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Mu-Law Companding

A companding operation compresses dynamic range on encode and expands dynamic range on decode. In digital telephone networks and voice modems (currently in use everywhere), standard CODECG.9chips are used in which audio is digitized in a simple 8-bit $ \mu$-law format (or simply ``mu-law'').

Given an input sample $ x(n)$ represented in some internal format, such as a short, it is converted to 8-bit mu-law format by the formula [48]

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

where $ Q_\mu[]$ is a quantizer which produces a kind of logarithmic fixed-point number with a 3-bit characteristic and a 4-bit mantissa, using a small table lookup for the mantissa.

As we all know from talking on the telephone, mu-law sounds really quite good for voice, at least as far as intelligibility is concerned. However, because the telephone bandwidth is only around 3 kHz (nominally 200-3200 Hz), there is very little ``bass'' and no ``highs'' in the spectrum above 4 kHz. This works out fine for intelligibility of voice because the first three formants (envelope peaks) in typical speech spectra occur in this range, and also because the difference in spectral shape (particularly at high frequencies) between consonants such as ``sss'', ``shshsh'', ``fff'', ``ththth'', etc., are sufficiently preserved in this range. As a result of the narrow bandwidth provided for speech, it is sampled at only 8 kHz in standard CODEC chips.

For ``wideband audio'', we like to see sampling rates at least as high as 44.1 kHz, and the latest systems are moving to 96 kHz (mainly because oversampling simplifies signal processing requirements in various areas, not because we can actually hear anything above 20 kHz). In addition, we like the low end to extend at least down to 20 Hz or so. (The lowest note on a normally tuned bass guitar is E1 = 41.2 Hz. The lowest note on a grand piano is A0 = 27.5 Hz.)


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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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