Lazinica A. (ed.) Particle Swarm Optimization
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A Particle Swarm Optimization technique used for
the improvement of analogue circuit performances
181
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11
Particle Swarm Optimization Applied for
Locating an Intruder by an Ultra-Wideband
Radar Network
Rodrigo M. S. de Oliveira, Carlos L. S. S. Sobrinho, Josivaldo S. Araújo and
Rubem G. Farias
Federal University of Pará (UFPA)
Brazil
1. Introduction
As it was shown by the authors in a previous work, the Finite-Difference Time-Domain
(FDTD) method is adequate to solve numerically Maxwell's Equations for simulating the
propagation of Ultra-Wideband (UWB) pulses in complex environments. These pulses are
important in practice in high-resolution radar and GPS systems and in high performance
(wideband) wireless communication links, because they are immune to selective frequency
fading related to complex environments, such as residences, offices, laboratories among
others. In this case, it is necessary to use spread spectrum techniques for transmission, in
order to avoid interferences to other wireless systems, such as cell phone networks, GPS,
Bluetooth and IEEE802.11. It is worth to mention that by combining these techniques to
UWB pulses; it is possible to obtain a signal with power spectrum density under noise
threshold, what is a very interesting characteristic for this application.
The proposed simulated environment is a building consisting of several rooms (laboratories)
separated by masonry. Internal and external walls are characterized by specific widths and
electrical parameters. Wood doors were included in the analysis domain. The analysis
region is then limited by U-PML (Uniaxial Perfectly Matched Layers) technique and the
system is excited by omni-directional antennas. In order to make the simulations more real,
Additive White Gaussian Noise was considered. Aiming at verifying the robustness of the
radar network, objects are included in the domain in a semi-random spatial distribution,
increasing the contribution of the wave scattering phenomena. Omni-directional antennas
were used to register transient electric field in specific points of the scenery, which are
adequate for the propose of this work. From those transient responses, it is possible to
determine the time intervals the electromagnetic signal requires to travel through the paths
transceiver-intruder-transceiver and transceiver-intruder-receivers, forming, this way, a
non-linear system of equations (involving circle and ellipses equations, respectively).
In order to estimate the intruder position, the PSO method is used and a new methodology
was conceived. The main idea is to apply PSO to determine the equidistant point to the
circle and to the two ellipses generated by using the data extracted from received transient
signals (those three curves usually does not have a single interception point for highly
Particle Swarm Optimization
184
scattering environments). The equidistant point, determined via PSO, is the position
estimative for single radar.
For a radar network, which is necessary for a large area under monitoring, the transmitters
should operate in TDM (Time-Division Multiplexing) mode in order to avoid interference
among them. For each possible transceiver-receivers combination, an estimate is obtained
and, from the set of estimations, statistical parameters are calculated and used in order to
produce a unique prediction of the intruder's position.
2. The FDTD Method and its Applications
The FDTD Method (Finite-Difference Time-Domain) had its first application in the solution
of Maxwell’s equations, in 1966, when Kane Yee used it in the analysis of spread of
electromagnetic waves through bidimensional structures (Yee, 1966). This technique defines
the spacial positioning of the components of the electric and magnetic fields in such a way
that Ampère and Faraday laws are satisfied, and it approaches the derivates, constituents of
those equations, by centered finite differences, in which the updating of the components of
the electric fields is alternately verified in relation to those of the magnetic fields, by forming
this way what is known as the algorithm of Yee. The method constitutes a solution of
complete wave, in which the reflection, refraction, and diffraction phenomena are implicitly
included.
Years passed by and, along with them, several scientific advances contributed to the
establishment of this method as an important tool in the analysis and synthesis of problems
in electromagnetism, among them it is noted: new high speed computers and auto-
performance computer networks; the expansion of the method for the solution of problems
in the 3D space, with the inclusion of complex materials, and the condition of stability
(Taflove & Brodwin, 1975); development of truncation techniques of the region of analysis,
known as ABC´s (Absorbing Boundary Conditions), such as the operators of Bayliss-Turkel
Bayliss, & Turkel, (1980), Mur of first and second orders (Mur, 1981), Higdon technique
(Ridon, 1987), Liao (Liao, 1987), PML of Berenger (Berenger, 1994), and the UPML of Sacks
(Sacks et al., 1995).
The FDTD method, for its simplicity of application, strength and application in all spectrum
of frequencies, has been used in the solution of antenna problems (Zhang et al., 1988), circuit
analysis in high frequencies (Fornberg et al., 2000), radars (Muller et al., 2005), photonic
(Goorjian & Taflove, 1992), communication systems (Kondylis et al., 1999), periodic
structures (Maloney & Kesler, 1998), medicine (Manteuffell & Simon, 2005) , electric
grounding system (Tanabe, 2001) , etc.
2.1 The Yee’s Algorithm
Equations (1) and (2) represent the equations of Maxwell in its differential form, where E
and H are the vectors intensity of electric and magnetic fields, respectively, μ is the magnetic
permeability, ε is the electric permittivity of the medium and J is the current density vector.
For the solution of these equations by the FDTD method, Kane Yee (Yee, 1966) proposed
that the components of E (E
x
, E
y
, E
z
) and H (H
x
, H
y
, H
z
) were positioned in the space as it
shown in Fig. 1.
xµ
∂
∂t
(1)
Particle Swarm Optimization Applied for Locating an Intruder
by an Ultra-Wideband Radar Network
185
xε
∂
∂t
.
(2)
Such procedure is justified by the necessity of agreement with the mathematical operators
indicated in the equations above.
Figure 1. The Yee’s Cell
This way, by expanding the curl operators in (1) and (2), it results in the following scalar
equations
1
,
(3.a)
1
,
(3.b)
1
,
(3.c)
and
1
y
xz
x
H
EH
E
tzy
σ
ε
∂
⎛⎞
∂∂
=−−
⎜⎟
∂∂∂
⎝⎠
(4.a)
1
,
(4.b)
1
,
(4.c)
respectively.
Particle Swarm Optimization
186
The derivates in (3) and (4) are then approximated by central finite differences, in the
following way
11
22
()()Fl l Fl l
F
ll
+Δ− −Δ
∂
∂Δ
(5)
where F represents any component of either electric or magnetic field and can be x, y, z or t.
By applying (5) in (3) and (4), it results in the updating equations of the components of fields
given by (6)-(7), as follows.
,,
,,
∆
,, 1
,,
∆
, 1,
,,
∆
,
(6.a)
,,
,,
∆
1,,
,,
∆
,, 1
,,
∆
,
(6.b)
,,
,,
∆
, 1,
,,
∆
1,,
,,
∆
,
(6.c)
and
,,
,,
1
∆
2
1
∆
2
∆
1
∆
2
/
,,
/
,1,
∆
/
,,
/
,, 1
∆
,
(7.a)
,,
,,
1
∆
2
1
∆
2
∆
1
∆
2
/
,,
/
,, 1
∆
/
,,
/
1,,
∆
,
(7.b)
Particle Swarm Optimization Applied for Locating an Intruder
by an Ultra-Wideband Radar Network
187
,,
,,
1
∆
2
1
∆
2
∆
1
∆
2
/
,,
/
1,,
∆
/
,,
/
,1,
∆
,
(7.c)
where i,j,k and n are integers; i,j,k are indexes for the spatial coordinates x, y and z and n is
the temporal index for the time t, in such way that x = i∆
x
, y = j∆
y
, z = k∆
z
and t = n∆
t
(∆
x
, ∆
y
and ∆
z
are the spatial increments and ∆
t
is the time step).
2.2 Precision and Stability
The precision represents how close the obtained result is to the exact result, and the stability
is the guarantee that the solution of the problem will not diverge. In order to precision and
stability to be guaranteed, the following criteria are adopted in this work (Taflove &
Hagness 2005):
∆
,,
λ
10
and
∆
1
1
∆
1
∆
1
∆
.
which means that the minimum wave length existing in the work environment has to be
characterized by, at least, 10 cells (Taflove & Hagness 2005). Depending on the application,
this number can be superior to 100, and the time increment will be limited by the maximum
distance to be travelled, by the respective wave, in the cell of Yee (Taflove & Hagness 2005).
2.3 The Sacks’ Uniaxial Perfecttly Matched Layers
One of the problems of the numeric methods is the fact that they do not offer resources that
permits the interruption of the spacial iterative process. This causes the method to be
limited, mainly when the solution of open problems is taken into account. In order to solve
this problem, several techniques have been developed, among them there is the UPML
(Uniaxial perfecttly matched layers) (Sacks et al., 1995), which was used in this work. This
method takes into account, around the region of analysis (Fig.2), layers perfectly matched,
constituted by anisotropic media and with loss, which are characterized by the following
equations of Maxwell, in the frequency domain.
xjωµ
s
(8)
xjωε
s
,
(9)
18
8
w
h
at
t
th
e
d
o
Fi
g
In
th
e
w
h
an
of
d
o
th
r
&
H
3.
3.
1
T
h
pe
8
h
ere the tensor
t
enuation electri
c
e
medium. E and
o
main, respective
l
g
ure 2. The anal
y
order to avoid t
e
time domain, o
n
h
ere D
x
, D
y
, D
z
a
n
d ma
g
netic (B),
r
the electric field,
o
ne b
y
the substi
t
r
ou
g
h eq. (10). T
h
H
a
g
ness, 2005).
Particle Swar
m
1
Basic Radar T
h
h
e main reason f
o
rformed b
y
tran
s
[]
0
0
0
0
yz
x
x
ss
s
s
s
s
⎡
⎢
⎢
⎢
=
⎢
⎢
⎢
⎢
⎣
c
conductivit
y
in
H are the vector
ly
.
sis re
g
ion limite
d
he use of convo
l
n
e defines the fol
,
y
x
x
x
s
DE
s
ε
⎛⎞
=
⎜⎟
⎝⎠
,
y
x
x
x
s
BH
B
s
μ
⎛⎞
=
⎜⎟
⎝⎠
n
d B
x
, B
y
, B
z
are
r
espectivel
y
. In
o
the updatin
g
eq
u
t
ution of [s] in (9
)
h
e same procedu
r
m
Optimizatio
n
h
eory Overview
o
r usin
g
radars
i
s
mittin
g
electro
m
0
0
0
0
x
z
y
xy
z
s
s
ss
s
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, wi
t
the directions α
s intensit
y
of ele
c
d
b
y
UPML
l
ution in the tra
n
lowin
g
constitut
i
,
z
yy
z
y
s
DED
s
ε
⎛⎞
=
⎜⎟
⎜⎟
⎝⎠
,
z
yy
z
y
s
B
HB
s
μ
⎛⎞
=
⎜⎟
⎜⎟
⎝⎠
the components
o
rder to find the
u
u
ations for D
x
, D
y
)
, considerin
g
(1
0
r
e is done in ord
e
n
and Radar A
p
i
s to determine t
m
a
g
netic pulses t
h
P
a
t
h
0
1
σ
s
j
α
α
ωε
=+
,
= x,
y
, z, introd
u
c
tric and ma
g
net
i
n
sformation of t
h
i
ve relations:
;
x
z
z
z
s
E
s
ε
⎛⎞
=
⎜⎟
⎝⎠
,
x
z
z
z
s
H
s
μ
⎛⎞
=
⎜⎟
⎝⎠
of the vectors e
u
pdatin
g
equati
o
y
and D
z
, have to
0
). Next, the com
e
r to find the co
m
p
plications
he spatial positi
o
h
rou
g
h space, w
h
a
rticle Swarm Opti
m
in which
σ
α
u
cin
g
the anisotr
i
c field in the fre
q
h
e equations abo
v
lectric flux dens
i
o
ns for the comp
o
be found first,
w
ponents of E are
m
ponents of H (
T
o
n of an ob
j
ect.
T
h
ich will be scatt
e
m
ization
is the
op
y
in
q
uenc
y
v
e into
(10)
(11)
i
t
y
(D)
o
nents
w
hich is
found
T
aflove
T
his is
e
red b
y
Particle Swarm Optimization Applied for Locating an Intruder
by an Ultra-Wideband Radar Network
189
the target and by other objects. Transient responses obtained at certain points in space are
used to determine the target’s position.
In particular, multistatic radars can be composed by a transceiver (transmitter and receiver
at the same point – Tx/Rx1) and, at least, two remote receivers (Rx2 and Rx3), such as
illustrate
d by Fig. 4. The signal leaves the transceiver and reaches the target, which reflects it
toward the receivers and the transceiver as well.
From the transceiver perspective, the signal takes a certain amount of time for returning.
This only means that the target could be at any point of a circle centered at the transceiver’s
coordinates (Fig. 4). Longer the time, larger is the circumference. From the receiver
perspective, the signal travels from the transmitter, it reaches the target and then it arrives at
the receiver coordinates. This only means that the target is at the locus defined by an ellipse
with foci at the transceiver’s and at the receiver’s coordinates (Fig. 3). Of course, longer the
propagation time, greater is the total path (calculated by using time and the propagating
speed) and larger is the ellipse’s semiminor axis. The solution of the system of equations this
way composed provides the target’s position.
Figure 3. Ideal multistatic radar
Figure 4. The ellipse’s parameters
Fig. 4 shows the ellipse’s basic parameters (T indicates the transceiver position, R indicates
the receiver’s, position and P defines the intruder’s location).
Target
Tx/Rx1
Rx 2
Rx3
Particle Swarm Optimization
190
The ellipse equation is given by
,
,
,
,
(12)
where
,
,
(13)
,
and
(14)
,
(15)
in which a is the semimajor axis, b is the semiminor axis, x
c
and y
c
are the coordinates of the
center C of the ellipse, and α is the angle from the x-axis to the ellipse’s semimajor axis.
Here, n is the receiver identifier (index). The parameters
n
C
x
,
n
C
y
,
n
a
,
n
b
and
n
α
are
calculated by
,
(16)
,
(17)
,
(18)
,
(19)
,
(20)
.
(21)
were x
T
and y
T
are the coordinates of the transmitter, x
R
and y
R
are the coordinates of the
receiver R and d
TR
is the distance from the receiver to the transmitter. Finally, d
TPR
is given