# Bilinear Transform, Introduction.

Cuthbert Nyack

In this approach, the digital "cut-off" frequency and sampling time are first
determined, then the analog filter transfer function is
prewarped by replacing s with s/omega_{a} to give a new T/F
G(s/omega_{a}).
The resulting T/F is now converted to a sampled T/F *H(z)* by using the
bilinear transform shown below. This transform maps the entire left half
of the s plane onto the unit circle in the Z plane. Because the left half
of the s plane is mapped to the interior of the unit circle in the Z
plane then stable analog systems give rise to stable sampled systems.
Since the entire imaginary axis of the s plane is mapped onto the circumference
of the unit circle in the Z plane, then there is considerable warping
of the frequency scale and this is why prewarping is necessary.

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

The Transfer function of a second order Butterworth lowpass filter is shown
above. Applying the above algorithm gives the following sampled T/F.
Despite its complicated appearance this is one of the simpler ways of
converting an analog T/F to a sampled one. An extra factor which makes it
relatively simple is the fact that if the analog T/F function is
available in factored form, then each factor can be converted separately.
For an nth order filter, the sampled T/F has an nth order zero at half
of the sampling frequency (z = -1).

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COPYRIGHT © 1999 Cuthbert A. Nyack.