Stationarity and Ergodic Random Processes - Signal Processing of Random Physiological Signals - page 50

By definition, a random signal is an unpredictable signal; therefore, there is always
some element of chance associated with it. Examples of random signals are given in (7.4),
(7.5), and (7.6).

x ( t ) = 10 sin (2 π t + θ ) (7.4)

where θ is a random variable uniformly distributed between 0 and 2 π .

x ( t ) = A sin (2 π t + θ ) (7.5)

where θ and A are independent random variables with known distributions.

x ( t ) = A (7.6)

where A is a noise-like signal with no particular deterministic structure.
Random signals are further divided into those signals that are classified as
“Stationary” and those that are classified as “Nonstationary.” Only stationary random
processes will be discussed, since available standard statistical analysis techniques are
applicable only to stationary random processes.
A random process is “Stationary” if its statistical properties (mean and autocorrela-
tion) are “Time Invariant.” The two methods used in obtaining the statistical properties
of any random process include the following:

1. Ensemble method : Where the assumption is that a large number of systems

equipped with identical instruments are available and that a set of instantaneous
values from multiple channels are obtained at the same time.

2. Extension-in-time method (Ergodic Testing) : Reduces the number of observa-

tions on a system to a single long record taken over an extended period of
time.

7.2 ENSEMBLE METHOD TO TEST
FOR STATIONARITY
Parameters derived from the Ensemble method will be expressed as x ( t ). As seen in
Fig. 7.1, parameters derived from the Ensemble method, such as the ensemble mean,