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Optimal Bark Warping

Figure 1 illustrates the surprisingly good match between the allpass transformation ${\cal A}_{\rho }$ and a Bark frequency warping when the map parameter $\rho $ is properly chosen. In the following, a simple direct-form expression is developed for the map parameter giving the best least-squares fit to a Bark scale for a chosen sampling rate. As Fig.1 shows, the error is so small that the solution is also very close to the optimal Chebyshev fit. In fact, the $L_2$ optimal warping is within 0.04 Bark of the $L_\infty$ optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [20,22,31,38], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically.



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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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