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Spectral Audio Signal Processing
Bilinear Frequency-Warping for Audio Spectrum Analysis over a Bark Frequency Scale
The Bilinear Transform

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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 [37, page 75]

$\displaystyle {(\zeta -\zeta _1)(\zeta _2-\zeta _3) \over (\zeta _2-\zeta _1)(\zeta -\zeta _3)} = {(z-z_1)(z_2-z_3) \over (z_2-z_1)(z-z_3)}.$

It can be seen that choosing three specific points and their images determines the mapping for all 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 `` 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 (E.6) leaves us with transformations of the form

$\displaystyle z= {\cal A}_{\rho }(\zeta ) = {\zeta + \rho \over 1 + \zeta \rho } , \qquad \rho = {\zeta _3 - z_3 \over 1 - z_3\zeta _3}.$

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

$\displaystyle \rho = {\sin\{[a(\omega )-\omega ]/2\} \over \sin\{[a(\omega )+\omega ]/2\} },$

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 plane to the $\zeta$ plane.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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