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Bilinear Transformation

To digitize via the bilinear transform, we make the substitution

$\displaystyle s = c \frac{1-z^{-1}}{1+z^{-1}}
$

where $ c$ is any positive real constant (typically $ 2/T$ ).

For the ideal mass reflectance

$\displaystyle S_m(s) = \frac{ms-R_0}{ms+R_0}
$

the bilinear transform yields

$\displaystyle \tilde{S}_m(z) = \frac{p_m-z^{-1}}{1-p_mz^{-1}}
$

with

$\displaystyle p_m \isdef \frac{mc-R_0}{mc+R_0}
$

Note that $ \vert p_m\vert<1$ and $ \vert\tilde{S}_m(e^{j\omega T})\vert=1$ . The stable allpass nature of the digitized mass reflectance is preserved by the bilinear transform, as always.

Important Observation:

If we choose $ R_0=mc$ , then $ p_m=0$ and $ \tilde{S}_m(z) =
-z^{-1}\,\,\Rightarrow\,\,$ no delay-free path through the mass reflectance


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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2017-06-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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