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
:
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