 Graph of μ-law and A-law algorithms Plot of F(x) for A-Law for A = 87.6

An A-law algorithm is a standard companding algorithm, used in European 8-bit PCM digital communications systems to optimize, i.e. modify, the dynamic range of an analog signal for digitizing. It is one of two versions of the G.711 standard from ITU-T, the other version being the similar μ-law, used in North America and Japan.

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

$F(x)=\operatorname {sgn}(x){\begin{cases}{\dfrac {A|x|}{1+\ln(A))),&|x|<{\dfrac {1}{A)),\\[1ex]{\dfrac {1+\ln(A|x|)}{1+\ln(A))),&{\dfrac {1}{A))\leq |x|\leq 1,\end{cases))$ where $A$ is the compression parameter. In Europe, $A=87.6$ .

A-law expansion is given by the inverse function:

$F^{-1}(y)=\operatorname {sgn}(y){\begin{cases}{\dfrac {|y|(1+\ln(A))}{A)),&|y|<{\dfrac {1}{1+\ln(A))),\\{\dfrac {e^{-1+|y|(1+\ln(A)))){A)),&{\dfrac {1}{1+\ln(A)))\leq |y|<1.\end{cases))$ The reason for this encoding is that the wide dynamic range of speech does not lend itself well to efficient linear digital encoding. 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.

## Comparison to μ-law

The μ-law algorithm provides a slightly larger dynamic range than the A-law at the cost of worse proportional distortion for small signals. By convention, A-law is used for an international connection if at least one country uses it.