We conclude discussion of the Bark bilinear transform with the filter
design example of Fig.E.9. A
th-order pole-zero filter was fit
using Prony's method [162] to the equalization function plotted in the
figure as a dashed line. Prony's method was applied normally over a
uniformly sampled linear frequency grid in the example of
Fig.E.9a, and over an approximate Bark-scale axis in the example
of Fig.E.9b. The procedure in the Bark-scale case was as
follows [257]:^{E.2}

- The optimal allpass coefficient
was found using
(E.3.5).
- The desired frequency response
defined on a linear
frequency axis
was warped to an approximate Bark scale
using the Bark bilinear transform,
.
- A parametric ARMA model
was fit to the desired
Bark-warped frequency response
over the unit circle
.
- Finally, the inverse Bark bilinear transform was used to ``unwarp'' the modeled system to a linear frequency axis.

Referring to Fig.E.9, it is clear that the warped solution provides a
better overall fit than the direct solution which sacrifices accuracy below
kHz to achieve a tighter fit above
kHz. In some part, the spacing
of spectral features is responsible for the success of the Bark-warped
model in this particular example. However, we generally recommend using
the Bark bilinear transform to design audio filters, since doing so weights
the error norm (for norms other than Chebyshev types) in a way which gives
equal importance to matching features having equal Bark bandwidths. Even
in the case of Chebyshev optimization, auditory warping appears to improve
the *numerical conditioning* of the filter design problem; this applies
also to optimization under the Hankel norm which includes an optimal
Chebyshev design internally as an intermediate step. Further filter-design
examples, including more on the Hankel-norm case, may be found in
[257].

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