### Autocorrelation

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

(3.24) |

From the

*correlation theorem*, we have

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
is that of a *lossless allpass filter*
[263] if and only if
.
In other words, the autocorrelation of the impulse-response of every
allpass filter is impulsive.

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Correlation Theorem for the DTFT