I'm trying to understand the similarities/differences between the following definitions for cross correlation)

Signal Processing: (or it's discrete equivalent) $$R_{XY}(\tau) = \int_{-\infty}^{+\infty} x^*(t) y(t+ \tau) dt$$

Random Processeses$$R_{XY}(t_1, t_2) = E\begin{Bmatrix} X_{t_1} Y^*_{t_2} \end{Bmatrix}$$

Are these related?

  • 1
    $\begingroup$ Yes, but one is deterministic and the other is probabilistic. They both are measures of similarity. $\endgroup$
    – MBaz
    Mar 17, 2021 at 22:16
  • $\begingroup$ See this question for some ideas. $\endgroup$ Mar 18, 2021 at 2:59

1 Answer 1


If the process is stationary it is ergodic, then time averages are equal to probabilistic averages.


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