.,1,Chapter2,LinearTime-invariantSystems,.,2,Consideralineartime-invariantsystem,Example1anLTIsystem,.,3,2.1Discrete-timeLTISystems:TheConvolutionSum（卷積和）,2.1.1TheRepresentationofDiscrete-TimeSignalsinTermsofimpulses,SiftingProperty,離散時(shí)間信號(hào)的沖激表示,.,4,2.1.2TheDiscrete-TimeUnitImpulseResponsesandtheConvolution-SumRepresentationofLTISystems,TheUnitImpulseResponses單位沖激響應(yīng),Let,Let,.,5,2.Convolution-Sum（卷積和）,TimeInvariant,Scaling,Additivity,Convolution-Sum（卷積和）,系統(tǒng)在n時(shí)刻的輸出包含所有時(shí)刻輸入脈沖的影響,k時(shí)刻的脈沖在n時(shí)刻的響應(yīng),.,6,3.卷積和的計(jì)算,圖解法,例2.3,圖解法步驟：,求乘積,對(duì)每一個(gè)n求和,.,7,Example2.4,Determinetheoutputsignal,Solution,(a)n0微分n0積分,.,28,.,29,Example,Solution1,.,30,.,31,ExampleConsidertheconvolutionofthetwosignals,.,32,6幾種典型系統(tǒng),恒等系統(tǒng),微分器,積分器,延遲器,累加器,.,33,2.3.4LTISystemswithandwithoutMemory,1.Discrete-timeSystem,Itismemoryless,AnLTIsystemwithoutmemory,2.Continuous-timeSystem,AnLTIsystemwithoutmemory,.,34,2.3.5InvertibilityofLTISystemsLTI系統(tǒng)的可逆性,.,35,2.3.6CausalityforLTISystemsLTI系統(tǒng)的因果性,1.Discrete-timeSystem,Causalsystem,2.Continuous-timeSystem,.,36,Consideralinearsystem,Foranytimet0,2.3.7StabilityforLTISystems(穩(wěn)定性）,1.Discrete-timeSystem,.,37,2.Continuous-timeSystem,absolutelySummable絕對(duì)可加,Thesystemisstable.,absolutelyIntegrable絕對(duì)可積,Thesystemisnotstable.,.,38,2.3.8TheUnitStepResponseofanLTISystemsLTI系統(tǒng)的單位階躍響應(yīng),Discrete-timeSystem,Continuous-timeSystem,UnitStepResponse,UnitStepResponse,.,39,2.4SingularityFunctions(奇異函數(shù)）,2.4.1TheUnitImpulseasanIdealizedShortPulse,.,40,Example2.16,.,41,Example2.17,.,42,2.4.2DefiningtheUnitImpulsethroughConvolution單位沖激的卷積定義,Foranyx(t),Foranynormal,whichiscontinuousattimet=0.,Lett=0,SiftingProperty,Theunitimpulsehasunitarea,.,43,2.4.3UnitDoubletsandOtherSingularityFunctions單位沖激偶和其它奇異函數(shù),1.,Ingeneral,.,44,.,45,3.PropertiesofUnitDoublets（沖激偶的性質(zhì)）,2.Defining,Forany,.,46,Ingeneral,Derivativesofdifferentordersoftheunitimpulse單位沖激的各階積分,.,47,2.5CausalLTISystemsdescribedbyDifferentialandDifferenceEquations用微分方程和差分方程描述的因果LTI系統(tǒng),2.5.1LinearConstant-coefficientDifferentialEquations線性常系數(shù)微分方程,一經(jīng)典解法,1.Homogeneoussolution齊次解,特征方程,.,48,特征方程有N個(gè)不同的單根,特征方程有1個(gè)r階重根，其余N-r個(gè)根各不相同,HomogeneousSolutionNaturalResponse齊次解自然響應(yīng),2.ParticularSolutionForcedResponse特解受迫響應(yīng),的函數(shù)形式取決于輸入的函數(shù)形式,.,49,不是特征單根是特征單根,或,.,50,齊次解：,特解：,全響應(yīng)：,.,51,若初始條件不為零，,若初始條件不躍變，,代入,全響應(yīng),N階系統(tǒng)，初始松弛：,初始松弛條件下：,.,52,二零輸入、零狀態(tài)解法,全響應(yīng)：,1.零輸入響應(yīng)：,具有齊次解的形式；,假定特征方程含有N個(gè)不同的單根,2.零狀態(tài)響應(yīng)：,.,53,由初始狀態(tài)唯一決定,由零初始狀態(tài)及輸入共同決定,由初始狀態(tài)及輸入決定,函數(shù)形式由輸入信號(hào)決定,1.零輸入零狀態(tài)解法,2.經(jīng)典解法,三兩種響應(yīng)分解形式的關(guān)系,.,54,Zero-inputresponse,Zero-stateresponse,3.Fullresponse,.,55,2.5.2LinearConstant-coefficientDifferenceEquations線性常系數(shù)差分方程,Nthorder,RecursiveSolution遞推算法,RecursiveEquation遞歸方程,.,56,ParticularlyN=0,NonrecursiveEquation非遞歸方程,Unitimpulseresponse,Finiteimpulseresponse(FIR)system,Example2.15,.,57,.,58,.,59,2.差分方程的經(jīng)典解法,FullHomogeneousParticular,Homogeneoussolution,Eigenequation特征方程,(a)特征方程有N個(gè)不同的單根,(b)特征方程有1個(gè)r階重根1其余N-r個(gè)根各不相同,.,60,Particularsolution(ForcedResponse),不是特征單根是特征單根,或,.,61,3.零輸入、零狀態(tài)解法,FullZero-inputZero-state,零輸入響應(yīng)：,具有齊次解的形式；,零狀態(tài)響應(yīng)：,假定特征方程含有N個(gè)不同的單根,.,62,由初始狀態(tài)唯一決定,由零初始狀態(tài)及輸入共同決定,由初始狀態(tài)及輸入決定,函數(shù)形式由輸入信號(hào)決定,4.兩種響應(yīng)分解形式的關(guān)系,.,63,Solution1,HomogeneousSolution,ParticularSolution,FullSolution,.,64,InitialState:,FullSolution,Solution2,Zero-inputSolution,.,65,Zero-stateSolution,FullSolution,Zero-inputresponse,Zero-stateresponse,Naturalresponse,Forcedresponse,.,66,2.5.3BlockDiagramRepresentationsofFirst-OrderSystem一階系統(tǒng)的方框圖表示,1.Discrete-timesystems,Addition,Multiplicationbyacoefficient,Delay,RecursiveEquation,.,67,2.Continuous-ti