Editing Pink noise (section)

== Properties ==
[[File:Pink noise 1d autocorrelation.png|thumb|upright=2|The autocorrelation (Pearson's correlation coefficient) of one-dimensional (top) and two-dimensional (bottom) pink noise signals, across distance d (in units of the longest wavelength comprising the signal); grey curves are the autocorrelations of a sample of pink noise signals (comprising discrete frequencies), and black is their average, red is the theoretically calculated autocorrelation when the signal comprises these same discrete frequencies, and blue assumes a continuum of frequencies<ref name="Das-thesis"/>]]

=== Power-law spectra ===
The power spectrum of pink noise is <math>\frac{1}{f}</math> only for one-dimensional signals. For two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as <math>\frac{1}{f^2}</math>, and in <math>d</math> dimensions, it falls as <math>\frac{1}{f^d}</math>. In every case, each octave carries an equal amount of noise power. 

The average amplitude <math>a_\theta</math> and power <math>p_\theta</math> of a pink noise signal at any orientation <math>\theta</math>, and the total power across all orientations, fall off as some power of the frequency. The following table lists these power-law frequency-dependencies for pink noise signal in different dimensions, and also for general power-law colored noise with power <math>\alpha</math> (e.g.: [[Brown noise]] has <math>\alpha=2</math>): <ref name="Das-thesis"/>

{| class="wikitable"
|+ Power-law spectra of pink noise
|-
! dimensions !! avg. amp. <math>a_\theta(f)</math> !! avg. power <math>p_\theta(f)</math> !! tot. power <math>p(f)</math>
|-
| 1 || <math>1/\sqrt{f}</math> || <math>1/f</math> || <math>1/f</math>
|-
| 2 || <math>1/f</math> || <math>1/f^2</math> || <math>1/f</math>
|-
| 3 || <math>1/f^{3/2}</math> || <math>1/f^3</math> || <math>1/f</math>
|-
| <math>d</math> || <math>1/f^{d/2}</math> || <math>1/f^d</math> || <math>1/f</math>
|-
| <math>d</math>, power <math>\alpha</math> || <math>1/f^{\alpha d /2}</math> || <math>1/f^{\alpha d}</math> || <math>1/f^{1+(\alpha-1)d}</math>
|}

=== Distribution of point values ===
Consider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean <math>\mu</math> and sd <math>\sigma</math>, then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter <math>\boldsymbol{a}</math>). Then the point values of the pink noise signal will also be normally distributed, with mean <math>\mu</math> and sd <math>\lVert \boldsymbol{a} \rVert \sigma</math>.<ref name="Das-thesis">{{cite thesis |last=Das |first=Abhranil |date=2022 |title=Camouflage detection & signal discrimination: theory, methods & experiments (corrected) |type=PhD |publisher=The University of Texas at Austin |url=http://dx.doi.org/10.13140/RG.2.2.10585.80487 |doi=10.13140/RG.2.2.10585.80487}}</ref>

=== 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]].