Frequency Warping via the Bilinear Transform

The bilinear conformal map, defined by the substitution

$$z= {\cal A}_{\rho }(\zeta ) \isdef {\zeta + \rho \over 1 + \zeta \rho }$$

takes the unit circle in the $z$ plane to the unit circle in the $\zeta$ plane^E.1 in such a way that, for $0<\rho<1$, low frequencies are stretched and high frequencies are compressed, as in a transformation from frequency in Hertz to the Bark scale. Because the conformal map ${\cal A}_{\rho }(\zeta )$ is identical in form to a first-order allpass transfer function (having a pole at $\zeta =-1/\rho$), we also call it the first-order allpass transformation, and $\rho$ the allpass coefficient.

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 $\rho$. It turns out, however, that the match is surprisingly good over a wide range of sampling rates, as illustrated in Fig.E.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 $\rho$ as a function of sampling rate.

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.