CH9-The-Laplace-Transform

1Chapter9TheLaplaceTransform29.0INTRODUCTION3TheLaplacetransform(拉普拉斯变换)isageneralizationofthecontinuous-timeFouriertransform.TheLaplacetransformprovidesuswitharepresentationforsignalsaslinearcombinationsofcomplexexponentialsoftheformwiths=σ+jωWithLaplacetransform,weexpandtheapplicationinwhichFourieranalysiscannotbeused.ste49.1TheLaplaceTransform5Lets=σ+jω,()()ttjtFxtexteedtForsomesignalswhichhavenotFouriertransforms,ifwepreprocessthembymultiplyingwitharealexponentialsignal,thentheymayhaveFouriertransforms.tebilateralLaplacetransform)()(txsXL)()(sXtxLtdetxsXst)()(tdetxtj)(tdetxsXst0)()(unilateralLaplacetransform6ifσ=0RegionofConvergence(ROC，收敛域):RangeofsforX(s)toconvergejsRelationshipbetweenFourierandLaplaceTransformdtetxjXdtetxsXtjts)()()()(sjjsjXsXorsXjX|)()(|)()(js7Example9.1Considerthesignal)()(tuetxtaaseaseasdtedtetuesXtastastassttat111)()(lim)(0)(0)(Forconvergence,werequirethatRe{s+a}0orRe{s}–aasROCastueLTta}{:,1)(Re(aisreal)–aReIms-plane8Example9.2Considerthesignal)()(tuetxtatastastassttateasaseasdtedttueesXlim)(0)(0)(111)()(Forconvergence,werequirethatRe{s+a}0,orRe{s}–aasROCastueLTta}{:,1)(Re–aReIms-plane(aisreal)9Example9.3Considerthesignal)(2)(3)(2tuetuetxtt2Re21)(2sstueLt1Re11)(sstueLt1Re2311223)(2)(322sssssstuetueLtt-2-1ReIms-plane10Example9.4Considerthesignal)()3(cos)()(2tutetuetxttUsingEuler’srelation,)(]2121)([)()31()31(2tueetuetxtjtjt2Re21)(2sstueLt1Re)31(1)()31(sjstueLtj1Re)31(1)()31(sjstueLtj11Consequently,2222cos(),{}0,sin(),{}0LLstutsstutssReReSomeusefulLTpairs:)2)(102(1252)31(1)31(121)()(22sssssjsjsssXtxL1Res12Thepole-zeroplot(零极点图)Generally,theLaplacetransformisrational,i.e.,itisaratioofpolynomialsinthecomplexvariables:TherootsofN(s)=0arethezeros(零点)ofX(s);TherootsofD(s)=0arethepoles(极点)ofX(s).TherepresentationofX(s)throughitspolesandzerosinthes-planeisreferredtoasthepole-zeroplot(零极点图)ofX(s).)()()(sDsNsXNumeratorpolynomialsDenominatorpolynomialsExceptforascalefactor,acompletespecificationofarationalLaplacetransformconsistsofthepole-zeroplotofthetransform,togetherwithitsROC.13Ingeneral,iftheorderofthedenominatorexceedstheorderofthenumeratorbyk,X(s)willhavekzerosatinfinity.Similarly,iftheorderofthenumeratorexceedstheorderofthedenominatorbyk,X(s)willhavekpolesatinfinity.Thepoleandzeroplotatinfinity11101110()()()mmmmnnnnbsbsbsbNsXsDsasasasa:nmlim()lim0mmnssnbsXsas:nmlim()limmmnssnbsXsas14Example9.5211213111341)(2ssssssX–112ReIms-planePole-zeroplotandROC)(31)(34)()(2tuetuettxtt1)()(dtettLst2}{:ROCplane-sentirethe2}{1}{sssReReReFTdoesnotexistPlotthepole-zeroplot(零极点图)ROC：theentires-planepoles:p1=-1,p2=2zeros:s=1(2ndorder)159.2TheRegionofConvergenceForLaplaceTransform16Property1:TheROCofX(s)consistsofstripesparalleltothejω-axisinthes-plane.Property2:ForrationalLaplacetransforms,theROCdoesnotcontainanypoles.Property3:Ifx(t)isoffinitedurationandisabsolutelyintegrable,thentheROCistheentires-plane.dteetxdtetxtjtts])([)(17Example9.6TasTtstaeasdteesX)(011)(()(1)lim()lim[]lim()saTaTsTsasasadedsXsTeeTdsadsThepoleats=-aisremovableInfact,s=-aarebothpoleandzeroofX(s).)()()(Ttutuetxta18Property4:Ifx(t)isrightsided,andifthelineRe{s}=σ0isintheROC,thenallvaluesofsforwhichRe{s}σ0willalsobeintheROC;andtheROCofaright-sidedsignalisaright-halfplane.Property5：theROCofaleft-sidedsignalisaleft-halfplane.ReImσRReImσL19Property6：Ifx(t)istwosided,andifthelineRe{s}=σ0isintheROC,thentheROCwillconsistofastripinthes-planethatincludesthelineRe{s}=σ0.σLσRReImProperty7：IftheLaplacetransformX(s)isrational,thenitsROCisboundedbypolesorextendstoinfinity.Inaddition,nopolesofX(s)arecontainedintheROC.Property8：IfX(s)isrational,thenIfx(t)isrightsided,theROCistotherightoftherightmostpole.Ifx(t)isleftsided,theROCistotheleftoftheleftmostpole.20Example9.7tbetx)()()()(tuetuetxbtbtbsbstueLbtRe1)(bsbstueLbtRe1)(b0,b0,LTdoesnotexistbsbbsbbsbstxL}Re{211)(2221Example9.8)2)(1(1)(sssXReIms-planeROCcorrespondingtoaright-sidedsignalROCcorrespondingtoaleft-sidedsignalROCcorrespondingtoatwo-sidedsignalTherearethreepossibleROCs,correspondingtothreedistinctsignals.22x(t)ROCfinitedurationsignalentires-planeleft-sidedsignalleft-halfplaneright-sidedsignalright-halfplanetwosidedsignalstripSummaryofROC239.3TheInverseLaplaceTransform24desXsXtj)(21)(1-FMultiplyingbothsidesbytedesXtxts)(21)(s=σ+jω，ds=jdωjjtsdsesXjtx)(21)(tetx)(ttjtetxFdteetxsX)()()(25TheformalevaluationoftheintegralforageneralX(s)requirestheuseofcontourintegration(围线积分)inthecomplexplane.Fortheclassofrationaltransforms,theinverseLaplacetransformcanbedeterminedbypartial-fractionexpansion.mii