Digital Signal Processing Reference
In-Depth Information
[ ]
(
)
=
………
[1.8]
P
F
x
,
,
x
,
t
,
,
t
1
n
1
n
The set of these three laws fully characterizes the random signal, but a partial
characterization can be obtained
via
the moments of order
M
of the signal, defined
(when these exist) by:
(
)
k
k
k
() ()
∫∫
k
(
)
E x t
1
..
x t
n
=
… ………
x
1
x
p x
,
,
x
,
t
,
,
t
dx
,
,
dx
n
1
n
n
1
n
1
n
1
n
1
n
ℜ
k
n
∫∫
∑
(
)
=
… ……
x
1
x
n
Fx
,
,
x
,
t
,
,
t
[1.9]
1
n
1
n
1
n
ℜ
with
Mk
=
i
i
It can be shown that these moments are linked in a simple way to the Taylor
series expansion of the characteristic function of the
n
-tuple
{
}
()
()
x
11
,
t
…
,
nn
x
t
defined by:
(
)
( )
T
(
) ( )
φ
u
,
…
,
u
=
φ
u
E
p
j
u
.
x
1
n
[1.10]
T
T
u
=
uu
…
,
x
=
xx
…
1
n
1
n
We see that the laws and the moments have a dependence on the considered
points {
ti,...,t
n
}
of the time axis. The separation of random signals into classes very
often refers to the nature of this dependence: if a signal has one of its characteristics
invariant by translation, or in other words, independent of the time origin, we will
call the signal stationary for this particular characteristic. We will thus speak of
stationarity in law
if:
(
)
(
)
px
,
…… … …
,
x t
,
,
,
t
=
px
,
,
x t
,
+
t
,
,
t
+
t
1
n
1
n
1
n
1
0
n
0
and of
stationarity for the moment of order M
if:
(
)
(
(
)
k
k
k
k
() ()
)
(
)
1
…
n
1
n
Ext
xt
=
Ext
+
t
..
xt
+
t
1
n
1
0
n
0
The interval
t
0
for which this type of equality is verified holds various
stationarity classes:
-
local
stationarity if this is true only for
[
]
t
∈
TT
,
,
0
0
1
- a
symptotic
stationarity if this is verified only when
0
→∞
, etc.
t
We will refer to [LAC 00] for a detailed study of these properties.
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