View source for A-law algorithm

{{Short description|Algorithm Global 64-bit PCM}}
[[File:Comparison of A-law and μ-law compression on an input signal.svg|thumb|Comparison of A-law (blue) and μ-law (red) compression on an input signal (green). Both axis use [[Logarithmic scale|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         = music
| header       = Audio quality comparison
| filename     = Larynx-HiFi-GAN speech sample-8k-Alaw.flac
| title        = 8-bit A-law
| filename2    = Larynx-HiFi-GAN speech sample-8k-8bit.flac
| title2       = 8-bit linear PCM
| filename3    = Larynx-HiFi-GAN speech sample-8k-16bit.flac
| title3       = 16-bit linear PCM
}}

An '''A-law algorithm''' is a standard [[Companding|companding]] algorithm, used in [[Earth:Europe|Europe]]an 8-bit PCM [[Digital data|digital]] [[Engineering:Telecommunication|communications]] systems to optimize, i.e. modify, the [[Dynamic range|dynamic range]] of an [[Engineering:Analog signal|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|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-[[Physics:Distortion|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|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.

== See also ==
* [[Μ-law algorithm|μ-law algorithm]]
* Audio level compression
* [[Signal compression]]
* [[Companding]]
* [[G.711]]
* DS0
* [[Tapered floating point]]

== External links ==
* [http://www.cisco.com/en/US/tech/tk1077/technologies_tech_note09186a00801149b3.shtml Waveform Coding Techniques] - Has details of implementation (but note that the A-law equation is incorrect)
* [https://github.com/deftio/companders A-law implementation in C-language with example code]
{{Compression Methods}}

[[Category:Audio codecs]]

{{Sourceattribution|A-law algorithm}}