# 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=49620
This article was adapted from an original article by R.L. DobrushinV.V. Prelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article