Brownian noise

For similar phrases, see Brown note (disambiguation).

Colors of noise
White
Pink
Red (Brownian)
Grey

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

In science, Brownian noise ( Sample ), 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" comes not 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.

Explanation

The graphic representation of the sound signal mimics a Brownian pattern. Its spectral density is inversely proportional to f², 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/f² frequency spectrum.

Power spectrum

A Brownian motion, also called a Wiener process, is obtained as the integral of a white noise signal, $dW(t)$ ,

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

meaning that Brownian motion is the integral of the white noise $dW(t)$ whose power spectral density is flat^[1]

$S_0 = \left|\mathcal{F}\left[\frac{dW(t)}{dt}\right](\omega)\right|^2 = \text{const}$

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^[2]

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

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

$S(\omega)= \left|\mathcal{F}[W(t)](\omega)\right|^2= \frac{S_0}{\omega^2}$ .

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.

Sample

	Brown noise 10 seconds of Brown noise
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References

↑ Gardiner, C. W. Handbook of stochastic methods. Berlin: Springer Verlag.
↑ 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
↑ "Integral of White noise". 2005.
↑ Bourke, Paul (October 1998). "Generating noise with different power spectra laws".

Noise (physics and telecommunications)

General	Acoustic quieting Distortion Noise cancellation Noise control Noise measurement Noise power Noise reduction Noise temperature Phase distortion

Noise in...	Audio Buildings Electronics Environment Government regulation Human health Images Radio Rooms Ships Sound masking Transportation Video

Class of noise	Additive white Gaussian noise (AWGN) Atmospheric noise Background noise Brownian noise Burst noise Cosmic noise Flicker noise Gaussian noise Grey noise Jitter Johnson–Nyquist noise (thermal noise) Pink noise Quantization error (or q. noise) Shot noise White noise Coherent noise Value noise Gradient noise Worley noise

Engineering terms	Channel noise level Circuit noise level Effective input noise temperature Equivalent noise resistance Equivalent pulse code modulation noise Impulse noise (audio) Noise figure Noise floor Noise shaping Noise spectral density Noise, vibration, and harshness (NVH) Phase noise Pseudorandom noise Statistical noise

Ratios	Carrier-to-noise ratio (C/N) Carrier-to-receiver noise density (C/kT) dBrnC E_b/N₀ (energy per bit to noise density) E_s/N₀ (energy per symbol to noise density) Modulation error ratio (MER) Signal, noise and distortion (SINAD) Signal-to-interference ratio (S/I) Signal-to-noise ratio (S/N, SNR) Signal-to-noise ratio (imaging) Signal to noise plus interference (SNIR) Signal-to-quantization-noise ratio (SQNR)

Related topics	List of noise topics Acoustics Colors of noise Interference (communication) Noise generator Radio noise source Spectrum analyzer Thermal radiation

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