Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 324

Then multiplying ðz þ 1Þ on both numerator and denominator leads to

ðz 1Þ þ 0:6682ðz þ 1Þ ¼ 0:6682 z þ 0:6682

0:6682ðz þ 1Þ

HðzÞ¼

1:6682z 0:3318

Dividing both numerator and denominator by ð1:6682$zÞ leads to
HðzÞ¼ 0:4006 þ 0:4006 z 1
1 0:1989z 1

b. Steps 2 and 3 can be carried out using MATLAB Program 8.3, as shown in the first three lines of the MATLAB
code. Figure 8.16 describes the filter frequency responses.

0

-5

-10

-15

-20

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hertz)

0

-20

-40

-60

-80

-100

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hertz)

FIGURE 8.16

Frequency responses of the designed digital filter for Example 8.7.

Program 8.3. MATLAB program for Example 8.7.
% Example 8.7
% Design of the digital lowpass Butterworth filter
format long
fs

¼ 8000; % Sampling rate

[B A] ¼ lp2lp([1],[1 1], 1.0691*10^4) % Complete step 2

bilinear(B,A,fs) % Complete step 3
% Plot the magnitude and phase responses j H(z) j
%b ¼ [0.4005 0.4005]; numerator coefficients from MATLAB
%a ¼ [1 -0.1989]; denominator coefficients from MATLAB
freqz(b,a,512,fs);
axis([0 fs/2 -20 1])

[b a] ¼