## Optimal Bilinear Bark Warping

It turns out that a first-order conformal map (bilinear transform) can provide a surprisingly close match to the Bark frequency scale [268,269]. This is shown in Fig.E.1.### Computing

Our goal is to find the allpass coefficient such that the frequency mapping**angle**

*equation error*is introduced, as is common practice in developing solutions to nonlinear system identification problems [152]. Consider mapping the frequency via the allpass transformation , Now, multiply (E.3.1) by the denominator , and substitute from (E.3.1), to get

*k*th diagonal element , then the weighted least-squares solution (E.3.1) reduces to

The

*k*th diagonal element of an optimal diagonal weighting matrix is given by [269] Note that the desired weighting depends on the unknown map parameter . To overcome this difficulty, we suggest first estimating using , where denotes the identity matrix, and then computing using the weighting (E.3.1) based on the unweighted solution. This is analogous to the

*Steiglitz-McBride algorithm*for converting an equation-error minimizer to the more desired ``output-error'' minimizer using an iteratively computed weight function [151].

### Optimal Frequency Warpings

In [269], optimal allpass coefficients were computed for sampling rates of twice the Bark band-edge frequencies by means of four different optimization methods:- Minimize the peak arc-length error at each sampling rate to obtain the optimal Chebyshev allpass parameter .
- Minimize the sum of squared arc-length errors to obtain the optimal least-squares allpass parameter .
- Use the closed-form weighted equation-error solution (E.3.1) computed twice, first with , and second with set from (E.3.1) to obtain the optimal ``weighted equation error'' solution .
- Fit the function to the optimal Chebyshev allpass parameter via Chebyshev optimization with respect to . We will refer to the resulting function as the ``arctangent approximation'' (or, less formally, the ``Barktan formula''), and note that it is easily computed directly from the sampling rate.

^{E.1}are plotted for all four cases (Chebyshev, least squares, weighted equation-error, and arctangent approximation). The conformal-map fit to the Bark scale is generally excellent in all cases. We see that the rms error is essentially identical in the first three cases, although the Chebyshev rms error is visibly larger below 10 kHz. Similarly, the peak error is essentially the same for least squares and weighted equation error, with the Chebyshev case being able to shave almost 0.1 Bark from the maximum error at high sampling rates. The arctangent formula shows up to a tenth of a Bark larger peak error at sampling rates 15-30 and 54 kHz, but otherwise it performs very well; at 41 kHz and below 12 kHz the arctangent approximation is essentially optimal in all senses considered. At sampling rates up to the maximum non-extrapolated sampling rate of kHz, the peak mapping errors are all much less than one Bark (0.64 Barks for the Chebyshev case and 0.67 Barks for the two least squares cases). The mapping errors in Barks can be seen to increase almost linearly with sampling rate. However, the irregular nature of the Bark-scale data results in a nonmonotonic relationship at lower sampling rates. The specific frequency mapping errors versus frequency at the kHz sampling rate (the same case shown in Fig.E.1) are plotted in Fig.E.5. Again, all four cases are overlaid, and again the least squares and weighted equation-error cases are essentially identical. By forcing equal and opposite peak errors, the Chebyshev case is able to lower the peak error from 0.67 to 0.64 Barks. A difference of 0.03 Barks is probably insignificant for most applications. The peak errors occur at 1.3 kHz and 8.8 kHz where the error is approximately 2/3 Bark. The arctangent formula peak error is 0.73 Barks at 8.8 kHz, but in return, its secondary error peak at 1.3 kHz is only 0.55 Barks. In some applications, such as when working with oversampled signals, higher accuracy at low frequencies at the expense of higher error at very high frequencies may be considered a desirable tradeoff. We see that the mapping falls ``behind'' a bit as frequency increases from zero to 1.3 kHz, mapping linear frequencies slightly below the desired corresponding Bark values; then, the mapping ``catches up,'' reaching an error of 0 Barks near 3 kHz. Above 3 kHz, it gets ``ahead'' slightly, with frequencies in Hz being mapped a little too high, reaching the positive error peak at 8.8 kHz, after which it falls back down to zero error at . (Recall that dc and half the sampling-rate are always points of zero error by construction.)

### Bark Relative Bandwidth Mapping Error

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.

### Error Significance

In one study, young normal listeners exhibited a standard deviation in their measured auditory bandwidths (based on notched-noise masking experiments) on the order of 10% of center frequency [178]. Therefore, a 20% peak error in mapped bandwidth (typical for sampling rates approaching 40 kHz) could be considered significant. However, the*range*of auditory-filter bandwidths measured in 93 young normal subjects at 2 kHz [178] was 230 to 410 Hz, which is -26% to +32% relative to 310 Hz. In [298], 40 subjects were measured, yielding auditory-filter bandwidths between -33% and +65%, with a standard deviation of 18%. It may thus be concluded that a worst-case mapping error on the order of 20%, while probably detectable by ``golden ears'' listeners, lies well within the range of experimental deviations in the empirical measurement of auditory bandwidth. As a worst-case example of how the 18% peak bandwidth-mapping error in Fig.E.7 might correspond to an audible distortion, consider one critical band of noise centered at the frequency of maximum negative mapping error, scaled to be the same loudness as a single critical band of noise centered at the frequency of maximum positive error. The systematic nature of the mapping error results in a narrowing of the lower band and expansion of the upper band by about 1.7 dB. As a result, over the warped frequency axis, the upper band will be effectively

*emphasized*over the lower band by about 3 dB.

### Arctangent Approximations for

This subsection provides further details on the arctangent approximation for the optimal allpass coefficient as a function of sampling rate. Compared with other spline or polynomial approximations, the arctangent form was found to provide a more parsimonious expression at a given accuracy level. The idea was that the arctangent function provided a mapping from the interval , the domain of , to the interval , the range of . The additive component allowed to be zero at smaller sampling rates, where the Bark scale is linear with frequency. As an additional benefit, the arctangent expression was easily inverted to give sampling rate in terms of the allpass coefficient :*slope*error), the arctangent formula optimization yields The performance of this formula is shown in Fig.E.8. It tends to follow the performance of the optimal least squares map parameter even though the peak parameter error was minimized relative to the optimal Chebyshev map. At 54 kHz there is an additional 3% bandwidth error due to the arctangent approximation, and near 10 kHz the additional error is about 4%; at other sampling rates, the performance of the RBME arctangent approximation is better, and like (E.3.5), it is extremely accurate at 41 kHz.

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The Bilinear Transform