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
[#!SporerAndBrandenburg95!#,#!Bosi96!#,#!MPEGSpec93!#,#!Matti85!#]. 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 [#!Strube80!#].

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'')
[#!ZwickerAndScharf65!#,#!ZwickerAndTerhardt80!#,#!Zwicker90!#,#!Moore97!#,#!Hartmann97!#,#!SmithAndAbel99!#].
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 [#!GlasbergAndMoore02!#], is based on different psychoacoustic experiments resulting in generally narrower critical bandwidth estimates.

This appendix, condensed from [#!SmithAndAbel99!#,#!SmithAbelMohonk95!#],
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
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.

- The Bark Frequency Scale
- The Bilinear Transform
- Optimal Bilinear Bark Warping
- Computing
- Optimal Frequency Warpings
- Bark Relative Bandwidth Mapping Error
- Error Significance
- Arctangent Approximations for

- Application to Audio Filter Design

- Equivalent Rectangular Bandwidth

- Directions for Improvements
- Summary

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University