The slope of the frequency versus warped-frequency curve can be interpreted as being proportional to critical bandwidth, since a unit interval (one Bark) on the warped-frequency axis is magnified by the slope to restore the band to its original size (one critical bandwidth). It is therefore interesting to look at the relative slope error, i.e., the error in the slope of the frequency mapping divided by the ideal Bark-map slope. We interpret this error measure as the relative bandwidth-mapping error (RBME). The RBME is plotted in Fig.E.6 for a kHz sampling rate. The worst case is 21% for the Chebyshev case and 20% for both least-squares cases. When the mapping coefficient is explicitly optimized to minimize RBME, the results of Fig.E.7 are obtained: the Chebyshev peak error drops from 21% down to 18%, while the least-squares cases remain unchanged at 20% maximum RBME. A 3% change in RBME is comparable to the 0.03 Bark peak-error reduction seen in Fig.E.5 when using the Chebyshev norm instead of the norm; again, such a small difference is not likely to be significant in most applications.
Similar observations are obtained at other sampling rates, as shown in Fig.E.8. Near a 10 kHz sampling rate, the Chebyshev RBME is reduced from 17% when minimizing absolute error in Barks (not shown in any figure) to around 12% by explicitly minimizing the RBME, and this is the sampling-rate range of maximum benefit. At 15.2, 19, 41, and 54 kHz sampling rates, the difference is on the order of only 1%. Other cases generally lie between these extremes. The arctangent formula generally falls between the Chebyshev and optimal least-squares cases, except at the highest (extrapolated) sampling rate 54 kHz. The rms error is very similar in all four cases, although the Chebyshev case has a little larger rms error near a 10 kHz sampling rate, and the arctangent case gives a noticeably larger rms error at 54 kHz.
Optimal Frequency Warpings