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Concept
Loi µ
Résumé
The μ-law algorithm (sometimes written mu-law, often approximated as u-law) is a companding algorithm, primarily used in 8-bit PCM digital telecommunication systems in North America and Japan. It is one of the two companding algorithms in the G.711 standard from ITU-T, the other being the similar A-law. A-law is used in regions where digital telecommunication signals are carried on E-1 circuits, e.g. Europe.
Companding algorithms reduce the dynamic range of an audio signal. In analog systems, this can increase the signal-to-noise ratio (SNR) achieved during transmission; in the digital domain, it can reduce the quantization error (hence increasing the signal-to-quantization-noise ratio). These SNR increases can be traded instead for reduced bandwidth for equivalent SNR.
At the cost of a reduced peak SNR, it can be mathematically shown that μ-law's non-linear quantization effectively increases dynamic range by 33 dB or bits over a linearly-quantized signal, hence 13.5 bits (which rounds up to 14 bits) is the most resolution required for an input digital signal to be compressed for 8-bit μ-law.
The μ-law algorithm may be described in an analog form and in a quantized digital form.
For a given input x, the equation for μ-law encoding is
where μ = 255 in the North American and Japanese standards, and sgn(x) is the sign function. It is important to note that the range of this function is −1 to 1.
μ-law expansion is then given by the inverse equation:
The discrete form is defined in ITU-T Recommendation G.711.
G.711 is unclear about how to code the values at the limit of a range (e.g. whether +31 codes to 0xEF or 0xF0).
However, G.191 provides example code in the C language for a μ-law encoder. The difference between the positive and negative ranges, e.g. the negative range corresponding to +30 to +1 is −31 to −2. This is accounted for by the use of 1's complement (simple bit inversion) rather than 2's complement to convert a negative value to a positive value during encoding.
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