# 1 / f² noise

1 / f² noise (also known as "Brownian", "Brown" or "red noise") denotes noise in which the power density is inversely proportional to the square of the frequency (~ 1 / f²). The noise power density decreases by a factor of four or 6  dB when the frequency doubles ( octave ) or by 20 dB per decade. The similar 1 / f noise , on the other hand, shows a decrease in the noise power density of 3 dB per octave or 10 dB per decade.

The names Brown and Brownian for 1 / f² noise relate to the Scottish botanist and namesake of the Brownian motion , Robert Brown , not "brown" in color ( english brown ). For example, Brownian motion corresponds to 1 / f² noise. Nevertheless, since other types of noise are also referred to with colors (“ white noise ” or 1 / f noise as “ pink noise ”), the term brown noise is also common.

Audio sample of 1 / f² noise

## Power density spectrum

Power density spectrum of 1 / f² noise. The double logarithmic plot of the power density over the frequency results in a straight line with a slope of -2
Temporal representation of an exemplary 1 / f² noise signal

The Brownian molecular motion can be described as a stochastic process in the context of the Wiener process as the integral of white noise : ${\ displaystyle W (t)}$ ${\ displaystyle dW (t)}$

${\ displaystyle W (t) = \ int _ {0} ^ {t} dW (s)}$

White noise has a constant power density:

${\ displaystyle S_ {0} = \ left | {\ mathcal {F}} \ left [{\ frac {dW (t)} {dt}} \ right] (\ omega) \ right | ^ {2} = { \ text {constant}}}$

with the Fourier transform . One property of the Fourier transform is that the derivative that occurs can be expressed as a product as: ${\ displaystyle {\ mathcal {F}}}$

${\ displaystyle {\ mathcal {F}} \ left [{\ frac {dW (t)} {dt}} \ right] (\ omega) = \ mathrm {j} \ omega {\ mathcal {F}} [W (t)] (\ omega)}$

with as the imaginary unit and the angular frequency . ${\ displaystyle \ mathrm {j}}$ ${\ displaystyle \ omega}$

This results in the absolute power density spectrum for 1 / f² noise from the constant absolute power density spectrum for white noise as: ${\ displaystyle S_ {0}}$

${\ displaystyle S (\ omega) = \ left | {\ mathcal {F}} [W (t)] (\ omega) \ right | ^ {2} = {\ frac {S_ {0}} {\ omega ^ {2}}}}$

1 / f² noise can clearly be generated by filtering white noise with a second-order low-pass filter with a cut-off frequency of 0 Hz.

1 / f² noise can also be made audible, but the frequency component is limited to low-frequency signal components due to the sharp drop in the power density spectrum of 20 dB per decade, so that infrasound occurs primarily not or only with difficulty for humans .

## Visualization

1 / f² noise can be visualized by inverse Fourier transforming a discrete two-dimensional complex function with a bi- hyperbolically decreasing amplitude and a random phase . The amount of the complex-valued inverse Fourier transform can be output both in one color (gray levels) and separately for the three color channels as an RGB signal .

1 / f² noise can theoretically be made audible by inversely Fourier transforming a discrete one-dimensional complex function with a bi- hyperbolically decreasing amplitude and a random phase . However, the frequency component is limited to very low-frequency signals so that the infrasound cannot be heard by humans.

1 / f² noise
Two-dimensional,
colored noise signals
Two-dimensional,
gray-scale noise signals

## Color analogy of the name

The term red noise was formed with a comparable color analogy to the terms white noise and pink noise . Since the lower frequencies dominate even more strongly in the power density spectrum of red noise than with pink noise, the resulting color impression - in the figurative sense - corresponds to something that is redder than pink.

## literature

• Rudolf Müller: Noise . 1st edition. Springer, 1979, ISBN 3-540-09379-6 .
• Michael Dickreiter, Volker Dittel, Wolfgang Hoeg, Martin Wöhr: Handbuch der Tonstudiotechnik, 2 volumes . Ed .: ARD.ZDF medienakademie. 7th edition. Saur, Munich 2008, ISBN 978-3-598-11765-7 .
• Thomas Görne: Sound engineering . Fachbuchverlag Leipzig in Carl Hanser Verlag, Munich 2006, ISBN 3-446-40198-9 .