In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, white noise is a random
signal having equal intensity at different
frequencies, giving it a constant
power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
acoustical engineering,
telecommunications
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that ...
, and
statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal. White noise draws its name from
white light
White is the lightest color and is achromatic (having no hue). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully reflect and scatter all the visible wavelengths of light. White on ...
, although light that appears white generally does not have a flat power spectral density over the
visible band.
In
discrete time, white noise is a
discrete signal whose
samples are regarded as a sequence of
serially uncorrelated random variables with zero
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
and finite
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
; a single realization of white noise is a random shock. Depending on the context, one may also require that the samples be
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
and have identical
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
(in other words
independent and identically distributed random variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usu ...
are the simplest representation of white noise). In particular, if each sample has a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with zero mean, the signal is said to be
additive white Gaussian noise
Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
* ''Additive'' because it is added to any nois ...
.
The samples of a white noise signal may be
sequential in time, or arranged along one or more spatial dimensions. In
digital image processing, the
pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device.
In most digital display devices, pixels are the ...
s of a ''white noise image'' are typically arranged in a rectangular grid, and are assumed to be independent random variables with
uniform probability distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies bet ...
over some interval. The concept can be defined also for signals spread over more complicated domains, such as a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
or a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
.
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. 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 an
audio signal, the relevant range is the band of audible sound frequencies (between 20 and 20,000
Hz). 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, the "sh" sound in "ash" is a colored noise because it has a formant structure. In
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
and
acoustics, the term "white noise" may be used for any signal that has a similar hissing sound.
The term white noise is sometimes used in the context of
phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data. It is sometimes used analogously in nontechnical contexts to mean "random talk without meaningful contents".
[
Claire Shipman (2005), '']Good Morning America
''Good Morning America'' (often abbreviated as ''GMA'') is an American morning television program that is broadcast on ABC. It debuted on November 3, 1975, and first expanded to weekends with the debut of a Sunday edition on January 3, 1993. ...
'': "The political rhetoric on Social Security
Welfare, or commonly social welfare, is a type of government support intended to ensure that members of a society can meet basic human needs such as food and shelter. Social security may either be synonymous with welfare, or refer specifical ...
is white noise." Said on ABC's ''Good Morning America
''Good Morning America'' (often abbreviated as ''GMA'') is an American morning television program that is broadcast on ABC. It debuted on November 3, 1975, and first expanded to weekends with the debut of a Sunday edition on January 3, 1993. ...
'' TV show, January 11, 2005.
Statistical properties
Any distribution of values is possible (although it must have zero
DC component
DC, D.C., D/C, Dc, or dc may refer to:
Places
* Washington, D.C. (District of Columbia), the capital and the federal territory of the United States
* Bogotá, Distrito Capital, the capital city of Colombia
* Dubai City, as distinct from t ...
). Even a binary signal which can only take on the values 1 or 0 will be white if the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
is statistically uncorrelated. Noise having a continuous distribution, such as a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, can of course be white.
It is often incorrectly assumed that
Gaussian noise (i.e., noise with a Gaussian amplitude distributionsee
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
) 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.
White noise is the generalized mean-square derivative of the
Wiener process or
Brownian motion.
A generalization to
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. The concept was introduced by who commented that the “development of probability theory and expansi ...
s on infinite dimensional spaces, such as
random field In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...
s, is the
white noise measure.
Practical applications
Music
White noise is commonly used in the production of
electronic music
Electronic music is a genre of music that employs electronic musical instruments, digital instruments, or circuitry-based music technology in its creation. It includes both music made using electronic and electromechanical means ( electro ...
, usually either directly or as an input for a filter to create other types of noise signal. It is used extensively in
audio synthesis
A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis and f ...
, typically to recreate percussive instruments such as
cymbals or
snare drums which have high noise content in their frequency domain. A simple example of white noise is a nonexistent radio station (static).
Electronics engineering
White noise is also used to obtain the
impulse response of an electrical circuit, in particular of
amplifier
An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost t ...
s 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.
Computing
White noise is used as the basis of some
random number generators. For example,
Random.org
Random.org (stylized as RANDOM.ORG) is a website that produces random numbers based on atmospheric noise.
In addition to generating random numbers in a specified range and subject to a specified probability distribution, which is the most comm ...
uses a system of atmospheric antennae to generate random digit patterns from white noise.
Tinnitus treatment
White noise is a common synthetic noise source used for sound masking by a
tinnitus masker.
White noise machine
A white noise machine is a device that produces a noise that calms the listener, which in many cases sounds like a rushing waterfall or wind blowing through trees, and other serene or nature-like sounds. Often such devices do not produce actual wh ...
s and other white noise sources are sold as privacy enhancers and sleep aids (see
music and sleep
Music and sleep involves the listening of music in order to improve sleep quality or improve sleep onset insomnia in adults (for infant use of music and sleep, see lullaby). This process can be either self-prescribed or under the guidance of a m ...
) and to mask
tinnitus. The Marpac Sleep-Mate was the first domestic use white noise machine built in 1962 by traveling salesman Jim Buckwalter. Alternatively, the use of an FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise. 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.
There is evidence that white noise exposure therapies may induce maladaptive changes in the brain that degrade neurological health and compromise cognition.
Work environment
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
Attention deficit hyperactivity disorder (ADHD) is a neurodevelopmental disorder characterised by excessive amounts of inattention, hyperactivity, and impulsivity that are pervasive, impairing in multiple contexts, and otherwise age-inap ...
(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.
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.
Mathematical definitions
White noise vector
A
random vector (that is, a partially indeterminate process that produces vectors of real numbers) is said to be a white noise vector or white random vector if its components each have a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
with zero mean and finite
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
, and are
statistically independent: that is, their
joint probability distribution must be the product of the distributions of the individual components.
[
Jeffrey A. Fessler (1998), ]
On Transformations of Random Vectors.
' Technical report 314, Dept. of Electrical Engineering and Computer Science, Univ. of Michigan. ( PDF)
A necessary (but,
in general, not sufficient) condition for statistical independence of two variables is that they be
statistically uncorrelated; that is, their
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
is zero. Therefore, the
covariance matrix ''R'' of the components of a white noise vector ''w'' with ''n'' elements must be an ''n'' by ''n''
diagonal matrix, where each diagonal element ''R
ii'' is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of component ''w
i''; and the
correlation matrix must be the ''n'' by ''n'' identity matrix.
If, in addition to being independent, every variable in ''w'' also has a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
with zero mean and the same variance
, ''w'' is said to be a Gaussian white noise vector. In that case, the joint distribution of ''w'' is a
multivariate normal distribution; the independence between the variables then implies that the distribution has
spherical symmetry in ''n''-dimensional space. Therefore, any
orthogonal transformation of the vector will result in a Gaussian white random vector. In particular, under most types of
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
, such as
FFT and
Hartley
Hartley may refer to:
Places Australia
*Hartley, New South Wales
* Hartley, South Australia
** Electoral district of Hartley, a state electoral district
Canada
*Hartley Bay, British Columbia
United Kingdom
* Hartley, Cumbria
* Hartley, Pl ...
, the transform ''W'' of ''w'' will be a Gaussian white noise vector, too; that is, the ''n'' Fourier coefficients of ''w'' will be independent Gaussian variables with zero mean and the same variance
.
The
power spectrum ''P'' of a random vector ''w'' can be defined as the expected value of the
squared modulus of each coefficient of its Fourier transform ''W'', that is, ''P
i'' = E(, ''W
i'',
2). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with ''P
i'' = ''σ''
2 for all ''i''.
If ''w'' is a white random vector, but not a Gaussian one, its Fourier coefficients ''W
i'' will not be completely independent of each other; although for large ''n'' 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.
[Eric Zivot and Jiahui Wang (2006)]
Modeling Financial Time Series with S-PLUS
Second Edition. ( PDF) Other authors use strongly white and weakly white instead.
Francis X. Diebold
Francis X. Diebold (born November 12, 1959) is an American economist known for his work in predictive econometric modeling, financial econometrics, and macroeconometrics. He earned both his B.S. and Ph.D. degrees at the University of Pennsylvani ...
(2007),
Elements of Forecasting
'' 4th edition. ( PDF)
An example of a random vector that is "Gaussian white noise" in the weak but not in the strong sense is