信號與系統SignalsandSystems吉林大學DefinitionsofLaplaceTransformandInverseLaplaceTransformDefinitionsofLaplacetransform(LT)andinverseLTFromFouriertransformtoLaplacetransformⅠTheLaplacetransform(LTforshort)ⅡOne-sided(unilateral)LT:Two-sided(bilateral)LT:DefinitionsofLaplacetransform(LT)andinverseLTⅢ信號與系統SignalsandSystems吉林大學TheRegionofConvergence(ROC)forLaplaceTransformsTheROCoftheLaplacetransform[Example]TheROCoftheLaplacetransformTherangeofvaluesofsforwhichtheintegralinconvergesisreferredtoastheregionofconvergence(ROC)oftheLaplacetransform(LT).Theregionofconvergence(ROC)oftheLTf(t)isintegrablewithin(a,b)(1)(0≤a<b<∞)Foranarbitraryσ0,(2)信號與系統SignalsandSystems吉林大學CommonLaplaceTransformpairsCommonLaplacetransformpairsCommonone-sidedLTpairsCommonLaplacetransformpairs[Example]信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:LinearityandShiftinthes-DomainProperty1：LinearityProof:Property2：Shiftinthes-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplacetransform:RightShiftinTimeProperty3:RightshiftintimeProof:Property3:Rightshiftintime(3)Property3:RightshiftintimeSolution:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:TimeScalingProperty4:TimescalingProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:ConvolutionTheoremsProperty5:Convolutioninthet-domainProof:Property5:Convolutioninthet-domainContinued:Property

6:Convolutioninthes-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:Differentiationinthet-domainProperty7:Differentiationinthet-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:Integrationinthet-domainProperty8:Integrationinthet-domainProof:Property8:Integrationinthet-domain信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform:DifferentiationandIntegrationinthes-DomainProperty9:Differentiationinthes-domainProof:Property10:Integrationinthes-domainProof:信號與系統SignalsandSystems吉林大學PropertiesofLaplaceTransform：InitialandFinal-ValuetheoremsProperty11：Initial-valuetheoremProof:Property12：Final-valuetheoremProof:信號與系統SignalsandSystems吉林大學ComputationoftheInverseLaplaceTransform(Ⅱ)PartialFractionExpansionComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(1)Conditions:ComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(2)ComputationoftheinverseLaplacetransform(Ⅱ)Partialfractionexpansion(3)信號與系統SignalsandSystems吉林大學SolvingtheDifferentialEquationsinthes-DomainSolvingthedifferentialequationsinthes-domain[Example]Given:Find：

Solvingthedifferentialequationsinthes-domainSolvingthedifferentialequationsinthes-domainSolvingthedifferentialequationsinthes-domain信號與系統SignalsandSystems吉林大學Thes-DomainRepresentationsofCircuits(I)Thes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThesameresistanceThes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThes-domainimpedanceThes-domainrepresentationsofcircuits(I)1Thes-domainequivalentcircuitelementsThes-domainimpedanceThes-domainrepresentationsofcircuits(I)2TheformsofKVLandKCLinthes-domain信號與系統SignalsandSystems吉林大學TheBlockDiagramofaSysteminthes-DomainTheblockdiagramofasysteminthes-domainScalarmultiplier1Adder/Subtractor2Theblockdiagramofasysteminthes-domainIntegrator3信號與系統SignalsandSystems吉林大學TheDefinitionofTransferFunctionanditsSolutionsThedefinitionoftransferfunctionanditssolutionsThetransferfunctionⅠHowtofind21.GiventhesystemdifferentialequationThedefinitionoftransferfunctionanditssolutionsHowtofind21.GiventhesystemdifferentialequationThedefinitionoftransferfunctionanditssolutions2.Giventheimpulseresponseh(t)Thedefinitionoftransferfunctionanditssolutions3.GiventhestructureofthecircuitUsingthedefinitioninthes-domainrepresentationofthecircuit.4.Usingthepole-zeroplot信號與系統SignalsandSystems吉林大學TheTransferFunctionandthePole-ZeroPlotThetransferfunctionandthepole-zeroplotPolesandzeros1Zeros:Poles:Thetransferfunctionandthepole-zeroplotThepole-zeroplot2Aplotinthecomplexplaneshowingthelocationsofallthepoles(markedby×)andallthezeros(markedby○)iscalledthepole-zeroplot.Zeros:Poles:信號與系統SignalsandSystems吉林大學ApplicationsofthePole-ZeroPlot:DeterminingtheFormofh(t)Thepoles

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left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

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left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

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