信号与系统奥本海姆原版PPT第九章

9TheLaplaceTransform9.TheLaplaceTransform9.1TheLaplaceTransform(1)DefinitiondtetxsXst)()()(jswhere(2)RegionofConvergence(ROC)ROC:RangeofforX(s)toconvergeRepresentation:A.InequalityB.RegioninS-plane9TheLaplaceTransformExampleforROCReReS-planeS-planeImIm-a-a9TheLaplaceTransform(3)RelationshipbetweenFourierandLaplacetransformsjjstjstjXsXorsXjXdtetxjXdtetxsX|)()(|)()()()()()(Example9.19.29.39.59TheLaplaceTransform9.2TheRegionofConvergenceforLaplaceTransformProperty1:TheROCofX(s)consistsofstripsparalleltoj-axisinthes-plane.Property2:ForrationalLaplacetransform,theROCdoesnotcontainanypoles.Property3:Ifx(t)isoffinitedurationandisabsolutelyintegrable,thentheROCistheentires-plane9TheLaplaceTransformProperty4:Ifx(t)isrightsided,andifthelineRe{s}=0isintheROC,thenallvaluesofsforwhichRe{s}0willalsointheROC.9TheLaplaceTransformProperty5:Ifx(t)isleftsided,andifthelineRe{s}=0isintheROC,thenallvaluesofsforwhichRe{s}0willalsointheROC.x(t)T2te-0te-1t9TheLaplaceTransformProperty6:Ifx(t)istwosided,andifthelineRe{s}=0isintheROC,thentheROCwillconsistofastripinthes-planethatincludesthelineRe{s}=0.9TheLaplaceTransformS-planeReReReImImImRLLR9TheLaplaceTransformProperty7:IftheLaplacetransformX(s)ofx(t)isrational,thenitsROCisboundedbypolesorextendstoinfinity.Inaddition,nopolesofX(s)arecontainedintheROC.Property8:IftheLaplacetransformX(s)ForrationalLaplacetransform,theROCdoesnotcontainanypoles.Property3:Ifx(t)isoffinitedurationandisabsolutelyintegrable,thentheROCistheentires-plane9TheLaplaceTransformProperty7:IftheLaplacetransformX(s)ofx(t)isrational,thenitsROCisboundedbypolesorextendstoinfinity.Inaddition,nopolesofX(s)arecontainedintheROC.Property8:IftheLaplacetransformX(s)ofx(t)isrational,thenifx(t)isrightsided,theROCistheregioninthes-planetotherightoftherightmostpole.Ifx(t)isleftsided,theROCistheregioninthes-planetotheleftoftheleftmostpole.Example9.79.89TheLaplaceTransformAppendixPartialFractionExpansionConsiderafractionpolynomial:)()()()(012211012211mnwhereasasasasbsbsbsbsbsDsNsXnnnnnmmmmmmDiscusstwocasesofD(s)=0,fordistinctrootandsameroot.9TheLaplaceTransform(1)Distinctroot:)())(()(21012211nnnnnnsssasasasassDniiinnnmmmmmmsAsAsAsAsssbsbsbsbsbsX1221121012211)())(()(thus9TheLaplaceTransformCalculateA1:Multiplytwosidesby(s-1):nnssAssAAsXs)()()()(1212111|)()(11ssXsALets=1,soisisXsAi|)()(Generally9TheLaplaceTransform(2)Sameroot:)())(()()(211012211nrrrnnnnnssssasasasassDnnrrrrrnrrrmmmmmmsAsAsAsAsAssssbsbsbsbsbsX11111112111211012211)()()()())(()()(thus),,2,1(|)()(nrrisXsAiisiForfirstorderpoles:9TheLaplaceTransformrnnrrrrrssAsAsAsAAsXs)]([)()()()(111111112111Multiplytwosidesby(s-1)r:Forr-orderpoles:1|)()(111srsXsASo1|)]'()[(112srsXsA1|)]()[()!1(1111srrrsXsrA9TheLaplaceTransform9.3TheInverseLaplaceTransformdejXtxordejXjXFetxetxFjsdteetxjXdtetxsXtjtjtttjtst)(1)(21)()(21)]([)(])([)()()()()(SojjstdsesXjtx)(21)(9TheLaplaceTransformThecalculationforinverseLaplacetransform:(1)Integrationofcomplexfunctionbyequation.(2)ComputebyFractionexpansion.GeneralformofX(s):niiinnsAsAsAsAsX12211)(Importanttransformpair:polerighttuepolelefttuesttiii),(),(1Example9.99.109.11