Pink noise

For the album by Laura Mvula, seePink Noise (album).

This article is about 1/f noise. For quantum 1/f noise, seequantum 1/f noise.

"Fractal noise" redirects here. For the novel, seeFractal Noise.

A two-dimensional pink noisegrayscaleimage, generated with a computer program; some fields observed in nature are characterized by a similar power spectrum

A 3D pink noise image, generated with a computer program, viewed as an animation in which each frame is a 2D slice

Pink noise,1⁄fnoise,fractional noiseorfractal noiseis asignalor process with afrequency spectrumsuch that thepower spectral density(power per frequency interval) isinversely proportionalto thefrequencyof the signal. In pink noise, eachoctaveinterval (halving or doubling in frequency) carries an equal amount of noise energy.

Pink noise sounds like awaterfall.It is often used to tuneloudspeakersystems inprofessional 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 withwhite noisewhich 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

$$S(f)\propto {\frac {1}{f^{\alpha }}},$$

wherefis 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$N$one-dimensional pink noise signal (i.e. aGaussian white noisesignal 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:

$$h(x)=\sigma {\sqrt {\frac {N}{2}}}\sum _{u}{\frac {\chi _{u}}{u}}\sin({\frac {2\pi ux}{N}}+\phi _{u}),\quad \chi _{u}\sim \chi (2),\quad \phi _{u}\sim U(0,2\pi ).$$

$\chi _{u}$areiidchi-distributed variables,and$\phi _{u}$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$N$can be written as: $$h(x,y)={\frac {\sigma N}{\sqrt {2}}}\sum _{u,v}{\frac {\chi _{uv}}{\sqrt {u^{2}+v^{2}}}}\sin \left({\frac {2\pi }{N}}(ux+vy)+\phi _{uv}\right),\quad \chi _{uv}\sim \chi (2),\quad \phi _{uv}\sim U(0,2\pi ).$$

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 fromcondensed-mattersystems inquasi-equilibrium,as discussed below.Noises with a broad range of α generally correspond to a wide range ofnon-equilibriumdrivendynamical systems.

Pink noise sources includeflicker noisein electronic devices. In their study offractional Brownian motion,Mandelbrotand Van Ness proposed the namefractional noise(sometimes since calledfractal noise) to describe 1/f^αnoises for which the exponent α is not an even integer,or that arefractional derivativesofBrownian(1/f²) noise.

Description

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) andwhite noise(right) on anFFTspectrogramwith the vertical axis being linear frequency

In pink noise, there is equal energy peroctaveof frequency. The energy of pink noise at each frequency level, however, falls off at roughly 3dBper octave. This is in contrast towhite noisewhich has equal energy at all frequency levels.

Thehuman auditory system,which processes frequencies in a roughly logarithmic fashion approximated by theBark scale,does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz soundloudestfor a given intensity. However, humans still differentiate between white noise and pink noise with ease.

Graphic equalizersalso 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, orcrest factor,is important for testing purposes, such as foraudio power amplifierandloudspeakercapabilities 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 ofdynamic range compressionin music signals. On some digital pink-noise generators the crest factor can be specified.

Generation

The spatial filter which is convolved with a one-dimensional white noise signal to create a pink noise signal

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$N$signal in one dimension, the filter has the following form:

$$a(x)={\frac {1}{N}}\left[1+{\frac {1}{\sqrt {N/2}}}\cos \pi (x-1)+2\sum _{k=1}^{N/2-1}{\frac {1}{\sqrt {k}}}\cos {{\frac {2\pi k}{N}}(x-1)}\right].$$

Matlab programs are available to generate pink and other power-law coloured noise inoneorany numberof dimensions.

Properties

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

Power-law spectra

The power spectrum of pink noise is${\frac {1}{f}}$only for one-dimensional signals. For two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as${\frac {1}{f^{2}}}$,and in$d$dimensions, it falls as${\frac {1}{f^{d}}}$.In every case, each octave carries an equal amount of noise power.

The 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. The 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 noisehas$\alpha =2$):

Power-law spectra of pink noise

dimensions	avg. amp.$a_{\theta }(f)$	avg. power$p_{\theta }(f)$	tot. power$p(f)$
1	$1/{\sqrt {f}}$	$1/f$	$1/f$
2	$1/f$	$1/f^{2}$	$1/f$
3	$1/f^{3/2}$	$1/f^{3}$	$1/f$
$d$	$1/f^{d/2}$	$1/f^{d}$	$1/f$
$d$,power$\alpha$	$1/f^{\alpha d/2}$	$1/f^{\alpha d}$	$1/f^{1+(\alpha -1)d}$

Distribution of point values

Consider 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${\boldsymbol {a}}$). Then the point values of the pink noise signal will also be normally distributed, with mean$\mu$and sd$\lVert {\boldsymbol {a}}\rVert \sigma$.

Autocorrelation

Unlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows.

1D signal

The 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: $$r(d)={\frac {\sum _{k}{\frac {\cos {\frac {2\pi kd}{N}}}{k}}}{\sum _{k}{\frac {1}{k}}}}.$$ If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from$k_{\textrm {min}}$to$k_{\textrm {max}}$,the autocorrelation coefficient is: $$r(d)={\frac {{\textrm {Ci}}({\frac {2\pi k_{\textrm {max}}d}{N}})-{\textrm {Ci}}({\frac {2\pi k_{\textrm {min}}d}{N}})}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},$$ where${\textrm {Ci}}(x)$is thecosine integral function.

2D signal

The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as: $$r(d)={\frac {\sum _{k}{\frac {J_{0}({\frac {2\pi kd}{N}})}{k}}}{\sum _{k}{\frac {1}{k}}}},$$ where$J_{0}$is theBessel function of the first kind.

Occurrence

Pink noise has been discovered in thestatistical fluctuationsof an extraordinarily diverse number of physical and biological systems (Press, 1978;see articles in Handel & Chung, 1993,and references therein). Examples of its occurrence include fluctuations intideand river heights,quasarlight emissions, heart beat, firings of singleneurons,resistivityinsolid-state electronicsand single-molecule conductance signalsresulting inflicker noise.Pink noise describes thestatistical structure of many natural images.

General 1/f^αnoises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.In physical systems, they are present in somemeteorologicaldata series, theelectromagnetic radiationoutput of some astronomical bodies. In biological systems, they are present in, for example,heart beatrhythms, neural activity, and the statistics ofDNA sequences,as a generalized pattern.

An accessible introduction to the significance of pink noise is one given byMartin Gardner(1978) in hisScientific Americancolumn "Mathematical Games".In this column, Gardner asked for the sense in which music imitates nature. 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.So music is like tides not in terms of how tides sound, but in how tide heights vary.

Precision timekeeping

Main article:Allan variance

The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.The derivation is based on.

A clock is most easily tested by comparing it with afar more accuratereference clock. During an interval of timeτ,as measured by the reference clock, the clock under test advances byτy,whereyis the average (relative) clock frequency over that interval.

Suppose that we have a timekeeping device (it could be anything fromquartz oscillators,atomic clocks,andhourglasses). Let its readout be a real number$x(t)$that changes with the actual time$t$.For concreteness, let us consider a quartz oscillator. In a quartz oscillator,$x(t)$is the number of oscillations, and${\dot {x}}(t)$is the rate of oscillation. The rate of oscillation has a constant component${\dot {x}}_{0}$and a fluctuating component${\dot {x}}_{f}$,so${\textstyle {\dot {x}}(t)={\dot {x}}_{0}+{\dot {x}}_{f}(t)}$.By selecting the right units for$x$,we can have${\dot {x}}_{0}=1$,meaning that on average, one second of clock-time passes for every second of real-time.

The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval$[k\tau,(k+1)\tau ]$as$$y_{k}={\frac {1}{\tau }}\int _{k\tau }^{(k+1)\tau }{\dot {x}}(t)dt={\frac {x((k+1)\tau )-x(k\tau )}{\tau }}$$Note that$y_{k}$is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock.

TheAllan varianceof the clock frequency is half the mean square of change in average clock frequency:$$\sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}$$where$K$is an integer large enough for the averaging to converge to a definite value. For example, a 2013 atomic clockachieved$\sigma (25000{\text{ seconds}})=1.6\times 10^{-18}$,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 40femtoseconds.

Now we have$$y_{k}-y_{k-1}=\int _{\mathbb {R} }g(k\tau -t){\dot {x}}_{f}(t)dt=(g\ast {\dot {x}}_{f})(k\tau )$$where$g(t)={\frac {-1_{[0,\tau ]}(t)+1_{[-\tau,0]}(t)}{\tau }}$is one packet of asquare wavewith height$1/\tau$and wavelength$2\tau$.Let$h(t)$be a packet of a square wave with height 1 and wavelength 2, then$g(t)=h(t/\tau )/\tau$,and its Fourier transform satisfies${\mathcal {F}}[g](\omega )={\mathcal {F}}[h](\tau \omega )$.

The Allan variance is then$\sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{2}}{\overline {(g\ast {\dot {x}}_{f})(k\tau )^{2}}}$,and the discrete averaging can be approximated by a continuous averaging:${\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}\approx {\frac {1}{K\tau }}\int _{0}^{K\tau }{\frac {1}{2}}(g\ast {\dot {x}}_{f})(t)^{2}dt$,which is the total power of the signal$(g\ast {\dot {x}}_{f})$,or the integral of itspower spectrum:

$\sigma ^{2}(1)$is approximately the area under the green curve; when$\tau$increases,$S[g](\omega )$shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; when$S[{\dot {x}}_{f}](\omega )\propto \omega ^{-\alpha }$,the combined effect of both is that$\sigma ^{2}(\tau )\propto \tau ^{\alpha -1}$

$$\sigma ^{2}(\tau )\approx \int _{0}^{\infty }S[g\ast {\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[g](\omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[h](\tau \omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega$$In words, the Allan variance is approximately the power of the fluctuation afterbandpass filteringat$\omega \sim 1/\tau$with bandwidth$\Delta \omega \sim 1/\tau$.

For$1/f^{\alpha }$fluctuation, we have$S[{\dot {x}}_{f}](\omega )=C/\omega ^{\alpha }$for some constant$C$,so$\sigma ^{2}(\tau )\approx \tau ^{\alpha -1}\sigma ^{2}(1)\propto \tau ^{\alpha -1}$.In particular, when the fluctuating component${\dot {x}}_{f}$is a 1/f noise, then$\sigma ^{2}(\tau )$is independent of the averaging time$\tau$,meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case$\sigma ^{2}(\tau )\propto \tau ^{-1}$,meaning that doubling the averaging time would improve the stability of frequency by${\sqrt {2}}$.

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

Humans

Inbrains,pink noise has been widely observed across many temporal and physical scales fromion channelgating toEEGandMEGandLFPrecordings in humans.In clinical EEG, deviations from this 1/f pink noise can be used to identifyepilepsy,even in the absence of aseizure,or during the interictal state.Classic models of EEG generators suggested that dendritic inputs ingray matterwere principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals. However, recent computational models usingcable theoryhave shown thataction potentialtransduction alongwhite mattertracts 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.

It has also been successfully applied to the modeling ofmental statesinpsychology,and used to explain stylistic variations in music from different cultures and historic periods.Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale ofpitches,will tend towards a pink noise spectrum.Similarly, a generally pink distribution pattern has been observed infilm shotlength by researcherJames E. CuttingofCornell University,in the study of 150 popular movies released from 1935 to 2005.

Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.Later, Gilden (1997) and Gilden (2001) found that time series formed fromreaction timemeasurement and from iterated two-alternative forced choice also produced pink noises.

Electronic devices

Main article:Flicker noise

The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.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.Since the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10¹⁴Hz), the exponential factors in theArrhenius equationfor the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because$\textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.$

There is no known lower bound to background pink noise in electronics. Measurements made down to 10⁻⁶Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.(Kleinpenning, de Kuijper, 1988)measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range of$[10^{-5.5}\mathrm {Hz},10^{4}\mathrm {Hz} ]$,a range of 9.5 decades.

A pioneering researcher in this field wasAldert van der Ziel.

Flicker noise is commonly used for the reliability characterization of electronic devices.It is also used for gas detection in chemoresistive sensorsby dedicated measurement setups.

In gravitational wave astronomy

Noise curves for a selection ofgravitational-wave detectorsas a function of frequency

1/f^αnoises with α near 1 are a factor ingravitational-wave astronomy.The noise curve at very low frequencies affectspulsar timing arrays,theEuropean Pulsar Timing Array(EPTA) and the futureInternational Pulsar Timing Array(IPTA); at low frequencies are space-borne detectors, the formerly proposedLaser Interferometer Space Antenna(LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initialLaser Interferometer Gravitational-Wave Observatory(LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve.

Climate dynamics

Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in theclimate system.

Diffusion processes

Many time-dependent stochastic processes are known to exhibit 1/f^αnoises with α between 0 and 2. In particularBrownian motionhas apower spectral densitythat equals 4D/f²,whereDis thediffusion coefficient.This type of spectrum is sometimes referred to asBrownian noise.Interestingly, the analysis of individual Brownian motion trajectories also show 1/f²spectrum, albeit with random amplitudes.Fractional Brownian motionwithHurst exponentHalso show 1/f^αpower spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5<H<1).

Origin

See also:Supersymmetry § Supersymmetry in dynamical systems

There are many theories about the origin of pink noise. Some theories attempt to be universal, while others apply to only a certain type of material, such assemiconductors.Universal theories of pink noise remain a matter of current research interest.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to thecentral limit theoremof statistics.TheTweedie convergence theoremdescribes the convergence of certain statistical processes towards a family of statistical models known as theTweedie distributions.These distributions are characterized by a variance to meanpower law,that have been variously identified in the ecological literature asTaylor's lawand in the physics literature asfluctuation scaling.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.Both of these effects can be shown to be the consequence ofmathematical convergencesuch as how certain kinds of data will converge towards thenormal distributionunder the central limit theorem. This hypothesis also provides for an alternative paradigm to explainpower lawmanifestations that have been attributed toself-organized criticality.

There are various mathematical models to create pink noise. Althoughself-organised criticalityhas been able to reproduce pink noise insandpilemodels, these do not have aGaussian distributionor other expected statistical qualities.It can be generated on computer, for example, by filtering white noise,inverse Fourier transform,or by multirate variants on standard white noise generation.

Insupersymmetric theory of stochastics,an approximation-free theory ofstochastic differential equations,1/fnoise is one of the manifestations of the spontaneous breakdown of topologicalsupersymmetry.This supersymmetry is an intrinsic property of all stochastic differential equations and its meaning is the preservation of the continuity of thephase spaceby continuous time dynamics. Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept ofdeterministic chaos,whereas the associated emergence of the long-term dynamical memory or order, i.e., 1/fandcracklingnoises, theButterfly effectetc., is the consequence of theGoldstone theoremin the application to the spontaneously broken topological supersymmetry.

Audio testing

Pink noise is commonly used to test the loudspeakers insound reinforcement systems,with the resulting sound measured with a testmicrophonein the listening space connected to aspectrum analyzeror a computer running a real-timefast Fourier transform(FFT) analyzer program such asSmaart.The sound system plays pink noise while the audio engineer makes adjustments on anaudio equalizerto obtain the desired results. Pink noise is predictable and repeatable, but it is annoying for a concert audience to hear. Since the late 1990s, FFT-based analysis enabled the engineer to make adjustments using pre-recorded music as the test signal, or even the music coming from the performers in real time.Pink noise is still used by audio system contractorsand by computerized sound systems which incorporate an automatic equalization feature.

In manufacturing, pink noise is often used as aburn-insignal foraudio amplifiersand other components, to determine whether the component will maintain performance integrity during sustained use.The process of end-users burning in theirheadphoneswith pink noise to attain higher fidelity has been called anaudiophile"myth".

See also

• Architectural acoustics
• Audio signal processing
• Brownian noise
• White noise
• Colors of noise
• Crest factor
• Fractal
• Flicker noise
• Johnson–Nyquist noise
• Noise (electronics)
• Quantum 1/f noise
• Self-organised criticality
• Shot noise
• Sound masking
• Statistics

References

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External links

• Coloured Noise: Matlab toolbox to generate power-law coloured noise signals of any dimensions.
• Powernoise: Matlab software for generating 1/fnoise, or more generally, 1/f^αnoise
• 1/f noise at Scholarpedia
• White Noise Definition Vs Pink Noise