Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 325

EXAMPLE 8.8
a. Design a first-order highpass digital Chebyshev filter with a cutoff frequency of 3 kHz and 1 dB ripple on the
passband using a sampling frequency of 8,000 Hz.
b. Use MATLAB to plot the magnitude and phase responses.
Solution:
a. First, we obtain the digital frequency in radians per second:

u d ¼ 2pf ¼ 2pð3; 000Þ¼6; 000p rad=sec; and T ¼ 1=f s ¼ 1=8; 000 sec

Following the steps of the design procedure, we have

1.

u d T
2

6; 000p=8; 000
2

u a ¼ T tan ¼ 3:8627 10 4 rad=sec ¼ 16; 000 tan

2. Since the filter order is given as 1, we select the first-order lowpass prototype from Table 8.5 :
H P ðsÞ¼ 1:9625
s þ 1:9625

Applying the prototype transformation (lowpass to highpass), we obtain

1:9625 1:9625s

1:9625s þ 3:8627 10 4

HðsÞ¼H P ðsÞj u s ¼

s þ 1:9625 ¼

u a

Dividing both numerator and denominator by 1.9625 gives

s
s þ 1:9683 10 4

HðsÞ¼

3. Using the BLT, we have

s¼16;000ðz1Þ=ðzþ1Þ

s
s þ 1:9683 10 4

HðzÞ¼

The algebra work proceeds as follows:

z 1
z þ 1

16; 000

HðzÞ¼

z 1
z þ 1 þ 1:9683 10 4

16; 000

Simplifying the transfer function yields
HðzÞ¼ 0:4484 0:4484 z 1
1 þ 0:1032z 1

b. Steps 2 and 3 and frequency response plots shown in Figure 8.17 can be carried out using MATLAB Program 8.4.

Program 8.4. MATLAB program for Example 8.8.
% Example 8.8
% Design of the digital highpass Butterworth filter