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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/c/c021/c021440/c0214401.png" /> are determined by the input signals transmitted at the times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214403.png" /> (and therefore do not depend on the signals transmitted prior to the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214404.png" />); the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214405.png" /> is called the size (or length) of the memory of the channel.
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More precisely, a discrete-time communication channel where the input and output signals are given, respectively, by random sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214407.png" /> with values in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c0214409.png" /> is called a channel with a finite memory if a compatible set of conditional distributions
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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/c/c021/c021440/c02144010.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 $
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are determined by the input signals transmitted at the times  $  t  ^  \prime  $,
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$  t - m \leq  t  ^  \prime  \leq  t $(
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and therefore do not depend on the signals transmitted prior to the time  $  t - m $);  
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the number  $  m $
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is called the size (or length) of the memory of the channel.
  
by means of which such a channel can be defined, satisfies for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144012.png" /> the conditions
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More precisely, a discrete-time communication channel where the input and output signals are given, respectively, by random sequences  $  \eta = ( .  .  .  , \eta _ {-} 1 , \eta _ {0} , \eta _ {1} , . . . ) $
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and $  \widetilde \eta  = ( \widetilde \eta  _ {-} 1 , \widetilde \eta  _ {0} , \widetilde \eta  _ {1} , . . . ) $
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with values in the spaces  $  ( Y, S _ {Y} ) $
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and  $  ( \widetilde{Y}  , S _ {\widetilde{Y}  }  ) $
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is called a channel with a finite memory if a compatible set of conditional distributions
  
<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/c/c021/c021440/c02144013.png" /></td> </tr></table>
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$$
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{\mathsf P}
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\{ \widetilde \eta  {} _ {k}  ^ {n} \in \widetilde{A}  \mid  \
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\eta _ {- \infty }  ^ {n} \} ,
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$$
  
<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/c/c021/c021440/c02144014.png" /></td> </tr></table>
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by means of which such a channel can be defined, satisfies for any  $  i, j, k, n $,
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and  $  \widetilde{A}  , \widetilde{B}  $
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the conditions
  
<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/c/c021/c021440/c02144015.png" /></td> </tr></table>
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$$
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{\mathsf P}
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\{ \widetilde \eta  {} _ {k}  ^ {n} \in \widetilde{A}  \mid  \
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\eta _ {- \infty }  ^ {n} \}  = {\mathsf P}
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\{ \widetilde \eta  {} _ {k}  ^ {n} \in \widetilde{A}  \mid  \
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\eta _ {k - m }  ^ {n} \} ,
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$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144018.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144019.png" />) is a set in the direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144020.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144021.png" />) copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021440/c02144022.png" />. A continuous-time channel with a finite memory is defined similarly.
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$$
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{\mathsf P} \{ \widetilde \eta  {} _ {i}  ^ {j} \in \widetilde{B}  ,
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\widetilde \eta  {} _ {k}  ^ {n} \in \widetilde{A}  \mid  \eta _ {- \infty }  ^ {n} \} =
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$$
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$$
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= \
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{\mathsf P} \{ \widetilde \eta  {} _ {i}  ^ {j} \in \widetilde{B}
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\mid  \eta _ {- \infty }  ^ {n} \}  \cdot  {\mathsf P}
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\{ \widetilde \eta  {} _ {k}  ^ {n} \in \widetilde{A}  \mid  \eta _ {- \infty }  ^ {n} \} .
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$$
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Here  $  \widetilde \eta  _ {k}  ^ {n} = ( \widetilde \eta  _ {k} \dots \widetilde \eta  _ {n} ) $,  
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$  \eta _ {- \infty }  ^ {n} = ( . . . , \eta _ {n - 1 }  , \eta _ {n} ) $,
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and $  \widetilde{A}  $(
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respectively, $  \widetilde{B}  $)  
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is a set in the direct product of $  n - k + 1 $(
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respectively, $  j - i + 1 $)  
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copies of $  ( \widetilde{Y}  , S _ {\widetilde{Y}  }  ) $.  
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A continuous-time channel with a finite memory is defined similarly.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "On the basic theorems of information theory"  ''Uspekhi Mat. Nauk'' , '''11''' :  1  (1956)  pp. 17–75  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Feinstein,  "Foundations of information theory" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Wolfowitz,  "Coding theorems of information theory" , Springer  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Ya. Khinchin,  "On the basic theorems of information theory"  ''Uspekhi Mat. Nauk'' , '''11''' :  1  (1956)  pp. 17–75  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Feinstein,  "Foundations of information theory" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Wolfowitz,  "Coding theorems of information theory" , Springer  (1964)</TD></TR></table>

Latest revision as of 16:43, 4 June 2020


A communication channel for which the statistical properties of the output signal at a time $ t $ are determined by the input signals transmitted at the times $ t ^ \prime $, $ t - m \leq t ^ \prime \leq t $( and therefore do not depend on the signals transmitted prior to the time $ t - m $); the number $ m $ is called the size (or length) of the memory of the channel.

More precisely, a discrete-time communication channel where the input and output signals are given, respectively, by random sequences $ \eta = ( . . . , \eta _ {-} 1 , \eta _ {0} , \eta _ {1} , . . . ) $ and $ \widetilde \eta = ( \widetilde \eta _ {-} 1 , \widetilde \eta _ {0} , \widetilde \eta _ {1} , . . . ) $ with values in the spaces $ ( Y, S _ {Y} ) $ and $ ( \widetilde{Y} , S _ {\widetilde{Y} } ) $ is called a channel with a finite memory if a compatible set of conditional distributions

$$ {\mathsf P} \{ \widetilde \eta {} _ {k} ^ {n} \in \widetilde{A} \mid \ \eta _ {- \infty } ^ {n} \} , $$

by means of which such a channel can be defined, satisfies for any $ i, j, k, n $, and $ \widetilde{A} , \widetilde{B} $ the conditions

$$ {\mathsf P} \{ \widetilde \eta {} _ {k} ^ {n} \in \widetilde{A} \mid \ \eta _ {- \infty } ^ {n} \} = {\mathsf P} \{ \widetilde \eta {} _ {k} ^ {n} \in \widetilde{A} \mid \ \eta _ {k - m } ^ {n} \} , $$

$$ {\mathsf P} \{ \widetilde \eta {} _ {i} ^ {j} \in \widetilde{B} , \widetilde \eta {} _ {k} ^ {n} \in \widetilde{A} \mid \eta _ {- \infty } ^ {n} \} = $$

$$ = \ {\mathsf P} \{ \widetilde \eta {} _ {i} ^ {j} \in \widetilde{B} \mid \eta _ {- \infty } ^ {n} \} \cdot {\mathsf P} \{ \widetilde \eta {} _ {k} ^ {n} \in \widetilde{A} \mid \eta _ {- \infty } ^ {n} \} . $$

Here $ \widetilde \eta _ {k} ^ {n} = ( \widetilde \eta _ {k} \dots \widetilde \eta _ {n} ) $, $ \eta _ {- \infty } ^ {n} = ( . . . , \eta _ {n - 1 } , \eta _ {n} ) $, and $ \widetilde{A} $( respectively, $ \widetilde{B} $) is a set in the direct product of $ n - k + 1 $( respectively, $ j - i + 1 $) copies of $ ( \widetilde{Y} , S _ {\widetilde{Y} } ) $. A continuous-time channel with a finite memory is defined similarly.

References

[1] A.Ya. Khinchin, "On the basic theorems of information theory" Uspekhi Mat. Nauk , 11 : 1 (1956) pp. 17–75 (In Russian)
[2] A.A. Feinstein, "Foundations of information theory" , McGraw-Hill (1968)
[3] J. Wolfowitz, "Coding theorems of information theory" , Springer (1964)
How to Cite This Entry:
Channel with a finite memory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Channel_with_a_finite_memory&oldid=46304
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