4.2 Stationarity

Stationarity in a Random Process implies that its statistical characteristics do not change with time . Put another way, if one were to observe a stationary random process at some time $t$ it would be impossible to distinguish the statistical characteristics at that time from those at some other time $t^{\prime }$ .

Strict sense stationarity (sss)

Choose a Random Vector of length $N$ from a Random Process:

Wide sense (weak) stationarity (wss)

If we are only interested in the properties of moments up to 2nd order (mean, autocorrelation, covariance, ...), which isthe case for many practical applications, a weaker form of stationarity can be useful:

$X(t)$ is defined to be Wide Sense Stationary (or Weakly Stationary) iff:

• The mean value is independent of $t$ , for all $t$
  $(X(t))=\mu$
• Autocorrelation depends only upon $\tau =t_2-t_1$ , for all $t_1$
  $(X(t_1)X(t_2))=(X(t_1)X(t_1+\tau ))=r_{XX}(\tau )$