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

Latest revision as of 14:55, 7 June 2020


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.

Comments

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=48921
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