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.
Power density spectrum
The Brownian molecular motion can be described as a stochastic process in the context of the Wiener process as the integral of white noise :
White noise has a constant power density:
with the Fourier transform . One property of the Fourier transform is that the derivative that occurs can be expressed as a product as:
with as the imaginary unit and the angular frequency .
This results in the absolute power density spectrum for 1 / f² noise from the constant absolute power density spectrum for white noise as:
1 / f² noise can clearly be generated by filtering white noise with a secondorder lowpass filter with a cutoff frequency of 0 Hz.
1 / f² noise can also be made audible, but the frequency component is limited to lowfrequency 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 twodimensional complex function with a bi hyperbolically decreasing amplitude and a random phase . The amount of the complexvalued 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 onedimensional complex function with a bi hyperbolically decreasing amplitude and a random phase . However, the frequency component is limited to very lowfrequency signals so that the infrasound cannot be heard by humans.
1 / f² noise  

Twodimensional, colored noise signals 

Twodimensional, grayscale 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 3540093796 .
 Michael Dickreiter, Volker Dittel, Wolfgang Hoeg, Martin Wöhr: Handbuch der Tonstudiotechnik, 2 volumes . Ed .: ARD.ZDF medienakademie. 7th edition. Saur, Munich 2008, ISBN 9783598117657 .
 Thomas Görne: Sound engineering . Fachbuchverlag Leipzig in Carl Hanser Verlag, Munich 2006, ISBN 3446401989 .
Web links
Individual evidence
 ^ JA Barnes, DW Allan: A statistical model of flicker noise . In: Proceedings of the IEEE . 54, No. 2, 1966, pp. 176178. and the references listed therein