Biomedical Engineering Reference
In-Depth Information
t
1
and
t
2
,
the implications are that the joint function is obtained for any two random
variables that describe the process at separate times. If a second-order random process
is stationary, the first-order is also stationary. Second-order statistics are described by
(7.11) and (7.12).
R
xx
(
τ
)
=
x
(
t
)
x
(
t
+
τ
)
d
τ
Autocorelation function
(7.11)
x
][[
x
(
t
L
xx
=
[
x
(
t
)
−
−
τ
)
−
x
]
Covariance
(7.12)
7.2.1.3
Wide-Sense Stationary
A random process is stationary in the wide sense if the mean from the first-order statistics
and the autocorrelation from the second-order statistics are time invariant. Therefore,
a process considered to be of second-order stationarity is also stationary in the wide
sense. However, wide-sense stationarity does not imply second-order stationarity since
covariance is not a statistic included in wide-sense stationarity.
7.2.2 Ensemble Method
7.2.2.1
Weakly Stationary
By definition, an ensemble is said to be a Stationary Random process as
Weakly Stationary
or
Stationary in the Wide Sense
, when the following two conditions are met:
1.
The mean
,
μ
x
(
t
)
1
, or the first moment of the random processes at all times,
t
,
and
2.
The autocorrelation function
,
R
xx
(
t
1
,
t
1
), or joint moment between
the values of the random process at two different times do not vary as time
t
1
varies.
+
λ
The conditions are shown in (7.13) and (7.14).
N
1
N
μ
x
(
t
1
)
=
μ
=
lim
N
x
k
(
t
1
)
(7.13)
x
→∞
k
=
1
N
1
N
R
xx
(
t
1
,
t
1
+
λ
)
=
R
xx
(
λ
)
=
lim
N
x
k
(
t
1
)
x
k
(
t
1
+
λ
)
(7.14)
→∞
k
=
1
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