4.3 Ergodicity

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 timeshifts of a single signal $X(t)$ .

Recall from our previous discussion of Expectation that the expectation of a function of a random variable is given by

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

• Mean Ergodic:
  $(X(t))=\int_{()} \,d x$∞ ∞ x f X ( t ) x T ∞ 1 2 T t T T X t
• Correlation Ergodic:
  $r_{XX}(\tau )=(X(t)X(t+\tau ))=\int_{()} \,d x_2$∞ ∞ x 1 ∞ ∞ x 1 x 2 f X ( t ) , X ( t + τ ) x 1 x 2 T ∞ 1 2 T t T T X t X t τ
  and similarly for other correlation or covariance functions.

In almost all practical situations, processes are stationary only over some limited time interval (say $T_1$ to $T_2$ ) rather than over all time. In that case we deliberately keep the limits of the integral finite and adjust $f_{X\left(t\right)}$ accordingly. For example the autocorrelation function is then measured using