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The Bilinear Transform

The formula for a general first-order (bilinear) conformal mapping of functions of a complex variable is conveniently expressed by [3, page 75]

\begin{displaymath}
{(\zeta -\zeta _1)(\zeta _2-\zeta _3) \over
(\zeta _2-\zeta ...
...\zeta -\zeta _3)} =
{(z-z_1)(z_2-z_3) \over
(z_2-z_1)(z-z_3)}.
\end{displaymath} (2)

It can be seen that choosing three specific points and their images determines the mapping for all $z$ and $\zeta $.

Bilinear transformations map circles and lines into circles and lines (lines being viewed as circles passing through the point at infinity). In digital audio, where both domains are ``$z$ planes,'' we normally want to map the unit circle to itself, with dc mapping to dc ( $z_1=\zeta _1=1$) and half the sampling rate mapping to half the sampling rate ( $z_2=\zeta _2=-1$). Making these substitutions in Eq. (2) leaves us with transformations of the form

\begin{displaymath}
z= {\cal A}_{\rho }(\zeta ) = {\zeta + \rho \over 1 + \zeta \rho } , \qquad
\rho = {\zeta _3 - z_3 \over 1 - z_3\zeta _3}.
\end{displaymath} (3)

The constant $\rho $ provides one remaining degree of freedom which can be used to map any particular frequency $\omega $ (corresponding to the point $e^{j\omega }$ on the unit circle) to a new location $a(\omega )$. All other frequencies will be warped accordingly. The allpass coefficient $\rho $ can be written in terms of these frequencies as
\begin{displaymath}
\rho = {\sin\{[a(\omega )-\omega ]/2\} \over \sin\{[a(\omega )+\omega ]/2\} },
\end{displaymath} (4)

In this form, it is clear that $\rho $ is real and that the inverse of ${\cal A}_{\rho }$ is ${\cal A}_{-\rho }$. Also, since $0\leq\{\omega ,a(\omega )\}\leq\pi$, and $a(\omega )\geq\omega $ for a Bark map, we have $\rho \in[0,1)$ for a Bark map from the $z$ plane to the $\zeta $ plane.


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Download bbt.pdf

``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.
Copyright © 2007-05-10 by Julius O. Smith III and Jonathan S. Abel
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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