Pink noise
Encyclopedia
Pink noise or 1/ƒ noise (sometimes also called flicker noise
) is a signal or process with a frequency spectrum
such that the power spectral density
(energy or power per Hz) is inversely proportional to the frequency
. In pink noise, each octave
carries an equal amount of noise power. The name arises from being intermediate between white noise
(1/ƒ0) and red noise (1/ƒ2) which is commonly known as Brownian noise
.
Within the scientific literature the term 1/ƒ noise is sometimes used a little more loosely to refer to any noise with a power spectral density
of the form
where ƒ is frequency and 0 < α < 2, with α usually close to 1. These "1/ƒ-like" noises occur widely in nature and are a source of considerable interest in many fields. The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction. The former (narrow sense) generally come from condensed matter systems in quasi-equilibrium, as discussed below. The latter (broader sense) generally correspond to wide range of non-equilibrium driven dynamical systems.
The term flicker noise
is sometimes used to refer to 1/ƒ 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.
There is equal energy in all octave
s (or similar log bundles). In terms of power at a constant bandwidth, 1/ƒ noise falls off at 3 dB
per octave. At high enough frequencies 1/ƒ 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 ƒ1 to ƒ2 is proportional to log(ƒ2/ƒ1), and if ƒ2 is infinity, so is the energy. Similarly, the energy of a pink noise signal would be infinite for ƒ1 = 0.
Practically, noise can be pink only over a specific range of frequencies. For ƒ2, 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 loudspeaker
s or power amplifier
s, because the signal power is a 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.
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, neural activity, and the statistics of DNA sequences. In financial systems it is often referred to as a long-term memory effect. Also, it describes the statistical structure of many natural images (images from the natural environment).
Recently, 1/f noise has also been successfully applied to the modeling of mental states in psychology.
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 1/f 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.
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/ƒ 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/ƒ noise remain a matter of current research interest.
.
In electronics, white noise
will be stronger than pink noise (flicker noise) above some corner frequency. 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.
A pink noise source is sometimes included on analog synthesizer
s (although a white noise source is more common), both as a useful audio sound source for further processing, and also as a source of random control voltages for controlling other parts of the synthesizer.
The principal sources of 1/f 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.
The explanation for the approximately 1/f spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes. 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 1/f spectrum, because d(ln(f))/df = 1/f.
Flicker noise
Flicker noise is a type of electronic noise with a 1/ƒ, or pink power density spectrum. It is therefore often referred to as 1/ƒ noise or pink noise, though these terms have wider definitions...
) is a signal or process with a frequency spectrum
Frequency spectrum
The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...
such that the power spectral density
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
(energy or power per Hz) is inversely proportional to the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
. In pink noise, each octave
Octave (electronics)
In electronics, an octave is a doubling or halving of a frequency. The term is derived from the musical octave which similarly describes such frequency ratios, but the prefix octa-, denoting eight, has no significance in physics...
carries an equal amount of noise power. The name arises from being intermediate between white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
(1/ƒ0) and red noise (1/ƒ2) which is commonly known as Brownian noise
Brownian noise
In science, Brownian noise , also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise...
.
Within the scientific literature the term 1/ƒ noise is sometimes used a little more loosely to refer to any noise with a power spectral density
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
of the form
where ƒ is frequency and 0 < α < 2, with α usually close to 1. These "1/ƒ-like" noises occur widely in nature and are a source of considerable interest in many fields. The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction. The former (narrow sense) generally come from condensed matter systems in quasi-equilibrium, as discussed below. The latter (broader sense) generally correspond to wide range of non-equilibrium driven dynamical systems.
The term flicker noise
Flicker noise
Flicker noise is a type of electronic noise with a 1/ƒ, or pink power density spectrum. It is therefore often referred to as 1/ƒ noise or pink noise, though these terms have wider definitions...
is sometimes used to refer to 1/ƒ noise, although this is more properly applied only to its occurrence in electronic devices due to a direct current
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
. Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
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
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
s (or similar log bundles). In terms of power at a constant bandwidth, 1/ƒ noise falls off at 3 dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
per octave. At high enough frequencies 1/ƒ noise is never dominant. (White noise is equal energy per hertz.)
Bark scale
The Bark scale is a psychoacoustical scale proposed by Eberhard Zwicker in 1961. It is named after Heinrich Barkhausen who proposed the first subjective measurements of loudness....
, 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
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
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 ƒ1 to ƒ2 is proportional to log(ƒ2/ƒ1), and if ƒ2 is infinity, so is the energy. Similarly, the energy of a pink noise signal would be infinite for ƒ1 = 0.
Practically, noise can be pink only over a specific range of frequencies. For ƒ2, 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
Crest factor
The crest factor or peak-to-average ratio or peak-to-average power ratio is a measurement of a waveform, calculated from the peak amplitude of the waveform divided by the RMS value of the waveform.C =...
, is important for testing purposes, such as for amplifier
Audio amplifier
An audio amplifier is an electronic amplifier that amplifies low-power audio signals to a level suitable for driving loudspeakers and is the final stage in a typical audio playback chain.The preceding stages in such a chain are low power audio amplifiers which perform tasks like pre-amplification,...
and loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
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 loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
s or power amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
s, because the signal power is a 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.
Generalization to more than one dimension
The spectrum of pink noise is 1/f only for one-dimensional signals. For two-dimensional signals, e.g., images, the spectrum is reciprocal to f2. In general, in an n-dimensional system, the spectrum is reciprocal to fn. For higher-dimensional signals it is still true (by definition) that each octave carries an equal amount of noise power. The frequency spectrum of two-dimensional signals, for instance, is also two-dimensional, and the area covered by succeeding octaves is four times as large.Occurrence
1/ƒ 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 radiationElectromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
output of some astronomical bodies, and in almost all electronic devices (referred to as flicker noise
Flicker noise
Flicker noise is a type of electronic noise with a 1/ƒ, or pink power density spectrum. It is therefore often referred to as 1/ƒ noise or pink noise, though these terms have wider definitions...
). In biological systems, it is present in heart beat rhythms, neural activity, and the statistics of DNA sequences. In financial systems it is often referred to as a long-term memory effect. Also, it describes the statistical structure of many natural images (images from the natural environment).
Recently, 1/f noise has also been successfully applied to the modeling of mental states in psychology.
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 1/f distribution pattern has been observed in film shot length by researcher James E. Cutting of Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
, in the study of 150 popular movies released from 1935 to 2005.
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/ƒ 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/ƒ noise remain a matter of current research interest.
Electronic devices
A pioneering researcher in this field was Aldert van der ZielAldert van der Ziel
Prof. Dr. Aldert van der Ziel, , was a Dutch physicist who studied electronic noise processes in materials such as semiconductors and metals.-Biography:...
.
In electronics, white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
will be stronger than pink noise (flicker noise) above some corner frequency. 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.
A pink noise source is sometimes included on analog synthesizer
Analog synthesizer
An analog or analogue synthesizer is a synthesizer that uses analog circuits and analog computer techniques to generate sound electronically. The earliest analog synthesizers in the 1920s and 1930s such as the Trautonium were built with a variety of vacuum-tube and electro-mechanical technologies...
s (although a white noise source is more common), both as a useful audio sound source for further processing, and also as a source of random control voltages for controlling other parts of the synthesizer.
The principal sources of 1/f 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.
The explanation for the approximately 1/f spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes. 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
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...
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 1/f spectrum, because d(ln(f))/df = 1/f.
See also
- Architectural acousticsArchitectural acousticsArchitectural acoustics is the science of noise control within buildings. The first application of architectural acoustics was in the design of opera houses and then concert halls. More widely, noise suppression is critical in the design of multi-unit dwellings and business premises that generate...
- Audio signal processingAudio signal processingAudio signal processing, sometimes referred to as audio processing, is the intentional alteration of auditory signals, or sound. As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain...
- Brownian noiseBrownian noiseIn science, Brownian noise , also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion, hence its alternative name of random walk noise...
- White noiseWhite noiseWhite noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
- Colors of noiseColors of noiseWhile noise is by definition derived from a random signal, it can have different characteristic statistical properties corresponding to different mappings from a source of randomness to the concrete noise. Spectral density is such a property, which can be used to distinguish different types of noise...
- Crest factorCrest factorThe crest factor or peak-to-average ratio or peak-to-average power ratio is a measurement of a waveform, calculated from the peak amplitude of the waveform divided by the RMS value of the waveform.C =...
- FractalFractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
- Flicker noiseFlicker noiseFlicker noise is a type of electronic noise with a 1/ƒ, or pink power density spectrum. It is therefore often referred to as 1/ƒ noise or pink noise, though these terms have wider definitions...
- Johnson–Nyquist noiseJohnson–Nyquist noiseJohnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage...
- Noise (physics)
- Quantum 1/f noiseQuantum 1/f noiseQuantum 1/f noise is claimed to be an intrinsic part of quantum mechanics . The model is based on the scattering of different particles off one another in solid state physics...
- Self-organised criticality
- Shot noiseShot noiseShot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...
- Sound maskingSound maskingSound masking is the addition of natural or artificial sound into an environment to cover up unwanted sound by using auditory masking. This is in contrast to the technique of active noise control...
- StatisticsStatisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
Notations
- A. Chorti and M. Brookes (2007), "Resolving near-carrier spectral infinities due to 1/ƒ phase noise in oscillators", ICASSP 2007, Vol. 3, 15–20 April 2007, Pages:III–1005 — III–1008, DOI 10.1109/ICASSP.2007.366852
External links
- Powernoise: Matlab software for generating 1/ƒ noise, or more generally, 1/ƒα noise
- A Bibliography on 1/ƒ Noise
- A collection of various test tones playable online (White/Pink/Brown Noises)
- Pink noise in wave(.wav) format (1 minute)
- Noisy, an open-source pink noise generator for Mac OS X Leopard
- The ALSA utility speaker-test under Linux defaults to producing pink noise
- The Audio Test File Generator Pink Noise and Sine Wave Generator utility that outputs Windows PCM audio files (*.wav)
- Simply Noise, a free online white, pink and brown/red noise generator, uses Flash
- 1/f noise at Scholarpedia