Editing Brownian noise

{{Short description|Type of noise produced by Brownian motion}}
{{Redirect|Brown noise|the hypothetical sound that affects the human bowel|Brown note|other uses}}
{{Listen|filename=Brownnoise.ogg|title=Brown noise|description=10 seconds of Brownian noise|pos=right}}
[[File:Red-noise-trace.svg|thumb|Sample trace of Brownian noise]]
{{Colors of noise}}

In [[science]], '''Brownian noise''', also known as '''Brown noise''' or '''red noise''', is the type of [[signal noise]] produced by [[Brownian motion]], hence its alternative name of '''[[random walk]] noise'''. The term "Brown noise" does not come from [[brown|the color]], but after [[Robert Brown (Scottish botanist from Montrose)|Robert Brown]], who documented the erratic motion for multiple types of inanimate particles in water.  The term "red noise" comes from the "white noise"/"white light" analogy; red noise is strong in longer wavelengths, similar to the red end of the [[visible spectrum]].

==Explanation==
The graphic representation of the sound signal mimics a Brownian pattern. Its [[spectral density]] is inversely proportional to ''f'' <sup>2</sup>, meaning it has higher intensity at lower frequencies, even more so than [[pink noise]]. It decreases in intensity by 6 [[Decibel|dB]] per [[Octave (electronics)|octave]] (20&nbsp;dB per [[Decade (log scale)|decade]]) and, when heard, has a "damped" or "soft" quality compared to [[white noise|white]] and [[pink noise|pink]] noise. The sound is a low roar resembling a [[waterfall]] or heavy [[rainfall]]. See also [[Colors of noise#Violet noise|violet noise]], which is a 6&nbsp;dB ''increase'' per octave.

Strictly, Brownian motion has a Gaussian probability distribution, but "red noise" could apply to any signal with the 1/''f'' <sup>2</sup> frequency spectrum.

==Power spectrum==
[[File:Brown noise spectrum.svg|thumb|right|Spectrum of Brownian noise, with a slope of –20 dB per decade]]
A Brownian motion, also known as a [[Wiener process]], is obtained as the integral of a [[white noise]] signal:
<math display="block"> W(t) = \int_0^t \frac{dW}{d\tau}(\tau) d\tau </math>
meaning that Brownian motion is the integral of the white noise <math>t\mapsto dW(t)</math>, whose [[Spectral density#Power spectral density|power spectral density]] is flat:<ref>{{Cite book|title=Handbook of stochastic methods|first= C. W. |last=Gardiner|publisher= Springer Verlag|location= Berlin}}</ref>
<math display="block">
   S_0 = \left|\mathcal{F}\left[t\mapsto\frac{dW}{dt}(t)\right](\omega)\right|^2 = \text{const}.
</math>

Note that here <math>\mathcal{F}</math> denotes the [[Fourier transform]], and <math>S_0</math> is a constant. An important property of this transform is that the derivative of any distribution transforms as<ref>{{ cite journal|title=A statistical model of flicker noise|author1=Barnes, J. A.  |author2=Allan, D. W.  |name-list-style=amp |journal=Proceedings of the IEEE| volume= 54 | issue= 2 |year= 1966| pages= 176–178 | doi=10.1109/proc.1966.4630|s2cid=61567385 }} and references therein</ref>
<math display="block">
    \mathcal{F}\left[t\mapsto\frac{dW}{dt}(t)\right](\omega) = i \omega \mathcal{F}[t\mapsto W(t)](\omega),
</math>
from which we can conclude that the power spectrum of Brownian noise is
<math display="block">
    S(\omega) = \big|\mathcal{F}[t\mapsto W(t)](\omega)\big|^2 = \frac{S_0}{\omega^2}.
</math>

An individual Brownian motion trajectory presents a spectrum <math>S(\omega) = S_0 / \omega^2</math>, where the amplitude <math>S_0</math> is a random variable, even in the limit of an infinitely long trajectory.<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|pages=023029|doi=10.1088/1367-2630/aaa67c|doi-access=free|arxiv=1801.02986|bibcode=2018NJPh...20b3029K}}</ref>

==Production==
[[File:2D Brown noise.png|thumb|right|A two-dimensional Brownian noise image, generated with a [https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise computer program]]]
[[File:3D Brown noise.gif|thumb|right|A 3D Brownian noise signal, generated with a [https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise computer program], shown here as an animation, where each frame is a 2D slice of the 3D array]]
Brown noise can be produced by [[integral|integrating]] [[white noise]].<ref>{{cite web|url=http://www.dsprelated.com/showmessage/46697/1.php|title=Integral of White noise|year=2005|access-date=2010-04-30|archive-date=2012-02-26|archive-url=https://web.archive.org/web/20120226024012/http://www.dsprelated.com/showmessage/46697/1.php|url-status=dead}}</ref><ref>{{cite web|url=http://paulbourke.net/fractals/noise/|title=Generating noise with different power spectra laws
|first= Paul |last=Bourke|date=October 1998
}}
</ref> That is, whereas ([[Digital data|digital]]) white noise can be produced by randomly choosing each [[sample (signal)|sample]] independently, Brown noise can be produced by adding a random offset to each sample to obtain the next one. As Brownian noise contains infinite spectral power at low frequencies, the signal tends to drift away infinitely from the origin. A [[leaky integrator]] might be used in audio or electromagnetic applications to ensure the signal does not “wander off”, that is, exceed the limits of the system's [[dynamic range]]. This turns the Brownian noise into [[Ornstein–Uhlenbeck process|Ornstein–Uhlenbeck]] noise, which has a flat spectrum at lower frequencies, and only becomes “red” above the chosen cutoff frequency.

Brownian noise can also be computer-generated by first generating a white noise signal, Fourier-transforming it, then dividing the amplitudes of the different frequency components by the frequency (in one dimension), or by the frequency squared (in two dimensions) etc.<ref name="Das-thesis">{{cite thesis |last=Das |first=Abhranil |date=2022 |title=Camouflage detection & signal discrimination: theory, methods & experiments (corrected) |type=PhD |publisher=The University of Texas at Austin |url=http://dx.doi.org/10.13140/RG.2.2.32016.07683 | doi=10.13140/RG.2.2.32016.07683}}</ref> Matlab programs are available to generate Brownian and other power-law coloured noise in one<ref>{{Cite web |last=Zhivomirov |first=Hristo |date=1 August 2013 |title=Pink, Red, Blue and Violet Noise Generation with Matlab |url=https://www.mathworks.com/matlabcentral/fileexchange/42919-pink-red-blue-and-violet-noise-generation-with-matlab |access-date=9 November 2024 |website=MathWorks}}</ref> or  any number<ref>{{Cite web |last=Das |first=Abhranil |date=23 November 2022 |title=Colored Noise |url=https://www.mathworks.com/matlabcentral/fileexchange/121108-colored-noise |access-date=9 November 2024 |website=MathWorks}}</ref> of dimensions.

==Experimental evidence==
Evidence of Brownian noise, or more accurately, of Brownian processes has been found in different fields including chemistry,<ref>{{cite journal |last=Kramers |first=H.A. |title=Brownian motion in a field of force and the diffusion model of chemical reactions |journal=Physica |volume=7 |issue=4 |year=1940 |pages=284–304 |issn=0031-8914 |doi=10.1016/S0031-8914(40)90098-2 |url=https://www.sciencedirect.com/science/article/pii/S0031891440900982|url-access=subscription }}</ref> electromagnetism,<ref>{{cite journal |last=Kurşunoǧlu |first=Behram |title=Brownian motion in a magnetic field |journal=Annals of Physics |volume=17 |issue=2 |year=1962 |pages=259–268 |issn=0003-4916 |doi=10.1016/0003-4916(62)90027-1 |url=https://www.sciencedirect.com/science/article/pii/0003491662900271|url-access=subscription }}</ref> fluid-dynamics,<ref>{{cite journal |last1=Hauge |first1=E.H. |last2=Martin-Löf |first2=A. |title=Fluctuating hydrodynamics and Brownian motion |journal=Journal of Statistical Physics |volume=7 |year=1973 |pages=259–281 |doi=10.1007/BF01030307 |url=https://doi.org/10.1007/BF01030307|url-access=subscription }}</ref> economics,<ref>{{cite journal |last=Osborne |first=M.F.M. |title=Brownian Motion in the Stock Market |journal=Operations Research |volume=7 |issue=2 |year=1959 |pages=145–173 |doi=10.1287/opre.7.2.145 |url=https://doi.org/10.1287/opre.7.2.145|url-access=subscription }}</ref> and human neuromotor control.<ref name="Tessari2024">{{cite journal |last1=Tessari |first1=F. |last2=Hermus |first2=J. |last3=Sugimoto-Dimitrova |first3=R. |title=Brownian processes in human motor control support descending neural velocity commands |journal=Scientific Reports |volume=14 |year=2024 |pages=8341 |doi=10.1038/s41598-024-58380-5 |url=https://doi.org/10.1038/s41598-024-58380-5|pmc=11004188 }}</ref>

===Human neuromotor control===
In human neuromotor control, Brownian processes were recognized as a biomarker of human natural drift in both postural tasks—such as quietly standing or holding an object in your hand—as well as dynamic tasks. The work by Tessari et al. highlighted how these Brownian processes in humans might provide the first behavioral support to the neuroscientific hypothesis that humans encode motion in terms of descending neural velocity commands.<ref name="Tessari2024" />

== References ==
{{Reflist|2}}

{{Noise}}

[[Category:Noise (electronics)]]