Editing Pink noise (section)

=== 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 <math>k</math>) with itself across a distance <math>d</math> in the configuration (space or time) domain is:<ref name="Das-thesis"/>
<math display="block">r(d)=\frac{\sum_k \frac{\cos \frac{2 \pi k d}{N} }{k}}{\sum_k \frac{1}{k}}.</math>
If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from <math>k_\textrm{min}</math> to <math>k_\textrm{max}</math>, the autocorrelation coefficient is:<ref name="Das-thesis"/>
<math display="block">r(d)=\frac{\textrm{Ci}(\frac{2 \pi k_\textrm{max}d}{N} )-\textrm{Ci}(\frac{2 \pi k_\textrm{min}d}{N} )}{\log \frac{k_\textrm{max}}{k_\textrm{min}}},</math>
where <math>\textrm{Ci}(x)</math> is the [[Trigonometric integral#Cosine integral|cosine integral function]].

==== 2D signal ====
The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:<ref name="Das-thesis"/>
<math display="block">r(d)=\frac{\sum_k \frac{J_0 (\frac{2 \pi k d}{N})}{k}}{\sum_k \frac{1}{k}},</math>
where <math>J_0</math> is the [[Bessel function#Bessel functions of the first kind|Bessel function of the first kind]].