Biomedical Engineering Reference
In-Depth Information
Some processes that are stationary may also be categorized as Ergodic, but not
“all” stationary processes are Ergodic. Yet, “all” Ergodic processes are considered weakly
stationary. Orders of ergodicity can also be established. First-order ergodicity requires
that all first-order statistics have equal ensemble and time series values. The same is true
for second-order and weak-sense ergodicity. For most analyses techniques, weak-sense
ergodicity is sufficient.
It should be noted that only Stationary Random Processes can be classified as
either being
Ergodic
or being
Nonergodic
. In addition, if the signal (random process) is
Ergodic, then the signal is also considered to be
Stationary in the Weak Sense
. It is also
important to note that in accordance to Bendat and Piersol, that for many applications
the verification of
Weak Stationarity
justifies the assumption of
Strong Stationarity
.
For example
, one can obtain the mean value
μ
x
(
k
) and the autocorrelation function
R
x
(
λ,
k
)ofthe
k
th sample function with (7.16) and (7.17).
T
1
T
μ
x
(
k
)
=
lim
T
x
k
(
t
)
dt
(7.16)
→∞
0
T
1
T
R
x
(
λ,
k
)
=
lim
T
x
k
(
t
)
x
k
(
t
+
λ
)
dt
(7.17)
→∞
0
For the random process
{
x
(
t
)
}
to be stationary, the time averaged mean value
μ
x
(
k
), and
the autocorrelation function
R
x
(
k
) of the segments should NOT differ (statistically)
when computed over different sample functions; in that case, the random process is
considered to be Ergodic.
λ,
7.4 REVIEW OF BASIC PARAMETRIC STATISTICS
AND MOMENT-GENERATING FUNCTIONS
7.4.1 Moment-Generating Function
Given a random variable for continuous distribution as in (7.18),
≡
e
tx
=
e
tx
P
(
x
)
dx
M
(
t
)
(7.18)
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