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Autocorrelation
The crosscorrelation of a signal with itself gives its autocorrelation:
The autocorrelation function is Hermitian:
When
is real, its autocorrelation is
real and
even
(symmetric about lag zero).
The unbiased crosscorrelation similarly reduces to an unbiased
autocorrelation when :

(8.2) 
The DFT of the true autocorrelation function
is the (sampled)
power spectral density (PSD), or power spectrum, and may
be denoted
The complete (not sampled) PSD is
, where the
DTFT is defined in Appendix
B (it's just an
infinitely long DFT). The DFT of
thus provides a samplebased
estimate of the PSD:
^{8.10}
We could call
a ``sampled sample power spectral
density''.
At lag zero, the autocorrelation function reduces to the average
power (mean square) which we defined in §5.8:
Replacing ``correlation'' with ``covariance'' in the above definitions
gives corresponding zeromean versions. For example, we may define
the sample circular crosscovariance as
where
and
denote the means of
and
,
respectively. We also have that
equals the sample
variance of the signal
:
Previous: Unbiased CrossCorrelationNext: Matched Filtering
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.