Editing Pink noise (section)

==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

<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 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 (i.e. a [[Gaussian white noise]] signal with zero mean and standard deviation {{mvar|σ}}, which has been suitably filtered), 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 phases are random:<ref name="Das-thesis"/>

<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>

{{mvar|χ{{sub|u}}}} are [[iid|independently and identically ({{abbr|iid}})]] [[chi distribution | chi-distributed variables]], and {{mvar|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:<ref name="Das-thesis"/>
<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/''f{{isup|0.2em|α}}''}}-like noises occur widely in nature and are a source of considerable interest in many fields. Noises with {{mvar|α}} near 1 generally come from [[condensed matter physics|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 = 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
      | author = Mandelbrot, B. B.
      | author-link = Benoit Mandelbrot
      |author2=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 }}</ref> [[Benoit Mandelbrot|Mandelbrot]] and Van Ness proposed the name ''fractional noise'' (sometimes since called ''fractal noise'') to describe {{math|1/''f{{isup|0.2em|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/''f''{{isup|0.2em|2}}}}) noise.