Autocorrelation

The autocorrelation of a signal $x$ is simply the cross-correlation of $x$ with itself:

$$(x \star x)(n) \isdef \sum_m\overline{x(m)}x(m+n).$$

From the correlation theorem, we have

$$\zbox {(x \star x) \leftrightarrow \vert X\vert^2}$$

Note that this definition of autocorrelation is appropriate for signals having finite support (nonzero over a finite number of samples). For infinite-energy (but finite-power) signals, such as stationary noise processes, we define the sample autocorrelation to include a normalization suitable for this case (see Chapter 5 and Appendix D).