Talk:Digital Signal Processing/Bilinear Transform

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[edit] 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 ^{− w * T} * [1 − A * (4 * w / w_s)^B]

       where Ws=sampling rate <Nyquist rate (rad/s)
               T=sampling period, sec =2PI/Ws
               w=s-term, real or complex
               4w/Ws <=1        (A,B)=(0.85,3.17)  

for example, 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)