RS-485: Imported from Wikipedia (overwrite)

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{{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]]. In acoustics, this translates to a sound that is heavily weighted towards low frequencies, producing a deep, muffled roar.

==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 dB per [[Decade (log scale)|decade]]). To put this into perspective, for every doubling of frequency, the power of Brownian noise drops by a factor of four.

When heard, it has a "damped" or "soft" quality compared to [[white noise|white]] and [[pink noise|pink]] noise. Because the higher frequencies are so severely attenuated, the sound is a low roar resembling a [[waterfall]], heavy [[rainfall]], or the distant rumble of thunder. See also [[Colors of noise#Violet noise|violet noise]], which is the exact opposite with a 6 dB ''increase'' per octave.

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

==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>

This mathematically demonstrates that integrating a white noise signal in the time domain corresponds to dividing its spectrum by the angular frequency <math>\omega</math> in the frequency domain, resulting in the characteristic <math>1/\omega^2</math> power spectral density. 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:Locale_RS6_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] {{pay}}{{efn|name=demo-gh|Also available freely on [[GitHub]]: [https://github.com/abhranildas/coloured_noise/blob/main/colored_noise.m]}}]]
[[File:Locale_RS6_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] {{pay}}{{efn|name=demo-gh}}, 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 is typically achieved in digital processing by multiplying the previous sample by a decay factor slightly less than one (e.g., 0.999) before adding the new random white noise offset.
* This slowly pulls the signal back towards zero over time, preventing continuous DC offset accumulation.
* 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 established by the decay factor.

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. 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 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> where it forms the mathematical basis for models like Black-Scholes used in option pricing, 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, rather than absolute position commands.<ref name="Tessari2024" />

== Notes ==
{{notelist}}

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

{{Noise}}

[[Category:Noise (electronics)]]

Brownian noise - Revision history

RS-485: Imported from Wikipedia (overwrite)