This subsection provides further details on the arctangent approximation
for the optimal allpass coefficient as a function of sampling rate.
Compared with other spline or polynomial approximations, the arctangent
form

(24) |

To obtain the optimal arctangent form
, the expression for
in Eq. (23) was optimized with respect to its free
parameters
to match the optimal
Chebyshev allpass coefficient as a function of sampling rate:

(25) |

where is expressed in units of kHz. This formula is plotted along with the various optimal curves in Fig.4a, and the approximation error is shown in Fig.4b. It is extremely accurate below 15 kHz and near 40 kHz, and adds generally less than 0.1 Bark to the peak error at other sampling rates. The rms error versus sampling rate is very close to optimal at all sampling rates, as Fig.5 also shows.

When the optimality criterion is chosen to minimize relative bandwidth
mapping error (relative map *slope* error), the arctangent
formula optimization yields

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