# A-law algorithm

Graph of μ-law & A-law algorithms

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,

${\displaystyle F(x)=\operatorname {sgn}(x){\begin{cases}{A|x| \over 1+\log(A)},&|x|<{1 \over A}\\{\frac {1+\log(A|x|)}{1+\log(A)}},&{1 \over A}\leq |x|\leq 1,\end{cases}}}$

where A is the compression parameter. In Europe, ${\displaystyle A=87.6}$.

A-law expansion is given by the inverse function,

${\displaystyle F^{-1}(y)=\operatorname {sgn}(y){\begin{cases}{|y|(1+\ln(A)) \over A},&|y|<{1 \over 1+\ln(A)}\\{\exp(|y|(1+\ln(A))-1) \over A},&{1 \over 1+\ln(A)}\leq |y|<1.\end{cases}}}$

The reason for this encoding is that the wide dynamic range of speech does not lend itself well to efficient linear digital encoding. 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.