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V
T
n
p
n
fa
Figure 2.8. Threshold
versus
for several values of
.
2.6. Probability of Detection Calculation
Marcum defined the probability of false alarm for the case when
n
P
>
1
as
P
fa
≈
ln
()
(
n
P
⁄
n
fa
)
(2.68)
The single pulse probability of detection for non-fluctuating targets is given in
Eq. (2.24). When , the probability of detection is computed using the
Gram-Charlier series. In this case, the probability of detection is
n
P
>
1
V
2
⁄
2
erfc V
(
⁄
2
)
e
V
2
V
2
P
D
≅
------------------------------
------------
--
C
3
[
(
1
)
+
C
4
V
3
(
)
(2.69)
2
2π
C
6
VV
4
10
V
2
(
+
15
)
]
where the constants
C
3
,
C
4
, and
C
6
are the Gram-Charlier series coefficients,
and the variable
V
is
V
T
n
P
1
(
+
SNR
)
V
=
--------------------------------------------
(2.70)
ϖ
In general, values for
C
3
,
C
4
,
C
6
, and
ϖ
vary depending on the target fluctu-
ation type.
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