# 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 of the channel is given by a density , , ( and are the spaces of the values of the signals at the input and output of the channel, respectively) depending only on the difference , i.e. . In this case the signal at the output of the channel can be represented as the sum of the input signal and a random variable independent of it, called additive noise, so that .

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 , where is in the given interval, , and are random processes representing the signals at the input and the output of the channel with additive noise, respectively; moreover, the process is independent of . In particular, if 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: , , where and are Wiener noise processes. The general situation of a stochastic differential equation of the form is referred to as having multiplicative noise.

**How to Cite This Entry:**

Additive noise. R.L. DobrushinV.V. Prelov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Additive_noise&oldid=18399