White noise

White noise

Overview
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 (cone
Cone cell
Cone cells, or cones, are photoreceptor cells in the retina of the eye that are responsible for color vision; they function best in relatively bright light, as opposed to rod cells that work better in dim light. If the retina is exposed to an intense visual stimulus, a negative afterimage will be...

s) are approximately equally stimulated.
In statistical sense, a time series rt is characterized as having weak white noise if {rt} is a sequence of serially uncorrelated random variables with zero mean and finite variance.
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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 (cone
Cone cell
Cone cells, or cones, are photoreceptor cells in the retina of the eye that are responsible for color vision; they function best in relatively bright light, as opposed to rod cells that work better in dim light. If the retina is exposed to an intense visual stimulus, a negative afterimage will be...

s) are approximately equally stimulated.
In statistical sense, a time series rt is characterized as having weak white noise if {rt} 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 rt is normally distributed with mean zero and standard deviation σ , the series is called a Gaussian white noise.

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


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
Spatial frequency
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance. The SI unit of spatial frequency is...

 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). 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
Gaussian noise
Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. A special case is white Gaussian noise, in which...

 (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.

We can therefore find Gaussian white noise, but also Poisson
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

, Cauchy
Cauchy distribution
The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous probability distribution. As a probability distribution, it is known as the Cauchy distribution, while among physicists, it is known as the Lorentz distribution, Lorentz function, or Breit–Wigner...

, 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
Additive white Gaussian noise
Additive white Gaussian noise is a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density and a Gaussian distribution of amplitude. The model does not account for fading, frequency selectivity, interference,...

 has a standard abbreviation: AWGN. Gaussian white noise has the useful statistical property that its values are independent (see Statistical independence
Statistical independence
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
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

 or Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

.

Applications


It is used by some emergency vehicle siren
Siren (noisemaker)
A siren is a loud noise making device. Most modern ones are civil defense or air raid sirens, tornado sirens, or the sirens on emergency service vehicles such as ambulances, police cars and fire trucks. There are two general types: pneumatic and electronic....

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
Electronic music
Electronic music is music that employs electronic musical instruments and electronic music technology in its production. In general a distinction can be made between sound produced using electromechanical means and that produced using electronic technology. Examples of electromechanical sound...

, 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
Cymbal
Cymbals are a common percussion instrument. Cymbals consist of thin, normally round plates of various alloys; see cymbal making for a discussion of their manufacture. The greater majority of cymbals are of indefinite pitch, although small disc-shaped cymbals based on ancient designs sound a...

s which have high noise content in their frequency domain.

It is also used to generate impulse response
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

s. To set up the equalization
Equalization
Equalization, is the process of adjusting the balance between frequency components within an electronic signal. The most well known use of equalization is in sound recording and reproduction but there are many other applications in electronics and telecommunications. The circuit or equipment used...

 (EQ) for a concert or other performance in a venue, a short burst of white or pink noise
Pink noise
Pink noise or 1/ƒ noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency. In pink noise, each octave carries an equal amount of noise power...

 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 used as the basis of some random number generators
Hardware random number generator
In computing, a hardware random number generator is an apparatus that generates random numbers from a physical process. Such devices are often based on microscopic phenomena that generate a low-level, statistically random "noise" signal, such as thermal noise or the photoelectric effect or other...

. For example, Random.org
Random.org
Random.org is a website that produces "true" random numbers based on atmospheric noise.In addition to generating random numbers, it has free tools to do things such as flip coins, shuffle cards, and roll dice...

 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
Tinnitus masker
Tinnitus maskers are a range of devices based on simple white noise machines which are used to add natural or artificial sound into a tinnitus sufferer's environment in order to suppress or mask the perceived ringing....

. White noise machine
White noise machine
A white noise machine is a device that produces a sound that is random in character, which sounds like a rushing waterfall or wind blowing through trees...

s and other white noise sources are sold as privacy enhancers and sleep aids and to mask tinnitus
Tinnitus
Tinnitus |ringing]]") is the perception of sound within the human ear in the absence of corresponding external sound.Tinnitus is not a disease, but a symptom that can result from a wide range of underlying causes: abnormally loud sounds in the ear canal for even the briefest period , ear...

. Alternatively, the use of a FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.

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. Other work indicates it is effective in improving the mood and performance of workers by masking background office noise, but decreases cognitive performance in complex card sorting tasks .

White random vector


A random vector is a white random vector if and only if its mean vector and autocorrelation
Autocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

 matrix are the following:

That is, it is a zero mean random vector, and its autocorrelation matrix is a multiple of the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

. 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 where is a white noise process if and only if its mean function and autocorrelation function satisfy the following:

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
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

.

The above autocorrelation function implies the following power spectral density:

since the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 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
Frequency spectrum
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...

 of white light
White
White is a color, the perception of which is evoked by light that stimulates all three types of color sensitive cone cells in the human eye in nearly equal amounts and with high brightness compared to the surroundings. A white visual stimulation will be void of hue and grayness.White light can be...

.

A generalization to random element
Random element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line...

s on infinite dimensional spaces, such as random field
Random field
A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....

s, is the white noise measure
Nuclear space
In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size...

.

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
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...

 of the transformed white random vector matches the mean and covariance matrix
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...

 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
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...

 and audio. These concepts are also used in data compression
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

.

Simulating a random vector


Suppose that a random vector has covariance matrix
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...

 . Since this matrix is Hermitian symmetric
Hermitian adjoint
In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...

 and positive semidefinite
Positive semidefinite
In mathematics, positive semidefinite may refer to:* positive-semidefinite matrix* positive-semidefinite function...

, by the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

 from linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, we can diagonalize or factor the matrix in the following way.

where is the orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

 of eigenvectors and is the diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

 of eigenvalues. Thus, the inverse equation also holds.

We can simulate the 1st and 2nd moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 properties of this random vector  with mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

  and covariance matrix via the following transformation of a white vector of unit variance:

where

Thus, the output of this transformation has expectation

and covariance matrix

Whitening a random vector


The method for whitening
Whitening transformation
The whitening transformation is a decorrelation method that converts the covariance matrix S of a set of samples into the identity matrix I. This effectively creates new random variables that are uncorrelated and have the same variances as the original random variables...

 a vector with mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

  and covariance matrix
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...

  is to perform the following calculation:

Thus, the output of this transformation has expectation

and covariance matrix

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

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 can simulate any wide-sense stationary
Stationary
Stationary can mean:* In statistics and probability: a stationary process.* In mathematics: a stationary point.* In mathematics: a stationary set.* In physics: a time-invariant quantity, such as a constant position or temperature....

, continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

-time random process  with constant mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

  and covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 function


and power spectral density

We can simulate this signal using frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

 techniques.

Because is Hermitian symmetric
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...

 and positive semi-definite, it follows that is real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 and can be factored as

if and only if satisfies the Paley-Wiener criterion.

If is a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

, we can then factor it into pole-zero
Zero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...

 form as

Choosing a minimum phase
Minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable....

  so that its poles and zeros lie inside the left half s-plane, we can then simulate with as the transfer function of the filter.

We can simulate by constructing the following linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

, time-invariant filter
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...



where is a continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

-time, white-noise signal with the following 1st and 2nd moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 properties:

Thus, the resultant signal has the same 2nd moment
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 properties as the desired signal .

Whitening a continuous-time random signal


Suppose we have a wide-sense stationary
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

, continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

-time random process  defined with the same mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 , covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 function , and power spectral density  as above.

We can whiten this signal using frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

 techniques. We factor the power spectral density as described above.

Choosing the minimum phase
Minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable....

  so that its poles and zeros lie inside the left half s-plane, we can then whiten with the following inverse filter

We choose the minimum phase
Minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable....

 filter so that the resulting inverse filter is stable
BIBO stability
In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

. Additionally, we must be sure that is strictly positive for all so that does not have any singularities
Mathematical singularity
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:

so that is a white noise random process with zero mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

 and constant, unit power spectral density


Note that this power spectral density corresponds to a delta function for the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 function of .

In music


White noise, pink noise
Pink noise
Pink noise or 1/ƒ noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency. In pink noise, each octave carries an equal amount of noise power...

, and brown noise
Brownian noise
In science, Brownian noise , also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise...

 are used as percussion in 8-bit (chiptune
Chiptune
A chiptune, also known as chip music, is synthesized electronic music often produced with the sound chips of vintage computers and video game consoles, as well as with other methods such as emulation. In the early 1980s, personal computers became cheaper and more accessible than they had previously...

) music.
They are also used in Electronic Music
Electronic music
Electronic music is music that employs electronic musical instruments and electronic music technology in its production. In general a distinction can be made between sound produced using electromechanical means and that produced using electronic technology. Examples of electromechanical sound...

 such as Trance
Trance
Trance denotes a variety of processes, ecstasy, techniques, modalities and states of mind, awareness and consciousness. Trance states may occur involuntarily and unbidden.The term trance may be associated with meditation, magic, flow, and prayer...

 and House music
House music
House music is a genre of electronic dance music that originated in Chicago, Illinois, United States in the early 1980s. It was initially popularized in mid-1980s discothèques catering to the African-American, Latino American, and gay communities; first in Chicago circa 1984, then in other...

, to create "sweeps".

External links