Bilinear Frequency-Warping for Audio Spectrum Analysis over a Bark Frequency Scale

Frequency warping is an important tool for audio digital filter design and other applications. For example, methods having no weighting function versus frequency, such as linear predictive coding (LPC), can be given an effective weighting function by means of a frequency warping. A good choice of frequency warp in audio applications is a ``Bark-scale'' warping; the Bark scale is defined so that critical bands of hearing are uniform (one critical bandwidth equals one Bark).

Frequency warping is generally employed in audio filter design by

1. warping the desired frequency response, thus ``stretching out'' the more important regions of the frequency axis,
2. performing a filter design over the warped frequency axis, and
3. transforming the resulting filter to eliminate the frequency warp, returning it to the normal frequency axis.

The third step is generally carried out using a conformal map (i.e., substituting some rational-function-of-$z$ for $z$ in the filter transfer function). In order for the unwarped filter to have the same order as the designed filter, this conformal map should be first order.

In this appendix, adapted from [245], we describe how a first-order conformal map can be used to obtain a very close match to the Bark frequency scale. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since the mapping is first order, filter design can be carried out over a Bark frequency scale, with the resulting filter transformed back to unwarped form without increasing its order.

A closed-form weighted-equation-error method is derived which computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0.8517*sqrt(atan(0.06583*Fs))-0.1916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate Fs in kHz for sampling rates greater than 1 kHz. A filter design example is included which illustrates improvements due to carrying out the design over a Bark scale.

Corresponding results are also given and compared for approximating the related ``equivalent rectangular bandwidth (ERB) scale'' of Moore and Glasberg using a first-order allpass transformation. Due to the higher frequency resolution called for by the ERB scale, particularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater than the Bark-scale case, depending on the sampling rate.