Correlation
The correlation operator for two signals and
in
is defined as
![$\displaystyle \zbox {(x\star y)_n \isdef \sum_{m=0}^{N-1}\overline{x(m)} y(m+n)}
$](http://www.dsprelated.com/josimages_new/mdft/img1208.png)
We may interpret the correlation operator as
![$\displaystyle (x\star y)_n = \left<\hbox{\sc Shift}_{-n}(y), x\right>
$](http://www.dsprelated.com/josimages_new/mdft/img1209.png)
![$ \vert\vert\,x\,\vert\vert ^2=N$](http://www.dsprelated.com/josimages_new/mdft/img1210.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ y$](http://www.dsprelated.com/josimages_new/mdft/img26.png)
![$ n$](http://www.dsprelated.com/josimages_new/mdft/img80.png)
![$ n$](http://www.dsprelated.com/josimages_new/mdft/img80.png)
![$ n$](http://www.dsprelated.com/josimages_new/mdft/img80.png)
![$ \overline{x(m)}
y(m+n)$](http://www.dsprelated.com/josimages_new/mdft/img1211.png)
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The correlation operator for two signals and
in
is defined as
We may interpret the correlation operator as