Stationarity and Ergodic Random Processes - Signal Processing of Random Physiological Signals - page 54

7.2.2.2 Strictly Stationary
If one could collect all the higher moments and all the joint moments, then one would have
a complete family of probability distribution functions describing the process. In the case
where all possible moments and joint moments are time invariant, the random process
is Strongly Stationary or Strictly Stationary . If a random process is strictly stationary, it
must be stationary for any lower case.

7.3 EXTENSION-IN-TIME METHOD
The second method to test for stationarity, which deals with those random processes
derived from the “Extension-in-Time Method.” The extension-in-time method requires
dividing a single long signal into segments of equal length of time, then the mean squared
value (magnitude) of each segment is obtained. If the signal is a continuous analog signal,
the average magnitude for the segment is given by (7.15).

t 1

X 1 ( t ) = x ( t ) dt (7.15)

0

If the signal were a discrete sampled signal, then the segment average value would be
obtained by dividing the sum of the data points in a segment by the total number of data
points in the segment. From the extension-in-time point of view, a stationary random
process is one in which the statistics measured from observations made on one system
over an extended period of time are independent of the time in which observed time
interval is located.

7.3.1 Ergodic Random Process
The previous section discussed how the properties of a random process can be determined
by computing ensemble averages at specific instances of time. Often, it is possible to
describe the properties of a stationary random process by computing time averages over
specific sample functions in the ensemble. However, for the case of the extension-in-
time method, the random process is referred to as an Ergodic process and the test is for
Ergodicity .