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Bark and ERB Bilinear Transforms

Julius O. Smith III

Center for Computer Research in Music and Acoustics

(CCRMA), Stanford University
Stanford, CA 94305 USA

Jonathan S. Abel1
Human Factors Research Division
NASA-Ames Research Center
Moffet Field, CA 94035 USA


(Final draft accepted for publication in the IEEE Transactions on Speech and Audio Processing, November, 1999.)

Abstract:

Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes 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 the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived which computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0.8517*sqrt(atan(0.06583*Fs))-0.1916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate Fs in kHz for sampling rates greater than 1 kHz. A filter design example is included which 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 transformation. Due to the higher frequency resolution called for by the ERB scale, particularly 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 than the Bark-scale case, depending on the sampling rate.




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Download bbt.pdf

``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.
Copyright © 2007-05-10 by Julius O. Smith III and Jonathan S. Abel
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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