信号教学课件(华中科技大学)cha

CHAPTER2LINEARTIME-INVARIANTSYSTEMS2.0INTRODUCTION,Representationofsignalsaslinearcombinationofdelayedimpulses.Convolutionsum(卷积和)orconvolutionintegral(卷积积分)representationofLTIsystems.ImpulseresponseandsystemspropertiesSolutionstolinearconstant-coefficientdifferenceanddifferentialequations(线性常系数差分或微分方程).,2.1DISCRETE-TIMESYSTEMS:THECONVOLUTIONSUM,Derivationsteps:Step1:Representingdiscrete-timesignalsintermsofunitsamples:,Step2:DefiningUnitsampleresponsehn:responseoftheLTIsystemtotheunitsamplen.nhn,Step3:Writinganyarbitraryinputxnas:,Step4:Bytakinguseoflinearityandtime-invariance,wecangettheresponseyntoxnwhichistheweightedlinearcombinationofdelayedunitsampleresponsesasfollowing:,TheConvolutionSumRepresentationofLTISystems,convolutionsumorsuperpositionsum:,Convolutionoperationsymbol:,LTIsystemiscompletelycharacterizedbyitsresponsetotheunitsample-hn.,Example2.1,1,1,0.5,GraphofyninExample2.1,FromExample2.1,wecandrawthefollowingtable:,Thus,weobtainamethodforthecomputationofconvolutionsum,thatissuitablefortwoshortsequences.,xn=1,1,10,hn=0.5,1,0.5,1,0.5-2,xn*hn=0.5,1.5,2,2.5,2,1.5,0.5-2,0.51.522.521.50.5,0.510.510.5,Example2.2,Consideraninputxnandaunitsampleresponsehngivenby,Determineandplottheoutput,Usingthegeometricsumformulatoevaluatetheequation,wehave,GraphofyninExample2.2,2.2CONTINUOUS-TIMELTISYSTEMS:THECONVOLUTIONINTEGRAL,TheRepresentationofContinuous-TimeSignalsinTermsofImpulses:,Mathematicalrepresentationfortherectangularpulses,as,thesummationapproachesanintegralandistheunitimpulsefunction,ComparedwiththeSamplingpropertyoftheunitimpulse:,Givetheastheresponseofacontinuous-timeLTIsystemtotheinput,thentheresponseofthesystemtopulseis,Thus,theresponsetois,As,inaddition,thesummingbecomesanintegral.Therefore,convolutionintegralorsuperpositionintegral:,unitimpulseresponseh(t):theresponsetotheinput.（单位冲激响应）,Convolutionintegralsymbol:,Acontinuous-timeLTIsystemiscompletelycharacterizedbyitsunitimpulseresponseh(t).,Example2.3,Considertheconvolutionofthefollowingtwosignals,whicharedepictedin(a):,Fromthedefinitionoftheconvolutionintegraloftwocontinuous-timesignals,2T,h(t),t-2T0tT,1,x(),For0tT,.Thus,for0tT,.,Interval2.For0tT,2T,h(t),t-2TTt,1,x(),ForTt2T,Thus,forTtTbutt-2T0,i.e.Tt0,butt-2TT,i.e.2Tt3T,Thus,for2Tt3T,.,For2T3T,thereisnooverlapbetweenthenonzeroportionsofand,hence,Summarizing,2.3PROPERTIESOFCONVOLUTIONOPERATION,2.3.1TheCommutativeProperty(交换律),2.3.2TheDistributiveProperty(分配律),Twoequivalentsystems:havingsameimpulseresponses,2.3.3TheAssociativeProperty(结合律),Fourequivalentsystems,2.3.4ConvolvingwithImpulse,2.3.5DifferentiationandIntegrationofConvolutionIntegral,Combiningthetwoproperties,wehave,2.3.6FirstDifferenceandAccumulationofConvolutionSum,2.4.1LTISystemswithandwithoutMemory,2.4.2InvertibilityofLTISystems,Since,2.4PROPERTIESOFLTISYSTEMS,2.4.3CausalityforLTISystems,2.4.4StabilityforLTISystems,Suppose,Proof:,Then,If,Then,Therefore,theabsolutelysummableisasufficientconditiontoguaranteethestabilityofadiscrete-timeLTIsystem.,Toshowthattheabsolutelysummableisalsoanecessaryconditionforthestabilityofadiscrete-timeLTIsystem,Let,where,isconjugate.,Then,xnisboundedby1,thatis,However,If,Then,2.5TheUnitStepResponse(单位阶跃响应)ofanLTISystem,Theunitstepresponse,snors(t),istheoutputofanLTIsystemwheninputxn=unorx(t)=u(t).,Theunitstepresponseofadiscrete-timeLTIsystemistherunningsumofitsunitsampleresponse:,Theunitsampleresponseofadiscrete-timeLTIsystemisthefirstdifferenceofitsunitstepresponse:,Theunitstepresponseofacontinuous-timeLTIsystemistherunningintegralofitsunitimpulseresponse:,Theunitimpulseresponseofacontinuous-timeLTIsystemisthefirstderivativeoftheunitstepresponse:,2.6CAUSALLTISYSTEMSDESCRIBEDBYDIFFERENTIALANDDIFFERENCEEQUATIONS,Linearconstant-coefficientdifferentialequation,Linearconstant-coefficientdifferenceequationisthemathematicalrepresentationofadiscrete-timeLTIsystem.,Linearconstant-coefficientdifferentialequationisthemathematicalrepresentationofacontinuous-timeLTIsystem.,Wemustspecifyoneormoreauxiliaryconditionstosolveadifferential(difference)equation.,Initialrest(初始静止):foracausalLTIsystem,ifx(t)=0fort0intotheoriginalequationyields,Thus,Sothesolutionofthedifferentialequationfort0is,InExample2.4,Takinguseoftheconditioninitialrest,weobtain,Consequently,or,fort0,Example2.5,Jacksavesmoneyeverymonth.ItisknownthatatthebeginningofthenthmonththeamounthesavesintothebankisRMBxnyuan,andtherateofinterestispermonth.SupposeJackwouldntwithdrawhisbankdepositsinwhateversituation,trytogivethedifferenceequationrelatingxnandyn,whichisthedepositsofJackattheendofthenthmonth.(beforethebankcalculatestheinterest),Solution:,ynisconsistsofthesumofthefollowingthreeparts:,xnsavedatthebeginningofthenthmonth,yn-1interestattheendofthe(n-1)thmonth,yn-1depositofthe(n-1)thmonth,Sothedifferenceequationis,also,Difference:,Forsequencexn,itsFirstforwarddifference(一阶前向差分)isdefinedasxn=xn+1xn,itsFirstbackwarddifference(一阶后向差分)isdefinedasxn=xnxn-1,GeneralNth-orderlinearconstant-coefficientdifferenceequation:,Firstresolution:,Nauxiliaryconditions:,Secondresolution:(recursivemethod(迭代法),Example2.6,Solvethedifferenceequationandtheinitialconditionisy0=1.,Theeigenequationis,Sotheeigenvalueisa=2,Wecanwrite,Let,Takingintotheoriginalequationyields,Thus,Thesolutionforthegivenequationis,Fromtheinitialconditionofy0=1,wehave,Consequently,Example2.7,Considerthedifferenceequation,Determinetheoutputrecursivelywiththeconditionofinitialrestand,Rewritethegivendifferenceequationas,Startingfrominitialcondition,wecansolveforsuccessivevaluesofynforn1:,Consideringyn=0forn0,thesolutionis,Question:,Example2.8,Consideracontinuous-timeLTIsystemdescribedbythefollowingdifferentialequationwithinitialconditionsofandinput,Fromthedefinitionofthezero-inputresponse,wehave,Inthecaseofzeroinput,Thuswecanwrite,Equivalently,Consequently,Next,solveforh(t).,Fromthedefinitionoftheunitimpulseresponse,wehave,Andfort0,itbecomes,h(t)isthesolutionforthehomogeneousequation.Thus,Andbecausethesystemisacausalone,thereshouldbe,TheinitialconditionsusedtodetermineA1andA2are,But,Let(2),Then(3),Takingequations(2)and(3)intoequation(1),wehave,Comparingthecoefficientsofthecorrespondingtermsoneachside,Computetheintegralintheintervalof0-,0+onbothsidesofequation(2)toobtain,Analogouslytoequation(3)toobtain,Consequently,Thenfrom,Weobtain,So,Then,Example2.9,ConsideracausalLTIsystemdescribedby,Determinetheunitsampleresponsehn.,Forn0,hnsatisfiesthedifferenceequation,Andthereshouldbe,Substitutinghnforynandnforxnintheoriginaldifferenceequation,andletn=0,weobtain,Itsobviousthat,Takinguseofh0,wemakeoutthecoefficientinhn:,So,Infact,forn=0,h0alsosatisfy,Thus,wecanwrite,Youmayalsotrytherecursivemethodtoobtainthehnforthissystem!,2.7BlockDiagramRepresentationsofFirst-OrderSystemsDescribedbyDifferentialandDifferenceEquations,First-orderdifferenceequation:,addition,delay,multiplication,Threebasicelementsinblockdiagram:adder,multiplieranddelayer.(方框图)(加法器)(乘法器)(延时器),Basicelementsfortheblockdiagramrepresentationofcausaldiscrete-timesystems.(a)anadder(b)amultiplier(c)adelayer.,Blockdiagramrepresentationforthecausaldiscrete-timesystemdescribedbythefirst-ord