Talk:Brownian noise

message: 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):
: \operatorname{cov}(W_s,W_t) = \min(s,t) \, , 
: \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  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:<BR>
: |\operatorname{Y}(\omega)|^2 = |\operatorname{X}(\omega)|^2 |\operatorname{H}(\omega)|^2 
where  \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:
: \operatorname{H}(s) = \frac{\omega_n}{\omega_n + s}  
: |\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 \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)
thread_title:  Derivation in Spectrum section is factually wrong 

message: 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)
thread_title:  Red or brown? 

message: 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)
thread_title:  WikiProject class rating

message: 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. <BR>
below is fourier fomula(frequency anlaysis)<BR>
 \mathcal{F}[f^\prime(t)](\omega) = \imath \omega \mathcal{F}[f(t)](\omega)  <BR>
W(t) = \int _{-\infty}^{+\infty} dW(t)  <BR>
this is white noise fomula.<BR>
S(\omega) =  S_0 .<BR>
brown noise downed 6db per octave. <BR>
 S(\omega)= \frac{S_0^2}{\omega^2} <BR>
 S(2\omega )= \frac{S_0^2}{\omega^2} -6db <BR>
 \frac{S_0^2}{(2\omega)^2} = \frac{S_0^2}{\omega^2} -6db <BR>
 \frac{S_0^2}{4\omega^2} = \frac{S_0^2}{\omega^2} -6db <BR>
 \frac{S_0^2}{\omega^2}-\frac{S_0^2}{4\omega^2} =  6db <BR>
 \frac{S_0^24\omega^2 - S_0^2\omega^2 }{4\omega^4} =  6db <BR>
 S_0^2 \frac{ 3\omega^2 }{4\omega^4} =  6db <BR>
 S_0^2 \frac{ 3 }{4\omega^2} =  6db <BR>
pink noise downed 3db per octave.<BR>
S(f) \propto 1/f^\alpha(ƒ is frequency and 0 < α < 2)  <BR>
 S(\omega)= \frac{S_0}{\omega}<BR>
 S(2\omega )= \frac{S_0}{\omega} - 3db<BR>
 \frac{S_0}{2\omega}= \frac{S_0}{\omega} - 3db<BR>
 \frac{S_0}{\omega} - \frac{S_0}{2\omega}=  3db<BR>
 S_0\frac{2\omega-\omega}{2\omega^2}=  3db<BR>
 S_0\frac{\omega}{2\omega^2}=  3db<BR>
 S_0\frac{1}{2\omega}=  3db<BR>
are you understand?
decibel is not equl power value of frequency spectrum.
211.204.114.196 (talk) 17:13, 24 November 2009 (UTC)
thread_title:   Brown noise spectrum graph in prev-page is wrong 

message: The notations for both processes are the same: W(t) -which creates inconsistent statements in the page. I have found η(t), ξ(t), of \dot{W}(t) (assuming white noise is the time-derivative of the Wiener process in an appropriate sense, and the dot signifies time-derivative), as notations for white noise, after a quick search. Can anyone correct that or explain to me why leave the page as is ? Thank you. Plm203 (talk) 05:22, 9 May 2023 (UTC)
thread_title:  Notations for white noise and Wiener process 

message: I suggest merging the article with the article "Brownian Surface", as both are one and the same. Pia novice (talk) 08:18, 22 January 2024 (UTC)
thread_title:  Suggest merging with article Brownian Surface 

message: First, "Brown noise can be produced by adding a random offset to each sample to obtain the next one" is only applicable to one dimension.
Second, the images and the link to the Matlab script by Das are nice. Unfortunately, they do only create noise which is visually similar to Brownian noise, but the correlation structure dissimilar from that of Brownian noise. This is because Brownian noise is not stationary which requires special considerations when generating it via the Fourier Transform. There are many simple 2D Brownian noise generators from the field of image processing, which unfortunately all suffer from similar issues. A method for generating Brownian noise with exact correlation structure is described in "Fast and Exact Simulation of Fractional Brownian Surfaces M.L. Stein (2002)". A Matlab implementation is available from Zdravko Botev, 2016 
Third, "generation" seems a more suitable heading than "production". Pia novice (talk) 08:38, 22 January 2024 (UTC)
thread_title:  Paragraph "Production" needs to be revised