Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
7,
No.
3,
June
2017,
pp.
1255
–
1261
ISSN:
2088-8708
1255
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
Localization
of
Distrib
uted
W
ir
eless
Sensor
Netw
orks
using
T
w
o
Sage
SDP
Optimization
Reza
Shahbazian
1
and
Sey
ed
Ali
ghorashi
2
1,2
Cogniti
v
e
T
elecommunication
Research
Group,
Department
of
Electrical
Engineering,
Shahid
Beheshti
Uni
v
ersity
G.
C.,
T
ehran,
Iran.
2
CyberSpace
Research
Institute,
Shahid
Beheshti
Uni
v
ersity
G.
C.,
T
ehran,
Iran.
Article
Inf
o
Article
history:
Recei
v
ed
Jan
2,
2017
Re
vised
May
9,
2017
Accepted
May
22,
2017
K
eyw
ord:
W
ireless
Sensor
Netw
ork
Localization
Optimization
Semi-Definite
Programming
(SDP)
V
irtual
Anchor
ABSTRA
CT
In
man
y
applic
ations
of
wireless
sensor
netw
ork
(WSN),
the
location
of
sensors
is
a
neces-
sity
to
e
v
aluate
the
sensed
dat
a
and
it
is
not
ener
gy
and
cost
ef
ficient
to
equip
all
sensors
with
global
positioning
systems.
In
WSN
localization,
some
sensors
(called
anchors)
ar
e
a
w
are
of
their
location.
Then,
the
distance
measurements
between
sensors
and
anchors
are
used
to
localize
the
whole
netw
ork.
WSN
localization
is
a
non-con
v
e
x
optim
ization,
ho
we
v
er
,
relaxation
techniques
such
as
semi-definite
programming
(SDP)
are
used
to
relax
the
opti-
mization.
T
o
solv
e
this
problem,
all
constraints
should
be
considered
simultaneously
and
the
solution
comple
xity
order
is
O
n
2
where
n
is
the
number
of
sensors.
The
comple
xity
of
SDP
pre
v
ents
solving
lar
ge
size
problems.
Therefore,
it
is
necessary
to
reduce
the
problem
size
in
lar
ge
and
distrib
uted
WSNs.
In
this
paper
,
we
propose
a
tw
o
stage
optimization
to
re-
duce
the
solution
time,
while
pro
vide
better
accurac
y
compared
with
original
SDP
method.
W
e
first
select
some
sensors
that
ha
v
e
the
maximum
connection
with
anchors
and
perform
the
localization.
Then,
we
select
some
of
these
sensors
as
virtual
anchors.
By
adding
the
virtual
anchors,
we
decrease
the
number
of
constraints.
W
e
propose
an
al
gorithm
to
select
virtual
anchors
so
that
the
total
solution
comple
xity
and
time
decrease
considerably
,
while
impro
ving
the
localization
accurac
y
.
Copyright
c
2017
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
S.
A.
Ghorshi
Department
of
T
elecommunications,
F
aculty
of
Electrical
Engineering,
Shahid
Beheshti
Uni
v
ersity
T
ehran,
1983963113,
IRAN
+9829904135
a
ghorashi@sb
u.ac.ir
1.
INTR
ODUCTION
W
ireless
sensor
netw
orks
(WSN)
pro
vide
f
ast,
quite
cheap
and
reliable
solutions
to
a
lar
ge
number
of
in-
dustrial,
commercial
and
military
applications,
ranging
from
surv
eillance
[1]
and
tracking
to
disaster
management,
robotics
and
other
tasks
[2,
3,
4]
.
Kno
wing
the
correct
positions
of
sensor
nodes
is
essential
to
man
y
applications
in
ne
xt-generation
sensor
netw
orks.
Location
a
w
areness
refers
to
de
vices
that
can
passi
v
ely
or
acti
v
ely
determine
their
location.
Location
a
w
areness
without
the
acti
v
e
participation
of
the
de
vice
is
kno
wn
as
non-cooperati
v
e
localization.
Location
a
w
areness
enables
ne
w
applications
for
ubiquitous
computing
systems
and
mobile
phones.
WSNs
present
no
v
el
trade-of
fs
in
system
design;
on
one
hand,
the
lo
w
cost
of
the
sensors
f
acilitates
massi
v
e
scale
and
highly
parallel
computation.
On
the
other
hand,
each
sensor
is
lik
ely
to
ha
v
e
limited
po
wer
,
limited
reliability
,
and
only
local
communication
with
a
modest
number
of
neighbors.
These
application
conte
xts
and
potential
massi
v
e
scale
mak
e
it
unrealistic
to
rely
on
careful
placement
or
uniform
arrangement
of
sensors.
Using
globally
accessible
beacons
or
to
equip
sensors
with
GPS
is
not
feasible
considering
cost
or
ener
gy
constraints.
Therefore,
ranging-base
localization
techniques
are
introduced
for
WSNs
[5].
The
goal
of
localization
is
to
determine
the
ph
ysical
coordinates
of
a
group
of
sensor
nodes.
These
coordi
nates
can
be
global,
meaning
that
the
y
are
aligned
with
some
e
xternally
meaningful
system
lik
e
GPS.
The
y
can
also
be
local,
meaning
that
the
sensor
nodes
only
ha
v
e
the
y
related
position
in
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v7i3.pp1255-1261
Evaluation Warning : The document was created with Spire.PDF for Python.
1256
ISSN:
2088-8708
constellation.
In
localization
problem
using
neighboring
dist
ance
measurements,
it
is
assumed
that
the
accurate
positions
of
some
nodes
are
kno
wn,
called
anchors.
Using
some
partial
pair
-wise
distance
measurements
and
the
localization
algorithm,
it
is
possible
to
locate
all
sensor
nodes
in
the
netw
ork.
The
dis
tance
measurement
between
nodes
could
be
performed
in
dif
ferent
methods.
By
kno
wing
the
transmitted
po
wer
,
using
the
recei
v
ed
signal
strength
(RSS),
and
considering
the
ef
fect
of
path
loss,
shado
wi
ng,
and
other
losses
the
distance
between
tw
o
sensors
could
be
found
[6].
Both
time
of
arri
v
al
(T
O
A)
and
time
dif
ference
of
arri
v
al
(TDO
A)
could
be
used
to
estimate
the
distances
between
sensors.
Ho
we
v
er
,
unsynchronized
sensors
with
multi-path
ef
fect,
highly
de
grades
the
accurac
y
of
this
es
timation
[7].
The
methods
such
as
angle
of
arri
v
al
(A
O
A)
or
direction
of
arri
v
al
(DO
A)
are
not
applicable
in
pair
-wise
distance
measurement
and
may
be
used
for
tar
get
localization
in
a
direct
formulation
[7].
In
the
past,
localization
problems
were
solv
ed
algebraically
and
computed
by
least
squares
solution
to
h
yper
-
bolic
equations
called
multi-lateration.
No
w
adays,
the
optimum
solution
for
sensor
netw
ork
localization
is
pro
vided
using
optimization.
One
of
the
first
proposed
con
v
e
x
optimization
models
for
wireless
sensor
netw
ork
localization
is
introduced
[8]
in
which
the
problem
is
relax
ed
to
a
second
order
cone
programming
(SOCP)
problem.
Ho
we
v
er
,
this
method
needs
a
lar
ge
number
of
anchors
on
the
area
boundary
to
ha
v
e
acceptable
performance.
In
2004,
Bisw
as
[9]
proposed
a
semi-definite
programming
(SDP)
relaxation
method
for
WSN
localization.
The
research
sho
ws
that
the
problem
of
finding
uni
qu
e
solution
to
a
noiseless
non-linear
system
describing
the
common
point
of
intersection
of
h
yper
spheres
in
real
Euclidean
v
ector
space,
can
be
e
xpressed
as
a
semi-definite
program
via
distance
geometry
.
This
method
outperforms
the
SOCP
[8]
in
terms
of
accurac
y
and
a
v
erage
estimation
error
.
Bisw
as
also
de
v
eloped
a
solution
[10]
to
deal
with
noisy
corrupted
data,
and
SDP
relaxation
used
to
transform
the
problem
into
a
con
v
e
x
optimization
scenario.
This
method
[10]
is
based
on
maximum
lik
elihood
(ML)
estimation.
Ho
we
v
er
,
ML
based
methods
are
usu-
ally
v
ery
time
consuming
in
comparison
with
other
estimation
methods.
T
o
reduce
the
solution
time,
a
method
called
smaller
SDP
(SSDP)
[11]
w
as
proposed
which
further
relax
es
the
original
SDP
.
In
this
method,
a
single
semi-definite
matrix
cone
is
relax
ed
into
a
set
of
small-size
semi-definite
matrix
cones.
The
non-con
v
e
xity
of
the
problem
co
v
ered
by
introducing
a
con
v
e
x
objecti
v
e
function
[12].
This
method
is
only
ef
fecti
v
e
for
noise-free
measurement
cases.
The
research
on
impro
ving
the
performance
of
SDP
is
still
an
open
issue.
In
recent
de
v
elopments,
researchers
impro
v
e
the
performance
of
WSNs
localization
in
terms
of
rob
ustness
and
accurac
y
especially
in
non-line
of
sight
(NLOS)
and
harsh
en
vironments
[13].
Ho
we
v
er
,
in
all
SDP
solutions,
when
the
size
of
SDP
problem
increases,
the
dimension
of
matrix
cone
increases,
simultaneously
and
the
amount
of
unkno
wn
v
ariables
increases,
non-linearly
.
It
is
kno
wn
that
the
arithmetic
operation
comple
xity
of
the
SDP
is
at
least
O
n
2
to
obtain
an
approximate
solution.
This
comple
xity
pre
v
ents
solving
lar
ge
size
problems.
Therefore,
it
w
ould
be
v
ery
beneficial
to
reduce
the
SDP
problem
size.
On
the
other
hand,
the
impact
of
noise
on
distance
measurement
and
est
imation
error
is
important
and
this
ef
fect
v
aries
in
v
ersely
with
problem
size.
The
moti
v
ation
of
this
w
ork
is
to
reduce
the
required
time
to
solv
e
the
optimization
problem,
while
increasing
the
localization
accurac
y
.
In
this
paper
,
we
propose
a
tw
o
stage
localization
algorithm
based
on
SDP
for
WSNs
that
is
applicable
in
an
y
m
odification
of
the
original
SDP
approach.
Simulation
results
demonstrate
that
the
proposed
method
is
v
ery
ef
fecti
v
e
and
significantly
decreases
the
solution
time
and
impro
v
es
the
a
v
erage
position
error
.
The
remainder
of
the
paper
is
or
g
anized
as
follo
ws.
In
section
2,
the
system
model
is
described
and
the
problem
is
formulated.
Section
3
proposes
a
sol
ution
for
the
problem.
In
section
4,
simulation
re
sults
are
presented
and
section
5
concludes
the
paper
.
2.
RESEARCH
MODEL
W
e
consider
a
WSN
with
m
anchors
(kno
wn
positions)
and
n
sensors
(unkno
wn
positions)
in
a
tw
o
dimen-
sional
(2D)
en
vironment.
The
e
xtension
of
this
localization
problem
to
hi
gh
e
r
dimensions
is
straightforw
ard.
Some
notations
used
in
this
paper
are
as
follo
ws.
I
,
e
and
0
denote
the
identity
matrix,
the
v
ector
of
all
ones
and
the
v
ector
of
all
zeros,
respecti
v
ely
.
The
2-norm
of
a
v
ector
x
is
denoted
by
k
x
k
.
A
positi
v
e
semi-definite
matrix
X
is
represented
by
X
0
.
The
position
of
anchor
nodes
are
presented
by
v
ector
V
a
=
f
a
1
;
a
2
;
:::;
a
m
g
and
the
Euclidean
distance
between
x
j
and
x
i
is
denoted
as
d
ij
and
between
a
k
and
x
j
is
denoted
by
d
j
k
as
follo
ws:
d
ij
=
k
x
i
x
j
k
and
d
j
k
=
k
x
j
x
k
k
(1)
3.
PR
OPOSED
METHOD
The
optimization
problem
can
be
e
xpressed
as
follo
ws:
IJECE
V
ol.
7,
No.
3,
June
2017:
1255
–
1261
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1257
Find
X
2
R
2
n
S.t.
Y
ii
2
Y
ij
+
Y
j
j
=
d
ij
2
;
8
(
i;
j
)
2
N
s
Y
j
j
2
X
T
j
a
k
+
a
k
2
=
d
j
k
2
;
8
(
j
;
k
)
2
N
a
N
s
=
f
(
i;
j
)
j
x
i
x
j
<
r
g
;
N
a
=
f
(
j
;
k
)
j
x
j
x
k
<
r
g
Y
=
X
T
X
(2)
where
r
is
the
communication
range,
and
X
=
[
x
1
;
:
:
:
;
x
n
]
,
d
ij
and
d
j
k
are
the
noisy
range
measurements.
The
problem
in
(2)
is
non-con
v
e
x
and
may
be
relax
ed
using
SDP
method
as
presented
in
(3).
Y
X
T
X
!
Z
0;
Z
=
I
2
X
T
X
Y
(3)
The
idea
behind
the
proposed
method
is
simple.
The
nearest
sensors
to
the
anchors
with
most
connections
to
the
anchors
ha
v
e
the
best
measurements.
W
e
first
localize
some
of
sensors
wit
h
best
measurements,
and
use
these
sensors
with
kno
wn
location
as
virtual
anchors.
In
the
second
stage,
the
real
and
virtual
anchors
are
both
used
to
localize
the
rest
of
sensors
in
the
netw
ork.
The
proposed
method
is
e
xplained
in
T
able
1.
T
able
1.
Proposed
Algorithm
Step
Number
Operation
Initialization
Anchors
with
kno
wn
and
sensors
with
unkno
wn
location
1
Calculate
all
the
sensor
-sensor
and
sensor
-anchor
distance
using
noisy
measurements
d
ij
,
d
j
k
2
Select
the
sensors
that
are
in
communication
range
of
at
least
tw
o
anchors
3
Estimate
the
location
of
selected
sensors
4
Choose
the
sensors
with
most
distance
to
e
xisting
anchors
a
s
virtual
anchor
5
Choose
the
sensors
that
are
not
selected
in
2
6
Calculate
all
the
measurements
d
ij
and
d
j
k
with
ne
w
virtual
anchors
7
Estimate
the
location
of
remained
sensors
4.
RESUL
T
AND
AN
AL
YSIS
Computer
simulations
are
used
to
e
v
aluate
the
perform
ance
of
the
proposed
algorithm.
W
e
consider
a
netw
ork
with
7
anchors
and
100
sensors
deplo
yed
randomly
in
a
normalized
area.
Simulation
is
performed
using
CVX
toolbox
in
MA
TLAB
softw
are
[14].
The
a
v
erage
estimation
error
is
calculated
as
follo
ws:
A
v
erage
Error
=
1
n
:
n
X
j
=1
k
x
j
a
j
k
(4)
W
e
perform
the
simulation
for
dif
ferent
communication
ranges.
W
e
assume
that
the
distance
measurements
are
corrupted
by
noise
denoted
by
noise
f
actor
as
calculated
using
(5)
and
(6).
W
e
compare
the
results
with
original
SDP
used
to
solv
e
the
WSN
localization
[13].
Simulation
parameters
are
summarized
in
table
2.
d
ij
=
d
ij
:
(1
+
r
andn
(1)
noise
f
actor
)
(5)
d
j
k
=
d
j
k
:
(1
+
r
andn
(1)
noise
f
actor
)
(6)
The
simulation
en
vironment,
the
location
of
anchors,
the
e
xact
and
estimated
locations
of
sensor
s
using
original
SDP
and
proposed
tw
o-stage
SDP
are
illustrated
in
figure
1
and
figure
2,
respecti
v
ely
.
In
figure
1
and
figure
2,
the
communication
range
and
noise
f
actor
are
set
to
0
:
3
and
0
:
15
,
respecti
v
ely
.
The
a
v
erage
e
x
ecution
time
to
solv
e
the
optimization
problem,
highly
depends
to
hardw
are
configuration.
Ho
we
v
er
,
the
relati
v
e
and
normalized
e
x
ecution
time
could
be
used
as
a
criterion
to
compare
the
computational
comple
xity
of
the
proposed
algorithm.
In
this
paper
,
we
ha
v
e
a
v
eraged
the
total
e
x
ecution
time
o
v
er
50
netw
orks
with
1000
realizations.
The
proposed
tw
o
stage
SDP
Localization
of
Distrib
uted
WSNs
using
T
wo
Sa
g
e
SDP
Optimization
(Shahbazian)
Evaluation Warning : The document was created with Spire.PDF for Python.
1258
ISSN:
2088-8708
T
able
2.
Simulation
parameters
used
for
localization
V
ariable
v
alue
Simulation
Area
Normalized
1
1
Number
of
sensors
100
Number
of
Anchors
7
Communication
range
0
:
3
0
:
35
0
:
4
0
:
45
Noise
f
actor
0
:
05
0
:
1
0
:
15
0
:
2
0
:
25
outperforms
the
original
SDP
[10,
13],
by
12
%
impro
v
ement
in
e
x
ecution
time.
One
the
other
hand,
the
a
v
erage
localization
error
sho
ws
30
%
impro
v
ement
on
a
v
erage.
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Estimated Location of sensros
Exact Location of sensos
Anchors
Figure
1.
Simulation
area,
the
e
xact
and
estimated
location
of
sensors
using
original
one-step
localization
In
figure
3,
the
ef
fect
of
noise
f
actor
in
e
x
ecution
time
is
e
v
aluated.
This
simulation
demonstrates
that
the
increase
in
noise
f
actor
,
highlights
the
dif
ference
between
original
SDP
and
the
proposed
algorithm
in
terms
of
e
x
ecution
time.
In
all
noise
f
actors,
the
solution
time
of
the
proposed
algorithm
is
much
less
than
the
original
SDP
.
The
ef
fect
of
communication
range
on
a
v
erage
localization
error
is
presented
in
figure
4.
As
this
simulation
sho
ws,
increasing
the
communication
range,
decreases
the
localization
error
.
The
proposed
tw
o
stage
algorithm
has
better
localization
error
in
comparison
with
original
SDP
in
all
communication
ranges.
Figure
5,
illustrates
the
ef
fect
of
dif
ferent
noise
f
actors
on
a
v
erage
localization
error
.
This
figure
demonstrates
that
in
all
noise
figures
and
corrupted
measurements,
our
proposed
tw
o
stage
localization
has
reduced
the
a
v
erage
loc
alization
error
in
compared
with
one
stage
original
SDP
localization
[10,
13].
5.
CONCLUSION
In
this
paper
we
studied
the
localization
problem
in
wireless
sensor
netw
orks.
The
localization
could
be
interpreted
as
a
non-con
v
e
x
optimization
problem.
Semi-definite
programming
ha
v
e
widel
y
been
used
to
relax
and
solv
e
the
optimization
problem.
Ho
we
v
er
,
the
computational
com
p
l
e
xity
increases
non-linearly
with
increasing
the
number
of
sensors.
W
e
proposed
a
tw
o
stage
SDP
solution
to
sol
v
e
the
localization
problem.
W
e
first,
performed
the
localization
with
some
selected
sensors
and
used
some
of
localized
sensors
as
virtual
anchors.
Simulation
results
confirm
that
the
proposed
algorithm
impro
v
es
the
localization
accurac
y
30%
on
a
v
erage,
while
decreases
the
a
v
erage
e
x
ecution
time
by
12%.
IJECE
V
ol.
7,
No.
3,
June
2017:
1255
–
1261
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1259
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Estimated Location of Sensors
Anchors
Exact Location of Sensors
Figure
2.
Simulation
area,
the
e
xact
and
estimated
location
of
sensors
using
proposed
tw
o-step
localization
0.05
0.1
0.15
0.2
0.25
Noise Factor (Normalized)
0
0.5
1
Total Time (Normalized)
Proposed Two Step SDP
One Step SDP [10], [13]
Figure
3.
The
ef
fect
of
noise
f
actor
on
the
e
x
ecution
time
of
localization
algorithms
Localization
of
Distrib
uted
WSNs
using
T
wo
Sa
g
e
SDP
Optimization
(Shahbazian)
Evaluation Warning : The document was created with Spire.PDF for Python.
1260
ISSN:
2088-8708
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Radio and Measurement Range (Normalized)
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Average Error (Normalized)
Figure
4.
The
ef
fect
of
radio
range
on
a
v
erage
localization
error
0.05
0.1
0.15
0.2
0.25
Noise Factor (Normalized)
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Average Error (Normalized)
Proposed Two Step SDP
One Step SDP [10], [13]
Figure
5.
The
ef
fect
of
noise
f
actor
on
a
v
erage
localization
error
IJECE
V
ol.
7,
No.
3,
June
2017:
1255
–
1261
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1261
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ersion
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BIOGRAPHIES
OF
A
UTHORS
Reza
Shahbazian
recei
v
ed
the
B.Sc.
de
gree
in
Electrical
Engineering
from
the
Iran
Uni
v
ersity
of
Science
and
T
echnology
(IUST),
T
ehran,
Iran,
in
2008
and
the
M.Sc.
de
gree
(with
honors)
in
T
elecommunicati
ons
Engineering
from
the
IUST
,
in
2011.
In
past,
he
has
been
a
member
of
the
W
ireless
Communications
Labora
tory
in
the
School
of
Electrical
Engineering
at
the
IUST
.
He
is
currently
a
member
of
Cogniti
v
e
Radio
Laboratory
at
the
Shahid
Beheshti
Uni
v
ersity
(SB
U),
T
ehran,
Iran,
to
w
ork
under
supervision
of
Dr
.
Ghorashi.
He
is
af
filiated
with
IEEE
as
student
member
.
In
journal
of
supercomputing,
sensor
re
vie
w
,
wireless
personal
communications,
and
other
scientific
publications,
he
has
serv
ed
as
in
vited
re
vie
wer
.
Further
info
on
his
homepage:
http://f
aculties.sb
u.ac.ir/shahbazian
Sey
ed
Ali
Ghorashi
recei
v
ed
his
B.Sc.
and
M.Sc.
de
grees
in
Electrical
Eng.
from
the
Uni
v
ersity
of
T
ehran,
Iran,
in
1992
and
1995,
respecti
v
ely
.
Then,
he
joined
SAN
A
Pro
Inc.,
where
he
w
ork
ed
on
modelling
and
simulation
of
OFDM
based
wireless
LAN
systems
and
interference
cancellation
methods
in
WCDMA
systems.
Since
2000,
he
w
ork
ed
as
a
research
assoc
iate
at
Kings
Colle
ge
London
on
capacity
enhancement
methods
in
multi-layer
W
-CDMA
systems
sponsored
by
Mobile
VCE.
In
2003
He
recei
v
ed
his
PhD
at
Kings
Colle
ge
and
since
then
he
w
ork
ed
at
Kings
Colle
ge
as
a
research
fello
w
.
In
2006
he
joined
Samsung
Electronics
(UK)
Ltd
as
a
s
enior
researcher
and
no
w
he
serv
es
as
an
associate
professor
at
Department
of
T
elecommunications,
F
aculty
of
Electrical
Engineering,
Shahid
Beheshti
Uni
v
ersity
at
T
ehran,
Iran,
w
orking
on
wireless
communications.
Localization
of
Distrib
uted
WSNs
using
T
wo
Sa
g
e
SDP
Optimization
(Shahbazian)
Evaluation Warning : The document was created with Spire.PDF for Python.