also for general power-law colored noise with power (e.g.: Brown noise has ): 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 (equivalent to spatially filtering it 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 (comprising discrete frequencies ) with itself across a distance in the configuration (space or time) 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 (Press, 1978; see articles in Handel & Chung, 1993, and references therein). 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 (1978) 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 (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth). The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed