NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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Definition: The circular cross-correlation
of two signals 
and 
in
may be defined by
(Note carefully above that ''
'' is an
integer variable, not the constant
.) The
term ''cross-correlation'' comes from statistics, and what
we have defined here is more properly called the ''sample
cross-correlation,'' i.e., it is an estimator of the true
cross-correlation which is a statistical property of the signal
itself. The estimator works by averaging lagged products
. The true statistical cross-correlation is the
so-called expected value of the lagged products in random
signals and
, which
may be denoted 
. In principle, the expected value must be computed by averaging
over many realizations of the stochastic
process and
. That is,
for each ''roll of the dice'' we obtain 
and 
for all time, and we can average
across all realizations to estimate the expected
value of. This is called an ''ensemble average'' across
realizations of a stochastic process. If the signals are
stationary (which primarily means their statistics are
time-invariant), then we mayaverage across time to
estimate the expected value. In other words, for stationary
noise-like
signals, time averages equal ensemble averages. The above
definition of the sample cross-correlation is only valid for
stationary stochastic processes.
The DFT
of the cross-correlation is called the cross-spectral
density, or ''cross-power spectrum,'' or even simply ''cross-spectrum.''
Normally in practice we are interested in estimating the true
cross-correlation between two signals, not the circular
cross-correlation which results naturally in a DFT setting. For
this, we may define instead
where we chose 
(e.g. 
) in order to have enough lagged products at the
highest lag that a reasonably accurate average is obtained. The
term ''unbiased'' refers to the fact that we are dividing the sum
by
rather than 
.
Note that instead of first estimating the cross-correlation
between signals
and and then
taking the DFT to estimate the cross-spectral density, we may
instead compute the sample cross-correlation for each block of a
signal, take the DFT of each, and average the DFTs to form a
final cross-spectrum estimate. This is called the periodogram
method of spectral estimation.
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