Pink noise

Pink noise, 1/f 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 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 that 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

S ( f ) ∝ 1 f α , {\displaystyle S(f)\propto {\frac {1}{f^{\alpha }}},}

where f is frequency, and 0 < α < 2, with exponent α usually close to 1. One-dimensional signals with α = 1 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 kmin to kmax, the autocorrelation coefficient is

r ( d ) = Ci ( 2 π k max d N ) − Ci ( 2 π k min d N ) log ⁡ k max k min , {\displaystyle r(d)={\frac {{\textrm {Ci}}\left({\frac {2\pi k_{\textrm {max}}d}{N}}\right)-{\textrm {Ci}}\left({\frac {2\pi k_{\textrm {min}}d}{N}}\right)}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},}

where Ci(x) is the cosine integral function.

If instead, the pink noise is approximated my a discrete sum of frequencies k , the Pearson autocorrelation coefficient is

r ( d ) = ∑ k 1 k cos ⁡ 2 π k d N ∑ k 1 k . {\displaystyle r(d)={\frac {\sum _{k}{\frac {1}{k}}\cos {\frac {2\pi kd}{N}}}{\sum _{k}{\frac {1}{k}}}}.}

2D signal The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:

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