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A [[Communication channel|communication channel]] whose transition function possesses some kind of symmetry. A homogeneous discrete time [[Memoryless channel|memoryless channel]] with finite alphabets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916002.png" /> of input and output letters, respectively, and defined by a matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916003.png" /> is called a symmetric channel if:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916004.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916005.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916007.png" />. The most studied example of a memoryless symmetric channel is the binary symmetric channel with matrix of transition probabilities
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A [[Communication channel|communication channel]] whose transition function possesses some kind of symmetry. A homogeneous discrete time [[Memoryless channel|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916008.png" /></td> </tr></table>
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$$ \tag{* }
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q ( y, \widetilde{y}  )  = \
 +
\left \{
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s0916009.png" /> of the form (*) the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s09160010.png" /> (cf. [[Transmission rate of a channel|Transmission rate of a channel]]) is given by the equation
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\begin{array}{ll}
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q  & \textrm{ when }  y = \widetilde{y}  , \\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091600/s09160011.png" /></td> </tr></table>
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\frac{1 - q }{n - 1 }
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  & \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
 +
 
 +
$$
 +
\left \|
 +
\begin{array}{cc}
 +
q  &1 - q  \\
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1 - q  & q  \\
 +
\end{array}
 +
\
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\right \| .
 +
$$
 +
 
 +
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|Transmission rate of a channel]]) is given by the equation
 +
 
 +
$$
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C  =  \mathop{\rm log}  n + q  \mathop{\rm log}  q + ( 1 - q)  \mathop{\rm log} 
 +
\frac{1 - q }{n - 1 }
 +
.
 +
$$
  
 
For references see ,
 
For references see ,
  
 
cited under [[Communication channel|Communication channel]].
 
cited under [[Communication channel|Communication channel]].
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gallager,  "Information theory and reliable communication" , Wiley  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gallager,  "Information theory and reliable communication" , Wiley  (1968)</TD></TR></table>

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=49462
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