Global Positioning System Reference
In-Depth Information
ε
2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
b
1
b
2
σ
σ
z
1
= σ
b
1
−
b
2
σ
b
2
−
σ
(7.163)
Th
is expression shows that the variance of the transformed variable decreases com-
pa
red to the original one, i.e.,
z
1
<
b
1
whenever
σ
b
1
b
2
σ
σ
σ
b
2
>
0
.
5
(7.164)
an
d that both are equal when
σ
b
1
b
2
/
b
2
= |
2
0
.
5. The property of decreasing
the variance while preserving the integer makes the transformation (7.161) a favorite
to resolve ambiguities because it minimizes the search. It is interesting to note that
z
1
and
z
2
would be completely decorrelated if one were to choose
σ
ε
| =
b
2
.
However, such a selection is not permissible because it would not preserve the integer
property of the transformed variables.
When implementing LAMBDA, the
Z
matrix is constructed from the
n
β =−σ
b
1
b
2
/
σ
[28
n
subma-
trix
Q
b
(7.135). There are
n
variables
b
that must be transformed. Using the Cholesky
decomposition we find
×
Lin
—
1
——
Nor
PgE
H
T
KH
Q
b
=
(7.165)
The matrix
H
is the modified Cholesky factor that contains 1 at the diagonal and
fo
llows (7.136).
K
is a diagonal matrix containing the diagonal squared terms of the
Ch
olesky factor. Assume that we are dealing with ambiguities
i
and
i
+
1 and partition
[28
th
ese two matrices accordingly,
1
.
.
.
.
---------------------------
H
11
OO
h
i,
1
···
1
H
=
=
H
21
H
22
O
(7.166)
h
i
+
1
,
1
···
h
i
+
1
,i
1
H
31
H
32
H
33
---------------------------
.
.
.
.
.
.
···
h
n,
1
···
h
n,i
h
n,i
+
1
···
1
k
1
,
1
.
.
.
---------------------------
K
11
OO
OK
22
O
OO
k
i,i
=
=
K
(7.167)
k
i
+
1
,i
+
1
---------------------------
33
.
.
.
k
n,n
