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.
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At moderate sound levels, the ERB in Hz is defined by [177]
where








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The ERB scale is defined as the number of ERBs below each frequency
for


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.
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The peak and rms mapping errors are plotted versus sampling rate in Fig.E.13. Compare these results for the ERB scale with those for the Bark scale in Fig.E.4. The ERB map errors are plotted in Barks to facilitate comparison. The rms error of the conformal map fit to the ERB scale increases nearly linearly with log-sampling-rate. The ERB-scale error increases very smoothly with frequency while the Bark-scale error is non-monotonic (see Fig.E.4). The smoother behavior of the ERB errors appears due in part to the fact that the ERB scale is defined analytically while the Bark scale is defined more directly in terms of experimental data: The Bark-scale fit is so good as to be within experimental deviation, while the ERB-scale fit has a much larger systematic error component. The peak error in Fig.E.13 also grows close to linearly on a log-frequency scale and is similarly two to three times the Bark-scale errors of Fig.E.4.
The frequency mapping errors are plotted versus frequency in
Fig.E.14 for a sampling rate of
kHz. Unlike the Bark-scale
case in Fig.E.5, there is now a visible difference between the
weighted equation-error and optimal least-squares mappings for the ERB
scale. The figure shows also that the peak error when warping to an ERB
scale is about three times larger than the peak error when warping to the
Bark scale, growing from 0.64 Barks to 1.9 Barks. The locations of the
peak errors are also at lower frequencies (moving from 1.3 and 8.8 kHz in
the Bark-scale case to 0.7 and 8.2 kHz in the ERB-scale case).
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.
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Figure E.16 shows the rms and peak ERB RBME as a function of sampling rate. Near a 10 kHz sampling rate, for example, the Chebyshev ERB RBME is increased from 12% in the Bark-scale case to around 37%, again a tripling of the peak error. We can also see in Fig.E.16 that the arctangent formula gives a very good approximation to the optimal Chebyshev solution at all sampling rates. The optimal least-squares and weighted equation-error solutions are quite different, with the weighted equation-error solution moving from being close to the least-squares solution at low sampling rates, to being close to the Chebyshev solution at the higher sampling rates. The rms error is very similar in all four cases, as it was in the Bark-scale case, although the Chebyshev and arctangent formula solutions show noticeable increase in the rms error at low sampling rates where they also show a reduction in peak error by 5% or so.
Arctangent Approximations for
, ERB Case
For an approximation to the optimal Chebyshev ERB frequency mapping, the arctangent formula becomes
where


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