Graph of μlaw & Alaw algorithms
An Alaw algorithm is a standard companding algorithm, used in European 8bit PCM digital communications systems to optimize, i.e., modify, the dynamic range of an analog signal for digitizing.
It is similar to the μlaw algorithm used in North America and Japan.
For a given input x, the equation for Alaw encoding is as follows,

F(x) = \sgn(x) \begin{cases} {A x \over 1 + \ln(A)}, & x < {1 \over A} \\ \frac{1+ \ln(A x)}{1 + \ln(A)}, & {1 \over A} \leq x \leq 1, \end{cases}
where A is the compression parameter. In Europe, A = 87.7; the value 87.6 is also used.
Alaw expansion is given by the inverse function,

F^{1}(y) = \sgn(y) \begin{cases} {y (1 + \ln(A)) \over A}, & y < {1 \over 1 + \ln(A)} \\ {\exp(y (1 + \ln(A))  1) \over A}, & {1 \over 1 + \ln(A)} \leq y < 1. \end{cases}
The reason for this encoding is that the wide dynamic range of speech does not lend itself well to efficient linear digital encoding. Alaw encoding effectively reduces the dynamic range of the signal, thereby increasing the coding efficiency and resulting in a signaltodistortion ratio that is superior to that obtained by linear encoding for a given number of bits.
Comparison to μlaw
The μlaw algorithm provides a slightly larger dynamic range than the Alaw at the cost of worse proportional distortion for small signals. By convention, Alaw is used for an international connection if at least one country uses it.
See also
External links

Waveform Coding Techniques  Has details of implementation (but note that the Alaw equation is incorrect)

ALaw and μlaw Companding Implementations Using the TMS320C54x (PDF)

Alaw implementation in Clanguage with example code
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