Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 31

We can modify the last equation in (2.20) as

y k ¼ 0 : 2596 ð 0 : 7961w k 1 0 : 8347w k 2 þ u k Þþ 0 : 5193w k 1 þ 0 : 2596w k 2 :

ð 2 : 21 Þ

Now the modified set is

w k ¼ 0 : 7961w k 1 0 : 8347w k 2 þ u k ;

w k 1 ¼

þ þ

0u k ;

ð 2 :

22 Þ w k 1 0w k 2

¼ 0 :

7260w k 1 þ 0

:

0429w k 2 þ 0

:

2596u k : y k

This can be represented in standard A; B; C; D matrix form as

¼

w k 1
w k 2

þ

u k ;

w k
w k 1

0 : 7961 0 : 8347 1

0

ð 2 : 23 Þ

1 0

þ 0

w k 1
w k 2

y k ¼ 0 ð : 7260 0 : 0429 Þ : 2596u k : ð 2 : 24 Þ

The above difference equation is shown in Figure 2.7, which gives the minimum
delay realisation. In brief, w k ¼ Aw k 1 þ B u k and y k ¼ C t w k 1 þ D u k :

Figure 2.7 Digital filter minimum delay realisation