Stochastic Signals and Systems
Random Variables
Virginia Tech
Fall 2008
Characteristic Function
The characteristic function of the rv
X
is, except for the sign of
j
, the Fourier transform of its probability density function,
Φ
X
(
ω
) =
E
[
e
j
ω
X
] =
Z
∞
∞
f
X
(
x
)
e
j
ω
x
dx
.
From the Fourier transform inversion formula, the probability
density function of
X
is given by
f
X
(
x
) =
1
2
π
Z
∞
∞
Φ
X
(
ω
)
e

j
ω
x
d
ω
Note: As we shall see later, the characteristic function of the
sum of a number of independent random variables is simply
related to the characteristic functions of each.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Characteristic Function
If
X
is a discrete random variable, the characteristic function is
given by
Φ
X
(
ω
) =
X
k
p
X
(
x
k
)
e
j
ω
x
k
(Note that the characteristic function in this case is a periodic
function of
ω
with period 2
π
.)
The pdf can be obtained from the characteristic function by
using
p
X
(
k
) =
1
2
π
Z
2
π
0
Φ
X
(
ω
)
e

j
ω
k
d
ω
k
=
0
,
±
1
,
±
2
, . . .
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 DASILVER
 Laplace, Probability theory, probability density function, Characteristic function

Click to edit the document details