hilbert transform 2

TheHilberttransform MathiasJohansson MasterThesis Mathematics/AppliedMathematics Supervisor:BÄorjeNilsson,VÄaxjÄoUniversity. Examiner:BÄorjeNilsson,VÄaxjÄoUniversity. Abstract TheinformationabouttheHilberttransformisoftenscatteredin booksaboutsignalprocessing.Theirauthorsfrequentlyusemathe- maticalformulaswithoutexplainingthemthoroughlytothereader. Thepurposeofthisreportistomakeamorestringentpresentationof theHilberttransformbutstillwiththesignalprocessingapplication inmind. Contents 1Introduction:::::::::::::::::::::::::::::::::1 2MathematicalmotivationsfortheHilberttransform:::::::::::2 2.1TheCauchyintegral:::::::::::::::::::::::::3 2.2TheFouriertransform::::::::::::::::::::::::7 2.3The§¼=2phaseshift:::::::::::::::::::::::::10 3PropertiesoftheHilberttransform::::::::::::::::::::11 3.1Linearity::::::::::::::::::::::::::::::::11 3.2MultipleHilberttransformsandtheirinverses:::::::::::12 3.3DerivativesoftheHilberttransform:::::::::::::::::13 3.4Orthogonalityproperties:::::::::::::::::::::::14 3.5EnergyaspectsoftheHilberttransform:::::::::::::::15 3.6TheHilberttransformofstronganalyticsignals::::::::::15 3.7Analyticsignalsinthetimedomain:::::::::::::::::17 4NumericalcalculationsoftheHilberttransform:::::::::::::18 4.1Continuoustime::::::::::::::::::::::::::::18 4.1.1Numericalintegration.::::::::::::::::::::18 4.1.2Hermitepolynomials:::::::::::::::::::::18 4.1.3Fourierseries:::::::::::::::::::::::::22 4.2DiscreteFouriertransform::::::::::::::::::::::22 5Anapplication::::::::::::::::::::::::::::::::27 AAppendix::::::::::::::::::::::::::::::::::30 A.1ImplementationoftheHermitepolynomial:::::::::::::30 A.2ImplementationoftheFourierseries:::::::::::::::::31 A.3ImplementationoftheFouriertransform::::::::::::::32 1.Introduction Signalprocessingisafastgrowingareatodayandadesirede®ectivenessinuti- lizationofbandwidthandenergymakestheprogressevenfaster.Specialsignal processorshavebeendevelopedtomakeitpossibletoimplementthetheoretical knowledgeinane±cientway.Signalprocessorsarenowadaysfrequentlyusedin equipmentforradio,transportation,medicineandproductionetc. In1743afamousSwissmathematiciannamedLeonardEuler(1707-1783)de- rivedtheformula ejz=cos(z)+jsin(z): 150yearslaterthephysicistArthurE:KennellyandthescientistCharlesP: Steinmetzusedthisformulatointroducethecomplexnotationofharmonicwave formsinelectricalengineering,thatis ej!t=cos(!t)+jsin(!t): Lateron,inthebeginningofthe20thcentury,theGermanscientistDavidHilbert (1862-1943)¯nallyshowedthatthefunctionsin(!t)istheHilberttransformof cos(!t).Thisgaveusthe§¼=2phase-shiftoperatorwhichisabasicpropertyof theHilberttransform. Arealfunctionf(t)anditsHilberttransformbf(t)arerelatedtoeachother insuchawaythattheytogethercreateasocalledstronganalyticsignal.The stronganalyticsignalcanbewrittenwithanamplitudeandaphasewherethe derivativeofthephasecanbeidenti¯edastheinstantaneousfrequency.The Fouriertransformofthestronganalyticsignalgivesusaone-sidedspectruminthe frequencydomain.ItisnothardtoseethatafunctionanditsHilberttransform alsoareorthogonal.Thisorthogonalityisnotalwaysrealizedinapplications becauseoftruncationsinnumericalcalculations.However,afunctionandits Hilberttransformhasthesameenergyandthereforetheenergycanbeusedto measurethecalculationaccuracyoftheapproximatedHilberttransform. TheHilberttransformde¯nedinthetimedomainisaconvolutionbetween theHilberttransformer1=(¼t)andafunctionf(t)[7].Thisismotivatedlateron. De¯nition1.1.TheHilberttransformbf(t)ofafunctionf(t)isde¯nedforallt by bf(t)=1 ¼PZ1 ¡1f(¿) t¡¿d¿; whentheintegralexists. ItisnormallynotpossibletocalculatetheHilberttransformasanordinary improperintegralbecauseofthepoleat¿=t.However,thePinfrontofthe 1 integraldenotestheCauchyprincipalvaluewhichexpandingtheclassoffunctions forwhichtheintegralinDe¯nition1.1exist.Itcanbede¯nedbythefollowing de¯nition[5]. De¯nition1.2.Let[®;¯]bearealintervalandletfbeacomplex-valuedfunc- tionde¯nedon[®;¯].Iffisunboundednearaninteriorpoint»of[®;¯],the integraloffover[®;¯]doesnotalwaysexist.However,thetwolimits lim²!0Z»¡² ®f(x)dxandlim²!0Z¯ »+²f(x)dx; stillmayexist,andiftheydotheirsumiscalledtheimproperintegraloffover [®;¯]andisdenotedbytheordinaryintegrationsymbol Z¯ ®f(x)dx: Evenifthesetwolimitsdonotexist,itmayhappenthatthe"symmetriclimit" lim²!0+ÃZ»¡² ®f(x)dx+Z¯ »+²f(x)dx! ; exists.Ifitdoes,itiscalledtheprincipalvalueintegralofffrom®to¯andis denotedbythesymbol PZ¯ ®f(x)dx: Example1.3.Anordinaryrealfunction1=xthatisintegratedfrom¡atoacan bewrittenasZa ¡a1 xdx=lim±!0+Z¡± ¡a1 xdx+lim"!0+Za "1 xdx, andweseethatitisnotpossibletocalculatetheseintegralsseparatelybecause ofthepoleinx=0:However,ifweapplytheCauchyprincipalvaluethen±and "tendtozeroatthesamespeed,thatis PZa ¡a1 xdx=lim"!0+µZ¡" ¡a1 xdx+Za "1 xdx¶ =0; andtheintegralconverges. 2.MathematicalmotivationsfortheHilberttransform InthischapterwemotivatetheHilberttransforminthreedi®erentways.First weusetheCauchyintegralinthecomplexplaneandsecondweusetheFourier transforminthefrequencydomainandthirdwelookatthe§¼=2phase-shift whichisabasicpropertyoftheHilberttransform. 2 Figure2.1:¡isapiecewisesmoothclosedcontouronanopendomain­ 2.1.TheCauchyintegral TheCauchyintegralisa¯gurativewaytomotivatetheHilberttransform.The complexviewhelpsustorelatetheHilberttransformtosomethingmoreconcrete andunderstandable. Consideranintegralinthecomplexz-planeontheform I ¡f(z) z¡adz, whichisknownasaCauchyintegral.Iffisanalyticand¡isapiecewisesmooth closedcontourinanopendomain­;seeFigure2.1,thentheCauchyintegral theoremisapplicableas I ¡f(z) z¡adz=( 2¼if(a)ifaisinside¡ 0ifaisoutside¡ Togetaresultwhenalieson¡wehavetocreateanewcontour¡0 "asin Figure2.2whereI ¡0 "f(z) z¡adz=2¼if(a). (2.1) Iftheradius"ofthesemicircle°"tendstozero,thecontributionfromthesemi- circle°"totheintegralalong¡0 "approaches¼if(a)accordingtoLemma2.1[7]. Lemma2.1.Ifghasasimplepoleatz=aand°risthecirculararcofFigure 2.3de¯nedby °r:z=a+rei£(£1·£·£2), 3 Figure2.2:¡0 "isapiecewisesmoothclosedcontouronanopendomain­. Figure2.3:°risacirculararcarounda 4 Figure2.4:¡isapiecewisesmoothclosedcontour. then limr!0+Z °rg(z)dz=i(£2¡£1)Resz=ag(z). FromLemma2.1andfromthede¯nitionoftheCauchyprincipalvalue,we seethat lim"!0I ¡0 "f(z) z¡adz=PZ ¡f(z) z¡adz+lim"!0Z °"f(z) z¡adz=2¼if(a), andthattheintegraloftheCauchyprincipalvalueis PZ ¡f(z) z¡adz´lim"!0Z ¡"f(z) z¡adz=¼if(a). (2.2) where¡"isanonclosedcontourwithouttheindentation°".By(2.2)wehave generalizedthede¯nitionoftheCauchyprincipalvaluecomparedtoDe¯nition 1.2. Aparticularlyusefulidentityariseswhenwehaveacontour¡asinFigure2.4 thatisclosedbythesemicircle°Randtherealx-axis.Iff(z)isafunctionthat isanalyticinanopenregionthatcontainstheupperhalf-planeandtendstozero atin¯nityinsucharatethatthecontributionfromthesemicircle°Rvanishesas R!1,thenwehave PZ1 ¡1f(») »¡xd»=¼if(x). (2.3) Theintegralover°RwilldisappearwhenR!1if jf(z)j<C jzj, 5 foranypositiveconstantC.Thesameyieldsif jf(z)j<C¯¯¯eimz¯¯¯, forpositivemaccordingtoLemma2.2[7]. Lemma2.2.(Jordan'slemma)Ifm>0andP=Qisthequotientoftwopoly- nomialssuchthat degreeQ¸1+degreeP, then lim½!1Z C+½P(z) Q(z)eimzdz=0, whereC+ ½istheupperhalf-circlewithradius½. Ifweexpressf(x)as f(x)=g(x)¡ih(x); onbothsidesof(2.3)withargumentsontherealx-axisandequatingrealand imaginarypartsthenweobtainfortherealpart g(x)=¡1 ¼PZ1 ¡1h(») x¡»d»=¡Hh(x); andfortheimaginarypart h(x)=1 ¼PZ1 ¡1g(») x¡»d»=Hg(x): (2.4) FromDe¯nition1.1wehavethath(x)in(2.4)istheHilberttransformofg(x) whereHistheHilberttransformoperator.Wealsonotethatg(x)=H¡1h(x) withH¡1astheinverseHilberttransformoperator.WeseethatHRef(x)= Imf(x)whichmotivatesthefollowingde¯nition. De¯nition2.3.Acomplexsignalf(x)thatful¯lstheconditionspreceding(2.3) iscalledastronganalyticsignal. Abovewehavejustprovedthefollowingtheorem. Theorem2.4.Forastronganalyticsignalf(x)wehavethatHRef(x)= Imf(x): 6 2.2.TheFouriertransform TheFouriertransformisimportantinthetheoryofsignalprocessing.Whena functionf(t)isreal,weonlyhavetolookonthepositivefrequencyaxisbecause itcontainsthecompleteinformationaboutthewaveforminthetimedomain. Therefore,wedonotneedthenegativefrequencyaxisandtheHilberttransform canbeusedtoremoveit.Thisisexplainedbelow. Letusde¯netheFouriertransformF(!)ofasignalf(t)by F(!)=Z1 ¡1f(t)e¡i!tdt: (2.5) Thisde¯nitionmakessenseiff2L1(<);thatisifR1 ¡1jf(t)jdtexists.Itis importanttobeabletorecoverthesignalfromitsFouriertransform.Todothat wede¯netheinverseFouriertransformas ef(t)=1 2¼Z1 ¡1F(!)ei!td!: IfbothfandFbelongtoL1(<)thenf(t)iscontinuousandboundedforallreal tandwehavethatef(t)=f(t);thatis[1] f(t)=1 2¼Z1 ¡1F(!)ei!td!: (2.6) ThisresultisaformoftheFourierinversiontheorem. Anothervariantoftheinversiontheorem[2]isthatiffbelongstoL1(<);f isofboundedvariationinaneighbourhoodoftandfiscontinuousattthen f(t)=limT!11 2¼ZT ¡TF(!)ei!td!: Thismeansthat(2.6)istobeinterpretedasatypeofCauchyPrincipalvalue. Furthermoregeneralvariantsoftheinversiontheoremexistforf2L1(<)but arenotmentionedhere. ThereisalsoatheoryfortheFouriertransformwhenf2L2(<)[2].Inthis casewede¯netheFouriertransformas F(!)=limN!1ZN ¡Nf(t)e¡i!tdt: Themeanlimit F(!)=limN!1FN(!); istobeinterpretedas limN!1kF(!)¡FN(!)k2=0: 7 NowF2L2(<)andtheinversionformulastatesthat f(t)=limN!11 2¼ZN ¡NF(!)ei!td!: OftenweneedtoconsidertheFouriertransformofasignalasitneitherbelongs toL1(<)nortoL2(<).Thedeltafunctionissuchanexample.Thereisatheory oftheFouriertransform,seee.g.[1],fortheresocalledtemperaturedistributions. Thistheoryis,however,outsidethescoopofthisreport,butoccasionallyweuse someresultsfromthistheory. AusefulrelationisthePlancherelformula. Theorem2.5.Iff,gandGbelongtoL1(<)oriffandgbelongtoL2(<)then Z1 ¡1f(t)g¤(t)dt=1 2¼Z1 ¡1F(!)G¤(!)d!: Proof.Foraproofwereferto[1]. Iff(t)isarealfunctionthatcanberepresentedbyaninverseFouriertransform thenwehavethefollowingrelationshipinthetimedomain f(t)=f¤(t)=1 2¼Z1 ¡1F(!)ei!td! =1 2¼Z1 ¡1F¤(!)e¡i!td! =1 2¼Z1 ¡1F¤(¡!)ei!td!: ThisgivesustherelationF(!)=F¤(¡!)orF(¡!)=F¤(!)inthefrequency domainandweseethatFfornegativefrequenciescanbeexpressedbyF¤for positiveones. Theorem2.6.Iff(t)isarealfunctionthen f(t)=1 2¼Z1 0h F¤(!)e¡i!t+F(!)ei!ti d!: Proof.IfweapplytheFouriertransformonarealfunctionf(t)then f(t)=1 2¼Z0 ¡1F(!)ei!td!+1 2¼Z1 0F(!)ei!td! =1 2¼Z1 0F(¡!)e¡i!td!+1 2¼Z1 0F(!)ei!td! =1 2¼Z1 0³ F¤(!)e¡i!t+F(!)ei!t´ d!: 8 Figure2.5:TheamplitudesofF(!)andZf(!)asfunctionsof!: Thismeansthatthepositivefrequencyspectraissu±cienttorepresentareal signal. Letusde¯neafunctionZf(!);thatiszeroforallnegativefrequenciesand 2F(!)forallpositivefrequencies Zf(!)=F(!)+sgn(!)F(!); (2.7) where sgn(!)=8 >< >:1for!>0 0for!=0 ¡1for!<0 andF(!)istheFouriertransformoftherealfunctionf(t).InFigure2.5wesee therelation(2.7)betweenF(!)andZf(!).TheinversetransformofZf(!)is thereforewrittenas zf(t)=1 2¼Z1 ¡1Zf(!)ei!td!=1 ¼Z1 0F(!)ei!td!; wherezf(t)isacomplexfunctionontheform zf(t)=f(t)+ig(t): (2.8) Wewillshowbelowthatg(t)isreal.From(2.7)and(2.8)wehavethat f(t)+ig(t)F,F(!)+sgn(!)F(!): (2.9) 9 Thede¯nitionoftheFouriertransformtellsusthatf(t)F,F(!)andtherefore weknowfrom(2.9)thatF(!)sgn(!)istheFouriertransformofig(t),thus g(t)F,F(!)(¡isgn(!)): Itisastandardresultthattheinversetransformof¡isgn(!)equals1=(¼t);that is g(t)=f(t)¤1 ¼t=1 ¼PZ1 ¡1f(¿) t¡¿d¿=Hf(t)=bf(t); andweseethatg(t)canbewrittenasbf(t)whichisknownastheHilberttransform off(t):Furthermoreg(t)isreal. 2.3.The§¼=2phaseshift The§¼=2phaseshiftisinterpretedinthefrequencydomainasamultiplication withtheimaginaryvalue§i,thus H(!)=( ¡i=e¡i¼ 2for!>0 i=ei¼ 2for!<0 H(!)isunfortunatelynotapropertyofFouriertransformbuttheproblemcan besolvedbyexpressingH(!)asalimitofaboundedfunctionG(!),thatis G(!)=( ¡ie¡¾!for!>0 ie¾!for!<0 where lim¾!0G(!)=H(!): (2.10) ItisnowpossibletousetheinverseFouriertransformonG(!),thus g(t)=F¡1G(!) =1 2¼Z0 ¡1ie¾!ei!td!+1 2¼Z1 0¡ie¡¾!ei!td! =i 2¼Z1 0(e¡(¾+it)!¡e¡(¾¡it)!)d! =i 2¼· ¡1 ¾+ite¡(¾+it)!+1 ¾¡ite¡(¾¡it)!¸1 0 =t ¼(¾2+t2): 10 whereg(t)!h(t)when¾!0andtheinverseFouriertransformoftheimpulse responseofH(!)is h(t)=lim¾!0g(t)=lim¾!0t ¼(¾2+t2)=1 ¼t: Aconvolutionbetweenf(t)andtheimpulseresponseh(t)givesus bf(t)=1 ¼Z1 ¡1f(¿) t¡¿d¿; wherebf(t)isknownastheHilberttransform.Noticethatthisintegralshallbe consideredasaprincipal-valuedintegralalimitthatcorrespondstothelimitin (2.10).Tomakearigorouspresentationofthisproblemweshouldapplythe distributiontheorybutthatisbeyondthescoopofthisreport. 3.PropertiesoftheHilberttransform InthischapterwelookatsomepropertiesoftheHilberttransform.Weassume thatF(!)doesnotcontainanyimpulsesfor!=0andthatf(t)isarealvalued function.Someoftheformulasaretobeinterpretedinadistributionalsense. 3.1.Linearity TheHilberttransformthatisaCauchyprinciple-valuedfunction,seeChapter1, isexpressedontheform Hf(t)=lim"!01 ¼Z jx¡tj>"f(¿) t¡¿d¿: Ifwewritethefunctionf(t)asc1f1(t)+c2f2(t)wheretheHilberttransformof f1(t)andf2(t)existsthen Hf(t)=H(c1f1(t)+c2f2(t)) =lim"!01 ¼Z jx¡tj>"c1f1(¿)+c2f2(¿) t¡¿d¿ =c1lim"!01 ¼Z jx¡tj>"f1(¿) t¡¿d¿+c2lim"!01 ¼Z jx¡tj>"f2(¿) t¡¿d¿ =c1Hf1(t)+c2Hf2(t): ThisisthelinearitypropertyoftheHilberttransform. 11 3.2.MultipleHilberttransformsandtheirinverses TheHilberttransformusedtwiceonarealfunctiongivesusthesamerealfunction butwithalteredsign H2=¡I; withIastheidentityoperator.TheHilberttransformusedfourtimesonthe samerealfunctiongivesustheoriginalfunctionback H2H2=H4=I: (3.1) AmoreinterestingpropertyofmultipleHilberttransformsarisesifweusethe Hilberttransform3times,thus H3H=I)H¡1=H3: ThistellsusthatitispossibletousethemultipleHilberttransformtocalculate theinverseHilberttransform. AsweseenbeforetheHilberttransformcanbeappliedinthetimedomain byusingthede¯nitionoftheHilberttransform.Inthefrequencydomainwe simplymultiplytheHilberttransformoperator¡isgn(!)tothefunctionF(!). BymultiplyingtheHilberttransformoperatorbyitselfwegetaneasymethodto domultipleHilberttransforms,thatis Hnf(t)F ,(¡isgn(!))nF(!); wherenisthenumberofHilberttransforms. Example3.1.WewanttocalculatetheinverseHilberttransformofthefunction f(t)byusingmultipleHilberttransformsinthefrequencydomain.Firstwehave toFouriertransformthefunctionf(t) F(!)=Z1 ¡1f(t)e¡i!tdt; andthenusetheHilberttransformthreetimesinthefrequencydomain,thatis H3=(¡isgn(!))3: FinallyweusetheinverseFouriertransform H¡1f(t)=1 2¼Z1 ¡1H3F(!)ei!td!: Fromaboveweseethatweonlyhavetocalculatetwoin¯niteintegralsin thefrequencydomaincomparedtothreein¯niteintegralsinthetimedomain. Anotheradvantageinthefrequencydomainisthatweformallycanchoosethe numberoftimeswewanttousetheHilberttransform. 12 3.3.DerivativesoftheHilberttransform Theorem3.2.TheHilberttransformofthederivativeofafunctionisequivalent tothederivativeoftheHilberttransformofafunction,thatis f0(t)H,d dtbf(t): (3.2) Proof.FromDe¯nition1.1wehavethat bf(t)=1 ¼PZ1 ¡1f(¿) t¡¿d¿: Ifwesubstitute¿witht¡s bf(t)=1 ¼PZ1 ¡1f(t¡s) sds; andthenapplythederivativeoftonbothsidesweget d dtbf(t)=1 ¼PZ1 ¡1f0(t¡s) sds: Thesubstitutions=t¡¿givesusthat d dtbf(t)=1 ¼PZ1 ¡1f0(¿) t¡¿d¿; andtherelationin(3.2)isvalid. Fromtheproofaboveweconcludethattherelationcanbeusedrepeatedly. LetuslookatanexamplewherewealsomakeuseofmultipleHilberttransforms, seeSection3.2. Example3.3.By(3.2)wemaycalculatetheHilberttransformofthedeltafunc- tion±(t)anditsderivatives.AtthesametimewegettheHilberttransformrep- resentationofthedeltafunction.ConsidertheHilberttransformofthedelta function H±(t)=1 ¼t: Thederivativeofthedeltafunctioniscalculatedto H±0(t)=¡1 ¼t2; (3.3) andifweapplytheHilberttransformonbothsidesthenweget ±0(t)=Hµ1 ¼t2¶ : 13 Thederivativeof(3.3)is H±00(t)=2 ¼t3; andwhenweapplytheHilberttransformonbothsidesweget ±00(t)=Hµ ¡2 ¼t3¶ : Thisprocedurecanbecontinued. 3.4.Orthogonalityproperties AsymmetryfoundinSection2.2abouttheFouriertransformF(!)ofareal functionf(t)leadsustodothefollowingde¯nition[6]. De¯nition3.4.AcomplexfunctioniscalledHermitianifitsrealpartiseven anditsimaginarypartisodd. FromthiswehavethattheFouriertransformF(!)ofarealfunctionf(t)is Hermitian. Theorem3.5.Arealfunctionf(t)anditsHilberttransformbf(t)areorthogonal iff;bfandFbelongtoL1(<)oriffandbfbelongtoL2(<): Proof.FromTheorem2.5wehavethat Z1 ¡1f(t)bf(t)dt=1 2¼Z1 ¡1F(!)(¡isgn(!)F(!))¤d! =i 2¼Z1 ¡1sgn(!)F(!)F¤(!)d! =i 2¼Z1 ¡1sgn(!)jF(!)j2d!; wheresgn(!)isanoddfunctionandthefactthatF(!)isHermitiangivesusthat jF(!)j2isanevenfunction.Weconcludethat Z1 ¡1f(t)bf(t)dt=0; andarealfunctionanditsHilberttransformareorthogonal. 14 3.5.EnergyaspectsoftheHilberttransform Theenergyofafunctionf(t)iscloselyrelatedtotheenergyofitsFouriertrans- formF(!).Theorem2.5withf(t)=g(t)iscalledtheRayleightheoremandit helpsustode¯netheenergyoff(t)andF(!)as Ef=Z1 ¡1jf(t)j2dt=1 2¼Z1 ¡1jF(!)j2d!: (3.4) Hereitisnaturaltoassumethatf2L2(<)whichmeansthatEfis¯nite.The sametheoremisusedtode¯netheenergyoftheHilberttransformoff(t)and F(!);thatis Ebf=Z1 ¡1¯¯¯bf(t)¯¯¯2 dt=1 2¼Z1 ¡1j¡isgn(!)F(!)j2d!; (3.5) wherej¡isgn(!)j2=1exceptfor!=0:But,sinceF(!)doesnotcontainany impulsesattheoriginwegetEbf=Ef. Aconsequenceof(3.5)isthatf2L2(<)inducesthatbf2L2(<):Theaccuracy oftheapproximatedHilberttransformoperatorcanbemeasuredbycomparing theenergyin(3.4)and(3.5).However,aminordi®erenceinenergyalwaysexists inrealapplicationsduetounavoidabletruncationerrors. 3.6.TheHilberttransformofstronganalyticsignals FromSection3.2wehavethattheHilberttransformofastronganalyticsignal z(t)is Hz(t)=H³ f(t)+ibf(t)´ =bf(t)¡if(t)=¡iz(t): (3.6) FromthisfollowstheresultoftheHilberttransformoftwomultipliedstrong analyticsignals. Theorem3.6.TheproductofH(z1(t))z2(t)isidenticalwiththeproductof z1(t)H(z2(t))ifz1(t)andz2(t)arestronganalyticsignals. Proof.Sincez1(t)andz2(t)arestronganalyticsignalsthen H(z1(t))z2(t)=³bf1(t)¡if1(t)´³ f2(t)+ibf2(t)´ (3.7) =¡i³ f1(t)+ibf1(t)´³ f2(t)+ibf2(t)´ =¡iz1(t)z2(t) =³ f1(t)+ibf1(t)´³bf2(t)¡if2(t)´ (3.8) =z1(t)H(z2(t)); (3.9) wherewemakeuseof(3.6)in(3.7)and(3.8). 15 Theorem3.7.Theproductofz1(t)z2(t)isidenticalwiththeproductiH(z1(t))z2(t)= iz1(t)H(z2(t))ifz1(t)andz2(t)arestronganalyticsignals. Proof.Sincez1(t)andz2(t)arestronganalyticsignalsthen z1(t)z2(t)=³ f1(t)+ibf1(t)´³ f2(t)+ibf2(t)´ =i³bf1(t)¡if1(t)´³ f2(t)+ibf2(t)´ =iH(z1(t))z2(t)=iz1(t)H(z2(t)); andthetheoremfollows. TheHilberttransformoftheproductoftwostronganalyticsignalsgivesus thesameresultasin(3.9).Toprovethiswe¯rstneedtoshowthattheproduct oftwostronganalyticsignalsisstronganalytic. Theorem3.8.Theproductoftwostronganalyticsignalsisstronganalytic. Proof.Letz1(t00)andz2(t00)beanalyticsignalsofthecomplexvariablet00=t+it intheopenupperhalf-plane.Thenz1(t00)z2(t00)isalsoananalyticsignalinthe sameregion.Assumethatz1(t00)andz2(t00)aredecreasinginsucharateat in¯nitythatthediscussioninSection2.1istruethenz1(t)andz2(t)arestrong analyticsignals.Ifz1(t00)andz2(t00)aredecreasingsu±cientlyrapidatin¯nity thenz1(t00)z2(t00)havetodecreasefasterthantheoneofz1(t00)andz2(t00)that isdecreasingwiththeleastrate.FromthiswehavethatH(Re(z1(t)z2(t)))= Im(z1(t)z2(t))andthatz1(t)z2(t)isastronganalyticsignal. Theorem3.9.H(z1(t)z2(t))=¡iz1(t)z2(t)ifz1(t)andz2(t)arestronganalytic signals. Proof.Sincez1(t)andz2(t)arestronganalyticsignalsthen H(z1(t)z2(t))=H³ f1(t)+ibf1(t)´³ f2(t)+ibf2(t)´ =H(f1(t)f2(t)¡bf1(t)bf2(t) +i³ f1(t)bf2(t)+bf1(t)f2(t)´ ) =f1(t)bf2(t)+bf1(t)f2(t) ¡i³ f1(t)f2(t)¡bf1(t)bf2(t)´ (3.10) =¡i(f1(t)f2(t)+if1(t)bf2(t) +ibf1(t)f2(t)¡ibf1(t)bf2(t)) =¡i³ f1(t)+ibf1(t)´³ f2(t)+ibf2(t)´ =¡iz1(t)z2(t); 16 wherewemakeuseofTheorem3.8in(3.10). ConsequentlyitispossibletoapplytheHilberttransformontheproductof twostronganalyticsignalsinseveraldi®erentways,thus H(z1(t)z2(t))=H(z1(t))z2(t)=z1(t)H(z2(t))=¡iz1(t)z2(t): ItdoesnotmatteronwhichstronganalyticsignalweapplytheHilberttransform. WeconcludethattheHilberttransformoftheproductofnstronganalyticsignals formtheequation Hzn(t)=H(z(t))zn¡1(t)=¡iz(t)zn¡1(t)=¡izn(t): 3.7.Analyticsignalsinthetimedomain TheHilberttransformcanbeusedtocreateananalyticsignalfromarealsignal. Insteadofstudyingthesignalinthefrequencydomainitispossibletolookata rotatingvectorwithaninstantaneousphase'(t)andaninstantaneousamplitude A(t)inthetimedomain,thatis z(t)=f(t)+ibf(t)=A(t)ei'(t): Thisnotationisusuallycalledthepolarnotationwhere A(t)=q f2(t)+bf2(t); and '(t)=arctanÃbf(t) f(t)! : IfweexpressthephasewithaTaylorseriesthen '(t)='(t0)+(t¡t0)'0(t0)+R; whereRissmallwhentisclosetot0.Theanalyticsignalbecomes z(t)=A(t)ei'(t)=A(t)ei('(t0)¡t0'0(t0))eit'0(t0)eiR; andweseethat'0(t0)hastheroleoffrequencyifRisneglected.Thismakesit naturaltointroducethenotionofinstantaneous(angular)frequency,thatis !(t)=d'(t) dt: 17 Example3.10.WehavearealsignalanditsHilberttransform f(t)=cos(!0t); bf(t)=sin(!0t): Togethertheyformananalyticsignalwheretheinstantaneousamplitudeis A(t)=q cos2(!0t)+sin2(!0t)=1: Theinstantaneousfrequencyiseasytocalculatefromthephase'(t)=!0t,that is !(t)=!0: Weseethatinthisparticularcasetheinstantaneousfrequencyisthesameasthe realfrequency. 4.NumericalcalculationsoftheHilberttransform Thepurposeofthischapteristostudydi®erenttypesofnumericalcalculation methodsfortheHilberttransform. 4.1.Continuoustime 4.1.1.Numericalintegration. Numericalintegrationworks¯neonsmoothfunctionsthatdecreaserapidlyat in¯nity.WhenwewanttocalculatetheHilberttransformbyDe¯nition1.1we arefacingsomeproblems.Innumericalintegrationweuse¯niteintervalsandit isthereforeimportanttoconsidertheintegrationregiontocontrolthecalculation error.Thisisthereasonwhyarapiddecreaseatin¯nityisanadvantage.Another problemisthattheintegrandinDe¯nition1.1isin¯nitewhenthenominatorvan- ishes.However,byusingmoreintegrationgridpointsinthenumericalintegration closetothisvaluewegetabetterapproximation. 4.1.2.Hermitepolynomials Thenumericalintegrationisine±cientwhenafunctiondecreasesinaslowrate atin¯nity.Itissometimesbettertouseaseriesoforthogonalpolynomialswhere thefunctiondoesnothavetodecreaserapidatin¯nity.Inthissectionweusethe HermitepolynomialstocalculatetheHilberttransform.Firstweneedtotakea lookatthede¯nitionoftheHermitepolynomials. 18 Thesuccessivedi®erentiationoftheGaussianpulsee¡t2 generatesthenth orderHermitepolynomialwhichisde¯nedbyRodrigues'formulaas[4] Hn(t)=(¡1)net2dn dtne¡t2 : ItisalsopossibletocalculatetheHermitepolynomialsbytherecursionformula [3] Hn(t)=2tHn¡1(t)¡2(n¡1)Hn¡2(t); (4.1) withn=1;2;3;::.andthestartcondition H0(t)=1: Letusalsode¯netheweightedHermitepolynomialsthatisweightedbythe generatingfunctione¡t2 ontheform[3] gn(t)=Hn(t)e¡t2 =(¡1)ndn dtne¡t2 : TheweightedHermitepolynomialsgn(t)donotrepresentanorthogonalsetinL2 sincethescalarproduct Z1 ¡1gn(t)gm(t)dt=Z1 ¡1Hn(t)Hm(t)e¡2t2 dt; isingeneraldi®erentfromzerowhenn6=m.Thesolutionistoreplacethe weightingfunctione¡t2 withe¡t2 2,thatis Z1 ¡1Hn(t)Hm(t)e¡t2 dt=( 0forn6=m 2nn!p¼forn=m BythattheweightedHermitepolynomialsful¯ltheconditionoforthogonality. Theweightedpolynomiale¡t2 2Hn(t)dividedbytheirnormq 2nn!p¼yieldsaset oforthonormalfunctionsinL2andiscalledtheHermitefunction 'n(t)=e¡t2 2Hn(t)q 2nn!p¼: (4.2) Ifwecombine(4.1)and(4.2)wegettherecursivealgorithm 'n(t)=s 2(n¡1)! n!t'n¡1(t)¡(n¡1)s (n¡2)! n!'n¡2(t);(4.3) whichcanbeusedtoderivetheHilberttransformoftheHermitefunctionsby applyingthe"multiplicationbyt"theorem[3]. 19 Theorem4.1.Ifweassumethatf(t)andbf(t)belongtoL1thentheHilbert transformoftf(t)isgivenbytheequation H(tf(t))=tbf(t)¡1 ¼Z1 ¡1f(¿)d¿: Theintegralisaconstantde¯nedbythefunctionf(t).Foroddfunctionsthis constantequalszero. Proof.ConsidertheHilberttransformoftf(t) H(tf(t))=1 ¼PZ1 ¡1¿f(¿) t¡¿d¿: Theinsertionofanewvariables=t¡¿yields H(tf(t))=1 ¼PZ1 ¡1(t¡s)f(t¡s) sds =1 ¼PZ1 ¡1tf(t¡s) sds¡1 ¼Z1 ¡1f(t¡s)ds =tH(f(t))¡1 ¼Z1 ¡1f(¿)d¿; andthetheoremfollows. FromTheorem4.1andfrom(4.3)wehavethat H('n(t))=b'n(t)=s 2(n¡1)! n!µ tb'n¡1(t)¡1 ¼Z1 ¡1'n¡1(´)d´¶ ¡(n¡1)s (n¡2)! n!b'n¡2(t); (4.4) wheren=1;2;3:::.The¯rstterm'0(t)canbecalculatedbyusingtheFourier transformonequation(4.2),thatis '0(t)=¼¡1 4e¡t2 2F,p 2¼1 4e¡!2 2: InthefrequencydomainwemultiplytheHermitefunction'0(t)bytheHilbert transformer¡isgn(!)and¯nallyweusetheinverseFouriertransformtogetthe HilberttransformoftheHermitefunction,thatis H('0(t))=b'0(t)=p 2¼1 4Z1 ¡1e¡!2 2(¡isgn(!))ei!td!: 20 Sincesgn(!)e¡!2 2isoddwehave b'0(t)=2p 2¼1 4Z1 0e¡!2 2sin(!t)d!; (4.5) whichcanbeusedin(4.4)toderivetherestoftheHilberttransforms. IthasbeenfoundthattheerrorislargeforhigherorderoftheHilberttrans- formedHermitefunctions(e.g.b'5(t))whenweusetherecursiveformula(4.4). Thise®ectiscausedbythe¯niteaccuracyinthecalculationoftheintegralin(4.5) wheretheerrorincreasesin(4.4).Itisthereforenotsuitabletousetherecursive methodtocalculatetheHilberttransformforhigherorderHermitefunctions. AnothermethodtocalculatetheHilberttransformbytheHermitefunctions istomultiplytheHilberttransformerbythespectrumoftheHermitefunction andusetheinverseFouriertransform.Noin¯niteintegralsisneededinthe calculationsoftheHermitefunctions.Thereforetheerrordoesnotpropagateas in(4.4). Theseriesexpansionoff(t)canbewrittenas f(t)=1X n=0an'n(t); where an=Z1 ¡1f(t)'n(t)dt: Iftheseriesexpansionf(t)islimitedatin¯nitytherewillbeanerror"N(t),that is f(t)=N¡1X n=0an'n(t)+"N(t): ThisseriesexpansioncanalsobeusedtocalculatetheHilberttransform Hf(t)=N¡1X n=0anb'n(t)+b"N(t); (4.6) andweseethatthiskindofmethodisusefulforfunctionswhereb"N(t)issmall. TocalculatetheHilberttransformof'n(t)byusingtheinverseFouriertrans- formontheproductoftheHermitefunctionspectraandtheHilberttransformer isaratherdemandingmethod.However,theHermitefunctionsneverchangeand wethereforeonlyhavetocalculatetheirHilberttransformsoncewhilean,which dependsonf;representsaneasyintegraltocalculate. 21 Figure4.1:ApproximationbyaFourierseriesef(t)withitsHilberttransform Hef(t),n=30: 4.1.3.Fourierseries IfwecreateaFourierseriesofafunctionf(t)andchangesthesinefunctionsto cosfunctionsandcosfunctionstosinefunctionswegettheHilberttransformof f(t).Thismethodgivesusatheability,inaneasyway,tostudytheunavoidable truncationerrors(a¯nitenumberoftermsintheFourierseries)fortheHilbert transformofaperiodicrectangularwave.InFigure4.1weseethatthepeaksof theHilberttransformonlyare1.5intheirmaximaandminimawhiletheyshould havebeenin¯nite.TheimplementationincomputercodeoftheFourierseriesin Figure4.1isfoundinAppendixA.2. 4.2.DiscreteFouriertransform ToderivethediscreteHilberttransformweneedthede¯nitionofthediscrete Fouriertransform(DFT),thatis F[k]=N¡1X n=0f[n]e¡i2¼ Nkn;k=0;1;:::;N¡1; (4.7) andtheinversionformula f[n]=1 NN¡1X k=0F[k]ei2¼ Nkn;n=0;1;:::;N¡1: (4.8) wherekisthediscretefrequencyandnisthediscretetime.Itiseasytoprove (4.8)byinserting(4.8)into(4.7).Notethat(4.8)de¯nesaperiodicfunctionwith 22 periodN.Letusexpand(4.7)initsrealandimaginarypartsonbothsides,thus F[k]=FRe[k]+iFIm[k]; andN¡1X n=0f[n]e¡i2¼ Nkn=N¡1X n=0f[n]cos(2¼ Nkn)¡iN¡1X n=0f[n]sin(2¼ Nkn): Therealandimaginarypartisnowidenti¯edas FRe[k]=N¡1X n=0f[n]cos(2¼ Nkn); FIm[k]=¡N¡1X n=0f[n]sin(2¼ Nkn); andweconcludethatFIm=0whenk=0andk=N=2:Aswehaveseenbefore theHilberttransformofthedeltapulse±(t)givesustheHilberttransformer 1=(¼t)andtheFouriertransformoftheHilberttransformergivesusthesignshift function¡isgn(!);thatis ±(t)H ,1 ¼tF ,¡isgn(!): (4.9) ThediscreteanalogueoftheHilberttransformerin(4.9)forevenNistherefore givenby H[k]=8 >< >:¡i 0 ifork=1;2;:::;N=2¡1 fork=0andN=2 fork=N=2+1;:::;N¡2;N¡1; andH[k]canbewrittenontheform H[k]=¡isgn(N 2¡k)sgn(k). (4.10) Herewehaveusedtheconventionthatsgn(0)=0.Thediscretefrequencykis calledpositiveintheinterval0<k<N=2andnegativeintheintervalN=2< k<NandalternatesignatN=2. ThediscreteinverseFouriertransformofthediscreteHilberttransformerin (4.10)givesusthediscreteimpulseresponseinthetimedomain,forevenN,thus h[n]=1 NN¡1X k=0H[k]ei2¼ Nkn =1 NN¡1X k=0¡isgn(N 2¡k)sgn(k)ei2¼ Nkn =2 NN 2¡1X k=1sin(2¼ Nkn); (4.11) 23 Figure4.2:ThediscreteimpulseresponseoftheHilberttransformforeven N(N=10)givenbythecotangentfunctionwitheverysecondsample(n= 0;2;4;:::)erasedbysin2(¼n=2): andh[n]canbeexpressedinclosedformas h[n]=2 Nsin2(¼n 2)cot(¼n N): (4.12) Thefunctionisgivenbythecotangentfunctionwitheverysecondsample(n= 0;2;4;:::)erasedbysin2(¼n=2);seeFigure4.2. ThesamederivationforoddNisgivenby H[k]=8 >< >:¡i 0 ifork=1;2;:::;(N¡1)=2 fork=0 fork=(N+1)=2;:::;N¡2;N¡1: Itiswrittenonthesameclosedformasintheevencasewiththedi®erencethat thereisnocancellationforsgn(N=2¡k)thatis H[k]=¡isgn(N 2¡k)sgn(k): (4.13) ThediscreteimpulseresponseforoddNoftheHilberttransformerin(4.9)is givenbythediscreteinverseFouriertransformofH[k]in(4.13),thus h[n]=2 NN¡1 2X k=1sin(2¼ Nkn); 24 Figure4.3:ThediscreteimpulseresponseoftheHilberttransformforoddN (N=11)witheverysecondsamplepositivewithnextsamplenegative. wheretheclosedformofh[n]canbeexpressedas h[n]=1 Nà cot(¼n N)¡cos(¼n) sin(¼n N)! : (4.14) AswementionedbeforewedonothavethesamecancellationforoddN(4.14) asforevenN(4.12),insteadh[n]ischangingsignbyoddandevenvaluesofn, seeFigure(4.3). ThediscreteHilberttransformofthesequencebf[n]isde¯nedbyconvolution ontheform bf[n]=N¡1X m=0h[n¡m]f[m]: (4.15) IfweinsteadchoosetousetheDFTalgorithmwethenhavethefollowingrelations f[n]DFT)F[k]DHT)bF[k]=¡isgn(N 2¡k)sgn(k)F[k]DFT¡1 )bf[n];(4.16) whereDFTdenotesthediscreteFouriertransform,DHTdenotesthediscrete HilberttransformandDFT¡1denotesthediscreteinverseFouriertransform. Thediscreteconvolutionalgorithm(4.15)isfasterthantheDFTalgorithm (4.16)becauseitinvolvesonlyasinglesummation.HowevertheDFTalgorithm maybereplacedbyafastFouriertransformalgorithm(FFT). TheaccuracyofthediscreteFouriertransformwhenappliedtocontinuous signalsisdependingonhowmanysampleswehaveinourcalculation(Asample isthesameasonesinglevalueofafunctionforonespeci¯ctime).Itisimportant thatthesamplefrequencyishigherthandoublethesignalfrequency,thatisthe Nyqvistfrequency. 25 Figure4.4:TheFouriertransformeF(w)byN=10termsofthefunctionf(t)= cos(2t): Example4.2.AssumethatwewanttocalculatethediscreteHilberttransform ofasine-wavef(t)=cos(2t)withN=10samplesusingtheDFTalgorithm accordingto(4.16)wherethesamplingfrequencyis5=¼andthesignalfrequency is1=¼:FirstweneedtocalculatethediscreteFouriertransform(4.7)off(t).Then weneedtouse(4.11)(Neven)toapplytheinversediscreteFouriertransform ontheproductofthediscreteHilberttransformoperatorH[k]in(4.13)andthe discreteFouriertransformoff(t):Thede¯nitionofthediscreteFouriertransform givesus(seeFigure4.4) eF[k]=N¡1X n=0f[n]e¡i2¼ Nkn=9X n=0cosµ2¼ 5n¶ e¡i¼ 5kn: Byusing(4.11)wegettheHilberttransformHf[n]inthetimedomain(Neven) Hf[n]=2 NN 2¡1X k=1eF[k]sin(2¼ Nkn) =1 54X k=1Ã9X n=0cosµ2¼ 5n¶ e¡i¼ 5kn! sin(¼ 5kn); Thefactthatthisisanharmonicperiodicsignal(ofsinandcos)andthatthe samplingrateismorethandoublethesignalfrequencygivesustheexactanswer, seeFigure4.5.TheimplementationincomputercodeisfoundinAppendixA.3. Example4.3.WewanttocalculatetheHilberttransformoff(t)=1=(t2+1) withN=2termsoftheHermitianpolynomialin(4.6)andusen=20termsof 26 Figure4.5:f(t)anditsHilberttransformHef(t). thediscreteFouriertransformin(4.7)and(4.11). Firstweneedtocalculate'0(t);:::;'2(t)by(4.2)or(4.3)andthenb'0(t);:::;b'2(t) asinExample4.2.anispossibletocalculatewithanumericalintegrationmethod becausetheintegralconvergessu±cientlyrapidatin¯nity.InFigure4.6wecan seethefunctionf(t)anditsapproximatedHilberttransformHy(t)comparedto theknownHilberttransformHf(t)=t=(t2+1).Theimplementationincomputer codeisfoundinAppendixA.1. 5.Anapplication Insignalsystemsweoftenmakeuseofmodulatedsignals.Onebigproblemwhen wemodulateasignalisthatitsspectrummirrorsitselfonbothsidesofthecarrier frequency.Onlyoneofthesespectra,orsidebands,isneededtodemodulatethe signalandthereforewehaveredundantinformation.Therearedi®erenttech- niquestoremoveoneofthesidebandsandtherebycreateasinglesidebandsignal (SSB).Themostsimpleoneistousealow-pass¯lterwiththemiddleofthe transitionbandonthecarrierfrequency.Thistechniquedemandsthatwehavea narrowtransitionbandandagoodsuppressionwhichishardtoobtain.Another techniqueistousetheHilberttransformwhereweremoveoneofthesidebands byaddingthesignaltoitself.Intheorywewillgetatransitionbandthatiszero andasuppressionthatisin¯nite. LetususetheHilberttransformtocreateanamplitudemodulatedSSBsignal (AM-SSB)withaPhase/Hilbertmodulator,seeFigure5.1. Theincomingsignal fm(t)=sin(!mt); (5.1) 27 Figure4.6:Thefunctionf(t)anditsHilberttransformHf(t)andHy(t)asthe approximatedHilberttransform. Figure5.1:Phase/Hilbertmodulator. 28 isdividedintwopartswhereoneispreservedandtheotheroneispassingthrow aHilbert¯lter,thatis bfm(t)=cos(!mt): Wemodulatethepreservedsignalfm(t)withasine-signalsin(!ct)withthecarrier frequency!c,thus fmc(t)=sin(!mt)sin(!ct): Tomodulatebfm(t)weneeda¼=2phaselagonthecarrierfrequencywhichgives usacos(!ct)function,seeFigure5.1,thatis bfmc(t)=cos(!mt)cos(!ct): Withsomehelpfromthetrigonometrywegetthat fmc(t)=1 2(cos((!m¡!c)t)¡cos((!m+!c)t)); and bfmc(t)=1 2(cos((!m¡!c)t)+cos((!m+!c)t)): Finallyweaddfmcandbfmcwhichgivesus s(t)=cos((!m¡!c)t); andwehavealowerAM-SSBsignal.Ifweinsteadsubtractfmcandbfmcweget anupperAM-SSBsignal. Thedelay-boxinFigure5.1hasthesamedelayastheHilbert¯lter.Thedelay isneededinrealapplicationstosynchronizethetwosignalsfm(t)andbfm(t). 29 A.Appendix A.1.ImplementationoftheHermitepolynomial ImplementedinMathematica. t:=t; x:=x; k:=k; Hy=0; f=1/(t^2+1); Hf=t/(t^2+1); a=-In¯nity; b=In¯nity; m=2; M=20; Do[ Fi=E^(-t^2/2)*HermiteH[n,t]/Sqrt[2^n*n!*Sqrt[Pi]]; FFi=Sum[Fi*E^(-I*2*Pi*k*t/M),ft,0,M-1g]; HFi=Re[2/M*Sum[FFi*Sin[(2*Pi*k*t)/M],fk,1,M/2-1g]]; Hy+=Integrate[f*Fi,ft,a,bg]*HFi, fn,0,mg] Plot[ff,Hf,Hyg,ft,-10,10g,AxesLabel->f"t","f(t),Hf(t),Hy(t)"g]; 30 A.2.ImplementationoftheFourierseries ImplementedinMathematica. t:=t; k:=k; y=Hy=0; f=1; a=-1; b=1; m=30; Do[ If[k<1, an=1/(2*Pi)*Integrate[f,ft,a,bg], an=1/Pi*Integrate[f*Cos[k*t],ft,a,bg]; bn=1/Pi*Integrate[f*Sin[k*t],ft,a,bg]; y+=an*Cos[k*t]+bn*Sin[k*t], fk,0,mg] Do[ Han=1/Pi*Integrate[f*Sin[k*t],ft,a,bg]; Hbn=1/Pi*Integrate[f*Cos[k*t],ft,a,bg]; Hy+=Han*Cos[k*t]+Hbn*Sin[k*t], fk,1,mg] Plot[fy,Hy,f*(Sign[b-t]+Sign[t-a])/2g,ft,-5,5g,AxesLabel->f"t","f(t)"g]; Print["f(t)=",f,"from",a,"to",b,"calculatedby",m,"terms."] 31 A.3.ImplementationoftheFouriertransform ImplementedinMathematica. n:=n; k:=k; M=10; Freq=2; f=Cos[2*Pi/M*n*Freq]; F=Sum[f*E^(-I*2*Pi*k*n/M),fn,0,M-1g]; Hf=2/M*Sum[F*Sin[(2*Pi*k*n)/M],fk,1,M/2-1g]; Plot[Abs[F],fk,-M/2,M/2g,AxesLabel->f"w","F(w)"g]; Plot[ff,Hfg],fn,-2*M/Pi,2*M/Pig,AxesLabel->f"t","f(t),Hf(t)"g]; 32 References [1]AnianssonJ.etal,Fouriermetoder,KTH,Stockholm,1989. [2]GoldbergR.R.,Fouriertransforms,Cambrigeuniversitypress,Cambridge. [3]HahnStefanL.,Hilberttransformsinsignalprocessing,ArtechHouse,Inc., Boston,1996. [4]LennartHellstrÄom,LinjÄaranalys,HÄogskolaniVÄaxjÄo,1995. [5]HenriciPeter,Appliedandcomputationalcomplexanalysis,Volume1,John Wiley&Sons,Inc.,NewYork,1988. [6]ProakisJohnG.,SalehiMasoud,Communicationsystemsengineering, Prentice-Hall,Inc.,NewJersey,1994. [7]Sa®E.B.,SniderA.D.,Complexanalysisformathematics,sienceanden- ginering,Prentice-Hall,Inc.,NewYork,1976. 33