Digital Signal Processing Reference
In-Depth Information
The joint density of variables
Y
1
and
Y
2
is found to be:
f
X
1
X
2
ð
y
1
;
y
2
Þ
jJðy
1
; y
2
Þj
f
Y
1
Y
2
ðy
1
; y
2
Þ ¼
:
(4.263)
The Jacobian is calculated by taking derivatives of
y
1
and
y
2
with respect to
x
1
and
x
2
:
2
p
p
cos
ð
2
px
2
Þ
@
y
1
@x
1
@
y
1
@x
2
p
2ln
ðx
1
Þ
2ln
ðx
1
Þ
sin
ð
2
px
2
Þ
x
1
2
p
x
1
:
J ¼
¼
¼
2
p
p
@
y
2
@x
1
@
y
2
@x
2
sin
ð
2
px
2
Þ
p
2ln
ðx
1
Þ
2ln
ðx
1
Þ
cos
ð
2
px
2
Þ
x
1
(4.264)
Since
x
1
is always positive (see (
4.259
)), then
2
p
x
1
:
jJj ¼
(4.265)
From (
4.261
) and (
4.262
), we find:
y
1
þ
y
2
2
x
1
¼
e
:
(4.266)
Placing (
4.266
) into (
4.264
) from (
4.263
), we arrive at:
e
y
1
þ
y
2
e
y
1
e
y
2
1
2
p
1
1
2
p
2
2
2
¼ f
Y
1
ðy
1
Þf
Y
2
ðy
2
Þ:
f
Y
1
Y
2
ðy
1
; y
2
Þ ¼
¼
p
2
p
p
(4.267)
Therefore, the transformation of two independent uniform random
variables produces two independent standard normal variables (with a
mean of zero and a variance of 1).
A.4.10. It is useful because of the fact that the sum
X
may have a PDF that is
closely approximated to a normal PDF for finite number
N
[PEE93,
p. 119].
A.4.11. In this case, the practical usefulness comes from the fact that the areas of
delta functions approximate Gaussian PDF curve. For more details see
[PAP65, pp. 268-272]. Also see Chap.
5
.
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