Pink noise, noise or fractional noise or fractal noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving or doubling in frequency) carries an equal amount of noise energy.
Pink noise sounds like a waterfall. It is often used to tune loudspeaker systems in professional audio. Pink noise is one of the most commonly observed signals in biological systems.
The name arises from the pink appearance of visible light with this power spectrum. This is in contrast with white noise which has equal intensity per frequency interval.
Definition
Within the scientific literature, the term 1/f noise is sometimes used loosely to refer to any noise with a power spectral density of the form
where f is frequency, and 0 < α < 2, with exponent α usually close to 1. One-dimensional signals with α = 1 are usually called pink noise.
The following function describes a length one-dimensional pink noise signal (i.e. a Gaussian white noise signal with zero mean and standard deviation , which has been suitably filtered), as a sum of sine waves with different frequencies, whose amplitudes fall off inversely with the square root of frequency (so that power, which is the square of amplitude, falls off inversely with frequency), and phases are random:
are iid chi-distributed variables, and are uniform random.
In a two-dimensional pink noise signal, the amplitude at any orientation falls off inversely with frequency. A pink noise square of length can be written as:
General 1/f α-like noises occur widely in nature and are a source of considerable interest in many fields. Noises with α near 1 generally come from condensed-matter systems in quasi-equilibrium, as discussed below. Noises with a broad range of α generally correspond to a wide range of non-equilibrium driven dynamical systems.
Pink noise sources include flicker noise in electronic devices. In their study of fractional Brownian motion, Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to describe 1/f α noises for which the exponent α is not an even integer, or that are fractional derivatives of Brownian (1/f 2) noise.
Description
In pink noise, there is equal energy per octave of frequency. The energy of pink noise at each frequency level, however, falls off at 3.1 dB per octave. This is in contrast to white noise which has equal energy at all frequency levels.
The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz sound loudest for a given intensity. However, humans still differentiate between white noise and pink noise with ease.
Graphic equalizers also divide signals into bands logarithmically and report power by octaves; audio engineers put pink noise through a system to test whether it has a flat frequency response in the spectrum of interest. Systems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer. Because pink noise tends to occur in natural physical systems, it is often useful in audio production. Pink noise can be processed, filtered, and/or effects can be added to produce desired sounds. Pink-noise generators are commercially available.
One parameter of noise, the peak versus average energy contents, or crest factor, is important for testing purposes, such as for audio power amplifier and loudspeaker capabilities because the signal power is a direct function of the crest factor. Various crest factors of pink noise can be used in simulations of various levels of dynamic range compression in music signals. On some digital pink-noise generators the crest factor can be specified.
Generation
Pink noise can be computer-generated by first generating a white noise signal, Fourier-transforming it, then dividing the amplitudes of the different frequency components by the square root of the frequency (in one dimension), or by the frequency (in two dimensions) etc. This is equivalent to spatially filtering (convolving) the white noise signal with a white-to-pink-filter. For a length signal in one dimension, the filter has the following form:
Matlab programs are available to generate pink and other power-law coloured noise in one or any number of dimensions.
