Digital Communications by John Proakis 4th Ed

PrefaceCONTENTS xix 1Introduction I 1-1Elements ofaDigitalCommunication System I 1-2Communication Channels 'andTheirCharacteristics 3 1-3Mathematical ModelsforCommunication Channels 11 1-4AHistorical Perspective intheDevelopment ofDigital Communications 13 1-5Overview oftheBook 16 1-6Bibliographical NotesandReferences 16 2Probability andStochastic Processes 17 2-1Probability 17 2-1-1Random Variables, Probability Distributions, andProbability Densities 22 2-1-2Functions ofRandom Variables 28 2-1-3Statistical Averages ofRandom Variables 33 2-1-4SomeUsefulProbability Distributions 37 2-1-5UpperboundsontheTailProbability 53 2-1-6SumsofRandom Variables andtheCentralLimit Theorem 58 2-2Stochastic Processes 62 2-2-1Statistical Averages 64 2-2-2PowerDensitySpectrum 67 2-2-3Response ofaLinearTime-Invariant SystemtoaRandom InputSignal 68 2-2-4Sampling Theorem forBand-Limited Stochastic Processes 72 2-2-5Discrete-Time Stochastic SignalsandSystems 74 2-2-6Cyclostationary Processes 75 2-3Bibliographic:al NotesandReferences 77 Problems 77 3SourceCoding 82 3-1Mathematical ModelsforInformation 82 3-2ALogarithmic Measure ofInformation 84 3-2-1Average MutualInformation andEntropy 87 3-2-2Information Measures forContinuous Random Variables 91 3-3CodingforDiscrete Sources 93 3-3-1CodingforDiscrete Memoryless Sources 94 3-3-2Discrete Stationary Sources 103 3-3-3TheLempel-Ziv Algorithm 106 3-4CodingforAnalogSources-Optimum Quantization 108 3-4-1Rate-Distortion Function 108 3-4-2ScalarQuantization 113 3-4-3VectorQuantization 118 3-5CodingTechniques forAnalogSources 125 3-5-1Temporal Waveform Coding 125 3-5-2Spectral Waveform Coding 136 3-5-3Model-Based SourceCoding 138 3-6Bibliographical NotesandReferences 144 Problems 144 4Characterization ofCommunication Signals andSystems 152 4-1Representation ofBandpass SignalsandSystems 152 4-1-1Representation ofBandpass Signals 153 4-1-2Representation ofLinear Bandpass Systems 157 4-1-3Response ofaBandpass SystemtoaBandpass Signal 157 4-1-4Representation ofBandpass Slationary Stochastic Processes 159 4-2SignalSpaceRepresentation 163 4-2-1VectorSpaceConcepts 163 4-2-2SignalSpaceConcepts 165 4-2-3Orthogonal Expansions ofSignals 165 4-3Represen tationofDigitally Modulated Signals 173 4-3-1Memoryless Modulation Methods 174 4-3-2LinearModulation WithMemory 186 4-3-3Nonlinear Modulation Methods withMemory 190 4-4Spectral Characteristics ofDigitally Modulated Signals 203 4-4-1PowerSpectJ"aofLinearly Modulated Signals 204 4-4-2PowerSpectraofCPFSKandCPMSignals 209 4-4-3PowerSpe'<lraofModulated SignalswithMemory 220 4-5Bibliographical NotesandReferenCes 223 Problems 224 5Optimum Receivers fortheAdditive White Gaussian NoiseChannel 233 5-1Optimum Receiver forSignalsqorrupted byAWGN 233 5-1-1Correlation Demodulalor 234 5-1-2Matched-Filter Demodulator 238 5-1-3TheOptimum Detector 244 5-1-4TheMaximum-Likelihood Sequence Detector 249 5-1-5ASymbol-by-Symbol MAPDetector forSignals withMemory 254 5-2Performance oftheOptimum Receiver forMemoryless Modulation 257 5-2-1Probability ofErrorforBinaryModulation 257 5-2-2Probability ofErrorforM-aryOrthogonal Signals 260 5-2-3Probability ofErrorforM-aryBiorthogonal Signals 264 5-2-4Probability ofErrorforSimplexSignals 266 5-2-5Probability ofErrorforM-aryBinary-Coded Signals 266 5-2-6Probability ofErrorforM-aryPAM 267 5-2-7Probability ofErrorforM-aryPSK 269 5-2-8Differential PSK(DPSK)anditsPerformance 274 5-2-9Probability ofErrorforQAM 278 5-2-10Comparison ofDigitalModulation Methods 282 5-3Optimum Receiver forCPMSignals 284 5-3-1Optimum Demodulation andDetection ofCPM 285 5-3-2Performance ofCfMSignals 290 5-3-3Symbol-by-Symbol Detection ofCPMSignals 296 5-4Optimum Receiver forSignalswithRandom PhaseinAWGN Channel 301 5-4-1Optimum Receiver forBinarySignals 302 5-4-2Optimum Receiver oforM-aryOrthogonal Signals 308 5-4-3Probability ofErrorforEnvelope Detection ofM-ary Orthogonal Signals 308 5-4-4Probability ofErrorforEnvelope Detection ofCorrelated BinarySignals 312 5-5Regenerative Repeaters andLinkBudgetAnalysis 313 5-5-1Regenerative Repeaters 314 5-5-2Communication LinkBudgetAnalysis 316 5-6Bibliographical NotesandReferences 319 Problems 320 6CarrierandSymbolSynchronization 6-1SignalParameter Estimation 6-1-1TheLikelihood Function 6-1-2CarrierRecovery andSymbolSynchronization inSignalDemodulation 6-2CarrierPhaseEstimation 6-2-1Maximum-Likelihood CarrierPhaseEstimation 6-2-2ThePhase-Locked Loop 6-2-3EffectofAdditive NoiseonthePhaseEstimate 6-2-4Decision-Directed Loops 6-2-5Non-Decision-Directed Loops 6-3SymbolTimingEstimation 6-3-1Maximum-Likelihood TimingEstimation 6-3-2Non-Decision-Directed TimingEstimation333 333 335 336 337 339 341 343 347 350 358 359 361 6-4JointEstimation ofCarrierPhaseandSymbolTiming 6-5Performance Characteristics orMLEstimators 6-6Bibliographical NotesandReferences Problems365 367 370 371 7Channel Capacity andCoding 374 7-1Channel ModelsandChannel Capacity 375 7-1-1Channel Models 375 7-1-2Channel Capacity 380 7-1-3Achieving Channel Capacity withOrthogonal Signals 387 7-1-4Channel Reliability Functions 389 7-2Random Selection ofCodes 390 7-2-1Random CodingBasedonM-aryBinary-Coded Signals 390 7-2-2Random CodiitgBasedonM-aryMUlliamplitude Signals 397 7-2-3Comparison ofR~withtheCapacity oftheAWGN Channel 399 7-3Communication SystemDesignBasedontheCutoffRate 400 7-4Bibliographical NotesandReferences 406 Problems 406 8BlockandConvolutional ChannelCodes 413 8-1LinearBlockCodes 413 8-1-1TheGenerator MatrixandtheParityCheckMatrix 417 8-1-2SomeSpecificLinearBlockCodes 421 8-1-3.CyclicCodes 423 8-1-4Opti",um Soft-Decision Decoding ofLinearBlockCodes 436 8-1-5Hard-Decision Decoding 445 8-1-6Comparison ofPerformance between Hard-Decision and Soft-Decision Decoding 456 8-1-7BoundsonMinimum Distance ofLinearBlockCod"s 461 8-1-8Nonbinary BlockCodesandConcatenated BlockCodes 464 8-1-9Inlerleaving ofCodedDataforChannels withBurst Errors 468 8-2Convolutional Codes 470 8-2-1TheTransfer Function ofaConvolutional Code 477 8-2-2Optimum Decoding ofConvohll;onal Codes- TheViterbiAlgorithm 483 8-2-3Probability ofErrorforSoft-Decision Decoding 486 8-2-4Probability ofErrorforHard-Decision Decoding 489 8-2-5Distance Properties ofBinaryConvolutional Codes 492 8-2-6Nonbinary Dual-kCodesandConcatenated Codes 492 8-2-7OtherDecodinl! Algorithms forConvolutional Codes 500 8-2-1'1Practical Considerations intheApplication of Convolutional Codes 506 8-3CodedModulation forBandwidth-Constrained Channels 511 8-4Bibliographical NotesandReferences 526 Problems 528 9SignalDesignforBand-Limited Channels 534 9-1Characterization ofBand-Limited Channels 534 9-2SignalDesignforBand-Umited Channels 540 9-2-1DesignofBand-Umited SignalsforNoIntersymbol Interference-The NyquistCriterion 542 9-2-2DesignofBand-Umited SignalswithControlled ISI- Partial-Response Signals 548 9-2-3DataDetection forControlled lSI 5S1 9-2-4SignalDesignforChannels withDistortion 5S7 9-3Probability ofErrorinDetection ofPAM 561 9-3-1Probability ofErrorforDetection ofPAMwithZerolSI 561 9-3-2Probability ofErrorforDetection ofPartial-Response Signals 562 9-3-3Probability ofErrorforOptimum SignalsinChannel withDistomon 56S 9-4Modulation CodesforSpectrum Shaping 566 9-SBibliographical NotesandReferences 576 Problems 576 10Communication throughBand-Limited Linear FilterChannels 583 10-1Optimum Receiver forChannels withlSIandAWON 584 10-1-1Optimum Maximum-Ukelihood Receiver 584 10-1-2ADiscrete-Time ModelforaChannel withlSI 586 10-1-3TheViterbiAlgorithm fortheDiscrete· TuneWhite NoiseFilterModel 589 10-1-4Performance ofMLSEforChannels withlSI 593 10-2LinearEqua1ization 601 10-2-1PeakDistortion Criterion 602 10-2-2MeanSquareError(MSE)Criterion 607 10-2-3Performance Characteriatics oftheMSEEqualizer 612 10-2-4Fractionally SpacedEqualizer 617 10-3Decision-Feedback Equalization 621 10-3-1Coefficient Optimization 621 10-3-2Performance Characteriatics ofDFE 622 10-3-3Predictive Deciaion-Feedback Equalizer 626 10-4Bibliographical NotesandReferences 628 Problems 628 11Adaptive· Equalization 636 11-1Adaptive LinearEqualizer 636 11-1-1TheZero-Forcing Algorithm 637 11-1-2TheLMSqorithm 639 11-1-3Convergence Propenies oftheLMSAlgorithm 642 11-1-4ExcesaMSEDuetoNoisyGradient Eatimates 644 11-1-SBasebaDd andPassband UnearEqualizers 648 11-2Adaptive Decision-Feedback Equalizer 649 11-2-1Adaptive Equalization ofTrellis-Coded Signals 650 11-3AnAdaptive Channel Estimator forMLSequence Detection 652 J1-4Recursive Least-Squares Algorithms forAdaptive Equalization 654 11-4-1Recursive Least-Squares (Kalman) Algorithm 656 11-4-2LinearPrediction andtheLallieeFilter 660 11-5Self-Recovering (Blind)Equalization 664 11-5-1BlindEqualization BasedonMaximum-Likelihood Criterion 664 11-5-2Stochastic Gradient Algorithms 668 )1-5-3BlindEqualization Algorithms BasedonSecond- andHigher-Order SignalStatistics 673 11-6Bibliographical NotesandReferences 675 Problems 676 12Multichannel andMulticarrier Systems 68{J 12-1Multichannel DigitalCommunication inAWGNChannels 680 12-1-1BinarySignals 682 12-1-2M-aryOrthogonal Signals 684 12-2Multicarrier Communications 686 12-2-1Capacity ofaNon-Ideal LinearFilterChannel 687 12-2-2AnFIT-BasedMulticarricr System 689 12-3Bibiliographical NotesandReferences 692 Problems 693 13SpreadSpectrum SignalsforDigitalCommunications 695 D-lModelofSpreadSpectrum DigitalCommunication System 697 13-2DirectSequence SpreadSpectrum Signals 698 13-2-1ErrorRatePerformance ofIheDecoder 702 13-2-2SomeApplications ofOSSpreadSpectrum Signals 712 13-2-3EffectofPulsedInterference onDSSpreadSpectrum Systems 717 13-2-4Generation ofPNSequences 724 13-3Frequcncy-Hoppped SpreadSpectrum Signals 729 13-3-1Performance ofFHSpreadSpcctrum SignalsinAWGN Channel 732 13-3-2Performance ofFHSpreadSpectrum Signal.inPartial- BandInterference 734 13-3-3ACDMASystemBasedonFHSpreadSpeclrum Signals 741 13-4OtherTypesofSpreadSpectrum Signals 743 13-5Synchronization ofSpreadSpectrum Signals 744 13-6Bibliographical NotesandReferences 752 Problems 753 14DigitalCommunication throughFading Multipath Channels 758 14-1Characlerization ofFadingMultipalh Channels 759 14-1-1Channel CorreialionFunctions andPowerSpeclra 762 14-1-2Statistical ModelsforFadingChannels 767 14-2TheEllectofCharacteristics ontheChoice ofaChannel Model no 14-3Frequency-Nonselective. SlowlyFadingChannel 772 14-4Diversity Techniques forFadingMultipath Channels 777 144-1BinarySignals . 778 14-4-2Multiphase Signals 785 144-3M-aryOrthogonal Signals 787 14-5DigitalSignaling overaFrequency-Selective, SlowlyFading Channel 795 14-5-1ATapped-Delay-Line Channel Model 795 14-5-2TheRAKEDemodulator 797 14-5-3Performance ofRAKEReceiver 798 14-6CodedWaveforms forFadingChannels 806 14-6-1Probability ofErrorforSoft-Decision Decoding ofLinear BinaryBlockCodes 808 14-6-2Probability ofErrorforHard-Decision Decoding of LinearBinaryBlockCodes 811 14-6-3UpperBoundsonthePerformance ofConvolutional CodesforaRaleighFadingChannel 811 14-64UscofConstant-Weight CodesandConcatenated Codes foraFadingChannel 814 14-6-5SystemDesignBasedontheCutollRate 825 14-6-6Trellis-Coded Modulation 830 14-7Bibliographical Notes andReferences 832 Problems 833 15Multiuser Communications 15-1Introduction toMultiple AccessTechniques 15-2Capacity ofMultiple AccessMethods 15-3Code-Division Multiple Access 15-3-1CDMASignalandChannel Models 15-3-2TheOptimum Receiver 15-3-3Suboptimum Detectors 15-34Performance Characteristics ofDetectors 154Random AccessMethods 15-4-1ALOHA SystemandProtocols 154-2CarrierSenseSystemsandProtocols 15-5Bibliographical NotesandReferences Problems840 840 843 849 849 851 854 859 862 863 867 872 873 Appendix A Appendix BTheLevinson-Durbin Algorithm ErrorProbability forMultichannel BinarySignals879 882 Appendix CErrorProbabilities forAdaptive Reception ofM-phaseSignals C-1Mathematical ModelforanM-phaseSignaling Communications System C-2Characteristic Function andProbability Density Function ofthePhase(I C-3ErrorProbabilities forSlowlyRayleigh Fading Channels C-4ErrorProbabilities forTime-Invariant andRicean FadingChannels887 887 889 891 893 Appendix DSquare-Root Factorization References andBibliography Index897 899 917 1 INTRODUCfION Inthisbook,wepresentthebasicprinciples thatunderlie theanalysisand designofdigitalcommunication systems. Thesubjectofdigitalcommunica­ tionsinvolvesthetransmission ofinfonnation indigitalformfromasource thatgenerates theinfonnation 10oneormoredestinations. Ofparticular importance intheanalysisanddesignofcommunication systemsarethe characteristics ofthephysical channels through whichtheinfonnation is transmitted. Thecharacteristics ofthechannelgenerally affectthedesignof thebasicbuildingblocksofthecommunication system.Below,wedescribethe elements ofacommunication systemandtheirfunctions. I-IELEMENTS OFADIGITAL COMMUNICATION SYSTEM Figure1-1-1illustrates thefunctional diagram andthebasicelements ofa digitalcommunication system.Thesourceoutputmaybeeitherananalog signal,suchasaudioorvideosignal,oradigitalsignal,suchastheoutputofa teletypemachine, thatisdiscreteintimeandhasafinitenumberofoutput characters. Inadigitalcommunication system,themessages produced bythe sourceareconverted intoasequence ofbinarydigits.Ideally,weshouldlike10 represent thesourceoutput(message) byasfewbinarydigitsaspossible. In olherwords,weseekanefficientrepresentation ofthesourceoutputthat resultsinlittleornoredundancy. Theprocessofefficiently converting the outputofeitherananalogordigitalsourceintoasequence ofbinarydigitsis calledsourceencoding ordatacompression. . Thesequence ofbinarydigitsfrom th~sourceencoder, whichwecallthe 1 Output .'iignal2DIGITAL COMMUNICATiONS InformationSOIln:e Channel Digital sourceandencoder encoder modulator inpu(transducer Channel Outptl~ Source Channel Oi.,ital transducer deco&r decoder demodulator FIGURE 1-1·1Basicelements of•digit.1commuOIcotion system. information sequence, ispassedtothechannelencoder. Thepurposeofthe channelencoder istointroduce, inacontrolled manner,someredundancy in thebinaryinformation sequence thatcanbeusedatthereceivertoovercome theeffectsofnoiseandinterference encountered inthetransmission ofthe signalthroughthechannel.Thus,theaddedredundancy servestoincreasethe reliability ofthereceiveddataandimproves thefidelityofthereceivedsignal. Ineffect,redundancy intheinformation sequence aidsthereceiverindecoding thedesiredinformation sequence. Forexample, a(trivial)formofencoding of thebinaryinformation sequence issimplytorepeateachbinarydigitmtimes, wheremissomepositiveinteger.Moresophisticated (nontrivial) encoding involvestakingkinformation bitsatatimeandmapping eachk-bitseqllence intoauniquen-bitsequence, calledacodeword.Theamountofredundancy introduced byencoding thedatainthismannerismeasured bytheration!k. Thereciprocal ofthisratio,namelykIn,iscalledtherateofthecodeor, simply,thecoderate. Thebinarysequence attheoutputofthechannelencoder ispassedtothe digitalmodulator, whichservesastheinterface tothecommunications channel. Sincenearlyallofthecommunication channels encountered inpracticeare capableoftransmitting electrical signals(waveforms), theprimarypurposeof thedigitalmodulator istomapthebinaryinformation sequence intosignal waveforms. Toelaborate onthispoint,letussuppose thatthecoded information sequence istobetransmitted onebitatatimeatsomeuniform rateRbits!s.Thedigitalmodulator maysimplymapthebinarydigit0intoa waveform so(t)andthebinarydigitIintoawaveform sJ(t).Inthismanner, eachbitfromthechannelencoderistransmitted separately. Wecallthisbinary modulation. Alternatively, themodulator maytransmit bcodedinformation bitsatatimebyusingM=2bdistinctwaveforms S,(/),i=0,I,...,M-I,one waveform foreach.ofthe2bpossible b-bitsequences. WecallthisM-ary modulation (M>2).Notethatanewb-bitsequence entersthemodulator CHAPTER I,INTRODUCTION 3 everyblRseconds. Hence,whenthechannelbitrateRisfixed,theal\lountof timeavailable totransmit oneoftheMwaveforms corresponding toab-bit sequence isbtimesthetimeperiodinasystemthatusesbinarymodulation. Thecommunication channelisthephysicalmedium thatisusedtosendthe signalfromthetransmitter tothereceiver. Inwireless transmission, the channelmaybetheatmosphere (freespace).Ontheotherhand,telephone channels usuallyemployavarietyofphysical media,including wirelines, opticalfibercables,andwireless(microwave radio).Whatever thephysical mediumusedfortransmission oftheinformation, theessential featureisthat thetransmitted signaliscorrupted inarandommannerbyavarietyofpossible mechanisms, suchasadditive thermalnoisegenerated byelectronic devices, man-made noise,e.g.,automobile ignitionnoise,andatmospheric noise,e.g., electrical lightning discharges duringthunderstorms. Atthereceiving endofadigitalcommunications system,thedigital demodulator processes thechannel-corrupted transmitted waveform andre­ ducesthewaveforms toasequence ofnumbers thatrepresent estimates ofthe transmitted datasymbols (bimiryorM-ary).Thissequence ofnumbers is passedtothechanneldecoder, whichattempts toreconstruct theoriginal information sequence fromknowledge ofthecodeusedbythechannel encoderandtheredundancy contained inthereceived data. Ameasure ofhowwellthedemodulator anddecoder perform isthe frequency withwhicherrorsoccurinthedecoded sequence. Moreprecisely. theaverageprobability ofabit-error attheoutputofthedecoderisameasure oftheperformance ofthedemodulator-decoder combination. Ingeneral, the probability oferrorisafunction ofthecodecharacteristics, thetypesof waveforms usedtotransmit theinformation overthechannel, thetransmitter power,thecharacteristics ofthechannel, i.e.,theamountofnoise,thenature oftheinterference, etc.,andthemethodofdemodulation anddecoding. These itemsandtheireffectonperformance willbediscussed indetailinsubsequent chapters. Asafinalstep,whenananalogoutputisdesired,thesourcedecoderaccepts theoutputsequence fromthechanneldecoder and,fromknowledge ofthe sourceencoding methodused,attempts toreconstruct theoriginalsignalfrom thesource.Duetochanneldecoding errorsandpossibledistortion introduced bythesourceencoder and,perhaps, thesourcedecoder, thesignalatthe outputofthesourcedecoderisanapproximation 10theoriginalsourceoutput. The'difference orsomefunction ofthedifference between theoriginalsignal andthereconstructed signalisameasure ofthedistortion introduced bythe digitalcommunication system. 1-2COMMUNICATION CHANNELS ANDmEIR CHARACTERISTICS Asindicated inthepreceding discussion, thecommunication channelprovides theconnection betweenthetransmitter andthereceiver. Thephysicalchannel 4DIGITALCOM"UNIC AnONS maybeapairofwiresthatcarrytheelectrical signal,oranopticalfiberthat carriestheinformation onamodulated lightbeam.oranunderwater ocean channelinwhichtheinformation istransmitted acoustically. orfreespaceover whichtheinformation-bearing signalisradiated byuseofanantenna. Other mediathatcanbecharacterized ascommunication channels aredatastorage media.suchasmagnetic tape,magnetic disks.andopticaldisks. Onecommon probleminsignaltransmission throughanychannelisadditive noise.Ingeneral,additivenoiseisgenerated internally bycomponents suchas resistorsandsolid-state devicesusedtoimplement thecommunication system. Thisissometimes calledthermalnoise.Othersourcesofnoiseandinterference mayariseexternally tothesystem,suchasinterference fromotherusersofthe channel.Whensuchnoiseandinterference occupythesamefrequency bandas thedesiredsignal,itselIectcanbeminimized byproperdesignofthe transmitted signalanditsdemodulator atthereceiver. Othertypesofsignal degradations thatmaybeencountered intransmission overthechannelare signalattenuation, amplitude andphasedistortion, andmultipath distortion. Theeffectsofnoisemaybeminimized byincreasing tbepowerinthe transmitted signal.However, equipment andotherpractical constraints limit thepowerlevelinthetransmitted signal.Another basiclimitation isthe available channel bandwidth. Abandwidth constraint isusuallyduetothe physical limitations ofthemedium andtheelectronic components usedto implement thetransmitter andthereceiver. Thesetwolimitations resultin constraining theamountofdatathalcanbetransmitted reliablyoverany communications channelasweshallobserveinlaterchapters. Below,we describe someoftheimportant characteristics ofseveralcommunication channels. WireliDe Channels Thetelephone network makesextensive useofwire Jinesforvoicesignaltransmission. aswellasdataandvideotransmission. Twisted-pair wirelinesandcoaxialcablearebasically guidedelectromagnetic channels thatproviderelatively modestbandwidths. Telephone wiregenerally usedtoconnect acustomer toacentralofficehasabandwidth ofseveral hundred kilohertz (kHz).Ontheotherhand,coaxialcablehasausable bandwidth ofseveralmegahertz (MHz).Figure1-2-1illustrates thefrequency rangeofguidedelectromagnetic channels. whichincludewaveguides and opticalfibers. Signalstransmitted throughsuchchannels aredistored inbothamplitude andphaseandfurthercorrupted byadditive noise.Twisted-pair wireline channels arealsopronetocrosstalk interference fromphysically adjacent channels. Because wireline channels carryalargepercentage ofourdaily communications aroundthecountryandtheworld,muchresearch hasbeen performed onthecharacterization oftheirtransmission properties andon methods formitigating theamplitude andphasedistortion encountered in signaltransmission. InChapter9,wedescribe methods fordesigning optimum transmitted signalsandtheirdemodulation: inChapters 10and11.we CHAPTER', INTRODUCT10N S Ulnviolcl10"liz Visible. 10-'..1._ 1014Hz IOOml'D Iell JOcm ~I..;;, j! ~s ~10m 100m Ikm 10km IOOkm nGURE 1·201Frequency rangeforguidedwirecbannel.---- Waveguide - -- Coaxialcable channels -~ Twisted-pllir wirelil'll: channels --1000Hz 100Hz IGHz ... ~ 100101Hzg [ "- IOMHz 1101Hz 100.Hz 10kHz 1kHz •consider thedesignofchannelequalizers thatcompensate foramplitude and phasedistortion onthesechannels. FiberOpticCh8mIeis Opticalfibersofferthecommunications system designer achannelbandwidth thatisseveralordersofmagnitude largerthan coaxialcablechannels. Duringthepastdecade.opticalfibercableshavebeen developed thathavearelatively lowsignalattenuation. andhighlyreliable photonic deviceshavebeendeveloped forsignalgeneration andsignal detection. Thesetechnological advances haveresultedinarapiddeployment of oplicalfiberchannels. bothindomestic telecommunication systemsaswellas fortrans-Allantic andtrans·Pacificcommunications. Wilhthelargebandwidth ()DIGITAL COMMUNICATIONS available onfiberopticchannels, itispossiblefortelephone companies tooffer subscribers awidearrayoftelecommunication services, including voice.data, facsimile, andvideo. Thetransmitter ormodulator inafiberopticcommunication systemisa lightsource,eitheralight-emitting diode(LED)oralaser.Information is transmitted byvarying(modulating) theintensity ofthelightsourcewiththe message signal.Thelightpropagates through thefiberasalightwaveandis amplified periodically (inthecaseofdigitaltransmission, itisdetected and regenerated byrepeaters) alongthetransmission pathtocompensate forsignal attenuation. Atthereceiver, thelightintensity isdetected byaphotodiode, whoseoutputisanelectrical signalthatvariesindirectproportion tothe powerofthelightimpinging onthephotodiode. Sourcesofnoiseinfiberoptic channels arephotodiodes andelectronic amplifiers. Itisenvisioned thatopticalfiberchannels willreplacenearlyallwireline channels inthetelephone networkbytheturnofthecentury. WIl'eJess Electromagnetic Channels Inwirelesscommunication systems, electromagnetic energyiscoupledtotbepropagation medium byanantenna whichservesastheradiator. Thephysicalsizeandtheconfiguration ofthe antenna dependprimarily onthefrequency ofoperation. Toobtainefficient radiation ofelectromagnetic energy,theantennamustbelongerthantoofthe wavelength. Consequently, aradiostationtransmitting intheAMfrequency band,sayatic=1MHz(corresponding toawavelength of,\.=elic=300m), requires anantenna ofatleast30m.Otherimportant characteristics and attributes ofantennas forwirelesstransmission aredescribed inChapter 5. Figure1-2-2illustrates thevariousfrequency bandsoftheelectromagnetic spectrum. Themodeofpropagation ofelectromagnetic wavesintheatmo­ sphereandinfreespacemaybesubdivided intothreecategories, namely, ground-wave propagation, sky-wave propagation, andline-oC·sight (LOS) propagation. IntheVLFandaudiofrequency bands,wherethewavelengths exceed10km,theearthandtheionosphere actasawaveguide forelectromag­ neticwavepropagation. Inthesefrequencyranges,communication signals practically propagate aroundtheglobe.Forthisreason,thesefrequency bands areprimarily usedtoprovidenavigational aidsfromshoretoshipsaroundthe world.-Thechannel bandwidths avaihible inthesefrequency bandsare relatively small(usually 1-10%ofthecenterfrequency). andhencethe information thatistransmitted throughthesechannels isoCrelatively slow speedandgenerally confined todigitaltransmission. Adominant typeoCnoise atthesefrequencies isgenerated fromthunderstorm activityaroundtheglobe. especially intropicalregions.Interference resultsfromthemanyusersofthese frequencybands.' • Ground-wave propagation, asillustrated inFig.1-2·3,isthedominant mode ofpropagatiori forfrequencies intheMFband(0.3-3MHz).Thisisthe frequency bandusedforAMbroadcasting andmaritime radiobroadcasting. In AMbroadcasting, therangewithgroundwave propagation oCeventhemore Frequency band UseCHAPTER I:INTRODUCTION 7 10-"m Icm 10em 1m ~10m;;,•u..>•il100m Ikm IOkm IOOkm---- -~ Ultraviolet VisibleIigl1l. Experimental Infrared -- -~ --- -- -Millimeter waves (EHF) Ex.perimental Navigation Satellite tosatellite Superhighfrequency Microwave relay (SHFj Earth-satellite Radar UhrahighfrequencyMobileradio (UHF) UHFTVandmobileradio Mobile.llC1'Onautical VeryhighfrequencyVHFTVandFMBroadcast(VHF)Mobileradio BusintssHipfrequencyAmatturradio(HF)International radio Citizen's band MediumfRquencyAMl:madcasl(MF) Lowfrequency Aerouutical (LF) Navigation RadioIeJetype Verylowfrequency (VLF) Audio band----IOI~Hz 100GHz--r- Microwave JOGHzradio IGHz1 Shortwave 100MHzradio ~y• 10MHz8 ~"- IMHz Lonawave radio 100kHz ~10kHz 1kHz FlGURE 1·2-2Frequency rangeforwirel.esselectromagnetic channels. [Adapted fromCarlson(1975). 2ndedition.©McGraw·HiII BookCompany Co.Reprinted withpermission ofthepublisher.J FlGURE 1·2-3IDuStralion ofground·wave propagation. •DIGITAL COMMUNICATIONS FIGURE 1·1-4IUustration ofsky·wave propagation.-'......,,,,,,, powerful radiostations islimitedtoaboutISOkID.Atmospheric noise, man-made noise,andthermalnoisefromelec\ronic components atthereceiver aredominant disturbances forsignaltransmission intheMFband. Sky-wave propagation, asillustrated inFig.1-2-4resultsfromtransmitted signalsbeingreflected (bentorrefracted) fromtheionosphere, whichconsists ofseverallayersofchargedparticles rangingin.altitude fromSOto400km abovethesurfaceoftheearth.Duringthedaytime hours,theheatingofthe loweratmosphere bythesuncausestheformation ofthelowerlayersat altitudes below120km.Theselowerlayers,especially theD-layer, serveto absorbfrequencies below2MHz,thusseverely limitingsky-wave propagation ofAMradiobroadcast. However, duringthenight-time hours,theelectron densityinthelowerlayersoftheionosphere dropssharplyandthefrequency absorption thatoccursduringthedaytime issignificantly reduced. Asa consequence, powerful AMradiobroadcast stationscanpropagate overlarge distances viaskywaveovertheF-Iayeroftheionosphere, whichrangesfrom 140to400kmabovethesurfaceoftheearth. Afrequently occurring problem withelectromagnetic wavepropagation via skywaveintheHFfrequency rangeissignalmultipath. Signalmultipath occurs ~henthetransmitted signalarrivesatthereceiver viamultiple propagation pathsatdifferent delays.Itgenerally resultsinintersymbol interference ina digitalcommunication system.Moreover, thesignalcomponents arrivingvia different propagation pathsmayadddestructively, resulting inaphenomenon cailedsignalfading,whichmostpeoplehaveexperienced whenlistening toa distantradiostationatnightwhenskywaveisthedominant propagation mode.Additive noiseatHFisacombination ofatmospheric noiseandthermal noise. Sky-wave ionospheric propagation ceasestoexistatfrequencies above approximately 30MHz,whichistheendoftheHFband.However, itis possible tohaveionospheric scatterpropagation atfrequencies intherange 30-60MHz,resulting fromsignalscattering fromthelowerionosphere. Itis alsopossibletocommunicate overdistances ofseveralhundred milesbyuseof tropospheric scattering atfrequencies intherange40-300MHz.Troposcatter resultsfromsignalscattering duetoparticles intheatmosphere ataltitudes of 10milesorless.Generally, ionospheric scatterandtropospheric scatter CHAPTER IINTRODllCnON I} involvelargesignalpropagation lossesandrequirealargeamount of transmitter powerandrelatively largeantennas. Frequencies above30MHzpropagatethrough theionosphere withrela­ tivelylittlelossandmakesatellite andextraterrestrial communications possible. Hence,atfrequencies intheVHFbandandhigher,thedominant modeofelectromagnetic propagation isline-of·sight (LOS)propagation. For terrestrial communication systems. thi~meansthatthetransmitter andreceiver antennas mustbeindirectLOSwithrelatively littleornoobstruction. Forthis reason,television stationstransmitting intheVHFandUHFfrequency bands mounttheirantennas onhightowerstoachieveabroadcoverage area. Ingeneral, thecoverage areaforLOSpropagation islimitedbythe curvature oftheearth.Ifthetransmitting antenna ismounted ataheighthm abovethesurfaceoftheearth,thedistance totheradiohorizon, assuming no physicalobstructions suchasmountains, isapproximately d=v'15hkm.For example, aTVantennamounted onatowerof300minheightprovides a coverage ofapproximalely 67km.Asanotherexample, microwave radiorelay systemsusedextensively fortelephone andvideotransmission atfrequencies above1GHzhaveantennas mounted ontalltowersoronthetopoftall buildings. Thedominant noiselimitingtheperformance ofacommunication systemin VHFandUHFfrequency rangesisthermalnoisegenerated inthereceiver frontendandcosmicnoisepickedupbytheantenna. Atfrequencies inthe SHFbandabove10GHz,atmospheric conditions playamajorroleinsignal propagation. Forexample, at10GHz,theattenuation rangesfromabout OJl03dB/kminlightraintoabout0.3dB/kminheavyrain.At100GHz,the attenuation rangesfromabout0.1dB/kminlightraintoabout6dBlkmin heavyrain.Hence,inthisfrequency range,heavyrainintroduces extremely highpropagation lossesthatcanresultinserviceoutages(totalbreakdown in thecommunication system). Atfrequencies abovetheEHF(extremely highfrequency) band,wehave theinfraredandvisiblelightregionsoftheelectromagnetic spectrum. which canbeusedtoprovideLOSopticalcommunication infreespace.Todate, thesefrequency bandshavebeenusedinexperimental communication systems,suchassatellite-to-satellite links. ynderwa'er Acoustic Channels Overthepastfewdecades, oceanex­ ploration activityhasbeensteadilyincreasing. Coupled withthisincrease isthe needtotransmit data,collected bysensorsplacedunderwater,tothesurface oftheocean.Fromthere,itispossibletorelaythedataviaasatellitetoadata collection center. Elee;tromagnetic wavesdonotpropagate overlongdistances underwater exceptatextremely lowfrequencies. However, thetransmission ofsignalsat suchlowfrequencies isprohibitively expensive because ofthelargeand powerful transmitters required. Theattenuation ofelectromagnetic wavesin watercanbeexpressed intermsoftheskindepth,whichisthedistanceasignal isattenuated bylIe.Forseawater,theskindepthli=250/v'].wherefis 10D!(ill';\l. COMMI-NICA 110l"l,;\ expressed inHzand[,isinm.Forexample. at10kHz.theskindepthis2.5m. Incontrast. acoustic signalspropagate overdistances oftensandeven hundreds ofkilometers. Anunderwater acousticchannel ischaracterized asamultipath channeldue tosignalreflections fromthesurfaceandthebottomofthesea.Because of wavemotion.thesignalmultipath components undergo time-varying propaga­ tiondelaysthatresultinsignalfading.Inaddition. thereisfrequency­ dependent attenuation. whichisapproximately proportional tothesquareof thesignalfrequency. Thesoundvelocity isnominally about1500m/s. butthe actualvaluewillvaryeitheraboveorbelowthenominal valuedepending on thedepthatwhichthesignalpropagates. Ambient oceanacoustic noiseiscausedbyshrimp. fish.andvarious mammals. Nearharbors. thereisalsoman-made acoustic noiseinaddition to theambient noise.Inspiteofthishostileenvironment. itispossible todesign andimplement efficient andhighlyreliableunderwater acoustic communica­ tionsystemsfortransmitting digitalsignalsoverlargedistances. StorageChannels Information storageandretrieval systemsconstitute a verysignificant partofdata-handling activities onadailybasis.Magnetic tape. including digitalaudiotapeandvideotape.magnetic disksusedforstoring largeamounts ofcomputer data.opticaldisksusedforcomputer datastorage. andcompact disksareexamples ofdatastorage systems thatcanbe characterized ascommunication channels. Theprocessofstoringdataona magnetic tapeoramagnetic oropticaldiskisequivalent totransmitting a signaloveratelephone oraradiochannel. Thereadback processandthe signalprocessing involved instoragesystemstorecoverthestoredinformation areequivalent tothefunctions performed byareceiver inatelephone orradio communication systemtorecoverthetransmitted information. Additive noisegenerated bytheelectronic components andinterference fromadjacent tracksisgenerally presentinthereadback signalofastorage system,justasisthecaseinatelephone oraradiocommunication system. Theamountofdatathatcanbestoredisgenerally limitedbythesizeofthe diskortapeandthedensity(number ofbitsstoredpersquareinch)thatcanbe achieved bythewrite/read electronic systems andheads.Forexample, a packing densityof1O~bitspersquareinchhasbeenrecently demonstrated in anexperimental magnetic diskstoragesystem.(Current commercial magnetic storageproducts achieveamuchlowerdensity.) Thespeedatwhichdatacan bewrittenonadiskortapeandthespeedatwhichitcanbereadbackarealso limitedbytheassociated mechanical andelectrical subsystems thatconstitute aninformation storagesystem. Channel codingandmodulation areessential components ofawell-designed digitalmagnetic oropticalstoragesystem.Inthereadbackprocess. thesignalis demodulated andtheaddedredundancy introduced bythechannelencoder is usedtocorrecterrorsinthereadback signal. CHAI'TER IINTRODUCTION 11 1-3MATHEMATICAL MODELS FOR COMMUNICATION CHANNELS Inthedesignofcommunication systemsfortransmitting information through physicalchannels, wetinditconvenient toconstruct mathematical modelsthat reflectthemostimportant characteristics ofthetransmission medium. Then, thernathematical modelforthechannelisusedinthedesignofthechannel encoder andmodulator atthetransmitter andthedemodulator andchannel decoder atthereceiver. Below,weprovide abriefdescription ofthe channelmodelsthatarefrequently usedtocharacterize manyofthephysical channels thatweencounter inpractice. TheAdditive NoiseChannel Thesimplest mathematical modelfora communication channelistheadditive noisechannel, illustrated inFig.1-3-1. Inthismodel,thetransmitted signalS(I)iscorrupted byanadditive random noiseprocessn(I).Physically. theadditive noiseprocess mayarisefrom electronic components andamplifiers atthereceiver ofthecommunication system,orfrominterference encountered intransmission (asinthecaseof radiosignaltransmission). Ifthenoiseisintroduced primarily byelectronic components andamplifiers atthereceiver, itmaybecharacterized asthermalnoise.Thistypeofnoiseis characterized stati~tically asagaussian noiseprocess. Hence,theresulting mathematical modelforthechannel isusuallycalledtheadditive gaussian noisechannel, Because thischannelmodelappliestoabroadclassofphysical communication channels andbecauseofitsmathematical tractability. thisis thepredominant channel modelusedinourcommunication systemanalysis anddesign.Channel attenuation iseasilyincorporated intothemodel.When thesignalundergoes attenuation intransmission through thechannel, the received signalis r(l)=as(I)+n{t) whereaistheattenuation factor.(1-3-1) TheLinearFilterChannel Insomephysical channels, suchaswireline telephone channels, filtersareusedtoensurethatthetransmitted signalsdo notexceedspecified bandwidth limitations andthusdonotinterfere withone another. Suchchannels aregenerally characterized mathematically aslinear filterchannels with'additive noise,asillustrated inFig.1-3-2.Hence.ifthe Channel ;:;'(;.;,"4---..( +}----1f-- rft)=.f(11+tilt) FIGURE 1·3-1Theadditivenoisechannel.n(O 12DIGITAL C()MMV~JCATIONS -_._-~----_._---------, ,, ,, , I Ulk"ar I \(1)I filter Irl.I)=.f(r)*nll""l(I) fin I~ ... ._JFIGURE t-3-2Thelinearfilterchannelwith additive noise.Channelnil) channelinputisthesignal5(1),thechanneloutpulisthesignal '(1)=S(/)*C(/)+n(/) ={,c(r)s(1-r)dr+11(1) (1-3-2) wherec(r)isIheimpulse response ofIhelinearfilterand*denotes convolution. (1-3-3)lbeLinearTime·Variant FilterChannel Physicalchannels suchasunder­ wateracoustic channels andionospheric radiochannels thatresultintime­ variantmultipath propagation ofthetransmitted signalmaybecharacterized matllematically astime-variant linearfilten;.Suchlinearfilten;arecharac­ terizedbyatime-variant channelimpulseresponse c(r;I),wherec(r;I)isthe response ofthechannelattimeIduetoanimpulseappliedattimeI-r.Thus, rrepresents Ille"age"(elapsed-time) variable. Thelineartime-variant tilter channelwithadditivenoiseisillustrated inFig.1·3-3.Foraninputsignals(t), thechanneloutputsignalis r(/)=S(/)*c(r;I)+n(/) =f,c(r;r)s(1-r)dr+n(/) Agoodmodelformultipatll signalpropagation throughphysicalchannels, suchas theionosphere (atfrequencies below30MHz)andmobilecellular radiochannels. isaspecialcaseof(1-3-3)inwhichthetime-variant impulse response Ilastilefonn I. c(r;I)=La.(I)IJ(r -T.) k=I(1-3-4) ChannelrillLinear Time-varianl f----<>{ fillerdt:11,,,, n(I) I•I ------~---_ ...._-~,----_._--------------·•·J(/JI FIGURE .-3-3Lineartime·variantIlIlerchannelwithadditivenoise. CHAPTER I:INTRODUCTION 13 wherethe{a.(t)}represents thepossiblytime-variant attenuation factorforthe Lmultipath propagation pathsand{'I".}arethecorresponding timedelays.If (1-3-4)issubstituted into(1-3-3),thereceivedsignalhastheform L r(t)=La.(t)s(t-'1".)+n(t).-1(1-3-5) Hence,thereceived signalconsistsofLmultipathcomponents, whereeach component isattenuated by{a.(t)}anddelayedby{'I"o}. Thethreemathematical modelsdescribed aboveadequately characterize the greatmajority ofthephysicalchannels encountered inpractice. Thesethree channelmodelsareusedinthistextfortheanalysisanddesignofcommunica­ tionsystems. 1-4AHISTORICAL PERSPECTIVE INTHE DEVELOPMENT OFDIGITAL COMMUNICATIONS Itisremarkable thattheearliestformofelectrical communication, namely telegraphy, wasadigitalcommunication system.Theelectrictelegraph was developed bySamuelMorseandwasdemonstrated in1837.Morsedevisedthe variable-length binarycodeinwhichlettersoftheEnglish alphabet are represented byasequence ofdotsanddashes(codewords).Inthiscode,more frequently occurring lettersarerepresented byshortcodewords,whileletters occurring lessfrequently arerepresented bylongercodewords.Thus,the Morsecode'was theprecursor ofthevariable-length sourcecodingmethods described inChapter 3. Nearly40yearslater,in1875,EmileBaudotdevisedacodefortelegraphy inwhicheveryletterwasencoded intofixed-length binarycodewordsoflength 5.IntheBaudotcode,binarycodeelements areofequallengthanddesignated asmarkandspace. Although Morseisresponsible forthedevelopment ofthefirstelectrical digitalcommunication system(telegraphy), thebeginnings ofwhatwenow regardasmodem digitalcommunications stemfromtheworkofNyquist (1924),whoinvestigated theproblem ofdetermining themaximum signaling ratethatcanbeusedoveratelegraph channelofagivenbandwidth without intersyrnbol interference. Heformulated amodelofatelegraph systemin whichatransmitted signalhasthegeneralform s(t)=Lang(t-nT) n(1-4-1) where'g(t)represents abasicpulseshapeand{an}isthebinarydatasequence of{±I}transmitted atarateoflITbits/s.Nyquistsetouttodetermine the optimum pulseshapethatwasbandlimited toWHzandmaximized thebitrate undertheconstraint thatthepulsecausednointersymbol interference atthe (1-4-2)14DIGITAL COMMl1N!CATJONS sampling timekJT.k=O.±1,±2,....Hisstudiesledhimtoconclude thatthe maximum pulseraleis2Wpulses/s. ThisrateisnowcalledtheNyquistrale. Moreover, thispulseratecanbeachieved byusingthepulsesg(l)= (sin21rWI)J2JrWt. Thispulseshapeallowsrecovery ofthedatawithout intersymbol interference atthesampling instants.Nyquist's resultisequivalent toaversionofthesampling theD!emforbandlimited signals,whichwaslater statedprecisely byShannon (19~8).Thesampling theoremstatesthatasignal ofbandwidth Wcanbereconstructed 'romsamplestakenattheNyquistrate of2Wsamples/s usingtheinterpolation formula st=2>('!!'-) sin[2JrW(t-n/2W)j ()"2W21rW(1-n/2W) InlightofNyquist's work,Hartley(1928)considered theissueofthe amountofdatathatcanbetransmitted reliablyoverabandlimited channel whenmultiple amplitude levelsareused.Duetothepresence ofnoiseand olherinterference, Hartleypostulated thatthereceiver canreHablyestimate thereceivedsignalamplitude tosomeaccuracy, sayA•.Thisinvestigation led Hartley toconclude thatthereisamaximum dataratethatcanbe communicated reliablyoverabandlimited channelwhenthemaximum signal amplitude islimitedtoAmRx(Iixedpowerconstraint) andtheamplitude resolutionisA,. Another signilicant advanceinthedevelopmen> ofcommunications wasthe workofWiener(1942),whoconsidered theproblem ofestimating adesired signalwaveform s(t)inthepresence ofadditive noisen(I),basedon observation ofthereceived signal1'(1)=S(I)+n(I).Thisproblem arisesin signaldemodulation. Wienerdetermined thelinearfilterwhoseoutputisthe bestmean-square approximation tothedesiredsignals(t).Theresulting filter iscalledtheoptimum linear(Wiener) filter. Hartley's andNyquist's resultsonthemaximum transmission rateofdigital information wereprecursors totheworkofShannon (1948a,b), whoestabl­ ishedthemathematical foundations forinformation transmission andderived thefundamental limitsfordigitalcommunication systems. Inhispioneering work.Shannon formulatedthebasicproblem ofreliable transmission of information instatistical terms,usingprobabilistic modelsforinformation sourcesandcommunication channels. Basedonsuchastatistical formulation, headoptedalogarithmic measure fortheinformation contentofasource.He alsodemonstrated thattheeffectofatransmitter powerconstraint. a bandwidth constraint, andadditivenoisecanbeassociated withthechannel andincorporated intoasingleparameter, calledthechannel capacity. For example, inthecaseofanadditive white(spectrally fiat)gaussian noise inJerference, anidealbandlimited channelofbandwidth Whasacapacity C givenby c=Wlog2(1+:N)bits/s (1-4-3) ClI.\P"I[-R I:l",rHO[ll.CIIO'" 15 wherePistheaveragetransmitted powerandNoisthepowerspectraldensity oftheadditivenoise.Thesignificance ofthechannelcapacity isasfollows: If theinformation rateRfromthesourceislessthanC(R<C)thenitis theoretically possibletoachievereliable(error-free) transmission throughthe channel byappropriate coding.Ontheotherhand,ifR>C..reliable transmission isnotpossible regardless oftheamountofsignalprocessing performed atthetransmitter nndreceiver. Thus,Shannon established basic limitsoncommunication ofinformation, andgavebirthtoanewfieldthatis nowcalledinformation theory. Another important contribution tothefieldofdigitalcommunication isthe .wor~ofKotelnikov (1947),whoprovided acoherent analysisofthevarious digitalcommunication systemsbasedonageometrical approach. Kotelnikov's approach waslaterexpanded byWozencraft andJacobs(1965). Following Shannon's publications, cametheclassicworkofHamming (1950)onerror-detecting anderror-correcting codestocombatthedetrimental effectsofchannelnoise.Hamming's workstimulated manyresearchers inthe yearsthatfollowed, andavarietyofnewandpowerful codeswerediscovered, manyofwhichareusedtodayintheimplementation ofmoderncommunica­ tionsystems. Theincrease indemand fordatatransmission duringthelastthreetofour decades, coupled withthedevelopment ofmoresophisticated integrated circuits,hasledtothedevelopment ofveryefficientandmorereliabledigital communication systems. Inthecourseofthesedevelopments, Shannon's originalresultsandthegeneralization ofhisresultsonmaximum transmission limitsoverachannelandonboundsontheperformance achieved haveserved asbenchmarks foranygivencommunication systemdesign.Thetheoretical limitsderivedbyShannon andotherresearchers thatcontributed tothe development ofinformation theoryserveasanultimategoalinthecontinuing effortstodesignanddevelopmoreefficientdigitalcommunication systems. Therehavebeenmanynewadvances intheareaofdigitalcommunications following theearlyworkofShannon, Kotelnikov, andHamming. Someofthe mostnotabledevelopments arethefollowing: •Thedevelopment ofnewblockcodesbyMuller(1954),Reed(1954), ReedandSolomon (1960),BoseandRay-Chaudhuri (1960a,b), andGappa (1970,1971). •Thedevelopment ofconcatenated codesbyForney(1966). •Thedevelopment ofcomputationally efficient decoding ofBCHcodes, e,g.,theBerlekamp-Massey algorithm (seeChien,1964;Berlekamp, 1968). •Thedevelopment ofconvolutional codesanddecoding algorithms by Wozencraft andReiffen (1961),Fano(1963),Zigangirov (1966),Jelinek (1969),Forney(1970,1972), andViterbi(1967,1971). •Thedevelopment oftrellis-coded modulation byUngerboeck (1982), Forneyetai.(1984),Wei(1987),andothers. •Thedevelopment ofefficient sourceencodings algorithms fordata 16DIGITAL COMMl;NICATIONS compression, suchasthosedevisedbyZivandLempel(1977,1978)andLinde etal.(1980). 1·5OVERVIEW OFTHEBOOK Chapter 2presents abriefreviewofthebasicnotionsinthetheoryof probability andrandomprocesses. Ourprimaryobjectives inthischapterare topresentresultsthatareusedthroughout thebookandtoestablish some necessary notation. InChapler 3,weprovideanintroduction tosourcecodingfordiscreteand analogsources. Included inthischapteraretheHuffman codingalgorithm and theLempel-Ziv algorithm fordiscretesources, andscalarandvectorquantiza­ tiontechniques foranalogsources. Chapter 4treatsthecharacterization ofcommunicati<:m signalsandsystems fromamathematical viewpoint. Included inthischapter isageometric representation ofsignalwaveforms usedfordigitalcommunications. Chapters 5-8arefocused onmodulation/demodulation andchannel coding/decoding fortheadditive, whitegaussian noisechannel. Theemphasis isonoptimum demodulation anddecoding techniques andtheirperformance. Thedesignofefficient modulators anddemodulators forlinearfilter channels withdistortion istreatedinChapters 9-11.Thefocusisonsignal designandonchannelequalization methods tocompensate forthechannel distortion. Thetinalfourchapters treatseveralmorespecialized topics.Chapter 12 treatsmultichannel andmulticarrier communication systems. Chapter 13is focused onspreadspectrum signalsfordigitalcommunications andtheir performance characteristics. Chapter 14provides ain-depth treatment of communication throughfadingmultipath channels. Included inthistreatment isadescription ofchannelcharacterization, signaldesignanddemodulation techniques andtheirperformance, andcoding!decoding techniques andtheir performance. Thelastchapterofthebookisfocusedonmultiuser communica­ tionsystemsandmultiple accessmethods. 1·6BIBLIOGRAPHICAL NOTES ANDREFERENCES Thereareseveralhistorical treatments regarding thedevelopment ofradioand telecommunications duringthepastcentury. Thesemaybefoundinthebooks byMcMahon (1984),Millman (1984),andRyderandFink(1984).Wehave alreadycitedtheclassical worksofNyquist(1924),Hartley(1928),Kotelnikov (1947),Shannon (1948),andHamming (1950),aswellassomeofthemore important advances thathaveoccurred inthefieldsince1950.Thecollected papersbyShannon havebeenpublished byIEEEPressinabookeditedby SloaneandWyner(1993).Othercollected workspublished bytheIEEEPress thatmightbeofinteresttothereaderareKeyPapersintheDevelopment of Coding Theory, editedbyBerlekamp (1974), andKeyPapersinthe Development ofInformation Theory,editedbySlepian(1974). 2 PROBABILITY AND STOCHASTIC PROCESSES Thetheoryofprobability andstochastic processes isanessential mathematical toolinthedesignofdigitalcommunication systems. Thissubjectisimportant inthestatistical modeling ofsourcesthatgenerate theinformation, inthe digitization ofthesourceoutput,inthecharacterization ofthechannelthrough whichthedigitalinformation istransmitted, inthedesignofthereceiver that processes theinformation-bearing signalfromthechannel, andinthe evaluation oftheperformance ofthecommunication system.Ourcoverage of thisrichandinteresting subject isbriefandlimitedinscope.Wepresenta numberofdefinitions andbasicconcepts inthetheoryofprobability and stochastic processes andwederiveseveralresultsthatareimportant inthe designofefficientdigitalcommunication systemsandintheevaluation oftheir performance. Weanticipate thatmostreadershavehadsomepriorexposure tothetheory ofprobability andstochastic processes, sothatourtreatment servesprimarily asareview.Somereaders, however, whohavehadnoprevious exposure may findthepresentation inthischapterextremely brief.Thesereaderswillbenefit fromadditional readingofengineering-level treatments ofthesubjectfoundin thetextsbyDavenport andRoot(1958),Davenport (1970),Papoulis (1984), Helstrom (1991),andLeon-Garcia (1994). 2-1PROBABILITY Letusconsider anexperiment. suchastherollingofadie,withanumberof possibleoutcomes. ThesamplespaceSoftheexperiment consistsofthesetof allpossibleoutcomes. Inthecaseofthedie, S={I,2,3,4,5,6} (2-1-1) 17 18DIGITAL COMMUNICATiONS wheretheintegers 1,...,6represent thenumberofdotsonthesixfacesofthe die.Thesesixpossible outcomes arethesamplepointsoftheexperiment. An eventisasubsetofS,andmayconsistofanynumber ofsamplepoints.For example, theeventAddinedas A={2,4} (2-1-2) consistsoftheoutcomes 2and4.Thecomplement oftheeventA,denoted by A,consistsofallthesamplepointsinSthatarenotinAand,hence, A={1,3,5,6} (2-1-3) Twoeventsaresaidtobemutallyexclusive iftheyhavenosamplepointsin common-that is,iftheoccurrence ofoneeventexcludes theoccurrence ofthe other. ~'arexample, ifAisdefinedasin(2-1-2)andtheeventBisdefinedas B={I,3,6} (2-1-4) thenAandBaremutually exclusive events.Similarly, AandAaremutually exclusive events. Theunion(sum)oftwoeventsisaneventthatconsistsofallthesample pointsinthetwoevents.Forexample, ifBistheeventdefinedin(2-1-4)andC istheeventdefinedas C={I,2,3} then,theunionofBandC,denoted byBUC,istheevent D=BUC ={I,2,3,6}(2-1-5) (2-1-6) Similarly, AUA=S,whereSistheentiresamplespaceorthecertainevent. Ontheotherhand,theintersection oftwoeventsisaneventthaIconsistsof thepointsthatarecommon tothetwoevents.Thus,ifE=BnCrepresents theintersection oftheeventsBandC,defined by(2-1-4)and(2-1-5), respectively, then E={I.3} Whentheeventsaremutually exclusive, theintersection isthenullevent, denoted as0.Forexample, AnB=0,andAnA=0.Thedefinitions of unionandintersection areextended tomorethantwoeventsinastraightfor­ wardmanner. Associated witheacheventAcontained inSisitsprobability P(A).Inthe assignment ofprobabilities toevents,weadoptanaxiomatic viewpoint. That CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 19 is,wepostulate thattheprobability oftheeventAsatisfies thecondition P(A);;' O.Wealsopostulate thattheprobability ofthesamplespace(certain event)ispeS)=1.Thethirdpostulate dealswiththe probability ofmutually exclusive events.Suppose thatA"i=I,2,...,area(possibly infinite)number ofeventsinthesamplespaceSsuchthat A,nA,=0 i>"j=I,2, ... Thentheprobability oftheunionofthesemutually exclusive eventssatisfies thecondition (2-1-7) Forexample, inarollofalairdie,eachpossible outcome isassigned the probability ~.TheeventAdefinedby(2-1-2)consistsoftwomutually exclusive subeve~ts oroutcomes, and,hence,P(A)=i=~.Also,theprobability ofthe eventAUB,whereAandBarethemutually exclusive eventsdefinedby (2-]-2)and(2-1-4),respectively, ispeA)+PCB)=~+!=~. JointEventsandJointProbabi6ties Instead ofdealing withasingle experiment, letusperform twoexperiments andconsider theiroutcomes. For example, thetwoexperiments maybetwoseparate tossesofasingledieora singletossoftwodice.Ineithercase,thesamplespaceSconsists 01the36 two-tuples (i,j)wherei,j=1,2,...,6.Ifthedicearefair,eachpointinthe samplespaceisassigned theprobabilityk.Wemaynowconsider jointevents, suchas{iiseven,j=3},anddetermine theassociated probabilities ofsuch eventsfromknowledge oftheprobabilities ofthesamplepoints. Ingeneral. ifoneexperiment hasthepossible outcomes A"i=1.2,...,n, andthesecondexperiment hasthepossible outcomes Bj,j=1,2,...,m,then thecombined experiment hasthepossible jointoutcomes (Ai'B,).i= 1,2,...,n,j=I,2,...,m.Associated witheachjointoutcome (A"BI)isthe jointprobability peA"~B)whichsatisfiestheconditiol) Assuming thattheoutcomes Bi,j=1,2,...,m,aremutually exclusive. it followsthat no 2:P(A"Bi)=peA,) j=l(2-1-8) Similarly, iftheoutcomes Ai,i=1,2,...,n,aremutually exclusive then n 2:peA"~Bi)=P(Bi) ;=1(2-1-9) 20DIGiTAl COMMUNICATIONS Furthermore, ifalltheoutcomes ofthetwoexperiments aremutually exclusive then n m 2":2:P(A"BI)=1 1=I)=I(2-1-10) Thegeneralization oftheabovetreatment tomorethantwoexperiments is straightforward. Conditional Probabilities Consider acombined experiment inwhicha jointeventoccurswithprobability P(A,B).Suppose thattheeventBhas occurred andwewishtodetermine theprobability ofoccurrence oftheevent A.Thisiscalledtheconditional probability oftheeventAgiventheoccurrence oftheeventBandisdefinedas P(AIB)=P(A,B) P(B)(2-1-11) provided P(B)>O.Inasimilarmanner, theprobability oftheeventB conditioned ontheoccurrence oftheeventAisdefinedas P(BIA)=P(A,B) PtA)(2-1-12) provided P(A)>O.Therelations in(2-1-11) and(2-1-12) mayalsobe expressed as P(A.B)=P(AIB)P(B)=P(BIA)P(A) (2-1-13) Therelations in(2-1-11), (2-1-12), and(2-1-13) alsoapplytoasingle experiment inwhichAandBareanytwoeventsdefinedonthesamplespaceS andP(A,B)isinterpreted astheprobability oftheAnB.Thatis,P(A,B) denotesthesimultaneous occurrence ofAandB.Forexample, consider the eventsBandCgivenby(2-}-4)and(2-1-5),respectively, forthesingletossof adie.Thejointeventconsistsofthesamplepoints{1,3}.Theconditional probability oftheeventCgiventhatBoccurred is z P(CIB)=~=~ 6 Inasingleexperiment, weobserve thatwhentwoeventsAandBare mutually exclusive,AnB=0and,hence,P(AIB)=O.Also,ifAisasubset ofBthenAnB=Aand,bence, P(AIB)=P(A) P(B) CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 21 Ontheotherhand,ifBisasubsetofA,wehaveAnB=Band,hence, _P(B)_1 PiAIB)-P(B)- Anextremely usefulrelationship forconditional probabilities isBayes' theorem, whichstatesthatifAi,i=1,2,...,n,aremutually exclusive events suchthat n UA,=S ;=1 andBisanarbitrary eventwithnonzero probability then PiAIB)=PtA"~B) ,P(B) =P(BIA;}P(A;} "LP(BIA;)P(AJ) j=1(2-1-14) Weusethisformula inChapter 5toderivethestructure oftheoptimum receiver foradigitalcommunication systeminwhichtheeventsAi.i= 1,2,...,n,represent thepossible transmitted messages inagiventime interval,PiA,)represent theiraprioriprobabilities, Brepresents thereceived signal,whichconsistsofthetransmitted message (oneoftheAi)corrupted by noise,andPiA,IB)istheaposteriori probability ofA,conditioned onhaving observed thereceived signalB. Statistical Independence Thestatistical independence oftwoormore eventsisanother important concept inprobability theory.Itusuallyarises whenweconsider twoormoreexperiments Drrepeated trialsofasingle experiment. Toexplainthisconcept, weconsider twoeventsAandBandtheir conditional probability PtAIB),whichistheprobability ofoccurrence ofA giventhatBhasoccurred. Suppose thattheoccurrence ofAdoesnotdepend ontheoccurrence ofB.Thatis, PiAIB)=PtA) Substitution of(2-1-15)into(2-1-13)yieldstheresult PtA,B)=P(A)P(B)(2-1-15) (2-1-16) Thatis,thejointprobability oftheeventsAandBfactorsintotheproductof 22!)I{iIIAL COMMl''''ICAno~s theelementary ormarginal probabilities P(A)andP(B).WhentheeventsA andBsatisfytherelation in(2-1-16). theyaresaidtobestatistically independent. Forexample. consider twosuccessive experiments intossingadie.LetA represent theeven-numbered samplepoints{2.4.6JinthefirsttossandB represent theeven-numbered possibleoutcomes {2.4,6}inthesecondtoss.In afairdie.weassigntheprobabilities PIA)=~andP(B)=~.Now.thejoint probability ofthejointevent"even-numbered outcome onthefirsttossand even-numbered outcome onthesecondtoss"isjustthe probability ofthenine pairsofoutcomes (i,j),i=2,4,6:j=2,4,6,whichis!.Also. PIA.B)=P(AlP(B) =l Thus.theeventsAandBarestatistically independent. Similarly. wemaysay that.theoutcomes ofthetwoexperiments arestatistically independent. Thedefinition ofstatistical independence canbeextended tothreeormore events.Threestatistically independent eventsAI'A1,andA,mustsatisfythe following conditions: P(AI,A,)=P(AI)P(A,) (2-1-17) P(AI,A2.A,)=P(AI)P(A2)P(A,) Inthegeneralcase.theeventsAi,i=I,2,...,n,arestatistically independent pmvided thattheprobabilities ofthejointeventstaken2.3.4,...•and11ata timefactorintotheproductofthepmbabilities oftheindividual events. 2-1-1Random Variables, Probability Distributions, and Probability Densities Givenanexperiment havingasamplespace5andelements SE5.wedefinea function X(s)whosedomain is5andwhoserangeisasetofnumbers onthe realline.Thefunction X(s)iscalledarandom variable. Forexample, ifweflip acointhepossible outcomes arehead(H)andtail(T),so5eontains two pointslabeledHandT.Suppose wedefineafunction X(s)suchthat X(S)={ I-I(s=H) (s=T)(2-1-18) Thuswehavemapped thetwopossible outcomes ofthecoin-flipping CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 23 experiment intothetwopoints(±i)ontherealline.Another experiment is thetossofadiewithpossibleoutcomes S={I,2,3,4,5, 6}.Arandomvariable definedonthissamplespacemaybeX(s)=s,inwhichcasetheoutcomes of theexperiment aremapped intotheintegers 1,...,6,'or,perhaps, X(s)=s', inwhichcasethepossible outcomes aremapped intotheintegers {I,4.9,16,25,36}.Theseareexamples ofdiscreterandomvariables. Although wehaveusedasexamples experiments thathaveafinitesetof possible outcomes, therearemanyphysical systems (experiments) that generate continuous outputs (outcomes). Forexample, thenoisevoltage generated byanelectronic amplifier hasacontinuous amplitude. Conse­ quently,thesamplespaceSofvoltageamplitudes vESiscontinuous andsois themapping X(v)=v.Insuch acase,therandomvariablet Xissaidtobea continuous randomvariable. GivenarandomvariableX,letusconsider theevent{X,,;;x}wherexisany realnumber intheinterval(-x,x).Wewritetheprobability ofthiseventas P(X,,;;x)anddenoteitsimplybyF(x).i.e., F(x)=P(X";;x) (-oo<x<x) (2-1-19) Thefunction F(x)iscalledtheprobability distribution functionoftherandom variable X.Itisalsocalledthecumulative distribution function (cdf).Since F(x)isaprobability, itsrangeislimitedtotheinterval0,,;;F(x),,;;1.Infact, F(-x)=0andF(x)=1.Forexample, thediscrete random variablegenerated byflipping afaircoinanddefinedby(2-1-18) hasthecdfshowninFig. 2-1-1(a). Therearetwodiscontinuities orjumpsinF(x),oneatx=-Iand oneatx=1.Similarly, therandomvariable X(s)=sgenerated bytossinga fairdiehasthecdfshowninFig.2-1-1(b). InthiscaseF(x)hassixjumps.one ateachofthepointsx=1,....6. FIGURE 2·1·1 Examples ofthecumulative distribution functions oftwodiscreterandom variables. FIn Fer)1----..-...__.... "''--__ 5 (;•(; 3 ~ 6 1 (; -nt---!---.,:---c;---;----~- ._--a 2 3 -l.'i6 16) tTherandomvariableX(J)willbewrittensimplyasX. 24Dl<ilTAI COMMUNICATIONS Fix) I------------ FIGURE 2-1-2 Anexample oftilecumulative distribution function ofa continuous randomvariable.o (2-1-20)Theedfofacontinuous randomvariable typically appearsasshowninFig. 2-1-2.Thisisasmooth, nondecreasing function ofx.Insomepractical problems, wemayalsoencounter arandomvariableofamixedtype_Thecdf ofsucharandomvariableisasmooth, nondecreasing function incertainparts ofthereallineandcontains jumpsatanumber ofdiscrete valuesofx.An example ofsuch acdfisillustrated inFig.2-1-3. Thederivative ofthecdfF(x),denoted asp(x),iscalledtheprobability densityfunction (pdf)oftherandomvariable X.Thus,wehave dF(x)p(x)=-- (-oo<x<;:cc)dx or,equivalently F(x)=r~p(U)dU (-oo<x<oc) (2-1-2I) SinceF(x)isanondecreasing function, itfollowsthatp(x);;.O.Whenthe randomvariable isdiscreteorofamixedtype,thepdfcontains impulses atthe pointsofdiscontinuity ofF(x).Insuchcases,thediscretepartofp(x)maybe expressed as n p(x)=2:P(X=x;}Il(x-x;} i=;I(2-1-22) where Xi,i=I,2,...,n,arethepossible discrete valuesoftherandom F(x) I-----------.:.-;.-"'-_- FIGURE 1-1·3Anexample oftltecumulative distribution functionofarandom variableofamixedtype.X,O X, CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 2S variable;P(X=x,),i=1,2,...,n,aretheprobabilities, ando(x)denotesan impulseatx=o. Oftenwearefacedwiththeproblem ofdetermining theprobability thata randomvariableXfallsinaninterval (XI'x2),wherex2>XI'Todetermine the probability ofthisevent,letusbeginwiththeevent{X'"X2}.Theeventcan alwaysbeexpressed astheunionoftwomutually exclusive events{X'"XIiand {XI<X'"X2}'Hencetheprobability oftheevent{X,""X2}canbeexpressed as thesumoftheprobabilities ofthemutually exclusive events,Thuswehave P(X"'X2)=P(X,""XI) +P(XI<X,""x,) F(x,)=F(x,)+P(xi<X,""X2) or,equivalently, P(XI<X,""X2)=F(X2)-F(xI) J"=p(x)dx "(2·1-23) (2-]-24) (2-1-26)Inotherwords,theprobability oftheevent{XI<X,""X2}issimplythearea underthepdfintherangeXI<X'"X2' Multiple Random Variables. JointProbability Distributions, andJoint Probability Densilies Indealingwithcombined experiments orrepeated trialsofasingleexperiment, weencounter multiple randomvariables andtheir cdfsandpdfs.Multiple random variables arebasically multidimensional functions definedonasamplespaceofacombined experiment. Letusbegin withtworandom variables X,andX2,eachofwhichmaybecontinuous, discrete,ormixed.Thejointcumulative distribution function (jointcdf)forthe tworandomvariables isdefinedas F(XI'X2)=P(XI'""X•.X2'"X2) =LLp(U" uz)du,duz wherep(x"x2)isthejointprobability densityfunction (jointpdf).Thelatter mayalsobeexpressed intheform a2 p(x"x2)=--F(x"X2) (2-1-25)ilx.ilX2 Whenthejointpdfp(xl,X2)isintegrated overoneofthevariables, we obtainthepdfoftheothervariable. Thatis, r~p(XI' x2)dx,=P(X2) [~P(XI' X2)dX2=p(x,) 26DIGITAL COMMUNICATIONS Thepdfsp(xI)andP(X2)obtained fromintegrating overoneofthevariables arecalledmarginal pdfs.Furthermore, ifP(XI'X2)isintegrated overboth variables, weobtain (2-1-27) WealsonotethatF(-00,-00)=F(-00,X2)=F(x"-(0)=O. Thegeneralization oftheaboveexpressions tomultidimensional random variables isstraightforward. Suppose thatXi'i=I,2,...,n,arerandom variables withajointcdfdefinedas (2-1-28) wherep(x"X2,...,xn)isthejointpdf.Bytakingthepartialderivatives of F(Xl,.x2,'..,xn)givenby(2-1-28), weobtain (2-1-29) Anynumberofvariables inp(xl•X2,'",xn)canbeeliminated byintegrating overthesevariables. Forexample, integration overX2andx,yields (2-1-30) ItalsofollowsthatF(x "00,"",x•....,xn)=F(XI,X4'X""" xn)and F(Xl1-00,-OO,X4,.'" x,.,)=o. Conditional Probability Distribution Functions Letusconsider tworan­ domvariables XIandX2withjointpdfp(xI,X2)'Suppose thatwewishto determine theprobability thattherandomvariableXI";XIconditioned on whereaX2issomepositiveincrement. Thatis,wewishtodetermine the probability oftheevent(XI";XlIX2-AX2<X2";X2)'Usingtherelations established earlierfortheconditional probability ofanevent,theprobability oftheevent(XI,,;x,lx2-AX2<X2";X2) canbeexpressed astheprobability CHAPTER 2,PROBABILITY ANDSTOCHASTIC PROCESSES 27 ofthejointevent(X,'"X"X2-l1X2<X2"X2)dividedbytheprobability of theevent(X2-l1x,<X2"X2)'Thus If';'I;;-Ax2P(U" u,)du,dU2P(X,"X ,X2-l1X,<X,"X2)= IX,()d Xl-'&X2P"2 U2 F(x,.x,)-F(x,.x,-l1x,) F(x,)-F(X2-l1X2)(2-1-31) (2-1-32)Assuming thatthepdfsp(x,.X2)andp(x2larecontinuous functions overthe interval(x,-l1X2.X2).wemaydividebothnumerator anddenominator in (2-1-31)byl1x,andtakethelimitasl1X2.....0.Thusweobtain I IiJF(X,.X,)/iJx,P(X,"x, X,=x2)=F(x, x,)=F()/iJx,iJx, iJ[f~xI"'.:xp(u" u,)du,du,]/iJx,=iJ[I:.>xp(u,) du,]/ax2 J"xp(u" X2)du, p(x,) whichistheconditional edfoftherandom variableX,giventherandom variable X2.Weobserve thatF(-00IX2)=0andF(ooIX2)=1.By differentiating (2-1-32)withrespecttoX"weobtainthecorresponding pdf p(x,IX2)intheform (I)_P(X"X2) (2-1-33)pX,X2-()PX2 Alternatively. wemayexpress thejointpdfp(x,.X2)intermsofthe conditional pdfs,p(x,Ix,)orp(x21x,),as p(x,.x,)=p(x,Ix,)p(x,) =p(x,lx,)p(x,) (2-1-34) Theextension oftherelations givenabovetomultidimensional random variables isalsoeasilyaccomplished. Beginning withthejointpdfofthe randomvariables Xi.i=1.2•...•n.wemaywrite p(x"x,•...,xn)=p(x,.X2•...•x,Ix.+,....•xn)P(Xk+l •...,xn)(2-1-35) wherekisanyintegerintherange1<k<n.Thejointconditional cdf corresponding tothepdfp(x"X2.'..•x.IXk+,.".,xn)is F(Xl,X2,'" ,XkIXk+b'" ,xn) f~lx'..f~<;r;P(Ub "2,".,Uk,Xlc+h'..,Xn)dUJdU2'..dUk P(Xk+"...•Xn)(2-1-36) (2-1-38)28DIGITAL COMMUNICATIONS Thisconditional cdfsatisfiestheproperties previously established forthese functions, suchas F(OO,X2"" ,Xk/Xk+b'" ~xn)=F(X1.,X3"'· ,XI</X.t+I"·" xn) F(-00,xz," .,x'"IXk+l,..••Xn)=0 Statistically Independent Random Variables. Wehavealready define$! statistical independence oftwoormoreeventsofasamplespaceS.The concept ofstatistical independence canbeextended torandom variables definedonasamplespacegenerated byacombined experiment orbyrepeated trialsofasingleexperiment. Iftheexperiments resultinmutually exclusive outcomes, theprobability ofanoutcome inoneexperiment isindependent of anoutcome inanyotherexperiment. Thatis,thejoint probability ofthe outcomes factorsintoaproduct oftheprobabilities corresponding toeach outcome. Consequently, therandomvariables corresponding totheoutcomes intheseexperiments areindependent inthesensethattheirjointpdffactors intoaproductofmarginal pdfs.Hencethemultidimensional randomvariables arestatistically independent ifandonlyif F(x"X2•...•x.)=F(x,)F(X2)" .F(xn) (2-1-37) or,alternatively, P(XloX2•...,x.)=p(X,)P(X2)" .p(x.) 2-1-2Functions ofRandom Variables Aproblem thatarisesfrequently inpractical applications ofprobability isthe following. GivenarandomvariableX,whichischaracterized byitspdfp(x), determine thepdfoftherandomvariable Y=g(X).whereg(X)issomegiven function ofX.Whenthemapping 15tromXtoYisone-to-one, the determination ofp(y)isrelatively straightforward. However. whenthe mapping isnotone-to-one, asisthecase,forexample, whenY=X2,wemust beverycarefulinourderivation ofp(y). Example 2-1-1 Consider therandomvariable Ydefinedas Y=aX+b (2-1-39) (2-1-40)whereaandbareconstants. Weassumethata>O.Ifa<0,theapproach is similar(seeProblem 2-3).Wenotethatthismapping, illustrated inFig. 2-1-4(a) islinearandmonotonic. LetFx(x)andFy(y)denotethecdfsforX andY.respectively.t Then Fy(y)=P(Y,,;;y) =P(aX+b,,;;y)=p(X,,;;Y:b) fIY",b)l.(b=..~Px(x)dx=Fxy:) tToavoidconfusion inchanging variables, subscripts areusedintherespective pdfsandcdfs. yCHAPTER 2,PROBABIUf\' ANDSTOCHASTIC PROCESSES 29 Px(x) oY=aX+b.a>O ---,.01-7"'-------- X (a) Py(Y) 1a , 0b-aLh........----.. -I (b) (2-1-41)Ie) tlGURE 2-1-4Alineartransformation ofarandomvariableXandanexample ofthecorresponding pdfsofX andY. Bydifferentiating (2-1-40) withrespecttoy,weobtaintherelationship between therespective pdfs.Itis py(y)=~px(Y:b) Thus(2-1-40)and(2-1-41)specifythecdfandpdfoftherandomvariable Y intermsofthecdfandpdfoftherandom variable Xforthelinear transformation in(2-1-39). Toillustrate thismapping foraspecificpdf Px(x),consider theoneshowninFig.2-1-4(b). Thepdfpy(y)thatresults fromthemapping in(2-1-39) isshowninFig.2-1-4(c). Example 2-1-2 Consider therandomvariableYdefinedas Y=aX'+b, a>O (2-1-42) AsinExample 2-1-1,themapping between XandYisone-to-one. Hence Fy(y)=P(Y""y)=P(aX3+b""y) (2-1-43) 30DIGITAL COM"MUN IeATIONS y Y=aX2+b FlGURE 2-1-5Aquadratic transformation oftherandomvariableX.b--o;r----X (2-1-44)Differentiation of(2-1-43) withrespecttoyyieldsthedesiredrelationship between thetwopdfsas 1[(Y_b)ll3] py(y)=3a[(y-b)/a]213Px -a- Example Z-I-3 TherandomvariableYisdefinedas Y=aX2+b,a>O (2-1-45) Incontrast toExamples 2-1-1and2-1-2,themapping between XandY, illustrated inFig.2-1-5,isnotone-lo-one. Todetermine theedfofY,we observethat Fy(y)=P(Y";y)=P(aX2+b,.;y) =P(IXI";~y~b) Hence (2-1-46) (2-1-47)Differentiating (2-1-46) withrespecttoy,weobtainthepdfofYintermsof thepdfofXintheform ()~Px{V(y-b)/a],,-,Px~[:-r.-v';:;=(===y=-~b#) /=?,a]pyy- +-2av'[(y-b)/a] 2aV[(y-b)/a] InExample 2-1-3,weobservethattheequation g(x)=ax2+b=yhastwo realsolutions, Iy-b Xl='J-a- X,= _~y:b (2-1-4H)CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 31 andthatpy(y)consists oftwotermscorresponding tothesetwosolutions. Thatis, Px[x, ~V(y-b)/a]Px[x,= -Y(y-b)/a] py(y)=Ig'[x, ~V(y-b)/a]1+19'[X2--V(y-b)/a]1 whereg'(x)denotes thefirstderivative ofg(x). Inthegeneral case,suppose thatx"x"...,x.,aretherealrootsofthe equation g(x)=y.Thenthepdfoftherandom variable Y=g(X)maybe expressed as "Px(x,) py(y)=2:-I'(.)1 1=1gX,(2-1-49) wheretherootsXi'i=1,2,...,n,arefunctions ofy. Nowletusconsider functions ofmultidimensional random variables. Suppose thatXi,i~1,2,...,n,arerandom variables withjointpdf Px(xloX2,'"•x,,).andlet1';,i=1.2,...,n,beanother setofnrandom variables relatedtotheXibythefunctions (2-1-50) Weassume thatthegi(X"X"...,X,,),i=1,2•...,n,aresingle-valued functions withcontinuous partialderivatives andinvertible. By"invertible" we meanthattheXi'i=1.2....,n.canbeexpressed asfunctions ofY,. i=1,2,...,n,intheform Xi=g,-'(YloY,•...,Y,,).i=1,2,....n (2-1-5]) wheretheinverse functions arealsoassumed tobesingle-valued with continuous partialderivatives. Theproblem istodetermine thejointpdfofY,. i=1,2•...,n,denoted byPY(YI.y"...,Yn),giventhejointpdf PX(XIJ X2,". ,xn)· Todetermine thedesiredrelation, letRxbetheregioninthen-dimensional spaceoftherandom variables Xi'i=1,2•...,n,andletRybethe (one-to-one) mapping ofRxdefinedbythefunctions Y,~gi(X"X"...,X,,). Clearly, f f...fPY(YIoY2,...,y,,)dy,dY2'..dy" Ry =ff···fpx(X"x, •...,Xn)dx,dX2···dX n(2-1-52) R, Bymakingachangeinvariables inthemultiple integralontheright-hand side of(2-1-52) withthesubstitution (2-1-53)32DIGITAL ('OMMlINICATIONS weobtain JJ...Jp,,(y,.y,.....y,,)dYIdv,...dy" 1<, =JJ..·Ip.-(X,=g,'.X,=g,' .....X"=g"')llldv,dv,"'dy,, 1<, whereIdenotesthejacobian ofthetransformation. definedbythedeterminant Ag,'A' Ag", g~ Avay, dYI ,, J= (2-1-54) Ag,'Ag,' ag/~I ay" aYII riYn Consequently. thedesiredrelationforthejointpdfofthe>':.i=I.2.....n.is p,.(y,.y,.....y,,)=Px(x, =g,'.x,=g,'...,.x"=g"')III(2-1-55) Example 2-1-4 Animportant functional relationbetween twosetsofn-dimensional random variables thatfrequently arisesinpractice isthelineartransformation n Y;=2:ailXI•i=I.2.....n 1=((2-1-56) wherethe{aiJareconstants. Itisconvenient toemploythematrixformfor thetransformation, whichis Y=AX (2-1-57) whereXandYaren-dimensional vectorsandAisannXnmatrix.We assumethatAisnonsingular. ThenAisinvertible and.hence, Equivalently, wehaveX=A-'y "Xi=2:bi/lj,i=I,2,....n 1=1(2-1-58) (2-1-59) where{b,Jaretheelements oftheinversematrixA-I.Thejacobian ofthis transformation isI=1/detA.Hence PY(YI.Y2.·..•Yn) (n n n) 1 =PxX,=2:bt,Y/,X,=2:b2jYj,·..•Xn=LbnjYjIdAI /~I /~t /~I et (2-1-60) (HAPTE-.R 2PR08AUILlTY ANDSTOnIA~TJ(' rROU:SSES 33 2-1-3Statistical Averages ofRandom Variables Averages playanimportant roleinthecharacterization oftheoutcomes of experiments andtherandom variables definedonthesamplespaceofthe expenments. Ofparticular interestarethefirstandsecondmoments ofasingle randomvariable andthejointmoments, suchasthecorrelation andcovari­ ance,between anypairofrandom variables inamultidimensional setof randomvariables. Alsoofgreatimportance arethecharacteristic function fora singlerandomvariableandtheJointcharacteristic function foramultidimen­ sionalsetofrandomvariables. Thissectionisdevotedtothedefinition ofthese important statistical averages. Firstweconsider asinglerandomvariable Xcharacterized byitspdfp(x}. Themeanorexpected valueofXisdefinedas E(X);;:mx=fxxp(x)dx (2-1-61) whereE()denotesexpectation (statistical averaging). Thisisthefirstmoment oftherandomvariableX.Ingeneral,thenthmoment isdefinedas E(X")=[xnp(x)dx (2-1-62) Now,suppose thatwedefinearandomvariable Y=g(X),whereg(X)is somearbitrary functionoftherandomvariable X.Theexpected valueofYis E(Y)=E[g(X») =[g(X)P(X) dx Inparticular, ifY=(X-mxrwherernxisthemeanvalueofX,then E(Y)=E[(X-m,)")=[(x-mx)"p(x) dx(2-1-63) (2-1-64) Thisexpected valueiscalledthenthcentralmoment oftherandomvariable X, because itisamoment takenrelativetothemean.Whenn=2,thecentral moment iscalledthevariance oftherandom variable anddenoted as0';. Thatis, (2-1-65) Thisparameter provides ameasureofthedispersion oftherandomvariable X. Byexpanding theterm(x-mx)2intheintegralof(2-1-65)andnotingthatthe expected valueofaconstant isequaltotheconstant, weobtaintheexpression thatrelatesthevariance tothefirstandsecondmoments, namely, <7~=E(X')-[E(X»)' =E(X')-m~ (2-1-66) 34DIGJTAl COMMVNICA'llONS Inthecaseoftworandomvariables, X,andX2,withjointpdfp(x"X2),we definethejointmomentas (2-1-67) andthejointcentralmomentas E[(X,-m,t(X, -m,tJ =f.f.(x,-mdk(x2-m2)"P(X"x,)dx, dx2(2-1-68) wheremi=E(X,).Ofparticular importance tousarethejointmoment and jointcentralmoment corresponding tok=n=1.Thesejointmoments are calledthecOfrelation andthecovariance oftherandomvariables XIandX2, respectively. . Inconsidering multidimensional random variables, wecandefinejoint moments ofanyorder.However, themoments thataremostusefulinpractical applications arethecorrelations andcovariances between pairsofrandom variables. Toelaborate, supposethatXi,i=I,2,...,n,arerandomvariables withjointpdfp(xI,X2,..,,xn),LetP(Xi'x)bethejointpdfoftherandom variables XiandXj'Thenthecorrelation between XiandXjisgivenbythe jointmoment (2-1-69) andthecovariance ofXiand~is p.'j==E{(Xi-m,)(Xj-mj)J =r~L~(Xi-m,)(Xj-mj)p(x i,Xi)dxidXj (2-1-70) ThenXnmatrixwithelements P.ijiscalledthecovariance matrixofthe randomvariables Xi'i=1,2,. ' .,n.Weshallencoull!eP thecovariance matrix inourdiscussion ofjointlygaussian randomvariables inSection2-1-4. Tworandom variables aresaidtobeuncorrelated ifE(XiXj)= E(Xi)E(X j)=mimj'Inthatcase,thecovariance P.ij=O.WenotethatwhenXi andXjarestatistically independent, theyarealsouncorrelated. However, ifXi andXjareunoorrelated, theyarenotnecessarily statistically independent. CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 35 Tworandomvariables aresaidtobeorthogonal ifE(XiX j)=O.Wenote thatthiscondition holdswhenXiandXjareuncorrelated andeitheroneor bothoftherandomvariables havezeromean. Characteristic Functions Thecharacteristic function ofarandomvariable Xisdefinedasthestatistical average (2-1-71) wherethevariable visrealandj~Y=1.Wenotethatl/f(jv)maybedescribed astheFouriertransformt ofthepdfp(x).HencetheinverseFouriertrans· formis 1IX.p(x)=-if1(jvV'"Xdv2Jr-x(2-1-72) Oneusefulproperty ofthecharacteristic function isitsrelation tothe moments oftherandomvariable. Wenotethatthefirstderivative of(2-1-71) withrespecttovyields Byevaluating thederivative atv~0,weobtainthefirstmoment (mean) E(X)=mx=_/I/1(jv)I dv"=0(2-1-73) Thedifferentiation processcanberepeated, sothatthenthderivative ofl/f(jv) evaluated atv=0yieldsthenthmoment E(X")=(-j)"d"l/f~v)I dv "=0(2-1-74) Thusthemoments ofarandom variable can~edetermined fromthe characteristic function. Ontheotherhand,suppose thatthecharacteristic functioncanbeexpanded inaTaylorseriesaboutthepointv~O.Thatis, l/1(jv)=1:[dnl/1~V)]v~ "=0dvv=on.(2-1-75) Usingtherelationin(2-1-74)toeliminate thederivative in(2-1-75), weobtain tUsuallytheFouriertransfonn ofafunctiong(u)isdefinedasG(u)~I:.g(u)e-j,,"du,which dillenfrom(2-1-71)bythenegative.sign intheexponential. Thisisa trivialdifference, however, so ...,call1heintesralin(2-1-71)aFouriertransform. J6LJI(iITAL COM~tl:SICAliONS anexpression forthecharacteristic function intermsofitsmoments inthe form o}J(jv)=iE(X")(jv/" "",0 fl.(2-1-76) Thecharacteristic function provides asimplemethodfordetermining the pdfofasumofstatistically independent randomvariables. Toillustrate this point.letXi'j=1,2.....n.beasetofnstatistically independent random variables andlet (2-1-77) Theproblem istodetermine thepdfofY.Weshalldetermine thepdfofYby firstfindingitscharacteristic function andthencomputing theinverseFourier transform. Thus . o}Jy(jv)=E(eI"') =E[exp(jv.t,Xi)] =E[U,(e1"X)] =fx'..Lx(Qe-'Vx'Jp(x, ..t"...•x,,)dx,dx,".dx"(2-1-78) Sincetherandom variables arestatistically independent. p(x,.x"...,x,,)= p(x,)p(x,)' ..p(x,,).and,hence.thenth-order integralin(2-1-78)reducestoa productofnsingleintegrals. eachcorresponding tothecharacteristic function ofoneoftheX,.Hence. "l/Jy(jv)=n>/Jx.(jv) ,=\(2-1-79) If.inaddition totheirstatistical independence.- theXiareidentically distributed thenalltheIjIx(jv)areidentical. Consequently, </Jy(jv)=['"x(jv)J" (2-1-80) Finally.thepdfofYisdetermined fromtheinverseFouriertransform of o}Jy(jv),givenby(2-1-72). Sincethecharacteristic function ofthesumofnstatistically independent randomvariables isequaltotheproductofthecharacteristic functions ofthe individual randomvariables Xi.i=I,2,...,n.itfollowsthat,inthetransform domain,thepdfofYisthen-foldconvolution ofthepdfsoftheX,.Usually then-foldconvolution ismoredifficulttoperform thanthecharacteristic functionmethoddescribed ahoveindetermining thepdfofY. Whenworking withn-dimensional random variables, itisappropriate to defineandn-dimensional Fouriertransform ofthejointpdf.Inparticular, if (2-1-82)CH"PTER', PROB"SILIn "NDSTOCHASTIC PROCESSES 37 Xi'i=1,2,...,n,arerandom variables withpdfp(Xt,X2,...,xn),the n-diTnellSional characteristic function isdefinedas .p(jv"jv"...,jvn) Ofspecialinterestisthetwo-dimensional characteristic function l/I(jVt,jV2)=fxfxei<v,x,+V,X'lp(X"X2)dx, dx2 Weobservethatthepartialderivatives of.p(jv,,jV2)withrespecttov,andv, canbeusedtogenerate thejointmoments. Forexample, itiseasytoshowthat ,P",(jvt>jV2)IE(X,X2)= (2-1-83) aVtaVl VI=tJ2=O Higher-order moments aregenerated inastraightforward manner. 2-1-4SomeUsefulProbability Distributions Insubsequent chapters, weshallencounter severaldifferent typesofrandom variables. Inthissectionwelistthesefrequently encountered random variables, theirpdfs,theircdfs,andtheirmoments. Webeginwiththebinomial distribution, whichisthedistribution ofadiscreterandomvariable, andthen wepresentthedistributions ofseveralcontinuous randomvariables. Binomial Distribution LetXbeadiscreterandomvariable thathastwo possible values,sayX=1orX=0,withprobabilities pand1-p, respectively. ThepdfofXisshowninFig.2-1-6.Now,suppose that n Y=LX, i=l wheretheXi'i=1,2,...,n,arestatistically independent andidentically flGURE 1-1-6Theprobability distribution functionofX.I_p p _Il~.xo t 38mOlTALCOMMUNICATIONS distributed randomvariables withthepdfshowninFig.2-1-6.Whatisthe probability distribution function ofY? Toanswerthisquestion, weobserve thattherangeofYisthesetof integersfrom0ton.Theprobability thatY=0issimplytheprobability that alltheX,=O.SincetheX,arestatistically independent, P(Y=0)=(1-pr Theprobability thatY=1issimplytheprobability thatoneXi=1andtherest oftheX,=O.Sincethiseventcanoccurinndifferent ways, P(Y=I)=np(l-pr' Togeneralize, theprobability thatY=kisthe probability thatkoftheXiare equaltooneandn-kareequaltozero.Sincethereare C)'"k!(nn~k)! different combinations thatresultintheevent{Y=k},itfollowsthat P(Y=k)=C)pk(1-pr-k(2-1-84) (2·1-85) where(:)isthebinomial coefficient. Consequently, thepdfofYmaybe expressed as n p(y)=LP(Y=k)5(y-k) k=O ThecdfofYis=±(n)pk(l_prkIi(y-k) k~Ok F(y)=P(Y";;y) [yJ =L(n)pk(l-pr* k~Ok(2-1-86) (2-1-87) where[y]denotesthelargestintegermsuchthatm,,;;y.Thecdfin(2-1-87) characterizes abinomially distributed randomvariable. Thefirsttwomoments ofYare E(Y)=np E(y2)=np(1-p)+n2p2 (T2=np(l-p) andthecharacteristic function is /fJ(jv)=(1-p+p~r(2-1-88) (2-1-89) pIx) Ifb(I -JaL-.."O+----+b----+ , (a)CHAPTER 2:PROBA81LlTY ANDSTOCHASTIC PROCESSES 39 F(x) -'o~'O.t-----tb--- ....x (b) FIGURE 2-1-7Thepdfandcd!ofauniformly distributed randomvariable. Uniform Distribution Thepdfandcdfofauniformly distributed random variableXareshowninFig.2-1-7.Thefirsttwomoments ofXare E(X)=Ha+b) E(X2)=~(a2+b2+ab) a2=f,<a-b)2 andthecharacteristic function is ell'"-eivtl o/J(jv)= .(b)-JV-a(2-1-90) (2-1-91) Gaussian (Normal) Distribution Thepdfofagaussian ornormally distributed randomvariable is p(x)=~e-(x--m,)2J2U" Y21Ca-(2-1-92) wherem,isthemeanandaZisthevariance oftherandomvariable. Thecdfis I 2f(Xm,)/"2"., =-- etdt 2vir x _ I I(X-mx)-2+2erf--:r;;:­v2a(2-1-93) 40DIGITAL COMMUNIC AllONS Fl.\") pIx) Ga----- -"""O+---m~,----- .. (CIt--,O+----n=-',------ .. (bl FIGURE 2-1-8Thepdfandedfofagaussian-distributed randomvariable. whereerf(x)denotestheerrorfunction, definedas 2LX,erf(x)=,Ie-,.dt v1t()(2-1-94) Thepdfandedfareillustrated inFig.2-1-8. ThecdfF(x)mayalsobeexpressed intermsofthecomplementary error function, Thatis, (X-m)F{x)=1-!erfeV2;' where 2JX,erfc(x)=virxe-,-de =1-erf{x) (2-1-95) (2-1-96)Wenotethaterf{-x)=-erf{x),erfc(-x)=2-erfc(x), erf(O)=erfc(oo)= 0,anderf(oo)=erfc(0)=1.Forx>m"thecomplementary errorfunction is proportional totheareaunderthetailofthegaussian pdf.Forlargevaluesof x,thecomplementary errorfunction erfc(x)maybeapproximated bythe asymptotic series e-X'(11·31'3-5 )erfc(x)=--1- -+- - + ...xVii lx222x·23x6 wheretheapproximation errorislessthanthelasttermused. Thefunction thatisfrequently usedfortheareaunderthetailofthe gaussian pdfisdenoted byQ{x)anddefinedas 1JX.Q{x)=--e""12dt,V21r. Bycomparing (2-1-95)with(2-1-97), wefind Q{x)=~erfc(;;)x""o (2-1-97) (2-1-98) CHAPTER~: PROIJAKII IIYANDSTOOIAS1K PRon'ssrs 41 Thecharacteristic function ofagaussian randomvariablewithmeanm,and variance (T'is ,!r(jv)=J'er,,[_l_e -(,-m,)''''''] dx <v'2i(1 Thecentralmoments ofagah,ian randomvariableare {I,3...(k~[)(1' E[(X-m,)']=Ii'= 0(evenk) (oddk)(2-1-99) (2-1-100) andtheordinary moments maybeexpressed intermsofthecentralmoments as (2-1-101) Thesumofnstatistically independent gaussian randomvariables isalsoa gaussian randomvariable. Todemonstrate thispoint,let n y=2:X, i.;1(2-1-102) wheretheXi'i=I,2,...,n,arestatistically independent gaussian random variables withmeansmjandvariances (1~.Usingtheresultin(2-1-79), wefind thatthecharacteristic functionofYis n o/Jy(jv)=no/Jx,(jv) j"='l n. 2!.,=neJVni,-v a,l_ i=1 where " m,V=Lmi i=1 "a2_'"2 _~.-LJUi i=1 Therefore, Yisgaussian-distributed withmeanm,andvariance a~.(2-1-103) (2-1-104) Chi-Square Distribution Achi-square-distributed random variable isre­ latedtoagaussian-distributed randomvariableinthesensethattheformercan beviewedasatransformation ofthelatter.Tobespecific,letY=X2,whereX isagaussian random variable. ThenYhasachi-square distribution. We distinguish between twotypesofchi-square distributions. Thefirstiscalleda 42DIGrJAl COMMlIN1CATIONS centralchi-square distriblllion andisobtained whenXhaszeromean.The secondiscalledanon-central chi-square distrioution. andisobtained whenX hasanonzero mean. Firstweconsider thecentralchi-square distribution. LetXbegaussian­ distributed withzeromeanandvariance u2SinceY=X'.theresultgivenin (2-1-47)appliesdirectlywitha=j~ndb=O.ThusweobtainthepdfofYin theform (2-1-105) (2-1-106)ThecdfofYis Fy(y)=J'p,.(U) d" tl 1J'1e1112(T=du =Y21ru tlYU whichcannotbeexpressed inclosedform.Thecharacteristic function, however, canbedetermined inclosedform.Itis op(jv)=(1-;'21Ju')"2 Now,suppose thattherandomvariable Yisdefinedas "Y=2:xi ;=1(2-1-107) (2-1-108) wheretheXi.i=1,2....,n.arestatistically independent andidentically distributed gaussian randomvariables withzeromeanandvariance (J'2.Asa consequence ofthestatistical independence oftheX"thecharacteristic functionofYis </J,.(jv)=(1-;'21JV')"12 Theinversetransform ofthischaracteristic function yieldsthepdf(2-1-109) (2-1-110) (2-1-111)( )_ 1 ,,12-f -"12,,'pyY-<T"2"12f(~n) ye·• wherer(p)isthegammafunction. definedas f(p}=ftp·'e-'dt. p>O r(p)=(p-I)!,panirtteger,p >0 rO)=v'1r,fa>=~VIr Thispdf,whichisageneralizationor(2-1-105), iscalledachi-square (or CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 43 ply) 0.5 n=1 0.4 FIGURE 2·1·9Thepdfofachi-square-dislribuled random variable forseveraldegreesoffreedom.n:::8 81012 14 gamma)pdfwithndegreesoffreedom.Itisillustrated inFig.2·1·9.Thecase n=2yieldstheexponential distribution. Thefirsttwomoments ofYare ThecdfofYisE(Y)=n(T' E(Y')=2n(T4+n'(T' (2·1·112) (2·1·1I3) Thisintegralcanbeeasilymanipulated intotheformoftheincomplete gamma function, whichistabulated byPearson(1965).Whenniseven,theintegralin (2·1·113) canbeexpressed inclosedform.Specifically, letm=!n,wheremis aninteger.Then,byrepeated integration byparts,weobtain m-I1 ()kFy(y)=1-e-y/'u'I..Y,, .~Ok.2(Ty;;'O (2·1·114) (2·I-Il5)Letusnowconsider anoncentral chi·square distribution, whichresultsfrom squaring agaussian randomvariable havinganonzero mean.IfXisgaussian withmeanm,andvariance (T',Iherandomvariable Y=X'hasthepdf 1.,~,(vYm)pyCv)= e',y·m,),"n cosh--,-', v2Jry(T (T whichisobtained byapplying theresultin(2·1·47)tothegaussian pdfgivenby (2·1·92). Thecharacteristic function corresponding tothispdfis (2·1·116) 44DI<i1lAL COMMl:~I(ATIONS Togeneralize theseresults,letYbethesumofsquaresofgaussian random variables asdefinedby(2-1-108). TheX"i=1,2,...,n,areassumed tobe statistically independent withmeansm"i=1,2,...,n,andidentical variances equalto1T2.Thenthecharacteristic function ofY,obtained from (2-1-116) by applying therelationin(2-1-79), is(JVim;) (2-1-117) .py(jv)=(1_j2~1T2)"12 exp1_;;~VU2 Thischaracteristic function canbeinverse-Fourier-transformed toyieldthepdf 1 ()(n-2)/4(s) y -(52+y)/2..,.2 ,py(y)=2u2:;> e I n12-Ivyu2' where,bydefinition,(2-1-118) (2-1-119) (2-1-120) (2-1-121)andla(x)istheath-order modified Besselfunction ofthefirstkind,whichmay berepresented bytheinfiniteseries x(x/2t+2' la(x)=t:ok!qa+k+1)' Thepdfgivenby(2-1-118) iscalledthenoncentra/ chi-square pdfwirhn degreesoffreedom. Theparameter S2iscalledthenoncentra/ity parameter of thedistribution. Thecdfofthenoncentral chisquarewithndegreesoffreedom is l'1(U)<n-2y4 -(s)F,(y)= - - e'(s'+u)l2u'l Vii-duy 2-2 2 n12-1 2ocrs IT Thereisnoclosed-form expression. forthisintegral. However, whenm=jnis aninteger,thecdfcanbeexpressed intermsofthegeneralized Marcum's Q function, whichisdefinedas (2-1-122) (2-1-123)where QI(a,b)=e-(a'+b')12 .~o(~r/.(ab), b>0>0 Ifwechangethevariable ofintegration in(2-1-121) fromutox,where CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCESSES 4S andlet02=s'fa',thenitiseasilyshownthat (2-1-124) Finally, westatethatthefirsttwomoments ofanoncentral chi-square­ distributed randomvariableare E(Y)=na'+52 E(y2)=2na4+4a's'+(na'+s')' a'=2na4+4<7'S2y(2-1-125) Rayleigh Distribution TheRayleigh distribution isfrequently usedto modelthestatistics ofsignalstransmitted through radiochannels suchas cellularradio.Thisdistribution iscloselyrelatedtothecentralchi-square distribution. Toillustrate thispoint,letY=X~+X~whereXIandX2are zero-mean statistically independent gaussian randomvariables, eachhavinga variance a2,Fromthediscussion above,itfollowsthatYischi-square­ distributed withtwodegreesoffreedom. Hence,thepdfofYis (2-1-126) Now,suppose wedefineanewrandomvariable (2-1-127) Makingasimplechangeofvariable inthepdfof(2-1-126), weobtainthepdf ofRintheform ( )_.!... -"12..'PRr-2e ,a(2-1-128) ThisisthepdfofaRayleigh-distributed randomvariable. Thecorresponding cdfis (2-1-129) Themoments ofRare •,(2-1-130) 46llllilTAL COMMUNICATIONS andthevariance is (J'~=(2-~1l")(T2 (2-1-13I) Thecharacteristic functionoftheRayleigh-distributed, randomvariable is (2-1-132) Thisintegralmaybeexpressed as o/JR(jV)=IX-;e-,'/20'cosVTdr+j(X~2e-r'(2o'sinVTdT [)a Jo(T =tFt(I,~;-~v2(2)+jv'firva'e.'0'/2 (2-1-133) where,Ft(l,t-0)istheconfluent hypergeometric function. wl:tichisdefined as <r(a+k)f(j3)x' ,Fi(a,j3;x)=t-or(a)fUHk)k!' (3#0,-I,-2"., (2-1-134) Beaulieu (1990)hasshownthatJFi(I,!;-0)maybeexpressed as (2-1-135) Asageneralization oftheaboveexpression. consider therandomvariable R=)~,Xf(2-1-136) (2-1-137)wheretheX;,i=I,2,. ,.,n,arestatistically independent, identically distrib­ utedzeromeangaussian random variables. Therandomvariable Rhasa generalized Rayleigh distribution. Clearly. Y=R2ischi-sQuare-distributed withndegreesoffreedom. I&tipdfisgivenby(2-1-110). Asimplechangein variable in(2-1-110)yieldsthepdfofRintheform r"-, PR(r)=2(n")l2unr(!n)e"'/20', r;;.0 Asaconsequence ofthefunctional relationship between thecentral chi-square andtheRayleigh distributions, thecorrespondiqg cdfsare'similar. Thus,foranyn,thecdfofRcanbeputintheformoftheincomplete gamma function. Inthespecialcasewhenniseven,i.e.;n=2m,thecdfofRcanbe expressed intheclosedform T;;'O (2-1-138) I CHAPTER 2:PROBABILITY ANDSTOCHASTIC PR()('ESSES 47 Finally,westatethatthekthmoment ofR"is (2-1-139) whichholdsforanyintegern, RiceDistribution JustastheRayleigh distribution isrelatedtothecentral chi-square distribution, theRicedistribution isrelatedtothenoncentral chi-square distribution. Toillustrate thisrelation, letY=xi+Xl,whereXI andX2arestatistically independent gaussian randomvariables withmeansmi, i=1,2,andcommon variance u2•Fromtheprevious discussion, weknowthat Yhasanoncentral chi-square distribution withnoncentrality parameter S2=m~+m~,ThepdfofY,obtained from(2-1-118) forn=2.is (2-1-140) Now,wedefineanewrandomvariableR=vY.ThepdfofR,obtained from(2-1-140) byasimplechangeofvariable, is (2-1-141) ThisisthepdfofaRicean-distributed randomvariable. Aswillbeshownin Chapter 5,thispdfcharacterizes thestatistics oftheenvelope ofasignal corrupted byadditive narrowband gaussian noise.Itisalsousedtomodelthe signalstatistics ofsignalstransmitted throughsomeradiochannels. Thecdfof Riseasilyobtained byspecializing theresultsin(2-1-124) tothecasem=1. Thisyields (2-1-142) whereQ,(a,b)isdefinedby(2-1-123). Asageneralization oftheexpressions givenabove,letRbedefinedasin (2-1-136) wheretheXi'i=1,2,.,. ,narestatistically independent gaussian randomvariables withmeansmi,i=1,2,...,n,andidentical variances equal tou2•TherandomvariableR2=Yhasanoncentral chi-square distribution withndegreesoffreedom andnoncentrality parameter S2givenby(2-1-119). Itspdfisgivenby(2-1-118). HencethepdfofRis (2-1-143) 48DltjlTAl COMMUNICATIONS andthecorresponding cdfis (2-1-144) whereFy(r')isgivenby(2-1-121). Inthespecialcasewherem=~nisan integer,wehave (2-1-145) whichfollowsfrom(2-1-124). Finally,westatethatthekthmoment ofRis E(Rk)=(2')kl2-,'IZ,,'fO(n+k»F(n+k~.~) k;;.0 ue f(~n)I'2'2'2U>' (2-1-146) where,F,(a,f3;x)istheconfluent hypergeometric function. NakalllDli m-Distribution BoththeRayleigh distribution andtheRice distribution arefrequently usedtodescribe thestatistical ftuctuations ofsignals received fromamultipath fadingchannel. Thesechannel modelsarecon­ sidered inChapter 14.Another· distribution thatisfrequently usedto characterize thestatistics ofsignalstransmitted through multipath fading channels istheNakagami m-distribUlion. Thepdfforthisdistribution isgiven byNakagami (1960)as (2-1-147) whereQisdefinedas (2-1-148) andtheparameter misdefinedastheratioofmoments, calledthefading figure, (2-1-149) Anormalized versionof(2-1-147) maybeobtained bydefining another randomvariableX=R/Yn(seeProblem 2-15).Thenthmoment ofRis f(m+~n)(Q)n,zE(Rn)=--r(m) m Bysettingm=I,weobservethat(2-1-147) reducestoaRayleigh pdf.For CHAPn;R 2,PROBABILIIT ANDSTOCHASTIC PROCESSES 49 f1GURE 2-1-11 Them-di,lribuled pdf,shownwilh o=I.misIhefadingfigure. (Miyagaki elal.1978.) valuesofmintherange!""m""1,weobtainpelfsthathavelargertailsthana Rayleigh-distributed randomvariable. Forvaluesofm>1.thetailofthepdf decaysfasterthanthatoftheRayleigh. Figure2-1-10illustrates thepdfsfor different valuesofm. Multivariate GaussiaB Distribution Ofthemanymultivariate ormulti­ dimensional distributions thatcanbedefined, themultivariate gaussian distribution isthemostimportant andtheonemostlikelytobeencountered in practice. Weshallbrieflyintroduce thisdistribution andstateitsbasic properties. LetusassumethatXi,i=I,2,...,n,aregaussian randomvariables with meansmi,i=I,2,...•n,variances uf,i=I,2,...•n.andcovariances Jiil' i,j=I,2,...,n.Clearly, lJ.ii=uf.i=I,2,,..•n.LetMdenotethenxn soDJGrrAL (UMMLN/CA nONS covariance matrix withelements {IL,),letXdenotethenx1columnvectorof randomvariables, andleIm,denotethenx 1columnvectorofmeanvalues m"i=I,2,...,n.Thejointpdfofthegaussian random variables X" i=1.2,...,n,isdefinedas (2-1-150) where1\1'IdenotestheinverseofMandx'denotesthetranspose ofx. Thecharacteristic function corresponding tothisn-dimensional jointpdfis wherevisann-dimensional vectorwithelements Vi'i=1,2,...,n. Evaluation ofthisn-dimensional Fouriertransform yieldstheresult "'(j1')=exp(jm;1'-11"Mvl (2-1-151) Animportant specialcaseof(2-1-150) isthebivariate ortwo-dimensional gaussian pdf.ThemeanDlxandthecovariance matrixMforthiscaseare [uiM- 1L12iL12] 17~(2-1-152) wherethejointcentralmoment iL12isdefinedas iLI2=E[(X,-m,)(X,-m,)] Itisconvenient todefineanormalized covariance i'1"j (2-1-153) where Pi,sati<fies thecondition 0""Ipi;1""I.Whendealingwiththetwo­ dimensional case,itiscustomary todropthesubscripts oniLl'andPI>'Hence thecovariance matrix Isexpressed as (2-1-154) Itsinverseis (2-1-155) CHAPTER 2PROBABtU'Y ANDSTOCHASTIC PROCESSES 51 anddetM=CTICT~(1- p2).Substitution forM-1into(2-1-150) yieldsthe desiredbivariate gaussian pdfintheform (2-1-156) Wenotethatwhenp=0,thejoinlpdfp(x"x2) in(2-1-156) factorsintotne productp(X')P(X2)' whereP(Xi)'i=1,2,arethemarginal pdfs.Sincepisa measure ofthecorrelation between X,andX2,wehaveshownthatwhenthe gaussian random variables X[andX2areuncorrelated. theyarealso statistically independent. Thisisanimportant property ofgaussian random variables, whichdoesnotholdingeneralforotherdistributions. Itextendsto n-dimensional gaussian randomvariables inastraightforward manner. Thatis, ifPij=0fori""jthentherandomvariables Xi'i=I,2,...,nareuncorrelated and,hence,statistically independent. Now,letusconsider alineartransformation ofngaussian randomvariables Xi'i=I,2,...,n,withmeanvectorm,andcovariance matrixM.Let Y=AX (2-1-157) whereAisanonsingular matrix.Asshownpreviously, thejacobian ofthis transformation isJ=IidetA.SinceX=A-Iy,wemaysubstitute forXin (2-1-150) and,thus,weobtainthejointpdfofYintheform p(y)=(2n')nl2(det M)112detAexp[-HA-'.y-m,)'M-I(A-ly-moll =(21r)"I2(~et Q)112exp[-!(y-my)'Q-'(y -m,.)] wherethevectormyandthematrixQaredefinedas my=Am, Q=AMA(2-1-158) (2-1-159) Thuswehaveshownthatalineartransformation ofasetofjointlygaussian randomvariables resultsinanothersetofjointlygaussian randomvariables. Suppose thatwewishtoperform alineartransformation thatresultsinn statistically independent gaussian ranct'omvariables. HowshouldthematrixA beselected? Fromourprevious discussion, weknowthatthegaussian random S2DIGITAL COMMUNICATIONS variables arestatistically independent iftheyarepairwise-uncorrelated, i.e.,if thecovariance matrixQisdiagonal. Therefore, wemusthave AMA'=D (2-1-160) whereDisadiagonal matrix.ThematrixMisacovariance matrix;hence,itis positivedefinite. Onesolution istoselectAtobeanorthogonal matrix (A'=A-I)consisting ofcolumns thataretheeigenvectors ofthecovariance matrixM.ThenDisadiagonal matrixwithdiagonal elements equaltothe eigenvalues ofM. Eumple 2-1-5 Consider thebivariate gaussian pdfwithcovariance matrix [1 l~]M=! Letusdetermine thetransformation Athatwillresultinuncorrelated randomvariables. First,wesolvefortheeigenvalues ofM.Thecharacteris­ ticequation is det(M-AI)=0 (1-A)2-~=0 Nextwedetermine thetwoeigenvectors. If8denotesaneigenvector, we have (M-.1.1)8=0 WithAI=~and.1.2=tweobtaintheeigenvectors Therefore, A=VI[11 1]-1 ItiseasilyverifiedthatA-I=A'andthat AMA'=D wherethediagonal elements ofDare~and!. CHAPTER 2:PROBABlLlTY ANDSTOCHASTIC PROCESSES 53 2·1-SUpperHOUDekontheTailProbability Inevaluating theperformance ofadigitalcommunication system,itisoften necessary todetermine theareaunderthetailofthepdf.Werefertothisarea asthetailprobability. Inthissection,wepresentfwoupperboundsonthetail probability. Thefirst,obtained fromtheChebyshev inequality, isratherloose. Thesecond,calledtheChernoff bound,ismuchtighter. Chebyshev Inequality Suppose thatXisanarbitrary randomvariablewith finitemeanmxandfinitevariance0";.Foranypositivenumber 6, (2-1-161) Thisrelation iscalledtheChebyshev inequality. Theproofofthisboundis relatively simple.Wehave \ 0";=L~(x-mx)2p(X)dx;;.I-m"..,(x-mx)2p(x)dx ;;.62Jp(x)dx=62p(lX-mxl;;'6) IX-m..I"6 Thusthevalidityoftheinequality isestablished. Itisapparent thattheChebyshev inequality issimplyanupperboundon theareaunderthetailsofthepdfp(y),whereY=X-mxoi.e.,theareaof p(y)intheintervals (-00,-8)and(6,00).Hence,theChebyshev inequality maybeexpressed as (2-1-162) or,equivalently, as (2-1-163) ThereisanotherwaytoviewtheChebyshev bound.Working withthezero meanrandom variable Y=X-m..forconvenience, suppose wedefinea function g(Y)as {I(IYI;;.6) g(Y)=0(IYI<8) (2-1-164) Sinceg(Y)iseither0orIwithprobabilities P(IYI<6)andP(IYI;;.8), respectively, itsmeanvalueis E[g(Y»)=P(IYI;;.8) (2-1-165) .54DIGITAL COMMUNICATIONS nGURt: 2-1·11 Aquadratic upperboundong(Y)usedin obtaining lhelailprobability (Chebyshev bound). Nowsupposethatweupper-bound g(Y)bythequadratic (YIS)',i.e., (2-1-166) Thegraphofg(Y)andtheupperboundareshowninFig,2-1-11.Itfollows that SinceE(g(Y)Jisthetailprobability, asseenfrom(2-1-165), wehaveobtained tileChebyshev bound. Formanypractical applications, theChebyshev boundisextremely loose. Thereasonforthismaybeattributed tothelooseness ofthequadratic (YIS)2 inoverbounding g(Y).Therearecertainly manyotherfunctions thatcanbe usedtooverbound g(Y).Below,weuseanexponential boundtoderivean upperboundonthetailprobability thatiseJCtremely tight. ClJemofr 80uDd TheChebyshev boundgivenaboveinvolves thearea underthetwotailsofthepdf.Insomeapplications weareinterested onlyin theareaunderonetail,eitherintheinterval(S,oc)orintheinterval(-:x:,l». Insuchacasewecanobtainanextremely tightupperboundbyoverbounding thefunctiong(Y)byanexponential havingaparameter thaIcanbeoptimized toyieldaslightanupperboundaspossible. Specifically, weconsider thelail probability intheinterval (I>,"").Thefunctiong(Y)isoverbounded as whereg(Y)isnowdefinedas g(Y)={OI(Y~S) (Y<S)(2-1-167) (2-1-168) CHAPTER 2:PROBABIUTY ANDSTOCHASTIC PROCESSES SS flGURE 1-1·U Anexponential upperboundong(Y)usedin oblaining thetailprobability (Chernoff bound).-=:=gf----+----,------- y andv;;o0istheparameter tobeoptimized. Thegraphofg(Y)andthe exponential upperboundareshowninFig.2-}-12. Theexpected valueofg(Y)is E[g(Y»)=P(Y;;O6)';;E(eV(Y-81) (2-1-169) Thisboundisvalidforanyv;;oO.Thetightestupperboundisobtained by selecting thevalueofvthatminimizes E(eY(Y-8». Anecessary condition fora minimum is !!.E(eY(Y-8» =0 dv(2-1-170) Buttheorderofdifferentiation andexpectation canbeinterchanged, sothat !!.E(eV(Y-6»=E(!!""eV(y-a») dv 'dv =E[(Y-8)eV(Y-3») =e-·a[E(YevY)-6E(evY»)=0 Therefore thevalueofvthatgivesthetightestupperboundisthesolution to theequation (2-1-171) Let~bethesolution of(2-1-171). Then,from(2-1-169), theupperboundon theone-sided tailprobability is P(Y~Ii)';;e-Y6E(eH) (2-1-172) ThisistheChernoff boundfortheuppertailprobability foradiscrete ora continuous randomvariable havingazeromean.tThisboundmaybeusedto showthatQ(x).;;e-x212 ,whereQ(x)istheareainthetailofthegaussian pdf (seeProblem 2-18). tNoteIhatE(evY)forrealvisnolthecharacteristic function ofY.Itiscalledthemomtnl gtneraling functionofY. S6DIGITAL COMMUNICATIONS I 2 fiGURE Z-l-lJ ThepdfofaLaplace-distributed randomvariable. o Anupperboundonthelowertailprobability canbeobtained inasimilar manner, withtheresultthat P(Y'"Il)'"e-V'E(eVY) whereifisthesolutionto(2-1-171) andIl<O. Example 2-1-6 Consider the(Laplace) pdf(2-1-173) (2-1-174) (2-1-176)whichisillustrated inFig.2-1,13.Letusevaluate theuppertailprobability fromtheChernoff boundandcompare itwiththetruetailprobability, whichis P(Y;:'5)=r!e-'dy=~e-' (2-1-175) Tosolve(2-1-171) forV,wemustdetermine themoments E(YevY)and E(e'Y).Forthepdfin(2-1-174), wefindlhat 2vE(Ye,Y)=------:~-_:: (v+If(v-If 1E(e'Y)----- (1+Y)(l-v) Substituting thesemoments into(2-1-171), weobtainthequadratic equation y21l+2v-5=0 whichhasthesolutions (2-1-177) Sincevmustbepositive, oneofthetwosolutions isdiscarded. Thus -1+~v= (2-1-178)Il CHAPTER 2,PROBABILITY ANDSTOCHASTIC PROCESSES 57 Finally, weevaluate theupperboundin(2-1-172) byeliminating E(eOY ) usingthesecondrelationin(2-1-176) andbysubstituting fori>from (2-1-178). Theresultis 132 P(y;oll).. el-VI+B2 2(-1+v'f'+'8') ForIl»I,(2-1-179) reducesto 6P(Y;OIl)"-e-B 2(2-1-179) (2-1-180) WenotethattheChernoff bounddecreases exponentially asIlincreases. Consequently, itapproximates closelytheexacttailprobability givenby (2-1-175). Incontrast, theChebyshev upperboundfortheuppertail probability obtained bytakingone-halfoftheprobability inthetwotails(due tosymmetry inthepdf)is 1P(Y;oil)..112 Hence,thisboundiselCtremeiy loose. Whentherandomvariablehasanonzero mean,theChernoff boundcanbe extended aswenowdemonstrate. IfY=X-mx,wehave P(Y;oil)=P(X-mx;Oil)=P(X;omx+Il)=P(X;oil..) where,bydefinition, Ilm=mx+6.SinceIl>0,itfollowsthatIlm>mx•Let g(X)bedefinedas andupper-bounded as(X)={I(X;0Ilm) g 0(X<Ilm)(2-1-181) (2-1-183)(2-1-182) (2-1-169)-g(X)..evex-6~) Fromthispoint,thederivation parallels thestepscontained in (2-1-172). Thefinalresultis P(X;ollm)"e-o,.E(eVX) whereIlm>mxandi>isthesolution totheequation E(Xe'X)-llmE(eVX)=0 (2-1-184) Inasimilarmanner, wecanobtaintheChernoff boundforthelowertail probability. ForIl<0,wehave P(X-mA"Il)=P(X"'"mx+13)=P(X""'Ilm)"E(ev(X-Bm»(2-1-185) Fromourprevious development, itisapparent that(2-1-185) resultsinthe bound P(X"'"8m)""'e-oa.E(eOX) where8m<mxand1)isthesolutionto(2-1-184).(2-1-186) 58l)1f,ITAL <OMM1INI('AlIONS 2·]·6SumsofRandom Variables andtheCentral LimitTheorem Wehavepreviously considered theproblemofdetermining thepdfofasumof nstatistically independent randomvariables. Inthissection,weagainconsider thesumofstatistically independent randomvariables, butourapproach is different andisindependent oftheparticular pdfoftherandomvariables in thesum.Tobespecific, suppose thatXi.i=1.2,...,n,arestatistically independent andidentically distributed randomvariables, eachhavingafinite meanm,andafinitevariance O'~.LetYbedefinedasthenormalized sum. calledthesamplemean: 1n Y=-LXi n;=1(2-1-187) Firstweshalldetermine upperboundsonthetailprobabilities ofYandthen weshallproveaveryimportant theorem regarding thepdfofYinthelimitas n~oc. TherandomvariableYdefinedin(2-1-187) isfrequently encountered in estimating themeanofarandomvariableXfromanumberofobservations Xi' i=1.2,....n.Inotherwords,theXi,i=1,2,...,n,maybeconsidered as independent samplesdrawnfromadistribution Fx(x),andYistheestimateof themeanmx• ThemeanofYis In E(Y)=my=-LE(Xi)ni=~ =m.x Thevariance ofYis Inn=22:LE(XiX)-m~ ni=Jj=1 1" 1n n =22:E(X;)+2LLE(Xi)E(X j)-m~n;=1 ni-I;....1 ;"'j 1 1=-(~+m;)+2n(n-l)m;-m;n n n WhenYisviewedasanestimate forthemeanm..wenotethatitsexpected valueisequaltomxanditsvariance decreases inversely withthenumberof CHAPTER 2:PROBABILITY ANDSTOCHASTIC PR')CE-SSES 59 samples n.Asnapproaches infinity, thevariance (T~approaches zero.An estimate ofaparameter (inthiscasethemeanmx)thatsatisfiestheconditions thatitsexpected valueconverges tothetruevalueoftheparameter andthe variance converges tozeroasn--+00issaidtobeaconsistent estimate. Thetailprobability oftherandomvariableYcanbeupper-bounded byuse oftheboundspresented inSection2-1-5.TheChebyshev inequality appliedto Yis ".2 P(IY-myI;;,,~),.;;~ (11"1)"..2P-LXi-mx;;.8,.;;~ ni~l n~(2-1-188) (2-1-189)Inthelimitasn--+00,(2-1-188) becomes ~~p(I~~Xi-mxI;;"5)=0 Therefore, theprobability thattheestimate ofthemeandiffersfromthetrue meanmxbymorethan8(Il>0)approaches zeroasnapproaches infinity.This statement isaformofthelawoflargenumbers. Sincetheupperbound converges tozerorelatively slowly,i.e.,inversely withn,theexpression in (2-1-188) iscalledtheweaklawoflargenumbers. TheChernoff boundappliedtotherandomvariable Yyieldsanexponential dependence ofn,andthusprovides atighterupperboundontheone-sided tail probability. Following theprocedure developed inSection 2-1-5,wecan determine thatthetailprobability foryis where8m=mx+8and ~>O.ButtheXi,i=1,2,...,n,arestatistically independent andidentically distributed. Hence, n =e-,n'~nE(e'X,) ;=1 (2-1-191) whereXdenotesanyone oftheXi'Theparameter vthatyieldsthetightest upperboundisobtained bydifferentiating (2-1-191) andsettingthederivative equaltozero. This yieldstheequation (2-1-192) (2-1-193)flODIGITAL COMMUNICATIONS Letthesolution of(2-1-192) bedenoted by~.Then,theboundontheupper tailprobability is p(~t.Xi~8m)0;;;[e~"mE(e"X)j", 8m>mx Inasimilarmanner, wefindthatthelowertailprobability isupper-bounded as P(Y0;;;8m)""[e~"·mE(e"X)j". 8m<mx (2-1-194) wherei>isthesolutionto(2-1-192). Example 2·1·7 LetXi'i=1,2,...,n,beasetofstatistically independent randomvariables definedas X.={1,-1withprobability p<~ withprobability 1-p (2-1-195)Wewishtodetermine atightupperboundontheprobability thatthesum oftheXiisgreaterthanzero.Sincep<!.wenotethatthesumwillhavea negative valueforthemean;henceweseektheuppertailprobability. With 8m=0in(2-1-193), wehave p(~X,~o)0;;;[E(e"XW where ~isthesolution totheequation E(XeVX)=0 Now Hence Furthermore, E(e"x)=pe"+(1-p)e-" Therefore theboundin(2-1-195) becomes p(t.Xi:;'0)0;;;[pe"+(1-p)e-"j" [(l=pIP]" 0;;;PYp+(I-p)y~ '"[4p(1-p))"12(2-1-196) (2-1-197) (2-1-198) Weobservethattheupperbounddecaysexponentially withn,asexpected. CHAPTER 2:PROBABILITY ANDSTOCHASTIC PROCI-SSrS 61 Incontrast, iftheChebyshev boundwereevaluated, thetailprobability woulddecrease inversely withn. CentralLimitTheorem Weconclude thissectionwithanextremely useful theorem concerning thecdfofasumofrandomvariables inthelimitasthe numberoftermsinthesumapproaches infinity.Thereareseveralversionsof thistheorem. Weshallprovethetheorem forthecaseinwhichtheranjom variables X"i=1,2,...,n,beingsummed arestatistically independent and identically distributed, eachhavingafinitemeanm,andafinitevariancea;. Forconvenience, wedefinethenormalized randomvariable lj.=Xi-m, I,i=1,2,... Jna, ThusV,hasazeromeanandunitvariance. Now,let 1nY=-LV i V;;i~l(2-1-199) Sinceeachterminthesumhasazeromeanandunitvariance, itfollowsthat thenormalized (by1/v';;)randomvariableYhaszeromeanandunitvariance. Wewishtodetermine thecdfofYinthelimitasn-+"". Thecharacteristic function ofYis [(.)]jv~Vi I/Jy(ju)=E(elvY)=Eexp:.;,; •(jV)=f1!/Iv,•r 1=1vn (2-1-200) whereVdenotesanyoftheVi'whichareidentically distributed. Now,letus expandthecharacteristic functionofUinaTaylorseries.Theexpansion yields l{1u(i.V,)=l+j.V,E(V)-u22 ,E(V2)+~~3E(V3)_'"vn vnn. (n)3! SinceE(U)=0andE(U2)=1,(2-1-201) simplifies to(2-1-201) (2-1-202) whereR(v,n)/ndenotes theremainder. WenotethatR(v,n)approaches 62DIGITAL COMMPNICo\TIONS zeroasn--+Xl.Substitution of(2-1-202) into(2-1-200) yieldsthecharacteristic function ofYintheform [V'R(vn)]"t/ty(jv) ~1 --+ '2nn Takingthenaturallogarithm of(2-1-203), weobtain [V'R(v,n)] In,py(jv) ~nIn1-2n+n Forsmallvaluesofx,In(I+xlcanbeexpanded inthepowerseries In(1+x)~x-h2+~X.'_ ... Thisexpansion appliedto(2-1-204) yields [V'R(v,n)1(v2R(v,n»)2 ]In,py(jv) ~n--+ - --+ +...2nn22n n(2-1-203) (2-1-204) (2-1-205) Finally, whenwetakethelimitasn-+"",(2-1-205) reduces to liTll,,_xIn,py(jv) ~-~v'.or,equivalently, lim,py(jv) ~e-.'t' n_x(2-1-206) But,thisisjustthecharacteristic function ofagaussian randomvariable with zeromeanandunitvariance. Thuswehavetheimportant resultthatthesum ofstatistically independent andidentically distributed randomvariables with finitemeanandvariance approaches agaussian cdfasn-+"".Thisresultis knownasthecentrallim;1theorem. Although weassumed thattherandomvariables inthesumareidentically distributed, theassumption canberelaxedprovided thatadditional restrictions areimposed ontheproperties oftherandomvariables. Thereisonevariation ofthetheorem, forexample, inwhichtheassumption ofidentically distributed randomvariables isabandoned infavorofacondition onthe third absolute moment oftherandomvariables inthesum.Foradiscussion ofthisandother variations ofthecentrallimittheorem, thereaderisreferred tothebookby Cramer(1946). 2·2STOCHASTIC PROCESSES Manyoftherandomphenomena thatoccurinnaturearefunctions oftime. Forexample, themeteorological phenomena suchastherandom fluctuations inairtemperature andairpressure arefunctions oftime.Thethermal noise voltages generated intheresistors ofanelectronic devicesuchasaradio receiver arealso"afunction oftime.Similarly, thesignalattheoutputofa sourcethatgenerates information ischaracterized asarandom signalthat CHAPTER 2:PROBABIliTY ANDSTOCliAST1C PROCESSES 6J varieswithtime.Anaudiosignalthatistransmitted overatelephone channel isanexample ofsuchasignal.Alltheseareexamples ofstochastic (random) processes. Inourstudyofdigitalcommunications, weencounter stochastic processes inthecharacterization andmodeling ofsignalsgenerated by information sources, inthecharacterization ofcommunication channels usedto transmit theinformation, inthecharacterization ofnoisegenerated ina receiver, andinthedesignoftheoptimum receiverforprocessing thereceived randomsignal. Atanygiventimeinstant,thevalueofastochastic process,whether itisthe valueofthenoisevoltagegenerated byaresistorortheamplitude ofthesignal generated byanaudiosource, isarandom variable. Thus,wemayviewa stochastic processasarandomvariableindexedbytheparameter I.Weshall denotesuchaprocessbyX(I).Ingeneral, theparameter Iiscontinuous, whereasXmaybeeithercontinuous ordiscrete, depending onthecharacteris­ ticsofthesourcethatgenerates thestochastic process. Thenoisevoltagegenerated byasingleresistororasingleinformation sourcerepresents asinglerealization ofthestochastic process. Hence,itis calledasamplefunclionofthestochastic process.Thesetofallpossiblesample functions, e.g.,thesetofallnoisevoltagewaveforms generated byresistors, constitute anensemble ofsamplefunctions or,equivalently, thestochastic processX(I).Ingeneral, thenumberofsamplefunctions intheensemble is assumed tobeextremely large;oftenitisinfinite. Havingdefined astochastic processX(I)asanensemble ofsample functions, wemayconsider thevaluesoftheprocessatanysetoftime.instants I,>12>I)>...>r.wherenisanypositiveinteger. Ingeneral, therandom variables X"""X(I;),i=1.2....,n.arecharacterized statistically bytheirjoint pdfp(X",x",...,x,J.Furthermore, alltheprobabilistic relations definedin Section2-1formultidimensional randomvariables carryovertotherandom variables X",i=1,2,...•n. Stationary Stochastic Processes Asindicated above,therandomvariables X",i=1,2,...,n,obtained fromthestochastic processX(I)foranysetof timeinstants 11>12>I)> >I.andanynarecharacterized statistically by thejointpdfp(x",x,",x,.).Letusconsider another setofnraildom variables X,,+I'"X(I;+I),i=I,2,...,n,whereIisanarbitrary timeshift. These random variables arech'iracterized bythejointpdf p(x/,+,.x,,+to.·.,x,,+,).Thejointpelfsoftherandomvariables X"andX,,+1O i=1,2....,n,mayormaynotbeidentical. Whentheyareidentical. i.e.. when (2-2-1) forallIandalln,thestochastic processissaidtobeslalionary inIhesrricl sense.Thatis,thestatistics gfastationary stochastic processareinvariant to anytranslation ofthetimeaxis.Ontheotherhand,whenthejointpdfsare different, thestochastic processisnonslalionary. 64DIGITAL COMMUNICATIONS Z.Z.1Statistical Averages Justaswehavedefinedstatistical averages forrandom variables, wemay similarly definestatistical averages forastochastic process. Suchaverages are alsocalledensemble averages. LetX(I)denotearandom process andlet X,==X(I.).ThenIhmoment oftherandomvariableXl,isdefinedas E(X7,):[x7,p(x,)dx" (2-2-2) (2-2-3)Ingeneral, thevalueofthenthmoment willdependonthetimeinstant I,ifthe pdfofX"depends onIi'Whentheprocessisstationary, however, p(x".,): p(x,)forallt.Hence,thepdfisindependent oftime,and,asaconsequence, thenthmoment isindependent oftime. Nextweconsider thetworandom variables X"==X(I,),i:1,2.The correlation between X"andX"ismeasured bythejointmoment E(X"X,,):fxfxX"x"p(X'" Xl,)dx" dx" Sincethisjointmoment depends onthetimeinstants IIandI,.itisdenoted by 4>(11,/2),Thefunction <1>(1,,12)iscalledtheaUlocorrelalion funclion ofthe stochastic process.WhentheprocessX(/)isstationary, thejointpdfofthepair (X",X,,)isidentical tothejointpdfofthepair(X".!,X".,)foranyarbitrary I. Thisimpliesthattheautocorrelation function ofX(I)doesnotdependonthe specifictimeinstants IIand12,but,instead, itdepends onthetimedifference 11- 12,Thus,f\Jrastationary stochastic process, thejointmoment in(2-2-3)is E(XI,X,,): cf>(lt•(2):<1>(1,-(2)=cf>("r) (2-2-4) wherer=II-12or,equivalently, 12:11-T.Ifwelet12=11+r.wehave cf>(-r):E(XI,X".,) :E(X,;XI;_,)=cf>(r) Therefore, <1>(r)isanevenfunction. Wealsonotethatcf>(0):E(X;)denotes theaveragepowerintheprocessX(/). Thereexistnonstationary processes withtheproperty thatthemeanvalue oftheprocessisindependent oftime(aconstant) andwheretheautocorrela­ tionfunction satisfiesthecondition that<1>(/,./ 2)=cf>(11-(2),Suchaprocessis calledwide-sense slalioTUlry. Consequently, wide-sense stationarity isaless stringent condition thanstrict-sense stationarity. Whenreference ismadetoa stationary stochastic processinanysubsequent discussion inwhichcorrelation functions areinvolved, thelessstringent condition (wide-sense stationarity) is implied. Relatedtotheautocorrelation function istheautocovariance function ofa stochastic process,whichisdefinedas ",(I"(2)=E{[X"-m(1,)][X l,-m(12)]} =<1>(1"12)-m(I,)m(1 2) (2-2-5) CHAPTER 2:PRO!JABILITY ANDSTOCHASTIC PROCESSES 6S wherem(tl)andm(t2)arethemeansofX/IandX",respectively. Whenthe processisstationary, theautocovariance function simplifies to (2-2-6) wherer=t,-t2. Higher-order jointmoments oftwoormorerandomvariables derivedfrom astochastic processX(I)aredefinedinanobviousmanner. Withthepossible exception ofthegaussian randomprocess,forwhichhigher-order moments can beexpressed intermsoffirstandsecondmoments, high-order moments are encountered veryinfrequently inpractice. Averages foraGaussian Process Suppose thatX(I)isagaussian random process.Hence,attimeinstants t=titi=I,2,...,n,therandomvariablesX,.' i=I,2,...,n,arejointlygaussian withmeanvaluesm(ti),i=1,2,...,n,and autocovariances JL(li,t;)=E[{X"-m(t,)(X'j -m(t)), i,j=1,2,...,n(2-2-7) (2-2-8)Ifwedenotethenxncovariance matrixwithelements JL(ti,t;)byMandthe vectorofmeanvaluesbym"thenthejointpdfoftherandomvariables X", i=I,2,...•nisgivenby(2-1-150). Ifthegaussian processisstationary thenm(ti)=mforall'iandJL(li•t,)= JL{li-til.Weobservethatthegaussian randomprocessiscompletely specified bythemeanandautocovariance functions. Sincethejointgaussian pdf depends onlyonthesetwomoments, itfollowsthatifthegaussian processis wide-sense stationary, itisalsostrict-sense stationary. Ofcourse,theconverse isalwaystrueforanystochastic process. Averages forJointStocbllstic Processes LetX(t)andY(t)denotetwo stochastic processes andletX,,'"X(li)'i=1,2,...,n,andY,;'"Y(tj),j= 1,2,...,m.represent therandomvariables attimest.>t2>t3>...>tnand I;>t;>...>t;",respectively. Thetwoprocesses arecharacterized statisti­ callybytheirjointpdf foranysetoftimeinstantsII't2,...,tn,t;,t;,...,t;"andforanypositive integervaluesofnandm. Thecross-correlation function ofX(t)andY(I),denoted by</lxy(tl>(2),is definedasthejointmoment 4>x,(11o12)=E(X/ IY,,)=f~[~x"y"p(JC" ,y,,}€ix"dy" andthecross-covariance is (2-2-9) 66DIUI"l'AL cor·.n~tl:N«·ATI()NS Whentheprocesses arejointlyandindividually stationary, wehave <1>"(1,,I,)=c/>x,.(r,-I,)andf.L"(I,,r,)=f.Lw(r,-I,).Inthiscase,wenotethat rf>".(-r)=E(X"Y".,)=E(X",Y,;)=c/>,Ar) (2-2-10) ThestochastiC processes X(I)andy(naresaidtobestatislically indepen­ dentifandonlyif p(x",x",...,x,,,,y",y"....,y,;.)=p(x".x,"...,x,,,lp(y,,,y';'....y,;,,) forallchoicesofI,andI:andforallpositiveintegersnandm.Theprocesses aresaidtobeuncorrelated if Hence, 1-""(/1,I,)=0 Acomplex-valued siochastic processZ(nisdefinedas Zit)=X(/)+jY(I) (2-2-11) (2·2-12)whereX(I)andY(I)arestochastic processes. Thejointpdfoftherandom variables Z,.==Z(I;),i=1,2....,isgivenbythejointpdfofthecomponents (X",Y,).i=I,2,....n.Thu~,thepdfthatcharacterizes Z",i=1,2,...,1'1,is p(X",XI.:"...•xt",Y,I,Yf.:'•..•,YI~) Thecomplex-valued stochastic processZ(t)isencountered intherepresen­ tationofnarrowband bandpass noiseintermsofitsequivalent lowpass components. Animportant characteristic ofsuchaprocessisitsautocorrela­ tionfunction. Thefunction isdefinedas <p"(I,,I,)=lE(Z"Z~) =~E[(X"+jY")(X,,-jY,,») =Hc/>u(tj,r,)+</1",(1"I,)+j[c/>vx(l" I,)-<p"(I,,I,))} wherec/>xxU,.I,)andc/>vv(t,,I,)aretheautocorrelation functions ofX(/)and Y(/),respectively, and'c/>y.,(/"I,)andc/>.n(/,,I,)arethecross-correlation functions. Thefactoroflinthedefinition oftheautocorrelation functionofa complex·valued stochastic process isanarbitrary butmathematically con­ venientnormalization factor,aswewilldemonstrate inourtreatment ofsuch processes inChapter 4. Whentheprocesses X(/)and.Y(I)arejointlyandindividually stationary, theautocorrelation function of"Z{t)becomes c/>"(/,,I,)=t/J,,(/,-I,)= c/>,,(c) whereI,=/,-r.Also,thecomplex conjugate of(2-2-12)is <t>:Ar)=lE(Z~Z"-,)=lE(Z~.,2,,)=c/>,,(-c) (2-2-13) Hence,cP,,(c)=c/>i,(-c). CHAPTER 2,PROBABILITY ANDsrOCHASTIC PROCESSES 67 Now,suppose thatZ(t)=X(t)+jY(t)andW(t)=V(t)+jV(r)aretwo complex-valued stochastic processes. Thecross-correlation function ofZ(t) andW(t)isdefinedas cP,w(t,,t2)=~E(Z" W~) =tE[(X"+jY,,)(V"-jV;,)] =~{<I>,"(t I't2)+<l>vv(t,,t2)+j[cPy,,(tI't2)-<I>,v(/"t2)]}(2-2-14) WhenX(t),Y(t),U(t),andV(t)arepairwise-stationary, thecross-correlation functions in(2-2-14) become functions ofthetimedifference r=/,-t2­ Furthermore, (2-2-15) 2-2-2Powel'Density Spectl'um Thefrequency contentofasignalisaverybasiccharacteristic thatdistin­ guishesonesignalfromanother. Ingeneral,asignalcanbeclassified ashaving eitherafinite(nonzero) averagepower(infiniteenergy)orfiniteenergy.The frequency contentofafiniteenergysignalisobtained astheFouriertransform ofthecorresponding timefunction.Ifthesignalisperiodic, itsenergy is infiniteand,consequently, itsFouriertransform doesnotexist.Themechanism fordealingwithperiodic signalsistorepresent theminaFourierseries.With sucharepresentation, theFouriercoefficients determine thedistribution of poweratthevariousdiscretefrequency components. Astationary stochastic process isaninfiniteenergysignal,and,hence,its Fouriertransform doesnotexist.Thespectral characteristic ofastochastic signalisobtained bycomputing theFouriertransform oftheautocorrelation function. Thatis,thedistribution ofpowerwithfrequency isgivenbythe function TheinverseFouriertransform relationship is <1>(r)=r<t>(f)ei'·t'df Weobservethat <1>(0)=J~<t>(f)df =E(IXf);.,o(2-2-16) (2-2-17) (2-2-18) Since<1>(0)represents theaveragepowerofthestochastic signal,whichisthe areaunder<t>(f),<t>(f)isthedistribution ofpowerasafunctionoffrequency. Therefore, <t>(f)iscalledthepowerdensityspectrum ofthestochastic process. 68DIGITAL COMMUNICATIONS Ifthestochastic processisreal,1/>(f)isrealandeven,and,hence<P(f)is realandeven.Ontheotherhand,iftheprocessiscomplex, 4>(r)=4>*(-r) and,hence <1l*(f)=[cP"(r)ei2""dr=f,<f>*(-1')e'i2 1<frd1' =r~cP(r)e-;2K/<dr=cf>(f) (2-2-19) Therefore, <PU)isreal. Thedefinition ofapower density spectrum canbeextended totwojointly stationary stochastic processes X(t)andY(I),whichhaveacross-correlation function 4>.y(f).TheFouriertransform ofcP...(r),i.e., (2-2-20) (2-2-21)iscaIledthecross-power densityspectrum.Ifweconjugate bothsidesof (2-2-20), wehave cf>~,(f)= r~<I>;,(r)el2K/Td1'=r~<p~v(-r)e-I~K(rdr =r~</IyA1')e-121<'rdl'=<P,Af) Thisrelationholdsingeneral. However, ifX(t)andY(t)arerealstochastic processes, (2-2-22) Bycombining theresultin(2-2-21)withtheresultin(2-2-22), wefindthatthe cross-power densityspectrum oftworealprocesses satisfiesthecondition (2-2-23) 2·2·3Response ofaLinearTime-Invariant Systemtoa Random InputSignal Consider alineartime-invariant system(filter)thatischaracterized byits impulseresponse h(l)or,equivalently, byitsfrequency response H(f).where h(t)andH(f)areaFouriertransform pair.LetX(I)betheinputsignaltothe systemandlety(t)denotetheoutputsignal.Theoutput of thesystemmaybe expressed intermsoftheconvolution integralas y(l)=f,h(r)x(t-r)41' (2-2-24) CHAI"TER 2'PROBABILITY ANDSTOCHASTIC PROCESSES 69 Now,supposethatx(/)isasamplefunction ofastationary stochastic process X(t).Then,theoutputy(l)isasamplefunction ofastochastic processY(I). Wewishtodetermine themeanandautocorrelation functions oftheoutput. Sinceconvolution isalinearoperation performed ontheinputsignalx(/), theexpected valueoftheintegralisequaltotheintegraloftheexpected value. Thus,themeanvalueofY(I)is my=E[Y(I)]=[~h(r)E[X(1 -r)]dr =mx[~h(r)dr=mxH(O) (2-2-25) whereH(O)isthefrequency response ofthelinearsystematf=O.Hence,the meanvalueoftheoutputprocessisaconstant. Theautocorrelation functionoftheoutputis 4>YAI,.(2)=!E(Y" Y~) =~[[h(tl)h*(a)E[X(I, -(J)X*(ll-a)]dadf3 =[~[~h(f3)h*(a)4>xx(/,-t 2+a-(3)dadf3 Thelaststepindicates thatthedoubleintegralisafunction ofthetime di1ference I,-12,Inotherwords,iftheinputprocessisstationary, theoutputis alsostationary. Hence (2-2-26) Byevaluating theFouriertransform ofbothsidesof(2-2-26), weobtainthe powerdensityspectrum oftheoutputprocessintheform 4>yy(f)=[~4>yy(r)e-j2>if<d1: =[~[~[~h*(a)h(f3)4>u(r+ a-(3)e-j2<f'drdadf3 =<l>u(f)IH(f)12(2-2-27) Thus,wehavetheimportant resultthatthepowerdensityspectrum ofthe outputsignalistheproductofthepowerdensityspectrum oftheinput multiplied bythemagnitude squaredofthefrequency response ofthesystem. 70DIGITA.L COMMUNICA.TIONS Whentheautocorrelation function c/>yy(r)isdesired,itisusuallyeasierto determine thepowerdensityspectrum <t>yy(f)andthentocompute theinverse transform. Thus,wehave cPyy(r)=[~<t>.,y(f)el2K!T df =[<t>xAf)IH(f)12ei2KfTdf Weobservethattheaveragepowerintheoutputsignalis c/>yy(O)=[<t>.Af)IH(f)12df Sincec/>yy(O)=£(1Y,12).itfollowsthat [<t>xx(f)IH(fWdf;'0(2-2-28) (2-2-29) Suppose weletIH(f)12=1foranyarbitrarily smallintervalf,~f~fi.and H(f)=0outsidethisinterval.Then, Butthisispossibleifandonlyif<t>.Af);.0forallf ElUIIIlple 2-2-1 Suppose thatthelowpass filterillustrated inFig.2-2-1isexcitedbya stochastic processx(t)havingapowerdensityspectrum Astochastic processhavingaflatpowerdensityspectrum iscalledwhite nGun 2-1-1AneXllIJlple afalawpassfilter. ('HAPTER 2:PR08ABILITY !\NDST(){'tfASrH- PRO(TSS!'.S 71 FIGURE 2·2-2Thepowerdensityspectrum ofthelowpassniteroutputwhen theinputi!lwhilenoise. II noise.Letusdetermine thepowerdensityspectrum oftheoutputprocess. Thetransferfunction ofthelowpassfilteris I+j21TfL/RR H(f)=R+j21TfL and,hence, , I IH(f)!-= 1+(21TLlR)'f'(2-2-30) Thepowerdensityspectrum oftheoutputprocessis <Jl=Nil I ...(f)2 I+(21TL/R)2f'(2-2-31) Thispowerdensityspectrum isillustrated inFig,2-2-2.ItsinverseFourier transform yieldstheautocorrelation function (2-2-32) Theautocorrelation function cb,,(r)isshowninFig.2-2-3.Weobservethat thesecondmoment oftheprocessY(t)isq".,.(O)=RN,,/4L. ~\\(t) FIGURE 2-2-3Theautocorrelation function oftheoutputofthelowpassfilter forawhite-noise input.II 72IJIG:TAL COMMl)f':ICATIOr-;S Asafinalexercise, wedetermine thecross-correlation function between y(r)andx(t),wherex(t)denotestheinputandy(t)denotestheoutputofthe linearsystem.Wehave If'· </>,,(t,.to)=~E(Y"X~) =2'.h(a)E[X(t,-alX*(t,)]cia =r~h(a)</>,,(t 1 -t,-a)cia=</>,,(1,-(2) Hence,thestochastic processes X(t)andY(t)arejointlystationary. With t,-t2=I.wehave </>,.(I)=L~h(a)</>x.(1 -a)da (2-2-33) Notethattheintegral in(2-2-33) isaconvolution integral. Henceinthe frequency domaintherelation(2-2-33) becomes rtJ,,(f) =eJ>,,(j)H(f) (2-2-34) Weobservethatiftheinputprocessiswhitenoise.thecrosscorrelation ofthe inputwiththeoutputofthesystemyieldstheimpulseresponse h(t)towithina scalefactor. 2-2-4Sampling Theorem forBand-Limited Stochastic Processes Recallthatadeterministic signals(t)thathasaFourier transform 5(f)is calledband-limited ifS(f)=0forItI>W,whereWisthehighestfrequency contained ins(t).Suchasignalisuniquely represented bysamplesofs(t)taken atarateoff,;;.2Wsamples!s. Theminimum ratetv=2Wsamples!s iscalled theNyquistrate,Sampling belowtheNyquistrateresultsinfrequency aliasing. Theband-limited signalsampled attheNyquistratecanbereconstructed fromitssamples byuseoftheinterpolation formula ~nsin[2;rW(t-2~)] s(t)=2s(-)------ n~-~2W(n)2n:Wt--2W(2-2-35) where{s(n/2W)} arethesamplesofs(t)takenatt=n/2W,n=O.±I,±2,.... Equivalently, s{t)canbereconstructed bypassingthesampled signalthrough CHAPTER 2,PROBABILITY ANDSTOCHASTIC PROCESSES 73 FIGURE 2-2....Signalreconstruction basedonideal interpolation. in-2)T (n-Of nT (It...OT-. anideallow-pass filterwithimpulseresponse h(t)=(sin2nWt)l2nWt. Figure 2-2-4illustrates thesignalreconstruction processbasedonidealinterpolation. Astationary stochastic processX(t)issaidtobeband-limited ifitspower densityspectrum lP(f)'"0forIfI>W.Sinceet>(f)istheFouriertransform of theautocorrelation function <1>(1:),itfollowsthat</1(r)canberepresented as (2·2-36) (2-2-37)where{.p(n/2W)} aresamplesof<1>(1:)takenatr=n/2W,n=0,±1,±2,.... Now,ifX(t)isaband-limited stationary stochastic processthenX(t)canbe represented as ~ IIsin[21tW(t-z':v)) X(/)=2:x(-)----.:. n~-~ZW21tW(t-2~) where{X(nI2W)} aresamplesofX(t)takenatt'"n/2W,n'"0,±I,±2,.... Thisisthesampling representation forastationary stochastic process.The samplesarerandomvariables thataredescribed statistically byappropriate jointprobability densityfunctions. Thesignalrepresentation in(2-2-37)is easilyestablished byshowingthat(Problem 2-17) (2-2-38) Hence,equality between thesa9\pling representation andthestochastic processX(/)holdsinthesensethatthemeansquareerroriszero. 74DICilTAL COM\1t:~I(";" nONs 2-2-5Discrete-Time Stothastic SignalsandSystems Thecharacterization ofcontinuous-time stochastic signalsgivenabovecanbe easilycarriedovertodiscrete-time stochastic signals.Suchsignalsareusually obtained byuniformly sampling acontinuous-time stochastic process. Adiscrete-time stochastic processX(n)consistsofanensemble ofsample sequences {x(n)}.Thestatistical properties ofX(n)aresimilartothe characterization ofX(t)withtherestriction thatnisnowaninteger(time) variable. Hence.themthmomentofX(n)isdefinedas E[X~'J=rX::'p(X")dX,, andtheautocorrelation sequence is </>(n.k)=~E(X"Xt) =rrX"X:p(X,,, X,JdX"dX. Similarly, theautocovariance sequence is J.L(n,k)=</>(n.k)-E(X,,)E(Xt)(2-2-39) (2-2-40) (2-2-41) Forastationary process, wehave</>(n.k)'=</>(n-k).j.t(n.k)'=J.L(n-k),and lL(n-k)=</>(n-k)-Im,r (2-2-42) wherem,=E(X,,)isthemeanvalue. Asinthecaseofcontinuous-time stochastic processes, adiscrete-time stationary processhasinfiniteenergybutafiniteaverage power.whichis givenas E(lX,l) =</>(0) (2-2-43) Thepowerdensitvspectrum forthediscrete-time process isobtained by computing theF(,~rier transform of</>(n).Since</>(n)isadiscrete-time sequence, theFouriertransform isdefinedas x <I>(f)=2:</>(n)e-,21<!" andtheinversetransform relationship is J'12 <f>(n)=<t>(f)eM"df -1/2(2-2-44) (2-2-45) Wemaketheobservation thatthepowerdensityspectrum <I>(f)isperiodic withaperiod!" =I.Inotherwords,<t>(f+k)=<I>(f)fork=±1.±2...,.This isacharacteristic oftheFouriertransform ofanydiscrete-time sequence such as<f>(n). Finally,letusconsider theresponse ofadiscrete-time, lineartime-invariant systemtoastationary stochastic inputsignal.Thesystemischaracterized in CHArTER.:!: PROBAB1LlTY ASDSTOCHASTIC PROCESSES 75 thetimedomainbyitsunitsampleresponse h(n)andinthefrequency domain bythefrequency response H(f),where x H(f)=2:h(n)eJ2"fn ,,=-:x(2-2-46) The response ofthesystemtothestationary stochastic inputsignalX(n)is givenbytheconvolution sum x y(n)=2:h(k)x(n-k) 1-,--ex. Themeanvalueoftheoutputofthesystemis x m,.=E[y(n»)=Lh(k)E[x(n -k») Ii.-'X m,=m,Lh(k)=m,H(O), whereH(O)isthezerofrequency (de)gainofthesystem. Theautocorrelation sequence fortheoutputprocessis eb,,(k)=~E[y*(n )y(n+kl] x =!2:2:h*(i)h(j)E[x*(n -i)x(n+k-j)) xx =LLh*(i)h(j)eb,,(k -j+i) "'-I-:x(2-2-47) (2-2-4ll) (2-2-49) Thisisthegeneralformfortheautocorrelation sequence ofthesystemoutput intermsoftheautocorrelation ofthesysteminputandtheunitsample response ofthesystem. BytakingtheFourier transform ofeb,,(k)and substituting therelationin(2-2-49), weobtainthecorresponding frequency domainrelationship <l>•.,(f)=<t>,,(f)IH(f)12(2-2-50) whichisidentical to(2-2-27)exceptthatin(2-2-50)thepowerdensityspectra <P"Cf)and<l>,,(f)andthefrequency response H(f)areperiodic functions of frequency withperiodJ;,=I. 2-2-6Cyclostationary Processes Indealingwithsignalsthatcarrydigitalinformation weencounter stochastic processes thathavestatistical averages thatareperiodic. Tobespecific,letus consider astochastic processoftheform x X(t)=2:a.g(t-nT) 11=-X(2-2-51) 76DIGITAL COMMUNICATIONS where{a"}isa(discrete-time) sequence ofrandom variables withmean ma=Elan)forallnandautocorrelation sequence </I..(k)=!£(a:anH)' The signalg(t)isdeterministic. Thestochastic processX(t)represents thesignalfor severaldifferent typesoflinearmodulation techniques whichareintroduced in Chapter 4.Thesequence {an}represents thedigitalinformation sequence (of symbols) thatistransmitted overthecommunication channel and1/T represents therateoftransmission oftheinformation symbols. Letusdetermine themeanandautocorrelation functionofX(I).First,the' meanvalueis x £[X(I») =2:£(a")g(1 -nT) n=-x x =m"2:g(1-nT) (2-5-52) Weobservethatthemeanistime-varying. Infact,itisperiodic withperiodT. Theautocorrelation function ofX(I)is cPxAI+r,I)=!£[X(I+r)X*(I») x x =!2:2:£(a:am)g*(t -nT)g(1+r-mT) n=-x.",=-x x x =2:2:<Pairn-n)g*(I-nT)g(l+ T-mT) (2-2-53) n=-xm=-x Again,weobservethat "'xx(t+r+kT,1+kT)=<P«(I+T,I) (2-2-54) fork=±1, ±2.....Hence,theautocorrelation function ofX(I)isalso periodic withperiodr. Suchastochastic processiscalledcycloslalionary orperiodically slalionary. Sincetheautocorrelation function depends onboththevariables Iandr.its frequency domain representation requires theuseofatwo-dimensional Fouriertransform. Sinceitishighlydesirable tocharacterize suchsignalsbytheirpower density spectrum, analternative approach istocompute thelime-average aUlocorrelalion funclion overasingleperiod,definedas - 1fTn<pxAr)=-T<Pxx(t+T,I)dl -Tn(2-2-55) Thus,weeliminate thetimedependence bydealing withtheaverage autocorrelation function. Now,thefouriertransform of;;;xx(r)yieldsthe CHAPTER 2:PROBABILITY ANDSTOCHASTIC ?ROCESSES 77 averagepowerdensityspectrum ofthecyclostationary stochastic process. This approach allowsustosimplycharacterize cyclostationary processes inthe frequency domainintermsofthepowerspectrum. Thatis,thepowerdensity spectrum is (2·2·56) 2-3BIBLIOGRAPHICAL NOTES ANDREFERENCES Inthischapter wehaveprovided areviewofbasicconcepts anddefinitions in thetheoryofprobability andstochastic processes. Asstatedintheopening paragraph, thistheoryisanimportant mathematical toolinthestatistical modeling ofinformation sources, communication channels, andinthedesignof digitalcommunication systems. Ofparticular importance intheevaluation of communication systemperformance istheChernoff bound.Thisboundis frequently usedinbounding the probability oferrorofdigitalcommunication systemsthatemploycodinginthetransmission ofinformation. Ourcoverage alsohighlighted anumber ofprobability distributions andtheirproperties, whicharefrequently encountered inthedesignofdigitalcommunication systems. ThetextsbyDavenport andRoot(1958),Davenport (1970),Papoulis (1984)Pebbles (1987),Helstrom (1991)andLeon-Garcia (1994)provide engineering-oriented treatments ofprobability andstochastic processes. A moremathematical treatment ofprobability theorymaybefoundinthetextby Loeve(1955).Finally, wecitethebookbyMiller(1964),whichtreats multidimensional gaussian distributions. PROBLEMS Z-tOneexperiment hasfourmutually exclusive outcomes A"i=1,2,3,4, anda secondexperiment hasthreemutuallyexclusive outcomes 8"j=1,2.3.Thejoint probabilities P(A"B,)arc . PtA"~B,)=0.10, P(A.,B,)~0.05, PtA"~B,)=0.05, P(A"B,)=O.II,PiA"~B,)=0.08, P(A"B,)=0.03, PiA"~B2)=O.12, PiA"~B,)=0.04,P(A"D,)=0.13 peA,.D,)=0.09 PiA"~D,)=0.14 peA"~D,)=0.06 Determine theprobabilities P(A,),i=1,2,3,4, andPCB,),j=1,2,3. Z·ZTherandomvariables X"i=1,2..,.,n,havethejointpdfpix"~x"...,x,,) Provethat p(x,.x,.x, •...,x,,) =p(x"Ix"."...,x,)p(x" ,Ix""...,x,)··'p(x,lx"x,)p(x,lx,)p(x,) 78DIGITAL COMMVN)CATJONS 2-3ThepdfofarandomvariableXisp(x).Arandomvariable Yisdefinedas Y=aX+b wherea<O.Determine thepdfofYintermsofthepdfofX. 2-4Suppose thatXisagaussian randomvariable withzeromeanandunitvariance. Let Y=aX'+b,a>0 Determine andplotthepdfofY. 2-5aLetX,andX,bestatistically independent zero·mean gaussian randomvariables withidentical variance. Showthata(rotational) transformation oftheform Y,+jY,=(X,+jX,)e''' resultsinanotherpair(1';,y)ofgaussian randomvariables thathavethesame jointpdfasthepair(X"X,). IINotethat whereAisa 2x2matrix.Asageneralization ofthetwo-dimensional transformation ofthegaussian random variables conSidered in(a).what property mustthelineartransformation AsatisfyifthepdfsforXandY.where Y=AX,X=(X,X,'.'Xn)andY=(Y,Y,'..Yn),areidentical? 2-6Therandomvariable Yisdefinedas n Y=LX, ;=1 wheretheX"i=1,2,...,n,arestatistically independent randomvariables with X={I• 0withprobability p withprobability 1-p •Determine thecharacteristic functionofY. bFromthecharacteristic function. determine themoments E(Y)andE(Y'). 2-7Thefourrandom variables X"X"X"X,arezero-mean jointlygaussian random variables withcovariance /l-'J=E(X,X,) andcharacteristic function tjJ(jv,,jv"jv,.jv.).Showthat 2-8Fromthecharacteristic functions forthecentralchi-square andnoncentral chi-square random variables givenby(2-1-109) and(2·1-117). respectively. n~,\PTI·.1{ 2:rROHAHIL.lr" ANI)SI-C)('HASr!C "HI)(TSSFS 79 determine thecorn:sponding firstandsecondmoments givenhy(2-1-112) and (2-1-125) 2-9ThepdfofaCauchydistrihuted randomvariahleXis a/Ifp(x)=-,-"x-+a--x<x<x aDetermine themeanandvariance ofX. bDetermine thetharacteristic functionofX. 2-)0Theral)domvariallie Yisdefinedas I"Y=-L;x,"iI whereX"i=1,2,...,n,arestatistically independent andidentically distrihuted randomvariables eachofwhichhastheCauchypdfgiveninProhlem 2-9 aDetermine thecharacteristic functionofY. bDetermine thepdfofY. cConsider thepdfofYinthelimitasn_x.Doesthecentrallimithold?Explain youranswer. 2-11Assumethatrandomprocesses X(I)andY(I)areinilividually andjointlyslalionary. aDetermine theautocorrelation functionof«I)=X(I)+y(t). bDetermine theautocorrelation function of«I)whenX(I)andY(f)are uncorrelated. cDetermine theautocorrelation function ofZ(I)whenX(I)and\'(1)arc uncorrelated andhavezeromeans. 2-UTheautocorrelation functionofastochastic processX(I)IS Suchaprocessiscalledwhilenoise.Suppose .'(1)istheinputtoanidealhandpass filterhavingthefrequency response characteristic showninFig.P2-12.Determine thetotalnoisepowerattheoutputofthefilter. 2·13Thecovariance matrixofthreerandomvariables X"X,andX,is [1'0," I''I FIGURE P.2-121HtfII _8-; I~~H~ ~'LLJ I,_, -f, () 1.- - FIGURE P2-1680DlGfTAl COMMVNICATJONS w-eIR Xli) YtnT Thelineartransformation Y=AXismadewhere Determine thecovariance matrixofV. lot4LetX(f)beastationary realBormalprocesswithzeromean.Letanewprocess Y(I)bedefinedby Y(I)=X'(I) Determine theautocorrelation function ofY(I)intermsoftheautocorrelation funclion ofX(f).Hint:Usetheresultongaussian variables derived inProblem 2-7. lotSFortheNakagami pdf,givenby(2-1-147). definethenormalized randomvariable X=RIVO.Determine thepdfofX. 2-16TheinputX(I)inthecircuitshowninFig.P2-16isastochastic processwith E[X(I»)=0and</I,,(r)=u'/)(r),i.e.,X(I)isawhitenoiseprocess. •Determine thespectraldensity¢>,,(f). bDetermine </1,.,(r)andE[Y'(f». 2-17Demonstrate thevalidityof(2-2-38). 2-18UsetheChernoff boundtoshowthatQ(x)'"e""whereQ(x)isdefinedby (2-1-97). 2-19Delermine themean.Iheautocorrelation sequence, andthepowerdensity spectrum oftheoutputofasystemwithunitsampleresponse {I(n=0) -2(n=I)h(n)-I(n=2) o(otherwise) whentheinputX(II)isawhite-noise processwithvarianceu;. loWTheautocorrelation sequence ofadiscrete-time stochastic processis</I(k)=(\)1'1. Determine itspowerdensityspectrum. 2-21Adiscrete-time stochaslic processX(n)'"X{IIT)isobtained byperiodicsampling ofacontinuous-lime zero-mean stationary processXU)whereTisthesampling interval.i.e.,t.=liTisthesampling rate. •Determine therelationship between theautocorrelation function ofX(I)and theautocorrelation sequence ofX(II). bExpressthepowerdensityspectrum ofX{n)intermsofthepowerdensity spectrum oftheprocessX(r).• CHAI'TER 2,PROBABILITY ANDSTOCHASTIC PROCESSES 81 cDetermine theconditions underwhichthepowerdensityspectrum ofX(n)is equaltothepowetdensityspectrum ofX(I). 2-UConsider aband-limited zero-mean stationary stochastic X(I)withpowerdensity spectrum {I(IfI"W) 4>(f)=0(IfI>W) X(I)issampledataratef,=lITtoyieldadiscrete-time processX(n).X(nT). aDetermine theexpression fortheautocorrelation sequence ofX(n). bDetermine theminimum valueofTthatresultsinawhite(spectraUy ftat) sequence. cRepeat (b) ifthepowerdensityspectrum ofX(I)is rt>(f)={1-lfllWOfl--W)o (lfl>W) 2.1)Showthatthefunctions k=0,±l,±2,... areorthogonal overtheinterval[-x,"l.i.e., J' {l/2W (k=j) .f.(I)t(I)dl= 0(k"j) Therefore, thesampling theorem reconstruction formulamaybeviewedasaseries expansion oftheband-limiled signalS(I),wheretheweightsaresamplesofS(I) andthe{f,(I)}arethesetoforthogonal functions usedintheseriesexpansion. 2·14Thenoiseequivalenl bandwidth ofasystemisdefinedas 1JZB,.=G"tH(f)I'df whereG=maxtH(f)I'.Usingthisdefinition. determine thenoiseequivalent bandwidth oftheidealbandpass filtershowninFig.Pl-t2andthelowpasssystem showninFig.P2-l6. 3 SOURCE CODING Communication systemsaredesigned totransmittheinformation generated by asourcetosomedestination. Information sourcesmaytakeavarietyof different forms.Forexample, inradiobroadcasting, thesourceisgenerally an audiosource(voiceormusic).InTVbroadcasting, theinformation sourceisa videosourcewhoseoutputisamovingimage.Theoutputsofthesesourcesare analogsignalsand,hence,thesourcesarecalledanalogsources.Incontrast, computers andstoragedevices,suchasmagnetic oropticaldisks,produce discrete outputs(usually binaryorASCIIcharacters) and,hence,theyare calleddiscretesources. Whether asourceisanalogordiscrete, adigitalcommunication systemis designed totransmitinformation indigitalfonn.Consequently, theoutputof thesourcemustbeconverted toaformatthatcanbetransmitted digitally. This conversion ofthesourceoutputtoadigitalfonnisgenerally performed bythe sourceencoder, whoseoutputmaybeassumed tobeasequence ofbinary digits. Inthischapter, wetreatsourceencoding basedonmathematical modelsof information sourcesandaquantitative measure oftheinformation emittedby asource.Weconsidertheencoding ofdiscretesourcesfirstandthenwediscuss theencoding ofanalogsources.Webeginbydeveloping mathematical models forinformation sources. 3·1MATHEMATICAL MODELS FORINFORMATION SOURCES Anyinformation sourceproduces anoutputthatisrandom, i.e.,thesource outputischaracterized instatistical terms.Otherwise, ifthesourceoutput 82 CHAPTER 3:SOURCE COOING83 wereknownexactly,therewouldbenoneedtotransmit it.Inthissection.we consider bothdiscrete andanaloginformation sources, andwepostulate mathematical modelsforeachtypeofsource. Thesimplest typeofdiscretesourceisonethatemitsasequenceofletters selected fromafinitealphabet. Forexample, abinarysourceemitsabinary sequence oftheformlOOIOlllO ...•wherethealphabet consists ofthetwo letters{O,I}.Moregenerally, adiscreteinformation sourcewithanalphabet of Lpossible letters,say{x"X2•.•••xLl.emitsasequence oflettersselected fromthealphabet. Toconstruct amathematical modelforadiscretesource,weassumethat eachletterinthealphabet {x,.X2".••XL}hasagivenprobability p,of occurrence. Thatis, (3-1-1)where Weconsider twomathematical modelsofdiscretesources. Inthefirst,we assumethattheoutputsequence fromthesourceisstatistically independent. Thatis.thecurrentoutputletterisstatistically independent fromallpastand futureoutputs. Asourcewhoseoutputsatisfies thecondition ofstatistical independence amongoutputlettersinthesequence issaidtobememoryless. Suchasourceiscalledadiscretememoryless source(DMS). Ifthediscrete sourceoutputisstatistically dependent, as,forexample, English text,wemayconstruct amathematical modelbasedonstatistical· stationarity. Bydefinition, adiscrete sourceissaidtobestCll;onary ifthe jointprobabilities oftwosequences oflengthn.sayaI.a,....•anand aI''O"a,+n"....an+m,areidentical foralln~1andforallshiftsm.Inother words,thejointprobabilities foranyarbitrary lengthsequence ofsource outputsareinvariant underashiftinthetimeorigin. Ananalogsourcehasanoutputwaveform x(t)thatisasamplefunction ofa stochastic processX(/).WeassumethatX(t)isastationary stochastic process withautocorrelation function ~xx(r)andpowerspectraldensity<II.Af).When X(/)isabandlimited stochastic process, Le.,<IIxAf)=0forIfI~W.the sampling theorem maybeusedtorepresent X(t)as ~nsin[21rW(/-2~)) X(/)=~X(-)---- n<-~2W(n)2lrWt-- 2W where{X(n/2W)} denotethesamples oftheprocessX(t)takenatthe sampling (Nyquist) rateoff,=2Wsamples/s. Thus.byapplying thesampling theorem, wemayconverttheoutputofananalogsourceintoanequivalent 84DIGIT....lCOMMUNICATIONS discrete-time source.Then,thesourceoutputischaracterized statistically by thejointpdfp(X.,X2'...,X",)forallm;;'1,whereX.=X(n/2W). 1...n...m, aretherandomvariables corresponding tothesamplesofX(t). .Wenotethattheoutputsamples{X(n/2W)} fromthestationary sourcesare generally contin.ous, and,hence,theycannotberepresented indigitalform withoutsomelossinprecision. Forexample, wemayquantize eachsampletoa setofdiscretevalues,butthequantization processresultsinlossofprecision, and,consequently, theoriginalsignalcannotbereconstructed exactlyfroptthe quantized samplevalues.Laterinthischapter, weshallconsider thedistortion resulting fromquantization ofthesamples fromananalogsource. 3-2ALOGAIUTHMIC MEASURE OFINFORMATION Todevelopanappropriate measureofinformation, letusconsider twodiscrete random variables withpossible outcomes Xi.i=1.2•...,n,andYi,i= I,2,...,m,respectively. Suppose weobserve someoutcome Y=Yiandwe wishtodetermine, quantitatively, theamount ofinformation thatthe occurrence oftheeventY=Yiprovides abouttheeventX=Xi>i=I,2,...,n. WeobservethatwilenXandYarestatistically independent, theoccurrence of Y=Yjprovides noinformation abouttheoccurrence oftheeventX=X,.On theotherhand,whenXandYarefullydependent suchthattheoccurrence of Y=Yidetermines theoccurrence ofX=Xi'theinformation contentissimply thatprovided bytheeventX=X,.Asuitable measure thatsatisfies these conditions isthelogarithm oftheratiooftheconditional probability P(X=XiIY=Y;)•P(xiIYi) dividedbytheprobability P(X=Xi)•P(Xi) Thatis,theinformation contentprovided bytheoccurrence oftheeventY=Yi abouttheeventX=Xiisdefinedas .-~[(Xi,Yj)-logP(Xi) (3-2-1) [(x,;Yi)iscalledthemutualinformation between XiandYi' Theunitsof[(Xi;Yi)aredetermined bythebaseofthelogarithm, whichis usuallyselectedaseither2ore.Whenthebaseofthelogarithm is2,theunits ofl(xi;Yj)arebits,andwhenthebaseise,theunitsofI(Xi;Y;)arecallednats (naturalunits).(Thestandard abbreviation forlog..isIn.)Since J Ina=In21082a=0.693151082 a theinformation measured innatsisequaltoIn2timestheinformation measured inbits. Whentherandom variables X.andYarestatistically independent, CHAPTER 3,SOURCE CODINGlIS P(XiIy)=P(Xi)and,hence,[(xi:.v;l=O.Ontheotherhand.whenthe occurrence oftheeventY=.viuniquely determines theoccurrence oftheevent )(=.ti•theconditional probability inthenumerator of(3-2-1)isunityand. hence. I [(Xi:Y/)=log-()=-logP(Xi)PXi(3-2-2) But(3-2-2)isjusttheinformation oftheeventX=Xi'Forthisreason.itis calledtheself-information oftheeventX=Xianditisdenoted as 1 [(Xi)=logP(Xi)=-logP(x;} (3-2-3) Wenotethatahigh-probability eventconveys lessinformation thana low-probability event.Infact.ifthereisonlyasingleeventXwithprobability P(x)=Ithen[(x)=O.Todemonstrate furtherthatthelogarithmic measureof information content istheappropriate onefordigitalcommunications. letus consider thefollowing example. Example 3-Z~1 Suppose wehaveadiscreteinformation sourcethatemitsabinarydigit, either0orI.withequalprobability everyr,seconds. Theinformation contentofeachoutputfromsourceis [(Xi)=-log,P(Xi). Xi=O.I =-log, ~=Ibit Nowsuppose thatsuccessive outputsfromthesourcearestatistically independent. i.e..thesourceismemoryless. Letusconsider ablockofk binarydigitsfromthesourcethatoccursinatimeintervalkr,.Thereare M=2kpossible k-bitblocks.eachofwhichisequally probable with probability 11M=2-k.Theself-information ofak-bitblockis [(xi)=-log22-'=kbits emilledinatimeintervalkr,.Thusthelogarithmic measure ofinformation contentpossesses thedesiredadditivity property whenanumberofsource outputsisconsidered asablock. Nowletusreturntothedefinition ofmutualinformation givenin(3-2-1) andmultiply thenumerator anddenominator oftheratioofprobabilities by P(yJSince ~ _P(XiIy;)P(Yj)_P(Xi•.vi)P(YiIXi) P(Xi)-P(x,)P(y) -P(x,)P(Yi) -Ply;) ..DIGITAL COMMUNICATiONS weconclude lhal (3-2-4) Therefore theinformation provided bytheoccurrence oflheevenlY=YI abouttheeventX=Xiisidentical totheinformation provided bythe occurrence oftheeventX=XiaboutlheeventY=YI' Euaple3-1-% Suppose thatXandYarebinary-valued {O,I}random variables that represent theinputandoutputofabinary-input, binary-output channel. Theinputsymbolsareequallylikelyandtheoutputsymbolsdependonthe inputaccording totheconditional probabilities P(Y=0IX=0)=I -Po P(Y=IIX=0)=Po P(Y=IIX=I)=I -Pt P(Y=OIX=I)=Pt Letusdetermine themutualinformation abouttheoccurrence oftheevents X=0andX=I,giventhatY=O. Fromtheprobabilities givenabove,weobtain P(y=0)=P(Y= 0IX=O)P(X=0)+P(Y=0IX=I)P(X=I) =!(l-PO+PI) P(y=I)=P(Y =Ilx=O)P(X=0)+P(Y=Ijx=I)P(X=I) =!(l-p,+po) Then,lhemutualinformation abouttheoccurrence oftheeventX=O. giventhatY=0isobserved, is I(x')=1(0'0)= IP(Y= 0IX=0)=102(1-Po) loYI ,~P(Y=O) ~1-Po+PI Similarly, givenlhatY=0isobserved, themutualinformalion aboutthe occurrence oftheeventX=Iis (3-2-5)CHAPTER s.SOURCE COOING87 LetUSconsider somespecialcases:First,ifPo=PI=0,thechanneliscalled noiseless and 1(0;0)=log,2= 1bit Hence,theoutputspecifies theinputwithcertainty. Ontheotherhand,if Po=PI=tthechannelisuselessbecause /(0:0)=log,1 = 0 However, ifPo=PI=tthen /(0;0)=log2~=0.587 1(0;1)=log2!=-1bit .Inaddition tothedefinition ofmutualinformation andself-information, itis usefultodefinetheconditional self-information as 1 I(xiIYj)=logP(XiIYj)=-logP(x,IYj) Then,bycombining (3-2-1),(3-2-3),and(3-2-5),weobtaintherelationship (3-2-6) Weinterpret I(xiIYj)astheself-information abouttheeventX=Xiafter havingobserved theeventY=Yj'SincebothI(xi);"0andI(Xi1Yj);;'0,it followsthatI(Xi;Yj)<0whenI(xiIYj)>l(x,).andl(x,;Yj)>0whenl(x,1y)< I(Xi)'Hence,themutualinformation between apairofeventscanbeeither positive,ornegative, orzero. 3-2-1Average MutualInformation andEntropy Havingdefinedthemutualinformation associated withthepairofevents (Xi'Yj),whicharepossible outcomes ofthetworandomvariables XandY,we canobtaintheaveragevalueofthemutualinformation bysimplyweighting I(xi;Yj)bytheprobability ofoccurrence ofthejointeventandsumming over allpossible jointevents.Thus,weobtain n m I(X;Y)=LLP(Xi.Yj)/(xj;Yj) i=I;"". (3-2-7) astheaverage mutualinformation between XandY.Weobserve that 88DIGITAL COMMUNICATIONS leX;Y)=0whenXandYarestatistically independent. Animportant characteristic oftheaveragemutualinformation isthatleX;y)..o(see Problem 3·4). Similarly, wedefinetheaverageself·information, denotedbyH(X).as n H(X)=LP(xi)l(xi) i=1 n = -LP(xi)logP(xi)i-'(3-2-8) WhenXrepresents thealphabetofpossibleoutputlettersfromasource,H(X) represents theaverageself-information persourceletter,anditiscalledthe entropyt ofthesource.Inthespecialcaseinwhichthelettersfromthesource areequallyprobable, P(Xi)=1/nforalli.and,hence, ..1 1H(X)=-L-log- ;'-"1nn =Iogn (3-2-9) Ingeneral,H(X)..;;logn(seeProblem3-5)foranygivensetofsourceletter probabilities. Inotherwords,theentropyofadiscretesourceisamaximum whentheoutputlettersareequallyprobable. Eum.ple3-1-3 Consider asourcethatemitsasequence ofstatistically independent letters, whereeachoutputletteriseither0withprobability qor1withprobability 1-q.Theentropyofthissourceis H(X)iEH(q)=-qlogq-(1-q)log(1-q) (3-2-10) (3-2-11)Thebinaryentropyfunction H(q)isillustrated inFig.3·2-1.Weobserve thatthemaximum valueoftheentropyfunction occursatq=!where H(!)=1. Theaverageconditional self-information iscalledtheevndition(l/ entropy andisdefined nm 1 H(XIY)=~~P(x;.Yj)logP(x iIYj) WeinterpretH(XIY)astheinformation oruncertainty inXafterYis tlbeterm.ntropyistakenfromstatislicaJ mechanics (thermodynamics). wheteafunction similarto(3-2-3)is«:ailed(thermodynamic) entropy. CHAPTER" SOURCE CODING19 HCq) 0.1 000.10.20.30.4O.S0.60.70.&0.91.09 P'r'obDlily q0.21.0 0.9 0.8 ~0.7 ~06 1.:0.5 f0.4 .il0.3 FIGURE 3-1·1Binaryentropyfunction. observed. Bycombining (3-2-7), (3-2-8), and(3-2-11) weobtainthe relationship I(X;Y)=H(X)-H(XIY) (3-2-12) SinceI(X;Y)...0,itfollowsthatH(X)'"H(XIY),withequalityifand onlyifXandYarestatistically independe!1t. IfweinterpretH(XIY)asthe averageamountof(conditional self-inforination) uncertainty inXafterwe observe Y,andH(X)astheaverageamountofuncertainty (self-informalion) priortotheobservation, thenI(X;Y)istheaverageamountof(mutual information) uncertainty provided aboutthesetXbytheobservation oftheset Y.SinceH(X)...H(XIYl,itisclearthatconditioning ontheobservation Y doesnotincreasetheentropy. Eumple3-Z-4 Letusevaluate theH(XIY)andI(X;Y)forthebinary-input, binary­ outputchanneltreatedpreviously inExample 3-2-2forthecasewhere Po=PI=p.Lettheprobabilities oftheinputsymbols beP(X=0)=qand P(X=I)=1-q.Thentheentropyis H(X)-H(q)=-qJogq-(1-q)Jog(1-q) whereH(q)isthebinaryentropyfunction andtheconditional CRtropy H(XIY)isdefinedby(3-2-11). AplotofH(XIY)asafunctionofqwith 90DIGITAL COMMUNICATIONS p=o.s 1.0 t:0.8 i ~06 iI0.4 ..":;;0.2 FIGURE l-1-1Conditional entropyforbinary-input, binary­ outputsymmetric channel.°0~-~0.::-2 --,o;!c.4:----:o~.6--;;0.';;-8--!'1.0,,- q q~probabilily ofsymbolX=0 pasaparameter isshowninFig.3·2-2.Theaveragemutualinformation I(X;y)isplottedinFig.3·2·3, Asinthepreceding' example, whentheconditional entropyH(XIY)is viewedintermsofachannel whoseinputisXandwhoseoutputisY, H(XIY)iscalledtheequivocation andisinterpreted astheamountofaverage uncertainty remaining inXafterobservation ofY. /(x,n FlGUU l-1-JA'leragemulUllIinformation forbinary·input, biDary-outpul sytllllletrie cbaMet.1.0p=O p-O.l p=0.2 p.O.3 p=O.5 0.20.40.60.3 ,-probobiIity ofsymbol"=01.0q CHA.PTER J:SOURCE('ODlN(j 91 Theresultsgivenabovecanbegeneralized tomorethantworandom variables. Inparticular, suppose wehaveablockofkrandom variables X,X2'..X"withjointprobability P(X,X2'" x.)'"P(X,=X,.X2= X2.'..•X.=x.).Then,theentropyfortheblockisdefinedas Iff".' "i H(X,X,' ..X.)=-LL'"LP(x;,x,,'..Xl.)10gP(Xj,XI,'•.X,.)(3-2-13) jf=Ih"'l ;,=1 Sincethejointprobability P(x,X2...x.)canbefactored as P(X,X,...x.)=P(x,1P(x2!X,)P(X,IX,X2)...P(x.IX,X,'..X.,) (3-2-14) itfollowsthat H(X,X,X,' ..X.)=H(X,)+H(X,IX,)+H(X31X,X,) +...+H(X.IX,...X.,) • =LH(XiIX,X"" Xi-,) ;-=I(3-2-15) Byapplying theresultH(X);;.H(XIY).where X=X",andY= X,X,'"X"".in(3-2-15)weobtain •H(X,X," .X.)";;LH(X",) HI--J(3-2-16) withequality ifandonlyiftherandom variables X,.X2•••••X.arc statistically independent. 3-2·2Infonnation Measures forContinuous Random Variables Thedefinition ofmutualinformation givenabovefordiscreterandomvariables maybeextended inastraightforward mannertocontinuous randomvariables. Inparticular, ifXandYarerandomvariables withjointpdfp(x.y)and marginal pdfsp(x)andp(y).theaveragemutualinformation between Xand Yisdefinedas IxIX p(Ylxw(x)I(X;y)= p(x)p(yjx)log dxdy -x-x p(x)p(y)(3-2-17) Although thedefinition oftheaveragemutualinformation carriesoverto 91DIGITAL COMMUNICA nONS continuous randomvariables, theconceptofself-information doesnot.The problem isthatacontinuous randomvariable requiresaninfinitenumberof binarydigitstorepresent itexactly.Hence,itsself-information isinfiniteand, therefore, itsentropyisalsoinfinite.Nevertheless, weshalldefineaquantity thatwecallthedifferential entropyofthecontinuous randomvariableXas H(X)= -(P(X)logP(X)dx (3-2-18) Weemphasize thatthisquantity doesnOIhavethephysical meaning of self-information, although itmayappeartobeanaturalextension ofthe definition ofentropyforadiscreterandomvariable(seeProblem 3-6). Bydefiningtheaverageconditional entropyofXgivenYas H(XIY)=-L(p(x,)I) logp(xIy)dxdy theaveragemutualinformation maybeexpressed as J(X;Y)=H(X)-H(XIY) or.alternatively. as I(X;Y)=H(Y)-H(YIX)(3-2-19) Insome.casesofpractical interest,therandomvariableXisdiscreteandY IScontinuous. Tobespecific, suppose thatXhas,pcssible outcomes Xi' j=1,2,...,n,andYisdescribed byitsmarginal pdfp(y).WhenXandYare statistically dependent, wemayexpressp(y)as n p()I)=~p(yIXi)P(Xi) i-I Themutualinformation provided abouttheeventX=Xibytheoccurrence of theeventY=)Iis J(x,;y)=logP(YIXi)P(X;} p(y)P(X i) =Iog~p(y) Then,theaveragemutualinformation between XandYis J(X;Y)=i[p(yIXi)P(Xi)10g~( X)i)dy i-I'-z P Y(3-2-20) (3-2-21) (3-2-22)CHAI'rER), SO[;Rfl CODING 93 Example 3-2·5 Suppose thatXisadiscreterandomvariablewithtwoequallyprobable outcomes xI=AandX2=-A.Lettheconditional pdfsp(yIx,),i=I.2,be gaussianwithmean Xiandvariance u2•Thatis, p(yIA)=_1_e-IY"A,2/2u'v'2Iru p(yI-A)=_1_e-IY+A"I2U2v'2Iru Theaveragemutualinformation obtained from(3-2-21)becomes(3-2-22) I(X;Y)=~[[P(yIA)logP~l:) +p(yI-A)IOgP(~L~A)] dy (3-2-23) p(y)=Hp(yIA)+p(yI-A)I (3-2-24) InChapter7,itwillbeshownthattheaveragemutualinformation I(X;Y) givenby(3-2-23)represents thechannelcapacityofabinary-input additive whitegaussiannoisechannel. 3-3CODING FORDISCRETE SOURCES InSection3-2weintroduced ameasurefurtheinformation contentassociated withadiscreterandomvariableX.WhenXistheoutputofadiscretesource, theentropyH(X)ofthesourcerepresents theaverageamountofinformation emittedbythesource.Inthissection,wel;9nsidertheprocessorencoding the outputofasource,i.e.,theprocessofrepresenting thesourceoutputbya sequence ofbinarydigits.Ameasure oftheefficiency ofasource-encoding methodcanbeobtained bycomparing theaveragenumberofbinarydigitsper outputletterfromthesourcetotheentropyH(X). Theencoding ofadiscretesourcehavingafinitealphabet sizemayappear, atfirstglance,tobearelatively simpleproblem. However, thisistrueonly whenthesourceismemoryless, i.e.,whensuccessive symbolsfromthesource arestatistically independent andeachsymbolisencoded separately. The discretememoryless source(DMS)isbyfarthesimplest modelthatcanbe devisedforaphysicalsource.Fewphysicalsources,however, closelyfitthis idealized mathematical model.Forexample, successive outputlettersfroma machine printingEnglishtextareexpected tobestatistically dependent. On theotherhand,ifthemachineoutputisacomputer programcodedinFortran, thesequence ofoutputlettersisexpected toexhibitamuchsmaller dependence. Inanycase,weshalldemonstrate thatitisalwaysmoreefficient toencodeblocksofsymbolinsteadofencoding eachsymbolseparately. By makingtheblocksizesufficiently large,theaveragenumberofbinarydigits 94DIGITAL COMMUNICATIONS peroutputletterfromthesourcecanbemadearbitrarily closetotheentropy ofthesource. 3-3-1CodingforDiscrete Memoryless Sources Suppose thataOMSproduces anoutputleiterorsymboleveryT,seconds. Eachsymbolisselected fromafinitealphabet ofsymbols Xi'i=1,2,...,L, occurring withprobabilities P(Xi),i=1,2,...,L.TheentropyoftheOMSin bitspersourcesymbolis L H(X)=-LP(Xi)log2P(Xi)..log;,L ;=1(3-3-1) whereequality holdswhenthesymbols areequallyprobable. Theaverage numberofbitspersourcesymbolisH(X)andthesourcerateinbits/sis definedasH(X)Ir,. Fixed-Length CodeWords Firstweconsider ablockencoding scheme thatassignsauniqueselofRbinarydigitstoeachsymbol. SincethereareL possible symbols, thenumberofbinarydigits.persymbolrequired forunique encoding whenLisapowerof2is R=log;,L (3-3-2) and,whenLisnotapowerof2,itis R=LIog2LJ+1 (3-3-3) whereLddenolesthelargestintegerlessthanx.ThecoderateRinbitsper symbolisnowRand,sinceH(X) ~log;,L,itfollowsthatR..H(X). Theefficiency oftheencoding forthe·OMSisdefinedastheratioH(X)IR. Weobserve thatwhenLisapowerof2andthesourcelettersareequally probable, R=H(X).Hence,afixed-length codeofRbitspersymbolattains 100%efficiency. However, ifLisnotapowerof2butthesourcesymbols are stillequallyprobable, RdiffersfromH(X)byatmost1bitper·symbol. When log2L»1,theefficiency ofthisencoding schemeishigh.Ontheotherhand, whenLissmall,theefficiency ofthefixed-length codecanbeincreased by encoding asequence ofJsymbols atatime.Toaccomplish thedesired encoding, werequireUuniquecodewords.Byusingsequences ofNbinary digits,wecanaccommodate 2Npossible codewords.Nmustbeselected such that N"Jlog;,L Hence,theminimum integervalueofNrequired is N=LJlog2LJ+1 (3-3-4) NowtheaveragenumberofbitspersourcesymbolisNIJ=R,and,thus,the CHAPTER]: SOU~CE ('ODING 95 inefficiency hasbeenreduced byapproximately afactorofIIJrelativetothe symbol-by-symbol encoding described above.BymakingJsufficiently large. theefficiency oftheencoding procedure, measured bytheratioJH(X)/N, can bemadeasclosetounityasdesired. Theencoding methods described aboveintroduce nodistortion sincethe encoding ofsourcesymbolsorblocksofsymbols intocodewordsisunique. Thistypeofencoding iscallednoiseless. Now,suppose wealtempt toreducethecoderateRbyrelaxing the condition thattheencoding processbe.unique.Forexample, suppose thatonly afractionoftheeblocksofsymbols isencoded uniquely. Tobespecific, let usselectthe2N-Imostprobable J-symbol blocksandencodeeachofthem uniquely, whiletheremainingL'-(2N-1)J-symbol blocksarerepresented bythesingleremaining codeword.Thisprocedure resultsinadecoding failure or(distortion) probability oferroreverytimealowprobability blockis mapped intothissinglecodeword.LetPedenotethisprobability oferror. Basedonthisblockencoding procedure, Shannon (1948a)proved the following sourcecodingtheorem. SourceCodingTheorem I LetXbetheensemble ofleitersfromaDMSwithfiniteentropyH(X). BlocksofJsymbols fromthesourceareencoded intocodewordsoflength Nfromabinaryalphabet. Foranye>0,theprobability Peofablock decoding failurecanbemadearbitrarily smallif N R=-~H(X) +e J andJissufficiently large.Conversely, if R,,;;H(X)-e thenp,.becomes arbitrarily closeto1asJismadesufficiently large.(3-3-5) (3-3-6) Fromthistheorem, weobserve thattheaveragenumberofbitspersymbol required toencodetheoutputofaDMSwitharbitrarily smallprobability of decoding failureislowerbounded bythesourceentropyH(X).Ontheother hand,ifR<H(X),thedecoding failurerateapproaches 100%asJis arbitrarily increased. VlIriable-Length CodeWords Whenthesourcesymbols arenotequally probable, amoreefficient encoding met~od istousevariable-length code !16DiGITAL COMMUNICATIONS TABLE 30301VARIABLE-LENGTH CODES LetIer Pta,) CodeICodenCodeDI a,1 1 0 02 a,100 10 614 aJ1 01 110 011•a,1 10 111 111• words.Anexample ofsuchencoding istheMorsecode,whichdatesbackto thenineteenth century. IntheMorsecode,thelettersthatoccurmore .frequently areassignedshortcodewordsandthosethatoccurinfrequently are assigned longcodewords.FolIowing thisgeneralphilosophy, wemayusethe probabilities ofoccurrence ofthedifferent sourcelettersintheselection ofthe codewords.Theproblem istodeviseamethodforselecting andassign­ ingthecodewordstosourceletters.Thistypeofencoding iscalIedentropy coding. Forexample. suppose thataOMSwithoutputlettersat.a,.a,.a4and corresponding probabilities Pia,)=!.P(a2):l.andP(a3)=P(04)=Ais encoded asshowninTable3-3-1.CodeIisavariable-length codethathasa basicflaw.Toseetheflaw,suppose wearepresented withthesequence 001001...,Clearly,thefirstsymbolcorresponding to00isa,.However, the nextfourbitsareambiguous (notuniquely decodable). Theymaybedecoded eitherasa.a3oraso,a2a,.Perhaps, theambiguity canberesolved bywailing foradditional bits,butsuchadecoding delayishighlyundesirable. Weshall onlyconsider codesthataredecodable instantaneously. thatis,withoutany decoding delay. CodeIIinTable3-3-1isuniquely decodable andinstantaneously decodable. Itisconvenient torepresent thecodewordsinthiscodegraphically asterminal nodesofatree,asshowninFig.3-3-1.Weobservethatthedigit0indicates the endofacodewordforthefirstthreecodewords.Thischaracteristic plusthe factthatnocodewordislongerthanthreebinarydigitsmakesthiscode instantaneously decodable. Notethatnocodewordinthiscodeisaprefixof anyothercodeword.Ingeneral,theprefixcondition requires thatforagiven codewordC.oflengthkhavingelements (bl>b,•...•bd.thereisnoother codewordoflength1<kwithelements (b,.0,•...•bl)for1"i,I"'"k-1.In flGURE 30301CodetreeforcodeIIinTable3-301.".• CHAPTER), SOURCE CODING 'Y7 •• ., •• FlGURE .J..J.2CodeIreeforcode11IInTable3-3-1. otherwords,thereisnocodewordoflengthI<kthatisidentical tothefirstI binarydigitsofanothercodewordoflengthk>/.Thisproperty makesthe codewordsinstantaneously decodable. CodeIIIgiveninTable3-3-1hasthetreestructure showninFig.3-3-2.We notethatinthiscasethecodeisuniquely decodable butnotinstantaneously decodable. Clearly,thiscodedoesnotsatisfytheprefixcondition. Ourmainobjective istodeviseasystematic predure forconstructing uniquely decodable variable-length codesthatareefficientinthesensethatthe averagenumberofbitspersourceletter,definedasthequantity L R=2:n.P(at) k~'(3-3-7) isminimized. Theconditions fortheexistence ofacodethatsatisfiesthe.prefix condition aregivenbytheKraftinequality. KraftInequality Anecessary andsufficient condition fortheexistence ofa binarycodewithcodewordshavinglengthsn,~n2~••.~nLthatsatisfythe prefixcondition is (3-3-8) First.weprovethat(3-3-8)isasufficient condition fortheexistence ofa codethatsatisfiestheprefixcondition. Toconstruct suchacode,webeginwith afullbinarytreeofordern=nLthathas2"terminal nodes andtwonodesof orderkstemming fromeachnodeoforderk-I,foreachk,1~k~n.Letus selectanynodeofordern,asthefirstcodewordC,.Thischoiceeliminates 2,,-n,terminal nodes(orthefractionrn,ofthe2"terminal nodes).Fromthe remaining available npdesofordern2'weselectonenodeforthesecondcode wordC2•Thischoiceeliminates 2"-n,terminal nodes(orthefractionz-n,of the2n.terminal nodes).Thisprocesscontinues untilthelastcodewordis assigned atterminal noden=n/..Since,atthenodeoforderj<L,the fractionofthenumberofterminal nodeseliminated is L•2:2-n,<2:2-n,~1 k-l k=1 98DIGITA.L COMMUNJ('ATIONS c o 0, ~ FIGURE 3-3-3Construction ofabinarytreecodeembedded inafulltree. thereisalwaysanodeoforderk>javailable tobeassigned tothenextcode word.Thus.wehaveconstructed acodetreethatisembedded inthefulltree of2"nodesasillustrated inFig.3-3-3.foratreehaving16terminal nodesand asourceoutputconsisting ofliveletterswithn.=I,112=2.n,=3,and II.=n,=4. Toprovethat(3-3-8)isanecessary condition, weobservethatinthecode treeofordern=Ill.•thenumberofterminal nodeseliminated fromthetotal numberof2"terminal nodesis LL2,,-n':!:f.2n k=! Hence, andtheproofof(3-3-8)iscomplete. TheKraftinequality maybeusedtoprovethefollowing (noiseless) source codingtheorem, whichappliestocodesthatsatisfytheprefixcondition. SOIll'CeCodingTheorem II LetXbetheensemble oflettersfromaDMSwithfiniteentropyH(X),and outputletters Xh.1""k""L.withcorresponding probabilities ofoccurrence Ph.1""k""L.Itispossible toconstruct acodethatsatisfies theprefix conditioll andhasanaveragelengthRthatsatisfiestheinequalities H(X)..R<H(X)+1 (3-3-9) CHAPTER 3'SOURCE CODING99 Toestablish thelowerboundin(3-3-9),wenotethatforcodewordsthat havelengthn.,1..k..L,thedifference H(X)-Rmaybeexpressed as (3-3-10) Useoftheinequality Inx..x-1in(3-3-10)yields L(2-n.)H(X)-R..(log2e)2:p,--1 '=1p, ..(lOg2e)(~l2-n.-])..0 wherethelastinequality followsfromtheKraftinequality. Equality holdsif andonlyifp,=rn•for1..k..L. Theupperboundin(3-3-9)maybeestablished undertheconstraint thatn" 1..k..L,areintegers, byselecting the{ndsuchthat2-n...p,<2-n,+I.Butif thetermsp, ~2-n.aresummedover1..k..L,weobtaintheKraftinequality, forwhichwehavedemonstrated thatthereexistsacodethatsatisfiestheprefix condition. Ontheotherhand,ifwetakethelogarithm ofP.<2--n,+I,we obtain logp,<-n.+1 or,equivalently, (3-3-11) Ifwemultiply bothsidesof(3-3-11)byp,andsumoverI..k"L,weobtain thedesiredupperboundgivenin(3-3-9).Thiscompletes theproofof(3-3-9). Wehavenowestablished thatvariablelengthcodesthatsatisfytheprefix condition areefficientsourcecodesforanyDMSwithsourcesymbolsthatare notequallyprobable. Letusnowdescribe analgorithm forconstructing such codes. 8........ CodbagAlgorithm Huffman (1952)devised avariable-length encoding algorithm, basedonthesourceletterprobabilities P(xi),j= 1,2,...,L.Thisalgorithm isoptimum inthesensethattheaveragenumberof binarydigitsrequiredtorepresent thesourcesymbolsisaminimum, subjectto theconstraint thatthecodewordssatisfytheprefixcondition, asdefined above,whichallowsthereceived sequence tobeuniquely andinstantaneously decodable. Weillustrate thisencoding algorithm bymeansoftwoexamples. 100 DIGITAL l'O..MliNICATIONS IUS O..lll 0.10 n.IO 0.()4 O.OI.l; 0.01.15tI Itl050 I, II II().J~ I IIH.15, 0I(l,U5IO.OJ'I I Lentr Probabilily Self-information Clltk .11 0.35 1.5146 00 .11 0..10 t7J70 01 "0.20 ~.3119 10 '.0.10 33219 110 "0.0< .J.6439 1110 "0.005 7.1>.139 11110 tlGURE J.l-4Anexampleofvariable-length-sOurce "0.005 7.1>.139 IIIII encoding foraDMS. H(X)=2.11 R=2.21 Eumple 3-3-1 Consider aOMSwithsevenpossible symbols x"Xl.'..•X7havingthe probabilities ofoccurrence illustrated inFig.3·3-4.Wehaveordered the sourcesymbols indecreasing orderoftheprobabilities, Le..P(x,)>P(XI)> ...>P(X7)'Webegintheencoding processwiththetwoleastprobable symbols X.andX7'Thesetwosymbolsaretiedtogether asshowninFig. 3-3-4.withtheupperbranchassigned a 0andthelowerbranchassigned a1. Theprobabilities ofthesetwobranches areaddedtogether atthenode wherethetwobranches meettoyieldtheprobability 0.01.Nowwehavethe sourcesymbolsx,•...•X5plusanewsymbol,sayx;'.obtained bycombining XI>andX7'Thenextstepistojointhetwoleastprobable symbolsfromthe setx..X2'XJ'X4.X,.x;'.ThesearelOSand x~.whichhaveacombined probability of0.05.ThebranchfromXsisassigned a 0andthebranchfrom x;'isassigned a1.Thisprocedure continues until'weexhaustthesetof possiblesourceletters.Theresultisacode"tree withbranches thatcontain thedesiredcodewords.Thecodewordsare.obtained bybeginning atthe rightmost nodeinthetreeandproceeding to.theleft.The resulting code wordsarelistedinFig.3-3-4.Theaveragenumberofbinarydigitsper symbolforthiscodeisR=2.21bits/symbol. Theentropyofthesourceis 2.11bits/symbol. Wemaketheobservation thatthecodeisnotnecessarily unique.For example, atthenexttothelaststepintheencoding procedure, wehaveatie betweenx,andX),sincethesesymbolsareequallyprobable. Atthispoint.we chosetopairx.withXl'Analternative istopairX2withX).Ifwechoosethis CHAPTER): SOURCE CODING101 o 0.35 0.30 0.20 0.10 0.04 0.005 0.005 FIGURE 3-3-5Analternative codefortheOMSin Example 3-3-1.0 0 U.6S I 0 0.35 I 0 0.15 I0.05I 0 0.01I1 I Lctu:r Code x, 0 Xz10 x) 110 x, 1110 Xs11110 ... 111110 "'1 JIIJJJ ii:2.21 pairing,theresulting codeisillustrated inFig.3-3-5.Theaverage number of bitspersourcesymbolforthiscodeisalso2.21.Hence,theresulting codesare equallyefficient. Secondly, theassignment ofa 0totheupperbranchanda 1 tothelower(lessprobable) branchisarbitrary. Wemaysimplyreversethe assignment ofa 0arid1andstillobtainanefficientcodesatisfying theprefix condition. Eu...ple3-3-2 Asasecondexample, letusdetermine theHuffman codefortheoutputofa DMSillustrated inFig.3-3-6.TheentropyorthissourceisH(X)= 2.63bits/symbol. TheHuffman codeasillustrated inFig.3-3-6hasan averagelengthofR=2.70bits/symbol. Hence,itsefficiency is0.97. FIGURE 3-3-6Huffman cooeforExample 3-3-2. 0.36 0 0.6)0leiter Code 0.141°0.27rlx, 000./.1x, 0'0 0.121°0.22x., 0" 0 x, 100 0.10x, 1010.37 0.091°'.110 0.15 "1'10 0.040I '.1111I0.06IR=2.70 0.02 /lIX):2.63 102 DIGITAL COMMUNICATIONS Thevariable-length encoding (Huffman) algorithm described intheabove examples generates aprefixcodehavinganRthatsatisfies(3-3-9).However, insteadofencoding onasymbol-by-symbol basis,amoreefficientprocedure is toencodeblocksofJsymbolsatatime.Insuchacase,theboundsin(3-3-9)of sourcecodingtheoremIIbecome JH(X) oe;;RI<JH(X)+1, (3-3-12) sincetheentropyofaJ-symbol blockfromaOMSisJH(X),andRJisthe averagenumberofbitsperJ-symbol blocks.Ifwedivide(3-3-12)byJ,we obtain R 1H(X)oe;;-!..<H(X)+-1 J(3-3-13) whereRJ/l-Ristheaveragenumberofbitspersourcesymbol.HenceRcan bemadeasclosetoH(X)asdesiredbyselectingJsufficiently large. Eumple 3-3-3 TheoutputofaOMSconsistsofletters XI>Xz,andXJwithprobabilities 0.45, 0.35,and0.20,respectively. Theentropy ofthissourceisH(X)= 1.518bits/symbol. TheHuffman codeforthissource,giveninTable3-3-2, requiresRI=1.55bits/symbol andresultsinanefficiency of97.9%.Ifpairs ofsymbolsareencodedbymeansoftheHuffman algorithm, theresulting codeisasgiveninTable3-3-3.Theentropyofthesourceoutputforpairsof leitersis2H(X)=3.036bits/symbol pair.Ontheotherhand,theHuffman coderequiresRz=3.0675bits/symbol pair.Thus,theefficiency ofthe encoding increases to2H(X)/R z=0.9'XJor,equivalently, to99.0%. Insummary, wehavedemonstrated thatefficientencoding foraOMSmay bedoneonasymbol-by-symbol basisusingavariable-length codebasedon TABLE l-:J.lHUFFMAN CODEFOREXAMPLE 3-3-3 Letter.............,SeIf~ X, 0.45 1.156 X, 0.35 1.520 X, 0.20 2.330Code I 00 01 H(X)=1.518bits/letter RI=1.55bils/letter Efficiency =97.9% CHAPTER J,SOURCE CODING103 TABLE J.J.3HUFFMAN CODEFORENCODING PAIRSOFLETIERS leiter..... ProbtlblIity Code X,X, XlX2 XlII X2X2 X1.l'":\ XJXJ X2X., X.,X2 .t.,X)0.2025 2312 0.1575 2.676 0.1575 2.676 0.1225 3.039 0.09 3.486 0.09 3.486 0.07 3.850 0.07 3.850 OJ)4 4.660 2H(X): 3.036bits/letter pair R,=3.0675bits/letter pairIR,:1.534bits/letter Efficiency: 99.0%10 001 010 011 III ססoo 0001 1100 IWI (3-3-14)theHuffman algorithm. Furthermore, theefficiency oftheencoding procedure isincreased byencoding blocksofJsymbols alatime.Thus,theoutputofa OMSwithentropyH(X)maybeencoded byavariable-length codewithan averagenumberofbitspersourceletterthatapproaches H(X)ascloselyas desired. 3-3-2Discrete Stationary Sources Intheprevious section,wedescribed theefficientencoding oftheoutputofa OMS.Inthissection,weconsider discretesourcesforwhichthesequence of outputlettersisstatistically dependent. Welimitourtreatment tosourcesthat arestatistically stationary. Letusevaluate theentropyofanysequence oflettersfromastationary source.Fromthedefinition in(3-2-13) andtheresultgivenin(3-2-15), the entropyofablockofrandomvariablesXIX"..X.is •H(X,X,··· Xd=2:H(X,'X,X,,"Xi'.) i'"I whereH(XiIX,X,'" Xi-I)istheconditional entropyoftheilhsymbolfrom thesourcegiventhepreviousi-Isymbols. Theentropyperleiterforthe k-symbol blockisdefinedas 1H.(X)=kH(X,X,·..X.) (3-3-15) (3-3-16)Wedefinetheinformation contentofastationary sourceastheentropyper letterin(3-3-15)inthelimitask~"'.Thatis, Hx(X)=limH.(X)=lim-k1H(XIX2'..X.) k_x k_:IC' 104 DIGITAL COMMLJNICATIONS TheeXistence ofthislimitisestablished below. Asanalternative. wemaydefinetheentropyperletterfromthesourcein terms()ftheconditional entropyH(X.IX,X,'" X._I)inthelimitask approaches infinity.Fortunately. thislimitalsoexistsandisidentical tothe limitin(3-3-16). Thatis. Hx(X)=limH(X,IXIX,'" X._,) k-..x(3-3-17) Thisresultisalsoestablished below.Ourdevelopment followstheapproach in Gallager (1968). First,weshowthat (3-3-18) fork;;.2.Fromourprevious resultthatconditioning onarandom variable cannotincrease entropy, wehave (3-3-19) Fromthestationarity ofthesource.wehave H(x.1X,X,'" X._,)=H(X._,IX,X,'..X._,)(3-3-20) Hence, (3-3·18) follows immediately. Thisresultdemonstrates that H(X.IX,X"..X._,)isanonincreasing sequence ink. Second,wehavetheresult H.(X);;'H(X.IX,X,'" X._,) (3-3-21) whichfollowsimmediately from(3-3-14)and(3-3-15)andthefactthatthelast terminthesumof(3-3-14) isalowerboundoneachoftheotherk- 1terms. Third,fromthedefinition ofH.(X).wemaywrite 1H.(X)=k:lH(X,X, ...X._,)+H(X.IXI'..X.-,)J 1="k[(k-l)H._,(X) +H(X.IX,...X._,») k-1 1';;-k-H.-I(X) +IeH.(X) whichreducesto H.(X).;;H._I(X) (3-3-22) Hence,H.(X)isanonincreasing sequence ink. SinceH.(X)andtheconditional entropyH(X.-jX,...X._,)areboth CHAI'H,R 3:SOURCE COOING lOS nonnegative andnonincreasing withk,bothlimitsmustexist.Theirlimiting formscanbeestablished byusing(3-3-14)and(3-3-15)toexpressHbj(X) as 1H"j(X)=-k .H(X,X 2···X._,)+J 1+k+j[H(x.1X""X.-I )+H(Xb,IX,'"X.) +...+H(X.+)IX,···Xbj-,)] Sincetheconditional entropy isnonincreasing. thefirstterminthesquare brackets servesasanupperboundontheotherterms.Hence, 1 j+1Hk+j(X)";-k ,H(X,X 2•••X._,)+-k.H(X.\X,X2'"X._,)+J +J (3-3-23) Forafixedk,thelimitof(3-3-23)asj-.00yields H~(X)";H(X.IX,X2••'X._,) But(3-3-24)isvalidforallk;hence,itisvalidfork-"';.Therefore, H~(X)"; limH(x.1X,X2"•X._I) ._x Ontheotherhand,from(3-3·21). weobtaininthelimitask_00, H~(X);;' limH(X.IX,X2" •X._,) ._x(3-3-24) (3-3-25) (3-3-26) whichestablishes (3-3-17). Nowsupposewehaveadiscretestationary source'thatemitsJletterswith HJ(X)astheentropyperletter.Wecanencodethesequence ofJletterswitha variable-length Huffman codethatsatisfiestheprefiXcondition byfollowing theprocedure described intheprevious section.Theresulting codehasan averagenumberofbitsfortheJ-Ietterblockthatsatisfiesthecondition H(X,'"XJ),.;R/<H(X,·· ·XJ)+1 (3-3-27) BydividiEg e.!1chtermof(3-3-27)byJ,weobtainthebou.son~heaverage numberR=RJIJofbitspersourceleiteras (3-3-28) Byincreasing theblocksizeJ,wecanapproach HAX)arbitrarily closely,and inthelimitasJ_00,Rsatisfies (3-3-29) lOtiDlii/fAl (D.'.fMl:~HXr)(JNS where fapproaches zeroas1/1.Thus,efficientencoding ofstationary sources isaccomplished byencoding largeblocksofsymbols intocodewords.We shouldemphasize, however, thatthedesignoftheHuffman coderequires knowledge ofthejointpdffortheJ-symbolblocks. leLempel-Ziv Algorithm Fromourpreceding discussion, wehaveobserved thattheHuffman coding algorithm yieldsoptimalsourcecodesinthesensethatthecodewordssatisfy theprefixcondition andtheaverageblocklengthisaminimum. Todesigna Huffman codeforaDMS,weneedtoknowtheprobabilities ofoccurrence of allthesourceletters.Inthecaseofadiscretesourcewithmemory, wemust knowthejointprobabilities ofblocksoflengthn;;'2.However, inpractice, thestatistics ofasourceoutputareoftenunknown. Inprinciple, itispossible toestimate theprobabilities ofthediscretesourceoutputbysimplyobserving alonginformation sequence emitted bythesourceandobtaining the probabilities empirically. Exceptfortheestimation ofthemarginal prob­ abilities{P.},corresponding tothefrequency ofoccurrence oftheindividual sourceoutputletters,thecomputational complexity involved inestimating jointprobabilities isextremely high.Consequently. theapplication ofthe Huffman codingmethodto'sourcecodingformanyrealsourceswithmemory isgenerally impractical. IncontrasttotheHuffman codingalgorithm, theLempeI-Ziv sourcecoding algorithm isdesigned tobeindependent ofthesourcestatistics. Hence,the Lempel-Ziv algorithm belongs totheclassofuniversal sourcecoding algorithms. Itisavariable-to-fixed-length algorithm, wheretheencoding is performed asdescribed below. IntheLempel-Ziv algorithm, thesequence attheoutputofthediscrete sourceisparsedintovariable-length blocks,whicharecalledphrases. Anew phraseisintroduced everytimeablockoflettersfromthesourcediffersfrom someprevious phraseinthelastletter.Thephrasesarelistedinadictionary, whichstoresthelocationoftheexistingphrases.Inencoding anewphrase,we simplyspecifythelocationoftheexistingphraseinthedictionary andappend thenewletter. Asanexample, consider thebinarysequence 10101101001001110101ססoo11001110101100011011 Parsingthesequence asdescribed aboveproduces thefollowing phrases: I,0,10,II,01,00,100, 111, 010,1000,Oil,001,110, 101, 10001,1011 Weobservethateachphraseinthesequence isaconcatenation ofaprevious phrasewithanewoutputletterfromthesource.Toencodethephrases, we CHAPTER.l SOURCE rODlN()107 TABLE 3-3-4DICTIONARY FORLEMPEL-ZIV ALGORITHM Dktianory Dkd.....ry Code location contents word I 0001 1 ססoo1 2 0010 0 0ססoo 3 0011 10 00010 4 0100 11 00011 5 OWl 01 00101 f> OlIO 00 00100 7 0111 100 ~11O R 1000 111 01001 9 1001 OW 01010 10 1010 1000 01110 11 1011 Oil 01011 12 IllJ(] 001 01101 13 1101 110 0)000 14 IIlQ 101 00111 15 1111 10001 10101 16 lOll 11101 construct adictionary asshowninTable3·3·4.Thedictionary locations are numbered consecutively, beginning with1andcounting up,inthiscaseto16, whichisthenumber ofphrasesinthesequence. Thedifferent phrases corresponding toeachlocationarealsolisted,asshown.Thecodewords are determined bylistingthedictionary location(inbinaryform)oftheprevious phrasethatmatchesthenewphraseinallbutthelastlocation. Then,thenew outputletterisappended tothedictionary location oftheprevious phrase. Initially, thelocation ססooisusedtoencodeaphrasethathasnotappeared previously. Thesourcede!=oder forthecodeconstructs anidentical tableatthe receiving endofthecommunication systemanddecodesthereceived sequence accordingly. 11shouldbeobserved thatthetableencoded 44sourcebitsinto16code wordsoffivebitseach,resulting in80codedbits.Hence,thealgorithm provided nodatacompression atall.However, theinefficiency isduetothe factthatthesequence wehaveconsidered isveryshort.Asthesequence is increased inlength,theencoding procedure becomes moreefficientandresults inacompressed sequence attheoutputofthesource. Howdoweselecttheoveralllengthofthetable?Ingeneral,nomatterhow largethetableis,itwilleventually overflow. Tosolvetheoverflow problem, thesourceencoderandsourcedecodermustagreetoremovephrasesfromthe respective dictionaries thatarenotusefulandsubstitute newphrasesintheir place. 108 DIGITAL COMMUNICATIONS TheLempel-Ziv algorithm iswidelyusedinthecompression ofcomputer files.The"compress" and"uncompress" utilitiesundertheUNIX©operating systemandnumerous algorithms undertheMS-DOS operating systemare implementations ofvariousversionsofthisalgorithm. 3-4CODING FORANALOG SOURCES-OPTIMUM QUANTIZATION Asindicated inSection3-1,ananalogsourceemitsamessage waveform x(/) thatisasamplefunction ofastochastic processX(/).WhenX(/)isa bandlimited, stationary stochastic process, thesampling theorem allowsusto represent XU)byasequence ofuniformsamplestakenattheNyquistrate. Byapplying thesampling theorem; theoutputofananalogsourceis converted toanequivalent discrete-time sequence ofsamples. Thesamplesare thenquantized inamplitude andencoded. Onetypeofsimpleencoding isto represent eachdiscreteamplitude levelbyasequence ofbinarydigits.Hence, ifwehaveLlevels,weneedR=log2LbitspersampleifLisapowerof2,or R=Llog,LJ+1ifLisnotapowerof2.Ontheotherhand,ifthelevelsare notequallyprobable, andtheprobabilities oftheoutputlevelsareknown,we mayuseHuffman coding(alsocalleden/ropycoding)toimprove theefficiency oftheencoding process. Quantization oftheamplitudes ofthesampled signalresultsindata compression butitalsointroduces somedistortion ofthewaveform oralossof signalfidelity.Theminimization ofthisdistortion isconsidered inthissection. Manyoftheresultsgiveninthissectionapplydirectlytoadiscrete-time, continuous amplitude, memo!'yless gaussian source.Suchasourceservesasa goodmodelfortheresidual errorinanumberofsourcecodingmethods described inSection3-5. 3-4-1Rate-Distortion Function Letusbeginthediscussion ofsignalquantization byconsidering thedistortion introduced whenthesamplesfromtheinformation sourcearequantized toa fixednumberofbits.Bytheterm"distortion," wemeansomemeasure ofthe difference between theactualsourcesamples{xdandthecorresponding quantized values.i••whichwedenotebyd{x••.id.Forexample, acommonly useddistortion measure isthesquared-errordistortion. definedas d(x".i,)=(X,-X,)2 (3-4-1) whichisusedtocharacterize thequantization errorinPCMinSection3-5-1. Otherdistortion measures maytakethegeneralform d(x.,x.)=Ix-.i.IP(3-4-2) whereptakesvaluesfromthesetofpositiveintegers. Thecasep=2hasthe advantage ofbeingmathematically tractable. CHAYTER J:SOURCE CODING 189 Ifd(x•.x.listhedistortion measure perleiter,thedistortion be!ween a sequence ofnsamplesXnandthecorresponding nquantized valuesx"isthe averageoverthensourceoutputsamples, i.e., _1" d(X,,,Xn)=-Ld(Xbi.) nk=J(3-4·3) Thesourceoutputisarandomprocess,and,hence,thensamples inXnare random variables. Therefore, d(Xn,X,,)isarandom variable. Itsexpected valueisdefinedasthedistortion D,i.e., _ I n D=E(d(X noXn)]=-LE[d(x.,i.)]=E[d(x,x)] n4""1(3-4-4) wherethelaststepfollowsfromtheassumption thatthesourceoutputprocess isstationary. Nowsuppose wehaveamemoryless sourcewithacontinuous-amplitude outputXthathasapdfp(x),aquantized amplitude outputalphabetX,anda perleiterdistortion measure d(x,x),wherexEXandxEX.Then,the minimum rateinbitspersourceoutputthatisrequired torepresent theoutput Xofthememoryless sourcewithadistortion lessthanorequaltoDiscalled therale-distortion function R(D)andisdefinedas R(D)=min_I(X,X) p(.'IXl,Eld(X.Xl)"'D(3-4-5) whereI(X;X)istheaveragemutualinformation between XandX.Ingeneral, therateR(D)decreases asDincreases or.conversely, R(D)increases asD decreases. Oneinteresting modelofacontinuous-amplitude, memoryless information sourceisthegaussian sourcemodel.Inthiscase,Shannon provedthefollowing fundamental theorem ontherate-distortion function. Theorem: Rate-Distortion FUhctioh foraMemoryless Gaussian Source (Shannoh, 19598) Theminimum information ratenecessary torepresent theoutput of a discrete-time, continuous-amplitude memoryless gaussian sourcebasedona mean-sQuare-error distortion measure persymbol(singleleiterdistortion measure) is (3-4-6) wherecr;isthevarianceofthegaussian sourceoutput. 110 I)\G1TAL COMMUNICATIONS F1GURE 3-4-1Ratedistortion functionforacontinuous-amplitude memoryless gaussian source.1)1.0 §.... 000.2 0.4 0.6 0.8 Dla; Weshouldnotethat(3-4-6)impliesthatnoinfonnation needbetransmitted whenthedistortion D;;;.<T~Specifically, D=~canbeobtained byusing zerosinthereconstruction ofthesignal.ForD>IT;,wecanusestatistically independent, zero-mean gaussian noisesampleswithavariance ofD-IT;for thereconstruction. R,(D)isplottedinFig.3-4-1. Theratedistortion functionR(D)ofasol/rceisassociated withthe following basicsourcecodingtheorem ininformation theory. Theorem: SourceCodingwithIIIDistortion Measure (Slwmon, 195911) Thereexistsanencoding schemethatmapsthesourceoutputintocode wordssuchthatforanygivendistortion D,theminimum rateR(D)bitsper symbol(sample) issufficient toreconstruct thesourceoutputwithan averagedistortion thatisarbitrarily closetoD. Itisclear,therefore, thattheratedistortion functionR(D)foranysource represents alowerboundonthesourceratethatispossibleforagivenlevelof distortion. Letusreturntotheresultin(3-4-6)fortheratedistortion function ofa memoryless gaussian source.Ifwereversethefunctional dependence between DandR.wemayexpressDintennsofRas D,(R)=2-2RIT; (3-4-7) Thisfuneion iscalledthedistortion-rate function forthediscrete-time, memoryless gaussian source. Whenweexpressthedistortion in(3-4-7)indB,weobtain 10IOg10D.(R)=-6R+1010gIOIT; (3-4-8) Notethatthemeansquaredistortion decreases atarateof6dB/bit. Explicitresultsontheratedistortion functions formemoryless non-gaussian sourcesarenotavailable. However, thereareusefulupperandlowerboundsI i I CHAPnR~: SOl'R<TCODINGIII ontheratedistortion function foranydiscrete-time, continuous-amplitude_ memoryless source.Anupperboundisgivenbythefollowing theorem. Theorem: UpperBoundonR(D) Therate-distortion function ofamemoryless, continuous-amplitude source withzeromeanandfinitevariancea;withrespecttothemean-square-error distortion measure isupperbounded as , 1a.\_R(D)" 210g2D(0"D.,(T;) (3-4-9) Aproofofthistheorem isgivenbyBerger(1971).Itimplies thatthe gaussian sourcerequires themaximum rateamongallothersources fora specified levelofmeansquaredistortion_ Thus,theratedistortion R(D)ofany continuous-amplitude. memoryless sourcewithzeromeanandfinitevariance a;satisfies thecondition R(D).,RK(D).Similarly. thedistortion-rate function ofthesamesourcesatisfies thecondition (3-4-10) Alowerboundontherate-distortion function alsoexists.Thisiscalledthe Shannon [owerboundforamean-square-error distortion measure, andisgiven as R*(D)=H(X)-llog,21CeD (3-4-11) whereH(X)isthedifferential entropyofthecontinuous-amplitude, memory­ lesssource_Thedistortion-rate function corresponding to(3-4-11) is D*(R) =_1_2-2IRIf(Xl/ 21Ce(3-4-12) Therefore, therate-distortion function foranycontinuous-amplitude, memory­ lesssourceisbounded fromaboveandbelowas andthecorresponding distortion-rate function isbounded as D*(R)"D(R)"Dg(R) Thedifferential entropy ofthememoryless gaussian sourceis(3-4-13) (3-4-14) (3-4-15) sothatthelowerboundR*(D)in(3-4-11) reduces toRK(D).Now,ifWe 112 DIG]rAlCOMML'NIt'ATlll~S ~Ipr~ssO*(R)intermsofdecibels andnormalize ilbys~lling(T;~1lor dividingO*(R)by(}'~J.weobtainfrom(3-4-12) 10logltlD*(R) ~-6R-6(H.(X)-H(X)J (3-4-16)TABU: .l-~-1 or,equivalently, IO(oglUD.(R)=6(H.(X)-H(X)JdB D*(R) =6(R,(O)-R*(O)JdB 0-4-17) (3-4-18)Therelations in(3-4-16)and(3-4-17)allowustocompare thelowerboundin thedistortion withtheupperboundwhichisthedistortion forthegaussian SOUTce.WenotethatO*(R)alsodecreases at-6dB/bit.Weshouldalso mention thatthedifferential entropyH(X)isupper-bounded byH.(X),as shownbyShannon (1948b). Table3-4-1listsfourpdfsthataremodelscommonly usedforsourcesignal distributions. Thetableshowsthedifferential entropies. thedifferences inrates inbits/sample, andthedifference indistortion between theupperandlower bounds. Notethatthegammapdfshowsthegreatest deviation fromthe gaussian. TheLaplacian pdfisthemostsimilartothegaussian, andthe uniformpdfrankssecondofthepdfssnowninthetable.Theseresultsprovide somebenchmarks onthedifference between theupperandlowerboundson distortion andrate. Beforeconcluding thissection, letusconsider aband-limited gaussian sourcewithspectraldensity ¢I(!)={U;/2W(IfI",;W) o(ifI>W) Whentheoutputofthissourceissampled attheNyquistrate,thesamplesare uncorrelated and,sincethesourceisgaussian, theyarealsostatistically TASlE 3-4-1DIFFERENTIAL ENTROPtES ANDRATEDISTORTION COMPARISONS OFFOUR COMMON PDFsFORSIGNAL MODELS Rg(D)-R*(D) D.(R)-D*(R) pdf PIx) HtX) (bits/sample) (dB) GaussianI-~2'2(r; ~log2(21reO'~) --e 0 0v'!1r(7.- 1\log,(12u;) Uniform2V3u.••~I'"V3u.•0.255 1.53 Laplacian_1_e-\/2"jxlJ'~~!log,(2.'u;) 0.104 062v'2u, VJ"-}1082(4;relJ.42Ju.;/3) Gammae-VJl,lr2.....0.709 4.25YSlrU,lfl CHAPTER 3,SOURCE CODING 113 independent. Hence,theequivalent discrete·timegaussian source ismemory­ less.Therate-distortion function foreachsampleisgivenby(3-4-6). Therefore, therate-distortion function fortheband-limited whitegaussian sourceinbits/sis 2a. 2)R.(D)= Wlog 2D(0.;:D.;:u. Thecorresponding distortion-rate function is D.(R)=Z-RIWa; which,w'henexpressed indecibels andnormalized byu;,becomes 10logD.(R)/rT; =-3R/W(3-4-19) (3-4-20) (3·4-21) Themoregeneralcaseinwhichthegaussian processisneitherwhitenor band-limited hasbeentreatedbyGallager (1968)andGoblick andHolsinger (1967). 3-4-2ScalarQuantization Insourceencoding, thequantizer canbeoptimized ifweknowtheprobability densityfunction ofthesignalamplitude attheinputtothequantizer. For example, suppose thatthesequence {xn}attheinputtothequantizer hasapdf p(x)andletL=2Rbethedesirednumber oflevels.Wewishtodesignthe optimum scalarquantizer thatminimizes somefunction ofthequantization errorq=i-x.wherexisthequantized valueofx.Toelaborate, supposethat f(i-x)denotes thedesiredfunction oftheerror.Then,thedistortion resulting fromquantization ofthesignalamplitude is D=[J(X-x)p(x)dx (3-4-22) Ingeneral, anoptimum quantizer isone:hatminimizes Dbyoptimally selecting theoutputlevelsandthecorresponding inputrangeofeachoutput level.Thisoptimization problem hasbeenconsidered byLloyd(1982)andMax (1960),andtheresulting optimum quantizer isusuallycalledtheLloyd-Max quantizer. Forauniform quantizer, theoutputlevelsarespecified asxk=~(2k-I)~, corresponding toaninputsignalamplitude intherange(k-I)~.;:x<k~. where ~isthestepsize.Whentheuniform quantizer issymmetric withan evennumberoflevels,theaveragedistortion in(3-4-22)maybeexpressed as U2-1Jk.> D=2L f(~(2k -1)~-x)p(x)dx k=I(k--l)A +2fx f(~(2k-I)~-x)p(x)dx (Ll2-1).1(3-4-23) 114 DI{IITAl CO!\1Ml ':-';ICAlIO;-';S TABLE 3·4·2OPTIMUM STEPSIZESFORUNIFORM QUANTIZ.ATION OFA GAUSSIAN RANDOM VARIABLE Number of Optimum step Minimum MSE tologD.... outputlevels sizeAopl D_ (dB) 2 1.5% 0.3634~4.4 4 0.9957 0.1188 -9.25 ~ 0.5860 0.03744 ~14.27 16 0.3352 0.01154 -I9.3R .12 0.1&\1 (WJ349 -IA.57 Inthiscase,theminimization ofDiscarriedoutwithrespecttothestep-size parameter d.Bydifferentiating Dwithrespectto11,weobtain IfI(2k_I)fUf(!(2k-1)Ll-x)p(x)dx J.-I (I..-IlJ. +(L-1)f~ 1'O(L-1)/l-x)p(x)dx=0(3-4-24) ~(LIZ1)4 wherel'(x)denotesthederivative ofI(x). Byselecting theerrorcriterion functionf(x),thesolutionof(3-4-24)forthe optimum stepsizecanbeobtained numerically onadigitalcomputer forany givenpdfp(x).Forthemean-square-error criterion, forwhichf(x)=x2,Max (1960)evaluated theoptimum stepsize/lop,andtheminimum meansquare errorwhenthepdfp(x)iszero·mean gaussian withunitvariance. Someof theseresultsaregiveninTable3-4-2.Weobserve thattheminimum mean squaredistortion Dm;ndecreases byalittlemorethan5dBforeachdoubling of thenumberoflevelsL.Hence,eachadditional bitthatisemployed ina uniformquantizer withoptimum stepsize/lop,foragaussian-distributed signal amplitude reducesthedistortion bymorethan5dB. Byrelaxing theconstraint thatthequantizer beuniform, thedistortion can bereduced further.Inthiscase,welettheoutputlevelbex=x,whenthe inputsignalamplitude isintherange X'~I';;;x<x,.ForanL-leveJquantizer, theendpointsareXo=-ocandXL=x.Theresulting distortion is D=,tr/(x,-x)p(x)dx (3-4-25) whichisnowminimized byoptimally selecting the{x.}and(x,}. Thenecessary conditions foraminimum distortion areobtained by differentiating Dwithrespect totile{x,}and{x,}.Theresultofthis minimization isthepairofequations I(x,-x.)=f(i'+1-x,).k=1.2L-1 (3-4-26)rf'(x,-x)p(x)dx=0,k=1.2, L (3-4-27)-,, CHAPTER JSOURCE CODING115 TABLE 3-4-3OPTIMUM FOUR·LEVEL QUANTIZER FORAGAUSSIAN RANDOM VARIABLE Levelk 2 3 4-0.9816 0.0 0.9816-1.510 -0.4528 0.4528 1.510 Dmin=0.1175 10logOm,"=-9.3dB Asaspecialcase,weagainconsider minimizing themeansquareva:ueof thedistortion. Inthiscase,f(x) =x2and,hence,(3-4-26)becomes x.=~(i.+i,>1)'k=I,2,...,L-1 (3-4-28) whichisthemidpoint betweeni.andih•.Thecorresponding equations determining {x.}are Ix, (i.-x)p(x)dx =0, XIc_1k=1,2,...,L (3-4-29) Thus,i.isthecentroid oftheareaofp(x)between x,_Iandx..These equations maybesolvednumerically foranygivenp(x). Tables3-4-3and3-4-4givetheresultsofthisoptimization obtained byMax TABLE 3-4-4OPT[MUM E[GHT·LEVEL QUANTIZER FORAGAUSS[AN RANDOM VAR[ABLE (MAX,[960) Levelk 1 2 3 4 5 6 7 8-I.748 -1.050 -0.5006 o 0.5006 1.050 1.748-2.152 -1.344 -0.7560 -0.2451 0.2451 0.7560 1344 2152 Dm...=0.03454 10logOm'"=-14.62dB 116 lJK,ITAL('OMMUNI(' ATIONS TABLE 3-4-SCOMPARISON OFOPTIMUM UNIFORM AND NONUNIFORM QUANTIZERS FORAGAUSSIAN RANDOM VARIABLE (MAX,1960;PAEZAND GLlSSON, 1972) R (bi../.....pIe) I 2 3 4 5 6 JUnifonn (dB) -4.4 -9.~5 -14.27 -19.38 -24.57 -29,83 -35,13-4.4 -9.30 -14.62 -20.22 -26.02 -31.89 -37.81 (1960)fortheoptimum four-level andeight-level quantizers ofagaussian distributed signalamplitude havingzeromeanandunitvariance. InTable 3-4-5,wecompare theminimum meansquaredistortion ofauniformquantizer tothatofanonuniform quantizer forthegaussian-distributed signalamplitude, Fromtheresultsofthistable,weobserve thatthedifference inthe performance ofthetwotypesofquantizers isrelatively smallforsmallvalues ofR(lessthan0,5dBforR,.;3),butitincreases asRincreases. Forexample, atR=5,thenonuniform quantizer isapproximately 1.5dBbetterthanthe' uniformquantizer. Itisinstructive toplottheminimum distortion asafunction ofthebitrate R=log2Lbitspersourcesample(letter)forboththeuniformandnonuniform quantizers, Thesecurvesareillustrated inFig.3-4-2,Thefunctional depen­ denceofthedistortion DonthebitrateRmaybeexpressed asD(R),the distortion-rate function, Weobservethatthedistortion-rate function forthe optimum nonuniform quantizer fallsbelowthatoftheoptimum uniform quantizer, Sinceanyquantizer reducesacontinuous amplitude sourceintoadiscrete amplitude source, wemaytreatthediscrete amplitude asletters,say X={x.,1,.;k,,;;L},withassociated probabilities {P.}.Ifthesignalampli­ tudesarestatistically independent, thediscretesourceismemoryless and, hence,itsentropyis L H(X)= -2:Pklog2Pk *~I(3-4-30) Forexample, theoptimum four-level nonuniform quantizer forthe gaussian-distributed signalamplitude resultsintheprobabilities P,=P.= 0.1635forthetwoouterlevelsandP2=P3=0.3365forthetwoinnerlevels. TheentropyforthediscretesourceisH(X)=1.911bits/letter. Hence,with 0 -5 -10 ~ :t-15 - -20 -25 -300CHArTER;\: SO\.lf,CE (,ODIN<. 117 , \ \ \ \ \.'. '-",, '-"," "''""'..'...- Optimum '-\'.",uniform quanti7Cr,,,', \'."Optimum nonuniform ,'., quantilcr \,, \.'.\. \',Entropy codin~\.', Disloniun-rale \.'" flJlk:liun for~.'. ".", gau"~lan SlllJr\:C '-\'." DIRI=:!lR '-...." \'- ~, \' • R=log:Lhib/sample l'IGURE 3-4-2Dish..lItionver&usratecurvesfordlscrete-tirne memOJ)'\ess gaussian source. entropycoding(Huffman coding)ofblocksofoutputletters,wecanachieve theminimum distortion of-9.30dBwith1.911bits/letter instead of 2bits/letter. Max(1960)hasgiventheentropyforthediscretesourceletters resulting fromquantization. Table3·4-6liststhevaluesoftheentropyforthe nonuniform quantizer. ThesevaluesarealsoplottedinFig.3-4·2andlabeled entropycoding. Fromthisdiscussion. weconclude thatthequantizer canbeoptimized when thepdfofthecontinuous sourceoutputisknown.Theoptimum quantizer of L=21<levelsresultsinaminimum distortion ofD(R),whereR=log,L TABLE 3-4-6ENTROPY OFTHEOl:TPUT OFANOPTIMUM NONUNtFORM QUANTIZER FORAGAUSSIAN RANDOM VARtABLE (MAX.1960) it Eolropy Distortion (bits/somple) (bits/""er) 1010110Dm.. 1.0 -4.4 2 1.911 -9.30 3 2.~25 -14.62 4 3.765 -20.22 5 4.730 -26.02 118 DIGITAL COMMUNICATIONS bits/sample. Thus,thisdistortion canbeachieved bysimplyrepresenting each quantized samplebyRbits.However, moreefficientencoding ispossible. The discretesourceoutputthatresultsfromquantization ischaracterized byaset ofprobabilities {Pk}thatcanbeusedtodesignefficientvariable-length codes forthesourceoutput(entropy coding).Theefficiency ofanyencoding method canbecompared withthedistortion-rate function or,equivalently, the rate-distortion fum:tion forthediscrete-time, continuous-amplitude sourcethat ischaracterized bythegivenpdf. Ifwecompare theperformance oftheoptimum nonuniform quantizer with the·distortion-rate function, wefind,forexample, thatatadistortion of -26dB,entropycodingis0.41bits/sample morethantheminimum rategiven by(3-4-8),andsimpleblockcodingofeachletterrequires 0.68bits/sample morethantheminimum rate.Wealsoobserve thatthedistortion rate functions fortheoptimaluniformandnonuniform quantizers forthegaussian sourceapproach theslopeof-6dB/bitasymptotically forlargeR. 3-4-3VectorQuantization Intheprevious section,weconsidered thequantization oftheoutputsignal fromacontinuous-amplitude sourcewhenthequantization isperformed ona sample-by-sample basis,i.e.,byscalarquantization. Inthissection,weconsider thejointquantization ofablockofsignalsamples orablockofsignal parameters. Thistypeofquantization iscalledblockorvectorquantization. It iswidelyusedinspeechcodingfordigitalcellularsystems. Afundamental resultofrate-distortion theoryisthatbetterperformance canbeachieved byquantizing vectorsinsteadofscalars, evenifthe continuous-amplitude sourceismemoryless. If,inaddition, thesignalsamples orsignalparameters arestatistically dependent, wecanexploitthedependency byjointlyquantizing blocksofsamplesorparameters and,thus,achievean evengreaterefficiency (lowerbitrate)compared withthatwhichisachieved byscalarquantization. . Thevectorquantization problem maybeformulated asfollows.Wehavean n-dimensional vectorX=[x1X2•••xn]withreal-valued, continuous­ amplitude components {Xk'1,..k,..n}thataredescribed byajointpdf e(Xt,X2"",xn).ThevectorXisquantized intoanothern-dimensional vector Xwithcomponents {.i\,l"'k"'n}. Weexpressthequantization asQ('), sothat X=Q(X) (3-4-31) whereXistheoutputofthevectorquantizer whentheinputvectorisX. Basically, vectorquantization ofblocksofdatamaybeviewedasapattern recognition probleminvolving theclassification ofblocksofdataintoadiscrete numberofcategories orcellsinawaythatoptimizes somefidelitycriterion, suchasmeansquaredistortion. Forexample, letusconsider thequantization CHAPTER.l SOlIRC(-: COI.)IS(i 119 FIGURE 3-4-3Anexample ofquantization intwo-dimensional SPi.ICC. oftwo-dimensional vectors X=[XIx,].Thetwo-dimensional spaceis' partitioned intocellsasillustrated inFig.3-4·3.wherewehavearbitrarilv selected hexagonal-shaped cells{e.}.AllinputvectorsthatfallincellCare quantized intothevectorX•.whichisshowninFig.3-4-3asthecenterof\hl.: hexagon. Inthisexample. thereareL=37vectors.oneforeachofthe37cells intowhichthetwo-dimensional spacehasbeenpartitioned. Wedenotetheset ofpossible outputvectorsas{X,.I,,;:k,,;:L}. Ingeneral, quantization oftheIl-dimensional vectorXintoan11­ dimensional vectorXintroduces aquantization errororadistortion d(X.XI. Theaveragedistortion overthesetofinputvectorsXis L D=LP(XEC,lE!d(X,X.l! XECd J..=l =±P(XEC,)1d(X.Xdp(X)dX J..I x,c.(3-4-32) whereP(XEC,)istheprobability thatthevectorXfallsinthecellC,and p(X)isthejointpdfofthenrandom variables. Asinthecaseofscalar quantization. wecanminimize Dbyselecting thecells{C.I""k,,;:L}fora givenpdfp(X). Acommonly useddistortion measure isthemeansquareerror(/,norm) definedas or.moregenerally. theweighted meansquareerror d,w(X,X)=(X-X)'W(X-Xj(3-4-33) (3-4-34) whereWisapositive-definite weighting matrix.Usually. Wisselected \0be theinverseofthecovariance matrixoftheinputdatavectorX. 120 DIGITAL COMMUNICATIONS Otherdistortion measures thataresometimes usedarespecialcasesofthelp normdefinedas (3-4-35) (3-4-36)Thespecialcasep=Iisoftenusedasanalternative top=2. Vectorquantization isnotlimitedtoquantizing ablockofsignalsamplesof asourcewaveform. Itcanalsobeappliedtoquantizing asetofparameters extracted fromthedata.Forexample, inlinearpredictive coding(LPC), described inSection 3-5-3,theparameters extracted fromthesignalarethe prediction coefficients, whicharethecoefficients intheall-polefiltermodelfor thesourcethatgenerates theobserved data.Theseparameters canbe considered asablockandquantized asablockbyapplication ofsome appropriate distortion measure. Inthecaseofspeechencoding, anappropriate distortion measure, proposed byItakura andSaito(1968,1975),isthe weighted squareerrorwheretheweighting matrixWisselected tobethe normalized autocorrelation matrixcI»oftheobserved data. Inspeechprocessing, analternative setofparameters thatmaybequantized asablockandtransmitted tothereceiver isthesetofreflection coefficients {aii'1,;;;i,;;;m}.Yetanothersetofparameters thatissometimes usedforvector quantization inlinearpredictive codingofspeechcomprises thelog-arearatios {r.}.whicharedefinedintermsofthereflection coefficients as 1+a..r.=log---,1,;;;k,;;;m1-a•• Now,letusreturntothemathematical formulation ofvectorquantization andletliSconsider thepartitioning ofthen-dimensional spaceintoLcells {C.,1,;;;k,;;;L}sothattheaverage distortion isminimized overallL-Ievel quantizers. Therearetwoconditions foroptimality. Thefirstisthatthe optimalquanlizer employs anearest-neighbor selection rule,whichmaybe expressed mathematically as Q(X)=X. lialldunlyif D(X,x.),,;;D(X,Xj),k¢j,I,;;;j,;;;L (3-4-37) Thesecondcondition necessary foroptimality isthateachoutputvectorX.be chosentominimize theaveragedistortion incellC•.[notherwords,X.isthe vectorinC.thatminimizes D.=E[d(X.X)IXEC.]=1d(X,X)p(X)dX XEC,(3-4-38) ThevectorX.thatminimizes D.iscalledthecentrQidofthecell.Thus,these conditions foroptimality canbeappliedtopartition then-dimensional space (3-4-39)CHAPTFR JSOURCF CODl'(i 121 intocells{ebI...k,;;;L}whenthejointpdfp(X)isknown.Itisclearthat thesetwoconditions represent thegeneralization oftheoptimum scalar quantization problem tothen-dimensional vectorquantization problem. In general, weexpectthecodevectorstobeclosertogether inregionswherethe jointpdfislargeandfartherapartinregionswherep(X)issmall. Asanupperboundonthedistortion ofavectorquantizer, wemayusethe distortion oftheoptimal scalarquantizer, whichcanbeappliedtoeach component ofthevectorasdescribed intheprevious section,Ontheother hand,thebestperformance thatcanbeachieved byoptimum vector quantization isgivenbytherate-distortion function or,equivalently, the distortion-ra tefunction. Thedistortion-rate function, whichwasintroduced intheprevious section, maybedefinedinthecontextofvectorquantization asfollows.Suppose we formavectorXofdimension nfromnconsecutive samples {xm},ThevectorX isthenquantized toformX=Q(X),whereXisavectorfromthesetof {X.,1,;;;k,;;;L}.Asdescribed above,theaveragedistortion Dresulting from representing XbyXisE[d(X,X)],whered(X,X)isthedistortion per dimension, e.g., _1" d(X,X)= -2:(x.-.£.)2 n4:=1 Thevectors{Xb1,;;;k...L}canbetransmitted atanaveragebitrateof H(X).R=--bits/sample n whereH(X)istheentropyofthequantized sourceoutputdefinedas L H(X)= -2:p(X,jlog2P(x,) i""l(3-4-40) ForagivenaveragerateR,theminimum achievable distortion Dn(R)is Dn(R)=minE[d(X,X)] Q{X)(3-4-41) whereR;;.H(X)/n andtheminimum in(3-441) istakenoverallpossible mappings Q(X).Inthelimitasthenumber ofdimensions nisallowedto approach infinity,weobtain D(R)=limDn(R) (3-442) whereD(R)isthedistortion-rate function thatwasintroduced intheprevious section.Itisapparent fromthisdevelopment thatthedistortion-rate function canbeapproachea arbitrarily closelybyincreasing thesizenofthevectors. Thedevelopment aboveispredicated ontheassumption thatthejointpdf p(X)ofthedatavectorisknown.However, inpractice, thejointpdfp(X)of thedatamaynotbeknown.Insuchacase,itispossible toselectthe 122 DIGITAL roMMUNI('ATIONS quantized outputvectorsadaptively fromasetoftrammg vectorsX(m). Specifically, suppose thatwearegivenasetofMtrainingvectorswhereMis muchgreaterthanL(M»L).Aniterative clustering algorithm, calledtheK meansalgorithm, whereinourcaseK=L,canbeappliedtothetraining vectors.Thisalgorithm iteratively subdivides theMtrainingvectorsintoL clusterssuchthatthetwonec;essary conditions foroptimality aresatisfied. The Kmeansalgorithm maybedescribed asfollows[Makhoul etoJ.(1985»). KMeansAlgorithm Step1Initialize bysettingtheiteration number i=0.Choose asetof outputvectorsX,.(O),I,,;;k,,;;L. Step2Classifythetraining vectors{X(m),1,,;;m,,;;M}intotheclusters {C.}byapplying thenearest-neighbor rule XEC.(i)iffD(X,x.(i»,,;;D(X,Xj(i))forallk...j Step3Recompute (setitoi+1)theoutputvectorsofeveryclusterby computing thecentroid -I'" X.(i)=- LJX(m), 1"<.k,,;;L MkXeC~ ofthetrainingvectorsthatfallineachcluster.Also,compute the resulting distortion D(i)attheithiteration. Step4Terminate thetestifthechangeD(i-I)-D(i)intheaverage distortion isrelatively small.Otherwise. gotoStep2. TheKmeansalgorithm converges toalocalminimum (seeAnderberg, 1973;Lindeetill.,1980).Bybeginning thealgorithm withdifferent setsof initialoutputvectors{XdO)}andeachtimeperforming theoptimization described intheKmeansalgorithm, itispossible tofindaglobaloptimum. However, thecomputational burdenofthissearchprocedure maylimitthe searchtoafewinitializations. Oncewehaveselectedtheoutputvectors{X.,I,,;;k,,;;L},eachsignalvector X(m) ;~quantized totheoutputvectorthatisnearesttoitaccording tothe di~lOrtjon measure thatisadopted.Ifthecomputation involves evaluating thedistancebetween X(m) andeachoftheLpossibleoutputvectors{X.},the procedure constitutes afullsearch.Ifweassumethateachcomputation requiresnmultiplications andadditions, thecomputational requirement fora fullsearchis '(J=nL (3-4-43) multiplication andadditions perinputvector. IfweselectLtobeapowerof2then10&2Listhenumberofbitsrequired torepresent eachveclor.Now,ifRdenoles thebitralepersample[per component ordimension ofX(m)],wehavenR=log2L,and,hence,the computational costis (3-4-44) (3-4-45)("HA~ER 3:SOliR\1 rOLJI:\(t 123 Notethatthenumber ofcomputations growsexponentially withthedimen sionality parameter nandthebitrateRperdimension. Because ofthis exponential increase ofthecomputational cost,vectorquantization hasbeen appliedtolow-bit-source encoding, suchascodingthereflection coefficients or logarearatiosinLPC. Thecomputational costassociated withfullsearchcanbereduced b\ slightlysuboptimum algorithms (seeChangetaI.,1984:Gersho, 1982). Inordertodemonstrate thebenefitsofvectorquantization compared with scalarquantization, wepresent thefollowing example takenfromMakhoul et al.(1985). Example 3-4-1 Letx,andx,betworandom variables withauniform jointpdf {I-(XEC)p(x"x,)=p(X)= ab o(otherwise) whereCistherectangular regionillustrated inFig.3-4-4.Notethatthe rectangle isrotatedby45°relativetothehorizontal axis.AlsoshowninFig. 3-4-4arethemarginal densitiesp(x,)andp(x,). FtGURE 3-4-4Auniformpdftot",odim<nsions. (Makhoul etal..1985.) j2!aof---!-=-- a-b 2/2(/+17 2/2 ~b " a+b '2ff ~ 0 x, ...L-L-----,-,_I,---~I" ~!~...:>-.... ", a+b u-ha+b -212 2II2[[ 114 DIGITAL COMMUNICATIONS IfwequantizexIandX2separately byusinguniformintervalsoflengthA, thenumberoflevelsneededis a+b L,=L2=V2A (3-4-46) (3-4-47) (3-4-48)Hence,thenumberofbitsneededforcodingthevectorX=[x1X2]is Rx=R,+R2=1082L,+1082L2 (a+b)2 Rx=log22l!2 Thus,scalarquantization ofeachcomponent isequivalent tovector quantization withthetotalnumberoflevels (a+b)2 Lx=L,L2=2A' Weobservethatthisapproach isequivalent tocovering thelargesquare thatenclosestherectangle bysquarecells,whereeachcellrepresents oneof theLxQuantized regions.Sincep(X)=0exceptforXEC.thisencoding is wastefulandresultsinanincreaseofthebitrate. l{weweretocoveronlytheregionforwhichp(X)¥0withsquares havingareaa2,thetotalnumberoflevelsthatwillresultistheareaofthe rectangle dividedbyA2,i.e., L'=ab xa2(3-4-49) Therefore, thedifference inbitratebetween thescalarandvector quantization methods is (a+b)'Rx-R;=10822ab(3-4-50) Forinstance, ifa=4b,thedifference inbitrateis Rx-R;=1.64bits/vector Thus,vectorquantization is0.82bits/sample betterforthesamedistortion. Itisinteresting tonotethatalineartransformation (rotation by45°)will decorrelate x,andX2andrenderthetworandom variables statistically independent_ Thenscalarquantizlltion andvectorquantization achievethe sameefficiency. Although alineartransformation candecorrelate avectorof random variables, itdoesnotresultinstatistically independent random variables, ingeneral. Consequently, vectorquantization willalwaysequalor exceedtheperformance ofscalarquantization (seeProblem 3-40). Vectorquantization hasbeenappliedtoseveraltypesofspeechencoding CHAPTER 3:SOURer COOfNG 12.5 methods including bothwavefonn andmodel-based methods whicharetreated inSection3-5.Inmodel-based methods suchasLPC,vectorquantization has madepossible thecodingofspeechatratesbelow1000bits/s(seeBUlDet01.. 1980;Roucosetal.•1982;Paul1983).Whenappliedtowaveform encoding methods, itispossible toobtaingoodqualityspeechat16000bits/s,or, equivalently, atR=2bits/sample. Withadditional computational complexity, itmaybepossible inthefuturetoimplement waveform encoders producing goodqualityspeechatarateofR=1bit/sample. 3-5CODING TECHNIQUES FORANALOG SOURCES Anumberofcodingtechniques foranalogsourceshavebeen developed over thepast40years.Mostofthesehavebeenappliedtotheencoding ofspeech andimages.Inthissection,webrieflydescribe severalofthesemethods and usespeechencoding asanexample inassessing theirperformance. Itisconvenient tosubdivide analogsourceencoding methods intothree types.Onetypeiscalledtemporal waveform coding.Inthistypeofencoding, thesourceencoder isdesigned torepresent digitally thetemporal characteris­ ticsofthesourcewaveform. Asecondtypeofsourceencoding isspectral waveform coding.Thesignalwaveform isusuallysubdivided intodifferent frequency bands.andeitherthetimewaveform ineachbandoritsspectral characteristics areencoded fortransmission. Thethirdtypeofsourceencoding isbasedonamathematical modelofthesourceandiscalledmodel-based coding. 3-5-1Temporal Waveform Coding Thereareseveralanalogsourcecodingtechniques thataredesigned to represent thetime-domain characteristics ofthesignal.Themostcommonly usedmethods aredescribed inthissection. PulseCodeModulationt (PCM) Letx(t)denoteasamplefunction emitted byasourceandletx"denotethesamples takenatasampling rate t.".2W.whereWisthehighestfrequency inthespectrum ofx(t).InpeM, eachsampleofthesignalisquantized tooneof2Ramplitude levels,whereRis thenumberofbinarydigitsusedtorepresent eachsample.Thustheratefrom thesourceisRf,bits/so Thequantization processmaybemodeled mathematically as itt=x"+q" (3-5-1) where.i"represents thequantized valueofx"andq"represents the quantization error,whichwetreatasanadditive noise.Assuming thata tPCM.DPCM.andADPCM aresourcecodinglechniques. Theyarenoldigilalmodulalion methods. 126 DIGITAl. COMMl'~ICAT10"S OuIpul III ----".1.'>,.-.--...,1C;-"---"';1-(-)I-I-;;()+--~":--let",----;3:-;"-~----... Input -~" FIGURE 3·5·1 Input-output cnaractcristic forauniform quantizer. uniformquantizer isused,havingtheinput-output characteristic illustrated in Fig.3-5-1,thequantization noiseiswellcharacterized statistically bythe uniformpdf 1 p(q)=:l' (3-5-2) wherethestepsizeofthequantizer is~=2R.Themeansquarevalueofthe quantization erroris E(q')=f,.:J.2=nx22R Measured indecibels, themeansquarevalueofthenoiseis tolog-b~2=10log(nx2-2R)=-6R-10.8dB(3-5-3) (3-5-4) Weobserve thatthequantization noisedecreases by6dB/bitusedinthe quantizer. Forexample, a7bitquanlizer resultsinaquantization noisepower of-52.8dB. Manysourcesignalssuchasspeechwaveforms havethecharacteristic that smallsignalamplitudes occurmorefrequently thanlargeones.However. a uniform quantizer provides thesamespacing between successive levels throughout theentiredynamic rangeofthesignal.Abetterapproach isto employ anonuniform quantizer.· Anonuniform quantizer characteristic is usuallyobtained bypassing thesignalthrough anonlinear devicethat compresses thesignalamplitude, followed byauniform quantizer. For CHAPTER J:SOURCE COlJlN(j127 10o.x OA (J.b I,d0.2 II0.1Iyl 0.4o_~ FIGURE )-5-2Inpul-output magnitude characteristic fora logarithmic compressor. example, alogarithmic compressor hasaninput-output magnitude characteristics oftheform log(I+J-LIxl) I.vI=log(I+!L)(3-5-5) whereIxl';;1isthemagnitude oftheinput,IYIisthemagnitude oftheoutput, and!Lisaparameter thatisselected togivethedesired compression characteristic. Figure3-5-2illustrates thiscompression relationship forseveral valuesofJ-L.Thevaluef-L=0corresponds tonocompression. Intheencoding ofspeechwaveforms, forexample, thevalueof!L=255has beenadopted asastandard intheUSAandCanada. Thisvalueresultsin abouta24dBreduction inthequantization noisepowerrelativetouniform quantization. asshownbylayant(1974).Consequently, a 7bitquantizer used inconjunction withaf-l=255logarithmic compressor produces aquantization noisepowerofapproximately -77dBcompared withthe-53.dBforuniform quantization. Inthereconstruction ofthesignalfromthequantized values.theinverse logarithmic relation isusedtoexpandthesignalamplitude. Thecombined compressor-expandoT pairistermedacOl1lpalldor. Ditrerentilll PulseCodeModulatio. (DPCM) InPCM,eachsampleof thewaveform isencoded ~dependently ofalltheothers.However, most sourcesignalssampled attheNyquist rateorfasterexhibitsignificant correlation between successive samples. Inotherwords.theaveragechangein amplitude between successive samples isrelatively small.Consequently, an encoding schemethatexploits theredundancy inthesamples willresultina lowerbitrateforthesourceoutput. Arelatively simplesolution istoencodethedifferences between successive samplesratherthanthesamples themselves. Sincedifferences between samples areexpected tobesmallerthantheactualsampled amplitudes, fewerbitsare required torepresent thedifferences. Arefinement ofthisgeneralapproach is 128 DIGITAL COMMliNICATIONS topredictthecurrentsamplebasedontheprevious psamples. Tobespecific. letXndenotethecurrentsamplefromthesourceandletindenotethe predicted valueofx'"definedas in=i0iXIl--i ;=1(3-5-6) (3-5-7)Thusinisaweighted linearcombination ofthepastpsamplesandthe{a;}are thepredictor coefficients. The{a,lareselectedtominimize somefunction of theerrorbetween Xnandin. Amathematically andpractically convenient errorfunction isthemean squareerror(MSE).WiththeMSEastheperformance indexforthepredictor. weselectthe{a,}tominimize ~p=E(e;,)=E[(xn-~aiX"-in =E(x~)-2iaiE(XnXn-i) +fiaiajE(xniXn-!) i=1 ;=1j=1 Assuming thatthesourceoutputis(wide-sense) stationary. wemayexpress (3-5-7)as '€p=</>(0)- 2ia;</>(i)+ifaiaj</>(i-j) i=1 ;=Ij=I(3-5-8) where</>(m)istheautocorrelation function ofthesampledsignalsequence Xn. Minimization of~pwithrespecttothepredictor coefficients {a,}resultsinthe setoflinearequations iaic/>(i-j)=</JU),j=I,2,...,p ;=-,1(3-5-9) Thus,thevaluesofthepredictor coefficients areestablished. Whenthe autocorrelation function <p(n)isnotknownapriori,itmaybeestimated from thesamples{xn}usingtherelationt 1N-n cJ,(n)=N~XiX,+n, n=0,1,2,...,p (3-5-10) andtheestimate 4>(n)isusedin(3-5-9)tosolveforthecoefficients {ail.Note thatthenormalization factoroflINin(3-5-10) dropsoutwhen4>(n)is substituted in(3-5-9). Thelinearequations in(3-5-9)forthepredictor coefficients arecalledthe normalequations ortheYule-Walker equations. Thereisanalgorithm developed byLevinson (1947)andDurbin(1959)forsolvingtheseequations efficiently. Itisdescribed inAppendix A.Weshalldealwiththesolution in greaterdetailinthesubsequent discussion onlinearpredictive coding. tTheestimation oftheautocorrelation functionfromalinitenumberofobservations Ix)isa separate issue,whichisbeyolidthescopeofthisdiscussion. Theestimate in(3-5-10)isonethatis frequently usedinpractice. SamplerCHAPTER J:SOURCE CODING129 ~<:""-@~~}-~rr Quanlil.er Tolran~mitter Prediclor .. in-f'"=i"-(Xn-.'i"J=~"+;~-x.. =i.,-x. =q"=quantization error (alEncoder ill=;Il+.e~ ~ •__..J1Tolowpa" filter L---ePr<d~ic~IO~r I.I t la,I (biDecoder FIGURE 3-S-3 (a)BlockdiagramofaDPCMencoder. (b)DPCMdecoderatthereceiver. Havingdescribed themethodfordetermining thepredictor coefficients, let usnowconsider theblockdiagramofapractical DPCMsystem, shown inFig. 3-5-3(0). Inthisconfiguration, thepredictor isimplemented withthefeedback looparoundthequantizer. Theinputtothepredictor isdenoted byin.which represents thesignalsample Xnmodified bythequantization process,andthe outputofthepredictor is Thedifference en=x"-in(3-5-11) (3-5-12) istheinputtothequantizer andendenotestheoutput.Eachvalueofthe quantized prediction errorenisencoded intoasequence ofbinarydigitsand transmitted overthechanneltothedestination. Thequantized enorenisalso addedtothepredicted valueintoyieldin. Atthedestination, thesamepredictor thatwasusedatthetransmitting end issynthesized anditsoutputinisaddedtoentoyieldin.Thesignalinisthe desiredexcitation forthepredictor andalsothedesiredoutputsequence from whichthereconstructed signali(t)isobtained byfiltering, asshowninFig. 3-5-3(b). Theuseoffeedback aroundthequantizer, asdescribed above,ensuresthat theerrorininissimplythequantization enorqn=en-enandthatthereisno 130 OIGITAL COMMUNICATiONS accumulation ofprevious quantization errorsintheimplementation ofthe decoder. Thatis, q"=:in-e" =en-(Xn-in) (3-5-13) Hencein=Xn+qn'Thismeansthatthequantized sampleindiffersfromtlie inputx"bythequantization errorqnindependent ofthepredictor used. Therefore, thequantization errorsdonotaccumulate. IntheDPCMsystemillustrated inFig.3-5-3,theestimate orpredicted valueinofthesignalsample Xnisobtained bytakingaHnarcombination of pastvaluesin-k•k=1,2•...•p,asindicated by(3-5-11). Animprovement in thequalityoftheestimate isobtained byincluding linearlyfilteredpastvalues Qfthequantized error.Specifically, theinestimate maybeexpressed as (3-5-14) where{b,}arethecoefficients ofthefilterforthequantized errorsequenceen' Theblockdiagrams oftheencoder aethetransmitter andthedecoderatthe receiverareshowninFig.3-5-4.Thetwosetsofcoefficients {a,}and{b,}are selectedtominimiZe somefunctionoftheerroren=Xn-in.suchasthemean squareerror. FIGURE 3-5-4DPCMmodified bytheadditionoflinearlyfilterederrorsequence. x(t) ~" e,Quantize.- •", Linear filler lbj} L.... filler t·,) (0)Encoder i, + i".+ Tolowpau flhec Li_ Linear filler filter Ib,t {G,I (6)Deccder CHAPTER]: SOURCE CUOING 131 Adaptive PCMandDPCM Manyrealsources arequasistationary in nature.Oneaspectofthequasistationary characteristic isthatthevariance and theautocorrelation function ofthesourceoutputvaryslowlywithtime.PCM andDPCMencoders, however, aredesigned onthebasisthatthesource outputisstationary. Theefficiency andperformance oftheseencoders canbe improved byhavingthemadapttotheslowlytime-variant statistics ofthe source. InbothPCMandDPCM,thequantization errorq"resulting froma.uniform. quantizer operating onaquasistationary inputsignalwillhaveatime-variant variance (quantization noisepower). Oneimprovement thatreduces the dynamic rangeofthequantiz~tion noiseistheuseofanadaptive quantizer. Although thequantizer canbemadeadaptive indifferent ways,arelatively simplemethod istouseauniform quantizer thatvariesitsstepsizein accordance withthevariance ofthepastsignalsamples. Forexample, a short-term runningestimate ofthevariance ofx"canbecomputed fromthe inputsequence {x,,}andthestepsizecanbeadjusted onthebasisofsuchan estimate. Initssimplest form,thealgorithm forthestep-size adjustment employs onlytheprevious signalsample. Suchanalgorithm hasbeen successfully usedbyJayant(1974)intheencoding ofspeechsignals.Figure 3-5-5illustrates sucha(3bit)quantizer inwhichthestepsizeisadjusted recursively according totherelation &"+,=&,,M(n) FlGURE 3-5-5Example ofaquantizer withanadaptive stepsize.(Juyanl,/974.) OutPUl(3-5-15) 132 DlGI1AL COWMUNICAflONS TABLE3-5-tMULTIPLICATION FACfORS FORADAPTIVE STEPSIZE ADJUSTMENT (JAYANT. 1974) PCM DPCM 2 3 4 2 3 4 M(I) 0.60 0.85 0.80 0.80 0.90 0.90 M(2) 220 1.00 0.80 1.60 0.90 0.90 M(3) 1.00 0.80 1.25 0.90 M(4) 1.50 0.80 1.70 0.90 M(5) 1.20 1.20 M(6) 1.60 1.60 M(7) 2.00 2.00 M(8) 2.40 2.40 whereM(n)isafactor,whosevaluedepends onthequantizer levelforthe sample Xn•andA"isthestepsizeofthequantizer forprocessing Xn•Valuesof themultiplication factorsoptimized forspeechencoding havebeengivenby Jayant(1974).ThesevaluesaredisplaYed inTable3-5-1for2,3,and4bit adaptive quantization. InDPCM,thepredictor canalsobemadeadaptive whenthesourceoutput inquasistationary. Thecoefficients ofthepredictor canbechanged periodically toreflectthechanging signalstatistics ofthesource.Thelinearequations given by(3-5-9)stillapply,withtheshort-term estimate oftheautocorrelation function ofXnsubstituted inplaceoftheensemble correlation function. The predictor coefficients thusdetermined maybetransmitted alongwiththe quantized errore(n)tothereceiver, whichimplements thesamepredictor. Unfortunately. thetransmission ofthepredictor coefficients resultsinahigher bitrateoverthechannel. offsetting, inpart,thelowerdatarateachieved by havingaquantizer withfewerbits(fewerlevels)tohandlethereduced dynamic rangeintheerrorenresulting fromadaptive prediction. Asanalternative, thepredictor atthereceiver maycompute itsown prediction coefficients fromenandin.where (3-5-16) Ifweneglectthequantization noise,inisequivalellt toXn.Hence,inmaybe usedtoestimate theautocorrelation function <ben)atthereceiver, andthe resulting estimates canbeusedin(3-5-9)inplaceof</I{n)tosolveforthe predictor coefficients. Forsufficiently finequantization, thedifference between Xnandinisverysmall.Hence,theestimateoft/J{n)obtained frominisusually adequate fordetenmining thepredictor c~fficients. Implemented inthis manller,theadaptive predictor resultsillalowersource<tatarate. Insteadofusingtheblockprocessing approach fordetermining the CHAPTER lSOURCE CODING133 Source encoder Source decoder~"+i,,::tlTolransminer Output (a' xli) x"+'"i,,=:t::1 Sampler Totran:-omiuer .r" Source encoder d, i"Output Source decoder d, (h, FIGURE 3-5-6(a)Blockdiagramofadeltamodulation system.(b)Anequivalent realization ofadelta modulation system. predictor coefficients {a,}asdescribed above,wemayadaptthepredictor coefficients 011asample-by-sample basisbyusingagradient-type algorithm, similarinformtotheadaptive gradient equalization algorithm thatisdescribed inChapter IJ.Similargradient-type algorithms havealsobeendevisedfor adapting thefiltercoefficients {ailand{b,}oftheDPCMsystemshowninFig. 3-5-4.Fordetailsonsuchalgorithms, thereadermayrefertothebookby JayantandNoll(1984). DeltaModulation (OM)Deltamodulation maybeviewedasasimplified formofDPCMinwhichatwo-level (1bit)quantizer isusedinconjunction withafixedfirst-order predictor. TheblockdiagramofaDMencoder-decoder isshowninFig.3-5-6(a). Wenotethat (3-5-17) 134 DIGITAL COMMUNICATIONS Since Itfollowsthat Thustheestimated (predicted) valueofx~isreallytheprevious sample XR-I modified bythequantization noise q~_I'Wealsonotethatthedifference equation (3-5-17) represents anintegrator withaninput i~.Hence,an equivalent realization oftheone-steppredictor isanaccumulator withaninput equaltothequantized errorsignali•.Ingeneral,thequantized errorsignalis scaledbysomevalue,sayAt.whichiscalledthestepsize.Thisequivalent realization isillustrated inFig.3-5-6(b). Ineffect,theencodershowninFig. 3-5-6approximates awaveform X(/)byalinearstaircase function. Inorderfor theapproximation toberelatively good,thewaveform x(t)mustchangeslowly relativetothesampling rate.Thisrequirement impliesthatthesampling rate mustbeseveral(afactorofatleast5)timestheNyquistrate. Atanygivensampling rate,theperformance oftheOMencoder.is limited bytwotypesofdistortion, asillustrated inFig.3-5-7.Oneiscalled slope-overload distortion. ItisduetotheuseofastepsizeAIthatistoosmall tofollowportionsofthewavefonn thathaveasteepslope.Thesecondtypeof distortion, calledgranular noise.resultsfromusingastepsizethatistoolarge inpartsofthewaveform havingasmallslope.Theneedtominimize bothof thesetwotypesofdistortion resultsinconflicting requirements intheselection ofthestepsizeAI'Onesolution istoselect ~ltominimize thesumofthe meansquarevaluesofthesetwodistortions. Evenwhen6\isoptimized tominimize thetotalmeansquarevalueofthe slope-overload distortion andthegranular noise,theperformance oftheOM encodermaystillbelessthansatisfactory. Analternative solutionistoemploy avariablestepsizethatadaptsitselftotheshort-term characteristics ofthe sourcesignal.Thatis,thestepsizeisincreased whenthewaveform hasasteep FIGURE 3-5-7Anexampleofslopeoverload distortion and"...,ular noisein•deltamodulation encoder. CHAPTER" SOURCE CQ[)JNG135 Siopc­ overload distonion FIGURE 3-5-8Anexampleofvariable-step-size deltamodulation encoding. slopeanddecreased whenthewaveform hasarelatively smallslope.This adaptive characteristic isillustrated inFig.3-5-8. Avarietyofmethods canbeusedtoadaptively setthestepsizeinevery iteration. Thequantized errorsequence enprovides agoodindication ofthe slopecharacteristics ofthewaveform beingencoded. Whenthequantized error e"ischanging signsbetween successive iterations, thisisanindication thatthe slopeofthewaveform inthatlocalityisrelatively small.Ontheotherhand, whenthewaveform hasasteepslope,successive valuesoftheerrorenare expected tohaveidentical signs.Fromtheseobservations, ilispossible to devisealgorithms thatdecrease orincrease thestepsizedepending on successive valuesofen'Arelatively simpleruledevisedbyJayant(1970)isto adaptively varythestepsizeaccording totherelation whereK""Iisaconstant thatisselected to minimize thetotaldistortion. A blockdiagram ofaDMencoder-decoder thatincorporates thisadaptive algorithm isillustrated inFig.3-5-9. Severalothervariations ofadaptive OMencoding havebeeninvestigated anddescribed inthetechnical literature. Aparticularly effective andpopular technique firstproposed byGreefkes (1970)iscalledcontinuously variable slopedeltamodulation (CVSO). InCVSDtheadaptive step-size parameter maybeexpressed as an=era,,_,+k, ife,,,en_I'ande"-2havethesamesign;otherwise, a"=era".,+k2 Theparameters Ci,kI.andk2areselectedsuchthat0<Ci<Iandk1»k2>O. Formorediscussion onthisandothervariations ofadaptive OM,the interested readerisreferred tothepapersbyJayant(1974)andFlanagan etal. (1979),whichcontainextensive references. Totnnsmiltcr136 DIGITAL COMMUNICATIONS Sampler Encoder t.AccumulatorQuanlizer i,,=%1::::F1---.-----.--- 1---- Output I'IGURE 3-5-9Anexampleofadeltamodulation systemwilbadaptive stepsize. PCM,DPCM,adaptive PCM,andadaptive DPCMandDMareallsource encoding lechniques thatattempttofaithfully represent theoutputwaveform fromthesource.Thefollowing classofwaveform encoding methods isbased onaspectraldecomposition ofthesourcesignal. 3-5-2SpeetnlWaveform Coding Inthissection,webrieflydescribe waveform codingmethods thatfilterthe sourceoutputsignalintoanumberoffrequency bandsorsubbands and separately encodethesignalineachsubband. Thewaveform encoding maybe CHAPTER 3,SOURCE COl3lNG 137 performed eitheronthetime-domain waveforms ineachsubband oronthe frequency-domain repr.:sentation ofth.:corresponding time-domain waveform ineachsubband. Subband Coding Insuhhand coding(SBC)ofspeechandimagesignals, thesignalisdivid.:dintoasmallnumberofsubbands andthetimewaveform in eachsubband isencoded separately. Inspeechcoding,forexample. the lower-frequency bandscontainmostofthespectralenergyinvoicedspeech.In addition, quantization noiseisnlOlcnoticeable totheearinthelower­ frequency bands.Consequently, morebitsareusedforthelower-band signals andfewerareusedforthehigher-frequency bands. Filterdesignisparticularly important inachieving goodperformance in SBe.Inpractice. quadrature-mirror filters(QMFs) aregenerally usedbecause theyyieldanalias-free response duetotheirperfectreconstruction property (seeVaidyanathan. 1993).ByusingQMFsinsubband coding,thelower­ frequency bandisrepeatedly subdivided byfactorsoftwo,thuscreating octave-band filters.TheoutputofeachQMFfilterisdecimated byafactorof two,inordertoreducethesampling rate.Forexample, suppose thatthe bandwidth ofaspeechsignalextends to3200Hz.ThefirstpairofQMFs dividesthespectrum intothelow(0-1600Hz)andhigh(1600-3200 Hz)bands. Then.thelowbandissplitintolow(0-800Hz)andhigh(800-1600 Hz)bands bytheuseofanotherpairofQMFs.Athirdsubdivision byanotherpairof QMFscansplitthe0-800Hzbandintolow(0-400Hz)andhigh(400-800 Hz) bands.Thus,withthreepairsofQMFs,wehaveobtained signalsinthe frequency bands0-400,400-800, 800-1600 and1600-3200 Hz.Thetime­ domainsignalineachsubband maynowbeencoded withdifferent precision. Inpractice, adaptive peMhasbeenusedforwaveform encoding ofthesignal ineachsubband. Adaptive Transform Coding Inadaptive transform coding(ATC), the sourcesignalissampled andsubdivided intoframesofNtsamples, andthedata ineachframeistransformed intothespectral domain Jorcodingand transmission. Atthesourcedecoder, eachframeofspectral samples is transformed backintothetimedomainandthesignalissynthesized fromthe time-domain samplesandpassedthroughaOfAconverter. Toachievecoding efficiency, weassignmorebitstothemoreimportant spectralcoefficients and fewerbitstothelessimportant spectralcoefficients. Inaddition, bydesigning anadaptive allocation intheassignment ofthetotalnumberofbitstothe spectralcoefficients, wecanadapttopossibly changing statistics ofthesource signal. Anobjective inselecting thetransformation fromthetimedomaintothe frequency domainistoachieveuncorrelated spectralsamples. Inthissense,the Karhunen-Loeve transform (KLT)isoptimal inthatityieldsspectralvalues thatareuncorrelated, buttheKLTisgenerally difficulttocompute (see 138 DIGITAL COMMUNICATIONS Wintz,1972).TheDFfandthediscretecosinetransform (Dcr)areviable alternatives, although theyaresuboptimum. Ofthesetwo,theDcryields goodperformance compared withtheKLT,andisgenerally usedinpractice (seeCampanella andRobinson, 1971;Zelinsky andNoll,1977). InspeechcodingusingATC,itispossibletoattaincommunication-quality speechatarateofabout9600bits/so 3-5-3Model-Based SourceCoding Incontrasttothewaveform encoding methods described above,model-based sourcecodingrepresents acompletely different approach. Inthis,thesourceis modeled asalinearsystem(filter)that,whenexcitedbyanappropriate input signal,resultsintheobserved sourceoutput.Insteadoftransmitting the samplesofthesourcewaveform tothereceiver, theparameters ofthelinear systemaretransmitted alongwithanappropriate excitation signal.Iftfte numberofparameters issufficiently small,themodel-based methods providea largecompression ofthedata. Themostwidelyusedmodel-based codingmethodiscalledlinearpredictive coding(LPC).Inthis,thesampledsequence, denotedbyx.,n=0,1,...,N ­ I,isassumed tohavebeengenerated byanall-pole(discrete-time) filter havingthetransferfunction GH(z)=---- 1-fa.z·· 11:=1(3-5-18) Appropriate excitation functions areanimpulse, asequence ofimpulses, ora sequence ofwhitenoisewithunitvariance. Inanycase,supposethattheinput sequence isdenotedbyv.,n=0,1,2,..,.Thentheoutputsequence ofthe all-polemodelsatisfiesthedifference equation X.=±a.x,,_.+Gv.,n=0,1,2,...'-1(3-5-19) Ingeneral, theobserved sourceoutputx..n=0,1,2,...,N-1,doesnot satisfythedifference equation (3-5-19), butonlyitsmodeldoes.Iftheinputis aWhite-noise sequence oranimpulse,wemayfonmanestimate (orprediction) ofx.bytheweighted linearcombination Thedifference betweenx.andin>namely, en=xn-i n =XII-fakXn-k'-1(3-5-20) (3-5-21) CHArTER'> SOltRCE CODl'fG 139 represents theerrorbetween theobserved valuex"andtheestimated (predicted) valueX'"Thefilterc.oefficients {ak}canbeselected tominimize the meansquarevalueofthiserror. Suppose forthemoment thattheinput{v,,}isawhite·noise sequence. Then, thefilteroutputx"isarandom sequence andsoisthedifference en=Xn-:in' Theensemble average ofthesquared erroris 'f "=E(e;,) I' I' I~ =cb(O)-22:akc/>(k)+2:2:aka",<b(k -m) J...I k~I111·-I(}-5-22) (3-5·23)where¢(111)istheautocorrelation function ofthesequence x"'n= 0,I....,N-I.Butf,.isidentical totheMSEgivenby(3-5-R)forapredictor usedinDPCM. Consequently. minimization of'lpin(3-5-22) yieldsthesetof normalequations givenpreviously by(3-5-9).Tocompletely specifythefilter H(~),wemustalsodetermine thefiltergainG.From(3-5-19), wehave [f' ']EI(Cv,,)') =C'E(v~)=C'=E(x"-f-,a,x"k) =-I" whereII'istheresidual MSEobtained from(3-5-22) bysubstituting the optimum prediction coefficients, whichresultfromthesolution of(3-5-9).With thissubstitution. theexpression for'f"and,hence,C'simplifies to I' 'fj,=C'=<b(0)-2:fI.<p(k) k=-"·:(3-5-24) Inpractice. wedonotusually knowapriorithetrueautocorrelation function ofthesourceoutput. Hence,inplaceof<p(n),wesubstitute an estimate J,(n)asgivenby(3-5-IO), whichisobtained fromthesetofsamples .tH,11=0,l....,N-l.emitted bythesource. Asindicated previously. theLevinson-Durbin algorithm derived inAppen­ dixAmaybeusedtosolveforthepredictor coefficients {ak}recursively. beginning withafirst-order predictor anditerating theorderofthepredictor uptoorderp.Therecursive equations forthe{admaybeexpressed as " iI ... <p(i)-Ia,Ikcb(i-k) k, i=2,3•....p f..=(1-a,,)f,,lliJ...=illIJ...-aI/a,. II".I,,;k,,;i-l (3-5-25) J,(I)a=--"4>(0), 140 DIGITAL COMMUNICATIONS wherea'bk=1,2,...,i,arethecoefficients oftheith-order predictor. The desiredcoefficients forthepredictor oforderpare andtheresidualMSEisak...api<,k=1,2,...,p t=G2=<P(O)-iak<P(k) k-1 =<P(O)Ii(1-af,) ;=1(3-5-26) (3-5-27) Weobserve thattherecursive relations in(3-5-25) giveusnotonlythe coefficients ofthepredictor fororderp,butalsothepredictor coefficients ofall orderslessthanp. Theresidual MSE ~;,i=1,2,...,p,formsamonotone decreasing se­ quence,i.e.~p";;~P_I,.;;..•,.;;~,";;~o.andtheprediction coefficients aiisatisfy thecondition laiil<1,i=1,2,...,p (3-5-28) Thiscondition isnecessary andsufficient forallthepolesofH(z)tobeinside theunitcircle.Thus(3-5-28)ensuresthatthemodelisstable. LPChasbeensuccessfully usedinthemodeling ofaspeechsource.Inthis case,thecoefficients aii,i=1,2,...,p,arecalledreflecrion coefficients asa consequence oftheircorrespondence tothereflection coefficients inthe acoustictubemodelofthevocaltract(seeRabinerandSchafer,1978;Dellerer al.,1993). Oncethepredictor coefficients andthegainGhavebeenestimated fromthe sourceoutput{x.l,eachparameter iscodedintoasequence ofbinarydigits andtransmitted tothereceiver. Sourcedecoding orwaveform synthesis may beaccomplished atthereceiver asillustrated inFig.3-5-10.Thesignal generator isusedtoproduce theexcitation function {vnl,whichisscaledbyG FIGURE 3-5-10 Blockdiagramofawaveform synthesizer (sourcedecoder) foranLPCsystem. Exc. Signal v,r---1-generatorH(:) Inpul Decoder Filterparameters Lowpass filter l Output CHAPTER 3,SOURCE CODING141 GainG.and1J:1\1oicedWhil~-noise generdlor1switch ..-----,1----' -i-o----G>-----I°IL..._A_:.;_:_e_L ~::~ Period,ci. I impulse generator Pilch periodlifo FIGURE 3-5-11 Blockdiagrammodelofthegeneration ofaspeechsignal. toproducethedesiredinputtotheall-polefiltermodelH(z)synthesized from thereceivedprediction coefficients. Theanalogsignalmaybereconstructed by passingtheoutputsequence fromH(z)throughananalogfilterthatbasically performs thefunctionofinterpolating thesignalbetweensamplepoints.Inthis realization ofthewaveform synthesizer, theexcitation function andthegain parameter mustbetransmitted alongwiththeprediction coefficients tothe receiver. . Whenthesourceoutputisstationary, thefilterparameters needtobe determined onlyonce.However, thestatisticsofmostsourcesencountered in practiceareatbestquasistationary. Underthesecircumstances, itisnecessary toperiodically obtainnewestimates ofthefiltercoefficients, thegainG,and thetypeofexcitation function, andtotransmittheseestimates tothereceiver. ElUIII1ple 3-5·1 Theblockdiagram showninFig.3-5-11illustrates amodelforaspeech source.Therearetwomutually exclusive excitation functions tomodel voicedandunvoiced speechsounds.Onashort-time basis,voiced·speechis periodicwithafundamental frequency foorapitchperiodlifothatdepends onthespeaker.Thusvoicedspeechisgenerated byexcitinganall·polefilter modelofthevocaltractbyaperiodicimpulsetrainwithaperiodequalto thedesiredpitchperiod.Unvoiced speechsoundsaregenerated byexciting theall-polefiltermodelbytheoutputofarandom-noise generator. The speechencoderatthetransmitter mustdetermine theproperexcitation function, thepitchperiodforvoicedspeech,thegainparameter G,andthe prediction coefficients. Theseparameters areencodedintobinarydigitsand transmitted tothereceiver. Typically, thevoicedandunvoiced information requires1bit,thepitchperiodisadequately represented by6bits,andthe gainparameter mayberepresented by5bitsafteritsdynamic rangeis compressed logarithmically. Theprediction. coefficients require 8­ 10bits/coefficient foradequate representation (seeRabiner andSchafer, 1978).Thereasonforsuchhighaccuracy isthatrelatively smallchangesin 142 DIGITAL COMMllNI('ATLONS Whlte­ noi-.e generator Output FIGURE 3-5-12 All-pole latticefilterforsynthesizing thespeechsignaL theprediction coefficients resultinalargechangeinthepolepositions of thefiltermodelH(z).Theaccuracy requirements maybelessened by transmitting thereflection coefficients ail.whichhaveasmallerdynamic range.Theseareadequately represented by6bits.Thus,forapredictor of orderp=10[fivepolesinH(z)),thetotal)lumberofbitsis72.Duetothe quasistationary natureofthespeechsignal,thelinearsystemmodelmustbe changed periodically, typically onceevery15-30ms.Consequently, thebit ratefromthesourceencoderisintherange4800-2400 bit/s. Whenthereflection coefficients aretransmitted tothedecoder, ItISnot necessary torecompute theprediction coefficients inordertorealizethe speechsynthesizer. Instead, thesynthesis isperformed byrealizing alattice filter,showninFig.3-5-12.whichutilizesthereflection coefficients directlyand whichisequivalent tothelinearprediction filter. Thelinearall-polefiltermodel,forwhichthefiltercoefficients areestimated vialinearprediction, isbyfarthesimplestlinearmodelforasource.Amore generalsourcemodelisalinearfilterthatcontains bothpolesandzeros.Ina pole-zero model,thesourceoutput Xnsatisfiesthedifference equation Xn=fakXn-k+±bkvn-k k=l k'=O where Vnistheinputexcitation sequence. Theproblem nowistoestimate the filterparameters {ak}and{bdfromthedataXi,i=O.1•...•N-1.emittedby thesource.However, theMSEcriterion appliedtotheminimization ofthe erroren=x.-i.,whereinisanestimate ofx.,resultsinasetofnonlinear equations fortheparameters {adand{bk}'Consequently, theevaluation ofthe {adand{bklbecomes tediousanddifficultmathematically. Toavoidhavingto solvethenonlinear equations, anumberofsuboptimum methods havebeen devisedforpole-zero modeling. Adiscussion ofthesetechniques wouldlead ustoofarafield,however. LPCasdescribed aboveformsthebaSISformorecomplex model-based sourceencoding methods. Whenappliedtospeechcoding,themodel-based QCflAPTER), SOURCE CODING143 methods aregenerally calledvocoders (forvoicecoders). Inaddition tothe conventional LPCvocoderdescribed above,othertypesofvocoders thathave beenimplemented includetheresidual excitedLPC(RELP) vocoder, the multipulse LPCvocoder, thecode-excited LPC(CELP) vocoder, andthe vector-sum-excited LPC(VSELP) vocoder. TheCELPandVSELPvocoders employ vector-quantized excitation codebookstoachieve communication qualityspeechatlowbitrates. Beforeconcluding thissection,weconsider theapplication ofwaveform encoding andLPCtotheencoding ofspeechsignalsandcompare thebitrates ofthesecodingtechniques. Encoding Methods Applied toSpeechSignals Thetransmission ofspeech signalsovertelephone lines,radiochannels, andsatellitechannels constitutes byfarthelargestpartofourdailycommunications. Itisunderstandable, therefore. thatoverthepastthreedecadesmoreresearch hasbeenperformed onspeechencoding thanonanyothertypeofinformation-bearing signal.In fact,alltheencoding techniques described inthissectionhavebeenappliedto theencoding ofspeechsignals.Itisappropriate, therefore, tocompare the efficiency ofthesemethods intermsofthebitraterequired totransmit the speechsignal. Thespeechsignalisassumed tobeband-limited tothefrequency range 200-3200 Hzandsampled atanominal rateof8000samples/s forallencoders exceptDM,wherethesampling rateisf,identical tothebitrate.ForanLPC encoder, theparameters giveninExample 3-5-1areassumed. Table3-5-2summarizes themaincharacteristics oftheencoding methods described inthissectionandtherequired bitrate.Intermsofthequalityofthe speechsignalsynthesized atthereceiverfromthe(error-free) binarysequence, allthewaveform encoding methods (PCM,DPCM, ADPCM, OM,ADM) providetelephone (toll)qualityspeech.Inotherwords,alistenerwouldhave difficulty discerning thedifference between thedigitized speechandtheanalog speechwaveform. ADPCM andADMareparticularly efficient waveform encoding techniques. WithCVSD,itispossibletooperatedownto9600bits/s TABLE 3-5·2ENCODING TECHNIQUES APPLIED TOSPEECH SIGNALS [nroell.1! method QlllDtizer Coder Tnnsmisllon rote(bluM PCM Linear 12bils 96000 LogPCM Logarithamic 7-8bils 56000-64 000 DPCM Logarithmic 4-6bits 32000-48000 ADPCM Adaptive 3-4bits 24000-32000 DM Binary 1bit 32000-64000 ADM Adaptive binary 1bit 16000-32000 LPC 2400-4800 144 DIGITAL ('OM~t:I"'I(AT1U;"<S withsomenoticeable waveform distortion. Infact,atratesbelow16000bits/so thedistortion produced bywaveform encoders increases significantly. Conse­ quently, thesetechniques arenotusedbelow9600bits/so Forratesbelow9600bits/s,encoding techniques, suchasLPCthatare basedonlinearmodelsofthesourceareusuallyemployed. Thesynthesized speechobtained fromthisclassofencoding techniques isintelligible. However. thespeechsignalhasasynthetic qualityandthereisnoticeable distortion. 3·6BIBLIOGRAPHICAL NOTES ANDREFERENCES Sourcecodinghasbeenanareaofintense research actlVlty since the publication ofShannon's classicpapersin1948andthepaperbyHuffman (1952).Overtheyears,majoradvances havebeenmadeinthedevelopment of highlyefficientsourcedatacompression algorithms. Ofparticular significance istheresearch onuniversal sourcecodinganduniversal quantization published byZiv(1985),ZivandLempel(1977,1978), Davisson (1973),Gray(1975),and Davisson etat.(1981). Treatments ofratedistortion theoryarefoundinthebooksbyGallager (1968).Berger(1971),ViterbiandOmura(1979),Blahut(1987)andGray (1990). Muchworkhasbeendoneoverthepastseveraldecadesonspeechencoding methods. Ourtreatment provides anoverview ofthisimportant topic.Amore comprehensive treatment isgiveninthebooksbyRabiner andSchafer(1978), JayantandNoll(1984),andDelleretai.(1993).Inaddition tothesetexts. therehavebeenspecialissuesoftheIEEETransactions onCommunicariol!s (April1979and~priI1982) and,morerecently, theIEEEJournalonSelected AreasinCommunications (February 1988)devoted tospeechencoding. We shouldalsomention thepublication byIEEEPressofabookcontaining reprintsofpublished papersonwaveform quantization andcoding.editedby Jayant(1976). Overthepastdecade, wehavealsoseenanumber ofimportant develop, mentsinvectorquantization. Ourtreatment ofthistopicwasbasedonthe tutorialpaperbyMakhoul etat.(1985).Acomprehensive treatment ofvector quantization andsignalcompression isprovided inthebookbyGershoand Gray(1992). PROBLEMS 3-1Consider thejointexperiment described inProblem 2-1withthegivenjoint probabilities P(A,.B,).Suppose weobservetheoutcomes A..i=I,2,3,4of experiment A. •Determine themutualinformation I(B,:A,)forj=I.2,3andi=1,2,3,4,in bits. bDetermine theaveragemutualinformation liB:A). CHAPTER 3,SOURCE CODING 145 3-2Suppose theoutcomes B,.j=1,2.3.inProblem 3-1represent thethreepossible outputlettersfromtheOMS.Determine theentropyofthesource. 3-3ProvethatIn""'''-1andalsodemonstrate thevalidityofthisinequality by plottingInuimd"- 1onthesamegraph. 3-4XandYaretwodiscreterandomvariables withprobabilities P(X=.t,Y=y)'"P(x,y) ShowthatI(X;Y)'"O.withequality ifandonlyifXandYarestatistically independent. [Him:Usetheinequality In"<"-I.for0<"<I,toshowthat-/(X;Y)'"0·1 3-5TheoutputofaOMSconsistsofthepossible lettersx,.x,.....x,,,whichoccur withprobabilities p,.p"...,p,,,respectively. ProvethattheentropyH(X)ofthe sourceisatmostlogn. 3-6Determine thedifferential entropyH(X)oftheuniformly distributed random variableXwithpdf p(X)={oa"(0'"Xo;a) (otherwise ) forthefollowing threecases: aa=I; ba=4; ca=~. Observe fromtheseresultsthatH(X)isnotanabsolute measure, butonlya relativemeasureofrandomness. 3-7AOMShasanalphabet ofeightletters, Xi'i=I.2•...•8.withprobabilities 0.25. 0.20,0.15,0.12,0.10,0.08,0.05, and0.05. aUsetheHuffman encoding procedure todetermine abinarycodeforthesource output. bDetermine theaveragenumberRofbinarydigitspersourceleiter. cDetermine theentropyofthesourceandcompare itwithR. 3-8AOMShasanalphabet offiveletters, X"i=1,2•...,5.eachoccurring with probability \.Evaluate theefficiency ofafixed-length binarycodeinwhich aeachletterisencoded separately intoabinarysequence; btwolettersatatimeareencoded intoabinarysequence; cthreelettersatatimeareencoded intoabinary sequence. 3-9Recall(3-2-6): I(x,;y,)=I(x,)-/(x.1y,) Provethat aI(xi;y,)=I(y,)-/(y,Ix,); bI(xi;y,)=I(x.)+I(y,)-/(xiy,). whereI(xiy,)=-logP(x"yJ 3-10LetXbeageometrically distributed randomvariable;. thatis, p(X=k)=p(l-p)"\ k=I.2.3.... aFindtheentropyofX. bKnowing thatX>K,whereKisapositiveinteger,whatistheentropyofX' 146 DJGITA.L COMMCNI(ATIONS 3-llLetXandYdenotetwojointlydistributed discrele valuedrandom variables . •Showthat H(X)= -~P(x,y)10gP(x) x.y H(Y)= -~P(x.y)logP(y) bUsetheaboveresulttoshowthat H(X.Y)"'H(X)+H(Y) Whendoesequality hold? cShowthat H(XIY)'"H(X) withequality ifandonlyifXandYareindependent. 3-12Twobinaryrandom variables XandYaredistributed according tothejoint distributions p(X=Y=0)=p(X=O.Y=I)=p(X=Y=I)=\.Compute H(X), H(V).H(XIV).H(YIX).andH(X.V). 3-BAMarkov processisaprocesswithone-step memory. i.e..aprocesssuchthat ptx"Ix",.x",.x",....)=p(x"Ix",) foralln.Showthat.forastationary Markov process. theentropyrateisgivenby H(X"!X",J 3-14LetY=g(X).wheregdenotes adeterministic function. Showthat.ingeneral. H(Y),",-H(X). Whendoesequality hold? 3-15Showthat/(X: Y)=H(X)+H(Y)-H(XY). 3-16Showthat.forstatistically independent events. "H(X,X," .X,,)=2:,H(X,) ,, )-17Foranoiseless channel. showthatH(XIY)=0. )-18Showthat I(X,:X,IX,)=H(X.IX,)-H(X,IX,X,) andthat H(X,IX,);>H(X,IX,X.) 3-11)LetXbearandom variahle withpdfp,(x)andletY=aX+hhealinear transformation ofX.whereaandharetwoconstants. Determine thedifferential entropyH(Y)intermsofH(X). )·20Theoutputsx"x,.andx,ofaDMSwithcorresponding. probahilities 1',=0,45. 1'1=0.35,andp,=0.20aretransformed bythelineartransformation Y=aX+h. whereaandbareconstants. Determine theentrop" H(Y)andcomment on",hal effectthetransformation hashadontheentropy ofX. )·21Theoptimum four-level nonuniform quantizer foragaussian-distributed signal amplitude results inthefourlevelsai'0:.OJ,and tJ~.with corr~srnnding prohahilities ofoccurrence I',=1',=0.3365andp,=p,=0.1635. CHAPTER 3,SOURCE CODI'G 147 FIGURE P3-Z2 aDesignaHuffman codethatencodesasinglelevelatatimeanddetermine the averagebitrate. bDesignaHuffman codethatencodestwooutputlevelsatatimeanddetermine theaveragebitrate. ~Whatistheminimum rateobtained byencoding Joutputlevelsatatimeas J---iooo? 3-22Afirst-order Markovsourceischaracterized bythestateprobabilities P(x,}, j=I,2...,.L.andthetransition probabilities p(x,lxi),k=1,2.",.L.and k'"i.TheentropyoftheMarkovsourceis l H(X)=2P(x,)H(XIx,),-, whereH(XIx,)istheentropyconditioned onthesourcebeinginstatex,. Determine theentropyofthebinary,first-order MarkovsourceshowninFig. P3-22,whichhasthetransition probabilities P(x,Ix,)=0.2andP(x,Ix2)=0.3. [Notethattheconoitional entropies H(XIx,)andH(XIx,)aregivenbythe binaryentropyfunctions H[P(x,Ix,)]andH[P(xlIx2)],respectively.] Howdoes theentropyoftheMarkovsourcecomllare withtheentropyofabinaryOMSwith thesameoutputletterprobabilities P(x,)andP(x,)? 3-23Amemoryless sourcehasthealphabet JlI1={-5,-3.-I,0,I,3,51.withcorre­ sponding probabilities {O.OS,0.1,0.1,0.15,0.05,0.25,0.3}. aFindtheentropyofthesource, bAssuming thatthesourceisquantized according tothequantization rule q(-5)=q(-3)=4 q(-I)=q(O)=q(I)=O q(3)=q(5)=4 tindtheentropyofthequantized source. 3-24DesignaternaryHuffman code,using0,I,and2asletters,forasourcewith outputalphabet probabilities givenby{O.OS,0.1,0,15,0.17,0.18,0.22, O.B}.What istheresulting averagecodeword length?Compare theaveragecodeword length withtheentropyofthesource.(Inwhatbasewouldyoucompute thelogarithms in theexpression fortheentropyforameaningful comparison?) 3-25FindtheLempel-Ziv sourcecodeforthebinarysourcesequence 0001001000ooo11 ססoo1 ס0ooooo1 o00ooo101 ססoo1o00ooo1101ס0ooooo1 100 Recover theoriginalsequence backfromtheLempel-Ziv sourcecode. [Him:Yourequiretwopassesofthebinarysequence todecideonthesizeofthe dictionary.] J-2liFindthedifferential entropyofthecontinuous randomvariableXinthefoUowing cases: 148 DIGITAL COMMUNICATiONS • Xisanexponential randomvariablewithparameter A>0,i.e., {rle-X" (x>0) fAx)=0 (otherwise) bXisaLaplacian randomvariablewithparameter ,\>0,i.e., 1fx(x)~2Ae-~" cXisatriangular randomvariablewithparameter A>O.i.e., {(x+.1.)/.1.' (-A,;;:x';;:0) (x(x)~ (-x+A)/A' (O<xEiA) o (otherwise) 3-27Itcanbeshownthattherate-distortion function foraLaplacian source, (Ax)=(2Arle-~" withanabsolute valueoferror-distortion measure d(x,x)= Ix-£/isgivenby ._{IOg(AID) (OEiD";A) R(D)- 0 (D>A) (seeBerger,1971). •Howmanybitspersamplearerequired torepresent theoutputsofthissource withanaveragedistortion notexceeding ~A? bPlotR(D)forthreedifferent valuesofAanddiscusstheeffectofchangesinA ontheseplots. 3-ZllItcanbeshownthatifXisazero-mean continuous randomvariablewithvariance u',itsratedistortion function, subjecttosquared errordistortion measure, satisfiesthelowerandupperboundsgivenbytheinequalities h(X)-~log21reD,;;:R(D)..~logla' whereh(X)denotesthedifferential entropyoftherandomvariable X(seeCover andThomas, 1991). •Showthat.foraGaussian randomvariable, thelowerandupperbounds coincide. bPlotthelowerandupperboundsforaLaplacian sourcewitha=1. cPlotthelowerandupperboundsforatriangular sourcewithiT=I. 3·2~Astationary randomprocesshasanautocorrelation function givenbyRx= IA'e-'rlcos2Jfj;,r anditisknownthattherandomprocessneverexceeds6in magnitude. Assuming A=6,howmanyquantization levelsarereqUired to guarantee asignal.to-quantization noiseratioofatleast60dB? 3·30Anadditivewhitegaussian noisechannelhastheoutputY~X+G,whereXis thechannelinputandGisthenoisewithprobabilily densityfunction 1 • •p(n)= :::t-wI2,.;, V1ia" IfXisawhitegaussian inputwithE(X)~0andE{Xz)=u;,determine •theconditional differential entropyH(XIG): btheaveragemutualinformation I(X:V). 3·31ADMShasanalphabet ofeightleiters.x"i~I.2.....8.withprobabilities •CHAPTER _"SOLIRCE CODING 149 giveninProblem 3-7.UsetheHuffman encoding procedure todetermine aternary code(usingsymbols0,I,and2)(orencoding thesourceoutput. [Hili/:Addasymbolx.withprobability p.=0,andgroupthreesymbols ata time.] 3-32Determine whether thereexistsabinarycodewithcodewordlengths (n"n"n"n.)=(I,2,2,3)thatsatisfytheprefiXcondition. 3-33Consider abinaryblockcodewith2·codewordsofthesamelengthn.Showthat theKrallinequality issatisfied forsuchacode. 3-34Showthattheentropy ofann-dimensional gaussian vectorX=[x,x,..,x.] withzeromeanandcovariance matrixMis H(X)=1log,(21re)"IMI 3-35Consider aDMSwithoutputbits(0,1)thatareequiprobable. Definethe distortion measure asJJ=Powherep.istheprobability oferrorintransmitting thebinarysymbols totheuseroveraBSC.Thentheratedistortion function is (Berger, 1971) R(D)=1+Dlog,D+(1-D)108,(1-D),0""D=P.""~ PlotR(D)(or0""D""~. 3-36Evaluate theratedistortion (unction foranM-arysymmetric channel where D=PMand 1-DR(D)=log,M+Dlog,D+(1-D)log,--­M-1 forM=2,4,8,and16.PMistheprobability oferror. 3-37Consider theuseoftheweighted mean-square-error (MSE)distortion measure definedas whereWisasymmetric, poJitive-definite wieghting matrix.Byfactorizing W,as W=P'P,showthatdn(X,X)iseqUivalent toanunweighted MSEdistortion measured,(X',X') involving transformed vectorsX"andX'. 3-38Consider astationary stochastic signalsequence {X(n)}withzeromeanand autocorrelation sequence ~(n)=fi(n=0) (n=±1) (otherwise) aDetermine theprediction coefficient ofthefirst-order minimum MSEpredictor for{X(n)}givenby i(n)=a,x(n-1) andthecorresponding minimum meansquareerror ~" bRepeat(a)forthesecond-order predictor i(n)=a,x(n-1)+a,x(n-2) 158 DIGITAL COMMUN(CAf(ONS FIGURE P3-39p(X.,J~15--l---l- 0 lb _lb, L_--l--t b---I;:~4---~------h-----::--+~:I-:-- .', _!b 2 pL.rr) ~t;;, ~---1l/,0 ~alo·'1 -~a2 3-39Consider theencoding oftherandomvariables X,andx,thatarecharacterized by thejointpdfp(x,,x,)givenby {15/7ab p(x"x,)= 0(x"x, EC) (otherwise ) FlGUREP~asshowninFig.P3-39.Evaluate thebitratesrequired foruniformquantization or X,andX,separately (scalarquantization) andcombined (vector) quantization of (x"x,).Determine thedifference inbitratewhena=4b. 3-40Consider theencoding oftworandom variables XandYthatareuniformly distributed ontheregionbetween twosquaresasshowninFig.P3-40. aFindIx(x)andIy(y). bAssume thateachoftherandomvariables XandYarequantized usingfour leveluniform quantizers. Whatistheresulting distortion? Whatistheresulting numberofbilsper(X,Y)pair? y 2 -2-1 -I -2 CHAPTER J:SO~;RrE CODlNG 151 y -2 nGURE P3-41-2 cNowassumethatinsteadofscalarquantizers forXandY.weemployavector quantizer toachievethesamelevelofdistortion asin(b).Whatistheresulting numberofbitspusourceoutputpair(X.Y)? 3-41Tworandomvariables XandYareuniformly distributed onthesquareshownin Fig.P3-41. aFindfx(x)andfy(y). bAssume thateachoftherandomvariables XandYarequantized usingfour leveluniformquantizers. Whatistheresulting distortion? Whatistheresulting numberofbitsper(X,Y)pair? cNowassumethat,insteadofscalarquantizers forXandY,weemployavector quantizer withthesamenumberofbitspersourceoutputpair(X,Y)asin(b). Whatistheresulting distortion forthisvectorquantizer? 4 CHARACTERIZATION OF COMMUNICATION SIGNALS ANDSYSTEMS Signalscanbecategorized inanumber ofdifferent ways,suchasrandom versusdeterministic, discretetimeversuscontinuous time,discreteamplitude versuscontinuous amplitude, lowpassversusbandpass, finiteenergyverslJs infiniteenergy,finiteaveragepowerversusinfiniteaverage power,etc.Inthis chapter. wetreatthecharacterization ofsignalsandsystemsthatareusually encountered in'thetransmission ofdigitalinformation overacommunication channel. Inparticular. weintroduce therepresenlation ofvariousformsof digitallymodulated signalsanddescribe theirspectralcharacteristics. Webeginwiththecharacterization ofbandpass signalsandsystems. including themathematical representation ofbandpass stationary stochastic processes. Then.wepresentavectorspacerepresentation ofsignals.We conclude withtherepresentation ofdigitally modulated signalsandtheir spectralcharacteristics. 4-1REPRESENTATION OFBANDPASS SIGNALS ANDSYSTEMS Manydigitalinformation-bearing signalsaretransmitted bysometypeof carriermodulation. Thechanneloverwhichthesignalistransmitted islimited inbandwidth toanintervaloffrequencies centered aboutthecarrier.asin double-sideband modulation, oradjacent tothecarrier,asinsingle-sideband modulation. Signalsandchannels (systems) thatsatisfythecondition thattheir bandwidth ismuchsmallerthanthecarrierfrequency aretermednarrowband Iwndpass .Iigna!.!andchannels (sysrems). Themodulation performed atthe 152 CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 1.53 IS(f)1 FIGURE: 4-1·1Spectrum ofabandpass signal. o transmitting endofthecommunication systemtogcmerate thebandpass signal andthedemodulation performed atthereceiving endtorecoverthedigital information involvefrequency translations. Withnolossofgenerality andfor mathematical convenience, itisdesirable toreduceallbandpass signalsand channels toequivalent lowpass signalsandchannels. Asaconsequence, the resultsoftheperformance ofthevariousmodulation anddemodulation techniques presented inthesubsequent chaplers areindependent ofcarrier frequencies andchannel frequency bands.Therepresentation ofbandpass signalsandsystems intermsofequivalent lowpass waveforms andthe characterization ofbandpass stationary stochastic processes arethemaintopics ofthissection. 4-1-1"Representation ofBandpass Signals Suppose thatareal-valued signalset)hasafrequency contentconcentrated in anarrowbandoffrequencies inthevicinityofafrequency fr.asshowninFig. 4-1-1.Ourobjective istodevelop amathematical representation ofsuch signals.First,weconstruct asignalthatcontains onlythepositivefrequencies in5(/).Suchasignalmaybeexpressed as (4-1·1) whereS(f)istheFouriertransform ot5(/)andu(f)istheunitstepfunction. Theequivalent time-domain expression for(4-1-1)is 5+(/)=rS+(f)ei2"ftdf =r'[2u(f»)*F"[S(f») (4-1-2) Thesignals+(t)iscalledtheanalyticsignalorthepre-envelope of5(/).We notethatr'[S(f»)=5(/)and F"[2u(f»)=8(/)+1. 1rI(4-1-3) 154DIGITAL COMMUNICATIONS Hence, Wedefinesit)asJ.(t)= [{j(t)+~]*sit) 1 =J(/)+j-*J(t)m(4-1-4) (4-1-7)1sit)=-*J(t)m =!rJ(r)dr (4-1-5)/rL.t-r Thesignalsit)maybeviewedastheoutputofthefilterwithimpulseresponse 1lI(t)=m'-oo<t<oo (4-1-6) whenexcited bytheinputsignals(t).SuchafilteriscalledaHilbert transformer. Thefrequency response ofthisfilterissimply H(f)=f.h(t)e-j2"fldl 1f~1=- - e-j2,qrdl 1C~:rl {-j(f>0) =0(f=0) j(f<0) Weobservethat'H(f)1=1andthatthephaseresponseelf)=-!/rforf>0 andelf)=!1rforf<O.Therefore, thisfilterisbasically a90"phaseshifterfor allfrequencies intheinputsignal. . Theanalyticsignals+(/)isabandpass signal.Wemayobtainanequivalent lowpass representation byperforming afrequency translation ofS+(f).Thus, wedefineSt(f)as. Stlf)=S+(f+Ie) Theequivalent time-domain relation is s/(/)=s+(t)e-j2J<j;, =[S(/)+j.f(t)]e -j2"", or,equivalently,(4-1-8) (4-1-9) s(t)+j.f(t)=St(t)ei"2q,J (4-1-10) Ingeneral, thesignals/(t)iscomplex-valued (seeProblem 4-5),andmaybe expressed as St(t)=x(t)+jy(t) (4-1-11) (4-1-14)CH~PTER 4CH~RACTERIZATION OFCOMMl'NICATION SIGNALS ANDS',TEM\ 155 Ifwesubstitute forS,(I)in(4-1-11) andequaterealandimaginary pari,on eachside,weobtaintherelations S(I)=r(l)cos2Trt-I- y(t)sin2Trj;.t (4-1-12) 5(1)=X(I)sin2Trhl+y(t)cos2Trfct (4-1-13) Theexpression (4-1-12) isthedesiredformfortherepresentation ofa bandpass signal.Thelow-frequency signalcomponents X(I)andy(l)maybe viewedasamplitude modulations impressed onthecarriercomponents cos21ifJandsin2nfct,respectively. Sincethesecarriercomponents arein phasequadrature, r(t)andy(t)arecalledthequadrature components ofthe bandpass signals(t), Another representation ofthesignalin(4-1-12) is s(t)=Re([x(t)+jy(I)]ei2</.,} =Re[S,(I)eJ2"f.J ] whereRedenotestherealpartofthecomplex-valued quantity inthebrackets following, Thelowpasssignals,(t)isusuallycalledthecomplex envelope ofthe realsignals(t),andisbasically theeqUivalent lowpass signal, Finally,athirdpossible representation ofabandpass signalisobtained by expressing s,(t)as where Thenl'(t)=v?(t)+/(t) -Iy(t)9(t)=tan­x(t)(4-1-15) (4-1-11) (4-1-17) (4-1-19)s(t)=Re[S,(t)e'2"f.'] =Re[a(t)eJI2"f.t+8(r»)j =a(l)cos[2Trj;t+9(t)] (4-1-18) Thesignalart)iscailedtheenvelope ofs(t),and9(1)iscalledthephaseofs(t). Therefore, (4-1-12), (4-1-14), and(4-1-18) areequivalent representations of bandpass signals. TheFouriertransform ofset)is S(f)=fxs(t)e_i'"!'dt =fx{Re[s,(t)e-'2rif,J]}e-,2><fi dr Useoftheidentity ReW=H~+ ~*) (4-1-20) 156 DIGITAL COMMUNICATIONS in(4-1-19)yieldstheresult S(n=![[SI(t)ei2"!.'+sf(t)e-j2lrj,'je-j21r"dt2-x =HSM-f,J+Sf(-[-fc)J (4-1-21) (4-1-23)whereS,(f)istheFourier transform ofStet).Thisisthebasicrelationship between thespectrum oftlierealbandpass signalS(f)andthespectrum ofthe equivalent lowpasssignal5/(/), Theenergyinthesignalset)isdefinedas '€=[xs\t)dt =[x{Re[St(t)ei2"r"Wdt (4-1-22) Whentheidentityin(4-1-20)isusedin(4-1-22), weobtainthefollowing result: '€=![ Is,(tWdt2-x 1IX+-/s,(t)12cos[41rfct+28(t)]dl2-x Consider thesecondintegralin(4-1-23). Sincethesignals(t)isnarrowband, therealenvelope aCt)==ISI(r)1or,equivalently, a2(t)variesslowlyrelativeto therapidvariations exhibited bythecosinefunction. Agraphical illustration of theintegrand inthesecondintegral of(4-1-23)·isshowninFig.4-1-2.The valueoftheintegralisjustthenetareaunderthecosinefunction modulated bya2(t).Sincethemodulating waveform a2(t)variesslowlyrelativetothe cosinefunction, thenetareacontributed bythesecondintegralisverysmall relative tothevalueofthefirstintegral in(4-1-23) and,hence,itcanbe a2{t) I nGURE 4-1.~Thesignal.'(1)COS[4>rfct+21/(1)J, CHAPTER 4:CHARACfERIZATION OFCOMMVNICATION S1GNAL5 ANDSYSTEMS 157 neglected. Thus,forallpractical pruposes, theenergyinthebandpass signal set),expressed intermsoftheequ:valent lowpasssignals,(t),is (4-1-24) whereIs,(t)1isjusttheenvelope aCt)ofs(t). 4-1·2Representation ofLinearBandpass Systems Alinearfilterorsystemmaybedescribed eitherbyitsimpulseresponse h(t) orbyitsfrequency response H(f),whichistheFouriertransform ofh(l). Sinceh(t)isreal, H*(-f) =H(f) LetusdefineH,(f-j;.)as H,(f-J;)={OH(f) ([>0)([<0) Then {o ([>0) m(-[ -J;)=H*(-f)(f<0) Using(4-1-25), wehave H(f)=H,(f-f.)+m(-f-j;.)(4-1-25) (4-1-26) (4-1-27) (4-1-28) whichresembles (4-1-21) exceptforthefactor!.Theinversetransform of H(f)in(4-1-28)yieldsh(t)intheform h(t)=h,(t)ei2nt'+hf(t)e-j2ntJ =2Re[hlt)ei2<J;'] (4-1-29) whereh,(t)istheinverseFouriertransform ofH,(!).Ingeneral, theimpulse response h,(t)oftheequivalent lowpasssystemiscomplex-valued. 4-1·3Response ofaBandpass SystemtoaBandpass Signal InSections 4-1-1and4-1-2,wehaveshownthatnarrowband bandpass signals andsystemscanberepresented byequivalent lowpasssignalsandsystems. In thissection, wedemonstrate thattheoutputofabandpass systemtoa 158 DIGIT"L COMMUNIC"nONS bandpass inputsignalissimplyobtained fromtheequivalent lowpass input signalandtheequivalent lowpassimpulseresponse ofthesystem. Suppose thatset)isanarrowband bandpass signalands,(t)istheequivalent lowpasssignal.Thissignalexcitesanarrowband bandpass systemcharacterized byitsbandpass impulse response h(t)orbyitsequivalent lowpassimpulse response h,(t).Theoutputofthebandpass systemisalsoabandpass signal, and,therefore, itcan 1-,~expressed intheform (4-1-30) whereret)isrelatedtotheinputsignalset)andtheimpulseresponse h(t)by theconvolution integral ret)=fxs(~)h(t-~)d~ (4-1-31) Equivalently, theoutputofthesystem,expressed inthefrequency domain, is R(f)=S(f)H(f) (4-1-32) Substituting from(4-1-21)forS(f)andfrom(4-1-28)forH(f),weobtainthe result R(f)=!fSt(f-f,.)+Sr(-f-.fc)J[H,U-fc)+H1(-f-/;»)(4-1-33) Whenset)isanarrowband signalandh(t)istheimpulseresponse ofa narrowband system,S,(f-fc)~0andH,(f-fc)=0for/<O.Itfollowsfrom Ihisnarrowband condition that S,(/-fc)Hf(-f-fc)=0,Sf(-f-fc)H,(f-fc)=0 Therefore, (4-1-33)simplifies to R(f)=US,(f-fc)H,(f-fc)+Sr(-f-fc)Hf(-/-fc)] =UR,(f-fc)+Rf(-f-fc)] where R,(f)=S,(f)H,(f)(4-1-34) (4-1-35) istheoutputspectrum oftheequivalent lowpass systemexcitedbythe equivalent lowpass signal.Itisclearthatthetimedomain relationforthe outputr,(r)isgivenbytheconvolution ofs,(t)withh,(t).Thatis, T,(t)=fxSI('()h,(!-r)dT (4-1-36) CHAPTER 4CHARA(TERLZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 159 Thecombination of(4-1-36) with(4-1-30)givestherelationship between thebandpass outputsignalr(t)andtheequivalent lowpasstimefunctions St(t) andh,(J).Thissimplerelationship allowsustoignoreanylinearfrequency translations encountered inthemodulation ofasignalforpurposes of matching itsspectral content tothefrequency allocation ofaparticular channel. Thus.formathematical convenience. weshalldealonlywiththe transmission ofequivalant lowpass signalsthrough equivalent lowpass channels. 4-1-4Representation ofBandpass Stationary Stochastic Processes Therepresentation ofbandpass signalspresented inSection4-1·1appliedto deterministic signals.Inthissection,weextendtherepresentation tosample functions ofabandpass stationary stochastic process. Inparticular, wederive theimportant relations between thecorrelation functions andpowerspectraof thebandpass signalandthecorrelation functions andpowerspectraofthe equivalent lowpasssignal. Suppose thatn(t)isasamplefunction ofawide-sl;nse stationary stochastic processwithzeromeanandpowerspectraldensityet>".(f).Thepowerspectral densityisassumed tobezerooutsideofanintervaloffrequencies centered around±f,.,wheref,.istermedthecarrierfrequency. Thestochastic process nIt)issaidtobeanarrowband fJandpass processifthewidthofthespectral densityismuchsmallerthan[...Underthiscondition, asamplefunctionofthe processnIt)canberepresented byanyofthethreeequivalent formsgivenin Section4-1-1,namely, nIt)=aIr)cos[2Jt'[.t+B(t)] =x(t)cos2Tif,.t-y(c)sin2Tif,.t =Re[z(r)e'2Kt.'](4-1-37) (4-1-38) (4-1-39) wherea(t)istheenvelope andB(t}isthephaseofthereal-valued signal.x(t) andy(t)arethequadrature components ofn(t).andzIt)iscalledthecomplex envelope ofnIt). Letusconsider theformgivenby(4-1-38)inmoredetail.First,weobserve thatifnIt)iszeromean.thenX(I)andy(t)mustalsohavezeromeanvalues. Inaddition, thestationarity ofn(t)impliesthattheautocorrelation and cross-correlation functions ofx(t)andY(I)satisfythefollowing properties: cPx.(r)=cPy,(r) cP.,.(r)=-cPyx(r)(4-1-40) (4.1-41) 160 DIGITAL COMMIJNICATIONS Thatthesetwoproperties followfromthestationarity ofn(t)isnow demonstrated. Theautocorrelation function c/JnAt")ofn(t)is E[n(t)n(t +r)]=E{[x(t)cos2nfct-y(t)sin21tj;.t] XIx(t+r)cos2nfc(t+r) -y(1+r)sin21tfc(t+r)]} ="'XI(r)cos21tfctcos21tfc(r+r) +<Py>'(r)sin2Jifctsin2nf(t+r) -<p",(r)sin2JrfJcos2nfc(t+r) -<p,,(r)cos2Itj;tsin2nfc(t +r) (4-1-42) Useofthetrigonometric identities COSAcosB=Hcos(A-B)+cos(A+B)] sinAsinB=Hcos(A-B)-cos(A+B)] sinAcosB=Hsin(A -B)+sin(A+B)] in(4-1-42)yieldstheresult E[n(t)n(t+r)1=H<t>xx(r)+c/Jy'(r)Icos21tj;r +U</>,,(r)-"'y,(r)1cos21tf.,(2t+r) -H</>.,(r)-"',,(r)]sin21tf.,r , , -H</>yAr)+c/J,y(r)}sin21tf.(2t+r)(4-1-43) (4-1-44) Sincen(t)isstationary, theright-hand sideof(4-1-44)mustbeindependent of t.Butthiscondition canonlybesatisfied if(4-1-40) and(4-1-41) hold.Asa consequence, (4-1-44)reducesto c/Jnn(r)=c/JxAr)cos21rfcr-c/Jvx(r)sin21tj;,r (4-1-45) Wenotethattherelationbetween theautocorrelation function <t>nn(r)ofthe bandpass process andtheautocorrelation andcross-correlation functions <t>xx(r)andc/JyX< r)ofthequadrature components isidentical informto (4-1-38), whichexpresses thebandpass processintermsofthequadrature components. Theautocorrelation functionoftheequivalent lowpassprocess isdefinedas,(I)=x(t)+jy(t) "',,(r)=!E[z*(t)z(t +r)}(4-1-46) (4-1-47) CHAPrER 4:CHARACTERIZATiON Of·COMMLJNICATION SIGNALS ANDSYSTEMS 161 Substituting (4-1-46) into(4-1-47) andperforming theexpectation operation. weobtain cP,,(r)=HcP,,(r)+cP,,(r)-jcPXy(r)+icP,,(r)] (4-1-48) Nowifthesymmetry properties givenin(4-1-40) and(4-1-41) areusedin (4-1-48), weobtain cP,,(r)=cPxAr)+j</JyxCr) (4-1-49) whichrelatestheautocorrelation function ofthecomplex envelope tothe autocorrelation andcross-correlation functions ofthequadrature components. Finally,weincorporate theresultgivenby(4-1-49)into(4-1-45), andwehave rf>",,(T)=Re[<I>,,(T)ei2<f.'] (4-1-50) Thus.theautocorrelation function cP".(r)ofthebandpass stochastic processis uniquely determined fromtheautocorrelation function cP,,(r)oftheequiv­ alentlowpassprocessz(t)andthecarrierfrequencyt. Thepowerdensityspectrum ct>",,(f)ofthestochastic processnCr)isthe Fouriertransform of"'"n(r).Hence, <p.,,(f)=fx{Re[rf>"(r)el2nr.']}e-i2nr,dr =H<P,,(f-j;.)+<P:.,(-f-}:» (4-1-51) where<P,,(f)isthepowerdensityspectrum oftheequivalent lowpassprocess z(t).Sincetheautocorrelation function ofz(t)satisfiestheproperty rf>,,(r)= <I>~J-r).itfollowsthat<P"(f)isareal-valued function offrequency. Properlies oftheQuadrature Components Itwasjustdemonstrated abovethatthecross-correlation function ofthequadrature components x(t) andy(t)ofthebandpass stationary stochastic process net)satisfies the symmetry condition in(4-1-41). Furthermore. anycross-correlation function satisfiesthecondition <1>,,(r)=cPx,(-r) Fromthesetwoconditions, weconclude that cP•.,(r)= -cP.;,(-r)(4-1·52) (4-1-53) Thatis.cP"(r)isanoddfunction ofr.Consequently. 4>".(0)=O.and.hence, x(t)andyet)areuncorrelated (forr=O.only).Ofcourse,thisdoesnolmean thattheprocesses x(r)andyet+r)areuncorrelated forallr.sincethatwould implythat<I>".(r)=0forallr.If,indeed,<I>.y(r)=0forallr.thencP,,(r)is realandthepowerspectraldensity<P,,(f)satisfiesthecondition (4-1-54) andviceversa.Thatis,<P,,(f)issymmetric aboutf=O. 162 DIGITAL COMMUNICATIONS Inthespecialcaseinwhichthestationary stochastic processn(t)isgaussian, thequadrature components x(t)andy(t+r)arejointlygaussian. Moreover, forT=O.theyarestatistically independent, and,hence,theirjointprobability densityfunction is ()I-ltZ-t.,'lI2CT2Px.y=--e . 21W2 wherethevariance (I'isdefined as(I'=4>x.,(O)=4>,,(0)=4>.,,(0).(4-1-55) (4-1-56)Representation ofWhiteNoiseWhitenoiseisastochastic processthatis defined tohaveaflat(constant) powerspectral density overtheentire frequency range.Thistypeofnoisecannotbeexpressed intermsofquadrature components. asaresultofitswideband character. Inproblems concerned withthedemodulation ofnarrowband signalsin noise,itismathematically convenient tomodeltheadditive noiseprocessas whiteandtorepresent thenoiseintermsofquadrature components. Thiscan beaccomplished bypostulating thatthesignalsandnoiseatthereceiving terminal havepassedthroughanidealbandpass filter,havingapassband that includes thespectrum ofthesignalsbutismuchwider.Suchafilterwill introduce negligible, ifany,distortion onthesignalbutitdoeseliminate the noisefrequency components outsideofthepassband. Thenoiseresulting frompassing thewhitenoiseprocess through a spectrally flatOdeal)bandpass filteristermedbandpass whitenoiseandhasthe powerspectral densitydepicted inFig.4-1-3.Bandpass whitenoisecanbe represented byanyoftheformsgivenin(4-1-37), (4-1-38), and(4-1-39). The equivalent lowpass noisez(t)hasapower'spectral density {N,}(IJI"'",iB) ct>,,(J)=0 (JJI>~B) anditsautocorrelation function is 4>)=N_si_n_Tr_B_r,,(r01fr Thelimitingformof</J,,(r)asBapproaches infinityis 4>Jr)=NoS(r) cJ),,,,I/1(4-1-57) (4-1-58) FIGURE 4-1-3Bandpass noisewithaftatspectrum. (HAPTEH 4CHARA(TERIZATl()~ OFCOMML~ICATIOf\; SIGNALS ANDS'rSTEMS 163 Thepowerspectral densityforwhitenoiseandbandpass whitenoiseis symmetric ahoutf=o.so<P"(r)=0forallr.Therefore. <p,,(r)=cPxx(r)=<p,,(r) (4-1-59) Thatis,thequadrature components x(t)andy(t)areuncorrelated foralltime shiftsrandtheautocorrelation functions ofz(t),x(r),andy(r)areallequal. 4-2SIGNAL SPACE REPRESENTATIONS Inthissection.wedemonstrate thatsignalshavecharacteristics thataresimilar tovectorsanddevelop avectorrepresentation forsignalwaveforms. Webegin withsomebasicdefinitions andconcepts involving vectors. 4-2-1VectorSpaceConcepts Avectorvinann-dimensional spaceischaracterized byitsncomponents [UIV2.•.v,,].Itmayalsoberepresented asalinearcombination ofunit vecrorsorbasisveerorse"I'"i'"n,i.e., "v=2:v,e, i"'-l(4-2-1) where,bydefinition. aunitvectorhaslengthunityandv,istheprojection of thevectorvontotheunitvectore,. Theinnerproduct oftwon-dimensional vectors VI=[v"Vl2...v,,,]and v,=[v"V22..•v2,,]isdefinedas ""I·V:c=2:V1iV2i j..--I(4-2-2) Twovectorsv,andV2areorthogonal ifv,.V,=O.Moregenerally. asetofm vectorsvk.1'"k'"m,areorthogonal if V,•VI=() (4-2-3) (4-2-4)forallI'"i,j'"mand,..j. Thenormofavectorvisdenoted byIIvIIandisdefinedas Ilvll=(v·V)112=~~,v,' whichissimplyitslength.Asetofmvectorsiss3idtobeorthonormal ifthe vectorsareorthogonal andeachvectorhasaunitnorm.Asetofmvectors is saidtobelinearlyindependent ifnooneveclorcanberepresented asalinear combination oftheremaining vectors. Twon-dimensional vectorsv,andV2satisfythetriangleinequality Ilv,+v,lI'",Iv,II+II'V,II (4-2-5) withequality ifv,andv,areinthesamedirection. i.e.,VI=U'V,whereaisa 164 DIGITAL COMMUNICATIONS positiverealscalar.Fromthetriangleinequality therefollowstheCou£hy­ Schwartz inequality (4-2-6) withequality if1'\=aV2'Thenormsquareofthesumoftwovectorsmaybe expressed as /Iv,+v2f=111',1/2+/11'2/12+21'1'1'2 If1'\and1'2areorthogonal thenv,•1'2=0and,hence, "v,+1'2112=111',112+111'2112(4-2-7) (4-2-8) ThisisthePythagorean relationfortwoorthogonal n-dimensional vectors. Frommatrixalgebra, werecallthatalineartransformation inann­ dimensional vectorspaceisamatrixtransformation oftheform v'=Av (4-2~) wherethematrixAtransforms thevectorvintosomevectorv'.Inthespecial casewherev'=Av,i.e., Av=Av (4-2-10) where,\issome(positive ornegative) scalar,thevectorviscalledan eigenvector ofthetransformation andAisthecorresponding eigenvalue. Finally,letusreviewtheGram-Schmidt procedure forconstructing asetof orthonormal vectorsfromasetofn-dimensional vectors Vi'1...i...m.We beginbyarbitrarily selecting avectorfromtheset,sayv,.Bynormalizing its length,weobtainthefirstvector,say (4-2-11) Next,wemayselect1'2and,first,subtracttheprojection of"2ontou,.Thus,we obtain u~=1'2-(1'2'udu, Then,wenormalize thevector u~tounitlength.Thisyields(4-2-12) (4-2-13) Theprocedure continues byselecting 1'3andsubtracting theprojections of 1'3intoU,andU2'Thus,wehave u~=v,-(",'U,)UI-(1'3'82)U2 Then,theorthonormal vectorU3is u~ 8--- 3-IIU~II(4-2-14) (4-2-15) CHAPTER 4:CHARACfERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS, 165 Bycontinuing thisprocedure, weshallconstruct asetofn,.orthonormal vectors,wheren,""n,ingeneral.Ifm<nthenn,";;m.andifm;;.nthen nl~n. 4-2-2SignalSpaceConcepts Asinthecaseofvectors, wemaydevelopaparalleltreatment forasetof signalsdefinedonsomeinterval [a,b).Theinnerproductoftwogenerally complex-valued signalsx,(t)andx,(t)isdenoted by(x,(t).X2(t»anddefined as (x,(t).X2(t)=rx,(t)x!(t) dt • Thesignalsareorthogonal iftheirinnerproductiszero. Thenormofasignalisdefinedas (rb)"2Ilx(t)11=LIx(t)12dt a(4-2-16) (4-2-17) Asetofmsignalsareorthonormal iftheyareorthogonal andtheirnormsare allunity.Asetofmsignalsislinearly independent, ifnosignalcanbe representedasalinearcombination oftheremaining signals. Thetriangle inequaliry fortwosignalsissimply Ilx,(t)+X2(t)II""'lx,(t)1I+IIx2(t)II andtheCauchy~Schwartz inequality is Ifx,(I)X!(I)dtl,,;;If/X,(t)/2dtl112If/X2(tWd,I'12 withequalitywhenX2(t)=ax,(t).whereaisanycomplex number.(4-2-18) (4-2-19) 4-2-3Orthogonal Expansions ofSignals Inthissection,wedevelopavectorrepresentation forsignalwaveforms, and, thus,wedemonstrate anequivalence betweenasignalwaveform anditsvector representation. Suppose thatS(I)isadeterministic, real-valued signalwithfiniteenergy (4-2-20) Furthermore, suppose thatthereexistsasetoffunctions {[.(t).n= 1,2,...,N}thatareorthonormal inthesensethat Ix {O(m""n)_}(t)f..(t)dt=1(m=n)(4-2-21) 166 OIGllAI. COMMUNICATiONS Wemayapproximate thesignals(t)byaweighted linearcombination of thesefunctions, i.e., K s(t)=2:s.t.(t) k=1(4-2-22) where{s..1'"k'"K}arethecoefficients intheapproximation ofs(t).The approximation errorincurred is e(t)=s(t)-s(t) (4-2-23) (4-2-24)Letusselectthecoefficients {s.}soastominimize theenergy ~,ofthe approximation error.Thus, ~,=[[s(t)-s(t)fdt =[[S(t) -~ls.f.(t)rdt Theoptimum coefficients intheseriesexpansion ofs(t)maybefoundby differentiating (4-2-24)withrespecttoeachofthecoefficients {s.}andsetting thefirstderivatives tozero.Alternatively, wemayuseawell-known result .fromestimation theorybasedonthemean-square-error criterion, which, simplystated,isthattheminimum of~withrespecttothe{s.}isobtained whentheerrorisorthogonal toeachofthefunctions intheseriesexpansion. Thus, [[S(t) -~ls.f.(t)Yn(t)dt=0,n=1,2,...,K Sincethefunctions {fn(t)}areorthonormal, (4-2-25)reducesto s.=[~S(t)[.(t) dt,n=1,2,...,K(4-2-25) (4-2-26) Thus,thecoefficients areobtained byprojecting thesignals(t)ontoeachofthe functions {[.(t)}.Consequently, 1(t)istheprojection ofs(t)ontothe K-dimensional signalspacespanned bythefunctions If.(t)}.Theminimum meansquareapproximation erroris ~mi.=[~e(t)s(t)dt =[[S(t»)2 dt-[~~1s.fk(t)s(t) dt (4-2-27) whichisnonnegative, bydefinition. CHAPTER 4.CHARACTERIZATION OFCOMMUNICAT[ON SIGNALS ANDSYSTEMS 167 Whentheminimum meansquareapproximation error ~mi"=0, 'f,=;±,S(=I",[s(eifdt Underthecondition thatlI'm'"=0,wemayexpresssit)as I< s(t)=2:s.!,(t) A:~-I(4-2-28) (4-2-29) (4-2-30) (4-2-31) (4-2-32)whereitisunderstood thatequalityofsit)toitsseriesexpansion holdsinthe sensethattheapproximation errorhaszeroenergy. Wheneveryfiniteenergysignalcanberepresented byaseriesexpansion of theformin(4-2-29)forwhichlI'mm=0,thesetoforthonormal functions {f,,(t)} issaidtobecomplete. Example 4·2-1:Trigonometric FourierSeries Afiniteenergysignals(t)thatiszeroeverywhere exceptintherange (J.,t"Tandhasafinitenumberofdiscontinuities inthisinterval. canbe represented inaFourierseriesas ,(2/tkt 2rrkf")s(t)=2:a.cos--+b,sin--"I T T wherethecoefficients {a,.b,}thatminimize themeansquareerroraregiven by 1[/ 2rrkea,=.r;;s(e)cos--dt vT\l T 1IT 2rrktb,=.r;;sit)sin--dtvT" T Thesetoftrigonometric functions {Y2/Tcos2rrke/T, V2/Tsin2rrke/T} is complete, and,hence,theseriesexpansion resultsinzeromeansquare error.Theseproperties areeasilyestablished fromthedevelopment given above, Gram-Schmidt Procedure Nowsuppose thatwehaveasetoffinite energysignalwaveforms {si(e),i=1,2,." ,M}andwewishtoconstruct aset oforthonormal waveforms. TheGram-Schmidt orthogonalization procedure allowsustoconstruct suchaset.Webeginwiththefirstwaveform slit),which isassumed tohaveenergy 11:,.Thefirstwaveform issimplyconstructed as ~Ie_s,(e)I)-~ Thus,/;(1)is.simply s,(t)normalized tounitenergy. 168 DIGITAL COMMUNICATIONS Thesecondwaveform isconstructed fromS2(t)byfirstcomputing the projection ofNt)ontoS2(t),whichis C'2=fxs,(I)/I(I)dl Then,c:zf,(t)issubtracted fromS2(1)toyield /;(t)=S2(1)-cI2NI)(4-2-33) (4-2-34) Thiswaveform isorthogonal to[,(t)butitdoesnothaveunitenergy.If ~denotestheenergyof1;(1),thenormalized waveform thatisorthogonal to [.(1)is 12(1)=~ Ingeneral,theorthogonalization ofthekthfunction leadsto !«t)=r:):J. where ,-I f~(t)=s,(t)-2:c,.[,(t) ;=1 and c"=fxs,(t)[,(t) dt,i=I,2,...,k-1(4-2-35) (4-2-36) (4-2-37) (4-2-38) Thus,theorthogonalization process iscontinued untilalltheMsignal waveforms {S;(I)}havebeenexhausted andN~Morthonormal waveforms havebeenconstructed. Thedimensionality Nofthesignalspacewillbeequal toMifallthesignalwaveforms arelinearlyindependent, i.e.,noneofthe signalswaveforms isalinearcombination oftheothersignalwaveforms. Example 4-2-2 LetusapplytheGram-Schmidt procedure tothesetoffourwaveforms illustrated iiiFig.4-2-1(0). Thewaveform SI(t)hasenergyg\=2,sothat f,(t)=VIs\(I).Next.weobserve thatC\2=0;hence,S2(1)and1,(1)are orthogonal. Therefore, f2(1)=sit)/Vi;. =VIS2(t).Toobtain[,(1),we compute C\landCn.whichareCI1=v'2andc"=O.Thus, {-I(2~I~3) f~(I)=S3(1)-Y2f,(t) = .o(otherwise) CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIG!"iALS ANDSYSTEMS 169 0 0 2'I-I ·{2((1 "'4((1 o -I~210 3 -I (a) hOI ff>-------~ o o -I I~(r)=0 oIf1 (hi Gram~Schmidt orthogonalization ofthesignals {Si(r).i==1.~.3.4}andthecorresponding orthogonal signals. Since[;(t)hasunitenergy,itfollowsthatJ,(r)=/;(1).Indetermining /.(1), wefindthatC'4=-V2.C24=0,andCJ•=I.Hence, /~(t)=s4(r)+V2f,(I)-f,(t)=0 Consequently, s,(I)isalinearcombination of[,(I)andh(l)and.hence, h(l)=0.Thethreeorthonormal functions areillustrated inFig,4-2-1(b). 170 D]<;rJAL. CO~l~W~1CATIONS Oncewehaveconstructed thesetoforthonormal waveforms (l,(I)}.wecan express theMsignals{s,,(I)}aslinearcombinations ofthe{f,,(I)l.Thus.wemay write and"I,(I)=.z::I",f;,(I),,,k=1,2.....M (4-2-39) f·"f,=[.1,(1)]'dl=.z::.Ii"=lis,II' .'. /II(4-2-40) Basedontheexpression in(4-2-39). eachsignalmayberepresented bythe vector s,=[5".\"...1,,] (4-2-41) or,equivalently. asapointinthelV-dimensional signalspacewithcoordinates {5,,,;=1,2,...,N}.Theenergyinthekthsignalissimplythesquareofthe lengthofthevectoror.equivalently_ thesquareoftheEuclidean distance from theorigintothepointintheN-dimensional space_Thus.anysignalcanbe represented geometrically asapointinthesignalspacespanned bythe orthonormal functions (l,(r)}. Example 4-2·3 Letusobtainthevectorrepresentation ofthefoursignalsshowninFig. 4-2-I(a) byusingtheorthonormal setoffunctions inFig.4-2-I(h). Sincethe dimensionality ofthesignalspaceislV=3,eachsignalisdescribed b~vthree components. Thesignals,(I)ischaracterized bythevectors,=(V2•O.OJ. Similarly. thesignals5,(1).s,(I).and"4(1)arecharacterized bvthevectors s,=(0.v1.0).5,=(V2.0,1).andS4=(-\,;'2.0.1),respectively. These vectors areshowninFig.4-2-2.Theirlengths arcIs,1=\11.15.1=\2. (. \. FIGURE 4-2-2Thefoursignalvectors r~prcst:ll(eJ asp()inh In three dim~nsional function 'pact', CHAPTER 4:CHARACTERIZATION OFCOMMU ....ICATION SIGNALS ANDSYSTEMS 171 Is,1=v'3,and15,1=v'3,andthecorresponding signalenergies areg.=Is.i2 , k=I,2,3,4. Wehavedemonstrated thatasetofMfiniteenergywaveforms {sn(t)}can berepresented byaweighted linearcombination oforthonormal functions {J,.(r)}ofdimensionality N,,;;M.Thefunctions {f,,(/)}areobtained byapplying theGram-Schmidt orthogonalization procedure on{sn(t)}.Itshouldbe emphasized, however, that'thefunctions {fn(I)}obtained fromtheGram­ Schmidt procedure arenotunique.Ifwealtertheorderinwhichthe orthogonalization ofthesignals{Sn(I)}isperformed, theorthonormal wave­ formswillbedifferent andthecorresponding vectorrepresentation ofthe signals{sn(t)}willdependonthechoiceoftheorthonormal functions {f,,(I)}. Nevertheless, thevectors{s.}willretaintheirgeometrical configuration and theirlengthswillbeinvariant tothechoiceoforthonormal functions {[n(I)}. Example 4-2-4 Analternative setoforthonormal functions forthefoursignalsinFig.4-2-1 isillustrated inFig.4-2-3(a). Byusingthesefunctions toexpand..{sn(1I},we FIGURE 4-2·3Analternative setoforthonormal functions forthefoursignalsinFig.4-2-1(a)andthe corresponding signalpoints. (a, Sl=l-l.-I.-l:,,, "--_~'-_"'" s!=:(I.-I.0) (h, 172 DIGITAL CO""L~ICAnONS obtainthecorresponding vectors 51=(1.1,0),52=(l,-I,0),53=(l,1,-1), andS4=(-L-L-1),whichareshowninFig.4-2-3(b). Notethatthe vectorlengths areidentical tothoseobtained fromtheorthonormal functions U;,(t)}. Theorthogonal expansions described aboveweredeveloped forreal-valued signalwaveforms. Theextension tocomplex-valued signalwaveforms isleftas anexercise forthereader(seeProblems 4-6and4-7). Finally.letusconsider thecaseinwhichthesignalwaveforms arebandpass andrepresented as S",(I)=Re[s,..,(t)e"""). m=1.2,....M (4-2-42) where{s,..,(t)}denotetheequivalent lowpass signals. Recallthatthesignal energies may.beexpressed eitherintermsof5",(1)orS,..,(I).as IfX =2.,IS,,,,(I)I'dl (4-2-43) Thesimilarity between anypairofsignalwaveforms. say5",(1)and5k(I).is measured bythenormalized cross-correlation ~r5",(I)S,(I) dl=Re{,vhJ'5,..,(l)slHl)dl} V'f'..mfl.: -x _~1lt-J. ..:c Wedefinethecomplex-valued cross-correlation coefficienr p,..,as Then, or,equivalently,(4-2-44) (4-2-45) (4-2-46) Sill•SA- Sill•SI.- 1IS",/IIiskII=V«;"ifk(4-2-47) Thecross-correlation coefficients between pairsofsignalwaveforms or signalvectorscomprise onesetofparameters thatcharacterize thesimilarity (4-2-48)CHAPTER 4:CHARACTERrZATfON OFCOMMUNfCATlON SiGNALS ANDS~'STEMS 173 ofasetofsignals.Another relatedparameter istheEuclidean distance d~'j, between apairofsignals,definedas d~'j,=Iism-s.1I ={[[sm(t)-s.(tlfdtr ={If",+iek-2v'~mgkRe(Pkm»)112 Whenii'",='if!.=gforallmandk.thisexpression simplifies to d~~:.={Ur1-Re(Pkm)j}"2 (4-2-49) (4-2-50)Thus,theEuclidean distance isanalternative measure {)fthesimilarity (or dissimilarity) ofIheselofsignalwaveforms orthecorresponding signal vectors. Inthefollowing section. wedescribe digitally modulated signalsandmake useofthesignalspacerepresentation forsuchsignals.Weshallobserve thaI digitally modulated signals, whichareclassified aslinear,areconveniently eJlpanded intermsoftwoorthonormal basisfunctions oftheform j,(t)=~~cos 21CfJ f,(t)=-~sin2Jrf,t Hence,ifs'm(t)isexpressed ass,..,(t)=x,(t)+jYi(t),itfollows thats..,(t)in (4-2-42) maybeexpressed as (4-2-51) wherex,(t)andy,(t)represent thesignalmodulations. 4-3REPRESENTATION OFDIGITALLY MODULATED SIGNALS Inthetransmission ofdigitalinformation overacommunications channel, the modulator istheinterface devicethatmapsthedigitalinformation intoanalog waveforms thatmatchthecharacteristics ofthechannel. Themapping is generally performed bytakingblocksofk=logzMbinarydigitsatatimefrom theinformation sequence {a,,}andselecting oneofM=2kdeterministic. finite energywaveforms (s..,(t).m=1.2,...,M}fortransmission overthechannel. Whenthemapping fromthedigitalsequence {an}towaveforms is performed undertheconstraint thatawaveform transmilled inanytime interval depends ononeormorepreviously transmilled waveforms. the modulator issaidtohavememory. Ontheotherhand,whenthemapping 174 DIGITAL COMMUNICATIONS. fromthesequence {a.}tothewaveforms {s",(t)}isperformed withoutany constraint onpreviously transmitted waveforms, themodulator iscalled memoryless. Inaddition toclassifying themodulator aseithermemoryless orhaving memory, wemayclassifyitaseitherlinearornonlinear. Linearity ofa modulation methodrequiresthattheprinciple ofsuperposition appliesinthe mapping ofthedigitalsequence intosuccessive waveforms. Innonlinear modulation, thesuperposition principle doesnotapplytosignalstransmitted in successive timeintervals. Weshallbeginbydescribing memoryless modulation methods. 4-3-1Memoryless Modulation Methods Asindicated above,themodulator inadigitalcommunication systemmapsa sequence ofbinarydigitsintoasetofcorresponding signalwaveforms. These waveforms maydifferineitheramplitude orinphaseorinfrequency, orsome combination oftwoormoresignalparameters. Weconsider eachofthese signaltypesseparately, beginning withdigitalpulseamplitude modulation (PAM).Inallcases,weassumethatthesequence ofbinarydigitsattheinput tothemodulator occursatarateofRbits/so PulseAmplitude Modu18ted (PAM)Signals IndigitalPAM,thesignal waveforms mayberepresented as s",(t)=Re[A",g(t)ei2""'] =A",g(/)cos21rj;.t,m=1,2•...,M.O.;;/.;;T (4-3-1) where{Am'1.;;m.;;M}denotethesetofMpossibleamplitudes corresponding toM=2kpossiblek-bitblocksorsymbols. Thesignalamplitudes Amtakethe discretevalues(levels) Am=(2m-l-M)d, m=I,2;...,M (4-3-2) where2disthedistance between adjacent signalamplitudes. Thewaveform g(t)isareal-valued signalpulsewhoseshapeinfluences thespectrum ofthe transmitted signal.asweshallobservelater.ThesymbolrateforthePAM signalisR/k.Thisistherateatwhichchangesoccurintheamplitude ofthe carriertoreflectthetransmission ofnewinformation. Thetimeinterval Tb=1/RiscalledthebitintervalandthetimeintervalT=k/R=kTbiscalled thesymbolinterval. TheMPAMsignalshaveenergies ~=fs~(t)dt =!A~fg2(t)dt =!A;'~g (4-3-3) CHAPTER 4CHARACTERIZATION OFCOMML'''ICATION SIGNALS ANDSYSTEMS 175 0• (il)!If=l (Xl UI II 10•• • •• (MM=4 000OUI 011010 110III 101100 FIGURE 4-3-1Signalspacediagram fordigitalPAMsignals. tdM=8 where't:.denotes theenergyinthepulseg(c).Clearly, thesesignalsare one-dimensional (N=I).and,hence,arerepresented bythegeneralform S,,,(c)=s".f(c) whereftc)isdefinedastheunit-energy signalwaveform givenas f(t)=/2g(c)cos2Jrf..1\j'{;K and Sm=A",~,m=1.2.....M(4-3-4) (4-3-5) (4-3-6) (4-3-7)IThecorresponding signalspacediagrams forM=2.M=4andM=Hare showninFig.4-3-1.DigitalPAMisalsocalledamplitude-shift keying(ASK). •Themapping orassignment ofkinformation bitstotheM=2'possihle signalamplitudes maybedoneinanumber ofways.Thepreferred assignment isoneinwhichtheadjacent signalsamplitudes differbyonebinarydigitas illustrated inFig.4-3-J.Thismapping iscalledGrayencoding. Itisimportant inthedemodulation ofthesignalbecause themostlikelyerrorscausedhy noiseinvolve theerroneous selection ofanadjacent amplitude tothe transmitted signalamplitude. Insuchacase,onlyasinglebiterroroccursin thek-bitsequence. WenotethattheEuclidean distance between anypairofsignalpointsis d~:r~=Vi(sm-sn)2 =V~jg.lAm-A"I =d~lm -/11 Hence,thedistance between apairofadjacent signalpoints.i.e..theminimum Euclidean distance, is (4-H) 176 DlOlTAlCOM"UNICA TIONS Thecarrier-modulated PAMsignalrepresented by(4-3-1)isadouble­ sideband (DSB)signalandrequires twicethechannel bandwidth ofthe equivalent lowpass signalfortransmission. Alternatively, wemayusesingle­ sideband (SSB)PAM,whichhastherepresentation (loweroruppersideband). sm(t)=Re{A",[g(t)±jg(t»)e1''!.'},m=1,2,...,M(4-3-9) whereget)istheHilberttransform ofget).Thus,thebandwidth oftheSSB signalishalfthatoftheDSBsignal. ThedigitalPAMsignalisalsoappropriate fortransmission overachannel thatdoesnotrequirecarriermodulation. Inthiscase,thesignalwaveform.may besimplyrepresented as Sm(t)=A..g(t),m=1.2,...•M (4-3-10) Thisisnowcalledabaseband signal.Forexample afour-amplitude level baseband PAM'signalisillustrated inFig.4-3-2(a). Thecarrier-modulated versionofthesignalisshowninFig.4-3-2(b). InthespecialcaseofM=2signals,thebinaryPAMwaveforms havethe specialproperty that FIGURE 4-3-2Baseband andbandpass PAMsignals. Signal amplitude oI I 0 T 2T JT14T S7 6T I Data:II 10 00 01 (a)llascband PAM.ignalII 00 ..---. I I I , I , I • I I OJtttHtitHIiftHtittHtiftt1f+tfttjltH-~------ (b)Ilondpass PAMsignal CHAPTER 4CHARACfERIZATION OFCOMMUNICAllON SiGNALS ANDSYSTEMS 177 Hence,thesetwosignalshavethesameenergyandacross-correlation coefficient of-I.Suchsignalsarecalledantipodal. Phase·Modulated Signals Indigitalphasemodulation, theMsignal waveforms arerepresented as s,.,(t)=Re[g(t)el''''I'''' (liMe!''''':'], m=1,2,...,M,0.;;t.;;T =g(t)cos[2Jr[.t+~(m-I)] 21r .21r . =g(t)cos-(m-I)cos21if,.t-g(t)Sin-(m-I)SIn21ifctM M (4·3-11) whereg(t)isthesignalpulseshapeand8m=21r(m-1)/M,m=I,2,...,M, aretheMpossible phasesofthecarrier'thatconveythetransmitted information. Digitalphasemodulation isusuallycalledphase-shift keying (PSK). Wenotethatthesesignalwaveforms haveequalenergy,i.e., 1IT=-g'(t)dt=1~.2\)(4-3-12) Furthermore, thesignalwaveforms mayberepresented asalinearcombination oftwo-orth~normal signalwaveforms, f,(r)and[,(r),i.e., where Nt)=~g(t)cos21ifct (2.[,(t)=-'Ji;,g(r)SID2trfct i andthetwo-dimensional vectors Sm=[sm(sm']aregivenby(4-3-13) (4-3-14) (4-3-15) m=I,2,...,M (4-3-16) 178 DUi!TAL COMl.1lJNIC'ATIONS II 01001100: M=2 • • 1\0 000 (JI III IIlU• • 101 II IlO M=R FIGURE 4-3-3Signalspacediagrams (orPSKsignals.10 M=:4 Signalspacediagrams forM=2,4,and8areshowninFig.4-3-3.Wenotethat M=2corresponds toone-dimensional signals,whichareidentical tobinary PAMsignals. AsisthecaseofPAM,themapping orassignment ofkinformation bitsto theM=2'possible phasesmaybedoneinanumberofways.Thepreferred assignment isGrayencoding, sothatthemostlikelyerrorscausedbynoisewill resultinasinglebiterrorinthek-bitsymbol. TheEuclidean distance between signalpointsis { [2Jr]}"2='t,I -cosM(m-n) (4-3-17) Theminimum Euclidean distance corresponds tothecaseinwhich1m-IIi=I, i.e.,adjacent signalphases.Inthiscase, (<'I_I.(_2Jr)d"H"-\jj(.1cosM (4-3·IH) Quadrature Amplitude Modulation Thebandwidth efficiency ofPAMI SSBcanalsobeobtained bysimultaneously impressing twoseparate k-bit symbols fromtheinformation sequence {au}ontwoquadrature carriers (4-3-20)(HAPTIR -lCIl,\!<MTI'RI/AIII)' ell'("{)MMt·"ICA/lO,," S/(iI'\AJ.S A....[)SY\I!-MS 179 cos2rr];'tandsin2rrft.Theresulting modulation technique iscalledquadrature PAMorQAM.andthecorresponding signalwaveforms maybeexpressed as s",(t)=Re[(A"" +jA",.)I!(t)e"K1JJ. /1/=1,2....•M.O""t""T =A"".g(l)cos2rrf.t-A""I(t)sin2rrf.r (4-3-\9) whereA""andA""aretheinformation-bearing signalamplitudes ofthe quadrature carriersandg(t)isthesignalpulse. Alternatively. theQAMsignalwaveforms maybeexpressed as s",(t)=Re[v,,,eO'''g(t)e''Kf') =Y,,,g(t)cos(2Jrf..t+8,,,) wherev."=v'A;",+A;",and8,,,=tan-,(A""IA"w)' Fromthisexpression, itis apparent thattheQAMsignalwaveforms maybeviewedascombined amplitude andphasemodulation. Infact.wemayselectanycombination ofM"level PAMandM"phase PSK toconstruct anM=M,Mocombined PAM-PSK signalCDnstellation, If M,=2"andM,=2"',thecombined PAM-PSK signalconstellation resultsin thesimultaneous transmission ofm-n=logM,M,binarydigit,occurring ata symbol rateRI(m+n).Examples ofsignalspacediagrams forcombined PAM-PSK areshowninFig.4·3-4,forM=8andM=\6. AsinthecaseofPSKsignals. theQAMsignalwaveforms may b~ represented asalinearcombination oftwoorthonormal signalwaveforms,f,(t) andf,(r),i.e., s",(r)=s",,!J(t)+s""h(t) where f,(t)=~~,g(t)cos2Jrj;t 12 [,U)=-\j'i"g(r)sin2Jr.f..t(4-3-2\) (4-3-22) FIGURE 4-:}-4Examples ofcombined PAM-PSK signalspacediagrams.M=R M='6 ..---.---.---4- o o, +•.---....-, ,, , +.'.---.- 0,,, 000 0,••+----t-0,, , •t...---.- , 00, 0,....--.-M==64....---.---.--- .., M=~2, ••---e••,, M=16,,,..---.,••, ,,,, M=8:,,-.-.-..,,,M=41 0,-.---.++,0,, -....--~,••,,, ,, 0,-e---.; •• FIGURE 4--3-5Severalsignalspacediagrams forrectangular QAM. and, o, • - - - • ---4-- -....--.- - - • - - -.-- -... =[An"Y~'lg Am,Y~'l.] 6Kistheenergyofthesignalpulseg(t). TheEuclidean distance between anypairofsignalvectorsis(4-3-23) (4-3-24) Inthespecialcasewherethesignalamplitudes takesthesetofdiscrete\alues {(2m-1-M)d,m=1.2,...,M),thesignalspacediagram isrectangular, as showninFig,4-3-5,Inthiscase,theEuclidean distance between adjacent points,i.e.,theminimum distance, is tI"·>=tlY2t.:. (4-3-25)mm g whichisthesameresultasforPAM. Multidimensional SignalsItisapparent fromthediscussion abovethatthe digitalmodulation ofthecarrieramplitude andphaseallowsustoconstruct signalwaveforms thatcorrespond totwo-dimensional vectorsandsignalspace diagrams, Ifwewishtoconstruct signalwaveforms corresponding tohigher­ dimensional vectors, wemayuseeitherthetimedomain orthefrequency domainorbothinordertoincrease thenumberofdimensions, Suppose wehaveN-dimensional signalvectors. ForanyN.wemay subdivide atimeinterval oflength T,=NTintoNsubintervals oflength T=TI/N.Ineachsubinterval oflengthT.wemayusebinaryPAM(a one-dimensional signal)totransmit anelement oftheN-dimensional signal CHAPTER... {-HARACTERIZ;-\TJO~ OFCOMMl"NICATION Sl(;r-.:ALS A!"DSYSTEMS 18t f .~I+4Af1--,--.,.--, ~)+3Aff----I--+--f !c)+2Af1--+--+----1 J;,'off----I--+--f FIGURE 4a3·6Subdivision oflimeandfrequency axesintodistinctslots.f"L-_-::---_:'::-_--= ....oT2T3T (4-3-26)vector.Thus.theNtimeslotsareusedtotransmit theN-dimensional signal vector.IfNiseven,atimeslotoflengthTmaybeusedtosimultaneously transmit twocomponents oftheN-dimensional vectorbymodulating the amplitude ofquadrature carriers independently bythecorresponding components. Inthismanner, theN-dimensional signalvector IStransmitted in INTseconds(INtimeslots). Alternatively, afrequency bandofwidthNt1fmaybesubdivided intoN frequency slotseachofwidtht1tAnN-dimensional signalvectorcanbe transmitted overthechannelbysimultaneously modulating theamplitude ofN carriers, oneineachoftheNfrequency slots.Caremustbetakentoprovide sufficient frequency separation t1tbetween successive carrierssothatthereis nocrosstalkinterference amongthesignalsontheNcarriers.Ifquadrature carriersareusedineachfrequency slot,theN-dimensional vector(evenN) maybetransmitted in~Nfrequency slots,thusreducing thechannelbandwidth utilization byafactorof2. Moregenerally, wemayuseboththetimeandfrequency domains jointlyto transmit anN-dimensional signalvector.Forexample, Fig.43-6illustrates a subdivision ofthetimeandfrequency axesinto12slots.Thus,anN=12­ dimensional signalvectormaybetransmitted byPAMoranN=24· dimensional signalvectormaybetransmitted byuseoftwoquadrature carriers (QAM)ineachslot. Orthogonal Multidimensional Signals Asaspecialcaseoftheconstruction ofmultidimensional signals,letusconsider theconstruction ofMequal-energy orthogonal signalwaveforms thatdifferinfrequency, andarerepresented as s,.,(t)=Re[s,,.,(I)e}2",,]. m=1.2....,M.0'"I'"T ru=Vrcos[21rf,1 +2nmMI] wheretheequivalent lowpasssignalwaveforms aredefinedas ~?~ S(I)=:-.::eJ2J'"·f' 10M0'"I,,;T I", T,nl=.~,.... • (4-3-27) Thistypeoffrequency modulation iscalledfrequency-shifl keying(FSK). 182 DIGITAL COMMUNICATIONS (4-3-28)Thesewaveforms arecharacterized ashavingequalenergyandcross- correlation coefficients P=2'lIT(ei2.(m-k)/;.{,dt 'm2'lJo =sinTCT(m-k)ilJei"nm-k)AI TCT(m-k)Ii.f TherealpartofPkrnis (4-3-29)sin[TCT(m-k)illl P,==Re(Pkno)=1CT(m_k)ilJcos[TCT(m-k)illl sin[21CT(m-k)illl=2/CT(m-k)li.f First,weobserve thatRe(Pkm)= 0whenIi.J=1/2Tandm"k.Since 1m-kl= 1corresponds toadjacent frequency slots,li.f=1/2Trepresents the minimum frequency separation between adjacent·signalsfororthogonality of theMsignals.PlotsofRe(Pom)versusilland!PkmIversusAfareshowninFig. 4-3-7.NotethatIP'ml=O formultiples oflITwhereas Re(p~",)=O for multiples ofI/2T. P. 2T (a) 3 T (b)aL.-----¥- .....,'--__40"'-_ Af 1 2T T FIGURE 4-3-7Cro..-correlation coefficient asafunction offrequency separation forFSKsign.ls. CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNA.LS ANDSYSTEMS· 183 f,(l) FIGURE 4-3-8Orthogonal signalsforM=N=3and M=N=2.~ ,,'"',IW' ./2l.,' ", , ""II:r-----,-i+-f.(l) ,~~~;-;-i; M=N=3 M=N=2 ForthecaseinwhichtV=1/2T,theMFSKsignalsareequivalent tothe N-dimensional vectors s,=[n 0000) Sz=[0n000)(4-3-30) SN=[000".0,WI whereN=M.Thedistance between pairsofsignalsis d~'~=mforallm,k (4-3-31) whichisalsotheminimum distance. Figure4-3-8illustrates thesignalspace diagram forM=N=2andM=N~3. Biortbogonal SipalsAsetofMbiorthogonal signalscanbeconstructed from!Morthogonal signalsbysimplyincluding thenegatives oftheorthogonal signals.Thus,werequireN=tMdimensions fortheconstruction ofasetofM biorthogonal signals.Figure4-3-9illustrates thebiorthogonal signalsforM=4 and6. . Wenotethatthecorrelation between anypairofwaveforms iseither p,=-lorO.Thecorresponding distances ared=2norm,withthelatter beingtheminimum distance. FIGURE 4-3-9Signalspacediagrams forM=4and M=6bionhogorlal signals.-5,52 M=45, s., I+-5, M=6 184 DIGITAL COMMU:-';I{' AnONS (4-3-35)1---M-lSimplex Signals Suppose wehaveasetofMorthogonal waveforms {sm(t)} or,equivalently, theirvectorrepresentation {sm}'Theirmeanis 1M 5= -LSm (4-3-32)Mm•1 Now,letusconstruct another setofMsignalsbysubtracting themeanfrom eachoftheMorthogonal signals.Thus, s~,=Sm-S,m=1,2,...,M (4-3-33) Theeffectofthesubtraction istotranslate theoriginofthemorthogonal signalstothepoints. Theresulting signalwaveforms arecalledsimplex signalsandhavethe following properties. Fir-st,theenergyperwaveform is Is;"I'=Is",-512 2 1=II:--l€+-iM M =11:(1-~) (4-3-34) Second,thecross-correlation ofanypairofsignalsis Re( )=s~.·s:. .Pm"I'II'ISmSn -11M I-IIM forallm.n.Hence.thesetofsimplex waveforms isequallycorrelated and requires lessenergy, bythefactorI-11M. thanthesetoforthogonal waveforms. Sinceonlytheoriginwastranslated, thedistance between anypair ofsignalpoints"maintained atd=m.whichisthesameasthedistance between anypairoforthogonal signals. Figure4-3-10illustrates thesimplexsignal>forM=2,3,and4.Notethat thesignaldimensionality isN=M-I. (4-3-37)SignalWaveforms fromBinaryCodesAsetofMsignaling waveforms canbegenerated fromasetofMbinarycodewordsoftheform C",=[C,.,I C,." c,.,..).m=I,2,...,M (4-3-36) wherec"',=0orJforallInandj.Eachcomponent ofacodewordismapped intoanelementary binaryPSKwaveform asfollows: If:'~em!=I::}sm,(t) = - cos2tr.f..r(0..t"7;) T; C'''i=0::}s,.,,(t)=-~2icos2rrfJ(0..t'"T;.) whereT,=T/Nand't,=t:1N.Thus,theMcodewords{e,.,}aremappedinto asetofMwaveforms {s",(t)}. CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 185 /,(/1 1,(1)., JU JUIf /,(11 1,1t).," " ') M=2 M=J 1,(1) FIGURE 4-3-10 Signalspacediagrams forM-arysimplex signals.~---\--\-- ..1,(1) Thewaveforms canberepresented invectorformas s",=[S"'1Sm2 S",N],m=1,2,...,M (4-3-38) where S",j=±y~/Nforallmandj.Niscalledtheblocklengthofthecode. anditisalsothedimension oftheMwaveforms. Wenotethatthereare2"possiblewaveforms thatcanbeconstructed from the2Npossible binarycodewords.WemayselectasubsetofM<2Nsignal waveforms fortransmission oftheinformation. Wealsoobservethatthe2N possiblesignalpointscorrespond totheverticesofanN-dimensional hyper­ cubewithitscenterattheorigin.Figure4-3-11illustrates thesignalpointsin N=2and3dimensions. FIGURE 4-3-11 Signalspacediagrams forsignals l,<t) generated frombinarycodes.1,(1) /,(1) .,.," "/,(/) '"'" /,(1) N=2 ., N=3 186 DIGITAL COMMl:~ICATIONS EachoftheMwaveforms hasenergy't:.Thecross-correlation between any pairofwaveforms depends onhowweselecttheMwaveforms fromthe2" possiblewaveforms. ThistopicistreatedinChapter 7.Clearly,anyadjacent signalpointshaveacross-correlation coefficient ~(I-2/N) p,= ~N-2 N(4-3-39) andacorresponding distanceof d(e)=V2~(1-p,) =V4f:/N Thisconcludes ourdiscussion ofmemoryless modulation signals.(4-3-40) 4-3·2LinearModulation withMemory Themodulation signalsintroduced intheprevious sectionwereclassified as memoryless, becausetherewasnodependence between signalstransmitted in non-overlapping symbolintervals. Inthissection,wepresentsomemodulation signalsinwhichthereisdependence between thesignalstransmitted in successive symbolintervals. Thissignaldependence isusuallyintroduced for thepurpose ofshaping thespectrum ofthetransmitted signalsothatit matchesthespectralcharacteristics ofthechannel. Signaldependence between signalstransmitted indifferent signalintervals isgenerally accomplished by encoding thedatasequence attheinputtothemodulator bymeansofa modulation code,asdescribed inChapter9., Inthissection,weshallpresentexamples ofmodulation signalswith memory andcharacterize theirmemory intermsofMarkovchains.Weshall confineourtreatment tobaseband signals.Thegeneralization tobandpass signalsisrelatively straightforward. Figure4-3-12illustrates threedifferent baseband signalsandthecorres­ ponding datasequence. Thefirstsignal,calledNRZ,isthesimplest. The binaryinformation digitIisrepresented byarectangular pulseofpolarity A andthebinary di~itzeroisrepresented byarectangular pulseofpolarity-A. NRZ NRZIuLJ FIGURE 4-3-t2 Baseband signals.Delay modulation (Millercode) Data,,, o :JI;0:f)0:I (-HAP'ITR 4('HARAC-rERI7Af)ON OFCOMM\:NICArlO!'l SI(jNAI5 AND"y",'I:\1S 187 Hence,theNRZmodulation ismemorylessandisequivalent toahinar\'PAM orabinaryPSKsignalinacarrier-modulated system. TheNRZIsignalisdifferent fromtheNRZsignalinthattransitions from oneamplitude leveltoanother occuronlywhena 1istransmitted. The amplitude levelremains unchanged whenazeroistransmitted. Thistypeof signalencoding iscalleddifferential encoding. Theencoding operation i, described mathematically bytherelation (4-]-41} where{a,}isthebinaryinformation sequence intotheencoder, {b,}isthe outputsequence oftheencoder, andEBdenotes addition modulo 2.When h,=1.thetransmitted waveform isarectangular pulseofamplitude A.and whenb,=O.thetransmitted waveform isarectangular pulseofamplitude -A Hence,theoutputoftheencoder ismapped intooneoftwowaveforms in exactlythesamemannerasfortheNRZsignal. Thedifferential encoding operation introduces memory inthesignal.The combination oftheencoder andthemodulator operations mayberepresented byastatediagram (aMarkovchain)asshowninFig.4-3-13.Thestatediagram maybedescribed bytwotransition matrices corresponding tothetwopossible inputbits{O.1}.Wenotethatwhenak=0,theencoderstaysinthesamestate. Hence,thestatetransition matrixforazeroissimply (4-3-42) where t"=Iifa,resultsinatransition fromstateitostatei,i=i.2,andi=1. 2:otherwise, t'l=O.Similarly, thestatetransition matrixfora,=1is T2=[~~] (4-3-43) Thus.thesetwostatetransition matrices characterize theNRZIsignal. Another waytodisplaythememory introduced bytheprecoding operation isbymeansofatrellisdiagram. Thetrellisdiagram fortheNRZIsignalis FIGURE 4-3-13 Sta'ediagram[or'heNRZIsignal. ()I~\lll Of.f(tJ 188 DIGITAL COMMLJNICA TlONS FIGURE 4-3-14 ThetrellisdiagramfortheNRZtsignal. (4-3-44)illustrated inFig.4-3-14.Thetrellisprovides exactlythesameinformation concerning thesignaldependence asthestatediagram, butalsodepictsatime evolution ofthestatetransitions. Thesignalgenerated bydelaymodulation alsohasmemory. Asshownin Chapter 9,delaymodulation isequivalent toencoding thedatasequence bya run-length-limited codecalledaMillercodeandusingNRZItotransmit the encoded data.Thistypeofdigitalmodulation hasbeenusedextensively for digitalmagnetic recording andincarriermodulation systemsemploying binary PSK.Thesignalmaybedescribed byastatediagram thathasfourstatesas showninFig.4-3-15(a). Therearetwoelementary.waveforms 5,(1)and5,(1) andtheirnegatives -5,(1)and-5,(1),whichareusedfortransmitting the binaryinformation. Thesewaveforms areillustrated inFig.4-3-15(b). The mapping frombitstocorresponding waveforms isillustrated inthestate diagram. Thestatetransition matrices thatcharacterize thememory ofthis encoding andmodulation methodareeasilyobtained fromthestatediagram in Fig.4-3-15.Whenak=0,wehave T'{~~iJ FIGURE 4-3-15 Stalediagram(a)andbasicwaveforms (b)fordelaymodulated (Miller-encoded) signal. A lu)JjU)=-Jt,fI051:5T -A J.ttI=-Jilt)051:5T Ib)oT CHAPTER 4CHARACTERIZATION OFCOMML!'oICATIOr-. SIGNALS ANDSY"i"TEM5 189 andwhenak=1,thetransition matrixis [0 1 o0 T2=0 1 o0o0]1 0 o0 1 0(4-3-45) Thus,thesetwo4x4statetransition matrices characterize thestatediaf;ram fortheMiller-encoded signal. Modulation techniques withmemory suchasNRZ1andMillercodingare generally characterized byaK-stateMarkov chainwithstationary state probabilities {Pi'i=1,2,...,K}andtransition probabilities {Pij,i,j= 1,2,...,K}.Associated witheachtransition isasignalwaveform s/(t), j=1,2,...,K.Thus,thetransition probability Pijdenotestheprobability that signalwaveform Sj(t)istransmitted inagivensignaling intervalafterthe transmission ofthesignalwaveform Si(t)intheprevious signaling interval.The transition probabilities maybearranged inmatrixformas P1K]P2K PKK(4-3-46) wherePiscalledthetransition probability matrix. Thetransition probability matrixiseasilyobtained fromthetransItion matrices {T,}andthecorresponding probabilities ofoccurrence oftheinput bits(or,equivalently, thestationary statetransition probabilities {p,}).The !,~ne.alrelationship maybeexpressed as 2 P=2:qiTi i=1(4-3-47) whereql=P(ak=0)andq2=P(ak=1). FortheNRZIsignalwithequalstateprobabilities PI=P2=!andtransition matrices givenby(4-3-42)and(4-3-43), thetransition probability matrixis p=un (4-3-48) Similarly, thetransItion probability matrixfortheMiller-coded signalwith equallylikelysymbols(qj=q2=!or,equivalently, PI=P2=P3=P4=1)is P=[~~~:] (4-3-49)!!0 0 !0!0 Thetransition probability matrixisusefulinthedetermination ofthespectral 190 DIGITAL COMMUNICATIONS characteristics ofdigitalmodulation techniques withmemory, asweshall observeinSection4-4. 4-3-3Nonlinear Modulation Methods withMemory Inthissection,weconsider aclassofdigitalmodulation methods inwhichthe phaseofthesignalisconstrained tobecontinuous. Thisconstraint resultsina phaseorfrequency modulator thathasmemory. Themodulation methodis alsononlinear. Continuous-PhlIse FSK(CPFSK) Aconventional FSKsignalisgenerated byshiftingthecarrierbyanamountf"=~AfIn>In=±1,±3,..., ±(M-1),to reflectthedigitalinformation thatisbeingtransmitted. ThistypeofFSKsignal wasdescribed inSection4-3-1,anditismemoryless. Theswitching fromone frequency toanother maybeaccomplished byhavingM=2*separate oscillators tunedtothedesiredfrequencies andselecting oneoftheM frequencies according totheparticular k-bitsymbolthatistobetransmitted in asignalintervalofduration T=k/Rseconds. However, suchabruptswitching fromoneoscillator outputtoanotherinsuccessive signaling intervals resultsin relatively largespectralsidelobesoutsideofthemainspectralbandofthe signaland,consequently, thismethodrequires alargefrequency bandfor transmission ofthesignal. Toavoidtheuseofsignalshavinglargespectralsidelobes,theinformation­ bearingsignalfrequency modulates asinglecarrierwhosefrequency ischanged continuously. Theresulting frequency-modulated signalisphase-continuous and,hence,itiscalledcontinuous-phase FSK(CPFSK). ThistypeofFSK signalhasmemory becausethephaseofthecarrierisconstrained tobe continuous. Inordertorepresent aCPFSKsignal,webeginwithaPAMsignal d(t)=LIng(t-nT) n(4-3-50) (4-3-52)where(In}denotesthesequence ofamplitudes obtained bymapping k-blt blocksofbinarydigitsfromtheinformation sequence {an}intotheamplitude levels±I,±3,...•±(M-1)andg(t) isarectanguhtr pulseofamplitude 1/2T andduration Tseconds. Thesignald(t)isusedtofrequency-modulate the carrier. Consequently, theequivalent lowpasswaveform vet)isexpressed as vet)=~expV[41l'TfdLd(T)dHc/Jo]} (4-3-51) wherefdisthepeakfrequency deviation andcf>oistheinitial phase ofthe carrier. Thecarrier-modulated signalcorresponding to(4-3-51)maybeexpressed as rnset)='JTCos[2nfct+«>(t;I)+«>0) CHAPTER~: CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 191 wherecP(t;I)represents thetime·varying phaseofthecarrier.whichisdefined as </>(1;1)=41CTf,J'~d(r)dr =41CT!.,L[~/"g(r-nT)]dr (4-3-53) Notethat.although d(t)contains discontinuities. theintegral ofd(t)is continuous. Hence.wehaveacontinuous-phase signal.Thephaseofthe carrierintheintervalnT""t,;;; (n+I)Tisdetermined byintegrating (4-3-53). 'Thus. n-I cP(t;I)=21C!.,T2:/k+21Cfd(t-nT)/" (4-3-57)(4-3-55) (4-3-56)(4-3-54) "--I e"=rrh2:/k k.=-?l {o(t<0) q(t)=t!2T(0""t""T) ~(t>T) Weobservethat9"represents theaccumulation (memory) ofallsymbolsupto time(n-I)T.Theparameter hiscalledthemodulation index.=8"+2rrh/"q(t -liT) whereh.8mandq(t)aredefinedas h=2fJ Continuous-Phase Modulation (CPM) Whenexpressed intheformof (4-3-54), CPFSKbecomes aspecialcaseofageneralclassofcontinuous-phase modulated (CPM)signalsinwhichthecarrierphaseis " <I>(t;I)=21C2:lkhkq(t-kT),nT""t""(n+I)T k=-x(4-3-58) where{Idisthesequence ofM-aryinformation symbols selected fromthe alphabet ±I,±3,...,±(M-I),{h.}isasequence ofmodulation indices,and q(t)issomenormalized waveform shape. Whenh.=hforallk,themodulation indexisfixedforallsymbols. When themodulationindexvariesfromonesymboltoanother, theCPMsignalis calledmulti-h. Insuchacase,the{h.)aremadetovaryinacyclicmanner throughasetofindices. Thewaveform q(t)mayberepresented ingeneralastheintegralofsome pulseg(t).i.e., q(t)=Lg(r)dr (4-3-59) 192 DIGITAL COMMUNICATIO!'JS ~(t) CIt/) IJ_ 2 2T o I )TCrI 'ul=-Il-co~~2T TT T10> qU)T T (h) FIGURE 4-3-16 Puheshape,forfullre'ponse CPM(a,b)andpartialresponse CPM(c,d). Ifg(l)=0fort>T,theCPMsignaliscalledfullresponse CPM.Ifg(t)""0for t>T,themodulated signaliscalledpartialrespbnse CPM.Figure4-3·16 illustrates severalpulseshapesforg(l),andthecorresponding q(r).Itis apparent thataninfinitevarietyofCPMsignalscanbegenerated bychoosing different pulseshapesg(t)andbyvaryingthemodulation indexhandthe alphabet sizeM. Itisinstructive tosketchthesetofphasetrajectories q,(I;I)generated byall possible valuesoftheinformation sequence {In}'Forexample, inthecaseof CPFSKwithbinarysymbols In=±I,thesetofphasetrajectories beginning at timet=0isshowninFig.4-3-17.Forcomparison, thephasetrajectories for quaternary CPFSKareillustrated inFig.4-3-18.Thesephasediagrams are calledphasetrees.WeobservethatthephasetreesforCPFSKarepiecewise linearasaconsequence ofthefactthatthepulseg(t)isrectangular. Smoother phasetrajectories andphasetreesareobtained byusingpulsesthatdonot containdiscontinuities, suchastheclassofraisedcosinepulses.Forexample, a phasetrajectory generated bythesequence (I,-1,-1,-1,I,1,-1.1)fora partialresponse, raisedcosinepulseoflength3Tisillustrated inFig.4-3-19. Forcomparison, thecorresponding phasetrajectory generated byCPFSKis alsoshown. Thephasetreesshowninthesefiguresgrowwithtime.However, thephase g(t)CHAPTER' CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 193 q(f) ! I 2" .!1-----------,4T o 8\1)=4'T[I-COS ~) .! 2T2T 0 2T (c) q(t) .! 2 FlGURE 4-3016o (Continued).T 2T (d)o 2T FlGURE 4-3017 Phasetrajectory forbinaryCPFSK.5hn--------.__--------- - _--- ----- - --- +1 4hll--------------------------- +1 o ,,,-2m L. _,,, ,-3M .L. .1. _,,, ,,,,-4Jrn L. I -' . : : : : -I I I I I-5h1f. L. .1 •.1. J.._••__ T 2T 3T 4T IT 194 DIGITAL COMMCNICATIONS r----·-,,,,,1._-----_.-,,,,, , !.•- - - ~--- --'-- - , ,, ,, ,, ,, ,, , r--- --- ----,--..- - -..,,,,, I._---------~----- .._.--41Jn-31m-21m··h1t -bhrc-51t1t6h7t..- - - - - - •- --,..- - -•- - -•-, ,, ,,., ,, , 5/1T!L - _ - - - - - - - - .....- - - - -- : :-t3, ,, ,, 4hn:~--- - - --- -.:-- -, ,,., ,, ,, , 31m~-- - --.---,,,,21mL _ :+-J,,,, 1mr-- (\ T !T 3T 4T FIGURE 4-3-18 Phasetrajectory forquaternary CPFSK. ofthecarrierisuniqueonlyintherangefromt/>=0tot/>=2!Cor.equivalently. fromt/>=-!Ctot/>=fr.Whenthephasetrajectories areplottedmodulo21t.say intherange(-!C.It).thephasetreecollapses intoastructure calledaphase tre//is.Toproperly viewthephasetrellisdiagram. wemayplotthetwo quadrature components x,.(t:I)=cost/>(t:I)andx.,(t:I)=sint/>(t:I)as functions oftime.Thus.wegenerate athree-dimensional plotinwhichthe quadrature components x,andx,appearonthesurfaceofacylinder ofunit radius.Forexample. Fig.4-3-20illustrates thephasetrellisorphasecylinder CHAPTER 4:CHARACTER1ZAT10N UFCOMMUNICATION SIGNALS ANDSYSTEMS 195 2h. -I -I -I +I,.. -21m+I.... SQ,~.!-+.!..I -'-...:-:.cl...:-.:.-_--'--~ ......-,....,.-,.....,~-,....,.--..- ... G-,' Tii'-_ 3T 7T.,'ST • -l..... -1....... ~+l ~x ~ y FIGURE 4-3·19 Phasetrajectories forhinaryCPFSK(dashed) andbinary,partialresponse CPMbasedonraised cosinepulseoflength3T(solid).{FromStlndberg (/9861.©1986I£EE.I (4-3-60)obtained withbinarymodulation, amodulation indexh=tandaraised cosinepulseoflength3T. Simpler representations forthephasetrajectories canbeobtained by displaying onlytheterminal valuesofthesignalphaseatthetimeinstants t=nT.Inthiscase,werestrictthemodulation indexoftheCPMsignaltobe rational. Inparticular, letusassumethath=m/p,wheremandpare relatively primeintegers, Then,afullresponse CPMsignalatthetimeinstants t=nTwillhavetheterminaL phasestates 0s={o,1tm,21tm,_..,(p-1)1tm} p p p whenmisevenand (4-3-61)0s={o,1tm,21tm,...,(2p-1)1tm} p p p whenmisodd.Hence,therearepterminal phasestateswhenmisevenand 2pstateswhenmisodd.Ontheotherhand,whenthepulseshapeextends nGURE 4-3-20 PhasecylinderforbinaryCPMwithh=~andaraisedcosine pulseoflength3T.[FromSundberg (/986),©1986I£EE.) 196 DIGITAL COMMUNICATIONS 1 11l1:=lit• 11l,=!11• o T 2T 3T 4T FIGURE 4-3-2t StatetrellisforbinaryCPFSKwithh~j. overLsymbolintervals (partialresponse CPM),thenumberofphasestates mayincreaseuptoamaximum ofSr,where S_{PML -t(evenm),-2pML -1(oddm) (4-3-62) whereMisthealphabet size.Forexample, thebinaryCPFSKsignal(full response, rectangular pulse)withh=thasS,=4(terminal) phasestates.The statetre/Jisforthissignalisillustrated inFig.4-3-21.Weemphasize thatthe phasetransitions fromonestatetoanotherareoottruephasetrajectories. Theyrepresent phasetransitions forthe(terminal) statesatthetimeinstants t=nT. Analternative representation tothestatetrellisisthestatediagram, which alsoillustrates thestatetransitions atthetimeinstantst=nT.Thisisaneven morecompact representation oftheCPMsignalcharacteristics. Onlythe possible(terminal) ;Jhasestatesandtheirtransitions aredisplayed inthestate diagram. Timedoes.lotappearexplicitly asavariable. Forexample, thestate diagramfortheCPFSKsignalwithh=~isshowninFig.4-3-22. Minimum-Shift Keying(MSK) MSKisaspecialformofbinaryCPFSK -I -I -I,...• 1 -I 11 J••FIGURE 4-3-21 Statediagram forbinaryCPFSKwithh=I CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 197 (and,therefore, CPM)inwhichthemodulation indexh=!.Thephaseofthe carrierintheintervalnT,.;t,.;(n~OTis "-I t/>(t;I)=!/r~I.+Ifl"q(t-nT) /(=-x (t-nT)=8n+!/rl"-T-, andthemodulated carriersignalisnT,.;t";(n +l)T (4-3-63) (4-3-64)s(t)=Acos[21rfct+0"+~1f/"(t-;T)] =Acos[21l{t;.+41T1n)t-1mrl n+8nJ.nT,.;t,.;(n+l)T Theexpression (4-3-64) indicates thatthebinaryCPFSK signalcanbe expressed asasinusQid havingoneoftwopossible frequencies intheinterval nT,.;t";(n+l)T.Ifwedefinethesefrequencies as 1 f,=fc-4T 1h=fc+4T(4-3-65) thenthebinary£PFSKsignalgivenby(4-3-64)maybewrittenintheform s;(t)=Acos[21r[;t+8"+1nlr(-IY-'j. i=I,2 (4-3-66) Thefrequency separation t1f=h-f,=1/2T.Recallthatt1f=1/2Tisthe minimum frequency separation thatisnecessary toensuretheorthogonality of thesignalss,(t)andsz(t)overasignaling intervaloflengthT.Thisexplains whybinaryCPFSKwithh=!iscalledminimum-shift keying(MSK).The phaseinthenthsignaling intervalisthephasestateofthesignalthatresultsin phasecontinuity between adjacent intervals. MSKmayalsoberepresented asaformoffour-phase PSK.Specifically, we mayelCpresstheequivalent lowpassdigitallymodulated signalintheform(see Problem 4-14) x v(t)=~[/lng(t-2nT)-j/ln+,g(t-2nT-T)J ,,=_:x: whereg(t)isasinusoidal pulsedefinedas {.msm-(O"t..2T) g(t)=02T (otherwise)(4,3-67) (4-3-68) 198 DIGITAL COMML'NICATIONS Thus,thistypeofsignalisviewedasafour-phase PSKsignalinwhichthe pulseshapeisone-half cycleofaJiinusoid. Theeven-numbered binary-valued (±1)symbols {I,,,}oftheinformation sequence {I,,}aretransmitted viathe cosineoLthecarrier,whiletheodd-numbered symbols{f,,,.,}aretransmitted viathesineofthecarrier.Thetransmission rateonthetwoorthogonal carrier components is1/2Tbitspersecondsothatthecombined transmission rateis liTbits/soNotethatthebittransitions onthesineandcosinecarrier components arestaggered oroffsetintimebyTseconds. I'"orthisreason,the signal S(I)=Ann~,J2ng(t-2nT)]cos21t!.' +[n~,f,,,.,g«(-2nT-T)]sin2iif..t} (4-3-09) iscalledoffsetquadrature PSK(OQPSK) orstaggered quadrature PSK (SQPSK). Figure4-3-23illustrates therepresentation oftheMSKsignalsastwo staggered quadrature-modulated binaryPSKsignals.Thecorresponding sum ofthetwoquadrature signalsisaconstant amplitude. frequency-modulated signal. Itisalsointeresting tocompare thewaveforms forMSKwithoffsetQPSK inwhichthepulseg(l)isrectangular for0,,;;I";;2T,andwithconventional -T T 3T 5T 7T l.u)In-pho1<;t"sigrlalcumpoocnl o 21 4T 6T 8T FIGURE 4-3-D Representation ofMSKsignalasaformoftwo staggered binaryPSKsignals.eachwitha sinusoidal envelope.1(blQuadrature signall'omponent .\rtPlfV\?VlJ oT2T3T4T5T6T7T (1"1MSKsignal(sumof(el'andIb)1• r CHAPTER 4:CHARACTERIZATION OFCOMMUNICATiON SIGNALS ANDSYSTEMS 199 -90"phaseshift+90"phase ~hift (a)MSK -90"phaseshift +90"phaseshift (b)Offset QPSK (c)QPSK+90"phaseshift Nodatatransilions -90"phaseshift FIGURE 4-l-24 Signalwaveforms for(a)MSK.(b)offsetQPSK(rectangular pulse).and(e)conventional QPSK (rectangular pulse).[FromGronemeyer andMcBride (/976);©J976JEEE.] quadrature (four-phase) PSK(QPSK)inwhichthepulseg(t)isrectangular for o~t""2T.Clearly,allthreeofthemodulation methods resultinidentical data rates.TheMSKsignalhascontinuous phase.TheoffsetQPSKsignalwitha rectangular pulseisbasically twobinaryPSKsignalsforwhichthephase transitions arestaggered intimebyTseconds. Thus,thesignalcontains phase jumpsof±90°thatmayoccurasoftenaseveryTseconds. Ontheotherhand, theconventional four-phase PSKsignalwithconstant amplitude willcontain phasejumpsof±180'or±90°every2Tseconds. Anillustration ofthesethree signaltypesisgiveninFig.4-3-24. SlplSpaceDi_crams forCPMIngeneral, continuous-phase signals cannotberepresented bydiscretepointsinsignalspaceasinthecaseofPAM, PSK,andQAM,because thephaseofthecarrieristime-variant. Instead, a continuous-phase signalisdescribed bythevariouspathsortrajectories from onephasestatetoanother. Foraconstant-amplitude CPMsignal,thevarious trajectories formacircle. 200 f>IGITAL CQMMLINKATIONS $$ h=!] FIGURE 4-3-25 SignalspacediagramforCPFSK. Forexample, Fig.4-3-25illustrates thesignalspace(phase trajectory) diagram forCPFSKsignalswithh=th=th=tandh=~.Thebeginning andendingpointsofthesephasetrajectories aremarkedinthefigurebydots. Notethatthelengthofthephasetrajectory increases withanincreaseinh.An increase inhalsoresultsinanincrease ofthesignalbandwidth, as demonstrated inthefollowing section. Multiamplitude CPMMultiamplitude CPMisageneralization ofordinary CPMinwhichthesignalamplitude isallowedtovaryoverasetofamplitude valueswhilethephaseofthesignalisconstrained tobecontinuous. For example, letusconsider atwo-amplitude CPFSKsignal,whichmaybe represented as whereS(I)=2Acos(21rfcl+</>,(1;I)]+Acos[21rfcr+oI>,(r;J»)(4-3-70) n-'trhl.(I-nT) 0I>2(1:1)=rrh.~xh+ T,nT'Er';(n+l)T (4-3-71) n-l trhJ,,(r-nT) 4>,(1:J)=trh.~xJ.+T,nT 'EI';(n+l)T(4-3-72) Theinformation isconveyed bythesymbolsequences {In}and{J,,},whichare relatedtotwoindependent binaryinformation sequences {an}and{bn}that takevalues{O,I}.Weobservethatthesignalin(4-3-70)isasuperposition of twoCPFSKsignalsofdifferent amplitude. However, thesequences {I,,}and {In}arenotstatistically independent, butareconstrained inordertoachieve phasecontinuity inthesuperposition ofthetwocomponents. Toelaborate, letusconsider thecasewhereh=tsothatwehavethe superposition oftwoMSKsignals.Atthesymboltransition points,thetwo CHAPTER 4;OiARAC'TERJZAT'ON OFCOMMUNJCA nONS·'GNALS ANDSYSTEMS 201 TABLE 4-3-1 ".b.I.J.AmpUIUcIe-pIuH relatioDl ()0-1-IAmplitude isconttant; phasedecreases ()1-1 1Amplitude changes; pl)asedecreases 10I 1Amplitude isconstant; phaseincreases I I 1-)Amplitude change.; phaseincreases amplilude components areeitherinphaseor180"outofphase.Thephase changeinthesignalisdetermined bythephaseofthelargeramplitude component, whiletheamplitude changeisdetermined bythesmaller component. Thus,thesmallercomponent isconstrained suchthatat"thestart andendofeachsymbolinterval, itiseitherinphaseor180"outofphasewith thelargercomponent, independent ofitsphase.·Underthisconstraint, the symbolsequences {In}and{in}maybeexpressed as In=2an-1 Theserelationships aresummarized inTable4-3-1. Asageneralization, amultiamplitude CPFSKsignal maybeexpressed as(4-3-73) withncomponents N-J s(t)=2N-'cos[2lflt+4>N(t;I)]+L2m-Jcos[21if.-t+4>m(t;Jm))(4.3.74) m"'l where andt-nT n-J I!>N(t;I)=IfhIn--+Ifh2:h.nT"'"t"'"(n+I)TT'.-ro t-nT4>m(t;Jm)=InJr[h+Wmn+1)]-T-(4-3-75) n-J +2:IfI.[h+Wm.+I)],nT"'"t",;;(n+I)T(4-3-76) k=~OO Thesequences {In}and{Jmn}arestatistically independent, binary-valued sequences thattakevaluesfromtheset{I,-l}. From(4-3-75)and(4-3-76), weobservethateachcomponent inthesum 202 DlGlTAl COMMUNICATiONS 3 -3 h=!2 3-3 h=!4 h=~3 FIGURE 4-3-26 Signalspacediagrams forlwo-component CPFSK. willbeeitherinphaseor18(foutofphasewiththelargestcomponent atthe endofthenthsymbolinterval, i.e.,att=(n+I)T.Thus,thesignalstatesare specifiedbyanamplitude levelfromthesetofamplitudes {l,3,5,...,2N-I} andaphaselevelfromtheset{G,71:8,271:8,...,271:-trh}.Thephaseconstraint isrequired tomaintain thephasecontinuity oftheCPMsignal. Figure4-3-26illustrates thesignalspacediagrams fortwo-amplitude (N=2) CPFSKwithh=t!,tandi.Thesignalspacediagrams forthree-component (N=3)CPFSKareshowninFig.4-3-27.Inthiscase,therearefouramplitude levels.Thenumberofstatesdepends onthemodulation indexhaswellasN. Notethatthebeginning andendingpointsofthephasetrajectories aremarked bydots. Additional multiamplitude CPMsignalformatsmaybeobtained byusing ('HAPTI:::R~: ("HARA<"llRIZArlOt..; (;t.COMMl'NICATION SI(iNALS .ASDSYS'IFMS203 h=!• It=1) FIGURE 4<\·27 Signalspacedi<lgrams forthn.:c--component CPFSK. pulseshapesotherthanrectangular, aswellassignalpulsesthatspanmore thanonesymhol(partialresponse). 4-4SPECTRAL CHARACTERISTICS OFDIGITALLY MODULATED SIGNALS Inmostdigitalcommunications systems. theavailahle channel bandwidth is limited. Consequently. thesystem designer mustconsider theconstraints imposed hythechannel handwidth limitation intheselection ofthemodula­ tiontechnique usedtotransmit theinformation. Forthisreason.itisimportant forustodetermine thespectral content ofthedigitally modulated signals descrihed inSection 4-.1. Sincetheinformation seyuence israndom. adigitally modulated signalisa stochastic process. Weareinterested indetermining thepowerdensity spectrum ofsuchaprocess. Fromthepowerdensity spectrum. wecan determine thechannel bandwidth required totransmit theinformation-bearing signal.Below.wetir~tderivethespectral characteristics oftheclassoflinearly 204 DIGITAL COMMUNICATIONS modulated signals.Then.we.consider thenonlinear CPFSK, CPM,and baseband modulated signalswithmemory. 4-4-1PowerSpectraofLinearly Modulated Signals Beginning withtheform s(t)=Re[V(t)e'2XJ./) whichrelatesthebandpass signals(t)totheequivalent lowpasssignalv(t),we mayexpresstheautocorrelation functionofs(t)as (4-4-1) where<1>",,(r)istheautocorrelation function oftheequivalent lowpasssignal v(r).TheFouriertransform of(4-4-1)yieldsthedesiredexpression forthe powerdensityspectrum$,,(/)intheform (4-4-2) where<P",,(f)isthepowerdensityspectrum ofv(t).Itsufficestodetermine the autocorrelation function andthepowerdensityspectrum oftheequivalent lowpasssignalv(t). Firstweconsider thelineardigitalmodulation methods forwhichlI(t)is represented inthegeneralform v> v(t)=2:Ing(t-nT) '1=-00(4-4-3) wherethetransmISSion rateisliT=Rlksymbols/s and{l,,}represents the sequence ofsYIPbolsthatresultsfrommapping k-bitblocksintocorresponding signalpointsselected fromtheappropriate signalspacediagram. Observe that inPAM,thesequence {In}isrealandcorresponds totheamplitude valuesof thetransmitted signal,butinPSK.QAM,andcombined PAM-PSK, the sequence {I,,}iscomplex-valued, sincethesignalpointshaveatwo-dimensional representation. Theautocorrelation function ofv(t)is cP"..(t+r;t)=~E[v'(t)lI(t +r)l == =~LLE[/~lmlg'(t -nT)g(t+r-mT)(4-4-4) n=-0:;0"''''--0:;0 Weassumethatthesequence ofinformation symbols {In}iswide-sense stationary withmeanIJ-iandautocorrelation function (4-4-5) CHAPTER 4:(HARA(lERIZATION OFCQMMUNIC.i\TION SIGNALS ANDSYSTEMS 20S Hence(4-4-4)canbeexpressed as ~ ~ cP~(1+r;I)=2:2:<I>;;(m-n)g*(1-nT)g(1+r-mT) n=-xffl=-X ~ ~ =L</>;;(m)Lg*(I-nT)g(l+r-nT-mT) (4-4-6) ",=-:10 n=-'X Thesecondsummation in(4-4-6),namely, ~ 2:g*(1-nT)g(1+r-nT-mT) isperiodic inthe1variable withperiodT.Consequently, <l>w(1+r;I)isalso periodic inthe1variablewithperiodT.Thatis, cP~(1+T+r;1+T)=cPnv(1+r;I) Inaddition, themeanvalueofV(I),whichis ~ E[II(t)J =jJ.;Lg(1-nT) n=-x(4-4-7) (4-4-8) isperiodic withperiodT.Therefore V(I)isastochastic processhavinga periodic meanandautocorrelation function. Suchaprocess iscalleda cycloslalionary processoraperiodically slalion'ary process inrhewidesense,as described inSection2-2-6. Inordertocompute thepowerdensityspectrum ofacyclostationary process,thedependence of4>",,(1+r;I)onthe1variable mUstbeeliminated. Thiscanbeaccomplished simplybyaveraging 4>",,(1+r;I)overasingle period.Thus, 11Tr2 ~~(r)=-</;",,(1+r;I)dt T-T12 ~ ~l1T12 =m~~4>;,(m)n'!-~T _mg*U-nT)g(/+r-nT-mT)dt ~ ~1IT12-nT m~~4>,,(m) n'!-~T-TI2-nTg*(r)g(1+r-mT)dl(4-4-9) Weinterpret theintegralin(4-4-9)asthetime-autocorrelation functionofg(r) anddefineitas 4>gg(r)=fxg*(I)g(t+r)dr (4-4-10) 106 DIGITALCOMMUNICATIONS Consequently (4-4-9)canbeexpressed as (4-4-11) TheFouriertransform oftherelationin(4-4-11)yieldsthe(average) power densityspectrum ofv(t)intheform (4-4-12) whereG(f)istheFouriertransform ofget),andct>ii(f)denotes thepower densityspectrum oftheinformation sequence, definedas ~ et>ii(f)=2:"'i,(m)e-jV,(mr m=-OX!(4-4-13) Theresult(4-4-12)illustrates thedependence ofthepowerdensityspectrum of v(t)onthespectral characteristics ofthepulseget)andtheinformation sequence {In}.Thatis,thespectralcharacteristics ofv(t)canbecontrolled by designofthepulseshapeg(t)andbydesignofthecorrelation characteristics of theinformation sequence. Whereas thedependence ofct>,.,(f)onG(t)iseasilyunderstood upon observation of(4-4-12), theeffectofthecorrelation properties ofthe information sequence ismoresubtle.Firstofall,wenotethatforanarbitrary autocorrelation cPii(m)thecorresponding powerdensityspectrum ct>ii(f)is periodic infrequency withperiodliT.Infact,theexpression (4-4-13)relating thespectrum ct>ii(f)totheautocorrelation ",,,(m)isintheformofan exponential Fourierserieswiththe{cPii(m)} astheFouriercoefficients. Asa consequence, theautocorrel"l;on sequence cPii(m)isgivenby jlf2T "'dm)=Tet>,,(f)ei21rrmT df -1/27(4-4-14) (4-4-15)Second, letusconsider thecaseinwhichtheinformation symbols inthe sequence arerealandmutually uncorrelated. Inthiscase.theautocorrelation function "',,(m)canbeexpressed as {ITT+JLT(m=0) ""i(m)=liT (m""0) whereIT;denotes thevariance ofaninformation symliol. When(4-4-15) is usedtosubstitute for"'dm)in(4-4-13), weobtain x 4>iI([) ==aT+IJ.}Le-/21Cfl"T JII~.-0:,(4-4-16) Thesummation in(4-4-16)isperiodic withperiodliT.Itmaybeviewedas CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 107 (4-4-17)theexponential Fourier seriesofaperiodictrainofimpulses witheachimpulse havinganarealiT.Therefore (4-4-16)canalsobeexpressed intheform IL' 00 (m)<t>i,(f)=CT1+~2:lif--Tm~-oo T Substitution of(4-4-17)into(4-4-12)yieldsthedesiredresultforthepower densityspectrum ofvet)whenthesequence ofinformation symbols is uncorrelated. Thatis, (4-4-18) <t>w(f)=CT7IG(f)12+IL;fIG(~)I'lJ(f_m)T T m~-roT T Theexpression (4-4-18) forthepowerdensityspectrum ispurposely separated intotwotermstoemphasize thetwodifferent typesofspectral components. Thefirsttermisthecontinuous spectrum, anditsshapedepends onlyonthespectralcharacteristic ofthesignalpulseget).Thesecondterm consistsofdiscretefrequency components spacedlITapartinfrequency. Each spectrallinehasapowerthatisproportional toIGU)I'evaluated atf=mIT. Notethatthediscretefrequency components vanishwhentheinformation symbolshavezeromean,i.e.,ILl=0_Thiscondition isusuallydesirable forthe digitalmodulation techniques underconsideration, anditissatisfied when theinformation symbolsareequallylikelyandsymmetrically positioned inthe complex plane.Thus,thesystemdesigner cancontrolthespectralcharacteris­ ticsofthedigitallymodulated signalbyproperselection ofthecharacteristics oftheinformation sequence tobetransmitted. Example 4-4-1 Toillustrate thespectralshapingresulting fromg(I),consider therectangu­ larpulseshowninFig.4-4-1(a). TheFouriertransform ofg(l)is G(f)=A Tsin1CfTe-,"'T 1CfT F1GURE 4-4-1Rectangular pulseandilsenergydensityspectrumIGUll'. IGIfH' liTj liT• tgU)I I 208 DIGITAL COMMUNICATIONS Hence (Sin!rfT)2IGU)12=(AT)2TifT (4-4-19) (4-4-20)Thisspectrum isillustrated inFig.4-4-1(b). Notethatitcontains zerosat mulliples oflITinfrequency andthatitdecaysinversely asthesquareof thefrequency variable. Asaconsequence ofthespectralzerosinC(f).all butoneofthediscretespectralcomponents in(4-4-18)vanish.Thus.upon substitution forIC(fl!2from(4-4-19), (4-4-18) re~llcesto $,v(f)=(J1A2Tci~rTr +A2p.fl3(f) Example 4-4-2 Asasecondillustration ofthespectral shaping resulting fromg(t),we consider theraisedcosinepulse (4-2-21) Thispulseisgraphically illustrated inFig.4-4·2(a). ItsFouriertransform is easilyderivedanditmaybeexpressed intheform _ATsinlifT -nfT C(f)-"2!rfT(l_f2neJ.(4-4-22) Thesquareofthemagnitude ofC(f)isshowninFig.4-4-2(b). Itis interesting tonotethatthespectrum haszerosatf=nIT,n=±2.±3, ±4,....Consequently, allthediscretespectralcomponents in(4-4-18), ex­ cepttheonesatf=0andf=±IIT,vanish.Whencompared withthe FIGURE 4-4-2Raisedcosinepulseanditsenergydensityspectrum IG(f)I'. g(1) A oT T2 lu)-4fT-3/T-liT-liT0lIT21T31T4fTI (b, CHAPTER 4:CHAR....CTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 209 spectrum oftherectangular pulse,thespectrum oftheraisedcosine pulsehasabroadermainlobebutthetailsdecayinversely asf". Example 4-4-3 Toillustrate thatspectral shapingcanalsobeaccomplished byoperations performed ontheinputinformation sequence, weconsider abinary sequence {bn}fromwhichweformthesymbols (4-4-23) The{bn}areassumed tobeuncorrelated randomvariables, eachhavingzero meanandunitvariance. Thentheautocorrelation function ofthesequence {In}is <l>Am)=E(l"In_m) {2(m=0) = I(m=±I) o(otherwise) Hence,thepowerdensityspectrum oftheinputsequence is <fJii(f)=2(1...cos21CfT) =4cos2rcfT(4-4-24) (4-4-25) andthecorresponding powerdensityspectrum forthe(Iowpass) modulated signalis 4 <fJvv(f)=TIG(f)J'cos'rcfT (4-4-26) 4·4-2PowerSpectraofCPFSKandCPMSignals Inthissection,wederivethepowerdensityspectrum fortheclassofconstant amplitude CPMsignalsthatweredescribed inSection 4-3-3.Webeginby computing theautocorrelation function anditsFouriertransform, aswasdone inthecaseoflinearlymodulated signals. Theconstant amplitude CPMsignalisexpressed as wheres(t;I)=Acos[2llj;.r+cb(t;I)) ~ <I>(t;I)=2rch~Ikq(t-kT) N=-OO(4-4-27) (4-4-28) Z10 DIGITAL COMMUNICATIONS Eachsymbolinthesequence {In}cantakeoneoftheMvalues{±1,±3,..., ±(M-1)1.Thesesymbols arestatistically independent andidentically distrib­ utedwithpriorprobabilities P,,=P(J.=n), n=±I,±3, ...,±(M-l) (4-4-29) where};nP.,=1.Thepulseg(/)=q'(I)iszerooutsideoftheinterval[0,LTJ, q(/)=0,I<0,andq(/)=~forI>LT. Theautocorrelation function oftheequivalent lowpasssignal V(/)=eJ"'CfOll is </>w(t+r;I)=~E[exp(j21fh.~~l.[q(1+r -kT)-q(l-kT)J)] (4-4-30) Firstweexpressthesumintheexponent asaproductofexponents. The resultis </>",,(1+r;I)=!E(fIexp{j2Jrhl.[q(r +r -kT)-q(l-kT)j}) (4-4-31) 1..=--0tJ Next,weperform theexpectation overthedatasymbols {J.}.Sincethese symbols arestatistically independent. weobtain </>",.(1+r;t)=~.TI x(,,~~[ ')P.,exp{j2Jrhn[q(t +r -kT)-q(1-kTJ]}) 11odd (4-4-32) Finally,theaverageautocorrelation function is - 117 <1>,..,(r)=TII<t>",,(1+r:I)dl (4-4-33) Although (4-4-32)impliesthatthereareaninfinitenumberoffactorsinthe product, thepulseg(I)=q·(t)=O for1<0andI>LT.andq(I)=Ofor1<0. Consequently onlyafinitenumberoftermsintheproduct havenonzero exponents. Thus,(4-4-32)canbesimplified considerably. Inaddition, ifwelet r=~+mT.where0'"~<Tandm=0,1,....theaverageautocorrelation in (4-4-33)reducesto J"",U+mT) IIf"'-,(,,-, )=2T II.J.1,.".~t1)P',exp{j2Jrhll[q(I+~-(k -m)T)-q(l-kT))} "odd (4-4-34) CHAPTER 4:CHARACTERIZATiON OFCOMMUNICATION SIGNALS ANDSYSTEMS 211 LetusfocusonJ","(~+mT)for~+mT;;.LT.Intbiscase.(4-4-34)maybe expressed as c!i",,(~+mT)=[.p(jh»)'" IA(~). m;;'L.0'"~<T(4-4-35) where.p(jh)isthecharactaistic function oftherandom sequence {I,,}.de­ finedas 'I-I2: ~f~fC',n ,,=-(,w-l) "odd(4-4-36) andA(~)istheremaining partoftheaverageautocorrelation function. which maybeexpressed as A(O=ZITr'~~L("~%_'I P"exp{j21lhn[l-q(t-kTm) "uoJd xk~~L(,,~1t:-I) P"exp[jZnhnq(t+~-kT)J)dl. m;;.L(4-4-37) 1/odd Thus,J,".,(r)maybeseparated intoaproductofA(~)and.p(jh)asindicated in (4-4-35)forr=~+mT;;.LTand0'"~<T.Thisproperty isusedbelow. TheFouriertransform ofCf>vvCr)yieldstheaveragepowerdensityspectrum as But<I>""(f)=fxCf>""(r)e-j2"trdr =ZRe[L'c!i",(r)e-j2"trdr] (4-4-38) (4-3-39) Withth.eaidof(4-4-35), tbeintegral intherang<-LT'"r<xmaybe expressed as (4-4-40) 212 DIGIT~L COMMUNIC~TIONS Now,Jetr=f+mT.Then(4-4-40)becomes (4-4-41) Aproperty ofthecharacteristic function is''''Uh)1.,;1.Forvaluesofhfor which1",(jh)1<1,thesummation in(4-4-41)converges andyields (4-4-42) Inthiscase,(4-4-41)reducesto [.;:(r)e-/2"fTdt= 1__fT;;.«+LT)e-j2"!lt+LT}dJ: 'l'vv 1_'''('h))'"'T 'I'~~ "LT '"/e l) (4-4-43) Bycombining (4-4-38), (4-4-39), and(4-4-43), weobtainthepowerdensity spectrum oftheCPMsignalintheform [fLT- - 1 i'L+!)T - ]ll>(f)=2Re ...(r)e-)2<!Tdr+ _ ...(r)e-/2"!'dr w 'l'vv I_.,,('h)-)2"fT 'l'vv " 'I'/e l.T (4-4-44) Thisisthedesired result whenIljJ(jh)1<I.Ingeneral, thepowerdensity spectrum isevaluated numerically from(4-4-44). Theaverageautocorrelation function 4>vv(r)fortherange0.,;r.,;(L+l)Tmaybecomputed numerically from(4-4-34). Forvaluesofhforwhich1y,(jh)1=1,e.g.,h=K,whereKisaninteger,we canset "'Uk)=e,2"",0,;;v<1 Then,thesumin(4-4-41)becomes(4-4-45) i:e,2"n/- ,-In..=~+_1io(f_!:.-~)-j!cotTTT(f_!:.)(4-4-46) "l) 2T,,~.. T T T CHArTER~. ('HARAnERIZATION of('()MMlI~ICATI()N SIGNALS .I\NDSYSTEMS 213 Thus,thepowerdensityspectrum nowcontains impulses locatedatfrequencies n+vt,=-r'0,;;:v<1,n=0,1,2,... (4-4-47) Theresult(4-4-46)canbecombined with(4-4-41)and(4-4-39)toobtainthe entirepowerdensityspectrum, whichincludes bothacontinuous spectrum component andadiscretespectrum component. Letusreturntothecaseforwhichlo/J(jh)1<1.Whenthesymbols are equallyprobable, i.e" 1P= -forallnnM thecharacteristic function simplifies totheform 1M-' o/J(jh)~-2: ei~h" Mn~-(M-II nodd 1sinMnh ~- Msinnh(4-4-48) Notethatinthiscaseo/J(jh)isreal.Theaverageautocorrelation function given by(4-4-34)alsosimplifies inthiscaseto i.)1iTl'nlTI1sin2nhM[q(t +r-kT)-q(t-kT)]'l'vv(r=- -. dt(4-4-49)2T0H-LMsm2nh[q(t +r-kT)-q(t-kT)] Thecorresponding expression forthepowerdensityspectrum reducesto ¢>v,(f)=2[1'T4>vv(r)cos2nfrdr 1-o/J(jh)cos2nfT jIL+IIT _ ] +1+f/12(jh)-2.p(jh)cos2nfT L T4>..(r)cos2nfrdr(4-4-50) PowerDensity Spectrum ofCPFSK Aclosed-form expression forthe powerdensityspectrum canbeobtained from(4-4-50) whenthepulseshape­ g(t)isrectangular andzerooutsidetheinterval[0,TJ.Inthiscase,q(t)is linearfor0,,;;t,;;:T.Theresulting powerspectrum maybeexpressed as 214 DIOITAL COMMLINICA nONS where sin]f[fT-H2n-1-M)h1 A,,(f)=!rIfT-~(2n-1-M)hI cos(2]ffT-it"m)-l/Jcosit"... B"m(f) =1+l/1'-2l/1cos2trfT it"m=trh(m+n-1-M) sinMTCh '"==l/1(jh)=M'hSInIf(4-4-52) (4-4-53)Thepowerdensityspectrum ofCPFSKforM=2,4,and8isplottedin Figs4-4-3,4-4-4,and4-4-5asafunction ofthenormalized frequency fT,with themodulation indexh=2j;,Tasaparameter. Notethatonlyone-half ofthe bandwidth occupancy isshowninthesegraphs.Theorigincorresponds tothe carrierj;.Thegraphsillustrate thatthespectrum ofCPFSK isrelatively smoothandwellconfined forh<1.Ashapproaches unity,thespectrabecome verypeakedand,forh=1when 11/11=1,wefindthatimpulses occuratM frequencies. Whenh>1thespectrum becomes muchbroader. [ncommunica­ tionsystems whereCPFSK isused.themodulation indexisdesigned to conserve bandwidth, sothath<1. ThespecialcaseofbinaryCPFSK withh=~(orf"=1/4T)and'"=0 corresponds toMSK.Inthiscase.thespectrum ofthesignalis 16A2T(cos21CfT)' <I>",,(f)=------;>1-16J'T 2 wherethesignalamplitude A=Iin(4·4-52). Incontrast thespectrum of four-phase offset(quadrature) PSK(OQPSK) witharectangular pulseg(l)of duration Tis ,(SintrfT)'<I>,.,,(f) =A-TTr[T (4-4-54) Ifwecompare thesespectral characteristics, weshouldnormalize the frequency variable bythebitrateorthebitinterval ~,.SinceMSKisbinary FSK,itfollowsthatT=T"in(4-4-53). Ontheotherhand.inOQPSK, T=2~, sothat(4-4-54) becomes ,(Sin2lffTb)2<I>,.,,(f)=2A-Tb2Tr[T b(4-4-55) ThespectraoftheMSKandOQPSK signalsareillustrated inFig.4-4-6. NotethatthemainlobeofMSKis50%widerthanthatforOQPSK. However. thesidelobesinMSKfalloffconsiderably faster.Forexample, ifwecompare the bandwidth Wthatcontains 99%ofthetotalpower.wefindthatW=1.2/T"for MSKandW=8/~,forOQPSK. Consequently. MSKhasanarrower spectral CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 215 1.6 1.60.4 0.8 12 Nonnalized frequencyIT ib) h=1.3 0.4 0.8 12 Normalized fl"CquclKyIT (<I)Spectral densityfortwo-level CPFSKh=2/,T .<.~h=0.9 h;0.95 I13 1.2 l.l 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 O.SpectraldensityflKtwo-Ievej CPFSK o l.l 1.0 D.9 0.8 ~0.7 .<;; c0.6u" ~0.5 !!.'"0.4 0.3 0.2 0.1 1.6 0 Normillized frequency[f (e)Spectral densityfortwo-level CPFSK 1.0 0.9 08 ~;050.7h"=2fdT ~0.6";;;cNf1.lje0.5 h=O.7 i04b---' '" ..-h=0.6 0.3\0.2 1).1 \. 0 1 2 Normalized frequencyIT i.1 Spectral den~i(yfortwo-level CPFSK 2.0 1.8 1.6 IA ~h;2/dT ;f1.2 u."1.0g ~0.8 '" 06 04 02 h=l.05 \l 0.0 0.8 1.2 0.4 FIGURE 4-4-3Powerdensityspectrum ofbinaryCPFSK. occupancy whenviewedintermsoffractional out-of-band powerabove fT"=1.Graphsforthefractional out-of-band.power forOQPSK andMSKare showninFig.4-4-7.NotethatMSKissignificantly morebandwidth-efficient thanQPSK.Thisefficiency accounts forthepopularity ofMSKinmanydigital communications systems. Evengreaterbandwidth efficiency thanMSKcanbeachieved byreducing 216 DIGITAL COM"L"'ICATJONS I Nonnalil.edfrequenc.:yIT ("Speclral densil)'forfour-level CPFSK o11=0.95 h=2f,T 1 ;~hr9A "=-,0.. O. O.0.1 ?:' ~0.6 <!IgOJ KO..J V>1.0 0.9 O.i I Norm;;'!;lcd frequency(r (tI)1.0 0.') {I~ o.? fo.()c"~(Ue"0.-+l:- V> OJ 0.2 0.1 Il I.i',----,------,-------, (J.~ H.KJ,.-=1.05 0.7 0.1 0.1~~~::~~~~:e~~~~~:...J (I :1 Nllrm<Jlized frequenq/T (c) FIGURE 444Powerdensityspectrum ofquaternary CPFSK. themodulation index.However, theFSKsignalswillnolongerbeorthogonal andtherewillbeanincreaseintheerrorprobability. Spectral Characteristics ofCPMIngeneral. thebandwidth occupancy of CPMdepends onthechoiceofthemodulation indexh.thepulseshapeg(t), andthenumberofsignalsM.Aswehmcobserved forCPFSK, smallvaluesof hresultinCPMsignalswithrelatively smallbandwidth occupancy, whilelarge valuesofhresultinsignalswitblargehandwidth occupancy. Thisisalsothe caseforthemoregeneralCPMsignals. CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSYSTEMS 217 Spectral densityforeight·level CPFSK 10,.--,---,--~,---,----,r---,r---r----, U.9 0.8h:2J.JSpectraldensityforclght·levc:1 CPFSK h=1.050.26r--r---r--.,--,----,----,-....---, 0.24 0.22 0.20 >..0.18 .~016 ~0.14I~:~ 0.08 0.06 0.04 0.G2 0.000'--'---~--'--""2:--'--~--'---' Normafizc:d frequencyIT (biI 2 3' Normalized frequencyfT (a)r--\-+..::.h.= OJ h=05h=0.4./ hOb F .h=0.7 h=08 oUI FIGURE 4-4-5Powerdensityspectrum ofoctalCPFSK FIGURE4-4-6PowerdensityspectraofMSKandoffsetQPSK.[FromGronemeyer andMcBride (1976);if) IEEE] -10.0I1-1 MSKI,-IITO Off""OPSK 10.0 9.UOtTsetOPSK MSK ~ 30 U ~ U ~ U Nonnalized rrequc:ncy offsetfromcarrier(j-l)TbI(Hz/bit)/s I-20.0 iii'IIIIII.. ;-30.0... c -l! ~-40.0 ~ ~•8.-50.0 ".~.. E ~-60.0 -70.0 -80.00 10 218 DIGITAL COMMUNICA.TlONS '"~ ~-10.0 8-;;-20.0 .~-30.0 i!l!-40.0 ~ o-50.0 ~OffsetQPSK -70.0 -80.002.04.06.08.010.012.014.0 16.018.0 lO.O !WT=two-<;idednormalizcd bandwidth ICHzlbit)/",;;;--60.0 2 "13 £F1GURE 4-4-7Fractional oUI-ol-band power(normalized two-sided bandwidth =2BTl.(From Gronemeyer andMcBnde (1976);<V1976 IEEE.] Theuseofsmoothpulsessuchasraisedcosinepulsesoftheform {_I~(I-cos2Trt)(O~1~LT) g(I)=02LT LT (4-4-56) (otherwise) whereL'=Iforfullresponse andL>Iforpartialresponse, resultinsmaller bandwidth occupancy and,hence,greaterbandwidth efficiency thantheuseof rectangular pulses.Forexample, Fig.4-4-8illustrates thepowerdensity spectrum forbinaryCPMwithdifferent partialresponse raisedcosine(LRC) pulseswhenh=~.Forcomparison, thespectrum ofbinaryCPFSKisalso shown.NotethatasLincreases thepulseg(l)becomes smoother andthe corresponding spectraloccupancy ofthesignalisreduced. dB F1GURE 4-4-lIPowerdensilyspectrum lorbinary('PMwithIr~l anddifferent pulseshapes.(FromAlilillela!.(/91111: <V19111IEEE]-so'----,":-_-'-_..L-:'--,l-n05I.n15 ~.o Nonnaliz~ ff"CquencyIT CHAPTER -t.CHARA.Cl"ERIZAriON OFCOMMUNICATION SIOl"ALS ANDSYSTf'''1S 219 dB , :I:~2() ~ E5 ~ ~ &. -60 FIGURE 4-4-9Powerdensityspectrum forM=4CPMwith3RCand different modulation indices. [FromAu/metal.(/98/); ©/981IEEE.).'1=4 3RC -80()!:---fU.~7,--;()C;'--;I::")7;;'--I:';O'­ NQrmalized frequencyIT Theeffectofvaryingthemodulation indexinaCPMsignalisillustrated in Fig.4-4-9forthecaseofM=4andaraisedcosinepulseoftheformgivenin (4-4-56) withL=3.Notethatthesespectral characteristics aresimilartothe onesillustrated previously forCPFSK, exceptthatthesespectraarenarrower duetotheuseofasmoother pulseshape. Finally,inFig.4-4-10,weillustrate thefractional out-of-band powerfor two-amplitude CPFSKwithseveraldifferent valuesofh. FIGURE 4-4-10 Fractional out~of-band powerfortwo-component CPFSK. (Mulligan./988.) 0 -~_()O -10.00 iO-IS.OO :3 0-20.00•8- ~-25.00cII ~-3(HKl?;;c-3S.(Xl-;; .~---40.(1()~e..---45.00 -50.00 III:::-13 '-::-';;:--;:-':=-::-:-=--:-':::--:--';:,-:-=~::-:--::-::-'::::_ 40.250.500.751.00 1.1~1.501.752.002.252.50 Normalized frequencyIT 220 DIGITAL COMMUNICATIONS 4-4-3PowerSpectraofModulated SignalswithMemory Inthelasttwosections, wehavedetermined thespectralcharacteristics forthe classoflinearly modulated signalswithout memory and'fortheclassof angle-modulated signalssuchasCPFSKandCPM,whicharenonlinear and possessmemory. InthissectIon, weconsider thespectral characteristics of linearlymodulated signalsthathavememory thatcanbemodeled byaMarkov chain.Weha\'ealreadyencountered suchsignalsinSection4-3-2,wherewe described severaltypesofbaseband signals. Thepowerdensity spectrum ofadigitally modulated signalthatis generated byaMarkovchainmaybederivedbyfollowing thebasicprocedure givenintheprevious section.Thus,wecandetermine theautocorrelation function andthenevaluate itsFouriertransform toobtainthepowerdensity spectrum. Forsignalsthataregenerated byaMarkov chainwithtransition probability matrixP,thepowerdensityspectrum ofthemodulated signalmay beexpressed inthegeneralform(seeTitsworth andWelch,1961) 1~IK(n)['(n)1Kcf>(f)=T2}=~p,S,T5f-T+Ti~P,IS;(f)12 2[KK ]+TRe'~J~'PiS;*(f)S;(f)P,j(f) whereS,(f)istheFouriertransform ofthesignalwaveform s,(t), K s;(I)=s,(t)-Lp,sdt) k.=I(4-4-57) P'J(f)istheFouriertransform ofthediscrete-time sequence piJ(n),definedas ~ P,/!)=Lp,,(n)e 'J2KnJT n=I(4-4-58) andKisthenumberofstatesofthemodulator. Thetermp'J(n)denotesthe probability thatthesignalsJ(t)istransmitted nsignaling intervals after thetransmission ofSi(t).Hence,{p'J(n)}arethetransition probabilities inthe transition probability matrixP".NotethatPij(1)=Pir Whenthereisnomemory inthemodulation method, thesignalwaveform transmitted oneachsignaling interval isindependent ofthewaveforms transmitted inprevious signaling intervals. Thepowerdensityspectrum ofthe resultant signalmaystillbeexpressed intheformof(4-4-57), ifthetransition probability matrixisreplaced by [PIP2 P=~'~2 P,P2...PI(] ...PK PI((4-4-59) CHAPTER~: CHARACTERIZATION OFCOMMUNICATION SIGNALS ANDSY"iTEMS 221 andweimposethecondition thatP"=Pforalln;;.1.Undertheseconditions. theexpression forthepowerdensityspectrum becomes afunction ofthe stationary stateprobabilities {Pitonly,and,hence,itreduces tothesimpler form 1~I'"(n)j2(n)CP(f)=T2n~~~PiSiT8f-T 1'"+Ti~p,(l-p,)ISi(fW 2'"K-T,~,}~PiP}Re[Si(f)St(/») ,'}(4-4-60) Weobserve thatourprevious resultforthepowerdensityspectrum of memoryless linearmodulation givenby(4-4-18) maybeviewedasaspecial caseof(4-4-60) inwhichallwaveforms areidentical exceptforasetofscale factorsthatconveythedigitalinformation (Problem 4-30). Wealsomaketheobservation thatthefirsttermintheexpression forthe powerdensityspectrum givenbyeither(4-4-57) or(4-4-60) consists ofdiscrete frequency components. Thislinespectrum vanishes when ±p,Si(!!-) =0 1=1T(4-4-61) Thecondition (4-4-61) isusuallyimposed inthedesignofpractical digital communications systems andiseasily ~atisfied byanappropriate choiceof signaling waveforms (Problem 4-31). Now,letusdetermine thepowerdensity spectrum ofthebaseband­ modulated signals described inSection 4-3-2.First.the:"'NRZsignalis characterized bythetwowaveforms s,(t)=g(t)andS,(I)= -g(I),whereg(l)is arectangular pulseofamplitude A.ForK=2,(4-4-60) reduces \0 CP(f)=(2p~1)'iIc(!!-)IO8(.r-'3...)+4p(1-p)IC(f)12(4-4-62) Tno-x T T T where (4-4-63) Observe thatwhenP=1,thelinespectrum vanishes and'1>(/)reducesto •1 CP(f)=T1C(f)l' (4-4-64) 222 DIGITAL COMMl'NJ(ATIONS TheNRZIsignalischaracterized bythetransition probability matrix p=[;D(4-4-65) NoticethatinthiscaseP"=Pforalln;;>LHence,thespecialformforthe powerdensityspectrum givenby(4-4-62)appliestothismodulation formatas well.Consequently, thepowerdensityspectrum fortheNRZIsignalis identical tothespectrum oftheNRZsignal. Delaymodulation hasatransition probability matrix [0!0!]001 1P= 2 2 ~!0 0 ~0~0(4-4-66) andstationary stiteprobabilities Pi=afor;=I,2,3,4.PowersofPmaybe obtained byuseoftherelation (4-4-67) wherepisthesignalcorrelation matrixwithelements (4-4-68) andwherethefoursignals{silt),;=1,2,3,4}areshowninFig.4-3-15.Itis easilyseenthat p=[~-~-~-~] -I00 1 Consequently, powersofPcanbegenerated fromtherelation P""p=-aP'p, k>1(4-4-69) (4-4-70) Useof(4-4.66). (4-4-69). and(4-4-70) in(4-4-57) yieldsthepowerdensity spectrum ofdelaymodulation. Itmaybeexpressed intheform cf>U)2.//(17.,.8cos81/1)[23- 2cos1/1-22cos21/1-12cos31/1+5cos41/, +12cos5</1+2cos6",-8cos71/1+2cos81/11(4-4-71) where I/J=Tr(r CH,\PTER ~..Cf-'ARACTfRIZAT/ON Of'COMMl-~IC/rnON S.!GNAlS A."iDSYSTEM.') 223 Delaymodulation (Millercode) ,5.2 48 4.4 40 ?:'3.6.;;; ~3.2 1'!2.8 [2.4 Vl2.0 16 1.2 0.8 0.4 o0t:::;:.,0:':.2,.......,.,O.,-4 """'0.""6':"'::0"'.8=1"'0~::Z1.:=2 -1."'4~1:i.,.6c>"1c':.8"""""2 () Normalized frequencyITPowerspectraldensity(one-sided) ofMillercode(delaymodulation) andNRZ/NRZI baseband signals. [FromHechrandGuida(1969); if)19691££E.]FIGURE 4-4-11 Thespectraofthesebaseband signalsareillustrated inFig.4-4-11.Observe thatthespectraoftheNRZandNRZIsignalspeakatf=O.Delaymodulation hasanarrower spectrum andarelatively smallzero-frequency content. Its bandwidth occupancy issignificantly smallerthanthatoftheNRZsignal. Thesetwocharacteristics makedelaymodulation anattractive choicefor channels thatdonotpassdc,suchasmagnetic recording media. 4-5BIBLIOGRAPHICAL NOTES ANDREFERENCES Thecharacteristics ofsignalsandsystemsgiveninthischapterareveryuseful inthedesignofoptimum modulation/demodulation andcoding/decoding techniques foravarietyofchannelmodels.Inparticular, thedigitalmodula­ tionmethods introduced inthischapterarewidelyusedindigitalcommunica­ tionsystems. Thenextchapter isconcerned withoptimum demodulation techniques forthesesignalsandtheirperformance inanadditive, white gaussian noisechannel. Ageneralreference forsignalcharacterization isthe bookbyFranks(1969)_ Ofparticular importance inthedesignofdigitalcommunications systems arethespectralcharacteristics ofthedigitally modulated signals,whichare presented inthischapterinsomedepth.Ofthesemodulation techniques, CPM isoneofthemostimportant duetoitseffiCientuseofbandwidth. Forthis reason,ithasbeenWidelyinvestigated bymanyresearchers, andalarge numberofpapershavebeenpublished inthetechnical literature. Themost comprehensive treatment ofCPM,inclUding itsperformance anditsspectral characteristics, canbefoundinthebookbyAnderson etat.(1986).Inaddition tothistext,thetutorialpaperbySundberg (1986)presents thebasicconcepts andanoverview oftheperformance characteristics ofvariousCPMtechniques. Thispaperalsocontainsover100references topublished papersonthistopic. Therearealargenumberofreferences dealingwiththespectralcharac­ teristicsofCPFSKandCPM.Asapointofreference, weshouldmention that MSKwasinvented byDoelzandHealdin1961.Theearlyworkonthepower 224 DIGITAL COMMUNICATIONS spectral densityofCPFSK andCPMwasdonebyBennett andRi£e(1963), Anderson andSalz(1965),andBennett andDavey(1965).ThebookbyLucky etal.(1968)alsocontains atreatment ofthespectral characteristics ofCPFSK. Mostoftherecentworkisreferenced inthepaperbySundberg (1986).We shouldalsocitethespecialissueonbandwidth-efficient modulation andcoding published bytheIEEETransactions onCommunications (March1981),which contains several papersonthespectral characteristics andperformance of CPM. Thegeneralization ofMSKtomultiple amplitudes wasinvestigated by Weberelal.(1978).Thecombination ofmultiple amplitudes withgeneralCPM wasproposed byMulligan (1988)whoinvestigated itsspectral characteristics anditserrorprobability performance ingaussian noisewithandwithout coding. 4-1Provethefollowing properties ofHilberttransforms: aIfx(t)=x(-t)theni(t)=-i(-I); bIfX(I)=-xl-I)theni(t)=i(-I); cIfX(I)=cosw,,1theni(l)=sinw"l; dIfx(t)=sinw,,1theni(l)=-cosw"l; ei(l)=-X(I); rr.X'(I)dl=r.i'(I)dl; gr.x(l)i(l)dl=O. 4-2IfX(I)isastationary randomprocesswithautocorrelation function <1>"(r)= E[x(I)x(1 +f)]andspectral density<I>,,(f)thenshowthat<I>".(r)=q",(f). <1>,,(r)=-ci>"(f),and<I>,,(f)=<p,,(n. 4-3Suppose thatn(l)isazero·mean stationary narrowband processrcpresented by either(4-1-37), (4-1-38), or(4-1-39). Theautocorrelation function ortheequiv­ alentlowpassprocessZ(I)=X(I)+jy(l)isdefinedas <I>....(r)=~Elz*(I)z(1 +r)] aShowthat E[Z(I)Z(I+rll=0 bSuppose <1>....(r)=Nolj(r).andlet v=f'Z(lldl " Determine E(V')andE(VV*) =E(iVl'). 4-4Determine theautocorrelation function ofthestochastic process X(I)=Asin(2Jrf,!+0) wheref,isaconstant and0isauniformly disllit'lbted phase,i.e.. 1 p(O)=211'O".0".2Ir 4-.5ProvethatS,(I)isgenerally acomplex-valued signalandgivethecondition under whichitisreal.Assumethats(t)isareal-valued bandpass signal. C1IAIYITR 4C1IARACIl-RIZATION OFCOMMUNICATION SIG~ALS ANDSY\TF.MS 225 4-6Supposc thats(r)iscltherarcal-orcomplex-valued signalthatisrepresented asa linearcomhination oforthonormal functions (f.,(I)},i.e., K S(I)=LS,j;(I) k·1 where Ix {O(m..n) J,(I)f,~(I) dl=1(m=n) Determine theexpressions forthecoefficients Is,}intheexpansion S(I)that minimize theenergy Z',=r~IS(I)-s(I)I'dl andthecorresponding residualerrorZ',. 4-7Suppose thatasetofMsignalwaveforms {S/m(l)}arecomplex·valued. Derivethe equations fortheGram-Schmidt procedure thatwillresultinasetof"'0;;;M orthonormal signalwaveforms. 4-8Determine thecorrelation coefficients P.mamongthefoursignalwaveforms {s,(lll showninFig.4-2-1,andthecorresponding Euclidean distances. 4-9Consider asetofMorthogonal signalwaveforms sm(t).10;;;m0;;;M.00;;;10;;;T.all ofwhichhavethesameenergy ~.DefineanewsetofMwaveforms as 1'" S';'(I)=Sm(l)- -LS,(I). 10;;;m""M,00;;;1""T M/C=I ShowthattheMsignalwaveforms {S';'(I)}haveequalenergy,givenby 'C'=(M-l)~/M andareequallycorrelated, withcorrelation coefficient 1r' 1 Pm"='f;'J"s:,,(I)S;'(I) dl= -M_I 4-10Consider thethreewaveforms [.(1)showninFig.P4-10. aShowthatthesewaveforms areorthonormal. bExpressthewaveform X(I)asaweighted linearcombination of[.(1).n=1,2,3,K . {-I(00;;;1<1) X(I)=1(10;;;I<3) -1(30;;;1<4) anddetermine theweighting coefficients. 4-11Consider thefourwaveforms showninFig.P4-11. aDetermine thedimensionality ofthewaveforms andasetofbasisfunctions. bUsethebasisfunctions torepresent thefourwaveforms byvectors 81•8"s" ands•. e:Determine theminimum distance between anypairofvectors. 4-12Determine asetoforthonormal functions forthefoursignalsshowninFig.P4-12. 226 l)lUITAL COMMUNICA.TIONS 12f------, 4f-----.., FIGURE P4-10o 2 41 Nt) 12 0 2 1-2o 44 .I.(t) $)(1) 2 0 04 34 -I -I .11(1) .I..(t) 2- 1- 030234 -2 -2 FlGUREP4-11 SI(t~1 3~Sl(t~b I 0 2 0I Sl(trc=rS'(~I oI 2 3 tI•FlGl.lRE P4-U 0 2 CHAPTER 4:CHARACTERIZATION OFCOMMUNICATION SIGNALS "NOSYSTEMS127 4-13Alowpassgaussian stochastic processX(I)hasapowerspectraldensity <I>(f)={No(lfl<B)o(lfl>B) Determine thepowerspectral densityandtheautocorrelation fOnction of y(l)=x'(I). 4-14Consider anequivalent lowpassdigitallymodulated signaloftheform U(I)=L[ang(1-2nT)-jb"g(l-2nT-T)] where{a,}and{b.}aretwosequences ofstatistically independent binarydigitsand g(l)isasinusoidal pulsedefinedas ({sin(m/2T) (0<1<2T)gI)=o (otherwise) Thistypeofsignalisviewedasafour-phase PSKsignalinwhichthepulseshapeis one-half cycleofasinusoid. Eachoftheinformation sequences {a,,}and{b.}is transmitted atarateof1/2Tbitslsand,hence,thecombined transmission rateis lITbits/s.Thetwosequences arestaggered intimebyTseconds intransmission. Consequently, thesignalu(t)iscalledstaggered four-phase PSK. •Showthattheenvelope lu(I)1isaconstant, independent oftheinformation a.on thein-phase component andinformation b.onthequadrature component. In otherwords,theamplitude ofthecarrierusedintransmitting thesignalis constant. bDetermine thepowerdensityspectrum ofu(t). eCompare thepowerdensityspectrum obtained from(b)withthepowerdensity spectrum oftheMSKsignal.Whatconclusion canyoudrawfromthis comparison? 4-15Consider afour-phase PSKsignalrepresented bytheequivalent lowpasssignal u(t)=Ll.g(t-nT) whereI,takesononeofthefourpossible valuesv'I(±1±j)withequal probability. Thesequence ofinformation symbols{I.}isstatistically independent . •Determine andsketchthepowerdensityspectrum ofu(l)when {A(O.,t'"T) g(I)=0(otherwise) bRepeat(a)when g(l)={Asin(It/IT)(0"t"T)o (otherwise) eCompare thespectraobtainedin(a)and(b)intermsofthe3dBbandwidth and thebandwidth tothetirstspectralzero. 228 D!<.iITAL l'OM~H:~ICA nONS ",ff}\,11) \-lm 1717111 0 0oII2I ~J7 _\~l/) riC/) hIll Ol-~-'" FIGURE P4-IK-/7 4-16Therandomprocessv(t)isdefinedas V(I)=Xcos21Cf.1-Ysin21Cf.r whereXandYarerandomvariables. ShowthatV(I)iswide-sense stationary if andonlyifE(X)=E(Y)=O.£(X')=E(Y').andE(XY)=0. 4-17CarryouttheGram-Schmidt orthogonalization ofthesignalsinFig.4-2-I(a) in theorderS"(I).S,(I).S,(I).and,thus.obtainasetoforthonormal functions {f.,,(I)}. Then.determine thevectorrepresentation ofthesignals{S,,(I)}byusingthe orthonormal functio~ (j;,,(I)}.Also,determine thesignalenergies. 4-18Determine thesignalspacerepresentation ofthefoursignalss,(I).k=I.2.3.4, showninFig.P4-18.byusingasbasisfunctions theorthonormal functions [.(1)and [,(I).Plotthesignalspacediagram andshowthatthissignalsetisequivalent to thatforafour-phase PSKsignal. 4-19Thepowerdensityspectrum ofthecyclostationary process V(I)=2:J"g(t-nT) wasderivedinSection4-4-1byaveraging theautocorrelation function "'",,(I+r.I) overtheperiodToftheprocessandthenevaluating theFouriertransform ofthe average autocorrelation function. Analternative approach istochangethe cyclostationary processintoastationary processv~(t)byaddingarandomvariable a.uniformly distributed over0",a<T.sothat V,(I)=LJ"g(t-"T-a) anddefining thespectral densityofV(I)'astheFourier transform ofthe autocorrelation function ofthestationary process V~(I).Derivetheresultin (4-4-11). byevaluating theautocorrelation function ofv,(t)anditsFourier transform. CII·\PTER" (HARA(·Tf.RJZATlO~ orn)MMl' .....I("Ano~ SI(j~ALS ANDSYSTlMS 229 Illput 'n d<llil II"::0:.II Output ~ I 0 I ~2T 1T 4·20APAMpartialresponse signal(PRS)isgenerated asshown InFig.P4·20by excitinganideallowpassfilterofbandwidth Wbythesequence atarateI/T=2Wsymbols/s. Thesequence {/..Iconsistsofbinarydigitsselected independently fromthealphabet {I.-I}withequalprohability. Hence,thefiltered signalhastheform u(1)=2:B.g(l-nT). ,,--<0IT=~ 2W aSketchthesignalspacediagram foru(1)anddetermine theprobability of occurrence ofeachsymbol. bDetermine thelwtocorreJation andpowerdensityspectrum ofthelhree·level sequence {E,,}. cThesignalpointsofthesequence {B.}formaMarkovchain.SketchthisMarkov chainandindicatethetransition probabilities amongthestates. 4-21Thelowpassequivalent representation ofaPAMsignalis u(l)=2:l.g(1-nT) Suppose gfl)isarectangular pulseand where{a.,}isasequence ofuncorrelated binary·valued (I.-I)random variables thatoccurwithequalprobability. aDetermine theautocorrelation function ofthesequence UHI bDetermine thepowerdensityspectrum of11(1). eRepeat(b)jfthepossible valuesofthe0"are(0,I). 4-22Showthatx(1)=5(1)cos21if,±i(r)sin2trf..risasingle·sideband signal,wheresIr) isband-limited toB""I.Hzand.1'(1)isitsHilberttransform. 230 DIGITALfOMMUl'iIC ATIONS 4-23UsetheresultsinSection4-4-3todetermine thepowerdensityspectrum ofthe binaryFSKsignalsinwhichthewaveforms are 5,(1)=sinw,l,i=1,2, O';I.;T wherew,=nklTandw,=m1l/T, II'"m,andmandIIarearbitrary positive integers, Assume thatp,=p,=j,Sketchthespectrum andcompare thisresult withthespectrum oftheMSKsignal. 4-24UselheresultsinSection4-4-3todetermine thepowerdensityspectrum of multitone FSK(MFSK) signalsforwhichthesignalwaveforms are 2Jrllls,,(I)=sinT,1I=I,2,.,.,M, O"'I.,;T Assume thattheprobabilities p,=1/Mforalli,Sketchthepowerspectraldensity. 4-25Aquadrature partialresponse signal(OPRS)isgenerated bytwoseparate partial response signalsofthetypedescribed inProblem 4·20placedinphasequadrature. Hence.theQPRSisrepr.esented as 5(1)=Re[u(I)e"""J where U(I)=v,(I)+jv,(t) =~B"II(I-liT)+j~C,,1I(t-nT) andB"=I"+I",andC"=i"+J",.Thesequences {B"}and{C,,}areuncorre· latedandI"= ±I,i"=±Iwithequalprobability. aSketchthesignalspacediagram fortheQPRSsignalanddetermine the probability ofoccurrence ofeachsymbol. bDetermine theautocorrelations andpowerspectradensityofu,(r),V,(I),and U(I). . . cSketchtheMarkovchainmodelandindicatethetransition probabilities forthe OPRS. 4-26Determine theautocorrelation functions fortheMSKandoffsetOPSKmodulated signalsbasedontheassumption thattheinformation sequences foreachofthe twosignalsareuncorrelated andzero-mean. 4-27SketchthephaseIree.thestatetrellis,andthestalediagram forparrialresponse CPMwithh=~and {1/4T(Oq';2T)1/(1)=o(otherwise) 4-;!8Determine thenumberofterminal phasestatesinIhestatetrellisdiagramfor a afullresponse binaryCPFSKwitheitherII=~orJ: b apartialresponse L=3binaryCPFSKwitheitherh=jorJ. 4-29Showthat16OAMcanberepresented asasuperposition oftwofour-phase constant envelope signalswhereeachcomponent isamplified separately before summing, i.e, 5(t)=CIA"cos2Jrf,1+B"sin2Jr[.IJ+IC,cos2Jrf.1+D"sin2trf..t) where{A..l.{B,,},{C,,}.and{D,,)arestatistically independent binarysequences (HAPTER 4nIARACH-RIZATION OF["OMHl:~ICATIOS Sl(i"iALS A~DS'l'STFMS 231 wilhelements tromtheset{+L-]}andGistheamplifier gain.Thus.showIhal theresulting signalisequivalent 10 J(l)=1"cos2JCJ:f+Q"sin2JCJ:/ anddetermine IIIandQ"intermsofA".8,1'C",andDOl' 4-:10Usetheresultin(4-4-60) toderivelheexpression torthepowerdensityspectrum ofmemorylesslinearmodulation givenby(4-4-18) underthecondition that J,(I)=I,J(I). k=1.2,'..•K where1,isoneoftheKpossible Iransmitted symbols thatoccurwilhequal probability. 4-31Showthatasufficient condition fortheabsenceofthelinespectrum component in (4-4-60) is ,, Isthiscondition necessary? Justifyyouranswer. 4-32Theinformation sequence {o,,}.~•isasequence ofiidrandom variables. eacli takingvalues+1and-Iwithequalprobability. Thissequence istobetransmitted atbaseband byabiphasecodingscheme. described by J(f)=2:o"l(r-nT) whereg(f)isshownmFig.P4-32. aFindthepowerspectraldensityof"(f). bAssumethatitisdesirable tohaveazerointhepowerspectrumall=liTTo thisend.weuseaprecoding schemebyintroducing boo=0"+ko""wherekis someconstant. andthentransmit the{b,,}sequence usingthesameI(r).Isit possibletochoosektoproduce afrequency nullatf= ]IT?Ifyes.whatarethe appropriate valueandtheresulting powerspectrum? eNowassumewewanttohavezerosatallmultiples oft,=1/4TIsitpossible to havethesezeroswithanappropriate choiceofkintheprevious part?Ifnot thenwhatkindofprecodingdoyousuggesttoresultinthedesirednulls" 4-33Starting withthedefinition ofthetransition probability matrixfordelav modulation givenin(4-4-66). demonstrate thattherelation P'p=-lp holds,and.hence. P"'p=-!P'P. k~I /:(1) FIGURE P4-32o -Ir , 232 DIGITAL COMMUNICATIONS 4-34Thetwosignalwaveforms forbinaryFSKsignaltransmission withdiscontinuous phaseare so(l)=.Jf!:cos[2/r(f-i}+80].0"'1<T S,(/)=.Jf!:cos[2/r(f+i)1+8,J0'"I'"T wheretif=1fT«t.and80and8,areuniformly distributed randomvariables on theinterval(0,2/r).Thesignals 50(/)and5,(/)areequallyprobable. aDetermine thepowerspectraldensityoftheFSKsignal. bShowthatthepowerspectraldensitydecaysasl/f'forf»[.. 5 OPTIMUM RECEIVERS FOR THEADDITIVE WHITE GAUSSIAN NOISE CHANNEL InChapter4,wedescribed varioustypesofmodulation methods thatmaybe .usedtotransmitdigitalinformation throughacommunication channel. Aswe haveobserved, themodulator atthetransmitter performs thefunction of mapping thedigitalsequence intosignalwaveforms. Thischapterdealswiththedesignandperformance characteristics of optimum receivers forthevariousmodulation methods, whenthechannel corrupts thetransmitted signalbytheaddition ofgaussian noise.InSection 5-1,wefirsttreatmemoryless modulation signals,followed bymodulation signalswithmemory. Weevaluate theprobability oferrorofthevarious modulation methods inSection5-2.Wetreattheoptimum receiverforCPM signalsanditsperformance inSection5-3.InSection5-4,wederivethe optimum receiver whenthecarrierphaseofthesignalsisunknown atthe receiver andistreatedasarandom variable. Finally,inSection5-5,we consider theuseofregenerative repeaters insignaltransmission andcarryout alinkbudgetanalysisforradiochannels. 5-1OPTIMUM RECEIVE~ FORSIGNALS CORRUPTED BYADDITIVE WHITE GAUSSIAN NOISE Letusbeginbydeveloping amathematical modelforthesignalattheinputto thereceiver. Weassumethatthetransmitter sendsdigitalinformation byuse ofMsignalwaveforms {sm(t),m=1,2,...,M}.Eachwaveform istransmitted withinthesymbol(signaling) intervalofdurationT:Tobespecific, weconsider thetransmission ofinformation overtheinterval0""t""T. 233 234 DIGITAL (llMML:NICATIONS Channel Trllsmitte1i }--I-+ Receiveds;@""I--I---<.,+/ .f"PI rlt)=JII,(11+",,} FIGURE 5-1·1Modelforreceivedsignalpassedthroughan AWONchannel.NoiSoC 11(1) Thechannel isassumed tocorruptthesignalbytheaddition ofwhite gauSsian noise.asillustrated inFig.5-1-1.Thus.thereceived signalinthe interval0.,;1'"Tmaybeexpressed as '(1)=S",(I)+n(/).0.,;1'"T (5-1-1) wheren(l)denotesasamplefunction oftheadditive whitegaussian noise (AWGN) processwithpowerspectraldensity 4>"..(f)=~NIIW1Hz.Basedon theobservation of,(1)overthysignalinterval. wewishtodesignareceiver thatisoptimum inthesense(hatitminimizes theprobability ofmakingan error. Itisconvenient tosubdivide Ihereceiver intotwoparts-the signal demodulator andthedetector-as showninFig.5-1-2.Thefunction ofthe signaldemodulator istoconvertthereceived waveform '(1)intoanN­ dimensional vectorr=[TIT2•••rNl.whereNisthedimension ofthe trli\nsmitted signalwaveforms. Thefunction ofthedetector istodecidewhich oftheMpossiblesignalwaveforms wastransmitted basedofthevectorr.. Tworealizations ofthesignaldemodulator aredescribed inthenexttwo sections. Oneisbasedontheuseofsignalcorrelators. Thesecondisbasedon theuseofmatched filters.Theoptimum detector thatfollowsthesignal demodulator isdesigned tominimize theprobability oferror. 5-1·1Correlation Demodulator Inthissection.wedescribe acorrelation demodulator thatdecomposes the received signalandthenoiseintoN-dimensional vectors.Inother'words.the signalandthenoiseareexpanded intoaseriesoflinearly weighted orthonormal basisfunctions {j;,(I)}.Itisassumed thattheNbasisfunctions {f,,(I)}spanthesignalspace.sothateveryoneofthepossible transmitted FIGURE 5-1·2Receiver con6guration. Received signalr(t)DelectorOulputt--- dc<:ision (5-1-2)CHAPTER S:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE G.AUSSIAN NOISECHANNEL 135 signalsoftheset{Sm(f),1,;;m';;M}canberepresented asaweighted linear combination of{fn(t)}.Inthecaseofthenoise,thefunctions {f,,(t)}donotspan thenoisespace.However, weshowbelowthatthenoisetenmsthatfalloutside thesignalspaceareirrelevant tothedetection ofthesignal. Suppose thereceived signalr(t)ispassedthrough aparallelbankofN crosscorrelators whichbasically compute theprojection ofr(t)ontotheNbasis functions {f.,(t)},asillustrated inFig.5-1-3.Thus,wehave fr(f){.(f) dt=f[s",(t)+n(t)]{.(t) dt r.=S""+n.,k=I,2,...,N where Sm'=rSm(t){.(I) dt,k=I,2,,N n.=rn(t){.(t)dt,k=I,2,,N(5-1-3) Thesignalisnowrepresented bythevector Smwithcomponents S"'" k=1.2,...,N.TheirvaluesdependonwhichoftheMsignalswastrans­ mitted.Thecomponents {n.}arerandomvariables thatarisefromthepresence oftheadditivenoise. Infact,wecanexpressthereceived signalr(t)intheinterval0",t,;;Tas N N r(l)=2:sm.{.(t)+2:n.t.(t)+n'(I) k=l k=1 N =2:rd.(l)+n'(t) k=1 Thetermn'«),definedas N n'(I)=nit)-2:nd.(t) k=l(5-1-4) (5-1-5) isazero-mean gaussian noiseprocessthatrepresents thedifference between theoriginalnoiseprocessnit)andthepartcorresponding totheprojection of nit)ontothebasis.functions {t.(t)}.Weshallshowbelowthatn'(f)is irrelevant tothedecision astowhichsignalwastransmitted. Consequently, the decision maybebasedentirely onthecorrelatar outputsignalandnoise components r.=Sm'+n.,k=I,2,...,N. Sincethesignals{s",(t)}aredeterministic, the·signalcomponents are deterministic. Thenoisecomponents {n.laregaussian. Theirmeanvaluesare E(n.)=frE[n{t»)t.(t) dt=0 o(5-1-6) 236DIGITAL COMMUNiCATIONS f,(t) nGURE 5-1-3Correlation-type demodulator. foralln.Theircovariances areReceived signal r(1) 'N,0-­,, Sample att=TTodetector E(n.nm)=rrE[n(t)n(·))fk(t)fm('r)dt d. =~NofrIl(t-r)f.(t)fm(.)dt dr ,!Noffk(t)fm(t) dt =!No8mk (5-1-7) where8mk=1whenm=kandzerootherwise. Therefore, theNnoise components {n.}arezero-mean uncorrelated gaussian randomvariables witha common variance (T~=!No. Fromtheabovedevelopment, itfollowsthatthecorrelator outputs{r.} conditioned onthemthsignalbeingtransmitted aregaussian randomvariables withmean (5-1-8) andequalvariance (5-1-9) Sincethenoisecomponents {nk}areuncorrelated gaussian randomvariables, theyarealsostatistically independent. Asaconsequence, thecorrelator outputs{r.}conditioned onthemthsignalbeingtransmitted arestatistically independent gaussian variables. Hence,theconditional probability density functions oftherandomvariables['I'2...'N)=raresimply N p(rISm)=flp(rkISmk), k-Jm=1,2,...,M (5-1-10) CHAPTER', OPTIMUM RECEiVERS FORTHEADDITIVE WHITEGAUSSIAN NOISECHANNEL 237 where 1[(ro-Smo)2]p(r,lsmo)=.r-;-;-eKp -No•k=\.2•...•N(5-1-\1) v1CNo 0 Bysubstituting (5-1-\1)into(5-\-10), weobtainthejointconditional pdfs I\[.:!:(ro-Smo)2] 2 M ( \\2)p(rsm)= (No)N12eKp -.LJ No•m=\,,...,5-• tr() k=I () Asafinalpointwewishtoshowthatthecorrelator outputs(rl''2,...,rN) aresufficient statistics forreaching adecision onwhichoftheMsignalswas transmitted, i.e.,thatnoadditional relevant inrormation canbeextracted from theremaining noiseprocessn'(I).Indeed,n'(t)isuncorrelated withtheN correrator outputsirk}.i.e., E[n'(I)r,) =E[n'(t»)Smk +E[n'(t)n.) =E[n'(t)nol =E{[n(t)-#,nih(t)]n.} =rE[n(t)n(r»)J.(r) dr-i~E(nJno)jj(t) =Vv,,!.(t)-~N"J.(t)=0 (5-\-13) Sincen'(I)andhIaregaussian anduncorrelated. theyarealsostatistically independent. Consequently, n'(t)doesnotcontainanyinformation thatis relevanttothedecision astowhichsignalwaveform wastransmitted. Allthe relevant information iscontained inthecorrelator outputs fro}·Hence.n'(t) maybeignored. Example 5-1-1 Consider anM-arybaseband PAMsignalsetinwhichthebasicpulseshape g(t)isrectangular asshowninFig.5-\·4.Theadditivenoiseisazero-mean whitegaussian noiseprocess. Letusdetermine thebasisrunction[(I)and theoutputorthecorrelation-type demodulator. Theenergyintherectangu. larpulseis nGURE 5·1·4SignalpulseforExample 5-1-1.~Il=+-'" orr 238 DIGITAL COMMUNICATIONS SincethePAMsignalsethasdimension·N =1,thereisonlyonebasis function{(I).Thisisgivenas 1((I)= .,.,,-;;;g(l)va-T ={I/YT (O~I~T) o(otherwise) Theoutputofthecorrelation-type demodulator is iT 1( r=0r(l)f(l)dl =YTJ or(l)dl Itisinteresting tonotethatthecorrelator becomes asimpleintegrator when 1(1)isrectangular. Ifwesubstitute forr(t).weobtain r=J:r{f[sm(r)+n(t)]}dl =J:r[fsm(t)dt+fn(l)dl] r=Sm+n .wherethenoisetermE(n)=0and (T~=E[~rfn(t)n(T)dld1'] ILTiT =-.£[n(l)n(1')] dld1'Too No(iT =2;Jo06(1-1')dldr= !No Theprobability densityfunctionforthesampledoutputis p(rISm)=_1_exp[ _.>...(r_-_S-",m,--)'] VtrNo No 5·1·2Matched·Filter Demodulator InsteadofusingabankofNcorrelators togenerate thevariablesirk},wemay useabankofNlinearfilters.Tobespecific,letussupposethattheimpulse responses oftheNfiltersare (5-1-14) c,HAI'T1-R 5:(WT1Mt-\1 HFlTl\TRS FORTIlEAUDITIVE WHITE GAl!S,IAN NOISECHASNEl 239 S(~~/1 hili:,(r-~~ Y-h~ FIGURE 5-1-5Signal,-(I)andfiltermatched tos(l)_iii.)SignalsIt) (b-lImpulseresponse offiltermatched (0,\"(/1 where([.(lllaretheNbasisfunctions andh.(I)=0outsideoftheinterval o~1~T.Theoutputsofthesefiltersare Ydl)=I'r(r)hk(l- r)dr o =f'r(r)!k(T-I+r)dr. k=I.2•....N o Now.ifwesampletheoutputsofthefiltersat1=T,weobtain Yk(T)=ITr(r)jk(r) dT=rk.k=1.2,...,N o(5-1-15) (5-1-16) Hence,thesampled outputsofthefiltersattimet=Tareexactlythesetof values{rk}obtained fromtheNlinearcorreIators. Afilterwho~eimpulseresponse h(t)=s(T-t).wheres(t)isassumed tobe confined tothetimeinterval0~t~T.iscalledthematched filtertothesignal S(I).Anexample ofasignalanditsmatched filterareshowninFig.5-1-5.The response ofh(t)=s(T-I)tothesignalS(I)is y(t)=I's(r)s(T-t+r)dr o(5-1-17) whichisbasically thetime-autocorrelation function ofthesignals(t).Figure 5-1-6illustrates y(t)forthetriangular signalpulseshowninFig.5-1-5.Note thattheautocorrelation functiony(l)isanevenfunction oft.whichattainsa peakatI=T. Inthecaseofthedemodulator described above,theNmatched filtersare y(ll-= J~S('t).I'(T-I"tld't FIGURE 5-1-6Thematched filteroutputistheautocorrelation function ofs(rl.-!:~--=---''!=---o T 2T I 240 DIGITAL COM.WUNICATJONS Received signal r(t) FIGURE 5-1-7Matched filterdemodulator. (5-1-18)matched tothebasisfunctions {A(t)}.Figure5-1-7illustrates thematched filter demodulator thatgenerates theobserved variables {r.}. Properties of,tbeMatched FilterAmatched filterhassomeinteresting properties. Letusprovethemostimportant property, whichmaybestatedas follows:Ifasignals(t)iscorrupted byAWGN,thefilterwithimpulseresponse matched tos(t)maximizes theoutputsignal-to-noise ratio(SNR), Toprovethisproperty, letusassumethatthereceivedsignalr(t)consistsof thesignals(t)andAWGNn(t)whichhaszero-mean andpowerspectral density4>nn(f)=!NoW1Hz,Suppose thesignalr(t)ispassedthroughafilter withimpulseresponse h(t),0,;;;t,;;;T,anditsoutputissampled attimet=T. Thefilterresponse tothesignalandnoisecomponents is y(t)=[r(-r)h(t--r)dr =[s(-r)h(t--r)d-r+I:n(-r)h(t--r)d-r Atthesampling instantt=T,thesignalandnoisecomponents are y(T)=rs(-r)h(T--r)dr+rn(-r)h(t--r)d-r =y,(T)+Yn(T) (5-1-19) (5-1-20)wherey,(T)represents thesignalcomponent andy.(T)thenoisecomponent. Theproblem istoselectthefilterimpulseresponse thatmaximizes theoutput signal-to-noise ratio(SNRo)definedas SNRo=y;(T) E[y~(T)l • CHAPTER _~()PTI\1l:~ RECEIVERS FORTHEADDITIVE WHiTE (;AL'SSlAN ~OISECHAN"'F.l 241 Thedenominator in(5-1-20) issimplythevariance ofthenoisetermatthe outputofthefilter.Letusevaluate E[y;,(T)). Wehave E[y;,(T)) =JTfTE[n(r)n(t»)h(T -r)h(T-t)drdr oIITr =!NoJJ8(r-r)h(T-r)h(T-t)dtdr o0 =~NoJTh'(T-I)dr o(5-1-21) Notethatthevariance depends onthepowerspectral densityofthenoiseand theenergyintheimpulseresponse h(t). Bysubstituting fory,(T)andE[y;,(T») into(5-1-20), weobtainthe expression fortheoutputSNRas [iT.s(r)h(T-r)dr)' SNRo=jfT',No 0h(T-t)dtifcih(r)s(T -r)drl' !NoJlh'(T-t)dt(5-1-22) Sincethedenominator oftheSNRdepends ontheenergyinh(t),the maximum outputSNRoverh(t)isobtained bymaximizing thenumerator subjecttotheconstraint thatthedemoninator isheldconstant. Themaximiza­ tionofthenumerator ismosteasilyperformed byuseoftheCauchy-Schwarz inequality, whichstates.ingeneral. thatifg,(t)andg2(t)arefinite-energy signalsthen [r~g,(t)g,(t) dtr""rg;(tldtr~gj(t)dr (5-1-23) withequality wheng,(t)=Cg,(t)foranyarbitrary constant C.Ifweset g,(I)="(1)andg,(r)=s(T- t),itisclearthattheSNRismaximized when h(t)=Cs(T-r),i.e.,h(t)ismatched tothesignals(t).ThescalefactorC' dropsoutoftheexpression fortheSNR,sinceitappears inboththe numerator andthedenominator. Theoutput(maximum) SNRobtained withthematched filteris 2fTSNRo=NS2(t)dt II0 =2'fINo (5-1-24) NotethattheoutputSNRfromthematched filterdepends ontheenergyof thewaveform s(t)butnotonthedetailedcharackristics ofset).Thisisanother interesting property ofthematched filter. Frequency-Domain Interpretation ortheMatched FilterThematched filterhasaninteresting frequency:domain interpretation. Sinceh(t)=s(T-f). 242 DIGITAL COMMUNICATIONS theFouriertransform ofthisrelationship is H(f)=rs(T_I)e~jz,../t dl =[iTs(r)eiZ"'ff dr]e~jZ"'fT =S*(f)e~jZ"'fT (5-1-25) Weobserve thatthematched filterhasafrequency response thatisthe complex conjugate ofthetransmitted signalspectrum multiplied bythephase factor e~iZ,.-JT,whichrepresents thesampling delayofT.Inotherwords, IH(f)1=IS(f)I.sothatthemagnitude response ofthematched filterisidentical tothetransmitted signalspectrum. Ontheotherhand,thephaseofH(f)isthe negative ofthephaseofS(f). Now,ifthesignalS(I)withspectrum S(f)ispassedthroughthematched filter,thefilteroutputhasaspectrum Y(f)=IS(fWe~jz"'fT. Hence,theoutput waveform is y.(l)=[Y(f)eiz,../r df Bysampling theoutputofthematched filterat1=T.weobtain y,(T)=[/5(fWdf=rSZ(I)dt=g'(5-1-26) (5-1-27) wherethelaststepfollowsfromParsevaJ's relation. Thenoiseattheoutputofthematched filterhasapowerspectraldensity cf>o(f)=!jH(fWNo Hence,thetotalnoisepowerattheoutputofthematched filteris Pn=rcf>o(f)df =~NofxIH(f)12df=~NorIS(f)12df=~~No TheoutputSNRissimplytheratioofthesignalpowerP"givenby(5-1-28) (5-1-29) (5-1-30) CHAFTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL 243 1,(1} [,(1) [[Jt JT 0ITTJT1 0ITTJT1 2 2 2 2 (0) hl(t)=f.<T-1) h2lf)=f2(T-r) rzJt J1' 0ITTJT1 0ITTJT1 2 2 2 2 (0) h,(t) .'~'!,(tl }VT 14"T 0!TTJT1 0ITTJTI 2 2 2 2 ,e) FIGURE 5-1-8Basisfunctions andmatched filterresponses forExample 5-]-2. tothenoisepowerPn.Hence, P,'t22'tSNRo=-=-=­Pn!'tNoNo whichagreeswiththeresultgivenby(5·1-24). Example 5-1-2(5-1-31) Consider theM = 4biorthogona\ signalsshowninFig.5-1-8fortransmitting information overanAWGNchannel. Thenoiseisassumed tohavezero meanandpowerspectraldensity!No.Letusdetermine thebasisfunctions forthissignalset,theimpulseresponses ofthematched-filter demodulators. andtheoutputwaveforms ofthematched-filter demodulators whenthe transmitted signalissl(r). TheM=4biorthogonal signalshavedimension N=2.Hence.twobasis functions areneededtorepresent thesignals.FromFig.5-1-8.wechoose f.(I)and/2(1)as {v'2/T(0";r,,;!T)ft(r)= .o(otherwIse) {v'2/T(!T,,;r,,; T)her)= .o(otherwIse)(5-1-32) (5-1-33) (0,;;;I";;!T) (otherwise)OT,,;;r';;;T) (otherwise)Thesewaveforms areillustrated inFig.5-I-S(a). Theimpulse responses of thetwomatched filtersare {Y2/T h,(t)=f,(7-t)=0 {Y2/Th,(r)=f,(T-r)= 0 (5-1-36)andareillustrated inFig.5-1-8(b). IfS,(I)istransmilled, the(noise-free) responses ofthetwomatched filtersareasshowninFig.5-l-S(c). Sincey,(I)andy,(t)aresampled at 1=T,weobservethaty,,(T)=v1A'fandyz,(T)=O.Notethat!A'T='t, thesignalenergy.Hence,thereceived vectorformedfromthetwomatched filteroutputsalthesampling instantI=Tis r=[" '2]=[~+n, n,] (5-1-34) wherenI=Y'n(T)andnz=Yz,,(T)arethenoisecomponents altheoutputs ofthematched filters.givenby Yk,,(T)=1.'n(llfk(l) dl.k=I,2 (5-1-35) Clearly,E(n.)=£(y,,,(T)) =O.Theirvariance is a;'=E[yi'.,(T)] =fFfFE!n(t)n(r)][.(I)[.( r)drdr ()n =!N"11'11'8(1-r)Mr)MI) dldr o() =!Nol'fZ(1)dl=!M, " (5-1-37)Observe thattheSNR"forthefirstmatched filteris (~'2'tSNR,,=-,-=- ,N"N" whichagreeswithourprevious result.Alsonotethatthefourpossible outputsofthetwomatched filters,corresponding tothefourpossible transmitted signalsinFig.5-1-8are(".,,)=(~+n,.n,). (n,.W+n,l. (-~+n,.n,) and(n"-~+n,). 5-1-3TheOptimum Detector Wehavedemonstrated that,forasignaltransmitted overanAWGNchannel, eitheracorrelation demodulator oramatched fillerdemodulator produces the vectorr=[""'"'N).whichcontains alltherelevant information inthe received signalwaveform. Inthissection,wedescribe theoptimum decision CHAPTER 5OPTIMU\l R[CEiVERS F'ORTHlADDITIVE' WHITE <JAUSSIAf'< NOISECHA~~EL 245 rulebasedontheobservation vectorr.Forthisdevelopment, weassumethat thereisnomemory insignalstransmitted insuccessive signalintervals. Wewishtodesignasignaldetector thatmakesadecision onthetransmitted signalineachsignalintervalbasedontheobservation ofthevectorrineach intervalsuchthattheprobability ofacorrectdecision ismaximized. Withthis goalinmind,weconsider adecision rulebasedonthecomputation ofthe posterior probabilities definedas P(signal s",wastransmittedIr),m=1,2..".M whichweabbreviate asP(s",Ir).Thedecision criterion isbasedonselecting thesignalcorresponding tothemaximum ofthesetofposterior probabilities {P(s",Ir)}.Later,weshowthatthiscriterion maximizes theprobability ofa correctdecision and,hence,minimizes theprobability oferror.Thisdecision criterion iscalledthemaximum aposteriori probability (MAP)criterion. UsingBayes'rule,theposterior probabilities maybeexpressed as P(s",Ir)p(rIs",lP(s",l p(r)(5-1-38) wherep(rIs",)istheconditional pdfoftheobserved vectorgivens"',and P(s",)istheaprioriprobability ofthemthsignalbeingtransmitted. The denominator of(5-1-38)maybeexpressed as M p(r)=2:p(rIs",)P(s",l m=1(5-1-39) (5·1-40)From(5-1-38)and(5-1·39), weobservethatthecomputation oftheposterior probabilities P(s",Ir)requires knowledge oftheaprioriprobabilities P(s",) andtheconditional pdfsp(rIs",)form=1,2,...,M. Somesimplification occursintheMAPcriterion whentheMsignalsare equallyprobable apriori,Le.,P(sn,)=1/MforallM.Furthermore, wenote thatthedenominator in(5-1-38) isindependent ofwhichsignalistransmitted. Consequently, thedecision rulebasedonfindingthesignalthatmaximizes P(s",Ir)isequivalent tofindingthesignalthatmaximizes p(rIs",). Theconditional pdfp(rIs",)oranymonotonic function ofitisusually calledthelikelihood function. Thedecision criterion basedonthemaximum of p(rIs",)overtheMsignalsiscalledthemaximum-likelihood (ML)criterion. Weobserve thatadetector basedontheMAPcriterion andonethatisbased ontheMLcriterion makethesamedecisions aslongastheaprion probabilities P(s",)areallequal,i.e.,thesignals{s"'}areequiprobable. InthecaseofanAWGNchannel, thelikelihood function p(rIs",)isgiven by(5-1·12). Tosimplify thecomputations, wemayworkwiththenatural logarithm ofp(rIs",),whichisamonotonic function. Thus. . IN Inp(rIs",)=-~NIn(rrN,,)- -:2:(rk-s",.)' NOk=-1 246 DIGITAL COMMUNICATIONS Themaximum ofInp(rI5,")over8misequivalent tofindingthesignal Smthat minimizes theEuclidean distance N D(r.sm)=L(r.-S...)2 *=1(5-1-41) WecallD(r,8",),m=I,2,...,M,thedistancemetrics.Hence,fortheAWGN channel, thedecision rulebasedontheMLcriterion reducestofindingthe signal 8mthatisclosest'indistance tothereceived signalvectorr.Weshall refertothisdecision ruleasminimum disrancedetection. Another interpretation oftheoptimum decision rulebasedontheML criterion isobtained byexpanding thedistance metricsin(5-1-41) as N N N D(r.sm)=Lr~-22:'"Smn+2:S~ln n""l "=1 "=1 (5-1-42) ThetermIrl'iscommon toalldecision metrics,and,hence,itmaybeignored inthecomputations ofthemetrics. Theresultisasetofmodified distance metrics D'(r,sm)=-2r·Sm+15",1' (5-1-43) Notethatselecting thesignal Smthatminimizes D'(r,sm) isequivalent to selecting thesignalthatmaximizes themetricC(r,sm)=-D'(r,5",),i.e., C(r,5",)=2r'5",-15",12(5-1-44) (5-1-45)Thetermr's'"represents theprojection ofthereceived signalvectoronto eachoftheMpossible transmitted signalvectors.Thevalueofeachofthese projections isameasure ofthecorrelation between thereceived vectorandthe mthsignal.Forthisreason,wecallC(r,sm), m=1,2•...,M.thecorrelation merricsfordeciding whichoftheMsignalswastransmitted. Finally.theterms 15".1'=~."m=1,2•....M,maybeviewedasbiastermsthatserveas compensation forsignalsetsthathaveunequal energies. suchasPAM.Ifall signalshavethesameenergy, 15",12mayalsobeignoredinthecomputation of thecorrelation metricse(r,sm)andthedistance metricsD(r.5",)orD'(r,5",). [tiseasytoshow(seeProblem 5-5)thatthecorrelation metricsC(r.8",)can alsobeexpressed as qr,s",)=2fTr(t)s",(r)d/- f:",.m=0,1....,M " Therefore. thesemetricscanbegenerated byademodulator thatcross­ correlates thereceived signalret)witheachoftheMpossible transmitted signalsandadjustseachcorrclator outputforthe!liasinthecaseofunequal signalenergies. Equivalently. thereceived signalmaybepassedthrough a bankofMfiltersmatched tothepossible transmitted sigmlls {.<,,,(t)land CHAPTER 5:OPTIMLM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISE CHA!\:~El 247 Olnpul de~'i'SionScIec.:1 ,11< largesl'--4-' +, :~ Rc-cei\'ed signalr(l),,,,,,.., : I ,J'",:-1,-0-,, Sample ai'=T FIGURE S-1-9Analternative re.ahzatiGn oftheoptimum AWGt\recei\'er. sampled att=T,theendofthesymbolinterval. Consequently, theoptimum receiver (demodulator anddetector) canbeimplemented inthealternative configuration illustrated inFig.5-1-9. Insummary, wehavedemonstrated thattheoptimum MLdetector computes asetofMdistances D(r,s",)orD'(r,s..,)andselectsthesignal corresponding tothesmallest (distance) metric.Equivalently, theoptimum ML detector computes asetofMcorrelation metricsC(r,s..,)andselectsthesignal corresponding tothelargestcorrelation metric. Theabovedevelopment fortheoptimum detector treatedtheimportant case inwhichallsignalsareequallyprobable. Inthiscase,theMAPcriterion is equivalent totheMLcriterion. However, whenthesignalsarenotequally probable, theoptimum MAPdetector basesitsdecision ontheprobabilities P(s..,Ir).m=I,2,...,M,"givenby(5-1·38) or,equivalently, onthemetrics. PM(r.s..,)=p(rIs",)P(s..,) Thefollowing example illustrates thiscomputation forbinaryPAMsignal,. Example 5·1·3 Consider thecaseofbinaryPAMsignalsinwhichthetwopossible signal pointsareSI=-S2=~.where ~histheenergyperbit.Theprior probabilities areP(sd=pandP(S2)=I -p.Letusdetermine themelric," fortheoptimum MAPdetector whenthetransmitted signaliscorrupted withAWGN. Thereceived signalvector(one-dimensional) forbinaryPAMis r=±~+y,,(T) (5-1-46) 248 DIGITAL COMMUNICATIONS whereYn(T)isazero-mean Gaussian random variable withvariance lT~=tNo.Consequently, theconditional pdfsp(r15m)forthetwosignals are 1[(r-y'W,;)2]p(rI5,)=v21rexp-22"2rrUn lTn 1[(r+V~Y]p(rI52)=v21rexp-22 21t'Un Un ThenthemetricsPM(r,SI)andPM(r,5,)are PM(r,Sl)=pp(rI51) P[(r-~)2]= exp- 2v'2irIT" 20'" 1-p[(r+~)2JPM(r,s2)=.J;;"""""exp-22v2nan ern.(5-1-47) (5-1-48) (5-1-49) (5-1-50) IfPM(r,S,)>PM(r,52),weselectSIasthetransmitted signal;otherwise, we select5,.Thisdecision rulemaybeexpressed as PM(r,5 1)"1---'---'-'- ~ PM(r,52)" But PM(r,s.)p[(r+vg,y-(r-~?] PM(r,52)1-Pexp 20'~ s.othat(5-1-51)maybeexpressed as (r+-...fti;,f-(r-vtg,;)2" 1-P, ~ln-- 2<7;, s~P orequivalently,(5-1-51) (5-1-52) (5-1-53) (5-1-54)-s,1-P1-PVgbrii"~O'~In--=lNoIn-- " p p Thisisthefinalformfortheoptimum detector. Itcomputes the correlation metricqr,SI)=r~andcompares itwiththreshold jNoIn((1-p)Ip].Figure5-1-10illustrates thetwosignalpoints SIand52' Thethreshold, denoted byrh,dividesthereallineintotworegions,sayRI andR2,whereRIconsistsofthesetofpointsthataregreaterthanrhand RegiorlR~-.--------_. RegionRIFIGURE 5-1·10 Signalspacerepresentation illustrating theoperation oftheoptimum detector forhinary(PAM)modulation.t, I\I::J;f;, CI-Lo\PTER 'i:OP'TIMlIM RECEIVERS FORTHEADDITIVE WHilE GA.USSlAN NOISECHANNEL 249 Rzconsists ofthesetofpointsthatarelessthanTh'Ifrw">Th'the decision ismadethat5\wastransmitted, andifrw"<Th'thedecision is madethat52wastransmitted. Thethreshold Thdepends onNoandp.If p=tTh=O.Ifp>tthesignalpoints\ismoreprobable and,hence, Th<O.Inthiscase,theregionR\islargerthanR2•sothat5 Iismorelikely tobeselectedthan5,.Ifp<!.theopposite isthecase.Thus,theaverage probability oferrorisminimized. Itisinteresting tonotethatinthecaseofunequal priorprobabilities, itis necessary toknownotonlythevaluesofthepriorpr:obabilities butalsothe valueofthepowerspectral density Noinordertocompute thethreshold. Whenp=tthethreshold iszero,andknowledge ofNoisnotrequired bythe detector. Weconclude thissectionwiththeproofthatthedecision rulebasedonthe maximum-likelihood criterion minimizes theprobability oferrorwhentheM signalsareequallyprobable apriori.LetusdenotebyRmtheregioninthe N-dimensional spaceforwhichwedecidethatsignal 5m(t)wastransmitted whenthevectorr=[r\'2...rN]isreceived. Theprobability ofadecision errorgiventhat5m(t)wastransmitted is (5-1-55) (5-1-56)whereR'",isthecomplement ofRm.Theaverageprobability oferroris MI pee)=2:-PeeISm) ",,,,1M m~l~JR~perIsm)dr m~\~[I- tP{rISrn)dr] NotethatP(e)isminimized byselecting thesignalSmifp(rIsm)islargerthan perISk)forallm,ek. WhentheMsignalsarenotequallyprobable, theaboveproofcanbe generalized toshowthattheMAPcriterion minimizes theaverageprobability oferror. 5-1·4TheMaximum-Likelihood Sequence Detector Whenthesignalhasnomemory, thesymbol-by-symbol detector described in thepreceding sectionisoptimum inthesenseofminimizing theprobahility of asymbolerror.Ontheotherhand,whenthetransmitted signalhasmemory, i.e.,thesignalstransmitted insuccessive symbolintervals areinterdependent, theoptimum detector isadetector thatbasesitsdecisions onobservation ofa (5-1-58)250 DJ(iITAL COMMUNICATIoNS sequence ofreceived signalsoversuccessive signalintervals. Below,we describe twodifferent typesofdetection algorithms. Inthissection, we describe amaximum-likelihood sequence detection algorithm thatsearches for theminimum euclidean distance paththroughthetrellisthatcharacterizes the memory inthetransmitted signaLInthefollowing section, wedescribe a maximum aposteriori probability algorithm thatmakesdecisions ona symbol-by-symbol basis,buteachsymboldecision isbasedonanobservation ofasequence ofreceived signalvectors. . Todevelop themaximum likelihood sequence detection algorithm, letus consider, asanexample, theNRZIsignaldescribed inSection4-3-2.Its memory ischaracterized bythetrellisshowninFig.4-3-14.Thesignal transmitted ineachsignalinterval isbinaryPAM.Hence,therearetwo possible transmitted signalscorresponding tothesignalpointssI=-52=vel;,, where'8histheenergyperbil.Theoutputofthematched-filter orcorrelation demodulator forbinaryPAMinthekthsignalintervalmaybeexpressed as 'k=±~+nk (5-1-57) wherenkisazero-mean gaussian randomvariable withvariance <T;'=No/2. Consequently, theconditional pdfsforthetwopossibletransmitted signalsare I1[(r k-yg,;)2] P('k5,)=vS exp- 22 Itu" u~ 1[('k+yg,;/]P(rkIs2)=V=-2exp-2tr. 1CUn ~ Now,suppose weobserve thesequence ofmatched-filter outputs',.'2....,rl<'Sincethechannelnoiseisassumed tobewhiteandgaussian, and f(t-iT),f(t-jT)forioFjareorthogonal, itfollowsthatE(nknj)=0,k,..j. Hence,thenoisesequence Il""2,...,nKisalsowhite.Consequently, forany giventransmitted sequence sCm!,thejointpdfof'"'"...,'Kmaybeexpressed asaproductofKmarginal pdfs,Le" K p('"'2.···.'KISCm)=nP('kIsimi) k=l K1 [('~_s(m»)2]=n exp_ k k~1Viii<Tn 2<T~ = (1rexp[-±('k-5;m1/] (5-1-59)ViiiUn k=12un whereeither 5k=vv.orSk=-~.Then,giventhereceived sequence '1.'2.....'Kattheoutputofthematched filterorcorrelation demodulator, the detector determines thesequence slm)={slm), s~m)•...•sir)}thatmaximizes theconditional pdfp(r,.'2•...,'KIslm»).Suchadetector iscalledthe maximum-likelihood (ML)sequence detecto,. Bytakingthelogarithm of(5-1-59) andneglecting thetermsthatare CHAPTER:'\: OPTIMUM RECEIVERS FORTHEADDITIVE WHITEGAL:SSIAN ~OISECHANNEL 251 5,•t0117;,t0117;,t0117;,t FIGURE: 5-)·11 TrellisforNRZIsignal 1=T 1=2T 1=3T (=4T (5-1-60)independent of(r"r"...,r.:l,wefindthatanequivalent MLsequence detector selectsthesequence sun,thatminimizes theeuclidean distance metric K D(r.simI)=2:(r,-sl,m')2 k=I Insearching through thetrellisforthesequence thatminimizes the euclidean distance D(r.simI),itmayappearthatwemustcompute thedistance D(r.s,m,)foreverypossible sequence. FortheNRZIexample. whichemploys binarymodulation. thetotalnumber ofsequences is2'.whereKisthe numberofoutputs obtained fromthedemodulator. However. thisisnotthe case.Wemayreducethenumberofsequences inthetrellissearchbyusingthe Viterbialgorithm toeliminate sequences asnewdataisreceived fromthe demodulator. TheViterbialgorithm isasequential trellissearchalgorithm forperforming MLsequence detection. Itisdescribed inChapter 8asadecoding algorithm forconvolutional codes.Wedescribe itbelowinthecontextoftheNRZI signal.Weassume thatthesearchprocess beginsinitially atstateSo.The corresponding trellisisshowninFig.5-1-11. Attimet=T,wereceiver,=s\ml+nfromthedemodulator. and3tt=2T. wereceive r2=s~nl+n2'Sincethesignalmemory isonebit.whichwedenote byL=I,weobserve thatthetrellisreaches itsregular(steadystate)form aftertwotransitions. Thus.uponreceiptof'2att=2T(andthereafter). we observe thattherearetwosignalpathsentering eachofthenodesandtwo signalpathsleaving eachnode.Thetwopathsentering node5"att=2T correspond totheinformation bits(0.0)and(I.I)or.equivalently. tothe signalpoints (~yy,;;,-Vt,;)and(v'jg,;,-v'jg,;), respectively. Thetwopaths entering node5,att=2Tcorrespond totheinformation bits(0.I)and(I.0) or.equivalently. tothesignalpoints(-yy,;;.v7':)and(~,~l, respectively, Forthetwopathsentering nodeSo.wecompute thetwoEuclidean distance metrics Do(O.0)'=(r,+v7':)'+(r,+yy,;;)2 Do(l.I)=(r,-~)2+(r,+yy,;;)'(5-1-61) (5-1-62) (5-1-63) (5-1-64)252 DIGITAL COMMUN'CATIONS byusingtheoutputsrlandr2fromthedemodulator. TheViterbialgorithm compares thesetwometricsanddiscards thepathhavingthelarger(greater­ distance) metric.tTheotherpathwiththelowermetricissavedandiscalled thesuroivor at/=2T.Theelimination ofoneofthetwopathsmaybedone without compromising theoptimality ofthetrellissearch, because any extension ofthepathwiththelargerdistance beyond t=2Twillalwayshavea largermetricthanthesurvivor thatisextended alongthesamepathbeyond t=2T. Similarly, forthetwopathsentering node5,att=2T,wecompute thetwo Euclidean distance metrics D1(0,1)=(r,+V'i",)2+(r2_~)2 D](1,0)=(r,-vi,;»+(r2-~)2 byusingtheoutputs r1andr2fromthedemodulator. Thetwometricsare compared andthesignalpathwiththelargermetriciseliminated. Thus,at t=2T,weareleftwithtwosurvivor paths,oneatnodeSoandtheotherat node5"andtheircorresponding metrics.ThesignalpathsatnodesSoandS, aretllenextended alongthetwosurvivor patlls. Uponreceipt'ofrJat/=3T,wecompute themetricsofthetwopaths entering stateSo.Suppose thesurvivors att=2Tarethepaths(0,0)atSoand (0,1)atS,.Then,thetwometricsforthepathsenteringSoat/=3Tare Do(O,0,0)=Do(O,0)+(r3+~)2 Do(O,I,1)=D,(O,1)+(r3+~)2 Thesetwometrics arecompared andthepathwiththelarger(greater­ distance) metriciseliminated. Similarly, themetricsforthetwopathsentering 5,att=3Tare D,(O,O,1)=Do(O,0)+(r3-v~;i D,(O,1,0)=D](O,1)+(r3-v'i,;)2 Thesetwometries arecompared andthepathwiththelarger(greater­ distance) metriciseliminated. Thisprocessiscontinued aseachnewsignalsample.isreceived fromthe demodulator. Thus,theViterbialgorithm computes twometricsforthetwo signalpathsentering anodeateachstageofthetrellissearchandeliminates oneofthetwopathsateachnode.Thetwosurvivor pathsarethenextended forward tothenextstate.Therefore, thenumber ofpathssearched inthe trellisisreduced byafactoroftwoateachstage. Itisrelatively easytogeneralize thetrellissearchperformed bytheViterbi algorithm forM-arymodulation. Forexample, delaymodulation employs tNotethat,forNRZI,thereception ofr2fromthedemodulator neitherincreases nordecreases therelativedilference between riletwometrics, DolO,0)andDoll,1).Attbispoint,onemay ponderontheimplication ofthisobservation. Inanycase.wecontinue withthedescription ofthe MLsequence detectorbasedontheViterbialgorithm. CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOiSECHANNEL 253 •5, 5, FIGURE 5-1-12 Onestageoftrellisdiagram fordelay modulation.5) 5. M=4signalsandischaracterized bythefour-state trellisshowninFig.5-1-12. Weobservethateachstatehastwosignalpathsentering andtwosignalpaths leavingeachnode.Thememory ofthesignalisL=1.Hence,theViterbi algorithm willhavefoursurvivors ateachstageandtheircorresponding metrics.Twometricscorresponding tothetwoentering pathsarecomputed at eachnode,andoneofthetwosignalpathsentering thenodeiseliminated at eachstateofthetrellis.Thus,theViterbialgorithm minimizes thenumberof trellispathssearched inperforming MLsequence detection. Fromthedescription oftheViterbialgorithm givenabove,itisunclearasto howdecisions aremadeontheindividual detected information symbolsgiven thesurviving sequences. Ifwehaveadvanced tosomestage,sayK,where K»Linthetrellis,andwecompare thesurviving sequences, weshalllindthat withprobability approaching oneallsurviving sequences willbeidentical inbit (orsymbol) positions K-5Landless.Inapractical implementation ofthe Viterbialgorithm, decisions oneachinformation bit(orsymbol) areforced afteradelayof5Lbits(orsymbols), andhence,thesurviving sequences are truncaled tothe5Lmostrecentbits(orsymbols). Thus,avariabledelayinbit orsymboldetection isavoided. Thelossinperformance resulting fromthe suboptimum detection procedure isnegligible ifthedelayisatleast5L. Example 5-1-4 Consider thedecision rulefordetecting thedatasequence inanNRZI signalwithaViterbialgorithm havingadelayof5Lbits.Th.etrellisforthe NRZIsignalisshowninFig.5-1-11.Inthiscase,L=I,herkethedelayin .bitdetection issettofivebits.Hence,atI=6T,weshallhavetwosurviving sequences, oneforeachofthetwostatesandthecorresponding metrics iJ.6(b"b2,b3,b4,bs,b6)and[.L6(b;,b"bi,b.,b:',b;').Atthisstage,with probability nearlyequaltoone,thebitb,willbethesameasb;;thatis. 2S4 DIGITAL COMMUNICATIONS bothsurviving sequences willhaveacommon firstbranch.Ifb1#b;,we mavselectthebit(b1orbi)corresponding tothesmallerofthetwometrics. Th~nthefirstbitisdropped fromthetwosurviving sequences. AtI:7T, thetwometrics/-Llb2'bJ•b.,Us,bo•b7)andJ.L7(bi.bj.b~,bs,u~.b7)willbe usedtodetermine thedecision onbitb2•Thisprocesscontinues ateach stageofthesearchthroughthetrellisfortheminimum distance sequence. Thusthedetection delayisfixedatfivebits.t 5-1·5ASymbol-by-Symbol Detector forSignals withMemory Incontrast tothemaximum-likelihood sequence detector fordetecting the transmitted information. wenowdescribe adetector thatmakessymbol-by­ symboldecisions basedonthecomputation orthemaximum aposteriori probability (MAP)foreachdetected symbol.Hence,thisdetector isoptimum inthesensethatitminimizes theprobability ofasymbolerror.Thedetection algorithm thatispresented belowisduetoAbendandFritchman (1970),who developed itasadetection algorithm forchannels withintersymbol inter­ ference,i.e.,channels withmemory. Weillustrate thealgorithm inthecontextofdetecting aPAMsignalwithM possible levels.Suppose thatitisdesiredtodetecttheinformation symbol transmitted inthekthsignalinterval, andlet',.'2,...•'k+Dbetheobserved received sequence, whereDisthedelayparameter whichischosentoexceed thesignalmemory. i.e.,D;;.L.whereListheinherent memory inthesignal. Onthebasisofthereceived sequence, wecompute theposterior probabilities (5-1-65) (5-1-MfortheMpossible symbolvaluesandchoosethesymbolwiththelargest probability. Since P(lk)_AI )_ P('HD•..., "IS(k):Am)P(slk) :Am) S-m'k+D,··.,'1 - p(rk+D''k+D-IJ···' rd andsincethedenominator iscommon forallMprobabilities, themaximum a posteriori probability (MAP)criterion isequivalent tochoosing thevalueof Slk)thatmaximizes thenumerator of(5-1-66). Thus,thecriterion fordeciding onthetransmitted symbolslk)is Slk)=arg{maXP('HD' ...,',ISlk)=Am)P(slk)=Am)} slot)(5-1-67) tOnemayhaveobserved hynowthattheMLsequence detector andthesymbol-by-symbol detector thatignoresthememory intheNRZIsignalreachthesamedecisions. Hence,thereisno needforadecisiondelay.Nevertheless, theprocedure described aboveappliesingeneral. CHAYfER 3:OPTIMUM RECEIVERS FORTHEADDITIVE WHITEGALSSIAN NOISECHANNEL 25S Whenthesymbols areequallyprobable, theprobability P(Slk)=Am)maybe dropped fromthecomputation. Thealgorithm forcomputing theprobabilities in(5-1-67)recursively begins withthefirstsymbol s(l).Wehave -11)-{( I(I)-A)P(O)-A)} s-argmaxp'k,+DI'..,'1S-mS-m .,(1) {"'"( I (1+0) (I»P((1+0) (')} =argmax"""...L.JPrl+DI'.."1S I'.•,s s ,... 1S 5(11 .f(1~nl s(2l (5-1-68) where S(I)denotesthedecision ons(l)and,formathematical convenience. we havedefined (0+0) (2)(1)= ( I11+0) (I»p( (1+0) II»PtS ,..•,s,s-P'1+D'.•. JTjS ,•••,S S ,..•,S (5-1-69) Thejointprobability P(S(l+O) •...•S(2).s(I» maybeomittedifthesymbols are equallyprobable andstatistically independent. Asaconsequence ofthe statistical independence oftheadditivenoisesequence. wehave (I(I'D) (I»P'l+l)•.•.,TIs ,...,S whereweassumethatSlk)=0fork,,;;;O. Fordetection ofthesymbol s(2).wehave -(2)_ ( I(2)-A)P((2)-A)}s-arg rr:(~xp '2+0.'.."1S-mS-m _ {"'..."( I (2+0) (21)p((2./») (21)} -argm~xL..L."PT2tO•..."1S , . , ,,S S • ' . , •S, tl-J,tl~'lIlt('l (5-1-711 Thejointconditional probability iiithemultiple summation canheexpressed as (I(2,lJ) .(2»P'2+lJ.···.'1S ,...•.s _ ( I (2'01 (HO-I.l) (-P'2+nS ,...,S P'1+J) I1I'f)) (2)(5172•."1S•...• $ --) 2S6 DIGITAL COMMUNICATIONS Furthermore, thejointprobability p(r,+D•...•r,IS(I+D),...•S(2)p(S(I+Dl, ...•S(2» canbeobtained fromtheprobabilities computed previously inthedetection of s(l).Thatis, (I(I+D) (2»P'I+D,...,'jS ,...,S -"( I II+DJ (I»P((I+D) (I»-LJp'l+D, ..."1S ,...,S S ,.•.,s ~.(I) =2:pt(S(I+D) ••..•S(2'.S(I» s(1)(5-1-73) Thus,bycombining (5-1-73) and(5-1-72)andthensubstituting into(5-1-71), weobtain (5-1-74) where,bydefinition, • =p(r2+fJISIHD'•...,S'2+D-I.')p(S'2+D') 2:PI(S'1'D'....•s(2l.~,l) .f(I' (5-1-75) Ingeneral,therecursive algorithm fordetecting thesymbolSl"isasfollows: uponreception ofr'+D.'..•r2.r,.wecompute -Ik,_{( I(")P(I'»} s-argm,axprHD....,r,S S f(.l) (5-1-76; where,bydefinition, ='p(rk+DIS'k+D),...•S,'+D-L»p(S('+ D»)2:p,-t(s" -,'D!,...,Slk-I») \.IlI) (5-1-77) Thus,therecursive natureofthealgorithm'is established bytherelations (5-1-76)and(5-1-77). CHAPTER 5:OPTIMUM RECEiVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL Z57 Themajorproblem withthealgorithm isitscomputational complexity. In' . . f d h b I (k+D) (k+,)(k"particular, theaveragmg perormeovert esym0sS•...•S,SIn (5-1-76) involves alargeamountofcomputation perreceived signal,especially ifthenumberMofamplitude levels{Am}islarge.Ontheotherhand,ifMis smallandthememory Lisrelatively short,thisalgorithm iseasily implemented. 5-2PERFORMANCE OFTHEOPTIMUM RECEIVER FORMEMORYLESS MODULATION Inthissection, weevaluate theprobability oferrorforthememoryless modulation signalsdescribed inSection4-3-1.First,weconsider binaryPAM signalsandthenM-arysignalsofvarioustypes. 5-2-1Probability ofErrorforHillaryModulation Letusconsider binaryPAMsignalswherethetwosignalwaveforms are 5,(1)=g(l)and5,(1)=-g(r),andg(r)isanarbitrary pulsethatisnonzero in theinterval0'"r'"Tnandzeroelsewhere. Since5,(r)=-sir),thesesignalsaresaidtobeantipodal. Theenergyinthe pulseg(r)is'€h'Asindicated inSection 4-3-1,PAMsignalsareone­ dimensional, and,hence,theirgeometric' representation issimplytheone­ dimensional vectors,=V'i;"s,=-~.Figure5-2-1illustrates thetwosignal points. Letusassumethatthetwosignalsareequallylikelyandthatsignals,(r)was transmitted. Then,thereceived signalfromthe(matched filterorcorrelation) demodulator is r=s,+n=~+n (5-2-1) wherenrepresents theadditive gaussian noisecomponent, whichhaszero meanandvariance (J"~=~N().Inthiscase,thedecision rulebasedonthe correlation metricgivenby(5-1-44) compares rwiththethreshold zero.If r>O.thedecision ismadeinfavorof5,(t),andifr<0,thedecision ismade thats,(r)wastransmitted. Clearly, thetwoconditional pdfsofrare Jp(rIs,)= _~(' vlrN" Ip(rIs,,)= _~e'",\0,)""" vrrN" FIGURE 5·2-1 Sign<ilpllintsforbinaryantipodal signals.(5-2-2) (5-2-3) F;. J7;. I• 'J II" ZS8 DIGITAL COMMUNICATIONS FIGURE S·Z·ZConditional pdfsoftwosignals. Thesetwoconditional pdfsareshowninFig.5-2-2. Giventhat$,(t)wastransmitted, theprobability oferrorissimplythe probability thatr<0,i.e., P(e1$,):[xp(r1$,)dr :_1_[exp[_(r-YG;;)2]dr VtrNo-x No 1J-V2~h!NO:__ e-x2J2dx Viir ~x 1IX -x'IZd:ViirV2~Noe x :Q(fk) (5-2-4) whereQ(x)istheQ·function definedin(2-1-97). Similarly, ifweassumethat $z(t)wastransmitted, r:-vg;;+nandtheprobability thatr>0isalso P(e1$2):Q(V2'€bINo). Sincethesignals$,(1)and$2(t)areequallylikelytobe transmitted, theaverageprobability oferroris Pb:~P(eIsd+lP(e1$2) :Q(fk) (5-2-5) Weshouldobserve twoimportant ch~racteristics ofthisperformance measure. First,wenotethattheprobability oferrordepends onlyontheratio '€blNoandnotonanyotherdetailedcharacteristics ofthesignalsandthenoise. Secondly, wenotethat2'€b!NoisalsotheoutputSNRo.fromthematched-filter (andcorrelation) demodulator. Theratio/CblNoisusuallycalledthesignal-to­ noiseratioperbit. Wealsoobservethattheprobability oferrormaybeexpressed intermsof thedistance between thetwosignals$,and$z.FromFig.5-2-1.weobserve thatthetwosignalsareseparated bythedistanced,z:2~.Bysubstituting '€b:ld~2into(5-2-5),weobtain Pb:Q(~:~) (5-2-6) ('1-2-71CHAPTER ~:OPTIMUM RECEIVERS FORTHEADDITIVE wHITEGAUSSlAN r-;O!SECHANNEL 259 FIGURE 5-Z-3Signalpointsforoinaryorthogonal signals Thisexpression illustrates thedependence oftheerrorprobability onthe distance between thetwosignalpoints. Next,letusevaluate theerrorprobability forbinaryorthogonal signals. Recallthatthesignalvectorss,ands,aretwo-dimensional, asshowninFig. 5-2-3,andmaybeexpressed, according to(4-3-30), as S,=[~0] 5,=[0~J where '(;hdenotestheenergyforeachofthewaveforms. Notethatthedistance between thesesignalpointsisd'2=Y2'f!h' Toevaluate theprobability oferror,letusassumethats,wastransmitted Then,thereceived vectorattheoutputofthedemodulator is r=[~+n, n,] (.'I-2-S) Wecannowsubstitute forrintothecorrelation metricsgivenby(5-1-44) to obtainC(r,5,)andC(r.S2)'Then.theprobability oferroristheprobability thatqr,S2)>qr,s,).Thus. Peels,)=P[qr,5,}>C(r,.S,)]=P[n2-n,>v\f,,] (5-2-9) Sincen,andn2arezero-mean statistically independent gaussian random variables eachwithvariance!N"therandomvariablex=n,-n,iszero-mean gaussian withvariance No.Hence. IJX=-- e-x2/2dx "V2'1rV2r"/NI (5-2-10) Duetosymmetry, thesameerrorprobability isobtained whenweassumethat 260 DIGITAL COMMUNICATIONS S,istransmitted. Consequently. theaverage errorprobability forbi"nary orthogonal signalsis (~'g"PI>=Q - )=Q(vY;;) No(5-2-11) where.bydefinition. y"istheSNRperbit. Ifwecompare theprobability oferrorforbinaryantipodal signalswiththat forbinaryorthogonal signals.wefindthatorthogonal signalsrequireafactor oftwoincrease inenergytoachievethesameerrorprobability asantipodal signals.Since10log",2=3dB.wesaythatorthogonal signalsare3dBpoorer thanantipodal signals.Thedifference of3dBissimplyduetothedistance between thetwosignalpoints.whichisdf2=2'l?"fororthogonal signals. whereas d12=4't'"forantipodal signals. Theerrorprobability versus10log,,)'t',,1Noforthesetwotypesofsignalsis showninFig.5-2-4.Asobserved fromthisfigure.atanygivenerror probability. the't;,,1Norequired fororthogonal signalsis3dBmorethanthat forantipodal signals. 5-2-2Probability ofErrorforM-aryOrthogonal Signals Forequalenergyorthogonal signals.theoptimum detector selectsthesignal resulting inthelargestcrosscorrelation between thereceived vectorrandeach oftheMpossible transmitted signalvectors{s,..}.i.e., "C(r.s",)=r·s",=L: '",Smk.m=1.2•...•M 1.:=1(5-2-12) , 1'0.......... I I i'. """"'\ p,""'\Orthogonal p.=-I\ i\ ~ignals Antipodal\X"=Q(JY;,I ,P,=Q'ny;)\ \1\ \ \ \ \ \ ! \ I\ 1\.\2 IO--t>O 24681012I~ SNRperbit.y,,(dBl\(J-5 5Itt­ 5 2 Hr· 5 ~22lO­ b5 -0 .22 ];10--4 B£5 FIGURE 5..24Probability oferrorforbinarysignals. ("HAP'IER~: OPTIVlt;\1 RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOiSECHANNEL 261 Toevaluate theprobability oferror,letussuppose thatthesignals,is transmitted. Thenthereceived signalvectoris (5-2-13) wheren"n"...,nMarezero-mean, mutually statistically independent gaus­ sianrandomvariables withequalvariance O"~=~No.Inthiscase,theoutputs fromthebankofMcorrelators are C(r,sd=~(~ +n,) C(r,s.z)=~n,(5-2-14) Notethatthescalefactorg,maybeelminated fromthecorrelator outputsby dividing eachoutputby~.Then,.withthisnormalization, thepdfofthefirst correlator output('1=yg;-n,)is andthepdfsoftheotherM-1correiatoroutputsare P(X)=_l_e-xi.,No m-23M'mmVnNo' - , , ...•(5-2-15) (5-2-16) Itismathematically convenient tofirstderivethe probability thatthe detector makesacorrectdecision. Thisisthe probability that'1islargerthan eachoftheotherM- Icorreiatoroutputsn"n"...nM'Thisprobability may beexpressed as (5-2-17) whereP(n,<'1,n,<'I,...,nM<'1/rt)denotes thejointprobability that n"n"...,nMarealllessthan'"conditioned onanygiven'1'Thenthisjoint probability isaveraged overall'1'Sincethe{Tm}arestatistically independent, thejointprobability factorsintoaproductofM-1marginal probabilities of theform m=2,3,...,M (5-2-18) Theseprobabilities areidentical form=2,3,...,M,and,hence,thejoint probability underconsideration issimplytheresultin(5-2-11\) raisedtothe (M~I)thpower.Thus,theprobability ofacorrectdecision is f~(1f"V2iN..,)MP=-' e "/2dx,-"v'2If", andthe probability ofa(k-bit)symbolerroris whereperl)dr, (5-2-19) (5-2-20) Ifoo[(If")M P--- 1--- e,li2dx M-v'2If ~v'21r_~']exp[-~(y-~ndy (5-2-21) Thesameexpression fortheprobability oferrorisobtained whenanyone oftheotherM-Isignalsistransmitted, SincealltheMsignalsareequally likely,theexpression forPMgivenin(5·2-21) istheaverageprobability ofa symbolerror.Thisexpression canbeevaluated numerically. Incomparing theperformance ofvariousdigitalmodulation methods, itis desirable tohavetheprobability oferrorexpressed intermsoftheSNRper bit,'lthlNil,insteadoftheSNRpersymbol,'It,(No_WithM=2"eachsymbol conveys kbitsofinformation, andhence'It,=kgh•Thus,(5-2-21) maybe expressed intermsof'thiN"bysubstituting for't,. Sometimes, itisalsodesirable toconverttheprobability ofasymbolerror intoanequivalent probability ofabinarydigiterror,Forequiprobable orthogonal signals, allsymbol errorsareequiprobable andoccurwith probability ~=~M-12k-1(5-2-22) Furthermore, thereare(~)waysinwhichnbitsoutofkmaybeinerror. Hence,theaveragenumberofbiterrors per k·bitsymbolis (5-2-23) and.theaveragebiterrorprobability isjusttheresultin(5-2-23)dividedbyk, thenumberofbitspersymboLThus, (5-2-24) Thegraphsoftheprobability ofabinarydigiterrorasafunction ofthe eIIApIL\{:i.: \ll'lIMi'\1 RI·(.JIVI·I{S HHtTHl-.ADI>ITIV[ WUITE C"iAl:SSIA""- ~(II\1CIIA...."..;lt263 ,'"M=4 2f--+-t--'ItH++---j--1 I'"f--+--:-+-\HjIf+-tl--j--1M=16- nGURE 5-2·5 Probability ofbiterrorforcoher~n[ detection of orthogonal signals.f-M=M 2f--+-+--IH+-t+t---j--1I II()~':-_(~)--'-4--'''!!M'-'--'c'1 2,-'-...:'16--:---:20 SNRp.:rbit.'f/,1dB, SNRperbit,'lhlNil.areshowninFig.5·2-5forM=2.4,8.16.32and64.This figureillustrates that,byincreasing thenumber Mofwayeforms, onecan reducetheSNRperbitrequired toachieveagivenprobability ofabiterror. Forexample. toachieveaP"=10',therequired SNRperbitisalittlemore than12dBforM=2.butifMisincreased to64signalwaveforms (k=6bits/symbol), therequired SNRperbitisapproximately 6dB.Thus,a savingsofover6dB(afactor-of-four reduction) isrealized intransmitter power(orenergy)required toachieveaPh=10-'byincreasing MfromM=2 toM=fi4. Whatistheminimum required 'fch/N"toachieve anarbitrarily small probability oferrorasM->x?Thisquestion isanswered below. AUnionBoundontheProbability ofErrol'Letusinvestigate theeffect ofincreasing Montheprobability oferrorfororthogonal signals.Tosimplify themathematical development, wefirstderiveanupperboundonthe probability ofasymbolerrorthatismuchsimplerthantheexactformgivenin (5-2·21). Recallthattheprobabilit~ oferrorforbinaryorthogonal signalsisgivenby (5-2-11). Now,ifweviewthedetector forMorthogonal signalsasonethat makesM-1binarydecisions between thecorrelator outputql',5,)that contains thesignalandtheotherM-1correlator outputs C(r,s'"l.m= 2,3...,.M.theprobability oferrorisupper-bounded bytheunionboundof theM-1events.Thatis,ifEjrepresents theeventthatC(r,s;)>C(r,5,)for i""1thenwehavePM=P(U'~,Ej)""~;·o1P(Ej).Hence, PM""(M-1)P,=(M-1)Q(V'tjN,,)<MQ(v''t,/ N,,) (5-2-25) 264 DIGITAL COMML:f\jICATlOr.;S Thisboundcanbesimplili~d furtherbyupper-bounding QC'llijN,,}. Wehave Q(Vl)NoJ<e it.i'N" (5-2-26) Thus, PM<lWe t,/?N"=i'e k{/,12NH PM<eJ,;,~,iiVIl 2In2)1'1.(5-2-27) Ask--+x,orequivalently, asM-.x,theprobability oferrorapproaches zero exponentially, provided thatthiN.,isgreaterthan2ln2.j,e., to->2In2'"l.39(1.42dB)N.,(5-2-28) Thesimpleupperboundontheprobability oferrorgivenby(5-2-27) impliesthat,aslongasSNR>1.42dB,wecanachieveanarbitrarily lowPM' However, thisunionboundisnotaverytightupperboundatasufficiently low SNRduetothefactthattheupperboundfortheQfunction in(5-2-26) is loose.Infact,bymoreelaborate bounding techniques, itisshowninChapter7 thattheupperboundin(5-2-27) issufficiently tightfor~o/No>4In2.For ~"INo<4In2,atighterupperboundonPMis Consequently, PH--+0ask-.x,provided that ~o- >In2=0.693(-1.6dB)N.,(5-2-29) (5<2-30) Hence,-1.6dBistheminimum required Sl\'Rperbittoachieveanarbitrarily sl1"ijlllprobability oferrorinthelimitask-.x(M.,-+x).Thisminimum SNR perbit(-1.6dB)iscalledtheShannon limitforanadditive whiteGaussian noisechannel. 5-2-3Probability ofErrorforM-aryBiorthogonal Signals Asindicated inSection 4-3,asetofM=2'biorthogonal signalsare constructed from~Morthogonal signalsbyincluding thenegatives ofthe orthogonal signals.Thus,weachieveareduction inthecomplexity ofthe demodulator forthebiorthogonal signalsrelativetothatfororthogonal signals, sincetheformerisimplemented with~Mcross-correia torsormatched filters, whereas the latter requiresMmatched filtersorcross-correlators. Toevaluate theprobability oferrorfortheoptimum detector, letusassume thatthesignalS,(t)corresponding tothevector 51=[Vi.0U...01was transmitted, Then,thereceived signalvectoris r=[~+n,n2'"nM12) (5-2-31) wherethe{n",}arezero-mean, mutually statistically independent andidenti­ callydistributed gaussian random variables withvarianceu;,=~N.).The. -. 1:., ,(= \' ----J magnitude ofthecross-correia tors MI~ C(r.SI!()=r'Sm=Lr"s'll},;' k=lm=I,2,,··,1 M (5-2-32) whilethesignofthislargesttermisusedtodecidewhether S",(I)or-5",(1)was transmitted. According tothisdecision rule,theprobability ofacorrect decision isequaltotheprobability thatr,=\~+n,>0andr,exceeds Ir",1=In",1form=2,3,....jM.But issimilartothatfororthogonal signals(seefig.5-2-5).However. inIhiscase, theprobability oferrorforM=4isgreaterthanthatforM=2.Thisisdueto thefactthatwehaveplottedthesymbolerrorprobability P",infig.5-2-6.If weplottedtheequivalent biterrorprobability. weshouldfindthatthegraphs forM=2andM=4coincide. Asinthecaseoforthogonal signals,asM--+x (ork->(0),theminimum required ~h/N"toachievearbitrarily smallprob­ abilityoferroris-1.6dB,theShannon limit. 5-2-4Probability ofErrorforSimplex Signals Nextweconsider the probability oferrorforMsimplexsignals.Recallfrom Section4-3thatsimplexsignalsareasetofMequallycorrelated signalswith mutualcross-correlation coefficient P"",=-1/(M-I).Thesesignalshavethe sameminimum separation of~between adjacent signalpointsinM­ dimensional spaceasorthogonal signals.Theyachievethismutualseparation withatransmitted energyof'(;,(M-1)/M.whichislessthanthatrequired for orthogonal signalsbyafactorof(M-1)/M.Consequently, theprobability of errorforsimplexsignalsisidentical totheprobability oferrorfororthogonal signals,butthisperformance isachieved withasavingof M10log(I-p)=10log--dBM-I(5-2-35) inSNR.forM=2,thesavingis3db.However. asMisincreased, thesaving inSNRapproaches 0dB. 5-2-5Probability ofErrorforM-aryBinary-Coded Signals WehaveshowninSection 4-3thatbinary-coded signalwaveforms are represented bythesignalvectors Sm=[S",I 5m2 S",N],m=1,2,...,M where S",j~±'1/'1:/Nforallmandj.Nistheblocklengthofthecode,andis alsothedimension oftheMsignalwaveforms, Ifd~lnistheminimum euclidean distance oftheMsignalwaveforms then theprobability ofasymbolerrorisupper-bounded as P,,,<(M-I)Ph=(M-I)Q(~~;:) <2k[(d::;!n)']exp---- 4N"(5-2-36) CHAI'TER 5,OPT/Ml:" RECEIVERS FORTHEADDlTJVE WHITEGAIISSIAN NOISECHANNEL 267 Thevalueoftheminimum euclidean distance willdepend alltheselectidn of thecodewords,i.e.,thedesignofthecode. S-2-6Probability ofErrorforM·aryPAM RecallthatM-aryPAMsignalsarerepresented geometrically asMone­ dimensional signalpointswithvalue s'"=~A"" m=I.2.....M (5-2-37) where ~.istheenergyofthebasicsignalpulseg(t).Theamplitude valuesmay beexpressed as Am=(2m-l-M)d, m=I.2.....M wheretheeuclidean distance between adjacent signalpointsisd~.(5-3-38) (5-2-39) Equivalently, wemaycharacterize thesesignalsintermsoftheiraverage power,whichis 19"12d2'fP=-=(M-I)=--:..!i"T h T(5-2-40) Theaverageprobability oferrorforM-aryPAMcanbedetermined from thedecision rulethatmaximizes thecorrelation metricsgivenby(5-1-44). Equivalently, thedetector compares thedemodulator outputrwithasetof M-Ithresholds, whichareplacedatthemidpoints ofsuccessive amplitude levels,asshowninFig.5-2-7.Thus,adecision ismadeinfavorofthe amplitude levelthatisclosesttor. Theplacingofthethresholds asshowninFig.5-2-7helpsinevaluating the probability oferror.Wenotethatifthemthamplitude levelistransmitted, the demodulator outputis (5-2-41) flGVRE 5-2-7Placernenl ofIhresholds atmidpainls of successive amplitude levels.,.\,..J,,..:,., I,>J I,I•I I, t, t,..I t,..! t,+.\ 268 DIGITAL COMMU"JICATIONS wherethenoisevariablenhaszeromeanandvariance a~=~No.Onthebasis thatallamplitude levelsareequallylikelyapriori,theaverageprobability ofa symbolerrorissimplytheprobability thatthenoisevariable nexceeds in magnitude one-half ofthedistance between levels.However, wheneitherone ofthetwooutsidelevels±(M-1)istransmitted, anerrorcanoccurinone direction only.Thus,wehave (5-2-42) Theerrorprobability in(5-2-42)canalsobeexpressed intermsoftheaverage transmitted power.From(5-2-40), wenotethat , 6dleg=,p.,TM-1(5-2-43) Bysubstituting ford'legin(5-2-42), weobtaintheaverage probability ofa symbolerrorforPAMintermsoftheaveragepoweras P_2(M-1)(M- QM6P.vT ) (M'-I)No(5-2-44) or,equivalently, p=2(M-1)Q(/6le.v) MMY(M'-I)N o(5-2-45) whereleO'=PayTistheaverageenergy. Inplottingtheprobability ofasymbolerrorforM-arysignalssuchasM-ary PAM,itiscustomary tousetheSNRperbitasthebasicparameter. Since T=kThandk=log,M,(5-2-45)maybeexpressed as 2(M-l) ( fwMQ(610~,M)gh") (M-I)No(5-2-46) whereleh"=PayThistheaveragebitenergyandleh,,/NoistheaverageSNR CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIA.N NOISECHANNEL 269 I"\. 1\ 1\\\ !\\\ \ \ 1\ l\-\ , II1M.'0\ III I I \\M'.\ I\ M=' 1tI:2 1, 5 ,1 10-°--6-4-10 24-6S101214161820.2 SNRperbit.y,,(dB)10'l<r ; FIGURE 5-Z-8Probability ofasymbolerrorforPAM. perbit.Figure5-2-8illustrates theprobability ofasymbolerrorasafunction of1Olog lO"'bavINo. withMasaparameter. Note,thatthecaseM=2 corresponds totheerrorprobability forbinaryantipodal signals.Alsoobserve thattheSNRperbitincreases byover4dBforeveryfactor-of-two increase in MForlargeM,theadditional SNRperbitrequired toincreaseMbyafactor oftwoapproaches 6dB. 5-2-7Probability ofErrorForM-aryPSK RecallfromSection4-3thatdigitalphase-modulated signalwaveforms maybe expressed as andhavethevectorrepresentation (5-2-48) where ~,=~~"istheenergyineachofthewaveforms andg(t)isthepulse shapeofthetransmitted signal.Sincethesignalwaveforms haveequalenergy. theoptimum detector fortheAWGNchannelgivenby(5-1-44)computes the correlation metrics C(r,sm)=r-s m•m=I.2".,.M,(5-2·49) Inotherwords,thereceived signalvectorr=[r1r2)isprojected ontoeachof 270 DIGITALCOM\tUNtCA TlONS theMpossible signalvectorsandadecision ismadeinfavorofthesignalwith thelargestprojection. Thecorrelation detector described aboveisequivalent toaphasedetector thatcomputes thephaseofthereceived signalfromrandselectsthesignal vector S",whosephaseisclosesttor.Sincethephaseofris r,e,=tan-'-" rl(5-2-50) wewilldetermine thepdfofe"fromwhichweshallcompute theprobability oferror. Letusconsider thecaseinwhichthetransmitted signalphaseise,=0, corresponding tothesignalSI(t).Hence,thetransmitted signalvectoris SO=[~ 0] andthereceived signalvectorhascomponents(5-2-51) (5-2-52) Because nIandn2arejointlygaussian random variables, itfollowsthatrl andr2arejointlygaussian random variables with£(r,)=~,£(r2)=0,and u~,=u~,=}No=u~.Consequently, (5-2-53) (5-2-54)Thepdfofthephasee,isobtained byachangeinvariables from(rl'r2)to. V=v'd+r~ e,=tan-I(r2/rIl Thisyieldsthejointpdf (Ve)=~ex (_V'+jgs-2v?;,Vcose,) Pv.e,' ,2 ' P 2 '!CO'r 0'r Integration ofPv.e,(V,e,)overtherangeofVyieldsPe,(e,). Thatis, Pe,(e,)=rPv.e,(V,8,)dV 1.[ - =_e~21', sm2e,Ve-(V-V 4y,cos6,)1/ 2dV 21f 0(5-2-55) CHAPTE.R~: OPTIMl'M RECEIVER5 FORTHEADDLTIVE WHITE GAlIS51A!"oj ;";OISECHAS\H 271 Pe«3,) I.fd 1.4~ 1.15y,=;10 1.07 0.89 0.71y,=4 y,=2 0.\4~......_y,=I .nGURE 5-2-9 Probability densityfunction PH,le,) lory,~I.2,4and10.0.18 -~.14-2.51-UI8-1.26-0.6:\ O.()O O.6~1.2bI,M82.51314 B, whereforconvenience, wehavedefinedthesymbolSNRas'Y,='to)No.Figure 5-2-9illustrates [e,(8,)forseveralvaluesoftheSNRparameter y,whenthe transmitted phaseiszero,Notethat[e,(8,)becomes narrower andmore peakedaboute,=0astheSNR'Y,increases, Whensltl)istransmitted, adecision errorismadeifthenoisecausesthe phasetofalloutsidetherange-tr/M""8,""tr/M.Hence,theprobability ofa symbolerroris f"IM PM=1-p'-l,(0,)de, -IfIM(5-2-56) Ingeneral,theintegralofPe,(8)doesnotreducetoasimpleformandmustbe evaluated numerically, exceptforM=2andM=4. Forbinaryphasemodulation, thetwosignalsS,(t)ands,(I)areantipodal, and,hence,theerrorprobability is (~2'fnh) p,=Q -• No(5-2-57) WhenM=4,wehaveIDeffecttwobinaryphase·modulation signalsinphase 272 D/{iIlAI, «lMMI·\/l'AIH}NS quadrature. Sincethereisnocrosstalk orinterference between thesignalson thetwoquadrature carriers. thebitarorprobability isidentical tothatin (5-2-57). Ontheotherhand.'thesymbolerrorprobability forM=4is determined bynotingthat ('i-2-SS) whereP,istheprobability ofacorrectdecision forthe2-bitsymbol.Theresult (5-2-5K) follows fromthestatistical independence ofthenoiseonthe quadrature carriers. Therefore. thesymbolerrorprobability forM=4is P,=l-p' (!2fh)[ I()2tih)J =2Q -1~;Q - \No ~ ,'Y"(5-2-59) ForM>4,thesymbolerrorprobability PMisobtained bynumerically integrating (5-2-55). Figure5-2-10illustrates thiserrorprobability asafunction oftheSNRperbitforM=2.4,8,16,and32.Thegraphsclearlyillustrate the penalty inSNRperbitasMincreases beyond .'vi=4.Forexample. at P"=III',thedifference between M=4andM=8isapproximately 4dB,and thedifference between M=KandM=IIIisapproximately 5dB.Forlarge valuesofM,doubling thenumherofphasesrequires anadditional 6dB/bitto achievethesameperformance. Anapproximation totheerrorprobability forlargevaluesofMandfor , I t'---..........1'..--'-.I ".\110 ,\\ \\I ~\1\M=,2 \ \\\\\ M=2~\ \ I\ \ M=.~\ AT~ \\ ;10~ ~04 ~12Ih_II ~~ S;-.Jkper011.'YillJHI10:'i10- =2 ~10J L =-) -:2 -=II ~.10 E) -?t. FIGURE 5-2-10 Probahllity ofasymholt:rrorforPSK 'iignal~. (5-2-61) (5-2-62)CHAI'IJ-!{ 5:<H'TIML\o1 IUTUVERS FORTHEAI)DITIVE WHITE(jAI;SSIAN NOISl: CHA~~E:L 273 largeSNRmaybeobtained byfirstapproximating pA,(e).For'i)N,,»Iand le,1""~1r.PA,(e,)iswellapproximated as Pe,(e,)= ~cose,e-2Y",n,e, (5-2-60) Bysubstituting forPe,(e,)in(5-2-56)andperforming thechangeinvariable frome,tou=~sine"wefindthat f",M~YPM=I- -"cose,e-ly,,'n2e,de, -TrIM -Tt 2f', =- e-..-ndu \.IJt\~sin[!rIM) =2Q(v1Y,sin;)=2Q(V2kYh sin;) wherek=log,Mand'Y,=kYh'Notethatthisapproximation totheerror probability isgoodforallvaluesofM.Forexample, whenM=2andM=4, wehaveP,=p.=2Qr\!2-Y:). whichcompares favorably (afactor-of-two difference) withtheexact·probability givenby(5-2-57). Theequivalent biterrorprobability forM-aryPSKisrathertediousto deriveduetoitsdependence onthemapping ofk-bitsymbols intothe corresponding signalphases.WhenaGraycodeisusedinthemapping, two k-bitsymbolscorresponding toadjacent signalphasesdifferinonlyasinglebit. Sincethemostprobable errorsduetonoiseresultintheerroneous selection of anadjacent phasetothetruephase,mostk-bitsymbolerrorscontainonlya single-bit error.Hence,theequivalent biterrorprobability forM-aryPSKis wellapproximated as I Pb=I:PM Ourtreatment ofthedemodulation ofPSKsignalsassumed thatthe demodulator hadaperfectestimateofthecarrierphaseavailable. Inpractice, however, thecarrierphaseisextracted fromthereceived signalbyperforming somenonlinear operation thatintroduces aphaseambiguity. Forexample, in binaryPSK,thesignalisoftensquaredinordertoremovethemodulation, and thedouble-frequency component thatisgenerated isfilteredanddividedby2 infrequency inordertoextractanestimate ofthecarrierfrequency andphase <J>.Theseoperations resultinaphaseambiguity of1800inthecarrierphase. Similarly, infour-phase PSK,thereceived signalisraisedtothefourthpower inordertoremovethedigitalmodulation, andtheresulting fourthharmonic of thecarrierfrequency isfilteredanddividedby4inordertoextractthecarrier component. Theseoperations yieldacarrierfrequency component containing theestimate ofthecarrierphase <J>.buttherearephaseambiguities of±90° and1800inthephaseestimate. Consequently, wedonothaveanabsolute estimate ofthecarrierphasefordemodulation. 5-2-8274 DIGITAL COMMUNICATIONS Thephaseambiguity problem resulting fromtheestimation ofthecarrier phase1>canbeovercome byencoding theinformation inphasedifferences between successive signaltransmissions asopposed toabsolute phaseencod­ ing.Forexample, inbinaryPSK,theinformation bit1maybetransmitted by shiftingthephaseofthecarrierby180"relativetotheprevious carrierphase, whiletheinformation bit0istransmitted byazerophaseshiftrelativetothe phaseintheprevious signaling interval. Infour-phase PSK,therelativephase shiftsbetween successive intervals are0,90°,180°,and-90°,corresponding to theinformation bits00,01,II,and10,respectively. Thegeneralization to M>4phasesisstraightforward. ThePSKsignalsresulting fromtheencoding processaresaidtobedifferentially encoded. Theencoding isperformed bya relatively simplelogiccircuitpreceding themodulator. Demodulation ofthedifferentially encoded PSKsignalisperformed as described above,byignoring thephaseambiguities. Thus,thereceived signalis demodulated anddetected tooneoftheMpossibletransmitted phasesineach signaling interval. Following thedetector isarelatively simplephasecom­ paratorthatcompares theghasesofthedemodulated signalovertwo consecutive intervals inordertoextracttheinformation. Coherent demodulation ofdifferently encoded PSKresultsinahigher probability oferrorthantheerrorprobability derivedforabsolute phase encoding. Withdifferentially encoded PSK,anerrorinthedemodulated phase ofthesignalinanygivenintervalwillusuallyresultindecoding errorSover twoconsecutive sjgnaling intervals. Thisisespecially thecaseforerror probabilities below0.1.Therefore, theprobability oferrorindifferentially encodedM-aryPSKisapproximately twicethe probability oferrorforM-ary PSKwithabsolute phaseencoding. However, thisfactor-of-two increase inthe errorprobability translates intoarelatively smalllossinSNR. Ditrerential PSK(DPSK) anditsPerformance Adifferentially encoded phase-modulated signalalsoallowsanothertypeof demodulation thatdoesnotrequiretheestimation ofthecarrierphase.t Instead,thereceived signalinanygivensignaling intervaliscompared tothe phaseofthereceived signalfromthepreceding signaling interval. To elaborate, suppose thatwedemodulate thedifferentially encoded signalby multiplying r(t)bycos2trfctandsin2trfctandintegrating thetwoproducts over theinterval T.Atthekthsignaling interval, thedemodulator outputis r.=[~cos(0.-1»+n",~sin(0.-tb)+n.2] or,equivalently, (5-2-63) tBecause nophaseestimation isrequired. DPSKisoftenconsidered tobeanoncoherent communicati0ll: technique. WetaketheviewthatDPSKrepresents afonnofdigitalphase modulation intbeextreme casewherethephaseestimate isderivedonlyfromtheprevious symbol interval. CHAPTE.R S:OPTIML"\.1: RECEIVERS FORTHEADDITIVE WHITE GAUSSIA:-.J NOiSE CHA;..;NEl 275 whereekisthephaseangleofthetransmitted signalatthekthsignaling interval, q,isthecarrierphase,andnk=nk,+jnk,isthenoisevector.Similarly, thereceived signalvectorattheoutputofthedemodulator inthepreceding signaling intervalis (5-2-64) Thedecision variable forthephasedetector isthephasedifference between thesetwocomplex numbers. Equivalently, wecanproject'kontork..'anduse thephaseoftheresulting complex number; thatis, (5-2-65) which,intheabsenceofnoise,yieldsthephasedifferenceek.ek.l.Thus.the meanvalueofrk,t-,isindependent ofthecarrierphase.Differentially encoded PSKsignaling thatisdemodulated anddetected asdescribed aboveis calleddifferential PSK(DPSK). Thedemodulation anddetection ofDSPKusingmatched filtersisillustrated inFigure5-2-11.Ifthepulseg(t)isrectangular, thematched filtersmaybe replaced byintegrate-and-dump filters. Letusnowconsider theevaluation oftheerrorprobability performance ofa DPSKdemodulator anddetector. Thederivation ofthee:<actvalueofthe probability oferrorforM-aryDPSKisextremely difficult, exceptforM=2. Themajordifficulty isencountered inthedetermination ofthepdfforthe phaseoftherandom variable'k,t-"givenby(5-2-65). However. an approximation totheperformance ofDPSKiseasilyobtained, aswenow demonstrate. Without lossofgenerality, suppose thephasedifference Ok-Ok-,=o. Furthermore, theexponential factorse'j(8,-,-<b)andei(O,-""in(5-2-65)canbe absorbed intothegaussian noisecomponents nk-'andnk,withoutchanging theirstatistical properties. Therefore,'krt-,in(5-2-65)canbeexpressed as 'k,t·,='1;,'+-~(nk+nt-,)+nknt-, (5-2-66) Thecomplication indetermining thepdfofthephaseisthetermnknt-I' However, atSNRsofpractialinterest, thetermnknt_,issmallrelativetothe dominant noiseterm~(nk+nt",),Ifweneglectthetermnknt_Iandwe nGURE 5-2·11 Blockdiagram ofDPSKdemodulator. Received signal Output decision (5-2-67)276 DIGITAL COMMUNICATIONS alsonormalize r,rZ~)bydividing through by~,thenewsetofdecision metricsbecomes x=~+Re(n,+nZ~,) y=1m(n,+nZ~I) Thevariables xandyareuncorrelated gaussian random variables with identical variances (T~=No.Thephaseis (5-2-68) Atthisstage,wehaveaproblem thatisidentical totheonewesolved previously forphase-coherent demodulation. Theonlydifference isthatthe noisevariance isnowtwiceaslargeasinthecaseofPSK.Thusweconclude thattheperformance ofDPSKis3dBpoorerthanthatforPSK.Thisresultis relatively goodforM;;04,butitispessimistic forM=2~thesensethatthe lossinbinaryDPSKrelativetobinaryPSKislessthan3dbatlargeSNR.This isdemonstrated below. InbinaryDPSK,thetwopossible transmitted phasedifferences are0and 1rrad.Asaconsequence, onlytherealpartofr,rZ~Iisneededforrecovering theinformation. Using(5-2-67), weexpresstherealpartas Re(r,rt~l) =Hr,rZ~1+rtr'~I) Because thephasedifference between thetwosuccessive signaling intervals is zero,anerrorismadeifRe(r,rZ_I)<O.Theprobability thatr,rZ_)+rZr,_,< oisaspecialcaseofaderivation, giveninAppendix Bconcerned withthe probability thatageneralquadratic formincomplex-valued gaussian random variables islessthanzero.Thegeneralformforthisprobability isgivenby (B-21)ofAppendix B,anditdepends entirelyonthefirstandsecondmoments ofthecomplex-valued gaussian randomvariables r,andr,_).Uponevaluating themoments andtheparameters thatarefunctions ofthemoments, weobtain theprobability oferrorforbinaryDPSKintheform (5-2-69) where ~blNoistheSNRperbit. Thegraph 1SshowninFig.5-2-12.Alsoshowninthatillustration isthe probability oferrorforbinary,coherent PSK.Weobserve thataterror probabilities ofPb,;;IO~Jthedifference inSNRbetween binaryPSKand binaryDPSKislessthan3dB.Infact,atPb,;;10-',thedifference inSNRis lessthan1dB. Theprobability ofabinarydigiterrorforfour-phase DPSKwithGray codingcanbeexpressed intermsofwell-known functions. butitsderivation is quiteinvolved. Wesimplystatetheresultatthispointandrefertheinterested readertoAppendix Cforthedetailsofderivation. Itisexpressed intheform Pb=Q,(a,b)-~lo(ab)exp [-Ha'+b2)] (5-2-70) CIIAPTER 5OPTIMl'M RECEIVERS FORTilEADDITIVE WIIITE GAUSSIAN NOh"CHA~NEL 277 10-1 5 I.()-l ..<5 ~2 -10-,1 ~?•~o.2fO~ )2 Io--~'\. I "-1\I \ 1\ Bina')'DPSK \ \p_1,-"'-2-!\ I \ BinaryPSK\1\I--'---p.=QlJF(,,) \ I 1 \ \ i\ \ I I I\ FIGURE 5·2·12 Probability oferrorforbinaryPSKandDPSK.4h810 12 16 SNRperbity~(dB) (5-2-71)whereQl(a,b)istheMarkum Qfunction definedby(2-[-122) and(2-1-123), ~)(x)isthemodified Besselfunction oforderzero,definedby(2-1-120), and theparameters aandbaredefinedas r----_=_ a=V2')'h(l-V!) b=V2')'h(1+"1) Figure5-2-13illustrates theprobability ofabinarydigiterrorfortwo-and 4n 101214 SNRperhi:.'(I,(dB], 1"",- "f\..'~,'\".\" ,\1\ , Two-andfnllr-\\ \.phasePSI(\\I Ir--r- TwO·rha~ ~\.\r--I-- DPSK , I1\\\ 1 \\ F.."u,-pha.,e-tf--!OPSI< 1\1\ \\,IO-s I(}-" "2\G­ 5 FIGURE 5·2-13 Probabilily ofbiIerrorforbinaryandfopr-phase PSK andDPSK. 278 DIGITAL COMMUNICATIONS four-phase DPSKandcoherent PSKsignaling obtained fromevaluating the exactformulas derivedinthissection. SincebinaryDPSKisonlyslightly inferiortobinaryPSKatlargeSNR,andDPSKdoesnotrequireanelaborate methodforestimating thecarrierphase,itisoftenusedindigitalcommunica­ tionssystems. Ontheotherhand,four-phase DPSKisapproximately 2.3dB poorerinperformance thanfour-phase PSKatlargeSNR.Consequently the choiceoetween thesetwofour-phase systemsisnotasclearcut.Onemust weighthe2.3dBlossagainstthereduction inimplementation complexity. 5-2-9Probability ofErrorforQAM RecallfromSection4-3thaIQAMsignalwaveforms maybeexpressed as s",(t)=A"".g(t)cos21if.t-A""g(r)sin2Trft.0'"t'"T(5-2-72) whereA",candA,,,,aretheinformation-bearing signalamplitudes ofthe quadrature carriersandg(r)isthesignalpulse.Thevectorrepresentation of thesewaveforms is Sm=[Am('~ Am.\~] (5-2-73) Todetermine theprobability oferrorforQAM,wemustspecifythesignal pointconstellation. WebeginwithQAMsignalsetsthathaveM=4points. Figure5-2-14illustrates twofour-point signalsets.Thefirstisafour-phase modulated signalandthesecondisaQAMsignalwithtwoamplitude levels. labeledAIandAz•andfourphases.Because theprobability oferroris dominated bytheminimum distance between pairsofsignalpoints.letus imposethecondition thatd~?n=2Aforbothsignalconstellations andletus evaluate theaveragetransmitter power,basedonthepremise thatallsignal pointsareequallyprobable. Forthefour-phase signal.wehave. P",=1(4)2A' =2A' (5-2-74) Forthetwo-amplitude, four-phase QAM.we,placethepointsoncirclesof radiiAandV3A.Thus.d:;;!n=2A.and (5-2-75) whichisthesameaverage powerastheM=4-phase signalconstellation. Hence.forallpractical purposes, theerrorrateperformance ofthetwosignal FIGURE5-2-t4 Twofour'point signalconstellations. CHAPTER S'OPTIMUM RECEIVERS FORmEADDITIVE WHITEGAUSSIAN NOISECHANNELm (C.C) C=/3.2/2 (b)(-3.I) (a)1-3.-I) (1•./3.0) Ie) ItlJ FIGURE 5-2-15 Foureight·point QAMsignalconstellations. setsisthesame.Inotherwords,thereisnoadvantage ofthetwo-amplitude QAMsignalsetoverM=4-phasemodulation. Next,letusconsider M=8QAM.Inthiscase,therearemanypossible signalconstellations. Weshallconsider thefoursignalconstellations shownin Fig.5-2-15,allofwhichconsistoftwoamplitudes andhaveaminimlim distance between signalpointsof2A.Thecoordinates (Ame>A~)foreach signalpoint,normalized byA,aregiveninthefigure.Assuming thatthesignal pointsareequallyprobable, theaveragetransmitted signalpoweris 1~ 2 ,Pay=-L..(Arne+Am,)Mm:1 A2M =M];,(a~+a~c) (5-2-76) where(am",am,)arethecoordinates ofthesignalpoints,normalized byA. Thetwosignalsets(a)and(c)inFig.5-2-15containsignalpointsthatfall onarectangular gridand'have Pay=6A'.Thesignalset(b)requires anaverage transmitted powerPa.=6.83A',and(d)requires PaY=4.73A2•Therefore, the fourthsignalsetrequires approximately 1dBlesspowerthanthefirsttwoand 1.6dBlesspowerthanthethirdtoachievethesameprobability oferror.This signalconstellation isknowntobethebesteight-point QAMconstellation because itrequires theleastpowerforagivenminimum distance between signalpoints. ForM""16,therearemanymorepossibilities forselecting theQAMsignal pointsinthetwo-dimensional space.Forexample, wemaychooseacircular multiamplitude constellation forM=16,asshowninFig.4-3-4.Inthiscase, 280 DlfllTAL COMMl'~ICATlOI'iS thesignalpoinlsatagivenamplitude levelarephase-rotated by~1frelativeto thesignalpointsatadjacent amplitude levels.Thisl6-QAM constellation isa generalization uftheoptimum 8-QAM constellation. However. thecircular l6-QAM constallation isnotthebest16·point QAMsignalconstellation for theAWGNchannel. Rectanguljlr QAMsignalconstellations havethedistinctadvantage ofbeing easilygenerated astwoPAMsignalsimpressed onphase·quadrature carriers. Inaddition. thevareeasilydemodulated. Although theyarenotthebestM-ary QAMsignalcGrlstellations forM'"16,theaveragetransmitted powerrequired toachieveaghenminimum distance isonlyslightlygreaterthantheaverage powerrequired forthebestM-aryQAMsignalconstellation. Forthese reasons. rectangular M-aryQAMsignalsaremostfrequently usedInpractice. Forrectangular signalconstellations inwhichM=2*,wherekiseven.the QAMsignalconstellation isequivalent totwoPAMsignalsonquadrature carriers. eachhaving\1M=Z*cvienalpoints.Sincethesignalsinthe phase-quadratUJ ecomponents canhe'perfectly separated atthedemodulator. the probability oferrorforQAMiseasilydetermined fromtheprobability of errorforPAM.Specifically, theprobability ofacorrectdecision fortheM-ary QAMsystemis (5-2-77) whereP''Mistbeprobability oferrorofa\1M-aryPAMwithone-half the averagepowerineachquadrature signaloftheequivalent QAMsystem.By appropriately modifying theprobability oferrorforM,aryPAM.weobtain (5-2-78) where(,JN"istheaverageSNRpersymbol.Therefore. theprobability ofa symbolerrorfortheM-aryQAMis (5-2-79) NotethatthisresultisexactforM=2*whenkiseven.Ontheotherhand. whenkisodd~thereisnoequivalent VM-aryPAMsystem.Thisisno problem, however. becauseitisrathereasytodetermine theerrorratefora rectangular signalset.Ifweemploytheoptimum detector thatbasesits decisions ontheoptimum distance metricsgivenby(5-1'43), itisrelatively straightforward toshowthatthesymbolerrorprobability istightlyupper­ bounded as P"'1_[l-2Q(-; 3%'0.)]2 M V{M-1)N" "'4Q( (5-2-80) CHAPTER \OPTIMUM RECEIVERS FORTHEADDITIVE WHilEGAUSSIAN NOiSECHANNEL 281 \\\ '\\ \ I--\ \~:M64I-QAM M=16~I-<f\-\ \1\\ QAMI M=4 ,2 lO­ S10-2 ..'S ~2 :g10­ E5 ~ ~2o .~10-" ::E5 ~210-1 S FIGURE 5-2·16 Probability ofasymbolerrorforQAM.2 10-"--6-4-::!O ~4h111(1121411'> 1!(2\l SNRperhil.'("fdB) (5-2-81)foranyk;"I,where '(;b,viNoistheaverageSNRperbit.Theprobability ofa symbolerrorisplottedinFig.5-2-16asafunction oftheaverageSNRperbit. Fornon-rectangular QAMsignalconstellations, wemayupper-bound the .errorprobability byuseofaunionbound.Anobviousupperboundis PM<(M-l)Q(V[d~ln12/2No) where d~!nintheminimum euclidean distance between signalpoints.This boundmaybeloosewhenMislarge.Insuchacase,wemayapproximate P" byreplacing M-1byMn,whereMnisthelargestnumber ofneighboring pointsthatareatdistance d~!nfromanyconstellation point. Itisinteresting tocompare theperformance ofQAMwiththatofPSKfor anygivensignalsizeM,sincebothtypesofsignalsaretwo-dimensional. Recall thatforM-aryPSK,theprobability ofasymbolerrorisapproximated as PM""2Q(V2y,Sin ~) wherey,istheSNRpersymbol.ForM-aryQAM,wemayusetheexpression (5-2-78). Sincetheerrorprobability isdominated bytheargument oftheQ function, wemaysimplycompare thearguments ofQforlhe twosignal formats. Thus,theratioofthesetwoarguments is 3/(M-1) PllM2sin2(11"/M) (5-2-82) Forexample, whenM=4,wehaverRM=I.Hence,4-PSKand4·QAM yield comparable performance forthesameSNRpersymbol.On'theotherhand, (5-2-83)282 DIGITAL COMMUf\;1CAT10NS TABLE 5·2·1SNRADVANTAGE OFM-ARY QAMOVERM-ARY PSK M 10log..91" 8 1.65 16 4.20 32 7.02 64 9.95 whenM>4wefindIhatrJlM>I,sothatM-aryQAMyieldsbetter performance thanM·aryPSK.Table5-2-1illustrales theSNRadvantage of QAMoverPSKforseveralvaluesofM.Forexample, weobserve that 32-QAM hasa 7dBSNRadvantage over32-PSK. 5-2-10Comparison ofDigitalModulation Methods Thedigitalmodulation methods described inthischaptercanbecompared ina numberofways.Forexample, onecancompare themonthebasisoftheSNR required toachieve aspecified probability oferror.However. sucha comparison wouldnotbeverymeaningful, unlessitweremadeonthebasisof someconstraint, suchasafixeddatarateoftransmission or, equivalently. on thebasisofafixedbandwidth. Withthisgoalinmind,letusconsider the bandwidth requirements forseveralmodulation methods. Formultiphase signals.thechannel bandwidth required issimplythe bandwidth oftheequivalent lowpasssignalpulseg(t),whichdepends onIts detailed characteristics. Forourpurposes. weassumethatg(t)isapulseof duration Tandthatitsbandwidth Wisapproximately equaltottlereciprocal ofT.Thus.W=lITand,sinceT=klR=(lOg2M)IR,itfollowsthat Rw=-­log2M Ttlerefore. asMisincreased. thechannelbandwidth required. whenttlebit rateRisfixed,decreases. Thebandwidth efficiency ismeasured bythebitrate tobandwidttl ratio,whichis Rw=log2M (5-2-84) (5-2-85)Thebandwidth-efficient methodfortransmitting PAMissingle-sideband. Ttlen,thechannelbandwidttl required totransmit thesignalisapproximately equal10Jl2Tand,sinceT=k IR=(I0g2M)IR,ilfollowsIhat RW=210g2M CtP..P1TRSOPIT\IUM Rl-CilyFRS I-HH.THfADDITI\E: WHlH:: GAUSSlA{\,' NOISECli\....'EI283 ThisisafactoroftwobetterthanPSK. InthecaseofQAM.wehavetwoorthogonal carriers. witheachcarric'r havingaPAMsignal.Thus.wedoubletheraterelativetoPAMHowc\cr.the QAMsignalmusthetransmitted viadoublesideband. C'onsequ.:ntly. QAM andPAMhavethesamebandwidth efficiency whenthebandwidth is referenced tothebandpass signal. Orthogonal signalshavetotallydifferent bandwidth requirements. Iflhe M=Z'orthogonal signalsareconstructed bymeansoforthogonal carrierswith minimum frequency separation ofI/ZTfororthogonality. thebandwidth required fortransmission ofk=logzMinformation bitsis M M Mw=-= = RZTZ(k/R) ZlogzM(5-2-Xfl) Inlhiscase.thebandwidth increases asMincreases. Similarrelationships obtainforsimplexandbiorthogonal signals.Inthecaseofbiothogonal signals. therequired bandwidth isonehalfofthatfororthogonal signals. Acompact andmeaningful comparison ofthesemodulation methods isone basedonthenormalized datarateRfW(bitspersecondperhertzof bandwidth) versustheSNRperbit('tIh/No)required toachieveagivenerror probability. Figure5-Z-17illustrates thegraphofR/WversusSNRperbilfor PAM.QAM.PSK.andorthogonal signals,forthecaseinwhichlheerror probability _sPM=10-5WeobservethatinthecaseofPAM,QAM,andPSK. increasing Mresultsinahigherbitrate-to-bandwidth ratioR/w.However. the costofachieving thehigherda'tarateisanincrease iniheSNRperbit. Consequently, thesemodulation methods areappropriate forcommunication channels thatarebandwidth limited,wherewedesireabitrate-to-bandwidth ratioR/W>Iandwherethereissufficiently highSNRtosupportincreases in M.Telephone channels anddigitalmicrowave radiochannels areexamples of suchbandlimited channels_ Incontrast, M-aryorthogonal signalsyieldabitrate-to-bandwidth ratioof R/W", 1.AsMincreases, R/Wdecreases duetoanincrease intherequired channel bandwidth. However. theSNRperbitrequired :toachieveagiven errorprobability (inthiscase,PM=to-')decreases asMincreases. Conse­ quently,M-aryorthogonal signalsareappropriate forpower-limited channels thathavesufficiently largebandwidth toaccommodate alarge'number of signals.Inthiscase,asM......x,theerrorprobability canbemadeassmall asdesired,provided that'lJh/No>0.693(-1.6dB). Thisistheminimum SNR perbitrequired toachieve reliable transmission inthelimitasthe channelbandwidth W......xandthecorresponding bitrate-to-bandwidth ratio R/W......O. AlsoshowninFig.5-2-17isthegraphforthenormalized capacity ofthe bandlimited AWGNchannel, whichisduetoShannon (1948).TheratioC/W, whereC (=R)isthecapacity inbits/s,represents thehighestachievable bit rate-to-bandwidth ratioonthischannel. Hence,itservesastheupperbound 284 DIGITAL COMMUNICATIONS 10 2Channel capacily limit~ .\1=16QAM M=4PI\M ISSBI M=4?SK M=2PAM ISSB)c IV M=64QAM M=8P'M(SSB) PSK Bandwidth-limited region:~>I -1.6, M=2M=2 , Asymplole : ,,,,,,,,,,,,,,,,,,,,0.5 0.3 0.2 0.110 15 M=8 M=16 M=32 M=64 Orthogonal signals Coherent detection20 25 SNRperbit.Yb=I/No(dB) FIGURE 5-2-17 Comparison ofseveralmodulalion methods at10-'symbolerrorprobability. onthebandwidth efficiency ofanytypeofmodulation. Thisboundisderived inChapter 7anddiscussed ingreaterdetailthere. 5-3OPTIMUM RECEIVER FORCPMSIGNALS WerecallfromSection4-3thatCPMisamodulation methodwithmemory. Thememory resultsfromthecontinuity ofthetransmitted carrierphasefrom onesignalintervaltothenext.Thetransmitted CPMsignalmaybeexpressed as rnS(I)=VTCOS[21ifct+4>(t;I)] (5-3-1) ("tiM'IH{~OPT1\H'M RrTF:IVH{S !'<)j{IIH ADDITIVE WHnf (jAl;"iS-IAl'O NOISl:..iA~~H285 wherecPrr;I)isthecarrierphase.Thefilteredreceived signalforanadditive gaussian noisechannelis whereret)=s(t)+net) nCr)=ne(t)cos2trfct-n,(t)sin2trfct(5-3-2) (5-3-3) 5-3-1Optimum Demodulation andDetection ofCPM Theoptimum receiver forthissignalconsistsofacorreiatorfollowed bya maximum-likelihood sequence detector thatsearches thepathsthrough the statetrellisfortheminimum euclidean distance path.TheViterbialgorithm is anefficient method forperforming thissearch.Letusestablish thegeneral statetrellisstructure forCPMandthendescribe themetriccomputations. RecallthatthecarrierphaseforaCPMsignalwithafixedmodulation index hmaybeexpressed as n cP(t;I)=2trh2:hq(t-kT) k=-·:x n-1- n =trhLI.+21th2:I.q(t-kT) k=-x k.=n-L+l =8,+8(r;I), nT""t""(n +1)7 wherewehaveassumed thatq(r)=0fort<0,q(t)=~fort;;.LT,and q(t)=[g(r)dr(5-3-4) (5-3-5) Thesignalpulseget)=0fort<Oandt;;.LT.ForL=1,wehaveafull response CPM,andforL>1,whereLisapositiveinteger;wehaveapartial response CPMsignal. Now,whenhisrational, i.e.,h=mlpwheremandparerelatively prime positiveintegers, theCPMschemecanberepresented byatrellis.Inthiscase, therearepphasestates e={1tm21tm -'.(P~--,1'-'.).:..:mn-,}s0,, ,...,p p p whenmiseven,and2pphasestates(5-3-6) (5-3-7) whenmisodd.IfL=1,thesearetheonlystatesinthetrellis.01\theother hand,ifL>1,wehaveanadditional numberofstatesduetotht"partial 286 DIGITAL COMMUNICATJO~S response character ofthesignalpulseget).Theseadditional statescanbe identified byexpressing O(t,I)givenby(5-3-4)as n-1 9(1;I)=2nhII.q(t-kT)+2nhlnq(t-nT) k=n-L+l(5-3-8) Thefirsttermontheright-handsideof(5-3-B)depends ontheinformation symbols (1n-1>In-".._,In-L+1),whichiscalledthecorrelative starevector, andrepresents thephasetermcorresponding tosignalpulsesthathavenot reached theirfinalvalue.Thesecondtermin(5-3-8)represents thephase contribution duetothemostrecentsymbolIn.Hence,thestateoftheCPM signal(orthemodulator) attimer=nTmaybeexpressed asthecombined phasestateandcorrelative state,denoted as (5-3-9) forapartialresponse signalpulseoflengthLT,whereL>1.Inthiscase,the numberofstatesis N={PML -1(evenm) ,2pML-1(oddm)(5-3-10) whenh=m/p. Now,supposethestateofthemodulator att=nTisSn'Theeffectofthe newsymbolinthetimeintervalnT.;;;t.;;;(n+l)Tistochangethestatefrom SntoSn+l'Hence,att=(n+l)T,thestatebecomes where Eumple 5-3-1 Consider abinaryCPMschemewithamodulation indexh=3/4anda partialresponse pulsewithL=2.Letusdetermine thestatesSnoftheCPM schemeandsketchthephasetreeandstatetrellis. First,wenotethatthereare2p=8phasestates,namely, e,={O,±in,±!n,±~n,nl Foreachofthesephasestates,therearetwostatesthatresultfromthe memory oftheCPMscheme.Hence,thetotalnumberofstatesisN,=16, namely, (0,1),(0,-1),(n,l),(n,-1),(in,1),(ill,-1),nn,1),(~Il,-1), air,1),alt',-1),(-llr,l),(-in,-1),(-tn,1),(-tn,-1), (-~n,1),(-~n,-1) CHAPTER S:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL FlGURE 5·3-1Slatetrellisforpartialresponse (L=2)CPM withh=i.(8"./..- 1) iO,I) 10,-1) (~x,I) (~.,-I) (!x,I) (~.,-1) (iX,I) (i"-I) (x,I) (x,-ll (ix,I) (~.,-I) (~.,I) (~.,-I) (~.,I) (~X,-I)(8..1,J..) (0,I) (0,-I) (!x,I) (!X,-I) (ix,I) IX,I) IX,-I) Ifthesystemisinphasestate9.=-~1l'andI'-I=-1then 9'+1=8.+tchl._ 1 =-in-~n=-n Thestatetrellisisillustrated inFig.5-3-1.Apa1hthroughthestatetrellis corresponding tothesequence (1,-1,-1,-1,1,1)isillustrated inFig. 5-3-2, Inordertosketchthephasetree,wemustknowthesignalpulseshape g(t),Figure5·3-3illustrates thephasetreewheng(t)isarectangular pulse ofduration 2T,withinitialstate(0,1). Havingestablished thestatetrellisrepresentation ofCPM,letusnow consider themetriccomputations performed intheViterbialgorithm. MetricComp.tatioDS Byreferring backtothemathematical development 288 D!GITAL COM\1U"lI('ATIONS ,,,,,,J (0.I)•··· (0,-1r• \ I~',). (~n.-;)• 1~,,)• , 112'.-,• I~'J)• (~n-'I. (It.II• 11t.··1)• (~,-,). (~n,)• (~n-,)• (ix.I).-, • ····,,.,.,..,·'·'·',',' \".:-,-,•• (0.-1)•",.,.,., ,, ,,., ,,.,,,.,,,..,,·•·,····,.,,,..,, • I~n.'.1• FIGURE 5-3oZAsingle signal paththroughthe trellis. ~(t:11 FIGURE 5-303PhasetreeforL=2partialresponse CPM withh=i--~T. -~Jt _J'J[ 4-. ~~x4 -~Jt CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL 289 forthederivation ofthemaximum likelihood demodulator giveninSection 5-1-4,itiseasytoshowthatthelogarithm oftheprobability oftheobserved signalr(l)conditioned onaparticular sequence oftransmitted symbols Iis proportional tothecross-correlation metric !(n+I)7 CMn(I)= -xr(l)cos[wJ+q,(I;I»)dl lln+IIT =CMn-t(l)+nrr(l)cos[wet+~(I;I)+9,,)dl(5-3-11) ThetermCMn_I(I) represents themetricsforthesurviving sequences upto timenT,andtheterm l(n+I)T v,,(I;9,,)= r(l)cos[weI+9(1;I)+9n)dl nT(5-3-12) represents theadditonal increments tothemetricscontributed bythesignalin thetimeinterval nT,,;;I'"(n+1)T.NotethatthereareMLpossiblesequences I=(In,In-I>'..,I"-L+I)ofsymbols andp(or2p)possible phasestates{9n}. Therefore, therearepML(or2pML)different valuesofun(l,9,,),computed in eachsignalinterval, andeachvalueisusedtoincrement themetrics corresponding tothepML-1surviving sequences fromtheprevious signaling interval. Ageneralblockdiagram thatillustrates thecomputations ofvn(l9n) fortheViterbidecoder isshowninFig.5-3-4. Notethatthenumberofsurviving sequences ateachstateoftheViterbi decoding processispML-1(or2pML-1).Foreachsurviving sequence, wehave Mnewincrements ofv,,(I;e,,)thatareaddedtotheexistingmetricstoyield pML(or2pM")sequences withpML(or2pML)metrics,However, thisnumber isthenreduced backtopML-1(or2pMl.-1)survivors withcorresponding metricsbyselecting themostprobable sequence oftheMsequences merging ateachnodeofthetrellisanddiscarding theotherM-1sequences. FIGURE: 5·3-4Computation ofmetrlcincrements v,,(I:0,,).rll)ToVilerhi del.:OOC'T 290 OIGITAL COMMl'NICATlO!'iS 5-3-2Performance ofCPMSignals Inevaluating theperformance ofCPMsignalsachieved withMLSE,wemust determine theminimum euclidean distance ofpathsthrough thetrellisthat separate atthenodeatt=0andre-emerge atalateitimeatthesamenode. Thedistance between twopathsthrough thetrellisisrelatedtothe corresponding signalsaswenowdemonstrate. Suppose thatwehavetwosignals5i(1)andSj(l)corresponding totwophase trajectories <pU:I,)and<p(t;I).Thesequences IiandIjmustbedifferent in theirfirstsymbol.Then,theeuclidean distance between thetwosignalsoveran intervaloflengthNT,wherel/Tisthesymbolrate,isdefinedas 1"T dfJ=0[s,(r)-5/1)/'dl INT fNC fNT =0sf(t)dl+0S;(l)dl-20Si(t)Sj(t)dt 2'€lNT=ZNg-2T0cos[wet+¢(t;I;)]cosIw"t+<p(t;I,)]dt 2\€lNT =2Ng-T0cos[¢(d,j-<p(t;I)]dt ''€INT=~T0{I-cos[I/>(l;Ii)-¢(I;I)]}dt (5-3-13) Hencetheeuclidean distance isrelatedtothephasedifference between the pathsirithestatetrellisaccording to(5-3-13). It,isdesirable toexpressthedistance d~intermsofthebitenergy.Since g='€blog,M,(5-3-13)maybeexpressed as (5-3-14) where o~isdefinedas (5-3-15) Furthermore, weobserve that<p(t;I,)-<p(t;Ij)=.p(t;Ii-Ij),sotliat,with ~=Ii-~,(5-3·15)maybewrittenas 10MjNT 8~=~ 0[1-cos.p(t;~)Jdt (5-3-16) whereanyelement of~cantakethevalues0,±2, ±4,±,..±2(M-Il, exceptthat~#O. CHAPTFR.~ OI'TI\Il'\1 RETEIVERSFORTHE:\DDITIVE WHITE GAViSL·\~ \"OISEO-lA:\"SH 291 Theerrorrateperformance forCPMisdominated bythetermcorrespond­ ingtotheminimum euclidean distance, anditmaybeexpressed as whereP-KQI'I~hs')II-{)n""\,N.min\V;l I 8~li~=limminS~, \~,- 1.1 {log,MJ'" }=limmin-.--- [l-coscI>(t:I,-I})]dt "~-,, 1,./r 0(5-3-17) (5-3-1H) Wenotethatforconventional binaryPSKwithnomemory, N=1and S~'"=S~,=2.Hence,(5-3-17) agreeswithourprevious result. SinceS;"",characterizes theperformance ofCPMwithMLSE,wecan invcstigatt; theeffecton8';'"resulting fromvarying thealphabet sizeM.the modulation indexh.andthelengthofthetransmitted pulseinpartialresponse CPM. First.weconsider fullresponse (L=I)CPM.Ifwetake.\1=2asa beginning, wenotethatthesequences I,=+I.-1.I,.I, I,= -I.+I.I,.I,(5-3-19) whichdifferfork=0.1andagreefork;.2.resultintwophasetrajectories thatmergeafterthesecond symbol. Thiscorresponds tothedifference sequence !;={2,-2.0,0,...} (5-3-20) Theeuclidean distance forthissequence iseasilycalculated from(5-3-16), and provides anupperboundon8;'"".ThisupperboundforM=2is ,(Sin2lCh')d;,(h)=2 I - . 2lChM=2 (5-3-21) Forexample, whereII=!,whichcorresponds toMSK,wehaved~n)=2,so that8;',,"(\)'"2. ForM>2andfullresponse (,PM,itisalsoeasilyseenthatphase trajectories mergeatt=2T.Hence,anupperboundon8;',,,,canbeobtained byconsidering thephasedifference sequence ~={a.-a.n.0,...}where a=±2. ±4,....±2(M-I).Thissequence yieldstheupperbound sin2klCh)} 2klCh(5-3-22) 292 DIGITAL COMMl'NKA nONS Q FIGURE S-J..STheupperboundd1asafunction ufthemodulat,on indexhforfullresponse CPMwithrectangular pulses. (FromAll/illandSundberg (1984).©1984.JohnWi/ev Ltd.Reprinted wirhperminilJn ofthepublisher.]d',M=/6 f)0.10.20.30.40.5()'60,7O.K1l.9J.() I, Thegraphsofd1(h)versushforM~2.4.8,16areshowninFig.5-3-5.Itis apparent fromthesegraphsthatlargegainsinperformance canbeachieved by increasing thealphabet sizeM.Itmustberemembered, however. that Il;";n(h)""d~(h).Thatis.theupperboundmaynotbeachievable forall valuesofh. Theminimum euclidean distance Il;";n(h)hasbeendetermined, byevaluat­ ing(5-3-16), foravarietyofCPMsignalsbyAulinandSundberg (198I).For example, Fig.5-3-6illustrates thedependence oftheeuclidean distance for binaryCPFSKasafunction ofthemodulation indexh.withthenumberNof 2.5 N=4 FIGURE 5-3-6Squared minimum euclidean distanceasafunctionofthe modulation indexforbinaryCPFSK.Theupper boundisil,.[FromAlliinandSIIndberg ({98/I. ©/98)IEEE.].2.0 1.00~--:!7---->M---­1.0 CHAnER 5:OPTIML"M RECEJVERS FORTHEADDJJrVE WHITE GAUSSJAN NOISE CHANNEL 293 bitobservation (decision) intervals(N=1.2.3.4)asaparameter. Alsoshown istheupperbound d~(h)givenby(5-3-21). Inparticular, wenotethatwhen h=~. O~in(j)=2,whichisthesamesquared distance asPSK(binaryor quaternary) withN=1.Ontheotherhand,therequired observation interval forMSKisN=2intervals, forwhichwehave 8~in(n=2.Hence,the performance ofMSKwithMLSEiscomparable to(binaryorquaternary) PSK aswehavepreviously observed. WealsonotefromFig.5-3-6thattheoptimum modulation indexforbinary CPFSK ish=0.715whentheobservation interval isN=3.Thisyields 8~in(0.715) =2.43,oragainof0.85dBrelativetoMSK. Figure5-3-7illustrates theeuclidean distance asafunction ofhfor M=4CPFSK, withthelengthoftheobservation intervalNasaparameter. Alsoshown(asadashedlinewhereitisnotreached) istheupperbound d~ evaluated from(5-3-22). Notethat c5~i"achieves theupperboundforseveral valuesofhforsomeN.Inparticular, notethatthemaximum valueofd~. whichoccursath=0.9.isapproximately reachedforN=8observed symbol intervals. Thetruemaximum isachieved ath=0.914withN=9.Forthiscase. o~m(0.914) =4.2,whichrepresents a3.2dBgainoverMSK.Alsonotethatthe euclidean distance contains minimaath=Lt~.1.etc.Thesevaluesofhare calledweakmodl/lation indicesandshouldbeavoided. Similarresultsare available forlargervaluesofM.andmaybefound-inthepaperbyAuIinand Sundberg (1981)andthetextbyAnderson eral.(1986). d~{") dj{h'l 4 ,/ ",,,,, 6 4 f-+-+---,,"::----,-L:---h10 FIGURE 5-3-7Squared minimum euclidean ;,Iistance asafunctionof themodulation indexforqualernaT)' CPFSK. Theupptrhoundisd~.[From,4lllillandSundherg (Nil/).if)!WI!IEEE.) II 294 DIGITAL COMMUNICATIONS ZdB -3dB FIGURE S-~Upperboundd';,ontheminimum distancefor partialresponse (raisedcosinepulse)binaryCPM. [FromSundberg (1986).©19861EEE.J h (5-3-23)Largeperformance gainscanalsobeachieved withMLSEofCPMbyusing partialresponse signals.Forexample, thedistance bound d~(h)forpartial response, raisedcosinepulsesgivenby {_l_(l-COS~) (O,;;,t".LT) g(r)=2LT 2LT o (otherwise) isshowninFig.5-3-8forM=2.Here,notethat,asLincreases, d~also achieves highervalues.Clearly, theperformance ofCPMimproves asthe correlative memory Lincreases, buthmustalsobeincreased inorderto achievethelargervaluesofd~.Sincealargermodulation indeximpliesa largerbandwidth (forfixedL).whilealargermemory lengthL(forfixedh) impliesasmallerbandwidth, itisbettertocompare theeuclidean distance asa function ofthenormalized bandwidth 2WTb•whereWisthe99%power bandwidth andTbisthebitinterval. Figure5-3-9illustrates thistypeof comparison withMSKusedasapointofreference (0dB).Notefromthis figurethatthereareseveraldecibels tobegainedbyusingpartialresponse signalsandhighersignaling alphabets. Themajorpricetobepaidforthis performance gainistheaddedexponentially increasing complexity inthe implementation oftheViterbidecoder. Theperformance resultsshowninFig.5-3-9illustrate that3-4dBgain relativetoMSKcanbeeasilyobtained withrelatively noincreaseinbandwidth bytheuseofraisedcosinepartialresponse CPMandM=4.Although these resultsareforraisedcosinesignalpulses,similargainscanbeachieved with otherpartialresponse pulseshapes.We'emphasize thatthisgaininSNRis achieved byintroducing memory intothesignalmodulation andexploiting the memory inthedemodulation ofthesignal.Noredundancy throughcodinghas CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL 295 -3'----4---'-----J-------'0.5 1.0 1.5 2WT/>FIGURE S·3-~Powerbandwidth tradeofflorpartialresponse CPM signalswithraisedcosinepulses.WistheIF} percentin-bandpowerbandwidth. [FromS,mdberg (1986).©1986IEEE.)6 5 4 2 dB o -I -21J' 10108102:;'un3RC.M=8 •MSK beenintroduced. Ineffect,thecodehasbeenbuiltintothemodulation andthe trellis-type (Viterbi) decoding exploitsthephaseconstraints intheCPMsignal. Additional gainsinperformance canbeachieved byintroducing additional redundancy through codingandincreasing thealphabet sizeasameansof maintaining afixedbandwidth. Inparticular, trellis-eoded CPMusingrelatively simpleconvolution codeshasbeenthoroughly investigated andmanyresults areavailable inthetechnical literature. TheViterbidecoder fortheconvolu­ tionallyencoded CPMsignalnowexploits thememory inherent inthecode andintheCPMsignal.Performance gainsoftheorderof4-6dB,relativeto uncoded MSKwiththesamebandwidth, havebeendemonstrated bycombin­ ingconvolutional codingwithCPM.Extensive numerical resultsforcoded CPMaregivenbyLindell(1985). MuIli-hCPMByvaryingthemodulation indexfromonesignaling interval toanother, itispossible toincrease theminimum euclidean distance 6~in between pairsofphasetrajectories and,thus,improve theperformance gain overconstant-h CPM.Usually, multi·hCPMemploys afixednumberHof modulation indicesthatarevariedcyclically insuccessive signaling intervals. Thus,thephaseofthesignalvariespiecewise linearly. Significant gainsinSNRareachievable byusingonlyasmallnumberof different valuesofh.Forexample, withfullresponse(L=1)CPMandH=2, itispossibletoobtainagainof3dBrelativetobinaryorqUll.ternary PSK,By increasing HtoH=4,againof4.5dBrelativetoPSKcanbeobtained, The performance gaincanalsobeincreased withanincrease inthesignalalphabet. Table5-3-1liststheperformance gainsachieved withM=2,4,and8for severalvaluesofH.Theupperboundsontheminimum euclidean distance are alsoshowninFig.5-3-10forseveralvaluesofMandH.Notethatthemajor gaininperformance isobtained whenHisincreased fromH=1toH=2.For H>2,theadditional gainisrelatively smallforsmallvaluesof{h,}.Onthe otherhand,significant performance gainsareachieved byincreasing the alphabet sizeM. Theresultsshownaboveholdtorfullresponse CPM.Onecanalsoextend theuseofmulti-hCPMtopartialresponse inanattempt tofurtherimprove performance. Itisanticipated thatsuchschemes willyieldsomeadditional performance gains,butnumerical resultsonpartialresponse, multi-hCPMare limited.Theinterested readerisreferred tothepaperbyAulinandSundberg (l982b). Multiamplitude CPMMultiamplilude CPM(MACPM) isbasically a combined amplitude andphasedigitalmodulation schemethatallowsusto increase thesignaling alphabet relativetoCPMinanother dimension and, thus,toachievehigherdataratesonaband-limited channel. Simultaneously, thecombination ofmultiple amplitude inconjunction withCPMresultsina bandwidth-efficient modulation technique. Wehavealreadyobserved thespectralcharacteristics ofMACPM inSection 4-3.Theperformance characteristics ofMACPM havebeeninvestigated by Mulligan (1988)forbothuncoded andtrellis-coded CPM.Ofparticular interest istheresultthattrellis-coded CPMwithtwoamplitude levelsachieves again of3-4dBrelativetoMSKwithoutasignificant increaseinthesignalbandwidth. 5-3-3Symbol-by-Symbol Detedion ofCPMSignals BesidestheMLsequence detector, thereareothertypesofdetectors thatcan beusedtorecovertheinformation sequence inaCPMsignal.Inthissection, weconsider symbol-by-symbol detectors. Onetypeofsymbol-by-symbol detector istheonedescribed inSection5-1-5,whichexploits thememory of CPMbyperforming matched filtering orcross-correlation overseveral signaling intervals. Because ofitscomputational complexity, however, this recursive algorithm hasnotbeendirectlyappliedtothedetection ofCPM. Instead, twosimilar,albeitsuboptimal, symbol-by-symbol detection methods havebeendescribed inthepapersbydeBuda (1972),Osborne andLuntz (1974),andSchonhofl (1976).Oneoftheseisfunctionally equivalent tothe algorithm giveninSection5-1-5,andthesecondisasuboptimum approxima­ tionofthefirst.Weshalldescribe thesetwomethods inthecontextof demodulation ofCPFSKsignals,forwhichthesedetection algorithms have beenapplieddirectly. Todescribe thesemethods, weassumethatthesignalisobserved overthe presentsignaling intervalandDsignaling intervals intothefutureindeciding ontheinformation symboltransmitted inthepresentsignaling interval. A CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WmTE (iAUSSIA~ NOISE CHA~NEL 297 TABLE 5-3-1MAXIMUM VALUES OFTHEUPPER BOUND d~FORMULTI·h LINEAR PHASE CPM" H=4, ,,, M=2.H=IM=4. H'22 2JM=8. H=I4 o~o-------::':,-----------,',- h05 1.05dBpin compared MHMlxd~ wltbMSK h,h,h,h. 10 212.43 0.85 0.715 0.715 224.0 3.0 0.5 0.5 0.5 2 3 4.88 3.87 0.620 0.686 0.714 0.673 2 4 5.69 4.54 0.73 0.55 0.73 0.55 0.64 4I4.23 3.25 0.914 0.914 426.54 5.15 0.772 0.772 0.772 4 3 7.65 5.83 0.795 0.795 0.795 0.795 816.14 4.87 0.964 0.964 8 2 7.50 5.74 0.883 0.883 0.883 8 3 8.40 6.23 0.879 0.879 0.879 0.879 aFromAulinandSundberg (1982b). d'8oul)dpeaka• M=8,H=3 8 Boundpeak M=4.H=3· 0 M=8.H= J Boundpeak 7M=8.H=2 •Boundpeak. 6M=4.H=2 FIGURE 5-3-10 Upperbounds on minimum squared euclidean distance forvariousMand Hvalues.[FromAutinandSundberg (/9li2b).(f)J982IEEE.] 298 [JI(aTAL CO'!MUNKAT'ONS r(l)..I,,=-(M~ IH !"r()dt1-__...exp() " L__J I,,;M-IJ exp( )exp( ) "",I,,;-(M-Il! 1"'lldl1----.1 "1"'(ldro ",f,,=M-11 FIGURE 5-3-11 Blockdiagram ofdemodulator fordetection ofCPFSK. blockdiagramofthedemodulator, implemented asabankofcross-correlators, isshowninFig.5-3-11.Recallthatthetransmitted CPFSKsignalduringthe nthsignaling intervalis where {[lrh[1-(n-l)T)/ n-I ]} v(I)=expj T n+Irh~oI.+<1>0 h~2fdTisthemodulation index,fdisthepeakfrequericy deviation, and<1>0is theinitialphaseangleofthecarrier. Indetecting thesymbolI"thecross-correlations showninFig.5-3-11are performed withthereference signals 5(1.I,.I"...,II+D)forallMD+Ipossible valuesofthesymbols 1"12",,,11+0transmitted overtheD+1signaling intervals. Butthesecorrelations ineffectgenerate thevariables T"T2•...,T,+D' whichinturnarethearguments oftheexponentials thatoccurinthepdf p(r"T2•...•r,+0II,.12,...•11+0) Finally, thesummations overtheMOpossible valuesofthesymbols 12,13,.••,I,+0represent theaveraging of p(r"T2•...•T,+0II,.I,.....Il+o)P(I,. 12,...•11+0) CHAPTtR':; OPTIMU~ RECEIVERS FORTHEADDITIVE WHITE GAUSSIA ........OISt:CHA"'SfL 299 overtheMDpossible value,ofthesesymbols. TheMoutputs ofthe demodulator constitute thedecision variables fromwhichthelargestisselected toformthedemodulated symbol. Consequently themetricsgenerated bythe demodulator showninFig.5-3-11areequivalent tothedecision variables given by(5-1-68)onwhichthedecision onI,isbased. Signalsreceived insubsequent signaling intervals aredemodulated inthe samemanner. Thatis.thedemodulator cross-correlates thesignalreceived overD+Isignaling intervals withtheM0+Ipossible transmitted signalsand formsthedecision variables asillustrated inFig.5-3-11.Thusthedecision madeonthemthsignaling interval isbasedonthecross-correlations performed overthesignaling intervals m,m+I,....m+D.Theinitialphase inthecorrelation intervalofduration (D+l)Tisassumed tobeknown.On theotherhand.thealgorithm described by(5-1-76) and(5-1-77) involves an additional averaging operation overthepreviously detected symbols. Inthis respect, thedemodulator showninFig.5-3-11differsfromtherecursive algorithm described above.However, thedifference isinsignificant. OnesUboptimum demodulation methodthatperforms almostaswellasthe optimum method embodied inFig.5-3-11basesitsdecision onthelargest outputfromthebankofMO+Icros>-correlators. Thustheexponential functions andthesummations areeliminated. But t~ismethod isequivalent toselecting thesymbol 1mforwhichtheprobability densityfunction p(r"" '111+\•..••'m+D11m.1m+ I'·..,lm+v)isamaximum. Theperformance ofthedetector showninFig.5-3·11hasbeenupper­ bounded andevaluated numerically. Figure5-3-12illustrates theperformance ofbinaryCPFSK withn=D+1asaparameter. Themodulation index h=0.715usedingenerating theseresultsminimizes theprobability oferroras 4-6 ~101214 SNRperbit,l/>(d81, Orthogonal FSKI--......'-.." =I,h=05 f'\\.'\, ,\1\'\ \ f--CPFSK\l'vC,~~S::K 11=4 ,'"\h"071S }~"=O_7\5\ ~1\ I--I\ \ CPFSK j\ \I--II=~ ;;\I--h"i,715 \1\!n---J 10''o15.1 S FIGURE 5-3-U Performance ofbinaryCPFSKwithcoherent detection. 300 DIGITAL COMMtJNICATIOJ't'S 2 4 6 81012 14 SNRperbit,'Yb(dB). It-.. f\\Orthogoni l \\\quaternary FSK- \n=\.II=0.5 -, , 1\II I 1\CPFSK, }.'n=2.11=1.75 CPFSK -n=31\\-h=0.8\ \ \\ ,rl111)- o.-5 ~-10-2 ~ i:' ~2 .~10--·i2 10-' 5 2 11)-0 Performance ofquaternary CPFSKwithcoherent detection. FIGURE 5-3013 shownbySchonhoff(1976).Wenotethatanimprovement ofabout2.5dBis obtained relativetoorthogonal FSK(n=1)byademodulator thatcross­ correlates overtwosymbols. Anadditional gainofapproximately 1.5dBis obtained byextending thecorrelation timetothreesymbols. Furtherextension ofthecorrelation timeresultsinarelatively smalladditional gain. Similarresultsareobtained withlargeralphabet sizes.Forexample, Figs 5-3-13and5-3-14illustrate theperformance improvements forquaternary and 4 6 810 12 14 SNRperbil,Yh(dB)1 2., IOrthogonal ~\octalFSK 2n.=I.h=0.5 \\1 \1\ \ 2 CPFSK 3 .......n=2.11=08791\ . CPFSK\ I-"=3, f-II=0.879 , 1\1\10 0::5 ~ :810 E 5i:'='0 .~10-I: 1()-4 5 2 IIr0 tlGURE 5-3ot4 Performance ofoctalCPFSKwith'coherent detection. CHAPTER 5OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSiAN r-<OlSE CHANNEL 301 octalCPFSK,respectively. Themodulation indicesgiveninthesegraphsare theonesthatminimize theprobability ofasymbolerror. Insteadofperforming coherent detection, whichrequires knowledge ofthe carrierphase</10'wemayassumethat<Poisuniformly distributed overthe interval0to2n,andaverageoveritinarrivingatthedecision variables. Thus coherent integration (cross-correlation) isperformed overthen=D+1 signaling intervals. buttheoutputsofthecorrelators areenvelope-detected. Thisiscallednoncoherent detection of CPFSK. Inthisdetection scheme, performance Isoptimized byselecting ntobeoddandmakingthedecision on themiddlesymbolinthesequence ofnsymbols. Thenumerical resultsonthe probability oferrorfornoncoherent detection ofCPFSKaresimilartothe resultsillustrated aboveforcoherent detection, Thatis,againof2-3dBin performance isachieved byincreasing thecorrelation interval fromn=Ito n=3andton=5. 5-4OPTIMUM RECEIVER FORSIGNALS WITH RANDOM PHASE INAWGNCHANNEL Inthissection, weconsider thedesignoftheoptimum receiver forcarrier modulated signalswhenthecarrierphaseisunknown atthereceiver andno attempt ismadetoestimate itsvalue.Uncertainty inthecarrierphaseofthe received signalmaybeduetooneormoreofthefollowing reasons: First,the oscillators thatareusedatthetransmitter andthereceiver togenerate thecarriersignalsaregenerally notphasesynchronous. Second,thetimedelav inthepropagation ofthesignalfromthetransmitter tothereceiver isn~t generally knownprecisely. Toelaborate onthispoint,atransmitted signalof theform thatpropagates through achannelwithdelayrowillbereceived as s(t-til)=Re[g(r-to)eJ2~r.«-''''1 =Re[g(t-Io)e J2~JJ"eJ2rrl'J Thecarrierphaseshiftduetothepropagation delaytois </J= -2nj,lo Notethatlargechanges inthecarrierphase </Jcanoccurduetorelatively small changes inthepropagation delay.Forexample, ifthecarrierfrequency j,=IMHz,anuncertainty orachangeinthepropagation delayof0.5J.I.swill causeaphaseuncertainity ofnrad.Insomechannels (e.g.,radiochannels) the timedelayinthepropagation ofthesignalfromthetransmitter tothereceiver 302 DIGITAL COMMUNICATIONS maychangerapidlyandinanapparently randommanner, sothatthecarrier phaseofthereceived signalvariesinanapparently randomfashion. Intheabsenceofknowledge ofthecarrierphase,wemaytreatthissignal parameter asarandom variable anddetermine theformoftheoptimum receiverforrecovering thetransmitted information fromthereceived signal. First,wetreatthecaseofbinarysignalsand,then,weconsider M-arysignals. 5-4-1Optimum Receiver forBinarySignals Weconsider abinarycommunication systemthatusesthetwocarrier modulated signalsSI(t)andS2(t)totransmit theinformation, where (5-4-1) ands/",(I),in=1,2aretheequivalent lowpasssignals.Thetwosignalsare assumed tohaveequalenergy iT 1ITl\'=S;,,(t)dr= -IS/m(r)12dr02 0 andarecharacterized bythecomplex-valued correlation coefficient lITPI2'"P=i0SMI)sI2(t) dr(5-4-2) (5-4-3) Thereceived signalisassumed tobeaphase-shifted versionofthe transmitted signalandcorrupted bytheadditivenoise n(t)=Re([nc(t)+jn,(t)]e'2"'''} =Re[z(t)ei2"'1 Hence,thereceived signalmaybeexpressed as r(t)=Re([SIm(t)el'"+z(t)]el2"'1 where r,(r)=SIm(t)ei'"+z(t),0""t,.:;T(5-4-4) (5-4-5) (5-4-6) istheequivalent lowpassreceived signal.Thisreceived signalisnowpassed through ademodulator whosesampled outputatt=Tispassedtothe detector. TheOptimum Demodulator InSection5-1-1,wedemonstrated thatifthe received signalwascorrelated withasetoforthonormal functions {!.(t)}that CH!\PTER 5:OPTI~1l:M RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISE CHANNEL 303 spanned thesignalspace,theoutputsfromthebankofcorreiatorsprovidea setofsufficient statistics forthedetector tomakeadecisionthatminimizes the probability oferror.Wealsodemonstrated thatabankofmatched filterscould besubstituted forthebankofcorrelators. Asimilarorthonormal decomposition canalsobeemployed forareceived signalwithanunknown carrierphase.However, itismathematically con­ venienttodealwiththeequivalent lowpasssignalandtospecifythesignal correiatorsormatched filtersintermsoftheequivalent lowpass signal waveforms. Tobespecific,theimpulseresponse h,lt)ofafilterthatismatched tothe complex-valued equivalent lowpass signals,(t),0,,;;t,,;;T,isgivenas(see Problem 5-6) h,(t)=5,*(T-I) andtheoutputofsuchafilteratI=Tissimply iT15,(1)12dl=2g II(5-4-7) (5-4-8) whereIEisthesignalenergy.Asimilarresultisobtained ifthesignal5,(1)is correlated withst(l)andthecorrelator issampled att=T.Therefore, the optimum demodulator fortheequivalent lowpassreceived signal5,(1)givenin (5-4-6)mayberealizedbytwomatched filtersinparallel,onematched toSI1(t) andtheotherto512(1),andshowninFig.5-4-1.Theoutputofthematched filtersDrcorreiatorsatthesampling instantarethetwocomplex numbers Suppose thatthetransmitted signalis5,(1).Then,itiseasilyshown(see Problem 5-35)that I " =2lCcos<f>+n"+j(U,SIn</>+IIi') " =nlplcos(</> +aO)+n2'+j[2l1'lplsin(</> +ao)+nc,] Filttr matched 10 ,\'/I{ll Dt'ledor Fillrr.(5-4-9) (5-4-10) Output dt'\isinn FI{;URF. 5-4-1Optimum n:ceivl..'r forhinarysignals. 304 DIGITAL COMMUNICATIONS wherepisthecomplex-valued correlation coefficient ofthetwosignalsS/I(I) andsdt),whichmaybeexpresse'd asp=Iplexp(ao).Therandom noise variables n,on,5'n2"andn2.<arejointlygaussian, withzeromeanandequal variance. TheOptimum Detector Theoptimum detector observes therandom variables [ric'"r2c'''']=r,where"='k+i'",and'2='2,·+i'""andbasesits decision ontheposterior probabilities P(5.,Ir),m=1,2.Theseprobabilities maybeexpressed as perI5.,)P(s",) per)m=1,2 (5-4-11) and,hence,theoptimurndecisiol\ rulemaybeexpressed as <, pes,Irl)'"P(s,Ir) " or,equivalently, ~~P(s,) perI5,)$,pes,)(5-4-12) TheratioofpdfsontheLeft-hand sideof(5-4-12) isthelikelihood rario,which wedenoteas A(r)=per[5,) p(rls,)(5-4-13) Theright-hand sideof(5-4-12)istheratioofthetwopriorprobabilities, which takesthevalueofunitywhenthetwosignalsareequallyprobable. Theprobability densityfunctions perI5,)andperIs,)canbeobtained by averaging thepdfsperI5n;,<1»overthepdfoftherandomcarrierphase,i.e., f,2" perISm)=perISm,<I»p(<I»d<l> ()(5-4-14) Weshallperform theintegration indicated in(5-4-14)forthespecialcasein whichthetwosignalsareorthogonal, i.e.,p=O.Inthiscase,theoutputsofthe demodulator are =21:cos<I>+n"+j(2~sin <I>+n,,) '2='It·+jr'b =n2c+jn2~(5-4-15) CHAPTER 5:OPTI\tl'\1 RECEIVERS FORTHEADDITIVE WHITEGAUSSIAN NOiSECHANl"El 30S where(nI,·'n".n,,,fl,,jaremutually uncorrelated and,hence,statistically independent. zero-mean gaussian randomvariables (seeProblem 5-25).Hence. thejointpdfofr=[r".r"r".'2,)maybeexpressed asaproduct ofthe marginal pdfs.Consequently. (5-4-16) whereu'=2'fN.,. Theuniform pdfforthecarrierphaset/Jrepresents themostignorance that canbeexhibited bythedetector. Thisiscalledtheleastfavorable pdffort/J. Withp(t/J)=1/21C.0.,;t/J.,;2Jr.substituted intotheintegral in(5-4-14). we obtain 1f" 2-p(r",r"Is,.t/J)dt/J IC" 1(ri,+ri,+4't:') 1J[2't:(rkcost/J+,,,sint/J)]=-exp- ,-exp dt/J2lt 2u- 2Jr u2 But-lI2 /r[2,€(r".cost/J+r\,sint/J)] _(2'€v'r1.+ rl,) 2exp , dt/J-I" 2 H(I U a(5-4-17) (5-4-18) wherelo(.r)isthemodified Besselfunction ofzerothorder.defined in (2-1-120). Byperforming asimilarintegration asin(5-4-17)undertheassumption that thesignalS2(t)wastransmitted. weobtaintheresult (5-4-19) When,wesubstitute theseresultsintothelikelihood ratiogivenby(5-4-13). weobtaintheresult (5-4-20) Thus,theoptimum detector computes thetwoenvelopesv'ri,+rioand Vr~,+,~,andthe corres"ondin~ values oftheBessel function /n(2~"c +d'/u')andlo(2l!Vrt+ r~/(2)toformthelikelihood ratio.We observe thatthiscomputation requires knowledge ofthenoisevariance (f'. 306 DIGITAL COMMUNICATIONS 5 4.5 4 3.5_3 ~2.5 --215 1 O.5 '---i~-7'--:;---.;';o.;--;;,;o-C.o0.511.522.533.54 FIGURE 5·4-2Graphofi,,(x). Thelikelihood ratio i~thencompared withthethreshold P(s,)1P(s,)to determine whichsignalwastransmitted. Asignificant simplification intheimplementation oftheoptimum detector occurswhenthetwosignalsareequallyprobable. Insuchacasethethreshold becomes unity.and,duetothemonotonicity oftheBesselfunction shownin Fig.5-4-2,theoptimum detection rulesimplifies to (5-4-21) Thus,theoptimum detector basesitsdecision onthetwoenvelopes Vri.+rL andYd,+ri,.and,hence,itiscalledanenvelope detector. Weobserve thatthecomputation oftheenvelopes ofthereceived signal samplesattheoutputofthedemodulator rendersthecarrierphaseirrelevant inthedecision astowhichsignalwastransmitted. Equivalently, thedecision maybebasedonthecomputation ofthesquared envelopes r~,+rT,and r},+rLinwhichcasethedetector iscalledasquare-law detector. BinaryFSKsignalsareanexample ofbinaryorthogonal signals.Recallthat inbinaryFSKweemploytwodifferent frequencies, sayItandfi=f,+:J.f,to transmit abinaryinformation sequence. Thechoiceofminimum frequency separation !::>f=f,-f,isconsidered below.Thus,thesignalwaveforms maybe expressed as s,(I)=Y2'({b1Tb cos2trf,t, 0~t~Th s,(t)=Y2'({blThcos2trht. 0~t,,;To andtheirequivalent lowpasscounterparts are Thereceived signalmaybeexpressed as f!!g'br(t)=-cos(2trf",t+r/>m)+net)Tb(5-4-22) (5-4-23) (5-4-24) CHAPTER s:OPTIMCM RECEIVERS FORTHEADDITIVE WHITE (jr\tISSIA~ ;-';Ol~l'CH,-\'\"'IH 307 cos~'It.!.' co~21((.t:,+~f)' 1-_.-'1~ L.::-_..J : ~in21t((+!:.f)1 : ,Received signal t S-Jmple 1=T FIGURE 5·~3 Demodulation andsquare-law detection ofhinaryFSKsignals. (5-4-25)where"''0isthephaseofthecarrierfrequency J,,,.Thedemodulation ofthe realsignalr(l)maybeaccomplished, asshowninFig.5-4-3.hyusingfour correlators withthebasisfunctions flm(r)=~:"cos[(2irfl+2irmi.f)!],m=0,1 f2m(r)=/2sin[(2irfl+2mllin!], m =0,I'JT" Thefouroutputs ofthecorrelators aresampled attheendofeachsignal intervalandpassedtothedetector.Ifthemthsignalistransmitted. thefour samplesatthedetector maybeexpressed as r=n[Sin[2ir(k -m)ifT] .,"2ir(k-m)DofTcos<Pm cos[2ir(k-m)ifT]-I, ] 2ir(k_m)t1fTSIn<Pm+11."k,m=\,2 (5-4-26) r=VCi:[COS2n(k-m)DofT-1 h "2n(k-m)DofT cos"'", +sin[2n(k-m)DofT], ] 2n(k-m)DofTSIn<Pm+n•."k.m=I,2 wheren.,andn•.,denotethegaussian noisecomponents inthesampled outputs, (5-4-27)30lI DIGITAL COMMUNICATIONS Weobservetbatwhenk=m,tbesampled valuestotbedetector are rm.=~cos"'m+nm• 'ms=~sin"'m+n"" Furthermore, weobserve thatwhenk".m,thesignalcomponents inthe samplesr••andrluwillvanish,independently ofthevaluesofthephaseshifts </I.,provided thatthefrequency separation between successive frequencies is af=11T.Insuchacase,theothertwocorrelator outputsconsistofnoise only,i.e., r••=n'nrlu=nlu.k".m (5-4-28) Withafrequency separation ofI1f=lIT,therelations (5-4-27) and(5-4-28) areconsistent withtheprevious result(5-4-15) forthedemodulator outputs. Therefore, weconclude thatforenvelope orsquare-law detection ofFSK signals,theminimum frequency separation required fororthogonality ofthe signalsisI1f=lIT.Thisseparation istwiceaslargeasthatrequired whenthe detection isphase-coherent. 5-4-2Optimum Receiver forM-aryOrthogonlll Sigallls Thegeneralization oftheoptimum demodulator anddetector tothecaseof M-aryorthogonal signalsisstraightforward. Iftheequalenergyandequally probable signalwaveforms arerepresented as sm(t)=Re[Slm(I)ei2nf,,), m=1,2,...,M,0",t0;;;T(5-4-29) wheres'm(t)aretheequivalent lowpasssignals,theoptimum correlation-type ormatched-filter-type demodulator produces theMcomplex-valu~d random variables rm=rm.+jrm•=rr,(t)st.(t) dt,m=1,2,...,M(5-4-30) wherer,(t)istheequivalent lowpass received signal.Then,theoptimum detector, basedonarandom, uniformly distributed carrierphaSe,computes the Menvelopes Irml="'?m.+?.....,m=1,2,...,M (5-4-31) or,equivalently, thesquared envelopesIrml2,andselectsthesignalwiththe largestenvelope (orsquaredenvelope). InthespecialcaseofM-aryorthogonal FSKsignals,theoptimum receiver hasthestructure illustrated inFig.5-4-4.Thereare2Mcorreiators:twofor eachpossible transmitted frequency. Theminimum frequency separation between adjacent frequencies tomaintain orthogonality is/;,f=1/T. 5-4-3Probllbility ofErrorforEnvelope Detectio. ofM-ary Orthogonal Signals Letusconsider thetransmission ofM-aryorthogonal equalenergysignalsover anAWGNchannel, whichareenvelope-detected atthereceiver. Wealso CHAPTER S:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOiSECHANNEL 309 COS2K!.,1 X j~()d, sin2ltf ...t x l~()dI COS2l't(j;.+tJ.f)/ X l~()d, sin2l't<J;_+Af)/ Received I~()d, x signal coslxlf,.+(M -1)M]r ,---{ x I~()dI1--_---- sin2Jt{fc+(rW-l)L1fl/Envelope orOutput square-law deeision deteclor FIGURE 5-4-4Demodulation ofM-aryFSKsignalsfornoncoherent detection. assumethattheMsignalsareequallyprobable aprioriandthatthesignals,(I) istransmitted inthesignalinterval0".;I"';T. TheMdecisionmetricsatthedetector aretheMenvelopes m=1,2,...,M (5-4-32) where andric=n,cos<1>,+n,,. I"=n,sin <1>,+n"(5-4-33) m=2,3,...,M (5-4-34) (5-4-35)Theadditive noisecomponents {nmcland{nmJaremutually statistically independent zero-mean gaussian variables withequalvariance u2=!N".Thus thepdfsoftherandomvariables attheinputtothedetector are (__1_(ri,+,f,+,€,-)(v'~,(rt+rLl) P.,.1'0r,,)-22e"p-22In 2 TeU U U m=2,3,...,M(5-4-36) 310 DIGITAL COMMUN)CATIONS Letusmakeachangeinvariables inthejointpdfsgivenby(5-4-35)and (5-4-36). Wedefinethenormalized variables Vr~c+r~s (5-4-37) Clearly, 'mo-=aRmcosemand'ms=uRmsinem'TheJacobian ofthistransfor­ mationis JIucosem usin8mI2R11=. =u m-uRmsmemuRmcosem Consequently, R,[I(22%'s)](rn:::)p(R"e')=2JrexP-2R,+No10Yli;,R" p(Rm,em)=~;exp(-!R;,), m=2,3,...,/If(5-4-38) (5-4-39) (5-4-40) Finally,byaveraging p(Rm,em)overem,thefactorofZiriseliminated from (5-4-39)and(5-4-40). Thus,wefindthatR,hasaRiceprobability distribution andRm,m=Z,3,__.,M,areeachRayleigh-distributed. Theprobability ofacorrectdecision issimplytheprobability thatRI>R2, andR,>R3,•_• ,andRI>Rm•Hence, Pc=P(Rz<R" R3<R"..-,RM<R,) =rP(Rz<RioR3<RI,·..,RM<R,IR,=X)PR,{X) dx(5·4-41) Because therandomvariables Rm,m=2,3,__...111,arestatistically iid,the jointprobability in(5-4-41)conditioned onRIfactorsintoaproductof.111-1 identical terms.Thus, where P(Rz<RIIR,=x)=rPR,('Z)drz =1 -e-xl/z The(M-l)thpowerof(5-4-43)maybeexpressed as (l_e-X212)M-' =Y'(_I)n(M -1)e-"-<'12 "=0 n(5-4-42) (5-4-43) (5-4-44) CHAPTER~: OPTI'-1l'\I1 R.ECEIVERS FORTHEADDITIVE WHITEGAU:\SIAN I\OISECHA:"JSEL 311 Substitution ofthisresultinto(5-4-42) andintegration overxyieldsthe probability ofacorrectdecision as (5-4-45) Pc=Y(-It(M-1)_1_exp[_n'(g,] n~o,nn+1 (n+1)N" where'l:,/N"istheSNRpersymbol. Then.theprobability ofasymbolerror, whichisPM=1-P,.becomes (5-4-46)'.1,1(M-l' 1 [nk'l:,,]PM=2:(-1)'"1 l-~exp--.-:-:~~n-' .n,n+l (n+l)lv" where'6,,1NoistheSNRperbit. Forbinaryorthogonal signals(M=2).(5-4-46)reducestothesimpleform (5-4-47) ForM>2.wemaycompute theprobability ofabiterrorbymakinguseof therelationship 2k-1 P,.=2'_1Pw (5-4-48) whichwasestablished inSection 5·2.Figure5-4-5showsthebit-error probability asafunction oftheSNRperbity"forM=2.4.8.16.and32.Just asinthecaseofcoherent detection ofM-aryorthogonal signals(seeSection 5·2-2),weobserve thatforanygivenbit-error probability. theSNRperbit decreases asMincreases. ItwillbeshowninChapter 7that.inthelimitas M->x(ork=log,M->x).theprobability ofabiterrorP"canbemade .,0 4 SIII S:\Rperbil.y~(JBl, ::....'"5~:;::'\,.\,·,H-2 ,~\\\ \s'\\\,, 2\\1\, J\1\"1='\, s·F\'~ ·.\1=10l-- 2\ \,•,\I~1\I-\-\-5I\1\ ~ M=~ ~~Channd cJpa..:it~I"IT,hmitl-loJBf\\s , ..'of=J2, I \,,."10 IIIIII ,: E"III E "C-. EIII .g :t III FIGURE 5-4·5 Prohahilily ofahit~rrorfornoncoher~nt detection (Jf orthogonal signal"i. (5-4-49)312m(;JTAL COMMUNICATIONS arbitrarily smallprovided that'theSNRperbit"is.greaferthantheShannon limitof-1:6dB.Thecost.(orinc~e8sing Misthebandwidth required to transmit thesignals.ForNt-aryFSK,thefrequency separation between adjacent frequencies isAf=1ITforsignalorthogonality. Thebandwidth required fortheMsignalsisW=M!If=MIT.AlSO,thebitrateisR=kIT, wherek=log,M.Therefore, thebit-rate-to-bandwidth ratiois R=10g,M W M 5-4-4Probability ofErrorforEnvelope Detection ofCorrelated BinarySignals Inthissection, we.consider theperformance oftheenvelope detector for binary,equal-energy correlated signals.Whenthetwosignalsarecorrelated, theinputtothedetector arethecomplex-valued randomvariables givenby (5-4-10). Weassumethatthedetector basesitsdecisionontheenvelopesI'd and1',1.whicharecorrelated (statistically dependent). Themarginal pdfsof R,=IrdandR,=1',1areRiceandistributed, andmaybeexpressed as (5-4-50) m=1,2.where(3,=2~and(3,=2~Ipl.basedontheassumption thatsignal s,(t)wastransmitted. SinceR,andR,arestatistically dependent asaconsequence ofthe nonorthogonality ofthesignals,theprobability oferrormaybeobtained by evaluating thedoubleintegral Pb=P(R,>R,)=rrp(x"x,)tit,dx, ox,(5-4-51) wherep(x,.x,)isthejointpdfoftheenvelopes R,andR,.Thisapproach was firstusedbyHelstrom (1955),whodetermined thejointpdfofR,andR,and evaluated thedoubleintegralin(5-4-51). Analternative approach isbasedontheobservation thattheprobability of errormayalsobeexpressed as Pb=P(R2>R,)=P(R~>Rl)=P(R~-R~>0) (5-4-52) ButR~-R~isaspecialcaseofageneralquadratic formincomplex-valued gaussian randomvariables, treatedlaterinAppendix B.Forthespecialcase underconsideration, thederivation yieldstheerrorprobability intheform Pb=Q,(a.b)-!e-(Q'+b'I12lo(ab) (5-4-53) CHAPTER 5~Or-TIMl'M RECEIVERS FORTHEADDITIVE WHITE GAl'SSIA~ ~[)ISE CHA,\;:,\U 313 FIGURE 5-4-6 Prohahility oferrorfornoncohercnt delection. whereJO-I 10~l:Iq10 1112U1415If) l~Ii) SNRperhi!.Y/,hIRl (5-4-54)a=Ifh(I-VI_Ipl') \/2N II h=I'f"(I+VI_Ip12) \/2NII Q,(a,b)istheQfunction definedin(2-1-123) andlll(x)isthemodified Bessel function oforderzero. Theerrorprobability P"isillustrated inFig.5-4-6forseveralvaluesofIpl. Phisminimized V'henp=0;thatis,whenthesignalsareorthogonal. Forthis case,a=0,h='€h/N...and(5-4-53)reducesto P.=Q(O~'lh)-le1,,1"''',, (5-4-55) h'" '1'" Fromthedefinition ofQ,(a,b)in(2-1-123), itfollowsthat Q,(O.~~)=e"f2N" Substitution oftheserelations into (5-4-55) yieldsthedesiredresullgiven previously in(5-4-47). Ontheotherhand,whenIpl=I.theerrorprobability in (5-4-53)becomes Ph=!.asexpected. 5-5REGENERATIVE REPEATERS ANDLINK BUDGET ANALYSIS Inthetransmission ofdigitalsignalsthrough anAWGNchannel. wehave observed thattheperformance ofthecommunication system,measured in termsoftheprobability oferror,depends solelyonthereceived SNR,'f:h/No, 314 DIGITAL COMMUNICATIONS Receiyed !;Ignal r(r)=as(,)+n(tl Noist nlf)Attenuation lXTrasmitted,...---------, Channel ~ignal tlGURE 5-5-1Mathematical modelofchannelwithatteouation andadditive noise. where'l!histhetransmitted energyperbitand1Nr,isthepowerspectraldensity oftheadditive noise.Hence, theadditive noiseultimately limitsthe performance 9fthecommunication system. Inaddition totheadditivenoise,anotherfactorthataffectstheperformance ofacommunication systemischannel attenuation. Allphysical channels, including wirelinesandradiochannels, arelossy.Hence,thesignal'is attenuated asittravelsthrough thechannel. Thesimplemathematical model fortheattenuation showninFig.5-5-1maybeusedforthechannel. Consequently, ifthetransmitted signaliss(t),thereceived signal,with 0<a'"Iis r(l)=as(t)+n(t) (5-5-1) Then,iftheenergyinthetransmitted signalis'f,h'theenergyinthereceived signalisa2l:h'Consequently, thereceived signalhasanSNRa2'f,h/NoHence, theeffectofsignalattenuation istoreducetheenergyinthereceived signal andthustorenderthecommunication systemmorevulnerable toadditive noise. Inanalogcommunication systems, amplifiers calledrepeaters areusedto periodically boostthesignalstrength intransmission through thechannel. However, eachamplifier alsobooststhenoiseinthesystem.Incontrast,digital communication systemsallowustodetectandregenerate aclean(noise-free) signalinatransmission channel. Suchdevices,calledregenerative repeaters, are frequently usedinwirelineandfiberopticcommunication channels. 5-5-1Regenerative Repeaters Thefrontendofeachregenerative repeater consistsofademodulator/detector lhatdemodulates anddetectsthetransmitted digitalinformation sequence sent bythepreceding repeater. Oncedetected, thesequence ispassedtothe transmitter sideoftherepeater, whichmapsthesequence intosignal waveforms thataretransmitted tothenexlrepeater. Thistypeofrepeater is calledaregenerative repeater. Sinceanoise-free signalisregenerated ateachrepeater, theadditive noise doesnotaccumulate. However, whenerrorsoccurinthedetector ofa repeater, theerrorsarepropagated forward tothefollowing repeaters in"the chal)nel. Toevaluate theeffectoferrorsontheperformance oftheoverall system,supposethatthemodulation isbinaryPAM,sothattheprobability of (5-5-2) (5-5-3)UfAPffl<.) ()Pll~H'\IIH-np.rRS FORTHEADOrJl\f wHITE (jA,U'iS1A1' :\OlSI'''H.\''''FI 3]5 ahiterrorforonehop(signaltransmission fromonerepeale'r totheileXl repeater inthechain)is Sinceerrorsoccurwithlowprobability, wemayignoretheprobability thatany onehitwillbedetected incorrectly morethanonceintransmission through a channel withKrepeaters. Consequently, thenumber oferrorswillinClease linearly withthenumber ofregenerative repeaters inthechannel. and therefore. theoverallprobability oferrormaybeapproximated as,-. (;2'f,b) Pb=KQ\jNo Incontrast, IheliseofKanalogrepeaters inthechannel reduces thereceived SNRbyK.andhence.thebiterrorprobability is (rii:)Pb=Q\/~ Clearly, forthesameprobability oferrorperformance. theuseofregenerallve repeaters resultsinasignificant savingintransmitter po'Vercompared with analog repeaters. Hence, indigitalcommunication systems. regenerative repeaters arepreferable. However, inwireline telephone channels thatare usedtotransmit bothanaloganddigitalsignals.analoj!repeaters aregenerally employed. Example 5-5-1 Abinarydigitalcommunication system transmits dataoverawireline channel oflength\000km.Repeaters areusedevery10kmtooffsetthe effectofchannel allenuation. Letusdetermine the'l:b/N"thatisrequired to achieve aprobability ofabiterrorof105if(a)analogrepeaters are employed, and(b)regenerative repeaters areemployed. Thenumber ofrepeaters usedinthesystem isK=J(Xl.Ifregenerative repeaters areused.thet:b/N"obtained from(5-5-2)is [()5=lOOQ(~)\jN:; 107=Q(!ff:) whichyieldsapproximately 11.3dB.Ifanalogrepeaters areused.the't,J'Vo obtained from(5-5-3)is 105=Q(IUb)'V100No whichyields ~b/No=29.6dB.Hence.thedifference intherequired SNRis (5-5-4) (5-5-5)316 DIGITAL CO....UNICATIONS ,'..',._-......'" I, --.... \,'.. "'-'t , , III..,~,.. I I \ I ' \I..Antenna, , \"........_......',.......................... " FlGURE 5-5-2lsotropically radialing antenna. about18.3dB,orapproximately 70timesthetransmitter powerofthe digitalcommunication system. 5-5-2Communication LinkBudgetAnalysis Inthedesignofradiocommunications systemsthattransmit overline-aI-sight microwave channels andsatellitechannels, thesystemdesigner mustspecify thesizeofthetransmit andreceiveantennas, thetransmitted power,andthe SNRrequired toachie,veagivenlevelofperformance atsomedesireddata rate.Thesystemdesignprocedure isrelatively straightforward andisoutlined below. Letusbeginwithatransmitantennathatradiatesisotropically infreespace atapowerlevelofPrwattsasshowninFig.5-5-2.Thepowerdensityata distance dfromtheantenna isPr/4Jrd2W1m2,Ifthetransmitting antennahas somedirectivity inaparticular direction, thepowerdensityinthatdirection is increased byafactorcalledtheantennagainanddenoted byGr.Insuch.a case,thepowerdensityatdistancedisPrGrl4Jrd2W1m2,TheproductPrGris usuallycalledtheeffectiveradiatedpower(ERPorElRP),whichisbasically theradiated powerrelativetoanisotropic antenna, forwhichGr=1. Areceiving antennapointedinthedirection oftheradiated powergathersa portionofthepowerthatisproportional toitscross-sectional area.Hence,the received powerextracted bytheantennamaybeexpressed as p_PrGrA R R-41Cd2 whereARistheeffective areaoftheantenna. Fromelectromagnetic field theory,weobtainthebasicrelationship between thegainGRofanantennaand itseffective areaas GRA22AR=--m 411" whereA=c/fisthewavelength ofthetransmitted signal,cisthespeedoflighl (3xIOSm/s),andfisthefrequency ofthetransmitted signal. Ifwesubstitute (5-5-5)forARillto(5·5-4),weobtainanexpression forthe received powerintheform (5-5-6) CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITEGAUSSIAN NOISECHANNEL 317 Thefactor.,\')2 L,=(4/td(5-5-7) iscalledthefree-space pathloss_Ifotherlosses,suchasatmospheric losses,are encountered inthetransmission ofthesignal,theymaybeaccounted forby introducing anadditional lossfactor,sayLa.Therefore, thereceived power maybewritteningeneralas (5-5-8) Asindicated above,theimportant characteristics ofanantennaareitsgain anditseffective area.Thesegenerally dependonthewavelength ofthe radiated powerandthephysical dimensions oftheantenna. Forexample. a parabolic (dish)antennaofdiameter Dhasaneffective area (5-5-9) wherelTrD'isthephysical areaand'IIistheillumination efficiency factor, whichfallsintherange0.5,;;;;'II,;;;;0.6.Hence,theantenna gainforaparabolic antennaofdiameter Dis (5-5-10) Asasecondexample, ahornantenna ofphysical areaAhasanefficiency factorof0.8,aneffective areaofAR=0.8A,andanantennagainof lOAG-­R-,\2 (5-5-11) Another parameter thatisrelatedtothegain(directivity) ofanantennais itsbeamwidth, whichwedenoteas0BandwhichisIllustrated graphically in Fig.5-5-3.Usually, thebeamwidth ismeasured asthe-3dBwidthofthe FlGURE 5-5-3Antenna beamwidth andpattern. Transmitter ((I)8camwidlh ofantenna..L::"::::"'-~~_...J..._-L...~---=>C::::>" ...e I 0 I-laB laB (b)Anlenn; pattern 318 DIGITAL COllotMllNIl'ATIONS antenna pattern.Forexample, the-3dBbeamwidth ofaparabolic antennais approximately 88=70(A/Dt (5-5-12) sothatGTisinversely proportional toe~.Thatis,adecrease ofthebeamwidth byafactoroftwo,whichisobtained bydoubling thediameter D.increases the antenna gainbyafactoroffour(6dB). Basedonthegeneralrelationship forthereceived signalpowergivenby (5-5-8),thesystemdesigner cancompute PRfromaspecification oftheantenna gainsandthedistance between thetransmitter andthereceiver. Such computations areusuallydoneonapowerbasis,sothat (5-5-13) (5-5-15)Example 5-5-2 Suppose thatwehaveasatellite ingeosynchronous orbit(36000kmabove theearth'ssurface) thatradiates 100Wofpower,i.e.,20dBaboveIW (20dBW).Thetransmit antenna hasagainof17dB,sothattheERP= 37dBW.Also,suppose thattheearthstationemploys a 3 mparabolic antenna andthatthedownlink isoperating atafrequency of4GHz.The efficiency factoris1)=0.5.Bysubstituting thesenumbers into(5·5-10), we obtainthevalueoftheantennagainas39dB.Thefree-space pathlossis L,=195.6dB Nootherlossesareassumed. Therefore, thereceived signalpoweris (PR)dB=20+17+39-195.6 =-119.6dBW or,equivalently, PR=1.1XlO-12W Tocomplete thelinkbudgetcomputation, wemustalsoconsider theeffect oftheadditivenoiseatthereceiverfrontend.Thermal noisethatarisesatthe receiver frontendhasarelatively fiatpowerdensityspectrum uptoabout 1012Hz,andisgivenas No=k8ToWlHz (5-5-14) wherek8isBoltzmann's constant (1.38x10-23Ws/K)andToisthenoise temperature inKelvin. Therefore, thetotalnoisepowerinthesignal bandwidth WisNoW. Theperformance ofthedigitalcommunications systemisspecified bythe ~b/Norequired tokeeptheerrorrateperformance belowsomegivenvalue. Since • ~bThPR1PR -=--=-- NoNoRNa (5-5-16)CHAPTER): OPTIMrM RECEIVERS FORTHEADDITIVE WHITE GAUSSIAN NOISECHANNEL 319 itfollowsthat PR('f:h)--R- No No<eq where('[;hlNo),eqistherequired SNRperbit.Hence,ifwehavePRINoandthe required SNRperbit,wecandetermine themaximum dataratethatis possible. Example 5-5-3 Forthelinkconsidered inExample 5·5-2,thereceived signalpoweris PR=1.lXlO'12W(-119.6dBW) Now,suppose thereceiver frontendhasanoisetemperature of300K, whichistypicalforreceiver inthe4GHzrange.Then No=4.1X10-21W/Hz or.equivalently, ~203.9dBW/Hz.Therefore. PR-= ~1l9.6+ 203.9=84.3dB Hz No Iftherequired SNRperbitis10dBthen,from(5-5-16). wehavethe available rateas RdB=84.3-10 =74.3dB(withrespectto1bit/s) Thiscorresponds toarateof26.9megabits/so whichisequivalent toabout 420peMchannels. eachoperating at64000bits/so Itisagoodideatointroduce somesafetymargin, whichweshallcallthe linkmarginMdS,intheabovecomputations forthecapacity ofthecom­ munication link.Typically. thismaybeselected asMdB=6dB.Then.thelink budgetcomputation forthelinkcapacity maybeexpressed inthesimpleform (PR)((I;h) RdH=NodBHI:-Non.."....-MdB =(P,),lIlW+(G')dB+(GR)dB +(LaldH+(L')dH-C::)-M"" or~q BIBLIOGRAPHICAL NOTES ANDREFERENCES Inthederivation oftheoptimum demodulator forasignalcorrupted t>y AWGN,weapplied mathematical techniques thatwereoriginallv usedin deriving optimum receiver slructures forradarsignals. Forexample. the PROBLEMS320 DIGITAL COMMl."NICATIONS matched tilterwastirstproposed byNorth(1943)foruseinradardetection, andissometimes calledtheNorthtilter.Analternative methodforderiving theoptimum demodulator anddetector istheKarhunen- Loeveexpansion, whichisdescribed intheclassical textsbyDavenport andRoot(1958), Helstrom (1968),andVanTrees(1968).Itsuseinradardetection theoryis described inthepaperbyKellyelal.(1960).Thesedetection methods are basedonthehypothesis testingmethods developed bystatisticians, e.g., Neyman andPearson(1933)andWald(1947). Thegeometric approach tosignaldesignanddetection, whichwaspresented inthecontextofdigitalmodulation andwhichhasitsrootsinShannon's original work,isconceptually appealing andisnowwidelyusedsinceits introduction inthete"tbyWozencraft and.Jacobs (1965). Designandanalysisofsignalconstellations fortheAWGNchannel have received considerable attention inthetechnical literature. Ofparticular significance istheperformance analysis oftwo-dimensional lQAM) signal constellations thathasbeentreatedinthepapersofCahn(1960),Hancock and Lucky(1960),Campopiano andGlazer(1962),LuckyandHancock (1962), Salzelal.(1971),SimonandSmith(1973),Thomas efal.(1974),andFoschini elat.(1974).Signaldesignbasedonmultidimensional signalconstellations has beendescribed andanalyzed inthepaperbyGershoandLawrence (1984). TheViterbialgorithm wasdevisedbyViterbi(J967)forthepurpose of decoding convolutional codes.Itsuseastheoptimal maximum·likelihood sequence detection algorithm forsignalswithmemory wasdescribed byForney (1972)andOmura(1971).Itsuseforcarriermodulated signalswasconsidered byUngerboeck (1974)andMacKenchnie (1973).Itwassubsequently applied tothedemodulation ofCPMbyAulinandSundberg (198Ia,b)andothers. 5-1Amatchedfilterhasthefrequency response HU)1-e-f1.1if7" j2lff •Determine theimpUlseresponse h(l)corresponding toH(f). bDetermine thesignalwaveform towhichthefiltercharacteristic ismatched. 5-2Consider thesignal S(I)={(AIT)fcos2tr/...t (O<>fEiT)o (otherwise) •Determine theimpulseresponseofthematchedfilterforthesignal. bDetermine theoutputofthematchedfilteratf=T. c:SupposethesignalS(I)ispassedthroughacorrelator thatcorrelates theinput S(f)withS(f).Determine thevalueoftbecorrelator outputatf=T.Compare yourresultwiththatin(b). CHAPTER 5:OPTIMUM RECf-IVERS FORTHEADDITIVE WHITE UAU~S[AN NOISECHANNEL 321 5-3Thisproblem dealswiththecharacteristics ofaDPSKsignal. aSuppose wewishtotransmit thedatasequence I101000101I 0 bybinaryDPSKLetS(I)=Acos(2tifcl+8)represent thetransmitted signalin anysignaling intervalofduration T.Givethephaseofthetransmitted sigllalfor thedatasequence. Beginwith8=0forthephaseofthelirstbittobe transmitted. bIfthedatasequence isuncorrelated, determine andsketchthepowerdensity spectrum ofthesignaltransmitted byDPSK. 5-4Abinarydigitalcommunication systememploys thesignals S,,(I)=0,O"I<;;T s,(t)=A, O<;;I<;;T fortransmitting theinformation. Thisiscalledon-offsignaling. Thedemodulator cross-correlates thereceived signalr(l)withset}alldsamples theoutputofthe 'correlator att=T. aDetermine theoptimum detector foranAWGN channel andtheoptimum threshold, assuming thatthesignalsareequallyprobable .. bDetermine theprobability oferrorasafunction oftheSNR.Howdoeson-01l signaling compare withantipodal signaling? 5-5Thecorrelation metricsgivenby(5-1-44)are N N C(r,sm)=22:rnSmn-2:s~",m=1,2J•••,M ,,=\ n-\ where 'n=r,(I}J.(t)dt Smn=rSm(I}!n(I) dl Showthatthecorrelation metricsareequivalent tothemetrics C(r,sm}=2rr(t}sm(l}dl -fs;,,(t)dl 5-6Consider theequivalent lowpass (complex-valued) signalS,(I},0",I<;;T,with energy 1iT~=2 01s,(I)I'dt Suppose thatthissignaliscorrupted byAWGN,whichisrepresented byits equivalent lowpassformZ(I).Hence,theobserved signalis ,,(I)=S,(I}+z(t},0<;;I<;;T Thereceived signalispassedthrough afilterthathasan(equivalent lowpass) impulse r~sponseh,(I).Determine h,(I)sothatthefiltermaximizes theSNRatits output(atI=T). 5-7LetZ(I}=x(t)+jy(l)beacomplex-valued, zero-mean whitegaussian noise FIGURE P5-3322 DIGITAL COMMUNiCATIONS S,(I) sz(tJ A A 0 0!rr T 2 -A --A processwithautocorrelation function .p,,(r)=NoS(r).Letfm(I),m=1,2,...,M, beasetofMorthogonal equivalent lowpasswaveforms definedontheinterval o,,;;t,,;;T.Define Nm,=Re[f2(t)f~(I) dlJ.m=1,2,.,. ,M aDetermine thevariance ofNm,' bShowthatE(Nm,N.,) =0fork""m. 5-8Thetwoequivalent lowpasssignalsshowninFig.P5·gareusedtotransmit a binarysequence overanadditivewhitegaussiannoisechannel. Thereceivedsignal canbeexpressed as T,(I)=Si(l)+2(1),0";;1";;T,i=1,2 where2(1)isazero-mean gaussian noiseprocesswithautocorrelation function 4>,,(r)=~E[2·{t)2(1 +r)J=No{;(r) aDetermine thetransmitted energyinS.(I)andS,(I)andthecross-correlation coefficient p". bSuppose thereceiverisimplemented bymeansofcoherent detection usingtwo matched filters,onematched toS,(I)andtheothertoS,(I).Sketchthe equivalent lowpassimpulseresponses ofthematched filters. cSketchthenoise-free response ofthetwomatched filterswhenthetransmitted signalisS,(I). dSuppose thereceiver isimplemented bymeansoftwocross-correia tors (multipliers followed byintegrators) inparallel. Sketchtheoutput0'each integrator asaf/Jnction oftimefortheinterval0,,;;t'"Twhenthetransmitted signalisS,(I). eCompare thesketches in(c)and(d).Aretheythesame?Explainbriefly: fFromyourknoWledge ofthesignalcharacteristics, givetheprobability oferror forthisbinarycommunications system. 5·9Suppose thatwehaveacomplex-valued gaussian random variable 2=x+jy, where(x,y)arestatistically independent variables withzeromeanandvariance E(x')=E(y')=,fl.Let r==z+m.wherem=:m"+jmi anddefineras r=a+jb Clearly, a=x+m,andb=y+mi'Determine thefollowing probability density flInetions: apta,b); CHAPH"R ~:OPTfML'\tf RECEIVERS FORTHEADDITIVE WHITl::GAUSSIAN NOISEC/iANNf::L 323 bp(Il,cjJ),whereIl~Va'+b'andcjJ~tan'b/a; Cp(Il). Note:In(b)itisconvenient todefine8=tan'(m,/mJ sothat m,=Ym;+m;cose,m,::::y'm~+m;sin9. Furthermore, youmustusetherelationIf'· ',,, 2-eO"'".'., dcjJ=lo(a)=L22n~f)'nII "_I)n. wherel,,(a)isthemodified Besselfunction oforderzero, 5-10Aternarycommunication systemtransmits oneofthreesignals,sIt),0,or-5(1), everyTseconds. Thereceived signaliseitherr,(t)=s(t)+;:(t),r,(t)=;:(1),or r,(I)=-sIt)+;:(t),wherezIt)iswhitegaussian noisewithE(;:(t)J =0and <p,,(r)=IE[z(r)z*(r») =N"S(t-r).Theoptimum receiver computes thecor­ reiationmetric U=Re[fr(t)s*(t)dt] andcompares Uwithathreshold Aandathreshold -A.IfU>A,Ihedecision;s madethatsIt)wassent.IfU<~A,thedecision ismadeinfavorof-5(1).If - A<U<A,thedecision ismadeinfavorof0, BDetermine thethreeconditional probabilities oferronP,giventhatS(I)was sent,P,giventhat-5(I)wassent,andP,giventhat0wassent. bDetermine theaverageprobability oferrorP,asafunction ofthethreshold A, assummg thatthethree'Symbols areequallyprobable apriori. cDetermine thevalueofAthatminimizes P,. s-nThetwoequivalent lowpass signalsshowninFig,P5-11areusedtotransmit a binaryinformation sequence, Thetransmitted signals,whichareequallyprobable. arecorrupted byadditive zero-mean whitegaussian noisehavinganequivalent lowpassrepresentation zIt)withanautocorrelation function cjJ,,(r)=IE[z*(t)z(t +1'») =NoS('I') aWhatisthetransmitted signalenergy? bWhatistheprobability ofabinarydigiterrorifcoherent detection isemployed atthereceiver? cWhatistheprobability ofabinarydigiterrorifnoncoherent detection is employed atthereceiver? 5-12InSection 4-3-1itwasshownthattheminimum frequency separation for orthogonality ofbinaryFSKsignalswithcoherent detection isIif=1/2T. I1(1) S!(I) A A 0 0r~TJT, FIGURE PS·ll-A -A 324 D1GfTAl COMMUNICATIONS UI(I) FIGURE PS-13A 01-1-'--3A oI-JL..o.- .,(~~n O~of-b--'----'- -A UJ')n ~A oI--+-'-~'- -A .4(~~n o~SetI Se'll SetIII However, alowererrorprobability ispossiblewithcoherent detection ofFSKifat isincreased beyond1/2T.Showthattheoptimum valueof!ifis0.715/Tand determine theprobability oferrorforthisvalueofa'f. 5-13Theequivalent lowpasswaveforms forthreesignalsetsareshowninFig.P5-13. Eachsetmaybeusedtotransmit oneoffourequallyprobable messages overan additivewhitegaussian noisechannel. Theequivalent lowpassnoisez(r)haszero meanandautocorrelation functioncP,,(r)=No~(r). 8Classifythesignalwaveforms insetsI,II,and111.Inotherwords,statethe category orclasstowhicheachsignalsetbelongs. bWhatistheaveragetransmitted energyforeachsignalset? cForsignalsetI,specifytheaverageprobability oferrorifthesignalsare detected coherently. dForsignalsetII,giveaunionboundontheprobability ofasymbolerrorifthe detoetion isperformed (i)coherently and(ii)noncoherently. eIsItpossibletousenoncoherent detection onsignalsetIII?Explain. rWhichsignalsetorsignalsetswouldyouselectifyouwishedtoachievearatio ofbitratetobandwidth (R/W)ofatleast2.Brieflyexplainyouranswer. 5·14Consider aquaternary (M=4)communication systemthattransmits, everyT seconds, oneoffourequallyprobable signals:S,(I),-S,(I).S,(I),-S,(I).The signalsS;(I)ands,(r)areorthogonal withequalenergy.Theadditivenoiseiswhite gaussian withzeromeanandautocorrelation function cP,,(r)=Noli(r).The demodulator consistsoftwofiltersmatched to,,(I)and,,(I),andtheiroutputsat thesampling instantareV,andV,.Thedetector basesitsdecision onthe following rule: v,>IV,I=>s,(t), V,>IVd=>S,(I),U,<-IV,I=>-S,(I) U,<-IV,I=>-s,(I) Sincethesignalsetisbiorthogonal, theerrorprobability isgivenby(1-P<)where p,.isgivenby(5-2-34). Expressthiserrorprobability intermsofasingleintegral CHAPTER 5:OPTIMUM RECEIVERS FORTHEADDITIVE WHITE GAU~SIAN NOISECHANNEL 325 hili t=+-,~, FIGURE PS-IS (,I (hI and,lhus,showthatthesymbolerrorprobability forabiorthogonal signalsetwith M=4isidentical tothatforfour-phase PSK.Hint:Achangeinvariables fromV, andV,toW,=V,+V,andW,=V,-V,simplifies theproblem. S·IsTheinputset)toabandpass filteris s(l)=Re[S,,(t)e'2'1,,) whereso(t)isarectangular pulse.asshowninFig.PS-IS(a). 8Determine theoutputY(I)ofthebandpass fillerforallI;;'0iftheimpulse response ofthefilteris g(l)=Re[2h(l)ei'K"'] whereh(I)isanexponential asshowninFig.S-IS(b). bSketchtheequivalent lowpassoulpulofthefilter. cWhenwouldyousampletheoutputofthefilterifyouwishedtohavethe maximum outpulatthesampling instant?Whatisthevalueofthemaximum output? dSuppose thatinaddition totheinputsignalS(I),thereisadditivewhitegaussian nOIse n(l)=Re[z(I)e"'f~l where <Pu(r)=No&(r).Atthesampling instantdetermined in(c),thesignal sampleiscorrupted byanadditivegaussian noiseterm.Determine itsmeanand variance. eWhatisthesignal-to-noise ratiol'ofthesampled output? rDetermine thesignal-to-noise ratiowhenh(l)isthematched filtertos(t)and compare thisresultwiththevalueofl'obtained in(e). S·16Consider theoctalsignalpointconstellations inFig.PS·16. FlGURE P5-l6 8-PSK 8-QAM FIGURE PS·19326 DIGITAtCOI,-iMLINIC ATIONS n aThenearest-neighbor signalpointsinthe8-QAM signalconstellation are separated indistance byAunits.Determine theradiia.ndboftheinnerand outercircles. bTheadjacent signalpointsinthe8-PSKareseparated byadistanceofAunits. Determine theradiusrofthecircle. cDetermine theaveragetransmitter powersforthetwosignalconstellations and compare thetwopowers. Whatistherelative poweradvantage ofone constellatiun overtheother?(Assume thatallsignalpointsareequally probable.) 5-17Consider the8-pointQAMsignalconstellation showninFig.P5-16. aIsitpossible toassignthreedatabitstoeachpoin!ofthesignalconstellation suchthatnearest(adjacent) pointsdifferinonlyonebitposition? bDetermine thesymbolrateifthedesiredbitrateis90Mbits/s. 5-18Suppose thatbinaryPSKisusedfortransmitting infonnation overanAWGNwith apowerspectraldensityof~No=10-10W1Hz.Thetransmitted signalenergyis 'l:.=\A2T,whereTisthebitintervalandAisthesignalamplitude. Determine thesignalamplitude required toachieveanerrorprobability of10-·whenthedata rateis(a)10kbits/s,(b)100kbits/s,and(c)1Mbit/s. 5-19Consider asignaldetector withaninput r=±A+n where+Aand-Aoccurwithequalprobability andthenoisevariable nis characterized bythe(Laplacian) pdfshowninFig.P5-19. aDetermine theprobability oferrorasafunction oftheparameters Aandrr bDetermine theSNRrequired toachieveanerrorprobability oflW'.Howdoes theSNRcompare withtheresultforaGaussian pdf? 5-20Consider thetwo8-pointQAMsignalconstellations showninFig.P5-Z0.The minimum distance between adjacent pointsis2A.Determine theaverage transmitted powerforeachconstellation, assuming thatthesignalpointsare equallyprobable. Whichconstellation ismorepower·efficient? 5-21FortheQAMsignalconstellation showninFig.P5-21,determine theoptimum decision boundaries forthedetector. assuming thattheSNRissufficiently highso thaterrorsonlyoccurbetween adjacent points. +0+0 ••• • FIGURE P5-20 (al (bl CHAPTER 5:OPTIMUM RE.CEIVERS FORTHEADDITIVE WHITE GALJSSIAt' NOISECHANNEL 327 FIGURE PS-21-5-J -III-I -J -5 5-22SpecifyaGraycodeforthe16-QAM signalconstellation showninFig.P5-21. 5-23Twoquadrature carriers cos211{Jandsin2trf..tareusedtotransmit digital information through anAWGNchannelattwodifferent datarates,10kbits/sand 100kbits/s. Determine therelativeamplitudes ofthesignalsforthetwocarriersso thatthe'ifhINhforthetwochannels isidentical. 5-24Threemessages m"m"andm,aretobetransmitted overanAWGN channel withnoisepowerspectral density\Nl1'Themessages are (O<;;t<;;T) (otherwise) {I(O<;;t<;;\T) S,{I)=-s,{t)=-1<!T<;;O<;;T) o(otherwise) aWhatisthedimensionality ofthesignalspace'! bFindanappropriate basisforthesignalspace.[Hint:Youcanfindthebasis withoutusingtheGram-Schmidt procedure.) cDrawthe signal constellation forthisproblem. dDeriveandsketchtheoptimaldecision regionsR"R"andR,. eWhichofthethreemessages ismorevulnerable toerrorsandwhy?Inother words,whichofP(errorIm,transmitted), i=1,2,3,islarger? S-2SWhentheadditive noiseattheinputtothedemodulator iscolored, thefilter matched tothesignalnolongermaximizes theoutputSNR.Insuchacasewemay consider theuseofaprefilter that"whitens" thecolored noise.Theprefilter is followed byafiltermatched toIheprefiltered signal.Towards thisend,cOlJsider theconfiguration showninFig.P5-25. aDetermine thefrequency response characteristic oftheprefilter thatwhitensthe noise. FIGURE P5-25 Tr.....---.1 Del~clorL_*I ~I..(11=_\(,')+11(1)Pr~whiteningr(rJ=.I(l)+ii(J)Filler-/,nU)j\niter matched colorednoi\cHI'(t') (0\(1f Sarnp ;JIr= 328 DIGITAL COMMUNICATIONS bDetermine thefrequency response characteristic ofthefiltermatched to5(1). eConsider theprefiherandthematched filterasasingle"generalized matched filter."Whatisthefrequency response characteristic ofthisfilter? dDetermine theSNRattheinputtothedetector. 5-26Consider adigitalcommunication systemthattransmits information viaQAM overavoice-band telephone channelatarate2400symbols/s. Theadditivenoise isassumed tobewhiteandgaussian. aDetermine the'l,!N"required toachieve anerrorprobability of10'at 48OObits/s bRepeat(a)forarateof9600bits!s. tRepeat(a)forarateof19200bits{s. dWhatconclusions doyoureachfromtheseresults? 5·27Consider thefour-phase andeight-phase signalconstellations showninFig.P5-27. Determine theradii"and'2ofthecirclessuchthatthedistance between two adjacent pointsinthetwoconstellations isd.Fromthisresult,determine Ihe additional transmitted energyrequired inthe8-PSKsignaltoachievethesame errorprobability asthefour-phase signalathighSNR,wheretheprobability of errorisdetermined byerrorsinseletting adjacent points. 5·28Digitalinformation istobetransmitted bycarriermodulation throughanadditive gaussian noisechannel withabandwidth of100kHzandNo,~10-'"W1Hz. Determine themaximum ratethatcanbetransmitted through thechannelfor four-phase PSK,binaryFSK,andfour.frequency orthogonal FSK,whichis detected noncoherently. 5-29InaMSKsignal,theinitialstateforthephaseiseither0orJrrad.Determine the terminalphasestateforthefollowing fourinputpairsofinputdata:(a)00:(b)OJ; tc)10;(d)II. 5-30Acontinuous-phase FSKsignalwithh=~isrepresented as /2'(;,(Jrt) /2'(;,(Jrl)5(1)=±V--:r:cos2T bcos2Jrf,.1±VT"sin2T"sin2trf,I. FIGURE P5-Z7wherethe±signsdependontheinformation bitstransmitted. aShowthatthissignalhasconstant amplitude. bSketchablockdiagramofthemodulator forsynthesizing thesignal. cSketchablockdiagram ofthedemodulator anddetector forrecovering the information. 5·31Sketchthephasetree,thestatetrellis.andthestatediagram forpartial-response CPMwithh=Jand {1/4T(0<;;1<>2T)l/(I)=o(otherwise) 11/=4 M=K CHArTfR 5:Orn\I14M Rf'CFIVER;$ fORfHEALmrnn,: WUITf GACSSfAN NOISECHANNEl. 329 5-32Determine thenumberofterminalphaseslatesintheslaletrellisdiagram for(a)" fullresp'lUse binaryCPFSKwitheither"=Ior~and(b)aparlial-response L=:I binan·CPFSKwilheilherh=~or1. 5.33Consider abiorthogonal signalselwilhM=8signalpoints.Delermine aunion boundfortheprol>abilily ofasyml>olerrorasafunclion of"!./N".Thesignal poinlsareequallylikelyapriori. 5.34Consider anM-arydigilalcommunication syslemwhereM=2".andNisthe dimension ofthesignalspace.Suppose thaiIheMsignalvectorslieonthevertices ofahypercul>e thatiscentered atIheorigin.Determine theaverageprobability of asrmbolerrorasafunctionoff:,/N"wherei,istheenergypersyml>ol.\N"isthe powerspectraldensityoftheAWGN.andallsignalpoinlsareequallyprobable. 5-35Consider thesignalwaveform n s(1)=2:e,p(t-k7;) i~1 wherep(t)isarectangular pulseofunitamplitude andduration 7;.The{e.}may beviewedasacodevectorC=Ie,c,...cn).wheretheelements c,=±1.Show Ihatthefiltermalched 10thewaveform '(1)mayberealizedasacascadeofafilter matched 10p(l)followed byadiscrete-time filtermatched tothevectorC. Determine thevalueoftheoutpulofthematched filteratthesampling instanl 1=nT,.. 5·:36Aspeechsignalissampled atarateofgkHz,logarithmically compressed and end>ded intoaPCMformalusinggbits/sample. ThePCMdalaistransmitted IhroughanAWGNbaseband channelviaM-levelPAM.Delermine theband­ widlhrequired forlransmission when(a)M=4.(b)M=8,and(c)M=16. 5-37AHadamard matrixisdefinedasamalrixwhoseelements are±1andwhoserow vectorsarepairwise orthogonal. Inthecasewhennisapowerof2,annXn Hadamard matrixisconstructed bymeansoftherecursion H=[Hn Hn] ;z,.H,l-H" aleICidenoletheithrowofannXnHadamard matrixasdefinedabove.Show thaiIhewaveforms constructed as n 'i(l)=2:c".p(t-kT,.), j=L2....,n 1<"'1 areorthogonal, wherep(l)isanarbilrary pUlseconfined tothelimeinterval O~t~T. .. bShowthatIhematched filters(orcross-correlators) forthenwaveforms {s,(lll canberealized byasinglefilter(orcorrelator) matched tothepulsep(l) followed byasetofncross-correlalors usingIhecodewordsIC,}. 5·38Thediscretesequence r,=~c,+n"k=1,2,_..,n represents Iheoutputsequence ofsamplesfromademodulator, wherec,=±1are elements ofoneoftwopossible codewords,C,=[I1...11andC,= [11.., 1-1'..-1].ThecodewordC,hasIVelemenls thatare+1andn-w 330 DIOJTAI. COMMUNICATIONS elements thatare-I,where'" issomepositiveinteger.Thenoisesequence {n.}is whitegaussian withvariance rr'. •Whatistheoptimum maximum likelihood detector forthetwopossible transmitted signals? bDetermine theprobability oferrorasafunction oftheparameters (rr',~.,",). cWhatisthevalueofwthatminimizes theerrorprObability? 5-39Derivetheoutputs"andr,ofthetwocorrelators showninFig.5-4-1.Assume thatasignalS,,(I)istransmilled andthat r,(I)=SII(I)""+Z(I) wherez(l)=nAI)+jnAI)istheadditive gaussian noise. 5-4/)Determine thecovariance. andvariances ofthegaussian randomnoisevariables II".,II".,n,,,andn"in(5-4-15)andthejointpdf. 5-41Derivethematched tilteroutputsgivenby(5-4·10). 5-4ZInon-offkeyingofacarrier-modulated signal,thetwopossible signalsare so(l)=0,0,.;I";T. rn.s,(I)=VT;cos2trf,t, Thecorresponding received signal.are r(t)=n(t),0,.;t'S;T" Iff~· r(/)=-co.(2trf,1+1/»+II{I),T. whereI/>isIhecarrierphaseandn{l)isAWGN. •Sketchablockdiagramofthereceiver (demodulator anddetector) Ihatemploys nonwherent (envelope) deleclion. bDetermine thepdfsforthetwopossible decision variables atthedetector corresponding toIhetwopossiblereceived signals. cDerivetheprobability oferrorforthedetector. 5-43Intwo-phase DPSK,thereceived signalinonesignaling inlerval isusedasaphase reference forthereceived signalinthefollowing signaling interval. Thedecision variable is "I" D=Re{V.Y~_/) ~0 "0" represents thecomplex-valued outputofthefiltermatched tothelTansmilled signalU(I).N.isacomplex-valued gaussian variable havingzeromeanand statistically independent components. aWritingV.=X.+jY..showthatDisequivalent to d=H(Xm+Xm_,)j'+[HY..+Y'n-')]'-[HX..-Xm-,)j'-[\(y.n-Ym-,»)' bFormathematical convenience; suppose that9.=9._,.Showthattherandom variables V"V"V"andU.arestatistically independent gaussian variables, whereV,=HXm+Xm_,),U,=HY..+Ym-,),V,=HX~-Xm_,),andU,= ~(Ym-Y,n-')' CH..\PlEH _~OPTIMl'M RFTLVERS fORTHEADDITIVE VYHITFCiAl'SSIAl" SOlSFCHAl"SEL 331 cDefinetherandom variables WI~vi+v;andW,~V~+U~.Then .., D~W,-W,,,,O ··11'· Determine theprobability densityfunctions forW,andW,. dDetermine theprobability oferrorPr"where Ph=P(D<0)=peW,-w,<0)=J'pew,>w,Iwl)pj",) <1\1", " 5-44RecallthatMSKcanberepresented asafour-phase offsetPSKmodulation having thelowpassequivalent form vCr)~:Z:[hll(r ~2kT,,)+il,u(1-2kT"~T,)], where {sin(m12Th) 11(1)~()(O~I~2T,,) (otherwise) and{I,}and{l,}aresequences ofinformation symbols(±I). aSketchtheblockdiagram ofanMSKdemodulator foroffsetOPSK. bEvaluate theperformance ofthefour-phase demodulator forAWGNifno account istakenofthememory inthemodulation. cCompare theperformance obtained in(b)withthatforViterbidecoding ofthe MSKsignal. dTheMSKsignalisalsoequivalent tobinaryFSK.Determine tbeperformance of noncoherent detection oftheMSKsignal.Compare yourresultwith(b) and(c). 5-45Consider atransmission linechannel thatemploys n-1regenerative repeaters plustheterminal receiver inthetransmission ofbinaryinformation. Assume that theprobability orerrOratthedetector ofeachreceiver ispandthaterrorsamong repeaters arestatistically independent. aShowthatthebinaryerrorprobability atthetermmal receiver is P"~\[1~(I-2p)"] bIfP~10handn=]00.determine anapproximate valueofP,. 5-46Adigitalcommunication systemconsists ofatransmission linewith100digital (regenerative) repeaters. Binaryantipodal signalsareusedfortransmitting the information. Iftheoverallend-to-end errorprobability isIOh •determine the probability oferrorforeachrepeater andtherequired tJ,/II:,toachievethis performance inAWGN. 5-47Aradiotransmiller hasapoweroutputofPT~]Watafrequency of1GHz.The transmitting andreceiving antennas areparabolic disheswithdiameter D~3m. aDetermine theantenna gains. bDetermine theEIRPforthetransmitter. cThedistance (freespace)between thetransmitting andreceiving antennas is 20km.Determine thesignalpowerattheoutputofthereceiving antenna in dEmo 332 DIGITAL COMMUNICATIONS 5-48Aradiocommunication systemtransmits atapowerlevelof0.1Wat1GHz.The transmitting andreceiving antennas areparabolic, eachhavingadiameter of1m. Thereceiver islocated30kmfromthetransmitter. aDetermine thegainsofthetransmitting andreceiving antennas. bDetermine theEIRPofthetransmitted signal. cDetermine thesignalpowerfromthereceiving antenna. 5-49Asatelliteinsynchronous orbitisusedtocommunicate withanearthstationata distance of40()()()km.Thesatellite hasanantenna withagainof15dBanda transmitter powerof3W.Theearthstationusesa10mparabolic antennawithan efficiency of0.6.Thefrequency bandisatf=10GHz.Determine thereceived powerlevelattheoutputofthereceiverantenna. S-SOAspacecraft located100()()()kmfromtheearthissending dataatarateof Rbits/soThefrequency bandiscentered at2GHzandthetransmitted poweris 10W.Theearthstationusesaparabolic antenna, 50mindiameter, andthe spacecraft hasanantenna withagainof10dB.Thenoisetemperature ofthe receiverfrontendisTo=300K. aDetermine thereceived powerlevel. bIfthedesired 'l:h/No=10dB,determine themaximum bitratethatthe spacecraft cantransmit. 5-51Asatelliteingeosynchronous orbitisusedasaregenerative repeater inadigital communication system.Consider thesatellite-to-earth linkinwhichthesatellite antennahasagainof6dBandtheearthstationantennahasagainof50dB.The downlink isoperated atacenterfrequency of4GHz,andthesignalbandwidth is IMHz.Iftherequired 'l:hlN"forreliablecommunication is15dB,determine the transmitted powerforthesatellitedownlink. AssumethatNil=4.1X10-21W1Hz. (6-1-2) t/J= -21ifcr 3336 CARRIER ANDSYMBOL SYNCHRONIZATION Wehaveobserved thatinadigitalcommunication system,theoutputofthe demodulator mustbesampled periodically, oncepersymbolinterval, inorder torecoverthetransmitted information. Sincethepropagation delayfromthe transmitter tothereceiverisgenerally unknown atthereceiver, symboltiming mustbederivedfromthereceived signalinordertosynchronously samplethe outputofthedemodulator. Thepropagation delayinthetransmitted· signalalsoresultsinacarrier offset,whichmustbeestimated atthereceiver ifthedetector isphase­ coherent. Inthischapter,weconsider methods forderivingcarrierandsymbol synchronization atthereceiver. 6-1SIGNAL PARAMETER ESTIMATION Letusbeginbydeveloping amathematical modelforthesignalattheinputto thereceiver. Weassumethatthechannel delaysthesignalstransmitted through itandcorrupts thembytheaddition ofgaussian noise.Hence,the received signalmaybeexpressed as r(/)=S(I-r)+n(l). where s(/)=Re[s,(/)ei2><!,.l] (6-1-1) andwhereristhepropagation delayandS/(I)istheequivalent lowpasssignal. Thereceived signalmaybeexpressed as r(/)=Re{[S/(I-r)ei4>+Z(I)]ei2><!,.l} wherethecarrierphase4>,duetothepropagation delay T,is 334 DIG!TALCOMMUNICATIONS Now,fromthisformulation, itmayappearthatthereisonlyonesignal parameter tobeestimated, namely, thepropagation delay,sinceonecan determine c/>fromknowledge ofIeandr.However, thisisnotthecase.Firstof all,theoscillator thatgenerates thecarriersignalfordemodulation atthe receiver isgenerally notsynchronous inphasewiththatatthetransmitter. Furthermore, thetwooscillators maybedriftingslowlywithtime,perhaps in different directions. Consequently, thereceived carrierphaseisnotonly dependent onthetimedelayr.Furthermore, theprecision towhichonemust synchronize intimeforpurpose ofdemodulating thereceived signaldepends onthesymbolinterval T.Usually, theestimation errorinestimating Tmustbe arelatively smallfraction ofT.Forexample, ±1%ofTisadequate for practical applications. However. thislevelofprecision isgenerally inadequate forestimating thecarrier phase, evenifc/>depends onlyonr.Thisisduetothe factthatfcisgenerally large,and,hence,asmallestimation errorinrcausesa largephaseerror. Ineffect,wemustestimate bothparameters Tandc/>inordertodemodulate andcoherently detectthereceived signal.Hence,wemayexpressthereceived signalas ret)=set;c/>,r)+n(/) (6-1-3) (6-1-4)where<pandTrepresent thesignalparameters tobeestimated. Tosimplifythe notation, welet111denotetheparameter vector{c/>,T}.sothatset;c/>,r)is simplydenoted byset;tJ,o). Therearebasically twocriteriathatarewidelyappliedtosignalparameter estimation: themaximum-likelihood (ML)criterion andthemaximum a posteriori probability (MAP)criterion. IntheMAPcriterion, thesignal parameter vector'"ismodeled asrandom, andcharacterized byanapriori probability densityfunction p(tJ,o).Inthemaximum-likelihood criterion, the signalparameter vector tJ,oistreatedasdeterministic butunknown. Byperforming anorthonormal expansion ofret)usingNorthonormal functions {!net)},wemayrepresent ret)bythevectorofcoefficients [r,r2...rN)""r.Thejointpdfoftherandomvariables [r,r2...TN)inthe expansion canbeexpressed asperItJ,o).Then,theMLestimate of'"isthe valuethatmaximizes perIIII).Ontheotherhand,theMAPestimate isthe valueof'"thatmaximizes theaposteriori probability densityfunction p(tJ,oIr)=perI"')p("') per) Wenotethatifthereisnopriorknowledge oftheparameter vectorIII,we mayassumethatp("')isuniform (constant) overtherangeofvaluesofthe parameters. Insuchacase,thevalueof'"thatmaximizes perIIII)also maximizes p('"Ir).Therefore, theMAPandMLestimates areidentical. Inourtreatment ofparameter estimation givenbelow,weviewthe parameters c/>andrasunknown, butdeterministic. Hence,weadopttheML criterion forestimating them. CHAPTER flo:(ARRIER ANDSYMBOL SYNCHRONIZATION 335 IntheMLestimation ofsignalparameters, werequirethatthereceiver extracttheestimate byobserving thereceived signaloveratimeinterval 4,~T,whichiscalledtheobservation interval. Estimates obtained froma singleobservation interval aresometimes calledone-shot estimates, In practice, however, theestimation isperformed onacontinuous basisbyusing trackingloops(eitheranalogordigital)thatcontinuously updatetheestimates. Nevertheless, one-shot estimates yieldinsightfortracking loopimplementa­ tion.Inaddition, theyproveusefulintheanalysisoftheperformance ofML estimation, andtheirperformance canberelatedtothatobtained witha trackingloop. 6-1-1TheLikelihood Function Although itispossibletoderivetheparameter estimates basedonthejointpdf oftherandomvariables [r,r2...rNIobt"lined fromtheexpansion ofr(oI),itis convenient todealdirectlywiththesignalwaveforms whenestimating their parameters. Hence,weshalldevelop acontinuous-time equivalent ofthe maximization ofp(rI"'). Sincetheadditivenoisen(l)iswhiteandzero-mean gaussian, thejointpdf p(rI"')maybeexpressed as p(rI"')=('-I-f exp{-~[r"-Sn;tlJW} (6-1-5)Vfica ,,~12a where r,=i'r(I)f,,(t)dl 7;, s,,(tlJ)=iS(I;IlI)!n(t)dt 7il(6-1-6) where4,represents theintegration intervalintheexpansion ofr(t)ands(t;"'). Wenotethattheargument intheexponent maybeexpressed intermsof thesignalwaveforms r(t)and$(t;"'),bysubstituting from(6-1-6)into(6-1-5). Thatis, 1,'I 1f-22: [r"-s,,(I;"'W= -[r(t)-$(t;tlJ)J2dt2a11=1 No7(\(6-1-7) wheretheproofisleftasanexercise forthereader(seeProblem 6-1).Now, themaximization ofp(rItlJ)withrespecttothesignalparameters tlJis equivalent tothemaximization ofthelikelihood function. A(",)=exp{-J.([r(t)-S(I;",)fdt} N,}J1;)(6-1-8) Below,weshallconsider signalparameter estimation fromtheviewpoint of maximizing A(I/I). 336 DIGITAL COMMUNICATiONS Received ~ignalOutput,..----, dala Detector CarTier recoverySignal pulse generator FIGURE 6-1-1BlockdiagramofbinaryPSKreceiver. 6-1-2CarrierRecovery andSymbolSynchronization inSignalDemodulation Symbolsynchronization isrequired ineverydigitalcommunication system whichtransmits information synchronously. Carrierrecovery isrequired ifthe signalisdetected coherently. Figure6-1-1illustrates theblockdiagramofabinaryPSK(orbinaryPAM) signaldemodulator anddetector. Asshown,thecarrierphaseestimate <i>is usedingenerating thereference signalg(t)cos(21ifct+4»forthecorrelator. Thesymbolsynchronizer controls thesampler andtheoutputofthesignal pulsegenerator. Ifthesignalpulseisrectangular thenthesignalgenerator can beeliminated. TheblockdiagramofanM-aryPSKdemodulator isshowninFig.6-1-2.In thiscase,twocorrelators (ormatched filters)arerequired tocorrelate the received signalwiththetwoquadrature carriersignalsg(t)cos(21ifJ+1»and g(t)sin(2rrfct+J»,where4>isthecarrierphaseestimate. Thedetector isnow aphasedetector, whichcompares thereceived signalphaseswiththepossible transmitted signalphases. TheblockdiagramofaPAMsignaldemodulator isshowninFig.6-1-3.In thiscase,asinglecorrelator isrequired, andthedetector isanamplitude detector, whichcompares thereceived signalamplitude withthepossible transmitted signalamplitudes. Notethatwehaveincluded anautomatic gain control(AGe)atthefront-end ofthedemodulator toeliminate channelgain variations, whichwouldaffecttheamplitude detector. TheAGChasa relatively longtimeconstant, sothatitdoesnotrespond tothesignal amplitude variations thatoccuronasymbol·by-symbol basis.Instead, the AGCmaintains afixedaverage(signalpIusnoise)poweratitsoutput. Finally,weillustrate theblockdiagram ofaQAMdemodulator inFig. 6-1-4.AsinthecaseofPAM,anAGCisrequired tomaintain aconstant averagepowersignalattheinputtothedemodulator. Weobservethatthe demodulator issimilartoaPSKdemodulator, inthatbothgenerate in-phase andquadrature signalsamples(X,Y)forthedetector. InthecaseofQAM, ("HAYTER 0,CARRIER ANDSYMBOL SYNCHRONIZATION 337 signalSignal plJlse generalor,_.l-_, Output dala 90' pha!>C shift x)4---------J FIGURE 6-1-2 Bloc~diagram 01M-aryPSKreceiver. thedetector computes theeuclidean distance between thereceived noise­ corrupted signalpointandtheMpossible transmitted points,andselectsthe signalclosesttothereceived point. 6·2CARRIER PHASE ESTIMATION Therearetwobasicapproaches fordealingwithcarriersynchronization atthe receiver. Oneistomultiplex, usuallyinfrequency, aspecialsignal,calleda pilotsignal,thatallowsthereceiver toextractand,thus,tosynchronize its localoscillator tothecarrierfrequency andphaseofthereceived signal.When FIGURE 6-1-3BlockdiagramofM-aryPAMreceiver. ReCti\'ed signalAutomatic galt!. control Signal pulse genera.tor OUfput 338 DIGITAL COMMUNICATIONS cos(2nj, t+~) Received sign<11Signal pul~ generatorCompute euclidean Output distance decision metrles W phase shift -sin(2xJ;r+~) x}4---------' FIGURE 6-1-4BlockdiagramofQAMreceiver. anunmodulated carriercomponent istransmitted alongwiththeinformation­ bearingsignal,thereceiveremploys aphase-locked loop(PLL)toacqLlireand trackthecarriercomponent. ThePLLisdesigned tohaveanarrowbandwidth sothatitisnotsignificantly affectedbythepresence offrequency components fromtheinformation-bearing signal. Thesecondapproach, whichappearstobemoreprevalent inpractice, isto derivethecarrierphase estimate directlyfromthemodLilated signal.This approach hasthedistinctadvantage thatthetotaltransmitter poweris allocated tothetransmission oftheinformation-bearing signal.Inour treatment ofcarrierrecovery, weconfineourattention tothesecondapproach; hence,weassumethatthesignalistransmitted viasuppressed carrier. Inordertoemphasize theimportance ofextracting anaccurate phase estimate, letusconsider theeffectofacarrierphaseerroronthedemodulation ofadouble-sideband, suppressed carrier(DSB/SC) signal.Tobespecific, suppose wehaveanamplitude-modulated signaloftheform s(t)=A(I)cos(21ifct+cP) (6-2-1) Ifwedemodulate thesignalbymultiplying s(t)withthecarrierreference c(l)=cos(2rrf,1+.{» (6-2-2) weobtain c(t)s(t) =~A(t)cos (4)-.{»+~A(t)cos(4Jl'j;.t +rP+4» (6-2-5)CHAPTER 6,CARRIER ANDSYMBOL SYNCHRONIZATION 339 Thedouble-frequency component mayberemoved bypassing theproduct signalc(t)s(r)through alowpass filter.Thisfiltering yieldstheinformation­ bearingsignal y(t)=M(t)cos(4)-<1» (6-2-3) Notethattheeffectofthephaseerror4>-<1>istoreducethesi$nallevelin voltagebyafactorcos(4)-<1»andinpowerbyafactorcos2(4)-c/J).Hence,a phaseerrorof10°resultsinasignalpowerlossof0.13dB,andaphaseerrorof 30°resultsinasignalpowerlossof1.25dBinanamplitude-modulated signal. TheeffectofcarrierphaseerrorsinQAMandmultiphase PSKismuch moresevere.TheQAMandM-PSKsignalsmayberepresented as s(t)=A(t)cos(211:[,.1+4»-B(t)sin(2Jrj;.I+4» (6-2-4) Thissignalisdemodulated bythetwoquadrature carriers c,.(t)=cos(2trfct+<1» c,(t)=-sin(211:[,1+<1» Multiplication ofsit)withcc(t)followed bylowpass filtering yieldsthein-phase component yltJ=~A(t)cos(d> -<1»-~B(t)sin(4> -(1,) (6-2-6) Similarly, multiplication ofS(I)byc,(t)followed bylowpass filtering yieldsthe quadrature component YQU)=~B(t)cos('"-(f,)+~A(t)sin(4)-(1,) (6-2-7) Theexpressions (6-2-6)and(6-2-7)clearlyindicate thatthephaseerrorinthe demodulation ofQAMandM-PSKsignalshasamuchmoresevereeffectthan inthedemodulation ofaPAMsignal.Notonlyisthereareduction inthe powerofthedesiredsignalcomponent byafactorcos2(c/J-J,),butthereis alsocrosstalk interference fromthein-phase andquadrature components. Sincetheaverage powerlevelsofA(t)andB(I)aresimilar.asmallphaseerror causesalargedegradation inperformance. Hence, thephaseaccuracy requirements forQAMandmultiphase coherent PSKaremuchhigherthan DSB/SC PAM. 6-2-1Maximum-Likelihood CarrierPhaseEstimation First,wederivethemaximum-likelihood carrierphaseestimate. Forsimplicity. weassume thatthedelayrisknownand,inparticular, wesetr=O.The function tobemaximized isthelikelihood function givenin(6-I-K). With¢ substituted forIII,thisfunction becomes 1\(<1»=exp{-J.J[r(t)-s(t:<t>Wdt} No 7{1 =exp{-J.Jr2(t)dt+3..Jr(t)s(t:4»dt-2.Js'(t; <1»dt}. No10 No7(J No7{) (6-2-8) (6-2-9)34Q Dl<.HTAL COMMI.)NlCATIONS Notethatthe/irsttermoftheexponential factordoesnotinvolvethesignal parameter </J.Thethirdterm,whichcontains theintegralofS2(1;"'},isa constant equaltothesignalenergyovertheobservation interval Toforany valueof<p.Onlythesecondterm,whichinvolves thecross-correlation ofthe received signalr(l}withthesignalS(I:</J),depends onthechoiceor</J. Therefore. thelikelihood functionA(</J}maybeexpressed as A(¢}=Cexp[~ir(l)s(l;</J}dl] NoTo whereCisaconstant independent of</J. TheMLestimate 4>MListhevalueor</Jthatmaximizes A(</J}in(6-2-9). Equivalently, thevalue~MLalsomaximizes thelogarithm ofA(<f»,i.e.,the log-likelihood function AL(</J}=.3.-rr(I)s(I;</J)dl NoJr, NOlethaIindefiningAL(</J)wehaveignoredtheconstant termInC.(6-2-10) Eumple 6-2-1 Asanexample oftheoptimization todetermine thecarrierphase,letus consider thetransmission oftheunmodulated carrierAcos21ifc1.The received signalis r(l)=Acos(2nj;.1+</J)+n(l) where </Jistheunknown phase.Weseekthevalue</J,say4>ML.that maximizes Ad</J)=z:1r(/}cos(21ifc1-+-</J)dl oTo Anecessary condition foramaximum isthat Thiscondition yields or,equivalently,ir(l)sin(2rifct+~Mddl=0 To(6-2-11) ~ML=-tan-1[fr(l)sin21ifcldtjir(l}cos21ifc1dl](6-2-12) To To Weobservethattheoptimality condition givenby(6-2-11)impliestheuse CHAPTER b:CARRIER ANDSYMBOL SYNCHRONIZATION 341 FIGURE 6-2-1APLLforobtaining theMLestimate ofthephaseofan unmodulated carrier. r(t) FIGURE 6-2-2A(one-shot) MLestimate oftheph.,eofan unmodulated carrier.x>---.J COS21Cf.,(Jlid'~_~x T" ---,(Y)4lMl-tan,X- Jtid'I---J T" y ofalooptoextracttheestimate asillustrated inFig.6-2-1.Theloopfilteris anintegrator whosebandwidth isproportional tothereciprocal ofthe integration interval To.Ontheotherhand,(6-2-12) implies an implementation thatusesquadrature carriers tocross-correlate withret). Then,4>Mlistheinversetangent oftheratioofthesetwocorrelator outputs, asshowninFig.6-2-2.Notethatthisestimation schemeyields4>Ml explicitly. Thisexample clearlydemonstrates thatthePLLprovides theMLestimate ofthephaseofanunmodulated carrier. 6-2-2ThePhase-Locked Loop ThePLLbasically consists ofamultiplier, aloopfilter,andavoltage­ controlled oscillator(yeO),asshowninFig.6-2-3.Ifweassumethattheinput tothePLListhesinusoid cos(21ifct+c/J)andtheoutputoftheyeOis sin(21ifct+4»,where4>represents theestimate ofc/J,theproduct ofthese twosignalsis e(l)=cos(21ifcl+"')sin(27t"fct+4» =hin(¢-c/J)+!sin(41ifct+c/J+¢) (6-2-13) FlGURE 6-2-3Basicelements ofaphase-located loop(PLL).DutPul~...L -lveo signal l-_--' 342 DIGITAL COMMUNICATIONS Theloopfilterisalowpassfilterthatresponds onlytothelow-frequency component !sin(<i>-tf;)andremoves thecomponent at2fc.Thisfilteris usuallyselectedtohavetherelatively simpletransferfunction G(s)=1+'2S(6-2-14) 1+'tS where"and'2aredesignparameters ('t~(2)thatcontrolthebandwidth of theloop.Ahigher-order filterthatcontains additional polesmaybeusedif necessary toobtainabetterloopresponse. Theoutputoftheloopfilterprovides thecontrolvoltagev(t)fortheYeo. TheyeOisbasically asinusoidal signalgenerator withaninstantaneous phase givenby 21Cfct+(j,(I)=21Cfct+KL~v(,)dr whereKisagainconstant inrad/Y.Hence, (j,(t)=KL~V(T)d,(6-2-15) (6-2-16) Byneglecting thedouble-frequency termresulting fromthemultiplication of theinputsignalwiththeoutputoftheYeO,wemayreducethePLLintothe equivalent closed-loop systemmodelshowninFig.6-2-4.Thesinefunctionof thephasedifference tf;-(j,makesthissystemnonlinear, and,asaconse­ quence,theanalysisofitsperformance inthepresence ofnoiseissomewhat involved but,nevertheless, itismathematically tractable forsomesimpleloop filters. Innormaloperation whentheloopistracking thephaseoftheincoming carrier,thephaseerrortf;-(j,issmalland,hence, (6-2-17) Withthisapproximation, thePLLbecomes linearandischaracterized bythe dosed-loop transferfunction H(s)=KG(s)/s 1+KG(s)/s(6-2-18) FIGURE 6-2-4Modelofphase-locked loop. CHAnER tl:CARRIER ANDSYMBOL SYNCHRONllATION 343 wherethefactorof~hasbeenabsorbed intothegainparameter K.By substituting from(6-2-14)forG(s)into(6-2-18), weobtain I+T,sH(s)=-,1+(T,+IIK)s+(TJK)s-(6-2-19) Hence.theclosed-loop systemforthelinearized PLLissecond-order when G(s)isgivenby(6-2-14). Theparameter T2controls theposition ofthezero. whileKandT,areusedtocontrolthepositionoftheclosed-loop systempoles. Itiscustomary toexpressthedenominator ofH(S)inthestand,ard form D(s)=s'+2(w"s+w;, (6-2-20) (6-2-22)where{iscalledtheloopdamping factorandw"isthenaturalfrequency ofthe loop.Intermsoftheloopparameters. w"=VKIT,.and(=(T2+IIK)/2wn• theclosed-loop transferfunction becomes (2(w"-w;,1K)s+w;, 2H(s)=, , (6-2-I)s-+2lw"s+w;/ The(one-sided) noise-equivalent bandwidth (seeProblem 2-24)oftheloopis B=GOIG+KIT,) <44(T,+11K) I+(T2W,,)' 8(w" Themagnitude response 20logIH(w)1asafunction ofthenormalized frequency wiw"isillustrated inFig.6-2-5.withthedamping factor,asa parameter andT,»1.Notethat(=Iresultsinacritically damped loop response.«1produces anunderdamped response, and(>1yieldsan overdamped response. Inpractice, theselection ofthebandwidth ofthePLLinvolves atrade-off between speedofresponse andnoiseinthephaseestimate. whichisthetopic considered below.Ontheonehand,itisdesirable toselectthebandwidth of thelooptobesufficiently widetotrackanytimevariations inthephaseofthe received carrier.Ontheother.awidebandPLLallowsmorenoisetopassinto theloop.whichcorrupts thephaseestimate. Below,weassesstheeffectsof noiseinthequalityofthephaseestimate. 6-2-3ElfectofAdditive NoiseonthePhaseEstimate Inordertoevaluate theeffectsofnoiseontheestimate ofthecarrierphase,let usassumethatthenoiseattheinputtothePLLisnarrowband. Forthis analysis, weassumethatthePLListracking asinusoidal signaloftheform s(t)=A,cos[2Jr[,.r+ct>(t)] (6-2-23) 344 DrGlTAl COMMliNICA T/ONS +61--_+---+_-+--+--+-+:'.f-~ of},,...-1--+'-+---+-+.+-+++1... '-....\ t=j=t=~'~=~0.~70~7i~a~'~"~tt::t:tt=tj1~+! ~ ........I-.f-i=,50 3'I----+---+-+--+-+f-+++I--N'I\~ \,,"<;:""'-f:::.....t~4--t,..::::r1""zIf.;l:\ :t~I---+---l---t--+--+-+-H-IH--+_\:>j-I\.,:>""',*,,,,"-*-+-'K.+-=:~I+t 't 1--+--+_-+--+--J..--j1---t-+-H1---+-.......p1\""""4~~l':::~;'-i==..:.;1.0;:....,...Pf,/g-8 \~"f"T -101--+--I---+---,hH-+-+.J+-+-+......:>rt~~~I>.d+H\.,0.707~-'zl-----t----1-+--+-+-+-++I----l-+--.::j,;"'--I';\._+-'I<"'+'k+- -141--I---J--I---I--I---+--JI--l---I-l--I---l--I-'''-+; \.0';:-.5*~1'J...P,=0.3',1\ -'61---J-+---I---+--HI-++++--I---+-~F-+,,+-N~~ -181-----t--+--1-+-++--H-HI----l-+--+-l-~hf4_d- 'I)L_.L--l_.-l_..L..-L.L.Ll~l-_l-...L_--L_L.._.L...J:>...LL...L -·0.1 0.20..30.4 0.5 0.7l.a ."\-I-5710 oo!w" FIGURE 6-2·5Frequf"ncy response ofasecond·order loop.[FromPhaselock. Techniques. 2ndedition,byF.,"-". Gardner,©/979byJuhnWileyandSOliS,Inc.Reprinted withpermission ofthepublishe.r. J thatiscorrupted bytheadditive narrowband noise n(r)=X(t)cos21!f..!-y(r)sin2Jif,t (6-2·24) Thein·phase andquadrature components ofthenoiseareassumed tobe statistically independent, stationary gaussian noiseprocesses with(two-sided) powerspectraldensity ~NoW1Hz.Byusingsimpletrigonometric identities. the noisetermin(6-2-24)canbeexpressed as where Wenote thatn(l)=n..(r)cos[2nt.I+q,(I)]-n,(t)sin[2nt.!+1/>(t)J n,(I)=X(I)cos<b(I)+y(l)sinI/>(t) n,(I)=-X(I)sincf>(t)+y(t)cos.p(I)(6-2-25) (6-2-26) n,(r)+jn,(t)=[x(t)+jy(t)]e~i~(') sothatthequadrature components nc(l)andn,(t)haveexactlythesame statistical characteristics asX(I)andy(t). IfS(I)+n(l)ismultiplied bytheoutputoftheveoandthedouble· frequency termsareneglected, theinputtotheloopfilteristhenoise­ corrupted signal e(l)=AI'sin!!J.1/>+n,(')sin!!J.cf>-n,(t)cos!!J.cf> =A,sin!!J.cf>+n,(t) (6-2-27) CHAPTER 0:CARRIER ANDSYMBOL SYNCHRONI7AlIO!' 345 FIGURE 6-2-6 Equivalent PLLmodelwithadditive noise VCr) (6-2-28) (6-2-2'1)where,bydefinition, A<p=<p-c/>isthephaseerror.Thus,wehavethe equivalent modelforthePLLwithadditive noiseasshowninFig.6-2-6. Whenthepowerp,=~A;oftheincoming signalismuchlargerthanthe noisepower,wemaylinearize thePLLand,thus,easilydetermine theeffectof theadditive noiseonthequalityoftheestimate <1>.Undertheseconditions. the modelforthelinearized PLLwithadditive noiseisillustrated inFig.6-2-7. Notethatthegainparameter A,maybenormalized tounity,provided thatthe noisetermsarescaledbyl/A"i.e.,thenoisetermsbecome n,(t) n,(t)n,(t)=--.sintJ.<t>---cos tJ.</JAc A, Sincethenoisen,(t)isadditive attheinputtotheloop,thevariance ofthe phaseerrorA<p,whichisalsothevariance oftheveooutputphase.is , _NOBeq CTJ,-A', whereBeqisthe(one-sided) equivalent noisebandwidth oftheloop,givenin (6-2-22). Notethata~issimplytheratiooftotalnoisepowerwithinthe bandwidth ofthePLLdividedbythesignalpowerA2Hence, ai=1/YL (6-2-30) whereYLisdefinedasthesignal-to-noise ratio A'SNR==YL=-'- (6-2-31)lV."Beq FIGURE 6·2·7 Linearized PLLmodelwith.<ldditive noise. veo 346 DIGITAL CO~MUNI('ATIONS FIGURE 6-2~8Comparison ofveophasevariance forexactandapproximate (linearmodel)first-order PLL.[FromPrinciples ofCoherent Communication. byA.J.Vi/rrbi;©/966byMcGraw-Hill BookCompany. Reprinted withpermission ofthepuhiisher.]1.6",;1.4 ~ .~1.2 1.0 ~."0.8~Q, 00.6u> '00.4 ~0.2<"<;;0 00.20.'0.60.81.01.2 NuBe/A~ (6-2-32)Theexpression forthevariance (T~oftheyeOphaseerrorappliestothe casewheretheSNRissufficiently highthatthelinearmodelforthePLL applies. Anexactanalysis basedonthenonlinear PLLismathematically tractable whenG(s)=1,whichresultsinafirst-order loop.Inthiscase,the probability densityfunction forthephaseerrormaybederived(seeYiterbi, 1966)andhastheform p(!J.cjJ)=exp(I'Lcos!J.cjJ) 21rfn(I'JJ where1'1.istheSNRgivenby(6-2-31)withB,qbeingtheappropriate noise bandwidth ofthefirst-order loop,and1,,(-)isthemodified Besselfunction of orderzero. Fromtheexpression forp(!J.cjJ),wemayobtaintheexactvalueofthe variance forthephaseerroronafirst-order PLLThisisplottedinFig.6-2-8as afunction of1/1'/.Alsoshownforcomparison istheresultobtained withthe linearized PLLmodel.Notethatthevariance forthelinearmodeliscloseto theexactvariance forI'L>3.Hence,thelinearmodelisadequate forpractical purposes. Approximate analyses ofthestatistical characteristics ofthephaseerrorfor thenonlinear PLLhavealsobeenperformed. Ofparticular importance isthe transient behavior ofthePLLduringinitialacquisition. Another important problem isthebehavior ofPLLatlowSNR.Itisknown,forexample, that whentheSNRatthe input tothePLLdropsbelowacertainvalue,thereisa rapiddeterioration intheperformance ofthePLL.Theloopbeginstolose lockandanimpulsive-type ofnoise,characterized asclicks,isgenerated which degrades theperformance oftheloop.Resultsonthesetopicscanbefoundin thetextsbyYiterbi(1966),Lindsey (1972),LindseyandSimon(1973),and Gardner (1979),andinthesurveypapersbyGupta(1975)andLindsey and Chie(1981). Uptothispoint,wehaveconsidered carrierphaseestimation whenthe carriersignalisunmodulated. Below,weconsider carrierphaserecovery when thesignalcarriesinformation. CHAPTER fi:CARRIER ANDSYMBOL SYNCHRO"IZATIO~ 347 6-2-4Decision-Directed Loops Aproblem arisesinmaxlmlzmg either(6-2-9)or(6-2-10) whenthesignal S(I;q,)carriestheinformation sequence {In}.Inthiscasewecanadoptoneof twoapproaches: eitherweassume that{In}isknownorwetreat{In}asa randomsequence andaverageoveritsstatistics. Indecision-directed parameter estimation, weassumethattheinformation sequence {Io}overtheobservation interval hasbeenestimated and,inthe absenceofdemodulation errors,t=1mwhere1.denotes thedetected valueof theinformation I".Inthiscases(t;¢)iscompletely knownexceptforthe carrierphase.Decision-directed phaseestimation wasfirstdescribed by Proakiselal.(1964). Tobespecific, letusconsider thedecision-directed phaseestimate forthe classoflinearmodulation techniques forwhichthereceivedequivalent lowpass signalmaybeexpressed as n =s,(I)e-j<J> +Z(I) (6-2-33) wheres,(t)isaknownsignalifthesequence {In}isassumed known. The likelihood function andcorresponding log-likelihood function fortheequiv­ alentlowpasssignalare A(q,)=Cexp{Re[~J,"r(l)s,*(t)ei<J> dt]} AL(4))=Re{[~ir(l)s,*(I)dl] e1<J>} ~OTo(6-2-34) (6-2-35) Ifwesubstitute fors,(I)in(6-2-35) andassumethattheobservation interval To=KT,whereKisapositive integer,weobtain {'1K-'1<n+l)T } AL(4»=RetI<J>No~oI::nTr(l)g*(t-nT)dl {IK-l }=Ree1<J>-2:I::Yn NOn~o where,bydefinition 1<n+l)T Yn= r(1)g*(t-nT)dl nT(6-2-36) (6-2-37) NotethatYnistheoutputofthematched filterinthenthsignalinterval. The MLestimate of4>iseasilyfoundfrom(6-2-36) bydifferentiating the log-likelihood (1K-I) (1K-')AL(4))=Re No2:I::Yncosq,-Im -2:I::Ynsinrt> on=O No11=0 348 DIGITAL COMMUNICATIONS sin(2x!..,+.)...------,Received signal vcoto(/) x'l---l FIGURE 6-2-9Carrierrecovery withadecision-feedback PLL. withrespecttoc/>andsettingthederivative equaltozero.Thus,weobtain "'ML=-tan-I[1m(~~I:Yn)/Re(~~I:Yn)] (6-2-38) Wecall"'MLin(6-2-38) thedecision-directed (ordecision-feedback) carrier phaseestimate. Itiseasilyshown(Problem 6-10)thatthemeanvalueofIf>MLis </I,sothattheestimate isunbiased. F"rthermore, thepdfofIf>MLcanbe obtained (Problem 6-11)byusingtheprocedure described inSection5-2-7. Adecision-feedback PLL(DFPLL) thatisappropriate foradouble­ sideband PAMsignaloftheformA(I)cos(2nfct+4»isshowninFig. 6-2-9. Thereceived signalismultiplied bythequadrature carriersce(t)andc,(t),as givenby(6-2-5),whicharederivedfromtheYeo.Theproductsignal r(t)cos(21ifct+If>)=HA(t)+ne(t)]costJ.4> -!n,(t)sintJ.4>+double-frequency terms(6-2-39) isusedtorecovertheinformation carriedbyA(t).Thedetector makesa decisiononthesymbolthatisreceived everyTseconds. Thus,intheabsence ofdecision errors,itreconstructs A(t)freeofanynoise.Thisreconstructed signalisusedtomuhiply theproductofthesecondquadrature multiplier, whichhasbeendelayed byTsecondstoallowthedemodulator toreacha decision. Thus,theinputtotheloopfilterintheabsenceofdecisionerrorsis theerrorsignal e(t)=!A(t){[A(t) +ne(t)]sin4</1-n,(t)cosa</l} +double-frequency terms =!A2(t)sintJ..p+!A(t)[n,(t) sintJ.</I-n,(t)cos~4>] +double-frequency terms (6-2-40) Theloopfilterislowpassand,hence,itrejectsthedouble-frequency termin e(t).Thedesiredcomponent isA2(t)sin~4>,whichcontainsthephaseerrorfor driving.theloop. Received ~ignalCHAPTER 6:CARRIER ANDSYMBOL SYNCHRONIZAriON 349 x FIGURE: 6-2·10 Carrierrecovery forM-aryPSKusingadecision-feedback PLL. InthecaseofM-aryPSK,theDFPLLhastheconfiguration showninFig. 6-2-10.Thereceived signalisdemodulated toyieldthephaseestimate ,2rre=-(m-1) tilM which.intheabsenceofadecision error,isthetransmitted signalphase.The twooutputsofthequadrature multipliers aredelayedbythesymbolduration Tandmultiplied bycosemandsinemtoyield r(l)cos(2rrj,.t+4»sinem =HAcos9m+n,(I)]sinemcos(c/>-4» -HAsinem+n,(I)]sin9..sin(c/J-1» +double-frequency terms r(l)sin(2rrf..t+$)cos9m =-HAcos9m+n,.(I)]cos6msin(c/J -$) -HAsinOm+n,(I)]cosOmcos(c/J-cf,) +double-frequence terms(6-2-41) 3SO DIGITAL COMMUNICATIONS Thetwosignalsareaddedtogenerate theerrorsignal e(t)=-~Asin (cP-cl»+~nc(t)sin(cP -?>-8m) +~n,(t)cos(cP-?>-8.,)+double-frequency terms (6-2-42) Thiserrorsignalistheinputtotheloopfilterthatprovides thecontrolsignal fortheYCO. Weobservethatthetwoquadrature noisecomponents in(6-2-42)appearas additiveterms.Thereisnoterminvolving aproductortwonoisecomponents asinanMth-power lawdevice,described inthenextsection.Consequently, thereisnoadditional powerlossassociated withthedecision-feedback PLL. ThisM-phasetrackingloophasaphaseambiguity of3600/M,necessitating theneedtodifferentially encodetheinformation sequence priortotransmis­ sionanddilferentially decodethereceived sequence afterdemodulation to recovertheinformation. TheMLestimate in(6-2-38)isalsoappropriate forQAM.TheMLestimate foroffsetQPSKisalsoeasilyobtained (Problem 6-12)bymaximizing the log-likelihood functionin(6-2-35), withs/(t)givenas s,(t)=Ll"g(t-nT)+jLI.g(r-nT-~T) (6-2-43) • • whereI.=±1andi"=±1. Finally,weshouldalsomention thatcarrierphaserecovery forCPMsignals canbeaccomplished in'adecision-directed mannerbyuseofaPLL.Fromthe optimum demodulator forCPMsignals,whichisdescribed inSection5-3,we cangenerate anerrorsignalthatisfilteredinaloopfilterwhoseoutputdrives aPLL. 6-2-5Non-Decision-Directed Loops Insteadofusingadecision-directed schemetoobtainthephaseestimate, we maytreatthedataasrandomvariables andsimplyaverageA(cP)overthese randomvariables priortomaximization. Inordertocarryoutthisintegration, wemayuseeithertheactualprobability distribution functionofthedata,ifit isknownor,perhaps, wemayassumesomeprobability distribution thatmight beareasonable approximation tothetruedistribution. Thefollowing example illustrates thefirstapproach. Example 6-2-2 Suppose therealsignals(t)carriesbinarymodulation. Then,inasignal interval, wehave s(t)=Acos21Cfct,0..t..T CHAPTER 0CARRIER ANDSYMBOL SyNCHRONIZATION 351 whereA=±Iwithequalprobability. Clearly,thepdfofAisgivenas p(A)=~o(A-1)+jo(A+I) Now,thelikelihood function A(r!»givenby(6-2-9)isconditional onagiven valueofAandmustbeaveraged overthetwovalues.Thus, A(r!»=rA(r!>)p(A)dA =~exp[~)rr(t)cos(2rrfct+cf»dt] +~exp[-~JTr(t)cos(2rr[,1+r!»lit] M."0 =cosh[~jTr(l)cos(2rrf;.t+r!»lil] NoI) andthecorresponding log-likelihood function is 5.,(r!»=Incosh[~jrr(t)cos(2rr[,t +r!»dt] !Vel0 .(6-2-44) Ifwedifferentiate Ar(cf»andsetthederivative equaltozero,weobtainthe MLestimate forthenon-decision-directed estimate. Unfortunately, the functional relationship in(6-2-44) ishighlynonlinear and,hence,anexact solution isdifficult toobtain.Ontheotherhand.approximations are possible. Inparticular, {~X2(Ixl~I)Incoshx= Ixl(Ixl'"1) Withtheseapproximations. thesolution for¢becomes tractable.(6-2-45) Inthisexample, weaveraged overthetwopossible valuesofthe information symbol.Whentheinformation symbols areM-valued, whereMis large,theaveraging operation yieldshighlynonlinear functions ofthe parameter tobeestimated. Insuch acase,wemaysimplify theproblem by assuming thattheinformation symbols arecontinuous random variables. For example, wemayassume thatthesymbols arezero-mean gaussian. The following example illustrates thisapproximation andtheresulting formforthe averagelikelihood function. (6-2-46)352 DIGITAL COM"UNICATlO'S EllllIIIple 6-2-3 Letusconsider thesamesignalasinExample 6-2-2,butnowweassume thattheamplitude Aiszero-mean gaussian withunitvariance. Thus, piA)=_1_e-A2"V2i Ifweaverage A(e/»overtheassumed pdfofA,weobtaintheaverage likelihood A(e/»intheform A(c/»=Cexp([~orT(I)cos(2tifcl+4»dtn andthecorresponding log-likelihood as AL(e/»=[~ofr(l)cos(21rfct+e/»dlr (6-2-47) WecanobtaintheMLestimate ofe/>bydifferentiating AL(e/»andsetting thederivative tozero. Itisinteresting tonotethatthelog-likelihood function isquadratic under thegavssian assumption andthatitisapproximately quadratic, asindicated in (6-2-45)forsmallvaluesofthecross-correlation ofr(t)withset;4».Inother words,ifthecross-correlation overasingleinterval issmall,thegaussian assumption forthedistribution oftheinformation symbols yieldsagood approximation tothelog-likelihood function. Inviewofthese results, wemayusethegaussian approximation onallthe symbolsintheobservation interval10=KT.Specifically, weassumethattheK information symbols arestatistically independent andidentically distributed. Byaveraging thelikelihood function A(e/»overthegaussian pdfforeachof theKsymbolsintheinterval To=KT,weobtaintheresult {K-I[2r.+llT ]2}A(e/»=Cexp2:- ret)cos(2tifct+4»dl ~""ONonT(6-2-48) Ifwetakethelogarithm of(6-2-48), differentiate theresulting log-likelihood function, andsetthederivative equaltozero,weobtainthecondition forthe MLestimate as K.-'J.(,+I)T It'+I)T n~o.T r(t)cos(21if,t+~)dt. Tr(t)sin(21rfct+$)dt'=0 (6-2-49) CHAPTER Il:CARRH:R A;'IODSYMBOL SYNnIRO~IZATION 353 Sampler x ! (I'"I,=nT r~')veo <;in(,!rrl~l+bl Sampler x j (Idl I J=/IT FIGURE 6-2-11 Non-decision-din:cted PLLforcarrierphaseestimations ofPAMsignals. Although thisequation canbemanipulated further,itspresentformsuggests thetracking loopconfiguration illustrated inFig.6-2-11.Thisloopresembles a Costasloop.whichisdescribed below.Wenotethatthemultiplication ofthe twosignalsfromtheintegrators destroys thesigncarriedbytheinformation symbols. Thesummer playstheroleoftheloopfilter.Inatracking loop configuration, thesummer maybeimplemented eitherasasliding-window digitalfilter(summer) orasalowpassdigitalfilterwithexponential weighting ofthepastdata. Inasimilarmanner, onecanderivenon-decision directed MLphase estimates forQAMandM-PSK.Thestartingpointistoaveragethelikelihood function givenby(6-2-9)overthestatistical characteristics ofthedata.Here again,wemayusethegaussian approximation (two-dimensional gaussian for complex-valued information symbols) inaveraging overtheinformation sequence. Squaring LoopThesquaring loopisanon-decision-directed loopthatis widelyusedinpractice to establish thecarrierphaseofdouble-sideband suppressed carriersignalssuchas PAM. Todescribe itsoperation, consider the problem ofestimating thecarrierphaseofthedigitallymodulated PAMsignal oftheform s(r)=A(t)cos(21if,r+c/J) (6-2-50) whereA(1)carriesthedigitalinformation. NotethatE[s(t»)=E[A(t»)=0 whenthesignallevelsaresymmetric aboutzero.Consequently, theaverage valueofset)doesnotproduce anyphasecoherent frequency components at anyfrequency. including thecarrier.Onemethodforgenerating acarrierfrom thereceived signalistosquarethesignaland,thus,togenerate afrequency component at2j;.,whichcanbeusedtodriveaphase-locked loop(PLL)tuned to2[..Thismethod isillustrated intheblockdiagram showninFig.6-2-12. 354 DIGITAL COMMUN ICATlONS ,(t)SqllllR-law device (full-wave rectifier),'(I)Bandpass filter lunedto 2/, sin(211J;,I+.> Output10coherentL.-_...J demodulator Frequency divider FIGURE 6-2-12 Carrierrecovery u.ingasquare-law device. Theoutputofthesquare-law deviceis S2(t)=A2(t)cos2(2iifct+</» =!A2(t)+!A2(t)cos(41rfJ+2<1» (6-2-51) Sincethemodulation isacyclostationary stochastic process,theexpected value ofS2(t)is (6-2-52) Hence,thereispoweratthefrequency 21,. Iftheoutputofthesquare-law deviceispassedthroughabandpass filter tunedtothedouble-frequency termin(6-2-51),themeanvalueofthefilterisa sinusoid withfrequency 2fc,phase2</>,andamplitude !E[A2(t»)H(2fc), where H(2fc)isthegainofthefilteratf=2tc.Thus,thesquare-law devicehas produced aperiodic component fromtheinputsignalset).Ineffect,the squaring ofset)hasremoved thesigninformation contained inA(t)and,thus, hasresultedinphase-coherent frequency components attwicethecarrier.The filteredfrequency component at2fcisthenusedtodrivethePLL. Thesquaring operation leadstoanoiseenhancement thatincreases the noisepowerlevelattheinputtothePLLandresultsinanincrease inthe variance ofthephaseerror. Toelaborate onthispoint,lettheinputtothesquarerbeset)+net),where set)isgivenby(6-2-50)andnet)represents thebandpass additive gaussian noiseprocess.Bysquaring set)+net),weobtain (6-2-53) whereS2(t)isthedesiredsignalcomponent andtheothertwocomponents are thesignalxnoiseandnoisexnoiseterms.Bycomputing theautocorrelation functions andpowerdensityspectraofthesetwonoisecomponents, onecan CHAPIER f,:['ARRIl'R A~DSY~fHOL ,YNctIRO:-.:lI.i\r!OI\ 355 easilvshowthatbothcomponents havespectralpowerinthefrequency band cellt~red at2f..Consequently. thebandpass filterwithbandwidth Bhpcentered at2f,.whichproduces thedesiredsinusoidal signalcomponent thatdrivesthe PLL.alsopassesnoiseduetothesetwoterms. Sincethebandwidth oftheloopisdesigned tobesignificantly smallerthan thebandwidth Bhl_ofthebandpass filter,thetotalnoisespectrum attheinput tothePLLmaybeapproximated asaconstant withintheloopbandwidth. This approximation allowsustoobtainasimpleexpression forthevariance ofthe phaseerroras (6-2-54) where51iscalledthesquaring lossandisgivenby (6-2-55) Since5/<I.5/'represents theincrease inthevariance ofthephaseerror causedbytheaddednoise(noisexnoiseterms)thatresultsfromthesquarer. Note,forexample. thatwhenYI=Bhp/2Boq,thelossis3dB. Finally,weobservethattheoutputoftheVCOfromthesquaring loopmust befrequency-divided by2togenerate thephase-locked carrierforsignal demodulation. Itshouldbenotedthattheoutputofthefrequency dividerhas aphaseambiguity oflSO°relativetothephaseofthereceived signal.Forthis reason,thebinarydatamustbedifferentially encoded priortotransmission anddifferentially decoded atthereceiver. CostasLoopAnother methodforgenerating aproperly phasedcarrierfor adouble-sideband suppressed carriersignalisillustrated bytheblockdiagram showninFig.6-2-13.Thisscheme wasdeveloped byCostas(1956)andis FIGURE 6-2-13 BlockdiagramofCostasloop.x}-__.J (6-2-56)356 DIGIiAtCOMMUNICATlONS calledthtCostasloop.Thereceived signalismultiplied bycos(21Cfct+cI»and sin(2tcfct+4J).whichareoutputsfromtheVCO.Thetwoproducts are yc(t)=[s(t)+net)]cos(21Cfct+J,) =!IA(t)+nc(t)]costi</>+!n,(t)sin6.</> +double-frequency terms y,(I)=[set)+net»)sin(21ifc1+J,) =!IA(t)+nc(t»)sinti.</>-!nAt)cos6.</> +double-frequency terms wherethephaseerrorti</>=cI>-</>.Thedouble-frequency termsareeliminated bythelowpassfiltersfollowing themultiplications. Anerrorsignalisgenerated bymultiplying thetwooutputsofthelowpass filters.Thus, e(t)=H[A(t)+nc(tW-n~(t)}sin(2ti</» -!n,(t)[A(t) +n,(I)]cos(26.</» (6-2-57) Thiserrorsignalisfilteredbytheloopfilter.whoseoutputisthecontrol voltagethatdrivestheVCO.Thereadershouldnotethesimilarity ofthe G:lstaslooptothePLLshowninFig.6-2-11. Wenotethattheerrorsignalintotheloopfilterconsistsofthedesiredterm A2(t)sin2(cI>-</»plustermsthatinvolvesignalxnoiseandnoisexnoise. ThesetermsaresimilartothetwonoisetermsattheinputtothePLLforthe squaring method.Infact,iftheloopfilterintheCostasloopisidentical tothat usedinthesquaring loop,thetwoloopsareequivalent. Underthiscondition, theprobability densityfunctionofthephaseerrorandtheperformance ofthe twoloopsareidentical. Itisinteresting tonotethattheoptimum lowpassfilterforrejecting the double-frequency termsintheCostasloopisafiltermatched tothesignal pulseintheinformation-bearing signal.Ifmatched filtersareemployed forthe lowpassfilters,theiroutputscouldbesampled atthebitrate,attheendof eachsignalinterval, andthediscrete-time signalsamplescouldbeusedtodrive theloop.Theuseofthematched filterresultsinasmallernoiseintotheloop. Finally,wenotethat,asinthesquaring PLL,theoutputoftheVCO contains aphaseambiguity of180·.necessitating theneedfordifferential encoding ofthedatapriortotransmission anddifferential decoding atthe demodulator. CurierEsti...tionforMultiplePbueSignals Whenthedigitalinforma­ tionistransmitted viaM-phasemodulation ofacarrier,themethods described abovecanbegeneralized toprovide theproperly phasedcarrierfor CHAl'rt ..'RroC,\H,KIER ANDSY~tBOL SYNCHH,()!"'IZATlOS 357 Rl·~·l·i\ ...',',.\lIh-l.....l\\l'f ,i~I1;'1 <int..',·BalldpJ.... fthl'r wned10 JII/ Frt-quen~')' divider ,,:-M Output fiGURE: (1·2-14 Carrier recO\'er~' "",·jrh.\1rhpowerlawde\'iceforM-aryPSK. demodulation. Thereceived M-phase signal,excluding theadditive noise.may beexpressed as s(t)=Acos[2Jrtl+'"+~(m-1)lm=1,2,...,M(6-2-58) where2Jr(m-1)/Mrepresents theinformation-bearing component ofthe signalphase.Theproblem incarrierrecovery is10remove theinformation­ bearingcomponent and.thus,toobtaintheunmodulated carriercos(2Jrj;.1+ "').Onemethodbywhichthiscanbeaccomplished isillustrated inFig.6-2-14. whichrepresents ageneralization ofthesquaring loop.Thesignalispassed through anMth-power-Iaw device.whichgenerates anumberofharmonics of J..Thebandpass filterseJectstheharmonic cos(2JrMfcl+Me/»fordrivingthe PLL.Theterm 2JrM(m-l)M=2Jr(m-l)==O (mod2Jr), m=1,2....,M Thus,theinformation isremoved. TheyeOoutputissin(2JrMf./+MJ,),so thisoutputisdivided infrequency byMtoyieldsin(2JrfJ+cf,),and phase·shifted by1Jrradtoyieldcos(2Jrfct+J,).Thesecomponents arethenfed tothedemodulator. Although notexplicitly shown,thereisaphaseambiguity inthesereference sinusoids of3600 /M,whichcanbeovercome bydifferential encoding ofthedataatthetransmitter anddifferential decoding after demodulation atthereceiver. Justasinthecaseofthesquaring PLL,theMth-power PLLoperates inthe presence ofnoisethathasbeenenhanced bytheMth-power-law device.which resultsintheoutput y(ll=[S(I)+n(I)]M 358 DIGITAL COMMUNICATIONS Thevariance ofthephaseerrorinthePLLresulting fromtheadditivenoise maybeexpressed inthesimpleform S-l 2MLu·=- '"'YL(6-2-59) where'YListheloopSNRands;;.listheM-phase powerloss.SMLhasbeen evaluated byLindseyandSimon(1973)forM=4and8. Another' method forcarrierrecovery inM·aryPSKisbasedona generalization oftheCostasloop.Thatmethod requires multiplying the receivedsignalbyMp!lase-shifted carriersoftheform sin[2nfcl+4>+;(k-1)].k=1,2,...,M lowpass-filtering eachproduct, andthenmultiplying theoutputsofthelowpass filterstogenerate theerrorsignal.Theerrorsignalexcitestheloopfilter, which,inturn,provides thecontrolsignalfortheVCO.Thismethod is relatively complex toimplement and,consequently, !lasnotbeengenerally usedinpractice. Comparison 01Decision-DiJeded withNon-DecisioD-Direded Loops Wenotethatthedecision-feedback p!lase-locked loop(DFPLL) differsfrom theCostaslooponlyinthemethod bywhichA(t)isrectified forthe purposeofremoving themodulation. IntheCostasloop,eachofthetwo quadrature signalsusedtorectifyA(t)iscorrupted bynoise.IntheDFPLL, onlyoneofthesignalsusedtorectifyA(I)iscorrupted bynoise.Onthe otherhand,thesquaring loopissimilartotheCostasloopintermsofthe noiseeffectontheestimate If,.Consequently, theDFPLL issuperior in performance toboththeCostasloopandthesquaring loop,provided that thedemodulator isoperating aterrorratesbelow10-2whereanoccasional decisionerrorhasanegligible effecton</>.Quantitative comparisons ofthe variance ofthephaseerrorsinaCostaslooptothoseinaDFPLL have beenmadebyLindseyandSimon(1973),ands!lowthatthevarianceofthe DFPLLis4-10timessmallerforsignal-to-noise ratiosperbitaboveOdb. 6-3SYMBOL TIMING ESTIMAnON Inadigitalcommunication system,theoutputofthedemodulator mustbe sampled periodically atthesymholrate,attheprecisesampling timeinstants 1m=mT+'1',whereTisthesymbolintervaland'l'isanominaltimedelaythat accounts forthepropagation timeofthesignalfromthetransmitter tothe receiver. Toperform thisperiodicsampling, werequireaclocksignalatthe CHAPlFR 6:CARRIEK\ND "Y\1BOI SYNCHRONI/ATlOi'l" 359 receiver. Theprocessofextracting suchaclocksignalatthereceiver isusual\, calledsymbolsynchronization ortimingrecovery. Timingrecovery isoneofthemostcriticalfunctions thatisperformed atthe receiver ofasynchronous digitalcommunication system.Weshouldnote that thereceiver mustknownotonlythefrequency (IJT)atwhichtheoutputsof thematched filtersorcorrelators arcsampled. butalsowhere10takethe samples withineachsymbolinterval. Thechoiceofsampling instantwithinthe symbolintervalofduration Tiscalledthelimingphase. Symbol synchronization canbeaccomplished inoneofseveralways.In somecommunication systems, thetransmitter andreceiver clocksaresyn­ chronized toamasterclock,whichprovides awryprecisetimingsignal.Inthis case.thereceiver mustestimate andcompensate fortherelative timedelal' between thetransmitted andreceived Signals.Suchmaybethecaseforradio communication systems thatoperate intheverylowfrequency (VLF)hand (below30kHz),wherepreciseclocksignalsarctransmitted fromamaster radiostation. Another methodforachieving symbolsynchronization isforthetransmitter tosimultaneously transmit theclockfrequency liToramultiple of1/Talong withtheinformation signal.Thereceiver maysimplyemploy anarrowhand filtertunedtothetransmitted clockfrequency and,thus,extracttheclock signalforsampling. Thisapproach hastheadvantage ofheingsimpleto implement. Thereareseveral disadvantages. however. Oneisthatthe transmitter mustallocatesomeofitsavailable powertothetransmission ofthe clocksignal.Another isthatsomesmallfraction oftheavailahle channel bandwidth mustbeallocated forthetransmission oftheclocksignal.Inspiteof thesedisadvantages, thismethod isfrequently usedintelephone transmission systemsthatemploylargebandwidths totransmit thesignalsofmanyusers.In suchacase,thetransmission ofaclocksignalissharedinthedemodulation of thesignalsamongthemanyusers.Through thisshareduseoftheclocksignal, thepenalty intransmiller powerandinbandwidth allocation isreduced proportionally bythenumberofusers. Aclocksignalcanalsobeextracted fromthereceived datasignal.Thereare anumber ofdifferent methods thatcanbeusedatthereceiver toachieve self-synchronization. Inthissection, wetreathothdecision-directed and non-decision-directed methods. 6-3-1Maximum-Likelihood TimingEstimation Letusbeginbyobtaining theMLestimalc ofthetimedelayr.Ifthesignalisa baseband PAMwaveform, itisrepresented as wherer(t)=.'(1;r)+11(1) s(l;r)=LIn);(1-nT--r) n(63-[) (6-3-c) 360 DIGITAL COMMUNICATIONS AsinthecaseofMLphaseestimation, wedistinguish between twotypesof timingestimators, decision-directed timingestimators andnon-decision­ directed estimators. Intheformer,theinformation symbolsfromtheoutputof thedemodulator aretreatedastheknowntransmitted sequence. Inthiscase, thelog-likelihood function hastheform AL(r)=CL(r(t)s(t;r)dtlTo Ifwesubstitute (6-3-2)into(6-3-3),weobtain AL(r)=CLLIn1r(t)g(t-nT-r)dt "ro =CLLInyir) n whereYn(t)isdefinedas Yn(r)=1r(t)g(t-nT-r)dt To Anecessary condition forttobetheMLestimate ofristhat(6-3-3) (6-3-4) (6-3-5) LIn:1r(t)g(t-nT-r)dt "t10 d =LIn-d[Yn(r)]=0 nr(6-3-6) Theresultin(6-3-6)suggests theimplementation ofthetracking loopshown inFig.6-3-1.Weshouldobservethatthesummation intheloopservesasthe loopfilterwhosebandwidth iscontrolled bythelengthoftheslidingwindowin thesummation. Theoutputoftheloopfilterdrivesthevoltage-controlled clock (Veq,orvoltage-controlled oscillator, whichcontrols thesampling timesfor theinputtotheloop.Sincethedetected information sequence {In}isusedin theestimation ofr,theestimate isdecision-directed. Thetechniques described aboveforMLtimingestimation ofbaseband nGURE 6-3-1Decision-directed MLestimation oftimingforbaseband PAM. I, r(1) Matd1ed filter g(-I).!.(.) dtSampler nT+tML CH"rTER h:CARRIER ANDSYMHOL SY!'CHRf)."IZAll()~ 361 PAMsignalscanbeextended tocarriermodulated signalformats suchas QAMandPSKinastraightforward manner, bydealingwiththeequivalent lowpassformofthesignals.Thus.theproblem ofMLestimation ofsymbol timingforcarriersignalsisverysimilartotheproblem formulation forthe baseband PAMsignal. 6-3-2Non-Decision-Directed TimingEstimation Anon-decision-directed timingestimate canbeobtained byaveraging the likelihood ratioA(f)·overthepdfoftheinformation symbols, toobtainA(f). andthendifferentiating eitherA(f)orIn}i(f)=}iLlf)toobtainthe-condition forthemaximum-likelihood estimate TML' Inthecaseofbinary(baseband) PAM,whereIn=±1withequalprob­ ability,theaverageoverthedatayields }iL(f)=L:IncosheYn(f) n(6-3-7) justasinthecaseofthephaseestimator, SinceIncoshx=~X2forsmallx.the square-law approximation (6-3-8) (6-3-9)n isappropriate forlowsignal-to-noise ratios.Formultilevel PAM,wemay approximate thestatistical characteristics oftheinformation symbols {In}by thegaussian pdf,withzeromeanandunitvariance. WhenweaverageA(r) overthegaussian pdf,thelogarithm ofA(r)isidentical toAL(r)givenby (6-3-8).Consequently, thenon-decision-directed estimateofrmaybeobtained bydifferentiating (6-3-8).Theresultisanapproximation totheMLestimate of thedelaytime.Thederivative of(6-3-8)is ddL:y~(f)=22:Yn('r)dYdn(r)=0 tn n t whereYn(r)isgivenby(6-3-5). Animplementation ofatracking loopbasedontnederivative of/\L(r) givenby(6-3-7)isshowninFig.6-3-2.Alternatively, animplementation ofa FIGURE 6-3-2Non-declSion-directed estimation 01timinglorbinarybaseband PAM. r(t)Malched filter g(-t)Nonlinear device (0)1 or1,1 orIncosh(o)Sampler 362 DIGITAL COMMUNICATIONS r\1} FIGURE 6·.J..3Non-decision-directed estimation oftiming forbaseband PAM.Matched filter g(-I)Sampler nT+t Sampler trackingloopbasedon(6-3-9)isillustrated inFig.6-3-3.Inbothstructures, we observethatthesummation servesastheloopfilterthatdrivestheVCe.Itis it1teresting tonotetheresemblance ofthetimingloopinFig.6-3-3tothe Costasloopforphaseestimation. Early-Late GateSynchronizers Another non-decision-directed timinges­ timatorexploitsthesymmetry properties ofthesignalattheoutput of the matched filterorcorrelator. Todescri':>e thismethod, letusconsider the rectangular pulses(t),0""t""T,showninFig.6-3-4(a). Theoutputofthefilter matched tos(t)attainsitsmaximum valueattimet=T,asshowninFig. 6-3-4(b). Thus,theoutputofthematched filteristhetimeautocorrelation function ofthepulses(t).Ofcourse,thisstatement holdsforanyarbitrary pulseshape,sotheapproach thatwedescribe appliesingeneraltoanysignal pulse.Clearly,thepropertimetosampletheoutputofthematched filterfora maximum outputisatt=T,Le..atthepeakofthecorrelation function. Inthepresence ofnoise.theidentification ofthepeakvalueofthesignalis generally difficult. Insteadofsampling thesignalatthepeak,suppose we sampleearly,att=T-{jandlateatt=T+l3.Theabsolute valuesofthe earlysamplesly(m(T-'0»)1andthelatesamplesly(m(T+{j))lwillbesmaller (ontheaverageinthepresence ofnoise)thanthesamplesofthepeakvalue ly(mT)l. Sincetheautocorrelation function isevenwithrespecttothe optimum sampling timet=T,theabsolutevaluesofthecorrelation functionat 1=T-{jandt=T+{jareequal.Underthiscondition, thepropersampling FIGURE 6-3-4Rectangular signalpulse(a!andils matched filteroutput(b).'1J=h'A oT, la'MatchedfilterOlXpul Early Optimum sample/'sample :~Latesample,',', oT-oTTT+or2T tb' CHAPTfR h:rARRlfR ANDSYMBOL SYNCHRONIZATION 363 RCCCIVl.:d ..ignal FJGURE 6-3-5 BlockdiJgram ofearly-late gatesynchromzer. timeisthemidpoint between t=T-8andt=T+8.Thiscondition formsthe basisfortheearly-late gatesymbolsynchronizer. Figure6-3-5illustrates theblockdiagram ofanearly-late gatesynchronizer. Inthisfigure,correlators areusedinplaceoftheequivalent matched filters. Thetwocorrelators integrate overthesymbolintervalT,butonecorrelator startsintegrating 8secondsearlyrelativetotheestimated optimum sampling timeandtheotherintegrator startsintegrating 8seconds laterelativetothe estimated optimum sampling time.Anerrorsignalisformedbytakingthe difference between theabsolute valuesofthetwo correIa toroutputs. To smooththenoisecorrupting thesignalsamples, theerrorsignalispassed throughalowpassfilter.Ifthetimingisoffrelativetotheoptimum sampling time,theaverageerrorsignalattheoutputofthelowpassfilterisnonzero, and theclocksignaliseitherretarded oradvanced, depending onthesignofthe error.Thus,thesmoothed errorsignalisusedtodriveavoltage-controlled clock(VCC),whoseoutputisthedesiredclocksignalthatisusedforsampling. Theoutputofthevceisalsousedasaclocksignalforasymbolwaveform generator thatputsoutthesamebasicpulsewaveform asthatofthe transmitting filter.Thispulsewaveform isadvanced anddelayedandthenfed tothetwocorreiators,asshowninFig.6-3-5.Notethatifthesignalpulsesare rectangular, thereisnoneedforasignalpulsegenerator withinthetracking loop. Weobservethattheearly-late gatesynchronizer isbasically aclosed-loop controlsystemwhosebandwidth isrelatively narrowcompared tothesymbol rateliT.Thebandwidth oftheloopdetermines thequalityofthetiming estimate. Anarrowband loopprovides moreaveraging overtheadditivenoise and,thus,improves thequalityoftheestimated sampling instants, provided thatthechannelpropagation delayisconstant andtheclockoscillator atthe transmitter isnotdriftingwith time (ordriftingveryslowlywithtime).Onthe otherhand,ifthechannelpropagation delayischanging with time and/orthe J64 DIGITAL COMMUNICATIONS Received signal FIGURE 6-3-6Blockdiagramofearly-late gatesynchronizer-an alternative form. transmitter clockisalsodriftingwithtimethenthebandwidth oftheloopmust beincreased toprovideforfastertrackingoftimevariations insymboltiming. Inthetrackingmode,thetwocorrelators areaffected byadjacent symbols. However, ifthesequence ofinformation symbolshaszeromean,asisthecase forPAMandsomeothersignalmodulations, thecontribution totheoutputof thecorreiatorsfromadjacentsymbolsaverages outtozerointhelowpassfilter. An"equivalent realization ofthe,early-late gatesynchronizer thatis somewhat easiertoimplement isshowninFig.6-3-6.Inthiscasetheclock signalfromtheveeisadvanced anddelayedby8,andtheseclocksignalsare usedtosampletheoutputsofthetwocorrelators. Theearly-late gatesynchronizer described aboveisanon-decision-directed estimator ofsymboltimingthatapproximates themaximum-likelihood es­ timator.Thisassertion canbedemonstrated byapproximating thederivative of thelog-likelihood functionbythefinitedifference, i.e., AL(r+8)-AL(r- 8) 28(6-3-10) Ifwesubstitute forAL(r)from(6-3-8)into(6-3-10), weobtaintheapproxima­ tionforthederivative as C' 48~[y~(r+8)-y~(r-8)] "=C22:{[f.r(t)g(t-nT-r-8)dr]' 45n7i, -[fTor(r)g(r-nT-r+8)dr]'} (6-3-11) CHAPTER ti:CARRIER I\NDSYMBOL SYNC'HRONI7.ATION 365 Butthemathematical expression in(6-3-11)basically describes thefunctions performed bytheearly-late gatesymbolsynchronizers illustrated inFigs6-3-5 and6-3-6. 6-4JOINTESTIMATION OFCARRIER PHASE ANDSYMBOL TIMING Theestimation ofthecarrierphaseandsymboltimingmaybeaccomplished separately asdescribed aboveorjointly.JointMLestimation oftwoormore signalparameters yieldsestimates thatareasgoodandusuallybellerthanthe estimates obtained fromseparate optimization ofthelikelihood function. In otherwords.thevariances ofthesignalparameters obtained fromjoint optimization arelessthanorequaltothevariance ofparameter estimates obtained fromseparately optimizing thelikelihood function. Letusconsider thejointestimation ofthecarrierphaseandsymboltiming. Thelog-likelihood function forthesetwoparameters maybeexpressed in termsoftheequivalent lowpasssignalsas AL(",.r)=Re[..!.fr(t)st(t; <1>.r)dt] NoTil(6-4-1) (6-4-2)wheres,(t;cP,r)istheequivalent lowpasssignal,whichhasthegeneralform S/(I;<1>,r)=e-id'[2:I"g(t-nT-r)+j2:J"w(t-nT-r)] • • where{I,,}and{J,,}arethetwoinformation sequences. Wenotethat,forPAM,wemaysetJ"=0foralln.andthesequence {I.}is real.ForQAMandPSK,wesetJ.=0forallnandthesequence {I.}is complex-valued. ForoffsetQPSK,bothsequences {L,}and{J,,}arenonzero andw(t)=get-~T). Fordecision-directed MLestimation ofcPandr.thelog-likelihood function becomes where{~. }AL(cP.r)=Re-:2: [I~y,,(r) +iJ~x,,(r)] ,~ln y,,(r)=Jr(t)g*(t -nT-r)dt r" x,,(r)=fr(t)w*(t -nT-r)dt 1;,(6-4-3) (6-4-4\ Necessary conditions fortheestimates ofcf>andrtobetheMLestimates are (6-4-5) 366DIGITAL COMMUNICUIONS Itisconvenient todefine A(r)+jB(r)=~2:[/~YnCr)+jJ~Xn(r») Non Withthisdefinition, (6-4-3)maybeexpressed inthesimpleform AL(cP,r)=A(r)coscP-B(r)sincP Nowtheconditions in(6-4-5)forthejointMLestimates become M(cf>,r) . )---'-'-'--'- = -A(r)smcf>-B(rcos.p=0a.p aA(cf>,r)aA(r) aB(r) .--cos cf>---sm.p =0ar ar iJr From(6-4-8),weobtain :i.=-t-I[B(iMd] ....MLanA(iMd Thesolutionto(6-4-9)thatincorporates (6-4-10) is [A(r)aA(r)+B(r)aB(r)] _=0 ar iJTl'=fML(6-4-6) (6-4-7) (6-4-8) (6-4-9) (6-4-10) (6-4-11) Thedecision-directed trackingloopforQAM(orPSK)obtained fromthese equations isillustrated inFig.6-4-1. FIGURE 6-4-1Decision-dire<ted jointtrackingloopforcarrierphaseandsymboltiminginQAMandPSK. r(t) Im(/,,)1 4, CHAPTER f'>:C'ARRIFR A~DSYMBOL SYS['HRO~11ATIO\ 367 OffsetQPSKrequires aslightlymorecomplex structure forjointestimation ofq,andr.Thestructure iseasilyderivedfrom(6-4-6)-(6-4-11). Inaddition tothejointestimates givenabove,itisalsopossible toderive non-decision·directed estimates ofthecarrierphaseandsymbol timing, although weshallnotpursuethisapproach. Weshouldalsomention thatDnecancombine theparameter estimation problem withthedemodulation oftheinformation sequence {I,,}.Thus,one canconsider thejointmaximum-likelihood estimation ofif,,},thecarrierphase q"andthesymbol timingparameter r.Results onthesejointesti'mation problems haveappeared inthetechnical literature, e.g.Kobayashi (1971), Falconer (1976),andFalconer andSalz(1977). 6-5PERFORMANCE CHARACTERISTICS OFML ESTIMATORS Thequalityofasignalparameter estimate isusuallymeasured intermsofits biasanditsvariance. Inordertodefinetheseterms,letusassumethatwehave asequence ofobservations [x,X2x,...x,,)=x,withpdfp(xI<1».from whichweextractanestimate ofaparameter q,.Thebiasofanestimate, say cl>(x),isdefinedas bias=E[<P(x)]- c/> (6-5-1) where'" isthetruevalueoftheparameter. WhenE[c/>(x»)=</J.wesaythatthe estimate isunbiased. Thevariance oftheestimate </>(x)isdefinedas a~=E{[<P(x)]'} -{E[cl>(X)]}2 (6-5-2) Ingeneral a~maybedifficulttocompute. However, awell-known resultin parameter estimation (seeHelstrom, 1968)istheCramer-Rao lowerboundon themeansquareerrordefinedas (6-5-3) Notethatwhentheestimate isunbiased, thenumerator Df(6-5-3)isunity andtheboundbecomes alowerboundonthevariance a~oftheestimate cl>(x),i.e., (6-5-4) SinceInp(xI<1»differsfromthelog-likelihood function byaconstant factor J68 lJf(iIf.-\L CU\I\Il'1'\IC-\TIONS independent ofcb.itfollowsthat E{[,,:Inp(xI<1»n=E{[,,~InA(<I»n =-EL~2In A(cb)} Therefore. thelowerboundonthevariance is 0"1;"1/E{[,,:In,'\(<1»n=-1/EL~2In A(<I>1](6-5-5) (6-5-6) Thislowerboundisaveryusefulresult.Itprovides abenchmark for comparing thevariance ofanypractical estimate tothelowerbound.Any estimate thatisunbiased andwhosevariance attainsthelowerboundiscalled anefficient estimate. Ingeneral,efficientestimates arerare.Whentheyexist.theyaremaximum­ likelihood estimates. Awell-known resultfromparameter estimation theoryis thatanyMLparameter estimate isasymptotically (arbitrarily largenumberof observations) unbiased andefficient. Toalargeextent,thesedesirable properties constitute theimportance ofMLparameter estimates. Italsoknown thatanMLestimate isasymptotically gaussian-distributed [withmean <I>and varianceequaltothelowerboundgivenby(6-5-6).) InthecaseoftheMLestimates described inthischapterforthetwosignal parameters. theirvariance isgenerally inversely proportional tothesignal-to­ noiseratio.or.equivalently, inversely proportional tothesignalpower multiplied bytheobservation interval To.FlIrthermore, thevariance ofthe decision-directed estimates, atlowerrorprobabilities, aregenerally lowerthan thevariance ofnon-decision-directed estimates. Infact.theperformance ofthe MLdecision-directed estimates forcbandrattainthelowerbound. Thefollowing example isconcerned withtheevaluation oftheCramer-Rao lowerboundfortheMLestimate ofthecarrierphase. Example 6-S-1 TheMLestimate ofthephaseofanunmodulated carrierwasshownin (6-2-11)tosatisfythecondition fr(t)sin(2Tlf,.t+¢MLldt=0 (6-5-7) 7;, where r(t)=s(t;¢»+n(t) =Acos(2Trj;.t+cf»+n(t) (6-5-8) Thecondition in(6-5-7)wasderivedbymaximizing theloglikelihood function Al(</J)=No21r(t)s(t:<jJ)dt (6-5-9) o7;, CHAPTl::R 6:CARRIER AND:\YMHOL SYNCHRONIZATION 369 Thevariance ofibMLislower-bounded as {2AJ }-Ic/>L~-E[r(t)]cos(2rifct+<p)dt NoTo {A'l}-INo~-dl=-- ~iTo A27;, ~~J2T"=NoBeg !A2~A2(6-5-10) Thefactor1/27;,issimplythe(one-sided) equivalent noisebandwidth ofthe idealintegrator. Fromthisexample, weobservethatthevariance oftheMLphaseestimate islower-bounded as (6-5-11) whereYL=A'/2N.,B eqistheloop SNR. Thisisalsothevariance obtained for thephaseestimate fromaPLLwithdecision-directed estimation. Aswehave alreadyobserved, non-decision-directed estimates donotperform aswelldue tolossesinthenonlinearities required toremovethemodulation, e.g"the squaring lossandtheMth-power loss. Similarresultscanbeobtained onthequalityofthesymboltimingestimates derivedabove.Inaddition totheirdependence ontheSNR,thequalityof symboltimingestimates isafunctionofthesignalpulseshape.Forexample, a pulseshapethatiscommonly usedinpracticeisonethathasaraisedcosine spectrum (seeSection9-2).Forsuchapulse,thermstimingerror(at)asa function ofSNRisillustrated inFig.6-5-1,forbothdecision-directed and 0.20 Non-decision-direcled------------i;ij----- K=25symbolsDecision-direCled ~es(imale ~o25-----_. 1520 S~R(dB)10~1 z0.02Performance ofbaseband symboltimingestimate for fixedsignalandloopbandwidths, [From Synchronization Subsystems: Analysis andDesign, byLFranks,1983.Reprinted withpermission of theauthor.JflGURE 6oS-t 370 DIGITAL COMMUNICATIONS 0.50 FlGURE 6·5-2Performance ofbaseband symbollimingestimate forfixed SNRandfixedloopbandwldth. [FromSynchronization Subsyslems: Analysis andDesign,byL.Franks,/98.J Reprinted withpermission ofthemahar.]!::: 0"0.20g "'".S0.10g ~ ~0.05 1z0.02Non-decision-dire.:tcd ~es.timate K=25symbol!. SNR~15dB Decision-direCled estimate o0.10.2OJOA05 Excessbandwidth fa-ctor ~ (Band.....idth=cI...~l/2T1 non-decision-directed estimates. Notethesignificant improvement in performance ofthedecision-directed estimate compared withthenon-decision­ directed estimate. Now,ifthebandwidth ofthepulseisvaried,thepulseshape ischanged and,hence,therrnsvalueofthetimingerroralsochanges. For example, whenthebandwidth ofthepulsethathasaraisedcosinespectrum is varied,thermstimingerrorvariesasshowninFig.6-5-2.Notethattheerror decreases asthebandwidth ofthepulseincreases. Inconclusion, wehavepresented theMLmethod forsignalparameter estimation andhaveappliedittotheestimation ofthecarrierphaseand symboltiming.Wehavealsodescribed theirperformance characteristics. 6-6BIBLIOGRAPHICAL NOTES ANDREFERENCES Carrierrecovery andtimingsynchronization aretwotopicsthathavebeen thoroughly investigated overthepastthreedecades. TheCostasloopwas invented in1956andthedecision-directed phaseestimation methods were described byProakisetal.(1964)andbyNataliandWalbesser (1%9).The workondecision-directed estimation wasmotivated byearlierworkofPrice (1962a,b).Comprehensive treatments ofphase-locked loopsfirstappeared in thebooksbyViterbi(1966)andGardner (1979).Booksthatcovercarrier phaserecovery andtimesynchronization techniques havebeenwrittenhy Stiffler(1971),Lindsey (1972),Lindsey andSimon(1973),andMeyrillld Ascheid (1990). Anumberoftutorialpapershaveappeared inIEEEjournals onIhePLL andontimesynchronization. Wecite,forexample, thepaperbyGUptil(1975), whichtreatsbothanaloganddigitalimplementation ofPLLs,andthepaperhy LindseyandChie (1981),whichisdevotedtotheanalysisofdigitalPLLs.In addition, thetutorialpaperbyFranks(1980)describes' bothcarrierphaseand symbolsynchronization methods, including methods basedonthemaximum­ likelihood estimation criterion. ThepaperbyFranksiscontained inaspecial PROBLEMS FIGURE P6-5CHAPTER 6:C~aRIER ANDSYMBOL SYi'iCHROJ\lZATIO/,\ 371 issueoftheIEEETransactions onCommunications (August 1980)devoted to synchronization. ThepaperbyMueller andMuller(1976)describes digital signalprocessing algorithms forextracting symboltiming. Application ofthemaximum-likelihood criterion toparameter estimation wasfirstdescribed inthecontextofradarparameter estimation (rangeand rangerate).Subsequently, thisoptimal criterion wasappliedtocarrierphase andsymboltimingestimation aswellastojointparameter estimation withdata symbols. Papersonthesetopicshavebeenpublished byseveralresearchers. including Falconer (1976),Mengali (1977),Falconer andSalz(1977), and MeyersandFranks(1980). TheCramer-Rao lowerboundonthevariance ofaparameter estimate is derived andevaluated inanumber ofstandard textsondetection and estimation theory,suchasHelstrom (1968)andVanTrees(1968).Itisalso described inseveralbooksonmathematical statistics, suchasthebookby Cramer(1946). 6-1Provetherelation(6-1-7). 6-2Sketchtheequivalent realization ofthebinaryPSKreceiver inFig.6-1-1thai employsamatchedfilterinsteadofacorrelator. 6-3Supposethattheloopfilter[see(6-2-14)] foraPLLhasthetransferfunction I G(s)=--~ s+v2 aDetermine theclosed-loop transferfunction H(s)andindicateiftheloopis stable. bDetermine thedamping factorandthenaturalfrequency oftheloop_ 6-4Consider thePLLforestimating thecarrierphaseofasignalinwhichtheloop filterisspecified as KG(s)=-­ 1+"s aDetermine theclosed-loop transferfunction H(s)anditsgainatf=O. bForwhatrangeofvaluesof"andKistheloopstable' 6-5TheloopfilterG(s)inaPLLisimplemented bythecircuitshowninFig.P6-5. Determine thesystemfunction G(s)andexpressthetimeconstants "and"in termsofthecircuitparameters. Q------wh-hn-r InputIR~OutPlJI o----L-o FlGURE P6-6 FIGURE P6-7372 DIGITAL COMMUNICATIONS R,r R -A'>-""--o 6-6TheloopfilterG(s)inaPLLisimplemented withtheactivefillershowninFig. P6-6.Determine thesystemfunctionG(s)andexpressthetimeconstants r\andr, intermsofthecircuitparameters. 6-7Showthattheearly-late gatesynchronizer illustrated inFig.6-3-5isaclose approximalion tothetimingrecovery systemillustrated inFig.P6-7. 6-8BasedonaMLcriterion, determine acarrierphaseestimation methodforbinary on-offkeyingmodulation. 6-9Inthetransmission andreception ofsignals 10andfrommovingvehicles, the transmitted signalfrequency isshiftedindirectproportion tothespeedofthe vehicle.Theso-called Doppler frequency shiftimparted toasignalthatisreceived inavehicletraveling atavelocityvrelativetoa(fixed)transmiller isgivenbythe formula whereAisthewavelength,.and thesigndepends onthedirection (moving toward ormovingaway)thatthevehicleistraveling relativetothetransmitter. Suppose thatavehicleistraveling ataspeedof100km/hrelativetoabaseslalionin amobilecellular communication system.Thesignalisanarrowband signal transmitted atacarrierfrequency of1GHz. IIDetermine theDoppler frequency shift. bWhatshouldbethebandwidth ofaDoppler frequency tracking loopiftheloop isdesigned totrackDoppler frequency shiftsforvehiclestraveling atspeedsup tolOOkm/h? cSuppose thetransmitted signalbandwidth is2MHzcentered atIGHz. Sampler Sampler ClIAPTFH 0:CARRIER ANDSYMBOL SYNCHROl"iIZATION 373 Determine theDoppler frequency spreadbetween theupperandlower frequencies inthesignal. 6.10ShowthatthemeanvalueoftheMLestimate in(6·2-38) is<p.i.e.•thatthe estimate isunbiased. 6-11Determine thepdfoftheMLphaseestimate in(6-2-38). 6·12Determine theMLphaseestimate foroffsetQPSK. 6·13Asingle·sideband PAMsignalmayberepresented as 1I",(t) ~A",[g1(1)cos2trf.t-gl(l)sin2trtl] whereg,(I)istheHilberttransform ofg,(l)andAmistheamplitude levelthat conveystheinformation. Demonstrate mathematically thataCostasloopcanbe usedtodemodulate theSSBPAMsignal. 6-14Acarriercomponent istransmined onthequadrature carrierinacommunication systemthattransmits information viabinaryPSK.Hence.thereceived signalhas theform r(l)=±vTP,cos(2trf.+</»+V2P."sin(2trf.+<p)+n(l) where d>isthecarrierphaseandn(l)isAWGN. Theunmodulated carrier component isusedasapilotsignalatthereceivertoestimate thecarrierphase. sSketchablockdiagramofthereceiver. including thecarrierphaseestimator. bIllustrate mathematically theoperations involved intheestimation ofthecarrier phase<p. cExpresstheprobability oferrorforthedetection ofthebinaryPSKsignalasa function ofthetotaltransmitted powerPI~P.+Pc.Whatisthelossin performance duetotheallocation ofaportionofthetransmitted powertothe pilotsignal?Evaluate thelossforPJPI=0.1. 6-15Determine thesignalandnoisecomponents attheinputtoafourth·power (M=4) PLLthatisusedtogenerate thecarrierphasefordemodulation ofQPSK.By ignoring allnoisecomponents exceptthosethatarelinearinthenoisen(l). determine thevariance ofthephaseestimate attheoutputofthePLL. 6·16Theprobability oferrorforbinaryPSKdemodulation anddetection whenthereis acarrierphaseerror<p,is Suppose thatthephaseerrorfromthePLLismodeled asazero-mean gaussian randomvariable withvariance u~« 11.Determine theexpression fortheaverage probability oferror(inintegralform). 7 CHANNEL CAPACITY ANDCODING InChapter 5,weconsidered theproblem ofdigitalmodulation bymeansof M'"2ksignalwaveforms, whereeachwaveform conveyskbitsofinformation. Weobserved thatsomemodulation methods providebetterperformance than others.Inparticular, wedemonstrated thatorthogonal signaling waveforms allowustomaketheprobability oferrorarbitrarily smallbylettingthe numberofwaveforms M.....00,provided thattheSNRperbitYb;;'-1.6dB. Thus,wecanoperateatthecapacity oftheadditive, whitegaussian noise channelinthelimitasthebandwidth expansion factorB,'"W/R.....00.Thisis aheavypricetopay,because Begrowsexponentially withtheblocklengthk. Suchinefficient useofchannelbandwidth ishighlyundesirable. Inthisandthefollowing chapter, weconsider signalwaveforms generated fromeitherbinaryornonbinary sequences. Theresulting waveforms are generally characterized byabandwidth expansion factorthatgrowsonly linearlywithk.Consequently, codedwaveforms offerthepotential forgreater bandwidth efficiency thanorthogonal M-arywaveforms. Weshallobservethat. ingeneral, codedwaveforms offerperformance advantages notonlyin power-limited applications whereR/W<1,butalsoinbandwidth-limited systemswhereR/W>I. Webeginbyestablishing severalchannel modelsthatwillbeusedto evaluate thebenefitsofchannelcoding,andweshallintroduce theconceptof channelcapacityforthevariouschannelmodels.Then,wetreatthesubjectof codedesignforefficientcommunications. 374 CHAPTER 7:CHA!'O\lL CAl-'I\CITY AN))CODING 375 7-1CHANNEL MODELS ANDCHANNEL CAPACITY Inthemodelofadigitalcommunication systemdescribed inSection1-1,we recallthatthetransmitter building blocksconsistofthediscrete-input, discrete-output channelencoder followed bythemodulator. Thefunction of thediscrete channelencoder istointroduce, inacontrolled manner, some redundancy inthebinaryinformation sequence, whichcanbeusedatthe receivertoovercome theeffectsofnoiseandinterference encountered inthe transmission ofthesignalthroughthechannel. Theencoding processgenerally involvestakingkinformation bitsatatimeandmapping eachk-bitsequence intoauniquen-bitsequence, calledacodeword.Theamountofredundancy introduced bytheencoding ofthedatainthismannerismeasured bytheratio n/k.Thereciprocal ofthisratio,namelykIn,iscalledthecoderate. Thebinarysequence attheoutput of thechannel encoder isfedtothe modulator, whichservesastheinterface tothecommunication channel. Aswe havediscussed, themodulator maysimplymapeachbinarydigitintooneof twopossible waveforms, i.e.,a 0ismapped intos,(t)anda Iismappedinto S2(t).Alternatively, the modulator maytransmit q-bitblocksatatimebyusing M=2qpossiblewaveforms. Atthereceiving endofthedigitalcommunication system,thedemodulator processes thechannel-corrupted waveform andreduceseachwaveform toa scalaroravectorthatrepresents anestimate ofthetransmitted datasymbol (binaryorM-ary).Thedetector, whichfollowsthedemodulator, maydecide onwhether thetransmitted bitisa 0ora1.Insuchacase,thedetector has madeaharddecision.Ifweviewthedecision processatthedetector asaform ofquantization, weobserve thataharddecision corresponds tobinary quantization ofthedemodulator output.Moregenerally, wemayconsider a detector thatquantizes toQ>2levels,i.e.,aQ-arydetector.IfM-arysignals areusedthenQ;;"M.Intheextreme casewhennoquantization isperformed, Q=00.InthecasewhereQ>M,wesaythatthedetector hasmadeasoft decision. Thequantized outputfromthedetector isthenfedtothechanneldecoder, whichexploitstheavailable redundancy tocorrectforchanneldisturbances. Inthefollowing sections, wedescribe threechannelmodelsthatwillbeused toestablish themaximum achievable bitrateforthechannel. 7-1-1Channel Models Inthissectionwedescribe channelmodelsthatwillbeusefulinthedesignof codes.Thesimplest isthebinarysymmetric channel(BSC),whichcorresponds tothecasewithM=2andharddecisions atthedetector. BinarySymmetric Channel Letusconsider anadditive noisechanneland letthemodulator andthedemodulator/detector beinduded aspartsofthe Ilemodulalor and detector376 DIGITAL COMMUNICATIONS ------·--------------------------~l,,,, •,, Composite discrete-input. discrete-output c:bannel :I. ...._....__.._..__...._....__........__I FlGURE 7-1-1Acomposite discrete-input, discrete-outpul channelformedbyincluding themodulator andthe demodulator/delector aspartofIhechannel. channel.Ifthemodulator employs binarywaveforms andthedetector makes harddecisions, thenthecomposite channel, showninFig.7-1-1,hasa discrete-time binaryinputsequence andadiscrete-time binaryoutput sequence. Suchacomposite channelischaracterized bythesetX={O,I}of possible inputs,thesetofY={O,I}ofpossible outputs, andasetof conditional probabilities thatrelatethepossibleoutputstothepossibleinputs. Ifthechannel noiseandotherdisturbances causestatistically independent errorsinthetransmitted binarysequence withaverageprobability pthen P(Y=0IX=1)=P(Y=11X=0)=p P(Y=llx=1)"P(Y=OIX=O}=l-p(7-1-1) Thus,wehavereduced thecascadeofthebinarymodulator, thewaveform channel, andthebinarydemodulator anddetectorintoanequivalent discrete­ timechannelwhichisrepresented bythediagram showninFig.7-1-2.This binary-input, binary-output, symmetric channel issimplycalled·abinary symmetric channel(BSC).Sinceeachoutputbitfromthechanneldepends only onthecorresponding inputbit,wesaythatthechannelismemoryless. Discrete Memoryless Cban.els TheBSCisaspecialcaseofamore generaldiscrete-input, discrete-output channel. Suppose thattheoutputfrom thechannelencoder areq-arysymbols, Le.,X={xv,Xv.•.,xq_,}andthe outputofthedetector consistsofQ-arysymbols, whereQ;;.M=zq.Ifthe I-p000;:-------,10 0 FIGURE 7-1-2Binal)'symmetric channel. I-p CHAPTER 7:CHANNEL CAPACITY ANDCODING 377 IXI IYI FIGURE 7-1-3 Discrete q-aryinput.Q-aryoutputchannel. channel andthemodulation arememoryless, thentheinput-output characteristics ofthecomposite channel, showninFig.7-1-1,aredescribed by asetofqQconditional probabilities (7-1-2) wherei=0,1,. ,_,Q -1andj=0,1,. ,.,q-1.Suchachannel iscalleda dicretememoryless channel(DMC), anditsgraphical representation isshown inFig.7-1-3.Hence,ifth~inputtoaDMCisasequence ofnsymbols UI,U2,' , ,,Unselected fromthealphabet Xandthecorresponding outputis thesequence VI'V2,' . ,,Unofsymbols fromthealphabet Y,thejoint conditional probability is P(YI=VioY2=V2,• , . ,Y,=VnIX=UI,• , , •X=un) n =nPlY=VkIX=Uk) k'"I(7-1-3) Thisexpression issimplyamathematical statement ofthememoryless condition. Ingeneral, theconditional probabilities {P(YiIXj)}thatcharacterize aDMC canbearranged inthematrixformP=[Pji],where, bydefinition, PJ-==P(YiIxJ)-Piscalledtheprobability transition matrixforthechannel. Discrete-Input. Continuous-Output Channel Now,suppose thattheinput tothemodulator comprises symbols selected fromafiniteanddiscrete input alphabet X={xo.XI'' . _ ,xq-t}andtheoutputofthedetector isunquantized (Q=x).Then,theinputtothechanneldecodercanassumeanyvalueonthe realline,i.e.,Y={-x,x}.Thisleadsustodefineacomposite discrete-time 378 DIGITAL COMMUNICATIONS memoryless channel thatischaracterized bythediscrete inputX,the continuous outputY,andthesetofconditional probability densityfunctions p(yIX=xd.k=0,1,...,q-1 Themostimportant channelofthistypeistheadditive whitegaussian noise channel(AWON),forwhich Y=X+G (7-1-4) whereGisazero-mean gaussian random variable withvariance 0'2and X=Xk,k=0,I,...,q-1.ForagivenX,itfollowsthatYisgaussian with meanXkandvariance a2•Thatis, I1 ( )'/"p(yX=xd=_~ e-Y-" _if v2TrO'(7-1-5) Foranygiveninputsequence, Xi'i=1,2.....n,thereisacorresponding outputsequence Y;=Xi+G i,i=I,2.....n Thecondition thatthechannel ismemoryless maybeexpressed as(7-1-6) .n p(Yt,y",..•YnIXI=Ut,X2=Uz,··.,X,=un)=np(y,IXi=U,) ;=1 (7-1-7) Waveform Channels Wemayseparate themodulator anddemodulator fromthephysical channel, andconsider achannel modelinwhichtheinputs arewaveforms andtheoutputsarewaveforms. Letusassumethatsucha channel hasagivenbandwidth W,withidealfrequency response C(f)=1 withinthebandwidth W.andthesignalatitsoutputiscorrupted byadditive whitegaussian noise.Suppose, thatx(t)isaband-limited inputtosucha channelandyet)isthecorresponding output.Then, yet)=x(t)+n(t) (7-1-8) wherenet)represents asamplefunction oftheadditive noiseprocess. A suitablemethodfordefiningasetofprobabilities thatcharacterize thechannel istoexpandx(t),y(t),andnet)intoacomplete setoforthonormal functions. Thatis,weexpressx(t),y(t),andnet)intheform yet)=~y,Ji(t) x(t)=~XJi(t) net)=~nJi(t)(7-1-9) CHAPTER i:CHA;o.rlNEL CAPA<TI'A1\DCODINe, 379 where{y;},{x;}.and{n;}arethesetsofcoefficients Inthecorresponding. eKpansions, e.g., Yi=(Y(t)fi*(t) dt J" =fT[x(t)+n(t)lf,*(t)dt " =-x,+ni (7-1-10) Thefunctions {j;(t)}formacomplete orthonormal setovertheinterval (0,7).i.e., iT {I(i=j) "j;(t)ft(t) dt=5,j=0(i""j)(7-1-11) where5,)istheKronecker deltafunction. Sincethegaussian noiseiswhite,any complete setoforthonormal functions maybeusedintheeKpansions (7-1-9). Wemaynowusethecoefficients intheexpansion forcharacterizing the channel. Since wheren,isgauss",n, itfollowsthat 1 P(YiIx,)=V2Jrifie i=1,2.... (7-1-12\ Sincethefunctions {j;(t)}intheexpansion areorthonormal, itfollowsthatthe {n,}areuncorrelated. Sincetheyaregaussian, theyarealsostatistically independent. Hence, N p(Y"Yz.· ··.YNIxl.x,....,xN)=np(Yi!X,) i=I(7-1-13) foranyN.Inthismanner, thewaveform channel isreduced toanequvalent discrele-time channelcharacterized bytheconditional pdfgivenin(7-I-(2). Whentheadditivenoiseiswhiteandgaussian withspectraldensityIN",the variances <r?=!Noforalliin(7-1-12). Inthiscase,samplesofx(t)andy(t) maybetakenattheNyquistrateof2Wsamples/s, sothatx,=x(i/2W) and Yi=y(i/2W). Sincethenoiseiswhite,thenoisesamples arestalislicallv independent. Thus,(7-1-12)and(7-1-13)describe thestatistics ofthesampled signal.WenotethatinatimeintervaloflengthT,thereareN=2WTsamples. Thisparameter isusedbelowinobtaining thecapacity oftheband-limited AWGNwaveform channel. Thechoiceofwhichchannelmodeltouseatanyone timedepends onour objectives. Ifweareinterested inthedesignandanalysisoftheperformance 380 DIGITAL COMMUNICATIONS ofthediscrete channelencoder anddecoder. itisappropriate toconsider channelmodelsinwhichthemodulator anddemodulator areapartofthe composite channel. Ontheotherhand,ifourintentistodesignandanalyze theperformance ofthedigitalmodulator anddigitaldemodulator, weusea channelmodelforthewaveform channel. 7·1·2Channel Capacity Nowletusconsider aDMChavinganinputalphabetX={xo.x, •...,xq_,}. anoutputalphabet Y={Yo.y,•...•YQ_I}.andthesetoftransition prob­ abilities P(YiIXj)asdefined in(7-1-2).Suppose thatthesymbolxiis transmitted andthesymbol Yiisreceived. Themutualinformation provided about.theeventX=xibytheoccurrence oftheeventY=Yiis log[P(YiIXJl/P(Yi»). where P(Yi)'"P(Y'"Yi)=~'P(X.)P(YiIx.) (7-1-14).-0 Hence,theaveragemutualinformation provided bytheoutputYaboutthe inputXis ~lQ-I pl(".lv\)~l(X;Y)=.2:P(XJP(YiIXi)logP('. . }~o,-0 y,)(7-1-15) Thechannelcharacteristics determine thetransition probabilities P(YiIXi)' buttheprobabilities oftheinputsymbolsareunderthecontrolofthediscrete channelencoder. ThevalueofI(X;Y)maximized overthesetofinputsymbol probabilities P(Xj)isaquantity thatdepends onlyonthecharacteristics ofthe DMCthroughtheconditional probabilities P(YiIXi)'Thisquantity iscalledthe capaci/yofthechannelandisdenotedbyC.Thatis,thecapacityofaDMCis definedas C=maxI(X;Y) P(x,) =max~I~1 P(Xj)P(YiIXi)log~(ix)2 P(xJ)j=O;=0 PYi Themaximization ofI(X;Y)isperformed undertheconstraints that P(Xi)"0 ~lP(xi)=1 )'=0(7-1-16) TheunitsofCarebitsperinputsymbolintothechannel(bits/channel use) cHArTER 7:CHANNEL GAPACIT" A"lDCODINf,381 FIGURE 7·1-4Thecapacity ofaBseasafunction oftheerror pwnanility /'.o02OA0.6 (J.~IJ Probi.ibilily oferror.r whenthelogarithm isbase2,andnats!input symbol when thenatural logarithm (basee)isused.Ifasymbolentersthechannelevery T,seconds. the channelcapacity inbitslsornatslsisCIT,. Example 7·1·1 FortheBSCwithtransition probabilities P(011)=P(1I0)=p theaveragemutualinformation ismaximized whentheinputprobabilities P(G)=P(I)=t.Thus,thecapacity oftheBSCis C=plog2p+(1-p)log2(1-p)=1-H(p) (7-1-17) whereH(p)isthebinaryentropy function. AplotofCversuspis illustrated inFig.7-1-4.Notethatforp=0,thecapacity isIbit}channel use. Ontheotherhand.forp=Lthemutualinformation between inputand outputiszero.Hence,thechannelcapacity iszero.For~<p'"I,wemay reversethepositionof0andIattheoutputoftheBSC,sothatCbecomes symmetric withrespecttothepointp=~.Inourtreatment ofbinary modulation anddemodulation giveninChapter 5,weshowedthatpisa monotonic function ofthesignal-lo-noise ratio(SNR)asillustrated inFig. 7-J·5(0). Consequently whenCisplottedasafunction oftheSNR,it increases monotonically astheSNRincreases. Thischaracteristic behavior ofCversusSNRisillustrated inFig.7-1-5(b). Nextletusconsider thediscrete-time AWGNrnemoryless channel de­ scribedbythelransition probability densityfunctions definedby(7-1-5).The FIGURE '·l·SGeneralbehavior oferrorprobability andchannetcapacity asa functionofSNR.1 Cj~lz oSNR 0SNR (ul (h) 382 D1GITAlCOMMUNiCATIONS averagemutualinformation between thediscreteinputX={XO,X"...•xq_l} andtheoutputY= {-OC.:xl}isgivenbythecapacity ofthischannel in bits/channeluseis where p(y)=~'p(yIXdP(Xk) *=0(7-1-18) (7-1-19) (7-1-20)Example 7·1-2 Letusconsider abinary-input AWGNmemoryless channelwithpossible inputsX=AandX=-A.Theaveragemutualinformation leX;Y)is maximized whentheinputprobabilities areP(X=A)=P(X=-A)=1. Hence,thecapacityofthischannelinbits/channel useis C=~[p(y1A)IOg2~( A»dy -~ PY +!f~P(YI-A)IO~P(Y(I-)A)dY _00 PY Figure7-1-6illustrates Casafunction oftheratioA2/2u2•NotethatC increases monotonically from0to1bit/symbol asthisratioincreases. ItisilUeresting tonotethatinthetwochannelmodelsdescribed above,the choiceofequallyprobable inputsymbols maximizes theaverage mutual information. Thus,thecapacity ofthechannelisobtained whentheinput symboisareequallyprobable. Thisisnotalwaysthesolutionforthecapacity formulas givenin(7-1-16)and(7-1-18), however. Nothing canbesaidin generalabouttheinputprobability assignment thatmaximizes theaverage mutualinformation. However. inthetwochannelmodelsconsidered above, 1.0 il08 Ii ~0.6 :SW0.4 FIGURE 7-1-6ChannelcapacityasafunctionofA'/2u'forbinary-input AWGN memorylesschannel.~20"""-::"'1'::-2 ---4~~4~..J'2­ 10log(A'I2,,'] (dB] CHAPTER 7:CHANNEL CAPACITY ANDCODING383 thechannel transition probabilities exhibit aformofsymmetry thatresultsin themaximum ofleX;Y)beingobtained whentheinputsymbols areequally probable. Thesymmetry condition canbeexpressed intermsoftheelements oftheprobability transition matrixPofthechannel. Wheneachrowofthis matrixisapermutation ofanyotherrowandeachcolumnisapermutation of anyothercolumn, theprobability transition matrixissymmetric andinput symbolswithequalprobability maximize I(X;V). Ingeneral, necessary andsufficient conditions forthesetofinputprob­ abilities{P(x))}tomaximize leX;y)and,thus,toachievecapacity onaDMC arethat(Problem 7-1) I(x);Y)=CforalljwithP(x)>0 lex];Y).:;;CforalljwithP(Xj)=0 whereCisthecapacity ofthechanneland ._Q-l ~ I(x"Y)-2:P(y,IXj)log( ,<0 PYi)(7-1-21) (7-1-22) Usually. itisrelatively easytocheckiftheequallyprobable setofinput symbols satisfytheconditions (7-1-21).Iftheydonot,thenonemust determine thesetofunequalprobabilities {P(x])}thatsatisfy(7-1-21). Nowletusconsider aband-limited waveform channel withadditive white gaussian noise.Formally, thecapacity ofthechannel perunittimehasbeen definedbyShannon(I948b)as C=limmax1.I(X;Y) (7-1-23) T_..,0p(x)T wheretheaveragemutualinformation leX;Y)isgivenin(3-2-17). Alterna­ tively,wemayusethesamplesorthecoefficients {y,},{x,),and{n,}intheseries expansions ofyet),X(/),andnet),respectively, todetermine theaverage mutualinformation between XN=[x,X2...XN]andYN=[y,Y2...YN]. whereN=2WT,Yi=Xi+ni,andP(YiIXi)isgivenby(7-1-12). Theaverage mutualinformation between 'TwandYNfortheAWONchannelis I(XN;YN)=f...·fi..·fP(YNIXN)P(XN) 10gP(Y(NIX)N)dXNdy" XN YN PY.AJ =i~[tP(YiIX,)P(X,)IOg~dYi dXi (7-1-24) where 1p(v·1x)=--e-(y,-x,j'INo.,,'l/trNo(7-1-25) 384 DIGITAL COMMUNICA TlONS Themaximum ofleX;Y)overtheinputpdfsp(x,) isobtained whenthe{xJ arestatistically independent zero-mean gaussian random variables, Le., where0-;is,thevariance ofeachX;.Then,itfollowsfrom(7-1-24)that (20-2) =WTlog1+No'(7-1-26) (7-1-27) Suppose thatweputaconstraint ontheaveragepowerinx(t).Thatis, PO'=.!.(E[x2(t)]dtTJo 1N=-2:E(xf) T;~I NO"; T Hence, 2TPav0-=-­,N 2W Substitution ofthisresultinto(7-1-27)foru;yields maxl(X N;YN)=WTlog(1+ Pav ) PI·') WNQ(7-1-28) (7-1-29) (7-1-30) Finally,thechannelcapacity perunittimeisobtained bydividing theresultin (7-1-30) byT.Thus (P") C=Wlog1+-WNo(7-1-31) Thisisthebasicformula forthecapacity oftheband-limited AWGN CHAP:fER~: CHAN!'IEL CAPACITY ANDCODING 385 10 FIGURE 7-1·7 Normalized channel capacity asafunction ofSNRforhand·limited AWGNchannel.-40..8121618 IOlog(Pa,iWNo) waveform channel withaband-limited andaverage power-limited input.Itwas originally derivedbyShannon (l948b). Aplotofthecapacity inbits/snormalized bythebandwidth Wisplottedin Fig.7-1-7asafunction oftheratioofsignalpowerP"tonoisepowerWNo. Notethatthecapacity increases monotonically withincreasing SNR.Thus,for afixedbandwidth. thecapacity ofthewaveform channel increases withan increase inthetransmitted signalpower.Ontheotherhand,ifP"isfixed,the capacity canbeincreased byincreasing thebandwidth W.Figure7-1-8 illustrates agraphofCversusW.NotethatasWapproaches infinity. the capacity ofthechannt:'!approaches theasymptotic value Pav Pav• /Coo=-log2e=--bitsS ~) Noln2 Itisinstructive toexpress thenormalized channel function oftheSNRperbit.SincePavrepresents the powerandCistheratioinbits/s,itfollowsthat(7-1-32) capacity C/Wasa average transmitted Pav=Cl:b where'thistheenergyperbit.Hence,(7-1-31) maybeexpressed as C(C'fb)-;=log2I+-- \-Ii WNo P~-.Nlog~t' "(7-1-33) (7-1-34) FIGURE 7-1-8Channel capacity asalunctionofbandwidth withafixed transmitted average power. 386 DIGITAL COMMUNIC AnONS Consequently, '€b2C1W-1 NoC/W WhenC/W=1,'€b/NO=1(0dB).AsC/W-."', '€b2C1w-=-- NoC/W =exp(~ln2-In~)(7-1-35) (7-1-36) Thus,'lblNoincreases exponentially asC/W-.ce.Ontheotherhand,as C/W-.O, 2C1W-1lim =In2 ClW_OC/W(7-1-37) whichis-1.6dB. AplotofC/Wversus'"Cb/NoisshowninFig.5-2-17. Thus,wehavederivedthechannel capacities ofthreeimportant channel modelsthatareconsidered inthisbook.Thefirstisthediscrete-input, discrete-output channel, ofwhichtheBSCisaspecialcase.Thesecondisa discrete-input, continuous-output memoryless additive whitegaussian noise channel. Fromthesetwochannelmodels,wecanobtainbenchmarks forthe codedperformance withhard-andsoft-decision decoding indigitalcom­ munications systems. Thethird·channel modelfocusesonthecapacity inbits/sofawaveform channel. Inthiscase,weassumed thatwehaveabandwidth limitation onthe channel, anadditive gaussian noisethatcorrupts thesignal,andanaverage powerconstraint atthetransmitter. Undertheseconditions, wederivedthe resultgivenin(7-1-31). Themajorsignificance ofthechannelcapacity formulas givenaboveisthat theyserveasupperlimitsonthetransmission rateforreliablecommunication overanoisychannel. Thefundamental ratethatthechannelcapacity playsis givenbythenoisychannelcodingtheoremduetoShannon (1948a). NoisyChannel CodingTheorem Thereexistchannelcodes(anddecoders) thatmakeitpossible toachieve reliablecommunication, withassmallanerrorprobability asdesired,ifthe transmission rateR<C,whereCisthechannelcapacity.IfR>C,itisnot possibletomaketheprobability oferrortendtowardzerowithanycode. Inthefollowing section,weexplorethebenefitsofcodingfortheadditive CHAPTFR 7;CHANNl'.L CAPACITY ""DCOOING 387 noisechannel modelsdescribed above,andusethechannel capacity asthe benchmark foraccessing codeperformance. 7-1-3Achieving Channel Capacity withOrthogonal Signals InSection5-2,weusedasimpleunionboundtoshowthat,fororthogonal signals,theprobability oferrorcanbemadeassmallasdesiredbyincreasing thenumberMofwaveforms, provlded that'th/No>2In2.Weindicated that thesimpleunionbounddocsnotproduce thesmallest lowerboundonthe SNRper'bit.Theproblem isthattheupperboundusedonQ(x)isvcryloose forsmallx. Analternative approach istousetwodifferent upperboundsforQ(x). depending onthevalueofx.Beginning with(5-2-21), weobserve that l-[I-Q(y)J"' E(M-I)Q(y)<:Me ,'/2 Thisisjusttheunionbound,whichistightwhenyislarge,i.e.,forv>vo, whereYodepends onM.Whenyissmall.theunionboundexceeds unityfor largeM.Since 1-[I-Q(y)j" 'EI (7-i-3lJ) forally,wemayusethisboundfory<:Yobecause itistighterthant~eunion bound.Thus(5-2-21)maybeupper-bounded as (7-1-40) ThevalueofYothatminmizes thisupperboundisfoundbydifferentiating theright-hand sideof(7-1-40) andsettingthederivative equaltozero.Itis easilyverifiedthatthesolution is or,equivalently, Yo=v'2InM=v'2In2log,M =v'ZkIn2(7-1-41) (7-1-42) Havingdetermined Yo,letusnowcompute simpleexponential upperbounds fortheintegrals in(7-1-40). Forthefirstintegral. wehave 1fYU IJ~(\'2Y·H.)/v2vz,r_~e-,r-V2;)212dy=Vir:.> e-"dx =Q(v'zY-Yo),YoEv'zY yuEV2y (7-1-43) 388 DIGITAL COMMUNICATIONS Thesecondintegralisupper-bounded asfollows: ~f~ e-"IZe-(,··V2;)'12dy=~e Y,2Ix e-"dx V2ii:", V2Jr v"--vY/2 {Me-yI2 (Yo";v'IY) <Me-Y'2e-(Y,,-vY/2I' (Yo;;'v'IY)(7-1-44) Combining theboundsforthetwointegrals andsubstituting eyI,J2forM.we obtain {e-(V2Y-v"J'12 +e(Yi,-y)/2 (0";Yo";vIr) PM<e-(V2Y-Y"I'12+e(YI,-Y)/2e-(y,,-vY/2)2(YfY,,;Yo";-vz.y) (7-1-45) Intherange0,,;Yll";vTY.theboundmaybeexpressed as PM<e<'Y~-Y)/2(1 +e-(YII-v:;i2)2) <2e(Vi,-yY2,0,,;Yu";vTY (7-1-46) (7-1-47)Intherangev'fY";y,,,,; -vz.y,thetwotermsin(7-1-45)areidentical. Hence, P<2e-(V2;:;~Yu)212 ~~~~...;;:':-2 M ,vl1'~Yo~ VL1' Nowwesubstitute fory"and1'.SinceYo='12InM=V2kIn2andl'=k1'b' theboundsin(7-J-46)and(7-1-47)maybeexpresed as (InM,,;h) (a.",,;InM,,;1')(7-1-48) Thefirstupperboundcoincides withtheunionboundpresented earlier,butit islooseforlargevaluesofM.ThesecondupperboundisbetterfOflarge valuesofM.WenotethatPM->0ask->cc(M->cc)provided that1'b>In2. But,In2isthelimiting valueoftheSNRperbitrequired forreliable transmission whensignaling atarateequaltothecapacity oftheinfinite­ bandwidth AWGN channel asshowninSection 7-1-2.Infact,whenthe substitutions Yo=V2kIn2=V2RTIn2 TP.. 1'=N.=TCxln2(7-1-49) aremadeintothetwoupperboundsgivenin(7-1-46) and(7-1-47), where C~=P.vI(NoIn2)isthecapacity oftheinfinite-bandwidth AWGNchannel, the resultis (7-1-50) CHAPTER 7CHANNEL CAPACITY ANDCODING389 Thuswehaveexpressed theboundsintermsofC~andthebitrateinthe channel. Thefirstupperboundisappropriate forratesbelow iC~,whilethe secondistighterthanthefirstforratesbetween ic~andC~.Clearly, the probability oferrorcanbemadearbitrarily smallbymakingT-->""(M.....'" forfixedR),provided thatR<C~=P•.I(NoIn2).Furthermore, weobserve thatthesetoforthogonal waveforms achieves thechannelcapacity boundas M-->00,whentherateR<C~. 7·1-4Channel Reliability Functions Theexponential bounds ontheerrorprobability forM-aryorthogonal signalsonaninfinite-bandwidth AWGN channel givenby(7-1-50) maybe expressed as Theexponential factorPM<2·2-TE(R) (7-1-51) (7-1-52)(0";R,.;iC) (~Cx";R,.;Cx) in(7-1-51) iscalledthechannel reliability function fortheinfinite-bandwidth AWGNcharinel. AplotofE(R)/C xisshowninFig.7-1-9.Alsoshownisthe exponential factorfortheunionboundonPM'givenby(5-2-27), whichmaybe expressed as (7-1-53) Clearly.theexponential factorin(7-1-53)isnotastightasE(R),duetothe looseness oftheunionbound. Theboundgivenby(7-1-51) and(7-1-52) hasbeenshownbyGallager (1965)tobeexponentially tight.Thismeansthattheredoesnotexistanother reliability function, sayE,(R),satisfying thecondition E,(R)>E(R)forany R.Consequently, theerrorprobability isbounded fromaboveandbelowas FIGURE '-1-9Channel reliability function fortheinfinite-bandwidth AWGN channel.(7-1-54) Exponent ror uniooboond ooL.-----'-, .--::::::..- ....... 2C- C. Transmission ratcR(bits/s) 390 DJ01TAL COMM(;!':J<.·Al'ION~ wheretheconstants haveonlyaweakdependence onT,i.e.,theyvaryslowly withT. Sinceorthogonal signalsprovideessentially thesameperformance asthe optimum simplexsignalsforlargeM,thelowerboundin(7-1-54)appliesfor anysignalset.Hence,thereliability function E(R)givenby(7-1-52) determines theexponential characteristics oftheerrorprobability fordigital signaling overtheinfinite-bandwidth AWONchannel. Although theerrorprobability canbemadearbitrarily smallbyincreasing thenumber ofeitherorthogonal, biorthogonal, orsimplex signals,with R<C~,forarelatively modestnumberofsignals,thereisalargegapbetween theactualperformance andthebestachievable performance givenbythe channelcapacity formula. Forexample, fromFig.5-2-17,weobservethataset ofM=16orthogonal signalsdetected coherently requires aSNRperbitof approximately 7.5dBtoachieveabiterrorrateofp~=10-5.Incontrast, the channelcapacity formulaindicates thatforaC/W=0.5,reliabletransmission ispossiblewithaSNRof-0.8dB.Thisrepresents aratherlargedifference of 8.3dB/bitandservesasamotivation forsearching formoreefficientsignaling waveforms. InthischapterandinChapter 8,wedemonstrate thatcoded waveforms canreducethisgapconsiderably. Similargapsinperformance alsoexistinthebandwidth-limited regionof Fig.5-2-17,whereR/W>1.Inthisregion,however, wemustbemoreclever inhowweusecodingtoimproveperformance, becausewecannotexpandthe bandwidth asinthepower-limited region.Theuseofcodingtechniques for bandwidth-efficient communication isalsotreatedinChapter8. 7-2RANDOM SELECfION OFCODES Thedesignofcodedmodulation forefficienttransmission ofinformation may bedividedintotwobasicapproaches. Oneisthealgebraic approach, whichis primarily concerned withthedesignofcodinganddecoding techniques for specificclassesofcodes,suchascyclicblockcodesandconvolutional codes. Thesecondistheprobabilistic approach, whichisconcerned withtheanalysis oftheperformance ofageneralclassofcodedsignals.Thisapproach yields boundsontheprobability oferrorthatcanbeattainedforcommunication over achannelhavingsomespecified characteristic. Inthissection,weadopttheprobabilistic approach tocodedmodulation. Thealgebraic approach, basedonblockcodesandonconvolutional codes,is treatedinChapter8. 7-2-1Random CodingBasedonM-aryBinaryCodedSignals Letusconsider asetofMcodedsignalwaveforms constructed fromasetof n-dimensional binarycodewordsoftheform Cj=[cjJCi2...C;n],i=1,2,...,M (7-2-1) CHAPTER 7:CHANNEL CAPACITY At'>ODCODING391 where C'I=0orI.Eachbitinthecodewordismapped intoabinaryPSK waveform, sothatthesignalwaveform corresponding tothecodewordC,may beexpressed as where"Si(t)=ISi;f,(t). i=I.2,...,M j==-1(7-2-2) whenc i;=1 whenc i}=0(7-2-3) and19,.istheenergypercodebit.Thus,thewaveforms Si(t)areequivalent to then-dimensional vectors (7-2-4) whichcorrespond totheverticesofahypercube inn-dimensional space. Now,suppose thattheinformation rateintotheencoder isRbits/sandwe encodeblocksofkbitsatatimeintooneoftheMwaveforms. Hence,k=RT andM=2'=2RTsignalsarerequired.Itisconvenient todefineaparameter D as nD=rdimensions/s (7-2-5) Thus.n=DTisthedimensionality ofthesignalspace. Thehypercube has2"=20Tvertices, ofwhichM=2RTmaybeusedto transmit theinformation. Ifweimposethecondition thatD>R,thefraction oftheverticesthatweuseassignalpointsis 2'2RTF=_=_=2- jD-R)T 2"20T (7-2-6) Clearly,ifD>R,wehaveF......0asT......x. Thequestion thatwewishtoposeisthefollowing. Canwechooseasubset M=2RTverticesoutofthe2"=20Tavailable verticessuchthattheprobability oferrorP......0asT......xor,equivalently, asn......x?SincethefractionFof verticesusedapproaches zeroasT--+"",itshouldbepossible toselectM signalwaveforms havingaminimum distance thatincreases asT--+xand, thus,P,......0. Insteadofattempting tofindasinglesetofMcodedwaveforms forwhich wecompute theerrorprobability, letusconsider theensemble of(2")'"distinct waysinwhichwecanselectMverticesfromthe2"available verticesofthe hypercube. Associated witheachofthe2"Mselections, thereisacommunica­ tionsystem,consisting ofamodulator, achannel, andademodulator, thatis optimum fortheselected setofMwaveforms. Thus,thereare2"'" OutputOutput OutputOutput392 DIGITAL COMMUNICATIONS II{/) !s,(1)}I Modulator n<O {S,(1)\2 InputModLllator sequence ls,(t)l... Modulator n(r) ls,(r)l:!". Modulator FIGURE 7-2-1Anensemble at2""communications system.Eachsystememploysadifferent setofMsignals fromthe~etof2"Mpossiblechoices. communication systems, oneforeachchoiceoftheMcodedwaveforms, as illustrated inFig.7-2-1.Eachcommunication systemischaracterized byits probability oferror. Suppose thatourchoiceofMcodedwaveforms isbasedonrandom selection fromthesetofznMpossiblesetsofcodes.Thus,therandomselection ofthemthcode,denotedby{silm.occurswithprobability (7-2-7) andthecorresponding conditional probability oferrorforthischoiceofcoded signalsisP,({silm)' Then,theaverageprobability oferrorovertheensemble of codesis 2""" P,=LP,({sJ",)P({sJm) m=l 2"M =2-nMLP,({SJ",) 111=1(7-Z-8) wheretheoverbaronP,denotesanaverageovertheensemble ofcodes. Itisclearthatsomechoicesofcodeswillresultinlargeprobability oferror. Forexample, thecodethatassignsallMk-bitsequences tothesamevertexof thehypercube willresultinalargeprobability oferror.Insuchacase, P,({sJm)>P,.However, therewillalsobechoices ofcodesforwhich P,({Si}m)<P,.Consequently, ifweobtainanupperboundonP"thisbound willalsoholdforthosecodesforwhichP,({sJm)<P,.Furthermore, ifP,.....0as T.....00thenweconclude that,forthesecodes,P({s,}m) ......OasT......oc. Inordertodetermine anupperboundonP"weconsider thetransmission CHAPTER 7:CHA.~NEl. CAPACITY AND('ODlNG393 ofak-bitmessage X.==[X,X2X, ...x.],wherex)=0orJforj=1.2•...•k. Theconditional probability oferroraveraged overtheensemble ofcodesis P,(X,)=LP,(X•.{Si}m)P({S;},,,) a.11codes(7-2-9) whereP,(X..{sJm)istheconditional probability oferrorforagivenk-bit message X•.whichistransmitted byuseofthecode{Si}""Forthemthcode, theprobability oferrorP,(X..{Si}"')isupper-bounded as M P,(X•.{Si}"')";;;LP2m(S/.s.) f=l I'"(7-2-10) whereP2m(SIos.)istheprobability oferrorforabinarycommunication system thatemploys thesignalvectors SIands.tocommunicate oneoftwoequally likelyk-bitmessages. Hence, M P,(X,)";;;LP,({Si}m)LP2",(s,.S.) allcodes f=l )... Ifweinterchange theorderofthesummations in(7-2-11)weobtain P,(X.)";;;#,C~e,P,({S,l",)P2"'(S,. Sd] f~k M-=-~~ ,,;;;2:P2(Shs.) 1=1,...(7·2-11) (7-2-12) whereP2(s"so)represents theensemble average ofP2m(S/.s.)overthe2"M codesorthe2nMcommunication systems. Fortheadditive whitegaussian noisechannel, thebinaryerrorprobability P2",(SIo5.)is (fdf:) gm(S,.5.)=Q\/m;, wheredi.=151-5.1'-If5,ands.differindcoordinates, n df.=Is,-5.12=L(5,)-S.Y=d(2~f =4d'€, j=l Hence,(7-2-13) (7-2-14) (7-2-15) Now,wecanaverage P2m(S/.5.)overtheensemble ofcodes.Sinceall thecodesareequallyprobable, thesignalvector 5,isequallylikelytobeany ofthe2npossible verticesofthehypercube anditisstatistically independent 394 DIGITAL COMMUNICATIONS ofthesignalvectors•.Therefore, P(Sfj=s.;)=1andP(Sfj~s.;)=tinde­ pendently foralli=I,2,..,,n.Consequently, theprobability that51ands. differindpositions issimply P(d)=orC) (7-2-16) Hence,theexpected valueofP2m(S/,S.)overtheensemble ofcodesmaybe expressed as (7-2-17) Theresult(7-2-17)canbesimplified ifweupper-bound theQ-function as Q(~2~;)<e~di,lN" Thus, P2(S/.s.)~2-ni(nd)e-'I"W" d=O ~2-"(1+e-f,fN u)" ~[W+e-I,IN"»),, (7-2-18) Weobservethattheright-hand sideof(7-2-18)isindependent oftheindices' andk.Hence,whenwesubstitute thebound(7-2-18)into(7-2-12), weobtain M p,.(X.)~2:""p2'7(s-/o-s.-c)=(M-lm(l+e~/JN")),, 1=1,,.. <M{W +e-i,lN,,»)" Finally, ~unconditional average errorprobability Feisobtained by averaging Pe(X.)overallpossible k-bitinformation sequences. Thus, Fr=2:Pe(X.)P(X.) <M[1(l+e-I,IN,,»)"2:P(X.)• • <M[W+e-t.IN,,»)" (7-2-19) Thisresultcanbeexpressed inamoreconvenient formbyfirstdefining a parameter Ro•whichiscalledthecutoffrateandhasunitsofbits/dimension, as 2 R"=IOg21-'IN+e...,0 =I -log2(1+eI,IN,,).antipodal signaling Then,(7-2-19)becomes(7-2-20) (7-2-21) CHAPTER 7:CHANNEL CAPACITY ANDCODING 395 I.V § ~0.8 ~V6 ~0.4 ~ ~V2 '" FtGURE 7-2-2ThecutoffrateR"asafunctionoftheSNRperdimension indecibels. Sincen=DT,(7-2-21)maybeexpressed asVL--->_'---'-_---'-_---' -10-j0510 ~.IN()(dB) (7-2-22) Theparameter R"isplottedasafunction of'lclNoinFig.7-2-2.Weobserve that0'"Ro'"1.Consequently, P,......0asT.......00,provided thattheinformation rateR<DRo. Alternatively, (7-2-2l)maybeexpressed as TheratioR;Dalsohasunitsofbitsldimension andmaybedefinedas R R RTkR=-=-=-=- cDniTn n Hence,Rcisthecoderateand(7-2-23) (7-2-24) (7-2-25) Weconclude thatwhenRc<Ro,theaverageprobability oferrorP,.......0asthe codeblocklengthn.......00.Sincetheaveragevalueofthe probability errorcan bemadearbitrarily smallasn--.00,itfollowsthatthereexistcodesinthe ensemble of20Mcodesthathaveaprobability oferrornolargerthanP,. Fromthederivation oftheaverage errorprobability givenabove,we conclude thatgoodcodesexisl.Although wedonotnormally selectcodesat random, itisinteresting toconsider thequestion ofwhether ornotarandomly selectedcodeislikelytobeagoodcode.Infact,wecaneasilyshowthatthere aremanygoodcodesintheensemble. First,wenotethatP,isanensemble averageoferrorprobabilities overallcodesandthatalltheseprobabilities are obviously positivequantities. Ifacodeisselected atrandom, theprobability thatitserrorprobability Pe>aP,islessthanIIa.Consequently, nomorethan 10%ofthecodeshaveanerrorprobability thatexceedslOP,andnomorethan 1%ofthecodeshaveanerrorprobability thatexceedslOOP,. 396 DIGITAL COMMUNICATIONS Weshouldemphasize thatcodeswitherrorprobabilities exceedingp.are notnecessarily poorcodes.Forexample, supposethatanaverageerrorrateof P,<10-10canbeattained byusingcodeswithdimensionality nowhen Ro>R,.Then,ifweselectacodewitherrorprobability 1000P.=10-1,wemay compensate forthisreduction inerrorprobability byincreasing nfromnoto n=lOno/7.Thus,byamodestincreaseindimensionality, wehaveacodewith f>.<10-10•Insummary, goodcodesareabundant and,hence,theyareeasily foundevenbyrandomselection. Itisalsointeresting toexpresstheaverageerrorprobability in(7-2-25) in termsoftheSNRperbit,Yb'Toaccomplish this,weexpresstheenergyper signalwaveform as (7-2-26) Hence,n=k'€b/'C" WealsonotethatR''€b/'i, =1.Therefore, (7-2-25)maybe expressed as (7-2-27) where 'Yoisanormalized SNRparameter, definedas R, 'Yo=R o')Ib (7-2-28) Now,wenotethatp.......0ask......co,provided thattheSNRperbit,'Yb>'Yo. Theparameter 'YoisplottedinFig.7-2-3asafunctionofR,'Yb'Notethatas R,')Ib......0,yo......2In2.Consequently, theerrorprobability forM-arybinary codedsignalsisequivalent tothe'errorprobability obtained fromtheunion boundforM-aryorthogonal signals,provided thatthesignaldimensionality is sufficiently largesothat'Yo~21n2. Thedimensionality parameter Dthatweintroduced in(7:2-5)ispropor­ tionaltothechannelbandwidth required totransmit thesignals.Recallfrom thesampling theorem thatasignalofbandwidth Wmayberepresented by samplestakenatarateof2Wsamples/so Thus,inthetimeintervaloflengthT 10 FIGURE 7·2-3LowerboundonSNRperbit,'rb.forbinaryantipodal signals.8.; ~6 oc' " 4,. 2F-__ ~ ~!-:IO:---':-5:---:0~~5 --:"0"" R,Y. R,y.(dB) CHAffER', CHAN~EL CAPACITY ANDCODING397 therearen=2WTsamples or,equivalently, ndegrees offreedom (dimen­ sions).Consequently, Dmaybeequatedwith2W. Finally,wenotethatthebinarycodedsignalsconsidered inthissectionare appropriate whentheSNRperdimension issmall,e.g.,'lJNo<10.However, when't,1No>10,Rosaturates atIbit/dimension. Sincethecoderateis restricted tobelessthanRo,binarycodedsignalsbecome inefficient al 'ifSJNo>lO.[nsuchacase,wemayusenonbinary-coded signals 10achievean increaseinthenumberofbitsperdimension. Forexample, multiple-amplitude codedsignalsetscanbeconstructed fromnonbinarycodesbymapping each codeelementintooneofqpossibleamplitude levels(asinPAM).Suchcodes areconsidered below. 7-2-2Random CodingBasedonM-ary Multiamplitude Signals Insteadofconstructing binary-coded signals,suppose weemploynonbinary codeswithcodewordsoftheformgivenby(7-2-1),wherethecodeelements Ci)areselected fromtheset{O,1,...,q-I}.Eachcodeelement ismapped. intooneofqpossible amplitude levels.Thus,weconstruct signalscorrespond­ ington-dimensional vectorslSi}asin(7-2-4),wherethecomponents {sillare selected fromamultiamplitude setofqpossible values.Now,wehaveq" possiblesignals,fromwhichweselect.'vi=2RTsignalstotransmit k-bitblocks ofinformation. Theqamplitudes corresponding tothecodeelements {O,1,...,q-I}maybedenoted by{a"a2,...,aq},andtheyareassumed to beselected according tosomespecified probabilities {p,}.Theamplitude levels areassumed tobeequallyspacedovertheinterval [-~, ~].ForeKample. Fig.7-2-4illustrates theamplitude valuesforq=4.Ingeneral, adjacent amplitude levelsareseparated by2~/(q-J).Thisassignment guarantees notonlythateachcomponent Si!ispeak-energy-iimited to~,but,also,each codewordisconstrained inaverageenergytosatisfythecondition ISil2<n'ifSc (7-2-29) Byrepeating thederivation givenaboveforrandomselection ofcodesinan AWONchannel, wefindthattheaverage probability oferrorisupper­ bounded as (7-2-30) whereR0isdefinedas Ro=-log,(f 1=1q,)m'2;\PIPme-d,,,,/4N!1 •(7-2-31) FIGURE 7-2-4Signalalphabet co!'slsting offouramplitude levels. 398 DIGITAL COMMUNICATIONS and d'm=la,-ami, I,m=I.2,...,q (7-2-32) Inthespecialcasewherealltheamplitude levelsareequallylikely, P,=Pm=l/qand(7-2-31)reducesto (1q R"=-log,-,2: q'~Iie~dl.I4N") m=1(7-2-33) Forexample, whereq=2and01=-Vi;, 02=Vi;,wehavedll=d22=0, d'2=d"=2Vi.",and,hence, q=2 whichagreeswithourprevious result.Whenq=4,01=.-Vi;, 02=-Vi;/3. oJ=Vi;/3,and04=~,wehavedmm= 0form=1,2,3,4,d12=d2J=d)4= d21=dJ2=d'J=2Vi."/3,d"=dJ,=d2•=d42=4Vi;/3, anddl4=d41=2~. Hence, 8R"=10"-------;=,,---==-----;~"'"2+3e""9N"+2e'-4~19N"+e',IN.,'q=4 (7-2-34) Clearly,R"nowsaturates at2bits/dimension as'leINoincreases. ThegraphsofR"asafunctionof'ff,lNoforequallyspacedandequallylikely amplitude levelsareshowninFig.7-2-5forq=2,3,4,8,16,32,and64.Note thatthesaturation levelnowoccursatIog2qbits/dimension. Consequently, for highSNR,1>,.....0asn-->00,provided thatR<DRo=2WRobits/so Ifweremovethepeakenergyconstraint oneachoftheelements, butretain·, FIGURE,·z-sCutoffrateR"forequallyspacedq-Ievelamplitude modulation withequalprobabilities p,.=I/q.[FromPrinciples ofCommunication Engineering, hy1.M.WauneTaft and I.M.Jacobs.© 1965byJohnWileyandSOliS.Illc.Reprinted .,irhpermission of'hepulilisher.]10.0 8.0640.032 4.0816 R· ·10 2.0• 3 ~ q=2 :!21.0 j0.8 -::.0.6 2 0<J0.'8 0.264 oI~w....-,:,......,,::-~-'---, -1001020304050 Energyratioperdimension . 10'og,o<6..IN.) • • CHAPTER 7:CHANNEL CAPACITY A~DCOOIN"(, 399 theaverageenergyconstraint percodewordasgivenby(7-2-29) i1ispossible toobtainalargerupperboundonthenumberofbitsperdimension. Forthis case,theresultobtained bySllannon (1959b)is 1 ['£c~('£c)Z]R~=-1+- - 1+ - logze 2Nil IV" +~logz[~(1+~1+(~n](7-2-35) ThegraphofR~asafunctionoftheSNRperdimension, 'l:,./No,isalsoshown inFig.7-2-5.Itisclearthatourselection oftheequallyspaced,equallylikely amplitude levelsthatresultinRoissuboptimum. However, thesecodedsignals areeasilygenerated andimplemented inpractice. Thisisanimportant advantage thatjustifiestheiruse. 7-2-3Comparison ofRtwiththeCapacity ofthe AWGNChannel Thechannelcapacityoftheband-limited additivewhitegaussian noisechannel withanaveragepowerconstraint ontheinputsignalwasderivedinSection 7-1-2,andisgivenby C=Wlogz(1+Pav )bitslsWNo(7-2-36) wherePa,.istheaverage poweroftheinputsignalandWisthechannel bandwidth. Itisinteresting toexpressthecapacityofthischannel intermsof bits/dimension andtheaverage powerintermsofenergy/dimension. With D=2Wand wehave (7-2-37) Bydefining Cn=C/2W=C/Dandsubstituting forWandPm(7-2-36)maybe expressed as I ( '#:,)Cn=,log21+2- No =~log,(l+2RcYh)bits/dimension (7-2-38) 400 DIGITAL c-OMMUNICATIONS FIGURE '-2-6Comparison ofcutoffrateR~withthechannelcapacityfor anAWGNchanne\..~.1.0 ~2.5 :E f;2.0 :is ~:;1.5 §1.0 ;,.;"0.5 t;; '"-50 5 1015 <INo(dB) Thisexpression forthenormalized capacity maybecompared withR:;,as showninFig.7-2-6.SinceCnistheultimate upperlimitonthetransmission rateRID.Rt<Cnase"pected. Wealsoobservethatforsmallvaluesof'€,./No. thedifference between R~andCnisapproximately 3dB.Therefore; theuseof randomly selected, optimum average power-limited, multiamplitude signals yieldsaratefunction R~thatiswithin3dBofthechannel capacity. More elaborate bounding techniques arerequired. toshowthattheprobability of errorcanbemadearbitrarily smallwhenR<DCn=2WCn=C. 7·3COMMUNICATION SYSTEM DESIGN BASED ONTHECUTOFF RATE Intheforegoing discussion, wecharacterized codingandmodulation per­ formance intermsoftheerrorprobability, whichiscertainly ameaningful criterion forsystemdesign.However, inmanycases,thecomputation ofthe errorprobability isextremely difficult. especially ifnonlinear operations such assignalquantization areperformed inprocessing thesignalatthereceiver, or iftheadditivenoiseisnongaussian. Insteadofattempting tocompute theexactprobability oferrorforspecific codes,wemayusetheensemble averageprobability oferrorforrandomly selected codewords.Thechannel isassumed tohaveqinputsymbols {O,1,...,q-l}andQoutputsymbols {O,1,...,Q -I},andtobecharac­ terizedbythetransition probabilities P(iIj).wherej=G.I,....q-Iand i=0,l,...,Q -J,withQ'"q.Theinputsymbolsoccurwithprobabilities {pJ andareassumed tobestatistically independent. Inaddition, thenoiseonthe channelisassumed tobestatistically independent intime,sothatthereisno dependence amongsuccessive received symbols. Undertheseconditions, Ihe ensemble averageprobability oferrorforrandomselectedcodewordsmaybe derivedbyapplying theChernoff bound(seeViterbiandOmura,1979). Thegeneralresultthatisobtained forthediscretememorylesschannel is (7-3-1) wherenistheblocklengthofthecode,Ristheinformation rateinbiIS/s,Dis CHAYfER 7:CHANNEL CAPACITY ANDCODING 401 p\r1j) FIGURE '-J-lExample ofquantization oflhe demodulator outputintofivelevels. thenumberofdimensions persecond,andRQisthecutoffrateforaquantizer withQlevels,definedas RQ=~p~f{-log2~l[%pjv'PVlJ)n (7-3-2) Fromtheviewpoint ofcodedesign,thecombination ofmodulator, waveform channel, anddemodulator constitutes adiscrete-time channelwithq inputsandQoutputs. Thetransition probabilities {P(iIj)}dependonthe channelnoisecharacteristics, thenumberofquantization levels,andthetype ofquantizer, e.g.,uniform ornonuniform. Forexample, inthebinary-input AWGNchannel, theoutputofthecorrelator atthesampling instantmaybe expressed as p(yIj)=_1_e-(y-m j)'l2u',j=0,1 (7-3-3)V2Ro' whererno=-Vi,:,m1=Vi,:,and0'2=!No•ThesetwopdfsareshowninFig. 7-3-l..Also illustrated inthefigureisaquantization schemethatsubdivides thereallineintofiveregions. Fromsuchasubdivision, wemaycompute the transition probabilities andoptimally selectthethresholds thatsubdivide the regionsinawaythatmaximizes RQforanygivenQ.Thus, P(ijj)=fp(yIj)dy ';(7-3-4) (7-3-5)wheretheintegralofp(ytj)isevaluated overtheregionrithatcorrespond!> to thetransition probability P(iIj). ThevalueoftherateRQinthelimitasQ.....""yieldsthecutoffrateforthe unquantized decoder.Itisrelatively straightforward toshowthatasQ.....0::. thefirstsummation (sumfromi=0toQ-1)in(7-3-2)becomes anintegral andthetransition probabilities arereplaced bythecorresponding pdfs.Thus, whenthechannelconsists ofqdiscrete inputsandonecontinuous outputy. whichrepresent!> theunquantized outputfromamatched filteroracross· correlator inasystemthatemploys eitherPSKoramultiamplitude (PAM) modulation, thecutoffrateisgivenby Ro=max{-log2[dy[~lPiv'PCYl7>]2} {PI} -x /=0 wherePj.0~j~q-I,isthe probability oftransmitting thejthsymboland 402 DIGITAL COMMUNICATIONS p(yIj)istheconditional probability densityfunction oftheoutputyfromthe matched filterorcross-correlator whenthejthsignalistransmitted. Thisisthe desiredexpression forunquantized (soft-decision) decoding. Weobserve thatwhentheinputsignalisbinaryPSKwithPo=P,=~and thenoiseisadditive, white,andgaussian, (7-3-5)reducestothefamiliarresult givenpreviously in(7-2-20). Thegeneralexpressions in(7-3-5)and(7-3-2)allowustocompare the performance ofvariousreceiver implementations basedonadifferent number ofquantization levels. Example 7-3-1 Letuscompare theperformance ofabinaryPSKinputsignalinanAWGN channelwhenthereceiverquantizes theoutputtoQ=2,4,and8levels.To simplify theoptimization problem forthequantization ofthesignalatthe outputofthedemodulator, thequantization levelsareplacedat0,±'h' ±21'h'.., ,±(2b-t-l)1'h'where 1'.isthequantizer step-size parameter, whichistobeselected, andbisthenumberofbitsofthequantizer. Agood strategy fortheselection of'histochooseittominimize theSNRperbit I'bthatisrequired foroperation atacoderateRo.Thisimpliesthatthe step-size parameter mustbeoptimized foreverySNR,whichinapractical implementation ofthereceiver meansthattheSNRmustbemeasured. Fortunately, 1'hdoesnotexhibithighsensitivity tosmallchangesinSNR,so thatitispossible tooptimize'hforoneSNRandobtaingoodperformance forawiderangeofSNRsaboutthisnominal valuebyusingafixed'h' Basedonthisapproach, theexpression forRQgivenby(7-3-2)was evaluated forb=1(hard-decision decoding), 2,and3bits,corresponding to Q=2,4,and8levelsofquantization. TheresultsareplottedinFig.7-3-2. ThevalueofRoforunquantized soft-decision decoding, obtained by evaluating (7-3-5)isalsoshowninFig.7-3-2.Weobserve thattwo-bit quantization with1'h=1.0gainsabout1.4dBoverhard-decision decoding, andthree-bit quantization with1'h=0.5yieldsanadditional 0.4dBimprove­ ment.Thus,withathree-bit quantizer, wearewithin0.2dBofthe oO!:-----!-I-U..J'!:-2-~3-L.--'4:---'5"" SNRperbil.lb(dB)FIGURE 7-3-2Effectofquantization ontheperformance ofacoded communications systemoperating atarateR=:Roor R=RQ•withbinaryPSKmodulation onanAWON channel.'"..0.75 1 <t <')0.5 ~ 0.25Q=oo Q=8 (3biu) til=0.5 Q=4 (2biu) t,,=1 (7-3-6)CHAPTER 7:CHANNEL CAPACITY ANDCODING 403 unquantized soft-decision decoding limit.Clearly,thereislittletobegained byincreasing theprecision anyfurther. Whenanonbinary codeisusedinconjunction withM-ary(M=q) signaling. thereceived signalattheoutputoftheMmatched filtersmaybe represented bythevectory=[y,y,...YM].ThecutoffrateforthisM-input, M-output (unquantized) channel is Ro='?P~Jx{-IOg2'%,'~' PiP;rv'p(yIj)p(yIT)dY} wherep(yIj)istheconditional probability densityfunction oftheoutput vectoryfromthedemodulator giventhatthejthsignalwastransmitted. Note that(7-3-6)issimilarinformto(7-3-5)exceptthatwenowhaveanM-fold integraltoperform becausethereareMoutputsfromthedemodulator. LetusassumethattheMsignalsareorthogonal sothattheMoutputs conditioned onaparticular inputsignalarestatistically independent. Asa consequence, M-1 p(yIj)=p,+,,(Yj)np,,(y,) i~O '~J(7-3-7) wherep,+,,(y) isthepdfofthematched filteroutputcorresponding tothe transmitted signaland{p,,(y,)}corresponds tothenoise-only outputsfromthe otherM-1matched filters.When(7-3-7)isincorporated into(7-3-6)we obtain Ro=max{-log,[YP~+Y M,2:._' O'PiPj(i_~, dyv'P,+n(Y)P,,(Y)rJ} {Pj} )=0 j=O ~ i~j(7-3-8) Themaximization ofRooverthesetofinputprobabilities yieldsP,=11Mfor 1",j",M.Consequently, (7-3-8)reducesto {M } Ro=log21+(M-1)[J"~v'Ps+n(Y)Pn(Y)dyI' =log2M-log2{I+(M-l)[f~ v'p,+,,(y)p"(y)dyn(7-3-9) ThisisthedesiredresultforthecutoffrateofanM-aryinput,M-aryvector outputunquantized channel. Forphasecoherent detection oftheM-aryorthogonal signalstheappropri­ atepdfsare 1 2/2'p,,(y)=--e'Y "v2ira(7-3-10) 404 DIGITAL COMMUNICATIONS wherem=Wand 0'2=!No.Substituting theserelations into(7-3-9)and evaluating theintegralyields (7-3-11) where'tisthereceived energyperwavefonn, Rwistheinformation ratein bits/waveform, andI'b='ib/NoistheSNRperbit. Weshouldemphasize thattherateparameter Rwhasimbedded initthe coderateRe.Forexample, ifM=2andthecodeisbinarythenRw=R"More generally. ifthecodeisbinaryandM=2'theneachM-arywaveform conveys Rw=vRcbitsofinformation. Itisalsointeresting tonotethatifthecodeis binaryandM=2then(7-3-11)reducesto (2) R-100-~1+e~R",12 •M=2orthogonal signals (7-3-12) whichis3dBworsethanthecutoffrateforantipodal signals.IfwesetRw•=Ro in(7-3-11)andsolvefor"Yb.weobtain 2 (M-1) "Yb=RoIn2~R"M- 1(7-3-13) GraphsofRoversus'YbforseveralvaluesofMareillustrated inFig.7-3-3. NotethatthecurveforanyvalueofMsaturates atR"=log2M. Itisalsointeresting toconsider thelimitingformof(7-3-11)inthelimitas M--+00.Thisyields '#limR"= bits/waveform M_x 2Noln2(7-3-14) 7:Limitas.M=8 :M~t>C -",,-- '(lAdS),, : M=4,,,,,,, ., 12345. SNRpe..-bit,Y,,(dBl312.5 i2 ~I.S atl .l< O!0.5SNRperbitrequited tooperateatarateR"withM-ary orthogonal signal,detected coherently inanAWGN channel.FIGURE'·3-3 (7-3-15) (7-3-17)CHAPTER 7:CHANNEL CAPACITY AND(ODIN(, 405 Since'l=P"T,whereTisthetimeinterval perwaveform, itfollowsthat limRll=Pa,=~C M-~T2NoIn2 x Hence,inthelimitasM--->"',thecutoffrateisone-half ofthecapacity forthe infinitebandwidth AWGNchannel. Alternatively, thesubstitution of'l:=Ro'l:, into(7-3-14) yields "JIh=2ln2(1.4dB),whichistheminimum SNRrequired to operateatRo(asM--->"').Hence,signaling atarateRorequires 3dBmore powerthantheShannon limit. ThevalueofRogivenin(7-3-11) isbasedontheuseofM-aryorthogonal signals, whichareclearlysuboptimal whenMissmall.Ifweattempt to maximize Robyselecting thebestsetofMwaveforms, weshouldnotbe surprised tofindthatthesimplex setofwaveforms isoptimum. Infact,Rllfor theseoptimum waveforms issimplygivenas Ro=log,L+(M_1)~-"Ul2(M-')N"] (7-3-16) Ifwecompare thisexpression with(7-3-11) weobserve thatRoin(7-3-16) simply reflects thefactthatthesimplexsetismoreenergy-efficient byafactor M/(M-1). Inthecaseofnoncoherent detection, theprobability density functions corresponding tosignal-plus-noise andnoisealonemaybeexpressed as Ps+n(Y)=ye'(Y'+a')/2/o(ay),y'"0 Pn(y)=ye,'/2y",O ;lh,re,bydefinition, a=Y2'l:/N ll.Thecomputation ofRogivenby(7-3-9)does notyieldaclosed-form solution. Instead, theintegral in(7-3-9)mustbe evaluated numerically. ResultsforthiscasehavebeengivenbyJordan(1966) andBucher(1980).Forexample, the(normalized) cutoff rateRllforM-arv orthogonal signalswithnoncoherent detection isshowninFig.7-3-4for FIGURE 7-J...4 SNRperbitrequired tooperateatarateR"with M·aryorthogonal signalsdetected noncoherently inanAWGN channel.M=It! M=H 0,75 ~eM=4"0 .~05- ~0z U.25 0 0 5 10 SNR~rOIL"'f/,IJBI(, 406 DIGITAL COMMUNICATIONS M=2,4,8,and16.Forpurposes ofcomparison wealsoplotthecutoffratefor hard-decision decoding (Q=M)oftheM-arysymbols. Inthiscase,wehave RQ=logz{P/(1_PM)+~(M_I)PMf}' Q=M(7-3-18) wherePMistheprobability ofasymbolerror.Forarelatively broadrangeof rates,thedifference betweensoft-andhard-decision decoding isapproximately 2dB. Themoststrikingcharacteristic oftheperformance curvesinFig.7-3-4is thatthereisanoptimum coderateforanygivenM.Unlikethecaseof coherent detection, wheretheSNRperbitdecreases monotonically witha decrease incoderate,theSNRperbitfornoncoherent detection reachesa minimum inthevicinityofanormalized rateof05,andincreases [orbothhigh andlowrates.Theminimum isratherbroad,sothereisreallyarangeofrates from0.2to0.9wheretheSNRperbitiswithin1dBoftheminimum. This characteristic behavior intheperformance withnoncoherent detection is attributed tothenonlinear characteristic ofthedetector. 7·4BIBLIOGRAPHICAL NOTES ANDREFERENCES Thepioneering workonchannelcharacterization intermsofchannelcapacity andrandom codingwasdonebyShannon (1948a,b,1949).Additional contributions weresubsequently madebyGilbert(1952),Elias(1955),Galla­ ger(1965),Wyner(1965),Shannon eral.(1967),Forney(1968)andViterbi (1969).Alloftheseearlypublications arecontained intheIEEEPressbook entitled KeyPapersintheDevelopment ofInformation Theory, editedby Slepian(1974). Theuseofthecutoffrateparameter asadesigncriterion wasproposed and developed byWozencraft andKennedy (1966)andbyWozencraft andJacobs (1965).ItwasusedbyJordan(1966)inthedesignofcodedwaveforms for M-aryorthogonal signalswithcoherent andnoncoherent detection. Following thesepioneering works,thecutoffratehasbeenwidelyusedasadesign criterion forcodedsignalsinavarietyofdifferent channelconditions. PROBLEMS 7-1ShowthatthefollOWing tworelations arenecessary andsufficient conditions for thesetofinputprobabilities (P(xj)}tomaximize leX;Y)and,thus,toachieve capacityforaDMC: l(x,:Y)=CforalljwithP(x,)>0 l(x,:y)",;CforalljwithP(xj)=0 whereCisthecapacityofthechanneland Q-' P(..Iv) l(xj:Y)=2:P(YiIx;)log~). ,-0 P(y, f'IGURE P7-ZCHArTER 7:CHAN!'EL CAPACITY :\~DlOD],\(· 407 Input O'ilpur X I_I' to......-----.:.~---_", ...() p/(M-I) "_ t:L-----=".t 1-Il 7-2FigureP7-2illustrates anM-arysymmetric DMCwithtransition probabilities P(vIx)~I-pwhenx=y=kfork~O,I,....M-I,andPlyix)~p/(M -Ii whenx~v. aShowthatthischannel satisfies thecondition giveninProblem 7-1when P(x,) ~1/M bDetermine andplotthechannelcapacity asafunction ofp. 7·3Determine thecapacities ofthechannels showninFig.P7-3, FIGURE P7.J (1I) thJ 7·4Consider thetwochannels withthetransition probabilities asshowninFig_P7-4 Determine ifequallyprobable inputsymbols maximize theinformation rate through thechannel. 0.6 0.6 ",.x, , , ., ., \"~ x~ .1'.,x, V1 0.6 0.6 FIGURE P7·4 (a) (h) FIGURE P7-6408 DIGITAL COMMUNICATiONS x,Je.0--""""""1-------" YJ 2(I-pi 7·5Atelephone channel hasabandwidth W=3000Hzandasignal-to-noise power ratio()f400(26dB).Suppose wecharacterize thechannel asaband-limited AWGNwaveform channelwithP.)WN o=400. aDetermine thecapacityofthechannel inbitsis. bIsthecapacity ofthechannelsufficient tosupportthetransmission ofaspeech signalthathasbeensampled andencoded bymeansoflogarithmic peM? cUsually, channelimpairments otherthanadditive noiselimitthetransmission rateoverthetelephone channel tolessthanthechannel capacity ofthe equivalent band-limited AWGN channel considered in(a).Suppose thata transmission rateofO.7Cisachievable inpractice withoutchannel encoding. Whichofthespeechsourceencoding methods described inSection3-5provide sufficient compression tofitthebandwidth restrictions ofthetelephone channel? 7-6Consider thebinary-input, quaternary-output DMCshowninFig.P7-6. aDetermine thecapacity ofthechannel. bShowthatthischannelisequivalent toaBSe. ,-,Detennine thechannelcapacity forthechannelshowninFig.P7·7. '-8Consider aBSCwithcrossover probability oferrorp.Suppose thatRi<the numberofbitsinasourcecodewordthatrepresents oneof2Rpossible levelsat theoutputofaquantizer. Determine atheprobability thatacodewordtransmitted overtheBSeisreceived correctly; btheprobability ofhavingatleastonebiterrorinacodewordtransmittLd over theBSC; ctheprobability ofhavingn,orlessbiterrorsinacodeword; 1-"x,.\"1 x,.\"2 x," FIGURE P7-7t-p FIGURE P7-10 FIGURE P7.1lCHAPTER 7:CHANNEL CAPACiTY ANDCODtNG409 Input Output X I-p Y 0 0 , r I-p dEvaluate theprobability in(a),(b),and(c)forR=5,p=0.01.andno=5. 7·9Showthat,foraDMC,theaverage mutualinformation between asequence XIX,...X"ofchannel inputsandthecorresponding channel outputssatisfythe condition "[(XIX,·..X,,;YI.Y"...Y,,)";2:[(X,;Y;) i=l withequality ifandonlyifthesetofinputsymbols isstatistically independent. 7-10FigureP7-10illustrates abinaryerasure channel withtransition probabilities P(O10)=P(I!1)=1-PandPee10)=Pee11)=p.Theprohabilities forthe inputsymbols areP(X=D)=aandP(X=I)=I-a. aDetermine theaveragemutualinformation [(X;Y)inhits. bDetermine tilevalueofathaImaximizes I(X;Y),i.e.,thechannelcapacity Cin bits/channel use,andplotCasafunction ofpfortheoptimum valueofa cForthevalueofafoundin(h),determine themutualinformation I(x;y)= 1(0;0),/(1;I),1(0;e),and1(1;e). 7-11Consider thehinary-input. ternary-output channel withtransition prohahIlities showninFig.P7-11,whereedenotesanerasure. FortheAWGNchannel. aandp aredefinedas 0':;::::1Jf3e-(.l+"i'J2/Nodx VtrN,.l -13 P=1J"'"e-(x+vij2/N,J dx VrrN" p aDetermine RQforQ=3asafunctionoftheprobabilities aandp. bTherateparameter RQdepends onthechoiceofthethreshold f3through the probabilities", andp.Forany'if,'!No,thevalueof{JthaImaximizes RQcanhe determined bytrialanderror.Forexample, itcanbeshownthatfor~/.V, below0dB,{Jop.=0.65vTN:,; for1,.;'if.!N,,";10,{Jop.·variesapproximately 1-r-~J0..,.___+-- ...() l-p-a 410 DIGITAL COMMUNICATIONS FIGURE P7-130---:'---.; ...a ---=---- ...b Channel Io0.5./' l~--'d Channel 2oa b L..__= d linearlybetween 0.65vTN" andLOvTN". Byusing13=0.65v1N., fortheentire rangeof't,!N".plotRQversus'telN"andcompare thisresultwithRv(Q=x). 7-12Findthecapacity ofthecascadeconnection ofnbinary-symmetric channels with thesamecrossover probability E.Whatisthecapacity whenthenumher of channels goestoinfinity? 7·13Channels 1,2.and3areshowninFig.P7-13. •Findthecapacityofchannell. Whatinputdistribution achieves capacity? bFindthecapacityofchannel2.Whatinputdistribution achieves capacity? cLetCdenotethecapacity ofthethirdchannel andC,andC,represent the capacities ofthefirstandsecondchannel. Whichofthefollowing relations holds trueandwhy? C<\(C,+C,) C=\(C,+C,) C>l<c,+C,)(i) (ii) (iii) FIGURE P7-157·14LetCdenotethecapacity ofadiscrete memorylesschannel withinputalphabet 't'={x"x,.....x,}andoutputalphabet 'i!I={y"y, ....,y,,}.ShowthatC,,;; min{logM,logN} 7·15ThechannelC(knownastheZchannel) isshowninFig.P7-15. •Findtheinputprobability distribution thatachieves capacity. bWhatistheinputdistribution andcapacity forthespecialcasesE=0,E=1,and E=0.5? cShowthatifIIsuchchannels arecascaded, theresulting channel willbe equivalent toaZchannelwithE,=E". dWhatisthecapacity oftheequivalent Zchannelwhenn-+x. 7·16Findthecapacity ofanadditive whiteGaussian noisechannel withabandwidth 1MHz,power10W,andnoisepowerspectraldensityIN"=100W1Hz. 7·17Channel C,isanadditive whitegaussian noisechannel withabandwidth W. averagetransmitter powerP,andnoisepowerspectraldensity\N".Channel C,is o •0 I~', CHAPTER 7CH.A.NNEL CAPACITY ANDCODI)\;(, 411 anadditive gaussian noisechannelwiththesamebandwidth andpoweraschannel C,butwithnoisepowerspectraldensityct>,,(f).Itisfurtherassumed thatthetotal noisepowerforbothchannels isthesame;thatis, Whichchanneldoyouthinkhasalargercapacity? Giveanintuitive reasoning. 7.18Adiscrete-time memoryless gaussian sourcewithmean0andvariancea'ist,)he transmitted overabinary-symmetric channel withcrossover probability p. •Whatistheminimum valueofthedistortion attainable atthedestination (distortion ismeasured inmean-squared error)? bIfthechannel isadiscrete-time memoryless additive gaussian noisechannel withinputpowerPandnoisepowerP,,,whatistheminimum attainable distortion? rNowassume thatthesourcehasthesamebasicproperties butisnot memoryless. Doyouexpectthedistortion intransmission overthebinarv­ symmetric channel tobedecreased orincreased? Why? 7-19Xisabinarymemoryless sourcewithp(X=0)=0.3.Thissourceistransmitted overabinary-symmetric channel withcrossover probability p=0.1. •Assume thatthesourceisdirectlyconnected tothechannel, i.e.,nocodingis employed. Whatistheerrorprobability atthedestination? bIfcodingisallowed, whatistheminimum possible errorprobability inthe reconstruction ofthesource cForwhatvaluesofpisreliabletransmission possible (withcoding.ofcourse)"' 7-20Plotthecapacity ofanAWGNchannel thatemploys binaryantipodal signaling. withoptimal bit-by-bit detection atthereceiver. asafunction of*./No_Onthe sameaxis,plotthecapacity ofthesamechannelwhenbinaryorthogonal signaling isemployed. 7-21Inacodedcommunication system,Mmessages I,2,...,M=2*aretransmitted byMbaseband signalsxl(I).x,(t),...,xM(r),eachofduration nT.Thegeneral formofx,(t)isgivenby ", x,(t)=2.1.,(t-jT) /=(\ wheref,,(t)canbeeitherofthetwosignalsf,(t)or[,(r),whereI,(r)=[,(t)==0for allt'1-[0,T].Wefurtherassumethatf,(t)andf,(t)haveequalenergy '{;andthe channel isideal(noattenuation) withadditive whitegaussian noiseofpower spectral density ~No'Thismeansthatthereceived signalisr(t)=x(t)+n(t), wherex(t)isoneofthex,(t)andn(r)represents thenoise, aWithf,(t) =-het),showthatN,thedimensionality ofthesignalspace,satisties N~n. bShowthat,ingeneral,N~m. eWithM=2,showthat.forgeneralf,(t)and[,(t), p(errorIx,(t)sent)~f-.-f\!p(rIx,)p(rIx,)dr ~, wherer,x"andx,arethevectorrepresentations ofr(l),x,(r),andx,(t)inthe N-dimensionaf space. 412 DIGITAL COMMUNICATIONS dUsingtheresultof(c),showthat,forgeneralM, p(errorIxm(t)sent)<;;1~~~Mr ~.Jv'JiTrlxm)p(r IXm)dr u,'...mR eShowthatr·JVp(rIxm)p(rIxm·)dr=exp(IXm4~:ml') RN and,therefore, I,,( IXm-Xm.I')p(error x",(t)sent)<;; £.,exp l....m·....M 4Nu m''''n 8 BLOCK AND CONVOLUTIONAL CHANNEL CODES InChapter 7,wetreatedchannel codinganddecoding fromageneral viewpoint, andshowedthatevenrandomly selected codes ontheaverageyield performances closetothecapacity ofachannel. Inthecaseoforthogonal signals,wedemonstrated thatthechannelcapcitylimitcanbeachieved asthe numberofsignalsapproaches infinity. Inthischapter, we"describe specificcodesandevaluate theirperformance fortheadditive whitegaussian noisechannel. Inparticular, wetreattwo classesofcodes,namely,linearblockcodesandconvolutional codes.Thecode performance isevaluated forbothhard-decision decoding andsoft-decision decoding. 8-1LINEAR BLOCK CODES Ablockcodeconsistsofasetoffixed-length vectorscalledcodewords.The lengthofacodewordisthenumberofelements inthevectorandisdenoted byn.Theelements ofacodewordareselected fromanalphabet ofq elements. Whenthealphabet consistsoftwoelements, 0and1,thecodeisa binarycodeandtheelements ofanycodewordarecalledbits.Whenthe elements ofacodewordareselected fromanalphabet havingqelements (q>2),thecodeisnonbinary. Itisinteresting tonotethatwhenqisapower of2,i.e.,q=2"wherebisapositive integer,eachq-aryelement hasan equivalent binaryrepresentation consisting ofbbits,and,thus,anonbinary codeotblocklengthNcanbemapped intoabinarycodeofblocklength n=bN. Thereare2"possible codewordsinabinaryblockcodeoflengthn.From 413 414 DIGITAL COMMUNICATIONS these2"codewords,wemayselectM=2'codewords(k<n)toformacode. Thus.ablockofkinformation bitsismapped intoacodewordoflengthn selected fromthesetofM=2'codewords.Werefertotheresulting block codeasan(n,k)code,andtheratiokin==Rcisdefinedtobetherateofthe code.Moregenerally, inacodehavingqelements, thereareqnpossible code words.AsubsetofM=2'codewordsmaybeselected to transmitk-bitblocks ofinformation. Besidesthecoderateparameter Reoanimportant parameter ofacodeword isitsweight,whichissimplythenumberofnonzero elements thatitcontains. Ingeneral,eachcodewordhasitsownweight.Thesetofallweightsinacode constitutes theweightdistribution ofthecode.WhenalltheMcodewords haveequalweight,thecodeiscalledafixed-weight codeoraconstant-weight code. Theencoding anddecoding functions involvethearithmetic operations of addition andmultiplication performed oncodewords.Thesearithmetic operations areperformed according totheconventions ofthealgebraic field thathasasitselements thesymbols contained inthealphabet. Forexample, thesymbolsinabinaryalphabet are0and1;hence,thefieldhastwoelements. Ingeneral, afieldFconsists ofasetofelements thathastwoarithmetic operations definedonitselements, namely, addition andmultiplication, that satisfythefollowing properties (axioms). Addition 1ThesetFisclosedunderaddition, i.e.,ifa,bEFthana+bEF. 2Addition isassociative, i.e.,ifa,/:I,andeareelements ofFthen a+(b+e)=(a+b)+e. 3Addition iscommutative, i.e.,a+b=b+a. 4Thesetcontains anelement calledzerothatsatisfies thecondition a+O=a. 5Everyelement inthesethasitsownnegative element. Hence,ifbisan element, itsnegative isdenoted by-b.Thesubtraction oftwoelements, such asa-b.isdefinedasa+(-b). Multip6cation 1ThesetFisclosedundermultiplication, i.e.,ifa.bEFthenabEF. 2Multiplication isassociative, i.e.,a(bc)=(ob)e. 3Multiplication iscommutative, i.e.,ab=bo. 4Multiplication isdistributive overaddition, i.e.,(a+b)e=ae+be. 5ThesetFcontains anelement, calledtheidentity, thatsatisfies the condition a(l)=a.foranyelementaEF. 6Everyelement ofF.exceptzero,hasaninverse.Hence,ifbEF(b~0) CHArTER KBLOCKANDCONVOLUTIONAL CHA'Nft. ('ODES415 thenitsinverseisdefinedasb-I,andbb-I=IThedivisionoftwoelements. suchasa-;-b,isdefinedasab-1 Weareveryfamiliarwiththeheldofrealnumbers andthefieldofcomplex numbers. Thesefieldshaveaninfinitenumber ofelements. However, as indicated above,codesareconstructed fromfieldswithafinitenumber of elements. Afinitefieldwithqelements isgenerally calledaGaloisfieldand denoted byGF(q). Everyfieldmusthaveazeroelement andaoneelement. Hence.the simplest fieldisGF(2).Ingeneral. whenqisaprime,wecanconstruct the finitefieldGF(q)consisting oftheelements {O.I,....q-I).Theaddition and multiplication operations ontheelements ofGF(q)aredefinedmoduloqand denoted as(modq).Forexample, theaddition andmultiplication tablesfor GF(2)are + a 1a1 a1 1a:tHa1 a a a 1a1 whichareoperations (mod2).Similarly, thefieldGF(5)isasetconsisting of theelements{a.1,2,3.4}.Theaddition andmultiplication tablesforGF(5)are +a1 2 3 4 01234 001 23 4a00a00 1I 2 3 4 0101234 2234012a2413 334a12303142 4401234043 21 Ingeneral.thefinitefieldGF(q)canbeconstructed onlyifqisaprimeora powerofaprime.Whenqisaprime,multiplication andaddition arebasedon modulo'q arithmetic asillustrated above.Ifq=pmwherepisaprimeandmis anypositive integer, itispossible toextendthefieldGF(p)tothefield GF(pm). Thisiscalledtheextension fieldofGF(p). Multiplication and addition oftheelements intheextension fieldarebasedonmodulo-p arithmetic. Withthisbriefintroduction tothearithmetic operations thatmaybe performed ontheelements ofcodewords,letusnowconsider somebasic characteristics ofblockcodes. Suppose C,andCjareanytwocodewordsinan(n,k)blockcode.A measure ofthedifference between thecodewordsisthenumber of corresponding elements orpositions inwhichtheydiffer.Thismeasure iscalled theHamming distance between thetwocodewordsandisdenoted asdir 416 DIGITAL COMMU.\'IC-\TWNS Clearly,d'jfori""jsatisfiesthecondition 0<d,]";;n.Thesmallest valueofthe set{d,JfortheMcodewordsiscalledtheminimum distanceofthecodeandis denoted asdrnin•SincetheHamming distance isameasure oftheseparation between pairsofcodewords,itisintimately relatedtothecross-correlation coefficient between corresponding pairsofwaveforms generated fromthecode words.Therelationship isdiscussed inSection8-1-4. Besidescharacterizing acodeasbeingbinaryornonbinary, onecanalso describe itaseitherlinearornonlinear. Suppose C,andC]aretwocodewords inan(n,k)blockcodeandleta,and00beanytwoelements selected from thealphabet. Thenthecodeissaid10belinearifandonlyifalC,+aoCjis alsoacodeword.Thisdefinition impliesthatalinearcodemustcontainthe all-zerocodeword.Consequently aconstant-weight codeisnonlinear. Suppose wehaveabinarylinearblockcode,andletC"i=1,2,...,M, denotetheMcodewords.Forconvenience, letC,denotetheall-zerocode word,i.e.,C,=[00...0],andletw,denotetheweightoftherthcodeword.It followsthatw,istheHamming distance between thecodewordsC,andC,. Thus,thedistanced"=W,.Ingeneral, thedistance d,]between anypairof codewordsC,andC]issimplyequaltotheweightofthecodewordformedby takingthedifference between C,andCrSincethecodeislinear,thedifference (equivalent totakingthemodul0-2 sumforabinarycode)between C,andC]i~ alsoacodewordhavingaweightincluded intheset{w,}.Hence.theweight distribution ofalinearcodecompletely characterizes thedistance properties of thecode.Theminimum distance ofthecodeis.therefore. dm1n=min{w,..! r.r#I(8-1-1) Anumberofelementary concepts fromlinearalgebraareparticularly useful indealingwithlinearblockcodes.SpecificalJy, thesetofalJn-tuples (vectors withnelements) 'formavectorspaceS.Ifweselectasetofk<nlinearly independent vectorsfrom5andfromtheseconstruct thesetofalJlinear combinations ofthesevectors, the resulting setformsasubspace of5,say5", ofdimension k.Anysetofklinearlyindependent vectorsinthesubspace 5, constitutes abasis.Nowconsider thesetofvectorsin5thatareorthogonal to everyvectorinabasisforS<(and,hence,orthogonal toallvectorsin5,).This setofvectorsisalsoasubspace of5andiscalledthenullspaceof5,.Ifthe dimension of5<isk,thedimension ofthenullspaceisn-k. Expressed intermsappropriate forbinaryblockcodes,thevectorspace5 consistsofthe2"binaryvaluedn-luples. Thelinear(n,k)codeisasetof2k n-tuples calledcodewords,whichformsasubspace Sooverthefieldoftwo elements. Sincethereare2kcodewordsin5,.,abasisfor5,haskcodewords. Thatis,klinearlyindependent codewordsarerequired toconstruct 2klinear combinations, thusgenerating theentirecode.Thenullspaceof5,isanother linearcode,whichconsists of2n-kcodewordsofblocklengthnandn-k information bits.Itsdimension isn-k.InSection8-1-1,weconsider these relationships ingreaterdetaiL CHAPTER K:BUKK A!'ODCO:--.;VOLUTIO!'lAl ("HAN~El. CODES417 8-1-1TheGenerator MatrixandtheParityCheckMalrix LetXmI'Xml'-_.,Xmkdenotethekinformation bitsencoded intothecode wordCm-Throughout thischapter. wefollowtheestablished convention in codingofrepresenting codewordsasrowvectors. Thus,thevectorofk inf<JTmation bitsintotheencoder isdenoted by andtheoutputoftheencoderisthevector Theencoding operation performed inalinearbinaryblockencoder canbe represented byasetofnequations oftheform (8-1-2) wheregi,=0or1andXmig"represents theproductofXmiandgij'Thelinear equations (8-1-2)mayalsoberepresented inamatrixformas whereG,calledthegenerator matrixofthecode,is [<-gl-] [gilgil...gin] G=<-gl- =g,1&1..-gln·.. .·.. .·.. . <-gk-g"gAl gkn(8-1-3) (8-1-4) Notethatanycodewordissimplyalinearcombination ofthevectors{g,}ofG, i.e., (8-1-5) Sincethelinear(n,k)codewith2kcodewordsisasubspace ofdimension k, therowvectors{g,}ofthegenerator matrixGmustbelinearlyindependent, i.e.,theymustspanasubspace ofkdimensions. Inotherwords,the{g,}must beabasisforthe(n,k)code.Wenotethatthesetofbasisvectorsisnot unique,and,hence,Gisnotunique.Wealsonotethat,sincethesubspace has dimension k,therankofGisk. Anygenerator matrixofan(n,k)codecanbereduced byrowoperations (andcolumnpermutations) tothe"systematic form," [100...0PliPil-..P•••] G=[Ik' 0 10'"0PliP22- -.P2nk :P]=~(8-1-6). 001PAlPk' PAn-k where-Ikisthekxkidentity matrixandPisakX(n-k) matrixthat 418 DIGITAL (O'-I\jU\!t-\r10\,S determines then-kredundant bitsorparitycheckbits.Notethatagenerator matrixofthesystematic formgenerates alinearblockcodeinwhichthefirstk bitsofeachcodewordareidentical 10theinformation bitstobetransmiUed, andtheremaining n-kbitsOfeachcodewordarelinearcombinations ofthe kinformation bits.Thesen-kredundant bitsarecalledparitycheckbits.The resulting (n.k)codeiscalledasystematic code. An(n,k)codegenerated byagenerator matrixthatisnotinthesystematic form(8-1-6)iscallednonsystemaric. However. suchagenerator matrixis equivalent toagenerator matrixofthesystematic forminthesensethatone canbeobtained fromtheotherbyelementary rowoperations andcolumn permutations. Thetwo(n.k)linearcodesgenerated bythetwoequivalent generator matrices aresaidtobeequivalent. andonecanbeobtained fromthe otherbyapermutation oftheplacesofeveryelement. Thus.everylinear (n.k)codeisequivalent toalinearsystematic (n,k)code. Example 8·1·1 Consider a(7.4)codewithgenerator matrix [1 0 0 0 I o1 0 0 IG=o0 I 0 1 o0 0 I 0o I I II]I , ~=[14:PI(8-1-7) Atypicalcodewordmaybeexpressed as wherethe{x",,}represents the'fourinformation bitsandthe{em)represent thethreeparitycheckbitsgivenby (8-1-8) Alinearsystematic (n.k)binaryblockencoder maybeimplemented by usingak-bitshiftregisterandn-kmodulo-2 adderstiedtotheappropriate stagesoftheshiftregister. Then-kaddersgenerate theparitycheckbits. whicharesubsequently storedtemporarily inasecondshiftregisteroflength n-k.Thek-bitblockofinformation bitsshiftedintothek-bitshiftregister andthen-kparitycheckbitsarecomputed. Thenthekinformation bits followed bythen-kparitycheckbitsareshiftedoutofthetwoshiftregisters FIGURE 8-1-1 Alinearshiftregister forgenerating a(7,41hinary code. andfedtothemodulator. Thi,encoding i,illu,trated inFig.8-1-1forthe(7.4) codeofExample 8-1-1. As,ociated withanylinear(n.k)codei,thedualcodeofdimen,ion n-k. Thedualcodeisalinear(n.n-k)codewith2"-'codevectors, whichi,the null,paceofthe(n.k)code.Thegenerator matrixforthedualcode.denoted byRcon,i,tsofn-klinearlyindependent codevectors,elected fromthenull space.Anycodewordemofthe(n,k)codeisorthogonal toanycodewordin thedualcode.Hence.anycodewordofthe(n.k)codei,orthogonal toevery rowofthematrixH,i.e" (8-1-9) where0denotes anall-zerorowvectorwithfJ-kelements, andemisacode wordofthe(n,k)code,Since(8-1-9)holdsforeverycodewordofthe(n,k) code.itfollowsthat GH'=O (8-1-10) where0isnowakX(n-k)matrixwithall-zeroelements. Nowsuppose thatthelinear(n,k)codeissystematic anditsgenerator matrixGisgivenbythesystematic form(8-1-6).Then.sinceGH'=O.it followsthat H=[-P':I".•] (8-1-11) Thenegative signin(8-1-11)maybedropped whendealingwithbinarycodes. sincemodulo-2 subtraction isidentical tomodul0-2 addition. Example 8-1-2 Forthesystematic (7,4)codegenerated bymatrixGgivenby(8-1-7).we have,according to(8-1-11), thematrixHintheform H=[~::~~~~] 1101001(8-1-12) 420 DIGITAL COMMlJNI('ArJONS Now,theproductCmH'yieldsthethreeequations Xml+XIII:?+X,rd+emS=0 Xml+XmJ+Xm.t+Cmh=0 Xml+Xm2+Xm4+Cm7=0(8-1-13) Thus,weobservethattheproductCmH'isequivalent toaddingtheparity checkbitstothecorresponding linearcombinations oftheinformation bits usedtocompute em"j=5,6,7.Thatis,(8-1-13)areequivalent to(8-1-8). ThematrixHmaybeusedbythedecodertocheckthatareceived code wordYsatisfies thecondition (8-1-13), i.e.,YH'=O.Insodoing,the decoder checksthereceived paritycheckbitswiththecorresponding linear combination ofthebitsy"Y2,y."andY.thatformedtheparitycheckbitsat thetransmitter. Itis,therefore, appropriate tocallHtheparitycheckmatrix associated withthe(n,k)code. Wemakethefollowing observation regarding therelationoftheminimum distance ofacodetoitsparitycheckmatrixH,TheproductCmH'withCm""0 represents alinearcombination ofthencolumns ofH'.SinceCmH'=0,the columnvectorsofHarelinearlydependent. Suppose Cidenotestheminimum weightcodewordofalinear(n,k)code.Itmustsatisfythecondition CiH'=O. Sincetheminimum weightisequaltotheminimum distance, itfollowsthat dm;.ofthec.olumns ofHarelinearlydependent. Alternatively, wemaysay thatnomorethandm;n-Icolumns ofHarelinearlyindependent. Sincethe rankofHisalmosln-k.wehaven-k~dm;n-1.Therefore, dm;nis upper-bounded as dm;n~n-k+ (8-1-14) Givenalinearbinary(n,k)codewithminimum distance dm;n'wecan construct alinearbinary(n+1,k)codebyappending oneadditional parity checkbittoeachcodeword.Thecheckbitisusuallyselected tobeacheckbit onallthebitsinthecodeword.Thustheaddedcheckbitisa 0iftheoriginal codewordhasanevennumberofIsanditisa 1ifthecodewordhasanodd numberofIs.Consequently, iftheminimum weightand,hence,theminimum distanceofthecodeisodd,theaddedparitycheckbitincreases theminimum distance byI.Wecallthe(n+1,k)codeanextended code.Itsparitycheck matrixis o o H=e H o.............................. ; 1 1 1 1 I(8·1-15) whereHisthepantycheckmatrixoftheoriginalcode. CHAPTER KBLOCK ANDCONVOLUTIONAL CHAf';NEL ('ODES421 Asystematic (n,k)codecanalsobeshortened bysettinganumberofthe information bitstozero.Thatis,alinear(n,k)codeconsisting ofk information bitsandn-kcheckbitscanbeshortened intoa(n-l,k-l) linearcodebysettingthefirstIbitstozero.TheseIbitsarenottransmitted. Then-kcheckbitsarecomputed intheusualmanner, asintheoriginalcode. Since Cm=XmG theeffectofsettingthefirstlbitsofXmto0isequivalent toreducing the numberofrowsofGbyremoving thefirstlrows.Equivalently, since CmH'=0 wemayremove thefirstlcolumns ofH.Theshortened (n-l,k-l)code consistsof2'-Icodewords.Theminimum distance ofthese2k-Icodewordsis atleastaslargeastheminimum distance oftheoriginal(n,k)code. 8-1·2SomeSpecificLinearBlockCodes Inthissubsection, weshallbrieflydescribe threetypesoflinearblockcodes thatarefrequently encountered inpractice andlisttheirimportant parameters. Hamming Codes TherearebothbinaryandnonbinaryHamming codes. Welimitourdiscussion totheproperties ofbinaryHamming codes.These comprise aclassofcodeswiththeproperty that (n,k)=(2m-1,2m-1-m) (8-1-16) wheremisanypositiveinteger.Forexample, ifm=3,wehavea(7,4)code. TheparitycheckmatrixHofaHamming codehasaspecialproperty that allowsustodescribe thecoderathereasily.Recallthattheparitycheckmatrix ofan(n,k)codehasn-krowsandncolumns. Forthebinary(n.k) Hamming code,then=2m -Icolumns consistofallpossible binaryvectors withn-k=melements, excepttheall-zero vector.Forexample. the(7,4) codeconsidered inExamples 8-1-1and8-1-2isaHamming code.Itsparit} checkmatrixconsists ofthesevencolumnvectors(001),(010), (011), (100). (101),(110),(111). Uwedesiretogenerate asystematic Hamming code,theparitycheck matrixHcanbeeasilyarranged inthesystematic form(8-1-11). Thenthe corresponding generator matrixG(;anbeobtained from(8-1-11), Wemaketheobservation thatnotwocolumnsofHarelinearlydependent, forotherwise thetwocolumns wouldbeidentical. However, form>I.itis possible tofindthreecolumns ofHthataddtozero.Consequently, dmin=3for an(n,k)Hamming code. Byaddinganoverallparitybit,aHamming (n,k)codecanbemodified to yieldan(n+I,k)codewithdmin=4.Ontheotherhand,an(n,k)Hamming codemaybeshortened to(n-t,k-l)byremoving lrowsofitsgenerator J;llatrixGor,eqUivalently, byremoving Icolumns ofitsparitycheckmatrixH. 422 DIGITALI'OMMliNICAT10t'-;S Theweightdistribution fortheclassofHamming (n.k)codesisknownand isexpressed incompact formbytheweightenumerating polynomial n A(z)=2:Aizi i={) =_1.[(l+z)"+n(l+Zlin-'1/2(1 -Zlin+ ,li2] n+I whereAiisthenumber ofcodewordsofweighti.(8-1-17) (8-1-18) (8-1-19) (8-1-20) (8-1-21)Hadamard Codes AHadamard codeisobtained byselecting ascode wordstherowsofaHadamard matrix.AHadamard matrixMnisannXn matrix(naneveninteger)ofIsandOswiththeproperty thatanyrowdiffers fromanyotherrowinexactlyInpositions.tOnerowofthematrixcontains all zeros.Theotherrowscontain!nzerosand!nones. Forn=2.theHadamard matrixis_[00] M2-oI Furthermore, fromMn•wecangenerate theHadamard matrixM2naccording totherelation M=[Mn~n] 2nM Mn n whereMndenotes thecomplement (Osreplaced byIsandviceversa)ofM". Thus,bysubstituting (8-1-18) into(8-1-19), weobtain [0000]o1 0 1 M4=0 0 I 1 oI 1 0 Thecomplement ofM.is U!J NowtherowsofM4andM4formalinear binary codeofblocklengthn= 4 having2n= 8codewords.Theminimum distance ofthecodeisdmi• =!n=2. Byrepeated application of(8-1-19), wecangenerate Hadamard codeswith blocklengthn=2"'.k=log,2n=log2zm+'=m+l. anddmi"=ln=zm-', wheremisapositiveinteger.Inaddition totheimportant specialcasewhere n=2m•Hadamard codesofotherblocklengthsarepossible, butthecodesare notlinear. tSometimes theelements oftheHadamard matrixaredenotedhy-Iand-1.Thentherows oftheHadamard matrixaremutually orthogonal. WealsonoteIhaltheM=i'signalwaveforms, conslructed fromHadamard codewordsbymapping eachhitinacodewordintoahinaryPSK signal.areorthogonal. CHAPTER KBLOCK ANDCONVOLl;nONAL CHAN""EL. CODES423 TABLE 8-1·1WEIGHT DISTRIBUTION OFGOLAY (23,12)ANDEXTENDED GOLAY (24.12) CODES Numberofcodewords Weight o 7 8 11 12 15 16 23 24(23,121code 1 253 SOIl 1288 1288 506 253 1 ()(24,12)code 1 o 759 () 2576 o 759 () 1 SO/lrce:Pderson andWeldon (1972]. GolayCodeTheGolaycodeisabinarylinear(23,12)codewithdrnin'=7. Theextended Golaycodeobtained byaddinganoverallparitytothe(23.12) isabinarylinear(24,12) codewithdrnin=8:Table8-1-1liststheweight distribution ofthecodewordsintheGolay(23,12)andtheeXlended Golay (24,12)codes.Wediscussthegeneration oftheGolaycodeinSection8-1-3. 8-1-3CyclicCod,es Cycliccodesareasubsetoftheclassoflinearcodesthatsatisfythefollowing cyclicshiftproperty: ifC=[e,,_lc,,_,...c,colisacodewordofacycliccode then[cn_,c" -.1'..CUC,,-Ij,obtained byacyclicshiftoftheelements ofC,is alsoacodeword.Thatis,allcyclicshiftsofCarecodewords.Asa consequence ofthecyclicproperty, thecodespossessaconsiderable amountof structure whichcanbeexploited intheencoding anddecoding operations. A numberofefficientencoding andhard-decision decoding algorithms havebeen devisedforcycliccodesthatmakeitpossible toimplement longblockcodes withalargenumber ofcodewordsinpractical communications systems. A description ofspecific algorithms isbeyond thescopeofthisbook.Our primary objective istobrieflydescribe anumber ofcharacteristics ofcyclic codes. Indealingwithcycliccodes,itisconvenient toassociate withacodeword C=[c"'Cn-2...c,Cll]apolynomial C(p)ofdegree";; n-1.definedas C()n-I ,,--2 .P=C,,_IP +c,,_,P +...-rCIP+C" (8-1-22) Forabinarycode.eachofthecoefficients ofthepolynomial iseitherzeroor one. Nowsuppose weformthepolynomial C() -"+",+'+ PP-C,,_IP C,,-2P ...-c,p CoP 424 DIGITAL COMMUNICATIONS Thispolynomial cannotrepresent acodeword,sinceitsdegreemaybeequal ton(when C"_I=1).However, ifwedividepC(p)byp"+1,weobtain pC(p)=c +C,(p) (8-1-23) p"+1"-1p"+1 where C1(p)=c"_,p,,-I+Cn-3P"- 2+...+CoP+C"_I Notethatthepolynomial C,(p)represents thecodewordC1= [c"2'"CoC"_d,whichisjustthecodewordCshiftedcycliclybyoneposition. SinceC1(p)istheremainder obtained bydividing pC(p)byp"+I,wesay that C1(p)=pC(p) mod(p"+1) (8-1-24) Inasimilarmanner,ifC(p)represents acodewordinacyclicr.odethen p'C(p)mod(p"+1)isalsoacodewordofthecycliccode.Thuswemaywrite piC(p) =Q(p)(p"+1)+Ci(p) (8-1-25) (8-1-26)wheretheremainder polynomial Ci(p)represents acodewordofthecyclic codeandQ(p)isthequotient. Wecangenerate acycliccodebyusingagenerator polynomial g(p)of degreen-k.Thegenerator polynomial ofan(n,k)cycliccodeisafactorof p"+ 1andhasthegeneralform (),,-k+ n-k-I IgP=p g"-k-IP +...+glP+ Wealsodefineamessage polynomial X(p)as X(p)=X,_lpk-1 +Xk_,pk-'+ ...+xIP+xo (8-1-27) where[Xk-lXk-'...X,XO]represent thekinformation bits.Clearly,theproduct X(p)g(p) isapolynomial ofdegreelessthanorequalton-I,whichmay represent acodeword.Wenotethatthereare2kpolynomials {Xi(p)},and, hence,thereare2kpossible codewordsthatcanbeformedfromagiveng(p). Suppose wedenotethesecodewordsas (8-1-28) Toshowthatthecodewordsin(8-1-28) satisfythecyclicproperty, consider anycodewordC(p)in(8-1-28). AcyclicshiftofC(p)produces C,(p)=pC(p)+C,,_I(P"+1) (8-1-29) and,sinceg(p)dividesbothp"+1andC(p),italsodividesC1(p),i.e.,C,(p) canberepresented as CI(p)=X,(p)g(p) Therefore, acyclicshiftofanycodewordC(p)generated by(8-1-28) yields anothercodeword. Fromtheabove,weseethatcodewordspossessing thecyclicproperty can (8-1-31)CHAPTER M:BLOCK ANDCONVOLlTIONAL CHANNEL CODES425 begenerated bymultiplying the2kmessage polynomials withaunique polynomial g(p),calledthegenerator polynomial ofthe(n,k)cycliccode, whichdividesp"+1.andhasdegreen-k.Thecycliccodegenerated inthis manner isasubspace 5,ofthevectorspaceS.Thedimension ofScisk. Example 8·1·3 Consider acodewithblocklengthn=7.Thepolynomial p7+Ihasthe following factors: p7+1=(p+j~(p'+p2+ ])(p'+p+1) (8-1-30) Togenerate a(7,4)cycliccode,wemaytakeasagenerator polynomial one ofthefollowing twopolynomials: g,(p)=p'+p2+1 g2(P)=p'+p+1 Thecodesgenerated bygl(p)andg2(P)areequivalent. Thecodewordsin the(7,4)codegenerated byg,(p)'"p3+p2+1aregiveninTable8-1-2. Ingeneral,thepolynomial p"+1maybefactored as p"+1=g(p)h(p) whereg(p)denotes thegenerator polynomial forthe(n,k)cycliccodeand TABLE8-H(7,4)CYCLIC CODE Generator Polynomial: gl(p)=pJ+p2+1 Inronnation bits Codeword. p'p'p'p" p'p',p'p' p' p" p 0 0 00 0 0 0 0 0 0 0 0 0 0 I 00 0 II0I 00I0 00JI0ID 0 0 II 00101I I 0I0 0 01l0ID0 01 (JI 0I1ID0I 0I I 0 0J01I I 0 011I 010DD11 J000 IID1000 10 0 1 IID0I01 I0I0 1I I 00I0 I0I1 1II1III 1I00 101I100 1101 10100 0 I I1I0 I000II0 1I1I I0 0 1011 426 DIGITAL COMMCNICATIONS h(p)denotestheparitypolynomial thathasdegreek.Thelattermaybeused togenerate thedualcode. Forthispurpose, wedefinethereciprocal polynomial ofh(p)as pkh(p-I) =p'(p-'+h._Ip-k+l+h._,p-k+' +...+h,P-'+1) =1+h,_IP+h._ zp'+...+h,P·-'+p· (8-1-32) Clearly,thereciprocal polynomial isalsoafactorofp'+1.Hence,p'h(p-l) is thegenerator polynomial ofan(n,n-k)cycliccode.Thiscycliccodeisthe dualcodetothe(n,k)codegenerated fromg(p).Thus,the(n,n-k)dual codeconstitutes thenullspaceofthe(n,k)cycliccode. Example 8-1-4 Letusconsider thedualcodetothe(7,4)cycliccodegenerated inExample 8-1-3.Thisdualcodeisa(7,3)cycliccodeassociated withtheparity polynomial h,(p)=(p'+1)(p3+P+1) =p4+p'+p'+1 (8-1-33) Thereciprocal polynomial is p4h,(p")=1+p+p'+p4 Thispolynomial generates the(7,3)dualcodegiveninTable8-1-3.The readercanverifythatthecodewordsinthe(7,3)dualcodeareorthogonal tothecodewordsinthe(7,4)cycliccodeofExample 8-1-3.Notethat neitherthe(7,4)northe(7,3)codesaresystematic. Itisdesirable toshowhowagenerator matrixcanbeobtained fromthe generator polynomial ofacyclic(n,k)code.Aspreviously indicated, the generator matrixforan(n,k)codecanbeconstructed fromanyset'ofk TABLE 8·1-3(1,3)DUALCODE Generalor Polynomial p4h,(p')=p4+ p'+p+I Information bits Codewords p'pipO p'p'p4pJp'pipo 000 [) [)[)[)0 0 0 0 0 I [) [) I[)I1I [)I[) [)I [)I I I [) 0II I)I I I[) ()I 1[)lJ IlJII I [)[) IlJI I 11 I)I[)II I I lJ I I I 11 I)1[) III I I 11 111[)1 CHAPTER N:BLOCKANDCO!\'VOLVTJONAL CHANNEL (ODES427 linearlyindependent codewords.Hence,giventhegenerator polynomial g(p). aneasilygenerated setofklinearlyindependent codewordsisthecodewords corresponding tothesetofklinearlyindependent polynomials pk-lg(p), pk-2g(p),...,pg(p), g(p) Sinceanypolynomial ofdegreelessthanorequalton-1anddivisible by g(p)canbeexpressed asalinearcombination ofthissetofpolynomials, the setformsabasisofdimension k.Consequently, thecodewordsassociated with thesepolynomials formabasisofdimension kforthe(n,k)cycliccode. Example 8-1-5 Thefourrowsofthegenerator matrixforthe(7,4)cycliccodewith generator polynomial gl(p)=p3+p2+1areobtained fromthepolynomials pig1(p)=pHi+p2+i+pi,i=3,2,1,° Itiseasytoseethatthegenerator matrixis [1 1°1°G_OII01 1-°°1 1 0 o0 0 1 1o0]o0 1°°1(8-1-34) (8-1-35)Similarly, thegenerator matrixforthe(7,4)cycliccodegenerated bythe polynomial g2(P)=p'+p+1is [1°1 1°°0]G=010110020010110 0001011 Theparitycheckmatrices corresponding toG1andGzcanbeconstructed in thesamemannerbyusingtherespective reciprocal polynomials (Problem 8-8). Notethatthegenerator matrixobtained bythisconstruction isnotin systematic form.Wecanconstruct thegenerator matrixofacycliccodeinthe systematic formG=[Ik:P]fromthegenerator polynomial asfollows.First, weobserve thatthelthrowofGcorresponds toapolynomial oftheform pn-I+R,(p),/=1,2,...,k,whereR,(p)isapolynomial ofdegreelessthan n-k.Thisformcanbeobtained bydividingpn-'byg(p).Thus,wehave pn-I +R,(p) g(p)=Q,(p)g(p),1=1,2,...,k or,equivalently, pn-'=Q,(p)g(p)+R,(p), 1=1,2,...,k (8-1-36) 428 DIGITAL mMMl'NI{'ATIONS whereQ,(p)isthequotient. Butp"~'+R,(p)isacodewordofthecycliccode sincep,,-t+R,(p)=Q,(p)g(p). Therefore thedesiredpolynomial correspond­ ingtothelthrowofGisp"-,+R,(p). Example 8-1-6 Forthe(7,4)cycliccodewithgenerator polynomial g2(P)=p-'+p+I. previously discussed inExample 8-1-5,wehave ph=(p-'+p+1)g2(P)+p2+1 p'=(p2+l)g2(p)+p2+P+1 p'=pg2(P) +p2+P p'=gip)+p+I Hence,thegenerator matrixofthecodeinsystematic formis [1 0 0 0 1 oI 0 0 I G2=0 0 1 0 1 o0 0 1 0 andthecorresponding paritycheckmatrixis [1 1 1 0 1 H2=0 1 1 1 0 1 1 0 1 0i!J o0]1 0 o1(8-1-37) (8-1-38) Itisleftasanexercise forthereadertodemonstrate thatthegenerator matrixG2givenby(8-1-35) andthesystematic formgivenby(8-1-37) generate thesamesetofcodewords(Problem 8·2). Themethod forconstructing thegenerator matrixGinsystematic form according to(8-1-36) alsoimpliesthatasystematic codecanbegenerated directlyfromthegenerator polynomial g(p).Suppose thatwemultiply the message polynomial X(p)bypn-k.Thus,weobtain pn-kx(p)=Xk_1pn-1+Xk_2P,,'2+...+x,pn-HI+XOp,,'k Inasystematic code,thispolynomial represents thefirstkbitsinthecode wordC(p).Tothispolynomial wemustaddapolynomial ofdegreelessthan n-krepresenting theparitycheckbits.Now,ifpn-.X(p)isdividedbyg(p), theresultis pn'kX(p) =Q(p)+r(p) g(p) g(p) or,equivalently, p"-kX(p) =Q(p)g(p)+r(p) (8-1-39) CHAPTER KBLOCK ANDCONVOLlITIOl'iAL CHA~NEL CODES429 wherertp)hasdegreelessthan11-k.Clearly.Q(p)g(p) isacodewordofthe cycliccode.Hence.byadding(modulo-2) r(p)tobothsidesof(R-I-]'!). we obtainthedesiredsystematic code. Tosummarize, thesystematic codemaybegenerated by Imultiplying themessage polynomial X(p)byp"'k; 2dividingp"-kX(p) byg(p)toobtaintheremainder rip):and 3addingr(p)top"kX(p). Belowwedemonstrate howthesecomputations canbeperformed byusing shiftregisters withfeedback. Sincep"+I=g(p)h(p) or.equivalently. g(p)h(p) =0mod(p"+I),we saythatthepolynomials g(p)andh(p)areorthogonal. Furthermore. the polynomials p'g(p)andp'h(p)arealsoorthogonal foralliandj.However. the vectorscorresponding tothepolynomials g(p)andh(p)areorthogonal onlyif theorderedelements ofoneofthesevectorsarereversed. Thesamestatemenl appliestothevectorscorresponding topig(p)andpih(p).Infact,iftheparity polynomial h(p)isusedasagenerator forthe(n,11-k)dualcode.thesetof codewordsobtained justcomprises thesamecodewordsgenerated bythe reciprocal polynomial exceptthatthecodevectorsarereversed. Thisimplies thatthegenerator matrixforthedualcodeobtained fromthereciprocal polynomial pkh(p·l) canalsobeobtained indirectly fromhlp).Sincethe paritycheckmatrixHforthe(n,k)cycliccodeisthegenerator matrixforthe dualcode,itfollowsthatHcanalsobeobtained fromh(p).Thefollowing example illustrates theserelationships. Example 8-1-' Thedualcodetothe(7,4)cycliccodegenerated bygl(p)=p'+p'+1is the(7,3)dualcodethatisgenerated bythereciprocal polynomial p4h,(p'l) =p4+p2+P+l.However, wemayalsousehttp)toobtainthe generator matrixforthedualcode.Then,thematrixcorresponding tothe polynomials p'hl(p), i=2,1,0,is [I I I GFd=0 I I o0 I Thegenerator matrixforthe(7,3)dualcode,whichistheparitycheck matrixforthe(7,4)cycliccode,consistsoftherowsofGhltakeninreverse order.Thus,11]1 0 o0 430 DIGITAL COMMUNICATIONS ThereadermayverifythatG1Hi=0. NotethatthecolumnvectorsofH,consistofallsevenbinaryvectorsof length3,excepttheall-zerovector.Butthisisjustthedescription ofthe paritycheckmatrixfora(7,4)Hamming code.Therefore, the(7,4)cyclic codeisequivalent tothe(7,4)Hamming codediscussed previously in Examples 8-1-1and8-1-2. Encoders forCyclicCodesTheencoding operations forgenerating a cycliccodemaybeperformed byalinearfeedback shiftregisterbasedonthe useofeitherthegenerator polynomial ortheparitypolynomial. First,letus consider theuseofg(p). Asindicated above,thegeneration ofasystematic cycliccodeinvolvesthree steps,namelymultiplying themessagepolynomial X(p)byp.-k,dividingthe productbyg(p),and,finally,addingtheremainder top.-kX(p). Ofthese threesteps,onlythedivisionisnontrivial. Thedivisionofthepolynomial A(p)=p'-kX(p)ofdegreen-Ibythe polynomial ()_.-k+ .-k-I+++g p-g.-kP gn-k-IP ...g,pgo maybeaccomplished bythe(n-k)stagefeedback shiftregisterillustrated in Fig.8-1-2.Initially,theshiftregistercontainsallzeros.Thecoefficients ofA(p) areclockedintotheshiftregisterone(bit)coefficient atatime,beginning with thehigher-order coefficients, i.e.,witha._"followed bya.-2,andsoon.After thekthshift,thefirstnonzero outputofthequotient isq.=g.-ka•. Subsequent outputsaregenerated asillustrated inFig.8-1-2.Foreachoutput coefficient inthequotient, wemustsubtractthepolynomial g(p)multiplied by thatcoefficient, asinordinary longdivision. Thissubtraction isperformed by meansofthefeedback partoftheshiftregister. Thus,thefeedback shift registerinFig.8-1-2performs divisionoftwopolynomials. Inourcase,g.-k=go=1,and,forbinarycodes,thearithmetic operations areperformed inmodulo-2 arithmetic. Consequently, thesubtraction opera­ tionsreducetomodulo-2 addition. Furthermore, weareonlyinterested in DGURE 8-1·2Afeedback sbiftresisterfordividingtbepolynomial A(p)byg(p). CHAPTER K:BLOCK ANDCONVOLUTIONAL CHANNEL rODES431 ..Tomodulator Message 0 polynomial---- .......L-.,.-e'" p"-lX(p) -, Message bits FIGURE 8·1·)Encoding ofacycliccodebyuseofthegenerator polynomial g(p). Igenerating theparitycheckbitsforeachcodeword,sincethecodeis systematic. Consequently, theencoder forthecycliccode·takestheform illustrated inFig.8-1-3.Thefirstkbitsattheoutputoftheencoderaresimply thekinformation bits.Thesekbitsarealsoclockedsimultaneously intothe shiftregister, sincetheswitch1isintheclosedposition. Notethatthe polynomial multiplication ofp"-kwithX(p)isnotperformed explicitly. After thekinformation bitsareallclockedintotheencoder, thepositions ofthetwo switches arereversed. Atthistime,thecontents oftheshiftregisteraresimply then-kparitycheckbits,whichcorrespond tothecoefficients ofthe remainder polynomial. Thesen-kbitsareclockedoutoneatatimeandsent tothemodulator. Example 8-1-8 Theshiftregister forencoding the(7,4)cycliccodewithgenerator polynomial g(p)=p'+p+1isillustrated inFig.8-1-4.Suppose theinput fiGURE 8·1·4Theencoderforthe(7.4)cycliccodewith generator polynnmial g(p)=p'+P+1.Message bits 0110----'- .....""Output 432 l)I(ilTAI COMMI;:-';WATIONS Mes~ge bih FIGURE 8-1·5Encoder foran(n,k)cycliccodebasedonparitypolynomial h(p). message bitsare0110.Thecontents oftheshiftregisterareasfollows: Input Shift Shiftregistercontents o 1 1oo 1 2 3 4000 o0 0110 1 0 1 1 0 0 Hence,thethreeparitycheckbitsare100,whichcorrespond tothecode bitsc,=0,Co=O.andC7=1. Insteadofusingthegenerator polynomial, wemayimplement theencoder forthecycliccodebymakinguseoftheparitypolynomial h(p)=p·+h._,p·-I+ ...+h,p+l TheencoderisshowninFig.8-1·5.Initially, thekinformation bitsareshifted intotheshiftregisterandsimultaneously fedtothemodulator. Afterallk information bitsareintheshiftregister, theswitchisthrownintoposition 2 andtheshiftregisterisclocked n-ktimestogenerate then-kparitycheck bitsasillustrated inFig.8-1-5. Example 8-1·9 Theparitypolynomial forthe(7,4)cycliccodegenerated byg(p)= p3+P+1ish(p)=p4+p2+P+1.Theencoder forthiscodebasedonthe paritypolynomial isillustrated inFig.8-1-6.Iftheinputtotheencoder is FIGURE 8-1-6Theencoderforthe(7.4)cycliccodebasedontheparitypolynomial hlp)=p4+p'+I. OulPUl"--'--i0/10001 CfjAPTER~: BLOCK ANDCONVOLl:'nONAL CHANNH CODES433 themessage bits0110.theparitycheckbitsarec,=o.Co=0,andC7=1.as iseasilyverified. Itshouldbenotedthattheencoder basedonthegenerator polynomial is simplerwhenn-k<k(k>~n),i.e.,forhighratecodes(Rc>~).whilethe encoder basedontheparitypolynomial issimplerwhenk<n-k(k<~n). whichcorresponds tolowratecodes(R,.<D· CyclicHamming CodesTheclassofcycliccodesincludetheHamming codes,whichhaveablocklengthn=2'"-Iandn-k=mparitycheckbits. wheremisanypositiveinteger.ThecyclicHamming codesareequivalent to theHamming codesdescribed inSection8-1-2. Cyclic(23,12)GolayCodeThelinear(23,12)Golaycodedescribed in Section8-1-2canbegenerated asacycliccodebymeansofthegenerator polynomial g(p)=PII+pO+P7+po+p'+p+1 Thecodewordshaveaminimum distance dmio=7.(8-1-40) Maximum.Length Shift·Register CodesMaximum-length shift-register codesareaclassofcycliccodeswith (n,k)=(2'"-1.m) (8-1-41) wheremisapositiveinteger.Thecodewordsareusuallygenerated bymeans ofanm-stagedigitalshiftregister withfeedback, basedontheparity polynomial. Foreachcodewordtobetransmitted, theminformation bitsare loadedintotheshiftregister. andtheswitchisthrownfromposition 1to position 2.Thecontents oftheshiftregisterareshiftedtotheleftonebitata timeforatotalof2'"- 1shifts.Thisoperation generates asystematic code withthedesiredoutputlengthn=2'"-I.Forexample, thecodewords generated bythem=3stageshiftregisterinFig.8-1-7arelistedinTable 8-1-4. Notethat,withtheexception oftheall-zerocodeword,allthecodewords generated bytheshiftregisteraredifferent cyclicshiftsofasinglecodeword. Thereasonforthisstructure iseasilyseenfromthestatediagramoftheshift register, whichisillustrated inFig.8-1-8form=3.Whentheshiftregister is loadedinitiallyandshifted2"'-1times,itwillcyclethroughallpossible 2'"-1 states.Hence,theshiftresgister isbacktoitsoriginalstatein2'"-1shifts. FIGURE 8·)·'Three-stage (m~3)shiftregisterwith feedback, Outpul 434 DIGITAL COMMt:NICATIONS TABLE8·).4MAXIMUM-LENGTH SHIFT-REGISTER CODEFORm=3 Information bits Codewords 0 (J0 0 0 (J0 (J00 00J 00 J1I0I (JJ0 0I00IJI (JII (JI1I0I0 1 (J0 100IIJa I01 I0100I1 I I 0 I10l00I 11I 1I10I00 Consequently, theoutputsequence isperiodic withlengthn=2"'-1.Since thereare2m-1possiblestates,thislengthcorresponds tothelargestpossible period.Thisexplains whythe2"'- 1codewordsaredifferent cyclicshiftsofa singlecodeword. Maximum-length shift-register codesexistforanypositive valueofm. FIGURE ll-J-8ThesevenSlalesforthem=3maximum lengthshift register. CHAPTER 8:BLOCK ANDCONVOLUTIONAL CHA~!\'EL_ CODES435 TABLE8-1-5SHIFf-REGISTER CONNECTIONS FORGENERATING MAXIMUM-LENGTH SEQUENCES Stagesconleded Stage.connected S._gesconnected mtomodulo-2 adder mtomodulo·2 adder mtomodulo·2 adder 2 1.2 13 1.10.11.13 24 1.18.23.24 3 1.3 14 1.5.9.14 25 1.23 4 1.4 15 1.15 26 1.21.25.26 ~ 1.4 16 1.5.14.16 27 1.23.26.27 6 1,6 17 1,15 28 1.26 7 1.7 18 1.12 29 1.28 8 1.5.6.7 19 I.15.18.19 30 I.8.29.30 9 1.6 20 I.18 31 1.29 10 1.8 21 1.20 32 1,11.31.32 11 1.10 22 1.22 33 1.21 12 1.7.9.12 23 1.19 34 1.8.33.34 SVlIrce:Forney(19701. Table8-1-5liststhestagesconnected tothemodulo-2 adderthatresultina maximum-length shiftregisterfor2,;;;m,;;;34. Another characteristic ofthecodewordsinamaximum-length shift-register codeisthateachcodeword,withtheexception oftheall-zerocodeword. contains 2mIonesand2m.Izeros.Henceallthesecodewordshaveidentical weights. namely, w=2m-I.Sincethecodeislinear,thisweightisalsothe minimum distanceofthecode,i.e.. Finally,notethatthe(7,3)maximum-length shift-register codeshownin Table8-1-4isidentical tothe(7,3)codegiveninTable8-1-3,whichisthedual ofthe(7,4)Hamming codegiveninTable8-1-2.Thisisnotacoincidence. The maximum-length shift-register codesarethedualcodesofthecyclicHamming (2m-I,2m-I-m)codes. Theshiftregisterforgenerating themaximum-length codemayalsobeused togenerate aperiodic binarysequence withperiodn=2m-I.Thebinary periodic sequence exhibits aperiodic autocorrelation c/>(m)withvalues c/>(m)=nform=O.±n,±2n,...,andc/>(m)=-Iforallothershiftsas described inSection13-2-4.Thisimpulse-like autocorrelation impliesthatthe powerspectrum isnearlywhiteand,hence,thesequence resembles white noise.Asaconsequence, maximum-length sequences arecalledpseudo-noise (PN)sequences andfinduseinthescrambling ofdataandinthegeneration of spreadspectrum signals. Bose-Chaudhuri-Hocquenghem (BCH)CodesBCHcodescomprise a largeclassofcycliccodesthatincludebothbinaryandnonbinaIy alphabets 436 DIGITAl COMMUN'CATlON~ BinaryBCHcodesmaybeconstructed withparameters n-k~mt dmin=2t+1(8-1-42) wherem(m;;.3)andtarearbitrary positive integers. Hence,thisclassof binarycodesprovides thecommunications systemdesigner withalarge selection ofblocklengthsandcoderates.Nonbinary BCHcodesincludethe powerful Reed-Solomon codesthataredescribed later. Thegenerator polynomials forBCHcodescanbeconstructed fromfactors ofp'"'-I+1.Table8-1-6liststhecoefficients ofgenerator polynomials for BCHcodesofblocklengths7~n~255,corresponding to3.,;m.,;8.The coefficients aregiveninoctalform,withtheleft-most digitcorresponding to thehighest-degree termofthegenerator polynomial. Thus,thecoefficients of thegenerator polynomial forthe(15,5)codeare2467,whichinbinaryformis 10100110Ill.Consequently, thegenerator polynomial isg(p)=P10+pH+ p'+p4+p2+P+1. Amoreextensive listofgenerator polynomials forBCHcodesisgivenby Peterson andWeldon(1972),whotabulate thepolynomial factorsofp2~'-I+I form~34. 8-1-4Optimum Soft-Decision Decoding ofLinearBlock Codes Inthissubsection, wederivetheperformance oflinearbinaryblockcodeson anAWGN channel whenoptimum (unquantized) soft-decision decoding is employed atthereceiver. Thebitsofacodewordmaybetransmitted byany oneofthebinarysignaling methods described inChapter 5.Forourpurposes. weconsider binary(orquaternary) coherent PSK,whichisthemostefficient method, andbinaryorthogonal FSKeitherwithcoherent detection or noncoherent detection. Let'tdenotethetransmitted signalenergypercodewordandlet'€,denote thesignalenergyrequired totransmit asingleelement (bit)inthecodeword. Sincetherearenbitspercodeword,g=ngnandsinceeachcodeword conveys kbitsofinformation, theenergyperinformation bitis gng,. 't.=k=k't,=R c(8-1-43) Thecodewordsareassumed tobeequallylikelyaprioriwithpriorprobability 11M. Suppose thebitsofacodewordaretransmitted bybinaryPSK.Thuseach codewordresultsinoneofMsignaling waveforms. FromChapter 5,weknow thattheoptimum receiver, inthesenseofminimizing theaverageprobability CHAPTER K:BLOCKANDCONVOLUTIONAL CHANNEL CODES-437 TABLE 8-1-6COEFFICIENTS OFGENERATOR POLYNOMIALS (INOcrAL FORM)FORBCH CODES OFLENGTHS 7<On<0255 n k t g(p) 7 4 1 13 15 11 1 23 7 2 721 5 3 2467 31 26 1 45 21 2 3551 16 3 107657 11 5 5423325 6 7 313365047 63 57 1 103 51 2 12471 45 3 1701317 39 1 166623567 36 5 1033500423 30 6 157464165547 24 7 17323260404441 18 10 1363026512351725 16 11 6331141367235453 10 13 472622305527250155 7 15 5231045543503271737 127 120 I 211 113 2 41567 106 3 11554743 99 4 3447023271 92 5 624730022327 85 6 130704476322273 78 7 26230002166130115 71 9 6255010713253127753 64 10 1206534025570773100045 57 11 335265252505705053517721 50 13 54446512523314012421501421 43 14 17721772213651227521220574343 36 15 3146074666522075044764574721735 29 21 403114461367670603667530141176155 22 23 123376070404722522435445626637647043 15 27 22057042445604554770523013762217604353 8 31 7047264052751030651476224271567733130217 255 247 1 435 239 2 267543 231 3 156720665 223 4 75626641375 215 5 23157564726421 207 6 16176560567636227 199 7 7633031270420722341 191 8 2663470176115333714567 187 9 52755313540001322236351 179 10 22624710717340432416300455 171 11 1541621421234235607706163067 4J8 DIGITAL COMMUNICATIONS TABLE 8-1-6(Continued) n k g(p) 163 12 7500415510075602551574724514601 155 13 3757513005407665015722506464677633 147 14 1642130173537165525304165305441011711 139 15 461401732060175561570722730247453567445 131 18 2157133314715101512612502774421420241 65471 123 19 1206140522420660037172103265161412262 72506267 II; 21 6052666557210024726363640460027635255 6313472737 107 22 2220577232206625631241730023534742017 6574750154441 99 23 1065666725347317422274141620157433225 2411076432303431 91 25 6750265030327444172723631724732511075 550762720724344561 87 26 1101367634147432364352316343071720462 06722545273311721317 79 27 6670003563765750002027034420736617462 1015326711766541342355 71 29 2402471052064432151555417211233116320 5444250362557643221706035 63 30 1075447505516354432531521735770700366 6111726455267613656702543301 55 31 7315425203501100133015275306032054325 414326755010557044426035473617 47 42 2533542017062646563033041377406233175 123334145446045005066024552543173 45 43 1520205605523416113110134637642370156 3670024470762373033202157025051541 37 45 5136330255067007414177447245437530420 735706174323432347644354737403044003 29 47 3025715536673071465527064012361377115 34224232420117411406025475741040356 5037 21 55 1256215257060332656001773153607612103 22734140565307454252115312161446651 3473725 13 59 4641732005052564544426573714250066004 33067744547656140317467721357026134 460500547 9 63 1572602521747246320103104325535513461 41623672120440745451127661155477055 61677516057 Source,Ste.bit(1964),©1964IEEE. CHAPTER ABLOCK AND(ONVQLUT10!'iAL (,HA~NEl CODES.439 ofacodeworderror,fortheAWGNchannel, canbereahzed asaparallel bankofMfiltersmatched totheMpossible transmitted waveforms. The outputsoftheMmatched filtersattheendofeachsignaling interval, which encompasses thetransmission ofnbitsinthecodeword.arecompared andthe codewordcorresponding tothelargestmatched filteroutputisselected. Alternatively, Mcross-correlators canbeemployed. Ineithercase.thereceiver implementation canbesimphfied. Thatis,anequivalent optimum receiver can berealized byuseofasinglefilter(orcross-correlator) matched tothebinary PSKwaveform usedtotransmit eachbitinthecodeword,followed hya decoder thatformstheMdecision variables corresponding totheMcode words. Tobespecific, let'1'j=1,2,...,n,represent thensampled outputsofthe matched filterforanyparticular codeword.Sincethesignaling isbinary coherent PSK,theoutput'1maybeexpressed eitheras 'f=~+nl whenthejthbitofacodewordisaLoras 'j=-~+n,(8-1-44) (8-1-45) whenthejthbitisa0.Thevariables {nJrepresent additive whitegaussian noiseatthesampling instants. EachnlhaszeromeanandvarianceIN".From knowledge oftheMpossible transmitted codewordsanduponreception of {'fl,theoptimum decoder formstheMcorrelation metrics "eM;=qr,C,)=2:(2c"-1)'1'i=1,2....,M J"I(8-1-46) where Cildenotes thebitinthejthposition oftheithcodeword.Thus,if c;J=1,theweighting factor2c,)-1=1,andifC;I=0,theweighting factor 2c,)-1=-1.Inthismanner, theweighting 2c;J-1alignsthL'signalcom­ ponents in{,,}suchthatthecorrelation metriccorresponding totheactual transmitted codewordwillhaveameanvalueV'l;n,whiletheotherM-I metricswillhavesmallermeanvalues. Although thecomputations involved informing thecorrelation metricsfor soft-decision decoding according to(8-1-46) arerelatively simple.itmaystillbe impractical tocompute (8-1-46) forallthepossible codewordswhenthe number ofcodewordsislarge,e.g.,M>210.Insuchacaseitisstillpossible to implement soft-decision decoding usingalgorithms whichemploy techniques fordiscarding improbable codewordswithoutcomputing theirentirecorrela­ tionmetrics asgivenby(8-1-46). Several different typesofsoft-decision decoding algorithms havebeendescribed inthetechnical literature. The interested readerisreferred tothepapersbyForney(1966b), Weldon (1971). Chase(1972), Wainberg andWolf(1973), Wolf(1978), andMatisand Modestino (1982). Indetermining theprobability oferrorforalinearblockcode.notethat 440 DIGtTAL COMMUNICATIONS whensuchacodeisemployed onabinary-input, symmetric channelsuchas theAWGNchannelwithoptimum soft-decision decoding, theerrorprobability forthetransmission ofthemthcodewordisthesameforallm.Hence,we assumeforsimplicity thattheall-zerocodewordC\istransmitted. Forcorrect decoding ofC"thecorrelation metricCM\mustexceedalltheotherM-I correlation metricsCM""m=Z,...,M.AlltheCMaregaussian distributed. ThemeanvalueofCM\is~n,whilethemeanvaluesofCM""m=2,...,M is~n(1- Zw",/n). Thevariance ofeachdecision variable is~N".The derivation oftheexactexpression fortheprobability ofcorrectdecoding or, equivalently, theprobability ofacodeworderroriscomplicated bythe correlations among theMcorrelation metrics. Thecross-correlation coeffiCIents between C\andtheotherM-Icodewordsare p",=I-Zw,,,/n, m=2....•M (8-1-47) wherew'"denotestheweightofthemthcodeword. lnsteadofattempting toderivetheelCacterrorprobability, weresorttoa unionbound.Theprobability thateM",>eM.,is p,(m)=Q( J~(I-Pmi) (8-1-48) 'V~) i whereg=k'<f!histhetransmitted energyperwaveform. Substitution forPm from(8-1-47)andforrgyields 'Z#P2(m)=Q(viR,wm) =Q(V2YhR,W",) (8-1-49) where 'floistheSNRperbitandR,oisthecoderate.Thentheaverage probability ofacodeworderrorisbounded fromabovebythesumofthe binaryerroreventsgivenby(8-1-49). Thus, IIr=2 M ~2:Q(V2'fhR,Wm) fl/=2(8-1-50) Thecomputation oftheprobability oferrorforsoft-decision decoding according to(8-1-50) requires knowledge oftheweightdistribution ofthe code.Weightdistributions ofmanycodesaregiveninanumberoftextson codingtheory,e.g.,Berlekamp (1968)andMacWilliams andSloane(1977). Asomewhat looserboundisobtained bynotingthat Q(V2y hR,w",)"; Q(V2YhR,dmin)<exp(-YhR,dmin)(8-1-51) Consequently, (8-1-52) CHAPTFR 1'i.HLC)('K ANDc(lNVOUTIONAL n",-\~M'l CODi-S441 Thisboundisparticularly usefulsinceitdoesnotrequireknowledge ofthe weightdistribution ofthecode.Whentheupperboundin(8-1-52) iscompared withtheperformance ofanuncoded binaryPSKsystem, whichisupper­ bounded asjexp(-Yh).wefindthatcodingyieldsagainofapproximatcly Wlog(R,dn'in-kIn2/'Yh)dB.Wemaycallthisthecodinggain.Wenotethat itsvaluedepends onthecodeparameters andalsoontheSNRperbitYo Theexpression fortheprobability oferrorforequicorrelated waveforms thatcanbeobtained forthesimplex signalsdescribed inSection5-2givesus vetathirdapproximation totheerrorprobabilities forcodedwaveforms. We knowthatthemaximum cross-correlation coefficient between apairofcoded waveforms is 2 PmB.'!(=1--dminn(8-1-53) (1'-1-54)IfweassumeasaworstcasethatalltheMcodewordshaveacross-correlation coefficient equaltoP,HXthenthecodeworderrorprobability caneasilybe manipulated. Sincesomecodewordsareseparated bymorethantheminimum distance, theerrorprobability evaluated forp,=Pmaxisactually anupper bound.Thus. IJX..(I["+V·."R,"m" .)M'PM""I -,-- e'"'/2-- e'd'dx dl'V211,xV211_x Theboundsontheperformance oflinearblockcodesgivenabovearein termsoftheblockerrororcodeworderrorprobability. Theevaluation ofthe equivalent biterrorprobability Phismuchmorecomplicated. Ingeneral. whcn ablockerrorismade,someofthekinformation bits intheblockwillbe correctandsomewillbeinerror,Fororthogonal waveforms, theconversion factorthatmultiplies PMtoyieldPhis2k"/(2k-I).Thisfactorisunityfor k=1andapproaches ~askincreases, whichisequivalent toassuming that.on theaverage, halfofthekbitswillbeinerrorwhenablockerroroccurs.The conversion factorforcodedwaveforms depends inacomplicatcd wayonthe distance properties ofthecode,butiscertainly noworsethanassuming that, ontheaverage, halfofthekbitswillbeinerrorwhenablockerroroccurs. Consequently, Pb""~PM' Theboundsonperformance givenby(8-1-50), (8-1-52), and(8-1-54) also applytothecaseinwhichapairofbitsofacodewordaretransmitted by quaternary PSK,sincequaternary PSKmaybeviewedasbeingequivalent to twoindependent binaryPSKwaveforms transmitted inphasequadrature. Furthermore. theboundsin(8-}-52) and(8-}-54), whichdepend onlyonthe minimum distance ofthecode,applyalsotononlinear binaryblockcodes. Ifbinaryorthogonal FSKisusedtotransmit eachbitofacodewordonthe AWGN channel, theoptimum receiver canberealized bymeansoftwo matched filters.onematched tothefrequency corresponding toatransmission ofa0,andtheothertothefrequency corresponding toatransmission ofaL followed byadecoder thatformstheMcorrelation metricscorresponding to 442 DIGITAL COM~Il'~ICATIONS theMpossible codewords.Thedetection attheTeceiver maybecoherent or noncoherent. Ineithercase.let'OJand'"denotetheinputsamples tothe combiner. Thecorrelation metricsformedbythedecoder maybeexpressed as. n CM,=L:[CiJr,!+(1-e'llr",], i"<1i=I.2,.".M (8-1-55) wheree,irepresents thejthbitintheithcodeword.Thecodeword corresponding tothelargestofthe{CM,}isselected asthetransmitted code word. Ifthedetection ofthebinaryFSKwaveforms iscoherent. therandom variables (r"/land{r)aregaussi~n and,hence.thecorrelation metrics{CM,} arealsogaussian. Inthiscase.boundsontheperformance ofthecodeare easilyobtained. Tobespecific. suppose thattheall-zero codewordC,is transmitted. Then, ro)_=W.o,+no!}j=1,2,...,n rlj-nlj(8-1-56) wherethe{n,J,i=O.I.j=1,2,...,n.aremutually statistically independent gaussian random variables withzeromeanandvariance ~NoConsequently CM,isgaussian withmeanV€;"nandvariance ~No.Ontheotherhand,the correlation metriceM""corresponding tothecodewordhavingweight W,n'is gaussian withmean~n(l-w",ln)andvariance hN".Sincethe{CM",}are correlated. weagainresorttoaunionbound,Thecorrelation coefficients are givenby P'"=1 -wmln Hence.theprobability thatCM",>eM,is P,(m)=Q(v'YbR·w",)(8-1-57) (8-1-58) (8-1-59)Comparison ofthisresultwiththatgivenin(8-1-49)forcoherent PSKreveals thatcoherent PSKrequires 3dBlessSNRtoachievethesameperformance. Thisisnotsurprising inviewofthefactthatuncoded PSKis3dBbetterthan binaryorthogonal FSKwithcoherent detection. Hence,theadvantage ofPSK overFSKismaintained inthecodedwaveforms. Weconclude. then,thatthe boundsgivenin(8-1-50), (8-1-52), and(8-1-54) applytocodedwaveforms transmitted bybinaryorthogonal coherent FSKwith'Ybreplaced by!-rh' Ifsquare-law detection ofthebinaryorthogonal FSKsignalisemployed at thereceiver,· theperformance isfurther degraded bythenoncoherent combining Joss,asshowninChapter 12.Suppose againthattheall-zerocode wordistransmitted. Thenthecorrelation metricsaregivenby(8-1-55), where theinputvariables 10thedecoder arenow rOj=IVfc+Nol} 2 j=1,2,...•n r'j=iN,) CHAPTER!\ BLOCK A"lDCO~VOLt;TIOSAL CHAI'!\EL CODES443 where{No)and{N,;!represent complex-valued mutually statistically indepen­ dentgaussian random variables withzeromeanandvariance No.The correlation metricCM,isgivenas n CMI=L'OJ j=1(8-1-60) whilethecorrelation metriccorresponding tothecodewordhavingweight "'",isstatistically equlvalent tothecorrelation metricofacodewordinwhich e""=1forI,,;;j,,;; W",ande",)=0forw'"+l,,;;j,,;; rI.Hence,eM",maybe expressed as W,,, n CM",=2:'Ij+LrOj J=l )=wm+1 Thedifference betweenCMIandCMmis CM,-CM",=L(roj-r,) j=l(8-1-61) (8-1-62) andtheprobability oferrorissimplytheprobability thatCM,-eMm<O.But thisdifference isaspecialcaseofthegeneralquadratic formincomplex-valued gaussian random variables considered inChapter 12andAppendix B.The expression fortheprobability oferrorindeciding betweeneM,andeMn,is (seeSection12-1-1) 1 "'."-I P,(m)=22.,,-,exp(-hbRcw",) ~oK,(hbR,W",l' (8-1-63) where,bydefinition, K=~"'m~-'(2wm-l ,.,L.. ) I.r=O r(8-1-64) Theunronboundobtained bysumming p,(m)over2,,;;m,,;;Mprovides uswith anupperboundontheprobability ofacodeworderror. Asanalternative, wemayusetheminimum distance insteadoftheweight distribution toobtainthelooserupperbound M-1 dmin-j. PMoS<22d"..-,exp(-hbRcdm") i~Kj(hbRcdminl' (8-1-65) Ameasure ofthenoncoherent combining lossinherent inthesquare-law detection andcombining ofthenelementary binaryFSKwaveforms inacode wordcanbeobtained fromFig.12-1-1,wheredminisusedinplaceofL.The lossobtained isrelative tothecaseinwhichthenelementary binaryFSK waveforms arefirstdetected coherently andcombined asin(8-1-55) andthen thesumsaresquare-law-detected orenvelope-detected toyieldtheMdecision variables. Thebinaryerrorprobability forthelattercaseis P,(m)=~exp(-!-YbR,wm) (8-1-66) 444 DIGITAL COMML'N!C'ATIQNS and,hence, M P.,s;.LP2(m) m=2 Ifdm;"isusedinsteadoftheweightdistribution, theunionboundforthecode worderrorprobability inthelattercaseis (8-1-67) Thechannel bandwidth required totransmit thecodedwaveforms canbe determined asfollows.1fbinaryPSKisusedtotransmit eachbitinacode word,therequired bandwidth isapproximately equaltothereciprocal ofthe timeintervaldevotedtothetransmission ofeachbit.Foraninformation rate ofRbits/s,thetimeavailable totransmit kinformation bitsandn-k redundant (parity)bits(ntotalbits)isT=k/R.Hence, 1n Rw=-=-=-T/nk/RR,(8-1-68) Therefore, thebandwidth expansion factorB,forthecodedwaveform is WB=­,R n1=-=- kR,(8-1-69) Ontheotherhand,ifbinaryFSKwithnoncoherent detection isemployed for transmitting thebitsinacodeword,W=2n/T,and,hence,thebandwidth expansion factorincreases byapproximately afactorof2relativetobinary PSK.Inanycase,B,increases inversely withthecoderate,or,equivalently, it increases linearlywiththeblocksizen. 'Wearenowinaposition tocompare theperformance characteristics and bandwidth requirements ofcodedsignaling waveforms withorthogonal signal­ ingwaveforms. Acomparison oftheexpression forPMgivenin(5-2-21)for orthogonal waveforms andin(8-1-54)forcodedwaveforms withcoherent PSK indicates thatthecodedwaveforms resultinalossofatmost 10log(n/2dm;n)dBrelativetoorthogonal waveforms havingthesamenumber ofwaveforms. Ontheotherhand,ifwecompensate forthelossinSNRdueto codingbyincreasing thenumber ofcodewordssothatcodedtransmission requires Me=2k,waveforms andorthogonal signaling requires M"=2'" waveforms then[fromtheunionbounds in(5-2-27) and(8-1-52)], the performance obtained withthetwosetsofsignaling waveforms athighSNRis aboutequalif (8-1-70) CHAPTER K:BLOCK ANDCONVOLUTIONAL CHANNEL CODES445 Underthiscondition, thebandwidth expansion factorfororthogonal signaling canbeexpressed as Mo8eo~--"---210g2Mo while,forcodedsignaling waveforms, givenin(8-1-71)to8mwhichis2k"22R,dm,"-= 2ko4Rcdmin we!lave8,c~1/Rc'(8-1-71) TheratioofBeo (8-1-72) provides ameasure oftherelativebandwidth between orthogonal signaling andsignaling wit!lcodedcoherent PSKwaveforms. Forexample, suppose weusea(63,33)binarycycliccodethathasa minimum distance dmm=12.Thebandwidth ratiofororthogonal signaling relative tothiscode,givenby(8-1-72), is127.Thisisindicative ofthe bandwidth efficiency obtained throughcodingrelativetoorthogonal signaling. 8-1-5Hard-Decision Decoding TheboundsgiveninSection 8·1-4ontheperformance ofcodedsignaling waveforms ontheAWGNchannelarebasedonthepremisethatthesamples fromthematched filterorcrosscorrelator arenotquantized. Although this processing yieldsthebestperformance, thebasiclimitation isthecomputa­ tionalburdenofforming Mcorrelation metricsandcomparing these toobtain thelargest.Theamountofcomputation becomes excessive whenthenumber Mofcodewordsislarge. Toreducethecomputational burden,theanalogsamplescanbequantized andthedecoding operations arethenperformed digitally. Inthissubsection, weconsider theextreme situation inwhicheachsamplecorresponding toa singlebitofacodewordisquantized totwolevels:zeroandone.T!latis,a (hard)decision ismadeastowhether eachtransmitted bitinacodewordisa0 ora1.Theresulting discrete-time channel(consisting ofthemodulator, the AWGNchannel, andthedemodulator) constitutes assewithcrossover probability p.Ifcoherent PSKisemployed intransmitting andreceiving the bitsineachcodewordthen P=Q(f!:) =Q(V2YbRc) (8-1-73) Ontheotherhand,ifFSKisusedtotransmit thebitsineachcodewordthen p=Q(v'YbRcl (8-1-74) forcoherent detection and (8-1-75) fornoncoherent detection. 446 DIGI1Al COMMlJNlfATIONS Minimum-Distance (Maximum. Likelihood) Decoding Thenbitsfromthe demodulator corresponding toareceived codewordarepassedtothedecoder, whichcompares thereceived codewordwiththeMpossibletransmitted code wordsanddecidesinfavorofthecodewordthatisclosestinHamming distance (number ofbitpositions inwhichtwocodewordsdiffer)tothe received codeword.Thisminimum distance decoding ruleisoptimum inthe sensethatitresultsinaminimum probability ofacodeworderrorforthe binarysymmetric channel. Aconceptually simple,albeitcomputationally inefficient, method for decoding istofirstadd(modulo 2)thereceived codewordvectortoalltheM possible transmitted codewordsCitoobtaintheerrorvectors ej'Hence,ej represents theerroreventthatmusthaveoccurred onthechannelinorderto transform thecodewordCjintotheparticular received codeword.The numberoferrorsintransforming Cjintothereceived codewordisjustequal tothenumberofisinei'Thus,ifwesimplycompute theweightofeachofthe Merrorvectors{e,}anddecideinfavorofthecodewordthatresultsinthe smallestweighterrorvector,wehave,ineffect,arealization oftheminimum distancedecoding rule. Amoreefficientmethodforhard-decision decoding makesuseoftheparity checkmatrixH.Toelaborate, suppose thatCooisthetransmitted codeword andYisthereceivedcodewordattheoutputofthedemodulator. Ingeneral, Ymaybeexpressed as Y=C m+e whereedenotesanarbitrary binaryerrorvector.TheproductYH'yields YH'=(Cm+e)H' =CmH'+eH' =eH'=5 (8-1-76) wherethe(n-k)-dimensional vector5iscalledthesyndrome oftheerror pattern.Inotherwords,thevector5hascomponents thatarezeroforallparity checkequations thataresatisfiedandnonzeroforallparitycheckequations thatarenotsatisfied. Thus,5contains thepatternoffailuresintheparity checks. Weemphasize thatthesyndrome 5isacharacteristic oftheerrorpattern andnotofthetransmitted codeword.Furthermore, weobservethatthereare 2"possibleerrorpatterns andonly2"-ksyndromes. Consequently, different errorpatternsresultinthesamesyndrome. Suppose weconstruct adecoding tableinwhichwelistallthe2kpossible codewordsinthefirstrow,beginning withtheall-zerocodewordinthefirst (left-most) column.Thisall-zerocodewordalsorepresents theall-zeroerror pattern.Wefillinthefirstcolumnbylistingfirstalln-1errorpatterns {e,}of weight1,Ifn<2"-k,wemaythenlistalldoubleerrorpatterns, thenalltriple CHAPTER KBLOCKANDCONVOLL'TIO'oiAL CHA:-';:--JEL CODES447 errorpatterns, etc.,untilwehaveatotalof2"~Kentriesinthefirstcolumn. Thus,thenumber ofmwsthatwecanhaveis2"~K,whichisequaltothe numberofsyndmmes. ~ext,weaddeacherrorpatterninthefirstcolumnto thecorresponding codewords.Thus,wefillintheremainder ofIheIt>.(n-k) tableasfollows: (,C, C, C" e,C,+eoC,+e, ("+e; eJC,+eJC,+e., C"+eJ e2"(C,+f,"',C,+e,,,-, C"+~,,,' Thistableiscalledastandard array.Eachrow,including thefirst,consistsofk possible received codewordsthatwouldresultfromthecorresponding error patterninthefirstcolumn. Eachrowiscalledacosetandthefirst(left-most) codeword(orerrorpattern)iscalledacosetleader.Therefore, acosetconsists ofallthepossiblereceived codewordsresulting fromaparticular errorpattern (cosetleader). Example 8·1·10 Letusconstruct thestandard arrayforthe(5,2),systematic codewith generator matrixgivenby G=[OI 0 1 0 1]) 0 ) ) Thiccodehasaminimum distance dmin=3.Thestandard arrayisgivenin Tableg-)·7.Notethatinthiscode,thecosetleadersconsistoftheall-zero errorpattern, fiveerrorpatterns ofweight1,andtwoerrorpatterns of TABLE 8·1·7STANDARD ARRAY FORTHE(5,2)CODE Codewords 00000 00001 00010 o0 100 o1000 10000 1 I 000 1001001011 o1010 o1 0 0J o1 1 1 I o0 0 1 J ItO1 1 I 0 0 1 1 J100J10101 10100 1 0 J 1 1 10001 1 1 1 0 1 o0 1 0 1 o1 1 0 1 0011J1 1 1 1 0 11111 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 o1 1 1 0 o0 1 1 0 o1 1 0 0 448 DIGITAL COMMUNICATIONS weight2.Although manymoredoubleerrorpatterns exist,thereisonly roomfortwotocomplete thetable.Thesewereselected suchthattheir corresponding syndromes aredistinct fromthoseofthesingleerror patterns. Now,suppose thate,isacosetleaderandthatCmwasthetransmitted code word.Then,theerrorpatterneiwouldresultinthereceived codeword Y=C m+ei Thesyndrome is S=(Cm+ei)H'=CmH'+eiH'=eiH' Clearly, allreceived codewordsinthesamecosethavethesamesyndrome, sincethelatterdepends onlyontheerrorpattern.Furthermore, eachcosethas adiffere.nt syndrome. Havingestablished thischaracteristic ofthestandard array.wemaysimplyconstruct asyndrome decoding tableinwhichwelistthe zn-ksyndromes andthecorresponding zn-kcosetleadersthatrepresent the minimum weighterrorpatterns. Then,givenareceived codevectorY,we compute thesyndrome S=YH' Forthecomputed S,wefindthecorresponding (mostlikely)errorvector,say em'ThiserrorvectorisaddedtoYtoyieldthedecoded word Cm=YE!:le m Example 8-1-11 Consider the(5,2)codewiththestandard arraygiveninTable8-1-7.The syndromes versusthemostlikelyerrorpatterns aregiveninTable8-1-8. Nowsuppose theactualerrorvectoronthechannelis e=[1°1°0] TABLE 8·1·8SYNDROME TABLE FORTHE (5.2)CODE Syndrome 000 001 010 100 011 I 0 I I I 0 I I 1Errorpattern 00000 o0001 o001 0 00100 o1000 I0000 1 1000 1 0 0 1 0 CHAPTER H:BLOCK ANDrONVOlUTIONAl CHANNEL CODES449 Thesyndrome computed fortheerrorisS=[00I].Hence,theerror determined fromthetableise=[0000I].WheneisaddedtoY.the resultisadecoding error.[notherwordsthe(5,2)codecorrects allsingle errorsandonlytwOdoubleerrors.namely[1I 0 00]and{10 0 IOJ. Syndrome Decoding ofCyclicCodesAsdescribed above.hard-decision decoding ofalinearblockcodemaybeaccomplished byfirstcomputing the syndrome S=YH',thenusingatablelookuptofindthemostprobable error patternecorresponding tothecomputed syndrome S.and,finally.addingthe errorpatternetothereceived vectorYtoobtainthemostprobable codeword Cm'Whenthecodeiscyclic,thesyndrome computation maybeperformed by ashiftregistersimilarinformtothatusedforencoding. Toelaborate, letusconsider asystematic cycliccodeandletusrepresent thereceived codevectorYbythepolynomial Y(p).Ingeneral, Y=C+e, whereCisthetransmitted codewordandeistheerrorvector.Hence,wehave Y(p)=C(p)+e(p) =X(p)g(p) +e(p) (8-1-77) Now,supposewedivideY(p)bythegenerator polynomial g(p).Thisdivision willyield Y(p)=Q(p)+R(p) g(p) g(p) or,equivalently. Y(p)=Q(p)g(p)+R(p) (8-1-78) Theremainder R(p)isapolynomial ofdegreelessthanorequalton-k-1. Ifwecombine (8-1-77)with(8-1-78). weobtain e(p)=[X(p)+Q(p)Jg(p)+R(p) (8-1-79) Thisrelationship illustrates thattheremainder R(p)obtained fromdividing Y(p)byg(p)depends onlyontheerrorpolynomial e(p),and.hence,R(p)is simplythesyndrome associated withtheerrorpatterne.Therefore. Y(p)=Q(p)g(p)+S(p) (8-1-80) whereS(p)isthesyndrome polynomial ofdegreelessthanorequllJto n-k-1.Ifg(p)dividesY(p)exactlythenS(p)=0andthereceived decoded wordisCm=Y. ThedivisionofY(p)bythegenerator polynomial g(p)maybecarriedout bymeansofashiftregisterwhichperforms divisionasdescribed previously. Firstthereceived vectorYisshiftedintoan(n-k)-stageshiftregisteras 450DIOITA, C'OMMUNICA nONS fiGURE 8-1·9An(n-k)·,tageshiftregisterforcomputing thesyndrome.Output ~yndTome--.0--- 2Received codevector illustrated inFig.8-1-9.Initially,alltheshift-register contents arezeroandthe switchisclosedinposition 1.Aftertheentiren-bitreceived vectorhasbeen shiftedintotheregister, thecontents ofthen-kstagesconstitute the syndrome withtheorderofthebitsnumbered asshowninFig.8-1-9.These bitsmaybeclockedoutbythrowing theswitchintoposition 2.Giventhe syndrome fromthe(n-k)-stageshiftregister, atablelookupmaybe performed toidentifythemostprobable errorvector. Example 8·1·U Letusconsider thesyndrome computation forthe(7,4)cyclicHamming codegenerated bythepolynomial g(p)=p'+p+1.Suppose thatthe received vectorisY=[10 0 1 1 0 1].Thisisfedintothethree-stage registershowninFig.8-1-10.Aftersevenshiftsthecontents oftheshift registerare110,whichcorresponds tothesyndrome S=[011].Themost probable errorvectorcorresponding tothissyndrome ise =[00 0 I 0 0 0] and,hence, Cm=Y+e=f1 000101] Theinformation bitsareIa0O. fiGURE 8-1·10 Syndrome computation forthe(7.4)cycliccndewithgenerator polynomial g(p)=p'+p+1and received vectorY=[10 0 1 1 0 1]. J::L:JIntput Output 1011001 + + --..0-syndrome Shift Registercontents o 000 I 100 2 010 J 001 4 010 5 101 5 100 1 110 CHAPTER KBLOCKANDCONVOLUTIONAL CHANNEL CODES451 Thetablelookupdecoding methodusingthesyndrome ispractical only whenn-kissmall,e,g.,n-k<10.Thismethod isimpractical formany interesting andpowerful codes.Forexample, ifn-k=20,thetablehas220 (approximately 1million)entries.Suchalargeamountofstorageandthetime required tolocateanentryinsuchalargetablerendersthetablelookup decoding methodimpractical forlongcodeshavinglargenumbers ofcheck bits. Moreefficient andpractical hard-decision decoding algorithms havebeen devisedfortheclassofcycliccodesand,morespecifically, theBCHcodes.A description ofthesealgorithms requires furtherdevelopment ofcomputational methods withfinitefields,whichisbeyondthescopeofourtreatment of codingtheory.Itsufficestoindi~ate thatefficient decoding algorithms exist whichmakeitpossible toimplement longBCHcodeswithhighredundancy in practical digitalcommunications systems. Theinterested readerisreferred to thetextsofPeterson andWeldon (1972).LinandCostello (1983),Blahut (1983),andBerlekamp (1968),andtothepaperbyForney(1965). ErrorDetection andErrorCorrection Capability Itisclearfromthe discussion abovethatwhenthesyndrome consistsofallzeros,thereceived codewordisoneofthe2'possibletransmitted codewords.Sincetheminimum separation between apairofcodewordsisdmi~:itispossible foranerror patternofweightdmintotransform oneofthese2'codewordsinthecodeinto anothercodeword.Whenthishappens wehaveanundetected error.Onthe otherhand,iftheactualnumberoferrorsislessthandmin>thesyndrome will haveanonzero weight.Whenthisoccurs,wehavedetected thepresence of oneormoreerrorsonthechannel. Clearly,the(n,k)blockcodeiscapableof detecting dmin-1errors.Errordetection maybeusedinconjunction withan automatic repeat-request (ARQ)schemeforretransmission ofthecodeword. Theerrorcorrection capability ofacodealsodepends ontheminimum distance. However, thenumberofcorrectable errorpatterns islimitedbythe number ofpossible syndromes orcosetleadersinthestandard array.To determine theerrorcorrection capability ofan(/I;k)code,itisconvenient to viewthe2'codewordsaspointsinan/I-dimensional space.Ifeachcodeword isviewedasthecenterofasphereofradius(Hamming distance) t,thelargest valuethattmayhavewithoutintersection (ortangency) ofanypairofthe2' spheresist=U(dmin-1)J,whereLddenotesthelargestintegercontained in x'Withineachspherelieallthepossiblereceived codewordsofdistance less thanorequaltoIfromthevalidcodeword,Consequently, anyreceived code vectorthatfallswithinasphereisdecoded intothevalidcodewordatthe centerofthesphere.Thisimpliesthatan(n.k)eadewithminimum distance dminiscapable ofcorrecting I=L!(dm,n-I)Jerrors.Figure8-1-11isa two-dimensional representation ofthecodewordsandthespheres. Asdescribed above,acodemaybeusedtodetectdmin-1errorsorto correctt=U(dmi•-l)jerrors.Clearly,tocorrectterrorimpliesthatwehave 452 DIGITAL COMMUNiCATIONS • •• • ••• •• • ••••• ••• • •••••• •••• ••• • • c, • •c~ •••••• ••• ••••••• fiGURE 8-1·11 Arepresentation ofcodewordsascenters ofspheresofradius1=Ll(dm;n-Ill. detected Ierrors.However, itisalsopossible todetectmorethanterrorsifwe compromise intheerrorcorrection capability ofthecode.Forexample, acode withdmin'"7cancorrectt'"3errors.Ifwewishtodetectfourerrors,wecan dosobyreducing theradiusofthespherearoundeachcodewordfrom3to2. Thus,patterns withfourerrorsaredetectable butonlypatternsoftwoerrors arecorrectable. Inotherwords,whenonlytwoerrorsoccur,theseare corrected, andwhenthreeorfourerrorsoccur,thereceiver mayaskfora retransmission. Ifmorethanfourerrorsoccur,theywillgoundetected ifthe codewordfallswithinasphereofradius2.Similarly, fordmin'"7,fiveerrors canbedetected andoneerrorcorrected. Ingeneral, acodewithminimum distance dmincandetectederrorsandcorrect e,.errors,where and ProbabIHty ofErrorBasedonErrorCorrection Weconclude thissection withthederivation oftheprobability oferrorforhard·decision decoding of linearbinaryblockcodesbasedonerrorcorrection only. Fromtheabovediscussion, itisclearthattheoptimum decoder forabinary CHAPTER~. BLOCK ANDCONVOLl'TIONAL CHANNEL CODES453 symmetric channel willdecodecorrectly if(butnotnecessarily onlyif)the numberoferrorsinacodewordislessthanhalftheminimum distance dm'nof thecode.Thatis,anynumberoferrorsupto arealwayscorrectable. Sincethebinarysymmetric channel ismemoryless. the biterrorsoccurindependently. Hence.theprobability ofmerrorsinablockof nbitsis P(m.n)=Ck'(l-p)"-m (8-1-1\1) and,therefore. theprobability ofacodeworderrorisupper-bounded bythe expression n PM";2:P(m,n) m=r+1(8-1-82) Equality holdsin(8-1-82)ifthelinearblockcodeisaperfectcode.Inorder todescribe thebasiccharacteristics ofaperfectcode.suppose weplacea sphereofradiusIaroundeachofthepossible transmitted codewords.Each spherearoundacodewordcontains thesetofallcodewordsofHamming distance lessthanorequaltoIfromthecodeword..Now.thenumberofcode wordsinasphereofradiusr=U(dm'n-l)Jis 1+(n)+(~)+..+(n)=t(n) ] .....) t.i=OI, SincethereareM=2kpossible transmitted codewords.thereare2k nonoverlapping sphereseachhavingaradius I.Thetotalnumber ofcode wordsenclosed inthe2kspherescannotexceedthe2"possible received code words.Thus,aI-errorcorrecting codemustsatisfytheinequality or,equivalently,,(n\2,,-k;.2:.J i=OI(8-1-83) Aperfectcodehastheproperty thatallspheres ofHamming distance 1=U(dm'n-l)JaroundtheM=2kpossible transmitted codewordsare disjointandeveryreceived codewordfallsinoneofthespheres. Thus,every received codewordisatmost,atdistanceIfromoneofthepossibletransmitted codewordsand(8-1-83) holdswithequality. Forsuchacode,allerror 4S4 DIGITAL COMMUNICATIQN:S patterns ofweightlessthanorequaltotarecorrected bytheoptimum (minimum distance) decoder. Ontheotherhand,anyerrorpatternofweight t+1orgreatercannotbecorrected. Consequently, theexpression fortheerror probability givenin(8-1-82)holdswithequality.The Golay(23,12)code, having dm;n=7andt=3,isaperfectcode.TheHamming codes,whichhave theparameters n=2"-*-1,dm;n=3,andt=1,arealsoperfectcodes.These twonontrivial codesandthetrivialcodeconsisting oftwocodewordsofodd lengthnand dm;n=naretheonlyperfectbinaryblockcodes.Thesecodesare optimum onlheBSCinthesensethattheyresultin,aminimum error probability amongallcodeshavingthesameblocklengthandthesame numberofinformation bits. Theoptimality properly definedabovealsoholdsforquasiperfect codes.A quasiperfect codeischaracterized bytheproperty thatallspheresofHamming radiustar6undtheMpossibletransmitted codewordsaredisjointandevery received codewordisatmostatdistance t+1fromoneofthepossible transmitted codewords.Forsuchacode,allerrorpatternsofweightlessthan orequaltotandsomeerrorpatternsofweightt+1arecorrectable, butany errorpatternofweightt+2orgreaterleadstoincorrect decoding ofthecode word.Clearly, (8-1-82)isanupperboundontheerrorprobability and "PM~LP(m,n) m""t+2(8-1-84) isalowerbound. Amoreprecisemeasure oftheperformance forquasiperfect codescanbe obtained bymakinguseoftheinequality in(8-1-83).ThaIis,thetotalnumber ofcodewordsoutsidethe2·spheresofradiustis N=2"_2k~(n)1+1 £J. i-OI Ifthesecodewordsareequallysubdivided into2*setsandeachsetis associated withoneofthe2*spherestheneachsphereisenlarged bythe additionof f3,+I=2n-*-±(~) i=OI(8-1-85) codewordshavingdistance I+1fromthetransmitted codeword.Conse­ quently,oftheC;1)errorpatternsofdistance t+1fromeachcodeword, wecancorrectf3,+1errorpatterns. Thus,theerrorprobability fordecoding the quasiperfect codemaybeexpressed as PM=±p(m,n)+[( n1)-f3'+I]P,+1(1-p)n-r-l (8-1-86) m=t+2 1+ Therearemanyknownquasiperfect codes,although theydonotexistfor CHAPTER R:BLOCK A~DCO~VOLL·TlONAL CHA~:-':EL CODES455 allchoicesofnandk.Sincesuchcodesareoptimum forthebinarysymmetric channel, any(Il,k)linearblockcodemusthaveanerrorprobability thatisat leastaslargeas(8-1-86). Consequently, (8-1-86) isalowerboundonthe probability oferrorforany(n,k)linearblockcode,where(isthelargest integersuchthat{3,.);.O. Another pairofupperandlowerboundsisobtained byconsidering two codewordsthatdifferbytheminimum distance. First.wenotethatPwcannot belessthanthe probability oferroneously decoding thetransmitted codeword asitsnearestneighbor, whichisatdistance dmmfromthetransmitted code word.Thatis, (8-1-87) Ontheotherhand,P"cannotbegreaterthanM-1timestheprobability of erroneously decoding thetransmitted codewordasitsnearestneighbor, which isatdistance dm;nfromthetransmitted codeword.Thatisaunionbound. whichisexpressed as dfl'''~"d)P"'-'<(M-I)..2_(mm;npm(l-p)"m,.m (8-1-88) m-[dm,n,.::j+l WhenMislarge,thelowerboundin(8-1-87)andtheupperboundin(8-1-88) areveryIpose. AtightupperboundonPMcanbeobtained byapplying theChernoff bound presented earlierinSection2-1-6.Weassumeagainthattheall-zerocodewas transmitted. Incomparing thereceived codewordtotheall-zerocodeword andtoacodewordofweightIV""theprobability ofadecoding error,obtained fromtheChernoff bound(Problem 8-22),isupper-bounded bytheexpression P,(Wm)'-'<[4p(1-p)]".,,12 Theunionofthesebinarydecisions yieldstheupperbound M P"'-'<2[4p(1-pW,,';2(8-1-89) (8-1-90) Asimplerversionof(8-1-90) isobtained ifweemploy dm;ninplaceofthe weightdistribution. Thatis, PM~(M-1)[4p(l- p)]""'"" (8-1-91) Ofcourse(8-1-90) isatighterupperboundthan(8-1-91). InSection8-1-6.wecompare thevariousboundsgivenaboveforaspecific code,namely,theGolay(23,12)code.Inaddition, wecompare theerrorrate performance ofhard-decision andsoft-decision decoding. 456 DIGITAL COMMUNICATIONS 8-1-6Comparison ofPerformance between Hard-Decision andSoft-Decision Decoding Itisbothinteresting andinstructive tocompare theboundsontheerrorrate perfonnance oflinearblockcodesforsoft-decision decoding andhard-decision decoding onanAWGNchannel. Forillustrative purposes, weshallusethe Golay(23,12)code,whichhastherelatively simpleweightdistribution given inTable8-1-1.Asstatedpreviously, thiscodehasaminimum distance dmin=7. Firstwecompute andcompare theboundsontheerrorprobability for nard-decision decoding. SincetheGolay(23,12)codeisaperfectcode,the exacterrorprobability forhard-decision decoding is 23(23)PM=];:.mpm(l-p)23-m =1-~o(~)pm(l_p?3-m (8-1-92) wherepistheprobability ofabinarydigiterrorforthebinarysymmetric channel. Binary(orfour-phase) coherent PSKisassumed tobethe modulation/demodulation technique forthetransmission andreception ofthe binarydigitscontained ineachcodeword.Thus,theappropriate expression for pis·givenby(8-1-73). Inaddition totheexacterrorprobability givenby (8-1-92), wehavethelowerboundgivenby(8-1-87) andthethreeupper boundsgivenby(8-1-88), (8-1-90), and(8-1-91). Numerical resultsobtained fromtheseboundsarecompared withtheexact errorprobability inFig.8-1-12.Weobservethatthe!I:'verboundisveryloose. I\ I Ij.f.-Uppe,bound 21\(Exaci\'(S-I-~I ,8--1-92)\ _Upperbound \':>-I-'fli ,\ Uppe,bound \'8--1·-91) '\1./ \ UJwer\ ~bound.\ -'S-I-S7), \ \2 •10­ o.S ~52 ~IV-• ~S ~2 '61Q-4 ~s ~2 lO- S 2 10-<'o2 4 b8IV.2 ,~ SNRperbit.Y,.(dB)lO­ S F1GURE 8-1-U Comparison ofboundswilhexacterrorprobability for hard-decision decoding ofGolay(23,12)code. CHAPTER K:BLOCK ANDC()"'lVOll'HONA.L ('HA~'El CODES457 4tl ~101:' I~ SNRperbit.'(1.(08)o,\\ I I I , I.11, -~-S~)II·decl~ltlll ,"decoding:,"\-Hard-de~i~it\n ,',\tlecoJin!!- ,, J,, ,,1\,,,\-- tipper botlml~hE:\iJcr---(8-1-50)"\II(K-I-·I.)~.,,. Upperhound,, (8-1-52)t"'.,,,, ,,1\ , ,10, II)~10- ~5 "]10 ~5o "10 FIGURE 8-1-13 Comparison ofsoft-decision decoding with hard~decision decoding fortheGolay(23.12)code AtPM=10-5,thelowerboundisoffbyapproximately 2dBfromtheexact errorprobability. AtPM=10-2 ,thedifference increases toapproximately 4dB.Ofthethreeupperbounds, theonegivenby(8-1-88) isthetightest: it differsbylessthan1dBfromtheexacterrorprobability atPM=10'-The Chernoff boundin(8-1-90), whichemploys theweightdistribution, isalso relatively tight.Finally,theChernoff boundthatemploys onlytheminimum distance ofthecodeisthepoorestofthethree.AtPM=10-'.itdiffersfrom theexacterrorprobability byapproximately 2dB.Allthreeupp.erboundsare verylooseforerrorratesabovePM=10-2• 11isalsointeresting tocompare theperformance between soft-and hard-decision decoding. Forthiscomparison, weusetheupperboundsonthe errorprobability forsoft-decision decoding givenby(8-1-52) andtheexact errorprobability forhard-decision decoding givenby(8-1-92). Figure8-1-13 illustrates theseperformance characteristics. Weobservethatthetwobounds forsoft-decision decoding differbyapproximately 0.5dBatPM=10"andby approximately 1dBatPM=10-'.Wealsoobserve thatthedifference in performance between hard-andsoft-decision decoding isapproximately 2dB intherange102<PM<10".IntherangeP",>10'2,thecurveoftheerror probability forhard-decision decoding crossesthecurvesforthebounds.This behavior indicates thattheboundsforsoft-decision decoding areloosewhen f",>10-2• The2dBdifference between hard-andsoft-decision decoding isacharac­ teristicthatappliesnotonlytotheGolaycode,butisafundamental resultthaI appliesingeneraltocodeddigitalcommunications overtheAWGNchannel. Thisresultisderivedbelowbycomputing thecapacity oftheAWGNchannel withhard-andsoft-decision decoding. I IVVSoft-decision decodingV/ V/Hard-decision 1/~~in81- //458 DJGlTAL CQMMlJNlCATIONS FIGURE8-t-t4 Coderateasafunction oftheminimum SNRperbitfor soft-andhard-decision decoding.1.0 0.8 "t ~0.6 "80.4 U 0.2 o-2-I0 1 2 3456 Minimum SNRpelbitl",(dB} Thechannelcapacity oftheBSeinbitspercodesymbol,derivedinSection 7-1-2,is C=1+PlogzP+(1-p)logz(I-p) (8-1-93) wheretheprobability ofabiterrorforbinary,coherent PSKonanAWGN channel isgivenby(8-1-73). Suppose weuse(8-1-73) forp.letC=R,in (8-1-93), andthendetermine thevalueof)'bthatsatisfiesthisequation. The resultisshowninFig.8-1-14asagraphofR,versus )'h'Forexample, suppose thatweareinterested inusingacodewithrateR,=~.Forthiscoderate,note thattheminimum SNRperbitrequired toachievecapacity withhard-decision decoding isapproxima tely1.6dB. Whatisthelimitontheminimum SNRasthecoderateapproaches zero? ForsmallvaluesofR,.theprobability pcanbeapproximated as (8·I-94) Whentheexpression forpissubstituted into(8-1-93) andthelogarithms in (8-1-93)areapproximated by logz(I+x)=(x-~x2)!ln2 thechannelcapacityformulareducesto (8-I-95) NowwesetC=R,.Thus,inthelimitasR,approaches zero,weobtainthe result )'b=~1t"In2(0.37dB) (8-1-96) Thecapacity ofthebinary-input AWGNchannelwithsoft·decision decod­ ingcanbecomputed inasimilarmanner. Theexpression forthecapacity in bitspercodesymbol,derivedinSection7-1-2,is (8-1-97) CHAPTER K:BLOCK ANDCONVOLUTIONAL CHANSEL CODES459 wherep(yIk),k=0,1,denotetheprobability densityfunctions ofthe demodulator outputconditioned onthetransmitted bitbeinga0anda1, respectively. FortheAWGNchannel, wehave (Ik)=_I_e' (y-m,)'/2ff' k=0,1PY V2!rlT '(8-1-98) wherem"=-Vi;,m\=Vi;,lT2=~NIl'andge=R,gb'Theunconditional probability densityp(y)issimplyone-half ofthesumofp(y11)andp(y1o). AsR,approaches zero,theexpression (8-1~97)forthechannelcapacity canbe approximated as (8-1-99) Again,wesetC=R,.Thus,asR,-+0,theminimum SNRperbittoachieve capacity is 'Y,=In2 (-1.6dB) (8-1-100) Byusing(8-1-98) in(8-1-97)andsettingC=Re,anumerical solution canbe obtained forcoderatesintherange0,,;;Re,,;;1.Theresultofthissolution is alsoshowninFig.8-1-14. Fromtheabove,weobserve thatinthelimitasReapproaches zero,the difference inSNR'Ybbetween hard-andsoft-decision decoding isIn,whichis approximately 2dB.Ontheotherhand,asReincreases towardunity,the difference in'Ybbetween thesetwodecoding techniques decreases. For example, atRe=0.8,thedifference isabout1.5dB. ThecurvesinFig.8-1-14providemoreinformation thanjustthedifference inperformance between soft·andhard-decision decoding. Thesecurvesalso specifytheminimum SNRperbitthatisrequired foragivencoderate.For example, acode rate ofRe=D.8canprovidearbitrarily smallerrorprobability atanSNRperbitof2dB,whensoft-decision decoding isused.Bycomparison, anuncoded binaryPSKrequires 9.6dBtoachieveanerrorprobability of10-' Hence,a7.6dBgainispossible byemploying arateRc=~code.Unfortun­ ately,toachieve suchalargecodinggainusuallyimpliestheuseofan extremely longblocklengthcode,whichleadstoaverycomplex receiver. Nevertheless, thecurvesinFig.8-1-14provideabenchmark forcomparing the codinggainsachieved bypractically implementable codeswiththeultimate limitsforeithersoft-orhard-decision decoding. Instead ofcomparing thedifference between hard-andsoft-decision decoding basedonthechannel capacity relations, wemayperform similar comparisons basedontherandomcodingrateparameters. InChapter 7,we demonstrated thattheensemble average probability oferrorforrandomly selectedbinarycodewordsisupper-bounded as (8-1-101) where'R c=k/nisthecoderateandthecutoffrateRorepresents theupper 460 mOrTAL COMMUNICA'TIONS boundonResuchthatp"-.0asn--.:>c.Forunquantized (soft-decision) decoding. Roisgivenas 2R=log----..,-,-cc(} 21_e--l,/Nu(8-1-102) where'l../No=R,'YbistheSNRperdimension, Thisresultwasderived in Section7-2. Ontheotherhand,iftheoutputofthedemodulator isquantized toQlevels priortodecoding, theChernoff boundmaybeusedtoupper-bound the ensemble averagebinaryerrorprobability P,(SI.sm)definedinSection7-2,The resultofthisderivation isthesameupperboundonPegivenin(8-1-101) but witnR"replaced byRQ•where (8-1-103) (8-1-104)In(8-1-103), {pJarethepriorprobabilities ofthetwosignalsattheinputto thechanneland{PC;Ij)}denotethetransition probabilities ofthecnanneL For example. inthecaseofabinarysymmetric channel, wehaveP,=po=t P(O10)=p(111) =1-p.andP(O11)=p(ll0)=p,Hence, 2 RQ=log21+v'4p(1_p) Q=2 where p=Q(v'2YhRcl (8-1-105) AploiofRQversus10log('teiND)isillustrated inFig,8-1-15forQ=2and Q=x;(soft-decision decoding), Notethatthedifference indecoder perfor­ mancebetween unquantized soft-decision decoding andhard-decision decod­ ingisapproximately 2dB.Infact,itiseasilydemonstrated againthatas 'lc!No-.O.tnelossinperformance duetohard-decision decoding is FlGURE 8-1-15 Comparison ofRo(soft-deciSion decoding) withRQ(hard­ decision decoding) asafunction oftheSNRperdimension.1.0 NO.9 110.8 ~0.7 oc.'>0.610.5 <l0.4g0.3 "0.2 0.1 ~lL.n--_5L-~0'----'5L---'IO'+ IOlogtl"lN n)(dB) CHAPlER KBLOCK ANDCONVOLUTIONAL CHANNEL (ODES4til 10log,o ~Ir=2dB,whichisthesamedecibeldifference thatwasobtained in ourcomparison ofthechannelcapacity relations. Wemention thataboutIdB ofthislosscanberecovered byquantizing theoutputofthedemodulator to threelevelsinsteadoftwo(seeProblem 7-11).Additional improvements are possible byquantizing theoutputintomorethanthreelevels,asshownin Section7-3. 8-1-7BoundsonMinimum Distance ofLinearBlockCodes Theexpressions fortheprobability oferrorderived inthischapter for soft-decision andhard-decision decoding oflinearbinaryblockcodesclearly indicate theimportance thattheminimum distance parameter playsinthe performance ofthecode.Ifweconsider soft-decision decoding, forexample, theupperboundontheerrorprobability givenby(8-1-52)indicates that,fora givencoderateR,=kin,theprobability oferrorinanAWONchannel decrease's exponentially witham;n'Whenthisboundisusedinconjunction with thelowerboundonam"givenbelow,weobtainanupperboundonPMthat canbeachieved bymanyknowncodes.Similarly, wemayusetheupperbound givenby(8-1-82) fortheprobability oferrorforhard-decision decoding in conjunction withthelowerboundondm;ntoobtainanupperboundonthe errorprobability forlinearbinaryblockcodesonthebinarysymmetric channel.· Ontheotherhand,anupperboundonam;ncanbeusedtodetermine a lowerboundontheprobability oferrorachieved bythebestcode.For example, suppose thathard-decision decoding isemployed. Inthiscase,we havethetwolowerboundsonPMgivenby(8-1-86) and(8-1-87). withthe formerbeingthetighter.Wheneitheroneofthesetwoboundsisusedin conjunction withanupperboundondm;ntheresultisalowerboundonPMfor thebest(n,k)code.Thus,upperandlowerboundsondm;nareimportant in assessing thecapabilities ofcodes. Asimpleupperboundontheminimum distance ofan(n,k)binaryor non-binary linearblockcodewasgivenin(8-1-14) asam;"';;n-k+1.Itis convenient tonormalize thisexpression bytheblocksizen.Thatis. dmin 1~,;;(l-Rcl+-n n(8-1-106) whereR,isthecoderate.Forlargen,thefactorlincanbeneglected. Ifacodehasthelargestpossibledistance, i.e.,am;n=n-k+I,itiscalleda maximum-distance-separable code.Exceptforthetrivialrepetition-type codes. therearenobinarymaximum-separable codes.Infact,theupperboundin (8-1-106) isextremely looseforbinarycodes.Ontheotherhand,nonbinary codeswitham;n=n-k+Idoexist.Forexample, theReed-Solomon codes. whichcomprise asubclassofBCHcodes,aremaximum-distance-separable. Inaddition totheupperboundgivenabove,thereareseveralrelatively 462 Dl(i1rAL.CO",1:\ll\ICATlOJ"o;S tightboundsontheminimum distance oflinearblockcodes.Weshallbriefly describe fourimportant bounds. threeofwhichareupperboundsandthe otheralowerbound.Thederivations oftheseboundsarelengthyandarenot ofparticular interestinoursubsequent discussion. Theinterested readermay r!?fer10Chapler 4ofthebookbyPeterson andWeldon (1972)forthose derivations. Oneupperboundontheminimum distance canbeoblained fromthe inequality in(8-1-83). Bytakingthelogarithm ofbothsidesof(8-1-83)and dividing by11.weobtain 1'(n) I-R,;.-logo2:.n joeOI(8-1-107) Sincetheerror-correcting capability ofthecode.measured byI,isrelatedto theminimum distance, theaboverelationisanupperboundontheminimum distance.ItiscalledtheHamming upperbound. J'heasymptotic formof(8-1-107) isobtained bylettingn->00.Now,forany n.lettobethelargestinteger Iforwhich(8-1-107) holds.Then,itcanbeshown (Peterson andWeldon, 1972)thatasn--"'.theratiotinforany(n.k)block codecannotexceedloin.wherelulnsatisfiestheequation 1-Rc=H(luln) (8-1-108) andH(x)isthebinaryentropyfunction definedby(3-2-10). Thegeneralization oftheHamming boundtononbinary codesissimply 1-R..;'llog.,[i(.~)(q-I)'J n (=01(8-1-109) Another upperbound,developed byPlotkin(1960),maybestatedas follows.Thenumberofcheckdigitsrequired toachieveaminimum distance dmi,inan(n.k)linearblockcodesatisfiestheinequality kqdm;,-In-~ 1 -lo~,dminq-I(8-1-110) whereqisthealphabet size.Whenthecodeisbinary,(8·1-110) maybe expressed as dmin(I ) 1( 2) --1--2d.log2dmi.,,;;-1-Rc+-n min 2 n Inthelimitasn->xwithdmin/n,,;;t(8-1-110) reducesto (8-1-111) CHAPT[-.R 1".BLOCK A~[)CO~VGLl~TJO;'\AL CHA'~F-:l. CODES463 Finally, thereisanother tightupperboundontheminimurndistance obtained byElias(Berlekamp, 196R).Itmaybeexpressed initsasymptotic formas dminlnE2A(1 ~A) (R-I-I12) wheretheparameter AISrelatedtothecoderatethrough theequation R,=I+Alog,A+11-A)log,(l-A), OEAE~ (R-I-II3) Lowerboundsontheminimum distance of(n,k)blockcodesalsoexist.In particular. binaryblockcodesexistthathaveanormalized minimum distance thatasymptotically satisfies theinequality (R-I-II4) whereaisrelatedLlthecoderatethrough theequation R,=I-J-/(o:) =1+alog,a+(I-(\')log,(1-0:). (8-1-115i ThiSlowerboundisaspecialcaseofalowerbounddeveloped byGilbert (1952)andVarsharmov (1957).whichappliestononbinaryandbinaryblock codes. Theasymptotic bounds givenaboveareplottedinFig.8-1-16forbinary codes.Alsoplottedintheligureforpurposes ofcomparison arecurvesofthe minimum distance asafunction ofcoderateforBCHcodesofblocklengths n=31and63.Weobserve thatforn=31and63,thenormalized minimum distance fallswellabovetheVarsharmov-Gilbert lowerbound.Astheblock lengthnincreases. (heefficiencv oftheBCHcodesdiminishes. Forexample. whenn=1023,thecurvcforthenormalized minimum distance fallscloseto U.5 II;::6.iHCHcodes Plutklnurrcrbound 0.2 :l~ O.M08 Coderatt'R,1.0/I:::31BCH l'lKJe~ ",,.......... ()rJia~~_~ iuppc:bl)Und ()IIrGilbert-Varsharmn .... In\.',erhoundn FIGL~RE 8-1-16 Lippt:randlowerhoundsonnormalizeJ mInimum Jistance asafunctiun ofcoderJtc. 464 DIGITAL COMMF''''I('ATIONS theVarsharmov-Gilbert bound.Asnincreases beyondn=1023,thenormal­ izedminimum distance oftheBCHcodescontinues todecrease andfallsbelow theVarsharmov-Gilbert bound.Thatis,dminlnapproaches zeroasntendsto infinity.Consequently theBCHcodes,whicharethemostimportant classof cycliccodes,arenotveryefficientatlargeblocklengths. 8-1·8Nonbinary BlockCodesandConcatenated Block Codes Anonbinary blockcodeconsistsofasetoffixed-length codewordsinwhich theelements ofthecodewordsareselected fromanalphabet ofqsymbols. denoted by{O,1,2.' ,..q-I}.Usually, q=2"sothatkinformation bitsare mapped intooneoftheqsymbols. Thelengthofthenonbinarycodewordis denoted byNandthenumberofinformation symbols encoded intoablockof Nsymbols isdenoted byK.Theminimum distance ofthenonbinary codeis· denoted byDm,,,,Asystematic (N,K)blockcodeconsistsofKinformation symbols andN-Kparitychecksymbols. Among thevarioustypesofnonbinary linearblockcodes,theReed­ Solomon codesaresomeofthemostimportant forpractical applications. As indicated previously, they comprise asubsetoftheBCHcodes,whichinturn areaclassofcycliccodes.Thesecodesaredescribed bytheparameters N=q-!=2"-1 K=1,2,3,...,N-1 (8-1-116) Dmin=N-K+1 R,=KIN Suchacodeisguaranteed 10correctupto t=U(Dm'n-l)J =U(N-K)J (8-1-117) symbolerrors.Ofcourse,thesecodesmaybeextended orshortened inthe mannerdescribed previously forbinaryblockcodes. Theweightdistribution {A,}oftheclassofReed-Solomon codesisknown. Thecoefficients intheweightenumerating polynomial aregivenas (8-1-118) whereD==Dm'nandq=2*. Onereasonfortheimportance oftheReed-Solomon codesistheirgood CHAPTER KBLOCK ANDCONVOLUTIOSAL CHANNEL CODES465 distance properties. Asecondreasonfortheirimportance istheexistence of efficient hard-decision decoding algorithms, whichmakeitpossible toimple­ mentrelatively longcodesinmanypractical applications wherecodingis desirable. Anonbinary codeisparticularly matched toanM-arymodulation technique fortransmitting the2'possible symbols. Specifically. M-aryorthogonal signaling. e.g.,M-aryFSK,isfrequently used.Eachofthe2'symbols inthe q-aryalphabet ismapped tooneoftheM=2'orthogonal signals.Thus,the transmission ofacodewordisaccomplished bytransmitting Northogonal signals.whereeachsignalisselected fromthesetofM=2'possiblesignals. Theoptimum demodulator forsuchasignalcorrupted byAWGNconsists ofMmatched filters(orcross-correlators) whoseoutputsarepassedtothe decoder, eitherintheformofsoftdecisions orintheformofharddecisions. If harddecisions aremadebythedemodulator, thesymbolerrorprobability PM andthecodeparameters aresufficient tocharacterize theperformance ofthe decoder. Infact,themodulator. theAWGNchannel, andthedemodulator formanequivalent discrete(M-ary)input,discrete(M-ary)output,symmetric memoryless channel characterized bythetransition probabilities ~=1-PM andPM/(M-1).Thischannelmodel,whichisillustrated inFig.8-1-17,isa generalization oftheBSe. Theperformance ofthehard-decision decoder maybecharacterized bythe following upperboundonthecodeworderrorprobability: (8·1-119) whereIisthenumberoferrorsguaranteed tobecorrected bythecode. Whenacodeworderrorismade,thecorresponding symbol error probability is [ N(N'P=-Lilp'(I-P)N' {\Ni=lfl jJ;\.I M(8-1-[20) FIGURE 8-1-17 M-aryinput.M-aryoutput,symmetric memoryles5 channel.1-P'f .-- M-I«=-----._----"'0 M-. 466 DIGITAL COMMUNIC,\TI01"S Furthermore, ifthesymbols areconverted tobinarydigits,thehiterror probability corresponding to(8-1-120) is (8-1-121) Example 8-.-13 Letusevaluate theperformance ofanN=2'-1=31Reed-Solomon code withDm'n=3,S,9,and17.Thecorresponding valuesofKare29,27. 23. and15.Themodulation isNt=32orthogonal FSKwithnoncoherent detection atthereceiver. Theprobability ofasymbolerrorisgivenby(5-4-46). andmayhe expressed as. 1 M'M)P"=-e Y2:(-1)"\ e""Al n"-c2 n(8-1-122) whereyistheSNRpercodesymbol. By using (8-1-122) in(8-I-t2()) and combining theresultwith(8-1-121),weobtainthebiterrorprobability. The resultsofthesecomputations areplottedinFig.8-1-18.Notethatthemore powerful codes(largeDm,,,)givepoorerperformance atlowSNRperbit thantheweakercodes.Ontheotherhand,athighSNR,themorepowerful codesgivebetterperf01mance.Hence,therearecrossovers amongthe variouscodes,asillustrated forexample inFig.8-1-18forthet=1and1=8 codes.Crossovers alsooccuramongtheI=I,2,and4codesatsmaller valuesofSNRperbit.Similarly. thecurvesfor1=4and8andfor1=8and 2crossintheregionofhighS~R.Thisisthecharacteristic behavior for noncoherent detection ofthecodedwaveforms. Ifthedemodulator docsnotmakeaharddeciSIon oneachsvmhol. but. ---FIGURE 8-1-18 Performance ofseveral..'\ =31.{-errmcorrecting Reed-Solomon codeswith32-aryFSKmodulation onanAWGN channel (noncoherent demodulation).10'\ 10'-----.l--...-:-'c---'--c!.+.0 5.0n,;17u S:-.IRPt'!hi!,l'1,'dB\ CHAPTE:R KBLOCK A~()(,O!'lVOLlT[O~AL CHA"Fl lODES467 Input dat;!OUler encodc-r 1\K,Inl1l,"r <:ncod('f (11).1 FIGURE 8-t-19 Blockdiagram ofacommunications systememploying aconcatenated code. instead. passestheunquantized matched filteroutputs tothedecoder. soft-decision decoding canbeperformed. Thisdecoding involves theformation ofq"=2'"correlation metrics. whereeachmetriccorresponds tooneofthe q"codewordsandconsists ofasumofNmatched filteroutputscorresponding totheNcodesymbols. Thematched filteroutputsmaybeaddedcoherently. or theymaybeenvelope-detected andthenadded,ortheymaybesquare-law detected andthenadded.Ifcoherent detection isusedandthechannel noiseis AWGN.thecomputation oftheprobability oferrorisastraightforward extension ofthebinarycaseconsidered inSection8-1-4.Ontheotherhand. whenenvelope detection orsquare-law detection andnoncoherent combining areusedtoformthedecision variables. thecomputation ofthedecoder performance isconsiderably morecomplicated. Concatenated BlockCodes Aconcatenated codeconsistsoftwoseparate codeswhicharecombined toformalargercode.Usuallyonecodeisselected tobenonbinaryandtheotherisbinary.Thetwocodesareconcatenated as illustrated inFig.8·1-19.Thenonbinary (N,K)codeformstheoutercodeand thebinarycodeformstheinnercode.Codewordsareformedbysubdividing a blockofkKinformation bitsintoKgroups,calledsymbols, whereeachsymbol consistsofkbits.TheKk-bitsymbols areencoded intoNk-bitsymbols bythe outerencoder. asisusuallydonewithanonbinary code,Theinnerencoder takeseachk-bitsymbolandencodes itintoabinaryblockcodeoflengthn. Thusweobtainaconcatenated blockcodehavingablocklengthofNnbitsand containing kKinformation bits.Thatis,wehavecreated aneyuivalent (Nn,Kk)longbinarycode.Thebitsineachcodewordaretransmitted over thechannel bymeansofPSKor,perhaps. byFSK. Wealsoindicate thattheminimum distance ofti,.cuncatenated codeis dm;oDmio•whereDmmistheminimum distance oftheoutercodeandd""oisthe minimum distance oftheinnercode.Furthermore, therateoftheconcaten­ atedcodeisKkINn,whichiseyualtotheproduct ofthetwocoderates. Ahard-decision decoder foraconcatenated codeisconveniently separated intoaninnerdecoder andanouterdecoder. Theinnerdecoder takesthehard decisions oneachgroupofnbits,corresponding toacodewordoftheinner code,andmakesadecision onthekinformation bitsbasedonmaximum­ likelihood (minimum-distance) decoding. Thesekbitsrepresent onesymbol,,! 468 IlI(iITI\L ('OMMUNJ('ATIONS theoutercode.Whenablock ofNk-bitsymbols arereceived fromtheinner decoder, Iheoulerdecoder makesaharddecision ontheKk-bilsymbols basedonmaximum-likelihood decoding. Soft-decision decoding isalsoapossible alternative withaconcatenated code.Usually, thesoft-decision decoding isperformed ontheinnercode,ifitis selected tohaverelatively fewcodewords,i.e.,if2'isnottoolarge.Theouter codeisuSlJallydecoded bymeansofhard-decision decoding, especially ifthe blocklengthislongandtherearemanycodewords.Ontheotherhand,there maybeasignificant gaininperformance whensoft-decision decoding isused onboththeouterandinnercodes,tojustifytheadditional decoding complexity. Thisisthecaseindigitalcommunications overfadingchannels, as weshalldemonstrate inChapter 14. Weconclude thissubsection withthefollowing example. EXlimple 8-1·14 Suppose thatthe(7,4)Hamming codedescr;.bed inExamples 8-1-1and 8-1-2isusedastheinnercodeinaconcatenated codeinwhichtheouter codeisaReed-Solomon code.Sincek=4,weselectthelengthofthe Reed-Solomon codetobeN=24-1=15.Thenumber ofinformation symbols Kperoutercodewordmaybeselected overtherange1,,;;K,,;;14 inordertoachieveadesiredcoderate. 8-1-9Interleaving orCodedDataforChannels withBurst Errors Mostofthewell-known codesthathavebeendevisedforincreasing the reliability inthetransmission ofinformation areeffective whentheerrors causedbythechannel arestatistically independent. Thisisthecaseforthe AWGN channel. However, therearechannels thatexhibitburstyerror characteristics. Oneexample istheclassofchannels characterized bymultipath andfading,whichisdescribed indetailinChapter 14.Signalfadingdueto time-variant multipath propagation oftencausesthesignaltofallbelowthe noiselevel,thusresulting inalargenumberoferrors.Asecondexample isthe classofmagnetic recording channels (tapeordisk)inwhichdefectsinthe recording mediaresultinclustersoferrors.Sucherrorclustersare110tusually corrected bycodesthatareoptimally designed forstatistically independent errors. Considerable workhasbeendoneontheconstruction ofcodesthatare capable ofcorrecting bursterrors.Probably thebestknownbursterror correcting codesarethesubclassofcycliccodescalledFirecodes,namedafler P.Fire(1959),whodiscovered them.Another classofcycliccodesforburst errorcorrection weresubsequently discovered byBurton(1969). CHAPTER X:BLOCKANDCONVOLUTIONAL CHAN~EL CODES469 FIGURE 8·1·20 Blockdiagram ofsystememploying inlerleaving forburst-error channel. Abursroferrorsoflengthbisdefinedasasequence ofb-biterrors,thefirst andlastofwhichare1'5.Thebursrerrorcorrection capabiliry ofacodeis defined asonelessthanthelengthoftheshortest uncorrecta bleburst.Itis relatively easytoshowthatasystematic (n,k)code,whichhasn-kparity checkbits,cancorrectburstsoflengthb,,;LHn-k)J. Aneffective methodfordealingwithbursterrorchannels istointerleave thecodeddatainsuchawaythattheburstychannel istransformed intoa channel havingindependent errors.Thus,acodedesigned forindependent channelerrors(shortbursts)isused. Ablockdiagram ofasystemthatemploys interleaving isshowninFig. 8-1-20.Theencoded dataarereordered bytheinterleaver andtransmitted overthechannel. Atthereceiver, after(eitherhard-orsoft-decision) demodulation, thedeinterleaver putsthedatainpropersequence andpassesit tothedecoder. Asaresultoftheinterleavingldeinterleaving, errorburstsare spreadoutintimesothaterrorswithinacodewordappeartobeindependent. Theinterleaver cantakeoneoftwoforms:ablockstructure ora convolutional structure. Ablockinterleaver formats theencoded dataina rectangular arrayofmrowsandncolumns. Usually, eachrowofthearray constitutes acodewordoflengthn.Aninterleaver ofdegreemconsistsofm rows(mcodewords)asillustrated inFig.8-1-21.Thebitsarereadout FIGURE 8-1-21 Abtockinterleaver forcodeddata. Readoulbitstomodulator wst tt t t t t t--I81522 29 36·. mn-6 ......2916233037·. mn5--31017243138 mn-4--4II18253239·.. mn-3 mm--512 19 263340·. mn- 2--61320273441 mn-I--71421283542 mn +-n-kparitybits kdatabits 470 DI(JIT.~L COMMUNICATIONS column-wise andtransmitted overthechannel. Atthereceiver, thedeinter­ leaverstoresthedatainthesamerectangular arrayformat,butitisreadout row-wise, onecodewordatatime.Asaresultofthisreordering ofthedata duringtransmission, aburstoferrorsoflengthI~mbisbrokenupintom burstsoflengthb.Thus,an(n,k)codethatcanhandlebursterrorsoflength b~U(n-k)Jcanbecombined withaninterleaver ofdegreemtocreatean interleaved (mn,mk)blockcodethatcanhandleburstsoflengthmb. Aconvolutional interleaver canbeusedinplaceofablockinterleaver in muchthesameway.Convolutional interleavers arebettermatched foruse withtheclassofconvolutional codesthatisdescribed inthefollowing section. Convolutional interleaver structures havebeendescribed byRamsey (1970) andForney(1971). 8-2CONVOLUTIONAL CODES Aconvolutional codeisgenerated bypassingtheinformation sequence tobe transmitted through alinearfinite-state shiftregister. Ingeneral, theshift registerconsistsofK(k-bit)stagesandnlinearalgebraic function generators, asshowninFig.8-2-1.Theinputdatatotheencoder, whichisassumed tobe binary,isshiftedintoandalongtheshiftregisterkbitsatatime.Thenumber ofoutputbitsforeachk-bitinputsequence isnbits,Consequently, thecode rateisdefinedasK=kin,consistent withthedefinition ofthecoderatefora blockcode.Theparameter Kiscalledtheconstraint lengthoftheconvolu­ tionalcode.t I' k infonnation '--,,--'-......,,-'---''-,,<'-' bits FIGURE 8,2·1Convolutionat encoder.Kks[ages---------- .....\ Encoded sequence lomodulator tInmanycases.theconstraint lengthofthecodeisgiveninbitsratherthank-bit ~ytes.Hence theshiftregistermaybecalledaI.-stageshiftregister,whereL=Kk.Furthermore. Lmaynotbea multiple ofk.ingeneral. CHAPTER KBLOCK ANDCO:,\VOLL:TION,<\L CH,\:'\'\El ((mrS471 F1GURE 8~2~2K=3,k=1,11=3convolutional encoder. Onemethod fordescribing aconvolutional codeistogiv~itsgenerator matrix,justaswedidforblockcodes.Ingeneral, thegenerator matrixfora convolutional codeissemi-infinite sincetheinputsequence issemi-infinite in length.Asanalternative tospecifying thegenerator matrix,weshallusca functionally equivalent representation inwhichwespecifyasetofnvectors. onevectorforeachofthenmodulo-2 adders.EachvectorhasKkdimensions andcontains theconnections oftheencodertothatmodul0-2 adder.A Iinthe ithposition ofthevectorindicates thatthecorresponding stageintheshift register isconnected tothemodulo-2 adderanda0inagivenposition indicates thatnoconnection existsbetween thatstageandthemodulo-2 adder. Tobespecific, letusconsider thebinaryconvolutional encoder with constraint lengthK=3,k=I,andn=3,whichisshowninFig.8-2-2.Initially. theshiftregisterisassumed tobeintheall-zerostate.Suppose thefirstinput bitisaI.Thentheoutputsequence of3bitsisIII.Suppose thesecondbitisa O.Theoutputsequence willthenbe001.IfthethirdbitisaI,theoutputwill be100,andsoon,Now,suppose wenumber theoutputsofthefunction generators thatgenerate eachthree-bit outputsequence asI,2.and3,from toptobottom. andsimilarly number eachcorresponding function generator, Then,sinceonlythefirststageisconnected tothefirstfunction generator (no modulo-2 adderisneeded). thegenerator is g=[100] Thesecondfunction generator isconnected tostagesIand3.Hence g,=(101] Finally, g,=[IlI] Thegenerators forthiscodearemoreconveniently giveninoctalformas (4,5,7). Weconclude that.whenk=I,werequire ngenerators. eachof dimension Ktospecifytheencoder. Foraratekinbinaryconvolutional codewithk>Iandconstraint lengthK. thengenerators areKk-dimensional vectors, asstatedabove,Thefollowing example illustrates thecaseinwhichk=2andn=3. 472 DIGITA.L COMMUNICATIONS FIGURE 8-2-3K~2.k=2.n=3convolutional encoder.I ZOutput Lf-----o2 '------<:J 3 Example 8-2-1 Consider therate2/3convolutional encoderillustrated inFig.8-2-3.Inthis encoder, twobitsatatimeareshiftedintoitandthreeoutputbitsare generated. Thegenerators are g,=[1011J, Inoctalform,thesegenerators are(13.15,12). Therearethreealternative methods thatareoftenusedtodescribe a convolutional code.Thesearethetreediagram, thetrellisdiagram, andthe statediagram. Forexample, thetreediagram fortheconvolutional encoder showninFig.8-2-2isillustrated inFig.8-2-4.Assuming thattheencoderisin theall-zerostateinitially, thediagram showsthat,ifthefirstinputbitisa0, theoutputsequence is000and,iftbefirstbitisa1;theoutputsequence is111. Now,ifthefirstinputbitisa 1andthesecondbitisa0,thesecondsetofthree outputbitsis001.Continuing throughthetree,weseethatifthethirdbitisa 000 o III FIGURE 8-2-4Treediagram forratetf3.K=3convolutional code. CHAPTER ItBLOCK ANDCONVOLUTIONAL CHANNEL CODES473 othentheoutputis011,whileifthethirdbitisa 1thentheoutputis100. Giventhataparticular sequence hastakenustoaparticular nodeinthetree, thebranching ruleistofollowtheupperbranchifthenextinputbitisa 0and thelowerbranchifthebitisaI.Thus,wetraceaparticular paththroughthe treethatisdetermined bytheinputsequence. Closeobservation ofthetreethatisgenerated bytheconvolutional encoder showninFig.8-2-2revealsthatthestructure repeatsitselfafterthethirdstage. Thisbehavior isconsistent withthefactthattheconstraint lengthK=3.That is,thethree-bit outputsequence ateachstageisdetermined bythe input bit andthetwoprevious inputbits,i.e.,thetwobitscontained inthefirsttwo stagesoftheshiftregister.Thebitinthelaststageoftheshiftregisterisshifted outattherightanddoesnotaffecttheoutput.Thuswemaysaythatthe three-bit outputsequence foreachinputbitisdetermined bytheinputbitand thefourpossible statesoftheshiftregister, denoted asa=00,b=01,C=10, d=11.Ifwelabeleachnodeinthetreetocorrespond tothefourpossible statesintheshiftregister, wefindthatatthethirdstagetherearetwonodes withthelabela,twowiththelabelb,twowiththelabelc,andtwowiththe labeld.Nowweobservethatallbranches emanating fromtwonodeshaving thesamelabel(samestate)areidentical inthesensethattheygenerate identical outputsequences. Thismeansthatthetwonodeshavingthesame labelcanbemerged.IfwedothistothetreeshowninFig.8-2-4,weobtain anotherdiagram, whichismorecompact, namely,atrellis.Forexample, the trellisdiagram fortheconvolutional encoder ofFig.8-2-2isshowninFig. 8-2-5.Indrawing thisdiagram, weusetheconvention thatasolidlinedenotes theoutputgenerated bytheinputbit0andadottedlinetheoutputgenerated bytheinputbit1.Intheexample beingconsidered, weobservethat,afterthe initialtransient, thetrelliscontains fournodesateachstage,corresponding to thefourstatesoftheshiftregister, D,b,c,andd.Afterthesecondstage,each nodeinthetrellishastwoincoming pathsandtwooutgoing paths.Ofthetwo FIGURE 8-2-5Trellisdiagram forrate1}3,K=3convolutional code. 000 000 000 000 000a,011 011.011 .III'. 111\ Ill'.. 111• b•111\.. 001 c•. ••110 d•-~_.__:. 101t101 101 Steadystate 474 DIC,lTAl CO\1\-IlJNICAT!O).;S 011 010 FH;URE ~·2M6 Slalediagr;;m forrate1/3.K'-=3convolutional code.....-- "10l'"III outgoing paths,onecorrespcnds totheinputbit0andtheothertothepath followed iftheinputbitisa1. Sincetheoutputottheencoder isdetermined bytheinputandthestateof theencoder, anevenmorecompact diagram thanthetrellisisthestate diagram. Thestatediagram issimplyagraphofthepossible statesofthe encoder andthepossible transitions fromonestatetoanother. Forexample thestatediagram fortheencoder showninFig.8-2-2isillustrated inFig.8-2-6. Thisdiagram showst.hatthepossible transitions are a~a,Q4C, bJl~a,b-l:rc, c~b, c~d. d~b. d~d. where Cl'J...{3denotes thetransition fromstate [jotof3whentheinputbitisa1. Thethreebitsshownnexttoeachbranchinthestatediagram represent the outputbits.Adottedlineinthegraphindicates thattheinputbitisa1.while thesolidlineindicates thattheinputbitisaO. Example 8-2-2 Letusconsider thek=2.rate2/3convolutional codedescribed inExample 8-2-1andshowninFig.8-2-3.Thefirsttwoinputbitsmaybe00,01,10,or 11.Thecorresponding outputbitsare000, 010, 111, 101. Whenthenextpair ofinputbitsentertheencoder, thefirstpairisshiftedtothesecondstage. Thecorresponding outputbitsdependonthepairofbitsshiftedintothe secondstageandthenewpairofinputbits.Hence,thetreediagram forthis code,showninFig.8-2-7,hasfourbranches pernode,corresponding tothe fourpossible pairsofinputsymbols. Sincetheconstraint lengthofthecode isK=2.thetreebeginstorepeatafterthesecondstage.Asillustrated in Fig.8-2-7,allthebranches emanating fromnodeslabeleda(statea)yield identical outputs.Bymerging thenodeshavingidentical labels,weobtain thetrellis,whichisshowninFig.8-2-8.Finally, thestatediagram forthis codeisshowninFig.8-2-9. "I\PlIR' IlLO,K "~Il,O"OUIIO'"L (II.\~"'" 'OlliS 47~ r---- I)()() "Il!n k)!1 h 1(1(\, III, iO! d 110i------- " r--~ ~-- c--0"'-h ,I; dOl ", ';'-E"___1_0_1_ "-- III __ III h (,'1 111(J, -~ d :)11~--, "~-- Ij,,--, ..• IiII\lUl h 100 110 ro~,:"er3iize, wcstatethataratekin,constraint lengthK,convolutional cudeischaracterized by2'branches emanating [romeachnodeofthetree' diagram. Thetrellisandthestakdiagrams eachhave2«1("possible stat"s, Therearc2'branches enterlllg each,tateand2'branches leavingeachstalc (inthetrelliSandtree,thisIStrueaftertheinitialtransient), Thethreetypesofdiagrams described abovearealsousedtorepresent nonbinary convolutional codes,Whenthenumber ofsymbols intheeodc alphabet isq=2',k>1,theresulting llOnbinary codemayalsoberepresented asanequivalent binarycode.Thefollowing example considers aconvolu.ional codeofthistype, Example 8-2-3 Letusconsider theconvolutional codegenerated bytheencoder shownin Fig,8-2-JO.Thiscodemaybedescribed asabinaryconvolutional codewith parameters K=2.k=2,n=4.R,=1/2,andhavingthegenerators gl'~[IOlD], g,=[0101]. g,=[IIIO], d 110 d 11.0 d FlGUo 8-2-lITrellisdiagramlorK;2,k;2,n;3convolutional code.476 DIGITAL COMMUNICATIONS 000 000 110a d FlGURE 8-:1-9StatediagramlorK=2,k=2,n=3convolutional code. CHAPTER l\BLOCKANDCONVOLUTlOl'OAL CHANNEL CODES477 FIGURE 8-2-10 K~2,k~2,n~4convolutional encoder.Input k=2 '------------'>4 Exceptforthedifference inrate,thiscodeissimilarinformtotherate2/3. k=2convolutional codeconsidered inExample 8-2-1. Alternatively, thecodegenerated bytheencoder inFig.8-2-10maybe described asanonbinary (q=4)codewithonequaternary symbolasan inputandtwoquaternary symbols asanoutput.Infact,iftheoutputofthe encoder istreatedbythemodulator anddemodulator asq-ary(q=4) symbols thataretransmitted overthechannel bymeansofsomeM-ary (M=4)modulation technique, thecodeisappropriately viewed as nonbinary. Inanycase,thetree,thetrellis,andthestatediagrams areindependent ofhowweviewthecode.Thatis,thisparticular codeischaracterized bya treewithfourbranches emanating fromeachnode,oratrelliswithfour possible statesandfourbranches entering andleavingeach stat~or, equivalently, byastatediagram havingthesameparameters asthetrellis. 8-2-1TheTransfer Function ofaConvolutional Code Thedistanceproperties andtheerrorrateperformance ofaconvolutional code canbeobtained fromitsstatediagram. Sinceaconvolutional codeislinear,the setofHamming distances ofthecodesequences generated uptosomestagein thetree,fromtheall-zerocodesequence, isthesameasthesetofdistances of thecodesequences withrespecttoanyothercodesequence. Consequently, we assumewithoutlossofgenerality thattheall-zerocodesequence istheinputto theencoder. Thestatediagram showninFig.8-2-6willbeusedtodemonstrate the methodforobtaining thedistanceproperties ofaconvolutional code.First.we labelthebranches ofthestatediagramaseitherDO=I,D1,D2,orD',where theexponent ofDdenotestheHamming distanceofthesequence ofoutput bitscorresponding toeachbranchfromthesequence ofoutputbitscorres­ pondingtotheall-zerobranch.Theself-loop atnodeacanbeeliminated, since itcontributes nothingtothedistance properties ofacodesequence relativeto 478 DlOITAI COMMUNICATIONS D' D ,D' DD 0'" FIGURE 8-2-11 Statediagramforrate1/3,K~3convolutional code. theall-zerocode sequence. Furthermore, nodeaissplitintotwonodes,oneof whichrepresents theinputandtheothertheoutputofthestatediagram. Figure8-2-11illustrates theresulting diagram. Weusethisdiagram, whichnow consistsoffivenodesbecause nodeawassplitintotwo,towritethefourstate equations Xb=DXc+DXd Xd=D'Xc+D2Xd X,=D'Xh(8-2-1) Thetransferfunctien forthecodeisdefinedasT(D)=X,IXo'Bysolving thestateequations givenabove,weobtain T(D)=1_2D2 = D6+2D"+4DIU+8D12+... (8-2-2) where,bydefinition, (evend) (oddd)(8-2-3) Thetransfer function forthiscodeindicates thatthereisasinglepathof Hamming distance d=6fromtheall-zeropaththatmergeswiththeall-zero pathatagivennode.Fromthestatediagram showninFig.8-2-6orthetrellis diagram showninFig.8-2-5,itisobserved thatt_hed=6pathisacbe.Thereis nootherpathfromnodeatonodeehavingadistance d=6.Thesecondterm in(8-2-2)indicates thattherearetwopathsfromnodeatonodeehavinga CHAPTER 8:BLOCK ANDCONVOLUTIONAL CHANNEL CODES479 ,,,,-­,, [iJ-----~_;;,------·C-+-- ....J~D--+ .......+----<J~D-' ~-+-e nGURE 8-2-12 Statediagram forrate1/3,K~3convolutional code. distance d=8.Again,fromthestatediagram orthetrellis,weobserve that thesepathsareacdbeandacbcbe.Thethirdtermin(8-2-2)indicates thatthere arefourpathsofdistance d=10,andsoforth.Thusthetransferfunction gives usthedistance properties oftheconvolutional code.Theminimum distanceof thecodeiscalledtheminimum freedistance anddenoted bydr,ee'Inour example, dr.ee=6. Thetransferfunction canbeusedtoprovidemoredetailed information than justthedistance ofthevariouspaths.Suppose weintroduce afactorNintoall branchtransitions causedbytheinputbit1.Thus,aseachbranchistraversed. thecumulative exponent onNincreases byoneonlyifthatbranchtransition is duetoaninputbit1.Furthermore, weintroduce afactorofJintoeachbranch ofthestatediagram sothattheexponent ofJwillserveasacounting variable toindicate thenumberofbranches inanygivenpath fr~mnodeatonodee. Fortherate1/3convolutional rodeinourexample, thestatediagram that incorporates theadditional factorsofJandNisshowninFig.8-2-12. Thestateequations forthestatediagram showninFig.8-2-12are Xc=JND"X. +JNDX h Xh=JOX,+JDXd Xd=JND'X, +JND'X" X•.=JO'X h(8-2-4) Uponsolvingtheseequations fortheratioX./X.,weobtainthetransfer function T(D,N,J)=1_JND'(1+J) =J3N06+rN'D"+J5N'D'+J'N'OHJ +21"N3D'"+J'N30'"+... (8-2-5) Thisformforthetransfer functions givestheproperties ofallthepathsin 480 DIGITAL COMMUNICATIONS theconvolutional code.Thatis,thefirsttermintheexpansion ofT(D,N,J) indicates thatthedistance d=6pathisoflength3andofthethree information bits,oneisa1.Thesecondandthirdtermsintheexpansion of T(D,N,J)indicate thatofthetwod=8terms,oneisoflength4andthe secondhaslength5.Twoofthefourinformation bitsinthepathhavinglength 4andtwoofthefiveinformation bitsinthepathhavinglength5areIs.Thus, theexponent ofthefactorJindicates thelengthofthepaththatmergeswith theall-zeropathforthefirsttime,theexponent ofthefactorNindicates the numberofIsintheinformation sequence forthatpath,andtheexponent ofD indicates thedistance ofthesequence ofencoded bitsforthatpathfromthe all-zerosequence. ThefactorJisparticularly important ifwearetransmitting asequence of finiteduration, saymbits.Insuchacase,theconvolutional codeistruncated aftermnodesormbranches. Thisimpliesthatthetransfer function forthe truncated codeisobtained bytruncating T(D,N,J)atthetermJm.Onthe otherhand,ifwearetransmitting anextremely longsequence, i.e.,essentially aninfinite-length sequence, wemaywishtosuppress thedependence of T(D,N,J)ontheparameter J.Thisiseasilyaccomplished bysettingJ=1. Hence,fortheexample givenabove,wehave ND6 T(D,N,I)=T(D,N)= D21-2N =ND6+2N2D8+4N3DlO+.., oc =~adN(d-4)nDd d:6(8-2-6) wherethecoefficients {ad}aredefinedby(8-2-3). Theprocedure outlined abovefordetermining thetransferfunction ofa binaryconvolutional codeiseasilyextended tononbinary codes.Inthe following example, wedetermine thetransfer function ofthenonbinary convolutional codepreviously introduced inExample 8-2-3. Example 8-2-4 Theconvolutional codeshowninFig.B-2-l0hastheparameters K=2, k=2,n=4.Inthisexample, wehaveachoiceofhowwelabeldistances andcounterrors,depending onwhether wetreatthecodeasbinaryor nonbinary. Suppose wetreatthecodeasnonbinary. Thus,theinputtothe encoder andtheoutputaretreatedasquaternary symbols. Inparticular, if wetreattheinput and outputasquaternary symbols00,01,10,and11,the distance measured insymbols between thesequences 0111andססoois2. Furthermore, supposethataninputsymbol00isdecoded asthesymbol11; thenwehavemadeonesymbolerror.Thisconvention appliedtothe CHAPTER ilBLOCK ANDCONVOLUTIOSAL CHANNEL CODES481 )NV NJlY(12) NJD' 123/ NJDe.(1) Stated II (02)00=0 01=I 10=2 II=J III)JOe. IND' 121) JV' (22) en)JDe. INV FIGURE 8-2·]3 St.atediagramfmK=2,k=2,rate1/2nonbinarycode. convolutional codeshowninFig.8-2-10resultsinthestatediagram illustrated inFig.8-2-13,fromwhichweobtainthestateequations Xh=NJD'X"+NJDX h+NJDX,+NJD'X" X,=NJD'X"+NJD'X h+NJDX,+NJDX" X"=NJD'X"+NJDX h+NJD'X,+NJDX" X,=JD'(Xh+X,+X,,) Solutionoftheseequations leadstothetransferfunction 3NJ'D' T(D,N,J)=-t---2-=-N':":J D.::.....::-=-.-NJ-D~'(8-2-7) (8-2-8) Thisexpression forthetransferfunction isparticularly appropriate whenthe quaternary symbols attheoutputoftheencoder aremapped intoa 482 DIGITAL ("OMMl,Nj('AIIONS JSl}' (]()II)I\D~\ (i!;)()) INn' (IJ101 I State L Ii.""JD~((Jj(jfJ iN/)2"'" (10011 (I0lll) ({lOll)) JV1[) FIGURE H-2-14 Statelhagram torK-"2,k-=-2,ratl'112convolutional codewithoutputtreatedasitbinary .'>equence. corresponding setofquaternary waveforms Sm(t),m~1,2,3,4,e.g.,four orthogonal waveforms. Thus,thereisaone-to-one correspondence between codesymbols andsignalwaveforms. Alternatively, forexample, theoutputoftheencoder maybetransmitted asasequence ofbinarydigitsbymeansofbinaryPSK.Insuchacase,itis appropriate tomeasure distance intermsofbits.Whenthisconvention is employed, thestatediagram islabeledasshowninFig.8-2-14.Solution of thestateequations obtained fromthisstatediagram yieldsatransfer function thatisdifferent fromtheonegivenin(8-2-8). Someconvolutional codesexhibita<;haracteristic behavior thatiscalled catastrophic errorpropagation. Whenacodethathasthischaracteristic isused onabinarysymmetric channel, itispossible forafinitenumber ofchannel errorstocauseaninfinitenumberofdecoding errors.Suchacodecanbe identified fromitsstatediagram. Itwillcontain azero-distance path(apath withmultiplier D""CI)fromsomenonzero statebacktothesamestate.ThlS meansthatonecanlooparoundthiszero-distance pathaninfinitenumber (If timeswithout increasing thedistance relative totheall-zeropath.But.ifthIS self-loop corresponds tothetransmission ofaLthedecoder willmakean :nttnik number ofCrr,,)fS. Sincesuch CL)dl.'~are L'asil~,rccllgnilcd. theyan.' easilyaH)ickJ inpractice. 8-2-2Optimum Decoding ofConvolutional Codes-The ViterbiAlgorithm Inthedecoding ofablockcodeforamemoryless channel. weCl1mputcd the' distances (Hamming distance forhard-decision decoding andeuclidean di,· lancefursoft·decision decoding) between thereceived codcwordandthe~' possible transmitted codewords.Thenweselected thecodewordthatwas closestindistance tothereceived codeword.ThisdecJsion rule.whichre<jlllres thecllmputation of='m<'Tries. isoptimum inthesensethatIIresulhilla minimum prohdbihlV ofcrrurforthebinary,ymmetric channel withI';and theadditiv(' whitegaussian noisechannel. Unlikeablockcode.whichhasafixedlength /I.aconvolutional enl'oder IS basicallv afinite·state machine. Hencetheoptimum decoder isamaxImum· likelihood sequence estimator (MLSE) ofthetypedescribed inSectIon 5·1·-1 forsignalswithmemory. suchasNRZIandCPM.Therefurc, Dptimull1 decoding ofaconvolutiunal codeinvolves asearchthrough thetrellislorthc mostprobahle se<juence. Depending onwhether thedetector following the demodulator performs hardorsoftdecisions. thecorresponding metricinthe trellissearchmaybeeitheraHamming metricoraeuclIdean metric, respectively. Weelaborate below, using thetrellisinFig.8·2-5forthe convolutional codeshownInFig.8-2-2. Consider thetwopathsinthetrellisthatbegmattheinitialstatc{(and remerge atstateIIafterthreestatetranSItions (threebranches). corresponding tothetwo information sequences 000andJOOandthetransmilted sequences 000000 000 andIII001OIl,respectively. Wedenotethetransmitted bitsh} {cl''''j=I.2.3:III=I.2,3},wheretheindexjindicates thejthbranchandthe index 111themthbitinthatbranch,Correspondingly. wedefine{r,m'j=1.2.3: III=1,2,3}astheoutputofthedemodulator. Ifthedetector performs hard-decision decoding, itsoutputforeachtransmitted bitiseither0or1.On theotherhand,ifsoft-decision decoding isemployed andthecodedseguenc.: istransmitted bybinarycoherent PSK.theinputtothedecoder is (8-2·lJ) where Il)mrepresents theadditive noiseandt,.isthetransmitted signalenergy foreachcodehit. Ametricisdefinedforthejthbranchoftheithpaththrough thetrellisas thelogarithm ofthejointprobahility ofthesequence {r,,,,.III=1.2..'} 484 DJOlTAl COMMlTNJ('ATI()NS conditioned onthetransmiued sequence {cj~,m=I.2.3}fortheithpath. Thatis. J.Lj')=logP(YJIen.j=1.2,3.... (8-2-10) Furthermore. ametricfortheithpathconsisting ofBbranches through the trellisisdefinedas B PM(i)=2:J.Lji) i=I(8-2-11) (8-2-12)Thecriterion fordeciding between twopaths.throughthetrellisistoselect theonehavingthelargermetric.Thisrulemaximizes theprobability ofa correctdecision or.equivalently. itminimizes theprobability oferrorforthe sequence, ofinformation bits.Forexample, suppose thathard-decision decoding isperformed bythedemodulator, yielding thereceived sequence {JOI000lOO}.Leti=0denotethethree-branch all-zeropathandi=1the secondthree-branch paththatbeginsintheinitialstateaandremerges with theall-zeropathatstateaafterthreetransitions. Themetriesforthesetwo pathsare PMO)=6log(I-p)+3logP PM(I)=4log(1-p)+510gp wherepistheprobability ofabiterror.Assuming thatp<twefindthatthe metric PM(O)islargerthanthemetric PM(I).Thisresultisconsistent with theobservation thattheall-zeropathisatHamming distanced=3fromthe received sequence, whilethei=1pathisatHamming distance d=5fromthe received path.Thus,theHamming distance isanequivalent metricfor hard-decision decoding. Similarly, suppose thatsoft-decision decpding isemployed andthechannel addswhitegaussian noisetothesignal.Thenthedemodulator outputis described statistically bytheprobability densityfunction (I(i»__1_{[r,."-~(2cj~ -IW} prj", Cjm-YficITexp- 2cr (8-2-13) whereu2=~Noisthevariance oftheadditivegaussian noise.Ifweneglectthe termsthatarecommon toallbranchmetrics, thebranchmetricforthejth branchoftheithpathmaybeexpressed as (8-2-14) where,inourexample, n=3.Thusthecorrelation Metricsforthetwopaths underconsideration are 3 3 CM(O)=2:2:Tjm(2cj2!-1) j=l"..,=1 3 3 CMII)=2:2:Tjm(2cj~-1) j=l"..,=1(8-2-15) UIAI'Tl'l{ 8:IU.O('I( A""lDCONVOl.llllONAL CHAN~FL CODES485 Havingdefined thebranchmetTlCSandpathmetricscomputed bythe decoder. wenowconsider theuseoftheViterbialgorithm foroptimum decoding oftheconvolution allyencoded information sequence. Weconsider thetwopathsdescribed ahove.whichmergeatstateaafterthreetransitions. Notethatanyparticular paththroughthetrellisthatstemsfromthisnodewill addidentical termstothepathmetricsCMlf)andCMII).Consequently. if CMI'">CMII)atthemergednodeaafterthreetransitions CMlf)willcontinue tobelargerthanCMII)foranypaththatstemsfromnodea.Thismeansthat thepathcorresponding toCMIIIcanbediscarded fromfurtherconsideration. Thepathcorresponding tothemetricCMIII>isthesurvivor. Similarly. oneof thetwopathsthatmergeatstatebcanbeelminated onthebasisofthetwo corresponding metrics.Thisprocedure isrepeated atstatecandstated.Asa result.afterthefirstthreetransitions. therearefoursurviving paths.one terminating ateachstate.andacorresponding metricforeachsurvivor. This procedure isrepeated ateachstageofthetrellisasnewsignalsarereceived in subsequcnt timeintervals. Ingeneral. whenabinaryconvolutional codewithk=Iandconstraint lengthKisdecoded bymeansoftheViterbialgorithm, thereare2"-1states. Hence.thereare2"-Isurviving pathsateachstageand2"-1metrics,onefor eachsurviving path.Furthermore. abinaryconvolutional codeinwhichkbits atatimeareshiftedintoanencoder thatconsistsofK(k-bit)shift-register stagesgenerates atrellisthathas2kl"-I)states.Consequently, thedecoding of suchacodebymeansoftheViterbialgorithm requireskeepingtrackof2klKI) surviving pathsand2k(K-:Jmetrics.Ateachstageofthetrellis,thereare2k pathsthatmergeateachnode.Sinceeachpaththatconverges atacommon noderequires thecomputation ofametric.thereare2kmetricscomputed for eachnode.Ofthe2kpathsthatmergeateachnode,onlyonesurvives, andthis isthemost-probable (minimum-distance) path.Thusthenumberofcomputa­ tionsindecoding performed ateachstageincreases exponentially withkand K.Theexponential increase incomputational burdenlimitstheuseofthe Viterbialgorithm torelatively smallvaluesofKandk. Thedecoding delayindecoding alonginformation sequence thathasbeen convolutionally encoded isusuallytoolongformostpractical applications. Moreover. thememory required tostoretheentirelengthofsurviving sequences islargeandexpensive. Asindicated inSection5-1-4.asolution to thisproblem istomodifytheViterbialgorithm inawaywhichresultsinafixed decoding delaywithoutsignificantly affecting theoptimalperformance ofthe algorithm. Recallthatthemodification istoretainatanygiventimeIonlythe mostrecent8decoded information bits(symbols) ineachsurviving sequence. Aseachnewinformation bit(symbol) isreceived, afinaldecision is'madeon thebit(symbol) received 8branches backinthetrellis,bycomparing the metricsinthesurviving sequences anddeciding infavorofthebitinthe sequence havingthelargestmetric.If8ischosensufficiently large,allsurviving sequences willcontaintheidentical decoded bit(symbol) 8branches backin time.Thatis,withhighprobability, allsurviving sequences attimetstemfrom 486 DIGITAL COMMUNICATIONS thesamenodeatr-S.Ithasbeenfoundexperimentally (computer simula­ tion)thatadelayS;;"5Kresultsinanegligible degradation intheperformance relativetotheoptimum Viterbialgorithm. 8-2-3Probability ofErrorforSoft-Decision Decoding Thetopicofthissubsection istheerrorrateperformance oftheViterbi algorithm onanadditive whitegaussian noisechannel withsoft-decision decoding. Inderiving the probability oferrorforconvolutional codes,thelinearity property forthisclassofcodesisemployed tosimplify thederivation. Thatis. weassumethattheall-zerosequence istransmitted andwedetermine the probability oferrorindeciding infavorofanothersequence. Thecodedbinary digitsforthejthbranch oftheconvolutional code,denoted askl"" In=1,2,...•n}anddefinedinSection8-2-2,areassumed tobetransmitted by binaryPSK(orfour-phase PSK)anddetected coherently atthedemodulator. Theoutputofthedemodulator, whichistheinputtotheViterbidecoder, is thesequence {r;m.m=1,2•....n;j=1,2,...}whererjmisdefinedin(8-2-9). TheViterbisoft-decision decoder formsthebranchmetricsdefined by (8-2-14)andfromthesecomputes thepathmetrics B Bn CWi)=2:/-Lj')=2:Lrjm(2c~-1) j==! j=1m=l(8-2-16) whereidenotesanyone ofthecompeting pathsateachnodeandBisthe numberofbranches (information symbols) inapath.Forexample, theall-zero path,denoted asi=0,hasapathmetric Bn CM(O)=2:L:(-~+njm)(-I) j=lm=1 Bn =~Bn+L:2:njm j=lm==1(8-2-17) Sincetheconvolutional codedoesnotnecessarily haveafixedlength,we deriveitsperformance fromtheprobability oferrorforsequences thatmerge withtheall-zerosequence forthefirsttimeatagivennodeinthetrellis.In particular, wedefinethefirst-event errorprobability astheprobability that another paththatmergeswiththeall-zeropathatnodeBhasametricthat exceedsthemetricoftheall-zeropathforthefirsttime.Suppose theincorrect path,calliti=1,thatmergeswiththeall-zeropathdiffersfromtheall-zero pathindbits,i.e.,therearedIsinthepathi=1andtherestareOs.The probability oferrorinthepairwise comparison ofthemetricsCM(O)andCM(I) is P2(d)=P(CM(l);;" CM(O»=P(CM(') -CM(O);;"0) PM)=P[2~ln~Irjm(c}:.i-c)~!);;"0](8-2-18) (8-2-19)nl.-'\PTEJ~ X:BLOCK ANlJCONVOLUTIONAL CHANNEL CODES487 Sincethecodedbitsinthetwopathsareidentical exceptinthedpositions, (8-2-18)canbewritteninthesimplerform P2(d)=p(~r;~0) wheretheindex[runsoverthesetofdbitsinwhichthetwopathsdifferand theset{r;}represents theinputtothedecoder forthesedbits. The{r;}areindependent andidentically distributed gaussian random variables withmean-~andvariance!No.Consequently theprobability of errorinthepairwise comparison ofthesetwopathsthatdifferindbitsis P,(d)=Q(~) =Q(V2'YbRcd) (8-2-20) where Yh='t5b/Noisthereceived SNRperbitandRcisthecoderate. Although wehavederived thefirst-event errorprobability forapathof distance dfromtheall-zeropath,therearemanypossible pathswithdifferent distances thatmergewiththeall-zeropathatagivennodeB.Infact,the transfer functionT(D)provides acomplete description ofallthepossible pathsthatmergewiththeall-zeropathatnodeBandtheirdistances. Thuswe cansumtheerrorprobability in(8-2-20)overallpossible pathdistances. Upon performing thissummation, weobtainanupperboundonthefirst-event error probability intheform ti=drrce x ,;;;LadQ(V2Yb Rcd) (8-2-21) whereaddenotesthenumberofpathsofdistancedfromtheall-zeropaththat mergewiththeall-zeropathforthefirsttime. . Therearetworeasonswhy(8-2-21) isanupperboundonthefirst-event errorprobability. Oneisthattheeventsthatre,ultintheerrorprobabilities {P2(d)}arenotdisjoint. Thiscanbeseenfromobservation ofthetrellis. Second,bysumming overallpossible d~d'n",wehaveimplicitly assumed that theconvolutional codehasinfinitelength.Ifthecodeistruncated periodically afterBnodes,theupperboundin(8-2-21)canbeimproved bysumming the erroreventsford'"e,;;;d.,B.Thisrefinement hassomemeritindetermining the performance ofshortconvolutional codes,buttheeffectonperformance is negligible whenBislarge. Theupperboundm(8-2-21)canbeexpressed inaslightlydifferent formif theQfunction isupper-bounded byanexponential. Thatis. (8-2-22) 488 DIGITAL COMMUNICATIONS Ifweuse(8-2-22) in(8·2-21), theupperboundonthefirst-event error probability canbeexpressed as (8-2-23) Although thefirst-event errorprobability provides ameasure ofthe performance ofaconvolutional code,amoreusefulmeasureofperformance is thebiterrorprobability. Thisprobability canbeupper-bounded bythe procedure usedinbounding thefirst-eventerrorprobability_Specifically, we knowthatwhenanincorrect pathisselected, theinformation bitsinwhichthe selectedpathdiffersfromthecorrectpathwillbedecoded incorrectly. Wealso knowthattheexponents inthefactorNcontained inthetransferfunction T(D,N)indicate thenumber ofinformation biterrors(number ofIs)in selecting anincorrect paththatmergeswiththeall-zeropathatsomenodeB. Ifwemultiply thepairwiseerrorprobability P2(d)bythenumberofincorrectly decoded information bitsfortheincorrect pathatthenodewheretheymerge, weobtainthebiterrorrateforthatpath.Theaveragebiterrotprobability is upper-bounded bymultiplying eachpairwise errorprobability P2(d)bythe corresponding number ofincorrectly decoded information bits,foreach possibleincorrect paththatmergeswiththecorrectpathattheBthnode,and summing overalld., Theappropriate multiplication factorscorresponding tothenumber of information biterrorsforeachincorrectly selected pathmaybeobtained by differentiating T(D,N)withrespecttoNIngeneral, T(D,N)canbe expressed as ~ T(D,N)=2:adDdNf(d) d=dfrec(8-2-24) wherefed)denotestheexponent ofNasafunction ofd.Takingthederivative ofT(D,N)withrespecttoNandsettingN=1,weobtain (8-2-25) where{3d=adf(d).Thusthebiterrorprobability fork=1isupper-bounded by ~ Pb<2:{3dP2(d) d=dfrcc ~ <2:{3dQ('V2Yb Rcd) d=drree(8-2-26) CHAPTER 8:BLOCKANDCONVOLUTIONAL CHANNEL CODES489 IftheQfunction isupper-bounded byanexponential asindicated in(8-2-22) then(8-2-26)canbeexpressed inthesimpleform Ph<d~,_l3dDdL~._Yb'" <dT~N)IN_I.D~'_"" (8-2-27) Ifk>1,theequivalent biterrorprobability isobtained bydividing (8-2-26) and(8-2-27)byk. Theexpressions fortheprobability oferrorgivenabovearebasedonthe assumption thatthecodebitsaretransmitted bybinarycoherent PSK.The resultsalsoholdforfour-phase coherent PSK,sincethismodulation} demodulation technique isequivalent totwoindependent (phase-quadrature) binaryPSKsystems. Othermodulation anddemodulation techniques, suchas coherent andnoncoherent binaryFSK,canbeaccommodated byrecomputing thepairwiseerrorprobability P,(d).Thatis,achangeintilemodulation and demodulation technique usedtotransmit thecodedinformation sequence affectsonlythecomputation ofP2(d).Otherwise, thederivation forPhremains thesame. Althougll theabovederivation oftileerrorprobability forViterbidecoding ofaconvolutional codeappliestobinaryconvolutional codes,itisrelatively easytog<;neralize ittononbinary convolutional codesinwhicheachnonbinary symbolismappedintoadistinctwaveform. Inparticular, thecoefficients {,Bd} intheexpansion ofthederivative ofT(D,N),givenin(8-2-25), represent the numberofsymbolerrorsintwo paths separated indistance(measured interms ofsymbols) bydsymbols. Again,wedenotetheprobability oferrorina pairwisecomparison oftwopathsthatareseparated indistance bydasP2(d). Thenthesymbolerrorprobability, forak-bitsymbol,isupper-bounded by " PM';;2:I3dP2(d) d=d,n:c Thesymbolerrorprobability canbeconverted intoanequivalent biterror probability. Forexample, if2'orthogonal waveforms areusedtotransmit the k-bitsymbols, theequivalent biterrorprobability isPMmultiplied byafactor 2'-'/(2'-I),asshowninChapter 5. 8-2-4Probability ofEnorforHard-Decision Decoding Wenowconsider theperformance acllieved bytheViterbidecoding algorithm onabinarysymmetric channel. Forhard-decision decoding oftheconvolu­ tionalcode,themetricsintheViterbialgorithm aretheHamming distances between thereceived sequence andthe2'(K-1lsurviving sequences ateach nodeofthetrellis. Asinourtreatment ofsoft-decision decoding, webeginbydetermining the 490 DIGITAL COMMl'r'\I[CATIONS first-event errorprobability. Theall-zeropathisassumed tobetransmitted. Suppose thatthepathbeingcompared withtheall-zeropathatsomenodeB hasdistance dfromtheall-zeropath.Ifdisodd,theall-zeropathwillbe correctly selected ifthenumberoferrorsinthereceived sequence islessthan ~(d+I);otherwise, theincorrect pathwillbeselected. Consequently, the probability ofselecting theincorrect pathis (8-2-28) wherepistheprobability ofabiterrorforthebinarysymmetric channel.Ifd iseven,theincorrect pathisselected whenthenumberoferrorsexceeds ~d.If thenumberoferrorsequals ~d,thereisatiebetween themetricsinthetwo paths,whichmayberesolved byrandomly selecting oneofthepaths:thus,an erroroccurshalfthetime.Consequently, theprobability ofselecting the incorrect pathis "d) d) P2(d)=L(p'(1-P)"-k+~Lpdf2(1_P)"12 "~dI2+1,k \2d(8-2-29) Asindicated inSection8-2-3,therearemanypossible pathswithdifferent distances thatmergewiththeall-zeropathatagivennode.Therefore, thereis nosimpleexactexpression forthefirst-event errorprobability. However, we canoverbound thiserrorprobability bythesumofthepairwise error probabilities P2(d)overallpossible pathsthatmergewiththeall-zeropathat thegivennode.Thus,weobtaintheunionbound x P,<La"P,(d) (8-2-30) wherethecoefficients {a,,}represent thenumberofpathscorresponding tothe setofdistances {d}.Thesecoefficients arethecoefficients intheexpansion of thetransferfunctionT(D)orT(D,N). Insteadofusingtheexpressions forP2(d)givenin(8-2-28) and(8-2-29), we c'lnusetheupperbound P,(d)<[4p(l-p)]d/2 (8-2-31) whichwasgiveninSection8-1-5.Useofthisboundin(8-2-30)yieldsalooser upperboundonthefirst-event errorprobability, intheform x p,.<LQ,,[4p(1-p)]"/2 d-=~J'rLC' (8-2-32) CHAPTER R:BLOCKANDCONVOLUTIONAL CHANNEL CODES491 Letusnowdetermine theprobability ofabiterror.Asin'thecaseof soft-decision decoding, wemakeuseofthefactthattheexponents inthe factorsofNthatappearinthetransfer functionT(D,N)indicate thenumber ofnonzero information bitsthatareinerrorwhenanincorrect pathisselected mertheall-zeropath.Bydifferentiating T(D,N)withrespecttoNandsetting N=I,theexponents ofNbecomemultiplication factorsofthecorresponding error-event probabilities P2(d).Thus,weobtaintheexpression fortheupper boundonthebiterrorprobability. intheform PI><2:(3"P2(d) J=d ffC>;(8-2-33) wherethe{{3,,}arethecoefficients intheexpansion ofthederivative of T(D,N),evaluated atN=l.ForP2(d),wemayuseeithertheexpressions givenin(8-2-28) and(8-2-29)ortheupperboundin(8-2-31).Ifthelatteris used,theupperboundonPhcanbeexpressed as dT(D,N)' Ph<I (8-2-34)dlv ,'\/c-"I.D='v'4p(]-p) Whenk>1,theresultsgivenin(8-2-33) and(8-2·34) forPhshouldbedivided byk. Acomparison oftheerrorprobability fortherate1/3,K=3convolutIOnal codewithsoft-decision decoding andhard-decision decoding ismadeinFig. 8-2-15.NotethattheChernoff upperboundgivenby(8-2-34) islessthanIdB abovethetighterupperboundgivenby(8-2-33) inconjunction with(8-2-28) and(8-2-29). Theadvantage oftheChernoff boundisitscomputational ..j {)X10I~11 S:'\Hperbi:.-1!,(dBI, , j 2: \"I , \ \Chernoff boundi 5"(8'~'J4' "I ,, 2,t-1',, 1\,Uppelboundt 5 I\-=-(8-2-33),,with (S-::·2'~))7 UpperbOllnd\,, ~\and 1~-2·::8): 18-2-261,, , ! ' , 2+-T--f--_.-~()fJ-c~'l"I~jdn\ j~ decoding d ,--Hard-dc-cisiotl \t± ~CCOd',~ \' ,IO-to10-lQ-10- Co."g v10­ E. ~o > to ~ '" FIGURE 8~2~15 Comparison ofsoft-decision andhard-decisIOn decoding {orK=3,k=:1.n=3convolutional code 492 DIGITAL COMMUNICATCONS simplicity. Incomparing theperformance between soft-decision andhard­ decision decoding, notethatthedifference obtained fromtheupperboundsis approximately 2.5dBfor10-6",Pb'"10-2 . Finally,weshouldmention thattheensemble average errorrateperfor­ manceofaconvolutional codeonadiscretememorylesschannel, justasinthe caseofablockcode,canbeexpressed intermsofthecutoffrateparameter Ril as(forthederivation, seeViterbiandOmura,1979). _(q_1)q-KR,{R, Ph<[1-q(R"R,)fR'l2' whereqisthenumberofchannelinputsymbols, Kistheconstraint lengthof thecode,R,isthecoderate,andRoisthecutoffratedefinedinSections 7-2 and8-1.Therefore, conclusions reachedbycomputing Roforvariouschannel conditions applytobothblockcodesandconvolutional codes. 8-2-5Distance Properties ofBinal'YConvolutional Codes Inthissubsection, weshalltabulate theminimum freedistance andthe generators forseveralbinary,short-constraint-length convolutional codesfor severalcoderates. These binary codes areoptimalinthesensethat,foragiven rateandagivenconstraint length,theyhavethelargestpossible dc«c.The generators andthecorresponding valuesofdc«etabulated belowhavebeen obtained byOdenwalder (1970),Larsen(1973),Paaske(1974),andDautetaf. (1982)usingcomputer searchmethods. Heller(1968)hasderivedarelatively simpleupperboundontheminimum freedistanceofaratelinconvolutional code.Itisgivenby l2f-IJ dC,ee,;;;min-,--(K+i-I)n''''I2 - 1(8-2-35) whereLxJdenotes thelargestintegercontained inx.Forpurposes of comparison, thisupperboundisalsogiveninthetablesfortheratelincodes. ForratekInconvolutional codes,Dautetai.(1982)hasgivenamodification to Heller'sbound.Thevaluesobtained fromthisupperboundforkincodesare alsotabulated. Tables8-2-1to8-2-7listtheparameter ofratelinconvolutional codesfor n=2,3,...,8.Tables8-2-8to8-2-11listtheparameters ofseveralratekIn convolutional codesfork'"4andn'"8. 8-2-6Nonbinary Dual-kCodesandConcatenated Codes Ourtreatment ofconvolutional codesthusfarhasbeenfocusedprimarily on binarycodes.Binarycodesareparticularly suitable forchannels inwhich binaryorquaternary PSKmodulation andcoherent demodulation ispossible. CHAPTER" BLOCK ANDCONVOLUTIONAL CHANNEL CODES493 TABLE 8-2-1RATE1f2MAXIMUM FREEDISTANCE CODE ConstnWtt Upperbound lengthKGeneraton inoctal d.... Oild.... 3 5 7 5 5 4 15 17 6 6 5 23 35 7 8 6 53 75 8 8 7 133 171 10 10 8 247 371 10 11 9 561 753 12 12 10 1,167 1,545 12 13 11 2,335 3,661 14 14 12 4,335 5,723 15 15 13 10,533 17,661 16 16 14 21,675 27,123 16 17 So-urct:Odenwalder (1970)andLarsen(1973). However, therearemanyapplications inwhichPSKmodulation andcoherent demodulation isnotsuitableorpossible. Insuchcases,othermodulation techniques, e.g.,M-aryFSK,areemployed inconjunction withnoncoherent demodulation, Nonbinary codesareparticularly matched toM-arysignalsthat aredemodulated noncoherently. Inthissubsection, wedescribe aclassofnonbinary convolutional codes, calleddual-kcodes,thatareeasilydecoded bymeansoftheViterbialgorithm usingeithersoft-decision orhard-decision decoding. Theyarealsosuitable eitherasanoutercodeorasaninnercodeinaconcatenated code,aswillalso bedescribed below, TABLESoHRATE1/3MAXIMUM FREEDISTANCE CODES ConSlraint Upperbound lengthK Generators inoctal d.... ODdrr.,e 3 5 7 7 8 8 4 13 15 17 10 10 5 25 33 37 12 12 6 47 53 75 13 13 7 133 145 175 15 15 8 225 331 367 16 16 9 557 663 711 18 18 10 1,117 1,365 1,633 20 20 11 2,353 2,671 3,175 22 22 12 4,767 5,723 6,265 24 24 13 10,533 10,675 17,661 24 24 14 21,645 35,661 37,133 26 26 SOI.rces: Odcnwalder (197mandLarsen(1973). 494 DlCiITAL COMMUf\;!CATlON$ TABLE 8-2·3RATE1/4MAXIMUM FREEDISTANCE CODES Constraint Upperbound lengthK Generators inoctal d,,_ ondrree 3 5 7 7 7 10 10 4 13 15 15 17 13 15 5 25 27 33 37 16 16 n 53 67 71 75 18 18 " 135 135 147 163 20 20 , 8 235 275 313 357 22 22 9 463 535 733 745 24 24 10 UJ7 1,365 1,1i33 1.653 27 27 11 2,387 2.353 2,671 3.175 29 29 12 4.767 5,723 6.265 7,455 32 32 13 11.145 12,477 15,537 16,727 33 33 14 21.113 23,175 35.527 35.537 36 36 SOUT('{'- LarseniJ971l. TABLE 8-2-4RAlE1/5MAXIMUM FREEDISTANCE CODES Constraint Upperbound lengthK Generators inoctal d.... ondtree 3 7 7 7 5 5 13 13 4 ~7 17 13 15 15 16 In 5 37 27 33 25 35 20 20 6 75 71 73 65 57 22 22 7 175 131 135 135 147 25 25 8 257 233 323 271 357 28 28 Source. Dau:('(al.(19R2). TABLE 8-2-5 RATE1/6MAXIMUM FREEDISTANCE CODES Cttnstraint Upperbound lengthK Generators inoctal drree ondrne 7 7 7 16 16 7 :) 5 4 17 17 13 cO 20 13 IS 15 37 .\5 27 24 24 33 25 35 6 73 75 55 27 27 65 J7 57 7 173 151 135 30 30 135 1n3 137 8 2j3 375.331 34 34 235 313 357 Source- Daul('(at.(14X2). • CHAPTER f(:BLOCK ANDCONVOLUTIONAL CHANNEL ('ODES495 TABLEIl-Z-6RATE1/7MAXIMUM FREEDISTANCE CODES COIUIrMII UppeI''"""'"Ie"""KG_on iII~ 4-011d_ 3 777718 18 55 5 4 17 17 13 13 23 23 13 IS IS 5 35272527 28 28 3335 37 6 537565 75 32 32 476757 7 165 145 173 135 36 36 135 147137 B 275 253 375 331 40 40 235313357 SOliTa:Dautrtal.(1982). TABLEIl-Z·7RATE1/8MAXIMUM FREEDISTANCE CODES IIC-u.l Upperboaad 1eJIIl~K GeD....onilloctal d......d.... 7I 3 7 5 5 21 21 j 5777 4 17 17 13 1326 26 13 15 15 17 5 3733 25 25 32 32 353327 37 6 5773 51 65 36 36 754767 57 7 153III165 173 40 40 135 135 147137 8 275 275253 371 45 45 331235313 357 SOUTee:Daulttal.(1982). TABLElI-URATE213MAXIMUM FREEDISTANCE CODES ConstrUal Upper00..... IeJIIlhK Geaeralors iDodlIId_ oDd.... 2 17 06 15 3 4 3 27 7572 5 6 4 236 155337 7 7 So"ree:Duatttal.(1982). 496 DIGITAL COMMUNICATIONS TABLE 8-1-9RATEk/5MAXIMUM FREEDISTANCE CODES CODStniJll UpperboUDd Rel. le.llbK Genenton IDoctal d".. ODdfrft 2/5 2 170711 1204 6 6 3 2771526557 10 10 4 247366171266373 12 12 3/5 2 352375 6147 5 5 4/5 2 237274156255337 3 4 SOllrce:Daute/al.(1982). TAILE11-1-10 RATEk/7MAXIMUM FREEDISTANCE CODES Conslr8ln1 Upperbound Rate lengthK Ge••ntorsinoctal d.... ond_ 2/7 2 05 06 12 15 9 9 15 13 17 3 33 55 72 47 14 14 25 53 75 4 312 125 247 366 18 18 171 266 373 3/7 2 45 21 36 62 8 8 57 43 71 4/7 2 130 067 237 274 6 7 156 255 337 SOl4rce:Dautet01.(1982). TABLE 11-1-11 RATES 3/4AND3/8MAXIMUM fREEDISTANCE CODES Conslr8ln1 Upperbound Ral. lengthK Generatol'5 iDoctal d.... ond.... 3/4 2 1325 6147 4 4 3/8 2 15422361 8 8 51367547 SO"Tce:Dautetill.(1982). Adual-krate1/2convolutional encoder mayberepresented asshownin Fig.8-2-16.Itconsistsoftwo(K=2)k·bitshift-register stagesandn=2k funclion generators. Itsoutputistwok-bitsymbols. Wenotethatthecode considered inExample 8-2-3isadual-2convolutional code. CHAPTER K:BLOCK ANDCONVOLUTIONAL CHANNEL CODES497 Funclion gCneralors !,:1;":11';":'.~..;.'..~=-..:-_ k 1"1'"' k·bits l~F~IlAC~'~i<m~"f;~=: ~ generators 14.+1'···'a FIGURE 8-2-16 Encoder forrate1/2dual-kcodes, The2kfunction generators forthedual-kcodeshavebeengivenbyViterbi andJacobs(1975).Thesemaybeexpressed intheform 100o100 o0 1 0... oo o o o1I.l ..o}I. o0 1 0 o1 0o1 o0 o0 o1 o0 0 0 0, o o.._--_ ----_. o1 .., 0 0[:~::]=[~~~ +--Ik..... 0 0 0 +--Ik+'..... 1 1 0 0 +-gk+2..... 0 0 1 0 o0 0 1 000 100 1 1 0 0 o0 1 0 o0 0 1 0---- ~...-_.--_..000 100 (8-2-36) T(D.N.J)=1_NJ[2D+(2'_3)D2)whereI.denotesthekxkidentitymatrix. Thegeneralformforthetransfer function ofarate1/2dual-kcodehas beenderivedbyOdenwalder (1976).Itisexpressed as (2k-1)D4J2N .=2:QiDiNfU)JhU) ;=4(8-2-37) 498 DIGITAL COMMUNICATIONS whereDrepresents theHamming distance fortheq-ary(q=Zk)symbols, the t(i)exponent onNrepresents thenumberofinformation symbolerrorsthat areproduced inselecting abranchinthetreeortrellisotherthana corresponding branchontheall-zeropath,andtheh(i)exponent onJisequal tothenumberofbranches inagivenpath.Notethattheminimum free distance isdr,ee=4symbols (4kbits). Lower-rate dual-kconvolutional codescanbegenerated inanumberof ways,thesimplest ofwhichistorepeateachsymbolgenerated bytherate1/2 codertimes,wherer=1,Z,...,m(,=1corresponds toeachsymbol appearing once).Ifeachsymbolinanyparticular branchofthetreeortrellis orstatediagram isrepeated ,times,theeffectistoincrease thedistance parameter fromDtoD'.Consequently thetransferfunction forarateI/Z, dual-kcodeis (8-2-38)(2'-I)D4'JW T(D,N,J)=1_NJ[ZD'+(Z'-3)D2'J 11-2D'-(Z'-3)D2']'Inthetransmission oflonginformation sequences, thepathlengthpala­ meterJinthetransfer function maybesuppressed bysetting j=1.The resulting transfer functionT(D,N)maybedifferentiated withrespecttoN, andNissettounity.Thisyields dT~~Nl~1 (8-2-39) ';=4... where13,represents thenumberofsymbolerrorsassociated withapathhaving distlince D'fromtheall-zeropath,asdescribed previously inSection 8-Z-3. Theexpression in(8-2-39)maybeusedtoevaluate theerrorprobability for dual-kcodesundervariouschannelconditions, Performance ofDual-kCodeswithM·ary Modulation Suppose thata dual-kcodeisusedinconjunction withM-aryorthogonal signaling atthe modulator, whereM=2'.Eachsymbolfromtheencoderismappedintoone oftheMpossible orthogonal waveforms. Thechannelisassumed toaddwhite gaussian noise.Thedemodulator consistsofMmatched filters. Ifthedecoder performs hard-decision decoding, theperformance ofthe codeisdetermined bythesymbolerrorprobability PM'Thiserrorprobability hasbeencomputed inChapter5forbothcoherent andnoncoherent detection. FromPM'wecandetermine P,(d)according to(8-2-Z8)or(8-Z-29), whichis theprobability oferrorinapairwise comparison oftheall-zeropathwitha paththatdiffersindsymbols. Theprobability ofabiterrorisupper-bounded as 2h-1 0:. Pb<Z._ 1d~'(3dP,(d) (8-2-40) ThefactorZ'-I/(2'-1)isusedtoconvertthesymbolerrorprobability tothe biterrorprobability. CHA.PTER 11:BLOCK ANDCONVOLUTIONAL CHANNEL. CODES499 Insteadofhard-decision decoding, suppose thatthedecoder performs soft-decision decoding usingtheoutputofademodulator thatemploys a square-law detector. Theexpression forthebiterrorprobability givenby (8-2-40)stillapplies,butnowP2(d)isgivenby(seeSection12-1-1) I d-I P,(d)= 2d-}exp(-hbRc d)LKiO-r.Rcd) (8-2-41)2 ,~O whereIJ-I-i(2d-I)K=-L,i!,~OI andK.=lilTisthecoderate.Thisexpression followsfromtheresult(8-1-63). Concatenated CodesInSection8-1-8,weconsidered theconcatenation of twoblockcodestoformalongblockcode.Nowthatwehavedescribed convolutional codes,webroaden ourviewpoint andconsider theconcatenation ofablockcodewithaconvolutional codeortheconcatenation oftwo convolutional codes. Asdescribed previously, theoutercodeisusuallychosentobenonbinary, witheachsymbolselected fromanalphabet ofq=2ksymbols. Thiscodemay beablockcode,suchasaReed-Solomon code,oraconvolutional code,such asadual-kcode.Theinnercodemaybeeitherbinaryornonbinary, andeither abloc,koraconvolutional code.Forexample, aReed-Solomon codemaybe selectedastheoutercodeandadual-kcodemaybeselectedas theinnercode. Insuchaconcatenation scheme, thenumber ofsymbols intheouter (Reed-Solomon) codeqequals2"sothateachsymboloftheoutercodemaps intoak-bitsymboloftheinnerdual-kcode.M-aryorthogonal signalsmaybe usedtotransmit thesymbols. Thedecoding ofsuchconcatenated codesmayalsotakeavarietyof different forms.Iftheinnercodeisaconvolutional codehavingashort constraint length,theViterbialgorithm provides anefficient meansfor decoding, usingeithersoft-decision orhard-decision decoding. Iftheinnercodeisablockcode,andthedecoder forthiscodeperforms soft-decision decoding, theouterdecoder mayalsoperform soft-decision decoding usingasinputsthemetricscorresponding toeachwordoftheinner code.Ontheotherhand,theinnerdecodermaymakeaharddecision after receiptofthecodewordandfeedtheharddecisions totheouterdecoder. Thentheouterdecodermustperform hard-decision decoding. Thefollowing example describes aconcatenation codeinwhichtheouter codeisaconvolutional codeandtheinnercodeisablockcode. Example 8-2-5 Suppose weconstruct aconcatenated codebyselecting adual-kcodeas the outercodeandaHada~ard codeastheinnercode.Tobespecific, weselect arate1/2dual-5codeandaHadamard (16,5)innercode.Thedual-Srate 500 DIGITAL COMMUNICATIONS 1/2codehasaminimum freedistanceD',ee=4andtheHadamard codehas aminimum distancedmin=8.Hence,theconcatenated codehasaneffective minimum distance of32.Sincethereare32codewordsintheHadamard codeand32possiblesymbols intheoutercode,ineffect,eachsymbolfrom theoutercodeismappedintooneofthe32Hadamard codewords. Theprobability ofasymbolerrorindecoding theinnercodemaybe determined fromtheresultsoftheperformance ofblockcodesgivenin Sections 8-1-4and8-1-5forsoft-decision and hard-decision decoding, respectively. First,supposethathard-decision decoding isperformed inthe innerdecoder withtheprobability ofacodeword(symbolofoutercode) errordenoted asP32'sinceM=32.Thentheperformance oftheoutercode and,hence,theperformance oftheconcatenated codeisobtained byusing thiserrorprobability inconjunction withthetransferfunction forthedual-5 codegivenby(8-2-37). Ontheotherhand,ifsoft-decision decoding isusedonboththeouter andtheinnercodes,thesoft-decision metricfromeachreceived Hadamard codewordispassedtotheViterbialgorithm, whichcomputes the accumulated metricsforthecompeting pathsthroughthetrellis.Weshall givenumerical resultsontheperformance ofconcatenated codesofthis typeinourdiscussion ofcodingforRayleigh fadingchannels. 8-2-7OtherDecoding Algorithms forConvolutional Codes TheViterbialgorithm described inSection8-2-2istheoptimum decoding algorithm (inthesenseofmaximum-likelihood decoding oftheentire sequence) forconvolutional codes.However, itrequires thecomputation of 2kKmetricsateachnodeofthetrellisandthestorageof2k(K-I)metricsand 2klK-1)surviving sequences, eachofwhichmaybeabout5kKbitslong.The computational burdenandthestoragerequired toimplement theViterbi algorithm makeitimpractical forconvolutional codeswithlargeconstraint length. Priortothediscovery oftheoptimum algorithm byViterbi,anumberof otheralgorithms hadbeenproposed fordecoding convolutional codes.The earliestwasthesequential decoding algorithm originally proposed byWozen­ craft(1957,1961),andsubsequently modified byFano(1963). TheFanosequential decoding algorithm searches forthemostprobable paththroughthetreeortrellisbyexamining onepathatatime.Theincrement addedtothemetricalongeachbranchisproportional totheprobability ofthe received signalforthatbranch,justasinViterbidecoding, withtheexception thatanadditional negative constant isaddedtoeachbranchmetric.Thevalue ofthisconstant isselected suchthatthemetricforthecorrectpathwill increaseontheaverage, whilethemetric.foranyincorrect pathwilldecrease ontheaverage. Bycomparing themetricofacandidate pathwithamoving (increasing) threshold, Fano'salgorithm detectsanddiscardsincorrect paths. Tobemorespecific,letusconsider amemoryless channel. Themetricfor CHAPTER~, BLOCKANDCONVOLUTIONAL CHANNEL CODES501 theithpaththroughthetreeortrellisfromthefirstbranchtobranchBmaybe expressed as whereB" eMU)=LLiLj~: j=I",=1 nl.1,(1)\ (il_1~-J(iLim-ogz( )prim(8-2-42) (8-2-43) (8-2-44) (8-2-45)In(8-2-43), 'imisthedemodulator outputsequence, P(1jmIcj~)denotesthepdf of'1mconditional onthecodebitcj:':forthemthbitofthejthbranchoftheith path,andJ(isapositiveconstant. J(isselected asindicated abovesothatthe incorrect pathswillhaveadecreasing metricwhilethecorrectpathwillhave anincreasing metricontheaverage. NotethatthetermP('i"')inthe denominator isindependent ofthecodesequence, and,hence,maybe subsumed intheconstant factor. Themetricgivenby(8-2-43) isgenerally applicable foreitherhard-or soft-decision decoding. However, itcanbeconsiderably simplified when hard-decision decoding isemployed. Specifically, ifwehaveaBSCwith transition (error)probability P,themetricforeachreceived bit,consistent with theformin(8-2-43) isgivenby (i)={IOgz[2(1-p)]-Rcifrjm=cJ.~ iL1mI 2 R 'f- (i)ogzP-c 1rim#-elm where rImisthehard-decision outputfromthedemodulator andcj:':isthemth codebitinthejthbranchoftheithpathinthetreeandRcisthecoderate. Notethatthismetricrequires some(approximate) knowledge oftheerror probability. Example 8-2-6 Suppose wehavearateRc=1/3binaryconvolutional codefortransmitting information overaBSCwithP=0.1.Byevaluating (8-2-44)wefindthat {052'f- - (I)(r)= . 1rim-Cjm IJ.lm _265'f--'(i).IT1m.,-elm Tosimplifythecomputations, themetricin(8-2·45)maybenormalized. Itis wellapproximated as {1'f- - (i) (i)= IT,.",-Cjm iLlm-5"f--J-(i)1',m"""Cjm(8-2-46) Sincethecoderateis1/3,therearethreeoutputbitsfromtheencoder for eachinputbit.Hence,thebranchmetricconsistent with(8-2-46)is iLj')=3 -6d 502DIGITAL COMMUNICA nONs 8, 7, :g6t ~~, ~4,F3, " _ 2t------------------~---------- FIGURE 8-2-17 Anexample ofthepath..archinsequential ,-- -- - - -- - - - -- - - - - -- - - -- - -- - - - - -- decoding. [FromJordon(l9ll6).©1966 0 2 4 6 8 1012141618 IEEE.I Symbolnumber or,equivalently, (8-2-47) wheredistheHamming distanceofthethreereceived bitsfromthethree branchbits.Thus,themetricJLJ')issimplyrelatedtotheHamming distance ofthereceived bitstothecodebitsinIhejlhbranchoftheithpath. Initially, thedecoder maybeforcedtostartonIhecorrectpathbythe transmission ofafewknownbitsofdata.Thenitproceeds forwardfromnode tonode,takingthemostprobable (largestmetric)branchateachnodeand increasing thethreshold suchthatthethreshold isnevermorethansome preselected value,sayT,belowthemetric.Nowsupposethattheadditivenoise (forsoft-decision decoding) ordemodulation errorsresulting fromnoiseonthe channel(forhard-decision decoding) causethedecoder totakeanincorrect pathbecauseitappearsmoreprobable thanthecorrectpath.Thisisillustrated inFig.8-2-17.Sincethemetricsofanincorrect pathdecrease ontheaverage, themetricwillfallbelowthecurrentthreshold, sayTo.Whenthisoccurs,the decoder backsupandtakesalternative pathsthrough thetreeortrellis,in orderofdecreasing branchmetrics,inanattempttofindanotherpaththat exceedsthethreshold To.Ifitissuccessful infindinganalternative path,it continues alongthatpath,alwaysselecting themostprobable branchateach node.Ontheotherhand,ifnopathexiststhatexceedsthethreshold To,the threshold isreducedbyanamountrandtheoriginalpathisretraced.Ifthe originalpathdoesnotstayabovethenewthreshold, thedecoder resumesits backward searchforotherpaths.Thisprocedure isrepeated, withthe threshold reducedbyTforeachrepetition, untilthedecoderfindsapaththat remains abovetheadjusted threshold. Asimplified flowdiagram ofFano's algorithm isshowninFig.8-2-18. Thesequential decoding algorithm requiresabuffermemory inthedecoder tostoreincoming demodulated dataduringperiodswhenthedecoder is searching foralternate paths.Whenasearchterminates, thedecodermustbe capableofprocessing demodulated bitssufficiently fasttoemptythebuffer priortocommencing anewsearch.Occasionally, duringextremely long searches, thebuffermayoverflow. Thiscauseslossofdata,acondition that CHAPTER Il:BLOCK ANDCONVOLUTIONAL CHANNEL rODES503 FIGURE 8·2·18 Asimplified flowdiagramof Fano'salgorithm. [FromJordan (/966),©19661£££.1Start Ll= 0 r=() rLowerthre<J1old b}rf ----------------------,Fail ,, Test, Fall , Test mostlikelytFilii, previous nooe branch ,, 1Pass Test, Pass , this, Paso;, Stepba(.'k branch,,L , ,-----------------•____ J CStepforward')Istherea nex.t !Nol...- mostlikely branch·.' NoFirsttime al(his Yes node? 1Yes CTightenthreshold) I canberemedied byretransmission ofthelostinformation. Inthisregard,we shouldmention thatthecutoffrateRohasspecialmeaning insequential decoding. Itistherateabovewhichtheaverage number ofdecoding operations perdecoded digitbecomes infinite, anditistermed the computational cutoffrateRcomp•Inpractice, sequential decoders usually operateatratesnearRo. TheFanosequential decoding algorithm hasbeensuccessfully implemented inseveralcommunication systems. Itserrorrateperformance iscomparable to thatofViterbidecoding. However, incomparison withViterbidecoding, sequential decoding hasasignificantly largerdecoding delay.Onthepositive side,sequential decoding requires lessstoragethanViterbidecoc':ng and, hence,itappears attractive forconvolutional codeswithalargeconstraint length.Theissuesofcomputational complexity andstoragerequirements for sequential decoding areinteresting andhavebeenthoroughly investigaled. For ananalysisofthesetopicsandothercharacteristics oftheFannalgorithm, the interested readermayrefertoGallager (1968),Wozencraft andJacobs(1965), Savage(1966),andForney(1974). Another typeofsequential decoding algorithm, calleda.I'/tIck1I!!i0rithm. has beenproposed independently byJelinek(1969)andZigangirov (1966). In contrast totheViterbialgorithm, whichkeepsIrackof2"I)kpathsand S04 DIGITAL COMMUNICATIONS 000 ode uk=I -2J ngdistance-8 000001 -3d III-7 0 , 010 -2110r-2 -I,J101 , -4 I001Rale1/3c III-4Blanchmeb 010 -I110r-3d=Hammi -2"101 -5 Received sequence: WI III llO Oll Stackwithaccumulated pathmetrics. flGURE 8-2·19 Aexample oftheslackalgorithm fordecodiog arale1/3 convolutional code.Step Step Step Step Step Step a b, d £f -I -2 -3 -2-I -2 -3 -3 -J -J -3 -3 -4 -4 -4 -4 -4 -5 -5 -5 -4 -8-7 -5 -8 -7 -8 corresponding metrics, thestacksequential decoding algorithm dealswith fewerpathsandtheircorresponding metrics. Inastackalgorithm, themore probable pathsareorderedaccording totheirmetrics,withthepathatthelop ofthestackhavingthelargestmetric.Ateachstepofthealgorithm, onlythe pathatthetopofthestackisextended byonebranch.Thisyields2ksuccessors andtheircorresponding metrics.These2'successors alongwiththeotherpaths arethenreordered according tothevaluesofthemetricsandallpathswith metricsthatfallbelowsomepreselected amountfromthemetricofthetop pathmaybediscarded. Thentheprocessofextending thepathwiththelargest metricisrepeated. Figure8-2-19illustrates thefirstfewstepsinastack algorithm. Itisapparent thatwhennoneofthe2'extensions ofthepathwiththe largestmetricremains atthetopofthestack,thenextstepinthesearch involves theextension ofanotherpaththathasclimbedtothetopofthestack. Itfollowsthatthealgorithm doesnotnecessarily advance byonebranch throughthetrellisineveryiteration. ConseQuently, someamountofstorage mustbeprovided {ornewlyreceived signalsandpreviously received signalsin ordertoallowthealgorithm toextendthesearchalongoneoftheshorter paths,whensuchapathreachesthetopofthestack. CHAPTER Ii:BLOCKANDCONVOLUTiONAL CHAi\NEl coDES50S InacomparISon ofthestackalgorithm withtheViterbialgorithm, thestack algorithm requires fewermetriccomputations, butthiscomputational savingis offsettoalargeextentbythecomputations involved inreordering thestack aftereveryiteration. Incomparison withtheFanoalgorithm, thestack algorithm iscomputationally simpler,sincethereisnoretracing overthesame pathasisdoneintheFanoalgorithm. Ontheotherhand,thestackalgorithm requires morestoragethantheFanoalgorithm. Athirdalternative totheoptimum Viterbidecoder isamethod called feedback decoding (Heller, 1975),whichhasbeenappliedtodecoding fora BSe(hard-decision decoding). Infeedback decoding, thedecoder makesa harddecisionontheinformation bitatstageibasedonmetricscomputed from stageitostagei+m,wheremisapreselected positive integer.Thus.the decision ontheinformation bitiseither0or1depending onwhether the minimum Hamming distance paththatbeginsatstagejandendsatstagei+m contains a 0or1inthebranchemanating fromstagej.Onceadecision ismade ontheinformation bitatstagej,onlythatpaT!ofthetreethatstemsfromthe bitselected atstagejiskept(halfthepathsemanating fromnodej)andthe remaining partisdiscarded. Thisisthefeedback featureofthedecoder. Thenextstepistoextendthepartofthetreethathassurvived tostage j+1+mandconsider thepathsfromstagej+1toj+1+mindeciding on thebitatstagej+1.Thus,thisprocedure isrepeated ateverystage.The parameter missimplythenumberofstagesinthetreethatthedecoderlooks aheadbeforemakingaharddecision. Sincealargevalueofmresultsinalarge amountofstorage,itisde~irable toselectmassmallaspossible. Ontheother hand,mmustbesufficiently largetoavoidaseveredegradation inperfor· mance.Tobalancethesetwownllicting requirements, misusuallyselected in therangeK,,;;m'"2K,whereKistheconstraint length.Notethatthis decoding delayissignificantly smallerthanthedecoding delayinaViterbi decoder, whichisusuallyabout5K. Example 8-2-7 Letusconsider theuseofafeedback decoderfortherate1/3convolutional codeshowninFig.8·2-2.Figure8-2-20illustrates thetreediagram andthe operation ofthefeedback decoder form=2.Thatis,indecoding thebitat branchj,thed&oder considers thepathsatbranches j,j+I,andj+2. Beginning withthefirstbranch, thedecoder computes eightmetries (Hamming distances), anddecidesthatthebitforthefirstbranchis0ifthe minimum distance pathiscontained intheupperpartofthetree,and1if theminimum distancepathiscontained inthelowerpartofthetree.Inthis example, thereceived sequence forthefirstthreebranches isassumed tobe 101111110, sothattheminimum distance pathisintheupperpartofthe tree.Hence,thefirstoutputbitisO. Thenextstepistoextendtheupperpartofthetree(thepartofthetree thathassurvived) byonebranch,andtocompute theeightmetricsfor 506 DIGITAL COMNlINICATIONS o Step1 101Received sequence StepI:Upper-tree metric§: 7.6,5.2·; lower·tree metrics: 5,4,3,4-+0 Step2:Upper-tree metric,:7.6,5.6;lower-tree metrics:3,6,I·,2~IFlGURE 8-2-ZO Anexample offeedback decoding forarate1/3convolutional code. branches 2,3,and4.Fortheassumed received sequence 111110011, the minimum-distance pathiscontained inthelowerpartofthesectionofthe treethatsurVived fromthefirststep.Hence,thesecondoutputbitis1.The thirdstepistoextendthislowerpartofthetreeandtorepeattheprocedure described forthefirsttwosteps. Insteadofcomputing metricsasdescribed above,afeedback decoder for theBSCmaybeefficiently implemented bycomputing thesyndrome fromthe received sequence andusingatablelookupmethodforcorrecting errors.This methodissimilartotheonedescribed fordecoding blockcodes.Forsome convolutional codes,thefeedback decoder simplifies toaformcalleda majority logicdecoderorathreshold decoder(Massey, 1963:Heller,1975). 8-2-8Pradical Considerations intheApplication of Convolutional Codes Convolutional codesarewidelyusedinmanypractical applications of communications systemdesign.Viterbidecoding ispredominantly usedfor shortconstraint lengths(K,,;;10),whilesequential decoding isusedforlong constraint lengthcodes,wherethecomplexity ofViterbidecoding becomes prohibitive. Thechoiceofconstraint lengthisdictated bythedesiredcoding gain. Fromtheerrorprobability resultsforsoft-decision decoding givenby CHAPTER H:BLOCK ANDU)!'VOLU1l0NAL CHANNEL CODESS07 TABLE 8-2-12 UPPER BOUNDS ONCODING GAINFORSOFT-DECISION DECODING OFSOME CONVOLUTION CODES Rate1/2cocIes Role1/3codes Co_lnt Upperbo.... Coastniat Upperboallll len'"Kd.... (db) lea'"Kd.... (dB) 3 5 3.98 3 8 4.26 4 6 4.77 4 10 5.23 5 7 5.44 5 12 6.02 6 8 6.02 6 13 6.37 7 10 6.99 7 15 6.99 8 10 6.99 8 16 7.27 9 12 7.78 9 18 7.78 10 12 7.78 10 20 8.24 (8-2-26)itisapparent thatthecodinggainachieved byaconvolutional code overanuncoded binaryPSKorQPSKsystemis codinggain""1010glO(R,drree) Wealsoknowthattheminimum freedistance dlreecanbeincreased eitherby decreasing thecoderateorbyincreasing theconstraint length,orboth.Table 8-2-12provides alistofupperboundsonthecodinggainforseveral convolutional codes.Forpurposes ofcomparison, Table8-2-13liststheactual codinggainsandtheupperbounds forseveral shortconstraint length convolutional codeswithViterbidecoding.Itshouldbenotedthatthecoding gainincreases towardtheasymptotic limitastheSNRperbitincreases. Theseresultsarebasedonsoft-decision Viterbidecoding.Ifhard-decision decoding isused,thecodinggainsarereduced byapproximately 2dBforthe AWGNchannel. Largercodinggainsthanthoselistedintheabove ~ablesareachieved by TABLE 8-2-13 CODING GAIN(dB)FORSOFT·DECISION VITERBI DECODING ''IN. R,=1/3 R,=1(1 R,=2/3 R,.3/4.......... p. (dB) K=7K=8K=5"=6K=7K=6 K=8 K=6K=9 10-)6.8 4.2 4.4 3.3 3.5 3.8 2.9 3.1 2.6 2.6 10-' 9.6 5.7 5.9 4.3 4.6 5.1 4.2 4.6 3.6 4.2 10-711.3 6.2 6.5 4.9 5.3 5.8 4.7 5.2 3.9 4.8 Source:Jacobs(1974); ~IEEE_ 508 DIGITAL COMI.1UNICATIONS I'1GURE 8-2-21 Performance ofrate1/2andrate1/3Viterbiand sequential decoding, [FromOmuraandLl!1Jitt (1982)©1982lEEE,]w-J JO-2 ..' ~1O~3 0 :5'-0 .~10-4:5 ~ 0:: 1O-~ lO-6()Rale113,K=41. sequential RateIn.K=41. sequential Rate:II2.K:? Viterbi.softded~ion R...112,K=7, Viterbi.harddetision RaleII3.K:? Viterbi. harddecision Uncoded BPSK 2 4 68101214 6,iN.<dB) employing longconstraint lengthconvolutional codes,e.g.,K=50,and decoding suchcodesbysequential decoding. Invariably, sequential decoders areimplemented forhard-decision decoding toreducecomplexity. Figure 8-2-21illustrates theerrorrateperformance ofseveralconstraint-length K=7 convolutional codesforrates1/2and1/3andforsequential decoding (with harddecisions) ofarate1/2andarate1/3constraint-length K=41 convolutional codes.NotethattheK=41codesachieveanerrorrateof10-6 at2.5and3dB,whicharewithin4-4.5dBofthechannelcapacitylimit,Le.,in vicinityofthecutoffratelimit.However, therate1/2andrate1/3,K=7codes withsoft-decision Viterbidecoding operate atabout5and4.4dBat10-6, respectively. Theseshort-constraint-length codesachieveacodinggainof about6dBat10-6,whilethelongconstraint codesgainabout7.5-8dB. Twoimportant issuesintheimplementation ofViterbidecoding are 1theeffectofpathmemory truncation, whichisadesirable featurethat ensuresafixeddecoding delay,and 2thedegreeofquantization oftheinputsignaltotheViterbidecoder. Asaruleofthumb,westatedthatpathmemory truncation toaboutfive constraint lengthshasbeenfoundtoresultinnegligible performance loss. Figure8-2-22illustrates theperformance obtained bysimulation forrate1/2, constraint-lengths K=3,5,and7codeswithmemory pathlengthof32bits.In addition topathmemory truncation, thecomputations wereperformed with eight-level (threebits)quantized inputsignalsfromthedemodulator. The brokencurvesareperformance resultsobtained fromtheupperboundinthe biterrorrategivenby(8-2-26). Notethatthe simulation resultsareclosetothe theoretical upperbounds, whichindicate thatthedegradation duetopath memory truncation andquantization oftheinputsignalhasaminorefIecton performance (0.20-0.30 dB). CHAPTER 8:BLOCK ANDCONVOLUTIONAL CHANNEL CODES509 K:J........Simulation - - - - .Upperbound 'Li.--'\r--- K=7 10-'.·.·.·.· .....,.·.·.·.·.10-'""3~-4':--'5~-6!-'--:7~- E,lIVo(dB)10-3 :E'010-5 f Biterrorprobability forrate112Viterbidecoding with eight-level quantized inputstothedecoderand32-bitpath memory. [FromHellerandJacohs(1971).©1971IEEE·IFIGURE 8-2-U Figure8-2-23illustrates thebiterrorrateperformance obtained via simulation forhard-decision decoding ofconvolutional codeswithK=3-8. NotethatwilhtheK=8code,anerrorrateof10-5requiresabout6dB,which represents acodinggainofnearly4dBrelativetouncaded QPSK. Theeffectofinputsignalquantization isfurtherillustrated inFig_8-2-24for arate1/2,K=5code.Notethatthree-bit quantization (eightlevels)isabout 2dBbetterthanhard-decision decoding, whichistheultimate limitbetween soft-decision decoding andhard-decision decoding ontheAWONchannel. Thecombined effectofsignalquantization andpathmemory trunction forthe rate1/2,K=5codewith8-,16-,and32-bitpathmemories andeitherone-or three-bit quantization isshowninFig_8-2-25.Itisapparent fromtheseresults thatapathmemory asshortasthreeconstraint lengthsdoesnotseriously degradeperformance. Whenthesignalfromthedemodulator isquantized to morethantwolevels, anotherproblem thatmustbeconsidered isthespacingbetween quantization levels.Figure8-2-26illustrates thesimulation resultsforaneight-level uniform quantizer asafunction ofthequantizer threshold spacing. Weobserve that 10-1 FIGURE 8-2-23 Performance ofrate112codeswithhard-decision Viterbi decoding and32-bitpathmemorytruncation. IFromHellerandJacobs(1971).©1971IEEE.)10-'3~~4--5~---!6~-!i-~ .6.lNo(dB) 510 DIGITAL COMMUNICATIONS FIGURE 8-2·24 Performance ofrate1/2,K~5codewitheight-,four-,and two-level quantization attheinputtotheViterbidecoder. Pathtruncation length=32bits.[FromHell"andJacobs (1971).©1971IEEE) FlGURE 8-2·25 Performance ofrate1/2,K=5codewith32-,16-,andg-bit pathmemory truncation andeight-andtwo-level quantization. [FromHellerandJacobs(1971).©1971IEEE.]8-level quafltlzation quantization FlGURE 8-2-26 Errorrateperformance ofrate1/2,K=5Viterbidecoder for'l.INo=3.5dBandeight-level quantization asafunction ofquantizer tnreshold levelspacingforequallyspaced thresholds [FromHellerandJacobs(1971).©1971IEEE.J.... 1.8X10-,' ~1.6x10--·\ j;l.4x10---''0 ~1.2x10-' :0iLOx10-·\ 0.30.4 0.5 0.60.7 Quantizer Ih.reshold spacing thereisanoptimum spacingbetween thresholds (approximately equalto0.5). However, theoptimum issufficiently broad(0.4-0,7) sothat,onceitisset, thereislittledegradation resulting fromvariations intheAGelevelofthe orderof±200/0. ('HAPTER~: BLOCK ANDCONVOLUTIONAL (,HA~!'IEL CODES511 FIGURE 8-2·27 Performance. ofarate112.K=7codewithViterbi decoding andeight-level quantization asafunction of thecarrierphase,rackingloopSNRy'.'[FromHeller andJacohs(1971).<Ei1971IEEE]10--' 10-'...g "IO-~E. ~'It=1218 c .~ ]10-" E£ 10-7 .)45 6 7 8 9101112131415 (VNl,(dB) Finally. weshouldpointoutsomeimportant resultsintheperformance degradation duetocarrierphasevariations. Figure8-2-27illustrates the performance ofarate1/2.K=7codewitheight-level quantization anda carrierphasetracking loopSNR"y/..RecallthatinaPLL.thephaseerrorhas avariance thatisinversely proportional to"fL'TheresultsinFig.8-2-27 indicatethatthedegradation islargewhentheloopSNRissmall(YL<12dB). andcausestheerrorrateperformance tobottomoutatrelatively higherror rate. 8-3CODED MODULA nONFORBANDWIDTH­ CONSTRAINED CHANNELS Inthetreatment ofblockandconvolutional codesinSections 8-1and8-2. respectively. performance improvement wasachieved byexpanding theband­ widthofthetransmitted signalbyanamountequaltothereciprocal ofthe coderate.Recallforexample thattheimprovement inperformance achieved byan(n,k)binaryblockcodewithsoft-decision decoding isapproximately 1010glO(Redmin-kIn2/'Yh)compared withuncoded binaryorquaternary PSK.Forexample. when'Yh=10the(24.12)extended Golaycodegivesa codinggainof5dB.Thiscodinggainisachieved atacostofdoubling the bandwidth ofthetransmitted signaland.ofcourse.attheadditional costin receiver implementation complexity. Thus.codingprovides aneffectiw methodfortradingbandwidth andimplementation complexity againsttran,­ mitterpower.Thissituation appliestodigitalcommunications systemsthatare designed tooperateinthepower-limited regionwhereRIW<1. Inthissection. weconsider theuseofcodedsignalsforbandwidth­ constrained channels. Forsuchchannels. thedigitalcommunications systemis SU DIGITAL CO....UNICATIONS designed tousebandwidth-efficient multilevel/phase modulation, suchas PAM,PSK,DPSK,orQAM,andoperates intheregionwhereR/W>1. Whencodingisappliedtothebandwidth-constrained channel, aperformance gainisdesiredwithoutexpanding thesignalbandwidth. Thisgoalcanbe achieved byincreasing thenumberofsignalsoverthecorresponding uncoded systemtocompensate fortheredundancy introduced bythecode. Forexample, suppose thatasystememploying uncoded four-phase PSK modulation achieves anR/W'"2(bits/s)!Hz atanerrorprobability of10-6 ForthiserrorratetheSNRperbitis'Yb'"10.5dB.Wemaytrytoreducethe SNRperbitbyuseofcodedsignals,butthismustbedonewithoutexpanding thebandwidth. IfwechoosearateR,'"2/3code,itmustbeaccompanied by anincreaseinthenumberofsignalpointsfromfour(twobitspersymbol)to eighi(threebitspersymbol). Thus,therate2/3codeusedinconjunction with eight-phasePSK,forexample, yieldsthesamedatathroughput asuncoded four-phase PSK.However, werecallthatanincreaseinthenumberofsignal phasesfromfourtoeightrequires anadditional 4dBapproximately insignal powertomaintain thesameerrorrate.Hence,ifcodingistoprovideabenefit, theperformance gainoftherate2/3codemustovercome this4dBpenalty. Ifthemodulation istreatedasaseparate operation independent ofthe encoding, theuseofverypowerful codes(Iarge-constraint-iength convolutional codesorlarge-block-length blockcodes)isrequired tooffsetthelossand providesomesignificant codinggain.Ontheotherhand,ifthemodulation is aninregralpartoftheencoding processandisdesigned inconjunction withthe codetoincrease theminimum euclidean distance between pairsofcoded signals,thelossfromtheexpansion ofthesignalsetiseasilyovercome anda significant codinggainisachieved withrelatively simplecodes.Thekeytothis integrated modulation andcodingapproach istodeviseaneffective methodfor mapping thecodedbitsintosignalpointssuchthattheminimum euclidean distance ismaximized. Suchamethodwasdeveloped byUngerboeck (1982), basedontheprinciple ofmapping bysetpartitioning. Wedescribe this principle bymeansoftwoexamples. . Example 8-3-1:An8-PSKSignalConstellation Letuspartition theeight-phase signalconstellation showninFig.8-3-1into subsetsofincreasing minimum euclidean distance. Intheeight-phase signal set,thesignalpointsarelocatedonacircleofradius·~andhavea minimum distanceseparation of do'"2\/~sin ~1r'"V(2-Y2)jg'"O.765~ Inthefirstpartitioning, theeightpointsaresubdivided intotwosubsetsof fourpointseach,suchthattheminimum distancebetween pointsincreases tod,'"mInthesecondlevelofpartitioning, eachofthetwosubsetsis subdivided intotwosubsetsoftwopoints,suchthattheminimum 'distance increases todz'"2~.Thisresultsinfoursubsetsoftwopointseach. CHAPTER 8:BLOCK ANDCONVOLUTIONAL CHANNEL CODES513 A=8-PSK B, +"." +++ ++++++++000 100 010 110 nGURE 1I-J.1Setpartitioning ofan8-PSKsignalset.001 101 011 111 Finally,thelaststageofpartitioning leadstoeightsubsets, whereeachsubset contains asinglepoint.Notethateachlevelofpartitioning increases the minimum euclidean distance between signalpoints.Theresultsofthesethree stagesofpartitioning areillustrated inFig.8-3-1.Thewayinwhichthecoded bitsaremapped intothepartitioned signalpointsisdescribed below. Example 8-3-2:A16-QAM SignalConstellation The16-point rectangular signalconstellation showninFig.8-3-2isfir,' dividedintotwosubsetsbyassigning alternate pointstoeachsubsetas 514 DIGITAL COMMLN'CATIONS 16-QAM A=16QI\M~::::~ o•••• Ie,.e ~2!26 B. .-II.- 8,/Aoeo. .0.0/ ''oeo 2fT 010. O/O.O.~ ,. 417r~'O'O""--I ,C~.oeo ~2 I'-',-rala. ~Cl 010' 0000 10.0 00000000 '0'0 0000 0.0.aloe 0000 10.0 0000 0/0000 \1 0/eo.o \1 0/0000 \1 10oeo.\'Do DJ D~ Db D1 DsD.1 0, 000. 0100 ooao 0000 .000 DOlO 0000 00000000 0000 ooeo 1000 0000 0000 0'00 000. 0Jg~gg\1 0Jggg~~1 °J~gggv 6Jgg~g~1 0Jgg~g\1 0J~ggg\1 0Jggg~\1 0Jg~gg\1 OOCO000'00000'0000000000000000000000.000000000.00000 0000 0000 OOCO00000000 0000 00000000 00.00000.0000000 0000 0000 000000000.000000000.0.000000000.0000OOCO0000 OOCO000000.00000.000000000000000aoog00000000 0000 00000000eooo000000.00000000000000000 0000 0001 0000 oeo0000 ססoo10000'00"0000'0'01001'011100001'001010'110'001110'10111I'll nGURE 8-3-ZSetpartitioning of16-QAM signal. illustrated inthefigure.Thus,thedistancebetween pointsisincreased from2nto2mbythefirstpartitioning. Further partitioning ofthetwo subsetsleadstogreaterseparation ineuclidean distance between signal pointsasillustrated inFig.8-3-2.Itisinteresting tonotethatforthe rectangular signalconstellations, eachlevelofpartitioning increases the minimum euclidean distance byv'2,i.e.,di+,ldi=v'2foralli. Inthesetwoexamples, thepartitioning wascarriedouttothelimitwhere eachsubsetcontains onlyasinglepoint.Ingeneral,thismaynotbenecessary. Forexample, the16-point QAMsignalconstellation maybepartitioned only twice,toyieldfoursubsetsoffourpointseach.Similarly, theeight-phasePSK signalconstellation canbe-partitioned twice,toyieldfoursubsetsoftwopoints each. Thedegreetowhichthesignalispartitioned dependsonthecharacteristics ofthecode.Ingeneral,theencoding processisperformed asillustrated inFig. 8·3-3.Ablockofminformation bitsisseparated intotwogroupsoflengthk, , :::=::I=12Select .-I-•II.Z•..,.2"\ ---~----- I Sekcl-UIlCOded 2point bitsfrom subset k, II.2.....2"1ulator.I ~ nGUKE 8-3-3Generalstructure ofcombined encoder/modk, CHAPTER x,BLOCK ANDCONVOLUTIONAL CHANNEL CODES51S ,----oe, FIGURE 8-3-4Four-state trellis-coded 8-PSK modulation.L----o e, "UIlC:::;OOe=d:.;b:.:ilOc, (tlJEncoder 001 (c)Mappin, of codedbilslc••c2•ell toiignalpointb=10 c:=01 d.II*::=:::;:=::::~ 011 (b)Four·stale trellts andk2•ThekIbitsareencoded intonbitswhilethek2bitsareleftuncoded. Then,thenbitsfromtheencoderareusedtoselectoneoftheZ"possible subsetsinthepartitioned signalsetwhilethek2bitsareusedtoselectoneof theZk,signalpointsineachsubset.Whenk2=0,allminformation bitsine encoded. Example 8-3-3 Consider theuseoftherate 1/2convolutional codeshowninFig.8-3-4to encodeoneinformation bitwhilethesecondinformation bitisleftuncoded. Whenusedinconjunction withaneight-point signalconstellation, e.g., eight-phase PSKoreight-point QAM,thetwoencoded bitsareusedto selectoneofthefoursubsetsinthesignalconstellation, whiletheremaining information bitisusedtoselectoneofthetwopointswithineachsubset. Inthiscase,k1=1andk2=1.Thefour-state trellis,whichisshowninFig. 8-3-4(b), isbasically thetrellisfortherate1/2convolution encoderwiththe additionofparallelpathsineachtransition toaccommodate theuncoded bit c).Thus,thecodedbits(c"C2)areusedtoselectoneofthefoursubsets thatcontaintwosignalpoint1ieach,whiletheuncoded bitisusedtoselect oneofthetwosignalpointswithineachsubset.Notethatsignalpoints withinasubsetareseparated indistancebyd2=zVi.Hence,theeuclidean distancebetween parallelpathsisd2•Themapping ofcodedbits(c"C2.C3) tosignalpointsisillustrated inFig.8-3-4(c). Asanalternative coding scheme, wemayusearateZ/3convolutional encoder, and,thus,encode 516 DI(iITAl COMMUNICATIONS r----<Jc, FIGURE 8-3-5Rate2/3convolutional encoderforencoding bothinformation bits. C----o., bothinformation bitsasshowninFig.8-3-5.Thisencoding leadstoan eight-state trellisandresultsinbetterperformance, butalsorequires amore complex implementation ofthedecoder asdescribed below. Eitherblockcodesorconvolutional codesmaybeusedinconjunction with thepartitioned signalconstellation. Ingeneral, convolutional codesprovide comparable codinggainstoblockcodesandtheavailability oftheViterbi algorithm resultsinasimplerimplementation forsoft-decision decoding. For thisreason,welimitourdiscussion toconvolutional codes(lineartrelliscodes) andmoregenerally to(nonlinear) trelliscodes. Trellis-Coded Modulation Letusconsider theuseofthe8-PSKsignal constellation inconjunction withtrelliscodes.Uncoded four-phase PSK (4-PSK) isusedasareference inmeasuring codinggain.Uncoded 4-PSK employs thesignalpointsineithersubsetBoorBIofFig.8-3-1,forwhichthe minimum distance ofthesignalpointsism.Notethatthissignalcorres­ pondstoatrivialone·state trelliswithfourparallelstatetransitions asshown inFig.8-3-6(a). ThesubsetsDo,D,.D4•andDoareusedasthesignalpoints forthepurposeofillustration. Forthecoded8-PSKmodulation, wemayusethefour-state trellisshownin Fig.8-3-6(b). Notethateachbranchinthetrelliscorresponds tooneofthe foursubsetsCo.CloC2,orC3.Fortheeight-point constellation, eachofthe subsets Co.Ct.C,.andC3•contains twosignalpoints.Hence,thestate transition Cocontains thetwosignalpointscorresponding tothebits(000,1(0) or(0,4)inoctalrepresentation. Similarly, C,contains thetwosignalpoints corresponding to(010,110), orto(2,6)inoctal,C,contains thepointscorre­ sponding to(001,101), or(1,5)inoctal,aridC3contains thepoints corresponding to(011,111), or(3,7)inoctal.Thus,eachtransition inthe four-state trelliscontains twoparallelpaths,asshowninmoredetailinFig. 8-3-6(c). Notethatanytwosignalpathsthatdiverge fromonestateand remerge atthesamestateaftermorethanone transition haveasquared euclidean distance ofd~+2d~=d~+d~between them.Forexample, the CHAPTER ItBLOCK ANDC<)NVOLUTlONAL CHANNEL CODES517 SirCll.1;, [)n@5, S, .\n D.. Su D, 5, 5, (II)One-slale trellis 5, 5, (b)Four-~tate trellis ••• ••o 0:...-.~'_·,·.----f:.- _':.~'_-_-,-_-? ,,,, 2,',,,--.'("IC, (OAl(2,61 C,C, (1.5)<:'.1) C,Cll (2,6)(0,4) Ie)Four-slate trellis FlGURE 8-3-6Uncoded 4·PSKandtrellis·coded 8·PSKmodulation. signalpaths0,0,0and2,1,2areseparated byd~+d~=[(0.765?+4])g~ 4.585~.Ontheotherhand,thesquaredeuclidean distance between parallel transitions isd~~4~.Hence,theminimum euclidean distance separation between pathsthatdivergefromanystateandremergeatthesamestateinthe four-state trellisisd2=2~.Thisminimum distanceinthetrelliscodeiscalled thefreeeuclidean distanceanddenotedbyDfed• Inthefour-state trellisofFig.8-3-6(b), Dfed=2W.Whencompared with theeuclidean distance do=mfortheuncoded 4-PSKmodulation, we observethatthefour-state trelliscodegivesacodinggainof3dB. Weshouldemphasize thatthefour-state trelliscodeillustrated inFig. 8-3-6(b)isoptimum inthesensethatitprovides thelargestfreeeuclidean distance. Clearly, manyotherfour-state trelliscodescanbeconstructed. including theoneshowninFig.8-3-7,whichconsistsoffourdistincttransitions fromeachstatetoallotherstates.However, neitherthiscodenoranyofthe otherpossiblefour-state trelliscodesgivesalargerDfed• Theconstruction oftheoptimum four-state trelliscodefortheeight-point constellation wasperformed onthebasisofthefollowing heuristic rules: (a)Parallel transitions (whentheyoccur)areassigned tosignalpoints separated bythemaximum euclidean distance, e.g.,d2=2~for8-PSKinthe foursubsetsCo,C"C2•C,. 518 DI(iITAL COMMl~NICATIONS FIGURE 8-3-7 Analternati\'e four·state trelliscode. (b)Thetransition originating fromandmerging intoanystateisassigned thesubsets(Co.C2)or(C,.C,),whichhaveamaximum distanced,=m, (c)Thesignalpointsshouldoccurwithequalfrequency. Notethatrules(a)and(b)guarantee thattheeuclidean distance associated withsingleandmultiple pathsthatdivergefromanystateandremerge inthat stateexceeds theeuclidean distance ofuncoded 4-PSK.Rule(c)guarantees thatthetrelliscodewillhavearegularstructure. Weshouldindicatethatthespecificmapping ofcodedbitsintosignalpoints. asillustrated inFig.8-3-1,wheretheeightsignalpointsarerepresented inan equivalent binaryform,isnotimportant. Othermappings canbedevisedby permuting subsetsinawaythatpreserves themainproperty ofincreased minimum distance amongthesubsets. Inthefour-state trelliscode,theparalleltransitions wereseparated bythe euclidean distance2n,whichisalsoDred•Hence,thecodinggainof3dBis limitedbythedistanceoftheparalleltransitions. Largergainsinperformance relativetouncoded 4-PSKcanbeachieved byusingtrelliscodeswithmore states,whichallowforthe elimination oftheparalleltransitions. Thus,trellis codeswitheightormorestateswouldusedistincttransitions toobtainalarger D't::do Forexample, inFig.8·3-8,weillustrate aneight-state trelliscodedueto Ungerboeck (1982)forthe8-PSKsignalconstellation. Thestatetransitions for maximizing thefreeeuclidean distance weredetermined fromapplication of thethreebasicrulesgivenabove.Inthiscase,notethattheminimum squared euclidean distance is Died=d~+2d~=4.585g which,whencompared withd~=2gforuncoded 4-PSK,represents againof 3.6dB.Ungerboeck (1982,1987)hasalsofoundrate2/3trelliscodeswith16, 32,64,128,and256statesthatachievecodinggainsrangingfrom4to5.75dB for8-PSKmodulation. CHAPTER X:BLOCK ANDt'ONVOUJTlONAl CHA!'IlNEL CODES519 FIGURE 8-3-8Eight·state trelliscodeforcoded g·PSKmodulation.o II Thebasicprinciple ofsetpartitioning iseasilyextended tolargerPSK.signal constellations thatyieldgreater bandwidth efficiency. Forexample, 3(bits/s)/Hz canbeachieved witheitheruncoded 8-PSKorwithtrellis-coded 16-PSKmodulation. Ungerboeck (1987)hasdevised trelliscodesandhas evaluated thecodinggainsachieved bysimplerate1/2andrate2/3 convolutional codesforthe16-PSKsignalconstellations. Theresultsare summarized below. Soft-decision Viterbidecoding fortrellis-coded modulation isaccomplished intwosteps.Sinceeachbranchinthetrelliscorresponds toasignalsubset,the firststepindecoding istodetermine thebestsignalpointwithineachsubset, i.e.,thepointineachsubsetthatisclosestindistancetothereceived point.We maycallthissubsetdecoding. Inthesecondstep,thesignalpointselectedfrom eachsubsetanditssquared distance metricareusedforthecorresponding branchintheViterbialgorithm todetermine thesignalpaththrough thecode trellisthathastheminimum sumofsquareddistances fromthesequence of received (noisychanneloutput)signals. Theerrorrateperformance ofthetrellis-coded signalsinthepresence of additivegaussian noisecanbeevaluated byfollowing theprocedure described inSection8-2forconvolutional codes.Recallthatthisprocedure involves the computation oftheprobability oferror'foralldifferent erroreventsand 520 DI(inA! (,O!\1:\1I ';-.;KATIO;-';S summing theseerroreventprobabilities toobtainaunionboundonthe first-event errorprobability. Note.however. that.athighSNR.thefirst-event errorprobability isdominated bytheleadingterm,whichhastheminimum distanceD"J'Consequently. athighSNR,thefirst-event errorprobability is wellapproximated as (8-3-1) whereNrcJdenotes thenumberofsignalsequences withdistance Die"that divergeatanystateandremerge atthatstateafteroneormoretransitions. Incomputing thecodinggainachieved bytrellis-coded modulation, we usuallyfocusonthegainachieved byincreasing DleJandneglecttheeffectof NrcJ.However; trelliscodeswithalargenumberofstatesmayresultinalarge NrcJthatcannotbeignoredinassessing theoverallcodinggain. Inaddition tothetrellis-coded PSKmodulations described above,powerful trelliscodeshavealsobeendeveloped forPAMandQAMsignalconstella­ tions.Ofparticular practical importance istheclassoftrellis-coded two­ dimensional rectangular signalconstellations. Figure8-3-9illustrates these signalconstellations forM-QAM whereM=16,32,64,and128.TheM~32 and128constellations haveacrosspattern andaresometimes called cross-conJtellations. Theunderlying rectangular gridcontaining thesignal pointsinM-QAM iscalledaJaniceofrypeZ,(thesubscript indicates the dimensionality ofthespace).Whensetpartitioning isappliedtothisclassof signalconstellations. theminimum euclidean distance between successive partitions isdj.,/d, ~V2foralli.aspreviously observed inExample 8-3-2. FIGURE S-J.9 Rt:ctangular t....u-dimt:nsionaJ (CAM)signalconstellations. 0000000 0 00 0 0000 0- 0 0 00 0 000o0~---a000 0 00007 0 0 00 0 000.-; 000- 0000 0 0000~ 00-~ a000 0 0000 0 00f...- a000 0 0000 0 00 0 0 00 0 0000 0 00 0 0 00 0 0000 0 00 i 00 0 0000 0 00aa0000M:1ft AI:3!(l"rlM) CHAPTER K:fJlOCK ANDCOJ'l.,·VOlUTION ....LCHANNEl. CODES521 FIGURE 8-3-10 Eight-state trellisforrectangular QAMsignal constellations. Figure8-3-10illustrates aneight-state trelliscodethaIcanbeusedwithany oftheM·QAM rectangular signalconstellations forwhichM=2',where k=4,5,6,...,etc.Withtheeight-state trellis,weassociate eightsignal subsets,sothatanyoftheM-QAMsignalssetsforM;;.16aresuitable. For M=2m+t,twoinputbits(k,=2)areencoded inton=3(n=k,+1)bitsthat areusedtoselectoneoftheeightstatescorresponding totheeightsubsets. Theadditional k2=m -k1inputbitsareusedtoselectsignalpointswithina subset,andresultinparalleltransitions iotheeight-state trellis.Hence, 16·QAM involves twoparalleltransitions ineachbranchofthetrellis.More generally, thechoiceofanM=2m+'-pointQAMsignalconstellation implies thattheeight-state trelliscontains 2m-2paralleltransitions ineachbranch. Theassignment ofsignalsubsetstotransitions isbased00thesamesetof basic(heuristic) rulesdescribed aboveforthe8-PSKsignalconstellation. Thus, thefour(branches) transitions originating fromorleadingtothesamestateare assigned eitherthesubsetsDo,D2,D4,D6orD"D3,Ds, D7.Parallel transitions areassigned signalpointscontained withinthecorresponding subsets.Thiseight-statetrelliscodeprovides acodinggainof4dB.The euclidean distanceofparalleltransitions exceedsthefreeeuclidean distance, and,hence,thecodeperformance isnotlimitedbyparalleltransitions. LargersizetrelliscodesforM-QAM provideevenlargercodinggains.For 522 ()I(ilTAL COMMUNH'ATIONS TABLE 8·3-tCODING GAINS FORTRELLIS-CODED PAMSIGNALS Code m=t m=2 m_'" Number rate codinggain(dBl codinggain(dB) asymptotic of ~ of4-PAM versus of8·PAM versus codinggainm-+x states k,k,+Iuncoded 2·PAM uncod.d 4-PAM (dB' Nr~d 4 1/2 2.55 3.31 3.52 4 X 1/2 3.1l1 3.77 H7 4 16 1/2 3.42 4.IX 4.3~ X -J1/2 4.15 4.91 5.11 11 .'- 64 1/2 4.47 5.23 5.44 36 12X 1/2 5.1l5 5.XI 6.1l2 66 Sfmn-t': Ungcrhoeck (19H7). example, trelliscodeswilh2"statesforanM=2m+'QAMsignalconstellation canbeconstructed byconvolutionally encoding k,inputbitsintok,+1output bits.Thus,arateR,=k,/(k,+I)convolutional codeisemployed forthis purpose. Usually, thechoiceofk,=2provides asignificant fractionofthetotal codinggainthatisachievable. Theadditional k2=m-k,inputbitsare uncoded, andaretransmitted ineachsignalinterval byselecting signalpoints withinasubset. Tables8·3·1to8·3,3,takenfromthepaperbyUngerboeck (1987),provide asummary ofcodinggainsachievable withtrellis-coded modulation. Table 8·3·1summarizes thecodinggainsachieved fortrellis·coded (one·dimensional) PAMmodulation withrate1/2trelliscodes.Notethatthecodinggainwitha 128·state trelliscodeis5.8dBforoctalPAM,whichisclosetothechannel cutoffrateRIIandlessthan4dBfromthechannelcapacity limitforerrorrates intherangeof10"_10". Weshouldalsoobserve thatthenumberofpaths TABLE lI-3-2CODING GAINS FORTRELLIS-CODED 16·PSK MODULATION m=3 Number Coderatecodinggain(dB) of ~ oft6·PSK versus m~'" states k, k,+1 uncoded 8-PSK N... 4 1 1/2 3.54 4 X I 1/2 401 4 Ih I 1/2 4.44 X I 1/2 5.13 X ,,-I 1 1/2 5.33 2 :~~ 1 1/2 5.33 2 ~56 2 2/3 5.51 X "UI/ri't'·Un!-':coocck (IYX7). CHAPTER 8,BLOCKANDCONVOLUTIONAL CHANNEL CODES523 TABLE 8-3-3CODING GAINS FORTRELLIS-CODED QAMMODULATION m=3 m=4 ...=5 Code pill(dB)ofpm(dB)ofpia(dB)of III==oc Number rale l6-QAM v....... 32-QAM versus 64-QAM versus osymplotic of -.!L UDcode<l uacoded UDcoOe<I coding states k,k,+1 8-QAM 16-QAM 32-QAM pin(dB) N,... 4 1 1/2 3.01 3.01 2.80 3.01 4 8 2 2/3 3.98 3.98 3.77 3.98 16 16 2 213 4.77 4.77 4.56 4.77 56 32 2 213 477 4.77 4.56 4.77 16 64 2 2/3 5.44 5.44 4.23 5.44 56 128 2 2/3 6.02 6.02 5.81 6.02 344 256 2 2/3 6.02 6.02 5.81 6.02 44 So/trc.."e:Ungerboeck (1987). Nt'dwithfreeeuclidean distance Dredbecomes largewithanincrease inthe numberofstates. Table8-3-2liststhecodinggainfortrellis-coded 16-PSK. Again,weobserve thatthecodinggainforeightormoretrellisslagesexceeds4dB,relativeto uncoded 8-PSK.Asimplerate1/2codeyields5.33dBgainwitha128-stage trellis. Table8-3-3contains thecodinggainsobtained withtrellis-coded QAM signals.Relatively simplerate2/3trelliscodesyieldagainof6dBwith128 trellisstagesform=3and4. Theresultsinthesetablesclearlyillustrate thesignificant codinggainsthat areachievable withrelatively simpletrelliscodes.A 6dBcodinggainisclose tothecutoffrateRoforthesignalsetsunderconsideration. Additional gains thatwouldleadtotransmission inthevicinityofthechannelcapacity bound aredifficult toattainwithout a significant increase incoding/decoding complexity. Sincethechannelcapacityprovides theultimate limitoncodeperformance, weshouldemphasize thatcontinued partitioning oflargesignalsetsquickly leadstosignalpointseparation withinanysubsetthatexceeds thefree euclidean distanceofthecode.Insuchcases,paralleltransitions arenolonger thelimitingfactoronDred•Usually, apartition toeightsubsetsissufficient to obtainacodinggainof5-6dBwithsimplerate1/2orrate2/3trelliscodes witheither64or128trellisstages,asindicated inTables8-3-1to8-3-3. Convolutional encoders forthelineartrelliscodeslistedinTables8-3-1to 8-3-3fortheM-PAM, M-PSK, andM-QAM signalconstellations aregivenin thepapersbyUngerboeck (1982,1987).Theencoders mayberealizedeither withfeedback orwithoutfeedback. Forexample Fig.8-3-11illustrates three feedback-free convolutional encoders corresponding to4-,8-,and16-state trelliscodesfor8-PSKand16-QAM signalconstellations. Equivalent realiza­ tionsofthesetrelliscodesbasedonsystematic convolutional encoders with S24 DlGITAl rOMMlINlCA.TfONS {l1_~.- ---------- - ------_.-----~:~.:ib'QAM-~ n~ t'! Output ",-....,...-\ M-PSK ('ncoder 'I (0)4-stateencoder ",-of------'4j (bj8-stateencoder8-PSK encoderOutput (d16·stateencoder8-PSK Output encoder FIGURE 8-3-11 Minimal feedback-free convolutional encoders for8-PSKand16-QAM signals_[FromUngerboeck (1982).©/982IEEE] feedback areshowninFig.8-3-12.Usually. thesystematic convolutional encoders arepreferred inpractical applications. Apotential problem withlineartrelliscodesisthatthemodulated signal setsarenotusuallyinvariant tophaserotations. Thisposesaproblem in practical applications wheredifferential encoding isusuallyemployed toavoid phaseambiguities whenareceiver mustrecover thecarrierphaseaftera tempOrary lossofsignal.Theproblem ofphaseinvariance anddifferential encoding/decoding wassolvedbyWei(1984a,b).whodevised linearand nonlinear trelliscodesthatarerotationally invariant undereither1800or900 phaserotations, respectively. Forexample, Fig.8-3-13illustrates anonlinear eight-state convolutional encoder fora32-QAM rectangular signalconstella­ tionthatisinvariant under900phaserotations. Thistrelliscodehasbeen adopted asaninternational standard for9600and14,000bits/s(high-speed) telephone linemodems. Trellis·coded modulation schemes havealsobeendeveloped formulti­ dimensional signals.In practical systems, multidimensional signalsaretrans­ mittedasasequence ofeitherone-dimensional (PAM)ortwo-dimensional (QAM) signals.Trelliscodesbasedon4-,8·,and16-dimensional signal constellations havebeenconstructed. andsomeofthesecodeshavebeen CHAI'TER' BLOCKANDCONVOLUTIONAL CHANNEL CODES525 a_-•- - - - - - - ~- - --- - - - - - - - - -- - - - -;~:jr:Q~M-:, SC't.t~-=-1 °2 ~ ~~a,------ .....-------~ 8.PSK ~..-__ --:;C,~ encoder (a)4-slateencoder 8-PSK 1-._',-;'.lencoder I:bl8-staleencoder a,-----~:w_IOutput a,---_---_---~--_".,.l 8-PSK OutPlIIt encoder (c)16-slatcencoder FIGURE 8-.J..12 Equivalent realizations ofsystematic convolutional encoders withfeedback for8-PSKand 16-QAM. [FromUngerboeck (1982)©1982iEEE.] implemented incommercially available modems. Apotential advantage of trellis-coded multidimensional signalsisthatwecanusesmallerconstituent two-dimensional signalconstellations thatallowforatrade-off between coding gainandimplementation complexity. ThepapersbyWei(1987),Ungerboeck (1987),Gersho andLawrence (1984), andForney etal.(1984)treat multidimensional signalconstellations fortrellis-coded modulation. Finally,weshouldmention thatanewdesigntechnique fortrellis-coded modulation basedonlatticesandcosetsofasublattice hasbeendescribed by Calderbank andSloane (1987)andForney (1988). Thismethod for constructing trelliscodesprovides analternative tothesetpartitioning method described above.However, thetwomethods arecloselyrelated. Inthis alternative method, ablock ofk,bitsisfedtoaconvolutional encoder. Each blockofkIinputbitsproduces anoutputsymbolthatisacosetofthe sublattice A',whichisasubsetofthechosenlattice.Asecondblockofk2input bitsisusedtoselectoneofthepointsinthecosetattheoutputofthe convolutional encoder.Itisapparent thatthecosetsofthesubJattice areakin tothesubsetsinsetpartitioning andtheelements ofthecosetsareakintothe signalpointswithinasubset.Thisnewmethodhasled10thediscovery ofnew powerful trelliscodesinvolving largersignalconstellations, manyofwhichare listedinthepaperbyCalderbank andSloane(1987). 526 DIGITAL COMMUNICATIONS fiJ •~ II\ I'(". '----------j'--------------------------------j I I' I , f-~'~--- -----~------_+-- ('J,..+o'+'---t------+-- .....----7- ..("! " " " ~~ I, C 1:', "L --', ~---------_I~--------------------------------jDifferential encoderNonlinear convolutional encoder la)bcoder 0 0 1I111 00011 0 0 0 00010 10100 01010 0 0 2~0 0 01001 10101 11001 00101 0 0 0 0 0 00000 11110 01000 10110 11000 -4-24Binarysequence below /001101/"OWlI /0/'/ ~ignalpoint C~C4C~C~CI 0 0 0 0 0 11100 10010 01100 1/010 00100 0 0_2[ 0 0 0000111101 1000101101 0 0 0 01110 10000 00110 (blJ2·point QAM(cross)signal I'1~URE 8-J.1J Eight-Slate nonlinear convolutional encoderfor32-QAM signalsetthatexhibitsinvariance under 9(fphaserotations. 8-4BIBLIOGRAPHICAL NOTESANDREFERENCES Thepioneering workoncodingandcodedwavefonns fordigitalcommunica­ tionswasdonebyShannon (l948a,b),Hamming (1950),andGolay(1949). Theseworkswererapidlyfollowed withpapersoncodeperformance by ICHAPTER~: BLOCK ANDCONVOLlJTlO"Al nIA~NFL nuns527 Gilbert(1952),newcodesbyMuller(1954)andReed(1954),andcoding techniques fornoisychannels byElias(1954.1955)andSlepian(1956). Duringtheperiod1960-1970, therewereanumberofsignificant contribu­ tionsinthedevelopment ofcodingtheoryanddecoding algorithms. In particular, wecitethepapersbyReedandSolomon (1960)onReed-Solomon codes,thepapersbyHocquenghem (1959)andBoseandRay-Chaudhuri (1960a.b)onBCHcodes,andthePh.Ddissertation ofForney(l966a)on concatenated codes.Theseworkswerefollowed bythepapersofGoppa(1970, 1971)ontheconstruction ofanewclassoflinearcycliccodes,nowcalled Goppacodes(seealsoBerlekamp, 1973),andthe'paperofJustesen (1972)on aconstructive technique forasymptotically goodcodes.Duringthisperiod, workondecoding algorithms wasprimarily focusedonBCHcodes.Thefirst decoding algorithm forbinaryBCHcodeswasdeveloped byPeterson (1960). Anumberofrefinements andgeneralizations byChien(1964),Forney(1965), Massey(1965),andBerlekamp (1968)ledtothedevelopment ofacomputa­ tionallyefficientalgorithm forBCHcodes,whichisdescribed indetailbyLin andCostello (1983). Inparallelwiththesedevelopments onblockcodesarethedevelopments in convolutional codes,whichwereinvented byElias(1955).Themajorproblem inconvolutional codingwasdecoding. Wozencraft andReiffen(1961)de­ scribedasequential decoding algorithm forconvolutional codes.Thisalgo­ rithmwaslatermodified andrefinedbyFano(1963),anditisnowcalledthe Fonoalgorithm. Subsequently, thestackalgorithm wasdevisedbyZiganzirov (1966)andJelinek(1969),andtheViterbialgorithm wasdevisedbyViterbi (1967).Theoptimality andtherelatively modest complexity forsmall constraint lengthshaveservedtomaketheViterbialgorithm themostpopular indecoding ofconvolutional codeswithK""10. Oneofthemostimportant contributions incodingduringthe19705wasthe workofUngerboeck andCsajka(1976)oncodingforbandwidth-constrained channels. Inthispaper,itwasdemonstrated thatasignificant codinggaincan beachieved through theintroduction ofredundancy inabandwidth­ constrained channelandtrelliscodesweredescribed forachieving' codinggains of3-4dB.Thisworkhasgenerated muchinterestamongresearchers andhas ledtoalargenumberofpublications overthepast10years.Anumber of references canbefoundinthepapersbyUngerboeck (1982,1987)andForney et01.(1984).Additional papersoncoded'modulation forbandwidth­ constrained channels mayalsobefoundintheSpecialIssueonVoiceband Telephone DataTransmission, IEEEJournal onSelected AreasinCom­ municOlion (September 1984).Acomprehensive treatment oftrellis-coded modulation isgiveninthebookbyBiglierietof.(1991). Inaddition tothereferences givenaboveoncoding,decoding, andcoded signaldesign,weshouldmention thecollection ofpaperspublished bythe IEEEPressentitledKeyPapersintheDevelopment ofCodingTheory,edited byBerlekamp (1974).Thisbookcontains important papersthatwere published inthefirst25yearsofcodingtheory.WeshouldalsocitetheSpecial PROBLEMSIssueonError-Correcting Codes,IEEETransactions onComtlllllllcatiollS (October 1971). 8-1Thegenerator matriKforalinearbinarycodeis [0 0 I I 1 G=0 I 0 0 1 1 0 0 I IoI]1 1 I 0 aExpress GinsystematicIIIPIform. bDetermine theparitycheckmatrixHforthecode. cConstruct thetableofsyndromes forthecode. dDetermine theminimum distance ofthecode. eDemonstrate thatthecodewordcorresponding totheinformatin sequence ]0] isorthogonal toH. 8-2Listthecodewordsgenerated bythematrices givenin(8-1-35) and(8-1-37), and. thus,demonstrale thatthesematrices generate thesamesetofcodewords. 8-3Theweightdistribution ofHamming codesisknown.Expressed asapolynomial in powersofx,lheweighIdistribution forthebinaryHamming codesofblocklength nIS "A(x)=LA,x' 1-0 I =--[II+x)"+n(l+x)'''""(I-x)"""")n+I whereA,isthenumberofcodewordsofweight i.Usethisformula todetermine theweighIdistribulion ofthe(7.4)Hamming codeandcheckyourresultwilhthe listofcodewordsgiveninTable8-1-2. 8-4Thepolynomial g(p)=p'+p +I isthegenerator forthe(15,]1)Hamming binarycode. aDelermine ageneralor matrixGforthiscodeinsystemalie form. bDetermine thegeneralor polynomial forthedualcode. 8-5Forthe(7,4)cyclicHamming codewithgenerator polynomial g(p)=p'+1"+I. eonstrucl an(8.4)extended Hamming codeandlistallthecodewords.Whatis d"'"fortheextended code'? 8-6An(8,4)linearhlockcodeise(mstrueted byshortening a(15.II)Hamming code generated bythegeneral<>r polynomial g(p)=,,'+"+I. aConSiruet thecodewordsofthe(X.4)codeandlistthem. bWhatistheminimum distanceofthe(8,4)code" 8-7ThepolynomialI''' +Iwhenfaclorcd yields r"+1=(,,"+,,'+1)(,,'+" ,+".'+"+I) X(,,'+"+1)("'+,,+1)(/+I) aConstruct asystematic (15.5)codeusingthegenerator pol.vllol1lial lif1')=(I"+IJ+1"+I'+1)(II'+-1'+1)(1'.'+1'+I) CHAPTER K:BLOCK ANDCONVOLUTIONAL CHANNEL CODES519 bWhatistheminimum distanceofthecode? cHowmanyrandomerrorspercodewordcanbecorrected? dHowmanyerrorscanbedetected bythiscode? eListthecodewordsofa(15,2)codeconstructed fromthegenerator polynomial g(p)=(p"+I)/(p'+p+I) anddetermine theminimum distance. 8-8Construct theparitycheckmatricesHIandH,corresponding tothegenerator matrices G,andG,givenby(8-1-34)and(8-1-35), respectively. 8-9Construct anextended (8,4)codefromthe(7,4)Hamming codebyspecifying the generator matrixandtheparitycheckmatrix. 8-10Asystematic (6.3)codehasthegenerator matrix [I 0 0 I I O~]G=0 I 0 0 I o0 I 1 0 Construct thestandard arrayanddetermine thecorrectable errorpatterns and theircorresponding syndromes. 8-11Construct thestandard arrayforthe(7,3)codewithgenerator matrix G=[~~~:~011] 00101 anddetermine thecorrectable patterns andtheircorresponding syndromes_ 8-12Determine thecorrectable errorpatterns(ofleastweight)andtheirsyndromes for thesystematic (7,4)cyclicHamming code. 8·13Provethatifthesumoftwoerrorpatterns e,ande,isavalidcodewordC,then eachpatternhasthesamesyndrome. 8-14Letg(p)=p"+p.+p'+p'+Ibeapolynomial overthebinaryfield. aFindthelowest-rate cycliccodewhosegenerator polynomial isg(p).Whatis therateofthiscode? bFindtheminimum distance ofthecodefoundin(a). eWhatisthecodinggainforthecodefoundin(a)_ 8-15Thepolynomial g(p)=p+loverthebinaryfieldisconsidered. aShowthatthispolynomial cangenerate acycliccodeforanychoiceofn.Find thecorresponding k. bFindthesystematic formofGandHfortheeodegenerated byg(p). eCanyousaywhattypeofcodethisgenerator polynomial generates? 8-16Designa(6,2)cycliccodebychoosing theshortestpossiblegenuatorpolynomial. aDetermine thegenerator matrixG(inthesystematic form)forthiscodeand findallpossible codewords. bHowmanyerrorscanbecorrected bythiscode? 8·17Provethatanytwon-tuples inthesamerowofastandard arrayaddtoproduce a validcodeword. 8-18Beginning witha(15.7)BCHcode,construct ashortened (12.4)code.Givethe generator matrixfortheshortened code. 8-19InSection8-1-2,itwasindicated thatwhenan(n,k)Hadamard codeismapped intowaveforms bymeansofbinaryPSK,thecorresponding M=2"waveforms 530 DI(JITAL COMMl'!':ICAlIONS Heorthogonal. Determine thebandwidth expansion factorfortheMorthogonal waveforms andcompare thiswiththebandwidth requirements oforthogonal FSK detected coherently. 8-20Showthatthesignaling waveforms generated fromamaximum-length shift­ registercodebymapping eachbitinacodewordintoabinaryPSKsignalare equicorrelated withcorrelation coefficient p,=-Ij(M-I),i.e., theMwaveforms lormasimplexset. 8-21Compute theerrorprobability obtained witha(7,4)Hamming codeonan AWGNchannel, bothforhard-decision andsofl-decision decoding. Use(H-I-50), (8-1-52). {8-1-82), (8-1-90), and(8-1-91). 8-22UsetheresultsinSection2-1-6toobtaintheChernoff boundforhard-decision decoding givenby(8-1-89) and(8-1-90). Assume thatIheall-zerocooewordis transmilled anddetermine anupperboundontheprobability thatcodewordc.... havingweightw""isselected. Thisoccursif)w",ormorebilsareinerror.To applytheChernol! bound.defineasequence ofw'"randomvariables as x=II,-1withprobability p wilhprobability I -P wherei=1,2.'...w""andpIStheprobability oferror.FortheESC.the{X,lare statistICally independent. 8-23Aconvolutional codeisdescribed by g,=[I00].g,=[I01].g,=[IIJ aDrawtheencoder corresponding tothiscode. bDrawthestale-transition diagram forthiscode. cDrawthetrellisdiagram forthiscode. dFindthetransferfunction andthefreedistance ofthiScode. eVerifywhether ornotthiscodeiscatastrophic. 8-24Theconvolutional codeofProblem 8-23isusedfortransmission overa AWGN channel withhard-decision decoding. Theoutputofthedemodulator detector is (101001011110111 ...).UsingtheViterbialgorithm, findthetransmitted sequence. 8-25RepeatProblem 8-23foracodewith g,=[10].g,=[10I],g,=[1II 8-16Theblockdiagramofabinaryconvolutional codeisshowninFig.P8-26. aDrawthestatediagram forthecode. bFindthetransferfunction ofthecode,T(D). cWhatisd"wtheminimum freedistanceofthecode',' FIGURE P8-26L..-__-.{+ FIGURE P8·27 FlGURE P8·2SCHAPTER l'l:BLOCK Af\;DCONVOUJTlO!'lAL CHANNEL CODES531 I ~1 2 ) dAssume thatamessage hasbeenencoded bythiscodeandtransmitted overa binary-symmetric channelwithanerrorprobability ofp=10-'.Ifthereceived sequence isr=(1l0,110, 110, 111,010,101,101), usingtheViterbialgorithm. findthetransmitted bitsequence. eFindanupperboundtothebiterrorprobability ofthecodewhentheabove binary-symmetric channelisemployed. Makeanyreasonable approximation. 8·27Theblockdiagram ofa(3,I)convolutional codeisshowninFig.P8-27. aDrawthestatediagramofthecode. bFindthetransferfunctionT(D)ofthecode. cFindtheminimum freedistance (dr,..)ofthecodeandshowthecorresponding path(atdistance dr".fromtheall-zerocodeword)onthetrellis. dAssume thatfourinformation bits(x,.x"x"x.). followed hytwozerobits. havebeenencoded andsentviaabinary-symmetric channel withcrossover probability equalto0.1.Thereceived sequence is(111,III.111,111,lli.Ill). UsetheVllerbidecoding algorithm tofindthemostlikelydatasequence. 8-28Intheconvolutional codegenerated bytheencodershowninFig.P8-28. aFindthetransferfunctionofthecodeintheformT(N,D). bFindd"..ofthecode. cIfthecodeisusedonachannelusinghard·decision Viterbidecoding, assuming thecrossover probability ofthechannel isp=10-·,usethehard-decision bound tofindanupperbDundontheaveragebiterrorprobability ofthecode. ~ ~,gl=IOOIIL'y g,={I10J ----' 8·29FigureP8-29depictsarate1/2,constraint lengthK=2,convolutional code. aSketchthetreediagram, thetrellisdiagram, andthestatediagram. bSolveforthetransferfunctionT(D,N.J)and,fromthis,specifytheminimum freedistance. FIGURE PJl.29Input binary sequenceOutput binary sequence FIGURE P8-JO FIGURE 1'8-36532 DlGJTAL COMMUf"lCATlONS 8-30Arate1/2,K=3,binaryconvolutional encoder isshowninFig.PS-30. aDrawthetreediagram, thetrellisdiagram, andthestatediagram. bDetermine thetransferfunctionT(D,N,J)and,fromthis,specifytheminimum freedistance. 8-31Sketchtheconvolutional encoders forthefollowing codes: arate1/2,K=S,maximum freedistancecode(Table8-2-1); brate1/3,K=S,maximum freedistancecode(Table8-2-2); crate2/3,K=2,maximum freedistancecodeCTable8-2-8). 8·32Drawthestatediagram fortherate'2/3,K=2,convolutional codeindicated in Problem 8-31(c)and,foreachtransition, showtheoutputsequence andthe distanceoftheoutputsequence fromtheall-zerosequence. 8-33Consider theK=3,rate1/2,convolutional codeshowninFig.P8-30.Suppose thatthecodeisusedonabinarysymmetric channelandthereceived sequence for thefirsteightbranches is0001100000001001.Tracethedecisions ona trellisdiagram andlabelthesurvivors' Hamming distance metricateachnode level.Ifatieoccursinthemetricsrequired foradecision, alwayschoosetheupper path(arbitrary choice). 8-34Usethetransfer function derived inProblem 8-30fortheR,=1/2.K=3, convolutional codetocompute theprobability ofabiterrorforanAWGN channel withCa)hard-decision and(b)soft-decision decoding. Compare the performance byplottingtheresultsofthecomputation onthesamegraph. 8-35Usethegenerators givenby(8-2-36)toobtaintheencoderforadual·3.rate1/2 convolutional code.Determine thestatediagram andderivethetransferfunction T(D,N,J). 8-36Drawthestatediagram fortheconvolutional codegenerated bytheencoder showninFig.P8-36and,thus,determine ifthecodeiscatastrophic or noncatastrophic. Also,giveanexample ofarate1/2,K=4.convolutional encoder thatexhibitscatastrophic errorpropagation. 8-37AtrelliscodedsignalisformedasshowninFig.P8-37byencoding onebitbyuse CHAPTER H:BLOCK ANDCONVOLUTIONAL CHANNEL ("ODES533 CD CD}Codedbib @ FIGUREP8-37@ 0~}UncodedCD--------<0 ~ bi~s @) 00 ofarate1/2convolutional code,whilethreeadditional information bitsareleft uncoded. Perform thesetpartitioning ofa32-QAM (cross)constellation and indicatethesubsetsinthepartition. Byhowmuchisthedistance between adjacent signalpointsincreased asaresultofpartitioning? 8-38LetXIandX,betwocodewordsoflengthnwithdistance dandassumethatthese twocodewordsaretransmitted viaabinary-symmetric channel withcrossover probability p.LetP(d)denotetheerrorprobability intransmission ofthesetwo codewords. aShowthat '"P(d),,;;2: v'P(YiIXI)P(YiIX2) i"'l wherethesummation isoverallbinarysequences Yi' bFromtheabove,conclude that P(d)"[4p(1-P)Jm 9 SIGNAL DESIGN FOR BAND-LIMITED CHANNELS Inprevious chapters, weconsidered thetransmission ofdigitalinformation throughanadditive gaussian noisechannel. Ineffect,nobandwidth constraint wasimposed onthesignaldesignandthecommunication systemdesign. Inthischapter, weconsider theproblem ofsignaldesignwhenthechannel isband-limited tosomespecified bandwidth WHz.Underthiscondition. the channel maybemodeled asalinearfilterhavinganequivalent lowpass frequency response C(f)thatiszeroforIfI>W. Thefirsttopicthatistreatedisthedesignofthesignalpulseg(t)ina linearlymodulated signal,represented as u(t)=2:I"g(t-nT) " thatefficiently utilizesthetotalavailable channelbandwidth W.Weshallsee thatwhenthechannelisidealforIfI.,W.asignalpulsecanbedesigned that allowsustotransmit atsymbolratescomparable toorexceeding thechannel bandwidth W.Ontheotherhand,whenthechannel isnotideal. signal transmission atasymbolrateequaltoorexceeding Wresultsinintersymbol interference (lSI)amonganumberofadjacent symbols. Thesecondtopicthatistreatedin[hischapteristheuseofcodingtoshape thespectrum ofthetransmitted signaland,thus,toavoidtheproblem oflSI. Webeginourdiscussion withageneralcharacterization ofband-limited. linearfilterchannels. 9-1CHARACTERIZATION OFBAND-LIMITED CHANNELS Ofthevarious channels available fordigitalcommunications, telephone channels arebyfarthemostwidelyused.Suchchannels arecharacterized as 534 CHAPTER 'I.SI{iNAI, DESIGN fURRASD-UMlTFO ('HA~NUS 535 band-limited linearfilters.Thisiscertainly thepropercharacterization when frequency-division ~ultiplexing (fDM) isusedasameansforestablishing channels inthetelephone network. Recentadditions tothetelephone network employpulse-code modulation (PCM)fordigitizing andencoding theanalog signalandtime-division multiplexing (TDM)forestablishing multiple chan­ nels.Nevertheless, filtering isstillusedontheanalogsignalpriortosampling andencoding. Consequently. eventhoughthepresent telephone network employs amixtureofFDMandTDMfortransmission, thelinearfiltermodel fortelephone channels isstillappropriate. Forourpurposes, aband-limited channel suchasatelephone channel will becharacterized asalinearfilterhavinganequivalent lowpass frequency response characteristic C(f).Itsequivalent lowpass impulse response is denoted bye(t}.Then,ifasignaloftheform (9-1-1) istransmitted overabandpass telephone channel, theequivalent lowpass received signalis r,(t)=fxv(r)e(r-r)dr+e(t) (9-1-2) wheretheintegral represents theconvolution ofe(t)withv(t).andz(t) denotestheadditive noise.Alternatively, thesignaltermcanberepresented in thefrequency domainasV(f)C(J). whereV(f)istheFourier transform of V(l). Ifthechannel isband-limited toWHzthenC(f)=0forIfI>W.Asa consequence, anyfrequency components inV(f)aboveIfI~Wwillnotbe passedbythechannel. Forthisreason, welimitthebandwidth ofthe transmitted signaltoWHzalso. Withinthebandwidth ofthechannel, wemayexpress thefrequency response C(f)as C(f}~lC(fllef8(f> (9-1-3) where1C(f)1istheamplitude response characteristic and6(f)isthephase response characteristic. Furthermore, theenvelope delaycharacteristic is definedas r(f)~_~d6(f) 2;rdf(9-1-4) AchannelissaidtobenondislOrting oridealiftheamplitude response 1C(f)1is constant forallIfIos;;Wand6(f)isalinearfunction offrequency, i.e.,r(f)isa constant forallIfIos;;w.Ontheotherhand,if1C(f}1isnotconstant forall IfI,;;;W,wesaythatthechannel distorts thetransmitted signalVU)in amplitude, and,ifr(f)isnotconstant forallifI,,;;;W,wesaythatthechannel distortsthesignalV(f)indelay. . 536 DIGITAl. COM:\-1!;.J'''I('ATI()NS -5T--47-:'T-2T -SF--4Tn-Tl{o/27:'7"47'5T 2T_IT4"~T-ST--4T-3T-27-T()T~Tn4T51' FIGURE 9-1-1 Effectofchannel distonion: (a)channellOpur: (h)ch,mne! oulpuC (£"1equalizer output. Asaresultoftheamplitude anddelaydistortion causedbythenonideal channel frequency response characteristic C(f),asuccession ofpulsestrans­ mittedthrough thechannel atratescomparable tothebandwidth It'an.: smeared tothepointthattheyarenolongerdistinguishable aswell-defined pulsesatthereceiving terminal. Instead, theyoverlap and,thus,wehave intersymbol interference. Asanexample oftheeffectofdelaydistortion ona transmitted pulse,Fig.9-1-1(a) illustrates aband-limited pulsehavingzeros periodically spacedintimeatpointslabeled±T,±2T,etc.Ifinformation is conveyed bythepulseamplitude, asinPAM.forexample, thenonecan transmit asetjuence ofpulses,eachofwhichhasapeakattheperiodic zerosof theotherpulses.However, transmission ofthepulsethrough achar.nel modeled ashavingalinearenvelope delaycharacteristic T(l)[quadratic phase 8(l)]resultsinthereceived pulseshowninFig.9-I-l(h) havingzero-crossings thatarcnolongerperiodically spaced.Consetjuently. asequence ofsuccessive pulseswouldbesmeared intooneanother andthepeaksofthepulseswould nolongerbedistinguishable. Thus.thechannel delaydistortion resultsin intersymbol interference. Aswillbediscussed inChapter 10.itispossible to compensate for thenonidealfretjuency response characteristic (Ifthechannel bymeofafilterorequalizer atthedemodulator, Figure9-1-1(e)illustrates the outputofalinearequalizer thatcompen",tes forthelincardistortion inthe channel. . Thcextcntoftheintersyrnbol intnfercnce onatelephone channcl canbe CHAPTER~: SIGNAL DESIGN FORBAND-LIMITED CHANNELS 537 1.25 1.00• § "0.a0.75~Q. •EU..05"0 !!. 0.250 U>c OJ4 3 2 o 1000 2000 3000 0 1000 2000 )000 Frequency (Hz) FrequenC)' (Hz) FIGURE 9-1-2Average amplitude anddelaycharacteristic5 ofmedium-range telephone channel. appreciated byobserving afrequency response characteristic ofthechannel. Figure9-1-2illustratesthemeasured averageamplitude anddelayasfunctions offrequency foramedium-range (180-725 miltelephone channel ofthe switched telecommunications network asgivenbyDuffyandTratcher (1971). Weobservethattheusablebandofthechannelextendsfromabout300Hzto about3000Hz.Thecorresponding impulseresponse ofthisaveragechannel is showninFig.9-1-3.Itsduration isabout10ms.Incomparison, thetransmitted symbolratesonsuchachannelmaybeoftheorderof2500pulsesorsymbols persecond.Hence,intersymbol interference mightextendover20-30symbols. Inaddition tolineardistortion, signalstransmitted through telephone channels aresubjecttootherimpairments, specifically nonlinear distortion. frequency offset,phasejitter,impulsenoiseandthermalnoise. Nonlinear distortion intelephone channels arisesfromnonlinearities in FIGURE 9-1·3Impulse response ofaveragechannelwithamplitude anddelayshowninFig.9-1-2. 05 0.4 ·0.3 0.2 •0.1 "C.3 'E.0 E 10 .. -0.1 "ijrneems) ---0.2 --f1.3 -0.4 -05 538 DIGITAL ('OM~flINl("ATlONS amplifiers andcompandors usedinthetelephone system. Thistype01 distortion isusuallysmallanditisverydifficulttocorrect. Asmallfrequency offset.usuallylessthan5Hz,resultsfromtheuseof carrierequipment inthetelephone channel. Suchanoffsetcannotbetolerated inhigh-speed digitaltransmission systemsthatusesynchronous phase-coherent demodulation. Theoffsetisusuallycompensated forbythecarrierrecovery loopinthedemodulator. Phasejitterisbasically alow-index frequency modulation ofthetransmitted signalwiththelowfrequency harmonics ofthepowerlinefrequency (50-60Hz).Phasejitterposesaseriousproblem indigitaltransmission ofhigh rates.However, itcanbetrackedandcompensate<l for,tosomeextent.atthe demodulator. Impulse noiseisanadditive disturbance. Itarisesprimarily fromthe switching equipment inthetelephone system.Thermal (gaussian) noiseisJlso presentatlevelsof20-30dBbelowthesignal. Thedegreetowhichonemustbeconcerned withthesechannelimpairments depends onthetransmission rateoverthechannel andthemodulation technique. Forratesbelow1800bits/s(R/W<1),onecanchooseamodula­ tiontechnique, e.g.,FSK,thatisrelatively insensitive totheamount of distortion encountered ontypicaltelephone channels fromallthesourceslisted above.Forratesbetween 1800and2400bits/s(R/W=1),amorebandwidth­ efficientmodulation technique suchasfour-phase PSKisusuallyemployed. At theserates,someformofcompromise equalization isoftenemployed to compensate fortheaverageamplitude anddelaydistortion intnechannel. In addition, thecarrierrecovery method isdesigned tocompensate forthe frequency offset.Theotherchannelimpairments arenotthatseriousIntneir effectsontheerrorrateperformance attheserates.Attransmission rates above2400bits/s(R/W>1),bandwidth-efficient codedmodulation techniques suchastrellis-coded QAM,PAM,andPSKareemployed. Forsuchrates, specialattention mustbepaidtolineardistortion, frequency offset,andphase jitter.Lineardistortion isusuallycompensated forbymeansofanadaptive equalizer. Phasejitterishandledbyacombination ofsignalco:signandsome typeofphasecompensation atthedemodulator. Atratesabove9600bits/s, specialattention mustbepaidnotonlytolineardistortion, phasejitter,and frequency offset,butalsototheotherchannelimpairments mentioned above. Unfortunately, achannelmodelthatencompasses alltheimpairments listed abovebecomes difficulttoanalyze. Formathematical tractability thechannel modelthatisadopted inthisandthenexttwochapters isalinearfilterthat introduces amplitude anddelaydistortion andaddsgaussian noise. Besides thetelephone channels, thereareotherphysical channels that exhibitsomeformoftimedispersIOn, andthus,introduce intersymbol interference. Radiochannels suchasshortwave ionospheric propagation (HF) andtropospheric scatteraretwoexamples oftime-dispersive channels. Inthese channels, timedispersion and,hence,intersymbol interferenj:e istheresultof multiple propagation pathswithdifferent pathdelays.Thenumberofpaths CHAPTER 'I:SIGNAL DESIGN FORBAND-LIMITED rHA~~FlS 539 ,{) 9 8 7~';r.c 6.:J, 5~ 110 Co 4 ~ '~ 3u '"2 6080 4020o/ / /7 / / /8>?-----------.,L..----------¥9./ 7 .~ '00~IOO ~80--60-40-20 FrequerKy (H7) FIGLIRE 9·1·4Scattering function ofamedium~range tropospheric scatterchannel. andtherelativetimedelaysamongthepathsvarywithtime,and,forthis reason,theseradiochannels areusuallycalledtime-variant multipath channels. Thetime-variant multipath conditions giverisetoawidevarietyoffrequency response characteristics. Consequently thefrequency response characterization thatisusedfortelephone channels isinappropriate fortime-variant multipath channels. Instead, theseradiochannels arecharacterized statistically, as explained inmoredetailinChapter 14,intermsofthescattering function. which,inbrief,isatwo-dimensional representation oftheaverage received signalpowerasafunction ofrelativetimedelayandDoppler frequency. Forillustrative purposes, ascattering function measured onamedium-range (150mi)trophospheric scatterchannel isshowninFig.9-1-4.Thetotaltime duration (multipath spread)ofthechannelresponse isapproximately 0.7p..son theaverage, andthespreadbetween "half-power points"inDoppler fre­ quencyisalittlelessthan1Hzonthestrongest pathandsomewhat largeron theotherpaths.Typically, ifoneistransmitting atarateofIll'symbols/s over suchachannel, themultipath spreadof0.7p.swillreslJltinintersymbol interference thatspansaboutsevensymbols. Inthischapter, wedealexclusively withthelineartime-invariant tilter modelforaband-limited channel. Theadaptive equalization techniques presented inChapters 10and11forcombating intersymbol interference are alsoapplicable totime-invariant multipath channels. underthecondition that 540 DI(jITAL COMMl:1''IlICATI01'o:S thetimevariations inthechannelarerelatively slowincomparison tothetotal channelhandwidth or,equivalently, tothesymboltransmission rateoverthe channel. 9-2SIGNAL DESIGN FORBAND-LIMITED CHANNELS ItwasshowninChapter 4thattheequivalent lowpasstransmitted signalfor severaldifferent typesofdigitalmodulation techniques hasthecommon form v(t)=LI"g(t-nT) "C~0(9-2-1) where{In}represents thediscreteinformation-bearing sequence ofsymbols and g(t)isapulsethat,forthepurposes ofthisdiscussion. isassumed tohavea hand-limited frequency response characteristic G(f).i.e"G(f)=0forIfI>w. Thissignalistransmitted overachannel havingafrequency response C(f), alsolimitedtoIfI,;;W.Consequently, thereceived signalcanberepresented as wherew T,(t)=Ll"h(t-nT)+z(t) "~() h(t)=fwg(T)C(t-T)dr(9-2-2) (9-2-3) andz(t)represents theadditive whiteGaussian noise. Letussuppose thatthereceived signalispassedfi'rstthrough afilterand thensampled atarateI/Tsamples/s. Weshallshowinasubsequent section thattheoptimum filterfromthepointofviewofsignaldetection isone matched tothereceived pulse.Thatis,thefrequency response ofthereceiving filterisH*(f),Wedenotetheoutputofthereceiving filteras = y(t)=LI"x(t-nT)+v(t) "=()(9-2-4) wherex(t)isthepulserepresenting theresponse ofthereceiving filtertothe inputpulseh(t)andv(t)istheresponse ofthereceiving filtertothenoisez(t). Now,ify(r)issampled attimest=kT+Tn.k=O.1.,,..wehave = y(kT+To)==Y.=Ll"x(kT-nT+To)+v(kT+r,,) 11=0 or,equivalently.(9-2-5) y"=2:l"xJ.;. -It+v);. 1/=0k=0,I,.,. (9-2-6) CHAPTER 9:SlG"':AL DESI(iN f-ORHAS[)-U\-111 (-DClt:-\V\I-oLS541 where Toisthetransmission delaythrough thechannel. Thesamplevaluescan beexpressed asI'Yk=x,,(lk+--2:luX'-n)+v,.k=0,I,_.. (1}-2-7) Xo,t=1) /I#!.: Weregardx"asanarbitrary scalefactor,whichwearbitrarily setequalto unityforconvenience. Then .\-'.4,=1.4,+Lll1x.4,-tI+VI<. lI,n /I~k(1}-2-H) Theterm1,represents thedesiredinformation symbolatthekthsampling. instant,theterm 2:fllxJ.;. /I IT(I """.4, represents theintersymbol interference (lSI),andv,istheadditive gaussian noisevariable atthekthsampling instant. Theamountofintersymbol interference andnoiseinadigitalcommunica­ tionssystemcanbeviewedonanoscilloscope. ForPAMsignals, wecan displaythereceived signaly(l)ontheverticalinputwiththehorizontal sweep ratesetatliT.Theresulting oscilloscope display iscalledaneve(>lI/Ii'n! because ofitsresemblance tothehumaneye.Forexample. Fig.lJ-2-1 illustrates theeyepatterns forbinaryandfour-level PAMmodulation. The? effectoflSIistocausetheeyetoclose.thereby reducing themarginfor additive noisetocauseerrors.Figure9-2-2graphically illustrates theeffectof intersymhol interference inreducing theopening ofabinaryeye.Notethat intersymbo! mterference distortstheposition ofthezero-crossings andcauses FIGURE 9~2·1 Examples ofeyepatterns forbinaryandquaternary amplitude shiftkt'ying(nfPAM). BINARY QUATERNARY 542 DIGITAL CDMMUNICATIONS Optimum ~ampling FIGURE 9w2~2Effectofintersymbol interference oneyeopening.Sensitivity toliming error Peakdhwrtionrime Di.~tonion of zerocrossings Noi~margin areduction intheeyeopening. Thus.itcausesthesystemtobemoresensitive toasynchronization error. ForPSKandQAMitiscustomary todisplaythe"eyepattern" asa two-dimensional scatterdiagram illustrating thesampled values{Yk}that represent thedecision variables atthesampling instants.Figure9-2-3illustrates suchaneyepatternforan8-PSKsignal.Intheabsence ofintersymbol interference andnoise,thesuperimposed signalsatthesampling instantswould resultineightdistinctpointscorresponding totheeighttransmitted signal phases.Intersymbol interference andnoiseresultinadeviation ofthereceived samples {y.}fromthedesired8-PSKsignal.Thelargertheintersymbol interference andnoise,thelargerthescallering ofthereceived signalsamples relativetothetransmilled signalpoints. Below,weconsider theproblem ofsignaldesignunderthecondition that thereisnointersymbol interference atthesampling instants. 9-2-1DESIGN OFBAND-LIMITED SIGNALS FORNO INTERSYMBOL INTERFERENCE-THE NYQUIST CRITERION Forthediscussion inthissectionandinSection9-2-2.weassumethatthe band-limited channelhasidealfrequency response characteristics, i.e..C(f)=I FIGURE 9-2-3Two-dimensional digital"eyepallerns'-'00!iF'iF. 0•... ~ 0•..... 0• #-if Tral'l';lTIi((ed Received ...ignal'ample ... eight-pha-.e signal attheoutputofdemodulator lui (h) CHAPlTR ')S1('!\Al.Dnj(jS FORBAND·LlMITED ('HANSELS 543 forIfI~W.Thenthepulsex(t)hasaspectral characteristic X(f)=IG(fW, where x(t)=r~X(/)e1,nr, df (9-2-9) Weareinterested indetermining thespectral properties ofthepulsex(t)and, hence,thetransmitted pulseget),thatresultsinnointersymbol interference, Since 00 YI<=Ik+LII/xk --/1+VI< IfC-O II"'/':' thecondition fornointersymbol interference is {I(k=0)x(t=kT)=x = •0(k"O)(9-2-10) (9-2-11) Below,wederivethenecessary andsufficient condition onX(/)inorderfor X(f)tosatisfytheaboverelation. Thiscondition isknownastheNyquist pulse-shaping criterion orNyquistcondition forzerolSIandisstatedinthe following theorem. Theorem (Nyquist) Thenecessary andsufficient condition forx(t)tosatisfy {I(n=O)xnT-( ) - 0 (n"0) isthatitsFouriertransform X(f)satisfy 2:XU+miT)=T m=-0:; Proof Ingeneral,x(t)istheinverseFouriertransform ofXC/).Hence, Atthesampling instants t=nT,thisrelation becomes x(nT)=[~X(/)e/2>rfnTdf(9-2-12) (9-2-13) (9-2-14) (9-2-15) S44 DJGJTALCOMMlJNJC ATJONS Letusbreakuptheintegral in(9-2-15) intointegrals covering thefinite rangeoflIT.Thus,weobtain xI(2m+1),'2T x(nT) ~~ X(f)ei2'fnT df m=-x (2m~l)J2T xJ'I2T ~m":f~-1'2TX(f+mlT)eJ2nfnT de JII2r[x ] ~.1"T m~_xX(f.,.mlT)eJ2nfnTdf fl/2T ~B(f)eJ'nfn Tdf -1'2T wherewehavedefinedB(f)as x B(f)~~X(f+mIT) m=-x;(9-2-16) (9-2-17) Obviously BU)isaperiodic function withperiodliT,and,therefore, itcanbe expanded intermsofitsFourierseriescoefficients ibn}as ~ BU>~L:bneJhnfT n=-ox; where bn~Tf'lorB(f)e-j2nnfTdf -l12T Comparing (9-2-19)and(9-2-16), weobtain bn~Tx(-nT)(9-2-18) (9-2-19) (9-2-20) Therefore, thenecessary andsufficient condition for(9-2-10) tobesatisfied is that b~{T(n=0) n°(n,eO) which,whensubstituted into(9-2-18), yields B(f)=T or,equivalently, x ~X(f+miT)=T m=-:lC(9-2-21) (9-2-22) (9-2-23) CHAPTER lJ:SIGNAL· DESIGN FORBAND-liMIlED CHANNELS S4S ,f[\. FIGURE 9-2-4!_!+w-W 0 W T T PlotofBif)forthecaseT<1/2W.Lw T! T!+W T• (9-2-24) (9·2·25)Thisconcludes theproofofthetheorem. Nowsuppose thatthechannelhasabandwidth ofW.ThenC(f)!E°for IfI>Wand,consequently, X(f)=°forIfI>w.Wedistinguish threecases. 1WhenT<1/2W,or,equivalently, liT>2W,sinceB(f)=L;:-~X(f+ niT)consistsofnonoverlapping replicasofX(f),separated byI/Tasshown inFig.9-2-4,thereisnochoiceforX(f)toensureB(f)...Tinthiscaseand thereisnowaythatwecandesignasystemwithnolSI. 2WhenT=1/2W,or,equivalently, liT=2W(theNyquist rate),the replications ofX(f),separated byI/T,areasshowninFig.9-2-5.Itisclear thatinthiscasethereexistsonlyoneX(f)thatresultsinB(f)=T,namely, XU)={OT(IfI<W) (otherwise) whichcorresponds tothepulse x(t)__sin(miT) (1U) IrtlTsineT ThismeansthatthesmallestvalueofTforwhichtransmission withzerolSIis possible isT=1/2W,andforthisvalue,x(t)hastobeasincfunction. The difficulty withthischoiceofx(t)isthatitisnoncausal andtherefore nonrealizable. Tomakeitrealizable, usuallyadelayed versionofit,i.e., sine[1r(t-to)/T]isusedandtoischosensuchthatfort<0,wehave sinc[1r(I-to)/T]""0.Ofcourse,withthischoiceofx(t),thesampling time nGURE 9-2·SPlotofB(f)forthecaseT=1/2W. t'~ ~I!\./:~r. _! 0w=-!- ! T 2TT I T FIGURE 9·Z-6546 DIGITAL COMMUNICA nONS x~xL;h~: ----L...C>X_----L.:.X'-------., _!-w_1+w! _wW T T T PlotafB(nfarthecaseT>1/2W. mustalsobeshiftedtomT+t".Aseconddifficulty withthispulseshapeisthat itsrateofconvergence tozeroisslow.Thetailsofx(t)decayaslit; consequently, asmallmistiming errorinsampling theoutputofthematched filteratthedemodulator resultsinaninfiniteseriesoflSIcomponents. Sucha seriesisnotabsolutely summable becauseofthe1/trateofdecayofthepulse. and,hence,thesumoftheresulting lSIdoesnotconverge. 3WhenT>1/2W,B(f)consists ofoverlapping replications ofX(f) separated by11T,asshowninFig.9-2-6.Inthiscase,thereexistnumerous choicesforXU)suchthatB(f)==T. Aparticular pulsespectrum, fortheT>I/2Wcase,thathasdesirable spectralproperties andhasbeenwidelyusedinpracticeistheraisedcosine spectrum. Theraisedcosinefrequency characteristic isgivenas(seeProblem 9-11) T X,,(f)=f{1+cos[1r;(If1-121)]} o(0,;;IfI.,;\-/) (1-{3 1+(3)2T':;IfI""IT (IfI>12+/) (9-2-26) where{3iscalledtherollofffactor,andtakesvaluesintherange0.,;{3""1.The bandwidth occupied bythesignalbeyondtheNyquistfrequency 1/2Tiscalled theexcessbandwidth andisusuallyexpressed asapercentage oftheNyquist frequency. Forexample, when{3=ttheexcessbandwidth is50%,andwhen (3=I,theexcessbandwidth is100%.Thepulsex(t),havingtheraisedcosine spectrum, is x(t)sin(miT)cos(1r{3tIT) miT1-4{32t21T2 _.(/T)cos(1C{3tIT)-sine1CI 2 2 21-4f3tiT(9-2-27) CHAPTER 9:SIGNAL DESIGN FORBAND-LIMITED CHAr-.:SELS 547 t:(I) 4T p=o P=O.5 lal I. T (inI 2T!f T FIGURE 9~2~7 Pulseshavingaraisedcosinespectrum. NotethatX(I)isnormalized sothatx(O)=1.Figure9-2-7illustrates theraised cosinespectralcharacteristics andthecorresponding pulsesfor{3=o.land1. Notethatfor(3=O.thepulsereducestox(r)=sinc(miT).andthesymbol rateliT=2W.When(3=1,thesymbolrateisliT=W.Ingeneral, thetails ofX(I)decayas1/13for(3>0.Consequently, amistiming errorinsampling leadstoaseriesoflSIcomponents thatconverges toafinitevalue. Duetothesmoothcharacteristics oftheraisedcosinespectrum, itispossible todesignpractical filtersforthetransmitter andthereceiver thatapproximate theoveralldesiredfrequency response. Inthespecialcasewherethechannel is ideal,i.e.,CU)=1.IfI~W.wehave (9-2-28) whereGT(f)andGR(!)arethefrequency responses ofthetwofilters.Inthis case,ifthereceiver filterismatched tothetransmitter filter.wehave X,,(f)=GT(f)GR(f) =ICTUW Ideally, (9-2-29) andGR(f)=G'f(f).where10issomenominaldelaythatisrequired toensure physical realizability ofthefilter.Thus,theoverallraisedcosinespectral characteristic issplitevenlybetween thetransmitting filterandthereceiving filter.Notealsothatanadditional delayisnecessary toensurethephysical realizability ofthereceiving filter. 548 DI(jITAL COMMl 'NICA"110NS 9·2-2DesignofBand-Limited SignalswithContl'oUed ISI­ Partial·Response Signals Aswehaveobserved fromourdiscussion ofsignaldesignforzerolSI,itIS necessary toreducethesymbolratel/TbelowtheNyquist rateof2W symbols/s torealizepractical transmitting andreceiving filcers.Ontheother hand.suppose wechoosetorelaxthecondition ofzerolSIand,thus,achievea symbol transmission rateof2Wsymbols/s. Byallowing foracontrolled amountoflSI,wecanachievethissymbolrate. Wehavealreadyseenthatthecondition forzerolSIis.f(nT)=0forn""O. However, suppose thatwedesigntheband-limited signaltohavecontrolled lSIatonetimeinstant.Thismeansthatweallowoneadditional nonzero value inthesamples {x(nT)}. ThelSIthatweintroduce isdeterministic or "controlled" and.hence,itcanbetakenintoaccount atthereceiver, as discussed below. Onespecialcasethatleadsto(approximately), physically realizable transmitting andreceiving filtersisspecified bythesamplest x(nT)={oj(n=0,I) (otherwise) Now,using(9-2-20), weobtain {T(n=0,-I) b"=0(otherwise) which,whensubstituted into(9-2-18), yields B(f)=T+TeJ"rfT(9-2-30) (9-2-31) (9-2-32) (9-2-33)Asinthepreceding section,itisimpossible tosatisfytheaboveequation for T<1/2W.However. forT=1/2W,weobtain {_I(I+e""'/I~)(If:<W)X(n=2W o (otherwise) {~e-j"fI2Wcosnf(IfI<W) =W 2W o (otherwise) Therefore, x(t)isgivenby X(I)=sinc(2nWt)+sinc[2n(Wt-!)l (9-2-34) Thispulseiscalledaduobinary signalpulse.Itisillustrated alongwithits tItisconvenient todealwithsamplesofx(t)thatarenormailzed tounity(orn:::::O.I. CHAPTER' SIONAL DESIGN FORBAND·LlMITED CHANNELS 549 -:IT-2~/n :IT~T WIX(l11 I/W Inf-cos­W2W -W fiGURE: 9·2·8 Timedomainandfrequency domaincharacteristics ofaduobinary signal. magnitude spectrum inFig.9-2-8.Notethatthespectrum decaystozero smoothly, whichmeansthatphysically realizable filterscanbedesigned that approximate thisspectrum veryclosely.Thus,asymbolrateof2Wisachieved. Another specialcasethatleadsto(approximately) physically realizable transmitting andreceiving filtersisspecified bythesamples (n=-1) (n=1) (otherwise)(9-2-35) Thecorresponding pulseX(I)isgivenas [n(/+Tl] [n(1--Tl]X(I)=sine T-sine----:r- anditsspectrum is(9-2-36) {_1_(el'rmv_e-prim)=1.sinJrf X(f)= 2W WW ofl~W /I>W(9-2-37) Thispulseanditsmagnitude spectrum areillustrated inFig.9-2-9.Itiscalleda modified duobinary signalpulse.Itisinteresting tonotethatthespectrum of FIGURE 9-2-9 Timedomain andfrequency domaincharacteristics ofamodified duobinary signal. x(rJ' ~T-37 lal4T/nJ W 550 DIGITAL COMMIJNICAT]()/',''S thissignalhasazeroatf=0,makingitsuitable fortransmission overa channelthatdoesnotpassd.c. Onecanobtainotherinteresting andphysically realizable filtercharacteris­ tics,asshownbyKretzmer (1966)andLuckyetal.(1968),byselecting different valuesforthesamples (x(n/2W)} andmorethantwononzero samples. However, asweselectmorenonzero samples, theproblem of unraveling thecontrolled lSIbecomes morecumbersome andimpractical. Ingeneral,theclassofbandlimited signalspulsesthathavetheform x(t)=fx(~)sinc[21l'W(t-nw)] n~-~2W 2(9-2-38) andtheircorresponding spectra {_Ifx(~)e-jnnfIW X(f)=~Wn~-~2W(lfl,,;;;W) (IfI,,;;;W)(9-2-39) arecalledpartial-response signalswhencontrolled lSIispurposely introduced byselecting twoormorenonzero samples fromtheset{x(n/2W)}. The resulting signalpulsesallowustotransmit information symbolsattheNyquist rateof2Wsymbols/s. Thedetection ofthereceived symbolsinthepresence of controlled lSIisdescribed below. Alternative Characterization ofPartial·Response Signals Weconclude thissubsection bypresenting another interpretation ofapartial-response signal.Suppose thatthepartial-response signalisgenerated, asshowninFig. 9-2-10,bypassingthediscrete-time sequence {In}throughadiscrete-time filter x xFIGURE '-2-10 Analternative methodforgenerating apartial-response signal. ~Jlla(l-nn Output H(f)III-w0w CHAPTER "/:SIGNAL DESIC", FORBAND-LIMITED CHA:"oll'iELS SSt withcoefficients Xn'=x(n/2W), n=0.1.....N-I.andusingtheoutput sequence {Bn)fromthisfiltertoexciteperiodically withaninputB,,8(1-nT) ananalogfilterhavinganimpulseresponse sine(211"Wt).Theresulting output signalisidentical tothepartial-response signalgivenby(9-2-38). Since N-I B"=2:X,J'I-1.; Ie0=0(9-2-40) thesequence ofsymbols {B,,}iscorrelated asaconsequence ofthefiltering performed onthesequence {In).Infact,theautocorrelation function ofthe sequence {Bn)is N--IN-I =2:2:xkx,E(ln-k1n+rn-l) k=O'=0 Whentheinputsequence iszero-mean andwhite,(9-2-41) (9-2-42) wherewehaveusedthenormalization E(l~)=1.Substitution of(9-2-42), into (9-2-41)yieldsthedesiredautocorrelation function for{Bn}intheform N--]-Iml cf>(m)=2:xkXk+lml' m=O,±I, ...,±(N-l) k=:O Thecorresponding powerspectraldensityis N-I 4>(/)=2:<b(m)e-J2",mT m=-(N-l) whereT=I/2WandIfI",I/2T=W.(9-2-43) (9-2-44) 9-2-3DataDetection forControlled lSI Inthissection,wedescribe twomethods fordetecting theinformation symbols atthereceiver whenthereceived signalcontains controlled lSI.Oneisa symbol-by-symbol detection methodtbatisrelatively easytoimplement. The second met~odisbasedonthemaximum-likelihood criterion fordetecting a seQuence ofsymbols. Thelattermethodminimizes theprobability oferrorbut isalittlemorecomplex toimplement. Inparticular, weconsider thedetection of-theduobinary andthemodified duobinary partialresponse signals.Inboth 552 DJ(iITAL COMMll!\'ICATIONS cases.weassumethatthedesiredspectralcharacteristic X(f)forthepartial response signalissplitevenlybetween thetransmitting andreceiving filters, i.e..IGT(f)1=IGR(f)1=IX(f)11;2 Thistreatment isbasedonPAMsignals,but itiseasilygeneralized toQAMandPSK. Symbol·by·Symbol Suboptimum Detection Fortheduobinary signal pulse.x(nT)=1.forn=O.I,andzerootherwise. Hence.thesamples atthe outputofthereceiving filter(demodulator) havetheform (9-2-45) where{I,,,}isthetransmitted sequence ofamplitudes and{vm}isasequence of additivegaussian noisesamples. Letusignorethenoiseforthemoment and consider thebinarycasewhere1m=±Iwithequalprobability. Then8mtakes ononeofthreepossible values,namely, 8m=-2,O.2withcorresponding probabilities 1/4,1/2.114.If1m-Iisthedetected symbolfromthe(m-1)th signaling interval, itseffecton8m,thereceived signalinthemthsignaling interval,canbeeliminated bysubtraction, thusallowing 1mtobedetected, This processcanberepeated sequentially foreveryreceived symbol. Themajorproblem withthisprocedure isthaterrorsarisingfromthe additivenoisetendtopropagate, Forexample, if1m-Iisinerror,itseffecton 8misnoteliminated but,infact,itisreinforced bytheincorrect subtraction. Consequently, thedetection of8misalsolikelytobeinerror. Errorpropagation canbeavoidedbyprecoding thedataatthetransmitter insteadofeliminating thecontrolled lSIbysubtraction atthereceiver. The precoding isperformed onthebinarydatasequence priortomodulation. From thedatasequence {Dn}ofIsandOsthatistobetransmitted, anewsequence {P,,}.calledtheprecoded sequence. isgenerated, Fortheduobinary signal,the precoded sequence isdefinedas Pm=Dm8Pm-',m=1,2,... (9-2-46) whereedenotesmodulo-2 subtraction.t Thenweset1m=-1ifPm=0and 1m=1ifPm=I.i.e.,1m=2Pm-1.Notethatthisprecoding operation is identical tothatdescribed inSection4-3-2inthecontextofourdiscussion of anNRZIsignal. Thenoise-free samplesattheoutputofthereceiving filteraregivenby Consequently.=(2Pm-1)+(2Pm-I -1) =2(Pm+Pm-I-1) (9-2-47) (9-2-48) tAlthough thisisidentical tomodulo-2 addition. ilisconvenient toviewtheprecoding operation forduobinary intermsofmodulo-2 subtraction. CHAPTER q:SIGNAL DESIGN FORRAND·lIMITED CHANNELS 553 TABLE 9-2-1BINARY SIGNALING WITHDUOBINARY PULSES Data sequence D" 10 0 0 100 0 10 Precoded sequence P" 0 0 1I000ItI0I10 Transmitted sequence 1m-1-I -1-1-1 11-I1I-I Received sequence B'1 00020-2-202220 020 Decoded sequence Dn 10 00 ()0() () SinceDm=PmEBPm"itfollowsthatthedatasequence Dmisobtained from B",usingtherelation Dm=~Bm+ 1(mod2) (9-2-49) Consequen1ly, ifBm=±2thenDm=0,andif8m=0thenDm=1.An example thatillustrates theprecoding anddecoding operations isgivenin Table9-2-1.Inthepresence ofadditive noise,thesampled outputsfromthe receiving filteraregivenby(9-2-45). Inthiscasey",=8m+Vmiscompared withthetwothresholds setat+1and-I.Thedatasequence {Dn}isobtained according to1hedetection rule D={I(ly",I<I) m0()Yml'"1)(9-2-50) Theextension frombinaryPAMtomultilevel. PAMsignaling usingthe duobinary pulsesisstraightforward. InthiscasetheM-Ievel amplitude sequence {1m}resultsina(noise-free) sequence Bm=lm+/",_', m=1,2,... (9-2-51) whichhas2M-1possible equallyspacedlevels.Theamplitudelevelsare determined fromtherelation lm=2P",-(M-I) (9-2-52) where{Pm}istheprecoded sequence thatisobtained fromanM-leveldata sequence {Dm}according totherelation Pm=Drn8Pm-1(modM) (9-2-53) wherethepossiblevaluesofthesequence {Dm}are0,1,2,...,M-1. Intheabsenceofnoise,thesamplesattheoutputofthereceiving filtermay beexpressed as 8m=1m+[",-I=2[Pm+Pm-I-(M-1)] (9-2-54) 554 DIGITAL COMMUNICATIONS TABLE 9-2-2FOUR-LEVEL SIGNAL TRANSMISSION WITHDUOBINARY PULSES Data sequence D", 0 () 3 2()332 0 0 Precoded sequence P,,, ()0 1I 233 2 322 Transmitted sequence I",-3-3-3-I 33-) -1-)3 Received sequence 8m -6-6-4 1I4620()-2242 Decoded sequence D", U 1I .3 2()3 3 20 0 Hence, P"l+Pm1:::::~Bm+(M-1) SinceDm=Pm+P,,,I(modM),itfollowsthat Dm=!Bm+(M-l) (modM)(9-2-55) (9-2-56) Anexample illustraling multilevel precoding anddecoding isgiveninTable 9-2-2. Inthepresence ofnoise,thereceived signal-plus-noise isquantized tothe nearestofthepossible signallevelsandtherulegivenaboveisusedonthe quantized valuestorecoverthedatasequence. Inthecaseofthemodified duobinary pulse.thecontrolled lSIisspecified bythevaluesx(n/2W) =-1,forn=1,x(n/2W) =1forn=-1,andzero otherwise. Consequently, thenoise-free sampled outputfromthereceiving filterisgivenas Bm=1m-/nr-2 (9-2-57) wheretheM-levelsequence {I",}isobtained bymapping aprecoded sequence according totherelation(9-2-52)and P,,,=Dm!IJp,,, 2(modM) (9-2-58) Fromtheserelations, itiseasytoshowthatthedetection ruleforrecovering thedatasequence {Dm}from{Bm}intheabsenceofnoiseis D",=!B",(modM) (9-2-59) Asdemonstrated above,theprecoding ofthedataatthetransmitter makes itpossible todetectthereceived dataonasymbol-by-symbol basiswithout havingtolookbackatpreviously detected symbols. Thus,errorpropagation is avoided. Thesymbol-by-symbol detection ruledescribed aboveisnottheoptimum detection schemeforpartialresponse signalsduetothememory inherent in CHAPTER qSIGNAL DESIGS FORBAND·L1MITED CHAl'SELS 555 1/2 1/2 1/2 FIGURE 9·2-11 Trellisforduobinary partialresponse signal.-I~......-....,.'-----,.,....---<-~t-11-2t-11--2 -11-2t 1=0 f=T 1=2T t=)T (9-2-60)thereceived signal.Nevertheless, symbol-by-symbol detection isrelatively simpletoimplement andisusedinmanypractical applications involving duobinary andmodified duobinary pulsesignals.Itsperformance isevaluated inthefollowing section. Maximum·Likelihood Sequence Detection Itisclearfromtheabove discussion thatpartial-response waveforms aresignalwaveforms withmemory. Thismemory isconveniently represented byatrellis.Forexample, thetrellis fortheduobinary partial-response signalforbinarydatatransmission is illustrated inFig.9-2-11.Forbinarymodulation, thistrelliscontains twostates. corresponding tothetwopossible inputvaluesof1m,i.e.,1m=±I.Each branchinthetrellisislabeledbytwonumbers. Thefirstnumber ontheleftis thenewdatabit,i.e.,1m+1=±I.Thisnumberdetermines thetransition tothe newstate.Thenumberontherightisthereceived signallevel. Theduobinary signalhasame'mory oflengthL=I.Hence,forbinary modulation thetrellishasSf=2states.Ingeneral, forM-arymodulation. the numberoftrellisstatesisML• Theoptimum maximum-likelihood (ML)sequence detector selectsthemost probable paththrough thetrellisuponobserving Ihereceived datasequence {Ym}atthesampling instants t=mT,m=I,2,....Ingeneral, eachnodeinthe trelliswillhaveMincoming pathsandMcorresponding metrics.Oneoutof theMincoming pathsisselected asthemostprobable, basedonthevaluesof themetricsandtheotherM-Ipathsandtheirmetricsarediscarded. The surviving pathateachnodeisthenextended toMnewpaths,oneforeachof theMpossible inputsymbols, andthesearchprocess continues. Thisis basically theViterbialgorithm forperforming thetrellissearch. Fortheclassofpartialresponse signals,thereceived sequence {Ym.I,;;;m,;;; N}isgenerally described statistically bythejointpdfllYNIIN).where YN=[y,y,'"YNI'andIN=[I,I,'"INI'andN>L.Whentheadditive noiseiszcro-mean gaussian. f(YNIv)is3multivariate gaussian pdf,i.e.. . I I 1 ,-,llYNIN)=(2JfdetC)Vt'exp[-,(Yv-B,,)C(YN-BN)] whereBN=[8,B,...8,,11,isthemeanofthevectory"andCistheNxN covariance matrixofY~.Then.theMLsequence detector selectsthesequence through thetrellisthatmaximizes thepdff(YNII,,). SS6 DIGITAL COMMUNICATIONS Thecomputation forfindingthemostprobable sequence throughthetrellis issimplified bytakingthenaturallogarithms off(YNIIN)'Thus, Inf(YNIIN)=-!Nln(21rdet C)-!(YN-BNYC'(YN-B N)(9-2-61) Giventhereceived sequence {Ym},thedatasequence {Im}thatmaximizes Inf(yNliN)isidentical tothesequence {IN}thatminimizes (yN­ BN)'C-'(YN - BN),i.e. iN=argmin[(YN-BNYC-'(YN - BN)] IN(9-2-62) Themetriccomputations inthetrellissearcharecomplicated bythe correlation ofthenoisesamples attheoutputofthematched filterforthe partialresponse signal.Forexample, inthecaseoftheduobinary signal waveform, thecorrelation ofthenoisesequence {vm}isovertwosuccessive signalsamples. Hence, VmandVm+karecorrelated fork=1anduncorrelated fork>1.Ingeneral, apartialresponse signalwaveform withmemory Lwill resultinacorrelated noisesequence attheoutput of thematched filter,which satisfiesthecondition E[vmvm+k] =0fork>L.Insuchacase,theViterbi algorithm forperforming thetrellissearchmaybemodified asdescribed in Chapter 10. Somesimplification inthemetriccomputations resultifweignorethenoise correlation byassuming thatE(vmvm+k) =0fork>0.Then,byassumption, thecovariance matrixC=(7~IN'where (7~=E[v;,]andINistheNxN identitymatrix.tInthiscase,(9-2-62)simplifies to where=argmin[f(Ym-±XkIm_.)2] IN m=1 k=O L Em=2x.Im_. k=O(9-2-63) (9-2-64)andx.=x(kT)arethesampled valuesofthepartialresponse signalwaveform. Inthiscase,themetriccomputations ateachnodeofthetrellishavetheform DMm(Im)=DMm-,(lm-tl +(Ym-JoX.Im-·r whereDMm(Im)arethedistancemetricsattimet=mT,DMm-,(l m-,)arethe distance metricsattime1=(m-I)Tandthesecondtermontheright-hand sideof(9-2-64)represents thenewincrements tothemetricsbasedonthenew receivecJ. sampleYm' tWeareusingINhc!retoa\'oidconfusion withIN' CHAPTER 9:SIGNAL DESIGN FORBAND-LIMITED CHANNELS 557 Asindicated inSection5-1-4,MLsequence detection introduces avariable delayindetecting eaclltransmitted information symbol. Inpractice, the variable delayisavoided bytruncating· thesurviving sequences toN,most recentsymbols, whereN,»5L.thusachieving afixeddelay.Inthecasethat theMLsurviving sequences attime(=mTdisagreeonthesymboll m-N,.the symbolinthemostprobable surviving sequence maybechosen.Thelossin performance resulting fromthistruncation isnegligible ifNt>5L. 9-2-4SignalDesignforChannels withDistortion InSections 9-2-1and9-2-2,wedescribed signaldesigncriteriaforthe modulation filteratthetransmitter andthedemodulation filteratthereceiver whenthechannelisideal.Inthissection,weperform thesignaldesignunder thecondition thatthechanneldistortsthetransmitted signal.Weassumethat thechannelfrequency response C(f)isknownforIfI..WandthatC(f)=0 forIfI>W.Thecriterion fortheoptimization ofthefilter.responses GT(f)and GR(f)isthemaximization oftheSNRattheoutputofthedemodulation filter orequivalently, attheinputtothedetector. Theadditive channel noiseis assumed tobegaussian withpowerspectral density <t>nn(f).Figure9-2-12 illustrates theoverallsystemunderconsideration. Forthesignalcomponent attheoutputofthedemodulator, wemustsatisfy thecondition (9-2-65) (9-2-66)whereXd(f)isthedesiredfrequency response ofthecascade ofthe modulator, channel, anddemodulator, andtoisatimedelaythatisnecessary toensurethephysicalrealizability ofthemodulation anddemodulation filters. Thedesiredfrequency response Xd(f)maybeselectedtoyieldeitherzerolSI orcontrolled lSIatthesampling instants. Weshallcarryouttheoptimization forzerolSIbyselecting Xd{f)=Xrc(f).whereXrc(f)istheraisedcosine spectrum withanarbitrary rollofffactor. Thenoiseattheoutputofthedemodulation filtermaybeexpressed as V(I)=r~n(l--r)gR(-r)d-r FIGURE 9-Z·1Z Systemmodelforthedesignofthemodulation anddemodulation filters. Input dataModulation filter Grit)Channel C{t) Gaussian noiseDemodulation filleT G.(I)DetectorOutput clata 558 DIGITAL COMMUNICATIONS wheren(t)istheinputtothefilter.Sincen(t)iszero-mean gaussian, v(t)is zero-mean gaussian, withapowerspectraldensity (9-2-67) Forsimplicity, weconsider binaryPAMtransmission. Then,thesampled outputofthernatchedfilteris (9-2-68) where Xoisnormalizedt tounity,1m=±d,andVmrepresents thenoiseterm, whichiszero-mean gaussian" withvariance (9-2-69) (9-2-70)Consequently, theprobability oferroris 1IXP,=- e~y2/'dY=Q(Vd2/u~) V21rdla, Theprobability oferrorisminimized bymaximizing theSNR=d'/~",or, equivalently, byminimizing thenoise-to-signal ratiou;/d'.Butd'isrelatedto thetransmitted signalpowerasfollows: E(l')f~ d'f~Pav=_m_ g~t)dt=- g~t)dtT _~ T_~ 1 1 f~d'=P,vT_~IGT(f)I'df(9-2-71) However, GT(f)mustbechosen tosatisfythezerolSIcondition. Consequently, Ifl""WG(f)IX,cU)1 ITI1C(f)1IGR(f)1, andGT(f)=0forIfI;;,W.Hence ~=_1_fWIX,c(f)l' df d'P.,T-wlC(f)I'IGR(f)I'(9-2-72) (9-2-73) (9-2-74)Therefore, thenoise-to-signal ratiothatmustbeminimized withrespectto IGR(f)1forIfI""Wis 17'1fW .fWIX(f)I' d;=P,vT-w<Pnn(f)IGR(f)I'df_wlC(f)I;cIGR(f)I' df tHysetting Xo=1and1m=±d,thescalingbyXoisincorporated intotheparameter d. CHAPTER 9~SIGNAL DESIGN FORBAND·LIMITED CHANNELS 559 Theoptimum IGR(f)1canbefoundbyapplying theCauchy-Schwartz inequality, whereIU,(f)1andIVAf)1aredefinedas IV,(f)1=1v'<I>nn(f)IIGR(f)1 (f)IX,,(f)! IV21=1C(f)IIGR(f)1(9·2·75) (9·2·76) Theminimum valueof(9-2-74) isobtained whenIVI(f)1isproportional to IV2(f)I.or,equivalently, when (9-2-77) whereKisanarbitrary constant. Thecorresponding modulation filterhasa magnitude characteristic !G(f)1=1IX".(f)1"2[¢nn(f)l'l4 IfI'"w TKIC(f)I'12 '(9-2-78) Finally, themaximum SNRachieved bytheseoptimum transmitting and receiving filtersis d2P.vT (1~={j",WIX,c(f)1[<I>nn(f)]I12IC(f)1 Idf}2(9-2-79) Wenotethattheoptimum modulation anddemodulation filtersare specified inmagnitude only.Thephasecharacteristics forGT(f)andGR(f) maybeselectedsoastosatisfythecondition in(9-2-65), i.e., (9-2-80) (9-2-81)whereST(f),Sc(f),andSR(f)arethephasecharacteristics ofthemodulation filter,thechannel, andthedemodulation filter,respectively. Inthespecialcasewheretheadditive noiseattheinputtothedemodulator iswhitegaussian .withspectraldensity!No•theoptimum filtercharacteristics specified by(9·2-77)and(9-2-78)reduceto IX,c(f)II12 IGR(f)1= K,lC(f)I'I2' IfI'"w IX,c(f)I'12 IGr(f)!=K2lC(f)II12'IfI'"w S60 DIOITALCOMMUNICA nONS whereKlandK2arearbitrary scalefactors.Notethat,inthiscase,jGR(f)1is thematched filtertoIGT(f)I·Thecorresponding SNRatthedetector, givenby (9-2-79)reducesto d2=2Pa.T[fWIX,,(f)1df]-2 a~No-w!C(f)1(9-2-82) Example 9-2-} Letusdetermine theoptimum transmitting andreceiving filtersforabinary communication systemthattransmits dataatarateof4800bits/sovera channelwithfrequency (magnitude) response le(f)!=VI+tfIW)2'IfI.,;;;W(9-2-83) whereW=4800Hz.Theadditive noiseiszero-mean, white,gaussian with spectraldensity ~No=10-15W IHz. SinceW=lIT=4800,weuseasignalpulsewitharaisedcosine spectrum andf3=1.Thus, Then,X,c(f)=~T(l+cos(JrTIfI)l =Tcos'(JrIfI) 9600(9-2-84) (9-2-85) andiGT(f)!=IGk(f)1=0,otherwise. Figure9-2-13illustrates thefilter characteristic GT(f). Onecannowusetheseoptimum filterstodetermine theamount of transmitted energy'Ifrequired toachieveaspecified errorprobability. This problem isleftasanexerciseforthereader. IGr(f)1 I FIGURE 9-2-13 Frequency response ofoptimum transmitter filter. o 4IlOOf CHAPTER II:SlliN,",L DESIG~ FORBAND-LIMITED CHANNELS 561 9-3PROBABILITY OFERROR INDETECTION OFPAM Inthissection,weevaluate theperformance ofthereceiverfordemodulating anddetecting anM-aryPAMsignalinthepresence ofadditive, white,gaussian noiseatitsinput.First.weconsider thecaseinwhichthetransmitter and receiverfiltersGr(f)andGR(f}aredesigned forzerolSI.Then,weconsider thecaseinwhichGr(f)andGR(f}aredesigned suchthatx(t)=gr(t)*gR(t) iseitheraduobinary signaloramodified duobinary signal. 9-3-1Probability ofErrorforDetection ofPAMwithZerolSI IntheabsenceoflSI,thereceived signalsampleattheoutputofthereceiving matched filterhastheform where Xo=L:IGr(fWdf=leg andVmistheadditivegaussian noisethathaszeromeanandvariance a~= ~legNo(9-3.1) (9-3-2) (9-3-3) (9-3-4)Ingeneral,1mtakesoneofMpossibleequallyspacedamplitude valueswith equalprobability. Givenaparticular amplitude level,theproblem isto determine theprobability oferror. Theproblem ofevaluating theprobability oferrorfordigitalPAMina band-limited, additive whitegaussian noisechannel, intheabsenceoflSI,is identical totheevaluation oftheerrorprobability forM-aryPAMasgivenin Section5-2.Thefinalresultthatisobtained fromthederivation is p=2(M-1)Q( I~) MM\jNo Butleg=3'€avl(M2 -I),lCav=k'€bavistheaverageenergypersymboland'€bav istheaverageenergyperbit.Hence, 6(10&2M)lCbav) (M1-l)No(9-3-5) Thisisexactlytheformfortheprobability oferrorofM-aryPAMderivedin Section5-2(see(5-2-46»). In'thetreatment ofPAMgiveninthischapter, we imposed theadditional constraint thatthetransmitted signalisband-limited to thebandwidth allocated forthechannel. Consequently, thetransmitted signal pulsesweredesigned tobeband-limited andtohavezerolSI. Incontrast, nobandwidth constraint wasimposed onthePAMsignals considered inSection5-2.Nevertheless, thereceivers (demodulators and detectors) inbothcasesareoptimum (matched filters)forthecorresponding 562 DIGITAL COMMUNICATiONS M-Ievel data {D.lTransmining filter lP",1 G1(f)l.-__ ...JReceving filter C;<f)[)etectorOutput AWCN FlGVRE !J.J..1Blockdiagramofmodulator anddemodulator lorpartial-response signals. transmitted signals.Consequently, nolossinerrorrateperformance results fromthebandwidth constraint whenthesignalpulseisdesigned forzerolSI andthechanneldoesnotdistortthetransmitted signal. 9-3-2Probability ofErrorforDetection ofPartial-Response Signals Inthissectionwedetermine theprobability oferrorfordetection ofdigital M-aryPAMsignaling usingduobinary andmodified duobinary pulses.The channel isassumed tobeanidealbandlimited channel withadditive white gaussian noise.Themodelforthecommunication systemisshowninFig. 9-3-l. Weconsider twotypesofdetectors. Thefirstisthesymbol-by-symbol detector andthesecondistheoptimum MLsequence detector described inthe previous section. Symbol·by-Symbol Delector Atthetransmitter, theM-leveldatase­ quence{D",}isprecoded asdescribed previously. Theprecoder outputis mapped intooneofMpossible amplitude levels.Thenthetransmitting filter withfrequency response GT(f)hasanoutput ~ v(t)~2:IngT(t-nT) n=-Xl(9-3-6) Thepartial-response functionXU)isdividedequallybetween thetransmitting andreceiving filters.Hence,thereceiving filterismatched tothetransmitted pulse,andthecascadeofthetwofiltersresultsinthefrequency characteristic (9-3-7) Thematched filteroutputissampled att~nT~n/2Wandthesamplesarefed tothedecoder. Fortheduobinary signal,theoutputofthematched filteratthe sampling instantmaybeexpressed as (9-3-8) wherev'"istheadditive noisecomponent. Similarly, theoutputofthematched filterforthemodified duobinary signalis (9-3-9) CHAPTER 9:SIGNAL DESIGN FORBAND·LlMlTED CHANNELS 563 Forbinarytransmission, let1m=±d,where2disthedistance between signal levels.Then,thecorresponding valuesofBmare(2d,0,-2d).ForM-aryPAM signaltransmission, where1m=±d,±3d,...•±(M-l)d,thereceived signal levelsareBm=0,±2d,±4d....,±2(M-l)d.Hence,thenumberofreceived levelsis2M-1.andthescalefactordisequivalent toXo=If.. Theinputtransmitted symbols {1m}areassumed tobeequallyprobable. Then,forduobinary andmodified duobinary signals,itiseasilydemonstrated that,intheabsence ofnoise,thereceived outputlevelshavea(triangular) probability distribution oftheform M'-ImlP(B=2md)=M2m=0,±1,±2....,±(M-1)(9-3-10) whereBdenotesthenoise-free received leveland2disthedistance between anytwoadjacent received signallevels. Thechannel corrupts thesignaltransmitted through itbytheaddition of whitegaussian noisewithzeromeanandpowerspectraldensity ~No. Weassumethatasymbolerroroccurswhenever themagnitude ofthe additive noiseexceedsthedistance d.Thisassumption neglects theTareevent thatalargenoisecomponent withmagnitude exceeding dmayresultina received signallevelthatyieldsacorrect symbol decision. Thenoise component Vmiszero-mean gaussian withvariance (T~=~NoJ:ICRU)I'df =~NoLW wIXU)Idf=2No/Jr (9-3-11) forbotl1theduobinary andthemodified duobinary signals.Hence,anupper boundonthesymbolprobability oferroris M-2 PM<2:P(ly-2mdl>dIB=2md)P(B =2md) m=-(M-2) +2P(y+2(M-l)d>dIB=-2(,1.1-l)d)P(B =-2(M-l)d) =P(lyl>dIb=0)[2~~P(B=2md)-P(B=0)-P(B=-2(M-1)d)J =(1-M-2)P(IYI>dIB=0) (9-3-12) But (9-3-13) S64DlGlTALCOMMUNICA noNS Therefore, theaverageprobability ofasymbolerrorisupper-bounded as PM<2(1-M-')Q(VmP!2N o) (9-3·14) Thescalefactordin(9·3-14)canbeeliminated byexpressing itintermsof theaveragepowertransmitted intothechannel. FortheM-aryPAMsignalin whichthetransmitted levelsareequallyprobable, theaveragepoweratthe outputofthetransmitting filteris Pa,=E~;")L:\Gr(fWdf =E(1;")JW\X(f)1df=~E(l;') (9-3-15) T-w trT whereE(l;")isthemeansquarevalueoftheMsignallevels,whichis (9-3-16) Therefore, d'=3trP•.r (9-3-17) 4(M'-1) Bysubstituting thevalueofd'from(9-3-17)into(9-3-14), weobtaintheupper boundonthesymbolerrorprobability as (9-3-18) where ~O¥istheaverageenergypertransmitted symbol,whichcanbealso expressed intermsoftheaveragebitenergyas~av=k~bav=(lOg2 M)~ba,' Theexpression in(9-3-18)fortheprobability oferrorofM-aryPAMholds forbothduobinary andmodified duobinary partial-response signals.Ifwe compare thisresultwiththeerrorprobability ofM-aryPAMwithzerolSI, whichcanbeobtained byusingasignalpulsewitharaisedcosinespectrum, we notethattheperformance ofpartialresponse duobinary ormodified duobinary hasalossofOtr)2,or2.1dB.ThislossinSNRisduetothefactthatthe detector forthepartialresponse signalsmakesdecisions onasym.bol-by symbolbasis,thusignoring theinherent memory contained inthereceived signalattheinputtothedetector. Maximum-Likelihood Sequence Detec:tar TheMLsequence detector searches throughthetrellisforthemostprobable transmitted sequence {1m}as previously described inSection9-2-3.Ateachstageofthesearchprocessthe detector compares themetciesofpathsthatmergeateachofthenodesand selectsthepaththatismostprobable ateachnode.Theperformance ofthe detector maybeevaluated bydetermining theprobability oferrorevents, basedonaeuclidean distancemetric,aswasdoneforsoft-decision decoding of convolutional codes.Thegeneralderivation isgiveninSectionto-l-4.Inthe CHAPTER 9:SIGNAL DESIGN FORBAND·LlMlTED CHANNELS 565 caseoftheduobinary andmodified duobinary signals,itisdemonstrated that the2.1dBlossinherent inthesuboptimum symbol-by-symbol detector is completely recovered bytheMLsequence detector. 9·3·3Probability ofErrorforOptimum SignalsinaChannel withDistortion InSection 9-2-4,wederived thefilterresponses for.themodulation and demodulation filtersthatmaximize theSNRattheinputtothedetector when' thereischanneldistortion. Whenthefiltersaredesigned forzerolSIatthe sampling instants, theprobability DferrorforM-aryPAMis Theparameterdisrelatedtotheaveragetransmitted poweras Pav=E[;;,]fwIGT(f)I'df =(M'-1)d'lW IGr(f)l'df 3T-w(9-3-19) (9-3-20) andthenoisevariance isgivenby(9-2-69). ForAWGN, (9-3-19) maybe expressed as 2(M-l) (PM= QM6~&V[fWIX",([)I]-') (M'-1)No-IVlc(fJldf(9-3-21) Finally,weobservethatthelossduetochanneldistortion is I[lWIXrc([)1 ]20oglQ-wlC(f)/df NotethatwhenC(f)=1forIfI'"W,thechannelisidealand(9-3-22) (9-3-23) sothatnolossisincurred. Ontheotherhand,whenthereisamplitude distortion, 1C(f)1<1forsomerangeoffrequencies inthebandIff'"Wand, hence,thereisalossinSNRincurred, asgivenby(9-3-22). Thislossis independent Dfchannelphasedistortion, because phasedistortion hasbeen perfectly compensated, asimpliedby(9-2-80). Thelossgivenby(9-3-22)isdue entirelytoamplitude distortion andisameasure ofthenoiseenhancement 566 DIGLTAL COMMlNICATlONS resulting fromthereceiving filter,whichcompensates forthechannel distortion. 9-4MODULA nONCODES FORSPECTRUM SHAPING Wehaveobserved thatthepowerspectraldensityofadigitalcommunication signalcanbecontrolled andshapedbyselecting thetransmitted sigpalpulse get)andbyintroducing correlation through coding,whichisusedtocombat channeldistortion andnoiseintransmission. Codinglorspectrum shaping is introduced following thechannel encoding sothatthespectrum ofthe transmitted signalmatches thespectral characteristics ofabaseband or equivalent lowpasschannel. Codesthatareusedforspectrum shaping aregenerally calledeither modulation codes,orlinecodes,ordatatranslation codes.Suchcodesgenerally placerestrictions onthesequence ofbitsintothemodulator and,thus. introduce correlation and.hence,memory intothetransmitted signal.Itisthis typeofcodingthatistreatedinth.issection. Modulation codesareusuallyemployed inmagnetic recording, inoptical recording, andindigitalcommunications overcablesystemstoachievespectral shapingandtoeliminate orminimize thed.c.contentinthetransmitted (or stored)baseband signal.Inmagnetic recording channels, themodulation code isdesigned toincrease thedistance between transitions intherecorded waveform and,thus,intersymbol interference effectsarealsoreduced. Asanexampleoftheuseofamodulation code,letusconsider amagnetic recording system,whichconsistsoftheelements shownintheblockdiagramof Fig.9-4-1.Thebinarydatasequence tobestoredisusedtogenerate awrite current.Thiscurrentmaybeviewedastheoutputforthe"modulator." The mostcommonly usedmethodtomaptheinformation sequence intothewrite currentwaveform isNRZI,whichwasdescribed inSection4-3-2.Recallthatin NRZI,atransition fromoneamplitude toanother(Ato-Aor-AtoA) occursonlywhentheinformation bitisa1.Notransition occurswhenthe information bitisa0,i.e.,theamplitude levelremains thesameasinthe previous signalinterval. Thepositive amplitude pulseresultsinmagnetizing FIGURE 1J-4.1Blockdiagramofmagnetic storageread/write system. Input da'"Write StorageWrite-currentdriverr---- ~ mediumr-bead(roodulator) (channel) OutpulDo'"f<>-Read-backda"'- demodulator head CHAPTER 9:SIGNAL DESIGN FORRA\iD-UMlTED CHANNELS 567 I71\ I\ 71 \ T50 / \ 1/ '\ 7 1"::11- -0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 o. oFIGURE 9~4·ZRead-back pulseInmagnet~( recording system. themedium inone(direction) polarityandthenegative pulsemagnetizes the medium intheopposite (direction) polarity. Sincetheinputdatasequence isbasically randomwithequallyprobable ls and0s.weshallencounter leveltransitions fromAto-Aor-AtoAwith probability 1/2foreverydatabit.Thereadbacksignalforapositive transition (-AtoA)isapulsethatiswell-modeled mathematically as 1(9-4-1) whereT",isdefinedasthewidthofthepulseatits50%amplitude level,as showninFig.9-4-2.Similarly, thereadback signalforanegative transition (A to-A)isthepulse-get).ThevalueofT,,,isdetermined bythecharacteristics orthemedium, theread/write heads,andthedistance oftheheadtothe medium. Now,suppose wewriteapositive transition followed byanegative transition. Let'svarythetimeintervalbetween thetwotransitions, whichwe denoteas7b(thebittimeinterval). Figure9-4-3illustrates thereadbacksignal pulses,whichareobtained byasuperposition ofpet)with-pet-T.).The parameter A=Tso/Tnisdefined asthenormalized density. Thecloserthebit transitions (Tbsmall),thelargerwillbethevalueofthenormalized density and,hence,thelargerwillbethepacking density. Wenoticethatas6.is 3 -2-10 rlThIIr ~-l I~=2I81".1l\7~=3 "",'7~ -.:-'I-,1/-~..,.:~ ,7~ 3{, , I0.9 o. o. 0.6 0.5 0.4 o. 0.• o. 0..-3 Read-back signalresponse loapulse. FIGURE 9-4-3 568 DIGITAL COMMUNICATIONS increased, thepeakamplitudes ofthereadbacksignalarereducedandarealso shiftedintimefromthedesiredtimeinstants. Inotherwords,thepulses interfere withoneanother, thuslimitingthedensitywithwhichwecanwrite. Thisproblem servesasamotivation todesignmodulation codes thattakethe originaldatasequence andtransform (encode) itintoanothersequence that resultsinawritewaveform inwhichamplitude transitions arespacedfarther apart.Forexample, ifweuseNRZI,theencoded sequence intothemodulator mustcontainoneormoreOsbetween Is. Thesecondproblem encountered inmagnetic recording istheneedtoavoid (orminimize) havingad.c.contentinthemodulated signal(thewritecurrent) duetothefrequency response characteristics ofthereadback systemand associated electronics. Thisrequirement alsoarisesindigitalcommunication overcablechannels. Thisproblem canbeovercome byaltering(encoding) the datasequence intothemodulator. Aclassofcodesthatsatisfytheseobjectives arethemodulation codesdescribed below. Runlength-Limited CodesCodesthathavearestriction onthenumberof consecutive IsorOsinasequence aregenerally calledrunlength-limited codes. Thesecodesaregenerally described bytwoparameters, saydandK,whered denotes theminimum number ofOsbetween twoIsinasequence, andK denotesthemaximum numberofOsbetween twoIsinasequence. Whenused withNRZImodulation, theeffectofplacingdzerosbetween successive Isisto spreadthetransitions fartherapart,thusreducing theoverlap inthechannel response duetosuccessive transitions andhencereducing theintersymbol interference. Settinganupperlimit Kontherunlength ofOsensures that transitions occurfrequently enoughsothatsymboltiminginformation canbe recovered fromthereceived modulated signal.Runlength-limited codesare usuallycalled(d,K)codes.t The(d,K)codesequence constraints mayberepresented byafinite-state sequential machine withK+1states,denoted as5i,J,.;;i,.;;K+1,asshownin Fig.9-4-4.Weobservethatanoutputdatabit0takesthesequence fromstate 5ito5i+l,i,.;;K.Theoutputdatabit1takesthesequence tostate51'The outputbitfromtheencodermaybea 1onlywhentbesequence isinstateS" d+1,.;;i,.;;K+1.Whenthesequence isinstate5.+btheoutputbitisalways 1. FIGURE 9·4-4Finite-Slale sequentl3t machine fora(d.K)-codedsequence. o tInfact.theyareusuallycalled(d,k)codes,wherekisthemaximum runlength ofzeros.We havesubstituted theGreekletterkappa I(fork,toavoidconfusion withourprevious us.eofk CI{APTER Q:SIGNAL DESiGN FORBAND-LIMITE!) ClIANNFLS 569 Thefinite-stale sequential machine mayalsoberepresented byas/ate trami/ion llIatrix.denoted asD,whichisasquare(K+I)X(K+I)with clementsd".where d'i=1 d,j={~(i;"d+I) (j=i+I) (otherwise)(9-4-2) (9-4-3)Example 9-4-1 Letusdetermine thestatetransition matrixfora(d,K)=(1,3)code.The (1,3)codehasfourstates.FromFig.9-4-4,weobtainitsstatetransition matrix,whichis D{~ff] Animportant parameter ofany(d,K)codeisthenumberofsequences of acertainlength,sayn,thaisatisfythe(d,K)constraints. Asnisallowedto increase, thenumber ofsequences N(n)thatsatisfythe(d,K)constraint alsoincreases. Thenumber ofinformation bitsthatcanbeuniquelv represented withN(n)codesequences is k=LIog,N(n)J whereLxJdenotes thelargestintegercontained inx.Themaximum code rateisthenR(,=kin. • Thecapacity ofa(d,K)codeisdefinedas C(d,K)=Jim.!.]og,N(n) n.......",on(9-4-4) Clearly,C(d,K)isthemaximum possibleratethatcanbeachieved withthe (d,K)constraints. Shannon (1948)showedthatthecapacity isgivenas C(d,K)=log,Am.. (9-4-5) whereAm..isthelargestrealeigenvalue ofthestatetransition matrixO. Example942 Letusdetermine thecapacity ofa(d,K)=(1,3)code.Usingthestate­ transition matrixgiveninExample 9-4-1forthe(1,3)code,wehave det(0_AI)=det[~A-\~~] ~~~A_IA = A4-A'-A-I=0 (9-4-6) 570 DIGITAL COMMUNICATIONS TABLE9-4-1CAPACITY C(d,K)VERSUS RUNLENGTH PARAMETERS dAND K Kd=O d=l d=Z d=3 d=4 d=5 d=6 2.8791 .4057 3.9468 .5515 .2878 4.9752 .6174 .4057 .2232 5.9881 .6509 .4650 .3218 .1823 6.9942 .6690 .4979 .3746 .2269 .1542 7.9971 .6793 .5174 .4057 .3142 2281 .1335 8.9986 .6853 .52g3 .4251 .3432 .2709 .1993 9.9993 .6888 .5369 .4376 .3620 .2979 .2382 10.9996 .6909 .5418 .4460 .3746 .3158 .2633 11.9998 .6922 .5450 .4516 .3833 .3285 .2804 12.9999 .6930 5471 .4555 .3894 .3369 .2924 13.9999 .6935 .5485 .4583 .3937 .3432 .3011 14.9999 .6938 .54g5 .4602 .3968 .3478 .3074 15.9999 .6939 .5501 .4615 .3991 .3513 .3122 x1.000 .6942 .5515 .4650 A057 .3620 .3282 Themaximum realrootofthispolynomial isfoundtobeAm"=1.4656. Therefore, thecapacity C(l,3)=logzAm.,=0.5515. Thecapacities of(d,K)codesfor0,;;;d,;;;6and2,;;;K,;;;15aregivenin Table9-4-1.WeobservethatC(d,K)<~foru''"3andanyvalueofK.The mostcommonly usedcodesformagnetic recording employd,;;;2;hence,their rateR,isatleast ~. Nowletusturnourattention totheconstruction ofsomerunlength-limited codes.Ingeneral,(d,K)codescanbeconstructed eitherasfixed-length 'codes orasvariable-length codes.Inafixed-length code,eachbitorblockofkbitsis encoded intoablockofn>kbits. Inprinciple, theconstruction ofafixed-length codeisstraightforward. Fora givenblocklengthn,wemayselectthesubsetofthe2ncodewordsthatsatisfy thespecified runlength constraints. Fromthissubset,weeliminate codewords thatdonotsatisfytherunlength constraints whenconcatenated. Thus,we obtainasetofcodewordsthatsatisfytheconstraints andcanbeusedinthe mapping oftheinputdatabitstotheencoder. Theencoding anddecoding operations canbeperformed byuseofalook-uptable, Example 9-4-3 Letusconstruct ad=0,K=2codeoflengthn=3,anddetermine its efficiency. Bylistingallthecodewords,wefindthatthefollowing fivecode wordssatisfythe(0,2)constraint: (010),(011),(10I),(110),(Ill).We mayselectanyfourofthesecodewordsandusethentoencodethepairsof CHAPTER 9,SIGNAL DESIGN FORBAND-LIMITED CHANNELS 571 databits(00,01,10,11). Thus,wehavearatekin=2/3codethatsatisfies the(0,2)constraint. Thefixed-length codeinthisexampleisnotveryefficient. Thecapacity is C(O,2)=0.8791,sothatthiscodehasanefficiency of . R, 2/3 efficiency =C(d,K)0.8791=0.76 Surely,better(0,2)codescanbeconstructed byincreasing theblocklength n. Inthefollowing example, weplacenorestriction onthemaximum runlength ofzeros. Example 9-4-4 Letusconstruct ad=1,K=00codeoflengthn=5.Inthiscase,weare placingnoconstraint onthenumberofconsecutive zeros.Toconstruct the code,weselectfromthesetof32possiblecodewordsthosethatsatisfythe d=1constraint. Thereareeightsuchcodewords,whichimpliesthatwecan encodethreeinformation bitswitheachcodeword.Thecodeisgivenin Table9-4-2.Notethatthefirstbitofeachcodewordisa0,whereas thelast bitmaybeeither0or1.Consequently, thed=1constraint issatisfied when thesecodewordsareconcatenated. ThiscodehasarateR,=3/5.When compared withthecapacity C(1,00)=0.6942obtained fromTable9-4-1,the codeefficiency is0.864,whichisquiteacceptable. Thecodeconstruction method described inthetwoexamples above produces fixed-length (d,K)codesthatarestate-independent. Bystate­ independent, wemeanthatfixed-length codewordscanbeconcatenated without violating the(d,K)constraints. Ingeneral, filted-length state­ independent (d,K)codesrequirelargeblocklengths, exceptincasessuchas thoseintheexamples abovewheredissmall.Simpler(shorter-length) codes TABLE 9-4-ZFIXEDLENGTH d=1,K='"CODE Inputdatabits 000 001 010 o1 1 1 0 0 1 0 1 1 1 0 1 1 1Outputcoded.....uence 00000 0000 1 00010 00100 001 0 1o1000 o1 0 0 1 o1 0 1 0 S72 DIGITAL COMMUNICATIONS aregenerally possiblebyallowing forstate-dependence andforvariablelength codewords.Below,weconsider codesforwhichboththeinputblockstothe encoderandtheoutputblocksmayhavevariablelength.Forthecodewordsto beuniquely decodable atthereceiver, thevariable-length codeshouldsatisfy theprefixcondition, described inChapter 3. Example 9-4-5 Averysimpleuniquely decodable variable-length d=0,K=2codeis 0--->01 10--->10 11--->11 Thecodeinthe above example hasafixedoutputblocksizebutavariable inputblocksize.Ingeneral,boththeinputandoutputblocksmaybevariable. Thefollowing example illustrates thelattercase. EXIIIIlpie 9-4-6 Letusconstruct a(2,7)variable blocksizecode.Thesolutiontothiscode construction iscertainly notunique,norisittrivial.Wepickedthisexample becausethe(2,7)c.odehasbeenwidelyusedbyIBMinmanyofitsdisk storagesystems.ThecodeislistedinTable9-4-3.Weobservethattheinput datablocksof2,3,and4bitsaremapped intooutputdatablocksof4,6, and8bits,respectively. Hence,thecoderateisRc=1/2.Sincethisisthe coderateforallcodewords,thecodeiscalledafixed-rate code.Thiscode hasanefficiency of0.5/0.5174 =0.966.Notethatthiscodesatisfiestheprefix condition. TABLE 9-4-3CODEBOOKFORVARIABLE­ LENGTH (2,7)CODE Inpatd.bbitsOutputcodedsequence 1 0 1 1 011o1 0 000 o0 1 1 o0 1 01000 ()..]0 0 000100 001000 1001 0 0o0 100100 00001000 CHAPTER 9:SIGNAL DESIGN FORBAND-LIMITED CHANNELS 573 TABLE 9-4-4ENCODER FOR(1,3)MILLER CODE InputdatabitsOutput(Odedsequence o x0 1 0 I x=0,ifpreceding inputbitisI x=1,ifpreceding inputbitis0 Another codethathasbeenwidelyusedinmagnetic recording istherate 1/2,(d,K)=(1,3)codeinTable9-4-4.Weobservethatwhentheinformation bitisa0,thefirstoutputbitis1iftheprevious inputbitwas0,Dra 0ifthe previous inputbitwasa1.Whentheinformation bitisa1,theencoderoutput is01.Decoding ofthiscodeissimple.Thefirstbitofthetwo-bitblockis redundant andmaybediscarded. Thesecondbitistheinformation bit.This codeisusuallycalledtheMillercode.Weobservethatthisisastate-dependent code,whichisdescribed bythestatediagram showninFig.9-4-5.Thereare twostateslabeled51andS,withtransitions asshowninthefigure.Whenthe encoder isastateSt.aninputbit1resultsintheencoderstayinginstate51and outputs01.Thisisdenoted as1/01.Iftheinputbitisa0,theencoder enters state5,andoutputs00.Thisisdenoted as0(00.Similarly, iftheencoder isin state5"aninputbit0causesnotransition andtheencoder outputis10.On theotherhand,iftheinputbitisa1,theencoderentersstate5\andoutputs 01.Figure9-4-6showsthetrellisfortheMillercode. TheMapping ofCodedBitsintoSignalWaveforms Theoutputsequence froma(d,K)encoder ismapped bythemodulator intosignalwaveforms for transmIssion overthechannel.Ifthebinarydigit1ismapped intoa rectangular pulseofamplitude Aandthebinarydigit0ismapped intoa FIGURE 9-4-5Statediagrams ford=1.K=3(Miller)code. FIGURE 9-4-6Trellisford~I.K~3(Miller)code.1i01 State 110\ 110) \101 5,~--<""'-;I'~-.-_-.:.,;..-~ 52e<---<.....-"'~-._-'¥'-- -"" 0110 0/10 0/10 574 DIGITAL CO'-1MUNICATIONS rectangular pulseofamplitude -A,theresultisa(d,K)codedNRZ modulated signal.Notethattheduration oftherectangular pulsesis ~.=ReiRb=R,Tb,whereRbistheinformation (bit)rateintotheencoder, Tb isthecorresponding (uncoded) bitinterval, andR,isthecoderateforthe (d,K)code. Whenthe(d,K)codeisastate-independent fixed-length codewithcode rateR,=kin,wemayconsider eachn-bitblockasgenerating onesignal waveform ofduration n~..Thus,wehaveM=2*signalwaveforms, onefor eachofthe2*possible k-bitdatablocks.Thesecodedwaveforms havethe generalformgivenby(4-3-6)and(4-3-38). Inthiscase,thereisnodependence between thetransmission ofsuccessive waveforms. Incontrast tothesituation considered above,themodulation signalisno longermemoryless whenNRZIisusedandlorthe(d,K)codeisstate­ dependent. Letusconsider theeffectofmapping thecodedbitsintoanNRZI signalwaveform. Recallthatthestatedependence inthe NRZI signalisduetothe differential encoding oftheinformation sequence. Thedifferential encoding is aformofprecoding, whichisdescribed mathematically as p*=d*$p*_l where{d*}isthebinarysequence intotheprecoder, {p*}istheoutputbinary sequence fromtheprecoder, andEBdenotesmodulo-2 addition. Thisencoding ischaracterized bythestatediagram showninFig.9-4-7(a). Then,the sequence {pdistransmitted byNRZ.Thus,whenp*=1,themodulator outputisarectangular pulseofamplitude A,andwhenP.=0,themodulator FlGURE 9-4-7Stateandtrellisdiagrams forNRZIsignal. ()/{) 011 CO:'"{) 110 1_)OI-s(t) Ols(t) ee~I/S~(r)-{;J 11-5(1) <I~ TI (b) (I/O 011 011 Ie)011 CHAPTER l,"!:SIGNAL DESIGN FORBAI'\D·l(MITED CHANNELS 575 outputisarectangular pulseofamplitude -A.Whenthesignalwaveforms are superimposed onthestatediagram ofFig.9-4-7(0), weobtainthecorrespond­ ingstatediagram showninFig.9-4-7(b). Thecorresponding trellisisshownin Fig.9-4-7(c). Whentheoutputofastate-dependent (d,K)encoder isfollowed byan NRZImodulator. wemaysimplycombine thetwo-state diagrams intoa single-state diagram forthe(d,K)codewithprecoding. Asimilarcombination canbeperformed withthecorresponding trellises. Thefollowing example illustrates theapproach forthe(1,3)Millercodefollowed byNRZI modulation. Example 9-4-7 Letusdetermine thestatediagram ofthecombined (1.3)Millercode followed bytheprecoding inherent inNRZImodulation. Sincethe(1,3) Millercodehastwostatesandtheprecoder hastwostates,thestate diagram forthecombined encoder hasfourstates.whichwedenoteas (SM,SN)=(UI'SI),(u"52)'(U2's,),(U2'S2),whereSM={u"0'2}represents thetwostatesoftheMillercodeandSN={SI.S2}represents thetwostates oftheprecoder forNRZI.ForeachdatainputbitintotheMillerencoder. weobtaintwooutputbitswhicharethenprecoded toyieldtwoprecoded outputbits.Theresulting statediagram isshowninFig.9-4-8,wherethe firstbitdenotestheinformation bitintotheMillerencoderandthenexttwo bitsrepresent thecorresponding outputoftheprecoder. Thetrellisdiagram fortheMillerprecoded sequence maybeobtained directlyfromthecombined statediagram orfromacombination ofthetrellises ofthetwocodes.Theresultofthiscombination isthefour-state trellis,one stageofwhichisshowninFig.9-4-9. Itisleftasanexerciseforthereadertoshowthatthefoursignalwaveforms obtained bymapping eachpairofbitsoftheMiller-precoded sequence intoan flGURE 9-4-llStatediagramoftheMillercodefollowedbytheprecoder. 576 DIGJTALCOMMUNIC AnONS FIGURE 9-4-9OnestageoftrellisdiagramfortheMillercodefollowedbytheprec<1der. (n,.") NRZsignalarebiorthogonal andthattheresulting modulated signalwaveform isidentical tothedelaymodulation thatwasdescribed inSeclion4-3-2. Fromthestatediagram ofastate-dependent runlength-limited code,one canobtainthetransition probability matrix,asdescribed inSection4-3-2. Then,thepowerspectraldensityofthecodemaybedetermined, asshownin Section4-4-3. 9-5BIBLIOGRAPHICAL NOTES ANDREFERENCES Thepioneering workonsignaldesignforbandwidth-constrained channels was donebyNyquist (1928).Theuseofbinarypartialresponse signalswas originally proposed byLender(1963),andwaslatergeneralized byKretzmer (1966).Otherearlyworkonproblems dealingwithintersymbol interference (lSI)andtransmitter· andreceiver optimization withconstraints onlSIwas donebyGerstandDiamond (1961),Tufts(1965),Smith(1965),andBerger andTufts(1967)."FasterthanNyquist" transmission hasbeenstudiedby Mazo(1975)andFoschini (1984). Modulation codeswerealsofirstintroduced byShannon (1948).Someof theearlyworkontheconstruction ofrunlength-limited codesisfoundinthe papersbyFreiman andWyner(1964),Gabor(1967),Franaszek (1968,1969, 1970),TangandBahl(1970),andJacoby(1977).Morerecentworkisfoundin papersbyAdlerCoppersmith andHassner (1983),andKarabed andSiegel (1991).Themotivation formostoftheworkonrunlength-limited codeswas provided byapplications tomagnetic andopticalrecording. Awell-written tutorialpaperonrunlength-limited codeshasbeenpublished byImmink (1990); PROBLEMS 9·1Achannel issaidtobedistortion/ess iftheresponsey(l)toaninputX(/)is KX(I- (0),whereKand10areconstants. Showthatifthefrequency response of thechannel isA(f)ei~(f), whereAU)and8(f)arereal,thenecessary and CHAPTER 9:SfGNAl DESIGN FORBAND-LfMITED CHANNELS S77 sufficient conditions fordistortion lesstransmission areA(f)=Kand8(f)= 2:rfto±nrr,n=0,1,2,.... 9-2Theraised-cosine spectralcharacteristic isgivenby(9-2-26). 8Showthatthecorresponding impuise response is sin(mIT)cos(13mIT) x(r)=mIT1-4/3't'11' bDetermine theHilberttransform ofx(t)when13=1. cDoesi(t)possessthedesirable properties ofx(t)thatmakeitappropriate for datatransmission? Explain. dDetermine theenvolope oftheSSBsuppressed-carrier signalgenerated from x(t). 9·38Showthat(Poisson sumformula) x(t)=J.g(l)h(t-k1)::>X(f) =L~.HG)C(t -~) Hint:MakeaFourier-series expansion oftheperiodic factor.Lh(l-kT) bUsingtheresultin(a),verifythe.following versions ofthePoissonsum: .l'() ,~.h(kT)=r}. H~ .l'()('2,'?.h(t-k1)=Tn"!-..H~exp];") . .l' ) ,~.h(kT)exp(-jZrrkTf) =rn~.H(t-~(i) (ii) (iii) cDerivethecondition fornointersymbol interference (Nyquist criterion) bv usingthePoissonsumformula. 9-4Suppose adigitalcommunications systememploys gaussian-shaped pulsesofthe form x(t)=exp(-rra't') Toreducethelevelofintersymbol interference toarelatively smallamount, we imposethecondition thatx(T)=0,01,whereTisthesymbolinterval. The bandwidth Wofthepulsex(t)isdefined asthatvalueofWforwhich X(W)IX(O)=0.01,whereX(f)istheFouriertransform ofx(t).Determine the valueofWandcompare thisvaluetothatofraised-cosine spectrum with100% rolloff. 9-5Aband-limited signalhaVingbandwidth Wcanberepresented as ( )~_sin-,-,=[2.:.c"W--,-,,( tc..-_n--,I-=Zc..W-,-,)1xt=LJXnn--' ZrrW(t-nI2W) 8Determine thespectrum X(f)andplotIX(f)1forthefollowing cases: xo=2,x,=I,x,=-I, xn=O,n"0,1,2 (i) L,=-I, xo=2,x,=-I, x.=O,n"'-I,O,1 (ii) 578 DIGITAL COMMUNICo\ TIONS bPlotx(t)forthesetwocases. rIfthesesignalsareusedforbinarysignaltransmission. determine thenumberof received levelspossible atthesampling instants 1=nT=n/2W.andthe probabilities ofoccurrence ofthereceived levels.Assume thatthebinarydigits atthetransmitter areequallyprobable. 9-6A 4kHzbandpass channel istobeusedfortransmission ofdataatarateof 9600bits/soIfIN"=10-10W/Hzisthespectraldensityoftheadditive, zero-mean gaussian noiseinthechannel, designaQAMmodulation anddetermine the ~veragepowerthatachieves abiterrorprobability of10".Useasignalpulsewith araised-cosine spectrum havingaroll-offfactorofatleast50%. 9·7Determine thebitratethatcanbetransmitted through a 4kHzvoice-band telephone (bandpass) channel ifthefollowing modulation methods areused:(a) binaryPAM;(b)four-phase PSK;(c)8-pointGAM;(d)binaryorthogonal FSK, withnoncoherent detection; (e)orthogonal four-FSK withnoncoherent detection; (f)orthogonal 8-FSKwithnoncoherent detection. For(a)-(c), assumethatthe transmitter pulseshapehasaraised·cosine spectrum witha50%roll-off. 9-8Anidealvoice-band telephone linechannel hasabandpass frequency response characteristic spanning thefrequency range600-3000 Hz. •DesignanM=4PSK(quadrature PSKorQPSK)systemfortransmitting data atarateof2400bits/s andacarrierfrequencyt.=1800Hz.Forspectral shaping, usearaised-cosine frequency-response characteristic. Sketchablock diagram ofthesystemanddescribe thefunctional operation ofeachblock. bRepeat(a)forabitrateR=4800bits/so 9·9Avoice-band telephone channel passesthefrequencies inthebandfrom300to 3300Hz.Itisdesiredtodesignamodemthattransmits atasymbolrateof2400 symbols/s, withtheobjective ofachieving 9600bits/soSelectanappropriate GAM signalconstellation, carrierfrequency, andtheroll-offfactorofapulsewitha raisedcosinespectrum thatutlJizestheentirefrequency band.Sketchthespectrum ofthetransmitted signalpulseandindicatetheimportant frequencies. 9·10Acommunication systemforavoice-band (3kHz)channel isdesigned fora received SNRatthedetector of30dBwhenthetransmitter powerisP,= ~3dBW.Determme thevalueofP,ifitisdesiredtoexpandthebandwidth ofthe systemto10kHz,whilemaintaining thesameSNRatthedetector. 9·11Showthatapulsehavingtheraisedcosinespectrum givenby(9-2-26)satisfiesthe Nyquistcriterion givenby(9-2-13)foranyvalueoftheroll·offfactor{3. 9-12Showthat,foranyvalueof(3,theraisedcosinespectrum givenby(9-2-26)satisfiesrX,,(f)df~1 [Hint:UsethefactthatX,,(f)satisfiestheNyquistcriterion givenby(9-2-13).] 9·13TheNyquistcriterion givesthenecessary andsufficient condition forthespectrum X(flofthepulseX(I)thatyieldszerolSI.Provethatforanypulsethatis band-limited toIII<liT.thezero-lSIcondition issatisfied ifRe[X(flj,forf>0, consistsofarectangular function plusanarbitrary oddfunction aroundI=1/2T. and1m[X(!)]isanyarbitrary evenfunction aroundJ=1/2T. 9·14Avoice-band telephone channel hasapassband characteristic mthefrequency range300Hz<f<3000Hz. aSelectasymbol rateandapowerefficient constellation sizetoachieve 9600bits/ssignaltransmission. CHAPTER 9:SIUI'Al DESIGN FORBAND·lIMITfD CH!\NNFLS 579 Filh:rwiill ;,/lOla!I-liT) raisedl'osilll" ~pectrum ,1.'(1)AWGN chanrlel FIGURE 1'9-16 FIGURE P9·17Carrier dO bIfasquare-root raisedcosinepulseisusedforthetransmitter pubeg(t).select theroll-olIfactor.Assume thatthechannel hasanidealfrequency response characteristic. 9·15DesignanM-aryPAMsystemthattransmits digitalinformation overanideal channel withbandwidth W=2400Hz.Thebitrateis14400bit/soSpecify the number oftransmitted points.thenumber ofreceived signalpointsusinga duobinary signalpulse.andtherequired't:.toachieveanerrorprobability of10'. Theadditive noiseiszero-mean gaussian withapowerspectral density 1O-'W/Hz. 9·16AbinaryPAMsignalisgenerated byexciting araisedcosineroll-offfilterwitha 50%roil-alIfactorandisthenDSB·SC amplitude-modulated onasinusoidal carrierasillustrated inFig.P9-16.Thebitrateis2400bit/so aDetermine thespectrum ofthemodulated binaryPAMsignalandsketchit. bDrawtheblockdiagram illustrating theoptimum demodulator/detector forthe received signal.whichisequ.altothetransmitted signalplusadditive white gaussiannoise. 9-17Theelements ofthesequence {a,,}:~ xareindependent binaryrandom variables takingvaluesof±Iwithequalprobability. Thisdatasequence isusedtomodulate thebasicpulseg(t}showninFig.P9-17(a}. Themodulated signalis X(tl=2:a"g(t-nT} aFindthepowerspectral densityofX(I}. bIfgI(t)(showninFig.9-17b)isusedinsteadofg(t}.howwouldthepower spectrum in(a)change? cIn(b)assumewewanttohaveanullinthespectrum atf=1/3T.Thisisdone byaprecoding oftheformb"=a"+r>a".,.Findther>thatprovides thedesired null. dIsitpossible toemployaprecoding oftheformb"=a"+L,'Iaja",forsome finiteNsuchthatthefinalpowerspectrum willbeidentical tozerofor 1/3T<;;lfl<;;1/2T? Ifyes.how?Ifno.why?[flint:Useproperties ofanalytic functions.] '(~L "'~L~-J---+ tal FIGUREN-n580 DIGITAL COMMUNICATIONS R 9.18Consider thetransmission ofdataviaPAMoveravoice-band telephone channel thathasabandwidtlI of3000Hz.Show1I0wthesymbolratevariesasafunction of theexcessbandwidth. Inparticular, determine thesymbolrateforanexcess bandwidth of25%,33%,50%,57%,75%,and100%. 9·19Thebinarysequence 10010110010 istheinputtoaprecoder whoseoutputisused tomodulate aduobinarytransmitting filter.Construct atableasinTable9-2-1 showing theprecoded sequence, thetransmitted amplitude levels,thereceived signallevelsandthedecoded sequence. 9-20RepeatProblem 9-19foramodified duobinarysignalpulse. 9·21Aprecoder forapartialresponse signalfailstoworkifthedesiredpartial response atn:0iszeromoduloM.Forexample, consider thedesiredresponse forM:2: x(nT):{ ~-1 o(n:0) (n:1) (n:2) (otherwise) SlIowwhythisresponse cannotbeprecoded. 9·22Consider theRClowpassfiltershowninFig.P9-22,whereT:RC:10-'. aDetermine andsketchtheenvelope (group)delayofthefilterasafunction of frequency. bSuppose thattheinputtothefilterisalowpasssignalofbandwidtht:.t:1kHz. Determine theeffectoftbeRCtilteronthissignal. 9-23Amicrowave radiochannel hasafrequency response C(f):I+0.3cos21tjT Determine thefrequency response characteristic oftheoptimum transmitting and receiving filtersthatyieldzerolSIatarateofliTsymbolsls andhavea50% eXcessbandwidth. Assumethattheadditive noisespectrum islIat. 9-24M:4PAMmodulation isusedfortransmitting atabitrateof9600bitlsona clIannelhavingafrequency response qf):I+i(t12400) forIfI,;;;2400,andCU):0otherwise. Theadditive noiseiszero-mean, white Gaussian withpowerspectral density ~NoW1Hz.Determine the(magnitude) frequency response characteristic oftheoptimum transmitting andreceiving filters. 9-25Determine thecapacity ofa(0,I)runlength-limited code.Compace itscapacity withthatofa(I,00)codeandexplaintherelationship. 9·26Aternarysignalformatisdesigned forachannel thatdoesnotpassd.c.The FIGURE P9-31CHAPTER IJ:SIGNAL DESIGN FORBAND·LJMITED CHANNELS SSI binaryinputinformation sequence istransmitted bymapping a 1intoeithera positive pulseoranegative pulse,andazeroistransmitted bytheabsence ofa pulse.Hence,forthetransmission ofIs.thepolarityofthepulsesalternate. Thisis calledanAMI(alternate markinversion) code.Determine thecapacity ofthe code. 9·27Giveanalternative description oftheAMIcodedescribed inProblem 9-26using therunning digitsum(RDS)withtheconstraint thattheRDScantakeonlythe values0and+I. 9-28(kBnTcodes)FromProblem 9-26,notethattheAMIcodeisa"pseudo-ternary" codeinthatittransmits onebitpersymbolusingaternaryalphabet, whichhasthe capacity oflog,3=1.58bits.Suchacodedoesnotprovide sufficient spectral shaping. Betterspectralshaping isachieved bytheclassofblockcodesdesignated askBnT.wherekdenotes thenumber ofinformation bitsandndenotes the numberofternarysymbols perblock.Byselecting thelargestkpossible foreach n.weobtainthefollowing table: k I 3 4 6n 1 2 3 4Code IBIT 3B2T 4B3T 6B4T FIGURE P9-32Determine theefficiency ofthesecodesbycomputing theratioofthecodein bits/symbol dividedbylog,3.NotethatIBITistheAMIcode. 9-29Thisprohlem dealswiththecapacity oftwo(d.K)codes. aDetermine thecapacity ofa(d.K)codethathasthefollowing statetransition matrix: D=[:~] bRepeat(a)for D=[II]oI cComment onthedifferences between (a)and(b). 9-30Asimplified modelofthetelegraph codeconsistsoftwosymbols (Blahut, 1990). Adotconsistsofonetimeunitoflineclosurefollowed byonetimeunitofline 582 DIGITAL COMMUNICATIONS open.Adashconsistsofthreeunitsoflineclosurefollowed byonetimeunitof lineopen. aViewingthiscodeasaconstrained codewithsymbolsofequalduration. givethe constraints. bDetermine thestate-transition matrix. c:Determine thecapacity. 9·31Determine thestate-transition matrixfortherunlength-constrained codedescribed bythestatediagram showninFig.P9-3l.Sketchthecorresponding trellis. 9·32Determine thestate·transition matrixforthe(2,7)runlength-limited code specified bythestatediagram showninFig.P9-32. 10 COMMUNICA TION THROUGH BAND-LIMITED LINEAR FILTER CHANNELS InChapter 9.wefocused onthedesignofthemodulator anddemodula tor filtersforband-limited channels. Thedesignprocedure wasbasedonthe assumption thatthe(idealornon-ideal) channel response characteristic C(f) wasknownapriori.However, inpractical digitalcommlmicacions systems thaI aredesigned totransmit athighspeedthrough band-limited channels. the frequency response C(f)ofthechannel isnotknownwithsufficient precision todesignoptimum filtersforthemodulator anddemodulator. Forexample. in digitalcommunication overthedial-uptelephone network, thecommunication channel willbedifferent everytimewedialanumber. because thechannel routewillbedifferent. Thisisanexample ofachannel whosecharacteristics areunknown apriori.Thereareolhertypesofchannels, e.g.•wireless channels suchasradiochannels andunderwater acoustic channels. whosefrequency response characteristics aretime-variant. Forsuchchannels. itisnotpossible todesignoptimum fixeddemodulation filters. Inthischapter. weconsider theproblem ofreceiver designinthepresence ofchannel distortion. whichisnotknownapriori.andAWGN.Thechannel distortion resultsinintersymbol interference. which,ifleftuncompensated. causeshigherrorrates.Thesolution tothelSIproblem istodesignareceiver thatemploys ameansforcompensating orreducing thelSIinthereceived signal.Thecompensator forthelSIiscalledanequalizer. Threetypesofequalization methods aretreatedinthischapter. Oneis basedonthemaximum-likelihood (ML)sequence detection criterion. whichis optimum fromaprobability oferrorviewpoint. Asecondequalization method isbasedontheuseofalinearfilterwithadjustable coefficients. Thethird equalization method thatisdescribed exploits theuseofprevious detected 583 S84 DIGITAL COMMt'NIC'ATIONS symbols tosuppress thelSIinthepresentsymbolbeingdetected, anditis calleddecision-feedback equalization. Webeginwiththederivation ofthe optimum detector forchannels withlSI. 10-1OPTIMUM RECEIVER FORCHANNELS WITH lSIANDAWGN Inthissection, wederivethestructure oftheoptimum demodulator and detector fordigitaltransmission throughanonideal, band-limited channelwith additive gaussian noise.Webeginwiththetransmitted (equivalent lowpass) signalgivenby(9-2-1).Thereceived (equivalent lowpass) signalisexpressed as (10-1-1) n whereh(t)represents theresponse ofthechanneltotheinputsignalpulseg(t) andz(t)represents theadditive whitegaussian noise. Firstwedemonstrate thattheoptimum demodulator canberealized asa filtermatched toh(t),followed byasampleroperating atthesymbolrateliT andasubsequent processing algorithm forestimating theinformation sequence {In}fromthesamplevalues.Consequently, thesamples attheoutputofthe matched filteraresufficient fortheestimation ofthesequence {In}. 10-1-1Optimum Maximum-Likelihood Receiver Letusexpandthereceived signalT,(t)intheseries N T,(t)=limLT,[.(t) N...........,.1<=1(10-1-2) where(f.(t)}isacomplete setoforthonormal functions and{r.}arethe observable randomvariables obtained byprojecting TI(t)ontotheset(f.(t)}.It iseasilyshownthat nk=1,2,... (10-1-3) whereh'nisthevalueobtained fromprojecting h(t-nT)onto[.(t),andz.is thevalueobtained fromprojecting z(t)onto[.(t).Thesequence {z.}is gaussian withzeromeanandcovariance (10-1-4) Thejointprobability density function oftherandom variables CHAPTER 10:COMMUNICATION THROUGH BAND·LlMITED CHANNELS S8S rN'"[r,'2...rN]conditioned onthetransmitted sequence Ip'"[I,J,...Ipl. wherep'"N,is p(rN31p)=(2:NJ exp(-2~otlIrk-~I.hkJ) (10-1-5) Inthelimitasthenumber Nofobservable randomvariables approaches infinity,thelogarithm ofp(rN3Ip)isproportional tothemetricsPM(lp), definedas PM(lp)= -r~1r,(I)-~l.h(1-nT)/'dl =-r~Ir{(1)12dl+2Re~[I~r~r{(I)h*(t-nT)dl] -~~1~/m[~h*(t-nT)h(t-mT)dt (10-1-6) Themaximum-likelihood estimates ofthesymbols11012,•••,Iparethosethat maximize thisquantity. Note,however, tllattheintegralofIr,(1)12iscommon to allmetrics,and,hence,itmaybediscarded. Theotherintegralinvolving r(l) givesrisetothevariables Y.'"y(nT)=r~rl(t)h"(t -nT)dl (10-1-7) Thesevariables canbegenerated bypassingr(l)throughafiltermatched to h(l)andsampling theoutputatthesymbolrateliT.Thesamples{Yn}forma setofsufficient statistics forthecomputation ofPM(lp)or,equivalently, ofthe correlation metrics CM(l p)=2Re(~I~Y.)-~~I~Imx._~ • •m(10-1-8) (10-1-9)where,bydefinition, X(I)istheresponse ofthematched filtertoh(l)and x.'"x(nT)=[~h*(I)h(1+nT)dl Hence,X(I)represents tileoutputofafilterhavinganimpulseresponseh*(-I) andanexcitation h(I).Inotherwords,X(I)represents theautocorrelation functionofh(I).Consequently, {x.}represents thesamplesoftheautocorrela­ tionfunction ofh(I),takenperiodically atliT.Wearenotparticularly concerned withthenoncausal characteristic ofthefiltermatched toh(I),since, inpractice, wecanintroduce asufficiently largedelaytoensurecausality ofthe matched filter. Ifwesubstitute forr/(I)in(10-1-7)using(10-1-1), weobtain Yk=2:IoXk-.+Vk •(10-1-10) S86 DIGITAL COMMUNICATIONS where Vkdenotestheadditive noisesequence oftheoutputofthematched filter,i.e., Vk=fxz(t)h*(1-kT)dl (lO-l-ff) Theoutputofthedemodulator (matched filter)atthesampling instantsis corrupted bylSIasindicated by(10-1-10). Inanypractical system, itis reasonable toassumethatthelSIaffectsafinitenumberofsymbols. Hence, wemayassumethatXn=0forinl>L.Consequently, thelSIobserved atthe outputofthedemodulator maybeviewedastheoutputofafinitestate machine. ThisimpliesthatthechanneloutputwithlSImayberepresented by atrellisdiagram, andthemaximum-likelihood estimate oftheinformation sequence (/"1,,...,lp)issimplythemostprobable paththrough thetrellis giventhereceived demodulator outputsequence {Yn}.Clearly, theViterbi algorithm provides anefficientmeansforperforming thetrellissearch. Themetricsthatarecomputed fortheMLSEofthesequence {Ik}aregiven by(10-1-8).Itcanbeseenthatthesemetricscanbecomputed recursively in theViterbialgorithm, according totherelation Figure10-1-1illustrates theblockdiagram oftheoptimum receiver foran AWGNchannelwithlSI. 10·1·2ADiscrete-Time ModelforaChannel withlSI Indealingwithband-limited channels thatresultinlSI,itisconvenient to develop anequivalent discrete-time modelfortheanalog(continuous-time) system. Sincethetransmitter sendsdiscrete-time symbols atarate I/Tsymbols/s andthesampled outputofthematched filteratthereceiveris alsoadiscrete-time signalwithsamplesoccurring atarateI/Tpersecond,it followsthatthecascadeoftheanalogfilteratthetransmitter withimpulse response g(I),thechannelwithimpulseresponse c(I),thematched filteratthe receiverwithimpulseresponse h*(-I),andthesamplercanberepresented by FlGURE 10.1-1 Optimum receiverforanAWGNchannelwithlSI. ReceivedMatched MLSEOutputfilter-Sampler-(Viterbi signalh-(-t) Algorithm)data TI(!) r Clock r=kT CHAPTER 10:COMMUNICATION THROUGH BA~D-LiMITED CHANNELS 587 X_L x x (10-1-13)Z-l=delayofT II'nI nGURE 18-1-2 Equivalent dis<:rete-time modelofchannelwithintersymbol interference. anequivalent discrete-time transversal filterhavingtapgaincoefficients {x,,}. Consequently, wehaveanequivalent discrete-time transversal filterthatspans atimeintervalof2LTseconds. Itsinputisthesequence ofinformation symbols {I,,}anditsoutputisthediscrete-time sequence {y,,}givenby (10-1-10). Theequivalent discrete-time modelisshowninFig.10-1-2. Themajordifficulty withthisdiscrete-time modeloccursintheevaluation of performance Dfthevariousequalization orestimation techniques thatare discussed in'hefollowing sections. Thedifficulty iscausedbythecorrelations inthenoisesequence {v,,}attheoutputofthematched filter.Thatis,thesetof noisevariables {v.}isagaussian-distributed sequence withzeromeanand autocorrelation function(seeProblem 10-5) 1£(• ) _{NoXh Ok-il,,;;L) 2Vicv·- .J0 ( otherwise) Hence,thenoisesequence iscorrelated unlessx"=0,k""O.Sinceitismore convenient todealwiththewhitenoisesequence whencalculating theerror rateperformance, itisdesirable' towhitenthenDisesequence byfurther filteringthesequence [y,,}.Adiscrete-time noise-whitening filterisdetermined asfDllows. LetX(z)denotethe(two-sided) ztransform ofthesampled autocorrelation function{x,,},i.e., L X(z)=2:x"z-" k=-L(10-1-14) SinceXk=x*..,itfollowsthatX(z)=X*(Z-I) andthe2LrootsofX(z)have thesymmetry thatifpisaroot,1/p*isalsoaroot.Hence,X(z)canbe factored andexpressed as X(z)=F(z)P(z -I) (10-1-15) S88 DIGITAL COMMUNICA nONS whereP(z)isapolynomial ofdegreeLhavingtherootsPI'P2,...,PLand P*(Z-I) isapolynomial ofdegreeLhavingtheroots1/pt,1/pr,.··, IIp!. Thenanappropriate noise-whitening filterhasaztransform l/p*(Z-I). Since thereare2Lpossible choicesfortherootsofP*(z-1),eachchoiceresulting in afiltercharacteristic thatisidentical inmagnitude butdifferent inphasefrom otherchoicesoftheroots,wepropose tochoosetheuniqueP*(z") having minimum phase,i.e.,thepolynomial havingallitsrootsinsidetheunitcircle. Thus,whenalltherootsofF*(Z") areinsidetheunitcircle,1/F*(i") isa physically realizable, stable,recursive discrete-time filter.tConsequently, passageofthesequence {Y.}through thedigitalfilter1/p*(z'l) resultsinan outputsequence {v.}thatcanbeexpressed as L v.=2:fnI.·n+'I. ,,=0(10-1-16) where{'1.lisawhitegaussian noisesequence andU.}isasetoftap coefficients ofanequivalent discrete-time transversal filterhavingatransfer function P(z).Ingeneral, thesequence {Vk}iscomplex-valued. Insummary, thecascadeofthetransmitting filterg(t),thechannelc(t),the matched filterh*(-t),thesampler, andthediscrete-time noise-whitening filter 1/F*(z-I)canberepresented asanequivalent discrete-time transversal filter havingtheset{lk}asitstapcoefficients. Theadditive noisesequence {'I.} corrupting theoutputofthediscrete-time transversal filterisawhitegaussian noisesequence havingzeromeanandvariance No.Figure10-1-3illustrates the modeloftheequivalent distrete-time systemwithwhitenoise.Werefertothis modelastheequivalent discrete-time whitenoisefiltermodel. FIGURE 10.1·3 Equivalent discrete-lime modelofintersymbol interference channelwithWON. II,I to x l~1=delayofTx x tByremDving thestability condition, wecanalsoshowF*{t-1)tohaverootsontheunitcircle. CHAPTER 10:COMMUNICATION THROUGH BAND·LIMITED CHANNELS 589 Example 18-1·1 Suppose thatthetransmitter signalpulseg(t)hasduration Tandunit energyandthereceived signalpulseish(/)=g(/)+ag(t-T).Letus determine theequivalent discrete-time white-noise filtermodel.Thesample autocorrelation function isgivenby {a* (k=-1) Xk=1+jal2(k=0) a (k=1) Theztransform ofXkis ! X(z)=2:Xk<-k k=-l =a*z+(1+lal2)+az·1 =(az·!+1)(a*z+1)(10-1-17) (l0-1-18) Undertheassumption thatlaJ>1,onechoosesF(z)=OZ"l+1.sothatthe equivalent transversal filterconsistsoftwotapshavingtapgaincoefficients fo=1,J,.=a.Notethatthecorrelation sequence {Xk}maybeexpressed in termsofthe{.f.}as L-k Xk=2:f~fn+k' k=0,1,2,...•L "=0(10-1-19) Whenthechannel impulse response ischanging slowlywithtime,the matched filteratthereceiverbecomes atime-variable filter.Inthiscase,the timevariations ofthechannel/matched-filter pairresultinadiscrete-time filter withtime-variable coefficients. Asaconsequence, wehavetime-variable intersymbol interference effects,whichcanbemodeled bythefilterillustrated inFig.10-1-3,wherethetapcoefficients areslowlyvaryingwithtime. Thediscrete-time whitenoiselinearfiltermodelfortheintersymbol interference effectsthatariseinhigh-speed digitaltransmission overnonideal band-limited channels willbeusedthroughout theremainder ofthischapterin ourdiscussion ofcompensation techniques fortheinterference. Ingeneral,the compensation methods arecalledequalization lechniques orequalization algorilhms. . 10·1·3TheViterbiAlgorithm fol'theDiscI'ete-Time White NoiseFilterModel MLSEoftheinformation sequence {/k}ismosteasilydescribed intermsofthe received sequence {vdattheoutputofthewhitening filter.Inthepresence of 590 DIGITAL COMMUNICATIONS intersymbol interference thatspansL+1symbols(Linterfering components), theMLSEcriterion isequivalent totheproblem ofestimating thestateofa discrete-time finite-state machine. Thefinite-state machine inthiscaseisthe equivalent discrete-time channel withcoefficients {[d.anditsstateatany instantintimeisgivenbytheLmostrecentinputs,i.e.,thestateattimekis whereI,=0fork,.;O.Hence.iftheinformation symbols areM-ary,the channel filter!JasMLstates.Consequently, thechannel isdescribed byan ML-state trellisandtheViterbialgorithm maybeusedtodetermine themost probable paththroughthetrellis. Themetricsusedinthetrellissearchareakintothemetricsusedin soft-decision decoding ofconvolutional codes.Inbrief,webeginwiththe samplesvI.V2•••••VL'10fromwhichwecompute theML+1metrics TheML+Ipossible sequences of1,.+101"...•/2•IIaresubdivided intoML groupscorresponding totheMLstates(lL+I.I,'...•12),NotethattheM sequences ineachgroup(state)differinIIandcorrespond to'thepathsthrough thetrellisthatmergeatasinglenode.FromtheMsequences ineachofthe MLstates,weselectthesequence withthelargestprobability (withrespectto I,)andassigntothesurviving sequence themetric PMI(IL+,)=PMI(lL+I, 1,•...,/2)L.'2:lnp(v,I1"/H,...,I,d k=\ L+I =max2:lnp(v,lIb1,-1•....1,_,) I,*=1(10-1-20) (10-1-21) (10-1-22), TheM-1remaining sequences fromeachoftheMLgroupsarediscarded. Thus,weareleftwithMLsurviving sequences andtheirmetrics. Uponreception ofVL+'.theMLsurviving sequences areextended byone stage,andthecorresponding ML+Iprobabilities fortheextended sequences arecomputed usingtheprevious metricsandthenewincrement, whichis Inp(vL+21/L+20 1,+10'..•1,).Again,theML+1sequences aresubdivided into MLgroupscorresponding totheMLpossiblestates(JL+2'...,/3)andthemost probable sequence fromeachgroupisselected, whiletheotherM-1 sequences arediscarded. Theprocedure described continues withthereception ofsubsequent signal samples. Ingeneral,uponreception ofvLH.themetricst tWeobserve thatthemetrics PMk(l)aresimplyrelatedtotheeuclidean distance metrics DM,(I)whentheadditivenoiseisgaussian. CHAPTER 10:COMMUNICATION THROUGH BAND·lIMITED CHANNELS 591 thatarecomputed givetheprobabilities oftheMLsurviving sequences. Thus, aseachsignalsampleisreceived, theViterbialgorithm involves firstthe computation oftheML+Iprobabilities Inp(vL+k I/uk>'..,I.)+PMk-I(IL+k-l) (10-1-24) corresponding totheML+Isequences thatformthecontinuations of.theML surviving sequences fromtheprevious stageoftheprocess. ThentheML+I sequences aresubdivided intoMLgroups,witheachgroupcontaining M sequences thatterminate inthesamesetofsymbolsIUk,'..,/HIanddifferin thesymbolI..FromeachgroupofMsequences, weselecttheonehavingthe largestprobability asindicated by(10-1-23), whiletheremaining M-1 sequences arediscarded. Thus,weareleftagainwithMLsequences havingthe metricsPMk(lL+k)' Asindicated previously, thedelayindetecting eachinformation symbolis variable. Inpractice, thevariable delayisavoidedbytruncating thesurviving sequences totheqmostrecentsymbols, whereqPL,thusachieving afixed delay.InthecasethattheMLsurviving sequences attimekdisagree onthe symbolh_q'thesymbolinthemostprobable sequence maybechosen.The lossinperformance resulting fromthissuboptimum decision procedure is negligible ifq;;.5L. Example 10-1·2 Forillustrative purposes, supposethataduobinary signalpulseisemployed totransmit four-level (M=4)PAM.Thus,eachsymbolisanumber selectedfromtheset{-3,-1,1,3}.Thecontrolled intersymbol interference inthispartialresponse signalisrepresented bytheequivalent discrete-time channelmodelshowninFig.10-1-4.Suppose wehavereceived VIandV2, where FIGURE 10-1-4 Equivalent discrete-time model forintersymbol interference resulting fromaduobinary pulse.VI=[I+1/1 V2=12+[I+'12 o T lalInput1" x (b)(10-1-25) Output lI,,=I"+11~1+TIt S92 DIGITAL COMMUNICA nONS and{1'J,}isasequence ofstatistically independent zero-mean gaussian noise. Wemaynowcompute the16metrics whereh=0fork,,;;O. Notethatanysubsequently receivedsignals{v;}donotinvolve 11.Hence, atthisstage,wemaydiscard12ofthe16possiblepairs{II,12}.Thisstepis illustrated bythetreediagram showninFig.10-1-5.Inotherwords,after computing the16metricscorresponding tothe16pathsinthetreediagram, /,~-.I --.:0-_1•=-J {~""J, \-{- I):::\ "'-~--IIJ =-I / "':'.) FIGURE IO-I·S Treediagram forViterb-idecoding ofthedouoinary pulse. rrf (10-1-27)CHAPTER 1(1:COMMUNICATION THROUGH BAND-LIMITED CHANNELS 593 wediscardthreeoutofthefourpathsthatterminate with12=3andsavethe mostprobable ofthesefour.Thus,themetricforthesurviving pathis PMt(I,=3,I,)=max[-±(Vk_±h-i)'] I, k=l }=o Theprocessisrepeated foreachsetoffourpathsterminating withI,=I, I,=-1,andI,= -3.Thusfourpathsandtheircorresponding metricssurvive afterv,andv,arereceived. WhenV3isreceived, thefourpathsareextended asshowninFig.10-1-5,to yield16pathsand16corresponding metrics,givenby PM,(l3'I"It)=PMt(l"I,)-(V3-±I3-i)' J-O Ofthefourpathsterminating withtheI,=3,wesavethemostprobable. This procedure isagainrepeated for13=1,I,=-1,and13=-3.Consequently, onlyfourpathssurviveatthisstage.Theprocedure isthenrepeated foreach subsequently received signal Ukfork>3. 10-1-4Performance ofMLSEforChannels withlSI Weshallnowdetermine theprobability oferrorforMLSEofthereceived information sequence whentheinformation istransmitted viaPAMandthe additivenoiseisgaussian. Thesimilarity between aconvolutional codeanda finite-duration intersymbol interference channelimpliesthatthemethodfor computing theerrorprobability forthelattercarriesoverfromtheformer.In particular, themethodforcomputing theperformance ofsoft-decision decod­ ingofaconvolutional codebymeansoftheViterbialgorithm, described in Section8-2-3,applieswithsomemodification. InPAMsignaling withadditivegaussian noiseandintersymbol interference, themetricsusedintheViterbialgorithm maybeexpressed asin(10-1-23), or equivalently, as PMk-L(lk)=PMk-L-t(lk-t) -(Uk-±fih-i)' (10-1-28) J-O wherethesymbols {In}maytakethevalues±d,±3d,...,±(M-l)d,and2d isthedistancebetween successive levels.ThetrellishasMLstates,definedat timekas (10-1-29) Lettheestimated symbols fromtheViterbialgorithm bedenoted by{In} andthecorresponding estimated stateattimekby (10-1-30) 594 DIGITAL COMMUNICATIONS Nowsuppose thattheestimated paththrough thetrellisdiverges fromthe correctpathattimekandremerges withthecorrectpathattimek+t.Thus, Sk=SkandShl=Sk+l,butSm¥Smfork<m<k+l.Asinaconvolutional code,wecallthisanerrorevent.SincethechannelspansL+1symbols, it followsthatI;;;.L+1. Forsuchanerrorevent,wehave1k¥1,andlk+I-L-l¥lk+I-L-h but1m=1m fork-L,,;;m.;;k-1andk+I-L,,;;m.;;k+t-1.Itisconvenient todefine anerrorvectorecorresponding tothiserroreventas (10-1-31) wherethecomponents ofearedefinedas (10-1-32) Thenormalization factorof2din(10-1-32) resultsinelements Ejthattakeon thevalues±I,±2,±3,...,±(M-1).Moreover, theerrorvectorischarac­ terizedbytheproperties thatEk¥0,EHI-L-I ""0,andthereisnosequence of Lconsecutive elements thatarezero.Associated withtheerrorvectorin (10-1-31) isthepolynomial ofdegree1-L-1, (10·1-33) Wewishtodetermine theprobability ofoccurrence oftheerroreventthat beginsattimekandischaracterized bytheerrorvectoregivenin(10-1-31), or,equivalently, bythepolymonial givenin(10-1-33). Toaccomplish this,we followtheprocedure developed byForney(1972).Specifically, fortheerror eventeto occur, thefolJowing threesubevents E"E2,and£3mustoccur: E,:attimek,Sk=Sk; E2:theinformation symbols lk'h+h"" 1H'-L-lwhenaddedtothe scalederrorsequence 2d(Ek> E,+".•.,EHI-L-l) mustresultinan allowable sequence, i.e.,thesequence1..1k+h...,lk+I-L-l musthave valuesselectedfrom±d,±3d,±...±(M-l)d; E,:fork.;;m<k+I,thesumofthebranchmetricsoftheestimated path exceedthesumofthebranchmetricsofthecorrectpath. TheprobabIlity ofoccurrence ofE,is But L Vi=2:h/i-/+7/, j=O(10-1-34) (10-1-35) CHAPTER 10:COMMUNICATION nlROUGH BAND-LIMITED CHANNELS 595 where{'Ij,}isareal-valued whitegaussian noisesequence. Substitution of (10-1-35) into(10-1-34) yields [HI-1 ( L)'Hr-I ] peE,)=P,~Tf,+2dtt,hEi-j<~7/J [HI-I(L) HI-I(L)2] =P4di~l1ij~.tEi-j<-4d2 i~j~JjE'_j where Ej=0fori<kandi>k+!-L-1.Ifwedefine L ai='itEi-j j=O then(10-1-36) maybeexpressed as(10-1-36) (10-1-37) (10-1-38) wherethefactorof4dcommon tobothtermshasbeendropped. Now (10-1-38) isjusttheprobability thatalinearcombination ofstatistically independent guassian random variables islesstllansomenegative number. Thus (10-1-39) Forconvenience, wedefine k+l-l k+l-l (L )2 c')2(E)=i~a}=,~~JjEi-j (l0-1-40) whereej=0forj<kandi>k+/-L-1.Notethatthe{a,}resulting from theconvolution or{.t:lwith{ej}arethecoefficients ofthepolynomial a(z)=F(z)e(z) (10-1-41) Furthermore, c')2(E)issimplyequaltothecoefficient ofZOinthepolynomial a(z)a(z-I) =F(z)F(z-l)e(Z)E(z-l) =X(z)E(z)e(z -I) Wecallc')2(E)theeuclidean wei$!hloftheerr')reventE.(10-1-42) 596 DIGITAL COMMUNICATIONS Analternative methodforrepresenting theresultofconvolving {Ji}with{e) isthematrixform a=ef whereaisanI-dimensional vector,fisan(L+1)-dimensional vector,andeis anIx(L+1)matrix,definedas o ooo o o(10-1-43) Then 02(E)=a'lX =r'e'er =fAf whereAisan(L+1)x(L+1)matrixoftheform(10-1.44) f30fll{32 (31fJo(31 A=e'e= f3,fllf30PI f3Lf3L (3L-I (3L-2 (10-1-45) flo and k+l-l-m 13m=LE;Ei+m i=k(10-1-46) (10-1-47) =Q(Wemayuseeither(10-1-40) and(10-1-41) or(10-1-45)-(10-1-46) inevaluating theerrorrateperformance. Weconsider thesecomputations later.Fornowwe conclude thattheprobability ofthesubevent £3'givenby(10-1-39), maybe expressed as wherewehaveusedtherelation 23 d=M2-1Tp"v (10-1-48) toeliminate d2and1'av=TP.JNo.Notethat,intheabsenceofintersymbol (-HAPTER 10:COMMUNICATION THROUGH BAND-LIMITED CHANNELS 597 interference, 02(E)=1andP(E3)isproportional tothesymbolerrorprob­ abilityofM-aryPAM. Theprobability ofthesubevent E2depends onlyonthestatistical properties oftheinputsequence. Weassumethattheinformation symbols areequally probable andthatthesymbols inthetransmitted sequence arestatistically independent. Then,foranerroroftheformIE;I=j,j=1,2,...,M-1,there areM-jpossiblevaluesofIisuchthat I;=Ii+2dE, Hence (-L-lM-Iii P(E2)=nM (10-1-49) (10-1-50)Theprobability ofthesubevent EIismuchmoredifficulttocompute exactly because ofitsdependence onthesubevent £3'Thatis,wemustcompute P(EIIE3)·However, p(EIIE3)=1-PM,where PMisthesymbolerror probability. HenceP(EIIE3)iswellapproximated (andupper-bounded) by unityforreasonably lowsymbolerrorprobabilities. Therefore, theprobability oftheerrorevent£iswellapproximated andupper-bounded as (6)/-L-lM-Iii P(E)";;Q M2-1'Ya.02(E)nM LetEbethesetofallerrorevents Estartingattimekandletw(E)bethe corresponding numberofnonzero components (Hamming weightornumberof symbolerrors)ineacherroreventE.Thentheprobability ofasymbolerroris upper-bounded (unionbound)as PM";;LW(E)P(e.) (10-1-51) NowletDbethesetofallO(E).Foreach0ED,let£8bethesubsetoferror eventsforwhichO(E)=O.Then(10-1-51) maybeexpressed as (I6 )[ I-L-IM-Iii] PM";;8~DQVM2-1'Yn02.~,W(E),I] M ,,;;LK8Q(I26_'Ya,52) (10-1-52) 8ED,/M1 where (l0-1-53) Theexpression fortheerrorprobability in(l0-1-52) issimilartotheformof theerrorprobability foraconvolutional codewithsoft-decision decoding given (10-1-54)S9IDIGITAL COMMUNICATIONS by(8-2-26). Theweighting factors{K.}maybedetermined bymeansofthe errorstatediagram, whichisakintothestatediagram ofaconvolutional encoder. Thisapproach hasbeenillustrated byForney(1972)andViterbiand Omura(1979). Ingeneral,however, theuseoftheerrorstatediagramforcomputing PMis tedious.Instead,wemaysimplify thecomputation ofPMbyfocusing onthe dominant terminthesummation of(10-1-52). Duetotheexponential dependence ofeachterminthesum,theexpression PMisdominated bythe termcorresponding totheminimum valueof[j,denoted as[jmin'Hencethe symbolerrorprobability maybeapproximated as 6 2 )M2_1)'av[jmin where (10-1-55) Ingeneral, [j~jn";:1.Hence,10log[j;"mrepresents thelossinSNRdueto intersymbol interference. Theminimum valueof[jmaybedetermined eitherfrom(10-1-40) orfrom evaluation ofthequadratic formin(10-1-44) fordifferent errorsequences. In thefollowing twoexamples weuse(10-1-40). ElUIIIlple 10-1-3 Consider atwo-path channel(L=1)witharbitrary coefficients faandIt satisfying theconstraintf5+f~=1.Thechannelcharacteristic is Foranerroreventoflengthn, ( )_+-\+ -(.-1)eZ-EoEIZ...+e._,z , Theproducta(z)=F(z)e(z) maybeexpressed as a(z)=00+£lIZ-I+...+anz-· whereao=eofo anda.=fIE.-,.Sinceeo;060,e._,,",,0,and • [j2(e)=2:a~ k=O itfollowsthat(l0-1-56) (10-1-57) (10-1·58) (10-1-59) 5~in;a,f5+fl=1 Indeed, [j~in=1whenasingleerroroccurs,i.e.e(z)=Eo.Thus,we conclude thatthereisnolossinSNRinmaximum-likelihood sequence CHAPTER 10:COMMUNICATION THROUGH BAND-LIMITED CHANNELS 599 estimation oftheinformation symbols whenthechannel dispersion has length2. Example 10-1·4 Thecontrolled intersymbol interference inapartialresponse signalmaybe viewedashavingbeengenerated byatime-dispersive channel. Thus,the intersymbol interference fromaduobinary pulsemayberepresented bythe (normalized) channelcharacteristic Similarly, therepresentation foramodified duobinary pulseis F(z)=Y!-Y!Z-2 Theminimum distance 8~in=1foranyerroreventofthe.form f(z)=±(I-z-1-z-2•••_z-(n-l), n;;;.1 forthechannelgivenby(10-1-60) since a(z)=±y!'fy!z-' Similarly, when E(Z)=±(1+Z·2_Z-4+...+z-2(n-l), n;;;.1 8~,;n=1forthechannelgivenby(10-1-61), since a(z)=±Y!'fY!z-2n(10-1-60) (10-1-61) (10-1-62) (10-1-63) HenceMLSEofthesetwopartialresponse signalsresultsinnolossinSNR. Incontrast, thesuboptimum symbol-by-symbol detection described pre­ viouslyresultedina2.1dBloss. TheconstantK.m,"iseasilyevaluated forthesetwosignals. With precoding, thenumber ofoutputsymbolerrors(Hamming weight) as­ sociated Vliththeerroreventsin(10-1-62) and(10-1-63) istwo.Hence, ,(MI)'K.m,"=2~l;;=2(M-1) (10-1-64) Ontheotherhand,withoutprecoding, theseerroreventsresultinnsymbol errors,and,hence, x(Ml)nK.",,"=22:n---=2M(M-1) n-lM(10-1-65) Asafinalexercise, weconsider theevaluation ofo~;nfromthequadratic 600 DIGITAL COMMUNICATIONS formin(10-1-44). ThematrixAofthequadratic formispositive-definite; hence,allitseigenvalues arepositive.If{/L,,(e)}aretheeigenvalues and{v.(e)} arethecorresponding orthonormal eigenvectors ofAforanerroreventethen thequadratic formin(10-1-44) canbeexpressed as L+l /)2(e)=2:IJok(e)[f'vk(E)y k=1(10-1-66) Inotherwords,/)2(E)isexpressed asalinearcombination ofthesquared projections ofthechannelvectorfontotheeigenvectors ofA.Eachsquared projection inthesumisweighted bythecorresponding eigenvalue Pk(E), k=1,2,...,L+1.Then <5~ln=min1J2(E)•(10-1-67) Itisinteresting tonotethattheworstchannel characteristic ofagiven lengthL+1canbeobtained byfindingtheeigenvector corresponding tothe minimum eigenvalue. Thus,ifIJomin(E)istheminimum eigenvalue foragiven erroreventeandvmln(e)isthecorresponding eigenvector then Pmin=min)Lmin(£;)• f=minVmin(E)• and Example 18-1-5 Letusdetermine theworsttime-dispersive channel oflength3(L=2)by lindingtheminimum eigenvalue ofAfordifferent errorevents.Thus, F(z)=/0+/,Z-1+fiz-2 wherefa,f"andfiarethecomponents oftheeigenvector ofA corresponding totheminimum eigenvalue. Anerroreventoftheform £(z)=I-z-1 resultsinamatrix A=[-~ -~-~] o-12 whichhastheeigenvalues ILl=2,IJo2=2+v'2,P3=2-v'2.Theeigenvec­ torcorresponding toJ.L3is (10-1-68) CHAPTER II):COMMUNICATION THROUGH BAN[).UMITED CHANNELS 601 Wemayalsoconsider thedualerrorevent E(z)=I+z-1 whichresultsinthematrix A=[~~~] o12 Thismatrixhaseigenvalues identical tothoseoftheoneforE(Z)=1-Z-1 Thecorresponding eigenvector forIJ-3=2 -V2is v;=[-!v1-!] (10-1-69) AnyothererroreventsleadtolargervaluesforILm'o'Hence, ILmin= 2-V2andtheworst-case channel iseither [!v1!]or[-!V!-~] ThelossinSNRfromthechannel is -10log8;"'0=-10logIJ-min=2.3dB Repetitions oftheabovecomputation forchannels withL=3,4,and5 yieldtheresultsgiveninTable10-1-1. 16-2LINEAR EQUALIZATION TheMLSEforachannelwithlSIhasacomputational complexity thatgrows exponentially withthelengthofthechanneltimedispersion. Ifthesizeofthe symbolalphabet isMandthenumberofinterfering symbols contributing to lSIisL,theViterbialgorithm computes ML+1metricsforeachnewreceived symbol. Inmostchannels ofpractical interest, suchalargecomputational complexity isprohibitively expensive toimplement. Inthisandthefollowing sections, wedescribe twosuboptimum channel equalization approaches tocompensate forthelSI.Oneapproach employs a lineartransversal filter,whichisdescribed inthissection. Thesefilter TABLE 10-1·t MAXIMUM PERFORMANCE LOSSANDCORRESPONDING CHANNEL CHARACTERISTICS Cunnel !enRth L+l 3 4 5 6Performance ..... -10log6;".(dB) Mlnimul8-dlstuce _Rei 2.3 0.50,0.71,0.50 4.2 0.38,0.60, 0.60,0.38 5.7 0.29,0.50,0.58,0.50,0.29 7.0 0.23,0.42,0.52,0.52,0.42,0.23 602 DIGITAL COMMUNICATIONS Unequalized input C! CI X X Equalized output '-- --;Algorithm fortap '-- ~g::.ai:n:ad:::ju:~t::::me::.:n::..1 J'~------..J FIGURE 16-2·1 LinearIransversal filler. structures haveacomputational complexity thatisIilinearfunction of'the channeldispersion lengthL. Thelinearfiltermostoftenusedforequalization isthetransversal filter showninFig.10-2-1.Itsinputisthesequence {II.}givenin(10-1-16) andits outputistheestimate oftheinformation sequence {I.}.Theestimate ofthe kthsymbolmaybeexpressed as K 1,=LCjll'_j j=-K(10-2-1) where{cJarethe2K+ 1complex-valued tapweightcoefficients ofthefilter. Theestimate1.isquantized tothenearest(indistance) information symbolto formthedecisioni•.Ifi.isnotidentical tothetransmitted information symbol I,.anerrorhasbeenmade. Considerable research hasbeenperformed onthecriterion foroptimizing thefiltercoefficients {col.Sincethemostmeaningful measure ofperformance foradigitalcommunications systemistheaverageprobability oferror,itis desirable tochoosethecoefficients tominimize thisperformance index. However, theprobability oferrorisahighlynonlinear function of{cJ Consequently, theprobability oferrorasaperformance indexforoptimizing thetapweightcoefficients oftheequalizer isimpractical. Twocriteriahavefoundwidespread useinoptimizing theequalizer coefficients (cJ.Oneisthepeakdistortion criterion andtheotheristhemean squareerrorcriterion. 10·2·1PeakDistortion Criterion Thepeakdistortion issimplydefinedastheworst-case intersymbol inter­ ferenceattheoutputoftheequalizer. Theminimization ofthisperformance indexiscalledthepeakdistortion criterion. Firstweconsider theminimization CHAPTER JO:COMMLJNICAT'ON THROt'GH BAND-LIMITED CHANNELS 603 ofthepeakdistortion assuming thattheequalizer hasaninfinitenumberof taps.Thenweshalldiscussthecaseinwhichthetransversal equalizer spansa finitetimeduration. Weobservethatthecascadeofthediscrete-time linearfiltermodelhaving animpulseresponse {fn}andanequalizer havinganimpulseresponse (e.}can berepresented byasingleequivalent filterhavingtheimpulseresponse x qn=LcJ.-j j=-~(10-2·2) Thatis,{q.Jissimplytheconvolution of{cn}and{[.}.Theequalizer isassumed tohaveaninfinitenumberoftaps.Itsoutputatthekthsampling instantcan beexpressed intheform x lk=qolk+LI.qk-.+Lc/Yh-j n"'/< j=-'I.::(10-2·3) Thefirsttermin(10-2-3)represents ascaledversionofthedesiredsymbol. Forconvenience, wenormalize qotounity.Thesecondtermistheintersymbol interference. Thepeakvalueofthisinterference, whichiscalledthepeak distortion, is n=-x =.~xIj~xcJ·-JI (10-2-4) ,,"0 Thus,!?i'(c)isafunctionoftheequalizer tapweights. Withanequalizer havinganinfinitenumberoftaps,itispossibletoselect the tapweightssothat!?i'(c)=0,Le.,q"=0forallnexceptn=O.Thatis,the intersymbol interference canbecompletely eliminated. Thevaluesofthetap weightsforaccomplishing thisgoalaredetermined fromthecondition ~ {I(n=0) q.=If:xcd'-l=0(n""0) Bytakingtheztransform of(10-2-5), weobtain Q(z)=C(z)F(z) =I or,simply, 1C(z)=­F(z)(10-2-5) (10-2-6) (10-2-7) whereC(z)denotestheztransform ofthe{cl}'Notethattheequalizer, with transferfunctionC(z),issimplytheinversefiltertothelinearfiltermodel F(z).Inotherwords,complete elimination oftheintersymbol interference requirestheuseofaninversefiltertoF(z).Wecallsuchafilterazero-forcing 604 DIGITAL COMMUNICATIONS FlGURF to.1-2 Blockdiagramofchannelwithzero-forcing equalizeT.AWGN '~.IEqualizer I~I ICt:l=F'(l) filter.Figure10-2-2illustrates in'blockdiagram theequivalent discrete-time channelandequalizer. Thecascade ofthenoise-whitening filterhavingthetransfer function IIF*(Z-I) andthezero-forcing equalizer havingthetransferfunctionIIF(z) resultsinanequivalent zero-forcing equalizer havingthetransferfunction '_ 1 1 C(z)-F(Z)F*(Z-I) =X-(z-)(10-2-8) (10-2-9)asshowninFig.10-2-3.Thiscombined filterhasasitsinputthesequence {yd ofsamplesfromthematched filter,givenby(10-1-10). Itsoutputconsistsof thedesiredsymbolscorrupted onlybyadditivezero-mean gaussian noise.The impulseresponse ofthecombined filteris c~=2~fC(z)zk-l dz 1fZk-I =2TrjX(z/z wheretheintegration isperformed onaclosedcontour thatlieswithinthe regionofconvergence ofC(z).SinceX(z)isapolynomial with2Lroots (Pt,P2,""PL,I/pf,IIp!,...,l/pt),itfollowsthatC(z)mustconverge inanannular regioninthezplanethaiincludes theunitcircle(z=ej8). Consequently, theclosedcontourintheintegralcanbetheunitcircle. Theperformance oftheinfinite-tap equalizer thatcompletely eliminates the intersymbol interference canbeexpressed intermsofthesignal-to-noise ratio (SNR) at itsoutput.Formathematical convenience, wenormalize thereceived FlGURE to.2-3 Blockofchannelwithequivalent zero-foTcing equalizer. ~Noise-whiteningEqullizcrA ChannelH IY.I fillerIv,} 1/,1 X(z)=F(l)F*(z-l) .. C(,)=_I_ C(,)=_I- F'(,-1) F(<) Gaussian noiseEquivalent equalizer C'(z) zJ=_1_ F(,)F'(,-') Xll) CHAPTER'aCOM/./UNICATION THROUGH BAND-LIMITED CHANNELS 60S signalenergytounity.tThisimpliesthatqo=1andthatthe'expected valueof II;12isalsounity.ThentheSNRissimplythereciprocal ofthenoisevariance ~attheoutputoftheequalizer. ThevalueofCT~canbesimplydetermined byobserving thatthenoise sequence {v.}attheinputtotheequivalent zero-forcing equalizer C(z)has zeromeanandapowerspectraldensity (10-2-10) where X(ei~') isobtained fromX(z)bythesubstitution z=e'~T.Since C'(z)=1/X(z), itfollowsthatthenoisesequence attheoutputofthe equalizer hasapowerspectraldensity IfIwl.o;; (10-2-11) (10-2-12) (10-2-14)Consequently, thevariance ofthenoisevariable attheoutputoftheequalizer is Tf'CIT CT~=2- <Il..(w)dw 1t-trlT TN0fKIT dw =21r_,rITX(eiw') andtheSNRforthezero-forcing equalizer is I'~=l/CT~ [TNof"iT dw]-1""- (to-2-13)21f_mX(eiw') wherethesubscript onI'indicates thattheequalizer hasaninfinitenumberof taps. Thespectralcharacteristics X(eiw')corresponding totheFouriertransform ofthesampled sequence {x.}hasaninteresting relationship totheanalogfilter H(w)usedatthereceiver. Since Xk=[~h*(t)h(t+kT)dt useofParseval's theorem yields Xk=21[IH(wWeiwkTdwIf_~ whereH(w)istheFouriertransform ofh(t).Buttheintegralin(10-2-14) can beexpressed intheform 1f"IT[~I(21rl1)12 ]X.=21f-.'Tn~~HW+Tei"""Tdw tThisnormalization isusedthroughout thischapterformathematical convenience.(10-2-15) 606 DIGITAL COMMUl\KATIONS Now,theFouriertransform of{x.}is x X(elwT)=2:xke-j~kT k=-x andtheinversetransform yields(10-2-16) ~1O-2-I7) Fromacomparison of(l0-2-I5) and(10-2-17), weobtainthedesired relationship between X(e;W~andH(w).Thatis, 1C Iwl"'­T(10-2-18) wheretheright-hand sideof(10-2-18) iscalledthefoldedspectrum ofIH(wW. WealsoobservethatIH(wW =X(w),whereX(w)istheFouriertransform of thewaveform x(t)andx(t)istheresponse ofthematched filtertotheinput her).Therefore theright-hand sideof(10-2-18) canalsobeexpressed interms ofX(w). Substitution forX(eiw7)in(10-2-13) usingtheresultin(10-2-18) yieldsthe desiredexpression fortheSNRintheform (10-2-19) Weobserve thatifthefoldedspectral characteristic ofH(w)possesses any zeros,theintegrand becomes infiniteandtheSNRgoestozero.Inother words,theperformance oftheequalizer ispoorwhenever thefoldedspectral characteristic possesses nullsortakesonsmallvalues.Thisbehavior occurs primarily because theequalizer, ineliminating theintersymbol interference, enhances theadditive noise.Forexample, ifthechannelcontains aspectral nullinitsfrequency response, thelinearzero-forcing equalizer attempts to compensate forthisbyintroducing aninfinitegainatthatfrequency. Butthis compensate~ forthechannel distortion attheexpense ofenhancing the additive noise.Ontheotherhand,anidealchannel coupled withan appropriate signaldesignthatresultsinnointersymbol interference willhavea foldedspectrum thatsatisfiesthecondition (10-2-20) Inthiscase,theSNRachieves itsmaximum value,namely, 1 ')'.=No(10-2-21) CHAPTER 10:COMMUNICA nONTHROUGH BAND-LlMIT£D CHANNELS -607 Finite-Length Equalizer Letusnowturnourattention toanequalizer having2K+1taps.SinceCj=0forIii>K,theconvolution of{fn}with{en}is zerooutsidetherange-K",;;n,;;;K+L-1.Thatis,qn=0forn< -Kand n>K+L-1.Withqonormalized tounity,thepeakdistortion is K,L~1 K,L-!I I 0l(c)=nf;KIqnl=nf;KtCdn-j n'J"l'O n¢O(10-2-22) Although theequalizer has2K+ 1adjustable parameters, thereare2K+L nonzero valuesintheresponse {qn}'Therefore, itisgenerally impossible to completely eliminate theintersymbol interference attheoutputofthe equalizer. Thereisalwayssomeresidual interference whentheoptimum coefficients areused.Theproblem istominimize 0l(c)withrespecttothe coefficients {cJ Thepeakdistortion givenby(10-2-22) hasbeenshownbyLucky(1965)to beaconvexfunction ofthecoefficients {Cj}'Thatis,itpossesses aglobal minimum andnorelative minima. Itsminimization canbecarriedout numerically using,forexample, themethodofsteepest descent. Littlemore canbesaidforthegeneralsolutiontothisminimization problem. However, for onespecialbutimportant case,thesolution fortheminimization of0l(c)is known.Thisisthecaseinwhichthedistortion attheinputtotheequalizer. definedas' 1L Do=1101~IIf,1(10-2-23) islessthanunity.Thisco.ndition isequivalent tohavingtheeyeopenpriorto equalization. Thatis,theintersymbol interference isnotsevereenoughtoclose theeye.Underthiscondition, thepeakdistortion 0l(c)isminimized by selecting theequalizer coefficients toforceqn=0for1,;;;Inl'"Kandqo=1.In otherwords,thegeneralsolutiontotheminimization of0l(c),whenDo<1,is thezero-forcing solutionfor{qn}intherange1",;;Inl",;;K.However, thevalues of{qn}forK+I",;;n,;;;K+L-1arenonzero, ingeneral. Thesenonzero valuesconstitute theresidual intersymbol interference attheoutputofthe equalizer. 10·2·2MeanSquareError(MSE)Criterion IntheMSEcriterion, thetapweightcoefficients {eJ}oftheequalizer are adjusted tominimize themeansquarevalueoftheerror E.=I.-1, (10-2-24) whereI,istheinformation symboltransmitted inthekthsignaling interval and1.istheestimate ofthatsymbolattheoutputoftheequalizer, defined in (10-2-25)60S DIGITAL COMMUNICATIONS (10-2-1). Whentheinformation symbols {I.}arecomplex-valued, theperfor­ mance index fortheMSEcriterion, denoted byJ,isdefinedas J=E1£.12 A2 =Ell.-1.1 Ontheotherhand,whentheinformation symbols arereal-valued, the performance indexissimplythesquareoftherealpartofEk'Ineithercase,J isaquadratic function oftheequalizer coefficients {c,}.Inthefollowing discussion, weconsider theminimization ofthecomplex-valued formgivenin (10-2-25). Infinite-Length Equalizer First,weshallderivethetapweightcoefficients thatminimize Jwhentheequalizer hasaninfinitenumberoftaps.Inthiscase, theestimate1.isexpressed as x 1.=2:(iv'"J j=::-x(10-2-26) Substitution of(10-2-26) intotheexpression forJgivenin(10-2-25) and expansion oftheresultyieldsaquadratic function ofthecoefficients {cJThis function canbeeasilyminimized withrespecttothe{ci}toyieldaset(infinite innumber) oflinearequations forthe{cJ}'Alternatively, the setoflinear equations canbeobtained byinvoking theorthogonality principle inmean squareestimation. Thatis,weselectthecoefficients {ci}torendertheerrorE. orthogonal tothesignalsequence {vr-,}for-<:Xl< /<00.Thus, E(EkVt_l) =0,-00</< 00 Substitution forE.in(10-2-27) yields E[(I.-i~xc;v.-;)vt-IJ=0 or,equivalently, x 2.:c;E(v._;vt_,) =E(I.vt_I), -00</<00 j=;-x.(10-2-27) (10-2-28) Toevaluate themoments in(10-2-28), weusetheexpression foru.givenin (10-1-16). Thus,weobtain L E(v.-ivt-I) =2:1:ln+l-i+NoB'in=O and={XI-i+NoBli(1/-jl,.;;L) o (othe~se) E(l*{/!I(-L,.;;/,.;;O).V._I)=0(otherwise)(10-2-29) (10-2-30) CHAPTER 10:COMMUNICATION THROUGH BAND·L1MITED CHANNELS 609 Now,ifwesubstitute (10-2-29) and(10-2-30) into(10-2-28) andtakethez transform ofbothsidesoftheresulting equation, weobtain (10-2-31) (10-2-32)C(z)=F(z)F*(z I)+NoTherefore, thetransferfunctionoftheequalizer basedontheMSEcriterion is F*(z ~I) Whenthenoise-whitening filterisincorporated intoC(z),weobtainan equivalent equalizer havingthetransferfunction (10-2-33)1 X(z)+NoC(z)=F(z)F*(Z-') +No I Weobservethattheonlydifference between thisexpression forC(z)and theonebasedonthepeakdistortion criterion isthenoisespectral density factorNothatappearsin(10-2-33). WhenNoisverysmallincomparison with thesignal,thecoefficients thatminimize thepeakdistortion 9iI(c)are approximately equaltothecoefficients thatminimize theMSEperformance indexJ.Thatis,inthelimitasNo-+0,thetwocriteriayieldttlesamesolution forthetapweights. Consequently, whenNo=0,theminimization oftheMSE resultsincomplete elimination oftheintersymbol interference. Ontheother hand,thatisnotthecasewhenNo""O.Ingeneral,whenNo""0,thereisboth residual intersymbol interference andadditive noiseattheoutput of the equalizer. Ameasure oftheresidual intersymbol interference andadditive noiseis obtained byevaluating theminimum valueofJ,denoted byJm;n,whenthe transferfunctionC(z)oftheequalizer isgivenby(10-2-32). SinceJ=EIfAI'= E(fAIt)-E(fAlnandsinceE(fAltJ=0byvirtueoftheorthogonality conditions givenin(10-2-27), itfollowsthat 1m;n=E(fAm , =E11.1'-2:CjE(UA_/V j=-:x; x =1-Lc;/-J j=-'X(10·2-34) Thisparticular formforJm;nisnotveryinformative. Moreinsightonthe performance oftheequalizer asafunction ofthechannel characteristics is obtained whenthesummation in(10-2-34) istransformed intothefrequency domain. Thiscanbeaccomplished byfirstnotingthatthesummation in (10-2-34) istheconvolution of{cJwith{fi},evaluated atashiftofzero.Thus, 610 DIGITAL COMM"NICATIONS if{bddenotes theconvolution ofthesetwosequences, thesummation in (10-2-34) issimplyequaltobooSincetheztransform ofthesequence {bk}is B(z)=C(z)F(z) F(z)F*(z') thetermboisF(Z)F*(Z~l)+No X(z) X(z)+No(10-2-35) (10-2-36) (10-2-37)b__1fB(Z)d- ()-27fj Z< =_1fX(z) dz 27fjz[X(z)+No] Thecontourintegralin(10-2-36) canbetransformed intoanequivalent line integralbythechangeofvariablez=eiwTTheresultofthischangeofvariable is Tf~/TX(e.iw1)bo=- . dw 27f~~ITX(e.iw1)+No Finally,substitution oftheresultin(10~2~37) forthesummation in(10-2-34) yieldsthedesiredexpression fortheminimum MSEintheform (10-2-39)TlWIT X(eiw~J.=1-- .dw mon 27f_wITX(elw')+No Tf~/T No =2/f_~ITX(ejw~ +Nodw Tf~/T No =2- T-I~c:cH( )2dw7f-~/T "'N~_.1 lJ)+21m!T'+No IntheabsenceofintersymboJ interference, X(eiwT)=1and,hence, NoJ.=-- monl+No(10-2-38) Weobservethat0<0Jm;n0;;1.Furthermore, therelationship between theoutput (normalized bythesignalenergy)SNR"1.andJm;nmustbe 1-Jm'n"1.=Jmin(10-2-40) Moreimportantly, thisrelationbetween YxandJm'nalsoholdswhenthereis residualintersymbol interference inaddition tothenoise. CHAPTER 10:COMMUNICATION THROUGH BAND-LIMITED CHANNELS 611 finlte-LeDgth Equalizer Letusnowtumouratrention tothecasein whichthetransversal equalizer spansafinitetimeduration. Theoutputofthe equalizer inthekthsignaling intervalis "I,=2:CjV'_j j=-K(1o-Z-41) (IO-Z-42)TheMSEfortheequalizer havingZK+1taps,denoted byJ(K),is J(K)=Ell,-1,12=Ell,-j~/(CjV,-jr Minimization ofJ(K)withrespecttothetapweights {Cj}or,equivalently, forcingtheerrorEk=I,-1.tobeorthogonal tothesignalsamplesvi-I>1/1,,;;K. yieldsthefollowing setofsimultaneous equations: where and/( 2:Cl'j=~" 1=-K•...•-l.0.l,...•K j=-K f,.={XI_j+Noo'j(\1-jl,,;;L)'0 (othef1Vise ) {f!1(-L,,;;/";;0) ~/=o(otherwise)(1o-Z-43) (10-Z-44) (1o-Z-45) Itisconvenient toexpressthesetoflinearequations inmatrixform.Thus, rc=~ (lo-Z-46) whereCdenotes thecolumn vectorof2K+1tapweightcoefficients, r denotes the(ZK+1)x(2K+1)Hermitian covariance matrixwithelements f,j.and~isa(ZK+I)-dimensional column vectorwithelements ~/The solutionof(10-2-46) is c=r-I~ opt "(1o-Z-47) Thus,thesolution forCoptinvolves inverting thematrixr.Theoptimum tap weightcoefficients givenby(10-2-47) minimize theperformance indexJ(K), withtheresultthattheminimum valueofJ(K)is o Jmin(K)=1-2:cd-i j--I( (10-2-48) where ~represents thetranspose ofthecolumn vector~. Jm;n(K)maybeused 612 DIGITAL COMMUNICATIONS in(10-2-40) tocompute theoutputSNRforthelinearequalizer with2K+1 tapcoefficients. 10-2-3Performance Characteristics oftheMSEEqlLll'izer Inthissection, weconsider theperformance characteristics ofthelinear equalizer thatisoptimized byusingtheMSEcriterion. Boththeminimum MSEandtheprobability oferrorareconsidered asperformance measures for somespecificchannels. Webeginbyevaluating theminimum MSEJm'nandthe outputSNR)Ixfortwospecificchannels. Then,weconsider theevaluation of theprobability oferror. Example 10-2-1 First,weconsider anequivalent discrete-time channel modelconsisting of twocomponents 10andf"whicharenormalized to1/012+1/.12=1.Then and X(z)=10f1z+1+nf,z' Thecorresponding frequency response is X(e-'wT)=lotre1wT+1+nj;e' jwT =1+2IfolII"cos(",T+6)(10-2-49) (10-2-50) (10-2-51) (10-2-52)where8istheangleofIon.Wenotethatthischannel characteristic possesses anullatOJ=KITwhen10=j;=vT Alinearequalizer withaninfinitenumberoftaps,adjusted onthebasis oftheMSEcriterion, willhavetheminimum MSEgivenby(10-2-38). Evaluation oftheintegralin(10-2-38) fortheX(eiwT)givenin(10-2-51) yieldstheresult No Jm'n=v'N~+2No01012+IAI2)+(11012-1M)2 No v'Na+2No+(IIi-IM/ Letusconsider thespecialcaseinwhichto=j;=v1.Theminimum MSE isJm;n=No/v'N5+2Noandthecorresponding outputSNRis )Ix=11+2-1V·No (10-2-53) Thisresultshouldbecompared withtheoutputSNRof11Noobtained in CHAPTER 10:roMMllNICATION THROUGH BAND-LIMITED CHANNELS 613 thecaseofnointersymbol interference. Asignificant lossinSNRoccurs fromthischannel. Example 10-2·2 Asasecondexample, weconsider anexponentially decaying characteristic oftheform f.=v'f=.7a·,k=0,1,... wherea<1.TheFouriertransform ofthissequence is . 1-a2 X(e'wT) =--;;----­1+a2-20coswT(10-2-54) (10-2-55)whichisafunction thatcontains aminimum atw=1ftT. TheoutputSNRforthischannelis,..-----;:--- (I 1+2)-1 "l'~='J1+2No1_:2+N~-1 1-a2 = No~l(J+a2)No' Therefore, thelossinSNRduetothepresence oftheinterference is 1OIoglOG:::) Probllbility orErrorPenonDllDce orUDelli'MSEEqualizer Above,we discussed theperformance ofthelinearequalizer intermsoftheminimum achievable MSEJot;"andtheoutputSNR"l'thatisrelatedtoJminthroughthe formulain(10-2-40). Unfortunately, thereisnosimplerelationship between thesequantities andtheprobability oferror.ThereasonisthatthelinearMSE equalizer contains someresidual intersymboI interference atitsoutput.This situation isunlikethatoftheinfinitely longzero-forcing' equalizer, forwhich thereisnoresidual interference, butonlygaussian noise.Theresidual interference attheoutputoftheMSEequalizer isnotwellcharacterized asan additional gaussian noiseterm,and,hence,theoutputSNRdoesnottranslate easilyintoanequivalent errorprobability. Oneapproach tocomputing theerrorprobability isabruteforcemethod thatyieldsanexactresult.Toillustrate thismethod, letusconsider aPAM signalinwhichtheinformation symbols areselected fromthesetofvalues 2n-M-1,n=1,2,...,M,withequalprobability. Nowconsider thedecision onthesymbolIn.Theestimate ofI.is K t=qo/.+2:hq.-k+2:'/'1.-; *""n j--l((10-2-56) 614 OIGnALCOMMlJr.;ICATI()~S where{q,,}represent theconvolution oftheimpulseresponse oftheequalizer andequivalent channel, i.e., "q"=Icd", /..=K andtheinputsignaltotheequalizer is l- V,=LhI,I+1), j~0(10·2-57) (10-2-58) Thefirsttermintheright-hand sideof(10-2-56) isthedesiredsymbol,the middletermistheintersymbol interference, andthelasttermisthegaussian noise.Thevariance ofthenoiseis "2_" "2(Tn-lYO~cJ I~-I<.(10-2-59) Foranequalizer with2K+Itapsandachannel response thatspansL+1 symbols, thenumberofsymbols involved intheintersymbol interference is 2K+L. Define '!IJ=Il,q", /..#-'1(10-2-60) (10-2-61)Foraparticular sequence of2K+Linformation symbols, saythesequence IJ­ theintersymbol interference termft==DJisfixed.Theprobability oferrorfor afixedDJis (M-1) PM(DJ)=2MpeN+DJ>qo) 2(M-I) ((qO-D J)2)= Q M fT~ whereNdenotes theadditive noiseterm.Theaverage probability oferroris obtained byaveraging PM(D J)overallpossible sequences IJ•Thatis, P"=IPM(DJ)P(I J) IJ (10-2-62) Whenallthesequences areequallylikely, (10-2-63) Theconditional errorprobability termsPM(DJ)aredominated bythe sequence thatyieldsthelargestvalueofDJ•ThisoccurswhenI"=±(M-1) CHAPTt:R 10.COMMU~ICATION UIROUUH BAND·L!MnED CHA;-.lI'\F.lS 615 andthesignsoftheinformation symbolsmatchthesignsofthecorresponding {qn}'Then, DJ=(M-1)2:Iqkl k"0 and 2(M-I) (~q6( M-I )2)f,,(DJ) =Q..., 1 ----2:Iqkl M er" q()k#()(10-2-64) Thus,anupperboundontheaverageprobability oferror(orequallylikely symbolsequences is f...""PM(DJ) (10-2-65) Ifthecomputation oftheexacterrorprobability in(10-2-62) provestobe toocumbersome andtootimeconsuming becauseofthelargenumberofterms inthesumandiftheupperboundistooloose,onecanresorttooneofa numberofdifferent approximate methods thathavebeendevised, whichare knowntoyieldtightboundsonPM'Adiscussion ofthesedifferent approaches wouldtakeustoofarafield.Theinterested readerisreferred tothepapersby Saltzberg (1968),Lugannani (1%9),HoandYeh(1970),ShimboandCelebiler (1971),Glave(1972),Yao(1972),andYaoandTobin (197~). Asanillustration oftheperformance limitations ofalinearequalizer inthe presence ofsevereintersymbol interference, weshowinFig.10-2-4the probability oferrorforbinary(antipodal) signaling, asmeasured byMonte Carlosimulation, forthethreediscrete-rime channel characteristic shownin JD"- Channelof 5Fig.10.2.5k) 2 Channelof IIC' Fig.1O.2.Srb} ~31tapsilltransversal equalizn ~ 0 2 .Q222Y=fIII,110.- IICl "' No interference FIGURE 16-2·4 Errorraleperformance oflinear MSEequalizer.Ch3nnelof Fig.10.2.5(0) I0-1O~--:5~--:'1 0;:--:':15:-----:2=0----:2 75----:}('-, ----:~5 SNR10Jog'((dBl 616 DIGITAL COMMUNICA.TlONS U.815 0.72 0.407 0.407 0.04 0.2270.D7 0.05 0.21 0.46005OJ6 O.M81.1/.•Y'"I I 0.460 IIfJ!=I, 0.227I--T.1.r--+l tb)IliJY=I I FIGURE IG-2..5Threediscrete-time channelcharacrerrsrics. Fig.10-2-5.Forpurposes ofcomparison, theperformance obtained fora channelwithnointersymbol interference isalsoillustrated inFig_10-2-4.The equivalent discrete-time channel showninFig.10-2-5(a) istypicalofthe response ofagoodqualitytelephone channel. Incontrast, theequivalent discrete-time channelcharacteristics showninFig.1O-2-5(b) and(c)resultin severeintersymbol interference. Thespectral characteristics IX(eIW)1forthe threechannels. illustrated inFig.10-2-6,clearlyshowthatthechannel inFig. 1O-2-5(c) hastheworstspectralcharacteristic. Hencetheperformance ofthe linearequalizer forthischannel isthepoorestofthethreecases.Nextin performance isthechannel showninFig.1O-2-5(b). andfinally,thebest performance isobtained withthechannelshowninFig.1O-2-5(a). [nfact,the errorrateofthelatteriswithin3dBoftheerrorrateachieved withno interference. Oneconclusion reachedfromtheresultsonoutputSNRy<andthelimited probability oferrorresultsillustrated inFig.10-2-4isthatalinearequalizer yieldsgoodperformance onchannels suchastelephone lines.wherethe spectral characteristics ofthechannels arewellbehaved anddonotexhibit spectral' nulls.Ontheotherhand,alinearequalizer isinadequate asa compensator fortheintersymbol imerference onchannels withspectral nulls, whichmaybeencountered inradiotransmission. CHAPTER 10:COMMUNICATION THROUGH BAND~UMITED CHANNELS 617 0.00 -0.00., ~-12.00"".~ C.·-18.00E« ~2400 -3000 L---'---'_-'----'-_.L-~----'-:-':-~~ 0.00D.310.630.99 1.261.571.882.202.512..833.14 Frequenc:y (t) (a) 000 -0.000.00r---__ -6()() ~_-12.00 ] i.-18.00E -< -24.00 -30.()() L-~----'_-':-~_-'--L_L---'---'-'--c' 0.000.310.630.99 1.261.57 1.88 2.202.512.833.14 Frequenq 00 (b) '"':3-12.00 ~ ~-18.00E« -24.00 -JO.oo L-~----'~-':-~_-'-'--'--_~.LL--'---J 0.000.31 0.63 0.991.261.511.882.202.512.833.14 frequel1cy (t) (t') FIGURE 10·2·6 Amplitude spectraforthechannels showninFigs10-2-5(a), (b).and(c),respectively. Thebasiclimitation ofthelinearequalizer tocopewithseverelSIhas motivated aconsiderable amountofresearch intononlinear equalizers with lowcomputational complexity. Thedecision-feedback equalizer described in Section10-3isshowntobeaneffective solution tothisproblem. 10-2-4Fractionally SpacedEqualizers Inthelinearequalizer structures thatwehavedescribed intheprevious section,theequalizer tapsarespacedatthereciprocal ofthesymbolrate,i.e., atthereciprocal ofthesignaling rate1/T.Thistapspacingisoptimum ifthe equalizer ispreceded byafiltermatched tothechanneldistorted transmitted pulse.Whenthechannel characteristics areunknown, thereceiver filteris usuallymatched tothetransmitted signalpulseandthesampling timeis optimized forthissuboptimum filter.Ingeneral, thisapproach leadstoan equalizer performance thatisverysensitive tothechoiceofsampling time. Thelimitations ofthesymbolrateequalizer aremosteasilyevidentinthe (10-2-66)618 DIGITAL COMMUNICATIO\lS frequency domain. From(9-2-5),thespectrum ofthesignalattheinputtothe equalizer maybeexpressed as YT(f)=~LX(t-~)e1,""n1n,,,TnT whereY,.(f)isthefoldedoraliasedspectrum, wherethefoldingfrequency is I/ZT.Notethatthereceived signalspectrum isdependent ontilechoiceofthe sampling delay To.Thesignalspectrum at·theoutputoftheequalizer is CT(f)Y,.(f). where K CT(f)=Lc.e-j'"fkT It.-~-K(10-2-67) Itisclearfromtheserelationships thatthesymbolrateequalizer canonly compensat'e forthefrequency response characteristics ofthealiasedreceived signal.Itcannotcompensate forthechanneldistortion inilerent inX(f)el'"f'". Incontrast totilesymbolrateequalizer, afractionally spacedequalizer (FSE)isbasedonsampling theincoming signalatleastasfastastheNyquist rate.Forexample, ifthetransmitted signalconsistsofpulseshavingaraised cosinespectrum witharoll-offfactor{3,itsspectrum extends toFmax= (I+{3)/2T.Thissignalcanbesampled atthereceiveratarate 1+132Fmax=T (10-2-68) andthenpassedtilrougil anequalizer witiltapspacingofT/(1+{3).For example, if13=1,wewouldhavea~T-spaceq equalizer. If13=0.5,wewould havea~T-spaced equalizer, andsoforth.Ingeneral, then,adigitally implemented fractionally spacedequalizer hastapspacingofMT/NwhereM andNareintegersandN>MUsually,a!T-spaced equalizer isusedinmany applications. Sincethefrequency response oftheFSEis K Cdfl=Lc.e-I'K{kr It.~-·-K(10-2-69) whereT'=MT/N,itfollowsthatCdflcanequalize thereceived signal spectrum beyondtileNyquistfrequencyf=I/ZTtof=(1+(3)/T=NIMT Theequalized spectrum is Cr(f)Yr(f) =Cr(f)~X(f-;,)eJ2Ku-nITJ,,, =Cr(f)Lx(t-nN)~'KU-nN'MT)'" (10-2-70) nMT SinceX(f)=O.forIfI>N/MT,(10-2-10) maybeexpressed as 1CdflYr(f) =Cdf)X(f)eP,qr", IfI';;;'ZT' (10-2-71) (10-2-72)UIAI'Tt-:R HI.UlMMU!\:lt'AT!ON l}{KtlL:'tiH B.\~D-L1~lIIFD t·H-\~'o;EI.S 619 Thus.weobserve thattheFSEcompensates forthechannel distortion inthe received signalbeforethealiasingeffectsduetosymbolratesampling. Inother words.Cr(f)cancompensate foranyarbitrary timingphase. TheFSEoutputissampled atthesymbolrate1/Tandhasthespectrum 2:CT(f-~)X(f-~)e""U A/F)" A •T T Ineffect.theoptimum FSEisequivalent totheoptimum linearreceiver consisting 01thematched filterfollowed byasymbolrateequalizer. Letusnowconsider theadjustment ofthetapcoefficients intheFSE.The inputtotheFSEmaybeexpressed as (kMT)(kMT) (HIT).v-- ~2:I"x---liT+v-- N"N N Ineachsymbolinterval, theFSEproduces anoutputoftheform " nMT fA="f=,c".v(kT-N)(10-2-73) (10-2-74) wherethecoefficients oftheequalizer areselected tominimize theMSE.This optimization leadstoasetoflinearequations fortheequalizer coefficients that havethesolution Cop.=A'Ia (10-2-75) whereAisthecovariance matrixoftheinputdataandaisthevectorof cross-correlations. Theseequations areidentical informtothoseforthe symbolrateequalizer. buttherearesomesubtledifferences. OneisthatAis Hermitian. butnotToeplitz. Inaddition. Aexhibits periodicities thatare inherent inacyclostationary process. asshownbyQureshi (1985).Asaresult ofthefractional spacing. someoftheeigenvalues ofAarenearlyzero. Attempts havebeenmadebyLongetal.(1988a.b)toexploitthisproperty in thecoefficient adjustment. Ananalysis oftheperformance offractionally spacedequalizers, including theirconvergence properties. isgiveninapaperbyUngerboeck (1976). Simulation resultsdemonstrating theeffectiveness oftheFSEoverasymbol rateequalizer havealsobeengiveninthepapersbyQureshi andForney (1977)andGitlinandWeinstein (1981).Wecitetwoexamples fromthese papers. First.Fig.10-2-7illustrates theperformance ofthesymbol rate equalizer anda!T-FSEforachannel withhigh-end amplitude distortion. whosecharacteristics arealsoshowninthisfigure.Thesymbol-spaced equalizer waspreceded withafiltermatched tothetransmitted pulsethathad a(square-root) raisedcosinespectrum witha20%roll-off(fJ=0.2).TheFSE didnothaveanyfilterpreceding il.Thesymbolratewas2400symbolsIsand themodulation wasQAM.Thereceived SNRwas30dB.Bothequalizers had 31taps:hence.the~T·FSE spanned one-half ofthetimeinterval ofthe 620 DIGITAL COMMUNICATiONS -15 -20 0 12 10iii-25-5~ c -10 8.2 1:-30 ~iii-15 6!~ ~ ;; c >- ~-35 <3-204"~ -25 2 -40 -30 0MSE ~TNRFMSETRF lSITRF -35 -2 -45 III ' , , ,o4008001200160020002400280032003600 910 15 16 1718 Frequency (Hz) Time(symbDImterval) (01Channel withhigh-end amplirude distortion (HAl (b)Equalizer performance FIGURE 16-2-7 Tand~Tequalizer performance asafunctionuf timingphasefor2400symbolspersecond.(NRF indicates noreceiverfilter.)[FromQureshiandForney(1977).©/977lEEE.) symbolrateequalizer. Nevertheless, tileFSEoutperformed thesymbolrate equalizer whenthelatterwasoptimized atthebestsampling time. Furthermore, theFSEdidnotexhibitanysensitivity totimingphase,as illustrated inFig.10-2-7. Similarresultswereobtained byGitlinandWeinstein. Forachannelwith poorenvelope delaycharacteristics, theSNRperformance ofthesymbolrate equalizer anda~T-FSEareillustrated inFig.10-2-g.Inthiscase,both equalizers hadthesametimespan.TheT-spaced equalizer had24tapswhile theFSEhad48taps.Thesymbolratewas2400symbols/s andthedatarate was9600bits/swith16-QAM modulation. TilesignalpUlsehadaraisedcosine spectrum with{3=0.12.NoteagainthattheFSEoutperformed theT-spaced equalizer byseveraldecibels, evenwhenthelatterwasadjusted foroptimum FIGURE 16-2-8 Performance ofTand~Tequalizers asafunctionof timingphasefor2400symbols/s 16-QAM onacnannel withpoorenvelope delay.[FromGitlinandWeinstein (l98/).Reprinted withpermission fromBeUSystem Technical Journal.©/981AT&T.]30'r482 25 iii' ~ !20 '"z '"15 10-!r-!70'7!r ,l,l Timingphase CHAPTER 10:COMMUNICATION THROUGH BAND-LIMITED CHANNELS 6Z1 Inputfrom Feedforward~Symbol-by- OuIPUImatched fillertransversal symbol ti,l lvd filter detector Feedback L-.. transversal-- filter FIGURE 10-3-1 Structure ofdecision-feedback equalizer. sampling. Theresultsinthesetwopapersclearlydemonstrate thesuperior performance achieved withafractionally spacedequalizer. 10-3DECISION-FEEDBACK EQUALIZATION Thedecision-feedback equalizer (DFE),depicted inFig.10-3-1,consistsoftwo filters,afeedforward filterandafeedback filter.Asshown,bothhavetaps spacedatthesymbolintervalT.Theinputtothefeedforward sectionisthe received signalsequence {Uk}'Inthisrespect,thefeedforward filterisidentical tothelineartransversal equalizer described inSection10-2.Thefeedback filter hasasitsinputthesequence ofdecisions onpreviously detected symbols. Functionally, thefeedback filterisusedtoremovethatpartoftheintersymbol interference fromthepresentestimate causedbypreviously detected symbols. 10-3-1Coefficient Optimization Fromthedescription givenabove,itfollowsthattheequalizer outputcanbe expressed as (10-3-1) whereJkisanestimate ofthekthinformation symbol, {cJarethetap coefficients ofthefilter,and{lk-h.._,lk-K,}arepreviously detected symbols. Theequalizer isassumed tohave(K,+1)tapsinitsfeedforward sectionand Kzinitsfeedback section.Itshouldbeobserved thatthisequalizer isnonlinear becausethefeedback filtercontains previously detected symbols {lk}' Boththepeakdistortion criterior andtheMSE crit~rion resultina mathematically tractable optimization oftheequalizer coefficients, ascanbe concl~ed fromthepapersbyGeorgeelal.(1971).Price(1972),Salz(1973), andProakis(1975).SincetheMSEcriterion ismoreprevalent-in practice, we focusourattention onit.Basedontheassumption thatpreviously detected symbolsinthefeedback filterarecorrect,theminimization ofMSE (10-3-2) 622 DIGITAL COMML;!"iI<ATIUNS leadstothefollowing setoflinearequations forthecoefficients ofthe feedforward filter: "L'hcJ=/*" J~-/(1!=-K"...,-I,O (10-3-3) where-, I/I,}=L!:"!m+"j+N,,8,1' !,j=-K,.".'-1,0 m=O(10-3-4) Thecoefficients ofthefeedback filteroftheequalizer aregivenintermsofthe coefficients ofthefeedforward sectionbythefollowing expression: I) Ck= -LCJk'" k=1,2,...,K1 }=--I<.,(10-3-5) Thevaluesofthefeedback coefficients resultincomplete elimination of intersymbol interference frompreviously detected symbols, provided that previous decisions arecorrectandthatK1;;;,L(seeProblem 10-9). 10-3-2Performance Characteristics ofDFE Wenowturnourattention totheperformance achieved withdecision­ feedback equalization. Theexactevaluation oftheperformance iscomplicated tosomeextentbyoccasional incorrect decisions madebythedetector, which thenpropagate downthefeedback section.Intheabsenceofdecision errors, theminimum MSEisgivenasI "Jmin(K,) =1 -LcJ-i j-=-K,(10-3-6) Bygoingtothelimit(K,~co)ofaninfinitenumberoftapsinthefeedforward filter,weobtainthesmallest achievable MSE,denoted asJm'n'Withsome effortIm'ncanbeexpressed intermsofthespectral characteristics ofthe channelandadditivenoise,asshownbySalz(1973).Thismoredesirable form for1m,.is {Tf~jT[No] } 1m'n=exp2"-"ITInX(eiw1)+Nodw Thecorresponding outputSNRis 1-1m;. ')Ix=Jrnin(10-3-7) (10-3-8) {Tf~IT.[No+X(eiw1)]}=-1+exp-In dw 21T-~IT No Weobserve againthat,in'theabsence ofintersymbol interference, X(eiwT)=1and,hence,Jm'n=Nol(1+No).Thecorresponding outputSNRis 'Yx=11No· CHArTER 10:COMMV"')CATION TIlROlJGH BAND-LIMITED CHASSElS 623 (10-3-9) 1+No+\/(1+N,i-41fof,I' NotethatIm'nismaximized whenlfi,l=If.I=VI.ThenExample 10-3-1 Itisinteresting tocompare thevalueofJm;nforthedecision-feedback equalizer withthevalueofJm'nobtained withthelinearMSEequalizer. For example, lerusconsider thediscrete-time equivalent channelconsisting of twotapsfoandf,.Theminimum MSEforthischannelis {TfRIT[ No ] }Jm'n=exp-In dO) 2%-RIT1+No+21J..llfdcos(wT+9) =Noexp[--~f"In(1+N,+21follf,1cosw)dW] 211_" 2No 2No Jm'n=1+No+\/(1+No)'-1 =2No,No4:1 Thecorresponding outputSNRis(10-3-10) (10-3-11)1 ¥x=2No' Therefore, thereisa 3dBdegradation inoutputSNRduetothepresence of intersymbol interference. Incomparison, theperformance lossforthelinear equalizer isverysevere.ItsoutputSNRasgivenby(10-2-53) is¥x= (2/No)'12forNo<l:1. Example 10-3·2 Consider theexponentially decaying channelcharacteristic oftheform A=(1-02)'120k,k=0,1,2,... (10-3-12) where 0<1.TheoutputSNRofthedecision-feedback equalizer is 1{Ifir1[1+02+(1-a2)/No-2acosW]}¥x= -+exp-n dw2n_" 1+02-2acosw 1 =-1+-2{I-02+No(1+02)+V[1- 02+No(l+a2))'-4a2N~}No (1-02)[1+No(1+02)/(1-02)]-No No (10-3-13) &24 OIGITAL ('OMMUNH-AlJONS Thus,thelossinSNRis10log,o(I-a2)dB.Incomparison, thelinear equalizer hasalossof10log",[(1-a2)/(1+a2»)dB. Theseresultsillustrate thesuperiority ofthedecision-feedback equalizer overthelinearequalizer whentheeffectofdecision errorsonperformance is neglected. Itisapparent thataconsiderable gaininperformance canbe achieved relativetothelinearequalizer bytheinclusion ofthedecision­ feedback section, wnicheliminates theintersymbol interference frompre­ viouslydetected symbols. Onemethod'ofassessing theeffectofdecision errorsontheerrorrate performance ofthedecision-feedback equalizer isMonteCarlosimulation ona digitalcomputer. Forpurposes ofillustration, weofferthefollowing resultsfor binaryPAMsignaling through theequivalent discrete-time cnannel models showninFigs1O-2-5(b) and(c). Tneresultsofthesimulation aredisplayed inFig.10-3-2.Firstofall,a comparison oftheseresultswiththosepresented inFig.10-2-4leadsusto conclude thattiledecision-feedback equalizer yieldsa significant improvement inperformance relativetothelinearequalizer havingthesamenumberoftaps. Second, theseresultsindicate thatthereisstillasignifiCant degradation in performance ofthedecision-feedback equalizer duetotheresidualintersymbol interference, especially onchannels withseveredistortion suchastheone FIGURE 1~)'2 Performanre ofdecision-feedback equalizer withandwithouterrorpropagation. 2510-1 5 2 ~10-2 '0 .~I 10-3 5 2 10"\\\ \i\\\\ Detected \\\\\~'7'symbols redback \V(\ i\\\\\Corre<t \\\\Ysymbols fedbackKl=15\y\\K,=IS No\\\\1~#.1:lr,1'interfereleCe \ \1\ Channelof-Chan",,1of Fig10.2.5(b)Fig10.2.5(e) 5 10 IS 20 SNR.I°loslldB)2S 30 lS CHAPTER Ill:COMMUNICATiON THROlJ(iH HANI)·L1MHU) CHANNELS 62S showninFig.1O-2-5(c). Finally, theperformance lossduetoincorrect decisions beingfedbackis2dB,approximately, forth.ech.annel responses underconsideration. Additional resultsontheprobability oferrorfora decision-feedback equalizer witherrorpropagation maybefoundinthepapers byOuttweiler elal.(1974)andBeaulieu (1992). Thestructure oftheOFEthatisanalyzed aboveemploys aT-spaced filter forthefeedforward section.Theoptimality ofsuchastructure isbasedonthe assumption thattheanalogfilterpreceding theOFEismatched tothe channel-corrupted pulseresponse anditsoutputissampled attheoptimum timeinstant.Inpractice, thechannelresponse isnotknownapriori,soitisnot possible todesignanidealmatched filter.Inviewofthisdifficulty, itis customary inpractical applications touseafractionally spacedfeedforward filter.Ofcourse,thefeedback filtertapspacingremainsatT.Theuseofthe FSEforthefeedforward filtereliminates thesystemsensitivity toatiming error. Performance Comparison withMLSE Weconclude thissubsection onthe performance oftheOFEbycomparing itsperformance againstthatofMLSE. Forthetwo-path channelwithto=I.=VI.wehaveshownthatMLSEsuffers noSNRlosswhilethedecision-feedback equalizer suffersa 3dBloss.On channels withmoredistortion, theSNRadvantage ofMLSEoverdecision­ feedback equalization isevengreater.Figure10-3-3illustrates acomparison of theerrorrateperformance ofthesetwoequalization techniques,obtained via MonteCarlosimulation, forbinaryPAMandthechannel characteristics showninFigs1O-2-5(b) and(c).Theerrorratecurvesforthetwomethods havedifferent slopes;hencethedifference inSNRincreases astheerror FIGURE 111-3-3 Comparison ofperformance between MLSEanddecision-feedl>ack equalizalion forchannel characteristics shown(a)inFig.1O-2-5(b) and(b)inFig.1O-2-5(c). MLSEsimulation 25 20 10 15 SNR.lOlogYCdB) Cb)5[)ech,ion-feedback equalizer f----".f----'-A''''-7''< :corTeCtbitsfedback I1----11- MLSEsimulalionI---f---i Hr'g•'0 .~10-' :si 10--' 10"0 25 20 10 15 SNR.10logrtdB) (6)Decision-feedback f---.,)~~~*-- equalizer10'g•'0 .~10' :si 10--1 MLsE lowerand upperbounds 10-'0 j 626 DIGITAL COMMVNJCATJQNS Received .ignal Output sa.mples JFeedforwardI + IDecisionIdecis.ion IfillerI-'IdeviceI -+ Oulput 1~'prediction signal Feedback ~~:Errorsignalfiher forfilter(predictor) adjustment ~ '1 FIGURE10-3-4 Blockdiagramofpredictive DFE. probability decreases. Asabenchmark, theerrorratefortheAWGNchannel withnointersymbol interference isalsoshowninFig.10-3-3. (10-3-14) (10-3-15)16-3-3Predictive Decision-Feedback Equalizer BelfioreandPark(1979)proposed anotherDFEstructure thatisequivalent to theoneshowninFig.10-3-1underthecondition thatthefeedforward filterhas aninfinitenumberoftaps.Thisstructure consistsofaFSEasafeedforward filterandalinearpredictor asafeedback filter,asshownintheconfiguration giveninFig.10-3-4.Letusbrieflyconsider theperformance characteristics of thisequalizer. Firstofall,thenoiseattheoutputoftheinfinitelengthfeedforward filter hasthepowerspectraldensity NoX(e1wT) INn+X(e1w7)12' Theresidualintersyrnbol interference hasthepowerspectraldensity IX(e1wT) 12N~ If 1-Nr,+X(eiwT) =INo+X(eiWTW' Iwl'"T Thesumofthesetwospectrarepresents thepowerspectraldensityofthetotal noiseandintersymbol interference attheoutputofthefeedforward filter. Thus,onadding(10-3-14) and(lO-3-1S), weobtain E(w)-Nil 1C (0 36)- Nn+X(e1wT)•Iwl";;T .I--I Aswehaveobserved previously, ifX(e'wT) =I,thechannelisidealand, (lIAvn:R 10"c.;()M~II~I(:ATH)N TliROtl(i11 BA~f)"UMIIF[) ("t1A1'iNI-.l.S 627 hence.itisnotpossibletoreducetheMSEanyfurther.Ontheotherhand,if thereischanneldistortion, thepowerintheerrorsequence attheoutputofthe feedforward filtercanbereducedbymeansoflinearprediction basedonpast valuesoftheerrorsequence. If~(w)represents thefrequency response oftheinfinitelengthfeedback predictor, i.e.. ~ ~(w)=2:bnei_T n'~I thentheerrorattheoutputofthepredictor is £(w)-£(w)9II(w) =£(w)(1- 9Il(w)] Theminimization ofthemeansquarevalueofthiserror.i.e.. 1fnlTJ=- 11-~(w)fl£(w)12dw 21r-niT(l0-3-17) (10-3-18) (10-3-19) overthepredictor coefficients {bn}yieldstheoptimum predictor intheform G(w)9//(w)=1--­ go whereG(w)isthesolution tothespectralfactorization G(w)C*(-w) =IE(wW and ~ C(w)=2:g"e-i-T n=O(10-3-20) (10-3-21) (10-3-22) Theoutput of theinfinitelengthlinearpredictor isawh.itenoisesequence with powerspectraldensity1/g~andthecorresponding minimum MSEisgivenby (10-3-7). Therefore, theMSEperformance oftheinfinite-length predictive DFEisidentical totheconventional OFE. Although thesetwoDFEstructures resultinequivalent performance iftheir lengthsareinfinite.thepredictive DFEissuboptimum ifthelengthsofthetwo filtersarefinite.Thereasonfortheoptimality oftheconventional DFEis relatively simple.Theoptimization ofitstapcoefficients inthefeedforward andfeedback filtersisdonejointly.Hence.ityieldstheminimum MSE.Onthe otherhand,theoptimizations ofthefeedforward filterandthefeedback predictor inthepredictive OFEaredoneseparately. Hence,itsMSEisatleast aslargeasthatoftheconventional OFE.InspiteofAtlissuboptimality ofthe predictive DFE.itissuitableasanequalizer fortrellis-coded signals.wherethe conventional DFEisnotassuitable, asdescribed inthenextchapter. 628 DJGJTAl COMMUNJCATJONS 10-4BIBLIOGRAPHICAL NOTES ANDREFERENCES Channel equalization fordigitalcommunications wasdeveloped byLucky (1965,1966),whofocusedonlinearequalizers thatwereoptimized usingthe peakdistortion criterion. Themeansquareerrorcriterion foroptimization of theequalizer coefficients wasproposed byWidrow(1966). Decision-feedback equalization wasproposed andanalyzed byAustin (1967).Analyses oftheperformance oftheDFEcanbefoundinthepapersby Monsen(1971),Georgeet01.(1971),Price(1972),Salz(1973),DuttweiIer et01. (1974),andAltekarandBeaulieu (1993). TheuseoftheViterbialgorithm astheoptimal maximum-likelihood sequence estimator forsymbolscorrupted bylSIwasproposed andanalyzed byForney(1972)andOmura(1971).Itsuseforcarrier-modulated signalswas considered byUngerboeck (1974)andMacKenchnie (1973). PROBLEMS 10·1InabinaryPAMsystem,theinputtothedetector is wheream=±Iisthedesiredsignal,n..isazero-mean Gaussian randomvariable withvariance ,,~,and;..represents thelSIduetochanneldistortion, ThelSI termisarandomvariable thattakesthevalues -~.0,andIwithprobabilities I. I,andk.respectively. Determine Iheaverageprobability oferrorasafunction ofu~, 10·2InabinaryPAMsystem,theclockthatspecifies thesampling ofthecorrelator outputisoffsetfromtheoptimum sampling timeby10%. aIfthesignalpulseusedisrectangular, determine thelossinSNRduetothe misliming. bDetermine theamountoflSIintroduced bythemistiming anddetermine its effectonperformance. 16-3Thefrequency response characteristic ofalowpasschannelcanbeapproximated by H(f)={I+acoSz,r ft"(lal<l,lfl""W)o (otherwise) whereWisthechannel bandwidth. Aninputsignals(l)whosespectrum is bandlimited toWHzispassedthroughthechannel. aShowthat y(J)=5(f)+~a(s(f-f,,)+5(f+f,,)] Thus,thechannelproduces apairofechoes. bSuppose thatthereceived signaly(f)ispassedthroughafiltermatched toS(f). Determine theoutputofthematched filteratf=kT.k=O.±I,±2.... whereTisthesymbolduration. eWhatisthelSIpatternreSUlting fromthechannelifI"=T? 10-4Awirelinechanneloflength1000kmisusedtotransmit databymeansofhinary CHAPTER IIICOM~tlINICATION rHROl.'(iH BAND-LIMITED CHANNEl.S 629 PAM.Regenerative repeaters arespaced50kmaparlalongthesystem.Each sel!ment ofIhechannel hasanideal(conslant) frequency response overIhe fr;quency band0~f~1200Hzandanattenuation ofIdB/km.Thechannel noiseisAWGN. •Whatisthehighestbitratethatcanbetransmitted withoutlSI'! bDetermine therequired'i.IN.,10achieveabiterrorofP,=10'foreach repeater. eDetermine thetransmitted powerateachrepeater toachievethedesired 'i.IN".whereN.,=4.1x10'"W/Hz. 10-5ProveIherelationship in(10-1-13) fortheautocorrelation ofthenoiseatthe outputofthematched filter. 18-6InthecaseofPAMwithcorrelated noise,thecorrelation metricsintheViterbi algorithm maybeeKpressed ingeneralas(Ungerboeck. 1974) eM(I)=2LI"r"-2:LI.,l",x"", wherex"=x(nT)isthesampled signaloutputofthematched filter.{I,,}isthe datasequence, and{r,,}isthereceived signalsequence attheoutput of the matched filter.Determine themetricfortheduobinary signal. 10-7Consider theuseofa(square-root) raisedcosinesignalpulsewitharoll-offfactor ofunityfortransmission ofbinaryPAMoveranidealbandlimited channelthat passesthepulsewithoutdistortion. Thus,thetransmitted signalis v(t)=2:I.gT(t-kT.) k~-% wherethesignalinterval1;,=~T.Thus.thesymbolrateisdoubleofthatforno lSI. •Determine thelSIvaluesattheoutputofamatched filterdemodulator, bSketchthetrellisforthemaximum-likelihood sequence detector andlabelthe states, Ill-l1Abinaryantipodal signalistransmitted overanonideal band-limited channel, whichintroduces lSIovertwoadjacent symbols. For'anisolated transmitted si8!!alpulses(r),the(noise-free) outputofthedemodulator is~at(=T, v''l,,14att=2T,andzerofort=kT,k>2.where ~.isthesignalenergyandTis thesignaling interval. •Determine Iheaverageprobability oferror,assuming thatthetwosignalsare equallyprobable andtheadditivenoiseiswhiteandgaussian. bByplottingtheerrorprobability obtained in(a)andthatforthecaseofnolSI. determine therelativedifference inSNRoftheerrorprobability of10-'. I'"Derivetheexpression in(10-3-5)forthecoefficients inthefeedback filterofthe DFE. 10-18BinaryPAMisusedtotransmit information overanunequalized linearfilter channel. Whena=1istransmitted, thenoise-free outputofthedemodulator is {0.3(m=I) 0.9(m=0)x='"0.3(m=-1) o (otherwise) 630 DIGlTAl COMMVNlCATlONS aDesignathree-tap zero-forcing linearequalizer sothattheoutputis bDetermine qmform=±2. ±3. byconvolving theimpulse response ofthe equalizer withthechannelresponse. 16-11Thetransmission ofasignalpulsewitharaisedcosinespectrum through a channel resultsinthefollowing (noise-free) sampled outputfromthe demodulator: -0.5(k=-2) 0.1(k=-I) 1(k=0) -0.2(k=1) 0.05(k=2)o(otherwise) BDetermine thetapcoefficients ofathree-tap linearequalizer basedonthe lero-fordng criterion .. bForthecoefficients determined in(a),determine theoutputoftheequalizer forthecaseoftheisolatedpulse.Thus.determine theresiduallSIanditsspan intime. 10-12Anonidell band-limited channelintroduces lSIoverthreesuccessive symbols. The(noisc-free) response ofthematched filterdemodulator sampled atthe sampling timekTis {ir'(k=OJ [0.9i!'.(k=±I) .•s(t)s(t-kT)dt=0.1i!'.(k=±2) o(otherwise) BDetermine the tapcoetlicients ofathree-tap linearequalizer thatequalizes the channel(received signal)response toanequivalent partialresponse (duobi­ nary)signal {il'. Y.=0(k=0.1) (otherwise ) bSuppose thatthelinearequalizer in(a)isfollowed byaViterbisequence detector forthepartialsignal.Giveanestimate oftheerrorprobability if.the additivenoiseiswhiteandgaussian, withpowerspectraldensity ~NoW1Hz. '10-13Determine thetapweightcoefficients ofathree-tap zero-forcing equalizer ifthe lSIspansthreesymbols andischaraC/erized bytbevaluesx(O)=1,x(-1)=0.3, x(l)=0.2.Alsodetermine theresiduallSIattheoutputoftheequalizer forthe optimum tapcoefficients. 10-14Inline-of-sight microwave radiotransmission. thesignalarrivesatthereceiver viatwopropagation paths:thedirectpathandadelayedpaththatoccursdueto signalreflection fromsurrounding lerrain.Suppose Ihallhereceived signalhas theform r(t)=sit)+as(t-T)+nit) CHAPTER Jl):COMMUNICATION THROUGH BAND·L1MrrED CHANNELS 631 where5(1)isthetransmitted signal,aistheattenuation (0<I)ofthesecondary pathandn(l)isAWGN. aDetermine theoutputofthedemodulator atI~Tand1~2Tthatemploys a filtermatched to5(1). bDetermine theprobability oferrorforasymbol-by-symbol detector ifthe transmitted signalisbinaryantipodal andthedetector ignoresthelSI. cWhatistheerror-rate performance ofasimple(one-tap) DFEthatestimates a andremoves thelSI?Sketchthedetector structure thatemploys aDFE. 18-15RepeatProblem 1()"10usingtheMMSEasthecriterion foroptimizing thetap coefficients. Assumethatthenoisepowerspectraldensityis0.1W1Hz. 18-16Inamagnetic recording channel, wherethereadback pulseresulting froma positivetransition inthewritecurrenthastheform p(l)~[I+(:}] I alinearequalizer isusedtoequalize thepulsetoapartialresponse. The parameter T..is·definedasthewidthofthepulseatthe50%amplitude level. ThebitrateislIT,andtheratioofT~,IT. ~Aisthenormalized densityofthe recording. Suppose thepulseisequalized .tothepartial-response values {I(n~-I.I) x(nT)=2(n~0) o(otherwise) whereX(I)represents theequalized pulseshape. aDetermine thespectrumXU)oftheband-limited equalized pulse. bDetermine thepossibleoutputlevelsatthedetector, assuming thatsuccessive transitions canoccurattherate1/7;,. cDetermine theerrorrateperformance ofthesymbol-by-symbol detector for thissignal,assuming thattheadditive noiseiszero-mean gaussian with variance u2. 1ll-17SketchthetrellisfortheViterbidetector oftheequalized signalinProblem 10-16 andlabelallthestates.Also.determine theminimum euclidean distance between merging paths. 1ll-18Consider theproblem ofequalizing thediscrete-time equivalent channelshown inFig.PIO-IS.Theinformation sequence if,,}isbinary(±I)anduncorrelated. FIGURE PIo.18I10=fiII,=fi FIGURE Pl..U631 DIGITAL COMMUNICATIONS If."IfIf,=If Theadditive noise{v,liswhiteandreal-valued, withvariance No.Thereceived sequence {y,lisprocessed byalinearthree-tap equaJizer thatisoptimized onthe basisoftbeMSEcriterion. •Determine theoptimum coefficients oftheequalizer asafunction ofNo. bDetermine thethreeeigenvalues A••A2•andA,ofthecovariance matrixrand tbecorresponding (normalized tounitlength)eigenvectors "I>"2'"" cDetermine theminimum MSEforthethree-tap equalizer asafunction ofNo. dDetermine tbeoutputSNRforthethree-tap equalizer asafunction ofNo. Howdoesthiscompare withtheoutputSNRfortheinfinite-tap equalizer? For example, evaluate theoutputSNR(orthesetwoequalizers wbenN.=0.1. 19-19Usetheorthogonality principle toderivetheequations forthecoefficients ina decision-feedback equalizer basedontheMSEcriterion andgivenby(10-3-3) and(10-3-5). 19-10Suppose thatthediscrete-time modelfortheintersymbol interference is characterized bythetapcoefficients 10.f"...,kFromtheequations forthetap coefficients ofadecision-feedback equalizer (DFE),showthatonlyLtapsare neededinthefeedback filteroftheOFE.Thatis,if(c.)arethecoefficients ofthe feedback filterthenc,=0fork;0L+I, 19-UConsider thechannel modelshowninFig.PIG-ZI.{v,}isareal-valued white-noise sequence withzeromeanand'variance No.Suppose thecbannelisto beequalized byOFEhavingatwo-tapfeedforward filter(co.c••)andaone-tap feedback filter(c,).The{ci}areoptimized usingtheMSEcriterion. •Determine tbeoptimum coefficients andtheirapproximate valuesforNo""I. bDetermine theexactvalueoftbeminimum MSEandafirst-order approxima­ tionappropriate tothecaseNo-<I. cDetermine theexactvalueoftheoutputSNRforthethree-tap equalizer asa functionofNoandafirst-orderapproximation appropriate tothecaseNo-<I. dCompare theresultsin(b)and(c)withtheperformance oftheinfinite-tap DFE. eEvaluate andcompare tbeexactvaluesoftheoutputSNRforthethree-tap andinfinite-tap OFEinthespecialcaseswbereN.=0.1andO.oJ.Comment on howwellthethree-tap equalizer performs relativetotheinfinite-tap equalizer. 19-22Apulseandits(raised-cosine) spectralcharacteristic areshowninFig_PI0-22. Thispulseisusedfortransmitting digitalinformation overaband-limited channelatarateliTsymbols/s. FIGURE Pl8-U-3T-2T8(1) (a)CHAPTER'" COMMUNiCATION THROUGH BAND-LIMITED CHANNELS 633 GIn o.s------~-- T3T 0 9001200 f(Hz) 1.0(b) 0.6 0.1 3T ~.4(ci BWhatistheroll-offfactor13? bWhatisthepulserate? cThechanneldistortsthesignalpulses.Suppose thesampled valuesofthe filteredreceived pulseX(I)areasshowninFig.PID-22(c) Itisobviousthat therearefiveinterfering signalcomponents. Givethesequence of+15and-Is thatwillcausethelargest(destructive orconstructive) interference andthe corresponding valueoftheinterference (thepeakdistortion). dWhatistheprobability ofoccurrence oftheworstsequence obtained in(c). assuming thatallbinarydigitsareequallyprobable andindependent? 18-23Atime-dispersive cbannelbavinganimpulseresponse h(l)isusedtotransmit four-phase PSKatarateR=liTsymbols/s. Theequivalent discrete-time channel isshowninFig.PIO-23.Thesequence {'I.}isawhitenoisesequence havingzeromeanandvariance<i'=No. BWhatistbesampled autocorrelation functionsequence {x.}definedby x.=f.h*(I)h(1+kT)dl FIGURE Pl8-23Iy.)• FlGURE PIo-24634 DIGITAL COMMUNICATIONS forthischannel? bTheminimum MSEperformance ofa·linearequalizer andadecision-feedback equalizer havinganinfinitenumberoftapsdependsonthefoldedspectrumof thechannel 1•I(21D1)I'T.?;.Hw+r whereH(w)istheFouriertransform ofh(t).Determine thefoldedspectrum ofthechannelgivenabove. eUseyouranswerin(b)toexpresstheminimum MSEofalinearequalizer in termsofthefoldedspectrum ofthechannel. (Youmayleaveyouranswerin integralform.) dRepeat(c)foraninfinite-tap decision-feedback equalizer. ]1).24Consider afour-level PAMsystemwithpossible transmitted levels,3,1,-I,and -3.Thechannelthroughwhichthedataaretransmitted introduces intersymbol interference overtwosuccessive symbols. Theequivalent discrete-time channel modelisshowninFig.PIG-24.{'I.}isasequence ofreal-valued independent zero-mean gaussian noisevariables withvariancecr=No.Thereceived sequence is y,=0.81,+n, y,=0.81,-0.61,+n, y,=0.81,-0_61,+n, y.=0.81.-0.610_,+n• •Sketchthetreestructure, showing thepossible signalsequences forthe received signalsy"y,andy,. bSuppose theViterbialgorithm isusedtodetecttheinfonmation sequence. How manyprobabilities mustbecomputed ateachstageofthealgorithm? eHowmanysurviving sequences arethereintheViterbialgorithm forthis channel? dSuppose thatthereceived signalsare y,=0.5,y,=2.0,y,=-1.0 CHAPTER Ill:COMMlJNKATION THROVGH BAND-LIMITED CHA-:"SELS 635 Determine thesurviving sequences through stagey,andthecorrespondin[ metrics. eGiveatightupperboundfortheprobability oferrorforfour-level PAM transmitted overthischannel. 10-25Atransversal equalizer withKtapshasanimpulse response ,I e(t)=2:e,~(t-kT)," whereTisthedelaybetween adjacent taps,andatransferfunction 1\-I E(z)=2:e,z' /.:;U ThediscreteFouriertransform (DFf)oftheequalizer coefficients {c,}isdefined as K, En'"£(z)l.,-".,.,=.2:e,e "--II TheinverseDFTisdefinedas 1KIb=-2:£e',..,'-K /.:K"-II" 'n=0,I.' _..K- I k=0.1....,K-I aShowthatb,=e"bysubstituting forEnintheaboveexpression. bFromtherelations givenabove.deriveanequivalent filterstructure havingthe ztransform 1_,-KKI E E() - '" "Z-KL.1_e""'''''---v---" 11"'1) ~,_~_z____ E2(z) cIfE(z)isconsidered astwoseparate filtersE,(z)andE,(z)incascade, sketch ablockdiagram foreachofthefilters,usingz"Itodenoteaunitofdelay. dInthetransversal equalizer, theadjustable parameters aretheequalizer coefficients Ie,}.Whataretheadjustable parameters oftheequivalent equalizer in(b),andhowaretheyrelatedto{e.}? 11 ADAPTIVE EQUALIZATION InChapter 10,weintroduced bothoptimum andsuboptimum receivers that compensate forlSIinthetransmission ofdigitalinformation through band­ limited, nonideal channels. Theoptimum receiver employed maximum­ likelihood sequence estimation fordetecting .theinformation sequence from thesamples ofthedemodulation filter.Thesuboptimum receivers employed eitheralinearequalizer oradecision-feedback equalizer. Inthedevelopment ofthethreeequalization methods, weimplicitly. assumed thatthechannelcharacteristics, eithertheimpulseresponse orthe frequency response, wereknownatthereceiver. However, inmostcom­ munication systems thatemployequalizers" thechannel characteristics are unknown aprioriand,inmanycases,thechannelresponse istime-variant. In suchacase,theequalizers aredesigned tobeadjustable tothechannel' response and,fortime-variant channels, tobeadaptive tothetimevariations inthechannelresponse. Inthischapter, wepresent algorithms forautomatically adjusting the equalizer coefficients tooptimize aspecified performance indexandto adaptively compensate fortimevariations inthechannelcharacteristics. We alsoanalyzetheperformance characteristics ofthealgorithm, including their rateofconvergence andtheircomputational complexity. 11-1ADAPTIVE LINEAR EQUALIZER Inthecaseofthelinearequalizer, recallthatweconsidered twodifferent criteriafordetermining thevaluesoftheequalizer coefficients {ckl.One criterion wasbasedontheminimiza tionofthepeakdistortion attheoutputof 636 CHAPTER II:AOAmVE EQUALIZATION 637 theequalizer, whichisdefined by(10-2-4). Theothercriterion wasbasedon theminimization ofthemean-square errorattheoutputoftheequalizer, whichisdefined by(10-2-25). Below, wedescribe twoalgorithms for performing theoptimization automatically andadaptively. 11·1·1TheZero·Forcing Alxoritbm Inthepeak-distortion criterion, thepeakdistortion ~(c),givenby(10-2-22), is' minimized byselecting theequalizer coefficients {e.}.Ingeneral, thereisno simplecomputational algorithm forperforming thisoptimization, exceptinthe specialcasewherethepeakdistortion attheinputtotheequalizer, definedas ~in(10-2-23), islessthanunity.When~<1,thedistortion ~(c)atthe outputoftheequalizer isminimized byforcingtheequalizer response q.=0, for1,.;Inl'"K,andqo=1.Inthiscase,thereisasimplecomputational algorithm, calledthezero-forcing algorithm, thatachieves theseconditions. Thezero-forcing solution isachieved byforcingthecross-correlation between theerrorsequence E.=I.-1.andthedesiredinformation sequence {I.}tobezeroforshiftsintherange0,.;InI,.;K.Thedemonstration thatthis leadstothedesiredsolution isquitesimple.Wehave E(E.It_) =E(I.-1.)/t-i) =E(Mt.)-EO.It-i)'j=-K,...,K(11-1-1) Weassumethattheinformation symbols areuncorrelated, i.e.,E(I.q) =8k1• andthattheinformation sequence{hiisuncorrelated withtheadditivenoise sequence {'I.}.For1.,weusetheexpression givenin(10-2-41). Then,after takingtheexpected valuesin(11-1-1), weobtain (11-1-2) Therefore, theconditions (11-1-3) arefulfilledwhenq"=1andq.=0,1,.;Inl'"K. Whenthechannel response isunknown, thecross-correlations givenby (11-1-1)arealsounknown. Thisdifficulty canbecircumvented bytransmitting aknowntrainingsequence {I.!tothereceiver, whichcanbeusedtoestimate thecross-correlation bysubstituting timeaverages fortheensemble averages givenin(11-1-1). Aftertheinitialtraining, whichwillrequirethetransmission ofatrainingsequence ofsomepredetermined lengththatequalsorexceedsthe equalizer length,theequalizer coefficients thatsatisfy(11-1-3) canbe determined. 638 DIGITAL COMMUNICATIONS Asimplerecursive algorithm foradjusting theequalizer coefficients is C(H1l=C(·l+ae 1*./ / kk~J'j=-K•...•-I.O.1•...•K(11-1-4) where C)klisthevalueofthejthcoefficient attimet=kT.e.=I.-1.isthe errorsignalattimet=kT.andaisascalefactorthatcontrols therateof adjustment, aswillbeexplained laterinthissection.Thisisthezero-forcing algorithm. Theterme.It-jisanestimate ofthecross-correlation (ensemble average) E(e.It-JTheaveraging operation ofthecross-correlation is accomplished bymeansoftherecursive first-order difference equation algorithm in(11-1-4), whichrepresents asimplediscrete-time integrator. Following thetrainingperiod,afterwhichtheequalizer coefficients have converged totheiroptimum values,thedecisions attheoutput of thedetector aregenerally sufficiently reliablesothattheymaybeusedtocontinue the coefficient adaptation process. Thisiscalledadecision-directed modeof adaptation. Insuchacase,thecross-correlations in(11-1-4)involvetheerror signall!. =1.-1.andthedetected outputsequenceI._j,j= -K,...•K.Thus, intheadaptive mode,(11-1-4)becomes (11-1-5) Figure11·1-1illustrates thezero-forcing equalizer inthetrainingmodeandthe adaptive modeofoperation. . Thecharacteristics ofthezero-forcing algorithm aresimilartothoseofthe LMSalgorithm, whichminimizes theMSEandwhichisdescribed indetailin thefollowing section. flGURE H-I-I Anadaptive zero-forcing equalizer. Input ~Output Tninios se<juence senerator CHAPTER· /I:ADAPTIVE EQUALIZATION 639 11-1-2TheLMSAlgorithm Intheminimization oftheMSE,treatedinSection10-2·2,wefoundthatthe optimum equalizer coefficients aredetermined fromthesolutionofthesetof linearequations, expressed inmatrixformas rc=~ (11-1-6) whereristhe(2K+1)x(2K+I)covariance matrixofthesignalsamples {v.},Cisthecolumnvectorof(2K+I)equalizer coefficients, and~isa (2K+I)-dimensional columnvectorofchannelfiltercoefficients. Thesolution fortheoptimum equalizer coefficients vectorCop,canbedetermined by inverting thecovariance matrixr,whichcanbeefficiently performed byuseof theLevinson-Durbin algorithm described inAppendix A. Alternatively, aniterative procedure thatavoidsthedirectmatrixinversion maybeusedtocompute Cop,.Probably thesimplest iterative procedure isthe methodofsteepest descent, inwhichonebeginsbyarbitrarily choosing the vectorC,sayasCo.Thisinitialchoiceofcoefficients corresponds tosomepoint onthequadratic MSEsurface inthe(2K+1).dimensional spaceof coefficients. Thegradient vectorG".havingthe2K+ Igradient components I(lJlac"",k= -K,...,-1,0,I,...,K.isthencomputed atthispointonthe MSEsurface,andeachtapweightischanged inthedirection opposite toits corresponding gradient component. Thechangeinthejthtapweightis proportional tothesizeofthejthgradientcomponent. Thus,succeeding values ofthecoefficient vectorCareobtained according totherelation C..I=Ck-AGk,k=O.1,2,... wherethegradient vectorGkis(l1-1-?) (11-1-8) ThevectorC.represents thesetofcoefficients atthekthiteration, Ek=fk-l, istheerrorsignalatthekthiteration, Vkisthevectorofreceived signal samplesthatmakeuptheestimate lk'i.e.,Vk=(Vk+1(.•.Vk'"Vk'KI'.and Aisapositivenumber chosensmallenoughtoensureconvergence ofthe iterative procedure. Iftheminimum MSEisreached forsomek=kothen Gk=0,sothatnofurtherchangeoccursinthe tapweights.Ingeneral, Jm;n(K) cannotbeattainedforafinitevalueofkowiththesteepest-descent method.It can,however, beapproached ascloselyasdesiredforsomefinitevalueofko. Thebasicdifficulty withthemethodofsteepestdescentfordetermining the optimum tapweightsisthelackofknowledge ofthegradient vectorG..which depends onboththecovariance matrixrandthevector ~ofcross-correlations. Inturn,thesequantities dependonthecoefficients {J.}oftheequivalent discrete-time channel modelandonthecovariance oftheinformation sequence andtheadditive noise,allofwhichmaybeunknown atthereceiver 640 DIGITAL COMMUNICATIONS ingeneral.Toovercome thedifficulty, estimates ofthegradient vectormaybe used.Thatis,thealgorithm foradjusting thetapweightcoefficients maybe expressed intheform (IJ-I-9) whereG.denotesanestimate ofthegradient vectorG.andC.denotesthe estimate ofthevectorofcoefficients. From(11-1-8)wenotethatG,isthenegative oftheexpected valueofthe e.Vr.Consequently, anestimateofG.is (11-1-10) SinceE(G.)=G••theestimateG.isanunbiased estimate ofthetruegradient vectorG•.Incorporation of(11-1-10) into(11-1-9)yieldsthealgorithm (ll-l-lJ) ThisisthebasicLMS(Ieast-mean-square) algorithm forrecursively adjusting thetapweightcoefficients oftheequalizer firstproposed byWidrowandHoff (1960).Itisillustrated intheequalizer showninFig.11-1-2. Thebasicalgorithm givenby(11-1-11) andsomeofitspossible variations havebeenincorporated intomanycommercial adaptive equalizers thatare FIGURE 11-1-2 Linearadaptive equalizer basedonMSEcriterion. InputIud x "---{ xJ-o---{ lEd Training sequence generatorOlAput CHArTER 11:ADAPTIVE EQUALIZATION 641 usedinhigh-speed modems. Threevariations ofthebasicalgorithm are obtained byusingonlysigninformation contained intheerrorsignalE.and!or inthecomponents ofVk'Hence,thethreepossible variations are C(k+I)J=C'i+~cSgn(Edvt-i' j=-K....•-1.O.l•....K (11-1-12) c('+I)i=c'i+~E.csgn(vt-J. j=-K•...,-I,O,I,...,K (II-I-B) cl.+I)j=c./+~csgn(E.)csgn(vtJl. j=-K,...,-1,0.1,...•K(11-1-14) wherecsgn(x)isdefinedas {I+j 1-jcsgn(x)= .-1+} -1-j(Re(x)>0,1m(x)>0) (Re(x)>O.1m(x)<0) (Re(x)<0,1m(x»0) (Re(x)<0,1m(x)<0)(11-1-15) (Notethatin(11-1-15), j==v'=l,asdistinctfromtheindexjin(11-1-12)-(11­ 1-14)_)Clearly, thealgorithm in(11-1-14) isthemosteasilyimplemented, but itgivestheslowestrateofconvergence totheothers. Severalothervariations oftheLMSalgorithm areobtained byaveraging or filtering thegradient vectorsoverseveraliterations priorto making adjust­ mentsoftheequalizer coefficients. Forexample, theaverageoverNgradient vectorsis (11-1-16) andthecorresponding recursive equation forupdating theequalizer coefficients onceveryNiterations is (11-1-17) Ineffect,theaveraging operation performed in(11-1-16) reducesthenoisein theestimate ofthegradient vector,asshownbyGardner (1984). Analternative approach istofilterthenoisygradient vectorsbyalowpass filterandusetheoutputofthefilterasanestimate ofthegradient vector.For example, asimplelowpassfilterforthenoisygradients yieldsasanoutput (11-1-18) wherethechoiceof0""w<Idefermines thebandwidth ofthelowpass filter. Whenwisclosetounity,thefilterbandwidth issmallandtheeffective averaging isperformed overmanygradient vectors.Ontheotherhand,when wissmall,thelowpassfilterhasalargebandwidth and,hence,itprovides little averaging ofthegradient vectors.Withthefilteredgradient vectorsgivenby 64Z O!(,ITAL C()MMV"ICATIG"S. (1l-1-18) inplaceofG"weobtainthefilteredgradient LMSalgorithm given by (11-1-19) Intheabovediscussion, ithasbeenassumed thatthereceiver has knowledge ofthetransmitted information sequence informingtheerrorsignal between thedesiredsymbolanditsestimate. Suchknowledge canbemade a,ailable duringashorttraining periodinwhichasignalwithaknown information sequence istransmitted tothereceiver forinitiallyadjusting the tapweights.Thelengthofthissequence mustbeatleastaslongasthelength oftheequalizer sothatthespectrum ofthetransmitted signaladequately coversthebandwidth ofthechannel being equalized. Inpractice, thetraining sequence isoftenselected tobeaperiodic pseudo-random sequence. suchasamaximum lengthshift-register sequence whoseperiodNisequaltothelengthoftheequalizer (N=2K+1).Inthis case.thegradient isusuallyaveraged overthelengthofthesequence as indicated in(11-1-16) andtheequalizer isadjusted onceaperiodaccording to (H-l-l7). Apractical schemeforcontinuous adjustment ofthetapweights maybeeitheradecision-directed modeofoperation inwhichdecisions onthe information symbols areassumed tobecorrectandusedinplaceofI,in forming theerrorsignale"oroneinwhichaknownpseudo-random-probe sequence isinserted intheinformation-bearing signaleitheradditively orby interleaving intimeandthetapweightsadjusted bycomparing thereceived probesymbols withtheknowntransmitted probesymbols. Inthedecision­ directed modeofoperation. theerrorsignalbecomes i!,=7.-1k,where1,is thedecisionofthereceiverbasedontheestimate1k•Aslongasthereceiveris operating atlowerrorrates.anoccasional errorwillh;lVeanegligible effecton theconvergence ofthealgorithm. Ifthechannelresponse changes. thischangeisreflected inthecoefficients {[.}oftheequivalent discrete-time channelmodel.Itisalsoreflected inthe errorsignalEk.sinceitdepends on{[d.Hence,thetapweightswillbechanged according to(11-1-11)10reflectthechangeinthechannel. Asimilarchangein thetapweightsoccursifthestatistics ofthenoiseortheinformation sequence change.Thus.theequalizer isadaptive. 11-1-3Convergence Properties oftheLMSAlgorithm Theconvergence properties oftheLMSalgorithm givenby(11-1-11) are governed bythestep-size parametera.Weshallnowconsider thechoiceof theparameteratoensureconvergence ofthesteepest-descent algorithm in (11-1-7). whichemploys t~e.exiCl valueofthegradient. From(11-1-7)and(11-1-8), wehave Ck•I=Ck-aG, :(I-aqC.+A~ (11-1-20) CHAPTER ll:ADAPTIVE EQUALIZATION 643 Filtet 7l,i=~/--..,­,.I rc. FIGURE 11-\-3 Closed-loop controlsystemrepresentation ofrecursive equation in(J1-1-20). whereIistheidentitymatrix,ristheautocorrelation matrixofthereceived signal,C.isthe(2K+1)-dimensional vectorofequalizer tapgains,and~isthe vectorofcross-correlations givenby(10-2-45). Therecursive relation in (11-1-20) canberepresented asaclosed-loop controlsystemasshowninFig. 11-1-3.Unfortunately, thesetof2K+1first-order difference equations in (11-1-20) arecoupledthroughtheautocorrelation matrixr.Inordertosolve theseequations and,thus,establish theconvergence properties oftherecursive algorithm, itismathematically convenient todecouple theequations by performing alineartransformation. The appropriate transformation is obtained bynotingthatthematrixrisHermitian and,hence,canbe represented as r=UAU" (11-1-21) whereUisthenormalized modalmatrixofrandAisadiagonal matrixwith diagonal elements equaltotheeigenvalues ofr. When(11-1-21) issubstituted into(11-1-20) andifwedefinethetrans­ formed(orthogonalized) vectorsG=U'·C.and~"=U/·~,weobtain (11-1-22) Thissetoffirstorderdifference equations isnowdecoupled. Theirconver­ genceisdetermined fromthehomogeneous equation G+i=(1-aA)G (11-1-23) Weseethattherecursive relationwillconverge provided thatallthepoleslie insidetheunitcircle,i.e., 11-aA.1<1,k=-K,...,-1,0,1,...,K (11-1-24) where{A.}isthesetof2K+1(possibly nondistinct) eigenvalues ofr.Sincer isanautocorrelation matrix,itispositive-definite and,hence,A.>0forallk. Consequently convergence oftherecursive relationin(11-1-22) isensuredifA. satisfiestheinequality 20<.1<­Amu(11-1-25) whereAmaxisthelargesteigenvalue ofr. Sincethelargesteigenvalue ofapositive-definite matrixislessthanthesum 644 DIGITAL COMMUNICATIONS ofalltheeigenvalues ofthematrixand,furthermore. sincethesumofthe eigenvalues ofamatrixisequaltoitstrace,wehavethefollowing simpleupper boundonAm.,: K Am"<2:A.=trr=(2K+l)r.. k-,.;-K =(2K+1)(xo+No) (11-1-26) From(11-1-23) and(11-1-24) weobserve thatrapidconvergence occurs whenII-/lA.1issmall,i.e.,whenthepolepositions arefarfromtheunit circle.Butwecannotachievethisdesirable condition andstillsatisfy(11-1-25) ifthereisalargedifference between thelargestandsmallesteigenvalues ofr. Inotherwords,evenifweselect/ltobeneartheupperboundgivenin (11-1-25), theconvergenc,e rateoftherecursive MSEalgorithm isdetermined bythesmallest eigenvalue Am;n'Consequently, theratioAm../Am;nultimately determines theconvergence rate.IfAmax!Aminissmall, ~canbeselectedsoas toachieverapidconvergence. However, iftheratioAm..!Am;nislarge,asisthe casewhenthechannel frequency response hasdeepspectral nulls,the convergence rateofthealgorithm willbeslow. 11-1-4ExcessMSEDuetoNoisyGradient Estimates Therecursive algorithm in(11-1-11) foradjusting thecoefficients ofthelinear equalizer employsunbiased noisyestimates ofthegradient vector.Thenoisein theseestimates causesrandom fluctuations inthecoefficients abouttheir optimalvaluesand,thus,leadstoanincrease intheMSEattheoutputofthe equalizer. Thatis,thefinalMSEis1min+J~,wherehisthevariance ofthe measurement noise.Thetermiii.duetotheestimation noisehasbeentermed excessmeans-square e"orbyWidrow(1966). ThetotalMSEattheoutputoftheequalizer foranysetofcoefficients C canbeexpressed as (11-1-27) where CoPtrepresents theoptimum coefficients, whichsatisfy(11-1-6). This expression fortheMSEcanbesimplified byperforming thelinearorthogonal transformation usedabovetoestablish convergence. Theresultoithis transformation appliedto(11-1-27) is K 1=1min+2:A.EleI:-cfoptl2 .~-K(11-1-28) wherethe{cl:larethesetoftransformed equalizer coefficients. Theexcess MSEistheexpected valueofthesecondtermin(11-1-28), i.e., /( 11i.=2:A.EleI:-cl:opl k~-K(11-1-29) CHAPTER II:ADAPTIVE EQUALIZATION 64S IthasbeenshownbyWidrow(1970,1975)thattheexcessMSEis (11-1-30) Theexpression in(11-1-30) canbesimplified when ~isselected suchthat ~Ak«Iforallk.Then K Jt>""~~Jm;n2:Ak 1c=-K ""~~Jmintrr (11-1-31) NotethatXo+Norepresents thereceived signalplusnoisepower. Itisdesirable tohaveJt><Jmin.Thatis,~shouldbeselectedsuchthat J~= ~~(2K+I)(xo+No)<1Jmin or,equivalently, 2 ~<-----'=---­(2K+I)(xo+No) Forexample, if~isselectedas __---'0::.;:.2=--__~=(2K+I)(xo+No)(11-1-32) (11-1-33) thedegradation intheoutputSNRoftheequalizer duetotheexcessMSEis lessthan1dB. Theanalysisgivenaboveontheexcessmeansquareerrorisbasedonthe assumption thatthemeanvalueoftheequalizer coefficients hasconverged to theoptimum valueCopt.Underthiscondition, thestepsize~shouldsatisfythe boundin(11-1-32). Ontheotherhand,wehavedetermined thatconvergence ofthemeancoefficient vectorrequires that~<2/Am...Whileachoiceof~ neartheupperbound2/A",axmayleadtoinitialconvergence ofthe deterministic (known) steepest-descent gradient algorithm, suchalargevalue of~willusuallyresultininstability oftheLMSstochastic gradient algorithm. Theinitialconvergence ortransient behavior oftheLMSalgorithm has beeninvestigated byseveralresearchers. Theirresultsclearlyindicatethatthe stepsizemustbereducedindirectproportion totbelengthoftheequalizer as specified by(11-1-32). Hence,theupperboundgivenby(11-1-32) isalso necessary toensuretheinitialconvergence oftheLMSalgorithm. Thepapers byGitlinandWeinstein (1979)andUngerboeck (1972)containanalysesofthe transient behavior andthe convergence properties oftheLMSalgorithm. FIGURE 11-1-4646 DIGITAL COMMUNICATIONS Initialconvergence characteristics oftheLMS algorithm withdifferent stepsizes.[FromDigital SignalProcessing, byJ.G.Pro.kisandD.G.Mano/akis. /988,Macmillan Publishing Company. Reprinred with permission ofthepublisher.J10-)l---,'_--::':--=':-:,..-~---::'.o100200300400500 Numberofiterations Thefollowing example servestoreinforce theimportant pointsmadeabove regarding theinitialconvergence oftheLMSalgorithm. Example 11-1-1 TheLMSalgorithm wasusedtoadaptively equalize acommunication channelforwhichtheautocorrelation matrixrhasaneigenvalue spreadof Amax/llmin=11.Thenumberoftapsselectedfortheequalizer was2K+1= 11.Theinputsignalplusnoisepower Xo+Nowasnormalized tounity. Hence,theupperboundon~givenby(11-1-32) is0.18.Figure11-1-4 illustrates theinitialconvergence characteristics oftheLMSalgorithm for t.=0.045,0.09,and0.115,byaveraging the(estimated) MSEin200 simulations. Weobserve thatbyselecting ~=0.09(one-half oftheupper bound)weobtainrelatively fastinitialconvergence. Ifwedivide ~bya factorof2to~=0.045,theconvergence rateisreduced buttheexcess meansquareerrorisalsoreduced, sothattheLMSalgorithm performs betterinsteadystate(inatime-invariant signalenvironment). Finally,we notethatachoiceof~=0.115,whichisstillfarbelowtheupperbound, causeslargeundesirable fluctuations intheoutputMSEofthealgorithm. Inadigitalimplementation oftheLMSalgorithm, thechoiceofthe step-size parameter becomes even morecritical.Inanattempttoreducethe excessmeansquareerror,itispossibletoreducethestep-size parameter tothe pointwherethetotalmeansquareerroractually increases. Thiscondition occurswhentheestimated gradient components ofthevector fAvtafter multiplication bythesmallstep-size parameter ~aresmallerthanone-half of theleastsignificant bitinthefixed-point representation oftheequalizer coefficients. Insuchacase,adaptation ceases.Consequently, itisimportant for thestepsizetobelargee'lOughtobringtheequalizer coefficients inthe vicinityofCOPI'Ifitisdesiredtodecrease thestepsizesignificimtly, itis necessary toincrease theprecision intheequalizer coefficients. Typically, 16 CI-(APrER II:ADAPTIVE EQUALIZATION 647 bitsofprecision maybeusedforthecoefficients, withabout10-12ofthemost significant bitsusedforarithmetic operations intheequalization ofthedata. Theremaining leastsignificant bitsarerequired toprovidethenecessary precision fortheadaptation process. Thus,thescaled,estimated gradient components AeVrusuallyaffectonlytheleast-significant bitsinanyone iteration. Ineffect,theaddedprecision alsoallowsforthenoisetobeaveraged out,sincemanyincremental changes intheleast-significant bitsarerequired beforeanychangeoccursintheuppermoresignificant bitsusedinarithmetic operations forequalizing thedata.Forananalysis ofroundoff errorsina digitalimplementation oftheLMSalgorithm, thereaderisreferred tothe papersbyGitlinandWeinstein (1979),Gitlinetal.(1982),andCaraiscos and Liu(1984). Asafinalpoint,weshouldindicatethattheLMSalgorithm isappropriate fortrackingslowlytime-invariant signalstatistics. Insuchacase,theminimum MSEandtheoptimum coefficient vectorwillbetime-variant. Inotherwords, Jmin(n)isafunction oftimeandthe(2K+1)-dimensional errorsurface is movingwiththetimeindexn.TheLMSalgorithm attempts tofellowthe movingminimum Jmin(n)inthe(2K+I)-dimensional space,butitisalways laggingbehindduetoitsuseof(estimated) gradient vectors. Asaconse­ quence,theLMSalgorithm incursanotherformoferror,calledthelagerror, whosemeansquarevaluedecreases withanincrease inthestepsizeA.The totalMSEerrorcannowbeexpressed 'as whereJ,denotesthemeansquareerrorduetothelag. Inanygivennonstationary adaptive equalization problem, ifweplotthe errorsJ..andJ,asafunction oft1,weexpecttheseerrorstobehaveas illustrated inFig.11-1-5.WeobservethatJ~increases withanincrease inA whileJ,decreases withanincrease int1.Thetotalerrorwillexhibita minimum, whichwilldetermine theoptimum choiceofthestep-size parameter. Whenthestatistical timevariations ofthesignaloccurrapidly,thelagerror Meansquareerror .......i/errorduetoI.,g .,', ,J.3..errOl dlJ~to """" ~,//_"" no",ygrad,en" .... ~.. -"'-<~......,,'-------:-'-' --------_+0 .. <\,.E)(cessmeansquareerrorJ~andlag errorJ,asafunction ofthestepsize. [FromDigitalSignalProcessing. byJ.G. ProakisandD.G.Man%kis. /988. Macmillan Publishing Company. Reprinted withpermissiun ofrhe publisherIFIGURE 11-1-5 648 DIGITAL COMMUNICATIONS cos(I),r Received signal sin00,r FIGURE 11-1-6 QAMsignaldemodulation., Rell,1f---""'--:"':,"-l Decision 1mIIIIIdevice + Output willdominate theperformance oftheadaptive equalizer. Insuchacase, J,»Jm'n+JA•evenwhenthelargestpossible valueof~isused.Whenthis condition occurs,tileLMSalgorithm isinappropriate fortheapplication and onemustrelyonthemorecomplex recursive least-squares algorithms described inSection11-4toobtainfasterconvergence andtracking. 11-1-5Baseband andPassband LinearEqualizers Ourtreatment ofadaptive linearequalizers hasbeenintermsofequivalent lowpasssignals.However, inapractical implementation, t/lelinearadaptive equalizer showninFig.11-1-2canberealized eitheratbaseband orat bandpass. Forexample Fig.11-1-6illustrates thedemodulation ofQAM(or multiphase PSK)byfirsttranslating thesignaltobaseband andequalizing the baseband signalwithanequalizer havingcomplex·valued coefficients. Ineffect, thecomplex equalizer withcomplex-valued (in-phase andquadrature com­ ponents) inputisequivalent tofourparallelequalizers withreal-valued tap coefficients asshowninFig.11-1-7. Asanalternative, wemayequalize thesignalatpassband. Thisis In+phase signal component Quadrature sillnal component FIGURE 11·1·7 Complex-valued baseband equalizer for QAMsignals. CHAPTER II,ADAPTIVE EOUALIZATION 649 r/(A,terrorsl.nal~Passband iDecision complex xdevice Received..--equalizersignal PII...'--splittin.I-- x + BPFPa..sboJld M, (Hilbenerrorsignal Baseband transformer) nGURE 11-1-8 QAM0.PSKsignalequalizalion alpassband. accomplished asshowninFig.11-1-8foratwo-dimensional signalconstellation suchasQAMandPSK.Thereceived signalisfilteredand,inparallel,itis passedthroughaHilberttransformer, calledaphase-splitting jilter.Thus,we havetheequivalent ofin-phase andquadrature components atpassband, whicharefedtoapassband complex equalizer. Following theequalization, the signalisdown-converted toabaseband anddetected. Theerrorsignal generated forthepurposeofadjusting theequalizer coefficients isformedat baseband andfrequency-translated topassband asillustrated inFig.11-1-8. 11-2ADAPTIVE DECISION·FEEDBACK EQUALIZER Asinthecaseofthelinearadaptive equalizer, thecoefficients ofthe feedforward filterandthefeedback filterinadecision-feedback equalizer may beadjusted recursively, insteadofinverting amatrixasimpliedby(10-3-3). Basedontheminimization oftheMSEattheoutputoftheDFE,the steepest-descent algorithm takestheform (11-2-1) whereC.isthevectorofequalizer coefficients inthekthsignalinterval, E(E.Vt)isthecross-correlation oftheerrorsignalEk=I.-1.withV.and V.=[U.+K,...U.1.-1...I.-K,]'.representing thesignalvaluesinthe feedforward andfeedback filtersattimet=kT.TheMSEisminimized when thecross-correlation vectorE(E.Vt)=0ask-+00. Sincetheexactcross-correlation vectorisunknown atanytime'instant, we useasanestimate thevector E.V:andaverageoutthenoiseintheestimate throughtherecursive equation (11-2-2) ThisistheLMSalgorithm fortheDFE. 650 DI(jITAL COMMl::-.iICATI()SS Output n(;URE 11-2·\ Decision-feedback equalizer. Asinthecaseofalinearequalizer, wemayuseatrainingsequence to adjustthecoefficients oftheDFEinitially. Uponconvergence tothe(near-) optimum coefficients (minimum MSE),wemayswitchtoadecision-directed modewherethedecisions attheoutputofthedetector areusedinfonningthe errorsignale.andfedtothefeedback filter.Thisistheadaptive modeofthe DFE,whichisillustrated inFig11-2-1.Inthiscase,therecursive equation for adjusting theequalizer coefficient is (11-2-3) wheref.=I.-1,andV.=[Vk+",...v.1._,...i.-K,]'. Theperformance characteristics oftheLMSalgorithm fortheDFEare basically thesameasthedevelopment giveninSections 11-1-3and11-1-4for thelinearadaptive equalizer. 11-2-1Adaptive Equalization ofTrellis-Coded Signals Bandwidth efficienttrellis-coded modulation thatwasdescribed inSection8-3 isfrequently usedindigitalcommunications overtelephone channels toreduce therequired SNRperbitforachieving aspecified errorrate.Channel distortion ofthetrellis-coded signalforcesustouseadaptive equalization in ordertoreducetheintersymbol interference. Theoutputoftheequalizer is thenfedtotheViterbidecoder, whichperforms soft-decision decoding ofthe trellis-coded signal. CHAPTER II:ADAmvE EQUALIZATION 651 Enorsig:nal + Received signal samplesTent...tive decisions flGURE 11-2·2 Adjustment ofequalizer basedontentative decisions.Final d«isiofts Thequestion thatarisesregarding suchareceiver ishowdoweadaptthe equalizer inadatatransmission mode?Onepossibility istohavetheequalizer makeitsowndecisions atitsoutputsolelyforthepurpose ofgenerating an errorsignalforadjusting itstapcoefficients, asshownintheblockdiagram in Fig.11-2-2.Theproblem withthisapproach isthatsuchdecisions aregenerally unreliable, sincethepre-decoding codedsymbolSNRisrelatively low.Ahigh errorratewouldcauseasignificant degradation intheoperation ofthe equalizer, whichwouldultimately affectthereliability ofthedecisions atthe outputofthedecoder. Themoredesirable alternative istousethepost­ decoding decisions fromtheViterbidecoder, whicharemuchmorereliable, to continuously adapttheequalizer. Thisapproach iscertainly preferable and viablewhenalinearequalizer isusedpriortotheViterbidecoder. The decoding delayinherent intheViterbidecoder canbeovercome byintroduc­ inganidentical delayinthetapweightadjustment oftheequalizer coefficients asshowninFig.11-2-3.Themajorpricethatmustbepaidfortheaddeddelay isthatthestep-size parameter intheLMSalgorithm mustbereduced. as described byLongetal.(1987.1989),inordertoachievestability inthe algorithm. Inchannels withoneormorein-bandspectralnulls,thelinearequalizer is FIGURE 11-2·3 Adjustment ofequalizer basedondecisions fromtheViterbidecoder. Errorsisnal + Received signal s.amples Adaptiv-e linear I-~-'" equalizerVertibi decoderDecisions 652 mGITAL COMMUNICATIONS DataJH H I Tochannel 11.-._Enc_ode_r....J Inlerleaver Modulator " • (a)Transmitter Received sigrtal samples I)(lay Feedback fileer Cpredtctor) (b)Receiver FIGURE ll-Z-4 Useofpredictive DFEwithinterleaving andtrellis-coded modulation. nolongeradequate forcompensating thechannel intersymbol interference. Instead.weshouldliketouseaDFE.ButtheDFErequires reliabledecisions initsfeedback filterinordertocancelouttheintersymbol interference from previously detected symbols. Tentative decisions priortodecoding wouldbe highlyunreliable and,hence,inappropriate. Unfortunately, theconventional DFEcannotbecascaded withtheViterbialgorithm inwhichpost-decoding decisions fromthedecoder arefedbacktotheDFE. Onealternative istousethepredictive DFEdescribed inSection10-3-3.In ordertoaccommodate forthedecoding delayasitaffectsthelinearpredictor, weintroduce aperiodicinterleaver/deinterleaver pairthathasthesamedelay astheViterbidecoderand,thus,makesitpossibletogenerate theappropriate errorsignaltothepredictor asillustrated intheblockdiagram ofFig.11-2-4. Thenovelwayinwhichapredictive DFEcanbecombined withViterbi decoding toequalize trellis-coded signalsiscescribed andanalyzed by Eyuboglu (1988).Thissameideahasbeencarriedovertotheequalization of fadingmultipath channels byZhouetaI.(1988,1990),butthestructure ofthe DFEwasmodified touserecursive least-squares lattice-type filters,which providefasteradaptation tothetimevariations encountered inthechannel. 11-3ANADAPTIVE CHANNEL ESTIMATOR FORMLSEQUENCE DETECTION TheMLsequence detection criterion implemented viatheViterbialgorithm as embodied inthemetriccomputation givenby(10-1-23) andtheprobabilistic symbol-by-symbol detection algorithm described inSection 5·1-5require knowledge oftheequivalent discrete-time channelcoefficients {I.}.Toaccom­ modateachannelthatisunknown orslowlytime-varying, onemayincludea CHAI'TER" ADAPTIVE EQUALIZATION 653 -/, ~.:-......Output Channelestimatt FIGURE 11-3-1 Blockdiagramofmethodforestimating thechannel characteristics fortheViterbialgorithm. channelestimator connected inparallelwiththedetection algorithm, asshown inFig.11-3-1.Thechannelestimator, whichisshowninFig.11-3-2isidentical instructure tothelineartransversal equalizer discussed previously inSection 11-1.Infact,thechannelestimator isareplicaoftheequivalent discrete-time channel filterthatmodelstheintersymbol interference. Theestimated tap coefficients, denoted by{l.},areadjusted recursively tominimize theMSE between theactualreceived sequence andtheoutputoftheestimator. For example, thesteepest-descent algorithm inadecision-directed modeof operation is (11-3-1) whereC.isthevectoroftapgaincoefficients atthekthiteration, Aisthestep size,".=v.-1).istheerrorsignal,andi.denotes thevectorofdetected information symbolsinthechannelestimator atthekthiteration. WenowshowthatwhentheMSEbetween v.andU.isminimized, the resulting valuesofthetapgaincoefficients ofthechannelestimator arethe valuesofthediscrete-time channelmodel.Formathematical tractability, we assumethatthedetected information sequence{I.}iscorrect, i.e.,{ldis nGUKE 11·3-2 Adaptive transversal filterforestimating thechanneldispen;ion. 654 DIGITAL COMMl''SI('.'\TI()~S identical tothetransmitted sequence {I.}.Thisisareasonable assumption whenthesystemisoperating atalowprobability oferror.Thus,theMSE between thereceived signalv.andtheestimateii.is (11-3-2) Thetapcoefficients i?.}thatminiroize 1(0in(11-3-2) satisfythesetofNlinear equations whereN-·1 2:/;"'.,=db jI)k=0.1.....N-1 (11-3-3) (11-3-4) From(11·3-3) and(11-3-4). weconclude that,aslongastheinformation seque~ce {J.}isullcorrelated, theoptim\lm coefficiellts areexactlyequaltothe respective valuesoftheequivalent discrete-time channel. 11isalsoapparent thatwhellthenumber oftapsNinthechannel estimator isgreaterthanor equaltoL+I,theoptimum tapgaincoefficientsif.}areequaltothe respective valuesofthe{!.}.evenwhentheinformation sequence iscorrelated. -Subject totheaboveconditions, theminimum MSEissimplyequaltothe noisevariance No. Intheabovediscussion, theestimated information sequence attheoutputof theViterbialgorithm ortheprobabilistic symbol-by-symbol algorithm was usedinmaking adjust'ments ofthechannel estimator. Forstartupoperation, onemaysendashorttrainingsequence toperform theinitialadjustment ofthe tapcoefficients, asisusually doneinthecaseofthelineartransversal equalizer. Inanadaptive modeofoperation, thereceiver simplyusesitsown decisions toformanerrorsignal. 11-4RECURSIVE LEAST-SQUARES ALGORITHMS FORADAPTIVE EQUALIZATION TheLMSalgorithm thatwedescribed inSections 11-1and11-2foradaptively adjusting thetapcoefficients ofalinearequalizer oraDFEisbasically a (stochastic) steepest-descent algorithm inwhichthetruegradient vectoris approximated byanestimate obtained directlyfromthedata. Themajoradvantage ofthesteepest-descent algorithm liesinitscomputa­ tionalsimplicity. However, thepricepaidforthesimplicity isslowconver­ gence,especially whenthechannel characteristics resultinanautocorrelation matrixrwhoseeigenvalues havealargespread,i.e.,Am,,/Am;,,»I.Viewed in another way,thegradient algorithm hasonlyasingleadjustable parameter for •• I rCHAPTER ",ADAPTIVE EQUALIZUION 655 controlling theconvergence rate,namely,theparameter ~.Consequently the slowconvergence isduetothisfundamental limitation. Inordertoobtainfasterconvergence, itisnecessary todevisemorecomplex algorithms involving additional parameters. Inparticular, ifthematrixris NxNandhaseigenvalues A"'\2,...•AN'wemayuseanalgorithm that contains Nparameters--one foreachoftheeigenvalues. Theoptimum selection oftheseparameters toachieverapidconvergence isatopicofthis section. Inderiving fasterconverging algorithms, w.eshalladoptaleast-squares approach. Thus,weshalldealdirectlywiththereceived datainminimizing the quadratic performance index,whereas previously weminimized theexpected valueofthesquarederror.Putsimply,thismeansthattheperformance index isexpressed intermsofatimeaverageinsteadofastatistical average. Itisconvenient toexpresstherecursive least-squares algorithms inmatrix form.Hence,weshalldefineanumberofvectorsandmatrices thatareneeded inthisdevelopment. Insodoing,weshallchangethenotation slightly. Specifically, theestimate oftheinformation symbolattimeI,where Iisan integer,fromalinearequalizer isnowexpressed as K l(t)=LCj{1-l)v'-J ;=-K Bychanging theindexjonCJ{I-1)torunfromj=0toj=N-1and .simultaneously defining theestimate I(t)becomes N-, 1(1)=LCj{1-1)Y{1-j) j~O (11-4-1) whereCN{I-1)andYN{I)are,respectively, thecolumn vectorsofthe equalizer coefficients Cj(1-1),j=0,1,...,N-1,andthe input signalsy(1­ j),j=0,1,2,...,N-1. Similarly, inthedecision-feedback equalizer, wehavetapcoefficients Cj(I), j=0,1,...,IV-1,wherethefirstK,+1arethecoefficients ofthefeedfor­ wardfilterandtheremaining K2=N-K,-1arethecoefficients ofthe feedba<:k filter.Thedataintheestimate1(1)isV'+K"...,V,+I,1,-1>'..,l'-K" whereI'_j,1.,;;j.,;;K2,denotethedecisions onpreviously detected symbols. In thisdevelopment, weneglecttheeffectofdecision errorsinthealgorithms. Hence,weassumethat1'_J=1,_j,I.,;;j.,;;K,.Fornotational convenience, we alsodefine (11-4-2) 656 DIGITAL COMMUNIC AllONS Thus, Y,,(t)=[y(l)y(t-1)."y(l-N+1)]' (11-4-3) 11-4-1Recursive Least-Squares (Kalman) Algorithm Therecursive least-squares (RLS)estimation of1(I)mayheformulated as follows.Suppose wehaveobserved thevectorsYN(n),n=0,I,...:I,andwe wishtodetermine thecoefficient vectorCN(I)oftheequalizer (linearor decision-feedback) thatminimizes thetime-average weighted squared error I g~S=2:w'-nleN(n,t)12 ,,=0 wheretheerrorisdefinedas(11-4-4) (11-4-5) andwrepresents aweighting factor0<w<1.Thusweintroduce exponential weighting intopastdata,whichisappropriate whenthechannelcharacteristics aretime-variant. Minimization of'C,;/withrespecttothecoefficient vector CN(I)yieldsthesetoflinearequations whereRN(I)isthesignalcorrelation matrixdefinedas I RN(I)=2:w'-nY~(n )Y:V(n) n=O andD.v(t)isthecross-correlation vector I D.v(t)=2:w,-nl(n)Y~(n) n=O Thesolutionof(11-4-6)is(11-4--6) (11-4-7) (l1-4-B) (11-4-9) ThematrixRN(t)isakintothestatistical autocorrelation matrixrN,while thevectorDN(t)isakintothecross-correlation vector ~N'definedpreviously. Weemphasize, however, thatRN(t)isnotaToeplitz matrix.Wealsoshould mention that,forsmallvaluesofI,RN(r)maybeillconditioned; hence,itis customary toinitiallyaddthematrixliINtoRN(I),where {)isasmallpositive CHAP'TER II:ADAPT1VE EQUAliZATION 657 constant andINistheidentitymatrix.Withexponential weighting intothe past,theeffectofaddingSINdissipates withtime. Nowsuppose wehavethesolution (11-4-9)fortimet-1.i.e.,CN(t-1), andwewishtocomputeCN(t).Itisinefficient and,hence,impractical tosolve thesetofNlinearequations foreach.newsignalcomponent thatisreceived. Toavoidthis,weproceed asfollows.First,RN(t)maybecomputed recursively as (11-4-10) Wecall(11-4-10) thetime-update equalion forRN(t). SincetheinverseofRN(I)isneededin(11-4-9), weusethematrix-inverse identity R-1(1)=~[RI(t_1)R;;'(t- 1)Y~(I)Y:' ..{t)RN'(r -1)] NWNW +y:.v(I)RN'(t _I)Y~(I) (11-4-11) ThusRNI(I)maybecomputed recursively according to(11-4·11). Forconvenience, wedefinePN(I)=RNI(I).Itisalsoconvenient todefinean N-dimensional vector,calledtheKalmangainveclor,as 1(11-4-12) #LN(I)=Y:.v(t)PN(t -I)Y~I) Withthesedefinitions, (11-4-11) becomes 1PN(t)=-[PN(t-1)-KN(t)y:.v.cI)PN(t -1)] W(11-4-13) (11-4-14) Suppose wepostmultiply bothsidesof(11-4-14) byY~(t).Then 1PN(l)n(l)=-[PN(I-1)Y~I)- KN(I)Y:.v(I)PN(I-I)Y~I)lw 1 =-([w+#LN(I)]KN(t) -K",(t)/L",(t)}w =KN(t) (11-4-15) Therefore, theKalmangainvectormayalsobedefinedasPN(I)YN(I). Nowweusethematrixinversion identity toderiveanequation for obtaining CN(I)fromCN(I-1).Since CN(t)=PN(I)DN(t) and (11-4-16) 658 PiGIrAicO:W,",U:\'!CAT1QNS C/I/(I)~~[P/I/(I-I)-K/I/(I)Y~(r)P~(r -1)111\'0,,(1 -I)+1(i)YW)] w I =P/I/(I-1)0,(1-I)+-/(I)P,(t -1)Yt(r) Ii' -K~(I)Y\(I)P/I/(I -I)Odl-I) 1-~l(r)IC(I)V'v(I)P,(1 -IlYt(l),,- =C,(I-1)+K,(i)[/(I) -Y'y-(l)C,(r -I)] NotethatY',(i)C,(I- I)istheoutputoftheequalizer attimer,i.e.. and e,(I.I-l)=/(t)-I(l)=e,(I)(11-4-17) (11-4-18) istheerrorbetween thedesired symbolandtheestimate. Hence.C",(i)is upd3ted recursively according totherelation C,(I)=C,,(r-I)+K,(Ik,,(r) Theresidual MSEresulting fromthisoptimization is , t:~~"n=2:1\""I/(n)/'-C'v(t)Ol(l)"-~()(I1-4-20) (11-4-21) Tosummarize. suppose wehaveC",(I-I) andP",(I-I).Whenanew signalcomponent isreceived. wehave Y~(I).Thentherecursive computation forthetimeupdateofC",(I)andP,,(I)proceeds asfollows: •compute output: 1(I)=Y'",(I)C,,(t -I) •compute error: e,(I)=1(1)-l(t) •compute Kalman gainvector: KN(I)=P",(I-I)Y',(I) Ii'+Y",(I)P,(I- I)n(i) •updateinverseofthecorrelation matrix: 1PN(I)=-[P",(t -I)-K/I/(t)Y",(t)P",(t-I)]w •updatecoefficients: C",(t)=C,,(t-I)+K,,(I)e,,(r) =C/I/(I-I)+p,,(nY*(t)e,,(t) (11-4-22) CHAPTER II:ADAPTIVE EQI.;ALJZATION 659 111' JilGURE 11-4-t Comparison ofconvergence rateforthe Kalmanandgradientalgorithms.10~2L~;;::'=-~.....,,..,..=;:=;;:::=~=-~00100200300400500600 700 Numt:erofiaerations Thealgorithm described by(11-4-22) iscalledtheRLSdirectformorKolman algorithm. Itisappropriate whentheequalizerhasatransversal (direct-form) structure. Notethattheequalizer coefficients changewithtimebyanamountequalto theerroreN(t)multiplied bytheKalmangainvectorK",(t).SinceKN(t)is N-dimensional, eachtapcoefficient ineffectiscontrolled hyoneofthe elements ofKN(t).Consequently rapidconvergence isobtained. Incontrast, thesteepest-descent algorithm, expressed inourpresentnotation, is (11-4-23) andtheonlyvariableparameter isthestepsize!:J.. Figure11-4-1illustrates theinitialconvergence rateofthesetwoalgorithms forachannelwithfixedparameters to=0.26,t.=0.93,fi=0.26,andalinear equalizer with11taps.Theeigenvalue ratioforthischannelisAm../Ami.=1l. Alltheequalizer coefficients wereinitialized tozero.Thesteepest-descent algorithm wasimplemented with!:J.=0.020.Thesuperiority oftheKalman algorithm isclearly evident. Thisisespecially important intracking a time-variant channel. Forexample, thetimevariations inthecharacteristics of an(ionospheric) high-frequency (HF)radiochannel aretoorapidtobe equalized bythegradient algorithm, buttheKalman algorithm adapts sufficiently rapidlytotracksuchvariations. Inspiteofitssuperior tracking performance, theKalman algorithm described abovehavetwodisadvantages. Oneisitscomplexity. Thesecondis itssensitivity toroundoff noisethataccumulates duetotherecursive computations. Thelattermaycauseinstabilities inthealgorithm. Thenumberofcomputations oroperations (multiplications, divisions, and subtractions) incomputing thevariables in(11-4-22) isproportional toN2. Mostoftheseoperations areinvolved intheupdating ofPN(t).Thispartofthe computation isalsosusceptible toroundoff noise.Toremedythatproblem, algorithms havebeendeveloped thatavoidthecomputation ofPN(t)according to(11-4-14). Thebasisofthesealgorithms liesinthedecomposition ofP..,(t)in theform (11-4-24) 660 DIGITAL COMMUNICATIONS whereSN(t)isalower-triangular matrixwhosediagonal elements areunity, andAN(t)isadiagonal matrix.Suchadecomposition iscalledasqlUlre-rool factorization (seeBierman, 1977).Thisfactorization isdescribed inAppendix D.Inasquare-root algorithm, PN(t)isnotupdatedasin(11-4-14) norisit computed. Instead,thetimeupdating isperformed onSN(t)andAN(t). Square-root algorithms arefrequently usedincontrolsystemsapplications inwhichKalmanfilteringisinvolved. Indigitalcommunications, thesquare­ rootKalmanalgorithm hasbeenimplemented inadecision-feedback-equalized PSKmodemdesigned totransmitathighspeedoverHFradiochannels witha nominal3kHzbandwidth. Thisalgorithm isdescribed inthepaperbyHsu (1982).Ithasacomputational complexity of1.5N2+6.5N(complex-valued multiplications anddivisions peroutputsymbol). Itisalsonumerically stable andexhibitsgoodnumerical properties. Foradetaileddiscussion ofsquare­ rootalgorithms insequential estimation, thereaderisreferredtothebookby Bierman (1977). ItisalsopossibletoderiveRLSalgorithms withcomputational complexities thatgrowlinearlywiththenumberNofequalizer coefficients. Suchalgorithms aregenerally calledfastRLSalgorithms andhavebeendescribed inthepapers byCarayannis etal.(1983),CioffiandKailath(1984),andSlockandKailath (1988). 11-4-2LinearPrediction andtheLatticeFilter InChapter3,weconsidered thelinearprediction ofasignal,inthecontextof speechencoding. Inthissection,weshallestablish theconnection between linearprediction andalatticefilter. Thelinearprediction problem maybestatedasfollows:givenasetofdata y(t-I),y(t-2),...,y(t-p),predictthevalueofthenextdatapointy(t). Thepredictor oforderpis Minimization oftheMSE,definedas ~p=E[y(t)-p(t)]2 =E[y(t)-iap/<y(t_k)]2 k-1(11-4-25) (11-4-26) withrespecttothepredictor coefficients {apt}yieldsthesetoflinearequations iapk<p(k-I)=<P(I),I=I,2,...,p k-I where <P(l)=E[y(t)y(t+I») ThesearecalledthenormaleqUlltions ortheYule-Walker equations.(11-4-27) CHAPTER ILADArnVE EQUALIZATION 661 Thematrixellwithelements </>(k-I)isaToeplitz matrix,and,hence,the Levinson- Durbinalgorithm described inAppendix Aprovides anefficient meansforsolvingthelinearequations recursively, startingwithafirst-order predictor andproceeding recursively tothesolutionofthecoefficients forthe predictor oforderp.Therecursive relations fortheLevinson-Durbin algorithm are </>(1) a"=</>(0)' ~=</>(0) </>(m)-A:"~;;'_,amm ~m-l ~m=~m-,(I- Q~m) form=I,2,...,p,wherethevectors Am-Iand~~_,aredefinedas Am-I=(am-lIam-12,..am-Im-d ' ~~_,={</>(m-I) ¢(m-2) ...¢(1)1'(11-4-28) Thelinearprediction filterofordermmayberealizedasatransversal filter withtransferfunction m Am(z)=1-Lamz-' .\=1(11-4-29) Itsinputisthedata{yet)}anditsoutputistheerrore(t)=yet)-J(t).The prediction filtercanalsoberealized intheformofalattice,aswenow demonstrate. OurstartingpointistheuseoftheLevinson-Durbin algorithm forthe predictor coefficients am'in(11-4-29). Thissubstitution yields m-' Am(z)=1-L(am-I.-ammQm-lm-.)Z-· -ammz-m '-1 (11-4-30) Thuswehavethetransferfunctionofthemth-order predictor intermsofthe transferfunction ofthe(m-l)th-order predictor. Nowsuppose wedefineafilterwithtransferfunction Gm(z)as Gm(Z)=Z-mAm(z·') Then(11-4-30) maybeexpressed as Am(z)=Am-I(z) -ammz-'Gm-,(z)'(11-4-31) (11-4-32) 662 DIGITAL COMMUNICATIONS NotethatG",-I(Z) represents atransversal filterwith.tapcoefficients (-O"'-lm-" -Om-lm-2> ...'-am-II,1),whilethecoefficients ofAm_,(z) areexactlythesameexceptthattheyaregiveninreverseorder. Moreinsightintotherelationship between Am(z)andGm(z)canbe obtained bycomputing theoutputofthesetwofilterstoaninputsequence y(t).Usingz-transform relations, wehave Wedefinetheoutputsofthefiltersas .F",(z)=Am(z)Y(z) B..(z)=Gm(z)Y(z) Then(11·4-33) becomes Fm(z)=Fm_,(z)-ommz-'Bm-I(z) Inthetimedomain,therelation in(11-4-35) becomes where m-'f..(t)=y(t)-Lomky(t-k) 1c:=1 m-I bm(t)=y(t-m)- Lomky(t-m+k) k=1(11-4-33) (11-4-34) (11-4-35) (11-4-36) (11-4-37) (11-4-38) Toelaborate, fm(t)in(11-4-37) represents theerrorofanmth-order forward predictor, whileb..(t)represents theerrorofanmth-order backward predictor. Therelationin(11-4-36) isoneoftwothatspecifies alattic~filter.The secondrelationisobtained fromGm(z)asfollows: Gm(z)=Z-mAm(z-') =Z-m[Am_l(z -')-ommzmAm-,(z») =z-IGm_,(z) -ommAm- dz) (11-4-39) Now,ifwemultiply bothsidesof(11-4-39) byY(z)andexpresstheresultin termsofFm(z)andBm(z)usingthedefinitions in(11-4-34), weobtain (11-4-40) Bytransforming (11-4-40) intothetimedomain,weobtainthesecondrelation thatcorresponds tothelatticefilter,namely, (11-4-41) CHAPTER 11:ADAPTIVE EQUALIZATION 663 {a) FIGURE 11-4-2 Alatticefilter.y(t) (b) Theinitialcondition is fo(t)=bo(t)=y(t) (11-4-42) Thelatticefilterdescribed bytherecursive relations in(11-4-36) and(11-4-41) isillustrate!! inFig.11-4-2.Eachstageischaracterized byitsownmultiplication factor{aii},i=I,2,...,m.whichisdefinedintheLevinson-Durbin algorithm. Theforward andbackward errorsIm(t)andbm(t)areusuallycalledthe residuals. Themeansquarevalueoftheseresiduals is ~m=E[/~(t)]=E[b~(t)] (11-4-43) ~misgivenrecursively, asindicated intheLevinson-Durbin algorithm, by ~m=~m-,(I- a~m) m =~n(1-a;;) ;=1(11-4-44) where ~=<1>(0). Theresiduals {/m(t)}and{bm(t)}satisfyanumberofinteresting properties, asdescribed byMakhoul (1978).Mostimportant ofthesearetheorthogonality properties m,n;a.0E[bm(t)bn(t)] =~m8mn E[fm(t+m)t.(t+n)]=~m8mn Furthermore, thecross-correlation between Im(t)andb.(t)is {a..~m(m<on) E[/m(t)b.(t)] =0(m<n)(11-4-45) (11-4-46) Asaconsequence oftheorthogonality properties oftheresiduals, the differentsectionsofthelatticeexhibitaformofindependence thatallowsusto addordeleteoneormoreofthelaststageswithoutaffecting theparameters of theremaining stages.Sincetheresidual meansquareerror ~mdecreases monotonically with'thenumberofsections, ~mcanbeusedasaperformance indexindetermining wherethelatticeshouldbeterminated. Fromtheabovediscussion, weobservethatalinearprediction filtercanbe implemented eitherasalineartransversal filterorasalatticefilter.Thelattice filterisorder-recursive, and,asaconsequence, thenumberofsections it containscanbeeasilyincreased ordecreased withoutaffecting theparameters 664 DIGITAL COMMUNICATIONS oftheremaining sections. Incontrast, thecoefficients ofatransversal filter obtained onthebasisoftheRLScriterion areinterdependent. Thismeansthat anincrease oradecrease inthesizeofthefilterresultsinachangeinall coefficients. Consequently, theKalmanalgorithm described inSection11-4-1is recursive intimebutnotinorder. Basedonleast-squares optimization, RLSlatticealgorithms havebeen developed whosecomputational complexity growlinearlywiththenumberN offiltercoefficients (latticestages).Hence,thelatticeequalizer structure is computationally competitive withthedirect-form fastRLSequalizer algo­ rithms.RLSlatticealgorithms aredescribed inthepapersbyMorleral. (1973),Satorius andAlexander (1979),Satorius andPack(1981).Lingand Proakis(1984),andLingetal.(1986). RLSlatticealgorithms havethedistinctfeatureofbeingnumerically robust toround-off errorinherent indigitalimplementations ofthealgorithm. A treatment oftheirnumerical properties maybefoundinthepapersbyLinget al.(1984,1986). 11-5SELF-RECOVERING (BLIND) EQUALIZATION Intheconventional zero-forcing orminimum MSEequalizers, weassumed that aknowntrainingsequence istransmitted tothereceiverforthepurposeof initiallyadjusting theequalizer coefficients. However, therearesomeapplica­ tions,suchasmultipoint communication networks, whereitisdesirable forthe receiver tosynchronize tothereceived signalandtoadjusttheequalizer withouthavingaknowntrainingsequence available. Equalization techniques basedoninitialadjustment ofthecoefficients withoutthebenefitofatraining sequence aresaidtobeself-recovering orblind. Beginning withthepaperbySato(1975),threedifferent classesofadaptive blindequalization algorithms havebeendeveloped overthepasttwodecades. Oneclassofalgorithms isbasedonsteepest descentforadaptation ofthe equalizer. Asecondclassofalgorithms isbasedontheuseofsecond-and higher-order (generally, fourth-order) statistics ofthereceived signalto estimatethechannelcharacteristics andtodesigntheequalizer. Morerecently, athirdclassofblindefUlIlization algorithms basedonthemaximum·likelihood criterion havebeeninvestigated. Inthissection, webrieflydescribe these approaches andgiveseveralrelevantreferences totheliterature. 11-5-1BlindEqualization BasedonMaximum-Likelihood Criterion Itisconvenient tousetheequivalent, discrete-time channelmodeldescribed in Section10-1-2.RecallthattheoutputofthischannelmodelwithlSIis L Vn'"LA/.-.+11. k-O(11-5-1) (11-5-2)CHAPTER II:ADAPTIVE EQUALIZATION 665 whereU.}aretheequivalent discrete-time channelcoefficients, {In}represents theinformation sequence, and{TIn}isawhitegaussian noisesequence. ForablockofNreceived datapoints,the(joint)probability density function ofthereceived datavectorv=[v,V2•.•VN)'conditioned on knowing theimpulse response vectorI=[faj;...ILl'andthedatavector 1=(1,/ 2",IN)'is 1 ( . 1NIL 12 )p(v1',1)=(22)NexP -2~LVn-LMn-. TeO' IT'1=1 k=O Thejointmaximum-likelihood estimates ofIandIarethevaluesofthese vectorsthatmaximize thejointprobability densityfunction p(vIr,I)or, equivalently, thevaluesofrandIthatminimize thetermintheexponent. Hence.theMLsolutionissimplytheminimum overfandIofthemetric NIL 12 DM(I,I)=~, Vn-.'5;/.l n-•. =lIv-AI1I2 wherethematrixAiscalledthedatamatrixandisdefinedas(11-5-3) I,0 0 0 12I,0 0 A=1312I, 0 (11-5-4) ININ-,IN-2IN-L Wemakeseveralobservations. Firstofall,wenotethatwhenthedata vectorI(orthedatamatrixA)isknown,asisthecasewhenatraining sequence isavailable atthereceiver, theMLchannel impulse response estimate obtained byminimizing (11-5-3)overfis 'ML(I)=(A/A)-'Arv (11-5-5) Ontheotherhand,whenthechannel impulse response Iisknown, the optimumMLdetector forthedatasequence Iperforms atrellissearch(ortree search)byutilizingtheViterbialgorithm forthelSIchannel. WhenneitherInorfareknown,theminimization oftheperformance index DM(I,I) maybeperformed jointlyoverIandI.Alternatively, fmaybe estimated fromtheprobability densityfunctionp(vII),whichmaybeobtained byaveraging p(v,III)overallpossibledatasequences. Thatis, p(vII)=Lp(v,I(m)II) m =Lp(vII(m),f)p(rm» '"(11-5-6) 666 DIGITAL C:()}.1M~:NKATI0NS wherep(I,m»istheprobability ofthesequence 1=I(m),form=1,2,...,MN andMisthesizeofthesignalconstellation. Channel Estimation BasedonAverage overDataSequences Asindi­ catedintheabovediscussion, whenbothIandfareunknown, oneapproach is toestimate theimpulse response fafteraveraging theprobability density p(v,IIf)over:,11possibledatasequences. Thus,wehave (11-5-7)m =",[ 1 (_llv-A(m)fll')]p(I(m»)::(2IrU2)Nexp 2IT2 Then,thees'imate offthatmaximizes p(vIf)isthesolutionoftheequation iJp(vIC)=2:'p(I(m» ar m(11-5-8) Hence,theLstimateoffmaybeexpressed as f=[~P(I""')A,m"A(mlg(V, A(m),C)r' X2:p(I(m»g(v, Aim),f)A(m),v(11-5-9) m wherethefunctiong(v,A(ml,f)isdefinedas (11-5-10) Theresulting solutionfortheoptimum fisdenotedbyfML. Equ:lllOn (11-5-9)isanonlinear equation fortheestimate oftoechannel impulseresponse, giventhereceived signalvectorv,Itisgenerally difficultto obtaintheoptimum solution bysolving(11-5-9)directly. Ontheotherhand,it isrelatively simpletodeviseanumerical method thatsolvesforfML recursively. Specifically, wemaywrite (0+1)=[~p(l(m»A(m)'A(mlg(V,Alm), rk»]-) X2:p(I(ml)g(v, A(m),f(k»A1m),v (11-5-11) m Once f,WLisobtained fromthesolution of(11-5-9)or(11-5-11), wemay CHAPTER II'ADAPTIVE EOlJA!1/.\TlO'\ 667 simplyusetheestimate intheminimization ofthemetricDM(I.C"tI),givenb\ (ll-5-}), overallthepossibledatasequences. Thus,IMi.isthesequence Ithai mimimizes DM(I,'ML),i.e., minDM(I,fML)=minIIv-AIMLII' I I(11-5-12) WeknowthattheViterbialgorithm isthecomputationally efficientalgorithm forperforming theminimization ofDM(I,fMdover1- Thisalgorithm hastwomajordrawbacks, First,therecursion for'LMgiven by(11-5-11) iscomputationally intensive_ Second,and,perhaps, moreimpor­ tantly,theestimateIMLisnotasgoodasthemaximum-likelihood estimate 'ML(I)thatisobtained whenthesequence 1isknown.Consequently, theerror rateperformance oftheblindequalizer (theViterbialgorithm) basedonthe estimatef"'Lispoorerthanthatbasedon''''L(I).Next,weconsider joint channelanddataestimation. JointChannel andDataEstimation Here,weconsider thejointoptimiza­ tionoftheperformance indexDM(I,f)givenby(11-5-3). Sincetheelements oftheimpulseresponse vector,1arecontinuous andtheelements ofthedata vector1arediscrete, oneapproach istodetermine themaximum-likelihood estimate ofIforeachpossible datasequence and,then,toselectthedata sequence thatminimizes DM(I,l)foreachcorresponding channel estimate, Thus,thechannelestimate corresponding toth&rnthdatasequence I(m)is (ll.5-B) Forthemthdatasequence, themetricDM(I,f)becomes (ll.5-14) Then,fromthesetofMNpossiblesequences, weselectthedatasequence that minimizes thecostfunction in(11-5-14), i.e.,wedetermine (11.5-15) Theapproach described aboveisanexhaustive computational search method withacomputational complexity thatgrowsexponentially withthe lengthofthedatablock.WemayselectN=L,and,thus,weshallhaveone channelestimate foreachoftheMLsurviving sequences. Thereafter, wemay continue tomaintain aseparate channelestimate foreachsurviving pathofthe Viterbialgorithm searchthroughthetrellis. Asimilarapproach hasbeenproposed bySeshadri (1991).Inessence. Seshadri's algorithm isatypeofgeneralized Viterbialgorithm (GVA)that retainsK;;oIbestestimates ofthetransmitted datasequence intoeachstate 668 DIGITAL COMMUNICATiONS ofthetrellisandthecorresponding channelestimates. InSeshadri's GVA,the searchisidentical totheconventional VAfromthebeginning uptotheLstage ofthetrellis,i.e.,uptothepointwherethereceived sequence (v),V2,...,vd hasbeenprocessed. Hence,uptotheLstage,anexhaustive searchis performed. Associated witheachdatasequence I(m,.thereisacorresponding channelestimate (ML(I(m,). Fromthisstageon,thesearchismodified, toretain K;;;.1surviving sequences andassociated channelestimates perstateinsteadof onlyonesequence perstate.Thus,theGVAisusedforprocessing the received signalsequence {vn.n;;;.L+I}.Thechannel estimate isupdated recursively ateachstageusingtheLMSalgorithm tofurtherreducethe computational complexity. Simulation resultsgiveninthepaperbySeshadri (1991)indicate thatthisGVAblindequalization algorithm performs rather wellatmoderate signal-to-noise ratioswithK=4.Hence,thereisamodest increaseinthecomputational complexity oftheGVAcompared withthatfor theconventional VA.However, thereareadditional computations involved withtheestimation andupdating ofthechannelestimates (I(mI)assoch.ted witheachofthesurviving dataestimates. Analternative jointestimation algorithm thatavoidstheleast-squares computation forchannelestimation hasbeendevisedbyZervasetal.(1991). Inthisalgorithm, theorderforperforming thejoint minimization ofthe performance index;DM(I,f)isreversed. Thatis,achannelimpulseresponse, sayr=t(l)isselected andthentheconventional VAisusedtofindthe optimum sequence forthischannelimpulseresponse. Then,wemaymodifyf") insomemannertofi2)=f')l+At')andrepeattheoptimization overthedata sequences ·{I(m)}. Basedonthisgeneral approach, Zervasdeveloped anewMLblind equalization algorithm, whichiscalledaquantized-channel algorithm. The algorithm operates overagridinthechannelspace.whichbecomes finerand finerbyusingtheMLcriterion toconfinetheestimated channel inthe neighborhood oftheoriginalunknown channel. Thisalgorithm leadstoan efficientparallelimplementation, anditsstoragerequirements areonlythose oftheVA. 11-5-2Stochastic Gradient Algorithm Another classofblindequalization algorithms arestochastic-gradient iterative equalization schemes thatapplyamemoryless nonlinearity intheoutputofa linearFIRequalizatiol1 filterinordertogenerate the"desired response" in eachiteration. Letusbeginwithaninitialguessofthecoefficients oftheoptimum equalizer, whichwedenoteby{c,,}.Then,theconvolution ofthechannel response withtheequalizer response maybeexpressed as {en}*it,}={Il,,}+{en} (11-5-16) where{Il,,}istheunitsamplesequence and {e,,}denotestheerrorsequence CHAPTER 1);ADAPTIVE EQUAL,,,'.ATION 669 thatresultsfromourinitialguessoftheequalizer coefficients. Ifweconvolvt; theequalizer impulseresponse withthereceived sequence {vnl.weobtain {In}={vn}*{en} ={In}*{tn}*{en}+{1)n}*{en} ={In}*({5n}+{en})+{1)n}*{en} ={In}+{In}*{en}+{1)n}*{cn} (11-5-17) Theterm{I..}in(11-5-17) represents thedesireddatasequence, theterm {In}*{en}represents theresiduallSI,andtheterm{7)n}*{cn}represents the additivenoise.Ourproblem istoutilizethedeconvolved sequence {I.}tofind the"best"estimate ofadesiredresponse, denotedingeneralby{dn}.Inthe caseofadaptive equalization usingatrainingsequence, {dn}={In}.Inablind equalization mode,weshallgenerate adesiredresponse from{l.}. Themeansquareerror(MSE)criterion maybeemployed todetermine the "best"estimate of{In}fromtheobserved equalizer output<I.}.Sincethe transmitted sequence {In}hasanongaussian pdf,theMSEestimate isa nonlinear transformation of{In}.Ingeneral,the"best"estimate{dn}isgiven by (memoryless) (mth-order memory)(U-S-I8) whereg()isanonlinear function. Thesequence {dn}isthenusedtogenerate anerrorsignal,whichisfedbackintotheadaptive equalization tilter,asshown inFig.11-5-1. Awell-known classicalestimation problem isthefollowing.Iftheequalizer output1.isexpressed as In=in+fin (11-5-19) wherefinisassumed tobezero-mean gaussian (thecentrallimittheorem may FIGURE 11·5·1 Adaptive blindequalization withstochastic gradientalgorithms.Input Adaptive ,Output=FDeci~ion equalizerI• -v.I. I. -Nonlinear Etrot(unction signal"+ g(/.)-.Ud. 670 DJ(";ITAL COM~lJNICATIONS TABLE 11.5-1 STOCHASTIC GRADIENT ALGORITHMS FORBLIND EQUALIZATION EquaJi2er tapcoefficients Received signalsequence Equalizer outputsequence Equalizer errorsequence Tapcoefficient updateequation/en.O<;n<;N-I} Iv..} lin}~lv,,1*{e,,} Ie..!=g(i,,)-i" c"+1=C"+6.v:,e" Algorithm Godard Saro Benvenisre-Goursat Slop-and-GoNonfuJe8rity: ,(1.) l..-", _E{II,l}iD(1/,,1+R,1/,,1-11,,/),R,-E{II"I'} •E{[Re(In)]'} (csgn(I,,),(=EIiRe(In)11 i"+k,(in-I,,)+k,li"-1,,/[(csgD(i.)-i.!. k,and k2arepositiveconstants i"+\A(in-I,,)+\B(in-I")'(A,B)=(2.0),(I,I), I],-]),or(0.0),depending onrhesignsofdecision­ directed errort-7"andtheerror'csgn(i,.)-7" beinvokedherefortheresiduallSIandIheadditivenoise),{I.}and{ii.}are statistically independent, and{In}arestatistically independent andidentically distributed randomvariables, thentheMSEestimate of{I.}is d"=E(l",j.) (11-5-20) whichisanonlinear functionoftheequalizer outputwhen{I.}isnongaussian. Table11-5-1illustrates thegeneralformofexisting blindequalization algorithms thatarebasedonLMSadaptalion. Weobserve thatthebasic difference amongthesealgorithms liesinthechoiceofthememoryless nonlinearity. Themostwidelyusedalgorithm inpractice istheGodard algorithm, sometimes alsocalledtheconstant-modulus algorithm (CMA). Itisapparent fromTable11-5-1thattheoutputsequence {dn}obtained by takinganonlinear functionoftheequalizer outputplaystheroleofthedesired response oratrainingsequence. Itisalsoapparent thatthesealgorithms are simpletoimplement, sincetheyarebasically LMS-type algorithms. Assuch, weexpectthattheconvergence characteristics ofthesealgorithms willdepend ontheautocorrelation matrixofthereceiveddata{v,,}. Withregardtoconvergence, theadaptive LMS-type algorithms converge in themeanwhen (11-5-'21) and,inthemeansquaresense,when(superscript Hdenotes theconjugate transpose) E[~1J"g*(I.)1 =E[c~1Jnl:) E[l.g*(I,,)) =E[ll.12](11-5-22) CHAPTER'"ADAPTIVE EOUALIZATION 671 Therefore, itisrequired thattileequalizer output{in}satisfy(11-5-22). Note that(11-5-22) statesthattheautocorrelation of{in}(theright-hand side)equals thecross-correlation between1.andanonlinear transformation of1n(left-hand side).Processes thatsatisfythisproperty arecalledBussgang (1952),asnamed byBellini(1986).Insummary, thealgorithms giveninTable11-5-1converge whentheequalizer outputsequence1.satisfiestheBussgang property. Thebasiclimitation ofstochastic gradient algorithms istheirrelatively slow convergence. Someimprovement intheconvergence ratecanbeachieved by modifying theadaptive algorithms fromLMS-type torecursive-least-square (RLS)type. Godard Algorithm Asindicated above,theGodard blindequalization algorithm isasteepest-descent algorithm thatiswidelyusedinpracticewhena trainingsequence isnotavailable. Letusdescribe thisalgorithm inmoredetail. Godardconsidered theproblemofcombined equalization andcarrierphase recovery andtracking. Thecarrierphasetracking isperformed atbaseband, following theequalizer asshowninFig.11-5-2.Basedonthisstructure, we mayexpresstheequalizer outputas /( 1.=2:C.V._n n=-K(11-5-23) (11-5-25)andtheinputtothedecision deviceas1.exp(-jib.).whereib.isthecarrier phaseestimate inthekthsymbolinterval. Ifthedesiredsymbolwereknown,wecouldformtheerrorsignal e.=h-1.e-j4>4 (11-5-24) andminimize theMSEwithrespecttoib.and{c.},i.e.. I!1inE(lh-1.e-j4>412) .k'C FIGURE 11-5-2 Godardschemeforcombined adaptive (blind)equalization andcarnerphasetracking. {I.1QAM modulatorChan...,PIlose splitterAdaptive equalizer sin00,.1 I..DecisionI----/".-:\ device x e-lit Carrier lnICIcing1-----' 672 DlGIlAI. COMMUNKAIION) Thiscritenon leadsustousetheLMSalgorithm forrecursively estimating C and<b•.TheLMSalgorithm basedonknowledge ofthetransmitted sequence is (11-5-26) (11-5-27) wheretl."and!:>.'"arethestep-size parameters forthetworecursive equations. Notethattheserecursive equations arecoupledtogether. Unfortunately, these equations willnotconverge. ingeneral, whenthedesiredsymbolsequence {f.} isunknown. Theapproach proposed byGodardistouseacriterion thatdepends onthe amountofintersymbol interference attheoutputoftheequalizer butonethat isindependent oftheQAMsignalconstellation andthecarrierphase.For example. acostfunction thatisindependent ofcarrierphaseandhasthe property thatitsminimum leadstoasmallMSEis (11-5-28) wherepisapositiveandrealinteger.Minimization ofGIP'withrespecttothe equalizer coefficients resultsintheequalization ofthesignalamplitude only. Basedonthisobservation, Godard selected amoregeneralcostfunction, calledthedispersi<>n oforderp.definedas (11-5-29) (11-5-30)whereRpisapositiverealconstant. AsinthecaseofCIP',weobservethat DIeIisindependent ofthecarrierphase. Minimization ofD(P)withrespecttotheequalizer coefficients canbe performed recursively according tothesteepest-descent algorithm dDIP) C,"=C.-l1pdC, where !:>.pisthestep-size parameter. Bydifferentiating D(p)anddropping the expectation operation, weobtainthefollowing LMS-type algorithm for adjusting theequalizer coefficients: (11-5-31) (11-5-32)where f),.pisthestep-size parameter andtheoptimum choiceofR"is R=£(11.12 ") "£(Il.l") Asexpected, therecursion in(11-5-31) forC.doesnotrequireknowledge ofthecarrierphase.Carrier phasetracking maybecarriedoutina decision-directed modeaccording to(11-5-27). CHAPTER 1~:ADAPTIVE EQUALIZATION 673 Ofparticular importance isthecasep=2,whichleadstotherelatively simplealgorithm c•.1=C.+I1pVtl.(R 2_11.12) 4>'+1=4>.+11..1m(1.1te1:i>,) where1.istheoutputdecisionbasedon1.,and(11-5-33) (11-5-34) (11-5-35)Convergence ofthealgorithm givenin(11-5-33) wasdemonstrated inthe paperbyGodard(1980).Initially, theequalizer coefficients weresettozero exceptforthecenter(reference) tap,whichwassetaccording tothecondition 2£1/.1' ICul>21xul2[£(11.12)]2 whichissufficient, butnotnecessary, forconvergence ofthealgorithm. Simulation resultsperformed byGodardonsimulated telephone channels with typicalfrequency response characteristics andtransmission ratesof7200­ 12000bits/s indicatethatthealgorithm in(11-5-31) performs wellandleadsto convergence in5000-20000 iterations, depending onthesignalconstellation. Initially, theeyepatternwasclosedpriortoequalization. Thenumber of iterations required forconvergence isaboutanorderofmagnitude greater thanthenumber required toequalize thechannels withaknowntraining sequence. Noapparent difficulties wereencountered inusingthedecision­ directed phaseestimation algorithm in(11-5-33) fromthebeginning ofthe equalizer adjustment process. 11-S-3BlindEqualization Algorithms BasedonSecond- and Higher-Order SignalStatistics Itiswellknownthatsecond-order statistics (autocorrelation) ofthereceived signalsequence provide information onthemagnitude ofthechannel characteristics, butnotonthephase.However, thisstatement isnotcorrectif theautocorrelation function ofthereceived signalisperiodic, asis·thecase foradigitally modulated signal.Insuchacase,itispossible toobtaina measurement oftheamplitude andthephaseofthechannelfromthereceived signal.Thiscyclostationarity property ofthereceived signalformsthebasisfor achannelestimation algorithm devisedbyTongetal.(1993). Itisalsopossibletoestimate thechannelresponse fromthereceived signal byusinghigher-order statistical methods. Inparticular, theimpulseresponse of alinear,discrete-time-invariant systemcanbeobtained explicitly from cumulants ofthereceived signal,provided thatthechannelinputisnongaus­ sian.Wedescribe thefollowing simplemethodforestimation ofthechannel 674 DI(jITAI. ("O\I\U'SWAnO\S impulseresponse fromfourth-order cumulants ofthereceived signalsequence. Thefourth-order cumulant isdefinedas c(VI..' v~+/11'VJ.''I't',t,:.,)=Ct,(nl,n,l} =E(v.lVk>-/IIV,I,;-fnvJ.'/) -E(Vkvk+",)E(Vk e"Vb/) -E(Vkv,.,,)E(v, e",Vk,I) ~-E(v,v,.,)E(v"",v,.,,) (11-5-36) (Thefourth-order cumulant ofagaussian signalprocessiszero.)Consequently, itfollowsthat "'Am.n,I)=c(!"h,"'.f,,,"f,,,):Lld..",fu,,f,,, (11-5-37) k---0 Forastatistically independent andidentically distributed inputsequence {I,,} tothechannel, c(l"1,.",.h""I,+/)=k,aconstant, calledthekurtosis. Then, ifthelengthofthechannelresponse isL+I,wemayletm=n=(=-Lso that LA-L,-L,-L)=kfJi; Similarly. ifweletm=0,n=Land1=p,weobtain c,(O,L,p)=kfJ~fp(11-5-38) (11-5-39) Ifwecombine (11-5~3H) and(11-5-39). weobtaintheimpulseresponse withina scalefactoras p=I,2....,L (11-5-40) Thecumulants c,(m,n,I)areestimated fromsampleaverages ofthereceived signalsequence {v,J Another approach basedonhigher-order statistics isduetoHatzinakos and Nikias(1991).Theyhaveintroduced thefirstpolyspectra-based adaptive blind equalization methodnamedthetricepstrum equalization algorithm (TEA).This methodestimates thechannelresponse characteristics byusingthecomplex cepstrum ofthefourth-order cumulants (tricepstrum) ofthereceived signal sequence {vn}.TEAdepends onlyonfourth-order cumulants of{vn}andis capableofseparately reconstructing theminimum-phase andmaximum-phase characteristics ofthechannel. Thechannel equalizer coefficients arethen computed fromthemeasured channelcharacteristics. Thebasicapproach used inTEAistocompute thetricepstrum ofthereceived sequence {vn},whichis theinverse(three-dimensional) Fouriertransform ofthelogarithm ofthe trispectrum of{vn}.(Thetrispectrum isthethree-dimensional discreteFourier transform ofthefourth-order cumulant sequence c,(m,II,!).Theequalizer coefficients arethencomputed fromthecepstralcoefficients. CHAPTtR II.ADAPTIVE FOl'AUZAflOJ\ 675 Byseparating thechannelestimation fromthechannel equalization. itis possible .touseanytypeofequalizer forthelSI,i.e.,eitherlinear.or decision-feedback, ormaximum-likelihood sequence detection. Themajor disadvantage withthisclassofalgorithms isthelargeamountofdataandthe inherent computational complexity involved intheestimation ofthehigher. ordermoments (cumulants) ofthereceived signal. Inconclusion, wehaveprovided anoverview ofthreeclassesofblind equalization algorithms thatfindapplications indigitalcommunications. Ofthe threefamiliesofalgorithms described. thosebasedonthemaximum-likelihood criterion forjointlyestimating thechannel impulse response andthedata sequence areoptimalandrequirerelatively fewreceived signalsamples for performing channelestimation. However, thecomputational complexity ofthe algorithms islargewhenthelSIspansmanysymbols. Onsomechannels. such asthemobileradiochannel, wherethespanofthelSIisrelatively short.these algorithms aresimpletoimplement. However, ontelephone channels. where thelSIspansmanysymbols butisusuallynottoosevere.theLMS-type (stochastic gradient) algorithms aregenerally employed. 11-6BIBLIOGRAPHICAL NOTES ANDREFERENCES Adaptive equalization fordigitalcommunications wasdeveloped byLucky (1965,1966).Hisalgorithm wasbasedonthepeakdistortion criterion andled tothezero-forcing algorithm. Lucky'sworkwasamajorbreakthrough, which ledtotherapiddevelopment ofhigh-speed modems withinfiveyearsof publication ofhiswork.Concurrently. theLMSalgorithm wasdevised by Widrow(1966),anditsuseforadaptive equalization forcomplex-valued (in-phase andquadrature components) signalswasdescribed andanalyzed ina tutorialpaperbyProakisandMiller(1969). Atutorialtreatment ofadaptive equalization algorithms thatwerede· velopedduringtheperiod1965-1975 isgivenbyProakis(1975).Amorerecent tutorialtreatment ofadaptive equalization isgiveninthepaperbyQureshi (1985).Themajorbreakthrough inadaptive equalization techniques, beginning withtheworkofLuckyin1965coupledwiththedevelopment oftrellis-eoded modulation, whichwasproposed byUngerboeck andCsajka(1976),hasledto thedevelopment ofcommercially available highspeedmodems witha capability ofspeedsof9600-28800 bits/sontelephone channels. Theuseofamorerapidlyconverging algorithm foradaptive equalization wasproposed byGodard (1974).Ourderivation oftheRLS(Kalman) algorithm, described inSection· 11-4-1,followstheapproach outlined by Picinbono (1978).RLSlatticealgorithms forgeneralsignalestimation applica· tionsweredeveloped byMorfetal.(1977,1979).Theapplications ofthese algorithms havebeeninvestigated byseveralresearchers, including Makhoul (1978),SatoriusandPack(1981),SatoriusandAlexander (1979),andLingand Proakis(1982.1984a-c, 1985).ThefastRLSKalman algorithm foradaptive equalization wasfirstdescribed byFalconer andLiung(1978).Theabove PROBLEMS nGURE PlI·)676 DIGITAL COMMUNICATIONS references arejustafewoftheimportant papersthathaveheenpublished on RLSalgorithms foradaptive equalization andotherapplications. Sato's(1975)original workonblindequalization wasfocused onPAM (one-dimensinal) signalconstellations. Subsequently itwasgeneralized to two-dimensional andmultidimensional signalconstellations inthealgorithms devised byGodard (1980),Benveniste andGoursat (1984),Sato(1986), Foschini (1985),PicchiandPrati(1987),andShalviandWeinstein (1990). Blindequalization methods basedontheuseofsecond·andhigher-order moments ofthereceived signalwereproposed byHatzinakos andNikias (1991)andTongetal.(1994).Theuseofthemaximum-likelihood criterion for jointchannelestimation anddatadetection hasbeeninvestigated andtreated inpapersbySeshadri (1991),GhoshandWeber(1991),Zervasetal.(1991) andRahelietat.(1995).Finally.theconvergence characteristics ofstochastic gradient blindequalization algorithms havebeeninvestigated byDing(1990), Dingetal.(1989).andJohnson (1991). 11-1Anequivalent discrete-time channel withwhitegaussian noiseisshowninFig. PII-!. aSuppose weusealinearequalizer toequalize thechannel. Determine thelap coefficients c_"c".c,ofathree-tap equalizer. Tosimplifythecomputation, let theAWONbezero. bThelapcoefficients ofthelinearequalizer in(alaredetermined recursively via thealgorithm CH,=C,-Ilg" C,=[e" Co.c,,]' whereg,=rc,-bisthegradient vectorandIlisthestepsize.Determine the rangeofvaluesofIltoensureconvergence oftherecursive algorithm. To simplify thecomputation, lettheAWGI\bezero. c:Determine thetapweightsofaDFEwithtwofeedforward lapsandone feedback gap.Tosimplify thecomputation, lettheAWGNbezero. 11·2RefertoProblem 10-18andanswerthefollowing questions. k=O.I.....N-lCHAPTER IIADAPTIVE EQUALIZATION (,77 aDetermine themaximum valueof~thaIcanbeusedtoensurethatthe equalizer coefficients converge duringoperation intheadaptive mode. bWhatisthevariance oftheself-noise generated bythethree-tap equalizer when operating inanadaptive mode.asafunctionofd?Suppose itisdesiredtolimit thevariance oftheself-noise to10%oftheminimum MSEforthethree-tap equaliLCr whenN"=0.1.WhatvalueofAwouldyouselect? cIftheoptimum coefficients attheequalizer arecompuled recursively b)'the methodofsteepestdescent. therecursive equation canbeexpressed intheform C,,,,,,=(I-Af)C,,,,+A~ whereI,istheidentity matrix.Theaboverepresents aselofthreecoupled first-order difference equations. Theycanbedecoup'led byalineartransforma­ tionthatdiagonalizes thematrixr.Thatis,r=UAU'whereAisthediagonal matrixhavingtheeigenvalues ofrasitsdiagonal elcments andVisthe (normalized) modalmatrixthatcanbeobtained fromyouranswerto1O-18(b). LetC'=U'Canddetermine thesteady-state solution forC'.Fromthis.evaluate C= (V')-'C'=ve'and.thus,showlhatyouransweragreeswilhlheresult obtained in10-18(a). 11·3Whenaperiodic pseudo-random sequence oflengthNisusedtoadjustthe coefficients ofanN-taplinearequalizer, thecomputations cal}beperformed efficiently inthefrequency clomain byuseofthediscrete fourier transform (OFT).Suppose that{y,,}isasequence ofNreceixed samples(tak'enatthesymbol rate)attheequalizer input.Thenthecomputation oftheequalizer coefficients is performed asfollows. aCompute theDFTofoneperiodoftheequalizer inputsequence {.v"}.i.e., N-l Y4=Ly"e-j21Ifll4IA "=U ...Compute thedesiredequalizer spectrum C_X.l1 , -IY,I'' where{Xi}istheprecomputed DFTofthetrainingsequence. CCompute theinverseDfTof{C,}toobtaintheequalizer coefficients Ie,,}.Show thatthisprocedure intheabsenceofnoiseyieldsanequalizer whosefrequency response isequaltothefrequency response oftheinversefoldedchannel spectrum attheNuniformly spacedfrequenCies!. =kINT,k=0,1.....II'-1. 11-4Showthatthegradient vectorintheminimization oftheMSEmaybeexpressed as G,=-E(e,Vn wheretheerror£,=I,-I,.andtheestimateofG"i.e.. G,l,;==-£.V: satisfiesthecondition thatE(G,)=G,. U-SThetap-leakage LMSalgorithm proposed inthepaperbyGillin 1'1al.(1982)may beexpressed as 678 DIGITAL COMMUNIC ATIO~S where0<w<I,.iisthestepsize,andVN(n)isthedatavectorattimen. Determine thecondition fortheconvergence ofthemeanvalueofCN(n). 11-6Consider therandomprocess x(n)=gv(n)+w(n). n=O.I....,M-I wherev(n)isaknownsequence, gisarandom variable with£(g)=O.and £(g')=G.Theprocessw(n)isawhitenoisesequence with Determine thecoefficients ofthelinearestimator forg.thatis, M--I g=Lh(n)x(n) ..=0 thatminimize themeansquareerror 11·7Adigitaltransversal filtercanberealized inthefrequency-sampling formwith systemfunction (seeProblem 10-25) =H,(z)H,(z) whereH,(z)isthecombfilter,H,(z)istheparallelbankofresonators, and{H.} arethevaluesofthediscreteFouriertransform (OFf). aSuppose thatthisstructure isimplemented asanadaptive filterusingtheLMS algorithm toadjustthefilter(OFf)parameters {H.}.Givethetime-update equation fortheseparameters. Sketchtheadaptive filterstructure. bSuppose thatthisstructure isusedasanadaptive channelequalizer inwhichthe desiredsignalis \1..] d(n)=2:A,cos"',n, .11:=0 Withthisformforthedesiredsignal,whatadvantages arethereintheLMS adaptive algorithm fortheOFfcoefficients {H,}overthedirect-form structure withcoefficients {h(n)}?(seeProakis. 1970). U-8Consider theperformance index J=h'+40h+28 Suppose ihatwesearchfortheminimum ofJbyusingthesteepest-de,cent algorithm h(n+I)=h(n)-11g(n) whereg(n)isthegradient. aDetermine therangeofvaluesof.ithatprovides anoverdamped systemforthe adjustment process. bPlottheexpression forJasafunction ofnforavalueof.iinthisrange. CHAPTER 11:ADAPTIVE EQl;ALlZATION 679 + FIGURE PII·9x(1l)-~-----------.{+ e(IIj 11·9Determine thecoefficients a,and0,forthelinearf'redictor showninFig.PI1-9, giventhattheautocorrelation 1',,(m)oftheinf'utsignalis 1',,(m)=b''''',O<b<1 11·10Determine thelatticefilteranditsoptimum reflection coefficients corresponding to thelinearpredictor inProblem 11-9. 11·11Consider theadaptive FIRfiltershowninFig.PII-II.ThesystemC(z)is characterized bythesystemfunction C() _ 1z-I-0.9z· I Determine theoptimum coefficients oftheadaf'tive transversal (FIR)filter B(z)=b,,+b,z"thatminimize themeansquareerror.Theadditive noiseis whitewithvariancea:=0.1. 11·12AnNx!IIcorrelation matrixfhaseigenvalues AI>A,>...>AN>0and associated eigenvectors VI'V".•••VN'Suchamatrixcanberepresented as N f='"Av,·'LJ III ;"'\ aIff=f"'f'",wheref'"isthesquarerootoff,showthatf'"canbe represented as N rl12=""A~i2vv~( LJ I I I ;....1 bUsingthisrepresentation, determine aprocedure forcomputingr'·'. w(n) FIGURE PII·1Ix(n)-~-<ol Ct:)Adaplive FIR filter+~+-_t'(n) + 12 MULTICHANNEL AND MULTICARRIER SYSTEMS Insomeapplications, itisdesirable totransmit thesameinformation-bearing signaloverseveralchannels. T~ismodeoftransmission isusedprimarily in situations wherethereisahighprobability thatoneormoreofthechannels willbeunreliable fromtimetotime.Forexample, radiochannels suchas ionospheric scatterandtropospheric scattersufferfromsignalfadingdueto multipath, whichrendersthechannels unreliable forshortperiodsoftime.As another example. multichannel signaling issometimes employed inmilitary communication systemsasameansofovercoming theeffectsofjamming ofthe transmitted signal.Bytransmitting thesameinformation overmultiple channels, weareproviding signaldiversity, whichthereceiver canexploitto recovertheinformation. Another formofmultichannel communications ismultiple carriertransmis­ sion,wherethefrequency bandofthechannel issubdivided intoanumberof subchannels andinformation istransmitted oneachofthesubchannels. A rationale forsubdividing thefrequency bandofachannelintoanumberof narrowband channels isgivenbelow. Inthischapler, weconsider bothmultichannel signaltransmission and multicarrier transmission. Webeginwithatrealment ofmultichannel transmission. 12-1MULTICHANNEL DIGITAL COMMUNICATION INAWGNCHANNELS Inthissection,weconfineourattention tomultichannel signaling overfixed channels thatdifferonlyinattenuation andphaseshift.Thespecificmodelfor 680 CHAPTER I:!:MUlTKHANNEL ANDMLJLTICARRIER SYSTEMS 681 themultichannel digitalsignaling systemmaybedescribed asfollows.The signalwaveforms. ingeneralareexpressed as s);;'(I)=Re(s);'(tk'~"ldl. 0,,;;I";;T "=1,2,...,L,m=I,2,....M(12-1-1) whereListhenumberofchannels andMisthenumberofwaveforms. The waveforms areassumed tohaveequalenergyandtobeequallyprobable a priori.Thewaveforms {S~::'(t)ftransmitted overtheLchannels arescaledby thefactors{au},phase-shifted by{</>u},andcorrupted l,yadditive noise.The equivalent lowpasssignalsreceived fromtheLchannels maybeexpressed as r~n'(t)=erne ,J>"s~::i(t) -z,,(t). 0,,;;I'"T n=I,2,...,L,m=I,2,...,M(12-1-2) where{sl::i(t)}aretheequivalent lowpasstransmitted waveforms and{zu(I)} represent theadditive noiseprocesses ontheLchannels. Weassumethat {Z,,(I)}aremutually statistically independent andidentically distributed gaus­ siannoiserandomprocesses. Weconsider twotypesofprocessing atthereceiver, namely, coherent detection andnoncoherent detection. Thereceiver forcoherent detection estimates thechannel parameters {au}and{</>,,}andusestheestimates in computing thedecision variables. Suppose wedefineg"=a"eiJ>"andletg"be theestimate ofg".Themllltichannel receivercorrelates eachoftheLreceived signalswithareplicaofthecorresponding transmitted signals,multiplies each ofthecorrelator outputsbythecorresponding estimates {g~},andsumsthe resulting signals.Thus,thedecision variables forcoherent detection arethe correlation metrics I r CMm=n~' Re[g~L dn'(t)sl::i*(t)dt]. m=1,2,...,M(12-1-3) Innoncoherent detection, noattempt ismadetoestimate thechannel parameters. Thedemodulator maybaseitsdecision eitheronthesumofthe envelopes (envelope detection) orthesumofthesquaredenvelopes (square­ lawdetection) ofthematched filteroutputs. Ingene.ral. theperformance obtained withenvelope detection differslittlefromtheperformance obtained withsquare-law detection inAWGN.However, square-law detection of multichannel signaling inAWGNchannels isconsiderably easiertoanalyze thanenvelope detection. Therefore. weconfineourattention tosquare-law detection ofthereceived signalsoftheLchannels. whichproduces the decisionvariables (12-1-4) Letusconsider binarysignaling first.andassumethatsj;".n=1,2,...,L 682 DlCill,\L C()M\ll·:'oJ[{'ATI()~S aretheLtransmitted waveforms. Thenanerroriscommitted ifeM,>eM,. or.equivalently, ifthedifference D=eM,-eM,<0,Fornoncoherent detection, thisdifference maybeexpressed as I. D=L(IX"I'-IY"I') Il-~1 wherethevariables {X,,}and{Y,,}aredefinedas X"=ITd"'(t)sl7'*(t) dt,n=1,2,...,L " •Y"=ITd"'(t)sl~'*(t) dt,n=I,2",,,L "(12-1-5) (12-1-6) The{X,,}aremutually independent andidentically distributed gaussian random variables. Thesamestatement appliestothevariables {Y,,}.However, forany n,X"andY"maybecorrelated. Forcoherent detection, thedifference D=eM,-eM,maybeexpressed as where,bydefinition,L D=~L(X"Y~+ X~Y,') I/~I t:,=gil'n=1.2.....L r X"=fr)"'(t)[s)j"*(t) -s!f'*(t)Jd, "(12-1-7) (12-1-8) Iftheestimates {gn}areobtained fromobservation ofthereceived signalevef oneormoresignaling intervals, asdescribed inAppendix C,theirstatistical characteristics aredescribed bythegaussian distribution. Thenthe{Y,,}are characterized asmutually independent andidentically distributed gaussian random variables. Thesamestatement appliestothevariables {X.},Asin noncaherent detection. weallowforcorrelation between X"andY",butnot between X",andY"form'"n. '12-1-1BinarySignals InAppendix B.wederivetheprobability thatthegeneralquadratic form l. D=2:(AIX"I'+BIy,l+eX"Y~+C*X~Y,.) 1I-~I(12-1-9) Incomplex-valued gaussian random variables islessthanzero.Thisprob­ ability,whichisgivenin(B-21)ofAppendix B,istheprobability oferrorfor CHA.PTER J2:MUtTJCHA'l!'iEL A~DMl!LT1CARRJER SYSTEMS 683 binarymultichannel signaling inAWGN.Anumber ofspecialcasesareof particular importance. Ifthebinarysignalsareantipodal andtheestimates of{g,,}areperfect,asin coherent PSK,theprobability oferrortakesthesimpleform Ph=Q(v'ZYh) (IZ-I-1O) where 't;L Yh=No2:Ig,,12 On-I '"/=~~a2 NOn=1 "(IZ-1-11) (12-1-12)istheSNRperbil.Ifthechannels areallidentical, a"=aforallnand,hence. L't 2 "Yo=-a No WeobservethatL\I;isthetotaltransmitted signalenergyfortheLsignals.The interpretation ofthisresultisthatthereceivercombines theenergyfromtheL -channels inanoptimum manner. Thatis,thereisnolossinperformance in dividing thetotaltransmitted signalenergyamongtheLchannels. Thesame performance isobtained asinthecaseinwhichasinglewaveform having energyL'l:istransmitted ononechannel. Thisbehavior holdstrueonlyifthe estimates gn=g",foralln.Iftheestimates arenotperfect, alossin performance occurs.theamountofwhichdepends onthequalityofthe estimates, asdescribed inAppendix C. Perfectestimates for{gn}constitute anextremecase.Attheotherextreme, wehavebinaryDPSKsignaling. InDPSK,theestimates {g,,}aresimplythe (normalized)'signal-plus-noise samplesattheoutputsofthematched filtersin theprevious signaling interval. Thisisthepoorestestimate thatonemight consider usinginestimating {gn}.ForbinaryDPSK,theprobability oferror obtained from(B-21)is where,bydefinition,I I...I Po=~e-"Yf>Len'Yh 2 nO'1(12-1-13) (12-1-14) andYhistheSNRperbitdefinedin(12-1-11) and.foridentical channels in (12-1-12). Thisresultcanbecompared withthesingle-channel (L=I)error probability. Tosimplifythecomparison, weassumethattheLchannels have identical attenuation factors.Thus,forthesamevalueofYb,theperformance ofthemultichannel systemispoorerthanthatofthesingle-channel system. Thatis,splittingthetotaltransmitted energyamongLchannels resultsinaloss inperformance, theamountofwhichdepends onL. 684 DIGITAL COMMUNICA nONS /4 12 _10 '"~.8..2.. 'a6:c E <34 2 02 102050/002005001000 Numberofchannels.L FIGURE U-l·l Combining lossinnoncohereDt detection andcombination ofbinarymultichannel signals. Alossinperfonnance alsooccursinsquare-law detection oforthogonal signalstransmitted overLchannels. Forbinaryorthogonal signaling, the expression fortheprobability oferrorisidentical infonntothatforbinary DPSKgivenin(12-1-13), exceptthat"Ybisreplaced byhb'Thatis, binaryorthogonal signaling withnoncoherent detection is3dBpoorerthan binaryDPSK.However, thelossinperformance duetononcoherent combina­ tionofthesignalsreceived ontheLchannels' isidentical tothatforbinary DPSK. Figure12-1-1illustrates thelossresulting fromnoncoherent (square-law) combining oftheLsignalsasafunctionofL.Theprobability oferrorisnot shown,butitcanbeeasilyobtained fromthecurveoftheexpression Pb==!e-" (12-1-15) whichistheerrorprobability ofbinaryDPSKshowninFig.5-2-12andthen degrading therequired SNRperbit,rb,bythenoncoherent combining loss corresponding tothevalueofL. (12-1-16)L Um==2:IN"nl,m==2,3,...,M n-Ill-1-2M-aryOrthogonal Signals Nowletusconsider M-aryorthogonal signaling withsquare-law detection and combination ofthesignalsontheLchannels. Thedecisionvariables aregiven by(12-1-4). Suppose thatthesignals si~)(/),n==1,2,...•L,aretransmitted overtheLAWGNchannels. Then,thedecisionvariables areexpressed as L U1==2:12~an+Nnl12 n-I ("HAPrER 12:Ml:lTICHANNEl ANDMUlTiCARRIFR SYSTEMS 68S wherethe{N,,,,,}arecomplex-valued zero-mean gaussian random variables withvariance u1=~E(JN"ml') =2,€NlI.HenceV,isdescribed statistically asa noncentral chi-square random variable with2Ldegrees offreedom and noncentrality parameter L L S2=2:(2~a,.)2=4l!'22:a~ (J2-I-l7) ,,=1 Using(2-1-118). weobtainthepdfofV,as"=1 1(1I,)IL.'1/2 (S2+1I,)(5~) p(lI,)=4~No 52 exp-4'ffNlIh-~2'€No'u,;;;>O (12-1-18) Ontheotherhand.the{U;,.},m=2,3,...,M,arestatistically independent andidentically chi-square-distributed random variables, eachhaving2L degreesoffreedom. Using(2-1-110), weobtainthepdfforVmas m=2,3,...,M(12-1-19) Theprobability ofasymbolerroris PM=1-Pc =1-P(U 2<U"U3<V,,»>, VM<V,) =1-f[P(U2<IIIIV,=UI»)M-Ip(II,)dUI But where L y='€La~/No n=1(12-1-20) (12-1-21) (12-1-22) 686 DIGITAL COMMUN1CATIONS Theintegralin(12-1-22) canbeevaluated numerically. Itisalsopossible to expandtheterm(1-X)M-Iin(12-1-22) andcarryouttheintegration termby term.Thisapproach yieldsanexpression forPMintermsoffinitesums. Analternative approach istousetheunionbound PM<(M-1)P,(L) (12-1-23) whereP,(L)istheprobability oferrorinchoosing between U,andanyone of theM-1decision variables {Urn},m=2,3,...,M.Fromourprevious discussion ontheperformance ofbinaryorthogonal signaling, wehave 1 L-I P.(L)--e-k,,12'"C(lk'Y)n (12-1-24),-2'L-1 n~On2 b where Cnisgivenby(12-1-14). Forrelatively smallvaluesofM,theunion boundin(12-1-23) issufficiently tightformostpractical applications. 12-2MULTICARRIER COMMUNICATIONS Fromourtreatment ofnonideallinear filterchannels inChapters 10and11,we haveobserved thatsuchchannels introduce lSI,whichdegra(fes performance compared withtheidealchannel. Thedegreeofperform'ance degradation depends onthefrequency response characteristics. Furthermore, thecom­ plexityofthereceiverincreases asthespanofthelSIincreases. Givenaparticular channelcharacteristic, thecommunication systemdesig­ nermustdecidehowtoefficiently utilizetheavailable channel bandwidth in ordertotransmit theinformation reliably withinthetransmitter power constraint andreceiver complexity constraints. Foranonideal linearfilter channel, oneoptionistoemploy asinglecarriersysteminwhichthe information sequence istransmitted seriallyatsomespecified rateRsymbols/so Insuch achannel, thetimedispersion isgenerally muchgreaterthanthe symbolrateand,hence,lSIresultsfromthenonideal frequency response characteristics ofthechannel. Aswehaveobserved, anequalizer isnecessary tocompensate forthechanneldistortion. Analternative approach tothedesignofabandwidth-efficient communica­ tionsysteminthepresence ofchanneldistortion istosubdivide theavailable channelbandwidth intoanumberofsubchannels, suchthateachsubchannel is nearlyideal.Toelaborate, suppose thatC(f)isthefrequency response ofa nonideal, band-limited channel withabandwidth W,andthatthepower spectraldensityoftheadditive gaussian noiseis<Pnn(f).Then,wedividethe bandwidth WintoN=W//::;.1subbands ofwidthaf,wheret!.fischosen sufficiently smallthat/C(f)/2/<P..(f)isapproximately aconstant wil;.hineach subband. Furthermore, weshallselectthetransmitted signalpowertobe distributed infrequency asP(f),subjecttotheconstraint that LP(f)df'"Pav (12-2-1) (12-2-2) (12-2-4) (12-2-8)12-2-1CHAPTER 12,MULTICHANNEl ANDMULTICARRIER SYSTEMS687 where,Pa•istheavailable averagepowerofthetransmitter. Letusevaluate thecapacity ofthenonideal additive gaussian noisechannel. Capacity ofaNonideal LinearFilterChannel Recallthatthecapacityofanideal,band-limited, AWONchannelis (Pa.)C=Wlog21+WN o whereCisthecapacity inbits/s,Wisthechannelbandwidth, andPa•isthe averagetransmitted power.Inamulticarrier system,withAfsufficiently small, thesubchannel hascapacity C=AtI[1+!::l.fP(f,))c(fW] (12-2-3),082 Afcf>nn(f,) Hence,thetotalcapacityofthechannelis C=±C;=Af±1082[1+P(/;)1C(J;)12 ] i=1 '=1 cf>",,(f;) InthelimitasAf......0,weobtainthecapacity oftheoverallchannelinbits/sas C=flog2[1+P(f)IC(f)f]df (12-2-5) w <l>..(f) Undertheconstraint onP(f)givenby(12-2-1), thechoiceofP(f)that maximizes Cmaybedetermined bymaximizing theintegral f{1082[I+P{f)1C(f)12 ]+AP(f)}df (12-2-6) w <l>nn(f) whereAisaLagrange multiplier, whichischosentosatisfytheconstraint. By usingthecalculus ofvariations toperform themaximization, wefindthatthe optimum distribution oftransmitted signalpoweristhesolution tothe equation lC(fWP(f) +<l>nn(f)+A=0 (12-2-7) Therefore, P(f)+<l>nn(f)/IC(f)l' mustbeaconstant, whosevalueisadjusted tosatisfytheaveragepowerconstraint in(12-2-1). Thatis, P(f)={K-<l>nn(f)/lC(f)!2 (fEW) o (f~W) Thisexpression forthechannelcapacity ofanonideal linearfilterchannelwith additive gaussian noiseisdueto Shannon (1949).Thebasicinterpretation of thisresultisthatthesignalpowershouldbehighwhenthechannel SNR 1C(f)12/cf>n,,(f)ishigh,andlowwhenthechannel SNRislow.Thisresulton 688 DIGITAL COMMUNICATIONS K~~ cIt••(/) lel!)I' ",1,--w--i'I FlGliRE U·Z·l Theoptimum water-pouring spectrum. l'req1lencyf thetransmitted powerdistribution isillustrated inFig.12-2-1.Observe thatif 4>nn(f)/lC(tW isinterpreted asthebottomofabowlofunitdepth,andwe pouranamountofwaterequaltop••intothebowl,thewaterwilldistribute itselfinthebowlsoas·toachievecapacity. Thisiscalledthewater-filling interpretation oftheoptimum powerdistribution asafunction offrequency. IIisinteresting tonotethatthechannelcapacityistbesmallestwhenthe channelSNR1C(f)12/4>nn(f)isaconstant forallfeW.Inthiscase,P(f)isa constant forallfeW.Equivalently, ifthechannelfrequency response isideal, Le.,C(f)=1forfEW,thentheworstgaussiannoisepowerdistribution, from theviewpoint ofmaximizing capacity, iswhitegaussian noise. Theabovedevelopment suggeststhatmulticarrier modulation thatdivides theavailable channelbandwidth intosubbands ofrelatively narrowwidth Af=W/Nprovides asolution thatcouldyieldtransmission ratescloseto capacity. Thesignalineachsubband maybeindependently codedand modulated atasynchronous symbolrateofI/Af,withtheoptimum power allocation P(f).IfAtissmallenoughthenC(f)isessentially constant across eachsubband, sothatnoequalizatiOll isnecessary becausethelSIisnegligible. Multicarrier modulation hasbeenusedinmodems forbothradioand telephone channels. Multicarrier modulation hasalsobeenproposed forfuture digitalaudiobroadcast applications. Aparticularly suitableapplication ofmulticarrier modulation isindigital transmission overcopperwiresubscriber loops.Thetypicalchannelattenua­ tioncharacteristics forsuchsubscriber linesareillustrated inFig.12-2-2.We FIGURE u.Z.Z AUenuation characterislic ofa24gauge12ttlPICloop. [FrQmWtmer(991)©IEEE.]10=.-------, 20 ~.10 840 .~50 j60 70 BOo«J16016O6«l1<n1 Frequency (kH,) CHAPTER I:':MULTICHANNEL ANDMOLTICARRIER SYSTEMS 689 observethattheattenuation increases rapidlyasafunction offrequency. This characteristic makesitextremely difficulttoachieveahightransmission rate withasinglemodulated carrierandanequalizer atthereceiver. ThelSI penalty inperformance isverylarge.Ontheotberhand,multicarrier modulation withoptimum powerdistribution provides thepotential fora highertransmission rate. Tbedominant noiseintransmission oversubscriber linesiscrosstalk interference fromsignalscarriedonothertelephone lineslocatedinthesame cable.Thepowerdistribution ofthistypeornoiseisalsofrequency­ dependent, whicbcanbetakenintoconsideration intheallocation oftbe available transmitted power. Adesignprocedure foramulticarrier QAMsystemforanonideal linear filterchannelhasbeengivenbyKalet(1989).Inthisprocedure, theoverallbit rate'ismaximized, throughthedesignofanoptimalpowerdivisionamongthe subcarriers andanoptimum selection ofthenumber 01bitspersymbol(sizes oftheQAMsignalconstellations) foreachsubcarrier, underanaveragepower constraint andundertheconstraint thatthesymbolerrorprobabilities forall subcarriers areequal. Below,wepresentanimplementation ofamulticarrier QAMmodulator anddemodulator thatisbasedonthediscreteFouriertransform (DIT)forthe generation ofthemultiple carriers. 12-2-2AnFIT-Based Multicarrier System Inthissection,wedescribe amulticarrier communication systemthatemploys thefastFouriertransform (FFT)algorithm tosynthesize thesignalatthe transmitter andtodemodulate thereceived signalatthereceiver. TheFITis simplytheefficient computational toolforimplementing thediscrete fourier transform (DIT). Figure12-2-3illustrates ablockdiagram ofamulticarrier communication FIGURE 12-1a3Multicarrier communication system. Inpul dataSerial·to­ parallel bufferMullicarrier modulator (inverseOFT)Addcyclic prefix,and parallel-lo­ serialconvertDlA Output converter Remow Paranel· cyclic OlilPUI 10- prefixand AID bits serial serial-oo- converter converter parallel ccnvel'\ 690 OIOPTAl COMMUNICATIONS system.Aserial-to-parallel buffersegments theinformatio'! sequence into framesofNtbils.TheNtbitsineachframeareparsedintoNgroups,where theithgroupisassigned flibits,and iV 2:ii,=Nt j=!(12-2-9) Eachgroupmaybeencoded separately, sothatthenumberofoutputbitsfrom theencoderfortheithgroupisni;;;.ii,. Itisconvenient toviewthemulticarner modulation asconsisting ofN independent QAMchannels, eachoperating atthesamesymbolratel/T,but eachchannel having a distinctQAMconstellation, i.e.,theithchannel will employMi=2"signalpoints.Wedenotethecomplex-valued signalpoints corresponding totheinformation symbols onthesubchannels byX.,,,= 0,I,...,N-1.Inordertomodulate theNsubcarriers bytheinformation symbols{X.},weemploytheinverseOFT(10FT). However, ifwecompute theN-pointIOFfof{Xd,weshallobtaina complex-valued timeseries,whichisnotequivalent toNQAM-mo<lulated subcarriers. Instead,wecreateN=2Ninformation symbolsbydefining X"-k=Xt,k=I,...,N-1 (12-2-10) andX;'=Re(Xo),XiV=1m(Xo).Thus,thesymbolXoissplitintotwoparts, bothreal.Then,theN-pointIDFTyieldsthereal-valued sequence 1N-I X,=_=N2:X.t!'nnkIN, n=0,1,...,N-1 vNk=()(12-2-11) wherellYNissimplyascalefactor. '!besequence {x"O,,;;n";;N-I}corresponds tothesamplesofthesumX(I) ofNsubcarrier signals,whichisexpressed as (12·2-12) where1isthesymbolduration. Weobservethatthesubcarrier frequencies are t.=kll,k=0,1,...,N.Furthermore, thediscrete-time sequence {xn}in (12-2-10) represents thesamples ofX(I)takenattimest=nlINwhere n=0,1,...,N-1. Thecomputation oftheIDITofthedata{X.}asgivenin(12-2-10) maybe viewedasmultiplication ofeachdatapointX,byacorresponding vector where v=_1_e(2"'N)kn k"YN(12-2-13) (12-2-14) CHAPlFR 12:Mlll,TJCHANNEL ANDMULTICARFiER SYSTfMS 691 flGURE 12-24 Signalsynthesis formullicarrier modulation basedoninverseDFT.Dala X" x,Addcyclic To prefix channel asillustrated inFig.12-2-4.Inanycase,thecomputation oftheOFTis performed efficiently bytheuseoftheFFTalgorithm. Inpractice, thesignalsamples {xn}arepassedthrough a01Aconverter whoseoutput,ideally,wouldbethesignalwaveform X(I).Theoutputofthe channelisthewaveform r(l)=x(t)*h(l)+n(l) (12-2-15) whereh(l)istheimpulseresponse ofthechanneland*denotesconvolution. Byselecting thebandwidthAtofeachsubchannel tobeverysmall,thesymbol duration T=IIAtislargecompared withthechanneltimedispersion. Tobe specific, letusassumethatthechanneldispersion spansv+Isignalsamples wherev«N.OnewaytoavoidtheeffectoflSIistoinsertatimeguardband ofdurationvTINbetween transmissions ofsuccessive blocks. Analternative methodthatavoidslSIistoappendacyclicprefixtoeach blockofNsignalsamples {xo,X"...,XN-'}'Thecyclicprefixforthisblockof samples consistsofthesamples XN-y,XN-Y+I," .,XN-"Thesenewsamples areappended tothebeginning ofeachblock.Notethattheaddition ofthe cyclicprefixtotheblockofdataincreases thelengthoftheblocktoN+v samples, whichmaybeindexed fromn=-v,...,N-1.wherethefirstv samples constitute theprefix.Then,if{hn,0""n""v}denotes thesampled channelimpulseresponse, itsconvolution with{xn,-v""n""N-I}produces {rn},thereceived sequence. Weareinterested inthesamples of{rn}for 0""n""N-I,fromwhichwerecoverthetransmitted sequence byusingthe N-pointOFffordemodulation. Thus,thefirstvsamplesof{r,,)arediscarded. Fromafrequency-domain viewpoint, whenthechannelimpulseresponse is {hn,O""n""v},itsfrequency response atthesubcarrier frequencies!. =kiNis (12-2-16) Duetothecyclicprefix,successive blocks(frames) ofthetransmitted 692 DIGITAL COMMUNICATION' information sequence donotinterfere and,hence,thedemodulated sequence maybeexpressed as kk=HkX.+Tlk,k=0,1....,N-1 (12-2-17) where{A'k}istheoutputofthe]y-pointOffdemodulator, andTI.isthe additivenoisecorrupting thesignal~WenotethatbyselectingN»v,therate lossduetothecyclicprefixcanberendered negligible. AsshowninFig.12-2-3,theinformation isdemodulated bycomputing the OFTofthereceivedsignalafterithasbeenpassedthroughanA/Oconverter. TheOFTcomputation maybeviewedasamultiplication ofthereceivedsignal samples {r.}fromtheAIDconverter byv:,wherev.isdefinedin(12-2-12). As inthecaseofthemodulator, theOFTcomputation atthedemodulator is performed efficiently byuseoftheFFfalgorithm. Itisasimplemattertoestimate andcompensate forthechannelfactors{H.} priortopassingthedatatothedetector anddecoder, Atraining signal consisting ofeitheraknownmodulated sequence oneachofthesubcarriers or unmodulated subcarriers maybeusedtomeasure the{H.latthereceiver.If thechannelparameters varyslowlywithtime,itisalsopossibletotrackthe timevariations byusingthedecisions attheoutputofthedetector orthe decoder, inadecision-directed fashion.Thus,themulticarrier systemcanbe rendered adaptive. Multicarrier QAMmodulation ofthetypedescribed abovehasbeen implemented foravarietyofapplications, including high-speed transmission overtelephone lines,suchasdigitalsubscriber lines. Othertypesofimplementation besidestheOffarepossible. Forexample, adigitalfilterbankthatbasically performs the.DFf maybesubstituted forthe FFf-basedimplementation whenthenumberofsubcarriers issmall,e.g., N.,;32.Foralargenumberofsubcarriers; e.g.,N>32,theFFf-based systems arecomputatively moreefficient. Onelimitation oftheOFT-type modulators anddemodulators arisesfrom therelatively largesidelobes infrequency thatareinherent inOFT-type filter banks.Thefirstsidelobe isonly13dBdownfromthepeakatthedesired subcarrier. Consequently, theOFT-based implementations arevulnerable to interchannel interference (ICI)unlessafullcyclicprefixisused.IfICIisa problem, duetochannelanomalies, onemayresorttoothertypesofdigital filterbanksthathavemuchlowersidelobes. Inparticular, theclassofmultirate digitalfilterbanksthathavetheperfectreconstruction property associated withwavelet-based filtersappeartobeanattractive alternative (seeTzannes el ai.,1994;Rizoselal.,1994). 12-3BIBLIOGRAPHICAL NOTES ANDREFERENCES Multichannel signaltransmission iscommonly usedontime·varyingchannels toovercome theeffectsofsignalfading.Thistopicistreatedinsomedetailin Chapter 14,whereweprovideanumberofreferences topublished work.Of PROBLEMSCHAPTER i2:MULTICHANNEL ANDMULTICARRIER SYSTEMS 693 particular relevance tothetreatment ofmultichannel digitalcommunications giveninthischapteraretbetwopublications byPrice(1962a,b). Thereisalargeamountofliterature onmulticarrier digitalcommunication systems.Suchsystemshavebeenimplemented andusedforover30years.One oftheearliestsystems, described byDoeltzetal.(1957)andcalledKineplex, wasusedfordigitaltransmission intheHFband.Otherearlyworkon multicarrier systemdesignhas.beenreported inthepapersbyChang(1966) andSaltzburg (1967).TheuseoftheDFfformodulation anddemodulation of multicarrier systemswasproposed byWeinstein andEtJert(1971). Ofparticular interestinrecentyearsistheuseofmulticarrier digital transmission fordata,facsimile, andvideoonavarietyofchannels, including thenarrowband (4kHz)switched telephone network, the48kHzgroup telephone band,digitalsubscriber lines,cellularradio,andaudiobroadcast. Theinterested readermayrefertothemanypapersintheliterature. Wecite asexamples thepapersbyHirosaki etal.(1981,1986),Chowetal.(1991),and thesurveypaperbyBingham (1990).ThepaperbyKalet(1989)givesadesign procedure foroptimizing th.rateinarnulticarrier QAMsystemgiven constraints ontransmitter powerandchannelcharacteristics. Finally,wecite thebookbyVaidyanathan (1993)andthepapersbyTzannes etal.(1994)and Rizosetal.(1994)foratreatment ofmultirate digitalfilterbanks. 12-1X\.X"...•XNareasetofNstatistically independent andidentically distributed realgaussianrandomvariables withmoments E(Xi)=mandvar(X.)=u'. •Define N U=2:X•.-\ Evaluate theSNRofU,whichisdefinedas (SNR)=[£(U»)' u2u~ where u~isthevarianceofU. bDefine N V=2:X~ Evaluate theSNRofV,whichisdefinedas (SNR)_[£(V»)' v2u~ where u~isthevarianceofV. rPlot(SNR)uand(SNR)vversusm'/u'pnthesamegraphand,thus,compare theSNRsgraphically. 694 DIGJTAL COMMVNJCATJONS dWhatdoestheresultin(c)implyregarding coherent detection andcombining versussquare-law detection andcombining ofmultichannel signals? 12-2Abinarycommunication systemtransmits tbesameinformation ontwodiversity cbannels. Thetworeceived signalsare " =±~+n, ,,=±~+n, whereE(n,)=E(n,)=O.E(nD=criandE(n~)=cr~,andn,andn,areuncorre­ latedgaussian-variables. Thedetector basesitsdecision onthelinearcombination ofr(andrz•i.e., r=',+k,~ aDetermine thevalueofkthatminimizes lheprobability oferror. bPloltheprobability oferrorforcri=I,cr~=3,andeitherkc=Iorkisthe optimum valuefoundin(a).Compare theresults. 12-3Assessthecostofthecyclicprefix(usedinmultitone modulation toavoidlSI)in termsof aextrachannelbandwidth: bextrasignalenergy. 12-4Letx(n)beafinite-duration signalwilhlengthNandletX(k)beitsN-pointOFf. Suppose wepadx(n)withLzerosandcompute the(N+L)-pointOFT,X'(k). Whatistherelationship between X(O)andX'(O)?IfweplotIX(k)1andIX'(k)1on thesamegraph,explaintherelationships between thetwographs. 12-5Showthatthesequence {x..lgivenby(12-2-11) corresponds tothesamplesofthe signalX(I)givenby(12-2·12). 12-6Showthatthe10FTofasequence {X,.0'"k'"N-I}canbecomputed bypassing thesequence (X.Jthrough abankofNlineardiscrete-time filterswithsystem­ functions H,,(Z)=l 12f("IN~1-e Z 12-7PlotP,(L)forL=1andL=2asafunctionof10logy,anddetermine thelossin SNRduetothecombining lossfory,=10 13 SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS Spreadspectrum signalsusedforthetransmission ofdigitalinformation are distinguished bythecharacteristic thattheirbandwidth Wismuchgreaterthan theinformation rateRinbits/soThatis,thebandwidth expansion factor B,=W/Rforaspreadspectrum signalismuchgreaterthanunity.Thelarge redundancy inherent inspreadspectrum signalsisrequired toovercome the severelevelsofinterference thatareencountered inthetransmission ofdigital information oversomeradioandsatellitechannels. Sincecodedwaveforms are alsocharacterized byabandwidth expansion factorgreaterthanunityand sincecodingisanefficientmethodforintroducing redundancy, itfollowsthat codingisanimportant elementinthedesignofspreadspectrum signals. Asecondimportant elementemployed inthedesignofspreadspectrum signalsispseudo-randomness, whichmakesthesignalsappearsimilarto randomnoiseanddifficultto,demodulate byreceivers otherthantheintended ones.Thiselementisintimately relatedwiththeapplication orpurposeofsuch signals. Tobespecific,spreadspectrum signalsareusedfor •combatting orsuppressing thedetrimental effectsofinterference dueto jamming, interference arisingfromotherusersofthechannel, andself­ interference duetomultipath propagation; •hidingasignalbytransmitting itatlowpowerand,thus,makingit difficultforanunintended listenertodetectinthepresence ofbackground noise; •achieving messageprivacyinthepresence ofotherlisteners. In'applications otherthancommunications, spreadspectrum signalsareused 696 DIGITALCOMMUN 1('ATlONS toobtainaccurate range(timedelay)andrangerate(velocity) measurements inradarandnavigation. Forthesakeofbrevity,weshalllimitourdiscussion to digitalcommunications applications. Incombatting intentional interference (jamming), itisimportant tothe communicators thatthejammerwhoistryingtodisruptthecommunication doesnothavepriorknowledge ofthesignalcharacteristics exceptforthe overallchannelbandwidth andthetypeofmodulation, (PSK,FSK,etc.)being used.Ifthedigitalinformation isjustencoded asdescribed inChapter 8,a sophisticated jammer caneasilymimicthesignalemitted bythetransmitter and,thus,confusethereceiver. Tocircumvent thispossibility, thetransmitter introduces anelementofunpredictability orrandomness (pseudo-randomness) ineachofthetransmitted codedsignalwaveforms thatisknowntothe intended receiverbutnottothejammer. Asaconsequence, thejammermust synthesize andtransmit aninterfering signalwithout knowledge ofthe pseudo-random pattern. Interference fromtheotherusersarisesinmultiple-access communication systemsinwhichanumberofusersshareacommon channelbandwidth. At anygiventime,asubsetoftheseusersmaytransmit information simul­ taneously overthecommon channeltocorresponding receivers. Assuming that alltheusersemploythesamecodefortheencoding anddecoding oftheir respective infomlation sequences, thetransmitted signalsinthiscommon spectrum maybedistinguished fromoneanotherbysuperimposing adifferent pseudo-random pattern,alsocalledacode,ineachtransmitted signal.Thus,a particular receivercanrecoverthetransmitted information intended foritby knowing thepseudo-random pattern,i.e.,thekey,usedbythecorresponding transmitter. Thistypeofcommunication technique, whichallowsmultiple users tosimultaneously useacommon channelfortransmission ofinformation, is calledcodedivisionmultiple access(CDMA). CDMAwillbeconsidered in Sections 13-2and13-3. Resolvable mIJltipath components resulting fromtime-dispersive propaga­ tionthroughachannelmaybeviewedasaformofself-interference. Thistype ofinterference mayalsobesuppressed bytheintroduction ofapseudo-random patterninthetransmitted signal,aswillbedescribed below. Amessage maybehiddeninthebackground noisebyspreading its bandwidth withcodingandtransmitting theresultant signalatalowaverage power.Because ofitslowpowerlevel,thetransmitted signalissaidtobe "covert." Ithasalowprobability ofbeingintercepted (detected) byacasual listenerand,hence,isalsocalledalow-probability-of-intercept (LPI)signal. Finally, message privacymaybeobtained bysuperimposing apseudo­ randompatternonatransmitted message. Themessage canbedemodulated bytheintended receivers, whoknowthepseudo-random patternorkeyused atthetransmitter, butnotbyanyotherreceivers whodonothaveknowledge ofthekey. Inthefollowing sections, weshalldescribe anumberofdifferent typesof spreadspectrum signals, theircharacteristics, andtheirapplication. The CHAPTER Ll:SPKEAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 697 Information sequence Pseuoo.random pattern gelleralorPseudo-random patlem JeneratorOtltUI data FIGURE 13-1-1 Modelofspreadspectrum digitalcommunication system. emphasis willbeontheuseofspreadspectrum signalsforcombatting, jamming (antijam orAJsignals), forCOMA, andforLPI.Beforediscussing thesignaldesignproblem, however, weshallbrieflydescribe thetypesof channelcharacteri6tics assumed fortheapplications citedabove. 13-1MODEL OFSPREAD SPECTRUM DIGITAL COMMUNICATION SYSTEM Theblockdiagram showninFig.13-1-1illustrates thebasicelements ofa spreadspectrum digitalcommunication systemwithabinaryinformation sequence atitsinputatthetransmitting endandatitsoutputatthereceiving end.Thechannelencoder anddecoder andthemodulator anddemodulator arebasicelements ofthesystem,whichweretreatedinChapters 5,7and8.In addition totheseelements, wehavetwoidentical pseudo-random pattern generators, onethatinterfaces withthemodulator atthetransmitting endand asecondthatinterfaces withthedemodulator atthereceiving end.The generators generate apseudo·random orpseudo-noise (PN)binary-valued sequence, whichisimpressed onthetransmitted signalatthemodulator and removed fromthereceived signalatthedemodulator. Synchronization ofthePNsequence generated atthereceiverwiththePN sequence contained intheincoming received signalisrequired inorderto demodulate thereceived signal.Initially, priortothetransmission ofinforma­ tion,synchronization maybeachieved bytransmitting afixedpseudo·random bitpatternthatthereceiverwillrecognize inthepresence ofinterference with ahighprobability. Aftertimesynchronization ofthegenerators isestablished, thetransmission ofinformation maycommence. Interference isintroduced inthetransmission oftheinformation-bearing signalthroughthechannel. Thecharacteristics oftheintederence dependtoa largeextentonitsorigin.Itmaybecategorized asbeingeitherbroadband or narrowband relativetothebandwidth oftheinformation-bearing signal,and eithercontinuous orpulsed(discontinuous) intime.Forexample, ajamming signalmayconsistofoneormoresinusoids inthebandwidth usedtotransmit theinformation. Thefrequencies ofthesinusoids mayremainfixedorthey maychangewithtimeaccording tosomerule.Asasecondexample, the interference generated inCOMAbyotherusersofthechannelmaybeeither (13-2-2)(13-2-1)698 DIGlTAl COMMUNICATIONS broadband ornarrowband, depending onthetypeofspreadspt:(:trum signal thatisemployed toachievemultiple access.Ifitisbroadband, itmaybe characterized asanequivalent additivewhitegaussian Roise.Weshallconsider thesetypesofinterference andsomeothersinthefollowing sections. Ourtreatment ofspreadspectrum signalswillfocusontheperformance of thedigitalcommunication systeminthepresence ofnarrowband andbroad­ bandinterference. Twotypesofmodulation areconsidered: PSKandFSK. PSKisappropriate inapplications wherephasecoherence between the transmitted signalandthereceived signalcanbemaintained overatime intervalthatisrelatively longcompared tothereciprocal ofthetransmitted signalbandwidth. Ontheotherhand,FSKmodulation isappropriate in applications wheresuchphasecoherence cannotbemaintained dueto time-variant effectsonthecommunications link.Thismaybethecaseina communications linkbetween twohigh-speed aircraftorbetween ahigh-speed aircraftandagroundterminal. ThePNsequence generated atthemodulator isusedinconjunction withthe PSKmodulation toshiftthephaseofthePSKsignalpseudo-randomly as described inSection 13-2.Theresulting modulated signaliscalledadirect sequence (DS)orapseudo-noise (PN)spreadspectrum signal.Whenusedin conjunction withbinaryorM-ary(M>2)FSK,thepseudo-random sequence selectsthefrequency ofthetransmitted signalpseudo-randomly. Theresulting signaliscalledafrequency-hopped (FH)spreadspectrum signal.Although a numberofothertypesofspreadspectrum signalswillbebrieflydescribed, the emphasis ofourtreatment willbeonPNandFHspreadspectrum signals. 13-2DlRECI SEQUENCE SPREAD SPECTRUM SIGNALS InthemodelshowninFig.13-1-1,weassumethattheinformation rateatthe inputtotheencoderisRbitslsandtheavailable channelbandwidth isWHz. Themodulation isassumed tobebinaryPSK.Inordertoutilizetheentire available channel bandwidth, thephaseofthecarrierisshiftedpseudo­ randomly according tothepatternfromthePNgenerator atarateWtimes/s. Thereciprocal ofW,denoted byTe,definestheduration ofarectangular pulse,whichiscalledachipwhile1;iscalledthechipinterout.Thepulseisthe basicelementinaDS'spread spectrum signal. IfwedefineTb=11Rtobetheduration ofarectangular pulsecorrespond­ ingtothetransmission timeofaninformation bit,thebandwidth expansion factorWIRmaybeexpressed as WTbB=-=­• R7:c Inpractical systems,theratioTbITeisaninteger, L=Tb c1;, CHAPTER IlSPREAD 5PECTRUM SIGNALS FORDIGITAL COMMUNICATIUNS 699 PNsignal L,1,-j Datasignal +1 -1 DataI !+--T,--. talPNanddata'i>igna.1s QPSK signal '----.----' (b)DS-QPSK modulator FIGURE 13-2-1 ThePNanddatasignals(a)andtheQPSKmodulator (b)foraDSspreadspectrum system. whichisthenumberofchipsperinformation bit.Thatis,Lcisthenumberof phaseshiftsthatoccurinthetransmitted signalduringthebitduration ~=llR.Figure13-2-I(a) illustrates therelationships between thePNsignal andthedatasignal. Suppose thattheencodertakeskinformation bitsatatimeandgenerates a binarylinear(n,k)blockcode.Thetimeduration available fortransmitting thencodeelements isk~s.Thenumberofchipsthatoccurinthistime intervaliskLc•Hence,wemayselecttheblocklengthofthecodeasn=kLc• Ifthe encoder generates abinaryconvolutional codeofratekin,thenumber o[chipsinthetimeintervalkTbisalson=kLc.Therefore, thefollowing discussion appliestobothblockcodesandconvolutional codes. Onemethodforimpressing thePNsequence onthetransmitted signalisto 700 DIGITAL COMMUNICATIONS alterdirectlythecodedbitsbymodulo-2 addition withthePNsequence.t Thus,eachcodedbitisalteredbyitsadditionwithabitfromthePNsequence. [fbirepresents theithbitofth~PNsequence and Ciisthecorresponding bit fromtheencoder, themodulo-2 sumis Oi=biEBCi (13-2-3) Hence,Gi=1ifeitherbi=1andCi=0orbi=0andCi=1;also,Oi=0ifeither bi=1andCi=1orb,=0andCi=O.WemaysaythatOi=0whenbi=Ciand ai=1whenbi",eei'Thesequence {aJismappedintoabinaryPSKsignalofthe forms(t)=±Re[g(t)e-'2"{.') according totheconvention (I)= {g(t-i1;)(OJ=0) g,-g(t-iTJ(Gj=1)(13-2-4) whereg(t)represents apulseofduration T...sandarbitrary shape. Themodulo-2 addition ofthecodedsequence {e;}andthesequence {b,} fromthePNgenerator mayalsoberepresented asamultiplication oftwo waveforms. Todemonstrate thispoint,suppose thattheelements ofthecoded sequence aremappedintoabinaryPSKsignalaccording totherelation Ci(t)=(2e,-1)g(t-iT...) Similarly, wedefineawaveform Pj(t)as Pi(t)=(2b;-1)p(t-iT.c)(13-2-5) (13-2-6) wherep(t)isarectangular pulseofduration 7;.Thentheequivalent lowpass transmitted signalcorresponding totheithcodedbitis gj(t)=Pi(t)Cj(t) =(2bi-1)(2cj-1)g(t-i7;) (13-2-7) Thissignalisidentical totheonegivenby(13-2-4), whichisobtained fromthe sequence {oJConsequently, modulo-2 addition ofthecodedbitswiththePN sequence followed byamapping thatyieldsabinaryPSKsignalisequivalent tomultiplying abinaryPSKsignalgenerated fromthecodedbitswitha sequence ofunitamplitude rectangular pulses,eachofduration T.c,andwitha polarity whichisdetermined fromthePNsequence according to(13-2-6). Although itiseasiertoimplement modulo-2 addition followed byPSK modulation insteadofwaveform multiplication, itisconvenient, forpurposes ofdemodulation, toconsider thetransmitted signalinthemultiplicative form tWhenfour-phase PSKisdesired,onePNsequence isaddedtotheinformation sequence carried onthein-phasesisnalcomponent andasecondPNsequence isaddedtotheinformation sequence carriedonthequadrature component. tnmanyPN-spread spectrum systems, thesamebinary information sequence isaddedtothetwoPNsequences toformthetwoquadrature components. Thus,afour-phase PSI{signalisgenerated withabinaryinformation stream. CHAPTER 13:SPREAD SPECTRUM SIGNALS FORD1G1TAL COMMUNI' ATlO~S 701 givenby(13-2-7). Afunctional blockdiagramofafour-phase PSKDSspread spectrum modulator isshowninFig.13-2-1(b). Thereceived equivalent lowpasssignalfortheithcodeelementist r;(t)=p;(t)c,(t)+z(t),i1;,'"I'"(i+1)1;, =(2b,-1)(2c;-l)g(t-i1;,)+z(l) (13-2-8) whereZ(I)represents theinterferenge orjamming signalcorrupting the information-bearing signal.Theinterference isassumed tobeastationary randomprocesswithzeromean. IfZ(I)isasample function fromacomplex-valued gaussian process, the optimum demodulator maybeimplemented eitherasafiltermatched tothe waveform g(l)orasacorrelator, asillustrated bytheblockdiagrams inFig. 13-2-2.Inthematched filterrealization, thesampled outputfromthematched filterismultiplied by2b;-I,whichisobtained fromthePNgenerator atthe "GURE 13-2-2 Possibledemodulator structures forPNspreadspectrum signals. r(1)Matched filter g.(T,-r)Sampler Chip.rate clod (n)PN seque"", generatorTo decoder PN!oignaf .geMralorCtIiJH8te clock. (b)Sampler)'jTo decoder r(t) Sampter Chip-rate doc. (e)To decoder tForsimplicity, weassumethatthechannel attenuation a=Iandthephaseshiftofthe channeliszero.Sincecoherent PSKdetection isassumed, anyarbitrary channelphaseshiftis compensated forinthedemodulation. 702 DIGITAL COMMt-:NICATIONS demodulator whenthePNgenerator isproperly synchronized. Since(2b,­ 1)'=1whenb,=0andb,=1,theeffectofthePNsequence onthereceived codedbitsisthusremoved. InFig.13-2-2,wealsoobservethatthecross-correlation canbeaccompl­ ishedineitheroneoftwoways.Thefirst,illustrated inFig.13-2-2(b), involves premultiplying r,(t)withthewaveform p,(t)generated fromtheoutputofthe PNgenerator andthencross-correlating withg*(t)andsampling theoutputin eachchipinterval. Thesecondmethod, illustrated inFig.13-2-2(e), involves cross-correlation withg*(t)first,sampling theoutput of thecorrelator and, then,multiplying thisoutputwith2b,-I,whichisobtained fromthePN generator. IfZ(t)isnotagaussian random process, thedemodulation methods illustrated inFig.13-2-2arenolongeroptimum. Nevertheless, wemaystilluse anyofthesethreedemodulator structures todemodulate thereceived signal. Whenthestatistical characteristics oftheinterference zit)areunknown a priori,thisiscertainly onepossibleapproach. Analternative method, whichis described later,utilizesanadaptive filterpriortothematched filteror correiatortosuppress narrowband interferenct:. Therationale forthissecond methodisalsodescribed later. InSection13-2·1,wederivetheerrorrateperformance oftheDSspread spectrum systeminthepresence ofwidebandandnarrowband interference. Thederivations arebasedontheassumption thatthedemodulator isanyof thethreeequivalent structures showninFig.13-2-2. 13-2-1ErrorRatePerformance oftheDecoder Lettheunquantized outputofthedemodulator bedenoted byYJ,1'"j"'"n. Firstweconsider alinearbinary(n,k)blockcodeand,without lossof generality, weassumethattheall-zerocodewordistransmitted. Adecoder thatemploys soft-decision decoding computes thecorrelation metrics n CM,=2:(2c'j-l)y;,i=1,2,...,2' j=l(13-2-9) wheree,)denotes thejthbitintheithcodeword.Thecorrelation metric corresponding totheall-zerocodewordis n CM.=2n~c+2:(2cI)-1)(2b)-l)v) j=l n =2n~c-2:(2bj-l)vj j=l(13-2-10) whereVj'1'"j"'"n.istheadditivenoisetermcorrupting thejthcodedbitand ~cisthechipenergy.Itisdefinedas Vj=Re{{ g*(t)Z[f+(j-l)7;,jdt}, j=I,2,..:,n(13-2-11) CHAPTER LlSPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 703 Similarly, tlIecorrelation metriccorresponding tocodewordemhaving weight Wmis (2W) nCMm=2~cn1-nm+~(2cmj-1)(2b j-l)vj(13-2-12) Following theprocedure usedinSection8-1-4,weshalldetermine the probability thatCMm>CM,.Thedifference betweenCM,andeMmis n =4~cwm-22:cmPb j-l)vj j=1(13-2-13) Sincethecodewordemhasweight Wm•thereareWmnonzerocomponents in thesummation ofnoisetermscontained in(13-2-13). Weshallassumethatthe minimum distance ofthecodeissufficiently largethatwecaninvokethe centrallimittheorem forthesummation ofnoisecomponents. Thisassumption isvalidforPNspreadspectrum signalsthathaveabandwidth expansion of20 ormore.tThus,thesummation ofnoisecomponents ismodeled asagaussian randomvariable. SinceE(2bj-1)=0andE(vj)=0,themeanofthesecond termin(13-2-13) isalsozero. Thevariance is nn <T~=42:2:Cmicmj£[(2bj -1)(2bi-1»)£(vivj) j~1;=1(13-2-14) Thesequence ofbinarydigitsfromtlIePNgenerator areassumed tobe uncorrelated. Hence, andE[(2b j-1)(2bi-1»)=Oij (13-2-15) (13-2-16) whereE(v2)isthesecondmomentofanyoneelementfromtheset{v).This moment iseasilyevaluated toyield E(Y)=rrg*(t)g(r)<f>,,(t -r)dtdr =[IG(fW 4.>,,(f)df (13-2-17) tTypically. thebandwidth expansion factorinaspreadspectrum signalisoftheorderof100 andhigher. 704 DIGITALCOMMUNICATIONS where ~,,(T)=!E[z*(t):z:(t +r)]istheautocorrelation function and¢>z,(f)is thepowerspectraldensityoftheinterference z(t). Weobserve thatwhentheinterference isspectrally flatwithinthe bandwidtht occupied bythetransmitted signal,i.e., ¢>,,(f)=JoIfI""'!W (13-2-18) thesecondmoment in(13-2-17) isE(v2)=2~.Jo.and,hence,thevariance of theinterference termin(13-2-16) becomes (13-2-19) (13-2-20)Inthiscase,theprobability thatD<0is ~- ((2'1:,)P2(m)=Q'J-.;;;w., Buttheenergypercodedbit'C"maybeexpressed intermsoftheenergyper infonnation bit'Chas k '1:,= -'ih=R"~hn Withthissubstitution, (13-2-20) becomes.------- (2'1:h)P2(m)=Q - R,w.,JlI(13-2-21) (13-2-22) where'Yh='ihlJoistheSNRperinformation bit.Finally,thecodeworderror probability maybeupper-bounded bytheunionboundas M Pm"'"2:Q(v'2'YhR,Wm) m=2(13-2-23) whereM=2'.Notethatthisexpression isidentical totheprobability ofacode worderrorforsoft-decision decoding ofalinearbinaryblockcodeinan AWGNchannel. Although wehaveconsidered abinaryblockcodeinthederivation given above,theprocedure issimilarforan(n,k)convolutional code.Theresultof suchaderivation isthefollowing upperboundontheequivalent biterror probability: 1~ Ph"""Ie2:!3dQ(v'2'YbR,d) (13-2-24) d=dftft Thesetofcoefficients {!3,,}isobtained fromanexpansion ofthederivative of thetransferfunctionT(D,N),asdescribed inSection8-2-3. Next,weconsider anarrowband interference centered atthecarrier(atd.c. tIfthebandwidth ofthebandpass channelisW,thatoftheequivalent low-pass channel is~W. (13-2-25) (13-2-26) (13-2-27)CHAPTER IJSPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 705 fortheequivalent lowpass signal).Wemayfixthetotal(average) jamming powertoJav=ioW,whereJoisthevalueofthepowerspectral densityofan equivalent wideband interference (jamming signal).Thenarrowband inter­ ferenceischaracterized bythepowerspectraldensity {Jav_JOW""lW <P,,(f)= WI-WI(If1~2I) o(IfI>!WI) wherew»WI' Substitution of (13-2-25) for<P,,(f)into(13-2-17) yields Jfw,nE(v')=~IG(fWdf WI~W1/2 ThevalueofE(v')depends onthespectralcharacteristics ofthepulseg(t).In thefollowing example, weconsider twospecialcases. Example 13-2·1 Suppose thatg(t)isarectangular pulseassllowninFig.13-2-3(a) and IG(f)I' isthecorresponding energydensity spectrum showninFig. 13-2-3(b). Forthenarrowband interference givenby(13-2-26), thevariance ofthetotalinterference is eT'=4wE(v2)m m =8%'cwmTelavfW"2('inTifTc)'df WI-W,121ifTc _8'6cwmJavf#12(SinlrX)2- --dx WI-(312lrX FlGURE 13-2·] Rectangular pulseanditsenergydensityspectrum. IGI/)I' 8(1) J2if/T,I--~ 0 T,32I0fI2:y T,T,T, T,T,T. lal Ibl 706 DIGlTAL COMMUNICATIONS 1.0 0.8 ff0.6 ~c ~0.4 ~ 0.2 FIGURE 13-2-4 Plotofthevalueoftheintegralin(13-2-27).oIL---;O"'.2:--0-:.4:---::0.':-6--:0:'::.8--;:-'1.0""'" ~ ~ where {3=W,T,..Figure13-2-4illustrates thevalueofthisintegral for o<S:{3<s:I.Weobservethatthevalueoftheintegral isupper-bounded by WIT,.Hence,IT;"'"8'€CWmTelav. InIhelimitasWIbecomes zero,Iheinterference becomes animpulseat thecarrier.Inthiscasetheinterference isapurefrequency toneanditis usuallycalledaCWjamming signal.Thepowerspectraldensityis (13-2-28) andthecorresponding variance forthedecision variableD=CM!-CMmis IT;"=4wmI"IG(O)12 (13-2-29) Theprobability ofacodeworderrorforCWjamming isupper-bounded as PM<s:,~,Q(l~~~wm) (13-2-30) ButIf:c=Rclf:b.Furthermore, Tc~!/WandJa,/W=lo.Therefore (13-2-30) maybeexpressed as (13-2-31) whichistheresultobtained previously forbroadband interference. This resultindicates thataCWjammerhasthesameeffectonperformance asan equivalent broadband jammer. Thisequivalence isdiscussed furtherbelow. Example 13·2-2 Letusdetermine theperformance oftheDSspreadspectrum systeminthe presence ofaCWjammerofaveragepowerIavwhenthetransmitted signal pulseg(t)isone-half cycleofasinusoidasillustrated inFig.13-2-5,i.e., I!:lf:,mg(t)=-Tsin-,0<s:t'"Tc (13-2-32) cT, CHAPTER D:SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 107 g(t) o FIGURE 13·2-5 Asinusoidal signalpulse. Thevariance oftheinterference ofthispulseis IT;"=4wmJ.,IG(O)12 64 =r~cTcla.wm Hence,theupperboundonthecodeworderrorprobability is(13-2·33) (13-2-34) WeobservethatIheperformance obtairted withthispulseis0.9dBbetter thanthatobtained witharectangular pulse.Recallthatthispulseshape whenusedinoffsetQPSKresultsinanMSKsignal.MSKmodulation is frequently usedinDSspreadspectrum systems. lbeProcessing GainandtheJamming Margin Aninteresting interpreta­ tionoftheperformance characteristics fortheDSspreadspectrum signalis obtained byexpressing thesignalenergyperbit~bintermsoftheaverage power.Thatis,~b=PayTbowherePa,istheaveragesignalpowerandTbisthe bitinterval. Letusconsider theperformance obtained inthepresence ofCW jamming fortherectangular pulsetreated inExample 13-2-1.Whenwe substitute for1\'bandJointo(13-2-31), weobtain (13-2-35) whereL,isthenumberofchipsperinformation bitandPa.!J..isthe signal-to-jamming powerratio. Anidentical resultisobtained withbroadband jamming forwhichthe performance isgivenby(13-2-23). Forthesignalenergyperbit,wehave (13-2-36) (13-2-37) (13-2-38) (13-2-40)708 DIGITAL COMMUNICATIONS whereRistheinformation rateinbits/s.Thepowerspectraldensityforthe jamming signalmaybeexpressed as 1._J.. 0-W Usingtherelation in(13-2-36) and(13-2-37), theratio"CblJomaybe expressed as 'ibPavlRWIR-=--=-- JoJavlWJ,vIPav TheratioJav/Payisthejamming-to-signal powerratio,whichisusually greaterthanunity.TheratioWIR=TblTe=B,=L,isjustthebandwidth expansion factor,or,equivalently, thenumberofchipsperinformation bit. Thisratioisusuallycalledtheprocessing gainoftheOSspreadspectrum system.Itrepresents theadvantage gainedoverthejammerthatisobtained by expanding thebandwidth ofthetransmitted signal.Ifweinterpret 'iblJoasthe SNRrequired toachieveaspecified errorrateperformance andWIRasthe available baftdwidth expansion factor,theratioJavlP,viscalledthejamming marginoftheOSspreadspectrum system.Inotherwords,thejamming margin isthelargestvaluethattheratioJavlPavcantakeandstillsatisfythespecified errorprobability. Theperformance ofasoft-decision decoder foralinear(n,k)binarycode, expressed intermsoftheprocessing gainandthejamming margin,is (13-2-39) Inaddition totheprocessing gainWIRandJ.vIPa"weobserve thatthe performance depends onathirdfactor,namely,Rcwm•Thisfactoristhecoding gain.AlowerboundonthisfactorisRcdm;n.Thusthejamming margin achieved bytheOSspreadspectrum signaldependsontheprocessing gainand thecodinggain. Unooded DSSpreadSpednuD Signals Theperformance resultsgiven aboveforOSspreadspectrum signalsgenerated bymeansofan(n,k)cod"e maybespecialized toatrivialtypeofcode,namely,abinaryrepetition code. Forthiscase,k=1andtheweightofthenonzerocodewordisw=n.Thus, Rcw=1and,hence,theperformance ofthebinarysignaling systemreducesto P2=Q(~2J:b) =Q(~~a:~~) CHAPTER IJ:SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 709 Notethatthetrivial(repetition) codegivesnocodinggain.Itdoesresultin aprocessing gainofW/R. Example 13-2-3 Suppose thatwewishtoachieveanerrorrateperformance of10-6orless withanuncoded DSspreadspectrum system.Theavailable bandwidth expansion factorisW/R=1000.Letusdetermine thejamming margin. The'lb/JOrequired toachieveabiterrorprobability of10-6with uncoded binaryPSKis10.5dB.Theprocessing gainis1010glO1000=30dB. H~ncethemaximum jamming-to-signal powerthatcanbetolerated, i.e.,the jamming margin,is Jav10log,o-p=30-10.5=19.5dB av Sincethisisthejamming marginaclIieved witlIanuncoded OSspread spectrum system,itmaybeincreased bycodingtheinformation sequence. Thereisanotherwaytoviewthemodulation anddemodulation processes fortheuncoded (repetition code)DSspreadspectrum system. Atthe modulator, thesignalwaveform generated bytherepetition codewith rectangular pulses,forexample, isidentical toaunitamplitude rectangular pulses(t)ofdurationT"oritsnegative, depending onwhethertheinformation bitis1or0,respectively. Thismaybeseenfrom(13-2-7), wherethecoded chips{e;}withinasingleinformation bitareeitherallIsorOs.ThePN sequence multiplies eithers(t)or-s(t).Thus,whentheinformation bitisa1, theLcPNchipsgenerated bythePNgenerator aretransmitted withthesame polarity. Ontheotherhand,whentheinformation bitisa0,theL,PNchips whenmultiplied by-s(t)arereversed inpolarity. Thedemodulator fortherepetition code,implemented asacorrelator, is illustrated inFig.13-2-6.Weobserve thattheintegration interval inthe integrator isthebitinterval 7;,.Thus,thedecoderfortherepetition codeis eliminated anditsfunction issubsumed inthedemodulator. Nowletusqualitatively assesstheeffectofthis-demodulation processon FlGURE 13-Z-II Correlation-type demodulator fora repetition code.r(1)I;'()d,---x", Sampler 1 PN Symbol- sequenceHCIIiP""l1Itemeclockeenemor clook 710 DIGITAL COMMUNICATIONS theinterference z(t).Themultiplication ofz(t)bytheoutputofthePN generator, whichisexpressed as wet)=2:(2b,-l)p(t-ire) yields vet)=w(t)z(t) Thewaveforms wet)andz(t)arestatistically independent randomprocesses eachwithzeromeanandautocorrelation functions tPww(f)andcl>zz(f), respectively. Theproductvet)isalsoarandomprocesshaving l\Ilautocorrela­ tionfunction equaltotheproductofcl>ww(T)with4>zzCf).Hence,thepower spectraldensityoftheprocessvet)isequaltotheconvolution ofthepower spectraldensityofwet)withthepowerspectraldensityofz(t). Theeffectofconvolving thetwospectra istospreadthepowerin bandwidth. Sincethebandwidth ofwet)occupies theavailable channel bandwith W,theresultofconvolution ofthetwospectraistospreadthepower spectral densityofz(t)overthefrequency bandofwidthW.Ifz(t)isa narrowband process,i.e.,itspowerspectraldensityhasa-widthmuchlessthan W,thepowerspectral densityoftheprocessvCr)willoccupyabandwidth equaltoatleastW. Theintegrator usedinthecross-correlation showninFig.13-2-6hasa bandwidth approximately equaltol/T•.Sincel/T.«W,onlyafractionofthe totalinterference powerappearsattheoutputofthecorrelator. Thisfractionis approximately equaltotheratioofbandwidths 1/TotoW.Thatis, 1/To=_1_=Tc=..!. WWT.T.Le Inotherwords,themultiplication oftheinterference withthesignalfromthe PNgenerator spreadstheinterference tothesignalbandwidth W,andthe narrowband integration following themultiplication seesonlythefraction1/Le ofthetotalinterference. Thus,theperformance oftheuncoded DSspread spectrum systemisenhanced bytheprocessing gainLe• Linelll'CodeConcatenated withaBinaryRepetition CodeAsillustrated above,abinaryrepetition codeprovides amarginagainstaninterference or jamming signalbutyieldsnocodinggain.Toobtainanimprovement in performance, wemayusealinear(nI'k)blockorconvolutional code,where n!""n=kLe•Onepossibi.:ty istoselectn!<nandtorepeateachcodebitn, timessuchthatn=nln,.Thus,wecanconstruct alinear(n"k)codeby concatenating the(nl'k)codewithabinary(n"1)repetition code.Thismay beviewedasatrivialformofcodeconcatenation wheretheoutercodeisthe (n!>k)c~andtheinnercodeistherepetition code. Sincetherepetition codeyieldsnocodinggain,thecodinggainachieved by thecombined codemustreducetothatachieved bythe(n!,k)outercode.It CHAPTER 13:SPREAD SPECTRl:M SIGNALS fORDIGITAL rOMMlJNICATIONS 711 isdemonstrated thatthisisindeedthecase.Thecodinggainoftheoverall combined codeis Buttheweights {w",}forthecombined codemaybeexpressed as where {w~~}aretheweightsoftheoutercode.Therefore, thecodinggainofthe combined codeis whichisjustthecodinggainobtained fromtheoutercode. Acodinggainisalsoachieved ifthe(n"k)outercodeisdecoded using harddecisions. Theprobability ofabiterrorobtained withthe(n"1) repetition code(basedonsoft-decision decoding) is P=Q(~)=Q~~:R~) =Q(!2W/RRO) (13-2-42)'/JavlPayc, Thenthecodeworderrorprobability foralinear(n"k)blockcodeis upper-bounded as where(=U(dmin-l)J,oras M PM";;2:[4p(l- p)r~,r2 "1=2(13-2-43) (13-2-44) wherethelatterisaChernoff bound.Foran(n"k)binaryconvolutional code. theupperboundonthebiterrorprobability is x Ph";;L(3"P2(d) clo;-dtn:o:(13-2-45) whereP2(d)isdefinedby(8-2-28)forodddandby(8-2-29)forevend. Concatenated CodingforDSSpreadSpedrum Systems Itisapparent fromtheabovediscussion thatanimprovement inperformance can be obtained byreplacing therepetition codebyamorepowerful codethatwill 712 DIGITAL CO....UNICATIONS yieldacodinggaininaddition totheprocessing gain.Basicafly, theobjective in aDSspreadspectrum systemistoconstruct along,low·ratecodehavinga largeminimum distance. Thismaybebestaccomplisbedby""using code concatenation. WhenbinaryPSKisusedinconjunction withDSspread spectrum, theelements ofaconcatenated codewordmustbeexpressed in binaryform. Bestperformance isobtained whensoft-decision decoding isusedonboth theinnerandoutercodes.However, analternative, whichusuallyresultsin reducedcomplexity forthedecoder, istoemploysoft-decision decoding onthe innercodeandhard-decision decoding ontheoutercode.Theexpressions for theerrorrateperformance ofthesedecoding schemes depend,inpart,onthe typeofcodes(blockorconvolutional) selectedfortheinnerandoutercodes. Forexample, theconcatenation oftwoblockcodesmaybeviewedasan overalllongbinary(n.k)blockcodehavingaperformance givenby(13-2-39). Theperformance ofothercodecombinations mayalsobereadilyderived. For thesakeofbrevity,weshallnotconsider suchcodecombinations. 13-2·2SomeApplications ofDSSpreadSpectrum Signals Inthissubsection, weshallbrieflyconsider theuseofcodedDSspread spectrum signalsforthreespecific applications. Oneisconcerned with providing immunity againstajamming signal.Inthesecond,acommunication signalishiddeninthebackground noisebytransmitting thesignalatavery lowpowerlevel.Thethirdapplication isconcerned withaccommodating a number ofsimultaneous signaltransmissions onthesamechannel, i.e., CDMA. Antijamming Application InSection13-2-1,wederivedtheerrorrate performance foraDSspreadspectrum signalinthepresence ofeither:a narrowbandorawidebandjamming signal.Asexamples toillustrate the performance ofadigitalcommunications systeminthepresence ofajamming signal,weshallselectthreecodes.OneistheGolay(24,12),whichis characterized bytheweightdistribution giveninTable8-1-1andhasa minimum distance dm;n=8.Thesecondcodeisanexpurgated Golay(24,11) obtained byselecting 2048codewordsofconstant weight12.Ofcoursethis expurgated codeisnonlinear. Thesetwocodeswillbeusedinconjunction with arepetition code.Thethirdcodetobeconsidered isamaximum-length shift-register code. Theerrorrateperformance oftheGolay(24,12)withsoft-decision decoding is PM';;[759Q(~8W//R) +2576Q( 12W/R) JavPav Ja.lPav 759Q(16W/R) (24W/R)]+ Jay'Pay+QJavlPav(13-2-46) CHAPTER 13:S~READ SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 713 whereW/Ristheprocessing gainandJ.vIp••isthejamming margin.Since "'""'"2=12WfRand",'"24,eachcodedbitis,ineffect,repeated"2'" W/2Rtimes.Forexample, ifW/R=100(aprocessing gainof20dB),the blocklengthoftherepetition codeis"2'"50. Uhard-decision decoding isused,theprobability oferrorforacodedbitis p'"Q( (13-2-47) (13-2-48)andthecorresponding probability ofacodeworderrorisupper-bounded as PM';;;J4(~)pn'(l- p)24-m Asanalternative, wemayusetheChernoff boundforhard-decision decoding, whichis PM""759[4p(l- pW+2576[4p(l- p)]6 +759[4p(l- p)]8+[4p(l_p)]'2 (13-2-49) Figure13-2-7illustrates theperformance oftheGolay(24,12)asafunctionof thejamming marginJ.vIp...withtheprocessing gainasaparameter. The Chernoff boundwasusedtocompute theerrorprobability forhard-decision decoding. Theerrorprobability forsoft-decision decoding isdominated bythe term 759Q(.J.8W/R)i.vlp•• andthatforhard-decision decoding isdominated bytheterm759[4p(1 -P)]4. Hence,thecodinggainforsoft-decision decodingtisatmost10log4=6dB. Wenotethatthetwocurvescorresponding toW/R=1000(30dB) are identical inshapetotheonesforW/R'"100(20dB),exceptthatthelatterare shiftedby10dBtotherightrelativetotheformer.Thisshiftissimplythe difference inprocessing gainbetween thesetwoDSspreadspectrum signals. Theerrorrateperformance oftheexpurgated Golay(24,11)isupper­ bounded as forsoft-decision decoding andas:tllWfR) i.vlp••(13-2-50) (13-2-51) tThecodinggainislessthan6dBduetothemultiplicative factorof159,whichincreases the errorprobability relativetotheperformance ofthebinaryunooded system.*Weremindtbereaderthattheunionboundisnotverytightforlargesignalsets. 714 DIGITAL COMMUNiCATIONS 12 14 16 18 20 22 24G(24.12) PC=20dB \\... G(24,12)\·1\-yG=20dB\··· · \\•·· · •· · • • G(24.II)1\•· ··PG=20dB ,•· •·• \\, G\24.12)\•• .·, \PG=30dB,:\,,·• , , •.·GI24.12)• \\, ,·PG=30dB, ,• • ,,·\.\•, , , ,,\,• ,··\, •\,·, , . , \, PG=20dB ,,-- Soft-decison G(24.II),decoding, ,~- -•.Hard-decison ·decoding,,PC=processing gairJ···,\252 10-­ 2610-'10-' 5 2 10-2 5 .:2 ~10-3"1:05• ~•'02 .£ ]1(j-4 ~"-5 Jamming marginJ.)Pn FlGURE 13-2-7 Performance ofGolaycodesusedinDSspread'pectrum signal. forhard-decision decoding, wherepisgivenas (13-2-52) Theperformance characteristics ofthiscodearealsoplottedinFig.13-2-7for WIR=100.Weobserve thatthisexpurgated Golay(24,11)codeperforms about1dBbetterthantheGolay(24,12)code. Insteadofusingablockcodeconcatenated withalow-rate(1/n2)repetition code,letusconsider usingasinglelow·rate code.Aparticularly suitablesetof low-rate codesisthesetofmaximum-length shift-register codesdescribed in Section8-1-3.WerecallthaIforIhissetofcodes, (n,k)=(2m_l,m) dmin=2m-1(13-2-53) CHAPTER P:SPREAD SPECTRUM SIG~ALS FORDIGITAL COMMUNICATIONS 715 Allcodewordsexcepttheall-zerowordhaveanidentical weightof2m -I Hence,theerrorrateforsoft-decision decoding isupper-bounded ast 2WIRm2m-,)----- JavlPa,2'"-I (WIRm2"'-'),,;2mexp----m-­ favlP"2-I'(13-2-54) Formoderate valuesofm,R,dmin=!mand,hence,(13-2-54) maybeexpressed as _W_IR_m),,;2mexp(__m_W-,-1R_) fa,IP., 2J.,l'a,(13-2-55) Hence,thecodinggainisatmost10log1m. Forexample, ifweselectm=10thenn=210-1=1023.Sincen=kWIR~ mWIR,itfollowsthatWIR=102.Thus,wehaveaprocessing gainofabout 20dBandacodinggainof7dB.Thisperformance iscomparable tothat obtained withtheexpurgated Golay(24,11)code.Highercodinggainscanbe achieved withlargervaluesofm. Ifhard-decision decoding isusedforthemaximum-length shift-register codes,theerrorrateisupper-bounded bytheChernoff boundas wherepisgivenas(13-2-56) p=Q(2WIR) (V2W1R m ) --R ~Q----f,jP., C favlp..2m-1(13-2-57) Form=10,thecodeworderrorratePMiscomparable tothatobtained with theexpurgated Golay(24,11)codeforhard-decision decoding. Theresltsgivenaboveillustrate theperformance thatcanbeobtained with asinglelevelofcoding.Greatercodinggainscanbeachieved withconcaten­ atedcodes. tTheM=rwaveforms generated byamaximum·length shift·register codeform3simplexset (seeProblem 8-13).Theexactexpression fortheerrorprobability, giveninSection5-2-4,maybe used{orlargevaluesofM,where(heunionboundbecomes \ieryloose. 716DIGITAL COMMUNICATiONS Low-Detedlbility SipalTnmsmissioD Inthisapplication, thesignalis purposely' transmitted atIiverylowpowerlevelrelativetothebackground channelnoiseandthermalnoisethatisgenerated inthefrontendofthe receiver.IftheOSspreadspectrum signaloccupies abandwidth Wandthe spectraldensityoftheadditivenoiseisNoW1Hz,theaveragenoisepowerin thebandwidth WisNov=WNo. Theaveragereceived signalpowerattheintended receiver isPav•Ifwewish tohidethepresence ofthesignalfromreceivers thatareinthevicinityofthe intended receiver, thesignalistransmitted atalowpowerlevelsuchthat PavlNav«1.Theintended receivercanrecovertheinformation-bearing signal withtheaidoftheprocessing gainandthecodinggain.However, anyother receiver thathasnopriorknowledge ofthePNse.quence isunabletotake advantage oftheprocessing gainandthecodinggain.Hence,thepresence of theinformation-bearing signalisdifficulttodetect.Wesaythatthesignalhasa lowprobability ofbeingintercepted (LPI)anditiscalledanLPlsignal. Theprobability oferrorresultsgiveninSection 13-2-1alsoapplytothe demodulation anddecoding ofLPIsignalsattheintended receiver. CodeDivision Multiple AccessTheenhancement inperformance ob­ tainedfromaOSspreadspectrum signalthrough theprocessing gainand ~inggaincanbeusedtoenablemanyOSspreadspectrum signalstooccupy thesamechannelbandwidth provided thateachsignalhasitsowndistinctPN sequence. Thus,itispossible tohaveseveraluserstransmit messages simultaneously overthesamechannel bandwidth. Thistypeofdigital communication inwhicheachuser(transmitter-receiver pair)hasadistinctPN codefortransmitting overacommon channelbandwidth iscalledeithercode divisionmultipleaccess(COMA) orspreadspectrum multiple access(SSMA). Inthedemodulation ofeachPNsignal,thesignalsfromtheother simultaneous usersofthechannelappearasanadditiveinterference. Thelevel ofinterference varies,depending onthenumberofusersatanygiventime.A majoradvantage ofCOMAisthatalargenumberofuserscanbeaccommod­ atedifeachtransmits messages forashortperiodoftime.Insuchamultiple accesssystem,itisrelatively easyeithertoaddnewusersortodecrease the numberofuserswithoutdisrupting thesystem. Letusdetermine thenumberofsimultaneous signalsthatcanbesupported inaCOMAsystem.t Forsimplicity, weassumethatallsignalshaveidentical average powers., Thus,ifthereareNusimultaneous users,thedesired signal-to-noise inteJ1erence powerratioatagivenreceiveris 1 Nu-1(13-2-58) tInthissectionlbeinterference fromotheruse",istreatedasarandom prOCCS$.Thisisthe caseifthereisnocooperation amongtheusers.InChapter 15weconsiderCDMAtransmission in whichinterference fromother\lienisknownandissuppressed bythereceiver. CHAPTER 11SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 717 Hence,theperfonnance forsoft-decision decoding upper-bounded as M(IZW/R) ( PM";;m2;2Q-VN"-1R,wm';;(M-1)Qatthegivenreceiver is 2W/RRd.) N.-1cmUl• (13-2-59) Inthiscase,wehaveassumed thattheinterference fromotherusersis gaussian. Asanexample, suppose thatthedesiredlevelofperformance (error probability of10-6)isachieved when W/R--Rcdmin=20N.-I Thenthemaximum numberofusersthatcanbesupported intheCDMA systemis (13-2-60) IfWIR=100andRcdmin=4,asobtained withtheGolay(24,12)code,the maximum number isIV"=21.IfWIR=1000andR,dm'n=4,thisnumber becomes N"=20l. Indetermining themaximum numberofsimultaneous usersofthechannel, wehaveimplicitly assumed thatthePNcodesequences aremutually orthogonal andtheinterference fromotherusersaddsonapowerbasisonly. However, orthogonality amonganumberofPNcodesequences isnoteasily achieved, especially ifthenumberofPNcodesequences required islarge.In fact,theselection ofagoodsetofPNsequences foraCDMAsystemisan important problem thathasreceived considerable attention inthetechnical literature. Weshallbrieflydiscussthisproblem inSection13-2-3. 13-2-3EifertofPulsedInterference onDSSpreadSpectrum Systems Thusfar,wehaveconsidered theeffectofcontinuous interference orjamming onaDSspreadspectrum signal.Wehaveobserved thattheprocessing gain andcodinggainprovideameansforovercoming thedetrimental effectsofthis typeofinterference. However, thereisajamming threatthathasadramatic effectontheperformance ofaDSspreadspectrum system.Thatjamming signalconsistsofpulsesofspectrally flatnoisethatcoverstheentiresignal bandwidth W.Thisisusuallycalledpulsedinterference orpartial-timejamming. Suppose thejammerhasanaveragepowerJa,il\thesignalbandwidth W. HenceJo=J.v1W.Insteadoftransmitting continuously, thejammertransmits pulsesatapowerJavlafora%ofthetime,i.e.,theprobability thatthe jammeristransmitting atagiveninstantisa.Forsimplicity, weassumethat 718 DIGITAL COMMUNICATIONS aninterference pulsespansanintegralnumberofsignaling intervals and,thus, itaffectsanintegralnumberofbits.Whenthejammerisnottransmitting, the transmitted bitsareassumed tobereceived error-free, andwhenthejammeris transmitting, theprobability oferrorforanuncoded DSspreadspectrum systemisQ(V2a't.IJ o).Hence,theaverageprobability ofabiterroris 2aWIR) J.,/p. v(13-2-61) Thejammer selectsthedutycycleatomaximize theerrorprobability. On differentiating (13-2-61) withrespecttoa,wefindthattheworst-case pulse jamming occurswhen ('f",IJo<0.71)('f",IJo>0.71){0.71*-"'If.('t.IJo;;;'O.71)a-fOb0 1('t.llo<0.71) andthecorresponding errorprobability is {O'083=0.083J.vIp.. P_'f,.IJoWIR 2-Q(.,pWIR) J.vlp. v(13-2-62) (13-2-63) Theerrorrateperformance givenby(13-2-61) fora=1.0,0.1,and0.01 alongwiththeworst-case performance basedona*isplottedinFig.13-2-8. 10' ,,,,,,,, ,,,, (l=0.01......,10-'10-1.... ....,....worst-case pulsejamming(a=a·) ,,, nGURE lJ.z..lI Performance ofDSbinaryPSKwithpulse jamming.10"'--~_"""_~......I~_......J'--~~oSWIIW~~~ .!.lJo(dB) .. CHAPTER D:SPREAD SPECTRLM SIG~ALS FORDIGITAL COMMUNICATIONS 719 Bycomparing theerrorrateforcontinuous gaussian noisejamming with worst-case pulsejamming, weobservealargedifference inperformance, which isapproximately 40dBatanerrorrateof10-6 • Weshouldpointoutthattheaboveanalysisapplieswhenthejammerpulse duration isequaltoorgreaterthanthebitduration. Inaddition, weshould indicatethatpractical considerations mayprohibitttiejammerfromachieving highpeakpower(smallvaluesofa).Nevertheless, theerrorprobability given by(13-2-63) servesasanupperboundontheperformance oftheuncoded binaryPSKinworst-case pulsejamming. Clearly,theperformance oftheDS spreadspectrum systeminthepresence ofsuchjamming isextremely poor. IfwesimplyaddcodingtotheDSspreadspectrum system,theimprove­ mentovertheuncodedsystemisthecodinggain.Thus,'tblloisreducedbythe codinggain,whichinmostcasesislimitedtolessthan10dB.Thereasonfor thepoorperformance isthatthejamming signalpulseduration maybe selected toaffectmanyconsecutive codedbitswhenthejamming signalis turnedon.Consequently, thecodeworderrorprobability ishighduetothe burstcharacteristics ofthejammer. Inordertoimprove theperformance, weshouldinterleave thecodedbits priortotransmission overthechannel. Theeffectoftheinterleaving, as discussed inSection8-1-9,istomakethecodedbitsthatarehitbythejammer statistically independent. Theblockdiagram ofthedigitalcommunication systemthatincludes interleaving/deinterieaving isshowninFig.13-2-9.Alsoshownisthepos­ sibilitythatthereceiver knowsthejammerstate,i.e.,thatitknowswhen thejammer isonoroff.Knowledge ofthejammer state(calledside information) issometimes available fromchannel measurements ofnoise powerlevelsinadjacent frequency bands.Inourtreatment, weconsider two FIGURE 13-2·9 BlockdiagramofAJcommunication 'ystem. Dala DataEncoder DecoderPN generator Cl\aoncl 720 DIGITAL CO~Jl,nINICATIONS extreme cases,namely, noknowledge ofthe.jammer stateorcomplete knowledge ofthejammer state.Inanycase,therandom variable !: represertting thejammerstateischaracterized bytheprobabilities P(;=1)=a,P(;=0)=1-a Whenthejammer ison,thechannel ismodeled asanAWONwithpower spectraldensityNo=Jola=J.JaW; andwhenthejammerisoff,thereisno noiseinthechannel. Knowledge ofthejammerstateimpliesthatthedecoder knowswhenI:=1andwhenI:=0,andusesthisinformation inthe computation ofthecorrelation metrics.Forexample, thedecoder mayweight thedemodulator outputforeachcodedbitbythereciprocal ofthenoisepower levelintheinterval.Alternatively. thedecodermaygivezeroweight(erasure) toajammedbit. First,letusconsider theeffectofjamming withoutknowledge ofthejammer state.Theinterleaver/deinterleaver pairisassumed toresultinstatistically independent jammerhitsofthecodedbits.Asanexampleoftheperformance achieved withcoding,wecitetheperformance resultsfromthepaperofMartin andMcAdam (1980).Theretheperformance ofbinaryconvolutional codesis evaluated forworst-case pulsejamming. Bothhardandsoft-decision Viterbi decoding areconsidered. Softdecisions areobtained byquantizing the demodulator outputtoeightlevels.Forthispurpose, auniform quantizer is usedforwhichthethreshold spacing isoptimized forthepulsejammernoise level.Thequantizer playstheimportant roleoflimiting thesizeofthe demodulator outputwhenthepulsejammer ison.Thelimitingactionensures thatanyhitonacodedbitdoesnotheavilybiasthecorresponding path metrics. Theoptimum dutycycleforthepulsejammer inthecodedsystemis generally inversely proportional totheSNR,butitsvalueisdifferent fromthat givenby(13-2-62) fortheuncoded system.Figure13-2-10illustrates graphi· callytheoptimaljammerdutycycleforbothhard-andsoft-decision decoding oftherate1/2convolutional codes.Thecorresponding errorrateresultsfor thisworst-case pulsejammerareillustrated inFigs13-2-11and13-2-12forrate 1/2codeswithconstraint lengths3""K""9.Forexample, notethatat g=10-6,theK=7convolutional codewithsoft·decision decoding requires 'If.IJo=7.6dB,whereashard-decision decoding requires 'lfblJo=ll.7dB.This 4.1dBdifference inSNRisrelatively large.Withcontinuous gaussian noise, thecorrespondinB SNRsforanerrorrateof10-6are5dBforsoft-decision decoding and7dBforhard·decision decoding. Hence,theworst-caseplllse jammerhasdegraded theperformance by2.6dBforsoft-decision decoding andby4.7dBforhard-decision decoding. TheselevelsofdeBradation increase astheconstraint lengthoftheconvolutional codeisdecreased. Theimportant point,however, isthatthelossinSNRduetojamming hasbeenreducedfrom 40dBfortheuncaded systemtolessthan5dBforthecodedsystembasedon aK=7,rate1/2convolutional code. CHAPTER D:SPREAD SPECTRUM SIGNALS FORDiGiTAL COMMUNlCAT10!'lS 711 10-1Pulsejamming Binaryphaseshifakeying 51015202530 ",po(dB)FIGURE 13-2-10 Optimaldutycycleforpulsejammer.[From MartinandMcAdam (/980).©1980IEEE.)10-'L--,_-,-_..L._-'-_'--'_~ -50 Ill-'L_-'-_-'-_~-'-~l-.J...._h._---, 5 7 9 II13 15 1719 ..!',po(dB)Asimplermelhod forevalUating theperformance ofa munication systemistousethecutoffrateparameter Ro OmuraandLevitt(1982).Forexample, withbinary-coded cutoffratemaybeexpressed as Ro=1-log(1 +Do) 10-.1 I<r' ..< g • :E•IO-s '0 .£ i Ilt" "CURE 13-2-11 Performance ofrale1/2convolutional codes wilhhard-decision Viterbidecoding i>inary PSKwithoptimalpulsejamming. [From MartinandMcAdam (1980).©1980IEEE.IcodedAJcom­ asproposed by modulation, the (13-2-64) Unionbound OpIimalpulsejamming Binaryphaseshiftkeying RaleIl2convoluional code wilhVilerbidecoding Harddecisions K=3 K=4 K=5 K=6 K=7 K=8 K=9 lIT'"-----'_-'-----" ......--"-.....................,,-.1--"- __ l 4 6 g1012 14 16 6,1J,<dB)722 DIGITAL COMMUNICATIONS FIGURE U-Z·U Performance ofraleli2convolulional codes withsofl·decision Vilerbideeoding binary PSKwilhoptimalpulsejamming. [From MartinandMcAdam (1980).©1980IEEE.]10-] 10-< ..' ~ ~ 1l ~)(r''"5i 10-<Unionbound Optimalpulsejamming Binaryph'"shiftkeying RaIe112cOilvoluional code withViterbidecc:ding Softdecisions. ~K=3 K=4 \-'....-''t- K=S K=6 K=7 K=g K=9 wherethefactorDadepends onthechannelnoisecharacteristics andthe decoderprocessing. RecallthatforbinaryPSKinanAWONchanneland soft-decision decoding, Do:::;;e-~/N(J where ~,istheenergypercodedbit;andforhard-decision decoding, Dn=V4p(1-p)(13-2-65) (13-2-66) wherepistheprobability ofacodedbiterror.Here,wehaveNo'"10, ForacodedbinaryPSK,withpulsejamming, OmuraandLevitt(1982)have shownthat forsoft-decision decoding with knowledge ofjammerstate (13-2-67) Oa=min{[aexp(A2'lcNola) +1-a1exp(-2A"%',)} A~O forsoft-decision decoding with noknowledge ofjammerstate (13-2-68) Oa=aV4p(1-p)forhard-decision decoding with knowledge ofthejammerstate (13-2-69) Da=V4ap(1-ap)forhard-decision decoding with noknowledge ofthejammerstate(13-2-70) CHAPTER JJ:SPREAD SPECfRUM SIGNALS FORDIGITAL COMMUNI('ATlONS 723 1.0 0.9 "-0.8 '"0.7 ~e0.6 ",'0.5 ~0.4 '"0O.lEu0.1 0.1 (I -20-16-\2-8-4 0• A,./No(dB) Key (0'1Sofl-decision decoding inAWGN(a;2"f) (I)Soft-decision willijammerslaleinformation (2)Hard-decision withjammerstateinformation (3)Soft-decision wilhnojammerslateinformation (4)Kard..dedsiOl\withnojammerstateinfonnalion flGURE 13-2-13 CutoffrateforcodedDSbinaryPSKmodulation. [FromOmuraandLevi"(1982)©1982IEEEJ wheretheprobability oferrorforhard-decision decoding ofbinaryPSKis ThegraphsforRoasafunction of'lfclNoareillustrated inFig.13-2-13for thecasesgivenabove.Notethatthesegraphsrepresent thecutoffrateforthe worst-case valueofa=a·thatmaximizes Da(minimizes Ro)fOTeachvalueof 'If,1No.Furthermore, notethatwithsoft-decision decoding andnoknowledge ofthejammer state,Ro=O.Thissituation resultsfromthefactthatthe demodulator outputisnotquantized. ThegraphsinFig.13-2-13maybeusedtoevaluate theperformance of codedsystems. Todemonstrate theprocedure, suppose thatwewishto determine theSNRrequired toachieveanerrorprobability of10-6withcoded binaryPSKinworst-case pulsejamming. Tobespecific, weassumethatwe havearate1/2,K=7convolutional code.Webeginwiththeperformance of therate1/2,K=7convolutional codewithsoft-decision decoding inan AWONchannel. AtP2=10-6,theSNRrequired isfoundfromFig.8-2-21to be 'i.INo=5dB Sincethecodeisrate1/2,wehave 3dBlOdB 'lfe5dB-=Jo724 DIGITAL COMMUNICATIONS Now,wegotothegraphsinFig.13-2-13andfindthatfortheAWGNchannel (reference system)with'lfe/No=2dB,thecorresponding valueofthecutolf rateis Ro=0.74bits/symbol Ifwehaveanotherchannelwithdifferent noisecharacteristics (a worst-case pulsenoisechannel) butwiththesamevalueofthecutoffrateRD.thenthe upperboundonthebiterrorprobability isthesame,i.e.,10-6inthiscase. Consequently, wecanusethisratetodetermine theSNRrequired forthe worst-case pulsejammerchannel. FromthegraphsinFig.13-2-13,wefindthat forhard-decision decoding with noknowledge ofjammerstate forhard-decision decoding with knowledge ofjammerstate forsoft-decision decoding with knowledge ofjammerstate Therefore, thecorresponding valuesof'lfb/J.fortherate1/2,K=7convolu­ tionalare13,8,and6dB,respectively. Thisgeneralapproach maybeusedtogenerate errorrategraphsforcoded binarysignalsinaworst-case pulsejamming channelbyusingcorresponding errorrategraphsfortheAWGNchannel. Theapproach wedescribe aboveis easilygeneralized toM-arycodedsignalsasindicated byOmuraandLevitt (1982). Bycomparing thecutoffrateforcodedDSbinaryPSKmodulation shownin Fig.13-2-13,wenotethatforratesbelow0.7,thereisnopenaltyinSNRwith soft-decision decoding andjammer stateinformation compared withthe performance ontheAWGNchannel (01=1).Ontheotherhand,atRo=0.7, thereisa 6dBdifference inperformance between theSNRinanAWGN channelandthatrequired forhard-decision decoding withnojammerstate information. Atratesbelow0.4,thereisnopenaltyinSNRwithhard-decision decoding ifthejammerstateisunknown. However, thereistheexpected 2dB lossinhard-decision decoding compared withsoft-decision decoding inthe AWGNchannel. 13-2-4Generation ofPNSequences Thegeneration ofPNsequences forspreadspectrum applications isatopic thathasreceived considerable attention inthetechnical literature. Weshall brieflydiscusstheconstruction ofsomePNsequences andpresentanumberof important properties oftheautocorrelation andcross-correlation functions of suchsequences. Foracomprehensive treatment ofthissubject,theinterested readermayrefertothebookbyGolomb (1967). CHAPTER 13SPREAD SPEC"TRUM SIGNALS FORDIGITAL COMMUNICATIONS 725 ...---------""tag<>--------- OutpUl + FIGURE13-2·14 Generalm-stageshiftregisterwithlinearfeedback. ByfarthemostwidelyknownbinaryPNsequences arethemaximum­ lengthshift-register sequences introduced inSection 8-1-3inthecontextof codingandsuggest.ed againinSection 13-2-2foruseaslow-rate codes.A maximum-length shift-register sequence, orm-sequence forshort,haslength n=2m-1bi1sandisgenerated byanm-stage shiftregi&terwithlinear feedback asillustrated inFig.13-2-14.Thesequence isperiodic withperiodn. Eachperiodofthesequence contains 2m-lanesand2m-I-1zeros. InDSspreadspectrum applications thebinarysequence withelements {a,I} ismapped into3corresponding sequence ofpositive andnegative pulses according totherelation Pi(t)=(2bi-I)p(t-iT) wherePi(t)isthepulsecorresponding totheelement biinthesequence wIth elements {O,I}.Equivalently, wemaysaythatthebinarysequence with elements {O,I}ismappedintoacorresponding binarysequence withelements {-I,I}.Weshallcalltheequivalent sequence withelements{-I,I}abipolar sequence, sinceitresultsinpulsesofpositiveandnegative amplitudes. Animportant characteristic ofaperiodic PNsequence isitsperiodic autocorrelation function, whichisusuallydefined intermsofthebipolar sequence as n rj>(j)=L(2bi-I)(2bi+i-1),O",;,i",;,n-1 i=l(13-2-711 (13-2-72)wherenistheperiod.Clearly, cf>(j+rn)=cJ>(j)foranyintegervaluer. Ideally,apseudo-random sequence shouldhaveanautocorrelation function withtheproperty thatcf>(0)=nandcf>(j)=0for1",;,i",;,n-1.Inthecaseofm sequences, theperiodicautocorrelation function is cf>(j)={n(j=0) -I(1",;,i",;,n-1) Forlargevaluesofn,i.e.,forlongmsequences, thesizeoftheoff-peak values ofcf>(j)relativetothepeakvaluec/J(j)/c/J(O)=-lInissmalland,froma practical viewpoint, inconsequential. Therefore, msequences arealmostideal whenviewedintermsoftheirautocorrelation function. 726 DIGITAL COMML'~ICATIONS Inantijamming applications ofPNspreadspectrum signals,theperiodof thesequence mustbelargeinordertopreventthejammerfromlearning the feedback connections ofthePNgenerator. However, thisrequirement is impractical inmostcasesbecause thejammer candetermine thefeedback connections byobserving only2mchipsfromthePNsequence. This vUlnerability ofthePNsequence isduetothelinearity property ofthe generator. Toreducethevulnerability toajammer, theoutputsequences from severalstagesoftheshiftregisterortheoutputs fromseveraldistinctm sequences arecombined inanonlinear waytoproduce anonlinear sequence thatisconsiderably moredifficultforthejammertolearn.Furtherreduction in vulnerability isachieved byfrequently changing thefeedback connections and/orthenumber ofstagesintheshiftregisteraccording tosomeprear­ rangedplanformulated between thetransmiller andtheintended receiver. Insomeapplications, thecross-correlation properties ofPNsequences are asimportant astheautocorrelation properties. Forexample, inCDMA, each userisassigned aparticular PNsequence. Ideally,thePNsequences among usersshould'bemutually orthogonal sothatthelevelofinterference experienced byanyoneuserfromtransmissions ofotherusersaddsonapower basis.However, thePNsequences usedinpracticeexhibitsomecorrelation. Tobespecific,weconsider theclassofmsequences. Itis'known(Sarwate andPursley, 1980)thattheperiodic cross-correlation function between any pairofmsequences ofthesameperiodcanhaverelatively largepeaks.Table 13-2-1liststhepeakmagnitude <Pmaxfortheperiodiccross-correlation between pairsofmsequences for30;;m'"12.Thetablealsoshowsthenumberofm sequences oflengthn=2m-1for30;;m0;;12.Aswecansee,thenumberof msequences oflengthnincreases rapidlywithm.Wealsoobservethat,for mostsequences, thepeakmagnitude cPmaxofthecross-correlation function isa largepercentage ofthepeakvalueoftheautocorrelation function. Suchhighvaluesforthecross-correlations areundesirable inCDMA. TABLE 13-2-1PEAKCROSS-CORRELATION OFmSEQUENCES ANDGOWSEQUENCES PelIk Number 01cfOSSoocorrelatiClb mn=2"'-1 mleftueaces 4>_ ",_14>(0) tIm) t(m)'4>(O) 3 7 2 5 0.71 5 0.71 4 15 2 9 060 9 0.60 5 31 6 t1 0.35 9 0.29 6 63 6 23 0.36 17 0.27 7 127 18 41 0.32 17 0.13 8 255 16 95 0.37 33 0.13 9 511 48 113 0.22 33 0.06 10 1023 60 383 0.37 65 0.06 II 2047 176 287 0.14 65 0.03 12 409S 144 1407 0.34 129 0.03 (13-2-73)CHAPTER J.l:SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIO!"S 727 Although itispossible toselectasmallsubsetofm'sequences thathave relatively smallercross-correlation peakvalues,thenumberofsequences inthe setisusuallytoosmallforCDMAapplications. PNsequences withbetterperiodic cross-correlation properties thanm sequences havebeengivenbyGold(1967,1968)andKasami(1966).Theyare derivedfrommsequences asdescribed below. GoldandKasamiprovedthatcertainpairsofmsequences oflengthn exhibitathree-valued cross-correlation function withvalues{-I,-t(m). t(m)-2},where {2(m+1112+ 1(oddm) t(m)= ,2(m+211.+1(evenm) Forexample, ifm=10then1(10)=26+ 1=6Sandthethreepossiblevaluesof theperiodic cross-correlation function are{-I,-65,63}. Hencethemaximum cross-correlation forthepairofmsequences is65,whilethepeakforthe familyof60possible sequences generated byalO-stage shiftregisterwith different feedback connections is<Pm..=383-about asixfolddifference in peakvalues.Twomsequences oflengthnwithaperiodic cross-correlation function thattakesonthepossible values{-I,-t(m),t(m)-2}arecalled preferred sequences. ". Fromapairofpreferred sequences, saya=[0,02'..an]andb~ [b,bz.."bnl,weconstruct asetofsequences oflengthnbytakingthe modulo-2 sumofawiththencycliclyshiftedversions ofborviceversa.Thus, weobtainnnewperiodic sequencest withperiodn=2m-1.Wemayalso includetheoriginalsequences 8andband,thus,wehaveatotalofn+2 sequences. Then+2sequences constructed inthis'mannerarecalledGold sequences. Example 13-2-4 Letusconsider thegeneration ofGoldsequences oflengthn=31=25-l. Asindicated aboveform=5,thecross-correlation peakis t(5)=23+1=9 Twopreferred sequences, whichmaybeobtained fromPeterson and Weldon(1972),aredescribed bythepolynomials g,(p)=p5+p'+1 gz(p)=p5+p'+p2+P+1 tAnequivalent methodforgenerating thennewsequences istoemployashiftregisterof.. length2mwithfeedback connections specified byIhepolynomial h(p}=K,(P}K,(P}, whereK,(P) andK'(P)arethepolynomials thatspecifythefeedback connections ofthem-stageshiftregisters thatgenerate tbemsequences 8andb. 728 DIGITAL COMMUNICATIONS Gold__-{+ sequence gf.pl=P'+P'+P'+P+) FIGURE 13-2-15 Generation ofGoldsequences oflenSlh31. Theshiftregisters forgenerating thetwomsequences andthe corresponding Goldsequences areshowninFig.13-2-15.Inthiscase,there are33different sequences, corresponding tothe33relativephasesofthe twomsequences. Ofthese,'31sequences arenon-maximal-length sequences. Withtheexception ofthesequences aandb,thesetofGoldsequences does notcomprise maximum-length shift-register sequences oflengthn.Hence, theirautocorrelation functions arenottwo-valued. Gold(1968)hasshownthat thecross-correlation function foranypairofsequences fromthesetofn+ 2 Goldsequences isthree-valued withpossible values{-I,-t(m),t(m)-2}, wheret(m)isgivenby(13-2-73). Similarly, theoff-peak autocorrelation function foraGoldsequence takesonvaluesfromtheset{-I,-t(m),t(m)­ 2}.Hence,theoff-peak valuesoftheautocorrelation function areupper­ bounded byt(m). Thevaluesoftheoff-peak autocorrelation function andthepeakcross­ correlation function, i.e.,t(m),forGoldsequences islistedinTable13-2-1. Alsolistedarethevaluesnormalized by4>(0). Itisint~resting tocompare thepeakcross-eorrelation valueofGold sequences withaknownlowerboundonthecross-correlation between any pairofbinarysequences ofperiodninasetofMsequences. Alowerbound developed byWelc:h(1974)for4>m..is </J,.ax~n):n--\ (13-2-74) which,forlargevaluesofnandM,iswellapproximated asvn.ForGold sequences, n=2"- 1and,hence,thelowerboundisq.max'"2,"12.Thisbound islowerbyv'2foroddmandby2forevenmrelativeto4>,...=t(rn)forGold sequences. CHAPTER l3:SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNlC ATIONS 729 Aprocedure similartothatusedforgenerating Goldsequences will generate asmallersetofM=2",/2binarysequences ofperiodn=2m -1. wheremiseven.Inthisprocedure, webeginwithanmsequence aandwe formabinarysequence bbytakingevery2m12+Ibitofa.Thus,thesequence bisformedbydecimating aby2",/2+1.Itcanbeverifiedthattheresulting bis periodic withperiod2m/2-1.Forexample, ifm=10,theperiodof8is II=1023andtheperiodofbis31.Hence,ifweobserve 1023bitsofthe sequence b,weshallsee33repetitions ofthe31-bitsequence. Now,bytaking II=2"'-1bitsofthesequences 8andb,weformanewsetofsequences by adding,modul0-2, thebitsfrom8andthebitsfrombandall20012 -2cyclic shiftsofthebitsfromb_Byincluding 8intheset,weobtainasetof2"'/2binary sequences oflengthn=2m-I.ThesearecalledKasami sequences. The autocorrelation andcross-correlation functions ofthesesequences takeon valuesfromtheset{-I,_(2m12+1),2ml2-I}.Hence,themaximum cross­ correlation valueforanypairofsequences fromthesetis <Pmax=2ml2+i (13-2-75) Thisvalueof<PmaxsatisfiestheWelchlowerboundforasetof2m!2sequences oflengthn=2m-I.Hence,theKasamisequences areoptimal. Besidesthewell-known GoldandKasamisequences, thereareotherbinary sequences appropriate forCDMA applications. Theinterested readermay refertotheworkofScholtz(1979),Olsen(1977),andSarwate andPursley (1980)_ Finally,wewishtoindicate that,although wehavediscussed theperiodic cross-correlation function between pairsofperiodic sequences, manypractical CDMA systems mayuseinformation bitdurations thatencompass only fractions ofaperiodic sequence. Insuchcases,itisthepartial-period cross-correlation between twosequences tllatisimportant. Anumber of papersdealwiththisproblem, including thosebyLi"ndbolm (1968),Wainberg andWolf(1970),Fredricsson (1975),Bekiretal.(1978),andPursley(1979). 13·3FREQUENCY·HOPPED SPREAD SPECTRUM SIGNALS Inafrequency-hopped (FH)spreadspectrum communications systemthe available channelbandwidth issubdivided intoalargenumberofcontiguous frequency slots.Inanysignaling interval, thetransmitted signaloccupies one ormoreoftheavailable frequency slots.Theselection ofthe frequency slot(s) ineachsignaling interval ismadepseudo-randomly according totheoutput fromaPNgenerator. Figure13-3-1illustrates aparticular frequency-hopped patterninthetime-frequency plane. Ablockdiagram ofthetransmitter andreceiver for3frequency-hopped spreadspectrum systemisshowninFig.13-3-2.Themodulation isuS\lally eitherbinaryorM-aryFSK.Forexample, ifbinaryFSKisemployed, the modulator selectsoneoftwofrequencies corresponding tothetransmission of 730 DIGITAL COMMUNICATIONS 3- 6- ~ucg [ 25"-c:J c:J 4- FlGURE 13-3-1 Anexample ofafrequency-hopped (FH)patrero.oT,2T.,3r,4T,.5T/,6T,7T( Timeinlerval eithera IoraO.Theresulting FSKsignalistranslated infrequency byan amountthatisdetermined bytheoutputsequence fromthePNgenerator, which,inturn,isusedtoselectafrequency thatissynthesized bythefrequency synthesizer. Thisfrequency ismixedwiththeoutputofthemodulator andthe resultant frequency-translated signalistransmitted overthechannel. For example, mbitsfromthePNgeneratmmaybeusedtospecify2m-1possible frequency translations. Atthereceiver, wehaveanidentical PNgenerator, synchronized withthe received signal,whichisusedtocontrol theoutputofthefrequency synthesizer. Thus,thepseudo-random frequency translation introduced atthe transmitter isremoved atthereceiver bymixingthesynthesizer outputwith thereceived signal.Theresultant signalisdemodulated bymeansofanFSK demodulator. Asignalformaintaining synchronism ofthePNgenerator with thefrequency-translated received signalisusuallyextracted fromthereceived signal. Although PSKmodulation givesbetterperformance thanFSKinan FlGURE 13-3-1 BlockdiagramofaFHspreadspectrum system. Information ~uenceEncoder PN sequence gelleralor PN scquence....__...J aenerator CHAPTER ",SPREAD SPECTRUM SIGNAlS FORDIGITAL COMMUNICATIONS 731 lnformat~on sequence sequence generator nGURE 13-3-3 Blockdiagramofanindependent toneFHspreadspectrum syslem. AWGNchannel, itisdifficulttomainlain phasecoherence inthesynthesis of thefrequencies usedinthehoppingpatternand,also,inthepropagation ofthe signaloverthechannelasthesignalishoppedfromonefrequency to-another overawidebandwidth. Consequently, FSKmodulation withnoncoherent detection isusuallyemployed withFHspreadspectrum signals. Inthefrequency-hopping systemdepicted inFig. 13~3-2,thecarrier frequency ispseudo-randomly hoppedineverysignaling interval. TheM information-bearing tonesarecontiguous andseparated infrequency by1/'Fe. where1;,isthesignaling interval.Thistypeoffrequency hopping iscalled blockhopping. Another typeoffrequency hoppingthatislessvulnerable tosomejamming strategies is.independent tonehopping. Inthisscheme,theMpossibletones fromthe.modulator areassignedwidelydispened frequency slots.Onemethod foraccomplishing thisisillustrated inFig.13-3-3.Here,thembitsfromthePN generator andthekinformation bitsareusedtospecifythefrequency slotsfor thetransmitted signal. Thefrequency-hopping rateisusuallyselectedtobeeitherequaltothe (codedoruneaded) symbolrateorfasterthanthatrate.Iftherearemultiple hopspersymbol,wehaveafast-hopped signal.Onthe-otherhand,ifthe hoppingisperformed atthesymbolrate,wehaveaslow-hopped signal. Fastfrequency hoppingisemployed inAJapplications whenitisnecessary topreventatypeofjammer,calledafollower jammer,f!omhavingsufficient timetointercept thefrequency andretransmit italongwithadjacent frequencies soastocreateinterfering signalcomponents. However, thereisa penaltyincurred insubdividing asignalintoseveralfrequency-hopped ele­ mentsbecausetheenergyfromtheseseparate elements iscombined non­ coherently. Consequently, thedemodulator incursapenaltyintheformofa noncoherent combining lossasdescribed inSection12-1. FHspreadspectrum signalsareusedprimarily indigitalcommunications systemsthatrequireAJprojection andinCDMA,wheremanyuserssharea common bandwidth. Inmostcases,aFHsignalispreferred overaDSspread spectrum signalbecause of-thestringent synchronization requirements 732 DIGITI\L COMMUNICATIONS inherent inOSspreadspectrum signals.Specifically, inaOSsystem,timing andsynchronization mustbeestablished towithinafraction ofthechip intervalTc=I/W.Ontheotherhand,inanFHsystem,thechipintervalisthe timespentintransmitting asignalinaparticular frequency slotofbandwidth B«W.Butthisiniervalisapproximately 1/B,whichismuchlargerthanI/W. Hencethetimingrequirements inaFHsystemarenotasstringent asinaPN system. InSections 13-3-2and13-3-3,weshallfocusontheAJandCDMA applications ofFHspreadspectrum signals.First,weshalldetermine theerror rateperformance ofanuncoded andacodedFHsignalinthepresence of broadband AWGNinterference. Thenweshallconsider amoreserioustypeof interference thatarisesinAJandCDMAapplications, calledpartial-band interference. Thebenefitsobtained fromcodingforthistypeofinterference are determined. Weconclude thediscussion inSection13-3-3withanexample of anFHCDMAsystemthatwasdesigned forusebymobile users withasatellite servingasthechannel. 13-3-1Performance ofFHSpreadSpectrum Signalsin AWGNChannel Letusconsider theperformance ofaFHspreadspectrum signalinthe presence ofbroadband interference characterized statistically asAWGNwith powerspectral densitylo.Forbinaryorthogonal FSKwithnoncoherent detection andslowfrequency hopping(lhop/bit), theprobability oferror, derivedinSection5-4-1,is (13-3-1) whereYb='l:b/lo.Ontheotherhand,ifthebitintervalissubdivided intoL subintervals andFHbinaryFSKistransmitted ineachsubinterval, wehavea fastFHsignal.Withsquare-law combining oftheoutputsignalsfromthe corresponding matched filtersfortheLsubintervals, theerrorrateperfor­ manceoftheFHsignal,obtained fromtheresultsinSection12-1,is 1 L~I . P2(L)=22L-1e-,·12B,Ki(hb)'(13-3-2) wheretheSNRperbitis'Yb='l:bIJO=L'InYcistheSNRperchipinthe L-chipsymbol,and 1L-l-;(2L-1) K=-"" ,°t£.Jl.,.=0r(13-3-3) Werecallthat,foragivenSNRperbit"Yb'theerrorrateobtained from (13-3-2)islargerthanthatobtained from(13-3-1).Thedifference inSNRfora givenerrorrateandagivenLiscalledthenoncoherent combining loss,which wasdescribed andillustrated inSection12-1. Codingimproves theperformance oftheFHspreadspectrum systembyan CHAPTER 13:SPREAD SPECTRUM SIGNALS FORDIGITAL COMMUfIrl'lCATIONS 733 amount,whichwecallthecodinggain,thatdepends onthecodeparameters. Suppose weusealinearbinary(n,k)blockcodeandbinaryFSKmodulation withonehoppercodedbitfortransmitting thebits.Withsoft-decision decoding ofthesquare-law -demodulated FSKsignal,theprobability ofacode worderrorisupper-bounded as (13-3-4) whereP,(rn)istheerrorprobability indeciding between thernthcodeword andtheall-zerocodewordwhenthelatterhasbeentransmitted. The expression forP,(m)wasderivedinSection8-1-4andhasthesameformas (13-3-2)and(13-3-3), withLbeingreplaced byWmand"Ybby"YbRcwm, where WmistheweightofthemthcodewordandRcisthecoderate.Theproduct Rcw."whichisnotlessthanRcdmin,represents thecodinggain.Thus,wehave theperformance ofablockcodedFHsystemwithslowfrequency hoppingin broadband interference. Theprobability oferrorforfastfrequency hoppingwithn2hopspercoded bitisobtained byreinterpreting thebinaryeventprobability P2(m)in(13-3-4). Then2hopspercodedbitmaybeinterpreted asarepetition code,which, whencombined withanontrivial (n"k)binarylinearcodehavingweight distribution {wml.yieldsan(n,n2'k)binarylinearcodewithweightdistribu­ tion{n2wm}.Hence,P2(m)hastheformgivenin(13-3-2), withLreplaced by n2Wmand"Yb'by"YbRcn2IVm' whereRc=k/n,n2' Notethat"YbR,n2wm= Ybwmk/n" whichisjustthecodinggainobtained fromthenontrivial (n"k) code.Consequently, theuseoftherepetition codewillresultinanincreasein tpenoncoherent combining loss. Withhard-decision decoding andslowfrequency hopping, theprobability of acodedbiterrorattheoutputofthedemodulator fornoncoherent detection is (13-3-5) Thecodeworderrorprobability iseasilyupper-bounded, byuseofthe Chernoff bound,as M PM":;L[4p(1-pW.,12 m-2(13-3-6) However, iffastfrequency hopping isemployed withn2hopspercodedbit, andthesquare-law-detected outputsfromthecorresponding matched filters forthen2hopsareaddedasinsoft-decisioD decoding toformthetwodecision variables forthecodedbits,thebiterrorprobability pisalsogivenby(13-3-2), withLreplaced byn2and"Ybreplaced byYbRcn2,whereRcistherateofthe nontrivial (n"k)code.Consequently, theperformance ofthefastFHsystem inbroadband interference isdegraded relativetotheslowFHsystembyan amountequaltothenoncoherent combining lossofthesignalsreceived from then2hops. (13-3-7)734 DIGITAL COMMUl\ICATIONS Wehaveobserved thatforbothhard-decision andsoft-decision decoding, theuseoftherepetition codeinafast-frequency-hopping systemyieldsno codinggain.Theonlycodinggainobtained comesfromthe(n\.k)blockcode. Hence,iherepetition codeisinefficient inafastFHsystemwithnoncoherent combining. Amoreefficientcodingmethodisoneinwhicheitherasingle low-ratebinarycodeoraconcatenated codeisemployed. Additional improve­ mentsinperformance maybeobtained byusingnonbinarycodesinconjunc­ tionwithM-aryFSK.Boundsontheerrorprobability forthiscasemaybe obtained fromtheresultsgiveninSection12-1. Although wehaveevaluated theperformance oflinearblockcodesonlyin theabovediscussion, itisrelatively easytoderivecorresponding performance resultsforbinaryconvolutional codes.Weleaveasanexercise forthereader thederivation ofthebiterrorprobability forsoft-decision Viterbidecoding andhard-decision Viterbidecoding ofFHsignalscorrupted bybroadband interference. Finally, weobserve that'(;,,,theenergyperbit,canbeexpressed as 'tb=PaJR,whereRistheinformation rateinbitspersecondand10=l"IW. Therefore, 'Yhmaybeexpressed as 'thWIRI'h=-=--­ 101,,1Pay Inthisexpression, werecognize WIRastheprocessing gainand1,vfPayasthe jamming marginfortheFHspreadspectrum signal. 13·3·2Performance ofFHSpreadSpectrum Signalsin Partial-Band Interference Thepartial-band interference considered inthissubsection ismodeled asa zero-mean gaussian randomprocesswithaflatpowerspectraldensityovera fraction aofthetotalbandwidth Wandzeroelsewhere. Intheregionor regionswherethepowerspectraldensityisnonzero, itsvalueis<IJ,,(f)=lola, D<a""1Thismodeloftheinterference maybeappliedtoajamming signal ortointerference fromotherusersinaFHCDMAsystem. Suppose thatthepartial-band interference comesfromajammerwhomay selectatooptimize theeffectonthecommunications system.Inanuncoded pseudo-randomly hopped(slow-hopping) FHsystemwithbinaryFSKmodula­ tionandnoncoherent detection, thereceived signalwillbejammed with probability aanditwillnotbejammed withprobability 1-Ct.Whenitis jammed, theprobability oferroris!exp(-'thaI2lo),andwhenitisnot jammed, thedemodulation iserror-free. Consequently, theaverageprobability oferroris \('Q''tb) P2(Ct)=,aexp -21 0 where'€hl10mayalsobeexpressed as(WIR)/(J.vI Pay).(13-3-8) CHAPTER IJSPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 735 I0"~--r---,--.----r---.-.---' 5I-+-+-~+-+-~--j 2..... 10-'1-~,-+Wof'St-case partial-band 5\,~ammillg - is2510-1I-+-'\\I--\. \~-+--+-1 ~51-+--t-~rl\I'-+-+-l q21-+--l\-H\+-''tt-r-1 1'0-;I-+--+-I-I-+---+\,' '''t--l ~ \" 2I-+---t-JH-----'\r-H-P-i 10-4I-+-++-+-t\-+-\t \-1 5I-+--+--+-~-++-+--ll--j Ct=1.0a=O.1a=O.OI 2I--t=---il"'+t---=I"t,-F-F+-i 10-\L-...L-L----'-L-....L.l-L_.lL---' FlGUlI.E 13-3-4 Performance ofbinaryFSKwithpartial-band inlerference.o51015:?O25303'i SNRperbit,Y...(dB) Figure13-3-4illustrates theerrorrateasafunction of'iblJoforseveral valuesofo.Thejammer's optimum strategy istoselectthevalueof0that maximizes theerrorprobability. Bydifferentiating P2(a)andsolvingforthe extremum withtherestriction that0,;;;0';;;1,wefindthat (13-3-9) Thecorresponding errorprobability fortheworst-case partial-band jammer is ~-I[(W/R)]-1 p---- ~-- 2 -'iblJo-JavlPay(13-3-10) Whereas theerrorprobability decreases exponentially forfull-band jamming, wenowfindthattheerrorprobability decreases onlyinversely with'lb/10for theworst-case partial-band jamming. Thisresultissimilarto'theerrorrate performance ofbinaryFSKinaRayleigh fadingchannel(seeSection14-3)and totheuncoded DSspreadspectrum systemcorrupted byworst-case pulse jamming (seeSection[3-2-3). Asweshalldemonstrate below,signaldiversity obtained bymeansof codingprovides asignificant improvement inperformance relativetouncoded signals.Thissameapproach tosignaldesignisalsoeffective forsignaling over afadingchannel, asweshalldemonstrate inChapter 14. Toillustrate thebenefitsofdiversity inaFHspreadspectrum signalwith partial-band interference, weassumethatthesameinformation symbolis 736 DIGITAL CO""LJNIfATIONS transmitted bybinaryFSKonLindependent frequency hops.Thismaybe accomplished bysubdividing thesignaling interval intoLsubintervals, as described previously forfastfrequency hopping. Afterthehopping patternis removed, thesignalisdemodulated bypassingitthroughapairofmatched filterswhoseoutputsaresquare-Iaw·detected andsampled attheendofeach subinterval. Thesquare-law-detected signalscorresponding totheLfrequency hopsareweighted andsummed toformthetwodecision variables (metrics), whicharedenoted asV,andV2• Whenthedecision variableV,contains thesignalcomponents, V,andV2 maybeexpressed as l- V,=2:13k12~+Nlk12 II:=1 I- V2=2:13kIN,.12 k=/(13-3-11) where{13.lrepresent theweighting coefficients, ~cisthesignalenergyperchip intheL-chipsymbol,and{N,.Jrepresent theadditivegaussian noisetermsat theoutputofthematched filters. Thecoefficients· areoptimally selected toprevent thejammer frolll saturating thecombiner shouldthetransmitted frequencies besuccessfully hit inoneqrmorehops.Ideally, 13kisselectedtobeequaltothereciprocal ofthe variance ofthecorresponding noiseterms{N.}.Thus,thenoisevariance for eachchipisnormalized tounitybythisweighting andthecorresponding signal isalsoscaledaccordingly. Thismeansthatwhenthesignalfrequencies ona particular hoparejammed, thecorresponding weightisverysmall.Inthe absenceofjamming onagivenhop,theweightisrelatively large.Inpractice, forpartial-bound noisejamming, theweighting maybeaccomplished byuseof anAGChavingagainthatissetonthebasisofnoisepowermeasurements obtained fromfrequency bandsadjacent tothetransmitted tones.Thisis equivalent tohavingsideinformation (knowledge ofjammer state)atthe decoder. Suppose thatwehavebroadband gaussian noisewithpowerspectraldensity Noandpartial-band interference, overaWofthefrequency band,whichisalso gaussian withpowerspectral densitylola.Inthepresence ofpartial-band interference, thesecondmoments ofthenoisetermsNJkandN,.are •(13-3-12) Inthiscase,weselect13.~1/ui~[2~,(No+io/a)J-I.Intheabsence of partial-band interference, ui~2'i:cN.and,hence,13k=(2~No)-I.Notethat13. isarandomvariable. Anerroroccursinthedemodulation ifV2>V"Although itispossible to determine theexacterrorprobability, weshallresorttotheChernoff bound, (13-3-13) (13-3-14) (13-3-16)C'H!\PTER 13:SPREAD SPECTRlJM SIGNALS FORDIGITAL COMMUNICATIONS 737 whichyieldsaresultthatismucheasiertoevaluate andinterpret. Specifically. theChernoff (upper)boundsintheerrorprobability is 1'2=P(U2-VI>0)"'E{exp[v(V 2-VI)]} =E{exp[-V.t,13.(lZ~,+ N1.12-IN2kI')]} wherevisavariable thatisoptimized toyieldthetightestpossiblebound. Theaveraging in(13-3-13) isperformed withrespecttothestatistics ofthe noisecomponents andthestatisticsoftheweighting coefficients {13.},whichare randomasaconsequence ofthestatistical naturt'oftheinterference. Keeping the{13k}fixedandaveraging overthenoisestatistics first,weobtain P,(~)=E[exp(-v.t,13.IZ'iC,+Nal2+V~113.IN,.12)] L =nE[exp(-VfJk 12~,+Nu/2)JE[exp(vfJ.IN,.I')] *=1 nLI (-4~fJ.V)= 2exp .~tl-4v l+Zv SincetheFSKtonesarejammed withprobability «,itfollowsthat13k= [Zjg(No+Jo/aW' withprobability aand(Z~,NO)-1 withprobability I -a Hence,theChernoff boundis L{a[ -Z~,v ]I-a [-Z~,V]}p",---ex + ex ,.II1-4v'P(No+Jo/a)(1+2v)I -4v'PNo(1+Zv) {a [ -2~v ] 1-a[-Z~ v]}L =1_4v,exp (No+Jo/a)(I+Zv) +1_4v,exp No(1+'ZV) (13-3-15) Thenextstepistooptimize theboundin(13-3-15) withrespectto'the variable v.Initspresentform,however, theboundismessytomanipulate. A significant simplification occursifweassumethatJo/a»No,whichrendersthe secondtermin(13-3-15) negligible compared withthefirst.Alternatively, we letNo=0,sothattheboundonP2reducesto P,'"L_u4v,exp[J~::~~)]r Theminimum valueofthisboundwithrespecttovandthemaximum with respecttoa(worst-case partial-band interference) iseasilyshowntoOccur whenu=3JO/'l;e'"1andv=l.Forthesevaluesoftheparameters, (13-3-16) reducesto (4)L(147)LP,"'P,(L)= -=-'-, e'Ye y,~e ~bl'=-=--;;.3, JoUo(13-3-17) 738 DIGITAL COMMUNICATIONS where ')I,istheSNRperchipintheL-chipsymbol.Equivalently, [1.47(1•.1P.v)]L Pz""W/R 'W/R--,---;;.3 L(J•.IP.v)(13-3-18) Theresultin(13-3-17) wasfirstderivedbyViterbiandJacobs(1975). Weobserve thattheprobability oferrorfortheworst-case partial-band interference decreases exponentially withanincrease intheSNRperchip'Y,. Thisresultisverysimilartotheperformance characteristics ofdiversity techniques forRayleigh fadingchannels (seeSection 14-4).Wemayexpress theright-hand sideof(13-3-17) intheform (13-3-19) wherethefunctionh('Y,)isdefinedas (13-3-20) Aplotofh('Ye>isgiveninFig.13-3-5.Weobserve thatthefunction hasa maximum valueof~at"Ye=4.Consequently, thereisanoptimum SNRper chipof10log'Ye=6dB.Attheoptimum SNR,theerrorrateisupper-bounded as P,.;:P(L)--,,142-":2opt- e (13-3-21) Whenwecompare theerrorprobability boundin(13-3-21) withtheerror probability forbinaryFSKinspectrally flatnoise,whichisgivenby(13-3-1), weseethatthecombined effectofworst-case partial-band interference andthe noncoherent combining lossinthesquare-law combining oftheLchipsis3dB. Weemphasize, however, thatforagiven'i./lo,thelossisgreaterwhenthe orderofdiversity isnotoptimally selected. 0.3 ~02~......-................r--.. FIGURE 13-3-5 Graphofthefunction h(y,.).6 7 Y,.8 9 10 CHAYfER 13:SPREAD SPEITRUM SIGNALS FORDIGITAL COM~UNICATIONS 739 Codingprovides ameansforimproving theperformance ofthefrequency­ hoppedsystemcorrupted bypartial-band interference. Inparticular, ifablock orthogonal codeisused,withM=2'codewordsandLth-order diversity per codeword,theprobability ofacodeworderrorisupper-bounded as PM""(2'-1)P2(L)=(2'-1)(1.47)L =(2'-1)(1.4(7)L (13-3-22) 'Ye k'YbL (13-3-23)andtheequivalent biterrorprobability isupper-bounded as '-1(.1.47)LPb""2--k'YbfL Figure13-3-6illustrates theprobability ofabiterrorforL=1,2,4,8and FIGURE 13--3-6 Performance ofbinaryandoctalFSKwithL·orderdiversity forachannel withworst-case partial-band interference. 28 24 20 16:1\\ \ \ \11\ \r\ '\'\\\\ ,\L=8 \ L=4 \ f--~~- Optimum :k=3 k=I diversity -::::~ 1\ k=1 •\- ·1.\\\\~~:fT-\-+----1 • \ \1,:.L=8 \ ~:\,.t=IItt'~_--'-_--''--_--'-~....IJ~..I.-l--'---l~ __.J o 4 8 121(1-' 5 2 10-2 5 2 .: §Io-·l U ]5 •'0 .~2:sf10-' 5 2 1(1-' 5 SNRperbit.y.(dB) 740 DIGITAL COMMlJ!"KATIO!\S k=1.3.Withanoptimum choiceofdiversity, theupperboundcanbe expressed as Phs;,2klexp(-~k'Yh)=~exp[-k(hh-ln2)1 (13-3-24) Thus,wehaveanimprovement inperformance byanamount equalto 10log[k(l-2.77hh)].Forexample. ifYh=10andk=3(octalmodulation) thenthegainis3.4dB,whiIeIfk=5thenthegainis5.6dB. Additional gainscanbeachieved byemploying concatenated codesin conjunction withsoft-decision decoding. Intheexample below.weemploya dual-kconvolutional codeastheoutercodeandaHadamard codeastheinner codeonthechannel withpartial-band interference. Example 13..3-1 Suppose weuseaHadamard H(n.k)constant weightcodewithon-off keying(OOK)modulation foreachcodebit.Theminimum distance ofthe codeisdm,"=~n.and,hence.theeffective orderofdiversity obtained with OOKmodulation is~dm;"=~nThereare~nfrequency-hopped tones transmitted percodeword.Hence. (13-3-25) whenthiscodeisusedalone.Thebiterrorrateperformance for soft-decision decoding ofthesecodesforthepartial-band interference channel isupper-bounded as 1(~)"'4 2R,'Yh(13-3-26) Now,ifaHadamard (n.k)codeisusedastheinnercodeandarate1/2 dual-kconvolutional code(seeSection8-2-6)istheoutercode.thebiterror performance inthepresence ofworst-case partial-band interference is(see (8-2-40» whereP2(L)isgivenby(13-3-17) with k 'Y<= -'Yb=R,,'Ybn(13-3-28) CHAPTER l.lSPREA.D SPECTRUM SIGNALS FORDIGITAL CQMMUNICA nONS741 9101112D1415 SNRperbit,y"(dBJ\ \\ ,\1\ \\ \1\\ \Dual-} \1\H(l2.})-I-- \\ ~DuaI-5\\ H(20.5) \1\j.%-DuaI-4-\\- \\T'6·tJ -,\1I2 Ilr' 5 10-' FIGURE 13-3-7 Performance ofdual-kcodesconcatenated withHadamard 8 codesforachannelwithworsH:ase partial-band interference. Figure13-3-7illustrates theperformance ofthedual-kcodesfork=5,4, and3concatenated withtheHadamard H(20.S), H(16,4),andH(12,3) codes,respectively. Intheabovediscussion, wehavefocusedonsoft-decision decoding. Onthe otherhand,theperformance achieved withhard-decision decoding issig­ nificantly (several decibels) poorerthanthatobtained withsoft-decision decoding. Inaconcatenated codingscheme, however, amixture involving soft-decision decoding oftheinnercodeandhard-decision decoding ofthe outercoderepresents areasonable compromise between decoding complexity andperformance. Finally,wewishtoindicate thatanotherseriousthreatinaFHspread spectrum systemispartial-band multitone jamming. Thistypeofinterference is similarineffecttopartial-band spectrally flatnoisejamming. Diversity obtained throughcodingisaneffective meansforimproving theperformance oftheFHsystem.Anadditional improvement isachieved byproperly weighting thedemodulator outputssoastosuppress theeffectsofthejammer. 13-3-3ACOMASystemBasedonFHSpreadSpectrum Signals InSection13-2-2,weconsidered aCDMAsystembasedonuseofDSspread spectrum signals.Aspreviously indicated, itisalsopossibletohaveaCDMA systembasedonFHspreadspectrum signals.Eachtransmitter-receiver pairin suchasystemisassigned itsown.pseudo-random frequency-hopping pattern. 742 DIGITAL COMMUl'IlCATIONS Asidefromthisdistinguishing feature,thetransmillers andreceivers ofallthe usersmaybeidentical inthattheymayhaveidentical encoders, decoders, modulators, anddemodulators. COMAsystems basedonFHspreadspectrum signalsareparticularly allractive formobile(land,air,sea)usersbecausetimingrequirements arenot asstringent asinaPNspreadspectrum signal.Inaddition, frequency synthesis techniques andassociated hardware havebeendeveloped thatmakeitpossible tofrequency-hop overbandwidths thataresignificantly largerthanthose currently possible withOSspreadspectrum systems. Consequently, larger processing gainsarepossiblewithFH.Thecapacity ofCOMAwithFHisalso relatively high.Viterbi(1978)hasshownthaIwithdual-kcodesandM-ary FSKmodulation, itispossibletoaccomodate upto~WIRsimultaneous users whotransmit ataninformation rateRbitslsoverachannelwithbandwidth W. OneoftheearliestCOMAsystemsbasedonFHcodedspreadspectrum signalswasbuilttoprovidemultiple-access tacticalsatellite communications forsmallmobile(land,sea,air)terminals eachofwhichtransmitted relatively shortmessages overthechannel intermittently. Thesystemwascalledthe Tactical Transmission System(TATS) anditisdescribed inapaperby Orouilhet andBernstein (1969). AnoctalReed-Solomon (7,2)codeisusedintheTATSsystem.Thus,two 3bitinformation symbolsfromtheinputtotheencoderareusedtogenerate a seven-symbol codeword.Each3bitcodedsymbolistransmitted bymeansof octalFSKmodulation. Theeightpossible frequencies arespaced111;.Hz apart,where 'Fe.isthetime(chip)duration ofasinglefrequency transmission. Inaddition tothesevensymbolsinacodeword,aneighthsymbolisincluded. Thatsymbolanditscorresponding frequency arefixedandtransmitted atthe beginning ofeachcodewordforthepurpose ofproviding timingand frequency synchronizationt atthereceiver. Consequently, eachcodewordis transmitted in81;s. TATSwasdesigned totransmit atinformation ratesof75and2400bits/s. Hence, 7;.=10msand312.5jLS,respectively. Eachfrequency tonecorres­ ponding toacodesymbolisf~equency-hopped. Hence,thehopping rateis 100hopslsatthe75bitslsrateand3200hopslsatthe2400bitslsrate. ThereareM=26=64codewordsintheReed-Solomon (7,2)codeandthe minimum distance ofthecodeisdmin=6.Thismeansthatthecodeprovides an effective orderofdiversity equalto6. Atthereceiver, thereceived signalisfirstdehopped andthendemodulated bypassingitthroughaparallelbankofeightmatched filters,whereeachfilter istunedtooneoftheeightpossible frquencies. Eachfilteroutputis envelope-detected, quantized to4bits(oneof16levels),andfedtothe decoder. Thedecoder takesthe56filteroutputs corresponding tothe tSincemobileusersareinvolved, thereisaDoppler frequency offsetassociated with transmissio,n. Thisfrequency offsetmustbetrackedandcompensated forinthedemodulation of thesignal.Thesyncsymbolisusedforthispurpose. CHAFIERIJ:SPREAlJ SPFClRl 'MSIGNAl.S FORDIGITAL COMML'ICATIOI\S 743 receptioll ofeachseven-symbol codewordandforms64decision variables corresponding tothe64possible codewordsin'the(7.2)codebylinearly combining theappropriate envelope detected outputs. Adecision ismadein favorofthecodewordhavingthelargestdecision variable. Bylimitingthematched filteroutputsto16levels.interference (crosstalk) fromotherusersofthechannelcausesarelatively smalllossinperformance (0.75dBwithstronginterference ononechipand1.5dBwithstrong interference ontwochipsoutoftheseven).TheAGCusedinTATShasa timeconstant greaterthanthechipinterval "T.sothatnoattempt ismadeto perform optimum weighting ofthedemodulator outputs asdescribed in Section13-3-2. Thederivation oftheerrorprobability fortheTATSsignalinAWGNand worst-casepartial-band interference isleftasanexercise forthereader (Problems 13-23and13-24). 13-4OTHER TYPES OFSPREAD SPECTRUM SIGNALS OSandFHarethemostcommon formsofspreadspectrum signalsusedin practice. However, othermethods maybeusedtointroduce pseudo­ randomness inaspreadspectrum signal.Onemethod, whichisanalogous to FH,istimehopping (TH).InTH,atimeinterval, whichisselected tobemuch largerthanthereciprocal oftheinformation rate,issubdivided intoalarge numberoftimeslots.Thecodedinformation symbols aretransmitted ina pseudo-randomly selectedtimeslotasablockofoneormorecodewords.PSK modulation maybeusedtotransmit thecodedbits. Forexample, suppose thatatimeintervalTissubdivided into!OOOtime slotsofwidthT!IOOOeach.Withaninformation bitrateofRbits/s.the numberofbitstobetransmitted inTsisRT.Codingincreases thisnumberto RT/R,bits,whereR,isthecodingrate.Consequently, inatimeintervalof Ti1000s,wemusttransmit RT/R,bits.IfbinaryPSKisusedasthe modulation method. thebitrateisIOOOR/R,andthebandwidth required is approximately W~l000R/R,. Ablockdiagram ofatransmitter andareceiver foraTHspreadspectrum systemisshowninFig.13-4-1.Ouetotheburstcharacteristics ofthe transmitted signal,bufferstoragemustbeprovided atthetransmitter inaTH system,asshowninFig,13-4-1.Abuffermayalsobeusedatthereceiver to provideauniformdatastreamtotheuser. Justaspartial-band interference degrades anuncoded FHspreadspectrum system,partial-time (pulsed) interference hasasimilareffectonaTHspread spectrum system.Codingandinterleaving areeffective meansforcombatting thistypeofinterference, aswehavealreadydemonstrated forFHandOS systems. Perhapsthemajordisadvantage ofaTHsystemisthestringent timing requirements compared notonlywithFHbut,also,withOS. Othertypesofspreadspectrum signalscanbeobtained bycombining OS. 744 l>I(jlTAl C{)MMUNICATIONS Buffer and deinteriea ...cPN !>C~nce generatorBuffer and interleave -.equence generatorPNEncoderInformation !->equence OUlpul FIGURE 13-4-1 Blockdiagramoftime-hopping (TH)spreadspectrum .ystem. FH,andTH.Forexample, wemayhaveahybridDS/FH,whichmeansthata PNsequence isusedincombination withfrequency hopping. Thesignal transmitted onasinglehopconsistsofaDSspreadspectrum signalwhichis demodulated coherently. However, thereceived signalsfromdifferent hopsare combined noncoherently (envelope orsquare-law combining). Sincecoherent detection isperformed withinahop,thereisanadvantage obtained relativeto apureFHsystem.However, thepricepaidforthegaininperformance isan increase incomplexity, greatercost,andmorestringent timingrequirements. Another possible hybridspreadspectrum signalisDS/TH. Thisdoesnot seemtobeaspractical asDS/FH,primarily becauseofanincrease insystem complexity andmorestringent timingrequirements. 13·5SYNCHRONIZATION OFSPREAD SPECTRUM SYSTEMS Timesynchronization ofthereceiver tothereceived spreadspectrum signal maybeseparated intotwophases.Thereisaninitialacquisition phaseanda tracking phaseafterthesignalhasbeeninitiallyacquired. Acquisition Inadirectsequence spreadspectrum system,thePNcode mustbetime-synchronized towithinasmallfraction ofthechipinterval T,.=I/W.Theproblem ofinitialsynchronization maybeviewedasonein whichweattempttosynchronize intimethereceiverclocktothetransmitter clock.Usually, extremely accurate andstablet{meclocksareusedinspread spectrum systems. Consequently, accurate timeclocksresultinareduction of thetimeuncertainty between thereceiverandthetransmitter_However, there isalwaysaninitialtiminguncertainty duetorangeuncertainty between the transmitter andthereceiver. Thisisespecially aproblem whencommunication istakingplacebetween twomobileusers.Inanycase,theusualprocedure for establishing initialsynchronization isforthetransmitter tosendaknown CHAPTI:R nSPREAD SPECTRUM SIGNALS FORDIGITAL COMMUNICATIONS 745 pseudo-random datasequence tothereceiver. Thereceiveriscontinuously ina searchmodelooking forthissequence inordertoestablish initial synchronization. Letussupposethattheinitialtiminguncertainty isTuandthechipduration is7;.Ifinitialsynchronization istotakeplaceinthepresence ofadditivenoise andotherinterference, itisnecessary todwellforTd;N7;.inordertotest synchronism ateachtimeinstant.Ifwesearchoverthetimeuncertainty intervalin(coarse)timestepsof17;thenthetimerequired toestablish initial synchronization is 7;, T;ni~liYnc=1T"'t/1;.=2NI;( 21c(13-5-1) Clearly,thesynchronization sequence transmitted tothereceivermustbeat leastaslongas2N7;inorderforthereceivertohavesufficient timetoperform thenecessary searchinaserialfashion. Inprinciple, matched filteringorcross-correlation areoptimum methods for establishing initialsynchronization. Afiltermatched totheknowndata waveform generated fromtheknownpseudo-random sequence continuously looksforexceedence ofapredetermined threshold. Whenthisoccurs,initial synchronization isestablished andthedemodulator entersthe"datareceive" mode. Alternatively, wemayuseaslidingcorrelalor asshowninFig.13-5-1.The correiatorcyclesthroughthetime uncertainty, usuallyindiscretetimeintervals of17;,andcorrelates thereceived signalwiththeknownsynchronization sequence. Thecross-correlation isperformed overthetimeintervalN7;(N chips)andthecorrelator outputiscompared withathreshold todetermine if theknownsignalsequence ispresent.Ifthethreshold isnotexceeded, the knownreference sequence isadvanced intimebyfl;·sandthecorrelation processisrepeated. Theseoperations areperformed untilasignalisdetected oruntilthesearchhasbeenperformed overthetimeuncertainty interval 7;,.In thelallercase,thesearchprocessisthenrepeated. AsimilarprocessmayalsobeusedforFHsignals.Inthiscase,theproblem istosynchronize thePNcodethatcontrols thehoppedfrequency pattern.To accomplish thisinitialsynchronization, aknownfrequency hoppedsignalis FIGURE 13-5·1 Aslidingcorrelalor forDSsignalacquisition. Recei....ed \ignal J'(ldr--Threshold,-x"detector- 1 PN Search code....- control g~nerator clockSyoc. pulse 746 D1(iITAL (UMMl;NICATIONS fillerEnvelopetunedtodetector II filterEnvelopetunedtodetector Rcrci\'cd h signal filterEn\elopetuned(0 +detector(, Iihrr tunedto r~1En.....elope dececturTnre!Ohold detectorSYrlC pul~ FIGURE 13-5-2 SystemfmacqUlsilion ofaFHsignal. transmitted tothereceiver. Theinitialacquisition systematthereceiverlooks forthisknownFHsignalpattern.Forexample, abankofmatched filterstuned tothetransmitted frequencies intheknownpatternmaybeemployed. Their outputsmustbeproperly delayed, envelope- orsquare-law-detected, weighted, ifnecessary, andadded(noncoherent integration) toproducethesignaloutput whichiscompared withathreshold. Asignalpresentisdeclared whenthe threshold isexceeded. Thesearchprocessisusuallyperformed continuously in timeuntilathreshold isexceeded. Ablockdiagram illustrating thissignal acquisition scheme isgiveninFig.13-5-2.Asanalternative, asingle matched-tilter-envelope detector pairmaybeused,preceded byafrequency­ hopping patterngenerator andfollowed byapost-detection integrator anda threshold detector. Thiscontiguration, showninFig.13-5-3,isbasedonaserial searchandisakintotheslidingcorreiatorforOSspreadspectrum signals. Theslidingcorrelator fortheOSsignalsoritscounterpart showninFig. 13-5-3forFHsignalsbasically perform aserialsearchthatisgenerally time-consuming. Asanalternative, onemayintroduce somedegreeof parallelism byhavingtwoormoresuchcorrelators operating inparalleland searching overnonoverlapping timeslots.Insuchacase,thesearchtimeis reducedattheexpenseofamorecomplex andcostlyimplementation. Figure 13-5-2represents suchaparallelrealization fortheFHsignals. Duringthesearchmode,theremaybefalsealarmsthatoccuratthe designed falsealarmrateofthesystem.Tohandletheoccasional falsealarms, itisnecessary tohaveanadditional methodorcircuitthatcheckstoconfirm CHAPTER 13:SPREAD SPECTRUM SIGNALS FORDIGITAL rOMMuslCATIONS 747 Clock PNcode sequence generator Received signal----{ x}---.j Post· detection ;rn.egral1oo Sync. pulse FIGURE lJ.-5-3 Alternative systemforacquisition ofafHsignal. thatthereceived signalattheoutputofthecorrelator remains abovethe threshold. Withsuchadetection strategy, alargenoisepulsethatcausesafalse alarmwillcauseonlyatemporary exceedence ofthethreshold. Ontheother hand,whenasignalispresent,thecorrelator ormatched filteroutputwillstay abovethethreshold fortheduration ofthetransmitted signal.Thus,if confirmation fails,thesearchisresumed. Another initialsearchstrategy, calledasequential search.hasbeen investigated byWard(1965,1977). Inthismethod,thedwelltimeateachdelay inthesearchprocessismadevariable byemploying acorrelator witha variable integration periodwhose(biased) outputiscompared withtwo thresholds. Thus,therearethreepossibledecisions: 1iftheupperthreshold isexceedbythecorreiatoroutput, initial synchronization isdeclared established; 2ifthecorreiatoroutputfallsbelowthelowerthreshold, thesignalis declared absentatthatdelayandthesearchprocessresumes atadifferent delay; 3ifthecorreiatoroutputfallsbetween thetwothresholds, theintegration timeisincreased byonechipandtheresulting outputiscompared witht'letwo thresholds again. Hence,steps1,2,and3arerepeated foreachchipintervaluntilthecorrelator outputeitherexceedstheupperthreshold orfallsbelowthelowerthreshold. 748 DIGITAL. COMMUNICATIONS Recei....ed signal PN !;cquence generatorTuned filterat /, Tuned fillerat /,+!if Tuned filleral /,+M!ifSelect ,he greatest outputSync. pulse Controlsignal-------------_.-----------.------------------- FIGURE 13-5-4 InitialsearchforDopplerfrequency offsetinaDSsystem. Thesequential searchmethodfallsintheclassofsequential estimation methods proposed byWald(1947),whichareknowntoresultinamore efficientsearchinthesensethattheaveragesearchtimeisminimized. Hence, thesearchtimeforasequential searchislessthanthatforthe'fixeddwelltime integrator. Intheabovediscussion, wehaveconsidered onlytimeuncertainty in establishing initialsynchronization. However, anotheraspectofinitialsynchro­ nization isfrequency uncertainty. Ifthetransmitter and/orthereceiver are mobile,therelativevelocitybetween them results inaDopplerfrequency shift inthereceived signalrelativetothetransmitted signal.Sincethereceiverdoes notusuallyknowtherelativevelocity. apriori,theDoppler frequency shiftis unknown andmustbedetermined bymeansofafrequency searchmethod. Suchasearchisusuallyaccomplished inparalleloverasuitablyquantized frequency uncertainty intervalandseriallyoverthetimeuncertainty interval. AblockdiagramofthisschemeisshowninFig.13-5-4,Appropriate Doppler frequency searchmethods canalsobedevisedforFHsignals. Tracking Oncethesignalisacquired, theinitialsearchprocessisstopped andfinesynchronization andtracking begins.Thetracking maintains thePN codegenerator atthereceiver insynchronism withtheincoming signal. Tracking includesbothfinechipsynchronization and,forcoherent demodula­ tion,carrierphasetracking. Thecommonly usedtracking loopforaDSspreadspectrum signalisthe C11.....nlR 13SI'RI.....DSPECTRl1MSJ(;NAlS FORDIGITAL COMMUN1C~\TIONS 749 Re~'ei"t'd sign",1 FIGURE 13-5·5 Delay-locked loop(Dll)forPNcodetracking. delay-locked loop(DLL),whichisshowninFig.13-5-5.Inthistracking loop, thereceived signalisappliedtotwomultipliers, whereitismultiplied bytwo outputsfromthelocalPNcodegenerator, whicharedelayedrelativetoeach otherbyanamount 20""Te.Thus,theproduct signalsarethecross­ correlations between thereceived signalandthePNsequence atthetwovalues ofdelay.Theseproducts arebandpass-filtered andenvelope- (orsquare-Iaw-) detected andthensubtracted. Thisdifference signalisappliedtotheloopfilter thatdrivesthevoltagecontrolled clock(Vee).Theveeservesastheclock forthePNcodesignalgenerator. Ifthesynchronism isnotexact,thefilteredoutputfromonecorrelator will exceedtheotherandtheveewillbeappropriately advanced ordelayed. At theequilibrium point.thetwofilteredcorrelator outputs willbeequally displaced fromthepeakvalue,andthePNcodegenerator outputwillbe exactlysynchronized tothereceived signalthatisfedtothedemodulator. We observe thatthisimplementation oftheDLLfortracking aDSsignalis equivalent totheearly-late gatebittrackingsynchronizer previously discussed inSection6-3-2andshowninFig.6-3-5. Analternative methodfortimetracking aDSsignalistouseatau-dither loop(TDL),illustrated bytheblockdiagraminFig.13-5-6.TheTDLemploys asingle"arm"insteadofthetwo"arms"showninFig.13-5-5.Byproviding a suitablegatingwaveform, itispossibletomakethis"single-arm" implementa­ tionappeartobeequivalent tothe"two-arm" realization. Inthiscase,the cross-correlation isregularly sampled attwovaluesofdelay,bystepping the codeclockforwardorbackward intimebyanamount 8.Theenvelope ofthe cross-correlation thatissampled at±8hasanamplitude modulation whose phaserelativetothetau-dither modulator determines thesignofthetracking error. 750 DIGITAL COMMLNICATIONS FIGURE 13·5-6 Tau-dither loop(TDL). ILoop filterEnvelope detector veeBandpa~~ fillerRe'eivcd ~ignal Amajoradvantage oftheTDListhelesscostlyimplementation resulting fromelimination ofoneofthetwoarmsthatareemployed intheconventional DLL.Asecondandlessapparent advantage isthattheTDLdoesnotsuffer fromperformance degradation thatisinherent intheDLLwhentheamplitude gaininthetwoarmsisnotproperly balanced. TheDLL(anditsequivalent. theTDL)generate anerrorsignalby sampling thesignalcorrelation function at±<5offthepeakasshowninFig. l3-5-7(a). Thisgenerates anerrorsignalasshowninFig.13-5-7(b). The analysisoftheperformance oftheDLLissimilartothatfor\hephase-locked loop(PLL)carriedoutinSection6-3.Ifitwerenotfortheenvelope detectors inthetwoarmsoftheDLL.theloopwouldresemble aCostasloop.In general. thevariance ofthetimeestimation errorintheDLLisinversely proportional totheloopSNR,whichdepends ontheinputSNRtotheloop andtheloopbandwidth. Itsperformance issomewhat degraded asinthe squaring PLLbythenonlinearities inherent intheenvelope detectors, butthis degradation isrelatively small. FIGURE 13·5-' Autocorrelation function andtracking errorsignalforDLL. ¢',,(t) ell) 2T, T,,..'\,"',o=T.l2"0' ..,", ..(,"Tt:!.<5<T..",',~ ( , ",•,.....-cS=T,,,,,, """.T, ,"" ' "" ' ,,,..,"",, ~'..,'",...,,,,".,--T,.I> (a)Auwcorrelation function (b)Tracking ~rrorsignal CHAPTER l!l:SPREAD SPECTR'L'M SIGNALS FORDIGiTAL COMVlUN\CATIOSS 751 r-----+ Todemoou.lat<l( Recli:iwd signal VICr],...... Bandpass10-En\lelope~Lowpass '-".J flher detector filter V~(1)VJr>1 Frli:quency PNcode rveel ~ynthe~izer generator I (a)Tr<lding loopf(lrFHsignals Received frequencyj" I, f, LocalFH~L__;..:JO'-_--L__..:.J,-I__L-__f~2 L... _ ~=±bL1_UL--U +1 -1-Timingoffset=t • I (b)Wavefront fortracking anFHsignal FIGURE 13-~-8 Trraeking methodforFHsignals.[FromPickholtz etat(1982).©1982IEEE) Atypicaltrackingtechnique forFHspreadspectrum signalsisillustrated in Fig.13-5-8(a). Thismethodisalsobasedonthepremisethat,although initial acquisition hasbeenachieved, thereisasmalltimingerrorbetween the received signalandthereceiverclock.Thebandpass filteristunedtoasingle intermediate frequency anditsbandwidth isoftheorderof1/7;,whereT.is thechipinterval. Itsoutputisenvelope-detected andthenmultiplied bythe clocksignaltoproduceathree-level signal,asshown.inFig.13-5·8(b), which 752 DIGITAL CO....UNICATIONS drivestheloopfilter.Notethatwhenthechiptransitions fromthelocally generated sinusoidal waveform donotoccuratthesametimeasthetransitions intheincoming signal,theoutput of theloopfilterwillbeeithernegative or positive, depending onwhethertheVCCislaggingoradvanced relativetothe timingoftheinputsignal.Thiserrorsignalfromtheloopfilterwillprovidethe controlsignalforadjusting theVCCtimingsignalsoastodrivethe frequency synthesized pulsedsinusoid topropersynchronism withthereceived signal. 13-6BIBLIOGRAPHICAL NOTES ANDREFERENCES Theintroductory treatment ofspreadspectrum signalsandtheirperformance thatwehavegiveninthischapter isnecessarily brief.Detailed andmore specialized treatments ofsignalacquisition techniques, codetracking methods, andhybridspreadspectrum systems, aswellasothergeneraltopicsonspread spectrum signalsandsystems, canbefoundinthevastbodyoftechnical literature thatnowexistsonthesubject. Historically, theprimary application ofspreadspectrum communications hasbeeninthedevelopment ofsecure(AJ)digitalcommunication systemsfor military use.Infact,priorto1970,mostoftheworkonthedesignand development ofspreadspectrum communications wasclassified. Sincethen, thistrendhasbeenreversed. Theopenliterature nowcontains numerous publications onallaspectsofspreadspectrum signalanalysis anddesign. Moreover, wehaverecentlyseenpublications dealingwiththeapplication of spreadspectrum signaling techniques tocommercial communications suchas interoffice radiocommunications (seePahlavan, 1985)andmobile-user radio communications (seeYue,1983). Ahistorical perspective onthedevelopment ofspreadspectrum com­ munication systems covering theperiod1920-1960 isgiveninapaperby Scholtz(1982).Tutorial treatments focusing onthebasicconcepts arefoundin thepapersbyScholtz(1977)andPickholtz etal.(1982).Thesepapersalso containalargenumberofreferences toprevious work.Inaddition, thereare twopapersbyViterbi (1979,1985)thatprovide abasicreviewofthe performance charact«ristics ofDSandFHsignaling techniques. Comprehensive treatments ofvariousaspectsofanalysis anddesignof spreadspectrum signalsandsystems, including synchronization techniques are nowavailable inthetextsbySimonetal.(1985),ZiemerandPeterson (1985), andHolmes(1982).Inaddition tothesetexts,thereareseveralspecialissues oftheIEEETransactions onCommunications devoted 10spreadspectrum communications (August 1977andMay1982)andtheIEEETransactions on SelectedAreasinCommunication (September 1985,May1989,May1990,and June1993).Theseissuescontainacollection ofpapersdevoted toavarietyof topics,including multiple accesstechniques, synchronization techniques, and performance analyses withvarious typesofinterference. Anumber of important papersthathavebeenpublished inIEEEjournals havealsobeen reprinted inbookformbytheIEEEPress(Dixon, 1976;Cooketal.1983). FIGURE P13·2 PROBLEMSCHAPTER 1.\:SPREAD SPECrRlIM SJ<iNALS FORDI(jrfAl COMMUNICATIONS 753 10---------------------S~C1rum of inlerferenl.'e ~W,-- S"-----------W1«W Signal speclr-um I'W'1 Finally. werecommend thebookbyGolomb (1967)asabasicreference on shiftregister sequences forthereaderwhowishestodelvedeeperintothis topic. 13-1Following theprocedure outlined inExample 13-2-2,delermine theerrorrate performance ofaDSspreadspectrum systeminthepresence ofCWjamming whenthesignalpulseis g(t)~!16't·cos'[!J:(t-\7;)]. 0-<;1",;;7;y37; • 13-2ThesketchinFig.P13-2illustrates thepowerspectraldensities ofaPNspread spectrum signalandnarrowband interference inanuncoded (trivialrepetition code)digitalcommunication system.Referring toFig.13-2-6.whichshowsthe demodulator forthi,signal,sketchthe(approximate) spectralcharacteristics of thesignalandtheinterference afterthemultiplication of'(1)withtheoutputof thePNgenerator. Determine thefractionofthetotalinterference thatappearsat theoutputofthecorrelator whenthenumberofPNchipsperbitisL,. 13-3Consider theconcatenation ofaReed-Solomon (31.3)(q=32-aryalphabet) as theoutercodewithaHadamard (16,5)binarycodeastheinnercodeinaDS spreadspectrum system.Assumethatsoft-decision decoding isperformed onboth codes.Determine anupper(union)boundontheprobability ofabiterrorbased ontheminimum distance oftheconcatenated code. 13-4TheHadamard (n,k)=(2"',m+I)codesarelow-rate codeswithd.."=2"'. Determine theperformance ofthisclassofcodesforDSspread.spectrum signals withbinaryPSKmodulation andeithersoft-decision orhard-decision decoding. 13-5Arate1/2convolutional codewithd,,«=10isusedtoencodeadatasequence occurring atarateof1000bits/s.Themodulation isbinaryPSK.TheDS spread-spectrum sequence hasachiprateoflOMHz. aDetermine thecodinggain. bDetermine theprocessing gain. eDetermine thejamming marginassuming an'l:.IJ"=10. 13-6Atotalof30equal-power usersaretoshareacommon communication channel hy CDMA. Eachusertransmits information atarateof10kbitslsviaDSspread­ spectrum andbinaryPSK.Determine theminimum chipratetoobtainabiterror 754 D1lill.-\1 CO~I\ll':\IC.-HI()~S probability ofI(r'.Additive nOiseatthereceiver maybeignored inthis computation. 13-7ACDMAsystemisdesigned basedonDSspreadspectrum withaprocessing gain of1000andbinaryPSKmodulation. Determine thenumberofusersifeachuser hasequalpowerandthedesiredlevelofperformance isanerrorprobability of 10".Repeatthecomputation iftheprocessing gainischanged to500. 13-8ADSspread-spectrum systemtransmits atarateof1000bitslsinthepresence of atonejammer. Thejammerpoweris20dBgreaterthanthedesiredsignalandthe required'/.,IJ"toachievesatisfactory performance is10dB. aDetermine thespreading bandwidth required tomeetthespecifications. bIfthejammerisapulsejammer. determine thepulsedutycyclethatresultsin worst-case jamming andthecorresponding probability oferror. 13-9ACDMAsystemconsistsofISequal-power usersthattransmit information ata rateof10000bits/s.eachusingaDSspreadspectrum signaloperating atachip rateofIMHz.Themodulation isbinaryPSK. aDetermine the'fhll".whereJ"isthespectral density ofthecombined interference. bWhatistheprocessing gain0 cHowmuchshouldIheprocessing gainbeincreased toallowfordoubling the numberofuserswithoutaffecting theoutputSNR? 13-10AOSbinaryPSKspreadspectrum signalhasaprocessing gainof500.Whatisthe jamming marginagainstacontinuous-tone jammer ifthedesirederrorprobability is10'0 13·11RepeatProblem13-10ifthejammerisapulsed-noise jammerwithadutycycleof 1%. 13-12Consider IheDSspreadspeclfum signal x C(I)=2:CnP(1-nT,) where Cnisaperiodic msequence withaperiodN=127andp(l)isarectangular pulseofduration T,=I/-,s.Determine thepowerspectral densityofthesignal C(I). 13-13Suppose that{c,,}and{c,,)aretwobinary(O,I)periodic sequences withperiodsN, andN,.respectively. Determine theperiodofthesequence obtained byforming themodulo-2 sumoflei,}and{c,,}. 13-14Anm=10MLshiftregisterisusedtogenerate thepseudorandom sequence ina DSspreadspectrum system.Thechipduration isT.=I/-,S.andthebitduration is Tb=NT.,whereNisthelength(period)ofthemsequence. aDetermine theprocessing gainofthesystemindB. bDetermine thejamming marginiftherequIred't,IJo=10andthejammer isa tonejammerwithanaveragepoweri". 13-15AFHbinaryorthogonal FSKsystememploys anm=ISstagelinearfeedback shiftregisterthatgenerates anMLsequence. Eachstateoftheshiftregisterselects oneofLnonoverlapping frequency bandsinthehopping pattern. Thebitrateis lOObits/s andthehoprateisonceperbit.Thedemodulator employs noncoherent detection. aDetermine thehopping bandwidth forthischannel. bWhatistheprocessing gain? cWhatistheprobability oferrorinthepresence ofAWGN? CHAPTER l.~:SPREAD SPECTRUM SIGNALS FORDJGiTAL COM\1L'~ICATIO:"<JS 755 U-16Consider theFHbinaryorthogonal FSKsystemdescribed inProblem 13-15_ Suppose thatthehoprateisincreased to2hops/bit. Thereceiver usessquare-law combining tocombine the signal overthetwohops. aDetermine thehopping bandwidth forthechannel. bWhatistheprocessing gain? cWhatistheerrorprobability inthepresence ofAWGN? U-17InafastFHspread-spectrum system,theinformation istransmilled viaFSK,with noncoherent detection. Suppose thereareN=3hops/bit, withhard-decision decoding ofthesignalineachhop. aDetermine theprobability oferrorforthissysteminanAWGNchannel with powerspectral density ~N"andanSNR=13dB(totalSNRoverthethree hops). bCompare theresultin(a)withtheerrorprobability ofaFHspread-spectrum systemthathopsonceperbit. 13-18AslowFHbmaryFSKsystemwithnoncoherent detection operates at"t:,/J"=10. withahopping bandwidth of2GHz,andabitrateof10kbits/s. aWhatistheprocessing gainforthesystem',' bIfthejammer operates asapartial-band jammer, whatisthebandwidth occupancy forworst-case jamming') cWhatistheprobability oferrorfortheworst-case partial-band jammerO 13-19Determine theerrorprobability foraFHspreadspectrum signalinwhichabina" convolutional codeisusedincombination withbinaryFSK.Theinterference on thechanneli"AWGN.TheFSKdemodulator outpulsaresquare-law detected and passedtothedecoder. whichperforms optimum soft-decision Viterbidecoding a, described inSection8-2.Assume thatthehopping rateis1hoppercodedbit. 13-20RepeatProblem 13-19forhard-decision Viterbidecoding. 13-21RepeatProblem 13-19whenfastfrequency hopping isperformed atJhopping rate ofLhopspercodedbit. 13-22RepeatProblem 13-19whenfastfrequency hopping isperformed withLhopsper codedbitandthedecoder isahard-decision Viterbidecoder. TheLchipsper codedbitaresquare-law-detected and combined priortotheharddecision. 13-23TheTATSsignaldescribed inSection13-3-3isdemodulated byaparallelbankof eightmatched filters(octalFSK),andeachfilteroutputissquare-law-detected. Theeightoutputsobtained ineachofsevensignalintervals (56totaloutputs) are usedtoformthe64possible decision variables corresponding tntheReed­ Solomon (7,2)code.Determine anupper(union)boundofthecodeworderror probability forAWGNandsoft-decision decoding. 13·24RepeatProblem 13-23fortheworst-case partial-band interference channel. 13·25Derivetheresultsin(13-2-62) and(13-2-63) from(13-2-61). 13-26Showthat(13-3-14) followsfrom(13-3-13). 13·27Derive(13-3-17) from(13-3-16), 13-28Thegenerator polynomials forconstructing Goldcodesequences oflengthn~~ are g,(p)=p'+p +I g,(p)=p'+p'+1 Generate alltheGoldcodesoflength7anddetermine thecross-correlations at onesequence witheachoftheothers. 756 DIGITAL COMMUNICATIONS Unionbound Continuous gaussian ooise Binaryphaseshiftkeying Rate113convolutiona~ ~ withVitemidecodina Softdecisicn.s10-) 10-< K=4.:gK=5 • :E.11r' K=7'0 .~ K=8] i Ilr' 11r'L,-_.L 2__-'--_--'_-1-WL-.L-.L.1._LL_.L--' FIGURE Pl3-29 13-29InSection13-2-3,wedemonstrated techniques forevaluating theerrorprobability ofacodedsystemwithinterleaving inpulseinterference byusingthecutoffrate parameter Ro.Usetheerrorprobability curvesgiveninFig.P13·29forrate1/2 and1/3convolutional codeswithsoft-decision Viterbidecoding todetermine the corresponding errorratesforacodedsysteminpulseinterference. Perform this computation forK=3,5,and7. 13-30Incodedandinterleaved DSbinaryPSKmodulation withpulsejamming and soft-decision decoding, thecutoffrateis R".=1-log,(1+ae-••,_'N") whereaisthefractionofthetimethesystemisbeingjammed,if.='l.R,Risthe bitrate,andNo;510, aShowthattheSNRperbit,'t.IN",canbeexpressed as 'l.1 a-=~In-,-;--;;--c­NoaR2\Ro_1 bDetermine thevalueofathatmaximizes therequired 'i.IN"(worst-case pulse jamming) andtheresulting maximum valueof'/;.1No_ bPlotthegraphof10logCC.lrNo)versusRo'wherer=R"IR,for""orst-case pulsejamming andforAWGN(a=1).Whatconclusions doyouleach regarding theeffectofworst-case pulsejamming? CHAPTE~ 13:SPREAD SPE.CTRUM SIGNALS FaiRDtGIT"'l. COMMU!'i1CA.TIONS 757 It)Unionbound Continuous. gaus:sian noise Binaryphaseshiftkeying Rate112convolutional code withViterbidtcoding Softdecisions 10-' ..<K=4 ~K=S .c•I<C' K=7'0 ~ K=8E• K=9~£ I~ I<C'L_---'-__-'-_--'_-'l.-'-'U_.L......L-I.-''-- 1 2 3 4 5 6 7 <"'No(dB) FIGURE Pl3-29 (Continued), 13-31Inacodedandinterleaved frequency-hopped q-aryFSKmodulation withpartial bandjamming andcoherent demodulation withsoft-decision decoding, thecutoff rateis Ro=log,[I+(q_I~ae.'<i2N,] wherecristhefractionofthebandbeingjammed, Z',isthechip(ortone)energy, andNo=Jo' •ShowthattheSNRperbitcanbeexpressed as Z'b=~ln (q-I)cr NoaRq2RO-1 bDetermine thevalueofathatmaximizes therequired Z'blNo(worst-case partial bandjamming) andtheresulting maximum valueofZ'bINo, £Definer=Ro/Rin theresultforZ',INofrom (b),andplotlOloglZ'blrNo) versus thenormalized cutoffrateRo/log,qforq=2,4,8,16,32,Compare these graphswiththeresultsofProblem 13-3O(c)_ Whatconclusions doyoureach regarding theeffectofworst-case partialbandjamming? Whatistheeffectof increasing thealphabet sizeq?WhatisthepenaltyinSNRbetween theresults inProblem 13-30(c) andq-aryFSKasq-+oo? 14 DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS Theprevious chapters havedescribed thedesignandperformance ofdigital communications systems fortransmission oneithertheclassical AWGN channeloralinearfilterchannelwithAWGN.Weobserved thatthedistortion inherent inlinearfilterchannels requires specialsignaldesigntechniques and rathersophisticated adaptive equalization algorithms inordertoachievegood performance. Inthischapter, weconsider thesignaldesign,receiver structure. and receiver performance formorecomplex channels, namely, channels having randomly time-variant impulse responses. Thischaracterization servesasa modelforsignaltransmission overmanyradiochannels suchasshortwave ionospheric radiocommunication inthe3-30MHzfrequency band(HFj, tropospheric scatter(beyond-the-horizon) radiocommunications inthe300­ 3000MHzfrequency band(UHF)and3000-30000 MHzfrequency band (SHF),andionospheric forward scatterinthe30-300MHzfrequency band (VHF).Thetime-variant impUlse responses ofthesechannels areaconse­ quenceoftheconstantly changing physical characteristics ofthemedia.For example, theionsintheionospheric layersthatreflectthesignalstransmitted intheHFfrequency bandarealwaysinmotion.Totheuserofthechannel, the motionoftheionsappearstoberandom. Consequently, ifthesamesignalis transmitted atHFintwowidelyseparated timeintervals, thetworeceived signalswillbedifferent. Thetime-varying responses thatoccuraretreatedin statistical terms. Weshallbeginourtreatment ofdigitalsignalling overfadingmultipath channels byfirstdeveloping astatistical characterization ofthechannel. Then weshallevaluate theperformance ofseveralbasicdigitalsignaling techniques forcommunication oversuchchannels. Theperformance resultswilldemons- 758 CIIAvn'l{ !~.IJI(i!Ii\l ("OMM,"!\J("AIlO!\ THKllC(iH FADlMi \.n:UWAIH (HA'lt-jElS 759 tratetheseverepenalty inSNRthatmustbepaidasaconsequence ofthe fadingcharacteristics ofthereceived signal.Weshallthenshowthatthc pcnalty inSNRcanbedramatically reduced bymeansofefficicnt modulation/coding anddemodulation/decoding techniques. 14-1CHARACTERIZATION OFFADING MULTIPATH CHANNELS Ifwetransmit anextremely shortpulse.ideallyanimpulse. ovcratime-varying multipathchannel. thereceived signalmightappearasatrainofpulses.as showninFig.14-1-1.Hence.one characteristic ofamultipath medium isthe timespreadintroduced inthesignalthatistransmitted throughthechannel. Asecondcharacteristic isduetothetimevariations inthestructure ofthe medium. Asaresultofsuchtimevariations. thenatureofthemultipath varies withtime.Thatis.ifwerepeatthepulse-sounding experiment overandover, weshallobservechanges inthereceived pulsetrain.whichwillincludechanges inthesizesoftheindividual pulses,changes intherelativedelaysamongthe pulses.and.quiteoften.changes inthenumber ofpulsesobserved inthe received pulsetrainasshowninFig.14-1-1.Moreover. thetimevariations appeartobeunpredictable totheuserofthechannel. Therefore. itis reasonable tocharacterize thetime-varian tmultipathchannel statistically. Transmined signal (al Ib) (e)t=IItI::::t1+"tI: !=Iitti. 1=11tI::::I,+t~:t I::::t1tt111=I,+t,\ n~nnn 1=/1tl=t1+~~1tt::::f1+tJ4 r=t1+'11 '=I,"'--::1' FIGURE 14-1·t Example oftheresponse ofatime-variant multipath channel 10averynarrowpulse.'d)nn 760 DIGITAL COMMllNlCAllONS Towardthisend,tetusexamine theeffectsofthechannelonatransmitted signalthatisrepresented ingeneralas (14-1-1) Weassumethattherearemultiple propagation paths.Associated witheach pathisapropagation delayandanattenuation factor.Boththepropagation delaysandtheattenulltion factorsaretime-variant asaresultofchangesinthe structure ofthemedium. Thus,thereceived bandpass signalmaybeexpressed intheform X(I)=2:a.(t)s(t-,At» n(14-1-2) (14-1-3)whereClIn(l)istheattenuation factorforthesignalreceived onthenthpath andrAt)isthepropagation delayforthenthpath.Substitution fors(t)from (14-1-1)into(14-1-2)yieldstheresult x(t)=Re{[~"'.(t)e-j2K;f.T_I'!S,(t -'n(tl)Jei2</.'} Itisapparent from(14-1-3)thattheequivalent lowpassreceived signalis (14-1-4) n Sincer,(t)istheresponse ofanequivalent lowpasschanneltotheequivalent lowpasssignals,(t),itfollowsthattheequivalent lowpasschannelisdescribed bythetime-variant impulseresponse (14-1-5) n Forsomechannels, suchasthetropospheric scatterchannel, itismore appropriate toviewthereceived signalasconsisting ofacontinuum of multipath components. Insuchacase,thereceived signalx(t)isexpressed in theintegralform x(t)=[~a(r;t)S(1-r)dr (14-1-7)wherea(r;I)denotestheattenuation ofthesignalcomponents atdelayrand attimeinstantI.Nowsubstitution forS(I)from(14-I-l)into(14-1-6)yields x(t)=Re{[[~a(r;l)e-j2'/'Ts,(t-r)dr]ei2Kf"} Sincetheintegral in(14-1-7) represents theconvolution ofS,(t)withan equivalent lowpasstime-variant impulseresponse c(r;I),itfollowsthat c(r;t)=er(r;t)e-j2.{.r (14-1-8) rHAPTER 14:[}ICiIlAL (,OMMlr~HCATION THROUGH FADING MUlTrrATH CHAN~FLS 761 wherec(r:t)represents theresponse ofthechannelattime1duetoanimpulse applied attime1-r.Thus(14-1-8) istheappropriate definition ofthe equivalent lowpassimpulseresponse whenthechannel resultsincontinuous multipath and(14-1-5) isappropriate forachannel thatcontains discrete multipathcomponents. Nowletusconsider thetransmission ofanunmodulated carrieratfrequency [..Thens,(I)=1forallI.and,.hence,thereceived signalforthecaseof discretemultipath, givenby(14-1 ~4),reducesto r,(I)=2:a"(I)e-j2.r.,,,(I) " (14-1-9) " where8,(1)=2lrj;.r,,(t).Thus.thereceived signalconsists ofthesumofa numberoftime-variant vectors(phasors) 'havingamplitudes 0,,(1)andphases e,,(r).Notethatlargedynamic changesinthemedium arerequired for£1,,(1)to changesufficiently tocauseasignificant changeinthereceived signal.Onthe otherhand,e,,(I)willchangeby2rrradwhenever rnchanges by1If,..ButIIf,is asmallnumber and,hence,encanchangeby21(radwithrelatively small motionsofthemedium. Wealsoexpectthedelaysrn(l)associated withthe different signalpathstochangeatdifferent ratesandinanunpredictable (random) manner. Thisimpliesthatthereceived signalr,(I)in(14-1-9) can be modeled asarandomprocess. Whentherearealargenumberofpaths,the centrallimittheorem canbeapplied. Thatis,r,(t)maybemodeled asa complex-valued gaussian random process. Thismeansthatthetime-variant impulseresponse c(r:I)isacomplex-valued gaussian randomprocessintheI variable. Themultipathpropagation modelforthechannelembodied inthereceived signalr,(I).givenin(14-1-9), resultsinsignalfading.Thefadingphenomenon isprimarily aresultofthetimevariations inthephases{9n(I)}.Thatis,the randomly time-variant phases{9n(t)}associated withthevectorslane-J.,,}at timesresultinthevectorsaddingdestructively. Whenthatoccurs,theresultant received signalr,(I)isverysmallorpracticaIly zero.Atothertimes,thevectors lane-j.,,}addconstructively, sothatthereceived signalislarge.Thus,the amplitude variations inthereceived signal,termedsignalfading,areduetothe time-variant multipath characteristics ofthechannel. Whentheimpulse response c(r:I)ismodeled asazero-mean complex­ valuedgaussian process, theenvelope Ic(r;1)1atanyinstant tisRayleigh­ distributed. InthiscasethechannelissaidtobeaRayleigh fadingchannel. In theeventthattherearefixedscaUerers orsignalreflectors inthemedium, in addition torandomly movingscaUerers, c(r;I)cannolongerbemodeled as havingzeromean.Inthiscase,theenvelope Ic(r;1)1hasaRicedistribution andthechannel issaidtobeaRiceanfadingchannel. Another probability distribution function thathasbeenusedtomodeltheenvelope offading 762 D!(;ITAI. COM.\1l"SJCATJONS signalsistheNakagami.m distrihution. Thesefadingchannel models are considered inSection14-1-2. 14-}-}Channel Correlation Functions andPowerSpectra Weshallnowdevelop anumher ofusefulcorrelation functions andpower spectral densityfunctions thatdefinethecharacteristics ofafadingmultipath channel. Ourstartingpointistheequivalent lowpass impulse response c(r;t), whichischaracterized asacomplex-valued random processinthetvariable. WeassumethaIc(r;t)iswide-sense-stationary. Thenwedefinetheautocor­ relation function ofc(r:t)as (14-1-10) Inmostradiotransmission media,theattenuation andphaseshiftofthe channel associated withpathdelayr\isuncorrelated withtheattenuation and phaseshiftassociated withpathdelayr2'Thisisusuallycalleduncorrelated scattering. Wemaketheassumption thatthescattering attwodifferent delays isuncorrelated andincorporate itinto(14-1-10) toobtain (14-1-11) IfweletAt=O.theresulting autocorrelation function <pe(r;0)'"<p,(r)is simplytheaveragepoweroutputofthechannel asafunctionofthetimedelay r.Forthisreason,<p,(r)iscalledthemultipath intensity profileorthedelay powerspectrum ofthechallnel. Ingeneral, cPe(r;At)givestheaverage power outputasafunction ofthetimedelayrandthedifference Atinobservation time. Inpractice, thefunction <p,.(r;At)ismeasured bytransmitting verynarrow pulsesor,equivalently, awidebandsignalandcross-correlating thereceived signalwithadelayed versionofitself.Typically, themeasured function <p,(r) mayappearasshowninFig.14-1-2.Therangeofvaluesofroverwhich<pe(r) FIGURE 14-1·2 Multipath intensity profile.-,;j- -=-_t T.._--I_I CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS 763 isessentially nonzero iscalledthemultipath spreadofIhechannelandis denoted byTm. Acompletely analDgDus characterization Dfthetime-variant multipath channelbeginsinthefrequency domain. BytakingtheFouriertransform Df c(r;I)weDbtainthetime-variant transfer functiDn C(f;t),wherefisthe frequency variable. Thus, C(f;t)= r~c(r;l)e-j2r[Tdr (14-1-12) Ifc(r;I)ismDdeled asacomplex-valued zerD-mean gaussian randomprocess inthetvariable, itfollowsthatC(f;t)alsohasthesamestatistics. Underthe assumption thatthechannel iswide-sense-stationary, wedefinetheautocor­ relationfunctiDn (14-1-13) SinceC(f;t)istheFouriertransfDrm ofc(f;t),itisnotsurprising tDfind thatcPdft.f2;Ilt)isrelatedtocPC<r;At)bytheFDurier transfDrm. The relationship iseasilyestablished bysubstituting (14-1-12) intD(14-1-13). Thus. (14-1-14) whereIlf=Ii-fl'From(14-1-14), weDbservethatcPc(tl.f;tl.t)istheFDurier transform Dfthemultipathintensity profile.Furthermore, theassumption of uncorrelated scattering impliesthattheautDcorrelation function ofC(f:I)in frequency isafunctionofonlythefrequency difference df=Ii-f,.Therefore. itisappropriate tDcallcPddf;tl.t)thespaced-frequency, spaced-time correla­ tionfunctionofthechannel.Itcanbemeasured inpracticebytransmitting a pairofsinusDids separated byAfandcross-correlating thetwoseparately received signalswitharelativedelayAt. Suppose weset4t=0in(14-1-14). Then,withcPdM:O)=<J>etAf) and <l>Af;0)=cPc(f).thetransform relationship issimply (14-1-15) 764 l>IGlTAL COMMUNICATIONS .c<A.[).. Fourier. +c(t) transform pair Spaced-frequency correlation function FIGURE 14-1-3 Relationship between 4>c(t:J.!)and4>,(f).Multipath illtensityprofile Therelationship isdepicted graphically inFig.14-1-3.Sincet/JcCtJ.j)isan autocorrelation function inthefrequency variable, itprovides uswitha measure ofthefrequency coherence ofthechannel. AsaresultoftheFourier transform relationship between t/Jc(lif) and</Jc(r),thereciprocal ofthe multipath spreadisameasureofthecoherence bandwidth ofthechannel. That is, (14-1-16) where(Af)cdenotes thecoherence bandwidth. Thus,twosinusoids with frequency separation greaterthan(Ancareaffecteddifferently bythechannel. Whenaninformation-bearing signalistransmitted through thechannel, if (Af)"issmallincomparison tothebandwidth ofthetransmitted signal,the channel issaidtobefrequency-selective. Inthiscase,thesignalisseverely distorted bythechannel. Ontheotherhand,if(Af),islargeincomparison withthebandwidth ofthetransmitted signal,thechannel issaidtobe frequency-nonselective. Wenowfocusourattention onthetimevariations ofthechannel as measured bytheparameter AIin4>c(Af;AI).Thetimevariations inthe channelareevidenced asaDoppler broadening and,perhaps, inaddition asa Doppler shiftofaspectralline.InordertorelatetheDoppler effectstothe timevariations ofthechannel, wedefinetheFouriertransform of"'dAf;lit) withrespecttothevariable Attobethefunction Sc(t1f;A).Thatis, (14-1-17) (14-1-18)WithAfsettozeroandSc(O;A)=Sc(A), therelationin(14-1-17) becomes Sc(A)=r~t/Jc(At)e-j2",,,dAt nlAPTER l.a:DIGITAL COMMUNICATION THROUGH fADING MVLTIPATH CHANNELS '65 o.(AT).Fourier. 5(0..1 ( transform pair -::.----01;----- dr +-- (At),=t------- d Spaced-lime correlation function FlGURE 14-1·4 Relationsnip between cl>C<~()andSetA),o_8./_ Doppler powerspectrum ThefunctionSd,l)isapowerspectrum thatgivesthesignalin\ensi'y asa function oftheDoppler frequency ,I,Hence,wecallSdA)theDoppler power spec/rum ofthechannel, From(14-1-18), weobserve thatifthechannelistime-in,variant, <pdAt)=I andSc(A)becomes equaltothedeltafunction 0(,1,),Therefore, whenthereare notimevariations inthechannel, thereisnospectralbroadening observed in thetransmission ofapurefrequency tone. TherangeofvaluesofAoverwhichSd,l)isessentially nonzero iscalledthe Doppler spreadBdofthechannel, SinceSc(,I)isrelatedtocbc{!1t)bythe Fouriertransform, thereciprocal ofBdisameasure ofthecoherence timeof thechannel. Thatis, (14-1-19) where(!1/),denotesthecoherence time,Clearly,aslowlychanging channelhas alargecoherence timeor,equivalently, asmallDoppler spread,Figure14-1-4 illustrates therelationship between </Jc(!1/)andSc(A), Wehavenowestablished aFourier transform relationship between <Pc(!1f;!11)and<p,(r;AI)involving thevariables (r,!1f),andaFourier transform relationship between tl>c(!1f;At)andSc(M;A)involving thevari­ ables(At,A),Therearetwoadditional Fouriertransform relationships thatwe candefine,whichservetorelate<Pe(r;At)toSdAf;A)and,thus,closethe loop.Thedesiredrelationship isobtained bydefining anewfunction, denoted byS(r;A),tobetheFouriertransform of<Pe(r;AI)intheAtvariable. Thatis, (14-1-20) ItfollowsthatS(r;A)andSc(!1f;A)areaFouriertransform pair.Thatis, (14-1-21) (14-1-22)766 DIGITAL COMMl!!'<!CATIONS Furthermore, S(r:A)andcf>dtJ.[:tJ.t)arerelated bythedouble Fourier transform S(r:A)=rooL~cPc<tJ.j:!1t)e-".Ule'2'":'ldt!J.tdtJ.[ 14Jisnewfunction S(r:A)iscalledthescattering junction ojtltechannel. It provides uswithameasure oftheaverage poweroutputofthechannel asa function ofthetimedelayrandtheDoppler frequency A. Therelationships among thefourfunctions cf>c(t:.[:tJ.t),cf>c(r:At), <t>c<tJ.[:A).andS(r:A)aresummarized inFig.14-1-5. FIGURE 14-1-5 Relationships amongthechannel correlation functions andpowerspectra. [FromGreen(1%2). withpermission.] Ifou,;e, transformFourier tran!i.((>rm SIt;A)1'1 -L..----c~---"- 1"o _1/7"'-----. 1Fum'., transform Scattering function (Ht\J'TLR 1-1:DI(jIlAL. ("OM\1I'SI(ATIO~ THROUiti t-A[)I~(j \1ITTIPA rH(H,\......"EL"i767 II" 9, ">. 7f"~6t S! f\ "- A~u,, "2'" 1 0./ 2./,....- ,/.1./ -= ./. .7' A""./'S .7' --;;7""'C ./6 .7 ./7./8 ..7'9./ , 710 -100-80--60-40-20020406080100 Frequency (Hz/ FIGURE 14-1-6 Scattering function ofamedium-range tropospheric scatterchannel. Thetapsdelayincrement is 0.1p.s. Thescattering function 5(,;A)measured ona150mitropospheric scatter linkisshowninFig.14-1-6.Thesignalusedtoprobethechannelhadatime resol\1tion ofO.lp.s.Hence,thetime-delay axisisquantized inincrements of O.1p.s.Fromthegraph,weobservethatthemultipath spreadTm=0.7p.s.On theotherhand,theDoppler spread,whichmaybedefined asthe3dB bandwidth ofthepowerspectrum foreachsignalpath,appearstovarywith eachsignalpath.Forexample, inonepathitislessthan1Hz,whileinsome otherpathsitisseveralhertz.Forourpurposes, weshalltakethelargestof these3dBbandwidths ofthevariouspathsandcallthattheDoppler spread. 14-1-2Statistical ModelsforFadingChannels Thereareseveralprobability distributions thatcanbeconsidered inattempting tomodelthestatistical characteristics ofthefadingchannel. Whentherearea largenumberofscatterers inthechannelthatcontribute tothesignalatthe receiver, asisthecaseinionospheric ortropospheric signalpropagation, application ofthecentrallimittheorem leadstoagaussian processmodelfor thechannelimpulseresponse.Iftheprocessiszero-mean, thentheenvelope of thechannelresponse atanytimeinstanthasaRayleigh probability distribution andthephaseisuniformly distributed intheinterval(0,2Jr).Thatis, ()2r-,'10 Pr=-e Rn'r;;'O (14-1-23) 768 DIGITAL COMMU1'tc ATlONS where (14-1-24) Weobserve thattheRayleigh distribution ischaracterized bythesingle parameter E(R2). Analternative statistical modelfortheenvelope ofthechannelresponse is theNakagami-m distribution givenbythepdfin(2-1-147). Inconstrast tothe Rayleigh distribution, whichhasasingleparameter thatcanbeusedtomatch thefadingchannelstatistics, theNakagami-m isatwo-parameter distribution, namely,involving theparameter mandthesecondmoment Q=E(R2).Asa consequence, thisdistribution provides moreflexibility andaccuracy in matching theobserved signalstatistics. TheNakagami-m distribution canbe usedtomodelfadingchannelconditions thatareeithermoreorlesssevere thantheRayleigh distribution, anditincludes theRayleigh distribution asa specialcase(m=1).Forexample, Turin(1972)andSuzuki(1977)haveshown thattheNakagami-m distribution isthebestfitfordatasignalsrect;ived in urbanradiomultipath channels. TheRicedistribution isalsoatwo-parameter distribution. Itmaybe expressed bythepdfgivenin(2-1-141), wheretheparameters aresandu2• Recallthats"iscalledthenoncentrality parameter intheequivalent chi-square distribution. Itrepresents thepowerinthenonfading signalcomponents, sometimes calledspecular components, ofthereceived signal. Therearemanyradiochannels inwhichfadingisencountered thatare basically line-of-sight (LOS)communication linkswithmultipath components arisingfromsecondary reflections, orsignalpaths,fromsurrounding terrain.In suchchannels, thenumberofmultipath components issmall,and,hence,the channelmaybemodeled inasomewhat simplerform.Wecitetwochannel modelsasexamples. Asthefirstexample, letusconsider anairplane togroundcommunication linkinwhichthereisthe direct pathandasinglemultipath component ata delaytorelativetothedirectpath.Theimpulseresponse ofsuchachannelmay bemodeled as cit;t)=a~(t)+j3(/)~(t-to(/» (14-1-25) whereaistheattenuation factorofthedirectpathandP(t)represents .the time-variant multipath signalcomponent resulting fromterrainreflections. Often,(3(t)canbecharacterized asazero-mean gaussian randomprocess.The transferfunction forthischannelmodelmaybeexpressed as C(f;t)=a+(3(t)e-j2K!r,/I) (14-1-26) TlrischannelfitstheRiceanfadingmodeldefinedpreviously. Thedirectpath withattenuation arepresents thespecular component and/3(t)represents the Rayleigh fadingcomponent. Asimilarmodelhasbeenfoundtoholdformicrowave LOSradiochannels CHAPTER 14:DIGlT-\L COMMUNICATION THROUGH FADING MLlLTIPATH CHA!'IINELS 769 usedforlong-distance voiceandvideotransmission bytelephone companies throughout theworld.Forsuchchannels, Rummier (1979)hasdeveloped a three-path modelbasedonchannelmeasurements perfonned OIltypicalLOS linksinthe6GHzfrequency band.Thedifferential delayonthetwomUltipath components isrelatively small,and,hence,themodeldeveloped byRummier isonethathasachanneltransferfunction (14-1-27) whereaistheoverallattenlJation parameter, {3iscalledashapeparameter whichisduetothemultipath components, IIIisthefrequency ofthefade mmimum, andTvistherelativetimedelaybetween thedirectandthe mlJltipath components. Thissimplified modelwasusedtofitdataderivedfrom channelmeasurements. Rumml~ foundthattheparameters aand{3maybecharacterized as randomvariables that,forpractical purposes, arenearlystatistically indepen­ dent.Fromthechannelmeasurements, hefOlJndthatthedistribution of{3has theform(1-(3),3Thedistribution ofaiswellmodeled bythelognormal distribution. i.e.,-logaisgaussian. For{3>0.5,themeanof-20logawas foundtobe25dBandthestandard deviation was5dB.Forsmallervaluesof (3,themeandecreases to15dB.Thedelayparameter determined fromthe measurements wasTil=6.3ns.Themagnitude-square response ofC(f)is 1C(f)I'=a2[1+(32-2{3cos2tr(f-fo)To] (14-1-28) 1C(f)1isplottedinFig.14-1-7asafunction ofthefrequencyf-tofor To=6.3ns.Notethattheeffectofthemultipathcompo.nent istocreateadeep attenuation at1=10andatmultiples oflITo""159MHz.Bycomparison, the typicalchannelbandwidth is30MHz.ThismodelwasusedbyLundgren and RlJmmler (1979)todetermine theerrorrateperformance ofdigitalradio systems. 250 200 1501--1 Channel bandwidth 100 !-fn<MHz)O.GI0.02Magnitude freqllency response ofLOSchannelmodel 0.03FlGURE 14-1·7 770 Dl(jJTAl COMMVNICATJONS 14-2THEEFFECT OFSIGNAL CHARACTERISTICS ONTHECHOICE OFACHANNEL MODEL Havingdiscussed thestatistical characterization oftime-variant mUltipalh channels generally intermsofthecorrelation functions described inSection 14-1,wenowconsider theeffectofsignalcharacteristics ontheselection ofa channelmodelthatisappropriate forthespecified signal.Thus,lets,(t)bethe equivalent lowpasssignaltransmitted overthechannelandletS,(f)denoteits frequency content. Thentheequivalent lowpassreceived signal,exclusive of additive noise,maybeexpressed eitherintermsofthetimedomainvariables c(r;t)andStet)as orintermsofthefrequency functions C(f;r)andS,(f)as ,,(t)=[~C(f;t)S,(/)ei2rf/df(14-2-1) (14-2-2) Suppose wearetransmitting digitalinformation overthechannel by modulating (eitherinamplitude, orinphase,orboth)thebasicpulses,(t)ata rateliT.whereTisthesignaling interval. Itisapparent from(14-2-2)thatthe time-varia»t channelcharacterized bythetransferfunctionC(f;t)distortsthe sigl1alS,(f).IfS,(/)hasabandwidth Wgreaterthanthecoherence bandwidth (!if)eofthechannel,S,(/)issubjected todifferent gainsandphaseshiftsacross theband.Insuchacase,thechannel issaidtobefrequency-selectiue. Additional distortion iscausedbythetimevariations incrf;t).Thistypeof distortion isevidenced asavariation inthereceived signalstrength, andhas beentermedfading.Itshouldbeemphasized thatthefrequency selectivity and fadingareviewedastwodifferent typesofdistortion. Theformerdepends on themultipath spreador,equivalently, onthecoherence bandwidth ofthe channelrelativetothetransmitted signalbandwidth W.Thelatterdepends on thetimevariations ofthechannel, whicharegrosslycharacterized bythe coherence time(at)eor,equivalently, bytheDoppler spreadBd• Theeffectofthechannelonthetransmiued signals,(t)isafunctionofour choiceofsignalbandwidth andsignalduration. Forexample, ifweselectthe signaling intervalTtosatisfytheconditionT»Tm•thechannel introduces a negligible amountofintersymbol interference. Ifthebandwidth ofthesignal pulses,(t)isw""liT,theconditionT»T",impliesthat 1W«T.'",,(an m(14-2-3) Thatis,thesignalbandwidth Wismuchsmallerthanthecoherence bandwidth ofthechannel. Hence,thechannelisfrequency-nonselective. Inotherwords, CHAPTER 14:DIGITAL COMMllNICAllON THROUGH FADING MULTIPATHCHANNELS 771 allofthefrequency components inS,(f)undergo thesameattenuation and phaseshiftintransmission through thechannel. Butthisimpliesthat,within thebandwidth occupied byS,(f),thetime-variant transferfunction C(f;I)of thechannelisacomplex-valued constant inthefrequency variable. SinceS,(!) hasitsfrequency contentconcentrated inthevicinityoff=0,C(f;I)=C(O;I). Consequently, (14-2-2)reducesto r,(c)=C(O;c)r~S,(f)ei2'if'df =C(o;l)s,(C) (14-2-4) Thus,whenthesignalbandwidth WismuchHnallerthanthecoherence bandwidth (tin,ofthechannel, thereceived signalissimplythetransmitted signalmultiplied byacomplex-valued random process C(O;f),whichrep­ resentsthetime-variant characteristics ofthechannel. Inthiscase,wesaythat themultipath components inthereceived arenotresolvable becauseW« (!1f).. Thetransferfunction C(O;c)forafrequency-nonselective channelmaybe expressed intheform C(O:c)=a(c)e'j</><o (14-2-5) whereaCt)represents theenvelope andc/>(I)represents thephaseofthe equivalent lowpasschannel. WhenC(O;c)ismodeled asazero-mean complex­ valuedgaussian randomprocess,theenvelope a(l)isRayleigh-distributed for anyfixedvalueofIandc/>(c)isuniformly distributed overtheinterval(-]f,]f). Therapidityofthefadingonthefrequency-nonselective channelisdetermined eitherfromthecorrelation function c/>e(!1c)orfromtheDoppler power spectrum SeCA).Alternatively, eitherofthechannelparameters (111),orBdcan beusedtocharacterize therapidityofthefading. Forexample, suppose itispossible toselectthesignalbandwidth Wto satisfythecondition W«(tifL·andthesignaling interval Ttosatisfy.the condition T«(til),..SinceTissmallerthanthecoherence timeofthecha~nel. thechannelattenuation andphaseshiftareessentially fixedfortheduration of atleastonesignaling interval. Whenthiscondition holds,wecallthechannela slowlyfadingchannel. Furthermore, whenW""liT,theconditions thatthe channelbefrequency-nonselective andslowlyfadingimplythattheproductof TmandB"mustsatisfythecondition TmBd<1. TheproductT."Bdiscalledthespreadfaclorofthechannel.IfTmBd<1,the channel issaidtobeunderspread; otherwise, itisoverspread. Themultipath spread,theDoppler spread,andthespreadfactorarelistedinTable14-2-1for severalchannels. Weobserve fromthistablethaIseveralradiochannels, including themoonwhenusedasapassivereflector, areunderspread. Consequently, itispossibletoselectthesignalS,(I)suchthatthesechannels arefrequency-nonselective andslowlyfading.Theslow-fading condition m DIGITAL COMMUNICATIONS TABLE 14-2-1 MULTIPATH SPREAD. DOPPLER SPREAD. ANDSPREAD FACTOR J;'ORSEVERAL TIME·VARIANT MUlTIPATH CHANNELS Multipatb Doppler Sprelld Typeofrhunel dutltion spre'" flldor Shortwave ionospheric 10''·1 10"·10' ,propagation (HF) 10"-10' , Ionospheric propagation underdisturbed auroral conditions (HF) 10'3·10'2 10·100 10"·1 Ionospheric forward scatter 10" (VHF) 10" 10 Tropospheric scatter(SHF) 10-610 10-' Orbitalscatter(Xband) 10" 10' 10-1 Moonatmax.libration (k=O.4kmc) 10'2 10 10-1 impliesthatthechannelcharacteristics varysufficiently slowlythattheycanbe measured. InSection14-3,weshalldetermine tileerrorrateperformance forbinary signaling overafrequency·nonselective slowlyfadingchannel. Thischannel modelis,byfar,thesimplest toanalyze. Moreimportantly, ityieldsinsight intotheperformance characteristics fordigitalsignaling onafadingchannel andservestosuggestthetypeofsignalwaveforms thatareeffective in overcoming thefadingcausedbythechannel. Sincethemultipath components inthereceived signalarenotresolvable whenthesignalbandwidth Wislessthanthecoherence bandwidth (!!if)cofthe channel, Ihereceived signalappearstoarriveatthereceiver viaasinglefading path.Ontheotherhand,wemaychooseW»(!:J.f)nsothatthechannel becomes frequency-selective. Weshallshowlaterthat,underthiscondition, themultipath components inthereceived signalareresolvable witha resolution intimedelayofl/W.Thus,weshallillustrate thatthefrequency­ selective channelcanbemodeled asatappeddelayline(transversal) filterwith time-variant tapcoefficients. Weshallthenderivetheperformanl:e ofbinary signaling oversuchafrequency-selective channelmodel. 14-3FREQ{}ENCY·NONSELECTIVE, SLOWLY FADING CHANNEL Inthissection,wederivetheerrorrateperformance ofbinaryPSKandbinary FSKwhenthesesignalsaretransmitted overafrequency-nonselective, slowly fadingchannel. Asdescribed inSection 14-2,thefrequency-nonselective channel resultsinmultiplicative distortion ofthetransmitted signal5,(1). Furthermore, thecondition thatthechannel fadesslowlyimpliesthatthe multiplicative processmayberegarded asaconstant duringatleastone CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS 773 (14-3-4)(14-3-1)signaling interval. Consequently, ifthetransmitted signaliss;{t),thereceived equivalent lowpasssignalinonesignaling intervalis r/(t)=ae-1'I's/(t)+z(t),0,;;;t,;;;T wherez(r)represents thecomplex-valued whitegaussian noiseprocess corrupting thesignal. Letusassumethatthechannelfadingissufficiently slowthatthephaseshift </>canbeestimated fromthereceivedsignalwithouterror.Inthatcase,wecan achieveidealcoherent detection ofthereceived signal.Thus,thereceived signalcanbeprocessed bypassingitthroughamatched filterinthecaseof binaryPSKorthroughapairofmatched filtersinthecaseofbinaryFSK.One method thatwecanusetodetermine thepedormance ofthebinary communications systems istoevaluate thedecision variables andfromthese determine theprobability oferror.However, wehavealreadydonethisfora fixed(time-invariant) channel. Thatis,forafixedattenuation a,wehave previously derivedtheprobability oferrorforbinaryPSKandbinaryFSK. From(5-2-5),theexpression fortheerrorrateofbinaryPSKasafunction of thereceived SNRY6is P2(')'6)=Q(V2Y6) (14-3-2) whereYb=a'~b/No.Theexpression fortheerroi'rate ofbinaryFSK,detected coherently, isgivenby(5-2-10)as Pbb)=Q(Y:Y;;) (14-3-3) Weview(14-3-2) and(14-3-3) asconditional errorprobabilities, wherethe condition isthataisfixed.Toobtaintheerrorprobabilities whenaisrandom, wemustaverageP'(')'b),givenin(14-3-2)and(14-3-3), overtheprobability densityfunctionofYb'Thatis,wemustevaluate theintegral P,= [P'(Yb)P(Yb)dYb wherep(1b)istheprobability densityfunction ofYbwhenaisrandom_ Rayleip Fading SinceaisRayleigh-distributed, a'hasachi-square probability distribulion withtwodegreesoffreedom. Consequently, 1balsois chi-square-distributed. Itiseasilyshownthat 1 -P(Yb)=-::-e'7./-n, 1'b""O (14-3-5) Yb where'jibistheaveragesignal-to-noise ratio,definedas ~. 'jib=-E(a') No(14-3-6) ThetermE(a')issimplytheaveragevalueof",'. 774 DIGITAL COMM[J!"dCATIONS Nowwecansubstitute (14-3-5)into(14-3-4) andcarryouttheintegration forPiYh)asgivenby(14-3-2)and(14-3-3). Theresultofthisintegration for binaryPSKis (14-3-7) (14-3-8)Ifwerepeattheintegration withP2(')'h)givenby(14-3-3), weobtain probability oferrorforbinaryFSK,detected coherently, intheform 1(~)P2=21-\/~the (14-3-10)Inamvlng attheerrorrateresultsin(14-3-7) and(14-3-8), wehave assumed thattheestimate ofthechannelphaseshift,obtained inthepresence ofslowfading,isnoiseless. Suchanidealcondition maynotholdinpractice. In suchacase,theexpressions in(14-3-7) and(14-3-8) shouldbeviewedas representing thebestachievable performance inth'epresence ofRayleigh fading.InAppendix Cweconsider theproblem ofestimating thephaseinthe presence ofnoiseandweevaluate theerrorrateperformance ofbinaryand multiphase PSK. Onchannels forwhichthefadingissufficiently rapidtopreclude the estimation ofastablephasereference byaveraging thereceived signalphase overmanysignaling intervals, DPSK,isanalternative signaling method, Since DPSKrequires phasestability overonlytwoconsecutive signaling intervals, thismodulation technique isquiterobustinthepresence ofsignalfading.In derivingtheperformance ofbinaryDPSKforafadingchannel, webeginagain withtheerrorprobabiJity foranonfading channel, whichis P2(Yh)=~e-" (14-3-9) Thisexpression issubstituted intotheintegral in(14-3-4) alongwithPh'b) obtained from(14-3-5). Evaluation oftheresulting integral yieldsthe probability oferrorforbinaryDPSK,intheform 1P2=--- 2(1+Yb) Ifwechoosenottoestimate thechannel phaseshiftatall,butinstead employanoncoherent (envelope orsquare-law) detector withbinary,orthogo­ nalFSKsignals,theerrorprobability foranonfading channelis (14-3-11) WhenweaveragePiYb)overtheRayleigh fadingchannel attenuation, the resulting errorprobability is (14-3-12) n-IAI-'"1FR J.J./)l(jlf.'\L j'()MM: '~JCAIIONrHROlJ(jH f-ADjMi Mt;UIPATIl (HAl"~l-:ls 775 ~ 1".'"~ .11'-."""f'....FSK rFSK"'"~noncohc:renl I.:oheren( '"detectiOn detection """~, DPSK/~-......, """psi-~'""'"-......,'",-......,-......,,~" FIGURE 14-3-1 Performance ofbinarysignaling ona Rayleigh fadingchannel.0.5 0.2 10I o.~2 ~10 :05, ~ 02 ~ :0IO-J Jl,50: I()"-l 2 10o 10 152025 Theerrorprobabilities in(14-3-7), (14-3-8), (14-3-10), and(14-3-12) are illustrated inFig.14-3-1.Incomparing theperformance ofthefourbinary signaling systems, wefocusourattention ontheprobabilities oferrorforlarge SNR,i.e.,Yo»I.Underthiscondition, theerrorratesin(14-3-7), (I4-3-H), (14-3-10), and(14-3-12) simplifyto {1/4Yh P=1/2y/, 2112Yh 1/Yhforcoherent PSK forcoherent, orthogonal FSK forDPSK fornoncoherent, orthogonal FSK(14-3-13 ) From(14-3-13), weobserve thatcoherent PSKis3dBbetterthanDPSK and6dBbetterthannoncoherent FSK.Morestriking, however. isthe observation thattheerrorratesdecrease onlyinversely withSNR.Incontrast, thedecrease inerrorrateonanonfading channelisexponential withSNR. Thismeansthat,onafadingchannel, thetransmitter musttransmit alarge amountofpowerinordertoobtainalowprobability oferror.Inmanycases,a largeamountofpowerisnotpossible, technically and/oreconomically. An alternative solution totheproblem ofobtaining acceptable performance ona fadingchannel istheuseofredundancy, whichcanbeobtained bymeansof diversity techniques, asdiscussed inSection14-4, n6 OIWT.-\L CO.\iMUNfCA nONS Nakagami Fading IfaischaracterIzed statistically bytheNakagami-m distribution, therandomvariable I'=a'~h/Nohasthepdf(seeProblem 14-15) m/1/ p(I')=r(m))'''' I'm'e-my1y(14-3-14) where 'Y=E(a')'€IN o. Theaverage probability oferrorforanyofthemodulation methods is simplyobtained byaveraging theappropriate errorprobability foranonfading channeloverthefadingsignalstatistics. Asanexample oftheperformance obtained withNakagami-m fading statistics, Fig.14-3-2illustrates theprobability oferrorofbinaryPSKwithmas aparameter. Werecallthatm=1corresponds toRayleigh fading.Weobserve thattheperformance improves asmisincreased abovem=I,whichis indicative ofthefactthatthefadingislesssevere.Ontheotherhand,when m<I,theperformance isworsethanRayleigh fading. OtherFadingSignalStatistics Following theprocedure described above, onecandetermine theperformance ofthevariousmodulation methods for othertypesoffadingsignalstatistics. suchastheRicedistribution. Errorprobability resultsforRice-distributed fadingstatistics canbeIound inthepaperbyLindsey (1964).whileforNakagami-m fadingstatistics. the FIGURE 14-3-2 Average errorprobability fortwo-phase PSK symbolinnondiversity reception.10-1 10-1 m=0.5 ~10-3 .c ~IO~ ~ €/0.5.c "~ E10-1> "" 10-7 m='" I(fR(Nofading) 10-'L-----,L...L.---:'--'--''--,L'---'----'o 10 20 3U 40 Average SNRYIJ(dB) CHAPTER l.s:DJOITAl COMMl:NICATJDN THRot;GH FADING MlJlTIPATH CHANNELSm readermayrefertothepapersbyEspOsito (1967),Miyagaki etat(1978). Charash (1979),Al-Hussaini etal.(1985),andBeaulieu etal.(1991). 14-4DIVERSITY TECHNIQUES FORFADING MULTIPATH CHANNELS Diversity techniques arebasedonthenotionthaterrorsoccurinreception whenthechannelattenuation islarge,i.e.,whenthechannelisinadeepfade. Ifwecansupplytothereceiverseveralreplicasofthesameinformation signal transmitted overindependently fadingchannels, theprobability thatallthe signalcomponents willfadesimultaneously isreducedconsiderably. Thatis,if pistheprobability thatanyonesignalwillfadebelowsomecriticalvaluethen pListheprobability thatallLindependently fadingreplicasofthesamesignal willfadebelowthecriticalvalue.Thereareseveralwaysinwhichwecan provide thereceiver withLindependently fadingreplicas ofthesame information-bearing signal. Onemethodistoemployfrequency diversity. Thatis,thesameinformation­ bearingsignalistransmitted onLcarriers. wheretheseparation between successive carriersequalsoreJCceedsthecoherence bandwidth (tif),-ofthe channel. Asecondmethodforachieving Lindependently fadingversionsofthesame information-bearing signalistotransmit thesignalinLdifferent timeslots, wheretheseparation between successive timeslotsequalsorexceeds the coherence time(tit)cofthechannel. Thismethodiscalledtime.diversity. Notethatthefadingchannel fitsthemodelofaburstyerrorchannel. Furthermore, wemayviewthetransmission ofthesameinformation eitherat different frequencies orindifference timeslots(orboth)asasimpleformof repetition coding.Theseparation ofthediversity transmissions intimeby(!it), orinfrequency by(Af),isbasically aformofblock-interleaving thebitsinthe repetition codeinanattempttobreakuptheerrorburstsand,thus,toobtain independent errors.Laterinthechapter, weshalldemonstrate that,ingeneral, repetition codingiswasteful ofbandwidth whencompared withnontrivial coding. Another commonly usedmethodforachieving diversity employs multiple antennas. Forexample, wemayemployasingletransmitting antenna and multiple receiving antennas. Thelattermustbespacedsufficiently farapart thatthemultipath components inthesignalhavesignificantly different propagation delaysattheantennas. Usually aseparation ofatleast10 wavelengths isrequired between twoantennas inordertoobtainsignalsthat fadeindependently. Amoresophisticated methodforobtaining diversity isbasedontheuseofa signalhavingabandwidth muchgreaterthanthecoherence bandwidth (!if),of thechannel. Suchasignalwithbandwidth Wwillresolvethemultipath components and,thus,providethereceiverwithseveralindependently fading signalpaths.Thetimeresolution isI/W.Consequently, withamultipath 778 Dim'!,\!. COMMLINICAnoJ',S spreadofT,,,s.thereareT,,,Wresolvable signalcomponents. SinceT,,,= 1/(tJ.[)...thenumberofresolvable signalcomponents mayalsobeexpressed as W/(!:if),.Thus.theuseofawideband signalmaybeviewedasjustanother method forobtaining frequency diversity oforderL=W/(Af)c'Theoptimum receiver forprocessing thewidebandsignalwillbederivedinSection14-5.Itis calledaRAKEcorrelator oraRAKEmatched jilterandwasinvented byPrice andGreen\1958). Thereareotherdiversity techniques thathavereceived someconsideration inpractice. suchasangle-of-arrival diversity andpolarization diversity. However. thesehavenotbeenaswidelyusedasthosedescribed above. 14-4-1BinarySignals Weshallnowdetermine theerrorrateperformance forabinarydigital communications systemwithdiversity. Webeginbydescribing the mathemati­ calmodelfor thecommunications systemwithdiversity. Firstofall,weassume thatthereareLdiversity channels. carrying thesameinformation-bearing signal.Eachchannel isassumed tobefrequency-nonselective andslowlyfading withRayleigh-distributed envelope statistics. Thefadingprocesses amongthe Ldiversity channels areassumed tobemutually statistically independent. The signalineachchannel iscorrupted byanadditive zero-mean whitegaussian noiseprocess. Thenoiseprocesses intheLchannels areassumed tobe mutually statistically independent. withidentical autocorrelation functions. Thus.theequivalent low-pass received signalsfortheLchannels canbe expressed intheform r'k(t)=OIke-jd>'Skm(t) +Zk(t). k=1.2,...•L,m=I,2(14-4-1) where{DIke-Jd>'}represent theattenuation factorsandphaseshiftsfortheL channels, s'm(t)denotes themthsignaltransmitted onthekthchannel, and Zk(t)denotes theadditive whitegaussian noiseonthekthchannel. Allsignals intheset(Skm(t)}havethesameenergy. Theoptimum demodulator forthesignalreceived fromthekthchannel consists oftwomatched filters,onehavingtheimpulse response bkl(t)=sl,(T-t) andtheotherhavingtheimpulse response b,,(t)=sl2(T-t)(14-4-2) (14-4-3) Ofcourse. ifbinaryPSKisthemodulation method usedtotransmit the information. thenSk,(t)=-Sk2(t).Consequently, onlyasinglematched filteris required forbinaryPSK.Following thematched filtersisacombiner that formsthetwodecision variables. Thecombiner thatachieves thebest performance isoneinwhicheachmatched filteroutputismultiplied bythe corresponding complex-valued (conjugate) channel gainOI.eld>,.Theeffectof thismultiplication istocompensate forthephaseshiftinlhechannel andto CHAPTE.R 14:DIGITAL COMMUNICATION THROUGH FADlNG MULTIPATHCHANNELS 779 Receiver Output d«isloo Re<e2iverf---G~~~~j variables J Combiner Receiver1------' L (14-4-4)'L(1) FIGURE 14-4-1 Modelofbinarydigitalcommunications systemwithdiversity. weightthesignalbyafaclorthatisproportional tothesignalstrength. Thus,a strongsignalcarriesalargerweightthanaweaksignal.Afterthecomplex­ valuedweighting operation isperformed, twosumsareformed.Oneconsistsof therealpartsoftheweighted outputsfromthematched filterscorresponding toatransmitted O.Thesecondconsistsoftherealpartoftheoutputsfromthe matched filters.corresponding toatransmitted 1.Thisoptimum combiner is calledamaximal raliocombiner byBrennan (1959).Ofcourse,therealization ofthisoptimum combiner isbasedontheassumption thatthechannel attenuations {Uk}andthephaseshifts{cf>k}areknownperfectly. Thatis,the estimates oftheparameters {ak}and{cf>k}containnonoise.(Theeffectofnoisy estimates ontheerrorrateperformance ofmultiphasePSKisconsidered in Appendix C. Ablockdiagram illustrating themodelforthebinarydigitalcommunica­ tionssystemdescribed aboveisshowninFig.14-4-1. Letusfirstconsider theperformance ofbinaryPSKwithLlh-order diversity. Theoutputofthemaximal ratiocombiner canbeexpressed asa singledecisionvariable intheform U=Re(2~ kt\a~+J\UkNk) L L =2~2:ai+2:UkNkr 11:=1 k=l whereNhdenotestherealpartofthecomplex-valued gaussian noisevariable N,=ei~·rz,(t)sZ(t) dt (14-4-5) Wefollowtheapproach usedinSection14·3inderiving theprobability of error.Thatis,theprobability oferrorconditioned onafixedsetofattenuation 780 DIGITAL COMMUI<'CATlOl<S factors{ak}isobtained first.Thentheconditional probability oferroris averaged overtheprobability densityfunction ofthe{a.}. Rayfeigh Fading Forafixedsetof{adthedecisionvariableUisgaussian withmean L E{U)=2~.2:a; k=1(14-4-6) andvariance L lTt=2"iNa.2:a; '~I(14-4-7) Forthesevaluesofthemeanandvariance, theprobability thatUislessthan zeroissimply P,("Y.)=Q(v'21'.) (14-4-8) wheretheSNRperbit,"Y••isgivenas ~L "Yb=No.2:aiok=1 L =.2:Y. k=l(14-4-9) (14-4-1O)1-jvycwhere1'.='lai!Noistheinstantaneous SNRonthekthchannel. Nowwe mustdetermine theprobability densityfunction p(Yb)'Thisfunction ismost easilydetermined viathecharacteristic functionofYo'Firstofall,wenotethat forL=1,Yb""1'1hasachi-square probability densityfunction givenin (14-3-5).Thecharacteristic functionof1'1iseasilyshowntobe l/Jy,(jv)=E(&VY') 1 whereYcistheaverageSNRperchannel, whichisassumed tobeidentical for allchannels. Thatis, (14-4-11)~ Yc=NoE(ai) independent ofk.Thisassumption appliesfortheresultsthroughout this section.SincethefadingontheLchannels ismutually statistically indepen­ dent,theh.}arestatistically independent, and,hence,thecharacteristic function forthesumYoissimplytheresultin(14-4-10) raisedtotheLth power,i.e., (14-4-12) (14.4-14)CHAPTER 14:D1GI1Al COMMUNICATION THROUGH FADING MUlTIPATH CHANNEU 781 Butthisisthecharacteristic function ofachi-square-distributed random variable with2Ldegrees offreedom. Itfollowsfrom(2-1-107) thatthe probability densityfunctionp('Yb)is ()_1 "L-Ie-~'I'Y, (14-4-13)P'Yb-(L-1)!y~ ,b Thefinalstepinthisderivation istoaverage theconditional errOl probability givenin(14-4-&) overthefadingchannel statistics. Thus,we evaluate theintegral P2=rP'(-yb)P( 'Yb)d'Yb Thereisaclosed-form solution for(14-4-14), whichcanbeexpressed as Lot(L_l+k) P,=B(I-J!Wk~Dk·.[W+J!)]k Where,bydefinition,(14-4-15) (14-4-16) (14-4-17)J!=~'Yc 1+Yc WhentheaverageSNRperchannel,Yosatisfiesthecondition y,»1,theterm HI+J!)=1andthetermHl-J!)=1/4i',.Furthermore, Lf(L-1+k)=[ZL-1) k~Dk,L Therefore, wheny,issufficiently large(greater than10dB),theprobability of errorin(14-4-15) canbeapproximated as (14-4-18) Weobservefrom(14·4-18) thattheprobability oferrorvariesas1/i',raisedto theLthpower.Thus,withdiversity, theerrorraledecreases inver~ely withthe UhpoweroftheSNR. Havingobtained theperformance ofbinaryPSKwithdiversity, wenowturn ourattention tobinary,orthogonal FSKthatisdetected coherently. Inthis case,thetwodecision variables attheoutputofthemaximal ratiocombiner maybeexpressed as VI=Re(2~JIa~+ktlQ'kNkl) V2=Re(±Q'kNk2) k=l wherewehaveassumed thatsignalSkl(t)wastransmitted andwhere{Nkl}and {Nularethetwosetsofnoisecomponent attheoutputofthematched filters. (14-4-20)782 DIGITAL COMMlINIC.o\T10NS Theprobability oferrorissimplytheprobability thatV2>VI'Thiscomputa­ tionissimilartotheoneperformed forPSK,exceptthatwenowhavetwice thenoisepower.Consequently, whenthe{adarefixed,theconditional probability oferroris Weuse(14-4-13) toaverage P'(Yb)overthefading.Itisnotsurprising tofind thattheresultgivenin(14-4-15) stillapplies,with"Y,replaced byHe-Thatis, (14-4-15) istheprobability oferrorforbinary.orthogonal FSKwithcoherent detection. wheretheparameter /.Lisdefinedas ~/.L=\j~ (14-4-21) Furthermore, forlargevaluesofy,.theperformance P,canbeapproximated as (14-4-22) Incomparing (14-4-22) with(14-4-18), weobservethatthe3dBdifference in performance between PSKandorthogonal FSKwithcoherent detection, which existsinanonfading, nondispersive channel. isthesamealsoinafading channel. Intheabovediscussion ofbinaryPSKandFSK,detected coherently, we assumed thatnoiseless estimates ofthecomplex-valued channel parameters {ake'j",,} wereusedatthereceiver. Sincethechannel istime-variant, the parameters {a.e'N>,} cannotbeestimated perfectly. Infact,onsomechannels, thetimevariations maybesufficiently fasttopreclude theimplementation of coherent detection. Insuchacase,weshouldconsider usingeitherDPSKor FSKwithnoncoherent detection. Letusconsider DPSKfirst.InorderforDPSKtobeaviabledigital signaling method, thechannelvariations mustbesufficiently slowsothatthe channel phaseshifts{<I>k}donotchangeappreciably overtwoconsecutive signaling intervals. Inouranalysis. weassumethatthechannel parameters {ake'j",,} remainconstant overtwosuccessive signaling intervals. Thusthe combiner forbinaryDPSKwillyieldasanoutputthedecision variable V=Re[k~1(2\\'ake 'J""+N.,)(2ga.e'"" +lVltl] (14-4-23) where{NkI}and{N.2}denotethereceived noisecomponents attheoutputof thematched filtersinthetwoconsecutive signaling intervals. Theprobability oferrorissimplytheprobability thatU<O.SinceVisaspecialcaseofthe generalquadratic formincomplex-valued gaussian randomvariables treatedin Appendix B, theprobability oferrorcanbeobtained directlyfromtheresults giveninthatappendix. Alternatively, wemayusetheerrorprobability givenin (12-1-3), whichappliestobinaryDPSKtransmitted overLtime-invariant CHAPTER 14,DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS 783 channels, andaverageitovertheRayleigh fadingchannelstatistics. Thus,we havetheconditional errorprobability L-I P2('Yb)=(~)'L-le-Y' Lbk'Y1 k=O where 'Yoisgivenby(14-4-9)and 1L-I-k(2L-l) bk=-2:k!n~O n(14-4-24) (14-4-25) Theaverage ofP,(y,)overthefadingchannel statistics givenbyp('Yb)In (14-4-13) iseasilyshowntobe 1 L-I ( - )k P,=2"t(L-1)!(1+yJLk~Obk(L-1+k)!1:'y,(14-4-26) Weindicatethattheresultin(14-4-26) canbemanipulated intotheformgiven in(14-4-15), whichappliesalsotocoherent PSKandFSK.ForbinaryDPSK, theparameter JLin(14-4-15) isdefinedas(seeAppendix C) (14-4-27) ForYc»1,theerrorprobability in(14-4-26) canbeapproximated bythe expression (14-4-28) Orthogonal FSKwithnoncoherent detection isthefinalsignaling technique thatweconsider inthissection.Itisappropriate forbothslowandfastfading. However, theanalysisoftheperformance presented belowisbasedonthe assumption thatthefadingissufficiently slowsothatthechannelparameters lake-No,} remainconstant fortheduration ofthesignaling interval. The combiner forthemultichannel signalsisasquare-law combiner. Itsoutput consistsofthetwodecision variables L VI=2:12~ake-j",+Nkl k=l L V2=2:IN.,!2 k=!(14-4-29} whereVIisassumed tocontainthesignal.Consequently theprobability of erroristheprobability thatU,>VI' 784 DIGITAL COMMUNICATIONS AsinDPSK,wehaveachoiceoftwoapproaches inderiving the performance ofFSKwithsquare-law combining. InSection12-1,weindicated thattheexpression fortheerrorprobability forsquare-law combined FSKis thesameasthatforDPSKwith'Ybreplaced by!r•.Thatis,theFSKsystem requires 3dBofadditional SNRtoachievethesameperformance ona time-invariant channel. Consequently, theconditional errorprobability for DPSKgivenin(14-4-24) appliestosquare-Iaw-combined FSKwhen"'I.is replaced byho•.Furthermore, theresultobtained byaveraging (14-4-24) over thefading,whichisgivenby(14-4-26), mustalsoapplytoFSKwith"Ye replaced by!'Ye.Butwealsostatedpreviously that(14-4-26) and(14-4-15) are equivalent. Therefore, theerrorprobability givenin(14-4-15) alsoappliesto square-law-combined FSKwiththeparameter J.I-definedas (14-4-30) (14-4-31)Analternative derivation usedbyPierce(1958)toobtaintheprobability thatthedecision variableV,>VIisjustaseasyasthemethoddescribed above_Itbeginswiththeprobability densityfunctions p(VI)andp(V2).Since thecomplex-valued randomvariables {a.e-i"'j, {Nk,},and{Nk2}arezero-mean gaussian-distributed, thedecisionvariables V,andV2aredistributed according toachi-square probability distribution with2Ldegreesoffreedom. Thatis, (_ 1 L-I(V,) PVI)-(2uT)L(L _I)!VIexp-2~ where u~=!E(l2'€Oke-i". +Nkd') =2'€ND(l+"Yc) Similarly, where(U)-2 L-l (V2)P2 -(2u1)L(L-I)!V2exp-2u1(14-4-32) u~=2CCNo Theprobability oferrorisjusttheprobability thatV2>VI'Itisleftasan exercise forthereadertoshowthatthisprobability isgivenby(14-4-15), wherep.isdefinedby(14-4-30). When'Ye»I.theperformimce ofsquare-law-detected FSKcanbesimpl­ ifiedaswehavedonefortheotherbinarymultichannel systems. Inthiscase, theerrorrateiswellapproximated bytheexpression (14-4-33) Theerrorrateperformance ofPSK,DPSK,andsquare-Iaw-detected CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTlPATH CHANNELS 7115 SNRperbit.Y.<dB) FIGURE 14-4-1 Performance ofbinarysignalswithdiversity. orthogonal FSKisillustrated inFig.14-4-2forL=I,2,and4.The performance isplottedasafunction oftheaverageSNRperbit,rb,whichis relatedtotheaverageSNRperchannel,re'bytheformula (14-4-34) TheresultsinFig.14-4-2clearlyillustrate theadvantage ofdiversity asa meansforovercoming theseverepenaltyinSNRcausedbyfading. 14-4-2Multipbase Signals Multiphase signaling overaRayleigh fadingchannelisthetopicpresented in somedetailinAppendix C.Ourmainpurpose inthissectionistocitethe generalresultfortheprobability ofasymbolerrorinM-aryPSKandDPSK systemsandtheprobability ofabiterrorinfour-phase PSKandDPSK. 786 DIGITAL COMMUNICATiONS Thegeneralresultfortheprobability ofasymbolerrorinM-aryPSKand DPSKis where (14-4-36) forcoherent PSKand (14-4-37) (14-4-38)forDPSK.Again,'Ycistheaveragereceived SNRperchannel. TheSNRper bitis'Yb:L'Yclk,wherek=log,M. Thebiterrorrateforfour-phase PSKandDPSKisderivedonthebasisthat thepairofinformation bitsismappedintothefourphasesaccording toaGray code.Theexpression forthebiterrorratederivedinAppendix Cis 1[ JLL-l(2k)(1 _p.2)*] Pb=2:I-~ *~ok4 -2JL2 where JLisagaingivenby(14-4-36) and(14-4-37) forPSKandDPSK, respectively. Figure14-4-3illustrates theprobability ofasymbolerrorofDPSKand coherent PSKforM=2,4,and8withL=1.Notethatthedifference in pe~ormance between DPSKandcoherent PSKisapproximately 3dBforall threevaluesofM.Infact,when'Yb»1andL=1,(14-4-35) iswell approximated as forDPSKandasM-l p= c:..:...-'- _ M(Mlog2M)[sin2(IfIM)J'Yb M-l (Mlog,M)[sin2(IfIM)]2'Yb(14-4-39) (14-4-40) forPSK.Hence,athighSNR,coherent PSKis3dBbetterthanDPSKona Rayleigh fadingchannel. Thisdifference alsoholdsasLisincreased. Biterrorprobabilities aredepicted inFig.14-4-4fortwo-phase, rour-phase, andeight-phase DPSKsignaling withL=1,2,and4.Theexpression forthe biterrorprobability ofeight-phase DPSKwithGrayencoding isnotgiven here,butitisavailableinthepaperbyProakis(1968).Inthiscase,weobserve CHAPTER 14:DIGITAL COMMUNICATION THROUGH fADING MULTIPATHCHANNELS 787 ""'"~ ~'"l'"'"'",,\~"I\. "'"~: ~"'"M=8 DPSK M=2j/\\"'\'"M=8 PSKDPSK\k'\,~\M=4 I'\.M=4 PSK M~r\'~P5K- PSKI\."~ ~'\,~~ ~,\,~" '"\~\>. V2 1O~1 5 g•]2 E ~10-'•'0 ~ :0 l 2 IO~·1 5 10-' o 5 10 15 20 25 30 35 SNRperbit,Yb(dBl FIGURE 14-4-3 Probability ofsymbolerrorforPSKandDPSKforRayleigb fading. thattheperformances fortwo-andfour-phase DPSKare(approximately) the same,whilethatforeight-phase DPSKisaboutJdBpoorer.Although we havenotshownthebiterrorprobability forcoherent PSK,itcanbe demonstrated thattwo-andfour-phase coherent PSKalsoyieldapproximately thesameperformance. 14-4-3M-aryOrthogonal Signals Inthissub-section, wedetermine theperformance ofM-aryorthogonal signals transmitted overaRayleigh fadingchannel andweassesstheadvantages of higher-order signalalphabets relativetoabinaryalphabet. Theorthogonal signalsmaybeviewedasM-aryFSKwithaminimum frequency separation of anintegermultiple ofliT,whereTisthesignaling interval. Thesame information-bearing signalistransmitted onLdiversity chann'els. Each diversity channel isassumed tobefrequency-nonselective andslowlyfading, andthefadingprocesses ontheLchannels areassumed tobemutually 788 DIGITAL COMMUNIC"TlONS ~ ~~ '\~'~, 1\\"\I'\..\\\l~"- \1\\'.:1'1'\.."", '\\\1\'\'M#8 \''\M.,~M#"L;I\\\\M.8~ \\\M=2" \"\M=4I\M:8 , \ \\\ M=2'\ \~=~-M=4 .-lJL=4 ~ 2Q:.-.D \\\ \2 510255 2 10'" 510 IS 20 25 30 35 40 SNRpub'••T.<dBl nGURE 14-4-4 Probability ofabiterrorforDPSKwithdiversity forRayleigh fading. statistically independent. Anadditivewhitegaussiannoiseprocesscorruptsthe signaloneachdiversity channel.Weassumethattheadditivenoiseprocesses aremutually statistically independent. Although itisrelatively easytoformulate thestructure andanalyzethe performance ofamaximal ratiocombiner forthediversity channels inthe M-arycommunication system,itismorelikelythatapractical systemwould employnoncoherent detection. Consequently, weconfineourattention to square-law combining ofthediversity signals.Theoutputofthecombiner containing thesignalis L Vt=L12~ake-j··+Nkl12 k-l(14-4-41) whiletheoutputsoftheremaining M-1combiners are L Um'"LINkml2,m=2,3,4 •...•M k-1(14-4-42) (14-4-43)CHAP'TER 14:DIGITAL COMMUNICAnON THROUGH fADING MULTIPATHCHANNELS 789 Theprobability oferrorissimply1minustheprobability thatVI>Vmfor m=2,3•...•M.Sincethesignalsareorthogonal andtheadditive noise processes aremutually statistically independent, therandom variables VI.V2•••••VMarealsomutually statistically independent. Theprobability densityfunction ofVIwasgivenin(14-4-31). Ontheotherhand,V2•••••VM areidentically distributed anddescribed bythemarginal probability density function in(14-4-32). WithVIfixed,thejointprobability P(V2<VI,V3< V,•...,Vm<VI)isequaltoP(V2<VI)raisedtotheM-1power.Now, LUI P(V2<VI)=0p(VZ)dV2 (VI)L-I1 (VI)'=1-exp--~-- 2u~'-0k!2O'~ where u~=2'lNo.TheM-1powerofthisprobability isthenaveraged over theprobability densityfunction ofVItoyieldtheprobability ofacorrect decision.Ifwesubtractthisresultfromunity,weobtaintheprobability of errorintheformgivenbyHahn(1962) P-1[ 1 VL-I (V,) M--0(2O'f)L(L _1)1Iexp-2O'~ [(V)L-I1 (V)']M.'X1-exp-~L-~ dVI 20'2'-0kl20'2 _ 1 [ 1 VL-I (VI) - - 0 (1+y,)L(L-I)! Iexp- 1+1, (L-IV')M-I Xl-e-u,2:-+dVI'-0k.(14-4-44) where1,istheaverageSNRperdiversity channel. TheaverageSNRperbitis Yb=Ly,/Io82M=LyJk. Theintegralin(14-4-44) canbeexpressed inclosedformasadouble summation. Thiscanbeseenifwewrite (L-Ivt)'" m(L-I) ~-k '=LPkmvt k=O' k=O(14-4-45) wherefJkmisthesetofcoefficients intheaboveexpansion. ThenitfoHowsthat (14-4-44) reducesto (14-4-46) 790 DIGITAL COMMUNICATIONS Whenthereisnodiversity(L=I),theerrorprobability in(14-4-46) reducesto thesimpleform (14-4-47) ThesymbolerrorratePMmaybeconverted toanequivalent biterrorraleby multiplying PMwith2'-I/(2'-1). Although theexpression forPMgivenin(14-4-46) isinclosedform,itis computationally cumbersome toevaluate forlargevaluesofMandL.An alternative istoevaluate PMbynumerical integration. usingtheexpression in (14-4-44). Theresultsillustrated inthefollowing graphsweregenerated from (14-4-44). Firstofall,letusobservetheerrorrateperformance ofM-aryorthogonal signaling withsquare-law combining asafunction oftheorderofdiversity. Figures14-4-5and14-4-6illus!1;iIte thecharacteristics ofPMforM=2and4as L---y=',. L.---1,-=b "- y,=10_I--t----L----' ~ VT,=15-I-- r--~r---.. ./1,=20 ~'"I'-.. \f\."---.I- It=30 \'\. 1\'\. \"!'.. \"\t---y,=50 \ .11=2 f\)',-=totalreceivedSNR \ \ \ 'Y/=100 fiGURE 14-4-5 Performance ofsquare-law-detected binaryorthogonal signalsasafunction ofdivt>Tsity.5 2 \0-1 5 2 10" 5..' ~2 u10-.1 :85E ~•2 ~ 0 :E104 :is5i2 10-5 5 2 10" 5 2 10-7 235 10 Orderofdiver!lily. L203050 CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTIPATHCHAS!'<OELS 791 y=3 ~1/_6--Y,=1(\ r-..... ....-- t--.................. -------'i/=15 '-~:--- L..------'",=20 "r---...'\'-...... Y,=30 \'"1\"- \'- \........... \'Y,=SO \ 1\ \II=4 \y,=totalreceivedSNR \ y,=100 FIGURE 14-4-li Pedormance ofsquare-law-detected M=4orthogonal signalsasafunction ofdiversity.5 2 10-l 5 2 10-2 .:5 ~2 ~10-3 :&e5 ~•'02 fJO-' 5 2 10-5 5 2 10-<\ 5 2 10-7 235 10 Orderofdiversil)'. L203050 afunctionofLwhenthetotalSNR,definedasii,=LiiC'remainsfixed.These resultsindicatethatthereisanoptimum orderofdiversity foreachYr'Thatis. foranyii"thereisavalueofLforwhichPMisaminimum. Acareful observation ofthesegraphsrevealsthattheminimum inPMisobtained when Yc=ii'/L""3.Thisresultappearstobeindependent ofthealphabet sizeM. Second,letusobservetheerrorratePMasafunctionoftheaverageSNR perbit,definedasYb=Liic/k.(Ifweinterpret M-aryorthogonal FSKasa formofcodingtandtheorderofdiversity asthenumberoftimesasymbolis repeated inarepetition codethen'Yb='YciRpwhereRc=k/Listhecode rate.)ThegraphsofPMversus'YbforM=2,4,8,16,32andL=1,2,4are showninFig.14-4-7.Theseresultsillustrate thegaininperformance asM increases andLincreases. First,wenotethatasignificant gaininperformance isobtained byincreasing L.Second, wenotethatthegaininperformance obtained withanincrease inMisrelatively smallwhenLissmall.However, tInSection14-6,weshowthatM-aryorthogonal FSKwithdiversity maybeviewedasablock orthogonal code. 792 D1GITf\l COMMlINICA"TIONS I(r1 ~ 2 10-2 ~ ..'2 ~10-1 ~ :g ~e ~ ~ '0 .~ 10~ ~M=2 £M=4 M=8 M=16 M=32 IO-~ ~ 2 10" 0 10 I~20 2~ 30 3~ SNRperbit.Yb(dB) FIGURE 14-4-7 Performaoce oforthogonal signaling withMandLasparameters. asLincreases, thegainachieved byincreasing Malsoincreases. Sincean increaseineitherparameter resultsinanexpansion ofbandwidth, i.e., B=LM •log,M theresultsillustrated inFig.14-4-7indicatethatanincrease inLismore efficientthanacorresponding increaseinM.AsweshallseeinSection14-6, codingisabandwidth-effective meansforobtaining diversity inthesignal transmitted overthefadingchannel. Chel'llOl' BoundBeforeconcluding thissection,wedevelopaChemot! upperboundontheerrorprobability ofbinaryorthogonal signaling with Llh-order diversity, whichwillbeusefulinourdiscussion ofcodingforfading channels, thetopicofSection14-6.OurstartingpointistheeKpression forthe twodecisionvariables VIandV2givenby(14-4-29), whereVIconsistsofthe CHAPTER 14:DIGITAL COMMUNICATION THROUGl-l FADING MULTIPATH CHANNELS 793 square-law-combined signal.plus-noise termsandU2consists ofsquare-Iaw­ combined noiseterms.Thebinaryprobability oferror,denoted herebyP2(L). as g(L)=P(U2-U,>0) =P(X>0)=rp(x)dx wheretherandomvariableXisdefinedas L X=U2-U\=2:(INk2i2-12'lak+Nu12) k=)(14-4-48) (14-4-49) Thephaseterms{4>dinUjhavebeendropped sincetheydonotaffectthe performance ofthesquare-law detector. LetS(X)denotetheunitstepfunction. Thentheerrorprobability in (14-4-48) canbeexpressed intheform P2(L)=E[S(X)] (14-4-50) Following thedevelopment inSection2-1-5,theChernoff boundisobtained by overbounding theunitstepfunction byanexponential function. Thatis, S(X)e;elX,{;..0 (14-4-51) wheretheparameter {isoptimized toyieldatightbound.Thus,wehave (14-4-52). Uponsubstituting fortherandomvariableXfrom(14-4-49) andnotingthat therandomvariables inthesummation aremutually statistically independent, weobtaintheresult ButL PiL)"nE(eIIN4lI')E(e-'12~ ...+N"") k=l(14-4-53) andE(eCiN"I') =12 ' 1-2{U21{<­201(14-4-54) E(e-'IUa.+N.d') =1{>-1 (14-4-55) 1+2{u~' 2u~ where u~=2'lNo•ui=2'lNo{1+1c).and1cistheaverageSNRperdiversity channel. Notethatu;andu~areindependent ofk,i.e.,theadditive noise termsontheLdiversity channels aswellasthefadingstatistics areidentically distributed. Consequently, (14-4-53) reducesto 1 O~{~-2 2U2(14-4-56) 794 DIGITAL COMMliNICATIONS Bydifferentiating theright-hand sideof(14-4-56) withrespecttoC.wetind thattheupperboundisminimized when (1'2-(1'2 I:=' 2 (14-4-57) 4<ri<T~ Substitution of(14-4-57) forI:into(14-4-56) yieldstheChernoff upperbound intheform (14-4-58) (14-4-60)P,(L)""[4(1+y,;]L (2+y,.) Itisinteresting tonotethat(14-4-58) mayalsobeexpressed as P,(L)""[4p(1-P)JL (14-4-59) wherep=1/(2+y,.)istheprobability oferrorforbinaryorthogonal signaling onafadingchannel withoutdiversity. Acomparison oftheChernoff houndin(14-4-58) withtheexacterror probability forbinaryorthogonal signaling andsquare-law combining oftheL diversity signals,whichisgivenbytheexpression (1)LLO'(L-I+k)(1+)i)'P(L)= -L-' 21+'Y, "'0 k2+y" L-'(L-l+k)=pLL (l-p)' "~O k revealsthetightness ofthebound.Figure(14-4-8) illustrates thiscomparison. 404,5 20253035 SNRperbit.y,,(dB), "-'"I I ~:"'-Exact Chernoffbound_\\'\ ,/L;I \I~........~ '\\"-~ \\'\'\ \\"-"-\\ \ I'\.l"'- I \\"-'\ \\ I"- 1\\\i"-ChernOff bo"~ lA\\L=2 u..:.'\\Exact_ Chernoffbound\'\ L.4\\ I I\\\5 2 10-' 10t522 10-2lO­ S .....]()-3525 FlGURE 1......, Comparison ofChernoff boundwirhexact errorprobability. CHAPTER 1.4:DI<i1TAL COMMUNICATION THROU(iH FADlN<i MUL.TIPA1H CHANNELS 795 WeobservethattheChernoff upperboundisapproximately 6dBfromthe exacterrorprobability forL=I,but,asLincreases, itbecomes tighter.For example, thedifference between theboundandtheexacterrorprobability is about2.5dBwhenL=4. Finallywemention thattheerrorprobability forM-aryorthogonal signaling withdiversity canbeupper-bounded bymeansoftheunionbound PM'"(M-l)P,(L) (14-4-61) wherewemayuseeithertheexactexpression givenin(14-4-60) orthe Chernoff boundin(14-4-58) forP2(L). 14-5DIGITAL SIGNALING OVERAFREQUENCY· SELECTIVE, SLOWLY FADING CHANNEL Whenthespreadfactorofthechannelsatisfiesthecondition T",Bd«I,itis possibletoselectsignalshavingabandwidth W«(A.!),andasignalduration T«(lit),.Thus,thechannelisfrequency-nonselective andslowlyfading.In suchachannel, diversity techniques canbeemployed toovercome thesevere consequences offading. Whenabandwidth W»(A.f),isavailable totheuser,thechannelcanbe subdivided intoanumberoffrequency-division multiplexed (FDM)subchan­ nelshavingamutualseparation incenterfrequencies ofatleast(A.!),.Then thesamesignalcanbetransmitted ontheFDMsubchanne1s, and,thus, frequency diversity isobtained. Inthissection, wedescribe analternative method. 14-5·1ATapped·Delay-Line Channel Model Asweshallnowdemonstrate, amoredirectmethodforachieving basically the same:resultistoemployawideband signalcovering thebandwidth W.The channel isstillassumed tobeslowlyfadingbyvirtueoftheassumption that T«(At)c' Nowsuppose thatWisthebandwidth occupied bythereal bandpass signal.Thenthebandoccupancy oftheequivalent lowpasssignal s/(t)isIfI,.;~W.Sinces,(t)isband-limited toIfI,.;~W,application ofthe sampling theorem resultsinthesignalrepresentation ()_~(!:)sin[JrW(t-n/W»)s,t-..:..-s, ,,~-wWJrW(t-n/W)(14-5-1) (IfI,.;~W) (If!>~W)(14-5-2) '796 DIGITAL COMMUNICATIONS Thenoiseless received signalfromafrequency-selective channel was previol,lSly expressed intheform '/(1)=[~C(f;I)StU)ei2"/'df (14-5-3) (14-5-4)whereC(f;I)isthetime-variant transferfunction. Substitution forSI(t)from (14-5-2)into(14-5-3)yields rl(l)=~n~~s/(n/W)[~C(f;l)ej2tr!C,-.IW)df 1 ~=-Lsl(n/W)c(I- n/W;t)W":_:lC wherec(r;t)isthetime-variant impulseresponse. Weobservethat(14-5-4) hastheformofaconvolution sum.Hence,itcanalsobeexpressed inthe alternative form 1~ r(l)= -LSI(I-n/W)c(n/W; I) Wn=-;>g(14-5-5) Itisconvenient todefineasetoftime-variable channelcoefficients as (14-5-6) Then(14·5-5)expressed intermsofthesechannelcoefficients becomes ~ ret)=LC.(I)s/(t-n/W) n=-QIl(14-5-7) Theformforthereceived signalin(14-5-7) impliesthatthetime-variant frequency-selective channelcanbemodeled orrepresented asatappeddelay linewithtapspacingI/Wandlapweightcoefficients {c.(t)}.Infact,wededuce from(14-5-7)thatthelowpassimpulseresponse forthechannelis ~ c(r;I)=Lc.(t)S(r -n/W) n=-:w;> andthecorresponding time-variant transferfunction is ~ C(f;l)=Lc.(I)e-i2"""W n=-~(14-5-8) (14-5-9) CHAPTER'" DIGITAL COMMUNICATION TlIROUOH FADING MULTIPATH CHANNELS 797 I iiiI iiiI iiiI iii ,,(,).i;ct(t)S/(,- -£;)+z(') r-"':":"!:::--,..IAdditive noise dt} nGuItE 14-5-1 Trappeddelaylinemodeloffrequency-seleclive channel Thus,withanequivalent lowpasssignalhavingabandwidth!W,where W»(Ii!)" weachievearesolution of1/Winthemultipath delayprofile. Sincethetotalmultipath spreadisTm,forallpractical purposes thetapped delaylinemodelforthechannelcanbetruncated atL=ITmW)+1taps.Then thenoiseless receivedsignalcanbeexpressed intheform Ii(/)=~Cn(/)s,(I-;) (14-5-10) Thetruncated tappeddelaylinemodelisshowninFig.14-5-1.In accordance withthestatistical characterization ofthechannelpresented in Section14-1,thetime-variant tapweights {C.(I)}are complex-valued stationary randomprocesses. InthespecialcaseofRayleigh fading,themagnitudes ICn(I)/-an(l)areRayleigb-distributed andthephases"'n(t)areuniformly distributed. Sincethe{cn(t}}represent thetapweightscorresponding totheL different delays 'r=n/W,n=I,2,...,L,theuncorrelated scattering assumption madeinSection 7-1impliesthatthe{en(I)}aremutually uncorrelated. Whenthe{cn(t)}aregaussian random processes, theyare statislically independent. 14-5-2TheRAKEDemodulator Wenowconsidertheproblem ofdigitalsignaling overafrequency-selective channelthatismodeled byalappeddelaylinewithstatistically independent time-variant tapweightstenet)}.Itisapparentattheoutset,however, thatthe tappeddelaylinemodelwithstatistically independent tapweightsprovides us 798 DIGITAL COMMUNICATiONS withLreplicasofthesametransmitted signalatthereceiver. Hence,areceiver thatprocesses thereceived signalinanoptimum manner willachieve the performmce ofanequivalent Lth·order diversity communications system. Letusconsider binarysignaling overthechannel. Wehavetwoequal­ energysignalsS,,(I)andSn{I).whichareeitherantipodal ororthogonal. Their timeduration Tisselected tosatisfytheconditionT»Tm•Thus,wemay neglectanyintersymbol interference duetomultipath. Sincethebandwidth of thesignalexceedsthecoherent bandwidth ofthechannel, thereceived signalis expressed as L T,(I)=Lc.(r)s,,(1-k/W)+Z(I) k""I =Vi(I)+Z(I). O";I,.;T, j=1,2 (14-5-11) whereZ(I)isacomplex-valued zero-mean whitegaussian noiseprocess. Assume forthemoment thatthechannel tapweightsareknown.Thenthe optimum receiverconsistsoftwofiltersmatched tov,(I)andv2(1).followed by samplers andadecision circuitthatselectsthesignalcorresponding tothe largestoutput.Anequivalent optimum receiver employs crosscorrelation insteadofmatched filtering. Ineithercase,thedecision variables forcoherent detection ofthebinarysignalscanbeexpressed as Um=Re[{T/(I)V;:'(t)dl) =Re[±ITT,(I)C:(I)St.,(t -k/W)dt).m=1.2(14-5-12) k""l0 Figure14-5-2illustrates theoperations involved inthecomputation ofthe decision variables. Inthisrealization oftheoptimum receiver, thetwo reference signalsaredelayedandcorrelated withthereceived signalT,(I). Analternative realization oftheoptimum receiver employs asingledelay linethroughwhichispassedthereceived signal7,(1).Thesignalateachtapis correlated withCk(I)s;';.(I). wherek=1.2•...•LandIn~1,2.Thisreceiver structure isshowninFig..14-5-3.Ineffect,thetappeddelaylinereceiver aUempts tocollectthesignalenergyfromallthereceived signalpathsthatfall withinthespanofthedelaylineandcarrythesameinformation. Itsactionis somewhat analogous toanordinary gardenrakeand,consequently, thename "RAKE receiver" hasbeencoinedforthisreceiver structure byPriceand Green(1958). 14-5·3Performance ofRAKEReceiver Weshallnowevaluate theperformance oftheRAKEreceiver underthe condition thatthefadingissufficiently slowtoallowustoestimate c.(t) perfectly (without noise).Furthermore, withinanyonesignaling interval, c.(t) CHAPTER 14,DIGITAL COMMUNICATION THROUGH FADING MUlTIPATH CHANNELS 799 U2=Re() FIGURE 14-5-1 Optimum demodulator forwideband binarysignals(delayed reference configuration). decision variables In istreatedasaconstant anddenoted asc..Thusthe (14-5-12) maybeexpressed intheform Urn=Re[~:CtLT r(l)si:,,(1-k/W)dl], k~l 0m=I,2(14-5-13) Suppose thetransmitted signalisSII(I);thenthereceived signalis L r,(I)=LCnSII(1-n/W)+z(t),0""I0;;T '1~1 Substitution of(14-5-14) into(14-5-13) yields Urn=Re[i:cd:CoLTSIl(t-n/W)s?.(1 -k/W)dtJ k~1,,-1 0 +Re[~1ctrz(t)st",(t -k/W)dt]'m=I,2(14-5-14) (14-5-15) 800 DIGITAL COMMUNICATIONS h--'----" Todetision ,--.,........., cirwitL..--_t--:- r FIGURE 14-5-3 Optimum demodulator forwideband binarysignals(delayed receivedsignalconfiguration). Usuallythewideband signalss/t(t)andSI2(t)aregenerated frompseudo­ randomsequences. whichresultinsignalsthaihavethepropertyrsu(t-n/W)sZ(t -k/W)dt=0,k'Fn,i=I,2(14-5-16) Ifweassumethatourbinarysignalsaredesigned tosatisfythisproperty then (14-5-15) simplifies tot Um=Re[±ICkl2iTs,,(t-k/W)st.,(t -k/W)dt] .t-J 0 +Re[±c:iT z(t)st.,(t-k/W)dt].m=1,2(14-5·17) k-IC tAlthough theorthogonality property specified by(14-5·16) canbesatisfied byproper selection ofthepseudo-random sequences, theeross-correlation ofS,,(I-n/W)withSl(l-k/W) givesrisetoasignal-dependent sell-noise. whichultimately limitslheperformance. Forsimplicity, wedonotconsider theself-noise terminthefollowing calculations. Consequently, the performance resultspresented belowshouldbeconsidered aslowerbounds(idealRAKE). An appTOJlimatiOll tolheperformance oflheRAKEcanbeobtained bytreatingtheself-noise asan additional gaussiannoisecomponent withnoisepowerequaltoitsvariance. (14-5-19)CHAPTER 14,DIGITAL CO....UNIC...TIONTHROUGH F...DlNG..ULTlP...THCH",NNELS 1101 Whenthebinarysignalsareantipodal, asingledecisionvariablesuffices.In thiscase,(14-5-17) reducesto VI=Re(2~~Ia~+~IakNk) (14-5-18) whereak=ICkland Nk=ei..·rz(t)sr(t-k/W)dt But(14-5-18) isidentical tothedecision variable givenin(14-4-4), which corresponds totheoutputofamaximal ratiocombiner inasystemwith Lth-order diversity. Consequently, theRAKEreceiverwithperfect(noiseless) estimates ofthechanneltapweightsisequivalent toamaximal ratiocombiner inasystemwithLth-order diversity. Thus,whenallthetapweightshavethe samemean-square value,i.e.,E(aDisthesameforalIk,theerrorrate performance oftheRAKEreceiverisgivenby(14-4-15) and(14-4-16). Onthe otherhand,whenthemeansquarevaluesE(a~)arenotidentical forallk,the derivation oftheerrorrateperformance mustberepeated since(14-4-15) no longeraIlPlies. Weshallderivetheprobability oferrorforbinaryantipodal andorthogonal signalsunderthecondition thatthemean-square valuesof{ak}aredistinct. Webeginwiththeconditional errorprobability (14-5-20) (14-5-21') ~4-5-22) (14-5-23)wherep,=-1forantipodal signals,p,=0fororthogonal signals,and '(,L 'Yb=-La~N ok_, L =L'Yk k-I Eachofthe{')'k}isdistributed according toachi-squared distribution with twodegreesoffreedom. Thatis, 1t· p(')'d=~e-"~' 'Yk whereYkistheaverageSNRforthekthpath,definedas -'(,'Y,=-E(aD No Furthermore, from (14-4-10) weknowthatthecharacteristic function of'Ykis . 1 t/J~.(]V)=1 .-]V')'.(14-5-24) 802 DIGITAL COMMUNICATIONS Since"IhisthesumofLstatistically independent components {Yd,the characteristic function ofYbis L1 "'n(jv)=[11. _ (14-5-25) k~l-JV"Ik TheinverseFouriertransform ofthecharacteristic function in(14-5-25) yields theprobability densityfunctionof'Ybintheform (14-5-26) where Jrkisdefinedas (14-5-27) Whentheconditional errorprobability in(14-5-20) isaveraged overthe probability densityfunction givenin(14-5-26), theresultis 'Yk(1-p,)] 2+'Yk(1-p,)(14-5-28) Thiserrorprobability canbeapproximated as('Yk»1) P2=(2L-1)n__1__ L k~'2'Y.(1-p,)(14-5-29) Bycomparing (14-5-29) forp,=-1with(14-4-18), weobservethatthesame typeofasymptotic behavior isobtained forthecaseofunequalSNRperpatn andthecaseofequalSNRperpath. Inthederivation oftheerrorrateperformance oftheRAKEreceiver, we assumed thattheestimates ofthechanneltapweightsareperfect.Inpractice, relatively goodestimates canbeobtained ifthechannelfadingissufficiently slow,e.g.,(at)elT'"100,whereTisthesignaling interval. Figure14-5-4 illustrates amethodforestimating thetapweightswhenthebinarysignaling waveforms areorthogonal. Theestimate istheoutputofthelowpassfilterat eachtap.Atanyoneinstantintime,theincoming signaliseithersll(r)orS/2(t). Hence,theinputtothelowpassfilterusedtoestimate ek(t)containssignalplus noisefromoneofthecorrelators andnoiseonlyfromtheothercorreIator_ Thismethodforchannelestimation isnotappropriate forantipodal signals, becausetheadditionofthetwocorreiatoroutputsresultsinsignalcancellation_ Instead,asinglecorrelator canbeemployed forantipodal signals.Itsoutputis fedtotheinputofthelowpassfilteraftertheinformation-bearing signalis removed. Toaccomplish this,wemustintroduce adelayofonesignaling intervalintothechannelestimation procedure, asillustrated inFig.14-5-5. CHAPTER 14,DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS 803 FIGURE 14-5-4 Channel tapweightestimation withbinary orthogonal signals.x Tosummer andintegralor--_.ill- Tosummer andinlegrat<r FIGURE 14-5-5 Channel tapweightestimation withbinaryantipodal signals. IWIW1if Delay T Lowpass filterof bandwidth B,Signalfrom pre·..ious dcci~ion it 8D4 DKlITAI.COMMUN'CA nONS Decision variable FIGURE 14-5-6 RAKEdemodulator forDPSKsignals. Thatis,firstthereceivermustdecidewhethertheinformation inthereceived signalis+1or-1and,then,itusesthedecisiontoremovetheinformation fromthecorrelator outputpriortofeedingittothelowpassfiter. Ifwechoosenottoestimate thetapweightsofthefrequency-selective channel, wemayuseeitherDPSKsignaling ornoncoherently detected orthogonal signaling. TheRAKEreceiverstructure forDPSKisillustrated in Fig14-5-6.Itisapparent thatwhenthetransmitted signalwaveform $/(1) satisfiestheorthogonality property givenin(14-5-16), thedecisionvariable is identical thatgivenin(14-4-23) foranLth-order diversity system.Conse­ quently,theerrorrateperformance oftheRAKEreceiverforabinaryDPSK isidenticaltothatgivenin(14-4-15) withII-='Ye/(l+'Ye).whenallthesignal pathshavethesameSNR'Ye.Ontheotherhand,whentheSNRs{'Yk}are distinct,theerrorprobability canbeobtainedoyaveraging (14-4-24), whichis theprobability oferrorconditioned onatime-invariant channel, overthe probability densityfunction of"Ybgivenby(14-5-26). Theresultofthis integration is P2=<D2L-1}:,Im!bm±~k('Yk_)'"+1 (14-5-30) m-O k-l"Yk1+Yk whereIrkisdefinedin(14-5-27) andbmin(14-4-25). Finally, weconsider binaryorthogonal signaling overthefrequency­ selective channelwithsquare-law detection atthereceiver. Thistypeofsignal CHAPTER 14:DIGITAL COMMUNICATION rnROUGH FADING MULTlPATlt CHANNELS 805 L__I-../_+-_ Todecision circuit nGURE 14-5-7 RAKEdemodulator lorsquare-law combination oforthogonal signals. isappropriate wheneitherthefadingisrapidenoughtopreclude agood estimate ofthechanneltapweightsorthecostofimplementing thechannel estimators ishigh.TheRAKEreceiverwithsquare-law combining ofthesignal fromeachtapisillustrated inFig.14-5-7.Incomputing itsperformance, we againassumethattheorthogonality property givenin(14-5-16) holds.Then thedecisionvariables attheoutputoftheRAKEare L V,=2:12~.+N.,12.-t L V2=2:INk212 .=1(14-5-31) wherewehaveassumed thatSIJ(t)wasthetransmitted signal.Againwe observethatthedecisionvariables areidentical totheonesgivenin(14-4-29), whichapplytoorthogonal signalswithLth-order diversity. Therefore, the performance oftheRAKEreceiverforsquare-law-detected orthogonal signals isgivenby(14-4-15) withp.=y,/(2+Yc)whenallthesignalpathshavethe sameSNR.IftheSNRsaredistinct,wecanaveragetheconditional error probability givenby(14-4-24), with1breplaced byhb'overtheprobability 80fi DllilT-\L CUM""ll~I(-ATIO"S densityfunctionp(Yh)givenin(14-5-26). Theresultofthisaveraging isgiven by(14-5-30). with1kreplaced by\5k' Intheaboveanalysis. theRAKEdemodulator showninFig.14-5-7for square-law combination oforthogonal signalsisassumed toconlainasignal component ateachdelay.Ifthatisnotthecase.itsperformance willbe degraded. sincesomeofthelapcorrelators willcontribute onlynoise.Under suchconditions. thelow-level. noise-only contributions fromthetapcor­ relalorsshouldbeexcluded fromthecombiner. asshownbyChyiet01.(1988). Thisconcludes ourdiscussion ofsignaling overafrequency-selective channel. Theconfigurations oftheRAKErcceiver presented inthissection canbeeasilygeneralized tomultilevel signaling. Infact.ifM-aryPSKor DPSKischosen. theRAKEstructures presented inthissectionremain unchanged. OnlythePSKandDPSKdetectors thatfollowtheRAKE correlator aredifferent. 14-6CODED WAVEFORMS FORFADING CHANNELS Uptothispoint.wehavedemonstrated thatdiversity techniques arevery effective inovercoming thedetrimental effectsoffadingcaused bythe time-variant dispersive characteristics ofthechannel. Time-and/orfrequency­ diversity techniques maybeviewedasaformofrepetition (block)codingof theinformation sequence. Fromthispointofview,thecombining techniques described previously represent soft-decision decoding oftherepetition code. Sincearepetition codeisatrivialformofcoding,weshallnowconsider the additional benefitsderivedfrommoreefficienttypesofcodes.Inparticular. we demonstrate thatcodingprovides anefficientmeansforobtaining diversity on afadingchannel. Theamountofdiversity provided byacodeisdirectlyrelated toitsminimum distance. Asexplained inSection14-4,timediversity isobtained bytransmitting the signalcomponents carrying thesameinformation inmultiple timeintervals mutually separated byanamountequaltoorexceeding thecoherence time (At),.ofthechannel. Similarly, frequency diversity isobtained hytransmitting thesignalcomponents carrying thesameinformation inmultiple frequency slotsmutually separated byanamountofatleastequaltothecoherence bandwidth (j.f).ofthechannel. Thus,thesignalcomponents carrying thesame information undergo statistically independent fading. Toextendthesenotionstoacodedinformation sequence, wesimplyrequire thatthesignalwaveform corresponding toaparticular codeorcodesymbol fadeindependently ofthesignalwaveform corresponding toanyothercodebit orcodesymbol. Thisrequirement mayresultininefficient utilization ofthe available time-frequency space.withtheexistence oflargeunusedportions in thistwo-dimensional signaling space.Toreducetheinefficiency. anumber of codewordsmaybeinterleaved intimeorinfrequency orboth.insucha mannerIhatthewaveform corresponding tothebitsorsymbolsofagivencode wordfadeindependently. Thus.weassumeIhatthetime-frequency signaling ("HAPIFRHt)[{illAI.{"O\1\1l :~l(ArlO;"; IIlKOl~{iHF"D1~{I~n:LTII'A III('HN';~i:T'" 807 spaceispartitioned intononoverlapping time-frequenc)' cells.Asignal waveform corresponding toacodebitorcodesymbolistransmitted within such acell. Inaddition totheassumption ofstatistically independent fadingofthe signalcomponents ofagivencodeword,wealsoassumethattheadditivenoise components corrupting thereceived signalsarewhitegaussian processes that arestatistically independent andidentically distributed amongthecellsinthe time-frequency space.Also,weassume thatthereissufficient separation between adjacent cellssothatintercell interference isnegligible. Animportant issueisthemodulation technique thatisusedtotransmit the codedinformation sequence. Ifthechannelfadesslowlyenough toallowthe establishment ofaphasereference thenPSKorDPSKmaybeemployed. If thisisnotpossible thenFSKmodulation withnoncoherent detection atthe receiver isappropriate. Inourtreatment, weassumethatitisnotpossible to establish aphasereference orphasereferences forthesignalsinthedifferent cellsoccupied bythetransmitted signal.Consequently, wechooseFSK modulation withnoncoherent detection. AmodeIofthedigitalcommunications systemforwhichtheerrorrate performance willbeevaluated isshowninFig.14-6-1.Theencoder maybe binary.nonbinary, oraconcatenation ofanonbinary encoder withabinary encoder. Furthermore. thecodegenerated bytheencoder maybeablock code,aconvolutional code,or,inthecaseofconcatenation, amixtureofa blockcodeandaconvolutional code. Inordertoexplain themodulation, demodulation, anddecoding for FSK-type (orthogonal) signals,consider alinearbinaryblockcodeinwhichk information bitsareencoded intoablockofnbits.Forsimplicity andwithout lossofgenerality, letusassumethatallnbitsofacodewordaretransmitted simultaneously overthechannel onmultiple frequency cells.AcodewordC, havingbits{cij}ismapped intoFSKsignalwaveforms inthefollowing way.If ciJ=0,thetonej;,istransmitted, andifC,j=I.thetonefijistransmitted. This meansthat2ntonesorcellsareavailable totransmit thenbitsofthecode word,butonlyntonesaretransmitted inanysignaling interval. Sinceeach codewordconveys kbitsofinformation, thebandwidth expansion factorfor FSKisBe=2n/k Thedemodulator forthereceived signalseparates thesignalinto2n InpUiEncoocrFSK modulluorRa.y\eig\'o fading AWGN r.:hann~l FIGURE 14-6·1 Modelofcommunications systemwithFSK modulation/demodulation andencoding/decoding.Output [)c-(oderFilter-ban~ demodulator 808 DIGITAL C'OMMLNIC ATIO~S spectralcomponents corresponding totheavailable tonefrequencies atthe transmitter. Thus.thedemodulator canberealizedasabankof2nfilters, whereeachfilterismatched tooneofthepossible transmitted tones.The outputsofthe2nfiltersaredetected noncoherently. SincetheRayleigh fading andtheadditive whitegaussian noisesint.he2nfrequency cellsaremutually statistically independent andidentically distributed random processes, the optimum maximum-likelihood soft-decision decoding criterion requires that thesefilterresponses besquare-law-detected andappropriately combined for eachcodewordtoformlheM=2kdecision variables. Thecodeword corresponding tothemaximum ofthedecision variables isselected.If hard-decision decoding isemployed, theoptimum maximum-likelihood de­ coderselectsthecodewordhavingthesmallest Hamming distancerelaliveto thereceived codeword. Although thediscussion aboveassumed theuseofablockcode,a convolutional encoder canbeeasilyaccommodated intheblockdiagram showninFig.14-6-1.Forexample, ifabinaryconvolutional codeisemployed, eachbitintheoutputsequence maybetransmitted bybinaryFSK.The maximum-likelihood soft-decision decoding criterion fortheconvolutional codecanbeefficiently implemented bymeansoftheViterbialgorithm, in whichthemetricsforthesurviving sequences atanypointinthetrellisconsist ofthesquare-law-combined outputsforthecorresponding pathsthrough the trellis.Ontheotherhand,ifhard-decision decoding isemployed, theViterbi algorithm isimplemented withHamming distanceasthemetric. 14-6-1Probability ofErrorforSoft-Decision Decoding of LinearBinaryBlockCodes Consider thedecoding ofalinearbinary(n,k)codetransmitted overa Rayleigh fadingchannel, asdescribed above.Theoptimum soft-decision decoder, basedonthemaximum-likelihood criterion, formstheM=2' decisionvariables "Vi=L[(1-c,)!YQ/+eijIY,iJ j=l n =L[lYol+cij(!y,l-jy"l)]. i=1,2,...,2k j=l04-6-1) wherejy,/,j=1,2,....n.andr=O.1represent thesquaredenvelopes atthe outputsofthe211filtersthataretunedtothe2npossibletransmitted tones.A decision ismadeinfavorofthecodewordcorresponding tothelargest decisionvariableoftheset{VJ Ourobjective inthissectionisthedetermination oftheerrorrate performance ofthesoft-decision decoder. Towardthisend,letusassumethat theall-zerocodewordC\istransmitted. Theaveragereceived signal-to-noise CHAPTER 14:[)IGITAl C()M~l'NICATION lliROlfGH FADING MlILTIPATH CHANNEtS 809 ratiopertone(cell)isdenotedbyy,.Thetotalreceived SNRlorthentonesin ny,and.hence.theaverageSNRperbitis 1.=- R•.(14-6-2) (14-6-3) (14-6-4) (14-6-5) (14-6-6)whereR,.isthecoderate. Thedecision variable VIcorresponding tothecodewordCIisgivenby (14-6-1)witheij=0forallj.Theprobability thatadecision ismadeinfavorof themthcodewordisjust Pz(m)=P(V",>VI)=P(V,-V,;,<0) =p[i(clj-cOOj)(IYI;l2-!Yo;f)<0] }>I =p[}':(/YOj/2-IYll)<0] }O' where W",istheweightofthemthcodeword.Buttheprobability in(14-6-3) is justtheprobability oferrorforsquare-law combining ofbinaryorthogonal FSKwithwmth-order diversity. Thatis, "',.,-1 1k~(Woo-+)•P2(m)=p"m" (l-p)'-0 k ·~'(W..-I+k)(2Woo-l)~p~ ~ = p~'.0 k Woo where 1 1p=--=2+Y,2 +R,Yh Asanalternative. wemayusetheChernoff upperboundderivedinSection 14-4,whichinthepresentnotation is Pz(m)..,;[4p(1-p)]"m (14-6-7) ThesumofthebinaryerroreventsovertheM-1nonzero-weight code wordsgivesanupperboundontheprobability oferror.Thus. (14-6-8) Sincetheminimum distance ofthelinearcodeisequaltotheminimum weight.itfollowsthat 810 DIGITAL COMMUNICATIONS Theuseofthisrelation inconjunction with(14-6-5) and(14-6-8) yieldsa simple,albeitlooser,upperboundthatmaybeexpressed intheform },(2W: m-1) P< (14-6-9) M(2+Rc)'h)'lmoo Thissimpleboundindicates thatthecodeprovides aneffective orderof diversity equaltodmin•Anevensimplerboundistheunionbound PM<(M-1)[4p(1- pWmoo (14-6-10) whichisobtained fromtheChernoff boundgivenin(14-6-7). Asanexample servingtoillustrate thebenefitsofcodingforaRayleigh fadingchannel, wehaveplottedinFig.14-6-2theperformance obtained with theextended Golay(24,12)codeandtheperformance ofbinaryFSKand quarternay FSKeachwithdualdiversity. Sincetheextended Golaycode requires atotalof48cellsandk=12,thebandwidth expansion factorB,=4. Thisisalsothebandwidth expansion factorforbinaryandquaternary FSK withL=2.Thus,thethreetypesofwaveforms arecompared onthebasisof thesame.bandwidth expansion factor.NotethatatPh=10-4,theGolaycode outperforms quaternary FSKbymorethan6dB,andatPh=10-',the difference isapproximately 10dB. 14Ib1820 22 24 26 SNRperbit.Yb(dB)\ !~i'.8,=4 '\~i'. 1\~BinaryFSK L=2\""-""""\M=4FSK'"L='"'-~ 1\ ........~ \Golay"'" '\ (24.121 soft-decision decoding \, 1\25 10-6 12210-1, 52 10-' FIGURE 14-6-2 Example ofperformance obtained withconventional diversity versuscoding forB.=4. CHAPTER 14:DIGITAL COMMUNlCATION THROUGH FADING MlJLTIPATH CHA:"INELS 811 Thereasonforthesuperior performance oftheGolaycodeisitslarge minimum distance(dm;"=8),whichtranslates intoanequivalent eighth-order (L=8)diversity. Incontrast, thebinaryandquaternary FSKsignalshaveonly second-order diversity. Hence,thecodemakesmoreefficient useofthe available channel bandwidth. Thepricethatwemustpayforthesuperior performance ofthecodeistheincreaseindecoding complexity. 14-6-2Probability ofErrorforHard-Decision Decoding of LinearBinaryBlockCodes Boundsontheperformance obtained withhard-decision decoding 9falinear binary(n,k)codehavealreadybeengiveninSection8-1-5.Theseboundsare applicable toageneralbinary-input binary-output memoryless (binarysym­ metric)channel and,hence,theyapplywithoutmodification toaRayleigh fadingAWGNchannelwithstatistically independent fadingofthesymbols in thecodeword.Theprobability ofabiterrorneededtoevaluate thesebounds whenbinaryFSKwithnoncoherent detection isused a~themodulation and demodulation technique isgivenby(14-6-6). Aparticularly interesting resultisobtained whenweusetheChernoff upper boundontheerrorprobability forhard-decision decoding givenby(8-1-89). Thatis, (14-6-11) andPMisupper-bounded by(14-6-8). Incomparison, theChernoff upper boundforP2(m)whensoft-decision decoding isemployed isgivenby(14-6-7). Weobserve thattheeffectofhard-decision decoding isareduction inthe distance between anytwocodewordsbyafactorof2.Whentheminimum distance ofacodeisrelatively small,thereduction ofthedistances byafactor of2ismuchmorenoticeable inafadingchannelthaninanonfading channel. Forillustrative pruposes wehaveplottedinFig.14-6-3theperformance of theGolay(23,12)codewhenhard-decision andsoft-decision decoding are used.Thedifference inperformance atPb=10-5isapproximately 6dB.Thisis asignificant difference inperformance compared withthe2dBdifference between soft-andhard-decision decoding inanonfading AWGNchannel. We alsonotethatthedifference inperformance increases asPbdecreases. Inshort, theseresultsindicate thebenefits ofasoft-decision decoding overhard­ decision decoding onaRayleigh fadingchannel. 14-6-3UpperBoundsonthePerformance ofConvolutional CodesforaRayleigh FadingChannel Inthissubsection, wederivetheperformance ofbinaryconvolutional codes whenusedonaRayleigh fadingAWGNchannel. Theencoder accepts k binarydigitsatatimeandputsoutnbinarydigitsatatime.Thus,thecode rateisRc=kin.Thebinarydigitsattheoutputoftheencoderaretransmitted 8U DIGIT.I\L COMMUNICATIONS 510-1 10-2 5 16 18 20222426 SNRpet'bil.I.ldB,,Golay(23.12)code I\. 1\\ \1'\Hard-dtcisia. decodilllg \\. \. r\Soft-dccision \. \decoding \ \ \ r-.. \ 1\10-.'1 5 2 Itr" 1214..- ~2•j Io--~ •'E5 fIO-~ 5 FIGURE.....3Comparison ofperformance between hard­ andsoft-decision decoding. (14-6-12)overtheRayleigh fadingchannel bymeansofbinaryFSK,whichis square-law-detected atthereceiver. Thedecoder foreithersoft-orhard­ decisiondecoding performs maximum-likelihood sequence estimation, whichis efficiently implemented bymeansoftheViterbialgorithm. First,weconsider soft-decision decoding. Inthiscase,themetricscomputed intheViterbialgorithm aresimplysumsofsquare-law-detected outputsfrom thedemodulator. Suppose theall-zerosequence istransmitted. Following the procedure outlined inSectionB-2-3,itiseasilyshownthattheprobability of errorinapairwise comparison ofthemetriccorresponding totheall-zero sequence withthemetriccorresponding toanothersequence thatmergesfor thefirsttimeattheall-zerostateis P2(d)=pd~:(d-~+k)(1-pt wheredisthenumberofbitpositions inwhichthetwosequences differandp isgivenby(14-6-6). Thatis,P2(d)isjusttheprobability oferrorforbinary FSKwithsquare-law detection anddth-order diversity. Alternatively, wemay usetheChernoff boundin(14-6-7) forP2(d).Inanycase,thebiterror probability isupperbounded, asshowninSection8-2-3bytheexpression (14-6-13) CHAYTER 14:DIGITAL COMMUNICATiON THROUGH FADING MULTlPATH CHANNELS 813 wheretheweighting coefficients {~d}inthesummation areobtained fromthe expansion ofthefirstderivative ofthetransferfunction T(D,N),givenby (8-2-25). Whenhard-decision decoding isperformed atthereceiver, theboundson theerrorrateperformance forbinaryconvolutjonal codesderivedinSection 8-2-4apply.Thatis,Pbisagainupper-bounded bytheexpression in(14-6-13), whereP2(d)isdefinedby(8-2-28)forodddandby(8-2-29)forevend,or upper-bounded (Chernoff bound)by(8-2-31), andpisdefinedby(14-6-6). Asinthecaseofblockcoding,whentherespective Chernoff boundsare usedforP2(d)withhard-decision andsoft-decision decoding, itisinteresting to notethattheeffectofhard-decision decoding istoreducethedistances (diversity) byafactorof2relativetosoft-decision' decoding. Thefollowing numerical resultsillustrate theerrorrateperformance of binary,ratelin,maximal freedistanceconvolutional codesforn=2,3,and4 withsoft-decision Viterbidecoding. Firstofall,Fig.14-6-4showsthe performance oftherate1/2convolutional codesforconstraint lengths3,4,and 5.Thebandwidth expansion factorforbinaryFSKmodulation isB.=2n. Sinceanincrease intheconstraint lengthresultsinanincrease inthe complexity ofthedecodertogoalongwiththecorresponding increaseinthe miJrimurn freedistance, thesystemdesigner canweighthesetwofactorsinthe selection ofthecode. Another waytoincrease thedistance withoutincreasing theconstraint 10 12 14 16 18 20 SNR.....bil.l.(dB)8Rc=t .. ~: ~Consuaint ImJlh=3 \\\I \\'Constraint lenath=4 \'\ COllStt8inlleftlth =5\\ \\\ \\\ \ \ \\\ \\ \\\10-1 5 2 10-' ...~5 ~2 :S10-3•'05 .~ t~ 5 2 10-' 5 2 nGURE 14-6-4 Performance ofrate1{2binaryconvolutional 10-'6 codeswithsoftdecisiondecoding. 814 DIGrrAL COMMC"CA nONS lengthofthecodeistorepeateachoutputbitmtimes.Thisisequivalent to reducing thecoderatebyafactorofmorexpanding thebandwidth bythe samefactor.Theresultisaconvolutional codethathasaminimum free distanceofmd,,,",whered'mistheminimum freedistance oftheoriginalcode withoutrepetitions. Suchacodeisalmostasgood,fromtheviewpoint of minimum distance, asamaximum freedistance, ratelimncode.Theerrorrate performance withrepetitions isupper-bounded by I~ Ph<kL(3dp,(md) d!r~"(14-6-14) whereP,(md)isgivenby(14-6-12). Figure(14-6-5)illustrates theperformance oftherate1/2codeswithrepetitions (m=I,2.3.4)forconstraint length5. 14-6-4UseofConstant-Weight CodesandConcatenated • CodesforaFadingChannel Ourtreatment ofcodingforaRayleigh fadingchanneltothis'pointwasbased ontheuseofbinaryFSKasthemodulation technique fortransmitting eachof thebinarydigitsinacodeword.Forthismodulation technique, al/the2'code 201816 1214 108, I R.=..L <~m COllSlrdint length=~-, \ \\ \ \ \ \m=1 '\--,\ m-4~=3\ :\\5 l<r' 6Performance ofrate112m,constraint length5.binaryconvolutional codes withsoft-decision decoding.10 10-5510-5 tlGURE }4+5 ICH,\PT1·.R 14:DIGITAL COMMLNICATION THROliGH FADING MULTIPATH CHANNELS 815 wordsinthe(n,k)codehaveidentical transmitted energy.Furthermore, under thecondition thatthefadingonthentransmitted tonesismutually statistically independent andidentically distributed, theaveragereceived signalenergyfor the!vi=2kpossible codewordsisalsoidentical. Consequently, inasoft­ decision decoder, thedecision ismadeinfavorofthecodewordhavingthe largestdecision variable. Thecondition thatthereceived codewordshaveidentical averageSNRhas animportant ramification intheimplementation ofthereceiver.Ifthereceived codewordsdonothaveidentical averageSNR,thereceiver mustprovidebias compensation foreachreceived codewordsoastorenderitequalenergy.In general, thedetermination oftheappropriate biastermsisdifficult to implement because itrequires theestimation oftheaverage received signal power:hence,theequal-energy condition onthereceived codewords considerably simplifies thereceiver processing. Thereisanalternative modulation method forgenerating equal-energy waveforms fromcodewordswhenthecodeisconstant-weight, i.e.,whenevery codewordhasthesamenumber ofIs.Notethatsuchacodeisnonlinear. Nevertheless, suppose weassignasingletoneorcelltoeachbitposition ofthe 2kcodewords.Thus,an(n,k)binaryblockcodehasntonesassigned. Waveforms areconstructed bytransmitting thetonecorresponding toa particular bitinacodewordifthatbitisa1;otherwise, thattoneisnot transmitted fortheduration oftheinterval. Thismodulation technique for transmitting thecodedbitsiscalledon-offkeying(OOK). Sincethecodeis constant-weight, sayw,everycodedwaveform consistsofWtransmitted tones thatdep~ndonthepositions oftheIsineachofthecodewords. AsinFSK,alltonesintheOOKsignalthataretransmitted overthe channelareassumed tofadeindependently acrossthefrequency bandandin timefromonecodewordtoanother. Thereceived signalenvelope foreach toneisdescribed statistically bytheRayleigh distribution. Statistically inde· pendent additive whitegaussian noiseisassumed tobepresent ineach frequency cell. Thereceiver employs maximum·likelihood (soft-decision) decoding tomap thereceived waveform intooneoftheMpossible transmitted codewords.For thispurpose, nmatched filtersareemployed, eachmatched tooneofthe'! frequency tones.Fortheassumed statistical independence ofthesignalfading forthenfrequency cellsandadditive whitegaussian noise,theenvelopes of thematched filteroutputsaresquared andcombined toformtheMdecision variables "Vi=Lc"ly,12 ,i=1,2.....2k j=l(14-6-15) where 1,v,12corresponds tothesquared envelope ofthefiltercorresponding to thejthfrequency, wherej=1.2,...,n. ltmayappearthattheconstant-weight condition severely restricts our choiceofcodes.Thisisnotthecase,however. Toillustrate thispoint.we 816 DIGITAL COMMUNICATIONS brieflydescribe somemethods forconstructing constant-weight codes,This discussion isbynomeansexhaustive. Method 1:Nonlinear Transformation ofaLinearCodeIngeneral, ifin eachwordofanarbitrary binarycodewesubstitute onebinarysequence for everyoccurrence ofa 0andanothersequence foreach1,aconstant-weight binaryblockcodewillbeobtained ifthetwosubstitution sequences areof equalweightsandlengths.Ifthelengthoftheseque'nce is11andtheoriginal codeisan(n,k)codethentheresulting constant-weight codewillbean(vn,k) code.Theweightwillbentimestheweightofthesubstitution sequence, and theminimum distancewillbetheminimum distances oftheoriginalcodetimes thedistances between thetwosubstitution sequences. Thus,theuseof complementary sequences whenvisevenresultsinacodewithminimum distancevdminandweight! vn. Thesimplest formofthismethodisthecasev=2,inwhichevery0is replaced bythepairOlandevery1isreplaced bythecomplementary sequence 10(orviceversa).Asanexample, wetakeastheinitialcodethe(24,12) extended Golaycode.Theparameters oftheoriginal andtheresultant constant-weight codearegiveninTable14-6-1. Notethatthissubstitution processcanbeviewedasaseparate encoding. Thissecondary encoding clearlydoesnotaltertheinformation contentofa codeword-it merelychanges theforminwhichitistransmitted. Sincethe newcodewordiscomposed ofpairsofbits-one "on"andone"off'-the use ofOOKtransmission ofthiscodewordproduces awaveform thatisidentical 10thatobtained bybinaryFSKmodulation fortheunderlying linearcode. Method 2:Expurption Inthismethod, westartwithanarbitrary binary blockcodeandselectfromitasubsetconsisting' ofallwordsofacertain weight.Severaldifferent constant-weight codescanbeobtained fromone initialcodebyvaryingthechoiceoftheweightw.Sincethecodewordsofthe resulting expurgated codecanbeviewedasasubsetofallpossible permuta­ tionsofanyonecodewordintheset,thetermbinaryexpurgated permutariol1 modulation (BEXPERM) hasbeenusedbyGaarder (1971)todescribe sucha code.Infact,theconstant-weight binaryblockcodesconstructed bytheother TABLE 14-6-1 EXAMPLE OFCONSTANT-WEIGHT CODEFORMED BY METHOD 1 Codeparameters Original Goloy Co.....nl-weipl 11 24 4H k 12 12 M 4096 40'16 (1mI" H 16 K' variable 24 CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS 817 TABLE14+1EXAMPLES UFCONSTANT-WEIGHT CODESFORMED BYEXPURGATION P.......eten OrfPulc_,welp'No.ICo_' ........'No.1 n 24 24 24 k 12 9 11 M 4096 759 2576 dolln 8 ~8 ~8 '"variable 8 12 methods mayalsobeviewedasBEXPERM codes.Thismethodofgenerating constant-weight codesisinasenseopposite tothefirstmethodinthatthe wordlengthnisheldconstant andthecodesizeMischanged. Theminimum distance fortheconstant-weight subsetwillclearlybenolessthanthatofthe originalcode.Asanexample, weconsider theGolay(24,12)codeandform thetwodifferent constant-weight codesshowninTable14-6-2. Method 3:Hadamard Matrices Thismethod mightappeartoforma constant-weight binaryblockcodedirectly, butitactually isaspecialcaseof themethodofexpurgation. Inthismethod, aHadamard matrixisformedas described inSection8-1-2,andaconstant-weight codeiscreatedbyselection ofrows(codewords)fromthismatrix.RecallthataHadamard matrixisan nXnmatrix:(neveninteger) ofIsandOswiththeproperty thatanyrow differsfromanyotherrowinexactly1npositions. Onerowofthematrixis normally chosenasbeingallOs. Ineachoftheotherrows,halfoftheelements areOsand',theotherhalfIs. AHadamard codeofsize2(n-1)codewordsisobtained byselecting these n-1rowsandtheircomplements. Byselecting M=2k..2(n-I)ofthese codewords,weobtainaHadamard code,whichwedenotebyH(n,k),where eachcodewordconveyskinformation bits.Theresulting codehasconstant weight!nandminimum distance dm;n=!n. Sincenfrequency cellsareusedtotransmit kinformation bits,the bandwidth expansion factorfortheHadamard H(n,k)codeisdefinedas nBe=Iecellsperinformation bit whichissimplythereciprocal ofthecoderate.Also,theaverage signal-to­ noiseratio(SNR)perbit,denoted by"Yb'isrelatedtotheaverageSNRper cell,Yobytheexpression (14-6-16) 818 DIGITAL C'OMMUNICATIONS Letuscompare theperformance oftheconstant-weight Hadamard codes underafixedbandwidth constraint withaconventional M-aryorthogonal set ofwaveforms whereeachwaveform hasdiversity L.TheMorthogonal waveforms withdiversity areequivalent to:Iblockorthogonal codehavinga blocklengthn=LMandk=log2M.Forexample, ifM=4andL=2,the codewordsoftheblockorthogonal codeare C1=[10 000 00] C,=[0010000] CJ=[00001 100] C4=[00000 01IJ Totransmit thesecodewordsusingOOKmodulation requires n=8cells,and sinceeachcodewordconveys k=2bitsofinformation, thebandwidth expansion factorBe=4.Ingeneral, wedenotetheblockorthogonal codeas O(n,k).Thebandwidth expansion factoris nLMB=-=- ek k(14-6-17) Also,theSNRperbitisrelatedtotheSNRpercellbytheexpression (k) Yb =M-y.=M-n Be(14-6-18) Nowweturnourattention totheperformance characteristics ofthese codes.First,theexactprobability ofacodeword(symbol) errorforM-ary orthogonal signaling overaRayleigh fadingchannelwithdiversity wasgivenir closedforminSection14-4.Aspreviously indicated, thisexpression israther cumbersome toevaluate, especially ifeitherLorMorbotharelarge.Instead, weshalluseaunionboundthatisveryconvenient. Thatis,forasetofM orthogonal waveforms, theprobability ofasymbolerrorcanbeupper­ bounded as PM'"(M-I)P2(L) =(2'-1)P2(L)<2'P2(L) (14-6-19) whereP,(L),theprobability oferrorfortwoorthogonal waveforms, eachwith diversity L,isgivenby(14-6-12) withP=1/(2+'U·Theprobability ofbit errorisobtained bymultiplying PMby2*'1/(2'-1),asexplained previously. CHAPTER 14:DIGITAL COMMUNICA.TION THROVGH FADING MUlTiPATH (HANNl:LS 819 Asimpleupper(union)boundontheprobability ofacodeworderrorfor theHadamard H(n,k)codeisobtained bynotingthe probability oferrorin deciding between thetransmitted codewordandanyothercodewordis bounded fromabovebyP,(~dmin). wheredministheminimum distance ofthe code.Therefore, anupperboundonPMis PM";;(M-I)P,(ld m1n)<2kP21~dmln) (14-6-20) Thusthe"effective orderofdiversity" ofthecodeforOOKmodulation is ldmin•Thebiterrorprobability maybeapproximated aslpM,orslightly overbounded bymultiplying PMbythefactor2k-1/(2k-1),whichisthefactor usedabovefororthogonal codes.Thelatterwasselected fortheerror probability computations givenbelow. Figures14-tj-6and14-6-7illustrate theerrorrateperformance ofaselected numberofHadamard codesandblockorthogonal codes,respectively. for severalbandwidth expansion factors.Theadvantage resulting fromanincrease inthesizeMofthealphabet (ork,sincek=log2M)andanincrease inthe bandwidth expansion factorisapparent fromobservation ofthesecurves. Note,forexample, thattheH(20,5)codewhenrepeated twiceresultsina codethatisdenoted by,H(20.5)andhasabandwidth expansion factorB,=8. Figure14-6-8showstheperformance ofthetwotypesofcodescompared on thebasisofequalbandwidth expansion factors.Itisobserved thattheerror ratecurvesfortheHadamard codesaresteeperthanthecorresponding curves '4IS2022, \\\'\ \\~1\ \\\ \\ ~\\ ,'\H('0.5i- ~i\'\8,=4- \1\ 1\\\ ,H(36.6)\\ \'--- 8,=,I"\,HI20.5)1\ '---,H(48.6)\f'I8,~S\ 8,=,16".\\\H(4S.~) \ IT'\f'=SI--- ,H(36.6) '---8 =24'-,I1\\\5, , 2 10-6 10 12 14 16:5•'0 ~JI~;....10' 510-2 FlGURE 14-6-6 Performance ofHadamard codes. 820 DIGITAL COMMUNICATIONS 12141618202224 SNRperbit.'Y,(dB)""'\ 0<8.2"0.- ~\\8,=4 "'"\. \'\\\. .......... 1\\\. \\\1\1\0(16.21 .\\\,=8_r--- 1\\\\. 0(96.3~\\\\ B,=32\\\0(24.2)f\- \1\V,=12 \\\'\0<32.2)_ 1\\\.-1-\B,=16 0(72,3)~\\ B=24.....\"1\\ I\1\\ \52 10-' 2 1lT' 10IO-J 2 1lT' 5 FIGURE 14-6-7 Performance ofblockorthogonal codes. 1416182022 24 SNRperbiroYI(dB)\\'..\I". \\'.0(8,2) 8,=4. \. \.\. \'.'".. K·.0(16,21 , \,\'.B,=8-, \,\I'., \,H(20J\'-.,, \,B,=~\, \ \ U(~8,6)\ ,H(48,6)\\¥V8=81\" ,i Br=16\..0(32,2)~\,8=16,' , \1\2 1lT' 101210-) 2 Hr' 5 ....2 ~m" 0 :55 •~ 02 .~ ]10-'i5 2 10-7 5 FIGURE 14-6-8 Comparison ofperformance between Hadamard codesandblockorthogonal codes. CHAPTER 14:DIGITAL COMMUNICATION THROUGH FADING MULTIPATHCHANNELS 821 fortheblockorthogonal codes.Thischaracteristic behavior isduesimplyto thefactthat,forthesamebandwidth expansion factor,theHadamard codes providemorediversity thanblockorthogonal codes.Alternatively, onemay saythatHadamard codesprovide betterbandwidth efficiency thanblock orthogonal codes.Itshouldbementioned, however, thatatlowSNR,a lower-diversity codeoutperforms ahigher-diversity codeasaconsequence of thefactthat,onaRayleigh fadingchannel, thereisanoptimum distribution of thetotalreceived SNRamongthediversity signals.Therefore, thecurvesfor theblockorthogonal codeswillcrossoverthecurvesfortheHadamard codes atthelow-SNR (high-error-rate) region. Method 4:Concatenation Inthismethod, webeginwithtwocodes:one binaryandtheothernonbinary.Thebinarycodeistheinnercodeandisan (n,k)constant-weight (nonlinear) blockcode.Thenonbimiry code,whichmay belinear,istheoutercode.Todistinguish itfromtheinnercode,weuse uppercase letters,e.g.,an(N,K)code,whereNandKaremeasured interms ofsymbols fromaq-aryalphabet. Thesizeqofthealphabet overwhichthe outercodeisdefinedcannotbegreaterthanthenumberofwordsintheinner code.Theoutercode,when defined intermsofthebinaryinnercodewords ratherthanq-arysymbols, isthenewcode. Animportant specialcaseisobtained whenq=2kandtheinnercodesizeis chosentobe2k•ThenthenumberofwordsisM=2'Kandtheconcatenated structure isan(nN,kK)code.Thebandwidth expansion factorofthis concatenated codeistheproductofthebandwidth expansions fortheinner andoutercodes. Nowweshalldemonstrate theperformance advantages obtained ona Rayleigh fadingchannel bymeansofcodeconcatenation. Specifically, we construct aconcatenated codeinwhichtheoutercodeisadual-k(nonbinary) convolutional codeandtheinnercodeiseitheraHadamard codeorablock orthogonal code.Thatis,weviewthedual-kcodewithM-ary(M=Z') orthogonal signalsformodulation asaconcatenated code.Inallcasestobe considered, soft-decision demodulation andViterbidecoding areassumed. Theerrorrateperformance ofthedual-kconvolutional codesisobtained fromthederivation ofthetransferfunction givenby(8-2-39). Forarate-liZ, dual-kcodewithnorepetitions, thebiterrorprobability, appropriate forthe caseinwhicheachk-bitoutputsymbolfromthedual-kencoder ismappedinto oneofM=2korthogonal codewords,isupper-bounded as (14-6-21) whereP2(m)isgivenby(14-6-12). Forexample, arate-lIZ, dual-2codemayemploya4-aryorthogonal code O(4,2)astheinnercode.Thebandwidth expansion factoroftheresulting concatenated codeis,ofcourse,theproduct ofthebandwidth expansion 822 DI(ilTALCOMML:NICATIONS factorsoftheinnerandoutercodes.Thus,inthisexample, therateofthe outercodeis1/2andtheinnercodeis1/2.Hence,B,=(4/2)(2) =4. Notethatifeverysymbolofthedual-kisrepeated rtimes,thisisequivalent tousinganorthogonal codewithdiversity L=r.Ifweselectr=2inthe example givenabove,theresulting orthogonal codeisdenoted as0(8,2)and thebandwidth expansion factorfortherate-l/2, dual-2codebecomes B,=8. Consequently, thetermP,(m)in(14-6-21) mustbereplaced byP2(mL)when theorthogonal codehasdiversity L.SinceaHadamard codehasan"effective diversity" !dm;mitfollowsthatwhenaHadamard codeisusedastheinner codewithadual-koutercode,theupperboundonthebiterrorprobability of theresulting concatenated codegivenby(14-6-21) stil1appliesifPz(m)is replaced byP,(1mdm'n)'Withthesemodifications, theupperboundonthebit errorprobability givenby(14-6-21) hasbeenevaluated forrate-I/2, dual-k convolutional codeswitheitherHadamard codesorblockorthogonal codesas innercodes.Thustheresulting concatenated codehasabandwidth expansion factorequaltotwicethebandwidth expansion factoroftheinnercode. First,weconsider theperformance gainsduetocodeconcatenation. Figure 14-6-9illustrates theperformance ofdual-kcodeswithblockorthogonal inner codescompared withtheperformance ofblockorthogonal codesforband­ widthexpansion factorsB,=4,8,16,and32.Theperformance gainsdueto concatenation areveryimpressive. Forexample, atanerrorrateof\0-6and B,=8,thedual-kcodeoutperforms theorthogonal blockcodeby75dB.In \.\', 0<8.2i', \."\:,8,=4.,,,. ,\,- , , 1\\""\,0(16.2) \\""i\• B=8,'. \\..\ ,Dual-2 \,"\ '/"0(4.2)\, ,B==4\\",,\//,' ,. \\\"\ \\\,,\ \\ IDuaJ-3~,\ 0(24.3)'0(128.'1, \\8,=l6t\D,=32.. \Du4).2\",\, Dual-4,\0(8.2)\ ' ,\ 0(64.•1\D,=8 0(48.3i, B,=32\\\..8,=.16" I(t-' 10 12 1416 I~20n24 SNRperm•.'i,(dBIIO-.l 5 2 llr' 5 ..<Z ~1O-~ •5:E•'02 .~ :B1(t-6 i5 Z 10-7 5 Z FlGURE 14-6-9 Comparison ofperformance between block orthogonal codesanddual-kwithblock orthogonal innercodes. CHAfYIER 14DI(TIAL COMMt:I\ICATION THROUGH FADING MUlTlI'AT~1 CHA~NElS 823 , ., I--I ,I ,\,, 1\ , ! ,I,: , ,i \, I \,, ; \, \,, \:B(20.5) ,8,=" \ ,,, Dua1·0,, ~.Dual-5 •,I t-H.36,6r- B~=121\11120'5):.......... " R,=i~~Il(.~fl.6) IB,=12 SNRperbit.y,,(dBIFIGURE 14-6-18 Comparison ofperformance between Hadamard codesanddual-kcodeswith Hadamard innercodes.10,, I()~, ..< ~10-~ t, 15. -0?>, :E10-'•£5 10-7 5 2 10-11 10IZ14161820 short,thisgainmaybeattributed totheincreased diversity (increase in minimum distance) obtained viacodeconcatenation, Similarly, Fig.14-6·10 illustrates theperformance oftwodual-kcodeswithHadamard innercodes compared withtheperformance oftheHadamard codesaloneforB,=8and 12.Itisobserved thattheperformance gainsduetocodeconcatenation are stillsignificant, butcertainly notasimpressive asthosei1!ustrated inFig. 14-6-9.ThereasonisthattheHadamard codesaloneyieldaLugediversity. so thattheincreased diversity arisingfromconcatenation doesnotresultinas largeagaininperformance fortherangeoferrorratescovered inFig.14-6-10. Next,wecompare theperformance for thetwotypesofinnercodesused withdual-koutercodes.Figure14-6-11showsthecomparison forB,=8,Note 'thatthezH(4,2)innercodehasdm;n=4,and,hence,ithasaneffective order ofdiversity equalto2.Butthisdualdiversity isachieved bytransmitting four frequencies percodeword.Ontheotherhand,theorthogonal code0(8,2) alsogivesdualdiversity, butthisisachieved bytransmitting onlytwo frequencies percodeword.Consequently, the0(8,2)codeis3dBbetterthan thezH(4,2).Thisdifference inperformance ismaintained whenthetwocodes areusedasinnercodesinconjunction withdual-2code.Ontheotherhand.for B,=8,onecanusetheH(20,S)astheinnercodeofadual-Scode,andits performance issignificantly betterthanthatofthedual-2codeatlowerror· rates.Thisimprovement inperformance isachieved attheexpense ofan increase indecoding complexity. Similarly, inFig.14-6-12, wecompare the 824DIGITAL COMMUNICATIONS ~ \\1\ T\ \1\\ \ \1\ 1\ \Dual-2 \\\1!f(4.2) B...=8 '"-\ ~. \\Dual-2\ Dual~~\\\8.2)\ I---H(20.5)8...=8 B.=8 \1\\ \"I: \ ~ 1\\\ FlGURE 14-6-11 Performance ofdual-kc<><leswitheither Hadamard orblockorthogonal innercode forBr=8.10-3 5 2 10-< 5 ..<2 ~10-' u :5•"02 f10-' 10-' 5 2 10" 10 12 14 16 18202224 SNRperbit,y,,(dB) 141618'202224 SNRperbit.'Yb1dB)12, \1\ \\ \\ 1\\ "'\, ,\" \"Dua/-3 ,Hl8.J) B...=16 \,DuaI-3 0(24.3) 8...=16 [\1- l-Dual-6\\ H(48.6 1--8...=16 I1\2 210­ 5 2 ,0-< 5 2 10-' 10-1 10..<2 ~10- 5 FlGURE 14-6-U Performance ofdual-kcodeswitheither Hadamard orblockorthogonal innercode forB.~16. ('HAPr,.,-R;.l 1)1(iITAl COMMl-'l\/l-Al'lON THROl~"H f'ADIN{, MlJLTJPATH ('HANNJ-:LS 815 performance ofthedual-kcodeswithtwotypesofinnercodesforB,.=16. Notethatthe,H(8,3) innercodehasdm;n=12.and,hence.ityieldsan effective diversity of6.Thisdiversity isachieved bytransmitting 12frequencies percodeword.Theorthogonal innercode0(24.3) givesonlythird-order diversity. whichisachieved bytransmitting threefrequencies percodeword. Consequently the0(24.3) innercodeismoreefficientatlowSNR.thatis.for therangeoferrorratescovered inFig.14-6-12.AtlargeSNR.thedual-3code withtheHadamard .,H(8,3)innercodeoutperforms itscounterpart withthe 0(24.3)innercodeduetothelargediversity provided bytheHadamard code. Forthesamebandwidth expansion factorB,==16,onemayuseadual-6code withaH(48,6)codetoachieveanimprovement overthedual-3codewiththe ,H(8,3) innercode.Again,thisimprovement inperformance (whichinthis caseisnotasimpressive asthatshowninFig.14-6-11). mustbeweighed againsttheincreased decoding complexity inherent inthedual-6code_ Thenumerical resultsgivenaooveillustrate theperformance advantages in usingcodeswithgooddistance properties andsoft-decision decoding ona Rayleigh fadingchannel asanalternative toconventional M-aryorthogonal signaling withdiversity. Inaddition, theresultsillustrate thebenefitsofcode concatenation onsuchachannel, usingadual-kconvolutional codeasthe outercodeandeitheraHadamard codeorablockorthogonal codeasthe innercode.Although dual-kcodeswereusedfortheoutercode.similarresults areobtained whenaReed-Solomon codeisusedfortheoutercode.Thereis anevengreaterchoiceintheselection oftheinnercode. Theimportant parameter intheselection ofboththeouterandtheinner codesistheminimum distance oftheresultant concatenated coderequired to achieveaspecified levelofperformance. Sincemanycodeswillmeetthe performance requirements, theultimate choiceismadeonthebasisof decoding complexity andbandwidth requirements. 14-6-5SystemDesignBasedontheCutotfRate Intheabovetreatment ofcodedwaveforms. wehavedemonstrated the effectiveness ofvariouscodesforfadingchannels. Inparticular. wehave observed thebenefitsofsoft-decision decoding andcodeconcatenation asa meansforincreasing theminimum distance and.hence,theamountofdiversity inthecodedwaveforms. Inthissubsection. weconsider randomly selected codewordsandderiveanupper(union)boundontheerrorprobability that depends onthecutoffrateparameter fortheRayleigh fadingchannel. Letusconsider themodelforthecommunication systemillustrated inFig. 14-6-].Themodulator hasaq-aryorthogonal FSKalphabet. Codewordsof blocklengthnaremapped intowaveforms byselecting ntonesfromthe alphabet ofqtones.Thedemodulation isperformed bypassingthesignal through a·bankofqmatched filtersfollowed bysquare-law detectors. The decoding isassumed tobesoft-decision. Thus,thesquare-law detected outputs ( 826 DIGITAL COMMUNICATIONS fromthedemodulator areappropriately combined (added)withequalweight­ ingtoformMdecision variables corresponding totheMpossibletransmitted codewords. Toevaluate theunionboundontheprobability oferrorinaRayleigh fadingchannel withAWGN,wefirstevaluate thebinaryerrorprobability involving thedecision variableVI,whichcorresponds tothetransmitted code word,andanyoftheotherM-1decisionvariables corresponding totheother codewords.LetV2betheotherdecision variableandsuppose thatVIandV2 haveItonesincommon. Hence,thecontributions toV,andV,fromtheseI tonesareidentical and,therefore, canceloutwhenweformthedifference Vi-V,.Sincethetwodecision variables differinn-Itones,the probability of errorissimplythatforabinaryorthogonal FSKsystemwithn-Iorder diversity. Theexactformforthisprobability oferrorisgivenby(14-6-4), wherep=1/(2+Yc)'and-YcistheaverageSNRpertone.Forsimplicity, we choosetousetheChernoff boundforthisbinaryeventerrorprobability, given by(14-6-7), i.e., (14-6-22) Now,letusaverageovertheensemble ofbinarycommunication systems. Thereareq"possible codewords,fromwhichwerandomly selecttwocode words.Thus.eachcodewordisselected withequalprobability. Then,the probability thattworandomly selectedcodewordshaveItonesincommon is (n)(l)'(1)"-'P(/)=Iq1 -q (14-6-23) Whenweaverage (14-6-22) overtheprobability distribution oflgivenby •(14-6-23), weobtain "P,(V"V,)=LP,(VI'V,[I)P(/) 1=0 ~{~(~)(~),H 1-~)P(I-p)r' ~Ul1+4(q-1)P(I-p)lr (14-6-24) Finally, theunionboundforcommunication systems thatuSeM=2k randomly selectedcodewordsissimply (14-6-25) Bycombining (14-6-24) with(14-6-25), weobtaintheupperboundonthe symbolerrorprobability as (14-6-26) CHAPTER 14:DlGITA.L COMMUNICATION THROUGH fADING MUlTlP'ATH CHANNELS 827 URE14-6-13 Cutoffrateasafunction ofy,for Rayleigh fadingchannel.10 9 8 7 6 5 4 q=8 q=4 "cc') :Eq=2 ~1.00.9 E08 '"0.7 00.6:;u05 04 0.3 0.2 2 4 6 10 12 14 16llol20 SNRper tone.Y~(dB) whereR=kInisthecoderateandRoisthecutoffratedefinedas qRo=logz-----'---­ I+4(q-1)p(1-p) with(14-6-27) (14-6-28)1p=-­ 2 +'lie GraphsofR"asafunction ofYeareshowninFig.14-6-13forq=2,4, and8. Amoreinteresting formof(14-6-26) isobtained ifweexpressPMinterms oftheSNRperbit.Inparticular, (14-6-26) maybeexpressed as (14-6-29) where.bydefinition, _Rog(q,".)=-=- 1'e 1 [ q ] =-!Og2Ye1+4(q-1)p(l-p)(14-6-30) 828 DIGITAL COMMUNICA.TIONS 0.22 0.21 020 0.19 0.18 0.17 0.16 0.15q=100 q=50 q=20 q=10 .=5 0.14 0.13 0.12 0.11 0.10 0.09 o.m;q=2 5 4 30.01L_~__-,-__~__-'-__..o.-__ o FlGURE 1406-14 Graphoffunction g(q.YJ· AverageSNRperlone,yc1dB) (14-6-31)Graphsofg(q,Yc)asafunction ofYcareplottedinFig.14-6-14, withqasa parameter. First,wenotethateisanoptimum Ycforeachvalueofqthat minimizes theprobability oferror.Forlargeq,thisvalueisapproximately Yc=3(5dB),whichisconsistent withourprevious observation forordinary square-law diversity combining. Furthermore, asq->"',tltefunctiong(q,Yc) approaches alimit,whichis . _ _1 [(2+Yc)2] 11mg(q,'Yc)=g~('Yc)=-:-log 24(1+ ) q_oo "ft' 'Yc Thevalueofg~(yc)evaluated at1c=3is g=(3)=max8=(yJ i'. =0.215 (14-6-32) Therefore, theerrorprobability in(14-6-29) forthisoptimum divisionoftotal SNRis (14-6-33) CHAPTER 14:DIGITAL C:OMMUNICATION THROUGH fADING MULTIPATH CHANNELS 829 Thisresultindicates thattheprobability oferrorcanbemadearbitrarily small withoptimum SNRpercodechip,iftheaverage SNRperbit'Yb>4.65 (6.7dB).Evenarelatively modestvalueofq=20comesclosetothisminimum value.AsseenfromFig.14-6-14,g(20.3)=0.2,sothatP",-+O, provided 'Yb>5(7dB).Ontheotherhand,ifq=2,themaximum valueofg(2,iiJ= 0.0%andthecorresponding minimum SNRperbitis10.2dB. InthecaseofbinaryFSKwaveforms (q=2),wemayeasilycompare the cutoffratefortheunquantized (soft-decision) demodulator outputwiththe cutoffrateforbinaryquantization, forwhich Q=2 aswasgivenin(8-1-104). Figure14-6-15illustrates thegraphsforRoandRQ. Notethatthedifference between RoandRQisapproximately 3dBforrates below0.3andthedifference increases rapidlyathighrates.Thislossmaybe reducedsignificantly byincreasing thenumberofquantization levelstoQ=8 (threebits). Similarcomparisons intherelative performance between unquantized soft-decision decoding andquantized decision decoding canalsobemadefor q>2. 1.0 FIGURE 14-6-15 Cutoflratefor(unquantized) soft­ decisionandhard-decision decoding of codedbinaryFSK.SNRpertone,i(dB) 83f) DIGITAL COMMVNICATIONS 14-6·6Trellis-Coded Modulation Trellis-coded modulation wasdescribed inSection8-3asameansforachieving acodinggainonbandwidth-constrained channels, wherewewishtotransmit at abit-rate-to-bandwidth ratioR/W> 1.Forsuchchannels, thedigitalcom­ munication systemisdesigned tousebandwidth-efficient multilevel ormulti­ phasemodulation (PAM,PSK,DPSK,orQAM),whichallowsustoachieve anR/W>1.Whencodingisapplied insignaldesignforabandwidth constrained channel, acodinggainisdesiredwithoutexpanding thesignal bandwidth. Thisgoalcanbeachieved, asdescribed inSection 8-3,by increasing thenumberofsignalpointsintheconstellation overthecorres­ ponding uncoded systemtocompensate fortheredundancy introduced bythe code,anddesigning thetrelliscodesothattheeuclidean distance inasequence oftransmitted symbolscorresponding topathsthatmergeatanynodeinthe trellisislargerthantheeuclidean distance persymbolinanuncoded system. Incontrast, thecodingschemes thatwehavedescribed aboveinconjunction withFSKmodulation expandthebandwidth ofthemodulated signalforthe purposeofachieving signaldiversity. Coupled withFSKmodulation, whichis notbandwidth-efficient, thecodingschemes wehavedescribed areinappropri­ ateforuseonbandwidth-constrained channels. Indesigning trellis-coded signalwaveforms forfadingchannels, wemayusc thesamebasicprinciples thatwehavelearnedandappliedinthedesignof conventional codingschemes. Inparticular, themostimportant objective in anycodedsignaldesignforfadingchannels istoachieveaslargeasignal diversity aspossible. Thisimpliesthatsuccessive outputsymbols fromthe encodermustbeinterleaved orsufficiently separated intransmission, eitherin timeorinfrequency, soastoachieveindependent fadinginasequence of symbolsthatequalsorexceedstheminimum freedistance ofthetrelliscode. Therefore, wemayrepresent suchatrellis-coded modulation systembythe blockdiagram inFig.14-6-16, wheretheinterleaver isviewedbroadlyasa devicethatseparates thesuccessive codedsymbols soastoprovideindepen­ dentfadingoneachsymbol(through frequency ortimeseparation ofsymbols) inthesequence. Thereceiverconsistsofasignaldemodulator whoseoutputis deinterleaved andfedtothetrellisdecoder. FIGURE 14-6-t6 Blockdiagramoftrellis-coded modulation systems. Cl-lAPn-:R l-l:I)l(ill-\I. CO\1M{r"'ICATIO~ THROUGH fAOING \tlJLT1PATH ('HAN~I::LS 831 Asindicated above,thecandidate modulation methods thatachieve high bandwidth efficiency areM-aryPSK,DPSK,QAMandPAM.Thechoice depends toalargeextentonthechannel characteristics. IftherearerapId amplitude variations inthereceived signal,QAMandPAMmaybeparticu­ larlyvulnerable, because awidebandautomatic gaincontrol(AGC)mustbe usedtocompensate forthechannel variations. Insuchacase,PSKorDPSK aremoresuitable, sincetheinformation isconveyed bythesignalphaseand notbythesignalamplitude. DPSKprovides theadditional benefitthatcarrier phasecoherence isrequired onlyovertwosuccessive symbols. However, there isanSNRdegradation inDPSKrelativetoPSK. Inthedesignofthetrelliscode,ourobjective istoachieve aslargeafree distance aspossible, sincethisparameter isequivalent totheamount of diversity inthereceived signal.Inconventional l:Jngerboeck trelliscoding, eachbranchinthetrelliscorresponds toasingleM-ary(PSK,DPSK,QAM) outputchannel symbol.Letusdefinetheshortesterrorevenrpathastheerror eventpathwiththesmallest numberofnonzero distances between itselfand thecorrectpath,andletLbeitslength.Inotherwords,ListheHamming distance between theM-arysymbols ontheshortesterroreventpathandthose inthecorrectpath.Hence,ifweassume thatthetransmitted sequence corresponds totheall-zeropathinthetrellis,Listhenumber ofbranches in theshortest-length pathwithanonzeroM-arysymbol.Inatrellisdiagram with parallelpaths,thepathsareconstrained tohaveashortest erroreventlength ofone branch, sothatL=1.Thismeansthatsuchatrelliscodeprovides no diversity inafadingchanneland,hence,theprobability oferrorisinverselv proportional totheSt\Rpersymbol.Therefore, inconventional trelliscoding forafadingchannel, itisundesirable todesignacodethathasparallelpathsin itstrellis,because such acodeyieldsnodiversity. Thisisthecaseina conventional rate-m/(m+I)trelliscode,whereweareforcedtohaveparallel pathswhenthenumberofstatesislessthan2'". Onepossible waytoincrease theminimum freedistance and,thus,the orderofdiversity inthecode,istointroduce asymmetry inthesignalpoint constellation. Thisapproach appears tobesomewhat effective, andhasbeen investigated bySimonandDivsalar (1985),Divsalar andYuen(1984),and Divsalar etal.(1987). Amoreeffective waytoincrease thedistance Land,thus,theorderof diversity istoemploymultiple trellis-coded modulation (MTCM). InMTCM. illustrated inFig.14-6-17, binputbitstotheencoder arecodedintocoutput bits,whicharethensubdivided intokgroups,eachofmbits,suchthatc=km. Eachm-bitgroupismappedintoanM-arysymbol.Thus,weobtaintheM-ary outputsymbols. Thespecialcasek=1corresponds totheconventional Ungerboeck codes.WithkM-aryoutputsymbols, itispossible todesigntrellis codeswithparallelpathshavingadistanceL=k.Thus,wecanachievean errorprobability thatdecaysinversely as('f,/No)". Animportant consideration inthedesignofthedecoder forthetrelliscode istheuseofanysideinformation regarding thechannel attenuation foreach 832 DIGITAL COMMUNICATIONS b input bilSTrellis Encoder output bitsmbits mbits mbitsMapper andM·ary Modulalot2M-ary output symbols FIGURE 14-6-17 BlockdiagramofMTCMIransmitter. symbol.InthecaseofFSKmodulation withsquare-law combination atthe. decoder toformthedecision metrics,itisnotnecessary toknowthechannel attenuation foedemodulated symbols. However, withcoherent detection, the optimum euclidean distancemetricforeachdemodulated symbolisoftheform Irn-ll<nsnl',whereanisthechannelattenuation forthetransmitted symbols. andTnisthedemodulation output.Hence,thesumofbeanchmetricsforany givenpaththroughthetrellisisoftheform D(r,s(l))=LJrn-ans~)I' n wherethesuperscript (i)indicates theithpaththroughthetrellis.Therefore, theestimati'on ofthechannel attenuation mustbeperformed inorderto realizetheoptimum trellisdecoder. Theestimation ofthechannelattenuation andphaseshift,isconsidered iiiAppendix CforthecaseofPSKmodulation anddemodulation. Theeffectofthequalityoftheattenuation andphase estimates onthepetformance ofPSK(uncoded) modulation isalsoassessed in Appendix C. 14-7BIBLIOGRAPHICAL NOTES ANDREFERENCES Inthischapter, wehaveconsidered anumberoftopicsconcerned withdigital communications overafadingmultipath channel. Webeganwithastatistical characterization ofthechannel andthendescribed theramifications ofthe channelcharacteristics onthedesignofdigitalsignalsandontheirperfor­ mance.Weobserved thatthe.reliability ofthecommunication systemis enhanced bytheuseofdiversity transmission andreception. Finallywe demonstrated thatchannel encoding andsoft-decision decoding provide a bandwidth-efficient meansforobtaining diversity oversuchchannels. Thepioneering workonthecharacterization offadingmultipathchannels andonsignalandreceiverdesignforreliabledigitalcommunications oversuclt channels wasdonebyPrice(1954,1956).Thisworkwasfollowed byadditional significant contributions fromPriceandGreen(1958,1960),Kailath(1960. 1961),andGreen(1962),Diversity transmission anddiversity combining techniques underavatietyofchannelconditions havebeenconsidered inthe papersbyPierce(1958),Brennan (1959),Turin(1961,1962),PierceandStein PROBLEMSCHAPHR 14DIGITAL COMMUNICATION THROtUU FADING MULTIPATH CHANNELS 833 (1%0),Barrow(1963),BelloandNelin(l962a,b,1963),Price(I962a,b),and Lindsey(1964). Ourtreatment ofcodingforfadingchannels hasreliedoncontributions fromanumber ofresearchers. Inparticular, theuseofdual-kcodeswith M-aryorthogonal FSKwasproposed inpublications byViterbiandJacobs (1975)andOdenwalder (1976).Theimportance ofcodingfordigitalcom­ munications overafadingchanneJ wasalsoemphasized inapaperbyChase (1976).Thebenefits derived fromconcatenated codingwithsoft-decision decoding forafadingchannel weredemonstrated byPieperetal.(1978). There,aReed-Solomon codewasusedfortheoutercodeandaHadamard codewasselected astheinnercode.Theperformance ofdual-kcodeswith eitherblockorthogonal codesorHadamard codesasinnercodewere investigated byProakisandRahman (1979).Theerrorrateperforma'1ce of maximal freedistance binaryconvolutional codeswasevaluated byRahman (1981).Finally,thederivation ofthecutoffrateforRayleigh fadingchannels is duetoWozencraft andJacobs(1965). Trellis-coded modulation forfadingchannels hasbeeninvestigated bymany researchers, whoseworkwasmotivated toalargeextentbyapplications to mobileandcellularcommunications. ThebookbyBiglierietal.(1991)givesa tutorialtreatment ofthistopicandcontains alargenumberofreferences tothe technical literature. Ourtreatment ofdigitalcommunications overfadingchannels focused primarily ontheRayleigh fadingchannel model.Forthemostpart,thisisdue 10thewideacceptance ofthismodelfordescribing thefadingeffectsonmany radiochannels andtoitsmathematical tractability. Although otherstatistical models,suchasaRiceanfadingmodelortheNakagami fadingmodelmaybe moreappropriate forcharacterizing fadingonsomerealchannels, thegeneral approach inthedesignofreliablecommunications presented inthischapter carriesover. 14·1Thescallering function 5(1:A)forafadingmultipath channelisnonzeroforthe rangeofvalues00;;;ro;;;1msand-0.1Hz0;;;A0;;;0.1Hz.Assumethatthescallering function isapproximately uniforminthetwovariables. aGivenumerical valuesforthefollowing parameters: (i)themultipath spreadofthechannel; (ii)theDopplerspreadofthechannel; (iii)thecoherence timeofthechannel; (iv)thecoherence bandwidth ofthechannel; (v)thespreadfactorofthechannel. bExplainthemeaning ofthefollowing, takingintoconsideration theanswers givenin(a): (i)thechannelisfrequency-nonselective; (ii)thechannelisslOWlyfading; (iii)thechannelisfrequency-selective. 834 DIGITAL ('O\otMUNICATIONS cSuppose thatwehaveafrequency allocation (bandwidth) of10kHzandwewish totransmit alarateof100bits/soverthischannel. Designabinary communications systemwithfrequency diversity. Inparticular. specify(i)the typeofmodulation. (ii)thenumber ofsubchannels, (iii)thefrequency separation between adjacent carriers, and(iv)thesignaling intervalusedinyour design.Justifyyourchoiceofparameters. 14-2Consider abinarycommunications systemfortransmitting abinarysequence over afadingchannel. Themodulation isorthogonal FSKwiththird-order frequency diversity (L~3).Thedemodulalor consists ofmatched filtersfollowed by square-law detectors. Assume thattheFSKcarriers fadeindependently and identically according toaRayleigh envelope distribution. Theadditive noiseson thediversity signals arezero-mean gaussian withaU1ocorrelation functions \E[zt(l)z,(1 +r)1=N,,8(r). Thenoiseprocesses aremutually statistically independent. aThetransmitted signalmaybe viewed asbinaryFSKWilhsquare-law detection. generated byarepetition codeofthelorm I-->C,=[I I].0-->e"=1000] Determine theerrorrateperformance P",forahard-decision decoder following thesquare-law-detected signals. bEvaluate P",fory,=100and1000. cEvaluate theerrorratep,.fory,=100andlOOOifthedecoder employs soft-decision decoding. dConsider thegeneralization oftheresultin(a).Ifarepetition codeofblock lengthL(Lodd)isused,dctermine theerrorprobability P,,,01the hard-decision decoder andcompare IhatwithP,,,theerrorrateofthe soft-decision decoder. Assumey»I. 14-3Suppose thatthebinarysignalS,(I)istransmitted overafadingchannel andthe received signalis ,,(1)=±<I.,,(I) +~(t). 0<;1<;T where;:(1)iszero-mean whilegaussian noisewithautocorrelation function Theenergyinthetransmitted signalis~=;J;Is,(I)I'dl.Thechannel gain<Iis specified bytheprobability densityfunelion p(a)=n.11>(<1)+0.91>(<1-2) aDetermine theaverageprobability oferrorP,forthedemodulator thatemploys afillermatched toS,(I). bWhatvaluedoesP,approach as'$/Noapproaches infinity. cSuppose thatIhesamesignalistransmitted ontwostatistically il/d,'pel/dellllv ftl/lillgchannels withgainsII,and(/"where . 1>(/.)=O.II>(a.l+O.91i«/.-2). h=1.2 ThenoisesontheIwochannels arestatistically independent andidentically distributed. Thedemodulator cmploys amatched tilterforeachchannel and CH,\PTER 14:DIGlTAL COMMUNICATION THROUGH FADING MliLTIPATH CHI\N~ELS 835 simplyaddsthetwofilteroutputs toformthedecision variable. Determine the averageP,. dForthecasein(c)whatvaluedoesP,approach as'lINnapproaches infinity. 14-4Amultipath fadingchannel hasamultipath spreadof7;"=1sandaDoppler spreadB"=0.01Hz.Thetotalchannel bandwidth atbandpass available forsignal transmission isW=5Hz.Toreducetheeffectsofintersymbol interference. the signaldesigner selectsapul.eduration T=10s. aDetermine thecoherence bandwidth andthecoherence time. bIsthechannel frequency selective? Explain. cIsthechannel fadingslowlyorrapidly? Explain. dSuppose thatthechannel isusedtotransmit binarydatavia(antipodal) coherently detected PSKinafrequency diversity mode.Explain howyouwould usetheavailable channel bandwidth toobtainfrequency diversity anddeter· minehowmuchdiversity isavailable. eForthecasein(d).whatistheapproximate SNRrequired perdiversity to achieveanerrorprobability of10.? fSuppose thatawidebandsignalisusedfortransmission andaRAKE·type receiver isusedfordemodulation. HowmanytapswouldyouuseintheRAKE receiver? IIExplainwhether ornottheRAKEreceiver canbeimplemented asacoherent receiver withmaximal ratiocombining. hIfbinaryorthogonal signalsareusedforthewideband signalwithsquare-law postdetection combining intheRAKEreceiver, whatistheapproximate SNR required toachieveanerrorprobability of10-·?(assume thatalltapshavethe sameSNR.) 14-5Inthebinarycommunications systemshowninFig.PI4-5,Z,(t)andz,(t)are statistically independent whitegaussian noiseprocesses withzeromeanand identical autocorrelation functions c/>,,(r)=N"lJ(r).Thesampled valuesV,andV, represent therealpartsofthematched filteroutputs. Forexample, iff,(t)is transmitted, thenwehave V,=2'C+N, V,=N,+N, where'listhetransmitted signalenergyand N.=Re[ff,*(t)Z.(t) dt]'k=L2 FIGURE PJ4-SOpimum combiner u=VI+J}U.lu FIGURE PI4-6836 DIGITAL COMMUNICATIONS Itisapparent thatVIandV,arecorrelated gaussian vanibles whileNIandN,are independent gaussian variables. Thus, I(n~)p(nl)=v2iruexp-2u' I(n~)p(n,)=v2iruexp-2eT' wherethevariance ofN.iscr'=2'€No• aShowthatthejointprobability densityfunction forVIandV,is P(VI'V,)=2:eT'exp{-2~'[(VI-2~'-V,(VI-2~+~Vn} ifS(I)istransmitted and if-S(I)istransmitted. bBasedonthelikelihood ratio,showthattheoptimum combination ofVIandV, resultsinthedecisionvariable V=VI+f3V, wheref3isaconstant. Whatistheoptimum valueoff3? cSuppose thatS(I)istransmitted. Whatistheprobability densityfunction ofV? dWhatistheprobability oferrorassuming thatS(I)wastransmitted? Express youranswerasafunction fortheSNR'i/No. eWhatisthelossinperformance ifonlyV=VIisthedecision variable? 14-6Consider tbemodelforabinarycommunications systemwithdiversity asshownin Fig.PI4-6.Thechannels havefixedattenuations andphaseshifts.The{Z.(I)}are CHAPTER 14;DtGITAL COMMUNICATlQN THROUGH FADING MULTlP~TH CHANNELS 837 complex-valued whitegaussian noiseprocesses withzeromeanandautocorrela­ tionfunctions (Notethatthespectraldensities {No,}arealldillerent.) Also,thenoiseprocesses {,,(I)}aremutually statistically independent, The{13.}arecomplex-valued weighting factorstobedetermined, Thedecisionvariable fromthecombiner is aDetermine thepdfp(U)when+1istransmitted. bDetermine theprobability oferrorp"asafunction oftheweights{13,}. cDetermine thevaluesof{13,}!hatminimize P,. 14-7Determine theprobability oferrorforbinaryorthogonal signaling withLth-order diversity overaRayleigh fadingchannel. Thepdfsofthetwodecision variaoles aregivenby(14-4-31) and(14-4-32). 14-8Therate-1/3, L=3,binaryconvolutional codewithtransfer function givenby (8-2-5)isusedfortransmitting dataoveraRayleigh fadingchannel viabinary PSK. aDetermine andplottheprobability oferrorforbard-decision decoding. Assume thatthetransmitted waveforms corresponding tothecodedbitsfade independently. bDetermine andplottheprobability oferrorforsoft-decision decoding. Assume thatthewaveforms corresponding tothecodedbitsfadeindependently. 14-9Abinarysequence istransmitted viabinaryantipodal signaling overaRayleigh fadingchannel withLth-order diversity. WhenSI(I)istransmitted, thereceived equivalent lowpasssignalsare ThefadingamongtheLsubchannels isstatistically independent. Theadditive noiseterms{z..(I)}arezero-mean, statistically independent andidentically distributed whitegaussian noiseprocesses withautocorrelation function 4>,,(r)= Noli(f).EachoftheLsignalsispassedthroughafiltermatched toS/(I)andthe outputisphase-corrected toyield U,=Re[ej·'IT r,(I)s,*(I)dl]'k=1,2,...,L The{U,}arecombined byalinearcombiner toformthedecision variable L U=LU.,-I •Determine thepdfofUconditional onfixedvaluesforthe{a,}. bDetermine theexpression fortheprobability oferrorwhenthe{a,}are statistically independent andidentically distributed Rayleigh randomvariables. 838 DIGITAL COMMUNICATIONS 14-10TheChernoff boundfortheprobability oferrorforbinaryFSKwithdiversity Lin Rayleigh fadingwasshowntobe [1+-]1. P,(L)<[4p(1-p»'=4(2+;')' where _1 [(2+iiJ']g('Y,)=-::-log,4(1+-)1', ')', sPlotg(ii,)anddetermine itsapproximate maximum valueandthevalueofy, wherethemaximum occurs. bForagivenYo,determine theoptimalorderofdiversity. cCompare P,(L),underthecondition thatg(y,)ismaximized (optimal diversity). withtheerrorprobability forbinaryFSKinAWGNwithnofading.whichis anddetermine thepenaltyinSNRduetofadingandnoncoherent (square-law) combining. 14-11ADSspread-spectrum systemisusedtoresolvethemultipath signalcomponents inatwo·path radiosignalpropagation scenario.Ifthepathlengthofthesecondary pathis300mlongerthanthatofthedirectpath,determine theminimum chiprate necessary toresolvethemultipathcomponents. 14-12Abaseband digitalcommunication systememploys thesignalsshowninFig. PI4-12(a) forthetransmission oftwoequiprobable messages. Itisassumed that thecommunication problem studiedhereisa"one-shot" communication problem: thatis,theabovemessages aretransmitted justonceandnotransmission takes placeafterward. Thechannel hasnoattenuation (a=I),andthenoiseisAWG withpowerspectraldensityINn. sFindanappropriate orthonormal basisfortherepresentation ofthesignals. bInablockdiagram. givethe·precisespecifications oftheoptimum receiver using matched filters.Labelthediagram carefully. cFindtheerrorprobability oftheoptimum receiver. dShowthattheoptimum receiver canbeimplemented byusingJustonefilter \~,I) A1-_-, o AWGNA o~T T FlGURE Pl4-U CHAPTER 14:DICilTAL C(l.'I,.IMUNICATION THROUGH FADING MULTIPATH CHANNELS 839 A1----..., FIGURE PI4-14o T o -AT (seetheblockdiagram inFig.PI4-l2(b). Whatarethecharacteristics ofthe matched filterandthesampler anddecision device? eNowassumethatthechannel isnotidealbuthasanimpulse response of e(t)~8(t)+\8(t-IT).Usingthesamematched filteras(d),designan optimum reciever. fAssuming thatthechannelimpulseresponse ise(l)~8(1)+a8(t-\n,wherea isarandomvariableuniformly distributed on[0,1J.andusingthesamematched filterasin(d),designtheoptimum receiver. 14-13Acommunication systememploys dualantenna diversity andbinaryorthogonal FSKmodulation. Thereceived signalsatthetwoantennas are r(l)~",S(I)+n,(t) r,(I)~",.(t)+n,(I) where'"and'"arestatistically iidRayleigh randomvariables, andn,(I)andn,(I) arestatistically independent, zero-mean whitegaussian random processes with power-spectral density\Nu.Thetwosignalsaredemodulated, squared andthen combined (summed) priortodetection. aSketchthefunctional blockdiagram oftheentirereceiver, induding the demodulator, thecombiner andthedetector. bPlottheprobability oferrorforthedetector andcompare theresultwiththe caseofnodiversity. 14-14Thetwoequivalent lowpasssignalsshowninFig.P14-14areusedtotransmit a binarysequence. Theequivalent lowpass impulse response ofthechannel is h(l)~48(1)-28(t-T).Toavoidpulseoverlapbetween successive transmissions, thetransmission rateinbits/sisselected tobeR~1/2T.Thetransmitted signals areequallyprobable andarecorrupted byadditive zero-mean whitegaussian noisehavinganequivalent lowpass representation Z(I)withanautocorrelation function </>,,(r) ~~E[z'(I)z(1 +r))~Nu8(r) aSketchthetwopossible equivalent lowpassnoise-free receivedwaveforms. bSpecify theoptimum receiver andsketchtheequivalent lowpass impulse responses ofallfiltersusedintheoptimum receiver. Assumecoherent detection ofthesignals. 14-15Verifytherelationin(14-3-14) bymakingthechangeofvariable y~,,'l./N uin theNakagami-m distribution. 15 MULTIUSER COMMUNICATIONS Ourtreatment ofcommunication systemsuptothispointhasbeenfocusedon asinglecommunication linkinvolving atransmitter andareceiver. Inthis chapter, thefocusshiftstomultiple usersandmultiple communication links. Weexplorethevariouswaysinwhichthemultiple usersaccessacommon channel totransmit information. Themultiple accessmethods thatare described inthischapterformthebasisforcurrentandfuturewirelineand wireless communication networks, suchassatellite networks, cellular and mobilecommunication networks, andunderwater acoustic networks. 15-1INTRODUCTION TOMULTIPLE ACCESS TECHNIQUES Itisinstructive todistinguish amongseveraltypesofmultiuser communication systems. Onetypeisamultiple accesssysteminwhichalargenumberofusers shareacommon communication channeltotransmit information toareceiver. Suchasystemisdepicted inFig.15-1-1.Thecommon channelmaybethe up-linkinasatellitecommunication system,oracabletowhichareconnected asetofterminals thataccessacentralcomputer, orsomefrequency bandin theradiospectrum thatisusedbymultiple userstocommunicate witharadio receiver. Forexample, inamobilecellularcommunication system,theusers arethemobiletransmitters inanyparticular cellofthesystemandthereceiver residesinthebasestationoftheparticular cell. Asecondtypeofmultiuser communication systemisabroadcast network in whichasingletransmitter sendsinformation tomultiple receivers asdepicted 840 CHAPTER 15MlLfil'SER COMMUNICATIONS 841 I.1Tran~mlller 1I rTram.miner 2~ ('hann~1HReceiverI r.1Transmltler J(I FIGURE 15-1-1 Amultiple accesssvstem. inFig.15-1-2.Examples ofbroadcast systemsincludethecommon radioand TVbroadcast systems, aswellasthedown-links inasatellitesystem. Themultiple accessandbroadcast networks areprobably themostcommon multiuser communication systems. Athirdtypeofmultiuser syslemisa store-and-forward network, asdepicted inFig.15-1-3.Yetafourthtypeisthe two-way communication systemshowninFig.15-1-4. Inthischapler. wefocusonmultiple accessmethods formultiuser communications. Ingeneral, thereareseveraldifferent waysinwhichmultiple userscansendinformation throughthecommunication channeltothereceiver. Onesimplemethod istosubdivide theavailable channel bandwidth intoa number, sayN.offrequency nonoverlapping subchannels, asshowninFig. 15-1-5,andtoassignasubchannel toeachuseruponrequestbytheusers.This FIGURE 15-1-2 Abroadcast network. 842 DIGITAL COMMUNICA nONS FIGURE: 15·1·3 Astore-and-torward communication network withsatellite relays. Tran~miller Rece-i\-er UserI Channel U~r2 Receiver Trammitter FIGURE 15·1·4 Atwo-way communication channel. FIGURE 15·1·5 Subdivisions ofthechannelinlo nonoverlapping frequency bands.Frequency -+_---'--'-..J--"-~ ---'..::..=.c.J-__"__.L-_( melhodisgenerally calledfrequency-dilJJsion multiple access(FDMA). andis commonly usedinwirelinechannels toaccommodate multiple usersforvoice anddatatransmission. Another methodforcreatingmultiple subchannels formultiple accessisto subdivide theduration 'ft.calledtheframeduration. into.say.N nonoverlapping subintervals. eachofduralion Tr/N.Theneachuserwho wishestotransmit information isassigned toaparticular timeslotwithineach frame..Thismultiple accessmethod iscalledtime-division multiple access (TDMA) anditisfrequently usedindataanddigitalvoicetransmission. WeobservethatinFDMAandTDMA.thechannel isbasically partitioned intoindependent single-user suhchannels. Inthissense.thecommunication CHAPTER l;'i:MULTIUSER COMMliNI('ATIONS 843 systemdesignmethods thatwehavedescribed forsingle-user communication aredirectlyapplicable a\ldnonewproblems areencountered inamultiple accessenvironment, exceptfortheadditional taskofassigning usersto a~ailable channels. Theinteresting problems arisewhenthedatafromtheusersaccessing the network isburstyinnature.Inotherwords,theinformation transmissions from asingleuserareseparated byperiodsofnotransmission, wheretheseperiods ofsilencemaybegreaterthantheperiodsoftransmission. Suchisthecase generally withusersatvariousterminals inacomputer communications network thatcontains acentralcomputer. Tosomeextent,thisisalsothecase inmobilecellularcommunication systemscarryingdigitized voice,sincespeech signalstypically containlongpauses. Insuchanenvironment wherethetransmission fromthevarioususersis burstyandlow-duty·cycle, FDMAandTDMAtendtobeinefficient becausea certainpercentage oftheavailable frequency slotsortimeslotsassigned to usersdonotcarryinformation. Ultimately, aninefficiently designed multiple accesssystemlimitsthenumberofsimultaneous usersofthechannel. Analternative toFDMAandTDMAistoallowmorethanoneuserto shareachannel orsubchannel byuseofdirect-sequence spreadspectrum signals.Inthismethod, eachuserisassigned auniquecodesequence or signature sequence thatallowstheusertospreadtheinformation signalacross theassigned frequency band.Thussignalsfromthevarioususersareseparated atthereceiver bycross-correlation ofthereceived signalwitheachofthe possibleusersignature sequences. Bydesigning thesecodesequences tohave relatively smallcross-correlations, thecrosstalk inherent inthedemodulation ofthesignalsreceived frommultiple transmitters isminimized. Thismultiple accessmethodiscalledcode-division multipleaccess(CDMA). InCDMA, theusersaccessthechannel inarandommanner. Hence.the signaltransmissions amongthemultiple userscompletely overlapbothintime andinfrequency. Thedemodulation andseparation ofthesesignalsatthe receiver isfacilitated bythefactthateachsignalisspreadinfrequency bythe pseudo·random codesequence. CDMA issOmetimes calledspread-spectrum multipleaccess(SSMA). Analternative toCDMAisnonspread randomaccess.Insuchacase,when twousersattempttousethecommon channelsimultaneously, theirtransmis­ sionscollideandinterfere witheachother.Whenthathappens, theinforma­ tionislostandmustberetransmitted. Tohandlecollisions, onemustestablish protocols forretransmission ofmessages thathavecollided. Protocols for scheduling theretransmission ofcollidedmessages aredescribed below. 15-2CAPACITY OFMULTIPLE ACCESS METHODS Itisinteresting tocompare FDMA, TDMA, andCOMA intermsofthe information ratethateachmultipleaccessmethodachieves inanidealAWGN channelofbandwidth W.Letuscompare thecapacity ofKusers,whereeach (15-2-1)844 DIGIT,,,L COMMUNICATIONS userhasaverage powerP,=P,forall1..i""K.Recallthatinanideal band-limited AWONchannelofbandwidth W.thecapacityofasingleuseris C=Wlog2(1+:NJ where ~N(Jisthepowerspectraldensityoftheadditivenoise. InFDMA,eachuserisallocated abandwidth W/K.Hence,Ihecapacityof eachuseris andthetotalcapacityfortheKusersis KCK=Wlog2(1+:~J(15-2-2) (15-2-3) (15-2-4)Therefore, thetotalcapacityisequivalent tothatofasingleuserwithaverage powerPay=KP. Itisinteresting tonotethatforafixedbandwidth W,thetotalcapacity goes toinfinityasthenumberofusersincreases linearlywithK.Ontheotherhand, asKincreases, eachuserisallocated asmallerbandwidth (WIK)and, consequently, thecapacity peruserdecreases. Figure15-2-1illustrates the capacity CKperusernormalized bythechannelbandwidth W,asafunctionof '€b/No,withKasaparameter. Thisexpression isgivenas CK=.llog [1+KCK(~b)]W K 2WNo Amorecompact formof(15-2-4)isobtained bydefining thenormalized K=2K=I flGURE 15-2-1 Normalized capacityasafunctionof 'i.IN.forFDMA.12 '"Jew '":>: 8-8 ~ ~ 8-b i4 U 2 00K=l K=5 K=10 20 25 CHAPTER 15:MULTIUSER COMMUNICATIONS 845 IIr-~--~---"---~--~ 10 ~9v· "8"<J7 "=6 R.5 ~4 .E ~3 ...2 flGURE IS·Z·Z Totalcapacityperhertzasafunction of'i.1NoforFDMA.o /5 20 (15-2-6) (15-2-7)IotaIcapacily Cn=KCK/W, whichisthetolalbitrateforallKusersperunit ofbandwidth. Thus,(15-2-4)maybeexpressed as en=log2(1+Cn~) (15-2-S) or,equivalently, it,2e•-1-=--- NoCn ThegraphofCnversus'€h/NoisshowninFig.15-2-2.Weobserve thatCn increases as'ChINoincreases abovetheminimum valueofIn2. InaTDMAsystem,eachusertransmits for11Kofthetimethroughthe channelofbandwidth W,withaveragepowerKP.Therefore, thecapacity per useris CK=(~)Wlog2(1+:~J whichisidentical tothecapacity ofanFDMAsystem.However, froma practical standpoint, weshouldemphasize that,inTDMA, itmaynotbe possibleforIhetransmitters tosustainatransmitter powerofKPwhenKis verylarge.Hence,thereisapractical limitbeyondwhichthetransmitter power cannotbeincreased asKisincreased. InaCDMA system,eachusertransmits apseudo-random signalofa bandwidth WandaveragepowerP.Thecapacity ofthesystemdepends onthe levelofcooperation amongtheKusers.Atoneextreme isnoncooperative CDMA,inwhichthereceiverforeachusersignaldoesnotknowthespreading waveforms oftheotherusers,orchoosestoignoretheminthedemodulation process.Hence,theotheruserssignalsappearasinterference atthereceiverof eachuser.Inthiscase,themultiuser receiver consists ofabankofK ~.., "~(J~ ~__----- A=5 FIGllRE 15-2-3 Normalized capacity asafunction of"t>/V~I fOTnoncooperati,,'c CDMA.o 10 15 rjNoIdB)20K=ItJ single-user receivers. Ifweassumethateachuser'spseudorandom signal waveform isgaussian theneachusersignaliscorrupted bygaussian interference ofpower(K-I)Pandadditive gaussian noiseofpowerWNo. Therefore, thecapacityperuseris or,equivalently,C=Wlog[I+---P---] K zWNo+(K-l)P(15-2-8) (15-2-9) CKI[CK 'Cn/No ] W=ogz1+WI+(K-1)(CKIW)~h/No Figure15-2-3illustrates thegraphofCK/Wversus ~hINo.withKasa parameter, Foralargenumberofusers,wemayusetheapproximation In(1+x)<:;x. Hence, (15-2-10) (15-2-11)or,equivalently, 1C<:;logze--- " ~blNo I 1 1<:;----<- In2~hlNoIn2 Inthiscase,weobservethatthetotalcapacity doesnotincrease withKasin TDMAandFDMA. Ontheotherhand,suppose thattheKuserscooperate bytransmitting synchronously intime,andthemultiuser receiver knowsthespreading CHAPTER 15:MULTIUSER CO\1Ml'NICATIO~S 847 waveforms ofallusersandjointlydemodulates anddetects alltheusers signals. Thus.eachuserisassigned arateR"1,,;i,,;K,andacodebook containing asetof2"R,codewordsofpowerP.Ineachsignalinterval. each userselectsanarbitrary codeword. sayX"fromitsowncodebook andallusers transmit theircodewordssimultaneously. Thus.thedecoder atthereceiver observes I "y=Ix,+z , I(15-2-12) whereZisanadditive noisevector.Theoptimum decoder looksfortheK codewords,onefromeachcodebook,thathaveavectorsumclosesttothe received vectorYineuclidean distance. Theachievable K-dimensional rateregionfortheKusersinanAWGN channel, assuming equalpowerforeachuser,isgivenbythefollowing equations:.P)R,<Wlog,(1+WN o' (2P. R,+RJ<Wlog,1+--),WN",1,,;i,j,,;K(15-2-13) (15-2-14) "(KP) IRj<Wlog,1+-- ,~I WN"(15-2-15) Inthespecialcasewhenalltheratesareidentical, theinequality (15-2-15) is dominant overtheotherK-1inequalities. Itfollows thatiftherates {R"1,,;i,,;K}fortheKcooperative synchronous usersareselected tofallin thecapacity regionspecified bytheinequalities givenabovethenthe probabilities oferrorfortheKuserstendtozeroasthecodeblocklengthn tendstoinfinity. Fromtheabovediscussion. weconclude thatthesumoftheratesoftheK usersgoestoinfinitywithK.Therefore, withcooperative synchronous users, thecapacity ofCDMAhasaformsimilartothatofFDMAandTDMA. Note thatifalltheratesintheCDMAsystemareselected tobeidentical toRthen (15-2-15) reducesto W(KP)R<-log,1+--K WNo(15-2-16) whichisidentical totherateconstraint inFDMAandTDMA. Inthiscase. CDMAdoesnotyieldahigherratethanTDMAandFDMA. However, ifthe ratesoftheKusersareselected tobeunequal suchthattheinequalities (15-2-13)-(15-2-15) aresatisfied thenitispossible tofindthepointsinthe achievable rateregionsuchthatthesumoftheratesfortheKusersinCDMA exceedsthecapacity ofFDMAandTDMA. (15-2-17)848 DIGITAL COMMUNICATIONS Example 15·2·1 Consider thecaseoftwousersinaCDMAsystemthatemploys coded signalsasdescribed above.Theratesofthetwousersmustsatisfythe inequalities R,<Wlog2(1+:N) R2<Wlog2(I+:,.) (2P' R,+R2<Wlog,1+WN) wherePistheaveragetransmitted powerofeachuserandWisthesignal bandwidth. Letusdetermine thecapacity regionforthetwo-user CDMA system. Thecapacity regionforthetwo-user CDMA systemwithcodedsignal waveforms hastheformillustrated inFig.15-2-4,where C,=Wlog 2(1+~). i=1.2WNo arethecapacities corresponding tothetwouserswithP,=P,=P.Wenote thatifuser1istransmitting atcapacity C,.user2cantransmit uptoa maximum rate R2m=Wlog2(1+2P)-c,WNo =Wlog2(1+P)P+WN o whichisillustrated inFig.15-2-4aspointA.Thisresulthasaninteresting R, C,t--_~B FIGURE 15-2-4 Capacity regionoftwo-user CDMAmultiple accessgaussian channel.I I I I R'!Mt- - - - t-__-_A I II I CHAPTER 15:MULTIUSER COMMUNICATIONS 849 interpretation. WenotethatrateRm,corresponds tothecaseinwhichthe signalfromuser1isconsidered asanequivalent additive noiseinthe detection ofthesignalofuser2.Ontheotherhand,user1cantransmit at capacity C"sincethereceiver knowsthetransmitted signalfromuser2and, hence,itcaneliminate itseffectindetecting thesignalofuser1. Oue tosymmetry, asimilarsituation existsifuser2istransmitting at capacity C,.Then,user1cantransmit uptoamaximum rateRIm=R2"" whichisillustrated inFig.15.2.4aspointB.Inthiscase,wehaveasimilar interpretation asabove,withaninterchange intherolesofuser1anduser 2. ThepointsAandBareconnected byastraightline.Itiseasilyseenthat thisstraightlineistheboundary oftheachievable rateregion,sinceany pointonthelinecorresponds tothemaximum rateWlog2(1+2P/WN o). whichcanbeobtained bysimplytime-sharing thechannelbetween thetwo users. Inthenextsection, weconsider theproblem ofsignaldetection fora multiuser COMAsystemandassesstheperformance andthecomputational complexity ofseveralreceiverstructures. 15-3CODE·DIVISION MULTIPLE ACCESS Aswehaveobserved, TOMA andFOMAaremultiple accessmethods in whichthechannelispartitioned intoindependent, single-user subchannels, i.e., nonoverlapping timeslotsorfrequency bands,respectively. InCOMA, each userisassigned adistinctsignature sequence (orwaveform), whichtheuser employs tomodulate andspreadtheinformation-bearing signal.Thesignature sequences alsoallowthereceivertodemodulate themessage transmitted by multiple usersofthechannel, whotransmit simultaneously and,generally, asynchronously. Inthissection, wetreatthedemodulation anddetection ofmultiuser COMAsignals.Weshallseethattheoptimum maximum-likelihood detector hasacomputational complexity thatgrowsexponentially withthenumber of users.Suchahighcomplexity servesasamotivation todevisesuboptimum detectors havinglowercomputational complexities. Finally,weconsider the performance characteristics ofthevariousdetectors. 15-3·1COMASipalandChannel Models Letusconsider aCOMAchannelthatissharedbyKsimultaneous users.Each userisassigned asignature waveform g.(t)ofduration T,whereTisthe symbolinterval. Asignature waveform maybeexpressed as L-I g.(t)=2:a.(n)p(t-nTcl,0"",t"'"T """0(15-3-1) (15·3-3) (15·3-4)850 DIGITAL COMMUNICATIONS where{ak(n),0.,n.,L-I}isapseudo-noise (PN)codesequence consisting ofLchipsthattakevalues{±1},pet)isapulseofduration Tnand~.isthe chipinterval. Thus,wehaveLchipspersymbolandT=L~..Without lossof generality, weassumethatallKsignature waveforms haveunitenergy,i.e., ITg~(t)dt=1 (15-3-2) o Thecross-correlations between pairsofsignature waveforms playan important roleinthemetricsforthesignaldetector andonitsperformance. Wedefinethefollowing cross-correlations: Pu(r)=rgi(t)g,(t-r)dl,i.,j p,,(r)=iTgi(t)gJ(t+T-T)dt,i"'"j Forsimplicity, weassumethatbinaryantipodal signalsareusedtotransmit theinformation fromeachuser.Hence,lettheinformation sequence ofthekth userbedenoted by{bk(m)}, wherethevalueofeachinformation bitmaybe ±1.Itisconvenient toconsider thetransmission ofablock of bitsofsome arbitrary length,sayN.Then,thedatablockfromthekthuseris (15-3·5) andthecorresponding equivalent lowpass, transmitted waveform maybe expressed as N Sk(t)=~2:bkU)gk(t -iT) i=I(15-3-6) (15-3-7)where 'il:kisthesignalenergyperbit.Thecomposite transmitted signalforthe Kusersmaybeexpressed as K set)=2:rdt-Tk) k=l K N =2:~2:bk(i)gk(t -iT-Tk) 4:=1 i=1 where{T.larethetransmission delays,whichsatisfythecondition 0,,;;Tk<T for1.,k,,;;K.Without lossofgenerality, weassumethat0"T,";;T2";;" .., Tk<T.Thisisthemodelforthemultiuser transmitted signalinanasynchro­ nousmode.Inthespecialcaseofsynchronous transmission, Tk=0for 1"k.,K.ThevaluesofTofinterestinthecross-correlations givenby(15-3-3) and(15-3-4)mayalsoberestricted to0"T<T,withoutJossofgenerality. Thetransmitted signalisassumed tobecorrupted byAWON.Hence,the received signalmaybeexpressed as ret)=set)+n(l) (15-3-8) wheres(t)isgivenby(15-3-7)andnet)isthenoise,withpowerspectraldensity !No. 15-3-2TheOptimum Receiver Theoptimum receiver isdefinedasthereceiver thatselectsthemostprobable sequence ofbits{b,(f1),1.,n,,;;N,I,,;;k,,;;K}giventhereceived signalr(t) observed overthetimeinterval0,,;;t,,;;NT+2T.First,letusconsider thecase ofsynchronous transmission; later,weshallconsider asynchronous transmission. Synchronous Transmission Insynchronous transmission, each(user)inter­ fererproduces exactlyDnesymbolwhichinterferes withthedesiredsymbol. In additive whitegaussian noise,itissufficient toconsider thesignalreceived in onesignalinterval, say0,,;;t";;T,anddetermine theoptimum receiver. Hence, r(t)maybeexpressed as "ret)=LVi,;b.(I)g.(t) +net),O,,;;t,,;;T 1<=1(15-3-9) Theoptimum maximum-likelihood receiver computes thelog-likelihood function A(b)=iT[ret)-f~b.(I)g,(t)]2 dt (15-3-10) o 1<=1 andselectstheinformation sequence {b,(I),I,,;;k,,;;K}thatminimizes A(b).If weexpandtheintegralin(15-3-10), weobtain A(b)=rr2(t)dt-2~1~b'<l)rr(t)g,(t) dt I(" iT+j~~lv'~jg,b,(I)bJ(I) 0g.(t)gj(t) dt (15-3-11) Weobserve thattheintegral involving r2(t)iscommon toallpossible sequences {b,(l)}andisofnorelevance indetermining whichsequence was transmitted. Hence,itmaybeneglected. Theterm r,=rr(t)g,(t) dt,1,,;;k,,;;K (15-3-12) represents thecross-correlation ofthereceived signalwitheachoftheK signature sequences. Insteadofcross-correIa tors,wemayemplDy matched filters.Finally,theintegralinvolving g,(t)andgj(t)issimply Pj'(O)=fgj(t)g,(t) dt (15-3-\3) Therefore, (15-3-11) maybeexpressed intheformofcorrelation metrics I( J(" C(rl(,b,,)=2L~b,(I)r, -2:Lv'~g,b,(I)bj(l)p!,(O) (15-3-14) 1<=1 j~lk=1 852 DIGITAL COMMUNICATIONS Thesecorrelation metricsmayalsobeexpressed invectorinnerproductform as (15-3-15) where andRsisthecorrelation matrix,withelements Pjk(O).Itisobserved thatthe optimum detector· musthaveknowledge ofthereceived signalenergies in ordertocompute thecorrelation metrics. Thereare2Kpossible choicesofthebitsintheinformation sequence ofthe Kusers.Theoptimum detector computes thecorrelation metricsforeach sequence andselectsthesequence thatyieldsthelargestcorrelation metric. Weobservethattheoptimum detector hasacomplexity thatgrowsexponen­ tiallywiththenumberofusers,K. Insummary, theoptimum receiver forsymbol-synchronous transmission consistsofabankofKcorreiatorsormatched filtersfollowed byadetector thatcomputes the2Kcorrelation metricsgivenby(15-3-15) corresponding to the2Kpossible transmitted information sequences. Then,thedetector selects thesequence corresponding tothelargestcorrelation metric_ Asynchronous Tnmsmission Inthiscase,thereareexactlytwoconseclltive symbols fromeachinterferer thatoverlapadesiredsymbol.Weassumethat thereceiver knowsthereceived signalenergies {~.}fortheKusersandthe transmission delays{fk}'Clearly, theseparameters mustbemeasured atthe receiver orprovided tothereceiverassideinformation bytheusersviasome controlchannel. Theopt.imum maximum-likelihood receiver computes thelog-likelihood function A(b)=fT+2T[r(t)-kt,~~b.(i)g.(r -iT-r.)rdt lNT+2T K NLNT+2T =0r2(t)dt-2:?~~b.{i) 0r(t)g.(t-iT-r.)dt K K N N LNT+2T +~I~\ICC.~I~~b.(i)Mi) 0g.(t-iT-Tk)g,(t-iT-TI)dt (15-3-16) wherebrepresents thedatasequences fromtheKusers.Theintegralinvolving r2(t)maybeignored, sinceitiscommon toallpossibleinformation sequences. Theintegral f(i+t)T+f~ T.(i):; r(t)8.(t-iT-f.)dt, iT+T*(15-3-17) CHAPTER 15:MULTIUSER COMMUNICATIONS 853 represents theoutputsofthecorreIatorormatched filterforthekthuserin eachofthesignalintervals. Finally,theintegral INT+2T g,(r-iT-,,)gl(r-jT-'I)dt o =L'IH1T' i7-"g,(t)gl(t+iT-jT+"-'I)dr(15-3-18) -IT-Tk maybeeasilydecomposed intotermsinvolving thecross-correlation Pk/(,)= /1.1("-'I)fork",1andp,,(,)fork>I.Therefore, weobserve thatthe log-likelihood function maybeexpressed intermsofacorrelation metricthat involves theoutputs{r.(i),I",k",K,1",i,,;;N}ofKcorrelators ormatched filters-<me foreachoftheKsignature sequences. Usingvectornotation, it canbeshownthattheNKcorrelator ormatched filteroutputs {,,(i)}canbe expressed intheform where,bydefinition r=[r'(I)r'(2) r'(N)1' r(i)=['1(i)'2(i) 'K(i)l' b=[b'(I)b'(2)...b'(N»)' b(i)=[v'i;b,(i) ~b2(i) ...~bK(i)1' n=[0'(1)0'(2) n'(N)1' o(i)=[nl(i)n2(i)...nK(i)1' a(O) R~(I) 0 0 Ra(l)Ra(O) R~(I) 0 0(15-3-19) (15-3-20) (15-3-21) (15-3-22) o oo oo oRa(l)Ra(O) R~(I) oRa(l)Ra(O)(15-3-23) andR,(m)isaKxKmatrixwithelements R./(m)=L~rog.(r-")g,(r+mT-£/)dr (15-3-241 Thegaussian noisevectorsn(i)havezeromeanandautocorrelation matnx E[n(k)n'(j)] =~NoRa(k -j) (15-3-25) Notethatthevectorrgivenby(15-3-19) constitutes asetofsufficient statistics forestimating thetransmitted bitsb.(i). Ifweadoptablockprocessing approach, theoptimum 111Ldetector must compute 2NKcorrelation metricsandselecttheKsequences oflengthNthat correspond tothelargestcorrelation metric.Clearly, suchanapproach is muchtoocomplex computationally tobeimplemented inpractice, especially 854 DIGITAL <'OMMlINICATIQ:"lS whenKandNarelarge.Analternative approach isMLsequence estimation employing. theViterbialgorithm. Inordertoconstruct asequential-type detector, wemakeuseofthefactthateachtransmitted symboloverlaps at mostwith2K-2symbols. Thus,asignificant reduction incomputational complexity isobtained withrespecttotheblocksizeparameter N,butthe exponential dependence onKcannotbereduced. Itisapparent thattheoptimum MLreceiver employing theViterbi algorithm involvessuchahighcomputational complexity thatitsuseinpractice islimitedtocommunication systemswherethenumberofusersisextremely small,e.g.,K<10.ForlargervaluesofK.oneshouldconsider asequential· typedetector thatisakintoeitherthesequential decoding orthestack algorithms described inChapter 8.Below,weconsider anumber ofsub­ optimums detectors whosecomplexity growslinearlywithK. 15·3·3Suboptimum Detectors Intheabovediscussion, weobserved thattheoptimum detector fortheK CDMAusershasacomputational complexity, measured inthenumber of arithmetic operations (additions andmultiplications/divisions) permodulated symbol, thaIgrowsexponentially withK.Inthissubsection wedescribe suboptimum detectors withcomputational complexities thatgrowlinearlywith thenumberofusers,K.Webeginwiththesimplest suboptimum detector, whichwecalltheconventional (single-user) detector. (15-3-26) (15-3-27) (15-3-28) (15-3-29)T E[nW)]=~Ntlfg~(t)dl=~Nn 11Conventional Single·UserDetector Inconventional single-user detection, thereceiver foreachuserconsists ofademodulator thatcorrelates (or match-filters) thereceived signalwiththesignature sequence oftheuserand passesthecorrelator outputtothedetector, whichmakesadecision basedon thesinglecorrelator output.Thus,theconventional detector neglects the presence oftheotherusersofthechannelor,equivalently, assumes thatthe aggregate noiseplusinterference iswhiteandgaussian. Letusconsider synchronous transmission. Then,theoutput of thecor­ relatorforthekthuserforthesignalintheinterval0",t'"Tis r.=fTr(l)g.(t) dl " K =~b.(I)+ L:~bJ(I)pj.(O)+n.(I) i=I j#-k wherethenoisecomponent '1.(1)isgivenas '1.(1)=ITn(l)g.(t) dl "Sincenet)iswhitegaussian noisewithpowerspectraldensity ~Nn.thevariance of'1.(1)is CHAPTER IS·.Mt'LTIt:SER CO\1\1tJ~ICATIO~S 855 Clearly. ifthesignature sequences areorthogonal, theinterference fromthe otherusersgivenbythemiddletermin(15-3-27) vanishes andtheconven­ tionalsingle-user detector isoptimum. Ontheotherhand,ifoneormoreof theothersignature sequences arenotorthogonal totheusersignature sequence, theinterference fromtheotheruserscanbecome excessive ifthe powerlevelsofthesignals(orthereceived signalenergies) ofoneormoreof theotherusersissufficiently largerthanthepowerlevelofthekthuser.This situation isgenerally calledthenear-far problem inmultiuser communications. andnecessitates sometypeofpowercontrolforconventional detection . .Inasynchronous transmission, theconventional detector ismorevulnerable tointerference fromotherusers.Thisisbecause itisnotpossible todesign signature sequences foranypairofusersthatareorthogonal foralltime offsets.Consequently, interference fromotherusersisunavoidable inasyn­ chronous transmission withtheconventional single-user detection. Insucha case,thenear-far problem resulting fromunequal powerinthesignalstrans­ mittedbythevarioususersisparticularly serious. Thepractical solution generally requires apoweradjustment method thatiscontrolled bythe receiver viaaseparate communication channel thatallusersarecontinuousl) monitoring. Another optionistoemployoneofthemultiuser detectors described below. Decorrellltiog Detector Weobserve thattheconventional detector hasa complexity thatgrowslinearlywiththenumberofusers,butitsvulnerability to thenear-far problem requires sometypeofpowercontrol. Weshallno\\ deviseanothertypeofdetector thatalsohasalinearcomputational complexity butdoesnotexhibitthevulnerability toother-user interference. Letusfirstconsider thecaseofsymbol-synchronous transmission. Inthis case,thereceived signalvectorrKthatrepresents theoutputoftheKmatched filtersis (15-3-30) wherebK=[~bl(l) ~b2(1) ...yg;;:bK(I)]' andthenoisevectorwith elements OK=[nl(l)n,(I)...nK(I)]'hasacovariance (15-3-31) Sincethenoiseisgaussian, rKisdescribed byaK-dimensional gaussian pdf withmeanR,bKandcovariance R,.Thatis, Thebestlinearestimate ofbKisthevalueofbKthatminimizes thelikelihood function (15-3-33) 8S6 DIGITAL COMMUNICATIONS ];()dl Received signalr(1) Sample att-=T FIGL'RE 15-3-1 Recei\'er structure fordecorrelation receiver. Theresultofthisminimization yields" Lineae Transformation.,d Det«"lorDecision (15-3-34) Then,thedetected symbols areobtained bytakingthesignofeachelementof b~,i,e, (15-3-35) Figure15-3-1illustrates thereceiver structure. Notefrom(15-3-34) and (15-3-35) thatthedecorrelator requires knowledge oftherelativedelays,in general, toformR,;noknowledge ofthesignalamplitudes isrequired. Sincetheestimate b~isobtained byperforming alineartransformation on thevectorofconelator outputs, thecomputational complexity islinearinK. Thereadershouldobserve thaIthebest(maximum-likelihood) linear estimate ofbKgivenby(15-3-34) isdifferent fromtheoptimum nonlinear ML sequence detector thatfindsthebestdiscrete-valued {±I}sequence that maximizes thelikeli~ood function. Itisalsointeresting tonotethatthe estimate b~isthebestjjnearestimate thatmaximizes thecorrelation metric givenby(15-3-15). Aninteresting interpretation ofthedetector thatcomputes b~asin (15-3-34) andmakesdecisions according to(15-3-35) isobtained byconsidering thecaseofK~2users.Inthiscase, R,~[~~] R-'~_1_[1-IP],1-p2-p(15-3-36) (15-3-37) CHAPTER 15MlTI'!l'SER COMMl:NICATIONS 857 where p=f'g,(t)g2(t) dt (I Then.ifwecorrelate thereceived signal rtf)=~b,g,(t) +vg,b 2g2(t)+n(t)(15-3-38) (15-3-39) (15-3-40) whereII,andII,arethenoisecomponents attheoutputofthecorrelators. Therefore, =[vIW;b l+(II,-pn,)/(I-P:)] V't,b,+(112-pnd/(1~p )(15,3-41) Thisisaveryinteresting result,becausethetransformation R:'haseliminated theinterference components between thetwousers.Consequently. the near-far problem iseliminated andthereisnoneedforpowercontrol. Itisinteresting tonotethataresultsimilarto(15-3-41) isobtained ifwe correlate r(t)givenby(15-3-39) withthetwomodified signature waveforms g:(I)=g,(t)-pg,(t) g~(t)=g,(t)-pg,(t)(15-3-42) (15-3-43 I Thismeansthat,bycorrelating thereceived signalwiththemodified signature waveforms,' wehavetunedoutordecorrelated themultiuser interference. Hence,thedetector basedon(15-3-34) iscalledadecorrelating detector. Inasynchronous transmission, thereceived signalattheoutputofthe correlators isgivenby(15-3-19). Hence,thelog-likelihood function isgivenas (15-3-44) whereRNisdefinedby(15-3-23) andbisgivenby(15-3-21). Itisrelatively easytoshowthatthevectorbthatminimizes i\(b)is (15-3-45) ThisistheMLestimate ofbanditisagainobtained byperforming alinear transformation oftheoutputsfromthebankofcorreIatorsofmatched filters. Sincer=RNb+n,itfollowsfrom(15-3-45) that bO=b+R,;;'n (15-3-46) Therefore, b"ISanunbiased estimate ofb.Thismeansthatthemultiuser 8S8 DIGITAL COMMUNICATIONS interference hasbeeneliminated, asinthecaseofsymbol-synchronous transmission. Hence,thisdetector forasynchronous transmission isalsocalled adecorre/ating detector. Acomputationally efficient method forobtaining thesolution givenby (15-3-45) isthesquare-root factorization methoddescribed inAppendix D.Of course,therearemanyothermethods thatmaybeusedtoinvertthematrix RN•Iterative methods todecorrelate thesignalshavealsobeenexplored. Minimum Mean-Square-Error Detector Intheabovediscussion, we showed thatthelinearMLestimate ofbisobtained byminimizing the quadratic log-likelihood function in(15-3-44). Thus,weobtained theresult givenby(15-3-45), whichisanestimate derived byperforming alinear transformation ontheoutputsofthebankofcorreIatorsormatched filters. Another, somewhat different, solution isobtained ifweseekthelinear transformation bO=Ar,wherethematrixAistobedetermined soasto minimize themeansquareerror(MSE) feb)=E[(b-bO)'(b-bD)] =E[(b-Ar)'(b-Ar)] (15-3.47) Itiseasilyshownthattheoptimum choiceofAthatminimizes feb)is (15-3-48) and,hence. b"=(RN+~Nol) Ir (15-3-49) Theoutputofthedetector isthenb=sgn(bO). Theestimate givenby(15-3-49) iscalledtheminimum MSE(MMSE) estimate ofb.Notethatwhen ~Noissmallcompared withthediagonal elements ofRN,theMMSEsolution approaches tileMLsolution givenby (15-3-45). Ontheotherhand,whenthenoiselevelislargecompared withthe signallevelinthediagonal elements ofRN,A°approaches theidentitymatrix (scaled by~No).Inthislow-SNR case,thedetector basically ignoresthe interference fromotherusers,becausetheadditive noiseisthedominant term. I!shouldalsobenotedthattheMMSEcriterion produces abiasedestimate of b.Hence,thereissomeresidualmultiuser interference. Toperformthecomputations rhatleadtothevaluesofb,wesolvethesetof linearequations (RN+~NoI)b=r (15-3-50) Thissolution maybecomputed efficiently usingasquare-root factorization of thematrixRN+~NoIasindicated above.Thus,todetectNKbitsrequires 3NK2multiplications. Therefore, thecomputational complexity is3K multiplications perbit,whichisindependent oftheblocklengthNandislinear inK. CHAPTER 15:MllLTIliSI-R COMMIINJCATIO"'iS 859 OtherTypesofDetectors Thedecorrelating detector andtheMMSE detector described aboveinvolveperforming lineartransformations onablock ofdatafromabankofKcorrelators ormatched filters.TheMMSEdetector is akintothelinearMSEequalizer described inChapter 10.Consequently. MMSEmultiuser detection canbeimplemented byemploying atapped-delay­ linefilterwithadjustable coefficients foreachuserandselecting thefilter coefficients tominimize theMSEforeachusersignal.Thus,thereceived information bitsareestimated sequentially withfinitedelay.insteadofasa block. Theestimate bOgivenby(15-3-46), whichisobtained byprocessing ablock ofNbitsbyadecorrelating detector, canalsobecomputed sequentially. Xieel 01.(1990)havedemonstrated thatthetransmitted bitsmayberecovered sequentially fromthereceived signal,byemploying aformofadecision­ feedback equalizer withfinitedelay.Thus,thereisasimilarity between the detection ofsignalscorrupted bylSIinasingle-user communication system andthedetection ofsignalsinamultiuser systemwithasynchronous transmission. 15-3-4Performance Characteristics ofDetectors Thebiterrorprobability isgenerally thedesirable performance measure in multiuser communications. Inevaluating theeffectofmultiuser interference on theperformance ofthedetector forasingleuser,wemayuseasabenchmark theprobability ofabiterrorforasingle-user receiver intheabsence ofother usersofthechannel, whichis (15-3-51) wherer,='(;dNil.'tf,isthesignalenergyperbitand~Noisthepowerspectral densityoftheAWGN. Inthecaseoftheoptimum detector foreithersynchronous orasynchronous transmission, theprobability oferrorisextremely difficult andtediousto evaluate. Inthiscase,wemayuse(15-3-51) asalowerboundandthe performance ofasuboptimum detector asanupperbound. Letusconsider. first,thesuboptimum, conventional single-user detector. Forsynchronous transmission, theoutputofthecorreiatorforthekthuseris givenby(15-3-27), Therefore, theprobability oferrorforthekthuser. conditional onasequence biofbitsfromotherusers,is Pdbi)=Q()2[~+j~yg;bJl)P1k(0)J'/No) jF-k, Then,theaverageprobability oferrorissimply K P,=OlK-'2:Pk(b,) 1=1 i#k,(15-3-521 (15-3-531 860 DIGJTAL COMMUNlCATIONS Theprobability in(15-3-53) willbedominated bythetermthathasthe smallest argument intheQfunction. Thesmallest argument willresultinan SNRof (15-3-54) Therefore, Asimilardevelopment canbeusedtoobtainboundsontheperformance for asynchronous transmission. Inthecaseofadecorrelating detector, theother-user interference is c~mpletely eliminated. Hence,theprobability oferrormaybeexpressed as (15-3·56) whereuiisthevariance ofthenoiseinthekthelementoftheestimate bOo Example 15-3-1 Consider thecaseofsynchronous, two-user transmission, where~isgiven by(15-3-41). Letusdetermine theprobability oferror. Thesignalcomponent forthefirsttermin(15-3-41) is~.Thenoise component is n,-pn2n=1-p2 wherepisthecorrelation between thetwosignature signals.Thevariance ofthisnoiseis and2E[(n,-pn2W (T-1-(1-p2)2 1No---]-p22(15-3-57) (15-3-58) Asimilarresultisobtained fortheperformance oftheseconduser. Therefore, thenoisevariance hasincreased bythefactor(1-p2)-'.This noiseenhancement isthepricepaidforthe elimination ofthemultiuser interference bythedecorrelation detector. Theerrorrateperformance oftheMMSEdetector issimilartothatforthe decorrelation detector whenthenoiselevelislow.Forexample, from CHAPTER 15:MULTIUSER COMMUNICATIONS 861 (15-3-49), weobserve thatwhenNoissmallrelativetothediagonal elements of thesignalcorrelation matrixRN, bO=R;;;'r (15-3-59) whichisthesolution forthedecorrelation detector. Forlowmultiuser interference, theMMSE detector resultsinasmaJler noiseenhancement compared withthedecorrelation detector, buthassomeresidual biasresulting fromtheotherusers.Thus,theMMSEdetector attempts tostrikeabalance between theresidual interference andthenoiseenhancement. Analternative totheerrorprobability asafigureofmeritthathasbeen usedtocharacterize theperformance ofamultiuser communication systemis theratioofSNRswithandwithout thepresence ofinterference. Inparticular. (15-3-51) givestheerrorprobability ofthekthuserintheabsence of other-user interference. Inthiscase,theSNRis'Yk='lk/NO'Inthepresence of multiuser interference. theuserthattransmits asignalwithenergy ',f;kwillhave anerrorprobability P,thatexceedsPk(Yk)'TheeffectiveSNRY..isdefinedas theSNRrequired toachievetheerrorprobability (15-3-60) Theefficiency isdefined astheratioYk,hkandrepresents theperformance lossduetothemultiuser interference. Thedesirable figureofmeritisthe asymptotic efficiency, definedas (I5-HI) Thisfigureofmeritisoftensimpler tocompute thantheprobability oferror. Example 15-3·2 Consider thecaseoftwosymbol-synchronous userswithsignalenergiestI andf',.Letusdetermine theasymptotic efficiency oftheconventional detector. Inthiscase.theprobability oferroriseasilyobtained from(15-3-52) and (15-3-53) as P,=~Q(y'2(V\ii; +pV~Y/!Vo) +~Q(v'2(~ -p~)'/N(l) However. theasymptotic efficiency ismucheasiertocompute. Itfollows fromthedefinition (15-3-61) andfrom(15-3-52) that '71=[max(0,I-~iIpl)r Asimilarexpression isobtained for'7,. Theasymptotic efficiency oftheoptimum andsuboptimum detectors that wchavedescribed hasbeenevaluated byVerdu(1986),LupasandVerdu 862 DJGITAL COMMUNICATIONS 1.5.-------r------,..------,------, ................................................................................ ..._. _ __Conventional detcctor ............ Optimum detector __LinearMLdetector__MMSEdetect(lrr:LO ~t::,.::. •e•i ~0.5,,,,, \, \ \ \ \ \, \,,0.0L --'- -'- --"'J. ---l -20 -10 00 10 20 10(og,o(62/6» FIGURE 15·3-2 Asymptotic efficiencies ofoptimum (Vilerbi) detector, conventional detector, MMSEdetector, andlinearMLdetector inatwo-user synchronous DS/SSMA system.[FromXitet.1.(1990), ©IEEE.] (1989),andXieetal.(1990).Figure15-3-2illustrates theasymptotic efficiencies ofthesedetectors whenK=2usersaretransmitting synchronously. These graphsshowthatwhentheinterference issmall (~2-+0),theasymptotic efficiencies ofthesedetectors arerelatively large(nearunity)andcomparable. As~2increases, theasymptotic efficiency oftheconventional detector deteriorates rapidly.However, theotherlineardetectors perform relatively wellcompared withtheoptimum detector. Similarconclusions arereached by computing theerrorprobabilities, butthesecomputations areoftenmore tedious. 15-4RANDOM ACCESS METHODS Inthissection,weconsider amultiuser communication systeminwhichusers transmit information inpacketsoveracommon channel. Incontrast tothe CDMAmethoddescribed inSection15-3,theinformation signalsoftheusers arenotspreadinfrequency. Asaconsequence, simultaneous transmission of signalsfrommultiple userscannotbeseparated atthereceiver. Theaccess methods described belowarebasically random, becausepacketsaregenerated according tosomestatistical model.Usersaccessthechannelwhentheyhave oneormorepackets 10transmit. Whenmorethanoneuserattempts to transmitpackets simultaneow:l~, thepacketsoverlapintime,i.e.,theycollide, CHAPTER 15:MULTIUSER COMMUNICATIONS 863 FlGURE 15-4-1 Random accesspackettransmission: (a)packetsfromatypicaluser: (b)packersfromseveraluSers.nn tal nDODO (hin Time--' oOOD~nol.L.ilo_ilio0.ll0L--- / T~me----. Overlap and,hence,aconflictresults,whichmustberesolved bydevising somechannel protocol forretransmission ofthepackets. Below,wedescribe severalrandom accesschannelprotocols thatresolveconflicts inpackettransmission. 15-4-1ALOHA Systems andProtocols Suppose thatarandomaccessschemeisemployed whereeachusertransmits a packetassoonasitisgenerated. Whenapacketistransmitted byauserand nootherusertransmits apacketfortheduration ofthetimeintervalthenthe packetisconsidered successfully transmitted. However, ifoneormoreofthe otheruserstransmits apacketthatoverlaps intimewiththepacketfromthe firstuser.acollision occursandthetransmission isunsuccessful. Figure15-4-1 illustrates thisscenario.Iftheusersknowwhentheirpacketsaretransmitted successfully andwhentheyhavecollided withotherpackets. itispossible to deviseascheme, whichwemaycallachannelaccessprotocol, forretransmis­ sionofcollided packets. Feedback totheusersregarding thesuccessful orunsuccessful transmission ofpackets isnecessary andcanbeprovided inanumberofways.Inaradio broadcast system,suchasonethatemploys asatellite relayasdepicted inFig. 15-4-2.thepacketsarebroadcast toalltheusersonthedown-link. Hence,all I_l_-L FtGURE t5·4-2 Broadcast system. 864 DIGITA.L COMMUNICATIONS thetransmitters canmonitor theirtransmissions and,thus,obtainthefollowing ternaryinformation: nopacketwastransmitted, orapacketwastransmitted successfully, oracollision occurred, Thistypeoffeedback tothetransmitters is generally denoted as(0,1,C)feedback. Insystemsthatemploywirelineor filter-optic channels, thereceiver maytransmit thefeedback signalona separate channel. 'TheALOHA systemdevisedbyAbramson (1973,1977)andothersatthe University ofHawaiiemploys asatellite repeater thatbroadcasts thepackets received fromthevarioususerswhoaccessthesatellite. Inthiscase,allthe userscanmonitor thesatellitetransmissions and,thus,establish whether ornot theirpacketshavebeentransmitted successfully. Therearebasically twotypesofALOHA systems: synchroilized orslotted andunsynchronized orunslotted. Inanunslotted ALOHA system,ausermay begintransmitting apacketatanyarbitrary time.InaslottedALOHA, the packetsaretransmitted intimeslotsthathavespecified beginning andending times. Weassumethatthestarttimeofpacketsthataretransmitted isaPoisson pointprocesshavinganaveragerateofApackets/so LetT"denotethetime duration ofapacket.Then,thenormalized channel trafficG,alsocalledthe offeredchanneltraffic,isdefinedas (15-4-1) Therearemanychannel accessprotocols thatcanbeusedtohandle collisions. Letusconsider theoneduetoAbramson (1973).InAbramson's protocol, packets thathavecollided areretransmitted withsomedelayr, whererisrandomly selected according tothepdf p(r)=ae-a, (15-4-2) whereaisadesignparameter. Therandomdelayrisaddedtothetimeofthe initialtransmission andthepacketisretransmitted atthenewtime.Ifa collision occursagain,anewvalueofrisrandomly selected andthepacketis retransmitted withanewdelayfromthetimeofthesecondtransmission. This process iscontinued untilthepacketistransmitted successfully. Thedesign parameter adetermines theaverage delaybetween retransmissions. The smallerthevalueofa,thelongerthedelaybetween retransmissions. Now,letA',whereA'<A,betherateatwhichpackets aretransmitted successfully. Then,thenormalized channelthroughput is (15-4-3) Wecanrelatethechannelthroughput StotheofferedchanneltrafficGby making useoftheassumed starttimedistribution. Theprobability thata packetwillnotoverlapagivenpacketissimplytheprobability thatnopacket CHAPTER IS:MULTIUSER COMMUNKATIONS 86S 1.0 0.8 '"O.b'5 ~ ~ ~, 2 !=0.4 0.2 0 0.01 01 FIGURE 15-4-3 Throughput inALOHA systems.Offeredchannel(ram,G10 100 beginsT"sbeforeorTpsafterthestarttimeofthetransmitted packet.Since thestarttimeofallpackets isPoisson-distributed, theprobability thatapacket willnotoverlapisexp(-2AT,,)=exp(-2G).Therefore, S=Ge-2G(15-4-4) Thisrelationship isplottedinFig.15-4-3.Weobserve thatthemaximum throughput isSmox=1/Ze=0.184packets perslot,whichoccursatG=I. WhenG>!.thethroughput Sdecreases. Theabovedevelopment illustrates thatanunsynchronized orunslotted random accessmethod hasarelatively smallthroughput andisinefficient. Throughput forslottedALOHA Todetermine thethroughput ina slottedALOHA system,letG,betheprobability thattheithuserwilltransmit apacketinsomeslot.IfalltheKusersoperateindependently andthereisno statistical dependence between thetransmission oftheuser'spacketinthe currentslotandthetransmission oft)leuser'spacketinprevious timeslots,the total(normalized) offeredchanneltrafficis (15-4-5) (15-4-6)Notethat,inthiscase,Gmaybegreaterthanunity. Now,letSj""G,betheprobability thatapackettransmitted inatimeslotis received withoutacollision. Then,thenormalized channelthroughput is K s=2;s, i=1 866 Dl(ilTAL (·O\1\H·ISI('·\TH)',:S Theprobability thatapacketfromtheithuserwillnothaveacollision with another packetis Therefore."Q,=IT(1-G,) , I I', 5,=G,Q,(15-4-7) (15-4-R) Asimpleexpression forthechannel throughput isohtained hyconsidering Kidentical users.Then. G(j=­, K and Then.ifweletK-z.weobtainthethroughput S=Ge"(15-4-LJ) (15-4-10) ThisresultisalsoplottedinFig.15-4-3.Weohserve thatSreaches amaximum throughput ofSn""=lie=0.368packetsperslotatG=I.whichistwicethe throughput oftheunslotled ALOHA system. Theperformance oftheslottedALOHA systemgivenahoveisbasedon Abramson's protocol forhandling collisions. Ahigherthroughput ispossible bydevising abetterprotocol. Abasicweakness inAbramson's protocol isthatitdoesnottakeinto account theinformation ontheamount oftrafficonthechannel thatis available fromobservation ofthecollisions thatoccur.Animprovement in throughput oftheslottedALOHA systemcanbeobtained byusingatree-type protocol devised byCapetanakis (1979). Inthisalgorithm, usersarenot allowed totransmit newpackets thatarcgenerated untilallearliercollisions areresolved. Ausercantransmit anewpacketinatimeslotimmediately following itsgeneration, provided thatallprevious packets thathavecollided havebeentransmilled .successfully. Ifanewpacketisgenerated whilethe channel isclearing theprevious collisions. thepacket isstoredinabuffer. Whenanewpacketcollides withanother. eachuserassigns itsrespective packettooneoftwosets.sayAorB.withequalprobability (byflipping a coin).Then,ifapacketisputinsetA.theusertransmits itinthenexttime slot.Ifitcollides again,theuserwillagaInrandomly assignthepackettoone oftwosetsandtheprocessoftransmission isrepeated. ThisprocesscontInues untilallpackets contained insetAaretransmitted successfully. Then.all packets insetBaretransmitted following thesameprocedure. Alltheusers CHAPTER l~:MULTIUSER COMMUNICATIONS 867 monitorthestateofthechannel, and,hence,theyknowwhenallthecollisions havebeenserviced. Whenthechannel becomes available fortransmission ofnewpackets, the earliestgenerated packetsaretransmitted first.Toestablish aqueue,thetime scaleissubdivided intosubintervals ofsufficiently shortduration suchthat,on average, approximately onepacketisgenerated byauserinasubinterval. Thus,eachpackethasa"timetag"thatisassociated withthesubinterval in whichitwasgenerated. Then,anewpacketbelonging tothefirstsubinterval is transmitted inthefirstavailable timeslot.Ifthereisnocollision thenapacket fromthesecondsubinterval istransmitted, andsoon.Thisprocedure continues asnewpacketsaregenerated andaslongasanybacklogofpackets fortransmission exists.Capetanakis hasdemonstrated thatthischannelaccess protocol achieves amaximum throughput of0.43packetsperslot. Inaddition tothroughput, another important performance measure ina random accesssystemistheaverage transmission delayintransmitting a packet.InanALOHA system,theaveragenumberoftransmissions perpacket isGIS.Tothisnumber wemayaddtheaverage waitingtimebetween transmissions and,thus,obtainanaveragedelayforasuccessful transmission. Werecallfromtheabovediscussion thatintheAbramson protocol, the parameter adetermines theaverage delaybetween retransmissions. Ifwe selectasmall,weobtainthedesirable effectofsmoothing outthechannelload attimesofpeakloading, buttheresultisalongretransmission delay.Thisis thetrade-off intheselection ofain(15-4-2). Ontheotherhand,the Capetanakis protocol hasbeenshowntohaveasmalleraverage delayinthe transmission ofpackets. Hence,itoutperforms Abramson's protocol inboth averagedelayandthroughput. Another important issueinthedesignofrandom accessprotocols isthe stability oftheprotocol. Inourtreatment ofALOHA-type channel access protocols, weimplicitly assumed thatforagivenofferedload,anequilibrium pointisreached wheretheaveragenumberofpacketsentering thechannel is equaltotheaveragenumberofpacketstransmitted successfully. Infact,itcan bedemonstrated thatanychanuel accessprotocol, suchastheAbramson protocol, thatdoesnottakeintoaccountthenumberofprevious unsuccessful transmissions inestablishing aretransmission policyisinherently unstable. On theotherhand,theCapetanakis algorithm differsfromtheAbramson protocol inthisrespectandhasbeenprovedtobestable.Athorough discussion ofthe stability issuesofrandom accessprotocols isfoundinthepaperbyMassey (1988). 15-4·2CarrierSenseSystems andProtocols Aswehaveobserved, ALOHA-type (slotted andunslotted) random-access protocols yieldrelatively lowthroughput. Furthermore, aslottedALOHA systemrequires thatuserstransmit atsynchronized timeslots.Inchannels wheretransmission delaysarerelatively small,itispossible todesignrandom 868 DIGITAL COMMUNICATIONS ~u""~~ •• mCfJ~LJJ I- ~~ (propagation delay) FIGURE 154-4 Localareanetwork with busarchitecture.o accessprotocols thatyieldhigherthroughput. Anexample ofsuchaprotocol is carriersensingwithcollision detection, whichisusedasastandard Ethernet protocol inlocalareanetworks. Thisprotocol isgenerally knownascarrier sensemultiple accesswithcollision detection (CSMA/CD). TheCSMA!CDprotocol issimple.Alluserslistenfortransmissions onthe channel. Auserwhowishestotransmit apacketseizesthechannelwhenit sensesthatthechannel isidle.Collisions mayoccurwhentwoormoreusers sensean idle channel andbegintransmission. Whentheusersthatare transmitting simultaneously senseacollision, theytransmit aspecialsignal, calledajamsignal,thatservestonotifyallusensofthecollision andaborttheir transmissions. Boththecarriersensingfeatureandtheabortion oftransmission whenacollision occursresultinminimizing thechanneldown-time and,hence, yieldahigherthroughput. Toelaborate ontheefficiency ofCSMA/CD, letusconsider alocalarea network havingabusarchitecture, asshowninFig.15-4-4.Consider twousers VIandV2atthemaximum separation, i.e.,atthetwoendsofthebus,andlet Tdbethepropagation delayforasignaltotravelthelengthofthebus.Then, the(maximum) timerequired tosenseanidlechannel isfd.Suppose thatVI transmits apacketofduration Tp•UserU2mayseizethechannel fdslaterby usingcarriersensing, andbeginstotransmit. However, userVIwouldnot knowofthistransmission untilfdsafterV2beginstransmission. Hence,we maydefinethetimeinterval 2fdasthe(maximum) timeinterval 10detecta collision.Ifweassumethatthetimerequired totransmit thejamsignalis negligible, theCSMA/CD protocol yieldsahighthroughput when2rd«Tp• Thereareseveralpossible protocols thatmaybeusedtoreschedule transmissions whenacollision occurs.Oneprotocol iscallednonpersistent CSMA,asecondiscalledi-persistent CSMA,andageneralization ofthelatter iscalledp-persistant CSMA. Nonpersistent CSMA Inthisprotocol, auserthathasapackettotransmit sensesthechannelandoperates according tothefollowing rule. (a)Ifthechannelisidle,theusertransmits apacket. (b)Ifthechannel issensedbusy,theuserschedules thepacket CHAPTER 15:MULTIUSER COMMUNICATIONS 869 transmission atalatertimeaccording tosomedelaydistribution. Attheendof thedelayinterval, theuseragainsensesthechanneland repeats steps(a)and (b). I-Persistent CSMA Thisprotocol isdesigned toachievehighthroughput bynotallowing thechanneltogoidleifsomeuserhasapackettotransmit. Hence,theusersensesthechannelandoperates according tothefollowing rule. (a)Ifthechannel issensedidle,theusertransmits thepacketwith probability 1. (b)Ifthechannelissensedbusy,theuserwaitsuntilthechannelbecomes idleandtransmits apacketwithprobability one.Notethatinthisprotocol, a collision willalwaysoccurwhenmorethanoneuserhasapackettotransmit. p-Persistent CSMAToreducetherateofcollisions inI-persistent CSMA andincrease thethroughput, weshouldrandomize thestarting timefor transmission ofpackets. Inparticular, uponsensingthatthechannel isidle,a userwithapackettotransmitsendsitwithprobability panddelaysitbyTwith probability 1-p.Theprobability pischoser!inawaythatreduces the probability ofcollisions whiletheidleperiodsbetween consecutive (nonover­ lapping) transmissions iskeptsmall.Thisisaccomplished bysubdividing the timeaxisintominislots ofduration Tandselecting thepackettransmission at thebeginning ofaminislot. Insummary, inthep-persistent protocol, auser withapackettotransmitproceeds asfollows. (a)Ifthechannelissensedidle,thepacketistransmitted withprobability p,andwithprobability I -Pthetransmission isdelayedby[s. (b)Ifatt=T,thechannelisstillsensedtebeidle,step(a)isrepeated.Ifa collision occurs,theusersschedule retransmission ofthepacketsaccording to somepreselected transmission delaydistribution. (c)Ifatt=T,thechannel issensedbusy,theuserwaitsuntilitbecomes idle,andthenoperates asin(a)and(b)above. Slottedversionsoftheaboveprotocol canalsobeconstructed. Thethroughput analysis ferthenonpersistent andthep-persistent CSMA/CD protocols hasbeenperformed byKleinrock andTobagi(1975), basedonthefollowing assumptions: Itheaverage retransmission delayislargecompared withthepacket duration T,,; 2theinterarrival timesofthepointprocessdefinedbythestarttimesof allthepackets plusretransmissions areindependent andexponentially distributed. 870 D1GIIAL COMMllNICATlONS III~-----------------------===9 II=()---....~,.:::::::....__ l/=().OOI ~I=11.01 o.a 100 10 0.1ob::=::::::::=-~---~-----=:~:::::::';:~~""":::::""".::::",."J 0.01;,;0.6 j II-OA 0.1 OfferedchanneltrafficG FIGURE IS-4-S Throughput innonpersistent CSMA.[FromKleinrock andTobagi(/975),©IEEE.) Forthenonpersistent CSMA,thethroughput is S=Ge-'G G(1+2a)+e-oG (15-4-11) (15-4-12)wheretheparameter a=Tdfr".Notethatasa.....0,S-+G/(1+G).Figure 15-4-5illustrates thethroughput versustheofferedtrafficG,witt..aasa parameter. WeobservethatS-+1asG.....xfora=0.Fora>0,thevalueof Smaxdecreases. FortheI-persistent protocol, thethroughput obtained byKleinrock and Tobagi(1975)is S=G[1+ G+aG(1+G+~aG)le-G(t+2') G(1+2a)-(1-e-aG)+(1+aG)e-G(l+a) Inthiscase, I. _G(1+G)e-G ImS- c (l--+OG+e-'(15-4-13) whichhasasmallerpeakvaluethanthenonpersistent protocol. Byadopting thep-persistent protocol, itispossible toincrease the throughput relative totheI-persistent scheme. Forexample, Fig.15-4-6 illustrates thethroughput versustheofferedtrafficwitha=Td17;,fixedand withpasaparameter. Weobserve thataspincreases towardunity,the maximum throughput decreases. Thetransmission delaywasalsoevaluated byKleinrock andTobagi(1975). Figure15-4-7illustrates thegraphsofthedelay(normalized by7',,)versusthe CHAPTER I~:MULTIUSER. COMMUNfCAHONS 871 10 I()]O.l I JO 100 Oftcredchanneltra.ffil.:(;10 (01 O.X '",0.6"~.., 2OA~f- 0,2 0 11.0/ 1.0 (hi 0.8 '",0.6 ~ ~.., 2OA ~f- 0.2 0 0.01 FIGURE t5-4-6 Channel throughput inp-persistent CSMA:(a)a~O;(b)a=O_OL (c)a=0.1 [FromKleinrock andTobagi(/975). I{)IEEE.],0 k) O.X '";0.6 "~.., ~0.4 f- 0.2 a O.Otp=0.6 p:=0.99 p=0,03 I'=0.0] 0.1 I 10 Offeredchannel(raffieG100 throughput Sfortheslottednonpersistent andp-persistent CSMAprotocols. Alsoshownforcomparison isthedelayversusthroughput characteristic ofthe ALOHA slottedandunslotted protocols. Inthissimulation, onlythenewly generated packetsarederived independently fromaPoisson distribution. Collisions anduniformly distributed random retransmissions arehandled withoutfurtherassumptions. Thesesimulation resultsillustrate thesuperior performance ofthep-persistent andthenonpersistent protocols relativetothe ALOHA protocols. Notethatthegraphlabeled"optimum p-persistent" is 872 DIGITAL COMMUNICATIONS ALOHASlotted l-Pel'!>istenl Slotted ALOHASlotted Non-Persistent 20 1.009 0.708 0.605Optimum p-Persistent I•I·- .f/ 0.4OJ0.2 0.1o2 ThroughplJl 5 FIGURE 15....'Throughput ve"usdelayfromsimulation (a~(101).[FromKleinrock andTobagi(1975), ©IEEE.] obtained byfindingtheoptimum valueofpforeachvalueofthethroughput. Weobservethatforsmallvaluesofthethroughput, theI-persistent (p=1) protocol isoptimal. 15-5BIBLIOGRAPHICAL NOTES ANDREFERENCES FDMAwasthedominant multiple accessschemethathasbeenusedfor decades intelephone communication systemsforanalogvoicetransmission, Withtheadventofdigitalspeechtransmission usingPCM,DPCM,andother speechcodingmethods, TDMAhasreplaced FDMAasthedominant multiple accessschemeintelecommunications. CDMAandrandomaccessmethods, in general,havebeendeveloped overthepastthreedecades, primarily forusein wirelesssignaltransmission andinlocalareawirelinenetworks. Multiuser information theorydealswithbasicinformation-theoretic limitsin sourcecodingformultiple sources, andchannelcodingandmodulation for multiple accesschannels. Alargeamountofliterature exisuonthesetopics,In thecontextofourtreatment ofmultiple accessmethods, thereaderwilllind PROBLEMSCHAPTER IS,MULTIUSER COMMUNICATIONS 873 thepapersbyCover(1972),EIGamalandCover(1980)Bergmans andCover (1974),andHui(1984)parti..:ularly relevant. ThecapacityofacellularCDMA systemhasbeenconsidered inthepaperbyGilhousen etal.(1991). Signaldemodulation anddetection formultiuser communications has received considerable attention inrecentyears.Thereaderisreferred tothe papersbyVerdu(1986a-c, 1989),LupasandVerdu(1990),Xieetai.(l990a, b),PoorandVerdu(1988),ZhangandBrady(1993),andZvonarandBrady (1995).Earlier workonsignaldesignanddemodulation formultiuser communications isfoundinthepapersbyVanEtten(1975,1976),Horwood andGagliardi (1975),andKayeandGeorge(1970). TheALOHA system,whichwasoneoftheearliestrandomaccesssystems. istreatedinthepapersbyAbramson (1970,1977)andRoberts (1975).These paperscontain thethroughput analysis forunslotted andslottedsystems. Stability issuesregarding theALOHA protocols may.befoundinthepapersby Carleial andHellman (1975),Ghezetai.(1988),andMassey(1988).Stable protocols basedontreealgorithms forrandomaccesschannels werefirstgiven byCapetanakis (1977).Thecarriersensemultiple accessprotocols thatwe described arp.duetoKleinrock andTobagi(1975).Finally,wemention the IEEEPressbookeditedbyAbramson (1993),whichcontains acollection of papersdealingwithmultiple access com~unications. 15·1Intheformulation oftheCDMAsignalandchannelmodelsdescribed inSection 15-3-1,weassumedthatthereceivedsignalsarereal.ForK>1.thisassumption impliesphasesynchronism atalltransmitters, whichisnotveryrealisticina practical system.Toaccommodate thecasewherethecarrierphasesarenot synchronous, wemaysimplyalterthesignature waveforms fortheKusers,given by(15-3-1),tobecomplex-valued, oftheform L-' g,(t)=('0,La,(n)p(t -nT,J,los;kos;K .,0 where8,represents theconstantphaseoffsetofthekthtransmitter asseenbythe common receiver. 8Giventhiscomplex-valued formforthesignature waveforms, determine the formoftheoptimum MLreceiverthatcomputes thecorrelation metries analogous to(15-3-15). bRepeatthederivation fortheoptimum MLdetectorforasynchronous transmis­ sionthatisanalogous to(15-3-19). 15-2Consider aTDMAsystemwhereeachuserislimitedtoatransmitted powerP, independent ofthenumberofusers.Determine thecapacityperuser,C",andthe totalcapacityKC".PlotC,andKC,asfunctions of'i.IN.andcomment onthe resultsasK->"'. 15-3Consider anFDMAsystemwithK=2users,inanAWGNchannel,whereuser1 isassigned abandwidth W,=aWanduser2isassigned abandwidth W,= (l-a)W,where0os;a.;;1.LetP,andP,betheaveragepowersofthetwousers. flGUREPlS-6874 DIGITAL COMMUNICATIONS •Determine thecapacities C.andC,ofthetwousersandtheirsumC=CI+C, asafunction ofa.Onatwo-dimensional graphoftheratesR,versusRI,plot thegraphofthepoints(C"CI)asavariesintherange0...a'"1. bRecallthattheratesofthetwousersmustsatisfytheconditions RI<W,log, (1+;~J R,<W,log,(I+WP,) ,No· R,+R,<Wlog,(1+PI+P,) WNo Determine thetotalcapacity CwhenP,/a=P,I(1-a)=P,+P"and,thus, showthatthemaximum rateisachieved whena/(1-a)=P,IP,=Will¥,. 15-4Consider aTDMAsystemwithK=2usersinanAWONchannel. Suppose that thetwotransmitters arepeak-power-Iimited toP,andP"andletuserItransmit forl00a%oftheavailable time anduser2transmit 100(1-Q)%ofthetime.The available bandwidth isW. •Determine thecapacities C"C"andC=C,+C,asfunctions ofQ. bPlotthegraphofthepoints(C"C)asavariesintherange0...a'"L 15·5Consider aTDMAsystemwithK=2usersinanAWONchannel. Suppose that thetwotransmitters areaverage-power·limited, withpowersP,andP,.UserI transmits 100a%ofthetimeanduser2transmits 100(1-a)%ofthetime.The channelbandwidth isW. •Determine thecapacities C"C"andC=C,+C,asfunctions ofQ. bPlotthegraphofthepoints(C"C,)asavariesintherange0...a'"L cWhatisthesimilarity between thissolutionandtheFDMAsysteminProblem 15-3. 15-6Consider thetwo-user, synchronous, multiple-access channel andthesignature sequences showninFig.PI5-6.Theparameter A~0describes tlterelative strength between thetwousers,and0'"B...1describes thedegreeofcorrelation between thewaveforms. Let , x r(l)=LLb.(i)s,(I- i)+n(l) k=Ii.._x A,....-__.. -AB12 I' CHAPTER IS:MUlT1USER COMMUNICATIONS 875 denotethereceived waveform attimeI,wheren(l)iswhitegaussian noisewith powerspectral density 0",andh.(i)E{-I,+1).Inthe.following problems, you willcompare thestructure oftheconventional multiuser detector tooptimimum receiverstructures forvariousvaluesofA,00<;B0<;I,and0". 8Showthat,giventheobservation {,(t),-00<I..I},asufficient statisticforthe datab,(O)andb,(O)istheobservation duringIE[0,I). bConventional (suboptimum) multiuser detection chooses thedatab,(O)accord­ ingtothefollowing rule: h.(Oj=sgn(Y.) where y,=I''(I).s,(t)dl " Determine anexpression fortheprobability ofbiterrorforuserI,usingthe notation w,=1.'S:(I)dl P"=Ls,(I)S,(I) dl. cWhatistheformofthisexpression forA----0,B<I,andarbitrary u'? dWhatistheformofthisexpression forarbitrarily largeA,B<I,andarbitrary ,,'?Whatdoesthissayaboutconventional detection? eWhatistheformofthisexpression forB=I,andarbitrary 0"andA?Why doesthisdifferfromtheresultin(d)? rDetermine theformofthisexpression forarbitrarily large,,',arbitrary A.and B<l. gDetermine theformofIhisexpression for0"--->0,arbitrary A,andB<1. 15·7RefertoProblem 15-6.Themaximum-likelihood sequence receiver forthis channel selectsthedatah,(0)andh,(O)transmitted duringtheinterval [0,1J according totherule «~NOD =argmaxA[{r(I), 0<I<I}Ih"h,J h,·h whereA[{'(I),0<I<I}Ih"b,]isthelikelihood function ofb,andb,givenan observation of{'(I),0<I<I}.Itwillbehelpfultowritethismaximization as «~b,(O)) =argmaxargmaxA[{r(I).0<I<I}Ih,.h,J "1": wherethevalueMthatsatisfiestheinnermaximization maydependonhi'Note thattheneedfor"sequence detection" isobviated. 8Express thismaximization inthesimplesl possible terms.usingthesame notation asinProblem 15-6(b). Reducethismaximization tosimplest form. usingfactslike argmaxKe"'"=argmaxf,(x) K z if.say.Kisindependent ofx. 876 DIGITAL C(}\(MUNIrATlONS Tn~....smiuern,I'(f) Ma(I...hed )------.., filterI / Communication linksPOSt­ proce~wr Po'!.t­ processorM<ltched )---~-J filter2 FIGURE PlS-8 bWhatisthesimpleststructure oftheMLSreceiverastherelativestrength ofthe interferer vanishes, A~O?Howdoesitcompare withconventional detection? cWhatisthesimplest structure oftheMLSreceiver forB=1andarbitrary A and(T'?Howdoesitcompare withconventional detection? Why? dWhatisthesimplest structure oftheMLSreceiverforarbitrarily large0'and arbitrary AandB?Howdoesitcompare withconventional detection? Determine theerrorrateforuser1inthiscase.[Hine:Usethefactthat sgn(y,)=sgn(y,±p,,)withhighprobability inthiscase.} eDetermine theerrorprobability ofuserIoftheMLSreceiver for(T<--->0,and arbitrarily largeAandB<I?Howdoesitcompare withconventional detection? fWhatisthestructure oftheMLSreceiverforarbitrarily largeA,andB<1,and arbitrary (T,?Howdoesitcompare withconventional detection? Whatdoesthis sayaboutconventional detection inthiscase?[Hine:UsethefactthatEly,lis roughlyAtimesgreaterthanEly,I.] 15-8Consider theasynchronous communication systemshowninFig.PI5-8.Thetwo receivers arenotcoloeated, andthewhitenoiseprocesses nCIl(t)andn!2'(t)maybe considered tobeindependent. Thenoiseprocesses areidentically distributed, with powerspectraldensity(T'andzeromean.Sincethereceivers arenotcolocated, therelativedelaysbetween theusersarenotthesame-denote therelativedelay ofuserkatreceiveribyrlo.Allothersignalparameters coincide forthereceivers, andthereceived signalatreceiveriis 2 rU'(t)=2:2:b.(l)s,(t-IT-Tf')+n"'(t) k"l1=_.",", wheres,hassupporton[0,T].Youmayassumethatthereceiver ihasfull knowledge ofthewaveforms, energies, andrelativedelaysr\i'andri'.Although receiveriiseventually interested onlyinthedatafromtransmitter'i, notethat thereisafreecommunication linkbetween thesamplerofonereceiver, andthe postprocessing circuitry oftheother.Following eachpostprocessor, thedecision is attained bythreshold detection. Inthisproblem; youwillconsider optionsfor postprocessing andiorthecommunication linkinordertoimprove performance. CHAPTER 15:MULTIUSER COMMU~ICATIONS 877 aWhatisthebiterrorprobability forusers1and2ofareceiver pairthatdoesnot utilizethecommunicalion link,anddoesnolperform poslprocessing. Usethe following notation: p~,'=J.,,(t~r\")s,(t+T-t{')<il w,=Js;(/-ri")<il=J.<i(t-ri")<il bConsidcr <lpostprocessor forreceiver Ithatacceptsy,(I-I)andy,(I)fromthe communication link.andimplements thefollowing postprocessing ony,(l) ~,(I)=y,(I)-p\,'sgnLv,(I-I)] -p\\'sgnLv,(I)]. Determine anexactexpression Forthebiterrorrateforuser1. cDetermine theasymptotic multiuser efficiency ofthereceiver proposed in(b). andcompare withthatin(a).Doesthisreceiver alwaysperform betterthanthai proposed in(a)" 15-9Thebaseband waveforms showninFig.P15-6areassigned totwouserswhoshare the~amcl/synchronolls. narrowband channel. Assume thatB=IandA=4.We shouldIikctocomparc Iheperformance ofseveralreceivers, withacrilerion of .f',(O).Sincethisexpression is100complicated insomecases,weshallalsobe interested incomparing Iheasymptotic multiuser efficiency 'I,ofeachreceiver. Assumc thatT,=[)butthat[)<T,<Tisfixedandknownatthereceiver, and assume thaIwehaveinfinitehorizon transmission, 2M+l----+x. aFortheconventional. multiuser detector: (i)Findtheexactbitprobability oferrortoruser1.Expressthisresultinterms of\\'"p".1'",and(T'.[Hint:Conditioning onb,(-I)andb,(O)willhelp.] (ii)Plottheasymptotic multiuser efficiency 'I,asafunction ofT2•IndicaIeand explainthemaximum andminimum valuesof'I,inthisplot. bForthcMLSreceiver: Ii)Plot1),asafunction ofT,.Explain maximum andminimum values.and compare with(a)(ii). (ii)Whicherrorsequences aremostlikelyforeachvalueofr;' cForthelimitingdecorrelating detector: !i)Findanexactexpression fortheprobability oferrorforuserI.withsimilar parameters asin(a)(i)[Hint:Don'tforgettonormalize 1'"andp".J (ii)Plot'I,asafunction of',.Explain theminimum valueof'I,inthiscase. andcompare with(a)(ii). 15-111Thesymbol-by·symbol deleclor thatminimizes theprobabilily ofasymbolerro, differsfromthemaximum-likelihood sequence detector. Theformer ismOTe completely described asthedeteclor thatselectseachb,(O)according totherule h..(O)=argmaxA[{r(t),0<I<1)Ib,(O») "k(lI) k=O,I,2, ...878 DIGITAL COMMUNICATIONS ----- aShowthatthisdecision ruleminimizes A[b,(O)"'b.(O») amongalldecision rules withobservation {r(t),0<t<I}.Subjecttothiscriteria, itissuperior tothe MLSreceiver. bShowthatthesimplest structure oftheminimum-probability-of-error receiver foruser1isgivenby ----- [(b,y,) (V'-b,P12)]b,(O)=argmax exp--,-cosh' 2 1>1 u. (J' c\Findthesimplest formoftheminimum-probability-of-error receiver forB=1 andarbitrary Aand0-'.Howdoesthiscompare withtheabovereceivers? dFindthelimitingformoftheminimum-probability-of-error receiver forarbit­ rarilylargea'andarbitrary AandB.Compare withtheabovereceivers. eFindthelimitingformoftheminimum-probability-of·error receiver forA»I andarbitrary0-'andB.Compare withtheabovereceivers. rFindthelimitingformoftheminimum-probability-of·error receiver forA»1 a'-->0andarbitrary B.Compare withtheabovereceivers. 15-11InapureALOHA system.thechannelbitrateis2400bits/soSuppose thateach terminal transmits a100bitmessage everyminuteontheaverage. aDetermine themaximum numberofterminals thatcanusethechannel. bRepeat(a)ifslottedALOHA isused. 15·12Determine themaximum inputtrafficforthepureALOHA andslottedALOHA protocols. 15-13ForaPoissonprocess,theprobability ofkarrivalsinatimeintervalTis P(k)=e'"(AT-)' k! aDetermine theaveragenumberorarrivalsintheintervalT. bDetermine thevariancea'inthenumberofarrivalsintheinterval T. cWhatistheprobability ofatleastonearrivalintheintervalT? dWhatistheprobability ofexactlyonearrivalintheintervalT? 15-14RefertoProblem 15-13.TheaverageamvalrateisA=10packets/s. Detenmine atheaveragetimebetween arrivals; btheprobability thatanotherpacketwillarrivewithin15;within100ms. 15-15Consider apureALOHA systemthatisoperating withathroughput G=0.1and packetsaregenerated withaPoissonarrivalrateA.Determine athevalueofA; btheaveragenumberofattempted transmissions tosendapacket. 15·16Consider aCSMA/CD systeminwhichthetransmission rateonthebusis 10Mbits/s. Thebusis2kmandthepropagation delayis5/Ls/km. Packetsare 1000bitslong.Determine atheend-to-end delayT,,; bthepacketduration Tp: ctheratioT"IT,,: dthemaximum utilization ofthebusandthemaximum hitrate. APPENDIXA THELEVINSON-DURBIN ALGORITHM The Levinson-Durbin algorithm isanorder-recursive method fordetermining the solution tothesetoflinearequations $I'B,,=et»J' (A-!) where «1>"isapXPToeplitz matrix,."isthevectorofpredictor coefficients expressed as and<1>"isap-dimensional vectorwithelements <I>;,=[.p(I) </>(2)._-</>(p») Forafirst-order (p=I)predictor, wehavethesolution <I>(O)a" =.p(l) a"="'(I)/cf>(0) Theresidualmeansquareerror(MSE)forthefirst-order predictor is 't,="'(0)-a".p(l) ="'(0)-a;,.p(O) ="'(0)(1-a;,)(A-2) (A-3) Ingeneral, wemayexpress thesolution forthecoefficients ofanmth-order 879 886 DIGITAL CQMMUr..rrCATIONS predictor intermsofthecoefficients ofthe(m-1)th-order predictor. Thus,weexpress amasthesumoftwovectors, namely, (A-4) wherethevectordm_,andthescalarkmaretobedetermined. Also, <1>00maybe expressed as (A-S) where <1>:"_,isjustthevector<1>.,_,inreverseorder. Now (A-6) From(A-6),weobtaintwoequations. Thefirstisthematrixequation (A-7) But<l>oo_,aoo_.,=<1>00-"Hence,(A-7)simplifies to (A-S) Thisequation hasthesolution (A-9) But",:.._,isjust"'00-'inreverseorder.Hence,thesolution in(A-9)issimplyam,in reverseordermultiplied by-km.Thatis, [am',m.,] d=-k Q,.,-lm-2,...-/ "I. a",_l 1 Thesecondequation obtained from(A-6)isthescalarequation <1>:"-,a m-,+<1>::,dm_,+cI>[O)k m=cI>(m)(A-IO) (A-Il) Weeliminate dm_,from(A-II)byuseof(A-lO).Theresulting equation givesuskm. Thatis, k=.p(m)-"'::-,a m-, mcP('O)-4»;,:-lc();;'~)4»;1/) cI>(m)-"':.-,am-, cI>(O)-a;,-,<1>00, .p(m)-"';•.,am_, '€m-J(A-12) APPENDIX A:THELEVINSON-DURBIN ALGORlTHM SSt where'l:",,istheresidualMSEgivenas '8.,-,~<1>(0)-a:"-,oI>m-1 (A-l3) Bysubstituting (A-lO)fordm-,in(A-4),weobtaintheorder-recursive relation arn",:::=am-1A-kmQm-lm.-b k=1,2,...,m-l, m=1.2.....p(A-14) and Theminimum MSEmayalsobecomputed recursively. Wehave m '8m=<1>(0)-2:am.<I>(k) A=I Using(A-14)in(A-IS),weobtain 'l",=<1>(0)-mi'am,,<I>(k)-amm[<I>(m) -"fa.._"m,<I>(k)] A=l ~~I(A-IS) (A-16) ButtheterminsquareQrackets in(A-16)isjustthenumerator ofk..in(A-12).Hence. (A-17) APPENDIXB ERROR PROBABILITY FORMULTICHANNEL BINARY SIGNALS Inmultichannel communication systems thatemploybinarysignaling fortransmitting information overtheAWGNchannel, thedecision variable atthedetector canbe expressed asaspecialcaseofthegeneralquadratic form I D=2:(AIX.!'+BIV,I'+ex,v:+C'X:V,j ,I(B·1) incomplex-valued gaussian randomvariables. A,B,andCareconstants; X,andY,are apairofcorrelated complex-valued gaussian random variables, Forthechannels considered, theLpairs(X..Y.Iaremutually statistically independent andidentically distributed. Theprobability oferroristheprobability thatD<0,Thisprobability isevaluated below, Thecomputation beginswiththecharacteristic funetion, denoted by1jJ,,(jv),ofthe generalquadratic form.Theprobability thatD<0,denoted hereastheprobability of errorPh'is p.=P(D<0)=ff(D)dD (B-2) wherep(D),theprobability densityfunction ofD,isrelatedto1jJ,,(jv)bytheFourier transform, i.e., 882 APPENDIX B:ERROR PROBABlLlTY FORMULTICHANNEL BINARY SIGNALS 883 Hence, ['1J- p.=dD-o/Jo(jv)e-I•Ddv _I»2n --<JO(B-3) Letusinterchange theorderofintegration andcarryoutfirsttheintegration with respecttoD.Theresultis 1["I'o/JD(iV)P,=-----dv 2trj -ao-"/l- V(B-4) whereasmallpositive numberehasbeeninsened inordertomovethepathof integration awayfromthesingularity atv=0andwhichmustbepositive inorderto allowfortheinterchange intheorderofintegration. SinceDisthesumofstatistically independent random variables, thecharacteristic function ofDfactorsintoaproductofLcharacteristic functions, witheachfunction corresponding totheindividual randomvariables d"where d,=AIX,I'+BIY,I'+CX,l1+C·X:Y, Thecharacteristic functionofd,is C) v,v, [v,v,(-v'a" +jva,,)l ,pd,IV(v+jv,)(v-jv,)exp[(v+jv,)(v-jv,)J(B-5) wheretheparameters v,,v"a",anda"dependonthemeansX,andV,andthe second(central) moments 1-'",I-'yy.and/L",ofthecomplex-vlaued gaussian variables X, andY,throughthefollowing definitions (lei'-AB>0): _~, 1VI-W+ 2 2 -w4(l-'ul-',. -11-',,1)(IC[-AB) ~, 1V2=W+ 2 2 +w4(l-'ul-'., -11-',,/)(ICI-AB) MI=Ap..u+BI-'u+CJ.L:~+c*1-';';.'1' 4(/Lxxl-'"-II-'",J')(IC[' -AB) 2 -2 -2-- --a,,=2(1C[ -AB)(IX.lI-',,+IY.1 /Axx-X:y'/LXy-X,Yt/L:') a"=AIX,I'+BIY.I'+CX:Y"+C·X,V: 1-'"=IE[(X,-X,)(Y,-V.)·](B-6) Now,asaresultoftheindependence oftherandomvariables d,.thecharacteristic functionofDis, o/JD(jV)=no/Jd,(iV)'-I wllere(v,v,)' v,v,(jva,-v'a,)]o/JD(jV) expI-c-'--'-'~~-'-"-'"(v+jv,)'(v-jv,)' v+jv,)(v-jv,)(B-7) (B-8) 884 DIGITAL COMMUNICATiONS Theresult(B-7)issubstituted foroP,,(jv)in(B-4),andweobtain (V,v')'fx ,,,, dv ~v,v,(jVO'-V'O')]!'=----- exp,21Cj-"'1'v(v+jv,)'(v-jv,)1. V+jv,)(u-jv,) Thisintegralisevaluated asfollows, Thefirststepistoexpressthe«ponential function intheform (jA,jA,)exp-A,+--,----:­v+fV,v-fv, whereonecaneasilyverifythattheconstants A,.A,.andAJare given as u~v~A,=---(o,v, +a,) VI+V2(B-9) (B-IO) Second, aconformal transformation ismadefromthevplaneontothepplanevia thechangeinvallable VIV-jtJ2P=----,- V2V+JVt Inthepplane,theintegralgivenby(B-9)becomes p,=exp[v,v,(-20v,v,+o,v,-o,v,)/(v,+v,)'1_1_rf)d • (1+V,/VI)2I. I 2TrjJr(pp where f[1+(v,/V,)pj'L-' [A,(v,/v,) AJ(v,!v,)IJ(p)= I. exp P+---""-'-'-=P(I-p) V,+v, V,+v,P andrisacircularcontourofradiuslessthanunitylhatencloses theorigin, Thethirdstepistoevaluate theintegral ~ff(p)dp=~f(1+(v,!v,)p)'1.-1 211]r 21CfrpL(l-p) [A,(v,lv,) A,(vt!v,)IJdxexp p+ p VI+112 U1+V2P(B-II) (B-12) (B-13) (B-14) Inordertofacilitate subsequent manipUlations, theconstants Q;;.0andb;;.0are introduced anddefinedasfollows: ,,_A,(v,!v,) i4- , VI+V2(B-15) (B-16) (B-18),-\PPENDIX IIFRROR PROHABILITY FORMlilTICHANNEl BINARY S\(iNALS 88S LetusalsoexpandthefunctionII+(UO!U,)PI" 'asabinomial series.Asaresult.we ootain,f"'(2L-1)(0')'21<j,t(p),Ip=~,ku; x-'-.I1p'exp(la'+lb'p)dp 21<J,.p(l-p) p Thecontour integralgivenIn(B-16)isonerepresentation oftheBesse'function. It canbesolvedbymakinguseoftherelations {2~jw··Lui.,exp(I;'+lh'p)dp I.,(ab)=Ib"J !,-.(-)p'"exp(,a+~b'p)dp 21<Jar P whereJ,,(x)isthenthordermodified Besselfunction ofthefirstkindandtheseries representation ofMarcum's Qfunction intermsofBesselfunctions, i.e., "(a)"Q,(a.b)=exp[-l(a'+b')]+LbI,,(ab) 'J~ll First,consider thecase0..k..L-2in(B-16).Inthiscase.theresulting contour integralcanbewrittenintheformt ](\a'',) , ', 1 I..(b)",)exp'----+,bopdp=Q,(a,b)exp[,(a-+b')]+L-I,,(ab) II-p P w'a (B,]7) Next,consider thetermk=L-I.Theresulting contour integralcanbeexpressed in termsoftheQfunction asfollows: 2~jip(]~p)expe;o+lb'p)dp=Q,(a.b)expn(a'+b')1 Finally.consider thecaseL..k..2L-1.Wehave 1fp'-I, ('la'')---,--exp-'-+lb-pdp 2JrJ,,]-P P "1J.(', ) =L-,p"L'"exp,a+lb'pdp ,,~u21<Jr p ,,~,tLW"I,,(ab) =Q,(a.b)exp[\(0'+b')J-~:W"I,,(ab) (B-19) Collecting thetermsthatareindicated ontheright-hand sideof(8·16)andusing tThiscontour integralisrelatedtothegeneralized Marcum Qfunction, definedas Qm(a,b)~fx(xla)m.' exp[-\(x'+a'))1",.,(ax)dx,m;>1 inthefollowing manner: Qm(a.b)exp[}(a'+b'»)=2~'Lpm(i'- p)exp(I;'+lb'p),Ip 886 DIGITAL COMMUNICATiONS theresultsgivenin(B-17)-(B-19), thefollowing expression forthecontourintegralis obtained aftersomealgebra: 1i()'L-' _.f(p)dp= \+v, [expH(a'+b'»)Q,(a,b)-/o(ab)j 2rrl'" v, +lo(ab)~~(2Lk-1)(::r L-'L-I-.(2L-l)[(b)"(V')' (O')"(V,)'L-'-']+LI.(ab)L-- --- """I k-OkaVIbVI(B-20) Equation (B·20)inconjunction with(B-12)givestheresultfortheprobability of error.Afurthersimplification resultswhenoneusesthefollowing identity, whichcan easilybeproved: rVIU2 ] l'2 2 )eXPL ,(-2a,v,v,+a,v,-a,v,) =exp[-,(a +b)(v,+v,) Therefore, itfoDowsthat p.=Q,(a,b)-lo(ab)exp[-Ha'+b'll ',,(ab)exp[-\(0'+b')]~'(2L-1)(V')' exp[-Ha'+b'»)+ 2LI£.+ /"L-I(1+v,/v,) ,~ok.'"(1+lI,.,)- x:~:I.(ob)L%.C\-l) (B.21) x[(~)"(:J-(~)tr"-'] (L>i) p.=Q,(a.b) (B-22)Thisisthedesiredexpression fortheprobability oferror.Itisnowasimplemailer torelatethepar.metersaandbtothemoments ofthepairs{K"Y,}.Substituting for A,andA,from(B-10)into(B-15),weobtain a=[2V~V2(alV2 -a2)]'" (v,+v,)' b=[2v,.~(alv, +a,l]'" (v,+v,)' Since11"v,.ai'andu,havebeengivenin(B-6)and(B-8)directlyintermsofthe moments ofthepairsX,andy.,ourtaskiscompleted. APPENDIXC ERROR PROBABILITIES FORADAPTIVE RECEPTION OFM-PHASE SIGNALS Inthisappendix, wederiveprobabilities oferrorfortwo-andfour-phase signaling over anL-diversity-braneh time-invariant additive guassian noisechannel andforM-phase signaling overanL-diversity-branch Rayleigh fadingadditive gaussian noisechannel. Bothchannels corruptthesignaling waveforms transmitted through thembyintroduc­ ingadditive whitegaussian noiseandanunknown orrandom multiplicative gainand phaseshiftinthetransmitted signal.Thereceiver processing consists ofcross­ correlating thesignalplusnoisereceived overeachdiversity branchbyanoisy reference signal,whichisderived eitherfromthepreviously received information­ bearingsignalsorfromthetransmission andreception ofapilotsignal,andaddingthe outputsfromallL-diversity branches toformthedecision variable. C-IMATHEMATICAL MODEL FORANM-PHASE SIGNALING COMMUNICATIONS SYSTEM InthegeneralcaseofM-phase signaling, thesignaling waveforms atthetransmitter aret tThecomplex representation ofrealsignalsisusedthroughout. Complex conjugatiot1 IS denotedbyanasterisk. 887 '888 DIGITAl. COMMUNICATIONS where [.21r ]s,.(t)=g(l)exp}M(n-I),n=I,Z,.",M,0'"I';;:T (C-I) andTisthetimeduration ofthesignaling interval. Consider thecaseinwhichoneoftheseMwaveforms ist.ansmitted. forthe duration ofthesignaling interval, overLchannels. Assume thateachofthechannels corrupts thesignaling waveform transmitted throughitbyintroducing amultiplicative gainandphaseshift,represented bythecomplex-valued numbergk,andanadditive noiseZk(e).Thus,whenthetransmitted waveform iss,.(e),thewaveform received over thekthchannelis ",(I)=gkS~(t)+z,(e), O';;:I,;;:T, k=I,Z,...,L (C-Z) Thenoises{z,(e)}areassumed tobesamplefunctions ofastationary whitegaussian randomprocesswithzeromeanandautocorrleation function c/>,(r)=NoS(f),whereNo isthevalueofthespectraldensity.Thesesamplefunctions areassumed tobemutually statistically independent. Atthedemodulator. ",(e)ispassedthrough afilterwhoseimpulse response IS matched tothewaveform g(e).Theoutputofthisfilter,sampled attimee=T,is denoted as (C-3) wheregisthetransmitted signalenergyperchannelandN,isthenoisesamplefrom thekthfilter.Inorderforthedemodulator todecidewhichoftheMphaseswas transmitted inthe'signaling interval 0,;;:e,;;:T,itattempts toundothephaseshift introduced byeachchannel. Inpractice, thisisaccomplished bymultiplying the matched filteroutputX.by thecomplex conjugate ofanestimateg.ofthechannelgain andphaseshift.Theresultisaweighted andphase-shifted sampled outputfromthe kth-channel filter,whichisthenaddedtotheweighted andphase-shifted sampled outputsfromtheotherL-Ichannelfilters. Theestimateg,ofthegainandphaseshiftofthekthchannel isassumed tobe derivedeitherfromthetransmission ofapilotsignalorbyundoing themodulation on theinformation-bearing signalsreceived inprevious signaling intervals. Asanexample oftheformer,suppose thatapilotsignal,denoted bys",,(e),0".t'"T,istransmitted overthekthchannelforthepurpose ofmeasuring thechannelgainandphaseshift. Thereceived waveform is g,Spk(f)+zp,(t),0".e'"T wherezp,(e)isasamplefunction ofastationary whitegaussian random processwith zeromeanandautocorrelation function c/>p(f)=Noli(f).Thissignalplusnoiseispassed throughafiltermatched toSp,(f).Thefilteroutputissampled attimef=Ttoyieldthe randomvariableXp,=2i!pg,+N"",wherei!'pistheenergyinthepilotsignal,whichis assumed tobeidentical forallchannels, andNp,istheadditive noisesample.An estimate ofg,isobtained byproperly normalizing Xp"i.e.,g,=g,+Np,/Z'Cp- Ontheotherhand,anestimate ofg.canbeobtained fromtheinformation-bearing signalasfollows.Ifoneknewtheinformation component contained inthematched filteroutputthenanestimate ofg,couldbeobtained byproperly normalizing this APPENDIX C:ERROR PROBABILITIES FORADAPTIVE RECEPTION OFM-PHASE SIGNALS889 output.Forexample, theinformation component inthefilteroutputgivenby(C-3)is 2'l:g,exp[j(2JrIM)(n-I)].andhence,theestimate is whereN;=N,exp[-j(2JrIM)(n-Il]andthepdfofN;isidentical tothepdfofN,. Anestimate thatisobtained fromtheinformation-bearing signalinthismanner is calledac1airooyant estimate. Although aphysically realizable receiver doesnotpo>sess suchclairvoyance, itcanapproximate thisestimate byemploying atimedelayofone signaling intervalandbyfeedingbacktheestimate ofthetransmitted phaseinthe previous signaling inlerval. Whether theestimateofg,isobtained fromapilotsignalorfromtheinformation­ bearingsignal.theestimate canbeimproved byextending thetimeintervaloverwhich itisformedtoincludeseveralpriorsignaling intervals inawaythathasbeendescribed byPrice(1962a, b).Asaresultofextending themeasurement interval, the signal-to-noise ratiointheestimate ofg,isincreased_ Inthegeneralcasewherethe estimation intervalistheinfinitepast,thenormalized pilotsignalestimateis (C-4) where C,istheweighting coefficient onthesubestimate ofg,derivedfromtheithprior signalintervalandNp"isthesampleofadditivegaussian noiseattheoutputofthefilter matched to5p,(t)intheithpriorsignaling interval. Similarly. theclairvoyant estimate thatisobtained fromtheinformation-bearing signalbyundoing themodulation over lheinfinitepastis (C-S) Asindicated, thedemodulator formstheproductbetweeng:andX,andaddsthisto theproducts oftheotherL-Ichannels. Therandomvariable thatresultsis L L Z=2:X,g:=LX,Y: k=I k"I =Zr+jz; (C-6) where,bydefinition, Y,=g"Z,=Re(z).andz,=1m(z).Thephaseofzisthedecision variable. Thisissimply (C-7) C-2CHARACTERISTIC FUNCTION ANDPROBABILITY DENSITY FUNCTION OFTHEPHASE 6 Thefollowing derivation isbasedontheassumption thatthetransmitled signalphase IS zero,i.e.,n=1.Ifdesired, thepdfof9conditional onanyothertransmitted signal phasecanbeobtained bytranslating p(9)by theangle2Jr(n-1)1M.Wealsoassume 890 DIGITAL COMMUNICATIONS thatthecomplex-valued numbers{g.l.whichcharacterize theLchannels, aremutually statistically independenl andidentically distribuled zero-mean gaussian random vari­ ables.Thischaracterization isappropriate forslowlyRayleigh fadingchannels. Asa consequence, therrand0m variables (X,.Y,)arecorrelated, complex-valued, zero­ mean,gaussian, andstalistically independent, butidentically distributed withanyother pair(X"~). Themethod thathasbeenusedinevaluating theprobability densityp(9)inthe generalcaseofdiversity reception isasfollows.First,thecharacteristic function ofthe jointprobabilily distribution function ofz,andz"whereZ,andz,are,wocomponents thatmakeupthedecision variablee,isobtained. Second, thedoubleFouriertransform ofthecharacteristic function isperformed 'andyieldsthedensityp(z"z·JThenthe transformation r=v'z;+zi.9=tan'U')z,(C-8) yieldsthejointpdfoftheenvelope randthephase9.Finally,integration ofthisjoint pdfovertherandomvariable ryieldsthepdfof9. Thejointcharacteristic functionoftherandomvariablesz,andz,canbeexpressed in theform [4 ",('V'v)= m"m,,(1-IJLI')'1'1,,12 211 2.JLcosev,-J :2(v'mumw(1-IILIJ ]L .2IJL:sinf: 4 +",-1y;;;-;;;- ,+ -,(m"m,)1-IILI») m"m,,(1-IILI') where,bydefinition,(C-9) m,.=E(lX,I') m,.=E(IY.I') m"=E(X,}1')identical forallk identical forallk identical forallk (C-IO) TheresultofFourier-transforming thefunction t/J(}v"jv,)withrespect tothe variables ",andv,is ()_(I-IILI')/. (v?+"?)L" PZ"z,-(L_1)!lr2' z,+Z, Xexp[IJLI(z,.case+z,sine)]K/._,(v' z;+zi) (C-II) whereK,,(x)isthemodifiel! Hankelfunction ofordern.Thenthetransformation of random variables, asindicated in(C-8)yieldsthejointpdfoftheenvelope Tandthe phase9intheform (C-12) (C-13) (C-14)-\PPESDIX C:ERROR PROBABILITIES FORADAPTIVE RECEPTION OFM·PHASE SIGNALS 891 Now,integration overthevariable, yieldsthemarginal pdfofthephaseII.Wehave evaluated theintegraltoobtainp(8)intheform (- I)I'(\-IJ.LI')'{a'-'[ I p(O)=2Jr(L-I)! ab',b-1J.LI'cos' (0-E) !J.LICOS(O-E), (1J.L1COS(0-E»)]}I.+" -.,-.COS , I[b-!J.LIcos(11-E)I"" b" , Inthisequation, thenotation denotestheLlhpartialderivative ofthefunctionI(b,J.L)evaluated atb=I. C·3ERROR PROBABILITIES FORSLOWLY RAYLEIGH FADING CHANNELS Inthissection,theprobability ofacharacter errorandtheprobability ofabinarydigit errorarederived forM-phase signaling. Theprobabilities areevaluated viathe probability densityfunction andtheprobability distribution function of8. ThePrObability Distribution Function ofthePhaseInordertoevaluate the probability oferror,weneedtoevaluate thedefiniteintegral P(8,..8..8,)=f'p(8)d8• ", whereIi,and8,arelimitsofintegration andp(8)isgivenby(CU).Allsubsequent calculations aremadeforarealcross-correlation coefficient J.L.Arcal-valued J.Limplies thatthesignalshavesymmetric spectra.Thisistheusualsituation encountered. Sincea complex-valued J.Lcausesa shiftofEinthepdfof8,i.e.,Eissimplyabiasterm,the resultsthataregivenforrealJ.Lcanbealteredinatrivialwaytocoverthemoregeneral caseofcomplex-valued J.L. Intheintegration ofp(8),onlytherange0..9..Jrisconsidered, becausep(8)isan evenfunction. Furthermore. thecontinuity oftheintegrand anditsderivatives andthe factthatthelimits8,and8,areindependent ofballowfortheinterchange of integration anddifferentiation_ Whenthisisdone,theresulting integral canbe evaluated quitereadilyandcanbeexpressed asfollows: fe, _(-l)I··'(I-J.L')' ).,p(lI)de-2Jr(L-I)! ilL-'{I[J.LVI-(b/J.L'-I)x' xabL-'b_J.L' b'" cot'x (xb'''J.L)]}"I --eot-I VI-(b/J.L'-I)x' ,,'~' where,bydefinition, -J.LcosIIix- ;=1,2•-Vb-J.L'cos8,' .(C-IS) 892 DIGITAL COMMUNICATIONS Probability ofaSymbol ErrorTheprobability ofasymbolerrorforanyM-phase signaling systemis When(C-14)isevaluated atthesetwolimits,theresultis (_l)L-I(I_ ....')LaL-'{ I[tr ~- ----(M-I)"'-tr(L-I)! abL-'b-I-<'M ....sin(trIM) -'(-cos(trIM))]}I -y'b-....2co,?(trIM) cotv'b-'cos'(tr/M) 0_'(C-16) (C-I?) (C-18)Probability oraBinaryDigitErrorFirst,letusconsider two-phase signaling. In thiscase,theprobability ofabinarydigiterrorisobtained byintegrating thepdfp(8) overtherangeltr<8<3tr.Sincep(8)isanevenfunction andthesignalsareapriori equallylikely,thisprobability canbewrittenas P,~2(p(8)d8 Itiseasilyverifiedthat8,=ltrimpliesx,=0and8,=1rimpliesx,=p./~. Thus, (_I)L-'(I_I-<')L aL-l[I 1-<]/ P,=2(L-I)! abL-'b-....'-b'l2(b-IL') h-' Afterperforming thedifferentiation indicated in(C-17)andevaluating theresulting function atb=I,theprobability ofabinarydigiterrorisobtained intheform I [ L-'(2k)(1-1-<')']P,=-1-1L2 -- 2'-0k 4 Next,weconsider thecaseoffour-phase signaling inwhichaGraycodeisusedtomap pairsofbitsintophases.Assuming againthatthetransmitted sjgnalisstI(r),itisclear thatasingleerroriscommitted whenthereceived phaseis~tr<9<~1r,andadouble erroriscommitted whenthereceived phaseis~1r<9<tr.Thatis,theprobability ofa binarydigiterroris p..=r~(lJ)d8+2(p(e)d8 Itiseasilyestablished from(C-14)and(C-19)that (_l)L-I(l_ ....')LaL-'[ I....]1 p..=2(L-l)f abL'b-p.'-(b-....')(2b-....')'12h-' Hence,theprobability ofabinarydigiterrorforfour-phase signaling is _I[ ....L-'(2k)(I+P.')'] p.h-21-~,~ k4-2/l'(C-19) (C-20) Notethatifonedefinesthequantity p=..../~, theexpression forp..interms ofpis (C-21) APl'l:-.lDIX CEKROR PROBAtilLlTIFS FORADAPTIVE RECEPTION OFM·i'HASE SI(iNAl.S 893 Inotherwords.P",hasthesameformasP,givenin(C-18).Furthermore. notethatp. justlikeJL.canbeinterpreted asacross-correlation coefficient, sincetherangeofpis 0'"p'"Ifor0'"JL'"l.ThissimplefactwillbeusedinSectionC-4. Theaboveprocedure forobtaining thebiterrorprobability foranM-phasesignal withaGraycodecanbeusedtogenerate resultsforM;8.16,etc.•asshownby Proakis(1968). Evaluation oftheCross-Correlation Coeficient Theexpressions fortheprob­ abilitiesoferrorgivenabovedepend onasingleparameter, namely, thecross­ correlation coefficient JL.Theclairvoyant estimate isgivenby(C-5),andthematched filteroutput,whensignalwaveform S,,(I)istransmitted, isX,;2'tg,+N,.Hence.the cross-correlation coefficient is where,bydefinition.VCr,'+I)(-y;'+v) v;I*.Cil'/#,1<.1' --,gE(2)kI y,-AI19d,;.2,...,L ;v"(C-22) (C-23) Theparameter vrepresents theeffective numberofsignaling intervals overwhichthe estimate isformed.and-y,istheaverageSNRperchannel. Inthecaseofdifferential phasesignaling, theweighting coefficients are<,;J,c,=0 fori..I.Hence,v;IandJ.L;-Y"!(l+-y,). WhenV;::C,theestimate isperfectand I·~"ImJL=-- "--ox )I,+1 Finally.inthecaseofapilotsignalestimate. givenby(C-4)thecross-correlation coefficient is _[(r+1)(r+1)]'''' J.L-1+-_-1+-.- "'1, Y"Y, where.bydefinition. .'t,E(')"f,;N u!gkt ';g,;g+'ll' r;grc" ThevaluesofJLgivenabovearesummarized inTableC-1. C-4ERROR PROBABILITIES FORTlME.INV ARIANT ANDRICEAN FADING CHANNELS(C-24l InSectionC-2.thecomplex-valued channelgains{g.}werecharacterized aszero-mean gaussian random variables, whichisappropriate forRayleigh fadingchannels. Inthis section, thechannel gains{g,}areassumed tobenonzero-mean gaussian random variables. Estimates ofthechannel gainsareformedbythedemodulator andareused 894 DIGITAL COMMUNICATiONS TABLE C-IRAYLEIGH FADING CHANNEL Typeofestimate Clairvoyant estimateV'(y,-'+I)(j', '+v) vTv Pilotsignalestimate DifferentlOl phasesignaling(r+I)~(t.+r;I)(t,+r:,) -.!r...- 10:+1 Perfectestimate asdescribed inSectionCol.Moreover, thedecisionvariable 9isdefilledagainby(C-7). However, inthiscase,thegaussian randomvariablesX.andY••whichdenotethe matched filteroutputandtheestimate, respectively, forthekthchallnel. havenonzero means,whicharedenoted byX.andY•.Furthermore, tltesecondmoments are mxx=£(lX.-X.n identical forallchannels-,myy=£(1Y.-Y.I) identical furallchannels m,y=£[(X.-X.)(Yt-Yt)]identical forallchannels andthenormalized covariance isdefinedas (C-2S) (L'"2)Errorprobabilities aregivenbelowonlyfortWO-andfour-pbase signaling withtbis channelmodel.Weareinterested inthespecialcaseinwhichthefluctuating component ofeachofthechannelgains{g.}iszero,sothatthechannels aretime-invariant. If,in additiontothistimeinvariance, thenoisesbetween theestimate andthematched filter outputareuncorrelated thenI'-=O. Inthegeneralcase,theprobability oferrorfortwo-pbase signaling overL statistically independent channels characterized intbemannerdescribed abovecanbe obtained fromtheresultsinAppendiK B.Initsmostgeneralform,tbeexpresssion for thebinaryerrorrateis P,=Q,(a,b)-/o(a)eKp[-Ha'+b'») +lo(ab)expH(~: ~b'))I'(2L-1)(1+1'-)'[2/(1-1'-)] '-0kI-I'- exp[-Ha'+b')] +{2/(1-I'-)]2L1 L-I L-'-.(2L-1)[(b)·(1 +1'-)'(a)"(1+I'-)'L-l-'] XLI.(ab)L - - - - -,_,'-0ka1-IJbI-p.. P,=Q,(a,b)-W+p..)/o(ab) exp(-~(a'+b'»)(L=I) (C-26)APPENDIX CERROR PROBABILITIES FORADAPTIVE RECEPTION OFM·PHASE SIGNALS 89S where,bydefinition, (LIXY,I')'" (J-1L-'----'- -1-'<=1vm:vm: b-(l±I~+ll')'" -2t=1vm:vm: Q,(a,b)=rxexp[-Ha'+x')]Jo(ax) dx I..(x)isthemodified Besselfunction ofthefirstkindandofordern. Letusevaluate theconstants aandbwhenthechannelistime-invariant, p.=0,and thechannelgainandphaseestimates arethosegiveninSectionC-1.Recallthatwhen signals,(r)istransmitted, thematched filteroutputisX,=2'€g,+N,.Theclairvoyant estimate isgivenby(C-S).Hence, for thisestimate, themoments areX,=2't!g" Y,=go,m...=4t:N'handm....=NoIt:v,where'iisthesignalenergy, Noisthevalueof thenoisespectraldensity,andvisdefinedin(C-23).Substitution ofthesemoments into (C-26)resultsinthefollowing expressions foraandb: a=vrr;1\Iv-II b~vrr;1\Iv+11 (C-27) Thisisaresultoriginally derivedbyPrice(1962). Theprobability oferrorfordifferential phasesignaling canbeobtained bysetting v=Iin(C-27). Next,consider apilotsignalestimate. Inthiscase,theestimate isgivenby(C-4)and thematched filteroutputisagainX,=2'ig,+N,.Whenthemoments arecaiculated andthesearesubstituted into(C-26),thefollowing expressions foraandbare obtained: (C-28) where t:I. 1',=N:,~19,I' 'i€,='i+'ip r='i/'fp Finally,weconsider theprobability ofabinarydigiterrorforfour-phase signaling overatime-invariant channelforwhichthecondition p.=0obtains.Oneapproach that canbeusedtoderivethiserrorprobability istodetermine thepdfof9andthento integrate thisovertheappropriate rangeofvaluesof9.Unfonunately, thisapproach provestobeintractable mathematically. Instead,asimpler, albeitroundabout, method maybeusedthatinvolves theLaplacetransform. Inshort,theintegral in(14-4-14) of thetextthatrelatestheerrorprobability P,('Yh)inanAWGNchanneltotheerror 896 DIGITAL COMMUNICATIONS TABLE C·2TIME-INVARIANT CHANNELI Typeofestimate a b Twn.pbase siguali.. Clairvoyantv'fY.1V;;- II V"!Y.(V;;+1)estimate Differential phase0 vz:y,:signaling Pilotsignal~l~ ~I ~(~+[h) estimate "2r+I-\lr+1 Fnur.phase sil"a1iDg Clairvoyant v1Y.1v'v+ 1+V7+J v1Y.( y;;-::-~-v;;r+J estimate -v'v+I -v'V'"+l1 +v'v+1-v?+!) Differential phasev'fY.(v'2+ V2-v'2-\/2) v1Y.(v'2+V2+v'2-V2)signaling PilorsignalVr,Iv'v+r+vv'+r' ~r,(v'v+r+v'7+?estimale 4(r+I) 4(r+1) -v'v+r-v'V'+?1 +v'v+r-v:vr+?) probability P,inaRayleigh fadingchannel isaLaplacetransfo1Tl1. Sincethebiterror probabilities P,andp••foraRayleigh fadingchannel, givenby(C-18)and(C-21), respectively, havethesameformbutdifferonlyinthecorrelation coefficient, itfollows thatthebiterrorprobabilities forthetime-invariant channelalsohavethesameform. Thatis,(C-25)withIJ.=0isalsotheexpression forthebiterrorprobability ofa four-phase signaling systemwiththeparameters aandbmodified toreflectthe difference inthecorrelation coefficient. Thedetailed derivation maybefoundinthe paperbyProakis(1968).Theexpressions foraandbaregiveninTableC-2. ( I APPENDIXD SQUARE-ROOT FACTORIZATION Consider thesolutionofthesetoflinearequations RNC.N=U, (D-l) whereRNisanNXNpositive-definite symmetric matrix,CNisan.IV-dimensional vector ofcoefficients tobedetermined, andUNisanarbitrary N-dimensional vector.The equations in(D-I)canbesolvedefficiently byexpressing RNinthefactored form (D-2) whereSNisalowertriangular matrixwithelements is,,}andONisadiagonal matrix ..ithdiagonal elements {d,}.Thediagonal elements ofSNaresettounity,i.e.,s"=J. Thenwehave 'ij=!Si/"d/tS,b 1~j:0:.;i-].j~2 k=I (D-3) where{r,Jartetheelements ofR,.Consequently, theelements {s,,}and{d,}are determined from(D-3)according totheequations j-l s'id,='if-2:Sikd"Sfk' k=I (D-4) ,-I di='Ii-Ls7"dk.'2~i!!f:N Ie=I Thus,(D-4)defineSNandONintermsoftheelements ofR.,. 897 898 naiITAl. ('()\I\I(-"'h'Aml~S Thesolution to(D-I)isperrormcd intwosteps.With(D-2)suhstituted into(D-I) wehave Let I Y"=D...,S',C, Then S,Y,=U, Firstwesolw(D-6)forY,.Becauseofthetriangular formofS,.wehave ,, y,=II,-LS".\',.,, Havingobtained Y,.thesecondstepistocompute C,.Thatis. D,S\-C, =Y, S'"C,=D,Iy, Beginning with c,::::.v....Jd, theremaining coefficients ofC,areohtained recursively asfollows:(D-5) (D-6) (D-7) (D-8) '''''i$,N-1 (D-9) Thenumberofmultiplications anddivisions required toperform thefactorization of K,.isproportional toN'.Thenumher ofmultiplications anddivisions required to compute C,.onceS,isdetermined, isproportional toN'.InCOnlrast. whenR.,is Toeplitz theLevinson-Durbin algorithm shouldbeusedtodetermine thesolution of (D-I),sincethenumherofmultiplications anddivisions isproportional toN'.Onthe otherhand,inarecursive least-squares formulation_ S,andDNarcnotcomputed asin (D-3),buttheyareupdated recursively. Theupdateisaccomplished withN'operations (multiplications anddivisions). Thenthesolution forthevectorC.,followsthesteps (D-5)-(D-9). Consequently. thecomputational burdenoftherecursive least-squares formulation isproportional toN'. , c: f REFERENCES ANDBIBLIOGRAPHY Ahcnd. KandFritchman. B.D.(1\)70)."Statistical Detection forCommunication Channels with Intersymhol Interlen:nce."Proc lEEf.-,pp.779-7X5. May. Anramson. N.(ll)h3).Infi,rmalilm Theorytl/u/Coding. McGra\\'-Hill, NewYnrk. Ahramson. ~.(1l.)70)."TheALOHA Syst~m-AnOlh~r Alternative forCompukI Communica~ tions."/fJ70FallJoinlCOlllpllf. COllI,AFIDSConI:Pmc,vo!.37.pp.2XI-2S5. AFIPSPress. Montvak'. N.J. Anramson. N.lll)77). "Tht:Throughput ofPacket Broadcasting Channels:' 1/:'f.:1::TrailS. CommitI/., vol.COM-:!). pp.1l7-12K, ];muary. Ahramson. N.()ij(.}J).Multiple AcCl's.\ComnlUlliCaliOlH. IEEEPress,l'\ewYork. Adler.R.L..Coppersmith. D..andHassner. M.(19H3)."Algorithms forSliding BlockCodes'" ,,.:EETrans.InformTheory. vol.IT-29.pp_5-22.January. AI-Hussaini. E.andAI-Bassiouni (J<:JH5)."Performance ofMRCDiversity Systems forthe Dekction ofSignals withNakagami Fading:' IEFFTrans.Commun.. \'01.COM-]."'. pp. 1315-1319. Decemhn. Altekar. S,A.andBeaulieu. N.C.(I9Y3)."Upper Bounds ontheErrorProoability ofDetision Feedhack Equalizution:' IFEETrans.I,,{orm. Theory, \"01.IT-39.pp.145-156. January. Anderoerg. M.R.(1973).ClusrerAnalysis forApp'ifllthms, AC<'H.lemic. NewYork. Ander'\on. J.B..Aulin.T..andSundberg. C.W.(I9X6).DiKiwlPIIlI.n:Muduimiotl. Plenum. New York. Anuerson. R.R.andSail.J.(1\165)."Spectra ofDigitalFM'"8ell5",,1.Tech.1..vol.44pr. Ilo5-IIH9. Julv-August. Ash.R.B.(IY65j.Iliforl11arioll Theon. Interscicnce. NewYork. Aulll1.T.(IQHO)."Viterbj Detection ofContinuous PhaseModulated Signals," ,Vat.felecommllfl. ConIRecord. pp.}..t2.1-14.2.7. Houston. Texas.Novemher. Aulln.T..Rydbcck. N..andSundberg. C.W.(I9HI)."Continuous PhaseModulation-Part II: PartialResponse Signaling," IEEETram.Commlt"-. \'01.COM~29. pp.210-225. March. Aulln.T.andSandberg. C.W.(19HI)."Continuous PhaseModulation-Part I:FullRe,ponse Signaling." ItEETrot/s.COmmllll., vol.COM-29. pp.196-209. March. Aulin.T.andSundherg. CW.(19H2a). "OntheMinimum Euclidean Distance foraClassof SignalSpaceCodes,"IEEETrans.Inform.Thfury. vol.IT-28.pp.43-55.January. 899 900 DlGlTAL CUMMUNICATIONS Aulin,T.andSundberg, C.W.(1982b). "Minimum Euclidean Distance andPowerSpeclTurn fora ClassofSmoothed PhaseModulation CodeswithConstant Envelope," IEEETrans.Commltn .. vol.COM-3D, pp.1721-1729, July. Aulin,T.andSundberg, C.W.(1984)."CPM-An Efficient Constant Amplitude Modulation Scheme,"lnl. J.Sate/lite Commltn. vol.2,pp. 161-186. Austin~ M.E.(1967)."Decision-Feedback Equalization forDigitalCommunication Over Dispersive Ch3nnels," MITLincoln Laborarory. Lexington, Mass..Tech.Repor1No.437, August Barrow, B.(1963)."Diversity Combining ofFadingSignalswilhUnequal MeanSlrengths," IEEE Trans.Comml/n., SYS/..vol.CS-ll,pp.73-78,March. Beaulieu, N.C.(1990)."AnInfiniteSeriesfortheComputation oftheComplemenlary Probabilily Distribution Function ofaSumofIndependent Random Variables andItsApplication tothe SumofRayleigh Random Variables." IEEETrans.Commltn.. vol.COM-38, pp.1463-1474, September. Beaulieu, N.C.andAbu-Dayya. A.A.(1991)."Analysis ofEqualG.inDiI'ersity 01.Nakagami FadingChannels," IEEETrans.Commit"" vol.COM·39, pp.225-234, February. Bebr.N.E..Scholtz.R.A..andWelch.L.R.(1978)."Parlial-Period Correlation Properties ofPN Sequences." 1978Nal.Telfcomml/n. Can!Record,pp.35.1.1-25.1.4. Birmingham, Alabama, November. Belfiore, C.A.andPark.J.H"Jr.(1979)."Decision-Feedback Equalization." Proc-IEEE,vol. 67,pp.1143-1156. August. Bellini,J.(1986)."Bussgang Techmques forBlindEqualization," Proc.GLOBECOM'86, pp. 46.1.1-46.1.7, Houston, Texas,December. Bello,P.A.andNelin,B.D.(1962a)."Predetection Diversity Combining withSelectivity Fading Channels,"IRE Trans.Commun. Syst.,vol.C5-IO,pp.32-42.March. Bello,P.A.andNelin.B.D.(1962b). "TheInfluence ofFadingSpectrum ontheBinaryError Probabilities ofIncoherent andDifferentially Coherent Matched FilterReceivers," IRETrans. Commun. Syst.,vol.CS-IO,pp.160-168. June. Bello,P.A.andNelin.B.D.(1963)."TheEffectofFrequency Selective FadingontheBinary ErrorProbabilities ofIncoherent andDilferentially Coherent Matched FilterReceivers," IEEE Trans.Commun. SySl..vol.CS-ll,pp.170-186. June. Bennett, W.R.andDavey,J.R.(1965).DataTransmission, McGraw-Hili, NewYork. Bennett, W.R.andRice,S.O.(1963)."Spectral Density andAutocorrelation Function, Associated withBinaryFrequency-Shift Keying," Be/ISyst.Tech.J.,vol.42,pp.2355-2385. September. Benveniste, A.andGoursat, M.(1984)."BlindEqualizers." IEEETrans.Commun., vol. COM-32, pp.871-883,August. Berger,T.(1971).RateDistortion Theory,Prentice-Hall, Englewood Cliffs,N.J. Berger,T.andTufts,D.W.(1967)."Optimum PulseAmplitude Modulation. PartI:Transmitter· Receiver DesignandBoundsfromInformation Theory," IEEETrans.Inform.Theory.vol. IT-13,pp.196-208. Bergmans, P.P.andCover,T.M.(1974)."Cooperative Broadcasting," IEEETrans.Inform. Theory,vol.IT-20,pp.317-324, May. Berlekamp. E.R.(1968).Algebraic CodingTheory.McGraw-Hill, NewYork. Berlekamp, E.R.(1973)."GoppaCodes,"IEEETrans.Inform.Theory,vol.IT-19,pp.590-592. Berlekamp, E.R.(1974).KeyPapersintheDevelopmem ofCodingTheory,IEEEPress,New York. Bierman, G.1.(1977).FactoriZQtion Methods forDiscrete Sequential Estimation, Academic, New York. Biglieri,E.,Divsalar, D.,McLane, P.J.,andSimon,M.K.(1991).Introduction 10Trellis-Coded ModulQtion withApplications, Macmillan, NewYork. Bingham, J.A.C.(1990)."Mullicanier Modulation forDataTransmission: AnIdeaWhoseTime HasCome,"IEEEComm"n. Mag.,vol.28,pp.5-14,May. BIJhut,R.E.(1l)~3).lh(,(dyand l~racD{'f'offrrorControl Code\,Ad<.h...on-\\\: ...lcy.Reading. Mas,;. Blahut,R.E.(PlX?).PrinClple,\' willPrtldiceofInformution Theor}', Addi...on-Wesley, Rtading. Mass. Bose.R.C.andRay-Chaudhuri, D.K.(1%Oa). "OnaClassofErrorCorrecting, BinaryGroup Codes." Itl[orm.Control. vol.3.pp.68-79.March. Bose.R.C.andRay·Chaudhuri. D.K.(196Gb). "Further Result!!:inErrorCorrecting Binary (:iroupCodes." Inform.Control, vol.3.pp.279-29(J, September. Brennan. D.G.(IY59)."Linear Diversity Combining Techn.iques,'· Proc.IRE., \'01.47,pp. ·1075-1102. Busgang.l. J.(llJ52)."Crosscorrclati,m Functions ofAmplitude-Dis.tored Gaussian Signats," M1T RLETech.Report216. Bucher, E.A.(l9kO)."Coding Options forEfficient Communications on~on·Slatlonary Channels." Rec.IEtEInt.Con!Commun ..pp.4.1.1-4.1.7. Burton.H.0.(1%9)."AClassofAsymptotically Optimal BurstCorrecting BlockCodes," Proc. Ieee.Boulucr. Col..June. BUlO,A..Gruy.A.H.,Jr.,Gray.R.M.,andMarkel,1D.(1980)."Speech CodingBasedUpon VectorQuantization:' IEEETrans. ACOUJt., Speech, SignalProcesj';ng. Vol.ASSP·28 pp. 562-574,OCloher. Cahn.CR.(1960)."Combined DigitalPhaseandAmplitude Modulation Communication Systems." IRI-.Trans.Commun. Syst..vol.CS-8,pp.150-155, September. Calderhank, AR,andSloane, KJ.A(1987)."NewTrellisCodesBasedonLattices and Cosels," IEEETrans.Inform.Theory.vol.IT-33,pp.177-195, March. Campanella, S.J.artdRobinson, G.S.(1971)."AComparison ofOrthogonal Transformations for DigitalSpeechProcessing," IEEETrans.Commun., voJ.COM·19. pp.1045-1049, December. Campopiano, CN.andGlazer, B.G.(1962)."ACoherent DigitalAmplitude andPhase Modulation Scheme," IRETrans.Commun. Syst.,vol.CS-IO,pp.90-95,June. Capetanakis, J.I.(1979)."TreeAlgorithms forPacketBroadcast Channels," IEEETrans.Inform. Theory.voI.IT·25,pp. 505-515, September. Caraiscos. CandLiu,B.(1984)."ARoundoff ErTOrAnalysis oftheLMSAdaptive Algorithm," IEEETransAcourt.,Speech,SignalProcessing, Vol.ASSP·32, pp.34-41,January. Carayannis, G.,Manolaleis, D.G.,andKalouptsidis, N.(1983)."Af.stSequential Algonthm for Least-Squares Filtering andPrediction," IEEETrans.Acoust.,Sptech,SignalProcessing, Vol. ASSP·3I, pp.1394-1402, December. Carayannis, G.Manolakis, D.G.,andKalouptsidis, N.(1986)."AUnifiedViewofParametric Processing Algorithms forPrewindowed Signals," SignalProcessing, vol.10,pp.335-368, June. Carleial, A.B.andHellman, M.E.(1975)."Bistable Behavior ofALOHA·Type Sl"'tems," IEEE Trans.Commun" vol.COM·23, pp.401-410, April1975. Carlson, A.B.(1975).Communication Systems, McGraw·HiII, NewYork. Chang,D.Y.,Gersho,A,Ramamurthi, B,andShohan, Y.(1984)."fastSearchAlgorithms for VectorQuantization andPatternMatching," Proc.IEEEInt.ContAcoust., Speech,Signal Processing, paper9.11,SanDiego,CaJif.,March. Chang,R.W.(1966)."Synthesis ofBand-Limited Orthogonal SignalsforMultichannel Dat. Transmission," BellSyst.Tech.J.•vol.45,pp.1775-1796, December. Charash, U.(1979)."Reception Through Nakagami fadingMultipath Channels withRandom Delays," IEEETrans.Commun., vol.COM-27, pp.657--ti70, April. Chase,D.(1972)."ACJassofAlgotlthms forDecoding BlockCodesWithChannel Measurement Information," IEEETrans.Inform.Theory,vol.IT-1S,pp.170-182, January. Chase,D.(1976)."Digital SignalDesignConcepts foraTime-Varying RiceanChannel," IEEE Trans.Commun., vol.COM-24, pp.164-172, february. Chien,R.T.(1964)."CyclicDecoding Procedures forBCHCodes,"IEEETrans.Inform.Theory, vol.IT-I0,pp.357-363, October. 902 mOrrAl. COMMUNICATIONS Chow,J.S.•Tu,J.C,andCioffi.J.M.(1991)."ADiscrete Multitone Transceiver Systemlor HDSLApplicalions," IEEE1.SelectedAreasCQmmun, vol.SAC-9,pp,895-908, August. Chyi,G.T.,Proakis, J.G.,andKeller,CM.(19&8)."Diversity Selection/Combining Schemes withExcessNoise·Only Diversity Receptions OveraRayleigh-Fading Multipath Channel." Pmc.Con!Inj"orm. Sci.SySI.,Princeton University. Princeton. N.J.March. Cioffi,J.M.andKailath. T.(1984)."FastRecursive-Least Squares Transversal Filters for Adaptive Filtering," IEEETran!.Acoasl., Speech.SignalProcersing, vol.ASSP-32, pp. 304-337, April. Cook,CE.,Ellersick, F.W.,Milstein, LB.,andSchilling, D.L.(1983).SpreadSpectrum CommuncationJ, IEEEPress.NewYork. Costas,J.P.(1956)."Synchronous Communications," Proc.IRE,vol.44,pp.1713-1718, December. Cover,T.M.(1972)."Broadcast Channels," IEEETrans.Inform.Theory,vol.IT-18,pp.2-14, January. Cramer. H.(1946).Mathematical Methods ofStatistics, Princeton University Press.Princeton. N.J. Daut,D.G.,Modestino, J.W.,andWismer, L.D.(1982)."NewShortConstraint Length Convolutional CodeConstruction forSelected Rational Rates,"IEEETrans.Inform.Theory, vol.IT·28,pp.793-799, September. . Davenport, W.B.,Jr.(1970).Probability amiRandom Processes, McGraw-Hili. NewYork. Davenporl, W.B.Jr.andRoot,W.L.(1958).Random SignalsandNoise,McGraw-Hili, New York. Davisson, L.D.(1973)."Universal Noiseless CodIng," IEEETrans.Inform.Theory,vol.IT-19, pp.783-795. Davisson, L.D.,McEliece, R.J.,Pursley, M.B.,andWallace, M.S.(1981)."Efficient Universal NoiselssSourceCodes,"IEEETrans.Inform.Theory,vol.1T-27,pp.269-279. deBuda, R.(1972)."Coherent Demodulation ofFrequency ShiftKeyingwithLowDeviation RatiO,"IEEETrans.Commun., vol.COM-20, pp.429-435, June. Deller,J.P.,Proakis,J.G.,andHansen,H.L.(1993).Discrete-Time Processing ofSpeechSignals, MacMillan, NewYork. Ding,Z.(1990).Application AspectsofBlindAdaptive Equalizers inQAMDataCommunications, Ph.D.Thesis,Department ofElectrical Engineering, CornellUniversity. Ding,Z.,Kennedy, R.A.,Anderson, B.D.O.,andJohnson, CR.(1989)."Existence and Avoidance ofIII-Convergence ofGodardBlindEqualizers inDataCommunication Systems," Proc.13rdConfonInform.Sci.Systems, Baltimore, Md. Divsalar, D.,Simon,M.K.,andYuen,J.H.(1987)."Trellis CodingwithAsymmetric Modulation," IEEETrans,Commun., vol.COM-35, pp.130-141, February. Divsalar, D.andYuen.J.H.(1984),"Asymmetric MPSKforTrellisCodes," Proc, GLOBECOM'84, pp.20.6.1-20.6.8, Atlanta,Georgia. November. Dixon,R.C(1976).SpreadSpectrum Techniques, IEEEPress,NewYork. Doelz,M.L.,Heald,E.T.,andMartin,D.L.(1957)."BinaryDataTransmission Techniques for LinearSystems," Proc.IRE,vol.45,pp.656-661, May. Drouilhel, P.R.,Jr.andBernstein, S.L.(1969)."TATS--A Bandspread Modulation· Demodulation SystemforMultiple Acee,ssTactical Satellite Communication," 1969IEEE Electronics andAersopace Systems(EASCON) Conv.Record,Washington, D.C.,pp.126-132, October27-29. Duffy,F.P.andTratcher, T.W.(1971)."Analog Transmission Performance ontheSwitched Telecommunications Network," BellSyst.Tech.J.,vol.50,pp.1311-1347, April. Durbin, J.(1959)."Efficient Estimation ofParameters inMoving-Average Models," Biometrika, vol.46,part,1and2,pp.306-316. Duttweiler, 0.-L,Mazo,J.E.,andMesserschmitt, D.G,(1974)."ErrorPropagation in Decision-Feedback Equalizers," IEEETrans.Inform. Theory, vol.IT-20,pp.490-497, July. Eleftheriou, E.andFalconer, D,D.(1987)."Adaptive Equalization Techniques forHF Channels," IEEE1.SeleCledAreasCommun., vol.SAC-5pp.238-247, February. REFERENCES ANDHIRU()(iRAPHY 903 EIGamal.A.andCover.T.M.(1980)."Multiple UserInformation Theory:' Proc.IEEE.vol.68­ pp.1466-14M3. Decemher. Eli".P.(1954)."Error·Free Coding:' IRETrans.Inform. Theon'. vol.IT·4.pp.29-37. September. Ellas.P.(1955)."Coding forNoisyChannels," IREConvention Rf!cord, vol.3.part4,37­ 46. Es.posito. R.(1967)."ErrorProbahilities fortheNakagami Channel:' I££ETrans.Inform. Theory.vol.IT·I3.pp.145-14M. January. Eyuboglu. V.M.(I9811)."Delection ofCodedModulation SignalsonLinear.Severely Distorted Channels UsingDecision-Feedback NoisePrediction withInterleaving," fEEETrons, Commlln ..vol.COM-36. pp.401-409. April. Falconer. D. D.(1976)."Jointly Adaptive Equalization andCarrierRecovery inTwo-Dimensional DigitalCommunication Systems." BellSyst.Tech.J.,vol.55,pp.317-334, March. Falconer. D. D.andLjung,I..(197M)."Application ofFastKalman Estimation toAdaptive Equalization:' IEEETram.Commlln., vol.COM-26, pp.1439-1446. October. Falconer, D. D.andSalz.J.(1977)."Oplilnal Reception ofDigitalDataOvertheGaussian Channel withUnknown DelayandPhaseJitter,"IEEETrans.Inform.Theory,vol.IT-23.pp. 117-126, January. Fano,R.M.(1961).Transmission ofInformation, MITPress,Cambridge. Mass. Fano.R.M.(1963)."AHeuristic Discussion ofProbabilistic Coding," IEEETrans.Inform. Theory.vol.IT-9,pp.64-74.April. Feinstein, A.(1958).Foundations ofInformation Theory,McGraw-Hili, NewYork. Fire.P.(1959)."AClassofMultiple-Error-Correcting BinaryCodesforNon-Independent Errors," Sylvania ReportNo.RSL-E-32. Sylvania Electronic Defense Laboratory, Mountain View,Calif..March. Flanagan, J.1...etal.(1979)."Speech Coding," IEEETrans.Commun., vol.COM-n, pp. 710-736, April. Fomey,G.D.,Jr.(1965)."OneDecoding BCHCodes," IEEETrans.Inform.Themv.vol.IT-II. pp.549-557, October. Forney,G.D.,Jr.(I966a).Concatennted Codes,MITPress,Cambridge, Mass. Forney, G.D.,Jr.(l966b). "Generalized Minimum Distance Decoding," IEEETrans.Inform. Theory.vol.IT-12,pp.125-131, April. Forney, G.D.,Jr.(1968)."Exponential ErrorBounds forErasure, List,andDecision-Feedback Schemes." IEEETrans.Inform.Theory.vol.1T-14,pp.206-Z20, March. Forney, G.D.,Jr.(1970a). "Coding andItsApplication inSpaceCommunications," IEEE Spectrum, vol.7,pp.47-58.June. Fomey, G.D.,Jr.(l970b). "Convolutional CodesI:Algebraic Structure," IEEETrans.Inform. Theory,vol.IT-16,pp.720-738, November. Fomey,G.D.,Jr.(1971)."Bu"tCorrecting CodesfortheClassicBu"tyChannel," IEEETrans. Comm"n. Tech.,vol.COM-19, pp.772-781, October. Forney,G.D.,Jr.(1972)."M..imum-Likelihood Sequence Estimation ofDigitalSequences inthe Presence ofIntersymbol Interference." IEEETrans.Inform.Theory.vol.1T-18,pp.363-378. May. Forney,G.D.,Jr.(1974)."Convolutional CodesIII:Sequential Decoding," Inform.Control. vol. 25,pp.267-297, July. Forney, G.D.,Jr.(19MB)."CosetCodesI:Introduction andGeometrical Classification," IEEE Trans.Inform.Theory.vol.IT-J4,pp.671-680, September. Forney, G.D.,Jr.,Gallager, R.G.,Lang,G.R.,Longstaff, F.M.,andQureshi, S.U.(1984). "Ellicient Modulation forBand-Limited Channels," IEEE1.Selected AreasCommun ..vol. SAC-2,pp.632-647, September. Foschini, G.J.(1984)."Contrasting Performance ofFaster-Binary Signaling withQAM," Bell Syst.Tech.J.•vol.63,pp.1419-1445, October. Foschini, G.J.(1985)."Equalizing Without Altering orDetecting Data,BellSysl.Tech.J.,vol.64. pp.1885-1911, October. 904 DIGITAL COMMUNICATIONS Fosrhini, G.J.,Gitlin,R.D.,andWeinstein. S.B.(1974)."Optimization orTwo-Dimensional SignalConstellations inthePresence ofGaussian Noise," IEEETrans.Commun., vol. COM-22, pp.28-38,January. Franaszek. P.A.(1968)."Sequence-State CodingrorDigitalTransmission," BellSySI.Tech.J., vol.27,p.143. Franaszek, P.A(1969)."OnSynchronous Variable LengthCodingforDiscrete Noiseless Channels." Tnform.Control,vol.IS,pp..L55-164. Franaszek, P.A.'(1970)."Sequence-State Methods forRun-Length-Lirnited Coding," IBM1.Res. Dro.,pp.376-383, July. Franks,LE.(1969).SignalTheory,Prentice-HaIL, Englewood Cliff,N.J. Franks, L.E.(1983)."Carrier and BitSynchronization inDataCommunication-A Tutorial Review," TEEETrans.Commun., vol.COM-28, pp.1107-1121. August. Franks: LE.(1981). "Synchronization Subsystems: Analysis andDesign," inDigital Communications, Satellite/Earth StationEngineering, K.Feher(ed.),Prentice-HaIL, Englewood Cliffs,NJ. Fredricsson, S.11975)."Pseudo-Randomness Properties ofBinaryShiftRegister Sequences," IEEETrans.Inform.Theory,vol.IT-21,pp.115-120, January. Freiman, C.E.andWyner.AD_(1964)."Optimum BlockCodesforNoiseless InputRestricted Channels," Tnform.Control,vol.7,pp.398-415. Gaarder, N.T.(1971)."SignalDesignforFast-Fading Gaussian Channels," IEEETrans.Inform. Theory,vol.JT-17,pp.247-256, May. Gabor,A.(I967).,"Adaptive CodingforSelfClocking Recording," TEEETrans.Electronic Camp. vol.EC-16,p.866. Gallager, R.G_(1965)."Simple Derivation oftheCodingTheorem andSomeApplications," IEEETrans.Inform.Theory.voLIT-II,pp.3-18,January. Gallager, R.G.(1968).Tnformation TheoryandReliableCommunication, Wiley,NewYork. Gardner, F.M.11979).Phaselock Techniques. Wiley.NewYork. Gardner, W.A.(1984)."Learning Characteristics 01Stochastic-Gradient Descent Algorithms: A GeneralStUdy,Analysis, andCritique," SignalProcessing, vol.6,pp.H3-133, April. George, D.A.,Bowen,R.R.,andStorey,J.R.(1971)."AnAdaptive Decision-Feedback Equalizer," JEEETrans.Commun. Tech.,vol.COM-19, pp.281-293, June. Gersho, A.(1982)."OntheStructure ofVectorQuantizers," IEEETrans.Inform.Theory,vol. IT-28,pp.157-166, March. Gersho, A.andGray,R.M.(1992).VectorQuantization andSignalCompression. Kluwer Academic Publishers, Boston. Gersho, A.andLawerence, V.B.(1984)."Multidimensional SignalConstellations forVoiceband DataTransmission;' TEEEJ.SelectedAreasCommun., voLSAC.2,pp.687-7CJ2, September. Gerst,J.andDiamond, J.(1961)."TheElimination ofIntersymbol Interference byInputPulse Shaping," Proc.IRE,vol.53,JUly. Ghez,S.,Verdu,S.,andSchwartz, S.C.(1988)."Stability Properties ofSlottedAloha withMUJtipacket Reception Capability." IEEETransAU/om.Control, voL33,pp.640-649, July. Ghosh,M.andWeber,C.L.(1991)."Maximum likelihood BlindEqualization," Proc.1991SPlE Conf,SanDiego,Calif.July. Gilbert, E.N.(1952)."AComparison ofSignaling Alphabets," BellSyst.Tech.J.,vol.31,pp. 504-522, May. Gilhousen, K.S.,Jacobs,J.M.,Podovani, R,Viterbi,A.J..Weaver, LA.,andWheatley, G.E. m(1991)."OntheCapacity ofaCellularCDMASystem," IEEETrans.Vehicular Tech.,voL 40,pp.303-312, May. Gitlin,RD.Meadors, H.c.,andWeinstein, S.B.(1982)."TheTapLeakage Algorithm: An Algorithm fortheSlableOperalion 01aDigitally Implemented Fractionally Spaced,Adapti.. Equalizer," BellSyst.Tech.J.,vol.61,pp.1817-1839, October. Gitlin,RD.andWeinstein, S.B.(1979)."OntheRequired Tap-Weight Preci.ion farDigitally' Implemented Mean-Squared Equalizers," BellSyst.Tech.J.,vol.58,pp.301-321, February. ~EFEREN(,ES ANDBIHlH)(JRAPHY 905 Gillin.R.D.amIWeinslcin. S.B.(1981)."Fractionally-Spaced Equalizalion: AnImproved Digilal Transversal Equalizer." BdlSyM.Tech.1..vol.60,pp.275-2'16, February. GI"e.F.E.(1972)."AnUpperBoundontheProbability ofErrorduetoIntersymbol Interference forCorrleated Digital Si~nals." IEEETrans.Inform. Theory. vol.IT·18.pp, 356-362, May. Goblick, T.J"Jr.andHolsinger, J.L.(1'167)."Analog SourceDigitization: AComparison 01 TheoryandPractice." IEEETrans.Inform.Th«>ry,vol.IT-B,pp.323-326, April. Godard. D.N.(1974)."Channel Equalization UsingaKalmanFilterforFastDataTransmission," IBMJ.R".Dev.,vol.18,pp.267-273. May. Godard. D.N.(19SO)."Self·Recovering Equalization andCarrierTracking inTwo·Dimensional DataCommunications SySlems." IEEETrans.Commun" vol.COM·28. pp.1867-1875, November. Golay.M.J.E.(1949)."NoteonDigitalCoding," Proc.IRE,vol.37,p.657,June. Gold.R.(1967)."Optimal BinarySequences forSpreadSpectrum Multiplexing," IEEETrans. Inform.Theory,vol.IT·B,pp.619-621, October. Gold.R.(1968)."Maximal Recursive Sequences with3-Valued Recursive CrossCorrelation Functions." IEEETrans.Inform.Theory,vol.IT-14,pp.154-156, January. Golomb, S.W.(1967).ShiftRegisterSequences, Holden-Day, SanFrancisco, Calif. Goppa.V.D.(1970)."NewClassofLinearCorrecting Codes,"Prob!.Peredach. Inform" vol.6, pp.24-3O. Goppa. V.D.(1971)."Rational Presentation ofCodesand(L,g)-Codes," Prahl.Peredach. Inform" vol7,pp.41-49. Gray,R.M.(1975)."Sliding BlockSourceCoding,"IEEETrans.Inform.Theory,vol.IT21,pp_ 357-368, July. Gray.R.M.(199().SourceCodingTheory,KluwerAcademic Publishers, Boston. Greefkes, J.A.(1970)."ADigitally Companded DeltaModulation Modem forSpeech Transmission," Proc.IEEEInt.COlI!onCommun. pp.7.33-7.48, June. Green.P.E.,Jr.(1962)."RadarAstronomy Measurement Techniques," MITLincolnLaborotory, Lexington, Mass"Tech.ReportNo.282,December. Gronemeyer, S.A.andMcBride, A.L.(1976)."MSKandOffsetQPSKModulation," IEEE Trans.Comm"n" vol.COM·24, pp.809-820, August. Gupta,S.C.(1975)."Phase-Locked Loops,"Proc.iEEE,vol.63,pp.291-306, February. Hahn,P.M.(1'162).''Theoretical Diversity Improvement inMultiple Frequency ShiftKeying," IRETrans.Commun. Syst.,vol.CS·10,pp.177-184, June. Hamming, R.W.(1950)."ErrOlDetecting andErrorCorrecting Codes,"B,IISyst.Tech.J.,vol. 29,pp.147-160, April. Hamming, R.W.(1986).CodingandInformation Theory,Prentice·Hall, Englewood Cliffs,N.J. Hancock, J.C.andLucky,R.W_(1960)."Perfonmance ofCombined Amplitude andPhase­ Modulated Communication Systems," IRETrans.Commun. Syst.,vol.CS·3,pp.232-237, December. Hartley,R.V.(1928)."Transmission ofInformation," BeltSyst.Tech.J..vol.7,p.535. Hatzinakos, D.andNikias,C.L.(1991)."BlindEqualization UsingaTricepstrum-Based Algorithm," IEEETrans.Commun" vol.COM·39, pp.669-682, May. Hecht,M.andGuida,A.(1969)."DelayModulation," Proc.IEEE,vol.57,pp.1314-1316, July. Heller,J.A.(1968)."ShortConstraint LengthConvolutional Codes," JetPropulsion Laboralory, California InstitoeofTechnology, Pasadena, Calif.,SpaceProgram Summary 37-54,vol.3,pp_ 171-174, December. Heller.J.A.(1975)."Feedback Decoding ofConvolutional Codes," inAdvances inCommunica· tionSystems, vol.4,A.J.Viterbi(ed.),Academic, NewYork. Heller,J.A.andJacobs,I.M.(1971)."Viterbi Decoding forSatellite andSpaceCommmunica­ tion,"IEEETrans.Commun. T«h.,vol.COM-19, pp.835-848, Octo""r. Helstrom, C.W.(1955)."TheResolution ofSignalsinWhiteGaussian Noise,"Proc.IRE,vol.43, pp.1111-1118, September. 906 DIGITAL (>O~M{INI('ATI()NS Helstrom. C.W.(1968).Statistical Theon'ofSignalDetectioll, Pergamon. London. Helstrom. CW.(1991),Probability a~dStochastic.: Proa:'sses forEngineers. Macmillan. New York. Hildebrand. F.B.(1%0).Metho«s ofApplied Mathematics. Prentice-Hall. Englewood Cliffs. N.J. Hirosaki. B.(198I)."AnOrthogonallty Mulliplexed OAMSystemUsingtheDiscrete Fourier Transform:' IEEETrans.Commul).. "01.COM-29. pp.982-989. July. Hirosaki. B..Hasegawa. S.,andSabaco. A.(1986)."Advanced Group-Band Modem Using Orthogonally Multiplexed QAMTechniques," IEEETransCommun., vol.COM-34. pp. 587-592. June. Ho.E.Y.andYeh,Y.S.(1970)."ANewApproach forEvaluating theErrorProbability inthe Presence ofInlersymbol lnterferen<:e andAdditive Gaussian Noise,"BellSysc.Tech.1.,vol.49. pp.2249-2265. November. Hocquenghem, A(1959)."CodesCorrecteutS d'Erreurs," Chiff"s, vol.2,pp.147-156. Holmes, J,K.(1982).Coherent SpreadSpectrum Systems, Wiley-Interscience, NewYork. Horwood, D.andGagliardi, R.(1975)."SignalDesignforDigitalMultiple AccessCommunica- tions,"IEEETrans.Cornman., vol.COM-23, pp,378-383, March. Hsu,F.M.(1982)."Square-Root KalmanFiltering forHigh-SPeed DataReceived overFading Dispersive HFChannels," IEEETrans.Inform.Theory,vol.IT·28.pp.753-763, September. Huffman, D.A(1952)."AMethodfortheConstruction ofMinimum Redundancy Codes:'Proc. IRE,vol.40.pp.1098-1101, September. Hui,J.Y.N.(1984)."Throughput Analysis forCodeDivision MUltiple Accessing oftheSpread Spectrum Channel." IEEEI.SelecledAreasCommun ..vol.SAC-2.pp.482--486, jUly. [mminl<, K.A.S.(199<l)."Runlength-Limited Sequences," Proc.IEE£,vol.78,pp.1745-1759, November. Itahra, F.(1975)."Mmimum Prediction Residual Principle Applied toSpeechRecognition," IEEETrans.Acoust., Spee-ch,SignalProcessing, vol.ASSP-23. pp,67-72,February. Italura. F.andSaito,S.(1968)."Analysis Synthesis Telephony BasedontheMaximum­ Likelihood Methods," Proe.6thInl.Congr.Acoust.,Tokyo,Japan,pp.el7-C20. Jacobs.I.M.(1974)."Practial Applications ofCoding," IEEETrans.Inform.Theory.vol.fT-20, pp.305-310, May. Jacoby, G.V.(1977)."ANewLook-Ahead CodeforIncreased DataDensity," IEEETrans. Magnt!lics, vol.MAG-I3, 1202-1204. Jayant,N.S.(1970)."Adaptive DellaModulation withaOne-Bit Memory," Bel!Syit.Tech.J., pp.321-342. March. Jayant,N.S.(1974)."DigitalCodingofSpeechWaveforms: PCM.DPCM.andDMOuantizers," Proc.IEEE,vol.62,pp.611-632, May. Jayant,N.S.(1976).Wat>eform Quantization andCoding.IEEEPress,NewYork. Jayant,N.S.andNoll,P.(1984).DigilalCodingofWaveforms, Prenlice-Hall, Englewood Cliffs. N,J. Jelinek,F.(1968).Probabilistic Informalion Theory,McGraw-Hili, NewYork. Jelinek, F,(1969)."FaslSequential Decoding Algorithm UsingaStack,"IBMJ.Res.Dro.,vol. 13,pp.675-685. November. Jol108On, C.R.(1991)."Admissibility inBlindAdaptive Cbannel Equalization," IEEECOn/rol Syst.Mag.,Il',3-J5,January. Jones,S,K.•Cavin.R.K.andReed,W.M,(1982)."Analysis ofError-Gradient Adaptive Linear Equalizers foraClass01Stationary-Dependent Pro<:esses," IEEETrans.Inform.Theory,voi. IT-28,pp.318-329, March. Jordan,K.L.Jr.(19(jj;)."ThePerformance ofSequential Decoding inConjunction withEllicient Modulation," IEEETrans.Cornmun. Sysl.,vol.CS-14,pp.283-287, June. Justesen, J.(1972)."AClassofConstructive Asymptotically GoodAlgebraic Codes," IEEE Trans.Inform.Theory,vol.IT-IS,pp.652-656, September. Kailath, T.(1960)."Correlation Detection ofSignalsPerturbed byaRandom Channel," IRE Trans.Inform.Theory,vol.IT-6.pp.361-366,June. REFERENCES AND81BUOCRAPHY 907 Kailath.T.(1961)."Channel Characterization: Time-Variant Dispen;ive Channels, InLectures on Communication SystemTheory,Chap_6.E.Baghdady (ed.),McGraw-Hili, NewYork. Kaler,I.(1989)."TheMultitone Channel," (EEETrans.Commun., vol.COM-37, pp.119-124, February. Karabed. R.andSiegel.P.H.(1991)."Matched-Spectral NullCodestorPartial-Response Channels." (EEETrans.(nform.Theory,vol.IT-37,pp.818-855, May. Kasami.T.(1966)."Weight Distribution Formula forSomeClassofCyclicCodes," Coordinated ScienceLaboratory, Univen;ity ofllIinois,Urbana, Ill.,Tech.ReportNo.R-285.April. Kaye.A.R.andGeorge, D.A.(1970)."Transmission ofMultiplexed PAMSignalsover Multiple Channel andDiven;;ty Systems," (EEETraru.Commun., vol.COM-18, pp.520-525, October. Kelly.E.J..Reed.I.S.,andRoot,W.L.(1960)."TheDetection ofRadarEchoesinNoise.Pt.I." 1.S(AM,vol.8.pp.309-341, September. Kleinrock. L.andTohagi. FA.(1975)."Packet Switching inRadioChannels: PartI-earrier SenseMultiple-Access ModesandTheirThroughput-Delay Characteristics," (EEETraru. Commun., vol.COM-23, pp.1400-1416, December. Kobayashi, H.(1971)."Simultaneous Adaptive Estimation andDecision Algorithm forCarrier Modulated DataTransmission Systems," (EEETrans.Commun. Tech.,vol.COM-19, pp. 268-280. June. Kotelnikov, V.A.(1947)."TheTheoryofOptimum NoiseImmunity," Ph.D.Dissertation, Molotov EnergyInstitute. Moscow. [Translated byR.A.Silverman, McGraw-Hill, New York.] Kretzmer. E.R.(1966)."Generalization ofaTechnique forBinaryDataCommunication," IEEE TransCommun. Tech.,vol.COM-14, pp.67-68,February. Lan;erl,K.J.(1973)."ShortConvolutional CodeswithMaximal FreeDistance forRates1/2. 1/3, and114,"(EEETrans.(nform.Theory,vol.IT-19,pp.371-372, May. Lender,A.(1963)."TheDuobinary Technique forHighSpeedDataTransmission," AlEETrans. Commun. ElectrOniCs, vol.82,pp.214-218_ Leon-Garcia, A.(1994).Probability andRandom Processes forElectrical Engineering, Addison­ Wesley,Reading, Mass. Levinson, N.(1947)."TheWienerRMS(RootMeanSquare)ErrorCriterion inFilterDesignand Prediction," J.Math.andPhys.,vol.25,pp.261-278. Lin,S.andCostello, D.J.,Jr.(1983).ErrorControlCoding: Fundamenra15 andApplications, PrentIce-Hail, Englewood Cliffs,N.J. Linde,Y,Buzo,A.,andGray,R.M.(198<1)."AnAlgorithm forVectorQuantizer Design." (EEE Trans.Commun. vol.COM-28, pp.84-95,lanuary. Lindell,G.(1985)."OnCodedContinuous PhaseModulation," Ph.D.Dissertation, Telecomm­ unication Theory, University ofLund, Lund, Sweden, May. Lindholm, J.H.(1968)."AnAnalysis ofthePseudo-Randomness Properties ofSubsequences of Longm-Sequences," IEEETrans.(nform.Theory,vol.IT-14,pp.569-576, July. Lindsey, W.C.(1964)."ErrorProbabilities forRiceanFadingMultichannel Reception ofBinary andN-ArySignals," (EEETrans.Inform.Theory,vol.IT-IO,pp.339-350, October. Lindsey. W.C.(1972).Synchronization SystemsinCommllnicatiofl.l, Prentice-Hall, Englewood Cliffs,N.J. Lindsey, W.C.andChie,C.M.(1981)."ASurveyofDigitalPhase-locked Loops,"Proc.(EEE, vol.69,pp.410-432. Lindsey, W.C.andSimon,M.K.(1973).Telecommunication Systetm En.gineering, Prentice-Hall. Englewood Cliffs,N.J. Ling,F.(1988)."Convergence Characteristics ofLMSandLSAdaptive Algorithms forSignals withRank-Deficient Correlation Matrices," Proc.Int.ContAcousf., Speech,SignalProcessing: NewYork,25.D.4.7, April. Ling.F.•Manolakis, D.G.•andProakis,J.G.(1986a)."FiniteWord,Length EffectsinRecursive LeastSquares Algorithms withApplication toAdaptive Equalization," Annales des Telecommunications, vol.41,pp.1-9,May-June. 908 DIGITAL COMMUNICATIONS ~ing,F.,Manolakis, D.G.,andProakis, J.G.(1986b)."Numerically RobustLeast-Squares Lattice.ladder Alghorithms withDirectUpdating oftheReflection Coefficients," IEEETrans. Aeoust.,Speech,SignalProcessing. vol.ASSP-34, pp.837-845,August. Ling,F.andProakis, 1.G.(1982)."Generalized LeastSquare,latticeandItsApplications to DFE,"Proc.1982,IEEEInt.ConfonAcoustics, Speech,SignalProcessing, Paris,France, May. Ling,F.andProakis,J.G.(19840),"Numerical Accuracy andStability: TwoProblems ofAdaptive Estimalion Algorithms CausedbyRound-Off Error,"Proc.Int.ConfAc"us/.,Speech,Signal Processing, pp.30.3.1-30.3.4, SanDiego,Calif.,March. Ling,F..andProakis, J.G.(1984b). "Non'tationary Learning Characteristics ofLeastSquares Adaptive Estimation Algorithms," Proc.Int.Can!Acoust..Speech, SigruJlProcessing, pp. 3.7.1-37.4, SanDiego,Calif"March. Ling,F.andProakis,J.G.(19840)."AGeneralized Multichannel Least·Squares latticeAlgorithm witbSequential Processing Stages," IEEETrans.Acoust., Speech,SignalProcessing, vol. ASSP-32, pp.381-389, April. Ling,F.andProaltis, J.G.(1985)."Adaptive latticeDecision-Feedback Equalizers-Their Performance andAppJication toTime-Variant Multipath Channels," IEEETrans.Commun., vol.COM-33, pp.348-356, April. Ling,F.andProakis, J.G.(1986)."ARecursive Modified Gram-Schmidt Algorithm," IEEE Trans.Acoust.,Speech,SignalProcessing, vol.ASSP-34, pp.829-836, August. Ling,F.andQuershi, S.U.H.(1986)."latticePredictive Decision-Feedback Equali,er forDigital Communication OverFadingMultipath Channels," Proc.GLOBECOM '86,Houston, Texas, December. Ljung,S.andLjung,L.(1985)."ErrorPropagation Properties ofRecursive Least-Squares Adaptation Algorithms," Au/omalica, vol.21,pp.159-167. Lloyd,S.P.(1982)."LeastSquaresQuantization inPCM,"IEEETrans.Inform.Theory,vol. IT-28,pp.129-137, March. Loeve,M.(1955).Probability Theory.VanNostrand, Princeton, N.J. Long,G.,Ling,F.,andProakis, 1.G.(1987)."Adaptive Transversal FilterswitbDelayed Coefficient Adaptation." Proc.Int.ContAcollSt., Speech, SignalProcessing, Dallas.Texas, March. Long,G.,Ling,F.,andProakis, I.G.(1988a). "Fractionally-Spaced Equalizers Basedon Singular-Value Decomposition," Proc.Inf.ConIAcoust., Speech, SignalProcessing, New York,25.0.4.10, April. Long,G.,Ling,F.,andProalis, J.G.(1988b). "Applications ofFractionally-Spaced Decision­ Feedback Equalizers toHFFadingChannels," Proc.MILCOM, SanDiego,Calif., October. Long,G.,Ling,F.,andProalis,J.G.(1989)."TheLMSAlgorithm withDelayed Coefficient Adaptation, "IEEETrans.ACOUSI.,Speech,SignalProcessing, vol.ASSP-37, October. Lucky,R.W.(1965)."Automatic Equalization forDigitalCommunications, BellSys/.Tech.I., vol.44,pp.547-588, April. Lucky,R.W.(1966).'Techniques forAdaptive Equalization ofDigitalCommunication," Bell Syst.Tech.I.,vol.45,pp.255-28li. Lucky,R.W.andHancock, J.C.(1962)."OntbeOptimum Perfonmance ofN·arySystemsHaving TwoDegreesofFreedom," IRETrans.CommlUl. SyS!,vol.CS-IO,pp.185-192, June. Lucky,R.W.,Salz,1.,andWeldon, E.J.,Jr.(1968).Principles ofDalaCommunication, McGraw-Hili, NewYork. Lugannan~ R.(1969)."Intersymbol Interference andProbability ofErrorinDigitalSystems," IEEETrans.Inform.Theory,vol.IT-IS,pp.682-688, November. Lundgren, C.W.andRummier, W.D.(1979)."Digital RadioOutageDuetoSelective Fading-Qbservation vs.Prediction fromlaboratory Simulation," BellSyst.Tech.I.,vol.58, pp.1074-1100, May-June. Lupas.R.andVerdu,S.(1989)."Linear Multiuser Detectors fotSynchronous Code-Division MUltiple-Access Channels," IEEETrans.Inform.Theory,vol.fT-35,pp.123-136, January. REfERENCES ANDBfBLlOORAPHY 909 Lupas,R.andVerdu,S.(1990)."Near-Far Resistance ofMultiuser Detectors inAsynchronous Channels," IEEET,ans.Commun., vol.COM·38, pp.496-508, April. MacKenchnie, L.R.(1973)."Maximum Likelihood Receivers forChannels HavingMemory," Ph.D.Dissertation, Department ofElectrical Engineering, University ofNotreDame,Notre Dame,Ind.,January. MacWilliams, F.J.andSloane,J. J.(1977).TheTheoryofErro,CorreCling Codes,NorthHolland. NewYork. Magee,F.R.andProakis,J.G.(1973)."Adaptive Maximum·Likelihood Sequence Estimation for DigitalSignaling inthePresence ofIntersymhol Interference," IEEETrans.Inform.Theory. vol.IT·19,pp.120-124, January. Makhoul, J.(1978)."AClassofAII·Zero LalliceDigitalFilters:Properties andApplications," IEEETrans.ACOUSI.,Speech,SignalProcessing, vol.ASSP·26, pp.304-314, August. Makhoul, J.,Roucos,S.,andGish,H.(1985)."Vector Quantization inSpeechCoding," Proc. IEEE,vol.73,pp.1551-1587, November. Martin,D.R.andMcAdam, P.L.(1980)."Convolutional CodePerformance withOptimal Jamming," ConfRtc.Inr.ConfCommun., pp.4.3.1-4.3.7, May. Massey, J.L.(1963).Threshold Decoding, MITPress,Cambridge, Mass. Massey, J.L.(1965)."Step-by·Step Decoding oftheBCHCodes,"IEEETrans.Inform.Theory, vol.IT-II,pp.580-585, October. Massey,J.L.(1988)."SomeNewApproaches toRandom AccessCommunications," Performanct '87,pp.551-569. [Reprinted 1993inMultipleAccessCommu;nications, N.Abramson (ed.), IEEEPress,NewYork.] Massey, J.L.andSain,M.(1968)."Inverses ofLinearSequential Circuits," IEEETrans.Comput., voLC-l7,pp.330-337, April. Matis,K.R.andModestino, J.W.(1982).·'Reduced·State Soft·Decision TrellisDecoding of LinearBlockCodes,"IEEETrans.Inform.Theory,vol.IT·28,pp.61-68,January. Max,J.(1960)."Quantizing forMinimum Distortion," IRETrans.Inform.Theory,vol.IT-6,pp. 7-12,March. Mala,J.E.(1975)."Faster-Than· NyquistSignaling," BellSrs,.Tech.J.,vol.54,pp.1451-1462. October. Mazo,J.E.(1979)."OntheIndependence TheoryofEqualizet Convergence," BellSrst.Tech.J. vol.58,pp.963-993, May. McMahon, M.A.(1984).TheMakingofaProfession-A CmturyofElectrical Engineering in Ameflca, IEEEPress,NewYork. Mengali,U.(1977)."JointPhaseandTimingAcquisition inDataTransmission," IEEETrans. Commun., vol.COM-25, pp.1174-1185, October. Meyers, M.H.andFranks,LE.(1980)."JointCarrierPhaseandSymbolTimingforPAM Systems," IEEETrans.Commun., vol.COM·28, pp.1121-1129, August. Meyr.H.andAscheid, G.(1990).Synchronization inDigitalCommunications. WileyInterscience, NewYork. Miller,K.S.(1964).Multidimensional Gaussian Distributions, Wiley,NewYork. Mlliman, S.(ed.)(1984).AHistoryofEngineering andScienceintheBellSystem-Communication' Sciences (/925-1980), AT&TBellLaboratoreis. Mlyagaki, Y.,Morinaga, N.,andNamekawa, T.(1978)."ErrorProbability Characteristics for CPSKSignalThrough m·Dislrib.led FadingChannel," IEEETrans.Commun., vol.COM'26, pp.88-100,January. Monsen, P.(1971)."Feedback Equalization forFadingDispersive Channels," IEEETrans. Inform.Theory,vol.IT-l7,pp.56-64,January. Mort,M,(1977)."Ladder FormsinEstimation andSystemIdentification," Proc.11thAnnua" Asilomar Con!onCircuits, SystemsandComputers, Monterey, CaHf.,Nov.7-9. Morf,M.,Dickinson, B.,Kailalh,T.,andVieira,A.(1977)."Efficient Solution ofCovariance Equations forLinearPrediction," IEEETrans. ACOUSI" Speech, SignalProcessing, vol. ASSP-25, pp.429-433, October. Mort,M.andLee,D.(1979)."Recursive LeslSquares LadderForm,forFastParameler 910DIGITAL COMMUNICATIONS Tracking," Proc.1978IEEECon!onDecision andControl, SanDiego,Calif.,pp.1362-1367, January12. Man,M.,Lee,D.,Nickolls, J.andVieira,A.(1977)."AClassification ofAlgorithms forARMA ModelsandLadderRealizations," Proc.1977IEEEInt.ConfonAcouslics, Speech,Signal Processing, Hartford, Conn., pp.13-19,May. Man,M.,Vieira,A.,andLee,D.(1977)."Ladder FormsforIdentification andSpeech Processing," Proc.1977IEEEConfonDecision andControl, NewOrleans, La,pp. 1074-1078, December. Mueller, K.H.andMuller, M.S.(1976)."Timing Recovery inDigitalSynchronous Data Receivers," IEEETrans.Commun., vol.COM-24, pp.516-531, May. Muller,D.E.(1954)."Application ofBoolean Algebra'to Switching CircuitDesignandtoError Detection," IRETrans.Electronic Comput., vol.£C-3,pp,6-12,September. Mulligan, M.G.(1988)."Multi-Amplitude Continuous PhaseModulation withConvolutional Coding," Ph.D.Dissertation. Department ofElectrical andComputer Engineering, North­ easternUniversity. June. Nakagami, M.(1960)."Them-Distribution-A General Formula ofIntensity Distribution of RapidFading," inStalistical MethodsofRadioWavePropagation, W.C.Hoffman (ed.),pp. .3-36,Pergamon Press,NewYork. Natali.F.D.andWalbesser, W.J.(1969)."Phase-Locked LoopDetection ofBinaryPSKSignals Utilizing Decision Feedback," IEEETrans.:Aerospace Electronic Syst.,vol.AES-5,pp.83-90, January. Neyman, J.andPearson, E.S.(1933)."OntheProblem oftheMostEfficient TestsofStatistical Hypotheses," Phil.Trans.Roy.Soc.London, SeriesA,vol.231,289-337. North,D.O.(1943)."AnAnalysis oftheFactorsWhichDetermine Signal/Noise Discrimination inPulse-Carrier Systems," RCATech.ReportNo.6PTR-6C. Nyquist,H.(1924)."Certain FactorsAffecting Telegraph Speed,"BellSyst.Tech.J.,vol.3,pp. 324. Nyquist,H.(1928)."Certain TopicsinTelegraph Transmission Theory," AlEETrans.,vol.47,pp. 617-644. Odenwalder, J.P.(1970)."Optimal Decoding ofConvolutional Codes," Ph.D.Dissertation, Department ofSystems Sciences, SchoolofEngineering andApplied Sciences, Un~versity of California, LosAngeles. Odenwalder, J,P.(1976),"Dual-k Convolutional CodesforNoncoherently Demodulated Channels," Proc.Int.Telemetering Can!vol.12,pp.165-174, September. Olsen,J.D.(1977)."Nonlinear BinarySequences withAsymptotically Optimum Periodic Cross Correlation," Ph.D.Dissertation, University ofSouthern California, December. Omura,J,(1971)."Optimal Receiver DesignforConvolutional CodesandChannels wi\!)Memory ViaControlTheoretical Concepts," Inform.Sci.,vol,3,pp.243-26(). Omura, J.K.andLeviu,B.K.(1982)."Code Error Probability Evaluation forAntijam Communication Systems," IEEETrans.Commun., vol.COM-30, pp.896-903, May. Osborne, W.P.andLuntz,M.B.(1974)."Coherent andNoncoherenl Detection of.CPFSK," IEEETrans.Common., vol.COM·22, pp.1023-1036, August. Paaske,E.(1974)."ShortBinaryConvolutional CodeswithMaximal FreeDistance forRates2/3 and3/4,"IEEETrans.Inform.Theory,vol.IT·20,pp.683-689, September. Pael,M.D.andGlisson, T.H.(1972)."Minimum MeanSquaredEnorQuantization inSpeech PCMandDPCMSystems," IEEETransCommun., vol.COM-20, pp.225-230, April. Pahlavan, K.(1985)."Wireless Communications forOfficeInformation Nelworks," IEEE Commun. Mag.,vol.23,pp.18-27,June. Papaulis, A.(1984).Probabilil)/, Random Variables, andSrochastic Processes, McGraw-Hili, New York. Paul,D.B.(1983),"An800bpsAdaptiv'- VectorQuantization Vocoder UsingaPreceptuaf Distance Measure," Proc.[EEEInt.ConfAcoust.,Speech,SignalProcessing, Boston,Mass, pp.73-76,April. Pearson, K.,(1965).TablesoftheIncomplete f-Function, Cambridge University Press,London. REFERENCES ANDBIBLIOGRAPHY 911 Peebles, P.Z.(1987).Probability. Random Variables, andRandom SignalPrinciples, McGraw­ Hill,NewYork. Peterson. W. W. (1960)."Encoding andError-Correction Procedures forBose-Chaudhurt Codes," IRETrans.Inform.Theory,'01.IT-6.pp.459-470. September. Peterson, W.W.andWeldon. E.J.,Jr.(1972).Error-Correcting Codes.2ded.•MITPress. Cambrdige, Mass. Picci,G.andPrati,G.(1987)."BlindEqualization andCarrier Reco-ery UsingaStop­ and-Go Decision Directed Algorithm," IEEETrans.Commun ..'01.COM-35, pp.877-887. September. Picinbono. B.(978). "Adaptive SignalProcessing forDetection andCommunication," in Communication Systems andRandom Process Theory, J.K.Skwirzynski (ed.),Sijthoff& Nordhoff, AlphenaandenRijn,TheNetherlands. Pickholtl. R.L.Schilling, D_L,andMilstein. LB.(1982)."Theory ofSpreadSpectrum Cornmunications-A Tutorial." IEEETrans.Commun ..vol.COM-30, pp.855-884, May. Pieper.J.F.•Proakis. J.G"Reed,R.R.•andWolf,J.K.(1978)."Design ofEfficient Codingand Modulation foraRayleigh FadingChannel,"' IEEETrans.Inform. Theory, vol.IT-24_pp. 457-468,July. Pierce.J.N.(1958)."Theoretical Diversity Improvement inFrequency-Shift Keymg," ProcIRE. vol.46.pp.903-910. May. Pierce,J.N.andStein.S.(1960)."Multiple Diversity withNon-Independent Fadmg," Proc.IRE. vol.48.pp.89-104.January. Plotkin. M.(1960)."Binary CodeswithSpecified Minimum Distance," IRETrans.Inform Theory.vol.IT-6,pp.445-450, September. Poor.H.V.andVerdu,S.(1988)."Single-User Detectors forMultiuser Cbannels," IEEETrans Commun ..vol.36.pp.50-60.January. Price.R.(1954)."TheDetection ofSignalsPerturbed byScatterandNoise."IRETrans.Inform Theory.vol.PGIT-4. pp.163-170. September. Price.R.(1956). "Optimum Detection ofRandom Signals inNoise,withApplication to Scatter-Multipath Communicaton." IRETrans.Inform. Theory. vol.IT-2.pp.125-135. December. Price.R.(1962a). "ErrorProbabalities forAdaptive Multichannel Reception ofBinarySignals." MITLincolnLaboratory. Lexington, Mass.,Tech.ReportNo.258.July. Price.R.(1962b). "ErrorProbabilities forAdaptive Multichannel Reception ofBinarySignals:' IRETrans.Inform.Theory,vol.IT-8.pp_305-316, September. Price.R.(1972)."Nonlinearly Feedback-Equalized PAMvs.Capacity," Proc.1972IEEE1m COllfonCommuJI. Philadelphia,'Penn ..pp.22.12-22.17. June. Price,R.andGreen,P.E.,Jr.(1958)."ACommunication Technique forMultipath Channels ..' ProcIRE,vol.46.pp.555-570. March. Price.R.andGreen,P.E..Jf.(1960)."Signal Processing inRadarAstronomy-Communication viaFluctuating Multipath Media." MITLincoln Laboratory. Lexington, Mass..Tech.Rep:>rt No.234,October. Proakis. J.G.(1968)."Probabilities ofEr~orforAdaptive Reception ofM-Phase Signals." lEE/: Trans.Commlln. Tech.,vol.COM-16, pp_71-81,February. Proakis, J.G.(1975)."Advances inEqualization forInters)'mbol Interference." inAdvances m Communication Systems. vol.4.A.J.Viterbi(ed.),Academic, NewYork. Proakis. J.G..Drouilhet, P.R.,Jr"andPrice,R.(1964)."Performance ofCoherent Detection Systems UsingDecision·Directed Channel Measurement," IEEETrans.Commun. Slst.,\'01. CS-12.pp.54-63.March. Proakis. J.G.andLing,F.(1984)."Recursive LeastSquares Algorithms forAdaptive Equalization ofTime-Variant Multipath Channels," Proc.Int.Con!Comml.ln. Amsterdam. TheNetherlands, May. Proakis, J.G.andManolakis. D.G.(1988).Inrroduction toDigitalProcessing, Macmillan, New York. • Proakis, J.G.andMiller.J.H.(J969). "Adaptive Receiver forDigitalSignaling through 912DIGITAL COMMUNICA nONS Channels withIntersymbol Interference," IEEETram.Inform. Theory, vol.IT-15,pp. 484-497, July. Proakis,J.G.andRahman, I.(1979)."Performance ofConcatenated Dual-kCodesonaRayleigh FadingChannel withaBandwidth Constraint," IEEETrans.Commun., vol.COM-27, pp. 801-806, May. Pursley,M.B.(1979)."OntheMean-Square PartialCorrelation ofPeriodic Sequences," I'roc. 1979ConrInform. ScienceandSy$lems, JohnsHopkins University, Baltimore, Md.,pp. 377-379, March. Qureshi, S.U.H.(1976)."Timing Recovery forEqualized PartialResponse Systems," IEEE Trans.Commun., vol.COM-24, pp.1326-1331, December. QureShi, S.U.H.(1985)."Adaptive Equalization," Proc.IEEE,vol.53,pp.1349-1387, September. QureShi, S.U.H.andForney, G.D.,Jr.(1977)."Performance andProperties ofaT/2 Equalizer," NaILTelecom. ConrRecord,pp.11.1.1-11.1.14, LosAngeles, Calif.,December. Rabiner, L.R.andSchafer, R.W.(1978).DigilalProcessing ofSpeechSignat.,Prentice·Hall, Englewood Clills,N.J. Rabeli,R.,Polydoros, A.•andTzou,C.·K.(1995).'ThePrinciple ofPer·Survivor Processing: A General Approach toApproximate andAdaptive MLSE," IEEETrans.Commun., vol. COM-43 (toappear). Rahman, .I.(1981)."Bandwidth Constrained SignalDesignforDigitalCommunication over Rayleigh Fading Channels andPartialBandInterference Channels," Ph.D.Dissertation, Department ofElectrical Engineering, Northeastern University, Boston,Mass. Ramsey, J.L.(1970)."Realization ofOptimum Interleavers," IEEETrans.Inform.Theory,vol. IT·16,pp.338-345. Reed,I.S.(1954)."AClassofMultiple-Error Correcting CodesandtheDecoding Scheme," IRE Trans.Inform.,vol.IT-4,pp.38-49,September. Reed,I.S.andSolomon, G.(1%0)."Polynomial CodesOverCertainFiniteFields,"SIAMJ..vol. 8,pp.300-304, June. Rizos,A.D.,Proakis, J.G.,andNguyen, T.Q.(1994)."Comparison ofDFTandCosine Modulated FilterBanksinMulticarrier ModUlation," I'ro<.Globecom '94,pp.687-691, San Francisco. Calif.,November. Roberts, L.G.(1975)."AlohaPacketSystemwithandwithoutSlotsandCapture." Camp. Commun. Rev.•vol.5,pp.28-42,April. Roucos, S.,Schwartz, R.,andMakhoul, J.(1982)."Segment Quantization forVery-Low. Rate SpeechCoding," Proc.Int.ConrAcousl., Speech, SignalProce"ing, !"aris,France,pp. 1565-569, May. Rowe,H.E.andPrabhu, V.K.(1975)."PowerSpectrum ofaDigitalFrequency Modulation Signal,"BellSySl.Tech.J.,vol.54,pp.1095-1125, July-August. Rummier, W.D.(1979)."ANewSelective FadingModel:Application toPropagation Data,"Bell SYSI.Tech.J.,vol.58,pp.1037-1071, May-June. Ryder,J.D.andFink,D.G.(1984).Engineers andElectronics, IEEEPress,NewYork. Saltzberg, B.R.(1967)."Performance ofanEfficient ParallelDataTninsmission System," IEEE Trans.Commun., vol.COM-15, pp.805-811, December. Saltzberg, B.R.(1968)."Intersymbol Interference ErrorBounds withApplication toIdeal Bandlimited Signaling," IEEETram.Inform.Theory,vol.IT-14,pp.563-568, July. Salz,J.(1973)."Optimum Mean-Square Decision Feedback Equalization," BellSysl.Tech.J.,vol. 52,pp.1341-1373, October. Salz,J.,Sheehan, J.R.,andParis,D.J.(1971)."DataTransmission byCominbed AMandPM," BellSysLTech.J.,vol.SO,pp.2399-2419, September. SaIWale, D.V.andPursley, M.B.(1980)."Crosscorrelation Properties ofPseudorandom and RelatedSequences," Proc.IEEE,vol.68,pp.593-619, May. Sato,Y.(1975)."AMethodofSelf-Recovering Equalization forMultilevel Amplitude·Modulation Systems," IEEETrans.Commun., vol.COM-23, pp.679-682, June. Sato,Y.elal.(1986)."BlindSuppression ofTimeDependency anditsExtension toMulti­ Dimensional Equalization," I'roc.ICC'86,pp.46.4.1-46.4.5. REFERENCES ANDBIBLIOGRAPHY 913 Satorius, E.H.andAlexander, S.T.(1979)."Channel Equalization UsingAdaptive Lallice Algorithms." IEEETrans.Commun" vol.COM-27, pp.899-905, June. Satorius, E.H.andPack,J.D.(1981)."Application ofLeastSquares latticeAlgorithms toAdaptive Equalization," IEEETrans.Commun., vol.COM·29, pp.136-142, February. Savage,1.E.(1966)."Sequential Decoding-The Computation Problem," BellSy".Tech.1.,vol. 45,pp.149-176, January. Scholtz,RA.(1977)."TheSpreadSpectrum Concept." IEEETrans.Commun., vol.COM·25. pp.748-755. August. Scholtz.RA.(1979)."Optimal CDMACodes:'1979NOI.Telecommun. Can!Rec.,Washington, D.c..pp.54.2.1-54.2.4. I\ovember. Scholtz,R.A.(1982)."TheOrigi..ofSpreadSpectrum:' IEEETrans.Commun., vol.COM·30, pp.822-854. May. Schonhoff, T.A.(1976)."Symbol ErrorProbabilities forM-aryCPFSK: Coherent and Noncoherent Detection," IEEETrans.Commun., vol.COM·24. pp.644-652, June. Seshadri. N.(1994)."JointDataandChannel Estimation UsingFastBlindTrellisSearch Techniques," IEEETrans.Commun ..vol.COM-42, pp.1000-1011, March. Shalvi.O.andWeins1ein. E.(1990)."NewCriteria forBlindEqualization ofNonminimum Phase Systems Channels." IEEETrans.Inform.Throry.vol.JT·36,pp.312-321, March. Shannon, C.E.(l948a)."AMathematical TheoryofCommunication," BellSysl.Tech.J.,vol.27. pp.379-423. July. Shannon, C.E.(l948b)."AMathematical TheoryofCommunication:' BellSysr.Tec.h.J.,vol.27. pp.623-656, October. Shannon, C.E.(1949)."Communication inthePresence ofNoise:'Proc.IRE.vol.37,pp.10-21. January. Shannon. C.E.(1959a). "Coding Theorems toraDiscrete SOllrcewithaFidelity Criterian," IRE Nat.Can•.Rec.,pt.4,pp.142-163. March. Shannon. C.E.(1959b). "Probability ofErrorforOptimal CodesinaGaussian Channel," Bell Sysr.Tech.1.,vol.38,pp.611-656, May. Shannon, C.E.,Gallager. R.G.,andBerlekamp, E.R.(1967)."Lower Bounds toError Probability forCodingonDiscrete Memoryless Channels, IandII,"Inform.Control., vol.10. pp.65-103,January; pp.527-552, May. Shlmbo. O.andCelebiler, M.(1971)."TheProbability ofErrorduetoIntersymbol Interference andGaussian NoiseinDigitalCommunication Systems." IEEETrans.Commun. Tech.,vol. COM-19, pp.113-119. April. Simon.M.K.andDivsalar, D.(1985)."Combined TrellisCodingwithAsymmetric MPSK Modulation:' JPLPub/.85·24,Pasadena, Calif.May. Simon,M.K.,Omura. J.K..Scholtz. R.A.,andLevitt,B.K.(1985).SpreadSpeclrum Communications Vol.I,1I,III,Computer SciencePress.RockvilJe. Md. Simon,M.K.andSmith,3.G.(1973)."Hexagonal Multiple Phase-and·Amplitude-Shift Keyed SignalSets,"IEEETrans.Commun., vol.COM-21, pp.1108-1115, October. Slepian, D.(1956)."AClassofBinarySignalmg Alphabets," BeUSysr.Tech.J.,vol.35,pp. 203-234, January. Slepian,D.(1974).KeyPapersinIheDevrlopmenr ofInformo/ion Theory,IEEEPress,NewYork. Slepian, D.andWolf.J.K.(1973)."ACodingTheorem forMultiple AccessChannels with Correlated Sources," BellSysl.Tech.J.,vol.52,pp.1037-1076. Sloane,N.J.A.andWyner,A.D.(1993).TheCollected PapersofShannon, IEEEPress,Ne" York. Siock.D.T.M.andKailath, T.(1988)."Numerically StableFastRecursive Leasl-Squares Transversal Filters," Proc.Inl.Can!ACOUSI,Speech.SignalProcessing, pp.1365-1368, New York,April. Smith,J.W.(1965)."TheJointOptimization ofTransmitted SignalandReceiving FilterforData Transmission Systems," BellSysr.Tech.J..vol.44,pp.1921-1942, December. Stenbit,J.P.(1964)."TableofGenerators forBCHCodes,"IEEETrons.Inform.Theory,vol. IT·10.pp.390-391. October. 914 DIGITAL n'M\1UNICATIOJ"<~ Stiffler,J.J.(1971).TheoryofSymhronous Communicution\', Prentice· Hall,Englewood Cliffs,N.J. Sundberg, C.E.09H6)."Continuous PhaseMuuulation,'· IEEECommun. Maf{.,vol.24.pp. 25-38,April. Suzuki,H.(1977)."AStatistical ModelforUrbanMultipath Channels withRandom Delay," IEEETrans.Comman ..vol.COM-2S. pp.673-6~O. July. Tang,D.L.andBahl.L.R.(1970)."B1ockCodesforaClassofConstrained Noiseless Channels." Inform.Control. vol.17.pp.436-461. Titsworth, R.C.andWelch,L.R.(1961)."PowerSpectraofSignalsModulated byRandom and Pseudorandom Sequences." IPLTechRep32-140.October 10. Thomas, C.M.,Weidner. M.Y..andDurrani, S.H.(1974)."Digital Amplitude-Phase-Keying withM-aryAlphahets," IEEETrans.Comman., vol.COM-22, pp.168-180, February. Tong,L.Xu,G.,andKailath, T.(994). "Blind Identification andEqualization Basedon Second-Order Statistics," IEEETrans.Inform. Theory, vol.IT-40,pp.340-349, March. Tufts,D.W.(1965).''"Nyquist's Problem-The JointOptimization ofTransmitter andReceiver in PulseAmplitude Modulation." Proc.IEEE,vol.53.pp.248-259, March. Turin.G.L.(1%1)."OnOptimal Diversity Reception," IRETrans.Inform.Theory,vol.IT-/,pp. 154-166, July. Turin,G.L.(1%2)."OnOptimal Diversity Reception II,"IRETrans.Comman. Syst.,vol.CS-12, pp.22-31.March. Turin,G.L.etal.(1972)."Simulation ofUrbanVehicle Monitoring Systems:' IEEETrans. Vehicalar Tech.,pp.9-16,February. Tzannes, M.A.,Tzannes. M.C,Proakis,J.G.,andHeller,P.N.(1994)."DMTSystems. DWMT Systems andDigitalFilterBanks," Proc.Int.ConfCommun., pp.311-315, NewOrleans, Louisiana, May1-5. Ungerboeck, G.(1972)."Theory ontheSpeedofConvergence inAdaptive Equalizers forDigital Communication," IBM1.Res.Dev.,vol.16,pp.546-555, Nmo'em>cr Ungerboeck, G.(1974)."Adaptive Maximum·Likelihood Receiver rOTCarrier-Modulated Data­ Transmission Systems," IEEETrans.Comman., vol.COM-22, pp.624-636. May. Ungerboeck, G.(1976)."Fractional Tap-Spacing Equalizer andConsequences forClockRecovery inDataModems," IEEETrans.Comman., vol.COM-24, pp.856-864, August. Ungerboeck, G.(1982)."Channel CodingwithMultilevel/Phase Signals," IEEETrans.Inform. Theory,voJ.IT-28,pp.55-6/.January. Ungerboeck, G.(1987)."Trellis-Coded Modulation withRedundant SignalSets,Parts1andII," IEEEComman. Mag.,voJ.25.pp.5-21,February. Ungerboeck, G.andCsajka, 1.(1976)."OnImproving Data-Link Performance byIncreasing the Channel Alphabet andIntroducing Sequence Coding, 1976Int.Conf.Inform. Theory, Ronneby, Sweden, June. Vaidyanathan, P.P.(1993).Multlrate SystemsandFillerBanks,Prentice-Hall, Englewood Cliffs, N.J. VanEtten,W.(1975)."AnOptimum LinearReceiver fOTMultiple Channel DigitalTransmission Systems," IEEETrans.Comman., voJ.COM-23, pp.828-834, August. VanEtten,W.(1976)."Maximum Likelihood Receiver forMultiple Channel Transmission Systems," IEEETrans.Cornman., voJ.COM-24, pp.276-283, February. VanTrees,H.L.(1968).Detection, Estimation, andModulation Theory, PartI,Wiley,NewYork. Varsharmov, R.R.(195/)."Estimate oftheNumber ofSignalsinErrorCorrecting Codes," Doklady Akad.Naak.S.S.S.R., vol.11/,pp.739-741. Verdu,S./1986a). "Minimum Probability ofErrorforAsynchronous Gaussian Multiple-Access Channels;" IEEETrans.Inform.Theory,voJ.IT-32,pp.85-%.January. Verdu,S.(I986b)."Multiple-Access Channels withPoint-Process Observalion: Optimum De­ modulation." IEEETrans.lnform.Theory,vol.IT-32,pp.642-651, September. Verdu,S.(19860). "Optimum Multiuser Asymptotic Efficiency," IEEETrans.Comman., voJ. COM-34, pp.890-897, September. Verdu,S.(1989)."Recent Progress inMuLtiuser Detection," Advances inCommunications and SignalProcessing, Springer-Verlag, Berlin.[Reprinted inMulriple AccessCommunications. N. Abramson (cd.),IEEEPress.NewYork.] REFERENCES ANDBIBLlO(,RAPH) 915 Viterbi.A.J.(1966).Principles ofCohnent Communication, McGraw-Hili. NewYork. Viterbi, A.1.(1%7)."ErrorBounds forConvolutional CodesandanAsymptotically Optimum Decoding Algonthm." IEEETrans.Inform. Theory,vol.IT-B.pp.260-269. April. Viterbl. A.J.(1969)."ErrorBounds forWhiteGaussian andOtherVeryNoisyMemoryless Channels withGeneralized Decision Regions." IEEETrans.Inform. Theory, vol.IT-15.pr· 279-287, March. Viterbi.A,1.(1971)."Convolutional CodesandTheirPerformance inCommunication Systems." IEEETrans.Commun. Tech.vol.COM-19, pp.751-772, October. Viterbi. A.J.(1978)."AProcessing Satellite Transponder forMultiple AccessbyLow-Rate MobileUsers:'Proc.FourthIn!.Conf.onDigitalSatellite Communications, Montreal. Canada. pp.166-174. October. Viterbi.A.1.(1979).'·Spread Spectrum Communication-Myths andRealities," IEEEComnwn. Mag..vol.17,pp.11-18.May. Viterbi. A.1.(1985)."WhenNottoSpreadSpectrum-A Seque!." IEEECommlln. Mag..vol.,,,. pp.12-17.April. Viterbi, A.J.andJacobs.I.M.(l975)."Ad,'ances inCodingandModulation forNoncoherenl Channels Affected byFading.PartialBand,andMultiple-Access Interference," inAdvances in Communication Systems, vol.4.A.J.Viterbi(ed.),Academic. NewYork. Viterbi.A.J.andOmura, J.K.(1979). Principles ofDigitalCommunication andCodinK. McGraw-HilI. NewYurko Wainberg,S.andWolt,J.K.(1970)."Subsequences ofPseudo-Random Sequences'" IEEETrailS Commlln. Tech.,vol.COM-lB. pp.606-612, October. Wainberg. S.andWolf,1.K.(1973)."Algebraic Decoding ofBlockCodesOveraq·aryInput. Q·arvOutputChannel. Q>q."Inform.Control, voL22,pp.232-247, April Wald.A.(1947).Sequential Anaf.vsis, Wiley,NewYork. Ward.R.B.(1965J."Acquisition ofPseudonoise SignalsbySequential Estimation." IEEEhailS Commun Tech.,vol.COM·13. pp.474-483, December. Ward.R.B.andYiu.KP.(1977)."Acquis.ition ofPseudonoise SignalsbyRecursion-Aided Sequential Estimation,"IEEE Trans.Commun., vol.COM-25, pp.784-794, August. Weber. W.J..1I1,Stanton. P.H..andSumida, JT.(1978)."ABandwidth Compressive Modulation SystemUsingMulti-Amplitude Minimum-Shift Keying(MAMSK)." IEEETrans Commun, voLCOM-2b. pp.543-551, May. Wei,LF.(1984a). ;'Rotationally Invariant Convolutional Channel Coding withExpanded SignalSpace.PartI:]&JO."IEEE1.Selecred ArtasCommun., voLSAC·2,pp.659-671, Sep. tember. Wei.L.F(l984b). "Rotationally Invariant Convolutional Channel CodingwithExpanded Signal Space,PartII::"Jonlinear Codes," IEEEJ.Selected AreasCommun., vol.SAC~2.pp.672-6xn. September. Wei.LF.(1987)."Trellis-Coded Modulation withMulti-Dimensional Constellations," IEFF Trans.Inform.Theory. voLIT-33.pp.483-501. July. Weinstein. S.B.andEbert.P.M.(1971)."DataTransmission byFrequency-Division Multiplex.i­ ingUsingtheDiscrete Fourier Transform." IEEETrans.Commutl., vol.COM-19. pp. 628-634. October. Welch.L.R.(1974)."LowerBoundsontheMaximum CrossCorrelation ofSignals," IEEETrail"". Inf(>rm.Theorv,voLIT·20,pp.397-399. May. Weldon. E.J..Jr.(19711."Decoding BinaryBlockCodesonQ·aryOutputChannels." IEFE bans.In.form.Theor}.vol.IT-17,pp.713-718, November. Widrow. B.(1966). "Adaptive Filters, 1:Fundamentals:' Stanford Electronics Laboratory. Stanford University. Stanford, Calif.,TechReportNo.6764~6.December. - Widrow. B.(1970)."Adaptive Filters," AspecrsofNetwork andS-"stemTheory, R.E.Kalmanand N.DeClaris (eds,),Holt.Rinehart andWinston, NewYork. Widrow, B.andHoff,M.E..Jr.(1960)."Adaptive Switching Circuits." IREWESCON (0'1t' Ree..pI.4.pp.96-104. Widrow, B.eraf.(l975}."Adaptive NoiseCancelling: Principles andApplications," Proc.lEFt.". vol.63.pp.1692-1716, December. 916 DIGITAL COMMUNICATIONS Wiener, N.(1949).TheExtrapolation, Inrerpolation, andSmoothing ofStationary TimeSerieswith Engineering Applications, Wiley,NewYork(reprint oforiginal workpublished asanMIT Radiation Laboratory Reportin1942). Wintz,P.A.(1972)."Transform PictureCoding," Proc.IEEE,vol.60,pp.880-920, July. Wolf,J.K.(1978)."Efficient Maximum Likelihood Decoding ofLinearBlockCodesUsinga Trellis," IEEETrans.Inform.Theory.vol.IT·24,pp.76-81,January. Wozencraft, J.M.(1957)."Sequential Decoding forRehable Communication," IRENat.Conv. Rec,vol.5,pI.2,pp.11-25. Wozencraft. J.M.andJacobs.I.M.(1%5).Principles ofCommunication Engineering, Wiley,f'ew York. Wozencraft, J.M.andKennedy, R.S.(1966)."Modulation andDemodulation forProbabilistic De<oding," IEEETrans.Inform.Theory,vol.1T·12,pp.291-297, July. Wozencraft,1. M.andRieffen, B.(1%1).Sequential Decoding, MITPress,Cambridge, Mass. Wyner.A.D.(1965)."Capacity oftheBand·Limited Gaussian Channel," Bull.Syst.Tech.J.,vol. 45,pp.359-371, March. Xie,Z.,Rushforth, C.K.,andShort,R.T.(l990a). "Multiuser SignalDetection UsingSequential Decoding," IEEETrans.Comman., vol.COM·38, pp.578-583, May. Xie,Z..Short,R.T.,andRushforth, C.K.(199Gb)."AfamilyofSuboptimum Detectors for Coherent Multiuser Communiccftions." IEEEJ.Selected AreasCommun., vol.SAC-8, pp. 683-690, May. Yao,K.(1972). "OnMinimum Average Probability ofErrorExpression forBinaryPulse­ Communication SystemwithIntersymbol Interference," IEEETrans.Inform. Theory, vol. 1T-18,pp.528-531, July. Yao,K.andTobin,R.M.(1976)."Moment SpaceUpperandLowerErrorBoundsforDigital Systems withIntersymbol Interference," IEEETrans.Inform. Theory. vol.IT-22,pp.65-74. January. Vue,O.(1983)."Spread Spectrum MobileRadio1977-1982," IEEETrans.Vehicalar Tech.,vol. VT·32,pp.98-105,February. Zelinslci, P.andNoll,P.(1977)."Adaptive Transform CodiogofSpeechSignals." IEEETrans. Acoustics, Speech,SignalProcessing, vol.ASSP·25, pp.299-309, August. Zervas,E.,Proakis, J.G.,andEyuboglu, V.(1991)."AQuantized Channel Approach toBlind Equalization," ProcICC'9I,Chicago,II,June. Zhang,X.andBrady,D.(1993)."Soft·Decision Multistage Detection ofAsynchronous AWGN Channels," Proc.3/s1Allerlon Con!onCommun., Contr.,CompoAllerton, II.,October. Zhou,K.and Proakis, J.G.(1988)."Coded Reduced·Bandwidth QAMwithDecision·Feedback Equalization," Con!Rec.IEEEInt.Con!Comman., Philadelphia, Penn.,pp.12.6.1-12.6.5, June. Zhou,K.,Proakis, J.G"andLing,F.(1987)."Decision·Feedback Equalization ofFading Dispersive Channels withTrellis-Coded Modulation," 1m.Con!Commun. Tech.,Nanjing. China,November. Zhou,K.,Proakis, J.G.,andLing,F.(1990),"Decision· Feedback Equalization ofTime· Dispersive Channels withCodedModulation," IEEETrans.Commun., vol.COM·38, pp. 18-24,January. Ziemer, R.E.andPeterson, R.L(1985).DigitalCommunications andSpreadSpectrum Systems, Macmillan, NewYork. Zigangirov, K.S.(1966)."SomeSequential Decoding Procedures," Probl.Peredach. Inform., vol. 2,pp.13~25. Ziv,J.(1985)."Uni"rsal Quantization," IEEETrans.Inform.Theory,vol.31,pp.344-347. Ziv,J.andLempel, A.(1977)."AUniversal Algorithm forSequential DataCompression," IEEE Trans.Inform.Theory,vol.IT·23,pp.337-343. Ziv,J.andLempel, A.(1978). "Compression ofIndividual Sequences viaVariable-Rate Coding," IEEETrans. Inf~rm.Theory,vol.IT·24,pp.530-536. Zvonar, Z.andBrady,D.(1995)."Differentially Coherent Multiuser Detection InAsynchronous CDMAFlatRayleigh FadingChannels," IEEETrans.Comman., vol.COM·43, to>ppear. Adaptive equalization, 636-676 Adaptive equalizers, 636-676 (SeearsoEqualizers) blind,664--675 decision-feedback, 621-625, 649-650 linear,584-601, 648-649 baseband, 648 passband, 648-649 maximum likelih<XKt sequence estimator, 607-616, 652-654 Adaptive transform coding,137 Algorithm: Constant-modulus, 670 Godard,670-673 Huffman, 99-103 Kmeans,122 Lempel-Ziv, 106-1OB Levinson-Durbin, 12&139,879-881 LMS(MSE),639-642 recursive least-squares (RLS),654-664 RLS(fast),660 RLS(Kalman), 656-658 RLSlall;ce,660-664 RLSsquare-rool, 660 stochastic gradient. 668 zero-forcing, 637-638 Amplitude distortion, 535 Analog sources. 82 quantization of,IOB-125 optimum, 113 scalar,113-118 vector,118 sampling of,72-73INDEX Antenna: beamwidth,317 ellectivearea,316 effective radiatedpower,316 iJlumination efficiency faclor.3J7 Aposteriori probabitity,21 Aprioriprobability, 21 Autocorrelation function, 64 atoutputoflinearsystem,68-70 ofcyclostationary process,75-76 Autocovariance function, 64 Automotic gaincontrol(AGC),336 Average powerdensityspectrum,n Averages, 33-37 centralmoments, 33 characteristic function, 35-37 forsumofstatistically independent random variables, 36 correlation, 34 covariance, 34 covariance matrix,34 expected value(mean),33 jointmoments, 34 ofstochastic processes, 64-67 variancp.. 33 AWGN(additive whiteGaussian noise)channel, 233­ 234 Band-limited channels, 534-540 (Seea/soChannels) Bandpass signals,152-157 complex envelope of,159 envelope of,155 917 918 INDEX Bandpass signals(Cont.): pbaseof,J55 quadrature components, 155 Bandpass system,157-159 response of.157-159 Bandwidth efficiency, 283-284 Bandwidth expansion factor.444.807 Baseband signals,176 delaymodulation, l&l Miller,188 NRZ,187 NRZI.187 powerspectraof,220-223 Baudotcode,13 Bayes'theorem, 21 BCH(Bose-Chaudhuri-Hocquenghemj codes,435-436 Bibliography, 899-916 Binarysymmetric channel(BSC),381 capacity of,381 transition probability, 376-377 Binomial distribution. 37-38 Biorthogon.1 signals,183 Bitinterval. 174 Blindequalization, 664-675 constant modulus algorithm, 670 Godardalgorithm, 670-673 jointdataandchannel estimation. 667-668 maximum-likelihood algorithms. 664-667 stochastic gradient algorithms. 668-669 withsecond~order moments, 673-675 Blockcodes,413-468 binary.4 concatenated. 467-468 cyclic,423-436 Bose-Chaudburi-Hocquenghem (BCH),435-436 encoders for,430-435 generator polynomial for,437-438 Golay,433 Hamming, 433 maximum-length shift-register (MLSR), 433-435 tableofMLSRconnections, 435 dualcode,426 equivalent, 418 errorcorrection capability, 451-452 errordetection capability, 451-452 extended, 420 fixed-weight, 414 generator matrix.4J7 generator polynomial, 424 Golay,423,433 extended, 423 generator polynomial of,433 performance onAWGNchannel, 454-455 weightdistribution, 423Blockcodes(Com.): Hadamard,422-423 Hamming, 421-422 hard-decision decoding. 44.'\-456 linear,413-468 maximum~distance-separable. 461 message polynomial. 424 minimum distance hounds. 461~4(,4 Elias,463 Gilhcrt-Varsharmov.46.1: Hamming, 462 Plotkin, 462 nonbinary.464-468 nonsystematic.418 nullspace,416 parity-check matrix.419 paritypolynomial, 426 perfect.453 quasi-perfect, 454 rate.2.414 reciprocal pOlynomial. 426 Reed-Solomon. 464-466 shortened. 421 sofl-decision decoding. 436-445 standard array.447 syndrome. 449-451 systematic.41X Blocklength,414 Bursterrors.469 Bursterrorcorrection capability. 469 Capacity (set'Channel capacity) Carrier, 159 Carrierphaseestimation Costasloop,355-356 decision-directed. 347-350 MLmethods, 339-341 nondecision directed. 350-358 phase-locked loop,341-346 squaring loop,353-.155 Carrierrecovery. 336-358 C.nchy-Schwartz inequality, 165 Centrallimittheorem. 61-62 Central momcms, 33 Channel: additive whitegaussian noise(AWGN),233-234 band-Ilmited,534-54O binarysymmetric. 315-376 "",,"city, 380-386 AWGN, 381-386 bandlimitedAWGN,383-386 DMC,376-377 infinitebandwidth AWGN,385 coherence bandwidth. 764 Chann\-,j (Corn): coherence time.765 l'ututfratc.Jt,l4 forS'fstl.'mdesign.4(()-406 d'iscrc(~ mcrnorylcss (OMe).376-377 discn...·(c-timc mod~l. 5X6-5~ distortion. 534-540 amplitude ..';35 c:nvdopc...· delay.5J5 frequency offset. 5J~ Impulsenoise.5.l-R nonlinear. 537 phasejiner.535 squared-error. lOS thermal noise.5JS Distortion-rate function. 110 Dopp)~r powerspectrum. 7f,S Doppler spread.765 encoder. 1-2 coderafoe.2.414 codeword.2 fadingmultipath: characterization of.759-769 correlation functions for.763-767 impulse response. 760-761 modelsfor,767-769 transfer fumction. 763 fiberoptic..5 frequency nonselective. 764.772-795 digitalsignaling over.772-795 frequency selective. 764.798-806 digitalsignaling over.795-806 errorratefor.798-806 RAKEdemodulator for,797-S06 tapweightestimation of.801-803 tappeddelaylinemodelof.795-797 microwave LOS.767-769 modelsfor,11-13,375-380 additive noise.11j binarysymmetric, 375-376 discrelememoryless, 376-377 discrete-time. 586-588 linearfilter.II linear.time-variant filter.12 waveform, 378-380 multipath spread,763 Nakagami fading,761 overspread~ 771 Rayleigh fading,761 binarysignaling over,772-776 codedwaveforms for.806-832 cutoffratefor,825-832 frequency nonselective. 764 M-aryorthogonal signaling over,787-792 multiphasesignaling over.785-787INDEX919 Channel (Cont.): Riccanfading,761 scattering function. 766 spreadfactor.771 lallie,772 storage. 10 underspread. 771 underwater acoustic. 9 wireless..5 wirdinc.4 Channel encoder. 2 Channel reliability function. 389 Characteristic function ..35-37 ofbinomial. JS ofchi-square. 42-44 ofgaussian. 41 ofmultivariate gaussian. 49-52 ofuniform,39 Chebyshev inequality, 52-54 Chernoff bound,53-57 forBSC,455 forRayleigh fadingchannel, 792-794 Chi·square distribution. 41~45 central.42-43 noncentra!' 42-44 Codedivisionmultiple access(CDMA) asynchronous. 852-854 effeetive SNR,861 efficiency of.861 optimum receiver for.851-854 suboptimum detectors for.854-861 decorrelating, S55-857 MMSE,858-859 performance. 859 singleuser.854 synchronous. 851-852 Coderate,2 Codeword,2 fixedlength.94 variable lenglh(Huffman), 96-103 Codedmodulation, 511-526 Codes: source: instantaneously decodable, 96 uniquely decodable, 96 (SeealsoBlockcodes;Convolutional codes) Coding: entropy, 97,117 forAWGNchannel: blockcodes.413-468 convolutional codes,470-511 forBSC(seeBlockcodes;Convolutional codes) forRayleigh fadingchannel, 806~832 concaten.ted, 814-825 constanl-weight codes,814-825 920INDEX Coding(COni.): forRayleigh fadingchannel(Conr.): convolutional codes.811-814 cutolfrate,825-829 linearblockcodes,808-814 trelliscodes.830-832 Huffman (entropyJ,96-103 noiseless. 93-108 speech,143-144 Codinggain,441.507,733 Compandor, 127 Comparison ofdigitalmodulation, 282-284 Complementary errorfunction. 40 Complete orthonormal functions, 165-168 Complex envelope, 155 ofnarrowband process.155 Computational cutoffrate,503 (Seealsocutoffrale) Concatenated blockcodes.467-468 Concatenated convolutional codes,449-500 Conditional cdf(cumUlative distribution function), 26-28 Conditional pdf(probability densityfunction), 25 Conditional probability, 20 Consistent estimate (seeEstimate) Constraint length.470 Continuous-phase frequency-shift keying(CPFSK), 190- 191 performance of,284-301 powerdensityspectrum of,209-219 representation 01,284-285 Continuous-phase modulation (CPM),191-203 demodulation: rnaximum·likelihood sequence estimation, 284-2&9 multiamplitude, 200-203 multi-h,295 performance of,290-296 symbol'by-symbol, 296-300 fullresponse, 192 minimum·shift keying(MSK),196-199 modulation inde\.191 multiamplitude, 200-203 multi-h. 295 partialresponse, 192 ph.secylinder, 195 phasetreesof,192 po",erspectrum of,209-219 representation of,190-196 signalspacediagram for,199-200 statetrellis,196 trellisof,195 Continuously variableslopedeltamodulation (CVSD), 135 Convolutional codes.470-511 applications of.506-511 hinary,410-476Convolutional codes(Cont.): catastrophic errorpropagation. 482 concatenated, 492.499-500 constraint length.410 decoding, 483-486 Fanoalgorithm. 5OD-503 feedback, 505-506 sequential, 500-502 stackalgorithm. 503-504 Viterbi.483-486 distance properties of.492-496 dual-k.492-499 encoder, 470-478 generators, 471-472 hard-decision decoding. 489-492 minimum freedistance. 479 nonbinary, 492-499 optimum decoding of.483-485 performance onAGWNchannel,486-492 performance onBSC.489-491 performance onRayleigh fadingchannel.811-814 quantized metrics,5<:8--510 soft-decision decoding. 486-489 statediagram, 474-477 tableofgenerators formaximum freedistance. 493-497 transfer function, 477-480 treediagram, 472 trellisdiagram. 473 Correlation demodulator. 234-238 metricsfor.246 Correlative statevector,286 Coset,447 Cosetleader,447 Covariance. 34 Covariance function. 65 Cross-correlation function. 65 Cross-power densityspectrum. 68 Cumulalivedistribution {unction (edO,23 Cutoffrate.394 comparison withchannel capacity. 399-400 forbinarycoded signals~396 forM-aryinput.M-aryoutputvectorchannel. 40J lormultiamplitude signals.397-399 fornoncoherenn channel. 405-406 forq-aryinputQ-aryoutputchannel, 400-401 systemdesignwith.400-406 CWjamming, 706 Cycliccodes(secBloclcodes,cydil') ('yclostalionary proceSS, 75-76.205 Datacompression. 1 Datatranslation codes~5fo6 Decision-feedback equalizer (st't'Equalilers, decislon­ leedhack) Decoding ofblockcodes: forfadingchannels: hard-decision, 81J soft-decision,808-81l hard-decision, 445-456 boundsonperformance forBSC,452-455 Chernoff bound,455 syndrome, 449-451 tablelookupmelhod, 447-448 soft-decision, 436-445 boundsonperformance forAWGN. 440-443 comparison withhard-decision decoding, 456-461 Decoding ofconvolutional codes: forfadingchannel, performance. 8JJ-814 feedback,505-506 hard-decision, 489-492 performance onAWGN channel, 486-492 performance onBSC.489-491 sequential, 500-502 SDfldecision, 486-489 slackalgorithm. 503-504 Vlterbialgorithm, 483-486 Delaydistortion, 535 Delaypowerspectrum, 762 Deltamodulation (seeSource,encoding) Demodulation/Detection carrierrecovery for,337-358 Costasloop,355-356 decision-directed, 347-350 MLmethods, 339-341 Jlon-decision-direcled.350-358 squaring PLL.353-355 coherent: ofbinarysignals,257-260 ofbiorlhogonal signals, 264~266 comparison of.282-284 ofDPSKsignals,274-278 ofequicorrelated signals,2M ofM-arybinarycodedsignals,266-267 optimum, 244-257 oforthogonal signals.260-264 ofPAMsignals,267-269 ofPSKsignals.269-274 ofQAMsignals,278-282 correlation-type, 234-238 ofCPFSK. 284-289 performance, 289-301 forintersymbol interfer.ence, 584-627 matched filter,type, 238-244 maXimum-likelihood, 244-254 maximum likelihood sequence, 249-254 noncoherent, 302-3/3 ofbinarysignals.302-308 ofM-aryorthogonal signals,308-312 multichannel,68O-686INDEX921 Demodulation/Deleclion (Cont,): noncoheTent (Coni.): optimum, 302-312 symbol-by-symbol,254-256 Differentia] encoding, 187 Differential entropy. 92 Differenlial phase-shift keying(DPSK), 274-278 Digita'communication systemmodel.1--'\ DigitaLmodulator, 2 D~rectsequence (seeSpreadspectrum signals) Discrete memoryless channel(DMC). 376-377 Drscretc random variable. 23 D~stance (seeBlockcodes;Convolutional codes. minimum freedistance) Distortion (SeealsoChannel distortion): fromquantization. 113-1"25 granular noise.134 slopeoverload, 134 Distortion ralefunction. 110 Distributions (seeProbability distributions) Diversity: antenna, 177 frequency, 777 performance of.777-795 polarization, 778 RAKE,778 time,777 Double-sideband modulation. 176 DPCM(Differemial puJsecodemodulalion) (seeSource, encoding) DPSK(differenlial phase-shift keyiOg), 274-278 Dualcode,426 Dual-kcodes,492-499 Duobinary signal,548-549 Early-Iatc gatesynchronizer. 362~36S Effective anlenna area,316 Effeclive radiated power,316 Eigenvalue, 164 Eigenvector, 164 Eliasbound,461-463 Encoding (seeBlockcodes;Conventional codes) Energy, 156 Ensemble averages, 64-65 Entropy, 88 conditional, 88 differential. 92 discrete memoryless sources, 94-103 discrete stationary sources. 103-106 Entropy coding.96,117 Envelope. 155 Envelope delection, 306 Equalizers (SeealjoAdaptive equalizers) decision-feedback, 621-627. 649-650 922 INDEX Equalizers (ConI.): decision-feedback (Cont.): adaptive, 649-652 examples ofperformance. 622-623 o(Irellis·coded signals,650-652 minimum MSE,622 predictive form.626-627 linear.601-620.648-649 adaptive, 636-644 convergence ofMSEalgorithm, t~2-644 errorprobability. 613-617 examples ofperformance, 613-617 excessMSE.644-648 fnctionally spaced.617-620 LMS(MSE) algorithm. 639-642 limitonstepsize.645-646 mean-square error(MSE)criterion, fiJ7-620 minimum MSE.610-611 outputSNR(or.605.610 peakdistortion, 602 peakdistortion criterion, 602-607 zero-folcing. 603-604. 637-638 ma:.imum-likeJihood sequence estimalion. 584-586, 589-593.607-616 sel(·recovering (blind).644-675 \\ithtrellis-coded modulation. 650-652 usingtheViterbialgorithm. 589-593 channelestimator for,652-654 performance of.593-601 Equivalent codes,418 Equivalent lowpassimpulse response. 157-1S8 Equivalent lowpasssignal.155 Equivocation. 90 Errorfunction. 40 Errorprobability: coherent demodulation: binarycoded.266-267 forbinarysignals.257-260 forDPSK.274-278 forM-arybiorthogonal. 264-265 forM-aryequicorrclaled. 266 forM-aryorthogonal, 260-263 (orM-aIYPAM.267-269 (orPSK.269-274 forQAM.278-282 unionboundfor.263-204 multichannel. 680-6H6 noncohcrcnt dcmodulatlon. 301-313 forbinarysignsls.301-308 forM-aryorthogonal. 308·-312 Estimate: biased.367 consistent ..'i9.36;": efficrenr, 368Estimate (ConI.) unbiased. 367 Estimate ofphase{SeealsoCarrierphaseestimation) clairvoyant. 889 pilotsignal.889 Estimation. maximum-likelihood sequence (MLSE). 249­ 254 Estimation: maximum likelihood. 334-335 o(carrierphase.337-358 ofsignalparameters. 333-335 o(symboltiming.358-365 ofsymboltimingandcarrierphase.365-371 performance of.367-370 Euclidean: distance, 251 weight.595 Events. 18 intersection of.19 joint.19 mutually exclusive. 19 null.\q plObabllity of.19 unionof.19 Excessbandwidth. 546 ExcessMSE.644-648 Expected value.33 Expurgated codes.816-817 Extended codc.420 Extcnsiun field.415 Eyepattcrn. 541 Fadingchannels. 8.758-839 (SeealsoChannels) Feedback decoding. 505-506 FHspreadspectrum signals(ueSpreadspectrum signals) Filter: integrator. 238 matched. 239 Foldcdspectrum. 606 Follower jammer. 731 Fouriertransform. 35 Freeeuclidian distance. 517 Free-space pathloss.317 Frequcnc~· diversity. 777 Frequl'ncy division multiple a:;cess(fDMA). 842~844 Frequency-hopped (FH)spreadspectrum (sct'Spread spectrum signals) Frequency-shift keying(FSK).181-183. 190-1Yl continuous-phase (CPFSK): performance of.284~301 powerdenSityspectrum of.213-217 representation of,190-1']1 FUnCliOJ1~ ofrandomvariables. 28-32 INDEX923 GaloIsfield,415 Gamma function, 42 Gaussian dislrrnution, 39-41 multivariate. 49-52 Gaussian noise,11 Gaussian random-process. 65 Gaussian randomvariables. lineartransformation of, 50-52 Generator matrix,417 GencralOr polynomial, 424 Gilberl-Varsharmov bound,463 Golaycodes,423,433 cXlendcd. 423 generator polynomial of.433 performance onAWGNchannel, 454-455 Goldsequences, 727 Gram-Schmidt procedure, 167-173 Granular noise.134 Grayencoding. 175 Hadamard codes,422-423, H17-821 Hamming boundonminimum d~stance. 462 Hamming codes,421-422, 433 Hamming distance. 415 Hard-decision decoding: hloekcodes,445-456 convolutional codes.489-492 Hilhcrttransform. {54 Huffman coding,96-103 H1uminahon efficiency f2ctvr,317 Impulse noise.53K Impulse response. hK Independent events, 21 Independent randomvariahks. 2H InforamflOn. H4-85 equivocation. 90 measure of.M-91 mutual. H4 average, H7 self-,H5 average(entropy). ~ sequence. 3,M3 Interleaving. 468-470 hlock,469 convoJwional.470 Intcrsymhol interference. 536-5':\7 controlled (seeParhalrl'sponsc signals) discrete-time modelfor.5H6-5~9 equivalent whitenoisi:filtermodel.5KH Gptimum demodulator for,584-593 InversetlItcr.603Jacobian, 32 Jamming margm.707 Jointedf(cumulative distribution function), 25 Jointpdf(probahiltiy densityfunction), 25 Jointprocesses. 65 Kalman(RLS)algorilhm, 656-65H fast,660 Kasamisequences. 729 Kraftinequality, 97-98 Laplaceprobability densityfunction, 56 Lattice: filter,660-664 recursive least-squares. 664 Lawoflargenumbers (weak).59 Lea.tfavorable pdf,305 Lca.")l-squarcs algoritl1ms. 654-664 Lempel-Zlv algorithm, 106-108 Levinson-Durhin algorithm, 12H.1.19,H79-881 Likelihood ratio,.104 Linecodes.566 Linearcodes(seeBlockcodes.linear; Convolutional codes) Linearequalb'ation (J"eeEqualizers. linear) Linear-feedback shift-register. maximal length.433-435, 124-727 . Linearprediction, 12H-130, 138-144, 600-664 hackward,6I>1-662 forward, 661-662 residuals. 663 Linearprcdiclivc coding(LPC): speech,138-144 Lineartime-invariant system. OH~69 n..'sponsctostochastic input.68-72 Lineartransfmmation ofrandom ....ariahles, 2X-29. 50-S~ Linkhudgetanalysis, 316-319 Linkmargin, :\19 Lloyd-Max qllantizcr.In Lowpai;s signal.155 Lowpass system, 157 Lowprohahillty ofintercept. 696.715-71(, Magnetic recording. 567-56M normalized density. 567 Majority logicdecoder, 50~i Mapping hysetpartitioning, 512 Marginal probahility density, 26 Marcum's Q-function, 44 Mark-Oil chain.1S9 transition probahility matrixof.IXl.} Matched filler,238-244 Maximal ratiocombining. 779 performance of,71«1-7H2 924 INDEX Maximum aposteriori probability (MAP) criterion, 245,254-257 Maximum freedistance codes,tablesof,492-496 Maximum lengthshih-register codes,433-435,724-727 Maximum Jikehhood: parameter estimation, 333-335, 339-341 forcarrierphase,339-341 forjointcarrierandsymbol, 365-367 forsymboltiming,358-364 performance of,367-370 Maximum-likelihood criterion. 245-246 Maximum-likelihood recei,er, 233-257 Maximum-likelihood sequence estimation (MLSE), 249- 254 Mean-square error(MSE)criterion, 607-617 Meanvalue,33 Microwave LOSchannel, 768-769 Millercode,188,575 Minimum distance: boundson,461-464 definition, 416 Euclidean, 173 Hamming, 416 Minimum-shift keying(MSK),196--199 powerspectrum of,213-219 Models: channel, 375-386 source,82-84,93-95 Modified duobinary signal,549-550 ModuJation: binary,257-260 biorthogonal, 264-266 comparison of,282-284 continuous-phase FSK(CPFSK), 190-191 powerspectrum, 213-219 DPSK,274-278 equicorrelated (simplex), 266 index,191 linear,174-186 powerspectrum of,204-209 M-aryorthogonal, 260--264 multichannel, 680-686 nonlinear, 190-203 offsetQPSK,198 PAM(ASK),267-269 PSK,269-274 QAM,278-282 Modulation codes,566-576 (SeealsoPartialresponse signals) capacity of,569 Millercode,573 NRZ,574 NRZI,566,568,574-575 run-length limited,568-576Modulation codes(Com.): run-length limlted(ConI.): fixedrate,572 statedependent, 571 stateindependent, 571 Modulator: binary,2 digital,2 M-ary,2 Moments, 33 MorseCode,V Multicarrier communications capacity of,687-689 FIT-based system,689-692 Multichannel communications~ 680-686 withbinarysignals,682-684 withM-aryorthogonal signals,684-686 Multipath channels, 8,758-839 Multipath intensity profile,762 Muftipath spread,763 Multiple accessmethods, 840-849 capacity of,843-849 CDMA, 843,849-862 FDMA,842 randomaccess,962-872 TDMA,842 Multiuser communications, 840-872 Multivariate gaussian distribution, 49-52 Mutualinformation, 84 average, 87-88 Mutually exclusive events.18 Narrowband interference. 704-706 Narrowband process. 152 carrierfrequency of,1.53 Narrowband signal,152 Noise: gaussian. 162 white,162-163 Noisychannelcodingtheorem, 386-387 Noncoherent combining loss683-684 Nonlinear distortion, 537 Nonlinear modulation, 190 Nonstationary stochastic process, 63 Norm,165 Normalequations. 128 Normalrandom variables (setGaussian distribution) Null"ent,18 Nullspace,416 Nyquistcriterion, 542-547 Nyquistrate,14.72 Offsetquadrature PSK(OQPSK), 198 On-offsignalling (OOK), 321 Oplimum demodulalion: (seeDemodulation/Delection) Orthogonal signals.165-166 Orthogonality principle. mean-square est\mation. 608 Orthonormal: expansion. 165-173 functions, 165-166 Paritycheck.417 matrix.419 Pari!)"polynomial. 426 Partial-band interference. 734-i41 Partialresponse signals.548-560 duobinary. 548-549 errorprobability of.562-565 modified duobinary, 549 precoding for,551-555 Partial-time (pulsed) jamming, 717-724 Peakdistortion criterion. 602-607 Peakfrequency deyjatJon. )90 Perfectcodes.453-454 Periodically stationary, widesense, 75-76~205 Phasejiller.538 Phase-locked loop(PLLj,341-346 Costas,355-356 decision·djrectcd.347-350 M-Iawtype.356-358 non·decision-dirccted,350-351 square-law type.353-355 Phase-shift keying(PSK).177-178.269-274 adaptive receptIon of,887-896 pdfofphase.270-271 performance forAWGNchannel, 271-274 performance forRayleigh fadingchannel, 780-787. 887-894 Plotkinboundonminimum distance, 462 Powerdensityspectrum, 67-68.204-223 atoutputoflinearsystem. 69 ofdigitally modulated signals.204-223 Predichon (seeLinearprediction) Preferred sequences., 727 Prefixcondition, 96 Probability: apriori,21 aposteriori. 21 conditional, 20,26-28 ofevents.18 joint,19,25-26 Probability densityfunclion (pdf),24 Probability distribution funchon, 23 Probability distributions, 37-52 binomial,37-38 chi-square. 41-45 central,42-43 noncentral, 42-44INDEX925 Probability distributions (Con'.): gamma, 43 gaussian, 39-41 multivariate gaussian. 49-52 Nakagami, 48-49 Rayleigh, 45-46 Rice,47-48 uniform. 39 Probability transition matrix.377 Processjng gain,707 Pseudo-noise IPN)sequences: autocorrelation function. 725-726 generation viashiftregister. 724-729 Gold.727 Kasami.729 maximal-Iength,725-726 peakcross-correlation, 726-727 preferred. 727 (SeealsoSpreadspeclrum s;gnals) Pulseamplitude modulation (PAM), 174-176,267-269 Pulsecodemodulation (PCM),125-133 adaptive (ADPCM). 131-133 differential (DPCM), 127-129 Pulsedinterferen-ee,717 effecfonerrorrateperformance. 717-724 Quadrature amplitude modulation (QAM), 178-180. 278-282 Quadrature components. 155 ofnarrowband process. 155-156 properties of,161-162 Quantization, 108-125 block,118-125 optimization (Lloyd-Max), 113-118 scalar,113-118 vector,118-125 Quantization error,125-133 Quasiperfect codes.454 Raisedcosinespectrum. 546 excessbandwidth. 546 rolloffparameter, 546 RAKEcarrel.tor,797-798 RAKEreceiver: forbinaryantipodal signals,798-803 forbinaryorthogonal signals,BOI-802 forDPSKSignals,804 fornoncoherent detection oforthogonal signals.805 RAKEmalched filler,799-800 Random access,862-872 ALOHA, 863-867 carriersense,867-872 withcollision detection. 868 nonpersistent, 868 926 INDEX Random access(Cone,): carrierse.nse(Cont.): I-persistent. 869 p-persistent, 869 offeredchannel traffic.864 slottedALOHA, 864 throughput. 865-867 unslotted. 864 Random coding.390-4()(J binarycodedsignals,390-397 multiamplitude signals,397-399 Random Processes (seeStochastic processes) Random variables, 22-28 function of.28-32 multiple, 25 orthogonal. Y; single,22-24 statistically independent. 2X sumsof,58-63 centrallimittheorem. 61-62 transformation of.28-32 Jacobian of,32 linear,28,32,49-52 uncorrelated. 34 Rate: code,2,414 ofencoded information (seeSuurceencoding) Ratedistortion function, 108-113 ofbandlimited gaussian source.112 ofmemorylessgaussian source, 109-110 tableof.112 Rayleigh distribution, 45-46 Rayleigh fading(seeChannel, fadingmultipath; Channel, Rayleigh fading) Reciprocal polynomial, 426 Recursive leastsquares (RLS)algorithms, 654-664 fastRLS,660 RLSKalman, 656-6Nl RLSlattice.660-664 Reed-Solomon codes.464-466 References, 899-916 Reflection coefficients. 140 Regenerative repeaters, 314-316 Residuals. 663 Ricedistribution. 47-4g Riceanfadingchannel, 761 Run-length limitedcodes, 568-576 fixedrate,572 statedependent. 571 stateindependent. 571 Sample function, 63 SampleOlean,58 Samplespace.17-18Sampling theorem, 72-73 Scattering function, 766 Self·information,85 average(entropy). 88 Sequential decoding. 501-503 Setpartitioning. 512 Sho.nnon limit.264 Shortened code,421 Signalconstellations: PAM.174-176 PSK,177-178 OAM.178-18O Signaldesign.540-576 forband-limited channel. 540-551 forchannels withdistortion. 557-560 fornointersymbol interference, 540-547 withpartialresponse pulses,548-551 withraisedcosinespectral pulse,546-547 Signal-to-noisc ratio(SNR),258 Signals: bandpass. 152-157 haseband. 176,186-189 binaryantipodal, 257 binarycoded,266-267 hinaryorthogonal. 258 hiorthogonal. 183-184.264-266 carrierof.1S9 characterization of.152-163 complex envelope of,155 digitally modulated, 173-209 cyclostationary. 204-206 representation of,173-202 spectral characteristics of,202-213 discrete-time, 74-76 energyof.156 envelope of.ISS equivalent lowpass. 155 lowpass.155 M-aryorthogonal. 181-183 multiamplitude. 174-176 multidimensional. 180-181 multiphase, 177-178 narrowband, 152 optimum demodulation of,233-257 quadrature amplitude modulated (OAM), 178-180 quadrature components of,155-156 properties of,161-162 simplex. 184,266 speech,143-144 slochastic, 62-77,159-163 autocorrelation of.64.68-70.75-76 autocovariance. 64 bandbass stationary. 159-163 crosscorrelation of,65 Signals(Cont.): stochastic (ConI,): ensemble averages of.64-65 powerdensityspectrum, 67-68.204-223 properties ofquadrature components. 161-162 whitenoisc.162-163 Signature sequence. 843 Simplex signals.266 Single-sideband modulation, 176 Skindepth,9 SJopeoverload dIstortion. 134 Slopeoverload distortion. 134 Softdecision decoding: hlockcodes,436-445 convolutional codes. 4~o~4~9 Source: analog,K2-K3 binary,83 discrete memoryless (DMS),82-83 discrete stationary. IU3-106 endoding, 93-144 adaplive DM,135-136 adaptive DPCM, 131-133 adaptive PC\!.131-1.13 deltamodulation (DM),133-136 differential pulsecodemodulation (DPCM), 127-129 discrete memory less.94-103 Huffman, 99-103 Lempel-Ziv, 106-I~ linea.predictive coding(LPC),UK-142 pulsecodemodulation (PCM), 125-127 models,82-84 speech,143-144 spectral, 136-138 waveform, 125-144 Sourcecoding,82-144 Spaced-frequency. spaced-time correlation function, 763 Spectrum: ofCPFSK andCPM,209-219 ofdigitalsignals,203-223 oflinearmodulation. 204-2U9 ofsignalswithmemory, 220-223 Spreadfactor.771 lat>1eof,771 Spreadspectrum multiple access(SSMA). 7\6 Spreadspectrum signals: acquisition of.774-74X (orantijamming. 712-715 forcodedivision mUlliple access(COMA). 69ft.716­ 717,741-74:1 concatenated codesfor,711-712, 740-741 directsequence, 697-700 applications of.712-717 codingfor,710-712tNDEX91:7 Spread5pcctrum signals(Conr.): directsequence (Cont.): demodulation of,701-702 performance of.702-712 withpulseinterference. 717-724 examples ofDS,712-717 frequency-hopped (FH),729-743 hlockhopping, 731 follower jammer (or.731 performance of.732-7.14 withpartial-band interference. 734.741 hyhridcombinations. 743-744 forlow·probat>ility ofintercept ILPI),696.715-716 formultipath channels, 795-806 synchronization of.744-7.52 time-hopped (TH),743 trackingof.748 uncoded PN,70S Spreadspectrum systemmodel,697-698 Square-raw detection. 306 Square-root factorization, 660.897-R98 Siaggered quadralure PSK(SQPSK), 198 Statediagram, 196,474-477 Stationary stochastic processes. h3-M strict-sense. 63-64 wide-sense, 64­ Statistical averages. 64-67 Steepest·descent (gradient) algorithm, 639-642 Stochastic process. 62~72.159-163 cyclostationary. 75-76 discrete-time. 74-76 narrowband. 159 "o"stationary.63 stricl·scnsc stationary. 63-64 wide-sense stationary. 64 Storagechannel. 10 Strict-sense stationary. 63-64 Sut>hand coding,13; Symbol,"tervaL 174 Synchronization: carrier,337-358 effectofnoise.343-346 formultiphasc ~ignals.356-35H withCostasroop.355-356 withdecision-feedback loop.347-.150 wilhphase·lockedloop(PLL),341-346 withsquaring loop,353-355 ofspreadspectrum signals,744-752 slidingcorrclator. 747 symbol, 336-137 Syndrnme. 446 Syndrome decoding, 446-451 System,linear. ~-72 autocorrelation function atoutput.69 928 INDEX System,linear(Cont.): bandpass, response of,157-159 powerdensityspectrum atoutput,69-70 Systematic code,418 Tailprobability bounds,53-57 Chebyshev inequality, 53-54 Chernoff bound,54-57 TATS(tacticaltransmiSSion system), 741-743 Telegraphy, 13 Telephone channels, 4,563-538 Thermal noise,3,II Threshold decoder, 506 Timediversity, 77'1 Timedivisionmultiple access(TDMAJ, 842-844 Toeplitzmatrix,879 Transfer function: ofconvolutional code,477-483 oflinearsystem,68-72 Transformation ofrandomvariables, 29-32,49-52 Transition probabilities, 189 Transition probability matrix,)89 forchannel, 375-378 fordelaymodulation, 189-190 Treediagram, 192-195,471-472 Trellis-co<1ed modulation. 511-526 freeEuclidean distance. 517 subsetdecoding, 519 tablesofcodinggainsfor.522-523 Trellisdiagram, 473 Unoorrelaled random variables, 34 Uniform distribution, 39Unionbound,263-264, 387-389 Unionofevents.18 Uniquely decodable, 96 Universal sourcecoding. 106 Variable-length encoding, 95-103 Variance, 33 Vectorspace,163-165 Vectorquantization, 118-125 Viterbialgorithm, 251,287-289, 483-486 Vocaltract,141-143 Voltage-controlled oscillator (VeO),341-343 Weaklawoflargenumbers, 59 Weight: ofcodeword,414 distribution, 414 forGolay,423 Welchbound,128 W.hitenoise,162-163 Whitening tilter,587-588 Wide~sense stationary, 64 Wienerfilter,14 Yule-Walker equations, 128 Ztransform, 587 Zero-forcing equalizer, 602-605 Zero-forcing tilter,603-604