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{{Colors of noise}}

'''Pink noise''' or '''{{frac|1|''f''}} noise''' (sometimes also called [[flicker noise]]) is a [[signal (information theory)|signal]] or process with a [[frequency spectrum]] such that the [[spectral density|power spectral density]] (energy or power per Hz) is [[inversely proportional]] to the [[frequency]].{{Technical-statement|date=September 2013}} In pink noise, each [[Octave (electronics)|octave]] (halving/doubling in frequency) carries an equal amount of noise power. 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|pages=79|url=http://greenteapress.com/complexity/html/book009.html|quote=Visible light with this power spectrum looks pink, hence the name.}}</ref>

Within the scientific literature the term pink noise is sometimes used a little more loosely to refer to any noise with a power spectral density of the form

:<math>S(f) \propto \frac{1}{f^\alpha}</math>

where ''f'' is frequency and 0 < α < 2, with exponent α usually close to 1. These pink -like noises occur widely in nature and are a source of considerable interest in many fields.  The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction.  The former (narrow sense) generally come from [[condensed matter]] systems in [[quasi-equilibrium]], as discussed below.<ref name="Kogan-1996">{{cite book
      | author = Kogan, Shulim
      | year = 1996
      | title = Electronic Noise and Fluctuations in Solids
      | publisher = [Cambridge University Press]
      | isbn = 0-521-46034-4
}}</ref>  The latter (broader sense) generally correspond to wide range of [[non-equilibrium]] driven [[dynamical system]]s.

The term ''[[flicker noise]]'' is sometimes used to refer to pink noise, although this is more properly applied only to its occurrence in electronic devices due to a [[direct current]].  [[Benoît Mandelbrot|Mandelbrot]] and Van Ness proposed the name ''fractional noise'' (sometimes since called ''fractal noise'') to emphasize that the exponent of the spectrum could take non-integer values and be closely related to [[fractional Brownian motion]], but the term is very rarely used.

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

There is equal energy in all [[octave]]s (or similar log bundles) of frequency. In terms of power at a constant bandwidth, pink noise falls off at 3 [[decibel|dB]] per octave.  At high enough frequencies pink noise is never dominant. ([[White noise]] is equal energy per hertz.)

The [[human auditory system]], which processes frequencies in a roughly logarithmic fashion approximated by the [[Bark scale]], does not perceive frequency octaves with equal sensitivity; signals in the 1–4-kHz octave sound loudest, and the [[loudness]] of other frequencies drops increasingly, depending both on the distance from the peak-sensitivity area and on the level. 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 has a tendency 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.

{{Listen
 | filename    = Pink noise.ogg
 | title       = Pink noise
 | description = 10 seconds of pink noise, [[audio normalization|normalized]] to −1 [[dBFS]] [[peak amplitude|peak]]
 | type        = sound
}}

== Generalization to more than one dimension ==
The spectrum of pink noise is {{frac|1|''f''}} only for one-dimensional signals. For two-dimensional signals (e.g., images) the spectrum is reciprocal to ''f'' <sup>2</sup>. In general, in an ''n''-dimensional system, the spectrum is reciprocal to ''f <sup>n</sup>''. For higher-dimensional signals it is still true (by definition) that each octave carries an equal amount of noise power. The frequency spectrum of two-dimensional signals, for instance, is also two-dimensional, and the area covered by succeeding octaves is four times as large.

== Occurrence ==
Pink noise occurs in many physical, biological and economic systems. Some researchers describe it as being ubiquitous.<ref name="Field-1987">{{cite journal
      | author = Bak, P. and Tang, C. and Wiesenfeld, K.
      |year= 1987
      | title = Self-Organized Criticality: An Explanation of 1/''ƒ'' Noise
      | journal = [[Physical Review Letters]]
      | volume = 59
      | pages = 381–384
      | doi = 10.1103/PhysRevLett.59.381
      | issue = 4
      | bibcode=1987PhRvL..59..381B
}}</ref> In physical systems, it is present in some [[meteorological]] data series, the [[electromagnetic radiation]] output of some astronomical bodies, and in almost all electronic devices (referred to as [[flicker noise]]). In biological systems, it is 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>

In [[financial system]]s, it is often referred to as a ''long-term memory effect''{{specify|date=January 2013}}.  Also, it describes the statistical structure of many natural images (images from the natural environment).<ref name="Field-1987">{{Cite journal
 | volume = 4
 | issue = 12
 | pages = 2379–2394
 | 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
 | year = 1987
 | doi = 10.1364/JOSAA.4.002379
 | pmid = 3430225
 |bibcode = 1987JOSAA...4.2379F
}}</ref>  Recently, pink noise 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. and Holden, J.G. and 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
}}</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">[http://mlab.uiah.fi/~eye/mediaculture/noise.html 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). [http://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]</ref>

Although [[self-organised criticality]] has been able to reproduce pink noise in [[sandpile]] models, there are no simple mathematical models to create pink noise.  It is usually generated by filtering white noise.<ref name="Kuittinen" /><ref>[http://www.firstpr.com.au/dsp/pink-noise/ DSP Generation of Pink Noise]</ref><ref>[http://linkage.rockefeller.edu/wli/moved.8.04/1fnoise/mcclain01.pdf]</ref>

There are many theories of the origin of pink noise. Some theories attempt to be universal, while others are applicable 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 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">Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and [[multifractality]]. ''Phys. Rev E'' 84, 066120</ref>  The Tweedie convergence theorem<ref>{{cite journal|last=Jørgensen, B|coauthors=Martinez, JR & Tsao, M|title=Asymptotic behaviour of the variance function|journal=Scand J Statist|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 distributions]].  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">Taylor LR (1961) Aggregation, variance and the mean. ''Nature'' 189, 732–735</ref> and in the physics literature as ''fluctuation scaling''.<ref name="Eisler2008">Eisler Z, Bartos I & Kertesz (2008) Fluctuation scaling in complex systems: Taylor’s law and beyond. ''Adv Phys'' 57, 89–142</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">Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and multifractality. ''Phys. Rev E'' 84, 066120</ref>  Both of these effects can be shown to be the consequence of [[convergence in distribution|mathematical convergence]] much as how certain kinds of data will converge towards the [[normal distribution]] under the [[central limit theorem]].

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

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

In electronics, [[white noise]] will be stronger than pink noise (flicker noise) above some [[corner frequency]]. There is no known lower bound to pink noise in electronics.  Measurements made down to 10<sup>−6</sup> Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.{{Citation needed|date=March 2008}}

A pink noise source is sometimes included on [[analog synthesizer]]s (although a white noise source is more common), both as a useful audio sound source for further processing, and also as a source of random control voltages for controlling other parts of the synthesizer.

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/''ƒ'' 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
      | unused_data = URL: http://link.aps.org/doi/10.1103/RevModPhys.60.537
      | bibcode=1988RvMP...60..537W
}}</ref>  The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes.<ref name="Dutta-1981">{{cite journal
      | author = Dutta, P. and Horn, P. M.
      |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> 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>

== See also ==
{{Div col|cols=3}}
* [[Architectural acoustics]]
* [[Audio signal processing]]
* [[Brownian noise]]
* [[White noise]]
* [[Colors of noise]]
* [[Crest factor]]
* [[Fractal]]
* [[Flicker noise]]
* [[Johnson–Nyquist noise]]
* [[Noise (physics)]]
* [[Quantum 1/f noise]]
* [[Self-organised criticality]]
* [[Shot noise]]
* [[Sound masking]]
* [[Statistics]]
{{Div col end}}

== Footnotes ==
{{Reflist|2}}

== References ==
* {{cite journal
      | author = Bak, P. and Tang, C. and Wiesenfeld, K.
      |year= 1987
      | title = Self-Organized Criticality: An Explanation of 1/''ƒ'' Noise
      | journal = [[Physical Review Letters]]
      | volume = 59
      | pages = 381–384
      | doi = 10.1103/PhysRevLett.59.381
      | issue = 4
| bibcode=1987PhRvL..59..381B
}}
* {{cite journal
      | author = Dutta, P. and Horn, P. M.
      |year= 1981
      | title = Low-frequency fluctuations in solids: 1/''ƒ'' noise
      | journal = [[Reviews of Modern Physics]]
      | volume = 53
      | pages = 497–516
      | doi = 10.1103/RevModPhys.53.497
      | issue = 3
| bibcode=1981RvMP...53..497D
}}
* {{cite journal
      | author = Field, D. J.
      |year= 1987
      | title = Relations Between the Statistics of Natural Images and the Response Profiles of Cortical Cells
      | journal = Journal of the [[Optical Society of America]] A
      | volume = 4
      | pages = 2379–2394
      | url = http://redwood.psych.cornell.edu/papers/field_87.pdf
|format=PDF
      | doi = 10.1364/JOSAA.4.002379
| pmid=3430225
      | issue = 12|bibcode = 1987JOSAA...4.2379F }}
* {{cite journal
      | author = Gisiger, T.
      |year= 2001
      | title = Scale invariance in biology: coincidence or footprint of a universal mechanism?
      | journal = [[Biological Reviews]]
      | volume = 76
      | pages = 161–209
      | doi = 10.1017/S1464793101005607
      | issue = 2
}}
* {{cite journal
      | author = Johnson, J. B.
      |year= 1925
      | title = The Schottky effect in low frequency circuits
      | journal = [[Physical Review]]
      | volume = 26
      | pages = 71–85
      | doi = 10.1103/PhysRev.26.71
|bibcode = 1925PhRv...26...71J }}

* {{cite book
      | author = Kogan, Shulim
      | year = 1996
      | title = Electronic Noise and Fluctuations in Solids
      | publisher = [Cambridge University Press]
      | isbn = 0-521-46034-4
}}

* {{cite journal
      | author = Press, W. H.
      |year= 1978
      | title = Flicker noises in astronomy and elsewhere
      | journal = Comments on Astrophysics
      | volume = 7
      | pages = 103–119
      | url = http://www.lanl.gov/DLDSTP/Flicker_Noise_1978.pdf
|format=PDF|bibcode = 1978ComAp...7..103P }}
* {{cite journal
      | author = Schottky, W.
      |year= 1918
      | title = Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern
      | journal = [[Annalen der Physik]]
      | volume = 362
      | pages = 541–567
      | doi = 10.1002/andp.19183622304
      | issue = 23
|bibcode = 1918AnP...362..541S
      | authorlink = Walter Schottky }}
* {{cite journal
      | author = Schottky, W.
      |year= 1922
      | title = Zur Berechnung und Beurteilung des Schroteffektes
      | journal = [[Annalen der Physik]]
      | volume = 373
      | pages = 157–176
      | doi = 10.1002/andp.19223731007
      | issue = 10
|bibcode = 1922AnP...373..157S
      | authorlink = Walter Schottky }}
* {{cite journal
      | author = Keshner, M. S.
      | year = 1982
      | title = 1/''ƒ'' noise
      | journal = [[IEEE|Proceedings of the IEEE]]
      | volume = 70
      | issue = 3
      | pages = 212–218
      | doi = 10.1109/PROC.1982.12282
}}
* {{cite web
      | author = Li, W.
      | date = 1996–present
      | title = A bibliography on 1/''ƒ'' noise
      | url = http://www.nslij-genetics.org/wli/1fnoise/
}}
* {{cite journal
      | author = [[Benoît Mandelbrot|Mandelbrot, B. B.]] and Van Ness, 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 }}
* A. Chorti and M. Brookes (2007), "Resolving near-carrier spectral infinities due to 1/''f'' phase noise in oscillators", [[Icassp|ICASSP 2007]], Vol. 3,  15–20 April 2007, Pages:III–1005 — III–1008, DOI 10.1109/ICASSP.2007.366852

== External links ==
* [http://www.maxlittle.net/software/ Powernoise: Matlab software for generating 1/''f'' noise, or more generally, 1/''f''<sup>α</sup> noise]
* [http://www.nslij-genetics.org/wli/1fnoise/ A Bibliography on 1/''f'' Noise]
* [http://www.scholarpedia.org/article/1/f_noise 1/f noise at Scholarpedia]

{{DEFAULTSORT:Pink Noise}}
[[Category:Noise]]

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