Signals and Systems - Electrical Engineering

8.3 Discrete-Time Systems 501

An LTI discrete-time system is said to be BIBO stable if its impulse responseh[n]is absolutely summable,

∑

k

|h[k]|<∞ (8.39)

Assuming that the inputx[n] of the system is bounded, or that there is a valueM<∞such
that|x[n]|<Mfor alln, the outputy[n] of the system represented by a convolution sum is also
bounded, or

|y[n]|≤

∣ ∣ ∣ ∣ ∣ ∣

∑∞

k=−∞

x[n−k]h[k]

∣ ∣ ∣ ∣ ∣ ∣

≤

∑∞

k=−∞

|x[n−k]||h[k]|

≤M

∑∞

k=−∞

|h[k]|≤MN<∞

provided that

∑∞

k=−∞|h[k]|<N<∞, or that the impulse response be absolutely summable.

Remarks

n Nonrecursive or FIR systems are BIBO stable. Indeed, the impulse response of such a system is of finite
length and thus absolutely summable.
n For a recursive or IIR system represented by a difference equation, to establish stability we need to find the
system impulse response h[n]and determine whether it is absolutely summable or not.
n A much simpler way to test the stability of an IIR system will be based on the location of the poles of the
Z-transform of h[n], as we will see in Chapter 9.

nExample 8.26

Consider an autoregressive system

y[n]=0.5y[n−1]+x[n]

Determine if the system is BIBO stable.

Solution

As shown in Example 8.24, the impulse response of the system ish[n]=0.5nu[n]. Checking the

BIBO stability condition, we have

∑∞

n=−∞

|h[n]|=

∑∞

n= 0

0.5n=

1

1 −0.5

= 2

Thus, the system is BIBO stable. n