《信号与线性系统分析基础》课件 刘秀环 1.1.1Fundamental concepts of signals-4.2.1Property 12-linearity -shift in the s-domain

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-ti