Section Outline

In §E.5, the Bark frequency scale is reviewed, followed by §E.6 reviewing the bilinear transformation and its specialization to the first-order allpass transformation. Section E.7 is concerned with optimally choosing the allpass parameter: A weighted equation-error solution is derived which is shown to be essentially equal to the optimal least-squares solution. The optimal Chebyshev solution is compared and found to be insignificantly different from the least-squares solutions. A variation on the error criterion which is only concerned with mapped bandwidth error, as opposed to the absolute error in the mapping of Hz to Barks, is introduced and evaluated. A simple closed-form expression relating the sampling rate to the optimal warping parameter is presented. Section E.7 concludes with a filter-design example illustrating the benefits of working over a Bark frequency scale. Finally, Section E.8 applies the methods of §E.7 to approximating the ERB scale, and the results, while potentially useful, are found to be significantly less accurate than for the Bark scale. The appendix concludes with a summary.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.