Digital Signal Processing Reference
In-Depth Information
2.3.3.2 Relationships between Cumulants and Moments
We can relate cumulants and moments via the following recursive rule [214]:
k
c
i
·
k
−
1
−
1
c
k
=
κ
k
−
κ
k
−
i
(2.76)
i
−
1
i
=
1
Therefore, the
k
th moment is a
k
th order polynomial built from the first
k
cumulants. For instance, we have for
k
up to 6:
κ
1
=
c
1
c
1
κ
2
=
c
2
+
c
1
κ
3
=
c
3
+
3
c
2
c
1
+
3
c
2
+
6
c
2
c
1
+
c
1
=
c
4
+
4
c
3
c
1
+
κ
4
(2.77)
10
c
3
c
1
+
15
c
2
c
1
10
c
2
c
1
=
c
5
+
5
c
4
c
1
+
10
c
3
c
2
+
+
κ
5
15
c
4
c
1
+
10
c
3
+
20
c
3
c
1
+
15
c
2
κ
6
=
c
6
+
6
c
5
c
1
+
15
c
4
c
2
+
60
c
3
c
2
c
1
+
45
c
2
c
1
+
15
c
2
c
1
+
c
1
+
Clearly, when zero-mean distributions are considered, the terms in
c
1
are
removed from (2.77). A more detailed description of these relationships can
be found in [214].
2.3.3.3 Joint Cumulants
The joint cumulant of several r.v.'s
X
1
,
...
,
X
k
is defined similarly to (2.67)
[214]
ω
1
=···=
ω
p
=
0
c
x
k
1
,
x
k
2
,
,
x
k
p
r
r
∂
ϒ(
ω
1
,
...
, ω
p
)
...
(
−
j
)
(2.78)
w
k
p
w
k
1
∂
...∂
where
ϒ(
ω
1
,
...
, ω
p
)
represents the second characteristic function of the
joint pdf
X
1
,
,
X
k
. If the variables are independent, their joint cumulant
is null, and if all
k
variables are equal, the joint cumulant is
c
k
(
...
.
In order to link the concepts we have presented with the notion of signal,
we now consider the evolution of r.v.'s in detail.
X
)
