Chapter-5-The-Discrete-Time-Fourier-Trans.ppt

Chapter5TheDiscrete-TimeFourierTransform,罗欣UESTC,2012-2013(1),Contents,TheDiscrete-TimeFourierTransform(DFT)TheFourierTransformforPeriodicSignalsPropertiesoftheDFT,Notation:CFS:(thecontinuousFourierseries,FS)CFT:(thecontinuoustimeFourierTransform,FT)DFS:(thediscreteFourierseries)DFT:(thediscretetimeFourierTransform),5.1RepresentationofAperiodicSignals:DFT,(1)Fourierseries(periodicsignal),5.1.1DevelopmentofDFT(fromDFStoDFT),Fourierseries(periodicsignal),ak,xn,Fouriertransform(aperiodicsignal),(2)Fouriertransform(aperiodicsignal),Sinceover,When,So,DFT,ThecoefficientsakareproportionaltoX(ej)samplesof,Whereisthespacingofthesamplesinthefrequencydomain.,When,Thediscrete-timeFouriertransformpair,Synthesisequation,Analysisequation,Thatmeans,xniscomposedofejnatdifferentfrequencieswith“amplitude”X(ej)(d/2);X(ej)isreferredtothespectrumofxn.,Note:1.RelationbetweenofDFTandDFS,DiscreteperiodicPeriodicdiscrete,DiscreteperiodicAperiodiccontinuous,2.ComparingDFTtoFS,Continuous-time,Discrete-frequency,Discrete-time,Continuous-frequency,Duality,Eq.(5.9)canberegardedtoFSinfrequency-domain;Eq.(5.8)isjustthatxnistheFScoefficientsofX(ej).,3.RelationbetweenDFTandFT,Let,and,so,Example:PoissonformulaIfthen,Proof:,Note:,Thatimplies,ItisaFourierSeriesinFrequency-Domain.,(TimeShifting),5.1.2ExamplesofDFT,Example5.1,Example5.2,Example5.3,Similartoasinc,butperiodicwithaperiod2.,Note:IftheperiodizationofxnwhichperiodNismade:,WecangetDFScoefficients:,Justsameas(3.104)inP218,Example5.4,Example5.5,IdealDTLPF,Adiscretesincintimedomain,5.1.3ConvergenceofDFT,Convergenceconditions:,or,Absolutelysummable,Finiteenergy,GibbsphenomenoninDFT?,W=/4,W=3/8,W=/2,W=3/4,W=7/8,W=,TherearenotanybehaviorsliketheGibbsphenomenoninevaluationthediscrete-timeFouriertransformsynthesisequation.(seepage368fig5.7),5.2DFTforperiodicsignals*,RecallCTresult:,WhataboutDT:,a)Weexpectanimpulse(ofarea2)at=0b)ButX(ej)mustbeperiodicwithperiod2,Infact,If0=0,Nowconsideraperiodicsequence,Seepage370Fig5.9,Example,1.,2.,3.,5.3PropertiesofDTFT,5.3.1Periodicity,5.3.3TimeShiftingandFrequencyShifting,5.3.2Linearity,Example,5.3.4ConjugationandConjugateSymmetry,5.3.5DifferencingandAccumulation,Ifxnisreal,5.3.6TimeReversal,5.3.7TimeExpansion,ifnisamultipleofkifnisnotamultipleofk.,5.3.8DifferentiationinFrequency,5.3.7Parseval