华南理工大学：《数字信号处理》（双语版） Chapter 5 Stability Condition of a Discrete-Time LTI System

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

Causality Condition of a Discrete-Time LTI System Let xi[n] and x2In]be two input sequences i[]=x2n]forn≤no The corresponding output samples at n=n of an lti system with an impulse response thin are then given by Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 6 Causality Condition of a Discrete-Time LTI System • Let and be two input sequences with • The corresponding output samples at of an LTI system with an impulse response {h[n]} are then given by x [n] 1 x [n] 2 x [n] x [n] 1 = 2 n  no for n = no

Causality Condition of a Discrete-Time LTI System n[m]=∑k]x[-k]=∑k]x1[-k k=-∞ k=0 +∑hk]x1{m-k] k y2[no=∑hk]x2[m-k]=∑hkx2[m-k k: k=0 +∑hk]x2{o-k] Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 7 Causality Condition of a Discrete-Time LTI System    =  =− = − = − 0 2 2 2 k o k o o y [n ] h[k]x [n k] h[k]x [n k]  − =− + − 1 2 k o h[k]x [n k]    =  =− = − = − 0 1 1 1 k o k o o y [n ] h[k]x [n k] h[k]x [n k]  − =− + − 1 1 k o h[k]x [n k]

Causality Condition of a Discrete-Time LTI System If the lti system is also causal, then yIno]=y2lnol ASx1m]=x2 n for n≤no ∑h[kxo-k]=∑hk]x2{r-k] k=0 k=0 This implies ∑k-k]=∑kx2[r-k k: k 8 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 8 Causality Condition of a Discrete-Time LTI System • If the LTI system is also causal, then • As • This implies x [n] x [n] 1 = 2 n  no for [ ] [ ] o no y n y 1 = 2    =  = − = − 0 2 0 1 k o k o h[k]x [n k] h[k]x [n k]   − =− − =− − = − 1 2 1 1 k o k o h[k]x [n k] h[k]x [n k]

Causality Condition of a Discrete-Time LTI System As xi[n]*x2[n] for n>no the only way the condition ∑k]x[-k]=∑hk]x2{To-k] k will hold if both sums are equal to zero which is satisfied if h[k]=o for k<0 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 9 Causality Condition of a Discrete-Time LTI System • As for the only way the condition will hold if both sums are equal to zero, which is satisfied if x [n] x [n] 1  2 n  no   − =− − =− − = − 1 2 1 1 k o k o h[k]x [n k] h[k]x [n k] h[k] = 0 for k < 0

Causality Condition of a Discrete-Time LTI System a discrete-time lti system is causal if and only if its impulse response hn is a causal sequence Example- The discrete-time system defined yn]=1x{]+x2x{n-1]+a3x[n-2]+4x[n-3] is a causal system as it has a causal impulse response {h4n]}={a1a234} 10 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 10 Causality Condition of a Discrete-Time LTI System • A discrete-time LTI system is causal if and only if its impulse response {h[n]} is a causal sequence • Example - The discrete-time system defined by is a causal system as it has a causal impulse response [ ] [ ] [ 1] [ 2] [ 3] y n = 1 x n +2 x n − +3 x n − +4 x n − { [ ]} { } h n = 1 2 3 4 