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Optimal Bilinear Bark Warping
It turns out that a firstorder conformal map (bilinear transform) can
provide a surprisingly close match to the Bark frequency scale
[268,269]. This is shown in
Fig.E.1.
Figure:
Bark and allpass frequency warpings
at a sampling rate of
kHz (the highest possible without
extrapolating the published Bark scale bandlimits). a) Bark
frequency warping viewed as a conformal mapping of the interval
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
used for each map. The
discrete bandedges which define the Bark scale are plotted as
circles. The optimal Chebyshev (solid), leastsquares (dashed), and
weighted equationerror (dotdashed) 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.

In the following, a simple directform expression is developed for the
map parameter
giving the best leastsquares fit to a Bark scale
for a chosen sampling rate. As Fig.E.1 shows, the error is so
small that the solution is also very close to the optimal Chebyshev
fit. In fact, the
optimal warping is within 0.04 Bark of the
optimal warping. Since the experimental uncertainty when
measuring critical bands is on the order of a tenth of a Bark or more
[178,181,251,299],
we consider the optimal Chebyshev and leastsquares maps to be
essentially equivalent psychoacoustically.
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