Gaussian channel
A communication channel whose transition function determines a conditional Gaussian distribution. More precisely, a communication channel is called a Gaussian channel on the finite interval
if the following conditions hold: 1) the spaces of values of the input and output signals,
and
, are spaces of real-valued functions
and
,
, with the usual
-algebras of measurable sets (that is, the input and output signals of a Gaussian channel are given by stochastic processes
and
, respectively); 2) for any fixed
the transition function
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 with second moments that are independent of the conditioning); and 3) the restriction
is imposed only on the second moment of the random variable
.
An example of a Gaussian channel on is a channel whose input signal is given by a stationary random sequence
and whose output signal is the stationary random sequence
, obtained according to the formulas
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where is a stationary Gaussian random sequence independent of
with
,
and with spectral density
,
. The restriction on the input signal has the form
![]() |
where is the spectral density of
,
is some function and
is a constant. The capacity of such a channel is given by the formula
![]() |
where and
is determined by the equation
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See also [1], ,
cited in Communication channel.
References
[1] | J.M. Wozencraft, I.M. Jacobs, "Principles of communication engineering" , Wiley (1965) |
Gaussian channel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gaussian_channel&oldid=47054