# A-law algorithm

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An a-law algorithm is a standard companding algorithm, used in European digital communications systems to optimize, i.e., modify, the dynamic range of an analog signal for digitizing.

It is similar to the mu-law algorithm used in North America and Japan.

For a given input x, the equation for A-law encoding is as follows,

[itex]

F(x) = \begin{cases} {A |x| \over 1 + \ln(A)}, & 0 \leq |x| < {1 \over A} \\ \sgn(x) \frac{1+ \ln(A |x|)}{1 + \ln(A)}, & {1 \over A} \leq |x| \leq 1 \end{cases} [itex],

where A is the compression parameter. In Europe, A = 87.7.

The reason for this encoding is that the wide dynamic range of speech does not lend itself well to efficient linear digital encoding. A-law encoding effectively reduces the dynamic range of the signal, thereby increasing the coding efficiency and resulting in a signal-to-distortion ratio that is superior to that obtained by linear encoding for a given number of bits.

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