# Channel with a finite memory

A communication channel for which the statistical properties of the output signal at a time are determined by the input signals transmitted at the times , (and therefore do not depend on the signals transmitted prior to the time ); the number 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 and with values in the spaces and is called a channel with a finite memory if a compatible set of conditional distributions

by means of which such a channel can be defined, satisfies for any , and the conditions

Here , , and (respectively, ) is a set in the direct product of (respectively, ) copies of . 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=11735