Matched Z, Introduction.

Cuthbert Nyack
This approach maps the poles and zeros in the s plane to poles and zeros in the Z plane using the following relations.

Application to a second order Butterworth low pass filter.

The T/F function of a Butterworth lowpass 2nd order filter is shown below.
Applying the above algorithm gives the following sampled T/F. A "normalising" constant may be required to make the gain at zero frequency equal to 0dB.
This is one of the simplest approaches to use. The Butterworth filter is often described as allpole, however if only the poles are matched then the sampled response as the sampling frequency is increased is poor. Butterworth filters effectively have zeros at infinity and if these are matched(shown in green above) then an improved response results.
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COPYRIGHT © 1999 Cuthbert A. Nyack.