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

(5) |

Using squared frequency errors to gauge the fit between and
its Bark warped counterpart, the optimal mapping parameter may
be written as

(6) |

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,

(9) |

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 [17]. Consider mapping
the frequency
via the allpass transformation
,

(13) |

where is an equation error defined by

Referring to Fig.3, note that the equation error is the
difference between the unit circle chord and the
circle chord
. The average of the input and
Bark warped angles
bisects both these chords,
and therefore the chords are parallel. The equation error may then be
interpreted as the difference in chord lengths, rotated to the angle
. This suggests defining a rotated
equation error which is real valued. Multiply Eq. (14) by
to obtain

The rotated equation error
is linear in the unknown , and
its norm minimizer is easily expressed in closed form. Denote by
a set of frequencies corresponding to Bark
frequencies , and by
and
the columns

Then Eq. (16) becomes

where is a column of rotated equation errors. Eq. (20) is now in the form of a standard least squares problem. As is well known [17,34], the solution which minimizes the weighted sum of squared errors, , the matrix being an arbitrary positive-definite weighting, may be obtained by premultiplying both sides of Eq. (20) by and solving for , noting that at , by the orthogonality principle. Doing this yields the optimal weighted least-squares conformal map parameter

If the weighting matrix is diagonal with

where we have used Equations Eq. (18) and Eq. (19), and the trigonometric identities

to simplify the numerator and denominator, respectively.

It remains to choose a weighting matrix
. Recall that we initially
wanted to minimize the sum of squared chord-length errors
.
The rotated equation error
is proportional to the chord length error
, viz. Eq. (17), and it is easily verified that when
is
diagonal with *k*th diagonal element

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 Eq. (22) 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 [16].

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