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
Causality Condition of a Discrete- Time LTI System et xi[n] and xln be two input sequences 21」forn O The corresponding output samples atn=no of an lti system with an impulse response hnd 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[=∑hx1[no-k]=∑ hk ] iln-k k=-00 k=0 +∑k]x[-k k y2m]=∑kx2[no-k]=∑hkx2{no-k k=-00 k=0 +∑kx2[no-k] k=-∞0 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 O Asx[n]=x2[m]forn≤mo ∑k]x1[n0-k]=∑kx2[n- k=0 k=0 This implies ∑1{m-]=∑hk]x2{m-k 〓一00 k 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 ∑ h(k]xno k]=∑k]x2{-h k=-0o k=-0o 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 hin is a causal sequence Example- The discrete-time system defined y]=a1x1n]+a2x[n-1]+3xn-2]+a4xn-3] is a causal system as it has a causal impulse response hn]}={(10234 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
