# Symmetric channel

A communication channel whose transition function possesses some kind of symmetry. A homogeneous discrete time memoryless channel with finite alphabets $ Y $
and $ \widetilde{Y} = Y $
of input and output letters, respectively, and defined by a matrix of transition probabilities $ \{ q ( y, \widetilde{y} ) : y, \widetilde{y} \in Y \} $
is called a symmetric channel if:

$$ \tag{* } q ( y, \widetilde{y} ) = \ \left \{ \begin{array}{ll} q & \textrm{ when } y = \widetilde{y} , \\ \frac{1 - q }{n - 1 } & \textrm{ when } y \neq \widetilde{y} , \\ \end{array} \right .$$

where $ n $ is the number of elements of $ Y $, $ 0 \leq q \leq 1 $. The most studied example of a memoryless symmetric channel is the binary symmetric channel with matrix of transition probabilities

$$ \left \| \begin{array}{cc} q &1 - q \\ 1 - q & q \\ \end{array} \ \right \| . $$

For symmetric channels, many important information-theoretic characteristics can either be calculated explicitly or their calculation can be substantially simplified in comparison with non-symmetric channels. For example, for a memoryless symmetric channel with matrix $ \{ q ( y, \widetilde{y} ) : y, \widetilde{y} \in Y \} $ of the form (*) the capacity $ C $( cf. Transmission rate of a channel) is given by the equation

$$ C = \mathop{\rm log} n + q \mathop{\rm log} q + ( 1 - q) \mathop{\rm log} \frac{1 - q }{n - 1 } . $$

For references see ,

cited under Communication channel.

#### References

[a1] | R.C. Gallager, "Information theory and reliable communication" , Wiley (1968) |

**How to Cite This Entry:**

Symmetric channel.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Symmetric_channel&oldid=53724