Computing $\rho$

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

$$a(\omega )=$$   angle$$\left\{{\cal A}_{-\rho }(e^{j\omega }) \right\}$$

best approximates the Bark scale $b(\omega )$ for a given sampling rate $f_s$. (Note that the frequencies $\omega$, $a(\omega )$, and $b(\omega )$ are all expressed in radians per sample, so that a frequency of half of the sampling rate corresponds to a value of $\pi$.)

Using squared frequency errors to gauge the fit between $a(\omega )$ and its Bark warped counterpart, the optimal mapping parameter $\rho ^*$ may be written as

$$\rho ^*= \hbox{Arg}\left[\min_{\rho }\left\{\left\Vert\,a(\omega )- b(\omega )\,\right\Vert\right\}\right],$$

where $\left\Vert\,\cdot\,\right\Vert$ represents the $L_2$ norm. (We use the superscript `$\ast$' to denote optimality in some sense.) Unfortunately, the frequency error

$$\epsilon _{\hbox{\tiny A}}\isdef a(\omega )- b(\omega )$$

is nonlinear in $\rho$, and its norm is not easily minimized directly. It turns out, however, that a related error,

$$\epsilon _{\hbox{\tiny C}}\isdef e^{ja(\omega )}- e^{jb(\omega )},$$

has a norm which is more amenable to minimization. The first issue we address is how the minimizers of $\left\Vert\,\epsilon _{\hbox{\tiny A}}\,\right\Vert$ and $\left\Vert\,\epsilon _{\hbox{\tiny C}}\,\right\Vert$ are related.

Figure E.2: Frequency Map Errors

Denote by $\zeta$ and $\beta$ the complex representations of the frequencies $a(\omega )$ and $b(\omega )$ on the unit circle,

$$\zeta = e^{ja(\omega )}, \qquad \beta = e^{jb(\omega )}.$$

As seen in Fig.E.2, the absolute frequency error $\vert\epsilon _{\hbox{\tiny A}}\vert$ is the arc length between the points $\zeta$ and $\beta$, whereas $\vert\epsilon _{\hbox{\tiny C}}\vert$ is the chord length or distance:

$$\vert\epsilon _{\hbox{\tiny C}}\vert = 2\sin(\vert\epsilon _{\hbox{\tiny A}}\vert/2).$$

The desired arc length error $\epsilon _{\hbox{\tiny A}}$ gives more weight to large errors than the chord length error $\epsilon _{\hbox{\tiny C}}$; however, in the presence of small discrepancies between $\zeta$ and $\beta$, the absolute errors are very similar,

$$\vert\epsilon _{\hbox{\tiny C}}\vert \approx \vert\epsilon _{\hbox{\tiny A}}\vert,$$   when $$\vert\epsilon _{\hbox{\tiny A}}\vert\ll1.$$

Accordingly, essentially the same $\rho ^*$ results from minimizing $\left\Vert\,\epsilon _{\hbox{\tiny A}}\,\right\Vert$ or $\left\Vert\,\epsilon _{\hbox{\tiny C}}\,\right\Vert$ when the fit is uniformly good over frequency.

The error $\epsilon _{\hbox{\tiny C}}$ is also nonlinear in the parameter $\rho$, and to find its norm minimizer, an equation error is introduced, as is common practice in developing solutions to nonlinear system identification problems [136]. Consider mapping the frequency $z=e^{j\omega}$ via the allpass transformation ${\cal A}_{-\rho }(z)$,

$$\zeta = {z- \rho \over 1- z\rho }.$$

Now, multiply (E.7.1) by the denominator $(1-z\rho )$, and substitute $\zeta =\beta +\epsilon _{\hbox{\tiny C}}$ from (E.7.1), to get

$$(\beta + \epsilon _{\hbox{\tiny C}}) (1 - z\rho ) = z- \rho .$$

Rearranging terms, we have

$$(\beta - z) - (\beta z- 1) \rho = \epsilon _{\hbox{\tiny E}},$$

where $\epsilon _{\hbox{\tiny E}}$ is an equation error defined by

$$\epsilon _{\hbox{\tiny E}}\isdef (z\rho - 1) \epsilon _{\hbox{\tiny C}}.$$

It is shown in [245] that the optimal weighted least-squares conformal map parameter estimate is given by

$$\rho ^*= {\hbox{\boldmath$s$}^\top \hbox{\boldmath$V$}\hbox{\boldmath$d$}\over \hbox{\boldmath$s$}^\top \hbox{\boldmath$V$}\hbox{\boldmath$s$}} .$$

If the weighting matrix $\hbox{\boldmath $V$}$ is diagonal with kth diagonal element $v(\omega_{k})>0$, then the weighted least-squares solution (E.7.1) reduces to

$$\rho ^* = \frac{\sum_{k=1}^K v(\omega _k) \sin\left[\frac{b(\omega_{k})-\om... ...\sum_{k=1}^K v(\omega _k)\sin^2\left[\frac{b(\omega_{k})+\omega_{k}}{2}\right]}$$

$$= \frac{\sum_{k=1}^{K} v(\omega _k) \left\{\cos\left[b(\omega_{k})\... ...} v(\omega_{k}) \left\{\cos\left[b(\omega_{k}) + \omega_{k}\right]- 1\right\}}.$$

The kth diagonal element of an optimal diagonal weighting matrix $\hbox{\boldmath $V$}$ is given by [245]

$$v(\omega_{k}) = {1\over 1 + \rho ^2 - 2\rho \cos\omega_{k}},$$

Note that the desired weighting depends on the unknown map parameter $\rho$. To overcome this difficulty, we suggest first estimating $\rho ^*$ using $\hbox{\boldmath $V$}= \hbox{\boldmath $I$}$, where $\hbox{\boldmath $I$}$ denotes the identity matrix, and then computing $\rho ^*$ using the weighting (E.7.1) 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 [135].