Talk:Digital Signal Processing/Bilinear Transform

Bilinear Conversion
Provided sampling rate is not far in excess of the largest frequency component, a quick term-to-term conversion can use as approximation:

$$ z-term= e^{-a*T}*[1-A*(4*a/w_s)^B]  $$

where Ws=sampling rate <Nyquist rate (rad/s) T=sampling period, sec =2PI/Ws a=s-term, real or complex 4a/Ws <=1       (A,B)=f{specific problem}

for example,(A,B)=(0.85,3.17) the s-term (s+5.0) with T=2PI/40 yields z-term (z-0.4129). Similarly, term s^2+2.31s+2.72 becomes z^2-1.64z+0.70

there are existing rules for introducing additional z-terms as needed to stabilize and improve magnitude match.

For moderate sampling rate the bilinear transform sees a divergence region for frequency response towards the Nyquist limit.

This can be improved by applying a slightly increased sampling rate up to the original Nyquist limit, at least with well conditioned transfer functions ie positive real.--Billymac00 21:05, 12 December 2006 (UTC)

Filter design
Perhaps an example could be added to illustrate how to use it? Especially the frequency prewarping technique.

Wording
"...The resulting filter will have the same characteristics of the original filter..." This is not precisely true, as the typical scenario is solid match on magnitude but not necessarily on phase unless the transfer function is positive real.--Billymac00 (discuss • contribs) 01:57, 7 January 2015 (UTC)

Al-Alaoui variant
Al-Alaoui has introduced a variant of the bilinear as follows:
 * $$s = \frac{2(z-1)}{T(z(1+q)+(1-q))}$$\where 0<=q=<1

I have not scrutinized it closely, especially for phase match... --Billymac00 (discuss • contribs) 01:48, 7 February 2015 (UTC)