# 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.

*Return to main page*

*Return to page index*

COPYRIGHT © 1999 Cuthbert A. Nyack.