Our goal is to find the
allpass coefficient
such that the
frequency mapping
angle
best approximates the
Bark scale
for a given
sampling rate
. (Note that the frequencies
,
, and
are all expressed in radians per sample, so that a frequency of half
of the
sampling rate corresponds to a value of
.)
Using squared frequency errors to gauge the fit between
and
its
Barkwarped counterpart, the optimal mappingparameter
may
be written as
where
represents the
norm. (The superscript
`
' denotes optimality in some sense.) Unfortunately, the
frequency error
is
nonlinear in
, and its norm is not easily minimized directly.
It turns out, however, that a related error,
has a norm which is more amenable to minimization. The first issue we
address is how the minimizers of
and
are
related.
Figure E.2:
Frequency Map Errors

Denote by
and
the complex representations of the
frequencies
and
on the unit circle,
As seen in Fig.
E.2, the absolute frequency error
is the
arc length between the points
and
, whereas
is the chord length or distance:
The desired arc length error
gives more weight to large errors
than the chord length error
; however, in the presence of small
discrepancies between
and
, the absolute errors are
very similar,
Accordingly, essentially the same
results from minimizing
or
when the fit is uniformly good over
frequency.
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
, and
substitute
from (
E.3.1), to get
Rearranging terms, we have
where
is an equation error defined by
It is shown in [
269] that the optimal weighted
leastsquares conformal map parameter estimate is given by
If the weighting
matrix
is diagonal with
kth diagonal
element
, then the weighted leastsquares
solution (
E.3.1) reduces to
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
SteiglitzMcBride
algorithm for converting an equationerror minimizer to the more
desired ``outputerror'' minimizer using an iteratively computed
weight function [
151].
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