Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 173

EXAMPLE 6.8
Given the digital transfer function

z 1 0:5z 2
1 þ 1:2z 1 þ 0:45z 2

HðzÞ¼

plot poles and zeros.
Solution:
Converting the transfer function to its advanced form by multiplying the numerator and denominator by z 2 ,it
follows that

ðz 1 0:5z 2 Þz 2
ð1 þ 1:2z 1 þ 0:45z 2 Þz 2 ¼

z 0:5
z 2 þ 1:2z þ 0:45

HðzÞ¼

By setting z 2 þ 1:2z þ 0:45 ¼ 0 and z 0:5 ¼ 0, we obtain two poles
p 1 ¼0:6 þ j0:3
p 2 ¼ p 1 ¼0:6 j0:3
and a zero z 1 ¼ 0:5, which are plotted on the z-plane shown in Figure 6.6 . According to the form of Equation
(6.7) , we also yield the pole-zero form as

z 1 0:5z 2
1 þ 1:2z 1 þ 0:45z 2 ¼

ðz 0:5Þ
ðz þ 0:6 j0:3Þðz þ 0:6 þ j0:3Þ

HðzÞ¼

FIGURE 6.6
The z-plane pole-zero plot of Example 6.8.

With zeros and poles plotted on the z-plane, we are able to study system stability. We first establish
the relationship between the s-plane in the Laplace domain and the z-plane in the z-transform domain,
as illustrated in Figure 6.7 .