# Cross-Correlation

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. 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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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## Cross-Correlation

Definition: The circular cross-correlation of two signals and in may be defined by

(Note carefully above that '''' is an integer variable, not the constant.) The term ''cross-correlation'' comes from statistics, and what we have defined here is more properly called the ''sample cross-correlation,'' i.e., it is an estimator of the true cross-correlation which is a statistical property of the signal itself. The estimator works by averaging lagged products . The true statistical cross-correlation is the so-called expected value of the lagged products in random signals and , which may be denoted . In principle, the expected value must be computed by averaging over many realizations of the stochastic process and . That is, for each ''roll of the dice'' we obtain and for all time, and we can average across all realizations to estimate the expected value of. This is called an ''ensemble average'' across realizations of a stochastic process. If the signals are stationary (which primarily means their statistics are time-invariant), then we mayaverage across time to estimate the expected value. In other words, for stationary noise-like signals, time averages equal ensemble averages. The above definition of the sample cross-correlation is only valid for stationary stochastic processes.

The DFT of the cross-correlation is called the cross-spectral density, or ''cross-power spectrum,'' or even simply ''cross-spectrum.''

Normally in practice we are interested in estimating the true cross-correlation between two signals, not the circular cross-correlation which results naturally in a DFT setting. For this, we may define instead

where we chose (e.g. ) in order to have enough lagged products at the highest lag that a reasonably accurate average is obtained. The term ''unbiased'' refers to the fact that we are dividing the sum by rather than .

Note that instead of first estimating the cross-correlation between signals and and then taking the DFT to estimate the cross-spectral density, we may instead compute the sample cross-correlation for each block of a signal, take the DFT of each, and average the DFTs to form a final cross-spectrum estimate. This is called the periodogram method of spectral estimation.

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