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

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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