Editing Autocorrelation (section)

===Properties===
In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases. These properties hold for [[Stationary process#Weak or wide-sense stationarity|wide-sense stationary processes]].<ref>{{cite book|last1=Proakis|first1=John|title=Communication Systems Engineering (2nd Edition)|date=August 31, 2001|publisher=Pearson|isbn=978-0130617934|page=168|edition=2}}</ref>

* A fundamental property of the autocorrelation is symmetry, <math>R_{ff}(\tau) = R_{ff}(-\tau)</math>, which is easy to prove from the definition. In the continuous case,
** the autocorrelation is an [[even function]] <math>R_{ff}(-\tau) = R_{ff}(\tau)</math> when <math>f</math> is a real function, and
** the autocorrelation is a [[Hermitian function]] <math>R_{ff}(-\tau) = R_{ff}^*(\tau)</math> when <math>f</math> is a [[complex function]].
* The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay <math>\tau</math>, <math>|R_{ff}(\tau)| \leq R_{ff}(0)</math>.<ref name=Gubner/>{{rp|p.410}} This is a consequence of the [[rearrangement inequality]]. The same result holds in the discrete case.
* The autocorrelation of a [[periodic function]] is, itself, periodic with the same period.
* The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all <math>\tau</math>) is the sum of the autocorrelations of each function separately.
* Since autocorrelation is a specific type of [[cross-correlation]], it maintains all the properties of cross-correlation.
* By using the symbol <math>*</math> to represent [[convolution]] and <math>g_{-1}</math> is a function which manipulates the function <math>f</math> and is defined as <math>g_{-1}(f)(t)=f(-t)</math>, the definition for <math>R_{ff}(\tau)</math> may be written as:<!--
--><math display=block>R_{ff}(\tau) = (f * g_{-1}(\overline{f}))(\tau)</math>