![2D Brown noise](https://upload.wikimedia.org/wikipedia/commons/d/da/2D_Brown_noise.png)
]
![3D Brown noise](https://upload.wikimedia.org/wikipedia/commons/f/f2/3D_Brown_noise.gif)
In
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
, Brownian noise, also known as Brown noise or red noise, is the type of
signal noise produced by
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, hence its alternative name of
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
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
The visible spectrum is the portion of the electromagnetic spectrum that is visual perception, visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' or simply light. A typical human eye wil ...
.
Explanation
The graphic representation of the sound signal mimics a Brownian pattern. Its
spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
is inversely proportional to ''f''
2, meaning it has higher intensity at lower frequencies, even more so than
pink noise
Pink noise or noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving ...
. It decreases in intensity by 6
dB per
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(20 dB per
decade
A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years.
Usage
Any period of ten years is a "decade". For example, the statement that "du ...
) and, when heard, has a "damped" or "soft" quality compared to
white
White is the lightest color and is achromatic (having no hue). It is the color of objects such as snow, chalk, and milk, and is the opposite of black. White objects fully reflect and scatter all the visible wavelengths of light. White on ...
and pink noise. The sound is a low roar resembling a waterfall or heavy rainfall. See also
violet noise
In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly ...
, 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
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
, is obtained as the integral of a
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
signal:
:
meaning that Brownian motion is the integral of the white noise
, whose
power spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
is flat:
:
Note that here
denotes the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, and
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
:
An individual Brownian motion trajectory presents a spectrum
, where the amplitude
is a random variable, even in the limit of an infinitely long trajectory.
Production
Brown noise can be produced by
integrating white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
.
[
] That is, whereas (
digital) white noise can be produced by randomly choosing each
sample
Sample or samples may refer to:
Base meaning
* Sample (statistics), a subset of a population – complete data set
* Sample (signal), a digital discrete sample of a continuous analog signal
* Sample (material), a specimen or small quantity of s ...
independently, Brown noise can be produced by adding a random offset to each sample to obtain the next one. A
leaky integrator
In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics, ...
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
Dynamic range (abbreviated DR, DNR, or DYR) is the ratio between the largest and smallest values that a certain quantity can assume. It is often used in the context of signals, like sound and light. It is measured either as a ratio or as a base-1 ...
.
Brown 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.
Matlab programs are available to generate brown and other power-law coloured noise i
oneor
any numberof dimensions.
Sample
References
{{Noise
Noise (electronics)