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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Bilinear Frequency-Warping for Audio Spectrum Analysis over a Bark Frequency Scale
       Simple Approximations to Various Frequency Warpings

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Simple Approximations to Various Frequency Warpings

In [31], Camacho and Harris propose a number of simple approximations for the most often-used frequency scales. The general formula is

$\displaystyle f' = \alpha \log\left(1+\frac{f}{s}\right) $

where $ f$ denotes unwarped frequency in Hz, $ f'$ is warped frequency, and $ s$ is set to approximate various warped frequency scales:

\begin{eqnarray*} s &=& 0 \,\,\Rightarrow\,\,\mbox{logarithmic (semitone) scale}... ...ale}\\ s &\to& \infty \,\,\Rightarrow\,\,\mbox{linear Hz scale} \end{eqnarray*}

Additionally, the Bark scale was approximated by

$\displaystyle f' \eqsp \frac{C}{1+\frac{s}{f}} $

with $ s=1960$.

More recently, the authors have compared results for an approximation of the Greenwood scale with $ s=165$.

In their study of pitch estimation over a warped frequency scale, the ERB setting of $ s=229$ was found to yield best performance. Moving $ s$ either up or down from that value degraded performance. The authors generally recommend $ s\in[165,700]$ for use with their SWIPE pitch estimation algorithm.

Note that the Camacho-Harris frequency-warping approximations for mel, ERB, and Greenwood scales are identical in form to mu-law amplitude compression:

$\displaystyle \vert\hat{x}(n)\vert = Q_\mu\left[\log_2\left(1 + \mu\left\vert x(n)\right\vert\right)\right] $

where $ x(n)$ denotes input signal amplitude at time $ n$, and $ \vert\hat{x}(n)\vert$ is the mu-law compressed output magnitude. (The sign bit is handled separately.)

All mappings of the form $ log(1+x/s)$, for $ x\ge0$, can be viewed as mappings which are linear for small $ x$ and logarithmic for large $ x$, where ``large'' and ``small'' are defined by the mapping parameter $ s$.

This linear-log mapping also arises in loudness perception (§6.3.3). For example, the sone amplitude scale is defined in terms of actual loudness perception experiments [259]. At 1kHz and above, loudness perception is approximately logarithmic above 50 dB SPL or so. Below that, it tends toward being more linear. Thus, the use of a dB scale for signal amplitude display can be viewed as a high-amplitude approximation to the sone scale.

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About the Author: 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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