Editing A-law algorithm (section)

{{Short description|Audio companding algorithm}}
{{no footnotes|date=February 2013}}

[[File:Comparison of A-law and μ-law compression on an input signal.svg|thumb|Comparison of [[A-law]] (blue) and [[μ-law]] (red) [[Dynamic range compression|compression]] on an input signal (green). Both axis use [[logarithmic scale]] in [[decibels]].|350x350px]]
[[File:Plot of F(x) for A-Law for A = 87.6.svg|thumb|Plot of ''F''(''x'') for A-Law for ''A'' = 87.6|350x350px]]
{{Listen
| type         = speech
| header       = Audio quality comparison
| filename     = Speech 12dB s16.flac
| title        = 16-bit linear PCM (reference/original)
| filename2    = Speech 12dB alaw8.flac
| title2       = 8-bit A-law PCM
| filename3    = Speech 12dB u8.flac
| title3       = 8-bit linear PCM
}}

An '''A-law algorithm''' is a standard [[companding]] algorithm, used in [[Europe]]an 8-bit [[PCM]] [[digital communications]] systems to optimize, i.e. modify, the [[dynamic range]] of an [[analog signal]] for digitizing. It is one of the two companding algorithms in the [[G.711]] standard from [[ITU-T]], the other being the similar [[μ-law algorithm|μ-law]], used in North America and Japan.

For a given input <math>x</math>, the equation for A-law encoding is as follows:
<math display="block">
F(x) = \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}
</math>

where <math>A</math> is the compression parameter. In Europe, <math>A = 87.6</math>.

A-law expansion is given by the inverse function:
<math display="block">
F^{-1}(y) = \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}
</math>

The reason for this encoding is that the wide [[dynamic range]] of [[Speech communication|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 [[Channel coding|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.