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A [[Communication channel|communication channel]] for which the statistical properties of the output signal at a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633901.png" /> are determined only by the input signal transmitted at this moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633902.png" /> of time (and consequently do not depend on the signal transmitted prior to or after the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633903.png" />). More precisely, a discrete-time communication channel whose input and output signals are given by random sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633905.png" /> with values in spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633907.png" />, respectively, is called a memoryless channel if for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633908.png" /> and any sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339011.png" />, the equality
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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/m/m063/m063390/m06339012.png" /></td> </tr></table>
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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/m/m063/m063390/m06339013.png" /></td> </tr></table>
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A [[Communication channel|communication channel]] for which the statistical properties of the output signal at a time  $  t $
 +
are determined only by the input signal transmitted at this moment  $  t $
 +
of time (and consequently do not depend on the signal transmitted prior to or after the moment  $  t $).
 +
More precisely, a discrete-time communication channel whose input and output signals are given by random sequences  $  \eta = ( \eta _ {1} , \eta _ {2} ,\dots ) $
 +
and  $  \widetilde \eta  = ( \widetilde \eta  _ {1} , \widetilde \eta  _ {2} ,\dots ) $
 +
with values in spaces  $  ( Y , S _ {Y} ) $
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and  $  ( \widetilde{Y}  , S _ {\widetilde{Y}  }  ) $,
 +
respectively, is called a memoryless channel if for any natural number  $  n $
 +
and any sets  $  \widetilde{A}  _ {1} \dots \widetilde{A}  _ {n} $,
 +
$  \widetilde{A}  _ {k} \in S _ {\widetilde{Y}  }  $,
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$  k = 1 \dots n $,
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the equality
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339014.png" />. If furthermore the conditional probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339015.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339016.png" />, then the channel is called a homogeneous memoryless channel.
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$$
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{\mathsf P} \{ \widetilde \eta  _ {1} \in
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\widetilde{A}  _ {1} \dots \widetilde \eta  _ {n} \in \widetilde{A}  _ {n} \mid  \eta  ^ {n} \} =
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$$
  
If one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339017.png" /> the transmission rate of the channel (cf. [[Transmission rate of a channel|Transmission rate of a channel]]) for a segment of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339018.png" /> of a homogeneous memoryless channel, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339021.png" /> are finite (or countable) sets, a homogeneous memoryless channel is completely determined by the matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339022.png" />, where
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$$
 +
= \
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{\mathsf P} \{ \widetilde \eta  _ {1} \in \widetilde{A}  _ {1} \mid  \eta _ {1} \} \dots {\mathsf P} \{ \widetilde \eta  _ {n} \in \widetilde{A}  _ {n} \mid  \eta _ {n} \}
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$$
  
<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/m/m063/m063390/m06339023.png" /></td> </tr></table>
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holds, where  $  \eta  ^ {n} = ( \eta _ {1} \dots \eta _ {n} ) $.
 +
If furthermore the conditional probabilities  $  {\mathsf P} \{ \widetilde \eta  _ {k} \in \widetilde{A}  _ {k} \mid  \eta _ {k} \} $
 +
do not depend on  $  k $,
 +
then the channel is called a homogeneous memoryless channel.
 +
 
 +
If one denotes by  $  C _ {n} $
 +
the transmission rate of the channel (cf. [[Transmission rate of a channel|Transmission rate of a channel]]) for a segment of length  $  n $
 +
of a homogeneous memoryless channel, then  $  C _ {n} = n C _ {1} $.
 +
If  $  Y $
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and  $  \widetilde{Y}  $
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are finite (or countable) sets, a homogeneous memoryless channel is completely determined by the matrix of transition probabilities  $  \{ {q ( y , \widetilde{y}  ) } : {y \in Y,  \widetilde{y}  \in \widetilde{Y}  } \} $,
 +
where
 +
 
 +
$$
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q ( y , \widetilde{y}  )  = {\mathsf P}
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\{ \widetilde \eta  _ {k} = \widetilde{y}  \mid  \eta _ {k} = y \} ,\ \
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k = 1 , 2 ,\dots .
 +
$$
  
 
For references see , – cited in [[Communication channel|Communication channel]].
 
For references see , – cited in [[Communication channel|Communication channel]].

Latest revision as of 08:00, 6 June 2020


A communication channel for which the statistical properties of the output signal at a time $ t $ are determined only by the input signal transmitted at this moment $ t $ of time (and consequently do not depend on the signal transmitted prior to or after the moment $ t $). More precisely, a discrete-time communication channel whose input and output signals are given by random sequences $ \eta = ( \eta _ {1} , \eta _ {2} ,\dots ) $ and $ \widetilde \eta = ( \widetilde \eta _ {1} , \widetilde \eta _ {2} ,\dots ) $ with values in spaces $ ( Y , S _ {Y} ) $ and $ ( \widetilde{Y} , S _ {\widetilde{Y} } ) $, respectively, is called a memoryless channel if for any natural number $ n $ and any sets $ \widetilde{A} _ {1} \dots \widetilde{A} _ {n} $, $ \widetilde{A} _ {k} \in S _ {\widetilde{Y} } $, $ k = 1 \dots n $, the equality

$$ {\mathsf P} \{ \widetilde \eta _ {1} \in \widetilde{A} _ {1} \dots \widetilde \eta _ {n} \in \widetilde{A} _ {n} \mid \eta ^ {n} \} = $$

$$ = \ {\mathsf P} \{ \widetilde \eta _ {1} \in \widetilde{A} _ {1} \mid \eta _ {1} \} \dots {\mathsf P} \{ \widetilde \eta _ {n} \in \widetilde{A} _ {n} \mid \eta _ {n} \} $$

holds, where $ \eta ^ {n} = ( \eta _ {1} \dots \eta _ {n} ) $. If furthermore the conditional probabilities $ {\mathsf P} \{ \widetilde \eta _ {k} \in \widetilde{A} _ {k} \mid \eta _ {k} \} $ do not depend on $ k $, then the channel is called a homogeneous memoryless channel.

If one denotes by $ C _ {n} $ the transmission rate of the channel (cf. Transmission rate of a channel) for a segment of length $ n $ of a homogeneous memoryless channel, then $ C _ {n} = n C _ {1} $. If $ Y $ and $ \widetilde{Y} $ are finite (or countable) sets, a homogeneous memoryless channel is completely determined by the matrix of transition probabilities $ \{ {q ( y , \widetilde{y} ) } : {y \in Y, \widetilde{y} \in \widetilde{Y} } \} $, where

$$ q ( y , \widetilde{y} ) = {\mathsf P} \{ \widetilde \eta _ {k} = \widetilde{y} \mid \eta _ {k} = y \} ,\ \ k = 1 , 2 ,\dots . $$

For references see , – cited in Communication channel.

How to Cite This Entry:
Memoryless channel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Memoryless_channel&oldid=47821
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