1-f noise

$1/f$ noise, or more accurately $1/f^{\alpha }$ noise, is a signal or process with a power spectral density proportional to $1/f^{\alpha }$,

$S(f)={\frac {\textrm {constant}}{f^{\alpha }}}$

where $f$ is the frequency. Typically use of the term focuses on noises with exponents in the range 0 < α < 2, that is, fluctuations whose structure falls in-between white ($\alpha =0$) and brown (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 2} ) noise. Such "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} -like" noises are widespread in nature and a source of great interest to diverse scientific communities.

The "strict Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} " case of α = 1 is also referred to as pink noise, although the precise definition of the latter term is not a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} spectrum per se but that it contains equal power per octave, which is only satisfied by a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} spectrum. The name stems from the fact that it lies in the middle between white (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^0} ) and red (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^2} , more commonly known as Brown or Brownian) noise.

The term flicker noise is sometimes used to refer to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} noise, although this is more properly applied only to its occurrence in electronic devices. Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to emphasise that the exponent of the spectrum could take non-integer values and be closely related to fractional Brownian motion, but the term is very rarely used.

Description

In the most general sense, noises with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^\alpha} spectrum include white noise, where the power spectrum is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^0} = constant, and Brownian noise, where it is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^2} . The term black noise is sometimes used to refer to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^\alpha} noise with an exponent α > 2.

Pink noise

Pink noise is a term used in acoustics and engineering for noise which has equal power per octave or similar log-bundle. That is, if we consider all the frequencies in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [f, \lambda f]} , the total power should depend only on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} and not on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . We can see that a strict $1/f$ spectrum satisfies this if we calculate the integral,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{f}^{\lambda f} \frac{1}{f'}\mathrm{d}f' = \left[ \log f' \right]_{f}^{\lambda f} = \log(\lambda f) - \log f = \log\lambda + \log f - \log f = \log\lambda}

Relationship to fractional Brownian motion

The power spectrum of a fractional Brownian motion of Hurst exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is proportional to: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f^{(2H+1)}}

References

Bibliography

• Dutta, P. and Horn, P. M. (1981). "Low-frequency fluctuations in solids: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} noise". Reviews of Modern Physics 53 (3): 497–516. DOI:10.1103/RevModPhys.53.497. Research Blogging.

• Keshner, M. S. (1982). "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/f} noise". Proceedings of the IEEE 70 (3): 212–218.

• Mandelbrot, B. B. and Van Ness, J. W. (1968). "Fractional Brownian motions, fractional noises and applications". SIAM Review 10 (4): 422–437.

• Press, W. H. (1978). "Flicker noises in astronomy and elsewhere". Comments on Astrophysics 7 (4): 103–119.