Difference between revisions of "Memoryless channel"
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| − | + | A [[Communication channel|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|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|Communication channel]]. | For references see , – cited in [[Communication channel|Communication channel]]. | ||
Latest revision as of 08:00, 6 June 2020
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.
Memoryless channel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Memoryless_channel&oldid=47821