# Difference between revisions of "Additive noise"

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+ | $#C+1 = 25 : ~/encyclopedia/old_files/data/A010/A.0100670 Additive noise | ||

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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 | + | {{TEX|auto}} |

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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 ) $ | ||

+ | 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|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> | ||

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====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: | + | 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. |

## 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