Our goal is to find the allpass coefficient such that the frequency mapping

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 Bark-warped counterpart, the optimal mapping-parameter 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.

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 least-squares conformal map parameter estimate is given by

If the weighting matrix is diagonal with

The *k*th 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].

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University