Cross-Correlation

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

$$r_{xy}(l) \isdef E\{\overline{x(n)}y(n+l)\}$$

I.e., it is the expected value of the lagged products in random signals $x$ and $y$ . In principle, the expected value must be computed by averaging $x(n) y(n+l)$ over many realizations of the stochastic process $x$ and $y$ . That is, for each ``roll of the dice'' we obtain $x(\cdot)$ and $y(\cdot)$ 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 means their statistics are time-invariant), then we may replace ensemble averaging by averaging across time. In other words, for stationary stochastic processes, time averages equal ensemble averages.