1Chapter2LinearTime-invariantSystems2Chapter2LTISystemsConsideralineartime-invariantsystemExample1anLTIsystem02t1012t1L024t1-1024t1L3Chapter2LTISystems§2.1Discrete-timeLTISystems:TheConvolutionSum（卷積和）§2.1.1TheRepresentationofDiscrete-TimeSignalsinTermsofimpulsesSiftingProperty離散時間信號的沖激表示Example2n4Chapter2LTISystems§2.1.2TheDiscrete-TimeUnitImpulseResponsesandtheConvolution-SumRepresentationofLTISystemsTheUnitImpulseResponses單位沖激響應LetLetTime-InvariantUnitImpulseResponses5Chapter2LTISystems2.Convolution-Sum（卷積和）TimeInvariantScalingAdditivityConvolution-Sum（卷積和）系統在n時刻的輸出包含所有時刻輸入脈沖的影響k時刻的脈沖在n時刻的響應6Chapter2LTISystems3.卷積和的計算①圖解法例2.3圖解法步驟：㈠反折㈡平移㈢求乘積㈣對每一個n求和循環7Chapter2LTISystemsExample2.4DeterminetheoutputsignalSolution(a)n<0(b)0≤n＜4

8Chapter2LTISystems(d)(c)(e)9Chapter2LTISystemsSummarizing,weobtainLy=11Lx=5Lh=7Ly=Lx+Lh-110Chapter2LTISystems不帶進位的普通乘法 適用于因果序列或有限長度序列之間的卷積11Chapter2LTISystemsExample3DetermineSolution3 1 4 2 h[n]2 1 5 x[n]15 5 20 103 1 4 26 2 8 46 5 24 13 22 10y[0]y[1]y[2]y[3]y[4]y[5]y[n]={6,5,24,13, 22,10}n=0,1,2,3,4,512Chapter2LTISystems③多項式算法（適用于有限長度序列）y[n]={6,5,24,13, 22,10}n=0,1,2,3,4,5利用多項式算法求卷積和的逆運算已知y[n]、h[n]→x[n]已知y[n]、x[n]→h[n]13Chapter2LTISystemsExample4Determinex[n]y[n]={6,5,24,13, 22,10}n=0,1,2,3,4,5y(t)014Chapter2LTISystems§2.2Continuous-TimeLTISystems:TheConvolutionIntegral

（卷積積分）§2.2.1TheRepresentationofContinuous-TimeSignalsinTermsofimpulses0△t——SiftingPropertyForexample:15Chapter2LTISystemsAccordingtoSamplingProperty=1§2.2.1TheContinuous-TimeUnitImpulseResponseandthe ConvolutionIntegralRepresentationofLTISystems——TimeInvariant——Scaling16Chapter2LTISystemsConvolutionIntegral卷積積分τ時刻的沖激t時刻的響應§2.3卷積的計算一由定義計算積分例2.6zero-stateoutput17Chapter2LTISystems二圖解法例2.7求下列兩信號的卷積其余t其余t解：①②③18Chapter2LTISystems④⑤19Chapter2LTISystems§2.3PropertiesofLTISystemsLTI系統的特性可由單位沖激響應完全描述Example2.9①LTIsystem——輸入輸出關系是唯一的20Chapter2LTISystems②NonlinearSystem非線性系統無法用單位沖激響應完全描述本課程主要研究線性時不變（LTI）系統21Chapter2LTISystems§2.3.1PropertiesofConvolutionIntegralandConvolutionSum1.TheCommutativeProperty(交換律）22Chapter2LTISystems2.TheDistributiveProperty(分配律）23Chapter2LTISystems3.TheAssociativeProperty(結合律）交換積分次序24Chapter2LTISystemsCommutativePropertyAssociativeProperty25Chapter2LTISystems4.含有沖激的卷積①②26Chapter2LTISystems5.卷積的微分、積分性質①微分②積分27Chapter2LTISystems③推廣n=1m=-1n>0微分n<0積分28Chapter2LTISystems29Chapter2LTISystemsExampleotherwiseotherwiseSolution10123t0123tIntegral30Chapter2LTISystemsSolution20123t0123t31ExampleConsidertheconvolutionofthetwosignals(1)(1)0t-101t-2-1012t-2-2-1012t-2(-2)32Chapter2LTISystems6幾種典型系統①恒等系統②微分器③積分器④延遲器⑤累加器33Chapter2LTISystems§2.3.4LTISystemswithandwithoutMemory1.Discrete-timeSystemItismemorylessAnLTIsystemwithoutmemory2.Continuous-timeSystemAnLTIsystemwithoutmemory34Chapter2LTISystems§2.3.5InvertibilityofLTISystemsLTI系統的可逆性SystemInverseSystemidentitysystem(恒等系統）35Chapter2LTISystems§2.3.6CausalityforLTISystemsLTI系統的因果性1.Discrete-timeSystem與n時刻以后輸入有關Causalsystem2.Continuous-timeSystemCausalsystem36Chapter2LTISystemsConsideralinearsystemCausalInitialRest(初始松弛）Foranytimet0§2.3.7StabilityforLTISystems(穩定性）1.Discrete-timeSystem37Chapter2LTISystemsStableSystem2.Continuous-timeSystemabsolutelySummable

絕對可加StableSystem①Thesystemisstable.②absolutelyIntegrable絕對可積Thesystemisnotstable.38Chapter2LTISystems§2.3.8TheUnitStepResponseofanLTISystemsLTI系統的單位階躍響應Discrete-timeSystemContinuous-timeSystemUnitStepResponseUnitStepResponse39Chapter2LTISystems§2.4SingularityFunctions(奇異函數）§2.4.1TheUnitImpulseasanIdealizedShortPulse0△t02△t40Chapter2LTISystemsExample2.1641Chapter2LTISystemsExample2.1742DefineChapter2LTISystems§2.4.2DefiningtheUnitImpulsethroughConvolution單位沖激的卷積定義Foranyx(t)Foranynormal,whichiscontinuousattimet=0.Lett=0SiftingPropertyTheunitimpulsehasunitarea43Chapter2LTISystems§2.4.3UnitDoubletsandOtherSingularityFunctions單位沖激偶和其它奇異函數DerivativeUnitDoublets1.Ingeneral,ktimes0△t0△t44Chapter2LTISystemsDiscrete-timeSystemContinuous-timeSystemCausalsystemCausalsystemStableSystemStableSystem45Chapter2LTISystems3.PropertiesofUnitDoublets（沖激偶的性質）2.DefiningForany①②46Chapter2LTISystemsIngeneral,③Derivativesofdifferentordersoftheunitimpulse

單位沖激的各階積分47Chapter2LTISystems§2.5CausalLTISystemsdescribedbyDifferential andDifferenceEquations

用微分方程和差分方程描述的因果LTI系統§2.5.1LinearConstant-coefficientDifferentialEquations

線性常系數微分方程一經典解法1.Homogeneoussolution齊次解特征方程48Chapter2LTISystems①特征方程有N個不同的單根②特征方程有1個r階重根，其余N-r個根各不相同HomogeneousSolution——NaturalResponse

齊次解自然響應2.ParticularSolution——ForcedResponse

特解受迫響應的函數形式取決于輸入的函數形式49Chapter2LTISystems輸入特解C（常數）B（常數）

α不是特征單根α是特征單根或50Chapter2LTISystemsExample2.14齊次解：特解：全響應：LTI因果系統初始松弛由系統初始條件確定51Chapter2LTISystems②若初始條件不為零，若初始條件不躍變，代入全響應N階系統，初始松弛：初始松弛條件下：52Chapter2LTISystems二零輸入、零狀態解法全響應：1.零輸入響應：具有齊次解的形式；假定特征方程含有N個不同的單根2.零狀態響應：齊次解的另一部分特解齊次解的一部分53Chapter2LTISystems由初始狀態唯一決定零輸入響應由零初始狀態及輸入共同決定零狀態響應由初始狀態及輸入決定自然響應函數形式由輸入信號決定受迫響應1.零輸入零狀態解法2.經典解法三兩種響應分解形式的關系54Chapter2LTISystemsExample2.14Zero-input responseZero-state response3.Fullresponse55Chapter2LTISystems§2.5.2LinearConstant-coefficientDifferenceEquations

線性常系數差分方程NthorderRecursiveSolution遞推算法RecursiveEquation遞歸方程InputsignalInitialstate56Chapter2LTISystemsParticularlyN=0NonrecursiveEquation非遞歸方程UnitimpulseresponseFiniteimpulseresponse(FIR)systemExample2.15Initialrest57Chapter2LTISystems58Chapter2LTISystems由初始狀態唯一決定零輸入響應由零初始狀態及輸入共同決定零狀態響應由初始狀態及輸入決定自然響應函數形式由輸入信號決定受迫響應1.零輸入零狀態解法2.經典解法兩種響應分解形式的關系59Chapter2LTISystems2.差分方程的經典解法FullHomogeneousParticular①HomogeneoussolutionEigenequation特征方程(a)特征方程有N個不同的單根(b)特征方程有1個r階重根γ1

其余N-r個根各不相同60Chapter2LTISystems②Particularsolution(ForcedResponse)輸入特解

α不是特征單根α是特征單根或61Chapter2LTISystems3.零輸入、零狀態解法FullZero-inputZero-state①零輸入響應：具有齊次解的形式；②零狀態響應：齊次解的另一部分特解齊次解的一部分假定特征方程含有N個不同的單根62Chapter2LTISystems由初始狀態