Pink noise, 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.
Acoustic pink noise sounds like a waterfall. It is often used to calibrate 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 that has equal intensity per frequency interval.
Definition Within the scientific literature, the term " noise" is sometimes used loosely to refer to any noise with a power spectral density of the form
where is frequency, and , with exponent usually close to . One-dimensional signals with are usually called pink 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.
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.
Properties
Autocorrelation Unlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows.
1D signal For pink noise constrained to continuous frequencies from to , the autocorrelation coefficient is
where is the cosine integral function.
If instead, the pink noise is approximated by a discrete sum of frequencies , the Pearson autocorrelation coefficient is
2D signal The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:
where is the Bessel function of the first kind.
Power-law spectra in multiple dimensions In a pink noise signal in any number of dimensions, the total power at each frequency, summed across all orientations, falls off inversely with frequency: . Each octave carries an equal amount of total noise power. The total power at each frequency is the power summed over a spherical shell in the Fourier domain, so it is the average power along any orientation , multiplied by the surface area of the shell: . This tells us that the average power at any orientation falls off as in one dimension, as in two dimensions (e.g. pink noise images), and as in general dimensions. The average amplitude at any orientation is the square root of the average power: , so it falls as in 1D, in 2D, and as in general dimensions.
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 (pink noise has and Brown noise has ):
+ Power‑law spectra of pink noise dimensions tot. power avg. power avg. amp. 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 standard deviation , 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 standard deviation .
Occurrence Pink noise has been discovered in the statistical fluctuations of 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 in tide and river heights, quasar light emissions, heart beat, firings of single neurons, resistivity in solid-state electronics and single-molecule conductance signals resulting in flicker noise. Pink noise describes the statistical structure of many natural images.
General noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous. 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.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.
An accessible introduction to the significance of pink noise is one given by Martin Gardner (1978) in his Scientific American column "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
The ubiquitous noise poses a "noise floor" to precision timekeeping. The derivation is based on.
Suppose that we have a timekeeping device (it could be anything from quartz oscillators, atomic clocks, and hourglasses). 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.
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 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 and wavelength , 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 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 .
The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.
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. In clinical EEG, deviations from this pink noise can be used to identify epilepsy, even in the absence of a seizure, or during the interictal state. Classic models of EEG generators suggested that dendritic inputs in gray matter were principally responsible for generating the 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 spectral density. Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings, particularly if the effects of ephaptic coupling are taken into consideration.
It has also been successfully applied to the modeling of mental states in psychology, and used to explain stylistic variations in music from different cultures and historic periods.Pareyon, G. (2011). On Musical Self-Similarity, International Semiotics Institute & University of Helsinki. 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. 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.Anger, Natalie (March 1, 2010). "Bringing New Understanding to the Director's Cut". The New York Times. Retrieved on March 3, 2010. See also original study
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 from reaction time measurement and from iterated two-alternative forced choice also produced pink noises.
Electronic devices