Graphics Programs Reference
In-Depth Information
Other slightly different definitions for the false alarm number exist in the liter-
ature, causing a source of confusion for many non-expert readers. Other than
the definition in Eq. (2.22), the most commonly used definition for the false
alarm number is the one introduced by Marcum (1960). Marcum defines the
false alarm number as the reciprocal of . In this text, the definition given in
Eq. (2.22) is always assumed. Hence, a clear distinction is made between Mar-
cumÓs definition of the false alarm number and the definition in Eq. (2.22).
P fa
2.3. Probability of Detection
The probability of detection
P D
is the probability that a sample
R
of
r ()
will exceed the threshold voltage in the case of noise plus signal,
-----  r 2
A 2
r
ψ 2
rA
ψ 2
+
2
P D
=
------
I 0
exp
–
-----------------
d
(2.23)
V T
If we assume that the radar signal is a sine waveform with amplitude
A
, then its
A 2
A 2
2
power is
2
. Now, by using (single-pulse SNR) and
, then Eq. (2.23) can be rewritten as
SNR
=
V 2
2
(
)
=
ln
(
1
P fa
)
 r 2
A 2
r
ψ 2
rA
ψ 2
+
2
------ I 0
----- 
P D
=
exp
–
-----------------
d
r
=
(2.24)
2
ln
(
1
p fa
)
A 2
ψ 2
1
P fa
------ 
Q
------2
,
ln
ζ 2
α 2
–
(
+
)
2
Q αβ
[
,
] I 0 αζ
=
(
) e
d
(2.25)
β
is called MarcumÓs Q-function. When is small and is relatively
large so that the threshold is also large, Eq. (2.24) can be approximated by
Q
P fa
P D
A
---- –
1
P fa
------ 
P D
F
2
ln
(2.26)
where is given by Eq. (2.16). Many approximations for computing Eq.
(2.24) can be found throughout the literature. One very accurate approximation
presented by North (see bibliography) is given by
F ()
P D
0.5
×
erfc
(
–
ln
P fa
–
SNR
+
0.5
)
(2.27)
where the complementary error function is
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