Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 172

6z
z 0:5 þ

5z
z 0:25

Y ðzÞ¼

Using Equation (6.11) and Table 5.1 in Chapter 5, we finally yield

n uðnÞ5ð0:25Þ n uðnÞ

yðnÞ¼Z 1 fY ðzÞg ¼ 6ð0:5Þ

The impulse response for (a), step response for (b), and system response for (c) are each plotted in Figure 6.4 .

6.3 THE Z-PLANE POLE-ZERO PLOT AND STABILITY
A very useful tool to analyze digital systems is the z-plane pole-zero plot. This graphical technique
allows us to investigate characteristics of the digital system shown in Figure 6.1 , including the system
stability. In general, a digital transfer function can be written in the pole-zero form as shown in
Equation (6.7) , and we can plot the poles and zeros on the z-plane. The z-plane is depicted in
Figure 6.5 and has the following features:

FIGURE 6.5

z-plane and pole-zero plot.

1. The horizontal axis is the real part of the variable z , and the vertical axis represents the imaginary
part of the variable z.
2. The z-plane is divided into two parts by a unit circle.
3. Each pole is marked on z-plane using the cross symbol x, while each zero is plotted using the small
circle symbol o.

Let's investigate the z-plane pole-zero plot of a digital filter system via the following example.