The
cross-correlation function is used extensively in
pattern
recognition and
signal detection. We know from Chapter
5
that projecting one signal onto another is a means of measuring how
much of the second signal is present in the first. This can be used
to ``detect'' the presence of known signals as components of more
complicated signals. As a simple example, suppose we record

which we think consists of a signal

that we are looking for
plus some additive measurement
noise 
. That is, we assume the
signal model

. Then the projection of

onto

is
(recalling §
5.9.9)

since the projection of random, zero-mean
noise 
onto

is small
with probability one. Another term for this process is
matched filtering. The
impulse response of the ``matched
filter'' for a real signal

is given by

.
8.11 By time-reversing

, we transform the
convolution implemented by filtering into a
sliding cross-
correlation operation between the input signal

and
the sought signal

. (For complex known signals

, the matched
filter is

.) We detect occurrences of

in

by
detecting peaks in

.
In the same way that
FFT convolution is faster than direct convolution
(see Table
7.1), cross-correlation and matched filtering are
generally carried out most efficiently using an FFT algorithm (Appendix
A).
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