信号教学课件(华中科技大学)cha

CHAPTER5THEDISCRETE TIMEFOURIERTRANSFORM5 0INTRODUCTION Discrete timeFouriertransformandinverseFouriertransform ApplicationofDTFTindiscrete timeLTIsystems Similaritiesanddifferencesbetweencontinuous timeanddiscrete timeFouriertransforms 5 1REPRESENTATIONOFNONPERIODICSIGNALS THEDISCRETE TIMEFOURIERTRANSFORM As Definingafunction Thus Consequently As Differencesbetweenthecontinuous timeanddiscrete timeFouriertransformare periodicityofthediscrete timetransformandthefiniteintervalofintegrationinthesynthesisequation Indiscretetime Lowfrequenciesarethevaluesof nearevenmultipleof highfrequenciesarethosevaluesof nearoddmultiplesof Example5 1Considerthesignal 1 0 0 1 Example5 2Considerthesignal for0 1 Example5 3Considertherectangularpulse ConvergenceIssuesoftheDiscrete TimeFourierTransform Ifx n isaninfinitedurationsignal wemustconsiderthequestionofconvergenceoftheinfinitesummationintheanalysisequation Theanalysisequationwillconvergeifx n isabsolutelysummable thatis Incontrasttothesituationfortheanalysisequation therearegenerallynoconvergenceissuesassociatedwiththesynthesisequation Considertheunitsamplex n n Approximating n byanintegralofcomplexexponentialswithfrequenciestakenovertheinterval W i e ThisisplottedinthefollowingFigureforseveralvaluesofW Approximationtotheunitsampleusingcomplexexponentialswithfrequencies W a W 4 b W 3 8 c W 2 d W 3 4 e W 7 8 f W NotethatforW 5 2THEFOURIERTRANSFORMFORPERIODICSIGNALS FirstconsidertheFouriertransformofthesequence Tocheckthevalidityofthisexpression Nowconsideranarbitraryperiodicsequencex n withperiodNandwiththeFourierseriesrepresentation ApplyingtheFouriertransformtobothsides weobtain Thus theFouriertransformofaperiodicsignalcanbedirectlyconstructedfromitsFouriercoefficients Example5 4Considertheperiodicsignal Thatis Example5 5Considertheperiodicsampletrain ThenwecanrepresenttheFouriertransformofthesignalas Choosingtheintervalofsummationas0 n N 1 5 3PROPERTIESOFTHEDISCRETE TIMEFOURIERTRANSFORM 5 3 1PeriodicityoftheDiscrete TimeFourierTransform 5 3 2LinearityoftheFourierTransform then 5 3 3TimeShiftingandFrequencyShifting If then and 5 3 4ConjugationandConjugateSymmetry If then ifx n isrealvalued 5 3 5DifferencingandAccumulation First difference Accumulation 5 3 6TimeReversal 5 3 7TimeExpansion If then 5 3 8DifferentiationinFrequency 5 3 9Parseval sRelation Example5 7Considertheunitstepsequenceu n Since and Thus Example5 8Considerthesequencex n whichisillustratedinthefollowingfigure 5 4THECONVOLUTIONPROPERTY If then TheconvolutionpropertyrepresentsthattheFouriertransformoftheresponseofanLTIsystemtoanonperiodicinputaresimplytheFouriertransformoftheinputmultipliedbythesystem sfrequencyresponseevaluatedatthecorrespondingfrequencies TheconvolutionpropertymapstheconvolutionoperationoftwotimesignalstothemultiplicationoperationoftheirFouriertransforms ThefrequencyresponsecapturesthechangeincomplexamplitudeoftheFouriertransformoftheinputateachfrequency Example5 8ConsideranLTIsystemwithsampleresponse Thefrequencyresponseis Thus foranyinputx n theFouriertransformoftheoutputis Consequently Example5 9ConsideranLTIsystemwithsampleresponse Theinputtothissystemis Theoutputy n If Thus If Example5 10Considerthesystem Whatisthefrequencyresponseoftheoverallsystem Whereisanideallow passfilterwithcutofffrequency 4andunitygaininthepassband Thekeystep Thus Since Consequently Fromtheconvolutionproperty theoverallsystemhasthefrequencyresponse Stopband 5 5THEMULTIPLICATIONPROPERTY ConsidertheFouriertransformofy n x1 n x2 n wheretheFouriertransformsofx1 n andx2 n areknown since Example5 11FindtheFouriertransformofasignalx n whichistheproductofx1 n andx2 n where Fromthemultiplicationproperty 5 6DUALITY SUMMARYOFFOURIERSERIESANDTRANSFORMEXPRESSIONS Forthediscrete timeFourierseries dualitybetweenthesequencex n inthetime domainanditsFourierseriescoefficientf k is Forthecontinuous timeFouriertransform dualitybetweenthesignalx t inthetime domainanditsFouriertransformX j is Dualityimpliesthateverypropertyhasadual Thereisalsoadualitybetweenthediscrete timeFouriertransformandthecontinuous timeFourierseries Example5 12Determinethediscrete timeFouriertransformofthesequence SincetheFouriertransformofx n isperiodicwithperiod2 andwiththeformofsquarewave soweconsidersignalg t whichisaperiodicsquarewavewithperiod2 andwith theFourierseriescoefficientsofg t are SeeTable4 2 LetT1 2 thenwehaveak x k takingakandg t intotheanalysisequation Renamingkasnandtas wehave Replacingnby n weobtain Thus 5 7SYSTEMSCHARACTERIZEDBYLINEARCONSTANT COEFFICIENTDIFFERENCEEQUATIONS isaratioofpolynomialsinthevariable coefficientsofthenumeratorpolynomial coefficientsappearingontherightsideofthedifferenceequation coefficientsofthedenominatorpolynomial coefficientsappearingontheleftsideofthedifferenceequation Example5 13ConsideracausalLTIsystemthatischaracterizedbythedifferenceequations andlettheinputtothissystembe Determinetheoutputy n Theformofthepartial fractionexpansioninthiscaseis Sothat Consequently 5 8DISCRETE TIMEFREQUENCY SELECTIVEFILTER 5 9SAMPLINGOFDISCRETE TIMESIGNALS Impulse trainsampling spectrumofsampledsignalwith s 2 M Indiscrete timecase theresultofsamplingtheoremalsoexist Example5 14Considerasequencex n whoseFouriertransform Determinethelowestrateatwhichx n maybesampledwithoutaliasing Since thecorrespondingsamplingfrequencyis2 4 2 Sothat Thus Fromthesamplingtheorem weknow 5 10SUMMARY TheFouriertransformforperiodicdiscrete timesignals TheFouriertransformforaperiodicdiscrete timesignals Convenientwa