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Pink noise spectrum graphed logarithmically

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. 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,

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.

Pink noise as sound

10 seconds of pink noise, normalized to -1 dBFS peak

Problems listening to the file? See media help.

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.)

Pink noise (left) and white noise (right) on an FFT spectrogram with linear frequency vertical axis (on a typical audio or similar spectrum analyzer the pink noise would be flat, not downward-sloping, and the white noise rising)

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 f₁ to f₂ is proportional to log(f₂ / f₁), and if f₂ is infinity, so is the energy. Similarly, the energy of a pink noise signal would be infinite for f₁ = 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 certain range of frequencies. For f₂, there is an upper limit to the frequencies that can be measured.

One important parameter of noise, the peak versus average energy contents, or crest factor, cannot be specified for pink noise, because it depends on f₁ and therefore on the time a device is running.

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.[1]

There are no simple mathematical models to create pink noise. It is usually generated by filtering white noise.[2]

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.

Main article: Flicker noise

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 (such a measurement takes several weeks!) have not shown a ceasing of pink-noise behaviour. Therefore one could state that in electronics, noise can be pink down to f₁ = 1 / T where T is the time the device is switched on.

• White noise
• Noise (physics)
• Red noise
• Statistics
• Audio signal processing
• Self-organised criticality
• Fractal
• Colors of noise
• Crest factor
• Architectural acoustics
• Sound masking
• Brownian noise

• Dutta, P. and Horn, P. M. (1981). "Low-frequency fluctuations in solids: 1 / f noise". Reviews of Modern Physics 53: 497–516. doi:10.1103/RevModPhys.53.497. 

• Field, D. J. (1987). "Relations Between the Statistics of Natural Images and the Response Profiles of Cortical Cells". Journal of the Optical Society of America A 4: 2379–2394. 

• Gisiger, T. (2001). "Scale invariance in biology: coincidence or footprint of a universal mechanism?". Biological Reviews 76: 161–209. 

• Johnson, J. B. (1925). "The Schottky effect in low frequency circuits". Physical Review 26: 71–85. doi:10.1103/PhysRev.26.71. 

• Press, W. H. (1978). "Flicker noises in astronomy and elsewhere". Comments on Astrophysics 7: 103–119. 

• Schottky, W. (1918). "Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern". Annalen der Physik 362: 541–567. 

• Schottky, W. (1922). "Zur Berechnung und Beurteilung des Schroteffektes". Annalen der Physik 373: 157–176. 

• Keshner, M. S. (1982). "1 / f noise". Proceedings of the IEEE 70 (3): 212–218. 

• Li, W. (1996–present). A bibliography on 1 / f noise.

• Mandelbrot, B. B. and Van Ness, J. W. (1968). "Fractional Brownian motions, fractional noises and applications". SIAM Review 10 (4): 422–437. 

• Press, W. H. (1978). "Flicker noises in astronomy and elsewhere". Comments on Astrophysics 7 (4): 103–119. 

• A Bibliography on 1/f Noise
• A collection of various test tones playable online (White/Pink/Brown Noises)
• Pink noise in wave(.wav) format (1 minute)
• DSP Generation of Pink (1/f) Noise — Discussion of various algorithms, with code samples
• Noise — an open-source program that creates pink noise on the Macintosh
• The ALSA utility speaker-test under Linux defaults to producing pink noise
• Pink Noise and Sine Wave Generator utility that outputs Window's PCM audio files (*.wav) that play in your audio player