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

$\displaystyle (x \star x)(n) \isdefs \sum_m\overline{x(m)}x(m+n).$ (3.24)

From the correlation theorem, we have

$\displaystyle \zbox {(x \star x) \;\longleftrightarrow\;\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 6 and Appendix C).

From the autocorrelation theorem we have that a digital-filter impulse-response $ h(n)$ is that of a lossless allpass filter [263] if and only if $ h\star h = \delta \,\leftrightarrow\, 1$ . In other words, the autocorrelation of the impulse-response of every allpass filter is impulsive.

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Correlation Theorem for the DTFT