CH4-The-Continuous-Time-Fourier-Transform

1Chapter4TheContinuous-TimeFourierTransform24.0INTRODUCTION3Representcontinuous-timeaperiodicsignalsaslinearcombinationsofcomplexexponentials.FouriertransformandinverseFouriertransform.（傅立叶变换）（傅立叶反变换）UseFouriermethodstoanalyzeandunderstandsignalsandLTIsystems.44.1RepresentationofAperiodicSignals:Continuous-TimeFourierTransform54.1.1DevelopmentoftheFouriertransformrepresentationofanaperiodicsignal0102sin()FSkkTxtakT012sinkkTTa---continuousvariable---theenvelopeofTak---equallyspacedsampleska12sinTT206谱线间隔T=4T1T=8T1T=16T1012sinkkTTaT20T增大，间隔变小T1固定时，包络Tak不变T→∞（单个矩形脉冲），谱线间隔→0，谱系数ak→包络线（连续谱）7TtjkkktjkkdtetxTaeatx00)(~1)(~02()00kTa0/2/2()TjktkTTaxtedt0()()0jktkjtTaxtedtxtedt2/2/0)(TTtjkdtetxperiodicaperiodickaT0再用表示频谱不合适，虽然各频谱幅度无限小，但相对大小仍有区别。引入spectrumdensity(频谱密度)函数。kaTπ20090,01kaTf22jde)(10TTtkkttxTakTTajXlim22jde)(0limTTtkTttx连续变量00dk22jde)(0TTtkkttxTaT有界函数kTaspetrumdensityorspectrum单位频带上的频谱值2T2TdtetxTtjk010dtetxjXtj)()(Define)(0jkXTak)(10jkXTaktjkkejkXTtx0)(1)(~0)(~txFourierseriesofktjkejkX000)(2102()T00:,,Tdk()lim()or()()TTxtxtxtxtdejXtxtj)(21)(InverseFouriertransformFouriertransform11Fouriertransformpair:()()1()()2jtjtXjxtedtxtXjed()()FxtXjTheaperiodicsignalscanstillberepresentedasalinearcombinationofcomplexexponentials.ThemagnitudeofcomponentwithfrequencyωisAlthoughX(jω)isoftenabbreviatedas“spectrum”,itisdifferentfromak,whichisthespectrumofperiodicsignals.X(jω)isinfactspectrum-densityfunction(频谱密度函数)()2kXjdatjefaTajkXkk)(012Anusefulrelationship:0)(1kkjXTa)(~tx)(10jkXTakx(t)isoneperiodoftheperiodicsignal()()FxtXjkFSatx)(~∴ak正比于傅立叶变换X(jω)的等间隔样本00Periodicsignal(Aperiodicsignal)1()|()|kkkk()aXjTXjTa1300Periodicsignal(Aperiodicsignal)1()|()|kkkk()aXjTXjTaExample:periodicsquarewave:0102sin()()FSkkTxtakTSoaperiodicsignalx(t):12sin()()()FTxtXj144.1.2ConvergenceofFouriertransformsDirichletconditions:(1)x(t)isabsolutelyintegrable.(2)x(t)haveafinitenumberofmaximaandminimawithinanyfiniteinterval.(3)x(t)haveafinitenumberofdiscontinuitywithinanyfiniteinterval.Furthermore,eachofthesediscontinuitiesmustbefinite.|()|xtdtIfimpulsefunctionsarepermittedinthetransform,somesignalswhicharenotabsolutelyintegrableoveraninfiniteinterval,canalsobeconsideredtohaveFouriertransforms.15.Example4.14.1.3ExamplesofContinuous-TimeFT)()(tuetxat0ajaejadteedtetxjXtjatjattj101)()()(0221)(ajX)arctan()(argajX16Example4.2taetx)(0a00)(tetetxatat17TheunitimpulsehasaFouriertransformconsistingofequalcontributionatallfrequencies.Thisspectrumisreferredtoaswhite-spectrum.(becausethewhitecolorhasthesamespectrum).Example4.3()()xtt1)()(dtetjXtj18Example4.4111,()0,tTxttT)sin(2)()(111TdtedtetxjXTTtjtj19Example4.5Considerthesignalx(t)whoseFouriertransformis1,()0,WXjWtWtdedejXtxWWtjtj)sin(21)(21)()(sin)(WtcWtx20tttsin)Sa(ttSa1ππ2π3OπSamplingSignal(抽样信号)ttsinc1123O1tttsin)(csin21:()1,WXj()()xttW//W)()(,W，主瓣宽度的峰值变宽；txjX224.2TheFourierTransformforPeriodicSignals23AsignalhastheFouriertransform0()jtxteThen)(2)(0jXtjtjedetx0)(221)(0TheFouriertransformofthecomplexexponentialsignalisanimpulselocatedatwithitsarea2π.tje00)(20F0tje24Foranarbitraryperiodicsignalx(t),representingx(t)withtheFourierseriesastjkkkeatx0)()(2)(2)(00kakajXkkkk)(20F0tjeTheFouriertransformofaperiodicsignalwithFourierseriescoefficients{ak}canbeinterpretedasatrainofimpulsesoccurringattheharmonicallyrelatedfrequencies；Andtheareaoftheimpulseatthekthharmonicfrequencykω0is2πtimesthekthFourierseriescoefficientak.00()()2()FjktkkkkxtaeXjak25Example4.6Considertheperiodicsquarewave.ItsFourierseriescoefficientsare010102sin()sin()kkTkTakTkthenitsFouriertransformiskkkkkTkkajX)(sin2)(2)(010026FouriertransformofasymmetricsquarewaveforT=4T1010100sin2sin()2()(),kkkTkTXjkkkk010100sin2sin()2()(),kkkTkTXjkkkk27Example4.7Considerandttx01sin)(20()cosxttTheFourierseriescoefficientsforx1(t)are1,0,21,2111kajajakTheFourierseriescoefficientsforx2(t)are1,0,2111kaaak)]()([)(2)(0001jkajXkk)]()([)(2)(0002kkkajX282/2/1)(10TTtjkkTdtetTakkkTkTjX)()2(2)(00Example4.8impulsetrainkkTttx)()(时频相反：时域周期T增大，频域周期2π/T减小。294.3PropertiesofContinuous-TimeFourierTransform30then()(),()()FFxtXjytYjIf4.3.1Linearity)()()()(jbYjaXtbytaxF()()1()()2jtjtXjxtedtxtXjed()()FxtXj)()(txFjX)()(1jXFtx314.3.2TimeShiftingIfF()()xtXjthenF00()()jtxtteXjdejXttxttj)(00)(21)(∴timesh