4. s-Domain Analysis of Continuous-Time Signals and Systems.pdf

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1、4.s-DomainAnalysisofContinuous-TimeSignalsandSystems4.1.DefinitionofLaplaceTransform(9.0-9.3)4.2.PropertiesofLaplaceTransform(9.5)4.3.SystemFunction(9.7)4.4.ClassificationofaLinearTime-InvariantContinuous-TimeSystembyitsSystemFunction(9.7)4.5.LinearConstant-CoefficientDi

2、fferentialEquations(9.7)4.6.ImplementationofContinuous-TimeSystems(9.8)4.1.DefinitionofLaplaceTransformTheLaplacetransformofacontinuous-timesignalx(t)isdefinedasX(s)x(t)exp(st)dt,(4.1)wheresisacomplexvariable.x(t)canberecoveredfromX(s)by1jx(t)X(s)exp(st)ds.(4.

3、2)2jj(4.2)iscalledtheinverseLaplacetransform.4.1.1.DerivationofInverseLaplaceTransformLettings=+jin(4.1),weobtainX(j)x(t)exp(t)exp(jt)dt.(4.3)Thatis,X(+j)istheFouriertransformofx(t)exp(t).Thus,1x(t)exp(t)X(j)exp(jt)d.(4.4)2Multiplyingbo

4、ththesidesof(4.4)byexp(t),weobtain1x(t)X(j)exp(j)td(4.5)2andthus(4.2).4.1.2.RelationofLaplaceTransformtoFourierTransformTheLaplacetransformofx(t)onthelineRe(s)=istheFouriertransformofx(t)exp(t),i.e.,X(+j)=FT[x(t)exp(t)].(4.6)Especially,theLaplacetr

5、ansformofx(t)onthelineRe(s)=0istheFouriertransformofx(t),i.e.,X(j)=FT[x(t)].(4.7)4.1.3.ZerosandPolesofLaplaceTransformSupposethatX(s)isrational,i.e.,X(s)=P(s)/P(s),whereP(s)121andP(s)aretwopolynomials.TherootsofP(s)=0arecalledthe21zerosofX(s),andtherootsofP(s)=0arecalle

6、dthepolesofX(s).2Zerosandpolesareindicatedwithandinthecomplexplane,respectively.ThealgebraicexpressionofX(s)canbespecifiedbyitszerosandpolesexceptforascalefactor.4.1.4.RegionofConvergenceofLaplaceTransformTheopenregionwhereX(s)convergesisreferredtoastheregionofconverge

7、nce(ROC)ofX(s).Inotherdiscussions,theROCmaymeantheregionwhereX(s)converges.NotethatboththealgebraicexpressionandtheROCofX(s)arerequiredtospecifyx(t)uniquely.LetusassumethatasignalhastheLaplacetransform.Then,fordifferenttypesofthesignal,theROCoftheLaplacetransformhasdiffe

8、renttypes.(1)Ifthesignalisoffiniteduration,theROCistheentireplane,i.e.,sC.(2)Ifthesignalisright-sided,