There are 2 main algorithms defined in the standard, mulaw algorithm (used in America) and alaw algorithm (used in Europe and the rest of the world). Both are logarithmic, but the later alaw was specifically designed to be simpler for a computer to process.
The equations are:
mulaw:
y = ln(1 + ux) / ln(1 + u) with u=255
Alaw:
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
alaw 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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