Stability Condition of a Discrete-Time LTI System BIBO Stability Condition-a discrete-time Lti System is bibo stable if the output sequence ln remains bounded for any bounded input sequencexn A discrete-time LTI system is BiBO stable if and only if its impulse response sequence thn is absolutely summable, i.e S=∑hn]<∞ Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 Stability Condition of a Discrete-Time LTI System • BIBO Stability Condition - A discrete-time LTI system is BIBO stable if the output sequence {y[n]} remains bounded for any bounded input sequence{x[n]} • A discrete-time LTI system is BIBO stable if and only if its impulse response sequence {h[n]} is absolutely summable, i.e. = n=− S h[n]
Stability Condition of a Discrete-Time LTI System Proof: Assume hn] is a real sequence Since the input sequence x[n]is bounded we ave xl≤Bx<∞ Therefore y小=∑m-≤∑klnk1 k k ≤Bx∑k]=B k=-o Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 2 Stability Condition of a Discrete-Time LTI System • Proof: Assume h[n] is a real sequence • Since the input sequence x[n] is bounded we have • Therefore x[n] Bx y[n] h[k]x[n k] h[k] x[n k] k k = − − =− =− x k Bx h k = B =− [ ] S
Stability Condition of a Discrete-Time LTI System Thu,S< implies v[n]≤B,<∞ indicating that yn is also bounded To prove the converse, assume that yn]is bounded, i. e,y川l≤B Consider the input given by xm=/Sm1-m)ifh-nl≠0 K fh[-n」]=0 Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 3 Stability Condition of a Discrete-Time LTI System • Thus, S < implies indicating that y[n] is also bounded • To prove the converse, assume that y[n] is bounded, i.e., • Consider the input given by y[n] By n By y[ ] − = − − = 0 0 , if [ ] sgn( [ ]), if [ ] [ ] K h n h n h n x n
Stability Condition of a Discrete-Time LTI System where sgn(c)=+l ifc>0 and sgn(c)=-1 ifc<0andK≤1 Note: Since x[n]<1,x[n])is obviously ounded For this input, yn at n=0 is y0]=∑gn(1]=S≤B<O Therefore ynl≤B, implies s<∞ Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Stability Condition of a Discrete-Time LTI System where sgn(c) = +1 if c > 0 and sgn(c) = if c < 0 and • Note: Since , {x[n]} is obviously bounded • For this input, y[n] at n = 0 is • Therefore, implies S < −1 K 1 n By y[ ] x[n] 1 =− = = k y[0] sgn(h[k])h[k] S By
Stability Condition of a Discrete-Time LTI System Example- Consider a causal discrete-time Lti System with an impulse response h{n]=(a)" For this system s=∑an=∑a if a< 0 Therefore S< oo if ak< 1 for which the system is BiBO stable If=l, the system is not BIBO stable Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 5 Stability Condition of a Discrete-Time LTI System • Example - Consider a causal discrete-time LTI system with an impulse response • For this system • Therefore if for which the system is BIBO stable • If , the system is not BIBO stable S | | 1 | | 1 = h[n] ( ) [n] n = − = = = = =− 1 1 n 0 n n n S [n] if 1