Optimal Bark Warping

Figure 1 illustrates the surprisingly good match between the allpass transformation and a Bark frequency warping when the map parameter 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.1 shows, the error is so small that the solution is also very close to the optimal Chebyshev fit. In fact, the L₂ optimal warping is within 0.04 Bark of the optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [20,22,31,38], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically.

• Computing
• Optimal Frequency Warpings
• Relative Bandwidth Mapping Error
• Error Significance
• Arctangent Approximations for
• Filter Design Example