Digital Signal Processing Reference
In-Depth Information
In this case, the cumulant-generating function is given by
ln
X
,
X
∗
(
ω, ω
∗
)
ϒ
Y
(
ω
)
(2.69)
2.3.3.1 Properties of Cumulants
Cumulants possess a number of properties that are interesting from a signal
processing standpoint. Some of these properties are described in the sequel.
•
Invariance and equivariance
The first-order cumulant is equivariant, while the others are invari-
ant to shifts, i.e.,
c
1
(
X
+
α
)
=
c
1
(
X
)
+
α
(2.70)
c
k
(
X
+
α
)
=
c
k
(
X
)
for an arbitrary constant α.
•
Homogeneity
The
k
-order cumulant is homogeneous of
k
degree. Thus, for the real
case
α
k
c
k
(
α
X
)
=
·
c
k
(
X
)
(2.71)
In the complex case, the
k
-order cumulant is defined by
X
,
X
∗
)
=
,
X
∗
,
,
X
∗
c
k
(
c
k
(
X
,
,
X
s
terms
...
...
)
∀
s
+
q
=
k
(2.72)
q
terms
According to (2.72), the homogeneity property for a complex r.v. is
given by [14,173]
·
α
∗
q
α
X
, α
X
∗
)
=
(
s
X
,
X
∗
)
c
k
(
α
)
·
c
k
(
∀
s
+
q
=
k
(2.73)
Hence, for even-order cumulants, we may consider
s
=
q
, so that the
homogeneity condition becomes
k
c
k
(
α
Y
)
=|
α
|
·
c
k
(
Y
)
(2.74)
•
Additivity
If
X
and
Y
are statistically independent r.v.'s, the following relation
holds:
c
k
(
X
+
Y
)
=
c
k
(
X
)
+
c
k
(
Y
)
(2.75)
