Global Positioning System Reference
In-Depth Information
elements of both matrices
Z
and
Z
−
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
1 assures
that the inverse contains only integer elements if
Z
contains integers. Simply consider
this: if all elements of
Z
are integers, then this is also true for the cofactor matrix
C
.
Th
erefore, the inverse
are integers. The condition
|
Z
| =±
C
T
|
Z
−
1
=
(7.157)
|
Z
has integer elements because of the condition
|
Z
| =±
1. The latter condition also
im
plies that
Z
T
Q
b
Z
=
Z
T
|
|
Q
z
| =
Q
b
||
Z
| = |
Q
b
|
(7.158)
[28
Th
e quadratic form also remains invariant with respect to the
Z
transformation.
Substituting (7.154) and (7.155) in (7.138) and using the inverse of (7.156) gives
Lin
—
*
2
——
Lon
*PgE
b
b
T
Q
−
b
b
b
v
T
Pv
∆
=
−
−
=
z
z
T
Z
−
1
T
Q
−
b
Z
−
1
z
z
(7.159)
−
−
=
z
z
T
Q
−
z
z
z
−
−
Consider the following example with two random integer variables
b
[
b
1
b
2
]
T
.
=
[28
Le
t the respective covariance matrix be
b
1
σ
σ
b
1
b
2
Σ
b
=
(7.160)
b
2
σ
b
2
b
1
σ
Z
T
b
utilizes a transformation matrix of the special form
Th
e transformation
z
=
1
01
Z
T
=
(7.161)
z
2
]
T
. Note that
where
z
=
[
z
1
ˆ
ˆ
|
Z
| =
1. The element
β
is obtained by rounding
int
−σ
b
1
b
2
/
b
2
. Because
b
2
−σ
b
1
b
2
/
is an integer, the
tra
nsformed
z
variables will also be integers. Variance-covariance propagation gives
σ
to the nearest integer
β =
σ
β
β
2
2
2
b
1
2
σ
b
2
+
2
βσ
b
1
b
2
+ σ
βσ
b
2
+ σ
b
1
b
2
Z
T
Σ
z
=
Σ
b
Z
=
(7.162)
βσ
b
2
+ σ
σ
b
2
b
1
b
2
b
2
+ β
Le
t
ε
denote the change due to the rounding, i.e.,
ε
= σ
b
1
b
2
/
σ
. Using (7.162),
z
of the transformed variable
z
1
can be written as
the variance
σ
