# Compression laws (A-law and u-law)

The laws used in compressor of a non-uniform Quantizer are known as compression laws.

two laws are available

1. $mu&is-pending-load=1$– law.
2. $A&is-pending-load=1$-law.

$mu&is-pending-load=1$-law:-  A particular form of compression law that is used in practice is the so called  $mu&is-pending-load=1$– law which is defined by

$left&space;|&space;v&space;right&space;sgn(x)fracln&space;(1_mu&space;left&space;x&space;|)(1_mu&space;),&space;for&space;0leq&space;|leq&space;1&is-pending-load=1$.

where $left&space;|&space;v&space;right&is-pending-load=1$  –  Normalized compressed output voltage.

$left&space;|&space;x&space;right&is-pending-load=1$   – Normalized input voltage to the compressor.

and  $mu&is-pending-load=1$  is a positive constant and its  value decides the curvature of $left&space;|&space;v&space;right&is-pending-load=1$.

$mu&space;=0&is-pending-load=1$   corresponds to no compression, which is the case of uniform quanization, the curve is almost linear as the value of $mu&is-pending-load=1$increases signal compression increases.

$mu&space;=255&is-pending-load=1$  is the north american standard for PCM voice telephony.

for a given value of $mu&is-pending-load=1$, the reciprocal slope of the compression curve, which defines the quantum steps is given by the derivative of  $left&space;|&space;x&space;right&is-pending-load=1$  with respect to $left&space;|&space;v&space;right&is-pending-load=1$

$fracdxdv&space;&space;fracln&space;(1_mu&space;)mu&space;&space;left&space;|&space;x&space;right&space;|)&is-pending-load=1$

we see that therefore $mu&is-pending-load=1$law is neither strictly linear nor strictly logarithmic.

i.e,

It is approximately linear at low levels of input corresponding to $mu&space;left&space;|&space;x&space;right<<1&is-pending-load=1$. and  is approximately logarithmic at high input levels corresponding to $mu&space;left&space;|&space;x&space;right>>1&is-pending-load=1$.

typical value of $mu&space;=255&is-pending-load=1$

A-law :- another compression law that is used in practice is the so called A- law defined by

$left&space;|&space;v&space;right&space;=\left\{\begin{matrix}&space;sgn(x)&space;(fracAleft&space;x&space;|1_ln&space;A),&space;0leq&space;left&space;|leq&space;frac1A_&space;(frac1_&space;ln&space;Aleft&space;A),frac1A&space;leq&space;1_&space;endmatrixright_&is-pending-load=1$

A=1 corresponds to the case of uniform Quantization. A-law has been plotted for three different values of A. A=1, A=2, A=87.56.

typical value of A is 87.56 in European Commercial PCM standard which is being followed in India.

for both the $mu&is-pending-load=1$-law and A-law, the dynamic range capability of the compander improves with increasing $mu&is-pending-load=1$and A respectively. The SNR for low-level signals increases at the expense of the SNR for high level signals.

to accommodate these two conflicting requirements (i.e, a reasonable SNR for  both low and high-level signals), a compromise is usually made in choosing the value of parameter $mu&is-pending-load=1$for the $mu&is-pending-load=1$-law and parameter A for the A-law. The typical values used in practice are $mu=255&is-pending-load=1$and $A=87.56&is-pending-load=1$.

It is also interested to note that in actual PCM systems, the companding circuitry does not produce an exact replica of the non-linear compression curves shown in the compression law characteristics.

[ratings]

## Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems.

Insert math as
$${}$$