Digital Signal Processing Reference
In-Depth Information
Fig. 2.36
Densities and transformation in Example 2.6.7. (
a
) Input PDF. (
b
) Transformation.
(
c
) Output PDF
is monotone, there is a unique correspondence between
X
and
Y
, and each value of
X
corresponds to only one value of
Y
, resulting in:
Pfy
i
g¼Pfx
i
g
(2.190)
and
f
Y
ðyÞ¼
X
i
Pfy
i
gdðy y
i
Þ;
F
Y
ðyÞ¼
X
i
Pfy
i
guðy y
i
Þ;
(2.191)
where
y
i
¼ gðx
i
Þ:
(2.192)
For the nonmonotone transformation,
N
values of the random variable
X
gener-
ally correspond to only one value of the random variable
Y
, where
N
is an integer,
Y ¼ y
i
if
X ¼ x
1
;
or
X ¼ x
2
;
or
...
X ¼ x
N
:
(2.193)
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