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Computing $\rho $

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

a(\omega )= \mbox{angle}\left\{{\cal A}_{-\rho }(e^{j\omega }) \right\}
\end{displaymath} (5)

best approximates the Bark scale $b(\omega )$ for a given sampling rate fs. (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 ^*= \mbox{Arg}\left[\min_{\rho }\left\{\left\Vert\,a(\omega )- b(\omega )\,\right\Vert\right\}\right],
\end{displaymath} (6)

where $\left\Vert\,\cdot\,\right\Vert$ represents the L2 norm. (We use the superscript `$\ast$' to denote optimality in some sense.) Unfortunately, the frequency error
\epsilon _{\mbox{\tiny A}}\mathrel{\stackrel{\mathrm{\Delta}}{=}}a(\omega )- b(\omega )
\end{displaymath} (7)

is nonlinear in $\rho $, and its norm is not easily minimized directly. It turns out, however, that a related error,
\epsilon _{\mbox{\tiny C}}\mathrel{\stackrel{\mathrm{\Delta}}{=}}e^{ja(\omega )}- e^{jb(\omega )},
\end{displaymath} (8)

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

Figure 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 )}.
\end{displaymath} (9)

As seen in Fig.2, the absolute frequency error $\vert\epsilon _{\mbox{\tiny A}}\vert$ is the arc length between the points $\zeta $ and $\beta $, whereas $\vert\epsilon _{\mbox{\tiny C}}\vert$ is the chord length or distance:
\vert\epsilon _{\mbox{\tiny C}}\vert = 2\sin(\vert\epsilon _{\mbox{\tiny A}}\vert/2).
\end{displaymath} (10)

The desired arc length error $\epsilon _{\mbox{\tiny A}}$ gives more weight to large errors than the chord length error $\epsilon _{\mbox{\tiny C}}$; however, in the presence of small discrepancies between $\zeta $ and $\beta $, the absolute errors are very similar,
\vert\epsilon _{\mbox{\tiny C}}\vert \approx \vert\epsilon _{\mbox{\tiny A}}\vert, \quad \mbox{when } \vert\epsilon _{\mbox{\tiny A}}\vert\ll1.
\end{displaymath} (11)

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

The error $\epsilon _{\mbox{\tiny C}}$ 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 [17]. Consider mapping the frequency $z=e^{j\omega }$ via the allpass transformation ${\cal A}_{-\rho }(z)$,

\zeta = {z- \rho \over 1- z\rho }.
\end{displaymath} (12)

Now, multiply Eq.(12) by the denominator $(1-z\rho )$, and substitute $\zeta =\beta +\epsilon _{\mbox{\tiny C}}$ from Eq.(8), to get
(\beta + \epsilon _{\mbox{\tiny C}}) (1 - z\rho ) = z- \rho .
\end{displaymath} (13)

Rearranging terms, we have
(\beta - z) - (\beta z- 1) \rho = \epsilon _{\mbox{\tiny E}},
\end{displaymath} (14)

where $\epsilon _{\mbox{\tiny E}}$ is an equation error defined by
\epsilon _{\mbox{\tiny E}}\mathrel{\stackrel{\mathrm{\Delta}}{=}}(z\rho - 1) \epsilon _{\mbox{\tiny C}}.
\end{displaymath} (15)

Figure 3: Geometric Interpretation of Equation Error

Referring to Fig.3, note that the equation error is the difference between the unit circle chord $(\beta -z)$ and the $\rho $ circle chord $(\beta z-1)\rho $. The average of the input and Bark warped angles $\left[b(\omega )+\omega \right]/2$ 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 $\left[b(\omega )+\omega \right]/2 + \pi/2$. This suggests defining a rotated equation error which is real valued. Multiply Eq.(14) by $(z\beta )^{-{1\over2}}/2j$ to obtain

\sin\left[\frac{b(\omega )-\omega }{2}\right]
- \rho \sin\left[\frac{b(\omega )+\omega }{2}\right] = \epsilon _{\mbox{\tiny R}},
\end{displaymath} (16)

where the rotated equation error $\epsilon _{\mbox{\tiny R}}$ is defined by
\epsilon _{\mbox{\tiny R}}\mathrel{\stackrel{\mathrm{\Delta}}{=}}{1\over 2j} (z\beta )^{-{1\over2}} \epsilon _{\mbox{\tiny E}}
= {1\over 2j} (z\beta )^{-{1\over2}} (z\rho - 1) \epsilon _{\mbox{\tiny C}}.
\end{displaymath} (17)

The rotated equation error $\epsilon _{\mbox{\tiny R}}$ is linear in the unknown $\rho $, and its norm minimizer is easily expressed in closed form. Denote by $\omega_{k}, k=1,\ldots,K$ a set of frequencies corresponding to Bark frequencies $b(\omega_{k})$, and by $\mbox{\boldmath$d$}$ and $\mbox{\boldmath$s$}$ the columns

$\displaystyle \mbox{\boldmath$d$}$ = $\displaystyle \left[\matrix{ \sin\{[b(\omega_{1})-\omega_{1}]/2\} \cr \vdots \cr
\sin\{[b(\omega_{K})-\omega_{K}]/2\} }\right]$ (18)
$\displaystyle \mbox{\boldmath$s$}$ = $\displaystyle \left[\matrix{ \sin\{[b(\omega_{1})+\omega_{1}]/2\} \cr \vdots \cr
\sin\{[b(\omega_{K})+\omega_{K}]/2\} }\right].$ (19)

Then Eq.(16) becomes
\mbox{\boldmath$d$}- \mbox{\boldmath$s$}\rho = \mbox{\boldmath$\epsilon$}_{\mbox{\tiny R}},
\end{displaymath} (20)

where $\mbox{\boldmath$\epsilon$}_{\mbox{\tiny R}}$ 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 $\rho ^*$ which minimizes the weighted sum of squared errors, $\mbox{\boldmath$\epsilon$}_{\mbox{\tiny R}}^\top \mbox{\boldmath$V$}\mbox{\boldmath$\epsilon$}_{\mbox{\tiny R}}$, the matrix $\mbox{\boldmath$V$}$ being an arbitrary positive-definite weighting, may be obtained by premultiplying both sides of Eq.(20) by $\mbox{\boldmath$s$}^\top \mbox{\boldmath$V$}$ and solving for $\rho $, noting that at $\rho =\rho ^*$, $\mbox{\boldmath$s$}^\top \mbox{\boldmath$V$}\mbox{\boldmath$\epsilon$}_{\mbox{\tiny R}}=0$ by the orthogonality principle. Doing this yields the optimal weighted least-squares conformal map parameter
\rho ^*= {\mbox{\boldmath$s$}^\top \mbox{\boldmath$V$}\mbox{\boldmath$d$}\over \mbox{\boldmath$s$}^\top \mbox{\boldmath$V$}\mbox{\boldmath$s$}} .
\end{displaymath} (21)

If the weighting matrix $\mbox{\boldmath$V$}$ is diagonal with kth diagonal element $v(\omega_{k})>0$, then the weighted least-squares solution Eq.(21) reduces to

\rho ^*&=& \frac{\sum_{k=1}^K
v(\omega _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})\right]- \cos(\omega_{k})\right\}}{%
\sum_{k=1}^{K} v(\omega_{k}) \left\{\cos\left[b(\omega_{k}) + \omega_{k}\right]- 1\right\}},

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

\cos(A+B) - \cos(A-B) &=& -2\sin(A)\sin(B) \\
\cos(A)-1 &=& -2\sin^2(A/2)

to simplify the numerator and denominator, respectively.

It remains to choose a weighting matrix $\mbox{\boldmath$V$}$. Recall that we initially wanted to minimize the sum of squared chord-length errors $\mbox{\boldmath$\epsilon$}_{\mbox{\tiny C}}^\top \mbox{\boldmath$\epsilon$}_{\mbox{\tiny C}}$. The rotated equation error $\epsilon _{\mbox{\tiny R}}$ is proportional to the chord length error $\epsilon _{\mbox{\tiny C}}$, viz. Eq.(17), and it is easily verified that when $\mbox{\boldmath$V$}$ is diagonal with kth diagonal element

v(\omega_{k}) = {1\over 1 + \rho ^2 - 2\rho \cos\omega_{k}},
\end{displaymath} (22)

the chord length error and the weighted equation error coincide. Thus, with this diagonal weighting matrix, the solution Eq.(21) minimizes the chord-length error norm.

Note that the desired weighting depends on the unknown map parameter $\rho $. To overcome this difficulty, we suggest first estimating $\rho ^*$ using $\mbox{\boldmath$V$}
= \mbox{\boldmath$I$}$, where $\mbox{\boldmath$I$}$ denotes the identity matrix, and then computing $\rho ^*$ 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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``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.
Copyright © 2020-07-19 by Julius O. Smith III and Jonathan S. Abel
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University