White noise is a random signal (information theory) (or process) with a flat power spectral density In other words, the signal contains equal power within a fixed bandwidth (signal processing) 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 eyes three color receptors (Cone cell ) are approximately equally stimulated. An infinite-bandwidth white noise signal is purely a theoretical construction. By having power at all frequencies, the total power of such a signal is infinite and therefore impossible to generate. In practice, however, a signal can be "white" with a flat spectrum over a defined frequency band.

White noise in a spatial context

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 autocorrelation 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

Image:white-noise.png 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 i.e. the probability that the signal has a certain given value, while the term white refers to the way the signal power is distributed over time or among frequencies. Image:Noise.jpg of pink noise (left) and white noise (right), shown with linear frequency axis (vertical).]] We can therefore find Gaussian white noise, but also Poisson distribution Cauchy distribution 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 Gaussian white noise has the useful statistical property that its values are independent (see Statistical independence . White noise is the generalized mean-square derivative of the Wiener process or Brownian motion

Applications

It is used by some emergency vehicle Siren (noisemaker) due to its ability to cut through background noise, which makes it easier to locate. 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 which have high noise content in their frequency domain. It is also used to generate impulse response . 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 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 a common synthetic noise source used for sound masking by a tinnitus masker Jastreboff PJ. Tinnitus Habituation Therapy (THT) and Tinnitus Retraining Therapy (TRT). Tinnitus Handbook. San Diego: Singular, 2000;357-76 White noise is a particularly good source signal for masking devices as it contains higher frequencies in equal volumes to lower ones, and so is capable of more effective masking for high pitched ringing tones most commonly perceived by tinnitus sufferers. White noise is used as the basis of some hardware random number generator For example, Random.org uses a system of atmospheric antennae to generate random digit patterns from white noise. White noise machine are sold as privacy enhancers and sleep aids and to mask tinnitus Some people claim white noise, when used with headphones, can aid concentration by masking irritating or distracting noises in a persons environment.http://www.articlesbase.com/mental-health-articles/how-white-noise-can-help-improve-concentration-and-relaxation-1569670.html How White Noise Can Help Improve Concentration and Relaxation]

Mathematical definition

White random vector

A random vector $\backslash mathbfw\}$ is a white random vector if and only if its mean vector and autocorrelation matrix are the following: :$\backslash mu\_w\; \backslash mathbbE\}\backslash \; \backslash mathbfw\}\; \backslash \}\; 0$ :$R\_ww\}\; \backslash mathbbE\}\backslash \; \backslash mathbfw\}\; \backslash mathbfw\}^T\backslash \}\; \backslash sigma^2\; \backslash mathbfI\}\; .$ 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)

A continuous time random process $w(t)$ where $t\; \backslash in\; \backslash mathbbR\}$ is a white noise process if and only if its mean function and autocorrelation function satisfy the following: :$\backslash mu\_w(t)\; \backslash mathbbE\}\backslash \; w(t)\backslash \}\; 0$ :$R\_ww\}(t\_1,\; t\_2)\; \backslash mathbbE\}\backslash \; w(t\_1)\; w(t\_2)\backslash \}\; (N\_0\}/2)\backslash delta(t\_1\; -\; t\_2).$ 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. :$S\_xx\}(\backslash omega)\; N\_0\}/2\; ,\backslash !$ 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 whiteas an analogy to the frequency spectrum of White A generalization to random element on infinite dimensional spaces, such as random field , is the nuclear space

Random vector transformations

Two theoretical applications using a white random vector are the simulationand whiteningof another arbitrary random vector. To simulatean 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 whitenan 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 telecommunication and sound reproduction These concepts are also used in data compression

Simulating a random vector

Suppose that a random vector $\backslash mathbfx\}$ has covariance matrix $K\_xx\}$. Since this matrix is Hermitian adjoint and positive semidefinite by the spectral theorem from linear algebra we can diagonalize or factor the matrix in the following way. :$\backslash ,\backslash !\; K\_xx\}\; E\; \backslash Lambda\; E^T$ where $E$ is the orthogonal matrix of eigenvector and $\backslash Lambda$ is the diagonal matrix of eigenvalue . We can simulate the 1st and 2nd Moment (mathematics) properties of this random vector $\backslash mathbfx\}$ with mean $\backslash mathbf\backslash mu\}$ and covariance matrix $K\_xx\}$ via the following transformation of a white vector $\backslash mathbfw\}$ of unit variance: :$\backslash mathbfx\}\; H\; \backslash ,\; \backslash mathbfw\}\; +\; \backslash mu$ where :$\backslash ,\backslash !H\; E\; \backslash Lambda^1/2\}.$ Thus, the output of this transformation has expectation :$\backslash mathbbE\}\; \backslash \backslash mathbfx\}\backslash \}\; H\; \backslash ,\; \backslash mathbbE\}\; \backslash \backslash mathbfw\}\backslash \}\; +\; \backslash mu\; \backslash mu$ and covariance matrix :$\backslash mathbbE\}\; \backslash (\backslash mathbfx\}\; -\; \backslash mu)\; (\backslash mathbfx\}\; -\; \backslash mu)^T\backslash \}\; H\; \backslash ,\; \backslash mathbbE\}\; \backslash \backslash mathbfw\}\; \backslash mathbfw\}^T\backslash \}\; \backslash ,\; H^T\; H\; \backslash ,\; H^T\; E\; \backslash Lambda^1/2\}\; \backslash Lambda^1/2\}\; E^T\; K\_xx\}.$

Whitening a random vector

The method for whitening a vector $\backslash mathbfx\}$ with mean $\backslash mathbf\backslash mu\}$ and covariance matrix $K\_xx\}$ is to perform the following calculation: :$\backslash mathbfw\}\; \backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; (\; \backslash mathbfx\}\; -\; \backslash mathbf\backslash mu\}\; ).$ Thus, the output of this transformation has expectation :$\backslash mathbbE\}\; \backslash \backslash mathbfw\}\backslash \}\; \backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; (\; \backslash mathbbE\}\; \backslash \backslash mathbfx\}\; \backslash \}\; -\; \backslash mathbf\backslash mu\}\; )\; \backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; (\backslash mu\; -\; \backslash mu)\; 0$ and covariance matrix :$\backslash mathbbE\}\; \backslash \backslash mathbfw\}\; \backslash mathbfw\}^T\backslash \}\; \backslash mathbbE\}\; \backslash \; \backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; (\; \backslash mathbfx\}\; -\; \backslash mathbf\backslash mu\}\; )(\; \backslash mathbfx\}\; -\; \backslash mathbf\backslash mu\}\; )^T\; E\; \backslash ,\; \backslash Lambda^-1/2\}\backslash ,\; \backslash \}$ :$\backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; \backslash mathbbE\}\; \backslash (\; \backslash mathbfx\}\; -\; \backslash mathbf\backslash mu\}\; )(\; \backslash mathbfx\}\; -\; \backslash mathbf\backslash mu\}\; )^T\backslash \}\; E\; \backslash ,\; \backslash Lambda^-1/2\}\backslash ,$ :$\backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; K\_xx\}\; E\; \backslash ,\; \backslash Lambda^-1/2\}.$ By diagonalizing $K\_xx\}$, we get the following: :$\backslash Lambda^-1/2\}\backslash ,\; E^T\; \backslash ,\; E\; \backslash Lambda\; E^T\; E\; \backslash ,\; \backslash Lambda^-1/2\}\; \backslash Lambda^-1/2\}\backslash ,\; \backslash Lambda\; \backslash ,\; \backslash Lambda^-1/2\}\; I.$ Thus, with the above transformation, we can whiten the random vector to have zero mean and the identity covariance matrix.

Random signal transformations

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

Image:simulation-filter.png to simulate the 1st and 2nd moments of an arbitrary random process.]] We can simulate any wide-sense stationary continuous function time random process $x(t)\; :\; t\; \backslash in\; \backslash mathbbR\}\backslash ,\backslash !$ with constant mean $\backslash mu$ and covariance function :$K\_x(\backslash tau)\; \backslash mathbbE\}\; \backslash left\backslash \; (x(t\_1)\; -\; \backslash mu)\; (x(t\_2)\; -\; \backslash mu)^*\}\; \backslash right\backslash \}\; \backslash mbox\; where\; \}\; \backslash tau\; t\_1\; -\; t\_2$ and power spectral density :$S\_x(\backslash omega)\; \backslash int\_-\backslash infty\}^\backslash infty\}\; K\_x(\backslash tau)\; \backslash ,\; e^-j\; \backslash omega\; \backslash tau\}\; \backslash ,\; d\backslash tau.$ We can simulate this signal using frequency domain techniques. Because $K\_x(\backslash tau)$ is hermitian and positive semi-definite it follows that $S\_x(\backslash omega)$ is Real number and can be factored as :$S\_x(\backslash omega)\; |\; H(\backslash omega)\; |^2\; H(\backslash omega)\; \backslash ,\; H^*\}\; (\backslash omega)$ if and only if $S\_x(\backslash omega)$ satisfies the Paley-Wiener criterion : If $S\_x(\backslash omega)$ is a rational function we can then factor it into Pole (complex analysis) Zero (complex analysis) form as :$S\_x(\backslash omega)\; \backslash frac\backslash Pi\_k1\}^N\}\; (c\_k\; -\; j\; \backslash omega)(c^*\}\_k\; +\; j\; \backslash omega)\}\backslash Pi\_k1\}^D\}\; (d\_k\; -\; j\; \backslash omega)(d^*\}\_k\; +\; j\; \backslash omega)\}.$ Choosing a minimum phase $H(\backslash omega)$ so that its poles and zeros lie inside the left half s-plane we can then simulate $x(t)$ with $H(\backslash omega)$ as the transfer function of the filter. We can simulate $x(t)$ by constructing the following linear time-invariant filter (signal processing) :$\backslash hatx\}(t)\; \backslash mathcalF\}^-1\}\; \backslash left\backslash \; H(\backslash omega)\; \backslash right\backslash \}\; *\; w(t)\; +\; \backslash mu$ where $w(t)$ is a continuous function time, white-noise signal with the following 1st and 2nd moment (mathematics) properties: :$\backslash mathbbE\}\backslash w(t)\backslash \}\; 0$ :$\backslash mathbbE\}\backslash w(t\_1)w^*\}(t\_2)\backslash \}\; K\_w(t\_1,\; t\_2)\; \backslash delta(t\_1\; -\; t\_2).$ Thus, the resultant signal $\backslash hatx\}(t)$ has the same 2nd moment (mathematics) properties as the desired signal $x(t)$.

Whitening a continuous-time random signal

Image:whitening-filter.png Suppose we have a wide-sense stationary process continuous function time random process $x(t)\; :\; t\; \backslash in\; \backslash mathbbR\}\backslash ,\backslash !$ defined with the same mean $\backslash mu$, covariance function $K\_x(\backslash tau)$, and power spectral density $S\_x(\backslash omega)$ as above. We can whiten this signal using frequency domain techniques. We factor the power spectral density $S\_x(\backslash omega)$ as described above. Choosing the minimum phase $H(\backslash omega)$ so that its poles and zeros lie inside the left half s-plane we can then whiten $x(t)$ with the following inverse filter :$H\_inv\}(\backslash omega)\; \backslash frac1\}H(\backslash omega)\}.$ We choose the minimum phase filter so that the resulting inverse filter is BIBO stability Additionally, we must be sure that $H(\backslash omega)$ is strictly positive for all $\backslash omega\; \backslash in\; \backslash mathbbR\}$ so that $H\_inv\}(\backslash omega)$ does not have any mathematical singularity The final form of the whitening procedure is as follows: :$w\; (t)\; \backslash mathcalF\}^-1\}\; \backslash left\backslash \; H\_inv\}(\backslash omega)\; \backslash right\backslash \}\; *\; (x(t)\; -\; \backslash mu)$ so that $w(t)$ is a white noise random process with zero mean and constant, unit power spectral density :$S\_w\}(\backslash omega)\; \backslash mathcalF\}\; \backslash left\backslash \; \backslash mathbbE\}\; \backslash \; w(t\_1)\; w(t\_2)\; \backslash \}\; \backslash right\backslash \}\; H\_inv\}(\backslash omega)\; S\_x(\backslash omega)\; H^*\}\_inv\}(\backslash omega)\; \backslash fracS\_x(\backslash omega)\}S\_x(\backslash omega)\}\; 1.$ Note that this power spectral density corresponds to a delta function for the covariance function of $w(t)$. :$K\_w(\backslash tau)\; \backslash ,\backslash !\backslash delta\; (\backslash tau)$

In music

White noise, pink noise and Brownian noise are used as percussion in 8-bit (chiptune music.

See also

External links

*http://www.digitalsignallabs.com/white.pdf Meaning of a White-Noise Process] - "proper" definition of the term white noise

References

Category:Stochastic processes Category:Noise Category:Time series analysis Category:Data compression ar:ضجيج أبيض be-x-old:Белы шум bg:Бял шум ca:Soroll blanc cs:Bílý šum da:Hvid støj de:Weißes Rauschen es:Ruido blanco eo:Blanka bruo eu:Hots zuri fr:Bruit blanc ko:화이트 노이즈 it:Rumore bianco he:רעש לבן lt:Baltasis triukšmas nl:Witte ruis ja:ホワイトノイズ no:Hvit støy pl:Szum biały pt:Ruído branco ru:Белый шум fi:Valkoinen kohina sv:Vitt brus uk:Білий шум zh:白雜訊