# Talk:Brownian noise

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## Derivation in Spectrum section is factually wrong

Brownian noise in this article is said to be the "spectrum" of a noise process generated by a Brownian Motion (Wiener Process). This is a non-stationary process, since the covariance and correlation function are known to be (taken from the Wiki page of Wiener Process):

${\displaystyle \operatorname {cov} (W_{s},W_{t})=\min(s,t)\,,}$
${\displaystyle \operatorname {corr} (W_{s},W_{t})={\mathrm {cov} (W_{s},W_{t}) \over \sigma _{W_{s}}\sigma _{W_{t}}}={\frac {\min(s,t)}{\sqrt {st}}}={\sqrt {\frac {\min(s,t)}{\max(s,t)}}}\,.}$

One should remember that the power spectral density is well defined only for wide sense stationary processes, those whose autocorrelation function only depends on the difference ${\displaystyle t-s}$. The Wiener Process clearly does not satisfy this requirement, thus it does not have a well-defined power spectrum. That's why if you google all over the internet you'll never find anywhere but here that states that the Wiener process has a 1/f^2 spectrum. The derivation of this section is not correct because it uses the formula:

${\displaystyle |\operatorname {Y} (\omega )|^{2}=|\operatorname {X} (\omega )|^{2}|\operatorname {H} (\omega )|^{2}}$

where ${\displaystyle \operatorname {Y} (\omega ),\operatorname {X} (\omega ),\operatorname {H} (\omega )}$ are the output (brownian motion), system (integrator), and input(white noise). This equation only holds true when the input is a stationary process (satisfied for white noise) and if the system is stable. The integrator system is however not stable for any inputs with a DC frequency component since integrators of DC blow up as t-> infinity, and the white noise process has a DC component (non-zero at f = 0) and thus does not satisfy the requirements for using this.

Now I do not dispute that brownian noise with a spectrum of 1/f^2 exists. However the explanation in this section is wrong. The correct reasoning is that 1/f^2 noise spectrum arises from viewing a leaky integrator of white noise after it reaches equilibrium (stationarity).

A leaky integrator has the form:

${\displaystyle \operatorname {H} (s)={\frac {\omega _{n}}{\omega _{n}+s}}}$
${\displaystyle |\operatorname {H} (\omega )|^{2}={\frac {\omega _{n}^{2}}{\omega _{n}^{2}+\omega ^{2}}}}$

This system, unlike the ideal integrator producing Brownian Motion, is stable, and the outputs are stationary when integrating white noise. Observing this process at frequencies far greater than ${\displaystyle \omega _{n}}$, you will find a 1/f^2 dependence in the spectrum, which is the characteristic of brownian noise. I would suggest those with access to IEEE refer to the work of

Demir: Computing Timing Jitter From Phase Noise Spectra for Oscillators and Phase-Locked Loops With White and 1/f Noise (2006) TCAS-I http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1703773

Column 1 pg 1873 where he analyzes the same problem and points out its logical fallacy. I will edit the spectrum section with the modified definition soon. SCfan84 (talk) 07:53, 31 July 2011 (UTC)

## Brown noise does not cause you to defecate

The brown noise was also a South Park joke, something the French had experimented in,[1] which has some basis in reality.

Acoustic, Infrasound. Very low-frequency sound which can travel long distances and easily penetrate most buildings and vehicles. Transmission of long wavelength sound creates biophysical effects; nausea, loss of bowels, disorientation, vomiting, potential internal organ damage or death may occur. Superior to ultrasound because it is "in band" meaning that its [sic] does not lose its properties when it changes mediums [sic] such as from air to tissue. By 1972 an infrasound generator had been built in France which generated waves at 7 hertz. When activated it made the people in range sick for hours. Nonlethal Weapons: terms and references INNS Occasional Paper 15, 1997, pp 2,3 Kwantus 05:17, 2004 Dec 22 (UTC) thanks to TMH
I remember the South Park episode calling it a 'brown note' not 'brown noise,' perhaps just a link to brown note is sufficient? --Morbid-o 20:15, 8 Apr 2005 (UTC)
Well, I'm watching it right now and they say "brown noise." (If I'd heard "brown note" I'd've put it in "brown note".) Episode 317 according to transcripts.[2][3] Kwantus 01:43, 2005 May 5 (UTC)
They have to mean the brown note though, because it exactly fits that disputed audio effect, and has nothing to do with brown noise. Probably a minor mistake on their part. -- Northgrove 23:01, 26 April 2007 (UTC)

Brown noise makes you defecate

This was the assertion on an episode of the British tongue-in-cheek popular science television show Brainiac: Science Abuse. __meco 09:21, 17 September 2006 (UTC)

We are well aware. See brown note. — Omegatron 16:29, 17 September 2006 (UTC)
That's the brown note, not brown noise you idiot.:

## Red or brown?

After this post brown noise is not exactly the same as red noise. There it is explained that the name "red" was already taken to name brown (after brownian motion) noise: "If we were going to pick a color, red might be good since pink noise lies between this noise and white noise. Unfortuantly, red is already taken. AKA "random walk" or "drunkard's walk" noise". Default007 16:23, 4 September 2006 (UTC)

Isnt Brownian noise 1/F^2 ? This page says that it is proportional to F^2. "Its spectral density is proportional to f²" —Preceding unsigned comment added by 75.36.43.183 (talk) 02:44, 15 November 2009 (UTC)

## WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:45, 10 November 2007 (UTC)

## Brown noise spectrum graph in prev-page is wrong

1(or n)/f^2 it's spectral graph is like 'L'. Brown noise spectrum graph in prev-page is like 1/f. brown noise's spectral graph in prev-page is wrong.

below is fourier fomula(frequency anlaysis)
${\displaystyle {\mathcal {F}}[f^{\prime }(t)](\omega )=\imath \omega {\mathcal {F}}[f(t)](\omega )}$
${\displaystyle W(t)=\int _{-\infty }^{+\infty }dW(t)}$

this is white noise fomula.
${\displaystyle S(\omega )=S_{0}}$.

brown noise downed 6db per octave.
${\displaystyle S(\omega )={\frac {S_{0}^{2}}{\omega ^{2}}}}$
${\displaystyle S(2\omega )={\frac {S_{0}^{2}}{\omega ^{2}}}-6db}$
${\displaystyle {\frac {S_{0}^{2}}{(2\omega )^{2}}}={\frac {S_{0}^{2}}{\omega ^{2}}}-6db}$
${\displaystyle {\frac {S_{0}^{2}}{4\omega ^{2}}}={\frac {S_{0}^{2}}{\omega ^{2}}}-6db}$
${\displaystyle {\frac {S_{0}^{2}}{\omega ^{2}}}-{\frac {S_{0}^{2}}{4\omega ^{2}}}=6db}$
${\displaystyle {\frac {S_{0}^{2}4\omega ^{2}-S_{0}^{2}\omega ^{2}}{4\omega ^{4}}}=6db}$
${\displaystyle S_{0}^{2}{\frac {3\omega ^{2}}{4\omega ^{4}}}=6db}$
${\displaystyle S_{0}^{2}{\frac {3}{4\omega ^{2}}}=6db}$
pink noise downed 3db per octave.
${\displaystyle S(f)\propto 1/f^{\alpha }}$(ƒ is frequency and 0 < α < 2)
${\displaystyle S(\omega )={\frac {S_{0}}{\omega }}}$
${\displaystyle S(2\omega )={\frac {S_{0}}{\omega }}-3db}$
${\displaystyle {\frac {S_{0}}{2\omega }}={\frac {S_{0}}{\omega }}-3db}$
${\displaystyle {\frac {S_{0}}{\omega }}-{\frac {S_{0}}{2\omega }}=3db}$
${\displaystyle S_{0}{\frac {2\omega -\omega }{2\omega ^{2}}}=3db}$
${\displaystyle S_{0}{\frac {\omega }{2\omega ^{2}}}=3db}$
${\displaystyle S_{0}{\frac {1}{2\omega }}=3db}$

are you understand? decibel is not equl power value of frequency spectrum.

211.204.114.196 (talk) 17:13, 24 November 2009 (UTC)

## Requested move 15 October 2016

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: Not moved (non-admin closure) Fuortu (talk) 22:20, 22 October 2016 (UTC)

Brownian noiseBrown noise – Per WP:COMMONNAME, since "Brown noise" is the common name for such noise despite technically not referring to the color (where the tern "Red noise" more accurately describes the color spectrum of such noise). ANDROS1337TALK 19:41, 15 October 2016 (UTC)

• Oppose – no evidence has been presented for the assertion that Brown noise is a more common term for it. Dicklyon (talk) 02:30, 16 October 2016 (UTC)
• Every video on YouTube calls it "Brown noise". Go see for yourself. ANDROS1337TALK 04:10, 16 October 2016 (UTC)
• Yes, I see that YouTube has more "brown noise" than "brownian noise". That's not exactly a survey of reliable sources, though, and as Binksternet says below, not clearly the same topic. Dicklyon (talk) 05:29, 16 October 2016 (UTC)
• Oppose. Brown noise and Brownian noise are two completely different things. Binksternet (talk) 05:02, 16 October 2016 (UTC)
• Oppose Google Books has more for "Brownian noise" and most of the "Brown noise" are either unreliable sources or referring to something else In ictu oculi (talk) 07:10, 16 October 2016 (UTC)
• Oppose – Brownian noise is a scientifically well-defined topic. — JFG talk 16:25, 16 October 2016 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.