Mu-law algorithm

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 two versions of the G.711 standard from ITU-T, the other version being the similar A-law, 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 signal to quantization noise ratio). These SNR increases can be traded instead for reduced bandwidth for equivalent SNR.

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 (8 bits) in the North American and Japanese standards. 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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