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 . 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
If the weighting matrix is diagonal with kth diagonal element , then the weighted least-squares solution (E.3.1) reduces to
The kth diagonal element of an optimal diagonal weighting matrix is given by 
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 .
Optimal Frequency Warpings
Gaussian Central Moments