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Insignal processing,white noise is a randomsignal having equal intensity at differentfrequencies, giving it a constantpower spectral density.[1] The term is used with this or similar meanings in many scientific and technical disciplines, includingphysics,acoustical engineering,telecommunications, andstatistical forecasting. White noise refers to a statistical model for signals and signal sources, not to any specific signal. White noise draws its name fromwhite light,[2] although light that appears white generally does not have a flat power spectral density over thevisible band.
Indiscrete time, white noise is adiscrete signal whosesamples are regarded as a sequence ofserially uncorrelatedrandom variables with amean of zero and a finitevariance; a single realization of white noise is arandom shock. In some contexts, it is also required that the samples beindependent and have identicalprobability distribution (in other wordsindependent and identically distributed random variables are the simplest representation of white noise).[3] In particular, if each sample has anormal distribution with zero mean, the signal is said to beadditive white Gaussian noise.[4]
The samples of a white noise signal may besequential in time, or arranged along one or more spatial dimensions. Indigital image processing, thepixels of a white noise image are typically arranged in a rectangular grid, and are assumed to be independent random variables withuniform probability distribution over some interval. The concept can be defined also for signals spread over more complicated domains, such as asphere or atorus.
Aninfinite-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. Thus, random signals are considered white noise if they are observed to have a flat spectrum over the range of frequencies that are relevant to the context. For anaudio signal, the relevant range is the band of audible sound frequencies (between 20 and 20,000Hz). Such a signal is heard by the human ear as a hissing sound, resembling the /h/ sound in a sustained aspiration. On the other hand, thesh sound/ʃ/ inash is a colored noise because it has aformant structure. Inmusic andacoustics, the termwhite noise may be used for any signal that has a similar hissing sound.
In the context ofphylogenetically based statistical methods, the termwhite noise can refer to a lack of phylogenetic pattern in comparative data.[5] In nontechnical contexts, it is sometimes used to mean "random talk without meaningful contents".[6][7]
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Any distribution of values is possible (although it must have zeroDC component). Even a binary signal which can only take on the values 1 or -1 will be white if thesequence is statistically uncorrelated. Noise having a continuous distribution, such as anormal distribution, can of course be white.
It is often incorrectly assumed thatGaussian noise (i.e., noise with a Gaussian amplitude distribution – seenormal distribution) necessarily refers to 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 falling within any particular range of amplitudes, while the term 'white' refers to the way the signal power is distributed (i.e., independently) over time or among frequencies.
One form of white noise is the generalized mean-square derivative of theWiener process orBrownian motion.
A generalization torandom elements on infinite dimensional spaces, such asrandom fields, is thewhite noise measure.
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White noise is commonly used in the production ofelectronic 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 ascymbals orsnare drums which have high noise content in their frequency domain.[8] A simple example of white noise is a nonexistent radio station (static).
White noise is also used to obtain theimpulse response of an electrical circuit, in particular ofamplifiers and other audio equipment. It is not used for testing loudspeakers as its spectrum contains too great an amount of high-frequency content.Pink noise, which differs from white noise in that it has equal energy in each octave, is used for testing transducers such as loudspeakers and microphones.
White noise is used as the basis of somerandom number generators. For example,Random.org uses a system of atmospheric antennas to generate random digit patterns from sources that can be well-modeled by white noise.[9]
White noise is a common synthetic noise source used for sound masking by atinnitus masker.[10]White noise machines and other white noise sources are sold as privacy enhancers and sleep aids (seemusic and sleep) and to masktinnitus.[11] The Marpac Sleep-Mate was the first domestic use white noise machine built in 1962 by traveling salesman Jim Buckwalter.[12] Alternatively, the use of an AM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.[13] However, white noise generated from a common commercial radio receiver tuned to an unused frequency is extremely vulnerable to being contaminated with spurious signals, such as adjacent radio stations, harmonics from non-adjacent radio stations, electrical equipment in the vicinity of the receiving antenna causing interference, or even atmospheric events such as solar flares and especially lightning.
The effects of white noise upon cognitive function are mixed. A small study published in 2007 found that white noise background stimulation improves cognitive functioning among secondary students withattention deficit hyperactivity disorder (ADHD), while decreasing performance of non-ADHD students.[14][15] Other work indicates it is effective in improving the mood and performance of workers by masking background office noise,[16] but decreases cognitive performance in complex card sorting tasks.[17]
Similarly, an experiment was carried out on sixty-six healthy participants to observe the benefits of using white noise in a learning environment. The experiment involved the participants identifying different images whilst having different sounds in the background. Overall the experiment showed that white noise does in fact have benefits in relation to learning. The experiments showed that white noise improved the participants' learning abilities and their recognition memory slightly.[18]
Arandom vector (that is, a random variable with values inRn) is said to be a white noise vector or white random vector if its components each have aprobability distribution with zero mean and finitevariance,[clarification needed] and arestatistically independent: that is, theirjoint probability distribution must be the product of the distributions of the individual components.[19]
A necessary (but,in general, not sufficient) condition for statistical independence of two variables is that they bestatistically uncorrelated; that is, theircovariance is zero. Therefore, thecovariance matrixR of the components of a white noise vectorw withn elements must be ann byndiagonal matrix, where each diagonal elementRii is thevariance of componentwi; and thecorrelation matrix must be then byn identity matrix.
If, in addition to being independent, every variable inw also has anormal distribution with zero mean and the same variance,w is said to be a Gaussian white noise vector. In that case, the joint distribution ofw is amultivariate normal distribution; the independence between the variables then implies that the distribution hasspherical symmetry inn-dimensional space. Therefore, anyorthogonal transformation of the vector will result in a Gaussian white random vector. In particular, under most types ofdiscrete Fourier transform, such asFFT andHartley, the transformW ofw will be a Gaussian white noise vector, too; that is, then Fourier coefficients ofw will be independent Gaussian variables with zero mean and the same variance.
Thepower spectrumP of a random vectorw can be defined as the expected value of thesquared modulus of each coefficient of its Fourier transformW, that is,Pi = E(|Wi|2). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, withPi = σ2 for all i.
Ifw is a white random vector, but not a Gaussian one, its Fourier coefficientsWi will not be completely independent of each other; although for largen and common probability distributions the dependencies are very subtle, and their pairwise correlations can be assumed to be zero.
Often the weaker condition statistically uncorrelated is used in the definition of white noise, instead of statistically independent. However, some of the commonly expected properties of white noise (such as flat power spectrum) may not hold for this weaker version. Under this assumption, the stricter version can be referred to explicitly as independent white noise vector.[20]: p.60 Other authors use strongly white and weakly white instead.[21]
An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is where is a normal random variable with zero mean, and is equal to or to, with equal probability. These two variables are uncorrelated and individually normally distributed, but they are not jointly normally distributed and are not independent. If is rotated by 45 degrees, its two components will still be uncorrelated, but their distribution will no longer be normal.
In some situations, one may relax the definition by allowing each component of a white random vector to have non-zero expected value. Inimage processing especially, where samples are typically restricted to positive values, one often takes to be one half of the maximum sample value. In that case, the Fourier coefficient corresponding to the zero-frequency component (essentially, the average of the) will also have a non-zero expected value; and the power spectrum will be flat only over the non-zero frequencies.
A discrete-timestochastic process is a generalization of a random vector with a finite number of components to infinitely many components. A discrete-time stochastic process is called white noise if its mean is equal to zero for all , i.e. and if the autocorrelation function has a nonzero value only for, i.e..[citation needed][clarification needed]
In order to define the notion of white noise in the theory ofcontinuous-time signals, one must replace the concept of a random vector by a continuous-time random signal; that is, a random process that generates a function of a real-valued parameter.
Such a process is said to be white noise in the strongest sense if the value for any time is a random variable that is statistically independent of its entire history before. A weaker definition requires independence only between the values and at every pair of distinct times and. An even weaker definition requires only that such pairs and be uncorrelated.[22] As in the discrete case, some authors adopt the weaker definition for white noise, and use the qualifier independent to refer to either of the stronger definitions. Others use weakly white and strongly white to distinguish between them.
However, a precise definition of these concepts is not trivial, because some quantities that are finite sums in the finite discrete case must be replaced by integrals that may not converge. Indeed, the set of all possible instances of a signal is no longer a finite-dimensional space, but an infinite-dimensionalfunction space. Moreover, by any definition a white noise signal would have to be essentially discontinuous at every point; therefore even the simplest operations on, like integration over a finite interval, require advanced mathematical machinery.
Some authors[citation needed][clarification needed] require each value to be a real-valued random variable with expectation and some finite variance. Then the covariance between the values at two times and is well-defined: it is zero if the times are distinct, and if they are equal. However, by this definition, the integral
over any interval with positive width would be simply the width times the expectation:.[clarification needed] This property renders the concept inadequate as a model of white noise signals either in a physical or mathematical sense.[clarification needed]
Therefore, most authors define the signal indirectly by specifying random values for the integrals of and over each interval. In this approach, however, the value of at an isolated time cannot be defined as a real-valued random variable[citation needed]. Also the covariance becomes infinite when; and theautocorrelation function must be defined as, where is some real constant and is theDirac delta function.[clarification needed]
In this approach, one usually specifies that the integral of over an interval is a real random variable with normal distribution, zero mean, and variance; and also that the covariance of the integrals, is, where is the width of the intersection of the two intervals. This model is called a Gaussian white noise signal (or process).
In the mathematical field known aswhite noise analysis, a Gaussian white noise is defined as a stochastic tempered distribution, i.e. a random variable with values in the space oftempered distributions. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space can be defined via its characteristic function (existence and uniqueness are guaranteed by an extension of the Bochner–Minlos theorem, which goes under the name Bochner–Minlos–Sazanov theorem); analogously to the case of the multivariate normal distribution, which has characteristic function
the white noise must satisfy
where is the natural pairing of the tempered distribution with the Schwartz function (i.e. we consider as a fixed linear function on analogous to above) and.
Instatistics andeconometrics one often assumes that an observed series of data values is the sum of the values generated by adeterministiclinear process, depending on certainindependent (explanatory) variables, and on a series of random noise values. Thenregression analysis is used to infer the parameters of the model process from the observed data, e.g. byordinary least squares, and totest the null hypothesis that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same Gaussian probability distribution – in other words, that the noise is Gaussian white (not just white). If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are stillunbiased, but estimates of their uncertainties (such asconfidence intervals) will be biased (not accurate on average). This is also true if the noise isheteroskedastic – that is, if it has different variances for different data points.
Alternatively, in the subset of regression analysis known astime series analysis there are often no explanatory variables other than the past values of the variable being modeled (thedependent variable). In this case the noise process is often modeled as amoving average process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.
These two ideas are crucial in applications such aschannel estimation andchannel equalization incommunications andaudio. These concepts are also used indata compression.
In particular, by a suitable linear transformation (acoloring transformation), a white random vector can be used to produce a non-white random vector (that is, a list of random variables) whose elements have a prescribedcovariance matrix. Conversely, a random vector with known covariance matrix can be transformed into a white random vector by a suitablewhitening transformation.
White noise may be generated digitally with adigital signal processor,microprocessor, ormicrocontroller. Generating white noise typically entails feeding an appropriate stream of random numbers to adigital-to-analog converter. The quality of the white noise will depend on the quality of the algorithm used.[23]
The term is sometimes used as acolloquialism to describe a backdrop of ambient sound, creating an indistinct or seamless commotion. Following are some examples:
The term can also be used metaphorically, as in the novelWhite Noise (1985) byDon DeLillo which explores the symptoms ofmodern culture that came together so as to make it difficult for an individual to actualize their ideas and personality.
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The best-known generalized process is white noise, which can be thought of as a continuous time analogue to a sequence of independent and identically distributed observations.
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