{"history":[],"article":{"id":"e7059845-8a88-4aa2-bbda-69fd48760caa","title":"Pink noise","slug":"pink-noise","content":":::summary\n:::\n\n# \n**Pink noise**, **1/f noise**, **fractional noise** or **fractal noise** is a [signal](https://bliptext.com/articles/signal-information-theory) or process with a [frequency spectrum](https://bliptext.com/articles/[frequency](https://bliptext.com/articles/frequency)-spectrum) such that the [power spectral density](https://bliptext.com/articles/power-spectral-density) (power per [frequency](https://bliptext.com/articles/frequency) interval) is [inversely proportional](https://bliptext.com/articles/inversely-proportional) to the [frequency](https://bliptext.com/articles/frequency) of the [signal](https://bliptext.com/articles/signal-information-theory).\nIn pink noise, each [octave](https://bliptext.com/articles/octave-electronics) interval (halving or doubling in frequency) carries an equal amount of noise energy.\nPink noise sounds like a [waterfall](https://bliptext.com/articles/waterfall).\nIt is often used to tune [loudspeaker](https://bliptext.com/articles/loudspeaker) systems in [professional audio](https://bliptext.com/articles/professional-audio).\nPink noise is one of the most commonly observed signals in biological systems.\nThe name arises from the pink appearance of visible light with this power spectrum.\nThis is in contrast with [white noise](https://bliptext.com/articles/white-noise) which has equal intensity per frequency interval.\n# [A two-dimensional pink noise grayscale image, generated with a computer program; some fields observed in nature are characterized by a similar power spectrum | upload.wikimedia.org/wikipedia/commons/2/25/2D///pink///noise.png]\n\n# Definition\nWithin 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\nwhere f is frequency, and 0 < α < 2, with exponent α usually close to 1.\nOne-dimensional signals with α = 1 are usually called pink noise.\nThe following function describes a length N one-dimensional pink noise signal (i.e. a [Gaussian white noise](https://bliptext.com/articles/gaussian-white-noise) signal with zero mean and standard deviation \\sigma, 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 u (so that power, which is the square of amplitude, falls off inversely with frequency), and phases are random:\n\\chi_u are [iid](https://bliptext.com/articles/iid)[ chi-distributed variables](https://bliptext.com/articles/chi-distribution), and \\phi_u are uniform random.\nIn a two-dimensional pink noise signal, the amplitude at any orientation falls off inversely with frequency.\nA pink noise square of length N can be written as:\nGeneral 1/fα-like noises occur widely in nature and are a source of considerable interest in many fields.\nNoises with α near 1 generally come from [condensed-matter](https://bliptext.com/articles/condensed-matter-physics) systems in [quasi-equilibrium](https://bliptext.com/articles/quasi-equilibrium), as discussed below.\nNoises with a broad range of α generally correspond to a wide range of [non-equilibrium](https://bliptext.com/articles/non-equilibrium) driven [dynamical systems](https://bliptext.com/articles/dynamical-system).\nPink noise sources include [flicker noise](https://bliptext.com/articles/flicker-noise) in electronic devices.\nIn their study of [fractional Brownian motion](https://bliptext.com/articles/fractional-brownian-motion), [Mandelbrot](https://bliptext.com/articles/benoit-mandelbrot) and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to describe 1/fα noises for which the exponent α is not an even integer, or that are [fractional derivatives](https://bliptext.com/articles/fractional-derivative) of [Brownian](https://bliptext.com/articles/brownian-noise) (1/f2) noise.\n\n# Description\nIn pink noise, there is equal energy per [octave](https://bliptext.com/articles/octave) of frequency.\nThe energy of pink noise at each frequency level, however, falls off at roughly 3 [dB](https://bliptext.com/articles/decibels) per octave.\nThis is in contrast to [white noise](https://bliptext.com/articles/white-noise) which has equal energy at all frequency levels.\nThe [human auditory system](https://bliptext.com/articles/human-auditory-system), which processes frequencies in a roughly logarithmic fashion approximated by the [Bark scale](https://bliptext.com/articles/bark-scale), does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz sound [loudest](https://bliptext.com/articles/loudness) for a given intensity.\nHowever, humans still differentiate between white noise and pink noise with ease.\n[Graphic equalizers](https://bliptext.com/articles/graphic-equalizer) 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.\nSystems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer.\nBecause pink noise tends to occur in natural physical systems, it is often useful in audio production.\nPink noise can be processed, filtered, and/or effects can be added to produce desired sounds.\nPink-noise generators are commercially available.\nOne parameter of noise, the peak versus average energy contents, or [crest factor](https://bliptext.com/articles/crest-factor), is important for testing purposes, such as for [audio power amplifier](https://bliptext.com/articles/audio-power-amplifier) and [loudspeaker](https://bliptext.com/articles/loudspeaker) capabilities because the signal power is a direct function of the [crest factor](https://bliptext.com/articles/crest-factor).\nVarious crest factors of pink noise can be used in simulations of various levels of [dynamic range compression](https://bliptext.com/articles/dynamic-range-compression) in music signals.\nOn some digital pink-noise generators the crest factor can be specified.\n# [Spectrum of a pink noise approximation on a log-log plot; power density falls off at 10 dB/decade of frequency | upload.wikimedia.org/wikipedia/commons/7/77/Pink///noise///spectrum.svg]\n# [Relative intensity of pink noise (left) and white noise (right) on an FFT spectrogram with the vertical axis being linear frequency | upload.wikimedia.org/wikipedia/commons/7/76/Noise.jpg]\n\n# Generation\nPink 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.\nFor a length N signal in one dimension, the filter has the following form:\nMatlab programs are available to generate pink and other power-law coloured noise in [one](https://bliptext.com/articles/undefined) or [any number](https://bliptext.com/articles/undefined) of dimensions.\n# [The spatial filter which is convolved with a one-dimensional white noise signal to create a pink noise signal | upload.wikimedia.org/wikipedia/commons/e/ed/White///to///pink///filter.png]\n\n# Properties\n# [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 | upload.wikimedia.org/wikipedia/commons/a/a6/Pink///noise///1d///autocorrelation.png]\n\n## Power-law spectra\nThe power spectrum of pink noise is \\frac{1}{f} only for one-dimensional signals.\nFor two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as, and in d dimensions, it falls as.\nIn every case, each octave carries an equal amount of noise power.\nThe average amplitude a_\\theta and power p_\\theta of a pink noise signal at any orientation \\theta, and the total power across all orientations, fall off as some power of the frequency.\nThe following table lists these power-law frequency-dependencies for pink noise signal in different dimensions, and also for general power-law colored noise with power \\alpha (e.g.: [Brown noise](https://bliptext.com/articles/brown-noise) has \\alpha=2):\n\n## Distribution of point values\nConsider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean \\mu and sd \\sigma, then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter ).\nThen the point values of the pink noise signal will also be normally distributed, with mean \\mu and sd.\n\n## Autocorrelation\nUnlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows.\n\n### 1D signal\nThe Pearson's correlation coefficient of a one-dimensional pink noise signal (comprising discrete frequencies k) with itself across a distance d in the configuration (space or time) domain is:\nIf instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from to, the autocorrelation coefficient is:\nwhere is the [cosine integral function](https://bliptext.com/articles/trigonometric-integral).\n\n### 2D signal\nThe Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:\nwhere J_0 is the [Bessel function of the first kind](https://bliptext.com/articles/bessel-function).\n\n# Occurrence\nPink noise has been discovered in the [statistical fluctuations](https://bliptext.com/articles/statistical-fluctuations) of an extraordinarily diverse number of physical and biological systems (Press, 1978; see articles in Handel & Chung, 1993, and references therein).\nExamples of its occurrence include fluctuations in [tide](https://bliptext.com/articles/tide) and river heights, [quasar](https://bliptext.com/articles/quasar) light emissions, heart beat, firings of single [neurons](https://bliptext.com/articles/neuron), [resistivity](https://bliptext.com/articles/resistivity) in [solid-state electronics](https://bliptext.com/articles/solid-state-electronics) and single-molecule conductance signals resulting in [flicker noise](https://bliptext.com/articles/flicker-noise).\nPink noise describes the [statistical structure of many natural images](https://bliptext.com/articles/scene-statistics).\nGeneral 1/fα noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.\nIn physical systems, they are present in some [meteorological](https://bliptext.com/articles/meteorological) data series, the [electromagnetic radiation](https://bliptext.com/articles/electromagnetic-radiation) output of some astronomical bodies.\nIn biological systems, they are present in, for example, [heart beat](https://bliptext.com/articles/cardiac-cycle) rhythms, neural activity, and the statistics of [DNA sequences](https://bliptext.com/articles/dna-sequence), as a generalized pattern.\nAn accessible introduction to the significance of pink noise is one given by [Martin Gardner](https://bliptext.com/articles/martin-gardner) (1978) in his Scientific American column \"Mathematical Games\".\nIn this column, Gardner asked for the sense in which music imitates nature.\nSounds 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).\nThe 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.\nSo music is like tides not in terms of how tides sound, but in how tide heights vary.\n\n## Precision timekeeping\nThe ubiquitous 1/f noise poses a \"noise floor\" to precision timekeeping.\nThe derivation is based on.\nSuppose that we have a timekeeping device (it could be anything from [quartz oscillators](https://bliptext.com/articles/crystal-oscillator), [atomic clocks](https://bliptext.com/articles/atomic-clock), and [hourglasses](https://bliptext.com/articles/hourglass) ).\nLet its readout be a real number x(t) that changes with the actual time t. For concreteness, let us consider a quartz oscillator.\nIn a quartz oscillator, x(t) is the number of oscillations, and \\dot x(t) is the rate of oscillation.\nThe rate of oscillation has a constant component \\dot x_0and a fluctuating component \\dot x_f, so.\nBy selecting the right units for x, we can have, meaning that on average, one second of clock-time passes for every second of real-time.\nThe stability of the clock is measured by how many \"ticks\" it makes over a fixed interval.\nThe more stable the number of ticks, the better the stability of the clock.\nSo, define the average clock frequency over the interval asNote that y_k is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock.\nThe [Allan variance](https://bliptext.com/articles/allan-variance) of the clock frequency is half the mean square of change in average clock frequency:where K is an integer large enough for the averaging to converge to a definite value.\nFor example, a 2013 atomic clock achieved, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 [femtoseconds](https://bliptext.com/articles/femtosecond).\nNow we havewhere is one packet of a [square wave](https://bliptext.com/articles/square-wave) with height 1/\\tau and wavelength 2\\tau.\nLet h(t) be a packet of a square wave with height 1 and wavelength 2, then, and its Fourier transform satisfies.\nThe Allan variance is then, and the discrete averaging can be approximated by a continuous averaging: , which is the total power of the signal , or the integral of its [power spectrum](https://bliptext.com/articles/spectral-density):\nIn words, the Allan variance is approximately the power of the fluctuation after [bandpass filtering](https://bliptext.com/articles/band-pass-filter) at with bandwidth.\nFor 1/f^\\alpha fluctuation, we have for some constant C, so.\nIn particular, when the fluctuating component \\dot x_f is a 1/f noise, then is independent of the averaging time \\tau, meaning that the clock frequency does not become more stable by simply averaging for longer.\nThis contrasts with a white noise fluctuation, in which case, meaning that doubling the averaging time would improve the stability of frequency by \\sqrt 2.\nThe cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.\n# [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. | upload.wikimedia.org/wikipedia/commons/d/dd/AllanDeviation.svg]\n# [\\sigma^2(1) is approximately the area under the green curve; when \\tau 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 | upload.wikimedia.org/wikipedia/commons/5/57/Illustration///for///Allan///variance///of///1-f///noise.png]\n\n## Humans\nIn [brains](https://bliptext.com/articles/brains), pink noise has been widely observed across many temporal and physical scales from [ion channel](https://bliptext.com/articles/ion-channel) gating to [EEG](https://bliptext.com/articles/eeg) and [MEG](https://bliptext.com/articles/magnetoencephalography) and [LFP](https://bliptext.com/articles/local-field-potential) recordings in humans.\nIn clinical EEG, deviations from this 1/f pink noise can be used to identify [epilepsy](https://bliptext.com/articles/epilepsy), even in the absence of a [seizure](https://bliptext.com/articles/seizure), or during the interictal state.\nClassic models of EEG generators suggested that dendritic inputs in [gray matter](https://bliptext.com/articles/gray-matter) were principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals.\nHowever, recent computational models using [cable theory](https://bliptext.com/articles/cable-theory) have shown that [action potential](https://bliptext.com/articles/action-potential) transduction along [white matter](https://bliptext.com/articles/white-matter) tracts in the brain also generates a 1/f spectral density.\nTherefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings,\nparticularly if the effects of ephaptic coupling are taken into consideration.\nIt has also been successfully applied to the modeling of [mental states](https://bliptext.com/articles/mental-representation) in [psychology](https://bliptext.com/articles/psychology), and used to explain stylistic variations in music from different cultures and historic periods.\nRichard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of [pitches](https://bliptext.com/articles/pitch-music), will tend towards a pink noise spectrum.\nSimilarly, a generally pink distribution pattern has been observed in [film shot](https://bliptext.com/articles/shot-filmmaking) length by researcher [James E. Cutting](https://bliptext.com/articles/james-e-cutting) of [Cornell University](https://bliptext.com/articles/cornell-university), in the study of 150 popular movies released from 1935 to 2005.\nPink noise has also been found to be endemic in human response.\nGilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.\nLater, Gilden (1997) and Gilden (2001) found that time series formed from [reaction time](https://bliptext.com/articles/reaction-time) measurement and from iterated two-alternative forced choice also produced pink noises.\n\n## Electronic devices\nThe principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices.\nIn many cases the specific sources of the fluctuations are known.\nThese include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.\nThe 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.\nSince the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic \"attempt frequencies\" (e.g., 1014 Hz), the exponential factors in the [Arrhenius equation](https://bliptext.com/articles/arrhenius-equation) for the rates are large.\nRelatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates.\nIn the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because\nThere is no known lower bound to background pink noise in electronics.\nMeasurements made down to 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.\n(Kleinpenning, de Kuijper, 1988) measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range of, a range of 9.5 decades.\nA pioneering researcher in this field was [Aldert van der Ziel](https://bliptext.com/articles/aldert-van-der-ziel).\nFlicker noise is commonly used for the reliability characterization of electronic devices.\nIt is also used for gas detection in chemoresistive sensors by dedicated measurement setups.\n\n## In gravitational wave astronomy\n1/fα noises with α near 1 are a factor in [gravitational-wave astronomy](https://bliptext.com/articles/gravitational-wave-astronomy).\nThe noise curve at very low frequencies affects [pulsar timing arrays](https://bliptext.com/articles/pulsar-timing-array), the [European Pulsar Timing Array](https://bliptext.com/articles/european-pulsar-timing-array) (EPTA) and the future [International Pulsar Timing Array](https://bliptext.com/articles/international-pulsar-timing-array) (IPTA); at low frequencies are space-borne detectors, the formerly proposed [Laser Interferometer Space Antenna](https://bliptext.com/articles/laser-interferometer-space-antenna) (LISA) and the currently proposed evolved [Laser Interferometer Space Antenna](https://bliptext.com/articles/laser-interferometer-space-antenna) (eLISA), and at high frequencies are ground-based detectors, the initial [Laser Interferometer Gravitational-Wave Observatory](https://bliptext.com/articles/ligo) (LIGO) and its advanced configuration (aLIGO).\nThe characteristic strain of potential astrophysical sources are also shown.\nTo be detectable the characteristic strain of a signal must be above the noise curve.\n# [Noise curves for a selection of gravitational-wave detectors as a function of frequency | upload.wikimedia.org/wikipedia/commons/a/af/Gravitational-wave///detector///sensitivities///and///astrophysical///gravitational-wave///sources.png]\n\n## Climate dynamics\nPink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the [climate system](https://bliptext.com/articles/climate-system).\n\n## Diffusion processes\nMany time-dependent stochastic processes are known to exhibit 1/fα noises with α between 0 and 2.\nIn particular [Brownian motion](https://bliptext.com/articles/brownian-motion) has a [power spectral density](https://bliptext.com/articles/spectral-density) that equals 4D/f2, where D is the [diffusion coefficient](https://bliptext.com/articles/mass-diffusivity).\nThis type of spectrum is sometimes referred to as [Brownian noise](https://bliptext.com/articles/brownian-noise).\nThe analysis of individual Brownian motion trajectories also show 1/f2 spectrum, albeit with random amplitudes.\n[Fractional Brownian motion](https://bliptext.com/articles/fractional-brownian-motion) with [Hurst exponent](https://bliptext.com/articles/hurst-exponent) H also show 1/fα power spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5

History