信号与系统第二章课件

*2LINEARTIME-INVARIANTSYSTEMS线性时不变系统*Maincontent：Discrete-TimeLTISystems:TheConvolutionSum（离散时间LTI系统：卷积和）Continuous-TimeLTISystems:TheConvolutionIntegral(连续时间LTI系统：卷积积分）PropertiesofLinearTime-InvariantSystems（线性时不变系统的性质）CausalLTISystemsDescribedbyDifferentialandDifferenceEquations（用微分和差分方程描述的因果LTI系统）SingularityFunctions(奇异函数)*2.1.1TheRepresentationofDiscrete-timeSignalsinTermsofImpulses(p75)(用脉冲表示离散时间信号)SiftingPropertyofUnitSample:2.1DISCRETE-TIMELTI:CONVOLUTIONSUM(p75)(离散时间LTI系统：卷积和)(p75),(2.2)*Ifx[n]=u[n],then*2.1.2TheDiscrete-timeUnitImpulseResponseandtheConvolution-SumRepresentationofLTISystems(p77)(离散时间LTI系统的单位脉冲响应及卷积和表示)LTIx[n]=[n]y[n]=h[n]UnitImpulseResponseh[n]:responseoftheLTIsystemtotheunitsampleδ[n].δ[n]→h[n]Whydoweneedit?*LTIx[n]y[n]=?Solution:Question:[n]h[n][n-k]h[n-k]x[k][n-k]x[k]h[n-k]Theresponsey[n]tox[n]istheweightedlinearcombinationofdelayedunitsampleresponses.(p78),(2.6)*ConvolutionSumSoRepresentingtheconvolutionoperationsymbolicallyas:y[n]=x[n]*h[n]---ConvolutionSumThatis,theunitimpulseresponse--h[n]canfullycharacterizeanLTIsystem.SummaryoncalculatingconvolutionsumTimeInversal:h[k]h[-k]TimeShift:h[-k]h[n-k]Multiplication:x[k]h[n-k]Summing:(p78),(2.7)*Example2.1c(complementary)ConsideraLTIsystemwithunitsampleresponseh[n]andinputx[n],asillustratedinFigure(a).Calculatetheconvolutionsum(convolution)ofthesetwosequencesgraphically.nx[n]012nh[n]-202(a)122kx[k]012kh[-k]-202(b)221*kx[k]0122kh[-k]-20221n=0kh[-1-k]-3-20121n=-1kh[1-k]-1012321n=1***Example2.3(complementary)Consideraninputx[n]andaunitsampleresponseh[n]givenbyDetermineandplottheoutputUsingthegeometricalsumformulatoevaluatelastequation,wehaven<0thereisnooverlapbetweenthenonzeropointx[k]andh[n-k]thusy[n]=0forn<0*Forn>0y[n]=0forn<0Foralln*2.2CONTINUOUS-TIMELTISYSTEMS:CONVOLUTIONINTEGRAL(p90)(连续时间LTI系统：卷积积分)2.2.1TheRepresentationofContinuous-timeSignalsinTermsofImpulses(p90)(用冲激表示连续时间信号)Discrete-time:Continuous-time:*Why?t┉┉-Δ0Δ2ΔkΔ┉x(t)Staircaseapproximationtoacontinuous-timesignalx(t)Define(p92),(2.24)*Figure2.12(p91)Staircaseapproximationtoacontinuous-timesignal.*Therefore:Whatisthis?Wehavetheexpression:asΔ→0,thesummingapproachesanδΔ(t)integralandδ(t)istheunitimpulsefunction(p92),(2.25)(p92),(2.26)(p92),(2.27)*LTIx(t)=(t)y(t)=h(t)UnitImpulseResponseh(t):theresponseoftheLTIsystemtotheinputδ(t).2.2.2TheContinuous-timeUnitImpulseResponseandtheConvolutionIntegralRepresentationofLTISystems(p94)(连续时间LTI系统的单位冲激响应及卷积积分表示)LTIx(t)y(t)=?*GivethehΔ(t)astheresponseofacontinuous-timeLTIsystemtotheinputδΔ(t),thentheresponseofthesystemtopulseδΔ(t-kΔ)ishΔ(t-kΔ).Thus,theresponsetoisAs(p94),(2.29)(p96),(2.30)*inaddition,thesummingbecomesanintegral.Therefore,---ConvolutionIntegral(p97),(2.33)ConvolutionIntegralRepresentconvolutionintegraloftwosignalsx(t)andh(t)symbolicallyas:(p97),(2.34)Acontinuous-timeLTIsystemiscompletelycharacterizedbyitsunitimpulseresponseh(t).*ComputationofConvolutionIntegral:TimeInversal:h()h(-)TimeShift:h(-)h(t-)Multiplication:x()h(t-)Integrating:*Example2.3c(complementary)Considertheconvolutionofthefollowingtwosignals,whicharedepictedin(a):2x(t)1h(t)012t0123t-1(a)x(τ)h(-τ)-20123τt=0x(τ)h(t-τ)-20123τt0t0.*AgeneralNth-orderlinearconstant-coefficientdifferentialequation:orandinitialcondition:y(t0),y’(t0),……,y(N-1)(t0)(Nvalues)(p120),(2.109)*ForacausalLTIsystem:(p121),(2.122)2.4.2LinearConstant-CoefficientDifferenceEquations(p115)(线性常系数差分方程)AgeneralNth-orderlinearconstant-coefficientdifferenceequation:(p121),(2.113)*orandinitialcondition:y[0],y[-1],……,y[-(N-1)](Nvalues)Underinitialrest,thesystemdescribedbylinearconstant-coefficientdifferential(difference)equationiscausalandLTI.Firstresolution:*Generalsolutionstosuchdifferenceequations:laterinChapter5or10.Secondresolution:(recursivemethod)Nauxiliaryconditions:(p122),(2.115)***2.4.3BlockDiagramRepresentationsofFirst-OrderSystemsDescribedbyDifferentialandDifferenceEquations(p124)(用微分和差分方程描述的一阶系统的方框图表示)(1)Dicrete-timesystemFirst-orderdifferenceequation:additiondelaymultiplication(p124),(2.126)*Basicelements:A.AnadderB.MultiplicationbyacoefficientC.AnunitdelayFigure2.27(p125).*Example:y[n]+ay[n-1]=bx[n](2)Continuous-timesystemFirst-orderdifferentialequation:differentiationThreebasicelementsinblockdiagram:adder,multiplierandintegrator.Figure2.28(p125).(p125),(2.128)*Example:y’(t)+ay(t)=bx(t)(p126),(2.131)Figure2.29(p126).Figure2.32(p127).*2.5SINGULARITYFUNCTIONS(p127)（奇异函数）2.5.1TheUnitImpulseasanIdealizedShortPulse(p124)(作为理想化短脉冲的单位冲激)(1)*Important:forsmallΔ,theybothbehavesthesamefromanLTIsystem,seeFigure2.34.(2)(p128),(2.134)Figure2.33(p128).*2.5.2DefiningtheUnitImpulsethroughConvolution(p131)(通过卷积定义单位冲激)δ(t)definitionOr，equivalently,Theprimaryimportanceoftheunitimpulseisnotwhatitisateachvalueoft,butratherwhatitdoesunderconvolution.(p131),(2.138)(p131),(2.139)*2.5.3UnitDoubletandOtherSingularityFunctions(p132)(单位冲激偶和其它的奇异函数)(p133),(2.144)(p133),(2.145)(p135),(2.153)(p135),(2.156)*SUMMARY1.Arepresentationofanarbitrarydiscrete-timesignalasweightedsumsofshiftedunitsamples;2.Convolutionsumrepresentationfortheresponseofadiscrete-timeLTIsystems;3.Arepresentationofanarbitrarycontinuous-timesignalaweightedintegralsofshiftedunitimpulses;4.Convolutionintegralrepresentationforcontinuous-timeLTIsystems;*5.RelatingLTIsystemproperties,includingcausality,stability,tocorrespondingpropertiesoftheunitimpulse(sample)response;6.Someofthepropertiesofsystemsdescribedbylinearconstant-coefficientdifferential(difference)equations;7.Understandingoftheconditionofinitialrest.SUMMARY*Problems:2.1(b)2.52.72.102.122.162.172.192.21(c)2.22(e)