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426 Chapter 8 IIR FILTER DESIGN
2. Design an analog filter H a (s) using the specifications Ω p ,Ω s , R p , and
A s . This can be done using any one of the three (Butterworth, Chebyshev,
or elliptic) prototypes of the previous section.
3. Using partial fraction expansion, expand H a (s) into
H a (s) =
N∑
k=1
R k
s − p k
4. Now transform analog poles {p k } into digital poles {e pkT } to obtain
the digital filter:
N∑ R k
H(z) =
1 − e p kT
z −1 (8.64)
k=1
□ EXAMPLE 8.9 Transform
H a(s) =
s +1
s 2 +5s +6
into a digital filter H(z) using the impulse invariance technique in which
T =0.1.
Solution
We first expand H a(s) using partial fraction expansion:
H a(s) =
s +1
s 2 +5s +6 = 2
s +3 − 1
s +2
The poles are at p 1 = −3 and p 2 = −2. Then from (8.64) and using T =0.1,
we obtain
H(z) =
2
1 − e −3T z −1 − 1
1 − e −2T z −1 = 1 − 0.8966z −1
1 − 1.5595z −1 +0.6065z −2
It is easy to develop a MATLAB function to implement the impulse invariance
mapping. Given a rational function description of H a(s), we can use the
residue function to obtain its pole-zero description. Then each analog pole is
mapped into a digital pole using (8.63). Finally, the residuez function can be
used to convert H(z) into rational function form. This procedure is given in the
function imp invr.
function [b,a] = imp_invr(c,d,T)
% Impulse Invariance Transformation from Analog to Digital Filter
% ---------------------------------------------------------------
% [b,a] = imp_invr(c,d,T)
% b = Numerator polynomial in z^(-1) of the digital filter
% a = Denominator polynomial in z^(-1) of the digital filter
% c = Numerator polynomial in s of the analog filter
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