Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 193

6.6.1 Direct-Form I Realization
As we know, a digital filter transfer function, HðzÞ , is given by

AðzÞ ¼ b 0 þ b 1 z 1

þ/þ b M z M

HðzÞ¼ BðzÞ (6.26)

1 þ a 1 z 1 þ/þ a N z N

Let xðnÞ and yðnÞ be the digital filter input and output, respectively. We can express the relationship in
z-transform domain as

YðzÞ¼HðzÞXðzÞ (6.27)

where XðzÞ and YðzÞ are the z-transforms of xðnÞ and yðnÞ , respectively. If we substitute Equation
(6.26) into HðzÞ in Equation (6.27) , we have

b 0 þ b 1 z 1

þ/þ b M z M

YðzÞ¼ XðzÞ (6.28)

1 þ a 1 z 1 þ/þ a N z N

Taking the inverse of the z-transform of Equation (6.28) , we yield the relationship between input xðnÞ
and output yðnÞ in the time domain, as follows:

yðnÞ¼b 0 xðnÞþb 1 xðn 1 Þþ/þ b M xðn MÞ

(6.29)

a 1 yðn 1 Þa 2 yðn 2 Þ/ a N yðn NÞ

This difference equation thus can be implemented by the direct-form I realization shown in
Figure 6.22 ( a). Figure 6.22 (b) illustrates the realization of the second-order IIR filter ( M ¼ N ¼ 2).
Note that the notation used in Figures 6.22 (a) and (b) are defined in Figure 22 (c) and will be applied for
discussion of other realizations.
Notice that any of the a j and b i can be zero, thus not all the paths need to exist for realization.

6.6.2 Direct-Form II Realization
Considering Equations (6.26) and (6.27) with N ¼ M , we can express

XðzÞ
AðzÞ

YðzÞ¼HðzÞXðzÞ¼ BðzÞ

AðzÞ XðzÞ¼BðzÞ

(6.30)

XðzÞ

¼ðb 0 þ b 1 z 1 þ/þ b M z M Þ

1 þ a 1 z 1 þ/þ a M z M

| {z }
WðzÞ