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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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s0916001.png
 
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$#A+1 = 11 n = 0
 
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$#C+1 = 11 : ~/encyclopedia/old_files/data/S091/S.0901600 Symmetric channel
 
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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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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 $
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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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and  $  \widetilde{Y}  = Y $
 
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of input and output letters, respectively, and defined by a matrix of transition probabilities $  \{ q ( y, \widetilde{y}  ) :  y, \widetilde{y}  \in Y \} $
 
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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/s0916008.png" /></td> </tr></table>
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q ( y, \widetilde{y}  )  = \
 
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\left \{
 
  
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where  $  n $
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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 <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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is the number of elements of  $  Y $,  
 
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$  0 \leq  q \leq  1 $.  
 
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The most studied example of a memoryless symmetric channel is the binary symmetric channel with matrix of transition probabilities
 
  
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$$
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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/s09160011.png" /></td> </tr></table>
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\left \|
 
  
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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 \} $
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For references see ,
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of the form (*) the capacity  $  C $(
 
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cf. [[Transmission rate of a channel|Transmission rate of a channel]]) is given by the equation
 
  
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$$
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cited under [[Communication channel|Communication channel]].
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C  =  \mathop{\rm log}  n + q  \mathop{\rm log}  q + ( 1 - q)  \mathop{\rm log} 
 
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\frac{1 - q }{n - 1 }
 
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.
 
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$$
 
  
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For references see ,
 
  
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cited under [[Communication channel|Communication channel]].
 
  
 
====Comments====
 
====Comments====
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====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>

Revision as of 14:53, 7 June 2020

A communication channel whose transition function possesses some kind of symmetry. A homogeneous discrete time memoryless channel with finite alphabets and of input and output letters, respectively, and defined by a matrix of transition probabilities is called a symmetric channel if:

(*)

where is the number of elements of , . The most studied example of a memoryless symmetric channel is the binary symmetric channel with matrix of transition probabilities

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 of the form (*) the capacity (cf. Transmission rate of a channel) is given by the equation

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
Retrieved from "https://encyclopediaofmath.org/index.php?title=Symmetric_channel&oldid=49462"