# Difference between revisions of "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)

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.