View source for Channel with a finite memory

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A [[Communication channel|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====
<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>