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alarm can be derived from the conditional false alarm probability, which is
averaged over all possible values of the threshold in order to achieve an uncon-
ditional false alarm probability. The conditional probability of false alarm
when
y
=
can be written as
T
2ψ
2
y
⁄
P
fa
V
T
(
=
y
)
=
e
(2.103)
It follows that the unconditional probability of false alarm is
∞
∫
P
fa
=
P
fa
(
V
T
=
y
)
f
()
d
(2.104)
0
where
f
()
is the
pdf
of the threshold, which except for the constant
K
0
is the
same as that defined in Eq. (2.102). Therefore,
2
K
0
ψ
2
(
y
⁄
)
y
M
1
e
f
()
=
---------------------------------------
;
y
≥
0
(2.105)
)
M
Γ ()
2
K
0
ψ
2
(
Performing the integration in Eq. (2.104) yields
1
P
fa
=
------------------------
(2.106)
)
M
(
1
+
K
0
Observation of Eq. (2.106) shows that the probability of false alarm is now
independent of the noise power, which is the objective of CFAR processing.
2.9.2. Cell-Averaging CFAR with Non-Coherent Integration
In practice, CFAR averaging is often implemented after non-coherent inte-
sum of squared envelopes. It follows that the total number of summed ref-
erence samples is . The output is also the sum of squared enve-
lopes. When noise alone is present in the CUT, is a random variable whose
pdf
is a gamma distribution with degrees of freedom. Additionally, the
summed output of the reference cells is the sum of
n
P
Mn
P
Y
1
n
P
Y
1
2
n
p
Mn
P
squared envelopes.
Thus,
Z
is also a random variable which has a gamma
pdf
with
2
Mn
P
degrees
of freedom.
The probability of false alarm is then equal to the probability that the ratio
exceeds the threshold. More precisely,
Y
1
⁄
Z
P
fa
=
Prob Y
1
{
⁄
Z
>
K
1
}
(2.107)
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