《信號(hào)與線性系統(tǒng)分析基礎(chǔ) 》 課件 第 2 章 連續(xù)時(shí)間信號(hào)與系統(tǒng)的時(shí)域分析

信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)ModelingandLinearDifferentialEquationsSystemmodeling1ModelingandlineardifferentialequationsFindtheoutputresponsetotheexcitation.Solving2ModelingandlineardifferentialequationsHomogeneoussolutions:Forannth-orderdifferentialequation:Thecharacteristicequation(orauxiliaryequation):ModelingandlineardifferentialequationsTheformsofhomogeneoussolutions--dependentonthecharacteristicrootsWithnsimple(ordistinct)roots:Witharepeatedrootλofmultiplicityrandn-rsimpleroots:Withconjugatecomplexroots:treatedassimplerootsModelingandlineardifferentialequationsParticularsolutions---determinedbytheinput----αisanoncharicteristicroot.----α

isasimpleroot.----α

isarepeatedrootofmultiplicityr.---Zeroisarepeatedrootofmultiplicityr.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheUnitImpulseandtheUnitStepFunctionTheunitimpulseandtheunitstepfunctionSingularityfunctionsⅠContinuous-timesignalsthatarenotcontinuousatallpointscan’tbedifferentiableatallpoints,buttheymayhaveaderivativeinthegeneralizedsense.AFunctionitselforitsfirstderivative(oritsintegral)hasseveraldiscontinuities.TheunitimpulseandtheunitstepfunctionTwotypicalsingularfunctionsⅡ1.TheintroductionofandTheunitimpulseandtheunitstepfunctionTheunitimpulseandtheunitstepfunction2.Definitions:Theunitimpulseandtheunitstepfunction3.TherelationshipbetweenandThesignalmustbediscontinuousatifitsfirstderivativeinvolves.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)ThePropertiesoftheUnitImpulse(I)Thepropertiesoftheunitimpulse(I)Propertyoftranslation1Samplingproperty2Thepropertiesoftheunitimpulse(I)Time-scaling3Proof:Supposethatisanarbitrarytrialfunction.Thepropertiesoftheunitimpulse(I)Multipliedbyanordinaryfunction4Parity5Thepropertiesoftheunitimpulse(I)Thegeneralizedderivatives6Proof:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheUnitImpulseResponse(I)Theunitimpulseresponse(I)DefinitionITheimpulseresponseofacausallineartime-invariantcontinuous-timesystemistheoutputresponsewhentheinputistheunitimpulsewithnoinitialenergyinthesystemattime[priortotheapplicationof].Theunitimpulseresponse(I)Discussion:IIWeareinterestedinthemathematicalformof.Theformoftheunitimpulseresponseisdeterminedbythesystemequation,independentoftheapplicationandtheinitialenergy.Theunitimpulseresponse(I)HowtofindⅢTofindviatheunitstepresponseMethod1信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheUnitImpulseResponse(II)Theunitimpulseresponse(II)ImpulseequilibriumformulationMethod2[Example]Findofthesystemdeterminedbythedifferentialequationwithconstantcoefficients,referencedbelow.Analysis:Thestatevariablesjump.Theunitimpulseresponse(II)Theunitimpulseresponse(II)Forannth-ordersystem,referencedbelow,

Ifisoneofthesimpleroots,theformofwillbe:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheUnitStepResponseTheunitstepresponseDefinition1Thestepresponseofacausallineartime-invariantcontinuous-timesystemisthezero-stateresponsetotheunitstepfunction.Howtofind2Method1TosolvethesystemequationMethod2TofindviaTheunitstepresponseMethod3Comparisonmethod(equilibriumformulation)Decomposition:Theunitstepresponse信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)ConvolutionIntegralConvolutionintegral

ToexpressintermsofthesumofinfiniteimpulsesConvolutionintegral信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheZero-StateResponsetotheExcitationThezero-stateresponsetotheexcitationLimitsofintegration:ForasignalofForacausalsystemForsignalsandThezero-stateresponsetotheexcitation信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheCommonOperationsofContinuous-TimeSignalsThecommonoperationsofcontinuous-timesignalsAddition1Thecommonoperationsofcontinuous-timesignalsMultiplication2Thecommonoperationsofcontinuous-timesignalsDifferentiation3Thecommonoperationsofcontinuous-timesignalsShift4Time-scaling5Folding6[Example]Thecommonoperationsofcontinuous-timesignalsTheprofileofisgivenbelow,plotasthefunctionoft.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)GraphicalRepresentationofConvolutionGraphicalRepresentationofConvolutionGraphicalRepresentationofConvolution信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofConvolutionPropertiesofconvolutionProof:1.CommutativityPropertiesofconvolution2.DistributivitywithadditionPropertiesofconvolution3.Associativity4.DifferentiationandintegrationDifferentiation(1)Prop