Editing Colors of noise

{{Short description|Power spectrum of a noise signal}}
{{Redirect|Black noise|other uses|Black Noise (disambiguation){{!}}Black Noise}}
{{Confusing|date=December 2021}}
{{Use dmy dates|date=October 2023}}
[[File:The Colors of Noise.svg|thumb|right|Simulated power spectral densities as a function of frequency for various colors of noise (violet, blue, white, pink, Brown/red).
The power spectral densities are arbitrarily normalized such that the value of the spectra are approximately equivalent near 1&nbsp;kHz.
Note the slope of the power spectral density for each spectrum provides the context for the respective electromagnetic/color analogy.]]
{{Colors of 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 processing)|noise signal]] (a signal produced by a [[stochastic process]]). Different colors of noise have significantly different properties. For example, as [[audio signal]]s they will sound different to [[hearing (sense)|human ears]], and as [[digital image|images]] they will have a visibly different [[texture (visual arts)|texture]]. 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]] (which is also called "tone color"; however, the latter is almost always used for [[sound]], and may consider detailed features of the [[spectrum]]).

The practice of naming kinds of noise after colors started with [[white noise]], a signal whose spectrum has equal [[power (physics)|power]] 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.{{Citation needed|date=April 2015}} 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''&nbsp;<sup>''β''</sup> and hence they are examples of ''power-law noise''.  For instance, the spectral density of [[white noise]] is flat (''β'' = 0), while [[flicker noise|flicker]] or [[pink noise]] has ''β'' = 1, and [[Brownian noise]] has ''β'' = 2. [[Blue noise]] has ''β'' = -1.

==Technical definitions==

Various noise models are employed in analysis, many of which fall under the above categories. [[AR noise]] or "autoregressive noise" is such a model, and generates simple examples of the above noise types, and more. The [[Federal Standard 1037C]] Telecommunications Glossary<ref>{{cite web|title=ATIS Telecom Glossary|url=https://www.atis.org/glossary/normative.aspx|website=atis.org|publisher=Alliance for Telecommunications Industry Solutions|access-date=16 January 2018}}</ref><ref>{{cite web|title=Federal Standard 1037C|url=https://its.ntia.gov/research-topics/federal-standard-1037c/|website=Institute for Telecommunication Sciences|publisher=Institute for Telecommunication Sciences, National Telecommunications and Information Administration (ITS-NTIA)|access-date=30 November 2022}}</ref> 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.{{Citation needed|date=January 2018}}
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===White noise===
{{Main|White noise}}

[[Image:White noise spectrum.svg|thumb|right|White noise has a flat power spectrum.]]

[[White noise]] is a [[Signal (electrical engineering)|signal]] (or process), named by analogy to [[Electromagnetic spectrum#Visible radiation (light)|white light]], with a flat [[frequency spectrum]] when plotted as a linear function of frequency (e.g., in Hz). In other words, the signal has equal [[power (physics)|power]] in any band of a given [[Bandwidth (signal processing)|bandwidth]] ([[power spectral density]]) when the bandwidth is measured in [[hertz|Hz]].  For example, with a white noise audio signal, the range of frequencies between 40 [[hertz|Hz]] and 60&nbsp;Hz contains the same amount of sound power as the range between 400&nbsp;Hz and 420&nbsp;Hz, since both intervals are 20&nbsp;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.<ref name="Peters-2012">{{cite web|url=https://physics.mercer.edu/hpage/psd-tutorial/psd.html|author=Randall D. Peters|date=2 January 2012|title=Tutorial on Power Spectral Density Calculations for Mechanical Oscillators}}</ref>

{{Listen|filename=White noise.ogg|title=10 seconds of white noise|description=}}
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===Pink noise===
{{Main|Pink noise}}

<!-- Image is wrong. Pink noise should fall at 10dB/dec. [[Image:Pink-noise-db-logf.png|thumb|right|Pink spectrum <br>(log frequency axis)]] -->
[[File:Pink noise spectrum.svg|thumb|right|Pink noise spectrum. Power density falls off at 10&nbsp;dB/decade (−3.01&nbsp;dB/octave).]]

The frequency spectrum of [[pink noise]] is linear in [[logarithmic scale]]; it has equal power in bands that are proportionally wide.<ref>{{cite web|url=https://www.its.bldrdoc.gov/fs-1037/dir-027/_4019.htm|url-status=dead|archive-url=https://web.archive.org/web/20210608074332/https://www.its.bldrdoc.gov/fs-1037/dir-027/_4019.htm|archive-date=8 June 2021|title=Definition: pink noise|website=its.bldrdoc.gov}}</ref> This means that pink noise would have equal power in the frequency range from 40 to 60&nbsp;Hz as in the band from 4000 to 6000&nbsp;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&nbsp;Hz is heard as the same interval and distance as 4000–6000&nbsp;Hz), every octave contains the same amount of energy and thus pink noise is often used as a reference signal in [[audio engineering]]. The [[spectral power density]], compared with white noise, decreases by 3.01&nbsp;[[decibel|dB]] per [[octave]] (10&nbsp;dB per [[Decade_(log_scale)|decade]]); 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.{{citation needed|date=May 2014}}

{{Listen|filename=Pink noise.ogg|title=10 seconds of pink noise|description=}}
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===Brownian noise===
{{Main|Brownian noise}}

[[Image:Brown noise spectrum.svg|thumb|right|Brown spectrum (−6.02&nbsp;dB per octave)]]

[[Brownian noise]], also called Brown noise, is noise with a power density which decreases 6.02&nbsp;dB per octave (20&nbsp;dB per decade) with increasing frequency (frequency density proportional to 1/''f''{{i sup|2}}) over a frequency range excluding zero ([[direct current|DC]]). It is also called "red noise", with pink being between red and white.

Brownian noise can be generated with temporal [[Integral|integration]] of [[white noise]]. "Brown" noise is not named for a power spectrum that suggests the color brown; rather, the name derives from [[Brownian motion]], also known as "random walk" or "drunkard's walk". 
{{Listen|filename=Brown noise.ogg|title=10 seconds of Brown noise|description=}}
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===Blue noise===
[[Image:Blue noise spectrum.svg|thumb|right|Blue spectrum (+3.01&nbsp;dB per octave)]]

Blue noise is also called azure noise. Blue noise's power density increases <math>10\log_{10}2 = </math> 3.01&nbsp;dB per octave with increasing frequency (density proportional to ''f''&nbsp;) over a finite frequency range.<ref>{{cite web|url=https://www.its.bldrdoc.gov/fs-1037/dir-005/_0685.htm|url-status=dead|archive-url=https://web.archive.org/web/20210608074403/https://www.its.bldrdoc.gov/fs-1037/dir-005/_0685.htm|archive-date=8 June 2021|title=Definition: blue noise|website=its.bldrdoc.gov}}</ref> 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]]ing.<ref>{{cite book | last1=Mitchell | first1=Don P. | title=Proceedings of the 14th annual conference on Computer graphics and interactive techniques | chapter=Generating antialiased images at low sampling densities | year=1987 | volume=21 | issue=4 | pages=65–72 | doi=10.1145/37401.37410| isbn=0897912276 | s2cid=207582968 }}</ref> [[Retina]]l cells are arranged in a blue-noise-like pattern which yields good visual resolution.<ref>{{cite journal | last1 = Yellott | first1 = John I. Jr | year = 1983 | title = Spectral Consequences of Photoreceptor Sampling in the Rhesus Retina | journal = Science | volume = 221 | issue = 4608| pages = 382–85 | doi = 10.1126/science.6867716 | pmid = 6867716 | bibcode = 1983Sci...221..382Y | url = https://escholarship.org/uc/item/0qq9b8zx }}</ref>

[[Cherenkov radiation]] 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]] greater than unity. Cherenkov radiation also appears as a bright blue color, for these reasons.



{{Listen|filename=Blue noise.ogg|title=10 seconds of blue noise|description=}}
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===Violet noise===
[[Image:Violet noise spectrum.svg|thumb|right|Violet spectrum (+6.02&nbsp;dB/octave)]]

Violet noise is also called '''purple noise'''. Violet noise's power density increases 6.02&nbsp;dB per octave with increasing frequency<ref>Transactions of the American Society of Heating, Refrigerating and Air-Conditioning Engineers 1968 [https://books.google.com/books?doi=aTJSAAAAMAAJ&q=%22purple+noise%22] Quote: 'A "purple noise," accordingly, is a noise the spectrum level of which ''rises'' with frequency.'</ref><ref>{{cite conference | doi=10.1109/PLANS.1996.509090 | first1=Q. J. | last1=Zhang | first2=K.-P. | last2=Schwarz | title=Estimating double difference GPS multipath under kinematic conditions| book-title=Proceedings of the Position Location and Navigation Symposium – PLANS '96 | publisher=[[IEEE]] | location= Atlanta, GA, USA | conference = Position Location and Navigation Symposium – PLANS '96 |pages=285–91 | date = April 1996}}</ref> "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''&nbsp;<sup>2</sup>) over a finite frequency range. It is also known as [[Derivative|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.{{Citation needed|date=March 2020}}

Acoustic thermal noise of water has a violet spectrum, causing it to dominate [[hydrophone]] measurements at high frequencies.<ref name="Hildebrand2009">{{cite journal | doi=10.3354/meps08353 | title=Anthropogenic and natural sources of ambient noise in the ocean | date=2009 | first1=John A. | last1=Hildebrand | pages = 478–480 | journal=Marine Ecology Progress Series| volume=395 | bibcode=2009MEPS..395....5H | doi-access=free }}</ref> "Predictions of the thermal noise spectrum, derived from classical statistical mechanics, suggest increasing noise with frequency with a positive slope of 6.02&nbsp;dB octave<sup>−1</sup>." "Note that thermal noise increases at the rate of 20&nbsp;dB decade<sup>−1</sup>"<ref>{{cite journal | last1 = Mellen | first1 = R. H. | date=1952 | title= The Thermal-Noise Limit in the Detection of Underwater Acoustic Signals | journal = The Journal of the Acoustical Society of America | volume = 24 | issue=5 | pages = 478–80 | doi=10.1121/1.1906924| bibcode = 1952ASAJ...24..478M }}</ref>

{{Listen|filename=Purple noise.ogg|title=10 seconds of violet noise|description=}}
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===Grey noise===
{{Main|Grey noise}}

[[Image:Gray noise spectrum.svg|thumb|right|Grey spectrum]]
[[Grey noise]] is random white noise subjected to a [[Psychoacoustics|psychoacoustic]] 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.{{Citation needed|date=March 2010}} 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]].

{{Listen|filename=Gray noise.ogg|title=10 seconds of grey noise|description=}}
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===Velvet noise===
[[Image:Velvet_Noise_Spectrum.jpg|thumb|right|Velvet noise spectrum]]
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".<ref name="Välimäki2013">{{cite journal | doi=10.1109/TASL.2013.2255281 | title=A Perceptual Study on Velvet Noise and Its Variants at Different Pulse Densities | date=2013 | first1=Vesa | last1=Välimäki | first2=Heidi-Maria | last2=Lehtonen |first3=Marko | last3=Takanen | pages = 1481–1488 | journal= IEEE Transactions on Audio, Speech, and Language Processing| volume=21 | issue=7 | s2cid=17173495 }}</ref> The sparse nature of velvet noise allows for efficient time-domain [[convolution]], making velvet noise particularly useful for applications where computational resources are limited, like real-time [[reverberation]] algorithms.<ref>{{cite conference | first1=Hanna | last1=Järveläinen | first2=Matti | last2=Karjalainen | title=Reverberation Modeling Using Velvet Noise | publisher=[[Audio Engineering Society|AES]] | location= Helsinki, Finland | conference = 30th International Conference: Intelligent Audio Environments | date = March 2007}}</ref><ref>{{cite web |url=https://ccrma.stanford.edu/~keunsup/screverb.html | title = The Switched Convolution Reverberator, Lee et. al.}}</ref> Velvet noise is also frequently used in decorrelation filters.<ref>{{cite conference | first1=Benoit | last1=Alary | first2=Archontis | last2=Politis | first3=Vesa | last3=Välimäki | title= Velvet-Noise Decorrelator | location= Edinburgh, UK | conference = 20th International Conference on Digital Audio Effects (DAFx-17) | date = September 2017}}</ref>

{{Listen|filename=Velvet noise.ogg|title=2 seconds of velvet noise.}}

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==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.<ref>{{Cite web|url=https://www.ears.dmu.ac.uk/rubrique.php3?id_rubrique=93|archive-url=https://web.archive.org/web/20060522073646/https://www.ears.dmu.ac.uk/rubrique.php3?id_rubrique=93|url-status=dead|title=Index: Noise (Disciplines of Study [DoS&#93;)<!-- Bot generated title -->|archive-date=22 May 2006}}</ref><ref>{{cite journal |last1=Gilman |first1=D. L. |last2=Fuglister |first2=F. J. |last3=Mitchell Jr. |first3=J. M. |title=On the power spectrum of "red noise" |journal=Journal of the Atmospheric Sciences |volume=20 |number=2|pages=182–84 |doi=10.1175/1520-0469(1963)020<0182:OTPSON>2.0.CO;2 |bibcode=1963JAtS...20..182G |year=1963 |doi-access=free }}</ref> That is, it is similar to pink noise, but with different spectral content and different relationships (i.e. 1/f for [[pink noise]], while 1/f<sup>2</sup> for red noise, or a decrease of 6.02&nbsp;dB per octave).
* In areas where terminology is used loosely, "red noise" may refer to any system where power density decreases with increasing frequency.<ref>{{Cite journal|title=Red noise and regime shifts|author=Daniel L. Rudnick, Russ E. Davis|journal=Deep-Sea Research Part I|volume=50|issue=6|year=2003|pages=691–99|url=https://www-pord.ucsd.edu/~rdavis/publications/Red_Noise.pdf|doi=10.1016/S0967-0637(03)00053-0|bibcode = 2003DSRI...50..691R }}</ref>
<!--
====Orange noise==== The definition at https://www.ptpart.co.uk/colors-of-noise/ is a joke, not a serious definition. -->

===Green noise===
{{Listen|filename=GreenNoise.ogg|title=10 seconds of Wisniewski's version of green noise|description=}}
* The mid-frequency component of white noise, used in [[halftone]] [[dithering]]<ref>{{cite journal
| first1 = Daniel Leo
| last1  = Lau
| first2 = Gonzalo R.
| last2 = Arce
| first3 = Neal C.
| last3 = Gallagher
| title = Green-noise digital halftoning
| journal = Proceedings of the IEEE
| volume  = 86
| number   = 12
| year    = 1998
| pages   = 2424–42
| doi=10.1109/5.735449
}}</ref>
* Bounded Brownian noise
* Vocal spectrum noise used for testing audio circuits<ref name="comp.dsp FAQ">{{cite newsgroup
 | title = Colors of noise pseudo FAQ, version 1.3
 | author = Joseph S. Wisniewski
 | date = 7 October 1996
 | newsgroup = comp.dsp
 | url = https://www.ptpart.co.uk/colors-of-noise/
 | access-date = 2011-03-01
 | url-status = dead
 | archive-url = https://web.archive.org/web/20110430151608/https://www.ptpart.co.uk/colors-of-noise
 | archive-date = 30 April 2011
 }}</ref>
* 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&nbsp;Hz.<ref name="comp.dsp FAQ"/>

===Black noise===

* [[Silence]]
* [[Infrasound]]<ref>{{Cite web|url=https://vigilantcitizenforums.com/threads/david-bowie-and-the-black-noise.579/|title=David Bowie and the Black Noise|website=The Vigilant Citizen Forums|date=21 May 2017 }}</ref>
* Noise with a 1/''f''{{i sup|''β''}} spectrum, where {{nowrap|''β'' > 2}}. This formula is used to model the frequency of natural disasters.<ref>{{cite book |first=Manfred |last=Schroeder |title=Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise |url=https://books.google.com/books?id=Qpa77Jl2rvQC&pg=PA129 |pages=129–30 |publisher=Courier Dover |year=2009 |isbn=978-0486472041}}</ref>{{clarify|reason=What is the relation between "natural disasters" and "noise"?|date=January 2018}}
* 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.<ref>{{cite web|url=https://glossary.its.bldrdoc.gov/fs-1037/dir-005/_0649.htm|title=Definition of "black noise" – Federal Standard 1037C|access-date=28 April 2008|archive-url=https://web.archive.org/web/20081212191754/https://glossary.its.bldrdoc.gov/fs-1037/dir-005/_0649.htm|archive-date=12 December 2008|url-status=dead}}</ref>
* Noise with a spectrum corresponding to the [[blackbody]] radiation (thermal noise). For temperatures higher than about {{val|3|e=-7|u=K}} the [[Black-body radiation#Wien's displacement law|peak of the blackbody spectrum]] is above the upper limit of human [[hearing range]]. In those situations, for the purposes of what is heard, black noise is well approximated as [[violet noise]]. At the same time, [[Hawking radiation]] of [[black hole]]s may have a peak in hearing range, so the radiation of a typical [[stellar black hole]] with a mass equal to 6 solar masses will have a maximum at a frequency of 604.5&nbsp;Hz – this noise is similar to green noise. A formula is: <math>f_{\text{max}} \approx 3627 \times {\text{M}_\odot \over \text{M}}</math>&nbsp;Hz.  Several examples of audio files with this spectrum can be found [https://physics.stackexchange.com/a/216688/92058 here].{{citation needed|reason=Looks like original research, or a niche interest.|date=October 2023}}

===Noisy white===

In [[telecommunication]], the term '''noisy white''' has the following meanings:<ref>{{cite web|url=https://www.its.bldrdoc.gov/fs-1037/dir-024/_3570.htm|url-status=dead|archive-url=https://web.archive.org/web/20210608074217/https://www.its.bldrdoc.gov/fs-1037/dir-024/_3570.htm|archive-date=8 June 2021|title=Definition: noisy white|website=its.bldrdoc.gov}}</ref>
* In [[Fax|facsimile]] or display systems, such as [[television]], a nonuniformity in the [[white area]] of the image, ''i.e.'', document or picture, caused by the presence of [[noise]] in the received [[signal]].
* A signal or [[signal level]] 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]], the term '''noisy black''' has the following meanings:<ref>{{cite web|url=https://www.its.bldrdoc.gov/fs-1037/dir-024/_3569.htm|url-status=dead|archive-url=https://web.archive.org/web/20210608074217/https://www.its.bldrdoc.gov/fs-1037/dir-024/_3569.htm|archive-date=8 June 2021|title=Definition: noisy black|website=its.bldrdoc.gov}}</ref>
* In [[Fax|facsimile]] or display systems, such as [[television]], a nonuniformity in the black area of the image, ''i.e.'', document or picture, caused by the presence of [[noise]] in the received signal.
* A signal or [[signal level]] 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.<ref name="Das-thesis">{{cite thesis |last=Das |first=Abhranil |date=2022 |title=Camouflage detection & signal discrimination: theory, methods & experiments (corrected) |type=PhD |publisher=The University of Texas at Austin |url=http://dx.doi.org/10.13140/RG.2.2.32016.07683 | doi=10.13140/RG.2.2.32016.07683}}</ref> Matlab programs are available to generate power-law colored noise in [http://www.maxlittle.net/software/ one] or  [https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise 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.<ref>{{Cite book |last1=Riley |first1=W.J. |last2=Greenhal |first2=C.A. |title=18th European Frequency and Time Forum (EFTF 2004) |chapter=Power law noise identification using the lag 1 autocorrelation |date=2004 |url=https://digital-library.theiet.org/content/conferences/10.1049/cp_20040932 |language=en |publisher=IEE |pages=576–580 |doi=10.1049/cp:20040932 |isbn=978-0-86341-384-1}}</ref> 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):

<math>R_1 = \frac{\frac{1}{N}\sum_{t=1}^{N-1}(z_t - \bar z)*(z_{t+1} - \bar z)}
{\frac{1}{N}\sum_{t=1}^{N}{(z_t - \bar z)}^{2}}</math>

where <math>N </math> is the number of data points in the time series, <math>z_t </math> are the phase or frequency values, and <math>\bar z </math> is the average value of the time series. If used for clock stability analysis, the <math>z_t </math> 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 <math>(2sin(\pi f))^{-2\delta} </math> which are stationary for <math>\delta < .25 </math>. The value of <math>\delta </math> is calculated using <math>R_1 </math>:

<math>\delta = \frac{R_1}{1+R_1} </math>

where <math>R_1 </math> is the lag(1) autocorrelation function defined above. If <math>\delta > .25 </math> then the first differences of the adjacent time series data are taken <math>d </math> times until <math>\delta < .25 </math>. The power law for the stationary noise process is calculated from the calculated <math>\delta </math> and the number of times the data has been differenced to achieve <math>\delta < .25 </math> as follows:

<math>p = -2(\delta + d)  </math>

where <math>p </math> is the power of the frequency noise which can be rounded to identify the dominant noise type (for frequency data <math>p </math> is the power of the frequency noise but for phase data the power of the frequency noise is <math>p+2 </math>).

=== 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:<ref>{{Cite book |last1=Zhou Chunlei |last2=Zhang Qi |last3=Yan Shuhua |chapter=Power law noise identification using the LAG 1 autocorrelation by overlapping samples |date=August 2011 |pages=110–113 |title=IEEE 2011 10th International Conference on Electronic Measurement & Instruments |chapter-url=http://dx.doi.org/10.1109/icemi.2011.6037776 |publisher=IEEE |doi=10.1109/icemi.2011.6037776|isbn=978-1-4244-8158-3 }}</ref>

<math>R_m = \frac{\frac{1}{N}\sum_{t=1}^{N-m}(z_t - \bar z)*(z_{t+m} - \bar z)}
{\frac{1}{N}\sum_{t=1}^{N}{(z_t - \bar z)}^{2}}</math>

where <math>m</math> is the "lag" or shift between the time series and the delayed version of itself. A major difference is that <math>z_t </math> are now the averaged values of the original time series computed with a moving window average and averaging factor also equal to <math>m</math>. The value of <math>\delta </math> is computed the same way as in the previous method and <math>\delta < .25 </math> 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 (<math>\delta < .25 </math>) is done between values that are spaced a distance <math>m</math> apart:

<math>z_1 = z_{1+m}-z_1, z_2 = z_{2+m} - z_2..., z_{N-m} = z_N - z_{N-m} </math>

The value of the power is calculated the same as the previous method as well.

==See also==
* [[Mains hum]] (also known as the [[Alternating current|AC]] power hum)
* [[Whittle likelihood]]

==References==
{{Reflist}}
{{FS1037C}}

==External links==
* [https://web.archive.org/web/20050720074308/https://mv.lycaeum.org/M2/noise_ahf.html Some colored noise definitions]
* [https://mynoise.net/NoiseMachines/whiteNoiseGenerator.php Online Colored Noise Generator] and [https://mynoise.net/NoiseMachines/greyNoiseGenerator.php True Grey Noise Generator]
* [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1689925/pdf/LUM75XB5B6U2B1W0_266_969.pdf Black Noise and Population Persistence]
{{Noise}}
{{Timbre}}

{{DEFAULTSORT:Colors of Noise}}
[[Category:Noise (electronics)]]
[[Category:Encodings]]