Arctangent Approximations for , ERB Case

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

where is in kHz. This formula is plotted along with the various optimal 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

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

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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.