Bark and ERB bilinear transforms

Use of a bilinear conformal map to achieve a frequency warping nearly ident ical to that of the Bark frequency scale is described. Because the map take s the unit circle to itself, its form is that of the transfer function of a first-order allpass filter, Since it is a first-order map, it preserves th e model order of rational systems, making it a valuable frequency warping t echnique for use in audio filter design. A closed-form weighted-equation-er ror method is derived that computes the optimal mapping coefficient as a fu nction of sampling rate, and the solution is shown to be generally indistin guishable from the optimal least-squares solution. The optimal Chebyshev ma pping is also found to be essentially identical to the optimal least-square s solution. The expression 0.8517 [arctan (0.06583f(s))](1/2) - 0.916 is sh own to accurately approximate the optimal allpass coefficient as a function of sampling rate f(s) in kHz for sampling rates greater than 1 kHz, A filt er design example is included that illustrates improvements due to carrying out the design over a Bark scale. Corresponding results are also given and compared for approximating the related "equivalent rectangular bandwidth ( ERB) scale" of Moore and Glasberg using a first-order allpass transformatio n. Due to the higher frequency resolution called for by the ERB scale, part icularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater tha n the Bark-scale case, depending on the sampling rate.