Finite-Dimensional discrete Time LTI Systerns If we assume the system to be causal then the output y[n] can be recursively computed using yn]=-∑y{n-k]+ pk xIn-k k=10 k=1a provided do≠0 yn can be computed for all n>n knowing xn] and the initial conditions {no-1y{no-2],…,y{no Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 16 Finite-Dimensional DiscreteTime LTI Systems • If we assume the system to be causal, then the output y[n] can be recursively computed using provided • y[n] can be computed for all , knowing x[n] and the initial conditions d0 0 n no y[n 1], y[n 2],..., y[n N] o − o − o − [ ] [ ] [ ] 1 0 1 0 x n k d p y n k d d y n M k k N k k = − − + − = =
Total Solution Calculation The output response yn] of the Lti system described by N ∑dkyn-k]=∑Pkxn-k] k=0 k=0 can be computed as yn]=ycn +yiN Where is the complementary solution to the In] homogeneous difference equation obtained by setting xn]=0 Is the particular solution resulting from 17 Vpln i the specified input signal xInI Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 17 • The output response y[n] of the LTI system described by can be computed as where Total Solution Calculation = = − = − M k k N k k d y n k p x n k 0 0 [ ] [ ] [ ] [ ] [ ] c p y n y n y n = + [ ] c y n[ ] p y n is the complementary solution to the homogeneous difference equation obtained by setting is the particular solution resulting from the specified input signal x[n] x n[ ] 0 =
Computing the Complementary solution We assume that it is of the form y n]=n By substitution in the homogeneous equation, It Is ∑4m-]=∑4knk= k=0 k=0 2nN(d0+d12-+…+d1+dx)=0 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 18 • We assume that it is of the form • By substitution in the homogeneous equation, it is Computing the Complementary Solution [ ] n c y n = 0 0 1 0 1 1 [ ] ( ) 0 N N n k k k k k n N N N N N d y n k d d d d d − = = − − − − = = = + + + + =
Characteristic Polynomial din n-k The polynomial is called the characteristic polynomial of the given Lti system Let M,n 2, ...,AN denote its N roots Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 19 • The polynomial is called the characteristic polynomial of the given LTI system • Let denote its N roots Characteristic Polynomial 0 N n k k k d − = 1 2 , , , N
Complementary Solution If the n roots are distinct the complementar solution is expressed b y[n]=a41+a2+…+aMN where al,,, ..,aN are constants determined by the specified initial conditions of the dt system 20 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 20 • If the N roots are distinct, the complementary solution is expressed by where are constants determined by the specified initial conditions of the DT system Complementary Solution 1 1 2 2 [ ] n n n c N N y n = + + + 1 2 , , , N