Optimal Bilinear Bark Warping

It turns out that a first-order 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 $31$ 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.

In the following, a simple direct-form expression is developed for the map parameter $\rho$ giving the best least-squares 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 $\ensuremath{L_2}$ optimal warping is within 0.04 Bark of the $\ensuremath{L_\infty}$ 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 least-squares maps to be essentially equivalent psychoacoustically.