A-law algorithm

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.

Graph of μ-law & A-law algorithms

Plot of F(x) for A-Law for A = 87.6

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

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

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

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.