View source for Gaussian channel

A [[Communication channel|communication channel]] whose transition function determines a conditional Gaussian distribution. More precisely, a communication channel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435801.png" /> is called a Gaussian channel on the finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435802.png" /> if the following conditions hold: 1) the spaces of values of the input and output signals, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435804.png" />, are spaces of real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435807.png" />, with the usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435808.png" />-algebras of measurable sets (that is, the input and output signals of a Gaussian channel are given by stochastic processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g0435809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358010.png" />, respectively); 2) for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358011.png" /> the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358012.png" /> of the channel determines a conditional Gaussian distribution (one says that a collection of random variables has a conditional Gaussian distribution if every finite subfamily has a conditional finite-dimensional [[Normal distribution|normal distribution]] with second moments that are independent of the conditioning); and 3) the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358013.png" /> is imposed only on the second moment of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358014.png" />.

An example of a Gaussian channel on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358015.png" /> is a channel whose input signal is given by a stationary random sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358016.png" /> and whose output signal is the stationary random sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358017.png" />, obtained according to the formulas

<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/g/g043/g043580/g04358018.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358019.png" /> is a stationary Gaussian random sequence independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358022.png" /> and with spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358024.png" />. The restriction on the input signal has the form

<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/g/g043/g043580/g04358025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358026.png" /> is the spectral density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358028.png" /> is some function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358029.png" /> is a constant. The capacity of such a channel is given by the formula

<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/g/g043/g043580/g04358030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043580/g04358032.png" /> is determined by the equation

<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/g/g043/g043580/g04358033.png" /></td> </tr></table>

See also [[#References|[1]]], ,

cited in [[Communication channel|Communication channel]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.M. Wozencraft,   I.M. Jacobs,   "Principles of communication engineering" , Wiley  (1965)</TD></TR></table>