Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 361

8.8.2 Realization of Higher-Order Infinite Impulse Response Filters via the
Cascade Form

EXAMPLE 8.23
Given a fourth-order filter transfer function designed as
HðzÞ¼ 0:5108 z 2 þ 1:0215 z þ 0:5108
z 2 þ 0:5654z þ 0:4776

0:3730 z 2 þ 0:7460 z þ 0:3730
z 2 þ 0:4129z þ 0:0790

realize the digital filter using the cascade (series) form via second-order sections.
Solution:
Since the filter is designed using the cascade form, we have two sections of the second-order filters, whose transfer
functions are

H 1 ðzÞ¼ 0:5108 z 2 þ 1:0215 z þ 0:5108
z 2 þ 0:5654z þ 0:4776

¼ 0:5180 þ 1:0215 z 1 þ 0:5108 z 2
1 þ 0:5654z 1 þ 0:4776z 2

and

H 2 ðzÞ¼ 0:3730 z 2 þ 0:7460 z þ 0:3730
z 2 þ 0:4129z þ 0:0790

¼ 0:3730 þ 0:7460 z 1 þ 0:3730 z 2
1 þ 0:4129z 1 þ 0:0790z 2

Each filter section is developed using the direct-form I realization, shown in Figure 8.39 .

0 51 08 . yn

1 ()

037 30 .

x ()

y(n)

1

05654 .

1 1

04129 .

1

z z z 0 7460 . z

1 0215 .

+ +

1 1 1 1

z z z z

0 5108 . 04776 .

03730 . 00790 .

FIGURE 8.39

Cascade realization of IIR filter in Example 8.23 in direct-form I.

There are two sets of DSP equations for implementation of the first and second sections, respectively.
First section:

y 1 ðnÞ ¼0:5654y 1 ðn 1Þ 0:4776y 1 ðn 2Þ
þ0:5108xðnÞþ1:0215xðn 1Þþ0:5108xðn 2Þ

Second section:

yðnÞ¼0:4129yðn 1Þ0:0790yðn 2Þ
þ0:3730y 1 ðnÞþ0:7460y 1 ðn 1Þþ0:3730y 1 ðn 2Þ

Again, after we use the direct-form II for realizing each second-order filter, the realization shown in Figure 8.40 is
developed.