Brownian noise

Not to be confused with Brown note (disambiguation).

Spectrum of Brownian noise, with a slope of –20 dB per decade

In science, Brownian noise (), also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after Robert Brown, who documented the erratic motion for multiple types of inanimate particles in water. The term "red noise" comes from the "white noise"/"white light" analogy; red noise is strong in longer wavelengths, similar to the red end of the visible spectrum.

Explanation

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.

Power spectrum

A Brownian motion, also called a Wiener process, is obtained as the integral of a white noise signal:

$W(t)=\int _{0}^{t}{\frac {dW(\tau )}{d\tau }}d\tau$

meaning that Brownian motion is the integral of the white noise $dW(t)$, whose power spectral density is flat:

$S_{0}=\left|{\mathcal {F}}\left[{\frac {dW(t)}{dt}}\right](\omega )\right|^{2}={\text{const}}.$

Note that here ${\mathcal {F}}$ denotes the Fourier transform, and $S_{0}$ is a constant. An important property of this transform is that the derivative of any distribution transforms as

${\mathcal {F}}\left[{\frac {dW(t)}{dt}}\right](\omega )=i\omega {\mathcal {F}}[W(t)](\omega ),$

from which we can conclude that the power spectrum of Brownian noise is

$S(\omega )={\big |}{\mathcal {F}}[W(t)](\omega ){\big |}^{2}={\frac {S_{0}}{\omega ^{2}}}.$

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

Production

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 to obtain the next one. A leaky integrator might be used in audio applications to ensure the signal does not "wander off". Note that while the first sample is random across the entire range that the sound sample can take on, the remaining offsets from there on are a tenth or thereabouts, leaving room for the signal to bounce around.