Autocorrelation

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

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 5 and Appendix C).

From the autocorrelation theorem we have that a digital-filter impulse-response is that of a lossless allpass filter [247] if and only if . In other words, the autocorrelation of the impulse-response of every allpass filter is impulsive.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.