2023學(xué)年完整公開(kāi)課版chapter4

1Chapter4

TheDiscreteFourierTransform2Outline1.1

DTFT1.2UnderstandingtheDFTEquation1.3DFTProperties1.4InverseDFT1.5

DFTLeakage1.6Windows1.7DFTResolution,ZeroPadding,andFrequency-DomainSampling4.1DTFT3IntheliteratureofDSPyou’llencounterthetopicsofcontinuousFouriertransform,Fourierseries,discrete-timeFouriertransform,discreteFouriertransform,andperiodicspectra.4Figure4-1-1(a)Infinite-lengthcontinuoustimesignalwithtransforms

5Figure4-1-1(b)Infinite-lengthsignalofperiodicpulseswithtransforms

6Figure4-1-1(c)(d)Infinite-lengthdiscretetimesequenceanddiscreteperiodicsampleswithtransforms

7

Itisageometricseriesandcanbeevaluatedas:8Xo(ω)iscontinuousandperiodicwithaperiodof2π,whosemagnitudeshowninthefollowingFigure.Thisisanexampleofasampled(ordiscrete)time-domainsequencehavingaperiodicspectrum.Figure4-1-24.2UnderstandingtheDFTEquation9Withtheadventofthedigitalcomputer,theeffortsofearlydigitalprocessingpioneersledtothedevelopmentoftheDFTdefinedasthediscretefrequency-domainsequenceX(m),where:Eq4-1x(n)isadiscretesequenceoftime-domainsampledvaluesofthecontinuousvariablex(t).10

Eq4-211Quiteoftenwe’reinterestedinboththemagnitudeandthepower(magnitudesquared)containedineachX(m)term,andthestandarddefinitionsforrighttrianglesapplyhereasdepictedinthefollowingFigure.Figure4-2-112IfwerepresentanarbitraryDFToutputvalue,X(m),byitsrealandimaginaryparts:ThemagnitudeofX(m)is:Eq4-4Eq4-513Bydefinition,thephaseangleofX(m),X?(m),isThepowerofX(m),referredtoasthepowerspectrum,isthemagnitudesquaredwhereEq4-6Eq4-74.3DFTProperties14DFTProperties： 1)Symmetry 2)DFTLinearity 3)DFTMagnitudes 4)DFTFrequencyAxis 5)DFTShiftingTheorem15SymmetryLookingatFigure(a)again,thereisanobvioussymmetryintheDFTresults.Whentheinputsequencex(n)isreal,asitwillbeforallofourexamples,thecomplexDFToutputsform=1tom=(N/2)–1areredundantwithfrequencyoutputvaluesform>(N/2).Themthand(N–m)thoutputsarerelatedbythefollowing:Eq4-8

16SymmetryFigure4-2-2DFTresults:(a)magnitudeofX(m);(b)phaseofX(m);(c)realpartofX(m);(d)imaginarypartofX(m).17SymmetryWecanstatethatwhentheDFTinputsequenceisreal,X(m)isthecomplexconjugateofX(N–m),orWherethesuperscript*symboldenotesconjugation.Eq4-918Symmetry

Eq.(4–1)’sexponentialsaremucheasiertomanipulatewhenwe’retryingtoanalyzeDFTrelationships.Wegettheexpressionforthe(N–m)thcomponentoftheDFT:Eq4-919Symmetry

Eq4-11WeseethatX(N–m)inEq.(4–11)ismerelyX(m)inEq.(4–1)withthesignreversedonX(m)’sexponent—andthat’sthedefinitionofthecomplexconjugate.20DFTLinearity

TheDFThasaveryimportantpropertyknownaslinearity.ThispropertystatethattheDFTofthesumoftwosignalsisequaltothesumofthetransformsofeachsignal:Eq4-1221DFTMagnitudes

Eq4-1222DFTMagnitudes

Eq4-1423DFTMagnitudes TheDFTmagnitudeexpressionsinEqs.(4–13)and(4–14)arewhyweoccasionallyseetheDFTdefinedintheliteratureas:Eq4-1524DFTMagnitudesTherearecommercialsoftwarepackagesusingEq4-14andfortheforwardandinverseDFTs.25DFTFrequencyAxis

26DFTShiftingTheorem

Eq4-174.4InverseDFT27 WecanreverseDFTprocessandobtaintheoriginaltimedomainsignalbyperformingtheIDFTontheX(m)frequency-domainvalues.ThestandardexpressionsfortheIDFTareEq4-1828andequally,Eq4-194.5DFTLeakage29 Acharacteristic,knownasleakage,causesourDFTresultstobeonlyanapproximationofthetruespectraoftheoriginalinputsignalspriortodigitalsampling. Althoughtherearewaystominimizeleakage,wecan’teliminateitentirely.Thus,weneedtounderstandexactlywhateffectithasonourDFTresults.30

Eq4-19

31we’lljustsaythat,forarealcosineinputhavingkcyclesintheN-pointinputtimesequence,theamplituderesponseofanN-pointDFTbinintermsofthebinindexmisapproximatedbythesincfunctionEq4-2032DFTpositivefrequencyresponseduetoanN-pointinputsequencecontainingkcyclesofarealcosine.Figure4-5-1(a)amplituderesponseasafunctionofbinindexmWe’lluseEq.(4-20),illustratedintheFigure4-5-1(a),tohelpusdeterminehowmuchleakageoccursinDFTs.33WeshowtheDFT’smagnituderesponsetoarealinputintermsoffrequency(Hz)inFigure4-5-1(b).Figure4-5-1(b)magnituderesponseasafunctionoffrequencyinHz34Figure4-5-2DFTbinpositivefrequencyresponses35TheFigure4-5-2shows:(a)DFTinputfrequency=8.0kHz;(b)DFTinputfrequency=8.5kHz;(c)DFTinputfrequency=8.75kHz.36Figure4-5-3SpectralreplicationwhentheDFTinputis3.4cyclespersampleintervalFigure4-5-3showssomeoftheadditionalreplicationsinthespectrumforthe3.4cyclespersampleintervalexample.37Figure4-5-4DFToutputmagnitude38TheFigure4-5-4shows:(a)whentheDFTinputis3.4cyclespersampleinterval;(b)whentheDFTinputis28.6cyclespersampleinterval.4.6Windows39 WereducesDFTleakagebyforcingtheamplitudeoftheinputtimesequenceatboththebeginningandtheendofthesampleintervaltogosmoothlytowardasinglecommonamplitudevalue.Figure4–6-1willshowhowthisprocessworks.40Figure4-6-1Minimizingsampleintervalendpointdiscontinuities41TheFigure4-6-1shows:infinitedurationinputsinusoid;(b)rectangularwindowduetofinite-timesampleinterval;(c)productofrectangularwindowandinfinite-durationinputsinusoid;(d)triangularwindowfunction;(e)productoftriangularwindowandinfinite-durationinputsinusoid;(f)Hanningwindowfunction;(g)productofHanningwindowandinfinite-durationinputsinusoid;(h)Hammingwindowfunction.42

Eq4-2143Thefollowingexpressionsdefineourwindowfunctioncoefficients:Rectangularwindow:(alsocalledtheuniform,orboxcar,window)Triangularwindow:(verysimilartotheBartlett[3],andParzen[4,5]windows)Hanningwindow:(alsocalledtheraisedcosine,Hann,orvonHannwindow)Hammingwindow:44Figure4-6-2Windowmagnituderesponses

45Figure4-6-3Hanningwindow46TheFigure4-6-3shows:(a)64-sampleproductofaHanningwindowanda3.4cyclespersampleintervalinputsinewave;(b)HanningDFToutputresponsevs.rectangularwindowDFToutputresponse.47Figure4-6-4Increasedsignaldetectionsensitivityaffordedusingwindowing48TheFigure4-6-4shows:(a)64-sampleproductofaHanningwindowandthesumofa3.4cyclesanda7cyclespersampleintervalsinewaves;(b)reducedleakageHanningDFToutputresponsevs.rectangularwindowDFToutputresponse.4.7DFTResolution,ZeroPadding,andFrequency-DomainSampling49 OnepopularmethodusedtoimproveDFTspectralestimationisknownaszeropadding. Thisprocessinvolvestheadditionofzero-valueddatasamplestoanoriginalDFTinputsequencetoincreasethetotalnumberofinputdatasamples.

50Figure4-7-1ContinuousFouriertransformTheFigure4-7-1shows:64-sampleproductofaHanningwindowandthesumofa3.4cyclesanda7cyclespersampleintervalsinewaves;(b)reducedleakageHanningDFToutputresponsevs.rectangularwindowDFToutputresponse.51

Toillustratethisidea,supposewewanttoapproximatetheCFTofthecontinuousf(t)functioninFigure4–7-1(a).

Thisf(t)waveformextendstoinfinityinbothdirectionsbutisnonzeroonlyoverthetimeintervalofTseconds.IfthenonzeroportionofthetimefunctionisasinewaveofthreecyclesinTseconds,themagnitudeofitsCFTisshowninFigure4–7-1(b).(BecausetheCFTistakenoveraninfinitelywidetimeinterval,theCFThasinfinitesimallysmallfrequencyresolution,resolutionsofine-grainedthatit’scontinuous.)

It’sthisCFTthatwe’llapproximatewithaDFT.52Figure4-7-2DFTfrequency-domainsamplingTheFigure4-7-2shows:(a)16inputdatasamplesandN=16;(b)16inputdatasamples,16paddedzeros,andN=32;(c)16inputdatasamples,48