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

• 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):

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