# Memoryless channel

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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=16884
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