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 ($\alpha =2$) 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 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.

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.

Description

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

Pink noise

Pink noise is a term used in acoustics and engineering for noise which has equal energy per octave or similar log-bundle. That is, if we consider all the frequencies in the range $[f,\lambda f]$, the total energy should depend only on $\lambda$ and not on $f$. We can see that a strict $1/f$ spectrum satisfies this if we calculate the integral,

$\int _{f}^{\lambda f}{\frac {1}{f'}}\mathrm {d} f'=\left[\log f'\right]_{f}^{\lambda f}=\log(\lambda f)-\log f=\log \lambda +\log f-\log f=\log \lambda$

Relationship to fractional Brownian motion

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

References

Bibliography

• Dutta, P. and Horn, P. M. (1981). "Low-frequency fluctuations in solids: $1/f$ noise". Reviews of Modern Physics 53 (3): 497–516. DOI:10.1103/RevModPhys.53.497. Research Blogging.

• Keshner, M. S. (1982). "$1/f$ 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.