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466 Chapter 8 IIR FILTER DESIGN
P8.24 In this problem we will develop a technique called the step invariance transformation. In
this technique, the step response of an analog prototype filter is preserved in the resulting
digital filter; i.e., if v a(t) isthe step response of the prototype and if v(n) isthe step
response of the digital filter, then
v(n) = v a(t)| t=nT
,
T : sampling interval
Note that the frequency-domain quantities are related by
V a(s) △ = L [v a(t)] = H a(s)/s
and
V (z) = △ 1
Z [v(n)] = H(z)
1 − z −1
Hence the step invariance transformation steps are: Given H a(s)
• divide H a(s) bys to obtain V a(s),
• find residues {R k } and poles {p k } of V a(s),
• transform analog poles {p k } into digital poles { e p kT } where T is arbitrary,
• determine V (z) from residues {R k } and poles { e p kT } , and finally
• determine H(z) bymultiplying V (z) by ( 1 − z −1) .
Use the above procedure to develop a MATLAB function to implement the step invariance
transformation. The format of this function should be
function [b,a] =stp_invr(c,d,T)
% Step Invariance Transformation from Analog to Digital Filter
% [b,a] =stp_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
% d = Denominator polynomial in s of the analog filter
% T = Sampling (transformation) parameter
P8.25 Design the lowpass Butterworth digital filter of Problem P8.22 using the step invariance
method. Plot the log-magnitude response in dB and compare it with that in Problem P8.22.
Plot the step response v(n) and the impulse response v a(t) ofthe analog prototype and
compare their shapes.
P8.26 In this chapter we discussed a filter transformation technique called the matched-z
transformation. Using (8.69) write a MATLAB function called mzt that maps the analog
system function H a(s) into the digital system function H(z). The format of the function
should be
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