The Bilinear Transform

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The Bark and ERB Bilinear Transforms

The Bark Frequency Scale

Optimal Bark Warping

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Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

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A conformal map is a complex-valued function of a complex variable that is an angle-preserving mapping from one complex plane to another. That is, the mapping preserves the angles between intersecting curves in the complex plane. — Click for http://mathworld.wolfram.com/ConformalMapping.html

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The Bark and ERB Bilinear Transforms

The Bark Frequency Scale

Optimal Bark Warping

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]

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

(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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``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 © 2020-07-19 by Julius O. Smith III and Jonathan S. Abel
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University