Stationarity and Ergodic Random Processes - Signal Processing of Random Physiological Signals - page 49

CHAPTER 7

Stationarity and Ergodic
Random Processes

7.1 INTRODUCTION
In Chapter 3, signals were classified as either deterministic or random. Random signals
can be further subdivided and classified as stationary and nonstationary. Properties nec-
essary to a stationary process were provided for first-order, second-order, wide-sense,
and strictly stationary. Two different methods of determining the functions necessary
for a random process to be classified as a “Stationary Process,” which include ensemble
and extension in time, are discussed in this chapter. In addition, the necessary conditions
for a random process to be an “Ergodic Process” are also presented. Nonparametric tests
and examples are also included to provide the reader with the necessary information and
tools to identify or classify random processes as being a “Stationary Process” and/or an
“Ergodic Process.”
In qualifying data, one must first determine whether the signal is deterministic or
random. If a signal can be predicted exactly for a particular time span, it is said to be
deterministic. Examples of deterministic signals are given in (7.1), (7.2), and (7.3).

x ( t ) = 10 sin 2 π t Sine wave (7.1)

x ( t ) = 1 , t > 0 , and Unit step (7.2)

x ( t ) = 0 , t ≥ 0

e − t

x ( t )

= 1 − , ≥ 0 , and Exponential Response (7.3)

t

x ( t ) = 0 , t < 0