Difference between revisions of "Additive noise"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
− | + | <!-- | |
+ | a0106701.png | ||
+ | $#A+1 = 25 n = 0 | ||
+ | $#C+1 = 25 : ~/encyclopedia/old_files/data/A010/A.0100670 Additive noise | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
− | 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}} |
+ | {{TEX|done}} | ||
+ | |||
+ | 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 | ||
+ | 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> | ||
− | |||
− | |||
====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.
Additive noise. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_noise&oldid=45027