1/F NOISE

'$1/f$ noise', or more accurately ' noise', is a signal or process with a power spectral density proportional to ,
:
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 () and brown () noise. Such "$1/f$-like" noises are widespread in nature and a source of great interest to diverse scientific communities.
The "strict $1/f$" case of ''α'' = 1 is also referred to as 'pink noise', although the precise definition of the latter term^[1] is not a $1/f$ spectrum per se but that it contains equal energy per octave, which is only satisfied by a $1/f$ spectrum. The name stems from the fact that it lies in the middle between white ($1/f^0$) and red ($1/f^2$, more commonly known as Brown or Brownian) noise^[2].
The term 'flicker noise' is sometimes used to refer to $1/f$ noise, although this is more properly applied only to its occurrence in electronic devices due to a direct current. 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.

Contents

Description

Pink noise

Relationship to fractional Brownian motion

See also

References

Notes

Bibliography

Description

In the most general sense, noises with a spectrum include white noise, where the power spectrum is proportional to $1/f^0$ = constant, and Brownian noise, where it is proportional to $1/f^2$. The term black noise is sometimes used to refer to noise with an exponent ''α'' > 2.

Pink noise

¹

Relationship to fractional Brownian motion

The power spectrum of a fractional Brownian motion of Hurst exponent H is proportional to: $1/f^\{2H+1\}$

See also

★ Colors of noise

★ Detrended fluctuation analysis

★ Hurst exponent

References

Notes

1. Federal Standard 1037C and its successor, American National Standard T1.523-2001.
2. 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 spectrum: as ''α'' increases, the light becomes darker and darker red.

Bibliography

★ Low-frequency fluctuations in solids: $1/f$ noise, Dutta, P. and Horn, P. M., , , Reviews of Modern Physics,

★ $1/f$ noise, Keshner, M. S., , , Proceedings of the IEEE,

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★ Fractional Brownian motions, fractional noises and applications, Mandelbrot, B. B. and Van Ness, J. W., , , SIAM Review,

★ Flicker noises in astronomy and elsewhere, Press, W. H., , , Comments on Astrophysics,