Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 197

FIGURE 6.26

Direct-form I realization for Example 6.13.

Solution:
a. To perform the filter realizations using direct-form I and direct-form II, we rewrite the given second-order transfer
function as

0:5 0:5z 2
1 þ 1:3z 1 þ 0:36 z 2

HðzÞ¼

and identify that

a 1 ¼ 1:3; a 2 ¼ 0:36; b 0 ¼ 0:5; b 1 ¼ 0; and b 2 ¼0:5

Based on the realizations in Figure 6.22 , we sketch the direct-form I realization in Figure 6.26 .
The difference equation for the direct-form I realization is given by

yðnÞ¼0:5xðnÞ0:5xðn 2Þ1:3yðn 1Þ0:36yðn 2Þ

Using the direct-form II realization shown in Figure 6.23 , we present the realization in Figure 6.27 . The difference
equations for the direct-form II realization are expressed as

wðnÞ¼xðnÞ1:3wðn 1Þ0:36wðn 2Þ

yðnÞ¼0:5wðnÞ0:5wðn 2Þ

b. To achieve the cascade (series) form realization, we factor HðzÞ into two first-order sections to yield

1 þ 1:3z 1 þ 0:36z 2 ¼ 0:5 0:5z 1

0:5ð1 z 2 Þ 1 þ z 1

1 þ 0:9z 1

HðzÞ¼

1 þ 0:4z 1

FIGURE 6.27
Direct-form II realization for Example 6.13.