Many stationary random processes are also Ergodic. For an Ergodic Random Process we can exchange Ensemble Averages for Time Averages. This is equivalent to assuming that our ensemble of random signals is just composed of all possible time shifts of a single signal Xt X t .

Recall from our previous discussion of Expectation that the expectation of a function of a random variable is given by This result also applies if we have a random function g. g . of a deterministic variable such as tt. Hence Because tt is linearly increasing, the pdf fTt fT t is uniform over our measurement interval, say -T T to TT, and will be 12T 1 2 T to make the pdf valid (integral = 1). Hence If we wish to measure over all time, then we take the limit as T→∞ T .

This leads to the following results for Ergodic WSS random processes:

• Mean Ergodic:
  EXt=∫-∞∞xf X ( t ) xdx=limT→∞12T∫-TTXtdt X t x x f X ( t ) x T 1 2 T t T T X t (4)
• Correlation Ergodic:
  r X X τ=EXtXt+τ=∫-∞∞∫-∞∞x1x2f X ( t ) , X ( t + τ ) x1x2dx1dx2=limT→∞12T∫-TTXtXt+τdt r X X τ X t X t τ x2 x1 x1 x2 f X ( t ) , X ( t + τ ) x1 x2 T 1 2 T t T T X t X t τ (5)
  and similarly for other correlation or covariance functions.

Ergodicity greatly simplifies the measurement of WSS processes and it is often assumed when estimating moments (or correlations) for such processes.

In almost all practical situations, processes are stationary only over some limited time interval (say T1 T1 to T2 T2 ) rather than over all time. In that case we deliberately keep the limits of the integral finite and adjust f X ( t ) f X ( t ) accordingly. For example the autocorrelation function is then measured using This avoids including samples of XX which have incorrect statistics, but it can suffer from errors due to limited sample size.