Arctangent Approximations for $\rho ^*(f_s)$

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

$$\rho _{\mathbf\gamma}(f_s) \isdef \max\left\{0,\gamma_1\left[{2\over\pi}\arctan(\gamma_2f_s)\right]^{{1\over2}}+\gamma_3 \right\}$$

was found to provide a more parsimonious expression at a given accuracy level. The idea was that the arctangent function provided a mapping from the interval $[0,\infty)$ , the domain of $f_s$ , to the interval $[0,1)$ , the range of $\rho (f_s)$ . The additive component $\gamma_3$ allowed $\rho _{\mathbf\gamma}(f_s)$ to be zero at smaller sampling rates, where the Bark scale is linear with frequency. As an additional benefit, the arctangent expression was easily inverted to give sampling rate $f_s$ in terms of the allpass coefficient $\rho _{\mathbf\gamma}$ :

$$f_s= {1\over \gamma_2}\tan\left[{\pi\over2} \left(\frac{\rho _{\mathbf\gamma}- \gamma_3}{\gamma_1}\right)^2\right].$$

To obtain the optimal arctangent form $\rho ^*_{\mathbf\gamma}(f_s)$ , the expression for $\rho _{\mathbf\gamma}(f_s)$ in (E.3.5) was optimized with respect to its free parameters ${\mathbf\gamma}=\{\gamma_1,\gamma_2,\gamma_3\}$ to match the optimal Chebyshev allpass coefficient as a function of sampling rate:

$$\rho ^*_{\mathbf\gamma}(f_s) \isdef \hbox{Arg}\left[\min_{{\mathbf\gamma}}\left\{\left\Vert\,\rho ^*_\infty(f_s) - \rho _{\mathbf\gamma}(f_s)\,\right\Vert _\infty\right\}\right].$$

For a Bark warping, the optimized arctangent formula was found to be

$$\rho ^*_{\mathbf\gamma}(f_s) = 1.0674\left[{2\over\pi}\arctan(0.06583f_s)\right]^{{1\over2}}-0.1916,$$

where $f_s$ is expressed in units of kHz. This formula is plotted along with the various optimal $\rho ^*$ curves in Fig.E.3a, and the approximation error is shown in Fig.E.3b. 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.E.4 also shows.

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

$$\rho ^*_{\mathbf\gamma}(f_s) = 1.0480\left[{2\over\pi}\arctan(0.07212f_s)\right]^{{1\over2}}-0.1957.$$

The performance of this formula is shown in Fig.E.8. It tends to follow the performance of the optimal least squares map parameter even though the peak parameter error was minimized relative to the optimal Chebyshev map. At 54 kHz there is an additional 3% bandwidth error due to the arctangent approximation, and near 10 kHz the additional error is about 4%; at other sampling rates, the performance of the RBME arctangent approximation is better, and like (E.3.5), it is extremely accurate at 41 kHz.