Global Positioning System Reference
In-Depth Information
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The matrices
H
q
and
K
q
are obtained as part of the consecutive transformations.
The permuting steps assure that
K
q
contains decreasing diagonal elements, the small-
est element being located at the lower right corner. As a measure of decorrelation
between the ambiguities, we might consider the scalar (Teunissen, 1994)
1
/
2
r
= |
R
|
0
≤
r
≤
1
(7.183)
where
R
represents a correlation matrix. Applying (7.183) to
Q
b
and
Q
z,q
will give
a relative sense of the decorrelation achieved. A value of
r
close to 1 implies a high
decorrelation. Therefore, we expect
r
b
<r
z,q
. The scalar
r
is called the ambiguity
decorrelation number.
The search step entails finding candidate sets of
z
i
given
(
z
,
Q
z,q
)
, which minimize
=
z
z
T
Q
−
1
z,q
z
z
[28
v
T
Pv
∆
−
−
(7.184)
A
possible procedure would be to use the diagonal elements of
Q
z,q
, construct a range
fo
r each ambiguity centered around
z
i
, form all possible sets
z
i
, evaluate the quadratic
fo
rm for each set, and keep track of those sets that produce the smallest
Lin
—
1
——
No
PgE
v
T
Pv
.A
m
ore organized and efficient approach is achieved by transforming the
z
variables
in
to variables
w
that are stochastically independent. First, we decompose the inverse
of
Q
z,q
as
∆
Q
−
1
MSM
T
z,q
=
(7.185)
[28
wh
ere
M
denotes the lower triangular matrix with 1's along the diagonal, and
S
is a
di
agonal matrix containing positive values that increase toward the lower right corner.
Th
e latter property follows from the fact that
S
is the inverse of
K
q
. The transformed
va
riables
w
,
M
T
z
z
w
=
−
(7.186)
ar
e distributed as
w
N(
o
,
S
−
1
)
. Because
S
is a diagonal matrix the variables
w
are
sto
chastically independent. Using (7.186) and (7.185) the quadratic form (7.184) can
be
written as
∼
n
v
T
Pv
w
T
S w
w
i
s
i,i
2
∆
≡
=
1
ˆ
≤ χ
(7.187)
i
=
2
acts as a scalar; additional explanations will be given below. Finally,
we introduce the auxiliary quantity, also called the conditional estimate,
The symbol
χ
n
m
j,i
ˆ
z
j
w
i
|
I
ˆ
=
z
j
−
(7.188)
j
=
i
+
1
