**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Applications of Cross-Correlation

The cross-correlation function is used extensively in

pattern recognitionandsignal detection. We know 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 which we are looking for plus some additive measurement noise. Then the projection of onto is

since the projection of any specific signal onto random, zero-mean noise is close to zero. Another term for this process is calledmatched filtering. The impulse response of the ''matched filter'' for a signal is given by . By time reversing , we transform theconvolution implemented by filtering into a cross-correlation operation.