Finite-Dimensional Discrete Time LTI Systems If we assume the system to be causal, then the output yIn can be recursively computed using yn]=-∑y{n-k]+∑kxn-k k=140 k=100 provided do≠0 yIn] can be computed for all n2n knowing x[n] and the initial conditions Inolylno-2],,ylno-N 16 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 yIn] of the Lti system described b 在=01n k]=∑pkx[n-k] k=0 can be computed as yn]=ycIn]+yiN where is the complementary solution to the ye[n] homogeneous difference equation obtained by setting x[n=0 piNt is the particular solution resulting from the specified input signal xn 17 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 yn=n By substitution in the homogeneous equation, It Is 少N ∑akyn-k]=∑ k=0 k=0 N (d60+d12N1 +…+dM-1+d)=0 18 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 ∑4k2nk The polynomial ki=0 is calle d the characteristic polynomial of the given lti system Let入1,2,…, y denote its N roots 19 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 complementary solution is expressed by y[n]=011+a212+…+M where aj,a2, ...,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