White noise
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226069868
2008-07-16T18:16:54Z
Drizzd
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[[WP:UNDO|Undid]] revision 225748392 by [[Special:Contributions/59.163.146.5|59.163.146.5]] ([[User talk:59.163.146.5|talk]]). Why abbreviate if the abbreviation is not used?
{{Otheruses2|White noise}}
{{Colors of noise}}
[[Image:White noise spectrum.png|thumb|Calculated spectrum of a generated approximation of white noise]]
'''White noise''' is a random [[signal (information theory)|signal]] (or process) with a flat [[power spectral density]]. In other words, the signal contains equal power within a fixed [[bandwidth (signal processing)|bandwidth]] at any center frequency. White noise is considered analogous to [[White#White light|white light]] which contains all frequencies.
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. In practice, a signal can be "white" with a flat spectrum over a defined frequency band.
== Statistical properties ==
[[Image:white-noise.png|thumb|right|An example realization of a Gaussian white noise process.]]
The term white noise is also commonly applied to a noise signal in the spatial domain which 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). 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, however, restrict the values a signal can take. Any distribution of values is possible (although it must have zero [[DC component]]). For example, on [[Linux]] white noise can be generated with the command {{nowrap|<code>[[cat (Unix)|cat]] [[/dev/random|/dev/urandom]] > /dev/dsp</code>}}, 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 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 that the signal has a certain value at a certain instant, while the term 'white' refers to the way the signal power (taken over time) is distributed among frequencies.
[[Image:Noise.jpg|thumb|250px|left|[[Pink noise]] (left) and white noise (right) on a FFT [[spectrogram]] 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]]. 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)|siren]]s 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 [[cymbals]] which have high noise content in their frequency domain.
It is also used to generate [[impulse response]]s. 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 EQ to ensure a balanced mix.
{{Sound sample box align right|Music sample:}}
{{Listen
|filename=Whitenoisesound.ogg
|title=White noise
|description=10 second sample of white sound.
|format=[[Ogg]]}}
{{sample box end}}
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 [[hardware random number generator|random number generators]].
White noise can be used to disorient individuals prior to [[interrogation]] and may be used as part of [[sensory deprivation]] techniques.{{Fact|date=March 2007}} [[White noise machine]]s are sold as privacy enhancers and sleep aids and to mask [[tinnitus]]. White noise CDs, when used with headphones, can aid concentration by blocking out irritating or distracting noises in a person's environment.
== Mathematical definition ==
=== White random vector ===
A random vector <math>\mathbf{w}</math> is a white random vector if and only if its [[mean vector]] and [[autocorrelation]] matrix are the following:
:<math>\mu_w = \mathbb{E}\{ \mathbf{w} \} = 0</math>
:<math>R_{ww} = \mathbb{E}\{ \mathbf{w} \mathbf{w}^T\} = \sigma^2 \mathbf{I} .</math>
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 <math>w(t)</math> where <math>t \in \mathbb{R}</math> is a white noise process if and only if its mean function and autocorrelation function satisfy the following:
:<math>\mu_w(t) = \mathbb{E}\{ w(t)\} = 0</math>
:<math>R_{ww}(t_1, t_2) = \mathbb{E}\{ w(t_1) w(t_2)\} = (N_{0}/2)\delta(t_1 - t_2)</math>.
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.
:<math>S_{xx}(\omega) = N_{0}/2 ,\!</math>
since the [[Fourier transform]] of the [[delta function]] and likewise 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|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 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 [[telecommunication|communications]] and [[sound reproduction|audio]]. These concepts are also used in [[data compression]].
=== Simulating a random vector ===
Suppose that a random vector <math>\mathbf{x}</math> has [[covariance matrix]] <math>K_{xx}</math>. Since this matrix is [[Hermitian adjoint|Hermitian symmetric]] and [[positive semidefinite]], by the [[spectral theorem]] from [[linear algebra]], we can diagonalize or factor the matrix in the following way.
:<math>\,\! K_{xx} = E \Lambda E^T</math>
where <math>E</math> is the [[orthogonal matrix]] of [[eigenvector]]s and <math>\Lambda</math> is the [[diagonal matrix]] of [[eigenvalue]]s.
We can simulate the 1st and 2nd [[Moment (mathematics)|moment]] properties of this [[random vector]] <math>\mathbf{x}</math> with [[mean]] <math>\mathbf{\mu}</math> and covariance matrix <math>K_{xx}</math> via the following transformation of a white vector <math>\mathbf{w}</math>:
:<math> \mathbf{x} = H \, \mathbf{w} + \mu</math>
where
:<math> \,\!H = E \Lambda^{1/2}</math>
Thus, the output of this transformation has expectation
:<math> \mathbb{E} \{\mathbf{x}\} = H \, \mathbb{E} \{\mathbf{w}\} + \mu = \mu</math>
and covariance matrix
:<math> \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}</math>
=== Whitening a random vector ===
The method for whitening a vector <math>\mathbf{x}</math> with [[mean]] <math>\mathbf{\mu}</math> and [[covariance matrix]] <math>K_{xx}</math> is to perform the following calculation:
:<math>\mathbf{w} = \Lambda^{-1/2}\, E^T \, ( \mathbf{x} - \mathbf{\mu} )</math>
Thus, the output of this transformation has expectation
:<math> \mathbb{E} \{\mathbf{w}\} = \Lambda^{-1/2}\, E^T \, ( \mathbb{E} \{\mathbf{x} \} - \mathbf{\mu} ) = \Lambda^{-1/2}\, E^T \, (\mu - \mu) = 0</math>
and covariance matrix
:<math> \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}\, \}</math>
:<math> = \Lambda^{-1/2}\, E^T \, \mathbb{E} \{( \mathbf{x} - \mathbf{\mu} )( \mathbf{x} - \mathbf{\mu} )^T\} E \, \Lambda^{-1/2}\,</math>
:<math> = \Lambda^{-1/2}\, E^T \, K_{xx} E \, \Lambda^{-1/2}</math>
By diagonalizing <math>K_{xx}</math>, we get the following:
:<math> \Lambda^{-1/2}\, E^T \, E \Lambda E^T E \, \Lambda^{-1/2} = \Lambda^{-1/2}\, \Lambda \, \Lambda^{-1/2} = I</math>
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|thumb|450px|right|White noise fed into a linear, time-invariant filter to simulate the 1st and 2nd moments of an arbitrary random process.]]
We can simulate any wide-sense [[stationary]], [[continuous function|continuous]]-time [[random process]] <math>x(t) : t \in \mathbb{R}\,\!</math> with constant [[mean]] <math>\mu</math> and [[covariance]] function
:<math>K_x(\tau) = \mathbb{E} \left\{ (x(t_1) - \mu) (x(t_2) - \mu)^{*} \right\} \mbox{ where } \tau = t_1 - t_2</math>
and [[power spectral density]]
:<math>S_x(\omega) = \int_{-\infty}^{\infty} K_x(\tau) \, e^{-j \omega \tau} \, d\tau</math>
We can simulate this signal using [[frequency domain]] techniques.
Because <math>K_x(\tau)</math> is [[hermitian|Hermitian symmetric]] and [[positive semi-definite]], it follows that <math>S_x(\omega) </math> is [[Real number|real]] and can be factored as
:<math>S_x(\omega) = | H(\omega) |^2 = H(\omega) \, H^{*} (\omega) </math>
if and only if <math>S_x(\omega)</math> satisfies the [[Paley-Wiener criterion]].
:<math> \int_{-\infty}^{\infty} \frac{\log (S_x(\omega))}{1 + \omega^2} \, d \omega < \infty </math>
If <math>S_x(\omega)</math> is a [[rational function]], we can then factor it into [[Pole (complex analysis)|pole]]-[[Zero (complex analysis)|zero]] form as
:<math>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)}</math>
Choosing a [[minimum phase]] <math>H(\omega)</math> so that its poles and zeros lie inside the left half [[s-plane]], we can then simulate <math>x(t)</math> with <math>H(\omega)</math> as the transfer function of the filter.
We can simulate <math>x(t)</math> by constructing the following [[linear]], [[time-invariant]] [[filter (signal processing)|filter]]
:<math>\hat{x}(t) = \mathcal{F}^{-1} \left\{ H(\omega) \right\} * w(t) + \mu </math>
where <math>w(t)</math> is a [[continuous function|continuous]]-time, white-noise signal with the following 1st and 2nd [[moment (mathematics)|moment]] properties:
:<math> \mathbb{E}\{w(t)\} = 0</math>
:<math> \mathbb{E}\{w(t_1)w^{*}(t_2)\} = K_w(t_1, t_2) = \delta(t_1 - t_2)</math>
Thus, the resultant signal <math>\hat{x}(t)</math> has the same 2nd [[moment (mathematics)|moment]] properties as the desired signal <math>x(t)</math>.
=== Whitening a continuous-time random signal ===
[[Image:whitening-filter.png|thumb|450px|right|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 process|stationary]], [[continuous function|continuous]]-time [[random process]] <math>x(t) : t \in \mathbb{R}\,\!</math> defined with the same [[mean]] <math>\mu</math>, [[covariance]] function <math>K_x(\tau)</math>, and [[power spectral density]] <math>S_x(\omega)</math> as above.
We can '''whiten''' this signal using [[frequency domain]] techniques. We factor the power spectral density <math>S_x(\omega)</math> as described above.
Choosing the [[minimum phase]] <math>H(\omega)</math> so that its poles and zeros lie inside the left half [[s-plane]], we can then whiten <math>x(t)</math> with the following inverse filter
:<math>H_{inv}(\omega) = \frac{1}{H(\omega)}</math>
We choose the [[minimum phase]] filter so that the resulting inverse filter is [[BIBO stability|stable]]. Additionally, we must be sure that <math>H(\omega)</math> is strictly positive for all <math>\omega \in \mathbb{R}</math> so that <math>H_{inv}(\omega)</math> does not have any [[mathematical singularity|singularities]].
The final form of the whitening procedure is as follows:
:<math>w (t) = \mathcal{F}^{-1} \left\{ H_{inv}(\omega) \right\} * (x(t) - \mu)</math>
so that <math>w(t)</math> is a white noise [[random process]] with zero [[mean]] and constant, unit [[power spectral density]]
:<math>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</math>
Note that this [[power spectral density]] corresponds to a [[delta function]] for the [[covariance]] function of <math>w(t)</math>.
:<math>K_w(\tau) = \,\!\delta (\tau)</math>
== See also ==
<div style="-moz-column-count:2; column-count:2;">
*[[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]]
</div>
== External links ==
{{CommonsCat|White noise}}
*[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 noise generator, uses Flash]
[[Category:Stochastic processes]]
[[Category:Noise]]
[[Category:Time series analysis]]
[[Category:Data compression]]
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