Stationarity

• A random signal X(t) is STATIONARY in STRICT SENSE, if its statistical properties do not depend on time, that is, X(t) and X(t+τ) have the same statistical properties (same CDFs, PDFs, second order CDFs and PDFs, etc.).
• A random signal X(t) is STATIONARY in WIDE SENSE, IFF:
  • The mean is constant (does not depend on time): $E\left[X\left(t\right)\right]=\eta$.
    
    Figure 7 Meaning of “stationarity” in the mean
  • The autocorrelation only depends on the time instants difference:
    
    $R_{xx}(t_1,t_2) = R_{xx}(t_2-t_2) = R_{xx}(\tau )$
  • The first order PDF is independent of time:
    
    $p(x,t) = \frac{\partial F_x(x,t)}{\partial x}=p_x(x)$