# 1-f noise

*6 January 2011*.

**1 / f noise**, or more accurately

**1 /**, is a signal or process with a power spectral density proportional to 1 /

*f*^{α}noise*f*

^{α},

where *f* is the frequency. Typical use of the term focuses on noises with exponents in the range 0 < *α* < 2, that is, fluctuations whose structure falls in-between white (α = 0) and brown (α = 2) 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 power 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. 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 |

## [edit] Description

In the most general sense, noises with a 1 / *f*^{α} 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 1 / *f*^{α} noise with an exponent *α* > 2.

### [edit] 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 [*f*,λ*f*], the total power should depend only on λ and not on *f*. We can see that a strict 1 / *f* spectrum satisfies this if we calculate the integral,

### [edit] Relationship to fractional Brownian motion

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

## [edit] References

- ↑
^{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 1 /
*f*^{α}spectrum: as*α*increases, the light becomes darker and darker red.

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