White noise
[1] The term is used with this or similar meanings in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting.
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.
[3] In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.
[4] The samples of a white noise signal may be sequential in time, or arranged along one or more spatial dimensions.
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).
In music and acoustics, the term white noise may be used for any signal that has a similar hissing sound.
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.
One form of white noise is the generalized mean-square derivative of the Wiener process or Brownian motion.
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.
White noise is also used to obtain the impulse response of an electrical circuit, in particular of amplifiers and other audio equipment.
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.
[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 AM 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.
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.
[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 experiments showed that white noise improved the participants' learning abilities and their recognition memory slightly.
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, Pi = E(|Wi|2).
Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with Pi = σ2 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.
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.
[21] An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is
[clarification needed] This property renders the concept inadequate as a model of white noise signals either in a physical or mathematical sense.
Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space
Generating white noise typically entails feeding an appropriate stream of random numbers to a digital-to-analog converter.
[23] The term is sometimes used as a colloquialism to describe a backdrop of ambient sound, creating an indistinct or seamless commotion.
Following are some examples: The term can also be used metaphorically, as in the novel White Noise (1985) by Don DeLillo which explores the symptoms of modern culture that came together so as to make it difficult for an individual to actualize their ideas and personality.
