Digital Signal Processing Reference
In-Depth Information
∞
X
(
ω
)
p
X
(
x
)
exp
(
j
ω
x
)
dx
−∞
E
exp
)
(
j
ω
x
(2.63)
(
)
If
ω
is expanded in a Taylor series about the origin, we get to [230]
X
∞
κ
k
k
k
X
(
ω
)
!
(
j
ω
)
(2.64)
k
=
0
where κ
k
is the
k
-order moment. Thus, one can obtain the
k
-order moment
using the following expression
ω
=
0
Another important statistical measure is given by the cumulants. Cumu-
lants are tailored by specific relationships between the moments of a random
variable in order to reveal certain aspects of its pdf as well as to present
some useful properties for statistical processing. The cumulants are gener-
ated by the second characteristic function or cumulant-generating function,
defined by
k
∂
(
)
ω
X
k
κ
k
=
(
−
j
)
ω
k
∂
ϒ
X
(
ω
)
ln [
X
(
ω
)
]
(2.65)
The Taylor series of
ϒ
X
(
ω
)
around the origin can be written as
∞
c
k
k
k
ϒ
X
(
ω
)
!
(
j
ω
)
(2.66)
k
=
0
and
ω
=
0
k
∂
ϒ
X
(
ω
)
k
c
k
=
(
−
j
)
(2.67)
ω
k
∂
is the
k
-order cumulant
c
k
[230,244].
We have described the first and second characteristic functions for the
real case. For complex r.v.'s, a straightforward extension is possible, so that
the characteristic function becomes [13]
ω, ω
∗
)
X
,
X
∗
(
exp
j
ω
x
∗
+
dx dx
∗
∞
ω
∗
x
x
,
x
∗
)
p
X
,
X
∗
(
2
−∞
E
exp
j
ω
x
∗
+
ω
∗
x
(2.68)
2
