Digital Signal Processing Reference
In-Depth Information
z
Unit circle
Stable
region
1
Figure 11.8
Stable region of the
z
-plane.
The stability criterion can be stated in a different way. If a system is stable, the
poles of its transfer function are restricted to the interior of the unit circle.
Because
h
[
n
] is a causal function, the region of convergence of includes the
unit circle and the entire region of the finite plane outside the unit circle, as illus-
trated in Figure 11.9. Hence, the stability criteria can also be stated as follows:
H(z)
H(z)
An LTI discrete-time causal system is BIBO stable, provided that the region of con-
vergence of its transfer function includes the unit circle.
The
system characteristic equation
is, by definition, the denominator polyno-
mial of the transfer function set to zero. Hence, the characteristic equation is the
denominator of (11.44) set to zero; that is,
Á
+ a
N- 1
z + a
N
= a
0
(z - p
1
)(z - p
2
)
Á
(z - p
n
) = 0
a
0
z
N
+ a
1
z
N- 1
+
(11.57)
z
Unit circle
Poles
of
H
(
z
)
1
Figure 11.9
Region of convergence of a
stable system.
