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– law.
  2. A-law.

\mu-law:-  A particular form of compression law that is used in practice is the so called  \mu– law which is defined by 

\left | v \right |= sgn(x){\frac{\ln (1+\mu \left | x \right |)}{(1+\mu )}}, for\ 0\leq \left | x \right |\leq 1.

where \left | v \right |  –  Normalized compressed output voltage.

             \left | x \right |   – Normalized input voltage to the compressor.

and  \mu  is a positive constant and its  value decides the curvature of \left | v \right |.

\mu =0   corresponds to no compression, which is the case of uniform quanization, the curve is almost linear as the value of \muincreases signal compression increases.

\mu =255  is the north american standard for PCM voice telephony.

for a given value of \mu, the reciprocal slope of the compression curve, which defines the quantum steps is given by the derivative of  \left | x \right |  with respect to \left | v \right | 

\frac{dx}{dv} = \frac{\ln (1+\mu )}{\mu } (1+\mu \left | x \right |) 

we see that therefore \mulaw is neither strictly linear nor strictly logarithmic.


It is approximately linear at low levels of input corresponding to \mu \left | x \right |<<1. and  is approximately logarithmic at high input levels corresponding to \mu \left | x \right |>>1.

typical value of \mu =255 

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

\left | v \right | =\left\{\begin{matrix} sgn(x) (\frac{A\left | x \right |}{1+\ln A}), 0\leq \left | x \right |\leq \frac{1}{A}.\\ sgn(x) (\frac{1+ \ln A\left | x \right |}{1+\ln A}),\frac{1}{A} \leq \left | x \right |\leq 1. \end{matrix}\right.

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-law and A-law, the dynamic range capability of the compander improves with increasing \muand 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 \mufor the \mu-law and parameter A for the A-law. The typical values used in practice are \mu=255and A=87.56.

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.


Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems.

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