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The Bark and ERB Bilinear Transforms

Auditory Filter Banks

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Audition is the act of listening. — Click for http://www.google.com/search?q=auditory

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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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

A frequency warp is a non-uniform scaling of frequency. Frequency warps are commonly used in audio signal processing in order to better mimic the non-uniform frequency resolution of human hearing. — Click for https://ccrma.stanford.edu/~jos/bbt/Filter_Design_Example.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

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Hertz means 'cycles per second', and is abbreviated 'Hz'. It is the standard name for frequency units in cycles per second. Another common frequency unit is 'radians per second', which is two-pi times frequency in Hertz. — Click for http://en.wikipedia.org/wiki/Hertz

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Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

Psychoacoustics is the science of the human perception of sound. — Click for http://en.wikipedia.org/wiki/Psychoacoustics

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

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JOS Home Page

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The Bark and ERB Bilinear Transforms

Auditory Filter Banks

Introduction

With the increasing use of frequency-domain techniques in audio signal processing applications such as audio compression, there is increasing emphasis on psychoacoustic-based spectral measures [36,2,11,12]. One of the classic approaches is to analyze and process signal spectra over the Bark frequency scale (also called ``critical band rate'') [41,42,39,21,7]. Based on the results of many psychoacoustic experiments, the Bark scale is defined so that the critical bands of human hearing have a width of one Bark. By representing spectral energy (in dB) over the Bark scale, a closer correspondence is obtained with spectral information processing in the ear.

The bilinear conformal map, defined by the substitution

$\begin{displaymath} z= {\cal A}_{\rho }(\zeta ) \mathrel{\stackrel{\mathrm{\Delta}}{=}}{\zeta + \rho \over 1 + \zeta \rho } \end{displaymath}$

(1)

takes the unit circle in the z plane to the unit circle in the $\zeta$ plane² in such a way that, for $0<\rho<1$ , low frequencies are stretched and high frequencies are compressed, as in a transformation from frequency in Hertz to the Bark scale. Because the conformal map ${\cal A}_{\rho }(\zeta )$ is identical in form to a first-order allpass transfer function (having a pole at $\zeta =-1/\rho$ ), we also call it the first-order allpass transformation, and $\rho$ the allpass coefficient.

Since the allpass mapping possesses only a single degree of freedom, we have no reason to expect a particularly good match to the Bark frequency warping, even for an optimal choice of $\rho$ . It turns out, however, that the match is surprisingly good over a wide range of sampling rates, as illustrated in Fig.1 for a sampling rate of 31 kHz. The fit is so good, in fact, that there is almost no difference between the optimal least-squares and optimal Chebyshev approximations, as the figure shows. The purpose of this paper is to spread awareness of this useful fact and to present new methods for computing the optimal warping parameter $\rho$ as a function of sampling rate.

**Figure 1:** Bark and allpass frequency warpings at a sampling rate of 31 kHz (the highest possible without extrapolating the published Bark scale bandlimits). a) Bark frequency warping viewed as a conformal mapping of the interval $[0,\pi ]$ to itself on the unit circle. b) Same mapping interpreted as an auditory frequency warping from Hz to Barks; the legend shown in plot a) also applies to plot b). The legend additionally displays the optimal allpass parameter $\rho$ used for each map. The discrete band-edges which define the Bark scale are plotted as circles. The optimal Chebyshev (solid), least-squares (dashed), and weighted equation-error (dot-dashed) allpass parameters produce mappings which are nearly identical. Also plotted (dotted) is the mapping based on an allpass parameter given by an analytic expression in terms of the sampling rate, which will be described. It should be pointed out that the fit improves as the sampling rate is decreased.
$\includegraphics[scale=0.8]{eps/fitlogf}$

Subsections

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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