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Bilinear Frequency-Warping for Audio Spectrum Analysis over Bark and ERB Frequency Scales

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 [274,17,113,118]. In particular, frequency warping is an important tool in spectral audio signal processing. For example, audio spectrograms (Chapter 7) can display signal energy versus time over a more perceptual, nonuniform, audio frequency axis (§7.3). Also, methods for digital filter design (Chapter 4) having no weighting function versus frequency, such as linear predictive coding (LPC) (§10.3), can be given an effective weighting function by means of frequency warping [278].

A common choice of audio frequency warping in audio applications is from a linear frequency scale to a Bark frequency scale (also called ``critical band rate'') [307,308,305,179,102,269]. The Bark scale is defined so that critical bands of hearing are uniformly spaced. (One critical bandwidth equals one Bark.)

A more recently developed psychoacoustic frequency scale, called the Equivalent Rectangular Bandwidth (ERB) scale [88], is based on different psychoacoustic experiments resulting in generally narrower critical bandwidth estimates.

This appendix, condensed from [269,268], describes a useful class approximate Bark/ERB frequency warpings that may be implemented using a bilinear transform (first-order conformal map of the unit circle to itself in the $ z$ plane). Such warpings preserve order in filter-design applications. That is, the warping can be undone by the inverse bilinear transform which, because its first order, does not change the order of the filter that was designed over the warped frequency axis.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University