- warping the desired frequency response, thus ``horizontally stretching'' the more important low-frequency region of the spectrum.
- performing a filter design over the warped frequency axis, and
- transforming the resulting filter to eliminate the frequency warp, returning it to the normal frequency axis.
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  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 :E.2
- The optimal allpass coefficient
was found using
- The desired frequency response
defined on a linear
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 .
Equivalent Rectangular Bandwidth
Optimal Bilinear Bark Warping