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)

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 .^8.8 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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Order a Hardcopy of Mathematics of the DFT

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.