# Gaussian channel

A communication channel whose transition function determines a conditional Gaussian distribution. More precisely, a communication channel $ ( Q , V ) $
is called a Gaussian channel on the finite interval $ [ 0 , T ] $
if the following conditions hold: 1) the spaces of values of the input and output signals, $ ( {\mathcal Y} , {\mathcal S} _ {\mathcal Y} ) $
and $ ( \widetilde {\mathcal Y} , {\mathcal S} _ {\widetilde {\mathcal Y} } ) $,
are spaces of real-valued functions $ y ( t) $
and $ \widetilde{y} ( t) $,
$ t \in [ 0 , T ] $,
with the usual $ \sigma $-
algebras of measurable sets (that is, the input and output signals of a Gaussian channel are given by stochastic processes $ \eta = \{ {\eta ( t) } : {t \in [ 0 , T ] } \} $
and $ \widetilde \eta = \{ {\widetilde \eta ( t) } : {t \in [ 0 , T ] } \} $,
respectively); 2) for any fixed $ y \in Y $
the transition function $ Q ( y , \cdot ) $
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 $ V $
is imposed only on the second moment of the random variable $ \eta $.

An example of a Gaussian channel on $ ( - \infty , \infty ) $ is a channel whose input signal is given by a stationary random sequence $ \eta = (\dots, \eta _ {-1} , \eta _ {0} , \eta _ {1} ,\dots ) $ and whose output signal is the stationary random sequence $ \widetilde \eta = ( \dots, \widetilde \eta _ {-1} , \widetilde \eta _ {0} , \widetilde \eta _ {1} ,\dots ) $, obtained according to the formulas

$$ \widetilde \eta _ {i} = \ \sum _ {k = - \infty } ^ \infty a _ {k} \eta _ {i-k} + \zeta _ {i} ,\ \ i = 0 , \pm 1 , \pm 2 \dots $$

where $ \zeta = ( \dots, \zeta _ {-1} , \zeta _ {0} , \zeta _ {1} ,\dots ) $ is a stationary Gaussian random sequence independent of $ \eta $ with $ {\mathsf E} \zeta _ {i} = 0 $, $ i = \pm 1 , \pm 2 \dots $ and with spectral density $ f _ \zeta ( \lambda ) $, $ - 1 / 2 \leq \lambda \leq 1 / 2 $. The restriction on the input signal has the form

$$ \int\limits _ {- 1 / 2 } ^ { {1 } / 2 } | \Phi ( \lambda ) | ^ {2} f _ \eta ( \lambda ) d \lambda \leq S , $$

where $ f _ \eta ( \lambda ) $ is the spectral density of $ \eta $, $ \phi ( \lambda ) $ is some function and $ S $ is a constant. The capacity of such a channel is given by the formula

$$ C = \frac{1}{2} \int\limits _ {- 1 / 2 } ^ { {1 } / 2 } { \mathop{\rm log} \max } \ \left [ \left | \frac{a ( \lambda ) }{\Phi ( \lambda ) } \right | ^ {2} \cdot \frac \mu {f _ \zeta ( \lambda ) } , 1 \right ] \ d \lambda = S , $$

where $ a ( \lambda ) = \sum _ {k = - \infty } ^ \infty e ^ {- 2 \pi i k \lambda } $ and $ \mu $ is determined by the equation

$$ \int\limits _ {-1/2} ^ { 1/2 } \max \left [ \mu - \left | \frac{\Phi ( \lambda ) }{a ( \lambda ) } \ \right | ^ {2} f _ \zeta ( \lambda ) , 0 \right ] d \lambda = S . $$

See also [1], ,

cited in Communication channel.

#### References

[1] | J.M. Wozencraft, I.M. Jacobs, "Principles of communication engineering" , Wiley (1965) |

**How to Cite This Entry:**

Gaussian channel.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Gaussian_channel&oldid=55158