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Chapters

Spectral Audio Signal Processing
    Bilinear Frequency-Warping for Audio Spectrum Analysis over a Bark Frequency Scale
       Optimal Bark Warping
          Computing

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Computing

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

$\displaystyle a(\omega )=$   angle$\displaystyle \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

$\displaystyle \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

$\displaystyle \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,

$\displaystyle \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
\includegraphics[width=3in]{eps/eaec}

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

$\displaystyle \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:

$\displaystyle \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,

$\displaystyle \vert\epsilon _{\hbox{\tiny C}}\vert \approx \vert\epsilon _{\hbox{\tiny A}}\vert,$   when $\displaystyle \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)$,

$\displaystyle \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

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

Rearranging terms, we have

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

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

$\displaystyle \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

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

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

$\displaystyle 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].

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Previous: Optimal Bark Warping
Next: Optimal Frequency Warpings
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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