《信號(hào)與線性系統(tǒng)分析基礎(chǔ)》課件 劉秀環(huán) 1.1.1Fundamental concepts of signals-4.2.1Property 12-linearity -shift in the s-domain

信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)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.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)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信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)DeterminationofSystemCharacteristicsDeterminationofsystemcharacteristics[Example]Determineifthesystemdescribedbyislinear,time-invariant,causalandstable.Linearornonlinear?(1)Time-invariantortime-varying?(2)Let?DeterminationofsystemcharacteristicsCausalornoncausal?(3)Stableorunstable?(4)Tobecontinued:Conclusion:Thesystemislinear,time-varying,noncausalandstable.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)PropertiesofconvolutionProof:Integration(2)PropertiesofconvolutionCombinationofdifferentiationandintegration(3)ItisrequiredthatDuhamel’sIntegralPropertiesofconvolution5.Shiftintime6.Replication(Convolutionwiththeunitimpulse)Proof:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)IntroductionandtheBasicRepresentationofFourierSeriesTheBackgroundofFourierseriesFourierseries(F.S.forshort)isnamedaftertheFrenchmathematicianandphysicistJeanBaptistFourier(1768-1830),whowasthefirstonetoproposethatperiodicwaveformscouldberepresentedbyasumofsinusoids(orcomplexexponentials)inthepaperonheatconductionwhichwaspresentedtoParisAcademyofScience.Fourierwasalsoveryactiveinthepoliticsofhistime.Forexample,heplayedanimportantroleinNapoleon’sexpeditionstoEgyptduringthelate1790s.TheFourierseriesofperiodicsignals--trigonometricseriesTheF.S.intermsoftrigonometricseriesTheFourierseriesofperiodicsignals--harmonicsTheFourierseriesofperiodicsignals--harmonicsTheF.S.intermsofharmonics信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)ContributionofSymmetryoftotheFourierSeriesContributionofsymmetryoftotheF.S.(1)ContributionofsymmetryoftotheF.S.(2)ContributionofsymmetryoftotheF.S.(3)ContributionofsymmetryoftotheF.S.(4)(5)[Example]ContributionofsymmetryoftotheF.S.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheFourierSeriesintermsofPeriodicComplexExponentialsTheF.S.intermsofperiodiccomplexexponentialsTheFourierseriesofperiodicsignals--periodiccomplexexponentialsTheF.S.intermsofperiodiccomplexexponentialsTheFourierseriesofperiodicsignals--periodiccomplexexponentials信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)FrequencySpectraofPeriodicSignalsFrequencyspectraofperiodicsignalsDefinition--Graphsthatfrequencycomponentsforaredisplayedbyverticallines.Description1Howtoplotfrequencyspectra2FrequencyspectraofperiodicsignalsUnilateralspectraBilateralspectraFrequencyspectraofperiodicsignalsExercise:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheFourierSeriesofaRectangularPulseTrainTheFourierseriesofarectangularpulsetrain1TheFourierseriesofarectangularpulsetrain2TheFourierseriesofarectangularpulsetrain2信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)FourierTransformandInverseFourierTransformFouriertransformofanaperiodicsignalISpectraldensityfunctionFouriertransformofanaperiodicsignalⅡFouriertransformandinverseFouriertransformFouriertransformofanaperiodicsignal信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)CommonFourierTransformPairs(1)CommonFouriertransformpairs12CommonFouriertransformpairs3CommonFouriertransformpairs4CommonFouriertransformpairs5CommonFouriertransformpairs6信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty1:LinearityProperty1:LinearityProof:[Example]Property1:Linearity信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty2:DualityProperty2:DualityProof:[Example]Property2:Duality信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty3:MultiplicationbyacomplexexponentialProof:Property3:MultiplicationbyacomplexexponentialMultiplicationbyacomplexexponential(shiftinfrequency)Property3:Multiplicationbyacomplexexponential(1)Inferences:ModulationtheoremModulatingsignalCarriersignalModulatedsignalProperty3:MultiplicationbyacomplexexponentialProperty3:Multiplicationbyacomplexexponential信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty4:ShiftintimeProperty5:TimescalingProperty4:ShiftintimeProof:Property5:TimescalingProof:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty6:ConjugationandConjugateSymmetryProperty6:ConjugationandConjugateSymmetry(1)Property6:ConjugationandConjugateSymmetry(1)Property6:ConjugationandConjugateSymmetry(2)Property6:ConjugationandConjugateSymmetry(2)Property6:ConjugationandConjugateSymmetry(3)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty7&8:ConvolutionTheoremsProperty7:Convolutioninthet-domainProof:Property8:Multiplicationinthet-domainMultiplicationinthet-domain(Convolutionintheω-domain)Proof:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty9:Differentiationinthetime-domainProperty9:Differentiationinthetime-domainProof:(Suitabletotime-limitedsignals)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty10:Integrationinthet-domainProperty10:Integrationinthet-domainProof:Property10:Integrationinthet-domainProperty10:Integrationinthet-domainProof:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)PropertiesofFourierTransformProperty11&12:DifferentiationandIntegrationintheω-DomainProperty11:Differentiationintheω-domainProof:Example:Property11:Differentiationintheω-domainProperty12:Integrationintheω-domain信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheFourierTransformofaPeriodicSignalTheFouriertransformofaperiodicsignalⅠF.T.ofanon-sinusoidalperiodicsignalⅡTherelationshipbetweenandExample:TheFouriertransformofaperiodicsignal信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Steady-StateResponsetoNon-SinusoidalPeriodicSignalsSteady-stateresponsetonon-sinusoidalperiodicsignalsExample:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)FrequencyResponseFunction(SystemFunction)Frequencyresponsefunction(systemfunction)1.DefinitionFrequencyresponsefunction(systemfunction)Example:Findthesystemfunctionofthecircuitgivenbelow.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)ResponsetoAperiodicSignalsResponsetoaperiodicsignalsExample:Responsetoaperiodicsignals信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)AnalysisofDistortionlessSystemsAnalysisofdistortionlesssystemsⅠDistortionlesssystemAnalysisofdistortionlesssystemsⅡThenecessaryandsufficientconditionofdistortionlesstransmission信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)AnalysisofIdealLowpassFilters(ILFs)Analysisofideallowpassfilters(ILFs)ⅠThecharacteristicofILFsⅡTheimpulseresponseofILFAnalysisofideallowpassfilters(ILFs)ⅢTheapproximatelydistortionlessconditionofILFsAnalysisofideallowpassfilters(ILFs)ⅣPhysicalrealizabilityofasystemAnalysisofideallowpassfilters(ILFs)——Paley-WienercriterionIntime-domainInfrequency-domain信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)SamplingandtheFourierTransforms(FTs)ofSampledContinuous-TimeSignalsSamplingandtheFouriertransformofⅠSamplingprocessAsamplingprocessisto“extract”aseriesofdiscretesamplevaluesfromacontinuous-timesignalbyusingasamplingimpulse(orpulse)train.ⅡClassification

Impulse-trainsampling(idealizedsampling)

Rectangularpulse-trainsamplingSamplingandtheFouriertransformofImpulse-trainsampling(idealizedsampling)ⅢTheFTsofsampledcontinuous-ti