Biomedical Engineering Reference
In-Depth Information
Most authors require further restraints for a process to be stationary. Some define
several types of stationarity. It should be noted at this point that if a process can be
proven to be stationary in even the weakest sense, stationarity of the random process
can be assumed and analysis techniques requiring stationarity can then be conducted.
Various types of stationarity are described in the subsequent section.
7.2.1 Statistics and Stationarity
7.2.1.1
First-Order Stationary Random Process
In statistics, the general method is to obtain a measure of a distribution by calculat-
ing its moments, that is, the 1st moment about the origin is often referred to as the
“Mean” or “Average Value” about the origin, which is a measure of central tendency or
centroid. The second moment is generally the mean about some point other than the
origin, and the third moment is the variance of the distribution, which is a measure of
disbursement.
If first-order statistics are independent of time (time invariant), it is implied that
the moments of the ensemble values do not depend on time. The function associated with
the random variable describing the values taken on by the different ensemble members
at time
t
1
is identical to the function associated with the random variable at any other
time
t
2
.
First-order statistics for mean, mean square value, and variance are shown in (7.8),
(7.9), and (7.10), respectively.
∞
x
(
t
)
=
xf
x
(
t
)
dt
=
x
Mean Value
(7.8)
∞
−∞
x
2
(
t
)
x
2
f
x
(
t
)
dt
x
2
=
=
Mean Squared Value
(7.9)
∞
−∞
2
x
(
t
)
x
(
t
)]
2
f
x
(
t
)
dt
2
T
=
[
x
−
=
σ
Variance
(7.10)
−∞
7.2.1.2
Second-Order Stationary Random Process
If all second-order statistics associated with random variables
X
(
t
1
) and
X
(
t
2
) are depen-
dent at most on the time differences defined as
τ
= |
t
1
−
t
2
|
and not on the two values,
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