NationMaster - Encyclopedia: White Noise

White noise is a random signal (or process) with a flat power spectral density. In other words, the signal's power spectral density has equal power in any band, at any centre frequency, having a given bandwidth. White noise is considered analogous to white light which contains all frequencies. There are many forms of noise with various frequency characteristics that are classified by color. Some have well-defined technical definitions, while others are colloquial or jokes. ... Pink noise spectrum Pink noise ( ), also known as 1/f noise or flicker noise, is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. ... Brown noise spectrum In science, Brownian noise ( ), also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion. ... Wikipedia does not have an article with this exact name. ... Image File history File links White_noise_spectrum. ... Image File history File links White_noise_spectrum. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ... A white rose. ...

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.

1 Statistical properties
2 Colors of noise
3 Applications
4 Mathematical definition
- 4.1 White random vector
- 4.2 White random process (white noise)
5 Random vector transformations
- 5.1 Simulating a random vector
- 5.2 Whitening a random vector
6 Random signal transformations
- 6.1 Simulating a continuous-time random signal
- 6.2 Whitening a continuous-time random signal
7 See also
8 External links

Statistical properties

Four thousandths of a second of white noise

An example realization of a 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. Graph of a fraction of a second of white noise as created by Audactiy File links The following pages link to this file: White noise ... Graph of a fraction of a second of white noise as created by Audactiy File links The following pages link to this file: White noise ... I generated this image using Octave on my computer with the following command plot(randn(1000,1)). Mwilde 17:50, 4 May 2005 (UTC) File links The following pages link to this file: White noise Categories: GFDL images ... I generated this image using Octave on my computer with the following command plot(randn(1000,1)). Mwilde 17:50, 4 May 2005 (UTC) File links The following pages link to this file: White noise Categories: GFDL images ... In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. ...

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, 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. The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ...

It is often incorrectly assumed that Gaussian noise (i.e. noise with a Gaussian amplitude distribution — see normal distribution) is necessarily white noise. However, neither property implies the other. Gaussianity refers to the way signal values are distributed, while the term 'white' refers to the shape of the flat power spectral density. Gaussian noise is noise that has a probability density function (pdf) of the normal distribution (a. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ...

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). Image File history File links Download high resolution version (1059x735, 273 KB) Sumario Pink noise (left) and white noise (right) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (1059x735, 273 KB) Sumario Pink noise (left) and white noise (right) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Pink noise spectrum Pink noise ( ), also known as 1/f noise or flicker noise, is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. ... It has been suggested that this article or section be merged with periodogram. ... ]] selectivity, interference, nonlinearity or dispersion. ... In communications, the additive white Gaussian noise (AWGN) channel is one in which the only impairment is the linear addition of wideband Gaussian noise with a constant spectral density (expressed as watts per hertz of bandwidth). ... In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. ...

White noise is the generalized mean-square derivative of the Wiener process or Brownian motion. A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

Colors of noise

Main article: Colors of noise

There are also other "colors" of noise, the most commonly used being pink, brown and blue. There are many forms of noise with various frequency characteristics that are classified by color. Some have well-defined technical definitions, while others are colloquial or jokes. ... Pink noise spectrum Pink noise ( ), also known as 1/f noise or flicker noise, is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. ... In science, red noise, Brownian noise, or brown noise â–¶(?) is the kind of signal noise produced by Brownian motion. ...

Applications

One use for white noise is in the field of architectural acoustics. In order to dissemble distracting, undesirable noises in interior spaces, a low level of constant white noise is generated. It has been suggested that Acoustic transmission be merged into this article or section. ...

It is used by some emergency vehicle sirens due to its ability to cut through background noise and its lack of echo, which makes it easier to locate. It has been suggested that Fire siren be merged into this article or section. ...

White noise has also been used in electronic music, where it is used either directly or as an input for a filter to create other types of noise signal. In this respect, it is the analog to the violin in classical music. 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 has been suggested that Electronica be merged into this article or section. ... The violin is a bowed string instrument with four strings tuned in perfect fifths. ... This article discusses classical music in the first sense (see below). ... Audio synthesis is the art and science of generating audio signals. ... It is also possible that you want to know about the Cymbalum instrument. ...

It is also used to generate impulse responses. 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. He or she can then adjust the overall EQ to ensure a balanced mix. The Impulse response from a simple audio system. ... For information about computer bandwidth management, see Equalization (computing). ...

Music sample:

White noise ( file info) — play in browser (beta)
- 10 second sample of white sound.
- Problems listening to the file? See media help.

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. Image File history File links Whitenoisesound. ... Software development stages In computer programming, development stage terminology expresses how the development of a piece of software has progressed and how much further development it may require. ...

White noise is used as the basis of some random number generators. In computing, a hardware random number generator is an apparatus that generates random numbers from a physical process. ...

White noise can be used to disorient individuals prior to interrogation and may be used as part of sensory deprivation techniques.^{[citation needed]} White noise machines 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. The examples and perspective in this article or section may not represent a worldwide view. ... A prisoner at the United States Camp X-ray facility at Guantanamo Bay in Cuba being subjected to sensory deprivation, through the use of ear muffs, visor, breathing mask and heavy mittens. ... A clock radio that includes a white noise machine. ... Tinnitus (IPA pronunciation: or ,[1] from the Latin word for ringing[2]) is the perception of sound in the human ear in the absence of corresponding external sound(s). ...

Mathematical definition

White random vector

A random vector $mathbf{w}$ is a white random vector if and only if its mean vector and autocorrelation matrix are the following: The mean vector consists of the means of each variable and the variance-covariance matrix consists of the variances of the variables along the main diagonal and the covariances between each pair of variables in the other matrix positions. ... A plot showing 100 random numbers with a hidden sine function, and an autocorrelation of the series on the bottom. ...

$mu_w = mathbb{E}{ mathbf{w} } = 0$

$R_{ww} = mathbb{E}{ mathbf{w} mathbf{w}^T} = sigma^2 mathbf{I}$

I. e., 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. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

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 Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

The above autocorrelation function implies the following power spectral density.

$S_{xx}(omega) = N_{0}/2 ,!$

since the Fourier transform of the delta function and likewise the 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. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... A white rose. ...

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. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a 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. Copy of the original phone of Alexander Graham Bell at the MusÃ©e des Arts et MÃ©tiers in Paris Telecommunication is the transmission of signals over a distance for the purpose of communication. ... Sound reproduction is the electrical or mechanical re-creation and/or amplification of sound, often as music. ... In computer science and information theory, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than an unencoded representation would use through use of specific encoding schemes. ...

Simulating a random vector

Suppose that a random vector $mathbf{x}$ has covariance matrix $K x x$ . 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. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

$,! K_{xx} = E Lambda E^T$

where $E$ is the orthogonal matrix of eigenvectors and $Λ$ is the diagonal matrix of eigenvalues. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

We can simulate the 1st and 2nd moment properties of this random vector $mathbf{x}$ with mean $mathbf{mu}$ and covariance matrix $K x x$ via the following transformation of a white vector $mathbf{w}$ :-1... A multivariate random variable or random vector is a vector X=(X1,...,Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ...

$mathbf{x} = H , mathbf{w} + mu$

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 $mathbf{mu}$ and covariance matrix $K x x$ is to perform the following calculation: In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

$mathbf{w} = Lambda^{-1/2}, E^T , ( mathbf{x} - mathbf{mu} )$

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 $K x x$ , 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

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-time random process $x(t) : t in mathbb{R},!$ with constant mean $μ$ and covariance function Created on my computer using the Openoffice drawing program. ... Created on my computer using the Openoffice drawing program. ... Stationary can mean: Look up stationary in Wiktionary, the free dictionary. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...

$K_x(tau) = mathbb{E} left{ (x(t_1) - mu) (x(t_2) - mu)^{*} right} mbox{ where } tau = t_1 - t_2$

and power spectral density In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ...

$S_x(omega) = int_{-infty}^{infty} K_x(tau) , e^{-j omega tau} , dtau$

We can simulate this signal using frequency domain techniques. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

Because $K x (τ)$ is Hermitian symmetric and positive semi-definite, it follows that $S x (ω)$ is real and can be factored as A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

$S_x(omega) = | H(omega) |^2 = H(omega) , H^{*} (omega)$

if and only if $S x (ω)$ satisfies the Paley-Wiener criterion.

$int_{-infty}^{infty} frac{log (S_x(omega))}{1 + omega^2} , d omega < infty$

If $S x (ω)$ is a rational function, we can then factor it into pole-zero form as In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

$S_x(omega) = frac{Pi_{k=1}^{N} (c_k - j omega)(c^{*}_k + j omega)}{Pi_{k=1}^{D} (d_k - j omega)(d^{*}_k + j omega)}$

Choosing a minimum phase $H (ω)$ so that its poles and zeros lie inside the left half s-plane, we can then simulate $x (t)$ with $H (ω)$ as the transfer function of the filter. In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ... The S plane is a mathematical domain where, instead of viewing processes in the time domain modelled with time-based functions, they are viewed as equations in the frequency domain. ...

We can simulate $x (t)$ by constructing the following linear, time-invariant filter The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ... In electronics and signal processing, a filter is a device or process that modifies a signal. ...

$hat{x}(t) = mathcal{F}^{-1} left{ H(omega) right} * w(t) + mu$

where $w (t)$ is a continuous-time, white-noise signal with the following 1st and 2nd moment properties: In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... -1...

$mathbb{E}{w(t)} = 0$

$mathbb{E}{w(t_1)w^{*}(t_2)} = K_w(t_1, t_2) = delta(t_1 - t_2)$

Thus, the resultant signal $hat{x}(t)$ has the same 2nd moment properties as the desired signal $x (t)$ .-1...

Whitening a continuous-time random signal

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 $x(t) : t in mathbb{R},!$ defined with the same mean $μ$ , covariance function $K x (τ)$ , and power spectral density $S x (ω)$ as above. Created on my computer using the Openoffice drawing program. ... Created on my computer using the Openoffice drawing program. ... In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ...

We can whiten this signal using frequency domain techniques. We factor the power spectral density $S x (ω)$ as described above. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

Choosing the minimum phase $H (ω)$ so that its poles and zeros lie inside the left half s-plane, we can then whiten $x (t)$ with the following inverse filter In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ... The S plane is a mathematical domain where, instead of viewing processes in the time domain modelled with time-based functions, they are viewed as equations in the frequency domain. ...

$H_{inv}(omega) = frac{1}{H(omega)}$

We choose the minimum phase filter so that the resulting inverse filter is stable. Additionally, we must be sure that $H (ω)$ is strictly positive for all $omega in mathbb{R}$ so that $H i n v (ω)$ does not have any singularities. In control theory and signal processing, a linear, time-invariant system is minimum-phase if the system and its inverse are causal and stable. ... In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

The final form of the whitening procedure is as follows:

$w (t) = mathcal{F}^{-1} left{ H_{inv}(omega) right} * (x(t) - mu)$

so that $w (t)$ is a white noise random process with zero mean and constant, unit power spectral density In the mathematics of probability, a stochastic process can be thought of as a random function. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ...

$S_{w}(omega) = mathcal{F} left{ mathbb{E} { w(t_1) w(t_2) } right} = H_{inv}(omega) S_x(omega) H^{*}_{inv}(omega) = frac{S_x(omega)}{S_x(omega)} = 1$

Note that this power spectral density corresponds to a delta function for the covariance function of $w (t)$ . In physics, the spectral density, or more correctly the power spectral density (PSD) of a given bandwidth of electromagnetic radiation is the total power in this bandwidth divided by the specified bandwidth. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...

$K_w(tau) = ,!delta (tau)$

External links

Results from FactBites:

White Noise (297 words)
	Electronically synthesized white noise can be filtered so as to produce combinations of frequencies not obtainable on traditional musical instruments; or the white noise itself may be used as an element of music.
	White noise is aperiodic sound (that is, its wave pattern is not uniform).
	In the audible spectrum, white noise is a hiss or a roar, such as that produced when a television set is tuned to a channel over which no station is broadcasting.

White Noise (660 words)
	White Noise is widely used in office environments to reduce the cross talk, or interference, caused from conversations in adjacent cubicles.
	White Noise is conducive to relaxation, as white noise inherently sounds like nature, the wind, the sea, etc. It is often used as a sleep aid.
	White Noise is random equally over the 20 - 20khz spectrum and has a 'smooth' color, as opposed to the cracking and popping associated with radio noise.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Statistical properties GA_googleFillSlot("encyclopedia_square");