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[[File:White noise.svg|thumb|right|Plot of a Gaussian white noise signal]]
In [[signal processing]], '''white noise''' is a random [[signal (information theory)|signal]] having equal intensity at different [[frequencies]], giving it a constant [[power spectral density]].<ref>{{cite book|last=Carter,Mancini|first=Bruce,Ron|title=Op Amps for Everyone|year=2009|publisher=Texas Instruments|isbn=978-0080949482|pages=10–11}}</ref> The term is used, with this or similar meanings, in many scientific and technical disciplines, including [[physics]], [[acoustical engineering]], [[telecommunications]], and [[statistical forecasting]]. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. White noise draws its name from [[White#Optics|white light]],<ref>{{cite book |last=Stein |first=Michael L. |date=1999 |title=Interpolation of Spatial Data: Some Theory for Kriging |publisher=Springer |page=40 |isbn=978-1-4612-7166-6 |quote=white light is approximately an equal mixture of all visible frequencies of light, which was demonstrated by Isaac Newton|doi=10.1007/978-1-4612-1494-6 |series=Springer Series in Statistics }}</ref> although light that appears white generally does not have a flat power spectral density over the visible band.

[[File:White-noise-mv255-240x180.png|thumb|A "white noise" image]]
In [[discrete time]], white noise is a [[discrete signal]] whose [[sample (signal)|samples]] are regarded as a sequence of [[serial correlation|serially uncorrelated]] [[random variable]]s with zero [[mean (statistics)|mean]] and finite [[variance]]; a single realization of white noise is a '''random shock'''. Depending on the context, one may also require that the samples be [[statistical independence|independent]] and have identical [[probability distribution]] (in other words [[i.i.d.|independent and identically distributed random variables]] are the simplest representation of white noise).<ref>{{cite book |last=Stein |first=Michael L. |date=1999 |title=Interpolation of Spatial Data: Some Theory for Kriging |publisher=Springer |page=40 |isbn=978-1-4612-7166-6 |quote=The best-known generalized process is white noise, which can be thought of as a continuous time analogue to a sequence of independent and identically distributed observations.|doi=10.1007/978-1-4612-1494-6 |series=Springer Series in Statistics }}</ref> In particular, if each sample has a [[normal distribution]] with zero mean, the signal is said to be [[additive white Gaussian noise]].<ref>{{Cite book|last=Diebold|first= Frank|title=Elements of Forecasting |edition=Fourth |year=2007}}</ref>

The samples of a white noise signal may be [[sequential]] in time, or arranged along one or more spatial dimensions. In [[digital image processing]], the [[pixel]]s of a ''white noise image'' are typically arranged in a rectangular grid, and are assumed to be independent random variables with [[Continuous uniform distribution|uniform probability distribution]] over some interval. The concept can be defined also for signals spread over more complicated domains, such as a [[sphere]] or a [[torus]].

[[File:White-noise-sound-20sec-mono-44100Hz.ogg|right|thumb|Some "white noise" sound (Loud)]]
An ''infinite-bandwidth white noise signal'' is a purely theoretical construction. The bandwidth of white noise is limited in practice by the mechanism of noise generation, by the transmission medium and by finite observation capabilities. Thus, random signals are considered "white noise" if they are observed to have a flat spectrum over the range of frequencies that are relevant to the context. For an [[audio signal]], the relevant range is the band of audible sound frequencies (between 20 and 20,000 [[hertz|Hz]]). Such a signal is heard by the human ear as a ''hissing sound'', resembling the /h/ sound in a sustained aspiration. On the other hand, the /sh/ sound in "ash" is a colored noise because it has a formant structure. In [[music]] and [[acoustics]], the term "white noise" may be used for any signal that has a similar hissing sound.<!-- Other uses of the term -->

The term white noise is sometimes used in the context of [[Phylogenetic comparative methods|phylogenetically based statistical methods]] to refer to a lack of phylogenetic pattern in comparative data.<ref>{{cite journal |title=Developmental trait evolution in trilobites |journal=Evolution |volume=66 |issue= 2|pages=314–329 |year=2011 |pmid= 22276531|doi=10.1111/j.1558-5646.2011.01447.x|url=http://www.biology.ucr.edu/people/faculty/Garland/Fusco_et_al_2011_trilobites.pdf |last1=Fusco |first1=G|last2=Garland |first2=T., Jr |last3=Hunt |first3=G  |last4=Hughes |first4=NC}}</ref>  It is sometimes used analogously in nontechnical contexts to mean "random talk without meaningful contents".<ref name=shipman>
  [[Claire Shipman]] (2005), ''[[Good Morning America]]'': "The political [[rhetoric]] on [[Social Security (United States)|Social Security]] is white noise.'' Said on [[American Broadcasting Company|ABC]]'s ''[[Good Morning America]]'' TV show, January 11, 2005.</ref><ref>
  [[Don DeLillo]] (1985), ''[[White Noise (novel)|White Noise]]''
</ref>

==Statistical properties==
[[File:Noise.jpg|thumb|240px|[[Spectrogram]] of [[pink noise]] (left) and white noise (right), shown with linear frequency axis (vertical) versus time axis (horizontal). ]]
Any distribution of values is possible (although it must have zero [[DC component]]). Even a binary signal which can only take on the values 1 or –1 will be white if the sequence is statistically uncorrelated. Noise having a continuous distribution, such as a [[normal distribution]], can of course be white.

It is often incorrectly assumed that [[Gaussian noise]] (i.e., noise with a Gaussian amplitude distribution{{snd}}see [[normal distribution]]) necessarily refers to white noise, yet neither property implies the other. Gaussianity refers to the probability distribution with respect to the value, in this context the probability of the signal falling within any particular range of amplitudes, while the term 'white' refers to the way the signal power is distributed (i.e., independently) over time or among frequencies.

White noise is the generalized mean-square derivative of the [[Wiener process]] or [[Brownian motion]].

A generalization to [[random element]]s on infinite dimensional spaces, such as [[random field]]s, is the [[nuclear space|white noise measure]].

==Practical applications==

===Music===
White noise is commonly used in the production of [[electronic music]], usually either directly or as an input for a filter to create other types of noise signal. It is used extensively in [[audio synthesis]], typically to recreate percussive instruments such as [[cymbal]]s or [[snare drum]]s which have high noise content in their frequency domain. A simple example of white noise is a nonexistent radio station (static).

===Electronics engineering===
White noise is also used to obtain the [[impulse response]] of an electrical circuit, in particular of [[amplifier]]s and other audio equipment. It is not used for testing loudspeakers as its spectrum contains too great an amount of high frequency content. [[Pink noise]], which differs from white noise in that it has equal energy in each octave, is used for testing transducers such as loudspeakers and microphones.

===Computing===
White noise is used as the basis of some [[hardware random number generator|random number generators]]. For example, [[Random.org]] uses a system of atmospheric antennae to generate random digit patterns from white noise.

===Tinnitus treatment===
White noise is a common synthetic noise source used for sound masking by a [[tinnitus masker]].<ref>{{ cite book|last=Jastreboff |first= P. J.|chapter= Tinnitus Habituation Therapy (THT) and Tinnitus Retraining Therapy (TRT)|title= Tinnitus Handbook|location= San Diego|publisher=Singular|year= 2000|pages=357–376}}</ref> [[White noise machine]]s and other white noise sources are sold as privacy enhancers and sleep aids and to mask [[tinnitus]].<ref>{{cite journal |title=Evidence based complementary intervention for insomnia |journal=Hawaii Med J |volume=61 |issue=9 |pages=192, 213 |date=September 2002 |pmid=12422383 |doi= |url=http://cogprints.org/5032/1/2002_H.M.J_White-noise_for_PTSD.pdf |last1=López |first1=HH |last2=Bracha |first2=AS |last3=Bracha |first3=HS}}</ref> Alternatively, the use of an FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.<ref>{{Cite journal | issn = 0016-867X | volume = 58 | issue = 2 | pages = 28–34 | last = Noell | first = Courtney A |author2=William L Meyerhoff  | title = Tinnitus. Diagnosis and treatment of this elusive symptom | journal = Geriatrics | date = February 2003 | pmid=12596495}}</ref> However, white noise generated from a common commercial radio receiver tuned to an unused frequency is extremely vulnerable to being contaminated with spurious signals, such as adjacent radio stations, harmonics from non-adjacent radio stations, electrical equipment in the vicinity of the receiving antenna causing interference, or even atmospheric events such as solar flares and especially lightning.

There is evidence that white noise exposure therapies may induce maladaptive changes in the brain that degrade neurological health and compromise cognition.<ref>{{Cite journal|last=Attarha|first=Mouna|last2=Bigelow|first2=James|last3=Merzenich|first3=Michael M.|date=2018-10-01|title=Unintended Consequences of White Noise Therapy for Tinnitus-Otolaryngology's Cobra Effect: A Review|url=https://www.ncbi.nlm.nih.gov/pubmed/30178067|journal=JAMA Otolaryngology–Head & Neck Surgery |volume=144|issue=10|pages=938–943|doi=10.1001/jamaoto.2018.1856|issn=2168-619X|pmid=30178067}}</ref>

===Work environment===
The effects of white noise upon cognitive function are mixed. Recently, a small study found that white noise background stimulation improves cognitive functioning among secondary students with [[attention deficit hyperactivity disorder]] (ADHD), while decreasing performance of non-ADHD students.<ref>{{Cite journal| volume = 6| issue = 1| pages = 55| last = Soderlund| first = Goran|author2=Sverker Sikstrom |author3=Jan Loftesnes |author4=Edmund Sonuga Barke | title = The effects of background white noise on memory performance in inattentive school children| journal = Behavioral and Brain Functions| year = 2010| doi=10.1186/1744-9081-6-55| pmid = 20920224| pmc = 2955636}}</ref><ref>{{Cite journal | doi = 10.1111/j.1469-7610.2007.01749.x | pmid = 17683456 | issn = 0021-9630 | volume = 48 | issue = 8 | pages = 840–847 | last = Söderlund | first = Göran |author2=Sverker Sikström |author3=Andrew Smart  | title = Listen to the noise: Noise is beneficial for cognitive performance in ADHD. | journal = Journal of Child Psychology and Psychiatry | year = 2007 | citeseerx = 10.1.1.452.530 }}</ref> Other work indicates it is effective in improving the mood and performance of workers by masking background office noise,<ref>{{Cite journal | doi = 10.1177/0013916592243006 | volume = 24 | issue = 3 | pages = 381–395 | last = Loewen | first = Laura J. |author2=Peter Suedfeld  | title = Cognitive and Arousal Effects of Masking Office Noise | journal = Environment and Behavior | date = 1992-05-01 }}</ref> but decreases cognitive performance in complex card sorting tasks.<ref>{{Cite journal | issn = 0022-1309 | volume = 120 | issue = 3 | pages = 339–355 | last = Baker | first = Mary Anne |author2=Dennis H. Holding  | title = The effects of noise and speech on cognitive task performance. | journal = Journal of General Psychology | date = July 1993 | doi=10.1080/00221309.1993.9711152| pmid = 8138798 }}</ref>

Similarly, an experiment was carried out on sixty six healthy participants to observe the benefits of using white noise in a learning environment. The experiment involved the participants identifying different images whilst having different sounds in the background. Overall the experiment showed that white noise does in fact have benefits in relation to learning. The experiments showed that white noise improved the participant's learning abilities and their recognition memory slightly.<ref>Rausch, V. H. (2014). White noise improves learning by modulating activity in dopaminergic midbrain regions and right superior temporal sulcus . Journal of cognitive neuroscience , 1469-1480</ref>

==Mathematical definitions==

===White noise vector===
A [[random vector]] (that is, a partially indeterminate process that produces vectors of real numbers) is said to be a white noise vector or white random vector if its components each have a [[probability distribution]] with zero mean and finite [[variance]], and are [[statistically independent]]: that is, their [[joint probability distribution]] must be the product of the distributions of the individual components.<ref name=fessler>
  Jeffrey A. Fessler (1998), [http://andywilliamson.org/_/wp-content/uploads/2010/04/White-Noise.pdf ''On Transformations of Random Vectors.''] Technical report 314, Dept. of Electrical Engineering and Computer Science, Univ. of Michigan.</ref>

A necessary (but, [[normally distributed and uncorrelated does not imply independent|in general, not sufficient]]) condition for statistical independence of two variables is that they be [[correlation|statistically uncorrelated]]; that is, their [[covariance]] is zero. Therefore, the  [[covariance matrix]] ''R'' of the components of a white noise vector ''w'' with ''n'' elements must be an ''n'' by ''n'' [[diagonal matrix]], where each diagonal element ''Rᵢᵢ'' is the [[variance]] of component ''wᵢ''; and the [[Correlation and dependence#Correlation matrices|correlation]] matrix must be the ''n'' by ''n'' identity matrix.

In particular, if in addition to being independent every variable in ''w'' also has a [[normal distribution]] with zero mean and the same variance <math>\sigma^2</math>, ''w'' is said to be a Gaussian white noise vector. In that case, the joint distribution of ''w'' is a [[multivariate normal distribution]]; the independence between the variables then implies that the distribution has [[elliptical distribution|spherical symmetry]] in ''n''-dimensional space. Therefore, any [[orthogonal transformation]] of the vector will result in a Gaussian white random vector. In particular, under most types of [[discrete Fourier transform]], such as [[fast Fourier transform|FFT]] and [[discrete Hartley transform|Hartley]], the transform ''W'' of ''w'' will be a Gaussian white noise vector, too; that is, the ''n'' Fourier coefficients of ''w'' will be independent Gaussian variables with zero mean and the same variance <math>\sigma^2</math>.

The [[power spectrum]] ''P'' of a random vector ''w'' can be defined as the expected value of the squared modulus of each coefficient of its Fourier transform ''W'', that is, ''Pᵢ'' = E(|''Wᵢ''|²). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with ''Pᵢ''&nbsp;=&nbsp;''&sigma;² for all&nbsp;''i''.

If ''w'' is a white random vector, but not a Gaussian one, its Fourier coefficients ''Wᵢ'' will not be completely independent of each other; although for large ''n'' and common probability distributions the dependencies are very subtle, and their pairwise correlations can be assumed to be zero.

Often the weaker condition "statistically uncorrelated" is used in the definition of white noise, instead of "statistically independent". However some of the commonly expected properties of white noise (such as flat power spectrum) may not hold for this weaker version. Under this assumption, the stricter version can be referred to explicitly as independent white noise vector.<ref name=ezivot>
  Eric Zivot and Jiahui Wang (2006), [http://faculty.washington.edu/ezivot/econ584/notes/timeSeriesConcepts.pdf Modeling Financial Time Series with S-PLUS]. Second Edition.
</ref>{{rp|p.60}} Other authors use strongly white and weakly white instead.<ref name=diebold>
  Francis X. Diebold (2007), ''Elements of Forecasting,'' 4th edition.
</ref>

An example of a random vector that is "Gaussian white noise" in the weak but not in the strong sense is ''x''=[''x''₁,''x''₂] where ''x''₁ is a normal random variable with zero mean, and ''x''₂ is equal to +''x''₁ or to −''x''₁, with equal probability. These two variables are uncorrelated and individually normally distributed, but they are not jointly normally distributed and are not independent. If ''x'' is rotated by 45 degrees, its two components will still be uncorrelated, but their distribution will no longer be normal.

In some situations one may relax the definition by allowing each component of a white random vector ''w'' to have non-zero expected value <math>\mu</math>. In [[image processing]] especially, where samples are typically restricted to positive values, one often takes <math>\mu</math> to be one half of the maximum sample value. In that case, the Fourier coefficient ''W''₀ corresponding to the zero-frequency component (essentially, the average of the ''w''_i) will also have a non-zero expected value <math>\mu\sqrt{n}</math>; and the power spectrum ''P'' will be flat only over the non-zero frequencies.

===Discrete-time white noise===
A discrete-time [[stochastic process]] <math>W[n]</math> is a generalization of random vectors with a finite number of components to infinitely many components. A discrete-time stochastic process <math>W[n]</math> is called white noise if its mean does not depend on the time <math>n</math> and is equal to zero, i.e. <math>\operatorname{E}[W[n]] = 0</math> and if the autocorrelation function <math>R_{W}[n] = \operatorname{E}[W[k+n]W[k]]</math> only depends on <math>n</math> but not on <math>k</math> and has a nonzero value only for <math>n = 0</math>, i.e. <math>R_{W}[n] = \sigma^2 \delta [n]</math>.

===Continuous-time white noise===
In order to define the notion of "white noise" in the theory of [[continuous-time]] signals, one must replace the concept of a "random vector" by a continuous-time random signal; that is, a random process that generates a function <math>w</math> of a real-valued parameter <math>t</math>.

Such a process is said to be white noise in the strongest sense if the value <math>w(t)</math> for any time <math>t</math> is a random variable that is statistically independent of its entire history before <math>t</math>. A weaker definition requires independence only between the values <math>w(t_1)</math> and <math>w(t_2)</math> at every pair of distinct times <math>t_1</math> and <math>t_2</math>. An even weaker definition requires only that such pairs <math>w(t_1)</math> and <math>w(t_2)</math> be uncorrelated.<ref name=econterms>
  [http://economics.about.com/od/economicsglossary/g/whitenoise.htm ''White noise process'']. By Econterms via About.com. Accessed on 2013-02-12.
</ref>  As in the discrete case, some authors adopt the weaker definition for "white noise", and use the qualifier independent to refer to either of the stronger definitions. Others use weakly white and strongly white to distinguish between them.

However, a precise definition of these concepts is not trivial, because some quantities that are  finite sums in the finite discrete case must be replaced by integrals that may not converge. Indeed, the set of all possible instances of a signal <math>w</math> is no longer a finite-dimensional space <math>\mathbb{R}^n</math>, but an infinite-dimensional [[function space]]. Moreover, by any definition a white noise signal <math>w</math> would have to be essentially discontinuous at every point; therefore even the simplest operations on <math>w</math>, like integration over a finite interval, require advanced mathematical machinery.

Some authors require each value <math>w(t)</math> to be a real-valued random variable with expectation <math>\mu</math> and some finite variance <math>\sigma^2</math>. Then the covariance <math>\mathrm{E}(w(t_1)\cdot w(t_2))</math> between the values at two times <math>t_1</math> and <math>t_2</math> is well-defined: it is zero if the times are distinct, and <math>\sigma^2</math> if they are equal. However, by this definition, the integral
: <math>W_{[a,a+r]} = \int_a^{a+r} w(t)\, dt</math>
over any interval with positive width <math>r</math> would be simply the width times the expectation: <math>r\mu</math>. This property would render the concept inadequate as a model of physical "white noise" signals.

Therefore, most authors define the signal <math>w</math> indirectly by specifying non-zero values for the integrals of <math>w(t)</math> and <math>|w(t)|^2</math> over any interval <math>[a,a+r]</math>, as a function of its width <math>r</math>. In this approach, however, the value of <math>w(t)</math> at an isolated time cannot be defined as a real-valued random variable{{Citation needed|reason=an authoritative work on white noise given one such example should be given|date=January 2017}}. Also the covariance <math>\mathrm{E}(w(t_1)\cdot w(t_2))</math> becomes infinite when <math>t_1=t_2</math>; and the [[autocorrelation]] function <math>\mathrm{R}(t_1,t_2)</math> must be defined as <math>N \delta(t_1-t_2)</math>, where <math>N</math> is some real constant and <math>\delta</math> is [[Dirac delta function|Dirac's "function"]].

In this approach, one usually specifies that the integral <math>W_I</math> of <math>w(t)</math> over an interval <math>I=[a,b]</math> is a real random variable with normal distribution, zero mean, and variance <math>(b-a)\sigma^2</math>; and also that the covariance <math>\mathrm{E}(W_I\cdot W_J)</math> of the integrals <math>W_I</math>, <math>W_J</math> is <math>r\sigma^2</math>, where <math>r</math> is the width of the intersection <math>I\cap J</math> of the two intervals <math>I,J</math>. This model is called a Gaussian white noise signal (or process).

==Mathematical applications==

===Time series analysis and regression===
In [[statistics]] and [[econometrics]] one often assumes that an observed series of data values is the sum of a series of values generated by a [[deterministic]] [[linear model|linear process]], depending on certain [[Dependent and independent variables|independent (explanatory) variables]], and on a series of random noise values. Then [[regression analysis]] is used to infer the parameters of the model process from the observed data, e.g. by [[ordinary least squares]], and to [[hypothesis testing|test the null hypothesis]] that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same Gaussian probability distribution{{snd}}in other words, that the noise is white. If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are still [[bias of an estimator|unbiased]], but estimates of their uncertainties (such as [[confidence interval]]s) will be biased (not accurate on average). This is also true if the noise is [[heteroskedasticity|heteroskedastic]]{{snd}}that is, if it has different variances for different data points.

Alternatively, in the subset of regression analysis known as [[time series analysis]] there are often no explanatory variables other than the past values of the variable being modeled (the [[dependent variable]]). In this case the noise process is often modeled as a [[Moving average model|moving average]] process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.

===Random vector transformations===
These two ideas are crucial in applications such as [[channel estimation]] and [[Mixing console#Channel equalization|channel equalization]] in [[telecommunication|communications]] and [[sound reproduction|audio]]. These concepts are also used in [[data compression]].

<!-- This does not seem to be incorrect but seems to be original research, sort of. Needs to be trimmed to the bare essentials. -->
In particular, by a suitable linear transformation (a [[coloring transformation]]), a white random vector can be used to produce a "non-white" random vector (that is, a list of random variables) whose elements have a prescribed [[covariance matrix]]. Conversely, a random vector with known covariance matrix can be transformed into a white random vector by a suitable [[whitening transformation]].

==Generation==
White noise may be generated digitally with a [[digital signal processor]], [[microprocessor]], or [[microcontroller]]. Generating white noise typically entails feeding an appropriate stream of random numbers to a [[digital-to-analog converter]]. The quality of the white noise will depend on the quality of the algorithm used.<ref>{{cite web |url=http://www.dspguru.com/dsp/howtos/how-to-generate-white-gaussian-noise |title=How to Generate White Gaussian Noise |author=Matt Donadio |accessdate=2012-09-19}}</ref>

==See also==
{{colors of noise}}
{{columns-list|colwidth=30em|
*[[Bochner–Minlos theorem]]
*[[Brownian noise]]
*[[Dirac delta function]]
*[[Electronic noise]]
*[[Independent component analysis]]
*[[Noise (physics)]]
*[[Noise (video)]]
*[[Principal components analysis]]
*[[Sound masking]]
}}

==References==
{{reflist|30em}}

==External links==
{{Commons category|White noise}}

{{Noise}}
{{Stochastic processes}}

{{DEFAULTSORT:White Noise}}
[[Category:Noise (electronics)]]
[[Category:Statistical signal processing]]
[[Category:Data compression]]