# Pink noise

Colors of noise
White
Pink
Red (Brownian)
Grey

Pink noise or 1f noise is a signal or process with a frequency spectrum such that the power spectral density (energy or power per frequency interval) is inversely proportional to the frequency of the signal. Pink noise is the most common signal in biological systems. [1] In pink noise, each octave (halving/doubling in frequency) carries an equal amount of noise energy. The name arises from the pink appearance of visible light with this power spectrum.[2] This is in contrast with white noise which has equal intensity per frequency interval.

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

${\displaystyle S(f)\propto {\frac {1}{f^{\alpha }}},}$

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.[3] The latter (broader sense) generally correspond to a wide range of non-equilibrium driven dynamical systems.

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. Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to emphasize that the exponent of the power spectrum could take non-integer values and be closely related to fractional Brownian motion, but the term is very rarely used.

## Description

Spectrum of a pink noise approximation on a log-log plot. Power density falls off at 10 dB/decade of frequency.
Pink noise (left) and white noise (right) on an FFT spectrogram with linear frequency (vertical) axis[clarification needed]

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

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 kHz sound loudest for a given intensity. However, humans still differentiate between white noise and pink noise with ease.

Graphic equalizers 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.

## Generalization to more than one dimension

The power spectrum of pink noise is 1/f only for one-dimensional signals. For two-dimensional signals (e.g., images) the power spectrum is reciprocal to f 2 In general, in an n-dimensional system, the power spectrum is reciprocal to f n. 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 of the power spectrum covered by succeeding octaves is four times as large.

## Occurrence

In the past quarter century, pink noise has been discovered in the statistical fluctuations of an extraordinarily diverse number of physical and biological systems (Press, 1978;[4] see articles in Handel & Chung, 1993,[5] and references therein). Examples of its occurrence include fluctuations in tide and river heights, quasar light emissions, heart beat, firings of single neurons, and resistivity in solid-state electronics.

An accessible introduction to the significance of pink noise is one given by Martin Gardner (1978) in his Scientific American column "Mathematical Games".[6] 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.[7][8] So music is like tides not in terms of how tides sound, but in how tide heights vary.

Because pink noise occurs in many physical, biological and economic systems, some researchers describe it as being ubiquitous.[9] 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, heart beat rhythms, neural activity, and the statistics of DNA sequences, as a generalized pattern.[10]

In financial systems, it is often referred to as a long-term memory effect[specify]. Also, it describes the statistical structure of many natural images.[11] Recently, pink noise has also been successfully applied to the modeling of mental states in psychology,[12] and used to explain stylistic variations in music from different cultures and historic periods.[13] Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of pitches, will tend towards a pink noise spectrum.[14] Similarly, a generally pink distribution pattern has been observed in film shot length by researcher James E. Cutting of Cornell University, in the study of 150 popular movies released from 1935 to 2005.[15]

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.[16] 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.[17][18]

### Electronic devices

A pioneering researcher in this field was Aldert van der Ziel.[19]

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 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.[20]

A pink-noise source is sometimes included on analog synthesizers (although a white-noise source is more common), both as a useful audio sound source for further processing and 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.[3][21] 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.[22] Since the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 1014 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 ${\displaystyle \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}$

### In gravitational wave astronomy

Noise curves for a selection of gravitational-wave detectors as a function of frequency.

1/f noise is a factor in gravitational-wave astronomy. The noise curve at very low frequencies affect pulsar timing arrays, 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 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.[23]

## Origin

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 semiconductors. 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.[24] The Tweedie convergence theorem[25] 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[26] and in the physics literature as fluctuation scaling.[27] 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.[24] Both of these effects can be shown to be the consequence of 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.[28]

There are no simple mathematical models to create pink noise. Although self-organised criticality has been able to reproduce pink noise in sandpile models, these do not have a Gaussian distribution or other expected statistical qualities.[29][30] It is usually generated by filtering white noise[31][32][33] or inverse Fourier transform.[34]

In supersymmetric theory of stochastics,[35] an approximation-free theory of stochastic differential equations, 1/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 deterministic chaos,[36] whereas the associated emergence of the long-term dynamical memory or order, i.e., 1/f and crackling noises, the Butterfly effect etc., is the consequence of the Goldstone theorem in the application to the spontaneously broken topological supersymmetry.

## Footnotes

1. ^ Szendro, P (2001). "Pink-Noise Behaviour of Biosystems". European Biophysics Journal. 30 (3): 227-231. doi:10.1007/s002490100143.
2. ^ Downey, Allen (2012). Think Complexity. O'Reilly Media. p. 79. ISBN 978-1-4493-1463-7. Visible light with this power spectrum looks pink, hence the name.
3. ^ a b Kogan, Shulim (1996). Electronic Noise and Fluctuations in Solids. [Cambridge University Press]. ISBN 0-521-46034-4.
4. ^ Press, W. H. (1978). "Flicker noises in astronomy and elsewhere". Comments in Astrophysics. 7 (4): 103–119.
5. ^ Handel, P. H., & Chung, A. L. (1993). Noise in Physical Systems and 1/"f" Fluctuations. New York: American Institute of Physics.
6. ^ Gardner, M. (1978). "Mathematical Games—White and brown music, fractal curves and one-over-f fluctuations". Scientific American. 238: 16–32.
7. ^ Voss, R. F., & Clarke, J. (1975). "'1/f Noise' in Music and Speech". Nature. 258: 317–318. Bibcode:1975Natur.258..317V. doi:10.1038/258317a0.
8. ^ Voss, R. F., & Clarke, J. (1978). "1/f noise" in music: Music from 1/f noise". Journal of the Acoustical Society of America. 63: 258–263. Bibcode:1978ASAJ...63..258V. doi:10.1121/1.381721.
9. ^ Bak, P. and Tang, C. and Wiesenfeld, K. (1987). "Self-Organized Criticality: An Explanation of 1/ƒ Noise". Physical Review Letters. 59 (4): 381–384. Bibcode:1987PhRvL..59..381B. doi:10.1103/PhysRevLett.59.381.
10. ^ 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.
11. ^ Field, D. J. (1987). "Relations between the statistics of natural images and the response properties of cortical cells" (PDF). J. Opt. Soc. Am. A. 4 (12): 2379–2394. Bibcode:1987JOSAA...4.2379F. doi:10.1364/JOSAA.4.002379. PMID 3430225.
12. ^ Van Orden, G.C. and Holden, J.G. and Turvey, M.T. (2003). "Self-organization of cognitive performance". Journal of Experimental Psychology: general. 132 (3): 331–350. doi:10.1037/0096-3445.132.3.331.
13. ^ Pareyon, G. (2011). On Musical Self-Similarity, International Semiotics Institute & University of Helsinki. "On Musical Self-Similarity" (PDF).
14. ^ Noise in Man-generated Images and Sound
15. ^ Anger, Natalie (March 1, 2010). "Bringing New Understanding to the Director's Cut". The New York Times. Retrieved on March 3, 2010. See also original study
16. ^ Gilden, David L and Thornton, T and Mallon, MW (1995). "1/ƒ Noise in Human Cognition". Science. 267 (5205): 1837–1839. Bibcode:1995Sci...267.1837G. doi:10.1126/science.7892611. ISSN 0036-8075.
17. ^ Gilden, D. L. (1997). "Fluctuations in the time required for elementary decisions". Psychological Science. 8 (4): 296–301. doi:10.1111/j.1467-9280.1997.tb00441.x.
18. ^ Gilden, David L (2001). "Cognitive Emissions of 1/ƒ Noise". Psychological Review. 108 (1): 33–56. doi:10.1037/0033-295X.108.1.33. ISSN 0033-295X.
19. ^ Aldert van der Ziel, (1954), Noise, Prentice–Hall
20. ^ Kleinpenning, T. G. M. & de Kuijper, A. H. (1988). "Relation between variance and sample duration of 1/f Noise signals". Journal of Applied Physics. 63: 43. Bibcode:1988JAP....63...43K. doi:10.1063/1.340460.
21. ^ Weissman, M. B. (1988). "1/ƒ Noise and other slow non-exponential kinetics in condensed matter". Reviews of Modern Physics. 60 (2): 537–571. Bibcode:1988RvMP...60..537W. doi:10.1103/RevModPhys.60.537.
22. ^ Dutta, P. & Horn, P. M. (1981). "Low-frequency fluctuations in solids: 1/f noise". Reviews of Modern Physics. 53 (3): 497–516. Bibcode:1981RvMP...53..497D. doi:10.1103/RevModPhys.53.497.
23. ^ Moore, Christopher; Cole, Robert; Berry, Christopher (19 July 2013). "Gravitational Wave Detectors and Sources". Retrieved 17 April 2014.
24. ^ a b 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
25. ^ Jørgensen, B; Martinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function". Scand J Statist. 21: 223–243.
26. ^ Taylor LR (1961) Aggregation, variance and the mean. Nature 189, 732–735
27. ^ Eisler Z, Bartos I & Kertesz (2008) Fluctuation scaling in complex systems: Taylor’s law and beyond. Adv Phys 57, 89–142
28. ^ Kendal, WS (2015). "Self-organized criticality attributed to a central limit-like convergence effect". Physica A. 421: 141–150. Bibcode:2015PhyA..421..141K. doi:10.1016/j.physa.2014.11.035.
29. ^ Milotti, Edoardo (2002-04-12). "1/f noise: a pedagogical review". arXiv:. Bibcode:2002physics...4033M.
30. ^ O’Brien, Kevin P.; Weissman, M. B. (1992-10-01). "Statistical signatures of self-organization". Physical Review A. 46 (8): R4475–R4478. Bibcode:1992PhRvA..46.4475O. doi:10.1103/PhysRevA.46.R4475.
31. ^ "Noise in Man-generated Images and Sound". mlab.uiah.fi. Retrieved 2015-11-14.
32. ^ "DSP Generation of Pink Noise". www.firstpr.com.au. Retrieved 2015-11-14.
33. ^ McClain, D (May 1, 2001). "Numerical Simulation of Pink Noise" (PDF). Preprint. Archived from the original (PDF) on 2011-10-04.
34. ^ Timmer, J.; König, M. (1995-01-01). "On Generating Power Law Noise". Astronomy and Astrophysics. 300: 707–710. Bibcode:1995A&A...300..707T.
35. ^ Ovchinnikov, I.V. (2016). "Introduction to supersymmetric theory of stochastics". Entropy. 18: 108. arXiv:. Bibcode:2016Entrp..18..108O. doi:10.3390/e18040108.
36. ^ Ovchinnikov, I.V.; Schwartz, R. N.; Wang, K. L. (2016). "Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics". Modern Physics Letters B. 30: 1650086. arXiv:. doi:10.1142/S021798491650086X.

## References

• Kogan, Shulim (1996). Electronic Noise and Fluctuations in Solids. [Cambridge University Press]. ISBN 0-521-46034-4.