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Arctangent Approximations for $ \rho ^*(f_s)$ , ERB Case

For an approximation to the optimal Chebyshev ERB frequency mapping, the arctangent formula becomes

$\displaystyle \rho ^*_{\mathbf\gamma}(f_s) = 0.7446\left[{2\over\pi}\arctan(0.1418f_s)\right]^{{1\over2}}+0.03237,

where $ f_s$ is in kHz. This formula is plotted along with the various optimal $ \rho ^*$ curves in Fig.E.12a, and the approximation error is shown in Fig.E.12b. The performance of the arctangent approximation can be seen in Fig.E.13.

When the optimality criterion is chosen to minimize relative bandwidth mapping error in the ERB case, the arctangent formula optimization yields

$\displaystyle \rho ^*_{\mathbf\gamma}(f_s) = 0.7164\left[{2\over\pi}\arctan(0.09669f_s)\right]^{{1\over2}}+0.08667.

The performance of this formula is shown in Fig.E.16. It follows the optimal Chebyshev map parameter very well.

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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