Random Processes - Random Signals and Processes Primer with MATLAB - page 398

ergodic process must be stationary. However, the opposite is not true (i.e., there are
stationary processes which are not ergodic processes). As an example of this,
consider the process of DC voltages, shown in Fig. 6.11 . This process is obviously
stationary because the statistical characteristics are independent of the time instants
in which the process is observed. However, because the amplitudes of the
realizations are different, the time averages will vary from one realization to
another. Therefore, the process is stationary but not ergodic.
The mutual relations between stationarity and ergodicity can be illustrated, as
shown in Fig. 6.12 .
The ergodicity makes it possible to describe a process knowing only one realization.
This is of practical interest because usually one has knowledge of only one time function
of a process and, based on ergodicity, can find a statistical average of the process by
finding the time average of just one realization of the process.
6.7.3 Explanation of Ergodicity
Ergodicity can be explained using the example of die rolling. If we consider an
experiment in which one die is rolled many times, then we intuitively can expect the
same result as in an experiment in which a large number of dice are rolled only once.
Similarly, we can expect the same result if the measurement of one single source
of random voltage is repeated many times. This can be seen as the simultaneous
measurements of the random voltages of a large number of independent, identical
random voltage sources. The set of all voltages generated by the independent
sources is an ensemble or process while the particular measured random voltages
are the realizations of the process.
Fig. 6.11 Example of a
stationary process that is not
ergodic