Digital Signal Processing Reference
In-Depth Information
EXPONENTIAL PDF
EXPONENTIAL DISTRIBUTION
0.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20 25
Range
30 35
40 45
50
0
5
10
15
20
25
30 35
40 45
50
x
Fig. 5.9
Exponential PDF and distribution for the parameter
l ¼
0.1
5.4.2 Characteristic Function and Moments
Using the definition of the characteristic function, we have:
1
e
jox
f
X
ðxÞ
d
x ¼ l
1
0
l
l jo
:
e
xðljoÞ
d
x ¼
f
X
ðoÞ¼
(5.64)
0
Applying the moment theorem, we obtain the mean and mean squared values,
o¼
0
¼
1
j
d
f
X
ð
o
Þ
d
o
1
l
;
X ¼
(5.65)
o¼
0
¼
d
2
f
X
ð
o
Þ
d
o
2
1
j
2
2
l
2
:
X
2
¼
(5.66)
From (
5.65
) and (
5.66
), the variance is given as:
2
l
2
1
l
2
¼
1
l
2
:
X
2
s
2
¼ X
2
¼
(5.67)
Example 5.4.1
The power reflected from the aircraft into the radar is described
using the exponential variable:
P
0
e
x=P
0
1
for
x
0
;
f
X
ðxÞ¼
(5.68)
0
otherwise,
where
P
0
is the mean value of the received power.
Find the probability that the received power is larger than its average value.
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