View source for Memoryless channel

A [[Communication channel|communication channel]] for which the statistical properties of the output signal at a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633901.png" /> are determined only by the input signal transmitted at this moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633902.png" /> of time (and consequently do not depend on the signal transmitted prior to or after the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633903.png" />). More precisely, a discrete-time communication channel whose input and output signals are given by random sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633905.png" /> with values in spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633907.png" />, respectively, is called a memoryless channel if for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633908.png" /> and any sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m0633909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339011.png" />, the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339012.png" /></td> </tr></table>

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holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339014.png" />. If furthermore the conditional probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339015.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339016.png" />, then the channel is called a homogeneous memoryless channel.

If one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339017.png" /> the transmission rate of the channel (cf. [[Transmission rate of a channel|Transmission rate of a channel]]) for a segment of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339018.png" /> of a homogeneous memoryless channel, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339021.png" /> are finite (or countable) sets, a homogeneous memoryless channel is completely determined by the matrix of transition probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063390/m06339022.png" />, where

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For references see , – cited in [[Communication channel|Communication channel]].