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436 Chapter 8 IIR FILTER DESIGN
□ EXAMPLE 8.16 Transform the system function H a(s) inExample 8.15 using the bilinear function.
Solution
MATLAB script:
>> c = [1,1]; d = [1,5,6]; T = 1; Fs = 1/T;
>> [b,a] = bilinear(c,d,Fs)
b = 0.1500 0.1000 -0.0500
a = 1.0000 0.2000 0.0000
The filter is
H(z) = 0.15+0.1z−1 − 0.05z −2
1+0.2z −1
as before.
□
8.4.4 DESIGN PROCEDURE
Given digital filter specifications ω p , ω s , R p , and A s ,wewant to determine
H(z). The design steps in this procedure are the following:
1. Choose a value for T . This is arbitrary, and we may set T =1.
2. Prewarp the cutoff frequencies ω p and ω s ; that is, calculate Ω p and Ω s
using (8.67):
Ω p = 2 (
T tan ωp
)
, Ω s = 2 (
2
T tan ωs
)
(8.68)
2
3. Design an analog filter H a (s) tomeet the specifications Ω p ,Ω s , R p ,
and A s .Wehave already described how to do this in the previous
section.
4. Finally, set
H(z) =H a
( 2
T
1 − z −1 )
1+z −1
and simplify to obtain H(z) asarational function in z −1 .
In the next several examples we demonstrate this design procedure
on our analog prototype filters.
□ EXAMPLE 8.17 Design the digital Butterworth filter of Example 8.11. The specifications are
ω p =0.2π, R p =1dB
ω s =0.3π, A s =15dB
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