Global Positioning System Reference
In-Depth Information
bution is used for the case where five satellites are in view. The general formulas for
the chi-square density functions are provided next.
For a central chi-square,
[
]
[
]
(
(
)
)
()
k
21
−
()
−
x
2
k
2
f
cent
x
=
x
e
2
Γ
k
2
,
x
>
0
,
=
0
,
x
≤
0
where
is the gamma function.
For the probability of missed detection, the noncentral chi-square density func-
tion is integrated from 0 to the chi-square detection threshold to determine
Γ
, the
noncentrality parameter that provides the desired
P
md
. The minimum detectable bias
based on the selected proba
bi
lities of false alert and missed detection is denoted as
pbias, where pbias
λ
UERE
.
For a noncentral chi-square,
= σ
λ
[
]
∞
∑
{
[
]
}
(
)
()
(
)
(
)
()
−+
x
λ
2
j
k
2
+−
j
1
2
j
f
x e
=
2
k
2
λ
x
Γ
k j
2
+
⋅
2
⋅
j
!,
x
>
0
NC
..
j
=
0
=
0
,
x
≤
0
where
is the noncentrality parameter. It is defined in terms of the normalized mean
m
and the number of degrees of freedom
k
, as
λ
km
2
.
The chi-square density functions for a case of six visible satellites (2 degrees of
freedom) are shown in Figure 7.21. These density functions are used to define the
detection threshold to satisfy the false alarm and missed detection probabilities. For
supplemental navigation, the maximum allowable false alarm rate is one alarm per
15,000 samples or 0.002/hour. One sample was considered a 2-minute interval
based on the correlation time of SA. The maximum false alarm rate for GPS primary
means navigation is 0.333
λ =
10
−6
per sample. The minimum detection probability for
both supplemental and primary means of navigation is 0.999, or a missed detection
rate of 10
−3
[45].
Figure 7.22 displays a linear no-noise model of the estimated horizontal position
error versus the test statistic, forming a characteristic slope line for each visible satel-
lite. These slopes are a function of the linear connection, or geometry matrix,
H
, and
vary slowly with time as the satellites move about their orbits. The slope associated
with each satellite is given by
×
()
2
2
SLOPE
i
=
AA S
+
,
i
=
12
, ,
L
n
1
i
2
i
ii
where
(
)
−
1
AHHH
≡
T
T
and
S
was defined previously in (7.71), but also can be computed directly from
P
as
SPP
=
T
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