# Additive noise

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