# 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/omegaa to give a new T/F G(s/omegaa). 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).