Global Positioning System Reference
In-Depth Information
signal at the BS
i
and BS
1
, where
i
∈ {
2, 3, . . . ,
n
}
. The difference in the distance is therefore
defined as follows:
d
i
1
≡
d
i
−
d
1
=(
−
)
−
(
−
)
t
i
t
m
c
t
1
t
m
c
(7)
=(
t
i
−
t
1
)
c
It can be seen that the difference is not affected by errors in the MS clock time
t
m
. Substituted
Equation 7 in Equation 3, and then expanding and rearranging the terms produce the
following set of equations:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
k
2
−
d
21
x
2
y
2
x
3
y
3
. .
x
n
y
n
−
d
21
x
m
y
m
k
3
−
d
31
−
d
31
.
1
2
=
d
1
(8)
.
k
n
−
d
n
1
−
d
n
1
which can be expressed in a matrix form as follows:
Hx
=
d
1
c
+
r
(9)
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
k
2
−
d
21
x
2
y
2
x
3
y
3
. .
x
n
y
n
−
d
21
k
3
−
d
31
−
d
31
.
2
where
H
=
,
c
=
, and
r
=
.
.
k
n
−
d
n
1
−
d
n
1
Similarly, Equation 9 can be solved using the following linear least squares formulation:
−
1
T
T
x
=
(
d
1
c
+
r
)
ˆ
H
H
H
(10)
The solution of Equation 10 is determined in two steps. First, the estimate of the MS is
determined in terms of
d
1
, which is substituted in the quadratic expression
d
1
=
x
2
m
+
y
2
m
to compute
d
1
. Second, the value of
d
1
is substituted back in Equation 10 to solve for ˆ
x
Sayed
et al. (2005).
4.1.3 AOA data fusion
AOA techniques estimate the location of an MS by measuring the angle of signal arrival from
the MS at several BS's by means of an antenna array. The MS location is then estimated
through the intersection of the straight paths leaving from at least two BS's, as depicted in
Figure 3. However, combining only two AOA measurements may introduce a large amount of
uncertainty with respect to the MS location estimate, especially when the MS is close to the line
connecting the two BS's. Moreover, this localization method requires the MS to be in LOS with
the participating BS's, since reflected or diffracted signals result in misleading information.
For this reason, it is preferable for the AOA to be combined with another localization method,
such as TOA or TDOA.
4.1.4 RSS data fusion
RSS based localization is a method that employs mathematical models that describe the path
loss as a function of distance. Since these models translate the received signal power into a
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