In science

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, Brownian noise , also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, hence its alternative name of random walk noise. The term "Brown noise" comes not from the color

Brown

Brown is a color term, denoting a range of composite colors produced by a mixture of orange, red, rose, or yellow with black or gray. The term is from Old English brún, in origin for any dusky or dark shade of color....

, but after Robert Brown

Robert Brown (botanist)

Robert Brown was a Scottish botanist and palaeobotanist who made important contributions to botany largely through his pioneering use of the microscope...

, the discoverer of Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

.

Explanation

The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

 is inversely proportional to f², meaning it has more energy at lower frequencies, even more so than pink noise

Pink noise

Pink noise or 1/ƒ noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency. In pink noise, each octave carries an equal amount of noise power...

. It decreases in power by 6 dB

Decibel

The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

 per octave

Octave

In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

 and, when heard, has a "damped" or "soft" quality compared to white

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

 and pink noise

Pink noise

Pink noise or 1/ƒ noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency. In pink noise, each octave carries an equal amount of noise power...

. The sound is a low roar resembling a waterfall or heavy rainfall. See also purple noise, which is a 6 dB increase per octave.

Power spectrum

A Brownian signal is expressed mathematically as the integral of a white noise

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

 signal, , namely a Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

White noise has a constant spectrum . The Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 has the general property



Since white noise is the derivative of Brownian motion (more accurately, Brownian motion is the integral of white noise, as the derivative of a random function is not defined cf.) we conclude that the spectrum for Brownian noise is , therefore the power spectrum is given by

Production

Brown noise can be produced by integrating

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 white noise

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

. That is, whereas (digital

Digital

A digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...

) 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.

Sample

External links

• Brown noise in wave(.wav) format, 1 minute long
• PlayNoise, a free online white, pink, and brown noise generator, uses Javascript/HTML5.