Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 177

(a) (b)

(c) (d)

FIGURE 6.10

Pole-zero plots for Example 6.9.

z þ 0:5
ðz 0:5Þðz 2 þ 1:4141z þ 1Þ

c. HðzÞ¼

z 2 þ z þ 0:5
ðz 1Þ

d. HðzÞ¼

2

ðz þ 1Þðz 0:6Þ

For each, sketch the z-plane pole-zero plot and determine the stability status for the digital system.
Solution:
a. A zero is located at z ¼0:5.
Poles: z ¼ 0:5, jzj¼0:5 < 1; z ¼0:5 j0:5,
jzj ¼

q

ð0:5Þ

2 2

þð0:5Þ ¼ 0:707 < 1.

The plot of poles and a zero is shown in Figure 6.10 . Since the outmost poles are inside the unit circle, the system
is stable.
b. Zeros are z ¼j0:5.
Poles: z ¼ 0:5, jzj¼0:5 < 1; z ¼1:5 j0:5
jzj ¼

q

ð1:5Þ

2 2

þð0:5Þ ¼ 1:5811 > 1.

The plot of poles and zeros is shown in Figure 6.10 . Since we have two poles at z ¼1:5 j 0:5 that are outside
the unit circle, the system is unstable.
c. A zero is located at z ¼0:5.