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)
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).
FIR System Identification