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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