White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. 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, not to any specific signal. White noise draws its name from white light, 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 a mean of zero and a finite variance; a single realization of white noise is a random shock. In some contexts, it is also required that the samples be independent and have identical probability distribution. In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.
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 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. 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.
In the context of phylogenetically based statistical methods, the term white noise can refer to a lack of phylogenetic pattern in comparative data. In nontechnical contexts, it is sometimes used to mean "random talk without meaningful contents".

Statistical properties

Any distribution of values is possible. 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 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 over time or among frequencies.
One form of white noise is the generalized mean-square derivative of the Wiener process or Brownian motion.
A generalization to random elements on infinite dimensional spaces, such as random fields, is the 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 cymbals or snare drums which have high noise content in their frequency domain. A simple example of white noise is a nonexistent radio station.

Electronics engineering

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.

Computing

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

Tinnitus treatment

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

Work environment

The effects of white noise upon cognitive function are mixed. A small study published in 2007 found that white noise background stimulation improves cognitive functioning among secondary students with attention deficit hyperactivity disorder, while decreasing performance of non-ADHD students. Other work indicates it is effective in improving the mood and performance of workers by masking background office noise, but decreases cognitive performance in complex card sorting tasks.
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.

Mathematical definitions

White noise vector

A random vector 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.
A necessary 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 R_ii is the variance of component w_i; 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, 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.
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, P_i = E. Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with P_i = σ² for all i.
If w is a white random vector, but not a Gaussian one, its Fourier coefficients W_i 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 may not hold for this weaker version. Under this assumption, the stricter version can be referred to explicitly as independent white noise vector. Other authors use strongly white and weakly white instead.
An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is where is a normal random variable with zero mean, and is equal to or to, with equal probability. These two variables are uncorrelated and individually normally distributed, but they are not jointly normally distributed and are not independent. If 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 to have non-zero expected value. In image processing especially, where samples are typically restricted to positive values, one often takes to be one half of the maximum sample value. In that case, the Fourier coefficient corresponding to the zero-frequency component will also have a non-zero expected value ; and the power spectrum will be flat only over the non-zero frequencies.