- White noise
**White noise**is a random signal (or process) with a flatpower 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 rather 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 adelta function over the relevant space dimensions. The signal is then "white" in thespatial 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**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 ). For example, onLinux white noise can be generated with the command nowrap|`cat /dev/urandom > /dev/dsp`

, feeding the kernel random number generator (uniformly distributed integers between 0 and 255) into the digital signal processor. Even a binary signal which can only take on the values 1 or 0 will be white if the sequence of zeros and ones is statistically uncorrelated. Noise having a continuous distribution, such as anormal distribution , can of course be white.It is often incorrectly assumed that

Gaussian noise (i.e., noise with a Gaussian amplitude distribution — seenormal 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.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 . Gaussian white noise has the useful statistical property that its values are independent (seeStatistical independence ).White noise is the generalized mean-square derivative of the

Wiener process orBrownian motion .**Applications**It is used by some emergency vehicle sirens 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 inaudio synthesis , typically to recreate percussive instruments such ascymbal s which have high noise content in their frequency domain.It is also used to generate

impulse response s. To set up the 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 EQ to ensure a balanced mix.Listen

filename=Whitenoisesound.ogg

title=White noise

description=10 second sample of white sound.

format=Ogg White noise can be used for frequency response testing of amplifiers and electronic filters.It is sometimes used with a flat response microphone and an automatic equalizer. The idea is that the system will generate white noise and the microphone will pick up the white noise produced by the speakers. It will then automatically equalize each frequency band to get a flat response.That system is used in professional level equipment, some high-end home stereo and some high-end car radios.

White noise is used as the basis of some random number generators.

White noise can be used to disorient individuals prior to

interrogation and may be used as part ofsensory deprivation techniques.Fact|date=March 2007White noise machine s are sold as privacy enhancers and sleep aids and to masktinnitus . White noise CDs, when used with headphones, can aid concentration by blocking out irritating or distracting noises in a person's environment. In open plan offices, large corporations such as ExxonMobil apply white noise to reduce the reach of speech, thus, by preventing office staff from being distracted by conversations in the background, safeguarding productivity.**Mathematical definition****White random vector**A random vector $mathbf\{w\}$ is a white random vector if and only if its

mean vector andautocorrelation matrix are the following::$mu\_w\; =\; mathbb\{E\}\{\; mathbf\{w\}\; \}\; =\; 0$:$R\_\{ww\}\; =\; mathbb\{E\}\{\; mathbf\{w\}\; mathbf\{w\}^T\}\; =\; sigma^2\; mathbf\{I\}\; .$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)**A continuous time random process $w(t)$ where $t\; in\; mathbb\{R\}$ is a white noise process if and only if its mean function and autocorrelation function satisfy the following::$mu\_w(t)\; =\; mathbb\{E\}\{\; w(t)\}\; =\; 0$:$R\_\{ww\}(t\_1,\; t\_2)\; =\; mathbb\{E\}\{\; w(t\_1)\; w(t\_2)\}\; =\; (N\_\{0\}/2)delta(t\_1\; -\; t\_2)$.

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.:$S\_\{xx\}(omega)\; =\; N\_\{0\}/2\; ,!$

since the

Fourier transform of thedelta function and likewise is equal to 1. Since thispower spectral density is the same at all frequencies, we call it "white" as an analogy to thefrequency spectrum of white light.**Random vector transformations**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 andcovariance 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 andchannel equalization in communications and audio. These concepts are also used indata compression .**Simulating a random vector**Suppose that a random vector $mathbf\{x\}$ has

covariance matrix $K\_\{xx\}$. Since this matrix is Hermitian symmetric andpositive semidefinite , by thespectral theorem fromlinear algebra , we can diagonalize or factor the matrix in the following way.:$,!\; K\_\{xx\}\; =\; E\; Lambda\; E^T$where $E$ is the

orthogonal matrix ofeigenvector s and $Lambda$ is thediagonal matrix ofeigenvalue s.We can simulate the 1st and 2nd moment properties of this

random vector $mathbf\{x\}$ withmean $mathbf\{mu\}$ and covariance matrix $K\_\{xx\}$ via the following transformation of a white vector $mathbf\{w\}$::$mathbf\{x\}\; =\; H\; ,\; mathbf\{w\}\; +\; mu$where:$,!H\; =\; E\; Lambda^\{1/2\}$

Thus, the output of this transformation has expectation:$mathbb\{E\}\; \{mathbf\{x\}\}\; =\; H\; ,\; mathbb\{E\}\; \{mathbf\{w\}\}\; +\; mu\; =\; mu$

and covariance matrix:$mathbb\{E\}\; \{(mathbf\{x\}\; -\; mu)\; (mathbf\{x\}\; -\; mu)^T\}\; =\; H\; ,\; mathbb\{E\}\; \{mathbf\{w\}\; mathbf\{w\}^T\}\; ,\; H^T\; =\; H\; ,\; H^T\; =\; E\; Lambda^\{1/2\}\; Lambda^\{1/2\}\; E^T\; =\; K\_\{xx\}$

**Whitening a random vector**The method for whitening a vector $mathbf\{x\}$ with

mean $mathbf\{mu\}$ andcovariance matrix $K\_\{xx\}$ is to perform the following calculation::$mathbf\{w\}\; =\; Lambda^\{-1/2\},\; E^T\; ,\; (\; mathbf\{x\}\; -\; mathbf\{mu\}\; )$Thus, the output of this transformation has expectation:$mathbb\{E\}\; \{mathbf\{w\}\}\; =\; Lambda^\{-1/2\},\; E^T\; ,\; (\; mathbb\{E\}\; \{mathbf\{x\}\; \}\; -\; mathbf\{mu\}\; )\; =\; Lambda^\{-1/2\},\; E^T\; ,\; (mu\; -\; mu)\; =\; 0$

and covariance matrix:$mathbb\{E\}\; \{mathbf\{w\}\; mathbf\{w\}^T\}\; =\; mathbb\{E\}\; \{\; Lambda^\{-1/2\},\; E^T\; ,\; (\; mathbf\{x\}\; -\; mathbf\{mu\}\; )(\; mathbf\{x\}\; -\; mathbf\{mu\}\; )^T\; E\; ,\; Lambda^\{-1/2\},\; \}$:$=\; Lambda^\{-1/2\},\; E^T\; ,\; mathbb\{E\}\; \{(\; mathbf\{x\}\; -\; mathbf\{mu\}\; )(\; mathbf\{x\}\; -\; mathbf\{mu\}\; )^T\}\; E\; ,\; Lambda^\{-1/2\},$:$=\; Lambda^\{-1/2\},\; E^T\; ,\; K\_\{xx\}\; E\; ,\; Lambda^\{-1/2\}$

By diagonalizing $K\_\{xx\}$, we get the following::$Lambda^\{-1/2\},\; E^T\; ,\; E\; Lambda\; E^T\; E\; ,\; Lambda^\{-1/2\}\; =\; Lambda^\{-1/2\},\; Lambda\; ,\; Lambda^\{-1/2\}\; =\; I$

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

**Random signal transformations**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**We can simulate any wide-sense

stationary , continuous-timerandom process $x(t)\; :\; t\; in\; mathbb\{R\},!$ with constantmean $mu$ andcovariance function:$K\_x(\; au)\; =\; mathbb\{E\}\; left\{\; (x(t\_1)\; -\; mu)\; (x(t\_2)\; -\; mu)^\{*\}\; ight\}\; mbox\{\; where\; \}\; au\; =\; t\_1\; -\; t\_2$

and

power spectral density :$S\_x(omega)\; =\; int\_\{-infty\}^\{infty\}\; K\_x(\; au)\; ,\; e^\{-j\; omega\; au\}\; ,\; d\; au$We can simulate this signal using

frequency domain techniques.Because $K\_x(\; au)$ is Hermitian symmetric and

positive semi-definite , it follows that $S\_x(omega)$ is real and can be factored as:$S\_x(omega)\; =\; |\; H(omega)\; |^2\; =\; H(omega)\; ,\; H^\{*\}\; (omega)$if and only if $S\_x(omega)$ satisfies the

Paley-Wiener criterion .:$int\_\{-infty\}^\{infty\}\; frac\{log\; (S\_x(omega))\}\{1\; +\; omega^2\}\; ,\; d\; omega\; <\; infty$If $S\_x(omega)$ is a

rational function , we can then factor it into pole-zero form as:$S\_x(omega)\; =\; frac\{Pi\_\{k=1\}^\{N\}\; (c\_k\; -\; j\; omega)(c^\{*\}\_k\; +\; j\; omega)\}\{Pi\_\{k=1\}^\{D\}\; (d\_k\; -\; j\; omega)(d^\{*\}\_k\; +\; j\; omega)\}$Choosing a

minimum phase $H(omega)$ so that its poles and zeros lie inside the left halfs-plane , we can then simulate $x(t)$ with $H(omega)$ as the transfer function of the filter.We can simulate $x(t)$ by constructing the following

linear ,time-invariant filter:$hat\{x\}(t)\; =\; mathcal\{F\}^\{-1\}\; left\{\; H(omega)\; ight\}\; *\; w(t)\; +\; mu$where $w(t)$ is a continuous-time, white-noise signal with the following 1st and 2nd moment properties::$mathbb\{E\}\{w(t)\}\; =\; 0$:$mathbb\{E\}\{w(t\_1)w^\{*\}(t\_2)\}\; =\; K\_w(t\_1,\; t\_2)\; =\; delta(t\_1\; -\; t\_2)$

Thus, the resultant signal $hat\{x\}(t)$ has the same 2nd moment properties as the desired signal $x(t)$.

**Whitening a continuous-time random signal**Suppose we have a wide-sense stationary, continuous-time

random process $x(t)\; :\; t\; in\; mathbb\{R\},!$ defined with the samemean $mu$,covariance function $K\_x(\; au)$, andpower spectral density $S\_x(omega)$ as above.We can

**whiten**this signal usingfrequency domain techniques. We factor the power spectral density $S\_x(omega)$ as described above.Choosing the

minimum phase $H(omega)$ so that its poles and zeros lie inside the left halfs-plane , we can then whiten $x(t)$ with the following inverse filter:$H\_\{inv\}(omega)\; =\; frac\{1\}\{H(omega)\}.$We choose the

minimum phase filter so that the resulting inverse filter is stable. Additionally, we must be sure that $H(omega)$ is strictly positive for all $omega\; in\; mathbb\{R\}$ so that $H\_\{inv\}(omega)$ does not have any singularities.The final form of the whitening procedure is as follows::$w\; (t)\; =\; mathcal\{F\}^\{-1\}\; left\{\; H\_\{inv\}(omega)\; ight\}\; *\; (x(t)\; -\; mu)$

so that $w(t)$ is a white noise

random process with zeromean and constant, unitpower spectral density :$S\_\{w\}(omega)\; =\; mathcal\{F\}\; left\{\; mathbb\{E\}\; \{\; w(t\_1)\; w(t\_2)\; \}\; ight\}\; =\; H\_\{inv\}(omega)\; S\_x(omega)\; H^\{*\}\_\{inv\}(omega)\; =\; frac\{S\_x(omega)\}\{S\_x(omega)\}\; =\; 1.$

Note that this

power spectral density corresponds to adelta function for thecovariance function of $w(t)$.:$K\_w(\; au)\; =\; ,!delta\; (\; au)$**See also**

*Colors of noise

*Electronics

*Electronic noise

*Delta function

*Hiss

*Independent component analysis

*Noise (physics)

*Principal components analysis

*Statistics

*White noise machine

*Architectural acoustics

*Sound masking

*Pink noise

*Brownian noise **External links*** [

*http://www.elec.qmul.ac.uk/staffinfo/markp/2003/OjaPlumbley03-ica2003.pdf A mathematical application of noise whitening of pictures - pdf*]

* [*http://whitenoisemp3s.com/?p=22 White noise in wave(.wav) format (1 minute)*]

* [*http://www.sengpielaudio.com/calculator-noise.htm White noise calculator, thermal noise - Voltage in microvolts, conversion to noise level in dBu and dBV and vice versa*]

* [*http://socrates.berkeley.edu/~phylabs/bsc/Supplementary/NoiseGenerator.html Noise generator - A generator to explore different types of noise*]

* [*http://www.audiocheck.net A free collection of online test tones (white noise and more)*]

* [*http://simplynoise.com A free online white, pink and brown/red noise generator, uses Flash*]

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