Autocorrelation
The
cross-correlation 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 cross-correlation 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 sample-based
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 zero-mean versions. For example, we may define
the
sample circular cross-covariance as
where

and

denote the means of

and

,
respectively. We also have that

equals the sample
variance of the signal

:
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