# Memoryless channel

A communication channel for which the statistical properties of the output signal at a time are determined only by the input signal transmitted at this moment of time (and consequently do not depend on the signal transmitted prior to or after the moment ). More precisely, a discrete-time communication channel whose input and output signals are given by random sequences and with values in spaces and , respectively, is called a memoryless channel if for any natural number and any sets , , , the equality

holds, where . If furthermore the conditional probabilities do not depend on , then the channel is called a homogeneous memoryless channel.

If one denotes by the transmission rate of the channel (cf. Transmission rate of a channel) for a segment of length of a homogeneous memoryless channel, then . If and are finite (or countable) sets, a homogeneous memoryless channel is completely determined by the matrix of transition probabilities , where

For references see , – cited in Communication channel.

**How to Cite This Entry:**

Memoryless channel. R.L. DobrushinV.V. Prelov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Memoryless_channel&oldid=16884