### Matched Filtering

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)

*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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Autocorrelation