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

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 . 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 as a
function of sampling rate.

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