The first-order ``allpass'' conformal map which maps the unit circle to itself was configured to approximate frequency warpings from a linear frequency scale to either a Bark scale or an ERB frequency scale for a wide variety of sampling rates. The accuracy of this warping is extremely good for the Bark-scale case, and fair also for the ERB case; the first-order conformal map shows significantly more error in the ERB case (about three times that of the Bark-scale case) due to its narrower resolution bandwidths at low frequencies.
A closed-form expression was derived for the allpass coefficient which minimizes the norm of the weighted equation error between samples of the allpass warping and the desired Bark or ERB warpings. The weighting function was designed to give estimates as close as possible to the optimal least-squares estimate, and comparisons showed this to be well achieved, especially in the Bark-scale case.
A simple, closed-form, invertible expression which comes very close to the optimal Chebyshev allpass coefficient vs. sampling rate was given in (E.3.5) for the Bark-scale case and in (E.5.2) for the ERB-scale case.
Three optimal conformal maps were defined based on Chebyshev, least squares, and weighted equation-error approximation, and all three mappings were found to be psychoacoustically identical, for most practical purposes, in the Bark-scale case. When using optimal maps, the peak relative bandwidth mapping error is about % in the Bark-scale case and % in the ERB-scale case.
We conclude that the first-order conformal map is a highly useful tool
for audio digital filter design and related applications in digital
audio signal processing which may benefit from an order-invariant
mapping of the unit circle from a linear frequency scale to an
approximate auditory frequency scale.
Matlab code for plots, optimizations, and the filter design example
presented here are available online at
Matlab for Spectrum Analysis Windows
Directions for Improvements