Optimal Bark Warping

Figure E.1 illustrates the surprisingly good match between the allpass transformation ${\cal A}_{\rho }$ and a Bark frequency warping when the map parameter $\rho$ is properly chosen. In the following, a simple direct-form expression is developed for the map parameter giving the best least-squares fit to a Bark scale for a chosen sampling rate. As Fig.E.1 shows, the error is so small that the solution is also very close to the optimal Chebyshev fit. In fact, the $L_2$ optimal warping is within 0.04 Bark of the $L_\infty$ optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [162,165,229,274], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically.