Stability Condition of a Discrete-Time LTI System BIBO Stability Condition-A discrete-time Lti System is bibo stable if the output sequence [n remains bounded for any bounded input sequence[nI a discrete-time LTI system is BiBO stable if and only if its impulse response sequence hn is absolutely summable, i.e S n<∞ 1=-0 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 LT System Proof: Assume h[n] is a real sequence Since the input sequence xn is bounded we have x{]≤Bx< Therefore yn=∑m-≤∑k]xn-k k Bx∑Hk=BxS k 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 Thus,S< oo implies v[n]≤B,<∞ indicating that y[n] is also bounded o prove the converse, assume that yn] is bounded,ie,ynl≤B Consider the input given by Sgn(H[-n]),ifh-n]≠0 K f hl-n=o 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 fc<0andK≤1 Note: Since xl叫]≤1,{x]} is obviously bounded For this input, yn] at n=0 is y0]=∑gn(]=S≤B,< k Therefore, y[n]<B, implies S< oo 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 hn]=(a)[m] For this system s=∑a"rn=∑ f a< =0 Therefore S<o if ak< l for which the system is bibo stable Ifal, the system is not biBO stable 5 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