Brownian noise a white noise signal: meaning that Brownian motion is the integral of the white noise formula_2, whose power spectral density is flat: Note that here formula_4 denotes the Fourier transform, and formula_5 is a constant. An important property of this transform is that the derivative of any distribution transforms as from which we can conclude that the power spectrum of Brownian noise is Brown noise can be produced by integrating white noise. That is, whereas (digital) white noise can be produced by randomly choosing each sample independently, Brown noise can be produced by adding a random offset to each sample
Brownian motion moment of probability of displacement formula_2, is interpreted as mass diffusivity "D": Then the density of Brownian particles "ρ" at point "x" at time "t" satisfies the diffusion equation: Assuming that "N" particles start from the origin at the initial time "t"=0, the diffusion equation has the solution This expression (which is a normal distribution with the mean formula_12 and variance formula_13 usually called Brownian motion formula_14) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to