THE z-TRANSFORM - Signals, Systems, and Transforms - page 578

is the system characteristic equation. The system is stable, provided that the roots of
the system characteristic equation are inside the unit circle. A similar development
shows that the same requirements apply if H(z) has repeated poles. We now illus-

trate system stability with an example.

Stability of a discrete system EXAMPLE 11.12

Suppose that the transfer function of an LTI system is given by

2z 2 - 1.6z - 0.90

z 3 - 2.5z 2 + 1.96z - 0.48 .

H(z) =

The characteristic equation for this system is seen to be

z 3 - 2.5z 2 + 1.96z - 0.48 = (z - 0.5)(z - 0.8)(z - 1.2) = 0.

The poles of the transfer function are at 0.5, 0.8, and 1.2, as illustrated in Figure 11.10.
Thus, the system is unstable because the pole at is outside the unit circle. The
modes of the system are and the system is unstable, since is
unbounded. The characteristic-equation roots can be calculated with the following MATLAB
program:

z = 1.2

1.2 n

lim
n: q

0.5 n , 0.8 n , 1.2 n ;

P = [1 -2.5 1.96 -.48];
r = roots(p)
result: r = 1.2 0.8 0.5

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Invertibility

Recall from Section 9.6 the definition of the inverse of a system:

Inverse of a System
The inverse of a system H(z) is a second system H i (z) that, when cascaded with H(z),

yields the identity system.

z

Unit circle

0.5 0.8 1 1.2

Figure 11.10 Pole locations for Example

11.12.