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## Cross-Correlation

Definition:Thecircular cross-correlationof two signals and in may be defined by

(Note carefully above that “” is an integer variable, not the constant.) The term “cross-correlation” comes fromstatistics, and what we have defined here is more properly called the “sample cross-correlation,” i.e., it is anestimatorof the true cross-correlation which is a statistical property of the signal itself. The estimator works byaveraging lagged products. The true statistical cross-correlation is the so-calledexpected valueof the lagged products in random signals and , which may be denoted . In principle, the expected value must be computed by averaging over manyrealizationsof the stochastic process and . That is, for each “roll of the dice” we obtain and forall 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 arestationary(which primarily means their statistics aretime-invariant), then we mayaverage across timeto 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 DFTsto form a final cross-spectrum estimate. This is called theperiodogram methodof spectral estimation.