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As mentioned above, we assume four ref-
erence nodes, and the target is to estimate the
three-dimensional positions of the on-body nodes.
Based on the acquired TOA estimates
p
ij
, the
linear-LS technique is as follows (Di-Benedetto
and Giancola 2004; Guvenc, Chong et al. 2007;
Gezici, Guvenc et al. 2008):
get metric space from the information about the
inter-point distances, and is referred to as MDS
when the target space is Euclidean. Classical
MDS (C-MDS) is the simplest form of MDS,
and when the reference absolute frame is avail-
able it gives a unique solution for the estimated
coordinates. Otherwise, the result from MDS
could be a translated, rotated, and scaled version
of the actual set. This is essentially because of
the absence of the absolute coordinate reference
(Zhang-Xin, He-Wen et al. 2009).
In our system, we consider C-MDS, and
overcome the problem of the missing refer-
ence coordinates by using preceding frames as
reference-frames, and obtain the initial-frame
coordinates from the initial range measurement
phase of our system (Shaban, El-Nasr et al. 2010).
We further increase the accuracy of estimated
coordinates by using FFT interpolation after the
ranging stage. This interpolation is performed in
time (frame-to-frame).
An
m×m
matrix
D
consisting of squared dis-
tances
d
ij
2
, where
m
= 4 nodes for a three-dimen-
sional absolute coordinate system. To recover the
m×d
matrix
X
of positions in
d
-dimensional space;
three-dimensional space in our case,
D
is expressed
as (Zhang-Xin, He-Wen et al. 2009):
2
2
2
2
p
= (
x
−
x
)
+
(
y
−
y
)
+ −
(
z
z
)
ij
i
j
i
j
i
j
(18)
=
1
2
AX
p
(19)
T
[
]
.
where,
X
i
=
x
y
z
i
i
i
x
−
x
y
−
y
z
−
z
k
k
k
1
1
1
x
−
x
y
−
y
z
−
z
A
=
2
k
2
k
2
k
(20)
x
−
x
y
−
y
z
−
z
N
k
N
k
N
k
p
2
− − + − + − +
− − + − +
p
2
x
2
x
2
y
2
y
2
z
2
z
2
i
ki
k
k
k
1
1
1
1
p
2
p
2
x
2
x
2
y
2
y
2
− +
z
2
z
2
p
=
π
i
ki
k
k
k
2
2
2
2
2
2
2
2
2
2
2
2
p
− −
p
x
+ −
x
y
+ − +
y
z
z
Ni
ki
N
k
N
k
N
k
(21)
2
2
2
0
d d d
12
13
14
The position of the target-node is estimated
as (Guvenc, Chong et al. 2007; Gezici, Guvenc
et al. 2008):
d
2
0
d d
2
2
D
=
21
23
24
(23)
2
2
2
d d
0
d
31
32
34
d d d
2
2
2
0
41
42
43
1
2
)
−
=
(
1
X
A A A p
T
T
(22)
where,
d
ij
is defined in terms of the absolute
coordinates
x
i
and
x
j
as (Zhang-Xin, He-Wen et
al. 2009):
2
2
2
2
d
= (
x
−
x
) =
x
−
2
x x
+
x
(24)
CORE LOCALIZATION STAGE
ij
i
j
i
i
j
j
Multidimensional scaling (MDS) includes a
family of methods. Scaling refers to the methods
that construct a configuration of points in a tar-
The elements
b
ij
of the dot-product matrix
B=XX' are defined as follows (Zhang-Xin, He-
Wen et al. 2009):
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