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Directions for Improvements

Audio conformal maps can be adjusted by using a more general error weighting versus frequency. For example, the weighting can be set to zero above some frequency limit along the unit circle. A more general weighting can also be used to obtain improved accuracy in specific desired frequency ranges. Again, these refinements would seem to be of interest primarily for the ERB-scale and other mappings, since the Bark-scale warping is excellent already. The diagonal weighting matrix $\mbox{\boldmath$V$}$ in the weighted equation error solution Eq.(21) can be multiplied by any desired application-dependent weighting.

As another variation, an auditory frequency scale could be defined based on the cochlear frequency-to-place function [6]. In this case, a close relationship still exists between equal-place increments along the basilar membrane and equal bandwidth increments in the defined audio filterbank. Preliminary comparisons [6, Fig. 9] indicate that the first-order conformal map errors for this case are qualitatively between the ERB and Bark-scale cases. The first-order conformal map works best when the auditory filter bandwidths level off to a minimum width at low frequencies, as they do in the Bark-scale case below 500 Hz. Thus, the question of the ``audio fidelity'' of the first-order conformal map is directly tied to the question of what is really the best frequency resolution to provide at low frequencies in the auditory filterbank.

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``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 © 2020-07-19 by Julius O. Smith III and Jonathan S. Abel
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University