Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 71

Notice that this impulse response hðnÞ contains an infinite number of terms in its duration due to the
past output term yðn 1 Þ . Such a system as described in the preceding example is called an infinite
impulse response (IIR) system, which will be studied in later chapters.

3.4 BOUNDED-IN AND BOUNDED-OUT STABILITY
We are interested in designing and implementing stable linear systems. A stable system is one for
which every bounded input produces a bounded output (BIBO). There are many other stability defi-
nitions. To find the stability criterion, consider the linear time-invariant representation with all the
inputs reaching the maximum value M for the worst case. Equation (3.15) becomes

yðnÞ¼Mð / þ hð 1 Þþhð 0 Þþhð 1 Þþhð 2 Þþ / Þ (3.16)

Using the absolute values of the impulse response leads to

yðnÞ < Mð / þ jhð 1 Þj þ jhð 0 Þj þ jhð 1 Þj þ jhð 2 Þj þ / Þ (3.17)

If the absolute sum in Equation (3.17) is a finite number, the product of the absolute sum and the
maximum input value is therefore a finite number. Hence, we have a bounded input and bounded
output. In terms of the impulse response, a linear system is stable if the sum of its absolute impulse
response coefficients is a finite number. We can apply Equation (3.18) to determine whether a linear
time-invariant system is stable or not stable, that is,

N

S ¼ jhðkÞj ¼ / þ jhð 1 Þj þ jhð 0 Þj þ jhð 1 Þj þ / < N (3.18)

k ¼ N

Figure 3.17 describes a linear stable system, where the impulse response decreases to zero in a finite
amount of time so that the summation of its absolute impulse response coefficients is guaranteed to be
finite.

h ()

( n )

n n

linear stable
system

FIGURE 3.17

Illustration of stability of the digital linear system.

EXAMPLE 3.9
Given the linear system in Example 3.8,
yðnÞ¼0:25yðn 1ÞþxðnÞ for n 0 and yð1Þ¼0