Global Positioning System Reference
In-Depth Information
N
n
o
,
Σ
b
=
N
n
o
,
0
P
−
1
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
2
v
∼
σ
(4.232)
Applying two orthogonal transformations we can conveniently derive
v
T
Pv
.If
Σ
b
is nondiagonal, one can always find observations that are stochastically independent
and have a unit variate normal distribution. As discussed in Appendix A, for a positive
definite matrix
P
there exists a nonsingular matrix
D
such that the following is valid,
Λ
−
1
/
2
D
=
E
(4.233)
D
T
P
−
1
D
=
I
(4.234)
D
T
v
D
T
Ax
D
T
=
+
(4.235)
[13
Ax
+
¯
v
=
(4.236)
¯
=
b
=
¯
0
−
¯
D
T
D
T
0
−
b
(4.237)
Lin
—
0.1
——
Nor
*PgE
D
T
E(
v
)
E(
v
)
=
=
o
(4.238)
2
0
D
T
P
−
1
D
2
Σ
v
= σ
= σ
0
I
(4.239)
N
n
o
,
0
I
2
v
¯
∼
σ
(4.240)
The columns of the orthogonal matrix
E
consist of the normalized eigenvectors of
P
−
1
;
is a diagonal matrix having the eigenvalues of
P
−
1
at the diagonal. The
quadratic form
v
T
Pv
remains invariant under this transformation because
[13
Λ
v
T
Pv
v
T
1
/
2
E
T
PE
1
/
2
v
v
T
1
/
2
Λ
−
1
1
/
2
v
v
T
v
R
≡
=
Λ
Λ
=
Λ
Λ
=
(4.241)
If the covariance matrix
Σ
b
has a rank defect, then one could use matrix
F
of (A.52)
for the transformation. The dimension of the transformed observations
¯
b
equals the
rank of the covariance matrix.
In the next step, the parameters are transformed to a new set that is stochastically
independent. To keep the generality, let the matrix
A
in (4.236) have less than full
column rank, i.e.,
R(
A
)
r
matrix whose
columns constitute an orthonormal basis for the column space of
A
. One such choice
for the columns of
F
may be to take the normalized eigenvectors of
A A
T
. Let
G
be
an
n
=
r<u
. Let the matrix
F
be an
n
×
r)
matrix, such that [
FG
] is orthogonal and such that the columns of
G
constitute an orthonormal basis to the
n
×
(n
−
r
-dimensional null space of
A A
T
. Such a
matrix always exists. There is no need to compute this matrix explicitly. With these
specifications we obtain
F
T
G
T
−
[
FG
]
F
T
FF
T
G
G
T
FG
T
G
r
I
r
O
=
=
(4.242)
O
r
I
n
−
r
n
−
