Global Positioning System Reference
In-Depth Information
[
FG
][
FG
]
T
FF
T
GG
T
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
=
+
=
I
(4.243)
A
T
G
=
O
(4.244)
G
T
A
=
O
(4.245)
Th
e required transformation is
F
T
G
T
v
F
T
G
T
Ax
F
T
G
T
¯
=
+
(4.246)
or, equivalently,
F
T
F
T
Ax
o
F
T
¯
v
[13
=
+
(4.247)
¯
G
T
G
T
v
Lin
—
1.9
——
No
PgE
Labeling the newly transformed observations by
z
, i.e.,
z
1
z
2
F
T
¯
z
=
=
(4.248)
¯
G
T
we can write (4.247) as
v
z
1
v
z
2
F
T
Ax
o
z
1
z
2
v
z
=
=
+
(4.249)
[13
Th
ere are
r
random variables in
z
1
and
n
r
random variables in
z
2
. The quadratic
fo
rm again remains invariant under the orthogonal transformation, since
−
v
T
FF
T
GG
T
v
v
z
v
z
=
+
(4.250)
v
T
v
=
=
R
according to (4.243). The actual quadratic form is obtained from (4.249):
v
z
v
z
=
F
T
Ax
z
1
T
F
T
Ax
z
1
+
z
2
z
2
R
=
+
+
(4.251)
The least-squares solution requires that
R
be minimized by variation of the parame-
ters. Generally, equating partial derivatives with respect to
x
to zero and solving the
resulting equations gives the minimum. The special form of (4.251) permits a much
simpler approach. The expressions on the right side of Equation (4.251) consist of the
sum of two positive terms (sum of squares). Because only the first term is a function
of the parameters
x
, the minimum is achieved if the first term is zero, i.e.,
−
r
F
nn
A
uu
x
1
=
z
1
(4.252)
