Our goal is to find the allpass coefficient
such that the
frequency mapping
best approximates the Bark scale
Using squared frequency errors to gauge the fit between
and
its Bark-warped counterpart, the optimal mapping-parameter
may
be written as
where
is nonlinear in
has a norm which is more amenable to minimization. The first issue we address is how the minimizers of
Denote by
and
the complex representations of the
frequencies
and
on the unit circle,
As seen in Fig.E.2, the absolute frequency error
The desired arc length error
Accordingly, essentially the same
The error
is also nonlinear in the parameter
, and to find
its norm minimizer, an 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
Rearranging terms, we have
where
It is shown in [269] that the optimal weighted least-squares conformal map parameter estimate is given by
If the weighting matrix
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The kth 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].