Random Processes - Random Signals and Processes Primer with MATLAB - page 397

6.7 Ergodic Processes

6.7.1 Time Averaging

In the previous discussion, we observed processes (all realizations) in one or more
points. The obtained average values, like mean value and autocorrelation function
are ensemble average values.
Next we consider time average values obtained by averaging particular
realizations of the process in time. Similarly, as we introduced the notation E {}
for the ensemble averaging, we introduce the notation A {} for the averaging of time.
A time average of a function g ( x ( t )) is defined as:

T=

ð

2

1
T

AgxðtðÞ f g ¼ lim

T!1

gxðtð d t: (6.136)

T= 2

Two time average values: mean time value and time autocorrelation function
have a special importance. Those mean values are obtained observing a realization
x ( t ) of the process in time.
A mean time value is defined as:

T= 2

ð

1
T

t

Axðtfg¼ lim
T!1

xðtÞ d t ¼ xðtÞ : (6.137)

T= 2

Similarly, we can define time autocorrelation function for a realization x ( t ) as:

T= 2

ð

1
T

t

R xx ðtÞ¼AxðtÞxðt þ tÞ f g ¼ xðtÞxðt þ tÞ ¼ lim

T!1

xðtÞxðt þ tÞ d t; (6.138)

T= 2

t presents the averaging of a realization x ( t ) of the process X ( t ) in time.

where xðtÞ

6.7.2 What Is Ergodicity?

A process is said to be ergodic if all of its statistical averages are equal to its
corresponding time averages. From ( 6.137 ), it is clear that for an ergodic process,
any time average cannot be a function of time. In other words, in ergodic processes
the corresponding statistical averages cannot be functions of time. Knowing that
only stationary processes possess such characteristics, we can conclude that an