Pink noise

Pink noise, noise, fractional noise or fractal noise is a signal or process with a frequency spectrum such that the power spectral density is inversely proportional to the frequency of the signal. In pink noise, each octave interval carries an equal amount of noise energy.
Pink noise sounds like a waterfall. It is often used to tune 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 which 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
where is frequency, and, with exponent usually close to 1. One-dimensional signals with are usually called pink noise.
The following function describes a length one-dimensional pink noise signal, as a sum of sine waves with different frequencies, whose amplitudes fall off inversely with the square root of frequency , and phases are random:
are independently and identically chi-distributed variables, and 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 can be written as:
General -like noises occur widely in nature and are a source of considerable interest in many fields. Noises with near 1 generally come from condensed-matter systems in quasi-equilibrium, as discussed below. Noises with a broad range of 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, Mandelbrot and Van Ness proposed the name fractional noise to describe noises for which the exponent is not an even integer, or that are fractional derivatives of Brownian 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.

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, or by the frequency etc. This is equivalent to spatially filtering the white noise signal with a white-to-pink-filter. For a length signal in one dimension, the filter has the following form:
Matlab programs are available to generate pink and other power-law coloured noise in or of dimensions.
One efficient algorithm for generation is the Voss–McCartney algorithm, an efficient method to generate discrete-time pink noise. It sums multiple independent random sequences, each updated at different rates, to approximate the 1/f power spectral density. Lower-frequency components are updated less frequently than higher-frequency components.
A simple pseudocode implementation is:

n_streams = number of random streams
streams =
output =
for i in range:
for j in range:
if i % 0:
streams = random
output.append

Each stream is updated at intervals that are powers of two, ensuring that slower-changing streams contribute low-frequency content and faster-changing streams contribute high-frequency content.

Properties

Power-law spectra of amplitude and power

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, 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 :

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 sd, then multiplying its spectrum with a filter. Then the point values of the pink noise signal will also be normally distributed, with mean and sd.

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 with itself across a distance in the configuration domain is:
If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from to, the autocorrelation coefficient is:
where is the cosine integral function.

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.

Occurrence

Pink noise has been discovered in the statistical fluctuations of an extraordinarily diverse number of physical and biological systems. 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 1/f^α 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.
An accessible introduction to the significance of pink noise is one given by Martin Gardner 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 or too chaotic. The answer to this question was given in a statistical sense by Voss and Clarke, 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 1/f noise poses a "noise floor" to precision timekeeping. The derivation is based on.
Suppose that we have a timekeeping device. 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 asNote 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 havewhere 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.
The cause of the noise floor is often traced to particular electronic components within the oscillator feedback.