====== Random processes ======

===== Properties of random processes =====

  * Expected value of signal at time $t$:

$$ \mu_X(t) = E[X(t)] $$

  * Autocorrelation of signal at times $t_1$ and $t_2$:

$$ R_{XX}(t_1, t_2) = E[X(t_1)X(t_2)] $$

  * Autocovariance of signal at times $t_1$ and $t_2$:

$$ C_{XX}(t_1, t_2) = E[\tilde{X}(t_1)\tilde{X}(t_2)] = R_{XX}(t_1, t_2) - \mu_X(t_1)\mu_X(t_2) $$

where $\tilde{X}(t) = X(t) - \mu_X(t)$

===== Properties of two random processes =====

  * Cross-correlation of $X(t_1)$ and $Y(t_2):

$$ R_{XY}(t_1, t_2) = E[X(t_1)Y(t_2)] $$

  * Cross-covariance of $X(t_1)$ and $Y(t_2):

$$ C_{XY}(t_1, t_2) = E[\tilde{X}(t_1)\tilde{Y}(t_2)] $$