Global Positioning System Reference
In-Depth Information
2
2
zA
+
z
−
zA
2
2
σ
e
I
,
z
≥
0
()
n
pz
=
(5.39)
0
σ
2
σ
2
s
n
n
0
,
z
<
0
where:
z
=
value of the random variable
2
σ
=
RMS noise power
A
=
RMS signal amplitude
zA
=
I
modified Bessel function of zero order
0
σ
n
x
e
Ix
()
≈
for
x
>>
1
0
2
π
x
0 can be expressed in terms of the predetection SNR as
presented to the envelope detector,
C
/
N
(dimensionless), as follows:
Equation (5.39) for
z
≥
2
z
−
+
CN
zCN
2
z
()
2
2
σ
pz
=
e
I
(5.40)
n
s
0
σ
2
σ
n
n
where:
C
/
N
=
predetection signal to noise ratio
2
2
C
/
N
=
A
/
2
σ
n
=
(
C
/
N
0
)
T
T
=
search dwell time
For the case where there is no signal present, then evaluating (5.39) for
A
=
0
yields a Rayleigh distribution for
p
n
(
z
), which is defined by:
2
−
z
z
2
()
2
σ
pz
=
e
(5.41)
n
n
σ
2
n
The result of integrating (5.38) using the pdf of (5.41) is:
2
−
V
t
2
2
σ
pe
fa
=
(5.42)
n
Rearranging (5.42) yields the threshold in terms of the desired single trial proba-
bility of false alarm and the measured 1-sigma noise power:
V
=
σ
−
2ln
P
=
X
σ
(5.43)
t
n
fa
n
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