## Equivalent Rectangular Bandwidth

It also turns out that a first-order conformal map (bilinear transform) can provide a good match to the ERB scale [269] as well. Moore and Glasberg [177] have revised Zwicker's loudness model to better explain (1) how equal-loudness contours change as a function of level, (2) why loudness remains constant as the bandwidth of a fixed-intensity sound increases up to the critical bandwidth, and (3) the loudness of partially masked sounds. The modification that is relevant here is the replacement of the Bark scale by the*equivalent rectangular bandwidth*(ERB) scale. The ERB of the auditory filter is assumed to be closely related to the critical bandwidth, but it is measured using the

*notched-noise*method [205,206,251,181,87] rather than on classical masking experiments involving a narrow-band masker and probe tone [306,307,304]. As a result, the ERB is said not to be affected by the detection of beats or intermodulation products between the signal and masker. Since this scale is defined analytically, it is also more smoothly behaved than the Bark scale data.

*place*along the basilar membrane [96, p. 2601].

*ERB scale*is defined as the number of ERBs below each frequency for in Hz [177]. An overlay of the normalized Bark and ERB frequency warpings is shown in Fig.E.11. The ERB warping is determined by scaling the inverse of (E.5), evaluated along a uniform frequency grid from zero to the number of ERBs at half the sampling rate, so that dc maps to zero and half the sampling rate maps to . Proceeding in the same manner as for the Bark-scale case, allpass coefficients giving a best approximation to the ERB-scale warping were computed for sampling rates near twice the Bark band edge frequencies (chosen to facilitate comparison between the ERB and Bark cases). The resulting optimal map coefficients are shown in Fig.E.12. The allpass parameter increases with increasing sampling rate, as in the Bark-scale case, but it covers a significantly narrower range, as a comparison with Fig.E.3 shows. Also, the Chebyshev solution is now systematically larger than the least-squares solutions, and the least-squares and weighted equation-error cases are no longer essentially identical. The fact that the arctangent formula is optimized for the Chebyshev case is much more evident in the error plot of Fig.E.12b than it was in Fig.E.3b for the Bark warping parameter.

### ERB Relative Bandwidth Mapping Error

The optimal relative bandwidth-mapping error (RBME) for the ERB case is plotted in Fig.E.15 for a kHz sampling rate. The peak error has grown from close to 20% for the Bark-scale case to more than 60% for the ERB case. Thus, frequency intervals are mapped to the ERB scale with up to three times as much relative error (60%) as when mapping to the Bark scale (20%). The continued narrowing of the auditory filter bandwidth as frequency decreases on the ERB scale results in the conformal map not being able to supply sufficient stretching of the low-frequency axis. The Bark scale case, on the other hand, is much better provided at low frequencies by the first-order conformal map.### Arctangent Approximations for , ERB Case

For an approximation to the optimal Chebyshev ERB frequency mapping, the arctangent formula becomes 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 The performance of this formula is shown in Fig.E.16. It follows the optimal Chebyshev map parameter very well.**Next Section:**

Directions for Improvements

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Application to Audio Filter Design