# Impulse Invariant Approximation, Introduction.

Cuthbert Nyack

In this approach, the sampled filter is designed so that its impulse response
is a sampled version of the analog filter's impulse response. The transfer
function G(s) is first represented as a sum of first order factors as shown below. The
decaying exponential impulse response resulting from each factor is sampled to
produce a set of sampled terms as shown for H(z). The factor *T* can be
put in so that the "area" under the analog impulse response is the same as that of the
sampled impulse response. Quadratic terms can also be used
or any approach which finds the Z transform of the inverse Laplace transform of
the analog T/F as shown by the Eqn, below.

## Application to a second order Butterworth low pass filter.

The analog T/F of a second order Butterworth Lowpass filter is shown below.
Applying the algorithm above gives the following sampled T/F.
This approach is very "laborious" and its frequency response is not particularly
good.

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COPYRIGHT © 1999 Cuthbert A. Nyack.