2. Time-Domain Analysis of Continuous-Time Signals and Systems .pdf

《2. Time-Domain Analysis of Continuous-Time Signals and Systems .pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、2.Time-DomainAnalysisofContinuous-TimeSignalsandSystems2.1.Continuous-TimeImpulseFunction(1.4.2)2.2.ConvolutionIntegral(2.2)2.3.Continuous-TimeImpulseResponse(2.2)2.4.ClassificationofaLinearTime-InvariantContinuous-TimeSystembyitsImpulseResponse(2.3)2.5.LinearC

2、onstant-CoefficientDifferentialEquations(2.4.1)2.1.Continuous-TimeImpulseFunction2.1.1.Continuous-TimeImpulseFunctionThecontinuous-timeimpulsefunction,(t),isdefinedby(t)dt1.(2.1)(t)0,t0Theillustrationofthecontinuous-timeimpulsefunctionisgiveninfi

3、gure2.1.(t)A(tt)0A1ttt0Figure2.1.IllustrationofContinuous-TimeImpulseFunction.(t)canbeconsideredas(t)lim(t),(2.2)0where1/,

4、t

5、/2(t)(2.3)0,

6、t

7、/2(figure2.2).(t)1/t/2/2Figure2.2.(t).(t)hasasamplingproperty,i.e.,x(t)(tt0)=x(t0)(tt0)

8、,(2.4)wheretisanarbitraryrealnumber.02.1.2.Continuous-TimeStepFunctionThecontinuous-timestepfunctionisdefinedas0,t0u(t)(2.5)1,t0(figure2.3).Att=0,u(t)isundefinedbutshouldbeconsideredtobefinite.u(t)1tFigure2.3.Continuous-TimeStepFunction.u(t)canbeexpressed

9、astherunningintegralof(t),i.e.,tu(t)()d.(2.6)Itfollowsfrom(2.6)that(t)canbeconsideredasthederivativeofu(t),i.e.,du(t)(t).(2.7)dt2.2.ConvolutionIntegralTheconvolutionintegralofx(t)andx(t)isdefinedas12x(t)x(t)x()x(t)d.(2.8)1212Notethattheinte

10、grationiscarriedoutwithrespecttoanintroducedvariable,,andthefinalresultisafunctionoft.Theconvolutionintegralsatisfiesthecommutativepropertyx1(t)x2(t)=x2(t)x1(t),(2.9)theassociativeproperty[x(t)x(t)]x(t)=x(t)[x(t)x(t)],(2.10)123123andthedistributiveproper

11、tyx(t)[x(t)+x(t)]=x(t)x(t)+x(t)x(t).(2.11)1231213Theconvolutionintegralcanbecalculatedinthefollowingsteps.(1)Reflectx2()abouttheorigintoobtainx2().(2)Shiftx()byttoobtainx(t).22(3)Calculatetheconvolutionintegralatt.Steps(2)and(3)oftenneedtobecarriedout

12、indifferentwaysfordifferentintervalsoft.Example.Findx(t)x(t),where12(1)x(t)=eatu(t)andx(t)=u(t).121,0tTt,0t2T(2)x1(t)andx2(t).0,otherwise0,otherwise(3)x(t)=e2tu