信号与系统信号与系统信号与系统 (20).ppt
BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Zero Input ResponseUnit Impulse ResponseZero State ResponseResponse of D-T LTI Systems1.Definition of the ZI ResponseMathematical model:In case the input signal is zero,the response caused solely by the initial state is defined as zero-input response,denoted by yzik.Steps:(1)find the characteristic roots of the difference equation(DE)(2)determine the general format of the ZI response(3)evaluate the undetermined coefficients using the initial state2.Solving the ZI Response(1)Case of distinct real roots r1,r2,rn(2)Case of repeated real roots r1=r2=rn(3)Case of complex rootsThe format of the ZI response yzik2.Solving the ZI ResponseExample 1:Given a D-T LTI systemyk+3yk-1+2yk-2=xk,k0,and the initial state y-1=0,y-2=1/2,find the ZI response yzik.Solutions:characteristic equationcharacteristic rootsC1=1，C22(distinct real roots)Solutions:characteristic equationcharacteristic rootsC1=C2=-2Two repeated rootsExample 2:Given a D-T LTI systemyk+4yk-1+4yk-2=xk,k0,and the initial state y-1=0,y-2=1/2,find the ZI response yzik.characteristic rootsSolutions:characteristic equation(complex roots)Example 3:Given a D-T LTI systemyk+2yk-1+2yk-2=xk,k0,and the initial state y-1=0,y-2=1/2,find the ZI response yzik.Unit Impulse Response of D-T LTI SystemsuDefinition of the unit impulse responseuEvaluation of the unit impulse response1.Definition of the Unit Impulse ResponseIn case the initial state is zero,the systems response to the unit impulse sequence k is the systems unit impulse response,denoted by hk.khkd(the initial state is zero)By definition,the unit impulse response hk satisfiesRecall that a D-T LTI system can be representedby a linear constant-coefficient difference equation1.Definition of the Unit Impulse Response2.Solving Unit Impulse ResponseExample 4:Given a D-T LTI systems DE Find the systems unit impulse response hk.Hint:In case xk is dk,the output yk is hk.Therefore,the unit impulse response is a homogeneous solutionSince dk=0 if k 0,the DE is2.Solving Unit Impulse ResponseWe call the method equivalent initial conditions.The initial conditions can be recursively obtained from the DE and the causality,i.e.,h1=h2=hn=0.How to get the undetermined coefficients?As usual,a feasible way is taking advantage of the initial conditions,i.e.,h0,h1,to determine the coefficients.Solutions:hk satisfies(1)Determine the format of hkCharacteristic equationCharacteristic roots2.Solving Unit Impulse ResponseExample 4:Given a D-T LTI systems DE Find the systems unit impulse response hk.(2)Find the equivalent initial conditions For a causal system,h1=h2=0,put into the DE,2nd order system needs two conditions,h0 and h1.2.Solving Unit Impulse ResponseSolutions:hk satisfiesExample 4:Given a D-T LTI systems DE Find the systems unit impulse response hk.(3)Determine the coefficientsThe initial conditionsC1=1，C2=22.Solving Unit Impulse ResponseSolutions:hk satisfiesExample 4:Given a D-T LTI systems DE Find the systems unit impulse response hk.SummarySteps for solving the ZI response:(1)find the characteristic roots of the difference equation(DE);(2)determine the general format of the ZI response;(3)evaluate the undetermined coefficients using the initial stateDistinct real roots Repeated real roots Complex rootsSummarySteps for solving the impulse response:(1)Determine the format of hk(2)Find the equivalent initial conditions(3)Determine the undetermined coefficientsThe initial conditions can be recursively obtained from the difference equation and the causality.AcknowledgmentsMaterials used here are accumulated by authors for years with help from colleagues,media or other sources,which,unfortunately,cannot be noted specifically.We gratefully acknowledge those contributors.Response of D-T LTI Systems
