Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 195

(a)

(b)

FIGURE 6.23

(a) Direct-form II realization; (b) direct-form II realization with M ¼ 2.

and

yðnÞ¼b 0 wðnÞþb 1 wðn 1 Þþ.þ b M wðn MÞ (6.34)

Realization of Equations (6.33) and (6.34) produces another direct-form II realization, which is
demonstrated in Figure 6.23 (a). Again, the corresponding realization of the second-order IIR filter is
described in Figure 6.23 (b). Note that in Figure 6.23 ( a), the variables wðnÞ , wðn 1 Þ , wðn 2 Þ , . ,
wðn MÞ are different from the filter inputs xðn 1 Þ , xðn 2 Þ , . , xðn MÞ .

6.6.3 Cascade (Series) Realization
An alternate way to filter realization is to cascade the factorized HðzÞ in the following form:
HðzÞ¼H 1 ðzÞ$H 2 ðzÞ/H k ðzÞ (6.35)

where H k ðzÞ is chosen to be the first- or second-order transfer function (section), which is defined by
H k ðzÞ¼ b k 0 þ b k 1 z 1

(6.36)

1 þ a k 1 z 1

or

H k ðzÞ¼ b k 0 þ b k 1 z 1

þ b k 2 z 2

(6.37)

1 þ a k 1 z 1 þ a k 2 z 2

respectively. The block diagram of the cascade, or series, realization is depicted in Figure 6.24 .