The cross-correlation of a signal with itself gives the autocorrelation function of that signal:

$\displaystyle r_{x}(l) \isdef r_{xx}(l) = E\{\overline{x(n)}x(n+l)\}$ (C.25)

Note that the autocorrelation function is Hermitian:

$\displaystyle r_x(-l) = \overline{r_x(l)}

When $ x$ is real, its autocorrelation is symmetric. More specifically, it is real and even.

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Sample Autocorrelation
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Cross-Power Spectral Density