Digital Signal Processing Reference
In-Depth Information
The joint output density function can be obtained from (
5.50
) and (
5.51
):
x
1
¼ f
1
f
X
0
1
X
2
ð
x
1
;
x
2
Þ
Jðx
1
; x
2
Þ
f
Xy
ðx; yÞ¼
;
(5.52)
ðx; yÞ
x
2
¼ f
1
ðx; yÞ
where
J
is a Jacobian of the transformation (
5.51
):
x
1
0
x
0
1
þ x
2
x
2
x
0
1
þ x
2
p
p
1
x
0
1
þ x
2
1
x
:
J ¼
¼
p
¼
(5.53)
x
2
x
0
1
þ x
2
x
1
x
0
1
þ x
2
Note that the Jacobian of (
5.53
) is positive for all values of
x
(see (
5.51
)).
From Fig.
5.6
, we have:
x
1
0
¼ x
cos
y:
(5.54)
Placing (
5.51
) and (
5.54
) into (
5.50
), we arrive at:
x
2
2
xA
cos
y þ A
2
2
s
2
1
2
ps
2
e
f
X
1
0
X
2
ðx; yÞ¼
:
(5.55)
Finally, from (
5.52
), (
5.53
), and (
5.55
), we get the joint density function of
X
and
y
:
for
x
0
x
2
2
xA
cos
y þ A
2
2
s
2
x
2
ps
2
exp
f
Xy
ðx; yÞ¼
; p y p:
(5.56)
Note that the joint density function (
5.56
) cannot be represented as a product of
the marginal densities, indicating that the variables
X
and
y
are dependent.
The marginal PDF of the variable
X
is:
2
4
3
5
:
x
2
þ A
2
2
s
2
xA
cos
y
s
2
ð
ð
p
p
x
1
2
p
s
2
e
e
f
X
ðxÞ¼
f
Xy
ðx; yÞ
d
y ¼
d
y
(5.57)
p
p
The expression in brackets in (
5.57
) is the
modified zero-order Bessel function I
0
s
2
xA
cos
y
s
2
ð
p
xA
1
2
p
e
I
0
¼
d
y:
(5.58)
p
Search WWH ::
Custom Search