ISSN: 2180 - 1843 Vol. 3 No. 1 January - June 2011 Journal of Telecommunication, Electronic and Computer Engineering
58
error namely ROS model consists of high probability of large NLOS error and of
near-zero NLOS error. Finally, the fourth model of NLOS error was modeled as
uniformly distribution random variable which gives equal probability of taking
low and high NLOS values.
B. Linear Least squares estimation
Linear least squares LLS [7, 15] approach is a suboptimal positioning technique
which provides a solution with low computational complexity. Therefore,
it can be employed for applications that require fast and low complexity
implementation with reasonable positioning accuracy. In addition, for
applications that require precise location estimation, LLS can be used to obtain
initial position estimate for initializing high-accuracy positioning algorithms,
such as non linear least squares NLLS [15] and linearization based on Taylor
series [8]. A good initialization can
signiicantly decrease the computational complexity and the inal location error
of a high accuracy technique. Therefore, performance analysis of the LLS is
important from multiple perspectives. In this paper, LLS is utilized for performance
comparison with the developed algorithm.
The LLS approach begins with the set of equations 4 where each distance
measurement is assumed to deine a circle of uncertain region. Based on trilateration
method with distance measurements of at least three available BS, the intersection
at one point can be obtained by using LLS approach. These equations can be
simpliied by solving intersections of circles and presenting in the matrix form
as following [15] and explained in Fig. 2.
= A
b x
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ =
− +
−
−
=
= 5
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ =
− +
−
−
=
where
2 1
2 1
21 3
1 3
1 31
1 1
1 1
1
, ,
n n
n
x x
y y
b x
x y
y b
x x
y y
x x
y y
b −
− ⎡
⎤ ⎡
⎤ ⎢
⎥ ⎢
⎥ −
− −
⎡ ⎤
⎢ ⎥
⎢ ⎥
= =
= ⎢
⎥ ⎢
⎥ ⎢
⎥ −
⎣ ⎦
⎢ ⎥
⎢ ⎥
− −
⎣ ⎦
⎣ ⎦
A b
M M
M
x
⎡ ⎤
= −
+ ⎣
⎦ =
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
6
⎡ ⎤
= −
+ ⎣
⎦ =
with
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
2 2
2
0.5
ij j
i ij
b r
r d
⎡ ⎤
= −
+ ⎣
⎦ =
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ 7 =
and
=
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦
2 2
ij i
j i
j
d x
x y
y =
− +
−
−
=
=
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ =
− +
− 8
−
=
where i=1,2,…,n and n denotes the number of available BSs. d
j
represents the distance between BS
i
and BS
j
and is the serving BS.
From 5, the LLS solution can be obtained as [15, 16]
=
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ =
− +
−
1
ˆ +
T T
−
= A A A b
X x
=
− −
⎡ ⎤
⎡ ⎤
⎢ ⎥
⎢ ⎥
− −
− ⎡
⎤ ⎢
⎥ ⎢
⎥ =
= =
⎢ ⎥
⎢ ⎥
⎢ ⎥
− ⎣
⎦ ⎢
⎥ ⎢
⎥ −
− ⎣
⎦ ⎣
⎦ M
M M
⎡ ⎤
= −
+ ⎣
⎦ =
− +
−
−
=
9
near- r
which al
low r
ty In
on te
hms, such as d
ly on
erefore, performance n
th circle
= −
− ⎡
⎤ ⎡
⎤ ⎢
⎥ ⎢
⎥ −
− −
⎡ ⎤
⎢ ⎥
⎢ ⎥
= =
= ⎢
⎥ ⎢
⎥ ⎢
⎥ −
⎣ ⎦
⎢ ⎥
⎢ ⎥
− −
⎣ ⎦
⎣ ⎦
M M
M ⎡
⎤ =
− +
⎣ ⎦
= −
+ −
−
=
Fig. 2 LLS Geometry Calculation Using TOA
where is the 2-D desired vector of MS
coordinates and position of serving BS, denoted by X= [
x
i
,y
i
]
T
.
C. Linear Lines of Position Algorithm
