Arctangent Approximations for
, 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,
$](http://www.dsprelated.com/josimages_new/sasp2/img2970.png)
where
is in kHz.
This formula is plotted along with the various optimal 
curves in
Fig.E.12a, and the approximation error is shown in
Fig.E.12b.
The performance of the arctangent approximation can be seen in
Fig.E.13.
When the optimality criterion is chosen to minimize relative bandwidth mapping error in the ERB case, the arctangent formula optimization yields
![$\displaystyle \rho ^*_{\mathbf\gamma}(f_s) = 0.7164\left[{2\over\pi}\arctan(0.09669f_s)\right]^{{1\over2}}+0.08667.
$](http://www.dsprelated.com/josimages_new/sasp2/img2971.png)
The performance of this formula is shown in Fig.E.16. It follows the optimal Chebyshev map parameter very well.
Next Section:
Blackman Window Example
Previous Section:
ERB Relative Bandwidth Mapping Error
