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One of the main constituents of systems for the transmission of information considered in the theory of information transmission. It is used in the mathematical description of an actual communication channel, that is, the complex of technical apparatus enabling one to transmit information from a transmitter to a receiver over physical communication lines and media in which signals are propagated from the transmitter to the receiver.

A communication channel with one transmitter and one receiver used for the transmission of information in one direction (from the transmitter to the receiver), is defined by a pair of measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233502.png" /> (spaces of signals transmitted by the transmitter and received by the receiver, respectively), a transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233505.png" />, which is measurable with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233506.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233507.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233508.png" /> and is a [[Probability measure|probability measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c0233509.png" /> for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335010.png" /> (the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335011.png" /> gives the conditional probability distribution of the signal received by the receiver under the condition that the transmitter has transmitted the signal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335012.png" />), and a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335013.png" /> of the space of all probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335015.png" /> defines a restriction on the distribution of the signal transmitted by the transmitter). Two random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335017.png" /> defined on some probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335018.png" /> are connected by the channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335019.png" /> if they take values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335021.png" />, respectively, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335022.png" /> the conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335023.png" /> with probability 1, and if the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335024.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335025.png" />.

In applications one often encounters communication channels for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335027.png" /> are spaces of measurable functions defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335028.png" /> and taking values, respectively, in measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335030.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335031.png" />-algebra generated by the measurable cylindrical sets. In this case one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335033.png" /> and talks about a continuous-time communication channel on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335034.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335036.png" /> are random processes with values in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335038.png" />, respectively; the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335040.png" /> are then regarded as the transmitted and received signals at the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335041.png" />. Similarly, in the discrete-time case, the input and output signals of the communication channel are given by random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335043.png" /> the components of which take values in the measurable spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335045.png" />, respectively; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335047.png" /> are to be regarded as the transmitted and received signals at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335049.png" /> is the time interval between two successive transmissions of signals over the communication channel.

Often continuous- or discrete-time communication channels are also considered on a semi-infinite interval or an interval which is infinite on both sides. For example, by a continuous-time communication channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335050.png" /> on the semi-infinite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335051.png" /> one means an ordinary family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335052.png" /> of communication channels as described above, defined on all finite intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335053.png" /> and satisfying certain compatibility conditions. In this case each communication channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335054.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335055.png" /> segment of the communication channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335056.png" />. One of the possible variants of the compatibility conditions can be stated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335057.png" /> be an arbitrary set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335058.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335059.png" /> be an arbitrary function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335060.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335061.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335065.png" />. Then the compatibility condition imposed on the transition functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335066.png" /> is that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335067.png" />, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335069.png" />, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335070.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335073.png" />,

<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/c/c023/c023350/c02335074.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335075.png" /> is the cylindrical set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335076.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335077.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335078.png" />. The restrictions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335080.png" />, imposed on the probability measures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335081.png" /> are also required to satisfy certain compatibility conditions.

The notion of a continuous-time communication channel on the infinite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335082.png" /> is defined similarly. The compatibility conditions, necessary for the definition of such a channel, are usually formulated specifically for each type of channel. In some concrete situations, the notion of a communication channel on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335083.png" /> or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335084.png" /> is introduced directly (that is, without considering finite segments of these channels) by means of an explicit description of the processes performed on the input signal of the communication channel upon receipt of the output signal (see, for example, [[Gaussian channel|Gaussian channel]]; [[Additive noise|Additive noise]]). Everything said concerning a continuous-time communication channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335085.png" /> on intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335086.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335087.png" /> holds also for the analogous discrete-time communication channels.

Communication channels are divided into various classes depending on the types of the conditional distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335088.png" /> and of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023350/c02335089.png" /> (see, for example, [[Memoryless channel|Memoryless channel]]; [[Channel with a finite memory|Channel with a finite memory]]; [[Channel with a finite number of states|Channel with a finite number of states]]; [[Gaussian channel|Gaussian channel]]; [[Symmetric channel|Symmetric channel]]). A fundamental characteristic of a communication channel is the [[Transmission rate of a channel|transmission rate of a channel]] (capacity), which characterizes the maximum possible transmission rate of information (cf. [[Information, transmission rate of|Information, transmission rate of]]) over this channel.

Various generalizations of communication channels, as defined above, are possible; these correspond to more general and complex systems of information transmission (see, for example, [[Channel with feedback|Channel with feedback]]; [[Channel with multiple directions|Channel with multiple directions]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Shannon,   "Papers on information theory and cybernetics" , Moscow  (1963)  pp. 243–332  (In Russian; translated from English)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.L. Dobrushin,   "A general formulation of the fundamental theorem of Shannon in information theory"  ''Uspekhi Mat. Nauk'' , '''14''' :  6  (1959)  pp. 3–104  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Wolfowitz,   "Coding theorems of information theory" , Springer  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.G. Gallager,   "Information theory and reliable communication" , Wiley  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Feinstein,   "Foundations of information theory" , McGraw-Hill  (1958)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.M. Fano,   "Transmission of information. Statistical theory of communications" , M.I.T.  (1963)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I. Csiszar,   J. Körner,   "Information theory. Coding theorems for discrete memoryless systems" , Akad. Kiado  (1981)</TD></TR></table>