《信号与线性系统分析基础 》 课件 第 3 章 连续时间信号的频域分析

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(idealize