Definitions

# Pink noise

Pink noise or 1/f noise is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. For pink noise, each octave carries an equal amount of noise power. The name arises from being intermediate between white noise (1/f0) and red noise (1/f2, more commonly known as Brownian noise).

Within the scientific literature the term 1/f noise is used a little more loosely to refer to any noise with a power spectral density of the form,

$S\left(f\right) propto 1/f^alpha$
where $f$ is frequency and 0 < α < 2, with α usually close to 1. These "1/f-like" noises occur widely in nature and are a source of considerable interest in many fields.

The term flicker noise is sometimes used to refer to $1/f$ noise, although this is more properly applied only to its occurrence in electronic devices due to a direct current. Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to emphasise that the exponent of the spectrum could take non-integer values and be closely related to fractional Brownian motion, but the term is very rarely used.

## Description

There is equal energy in all octaves (or similar log bundles). In terms of power at a constant bandwidth, 1/f noise falls off at 3 dB per octave. At high enough frequencies 1/f noise is never dominant. (White noise is equal energy per hertz.)

The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive them with equal sensitivity; signals in the 2-4-kHz octave sound loudest, and the loudness of other frequencies drops increasingly, depending both on the distance from the peak-sensitivity area and on the level. 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 a "mirror image" using a graphic equalizer. Because pink noise has a tendency 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.

From a practical point of view, producing true pink noise is impossible, since the energy of such a signal would be infinite. That is, the energy of pink noise in any frequency interval from $f_1$ to $f_2$ is proportional to $log\left(f_2/f_1\right)$, and if $f_2$ is infinity, so is the energy. Similarly, the energy of a pink noise signal would be infinite for $f_1=0$. This is not a surprise, though, because a signal containing frequencies down to zero extends infinitely in time.

Practically, noise can be pink only over a specific range of frequencies. For $f_2$, there is an upper limit to the frequencies that can be measured.

One parameter of noise, the peak versus average energy contents, or crest factor, is important for testing purposes, such as for amplifier and loudspeaker capabilities. Various crest factors of pink noise can be used in simulations of various levels of dynamic range compression in music signals. A defined crest factor is also important for durability or heat tests on loudspeakers or power amplifiers, because the signal power is direct function of the crest factor. On some digital pink noise generators the crest factor can be specified because the algorithm can be adjusted to never exceed certain levels.

## Occurrence

1/f noise occurs in many physical, biological and economic systems. Some researchers describe it as being ubiquitous. In physical systems it is present in some meteorological data series, the electromagnetic radiation output of some astronomical bodies, and in almost all electronic devices (referred to as flicker noise). In biological systems, it is present in heart beat rhythms and the statistics of DNA sequences. In financial systems it is often referred to as a long memory effect. Also, it is the statistical structure of all natural images (images from the natural environment), as discovered by David Field (1987) .

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.

There are no simple mathematical models to create pink noise. It is usually generated by filtering white noise.

There are many theories of the origin of 1/f noise. Some theories attempt to be universal, while others are applicable to only a certain type of material, such as semiconductors. Universal theories of 1/f noise are still a matter of current research.

### Electronic devices

A pioneering researcher in this field was Aldert van der Ziel.

In electronics, white noise will be stronger than pink noise (flicker noise) above some corner frequency. Interestingly, there is no known lower bound to pink noise in electronics. Measurements made down to 10−6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour. Therefore one could state that in electronics, noise can be pink down to $f_1=1/T$ where $T$ is the time the device is switched on.

A pink noise source is sometimes included on analog synthesizers (although a white noise source is more common), both as a useful audio sound source for further processing, and also a source of random control voltages for controlling other parts of the synthesizer.

## References

### Notations

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• A. Chorti and M. Brookes (2007), "Resolving near-carrier spectral infinities due to $1/f$ phase noise in oscillators", ICASSP 2007, Vol. 3, 15-20 April 2007, Pages:III-1005 - III-1008, DOI 10.1109/ICASSP.2007.366852