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G.711 is an ITU-T standard for for audio compression. It is primarily used in telephony.

There are 2 main algorithms defined in the standard, mu-law algorithm (used in America) and a-law algorithm (used in Europe and the rest of the world). Both are logarithmic, but the later a-law was specifically designed to be simpler for a computer to process.

The equations are:
y = ln(1 + ux) / ln(1 + u) with u=255
y = Ax / (1 + ln A) for x <= 1/A where A=87.6
y = (1 + ln Ax) / (1 + ln A) for 1/A <= x <= 1

a-law encoding thus takes a 12 or 16 bit audio sample as input and converts it to an 8 bit value as follows:

Linear Input Code Compressed Code
s0000000wxyza... s000wxyz
s0000001wxyza... s001wxyz
s000001wxyzab... s010wxyz
s00001wxyzabc... s011wxyz
s0001wxyzabcd... s100wxyz
s001wxyzabcde... s101wxyz
s01wxyzabcdef... s110wxyz
s1wxyzabcdefg... s111wxyz

Where s is the sign bit.

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