigure10.13a.Following theprocedure justdescribed, wefindthatthecorresponding encoded sequence is (11,10, 11, 11, 01),whichagreeswiththefirst5pairsofbitsintheencoded sequence {Ci} derivedinExample 10.5. Fromthediagram ofFigure10.14,weobservethatthetreebecomes repetitive after thefirstthreebranches. Indeed,beyondthethirdbranch,thetwonodeslabeledaare identical, andsoarealltheothernodepairsthatareidentically labeled.Wemayestablish thisrepetitive property ofthetreebyexamining theassociated encoderofFigure10.13a. Theencoderhasmemory M=K-1=2message bits.Hence,whenthethirdmessage bitenterstheencoder, thefirstmessage bitisshiftedoutoftheregister. Consequently, afterthethirdbranch,themessagesequences (100 m3m4'..)and(000m3m4'..)generate thesamecodesymbols, andthepairofnodeslabeledamaybejoinedtogether. Thesame reasoning appliestoothernodes.Accordingly, wemaycollapse thecodetreeofFigure 10.14intothenewformshowninFigure10.15,whichiscalledatrellis.sItissocalled sinceatrellisisatreelikestructure withremerging branches. Theconvention usedinFigure 10.15todistinguish betweeninputsymbols 0and1isasfollows.Acodebranchproduced byaninput0isdrawnasasolidline,whereas acodebranchproduced byaninput1is drawnasadashedline.Asbefore,eachinput(message) sequence corresponds toaspecific paththroughthetrellis.Forexample, wereadilyseefromFigure10.15thatthemessage sequence (10011)produces theencoded outputsequence (11,10,11,11,01),whichagrees withourprevious result. Atrellisismoreinstructive thanatreeinthatitbringsoutexplicitly thefactthat theassociated convolutional encoder isafinite-state machine. Wedefinethestateofa convolutional encoderofrate1/nasthe(K-1)message bitsstoredintheencoder's shift register.Attimej,theportionofthemessage sequence containing themostrecentKbits iswrittenas(mj-K-i-l,...,mj-bmj),wheremjisthecurrentbit.The(K-l)-bitstateof theencoderattimejistherefore writtensimplyas(mj-b•..,mj-K+2'mj-K+')'Inthe 658 CHAPTER 10"ERROR-CONTROL CODING 00 00 11 00 00 11 11 a t ~ 1110 0100 11 00 FIGURE 10.14 Codetreefortheconvolutional encoder ofFigure1O.13a. caseofthesimpleconvolutional encoderofFigure10.13awehave(K1)=2.Hence, thestateofthisencodercanassumeanyoneoffourpossiblevalues,asdescribed inTable 10.7.Thetrelliscontains(L+K)levels,whereListhelengthoftheincoming message sequence, andKistheconstraint lengthofthecode.Thelevelsofthetrellisarelabeledas j=0,1,...,L+K-1inFigure10.15forK=3.Leveljisalsoreferredtoasdepthj; bothtennsareusedinterchangeably. Thefirst(K-1)levelscorrespond totheencoder's departure fromtheinitialstatea,andthelast(K-1)levelscorrespond totheencoder's 10.5Con'Vol..tiomdCodes 659 Level}=0 FIGURE10.15 Trellisfortheconvolutional encoderofFigure1O.13a. returntothestatea.Clearly,notallthestatescanbereachedinthesetwoportions ofthe trellis.However, inthecentralportionofthetrellis,forwhichtheleveljliesintherange K-1:5j:5L,allthestatesoftheencoderarereachable. Notealsothatthecentral portionofthetrellisexhibitsafixedperiodicstructure. Consider nextaportionofthetrelliscorresponding totimesjandj+1.Weassume thatj2:2fortheexampleathand,sothatitispossibleforthecurrentstateoftheencoder tobea,b,c,ord.Forconvenience ofpresentation, wehavereproduced thisportionof thetrellisinFigure10.16a.Theleftnodesrepresent thefourpossiblecurrentstatesofthe encoder, whereas therightnodesrepresent thenextstates.Clearly,wemaycoalesce the leftandrightnodes.Bysodoing,weobtainthestatediagram oftheencoder, shownin Figure10.16b.Thenodesofthefigurerepresent thefourpossiblestatesoftheencoder, witheachnodehavingtwoincoming branches andtwooutgoing branches. Atransition fromonestatetoanotherinresponse toinput0isrepresented byasolidbranch,whereas atransition inresponse toinput1isrepresented byadashedbranch.Thebinarylabelon eachbranchrepresents theencoder's outputasitmovesfromonestatetoanother.Suppose, forexample, thecurrentstateoftheencoderis(01),whichisrepresented bynodec.The application ofinput1totheencoderofFigure10.13aresultsinthestate(10)andthe encoded output(00).Accordingly, withthehelpofthisstatediagram, wemayreadily determine theoutputoftheencoderofFigure1O.13aforanyincoming messagesequence. Wesimplystartatstatea,theall-zeroinitialstate,andwalkthroughthestatediagramin accordance withthemessagesequence. Wefollowasolidbranchiftheinputisa 0anda dashedbranchifitisa1.Aseachbranchistraversed, weoutputthecorresponding binary labelonthebranch.Consider, forexample, themessagesequence (10011). Forthisinput wefollowthepathabcabd,andtherefore outputthesequence (11,10,11,11,01),which TABLE10.7Statetable fortheconvolutional encoderofFigure10.13a State BinaryDescription a 00 b 10 c 01 d 11 660 CHAPTER 10'"ERROR-CONIROLCODING 01/ / / /OIl / / / 10 / \.\.......------E-----''' \ 00 \ \ 11~ 11 \ \ \ \ 00 (b)10 (')- I I \ I \ / d00 "-"-""- 11»b / /po / /\ \ \ \ \ FIGURE10.16(a)AportionofthecentralpartofthetrellisfortheencoderofFigure1O.13a (h)Statediagramoftheconvolutional encoderofFigure1O.13a. agreesexactlywithourprevious result.Thus,theinput-output relationofaconvolutional encoderisalsocompletely described byitsstatediagram. 10.6Maximum Likelihood Decoding ofConvolutional Codes Nowthatweunderstand theoperation ofaconvolutional encoder, thenextissuetobe considered isthedecoding ofaconvolutional code.Inthissectionwefirstdescribethe underlying theoryofmaximum likelihood decoding, andthenpresentanefficientalgo rithmforitspractical implementation. Letmdenoteamessagevector,andcdenotethecorresponding codevectorapplied bytheencodertotheinputofadiscretememoryless channel. Letrdenotethereceived vector,whichmaydifferfromthetransmitted codevectorduetochannelnoise.Giventhe received vectorc,thedecoderisrequired tomakeanestimatem.ofthemessage vector. Sincethereisaone-to-one correspondence between themessage vectormandthecode vectorc,thedecodermayequivalently produce anestimatecofthecodevector.Wemay thenputm=mifandonlyifc=c.Otherwise, adecoding erroriscommitted inthe receiver. Thedecoding ruleforchoosing theestimatec,giventhereceived vectorr,issaid tobeoptimum whentheprobability ofdecoding errorisminimized. Fromthematerial presented inChapter 6,wemaystatethatforequiprobable messages, theprobability of decoding errorisminimized iftheestimatecischosentomaximize thelog-likelihood function. Letp(rIc)denotetheconditional probability ofreceiving r,giventhatcwassent. (10.56)10.6Maximum Likelihood Decoding ofCon1101"tw..a1 Codes 661 Thelog-likelihood function equalslogp(rIc).Themaximum likelihood decoderordeci sionruleisdescribed asfollows: Choosetheestimatecforwhichthe log-likelihood function logp(rIc)ismaximum. Consider nowthespecialcaseofabinarysymmetric channel. Inthiscase,boththe transmitted codevectorcandthereceivedvectorrrepresent binarysequences oflength N,say.Naturally, thesetwosequences maydifferfromeachotherinsomelocations be causeoferrorsduetochannelnoise.Letc;andr;denotetheithelements ofcandr, respectively. Wethenhave N p(rIc)=ITp(r;ICi) i=1 Correspondingly, thelog-likelihood is N logp(rIc)=Llogp(riICi) i=l(10.57) (10.58) Letthetransition probability p(riICi)bedefinedas p(rilc;)=g'_p,ifri"*Ci ifr;=C;(10.59) (10.60) (10.61)Supposealsothatthereceivedvectorrdiffersfromthetransmitted codevectorcinexactly dpositions. ThenumberdistheHamming distancebetweenvectorsrandc.Then,we mayrewritethelog-likelihood function inEquation (10.58)as logp(rlc)=dlogP+(N-d)log(l-p) =dlog(l~p)+Nlog(l-p) Ingeneral,theprobability ofanerroroccurring islowenoughforustoassumep<112. Wealsorecognize thatNlog(l-p)isaconstant forallc.Accordingly, wemayrestate themaximum-likelihood decoding ruleforthebinarysymmetric channelasfollows: ChoosetheestimateCthatminimizes theHamming distance betweenthereceivedvectorrandthetransmitted vectorc. Thatis,forthebinarysymmetric channel, themaximum"likelihood decoderreducestoa minimum distancedecoder. Insuchadecoder,thereceivedvectorriscompared witheach possibletransmitted codevectorc,andtheparticular oneclosesttorischosenasthe correcttransmitted codevector.Theterm"closest" isusedinthesenseofminimum num berofdiffering binarysymbols(i.e.,Hamming distance) betweenthecodevectorsunder investigation. EllTHEVITERBI ALGORITHM9 Theequivalence betWeenmaximum likelihood decoding andminimum distancedecoding forabinarysymmetric channelimpliesthatwemaydecodeaconvolutional codebychoos ingapathinthecodetreewhosecodedsequence differsfromthereceivedsequence inthe fewestnumberofplaces.Sinceacodetreeisequivalent toatrellis,wemayequallylimit ourchoicetothepossiblepathsinthetrellisrepresentation ofthecode.Thereasonfor preferring thetrellisoverthetreeisthatthenumberofnodesatanylevelofthetrellis 662 CHAPTER 10rnERROR-CONTII.OL CODING doesnotcontinue togrowasthenumberofincoming message bitsincreases; rather,it remainsconstantat2K-t,whereKistheconstraint lengthofthecode. Consider, forexample, thetrellisdiagram ofFigure10.15foraconvolutional code withrater=1/2andconstraint lengthK=3.Weobservethatatlevelj=3,thereare twopathsentering anyofthefournodesinthetrellis.Moreover, thesetwopathswillbe identical onwardfromthatpoint.Clearly,aminimum distance decodermaymakeade cisionatthatpointastowhichofthosetwopathstoretain,withoutanylossofperfor mance.Asimilardecisionmaybemadeatlevelj=4,andsoon.Thissequence ofdecisions isexactlywhattheViterbialgorithm doesasitwalksthroughthetrellis.Thealgorithm operates bycomputing ametricordiscrepancy foreverypossiblepathinthetrellis.The metricforaparticular pathisdefinedastheHamming distancebetweenthecodedsequence represented bythatpathandthereceivedsequence. Thus,foreachnode(state)inthetrellis ofFigure10.15thealgorithm compares thetwopathsentering thenode.Thepathwith thelowermetricisretained, andtheotherpathisdiscarded. Thiscomputation isrepeated foreveryleveljofthetrellisintherangeM:Sj:SL,whereM=K-1istheencoder's memoryandListhelengthoftheincoming messagesequence. Thepathsthatareretained bythealgorithm arecalledsurvivor oractivepaths.Foraconvolutional codeofconstraint lengthK=3,forexample, nomorethan2K-1=4survivor pathsandtheirmetricswill everbestored.Thislistof2K-1pathsisalwaysguaranteed tocontainthemaximum likelihood choice. Adifficulty thatmayariseintheapplication oftheViterbialgorithm isthepossibility thatwhenthepathsentering astatearecompared, theirmetricsarefoundtobeidentical. Insuchasituation, wemakethechoicebyflippingafaircoin(i.e.,simplymakeaguess). Insummary, theViterbialgorithm isamaximum-likelihood decoder, whichisop timumforanAWGNchannel.Itproceeds inastep-by-step fashionasfollows: Initialization Labeltheleft-most stateofthetrellis(i.e.,theall-zerostateatlevel0)as0,since thereisnodiscrepancy atthispointinthecomputation. Computation stepj+1 Letj=0,1,2,...,andsupposethatattheprevious stepjwehavedonetwothings: I>-Allsurvivor pathsareidentified. i>Thesurvivor pathanditsmetricforeachstateofthetrellisarestored. Then,atlevel(clocktime)j+1,compute themetricforallthepathsentering eachstate ofthetrellisbyaddingthemetricoftheincoming branches tothemetricoftheconnecting survivor pathfromlevelj.Hence,foreachstate,identifythepathwiththelowestmetric asthesurvivor ofstepj+1,therebyupdating thecomputation. FinalStep Continue thecomputation untilthealgorithm completes itsforwardsearchthrough thetrellisandtherefore reachesthetermination node(i.e.,all-zerostate),atwhichtimeit makesadecision onthemaximum likelihood path.Then,likeablockdecoder, these quenceofsymbols associated withthatpathisreleasedtothedestination asthedecoded versionofthereceived sequence. Inthissense,itistherefore morecorrecttorefertothe Viterbialgorithm asamaximum likelihood sequence estimator. However, whenthereceivedsequence isverylong(nearinfinite), thestoragerequire mentoftheViterbialgorithm becomes toohigh,andsomecompromises mustbemade. 10.6Maxi.....mLikelilwod Decoding ofCOIWolutional Codes 663 Theapproach usuallytakenisto"truncate" thepathmemory ofthedecoder asdescribed here.Adecoding window oflengtheisspecified, andthealgorithm operates onacorre sponding frameofthereceived sequence, alwaysstopping afteresteps.Adecision isthen madeonthe"best"pathandthesymbolassociated withthefirstbranchonthatpathis released totheuser.Thesymbolassociated withthelastbranchofthepathisdropped. Next,thedecoding window ismovedforward onetimeinterval, andadecision onthe nextcodeframeismade,andsoon.Thedecoding decisions madeinthiswayarenolonger trulymaximum likelihood, buttheycanbemadealmost:asgoodprovided thatthedecod ingwindow islongenough. Experience andanalysishaveshownthatsatisfactory results areobtained ifthedecoding window lengtheisontheorderof5timestheconstraint lengthKoftheconvolutional codeormore. II'>EXAMPLE 10.6Correct Decoding ofReceived All-Zero Sequence SupposethattheencoderofFigure10.13agenerates anall-zerosequence thatissentovera binarysymmetric channel,andthatthereceivedsequence is(0100010000 ...).Therearetwo errorsinthereceivedsequence duetonoiseinthechannel:oneinthesecondbitandtheother inthesixthbit.Wewishtoshowthatthisdouble-error patterniscorrectable throughthe application oftheViterbidecoding algorithm. InFigure10.17,weshowtheresultsofapplyingthealgorithm forlevelj=1,2,3,4, 5.Weseethatforj=2thereare(forthefirsttime)fourpaths,oneforeachofthefourstates oftheencoder. Thefigurealsoincludesthemetticofeachpathforeachlevelinthe computation. IntheleftsideofFigure10.17,forj=3weshowthepathsenteringeachofthestates, togetherwiththeirindividual metrics.Intherightsideofthefigure,weshowthefoursurvivors thatresultfromapplication ofthealgorithm forlevelj=3,4,5. Examining thefoursurvivorsinFigure10.17forj=5,weseethattheall-zeropath hasthesmallestmetricandwillremainthepathofsmallestmetricfromthispointforward. Thisclearlyshowsthattheall-zerosequence isthemaximum likelihood choiceoftheViterbi decoding algorithm, whichagreesexactlywiththetransmitted sequence. <ll EXAMPLE 10.7Incorrect Decoding ofReceived All-Zero Sequence Supposenextthatthereceivedsequence is(1100010000 ...),whichcontainsthreeerrors compared tothetransmitted all-zerosequence. InFigure10.18,weshowtheresultsofapplyingtheViterbidecoding algorithm forj= 1,2,3,4. Weseethatinthisexample thecorrectpathhasbeeneliminated bylevelj=3. Clearly,atriple-error patternisuncorrectable bytheViterbialgorithm whenappliedtoa convolutional codeofrate112andconstraint lengthK=3.Theexception tothisruleisa triple-error patternspreadoveratimespanlongerthanoneconstraint length,inwhichcase itisverylikelytobecorrectable. <ll illFREEDISTANCE OFACONVOLUTIONAL CODE Theperformance ofaconvolutional codedepends notonlyonthedecoding algorithm usedbutalsoonthedistance properties ofthecode.Inthiscontext, themostimportant singlemeasure ofaconvolutional code'sabilitytocombatchannelnoiseisthefreedistance, denoted bydf"e'Thefreedistance ofaconvolutional codeisdefinedastheminimum Hamming distancebetweenanytwocodewordsinthecode.Aconvolutional codewith freedistancedheecancorrectterrorsifandonlyifdf,eeisgreaterthan2t. Thefreedistance canbeobtained quitesimplyfromthestatediagram ofthecon volutional encoder. Consider, forexample, Figure10.16b,whichshowsthestatediagram 664 CHAPTER 10IIERROR-CONTROL CODING Received sequence j=l01 10_______ ,,,, '~ 1 0... ".1 ,,,,,,,,'\'.3 \~2 \ \ \ \·2Received sequence j=201 00 2.2.1...,,,,, '02 2. SurvivorsSurvivorsSurvivors10... ..,,,,,,,, ,1'3'\ '", \ \ \ \ \ \ \\)\\\ 2 \ \\2 \ '03 10... ...,,""\."x2/)<2/,,3 \ \ \ /1\\/23 \ 3 \ \ • • '03Received sequence 01 00 01 0,,,, ,1 j=3\ \ \ \ \ \ \ \\2 \----.. 4 Received sequence 01000100 2 2 0 j=4 2 Received sequence 0100010000 2 2 0,,, ,1,,2.. \j=5 \ FIGURE10.17 Illustrating stepsintheViterbialgorithm forExample 10.6. 10.6Maximum Likelihood Decoding ofCrnwolutitmal Codes 665 j=1Received sequence Received sequence11 2 O~ ''''...a.. 1100 j=2 Received sequence 110001 2 j=3a-", '......0 \,/"2,, \,~//1 \L-.. 1 3 Received sequence 11000100 2 3 j=42 '"• 3\ ,",... .I~ \ ~ \'( \/~~ 1''';.\",_3 \ 1\ ,". . 1 3 FIGURE10.18Illustrating breakdown oftheViterbialgorithm inExample 10.7. oftheencoderofFigure10.13a.Anynonzerocodesequence corresponds toacomplete pathbeginning andendingatthe00state(i.e.,nodea).Wethusfinditusefultosplitthis nodeinthemannershowninthemodified statediagram ofFigure10.19,whichmaybe viewedasasignal-flow graphwithasingleinputandasingleoutput.Asignal-flow graph consistsofnodesanddirectedbranches; itoperates bythefoHowing rules: 1.Abranchmultiplies thesignalatitsinputnodebythetransmittance characterizing thatbranch. 2.Anodewithincoming branches sumsthesignalsproduced byallofthosebranches. 3.Thesignalatanodeisappliedequallytoallthe branches outgoing fromthatnode. 4.Thetransferfunctionofthegraphistheratiooftheoutputsignaltotheinputsignal. 666 CHAPTER 10IIIERROR-CONTROL CODING DLf'\ I I \ I\, d .,DL/, I I DLj I I I I D2L I DL D2L....--->-__,<i---I__~~__ --+_-;;,~----,> flo b'''--.----E------/ C L FIGURE10.19 Modified statediagramofconvolutional encoder. Returning tothesignal-flow graphofFigure10.19,wenotethattheexponent ofDona branchinthisgraphdescribes theHamming weightoftheencoderoutputcorresponding tothatbranch.Theexponent ofLisalwaysequaltoone,sincethelengthofeachbranch isone.LetT(D,L)denotethetransferfunction ofthesignal-flow graph,withDandL playingtheroleofdummyvariables. FortheexampleofFigure10.19,wemayreadilyuse rules1,2,and3toobtainthefollowing input-output relations: b=D2LaO+LC} C=DLb+DLd d=DLb+DLd (10.62) a,=D2Lc whereao,b,c,d,andatdenotethenodesignalsofthegraph.SolvingthesetofEquations (10.62)fortheratioa,lao,wefindthatthetransferfunction ofthegraphinFigure10.19 isgivenby T{D,L)=1_DL(l+L)(10.63) Usingthebinomial expansion, wemayequivalently write T(D,L)=DSL'2:(DL{l+L))i i=O(10.64) SettingL=1inEquation (10.64),wethusgetthedistancetransferfunction expressed in theformofapowerseries: T(D,1)=D5+2D6+4D7+... (10.65) Sincethefreedistance istheminimum Hamming distancebetween anytwocodewords inthecodeandthedistancetransferfunctionT(D,1)enumerates thenumberofcode wordsthatareagivendistanceapart,itfollowsthattheexponent ofthefirstterminthe expansion ofT{D,1)definesthefreedistance. Thus,onthebasisofEquation (10.65),the convolutional codeofFigure10.13ahasafreedistancedire.=5. Thisresultindicates thatuptotwoerrorsinthereceivedsequence arecorrectable, fortwoorfewertransmission errorswillcausethereceivedsequence tobeatmostata Hamming distanceof2fromthetransmitted sequence butatleastataHamming distance of3fromanyothercodesequence inthecode.Inotherwords,inspiteofthepresence of 10.6Maximum Likelihood Decoding ofConvolutional Codes 667 TABLE10.8Maximumfree distances attainable withsystematic and nonsystematic convolutional codes ofrate1/2 Constraint LengthK Systematic Nonsystematic 2 3 3 3 4 5 4 4 6 5 5 7 6 6 8 7 6 10 8 7 10 anypairoftransmission errors,thereceived sequence remainsclosertothetransmitted sequence thananyotherpossiblecodesequence. However, thisstatement isnolongertrue iftherearethreeormorecloselyspacedtransmission errorsinthereceivedsequence. These observations confirmtheresultsreported earlierinExamples 10.6and10.7. Inusingthedistance transferfunction T(D,1)tocalculate thefreedistance ofa convolutional code,itisassumed thatthepowerseriesintheunit-delay variableDrep resenting T(D,1)isconvergent (i.e.,itssumhasa"finite"value).Thisassumption is required tojustifytheexpansion giveninEquation (10.65)fortheconvolutional codeof Figure10.13a.However, thereisnoguarantee thatT(D,1)isalwaysconvergent. When T(D,1)isnonconvergent, aninfinitenumberofdecoding errorsarecausedbyafinite numberoftransmission errors;theconvolutional codeisthensubjecttocatastrophic error propagation, andthecodeiscalledacatastrophic code.'°Inthiscontextitisnoteworthy thatasystematic convolutional codecannotbecatastrophic. Unfortunately, forapre scribedconstraint lengthK,thefreedistances thatcanbeattained withsystematic con volutional codesusingschemes suchasthoseshowninFigure10.13areusuallysmaller thanforthecaseofnonsystematic convolutional codes,asindicated inTable10.8. AsYMPTOTIC CODING GAINll Thetransferfunction oftheencoderstatediagram, modified inamannersimilartothat illustrated inFigure10.19,maybeusedtoevaluate aboundonthebiterrorratefora givendecoding scheme;detailsofthisevaluation are,however, beyondthescopeofour presentdiscussion. Herewesimplysummarize theresultsfortwospecialchannels, namely, thebinarysymmetric channelandthebinary-input additivewhiteGaussian noise(AWGN) channel, assuming theuseofbinaryphase-shift keying(PSK)withcoherent detection. 1.Binarysymmetric channel. Thebinarysymmetric channelmaybemodeled asanad ditivewhiteGaussian noisechannelwithbinaryphase-shift keying(PSK)asthemodula tionandwithhard-decision demodulation. Thetransition probability pofthebinarysym metricchannelisthenequaltothebiterrorrate(BER)fortheuncoded binaryPSKsystem. FromChapter6werecallthatforlargevaluesofEb/No,theratioofsignalenergyperbit to-noisepowerspectraldensity,thebiterrorrateforbinaryPSKwithoutcodingisdom inatedbytheexponential factorexp(-Eb/NoJ. Ontheotherhand,thebiterrorratefor thesamemodulation schemewithconvolutional codingisdominated bytheexponential 668 CHAPTER 10IIIERROR-CONTROL CODING factorexp(-df"erEb/2No), whereristhecoderateanddf,,"isthefreedistance ofthe convolutional code.Therefore, asafigureofmeritformeasuring theimprovement inerror performance madebytheuseofcodingwithhard-decision decoding, wemayusethe exponents todefinetheasymptotic codinggain(indecibels) asfollows: Ga=1010glO(df~er) dB (10.66) 2.Binary-input AWGNchannel. Consider nextthecaseofamemoryless binary-input AWGNchannelwithnooutputquantization [i.e.,theoutputamplitude liesintheinterval (-00,00)]. Forthischanne~ theoryshowsthatforlargevaluesofEblNothebiterrorrate forbinaryPSKwithconvolutional ~odingisdominated bytheexponential factor exp(-dheerEbINo), wheretheparameters areaspreviously defined.Accordingly, inthis case,wefindthattheasymptotic codinggainisdefinedby (10.67) FromEquations (10.66)and(10.67)weseethattheasymptotic codinggainforthe binary-input AWGNchannelisgreaterthanthatforthebinarysymmetric channelby3 dB.Inotherwords,forlargeEblNo,thetransmitter forabinarysymmetric channelmust generateanadditional 3dBofsignalenergy(orpower)overthatforabinary-input AWGN channelifwearetoachievethesameerrorperformance. Clearly,thereisanadvantage to begainedbypermitting anunquantized demodulator outputinsteadof making hardde cisions.Thisimprovement inperformance, however, isattainedatthecostofincreased decodercomplexity duetotherequirement foraccepting analoginputs. Theasymptotic codinggainforabinary-input AWGNchannelisapproximated to withinabout0.25dBbyabinaryinputQ-aryoutputdiscretememoryless channelwith thenumberofrepresentation levelsQ=8.Thismeansthatwemayavoidtheneedfor ananalogdecoderbyusingasoft-decision decoderthatperforms finiteoutputquantization (typically, Q=8),andyetrealizeaperformance closetotheoptimum. I10.7Trellis-Coded Modulation 12 Inthetraditional approach tochannelcodingdescribed inthepreceding sectionsofthe chapter,encoding isperformed separately frommodulation inthetransmitter; likewisefor decoding anddetection inthereceiver.Moreover, errorcontrolisprovided byttansmitting additional redundant bitsinthecode,whichhastheeffectoflowering theinformation bit rateperchannelbandwidth. Thatis,bandwidth efficiency istradedforincreased power efficiency. Toattainamoreeffectiveutilization oftheavailable bandwidth andpower,coding andmodulation havetobetreatedasasingleentity.Wemaydealwiththisnewsituation byredefining codingastheprocessofimposing certainpatternsonthetransmitted signal. Indeed,thisdefinition includesthetraditional ideaofparitycoding. Trelliscodesforband-limited channels resultfromthetreatment ofmodulation and codingasacombined entityratherthanastwoseparateoperations. Thecombination itself isreferredtoastrellis-coded modulation (TCM).Thisform ofsignaling hasthreebasic features: 1.Thenumberofsignalpointsintheconstellation usedislargerthanwhatisrequired forthemodulation formatofinterestwiththesamedatarate;theadditional points allowredundancy forforwarderror-control codingwithoutsacrificing bandwidth. 10.7Trellis-Coded Modulation 669 2.Convolutional codingisusedtointroduce acertaindependency betweensuccessive signalpoints,suchthatonlycertainpatterns orsequences ofsignalpointsare permitted. 3.Soft-decision decoding isperformed inthereceiver,inwhichthepermissible sequence ofsignalsismodeled asatrellisstructure; hence,thename"trelliscodes." Thislatterrequirement istheresultofusinganenlarged signalconstellation. Byincreasing thesizeoftheconstellation, theprobability ofsymbolerrorincreases forafixedsignal to-noiseratio.Hence,withhard-decision demodulation wewouldfaceaperformance loss beforewebegin.Performing soft-decision decoding onthecombined codeandmodulation trellisameliorates thisproblem. Inthepresence ofAWGN,maximum likelihood decoding oftrelliscodesconsistsof findingthatparticular paththroughthetrelliswithminimum squaredEuclidean distance tothereceivedsequence. Thus,inthedesignoftrelliscodes,theemphasis isonmaximizing theEuclidean distance between codevectors(or,equivalently, codewords)ratherthan maximizing theHamming distance ofanerror-correcting code.Thereasonforthisap proachisthat,exceptforconventional codingwithbinaryPSKandQPSK,maximizing theHamming distance isnotthesameasmaximizing thesquaredEuclidean distance. Accordingly, inwhatfollows,theEuclidean distance isadopted asthedistancemeasure ofinterest. Moreover, whileamoregeneraltreatment ispossible, thediscussion is(by choice)confined tothecaseoftwo-dimensional constellations ofsignalpoints.Theim plication ofsuchachoiceistorestrictthedevelopment oftrelliscodestomultilevel am plitudeandlorphasemodulation schemessuchasM-aryPSKandM-aryQAM. Theapproach usedtodesignthistypeoftrelliscodesinvolvespartitioning anM-ary constellation ofinterestsuccessively into2,4,8,...subsetswithsizeMl2,Ml4,Ml8,..., andhavingprogressively largerincreasing minimum Euclidean distancebetweentheirre spectivesignalpoints.Suchadesignapproach bysetpartitioning represents the"keyidea" intheconstruction ofefficientcodedmodulation techniques forband-limited channels. InFigure10.20,weillustrate thepartitioning procedure byconsidering acircular constellation thatcorresponds to8-PSK.Thefiguredepictstheconstellation itselfandthe 2and4subsetsresulting fromtwolevelsofpartitioning. Thesesubsetssharethecommon ;I ;Ido=2Sin(if)=.,j2.v'2 .%;0 o • • 0••:dol•• • 0o • • 0o • d,=2 Signal number00 ....d0 o'--.00 00 oO.o 0 o 0 • 0 10 2o0o • • 00,0 01 1.0o 0 o 0o • 11 3 FIGURE10.20 Partitioning of8-PSKconstellation, whichshowsthatdo<d,<d,. 670 CHAPTER 10IiERROR-CONTROL CODING •••Id ••• 0 o0 •0\1 o0 0 0 • 000 o0 0 0 o0 • 0o0 •001 a0 0 0 o0•0 000 0 • 0 0 0•O~ o0 0 0 • 0 •ayeo o •0 • 000 0 o •0• 0/00 00\1 000.o.0 0 0000 0000 o•0 0000. o0 0 0 0 0 0 0••••...~ o •0 •.0.0 o •0 • .~ o0 0 0 o.0. o0 0 0 o.0. 01\1 o0 0 0 a0 0 0 o.00 000. 000 0 0 0 a0 000.0.00~. d!:,.0 • 0 l~O..0.0 y~.O o•0 o0 0 o•0 000\1 o0 • 0 000 0 • 0a0 o0 0a Signal number000o100 4010 2110 6001 1101 5011 3111 7 FlGlJRE10.21 Partitioning of16-QAM constellation, w!richshowsthatdo<d]<d2<£1,. property thattheminimum Euclidean distances betweentheirindividual pointsfollowan increasing pattern: do<d1<d2• Figure10.21illustrates thepartitioning ofarectangular constellation corresponding to16-QAM. Hereagainweseethatthesubsetshaveincreasing within-subset Euclidean distances: do<d1<d2<d3• Basedonthesubsetsresulting fromsuccessive partitioning ofatwo-dimensional constellation, wemaydeviserelatively simpleandyethighlyeffective codingschemes. Specifically, tosendnbits/symbol withquadrature modulation (i.e.,onethathasin-phase andquadrature components), westartwithatwo-dimensional constellation of2"+1signal pointsappropriate forthemodulation formatofinterest;acirculargridisusedforM-ary PSK,andarectangular oneforM-aryQAM.Inanyevent,theconstellation ispartitioned into4or8subsets.Oneortwoincoming bitspersymbolenterarate-1I2 orrate-2/3 binaryconvolutional encoder, respectively; theresulting twoorthreecodedbitspersymbol determine theselection ofaparticular subset.Theremaining uncoded databitsdetermine whichparticular pointfromtheselectedsubsetistobesignaled. Thisclassoftrelliscodes isknownasUngerboeck codes. Sincethemodulator hasmemory, wemayusetheViterbialgorithm toperform maximum likelihood sequence estimation atthereceiver. Eachbranchinthetrellisofthe Ungerboeck codecorresponds toasubsetratherthananindividual signalpoint.Thefirst stepinthedetection istodetermine thesignalpointwithineachsubsetthatisclosestto thereceived signalpointintheEuclidean sense.Thesignalpointsodetermined andits metric(i.e.,thesquaredEuclidean distancebetweenitandthereceivedpoint)maybeused thereafter forthebranchinquestion, andtheViterbialgorithm maythenproceedinthe usualmanner. IIIUNGERBOECK CODES FOR8-PSK TheschemeofFigure10.22adepictsthesimplestUngerboeck 8-PSKcodeforthetranS missionof2bits/symbol. Theschemeusesarate-l/2 convolutional encoder; thecorre- 10.7Treliis-CodedModulatWn 671 Modulo-2 adderr--------------------- Input Flip-flop I I I I I I I I I l ~ Rate-lfZ convo!utional encoder (0)8-PSK signalmapper aa a a 11 1 a a11a a1 a1a1a1a a12345 6 7 SignalnumberMostsignificant bit Output Encoder state 00 10 01 11 (b) FIGURE10.22(a)Four-state Ungerboeck codefor8-PSK;themapperfollows Figure10.20. (b)Trellisofthecode. sponding trellisofthecodeisshowninFigure10.22b,whichhasfourstates.Notethat themostsignificant bitoftheincoming binarywordisleftuncoded. Therefore, eachbranch ofthetrellismaycorrespond totwodifferent outputvaluesofthe8-PSKmodulator or, equivalently, tooneofthefour2-pointsubsetsshowninFigure10.20.ThetrellisofFigure 10.22balsoincludestheminimum distancepath. TheschemeofFigure10.23adepictsanotherUngcrbocck 8-PSKcodefortrans mitting2bits/sample; itisnextinthelevelofcomplexity. Thissecondschemeusesa rate-2/3convolutional encoder. Therefore, thecorresponding trellisofthecodehaseight states,asshowninFigure10.23b.Inthiscase,bothbitsoftheincoming binarywordare encoded. Hence,eachbranchofthetrelliscorresponds toaspecificoutputvalueofthe 8-PSKmodulator. ThetrellisofFigure10.23balsoincludestheminimum distancepath. Figures10.22band10.23balsoincludetheencoderstates.InFigure10.22,thestate oftheencoderisdefinedbythecontents ofthetwo-stage shiftregister.Ontheotherhand, inFigure10.23,itisdefinedbythecontentofthesingle-stage (top)shiftregisterfollowed bythatofthetwo-stage (bottom) shiftregister. 672 CHAPI'ER 10"ERROR-CONTROL CODING Modulo-2 adder Output8-PSK signalmapper 0 0 0 0 0 0 0 0 0 0 0 0 0123 456 SignalnumberRate-2/3 convolutional encoderr---------------------- IFlip-flop : I I I III I I I II I I I II I I I II II :'-------+-----;*-H'-i-~ : I ~ JInput (a) Encoder state 000 010 100 110 001 011 101 111 (b) FIGURE10.23(a)Eight-state Ungerboeck codefor8-PSK;themapperfollowsFigure10.20. (b)Trellisofthecodewithonlysomeofthebranches shown. 10.7Trellis-CtHkdMod ..lati.... 673 mlAsYMPTOTIC CODING GAIN Following thediscussion inSection10.6,wedefinetheasymptotic codinggainofUnger boeckcodesas Ga=10IOglO(~ee) d(10.68) wherediceeisthefreeEuclidean distanceofthecodeanddreiistheminimum Euclidean distanceofanuncoded modulation schemeoperating withthesamesignalenergyperbit. Forexample, byusingtheUngerboeck 8-PSKcodeofFigure1O.22a,thesignalconstel lationhas8messagepoints,andwesend2messagebitsperpoint.Hence,uncoded trans missionrequiresasignalconstellation with4messagepoints.Wemaytherefore regard uncoded 4-PSKasthereference fortheUngerboeck 8-PSKcodeofFigure10.22a. TheUngerboeck 8-PSKcodeofFigure1O.22aachieves anasymptotic codinggain of3dB,calculated asfollows: 1..EachbranchofthetrellisinFigure10.22bcorresponds toasubsetoftwoantipodal signalpoints.Hence,thefreeEuclidean distance direeofthecodecanbenolarger thantheEuclidean distance d2betweentheantipodal signalpointsofsuchasubset. Wemaytherefore write wherethedistance d2isdefinedinFigure1O.24a;seealsoFigure10.20. 2.Theminimum Euclidean distanceofanuncoded QPSK,viewedasareference op eratingwiththesamesignalenergyperbit,equals(seeFigure10.24b) Hence,aspreviously stated,theuseofEquation (10.68)yieldsanasymptotic codinggain of10loglo2=3dB. Theasymptotic codinggainachievable withUngerboeck codesincreases withthe numberofstatesintheconvolutional encoder. Table10.9presentstheasymptotic coding gain(indB)forUngerboeck 8-PSKcodesforincreasing numberofstates,expressed with Quadrature (a)In-phaseQuadrature o (b)In-phase FIGURE10.24 Signal-space diagrams forcalculation ofasymptotic codinggainofUngerboeck 8-PSKcode.(a)Definition ofdistanced2•(b)Definition ofreference distancedref• 674 CHAPTER 10'"EUROR-CONTROL CODING TABLE10.9Asymptotic codinggainofVngerboeck 8-PSKcodes, withrespecttouncoded 4-PSK Numberofstates Codinggain(dB)A 33.616 4.132 4.664 4.8128 5256 5.4512 5.7 respecttouncoded 4-PSK.Notethatimprovements ontheorderof6dBrequirecodes withaverylargenumberofstates. I10.8TurboCodes13 Traditionally, thedesignofgoodcodeshasbeentackledbyconstructing codeswithagreat dealofalgebraic structure, forwhichtherearefeasibledecoding schemes. Suchanap proachisexemplified bythelinearblockcodesandconvolutional codesdiscussed inpre cedingsections. Thedifficulty withthesetraditional codesisthat,inanefforttoapproach thetheoretical limitforShannon's channelcapacity, weneedtoincreasethecode-word lengthofalinearblockcodeortheconstraint lengthofaconvolutional code,which,in turn,causesthecomputational complexity ofamaximum likelihood decodertoincrease exponentially. Ultimately, wereachapointwherecomplexity ofthedecoder issohigh thatitbecomes physically unrealizable. Variousapproaches havebeenproposed fortheconstruction ofpowerful codeswith large"equivalent" blocklengthsstructured insuchawaythatthedecoding canbesplit intoanumberofmanageable steps.Building ontheseprevious approaches, thedevelop mentofturbocodesandlow-density parity-check codeshasbeenbyfarmostsuccessful. Indeed,thisdevelopment hasopenedabrandnewandexcitingwayofconstructing good codesanddecoding themwithfeasiblecomplexity. Turbocodesarediscussed inthissec tionandlow-densityparity-check codesarediscussed inSection10.10. I!jTuRBO CODING Tnitsmostbasicform,theencoderofaturbocodeconsistsoftwoconstituent systematic encoders joinedtogether bymeansofaninterleaver, asillustrated inFigure10.25. Aninterleaver isaninput-output mapping devicethatpermutes theordering ofa sequence ofsymbolsfromafixedalphabet inacompletely deterministic manner; thatis, ittakesthesymbolsattheinputandproduces identical symbolsattheoutputbutina different temporal order.Theinterleaver canbeofmanytypes,ofwhich theperiodicand pseudo-random aretwo.Turbocodesuseapseudo-random interleaver, whichoperates Message bitsxf--------;;... Parity-check L-__ ----J bits'lOutput Parity-check bitsZ2 FIGURE10.25Blockdiagramofturboencoder. 10.8TurboCodes 675 onlyonthesystematic bits.Therearetworeasonsfortheuseofaninterleaver inaturbo code: ~Totietogether errorsthatareeasilymadeinonehalfoftheturbocodetoerrors thatareexceptionally unlikelytooccurintheotherhalf.Thisisindeedthemain reasonwhytheturbocodeperforms betterthanatraditional code. ~Toproviderobustperformance withrespecttomismatched decoding, whichisa problem thatariseswhenthechannelstatistics arenotknownorhavebeenincor rectlyspecified. Typically, butnotnecessarily, thesamecodeisusedforbothconstituent encoders inFigure10.25.Theconstituent codesrecommended forturbocodesareshortconstraint lengthrecursive systematic convolutional (RSC)codes.Thereasonformakingthecon volutional codesrecursive (i.e.,feedingoneormoreofthetapoutputsintheshiftregister backtotheinput)istomaketheinternalstateoftheshiftregisterdependonpastoutputs. Thisaffectsthebehavior oftheerrorpatterns(asingleerrorinthesystematic bitsproduces aninfinitenumberofparityerrors),withtheresultthatabetterperformance oftheoverall cod.ingstrategyisattained. ~ExAMPLE 10.8Eight-state RSCEncoder Figure10.26showsanexample eight-state RSCencoder. Thegenerator matrixforthisre cursiveconvolutional codeis (D)=[11+D+D2 +D3 ] (10.69)g ,1+D+D3 whereDisthedelayvariable. Thesecondentryofthematrixg(D)isthetransferfunctionof thefeedback shiftregister,definedasthetransform oftheoutputdividedbythetransform oftheinput.LetM(D)denotethetransform ofthemessagesequence [mi}~~landB(D)denote thetransform oftheparitysequence [bi17':-{Bydefinition, wehave B(D)1+D+D2+D3 M(D) 1+D+D3 Cross-multiplying, weget: (1+D+D2+D3)M(D)=(1+D+D3)B(D) which,oninversion intothetimedomain,yields mi+mi-l+mi-2+mi-3+hi+bi-1+bi-3=0 M~a:eo-...".. .."..Sy~~~;tic(10.70) '--------;0.. par::r~~eCk FIGURE10.26Example eight-state recursive systematic convolutional (RSC)encoder. 676 CHAPTER 10l!lERROR-CONTROL CODING wheretheaddition ismodulo-2. Equation (10.70)istheparity-check equation, whichthe convolutional encoderofFigure10.26satisfiesateachtimestepi. -<II InFigure10.25theinputdatastreamisapplieddirectlytoencoder1,andthepseudo randomly reordered versionofthesamedatastreamisappliedtoencoder2.Thesystematic bits(i.e.,originalmessagebits)andthetwosetsofparity-check bitsgenerated bythetwo encoders constitute theoutputoftheturboencoder. Although theconstituent codesare convolutional, inrealityturbocodesareblockcodeswiththeblocksizebeingdetermined bythesizeoftheinterleaver. Moreover, sincebothRSCencoders inFigure10.25are linear,wemaydescribeturbocodesaslinearblockcodes. Theblocknatureoftheturbocoderaisesapractical issue:Howdoweknowthe beginning andtheendofacodeword?Thecommon practiceistoinitialize theencoder totheall-zerostateandthenencodethedata.Afterencoding acertainnumberofdata bitsanumberoftailbitsareaddedsoastomaketheencoderreturntotheall-zerostate attheendofeachblock;thereafter thecycleisrepeated. Thetermination approaches of turbocodesincludethefollowing: ~Asimpleapproach istoterminate thefirstRSCcodeintheencoderandleavethe secondoneunterminated. Adrawback ofthisapproach isthatthebitsattheendof theblockduetothesecondRSCcodearemorevulnerable tonoisethantheother bits.Experimental workhasshownthatturbocodesexhibitalevelingoffinperfor manceastheSNRincrease~. Thisbehavior isnotlikeanerrorfloor,butithasthe appearance ofanerrorfloorcompared tothesteepdropinerrorperformance atlow SNR.Thiserrorfloorisaffectedbyanumberoffactors,thedominant oneofwhich isthechoiceofinterleaver. i>-Amorerefinedapproach14istoterminate bothconstituent codesintheencoderin asymmetric manner. Through thecombined useofagoodinterleaver anddual termination, theerrorfloorcanbereducedbyanorderofmagnitude compared to thesimpletermination approach. Intheoriginalversionoftheturboencoder, theparity-check bitsgenerated bythe twoencoders inFigure10.25werepunctured priortodatatransmission overthechannel tomaintain therateat1/2.Apunctured codeisconstructed bydeletingcertainparity checkbits,therebyincreasing thedatarate.Puncturing istheinverseofextending acode. Itshould,however, beemphasized thattheuseofapuncture mapisnotanecessary requirement forthegeneration ofturbocodes. Thenoveltyoftheparallelencoding schemeofFigure10.25isintheuseofrecursive systematic convolutional (RSC)codesandtheintroduction ofapseudo-random interleaver between thetwoencoders. Thusaturbocodeappearsessentially randomtothechannel byvirtueofthepseudo-random interleaver, yetitpossesses sufficient structure forthe decoding tobephysically realizable. Codingtheoryassertsthatacodechosenatrandom iscapableofapproaching Shannon's channelcapacity, provided thattheblocksizeis sufficiently large."5Thisisindeedthereasonbehindtheimpressive performance ofturbo codes,asdiscussed next. IillPERFORMANCE OFTURBO CODES Figure10.27showstheerrorperformance ofa1/2rate,turbocodewithalargeblocksize forbinarydatatransmission overanAWGNchannel."6Thecodeusesaninterleaver of 10.8TurboCodes 677 100r--~-~--~-~--~-~----., 10-2 ~ ~10-3 10-4.":-::-:"l' I I I I I I II I I1-Shannonlimit 1....Uncoded - -Turbocode 10-6_L4-~_~2-~0--~2--~4--6~---'8~--.JlO EbiNo•dB FIGURE10.27 Noiseperformances of1/2rate,turbocodeanduncoded transmission for AWGNchannel; thefigurealsoincludes Shannon's theoretical limitonchannel capacity forcode rater.= 1/2. size65,536andaBCJR-based decoder; detailsofthisdecoderarepresented laterinthe section.Eighteen iterations ofturbodecoding wereusedinthecomputation. Forthepurposeofcomparison, Figure10.27alsoincludestwoothercurvesforthe sameAWGNchannel: ..Uncoded transmission (i.e.,coderater=1). I!>Shannon's theoretical limitforcoderate1/2,whichfollowsfromFigure9.18b. FromFigure10.27,wemaydrawtwoimportant conclusions: 1.Although thebiterrorratefortheturbo-coded transmission issignificantly higher thanthatforuncoded transmission atlowEblNo,thebiterrorratefortheturbo codedtransmission dropsveryrapidlyonceacriticalvalueofEblNohasbeen reached. 2.AtabiterrorrateoflO-s,theturbocodeislessthan0.5dBfromShannon's theo reticallimit. Note,however, attaining thishighlyimpressive performance requiresthatthesizeof theinterleaver, or,equivalently, theblocklengthoftheturbocode,belarge.Also,the largenumberofiterations neededtoimproveperformance increases thedecoderlatency. Thisdrawback isduetothefactthatthedigitalprocessing ofinformation doesnotlend itselfreadilytotheapplication offeedback, whichisadistinctive featureoftheturbo decoder. Nowthatwehaveanappreciation fortheimpressive performance ofturbocodes, thestageissetforadiscussion ofhowturbodecoding isactuallyperformed. illTuRBO DECODING Turbocodesderivetheirdistinctive namefromanalogyofthedecoding algorithm tothe "turboengine"principle. Figure10.28ashowsthebasicstructure oftheturbodecoder.It operates onnoisyversionsofthesystematic bitsandthetwosetsofparity-cheek bitsin twodecoding stagestoproduceanestimateoftheoriginalmessagebits. 678 CHAPTER 10illERROR-CONTROL CODING Noisy sustematic o---~ bitsu Noisy parity-checko---- .....J bits., (a) Closeswitchattimestepn=0and set1,(1)=0Decoderbits i (b) FIGURE10.28(a)Blockdiagramofturbodecoder.(b)Extrinsicformofturbodecoder,whereI standsforinterleaver, Dforde-interleaver, andBCJRforBCJRalgorithm forlog-MAPdecoding. Eachofthetwodecoding stagesusesaBC]Ralgorithm,17 whichwasoriginally invented byBahl,Cocke,Jelinek,andRaviv(hencethename)tosolveamaximum a posteriori probability (MAP)detection problem. TheBCJRalgorithm differsfromthe Viterbialgorithm intwofundamental respects: 1.TheBCJRalgorithm isasoftinput-soft outputdecoding algorithm withtworecur sions,oneforwardandtheotherbackward, bothofwhichinvolvesoftdecisions. In contrast, theViterbialgorithm isasoftinput-hard outputdecoding algorithm, with asingleforward recursion involving softdecisions; therecursion endswithahard decision, whereby aparticular survivorpathamongseveralonesisretained. Incom putational terms,theBCJRalgorithm istherefore morecomplex thantheViterbi algorithm becauseofthebackward recursion. 2.TheBCJRalgorithm isaMAPdecoderinthatitminimizes thebiterrorsbyesti matingtheaposteriori probabilities oftheindividual bitsinacodeword;torecon structtheoriginaldatasequence, thesoftoutputsoftheBCJRalgorithm arehard limited.Ontheotherhand,theViterbialgorithm isamaximum likelihood sequence estimator inthatitmaximizes thelikelihood function forthewholesequence, not eachbit.Assuch,theaveragebiterrorrateoftheBCJRalgorithm canbeslightly betterthantheViterbialgorithm; itisneverworse. Mostimportant, formulation oftheBe]Ralgorithm restsonthefundamental assumptions that(1)thechannelencoding, namely,theconvolutional encoding performed inthetranS mitter,ismodeled asaMarkovprocess,and(2)thechannelismemoryless. Inthecontext ofourpresentdiscussion, theMarkovian assumption meansthatifacodecanberepre- 10.8TurboCo.us 679 sentedasatrellis,thenthepresentstateofthetrellisdepends onlyonthepaststateand theinputbit.(Amathematical treatment oftheBCJRalgorithm isgivenlaterinthis section.) Beforeproceeding todescribetheoperation ofthetwo-stage turbodecoderinFigure 10.28a,wefinditdesirable tointroduce thenotionofextrinsic information. Themost convenient representation forthisconceptisasalog-likelihood ratio,inwhichcaseex trinsicinformation iscomputed asthedifference between twolog-likelihood ratiosas depicted inFigure10.29.Formally, extrinsicinformation, generated byadecoding stage forasetofsystematic (message) bits,isdefinedasthedifference betweenthelog-likelihood ratiocomputed attheoutputofthatdecoding stageandtheintrinsicinformation repre sentedbyalog-likelihood ratiofedbacktotheinputofthedecoding stage.Ineffect, extrinsic information istheincremental information gainedbyexploiting thedependencies thatexistbetween amessage bitofinterestandincoming rawdatabitsprocessed bythe decoder. Onthisbasis,wemaydepicttheflowofinformation inthetwo-stage turbodecoder ofFigure10.28ainasymmetric extrinsic mannerasshowninFigllre10.28b.Thefirst decoding stageusestheBCJRalgorithm toproduce asoftestimate ofsystematic bitx;, expressed asthelog-likelihood ratio (10.71) j=1,2,...,k /(.)=I(P(X;=llu,t"12(X)))1x,og -, PIx;=0Iu,t"/2(X)) whereuisthesetofnoisysystematic bits,tlisthesetofnoisyparity-check bitsgenerated byencoder1,and12(x)istheextrinsic information aboutthesetofmessagebitsxderived fromtheseconddecoding stageandfedbacktothefirststage.Assuming thatthekmessage bitsarestatistically independent, thetotallog-likelihood ratioattheoutputofthefirst decoding stageistherefore k /,(x)=2:/,(x;) j=l(10.72) Hence,theextrinsic information aboutthemessage bitsderivedfromthefirstdecoding stageis 1,(x)=/,(X)-12(x) (10.73) where12(x)istobedefined. Beforeapplication totheseconddecoding stage,theextrinsic information II(x)isre orderedtocompensate forthepsuedo-random interleaving introduced intheturboen coder.Inaddition, thenoisyparity-check bitst2generated byencoder2areusedasinput. ThusbyusingtheBCJRalgorithm, theseconddecoding stageproduces amorerefined Other information}-__..,..,sOfl-in~~~~Zr-output Extrinsic information Raw data FIGURE10.29 Illustrating theconceptofextrinsic information. 680 CHAPTER 10"ERROR-CONTROL CODING softestimateofthemessagebitsx.Thisestimateisre-interleaved toproducethetotallog likelihood ratiolz(x).Theextrinsic information Iz(x)fedbacktothefirstdecoding stage istherefore (10.74) whereI,(x)isitselfdefinedbyEquation (10.73),andlz(x)isthelog-likelihood ratiocom putedbythesecondstage.Specifically, forthejthelementofthevectorx,wehave 1()-I(P(Xj=11u,tz,I,(X)))ZXj-ogz -, P(Xj=olu,tz,l,(x))j=1,2,...,k (10.75) Through theapplication ofIz(x)tothefirStstage,thefeedback looparoundthepairof decoding stagesistherebyclosed.Notethatalthough inactualfactthesetofnoisysys tematicbitsuisonlyappliedtothefirstdecoding stageasinFigure10.28a,byformulating theinformation flowinthesymmetric extrinsic mannerdepictedinFigure10.28bwefind thatuis,ineffect,alsoappliedtotheseconddecoding stage. Anestimate ofthemessage bitsxiscomputed byhard-limiting thelog-likelihood ratio12(x)attheoutputofthesecondstage,asshownby i=sgn(1z(x)) (10.76) wherethesignumfunction operates oneachelementof12(x)iJ:!dividually. Toinitiatetheturbodecoding algorithm, wesimplysetlz(x)=0onthefirstitera tionofthealgorithm; seeFigure10.28b. Themotivation forfeedingonlyextrinsic information fromonestagetothenextin theturbodecoderofFigure10.28istomaintain asmuchstatistical independence between thebitsaspossiblefromoneiteration tothenext.Thefeedbackdecoding strategydescribed hereinimplicitly reliesonthisassumption. Ifthisassumption ofstatistical independence isstrictlytrue,itcanbeshownthattheestimateidefinedinEquation (10.76)approaches theMAPsolution asthenumberofiterations approaches infinity.'s Theassumption of statistical independence appearstobeclosetothetruthinthevastmajority ofcasesen countered inpractice. IIITHEBeJRALGORITHM Foradiscussion ofturbodecoding tobecomplete, amathematical exposition oftheBCJR algorithm forMAPestimation isinorder. Letx(t)betheinputtoatrellisencoderattimet.Lety(t)bethecorresponding output observed atthereceiver. Notethaty(t)mayincludemorethanoneobservation; forex ample,aratelincodeproduces nbitsforeachinputbit,inwhichcasewehavean n-dimensional observation vector.Lettheobservation vectorbedenotedby y(','1=[y(l),y(2),...,y(t)] LetAm(t)denotetheprobability thatastatesIt)ofthetrellisencoderequalsm,where m=1,2,...,M.Wemaythenwrite A(t)=P[s(t)Iy] (10.77) 10.8TurboCodes 681 wheresit)andA(t)arebothM-by-lvectors.Then,foraratelInlinearconvolutional code withfeedback asintheRSCcode,theprobability thatasymbol"1"wasthe message bit isgivenby P(x(t)=11y)=2:A.(t) 5E9'A(10.78) where ~Aisthesetoftransitions thatcorrespond toasymbol"1"attheinput,andA.(t) isthes-component ofA(t). Definetheforwardestimation ofstateprobabilities astheM-by-lvector a(t)=P(s(t)Iy(l,tj) (10.79) wheretheobservation vectory(1,t)isdefinedabove.Alsodefinethebackward estimation ofstateprobabilities astheM-by-lvector wherel3(t)=P(s(t)Iy(t,k)) (10.80) (10.81)y(t,k)=[y(t),y(t+1),...,y(k)] Thevectorsa(t)andl3(t)areestimates ofthestateprobabilities attimetbasedonthepast andfuturedata,respectively. Wemaythenformulate theseparability theoremasfollows: Thestateprobabilities attimetarerelatedtotheforwardestimator a(t)andback wardestimator (3(t)bythevector a(t)•(3(t) A(t)=IIa(t)•(3(t)II, wherea(t)•(3(t)isthevectorproductofa(t)and(3(t),andIIa(t)•(3(t)II,isthe LInormofthisvectorproduct. Thevectorproductart)•l3(t)(nottobeconfused withtheinnerproduct) isdefinedin termsoftheindividual elements ofa(t)andl3(t)by rCl',(t)I3,(t) ] Cl'2(t)/32(t) a(t)•l3(t)= . Cl'M(t)I3M(t) andtheL,normofa(t)•l3(t)isdefinedby M IIa(t)•(3(t)II,=2:C>m(t)/3m(t) m=l(10.82) (10.83) Theseparability theorem saysthatthestatedistribution attimetgiventhepastis mdependent ofthestatedistribution attimetgiventhefuture,whichisintuitively satisfying recalling theMarkovian assumption forchannelencoding, whichisbasictotheBCJR algorithm. Moreover, thistheorem provides thebasisofasimplewayofcombining the forwardandbackward estimates toobtainacomplete description ofthestateprobabilities. Toproceedfurther,letthestatetransition probability attimetbedefinedby 'Ym',m(t) =P(s(t)=m,y(t)Is(t1)=m'l (10.84) 682 CHAPI'ER 1010ERROR-CONTROL CODING anddenotetheM-by-M matrixoftransition probabilities as r(t)={I'm',m(t)] Wemaythenformulate therecursion theorem asfollows:(10.85) Theforwardestimatea(t)andbackward estimatej}(t)arecomputed recursively as and f(t+1)j}(t+1) j}(t)=IIf(t+1)j}(t+1)II, wherethesuperscript Tdenotesmatrixtransposition.(10.86) (10.87) Theseparability andrecursion theorems together definetheBCJRalgorithm forthe computation ofaposteriori probabilities ofthestatesandtransitions ofacodetrellis, giventheobservation vector.Usingtheseestimates, thelikelihood ratiosneededforturbo decoding maythenbecomputed byperforming summations overselectedsubsetsofstates asrequired. 10.9Computer Experiment: TurboDecoding Twoproperties constitute thehallmark ofturbocodes: Property 1: Theerrorperformance oftheturbodecoderimproves withthenumberofiterations ofthe decoding algorithm. Thisisachieved byfeedingextrinsic information fromtheoutputDf thefirstdecoding stagetotheinputoftheseconddecoding stageintheforwardpathand feedingextrinsic information fromtheoutputofthesecondstagetotheinputofthefirst stageinthebackward path,andthenpermitting theiterativedecoding processtotakeits naturalcourseinresponse tothereceivednoisymessageandparitybits. Property 2 Theturbodecoder iscapableofapproaching theShannon theoretical limitofchannel capacity inacomputationally feasiblemanner;thisproperty hasbeendemonstrated ex perimentally butnotyetproventheoretically. Property 2requires thattheblocklengthoftheturbocodebelarge.Unfortunately, a demonstration ofthisproperty requires theuseofsophisticated implementations ofthe turbodecoding algorithm thatarebeyondthescopeofthisbook.Accordingly, wefocus ourattention onademonstration ofProperty 1inthiscomputer experiment. So,astheprimary objective ofthiscomputer experiment, wewishtousethelog MAPimplementation oftheBCJRalgorithm todemonstrate Property 1ofturbodecoding. 10.10 Low-Density Parity-Check Codes 683 10--3 c::w '" 10-4 10-58iterations 10iterations 10-6L- ~ ~ _ 1 15 ~5 EbINO'dB FIGURE10.30 Result.ofthecomputer experiment onturbodecoding, forincreasing numberof iterations. Theonlychannelimpairment assumed intheexperiment isadditivewhiteGaussian noise. Detailsoftheturboencoderanddecoderareasfollows: TurboEncoder (described inFigure10.25): Encoder 1:convolutional encoder[1,1,1] Encoder 2:convolutional encoder[1,0,1] Block(i.e.,interleaver) length:1,200bits TurboDecoder (described inFigure10.28): TheBCJRalgorithm forlog-MAP decoding. Theexperiment was carried outforEb/No=1,1.5,2,and2.5dB,withvarying numberofiterations ateachEb/No.Foreachtrialoftheexperiment, thenumberofbit errorswascalculated afteraccumulating atotalof20blocksofdata(each1,200bitslong) thatwerenoise-corrupted. Theprobability oferrorwasthenevaluated astheratioofbit errorstothetotalnumberofencodedbits.Notethatinthiscalculation, manyoftheblocks ofencoded bitswerecorrectly decoded. Theresultsoftheexperiment areplottedinFigure10.30.Thefollowing observations canbemadefromthisfigure: 1.ForagivenEb/No,theprobability oferrordecreases withincreasing numberof iterations, confirming Property 1ofturbodecoding. 2.Aftereightiterations, thereisnosignificant improvement indecoding performance. 3.Forafixednumberofiterations, theprobability oferrordecreases withincreasing Eb/N(),whichistobeexpected. I10.10 Low-Density Parity-Check Codes19 Turbocodes,discussed inSection10.8,andlow-density parity-check (LDPC)codes,dis cussedinthissection,belongtoabroadfamilyof error-eontrol codingtechniques called 684 CHAPTER 10'"ERROR-CONTROL CODING compound codes.Thetwomostimportant advantages ofLDPC ~odesoverturbocodes are: ""Absence oflow-weight codewords. ~Iterativedecoding oflowercomplexity. Withregardtotheissueoflow-weight codewords,weusuallyfindthatasmall numberofcodewordsinaturbocodeareundesirably closetothegivencodeword.Due tothiscloseness inweights, onceinawhilethechannelnoisecausesthetransmitted code wordtobemistaken foranearbycodeword.Indeed,itisthisbehavior thatisresponsible fortheerrorfloor(typically aroundabiterrorrateof10-5to10-6)thatwasmentioned earlier.Incontrast, LDPCcodescanbeeasilyconstructed sothattheydonothavesuch low-weight codewords,andtheycantherefore achievevanishingly smallbiterrorrates. Theerror-floor problem inturbocodescanbealleviated bycarefuldesignofthe interleaver. Turning nexttotheissueofdecoding complexity, wenotethatthecomputational complexity ofaturbodecoderisdominated bytheBCJRalgorithm, whichoperates on thetrellisfortheconvolutional codeusedintheencoder. Thenumberofcomputations in eachrecursion oftheBCJRalgorithm scaleslinearlywiththenumberofstatesinthetrellis. Commonly usedturbocodesemploytrelliseswith16statesormore.Incontrast, LDPC codesuseasimpleparity-check trellisthathasjusttwostates.Consequently, thedecoders forLDPCcodesaresignificantly simplerthanthoseforturbodecoders. Moreover, being parallelizable, LDPCdecoding maybeperformed atgreaterspeedsthanturbodecoding. However, apractical objection totheuseofLDPCcodesisthatforlargeblock lengths,theirencoding complexity ishighcompared toturbocodes. !i'lCONSTRUCTION OFLDPC CODES LDPCcodesarespecified byaparity-check matrixdenotedbyA,whichissparse;thatis, itconsistsmainlyofOsandasmallnumberof1s.Inparticular, wespeakof(n,tc)t,) LDPCcodes,wherendenotestheblocklength,tcdenotestheweight(i.e.,numberof1s) ineachcolumnofthematrixA,andtrdenotestheweightofeachrowwithtr>te•The rateofsuchaLDPCcodeis r=1 (10.88) whosevaliditymaybejustifiedasfollows. Letpdenotethedensityof1sintheparity checkmatrixA.Then,following theterminology introduced inSection10.3,wemayset tc=p(n-k) and tr=pn where(n-k)isthenumberofrowsinAandnisthenumberofcolumns (i.e.,theblock length).Therefore, dividingtcbyt"weget 1k tr n 10.10 Low-Density Parity-Check Codes 685 Bydefinition, thecoderateofablockcodeiskin,hencetheresultofEquation (10.88) follows.Forthisresulttohold,however, therowsofAmustbelinearlyindependent. Thestructure ofLDPCcodesiswellportrayed bybipartite graphs.Figure10.31 showssuchagraphfortheexample codeofn=10,tc=3,andt,=5.Theleft-hand nodesinthegraphofFigure10.31arevariablenodes,whichcorrespond toelements of thecodeword.Theright-hand nodesofthegrapharechecknodes,whichcorrespond to thesetofparity-check constraints satisfiedbycodewordsinthecode.LDPCcodesofthe typeexemplified bythegraphofFigure10.31aresaidtoberegularinthatallthenodes ofasimilarkindhaveexactlythesamedegree.IntheexamplegraphofFigure10.31,the degreeofthevariablenodesistc=3,andthedegreeofthechecknodesistr=5.Asthe blocklengthnapproaches infinity,eachchecknodeisconnected toavanishingly small fractionofvariablenodes,hencethetermlow-density. ThematrixAisconstructed byputtinglsinAatrandom, subjecttotheregularity constraints : I>-Eachcolumncontains asmallfixednumber, toofls. ~Eachrowcontains asmallfixednumber,t"ofls. Inpractice, theseregularity constraints areoftenviolatedslightlyinordertoavoidhaving linearlydependent rowsintheparity-check matrixA. Unlikethelinearblockcodesdiscussed inSection10.3,theparity-check matrixAof LDPCcodesisnotsystematic (i.e.,itdoesnothavetheparity-check bitsappearing in diagonal form),hencetheuseofasymboldifferent fromthatusedinSection10.3.Nev ertheless, forcodingpurposes, wemayderiveagenerator matrixGforLDPCcodesby meansofGaussian elimination performed inmodulo-2 arithmetic; thisprocedure isillus tratedlaterinExample 10.9.Following theterminology introduced inSection10.3,the 1-by-ncodevectorcisfirstpartitioned as c=[b;m] Variable nodes FIGURE10.31 Bipartitegraph ofthe(l0,3,5)LDPecode. 686 CHAPTER 10"ERROR.CONTROL CODING wheremisthek-by-1messagevector,andbisthe(n-k)-by-1 parityvector;seeEquation (10.9).Correspondingly, theparity-check matrixAispartitioned as (10.89) whereAlisasquarematrixofdimensions (n-k)X(n-k),andAzisarectangular matrix ofdimensions kX(n-k);transposition symbolized bythesuperscript Tisusedinthe partitioning ofmatrixAforconvenience ofpresentation. Imposing theconstraint ofEqua tion(10.16)ontheLDPCcode,wemaywrite or,equivalently, bAl+mAz=0 RecallfromEquation (10.7)thatthevectorsmandbarerelatedby b=mP(10.90) wherePisthecoefficient matrix.Hence,substituting thisrelationintoEquation (10.90), wereadilyfindthat,foranynonzeromessagevectorm,thecoefficient matrixofLDPC codessatisfiesthecondition PAl+Az=0 whichholdsforallnonzero message vectorsand,inparticular, formintheform [0...0 1 0...0]thatwillisolateindividual rowsofthegenerator matrix. Solvingthisequation formatrixP,weget (10.91) (10.92)whereA,listheinverseofmatrixAI,whichisnaturally definedinmodulo-2 arithmetic. Finally,thegenerator matrixofLDPCcodesisdefinedby G=[p,Ik] =[A2A,1:Ik] whereIkisthek-by-kidentitymatrix;seeEquation (10.12). Itisimportant tonotethatifwetaketheparity-check matrixAforsomearbitrary LDPCcodeandjustpick(n-k)columns ofAatrandomtoformasquarematrixAI> thereisnoguarantee thatAiwillbenonsingular (i.e.,theinverseA,lwillexist),evenif therowsofAarelinearlyindependent. Infact,foratypicalLDPCcodewithlargeblock lengthn,sucharandomly selectedAlishighlyunlikelytobenonsingular, becauseitis verylikelythatatleastonerowofAlwillbeallOs.Ofcourse,whentherowsofAare linearlyindependent, therewillbesomesetof(n-k)columns ofAthatwillmakea nonsingular AI>asillustrated inExample 10.9.Forsomeconstruction methods forLDPC codesthefirst(n-k)columnsofAmaybeguaranteed toproduceanonsingular Ai,or atleastdosowithahighprobability, butthatisnottrueingeneral. 10.10 Low-Density Parity-Check Codes 687 P.EXAi\1PLE 10.9(10,3,5)LDPCCode Consider thebipartite graphofFigure10.31pertaining roa(10,3,5)LDPCcode.Thepariry- checkmatrixofrhecodeisdefinedby 11a1a1a a1 i]a1 1a1a11a 1a a a 1 1a a1 A=a11 1a1 1a a 1a1a1a a1a a a a 1a a ,11 1~ Ai Ai whichappearstoberandom, whilemaintaining theregularity constraints: tc=3andt,=S. Partitioning thematrixAinthemannerdescribed inEquation (10.89): 1a1a1 i]1 1a1a a1a1 1A, 1a a1a a1 1a1 1a1 1a A,~[11a1a 1]1a a1 a1aa a1a1 ToderivetheinverseofmatrixA"wefirstuseEquation (10.90)towrite 1a1010] 1 1a1a a a1a1 1a [po,b"b2"h,b.,bs)1aa1a1 [~o,UhU2',U3, U4,usJ ba1 1a1a u=mA2 1a1100~ A, wherewehaveintroduced thevectorutodenotethematrixproductmA2•ByusingGaussian elimination, thematrixA,istransformed intolowerdiagonal form(i.e.,alltheelements above themaindiagonal arezero),asshownby 1a a a a a 11aa a a a11aaaA,->1a1 1aa a1a1 1a 1aa1a1 688 CHAPTER 10II!ERROR-CONTROL CODING Thistransformation isachieved bythefollowing modulo-2 additions performed onthecol umnsofsquarematrixA,: I>Columns 1and2areaddedtocolumn3. l>-Column2isaddedtocolumn4. I>Columns 1and4areaddedtocolumn5. I>-Columns 1,2and5areaddedtocolumn6. Correspondingly, thevectoruistransformed as Accordingly, premultiplying thetransformed matrixA,bytheparityvectorb,usingsuccessive eliminations inmodulo-2 arithmetic working backwards, andputtingthesolutions forthe elements oftheparityvectorbintermsoftheelements ofthevectoruinmatrixform,weget [~~~~~~ 1 1 1000 [uo,U"U2, U3, U4,U5]1 1 0 0 1 0 ~010011 1 1 1 1 0 1~A,' TheinverseofmatrixA,istherefore 0 01011 101001 A-'-1 1 1 0 0 0,-1 10 010 0 10011 1 1 1 1 0 1 ThematrixproductA2A,'is(usingthegivenvalueofA2andthevalueofA,'justfound) [1 0 0 1 AA-'_ 0 0 0 12'-0011 o1 0 1 Finally,usingEquation (10.92),thegenerator ofthe(10,3,5)LDPCcodeis G-[!001o:10 0 !]0 01 1 :01 0 0 110:001 10110:00 0~. A2A,' Ik Itisimportant torecognize thattheLDPCcodedescribed inthisexample isintended onlyforthepurposeofillustrating theprocedure involved inthegeneration ofsuchacode. Inpractice, theblocklengthnisordersofmagnitude largerthanthatconsidered inthis example. Moreover, inconstructing thematrixA,wemayconstrain allpairsofcolumnsto 10.10Low-Density Pa...ty-Check Codes 689 haveamatrixoverlap(i.e.,innerproductofanytwo columns inmatrixA)nottoexceed1; suchaconstraint, overandabovetheregularity constraints, isexpected toimprovetheper formance ofLDPCcodes.Unfortunately, withasmallblocklengthasthatconsidered inthis example,itisdifficulttosatisfythisaddirional requirement. .., MINIMUM DISTANCE OFLDPC CODES Inpractice, theblocklengthofaLDPCcodeislarge,rangingfrom103to106,which meansthatrhenumberofcodewordsinaparticular codeiscorrespondingly large.Con sequently, thealgebraic analysisofLDPCcodesisratherdifficult. Itismuchmorepro ductivetoperform astatistical analysisonanensemble ofLDPCcodes.Suchananalysis permitsustomakestatistical statements aboutcertainproperties ofmember codesinthe ensemble. Moreover, anLDPCcodewiththeseproperties canbefoundwithhighprob abilitybyarandomselection fromtheensemble. Amongtheseproperties, the minimum distanceofthemembercodesisofparticular interest.FromSection10.3werecallthattheminimum distanceofalinearblockcodeis, bydefinition, thesmallestHamming distancebetweenanypairofcodevectorsinthecode. Overanensemble ofLDPCcodes,theminimum distanceofamembercodeisnaturally a randomvariable. Elsewhere20itisshownthatastheblocklengthnincreases, forfixe.d tc2:3andt,>tctheprobability distribution oftheminimum distancecanbeoverbounded byafunction thatapproaches aunitstepfunction atafixedfraction Ll,toftheblock lengthn.Thus,forlargen,practically alltheLDPCcodes intheensemble h;;eaminimum distance ofatleastnLl'J,Table10.10presents theraterandLl'J,ofLDPCcodesfor different valuesoftheweight-pair (teottl.Fromthistableweseethatfortc=3andtt=6 thecoderaterattainsitshighestvalueof1/2andthefraction Lltc"attainsitssmallestvalue, hencethepreferred choiceoftc=3andt,=6inthedesignofLDPCcodes. llJlPROBABILISTIC DECODING OFLDPC CODES Atthetransmitter, amessage vectormisencoded intoacodevector C=mG,whereGis thegenerator matrixforaspecified weight-pair (teot,)andtherefore minimum distance dmin•Thevectorcistransmitted overanoisychanneltoproducethereceived vector r=c+e whereeistheerrorvectorduetochannelnoise;seeEquation (10.17).Byconstruction, thematrixAisaparitymatrixoftheLDPCcode;thatis,AGT=O.Giventhereceived TABLEIO.loa Theraterand fractional term..dv,ofLDPCcodes forvaryingweights tcandtr 5 4 3 4 3 3t, 6 5 4 6 5 6Rater 0.167 0.2 0.25 0.333 0.4 0.50.255 0.210 0.122 0.1290.044 0.023 'AdaptedfromGallager(1962)withpermission oftheIEEE. 690 CHAPTER 10IIIERROR-CONTROL CODING vectorr,thebit-by-bit decoding problem istofindthemostprobable vectorcthatsatisfies thecondition CATO. Inwhatfollows,abitreferstoanelementofthereceivedvectorr,andacheckrefers toarowofmatrixA.Let:J(i)denotethesetofbitsthatparticipate inchecki.Let:JlU) denotethesetofchecksinwhichbitjparticipates. Aset:J(i)thatexcludes bitjisdenoted by:J(i)\j.Likewise, asetj(j)thatexcludes checkiisdenoted by!J'(j)\i. Thedecoding algorithm hastwoalternating steps:horizontal stepandverticalstep, whichrunalongtherowsandcolumns ofmatrixA,respectively. Inthecourseofthese steps,twoprobabilistic quantities associated withnonzeroelements ofmatrixAarealter natelyupdated. Onequantity, denoted byPit,definestheprobability thatbitjissymbol x(i.e.,symbol0or1),giventheinformation derivedviachecksperformed inthehorizontal step,exceptforchecki.Thesecondquantity, denoted byQij,definestheprobability that checkiissatisfied, giventhatbitjisfixedatthevaluexandtheotherbitshavethe probabilities Pi;':j'E:J(i)\j. TheLDPCdecoding algorithm thenproceeds asfollows:21 InitUdi%ation ThevariablesPgandpi;aresetequaltotheaprioriprobabilities PYandpJofsymbols oand1,respectively, withPY+pJ=1. Horizontal Step Inthehorizontal stepofthealgorithm, werunthroughthechecksi.Define Foreachweight-pair (i,j),compute dQi;=IIdPij' ;'E9('JV Hence,set Vertical Step Intheverticalstepofthealgorithm, thevaluesoftheprobabilities Pgandpi;are updated usingthequantities computed inthehorizontal step.Inparticular, foreachbitj, compute Pg=OI.i;p7IIQ?-; i'E:J(j)\i P~=OI.i;PJIIQt,; j'E'J{j)\i wherethescalingfactor 01.#ischosentomake Pg+Pi;=l 10.11 I....egularCodes 691 Intheverticalstep,wemayalsoupdatethepseudo-posterior probabilities: P7=D:;p7ITQ~ iE§(j) PJ=D:;pJITQ!; iE:J(j) whereD:;ischosentomake P7+PJ=1 Thequantities obtained intheverticalstepareusedtocompute atentative estimate c.IftheconditioncAT=0issatisfied, thedecoding algorithm isterminated. Otherwise, thealgorithm goesbacktothehorizontal step.Ifaftersomemaximum numberofiterations (e.g.,100or200)thereisnovaliddecoding, adecoding failureisdeclared. Thedecoding procedure described hereinisaspecialcaseofthegenerallow-complexity sum-product algorithm. Simplystated,thesum-product algorithm passesprobabilistic quantities betweenthe checknodesandvariable nodesofthebipartite graph.Byvirtueofthefactthateach parity-check constraint canberepresented byasimpleconvolutional coderwithonebit ofmemory, wefindthatLDPCdecoders aresimplertoimplement thanturbodecoders, asstatedearlier. Intermsofperformance, however, wemaysaythefollowing inlightofexperimental resultsreported intheliterature: RegularLDPC codes donotappeartocomeascloseto Shannon's limitasdotheirturbocodecounterparts. L!0.•11Irregular Codes Theturbocodesdiscussed inSection10.8andtheLDPC codes discussed inSection10.10 arebothregularcodes,eachinitsownindividual way.Theerror-correcting performance ofbothofthesecodesoveranoisychannelcanbeimproved substantially byusingtheir respective irregular forms. Inastandard turbocodewithitsencoderasshowninFigure10.25,theinterleaver mapseachsystematic bittoauniqueinputbitofconvolutional encoder2.Incontrast, irregular turbocodes22useaspecialdesignofinterleaver thatmapssomesystematic bits tomultiple inputbitsoftheconvolutional encoder. Forexample, eachof10percentof thesystematic bitsmaybemappedtoeightinputsoftheconvolutional encoderinsteadof r------------...,... Sy~~~;tic Message bits xParity-check bitsZI Parity-check bitsZzOutput FIGURE 10.32 Blockdiagramofirregular turboencoder. 692 CHAPrER 10'"EnnOR-CONTROL CODING asingleone.AsshowninFigure10.32,similarirregular interleavers areusedinboth convolutional encoding pathstogenerate theparity-check bitsz,andZ2inresponse tothe message bitsx.Irregular turbocodesaredecoded inasimilarfashiontoregularturbo codes. Toconstruct anirregular LDPCcode/3thedegreesofthevariableandchecknodes inthebipartite grapharechosenaccording tosomedistribution. Forexample, wemay haveanirregular LDPCcodewiththefollowing graphical representation: ~Onehalfofthevariablenodeshavedegree5andtheotherhalfofthevariablenodes havedegree3. ~Onehalfofthechecknodeshavedegree6andtheotherhalfofthechecknodes havedegree8. Foragivenblocklengthandagivendegreesequence, wedefineanensemble ofcodesby choosing theedges(i.e.,theconnections betweenthevariableandchecknodes)inaran domfashion.Specifically, theedgesemanating fromthevariablenodesareenumerated in somearbitrary order,andlikewisefortheedgesemanating fromthechecknodes. Figure10.33plotstheerrorperformances ofthefollowing codes:24 l>Irregular LDPCcode:k=50,000,n=100,000, rate=1/2 ~Turbocode(regular): k=65,536,n=131,072, andrate=1/2 1>0Irregular turbocode:k=65,536,n=131,072, andrate=1/2 wherekisthenumberofmessagebitsandnistheblocklength.Thegenerator polynomials forthetwoconvolutional encoders intheregularlirregular turbocodesareasfollows: Encoder1:g(D)=1+D4 Encoder2:g(D)=1+D+D2+D3+D4 Figure10.33alsoincludesthecorresponding theoretical limitonchannelcapacityforcode rater=1/2. 10-2 .l!! ~e10-3 l;; 10-51-Shannon limit11. - .Regularturbocode ...Irregularturbocode - -IrregularLDPecode\ \ \ I \ \ 10-5 :------:"'::--7::-~c--:-'-:c_--=---:c'-:-c_::'-:'---=:'_::_____,J-1-D.8-D.6-0.4-D.200.2 0.4 0.60.8 EbiNo.dB FIGURE 10.33 Noiseperformances ofregularturbocode,irregular turbocodeandirregular low-density parity-check (LDPCj code,compared totheShannon limitforcoderater=1/2. 10.12Summary andDiscussion 693 Basedontheresultspresented inFigure10.33,wemaymakethefollowing observations: ..Theirregular LDPCcodeoutperforms theregularturbocodeinthatitcomescloser toShannon's theoretical limitby0.175dB. l»Amongthethreecodesdisplayed therein,theirregular turbocodeisthebestinthat itisjust0.213dBawayfromShannon's theoretical limit. l~0.12Summary andDiscussion Inthischapter,westudiederror-control codingtechniques thathaveestablished themselves asindispensable toolsforreliabledigitalcommunication overnoisychannels. Theeffect oferrorsoccurring duringtransmission isreducedbyaddingredundancy tothedataprior totransmission inacontrolled manner.Theredundancy isusedtoenableadecoderinthe receivertodetectandcorrecterrors. Error-control codingtechniques maybedividedintotwobroadlydefinedfamilies: 1.Algebraic codes,whichrelyonabstract algebraic structure builtintothedesign ofthecodesfordecoding atthereceiver. Algebraic codesincludeHamming codes, maximal-length codes,BCHcodes,andReed-Solomon codes.Theseparticular codessharetwoproperties: Linearity property, thesumofanytwocodewordsinthecodeisalsoacodeword. Cyclicproperty, anycyclicshiftofacodewordisalsoacodewordinthecode. Reed-Solomon codesareverypowerful codes,capableofcombatting bothrandom andbursterrors;theyfindapplications indifficultenvironments suchasdeep-space communications andcompact discs. 2.Probabilistic codes,whichrelyonprobabilistic methods fortheirdecoding atthe receiver.Probabilistic codesincludetrelliscodes,turbocodes,andlow-density parity checkcodes.Inparticular, thedecoding isbasedononeortheotheroftwobasic methods, assummarized here: Softinput-hard output,whichisexemplified bytheViterbialgorithm thatperforms maximum likelihood sequence estimation inthedecoding oftrelliscodes. Softinput-soft output,whichisexemplified bytheBCJRalgorithm thatperforms maximum aposteriori estimation onabit-by-bit basisinthedecoding ofturbocodes, oraspecialformofthesum-product algorithm inthedecoding oflow-density parity checkcodes. Trelliscodescombine linearconvolutional encoding andmodulation topermitsignificant codinggainsoverconventional uncoded multilevel modulation withoutsacrificing band widthefficiency. Turbocodesandlow-density parity-check codessharethefollowing properties: ..Random encoding ofalinearblockkind. ..Errorperformance withinahair'sbreadthofShannon's theoretical limitonchannel capacityinaphysically realizable fashion. Inpractical terms,turbocodesandlow-density parity-check codeshavemadeitpossible toachievecodinggainsontheorderof10dB,whichisunmatched previously. These codinggainsmaybeexploited todramatically extendtherangeofdigitalcommunication receivers, substantially increasethebitratesofdigitalcommunication systems,orsignifi- 694 CHAPTER 101lIERROR-CONTROL CODING candydecrease thetransmitted signalenergypersymbol. Thesebenefitshavesignificant implications forthedesignofwireless communications anddeep-space communications, justtomentiontwoimportant applications ofdigitalcommunications. Indeed,turbocodes havealreadybeenstandardized foruseondeep-space communication linksandwireless communication systems. INOTES ANDREFERENCES 1.Foranintroductory discussion oferrorcorrection bycoding,seeChapter 2ofLucky (1989);seealsothebookbyAdamek (1991),andthepaperbyBhargava (1983).Theclassic bookonerror-control codingisPeterson andWeldon(1972).Error-control codingisalso discussed intheclassicbookofGallager (1968).ThebooksofLinandCostello (1983), Micheleson andLevesque (1985),MacWilliams andSloane(1977),andWilson(1998)are alsodevotedtoerror-control coding.Foracollection ofkeypapersonthedevelopment of codingtheory,seethebookeditedbyBerlekamp (1974). 2.ForasurveyofvariousARQschemes, seeLin,Costello, andMiller(1984). 3.Inmedicine, thetermsyndrome isusedtodescribeapatternofsymptoms thataidsinthe diagnosis ofadisease.Incoding,theerrorpatternplaystheroleofthediseaseandparity checkfailurethatofasymptom. Thisuseofsyndrome wascoinedbyHagelbarger (1959). 4.Thefirsterror-correcting codes(knownasHamming codes)wereinvented byHamming ataboutthesametimeastheconception ofinformation theorybyShannon; fordetails, seetheclassicpaperbyHamming (1950). 5.Foradescription ofBCHcodesandtheirdecoding algorithms, seeLinandCostello(1983, pp.141-183) andMacWilliams andSloane(1977,pp.257-293). Table10.6onbinary BCHcodesisadaptedfromLinandCostello (1983). 6.TheReed-Solomon codesarenamedinhonoroftheirinventors: seetheirclassic1960 paper.FordetailsofReed-Solomon codes,seeMacWilliams andSloane(1977,pp.294 306).ThebookeditedbyWickerandBhargava (1994)contains anintroduction toReed Solomon codes,ahistorical overview ofthesecodeswrittenbytheirinventors, IrvingS. ReedandGustaveSolomon, andtheapplications ofReed-Solomon codestotheexplora tionofthesolarsystem,thecompact disc,automatic repeat-request protocols, andspread spectrum multiple-access communications, andchapters onotherrelatedissues. 7.Convolutional codeswerefirstintroduced, asanalternative toblockcodes,byP.Elias (1955). 8.Thetermtrelliswasintroduced byForney(1973). 9.Inaclassicpaper,Viterbi(1967)proposed adecoding algorithm forconvolutional codes thathasbecomeknownastheViterbialgorithm. Thealgorithm wasrecognized byForney (1972,1973)tobeamaximum likelihood decoder. Readable accountsoftheViterbial gorithmarepresented inLinandCostello (1983),Blahut(1990),andAdamek (1991). 10.Catastrophic convolutional codesarediscussed inBenedetto, Biglieri, andCastellani (1987).Table10.8isadaptedfromtheirbook. 11.Fordetailsoftheevaluation ofasymptotic codinggainforbinarysymmetric andbinary inputAWGNchannels, seeViterbiandOmura(1979,pp.242-252) andLinandCostello (1983,pp.322-329). 12.Trellis-coded modulation wasinvented byG.Ungerboeck; itshistorical evolution isde scribedinUngerboeck (1982).Table10.9isadaptedfromthislatterpaper. Trellis-coded modulation maybeviewedasaformofsignal-space coding-a view pointdiscussed atanintroductory levelinChapter 14ofthebookbyLeeandMesser- NatesandReferem;es 695 schmitt(1994).Foranextensive treatment oftrellis-<oded modulation, seethebooksby Biglieri,Divsalar, McLane, andSimon(1991),andSchlegel(1997). 13.Turbocodeswereoriginated byC.BerrouandA.Glavieux. Workonthesecodeswas motivated bytwopapersonerror-correcting codes:Battail(1987),andHagenauer and Hoecher (1989).Thefirstdescription ofturbocodesusingheuristic arguments waspre sentedataconference paperbyBerrou,Glavieux, andThitimajshima (1993);seealso BerrouandGlavieux (1996).Forreflections ontheearlyworkonturbocodesandsubse quentdevelopments, seeBerrouandGlavieux (1998). Forabookonthebasicsofturbocodes,seeHeegard andWicker(1999).Usinga procedure reminiscent ofrandomcoding(seeNote15),Benedetto andMontorosi (1996) haveprovided partialexplanations fortheimpressive performance ofturbocodes. Intwoindependent studiesreported inthepapersbyMcEliece, MacKay, andCheng (1998),andKschischang andFrey(1998),itisshownthatturbodecoding duplicates an algorithm inartificial intelligence duetoPearl(1982),whichinvolvesthepropagation of belief.Thetermbeliefisanotherwayofreferring toaposteriori probability. Thesetwo papershaveopenedanewavenueofresearch, whichlinksturbodecoding andlearning machines. Foraninsightful discussion ofturbocodes,seethebookbyFrey(1998). Apseudo-random interleaver isbasictotheoperation ofturbocodes.Denenshgaran andMondin (1999)presentasystematic procedure fordesigning interleavers (i.e.,per muters)forturbocodes. 14.Thedualtermination ofturbocodesisdiscussed inGuinand andLodge(1996). 15.Random codingisdiscussed inCoverandThomas (1991),Section 8.7. 16.Theplotspresented inFig.10.27followthoseinFig.6.8ofthebookbyFrey(1998). 17.Intheearly1960s,BaumandWelchderivedaniterativeprocedure forsolvingtheparam eterestimation problem, hencethenameBaum-Welchalgorithm (BaumandPetrie(1966); Baumetal.(1970)).IntheBC]Ralgorithm, namedafterBahl,Cocke,Jelinek,andRaviv (1974),theBaum-Welch algorithm isappliedtotheproblem ofsoftoutput,maximum likelihood decoding ofconvolutional codes. 18.TheproofthattheestimatexinEq.(10.76)approaches theMAPsolutionasthenumber ofiterations approaches infinityisdiscussed inthepaperbyMoherandGulliver(1998). 19.Low-density parity-check (LDPC)codeswereoriginally discovered byGallager (1962, 1963).Theywererediscovered independently byMacKay andNeal(1995);seealso MacKay (1999). Inthe1960sandforagoodwhilethereafter, thecomputers available atthattime werenotpowerful enoughtoprocessthelongblocklengthsthatareneededtoachieve excellent performance withLDPCcodes,hencethelackofinterestintheiruseforover twentyyears. 20.Foradetailedtreatment ofthestatement thattheprobability disttibution oftheminimum distanceofanLDPCcodeapproaches aunitstepfunction oftheblocklengthforcertain valuesofweight-pair (tot,),seeGallager (1962,1963). 21.Thedecoding algorithm ofLDPCcodesdescribed hereinfollowsMacKay andNeal(1996, 1997). 22.Irregular turbocodeswereinvented byFreyandMacKay (1999). 23.Irregular LDPCcodeswereinvented independently byMaKayetal.(1999)andRichardson etal.(1999). 24.Thecodes,whoseperformances areplottedinFig.10.34,areduetothefollowing originators: '"Regularrurbocodes:BerrouandGlavieux (1996);Berronetal.(1995). '"Irregular turbocodes:FreyandMacKay(1999). '"Irregular LDPCcodes:Richardson etal.(1999). j=0,1,...,Q -1696 CHAPTER 10IIERROR-CONTROL CODING IPROBLEMS Soft-Decision Coding 10.1Consider abinaryinputQ-aryoutputdiscretememoryless channel. Thechannelissaid tobesymmetric ifthechanneltransition probability pUIi)satisfiesthecondition: PUIO)=p(Q-1 -ill),i=0,1,...,Q-1 Supposethatthechannelinputsymbols0and1areequallylikely.Showthatthechannel outputsymbolsarealsoequallylikely;thatis, . 1 p(j)=Q' 10.2Consider thequantized demodulator forbinaryPSKsignalsshowninFig.10.3a.The quantizer isafour-level quantizer, normalized asinFig.PI0.2.Evaluate thetransition probabilities ofthebinaryinput-quarternary outputdiscretememoryless channelso characterized. Hence,showthatitisasymmetric channel. Assumethatthetransmitted signalenergyperbitisEb,andtheadditivewhiteGaussian noisehaszeromeanand powerspectraldensityNo/2. Quantizer output +3 -1 -3 FIGUREP10.2Quantizer input 10.3Consider abinaryinputAWGNchannel,inwhichthebinarysymbols 1and0arcequally likely.Thebinarysymbols aretransmitted overthechannelbymeansofphase-shift keying.ThecodesymbolenergyisE,andtheAWGNhaszeromeanandpowerspectral densityNo/2.Showthatthechanneltransition probability isgivenby p(yIO)=_1_exp[_l(y+f2E)2J,-00<Y<00VIiT 2 ~Jia LinearBlockandCyclicCodes 10.4Inasingle-parity-check code,asingleparitybitisappended toablockofkmessagebits (ml>m2,...,mk)'Thesingle"paritybith,ischosensothatthecodewordsatisfiesthe evenparityrule: m,+m2+...+mk+h,=0,mod2 Fork=3,setupthe2kpossiblecodewordsinthecodedefinedbythisrule. 10.5Compare theparity-check matrixofthe(7,4)Hamming codeconsidered inExample 10.2withthatofa(4,1)repetition code. Problems 697 10.6Consider the(7,4)Hamming codeofExample 10.2.Thegenerator IrultrixGandthe parity-check matrixHofthecodearedescribed inthatexample. Showthatthesetwo matrices satisfythecondition HGT=0 10.7(a)Forthe(7,4)Hamming codedescribed inExample 10.2,construct theeightcode wordsinthedualcode. (b)Findtheminimum distance ofthedualcodedetermined inpart(a). 10.8Consider the(5,1)repetition codeofExample 10.1.Evaluate thesyndrome sforthe following errorpatterns: (a)Allfivepossiblesingle-error patterns (b)All10possibledouble-error patterns 10.9Foranapplication thatrequireserrordetection only,wemayuseanonsystematic code. Inthisproblem, weexplorethegeneration ofsuchacycliccode.Letg(X)denotethe generator polynomial, andm(X)denotethemessage polynomial. Wedefinethecode polynomial c(X)simplyas c(X)=m(X)g(X) Hence,foragivengenerator polynomial, wemayreadilydetermine thecodewordsin thecode.Toillustrate thisprocedure, consider thegenerator polynomial fora(7,4) Hamming code: g(X)=1+X+X3 Determine the16codewordsinthecode,andconfirmthenonsystematic natureofthe code. 10.10Thepolynomial 1+X7has1+X+X3and1+X'+X3asprimitive factors.In Example 10.3,weused1+X+X3asthegenerator polynomial fora(7,4)Hamming code.Inthisproblem, weconsider theadoption of1+X'+X3asthegenerator polynomial. Thisshouldleadtoa(7,4)Hamming codethatisdifferent fromthecode analyzed inExample 10.3.Develop theencoderandsyndrome calculator forthegen eratorpolynomial: g(X)=1+X'+X3 Compare yourresultswiththoseinExample 10.3. 10.11Consider the(7,4)Hamming codedefinedbythegenerator polynomial g(X)=1+X+X3 Thecodeword0111001 issentoveranoisychannel, producing thereceived word 0101001 thathasasingleerror.Determine thesyndrome polynomial s(X)forthisre ceivedword,andshowthatitisidentical totheerrorpolynomial e(X). 10.12Thegenerator polynomial ofa(15,11)Hamming codeisdefinedby g(X)=1+X+X4 Develop theencoderandsyndromecalculator forthiscode,usingasystematic formfor thecode. 10.13Consider the(15,4)maximal-length codethatisthedualofthe(15,11)Hamming code ofProblem 10.12.Dothefollowing: (a)Findthefeedback connections oftheencoder, andcompare yourresultswiththose ofTable7.1onmaximal-length codespresented inChapter 7. (b)Findthegenerator polynomial g(X);hence,determine theoutputsequence assuming theinitialstate0001.Confirm thevalidityofyourresultbycyclingtheinitialstate throughtheencoder. 698 CHAPTER 10'"ERROR-CONIROL CODING 10.14Consider the(31,15)Reed-Solomon code. (a)Howmanybitsarethereinasymbolofthecode? (b)Whatistheblocklengthinbits? (c)Whatistheminimum distanceofthecode? (d)Howmanysymbolsinerrorcanthecodecorrect? Convolutional Codes 10.15Aconvolutional encoderhasasingle-shift registerwithtwostages,(i.e.,constraintlength K=3),threemodulo-2 adders,andanoutputmultiplexer. Thegenerator sequences of theencoderareasfollows: gil)=(1,0,1) g(2)=(1,1,0) g'31=(1,1,1) Drawtheblockdiagramoftheencoder. Note:ForProblems 10.16-10.23, thesamemessagesequence 10111...isusedsothat wemaycompare theoutputsofdifferent encoders forthesameinput. 10.16Consider therater=1/2,constraint lengthK=2convolutional encoderofFig.PI0.16. Thecodeissystematic. Findtheencoderoutputproduced bythemessage sequence 10111.... Input0--+-,,>-1Output Flip-flop FIGUREPI0.16 10.17FigurePI0.17showstheencoderforarater=1/2,constraint lengthK=4convolu tionalcode.Determine theencoderoutputproduced bythemessagesequence 10111.... Input FIGUREPI0.17Output Problems 699 10.18Consider theencoderofFig.10.13bforarater=2/3,constraint lengthK=2con volutional code.Determine thecodesequence produced bythemessage sequence 10111.... 10.19Construct thecodetreefortheconvolutional encoderofFig.PI0.16.Tracethepath throughthetreethatcorresponds tothemessagesequence 10111...,andcompare the encoderoutputwiththatdetermined inProblem 10.16. 10.20Construct thecodetreefortheencoderofFig.PI0.17.Tracethepaththroughthetree thatcorresponds tothemessage sequence lO11L...Compare theresulting encoder outputwiththatfoundinProblem 10.17. 10.21Construct thetrellisdiagramfortheencoderofFig.PI0.17,assuming amessagesequence oflength5.Tracethepaththroughthetrelliscorresponding tothemessagesequence 10111....Compare theresulting encoderoutputwiththatfoundinProblem 10.17. 10.22Construct thestatediagram fortheencoderofFig.PI0.17.Startingwiththeall-zero state,tracethepaththatcorresponds tothemessagesequence 10111...,andcompare theresulting codesequence withthatdetermined inProblem 10.17. 10.23Consider theencoderofFig.10.13b. (a)Construct thestatediagramforthisencoder. (b)Startingfromtheall-zerostate,tracethepaththatcorresponds tothemessagese quence10111....Compare theresulting sequence withthatdetermined inProblem 10.18. 10.24Byviewingtheminimum shiftkeying(MSK)schemeasafinite-state machine, construct thetrellisdiagramforMSK.(Adescription ofMSKispresented inChapter6.) 10.25Thetrellisdiagram ofarate-ll2, constraint length-3 convolutional codeisshownin FigurePI0.25.Theall-zerosequence istransmitted, andthereceived sequence is 100010000 ....UsingtheViterbialgorithm, compute thedecodedsequence. State 00 01 10 1100 ',)1,, FIGUREPIO.25 10.26Consider arate-1I2, constraint length-7convolutional codewithfreedistancedf,..=10. Calculate theasymptotic codinggainforthefollowing twochannels: (a)Binarysymmetric channel (b)BinaryinputAWGNchannel 10.27InSection10.6wedescribed theViterbialgorithm formaximum likelihood decoding of aconvolutional code.Another application oftheViterbialgorithm isformaximum likelihood demodulation ofareceivedsequence corrupted byintersymbol interference duetoadispersive channel. FigurePI0.27showsthetrellisdiagram forintersymbol interference, assuming abinarydatasequence. Thechannelisdiscrete,described bythe 700 CHAPTER 10IIIERROR-CONTROL CODING finiteimpulseresponse (1,0.1).Thereceived sequence is(1.0,-0.3,-0.7,0,...).Use theViterbialgorithm todetermine themaximum likelihood decoded versionofthis sequence. -1.1 FIGUREPI0.27-1.1 10.28FigurePI0.28 depicts32-QAM crossconstellation. Partition thisconstellation intoeight subsets.Ateachstageofthepartitioning, indicatethewithin-subset (shortest) Euclidean distance. FIGUREPI0.28 10.29Asexplained intheIntroduction tothischapter,channelcodingcanbeusedtoreduce theEb/Norequired foraprescribed errorperformance orreducethesizeofthereceiving antennaforaprescribed EbiNo.Inthisproblem weexplorethesetwopractical benefits ofcodingbyrevisiting Example 8.2inChapter 8onthedownlink powercalculations foradomestic satellitecommunication system.Inparticular, wenowassumethatthe designofthedownlink includes theuseofacodingschemeconsisting ofarate-Ill convolutional encoderwithlengthK=7andViterbidecoding. Thecodinggainofthis schemeis5.1dB,assuming theuseofsoftquantization. Hencedothefollowing: (a)Recalculate therequiredEJNoratioofthesystem. (b)Assuming thattherequired EblNoratioremainsunchanged, calculate thereduction inthesizeofthereceiving dishantennathatismadepossible bytheuseofthis codingschemeinthedownlink. 10.30Unliketheconvolutional codesconsidered inthischapter,werecallfromChapter6that theconvolutional codeusedinthevoiceband modemV.3lmodemisnonlinear. Figure PIO.30showsthecircuitdiagramoftheconvolutional encoderusedinthismodem;it usesmodulo-l multiplication andgatesinadditiontomodulo-2 additions anddelays. Explainthereasonfornonlinearity oftheencoderinFig.PIO.30,anduseanexample toillustrate yourexplanation. InputProblems 701 o14,n)Y4,n C[3,,, )Y3,1f 12,n Y2,n Output Flip-flop Mod-2 adder L... Yo,. FIGUREPIO.30 TurboCodes 10.31Letr~')=plq,andr~2)=p/q2bethecoderatesofRSCencoders 1and2intheturbo encoderofFig.10.25.Findthecoderateoftheturbocode. 10.32Thefeedback natureoftheconstituent codesintheturboencoderofFig.10.25hasthe following implication: Asinglebiterrorcorresponds toaninfinitesequence ofchannel errors.Illustrate thisphenomenon byusingamessagesequence consisting ofsymbol1 followed byaninfinitenumberofsymbols O. 10.33Consider thefollowing generator matrices forrate1/2turbocodes: [1+D+D2 ]4-stateencoder: g(D)=1,1+D2 [1+D2+D3]8-stateencoder: g(D)=1,1+D+D2+D3 16-stateencoder: g(D)-[1 1+D4 ]- , 1+D+D2+D3+D4 (a)Construct theblockdiagram for each oneoftheseRSCencoders. (b)Setuptheparity-check equation associated witheachencoder. 10.34TheturboencoderofFig.10.25involvestheuseoftwoRSCencoders. (a)Generalize thisencodertoencompass atotalofMinterleavers. (b)Construct theblockdiagramoftheturbodecoderthatexploitstheMsetsofparity checkbitsgenerated bysuchageneralization. 10.35Turbodecoding reliesonthefeedback ofextrinsic information. Thefundamental prin cipleadheredtointheturbodecoderistoavoidfeedingadecoding stageinformation thatstemsfromthestageitself.Explainthejustification forthisprinciple inconceptual terms. 10.36Suppose acommunication receiverconsistsoftwocomponents, ademodulator anda decoder. Thedemodulator isbasedonaMarkovmodelofthecombined modulator and 702 CHAPTER 10"ERROR-CONTROL CODING channel, andthedecoderisbasedonaMarkov modelofaforward errorcorrection code.Discusshowtheturboprinciple maybeappliedtoconstruct ajointdemodulatorl decoderforthissystem. Computer Experiment 10.37Inthisexperiment wecontinue theinvestigation intoturbocodespresented inSection 10.9byevaluating theeffectofblocksizeonthenoiseperformance ofthedecoder. Asbefore,thetwoconvolutional encoders oftheturboencoderareasfollows: Encoder 1:[1,1,1] Encoder 2:[1,0,1] Thetra)lsmitted E&INois1dB.Theblockerrorstotermination areprescribed notto exceed15. Withthisback!70und information, plotthebiterrorrateoftheturbodecoder versusthenumberofiterations fortwodifferent block(i.e.,interleaver) sizes:200and 400. PROBABILITY THEORY IAI.IProbabilistic Cotreepts Probability theoryisrootedinphenomena that,explicitly orimplicitly, canbemodeled byanexperiment withanoutcome thatissubjecttochance.Moreover, iftheexperiment isrepeated, theoutcome candifferbecauseoftheinfluence ofanunderlying random phenomenon orchancemechanism. Suchanexperiment isreferredtoasarandomexper iment.Forexample, theexperiment maybetheobservation oftheresultoftossingafair coin.Inthisexperiment, thepossibleoutcomes ofatrialare"heads"or"tails." Tobemorepreciseinthedescription ofarandomexperiment, weaskforthree features: 1.Theexperiment isrepeatable underidentical conditions. 2.Onanytrialoftheexperiment, theoutcome isunpredictable. 3.Foralargenumberoftrialsoftheexperiment, theoutcomes exhibitstatistical reg Illarity;thatis,adefiniteaveragepatternofoutcomes isobserved iftheexperiment isrepeated alargenumberoftimes. III!RElATIVE-FREQUENCY APPROACH LeteventAdenoteoneofthepossible outcomes ofarandomexperiment. Forexample, inthecoin-tossing experiment, eventAmayrepresent "heads." Supposethatin'ntrialsof theexperiment, eventAoccursNn(A)times.WemaythenassigntheratioNn(A)lntothe eventA.Thisratioiscalledtherelativefreqllency oftheeventA.Clearly,therelative frequency isanonnegative realnllmberlessthanoreqllaltoone.Thatistosay, (ALl) (Al.2)IfeventAoccursinnoneofthetrials,Nn(A)ln =O.If,ontheotherhand,eventAoccurs inallthentrials,Nn(A)In=1. Wesaythattheexperiment exhibitsstatistical reglilarity ifforanysequence ofn trialstherelativefrequency Nn(A)lnconverges tothesamelimitasnbecomes large.Itthus seemsnaturalforustodefinetheprobability ofeventAas P(A)=lim(Nn(A)) n--wn ThelimitshowninEquation (A1.l)shouldnotbeviewedinamathematical sense.Rather, wethinkofEquation (Al.l)asastatement thattheprobability ofaneventisthelong termproportion oftimesthataparticular eventAoccursinalongsequence oftrials.For example, inthecoin-tossing experiment, wemayexpect.thatoutofamilliontossesofa faircoin,aboutonehalfofthemwillshowupheads. Theprobability ofaneventisintended torepresent thelikelihood thatatrialofthe experiment willresultintheoccurrence ofthatevent.Formanyengineering applications andgamesofchance,theuseofEquation (Al.2)todefinetheprobability ofaneventis acceptable. However, formanyotherapplications, thisdefinition isinadequate. Consider, 703 704 APPENDIX IiiiPROBABILI'IY THEORY forexample, thestatistical analysisofthestockmarket:Howarewetoachieverepeata_ bilityofsuchanexperiment? Amoresatisfying approach istostatetheproperties that anymeasure ofprobability isexpected tohave,postulate themasaxioms,andthenUSe relative-frequency interpretations tojustifythem. iii!AxlOMS OFPROBABILI'IY Whenweperform arandomexperiment, itisnaturalforustobeawareofthevarious outcomes thatarelikelytoarise.Inthiscontext,itisconvenient tothinkofanexperiment anditspossibleoutcomes asdefiningaspaceanditspoints.Withthekthoutcome ofthe experiment, say,weassociate apointcalledthesamplepoint,whichwedenotebySk.The totalityofsamplepointscorresponding totheaggregate ofallpossible outcomes ofthe experiment iscalledthesamplespace,whichwedenotebyS.Aneventcorresponds to eitherasinglesamplepointorasetofsamplepoints.Inparticular, theentiresamplespace Siscalledthesureevent;thenullset0iscalledthenullorimpossible event;andasingle samplepointiscalledanelementary event. Consider, forexample, anexperiment thatinvolvesrhethrowofadie.Inthisex periment therearesixpossible outcomes: theshowing ofone,two,three,four,five,and sixdotsontheupperfaceofthedie.Byassigning asamplepointtoeachofthesepossible outcomes, wehaveaone-dimensional samplespacethatconsistsofsixsamplepoints,as showninFigureA1.1.Theelementary event describing thestatement "asixshows"cor responds tothesamplepoint{6}.Ontheotherhand,theeventdescribing thestatement "anevennumberofdotsshows"corresponds tothesubset{2,4,6}ofthesamplespace. Notethatthetermeventisusedinterchangeably todescribethesubsetorthestatement. Wearenowreadytomakeaformaldefinition ofprobability. Aprobability system consistsofthetriple: 1.AsamplespaceSofelementary events(outcomes). 2.Aclass~ofeventsthataresubsetsofS. 3.Aprobability measure P(·)assigned toeacheventAintheclass ~,whichhasthe following properties: (i) (ii)P(S)=1 O:=s;P(A):=s;1(AU) (AlA) (iii)IfA+Bistheunionoftwomutually exclusive eventsintheclass'&,then PtA+B)=PtA)+P(B) (AU) Properties (i),(ii),and(iii)areknownastheaxiomsofprobability. Axiom(i)statesthat theprobability ofthesureeventisunity.Axiom(ii)statesthattheprobability ofanevent Samplepaint./. 3 4 6 One-dimensional samplespace FIGUREAI.ISamplespacefortheexperiment ofthrowing adie. AI.IProbabilistic Concepts 705 isanonnegative realnumberthatislessthanorequaltounity.Axiom(iii)statesthatthe probability oftheunionoftwomutually exclusive eventsisthesumoftheprobabilities oftheindividual events.Thesethreeaxiomsaresufficient todealwithexperiments with finitesamplespaces. Although theaxiomatic approach toprobability theoryisabstractinnature,allthree axiomshaverelative-frequency interpretations oftheirown.Axiom(ii)corresponds to Equation (A1.l).Axiom(i)corresponds tothelimitingcaseofEquation (A1.l)whenthe eventAoccursinallthentrials.Tointerpret axiom(ui),wenotethatifeventAoccurs Nn(A)timesinntrialsandeventBoccursNn(B)times,thentheunionevent"AorB" occursinNn(A)+Nn(B)trials(sinceAandBcanneveroccuronthesametrial).Hence, Nn(A+B)=Nn(A)+Nn(B),andsowehave _N=n(_A_+_B--,-) =_Nn_(A_)+N_"_(B_) n n n whichhasamathematical formsimilartothatofaxiom(iii). Axioms(i),(ii),and(iii)constitute animplicitdefinition ofprobability. Wemayuse theseaxiomstodevelopsomeotherbasicproperties ofprobability, asdescribed next. Property 1 P(ji)=1 -P(A) whereA(denoting "notA")isthecomplement ofeventA.(A1.6) Theuseofthisproperty helpsusinvestigate thenonoccurrence ofanevent.Toprove it,weexpressthesamplespaceSastheunionoftwomutually exclusive eventsAandA: S=A+A Then,theuseofaxioms(i)and(iii)yields 1=PtA)+P(A) fromwhichEquation (A1.6)followsdirectly. Property 2 IfMmutually exclusive eventsAi>A",••.,AMhavetheexhaustive property Al+A2•••+AM=S then Toprovethisproperty, wefirstuseaxiom(i)inEquation (A1.7),andsowrite Next,wegeneralize axiom(iii)bywriting(A1.7) (A1.8) 706 APPENDIX 1"PROBABILfIYTHEORY Hence,theresultofEquation (A1.8)follows.WhentheMeventsareequallylikely(i.e., theyhaveequalprobabilities ofoccurrence), thenEquation (A1.8)simplifies as 1 P(Aj)=M' i=1,2,..., M Property 3 WheneventsAandBarenotmutually exclusive, thentheprobability oftheunionet'ent "AorB"equals P(A+B)=P(A)+P(B)-P(AB) whereP(AB)istheprobability ofthejointevent"AandB."(A1.9) Theprobability P(AB)iscalledajointprobability. Ithasthefollowing relative frequency interpretation: P(AB)=lim(N,,(AB)) n-->~n(ALlO) (A1.11)whereN,,(AB)denotesthenumberoftimestheeventsAandBoccursimultaneously inn trialsoftheexperiment. Axiom(iii)isaspecialcaseofEquation (A1.9);whenAandB aremutually exclusive, P(AB)is"zero,andEquation (A1.9)reducestothesameformas Equation (A1.5). IICONDITIONAL PROBABILITY Suppose weperform anexperiment thatinvolves apairofeventsAandB.LetP(BIA) denotetheprobability ofeventB,giventhateventAhasoccurred. Theprobability P(BIA)iscalledtheconditional probability ofBgivenA.Assuming thatAhasnonzero probability, theconditional probability P(BIA)isdefinedby P(BIA)=P(AB) P(A) whereP(AB)isthejointprobability ofAandB. Wejustifythedefinition ofconditional probability giveninEquation (A1.11)by presenting arelative-frequency interpretation ofit.Supposethatweperformanexperiment andexamine theoccurrence ofapairofeventsAandB.LetN,,(AB)denotethenumber oftimesthejointeventABoccursinntrials.Suppose thatinthesamentrials,theevent AoccursN,,(A)times.SincethejointeventABcorresponds tobothAandBoccurring, it followsthatN,,(A)mustincludeN,,(AB). Inotherwords,wehave Nll(AB):s:1 Nn(A) TheratioNn(AB)/Nn(A) represents therelativefrequency ofBgiventhatAhasoccurred. Forlargen,thisratioequalstheconditional probability P(BIA);thatis, P(BIA)=lim(N,,(AB)) n-->~N,,(A) ALlProbabilistic Concepts 707 or,equivalently, P(BIA)=lim(Nn(AB)ln) n-roNn(A)ln Recognizing that P(AB)=lim(Nn(AB)) n-ron and P(A)=lim(Nn(A)) n_oon theresultofEquation (A1.l1)follows. WemayrewriteEquation (Al.l1)as P(AB)=P(BIA)P(A) Itisapparent thatwemayalsowrite P(AB)=P(AIB)P(B)(A1.12) (A1.B) (A1.14)Accordingly, wemaystatethatthejointprobability oftwoeventsmaybeexpressed as theproductoftheconditional probability ofoneeventgiventheother,andtheelementary probability oftheother.Notethattheconditional probabilities P(BIA)andP(AIB)have essentially thesameproperties asthevariousprobabilities previously defined. Situations mayexistwheretheconditional probability P(AIB)andtheprobabilities P(A)andP(B)areeasilydetermined directly, buttheconditional probability p(BIA)is desired.FromEquations (Al.12)and(A1.B),itfollowsthat,provided P(A)*-0,wemay determine P(BIA)byusingtherelation p(BIA)=P(AIB)P(B) P(A) ThisrelationisaspecialformofBayes'rule. Suppose thattheconditional probability P(BIA)issimplyequaltotheelementary probability ofoccurrence ofeventB,thatis, P(BIA)=P(B) Underthiscondition, theprobability ofoccurrence ofthejointeventABisequaltothe productoftheelementary probabilities oftheeventsAandB: P(AB)=P(A)P(B) sothat P(AIB)=P(A) Thatis,theconditional probability ofeventA,assuming theoccurrence ofeventB,is simplyequaltotheelementary probability ofeventA.Wethusseethatinthiscasea knowledge oftheoccurrence ofoneeventtellsusnomoreabouttheprobability ofoc currence oftheothereventthanweknewwithoutthatknowledge. EventsAandBthat satisfythiscondition aresaidtobestatistically independent. 708 APPENDIX 1IIIPROBABILI'IY THEORY IAI.2Random Variables Itiscustomary, particularly whenusingthelanguage ofsamplespace,tothinkofthe outcome ofanexperiment asavariablethatcanwanderoverthesetofsamplepointsand whosevalueisdetermined bytheexperiment. Afunction whosedomainisasamplespace andwhoserangeissomesetofrealnumbers iscalledarandomvariableoftheexperiment. However, thetermrandomvariableissomewhat confusing. First,thewordrandomisnot usedinthesenseofequalprobability ofoccurrence, forwhichitshouldbereserved. Second,thewordvariable doesnotimplydependence (ontheexperimental outcome), whichisanessentialpartofthemeaning. Nevertheless, thetermissodeeplyimbedded in theliterature ofprobability thatitsusagehaspersisted. Whentheoutcome ofanexperiment iss,therandomvariableisdenotedasX(s)or simplyX.Forexample, thesamplespacerepresenting theoutcomes ofthethrowofadie isasetofsixsamplepointsthatmaybetakentobetheintegers1,2,...,6.Thenifwe identifythesamplepointkwiththeeventthatkdotsshowwhenthedieisthrown,the functionX(k)=kisarandomvariablesuchthatX(k}equalsthenumberofdotsthat showwhenthedieisthrown.Inthisexample, therandomvariabletakesonlyadiscrete setofvalues.Insuchacase,wesaythatwearedealingwithadiscreterandomvariable. Moreprecisely, therandomvariableXcantakeonlyafinitenumberofvaluesinanyfinite observation interval.If,however, therandomvariableXcantakeanyvalueinawhole observation interval,Xiscalledq.continuous randomvariable. Forexample, therandom variablethatrepresents theamplitude ofanoisevoltageataparticular instantoftimeis acontinuous randomvariable becauseitmaytakeanyvaluebetween plusandminus infinity. Toproceedfurther,weneedaprobabilistic description ofrandomvariables that worksequallywellfordiscreteaswellascontinuous randomvariables. Letusconsider therandomvariableXandtheprobability oftheeventX:$x.Wedenotethisprobability byP(X :$x).Itisapparent thatthisprobability isafunction ofthedummyvariablex.To simplifythenotation, wewrite Fx(x)=P(X:$x) (A1.15) Thefunction Fx(x)iscalledthecumulative distribution function (cdf)orsimplythedis tribution functionoftherandomvariableX.NotethatFx(x)isafunction ofx,notofthe randomvariableX.However, itdepends ontheassignment oftherandomvariableX, whichaccounts fortheuseofXassubscript. Foranypointx,thedistribution function Fx(x)expresses aprobability. Thedistribution function Fx(x)hasthefollowing properties, whichfollowdirectly fromEquation (A1.15): 1.Thedistribution function Fx(x)isbounded betweenzeroandone. 2.Thedistribution function Fx(x)isanondecreasing function ofx;thatis, (A1.16) Analternative description oftheprobability oftherandomvariableXisoftenuseful. Thisisthederivative ofthedistribution function, asshownby (ALl?) (AU8)A1.2Random Va..u.bles 709 whichiscalledtheprobability densityfunction (pdf)oftherandomvariableX.Notethat thedifferentiation inEquation (Al.l7)iswithrespecttothedummyvariablex.Thename, densityfunction, arisesfromthefactthattheprobability oftheeventXl<X:5X2equals P(XI<X:5X2)=P(X:5X2)-P(X:5Xl) =FX(X2)-FX(Xl) =rX2 fx(x)dxJXl Theprobability ofanintervalistherefore theareaundertheprobability densityfunction inthatinterval. Putting Xl=-00inEquation (A1.l8), andchanging thenotation some what,wereadilyseethatthedistribution function isdefinedintermsoftheprobability densityfunction asfollows: (AU9) SinceFx(oo)=1,corresponding totheprobability ofacertainevent,andFx(-oo)=0, corresponding totheprobability ofanimpossible event,wereadilyfindfromEquation (AU8)that roofx(x)dx=1 (AUO) Earlierwementioned thatadistribution function mustalwaysbenondecreasing. This meansthatitsderivative ortheprobability densityfunction mustalwaysbenonnegative. Accordingly, wemaystatethataprobability densityfunction mustalwaysbeanonneg ativefunction, andwithatotalareaofone. Thusfarwehavefocusedattention onsituations involving asinglerandomvariable. However, wefindfrequently thattheoutcome ofanexperiment requiresseveralrandom variables foritsdescription. Wenowconsider situations involving tworandomvariables. Theprobabilistic description developed inthiswaymaybereadilyextended toanynumber ofrandomvariables. Consider tworandomvariables XandY.Wedefinethejointdistribution function Fx,Y(x,y)astheprobability thattherandomvariableXislessthanorequaltoaspecified value XandthattherandomvariableYislessthanorequaltoaspecified valuey.The variables XandYmaybetwoseparate one-dimensional randomvariables orthecom ponentsofasingletwo-dimensional randomvariable. Ineithercase,thejointsamplespace isthexy-plane. Thejointdistribution functionFx,y(x,y)istheprobability thattheoutcome ofanexperiment willresultinasamplepointlyinginsidethequadrant(-00<X:5X, -00<Y:5y)ofthejointsamplespace.Thatis, Fx,Y(x,y)=P(X:5x,Y:5y) (AUl) (A1.22)Supposethatthejointdistribution functionFx,y(x,y)iscontinuous everywhere, and thatthepartialderivative f(X)=a2Fx,Y(x,y) X,Y,y axay existsandiscontinuous everywhere. Wecallthefunction !x,Y(x,y)thejointprobability densityfunction oftherandom variables XandY.Thejointdistribution function 710 APPENDIX 1..PROBABILITY THEORY Fx,y(x,y)isanondecreasing function ofbothxandy.Therefore, fromEquation (A1.22) itfollowsthatthejointprobability densityfunctionfx,y(x,y)isalwaysnOlU1egative. Also thetotalvolumeunderthegraphofajointprobability densityfunction mustbeunity,as shownby (A1.23) Theprobability densityfunction forasinglerandomvariable(X,say)canbeob tainedfromitsjointprobability densityfunction withasecondrandomvariable (Y,say) inthefollowing way.Wefirstnotethat (A1.24) Therefore, differentiating bothsidesofEquation (A1.24)withrespecttox,wegetthe desiredrelation; (A1.25) (A1.26)Thustheprobability densityfunction fx(x)isobtained fromthejointprobability density functionfx,y(x,y)bysimplyintegrating itoverallpossiblevaluesoftheundesired random variable Y.Theuseofsimilararguments intheotherdimension yieldsfy(y).Theproba bilitydensityfunctions fx(x)andfy(y)arecalledmarginal densities. Hence,thejoint probability densityfunctionfx,y(x,y)containsallthepossibleinformation aboutthejoint randomvariables XandY. SupposethatXandYaretwocontinuous randomvariables withjointprobability densityfunction fx,y(x,y).Theconditional probability densityfunction ofYgiventhat X=xisdefinedby f(I )=fx,y(x,y) yy x fx(x) provided thatfx(x)>0,wherefx(x)isthemarginal densityofX.Thefunction fy(ylx) maybethoughtofasafunction ofthevariabley,withthevariablexarbitrary, butfixed. Accordingly, itsatisfiesalltherequirements ofanordinary probability densityfunction, asshownby fy(yIx)2:0 and Iftherandomvariables XandYarestatistically independent, thenknowledge ofthe outcome ofXcaninnowayaffectthedistribution ofY.Theresultisthattheconditional probability densityfunction fy(yIx)reducestothemarginal densityfy(y),asshownby fy(Ylx)=fy(y) Insuchacase,wemayexpressthejointprobability densityfunction oftherandomvari ablesXandYastheproductoftheirrespective marginal densities, asshownby fx,y(x,y)=fx(x)fy(y) A1.3Statist",al Averages 711 Inwords,wemaystatethatifthejointprobability densityfunctionoftherandomvariables XandYequalstheproductoftheirmarginal densities, thenXandYarestatistically independent. IA1.3Statistical Averages Havingdiscussed probability andsomeofitsramifications, wenowseekwaysfordeter miningtheaveragebehavior oftheoutcomes arisinginrandomexperiments. Theexpected valueormeanofarandomvariableXisdefinedby J-tx=E[X]=rooxfx(X)dx (A1.2?) (A1.28) k=0,±1,±2,...whereEdenotesthestatistical expectation operator. Thatis,themeanJ-txlocatesthecenter ofgravityoftheareaundertheprobability densitycurveoftherandomvariableX.To interpret theexpected valueJ-tx,wewritetheintegralinthedefiningEquation (Al.2?)as thelimitofanapproximating sumformulated asfollows.Let[XkIk=0,±1, ±2,...} denoteasetofuniformly spacedpointsontherealline: Xk=(k+~)~, where ~isthespacingbetween adjacent points.WemaythenrewriteEquation (Al.2?) asthelimitingformofasum: (A1.29) Foraphysicalinterpretation ofthesumontheright-hand sideofEquation (Al.29),sup posethatwemakenindependent observations oftherandomvariableX.LetNn(k)denote thenumberoftimesthattherandomvariableXfallsinsidethekthbin: a ~Xk--<XSXk+-2 2'k=0,±1, ±2,... Then,asthenumberofobservations, n,ismadelarge,theratioNn(k)/napproaches the probability P(Xk-al2<XSXk+al2).Accordingly, wemayapproximate theexpected valueoftherandomvariableXas (A1.30) nlarge Wenowrecognize thequantity ontheright-hand sideofEquation (Al.30)simplyasthe sampleaverage.ThesumistakenoverallthevaluesXbeachofwhichisweighted bythe numberoftimesitoccurs;thesumisthendividedbythetotalnumberofobservations to givethesampleaverage.Indeed,Equation (Al.30)provides thebasisforcomputing the expected valueE[X]. 712 ApPENDIX 1"PROBABILI'IY'fHEORY Wenextconsider amoregeneralsituation. LetXdenotearandomvariable, andlet g(X)denoteafunction ofXdefinedontherealline.Thequantity obtained bylettingthe argument ofthefunctiong(X)bearandomvariableisalsoarandomvariable, whichWe denoteas Y=g(X) (A1.31) Tofindtheexpected valueoftherandomvariable Y,wecouldofcoursefindtheprobability densityfunction fy(y)andthenapplythestandard formula E[Y]=r~yfy(y)dy Asimplerprocedure, however, istowrite E[g(X)] =roog(x)fx(x) dx (A1.32) Indeed,Equation (A1.32)maybeviewedasgeneralizing theconceptofexpected valueto anarbitrary furtctiong(X)ofarandomvariableX. illlMOMENTS Forthespecialcaseofg(X)=Xn,usingEquation (A1.32)weobtainthenthmomentof theprobability distribution oftherandomvariableX;thatis, (A1.33) Byfarthemostimportant moments ofXarethefirsttwomoments. Thusputtingn=1 inEquation (A1.33)givesthemeanoftherandomvariable asshowninEq.(A1.2?), whereasputtingn=2givesthemean-square valueofX: (A1.34) Wemayalsodefinecentralmoments, whicharesimplythemoments ofthedifference between arandomvariableXanditsmean/Lx.Thus,thenthcentralmoment is E[(X-/LxtJ=r~(x-/Lxtfx(x) dx (A1.3S) Forn=1,thecentralmoment is,ofcourse,zero,whereasforn=2thesecondcentral moment isreferredtoasthevariance oftherandomvariableX,whichiswrittenas (A1.36) Thevariance ofarandomvariableXiscommonly denotedas~.Thesquarerootofthe variance, namely, U"x,iscalledthestandard deviation oftherandomvariableX. Thevariance ~ofarandomvariableXinsomesenseisameasureofthevariable's "randomness." Byspecifying thevariance ~,weessentially constrain theeffectivewidth oftheprobability densityfunction fx(x)oftherandomvariableXaboutthemean/LX· A1.3StatisticalA1Jerages 713 Aprecisestatement ofthisconstraint isduetoChebyshev. TheChebyshev inequality states thatforanypositivenumber €,wehave (A1.3?) (A1.3S) (A1.39)Fromthisinequality weseethatthemeanandvarianceofarandomvariablegiveapartial description ofitsprobability distribution, hencetheircommon useinpractice. WenotefromEquations (A1.34)and(A1.36)thatthevarianceaiandmean-square valueE[X2]arerelatedby ai=E[X2-2J.LxX+J.Lk] =E[X2]-2J.LxE[X]+J.L3r. =E[X2]-J.L3r. where,inthesecondline,wehaveusedthelinearityproperty ofthestatistical expectation operatorE.Equation (A1.38)showsthatifthemeanJ.Lxiszero,thenthevariance u3r.and themean-square valueE[X2]oftherandomvariableXareequal. !IICHARACTERISTIC FUNCTION Anotherimportant statistical averageisthecharacteristic function cf>x(v)oftheprobability distribution oftherandomvariableX,whichisdefinedastheexpectation ofthecomplex exponential function exp(jvX),asshownby cPx(v)=E[exp(jvX)] =roofx(x)exp(jvx)dx wherevisrealandj=v=I.Inotherwords,thecharacteristic function cPx(v)is(except forasignchangeintheexponent) theFouriertransform oftheprobability densityfunction fx(x);theFouriertransform isreviewed inAppendix 2.Inthisrelationwehaveused exp(jvx) ratherthanexp(-jvx), soastoconform withtheconvention adoptedinproba bilitytheory.Recognizing thatvandxplayanalogous rolestothevariables2nfandtof Fouriertransforms, respectively, wededucethefollowing inverserelationfromanalogy withtheinverseFouriertransform: 1foofx(x)=2n -00cPx(v)exp(-jvx) dv (Al.40) Thisrelationmaybeusedtoevaluatetheprobability densityfunctionfx(x)oftherandom variableXfromitscharacteristic function cPx(v). IIIJOINTMOMENTS Consider nextapairofrandomvariables XandY.Asetofstatistical averagesofimpor tanceinthiscaseisthejointmoments, namely,theexpected valueofXiy\whereiand kmayassumeanypositiveintegervalues.Wemaythuswrite (A1.41) (A1.44)714 APPENDIX 1l!lPRoBABILlrrTHEORY Ajointmomentofparticular importance isthecorrelation deifnedbyE[XY],whichcor responds toi=k=1inEquation (A1.41). Thecorrelation ofthecentered randomvariables X -E[X]andY -E[Y],thatis thejointmoment ' cov[XY] =E[(X-E[X])(Y-E[Y])] (Al.42) iscalledthecovariance ofXandY.Letting/kx=E[X]and/ky=E[Y],wemayexpand Equation (A1.42)toobtaintheresult cov[XY] =E[XY]-/kx/ky (A1.43) Letaiand~denotethevariances ofXandY,respectively. Thenthecovariance ofX andY,normalized withrespectto(J"x(J"y,iscalledthecorrelation coefficient ofXandY: cov[XY]p=-- (J"x(J"y Wesaythatthetworandomvariables XandYareuncorrelated ifandonlyiftheir covariance iszero,thatis,ifandonlyif cov[XY] =0 Wesaythattheyareorthogonal ifandonlyiftheircorrelation iszero,thatis,ifand onlyif E[XY]=0 FromEquation (A1.43)weobservethatifoneoftherandomvariables XandYorboth havezeromeans,andiftheyareorthogonal, then they areuncorrelated, andviceversa. NotealsothatifXandYarestatistically independent, thentheyareuncorrelated; however, theconverse ofthisstatement isnotnecessarily true. REPRESENTATION OF SIGNALS ANDSYSTEMS IA2.1FourierAnalysis Letg(t)denoteanonperiodic deterministic signal,expressed assomefunction oftimet. Bydefinition, theFouriertransform ofthesignalg(t)isgivenbytheintegral G(f)=r~g(t)exp(-j27rft) dt (Al.i) wherej=v=I,andthevariable fdenotesfrequency. GiventheFouriertransform G(f), theoriginalsignalg(t)isrecovered exactlyusingtheformula fortheinverseFourier transform: g(t)=r~G(f)exp(j27Tft) df (Al.2) NotethatinEquations (Al.i)and(Al.2)wehaveusedalowercase lettertodenotethe timefunctionandanuppercase lettertodenotethecorresponding frequency function. The functions g(f)andG(f)aresaidtoconstitute aFourier-transform pair. FortheFouriertransform ofasignalg(t)toexist,itissufficient butnotnecessary thatg(t)satisfiesthreeconditions knowncollectively asDirichlet's conditions: 1.Thefunction g(t)issingle-valued, withafinitenumberofmaximaandminimain anyfinitetimeinterval. 2.Thefunction g(t)hasafinitenumberofdiscontinuities inanyfinitetimeinterval. 3.Thefunction g(t)isabsolutely integrable, thatis, r~Ig(t)Idt<00 Wemaysafelyignorethequestion oftheexistence oftheFouriertransform ofatime function g(t)whenitisanaccurately specified description ofaphysically realizable signal. Inotherwords,physicalrealizability isasufficient condition fortheexistence ofaFourier transform. Indeed,wemaygoonestepfurtherandstatethatallenergysignals,thatis, signalsg(t)forwhich areFouriertransformable. TheFouriertransform provides themathematical toolformeasuring thefrequency content,orspectrum, ofasignal.Forthisreason,thetermsFouriertransform andspectrum areoftenusedinterchangeably. Thus,givenasignalg(t)withFouriertransform G(f),we mayrefertoG(f)asthespectrum ofthesignalg(t).Bythesametoken,werefertoIG(f)1 asthemagnitude spectrum ofthesignalg(t),andrefertoarg(G(f)}asitsphasespectrum. 715 716 APPENDIX 2'"REPRESENTATION OFSIGNALS ANDSYSTEMS illPROPERTIES OFTHEFOURIER TRANSFORM Itisusefultohaveinsightintotherelationship betweenatimefunctiong(t)anditsFourier transform G(f),andalsointotheeffectsthatvariousoperations onthefunction g(t)have onthetransform G(f).Thismaybeachieved byexamining certainproperties oftheFourier transform, whicharesummarized inTableA6.2. Ii!IDIRAC DELTA FUNCTION Strictlyspeaking, thetheoryoftheFouriertransform isapplicable onlytotimefunctions thatsatisfytheDirichlet conditions. Suchfunctions includeenergysignals.However, it wouldbehighlydesirable toextendthistheoryintwoways: 1.Tocombine theFourierseriesandFouriertransform intoaunifiedtheory,sothat theFourierseriesmaybetreatedasaspecialcaseoftheFouriertransform. 2.Toincludepowersignals(i.e.,signalsforwhichtheaveragepowerisfinite)inthe listofsignalstowhichwemayapplytheFouriertransform. Itturnsoutthatbothoftheseobjectives canbemetthroughthe"properuse"oftheDirac deltafunction, orunitimpulse. TheDiracdelta'function orjustdeltafunction, denotedbylilt),isdefinedashaving zeroamplitude everywhere exceptatt=0,whereitisinfinitely largeinsuchawaythat itcontains unitareaunderitscurve;thatis, lilt)=0,t"*°(Al.3) and r~B(t)dt=1 (AlA) Animplication ofthispairofrelations isthatthedeltafunction li(t)mustbeaneven function oftimet,whichiscentered att=0. Forthedeltafunction tohavemeaning, however, ithastoappearasafactorinthe integrand ofanintegralwithrespecttotimeandthen,strictlyspeaking, onlywhenthe otherfactorintheintegrand isacontinuous function oftime.Letg(t)besuchafunction, andconsider theproductofg(t)andthetime-shifted deltafunction li(t-to).Inlightof thetwodefiningequations, Equations (Al.3)and(AlA),wemayexpresstheintegralof thisproductasfollows: r~g(t)li(t-to)dt=g(to) (Al.5) Theoperation indicated ontheleft-hand sideofthisequation siftsoutthevalueg(to)of thefunction g(t)attimet=to,where -00<t<00.Accordingly, Equation (Al.5)is referredtoasthesiftingproperty ofthedeltafunction. Thisproperty issometimes used asthedefiningequation ofadeltafunction; ineffect,itincorporates Equations (A2.3)and (AlA)intoasinglerelation. Notingthatthedeltafunction lilt)isanevenfunction oft,wemayrewriteEquation (Al.5)soastoemphasize itsresemblance totheconvolution integral, asshownby r~g(T)li(t-T)dT=g(t) (Al.6) A2.1Fourie..A....lysis717 Inwords,theconvolution ofanyfunction withthedeltafunction leavesthatfunction unchanged. Werefertothisstatement asthereplication property ofthedeltafunction. Itisimportant torealizethatnofunctionintheordinary sensehasthetwoproperties ofEquations (Al.3)and(A2A)ortheequivalent siftingproperty ofEquation (Al.5). However, wecanimagineasequence offunctions thathaveprogressively tallerandthinner peaksatt=0,withtheareaunderthecurveremaining equaltounity,whereasthevalue ofthefunction tendstozeroateverypointexceptt=0,whereittendstoinfinity.That is,wemayviewthedeltafunction asthelimiting formofapulseofunitareaasthe durationofthepulseapproaches zero.Itisimmaterial whatsortofpulseshapeisused. I!!FOURIER TRANSFORMS OFPERIODIC SIGNALS ItiswellknownthatbyusingtheFourierseries,aperiodicsignalcanberepresented asa sumofcomplex exponentials. Also,inalimitingsense,Fouriertransforms canbedefined bycomplex exponentials. Therefore, itseemsreasonable torepresent aperiodicsignalin termsofaFouriertransform, provided thatthistransform ispermitted toincludedelta functions. Consider thenaperiodicsignalgTo(t)ofperiodTo.Wecanrepresent gTo(t)interms ofthecomplex exponential Fourierseries: (Al.?) (Al.S) (Al.9)where Cnisthecomplex Fouriercoefficient definedby 1JTOl2 Cn=-To gTo(t)exp(-j27Tnfo) dto-To/2 andfoisthefundamental frequency definedasthereciprocal oftheperiodTo;thatis, 1fo= To Letg(t)bea pulselike function, whichequalsgTo(t)overoneperiodandiszeroelsewhere; thatis g(t)={~TO(t),To To-2<t$2 elsewhere(Al.I0) TheperiodicsignalgTo(t)maynowbeexpressed intermsofthefunction g(t)asaninfinite summation, asshownby (Al.11) Basedonthisrepresentation, wemayviewg(t)asagenerating function, whichgenerates theperiodicsignalgTo(t). Thefunction g(t)isFouriertransformable. Accordingly, wemayrewritetheformula forthecomplex Fouriercoefficient asfollows: cn=for~g(t)exp(-"27Tnf ot)dt =foG(nfo) (Al.12) 718 APPENDIX 2illREPRESENTATION OFSIGNALS ANDSYSTEMS whereG(nfo)istheFouriertransform ofg(t)evaluated atthefrequency nfo.Wemaythus rewritetheformulaforthereconstruction oftheperiodicsignalgTo(t)as gTO(t)=fo2:G(nfo)exp(j2'TT'nf ot) or,equivalently, inlightofEquation (A2.11) ~ ~ 2:g(t-mTo)=fo2:G(nfo)exp(j2'TT'nf ot)(A2.13) (A2.14) (A2.15)Equation (A2.14)isoneformofPoisson's sumformula. Itisofinteresttoobservethatthefunction g(t),whichconstitutes oneperiodofthe periodicsignalgTo(t),hasacontinuous spectrum definedbyG(f).Ontheotherhand,the periodsignalgTo(t)itselfhasadiscretespectrum. Weconclude, therefore, thatperiodicity inthetimedomainhastheeffectofchanging thefrequency-domain description orspec trumofthesignalintoadiscreteformdefinedatintegermultiples ofthefundamental frequency. FOURIER-TRANSFORM PAIRS TableA6.3presents alistingofsomecommonly usedFourier-transform pairs,thederi vationsofwhich followfromthematerial justpresented. illTRANSMISSION OFSIGNALS THROUGH LINEAR SYSTEMS Asystemreferstoanyphysical devicethatproduces anoutputsignalinresponse toan inputsignal.Itiscus~omary torefertotheinputsignalastheexcitation andtotheou~ut signalastheresponse. Inalinearsystem,theprincipleofsuperposition holds;thatis,the response ofalinearsystemtoanumberofexcitations appliedsimultaneously isequalto thesumoftheresponses ofthesystemwheneachexcitation isappliedindividually. Inthetimedomain, alinearsystemisdescribed intermsofitsimpulseresponse, whichisdefinedastheresponseofthesystem(withzeroinitialconditions) toaunitimpulse ordeltafunction 6(t)appliedtotheinputofthesystem.Ifthesystemistimeinvariant, thentheshapeoftheimpulseresponse isthesamenomatterwhentheunitimpulse is appliedtothesystem.Thus,assuming thattheunitimpulseordeltafunction isappliedat timet=0,wemaydenotetheimpulseresponse ofalineartime-invariant systembyh(t). Letthissystembesubjected toanarbitrary excitation x(t).Theresponse, y(t),ofthesystem isdefinedintermsoftheimpulseresponse h(t)by y(t)=r~x(T)h(t-T)dT whichiscalledtheconvolution integral. Equivalently, wemaywrite y(t)=r~h(T)x(t-7)dT (A2.16) Hence,convolution iscommutative. Intheconvolution integral, threedifferent timescalesareinvolved: excitation time T,response timet,andsystem-memory timet-T.Thisrelationisthebasisoftime-domain analysisoflineartime-invariant systems.According toEquation (A2.15),thepresentvalue oftheresponse ofalineartime-invariant systemisaweighted integraloverthepasthistory A2.1F....rierAnalysis 719 oftheinputsignal,weighted according totheimpulseresponse ofthesystem.Thusthe impulseresponse actsasamemory function forthesystem. !iiiFREQUENCY RESPONSE OFLINEAR TIME-INVARIANT SYSTEMS Consider alineartime-invariant systemofimpulseresponse hit)drivenbyacomplex exponential inputofunitamplitude andfrequencyf,thatis, x(t)=exp(j211ft) Usingthisexcitation inEquation (A2.16), theresponse ofthesystemisobtained as y(t)=rooh(7)exp[j211/(t -7)]dr =exp(j211ft)rooh(r)exp(-j211fr) dr(A2.17) Definethefrequency response ofthesystemastheFouriertransform ofitsimpulsere sponse,asshownby H(f)=roohit)exp(-j21Tft) dt (A2.18) TheintegralinthelastlineofEquation (A2.17)isthesameasthatofEquation (A2.18), exceptthat7isusedinplaceoft.Hence,wemayrewriteEquation (A2.17)intheform y(t)=H(f)exp(j21Tft) (A2.19) Theresponse ofalineartime-invariant systemtoacomplex exponential function offre quencyfis,therefore, thesamecomplex exponential function multiplied byaconstant coefficient H(f). Thefrequency response H(f)is,ingeneral,acomplex quantity, sowemayexpress itintheform H(f)=IH(f)Iexp[jf3(f)] (A2.20) whereIH(f)Iiscalledthemagnitude response, andf3(f)isthephase,orphaseresponse. Inthespecialcaseofalinearsystemwithareal-valued impulseresponse hit),thefrequency response H(f)exhibitsconjugate symmetry, whichmeansthat IH(f)I=IH(-f)I and f3(f)=-f3(-f) Thatis,themagnitude response IH(f)Iofalinearsystemwithreal-valued impulseresponse isanevenfunction offrequency, whereasthephasef3(f)isanoddfunction offrequency. Insomeapplications, itispreferable toworkwiththelogarithm ofH(f)expressed inpolarformratherthanwithH(f)itself.Definethenaturallogarithm wherelogH(f)=a(f)+jf3(f) a(f)=logIH(f)I(A2.21) (A2.22) 720 APPENDIX 2..REPRESENTATION OFSIGNALS ANDSYSTEMS Thefunctiona(f}iscalledthegainofthesystem.Itismeasured innepers,whereasf3(f) ismeasured inradians.Equation (A2.21)indicates thatthegaina(f}andphasef3(f)are therealandimaginary partsofthe(natural) logarithm ofthefrequency response H(f), respectively. Thegainmayalsobeexpressed indecibels(dB)byusingthedefinition a'(f)=20logloIH(f}I Thetwogainfunctions a(f}anda'(f}arerelatedby a'(f)=8.69a(f} Thatis,1neperisequalto8.69dB. IA2.2Bandwidth(A2.23) (A2.24) Thetime-domain andfrequency-domain descriptions ofasignalareinversely related.In particular, wemaymakethefollowing important statements: 1.Ifthetime-domain descriptio~ ofasignalischanged, thefrequency-domain descrip tionofthesignalischanged inaninversemanner, andviceversa.Thisinversere lationship prevents arbitrary specifications ofasignalinbothdomains. Inother words,wemayspecifyanarbitrary functionoftimeoranarbitrary spectrum, but wecannotspecifybothofthemtogether. 2.Ifasignalisstrictlylimitedinfrequency, thetime-domain description ofthesignal willtrailonindefinitely, eventhoughitsamplitude mayassumeaprogressively smallervalue.Wesayasignalisstrictlylimitedinfrequency orstrictlybandlimited ifitsFouriertransform isexactlyzerooutsideafinitebandoffrequencies. Thesine pulse sinc(t)=sin(m)m isanexample ofastrictlyband-limited signal.Itisalsoasymptotically limitedin time,whichconfirms theopening statement wemadeforastrictlyband-limited signal.Inaninversemanner, ifasignalisstrictlylimitedintime(i.e.,thesignalis exactlyzerooutsideafinitetimeinterval), thenthespectrum ofthesignalisinfinite inextent,eventhoughtheamplitude spectrum mayassumeaprogressively smaller value.Thisbehavior isexemplified byarectangular pulse.Accordingly, wemaystate thatasignalcannotbestrictlylimitedinbothtimeandfrequency. Thebandwidth ofasignalprovides ameasure oftheextentofsignificant spectral contentofthesignalforpositivefrequencies. Whenthesignalisstrictlybandlimited,the bandwidth iswelldefined.Forexample, thesinepulsesinc(2Wt) hasabandwidth equal toW.However, whenthesignalisnotstrictlybandlimited,asisgenerally thecase,we encounter difficulty indefiningthebandwidth ofthesignal.Thedifficulty arisesbecause themeaning of"significant" atrached tothespectralcontentofthesignalismathematically imprecise. Consequently, thereisnouniversally accepted definition ofbandwidth. Nev ertheless, therearesomecommonly useddefinitions forbandwidth, asdiscussed next. Whenthespectrum ofasignalissymmetric withamainlobebounded bywell-defined nulls(i.e.,frequencies atwhichthespectrum iszero),wemayusethemainlobeasthe basisfordefiningthebandwidth ofthesignal.Specifically, ifthesignalislow-pass (i.e., itsspectralcontentiscenteredaroundtheorigin),thebandwidth isdefinedasonehalfthe totalwidthofthemainspectrallobesinceonlyonehalfofthislobeliesinsidethepositive frequency region.Forexample, arectangular pulseofduration Tsecondshasamain (A2.25)A2.2Bandwidth 721 spectrallobeoftotalwidth2fThertzcentered attheorigin.Accordingly, wemaydefine thebandwidth ofthisrectangular pulseas1fThertz.If,ontheotherhand,thesignalis band-pass withmainspectrallobescentered around±I"whereIeislargeenough,the bandwidth isdefinedasthewidthofthemainlobeforpositivefrequencies. Thisdefinition ofbandwidth iscalledthenull-to-null bandwidth. Forexample, anRFpulseofduration Tsecondsandfrequency Iehasmainspectrallobesofwidth21Thertzcentered around ±I"whereitisassumed thatIeislargecompared toliT.Hence,wemaydefinethenull to-nullbandwidth ofthisRFpulseas21Thertz.Onthebasisofthedefinitions presented here,wemaystatethatshiftingthespectralcontentofalow-pass signalbyasufficiently largefrequency hastheeffectofdoubling thebandwidth ofthesignal;suchafrequency translation isattainedbyusingmodulation. Anotherpopulardefinition ofbandwidth isthe3-dBbandwidth. Specifically, ifthe signalislow-pass, the3-dBbandwidth isdefinedastheseparation betweenzerofrequency, wherethemagnitude spectrum attainsitspeakvalue,andthepositivefrequency, atwhich theamplitude spectrum dropsto1/\/2ofitspeakvalue.Forexample, thedecaying ex ponential exp(-at)hasa3-dBbandwidth ofa12'Trhertz.If,ontheotherhand,thesignal isband-pass, centered at±Ie,the3-dBbandwidth isdefinedastheseparation (alongthe positivefrequency axis)betweenthetwofrequencies atwhichthemagnitude spectrum of thesignaldropsto1/0ofthepeakvalueofIe.The3-dBbandwidth hastheadvantage inthatitcanbereaddirectlyfromaplotofthemagnitude spectrum. However, ithasthe disadvantage inthatitmaybemisleading ifthemagnitude spectrum hasslowlydecreasing tails. Yetanothermeasure forthebandwidth ofasignalistherootmeansquare(rms) bandwidth, whichisdefinedasthesquarerootofthesecondmoment ofaproperly nor malizedform ofthesquaredmagnitude spectrum of thesignalaboutasuitablychosen point.Weassumethatthesignalislow-pass, sothatthesecondmoment maybetaken abouttheorigin.Asforthenormalized formofthesquaredmagnitude spectrum, weuse thenonnegative function IG(IW r~IG(f)12dl inwhichthedenominator appliesthecorrectnormalization inthesensethattheintegrated valueofthisratioovertheentirefrequency axisisunity.Wemaythusformally definethe rmsbandwidth ofalow-pass signalg(t)with Fouriertransform G(f)asfollows: W rUls=([.~F1G(f)12 dl)1/2 L~ 1G(f) 12dl Anattractive featureofthermsbandwidth WrUlsisthatitlendsitselfmorereadilyto mathematical evaluationthantheothertwodefinitions ofbandwidth, butitisnotaseasily measurable inthelaboratory. IIITIME-BANDWIDTH PRODUCT Foranyfamilyofpulsesignalsthatdifferbyatime-scaling factor,theproductofthe signal'sduration anditsbandwidth isalwaysaconstant, asshownby (duration Xbandwidth) =constant 722 APPENDIX 2IIIREPRESENTATION OFSIGNALS ANDSYSTEMS Theproductiscalledthetime-bandwidth productorbandwidth-duration product. The constancy ofthetime-bandwidth productisanothermanifestation oftheinverserelation shipthatexistsbetweenthetime-domain andfrequency-domain descriptions ofasignal. Inparticular, iftheduration ofapulsesignalisdecreased byreducing thetimescalebya factora,thefrequency scaleofthesignal'sspectrum, andtherefore thebandwidth ofthe signal,isincreased bythesamefactora,byvirtueofthetime-scaling property oftheFourier transform, andthetime-bandwidth productofthesignalistherebymaintained constant; seeitem2ofTableA6.2.Forexample, arectangular pulseofduration Tsecondshasa bandwidth (definedonthebasisofthepositive-frequency partofthemainlobe)equalto lIThertz,makingthetime-bandwidth productofthepulseequalunity.Whatever defini tionweuseforthebandwidth ofasignal,thetime-bandwidth productremainsconstant overcertainclassesofpulsesignals.Thechoiceofaparticular definition forbandwidth merelychangesthevalueoftheconstant. Tobemorespecific,consider thermsbandwidth definedinEquation (A2.25).The corresponding definition forthermsduration ofthesignalg(t)is (A2.26) whereitisassumed thatthesignalg(t)iscentered aroundtheorigin.Itmaybeshown that,usingthermsdefinitions ofEquations (A2.25)and(A2.26), thetime-bandwidth producthasthefollowing form: (A2.27) wheretheconstant is1I4'1T.TheGaussian pulseexp(_m2)satisfiesthiscondition withthe equalitysign. !IiNOISE EQUIVALENT BANDWIDTH Thedefinitions ofbandwidth justpresented (i.e.,3-dBbandwidth, null-to-null bandwidth, andrIDSbandwidth) areallformulated intermsofdeterministic signals.Anotherdefinition ofbandwidth thatpresentsitselfinthestudyofrandomsignalsandsystemsisthenoise equivalent bandwidth. Suppose thatawhitenoisesourceofpowerspectraldensityNo/2 isconnected rotheinputofthesimpleRClow-pass filterofFigureA2.1;thecorresponding valueofrheaverageoutpurnoisepowerisequaltoNo/(4RC).Forthisfilter,thehalf poweror3-dBbandwidth isequalto1I(2'1TRC). Hereagainwefindthattheaverageoutput noisepowerofthefilterisproportional tothebandwidth. Wemaygeneralize thisstatement toincludeallkindsoflow-pass filtersbydefining anoiseequivalent bandwidth asfollows.Supposethatwehaveasourceofwhitenoiseof R White~colored noise CT noise w(r)~n{t) FIGUREA2.1Relow-pass filter. A2.3HilbertTransform 723 lli(f)I, FP(O) ---:Ideallow-pass Vfilter I II -B FIGUREA2.2Illustrating thedefinition ofnoise-equivalent bandvvidth foralow-pass filter. zeromeanandpowerspectraldensityNo/2connected totheinputofanarbitrary low passfilteroftransferfunction H(f).Theresulting averageoutputnoisepoweristherefore Nou,=~orooIH(f) 12df =NorIH(f) 12df(A2.28) where,inthelastline,wehavemadeuseofthefactthatthemagnitude responseIH(f)I isanevenfunction offrequency. Consider nextthesamesourceofwhitenoiseconnected totheinputofanideallow passfilterofzero-frequency response H(O)andbandwidth B.Inthiscase,theaverage outputnoisepoweris (A2.29) Therefore, equating thisaverageoutputnoisepowertothatinEquation (A2.28),wemay formally definethenoiseequivalent bandwidth asrIH(fWdf B=H2(0)(A2.29) Thustheprocedure forcalculating thenoiseequivalent bandwidth consistsofreplacing thearbitrary low-pass filteroftransferfunctionH(f)byanequivalent ideallowcpass filter ofzerofrequency response H(O)andbandwidth B,asillustrated inFigureA2.2.Inasimilar way,wemaydefineanoiseequivalent bandwidth forbandpass filters. IA2.3Hilbert Transform TheFouriertransform isparticularly usefulforevaluating thefrequency contentofan energysignalor,inalimitingsense,thatofapowersignal.Assuch,itprovides themath ematical basisforanalyzing anddesigning frequency-selective filtersfortheseparation of signalsonthebasisoftheirfrequency content. Another methodofseparating signalsis basedonphaseselectivity, whichusesphaseshiftsbetweenthepertinent signalstoachieve thedesiredseparation. Thesimplestphaseshiftisthatof180degrees,whichismerelya polarityreversalinthecaseofasinusoidal signal.Shiftingthephaseanglesofallcom ponentsofagivensignalby180degreesrequirestheuseofanidealtransformer. Another phaseshiftofinterestisthatof±90degrees.Inparticular, whenthephaseanglesofall ~4APPENDIX 2!ilREPRESENTATION OFSIGNALS ANDSYSTEMS components ofagivensignalareshiftedby±90degrees,theresulting function oftimeis knownastheHilberttransform ofthesignal. Tobespecific,consider asignalg(t) withFouriertransform C(f).TheHilberttrans formofg(t),whichweshalldenotebyg(t),isdefinedby g(t)=.!.Joog(T)dT (A2.31) 7T-00t-T Clearly,theHilberttransformation ofg(t)isalinearoperation. TheinverseHilberttrans form,bymeansofwhichtheoriginalsignalg(t)isrecovered fromg(t),isdefinedby 1J~g(T)g(t)=----dT (A2.32) 7T-00t-T Thefunctions g(t)andg(t)aresaidtoconstitute aHilbert-transform pair.Ashorttable ofHilbert-transform pairsisgiveninTableA6.4. Wenotefromthedefinition oftheHilberttransform thatg(t)maybeinterpreted as theconvolution ofg(t)withthetimefunction lI7Tt.Wealsoknowfromtheconvolution theoremthattheconvolution oftwofunctions inthetimedomainistransformed intothe multiplication oftheirFouriertransforms inthefrequency domain; seeitem12ofTable A6.2.Forthetimefunction1/7Tt, wehave(seeTableA6.3) ~~-jsgn(f) 7T,t wheresgn(f)isthesignum function definedinthefrequency domainas {1,f>0 sgn(f)=0,f=0 -1,f<0 Itfollowstherefore thattheFouriertransform G(f)ofg(t)isgivenby C(f)=-jsgn(f)C(f)(A2.33) (A2.34) (A2.35) Equation (A2.35)statesthatgivenasignalg(t),wemayobtainitsHilberttransform g(t)bypassingg(t)throughalineartwo-port devicewhosefrequency response isequalto -jsgn(f).Thisdevicemaybeconsidered asonethatproduces aphaseshiftof-90degrees forallpositivefrequencies oftheinputsignaland+90degreesforallnegativefrequencies, asinFigureA2.3.Theamplitudes ofallfrequency components inthesignal,however, ate arg{H(!lJ ------j +90' ~------If-:o-----! _90't------ FIGUREA2.3Phasecharacteristic oflineartwo-port deviceforobtaining theHilberttransform ofareal-valued signal. A2.4Complex Representation ofSignalsandSystems 725 unaffected bytransmission through thedevice.Suchanidealdeviceisreferredtoasa Hilberttransformer. IIIlPROPERTIES OFTHEHILBERT 'TRANSFORM 'TheHilberttransform differsfromtheFouriertransform inthatitoperates exclusively in thetimedomain.Ithasanumberofusefulproperties, someofwhicharelistednext.The signalg(t)isassumed toberealvalued,whichistheusualdomainofapplication ofthe Hilberttransform. Forthisclassofsignals,wemaystatethefollowing: 1.Asignalg(t)anditsHilberttransform g(t)havethesamemagnitude spectrum. 2.Ifg(t)istheHilberttransform ofg(t),thentheHilberttransform ofg(t)is-g(t). 3.Asignalg(t)anditsHilberttransformg(t) areorthogonal overtheentiretimeinterval (-00,00), asshownby f_g(t)g(t)dt =0 Proofsoftheseproperties areleftasexercises forthereader;theproofsfollowfromEqua tions(A2.31),(A2.32)and(A2.35). A2.4Complex Representation ofSignalsandSystems IIIPRE-ENVELOPE Consider areal-valued signalg(t).Wedefinethepre-envelope, oranalytic signal,ofthe signalg(t)asthecomplex-valued function g+(t)=g(t)+jg(t) (A2.36) (A2.37)whereg(t)istheHilberttransform ofg(t).Wenotethatthegivensignalg(t)isthereal partofthepre-envelope g+(t),andtheHilberttransform ofthesignalistheimaginary partofthepre-envelope. Justastheuseofphasorssimplifies manipulations ofalternating currents andvoltages, sowefindthatthepre-envelope isparticularly usefulinhandling band-pass signalsandsystems. Oneoftheimportant featuresofthepre-envelope g+(t)isthebehavior ofitsFourier transform. LetG+(f)denotetheFouriertransform ofg+(t).Thenwemaywrite G+(f)=G(f)+sgn(f)G(f) fromwhichwereadilyfindthat {2G(f),f>0 G+(f)=G(O),f=0 0,f<0 whereG(O)isthevalueofG(f)atfrequencyf=O.Thismeansthatthepre-envelope of asignalhasnofrequency content(i.e.,itsFouriertransform vanishes) forallnegative frequencies. 726 APPENDIX 2"REPRESENTATION OFSIGNALS ANDSYSTEMS Fromtheforegoing analysisitisapparent thatforagivensignalg(t)wemaydeter mineitspre-envelope g+(t)inoneoftwoequivalent ways: 1.Wedetennine theHilberttransform g(t)ofthesignalg(t),andthenuseEquation (A2.36)tocompute thepre-envelope g+(t). 2.Wedetermine theFouriertransform G(f)ofthesignalg(t),useEquation (A2.37)to determine G+(f),andthenevaluatetheinverseFouriertransform ofG+(f)toobtain g+(t)=2rG(f)exp(j21Tft) df (A2.38) Foraparticular signalg(t)ofFouriertransform G(f),oneofthesetwowaysmaybebetter thantheother. Equation (A2.36)definesthepre-envelope g+(t)forpositivefrequencies. Symmetri cally,wemaydefinethepre-envelope fornegativefrequencies as g_(t)=g(t)-jg(t) (A2.39) Thetwopre-envelopes g+(t)andg-(t)aresimplythecomplex conjugate ofeachother,as shownby (A2AO) (A2A!)wheretheasteriskdenotescomplex conjugation. Thespectrum ofthepre-envelope g+(t) isnonzeroonlyforpositivefrequencies, asemphasized inEquation (A2.37); hence,the useofaplussignasthesubscript. Incontrast, thespectrum oftheotherpre-envelope g_(t) isnonzero only fornegativefrequencies, asshownbytheFouriertransform {a,f>0 G_(f)=G(O),f=0 2G(!),f<0 Thusthepre-envelopes g+(t)andg_(t)constitute acomplementary pairofcomplex-valued signals.Notealsothatthesumofg+(t)andg_(t)isexactlytwicetheoriginalsignalg(t). I!!ICANONICAL REPRESENTATIONS OFBAND-PASS SIGNALS Consider aband-pass signalg(t)whoseFouriertransform G(f)isnonnegligible onlyina bandoffrequencies oftOtalextent2W,say,centered aboutsomefrequency ±fC'Thisis illustrated inFigureA2Aa.Werefertofcasthecarrierfrequency. Inthemajority of communication signals,wefindthatthebandwidth 2Wissmallcompared withf"and sowerefertosuchasignalasanarrowband signal.However, aprecisestatement about howsmallthebandwidth mustbeforthesignaltobeconsidered narrowband isnot necessary forourpresentdiscussion. Letthepre-envelope ofanarrowband signalg(t),withitsFouriertransform G(f) centered aboutsomefrequency ±fe,beexpressed intheform g+(t)=g(t)exp(j21TfJ) (A2,42) Werefertog(t)asthecomplex envelope ofthesignaLEquation (A2.42)maybeviewed asthebasisofadefinition forthecomplex envelope g(t)intermsofthepre-envelopeg+(t). Wenotethatthespectrum ofg+(t)islimitedtothefrequency bandfc-W::::f::::fe+W, asillustrated inFigureA2.4b.Therefore, applying thefrequency-shifting property ofthe A2.4Complex Representation ofSignalsandSystems 727 lG(fll K •+'I~<&l~---- -+---L---'~----:'----+---:'-~-f <aJ IG+<!ll 2]G(f,lll--·----·---. ---------:'---.,.-L:::---'........,~-f (bl IG(fll ----_....L.-.-l_.l.- f w (c) FIGUREA2.4(a)Magnitude spectrum ofband-pass signalg(t).(h)Magnitude spectrum ofpre envelope g.,(t).(c)Magnitude spectrum ofcomplex envelope g(t). Fouriertransform toEquation (A2.42), whichisdescribed asitem5inTableA6.2,we findthatthespectrum ofthecomplex envelope g(t)islimitedtotheband-W:Sf:sW andcentered attheoriginasillustrated inFigureAl.4e.Thatis,thecomplex envelope g(t)ofaband-pass signalg(t)isalow-pass signal,whichisanimportant result. Bydefinition, thegivensignalg(t)istherealpartofthepre-envelope g+(t).Wemay thusexpresstheoriginalband-pass signalg(t)intermsofthecomplex envelope g(t)as follows: g(t)=Re[g(t)exp(j21T!ct)] (Al.43) Ingeneral,g(t)isacomplex-valued quantity; toemphasize thisproperty, wemayexpress itintheform (A2.44) 728 APPENDIX 2IIIREPRESENTATION OFSIGNALS ANDSYSTEMS wheregr(t)andgdt)arebothreal-valued low-pass functions; theirlow-pass property is inherited fromthecomplex envelope g(t).Wemaytherefore useEquations (Al.43)and (A2.44)toexpresstheoriginalband-pass signalg(t)inthecanonical, orstandard, form: g(t)=gr(t)COS(21Tfct) -gQ(t)sin(21Tfct) (A2.4S) Werefertogr(t)asthein-phase component oftheband-pass signalg(t)andtogQ(t)as thequadrature component ofthesignal;thisnomenclature recognizes thatsin(21Tfct) [i.e., themultiplying factorofgQ(t)]isinphase-quadrature withrespecttoCOS(21Tfct) [i.e.,the multiplying factorofgr(t)]andCOS(21Tfct) isviewedasthereference. According toEquation (Al.44),thecomplex envelopeg(t) maybepictured asatime varyingphasorpositioned attheoriginofthe(gr,gQ)-plane, asindicated inFigureA2.Sa. Withtimetvarying,theendofthephasormovesaboutintheplane.FigureA2.Sbshows thephasorrepresentation ofthecomplex exponential exp(j21Tfct). Inthedefinition given inEquation (Al.43), thecomplex envelope g(t)ismultiplled bythecomplex exponential exp(j21Tfct).Theanglesofthesetwophasorstherefore add andtheirlengthsmultiply, as showninFigureAl.Sc.Moreover, inthislatterfigure,weshowthe(gr,gQ)-plane rotating withanangularvelocityequalto21Tfcradianspersecond.Thus,inthepictureportrayed here,thephasorrepresenting thecomplex envelope g(t)movesinthe(gr,gQ)-plane and atthesametimetheplaneitselfrotatesabouttheorigin.Theoriginalband-pass signal g(t)istheprojection ofthistime-varying phasoronafixedlinerepresenting therealaxis, asindicated inFigureAl.Sc. 0""-----'--------- g/Imaginary axis 0""----'-------- ~:~ (a) (b) Rotateatthe ~2"f, gQ\ \ \ \ \ \ \ \ \ \ \ \ \ Real ~=::='=~------ axis get) (c) FIGUREA2.5Illustrating aninterpretation ofthecomplex envelope g(t)anditsmultiplication byexp(j271'fc t). A2.4Ctnnplex Representation ofSignalsandSystems 729 Sincebothgrit)andgg(t)arelow-pass signalslimitedtotheband- Ws:fs:W, theymaybederivedfromtheband-pass signalg{t)usingtheschemeshowninFigure A2.6a.Bothlow-pass filtersinthisfigureareidentical, eachofwhichhasabandwidth equaltoW.Toreconstruct g{t)fromitsin-phase andquadramre components, wemay usetheschemeshowninFigureA2.6b. Thetwoschemes showninFigureA2.6arebasictothestudyoflinearmodulation systems. Themultiplication ofthelow-pass in-phase component gr(t)byCOS(27Tlt) and themultiplication ofthelow-pass quadrature component gg{t)bysin(27T.fct) represent linearformsofmodulation. Giventhatthecarrierfrequency .fcissufficiently large,the resulting band-pass function g{t)definedinEquation (A2AS)isreferredtoasapassband signaling waveform. Correspondingly, themapping fromgr(t) andgg{t)intog(t)isknown aspassband modulation. Equation (A2A4)istheCartesian formofexpressing thecomplex envelope g(t). Alternatively, wemayexpressitinthepolarform g(t)=a(t)exp[i<P(t)] (A2A6) wherea(t)andcf>(t)arebothreal-valued low-pass functions. Basedonthispolarrepresen tation,theoriginalband-pass signalg(t)isdefinedby g{t)=a{t)COS[27T.fct+cf>(t)] (A2A7) Werefertoa{t)asthenaturalenvelope orsimplytheenvelope oftheband-pass signalg(t) andtocf>(t)asthephaseofthesignal.Equation (A2A7)represents ahybridformof amplitude modulation andanglemodulation; indeed,itincludes amplitude modulation, frequency modulation, andphasemodulation asspecialcases. Fromthisdiscussion itisapparent that,whether werepresent aband-pass (modu lated)signalg{t)intermsofitsin-phase andquadrature components asinEquation (A2AS)orintermsofitsenvelope andphaseasinEquation (A2A7), theinformation contentofthesignalg{t)iscompletely preserved inthecomplex envelope g(t). (a) (b)+ FIGUREA2.6(a)Schemeforderiving thein-phase andquadrature components ofaband-pass signal.(b)Scheme forreconstructing theband-pass signalfromitsin-phase andquadrature components. 730 APPENDIX 2IIIREPRESENTATION OFSIGNALS ANDSYSTEMS IIITERMINOLOGY Thedistinctions amongthethreedifferent envelopes thatwehaveintroduced todescribe aband-pass signalg(t)shouldbecarefully noted.Wesummarize theirdefinitions here: 1.Thepre-envelope g+(t)forpositivefrequencies isdefinedby g+(t)=g(t)+jg(t) whereg(t)istheHilberttransform ofthesignalg(t).According tothisrepresentation, g(t)maybeviewedasthequadrature function ofg(t).Correspondingly, inthe£Ie: quencydomainwehave {2G(f),f>0 G+(f)=G(O),f=0 0,f<0 2.Thecomplex envelope g(t)equalsafrequency-shifted versionofthepre-envelope g+(t),asshownby g(t)=g+(t)exp(-j2Trj;t) wherej;isthecarrierfrequency oftheband-pass signalg(t). 3.Theenvelope a(t)equalsthe,magnitude ofthecomplex envelope g(t)andalsothat ofthepre-envelope g+(t),asshownby a(t)=Ig(t)I=Ig+(t)I Notethatforaband-pass signalg(t),thepre-envelope g+(t)isacomplex band-pass signal whosevaluedepends onthecarrierfrequency j;.Ontheotherhand,theenvelope a(t)is alwaysareallow-pass signaland,ingeneral,thecomplex envelope g(t)isacomplex low passsignal;thevaluesofthelattertwoenvelopes areindependent ofthechoiceofthe carrierfrequency j;.Thisproperty givesthecomplex envelope g(t)ananalyticadvantage overtheoriginalsignalg(t). Theenvelope a(t)andphaseq,(t)ofg(t)arerelatedtothequadrature components gI(t)andgQ(t)asfollows(seethetime-varying phasorrepresentation ofFigureA2.5a): a(t)=Vgy(t)+ib(t) q,(t)=tan-1(gQ(t)) gr(t) Conversely, wemaywrite gdt)=a(t)cos[q,(t)] gQ(t)=a(t)sin[q,(t)] Thus,eachofthequadrature components ofaband-pass signalcontains bothamplitude andphaseinformation. Bothcomponents arerequired forauniquedefinition ofthephase q,(t),modulo2Tr. illBAND-PASS SYSTEMS Nowthatweknowhowtohandlethecomplex low-pass representation ofband-pass signals,itislogicalthatwedevelopacorresponding procedure forhandling theanalysis ofband-pass systems. Specifically, wewishtoshowthattheanalysisofband-pass systems canbegreatlysimplified byestablishing ananalogy(or,moreprecisely, anisomorphism) A2.4Complex Representation ofSignalsandSystems 731 between low-pass andband-pass systems. ThisanalogyisbasedontheuseoftheHilbert transform fortherepresentation ofband-pass signals. Consider anarrowband signalx(t),withitsFouriertransform denotedbyX(f).We assumethatthespectrum ofthesignalx(t)islimitedtofrequencies within±WHzofthe carrierfrequency fc.Also,weassumethatW<fc.Letthissignalberepresented interms ofitsin-phase andquadrature components asfollows: x(t)=XI(t)COS(27Tfct) -xQ(t)sin(27Tfct) (A2AS) whereXI(t)isthein-phase component andxQ(t)isthequadrature component. Then,using x(t)todenotethecomplex envelope ofx(t),wemaywrite itt)=XI(t)+jxdt) (A2.49) Letthesignalx(t)beappliedtoalineartime-invariant band-pass systemwithimpulse response h(t)andfrequency response H(f).Weassumethatthefrequency response ofthe systemislimitedtofrequencies within±Bofthecarrierfrequency fc.Thesystemband width2Bisusuallynarrower thanorequaltotheinputsignalbandwidth 2W.Wewish torepresent theband-pass impulseresponse h(t)intermsoftwoquadrature components, denoted byhI(t)andhdt).Thus,byanalogytotherepresentation ofband-pass signals, wemayexpressh(t)intheform h(t)=hilt)COS(27Tfct) ~hQ(t)sin(27Tfct) Definethecomplex impulseresponse oftheband-pass systemas Hence,wehavethecomplex representation h(t)=Re[h(t)exp(j27Tfct)](A2.50) (Al.51) (A2.52) NotethathIlt),ha(t),andh(t)arealllow-pass functions limitedtothefrequency band -B$f$B. - Wemaydetermine thecomplex impulseresponse h(t)intermsofthequadrature components hI(t)andhQ(t)oftheband-pass impulseresponse h(t)byusingEquation (A2.51). Alternatively, wemaydetermine itfromtheband-pass frequency response H(f) inthefollowing way.WefirstnotefromEquation (A2.52)that 2h(t)=h(t)exp(j27Tfct) +h*(t)exp(-j27Tfct) (A2.53) whereh*(t)isthecomplex conjugate ofh(t).Therefore, applying theFouriertransform toEquation (A2.53), andusingthecomplex-conjugation property oftheFouriertrans form,whichisdescribed initem10inTableA6.2,weget 2H(f)=H(ffc)+W(-f fc) (A2.54) whereH(f)istheFouriertransform ofh(t),andR(f)istheFouriertransform ofh(t). Equation (AJ..54)satisfiestherequirement thatH*(f)=H(-f)forarealimpulseresponse h(t).SinceH(f)represents alow-pass frequency response limitedtoIfI$BwithB<fc, wededucefromEquation (A2.54)that H(f-fc)=2H(f), f>O (A2.55) Equation (A2.55)indicates thatforaspecified band-pass frequency response H(f),we maydetermine fI(f)bytakingthepartofH(f)corresponding topositivefrequencies, 732 APPENDIX 2'"REPRESENTATION OFSIGNALS ANDSYSTEMS shiftingittotheoriginandthenscalingitbythefactor2.Todetermine thecomplex impulseresponse h(t),wetaketheinverseFouriertransform ofH(f),obtaining h(t)=fooH(f)exp(j271'ft) df (A2.56) Therepresentations justdescribed forband-pass signalsandsystemsprovidethebasis ofanefficientmethodfordetermining theoutputofaband-pass systemdrivenbyaband passsignal.Weassumethatthespectrum oftheinputsignalx(t)andthefrequency re sponseH(f)ofthesystemarebothcentered aroundthesamefrequency fc.Inpractice, thereisnoneedtoconsider asituation inwhichthecarrierfrequency oftheinputsignal isnotalignedwiththemidband frequency oftheband-pass system,sincewehaveconsid erablefreedom inchoosing thecarrierormidband frequency. Thus,changing thecarrier frequency oftheinputsignalbyanamount!:ifc,say,simplycorresponds toabsorbing (or removing) thefactorexp(:<::j271' !:if~t)inthecomplex envelope oftheinputsignalorthe complex impulseresponse oftheband-pass system.Wearetherefore justifiedinproceeding ontheassumption thatX(f)andH(f)arebothcentered aroundfc.Suppose thenweuse y(t)todenotetheoutputsignalofthesystem.Itisclearthaty(t)isalsoaband-pass signal, sothatwemayrepresent itintermsofitslow-pass complex envelope y(t),asfollows: y(t)=Re[y(t)exp(j271'fct)] (A2.57) Theoutputsignaly(t)isrelatedtotheinputsignalx(t)andimpulseresponseh(t)of thesystemintheusualwaybythe'convolution integral y(t)=fooh(T)X(t T)dT (Al.58) (A2.59) (A2.61)Intermsofpre-envelopes, wehaveh(t)=Re[h+(t)] andx(t)=Re[x+(t)]. Wemaytherefore rewriteEquation (A2.58)intermsofthepre-envelopes x+(t)andh+(t)asfollows: y(t)=fooRe[h+(T)] Re[x+(t-T)]dT Toproceedfurther,wemakeuseofabasicproperty ofpre-envelopes thatisdescribed by thefollowing relation(presented herewithoutproof): fooRe[h+(T)] Re[x+(T)] dT=~Re[fooh+(T)X:(T) dT] (A2.60) wherewehaveusedTastheintegration variabletobeconsistent withthatinEquation (Al.59). Next,wenotethatusingX(-T)inplaceofX(T)hastheeffectofremoving the complex conjugation ontheright-hand sideofEquation (Al.60). Hence,bearinginmind thealgebraic difference between theargument ofX+(T)inEquation (A2.60)andthatof x+(t-T)inEquation (A2.59), andusingtherelationship between thepre-envelope and complex envelope ofaband-pass function, weget y(t)=~Re[fooh+(T)X+(t -T)dT] =~Re[fooh(T)exp(j271'fcT)X(t -T)exp(j271'fc(t -T))dT] =~Re[eXJ>(j271'fct)rooh(T)X(t-T)dT] (A2.62)A2.4Cmnplex Representation ofSignalsandSystems 733 Thuscomparing theright-hand sidesofEquations (A2.57)and(A2.61),wereadilydeduce thatforalargeenoughcarrierfrequencyfc,thecomplex envelope jilt)oftheoutputsignal i!,relatedtothecomplex envelope x(t)oftheinputsignalandthecomplex impulseresponse h(t)oftheband-pass systemasfollows: 2ji(t)=f=h(T)X(t-T)dT or,usingtheshorthand notation for convolution, 2ji(t)=h(t)*x(t) (A2.63) where*denotesconvolution. Inotherwords,exceptforthescalingfactor2,thecomplex envelope ji(t)oftheoutpuf.-signal ofaband-pass systemisobtained byconvolving the complex impulseresponse h(t)ofthesystemwiththecomplex envelope x(t)oftheinput band-pass signal.Equation (A2.63)istheresultoftheisomorphism, forconvolution, betweenaband-pass function andthecorresponding low-pass function. Thesignificance ofthisresultisthatindealingwithband-pass signalsandsystems, weneedonlyconcernourselves withthelow-pass functionsx(t), ji(t),andh(t),representing theexcitation, theresponse, andthesystem,respectively. Thatis,theanalysisofaband passsystem,whichiscomplicated bythepresence ofthemultiplying factorexp(j2'rrfct), isreplaced byanequivalent butmuchsimplerlow-pass analysisthatcompletely retains theessenceofthefilteringprocess.Thisprocedure isillustrated schematically inFigure A2.7. The_complex envelope x(t)oftheinputband-pass signalandthecomplex impulse response h(t)oftheband-pass systemaredefinedintermsoftheirrespective in-phase and quadrature components byEquations (A2.49)and(A2.51),respectively. Substituting these relations inEquation (A2.63), weget (A2.64) Becauseconvolution isdistributive, wemayrewriteEquation (A2.64)intheequivalent form 2ji(t)=[hIlt)*XI(t)-hdt)*xdt)]+j[hdt)*xr(t)+hI*xdt)](A2.65) Letthecomplex envelope ji(t)oftheresponse bedefinedintermsofitsin-phase and quadrature components as ji(t)=YI(t)+jYdt) (A2.66) Comparing therealandimaginary partsinEquations (A2.65)and(A2.66), wehavefor thein-phase component YI(t)therelation (A2.67) x(l)=Re[i(t)exp(j27TfctlJ h(t) (a)yet)=Re[jilt)exp(j27TfctlJ ;(1)~ 2y(t) ---~"~ " (b) FIGUREA2.7(a)Narrowband filterofimpulse response h(t)withnarrowband inputsignalx(t). (b)Equivalent low-pass filterofcomplex impulse responseh(t)withcomplex low-pass inputx(t). 734 APPENDIX 2OJ.REPRESENTA'fION OFSIGNALS ANDSYSTEMS FIGUREA2.8Blockdiagram illustrating therelationships between thein-phase andquadrature components oftheresponse ofaband-pass filterandthoseoftheinputsignal. andforthequadrature component YQ(t}therelation 2YQ(t}=hQ(t)*xdt)+h,(t}*xQ!t} (A2.68) Thus,forthepurposeofevaluating thein-phase andquadrature components ofthecom plexenvelope )itt)ofthesystemoutput,wemayusethelow-pass equivalent modelshown inFigureA2.8.Allthesignalsandimpulseresponses showninthismodelarereal-valued low-pass functions. Accordingly, thisequivalent modelprovides apractical basisforthe efficientsimulation ofband-pass filtersorcommunication channels onadigitalcomputer. Tosumup,theprocedure forevaluating theresponse ofaband-pass system(with mid-band frequency fc)toaninputband-pass signal(ofcarrierfrequency fc)isasfollows: 1.Theinputband-pass signalx(t)isreplaced byitscomplex envelopeitt},whichis relatedtox(t}by x(t}=Re[i(t}exp(j21Tfct)] 2.Theband-pass system,withimpulseresponse h(t),isreplaced byalow-pass analog, whichischaracterized byacomplex impulseresponse h(t)relatedtoh(t}by h(t)=Re[h(t)exp(j21Tfct}] 3.Thecomplex envelope jilt}oftheoutputband-pass signaly(t)isobtained bycon volvingh(t}withitt),asshownby 2)i(t)=h(t)*itt) 4.Thedesiredoutputy(t)isfinallyderivedfromthecomplex envelope )itt}byusing therelation y(t}=ReI)i(t)exp(j21Tfct)] BESSEL FLINCTIONS IA3.1SeriesSolution ofBessel'sEquotion Initsmostbasicform,Bessel'sequationoforderniswrittenas (A3.1) (A3.2)whichisoneofthemostimportant ofallvariable-coefficient differential equations.1For eachn,asolutionofthisequation isdefinedbythepowerseries 00(-l)m(~ xr+2m fn(x)=~om!(n+m)! Thefunctionfn{x) iscalledaBesselfunctionofthefirstkindofordern.Equation (A3.1) hastwocoefficient functions, namely,llxand(I-n2Ix2).Hence,ithasnofinitesingular pointsexcepttheorigin.Itfollowstherefore thattheseriesexpansion ofEquation (A3.2) converges forallx>O.Equation (A3.2)maythusbeusedtonumerically calculate fn(x) forn=0,1,2,....TableA6.5presentsvaluesoffn{x)fordifferent ordersnandvarying x.Itisofinteresttonotethatthegraphsoffo(x)andfl(X)resemble thegraphsofcosx andsinx,respectively; seethegraphsofFigure2.23inChapter2. Thefunctionfn(x) mayalsobeexpressed intheformofanintegralas 1f" fn{x)=~cos{xsin8-n8)d81r0 or,equivalently, fn(x)=~J"exp(jxsin0-jn8)dO21r-" IA3.2Properties oftheBesselFundion TheBesselfunction fn{x)hasthefollowing properties:(A3.3) (A3.4) 1. fn(x)=(-I)"f-n(x) (A3.5) Toprovethisrelation,wereplace (Jby(1r-8)inEquation (A3.3).Then,notingthat sin(1r-8)=sin8,weget 1f" fn{x)= -cos(xsin8+n8n1r)d81r0 =1.("[cos(n1r) cos(xsin8+n8)+sin(n1r)sin(xsin8+n8)]d8.1rJo 735 736 APPENDIX 3..BESSEL FUNCTIONS Forintegervaluesofn,wehave cos(mT) =(-It sin(mT)=0 Therefore, (-ltl" fn(x)=-- cos(xsin&+n&)d& 7T0 FromEquation (A3.3),wealsofindthatbyreplacing nwith-n: 11" f-n(x)= -cos(xsin&+n&)d& 7Ta Thedesiredresultfollowsimmediately fromEquations (A3.6)and(A3.7). 2. fn(x)=(-l)nfn(-x)(A3.6) (A3.7) (A3.8) Thisrelationisobtained byreplacing xwith-xinEquation (A3.3),andthenusing Equation (A.3.6). 3. (A3.9) Thisrecurrence formulaisusefulinconstructing tablesofBesselcoefficients; itsder ivationfollowsfromthepowerseriesofEquation (A3.2). 4.Forsmallvaluesofx,wehave (A3.10) Thisrelationisobtained simplybyretaining thefirstterminthepowerseriesof Equation (A3.2)andignoring thehigher-order terms.Thus,whenxissmall,wehave fa(x)=1 f,(X)=~2 fn(x)=0forn>1 5.Forlargevaluesofx,wehave(A3.ll) (A3.12) fn(x)=(2cos(x_~_n7T).j-;; 4 2 Thisshowsthatforlargevaluesofx,theBesselfunction fn(x)behaveslikeasine wavewithprogressively decreasing amplitude. 6.Withxrealandfixed,fn(x)approaches zeroastheorderngoestoinfinity. 7. 2:fn(x)exp(jntp) =exp(jxsin,p) (A3.13) Toprovethisproperty, consider thesum2:;;~-00fn(x)exp(jn,p) andusetheformula ofEquation (A3.4)forfn(x)toobtain 00 100J"2Lfn(x)exp(jn,p) =27T2Lexp(jn,p) _"exp(jxsin&-jn&)d& A3.3Modified BesselFunction 737 Interchanging theorderofintegration andsummation: 00 1fTC 002:In(x)exp(jnt/J) = - _ deexp(jxsine)2:exp[jn(t/J -e)] n=-oo 21r 7r n=-oo Wenowinvokethefollowing relationfromFouriertransform theory:(A3.14) 1 00 o(t/J)=27Tnl-exp[jn(t/J)], (A3.15) where5(t/J)isadeltafunction. Therefore, usingEquation (A3.15)in(A3.14)and thenapplying thesiftingproperty ofthedeltafunction, weget i:In(x)exp(jnt/J)=rexp(jxsine)o(t/J-eJde n~-oo =e;;(jxsint/J) whichisthedesiredresult. 8.00 2:J~(x)=1forallx (A3.16) (A3.17)Toprovethisproperty, wemayproceedasfollows.WeobservethatIn(X)isreal. Hence,multiplying Equation (A3.4)byitsowncomplex conjugate andsumming overallpossiblevaluesofn,weget hlterchanging theorderofdoubleintegration andsummation: 00 2:J~(x)= n=-oo 1fTCfTC. -(27Tf -rr-rrdedt/Jexp[jx(sin e-sint/J)]n~ooexp[jn(t/J -e)] UsingEquation (A3.15)in(A3.17)andthenapplying thesiftingproperty ofthedelta function, wefinallyget whichisthedesiredresult. Manyoftheseproperties oftheBesselfunctionIn(x)mayalsobeillustrated innu mericaltermsbyreferring toTableA6.5. l~~.!»odified BesselFunction Themodified Besselequationoforderniswrittenas 2d2ydy Xdx2+xdx(x2+n2)y=0 (A3.18) (A3.19)738 APPENDIX 3.,BESSEL FUNCTIONS WithF=-1,wherejisthesquarerootof-1,wemayrewritethisequation as d2ydy.x2-+X-+(lx2-n2)y=0dx2dx FromthisrewriteitisevidentthatEquation (A3.1S)isnothingbutBessel'sequation, namely,Equation (A3.1),withxreplaced byjx.Thusreplacing xbyjxinEquation (A3.2), weget =(_W(~r+2m In(jx)=];0 m!(n+m)! (~r+2m =j"2- m~Om!(n+m)! NextwenotethatIn(jx)multiplied byaconstant willstillbeasolutionofBessel'sequa tion.Accordingly, wemultiply In(jx)bytheconstant rn,obtaining ~Gx)"+2m rnIn(jx) =];0m!(n+m)! Thisnewfunction iscalledthemodified Besselfunctionofthefirstkindofordern,denoted byIn(x).Wemaythusformally expressasolutionofthemodified Besselequation, Equa tion(A3.1S), as In(x)=rnIn(jx) =Gx)"+2m ];0m!(n+m)! Themodified Besselfunction In(x)isamonotonically increasing realfunction ofthear gumentxforalln,asshowninFigureA3.1forn=0,1,2. 30 n=O .:; ~< c 0i20 ... 1 '"1l ~10 ~ x FIGUREA3.1Modified Besselfunction In(x)ofvaryingordern. NotesamlReferences 739 Themodified Besselfun(;tion In(x)isidenti(;al totheoriginalBesselfun(;tion] nix) ex(;eptforanimportant differen(;e: Thetermsintheseriesexpansion ofEquation (A3.19) areallpositive, whereastheyalternate insignintheseriesexpansion ofEquation (A3.2). Therelationship berween]n(x) andIn(x)isanalogous tothewayinwhi(;hthetrigonometri(; functions wsxandsinxarerelatedtothehyperbolk fun(;tions wshxandsinhx. Aninteresting property ofthemodified Besselfun(;tion In(x)isderivedfromEquation (A3.13). Spe(;ukally, repladng xbyjxandtheangle4>bye-'TT/2inthisequation, and theninvoking thedefinition ofIn(x)inthefirstlineofEquation (A3.19), weobtain 2:In(x)exp(jne) =exp(xwse) (A3.20) Fromthisrelationitfollowsthat (A3.21) 1I'" In(x)=2'TT_"exp(xws(})ws(ne)de ThisintegralformulaforIn(x)may,ofwurse,alsobederivedfromEquation (A3.4)by makingtheappropriate (;hanges. Whentheargument xissmall,weobtainthefollowing asymptotk estimates dire(;tly fromtheseriesrepresentation ofEquation (A3.19): Io(x) ---?1forx---?0 (A3.22) and forn2:1andx---?0 (A3.23) Forlargevaluesofxwehavethefollowing asymptotk estimateforIn(x),whkhisvalid forallintegersn2:0: (A3.24) forx---?00In(x)=exp(x) V27TX Notethatthisasymptoti(; behavior ofIn(x)isindependent oftheordernforlargevalues ofx. INO~ES ANDREFERENCES 1.Equation (A3.1)isnamedfortheGermanmathematician andastronomer FriedrichWilhelm Bessel(1784-1846). Fordetailedtreatments ofthesolutiontothisequation andrelated issues,seeWylieandBarrett(1982)andWatson(1966). CONFLUENT HYPERGEOMETRIC FUNCTIONS IA4.1Kummer's Equation Thec:onfluent hypergeometric: func:tion' isasolutionofKummer's differential equation: d2y dyx-+(b-x)- -ay=°dx2dx(A4.1) (A4.2)where,ingeneral,theparameters aandbarecomplex numbers. Forthecasewhen boF0,-1,-2,...,thesolutionofKummer's equation isdefinedbytheseries a xa(a+1)x2 ,F,(a'b·x)=1+ --+~~~- +..., , b11bIb+1)2! where,F,(a;b;x)denotesaconfluent hypergeometric function parameterized byaandb. Inthisnotation, thefirstsubscript denotesthenumberoffactorials inthenumerator of thegeneralterminEquation (A4.2),thesecondsubscript denotesthenumberoffactorials, apartfromnt,inthedenominator. InEquation (A4.2),bothsubscripts areclearly1. A4.2Properties oftheConfluent Hypergeometric Function Property 1 Forsmallvaluesofx,thec:onfluent hypergeometric: function approximates as a,F,(a;b;x)=1+bxforx......° (A4.3) Thisproperty followsdirectlyfromtheseriesexpansion ofEquation (A4.2). Property 2 Fora=-1andb=1wehavetheexactidentity: ,F,(-l;1;x)=1 -xforallx (A4.4) 740Thisproperty alsofollowsdirectlyfromtheseriesexpansion ofEquation (A4.2). NotesandReferences 741 Property 3 Theconfluent hypergeometric function fora=-112andb=1isrelatedexactlytothe modified Besselfunction forallxasfollows: (A4.5) whereIn(x)isthemodified Besselfunctionofordern. AspecialcaseofEquation (A4.5)occurswhenxislarge.Fromthedefinition ofthe modified Besselfunction giveninAppendix 3,wehavethefollowing asymptotic formula forlargex: forx_00 In(x)=exp(x)~ Hence,combining Equations (A4.5)and(A4.6),weobtainthesimpleresult(A4.6) INOTES ANDREFERENCESF(_.!.1·-x)=2~ 112''~;forx_00(A4.7) 1.Foradiscussion ofconfluent hypergeometric functions, seeJeffreysandJeffreys(1956). Tabulated valuesofthesefunctions arepresented inAbramowitz andStegun(1965). CRYPTOGRAPHY Secrecyiscertainly important tothesecurityorintegrity ofinformation transmission. Indeed,theneedforsecurecommunications ismoreprofound thanever,recognizing that theconductofmuchofourcommerce, business, andpersonal mattersisbeingcarriedout todaythrough themedium ofcomputers, whichhasreplaced thetraditional mediumof papers. Cryptology istheumbrella termusedtodescribethescienceofsecretcommunica tions;itisderivedfromtheGreekkryptosandlogoswhichmean"hidden" and"word," respectively.! Thesubjectmatterofcryptology maybepartitioned neatlyintocryptogra phyandcryptanalysis. Cryptography dealswiththetransformations ofamessage into codedformbyencryption andtherecovery oftheoriginalmessage bydecryption. The originalmessagetobeencrypted (enciphered) iscalledtheplaintext, andtheresultpro ducedbyencryption iscalledacryptogram orciphertext; thelattertwotermsareused interchangeably. Thesetofdatatransformations usedtodotheencryption iscalleda cipher;normally, thetransformations areparameterized byoneormorekeys.Cryptanal ysis,ontheotherhand,dealswithhowtoundocryptographic communications bybreak ingacipherorforgingcodedsignalsthatmaybeaccepted asgenuine. Cryptographic systemsofferthreeimportant services: 1.Secrecy,whichreferstothedenialofaccesstoinformation byunauthorized users. 2.Authenticity, whichreferstothevalidation ofthesourceofamessage. 3.Integrity, whichreferstotheassurance thatamessagewasnotmodified byaccidental ordeliberate meansintransit. Aconventional cryptographic systemreliesontheuseofasinglepieceofprivateand necessarily secretinformation knownasthekey;hence,conventional cryptography isre ferredtoassingle-key cryptography orsecret-key cryptography.2 Thisformofcryptog raphyoperates onthepremisethatthekeyisknowntotheencrypter (sender)andbythe decrypter (receiver) buttonoothers;theassumption isthatoncethemessage isencrypted, itis(probably) impossible todothedecryption withoutknowledge ofthekey. Public-key cryptography,3 alsocalledtwo-key cryptography, diHersfromconven tionalcryptography inthatthereisnolongerasinglesecretkeysharedbytwousers. Rather,eachuserisprovided withkeymaterial ofone'sown,andthekeymaterial is dividedintotwoportions: apubliccomponent andaprivatecomponent. Thepubliccom ponentgenerates apublictransformation, andtheprivatecomponent generates aprivate transformation. But,ofcourse,theprivatetransformation mustbekeptsecretforsecure communication betweenthetwousers. IA5.1Secret-Key Cryptography Basically, theflowofinformation inasecret-key cryptographic systemisasshownin FigureAS.1.Themessage sourcegenerates aplaintext message, whichisencrypted intoa cryptogram atthetransmitting endofthesystem.Thecryptogram issenttoanauthorized useratthereceiving endoveran"insecure" channel; achannel isconsidered insecureif 742 A5.1Secret-Key Cryptography 743 Enemy MessageX FIGUREA5.1Blockdiagramofsecret-key cryptographic system. itssecurityisinadequate fortheneedsofitsusers.Itisassumed thatinthecourseof transmission thecryptogram maybeintercepted byanenemycryptanalyst4(i.e.,would beintruderintoacryptographic system).Therequirement istodotheencryption insuch awaythattheenemyisprevented fromlearningthecontents oftheplaintext message. Inabstractterms,acryptographic systemorcipher(forshort)isdefinedasasetof invertible transformations oftheplaintext space(i.e.,thesetofpossibleplaintext messages) intothecryptogram space(i.e.,thesetofallpossiblecryptograms). Eachparticular trans formation corresponds toencryption (enciphering) ofaplaintext withaparticular key. Theinvertibility ofthetransformation meansthatuniquedecryption (deciphering) ofthe cryptogram ispossible whenthekeyisknown.LetXdenotetheplaintext message, Y denotethecryptogram, andZdenote the key.LetFdenotetheinvertible transformation producing thecryptogram Y,asfollows: Y=F(X,Z)=FzlX) (AS.!) Thetransformation isintended tomakethecryptogram Yuselesstotheenemy.Atthe receiving endofthesystem,thecryptogram Yisdecrypted withtheinversetransformation F-1torecovertheoriginalplaintext messageX,asshownby (AS.2) Inphysicalterms,thecryptographic systemconsistsofasetofinstructions, apiece ofphysicalhardware, oracomputer program.Inanyevent,thesystemisdesigned tohave thecapability ofencrypting theplaintext (and,ofcourse,decrypting theresulting cryp togram) inavarietyofways;theparticular waychosentodotheactualencryption is determined bythespecifickey. Thesecurityofthesystemresidesinthesecretnatureofthekey,whichrequiresthat thekeymustbedelivered tothereceiveroverasecurechannel(e.g.,registered mail,courier service)asimpliedinFigureAS.1.Thecryptographic systemdepictedinthisfigureprovides asolutiontothesecrecyproblem, preventing anenemyfromextracting information from messages transmitted overaninsecure communication channel. Cryptography alsopro videsasolution totheauthentication problem, preventing anenemycryptanalyst from impersonating themessagesender.Inthissecondsituation, theenemycryptanalyst isthe onewhooriginates a"fraudulent" cryptogram Y'thatisdelivered tothereceiver(decryp ter),asshowninFigureAS.2.Theauthentic cryptogram Yisshownasadashedinputto theenemycryptanalyst, indicating thattheenemyproduces thefraudulent cryptogram Y' withouteverseeingtheauthentic one.Thereceivermaybeabletorecognize Y'asfraud ulentbydecrypting itwiththecorrectkeyZ;hence,thelinefromthereceiveroutputto thedestination isshowndashedtosuggestrejection ofthefraudulent cryptogram Y'by thereceiving user. 744 APPENDIX 5IIICRYPTOGRAPHY -~Destination FIGUREA5.2Illustrating theintrusion ofanenemycryptanalyst. IA5.2BlockandStreamCiphers Muchaserror-correcting codesareclassified intoblockcodesandconvolutional codes, cryptographic systems(ciphers) maybeclassified intotwobroadclasses:blockciphersand streamciphers.Blockciphersoperateinapurelycombinatorial fashiononlargeblocksof plaintext, whereasstreamciphersprocesstheplaintext insmallpieces(i.e.,characters or bits). FigureAS.3showsthegenericformofablockcipher.Theplaintext (c~nsisting of serialdata)isdividedintolargeblocks,eachofwhichisusuallymadeupofafixednumber ofbits.Successive blocksoftheplaintext areenciphered (encrypted) usingthesamesecret key,otherwise independently; theresulting enciphered blocksarefinallyconverted into serialform.Thus,aparticular plaintext blockidentical toaprevious suchblockgivesrise toanidentical ciphertext block.Specifically, eachbitofaparticular ciphered blockis chosentobeafunctionofallthebitsoftheassociated plaintext blockandthekey;the goalofablockcipheristohavenospecificbitoftheplaintext everappearintheciphertext directly. Blockciphersoperatewithafixedtransformation appliedtolargeblocksofplaintext data,onablock-by-block basis.Incontrast, astreamcipheroperates onthebasisofa time-varying transformation appliedtoindividual bitsoftheplaintext. Themostpopular streamciphersaretheso-called binaryadditivestreamciphers,thegenericform ofwhich isshowninFigureASA.Insuchacipher,thesecretkeyisusedtocontrolakeystream generator thatemitsabinarysequence calledthekeystream, whoselengthismuchlarger thanthatofthekey.LetXmYmandZndenotetheplaintext bit,ciphertext bit,andkey streambitattimen,respectively. Theciphertext bitsarethendetermined bysimplemod ulo-2addition oftheplaintext bitsandthekeystream bits,asshownby n=1,2,..., N (AS.3) whereNisthelengthofthekeystream. Becauseaddition andsubtraction inmodulo-2 arithmetic areexactlythesame,Equation (AS.3)alsoimpliesthefollowing relation n=1,2,..., N (ASA) Plainte>ct inserial form Key FIGUREA5.3Blockdiagramofablockcipher.Ciphertext A5.2BlockandStreamCiphers 745 Key bitsKey bits Keystream Plaintexto--i3>H-j---;'" Ciphertext YnY, Ciphertext o-~t-t-j---;"'Plajntext Xn Encrypter Decrypter FIGUREA5.4Binaryadditivestreamcipher. Wethusseethatinbinaryadditivestreamciphers,identical devicescanbeusedtoperform encryption anddecryption, asshowninFigureA5.4.Thesecretkeyischosenaccording tosomeprobability distribution. Toprovidesecureencryption, thekeystream shouldre sembleacoin-tossing (i.e.,completely random) sequence ascloselyaspossible. Blockciphersarenormally designed insuchawaythatasmallchangeinaninput blockofplaintext produces amajorchangeintheresulting output.Thiserrorpropagation property ofblockciphersisvaluable inauthentication inthatitmakesitimprobable for anenemycryptanalyst tomodifyencrypted data,unlessknowledge ofthekeyisavailable. Ontheotherhand,abinaryadditivestreamcipherhasnoerrorpropagation; thedecryp tionofadistorted bitintheciphertext affectsonlythecorresponding bitoftheresulting output. Streamciphersaregenerally bettersuitedforthesecuretransmission ofdataover error-prone communication channels; theyareusedinapplications wherehighdatarates arearequirement (asinsecurevideo,forexample) orwhenaminimaltransmission delay isessentiaL-' REQUIREMENT FORSECRECY Incryptography, afundamental assumption isthatanenemycryptanalyst hasknowledge oftheentiremechanism usedtoperform encryption, exceptforthesecretkey.Wemay identifythefollowing formsofattackthatmaybeattempted bytheenemycryptanalyst, depending ontheavailability ofadditional knowledge: 1.Ciphertext-only attackisacryptanalytic attackinwhichtheenemycryptanalyst has accesstopartoralloftheciphertext. 2.Known-plaintext attackisacryptanalytic attackinwhichtheenemycryptanalyst hasknowledge ofsomeciphertext-plaintext pairsformedwiththeactualsecretkey. 3.Chosen-plaintext attackisacryptanalytic attackinwhichtheenemycryptanalyst is abletosubmitanychosenplaintext message andreceiveinreturnthecorrect ciphertext fortheactualsecretkey. 4.Chosen-ciphertext attackisacryptanalytic attackinwhichtheenemycryptanalyst isabletochooseanarbitrary ciphertext andfindthecorrectresultforitsdecryption. Aciphertext-only attackoccursfrequently inpractice.Inthisformofattack,an enemycryptanalyst usesonlyknowledge ofthestatistical sttucture ofthelanguage inuse (e.g.,inEnglishthelettereoccurswithaprobability of13percent, andtheletterqis alwaysfollowed byu)andknowledge ofsomeprobable words(e.g.,aletterprobably beginswith"DearSir/Madam:"). Aknown-plaintext attackmaytakeplacebyvirtueof thestandard computer formatsusedinprogramming languages anddatageneration. In anycase,theciphertext-only attackisviewedastheweakestthreattowhichacrypto- 746 APPENDIX 5"CRYPTOGRAPHY graphicsystemcanbesubjected, andanysystemthatsuccumbs toitistherefore considered totallyinsecure. Thus,foracryptographic systemtoprovidesecrecy,attheminimum it shouldbeimmunetociphertext-only attacks;ideally,itshouldalsobeimmunetoknown plaintext attacks. IA5.3Informatian-TheoreticApproach IntheShannon modelofcryptography, namedinrecognition ofShannon's 1949landmark paperontheinformation-theoretic approach tosecrecysystems,theenemycryptanalyst isassumed tohaveunlimited timeandcomputing power.Butthe enemy ispresumably restricted toaciphertext-only attack.Cryptanalysis intheShannon modelisdefinedasthe processoffindingthesecretkey,giventhecryptogram (ciphertext) andtheaprioriprob abilitiesofthevariousplaintexts andkeys.Thesecrecyofthesystemisconsidered broken whentheenemycryptanalyst performs decryption successfully, obtaining auniquesolution tothecryptogram.6 LetX=(Xl>X2,•••,XN)denoteanN-bitplaintext message, andY=(YhY2,•••, YN)denotethecorresponding N-bitcryptogram; thatis,boththeplaintext andthecryp togramhavethesamenumberofbits.Itisassumed thatthesecretkeyZusedtoconstruct thecryptogram isdrawnaccording tosomeprobability distribution. Theuncertainty about Xisexpressed bytheentropyH(X),andtheuncertainty aboutXgivenknowledge ofY isexpressed bytheconditional entropyH(XIY).Themutualinformation betweenXand Yisdefinedby I(X;Y)=H(X)-H(XIY) (AS.5) Themutualinformation I(X;Y)represents abasicmeasure ofsecurity(secrecy) inthe Shannon model. iiiPERFECT SECURITY Assuming thatanenemycryptanalyst canobserveonlythecryptogram Y,itseemsappro priatethatwedefinetheperfectsecurityofacryptographic systemtomeanthattheplain textXandthecryptogram Yarestatistically independent. Inotherwords,wehave I(X;Y)=0 (AS.6) Then,usingEquation (AS.S),wefindthatthecondition forperfectsecuritymaybere writtenas H(XIY)=H(X) (AS.?) (AS.8)Equation (AS.?)statesthatthebestanenemycryptanalyst cando,giventhecryptogram Y,istoguesstheplaintext message Xaccording totheprobability distribution ofall possiblemessages. GiventhesecretkeyZ,werecognize that H(XIY):5 H(X,ZIY) =H(ZIY)+H(XIY,Z) Theconditional entropyH(XIY,Z)iszeroif,andonlyif,YandZtogether uniquely determine X;thisisindeedavalidassumption whenthedecryption processisperformed withknowledge ofthesecretkeyZ.Hence,wemaysimplifyEquation (AS.8)asfollows: H(XIY) :5H(ZIY) :5H(Z)(AS.9) A5.3InfornuJtion- Theoretic Approach 747 Thus,substituting Equation (AS.9)into(AS.?),wefindthatfor acryptographic systemto provideperfectsecurity,thefollowing condition mustbesatisfied: H(Z)2:H(X) (AS.I0) Theinequality ofEquation (AS.I0)isShannon's fundamental boundforperfectsecurity; itstatesthatforperfectsecurity, theuncertainty ofasecretkeyZmustbeatleastaslarge astheuncertainty oftheplaintextXthatisconcealed bythekey. Forthecasewhentheplaintext andkeyalphabets areofthesamesize,theuseof Shannon's boundforperfectsecurityyieldsthefollowing result:Thekeymustbeatleast aslongastheplaintext. Theconclusion tobedrawnfromthisresultisthatthelengthof thesecretkeyneededtobuildaperfectly securecryptographic systemmaybeimpractically largeformostapplications. Nevertheless, perfectsecurityhasaplaceinthepractical pic ture:Itmaybeusedwhenthenumberofpossiblemessages issmallorincaseswherethe greatestimportance isattached toperfectsecurity. Awell-known, perfectly securecipheristheone-time pad7(sometimes calledthe Vernamcipher),whichisusedforunconventional applications suchastwouserscom municating onahotlinewithhighconfidentiality requirements. Theone-time padisa streamcipherforwhichthekeyisthesameasthekeystream, asshowninFigureAS.5. Forencryption theinputconsistsoftwocomponents: amessagerepresented byasequence ofmessage bits[xnIn=1,2,...j,andakeyrepresented byasequence ofstatistically independent anduniformly distributed bits[znln=1,2,...j.Theresultant cipher [YnIn=1,2,...jisobtained bythemodulo-2 addition ofthetwoinputsequences, as shownby n=1,2,... Consider, forexample, thebinarymessage sequence 00011010 andthebinarykeyse quence01101001. Themodulo-2 addition ofthesetwosequences iswrittenasfollows: Message: Key: Cipher:00011010 01101001 01110011 Intheencryption ruledescribed here,keybit1interchanges Osand1sinthemessage sequence, andkeybit0leavesthemessage bitsunchanged. Themessagesequence isre coveredsimplybymodulo-2 addition ofthebinarycipherandkeysequences, asshown by Cipher: Key: Message:01110011 01101001 00011010 Theone-time padisperfectly secure,becausethemutualinformation betweenthemessage andthecipheriszero;itistherefore completely undecipherable. ~Ke:" Message .Cipher Xn Yn EncrypterrikKe~" Cipher Message Yn xn Decrypter FIGUREA5.5One-time pad(VernaIn cipher). (AS.11) (AS.12)748 APPENDIX 5'"CRYPTOGRAPHY 111lJNICnY DISTANCE Consider nowthepractical caseofanimperfect cipherandaskthequestion: Whencan anenemycryptanalyst breakthecipher?Astheamountofintercepted textincreases, intuitively weexpectthatapointmaybereachedatwhichitbecomes possible for anenemycryptanalyst withunlimited timeandcomputing powertofindthekeyand thusbreakthecipher.ThiscriticalpointintheShannon modeliscalledtheunicitydis tance,whichisformally definedasthesmallest Nsuchthattheconditional entropy H(ZIY"Y2, •••,YN)isapproximately zero.Foraparticular kindof"random cipher," theunicitydistance isapproximately givenby' No""H(Z) rlogLy whereH(Z)istheentropyofthekeyZ,andLyisthesizeoftheciphertext alphabet. The parameter risthepercentage redundancy ofthemessage information contained inthe N-bitciphertext; itisitselfdefinedby r=1 _H(X) NlogLy whereH(X)istheentropyoftheplaintext X.Inmostcryptographic systems, thesizeL, oftheciphertext alphabet isthesameasthesizeLxoftheplaintext alphabet; insucha case,risjustthepercentage redundancy oftheplaintext itself.Although thederivation of Equation (AS.11)assumesacertainwell-defined "random cipher,"itcanbeusedtoesti matetheunicitydistanceforordinary typesofciphers,whichistheroutinepracticetoday. LetKbethenumberofdigitsinthekeyZthatarechosen from analphabet ofsize L.;thenwemayexpresstheentropyofthekeyZasfollows: H(Z)~10g(L~) =KlogLz (AS.B) withequalityifandonlyifthekeyiscompletely random.LetthesizeLzofthekeyalphabet bethesameasthesizeLyoftheciphertext alphabet, andletthekeybechosencompletely atrandomtomaximize theunicitydistance. Then,substituting Equation (AS.B)with equalityintoEquation (AS.11),wegetthesimpleresult KNo=r(AS.14) Toillustrate theapplication ofEquation (AS.14), consider acryptographic systemwith Lx=L,=Lz>whichisusedfortheencryption ofEnglishtext.Thepercentage redundancy rfortypicalEnglishtextisabout7Spercent.Hence,according toEquation (AS.14),an enemycryptanalyst canbreakthecipherafterintercepting onlyabout1.333Kbitsof ciphertext data,whereKisthekeysize. However, itisimportant tonotethatanimperfect cipherthatispotentially breakable canstillbeofpractical value.Whentheintercepted ciphetext contains sufficient infor mationtosatisfyEquation (AS.II),thereisnoguarantee thatanenemycryptanalyst with limitedcomputational resources canactuallybreakthecipher.Specifically, itispossible fortheciphertobedesigned insuchawaythatthetaskofthecryptanalysis, though knowntobeattainable withafiniteamountofcomputation, issooverwhelming thatit willliterallyexhaustthephysicalcomputing resources oftheuniverse. Insuchacase,the imperfect cipherissaidtobecomputationally secure. A5.3Information-TheoreticApproach 749 ROLEOFDATACOMPRESSION INCRYPTOGRAPHY Lossless datacompression ordatacompaction isausefultoolincryptography. Wesay thisbecausedatacompaction removesredundancy, therebyincreasing theunicitydistance Noinaccordance withEquation (AS.il). Toexploitthisidea,datacompaction isused priortoencryption inthetransmitter, andtheredundant information isreinserted after decryption inthereceiver; thenetresultisthattheauthorized useratthereceiveroutput seesnodifference, andyettheinformation transmission hasbeenmademoresecure.It wouldbetempting toconsider theuseofperfectdatacompaction toremoveallredun dancy,therebytransforming amessage sourceintoacompletely randomsourceandre sultinginNo=00withanykeysize.Unfortunately, wedonothaveadevicecapableof performing perfectdatacompaction onrealisticmessagesources,norisitlikelythatthere willeverbesuchadevice.Itistherefore futiletorelyondatacompaction alonefordata security. Nevertheless, limiteddatacompaction tendstoincrease security, whichisthe reasonwhycryptographers viewdatacompression asausefultrick. I!lDIFFUSION ANDCONFUSION IntheShannon modelofcryptography, twomethods suggestthemselves asgeneralprin ciplestoguidethedesignofpractical ciphers.Themethods arecalleddiffusion andcon fusion,theaimsofwhich(bythemselves ortogether) aretofrustrate astatistical analysis ofciphertext bytheenemyandtherefore makeitextremely difficulttobreakthecipher. Inthemethodofdiffusion, thestatistical structure oftheplaintext ishiddenby spreading outtheinfluence ofasinglebitintheplaintext overalargenumberofbitsin theciphertext. Thisspreading hastheeffectofforcingtheenemytointercept atremendous amountofmaterial forthedetermination ofthestatistical structure oftheplaintext, since thestructure isevidentonlyinmanyblocks,eachoneofwhichhasaverysmallprobability ofoccurrence. Inthemethodofconfusion, thedatatransformations aredesigned tocom plicatethedetermination ofthewayinwhichthestatistics oftheciphertext dependonthe statistics oftheplaintext. Thus,agoodcipherusesacombination ofdiffusion and confusion. Foraciphertobeofpractical value,however, itmustnotonlybedifficulttobreak thecipherbyanenemycryptanalyst, butalsoitshouldbeeasytoencryptanddecrypt datagivenknowledge ofthesecretkey.Wemaysatisfythesetwodesignobjectives using aproductcipher,basedonthenotionof"divideandconquer." Specifically, theimple mentation ofastrongcipherisaccomplished asasuccession ofsimplecomponent ciphers, eachofwhichcontributes amodestamountofdiffusion andconfusion totheoverall makeupofthecipher.Productciphersareoftenbuiltusingsubstitution ciphersandtrans position ciphersasbasiccomponents; thesesimpleciphersaredescribed next. 1.Substitution cipher. Inasubstitution ciphereachletteroftheplaintext isreplaced byafixedsubstitute, usually alsoaletterfromthesamealphabet, withtheparticular substitution rulebeingdetermined bythesecretkey.Thustheplaintext x=(Xl>X2' X3'X4,•..) whereXl>X2, X3,•..arethesuccessive letters,istransformed intotheciphertext y(Y"Y2'Y3' Y4'...) =(f(x,),f(x2), f(x3), f(x4),...)(AS.IS) 750 APPENDIX 5IIICRYPTOGRAPHY Plaintext letters Ciphertext lettersABCDEFGHIJKLMNDPQRSTUVWXYZ YDUBHNACSVXELPFMKQJRWGOZIT FIGUREA5.6Substitution cipher. wheref(·)isafunction withaninverse.Whenthesubstitutes areletters,thekeyisa permutation ofthealphabet. Consider, forexample, theciphertext alphabet ofFigure AS.6,whereweseethatthefirstletterYisthesubstitute forA,thesecondletterDisthe substitute forB,andsoon.Theuseofasubstitution cipherresultsinconfusion. 2.Transposition cipher.Inatransposition cipher,theplaintext isdividedintogroupsof fixedperioddandthesamepermutation isappliedtoeachgroup,withtheparticular permutation rulebeingdetermined bythesecretkey.Forexample, consider thepermu tationruledescribed inFigureAS.7,forwhichtheperiodisd=4.According tothis cipher,letterX,ismovedfromposition 1intheplaintext toposition 4intheciphertext. Thus,theplaintext istransformed intotheciphertext Although thesingle-letter statistics oftheciphertext Yarethesameasthoseoftheplaintext X,thehigher-order statistics arechanged. Theuseofatransposition cipherresultsin diffusion. Byinterleaving thesimplesubstitutions andtranspositions andrepeating theinterleaving processmanytimes,itispossible tobuildastrongcipherequipped withgooddiffusion andconfusion. ~EXAMPLE A5.l Considertheplaintextmessage THEKINGISDEADLONGLIVETHEKING Usingthepermuted alphabetdescribeclin FigureAS.6forthesubstitution cipher,thisplaintext istransformed intotheciphertext RCHXSPASJBHYBEFP AESGHRCHXSPA Supposenextweapplythepermutation ruledescribed inFigureAS.7forthetransposition cipher;accordingly, theciphertext resultingfromthesubstitution cipherisfurthertransformed into HXCRASPSHYBJFBEBSGEACHRHPASX whichhasnoresemblance totheoriginalplaintext. PlaintextXlX,X,X4letters Ciphertextx3x4"Xlletters FIGUREA5.7 Transposition cipher. A5.4DataEncryption Standard 751 IA5.4DatuEncryption Stundard Thedataenc,yption standa,d (DES)9iscertainly thebestknown,andarguably themost widelyused,secret-key cryptoalgorithm; thetermalgo,ithm isusedtodescribeasequence ofcomputations. ThebasicDESalgorithm canbeusedforbothdataencryption anddata authentication. Itisthestandard cryptoalgorithm fordatastorageandmailsystems,elec tronicfundstransfers (retailandwholesale), andelectronic businessdatainterchange. TheDESalgorithm isastrongblockcipherthatoperateson64-bitblocksofplaintext dataandusesaS6-bitkey;itisdesigned inaccordance withShannon's methods ofdif fusionandconfusion. Essentially thesamealgorithm isusedforencryption anddecryption. Theoveralltransformations employed intheDESalgorithm maybewrittenas P-1{F[P(X)j}, whereXistheplaintext, Pisacertainpermutation, andthefunctionF combines substitutions andtranspositions. Thefunction Fisitselfobtained bycascading acertainfunctionf,witheachstageofthecascadereferredtoasa,ound. Theflow-chart ofFigureAS.8showsthedetailsoftheDESalgorithm forencryption. Afteracertaininitialpermutation, aplaintext of64bitsisdividedintoaleft-halfLoand aright-half Ro,eachofwhichis32bitslong.Thealgorithm thenperforms 16roundsof akey-dependent computation, withtheithroundofthecomputation described asfollows: Li=Ri-1 Ri=Li-1®f(Ri-bZi)i=1,2, ,16 i=1,2, ,16(AS.16) (AS.l?) Ontheright-hand sideofEquation (AS.1?), theaddition ismodulo-2 andeachZiisa different 48-bitblockofthekeyusedinroundi.Thefunction f(',.)isafunction witha 32-bitoutput.Theresultofthe16throundisreversed, obtaining thesequence R,6L,6. This32-bitsequence isinputintoafinalpermutation p-1toproducethe64-bitciphertext. Theaimisthatafter16roundsofkey-dependent computations, thepatternsintheoriginal plaintext areundetectable intheciphertext. FromEquations (AS.16)and(AS.1?),wenote thatfordecryption thefunction f(-,.)neednotbeinvertible, because(Li-bRi-1)canbe recovered from(Li,Ri)simplyasfollows: Ri-1=Li Li-1=Ri®f(Li,Zi)i=1,2, ,16 i=1,2,,16(AS.18) (AS.19) Equation (AS.19)holdsevenifthefunctionf(',.)isamany-to-one function (i.e.,itdoes nothaveauniqueinverse). FigureAS.9showstheflowchart forcomputing thefunction f(',.).The32-bitblock Risfirstexpanded intoanew48-bitblockR'byrepeating theedgebitsofeachsuccessive 4-bitword(Le.,thebitsnumbered 1,4,S,8,9,12, 13,16,...,28,29,32).Thus,given the32-bitblockRwrittenas R='1r2'3'4 '---y-----J first 4-bitwordrs"6r7ra '---y-----J second 4-bitword'29'30'31 '32 ei!ihth 4-bitword weconstruct theexpanded 48-bitblockR'asfollows: R'='32'1'2'3'4'5 fi~st 6-bitword'4'S'6'7'8'9'----.,-------- second 6-bitword'28'29'30'3,'32', eighth 6-bitword 752FIGUREA5.8Dataencryption standard. (FromDiffieandHelhnan, 1979,withpermission of theIEEE.) A5.4DataEncryption Standard 753 FIGUREA5.9f(R,K)flowchart. (FromDiffieandHellman, 1979,withpermission ofthe IEEE.) The48-bitblocksR'andZ;areaddedmodulo-2, andtheresultant isdividedintoeight 6-bitwords.LetthesewordsbedenotedbyB"B2,•••,BB'Wethuswrite (AS.20) Each6-bitwordBiisinputtoasubstitution boxSiintheformofalook-uptable,pro ducinga4-bitoutputSi(B;).Eachoutputbitofthesubstitution boxSi(Bi)isaBoolean function ofthe6-bitwordBi.TheeightoutputsS,(B,),S2(B2),•••,Sg(Bg)arearranged intoasingle32-bitblockthatisinputtothepermutation boxdenotedbyP[·].Theper mutedoutputsoproduced isthedesired32-bitfunction f(R,Zi),asshownby (AS.21) The48-bitblockZifortheithiteration usesadifferent subsetofthe64-bitkeyZoo Theprocedure usedtodetermine eachZiiscalledthekey-schedule calculation, theflow chartofwhichisshowninFigureAS.10.ThekeyZahaseightparitybitsinpositions 8, 16,...,64,whichareusedforerrordetection intheirrespective 8-bitbytes;theerrors 754 APPENDIX 5'"CRYPTOGRAPHY Shiftregisters FIGUREA5.10 Flowchart forthekey-schedule calculation. (FromDiffieandHellman, 1979, withpermission ofthelEEK) ofconcernmayariseinthegeneration, distribution, andstorageofthekeyZooTheper mutedchoice1disregards theparitybitsofZoandthenpermutes theremaining 56bits thatareloadedintotwo28-bitshiftregisters, eachwith24taps.The48tapsofthetwo shiftregistersaresubjectedto16iterations ofcomputation, witheachiterationinvolving oneortwocyclicleftshiftsfollowed byapermutation, referredtoaspermuted choice2. A5.5Public-Key Cryptography 755 Theoutputsresulting fromthese16iterations providethedifferent 48-bitblocks2" 22,•••,216ofthekeyusediniteration1,2,...,16,respectively. Despitealltheclaimstothecontrary, itappearsthatnoonehasyetdemonstrated a fundamental weakness oftheDESalgorithm. Notwithstanding allthecontroversy sur rounding itsuse,perhapsthemostsignificant contribution oftheDESalgorithm isthefact thatithasbeeninstrumental inraisingthelevelofinterestinusingcryptography asa mechanism forsecurecomputer networks. IA5.5Public-Key Cryptograph-ylO Forapairofuserstoengageincryptographic communication overaninsecurechannel, itisnecessary fortheuserstoexchange keyinformation priortocommunication. The requirement forasecuredistribution ofkeysamongauthorized usersappliestoallcryp tographic systems,regardless oftheirtype.Inconventional cryptography, theusersemploy aphysically securechannel(e.g.,courierserviceorregistered mail)forkeydistribution. However, theuseofsuchasupplementary channelpointstoamajorlimitation ofcon ventional cryptography. Needless tosay,theuseofcourierserviceorregistered mailfor keydistribution iscostly,inconvenient, low-bandwidth, andslow;also,itisnotalways secure. Theproblem ofkeydistribution isparticularly accentuated inlargecommunication networks, wherethenumberofpossibleconnections growsas(n2-n)/2fornusers.For largen,thecostofkeydistribution becomesprohibitive. Thus,inthedevelopment oflarge, securecommunication networks, wearecompelled torelyontheuseofinsecurechannels forbothexchange ofkeyinformation andsubsequent securecommunication. Thiscon straintraisesafundamental question: Howcankeyinformation beexchanged securely overaninsecurechannel? Inpublic-key cryptography, thisseemingly difficultissueisre solvedbymakingsomekeymaterial "public" andtherebyconsiderably simplifying the taskofkeymanagement. Thisisindirectcontrasttoconventional cryptography, where thekeyiskeptcompletely secretfromanenemycryptanalyst. Apublickeycryptographic systemisdescribed bytwosetsofalgorithms thatcom puteinvertible functions (transformations). Letthesetwosetsofalgorithms bedenotedby [E.!and{D.lthatareindexedbyz.Theinvertible transformations computed bythese algorithms maybewrittenasfollows E.:f.(x)=y (A5.22) D.:r;l(y)=x (A5.23) wherexisacertaininputmessageinthedomainofsomefunctionf.indexedbyz,andy isthecorresponding cryptogram intherangeoff•.Afundamental requirement ofthe systemisthatthefunctionf.mustbeatrapdoor one-way function. Theterm"one-way" referstothefactthatforxinthedomainoff.,itmustbeeasytocompute f.(x)from knowledge ofthealgorithm E.,butforacertaincryptogram yintherangeoff.,anenemy cryptanalyst mustfinditextremely difficulttocompute theinversef;l(y).Ontheother hand,anauthorized userinpossession oftheassociated algorithm D.wouldfinditeasy tocompute theinverser;l(y).Thustheprivatekey(algorithm) D.provides a"trapdoor" thatmakestheproblem ofinverting thefunctionf.appearextremely difficultfromthe viewpoint ofthecryptanalyst, buteasyforthe(soleauthorized) possessor ofD•.Since knowledge ofthekey(algorithm) E.doesnotbyitselfmakeitpossibletocompute the inverseoff.,itmaybemadepublic;hence,thename"public-key cryptography." 756 APPENDIX 5..CRYPTOGRAPHY Thenotionemerging fromthedescription ofapublic-key cryptographic systempre sentedhereinisthatthekeyscomeininversepairs(i.e.,publickeyandprivatekey),and thateachpairofkeyshastwobasicproperties: 1.Whatever messageisencrypted withoneofthekeyscanbedecrypted withtheother key. 2.Givenknowledge ofthepublickey,itiscomputationally infeasible tofindthesecret key. Theuseofpublic-key cryptography asdescribed hereinmakesitpossibletosolve thesecrecyproblem asfollows.Subscribers toasecurecommunication systemlisttheir publickeysina"telephone directory" alongwiththeirnamesandaddresses. Asubscriber canthensendaprivatemessagetoanothersubscriber simplybylookingupthepublickey oftheaddressee andusingthekeytoencryptthemessage. Theencrypted message(i.e., ciphertext) canonlybereadbytheholderofthatparticular publickey.Infact,shouldthe originalmessage(i.e.,plaintext) belost,evenitssenderwouldfinditextremely difficultto recoverthemessagefromtheciphertext. Thekeymanagement ofpublic-key cryptography makesitwellsuitedforthedevel opmentoflarge,securecommunication networks. Indeed,ithasevolvedfromasimple concepttoamainstay ofcryptographic technology. DIFFIE-HELLMAN PVBLIcKEV DISTRIBUTION. InasimpleandyetelegantsystemknownastheDiffie-Hellman publickey-distribution system,useismadeofthefactthatitiseasytocalculate adiscreteexponential butdifficult tocalculate adiscretelogarithm. Tobemorespecific,consider thediscreteexponential function Y=aXmodpfor1,;:;X,;:;p-1 (AS.24) wherethearithmetic isperformed modulo-po Theaisanintegerthatshouldbepril1litive (i.e.,aUpowersofagenerate alltheelements modprelatively primetop-1).Corre spondingly, Xisreferredtoasthediscretelogarithm ofYtothebaseCt,modp,asshown by for1,;:;Y,;:;P-1 (A5.25) Thecalculation ofYfromXiseasy,usingthetrickofsquare-and-multiply. Forexample, forX=16wehave Ontheotherhand,theproblem ofcalculating XfromYismuchmoredifficult. IntheDiffie-Hellman publickey-distribution system,allusersarepresumed toknow bothCtandp.Auseri,say,selectsanindependent randomnumberXiuniformly fromthe setofintegers{1,2,...,p}thatiskeptasaprivatesecret.Butthediscreteexponential Yi=~imodp (AS.26) isdeposited inapublicdirectory withtheuser'snameandaddress.Everyotheruserof thesystemdoesthesamething.Now,supposethatusersiandjwishtocommunicate A5.6JU"est--Shamir-AdlettuJn System 757 privately. Toproceed, userifetchesYjfromthepublicdirectory andusestheprivatesecret Xitocompute Kji=(yj)X,modp =(cri)x,modp =aXix,modp Inasimilarway,userjcomputes K;;.Butwehave Kji=K;;(AS.2?) (AS.2S) Accordingly, usersiandjarriveatK;;asthesecretkeyinaconventional cryptosystem. Another usermustcompute Kjiusingtheinformation YiandYjobtained fromthepublic directory, applying thealternative formula Kji=(yj)IogY,modp (AS.29) Apparently, thereisnoothermethodforanenemytofindthesecretkeyKji;however, thereisnoproofforit.Inlightofwhatwesaidearlier,Equation (AS.29)isdifficultto calculate asitinvolvesadiscretelogarithm, whereasEquation (AS.2?)iseasytocalculate asitinvolvesadiscreteexponentiaL Thus,securityofthesystemdependsonthedifficulty encountered incomputing adiscretelogarithm. TheDiffie-Hellman publickey-distribution systemistheoldestsysteminitsclass; nevertheless, itisstillgenerally considered tobeoneofthemostsecureandpractical public key-distribution systems. IA5.6Rivest-Shamir-Adleman System Todevelopapublic-key cryptographic systemisnoeasytask.Indeed,numerous such systemshavebeenproposed intheliterature, butunfortunately mostofthemhaveproven tobeinsecure. Todate,themostsuccessful implementation ofpublic-key cryptography is theRivest-Shamir-Adleman (RSA)system,!1 whichusesideasfromclassicalnumberthe ory.Itisconsidered tobeoneofthemostsecurecryptographic systemsinthatithas withstood manyattempts byexpertsinthefieldtobreakit. TheRSAalgorithm isablockcipherbasedonthefactthatfindingarandomprime numberoflargesize(e.g.,100digit)iscomputationally easy,butfactoring theproductof twosuchnumbers iscurrently considered computationally infeasible. Specifically, thecom putation ofparameters specifictotheRSAalgorithm proceeds asfollows: 1.Choosetwoverylargeprimenumbers, pandq,atrandom; theprimenumbers have tobefairlycarefully chosenassomeprimenumbers leadtoaveryweaksystem. 2.Multiply thenumberspandq,obtaining theproduct pq=n FindtheEulertotientfunction ofn,usingtheformula <{J(n)=(p-l)(q-1)(AS.3D) (AS.31) Equation (AS.31)followsfromthedefinition oftheEulertotientfunction <{J(n)as thenumberofpositiveintegersilessthann,suchthatthegreatestcommon divisor ofiandnisequaltoone. 758 APPENDIX 5..CRYPTOGRAPHY 3.Letebeapositiveintegerlessthan4>(n),suchthatthegreatestcommon divisorof eand4>(n)isequaltoone.Hence,findapositiveintegerdlessthan4>(n),suchthat de=1mod4>(n) (AS.32) TheRSAtrapdoor one-way function isthendefinedsimplybycomputing thediscrete exponentiation fAx)=x'=ymodn (AS.32) Thevaluesofnandeconstitute thepublickey;hence,publishing theeasy-to-find algorithm Eztocompute thefunction fzamounts justtopublishing thenumbersnande. Theprimenumberspandqconstitute theprivatekey.Sincedisrelatedtopandq, possession oftheeasy-to-find (whenoneknowsthetrapdoor z)algorithm Dztocompute theinversefunctionr;1amounts justtoknowingpandq.Inparticular, theinversefunc tionisdefinedby f;1(y)=Imodn (AS.34) Thedecrypting exponent disfoundusingEquation (AS.32), whichisequivalent tothe statement (inordinary integerarithmetic) that de=4>(n)Q+1 (AS.3S) forsomeintegerQ.Notethat4>(n) i~itselfrelatedtopandqbyEquation (AS.31).Since y=xe,wemayuseEquations (AS.32)and(AS.33)towrite yd=xde =x<p(n)Q+1(AS.36) Wenowmakeuseofacelebrated theorem ofEuler,whichstatesthatforanypositive integersxandnwithx<n,wehave x<p(n)=1modn Hence,theuseofEquation (AS.3?)in(AS.36)yieldsthedesireddecryption: yd=x(AS.3?) (AS.38) Wethusseethatfindingtheinversefunctionf;1iseasy,givenknowledge oftheprime numberspandq. ThesecurityoftheRSAcryptoalgorithm restsonthepremisethatanymethodof inverting thefunction fzisequivalent tofactoring n=pq.Thisequivalence raisesthe question: Isanattackbyfactoring ncomputationally feasible?Itappearsthattheanswer isno,provided thattheprimenumberspandqareontheorderof100decimaldigits eachandthatthereisnorevolutionary breakthrough infactoring algorithms. IIIDIGITAL SIGNATURES12 Foranelectronic mailsystemtoreplacetheuseofordinary papermailforbusinesstrans actions,itmustbepossibleforauserofthesystemto"sign"anelectronic message. The A5.7Summary andDiscussion 759 useofadigitalsignature provides proofthatthemessage didoriginate fromthesender. Tosatisfythisrequirement, thedigitalsignature musthavethefollowing properties: ~Thereceiverofanelectronic message isabletoverifythesender'ssignature . ..Thesignature isnotforgeable. J;>Thesenderofasignedelectronic message isunabletodisclaim it. Toimplement digital ~ignatures usingtheRSAalgorithm, wemayproceedasfollows. Auserinpossession oftheprivatekeydmaysignagivenmessage blockmbyforming thesignature s=mdmodn (AS.39) Itisdifficulttocompute sunlesstheprivatekeydisknown.Hence,adigitalsignature definedinaccordance withEquation (AS.39)isdifficulttoforge.Moreover, thesenderof messagemcannotdenyhavingsentit,sincenooneelsecouldhavecreatedthesignature s.Thereceiverproceeds byusingthepublickeyetocompute se=(md)'modn =mdemodn (AS.40) =mmodn where,inthelastline,useismadeofEquation (AS.32). Hence,thereceiverisableto validatethesender'ssignature byestablishing thatthecomputation ofsemodnproduces thesameresultasthedeciphered messagem.Thus,theRSAalgorithm satisfiesallthe threenecessary properties ofadigitalsignature. IA5.7Summary andDiscussion Cryptography isa"hot"research area.Thisstatement shouldnotcomeasasurprise. Considering thefactthatweareinaninformation society,theimportance ofcryptography asasecuritymechanism willcontinue togrow.Inthisappendix, wehavepresented an introductory treatment ofthishighlyimportant subject. Wemayclassifycryptography intosecret-key cryptography andpublic-key cryptog raphy,depending onwhetherthekeyusedfortheencryption ofamessageanditsdecryp tioniscompletely secretorpartlypublic.Alternatively, wemayclassifyacryptographic systemintoablockcipherorstreamcipher,depending onthemethodofimplementation. Ablockcipherexhibits errorpropagation, whichcanprovehighlyvaluable in authentication. Amongthemanycryptographic systemsdeveloped todate,thedataencryption stan dard(DES)andtheRivest-Shamir-Adleman (RSA)algorithms standoutasthemostsuc cessfulones.Bothofthesecryptoalgorithms areblockciphers.Theydifferfromeachother inthattheDESalgorithm involves theuseofasecretkeywhereas theRSAalgorithm involvestheuseofapublickey.Inasecret-key system,thesamekeyissharedbothbythe senderandthereceiver. Ontheotherhand,inapublic-key system,thekeyissplitinto twoparts:apublickeylocatedinthetransmitter andaprivate(secret)key located inthe receiver; inthelattersystem,itiscomputationally infeasible torecovertheplaintext mes sagefromitsencrypted versionwithoutknowledge oftheprivatekey. Although public-key cryptosystems suchasRSAprovideaneffectivemethodforkey management, theyareinefficient forthebulkencryption ofdataduetolowbandwidths. Incontrast, conventional cryptosystems suchasDESprovidebetterthroughput, butthey 760 APPENDIX 5illCRYPTOGRAPHY requirekeymanagement. Thissuggests thepossible useofahybridapproach exploiting thebestelements ofbothcryptosystems asthebasisforthepractical designofasecure communication system.Forexample, theRSAalgorithm maybeusedforauthentication, andtheDESalgorithm forencryption. INOTES ANDREFERENCES 1.Foranintroductory treatment ofcryptography, seeChapter15ofthebookbyAdamek (1991).Foracomprehensive treatment ofthemanyfacetsofcryptology, seethebook editedbySimmons (1992);thisbookisanexpanded editionofaSpecialIssueofthe Proceedings oftheIEEE(1988)oncryptology. Thechaptercontributions ofthebookby Simmons arewrittenbyleadingauthorities onthesubjectofcryptology. Anicetreatment ofcryptology isalsopresented inthebookbyvanTilborg(1988). 2.Theeraofscientific secret-key cryptography wasusheredinwiththepublication ofa landmark paperbyShannon (1949),whichestablished theconnection betweencryptog raphyandinformation theory. 3.Theeraofpublic-key cryptography wasestablished withthepublication ofanotherland markpaperbyDiffieandHeHman (1976),whichshowedforthefirsttimethatitispossible tohavesecretcommunications withoutanytransferofakeybetweensenderandreceiver. ItwasthepaperbyDiffieandHeHman thatsparkedtheexplosion ofresearchinterestin cryptology, whichhascontinued eyersince. 4.Thetermenemycryptanalyst iscommonly usedincryptology torefertoacryptogram interceptor (eavesdropper); itsusageoriginates frommilitaryapplications. 5.Foracomprehensive treatment ofstreamciphers,seeChapter2writtenbyR.A.Rueppel inthebookContemporary Cryptology, editedbySimmons (1992). 6.Forahighlyreadable accountoftheShannon modelofcryptography, seetheopening chapterbyJ.1.MasseyinthebookeditedbySimmons (1992). 7.Theone-time padderivesitsnamefromitsuse(shortlybefore,during,andafterWorld WarII)byspiesofseveralgovernments, whoweregivenapadofpaperwitharandomly chosenkeyandtoldtouseitonlyforasingleencryption. Theone-time padisalsoknown asVernam's cipher,sonamedinrecognition ofitsoriginator, G.S.Vernam. 8.Foraderivation ofEquation (A5.11),seetheoriginalpaperbyShannon (1949). 9.ThehistoryoftheDESalgorithm isrecounted byM.E.SmidandD.K.Branstad inChapter 1ofthebookeditedbySimmons (1992).Foradescription oftheDESalgorithm, seeDiffie andHeHman (1979).SeealsothebooksbyMeyerandMatyas(1982)andTorrieri(1992, Chapter6). 10.Foracomprehensive treatment ofpublic-key cryptography, seeChapter4byJ.Nechvatal inthebookeditedbySimmons (1992).Thisbookalsoincludesachaptercontribution by W.Diffiethatdescribes theseveralattempts todevisesecurepublic-key cryptoalgorithms andthegradualevolution ofavarietyofprotocols basedonthem. 11.TheRSAsystemispatented; itisnamedinrecognition ofitsoriginators R.1.Rivest, A.Shamir,and1.Adleman. Theoriginalreference forthiscryptosystem isRivest,Shamir, andAdleman (1978). 12.Theideaofadigitalsignature wasfirstdiscussed byDiffieandHellman (1976).Itsimple mentation usingtheRSAalgorithm isdescribed byRivest,Shamir,andAdleman (1978). Foradetailedtreatment ofdigitalsignatures, seeChapter6byC.J.MicheH, F.Piper,and R.WildinthebookeditedbySimmons (1992). Thetwelvetablescompiled inthisfinalappendix coverthefollowing: ~ASCIIcode ~FourierandHilberttransforms ~Besselfunctions ~Errorfunction ~Selectedmodemstandards ~Trigonometric identities, seriesexpansions, andintegrals ~Usefulconstants andrecommended unitprefixes 761 762 APPENDIX 6..TABLES ITABLEA6.1 ASCIIcode BitPosition 70 0 0 01 1 1 60 0 11001 432 50 1 01010 aa a aNUL DLE SPa @ PPaa a 1SOH DCl 1 A Q aqa a 1aSTX DC2 2 B Rb aa11ETX DC3 # 3 C S a1aaEOT DC4 $4D Td a1a1ENQ NAK %5 EU e u a11aACK SYN &6 FVf v a1 1 1BEL ETB 7 GW gw 1a a a BS CAN 8HXh x 1a a 1HT EM 9 IY y 1a1a LF SUB Jzj z 1a11VT ESC+ K[k 1 1 aa FF FS < L"-I 1 1 a1CR GS M ]m 111a SO RS > N 1\n 1 1 1 1 SI ?0 0DEL ACK Acknowledge ENQ Enquiry NULNullorallzeros BEL Belloralarm EaTEndoftransmission RS Recordseparator BS Backspace ESC Escape SIShiftin CAN Cancel ETB Endoftransmission block soShiftout CR Carriage return ETX Endoftext SOHStartofheading DCl Devicecontrol1 FF Formfeed SPSpace DCl Devicecontrol1 FS Fileseparator STXStartoftext DC3 Devicecontrol3 GS Groupseparator SUBSubstitute DC4 Devicecontrol4HT Horizontal tab SYNSynchronous idle DEL Delete LF Linefeed US Unitseparator DLE Datalinkescape NAK Negative acknowledge VTVerticaltab EM Endofmedium (FromCouch,1990,withpermission ofMacmillan.) Tables 763 ITABLEA6.2Summary ofproperties oftheFourier transform Property 1.Linearity 2.Timescaling 3.Duality 4.Timeshifting 5.Frequency shifting 6.Areaunderg(t) 7.AreaunderG(f) 8.Differentiation inthetimedomain 9.Integration inthetimedomain 10.Conjugate functions 11.Multiplication inthetimedomain 12.Convolution inthetimedomainMathematical Description agdt)+bg2(t)~aG,(f)+bG2(f) whereaandbareconstants g(at)~ThG(~) whereaisaconstant If g(t)~G(f), then G(t)~g(-f) g(t-to)~G(f)exp(-j27Tfto) exp(j27Tf,t)g(t) ~G(f-f,) r~g(t)dt=G(O) g(O)=[G(f)df 1,g(t)~j27TfG(f) Jt 1 G(O) g(T)dT~-2 fG(f)+-8(f) -~ J7T 2 If g(t)~G(f), then g*(t)~G'(-f) g,(t)g2(t)~r~G,(A)G2(f-A)dA roog,(T)g2(t -T)dT~G,(f)G 2(f) 764 APPENDIX 6iiiTABLES ITABLEA6.3Fourier-transform pairs TimeFunction FourierTransform rect(~) Tsinc(fT) sinc(2Wtl2~rect(2\v) exp(-at)u(t),1a>0a+j2rrf exp(-aItl),2aa>0a2+(2rrf)2 exp(-mZ) exp(-rrjll {I11 ItI<TTsincVT) T ' 0, Itl2:T 8(t) 1 1 8(f) 8(t-to) exp(-j2rrfto) exp(j2rrtt) 8(f-i) cos(2rrit) H8(f-tJ+li(f+ill sin(2rrtt)12j[li(f-il-8(1+i)l sgn(t)1 jrrf 1-jsgn(f) rrt u(t) 1.8(f)+_1_ 2 j2rrf L8(t-iTo)tn~~8(f;Ji=-"" Notes:u(t)=unitstepfunction B(t)=deltafunction, orunitimpulse rect(t)=rectangular functionofunitamplirude andunit duration centered ontheorigin sgn(l)=signumfunction sinc(t)=sinefunction Tubles 765 ITABLEA6.4 Hilberttransform pairs" TimeFunction HilbertTransform m(t)cos(271"j;t) m(t)sin(271"fct) m(t)sin(271"fct) -mit)cos(271"fct) cos(271"fct) sin(271"j;t) sin(271"fct) ~cos(271"fct) sint 1 -cost 1t--1 2reet(t) --log1 71"t+2 a(t) 71"t 1 t 1+r2 1+r2 1-71"a(t) 'Inthefirsttwopairs,itisassumedthatm(t)isband- limitedtotheintetval-W""f""W,whereW<f,. Notes:6(t):deltafunction rect(t):rectangular functionofunitamplitude and unitduration centered ontheorigin log:naturallogarithm ITABLEA6.5TableofBesselfunctions· In(x) n\x0.5 2 3 4 6 8 10 12 00.9385 0.7652 0.2239 -0.2601 -0.3971 0.1506 0.1717 -0.2459 0.0477 10.2423 0.4401 0.5767 0.3391 -0.0660 -0.2767 0.2346 0.0435 -0.2234 20.0306 0.1149 0.3528 0.4861 0.3641 -0.2429 -0.1130 0.2546 -0.0849 30.0026 0.0196 0.1289 0.3091 0.4302 0.1148 -0.2911 0.0584 0.1951 40.0002 0.0025 0.0340 0.1320 0.2811 0.3576 -0.1054 -0.2196 0.1825 5 0.0002 0.0070 0.0430 0.1321 0.3621 0.1858 -0.2341 -0.0735 6 0.0012 0.0114 0.0491 0.2458 0.3376 -0.0145 -0.2437 7 0.0002 0.0025 0.0152 0.1296 0.3206 0.2167 -0.1703 8 0.0005 0.0040 0.0565 0.2235 0.3179 0.0451 9 0.0001 0.0009 0.0212 0.1263 0.2919 0.2304 10 0.0002 0.0070 0.0608 0.2075 0.3005 11 0.0020 0.0256 0.1231 0.2704 12 0.0005 0.0096 0.0634 0.1953 13 0.0001 0.0033 0.0290 0.1201 14 0.0010 0.0120 0.0650 "FormoreextensivetablesofBesselfunctions, seeWatson(1966,pp.666-697), andAbramowitz andStegun(1965,pp. 358-406). 766 APPENDIX 6iiiTABLES TABLEA6.6 Theerrorfunction" u erf(u) u erf(u) 0.00 0.00000 1.10 0.88021 0.05 0.05637 1.15 0.89612 0.10 0.11246 1.20 0.91031 0.15 0.16800 1.25 0.92290 0.20 0.22270 1.30 0.93401 0.25 0.27633 1.35 0.94376 0.30 0.32863 1.40 0.95229 0.35 0.37938 1.45 0.95970 0.40 0.42839 1.50 0.96611 0.45 0.47548 1.55 0.97162 0.50 0.52050 1.60 0.97635 0.55 0.56332 1.65 0.98038 0.60 0.60386 1.70 0.98379 0.65 0.64203 1.75 0.98667 0.70 0.67780 1.80 0.98909 0.75 0.71116 1.85 0.99111 0.80 0.74210 1.90 0.99279 0.85 0.77067 1.95 0.99418 0.90 0.79691 2.00 0:99532 0.95 0.82089 2.50 0.99959 1.00 0.84270 3.00 0.99998 1.05 0.86244 3.30 0.999998 aTheerrorfunction istabulated extensively inseveral references; seeforexample, Abramowitz andStegun (1965,pp.297-316). Tables 767 ITABLEA6.7 Selection oflTVvoiceband (telephone line)modemstandards lTU StandardaTypeofmodulation Bitrate,blsSymbolrate,bauds (a)Symmetric modems: V.21 BinaryFSK 300 300 V.22bis QPSK 1,200 600 V.26 QPSK 2,400 1,200 V.2? 8-PSK 4,800 2,400 V.32 16-QAM 9,600 2,400 V.34 1024-QAM 28,800 3,429 V.34HighSpeed Nested-constellation 33,600 offour960-QAM constellations (b)Asymmetric moderns: V.90:Downstream Digital 56,000 Upstream V.34HighSpeed 33,600 aThesuffix"bIS"designates thesecondverSIonofapartIcular standard. ITABLEA6.8Trigonometric identities exp(±jli)=cosIi±jsinIi cosIi=Hexp(jli)+exp(-jli)] sinIi=-it[exp(jli)-exp{-jli)] sin'Ii+cos'Ii=1 cos'Ii-sin'Ii=cos{21i) cos2Ii=![1+cos(21i)] sin2Ii=![1-cos(21i)] 2sinIicosIi=sin(21i) sin(a±13)=sinacos13±cosasin13 cos(a±13)=cosacos13:;:sinasin13 ( )--=t=an=..=a-=±:....:::ta=n:.!:f3=-::tana±f3 =c-1+tanatan13 sinasin13=Hcos{a-13)-cos{a+13)] cosacos13=Hcos{a-13)+cos(a+13)] sinacos13=Hsin{a-13)+sin(a+13)] 768 APPENDIX 6illTABLES ITABLEA6.9Seriesexpansions Taylorseries [(x)=[(a)+['(a)(x_a)+"(a)(x_af+...+[(nl(a)(x_a)n+ 1! 2! n! where MacLaurin series ['(0) ["(0) [(nl(o) [(x)=[(0)+ux+2!x'+...+----;rxn+... where Binomial series Exponential seriesn(n-1) (1+x)n=1+nx+--2-!-x'+"',Inxl<1 Logarithmic series Trigonometric serieslog(1+x)=x-!X'+tx3-••• sinx=x-.!.x3+.!.x5-••• 3!5! cosx=1 -.!.x'+.!.x4-•••2!4! tanx=x+!x3+l:-XS+...315 sin-tx=X+!x3+1-x5+...640 tan-tX=X_!X 3+1xs_...Ixl<1 3 5 ' sincx=1 -.!.('lTX)'+.!.(=)4-...3! 5! Tables 769 ITABLEA6.10 Integrals Indefinite integralsJxsin(ax)dx=~[sin(ax) axcos(ax)] Jxcos(ax)dx=~[eos(ax)+axsin(ax)] Jxexp(ax)dx=~exp(ax)(ax -1)' Jxexp(ax2)dx=2.exp(ax2) 2a Jexp(ax)sin(bx)dx=a2:b2exp(ax)[a sin(bx) beos(bx)] Jexp(ax)eos(bx)dx=a2:b2exp(ax)[a cos(bx)+bsin(bx)] Ja2:xb2X2=~tan-1(b:) J/~~~X2=P-Ptan-1(b:) a>0,b>0 a>0,b>0Definite integrals (00xsin(ax)d=:!!(_b) Job2+x2X2expa , (00cos(ax) 7rJob2+x2dx=2bexp(-ab), (00eos(ax) 7r.Jo(b2_x2fdx=4b3[sm(ab) abcos(ab)], (00 (00 1Josinexdx=Josine2xdx=2:a>0,b>0 770 APPENDIX 6.,TABLES ITABLEA6.11 Usefulconstants Physical Constants Boltzmann's constant Planck's constant Electron (fundamental) charge Speedoflightinvacuum Standard (absolute) temperature Thermal voltage Thermal energykTatstandard temperature Onehertz(hz)=1cycle/second; 1cycle=21Tradians Onewatt(W)= 1joule/second Mathematical Constants Baseofnaturallogarithm Logarithm ofetobase2 Logarithm of2tobasee Logarithm of2tobase10 Pi ITABLEA6.12 Recommended unitprefiXeSk=1.38X10-23joule/degree Kelvin h6.626X10-34joule-second q=1.602X10-19coulomb c=2.998X108meters/second To=273degreesKelvin Vr=0.026voltatroomtemperature kTo=3.77X10-21joule e=2.7182818 log2e=1.442695 log2=0.693147 loglo2=0.30103 1T=3.1415927 Multiples andSubmultiples Prefixes Symbols 1012tera T 109giga G 106mega M 103kilo K(k) 10-3milli m 10-6micro I" 10-9nano n 10-12pica p IConventions andNotations 1.ThesymbolIImeanstheabsolute value,ormagnitude, ofthecomplex quantity contained within. 2.Thesymbolarg()meansthephaseangleofthecomplex quantity contained within. 3.ThesymbolRe[]meansthe"realpartof,"andIm[]meansthe"imaginary partof." 4.Unlessstatedotherwise, thenaturallogarithm isdenotedbylog.Logarithms tobases 2and10aredenotedbylOg2andloglo,respectively. 5.Theuseofanasteriskassuperscript denotescomplex conjugate, e.g.,x·isthecom plexconjugate ofx. 6.Thesymbol;;=: indicates aFourier-transform pair,e.g.,g(t)¢C(f),wherealow ercaseletterdenotesthetimefunction andacorresponding uppercase letterdenotes thefrequency function. 7.ThesymbolF[]indicates theFourier-transform operation, e.g.,F[g(t)]=C(f), andthesymbolp-l[ ]indicates theinverseFourier-transform operation, e.g., P-l[C(f)] =g(t). 8.Thesymbol*denotesconvolution, e.g., x(t)*h(t)=roox(7)h(t 7)d7 9.ThesymbolE8denotesmodulo-2 addition, exceptinChapter10wherebinaryarith meticisusedandmodulo-2 addition isdenotedbyanordinary plussignthroughout thatchapter. 10.Theuseofsubscript Toindicates thatthepertinent functiongTo(t), say,isaperiodic function oftimetwithperiodTo. 11.Theuseofahatoverafunction indicates oneoftwothings: (a)theHilberttransform ofafunction, e.g.,thefunctiong(t)istheHilberttransform ofg(t),or (b)theestimate ofanunknown parameter, e.g.,thequantity &(x)isanestimate of theunknown parameter a,basedontheobservation vectorx. 12.Theuseofatildeoverafunction indicates thecomplex envelope ofanarrowband signal,e.g.,thefunctiong(t)isthecomplex envelope ofthenarrowband signalg(t). Theexception tothisconvention isinSection10.8,where,inthedescription ofturbo decoding, thetildeisusedtosignifyextrinsic information andtherebydistinguish it fromlog-likelihood ratio. 13.Theuseofsubscript+indicates thepre-envelope ofasignal,e.g.,thefunction g+(t)isthepre-envelope ofthesignalg(t).Wemaythuswriteg+(t)=g(t)+jg(t), whereg(t)istheHilberttransform ofg(t).Theuseofsubscript -indicates that g_(t)=g(t)-jg(t)=g+·(t). 14.Theuseofsubscripts IandQindicates thein-phase andquadrature components of anarrowband signal,anarrowband randomprocess,ortheimpulseresponse ofa narrow-band filter,withrespecttothecarriercos(27T'jj). 771 772 GLOSSARY 15.Foralow-pass messagesignal,thehighestfrequency component ormessage band widthisdenotedbyW.Thespectrum ofthissignaloccupies thefrequency interval -Ws;fs;Wandiszeroelsewhere. Foraband-pass signalwithcarrierfrequency !C,thespectrum occupies thefrequency intervals, !c-Ws;fs;!c+Wand -!c-Ws;fs;-k+W,andso2Wdenotesthebandwidth ofthesignal.The (low-pass) complex envelope ofthisband-pass signalhasaspectrum thatoccupies thefrequency interval-Ws;fs;W. Foralowpassfilter,thebandwidth isdenotedbyB.Acommon definition offilter bandwidth isthefrequency atwhichthemagnitude response ofthefilterdropsby 3dBbelowthezero-frequency value.Foraband-pass filterofmid-band frequency Iethebandwidth isdenotedby2B,centered onfc.Thecomplex low-pass equivalent ofthisband-pass filterhasabandwidth equaltoB. Thetransmission bandwidth ofacommunication channel, required totransmit a modulated wave,isdenotedbyBT• 16.Random variables orrandomvectorsareuppercase (e.g.,XorX),andtheirsample valuesarelowercase (e.g.,xorx). 17.Averticalbarinanexpression means"giventhat,"e.g.,fx(xIHa)istheprobability densityfunction oftherandomvariableX,giventhathypothesis Hoistrue. 18.ThesymbolE[]meanstheexpected valueoftherandomvariableenclosed within; theEactsasanoperator. 19.Thesymbolvar[]meansthevariance oftherandomvariableenclosed within. 20.Thesymbolcov[]meansthecovariance ofthetworandom variables enclosed within. 21.Theaverageprobability ofsymbolerrorisdenotedbyPe. Inthecaseofbinarysignaling techniques, PIOdenotestheconditional probability oferrorgiventhatsymbol0wastransmitted, andPmdenotestheconditional prob abilityoferrorgiventhatsymbol1wastransmitted. Theaprioriprobabilities of symbols0and1aredenotedbypoandp"respectively. 22.Thesymbol()denotesthetimeaverageofthesamplefunction enclosed within. 23.Boldface letterdenotesavectorormatrix.TheinverseofasquarematrixRisdenoted byR-\Thetranspose ofavectorwisdenotedbywT•TheHermitian transpose of acomplex-valued vectorxisdenotedbyxH;Hermitian transposition involvesboth transposition andcomplex conjugation. 24.Thelengthofavectorxisdenoted byIIxII.TheEuclidean distance between the vectors XiandXiisdenotedbydii=IIXi-XiII. 25.Theinnerproductoftworeal-valued vectorsxandyisdenotedbyxTy;theirouter product isdenoted byxyT.Ifthevectors Xandyarecomplex valued,theirinner productisxHy,andtheirouterproduct isxr. 26.ThevectorproductoftwoM-by-lvectorsaand13isanM-by-lvectordefinedby a•13= [::~:1 (XM13M where (Xkand13karethekthelements ofaand13,respectively. TheL1normofthe vectorproducta.13isdefinedby M IIa°13111=2:aml3m m=1 Abbreviations 773 IFunctions 1.Rectangular function: 2.Unitstepfunction: 3.Signumfunction: 4.(Dirac)deltafunction: or,equivalently, 5.Sinefunction: 6.Sineintegral: 7.Errorfunction: Complementary errorfunction: 8.Binomial coefficient 9.Besselfunction ofthefirstkind ofordern: 10.Modified Besselfunction ofthe firstkindofzeroorder: 11.Confluent hypergeometric function{1,-!<t<!rect(t)= 0,ItI>! {1,t>0 u(t)=0,t<0 {1,t>0 sgn(t)=0,t=0 -1,t<0 8(t)=0,t*0 roo8(t)dt=1 roog(t)8(t-to)dt=g(to) .()sin(1Tx)smcx=--. 1TX .fUsinxSI(U)=--dx ox 2("erf(u)=y:;;:Joexp(-r) dz erfc(u)=1 -ed(u). (:)=(n_n~)!k! In(x)=21 1Tfwexp(jxsine-ine)de 1fW[o(X)=21T-wexp(xcose)de a xala+1)x2 IFI(a;b;x)=1+b11+b(b+1)2!+... IAbbreviations A: AC: ADC: ADM: ADPCM: ADSL: AM: ANSI: APB: APF: AQB:ampere alternating current analog-to-digital converter adaptive delta modulation adaptive differential pulse-eode modulation asymmetric digitalsubscriber line amplitude modulation American National Standards Institute adaptive prediction withbackward estimation adaptive prediction withforwardestimation adaptive quantization withbackward estimation 774 GLOSSARY AQF: ARQ: ASCll: ASK: ATM: AWGN: hIs: BER: BISDN: BPF: BSC: CAP: CCITT: CDM: CDMA:CELP: CO: codec: CPFSK: ·CRC: CW: DAC: dB: dBW: dBmW: DC: DEM:DES: DFT: DM: DMT: DPCM: DPSK: DSB-SC: DSIBPSK: DSL: exp: FDM: FDMA: FEXT: FFT: FH:adaptive quantization withforwardestimation automatic-repeat request American National Standard CodeforInformation Interchange amplitude-shift keying asynchronous transfermode additivewhiteGaussian noise bits/second biterrorrate broadband ISDN band-pass filter binarysymmetric channel carrierless amplitude/phase modulation Consultative Committee forInternational Telephone andTelegraph (Now renamed theflU) code-division multiplexing code-division multiple access codeexcitedlinearpredictive (model) centraloffice coder/decoder continuous-phase frequency-shift keying cyclicredundancy check continuous wave digital-to-analog converter decibel decibelreferenced to1watt decibelreference to1milliwatt directcurrent demodulator dataencryption standard discreteFouriertransform deltamodulation discretemultitone differential pulse-code modulation differential phase-shift keying doublesideband-suppressed carrier directsequence/binary phase-shift keying digitalsubscriber line exponential frequency-division multiplexing frequency-division multiple access far-endcrosstalk fastFouriertransform frequency hop FH/MFSK: FMFB: FSK: GMSK:GSM: HDTV: Hz: IDFT: IF: I/O: IF: IS-95: ISDN: lSI: ISO: lTV: JPEG: LAN: LDM: LMS: log: log2: loglO: LPC: LPF: MAP: ML: mmse: modem: MPEG: ms: p,s: MSK: NCO: NEXT: nm: NRZ: NTSC: OC: OFDM: OOK: OSI: PAM:frequency hop/M-ary frequency-shift keying frequency modulator withfeedback frequency-shift keying Gaussian filteredMSK globalsystemformobilecommunication highdefinition television Hertz inversediscreteFouriertransform intermediate frequency input/output internetprotocol intermediate standard-95 integrated servicesdigitalnetwork intersymbol interference International Organization forStandardization International Telecommunications Union jointphotographic expertsgroup local-area network lineardeltamodulation least-mean-square naturallogarithm logarithm tobase2 logarithm tobase10 linearpredictive coding(model) low-pass filter maximum aposteriori probability maximum likelihood minimum mean-square error modulator-demodulator motionphotographic expertsgroup millisecond microsecond minimum shiftkeying number-eontrolled oscillator near-end crosstalk nanometer nonreturn-to-zero National Television SystemsCommittee opticalcarrier orthogonal frequency-division multiplexing on-offkeying opensystemsinterconnection pulse-amplitude modulationAbbreviatUms 775 776 GLOSSARY PCM: PDM: PG: PLL: PN: POTS: PPM: PSK: PSTN: PWM: QAM: QoS: QPSK: RF: nns: RS: RS-232 RSA: RSC: RZ: s: SDH:SDMA: SDR:SNR: SONET: STFT: STM: TC: TCM: TDM: TDMA: TV: UHF: V: VCO: VHF: VLSI: W:WDM:pulse-code modulation pulse-duration modulation processing gain phase-locked loop pseudo-noise plainoldtelephone service pulse-position modulation phase-shift keying publicswitched telephone network pulse-width modulation quadrature amplitude modulation qualityofservice quadriphase-shift keying radiofrequency root-mean-square Reed-Solomon Recommended standard-232 (port) Rivest-Shamir-Adelman recursive systematic conyolutional (code) return-to-zero secondsynchronous digital hierarchy space-division multiple access signal-to-distortion ratio signal-to-noise ratio synchronous opticalnetwork short-time Fouriertransform synchronous transfermode timecompression trellis-coded modulation time-division multiplexing time-division multiple access television ultrahighfrequency volt voltage-controlled oscillator veryhighfrequency very-large-scale integration watt wavelength divisionmultiplexing BOOKS M.Abramowitz andLA.Stegun,Handbook ofMathematical Functions withFormulas, Graphs, andMathematical Tables(NewYork:DoverPublications, 1965). N.Abramson, Information TheoryandCoding(NewYork:McGraw-Hill, 1963). ].Adamek, Foundations ofCoding(NewYork:Wiley,1991). Y.Akaiwa, Introduction toDigitalMobileCommunication (NewYork:Wiley,1997). ].B.Anderson, T.Aulin,andG.E.Sundberg, DigitalPhaseModulation (NewYork:PlenumPub lishers,1986). J.B.Anderson andS.Mohan,SourceandChannelCoding:AnAlgorithmic Approach (Boston,Mass: KluwerAcademic, 1991). . ].B.Anderson, DigitalTransmission Engineering (Piscataway, N.J.:IEEEPress,1999). R.B.Ash,Information Theory(NewYork:Wiley,1965). BellLaboratories Technical Staff,AHistoryofEngineering ScienceintheBellSystem:TheEarly Years(1875-1925), (BooksonDemand, AnnArbor,Michigan: 1975). J.G.Bellamy, DigitalTelephony, SecondEdition(NewYork:Wiley,1991). S.Benedetto andE.Biglieri,PrinciplesOfDigitalTransmission withWireless Applications (New York:KluwerAcademiciPlenum Publishers, 1999). S.Benedetto, E.Biglieri,andV.Castellani, DigitalTransmission Theory(Englewood Cliffs,N.J.: Prentice-Hall, 1987). W.R.Bennett,Introduction toSignalTransmission (NewYork:McGraw-Hill, 1970). K.B.Bensonand].C.Whitaker, Television Engineering Handbook, rev.ed.(NewYork:McGraw Hill,1992). T.Berger,RateDistortion Theory:AMathematical BasisforDataCompression (Englewood Cliffs, N.J.:Prentice-Hall, 1971). E.R.Berlekamp, Algebraic CodingTheory(NewYork:McGraw-Hill, 1968). E.R.Berlekamp (editor),KeyPapersintheDevelopment ofCodingTheory(Piscataway, N.J.:IEEE Press,1974). V.K.Bhargava, D.Haccoun, R.Matyas,andP.Nuspl,DigitalCommunications bySatellite: Mod ulation,Multiple Access,andCoding(NewYork:Wiley,1981). E.Biglieri,D.Divsalar, P.].McLane, andM.K.Simon,Introduction toTrellis-Coded Modulation withApplications (NewYork:Macmillan, 1991). J.A.C.Bingham, TheTheoryandPracticeofModemDesign(NewYork:Wiley,1988). R.B.Blachman andJ.W.Tukey,TheMeasurement ofPowerSpectra,fromthePointofViewof Communication Engineering (NewYork:Dover,1958). H.S.Black,Modulation Theory(Princeton, N.J.:VanNostrand, 1953). R.E.Blahut,Principles andPracticeofInformation Theory(Reading, Mass.:Addison-Wesley, 1987). R.E.Blahut,DigitalTransmission ofInformation (Reading, Mass.:Addison-Wesley, 1990). G.E.P.BoxandG.M.Jenkins,TimeSeriesAnalysis: Forecasting andControl(SanFrancisco: Holden Day,1976). R.N.Bracewell, TheFourierTransform andItsApplications, 2nded.,rev.(NewYork:McGraw- Hill,1986). L.Brillouin, ScienceandInformation Theory,2nded.(NewYork:Academic Press,1962). K.W.Cattermole, Principles ofPulse-code Modulation (NewYork:American Elsevier, 1969). W.Y.Chen,DSL:Simulation Techniques andStandards Development forDigitalSubscriber Line Systems(Indianapolis, Ind.:Macmillan Technical Publishing, 1998). J.M.Cioffi,DigitalDataTransmission, EE379C CourseTextbook, Stanford University, 1998. 777 778 BIBLIOGRAPHY G.e.Clark,Jr.,andJ.B.Cain,Error-correction CodingforDigitalCommunications (NewYork: PlenumPublishers, 1981). L.W.Couch,DigitalandAnalogCommunication Systems,5thed.(Englewood Cliffs,N.].:Prentice Hall,1997). T.M.CoverandJ.A.Thomas, Elements ofInformation Theory(NewYork:Wiley,1991). H.CramerandM.R.Leadbetter, Stationary andRelatedStochastic Processes: SampleFunction Properties andTheirApplications (NewYork:Wiley,1967). W.B.Davenport, Jr.,andW.I.Root,AnIntroduction totheTheoryofRandom SignalsandNoise (NewYork:McGraw-Hill, 1958). R.e.Dixon,SpreadSpectrum Systems, 2nded.(NewYork:Wiley,1984). R.C.Dixon(editor),SpreadSpectrum Techniques (NewYork,IEEEPress,1976). L.J.Doob,Stochastic Processes (NewYork:Wiley,1953). J.J.Downing, Modulation SystemsandNoise(Englewood Cliffs,N.].:Prentice-Hall, 1964). W.Feller,AnIntroduction toProbability TheoryanditsApplication, vol.1,3rded.(NewYork: Wiley,1968). T.L.Fine,TheoriesofProbability: AnExamination ofFoundations (NewYork:Academic Press, 1973). L.E.Franks(editor),DataCommunication: Fundamentals ofBaseband Transmission (Dowden, Hutchison, andRoss,1974). L.E.Franks,SignalTheory(Englewood Cliffs,N.J.:Prentice-Hall, 1969). R.L.Freeman, Telecommunications Transmission Handbook, 4thed.(NewYork:Wiley,1998). R.M.Gagliardi, Introduction toCommunications Engineering, 2nded.(NewYork:Wiley,1988). R.G.Gallager, Information TheoryandReliableCommunication (NewYork:Wiley,1968). R.G.Gallager, Low-Density Parity-Check Codes(Cambridge, Mass.:MITPress,1963). F.M.Gardner, Phaselock Techniques, 2nded.(NewYork:Wiley,1979). V.K.Garyand].E.Wilkes,Principles &Applications ofGSM(Englewood Cliffs,N.].:Prentice Hall,1999). A.GershoandR.M.Gray,VectorQuantization andSignalCompression (Boston,Mass.:Kluwer Academic, 1992). J.D.Gibson(editor),TheMobileCommunications Handbook (Piscataway, N.J.:IEEEPress,1996). R.D.Gitlin,J.F.Hayes,andS.B.Weinstein, DataCommunications Principles (NewYork:Plenum, 1992). B.Goldberg andH.S.Bennett(editors), Communication Channels: Characterization andBehavior (NewYork:IEEEPress,1976). S.W.Golomb (editor),DigitalCommunications withSpaceApplications (Englewood Cliffs,N.J.: Prentice-Hall, 1964). S.W.Golomb, ShiftRegisterSequences (SanFrancisco: Holden-Day, 1967). R.M.GrayandL.D.Davisson, Random Processes: AMathematical Approach forEngineers (En- glewood Cliffs,N.J.:Prentice-Hall, 1986). P.E.Green,Jr.,Computer Network Architectures andProtocols (NewYork:Plenum,1982). P.E.Green,Jr.,FiberOpticNetworks (Englewood Cliffs,N.].:Prentice-Hall, 1993). M.S.Gupta(editor),Electrical Noise:Fundamentals andSources(NewYork:IEEEPress,1977). R.W.Hamming, CodingandInformation Theory(Englewood Cliffs,N.J.:Prentice-Hall, 1980). S.Haykin,Communication Systems, 3rded.(NewYork:Wiley,1994). S.Haykin,Adaptive FilterTheory,3rded.(Englewood Cliffs,N.J.:Prentice-Hall, 1996). S.HaykinandB.VanVeen,SignalsandSystems(NewYork:Wiley,1999). e.Heegard andS.B.Wicker,TurboCoding(Boston,Mass.:KluwerAcademic Publishers, 1999). G.Held,TheComplete ModemReference, 3rded.(NewYork:Wiley,1997). e.W.Helstrom, Statistical TheoryofSignalDetection (Elmsford, N.Y.:Pergamon Press,1968). e.W.Helstrom, Probability andStochastic Processes forEngineers, 2nded.(NewYork:Macmillan, 1990). K.Henney(editor),RadioEngineering Handbook (NewYork:McGraw-Hill, 1959). J.K.Holmes, Coherent SpreadSpectrum Systems (NewYork:Wiley,1982). W.e.Jakes,Jr.(editor),Microwave MobileCommunications (NewYork:Wiley,1974). Books 779 N.S.JayantandP.Noll,DigitalCodingofWaveforms: Principles andApplications toSpeechand Video(Englewood Cliffs,N.J.:Prentice-Hall, 1984). N.S.Jayant(editor),Waveform Quantization andCoding(NewYork:IEEEPress,1976). H.Jeffreys,Sir,TheoryofProbability, 3rded.(Oxford: Clarendon Press,1967). H.Jeffreys,Sir,andB.S.Jeffreys,MethodsofMathematical Physics,3rded.(Cambridge University Press,1956). M.e.Jeruchim, B.Balaban, andJ.S.Shanmugan, Simulation ofCommunication Systems(NewYork: Plenum,1992). E.C.JordanandK.G.Balmain, Electromagnetic WavesandRadiating Systems,2nded.(Englewood Cliffs,N.J.:PrenticeHall,1968). A.Khintehin, Mathematical Foundations ofInformation Theory(NewYork:Dover,1957). A.N.Kolmogorov, Foundations oftheTheoryofProbability (NewYork:ChelseaPublishing, 1956). V.A.Kotel'nikov, TheTheoryofOptimum NoiseImmunity (NewYork:McGraw-Hill, 1960). J.D.Kraus,Antennas (NewYork:McGraw-Hill, 1950). S.Kullback, Information TheoryandStatistics (NewYork:Dover,1968). P.Lafrance, Fundamental Concepts inCommunication (Englewood Cliffs,N.J.: Prentice-Hall, 1990). B.P.Lathi,ModernDigitalandAnalogCommunication Systems, 2nded.(OxfordUniversity Press, 1995). I.Lebow,Information Highways andByways(Piscataway, N.J.:IEEEPress,1995). E.A.LeeandD.G.Messerschmitt, DigitalCommunication, 2nded.(Boston,Mass.:KluwerAca demic,1994). J.S.LeeandI.E.Miller,CDMASystemsEngineering Handbook (Boston, Mass.:ArtechHouse Publishers, 1998). Y.W.Lee,Statistical TheoryofCommunication (NewYork:Wiley,1960). W.e.(Y.) Lee,MobileCommunications Engineering (NewYork:McGraw-Hili, 1982). A.Leon-Garcia, Probability andRandom Processes forElectrical Engineering, 2nded.(Reading, Mass.:Addison-Wesley, 1994). e.R.Lewart,TheUltimate ModemHandbook (Englewood Cliffs,N.J.:Prentice-Hall, 1998). S.LinandD.J.Costello, Jr.,ErrorControlCoding:Fundamentals andApplications (Englewood Cliffs,N.J.:Prentice-Hall, 1983). W.e.Lindsey, Synchronization SystemsinCommunication andControl(Englewood Cliffs,N.J.: Prentice-Hall,1972). W.e.LindseyandM.K.Simon(editors), Phase-locked LoopsandTheirApplications (NewYork: IEEEPress,1978). W.e.LindseyandM.K.Simon,Telecommunication SystemsEngineering (Englewood Cliffs,N.J.: Prentice-Hall, 1973). M.Loeve,Probability Theory(Princeton, N.J.:VanNostrand, 1963). R.W.Lucky,SiliconDreams:Information, Man,andMachine (NewYork:St.Martin's Press,1989). R.W.Lucky,J.Salz,andE.J.Weldon,Jr.,Principles ofDataCommunication (NewYork:McGraw- HiII,1968). F.J.MacWilliams andN.J.A.Sloane,TheTheoryofError-correcting Codes(Amsterdam: North Holland, 1977). V.K.Madisetti andD.B.WiIliams (editors), TheDigitalSignalProcessing Handbook (Piscataway, N.J.:IEEEPress,1998). R.J.MarksII,Introduction toShannon Sampling andInterpolation Theory(NewYorklBerlin: Springer-Verlag, 1991). J.e.McDonald (editor),Fundamentals ofDigitalSwitching, 2nded.(NewYork:Plenum,1990). R.McDonough andA.D.Whalen, Detection ofSignalsinNoise,2nded.(NewYork:Academic Press,1995). R.J.McEliece, TheTheoryofInformation andCoding:AMathematical Framework forCommu nication(Reading, Mass.:Addison-Wesley, 1977). A.MengaliandN.D'Andrea, Synchronization Techniques forDigitalReceivers (NewYork:Plenum, 1997). 780 BIBUOGRAPHY D.J.G.Mestdagh, Fundamentals ofMultiaccess OpticalFiberNetworks (Boston, Mass.:Artech HousePublishers, 1995). c.H.MeyerandS.M.Matyas,Cryptography: ANewDimension inComputer DataSecurity(New York:Wiley,1982). H.MeyrandG.Ascheid, Synchronization inDigitalCommunications, vol.1(NewYork:Wiley, 1990). H.Meyr,M.Moeneclaey, andS.A.Fechtel,DigitalCommunication Receivers: Synchronization, ChannelEstimation andSignalProcessing (NewYork:Wiley,1998). A.M.Michelson andA.H.Levesque, Error-control Techniques forDigitalCommunication (New York:Wiley,1985). D.Middleton, AnIntroduction toStatistical Communication Theory(NewYork:McGraw-Hill, 1960). J.G.Nellist,Understanding Telecommunications andLightwave Systems: AnEntryLevelGuide (Piscataway, N.J.:IEEEPress,1992). C.F.J.Overhage (editor),TheAgeofElectronics (NewYork:McGraw-Hill, 1962). P.F.Panter,Modulation, NoiseandSpectralAnalysis, AppliedtoInformation Transmission (New York:McGraw-Hill, 1965). A.Papoulis, Probability, Random Variables, andStochastic Processes, 2nded.(NewYork: McGraw-Hill,1984). J.D.Parsons, TheMobileRadioPropagation Channel(NewYork:Wiley,1992). K.Pahlavan andA.H.Levesque, WirelessInformation Networks (NewYork:Wiley,1996). J.Pearl,Probabilistic Reasoning inIntelligent Systems:Networks ofPlausible Inference (SanMateo, Calif:MorganKaufman Publishers, 1988). W.W.Peterson andE.J.Weldon, Jr.,ErrorCorrecting Codes,2nded.(Cambridge, Mass.:MIT Press,1972). J.R.PierceandA.M.Noll,Signals:TheScienceofTelecommunications (NewYork:Scientific Amer icanLibrary,1990). J.R.Pierce,Symbols, SignalsandNoise:TheNatureandProcessofCommunication (NewYork: Harper,1961). H.V.Poor,AnIntroduction toSignalDetection andEstimation, 2nded.(NewYorkJBerlin: Springer- Verlag,1994). T.PrattandC.W.Bostian,SatelliteCommunications (NewYork:Wiley,1986). J.G.Proakis,DigitalCommunications, 3rded.(NewYork:McGraw-Hill, 1995). L.R.RabinerandR.W.Schafer,DigitalProcessing ofSpeechSignals(Englewood Cliffs,N.J.:Pren tice-Hall, 1978). K.R.RaoandP.Yip,DiscreteCosineTransform: Algorithms, Advantages, Applications (NewYork: Academic Press,1990). T.S.Rappaport, SmartAntennas (Piscataway, N.J.:IEEEPress,1998). T.S.Rappaport, Wireless Communications: Principles andPractice(Piscataway, N.J.:IEEEPress, 1996). S.O.Rice,"NoiseinFMreceivers," inM.Rosenblatt (editor),Proceedings oftheSymposium on TimeSeriesAnalysis(NewYork:Wiley,1963),pp.395-411. J.H.Roberts, AngleModulation: TheTheoryofSystemsAssessment, IEECommunication Series5 (London: Institution ofElectricalEngineers, 1977). H.E.Rowe,SignalsandNoiseinCommunicatioll Systems(Princeton, N.J.:VanNostrand, 1965). D.J.Sakrison, Communication Theory:Transmission ofWaveforms andDigitalInformation (New York:Wiley,1968). C.Schlegel, TrellisCoding(Piscataway, N.J.:IEEEPress,1997). M.Schwartz, W.R.Bennett, andS.Stein,Communication SystemsandTechniques (NewYork: McGraw-Hill, 1966). M.Schwartz, Information Transmission, Modulation andNoise:AUnifiedApproach, 3rded.(New York:McGraw-Hill, 1980). M.Schwartz, Telecommunication Networks: Protocols, Modeling, andAnalysis (Reading, Mass.: Addison Wesley,1987). Books 781 K.S.Shanmugan, DigitalandAnalogCommunication Systems(NewYork:Wiley,1979). G.E.Shannon andW.Weaver, TheMathematical TheoryofCommunication (Urbana: University ofIllinoisPress,1949). G.J.Simmons (editor),Contemporary Cryptology: TheScienceofInformation Integrity (Piscataway, N.J.:IEEEPress,1992). M.K.Simon,J.K.Omura,R.A.Scholtz,andB.K.Levitt,SpreadSpectrum Communications, vols.I, II,andill(NewYork:Computer SciencePress,1985). B.Sklar,DigitalCommunications: Fundamentals andApplications (Englewood Cliffs,N.J.:Prentice Hall,1988). D. Slepian (editor),KeyPapersintheDevelopment ofInformation Theory(NewYork:IEEEPress, 1974). N.J.A.SloaneandA.D.Wyner,ClaudeShannon: Collected Papers(Piscataway, N.J.:IEEEPress, 1993). D.R.Smith,DigitalTransmission Systems(Princeton, N.J.:VanNostrand Reinhold, 1985). I.S.Sokolnikoff andR.M.Redheffer, Mathematics ofPhysicsandModernEngineering (NewYork: McGraw-HilI, 1966). J.J.Spilker,Jr.,DigitalCommunications bySatellite(Englewood Cliffs,N.J.:Prentice-Hall, 1977). W.Stallings, ISDNandBroadband ISDN,2nded.(NewYork:Macmillan, 1992). T.Starr,J.M.Cioffi,andP.J.Silverman, Understanding DigitalSubscriber LineTechnology (En- glewood Cliffs,N.J.:Prentice-Hall, 1999). R.Steele,DeltaModulation Systems(NewYork:Wiley,1975). R.Steeleand1.Hanzo(editors), MobileRadioCommunications, 2nded.(NewYork:Wiley,1999). J.J.Stiffler,TheoryofSynchronous Communications (Englewood Cliffs,N.J.:Prentice-Hall, 1971). E.D.Sunde,Communications SystemsEngineering Theory(NewYork:Wiley,1969). A.S.Tanenbaum, Computer Networks, 2nded.(Englewood Cliffs,N.J.:Prentice-Hall, 1995). S.Tantaratana andK.M.Ahmed,WirelessApplications ofSpreadSpectrum Systems: SelectedRead ings(Piscataway, N.J.:IEEEPress,1998). T.M.Thompson, FromError-correcting CodesThrough SpherePackings toSimpleGroups(The Mathematical Association ofAmerica, Washington D.G.:1983). D.J.Torrieri, Principles ofMilitaryCommunication Systems, 2nded.(Boston,Mass.:ArtechHouse Publishers, 1992). G.L.Turin,NotesonDigitalCommunications (Princeton, N.J.:VanNostrand-Reinhold, 1969). A.VanderZieI.Noise:Source,Characterization, Measurement (Englewood Cliffs,N.J.:Prentice- Hail,1970). H.F.Vanlandingham, Introduction toDigitalControlSystems(NewYork:Macmillan, 1985). H.G.A.vanTilborg,AnIntroduction toCryptology (Boston,Mass.:Kluwer,1988). H.L.VanTrees,Detection, Estimation, andModulation Theory,PartI(NewYork:Wiley,1968). A.J.Viterbi,Principles ofCoherent Communication (NewYork:McGraw-Hili, 1966). A.J.ViterbiandJ.K.Omura, Principles ofDigitalCommunication andCoding(NewYork: McGraw-HilI, 1979). G.N.Watson, ATreatiseintheTheoryofBesselFunctions, 2nded.(NewYork:Cambridge Uni versityPress,1966). N.Wax(editor),SelectedPapersonNoiseandStochastic Processes (NewYork:DoverPublications, 1954). E.T.Whirtaker andG.N.Watson, ACourseinModernAnalysis, 4thed.(NewYork:Cambridge University Press,1952). S.B.WickerandV.K.Bhargava (editors), Reed-Solomon Codes(Piscataway, N.J.:IEEEPress,1994). B.Widrow andS.D.Stearns,Adaptive SignalProcessing (Englewood Cliffs,N.J.:Prentice-Hall, 1985). N.Wiener,TheExtrapolation, Interpolation, andSmoothing ofStationary TimeSeries,withEn gineering Applications (NewYork:Wiley,1949). S.G.Wilson,DigitalModulation andCoding(Englewood Cliffs,N.J.:Prentice-Hall, 1996). E.Wong,Stochastic Processes inInformation andDynamical Systems(NewYork:McGraw-Hili, 1971). 782 RIBLIOGBAPHY P.M.Woodward, Probability andInformation Theory,withApplications toRadar,2nded.(Elms ford,N.Y.:Pergamon Press,1964). J.M.Wozencraft andI.M.Jacobs,Principles ofCommunication Engineering (NewYork:Wiley, 1965). W.W.Wu,Elements ofDigitalSatelliteCommunication, vol.I(NewYork:Computer SciencePress, 1984). C.R.WylieandL.C.Barrett,Advanced Engineering Mdthematics, 5thed.(NewYork:McGraw Hill,1982). R.D.YatesandD.J.Goodman, Probability andStochastic Processes: AFriendly Introduction for Electrical andComputer Engineers (NewYork:Wiley,1999). J.B.Yuen(editor), DeepSpaceTelecommunications SystemsEngineering (NewYork:Plenum, 1983). R.E.ZiemerandR.L.Peterson, DigitalCommunications andSpreadSpectrum Systems(NewYork: Macmillan, 1985). R.E.ZiemerandW.H.Tranter, Principles ofCommunications, 3rded.(Boston,Mass.:Houghton Miflin,1990). PAPERS, REPORTS, PATENTS I M.R.AaronandD.W.Tufts,"Intersymbol interference anderrorprobability," IEEETrans.on Information Theory,vol.IT-ll,pp.26-34,1966. J.E.Abate,"Linearandadaptive deltamodulation," Proceedings oftheIEEE,vol.55,pp.298 308,1967. A.N.Akansu, P.Duhamel, X.Lin,and·M.deCourville, "Orthogonal transmultiplexers incom munications: Areview,"IEEETransactions onSignalProcessing, vol.46,pp.979-995, 1998. Y.AkaiwaandY.Nagata,"Highlyefficientdigitalmobilecommunications withalinearmodulation method," IEEEJournalonSelectedAreasinCommunications, vol.SAC-5,pp.890-895, 1987. O.Al-Shaykh, R.Neff,D.Taubman, andA.Zakhour, "Videosequence compression." InV.K. Madisetti andD.B.Williams (editors), TheDigitalSignalProcessing Handbook, CRCPress, pp.55-1-55-19, 1998. F.Amoroso, "Thebandwidth ofdigitaldatasignals," IEEECommunications Magazine, vol.18, no.6,PP.13-24,1980. J.B.Anderson andD.P.Taylor,"Abandwidth-efficient classofsignalspacecodes,"IEEETrans actionsonInformation Theory,vol.IT-24,pp.703-712, 1978. R.R.Anderson andJ.Salz,"Spectra ofdigitalFM,"BellSystemTech.J.,vol.44,pp.1165-1189, 1965. R.Arens,"Complex processes forenvelopes ofnormalnoise,"IRETrans.onInformation Theory, vol.IT-3,pp.204-207,1957. E.H.Armstrong, "Amethodofreducing disturbances inradiosignaling byasystemoffrequency modulation," Proceedings oftheIRE,vol.24,pp.689-740,1936. E.ArthursandH.Dym,"Ontheoptimum detection ofdigitalsignalsinthepresence ofwhite Gaussian noise-A geometric interpretation andastudyofthreebasicdatatransmission sys tems,"IRETrans.onCommunication Systems, vol.CS-10,pp.336-372, 1962. B.S.AtalandJ.R.Remde,"AnewmodelofLPCexcitation forproducing natural-sounding speech atlowbitrates,"Proc.ICASSP'82,pp.614-17, 1982. B.S.AtalandM.R.Schroeder, "Stochastic codingofspeechsignalsatverylowbitrates,"IEEE International Conference onCommunications, May1984. M.Austin,"Decision-feedback equalization fordigitalcommunication overdispersive channels," MITResearch Laboratory ofElectronics Technical Report461,1967. E.Ayanoglu, N.R.Dagdeviren, J.E.Mazo,andR.Saltzberg, "High-speed modemsynchronized to aremotecodec,"UnitedStatesPatent5,394,437, February 28,1995. E.Ayanoglu, N.R.Dagdeviren, G.D.Golden,andJ.E.Mazo,"Anequalizer designtechnique for thePCMmodem:anewmodemforthedigitalpublicswitched network," IEEETransactions onCommunications, vol.46,pp.763-774,1998. Papers,Reports, Patents 783 L.R.Bahl,J.Cocke,F.Jelinek,and].Raviv,"Optimal decoding oflinearcodesforminimizing symbolerrorrate,"IEEETransactions onInfonnation Theory,vol.IT-20,pp.284-287, 1974. G.Battail,"Coding fortheGaussian channel: thepromiseofweighted outputdecoding," Interna tionalJ.SatelliteCommunications, vol.7,pp.183-192, 1989. G.Battail,"Ponderation dessymbolsdecodesparI'algorithme deViterbi," Ann.Telecommunica tion,vol.42,pp.31-38,1987. E.Bedrosian, "Theanalyticsignalrepresentation ofmodulared waveforms," Proceedings oftheIRE, vol.50,pp.2071-2076, 1962. P.A.Bello,"Characterization ofrandomly time-variant linearchannels," IEEETransactions on Communication Systems, vol.CS-ll,pp.360-393, 1963. S.Benedetto andG.Montorsi, "Unveiling turbocodes:Someresultsonparallelconcatenated coding schemes," IEEETransactions onInfonnation Theory,vol.42,pp.409-428, 1996. W.R.Bennett,"Spectra ofquantized signals," BellSystemTech.J.,vol.27,pp.446-472, 1948. N.Benvenuto, etaI.,"The32kb/sADPCM codingstandard," AT&TTechnical Journal,vol.65, pp.12-22,Sept.lOct. 1986. C.BerrouandA.Glavieux, "Nearoptimum errorcorrecting codinganddecoding: turbocodes," IEEETransactions onCommunications, vol.44,pp.1261-1271,1996. C.BerrouandA.Glavieux, "Reflections onthePrizePaper:Nearoptimum error-correcting coding anddecoding turbocodes,"IEEEInformation TheorySocietyNewsletter, vol.48,no.2,p. 1andpp.24-31,June1998. C.Berrou,A.Glavieux, andP.Thitrnajshima, "NearShannon limiterror-correction codingand decoding: turbocodes,"International Conference onCommunications, pp.1064-1090, Ge neva,Switzerland, May1993. V.K.Bhargava, "Forward errorcorrection schemesfordigitalcommunications," IEEECommuni cationsMagazine, vol.21,no.1,pp.11-19,1983. R.C.BoseandD.K.Ray-Chaudhuri, "Onaclassoferrorcorrecting binarygroupcodes,"Infor mationandControl,vol.3,pp.68-79,1960. K.Brandenburg andG.Stoll,"ISO-MPEG-l Audio:Agenericstandard forcodingofhigh-quality digitalaudio,"JournaloftheAudioEngineering Society,vol.42,pp.780-792, 1994. D.G.Brennan, "Lineardiversitycombining techniques," Proceedings oftheIRE,vol.47,pp.1075 1102,1959. A.BuZQ,A.H.Gray,Jr.,R.M.Gray,andJ.D.Markel,"Speechcodingbaseduponvectorquanti zation,"IEEETransactions onAcoustics, Speech,andSignalProcessing, vol.ASSP-28, pp. 562-574, 1980. C.R.Cahn,"Combined digitalphaseandamplitude modulation communication systems," IRE Transactions onCommunication Systems, vol.CS-8,pp.150-155, 1960. J.R.Carson,"Notesonthetheoryofmodulation," Proceedings oftheIRE,vol.10,pp.57-64, 1922. J.R.CarsonandT.C.Fry,"Variable frequency electriccircuittheorywithapplication tothetheory offrequency modulation," BellSystemTech.].,vol.16,pp.513-540, 1937. E.F.CasasandC.Leung,"OFDM fordatacommunication overmobileradioFMchannels," IEEE Transactions onCommunications, vol.39,pp.783-793, 1991. J.G.Chaffee,"Theapplication ofnegativefeedback tofrequency-modulation systems," BellSystem Tech.].,vol.18,pp.404-437,1939. R.W.Chang,"Synthesis ofband-limited orthogonal signalsformultichannel datatransmission," BellSystemTech.J.,vol.45,pp.1775-1796, 1996. W.Y.Chen,G.H.1m,and].J.Werner,"Designofdigitalcarrierless AM!PMtransceivers," Standard Pro;ect,TIE1.4192-149, AT&TandBellcore, August19,1992. S.Chennakeshu andG.J.Sauliner, "Differential detection of1T14-shifted-DQPSK fordigitalcellular radio,"IEEETransactions onVehicular Technology, vol.42,pp.46-57,1993. J.M.Cioffi,V.Oksman, ].-].Werner,T.Pollet,P.M.P.Spruyt,J.S.Chow,andK.S.Jacobsen, "Very high-speed digitalsubscriber lines,"IEEECommunications Magazine, vol.37,pp.72-79, April,1999. L.].Cimini,Jr.,andY.Li,"Orthogonal frequency divisionmultiplexing forwirelesscommunica- 784 BIBLIOGRAPHY tions,"tutorialnotes,TU18,International Conference onCommunications '99,Vancouver, BritishColumbia, Canada,June,1999. A.C.Clarke,"Extraterrestrial relays,"Wireless World,vol.51,pp.305-308, October1945. C.E.CookandH.S.Marsh,"Anintroduction tospreadspectrum," IEEECommunications Maga zine,vol.21,no.2,pp.8-16,1983. J.P.Costas, "Synchr~Jnous communications," Proceedings oftheIRE,vol.44,pp.1713-1718, 1956. J.P.Costas,"Poisson, Shannon, andtheradioamateur," Proceedings oftheIRE,vol.47,pp.2058 2068,1959. M.G.Crosby,"Frequency modulation noisecharacteristics," Proceedings oftheIRE,vol.25,pp. 472-514, April1937. C.C.Cutler,"Differential quantization ofcommunication signals," UnitedStatesPatent2-505-361, 1952. C.L.Dammann, L.D.McDaniel andC.L.Maddox, "D2channelbank-Multiplexing andcoding," BellSystemTech.J.vol.51,pp.1675-1699, 1972. F.Daneshgaran andM.Mondin, "Designofinterleavers forturbocodes:Iterativeinterleaver growth algorithms ofpolynomial complexity," IEEETransactions onInformation Theory,vol.45, pp.1845-1859, 1999. R.deBuda, "Coherent demodulation offrequency-shift keyingwithlowdeviation ratio,"IEEE Trans.onCommunications, vol.COM-20, pp.429-435, 1972. F.E.Dejager, "Deltamodulation, amethodofPCMtransmission usingthe1-unitcode,"Phillips Research Reports, vol.7,pp.442-46, 1952. F.E.DejagerandC.B.Dekker,"Tamedfrequency modulation: Anovelmethodtoachievespectrum economy indigitaltransmission," IEEETransactions onCommunications, vol.COM-26, pp. 534-542, 1978. lA.Develet,"Athreshold criterion forphase-lock demodulation," Proceedings oftheIEEE,vol. 51,pp.349-356, 1963. W.Diffie.andM.E.Hellman, "Newdirections incryptography," IEEETransactions onInformation Theory,vol.IT-22,pp.644-654, 1976. W.DiffieandM.E.Hellman, "Privacy andauthentication: Anintroduction tocryptography," Pro ceedingsoftheIEEE,vol.67,pp.397-427, 1979. D.Divsalar, "Turbocodes,"MILCOM 96tutorial,SanDiego,November 1996. M.1.DoelzandE.H.Heald,"Minimum shiftdatacommunication system," U.S.Patent2977417, March1961. R.M.Dolby,"Anaudioreduction system," JournaloftheAudioEngineering Society,vol.15,p. 383,1967. J.Dungundji, "Envelopes andpre-envelopes ofrealwave-forms," IRETransactions onInformation Theory,vol.IT-4,pp.53-57,1958. P.Elias,"Coding fornoisychannels," IREConvention Record,Part4,pp.37-46,March1955. L.H.Enloe,"Decreasing thethreshold inFMbyfrequency feedback," Proceedings oftheIRE,vol. 50,pp.18-30,1962. V.M.Eyuboglu, "Detection ofcodedmodulation signalsonlinear,severelydistorted channels using decision-feedback noisepJediction withinterleaving," IEEETransactions onCommunica tions,vol.COM-36, pp.401-409, 1988. D.D.Falconer, "Carrierless AMJPM," BellLaboratories, InternalMemorandum, July3,1975. K.Feher,"MODEMS foremerging digitalcellular-mobile radiosystem," IEEETransactions on Vehicular Technology, vol.40,pp.355-365, 1991. J.L.Flanagan, M.R.Schroeder, B.S.Atal,R.E.Crochiere, N.S.Jayant,andJ.M.Tribolet, "Speech coding," IEEETransactions onCommunications, vol.COM-27, pp.710-737,1979. B.LeFloch, R.Halbert-Lassalle, andD.Castelain, "Digitalsoundbroadcasting tomobilereceivers," IEEETransactions ofBroadcasting, vol.35,pp.493-503, 1989. G.D.Forney,Jr.,"Maximum likelihood sequence estimation ofdigitalsequences inthepresence of intersymbol interference," IEEETransactions onInformation Theory,vol.IT-18,pp.363 378,1972. G.D.Forney,Jr."TheViterbialgorithm," Proceedings oftheIEEE,vol.61,pp.268-278, 1973. Papers, Reports, Patents 785 G.D.Forney,Jr.,andM.V.Eyuboglu, "Combined equalization andcodingusingprecoding," IEEE Communications Magazine, vol.29,no.12,pp.25-34,1991. G.D.Forney,Jr.,L.Brown,M.V.Eyuboglu, and].L. MoranIII,"TheV.34high-speed modem standard," IEEECommunications Magazine, pp.28-93,December 1996. L.E.Franks,"Carrier andbit synchronization indatacommunications-A tutorialreview," IEEE Transactions onCommunications, vol.COM-28, pp.1107-1121, 1980. B.].FreyandD.].e.MacKay, "Irregnlar turbocodes," Proceedings ofthe37thAnnualAllerton Conference onCommunication, Control,andComputing, AllertonHouse,illinois,September 1999. H.T.Friis,"Noisefiguresinradioreceivers," Proceedings oftheIRE,vol.32,pp.419-422, 1944. K.E.FulzandD.B.Penick,"T1carriersystem," BellSystemTech.j.,vol.44,pp.1405-1451, 1965. D.Gabor,"Theory ofcommunications," journalofIEE(London), vol.93,PartIII,pp.429-457, 1946. D.L.Gall,"MPEG: avideocompression standard formultimedia applications," Communications oftheACM,vol.34,pp.47-58,1991. R.G.Gallager, "Low-density parity-check codes,"IRETransactions onInformation Theory,vol. 8,pp.21-28,1962. W.A.Gardner, "Introduction toEinstein's contribution totime-series analysis," IEEEASSPMag azine,vol.4,pp.4-5,October 1987. W.A.Gardner andL.E.Franks,"Characterization ofcyclostationary randomsignalprocesses," IEEETransactions onInformation Theory,vol.IT-21,pp.4-14,1975. D.A.George,"Matched filtersforinterfering signals," IEEETransactions onInformation Theory, vol.IT-11,pp.153-154, 1965. A.Gersho,"Adaptive equalization ofhighlydispersive channels fordatatransmission," BellSystem Tech.j.,vol.48,pp.55-70,1969. R.A.Gibbyand].W.Smith,"Someextensions ofNyquist'S telegraph transmission theory," Bell SystemsTech.j.,vol.44,pp.1487-1510, 1965. R.D.GitlinandE.Y.Ho,"Theperformance ofstaggered quadrature amplitude modulation inthe presence ofphasejitter,"IEEETransactions onCommunications, vol.COM-23, pp.348 352,1975. M.].E.Golay,"Noteondigitalcoding," Proceedings oftheIRE,vol.37,p.657,1949. M.J.E.Golay,"Binarycoding," IRETransactions onInformation Theory,vol.PGIT-4,pp.23-28, 1954. R.Gold,"Optimal binarysequences forspreadspectrum multiplexing," IEEETransactions on Information Theory,vol.IT-B,pp.619-621, 1967. R.Gold,"Maximal recursive sequences with3-valued recursive crosscorrelation functions," IEEE Transactions onInformation Theory,vol.IT-14,pp.154-156,1968. B.Goode,"Scanning theissue:Specialissueonglobalinformation infrastructure," Proceedings of theIEEE,vol.85,pp.1883-1886, 1997. R.M.Gray,"Vectorquantization," IEEEASSPMagazine, vol.1,no.2,pp.4-29,1984. W.J.Gruen,"Theory ofAFCsynchronization," Proceedings oftheIRE,vol.41,pp.1043-1048, 1953. P.Guinand and].Lodge,"Trellistermination forturboencoders," Proceedings of18thBiennial Symposium onCommunications, Queen'sUniversity, Kingston, Canada,June1996. D.W.Hagelbarger, "Recurrent codes:Easilymechanized, burst-correcting binarycodes,"BellSys temTech.j.,vol.38,pp.969-984, 1959. J.Hagenauer, E.Offer,and1.Papke,"Iterative decoding ofbinaryblockandconvolutional codes," IEEETransactions onInformation Theory,vol.42,pp.429-445, 1996. J.Hagenauer andP.Hoeher,"AViterbialgorithm withsoft-decision outputsanditsapplications," IEEEGlobecom 89,pp.47.11-47.17, November 1989,Dallas,Texas. R.W.Hamming, "Errordetecting anderrorcorrecting codes,"BellSystemTech.j.,vol.29,pp. 147-160,1950. J.e.Hancock andR.W.Lucky,"Performance ofcombined amplitude andphase-modulated com munication systems," IRETransactions onCommunication Systems, vol.CS-8,pp.232-237, 1960. 786 BIBUOGRAPHY H.H.Hanning andJ.W.Pan,"02channelbanksystemaspects," BellSystemTech.J.,vol.51,pp. 1641-1657,1972. H.Harashima andH.Miyakawa, "Matched-transmission technique forchannelswithintersymbol interference," IEEETransactions onCommunications, vol.COM-20, pp.774-779, 1972. T.V.L.Hartley,"Transmission ofinformation," BellSystemTech.J.,vol.7,pp.535-563, 1928. F.S.Hill,Jr.,"Ontime-domain representations forvestigialsideband signals," Proceedings ofthe IEEE,vol.62,pp.1032-1033, 1974. D.A.Huffman, "Amethodfortheconstruction ofminimum redundancy codes,"Proceedings ofthe IRE,vol.40,pp.1098-1101, 1952. P.A.Humblet andM.G.Troulis,"Theinformation driveway," IEEECommunications Magazine, pp.64-68,December, 1996. G.-H.1mandJ.-J.Werner,"Bandwidth-efficient digitaltransmission overunshielded twisted-pair wiring," IEEEJournalonSelectedAreasinCommunications, vol.13,pp.1643-1655, 1995. H.Insoe,Y.Yasuda,andJ.Murakami, "Atelemetering systembycodemodulation: 6.-~modula tion,"IRETransactions onSpaceElectronics andTelemetry, vol.SET-8,pp.204-209, 1962. M.Ishizuka andK.Hirade,"Optimum Gaussian filteranddeviated-frequency lockingschemefor coherentdetection ofMSK,"IEEETransactions onCommunications, vol.COM-28, pp.850 857,1980. 1.M.Jacobs,"Practical applications ofcoding," IEEETransactions onInformation Theory,vol.IT 20,pp.305-310, 1974. N.S.Jayant,"Adaptive deltamodulation withaone-bitmemory," BellSystemTech.J.,vol.49,pp. 321-342, 1970. N.S.Jayant,"Digitalcodingofspeechwaveforms, PCM,DPCMandOMquantizers," Proceedings oftheIEEE,vol.62,pp.611-632, 1974. N.S.Jayant,"Coding speechatlowbitrates,"IEEESpectrum, vol.23,no.8,pp.58-63,1986. A.J.Jerri,"TheShannon sampling theorem-its variousextensions andapplications: Atutorial review," Proceedings oftheIEEE,vol.65,no.11,pp.1565-1596, 1977. J.B.Johnson, "Thermal agitation ofelectricity inconductors," PhysicalReview,secondseries,vol. 32,pp.97-109,1928. P.KabalandS.Pasupathy, "Partial-response signaling," IEEETransactions onCommunications, vol.COM-23, pp.921-934, 1975. 1.Kalet,"Themultitone channel," IEEETransactions onCommunications, vol.37,pp.119-124, 1989. 1.Kalet,J.E.Mazo,andB.R.Saltzberg,"ThecapacityofPCMvoiceband channels," IEEEInter national Conference onCommunications, pp.507-511, Geneva,Switzerland, 1993. H.Kaneko,"Aunifiedformulation ofsegmentcompanding lawsandsynthesis ofcodesanddigital companders," BellSystemTech.J.,vol.49,pp.1555-1588, 1970. A.I.Khintchine, "Korrelationstheorie derstationiiren stochastischen Prozese," Mathematiche An nalen,vol.1,109,pp.415-458, 1934. H.Kobayashi, "Correlative levelcodingandmaximum-likelihood decoding," IEEETransactions onInformation Theory,vol.IT-17,pp.586-594, 1971. R.Kohno,"Spatialandtemporal communication theoryusingadaptiveantennaarray,"IEEEPer sonalCommunications, pp.28-35,February, 1998. E.T.Kretzmer, "Generalization ofatechnique forbinarydatacommunications," IEEETransactions onCommunication Technology, vol.COM-14, pp.67-68,Feb.1966. F.R.Kschischang andB.J.Frey,"Interactive decoding ofcompound codesbyprobability propaga tioningraphical models," IEEEJournalonSelectedAreasinCommunication, vol.16,pp. 219-230, 1998. J.W.Lechleider, "Linecodesfordigitalsubscriber lines,"IEEECommunications Magazine, vol.27, pp.25-32,September 1989. B.M.Leiner,V.G.Cerf,D.O.Clark,R.E.Kohn,1.Kleinrock, D.C.Lynch,J.Postel,L.G.Roberts, andS.Wolff,"AbriefhistoryoftheInternet," Commun. ACM,vol.40,pp.102-108, Feb ruary1997. A.Lender,"Theduobinary technique forhigh-speed datatransmission," IEEETransactions on Communications andElectronics, vol.82,pp.214-218, May1963. Papen,Reporls,Parenb 787 A.Lender,"Correlative digitalcommunication techniques," IEEETransactions onCommunication Technology, vol.COM-12, pp.128-135,1964. A.Lender,"Correlative levelcodingforbinary-data transmission," IEEESpectrum, vol.3,no.2, pp.104-115, 1966. N.-S.Linande.-P.J.Tzeng,"Full-duplex dataoverlocalloops,"IEEECommunications Magazine, vol.26,pp.31--42,February 1988. S.Lin,D.J.Costello, andM.J.Miller,"Automatic-repeat-request errorcontrolschemes," IEEE Communications Magazine, vol.22,no.12,pp.5-16,1984. Y.Linde,A.BuzoandR.M.Gray,"An'algorithm forvectorquantizer design," IEEETrans.on Communications, vol.COM-28, pp.84-95,1980. D.Linden,"Adiscussion ofsampling theorems," Proceedings oftheIRE,vol.47,pp.1219-1226, 1959. e.L.LiuandK.Feher,"Noncoherent detection of7T/4-shifted systemsinaCCI-AWGNcombined interference environment," Proceedings oftheIEEE40thVehicular Technology Conference, SanFrancisco, 1989. S.P.Lloyd,"Leastsquaresquantization inPCM,"unpublished BellLaboratories Technical Note, 1957.Thisreportwasreprinted inIEEETransactions onInformation Theory,vol.IT-28, pp.129-137, 1982. J.Lodge,R.Young,P.Hoeher, andJ.Hagenauer, "Separable MAP'filters'forthedecoding of productandconcatenated codes,"Proceedings oftheIEEEInternational Conference onCom munications, pp.1740-1745, Geneva,Switzerland, May1993. R.W.Lucky,"Automatic equalization fordigitalcommunication," BellSystemTech.J.,vol.44, pp.547-588, 1965. R.W.Lucky,"Techniques foradaptive equalization ofdigitalcommunication systems," BellSystem Tech.J.,vol.45,pp.255-286, 1966. R.Lugannani, "Intersymbol interference andprobability oferrorindigitalsystems," IEEETrans actionsonInformation Theory,vol.IT-15,pp.682-688, 1969. V.H.MacDonald, "Advanced mobilephoneservice:thecellularconcept," BellSystemTech.J.,vol. 58,pp.15--41,1979. D.J,C.MacKay, "Gooderror-correcting codesbasedonverysparsematrices," IEEETransactions onInformation Theory,vol.45,pp.399--431, 1999. D.J.C.MacKay andR.M.Neal,"NearShannon limitperformance oflowdensityparitycheck codes,"Electronics Letters,vol.33,No.6,pp.457-458, 1997;andvol.32,no.18,pp.1645 1646,1996. D.J.C.MacKay, S.T.Wilson,andM.e.Davey,"Comparison ofconstructions ofirregular Gallager codes,"IEEETransactions onCommunications, vol.47,pp.1449-1454, 1999. J.Max,"Quantizing forminimum distortion," IRETransactions onInformation Theory,vol. IT-6,pp.7-12,1960. K.Maxwell, "Asymmetric digitalsubscriber line:Interimtechnology forthenextfortyyears,"IEEE Communications Magazine, vol.34,pp.100-106, October1996. R.J.McEliece, D.J.e.MacKay, andJ.-F.Cheng,"TurbocodingasaninstanceofPearl'sbelief propagation algorithm," IEEEJournalonSelectedAreasofCommunication, vol.16,pp. 140-152,1998. D.Mennie,"AMstereo:Fivecompeting options," IEEESpectrum, vol.15,no.6,pp.24-31,1978. M.L.Moher,"Cross-entropy anditerative detection," Ph.D.thesis,Deparunent ofSystemsand Computer Engineering, Carleton University, Ottawa,Canada,May1997. M.L.MoherandT.A.Gulliver, "Cross-entropy anditerative decoding," IEEETransactions on Information Theory,vol.44,pp.3097-3104, 1998. P.Monsen, "Feedback equalization forfadingdispersive channels," IEEETransactions onInfor mationTheory,vol.IT-17,pp.56-64,1971. K.H.MuellerandJ.J.Werner, "Ahardware efficientpassband equalizer structure fordatatrans missions," IEEETransactions onCommunications, vol.COM-30, pp.538-541, 1982. K.MurotaandK.Hirade,"GMSK modulation fordigitalmobileradiotelephone," IEEETrans actionsonCommunications, vol.COM-29, pp.1044-1050, 1981. E.Murphy, "Whatever happened toAMstereo?"IEEESpectrum, vol.25,p.17,1988. 788 BIBLIOGRAPHY P.Noll,"MPEGdigitalaudiocodingstandards." InV.K.Madisetti andD.B.Williams (editors), TheDigitalSignalProcessing Handbook, Piscataway, N.J.:IEEEPress,pp.40-1-40-28, 1998. D.O.North,"Ananalysisofthefactorswhichdetermine signal/noise discrimination inpulsedcarrier systems," Proceedings oftheIEEE,vol.51,pp.1016-1027, 1963;thispaperisareprintof aclassified RCAReportpublished in1943. H.Nyquist, "Certain factorsaffecting telegraph speed,"BellSystemTech.J.,vol.3,pp.324-346, 1924. H.Nyquist, "Thermal agitation ofelectricchargeinconductors," PhysicalReview,secondseries, vol.32,pp.110-113, 1928. H.Nyquist, "Certain topicsintelegraph transmission theory," Transactions oftheAlEE,vol.47, pp.617-644, Feb.1928. M.W.Oliphant, "ThemobilephonemeetstheInternet," IEEESpectrum, vol.36,pp.20-28, August,1999. B.M.Oliver,J.R.Pierce,andC.E.Shannon, "Thephilosophy ofPCM,"Proceedings oftheIRE, vol.36,pp.1324-1331, 1948. D.Y.Pan,"Digitalaudiocompression," DigitalTechnical Journal,vol.5,pp.1-14,1993. S.Pasupathy, "Nyquist's thirdcriterion," Proceedings oftheIEEE,vol.62,pp.860-861, 1974. S.Pasupathy, "Correlative coding-A bandwidth-efficient signaling scheme," IEEECommunica tionsMagazine, vol.15,no.4,pp.4-11,1977. S.Pasupathy, "Minimum shiftkeying-A spectrally efficientmodulation," IEEECommunications Magazine, vol.17,no.4,pp.14-22,1979. A.J.PaulrajandB.G.Ng,"Space-time modems forwirelesspersonal communications," IEEEPer sonalCommunications, pp.36-48,February, 1998. A.J.PaulrajandG.B.Papadias, "Space-tiIDe processing forwirelesscommunications," IEEESignal Processing Magazine, pp.49-83,November, 1997. R.L.Pickholtz, D.L.Schilling, andL.B.Milstein, "Theory ofspread-spectrum communications-A tutorial," IEEETransactions onCommunications, vol.COM-30, pp.855-884, 1982. R.Price,"Nonlinearly feedback-equalized PAMvs.capacityfornoisyfilterchannels," International Conference onCommunications, ICC'72,pp.22.12-22.17, June1972,Philadelphia. R.PriceandP.E.Green,Jr.,"Acommunication technique formultipath channels," Proceedings of theIRE,vol.46,pp.555-570, 1958. J.G.Proakis,"Advances inequalization forintersymbol interference," Advances inCommunications Systems, editedbyA.J.Viterbi,vol.4,pp.123-198, Academic Press,1975. S.Qureshi, "Adaptive equalization," IEEECommunications Magazine, vol.20,no.2,pp.9-16, March1982. S.Qureshi, "Adaptive equalization," Proceedings oftheIEEE,vol.73,pp.1349-1387, 1985. T.A.Ramstad, "Stillimagecompression." InV.K.Madisetti andD.B.Williams (editors), TheDigital SignalProcessing Handbook, Piscataway, N.J.:IEEEPress,pp.52-1-52-27, 1998. I.S.ReedandG.Solomon, "Polynomial codesovercertainfinitefields,"JournalofSIAM,vol.8, pp.300-304, 1960. A.H.Reeves,"Thepast,presentandfutureofPCM,"IEEESpectrum, vol.12,no.5,pp.58-63, 1975. S.A.Rhodes,"Effectofnoisyphasereference oncoherent detection ofoffset-QPSK signals," IEEE Transactions onCommunications, vol.COM-22, pp.1046-1055, 1974. S.O.Rice,"Mathematical analysisofrandomnoise,"BellSystemTech.J.,vol.23,pp.282-332, 1944;vol.24,pp.46-156,1945. S.O.Rice,"Statistical properties ofasine-wave plusrandomnoise,"BellSystemTech.j.,vol.27, pp.109-157,1948. S.O.Rice,"Envelopes ofnarrow-band signals," Proceedings oftheIEEE,vol.70,pp.692-699, 1982. T.Richardson, A.Shokrollahi, andR.Urbanke, "Designofprovably goodlow-density paritycheek codes,"submitted in1999toIEEETransactions onInformation Theory. R.L.Rivest,A.Shamir,andL.Adleman, "Amethodforobtaining digitalsignatures andpublickey cryptosystems," Communications oftheACM,vol.21,pp.120-126, 1978. W.L.Root,"Remarks, mostlyhistorical, onsignaldetection andsignalparameter estimation," Pro ceedingsoftheIEEE,vol.75,pp.1446-1457, 1987. Papers, Reports, Patents 789 A.Ruiz,].M.Cioffi,andS.Kasturia, "Discrete multipletonemodulation withcosetcodingforrhe spectrally shapedchannel," IEEETransactions onCommunications, vol.40,pp.1012-1029, 1992. W.D.Rummier, "Anewselective fadingmodel-Application topropagation data,"BellSystem Tech.j.,vol.58,pp.1037-1071,1979. B.R.Saltzberg, "Comparison ofsingle-carrier andmultitone digitalmodulation forADSLapplica tions,"IEEECommunications Magazine, vol.36,pp.114-121, November, 1998. B.R.Saltzberg, "Performance ofanefficientparalleldatatransmission system," IEEETransactions onCommunication Technology, vol.COM-15, pp.805-811, 1967. S.D.Sandberg andM.A.Tzannes, "Overlapped discretemultitone modulation forhighspeedcopper wirecommunications," IEEEJournalonSelectedAreasinCommunications, vol.13,pp. 1571-1585, 1995. D.V.Sarwate andM.B.Pursley,"Crosscorrelation properties ofpseudorandom andrelatedse quences," Proceedings oftheIEEE,vol.68,pp.593-619, 1980. B.SayarandS.Pasupathy, ''Nyquist 3pulseshapingincontinuous phasemodulation," IEEETrans actionsonCommunications, vol.COM-35, pp.57-67,1987. H.R.Schindler, "Deltamodulation," IEEESpectrum, vol.7,no.10,pp.69-78,1970. R.A.Scholz,"Theoriginsofspread-spectrum communications," IEEETransactions onCommu nications, vol.COM-30, pp.822-854, May1982. R.A.Scholz,"Notesonspread-spectrum history," IEEETransactions onCommunications, vol. COM-31, pp.82-84,1983. l.S.Schouten, F.Delager, andl.A.Greefkes, "Deltamodulation, anewmodulation systemfor telecommunication," PhillipsTechnical Review,vol.13,pp.237-245, 1952. CE.Shannon, "Amathematical theoryofcommunication," BellSystemTech.J.,vol.27,pp.379 423,623-656,1948. CE.Shannon, "Communication theoryofsecrecysystems," BellSystemTech.J.,vol.28,pp.656 715,1949. CE.Shannon, "Communicarion inthepresence ofnoise,"Proceedings oftheIRE,vol.37,pp.10 21,1949. M.K.SimonandD.Divsalar, "Ontheimplementation andperformance ofsingleanddoubledif ferential detection schemes," IEEETrans.onCommunications, vol.40,pp.278-291, 1992. B.Sklar,"Aprimeronturbocodeconcepts," IEEECommunications Magazine, vol.35,pp.94 102,December 1997. B.Sklar,"Astructural overview ofdigitalcommunications-A tutorialreview," PartI,IEEECom munications Magazine, vol.21,no.5,pp.4-17,1983;PartII,vol.21,no.7,pp.6-21,1983. D.Slepian,"Onbandwidth," Proceedings oftheIEEE,vol.64,pp.292-300, 1976. B.Smith,"Instantaneous companding ofquantized signals," BellSystemTech.j.,vol.36,pp.653 709,1957. E.S.SousaandS.Pasupathy, "Pulseshapedesign for teletextdatatransmission," IEEETrans.on Communications, vol.COM-31, pp.871-878, 1983. S.Stein,"Unified analysisofcertaincoherent andnoncoherent binarycommunication systems," IEEETransactions onInformation Theory,vol.IT-I0,pp.43-51,1964. CE.Sundberg, "Continuous phasemodulation," IEEECommunications Magazine, vol.24,no.4, pp.25-38,1986. M.Tomlinson, "Newautomatic equaliser employing moduloarithmetic," Electronics Letters,vol. 7,pp.138-139, March1971. D.W.Tufts,"Nyquist's problem-The jointoptimization oftransmitter andreceiverinpulseam plitudemodulation," Proceedings oftheIEEE,vol.53,pp.248-259, 1965. G.L.Turin,"Anintroduction tomatched filters,"IRETransactions onInformation Theory,vol. IT-6,pp. 311-329, 1960. G.L.Turin,"Anintroduction todigitalmatched filters,"Proceedings oftheIEEE,vol.64,pp.1092 1112,1976. G.Ungerboeck, "Channel codingwitbmultilevelfphase signals,"IEEETransactions onInformation Theory,IT-28,pp.55-67,1982. 790 BIBLIOGRAPHY G.Ungerboeck, "Trellis-coded modulation withredundant signalsets,"Parts1and2,IEEECom munications Magazine, vol.25,no.2,pp.5-21,1987. M.e.Valenti,"Anintroduction toturbocodes,"EEDepartment, VirginiaPolytechnic Institute& StateUniversity, Blacksburg, Virginia, unpublished, 1998. B.vanderPol,"Thefundamental principles offrequency modulation," JournaloflEE(London), vol.93,partIII,pp.253-258, 1946. J.H.VanVleckandD.Middleton, "Atheoretical comparison ofvisual,aural,andmeterreception of pulsedsignalsinthepresenceofnoise,"JournalofAppliedPhysics,vol.17,pp.940-971, 1946. A.J.Viterbi,"Errorboundsforconvolutional codesandanasymptotically optimum decoding al gorithm," IEEETrans.onInformation Theory,vol.IT-B,pp.260-269, 1967. A.J.Viterbi,"Spread-spectrum communications-Myths andrealities," IEEECommunications Magazine, vol.17,no.3,pp.11-18,May1979. A.J.Viterbi,"Whennottospreadspectrum-A sequel,"IEEECommunications Magazine, vol.23, no.4,pp. 12-17,1985. A.J.Viterbi,"Wireless digitalcommunication: Aviewbasedonthreelessonslearned," IEEECom munications Magazine, vol.29,no.9,pp.33-36,1991. G.K.Wallace, "The]pEGstillpicturecompression standard," Communications oftheACM,vol. 34,pp.31-44,1991. D.K.Weaver,Jr.,"Athirdmethodofgeneration anddetection ofsingle-sideband signals," Pro ceedingsoftheIRE,vol.44,pp.1703-1705,1956. J.WeissandD.Schremp, "Putting dataonadiet,"IEEESpectrum, vol.30,pp.36-39,August 1993. T.A.Welch,"Atechnique forhighperformance datacompression," Computer, vol.17,no.6,pp. 8-19,1984. L.-F.Wei,"Rotationally invariant convolutional channelcodingwithexpanded signalspace-part I:180°,"IEEEJournalonSelectedAreasinCommunications, vol.SAC-2,pp.659-671, 1984. L.-F.Wei,"Rotationally invariant convolutional channelcodingwithexpanded signalspace-part II:nonlinear codes,"IEEEJournalonSelectedAreasinCommunications, vol.SAC-2,pp. 672-686, 1984. L.-F.Wei,"Trellis-coded modulation withmultidimensional constellations," IEEETransactions on Information Theory,vol.IT-33,pp.483-501, 1987. S.B.Weinstein, "Echocancellation inthetelephone network," IEEECommunications Magazine, vol.15,no.1,pp.8-15,1977. S.B.Weinstein andP.M.Ebert,"Datatransmission byfrequency-division multiplexing usingthe discreteFouriertransform," IEEETransactions onCommunications, vol.COM-19, pp.628 634,1971. J.J.Werner,"Tutorial oncarrierless AMlPM-Part I:Fundamentals anddigitalCAPtransmitter," AT&TBellLaboratories Report,Minneapolis, June23,1992. J.J.Werner, "Tutorial oncarrierless AMlPM-Part II:Performance ofbandwidth-efficient line codes,"AT&TBellLaboratories Report,Middletown, February 6,1993. B.WidrowandM.E.Hoff,Jr.,"Adaptive switching circuits," WESCON Convention Record,Pt. 4,pp.96-104, 1960. J.H.Winters, "Smartantennas forwirelesssystems," IEEEPersonal Communications, pp.23-27, February, 1998. J.H.Winters, "Adaptive antennas forwirelesscommunications," International Conference onCom munications '99,TutorialNotesTU5,Vancouver, June6,1999. A.D.Wyner,"Fundamental limitsininformation theory," Proceedings oftheIEEE,vol.69,pp. 239-251, 1981. J.L.Yen,"Onthenon-uniform sampling ofbandwidth-limited signals,"IRETransactions onCircuit Theory,vol.CT-3,pp.251-257, 1956. O.e.Yue,R.Luganani, andS.O.Rice,"Seriesapproximations fortheamplitude distribution and densityofshotprocesses," IEEETransactions onCommunications, vol.COM-26, pp.45 54,1978. INOTESNotes 791 N.ZervosandI.Kalet,"Optimized decisionfeedback equalization versusoptimized orthogonal frequency divisionmultiplexing forhigh-speed datatransmission overthelocalcablenet work,"International Conference onCommunications, ICC'89,pp.35.2.1-35.2.6, June1989. J.ZivandA.Lempel,"Auniversal algorithm forsequential datacompression," IEEETransactions onInformation Theory,vol.IT-23,pp.337-343, 1977. J.ZivandA.Lempel,"Compression ofindividual sequences viavariable-rate coding," IEEETrans actionsonInformation Theory,vol.IT-24,pp.530-536, 1978. W.Y.ZouandY.Wu,"COFDM: Anoverview," IEEETransactions onBroadcasting, vol.41,pp. 1-5,1995. 1.Thefollowing abbreviations areusedforsomeofthejournalpapers: ACM:Association forComputing Machinery AIEE:American Institute ofElectrical Engineers IEEE:InstituteofElectrical andElectronics Engineers IEE:Institution ofElectrical Engineers (London) IRE:Institute ofRadioEngineers SIAM:SocietyforIndustrial andAppliedMathematics A absolute entropy, 594 accumulative error,221-222 acquisition, 493 adaptive, 230 adaptive antennaattay,557 optimizing performance of,558 asspecialized technique, 559 structure 0f,557 useof,559 adaptive deltamodulation (ADM), 232-235 adaptive delta,modulation algorithm, 232 adaptive deltamodulation system, 234-235 adaptive diffetential pulse-code modulation (ADPCM), 229-232 desctiption of,230-231 useof,231 fotvoicesignals,232 adaptive equalization algorithm for,287-288 asthemethodofchoice,379 adaptive equalizer modesofoperation for,290 prefetted approach of,297 trackingcapability of,291 adaptive filtering, 297 adaptive prediction withbackward estimation, 231 disadvantages of,231 withforwardestimation, 231 prefetted methodof,231 schemesfor,230 adaptive predictor, 225 adaptive quantization, 230 withbackward estimation, 230 withforwardestimation, 230 problems of,230 adaptive quantizer, 230 792adaptive receiver,287 adaptive spatialprocessing, 557 adaptive synchronous equalizer, 287 adders,646 additivecode-modulated interference, 493 additivewhiteGaussian noise (AWGN),378 aschannelimpairment, 379 andreceiverdesign,337 signaldetection in,349 additivewhiteGaussian noise (AWGN)channel, 559 capacity, 431-432 characterizati~n of,322 andettorprobability, 332 receivedsignalfor,403 andsignaldetection, 329 andsignaltransmission, 309 techniques in,464 additivewhiteGaussian noise (AWGN)model,516 adjustment signal,453 ADM.Seeadaptive delta modulation ADSL.Seeasymmetric digital subscriber lines Advanced Research ProjectAgency Network (ARPANET) andimpactoncomputer communications, 28 andpioneering work,28 A-law capabilities of,203 definedas,202,203 algebraic code properties of,641-642 typesof,693 algebraic decoder, 629-630 aliasedspectrum, 187 aliasing,187allowedfrequency band,154-155 alternate markinversion (AMI) signaling, 207 AM.Seeamplitude modulation American Standatd Codefor Information Interchange (ASCII),6,762 amplification, 128 amplitude distortion, 191 amplitude limiter,129, 142, 143 amplitude-modulated signal, 89-90 amplitude modulation (AM),20, 89-90,422,729 definition of,90 limitations of,92-93 meritsof,162-163 typesof,162-163 virtuesof,92-93 amplitude modulation system noiseanalysisof,135 processof,90-91 amplitude quantization, 194 amplitude sensitivity, 90 amplitude-shift keying(ASK) basicsignaling scheme,344-345 signals,345 amplitude-shift keying(ASK) modulation, 345 AMreceiver comparison of,136-137 envelope detection, 135-137 modelof,135 performance of,136-137 AMsignal demodulation of,162-163 andFMsignalcomparison, 164 AM-to-PM conversion, 127-128 analogcommunication system designof,22 reasonsforstudyof,23 useof,21 analoginformation-bearing signal, 184 analogmodem designphilosophy of,429-430 limitedoperation of,429 noiseperformance of,429 analogpulsemodulation, 20 featureof,236 optimum formof,193 transmission of,183 variations of,236 analogtelevision, 5 analog-to-digital (AID)converter, 201,379-380,445 analysis-by-synthesis codec,552 analysisequation, 442-443 analyticsignal.Seepre-envelope angle,107,312 angle-modulated signal,387 interpretation of,108 waveform of,89-90 anglemodulation, 20,89-90 classification of,163 formsof,108-109 important featureof,107 provisions of,107 angularvelocity, 108 antenna beamwidth,521 designing of,17-18 multibeam use,514 receiving endof,18 antenna, multibeam, 514 antennaarrays,553 anti-aliasing filter,187 antijamcharacteristics, 498-499 antipodal signal,349 apertureeffect,191 aprioriprobabilities, 323-'324, 583 arctangent computer, 364 argument function, 54 Armstrong, EdwinH.,27 ARQ.Seeautomatic repeatrequest arrayoutputsignal,557 arraysignalprocessor, 553,556 ASCII.SeeAmerican Standard CodeforInformation Interchange asymmetric digitalsubscriber lines (ADSL) advantages of,446 motivation for,282 servicessupported by,281 useof,446asymmetric modem configuration of,425-426 designof,426 asymmetry ratio,282 asymptotic codinggain,668,673 asynchronous transfermode (ATM),14-15 asynchronous transmission, 7 ATM.Seeasynchronous transfer mode auditory masking, 9 auditory maskingphenomenon, 234 auditory system,235 autocorrelation function, 36,43, 482 definition of,35 evaluation of,51 graphical summary of,75-76 properties of,36-37 significance of,37 autocovariance function, 36 automatic-repeat request(ARQ), 628-629 forerrordetection, 628 philosophy of,628 typesof,628-629 AWGN.Seeadditivewhite Gaussian noise B band-limited channel,3 band-limited signal definedas,427 sampling theoremfor,186-187 band-limited whiteGaussian noise, 608 bandpass communication channel, 348-349 band-pass filter,98-99,515 band-pass signal components of,728,730 Hilberttransform and,731 representation of,113,726-729 band-pass system analysi;of,730-734 impulseresponse, 731 bandwidth, 720-723 definitions of,720-721 efficiency of,347,348 bandwidth-duration product, 721-722 bandwidth efficiency, 347,348 definedas,347INDEX 793 diagramof,601-602 productof,348 bandwidth-limited channel,16 bandwidth-noise trade-off, 193 Bardeen, John,28 barragenoisejammer,508 baseband, 88 baseband binarydatatransmission system,259 baseband binaryPAMsystem,259 baseband channel channelrequirements, 247 anddigitaldatatransmission, 247 baseband M-aryPAM transmission, 275-277 baseband powerspectraldensity ofabinaryPSKsignal,353 toevaluate, 347 basebandpulse,374 baseband-pulse transmission system andfixedcharacteristics, 297 performance of,296 andsignal-to-noise ratio,297 sourceofbiterrors,259 baseband signal,88,95 baseband signal-to-noise ratio,154 baseband space-time processor, 556 baseband spread-spectrum system, 488 basestation,530 basisfunction, 451 Baudot,Emile,26 bauds,276 Bayes'rule,585-587, 707 BCHcodes,653-654 BCJRalgorithm formulation of,678 mathematical exposition of,680 purposeof,678 versustheViterbialgorithm, 678 Bell,Alexander Graham, 27 Berners-Lee, Tim,28-29 Bessel'sequation, 735 Besselequation, modified, 738 Besselfunction, 735-739 behavior of,114-115 versusthemodulation index,114 properties of,735-737 Besselfunction, modified, 737-739 "besteffortservice", 14 binaryadditivestreamciphers,744 binaryBCHcode common typesof,653 794INDEX binaryBCHcode(Continued) versusnonbinary, 654 binarycode efficiency of,197 symbolsof,204 binaryCRCcodes capabilities of,652 anderrordetection, 652-653 binarydatasequence, 359 binarydatatransmission system, 285 binarydetection problem, 403 binarydifferential phase-shift keying(DPSK),415 binarydigit,204,569 binarydigitalcommunication system,403-405 binaryfrequency modulation, 397 binaryfrequency-shift keying(FSK) biterrorratefor,384 errorprobability of,382-384 binaryFSKsignal baseband powerspectraldensity of,386 withcontinuous phase,385-386 detection of,387 togenerate, 384 powerspectraof,353,385-386 binaryFSKsystem,381-386 binaryFSKtransmitter, 384-385 binaryhypothesis test,405 binary-input additivewhite Gaussian noise(AWGN) channel, 667,668 binaryphase-shift keying(PSK), 349-353 errorprobability of,350-352 asalinearoperation, 492 modulator, 490 signals,353 andspread-spectrum modulation, 550 transmitter, 352 useof,550 binarypulse,coded,193 binarypulsecodemodulation (PCM)wave,193 binarysignaling biterrorrate,543-544 scheme,407 binarysymmetric channel(BSC), 258,629,667 bipartite graphs,685bipolarcode,281 bipolarretum-to-zero (BRZ) signaling, 207 B-ISDN. Seebroadband integrated servicesdigitalnetwork bit,204,569 bit-by-bit interleaving procedure, 214 bitduration, 253 bitenergy-to-noise densityratio, 384 biterrorrate(BER) assumptions of,209 forcoherent binaryFSK,384 ofdigitalmodulation schemes, 417 probability of,23 insignalregeneration, 208 and.symbol errorprobability, 335-336 bit-ratereduction, 218 bitstuffing,215 blanking pulse,S blockcipher,744,745,746 blockcode,590,632 distinguishing featureof,627 rateof,685 blocks,572 Bose-Chaudhuri-Hocquenghem (BCH)codes,653-654 bo.{nds forprediction, 332 useof,332 Brattain, WalterH.,28 BritishBroadcasting Corporation (BBC),27 broadband integrated services digitalnetwork (B"ISDN), 14-15 cellsin,15 qualityofservice,14 andthetelephone network, 14 broadband networks, 14-15 broadcasting mode,2-3 broadcasting system,128 burstiness, 7 byte,7 C cable-television systems,17 Campbell's theorem, 60 CAP.Seecarrierless amplitude/ phasecapacity, 568 capacityboundry, 602 captureeffect,148-149 carrierandtimingsynchronization systems,449 carriercomponent, 135 carrierfrequency, 92 carrier-frequency tuning,128 carrierless amplitude/phase (CAP), 369,380 carrierless amplitude/phase (CAP) modulation, 373,431 bandwidth for,375 ideabehind,369 carrierless amplitude/phase (CAP) receiver digitalimplementation of, 379-380 improved performance of,379 inanunknown environment, 379 carrierless amplitude/phase (CAP) system application of,380 basicstructure of,378-379 modulation, 369,373,375 receiver,379-380 structure of,378-379 transmitter, 378 carrierless amplitude/phase (CAP) transmitter, 378 carrierphase,403 carrierphaserecovery, 448,458, 459-463 carriersynchronization, 448 carrier-to-noise ratio,137-139, 144 definedas,138,150 largeversussmall,141-142 levelofoperation, 138 lowversushigh,149 versussignal-to-noise ratio,151 carrierwave,19-20 Carson's rule forapproximate evaluation, 163 andtheuniversal curve,119 Cartesian product, 369-370 cascadeconnection noisefigureof,526 oftwo-port networks, 524-525 catastrophic code,667 CDMA. Seecodedivisionmultiple access COMsystems,505 celldelay definedas,14 variation of,14 celllossratio,14 cells,14-15 cellsplitting, 531 cell-switching technology, 14 cellularconcept,530 cellularradio idealized modelof,530 propagation problems of,532 wireless communications inthe contextof,530 CELP codecimplementation, 553 distinguishing featureof,552 encoderfor,552 modeling of,616 Seealsocode-excited LPC centrallimittheorem, 60,498 definition of,56 andtheGaussian process,55 centralmoments, 712 channel, 2 characteristics of,309-310 frequency response of,261 andrandomnoise,259 channelbandwidth definition of,3 occupancy of,347 primarycommunication resource, 92 usedinNorthAmerica, 102 channelcapacity, 587-589 conceptof,630-631 ofadiscretememoryless channel, 588 channelcapacitytheorem, 599 channelcodeword,21 channelcoding designgoalof,589 andmapping, 589 techniques for,628 channelcodingtheorem, 589-593, 616,630-631 application of,591-592 operations of,628 Shannon's secondtheorem, 590-591 significance of,592 unsatisfactory featureof,631 channeldatarate,627channeldecoder underdesigner's control,590 goalof,627 inversemapping operation, 590 channelencoder, 626 underdesigner's control,590 goalof,627 introduction ofredundancy, 590 mapping operation, 590 Markovian assumption for,681 channelimperfections, 2 channelimpulseresponse, 291 channelinput,597,610 channelmatrix,S 82 channelmodel,130 channelnoise,31-32 absenceof,228 andbiterror,248 condition ofactingalone, 282-283 effectsof,209,296 inPCMsystems,209,253 reducing theeffectof,210 sourceof,32 uncertainty dueto,403 channeloutput,72 channelparameters, 587 channelsignal-to-noise ratio,134 forAM,135 definition of,132 formulafor,147 characteristic function, 713 Chebyshev inequality, 713 checknode,685 chip,488,501 chipduration, 494 chiprate,501 chrominance signal,6 cipher,742 ciphertext, 742 circuit,10-11 circuit-switched network, 11 controlled by,11 establishing aconnection, 11 circuitswitching, 10-11 circulant matrix,442,443 circularconstellation, 368 Clark,Arthurc.,29 clockrecovery, 448 closed-loop optimization procedure, 551 coaxialcable application of,17INDEX 795 consistsof,17 versustwisted-pairs, 17 co-channel cells determination of,531 findingof,531 asinterference,S 53 code,catastrophic, 667 codebook,580 codec,552 codedbinarypulses,193 codedivisionmultipleaccess (COMA) advantage of,514 codesfor,505 systems,548 code-division multiplexing (COM) asanalternative method,505 bandwidth requirements of,505 definedas,21 codedpulse inanalogmodulation, 217 indigitalpulsemodulation, 183 useof,184 codeelements, 203-204, 212 code-excited LPC,552 coderate,590,627,654 codetree,657,657-658 codevectOr,660 codeword,627 averagelength,574 inbinaryform,574 duration of,212 code-word length,average,574 codingefficiency, 574 codinggain,425 codingtheory,28,676 coherence bandwidth, 540 coherent binaryfrequency-shift keying(FSK),380 coherent binaryFSKsystem,384 characterized by,381 generation anddetection of,384 receiver,384-385 coherent binaryphase-shift keying (PSK) biterrorrate,352,417 characteristics of,350 coherent binaryPSKsystem,384 biterrorof,357 characteristics of,350 receiver,352 signals,352 coherent detection, 98,131,133 796 INDEX coherent detection (Continued) anddemodulation, 95 effectof,97-98 useof,132 coherent M-aryPSK,367 coherent M-aryquadrature amplitude modulation (QAM), 464 coherent MSK biterrorratefor,394 expressions for,417 coherent phase-shift keying(PSK), 349 coherent QPSK biterrorratefor,417 symbolerrorprobability, 358 coherent QPSKsystem signalsof,359 specifications of,458 coherent quadriphase-shift keying (QPSK),354 colorednoisechannel, 607-611 colorreceptors inthehumaneye,6 typesof,6 communication, 1 applications of,1 frequencies for,4 fundamentals of,2-3 typesof,21-23 communication, error-free, 568 communication channel, 7,15-19, 277 classification of,3,15,19 description of,15-19 simultaneous useof,512 useof,88 communication link analysisof,517 incircuitswitching, 11 communication networks, 10-15 communication process,1-3 communication resources, 3 communications satellites ingeostationary orbit,19 historical notes,26-29 roleof,19 "second generation", 515 communication system common featureof,3 designof,21 elements of,2 noiseanalysisof,3,64,523 primaryresources of,3purposeof,19,88 sourceoflimitations, 248-252 transition fromanalogtodigital, 183 communication systemdesigner, 23 community-antenna television (CATV)system,17 commutator, 211 compander, 203 companding circuitry, 203 companding law,fifteen-segment, 426 complementary errorfunction, 255, 256,334 complex envelope, 347,727,728, 729,730 complex exponential Fourierseries, 717 complex Fouriercoefficient, 717 complex least-mean-square (lMS) algorithm advantages of,558 limitations of,558 Seealsoleast-mean-square algorithm composite signal,105,149 compound codes,683-684 compression algorithms, standard, 8 compression laws,202-203 compressor, 203 computer communications, 2 computer-generated data,7 conditional likelihood function, 403-404 conditional meanrequirement, 200 conditional probability, 706-707 conditional probability density funcrion, 320-321, 383, 451-452,710 conditional probability oferror, 255,256 conditional probability ofsymbol error,333 confluent hypergeometric function, 740-741 confusion, 749 conservation oftime,211-212 constant angularvelocity, 108 constant envelope, 111 constellation encoder, 444 constrained optimization problem definition of,437 solvingof,609continuous AWGNchannel, 318-319 continuous-phase frequency-shift keying(CPFSK), 381,387 continuous-phase frequency-shift keying(CPFSK) signal components of,389 deviation ratioof,388 phaseof,388 representation of,387 continuous randomvariable, 594, 708 continuous source,615 continuous-time channel partitioning, 432-436 continuous-wave (CW)modulation 20 effectsonreception, 130 familiesof,88-90 principles of,162 techniques for,130 continuous-wave (CW)modulation system,89-90 comparison of,132 components of,88-89 noisein,130 perfonrumce of,164 controlsymbols inASCII,6 forcommunication purposes, 6 forprintingofcharacters, 6 conventional coherent binaryFSK biterrorrateexpressions for, 417 withone-bitdecoding, 417 convolutional code,654-656 constraint lengthof,655 distauceproperties of,663 distinguishing featureof,627 maximum likelihood decoding of,660-663 performance of,663 useof,654-656 convolutional coding,669 convolutional encoder codetreefor,657-658 input-output relationof,660 stateof,657-658 trellisfor,658-659 useof,425 convolution integral,42-44,718 correlation coefficient, 342,471 correlation functions, 35-41 correlation Irultrix,40 correlation receiver,326-328, 329 correlative-level coding,266-271 basisof,268 generalized formof,274-275 ideailJustrated, 267 premiseof,267 correlatar,24-25 inputsof,549-550 outputsof,319-322 cosetleaders,638-639 Costasloop generalization of,454 forphaserecovery, 454 Costasreceiver consistsof,96-97 phasecontrolin,97 useof,96-97 costfunction, 558 covariance function, 35-36 CPFSK.Seecontinuous-phase frequency-shift keying Cramer-Rao bound definedas,462 modification of,462 CRCcode.Seecyclicredundancy check(CRC)code criticalband,234 crossconstellations, 369-370, 372 cross-correlation functions, 40,52 cross-spectral densities, 52 crosstalk, 21 causeof,279 definedas,501 asimpairment, 279 rypesof,279-280 cryptanalysis, 742 andauthorized user,742-743 definition of,746 description of,742 cryptogram, 742-743 cryptographic system classesof,744 classifications of,759 consistsof,743 definition of,743 servicesof,742 cryptography, 617,742 andauthentication problem, 743 classifications of,759 datacompression in,749 fundamental assumption, 745 importance of,759 andsecrecyproblem, 743 cryptology, 742crystal-controlled oscillator, 120 cumulative distribution function, 708 cycliccode,641-643 advantage of,641-642 characteristics of,652-654 classesof,652-654 encoderfor,645-646 generation of,643 properties of,642 insystematic form,645,646 cyclicprefix,441 cyclicproperty, 642 cyclicredundancy check(CRC) code forerrordetection, 652 generator polynomials of,653 D damping factor,160 data-aided synchronization, 449 databits binarypatternof,6 forerrordetection, 7 datacommunication, 7 datacompaction, 8,575 achieved by,575 assessing, 616 schemesfor,575 datacompression, 7,614-616 incryptography, 749 formsof,7-8 ideaof,614 asalossyoperation, 614-615 reasonforusing,615 systemcomponents, 749 techniques for,218 datacompressor, 614 dataencryption, 617 dataencryption standard (DES), 751-755, 759 dataencryption standard (DES) algorithm, 754 data-modulated carrier,500 datamultiplexers, 7 datanetwork, 11 datasignaling rate,426 datatransmission system asynchronous versus synchronous, 7 capabilities of,446 performance of,293 decisiondevice,259 decision-directed mode,290-291INDEX 797 decision-directed recursive algorithm, 450 decisionerrors,210 decisionfeedback, 270 decisionfeedback equalization, 291-293,379,430 decisionfeedback equalizer (DFE) consistsof,292 errorpropagation in,292-293 feedback sectionof,292 feedforward sectionof,292 decision-making criterion, 323 decision-making device designsof,277 operation of,25 decisionrule,350-351, 357 applying, 382 definedas,661 astheMAPrule,323-324 asthemaximum likelihood rule, 324 usedbythecoherent detector, 497 decisionthreshold, 194 decisiontree,575-576 decoder, 446 condition foroptimality, 200-201 consistsof,552 functionof,S52 unitsof,235 decoding algorithms, 678 decoding complexity, 684 decoding decisions, 663 decoding error,660 decoding process methods of,693 andpulsegeneration, 208 requirements of,261 decoding rule,660 decoding spheres maximum numberof,600 packingof,600 decoding window, 663 decommutator,211decorrelation time, 37 decryption, 742 deForest,Lee,27 delay,average,540 delaypowerspectrum, 539 delayspread,539 asachannelimpairment, 553 definedas,540 effectof,556 798 INDEX deltafunction, 62 property of,716-717 siftingproperty of,190,320 deltamodulation (DM),218-221 advantage of,219,237 anddigitalpulsemodulation, 237 principles of,218-219 quantization errorof,220 simplicity of,223 andtransmitters, 228 delta-sigma modulation, 221-223 demodulation, 20,88 methodof,99 stagesof,491 demodulation scheme,132 demodulator output,92 demultiplexer inreceiver, 359,445-446 intransmitter, 444 demultiplexing system,125-126 DES.Seedataencryption standarddetection anderrorcorrection, 628 ofapulsesignal,248 detector, 326, ~349 deviation ratio,387 definition of,119 versusmodulation index,119 DFT.SeediscreteFouriertransform diagonal matrix,443 dibits,276 difference-frequency term,158 differential detector components of,364 tangenttype,364-365 differential encoder ofthebinarywave,414 consistsof,421-422 methodused,207 requirement of,207 differential entropy,593-597 differential phaseencoder, 415 differential phasemodulation, 422 differential phase-shift keying (DPSK),407,41+-417 biterrorrateof,417 generation anddetection of,415 receiver,416 transmittcr,415-416 differential pulse-code modulation (DPCM),227-229 basicideaof,227 anddigitalpulsemodulation, 237systemcomparison, 228 andtransmitters, 228 differential quantization scheme, 227,228 differentiator,143 Diffie-Hellman publickey- distribution system,756 diffraction, 17-18 diffusion, 749 digitalaudiobroadcasting, 448 digitalcircuittechnology, 189 digitalcommunication basicformof,309-310 andbiterrorrate,24 anddesigngoals,354 elements of,24-25 receiver,337 reliability of,23,24-26 requirements of,23 andsystemdesign,22 taskofdesigner, 626 useof,21 digitaldatatransmission, 247 digitalfilter,second-order, 454-455 digitalhierarchy, 214 digitalmodem bidirectional, 428 capabilities of,428 anddatarates,429 designconstraints of,426 fundamental designphilosophy of,426 onerealization of,426-427 signaling schemefor,427 solutiontodesignproblems, 428 theoretical basisforthedesign of,429 digitalmodulation schemes comparison of,417-420 probability oferror,417 typesof,346 usingasinglecarrier,417-420 virtuesof,347 digitalmodulation techniques operation of,448 typesof,345-346 digitalmultiplexers, 214-215 designproblems, 215 majorgroupsof,214 digitalmultiplexing-demultiplexing operation, 214 digitalpassband transmission system,344 assessing performance of,335performance degradation of,544 digitalPSTN,426 digitalpulsemodulation basicformof,193 featureof,236 transmission of,183 digitalsatellitecommunication, 419 digitalsignals,214 digitalsignalzero(DSO),214 digitalsignature forelectronic mailsystems, 758-759 properties of,759 useof,758-759 digitalsubscriber line(DSL) asagrowingapplication, 277 linecodesfor,280-281 operational environment of,277, 447 andtwistedpairs,277,297 versusvoiceband modems, 446 digitalswitch,215,446 digital-to-analog converter (DAC), 445 digitaltransmission facility,215 digitalwirelesscommunication systems,550-551 Diracdeltafunction. Seedelta function directbroadcast satellite(DBS) simplicity andaffordability of, 517 directbroadcast satellites (DBS) useof,517 directfrequency modulation, 120-121,396 directivegain,520 directivity, 520 directmatrixinversion (DMI),558, 559 direct-sequence M-aryphaseshift ke0ng(DS~PSK),508 direct-sequence spreadbinary phase-shift-keyed (DSIBPSK) signal,490,492 direct-sequence spreadspectrum withcoherent BPSK,490-493 principles of,480 systems,498 DiricWet's conditions, 715 discretecosinetransform (DCT),8 discrete cosine transform coefficients, 8 discreteFouriertransform (DFT), 445 definedas,442 anddigitalsignalprocessing, 443 discretememoryless channel, 581-584,629-631 channelcapacityof,588 definedas,581-582 discretememoryless source,570 extension of,572 properties of,568 discretemultitone (DMT),431, 440-443,444-446 applications of,446 basicideaof,441 andmultichannel modulation, 447-448 useof,441 discretepulse-amplitude modulation (PAM),259 discretepulsemodulation, 259 discreterandomvariable, 708 discretesource,615 discrete-time, memorylessGaussian channel, 597 discrete-time channel, 291 discrete-time convolution, 627 discrete-time Fouriertransform, 185 discriminator, 144 discriminator output,14S,ISS dispersive channel, doubly,542 distancetransferfunction, 666 distortion, 2 acceptable, 614 methods ofreduction, 103 produced by,102-103 unavoidable, 611-612 distortion, amplitude, 191 distortionless baseband binary transmission, 261-262 distortion measure, 199 distribution function properties of,708 ofastationary randomprocess, 34 diversity techniques, 544-547 performance with,546-547 specialized techniques, 559 "divideandconquer", 431 DMI.Seedirectmatrixinversion DMT.Seediscretemultitone DonaldDuckvoiceeffect,99-100 Dopplershift,535 Dopplerspectrum, 540-541 Dopplerspread,539,541 double-frequency term,158doublesideband-suppressed carrier (DSB-SC) modulated signal (wave),95,96 doublesideband-suppressed carrier (DSB-SC) modulation, 133, 134 definition of,93 generated by,94 transmission ofsidebands, 163 doublesideband-suppressed carrier (DSB-SC) receiver compared toanAMreceiver, 136-137 modelof,132-133 doublydispersive channel, 542 downconvetsion, 105 downconverter, 448 downlink, 19,514-515 downstream datattansmission, 281-282 DPSK.Seediffetential phase-shift keying DSIBPSK waveform, 490,492 DSLenvitonment, 447 DSIMPSK system,508 dualcode,641 duobinary code,modified, 281 duobinarycoding,270 duobinaty convetsion filter, 268-269 duobinary encoder, 267-268 duobinary signaling scheme, 267-271 frequency response of,268 technique, 271-272 duobinarytechnique, 274-275 "dynamic" multipath environment, 532-533 E echocancellation, 277-278 compatison ofschemes, 278-279 modeofoperation for,277-278 echocanceller intransceiver, 278-279 useof,516 effectiveaperture, 521 effectiveradiatedpowerteferenced toanisotropic source(EIRP), 521 Einstein-Wiener-Khintchine telations, 46 EIRP.Seeeffectiveradiatedpower referenced toanisotropic sourceelastic store,215INDEX 799 electromagnetic interference (EMI), 17 electronbeam,4 elementary event,704 encodedtext,6 encoder condition foroptimality of, 199-200 functional unitsof,235 mainpartsof,551 opetation of,646 statesof,659,667 encoding process opetations of,9 stepsof,551 useof,203-204 encryption, 742 enemyctyptanalyst formsofattackby,745 intrusion of,743,744 energygap,99 energysignals,312 energyspectraldensity,48,353 ENIAC,28 ensemble average autocorrelation function, 284 estimation of,41 parametet, 75 substituting timeavetages for,41 entropiccodingredundancy, 9 entropy conditional, 584 definition of,569 formulafor,568 properties of,570-571 entropy, conditional, 584 envelope definedas,730 andphasecomponents, 67-69 typesof,730 envelope delay,16 envelope detection, 102-103, 131 envelope detector, 123,143 consistsof,92 foundin,92 lossofmessagein,138 needfot,406 performance of,137 signalcomparison, 141-142 envelope distortion, 90-91 equalizer, 191 equiprobable symbols, 571 equivalent noisetemperature, 61, 524-525 ergodicprocess,41-42,51 800INDEX error minimization of,551 possiblekinds,254 probability of,497-499 errorburst,652 errorcontrol,626 error-control code tbeoryof,485 typesof,627 error-control coding classesof,626 forreliablecommunication,S 67 techniques, 626 techniques for,683,693 useof,626,627 error-detection bit,7 error-free communication, 568 errorfunction, complementary, 255 errorminimization, 551 errorpattern,635 errorprobabilities, conditional, 352 errorpropagation e1imination ofpossibility, 273 phenomenon, 270 property, 745 errorrate,253 errorsignal calculation of,457 definition of,288,453 fortimingrecovery, 457 useof,557 error-syndrome vector,635-636 errorthreshold, 209-210 errorvector,635 estimation procedure, 517 Euler'sformula, 453 excessbandwidtb factor,441 excessmean-square error,291 excitation generator, 551,552 excitation time,718 expander, 203 expansion laws,203 exponential law,193 extended code averagecode-word lengtb,578 useof,577 extended prefixcode,578 extended source,572 extended-threshold demodulators, 152,153 extraction, 261 extrinsic information, 679 eyeopening, 293 eyepatterndefinition of,293 asanexperimental too~293 interpretation of,293 andperformance information, 293 F facsimile (fax)machine basicprinciple of,6 purposeof,6 inareceiving modeofoperation, 420 faderate,541 fadingchannel characteristics of,541 effectsof,545 fadingmultipath channel,536-539 far-endcrosstalk (FEXT),279-280 Farnsworth, PhiloT.,27 fastFouriertransform (FFT) algoritbm, 443-444 fast-frequency hopping, 502-503, 504 FDM.Seefrequency division multiplexing FDMA.Seefrequency division multipleaccess FDMAsystem,516 FDMsystem blockdiagramof,105-106 modulation stepsin,107 FEe.Seefeed-forward error correction feedback shifrregister,480,481 feedback system,second-order, 160 feed-forward errorcorrection (FEC),628,629 Fessenden, Reginald, 27 FHJMFSK system fastversusslow,502-503 jamming effectonreceiver, 502 symbolerrorin,502 field-power pattern,521 figureofmerit,134 foramplitude modulation, 136 definition of,132,193 forfrequency modulation, 147 fihering,128,208-209 fiheringscheme,100 finesynchronization, 493 finite-duration impulseresponse (FIR)filter,379 finite-state machine, 654 fixedchannelinput,582fixedchanneloutput,582 fixedmodulation scheme,627 fixedpoint-to-point links,18-19 fixedscatterers, 536 flat-fading channel, 71-72 flat-flatchannel,542 flatRayleigh fadingchannel,554 flat-topsamples, 191 Fleming, JohnAmbrose, 27 flip-flops, 646 flyback.Seehorizontal retrace FMdemodulator witbnegativefeedback, 153 andoscillator types,152 FMFBdemodulator, 152,153,154 FMFBreceiver,154 FMreceiver breaking pointof,149 interference suppression in, 148-149 modelof,142-143 noiseanalysisof,146 noisein,142 tbreshold effectsin,152 FMsignal averagepowerof,115 complex envelope of,114 demodulation of,121-124 desirable properties, 397 detection of,397 distinguishing fromAMsignal, 109 effectivebandwidtb for,117-119 fundamental characteristic of, 110 generation of,120-121 sidefrequencies of,117 spectralanalysisof,110 spectrum of,115 intheoryandpractice, 117 FMsignal,single-tone, 112-113 FMstereo multiplexing, 124-126 specification ofstandards, 124 transmission, 124 FMsystem emphasis in,154-156 nonlinear effectsin,126-128 Seealsofrequency modulation (FM)system FMthreshold effect,149-152 FMthreshold reduction, 152-154 FMwave bandwidth requirement, 118 withreducedmodulation index, 152-153 forward error-control coding,628, 668 forward errorcorrection (FEC), 626 forwardestimation, 681 forwardlink,547 Fourieranalysis, 715-720 Fourierseriesexpansion, 317 Fourierseriesrepresentation, 114 Fouriertransform definition of,715 inverse,715 ofperiodicsignals,717-718 properties of,716 theoryof,716 fractionally spacedequalizer (FSE), 287 frame,552 make-up of,S methodofsynchronization, 215-216 srructure, 547 framepacking, 9 frame-packing unit,236 freedistance, 663 freepropagation channels basedon,15 typesof,15 free-space loss,522 free-space propagation model, 518-523 frequency demodulation definedas,121 methods of,121 frequency deviation, 110,152 frequency-discrimination method stagesof,98-99 useof,100 frequency discriminator, 121, 124 consistsof,121-122 input,149 requirements of,99 frequency diversity, 544-545 frequency divisionduplexing (FOO),547 frequency divisionmultiple access (FOMA), 513,516 frequency-division multiplexing (FOM) definedas,20-21,105 methodofmodulation in,106frequency-domain description, 444, 720 frequency downconverter, 105, 516 frequency flat,541 frequency-flat channel, 542 frequency-hop M-aryfrequency shift-keying (FHlMFSK), 508 frequency hopping, 500 frequency-hop spreadspectrum, 499,500 communication systems, 500 principles of,480 frequency-modulated wave,413 frequency modulation (FM),20, 27,729 capability of,165 casesof,111 characteristic of,149 definition of,108-109 direct,120-121 andmixing,500 asanonlinear process,109 theoryof,126 frequency modulation (FM)system noiseanalysisof,142-147 similarities toPPMsystem,193 frequency multiplication ratio, 121 frequency multiplier consistsof,120 diagram of,120-121 frequency parameters, 128 frequency response choiceof,155 todenote,44-45 frequency reuse,S30 frequency-shift keying(FSK) basicsignaling scheme,344-345 anddesignofmodems, 421 andfrequency modulation, 345 represented by,464 frequency-shift keying(FSK) schemes, 418 frequency-shift keying(FSK)signal, 386 frequencytranslation, 103, 103-105 frequency upconverrer, 105 Friisformula, 526 Friisfree-space equation definedas,522 usedfor,522 FSK.Seefrequency-shift keyingINDEX 801 fullamplitude modulation, 162-163 full-cosine rolloffcharacteristic, 266 full-duplex link,628 functional definedas,54 versusfunction, 54 fundamental frequency, 717 fundamental inequality, 571 G gain,720 gap,432 Gaussian assumption, 498 Gaussian channel,S 97 Gaussian-distributed random variable, 54 Gaussian distribution, 54,72-73 Gaussian filter,397 Gaussian-filtered minimum shift keying(GMSK) asaspecialkindofbinary frequency modulation, 398 undesirable featureof,398 Gaussian-filtered minimum shift keying(GMSK) modulator frequency shapingpulseof,397 andintersymbol interference, 398 Gaussian-filtered minimum shift keying(GMSK) signal,397 powerspectrum of,400 spectralcompactness of, 398-400 Gaussian-filtered MSK,396-400 Gaussian function, 397 Gaussianity, 75 Gaussian model,S5 Gaussian process,54-58 definition of,57 mathematical justification, 55-56 inthestudyofcommunications, 55 usefulproperties of,56-58 virtuesof,55 Gaussian randomvariable, 54,58 generator equation, 634 generator polynomial, 645 ofacycliccode,643 definition of,656 geometric mean,436 geometric representation ofsignals, 311 802INDEX geometric signal-to-noise ratio,436 geostationary satellite communications system, 514-515 globalcoverage, 512 GlobalSystemforMobile Communications (GSM),548 frameefficiency of,548 wirelesscommunication system, 548 glottis,4 GMSK.SeeGaussian-filtered minimum shiftkeying Gold'stheorem, 505 Goldsequences (codes) classof,505 correlation properties of,507 goodcodes,631 Gram-Schmidt orthogonalization procedure, 315-317 granular noise,220 anddistortion, 221 versusquantization noise,221 Graycodingscheme,422-423 Grayencoder,276 Gray-encodeddibits, 363 GSM.SeeGlobalSystemfor MobileCommunications guardbands,513 guardinterval,441 guardtime,547 guidedpropagation channels basedon,15 typesof,15 H half-cycle cosinepulse,389 half-cycle sinepulse,389-390 half-duplex link,628. Hamming distance, 637,661, 666 Hamming single-error correcting code,653 Hamming weight,637,666 handover, 530 hard-decision coding,630 hard-decision decoders, 629-630 hard-decision demodulation, 669 harddecisions, 630 harmonic distortion, 112 harmonic structure, 4 headend,17 hearingmechanism, 4 Hermitian transposition, 443 Hertz,Heinrich, 26-27heterodyning function, 128 hexagonal cellulargeometry, 531 high-performing CAPsystem,375 Hilberttransform, 374,408, 723-725 properties of,725 ofasignal,724 Hilbert-transform pair,376,724 Hockham, G.A.,29 hoprate,501 horizontal retrace,5 host,13 Huffman code algorithm usedtosynthesize, 578 asaclassofprefixcodes,578 drawback of,580 nonuniqueness of,579 Huffman coding,578-580, 616 basicideaof,578 compared toLempel-Ziv algorithm, 581 anddatacompression, 8 asentropiccoding,8 Huffman decoding, 8-9 Huffman encoding process,578, 579 humanauditory system,234 humancommunication, 4 hybrid-modulated signal,123 hybridmodulation process,374 hybridtransformer definition of,278 simplified circuitof,278-279 I I-channel, 97 idealbaseband pulsetransmission, 262 idealdelayelement,268 idealenvelope detector, 135 idealfrequency discriminator, 124 idealnarrowband filter,45 idealNyquistchannel,262-264, 265-266 difficulties of,263-264 useof,263-264 idealsampledsignal,184 idealslopecircuit characterized by,121-122 frequency response of,122 idealsystem,601 identitymatrix,329 imageinterference, 129 impossible event,704impulsefunction, 62 impulsenoise,446 impulseresponse, 656,718 indexofperformance, 224 indirectfrequency modulation, 120-121 individual demodulators, 106 infinitebandwidth, 602 information, 2 information-bearing signal,31-32, 88 inthedigitaldomain,277 multiplying bythePNsignal, 488 information capacity, 598,616 ofachannel,597-598 definedas,23,598 evaluation of,598 increasing of,599 information capacitytheorem, 616 application of,607 argument for,599-600 asacolorednoisechannel,607 andGaussian channels, 597 implications of,601-603 systemparameters of,599 water-filling interpretation of, 610 information-theoretic concepts, 572 information theory fundamental limitsin,567 important resultof,591 asamathematical discipline, 567 Shannon's landmark paper,567 information transmission, 581 information vector,633 innerconductor, 17 innerproduct, 313,314 innovation symbol,581 in-phasechannel,408 in-phasecoherent detector, 97 in-phasecomponent, 93 powerspectraldensityof, 395-396 properties of,65-66 representation of,67-69 in-phasenoisecomponent, 131 inputalphabet, 582 inputsignal-to-noise ratio,134 definition of,131 equation for,497 insertion loss,16 instantaneous codes,577 instantaneous frequency, 110 definition of,163 equation for,108 instantaneous sampling, 184 integration beneficial effectsof,221-222 asalinearoperation, 223 interface, 11 interference averagepowerof,495 effectof,490 andfading,71-72 strengthof,148-149 asunintentional orintentional, 479 interference suppression, 148-149 interframe redundancy, 9 interlaced fields,S interlaced rasterscan,S interleaver definition of,674 typesof,674 useof,675 intermediate frequency (IF),128 intermediate frequency (IF)band, 18 Internet, 13-14 architecture of,13-14 evolution of,28-29 growthof,29 protocols for,13-14 Internetarchitecture functional blocksof,13-14 Internetprotocol (IP),13-14 InternetServiceProvider (ISP),420 andcommunication between PSTN,425 andpublicswitched telephone network (PSTN),420-425 andvoicemodems, 420-422 interpixel redundancy, 9 interpolation formula, 186 interpolation function, 186,427 intersymbol interference (lSI), 259-261,398 andbiterrors,247 aschannelimpairment, 379 condition of,282-283 underdesigner's control,268 asadominant impairment, 279 effectsof,294,296 asaformofinterference, 296 minimizing effectsof,260 andnoisepresence, 294 overcoming effectsof,441inpeakdistortion, 288 andtimingerror,266 asanundesirable effect,267 intrinsicinformation, 679 invariance, 331 inversediscreteFouriertransform (IDFT),442,445 inverseFouriertransform, 186, 715 inversemapping, 589 inverse-square law,519 irreducible polynomial, 505 irregular codes,691 irregular interleavers, 691-692 irregular LDPecode,692 irregular turbocode,691,692 J jammer,493 strategyof,495 typesof,508 waveforms of,508 jammer,barragenoise,508 jammer,multitone, 508 jammer,pulsenoise,508 jammer,single-tone, 508 jamming margin,499 jamming signal,488 jamming waveforms, 488 jitter,208 jointdistribution function, 33,709 jointmoments, 713-714 JointPhotographic ExpertsGroup (JPEG),8 jointprobability, 706,707 jointprobability densityfunction, 594,709-710 jointprobability distribution, 583 ]pEGimagecodingstandard, 8 K Kao,K.c.,29 keys,756 key-schedule calculation, 753,754 keystream, 744,745 Kotel'nikov, V.A.,27 Kraft-McMillan inequality, 576-577 Kummer's differential equation, 740 L Lagrange multipliers methodof,437 useof,609INDEX 803 laser,29 layer,11 layeredarchitecture, 11 layer-to-Iayer interface, 13 least-mean-square (LMS) algorithm, 288-290,557 foradaptive equalization, 288-289 andcombined use,297 equations for,289 forlinearadaptive prediction, 226 popularity of,226,227 similarities, 290 simplification of,289 summary of,289 usesof,292 usingmatrixnotation, 289 Leibniz's rule,257 Lempel-Ziv algorithm, 8,616 compared toHuffman coding, 581 definition of,580 encoding processperformed by, 580 standard forfilecompression, 581 Lempel-Ziv coding,580-581 light,6 likelihood functions, 322 linearadaptive prediction, 225-227 lineararraysignalprocessor todesign,554 forthereceiver, 554 requirements of,S54 linearblockcode basicproperty of,634 classesof,653-654 decoding procedure for,639 definition of,632 mathematical structure of, 632-633 minimum distanceof,637 standard arrayof,638 linearcombiner, 547 lineardeltamodulator, 221, 232-233 lineardiversity combining structure, 545 linearequalization, 379,556 linearfunction, 54 linearityproperty, 642 linearmodulation definition of,93 examples of,163 804INDEX linearmodulation (Continued) formsof,93-94 typesof,93 linearmodulation systems,729 linearprediction, 223-227 linearpredictive coding(LPC),551 linearpre-emphasis andde- emphasis filters applications in,157 useof,157 linearreceiver,248 designof,283 performance of,132 usingcoherent detection, 132 linearsystem,718 lineartime-invariant filter definedas,250-251 impulseresponse of,44 asamatchedfilter,250-251 useof,248 lineartime-invariant system, 719-720 linecodes candidates for,281 comparison of,281 forelectrical representation, 204-207 powerspectraof,206 selection of,280-281 typesof,205-207 useof,204-207 line-scanning frequency, 5-6 linkbudget,518 linkbudgetanalysis, 517 linkbudgetbalancesheet,517 Lloyd-Max quantizer, 198, 200-201 loadingproblem, 438 loadingprocess,438 localloop,420 localoscillator, 97 Lodge,Oliver,27 logarithmic function, 322 log-likelihood function, 452 definedas,322 definedforAWGNchannel,325 relationship of,322 loopfilter,157,159-160 loop-gain parameter, 159 losslesscompression definition of,7-8 fordigitaltext,8 versuslossycompression, 8 losslessdatacompression, 575lossycompression definition of,8 thepreferred approach, 8 low-density parity-check (LDPC) codes,683-686 advantages of,684 blocklengthof,689 construction of,684-685 decoding algorirhm, 690 decoding of,689-690 initialization of,690 minimum distanceof,689 sharedproperties of,693 statistical analysisof,689 srepsof,690-691 useof,684 low-noise amplifier, 515 low-weight codewords,684 LPC.Seelinearpredictive coding Lucky,Robert,28 luminance signal,6 M magnitude response,AS,608,719 magnitude spectrum, 715 mainlobe,368,720 Manchester code,207,281 many-to-one mapping, 8 MAPdecoder, 678 Marconi, Guglielmo, 27 marginal densiries, 710 marginal probability distribution, 583 Markovprocess,678 M-arydigitalmodulation techniques, 419-420 M-aryfrequency-shift keying (MFSK), 398-400 consistsof,401 forfrequency hoppingsystems, 500 property of,400 M-aryFSKsignal bandwidth efficiency of, 401-402 bandwidth requirements, 401 orthogonal signalsof,401-402 powerspectraof,401 spectralanalysisof,401,402 M-aryPAMsystem inachannelbandwidth., 276 consideration of,276 designcomplexity of,277 powerrequirements of,276-277M-aryPSK comparison of,419 likelihood functionfor,452 power-bandwidth requirements for,419-420 signalconstellations of,365, 420 similarspectralandbandwidth characteristics, 419 specialcase,365 symbolduration of,367 symbolerrorequation for, 365-366 M-aryPSKsignal bandwidth efficiency of,368 baseband powerspectraldensity of,367 powerspectraof,367 asspectrally efficient, 402 M-aryPSKsystems,449-450 M-aryQAM detection for,371 functions in,369 performance of,420 symbolerrorprobability for,371 transmitted energyin,371 M-aryQAMsignal,372 M-aryquadtature amplitude modulation (QAM),369-373 M-arysignal,402 M-arysignaling scheme,345-346 M-arysystem,276 maskingthreshold, 9,234, 234-235 matched filter,248-252, 286 inthefrequency domain,251 output,406 properties of,251-252 matchedfilterreceiver correlation and,326-327 detectorpartof,328 mathematical models classesof,31 inprobabilistic terms,31 matrixer difference signalgeneration, 125 sumsignalgeneration, 125 maximal-length sequence autocorrelation functionof,482 balanceproperty of,482 choosing a,484 definedas,482 properties of,482-484 maximal-ratio combiner, 547 maximal ratiocombining principle, 550 maximum aposteriori probability (1iAP)detection,678 maximum aposteriori probability (MAP)rule,323-324 maximum likelihood decisionrule foranAWGNchannel, 325 purposeof,325 maximum likelihood decoder, 322-326 forcomputation, 324 definedas,661 asanimplementation device,324 theoryof,660-661 maximum likelihood decoding rule, 661 maximum likelihood detection, 330,337 maximum likelihood detectors, 435 maximum likelihood estimation ofthecarrierphase,453-458 forproblem solving,449 maximum likelihood rule,324 maximum likelihood signal detection, 346 maximum-power transfertheorem applying, 61 useof,61 Maxwell, JamesClerk,26 mean,35 meanDoppler shift,541 meanfunctions, 35-39 meanoutputnoisepower definition of,139 equation for,141 meanoutputsignal,139 mean-square distortion, 199 mean-square error,285 asthecostfunction,S 58 definition of,284 mean-square errorcriterion forreceiverdesign,283 usesof,288 mediansignalstrength, 530 melodicstructure, 4 memoryless channel, 321 memoryless Gaussian channel, discrete-time, 597 memorylessquantizer, 194 Mersenne primelengthsequences, 485 message, 155 messagebandwidth, 90messagepoint,322-323 messagepolynomial, 643 messagesignal,132-133 description of,2 generation of,2 messagesource,348 messagespectrum fornegative frequencies, 103 requirement of,99 messagevector,660 methodofsteepestdescent, 225-226 microcells, 531 microphone, 15-16 Middleton, D.,27 minimum averageenergy,332 minimum distance considerations of,637-638 definition of,637 minimum distancedecoder, 661 minimum energysignals,331-332 minimum energytranslate, 332 minimum meansquareerfor (MMSE),558 criterion for,S58 equilizer, 285 receiver,286-287 minimum shiftkeying(MSK),387 asaformofbinaryFSK,361 signal-space diagram of, 389-392 minimum shiftkeying(MSK)signal powerspectraof,360 mixer consistsof,103-104 function of,129 operation of,105 mobileradio,18,529 mobileradiochannel capability of,18 asalineartime-varying channel, 18 propagation effectsof,18 mobileswitching center,530 mobility, 18 modem,7 configuration of,421 asaconversion device,420 designof,421 andtheInternet, 420 portions of,420 modem,facsimile, 420 modemconfiguration, 421 modified Besselequation, 738INDEX 805 modified Besselfunction, 737-739 modified duobinary code,281 modified duobinarycoder responses of,272,273 usefulfeatureof,272-273 modified duobinaryconversion filter,273 modulated signal(wave),95 modulating signal(wave),88 modulation, 19 operations of,628 stagesof,490 modulation format,101-102 modulation index definedas,110 restriction of,112 smallvaluesof,119 valuesof,117-118 modulation process,19-21 classification of,20 definition of,88 modulation scheme,bandwidth conserving, 354 modulation system,binary-coded, 197 modulator-demodulator, 420 modulo-211" correction logic,364 moments, 712 Morse,Samuel,26 Morsecode,26,574 MotionPictureExpertsGroup (MPEG), 9,234 moving-coil receiver,15-16 MPEG-1 audiocodingstandard, 9 capabilities of,234 operation of,235 performance of,234 suitablefor,10 MPEG-1 videocodingstandard, 9 MPEGaudiocodingstandard, 234, 237 MSKreceiver,394,395 MSKsignal,391 characteristics of,396 demodularion of,394 detection of,394 errorprobability of,392 generation of,394,396 possibleformsof,390 powerspectraof,394-396, 398 properties of,396 MSKsystem,391 MSKtransmitter, 394,395 11-law,202-203 806 INDEX multibeam antennas, 514 multichannel datatransmission system,433,434 multichannel modulation, 440-441 basicideaof,431 formof,431,465 multichannel transmission system, 437-438 multilevel encoding, 348,378 multiloop feedback citcuit,480 multipath physicalphenomenon of,513 presence of,513 multipath autocorrelation profile, 537 multipath channel asachannelimpairment, 553 classification of,541-542 frequency selection, 541-542 modelof,71-72 statistical characterization of, 535-542,554 typeoffadingexhibited, 72 multipath component, 549 multipath intensity profile,539 multipath phenomenon formsof,532-533 inamobileradioenvironment, 18 natureof,532 multipleaccess basictypesof,513-514 versusmultiplexing, 513 multiple-access interference (MAl), 548 multiple-access system goalof,549 interference, 548 multiple-access techniques conimon featureof,514 definedas,513-514 ideasbehind,513,514 andsharedcommunication resources, 513 multiple-receiver combining techniques, 544 multiplexed signal,125-126 multiplexed systems,347 multiplexer, 446 multiplexing, 105 definition of,20 ofdigitalsignals,214 typesof,20-21 multiplier, 157,646multi-pulse excitedLPC,551-552 multitone jammer,508 multiuser communications, 512 environment, 396 typesof,559 music asasourceofinformation, 4 structures of,4 musicalsignals andchannelbandwidth, 4 versusspeechsignals,4 mutualinformation, 584-585 forcontinuous ensemble, 593-597 definedas,596 properties of,585-587, 596 intheShannon model,746 N narrowband AMsignal(wave), 112-113 narrowband FMsignal(wave), 111-112,113 narrowband FMwaves,112-113 narrowband frequency modulation, 111-113 narrowband noise,64 characterization of,64 components of,64,65~66 effectsof,64 representation andcoordinate systemfor,67-68 representation of,64-66,67-69 narrowband noiseanalyzer, 64-66 narrowband noisesynthesizer, 64-66 narrowband phasemodulator, 120 narrowband process,65-66 National Television System Committee (NTSC), 6 naruralfrequency, 160 N-dimensional Euclidean space anglesin,312 lengthsofvectors,312 vectorsin,311,312 N-dimensional vector,311 near-end crosstalk (NEXT), 279-280,380 nearestneighbor condition, 200 near-farproblem, 548,549 negative-going click,150 network, 10 network, interconnected, 13 networkresources, 11,1490degreesrotational invariance, 422-423 noise absenceof,261 calculations, 61 incommunications systems,S 8 definition of,3 effectof,3 minimizing theeffectsof,157, 260 presence of,589 signalsin,322-326 sourcesof,3,58 asunwanted signals,58 noiseanalysis comparison of,163-164 atthereceiver,523 noisecalculations, 61 noisechannel,colored,607-611 noiseenhancement, 283 noiseequivalent bandwidth, 722-723 forbandpass filters,723 definedas,723 noisefigure,523-526 noise-free estimates, 545 noisejammer,barrage,508 noisejammer,partial-band, 508 noisemargin,281 noisemasking, 234 noisepower,3,61 noisepower,average outputcalculation of,151 determining, 145-147 noise-quieting effect,147 noise-co-mask ratio(NMR),234 noisevector covariance matrixof,330 particular realization of,323 statistical characteristics of,330 noisyreceivermodel,130 noisyresistor,60 noncoherent binaryDPSK,407 noncoherent binaryfrequency-shifr keying(FSK),407,413-414 biterrorratefor,414 consideration of,413 expressions for,417 noncoherent M-aryFSKdetector, 500 noncoherent matchedfilter,406 noncoherent orthogonal modulation, 407-409 noiseperformance of,407 receiverfor,408 specialcaseof,414 noncoherent receiver,405-406, 409 nondara-aided early-late delay (NDA-ELD) synchronizer, 458 nondata-aided recursive algorithm, 455 nondata-aidedsynchronization, 449 nonflatchannel, 542 nonlinearity basicformsof,126 effectsof,126 thepresence of,126 nonlinear modulation process,163 nonlinear pre-emphasis and de-emphasis techniques, 157 nonredundant coding,421 nonreturn-ro-zero levelencoder, 359 nonsystematic code,656 nonuniform quantizer, 202-203 normalized Gaussian distribution, 54-55,56 normalized transmission bandwidth, 164 North,D.O.,27 NorthAmerican digitalTDM hierarchy, 214-215 Nortonequivalent circuit,60 Noyce,Robert,28 NRZbinarydata,398 NTSCsystem,6 nulleveD[,704 null-to-null bandwidth, 368,721 number-controlled oscillator (NCO),458 Nyquist, Harry,27 Nyquistbandwidth, 262 Nyquistcriterion, 262 Nyquistinterval,186-187 Nyquistrate,186-187,262 o observable element,321 observation space,324,325-326 observation vector,321,331 octaphase-shift-keying,365-366 octet,7 offsetQPSK,362 onboard switching, 516 one-time pad,747 on-offlevelencoder, 384on-offsignaling, 205 opensystemsinterconnection (OSI) reference model,11 opticalcommunication, 29 opticalfiber advantages of,17 consistsof,17 properties of,17 asatransmission medium, 17 opticaltransmission system,15 optimization problem, 438-440 optimum CAPreceiver,379 optimum decisionrule,323-324 optimum equalizer, 288 optimum filter consistsof,378-379 impulseresponse of,250-251 optimum information capacity, 610 optimum in-phasefilter,378-379 optimum linearreceiver,282-287 interpretation of,286 andtransmitter, 286 optimum quadratic receiver, 403-405 optimum quadrature filter, 378-379 optimum quantization problem, 199-201 optimum receiver asacorrelation receiver, 326-327 designof,310 detectorpartof,328 subsystems of,326-327 optimum receiversubsystems, 326-327 optimum threshold, 257 orthogonal frequency-division multiplexing (OFDM), 447-448 applications of,448 techniques forbroadcasting, 344 useof,447-448 orthonormal basisfunctions, 311, 315,494 asadesirable property, 440-441 forMSK,390 shortcomings of,441 orthonormal matrix,443 oscillator, crystal-controlled, 120 OSImodel,11-13 outerconductor, 17 outputalphabet, 582 outputcurrent,4-5INDEX 807 outputnoisepower,523 outputsignal-to-noise ratio ofAMreceiver, 136 calculation of,151 versuscarrier-to-noise ratio,151 definition of,131,145 determining, 133 equation for,139,497 toevaluate, 135 improvement factorin,155-156 increasing of,155 ofauniformquantizer, 197 overrnodulated, 90 p packetswitching network, 11 principle of,11 pairwiseerrorprobability, 334 PAM.Seepulseamplitude modulation (PAM) PAMsignal generation of,188-189 performance of,191 sampling of,189 transmission of,191 waveform of,188~189 PAMsystem,191 parallelencoding scheme,676 parallel-to-serial converter, 445 paritybit,7,632 parity-check equations, 634 parity-check matrix,634 parity-check polynomial definedas,644 reciprocal of,645 partial-band noisejammer,508 partial-response signaling, 269, 274-275 partial-response signaling scheme, 267 achieved by,274-275 classesof,275 usefulcharacteristics of,275 partitioning, 670 passband basisfunctions properties of,434-436 timevariations of,373 passband datatransmission, 344 alternative techniques for,465 applications of,344 communication channelusedfor, 344 overnonlinear channels, 345 808 INDEX passband datatransmission systems determining thebandwidth efficiency of,348 goalof,346 passband in-phase filter,375,378 passband in-phase pulse,375 passband linecode,344 passband modulation, 729 passband pulse,375 passband quadrature pulse,374 passband signaling waveform, 729 passband transmission model, 348-349 pathloss,522 patternmatching operation, 615 PCM.Seepulse-code modulation peakdistortion, 288 peakpulsesignal-to-noise ratio definedas,248-249, 250 ofamatched filter,251 peerprocess,13 percentage modulation, 90 perception, 4 perceptual coding,9 perfectsecurity,746-747 periodicity, 718 periodicsignals,717-718 periodogram, 51 persistence ofvision,5 personal computers (PCs),6,6-7 "phaseandgainadjustors", 550. phasecontinuity, 388 phasecorrection, 365 phasedecisions, 394 phasedemodulation, 492 phase-difference computer, 364 phasediscriminator, 97 phasedistortion andthehumanear,99-100 presence of,99-100 phaseerror definedas,158 effectof,99 phase-error generator, 459 phase-locked loop,121 complexity of,159-160 components of,157-160 limitation of,159-160 loopfilterin,158 modelof,158-160 simplestformof,159-160 understanding, 157,158 useof,157-160phase-locked loopdemodulator, 152 andthreshold extension capacity, 154 asatrackingfilter,154 phase-locked looptheory,157 phasemodulation (PM),20,108, 729 phasemodulation schemes, 368 phasenonlinearity, 127 phaserecovery, 450 phase-recovery circuit,345 phaseresponse, 719 phaseselectivity, 723-725 phasesensitivity, 108 phase-shift keying(PSK),24 andcoherent systems,490 ofphasemodulation, 345 represented by,464 signaling scheme,344-345 phase-shift keying(PSK)schemes, 418 phasespectrum, 715 phasetree,388 phasetrellis,388-389 phasors, 532 photocathode, 4 photodetector circuit,58-59 physicallayer,13 TII4-shifted DQPSKsignals,364 TII4-shifted DQPSKsymbols, 363 TII4-shifted QPSKscheme,363 TI/4-shifted QPSKsignal demodulation of,365 residinginoneofeightpossible phasestates,362 pictures andthehumanvisual system, 4 perception of,4 as'asourceofinformation, 4 piecewise linearapproximation, 203 Pierce,JohnR.,29 pilotcarrier,99 plainoldtelephone service(POTS), 281-282 plaintext, 742 PMsignal,109 PNsequence correlation properties of,506 asanindependent andidentically distributed (iid)binary sequence, 496 asareference signal,550pointer,581 point-to-point communication, 2-3 Poisson's sumformula, 718 Poissondistribution, 59 polarnonreturn-to-zero (NRZ) levelencoder, 352 polarnonreturn-to-zero (NRZ) signaling binaryPCMsystembasedon, 253 disadvantages of, 205-206 polyvinylchloride (PVC)sheath,16 positive-going click,150 postdetection filter,143 power,available, 61 powercontrol inCDMAsystems,549 useof,549 powergain,523 ofanantenna, 520 conceptof,521 definition of,520 power-limited channel,3 powerspectra,347 powerspectraldensity,44-46, 347 andamplitude spectrum, 50-52 frequency portionsof,155 graphical summary of,75-76 properties of,46-47 ofrandomprocess,50,52 significance of,45 powerspectrum, 4,45 powertheorem, 520 Poynting vector,519 PPM.Seepulse-position modulation precoded duobinaryscheme, 270-271 prediction, 9 prediction filter,228 pre-envelope basicproperty of,732 definedas,725,730 determining, 726 quadrature components of,374 prefixcode definition of,575-576 distinguished by,577 property of,576 prefixcoding,575 prefixcondition, 575 premodulation low-pass filter, 396-397 presetthreshold values,277 primarycolors,6 represented byvideosignals,6 translnission of,6 primaryrate,214-215 primitive BCHcodes,653 primitive polynomial, 505 principle ofanalysisbysynthesis, 551-552 principle ofrotational invariance illustration of,330-331 statedas,330 principle ofsuperposition, 718 principle oftranslational invariance application of,331-332 statedas,331 probabilistic code,693 probabilistic concepts, 703-707 probabilistic decoder, 630 probability axiomsof,704-706 basicproperties of,705-706 ofbiterror,384 ofacorrectdecision, 357 oferror,254,409 ofsymbolerror,258,328-329, 352 probability, conditional, 706-707 probability densityfunction, 67-68,255,594,708-709, 710 probability distribution, 583 probability oferror,328-329, 497-499 invariance of,329-331 foranoisychannel,S 89 unionboundonthe,332-335 probability ofoccurrence, 568 probability ofsymbolerror,334 determination of,373 evaluation of,346-347 fonnulafor,401 forsignalconstellation, 337 probability system,704 probability theory,703,705 processing gain(PG) definedas,229,497 produced by,229 productcipher,749 producrmodulator, 94,98-99, 111,490 propagation, 4 propagation effects,532-535 propagation timedelay,516protocol oftheInternet, 13-14 typesof,13-15 pseudo-noise (PN)sequence, 480, 488 consistsof,288 generation andproperties of, 480 asatrainingsequence, 288 pseudo-random-ordered sequence, 500 PSK.Seephase-shift keying psychoacoustic modeling, 9 psychovisual redundancy, 9 public-key cryptographic system, 755 public-key cryptography, 742, 755-757 public-key system,759 publicswitched telephone network (PSTN),237,420 asananalognetwork, 420,421 disrortion on,286-287 efficientuseof,425 pulse,s pulse-amplitude modulated signal, 429 pulse-amplitude modulation (PAM),20,188-191,236 definition of,188 andmodulator design,277 andnaturalsampling, 188 pulse-code modulation (PCM), 193,615 advantages of,217,237 bandwidth requirement of,218 basiccondirion of,194 costofadvantages, 217-218 definition of,201 asaformofdigitalpulse modulation, 237 performance of,227 asthepreferred method, 20 forspeechcoding,229-230 useof,217-218, 560 pulse-code modulation (PCM)link, 210 pulse-code modulation (PCM) receiver, 258 pulse-code modulation (PCM) signal,208 pulse-eode modulation (peM) system,218 basicoperations of,201INDEX 809 characteristic of,210 description of,201-209 andinterference, 210 noiseconsiderations in,209-210 operation of,212 performance influenced bynoise, 209 pulsedemodulator, 211 pulse-duration modulation (PDM), 20,191-192,236 pulse-modulated signal,237 pulsemodulation familiesof,183 formsof,191-193, 237 lossynatureof,237-238 methodusedtotransmit, 211 assourcecodingtechniques, 237 standard digitalformof,20 typesof,20 pulse-modulation process incurred information lossof,238 lossofinformation anddesigner control,238 pulsemodulation systems, 236 pulsemodulator, 211 pulsenoisejammer, 508 pulse-position modulation (PPM), 20,192,236-237 pulse-position modulation (PPM) system versusfrequency modulation system,193 noiseanalysisof,193 performance of,193 pulseshaping, 247 pulse-shaping filter desirable properties of,396-397 Gaussian impulseresponse of, 398 pulse-shaping function, 373 pulse-width modulation, 191 punctured code,676 puncturing, 676 2 QAM.Seequadrature-amplitude modulation Q-channel, 97 QPSK.Seequadriphase-shift keying quadbits,421 quadratic receiver equation for,405 fonnsof,405 810INDEX quadrature-amplitude modulatiou (QAM),97-98 versusCAP,369 crossconstellation, 371 quadrature-amplitude modulation (QAM)constellations, 369-370 quadrature-amplitude modulators, 433-434 quadrature-earrier multiplexing, 97-98,354 quadrature-carrier multiplexing system,98 quadrature channel,408-409 quadrature component, 93 powerspectraldensityof,386 properties of,65-66 roleof,93,101 quadrature modulation, 670 quadrature noisecomponent, 131 quadrature nulleffect,97 quadrature-phase coherent detector, 97 quadrature receiver channels of,408 usingcorrelators, 405-406 usingmatched filters,405-406 quadriphasNhift keying(QPSK), 354,354-361 characterization of,354-355 errorprobability of,356-358 modeofoperation, 425 motivation forusing,508 signal-space diagram, 354-355 quadriphase-shift keying(QPSK) receiver,360 quadriphase-shift keying(QPSK) signal amplitude fluctuations, 362 andbinaryPSKsignal,360 commonly usedconstellations for,362 filtered,361 interference production, 396 observations of,360 phasetransitions of,361 powerspectraof,360-361 quadriphase-shift keying(QPSK) transmitter, 359 qualityofservice(QoS),14 quantization application of,202-203 andcoding,9 purposeof,8 typesof,194-195useof,195 functionof,196 typesof,220 quantization noise,195-197,228 designer's controlof,209 indeltamodulation, 221 asafunctionoftime,195 andhumanearperception, 9 inPCMsystems,209 quantization process,193-195, 236 inthegeneration ofabinary PCMwave,193 illustration of,195 nonlinear narnreof,198 resultsof,20 quantization table,8 quantized excitation, 552 quantized filterparameters, 552 quantizer characteristics of,194 classesof,615 components of,199 asasignalcompressor, 615 typesof,194 quantizer, nonuniform, 202-203 quantizer input,221 quantum, 194 quaternary system eyediagramfor,294-295 outputof,276 R radiation efficiency factor,520 radiation intensity, 519 radiation-intensiry pattern,520 radiocommunication link,522 radiocommunication system,31 r",diolinkanalysis, 517~523 radiopropagation infreespace,512 inurbanareas,532-533 radix,570,576-577 raisedcosinespectrum flatportionof,264-265 rolloffportionof,264-265 RAKEreceiver,549-550 basicideaof,549 consisfSof,549 asadiversity receiver, 549 techniques of,559 randombinarysequence, 482 randomexperiment description of,703 featuresof,703randomhopping, 500 randominterference, 31-32 randomprocess averagepowerof,610 classesof,75 definition of,33 ensemble averages of,41 expectations of,41 inlinearsystems,42-44 mathematical definition of, 32-33 parameter of,75 properties of,32 throughalineartime-invariant filter,42-44 randomvariable, 33,594,708-710 definition of,708 description of,708 distribution of,55-56 expected valueof,711 meanof,711 standard deviation of,712 varianceof,712 randomvectors,594 rasterscanning, 4-5 ratedistortion function, 612-613, 616 application of,616 definition of,613 ratedistortion theory,611-613 application of,612 mainparameters of,613 andShannon's codingtheorems, 612 Rayleigh's energytheorem definition of,251 useof,251-252 Rayleigh distribution, 68-69,70, 74-75 Rayleigh fadingchannel,536,541 binarysignaling over,542-547 performance of,545 receivedsignal components of,31-32 meanvalueofenergy,543 receivedsignalpoint,323 receivedvector,660' receivefilter,259 receiver ofananalogcommunication system,88-89 assumptions of,403 de-emphasis in,154-155 model,130 moving-coil, 15-16 noiseperformance of,387 asanoptimum maximum likelihood detector, 436 andpreprocessing thereceived signal,64 receivermodel,130 receiving antenna, 518 reciprocity principle inantennas, 521 reconstruction filter,187-188 reconstruction levels,194 rectangular function, 262 recursion theorem, 682 recursive algorithm forphaserecovery, 454 fortimingrecovery, 457-458 recursive symmetric convolutional (RSC)code,675 recursive Costasloop,454 convergence behavior of,461, 462 operations of,458 phase-acquisition behavior of, 459 forphasesynchronization, 454 recursive early-late-delay synchronizer, 463-464 redundancy addition of,626 basicformsof,9 controlled useof,628 redundant information, 227,575 Reed-Solomon codes,654,693 Reeves,Alec,27 reference antenna definition of,519 asanisotropic source,519 reference signal,557 reflectorantenna, 522 regeneration, 208 regenerative repeater, 208 regionofintegration, 332 regular-pulse excitation, 552 regularrurbocode,692 relative-frequency approach, 703-704 relativephasedifference, 414 relativephaseshift,532 replication ofdeltafunction property, 717 representation levels,194 reproduction quality,5-6 residualamplitude modulation, 112 resolurion, 6 resolution ofuncertainty, 568response time, 718 reverselink versusforwardlink,559 subbands for,547 Riciandistribution graphical presentarion of,70-71 normalized formof,71 Ricianfadingchannel, 536 Rivest-Shamir-Adleman (RSA) algorithm, 759 Rivest-Shamir-Adleman (RSA) system,757 rmsduration, 722 robustsystem,22 rollofffactor,265 frequency response for,265-266 timeresponse for,265-266 rootmeansquare(rms)bandwidth, 721 rotatednoisevector,330 router definedas,13 andhostdevices,10 primarypurposeof,10 rowvectors,633 RS-232standard, 6-7 RSAalgorithm, 757-758 RSAcryptoalgorithm, 758 RSAtrapdoor one-way function, 758 s samplefunctions, 32-33 samplepoint,32,704 samplespace,32,704 sampling, 201-202 sampling period,184 sampling process,236 anddigitalsignalprocessing and digitalcommunications, 184 inthegeneration ofabinary PCMwave,193 andpulsemodulation systems, 184,236 useof,184 sampling rate,184,201 sampling theorem, 201,236 forband-limited signals, 186-187 derivation of,186-187 essenceof,184 ofapulse-modulation system, 186-187 recurrent nonuniform equivalent formof,427INDEX 811 satellite forcommunication, 29 ingeostationary orbit,18 servicesof,18-19 satellitechannel capabilitiesof,516 coverage of,18 remoteareaaccess,18 satellitecommunications, 514-517 frequency bandfor,19 globalcoverage, 512 mostpopularfrequency band for,515 asatypeofmultiuser communications, 512 satellitecommunication system,18 designof,517 globalcoverage, 559 relyon,512 uniquesystemcapabilities of, 18-19 scalarquantization formof,194 useof,194 scalarquantizer conditions foroptimality of, 198-201 designing of,198 asasimplesignalcompressor, 615 scanning, 4-5,4-6 scanning spot,5 scattered beams,71 scatterers, 71-72 scattering function, 539 Schwarz's inequality asamathematical result, 249-250 proving, 313,314 SDMA,516 secondflyback.Seevertical retracesecond-order digital filter,454-455 second-order feedback system,160 secrecy,745 secretkey versuspublickey,759 selection of,745 secret-key cryptoalgorithm, 751 secret-key cryptography, 742-743 secret-key system,759 securechannel, 743 securecommunications inahostileenvironment, 479 needfor,742 812 INDEX securityoftransmission, 490 segments, 11 Seealsopackets separability theorem, 681 sequential scanning ofpictures, 4 processof,4-6 serial-to-parallel converter, 445 Shannon, Claudecapacity theorem, 611 and"TheMathematical Theory ofCommunication", 27-28 andthetheoretical foundations ofdigitalcommunications, 27-28 Shannon's capacity theorem, 611 Shannon's fundamental boundfor perfectsecurity, 747 Shannon's information capacity theorem, 23-24,433 Shannon's information theory,617 Shannon's secondtheorem, 616 Shannon's thirdremarkable theorem, 616 Shannon's thirdtheorem, 599 Shannon limit,602 Shannon modelofcryptography methodofconfusion, 749 methodofdiffusion, 749 methods ofdesigning, 749 shutparameters,S 31 shiftregister,481 Shockley, William, 28 shotnoise,58-60 sideband, upperandlower,91 sideinformation, 230 sigma-delta modulation, 222 signal definition of,3-4 detection innoise,322-326 dimensions of,3-4 receivedversustransmitted, 2 signalbandwidth, 3 signalconstellation, 322-323, 337 ascircularly symmetric, 335 constructed fromone dimensional PCMsymbols, 429 definingminimum distanceof, 335 signaldetection problem, 322 likelihood function for,405 statedas,323 signalenergy-to-noise spectral densityratio,252signalfading,532-533 signal-flow graph,665-666 signaling binaryinformation, 345 signaling interval,568 signaling rate,276 signalparameters, 403 signalpoweraverage, 3 signalregeneration, 208 signal-space analysis, 337 signal-space dimensionality, 312, 493 signal-space representations oftheinterfering signal (jammer), 493 ofthetransmitted signal,493 signalswithunknown phase, 403-406 signal-to-mask ratio(SMR),234 signal-to-noise ratio basicdefinitions of,3,130-132 atthedeviceoutput,524 ofanFMFBreceiver, 153 limitation of,261 ofthesource,524 signal-to-noise ratiogap,432 signal-to-quantization noiseratio, 229 signaltransitions, 207 signaltransmission decoder, 326, 349 signaltransmission encoder, 348, 352 signalvariability, 530 signalvector,311 simplexsignals,342 signumfunction, 724 sinefunction, 262 sinewaveplusnarrowband noise, 69,69-71 single-key cryptography, 742 singlekeyedoscillator, 384 single-letter distortion measure, 612 singlesideband (SSB)modulation, 98-100,163 basicoperation in,103 definition of,93 infrequency-division multiplexing, 106 single-sideband modulated signal, 98-99 single-tone FMsignal,112-113 single-tone jammer, 508 single-tone modulation andanarrowband FMsignal, 110andawideband FMsignal,110 sinusoidal carrierwave definedas,90 waveform of,490,492 sinusoidal modulating signal (wave),110 sinusoidal modulation, 112-113 sinusoidal wave,88 slicinglevels,277 slopecircuit,121-122 slopenetwork, 143 slopeoverload distortion, 220, 221 slowFHlMFSK signal,501 slow FHlMFSK system,502 slow-frequency hopping, 500-502 smoothness, 223 SNRratio.Seesignal-to-noise ratiosoft-decision coding,630 soft-decision decoding, 669 softdecisions, 630 softinput-hard output,693 softinput-soft output,693 SONET,15 soutcecode,574 typeof,575 variability inlengthsof,579 sourcecodeword,21 sourcecoding,574 dissection of,616~617 forefficient communication, 567 withafidelitycriterion, 611-612 source-coding theorem, 574-575, 612,616 averagecode-word lengthof, 611 inShannon's firsttheorem, 574-575 sourcedecoder, 575 sourceencoder, 21,574 functional requirements of,574 purposeof,21 spaced-frequency spaced-time correlation function, 538 spacediversity, 544-545 spacediversity technique, 546 space-division multiple access (SDMA),514 space-time processor, 557 spatialphenomenon, 534 spatialsampling, 4-5 spectralanalysis, 110 spectralcontent,492,493 spectraldecomposition, 443 spectrally efficientmodulation, 347 spectrally efficientschemes, 347 spectralnulls,368 spectralshaping, 348 spectrum, 3,4,715 spectrum despreading indemodulation, 491 asalinearoperation, 492 spectrum spreading asalinearoperation, 492 andphasemodulation, 491 speechcoding applications of,229-230 designphilosophy of,230 atlowbitrates,229-230 techniques for,551 speechcommunication process,4 speech-production process,4 speechsignal,4 asbipolar,6 limitsof,16 spherepacking, 599-600 split-phase signaling, 207 splitter,282 spontaneous fluctuations, 58-61 spreading code withpseudo-random properties, 490 useof,490 spreadspectrum communications, 508 important attribute of,488 notionof,488-490 spread-spectrumcommunication system advantage of,479 rejection ofinterference, 479 requirements of,493 spread-spectrum modulation definition of,479-480 formilitaryapplications, 480 principles of,480 toprovidemultipath rejection, 480 securecommunications of,479, 480 signaling techniques knownas, 479 spread-spectrum techniques asdirect-sequence spread spectrum, 490 inpassband transmission, 490 versusstandard modulation techniques, 480 squareconstellations, 369-370squarelaw,193 SSBmodulated signal,99 SSBmodulation, 134 standard modulation techniques, 480 statediagram, 657-660, 659 stateprobabilities, 681 staticpicture,4 stationary process,33-34 versusstrictlystationary, 33 variousnamesfor,36 statistical average,711-714 statistical expectation operator, 711 statistical regulariry, 703 step-size, 194 step-sizeparameter, 225-226 stereomultiplexing inFMradiobroadcasting, 124 asaformoffrequency-division multiplexing, 124 stochastic process,32 stop-and-wait automatic repeat request,628 stop-and-wait strategy, 628 streamciphers,744-746 operation of,744 usedin,745 strictlystationary process,35 Strowger, A.B.,27 Strowger switch,27 subframes, 552 subnets,13 substitution cipher description of,749-750 useof,749-750 successive errors,232 sufficient statistics, 321 sum-product algorithm, 691 Sunde'sFSK,381,385-386, 388 Sunde'sFSKsignal,386 superhet, 128 superheterodyne receiver, 27, 128-129 consistsof,128 differences betweenAM:andFM, 129 survivorpaths,662 switching center,mobile,530 symbol,2 symbolenergy-to-noise spectral densiryratio,502 symbolerror averageprobability of,276 conditional probability of,333INDEX 813 symbolerrorprobabiliry versusbiterrorrate(BER), 335-336 tocalculate, 357 definition of,209,310 evaluation of,543 formulafor,256-257 tominimize, 310,346 asaratio,358 simplification of,335 symbolrate,501 symbolshapingfunction definedas,353 energyspectraldensityof,386, 395 symbolsynchronization, 448 symboltiming,455, 458, 463-464 symboltimingrecovery, 463-464, 465 symmetric modemconfigurations, 421-425 synchronization, 448-450, 493 algorithmic (modern) approach, 449 basicmodesof,448 classicalapproach to,449 implementation of,449 processof,448 asastatistical parameter estimation problem, 449 inaTDMsystem,212 oftransmitrer andreceiver clocks,212 synchronization problem approaches forsolving,449 solutionto,493 synchronizing pulses,5,212 synchronous demodulation quadrature nulleffect,96 synchronous opticalnetwork (SONET),15 syndrome calculation of,646-648 importance of,635 properties of,635-636 syndrome calculator, 647 syndrome decoding, 635-636, 638, 639 syndrome polynomial, 647 synthesis equation, 442-443 synthesis filter,552 consistsof,552 aspartoftheencoder, 551 systematic blockcode,632 814INDEX systemcapacity definition of,549 description of,718 tomaximize, 549 system-dependent scalingfactor, 132-133 systemdesignobjectives, 3 system-memory time,718 T TDMA.Seetime-division multiple access telecommunications environment, 3-7 telegraph,26 telegraphic code,Baudot's, 26 telephone channel,15-17,287 telephone circuitfrequencies, 3 telephone network, 27 asacommunication network, 10 firstcommercial service,28 primarypurposeof,15 television network, 14 television picture,5-6 television signals,101-103, modulation formatof,101-102 aswideband signals,7 Telstarsatellite,29 temporal autocorrelation function, 284 ternarycode,204 t-error-correcting RScode,654 theorem ofirrelevance, 321, 321-322 theoryoferror-control codes,485 theoryofspectralanalysisof randomprocesses, 46 thermalnoise,32,60 Thevenin equivalent circuit,60 3-dBbandwidth, 721 three-level output,267-268 threshold effect,137-138 AMandFM,164 clicksheardin,149-150 definition of,138,149 inanenvelope detector, 138 threshold extension, 154 threshold reduction, 152 time-averaged autocorrelation function,42,51 time-bandwidth product, 721-722 choiceof,398' asadesignparameter, 397timecompression (TC) multiplexing scheme,278-279 useof,277-278 timediversity, 544-545 time-division multiple access (TOMA), 211,211-212, 513 efficientsystem,516 aswireless communications systems,547-550 time-division multiplexing (TOM) conceptof,211 definedas,21,105 useof,211-212 time-domain description, 720 time-flat channel, 542 time-frequency mapping, 9 timeresponse, 266 time-scaling property, 722 timeslot,513 time-to-frequency mapping network, 235 time-varying phasqr,728 time-varying transferfunction, 536 timingerror,264 timingsynchronization, 455 Toeplitzproperty, 225 tollconnection, 16 TomJinson-Harashima precoding, 430 tone-modulation analysis, 119 tracking, 493 trackingfilter,154 trade-offs, 602 trainingmode,290 transceiver, 278 transistor, 28 transition matrix,S 82 transition probability, 582,616 transmission bandwidth definedas,118 instantaneous spreading, 499 transmission delay,260 transmission path,212 transmission security, 490 transmit filter,259 transmitted codevector,660 transmitted FHlMFSK signal,500 transmitted power definition of,3 primary communication resource, 92 transmitted pulseamplitude, 253 transmitted pulseshape,261transmitted signal,280-281 transmitted signalenergyperbit, 258 transmitted signalpoint,322-323 transmitted TVsignal,102 transmitter ofananalogcommunication system,88-89 bycombining operations, 414 locationof,2 powerlimited,597 purposeof,2 useofpre-emphasis in,154-155 transmitting antenna function of,518 mounting of,17-18 asapointsource,519 andpowerdensity,521 transorthogonal signals.See simplexsignals transponder, 19,514-515 transponder channel, 515 transposition cipher description of,750 useof,750 transversal equalizer, 286 trapdoor one-way function, 755 traveling-wave tubeamplifier, 516 trellis,657-660, 661-662 ttellis-coded modulation, 668-669 trelliscodes forband-limited channels, 668-669 designof,669 trelliscoding asanerror-control coding technique, 430 asaforward-error correction scheme,424 trellisencoder, 425 turbocodes consistof,674 development of,674 performance of,676-677 properties of,682,693 termination approaches of,676 turbocoding,674 turbodecoder basicstructure of,677-678 complexity of,684 detailsof,683 turbodecoding, 677-680, 682-683 turboencoder, 683 twisted-pair cable versuscoaxialcables,17 consistsof,16 susceptible to,16-17 usesof,277 2B1Qcode,281 amplitude levelsof,317 compared tootherlinecodes, 281 desirable properties of,281 astheNorthAmerican standard, 281 two-dimensional matched filter, 378-379 two-dimensional optimum receiver, 378 two-keycryptography, 742 two-stage spectralanalysis, 110 two-step subspace procedure, 556 two-way transmission, 106 U Ungerboeck, G.,28 Ungerboeck codes,670 for8-PSK,670-672 asymptotic coding gainof,673 unicity distance, 748 uniformquantizer, 196 unionbound,337 illustration of,333 simplification of,335 asausefulupperbound, 332-333 useof,401 unionofevents,333 unipolar nonreturn-to-zero (NRZ) signaling, 205 unipolarreturn-to-zero (RZ) signaling disadvantages of, 207 featureof,207 uniquely decodable, 574 unitarymatrix,443 unitdelay,219 unit-delay elements, 646 universal curve,118-119 unmodulated carrier,108 upconversion, 104-105 upconverter,448 uplink,19,514-515 upperbound,401 upstream datatransmission, 281-282V V.32modemand nonredundant coding,423--424 phasechangesin,422--423 switching toQPSKmode,424 andtrelliscoding,423--424, 425 V.32modemstandardand alternative modulation schemes, 421 characteristics of,421 V.90modem,431 vanDuuren, H,C.A.,28 VanVleck,J.H.,27 variable-length code,574 variablenodes,685 variance, 579 VDSL.Seevery-high-rate digital subscriber lines vectorproduct, 681 vectorquantizer advantage of,615 encoding processin,615 versusscalarquantizer, 615 signal-to-quantization noiseratio for,615 vectors,633 vectorspace,554-555 Vernamcipher,747 verychigh-rate digitalsubscriber lines(VDSL) advantages of,446 useof,446 very-large-scale integrated (VLSI) circuits development of, 28 vestigialsideband (VSB)filter frequency response of,102 magnitude response of,100-101 vestigialsideband (VSB)modulated wave methods ofgenerating, 100 quadrature component of, 102-103 vestigialsideband(VSB) modulation, 100-101, 163 vestigialsideband (VSB)shaping filter,102 vestigialsideband modulation definition of,93 anditsroleincommercial TV broadcasting, 101-102 videobandwidth, 6 video-on-demand, 9,282 videosignal,4-5,6INDEX 815 virtualcommunication, 13 Viterbialgorithm, 661-663, 670 difficulty intheapplication of, 662 asamaximum likelihood decoder, 662 asamaximum likelihood sequence estimator, 662 VLSI.Seevery-large-scale integrated (VLSI)circuits vocaltract,4 voiceband modem,420 versusdigitalsubscriber lines, 446 operational environment of,447 voiceconununication, 22 voiceeffect,DonaldDuck,99-100 voicesignals,99 voicespectrum, 3 voltage-eontrolled oscillator (VeO),152-153, 157-158 vonNeumann, John,28 VSB.Seevestigialsideband W water-filling interpretation, 610-611 water-filling solution, 438 waveform, 21,276 ofimportant linecodes,204-207 inmodulation, 490,492 waveform distortion, 102-103 wavelength-division multiplexing (WDM),21 wavemotion,18 weaksignalsuppression, 142 weightvector,554 whiteGaussian noise,62 identically distributed, 334 process,392 whitenoise,61-63 autocorrelation functionof, 61-62 characteristics of,61-62 mathematical properties of,62 powerspectraldensityof,61-63 whitenoiseprocess,62 widebandcommunication channels, 218 widebandFMsignal,118 wideband frequency modulation, 113-115 widebandsignal,7 widebandtransmitted signal,490 816 INDEX Wiener-Hopf equations, 224 wiredcommunications, 559-560 wirelessbroadcast channels, 17-18 wirelesscommunications, 529-535 adaptive antennaarraysfor, 553 featuresof,512 goalof,553 majorchannelimpairments of, 553 andmobility, 512andOFDM,448 sourcecodingfor,550-553 asatypeofmultiuser communication, 512 asatypeofmultiuser radio communication system, 529-530 versuswiredcommunications, 559-560 wirelesscommunication system mobility of,559practical requirements of,396 problems usingMSK,396 WorldWideWeb,28-29 Z zero-forcing equalizer, 283 zero-forcing kind,556 zero-mean whiteGaussian noise process,310 zerostate,481 Zworykin, Vladimir K.,27 Newfeaturesinclude: •MAHARcomputerexperiments thatdemonstrate importantaspectsof communication theory •Expandedcoverageofemergingdigitaltechnologies, suchasdigital subscriberlines(DSL),carrierlessamplitudemodulation/phase modulation (CAP),anddiscretemulti-tone (DMT) •Dozensofexamplesthatrelatetheorytoreal-worldcommunication systems Superblyorganized,thetextskillfullyguidesstudentsthroughtopicsranging frompulsemodulation topassbanddigitaltransmission, andfromrandom processestoerror-control coding.Throughout, Haykinpresentsdifficultcon ceptsinlanguagethatstudentscaneasilyunderstand. ISBN0-471-17869-1 Wile~&Sons.Inc. 'ark/Chichester/Weinheim ne/Singapore/Taronlo viley.com/college90000> 9780471 178699