Memoryless channel

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.