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424 Chapter 8 IIR FILTER DESIGN
the literature. These transformations are derived by preserving different
aspects of analog and digital filters. If we want to preserve the shape
of the impulse response from analog to digital filter, then we obtain a
technique called impulse invariance transformation. If we want to convert
a differential equation representation into a corresponding difference
equation representation, then we obtain a finite difference approximation
technique. Numerous other techniques are also possible. One technique,
called step invariance, preserves the shape of the step response; this is
explored in Problem P8.24. Another technique that is similar to the
impulse invariance is the matched-z transformation, which matches the
pole-zero representation. It is described at the end of this section and is
explored in Problem P8.26. The most popular technique used in practice
is called a Bilinear transformation, which preserves the system function
representation from analog to digital domain. In this section we will study
in detail impulse invariance and bilinear transformations, both of which
can be easily implemented in MATLAB.
8.4.1 IMPULSE INVARIANCE TRANSFORMATION
In this design method we want the digital filter impulse response to look
“similar” to that of a frequency-selective analog filter. Hence we sample
h a (t) atsome sampling interval T to obtain h(n); that is,
h(n) =h a (nT )
The parameter T is chosen so that the shape of h a (t) is“captured” by
the samples. Since this is a sampling operation, the analog and digital
frequencies are related by
ω =ΩT or e jω = e jΩT
Since z = e jω on the unit circle and s = jΩ onthe imaginary axis, we
have the following transformation from the s-plane to the z-plane:
z = e sT (8.63)
The system functions H(z) and H a (s) are related through the frequencydomain
aliasing formula (3.27):
H(z) = 1 T
∞∑
k=−∞
H a
(
s − j 2π
T k )
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