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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" />.
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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
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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>
+
A [[Communication channel|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|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 $.
  
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
+
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
  
<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>
+
$$
 +
\widetilde \eta  _ {i}  = \
 +
\sum _ {k = - \infty } ^  \infty 
 +
a _ {k} \eta _ {i-} k + \zeta _ {i} ,\ \
 +
i = 0 , \pm  1 , \pm  2 \dots
 +
$$
  
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
+
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
  
<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>
+
$$
 +
\int\limits _ {- 1 / 2 } ^ { {1 }  / 2 }
 +
| \Phi ( \lambda ) |  ^ {2}
 +
f _  \eta  ( \lambda )  d \lambda
 +
\leq  S ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
=
 +
\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 [[#References|[1]]], ,
 
See also [[#References|[1]]], ,

Latest revision as of 19:41, 5 June 2020


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=15220
This article was adapted from an original article by R.L. DobrushinV.V. Prelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article