Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 360

x () 0 7 157 .

w () y ()

z

1

1349 . 14314 .

wn ( 1

1

z

0514 . wn ( 2 ) 0 7157 .

FIGURE 8.38

Realization of IIR filter in Example 8.22 in direct-form II.

Solution:
First, we can identify

b 0 ¼ 0:7157; b 1 ¼ 1:4314; b 2 ¼ 0:7151

and a 1 ¼ 1:3490; a 2 ¼ 0:5140

Applying the direct-form II realization developed in Chapter 6 leads to Figure 8.38 .
There are two difference equations required for implementation:
wðnÞ¼xðnÞ1:3490wðn 1Þ0:5140wðn 2Þ

yðnÞ¼0:7157wðnÞþ1:4314wðn 1Þþ0:7157wðn 2Þ

The MATLAB implementation is listed in Program 8.15.
Program 8.15. MATLAB code for Example 8.22.
% Sample MATLAB code
sample ¼ 2:2:20; % Input test array
x ¼ [0]; % Input buffer [x(n) ]
y ¼ [0]; % Output buffer [y(n)]
w ¼ [0 0 0]; % Buffer for w(n) [w(n) w(n-1) . ]
b ¼ [0.7157 1.4314 0.7157]; % Numerator coefficients [b0 b1 . ]

a ¼ [1 1.3490 0.5140]; % Denominator coefficients [1 a1 a2 . ]

for n ¼ 1:1:length(sample) % Processing loop

for k ¼ 3:-1:2
w(k) ¼ w(k-1); % Shift w(n) by one sample

end
x(1) ¼ sample(n); % Get new sample

w(1) ¼ x(1); % Perform IIR filtering

for k ¼ 2:1:3

w(1) ¼ w(1)-a(k)*w(k);

end
y(1) ¼ 0; % Perform FIR filtering

for k ¼ 1:1:3
y(1) ¼ y(1) þ b(k)*w(k);
end
out(n) ¼ y(1); % Send the filtered sample to output array

end
out