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In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density.[1] 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 light,[2] although light that appears white generally does not have a flat power spectral density over the visible band.

In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero 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 independent and have identical probability distribution (in other words independent and identically distributed random variables are the simplest representation of white noise).[3] In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.[4]

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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 pixels of a white noise image are typically arranged in a rectangular grid, and are assumed to be independent random variables with 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.

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 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.

The term white noise is sometimes used in the context of phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data.[5] It is sometimes used analogously in nontechnical contexts to mean "random talk without meaningful contents".[6][7]

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.

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 cymbals or snare drums which have high noise content in their frequency domain.[8] A simple example of white noise is a nonexistent radio station (static).

White noise is also used to obtain the impulse response of an electrical circuit, in particular of amplifiers 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.

White noise is used as the basis of some random number generators. For example, Random.org uses a system of atmospheric antennae to generate random digit patterns from sources that can be well-modeled by white noise.[9]

White noise is a common synthetic noise source used for sound masking by a tinnitus masker.[10] White noise machines and other white noise sources are sold as privacy enhancers and sleep aids (see music and sleep) and to mask tinnitus.[11] The Marpac Sleep-Mate was the first domestic use white noise machine built in 1962 by traveling salesman Jim Buckwalter.[12] Alternatively, the use of an FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.[13] 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.

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.[14][15] Other work indicates it is effective in improving the mood and performance of workers by masking background office noise,[16] but decreases cognitive performance in complex card sorting tasks.[17]

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 participants' learning abilities and their recognition memory slightly.[18]

A random vector (that is, a random variable with values in Rn) 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,[clarification needed] and are statistically independent: that is, their joint probability distribution must be the product of the distributions of the individual components.[19]

A necessary (but, in general, not sufficient) condition for statistical independence of two variables is that they be 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 matrix must be the n by n identity matrix.

If, in addition to being independent, every variable in w also has a normal distribution with zero mean and the same variance 2 {\displaystyle \sigma ^{2}} , 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 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 FFT and 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 2 {\displaystyle \sigma ^{2}} .

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 = tag_hash_1372 for all 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.

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

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 w {\displaystyle w} of a real-valued parameter t {\displaystyle t} .

Such a process is said to be white noise in the strongest sense if the value w ( t ) {\displaystyle w(t)} for any time t {\displaystyle t} is a random variable that is statistically independent of its entire history before t {\displaystyle t} . A weaker definition requires independence only between the values w ( t 1 ) {\displaystyle w(t_{1})} and w ( t 2 ) {\displaystyle w(t_{2})} at every pair of distinct times t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} . An even weaker definition requires only that such pairs w ( t 1 ) {\displaystyle w(t_{1})} and w ( t 2 ) {\displaystyle w(t_{2})} be uncorrelated.[22] 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. ff782bc1db