"Fractal noise" redirects here. For the novel, see
Fractal Noise.
Pink noise, 1⁄f noise, 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.[2] It is often used to tune
loudspeaker systems in
professional audio.[3] Pink noise is one of the most commonly observed signals in biological systems.[4]
The name arises from the pink appearance of visible light with this power spectrum.[5] This is in contrast with
white noise which has equal intensity per frequency interval.
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.[6]
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:[7]
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:[7]
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.[8] 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,[9]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,[10] 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 roughly 3
dB per octave. This is in contrast to
white noise which has equal energy at all frequency levels.[11]
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. [7] 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:[7]
Matlab programs are available to generate pink and other power-law coloured noise in
one or
any number of dimensions.
Properties
Power-law spectra
The power spectrum of pink noise is only for one-dimensional signals. For two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as , and in dimensions, it falls as . In every case, each octave carries an equal amount of noise power.
The average amplitude and power of a pink noise signal at any orientation , 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 (e.g.:
Brown noise has ): [7]
Power-law spectra of pink noise
dimensions
avg. amp.
avg. power
tot. power
1
2
3
, power
Distribution of point values
Consider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean and sd , then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter ). Then the point values of the pink noise signal will also be normally distributed, with mean and sd .[7]
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 ) with itself across a distance in the configuration (space or time) domain is:[7]
If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from to , the autocorrelation coefficient is:[7]
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".[17] 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.[18][19] So music is like tides not in terms of how tides sound, but in how tide heights vary.
The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.[12] The derivation is based on.[20]
Suppose that we have a timekeeping device (it could be anything from
quartz oscillators,
atomic clocks, and
hourglasses[21]). Let its readout be a real number that changes with the actual time . For concreteness, let us consider a quartz oscillator. In a quartz oscillator, is the number of oscillations, and is the rate of oscillation. The rate of oscillation has a constant component and a fluctuating component , so . By selecting the right units for , we can have , 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 as
Note that is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock[note 1].
The
Allan variance of the clock frequency is half the mean square of change in average clock frequency:
where is an integer large enough for the averaging to converge to a definite value.
For example, a 2013 atomic clock[22] 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.
Now we have
where is one packet of a
square wave with height and wavelength . Let be a packet of a square wave with height 1 and wavelength 2, then , and its Fourier transform satisfies .
The 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:
In words, the Allan variance is approximately the power of the fluctuation after
bandpass filtering at with bandwidth .
For fluctuation, we have for some constant , so . In particular, when the fluctuating component is a 1/f noise, then is independent of the averaging time , 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 , meaning that doubling the averaging time would improve the stability of frequency by .[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]
Humans
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.[25] Classic models of EEG generators suggested that dendritic inputs in
gray matter were principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals. 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]
It has also been successfully applied to the modeling of
mental states in
psychology,[27] and used to explain stylistic variations in music from different cultures and historic periods.[28] 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.[29] 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.[30]
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.[31] Later, Gilden (1997) and Gilden (2001) found that time series formed from
reaction time measurement and from iterated two-alternative forced choice also produced pink noises.[32][33]
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.[8][34] 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.[35] 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., 1014 Hz), the exponential factors in the
Arrhenius equation for 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
There is no known lower bound to background pink noise in electronics. Measurements made down to 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.[36] (Kleinpenning, de Kuijper, 1988)[37] 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.
Flicker noise is commonly used for the reliability characterization of electronic devices.[39] It is also used for gas detection in chemoresistive sensors [40] by dedicated measurement setups.[41]
Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the
climate system.[43][44]
Diffusion processes
Many time-dependent stochastic processes are known to exhibit 1/f α noises with α between 0 and 2. In particular
Brownian motion has a
power spectral density that equals 4D/f 2,[45] where D is the
diffusion coefficient. This type of spectrum is sometimes referred to as
Brownian noise. Interestingly, the analysis of individual Brownian motion trajectories also show 1/f 2 spectrum, albeit with random amplitudes.[46]Fractional Brownian motion with
Hurst exponentH also show 1/f α power spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5<H<1).[47]
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 as
semiconductors. 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 the
central limit theorem of statistics.[48] The
Tweedie convergence theorem[49] describes the convergence of certain statistical processes towards a family of statistical models known as the
Tweedie distributions. These distributions are characterized by a variance to mean
power law, that have been variously identified in the ecological literature as
Taylor's law[50] and in the physics literature as fluctuation scaling.[51] 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.[48] Both of these effects can be shown to be the consequence of
mathematical convergence such as how certain kinds of data will converge towards the
normal distribution under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain
power law manifestations that have been attributed to
self-organized criticality.[52]
In
supersymmetric theory of stochastics,[59] an approximation-free theory of
stochastic differential equations, 1/f noise is one of the manifestations of the spontaneous breakdown of topological
supersymmetry. This supersymmetry is an intrinsic property of all stochastic differential equations and its meaning is the preservation of the continuity of the
phase space by continuous time dynamics. Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept of
deterministic chaos,[60] 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.
Audio testing
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. The sound system plays pink noise while the audio engineer makes adjustments on an
audio equalizer to 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.[61] Pink noise is still used by audio system contractors[62] and by computerized sound systems which incorporate an automatic equalization feature.[63]
In manufacturing, pink noise is often used as a
burn-in signal for
audio amplifiers and other components, to determine whether the component will maintain performance integrity during sustained use.[64] The process of end-users burning in their
headphones with pink noise to attain higher fidelity has been called an
audiophile "myth".[65]
^Downey, Allen (2012).
Think Complexity. O'Reilly Media. p. 79.
ISBN978-1-4493-1463-7. Visible light with this power spectrum looks pink, hence the name.
^Josephson, Brian D. (1995). "A trans-human source of music?" in (P. Pylkkänen and P. Pylkkö, eds.) New Directions in Cognitive Science, Finnish Artificial Intelligence Society, Helsinki; pp. 280–285.
^
abGardner, M. (1978). "Mathematical Games—White and brown music, fractal curves and one-over-f fluctuations". Scientific American. 238 (4): 16–32.
doi:
10.1038/scientificamerican0478-16.
^
abVoss, R. F.; Clarke, J. (1978). "1/f noise" in music: Music from 1/f noise". Journal of the Acoustical Society of America. 63 (1): 258–263.
Bibcode:
1978ASAJ...63..258V.
doi:
10.1121/1.381721.
^Norton, M. P. (2003). Fundamentals of noise and vibration analysis for engineers. Karczub, D. G. (Denis G.) (2nd ed.). Cambridge, UK: Cambridge University Press.
ISBN9780511674983.
OCLC667085096.
Chorti, A.; Brookes, M. (2007). "Resolving near-carrier spectral infinities due to 1/f phase noise in oscillators". 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07. Vol. 3. pp. III–1005–III–1008.
doi:
10.1109/ICASSP.2007.366852.
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