# Channel with a finite memory

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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