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
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.
Normally in practice we are interested in estimating the true 