Editing Cross-correlation (section)

===Properties===
{{unordered list
| The cross-correlation of functions <math>f(t)</math> and <math>g(t)</math> is equivalent to the [[convolution]] (denoted by <math>*</math>) of <math>\overline{f(-t)}</math> and <math>g(t)</math>. That is:
: <math>[f(t) \star g(t)](t) = [\overline{f(-t)} * g(t)](t).</math>
| <math>[f(t) \star g(t)](t) = [\overline{g(t)} \star \overline{f(t)}](-t).</math>
| If <math>f</math> is a [[Hermitian function]], then <math>f \star g = f * g.</math>
| If both <math>f</math> and <math>g</math> are Hermitian, then <math>f \star g = g \star f</math>.
| <math>\left(f \star g\right) \star \left(f \star g\right) = \left(f \star f\right) \star \left(g \star g\right)</math>.
| Analogous to the [[convolution theorem]], the cross-correlation satisfies
: <math>\mathcal{F}\left\{f \star g\right\} = \overline{\mathcal{F} \left\{f\right\}} \cdot \mathcal{F}\left\{g\right\},</math>

where <math>\mathcal{F}</math> denotes the [[Fourier transform]], and an <math>\overline{f}</math> again indicates the complex conjugate of <math>f</math>, since <math>\mathcal{F}\left\{\overline{f(-t)}\right\}=\overline{\mathcal{F}\left\{f(t)\right\}}</math>. Coupled with [[fast Fourier transform]] algorithms, this property is often exploited for the efficient numerical computation of cross-correlations<ref name="KAP">{{cite book|doi=10.1109/ICSPCS.2015.7391783|isbn=978-1-4673-8118-5|chapter = GPU implementation of cross-correlation for image generation in real time|title = 2015 9th International Conference on Signal Processing and Communication Systems (ICSPCS)|pages = 1–6|year = 2015|last1 = Kapinchev|first1 = Konstantin|last2 = Bradu|first2 = Adrian|last3 = Barnes|first3 = Frederick|last4 = Podoleanu|first4 = Adrian|s2cid=17108908 }}</ref> (see [[circular cross-correlation]]).
| The cross-correlation is related to the [[spectral density]] (see [[Wiener–Khinchin theorem]]).
| The cross-correlation of a convolution of <math>f</math> and <math>h</math> with a function <math>g</math> is the convolution of the cross-correlation of <math>g</math> and <math>f</math> with the kernel <math>h</math>:
: <math>g \star \left(f * h\right) = \left(g \star f\right) * h</math>.
}}