信号与系统英文版课件Chapter 4

Contents§4.1Representationofaperiodicsignal: Thecontinuous-timeFouriertransform(FT)§4.2TheFouriertransformforperiodicsignals§4.3Propertiesofcontinuous-timeFT§4.4Theconvolutionproperty§4.5Themultiplicationproperty§4.6TablesofFTpropertiesandbasicFTpairs(seeP328-329)§4.7Systemscharacterizedbylinearconstant-coefficientdifferentialequationDevelopmentofFTConsider00T-TtDevelopmentofFTDevelopmentofFTDevelopmentofFTthen(4.3)and(4.4)respectivelybecomeDevelopmentofFTEq.(4.8)(4.9)—Fouriertransformpair.X(j)—FTofx(t),alsoasspectrumofx(t).Eq.(4.8)—originalfunctionorinverseFTofX(j)DevelopmentofFTTherelationwiththenotation:Someexplains1)Eq(4.8),aperiodic

x(t)isrepresentedasthecontinuoussumofejtDevelopmentofFT2)X(j)

representcomplexamplitudeinunitfrequencywide,inessence,“complexamplitudedensity”

X(j)∼,representoppositeamplitudeoffrequencyexponentials.00DevelopmentofFT3)ComparingX(j)

ofx(t)withakof,wehaveExampleConvergenceofFTAsetofconditionslikethoserequiredfortheconvergenceofFS1)Finiteenergycondition2)DirichletconditionExamplesofFTToplotitasafunctionof,expressX(j)intermsofitsmagnitudeandphase.01tExamplesofFTX(j)=X(j)=1/aExamplesofFTa-a2/a1/aExamplesofFT3.x(t)=δ(t)andx(t)=1(orconstant)Thatis,(t)

hasequalspectrumatallfrequencies.(uniformspectrum)Fromeq.(4.15),canobtainausefulequation.FFExamplesofFTHence,weobtainFTofconstant:ExamplesofFTF1001ExamplesofFTX(j)asshowninfigure.1ExamplesofFTFromdrawingofx(t)andX(j),wecansee:(uniformspectrum)1ExamplesofFTExamplesofFT1x(t)20t02TheFouriertransformforperiodicsignalsFFTheFouriertransformforperiodicsignalsExample(4.7-1)x(t)=sin0tFFTheFouriertransformforperiodicsignalsExample(4.7-2)x(t)=cos0tFTheFouriertransformforperiodicsignalsuniformimpulsetrainwasgivenX(j)PropertiesofFT:LinearitythenTimeshiftingthenProve:Atimeshiftofsignalisequivalenttophaseshiftinfrequencydomain.FTimeshiftingExampleLetx(t)X(j),thenExample4.9CalculateX(j)ofx(t),

whichshowninFigure000ConjugationandconjugatesymmetrythenExampleFFFFConjugationandconjugatesymmetrySpecify1.WhenexpressX(j)inrectangularorpolarformasConjugationandconjugatesymmetryIfx(t)isreal,thentherealpartand|X(j)|isevenfunctionof,andtheimaginarypartandX(j)isodd.2.Ifx(t)isbothrealandeven,thenX(j)willalsoberealandeven,ifx(t)isrealandodd,thenX(j)ispurelyimaginaryandodd.ExampleConjugationandconjugatesymmetry3.∵realx(t)canalwaysbeexpressedaswherethenConjugationandconjugatesymmetryExample4.10ConsideragaintheFTofexample4.2forx(t)=e-a|t|,a>0.0t20t10t+1-1ConjugationandconjugatesymmetryandAssignment(p334):4.1,4.2,4.5,4.7X1(j)DifferentiationandintegrationLetx(t)

X(j),thenwherethederivationorproofinexample4.17(4.31)F(4.32)FFDifferentiationandintegrationExample:Eq(4.32)rewriteasmoreusefulform:Letx(t)

X(j),andx(-),x()areconstant∴δ'(t)=u1(t)j

∵δ(t)1(4.32')Ifx(n)(t)Xn(j)DifferentiationandintegrationExample4.11CalculateX(j)ofx(t)=u(t)again.(4.32')(4.32’’)DifferentiationandintegrationExample:CalculateX(j)ofx(t),displayedinfigure.(refertoproblem4.8)∴F1DifferentiationandintegrationAssignment(p335):4.9andx()=1,x(-)=0Timeandfrequencyscaling

Ifx(t)

X(j)Thenx(at)Whereaisnonzerorealnumber.forx(at),if|a|>1,compressed|a|<1,stretched(orexpansion)

a<0,reversedFFTimeandfrequencyscaling

Asidefromtheamplitudefactor1/|a|,compressioninthetimeisequivalenttoexpansioninthefrequency,andviceversa.Also,leta=-1x(-t)X(-j)

x(t)reversed,FTalsoreversed.2ππx(t)=sin(t)t2πx(3t)=sin(3t)tπTimeandfrequencyscalingExample:CalculateX(j)ofthefollowingx(t)FFF1TimeandfrequencyscalingExample:Letx(t)X(j)CalculateFTofthefollowingsignalsF∴x(1-t)X(-j)e-jFDualityLetx(t)X(j)Ifx(t)isevenfunction,thentheyarecompleteduality.FDualityFFFt∵tThatisX(j)=DualityThedualitypropertycanalsoderivedifferentiationinfrequencyandfrequencyshifting.Parseval’sRelationIfx(t)

X(j),then|X(j)|2—EnergydensityspectrumParseval’sRelationAssignment:P3364.10(a,b),4.21(g,h)x(t)G(j)TheconvolutionpropertyConsidery(t)=x(t)*h(t)andY(j)=F[y(t)]whereH(j)=F[h(t)]TheconvolutionpropertyThatisTimedomainconvolutioncorrespondstospectrummultiplication.Theconvolutionpropertyisnotonlyusefultoolinfrequencydomain,butalsocanderivemanyimportantconclusion.(seeexample4.15-4.17)FTheconvolutionpropertyExample:proveequation(4.32)FromtheconvolutionpropertyTheconvolutionpropertyExample4.19ConsiderLTIsystem,h(t)=e-atu(t)a>0,x(t)=e-btu(t)b>0,y(t)=?TimedomainproblemstranslateintofrequencydomainproblemsTheconvolutionproperty[partial-fractionexpansion(seeappendix)]Whenb=a(ifba)TheconvolutionpropertyfromdifferentiationinfrequencyAssignment:P3364.11,4.15,4.19ThemultiplicationpropertyThemultiplicationintimedomaincorrespondstoconvolutioninfrequencydomain.s(t)*p(t)isoftenreferredtoamplitudemodulation(AM).Why?ThemultiplicationpropertyExample

4.21Letr(t)=s(t)·p(t),S(j)ofs(t)isdepictedinFig.Also,considerp(t)=cos0t,andobtainthespectrumP(j).AThemultiplicationproperty

spectrumcarryorshiftAR(j)ThemultiplicationpropertyExample:howwecanrecovertheoriginalsignals(t)fromtheamplitudemodulationsignalr(t)?H(j)02-0IdeallowpassfilterThemultiplicationproperty

Forideallowpassfilter,if01,thenY(j)=G(j)H(j)=S(j)A/4A/4A/220-20-11G(j)H(j)02-0Themultiplicationproperty

achievey(t)=s(t)

Thisprocessisreferredtoasdemodulation.(see§8)A/4A/4A/220-20-11G(j)A-11Y(j)Addition:Partial-fractionexpansionConsiderrationalfunctionsofageneralvariableu:Therearetwopossibilitiesofmandn:(1)m>n,thefunctionisknownasimproperfraction.(2)m<n,H(u)isproperfraction.Addition:Partial-fractionexpansion

IfH(u)isimproper,itcanalwaysbeseparatedintoasumofapolynomialinuandaproperfraction.

ExamplethefunctionAddition:Partial-fractionexpansionAddition:Partial-fractionexpansion

Consequently,thefollowingconcernswithonlytechniquesofexpandingaproperfraction(m<n).Simpleroots(Nomultipleroots)

WeshallfactorizeD(u)intonfirst-orderfactorsAddition:Partial-fractionexpansionoringeneralAddition:Partial-fractionexpansionExampleAddition:Partial-fractionexpansionComplexrootsTheprocedureoutlinedaboveholds.Example:A1isreal,A2andA3areconjugatecomplex.Addition:Partial-fractionexpansionRepeated(multiple)rootsThecoefficientsAr+1,···,An

arefoundasusual.Addition:Partial-fractionexpansionThecoefficientsa0,a1,…,ar-1correspondingto