Noise Spectrum
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In
audio engineering Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound * Audio frequency, a frequency in the audio spectrum *Digital audio, representation of soun ...
,
electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
,
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, and many other fields, the color of noise or noise spectrum refers to the
power spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...
of a noise signal (a signal produced by a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
). Different colors of noise have significantly different properties. For example, as
audio signal An audio signal is a representation of sound, typically using either a changing level of electrical voltage for analog signals or a series of binary numbers for Digital signal (signal processing), digital signals. Audio signals have frequencies i ...
s they will sound different to human ears, and as
images An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a project ...
they will have a visibly different
texture Texture may refer to: Science and technology * Image texture, the spatial arrangement of color or intensities in an image * Surface texture, the smoothness, roughness, or bumpiness of the surface of an object * Texture (roads), road surface c ...
. Therefore, each application typically requires noise of a specific color. This sense of 'color' for noise signals is similar to the concept of
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
in
music Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
(which is also called "tone color"; however, the latter is almost always used for
sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
, and may consider detailed features of the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
). The practice of naming kinds of noise after colors started with
white noise In signal processing, 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, i ...
, a signal whose spectrum has equal
power Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Math ...
within any equal interval of frequencies. That name was given by analogy with white light, which was (incorrectly) assumed to have such a flat power spectrum over the visible range. Other color names, such as ''pink'', ''red'', and ''blue'' were then given to noise with other spectral profiles, often (but not always) in reference to the color of light with similar spectra. Some of those names have standard definitions in certain disciplines, while others are informal and poorly defined. Many of these definitions assume a signal with components at all frequencies, with a power spectral density per unit of bandwidth proportional to 1/''f'' ''β'' and hence they are examples of ''power-law noise''. For instance, the spectral density of
white noise In signal processing, 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, i ...
is flat (''β'' = 0), while flicker or
pink noise Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequenc ...
has ''β'' = 1, and
Brownian noise In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from brown, the color, but after ...
has ''β'' = 2.
Blue noise In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly ...
has ''β'' = -1.


Technical definitions

Various noise models are employed in analysis, many of which fall under the above categories.
AR noise In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
or "autoregressive noise" is such a model, and generates simple examples of the above noise types, and more. The
Federal Standard 1037C Federal Standard 1037C, titled Telecommunications: Glossary of Telecommunication Terms, is a United States Federal Standard issued by the General Services Administration pursuant to the Federal Property and Administrative Services Act of 1949, ...
Telecommunications Glossary defines white, pink, blue, and black noise. The color names for these different types of sounds are derived from a loose analogy between the spectrum of frequencies of sound wave present in the sound (as shown in the blue diagrams) and the equivalent spectrum of light wave frequencies. That is, if the sound wave pattern of "blue noise" were translated into light waves, the resulting light would be blue, and so on.


White noise

White noise In signal processing, 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, i ...
is a
signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...
(or process), named by analogy to
white light White is the lightest color and is achromatic (having no chroma). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully (or almost fully) reflect and scatter all the visible wavelen ...
, with a flat
frequency spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed int ...
when plotted as a linear function of frequency (e.g., in Hz). In other words, the signal has equal
power Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Math ...
in any band of a given
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
(
power spectral density In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be decomposed into ...
) when the bandwidth is measured in Hz. For example, with a white noise audio signal, the range of frequencies between 40 Hz and 60 Hz contains the same amount of sound power as the range between 400 Hz and 420 Hz, since both intervals are 20 Hz wide. Note that spectra are often plotted with a logarithmic frequency axis rather than a linear one, in which case equal physical widths on the printed or displayed plot do not all have the same bandwidth, with the same physical width covering more Hz at higher frequencies than at lower frequencies. In this case a white noise spectrum that is equally sampled in the logarithm of frequency (i.e., equally sampled on the X axis) will slope upwards at higher frequencies rather than being flat. However it is not unusual in practice for spectra to be calculated using linearly-spaced frequency samples but plotted on a logarithmic frequency axis, potentially leading to misunderstandings and confusion if the distinction between equally spaced linear frequency samples and equally spaced logarithmic frequency samples is not kept in mind.


Pink noise

The frequency spectrum of
pink noise Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequenc ...
is linear in
logarithmic scale A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved. Unlike a linear Scale (measurement) ...
; it has equal power in bands that are proportionally wide. This means that pink noise would have equal power in the frequency range from 40 to 60 Hz as in the band from 4000 to 6000 Hz. Since humans hear in such a proportional space, where a doubling of frequency (an octave) is perceived the same regardless of actual frequency (40–60 Hz is heard as the same interval and distance as 4000–6000 Hz), every octave contains the same amount of energy and thus pink noise is often used as a reference signal in
audio engineering Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound * Audio frequency, a frequency in the audio spectrum *Digital audio, representation of soun ...
. The spectral power density, compared with white noise, decreases by 3.01  dB per
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
(10 dB per
decade A decade (from , , ) is a period of 10 years. Decades may describe any 10-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement ...
); density proportional to 1/''f''. For this reason, pink noise is often called "1/''f'' noise". Since there are an infinite number of logarithmic bands at both the low frequency (DC) and high frequency ends of the spectrum, any finite energy spectrum must have less energy than pink noise at both ends. Pink noise is the only power-law spectral density that has this property: all steeper power-law spectra are finite if integrated to the high-frequency end, and all flatter power-law spectra are finite if integrated to the DC, low-frequency limit.


Brownian noise

Brownian noise In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from brown, the color, but after ...
, also called Brown noise, is noise with a power density which decreases 6.02 dB per octave (20 dB per decade) with increasing frequency (frequency density proportional to 1/''f'') over a frequency range excluding zero ( DC). It is also called "red noise", with pink being between red and white. Brownian noise can be generated with temporal
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
of
white noise In signal processing, 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, i ...
. "Brown" noise is not named for a power spectrum that suggests the color brown; rather, the name derives from
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
, also known as "random walk" or "drunkard's walk".


Blue noise

Blue noise is also called azure noise. Blue noise's power density increases 10\log_2 = 3.01 dB per octave with increasing frequency (density proportional to ''f'' ) over a finite frequency range. In computer graphics, the term "blue noise" is sometimes used more loosely as any noise with minimal low frequency components and no concentrated spikes in energy. This can be good noise for
dither Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither is routinely used in processing of both digital audio and video data, and is ofte ...
ing.
Retina The retina (; or retinas) is the innermost, photosensitivity, light-sensitive layer of tissue (biology), tissue of the eye of most vertebrates and some Mollusca, molluscs. The optics of the eye create a focus (optics), focused two-dimensional ...
l cells are arranged in a blue-noise-like pattern which yields good visual resolution.
Cherenkov radiation Cherenkov radiation () is electromagnetic radiation emitted when a charged particle (such as an electron) passes through a dielectric medium (such as distilled water) at a speed greater than the phase velocity (speed of propagation of a wavefro ...
is a naturally occurring example of almost perfect blue noise, with the power density growing linearly with frequency over spectrum regions where the permeability of index of refraction of the medium are approximately constant. The exact density spectrum is given by the Frank–Tamm formula. In this case, the finiteness of the frequency range comes from the finiteness of the range over which a material can have a
refractive index In optics, the refractive index (or refraction index) of an optical medium is the ratio of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of light is bent, or refrac ...
greater than unity. Cherenkov radiation also appears as a bright blue color, for these reasons.


Violet noise

Violet noise is also called purple noise. Violet noise's power density increases 6.02 dB per octave with increasing frequency "The spectral analysis shows that GPS acceleration errors seem to be violet noise processes. They are dominated by high-frequency noise." (density proportional to ''f'' 2) over a finite frequency range. It is also known as differentiated white noise, due to its being the result of the differentiation of a white noise signal. Due to the diminished sensitivity of the human ear to high-frequency hiss and the ease with which white noise can be electronically differentiated (high-pass filtered at first order), many early adaptations of dither to digital audio used violet noise as the dither signal. Acoustic thermal noise of water has a violet spectrum, causing it to dominate
hydrophone A hydrophone () is a microphone designed for underwater use, for recording or listening to underwater sound. Most hydrophones contains a piezoelectric transducer that generates an electric potential when subjected to a pressure change, such as a ...
measurements at high frequencies. "Predictions of the thermal noise spectrum, derived from classical statistical mechanics, suggest increasing noise with frequency with a positive slope of 6.02 dB octave−1." "Note that thermal noise increases at the rate of 20 dB decade−1"


Grey noise

Grey noise Grey noise is random noise whose frequency spectrum follows an equal-loudness contour (such as an ''inverted'' A-weighting curve). The result is that grey noise contains all frequencies with equal loudness, as opposed to white noise, which ...
is random white noise subjected to a
psychoacoustic Psychoacoustics is the branch of psychophysics involving the scientific study of the perception of sound by the human auditory system. It is the branch of science studying the psychological responses associated with sound including noise, speech, ...
equal loudness curve (such as an inverted A-weighting curve) over a given range of frequencies, giving the listener the perception that it is equally loud at all frequencies. This is in contrast to standard white noise which has equal strength over a linear scale of frequencies but is not perceived as being equally loud due to biases in the human
equal-loudness contour An equal-loudness contour is a measure of sound pressure level, over the frequency spectrum, for which a listener perceives a constant loudness when presented with pure steady tones. The unit of measurement for loudness levels is the phon an ...
.


Velvet noise

Velvet noise is a sparse sequence of random positive and negative impulses. Velvet noise is typically characterised by its density in taps/second. At high densities it sounds similar to white noise; however, it is perceptually "smoother". The sparse nature of velvet noise allows for efficient time-domain
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
, making velvet noise particularly useful for applications where computational resources are limited, like real-time
reverberation In acoustics, reverberation (commonly shortened to reverb) is a persistence of sound after it is produced. It is often created when a sound is reflection (physics), reflected on surfaces, causing multiple reflections that build up and then de ...
algorithms. Velvet noise is also frequently used in decorrelation filters.


Informal definitions

There are also many colors used without precise definitions (or as synonyms for formally defined colors), sometimes with multiple definitions.


Red noise

* A synonym for Brownian noise, as above. That is, it is similar to pink noise, but with different spectral content and different relationships (i.e. 1/f for
pink noise Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequenc ...
, while 1/f2 for red noise, or a decrease of 6.02 dB per octave). * In areas where terminology is used loosely, "red noise" may refer to any system where power density decreases with increasing frequency.


Green noise

* The mid-frequency component of white noise, used in
halftone Halftone is the reprographic technique that simulates continuous tone, continuous-tone imagery through the use of dots, varying either in size or in spacing, thus generating a gradient-like effect.Campbell, Alastair. ''The Designer's Lexicon''. ...
dithering Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither is routinely used in processing of both digital audio and video data, and is often ...
* Bounded Brownian noise * Vocal spectrum noise used for testing audio circuits * Joseph S. Wisniewski writes that "green noise" is marketed by producers of ambient sound effects recordings as "the background noise of the world". It simulates the spectra of natural settings, without human-made noises. It is similar to pink noise, but has more energy in the area of 500 Hz.


Black noise

*
Silence Silence is the absence of ambient hearing, audible sound, the emission of sounds of such low sound intensity, intensity that they do not draw attention to themselves, or the state of having ceased to produce sounds; this latter sense can be exten ...
*
Infrasound Infrasound, sometimes referred to as low frequency sound or incorrectly subsonic (subsonic being a descriptor for "less than the speed of sound"), describes sound waves with a Audio frequency, frequency below the lower limit of human audibility ...
* Noise with a 1/''f'' spectrum, where . This formula is used to model the frequency of natural disasters. * Noise that has a frequency spectrum of predominantly zero power level over all frequencies except for a few narrow bands or spikes. ''Note:'' An example of black noise in a facsimile transmission system is the spectrum that might be obtained when scanning a black area in which there are a few random white spots. Thus, in the time domain, a few random pulses occur while scanning. * Noise with a spectrum corresponding to the
blackbody A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The radiation emitted by a black body in thermal equilibrium with its environment is ...
radiation (thermal noise). For temperatures higher than about the peak of the blackbody spectrum is above the upper limit of human
hearing range Hearing range describes the frequency range that can be heard by humans or other animals, though it can also refer to the range of levels. The human range is commonly given as 20 to 20,000 Hz, although there is considerable variation bet ...
. In those situations, for the purposes of what is heard, black noise is well approximated as violet noise. At the same time,
Hawking radiation Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. The radiation was not predicted by previous models which assumed that onc ...
of
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s may have a peak in hearing range, so the radiation of a typical
stellar black hole A stellar black hole (or stellar-mass black hole) is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. They are the remnants of supernova explosions, which may be ...
with a mass equal to 6 solar masses will have a maximum at a frequency of 604.5 Hz – this noise is similar to green noise. A formula is: f_ \approx 3627 \times  Hz. Several examples of audio files with this spectrum can be foun
here


Noisy white

In
telecommunication Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
, the term noisy white has the following meanings: * In
facsimile A facsimile (from Latin ''fac simile'', "to make alike") is a copy or reproduction of an old book, manuscript, map, art print, or other item of historical value that is as true to the original source as possible. It differs from other forms of r ...
or display systems, such as
television Television (TV) is a telecommunication medium for transmitting moving images and sound. Additionally, the term can refer to a physical television set rather than the medium of transmission. Television is a mass medium for advertising, ...
, a nonuniformity in the
white area White is the lightness, lightest color and is achromatic (having no colorfulness, chroma). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully (or almost fully) diffuse reflection, ref ...
of the image, ''i.e.'', document or picture, caused by the presence of
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
in the received
signal A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology. In ...
. * A signal or
signal level Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
that is supposed to represent a white area on the object, but has a noise content sufficient to cause the creation of noticeable black spots on the display surface or record medium.


Noisy black

In
telecommunication Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
, the term noisy black has the following meanings: * In
facsimile A facsimile (from Latin ''fac simile'', "to make alike") is a copy or reproduction of an old book, manuscript, map, art print, or other item of historical value that is as true to the original source as possible. It differs from other forms of r ...
or display systems, such as
television Television (TV) is a telecommunication medium for transmitting moving images and sound. Additionally, the term can refer to a physical television set rather than the medium of transmission. Television is a mass medium for advertising, ...
, a nonuniformity in the black area of the image, ''i.e.'', document or picture, caused by the presence of
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
in the received signal. * A signal or
signal level Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
that is supposed to represent a black area on the object, but has a noise content sufficient to cause the creation of noticeable non-black spots on the display surface or record medium.


Generation

Colored noise can be computer-generated by first generating a white noise signal, Fourier-transforming it, then multiplying the amplitudes of the different frequency components with a frequency-dependent function. Matlab programs are available to generate power-law colored noise i
one
or
any number
of dimensions.


Identification of power law frequency noise

Identifying the dominant noise type in a time series has many applications including clock stability analysis and market forecasting. There are two algorithms based on autocorrelation functions that can identify the dominant noise type in a data set provided the noise type has a power law spectral density.


Lag(1) autocorrelation method (non-overlapped)

The first method for doing noise identification is based on a paper by W.J Riley and C.A Greenhall. First the lag(1) autocorrelation function is computed and checked to see if it is less than one third (which is the threshold for a stationary process): R_1 = \frac where N is the number of data points in the time series, z_t are the phase or frequency values, and \bar z is the average value of the time series. If used for clock stability analysis, the z_t values are the non-overlapped (or binned) averages of the original frequency or phase array for some averaging time and factor. Now discrete-time fractionally integrated noises have power spectral densities of the form (2sin(\pi f))^ which are stationary for \delta < .25 . The value of \delta is calculated using R_1 : \delta = \frac where R_1 is the lag(1) autocorrelation function defined above. If \delta > .25 then the first differences of the adjacent time series data are taken d times until \delta < .25 . The power law for the stationary noise process is calculated from the calculated \delta and the number of times the data has been differenced to achieve \delta < .25 as follows: p = -2(\delta + d) where p is the power of the frequency noise which can be rounded to identify the dominant noise type (for frequency data p is the power of the frequency noise but for phase data the power of the frequency noise is p+2 ).


Lag(m) autocorrelation method (overlapped)

This method improves on the accuracy of the previous method and was introduced by Z. Chunlei, Z. Qi, Y. Shuhuana. Instead of using the lag(1) autocorrelation function the lag(m) correlation function is computed instead: R_m = \frac where m is the "lag" or shift between the time series and the delayed version of itself. A major difference is that z_t are now the averaged values of the original time series computed with a moving window average and averaging factor also equal to m. The value of \delta is computed the same way as in the previous method and \delta < .25 is again the criteria for a stationary process. The other major difference between this and the previous method is that the differencing used to make the time series stationary (\delta < .25 ) is done between values that are spaced a distance m apart: z_1 = z_-z_1, z_2 = z_ - z_2..., z_ = z_N - z_ The value of the power is calculated the same as the previous method as well.


See also

*
Mains hum Mains hum, electric hum, cycle hum, or power line hum is a sound associated with alternating current which is twice the Utility frequency, frequency of the mains electricity. The fundamental frequency of this sound is usually double that of the l ...
(also known as the AC power hum) *
Whittle likelihood In statistics, Whittle likelihood is an approximation to the likelihood function of a stationary Gaussian time series. It is named after the mathematician and statistician Peter Whittle, who introduced it in his PhD thesis in 1951. It is commonly ...


References


External links


Some colored noise definitions

Online Colored Noise Generator
an
True Grey Noise Generator

Black Noise and Population Persistence
{{DEFAULTSORT:Colors of Noise Noise (electronics) Encodings