Pink noise

General 1/f α-like noises occur widely in nature and are a source of considerable interest in many fields.

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.

The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:[7]

Pink noise has been discovered in the statistical fluctuations of an extraordinarily diverse number of physical and biological systems (Press, 1978;[12] see articles in Handel & Chung, 1993,[13] and references therein).

Examples of its occurrence include fluctuations in tide and river heights, quasar light emissions, heart beat, firings of single neurons, resistivity in solid-state electronics and single-molecule conductance signals[14] resulting in flicker noise.

[1] General 1/f α noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.

[15] In physical systems, they are present in some meteorological data series, the electromagnetic radiation output of some astronomical bodies.

In biological systems, they are present in, for example, heart beat rhythms, neural activity, and the statistics of DNA sequences, as a generalized pattern.

[16] An accessible introduction to the significance of pink noise is one given by Martin Gardner (1978) in his Scientific American column "Mathematical Games".

Sounds in nature are not musical in that they tend to be either too repetitive (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth).

The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed that pitch and loudness fluctuations in speech and music are pink noises.

[12] The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.

[23] In brains, pink noise has been widely observed across many temporal and physical scales from ion channel gating to EEG and MEG and LFP recordings in humans.

[24] In clinical EEG, deviations from this 1/f pink noise can be used to identify epilepsy, even in the absence of a seizure, or during the interictal state.

However, recent computational models using cable theory have shown that action potential transduction along white matter tracts in the brain also generates a 1/f spectral density.

Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings, [26] particularly if the effects of ephaptic coupling are taken into consideration.

[27] It has also been successfully applied to the modeling of mental states in psychology,[28] and used to explain stylistic variations in music from different cultures and historic periods.

[29] Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of pitches, will tend towards a pink noise spectrum.

[30] Similarly, a generally pink distribution pattern has been observed in film shot length by researcher James E. Cutting of Cornell University, in the study of 150 popular movies released from 1935 to 2005.

Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.

[8][35] The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes.

[37] (Kleinpenning, de Kuijper, 1988)[38] measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range of

The noise curve at very low frequencies affects pulsar timing arrays, the European Pulsar Timing Array (EPTA) and the future International Pulsar Timing Array (IPTA); at low frequencies are space-borne detectors, the formerly proposed Laser Interferometer Space Antenna (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial Laser Interferometer Gravitational-Wave Observatory (LIGO) and its advanced configuration (aLIGO).

The analysis of individual Brownian motion trajectories also show 1/f 2 spectrum, albeit with random amplitudes.

[52] When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa.

This hypothesis also provides for an alternative paradigm to explain power law manifestations that have been attributed to self-organized criticality.

[54] In contrast, the sandpile model of self-organized criticality, which exhibits quasi-cycles of gradual stress accumulation between fast rare stress-releases, reproduces the flicker noise that corresponds to the intra-cycle dynamics.

Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept of deterministic chaos,[62] whereas the associated emergence of the long-term dynamical memory or order, i.e., 1/f and crackling noises, the Butterfly effect etc., is the consequence of the Goldstone theorem in the application to the spontaneously broken topological supersymmetry.

Pink noise is commonly used to test the loudspeakers in sound reinforcement systems, with the resulting sound measured with a test microphone in the listening space connected to a spectrum analyzer[3] or a computer running a real-time fast Fourier transform (FFT) analyzer program such as Smaart.

[66] The process of end-users burning in their headphones with pink noise to attain higher fidelity has been called an audiophile "myth".

A two-dimensional pink noise grayscale image, generated with a computer program; some fields observed in nature are characterized by a similar power spectrum [ 1 ]
A 3D pink noise image, generated with a computer program, viewed as an animation in which each frame is a 2D slice
Spectrum of a pink noise approximation on a log-log plot; power density falls off at 10 dB/decade of frequency
Relative intensity of pink noise (left) and white noise (right) on an FFT spectrogram with the vertical axis being linear frequency
The spatial filter which is convolved with a one-dimensional white noise signal to create a pink noise signal [ 7 ]
The autocorrelation (Pearson's correlation coefficient) of one-dimensional (top) and two-dimensional (bottom) pink noise signals, across distance d (in units of the longest wavelength comprising the signal); grey curves are the autocorrelations of a sample of pink noise signals (comprising discrete frequencies), and black is their average, red is the theoretically calculated autocorrelation when the signal comprises these same discrete frequencies, and blue assumes a continuum of frequencies [ 7 ]
A clock is most easily tested by comparing it with a far more accurate reference clock. During an interval of time τ , as measured by the reference clock, the clock under test advances by τy , where y is the average (relative) clock frequency over that interval.
is approximately the area under the green curve; when increases, shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; when , the combined effect of both is that
Noise curves for a selection of gravitational-wave detectors as a function of frequency