Step Invariant Approximation, Introduction.

Cuthbert Nyack

In this approach the sampled filter is designed so its step response is a sampled version of the step response of the analog filter. The general equation used is shown below. G(s) is multiplied by 1/s to get the step response of the analog filter. The Inverse Laplace transform of G(s)/s is then found. The final step is to find the Z transform of the step response and multiply it by (z - 1)/z to get the sampled T/F. Multiplication by (z - 1)/z corresponds to passing the samples through a sample and hold.

Application to a second order Butterworth low pass filter.

THe Transfer Function of a second order Butterworth low pass filter is shown below.

Applying the above algorithm gives the following expression for H(z).

This is an improvement over the impulse invariant response approach but is one of the most laborious approaches mentioned here.

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