View source for Pink noise

{{short description|Signal with equal energy per octave}}
{{about||the album by Laura Mvula|Pink Noise (album)}}
{{about|1/f noise|quantum 1/f noise|quantum 1/f noise}}{{Redirect|Fractal noise|the novel|Fractal Noise}}
{{listen
 | filename    = Pink noise.ogg
 | title       = Pink noise
 | description = 10 seconds of pink noise, [[audio normalization|normalized]] to −1&nbsp;[[dBFS]] [[peak amplitude]]
 | type        = sound
}}
[[File:2D pink noise.png|thumb|upright|A two-dimensional pink noise [[grayscale]] image, generated with a computer program; some fields observed in nature are characterized by a similar power spectrum<ref name="Field-1987"/>]]
[[File:Pink noise cube.gif|thumb|upright|A 3D pink noise image, generated with a computer program, viewed as an animation in which each frame is a 2D slice]]
{{colors of noise}}

'''Pink noise''', '''{{math|1/{{itco|''f''}}}} 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 [[frequency]] of the signal. In pink noise, each [[Octave (electronics)|octave]] interval (halving or doubling in frequency) carries an equal amount of noise energy.

Pink noise sounds like a [[waterfall]].<ref>{{cite web |url=https://www.soundonsound.com/glossary/pink-noise |title=Glossary: Pink Noise |website=[[Sound on Sound]] |access-date=November 22, 2022}}</ref> It is often used to tune [[loudspeaker]] systems in [[professional audio]].<ref name=Sound>{{cite book |last1=Davis |first1=Gary |last2=Jones |first2=Ralph |date=1987 |title=The Sound Reinforcement Handbook |publisher=Hal Leonard |page=107 |isbn=0-88188-900-8}}</ref> Pink noise is one of the most commonly observed signals in biological systems.<ref>{{cite journal |last1=Szendro |first1=P |title=Pink-Noise Behaviour of Biosystems |journal=European Biophysics Journal |date=2001 |volume=30 |issue=3 |pages=227–231 |doi=10.1007/s002490100143 |pmid=11508842 |s2cid=24505215 |url=https://www.tandfonline.com/doi/abs/10.1081/JBC-100104145|url-access=subscription }}</ref>

The name arises from the pink appearance of visible light with this power spectrum.<ref name="Downey-2012">{{cite book |last=Downey |first=Allen |title=Think Complexity |year=2012 |publisher=O'Reilly Media |isbn=978-1-4493-1463-7 |page=79 |url=http://greenteapress.com/complexity/html/thinkcomplexity010.html#toc57 |quote=Visible light with this power spectrum looks pink, hence the name. }}</ref> This is in contrast with [[white noise]] that has equal intensity per frequency interval.

== Definition ==
Within the scientific literature, the term "{{math|1/{{itco|''f''}}}} noise" is sometimes used loosely to refer to any noise with a power spectral density of the form
<math display="block">S(f) \propto \frac{1}{f^\alpha},</math>
where {{mvar|f}} is frequency, and {{math|0 < ''α'' < 2}}, with exponent {{mvar|α}} usually close to {{math|1}}. One-dimensional signals with {{math|1=''α'' = 1}} are usually called pink noise.<ref>{{cite magazine |last=Baxandall |first=P. J. |date=November 1968 |title=Noise in Transistor Circuits: 1 - Mainly on fundamental noise concepts |url=http://www.keith-snook.info/wireless-world-magazine/Wireless-World-1968/Noise%20in%20Transistor%20Circuits%20-%20P%20J%20Baxandall.pdf |archive-url=https://web.archive.org/web/20160423103648/http://www.keith-snook.info/wireless-world-magazine/Wireless-World-1968/Noise%20in%20Transistor%20Circuits%20-%20P%20J%20Baxandall.pdf |archive-date=2016-04-23 |url-status=live |magazine=Wireless World |pages=388–392 |access-date=2019-08-08 }}</ref>

The following function describes a length {{mvar|N}} one-dimensional pink noise signal as a sum of sine waves with different frequencies, whose amplitudes fall off inversely with the square root of frequency {{mvar|u}} (so that power, which is the square of amplitude, falls off inversely with frequency) and whose phases are random:
<math display="block">h(x)=\sigma \sqrt{\frac{N}{2}} \sum_u \frac{\chi_u}{\sqrt{u}} \sin \left( \frac{2 \pi u x}{N} +\phi_u \right), \quad \chi_u \sim \chi(2), \quad \phi_u \sim U(0,2\pi).</math>

{{math|''χ''{{sub|''u''}}}} are [[iid|independently and identically]] ({{abbr|iid}}) [[chi distribution|chi-distributed variables]], and {{math|''ϕ''{{sub|''u''}}}} are uniform random.

In a two-dimensional pink noise signal, the amplitude at any orientation falls off inversely with frequency. A pink noise square of length {{mvar|N}} can be written as:
<math display="block">h(x,y)= \frac{\sigma N}{\sqrt{2}} \sum_{u,v}  \frac{\chi_{uv}}{\sqrt{u^2+v^2}} \sin \left(\frac{2 \pi}{N}(ux+vy) +\phi_{uv} \right), \quad \chi_{uv} \sim \chi(2), \quad \phi_{uv} \sim U(0,2\pi).</math>

General {{math|1/{{itco|''f''}}{{sup|''α''}}}}-like noises occur widely in nature and are a source of considerable interest in many fields. Noises with {{mvar|α}} near {{math|1}} generally come from [[condensed matter physics|condensed-matter]] systems in [[quasi-equilibrium]], as discussed below.<ref name="Kogan-1996">
{{cite book
 | last = Kogan |first = Shulim
 | year = 1996
 | title = Electronic Noise and Fluctuations in Solids
 | publisher = [Cambridge University Press]
 | isbn = 978-0-521-46034-7
}}</ref> Noises with a broad range of {{mvar|α}} generally correspond to a wide range of [[non-equilibrium thermodynamics|non-equilibrium]] driven [[dynamical system]]s.

Pink noise sources include ''[[flicker noise]]'' in electronic devices. In their study of [[fractional Brownian motion]],<ref name="Mandelbrot1968">
{{cite journal
 | last1 = Mandelbrot | first1 = B. B. | author1-link = Benoit Mandelbrot
 | last2 = Van Ness |first2 = J. W.
 | year = 1968
 | title = Fractional Brownian motions, fractional noises and applications
 | journal = [[SIAM Review]]
 | volume = 10
 | issue = 4
 | pages = 422–437
 | doi = 10.1137/1010093 |bibcode = 1968SIAMR..10..422M
}}</ref> [[Benoit Mandelbrot|Mandelbrot]] and Van Ness proposed the name ''fractional noise'' (sometimes since called ''fractal noise'') to describe {{math|1/{{itco|''f''}}{{sup|''α''}}}} noises for which the exponent {{mvar|α}} is not an even integer,<ref>{{cite journal |last1=Mandelbrot |first1=Benoit B. |last2=Wallis |first2=James R. |date=1969 |title=Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix |journal=Water Resources Research |volume=5 |issue=1 |pages=260–267 |doi=10.1029/WR005i001p00260 |bibcode=1969WRR.....5..260M }}</ref> or that are [[fractional derivative]]s of [[Brownian noise|Brownian]] ({{math|1/{{itco|''f''}}{{sup|2}}}}) noise.

== Description ==
[[File:Pink noise spectrum.svg|thumb|left|Spectrum of a pink noise approximation on a log-log plot; power density falls off at 10&nbsp;dB/decade of frequency]]
[[File:Noise.jpg|thumb|left|Relative intensity of pink noise (left) and [[white noise]] (right) on an [[fast Fourier transform|FFT]] [[spectrogram]] with the vertical axis being linear frequency]]

In pink noise, there is equal energy per [[octave]] of frequency. The energy of pink noise at each frequency level, however, falls off at roughly 3&nbsp;[[decibels|dB]] per octave. This is in contrast to [[white noise]] which has equal energy at all frequency levels.<ref>{{cite web |title=Noise |url=https://www.sfu.ca/~gotfrit/ZAP_Sept.3_99/n/noise.html |access-date=2024-02-06 |website=www.sfu.ca }}</ref>

The [[human auditory system]], which processes frequencies in a roughly logarithmic fashion approximated by the [[Bark scale]], does not perceive different frequencies with equal sensitivity; signals around 1–4&nbsp;kHz sound [[loudness|loudest]] for a given intensity. However, humans still differentiate between white noise and pink noise with ease.

[[Graphic equalizer]]s also divide signals into bands logarithmically and report power by octaves; audio engineers put pink noise through a system to test whether it has a flat frequency response in the spectrum of interest. Systems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer. Because pink noise tends to occur in natural physical systems, it is often useful in audio production. Pink noise can be processed, filtered, and/or effects can be added to produce desired sounds. Pink-noise generators are commercially available.

One parameter of noise, the peak versus average energy contents, or [[crest factor]], is important for testing purposes, such as for [[audio power amplifier]] and [[loudspeaker]] capabilities because the signal power is a direct function of the crest factor. Various crest factors of pink noise can be used in simulations of various levels of [[dynamic range compression]] in music signals. On some digital pink-noise generators the crest factor can be specified.

== Generation ==
[[File:White to pink filter.png|thumb|500px|The spatial filter which is convolved with a one-dimensional white noise signal to create a pink noise signal]]
Pink noise can be computer-generated by first generating a white noise signal, Fourier-transforming it, then dividing the amplitudes of the different frequency components by the square root of the frequency (in one dimension), or by the frequency (in two dimensions) etc. This is equivalent to spatially filtering (convolving) the white noise signal with a white-to-pink-filter. For a length {{math|''N''}} signal in one dimension, the filter has the following form:
<math display="block">a(x)=\frac{1}{N} \left[ 1+ \frac{1}{\sqrt{N/2}} \cos \pi (x-1) + 2 \sum_{k=1}^{N/2-1} \frac{1}{\sqrt{k}} \cos {\frac{2\pi k}{N}(x-1)} \right].</math>

Matlab programs are available to generate pink and other power-law coloured noise in [http://www.maxlittle.net/software/ one] or  [https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise any number] of dimensions.

One efficient algorithm for generation is the '''Voss–McCartney algorithm''', an efficient method to generate discrete-time pink noise ({{math|1/{{itco|''f''}}}} noise). It sums multiple independent random sequences (white noise sources), each updated at different rates, to approximate the {{math|1/{{itco|''f''}}}} power spectral density. Lower-frequency components are updated less frequently than higher-frequency components.<ref name="VossClarke1975">Voss, R. F.; Clarke, J. (1975). "1/f Noise in Music and Speech". *Nature*. 258 (5533): 317–318. doi:10.1038/258317a0.</ref>

A simple pseudocode implementation is:
<syntaxhighlight lang="text">
n_streams = number of random streams
streams = [random() for _ in range(n_streams)]
output = []

for i in range(total_samples):
    for j in range(n_streams):
        if i % (2**j) == 0:
            streams[j] = random()
    output.append(sum(streams))
</syntaxhighlight>

Each stream is updated at intervals that are powers of two, ensuring that slower-changing streams contribute low-frequency content and faster-changing streams contribute high-frequency content.

== Properties ==
[[File:Pink noise 1d autocorrelation.png|thumb|upright=2|The autocorrelation (Pearson's correlation coefficient) of one-dimensional (top) and two-dimensional (bottom) pink noise signals, across distance d (in units of the longest wavelength comprising the signal); grey curves are the autocorrelations of a sample of pink noise signals (comprising discrete frequencies), and black is their average, red is the theoretically calculated autocorrelation when the signal comprises these same discrete frequencies, and blue assumes a continuum of frequencies]]

=== Power-law spectra of amplitude and power ===
In a pink noise signal in any number of dimensions, the total power {{math|''p''}} at each frequency, summed across all orientations, falls off inversely with frequency: {{math|''p''({{itco|''f''}}) ~ 1/{{itco|''f''}}}}. Each octave carries an equal amount of total noise power. The total power at each frequency is the power summed over a spherical shell in the Fourier domain, so it is the average power {{math|''p''{{sub|''θ''({{itco|''f''}})}}}} along any orientation {{math|''θ''}}, multiplied by the surface area of the shell: {{math|''p''({{itco|''f''}}) ~ {{itco|''f''}}{{sup|''d''−1}}''p''{{sub|''θ''({{itco|''f''}})}}}}. This tells us that the average power at any orientation falls off as {{math|''p''{{sub|''θ''({{itco|''f''}})}} ~ 1/{{itco|''f''}}}} in one dimension, as {{math|1/{{itco|''f''}}{{sup|2}}}} in two dimensions (e.g. pink noise images), and as {{math|1/{{itco|''f''}}{{sup|''d''}}}} in general {{math|''d''}} dimensions. The average amplitude at any orientation is the square root of the average power: {{math|1=''a''{{sub|''θ''}} = {{sqrt|''p''{{sub|''θ''}}}}}}, so it falls as {{math|1/{{sqrt|{{itco|''f''}}}}}} in 1D, {{math|1/{{itco|''f''}}}} in 2D, and as {{math|{{itco|''f''}}{{sup|−''d''/2}}}} in general {{math|''d''}} dimensions.

The following table lists these power-law frequency-dependencies for pink noise signal in different dimensions {{math|''d''}}, and also for general power-law colored noise with power {{math|''α''}} (pink noise has {{math|1=''α'' = −1}} and [[Brown noise]] has {{math|1=''α'' = −2}}):

{| class="wikitable"
|+ Power‑law spectra of pink noise
|-
! dimensions
! tot. power <math>p(f)</math>
! avg. power <math>p_\theta(f) \sim f^{1-d}p(f)</math>
! avg. amp. <math>a_\theta(f)=\sqrt{p_\theta(f)}</math>
|-
| 1
| <math>f^{-1}</math>
| <math>f^{-1}</math>
| <math>f^{-\frac{1}{2}}</math>
|-
| 2
| <math>f^{-1}</math>
| <math>f^{-2}</math>
| <math>f^{-1}</math>
|-
| 3
| <math>f^{-1}</math>
| <math>f^{-3}</math>
| <math>f^{-\frac{3}{2}}</math>
|-
| <math>d</math>
| <math>f^{-1}</math>
| <math>f^{-d}</math>
| <math>f^{-\frac{d}{2}}</math>
|-
| <math>d</math>, power <math>\alpha</math>
| <math>f^\alpha</math>
| <math>f^{\alpha+1-d}</math>
| <math>f^{\frac{\alpha+1-d}{2}}</math>
|}

=== Distribution of point values ===
Consider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean {{math|''μ''}} and standard deviation {{math|''σ''}}, then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter <math>\boldsymbol{a}</math>). Then the point values of the pink noise signal will also be normally distributed, with mean {{math|''μ''}} and standard deviation <math>\lVert \boldsymbol{a} \rVert \sigma</math>.

=== Autocorrelation ===
Unlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows.

==== 1D signal ====
The Pearson's correlation coefficient of a one-dimensional pink noise signal (comprising discrete frequencies {{math|''k''}}) with itself across a distance {{math|''d''}} in the configuration (space or time) domain is:
<math display="block">r(d)=\frac{\sum_k \frac{\cos \frac{2 \pi k d}{N} }{k}}{\sum_k \frac{1}{k}}.</math>
If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from {{math|''k''{{sub|min}}}} to {{math|''k''{{sub|max}}}}, the autocorrelation coefficient is:
<math display="block">r(d)=\frac{\textrm{Ci}(\frac{2 \pi k_\textrm{max}d}{N} )-\textrm{Ci}(\frac{2 \pi k_\textrm{min}d}{N} )}{\log \frac{k_\textrm{max}}{k_\textrm{min}}},</math>
where {{math|Ci(''x'')}} is the [[Trigonometric integral#Cosine integral|cosine integral function]].

==== 2D signal ====
The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:
<math display="block">r(d)=\frac{\sum_k \frac{J_0 (\frac{2 \pi k d}{N})}{k}}{\sum_k \frac{1}{k}},</math>
where {{math|''J''{{sub|0}}}} is the [[Bessel function#Bessel functions of the first kind|Bessel function of the first kind]].

== Occurrence ==
Pink noise has been discovered in the [[statistical fluctuations]] of an extraordinarily diverse number of physical and biological systems (Press (1978);<ref name="Press-1978">
{{cite journal
 | last1 = Press |first1 = W. H.
 | year = 1978
 | title = Flicker noises in astronomy and elsewhere
 | journal = Comments in Astrophysics
 | volume = 7
 | pages = 103–119
 | issue = 4
 | bibcode = 1978ComAp...7..103P
}}</ref> see articles in Handel & Chung (1993)<ref name="Handel-1993">
{{cite book
 | last1 = Handel | first1 = P. H. | last2 = Chung | first2 = A. L.
 | year = 1993
 | title = Noise in Physical Systems and 1/"f" Fluctuations
 | publisher = American Institute of Physics 
 | location = New York 
}}</ref> and references therein). Examples of its occurrence include fluctuations in [[tide]] and river heights, [[quasar]] light emissions, heart beat, firings of single [[neuron]]s, [[resistivity]] in [[solid-state electronics]] and single-molecule conductance signals<ref>{{cite journal |last1=Adak |first1=Olgun |last2=Rosenthal |first2=Ethan |last3=Meisner |first3=Jeffery |last4=Andrade |first4=Erick F. |last5=Pasupathy |first5=Abhay N. |last6=Nuckolls |first6=Colin |last7=Hybertsen |first7=Mark S. |last8=Venkataraman |first8=Latha |date=2015-05-07 |title=Flicker Noise as a Probe of Electronic Interaction at Metal–Single Molecule Interfaces |url=https://pubs.acs.org/doi/pdf/10.1021/acs.nanolett.5b01270?rand=4a25iu4o |journal=Nano Letters |volume=15 |issue=6 |pages=4143–4149 |doi=10.1021/acs.nanolett.5b01270 |pmid=25942441 |bibcode=2015NanoL..15.4143A |issn=1530-6984 |url-access=subscription }}</ref> resulting in [[flicker noise]]. Pink noise describes the [[Scene statistics|statistical structure of many natural images]].<ref name="Field-1987">
{{cite journal
 | last = Field | first = D. J.
 | title = Relations between the statistics of natural images and the response properties of cortical cells
 | journal = J. Opt. Soc. Am. A
 | volume = 4
 | issue = 12
 | pages = 2379–2394
 | year = 1987
 | doi = 10.1364/JOSAA.4.002379
 | pmid = 3430225
 | bibcode = 1987JOSAA...4.2379F
 | url = https://redwood.psych.cornell.edu/papers/field_87.pdf
 | citeseerx = 10.1.1.136.1345
}}</ref> 

General {{math|1/{{itco|''f''}}<sup>''α''</sup>}} noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.<ref name="Bak-1987">
{{cite journal
 | last1 = Bak | first1 = P. | last2 = Tang | first2 = C. | last3 = Wiesenfeld | first3 = K.
 | year= 1987
 | title = Self-Organized Criticality: An Explanation of 1/''f'' Noise
 | journal = [[Physical Review Letters]]
 | volume = 59
 | pages = 381–384
 | doi = 10.1103/PhysRevLett.59.381
 | pmid= 10035754
 | issue = 4
 | bibcode=1987PhRvL..59..381B
 | s2cid=7674321
}}</ref> In physical systems, they are present in some [[meteorological]] data series, the [[electromagnetic radiation]] output of some astronomical bodies. In biological systems, they are present in, for example, [[cardiac cycle|heart beat]] rhythms, neural activity, and the statistics of [[DNA sequence]]s, as a generalized pattern.<ref>Josephson, Brian D. (1995). "A trans-human source of music?" in (P. Pylkkänen and P. Pylkkö, eds.) ''New Directions in Cognitive Science'', Finnish Artificial Intelligence Society, Helsinki; pp. 280–285.</ref>

An accessible introduction to the significance of pink noise is one given by [[Martin Gardner]] (1978) in his ''Scientific American'' column "Mathematical Games".<ref name="Gardner-1978">
{{cite journal
 | last1 = Gardner | first1 = M.
 | year = 1978
 | title = Mathematical Games—White and brown music, fractal curves and one-over-f fluctuations
 | journal = Scientific American
 | volume = 238
 | issue = 4
 | pages = 16–32
 | doi = 10.1038/scientificamerican0478-16
}}</ref> In this column, Gardner asked for the sense in which music imitates nature. Sounds in nature are not musical in that they tend to be either too repetitive (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth). The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed that pitch and loudness fluctuations in speech and music are pink noises.<ref name="Voss-1975">
{{cite journal
 | last1 = Voss | first1 = R. F. | last2 = Clarke | first2 = J.
 | year = 1975
 | title = '1/f Noise' in Music and Speech
 | journal = Nature
 | volume = 258
 | issue = 5533
 | pages = 317–318
 | doi=10.1038/258317a0
 | bibcode = 1975Natur.258..317V |s2cid=4182664
}}</ref><ref name="Voss-1978">
{{cite journal
 | last1 = Voss | first1 = R. F. | last2=Clarke | first2 = J.
 | year = 1978
 | title = 1/f noise" in music: Music from 1/f noise
 | journal = Journal of the Acoustical Society of America
 | volume = 63 | issue = 1
 | pages = 258–263
 | doi=10.1121/1.381721
 | bibcode = 1978ASAJ...63..258V
}}</ref> So music is like tides not in terms of how tides sound, but in how tide heights vary.

=== Precision timekeeping ===
{{main page|Allan variance}}
The ubiquitous {{math|1/{{itco|''f''}}}} noise poses a "noise floor" to precision timekeeping.<ref name="Press-1978" /> The derivation is based on.<ref>{{cite book |last=Voss |first=R.F. |title=33rd Annual Symposium on Frequency Control |chapter=1/F (Flicker) Noise: A Brief Review |date=May 1979 |pages=40–46 |doi=10.1109/FREQ.1979.200297|s2cid=37302662 }}</ref>
[[File:AllanDeviation.svg|right|thumb|300x300px|A clock is most easily tested by comparing it with a ''far more accurate'' reference clock. During an interval of time {{math|''τ''}}, as measured by the reference clock, the clock under test advances by {{math|''τy''}}, where {{math|''y''}} is the average (relative) clock frequency over that interval.]]
Suppose that we have a timekeeping device (it could be anything from [[Crystal oscillator|quartz oscillators]], [[Atomic clock|atomic clocks]], and [[Hourglass|hourglasses]]<ref>{{cite journal |last1=Schick |first1=K. L. |last2=Verveen |first2=A. A. |date=October 1974 |title=1/f noise with a low frequency white noise limit |url=https://www.nature.com/articles/251599a0 |journal=Nature |language=en |volume=251 |issue=5476 |pages=599–601 |doi=10.1038/251599a0 |bibcode=1974Natur.251..599S |s2cid=4200003 |issn=1476-4687|url-access=subscription }}</ref>). Let its readout be a real number {{math|''x''(''t'')}} that changes with the actual time {{math|''t''}}. For concreteness, let us consider a quartz oscillator. In a quartz oscillator, {{math|''x''(''t'')}} is the number of oscillations, and <math>\dot x(t)</math> is the rate of oscillation. The rate of oscillation has a constant component <math>\dot x_0</math>and a fluctuating component <math>\dot x_f</math>, so <math display="inline">\dot x(t) = \dot x_0 + \dot x_f(t)</math>. By selecting the right units for {{math|''x''}}, we can have <math>\dot x_0 = 1</math>, meaning that on average, one second of clock-time passes for every second of real-time.

The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval {{math|[''kτ'', (''k'' + 1)''τ'']}} as <math display="block">y_k = \frac{1}{\tau}\int_{k\tau}^{(k+1)\tau}\dot x(t)dt = \frac{x( (k+1 ) \tau) - x(k\tau)}{\tau}</math> Note that <math>y_k</math> is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock.{{NoteTag|Though in practice, since there are no ideal clocks, {{math|''t''}} is actually the ticks of a much more accurate clock.}}

The [[Allan variance]] of the clock frequency is half the mean square of change in average clock frequency:<math display="block">\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2</math>where <math>K</math> is an integer large enough for the averaging to converge to a definite value.
For example, a 2013 atomic clock<ref>{{cite journal |last1=Hinkley |first1=N. |last2=Sherman |first2=J. A. |last3=Phillips |first3=N. B. |last4=Schioppo |first4=M. |last5=Lemke |first5=N. D. |last6=Beloy |first6=K. |last7=Pizzocaro |first7=M. |last8=Oates |first8=C. W. |last9=Ludlow |first9=A. D. |date=2013-09-13 |title=An Atomic Clock with 10<sup>–18</sup> Instability |url=https://www.science.org/doi/10.1126/science.1240420 |journal=Science |language=en |volume=341 |issue=6151 |pages=1215–1218 |doi=10.1126/science.1240420 |pmid=23970562 |arxiv=1305.5869 |bibcode=2013Sci...341.1215H |s2cid=206549862 |issn=0036-8075 }}</ref> achieved <math>\sigma(25000\text{ seconds}) = 1.6 \times 10^{-18}</math>, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 [[Femtosecond|femtoseconds]].

Now we have <math display="block">y_{k} - y_{k-1} = \int_\R g(k\tau - t) \dot x_f(t) dt = (g\ast \dot x_f)(k\tau) </math>where <math>g(t) = \frac{-1_{[0, \tau]}(t) + 1_{[-\tau, 0]}(t)}{\tau}</math> is one packet of a [[Square wave (waveform)|square wave]] with height {{math|1/''τ''}} and wavelength {{math|2''τ''}}. Let {{math|''h''(''t'')}} be a packet of a square wave with height {{math|1}} and wavelength {{math|2}}, then {{math|1=''g''(''t'') = ''h''(''t''/''τ'')/''τ''}}, and its Fourier transform satisfies <math>\mathcal F[g](\omega) = \mathcal F[h](\tau\omega)</math>.

The Allan variance is then <math>\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac 12 \overline{(g\ast \dot x_f)(k\tau)^2}
</math>, and the discrete averaging can be approximated by a continuous averaging: <math display="block"> \frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2 \approx \frac{1}{K\tau}\int_0^{K\tau} \frac 12(g\ast \dot x_f)(t)^2 dt ,</math> which is the total power of the signal <math>(g\ast \dot x_f)</math>, or the integral of its [[Spectral density|power spectrum]]:
[[File:Illustration for Allan variance of 1-f noise.png|thumb|315x315px|<math>\sigma^2(1)</math> is approximately the area under the green curve; when <math>\tau</math> increases, <math>S[g](\omega) </math> shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; when <math>S[\dot x_f](\omega) \propto \omega^{-\alpha}</math>, the combined effect of both is that <math>\sigma^2(\tau) \propto \tau^{\alpha-1}</math>]]
<math display="block">\sigma^2(\tau) \approx \int_0^\infty S[g\ast \dot x_f](\omega) d\omega = \int_0^\infty S[g](\omega) \cdot S[\dot x_f](\omega)  d\omega = \int_0^\infty S[h](\tau \omega) \cdot S[\dot x_f](\omega) d\omega</math>In words, the Allan variance is approximately the power of the fluctuation after [[Band-pass filter|bandpass filtering]] at {{math|''ω'' ~ 1/''τ''}} with bandwidth {{math|Δ''ω'' ~ 1/''τ''}}.

For {{math|1/{{itco|''f''}}{{sup|''α''}}}} fluctuation, we have <math>S[\dot x_f](\omega) = C/\omega^\alpha</math> for some constant <math>C</math>, so <math>\sigma^2(\tau) \approx \tau^{\alpha-1} \sigma^2(1) \propto \tau^{\alpha-1}</math>. In particular, when the fluctuating component <math>\dot x_f</math> is a {{math|1/{{itco|''f''}}}} noise, then {{math|{{itco|''σ''}}{{sup|2}}(''τ'')}} is independent of the averaging time {{math|''τ''}}, meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case <math>\sigma^2(\tau) \propto \tau^{-1}</math>, meaning that doubling the averaging time would improve the stability of frequency by {{math|{{sqrt|2}}}}.<ref name="Press-1978" />

The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.<ref>{{citation |last=Vessot |first=Robert F. C. |title=5.4. Frequency and Time Standards††This work was supported in part by contract NSR 09-015-098 from the National Aeronautics and Space Administration. |date=1976-01-01 |url=https://www.sciencedirect.com/science/article/pii/S0076695X08607103 |work=Methods in Experimental Physics |volume=12 |pages=198–227 |editor-last=Meeks |editor-first=M. L. |access-date=2023-07-17 |series=Astrophysics |publisher=Academic Press |doi=10.1016/S0076-695X(08)60710-3 |language=en| url-access=subscription }}</ref>

=== Humans ===
In [[brains]], pink noise has been widely observed across many temporal and physical scales from [[ion channel]] gating to [[EEG]] and [[Magnetoencephalography|MEG]] and [[Local field potential|LFP]] recordings in humans.<ref>{{citation |last1=Destexhe |first1=Alain |title=Local Field Potentials: LFP |date=2020 |encyclopedia=Encyclopedia of Computational Neuroscience |pages=1–12 |editor-last=Jaeger |editor-first=Dieter |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-1-4614-7320-6_548-2 |isbn=978-1-4614-7320-6 |last2=Bédard |first2=Claude |s2cid=243735998 |editor2-last=Jung |editor2-first=Ranu}}</ref> In clinical EEG, deviations from this {{math|1/{{itco|''f''}}}} pink noise can be used to identify [[epilepsy]], even in the absence of a [[seizure]], or during the interictal state.<ref name="Kerr-2012">
{{cite journal
 | last1 = Kerr | first1 = W.T. 
 | display-authors=etal
 | title = Automated diagnosis of epilepsy using EEG power spectrum
 | journal = Epilepsia
 | year = 2012
 | volume = 53
 | issue = 11
 | pages = e189–e192
 | doi=10.1111/j.1528-1167.2012.03653.x |pmc=3447367
 | pmid = 22967005
}}</ref> Classic models of EEG generators suggested that dendritic inputs in [[gray matter]] were principally responsible for generating the {{math|1/{{itco|''f''}}}} power spectrum observed in EEG/MEG signals.  However, recent computational models using [[cable theory]] have shown that [[action potential]] transduction along [[white matter]] tracts in the brain also generates a {{math|1/{{itco|''f''}}}} spectral density.  Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings,<ref name="Douglas-2019">
{{cite book
 |last1=Douglas |first1=PK
 |title=2019 7th International Winter Conference on Brain-Computer Interface (BCI)
 |chapter=Reconsidering Spatial Priors in EEG Source Estimation : Does White Matter Contribute to EEG Rhythms?
 |display-authors=etal
 |publisher = IEEE 
 |year=2019
 |pages=1–12
 |doi=10.1109/IWW-BCI.2019.8737307 |arxiv=2111.08939
 |isbn=978-1-5386-8116-9
 |s2cid=195064621
}}</ref>
particularly if the effects of ephaptic coupling are taken into consideration.<ref name="Douglas-2024">
{{cite book
 |last1= Douglas |first1=PK |last2=Blair |first2=G.
 |title=2024 12th International Winter Conference on Brain-Computer Interface (BCI)
 |chapter=Towards a white matter ephaptic coupling model of 1/f spectra
 |display-authors=etal
 |publisher = IEEE 
 |year = 2024
 |pages=1–3
 |doi =10.1109/BCI60775.2024.10480498
 |isbn= 979-8-3503-0943-0
}}</ref> 

It has also been successfully applied to the modeling of [[mental representation|mental states]] in [[psychology]],<ref name="cognitive_2003">
{{cite journal
 | volume = 132
 | issue = 3
 | pages = 331–350
 | author = Van Orden, G.C. |author2=Holden, J.G. |author3=Turvey, M.T.
 | title = Self-organization of cognitive performance
 | journal = Journal of Experimental Psychology: General
 | year = 2003
 | doi = 10.1037/0096-3445.132.3.331
 | pmid = 13678372
}}</ref> and used to explain stylistic variations in music from different cultures and historic periods.<ref>Pareyon, G. (2011). ''On Musical Self-Similarity'', International Semiotics Institute & University of Helsinki. {{cite web |title=On Musical Self-Similarity |url=https://helda.helsinki.fi/bitstream/handle/10138/29824/Pareyon_Dissertation.pdf?sequence=2 }}</ref> Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of [[pitch (music)|pitches]], will tend towards a pink noise spectrum.<ref name="Kuittinen">{{cite web |url=http://mlab.uiah.fi/~eye/mediaculture/noise.html |title=Noise in Man-generated Images and Sound }}</ref> Similarly, a generally pink distribution pattern has been observed in [[Shot (filmmaking)|film shot]] length by researcher [[James E. Cutting]] of [[Cornell University]], in the study of 150 popular movies released from 1935 to 2005.<ref>Anger, Natalie (March 1, 2010). [https://www.nytimes.com/2010/03/02/science/02angi.html "Bringing New Understanding to the Director's Cut"]. ''The New York Times''. Retrieved on March 3, 2010. See also [http://pss.sagepub.com/content/early/2010/02/04/0956797610361679.full.pdf+html original study] {{webarchive |url=https://web.archive.org/web/20130124170244/http://pss.sagepub.com/content/early/2010/02/04/0956797610361679.full.pdf+html |date=2013-01-24 }}</ref>

Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.<ref name="Gilden-1995">
{{cite journal
 | author1 = Gilden, David L |author2=Thornton, T |author3=Mallon, MW
 | year = 1995
 | title = 1/''f'' Noise in Human Cognition
 | journal = Science
 | volume = 267
 | pages = 1837–1839
 | doi = 10.1126/science.7892611
 | pmid = 7892611
 | issn = 0036-8075
 | issue = 5205
 |bibcode = 1995Sci...267.1837G
}}</ref> Later, Gilden (1997) and Gilden (2001) found that time series formed from [[reaction time]] measurement and from iterated two-alternative forced choice also produced pink noises.<ref name="Gilden-1997">
{{cite journal
 | author = Gilden, D. L. 
 | year = 1997
 | title = Fluctuations in the time required for elementary decisions
 | journal = Psychological Science
 | volume = 8
 | pages = 296–301
 | doi = 10.1111/j.1467-9280.1997.tb00441.x
 | issue = 4
 | s2cid = 145051976
}}</ref><ref name="Gilden-2001">
{{cite journal
 | author = Gilden, David L
 | year = 2001
 | title = Cognitive Emissions of 1/''ƒ'' Noise
 | journal = [[Psychological Review]]
 | volume = 108
 | pages = 33–56
 | doi = 10.1037/0033-295X.108.1.33
 | issn = 0033-295X
 | issue = 1
 | pmid = 11212631
 | citeseerx = 10.1.1.136.1992
}}</ref>

=== Electronic devices ===
{{main|Flicker noise}}

The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.<ref name="Kogan-1996" /><ref name="Weissman-1988">
{{cite journal
 | author = Weissman, M. B.
 |year= 1988
 | title = 1/''f'' Noise and other slow non-exponential kinetics in condensed matter
 | journal = [[Reviews of Modern Physics]]
 | volume = 60
 | pages = 537–571
 | doi = 10.1103/RevModPhys.60.537
 | issue = 2
 | bibcode=1988RvMP...60..537W
}}</ref> Approximately pink spectral noise may arise from a distribution of kinetic activation energies of the fluctuating processes.<ref name="Dutta-1981">
{{cite journal
 |author1=Dutta, P. |author2=Horn, P. M.
 |name-list-style=amp |year= 1981
 | title = Low-frequency fluctuations in solids: 1/''f'' noise
 | journal = [[Reviews of Modern Physics]]
 | volume = 53
 | pages = 497–516
 | doi = 10.1103/RevModPhys.53.497
 | issue = 3
 | bibcode=1981RvMP...53..497D
}}</ref> Since the frequency range of the typical noise experiment (e.g., 1&nbsp;Hz – 1&nbsp;kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10<sup>14</sup>&nbsp;Hz), the exponential factors in the [[Arrhenius equation]] for the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because <math>\textstyle \frac{d}{df}\ln f = \frac{1}{f}.</math>

There is no known lower bound to background pink noise in electronics. Measurements made down to 10<sup>−6</sup>&nbsp;Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.<ref name="Kleinpenning-1988">
{{cite journal
 |author1=Kleinpenning, T. G. M. |author2=de Kuijper, A. H.
 |name-list-style=amp |year= 1988
 | title = Relation between variance and sample duration of 1/f Noise signals
 | journal = [[Journal of Applied Physics]]
 | volume = 63
 |issue=1
 | page = 43
 | doi = 10.1063/1.340460
 |bibcode = 1988JAP....63...43K
}}</ref> (Kleinpenning, de Kuijper, 1988)<ref>{{cite journal |last1=Kleinpenning |first1=T. G. M. |last2=de Kuijper |first2=A. H. |date=1988-01-01 |title=Relation between variance and sample duration of 1/ f noise signals |url=https://pubs.aip.org/jap/article/63/1/43/174101/Relation-between-variance-and-sample-duration-of-1 |journal=Journal of Applied Physics |language=en |volume=63 |issue=1 |pages=43–45 |doi=10.1063/1.340460 |bibcode=1988JAP....63...43K |issn=0021-8979 |url-access=subscription }}</ref> measured the resistance in a noisy carbon-sheet resistor, and found {{math|1/{{itco|''f''}}}} noise behavior over the range of {{val|e=-5.5|u=Hz}} to {{val|e=4|u=Hz}}, a range of 9.5 decades.

A pioneering researcher in this field was [[Aldert van der Ziel]].<ref>Aldert van der Ziel, (1954), ''Noise'', Prentice–Hall</ref>

Flicker noise is commonly used for the reliability characterization of electronic devices.<ref>{{cite journal |title=Low-frequency noise study in electron devices: review and update|journal= Microelectronics Reliability|author=Hei Wong|date=2003 |volume=43 |issue=4 |pages=585–599 |doi=10.1016/S0026-2714(02)00347-5 |bibcode=2003MiRe...43..585W }}</ref> It is also used for gas detection in chemoresistive sensors<ref>{{cite journal |title=Low-frequency 1/''f'' noise in graphene devices |journal= Nature Nanotechnology |author=Alexander A. Balandin |date=2013 |volume=8 |issue=8 |pages=549–555 |doi=10.1038/nnano.2013.144|pmid=23912107 |arxiv=1307.4797 |bibcode=2013NatNa...8..549B |s2cid=16030927 }}</ref> by dedicated measurement setups.<ref>{{cite journal |title=Flicker Noise in Resistive Gas Sensors—Measurement Setups and Applications for Enhanced Gas Sensing |journal=Sensors |date=2024 |volume=24 |issue=2 |page=405 |doi=10.3390/s24020405 |doi-access=free |last1=Smulko |first1=Janusz |last2=Scandurra |first2=Graziella |last3=Drozdowska |first3=Katarzyna |last4=Kwiatkowski |first4=Andrzej |last5=Ciofi |first5=Carmine |last6=Wen |first6=He |pmid=38257498 |pmc=10821460 |bibcode=2024Senso..24..405S }}</ref>

=== In gravitational wave astronomy ===

[[Image:Gravitational-wave detector sensitivities and astrophysical gravitational-wave sources.png|thumb|upright=2|Noise curves for a selection of [[gravitational-wave detector]]s as a function of frequency]]
{{math|1/{{itco|''f''}}{{sup|''α''}}}} noises with {{math|''α''}} near {{math|1}} are a factor in [[gravitational-wave astronomy]]. The noise curve at very low frequencies affects [[pulsar timing array]]s, the [[European Pulsar Timing Array]] (EPTA) and the future [[International Pulsar Timing Array]] (IPTA); at low frequencies are space-borne detectors, the formerly proposed [[Laser Interferometer Space Antenna]] (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial [[LIGO|Laser Interferometer Gravitational-Wave Observatory]] (LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve.<ref>{{cite web |title=Gravitational Wave Detectors and Sources |url=http://rhcole.com/apps/GWplotter/ |access-date=17 April 2014 |author=Moore, Christopher |author2=Cole, Robert |author3=Berry, Christopher |date=19 July 2013 }}</ref>

=== Climate dynamics ===
Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the [[climate system]].<ref>{{cite web|url=https://news.yale.edu/2018/09/04/think-pink-better-view-climate-change|title=Think pink for a better view of climate change|work=YaleNews|access-date=5 September 2018|author=Jim Shelton|date=2018-09-04}}</ref><ref>{{Cite journal |last1=Moon |first1=Woosok |last2=Agarwal |first2=Sahil |last3=Wettlaufer |first3=J. S. |date=2018-09-04 |title=Intrinsic Pink-Noise Multidecadal Global Climate Dynamics Mode |url=https://link.aps.org/doi/10.1103/PhysRevLett.121.108701 |journal=Physical Review Letters |volume=121 |issue=10 |article-number=108701 |doi=10.1103/PhysRevLett.121.108701|pmid=30240245 |arxiv=1802.00392 |bibcode=2018PhRvL.121j8701M |s2cid=52243763 }}</ref>

=== Diffusion processes ===
Many time-dependent stochastic processes are known to exhibit {{math|1/{{itco|''f''}}<sup>''α''</sup>}} noises with {{math|''α''}} between {{math|0}} and {{math|2}}. In particular [[Brownian motion]] has a [[Spectral density|power spectral density]] that equals {{math|4''D''/{{itco|''f''}}<sup>2</sup>}},<ref>{{cite book |title=Fundamentals of noise and vibration analysis for engineers |last=Norton, M. P. |date=2003 |publisher=Cambridge University Press |others=Karczub, D. G. (Denis G.) |isbn=978-0-511-67498-3 |edition=2nd |location=Cambridge, UK |oclc=667085096 }}</ref> where {{math|''D''}} is the [[Mass diffusivity|diffusion coefficient]]. This type of spectrum is sometimes referred to as [[Brownian noise]]. The analysis of individual Brownian motion trajectories also show {{math|1/{{itco|''f''}}<sup>2</sup>}} spectrum, albeit with random amplitudes.<ref>{{cite journal |last1=Krapf |first1=Diego |last2=Marinari |first2=Enzo |last3=Metzler |first3=Ralf |last4=Oshanin |first4=Gleb |last5=Xu |first5=Xinran |last6=Squarcini |first6=Alessio |date=2018-02-09 |title=Power spectral density of a single Brownian trajectory: what one can and cannot learn from it |journal=New Journal of Physics |volume=20 |issue=2 |page=023029 |doi=10.1088/1367-2630/aaa67c |arxiv=1801.02986 |bibcode=2018NJPh...20b3029K |issn=1367-2630 |doi-access=free }}</ref> [[Fractional Brownian motion]] with [[Hurst exponent]] {{math|''H''}} also show {{math|1/{{itco|''f''}}<sup>''α''</sup>}} power spectral density with {{math|1=''α'' = 2''H'' + 1}} for subdiffusive processes ({{math|''H'' < 0.5}}) and {{math|1=''α'' = 2}} for superdiffusive processes ({{math|0.5 < ''H'' < 1}}).<ref>{{cite journal |last1=Krapf |first1=Diego |last2=Lukat |first2=Nils |last3=Marinari |first3=Enzo |last4=Metzler |first4=Ralf |last5=Oshanin |first5=Gleb |last6=Selhuber-Unkel |first6=Christine |last7=Squarcini |first7=Alessio |last8=Stadler |first8=Lorenz |last9=Weiss |first9=Matthias |last10=Xu |first10=Xinran |date=2019-01-31 |title=Spectral Content of a Single Non-Brownian Trajectory |journal=Physical Review X |language=en |volume=9 |issue=1 |article-number=011019 |doi=10.1103/PhysRevX.9.011019 |arxiv=1902.00481 |bibcode=2019PhRvX...9a1019K |issn=2160-3308 |doi-access=free }}</ref>

== Origin ==
{{see also|Supersymmetry#Supersymmetry in dynamical systems}}
There are many theories about the origin of pink noise. Some theories attempt to be universal, while others apply to only a certain type of material, such as [[semiconductor]]s. Universal theories of pink noise remain a matter of current research interest.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the [[central limit theorem]] of statistics.<ref name="Kendal2011b">{{cite journal |vauthors=Kendal WS, Jørgensen BR | year = 2011 | title = Tweedie convergence: a mathematical basis for Taylor's power law, 1/''f'' noise and multifractality. | journal = Phys. Rev. E | volume = 84 | issue = 6 |article-number = 066120 | doi = 10.1103/physreve.84.066120 | pmid = 22304168 | bibcode = 2011PhRvE..84f6120K | url = https://findresearcher.sdu.dk:8443/ws/files/55639035/e066120.pdf }}</ref> The [[Tweedie convergence theorem]]<ref>{{cite journal |author1=Jørgensen B |author2=Martinez JR |author3=Tsao M |name-list-style=vanc |title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics |year=1994 |volume=21 |pages=223–243}}</ref> describes the convergence of certain statistical processes towards a family of statistical models known as the [[Tweedie distribution]]s. These distributions are characterized by a variance to mean [[power law]], that have been variously identified in the ecological literature as [[Taylor's law]]<ref name="Taylor1961">{{cite journal |vauthors=Taylor LR | year = 1961 | title = Aggregation, variance and the mean | journal = Nature | volume = 189 | issue = 4766| pages = 732–735 | doi = 10.1038/189732a0 | bibcode = 1961Natur.189..732T | s2cid = 4263093 }}</ref> and in the physics literature as ''fluctuation scaling''.<ref name="Eisler2008">{{cite journal |vauthors=Eisler Z, Bartos I, Kertesz J | year = 2008 | title = Fluctuation scaling in complex systems: Taylor's law and beyond | journal = Advances in Physics | volume = 57 | issue = 1| pages = 89–142 | doi = 10.1080/00018730801893043 | bibcode = 2008AdPhy..57...89E | arxiv = 0708.2053 | s2cid = 119608542 }}</ref> When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa.<ref name="Kendal2011b" /> Both of these effects can be shown to be the consequence of [[convergence in distribution|mathematical convergence]] such as how certain kinds of data will converge towards the [[normal distribution]] under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain [[power law]] manifestations that have been attributed to [[self-organized criticality]].<ref name=Kendal2015>
{{cite journal
 | author = Kendal, WS
 | year = 2015
 | title = Self-organized criticality attributed to a central limit-like convergence effect
 | journal =Physica A 
 | volume = 421
 | pages =141–150
 | doi = 10.1016/j.physa.2014.11.035
 |bibcode = 2015PhyA..421..141K
}}</ref>

There are various mathematical models to create pink noise.  The superposition of exponentially decaying pulses is able to generate a signal with the {{math|1/{{itco|''f''}}}}-spectrum at moderate frequencies, transitioning to a constant at low frequencies and {{math|1/{{itco|''f''}}{{sup|2}}}} at high frequencies.<ref>{{cite arXiv |title=1/f noise: a pedagogical review |eprint=physics/0204033 |date=2002-04-12 |first=Edoardo |last=Milotti }}</ref>  In contrast, the sandpile model of self-organized criticality, which exhibits quasi-cycles of gradual stress accumulation between fast rare stress-releases, reproduces the flicker noise that corresponds to the intra-cycle dynamics.<ref>{{cite journal | first1 = Alexander | last1 = Shapoval | first2 = Mikhail | last2 = Shnirman | title = Explanation of flicker noise with the Bak-Tang-Wiesenfeld model of self-organized criticality | journal = Physical Review E | volume = 110 | article-number = 014106 | year = 2024 | issue = 1 | doi = 10.1103/PhysRevE.110.014106 | pmid = 39160903 | url = https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.014106 | arxiv = 2212.14726 | bibcode = 2024PhRvE.110a4106S }}</ref> The statistical signature of self-organization is justified in O'Brien & Weissman (1992)<ref>{{cite journal |title = Statistical signatures of self-organization |journal = Physical Review A|date = 1992-10-01 |pages = R4475–R4478 |volume = 46 |issue = 8 |doi = 10.1103/PhysRevA.46.R4475 |first1 = Kevin P. |last1 = O'Brien |first2 = M. B. |last2 = Weissman |pmid = 9908765 |bibcode = 1992PhRvA..46.4475O }}</ref>  It can be generated on computer, for example, by filtering white noise,<ref>{{cite web |title = Noise in Man-generated Images and Sound |url = http://mlab.uiah.fi/~eye/mediaculture/noise.html |website = mlab.uiah.fi |access-date = 2015-11-14 }}</ref><ref>{{cite web |title = DSP Generation of Pink Noise |url = http://www.firstpr.com.au/dsp/pink-noise/ |website = www.firstpr.com.au |access-date = 2015-11-14 }}</ref><ref>{{cite journal |url = http://linkage.rockefeller.edu/wli/moved.8.04/1fnoise/mcclain01.pdf |title = Numerical Simulation of Pink Noise |last = McClain |first = D. |date = May 1, 2001 |journal = Preprint |archive-url = https://web.archive.org/web/20111004100713/http://linkage.rockefeller.edu/wli/moved.8.04/1fnoise/mcclain01.pdf |archive-date = 2011-10-04 }}</ref> [[inverse Fourier transform]],<ref>{{cite journal |title = On Generating Power Law Noise |journal = Astronomy and Astrophysics |date = 1995-01-01 |pages = 707–710 |volume = 300 |first1 = J. |last1 = Timmer |first2 = M. |last2 = König |bibcode= 1995A&A...300..707T }}</ref> or by multirate variants on standard white noise generation.<ref name="Voss-1978" /><ref name="Gardner-1978" />

In [[Supersymmetric theory of stochastic dynamics|supersymmetric theory of stochastics]],<ref>{{cite journal |year=2016 |title=Introduction to supersymmetric theory of stochastics |journal=Entropy |volume=18 |issue=4 |page=108 |doi=10.3390/e18040108 |author=Ovchinnikov, I.V. |arxiv=1511.03393 |bibcode=2016Entrp..18..108O |s2cid=2388285 |doi-access=free }}</ref> an approximation-free theory of [[stochastic differential equation]]s, {{math|1/{{itco|''f''}}}} noise is one of the manifestations of the spontaneous breakdown of topological [[supersymmetry]]. This supersymmetry is an intrinsic property of all stochastic differential equations and its meaning is the preservation of the continuity of the [[phase space]] by continuous time dynamics. Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept of [[chaos theory|deterministic chaos]],<ref>{{cite journal | author1 = Ovchinnikov, I.V.| author2 = Schwartz, R. N. | author3 = Wang, K. L. | title = Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics | journal = Modern Physics Letters B | volume = 30 | issue = 8 | year = 2016 | page = 1650086 | doi = 10.1142/S021798491650086X | arxiv = 1404.4076 | bibcode = 2016MPLB...3050086O | s2cid = 118174242 }}</ref> whereas the associated emergence of the long-term dynamical memory or order, i.e., {{math|1/{{itco|''f''}}}} and [[Crackling noise|crackling]] noises, the [[Butterfly effect]] etc., is the consequence of the [[Goldstone boson|Goldstone theorem]] in the application to the spontaneously broken topological supersymmetry.

== Audio testing ==
Pink noise is commonly used to test the loudspeakers in [[sound reinforcement system]]s, with the resulting sound measured with a test [[microphone]] in the listening space connected to a [[spectrum analyzer]]<ref name=Sound/> or a computer running a real-time [[fast Fourier transform]] (FFT) analyzer program such as [[Smaart]]. The sound system plays pink noise while the audio engineer makes adjustments on an [[Equalization (audio)|audio equalizer]] to obtain the desired results. Pink noise is predictable and repeatable, but it is annoying for a concert audience to hear. Since the late 1990s, FFT-based analysis enabled the engineer to make adjustments using pre-recorded music as the test signal, or even the music coming from the performers in real time.<ref>{{cite book |last=Loar |first=Josh |date=2019 |title=The Sound System Design Primer |pages=274–276 |publisher=Routledge |isbn=978-1-351-76818-4}}</ref> Pink noise is still used by audio system contractors<ref>{{cite web |url=http://www.aedesign-inc.com/blog/2018/8/21/sound-system-commissioning-say-what |last=Eckstein |first=Matt |title=Sound System Commissioning - Say What? |website=AE Design |date=30 August 2018 |access-date=November 22, 2022}}</ref> and by computerized sound systems which incorporate an automatic equalization feature.<ref>{{cite web |url=https://uc.yamaha.com/insights/blog/2021/january/what-is-pink-noise-and-what-does-it-do/ |title=What is Pink Noise and What Does It Do? |last=Cox |first=Tyler |website=Yamaha Insights |publisher=[[Yamaha Pro Audio]] |access-date=November 22, 2022 }}</ref>

In manufacturing, pink noise is often used as a [[burn-in]] signal for [[audio amplifier]]s and other components, to determine whether the component will maintain performance integrity during sustained use.<ref>{{cite magazine |last=Lacanette |first=Kerry |date=1990 |title=Create an Accurate Noise Generator |page=108 |magazine=[[Electronic Design (magazine)|Electronic Design]] |volume=38 |publisher=Hayden }}</ref> The process of end-users burning in their [[headphones]] with pink noise to attain higher fidelity has been called an [[audiophile]] "myth".<ref>{{cite web |url=https://www.soundguys.com/headphone-burn-in-isnt-real-17463/ |title=Headphone burn-in isn't real |last=Thomas |first=Christian |date=April 30, 2021 |website=Soundguys |access-date=November 22, 2022}}</ref>

== See also ==
{{div col|colwidth=20em}}
* [[Architectural acoustics]]
* [[Audio signal processing]]
* [[Brownian noise]]
* [[White noise]]
* [[Colors of noise]]
* [[Crest factor]]
* [[Fractal]]
* [[Flicker noise]]
* [[Johnson–Nyquist noise]]
* [[Noise (electronics)]]
* [[Quantum 1/f noise]]
* [[Self-organised criticality]]
* [[Shot noise]]
* [[Sound masking]]
* [[Statistics]]
{{div col end}}

== Footnotes ==
{{reflist|30em}}
{{reflist|group=note}} 

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* {{cite journal
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* {{cite journal
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* {{cite journal
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* {{cite book
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== External links ==
* [https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise Coloured Noise: Matlab toolbox to generate power-law coloured noise signals of any dimensions.]
* [http://www.maxlittle.net/software/ Powernoise: Matlab software for generating 1/''f'' noise, or more generally, 1/''f''<sup>α</sup> noise]
* [http://www.scholarpedia.org/article/1/f_noise 1/f noise at Scholarpedia]
* [http://www.acousticfields.com/white-noise-definition-vs-pink-noise/ White Noise Definition Vs Pink Noise]
* [https://www.dgkingpo.com/iec-62368-1-audio-amplifier-testing-annex-e/ Application of pink noise in IEC 62368-1 audio amplifier testing (Annex E)]
{{noise}}
{{DEFAULTSORT:Pink Noise}}

[[Category:Noise (electronics)]]
[[Category:Sound]]
[[Category:Acoustics]]