信號與系統(tǒng)課件及習(xí)題解答

ChapterTheChapterTheContinuous-TimeFourierFrequencyDomainAnalysisofLTIRepresentationofAperiodicTheResponseofLTISystemtoAperiodic3,3,4(a),10,11,14,15,25,32(a,b),35,36,37,4.0<ExampleX(2sin(k0T12sin(k0T12sin(4.0<ExampleX(2sin(k0T12sin(k0T12sin(xkkk00periodic1Xj|T02sin1210TT,20T1AperiodicT,20T1AperiodictX(2sin1signalofPeriodicAperiodicTx(t)1jk1Txk02PeriodicAperiodicTx(t)1jk1Txk020X(j)ex(t)k 0 x(t)eTX(j)kX(jk0)X(j)04.1RepresentationofAperiodicTheContinuous-TimeFourierDevelopmentoftheFourier4.1RepresentationofAperiodicTheContinuous-TimeFourierDevelopmentoftheFourierTransformfromFourierCorrespondingPeriodicSignaloftheAperiodicSignalPeriodicT0tT2T20TTAperiodicCorrespondingAperiodicSignalofthePeriodicSignalt01Txejkjk0Tx00kk0kTk1Tjkjkx020kkT2X(X(j)X(jk01Txejkjk0Tx00kk0kTk1Tjkjkx020kkT2X(X(j)X(jk0)201T1T3X(X(jk010kx(t)面X(j)e0x(t))ej0X(TtX(00X(j)ejtx(t)T0100X(jk)ejk0t So,foraaperiodicsignal,wehaveFourier復(fù)雜信號＝系數(shù)（）基本信號jtx(t)X(d反1x(So,foraaperiodicsignal,wehaveFourier復(fù)雜信號＝系數(shù)（）基本信號jtx(t)X(d反1x(t)ejtX(j)2(ejt正系數(shù)（）＝復(fù)雜信號（與）基本信號AndanotherwaytocalculatethecoefficientFourierT31T1T0X(X(jk)FxX(0kx(t)的頻FourierTransformofcorrespondingaperiodiccoefficientofFourierseriesofaperiodic4.1.2ConvergenceofFourier4.1.2ConvergenceofFourier|x(t)|dtx(t)havefinitenumberofmaximaandminimawithinanyfinitex(t)haveafinitenumberofdiscontinuitieswithinanyfiniteinterval,andeachofthesediscontinuitiesmustbefinite.4.1.3ExamplesofFourier(BasicFourierTransformPairsusedx(t)eatu(t)(j)21jX(j)稱為x(t)的頻4.1.3ExamplesofFourier(BasicFourierTransformPairsusedx(t)eatu(t)(j)21jX(j)稱為x(t)的頻dtX(j)dteee2 (ja)ta)t((e001j1je(ja)ta1X(X(j) ;(Figureshowninpageax(t)eatu(t)(j)2;(Figureshowninpage1X(j)x(t)eatu(t)(j)2;(Figureshowninpage1X(j)X(相頻特性時(shí)域信號不x(t)(t)(j)21X(1Ft00ej0jjX(tt)dt2|t|(j)2sin|t|21X(1F2t|t|(j)2sin|t|21X(1F2t (12T(f11 deejt11X(22sin212(eejtjTe)111 21 2||||sinFx(t)X(1BX(1F2B1tBB(f1 1||||sinFx(t)X(1BX(1F2B1tBB(f1 111BBx(t)jtjtjteX(1ejBt)sin111ejt(et2t2Examplesforgetting 32sinxkkT0TT10FsX(2sinExamplesforgetting 32sinxkkT0TT10FsX(2sin11F4.412sin2sin1TX(j)x3(同kkkT0004.2TheTransformofPeriodicjk2x(kxX(x(t)0kk0k4.2TheTransformofPeriodicjk2x(kxX(x(t)0kk0kk1x(t)112xk(k0)e X(j)e k)ejtdxkx0kk0kk()抽取jk2x(kxX(x(t)0kk0kk21(exampleT01T2jk2x(kxX(x(t)0kk0kk21(exampleT01T21tTT11FxkX(2122T1 T21T12T22sin(T10TT1CoefficientX(2sin111F4.4tT0111X(j)1TT01TT1X(2sin111F4.4tT0111X(j)1TT01TT1T3.5t0T0Tk11x2X(2k0012sin TT11TF4.6t0TT011e2(k1:x0(k1(k(k(k11eesine2(k1:x0(k1(k(k(k11eesintFj[()(00X(j)022k1:122k1:202(k(k(k(k1t1eF[(0)(0cosje0022X(j)122k1:00(examplex(t)(tkTkt-0TFX(j)((examplex(t)(tkTkt-0TFX(j)(k2k1T1TTTT1T1(f T(f TTFk(k(tkT))T4.3PropertiesoftheContinuous-TimeFourier eatu(t)F 14.3PropertiesoftheContinuous-TimeFourier eatu(t)F 1x(t)F1[X(1x(t)j[eatF FX(j)X(j) tjj2F時(shí)頻F變變?y(t)性y(t)j)性√?y(t)性y(t)j)性√表示表示表示表示質(zhì)質(zhì)(t)xi(t)i((xii√x(t)?h(t)(j)?H(??Y(y(t)4.3.1ix(t)X(iFax(t)by(t)4.3.1ix(t)X(iFax(t)by(t)(j)bY(tje012j0(22(je001(i)11(i)1222F21(i)1(i) 2j 2jcost[()(22000Fsin(0t)j[(0)(0t0e4.3.2Timex(t)X(F0X(x(tt0)t0e4.3.2Timex(t)X(F0X(x(tt0)1Fjtx(t)X(dtt1X(j)ej(tt0)x(tt)0ej(tt0)ejt0eF1X(j)ejt0ex(tt)0x2111ttt21223212 2sin(2sin(3FF)(t)X(x2111ttt21223212 2sin(2sin(3FF)(t)X((x1122x(t)1x(t2.5)x(t1222sin(2)2sin(31ejFj2ee4.3.2'Frequency0x(t)X(2ej0tx(t)X[j(0FFX[j()] x(t)ej0e4.3.2'Frequency0x(t)X(2ej0tx(t)X[j(0FFX[j()] x(t)ej0teX(j) x(t)e0ej0t2(20Reversal4.3.2''Timex(t)X(x(t)X(x(t)12jdjtX(dX( letjt)tXjd4.3.3Conjugatex(t)X(x*(t)X*(14.3.3Conjugatex(t)X(x*(t)X*(1jX(d*X*(j)ejtx(t)*'1Fjx(t)**X('<Especiallyforreal(1)x(t)isreal,X(j)is<Especiallyforreal(1)x(t)isreal,X(j)isaConjugate*X(x(t)X(X(j)X*(X*(j)X(共共x*(t)X*(FConjugateConjugaterealX(j)|X(j)|ejX(j)realX(j)|X(j)|ejX(j)R()jIX(j)|X(j)|ejX(j)R()jI*X*(j)|X(j)|ejX(jR()jIeeX(j)|X(j)|ejX(jR()jI(2)x(t)isrealeven,X*(j)isarealx*(t)X*(共共x(t)X((2)x(t)isrealeven,X*(j)isarealx*(t)X*(共共x(t)X(反反x(t)X(F(3)x(t)isrealodd,X*(j)isapureimaginaryx*(t)X*(共共x(t)X(反反x(t)X(F(4)Evenandoddpartofrealx(t)~~RealandimaginarypartofX(x(4)Evenandoddpartofrealx(t)~~RealandimaginarypartofX(x(t)x(t)Frealrealxe(t)xoFx2x(t)2ChaptereF(t)oX(j)R()jIrealevenimaginaryForrealsignal=TransformofEvenPart=RealPart奇部的F=TransformofOddPart=ImaginaryPartofx(t)ea|t|X(real?reale1x(t)ea|t|X(real?reale1ax(t)eatu(t)X(t01√1x(t)xx(t)FX( u(t)e2112F2Re[X(1偶部的=FTransformofEvenPart=RealPart4.3.5TimeandFrequencyFx(t)X(A.TimeX(j11x(at)|aa4.3.5TimeandFrequencyFx(t)X(A.TimeX(j11x(at)|aa1j'atx(at) X(jda0:a1d' 1aaaX(j)ex(at)a|aFX(jF拉壓11t(whena=-B.TimeF(whena=-B.TimeFx(t)X(x(t)X(1X(Ft4.3.4DifferentiationandIntegrationinTime(k)A.x(t)X4.3.4DifferentiationandIntegrationinTime(k)A.x(t)X(x'(t)jX(1Fjtx(t)X(dFjX(j)ejtx'(t)tB.0x(t)X(dttB.0x(t)X(dtt(不常用x()d1X(j)X(0)tx(t)X(tx()d(常用x()d1X(C.(t)tC.(t)t11tu(t)(t)u'(t)ju(t)1x(t?2dxx1211ttt11x(t?2dxx1211ttt11FtFejej22sinF(2sin22sin21X(j)j[x(t)x(t)]dt124.3.4'DifferentiationandIntegrationinFrequencyki(A.X(4.3.4'DifferentiationandIntegrationinFrequencyki(A.X(2jtdX(jtX(dX(jtjt)x(t)e(FB.(x(t)X(Canbe1x(t)x(0)B.(x(t)X(Canbe1x(t)x(0)(t)X()0X(dX(x(t)X(x(t)X(4.3.6Duality(對偶性x(t)X(CanbeX(24.3.6Duality(對偶性x(t)X(CanbeX(2(t)2()2|t|2sin(TFigurePage11|t|令T1||||||||,sin(Bt)2sin(Bt)t?2G(j)2g?2G(j)2g(t)?1t2?214.2a221t4.3.7Paserval’sx(t)X(|X(j)eCanbeF)|x(t)|dt24.3.7Paserval’sx(t)X(|X(j)eCanbeF)|x(t)|dt22|X(信號能各頻率成分能量之 (1t2dt11tPaserval’s12 (1t22dteed04.4TheConvolutionx(t)4.4TheConvolutionx(t)X(Fh(t)H(Canbex(t)h(t)X(j)H(?Y(4.5TheMultiplicationA.TheMultiplication1F4.5TheMultiplicationA.TheMultiplication1FAmplitudes(t)S(FCanbe1s(t)ip(t)[S(j)P(B.X1(1||1Xsint(11sin||B.X1(1||1Xsint(11sin||1X( 222X(2||2212X(sintsin2101(j)(j)X(223212t1224.6TablesofFourierPropertiesandBasicFourierTransformPairs<Seethebook4.6TablesofFourierPropertiesandBasicFourierTransformPairs<SeethebookUsageofthex(t)X(ii√性表表y(t)(?(1)Known(2)4.7ResponseofLTISystemtoAperiodice4.7.1ResponseofLTI4.7ResponseofLTISystemtoAperiodice4.7.1ResponseofLTIsystemAsshowineH(j)eF)jtH(ImpulseFrequency4.7.2ResponseofLTIsystemtoAperiodicA.11jtx(t)X(d√?2LLL1)H(jtX(dY(3B.4.7.2ResponseofLTIsystemtoAperiodicA.11jtx(t)X(d√?2LLL1)H(jtX(dY(3B.periodperiodH(2H(H(2FFFF1313ssX(1Y(H(jk0ii:FourierTransform3C.(tt0x(t)表x(ttxxii0e2e√√FX(C.(tt0x(t)表x(ttxxii0e2e√√FX(13FFFX(Xi(Xi(ei表0X(j)'xFF31F√2X(jX(X(i表tx(12F3表tx(12F3F1FX(j)X(0)1X(X(i[表 aebtu(t)1[eatu(t)bFFF1b1a aebtu(t)1[eatu(t)bFFF1b1a1(aj)(b111bi[]baa(aForj(teatu(t)eatu(t)?F1F1性質(zhì)(aadjif(難,sin(icisin(if(難,sin(icisin(