u-LAW and A-LAW

u-LAW and A-LAW definitions

A-law and u-law are companding schemes used in telephone network to get more dynamics to the 8 bit samples that is available with linear coding. Typically 12..14 bit samples (linear scale) sampled at 8 kHz sample are companded to 8 bit (logarithmic scale) for transmission over 64 kbit/s data channel. In the receiving end the data is then converter back to linear scale (12..14 bit) and played back. converted back


u-law definition
u-LAW (pronounced mu-LAW) is

sgn(m) ( |m |) |m |
y= ------- ln( 1+ u|--|) |--| =< 1
ln(1+u) ( |mp|) |mp|

Another definition for mu-law I have seen
ln(1+255 |x|)
output = sgn(x) ---------------------
ln(1+255)

x = normalized input ( between -1 and 1)
255 = compression parameter
sgn(x) = sign (+/-) of x


a-law definition
A-LAW is
| A (m ) |m | 1
| ------- (--) |--| =< -
| 1+ln A (mp) |mp| A
y=|
| sgn(m) ( |m |) 1 |m |
| ------ ( 1+ ln A|--|) - =< |--| =< 1
| 1+ln A ( |mp|) A |mp|

Values of u=100 and 255, A=87.6, mp is the Peak message value, m is the current quantised message value. (The formulae get simpler if you substitute x for m/mp and sgn(x) for sgn(m); then -1 <= x <= 1.)

Converting from u-LAW to A-LAW is in a sense "lossy" since there are quantizing errors introduced in the conversion.

"..the u-LAW used in North America and Japan, and the A-LAW used in Europe and the rest of the world and international routes.."

References:

Modern Digital and Analog Communication Systems, B.P.Lathi., 2nd ed. ISBN 0-03-027933-X