Cross-Correlation

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Cross-Correlation

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Cross-Correlation



Definition: The circular cross-correlation of two signals $x$ and $y$ in ${\ may be defined by

\

(Note carefully above that “$l$” is an integer variable, not the constant$1$.) 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 $x$ and $y$, which may be denoted ${\. In principle, the expected value must be computed by averaging $(n) y(n+l)$ over many realizations of the stochastic process $x$ and $y$. That is, for each “roll of the dice” we obtain $x(\ and $y(\ for all time, and we can average$x(n) y(n+l)$ across all realizations to estimate the expected value of$x(n) y(n+l)$. 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 $L<<N$ (e.g. $L=\) 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$N-l$ rather than $N$.

Note that instead of first estimating the cross-correlation between signals$x$ and $y$ 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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