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Brownian noise
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==Power spectrum== [[File:Brown noise spectrum.svg|thumb|right|Spectrum of Brownian noise, with a slope of β20 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= 176β178 | 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>
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