1-f noise: Difference between revisions
imported>Joseph Rushton Wakeling (→Pink noise: Some stuff on how 1/f satisfies the equal-energy-per-octave argument) |
imported>Joseph Rushton Wakeling (Edited definition of pink noise to bring it in line with references.) |
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where <math>f</math> 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 noise|white]] (<math>\alpha = 0</math>) and [[Brownian noise|brown]] (<math>\alpha = 2</math>) noise. Such "<math>1/f</math>-like" noises are widespread in nature and a source of great interest to diverse scientific communities. | where <math>f</math> 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 noise|white]] (<math>\alpha = 0</math>) and [[Brownian noise|brown]] (<math>\alpha = 2</math>) noise. Such "<math>1/f</math>-like" noises are widespread in nature and a source of great interest to diverse scientific communities. | ||
The "strict <math>1/f</math>" case of ''α'' = 1 is also referred to as '''pink noise''', although the precise definition of the latter term<ref name="ANSI-pink">[[Federal Standard 1037C]] and its successor, [[American National Standards Institute|American National Standard]] [http://www.atis.org/tg2k/ T1.523-2001].</ref> is not a <math>1/f</math> spectrum per se but that it contains equal | The "strict <math>1/f</math>" case of ''α'' = 1 is also referred to as '''pink noise''', although the precise definition of the latter term<ref name="ANSI-pink">[[Federal Standard 1037C]] and its successor, [[American National Standards Institute|American National Standard]] [http://www.atis.org/tg2k/ T1.523-2001].</ref> is not a <math>1/f</math> spectrum per se but that it contains equal power per octave, which is only satisfied by a <math>1/f</math> spectrum. The name stems from the fact that it lies in the middle between [[white noise|white]] (<math>1/f^0</math>) and [[Brownian noise|red]] (<math>1/f^2</math>, more commonly known as Brown or Brownian) noise<ref>Confusingly, the term "red noise" is sometimes used instead to refer to pink noise. In both cases the name springs from analogy to light with a <math>1/f^\alpha</math> spectrum: as ''α'' increases, the light becomes darker and darker red.</ref>. | ||
The term '''[[flicker noise]]''' is sometimes used to refer to <math>1/f</math> noise, although this is more properly applied only to its occurrence in electronic devices. [[Benoît Mandelbrot|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. | The term '''[[flicker noise]]''' is sometimes used to refer to <math>1/f</math> noise, although this is more properly applied only to its occurrence in electronic devices. [[Benoît Mandelbrot|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. | ||
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===Pink noise=== | ===Pink noise=== | ||
'''Pink noise''' is a term used in acoustics and engineering for noise which has equal | '''Pink noise''' is a term used in acoustics and engineering for noise which has equal power per octave or similar log-bundle<ref name="ANSI-pink" />. That is, if we consider all the frequencies in the range <math>[f, \lambda f]</math>, the total power should depend only on <math>\lambda</math> and not on <math>f</math>. We can see that a strict <math>1/f</math> spectrum satisfies this if we calculate the integral, | ||
:<math>\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</math> | :<math>\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</math> | ||
Revision as of 06:10, 10 February 2007
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, or more accurately 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, is a signal or process with a power spectral density proportional to ,
where 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 () and brown () noise. Such "-like" noises are widespread in nature and a source of great interest to diverse scientific communities.
The "strict " case of α = 1 is also referred to as pink noise, although the precise definition of the latter term[1] is not a spectrum per se but that it contains equal power per octave, which is only satisfied by a spectrum. The name stems from the fact that it lies in the middle between white () 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[2].
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[1]. 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 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 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 proportionnal 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
Notes
- ↑ 1.0 1.1 Federal Standard 1037C and its successor, American National Standard T1.523-2001.
- ↑ Confusingly, the term "red noise" is sometimes used instead to refer to pink noise. In both cases the name springs from analogy to light 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: as α increases, the light becomes darker and darker red.
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.