# 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
- Transmission Systems for Communications, Fifth Edition, by Members of the Technical Staff at Bell Telephone Laboratories, Bell Telephone Laboratories, Incorporated, Copyright 1959, 1964, 1970, 1982