# Brownian noise

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, the discoverer of Brownian motion. 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.

Colors of noise
White
Pink
Red (Brownian)
Grey
Spectrum of Brownian noise, with a slope of –20 dB per decade

## Explanation

The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to f 2, meaning it has more energy at lower frequencies, even more so than pink noise. It decreases in power 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:

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

meaning that Brownian motion is the integral of the white noise ${\displaystyle dW(t)}$, whose power spectral density is flat:[1]

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

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]

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

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

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

## Production

Brown noise can be produced by integrating white noise.[3][4] 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 there abouts, leaving room for the signal to bounce around.

## References

1. ^ Gardiner, C. W. Handbook of stochastic methods. Berlin: Springer Verlag.
2. ^ Barnes, J. A. & Allan, D. W. (1966). "A statistical model of flicker noise". Proceedings of the IEEE. 54 (2): 176–178. doi:10.1109/proc.1966.4630. and references therein
3. ^ "Integral of White noise". 2005.
4. ^ Bourke, Paul (October 1998). "Generating noise with different power spectra laws".
Acoustic quieting

This article primarily discusses mechanical and acoustic noise. See Noise reduction for electronic noise.

For signature reduction in general see Stealth technology.

Acoustic quieting is the process of making machinery quieter by damping vibrations to prevent them from reaching the observer. Machinery vibrates, causing sound waves in air, hydroacoustic waves in water, and mechanical stresses in solid matter. Quieting is achieved by absorbing the vibrational energy or minimizing the source of the vibration. It may also be redirected away from the observer.

One of the major reasons for the development of acoustic quieting techniques was for making submarines difficult to detect by sonar. This military goal of the mid- and late-twentieth century allowed the technology to be adapted to many industries and products, such as computers (e.g. hard drive technology), automobiles (e.g. motor mounts), and even sporting goods (e.g. golf clubs).

Active noise control

Active noise control (ANC), also known as noise cancellation, or active noise reduction (ANR), is a method for reducing unwanted sound by the addition of a second sound specifically designed to cancel the first.

Bilateral filter

A bilateral filter is a non-linear, edge-preserving, and noise-reducing smoothing filter for images. It replaces the intensity of each pixel with a weighted average of intensity values from nearby pixels. This weight can be based on a Gaussian distribution. Crucially, the weights depend not only on Euclidean distance of pixels, but also on the radiometric differences (e.g., range differences, such as color intensity, depth distance, etc.). This preserves sharp edges.

Block-matching and 3D filtering

Block-matching and 3D filtering (BM3D) is a 3-D block-matching algorithm used primarily for noise reduction in images.

Brown noise (disambiguation)

Brown noise can refer to:

Brownian noise, signal noise with a 1/f2 power spectrum

Brown note, a tone at a certain frequency said to cause loss of bowel control

World Wide Recorder Concert, also known as The Brown Noise, an episode of South Park

Brown sound

Brown sound may refer to:

Brown noise or Brownian noise, a random signal

Brown note, a hypothetical sound wave that would cause involuntary defecation

Dave Baksh (born 1980), a guitarist known as "Brownsound"

Brown sound, a guitar sound style of Eddie Van Halen

Bruitparif

Bruitparif is a non-profit environmental organization responsible for monitoring the environmental noise in the Paris agglomeration. It was founded in 2004.

Colors of noise

In audio engineering, electronics, physics, and many other fields, the color of noise refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly different properties: for example, as audio signals they will sound different to human ears, and as images they will have a visibly different texture. Therefore, each application typically requires noise of a specific color. This sense of 'color' for noise signals is similar to the concept of timbre in music (which is also called "tone color"); however the latter is almost always used for sound, and may consider very detailed features of the spectrum.

The practice of naming kinds of noise after colors started with white noise, a signal whose spectrum has equal power within any equal interval of frequencies. That name was given by analogy with white light, which was (incorrectly) assumed to have such a flat power spectrum over the visible range. Other color names, like pink, red, and blue were then given to noise with other spectral profiles, often (but not always) in reference to the color of light with similar spectra. Some of those names have standard definitions in certain disciplines, while others are very informal and poorly defined. Many of these definitions assume a signal with components at all frequencies, with a power spectral density per unit of bandwidth proportional to 1/f β and hence they are examples of power-law noise. For instance, the spectral density of white noise is flat (β = 0), while flicker or pink noise has β = 1, and Brownian noise has β = 2.

Fluid queue

In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term dam theory was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread of wildfires, in ruin theory and to model high speed data networks. The model applies the leaky bucket algorithm to a stochastic source.

The model was first introduced by Pat Moran in 1954 where a discrete-time model was considered. Fluid queues allow arrivals to be continuous rather than discrete, as in models like the M/M/1 and M/G/1 queues.

Fluid queues have been used to model the performance of a network switch, a router, the IEEE 802.11 protocol, Asynchronous Transfer Mode (the intended technology for B-ISDN), peer-to-peer file sharing, optical burst switching, and has applications in civil engineering when designing dams. The process is closely connected to quasi-birth–death processes, for which efficient solution methods are known.

Median filter

The Median Filter is a non-linear digital filtering technique, often used to remove noise from an image or signal. Such noise reduction is a typical pre-processing step to improve the results of later processing (for example, edge detection on an image). Median filtering is very widely used in digital image processing because, under certain conditions, it preserves edges while removing noise (but see discussion below), also having applications in signal processing.

Noise (signal processing)

In signal processing, noise is a general term for unwanted (and, in general, unknown) modifications that a signal may suffer during capture, storage, transmission, processing, or conversion.Sometimes the word is also used to mean signals that are random (unpredictable) and carry no useful information; even if they are not interfering with other signals or may have been introduced intentionally, as in comfort noise.

Noise reduction, the recovery of the original signal from the noise-corrupted one, is a very common goal in the design of signal processing systems, especially filters. The mathematical limits for noise removal are set by information theory, namely the Nyquist–Shannon sampling theorem.

Noise and vibration on maritime vessels

Noise and vibration on maritime vessels are not the same but they have the same origin and come in many forms. The methods to handle the related problems are similar, to a certain level, where most shipboard noise problems are reduced by controlling vibration.

Noise pollution

Noise pollution, also known as environmental noise or sound pollution, is the propagation of noise with harmful impact on the activity of human or animal life. The source of outdoor noise worldwide is mainly caused by machines, transport and propagation systems. Poor urban planning may give rise to noise pollution, side-by-side industrial and residential buildings can result in noise pollution in the residential areas. Some of the main sources of noise in residential areas include loud music, transportation noise, lawn care maintenance, nearby construction, or young people yelling (sports games). Noise pollution associated with household electricity generators is an emerging environmental degradation in many developing nations. The average noise level of 97.60 dB obtained exceeded the WHO value of 50 dB allowed for residential areas. Research suggests that noise pollution is the highest in low-income and racial minority neighborhoods. Documented problems associated with urban environment noise go back as far as ancient Rome.High noise levels can contribute to cardiovascular effects in humans and an increased incidence of coronary artery disease. In animals, noise can increase the risk of death by altering predator or prey detection and avoidance, interfere with reproduction and navigation, and contribute to permanent hearing loss. While the elderly may have cardiac problems due to noise, according to the World Health Organization, children are especially vulnerable to noise, and the effects that noise has on children may be permanent. Noise poses a serious threat to a child’s physical and psychological health, and may negatively interfere with a child's learning and behavior.

Non-local means

Non-local means is an algorithm in image processing for image denoising. Unlike "local mean" filters, which take the mean value of a group of pixels surrounding a target pixel to smooth the image, non-local means filtering takes a mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. This results in much greater post-filtering clarity, and less loss of detail in the image compared with local mean algorithms.If compared with other well-known denoising techniques, non-local means adds "method noise" (i.e. error in the denoising process) which looks more like white noise, which is desirable because it is typically less disturbing in the denoised product. Recently non-local means has been extended to other image processing applications such as deinterlacing, view interpolation, and depth maps regularization .

Shrinkage Fields (image restoration)

Shrinkage fields is a random field-based machine learning technique that aims to perform high quality image restoration (denoising and deblurring) using low computational overhead.

Soundproofing

Soundproofing is any means of reducing the sound pressure with respect to a specified sound source and receptor. There are several basic approaches to reducing sound: increasing the distance between source and receiver, using noise barriers to reflect or absorb the energy of the sound waves, using damping structures such as sound baffles, or using active antinoise sound generators.

Two distinct soundproofing problems may need to be considered when designing acoustic treatments - to improve the sound within a room (see reverberation), and reduce sound leakage to/from adjacent rooms or outdoors (see sound transmission class and sound reduction index). Acoustic quieting and noise control can be used to limit unwanted noise. Soundproofing can suppress unwanted indirect sound waves such as reflections that cause echoes and resonances that cause reverberation. Soundproofing can reduce the transmission of unwanted direct sound waves from the source to an involuntary listener through the use of distance and intervening objects in the sound path.

Total variation denoising

In signal processing, total variation denoising, also known as total variation regularization, is a process, most often used in digital image processing, that has applications in noise removal. It is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute gradient of the signal is high. According to this principle, reducing the total variation of the signal subject to it being a close match to the original signal, removes unwanted detail whilst preserving important details such as edges. The concept was pioneered by Rudin, Osher, and Fatemi in 1992 and so is today known as the ROF model.This noise removal technique has advantages over simple techniques such as linear smoothing or median filtering which reduce noise but at the same time smooth away edges to a greater or lesser degree. By contrast, total variation denoising is remarkably effective at simultaneously preserving edges whilst smoothing away noise in flat regions, even at low signal-to-noise ratios.

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 process or Brownian motion due to its historical connection with the physical process known as Brownian movement or Brownian motion originally observed by Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and unknown forces in control theory.

The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.

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