# Cross-Correlation

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

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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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