Course 04_Discrete Time Signals, Discrete Time Fourier Transform

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1、Parseval’sTheoremofCFT•if??and??areatransformpairofCFT,then∞21∞2????=????,−∞2?−∞(??and

2、??

3、arethemagnitudes)•Thisisparseval’stheorem,meansthattheenergyinthetimedomainisequaltotheenergyinthefrequencydomain(dividedbyascalefactor2).Discrete-timeSignalsinCo

4、ntinuousDomain•Howtorepresentadiscrete-timesignalinthetimedomainforcontinuousFouriertransform?•Adiscrete-timesignalcanberepresentedasasequenceofimpulsefunctions(animpulsetrain)occurredatequallyspacedtimeinstances,inthecontinuous-functionaldomain.Discret

5、e-timeSignals•Acommonwaytoobtainadiscrete-timesignalistosampleacontinuous-timesignalatequallyspacedtimeinstances.•Referredtoasuniformsampling:ImpulseSequenceContinuousFourierTransformofanImpulseSequenceRecall•Whenthefrequencydomainspectrumisanequally-sp

6、acedimpulsesequence,whatisthetimedomainsignal?•Thetime-domainsignalisaperiodicfunction.ContinuousFourierTransformofanImpulseSequenceNotethat•ContinuousFouriertransformistime-frequencyanalogous.ForwardBackward1jwtjwtftFjwedwFjwftedt2C

7、ontinuousFourierTransformofanImpulseSequence•Ifwe(purposively)performforwardtransformforthecomplexconjugateofthefrequencydomainspectrumF*(jw)andexchangetherolesofwandt,wehavethefollowingsymmetrypropertyoftheFouriertransform:***jwtjwtFjwedwF

8、jwedw2ft()ForwardtransformapplytheruleofofF*(jw)BackwardtransformContinuousFourierTransformofanImpulseSequence•Byexchangingthevariableswandt,wehave**jtwFjtedt2f()w•Thatis,whenF*(jt)isatime-domainsignal,itsfrequencydomainspectrumis2f*(

9、w).ContinuousFourierTransformofanImpulseSequence•Takingthecomplexconjugateoff(t)orF(jw)doesnotaffectthepropertiesofbeingaperiodicfunctionoranimpulsesequence.•So,whenthetimedomainsignalisanequally-spacedimpulsesequence,thefrequencydomainspectrumisaperiod

10、icfunction.SamplingaContinuousFunction•Whatistherelationshipofthecontinuous-timesignalandthesampledsignalinthefrequencydomain?SamplingTheorem•Givenananalog(i.e.,continuous-time)signalx(t)a•Letusassumethatadiscrete-timesignalx(nT)