Ch2(3) Stability Condition of an LTI Discrete-Time System.pdf

Stability Condition of an LTI Stability Condition of an LTI
Discrete-Time System Discrete-Time System
• Proof: Assume h[n] is a real sequence
• BIBO Stability Condition - A discrete-
time is BIBO stable if the output sequence • Since the input sequence x[n] is bounded we
{y[n]} remains bounded for all bounded have
input sequence {x[n]} x[n] £ Bx < ¥
• An LTI discrete-time system is BIBO stable • Therefore
if and only if its impulse response sequence ¥ ¥
y[n] = å h[k]x[n ­ k] £ å h[k] x[n ­ k]
{h[n]} is absolutely summable, . k=­¥ k=­¥
¥ ¥
S = h[n] < ¥
å £ Bx å h[k] = Bx S
1 n=­¥ 2
Copyright © 2001, S. K. Mitra k =­¥ Copyright © 2001, S. K. Mitra
Stability Condition of an LTI Stability Condition of an LTI
Discrete-Time System Discrete-Time System
• Thus, S < ¥ implies y[n] £ By < ¥ where sgn(c) = +1 if c > 0 and sgn(c) = ­1
indicating that y[n] is also bounded if c < 0 and K £1
• To prove the converse, assume y[n] is • Note: Since x [ n ] £ 1 , {x[n]} is obviously
bounded, ., y[n] £ By bounded
• Consider the input given by • For this input, y[n] at n = 0 is
¥
ìsgn(h[­n]), if h[­n] ¹ 0
x[n] = í y[0] = åsgn(h[k])h[k] = S £ B y < ¥
î K , if h[­n] = 0 k=­¥
• Therefore, y [ n ] £ B y implies S < ¥
3 Copyright © 2001, S. K. Mitra 4 Copyright © 2001, S. K. Mitra
Stability Condition of an LTI Causality Condition of an LTI
D