《信号与线性系统分析基础 》 课件全套 刘秀环 第1-6章 信号与系统的基本概念-离散时间信号与系统的时域分析

信号与系统SignalsandSystems吉林大学FundamentalConceptsofSignalsFundamentalConceptsofsignals1.Definition:AnalyticrepresentationAsignalisareal-valuedorscalar-valuedfunctionofthetimevariable.2.Description:GraphicalrepresentationFrequency-domainanalysisFundamentalConceptsofsignalsTime-domainrepresentationAnalyticrepresentationGraphicalrepresentationFrequency-domainrepresentationFundamentalConceptsofsignals3.Classification:DeterminatesignalRandomsignal(1)One-dimensionalsignalMulti-dimensionalsignal(2)PeriodicsignalAperiodicsignal(3)EnergysignalPowersignal(4)FundamentalConceptsofsignalsSampledsignalContinuous-timesignal(5)Discrete-timesignalAnaloguesignalPiecewise-continuoussignalDigitalsignalDecompositionmethodFundamentalConceptsofsignals4.Signalsprocessing:(1)DirectcurrentcomponentAlternatingcurrentcomponent(2)EvensignalOddsignalFundamentalConceptsofsignals(3)PulsesStepfunctions(4)RealcomponentImaginarycomponent(5)FundamentalConceptsofsignalsOrthogonalfunctions(6)SuchasIffunctionsandareintegrableontheinterval,andsatisfythenthetwofunctionsandaresaidtobeorthogonalontheinterval.信号与系统SignalsandSystems吉林大学FundamentalConceptsofSystemsFundamentalconceptsofsystems1.Definition:Asystemisaninterconnectionofcomponentswithterminalsoraccessportsthroughwhichmatter,energy,orinformationcanbeappliedorextracted.Asystemisamathematicalmodelforaphysicalprocessthatrelatestheinputtotheoutput.Fundamentalconceptsofsystems2.Blockdiagramrepresentation:ScalarmultiplierUnit-delayelementFundamentalconceptsofsystemsSummator/adder/subtracterIntegratorFundamentalconceptsofsystems3.Classification:(1)CausalsystemNoncausalsystem(2)Continuous-timesystemDiscrete-timesystem---Describedbyalgebraicequationsordifferentialequations.---Describedbydifferenceequations.Fundamentalconceptsofsystems(3)Time-varyingsystemTime-invariantsystem---Describedbydifferentialequationswithvariablecoefficients.---Describedbydifferentialequationswithconstantcoefficients.Fundamentalconceptsofsystems---Describedbyordinarydifferentialequations.---Describedbypartialdifferentialequations.Distributedparametersystem(5)Lumpedparametersystem(6)StablesystemUnstablesystem(Boundedinputboundedoutput,BIBO)(4)InstantaneoussystemDynamicsystem---Describedbyalgebraicequations.---Describedbydifferentialequations.Fundamentalconceptsofsystems(b)Superposition/Additivity(a)Homogeneity(c)Decomposition(8)LinearsystemNonlinearsystem(7)ReversiblesystemIrreversiblesystemInitialcondition信号与系统SignalsandSystems吉林大学DeterminationofSystemCharacteristicsDeterminationofsystemcharacteristics[Example]Determineifthesystemdescribedbyislinear,time-invariant,causalandstable.Linearornonlinear?(1)Time-invariantortime-varying?(2)Let?DeterminationofsystemcharacteristicsCausalornoncausal?(3)Stableorunstable?(4)Tobecontinued:Conclusion:Thesystemislinear,time-varying,noncausalandstable.信号与系统SignalsandSystems吉林大学ModelingandLinearDifferentialEquationsSystemmodeling1ModelingandlineardifferentialequationsFindtheoutputresponsetotheexcitation.Solving2ModelingandlineardifferentialequationsHomogeneoussolutions:Forannth-orderdifferentialequation:Thecharacteristicequation(orauxiliaryequation):ModelingandlineardifferentialequationsTheformsofhomogeneoussolutions--dependentonthecharacteristicrootsWithnsimple(ordistinct)roots:Witharepeatedrootλofmultiplicityrandn-rsimpleroots:Withconjugatecomplexroots:treatedassimplerootsModelingandlineardifferentialequationsParticularsolutions---determinedbytheinput----αisanoncharicteristicroot.----α

isasimpleroot.----α

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

Ifisoneofthesimpleroots,theformofwillbe:信号与系统SignalsandSystems吉林大学TheUnitStepResponseTheunitstepresponseDefinition1Thestepresponseofacausallineartime-invariantcontinuous-timesystemisthezero-stateresponsetotheunitstepfunction.Howtofind2Method1TosolvethesystemequationMethod2TofindviaTheunitstepresponseMethod3Comparisonmethod(equilibriumformulation)Decomposition:Theunitstepresponse信号与系统SignalsandSystems吉林大学ConvolutionIntegralConvolutionintegral

ToexpressintermsofthesumofinfiniteimpulsesConvolutionintegral信号与系统SignalsandSystems吉林大学TheZero-StateResponsetotheExcitationThezero-stateresponsetotheexcitationLimitsofintegration:ForasignalofForacausalsystemForsignalsandThezero-stateresponsetotheexcitation信号与系统SignalsandSystems吉林大学TheCommonOperationsofContinuous-TimeSignalsThecommonoperationsofcontinuous-timesignalsAddition1Thecommonoperationsofcontinuous-timesignalsMultiplication2Thecommonoperationsofcontinuous-timesignalsDifferentiation3Thecommonoperationsofcontinuous-timesignalsShift4Time-scaling5Folding6[Example]Thecommonoperationsofcontinuous-timesignalsTheprofileofisgivenbelow,plotasthefunctionoft.信号与系统SignalsandSystems吉林大学GraphicalRepresentationofConvolutionGraphicalRepresentationofConvolutionGraphicalRepresentationofConvolution信号与系统SignalsandSystems吉林大学PropertiesofConvolutionPropertiesofconvolutionProof:1.CommutativityPropertiesofconvolution2.DistributivitywithadditionPropertiesofconvolution3.Associativity4.DifferentiationandintegrationDifferentiation(1)PropertiesofconvolutionProof:Integration(2)PropertiesofconvolutionCombinationofdifferentiationandintegration(3)ItisrequiredthatDuhamel’sIntegralPropertiesofconvolution5.Shiftintime6.Replication(Convolutionwiththeunitimpulse)Proof:信号与系统SignalsandSystems吉林大学IntroductionandtheBasicRepresentationofFourierSeriesTheBackgroundofFourierseriesFourierseries(F.S.forshort)isnamedaftertheFrenchmathematicianandphysicistJeanBaptistFourier(1768-1830),whowasthefirstonetoproposethatperiodicwaveformscouldberepresentedbyasumofsinusoids(orcomplexexponentials)inthepaperonheatconductionwhichwaspresentedtoParisAcademyofScience.Fourierwasalsoveryactiveinthepoliticsofhistime.Forexample,heplayedanimportantroleinNapoleon’sexpeditionstoEgyptduringthelate1790s.TheFourierseriesofperiodicsignals--trigonometricseriesTheF.S.intermsoftrigonometricseriesTheFourierseriesofperiodicsignals--harmonicsTheFourierseriesofperiodicsignals--harmonicsTheF.S.intermsofharmonics信号与系统SignalsandSystems吉林大学ContributionofSymmetryoftotheFourierSeriesContributionofsymmetryoftotheF.S.(1)ContributionofsymmetryoftotheF.S.(2)ContributionofsymmetryoftotheF.S.(3)ContributionofsymmetryoftotheF.S.(4)(5)[Example]ContributionofsymmetryoftotheF.S.信号与系统SignalsandSystems吉林大学TheFourierSeriesintermsofPeriodicComplexExponentialsTheF.S.intermsofperiodiccomplexexponentialsTheFourierseriesofperiodicsignals--periodiccomplexexponentialsTheF.S.intermsofperiodiccomplexexponentialsTheFourierseriesofperiodicsignals--periodiccomplexexponentials信号与系统SignalsandSystems吉林大学FrequencySpectraofPeriodicSignalsFrequencyspectraofperiodicsignalsDefinition--Graphsthatfrequencycomponentsforaredisplayedbyverticallines.Description1Howtoplotfrequencyspectra2FrequencyspectraofperiodicsignalsUnilateralspectraBilateralspectraFrequencyspectraofperiodicsignalsExercise:信号与系统SignalsandSystems吉林大学TheFourierSeriesofaRectangularPulseTrainTheFourierseriesofarectangularpulsetrain1TheFourierseriesofarectangularpulsetrain2TheFourierseriesofarectangularpulsetrain2信号与系统SignalsandSystems吉林大学FourierTransformandInverseFourierTransformFouriertransformofanaperiodicsignalISpectraldensityfunctionFouriertransformofanaperiodicsignalⅡFouriertransformandinverseFouriertransformFouriertransformofanaperiodicsignal信号与系统SignalsandSystems吉林大学CommonFourierTransformPairs(1)CommonFouriertransformpairs12CommonFouriertransformpairs3CommonFouriertransformpairs4CommonFouriertransformpairs5CommonFouriertransformpairs6信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty1:LinearityProperty1:LinearityProof:[Example]Property1:Linearity信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty2:DualityProperty2:DualityProof:[Example]Property2:Duality信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty3:MultiplicationbyacomplexexponentialProof:Property3:MultiplicationbyacomplexexponentialMultiplicationbyacomplexexponential(shiftinfrequency)Property3:Multiplicationbyacomplexexponential(1)Inferences:ModulationtheoremModulatingsignalCarriersignalModulatedsignalProperty3:MultiplicationbyacomplexexponentialProperty3:Multiplicationbyacomplexexponential信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty4:ShiftintimeProperty5:TimescalingProperty4:ShiftintimeProof:Property5:TimescalingProof:信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty6:ConjugationandConjugateSymmetryProperty6:ConjugationandConjugateSymmetry(1)Property6:ConjugationandConjugateSymmetry(1)Property6:ConjugationandConjugateSymmetry(2)Property6:ConjugationandConjugateSymmetry(2)Property6:ConjugationandConjugateSymmetry(3)信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty7&8:ConvolutionTheoremsProperty7:Convolutioninthet-domainProof:Property8:Multiplicationinthet-domainMultiplicationinthet-domain(Convolutionintheω-domain)Proof:信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty9:Differentiationinthetime-domainProperty9:Differentiationinthetime-domainProof:(Suitabletotime-limitedsignals)信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty10:Integrationinthet-domainProperty10:Integrationinthet-domainProof:Property10:Integrationinthet-domainProperty10:Integrationinthet-domainProof:信号与系统SignalsandSystems吉林大学PropertiesofFourierTransformProperty11&12:DifferentiationandIntegrationintheω-DomainProperty11:Differentiationintheω-domainProof:Example:Property11:Differentiationintheω-domainProperty12:Integrationintheω-domain信号与系统SignalsandSystems吉林大学TheFourierTransformofaPeriodicSignalTheFouriertransformofaperiodicsignalⅠF.T.ofanon-sinusoidalperiodicsignalⅡTherelationshipbetweenandExample:TheFouriertransformofaperiodicsignal信号与系统SignalsandSystems吉林大学Steady-StateResponsetoNon-SinusoidalPeriodicSignalsSteady-stateresponsetonon-sinusoidalperiodicsignalsExample:信号与系统SignalsandSystems吉林大学FrequencyResponseFunction(SystemFunction)Frequencyresponsefunction(systemfunction)1.DefinitionFrequencyresponsefunction(systemfunction)Example:Findthesystemfunctionofthecircuitgivenbelow.信号与系统SignalsandSystems吉林大学ResponsetoAperiodicSignalsResponsetoaperiodicsignalsExample:Responsetoaperiodicsignals信号与系统SignalsandSystems吉林大学AnalysisofDistortionlessSystemsAnalysisofdistortionlesssystemsⅠDistortionlesssystemAnalysisofdistortionlesssystemsⅡThenecessaryandsufficientconditionofdistortionlesstransmission信号与系统SignalsandSystems吉林大学AnalysisofIdealLowpassFilters(ILFs)Analysisofideallowpassfilters(ILFs)ⅠThecharacteristicofILFsⅡTheimpulseresponseofILFAnalysisofideallowpassfilters(ILFs)ⅢTheapproximatelydistortionlessconditionofILFsAnalysisofideallowpassfilters(ILFs)ⅣPhysicalrealizabilityofasystemAnalysisofideallowpassfilters(ILFs)——Paley-WienercriterionIntime-domainInfrequency-domain信号与系统SignalsandSystems吉林大学SamplingandtheFourierTransforms(FTs)ofSampledContinuous-TimeSignalsSamplingandtheFouriertransformofⅠSamplingprocessAsamplingprocessisto“extract”aseriesofdiscretesamplevaluesfromacontinuous-timesignalbyusingasamplingimpulse(orpulse)train.ⅡClassification

Impulse-trainsampling(idealizedsampling)

Rectangularpulse-trainsamplingSamplingandtheFouriertransformofImpulse-trainsampling(idealizedsampling)ⅢTheFTsofsampledcontinuous-timesignalsSamplingandtheFouriertransformofSamplingandtheFouriertransformof信号与系统SignalsandSystems吉林大学SamplingTheorem(intheTime-Domain)Samplingtheorem(inthetime-domain)Samplingtheorem(inthetime-domain)Asignalwithbandwidthcanbereconstructedcompletelyandexactlyfromthesampledsignalbylowpassfilteringwithcutofffrequencyifthesamplingfrequencyischosentobegreaterthanorequalto.TheminimumsamplingfrequencyiscalledtheNyquistsamplingfrequency.

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

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

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

beinglocatedintheopen

left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedintheopen

left-halfcomplexplane1Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedattheorigin2Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedontheimaginaryaxis3Applicationsofthepole-zeroplot:Determiningtheformofh(t)Thepoles

beinglocatedintheopen

right-halfcomplexplane4Applicationsofthepole-zeroplot:Determiningtheformofh(t)信号与系统SignalsandSystems吉林大学Time-DomainAnalysisofDiscrete-TimeSystemsDiscrete-TimeSignalsDiscrete-TimeSignalsAdiscrete-timesignalf(k)hasvaluesforsomediscontinuouspointwhilehasnotdefinitionforotherpoints.k—integerDefinitionDiscrete-TimeSignalsAnalyticalmethod:Graphicalmethod:Sequencemethod:k=0RepresentationDiscrete-TimeSignalEnergyandPowerEnergy:Power:OperationofDiscrete-TimeSignalsAddition：Multiplication：Difference：forwarddifference:backwarddifferenceRunningsum：OperationofDiscrete-TimeSignalsTimeshift(m>0)RightshiftLeftshiftTransformationsoftheIndependentVariableOperationofDiscrete-TimeSignalsTimereversalTransformationsoftheIndependentVariablef(-k)isobtainedfromthesignalf(k)

byareflectionaboutk=0.BasicDiscrete-TimeSignalsUnitImpulseSequence(UnitSampleSequence)BasicDiscrete-TimeSignalsUnitStepSequenceBasicDiscrete-TimeSignalsRelationshipbetweend(k)ande(k)BasicDiscrete-TimeSignalsRectangularSequenceBasicDiscrete-TimeSignalsUnilateralexponentialsequenceswithrealvalues：f(k)=ak

(k)(aisarealnumber)BasicDiscrete-TimeSignalsUnitrampsequenceSinusoidalSequencesComplexExponentialSequences:Canda:complexnumbers信号与系统SignalsandSystems吉林大学RepresentationsofDiscrete-TimeSystemsRepresentationsofDiscrete-TimeSystemsAdiscrete-timesystemisasystemthattransformsdiscrete-timeinputsintodiscrete-timeoutputs.Definitionf(k):inputy(k):outputInput-outputrelation:f(k)→

y(k)RepresentationsofDiscrete-TimeSystemsnth-orderLinearConstant-CoefficientDifferenceEquation:LTISystemsDescribedbyDifferenceEquatioconstantsRepresentationsofDiscrete-TimeSystemsBlockDiagramRepresentationBasicelementsMultiplicationbyacoefficientAdderUnitDelayElementRepresentationsofDiscrete-TimeSystemsInterconnectionsofSystemsSeries(Cascade)interconnectionParallelinterconnectionFeedbackinterconnection信号与系统SignalsandSystems吉林大学Linearinput/outputdifferenceequationswithconstantcoefficientsInput:f(k)=0fork<0InitialCondition：y(0),y(1),y(2),…,y(n-1)InitialState：y(-1),y(-2),…,y(-n)Linearinput/outputdifferenceequationswithconstantcoefficientsEquation:Solution:Linearinput/outputdifferenceequationswithconstantcoefficientsTheHomogeneousSolutionHomogeneousequation

CharacteristicequationCharacteristicroot

j(j=1,2,3,

,n)HomogeneoussolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheHomogeneousSolutionExample:y(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2k,k

≥0,y(0)=0,y(1)=2.Findyh(k)

.Characteristicequation:Homogeneousequation

CharacteristicequationCharacteristicroot

j(j=1,2,3,

,n)HomogeneoussolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheParticularSolutionLinearinput/outputdifferenceequationswithconstantcoefficientsTheParticularSolutionExample:y(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2k,k

≥0,y(0)=0,y(1)=2.Findyp(k),k

≥0.Letyp(k)=P·2k,k

≥0Substitutethesystemequation:信号与系统SignalsandSystems吉林大学TheZero-InputResponse

and

TheZero-StateResponseTheZero-InputResponse

Characteristicequation

j,(j=1,2,3,

,n)CharacteristicrootZero-InputResponse

yzi

(0)，yzi

(1)，…，yzi

(n-1)y(-1)，y(-2)，…，y(-n)

yzi(k)=y(k)-

yzs(k)=y(k)，k<0InitialconditionCharacteristicequation:Characteristicroots:Zero-InputResponse:Example:TheZero-InputResponsey(k)+3y(k-1)+2y(k-2)=f(k),f(k)=2kε(k),y(-1)=0,y(-2)=1/2.Findyzi(k),k

≥0.yzi(k)+3yzi(k-1)+2yzi(k-2)=0TheZero-StateResponseCharacteristicequation

j

(j=1,2,3,

,n)Characteristicroot(distinctroots

j

)Zero-StateResponseyzs(-1)=yzs(-2)=…=yzs

(-n)=0Initialstateyzs(0)，yzs

(1)，…，yzs

(n-1)Initialcondition信号与系统SignalsandSystems吉林大学TheUnitSampleResponse

and

TheUnitStepResponseTheUnitSampleResponseDefinitionTheunitsampleresponseisthezero-stateresponseofthesystemresultingfromtheapplicationoftheunitpulse

(k).Denotedh(k)Initialstateh(-1)=h(-2)=…=h(-n)=0Initialconditionh(0),h(1),h(2),…,h(n-1)HowtofindSolvingadifferenceequationZ-transformTheUnitSampleResponseDeterminationk<0：

(k)

=0，h(k)=0k=0：

(k)

=1，h(0)——recursionk>0：

(k)

=0，h(k)——solutionofahomogeneousequationLTI

system:LetthenCi：determinedbyh1(1),h1(2),…,h1(n)TheUnitStepResponseDefinitionTheunitstepresponseisthezero-stateresponseofthesystemresultingfromtheapplicationoftheunit