Brownian noise

The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to f ², meaning it has higher intensity at lower frequencies, even more so than pink noise. It decreases in intensity by 6 dB per octave (20 dB per decade) and, when heard, has a "damped" or "soft" quality compared to white and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also violet noise, which is a 6 dB increase per octave.

Strictly, Brownian motion has a Gaussian probability distribution, but "red noise" could apply to any signal with the 1/f ² frequency spectrum.

A Brownian motion, also known as a Wiener process, is obtained as the integral of a white noise signal: $${\displaystyle W(t)=\int _{0}^{t}{\frac {dW}{d\tau }}(\tau )d\tau }$$ meaning that Brownian motion is the integral of the white noise ${\displaystyle t\mapsto dW(t)}$, whose power spectral density is flat:^[1] $${\displaystyle S_{0}=\left|{\mathcal {F}}\left[t\mapsto {\frac {dW}{dt}}(t)\right](\omega )\right|^{2}={\text{const}}.}$$

Note that here ${\displaystyle {\mathcal {F}}}$ denotes the Fourier transform, and ${\displaystyle S_{0}}$ is a constant. An important property of this transform is that the derivative of any distribution transforms as^[2] $${\displaystyle {\mathcal {F}}\left[t\mapsto {\frac {dW}{dt}}(t)\right](\omega )=i\omega {\mathcal {F}}[t\mapsto W(t)](\omega ),}$$ from which we can conclude that the power spectrum of Brownian noise is $${\displaystyle S(\omega )={\big |}{\mathcal {F}}[t\mapsto W(t)](\omega ){\big |}^{2}={\frac {S_{0}}{\omega ^{2}}}.}$$

An individual Brownian motion trajectory presents a spectrum ${\displaystyle S(\omega )=S_{0}/\omega ^{2}}$, where the amplitude ${\displaystyle S_{0}}$ is a random variable, even in the limit of an infinitely long trajectory.^[3]

Brown noise can be produced by integrating white noise.^[4][5] 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 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 turns the Brownian noise into Ornstein–Uhlenbeck noise, which has a flat spectrum at lower frequencies, and only becomes “red” above the chosen cutoff frequency.

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.^[6] Matlab programs are available to generate Brownian and other power-law coloured noise in one^[7] or any number^[8] of dimensions.

Evidence of Brownian noise, or more accurately, of Brownian processes has been found in different fields including chemistry,^[9] electromagnetism,^[10] fluid-dynamics,^[11] economics,^[12] and human neuromotor control.^[13]

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.^[13]