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White noise is a random signal (or process) with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency. White noise draws its name from white light in which the power spectral density of the light is distributed over the visible band in such a way that the eye's three color receptors (cones) are approximately equally stimulated. In statistical sense, a time series r_t is characterized as having weak white noise if {r_t} is a sequence of serially uncorrelated random variables with zero mean and finite variance. Strong white noise also has the quality of being independent and identically distributed, which implies no autocorrelation. In particular, if r_t is normally distributed with mean zero and standard deviation σ, the series is called a Gaussian white noise.^[1]

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. A random signal is considered "white noise" if it is observed to have a flat spectrum over a medium's widest possible bandwidth.

White noise in a spatial context [link]

While it is usually applied in the context of frequency domain signals, the term white noise is also commonly applied to a noise signal in the spatial domain. In this case, it has an auto correlation which can be represented by a delta function over the relevant space dimensions. The signal is then "white" in the spatial frequency domain (this is equally true for signals in the angular frequency domain, e.g., the distribution of a signal across all angles in the night sky).

Statistical properties [link]

An example realization of a Gaussian white noise process.

The image to the right displays a finite length, discrete time realization of a white noise process generated from a computer.

Being uncorrelated in time does not restrict the values a signal can take. 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 — see normal distribution) is necessarily 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 reaching an amplitude, while the term 'white' refers to the way the signal power is distributed over time or among frequencies.

Spectrogram of pink noise (left) and white noise (right), shown with linear frequency axis (vertical).

We can therefore find Gaussian white noise, but also Poisson, Cauchy, etc. white noises. Thus, the two words "Gaussian" and "white" are often both specified in mathematical models of systems. Gaussian white noise is a good approximation of many real-world situations and generates mathematically tractable models. These models are used so frequently that the term additive white Gaussian noise has a standard abbreviation: AWGN.

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

Applications [link]

It is used by some emergency vehicle sirens due to its ability to cut through background noise, which makes it easier to locate.^{[citation needed]}

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 which have high noise content in their frequency domain.

It is also used to generate impulse responses. To set up the equalization (EQ) for a concert or other performance in a venue, a short burst of white or pink noise is sent through the PA system and monitored from various points in the venue so that the engineer can tell if the acoustics of the building naturally boost or cut any frequencies. The engineer can then adjust the overall equalization to ensure a balanced mix.

	White noise 10 second sample of white sound.
Problems listening to this file? See media help.

White noise can be used for frequency response testing of amplifiers and electronic filters. It is not used for testing loudspeakers as its spectrum contains too great an amount of high frequency content. Pink noise 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 white noise.

White noise is a common synthetic noise source used for sound masking by a tinnitus masker.^[2] White noise machines and other white noise sources are sold as privacy enhancers and sleep aids and to mask tinnitus.^[3] Alternatively, the use of a FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.^[4] 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.^[5][6] Other work indicates it is effective in improving the mood and performance of workers by masking background office noise,^[7] but decreases cognitive performance in complex card sorting tasks.^[8]

Mathematical definition [link]

White random vector [link]

A random vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w}

is a white random vector if and only if its mean vector and autocorrelation matrix are the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_w = \mathbb{E}\{ \mathbf{w} \} = 0

Failed to parse (Missing texvc executable; please see math/README to configure.): R_{ww} = \mathbb{E}\{ \mathbf{w} \mathbf{w}^T\} = \sigma^2 \mathbf{I} .

That is, it is a zero mean random vector, and its autocorrelation matrix is a multiple of the identity matrix. When the autocorrelation matrix is a multiple of the identity, we say that it has spherical correlation.

White random process (white noise) [link]

A continuous time random process Failed to parse (Missing texvc executable; please see math/README to configure.): w(t)

where Failed to parse (Missing texvc executable; please see math/README to configure.): t \in \mathbb{R}
is a white noise process if and only if its mean function and autocorrelation function satisfy the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_w(t) = \mathbb{E}\{ w(t)\} = 0

Failed to parse (Missing texvc executable; please see math/README to configure.): R_{ww}(t_1, t_2) = \mathbb{E}\{ w(t_1) w(t_2)\} = (N_{0}/2)\delta(t_1 - t_2).

i.e. it is a zero mean process for all time and has infinite power at zero time shift since its autocorrelation function is the Dirac delta function.

The above autocorrelation function implies the following power spectral density:

Failed to parse (Missing texvc executable; please see math/README to configure.): S_{xx}(\omega) = N_{0}/2 ,\!

since the Fourier transform of the delta function is equal to 1. Since this power spectral density is the same at all frequencies, we call it white as an analogy to the frequency spectrum of white light.

A generalization to random elements on infinite dimensional spaces, such as random fields, is the white noise measure.

Random vector transformations [link]

Two theoretical applications using a white random vector are the simulation and whitening of another arbitrary random vector. To simulate an arbitrary random vector, we transform a white random vector with a carefully chosen matrix. We choose the transformation matrix so that the mean and covariance matrix of the transformed white random vector matches the mean and covariance matrix of the arbitrary random vector that we are simulating. To whiten an arbitrary random vector, we transform it by a different carefully chosen matrix so that the output random vector is a white random vector.

These two ideas are crucial in applications such as channel estimation and channel equalization in communications and audio. These concepts are also used in data compression.

Simulating a random vector [link]

Suppose that a random vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{x}

has covariance matrix Failed to parse (Missing texvc executable; please see math/README to configure.): K_{xx}

. Since this matrix is Hermitian symmetric and positive semidefinite, by the spectral theorem from linear algebra, we can diagonalize or factor the matrix in the following way.

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\! K_{xx} = E \Lambda E^T

where Failed to parse (Missing texvc executable; please see math/README to configure.): E

is the orthogonal matrix of eigenvectors and Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda
is the diagonal matrix of eigenvalues.  Thus, the inverse equation Failed to parse (Missing texvc executable; please see math/README to configure.):  E^T K_{xx} E = \Lambda 
also holds.

We can simulate the 1st and 2nd moment properties of this random vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{x}

with mean Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\mu}
and covariance matrix Failed to parse (Missing texvc executable; please see math/README to configure.): K_{xx}
via the following transformation of a white vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w}
of unit variance:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{x} = H \, \mathbf{w} + \mu

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \,\!H = E \Lambda^{1/2}.

Thus, the output of this transformation has expectation

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E} \{\mathbf{x}\} = H \, \mathbb{E} \{\mathbf{w}\} + \mu = \mu

and covariance matrix

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E} \{(\mathbf{x} - \mu) (\mathbf{x} - \mu)^T\} = H \, \mathbb{E} \{\mathbf{w} \mathbf{w}^T\} \, H^T = H \, H^T = E \Lambda^{1/2} \Lambda^{1/2} E^T = K_{xx}.

Whitening a random vector [link]

The method for whitening a vector Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{x}

with mean Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\mu}
and covariance matrix Failed to parse (Missing texvc executable; please see math/README to configure.): K_{xx}
is to perform the following calculation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{w} = \Lambda^{-1/2}\, E^T \, ( \mathbf{x} - \mathbf{\mu} ).

Thus, the output of this transformation has expectation

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E} \{\mathbf{w}\} = \Lambda^{-1/2}\, E^T \, ( \mathbb{E} \{\mathbf{x} \} - \mathbf{\mu} ) = \Lambda^{-1/2}\, E^T \, (\mu - \mu) = 0

and covariance matrix

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E} \{\mathbf{w} \mathbf{w}^T\} = \mathbb{E} \{ \Lambda^{-1/2}\, E^T \, ( \mathbf{x} - \mathbf{\mu} )( \mathbf{x} - \mathbf{\mu} )^T E \, \Lambda^{-1/2}\, \}

Failed to parse (Missing texvc executable; please see math/README to configure.): = \Lambda^{-1/2}\, E^T \, \mathbb{E} \{( \mathbf{x} - \mathbf{\mu} )( \mathbf{x} - \mathbf{\mu} )^T\} E \, \Lambda^{-1/2}\,

Failed to parse (Missing texvc executable; please see math/README to configure.): = \Lambda^{-1/2}\, E^T \, K_{xx} E \, \Lambda^{-1/2}.

So, from the inverse equation shown above, we get the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): = \Lambda^{-1/2}\, \Lambda \, \Lambda^{-1/2} = I.

Thus, with the above transformation, we can whiten the random vector to have zero mean and the identity covariance matrix.

Random signal transformations [link]

We cannot extend the same two concepts of simulating and whitening to the case of continuous time random signals or processes. For simulating, we create a filter into which we feed a white noise signal. We choose the filter so that the output signal simulates the 1st and 2nd moments of any arbitrary random process. For whitening, we feed any arbitrary random signal into a specially chosen filter so that the output of the filter is a white noise signal.

Simulating a continuous-time random signal [link]

White noise fed into a linear, time-invariant filter to simulate the 1st and 2nd moments of an arbitrary random process.

White noise can simulate any wide-sense stationary, continuous-time random process Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) : t \in \mathbb{R}\,\!

with constant mean Failed to parse (Missing texvc executable; please see math/README to configure.): \mu
and covariance function

Failed to parse (Missing texvc executable; please see math/README to configure.): K_x(\tau) = \mathbb{E} \left\{ (x(t_1) - \mu) (x(t_2) - \mu)^{*} \right\} \mbox{ where } \tau = t_1 - t_2

and power spectral density

Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega) = \int_{-\infty}^{\infty} K_x(\tau) \, e^{-j \omega \tau} \, d\tau.

We can simulate this signal using frequency domain techniques.^{[clarification needed]}

Because Failed to parse (Missing texvc executable; please see math/README to configure.): K_x(\tau)

is Hermitian symmetric and positive semi-definite, it follows that Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega) 
is real and can be factored as

Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega) = | H(\omega) |^2 = H(\omega) \, H^{*} (\omega)

if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega)

satisfies the Paley-Wiener criterion.

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\infty}^{\infty} \frac{\log (S_x(\omega))}{1 + \omega^2} \, d \omega < \infty

If Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega)

is a rational function, we can then factor it into pole-zero form as

Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega) = \frac{\Pi_{k=1}^{N} (c_k - j \omega)(c^{*}_k + j \omega)}{\Pi_{k=1}^{D} (d_k - j \omega)(d^{*}_k + j \omega)}.

Choosing a minimum phase Failed to parse (Missing texvc executable; please see math/README to configure.): H(\omega)

so that its poles and zeros lie inside the left half s-plane, we can then simulate Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)
with Failed to parse (Missing texvc executable; please see math/README to configure.): H(\omega)
as the transfer function of the filter.

We can simulate Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)

by constructing the following linear, time-invariant filter

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{x}(t) = \mathcal{F}^{-1} \left\{ H(\omega) \right\} * w(t) + \mu

where Failed to parse (Missing texvc executable; please see math/README to configure.): w(t)

is a continuous-time, white-noise signal with the following 1st and 2nd moment properties:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}\{w(t)\} = 0

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}\{w(t_1)w^{*}(t_2)\} = K_w(t_1, t_2) = \delta(t_1 - t_2).

Thus, the resultant signal Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{x}(t)

has the same 2nd moment properties as the desired signal Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)

.

Whitening a continuous-time random signal [link]

An arbitrary random process x(t) fed into a linear, time-invariant filter that whitens x(t) to create white noise at the output.

Suppose we have a wide-sense stationary, continuous-time random process Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) : t \in \mathbb{R}\,\!

defined with the same mean Failed to parse (Missing texvc executable; please see math/README to configure.): \mu

, covariance function Failed to parse (Missing texvc executable; please see math/README to configure.): K_x(\tau) , and power spectral density Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega)

as above.

We can whiten this signal using frequency domain techniques. We factor the power spectral density Failed to parse (Missing texvc executable; please see math/README to configure.): S_x(\omega)

as described above.

Choosing the minimum phase Failed to parse (Missing texvc executable; please see math/README to configure.): H(\omega)

so that its poles and zeros lie inside the left half s-plane, we can then whiten Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)
with the following inverse filter

Failed to parse (Missing texvc executable; please see math/README to configure.): H_{inv}(\omega) = \frac{1}{H(\omega)}.

We choose the minimum phase filter so that the resulting inverse filter is stable. Additionally, we must be sure that Failed to parse (Missing texvc executable; please see math/README to configure.): H(\omega)

is strictly positive for all Failed to parse (Missing texvc executable; please see math/README to configure.): \omega \in \mathbb{R}
so that Failed to parse (Missing texvc executable; please see math/README to configure.): H_{inv}(\omega)
does not have any singularities.

The final form of the whitening procedure is as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): w (t) = \mathcal{F}^{-1} \left\{ H_{inv}(\omega) \right\} * (x(t) - \mu)

so that Failed to parse (Missing texvc executable; please see math/README to configure.): w(t)

is a white noise random process with zero mean and constant, unit power spectral density

Failed to parse (Missing texvc executable; please see math/README to configure.): S_{w}(\omega) = \mathcal{F} \left\{ \mathbb{E} \{ w(t_1) w(t_2) \} \right\} = H_{inv}(\omega) S_x(\omega) H^{*}_{inv}(\omega) = \frac{S_x(\omega)}{S_x(\omega)} = 1.

Note that this power spectral density corresponds to a delta function for the covariance function of Failed to parse (Missing texvc executable; please see math/README to configure.): w(t) .

Failed to parse (Missing texvc executable; please see math/README to configure.): K_w(\tau) = \,\!\delta (\tau)

In music [link]

White noise, pink noise, and brown noise are used as percussion in 8-bit (chiptune) music. They are also used in electronic music such as trance and house music, to create "sweeps".

See also [link]

References [link]

External links [link]

Wikimedia Commons has media related to: White noise

• Meaning of a White-Noise Process - "proper" definition of the term white noise