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