Digital Signal Processing Reference
In-Depth Information
f
X
1
X
2
ðx
1
; x
2
Þ
. This means that using the inverse two-dimensional Fourier transform,
one can obtain the joint PDF from its joint characteristic function, as shown in the
following expression:
1
1
1
ð
2
pÞ
f
X
1
X
2
ðo
1
; o
2
Þ
e
jðo
1
x
1
þo
2
x
2
Þ
d
o
1
d
o
2
:
f
X
1
X
2
ðx
1
; x
2
Þ¼
(3.196)
2
1
1
If random variables
X
1
and
X
2
are independent, then their joint density is equal to
the product of the marginal densities. Thus the joint characteristic function (
3.195
)
becomes:
1
1
e
jo
1
x
1
e
jo
2
x
2
f
X
1
ðx
1
Þf
X
2
ðx
2
Þ
d
x
1
d
x
2
¼
f
X
1
X
2
ðo
1
; o
2
Þ¼
1
1
1
1
e
jo
1
x
1
f
X
1
ðx
1
Þ
d
x
1
e
jo
2
x
2
f
X
2
ðx
2
Þ
d
x
2
¼ f
X
1
ðo
1
Þf
X
2
ðo
2
Þ:
(3.197)
1
1
Therefore, for the independent random variables, the joint characteristic function
is equal to the product of the
marginal characteristic functions
. The reverse is also
true, that is if the joint characteristic function is equal to the product of marginal
characteristic functions, then the corresponding random variables are independent.
The marginal characteristic functions are obtained by making
o
1
or
o
2
equal to
zero in the joint characteristic function, as demonstrated in (
3.198
):
f
X
1
ðo
1
Þ¼f
X
1
X
2
ðo
1
;
0
Þ;
f
X
2
ðo
2
Þ¼f
X
1
X
2
ð
0
; o
2
Þ:
(3.198)
3.5.2 Characteristic Function of the Sum
of the Independent Variables
Consider the sum of two random variables
X
1
and
X
2
,
Y ¼ X
1
þ X
2
:
(3.199)
The characteristic function of the variable
Y
, according to (2.386) and the
definition (
3.194
), is equal to:
f
Y
ðoÞ¼Ef
e
joðX
1
þX
2
Þ
g¼f
X
1
X
2
ðo; oÞ:
(3.200)
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