Statistical Description of Signals

Realisations of random process $S$ : Ensemble of random signals $s_{k} (t)$
For each $t$ , the linear mean is :

E {s (t_{1})} = lim_{M - > \infty} \frac{1}{M} \sum_{k = 1}^{M} s_{k} (t_{1})

similarly and generally for any function $F$ :

E {F [s (t_{1})]} = lim_{M - > \infty} \frac{1}{M} \sum_{k = 1}^{M} F [s_{k} (t_{1})]

Stationary process: All ensemble averages are time invariant

i.e. $E {F [s (t_{1})]} = E {s (t_{1} + t_{0})}$ for all $t_{0}$
$E {F [s (t_{1}), s (t_{2})]} = E {s (t_{1} + t_{0}), s (t_{2} + t_{0})}$ for all $t_{0}$
...
Therefore, the mean of a stationary process is:

E {s (t_{1})} = E {s (t)} = m_{s}

The autocorrelation function of a stationary process:

φ_{s s} (τ) = E {s (t) s (t + τ)}

Ergodic process

"Ergodic" means the same behaviour averaged over time as averaged over the space. So, an ergodic process show time averages equal to ensemble averages.