Graphics Programs Reference
In-Depth Information
To obtain
f
()
use the relations
f z
σ
(
,
)
=
fz
σ
(
⁄
)
f
()
(2.56)
∫
f
()
fz
σ
=
(
,
) σ
d
(2.57)
Finally, using Eq. (2.56) in Eq. (2.57) produces
∫
f
()
fz
σ
=
(
⁄
)
f
()σ
d
(2.58)
where is defined in Eq. (2.55) and is in either Eq. (2.53) or
(2.54). The probability of detection is obtained by integrating the
pdf
derived
from Eq. (2.58) from the threshold value to infinity. Performing the integration
in Eq. (2.58) leads to the incomplete Gamma function.
f z
σ
(
⁄
)
f
()
2.5.1. Threshold Selection
When only a single pulse is used, the detection threshold is related to the
probability of false alarm as defined in Eq. (2.19). DiFranco and Rubin
1
derived a general form relating the threshold and
V
T
P
fa
P
fa
for any number of pulses
when non-coherent integration is used. It is
V
T
n
P
P
fa
=
1 Γ
I
----------
n
P
,
1
(2.59)
where
Γ
I
is used to denote the incomplete Gamma function. It is given by
V
T
⁄
∫
n
P
γ
n
P
(
1
)
1
γ
V
T
n
P
e
Γ
I
----------
n
P
,
1
=
-----------------------------------
d
(2.60)
(
(
n
P
1
)
1
)!
0
Note that the limiting values for the incomplete Gamma function are
Γ
I
(
0
N
,
)
=
0
Γ
I
∞
N
(
,
)
=
1
(2.61)
For our purposes, the incomplete Gamma function can be approximated by
n
P
1
V
T
V
T
n
P
V
T
e
n
P
V
T
1
(
n
P
1
)
n
P
(
2
)
----------
n
P
---------------------------
Γ
I
,
1
=
1
1
+
--------------
+
---------------------------------------
+
(2.62)
(
n
P
1
)!
V
2
(
n
P
1
)!
…
+
---------------------
n
P
1
V
T
1. DiFranco, J. V. and Rubin, W. L.,
Radar Detection
, Artech House, 1980.
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