# Pink noise

Pink noise or 1f noise is a 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 (halving or doubling in frequency) carries an equal amount of noise energy.

Colors of noise
White
Pink
Red (Brownian)
Grey

Pink noise is one of the most common signals in biological systems.[1]

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.

## Definition

Within the scientific literature the term 1/f noise is sometimes used 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. The canonical case with α = 1 is called pink noise.[3] General 1/f α-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.[4] The latter (broader sense) generally correspond to a wide range of non-equilibrium driven dynamical systems.

Pink noise sources include flicker noise in electronic devices. In their study of fractional Brownian motion,[5] Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to describe 1/f α noises for which the exponent α is not an even integer,[6] or that are fractional derivatives of Brownian (1/f 2) noise.

## Description

Spectrum of a pink noise approximation on a log-log plot. Power density falls off at 10 dB/decade of frequency.

Relative intensity of pink noise (left) and white noise (right) on an FFT spectrogram with the vertical axis being linear frequency.

In pink noise, 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;[7] see articles in Handel & Chung, 1993,[8] 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 resulting in flicker noise.

General 1/f α noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.[9] 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, heart beat rhythms, neural activity, and the statistics of DNA sequences, as a generalized pattern.[10] In financial systems, they are often referred to as a long-term memory effect[specify].

An accessible introduction to the significance of pink noise is one given by Martin Gardner (1978) in his Scientific American column "Mathematical Games".[11] 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.[12][13] So music is like tides not in terms of how tides sound, but in how tide heights vary.

Pink noise describes the statistical structure of many natural images.[14] Recently, it has also been successfully applied to the modeling of mental states in psychology,[15] and used to explain stylistic variations in music from different cultures and historic periods.[16] 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.[17] 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.[18]

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.[19] 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.[20][21]

### Electronic devices

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.[4][22] 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.[23] 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 ${\displaystyle \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}$

There is no known lower bound to background pink noise in electronics. Measurements made down to 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.[24]

A pioneering researcher in this field was Aldert van der Ziel.[25]

A pink-noise source is sometimes deliberately 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.[citation needed]

### In gravitational wave astronomy

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

1/f α noises with α near 1 are 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.[26]

### Climate change

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.[27]

### Diffusion processes

Many time-dependent stochastic processes are known to exhibit 1/f α noises with α between 0 and 2. In particular Brownian motion has a power spectral density that equals 4D/f 2,[28] where D is the diffusion coefficient. This type of spectrum is sometimes referred to as Brownian noise. Interestingly, the analysis of individual Brownian motion trajectories also show 1/f 2 spectrum, albeit with random amplitudes.[29] Fractional Brownian motion with Hurst exponent H also show 1/f α power spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5<H<1).[30]

## 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.[31] The Tweedie convergence theorem[32] 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[33] and in the physics literature as fluctuation scaling.[34] 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.[31] 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.[35]

There are various 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.[36][37] It can be generated on computer, for example, by filtering white noise,[38][39][40] inverse Fourier transform,[41] or by multirate variants on standard white noise generation.[13][11]

In supersymmetric theory of stochastics,[42] 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,[43] 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. PMID 11508842. S2CID 24505215.
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. ^ Baxandall, P. J. (November 1968). "Noise in Transistor Circuits: 1 - Mainly on fundamental noise concepts" (PDF). Wireless World. pp. 388–392. Retrieved 2019-08-08.
4. ^ a b Kogan, Shulim (1996). Electronic Noise and Fluctuations in Solids. [Cambridge University Press]. ISBN 978-0-521-46034-7.
5. ^ Mandelbrot, B. B.; Van Ness, J. W. (1968). "Fractional Brownian motions, fractional noises and applications". SIAM Review. 10 (4): 422–437. Bibcode:1968SIAMR..10..422M. doi:10.1137/1010093.
6. ^ Mandelbrot, Benoit B.; Wallis, James R. (1969). "Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix". Water Resources Research. 5 (1): 260–267. Bibcode:1969WRR.....5..260M. doi:10.1029/WR005i001p00260.
7. ^ Press, W. H. (1978). "Flicker noises in astronomy and elsewhere". Comments in Astrophysics. 7 (4): 103–119. Bibcode:1978ComAp...7..103P.
8. ^ Handel, P. H.; Chung, A. L. (1993). Noise in Physical Systems and 1/"f" Fluctuations. New York: American Institute of Physics.
9. ^ Bak, P.; Tang, C.; 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. PMID 10035754.
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. ^ a b Gardner, M. (1978). "Mathematical Games—White and brown music, fractal curves and one-over-f fluctuations". Scientific American. 238 (4): 16–32. doi:10.1038/scientificamerican0478-16.
12. ^ Voss, R. F.; Clarke, J. (1975). "'1/f Noise' in Music and Speech". Nature. 258 (5533): 317–318. Bibcode:1975Natur.258..317V. doi:10.1038/258317a0. S2CID 4182664.
13. ^ a b Voss, R. F.; Clarke, J. (1978). "1/f noise" in music: Music from 1/f noise". Journal of the Acoustical Society of America. 63 (1): 258–263. Bibcode:1978ASAJ...63..258V. doi:10.1121/1.381721.
14. ^ 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. CiteSeerX 10.1.1.136.1345. doi:10.1364/JOSAA.4.002379. PMID 3430225.
15. ^ Van Orden, G.C.; Holden, J.G.; 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. PMID 13678372.
16. ^ Pareyon, G. (2011). On Musical Self-Similarity, International Semiotics Institute & University of Helsinki. "On Musical Self-Similarity" (PDF).
17. ^ Noise in Man-generated Images and Sound
18. ^ 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 Archived 2013-01-24 at the Wayback Machine
19. ^ Gilden, David L; Thornton, T; 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. PMID 7892611.
20. ^ 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. S2CID 145051976.
21. ^ Gilden, David L (2001). "Cognitive Emissions of 1/ƒ Noise". Psychological Review. 108 (1): 33–56. CiteSeerX 10.1.1.136.1992. doi:10.1037/0033-295X.108.1.33. ISSN 0033-295X. PMID 11212631.
22. ^ 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.
23. ^ 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.
24. ^ 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 (1): 43. Bibcode:1988JAP....63...43K. doi:10.1063/1.340460.
25. ^ Aldert van der Ziel, (1954), Noise, Prentice–Hall
26. ^ Moore, Christopher; Cole, Robert; Berry, Christopher (19 July 2013). "Gravitational Wave Detectors and Sources". Retrieved 17 April 2014.
27. ^ Jim Shelton (2018-09-04). "Think pink for a better view of climate change". YaleNews. Retrieved 5 September 2018.
28. ^ Norton, M. P. (2003). Fundamentals of noise and vibration analysis for engineers. Karczub, D. G. (Denis G.) (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 9780511674983. OCLC 667085096.
29. ^ Krapf, Diego; Marinari, Enzo; Metzler, Ralf; Oshanin, Gleb; Xu, Xinran; Squarcini, Alessio (2018-02-09). "Power spectral density of a single Brownian trajectory: what one can and cannot learn from it". New Journal of Physics. 20 (2): 023029. doi:10.1088/1367-2630/aaa67c. ISSN 1367-2630.
30. ^ Krapf, Diego; Lukat, Nils; Marinari, Enzo; Metzler, Ralf; Oshanin, Gleb; Selhuber-Unkel, Christine; Squarcini, Alessio; Stadler, Lorenz; Weiss, Matthias; Xu, Xinran (2019-01-31). "Spectral Content of a Single Non-Brownian Trajectory". Physical Review X. 9 (1): 011019. doi:10.1103/PhysRevX.9.011019. ISSN 2160-3308.
31. ^ 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 (6): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. PMID 22304168.
32. ^ Jørgensen, B; Martinez, JR; Tsao, M (1994). "Asymptotic behaviour of the variance function". Scand J Statist. 21: 223–243.
33. ^ Taylor LR (1961). "Aggregation, variance and the mean". Nature. 189 (4766): 732–735. Bibcode:1961Natur.189..732T. doi:10.1038/189732a0. S2CID 4263093.
34. ^ Eisler Z, Bartos I, Kertesz (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond". Adv Phys. 57 (1): 89–142. arXiv:0708.2053. Bibcode:2008AdPhy..57...89E. doi:10.1080/00018730801893043. S2CID 119608542.
35. ^ 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.
36. ^ Milotti, Edoardo (2002-04-12). "1/f noise: a pedagogical review". arXiv:physics/0204033.
37. ^ 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. PMID 9908765.
38. ^ "Noise in Man-generated Images and Sound". mlab.uiah.fi. Retrieved 2015-11-14.
39. ^ "DSP Generation of Pink Noise". www.firstpr.com.au. Retrieved 2015-11-14.
40. ^ McClain, D (May 1, 2001). "Numerical Simulation of Pink Noise" (PDF). Preprint. Archived from the original (PDF) on 2011-10-04.
41. ^ Timmer, J.; König, M. (1995-01-01). "On Generating Power Law Noise". Astronomy and Astrophysics. 300: 707–710. Bibcode:1995A&A...300..707T.
42. ^ Ovchinnikov, I.V. (2016). "Introduction to supersymmetric theory of stochastics". Entropy. 18 (4): 108. arXiv:1511.03393. Bibcode:2016Entrp..18..108O. doi:10.3390/e18040108. S2CID 2388285.
43. ^ 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 (8): 1650086. arXiv:1404.4076. Bibcode:2016MPLB...3050086O. doi:10.1142/S021798491650086X. S2CID 118174242.