Topic EnergyPowerSpectraandCorrelation In Lecture we reviewed the notion of average signal - PDF document

5/25.1.1AquickcheckItisworthcheckingthisusingtherelationshipsfoundinLecture1:Cm=8:1 2(Am�iBm)form�0A0=2form=01 2(Ajmj+iBjmj)form0Forn0thequantitiesarejC0j2=1 2A022jCnj2=21 2(Am�iBm)1 2(Am+iBm)=1 2�A2n+B2n
inagreementwiththeexpressioninLecture1.5.2EnergysignalsvsPowersignalsWhenconsideringsignalsinthecontinuoustimedomain,itisnecessarytodis-tinguishbetween\niteenergysignals",or\energysignals"forshort,and\nitepowersignals".FirstletusbeabsolutelyclearthatAllsignalsf(t)aresuchthatjf(t)j2isapower.Anenergysignalisonewherethetotalenergyisnite:ETot=Z1�1jf(t)j2dt0ETot1:Itissaidthatf(t)is\squareintegrable".AsETotisnite,dividingbytheinnitedurationindicatesthatenergysignalshavezeroaveragepower.Tosummarizebeforeknowingwhatallthesetermsmean:AnEnergysignalf(t)alwayshasaFouriertransformF(!)alwayshasanenergyspectraldensity(ESD)givenbyEff(!)=jF(!)j2alwayshasanautocorrelationRff()=R1�1f(t)f(t+)dtalwayshasanESDwhichistheFToftheautocorrelationRff(),Eff(!)alwayshastotalenergyETot=Rff(0)=1 2R1�1Eff(!)d!alwayshasanESDwhichtransfersasEgg(!)=jH(!)j2Eff(!) 5/4Nowsupposeg(t)=f(t).WeknowthatZ1�1f(t)e�i!tdt=F(!))Z1�1f(t)e+i!tdt=F(!))Z1�1f(t)e�i!tdt=F(�!)Thisis,ofcourse,aquitegeneralresultwhichcouldhavebeenstuckinLecture2,andwhichisworthhighlighting: TheFourierTransformofacomplexconjugateisZ1�1f(t)e�i!tdt=F(�!)Takecarewiththe�!. Backtotheargument.IntheearlierexpressionwehadZ1�1f(t)g(t)dt=1 2Z1�1F(p)G(�p)dp)Z1�1f(t)f(t)dt=1 2Z1�1F(p)F(p)dpNowpisjustanyparameter,soitispossibletotidytheexpressionbyreplacingitwith!.Thenwearriveatthefollowingimportantresult Parseval'sTheorem:ThetotalenergyinasignalisETot=Z1�1jf(t)j2dt=1 2Z1�1jF(!)j2d!=Z1�1jF(!)j2df NB!Thedf=d!=2,andisnothingtodowiththesignalbeingcalledf(t).5.4TheEnergySpectralDensityIftheintegralgivesthetotalenergy,itmustbethatjF(!)j2istheenergyperHz.Thatis: TheENERGYSpectralDensityofasignalf(t),F(!)isdenedasEff(!)=jF(!)j2 5/65.6CorrelationCorrelationisatoolforanalysingwhetherprocessesconsideredrandomaprioriareinfactrelated.Insignalprocessing,cross-correlationRfgisusedtoassesshowsimilartwodierentsignalsf(t)andg(t)are.Rfgisfoundbymultiplyingonesignal,f(t)say,withtime-shiftedvaluesoftheotherg(t+),thensumminguptheproducts.IntheexampleinFigure5.1thecross-correlationwilllowiftheshift=0,andhighif=2or=5. Figure5.1:Thesignalf(t)wouldhaveahighercross-correlationwithpartsofg(t)thatlooksimilar.Onecanalsoaskhowsimilarasignalistoitself.Self-similarityisdescribedbytheauto-correlationRff,againasumofproductsofthesignalf(t)andacopyofthesignalatashiftedtimef(t+).Anauto-correlationwithahighmagnitudemeansthatthevalueofthesignalf(t)atoneinstantsignalhasastrongbearingonthevalueatthenextinstant.Correlationcanbeusedforbothdeterministicandrandomsignals.WewillexplorerandomprocessesthisinLecture6.Thecross-andauto-correlationscanbederivedforbothniteenergyandnitepowersignals,buttheyhavedierentdimensions(energyandpowerrespectively)anddierinothermoresubtleways.Wecontinuebylookingattheauto-andcross-correlationsofniteenergysignals.5.7TheAuto-correlationofaniteenergysignalTheauto-correlationofaniteenergysignalisdenedasfollows.Weshalldealwithrealsignalsf,sothattheconjugatecanbeomitted. 5/7 Theauto-correlationofasignalf(t)ofniteenergyisdenedRff()=Z1�1f(t)f(t+)dt=(forrealsignals)Z1�1f(t)f(t+)dtTheresultisanenergy. Therearetwowaysofenvisagingtheprocess,asshowninFigure5.2.Oneistoshiftacopyofthesignalandmultiplyvertically(sotospeak).Forpositivethisisashifttothe\left".Thisismostusefulwhencalculatinganalytically. Figure5.2:g(t)andg(t+)forapositiveshift.5.7.1Basicpropertiesofauto-correlation1.Symmetry.Theauto-correlationfunctionisanevenfunctionof:Rff()=Rff(�):Proof:Substitutep=t+intothedenition,andyouwillgetRff()=Z1�1f(p�)f(p)dp:Butpisjustadummyvariable.ReplaceitbytandyourecovertheexpressionforRff(�).(Infact,insometextsyouwillseetheautocorrelationdenedwithaminussigninfrontofthe.) 5/82.Foranon-zerosignal,Rff(0)�0.Proof:Foranynon-zerosignalthereisatleastoneinstantt1forwhichf(t1)6=0,andf(t1)f(t1)�0.HenceR1�1f(t)f(t)dt�0.3.Thevalueat=0islargest:Rff(0)Rff().Proof:Consideranypairofrealnumbersa1anda2.As(a1�a2)20,weknowthata21+a22a1a2+a2a1.Nowtakethepairsofnumbersatrandomfromthefunctionf(t).Ourresultshowsthatthereisnorearrangement,randomorordered,ofthefunctionvaluesinto(t)thatwouldmakeRf(t)(t)dt�Rf(t)2dt.Using(t)=f(t+)isanorderedrearrangement,andsoforanyZ1�1f(t)2dtZ1�1f(t)f(t+)dt5.8|Applications5.8.1|SynchronisingtoheartbeatsinanECG(DIYsearchandread)5.8.2|ThesearchforExtraTerrestrialIntelligence Figure5.3:ChattyaliensForseveraldecades,theSETIorganizationhavebeenlook-ingforextraterrestrialintelligencebyexaminingtheauto-correlationofsignalsfromradiotelescopes.Oneprojectscanstheskyaroundnearby(200lightyears)sun-likestarschoppingupthebandwidthbetween1-3GHzinto2billionchannelseach1Hzwide.(Itisassumedthatanattempttocommunicatewoulduseasinglefrequency,highlytuned,signal.)Theydeterminetheautocorrelationeachchan-nel'ssignal.Ifthechannelisnoise,onewouldobserveaverylowautocorrelationforallnon-zero.(SeewhitenoiseinLecture6.)Butifthereis,say,arepeatedmessage,onewouldobserveaperiodicriseintheautocorrelation. Figure5.4:Rffat=0isalwayslarge,butwilldroptozeroifthesignalisnoise.Ifthemessagesaligntheautocorrelationwithrise. 5/95.9TheWiener-KhinchinTheoremLetustaketheFouriertransformofthecross-correlationRf(t)g(t+)dt,thenswitchtheorderofintegration,FTZ1�1f(t)g(t+)dt=Z1=�1Z1t=�1f(t)g(t+)dte�i!d=Z1t=�1f(t)Z1=�1g(t+)e�i!ddtNoticethattisaconstantfortheintegrationwrt(that'showf(t) oatedthroughtheintegralsign).Substitutep=t+intoit,andtheintegralsbecomeseparableFTZ1�1f(t)g(t+)dt=Z1t=�1f(t)Z1p=�1g(p)e�i!pe+i!tdpdt=Z1�1f(t)ei!tdtZ1�1g(p)e�i!pdp=F(!)G(!):Ifwespecializethistotheauto-correlation,G(!)getsreplacedbyF(!).Then ForaniteenergysignalTheWiener-KhinchinTheoremasaysthatTheFToftheAuto-CorrelationistheEnergySpectralDensityFT[Rff()]=jF(!)j2=Eff(!) aNorbertWiener(1894-1964)andAleksandrKhinchin(1894-1959) (Thismethodofproofisvalidonlyforniteenergysignals,andrathertrivializestheWiener-Khinchintheorem.Thefundamentalderivationliesinthetheoryofstochasticprocesses.)5.10CorollaryofWiener-KhinchinThiscorollaryjustconrmsaresultobtainedearlier.WehavejustshownthatRff(),Eff(!).ThatisRff()=1 2Z1�1Eff!)ei!d!whereisusedbyconvention.Nowset=0 5/115.13|ExampleandApplication[Q]Determinethecross-correlationofthesignalsf(t)andg(t)shown. [A]Startbysketchingg(t+)asfunctionoft. f(t)ismadeofsectionswithf=0,f=t 2a,thenf=0.g(t+)ismadeofg=0,g=t a��2+ a
,g=1,theng=0.Theleft-mostnon-zerocongurationisvalidfor04a+a,sothatFor�4a�3a:Rfg()=Z1�1f(t)g(t+)dt=Z4a+0t 2a
1dt=(4a+)2 4aFor�3a�2a:Rfg()=Z1�1f(t)g(t+)dt=Z3a+0t 2a
t a�2+ adt+Z4a+3a+t 2a
1dtFor�2a�a:Rfg()=Z1�1f(t)g(t+)dt=Z3a+2a+t 2a
t a�2+ adt+Z2a3a+t 2a
1dtFor�a0:Rfg()=Z1�1f(t)g(t+)dt=Z2a2a+t 2a
t a�2+ adtWorkingouttheintegralsandndingthemaximumisleftasaDIYexercise. 5/12 Figure5.5:5.13.1ApplicationItisobviousenoughthatcross-correlationisusefulfordetectingoccurencesofa\model"signalf(t)inanothersignalg(t).Thisisa2Dexamplewherethemodelsignalf(x;y)isthebackviewofafootballer,andthetestsignalsg(x;y)areimagesfromamatch.Thecrosscorrelationisshowninthemiddle.5.14Cross-EnergySpectralDensityTheWiener-KhinchinTheoremwasactuallyderivedforthecross-correlation.Itsaidthat TheWiener-KhinchinTheoremshowsthat,foraniteenergysignal,theFToftheCross-CorrelationistheCross-EnergySpectralDensityFT[Rfg()]=F(!)G(!)=Efg(!) 5.15FinitePowerSignalsLetususef(t)=sin!0ttomotivatediscussionaboutnitepowersignals.Allperiodicsignalsarenitepower,inniteenergy,signals.OnecannotevaluateR1�1jsin!0tj2dt.However,bysketchingthecurveandusingthenotionofself-similarity,onewouldwishthattheauto-correlationispositive,butdecreasing,forsmallbutincreasing;thennegativeasthethecurvesareinanti-phaseanddissimilarinan\organized"way,thenreturntobeingsimilar.Theautocorrelationshouldhavethesameperiodasitsparentfunction,andlargewhen=0|soRffproportionaltocos(!0)wouldseemright.Wedenetheautocorrelationasanaveragepower.Notethatforaperiodic 5/13 Figure5.6:functionthelimitoveralltimeisthesameasthevalueoveraperiodT0Rff()=limT!11 2TZT�Tsin(!0t)sin(!0(t+))dt!1 2(T0=2)ZT0=2�T0=2sin(!0t)sin(!0(t+))dt=!0 2Z=!0�=!0sin(!0t)sin(!0(t+))dt=1 2Z�sin(p)sin(p+!0))dp=1 2Z�sin2(p)cos(!0)+sin(p)cos(p)sin(!0)dp=1 2cos(!0)Foraniteenergysignal,theFourierTransformoftheautocorrelationwastheenergyspectraldensity.Whatistheanalogousresultnow?Inthisexample,FT[Rff]= 2[(!+!0)+(!�!0)]Thisisactuallythepowerspectraldensityofsin!0t,denotedSff(!).The-functionsareobviousenough,buttocheckthecoecientletusintegrateoverallfrequencyf:Z1�1Sff(!)df=Z1�1 2[(!+!0)+(!�!0)]df=Z1�1 2[(!+!0)+(!�!0)]d! 2=1 4[1+1]=1 2: 5/14Thisdoesindeedreturntheaveragepowerinasinewave.WecanuseFourierSeriestoconcludethatthisresultsmustalsoholdforanyperiodicfunction.Itisalsoapplicabletoanyinniteenergy\nonsquare-integrable"function.WewilljustifythisalittlemoreinLecture61.Tonisho,weneedonlystatetheanalogiestotheniteenergyformulae,replacingEnergySpectralDensitywithPowerSpectralDensity,andreplacingTotalEnergywithAveragePower. TheautocorrelationofanitepowersignalisdenedasRff()=limT!11 2TZT�Tf(t)f(t+)dt: TheautocorrelationfunctionandPowerSpectralDensityareaFourierTransformPairRff(),Sff(!) TheaveragepowerisPAve=Rff(0) ThepowerspectrumtransfersacrossasystemasSgg(!)=jH(!)j2Sff(!)Thisresultisprovedinthenextlecture. 5.16Cross-correlationandpowersignalsTwopowersignalscanbecross-correlated,usingasimilardenition:Rfg()=limT!11 2TZT�Tf(t)g(t+)dtRfg(),Sfg(!)5.17InputandOutputfromasystemOneverylastthought.Ifoneappliesannitepowersignaltoasystem,itcannotbeconvertedintoaniteenergysignal|orviceversa. 1ToreallynailitwouldrequireustounderstandWiener-Khinchinintoomuchdepth.