Brownian noise

1 / f ² noise (also called " Brownian ", "brown " or " red noise" called ) refers to a noise with an amplitude curve is inversely proportional to the square of the frequency ( ~ 1 / f ²). The noise power density thus drops by 6 dB per octave or 20 dB per decade. The similar 1/f-noise has a drop of the noise power density of 3 dB per octave or 10 dB per decade.

The names Brown and Brownian 1 / f ² noise relate to the Scottish botanist and namesake of the Brownian motion, Robert Brown, not on the color " brown ". Brownian motion corresponds to, for example, a 1 / f noise ².

Power spectral density

Brownian 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. A property of the Fourier transform is that can be expressed as the product of the apparent derivation as:

As with the imaginary unit and the angular frequency.

From this the amount of the power density spectrum obtained for 1 / f ² noise from the constant amount power density spectrum for white noise:

Clearly, 1 / f ² noise generated by filtering white noise with a low pass second order filter with a cutoff frequency of 0 Hz.

1 / f ² noise may be audible, but the frequency component is limited by the strong decrease of the power density spectrum of 20 dB per decade on low frequency signals, so primarily for humans is difficult or impossible perceptible infrasound occurs.

Visualization

1 / f ² noise can be visualized by a two-dimensional discrete complex function with bihyperbolisch decreasing amplitude and random phase is inverse Fourier transformed. The magnitude of the complex-valued Fourier inverse transform can both monochrome (grayscale ) and also be issued separately for the three color channels as RGB signal.

1 / f ² noise can theoretically be made audible by a discrete one-dimensional complex function with a sloping with bihyperbolisch amplitude and random phase is inverse Fourier transformed. However, the frequency content is limited to very low-frequency signals, so that the infrasound is inaudible to humans.