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An interference added to the signal during its transmission over a [[Communication channel|communication channel]]. More precisely, one says that a given communication channel is a channel with additive noise if the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106701.png" /> of the channel is given by a density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106704.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106706.png" /> are the spaces of the values of the signals at the input and output of the channel, respectively) depending only on the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106707.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106708.png" />. In this case the signal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a0106709.png" /> at the output of the channel can be represented as the sum of the input signal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067010.png" /> and a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067011.png" /> independent of it, called additive noise, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067012.png" />.
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If one considers channels with discrete or continuous time over finite or infinite intervals, the notion of a channel with additive noise is introduced by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067014.png" /> is in the given interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067017.png" /> are random processes representing the signals at the input and the output of the channel with additive noise, respectively; moreover, the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067018.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067019.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067020.png" /> is a Gaussian random process, then the considered channel is called a [[Gaussian channel|Gaussian channel]].
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An interference added to the signal during its transmission over a [[Communication channel|communication channel]]. More precisely, one says that a given communication channel is a channel with additive noise if the transition function  $  Q(y, \cdot ) $
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of the channel is given by a density  $  q(y, \widetilde{y}  ) $,
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$  y \in {\mathcal Y} $,
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$  \widetilde{y}  \in \widetilde {\mathcal Y}  = {\mathcal Y} $(
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$  {\mathcal Y} $
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and  $  \widetilde {\mathcal Y}  $
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are the spaces of the values of the signals at the input and output of the channel, respectively) depending only on the difference  $  \widetilde{y}  - y $,
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i.e.  $  q(y, \widetilde{y}  ) = q( \widetilde{y}  -y) $.
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In this case the signal  $  \widetilde \eta  $
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at the output of the channel can be represented as the sum of the input signal  $  \eta $
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and a random variable  $  \zeta $
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independent of it, called additive noise, so that  $  \widetilde \eta  = \eta + \zeta $.
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If one considers channels with discrete or continuous time over finite or infinite intervals, the notion of a channel with additive noise is introduced by the relation $  \widetilde \eta  (t) = \eta (t) + \zeta (t) $,  
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where $  t $
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is in the given interval, $  \eta (t) $,  
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$  \widetilde \eta  (t) $
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and $  \zeta (t) $
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are random processes representing the signals at the input and the output of the channel with additive noise, respectively; moreover, the process $  \zeta (t) $
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is independent of $  \eta (t) $.  
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In particular, if $  \zeta (t) $
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is a Gaussian random process, then the considered channel is called a [[Gaussian channel|Gaussian channel]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Gallager,  "Information theory and reliable communication" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kharkevich,  "Channels with noise" , Moscow  (1965)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Gallager,  "Information theory and reliable communication" , McGraw-Hill  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kharkevich,  "Channels with noise" , Moscow  (1965)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
More generally, especially in system and control theory and stochastic analysis, the term additive noise is used for describing the following way noise enters a stochastic differential equation or observation equation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067024.png" /> are Wiener noise processes. The general situation of a stochastic differential equation of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010670/a01067025.png" /> is referred to as having multiplicative noise.
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More generally, especially in system and control theory and stochastic analysis, the term additive noise is used for describing the following way noise enters a stochastic differential equation or observation equation: $  d x = f ( x , t )  d t + d w $,
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$  d y = h ( x , t )  d t + d v $,  
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where $  w $
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and $  v $
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are Wiener noise processes. The general situation of a stochastic differential equation of the form $  d x = f ( x , t )  d t + g ( x , t )  d w $
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is referred to as having multiplicative noise.

Latest revision as of 16:09, 1 April 2020


An interference added to the signal during its transmission over a communication channel. More precisely, one says that a given communication channel is a channel with additive noise if the transition function $ Q(y, \cdot ) $ of the channel is given by a density $ q(y, \widetilde{y} ) $, $ y \in {\mathcal Y} $, $ \widetilde{y} \in \widetilde {\mathcal Y} = {\mathcal Y} $( $ {\mathcal Y} $ and $ \widetilde {\mathcal Y} $ are the spaces of the values of the signals at the input and output of the channel, respectively) depending only on the difference $ \widetilde{y} - y $, i.e. $ q(y, \widetilde{y} ) = q( \widetilde{y} -y) $. In this case the signal $ \widetilde \eta $ at the output of the channel can be represented as the sum of the input signal $ \eta $ and a random variable $ \zeta $ independent of it, called additive noise, so that $ \widetilde \eta = \eta + \zeta $.

If one considers channels with discrete or continuous time over finite or infinite intervals, the notion of a channel with additive noise is introduced by the relation $ \widetilde \eta (t) = \eta (t) + \zeta (t) $, where $ t $ is in the given interval, $ \eta (t) $, $ \widetilde \eta (t) $ and $ \zeta (t) $ are random processes representing the signals at the input and the output of the channel with additive noise, respectively; moreover, the process $ \zeta (t) $ is independent of $ \eta (t) $. In particular, if $ \zeta (t) $ is a Gaussian random process, then the considered channel is called a Gaussian channel.

References

[1] R. Gallager, "Information theory and reliable communication" , McGraw-Hill (1968)
[2] A.A. Kharkevich, "Channels with noise" , Moscow (1965) (In Russian)

Comments

More generally, especially in system and control theory and stochastic analysis, the term additive noise is used for describing the following way noise enters a stochastic differential equation or observation equation: $ d x = f ( x , t ) d t + d w $, $ d y = h ( x , t ) d t + d v $, where $ w $ and $ v $ are Wiener noise processes. The general situation of a stochastic differential equation of the form $ d x = f ( x , t ) d t + g ( x , t ) d w $ is referred to as having multiplicative noise.

How to Cite This Entry:
Additive noise. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_noise&oldid=18399
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