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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 $ \hbox{\boldmath $V$}$ in the weighted equation error solution (E.3.1) 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 [96]. In this case, a close relationship still exists between equal-place increments along the basilar membrane and equal bandwidth increments in the defined audio filter bank. Preliminary comparisons [96, 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 filter bank.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2021-11-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University