Applications of Cross-Correlation

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Applications of Cross-Correlation

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 recognition and signal 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 $x(n)$ which we think consists of a signal$s(n)$ which we are looking for plus some additive measurement noise$e(n)$. Then the projection of $x$ onto $s$ is

\

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

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