Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 198

where H 1 ðzÞ and H 2 ðzÞ are chosen to be

H 1 ðzÞ¼ 0:5 0:5 z 1
1 þ 0:4z 1

H 2 ðzÞ¼ 1 þ z 1
1 þ 0:9z 1

Notice that the obtained H 1 ðzÞ and H 2 ðzÞ are not the unique selections for realization. For example, there is another
way of choosing H 1 ðzÞ¼ 0:5 0:5 z 1

1 þ 0:9z 1 and H 2 ðzÞ¼ 1 þ z 1 1 þ 0:4z 1 to yield the same HðzÞ. Using the H 1 ðzÞ and

H 2 ðzÞ we have obtained, and with the direct-form II realization, we achieve the cascade formdepicted in Figure 6.28 .

FIGURE 6.28
Cascade realization for Example 6.13.

The difference equations for the direct-form II realization have two cascaded sections, expressed as
Section 1:

w 1 ðnÞ¼xðnÞ0:4wðn 1Þ

y 1 ðnÞ¼0:5w 1 ðnÞ0:5w 1 ðn 1Þ

Section 2:

w 2 ðnÞ¼y 1 ðnÞ0:9w 2 ðn 1Þ

yðnÞ¼w 2 ðnÞþw 2 ðn 1Þ

c. In order to yield the parallel form of realization, we need to make use of the partial fraction expansion, and we
first let

0:5ðz 2 1Þ
zðz þ 0:4Þðz þ 0:9Þ ¼

H ð z z ¼ A

z þ

B
z þ 0:4 þ

C
z þ 0:9

where

z¼0 ¼

z¼0 ¼1:39

A ¼ z

0:5ðz 2 1Þ
zðz þ 0:4Þðz þ 0:9Þ

0:5ðz 2 1Þ
ðz þ 0:4Þðz þ 0:9Þ

z¼0:4 ¼ 0:5ð z 2 1Þ z¼0:4 ¼ 2:1

0:5ðz 2 1Þ
zðz þ 0:4Þðz þ 0:9Þ

B ¼ðz þ 0:4Þ

zðz þ 0:9Þ

0:5 z 2 1
zðz þ 0:4Þðz þ 0:9Þ

z¼0:9 ¼ 0:5 z 2 1

z þ 0:9 z¼0:9 ¼0:21

C ¼

zðz þ 0:4Þ