Design &
Optimisation
of a
PIFA Antenna using
Genetic Algorithms
Ameerudden M. Riyad
Prof. H.C.S. Rughooputh
Electronics & Communication Engineering
Mphil / PhD Project
Abstract
Nowadays,
the
development
of
mobile
communications
and
the
miniaturization
of
radio
frequency
transceivers
are
experiencing
an
exponential
growth,
hence
increasing
the
need
for
small
and
low
profile
antennas
.
As
a
result,
new
antennas
have
to
be
developed
to
provide
larger
bandwidth
and
this,
within
small
dimensions
.
The
challenge
which
arises
is
that
the
gain
and
bandwidth
performances
of
an
antenna
are
directly
related
to
its
dimensions
.
The
objective
is
to
find
the
best
geometry
and
structure
giving
best
performance
while
maintain
the
overall
size
of
the
antenna
small
.
This
project
presents
the
optimisation
of
a
Planar
Inverted

F
Antenna
(PIFA)
in
order
to
achieve
an
optimal
bandwidth
in
the
2
GHz
band
.
Two
optimisation
techniques
based
upon
Genetic
Algorithms
(GA),
namely
the
Binary
Coded
GA
(BCGA)
and
Real

Coded
GA
(RCGA)
have
been
experimented
.
The
optimisation
process
has
been
enhanced
by
using
a
Hybrid
Genetic
Algorithm
by
Clustering
.
During
the
optimisation
process,
the
different
PIFA
models
are
evaluated
using
the
finite

difference
time
domain
(FDTD)
method

a
technique
belonging
to
the
general
class
of
differential
time
domain
numerical
modelling
methods
.
2
Desi gn & Opti mi sati on of a PIFA usi ng GA
Agenda
Problem Formulation
Process Overview
PIFA Modelling
FDTD Implementation
GA Optimisation
Simulation & Results
Future Work
Desi gn & Opti mi sati on of a PIFA usi ng GA
3
Problem Formulation
The objective of this project is to optimise the bandwidth of a PIFA antenna while keeping its
overall size small.
The
introduction
of
cellular
communications
and
mobile
satellite
technology
has
led
to
a
growing
awareness
of
the
vital
role
wireless
systems
are
playing
in
communication
networks
.
With
the
advent
of
the
third
and
nowadays
fourth
generation
of
the
mobile
systems
and
the
Universal
Mobile
Telecommunication
System
(UMTS),
efficient
antenna
design
has
been
the
target
of
many
engineers
during
the
past
recent
years
.
The
engineer
nowadays
must
therefore
develop
highly

efficient
and
low
profile
antennas
which
can
be
mounted
on
hand

held
transceivers
4
Desi gn & Opti mi sati on of a PIFA usi ng GA
Process Overview
PIFA
Modelling
Antenna
Evaluation
Performance
Optimisation
5
Desi gn & Opti mi sati on of a PIFA usi ng GA
PIFA Modelling
The
increase
in
the
capacity
and
quality
of
the
new
services
provided
by
mobile
communications
and
wireless
applications
requires
the
development
of
new
antennas
with
wider
bandwidths
.
At
the
same
time,
due
to
the
miniaturisation
of
the
transceivers,
the
antennas
should
have
small
dimensions,
low
profile
and
the
possibility
to
be
embedded
in
the
terminals
.
In
this
context,
PIFA
antennas
are
able
to
respond
to
such
demands
.
Its
conventional
geometry,
that
is,
the
simple
PIFA
is
shown
in
Fig
.
1
below
.
Fig 1. Geometry of a simple PIFA
Geometry of PIFA to be modelled
6
Desi gn & Opti mi sati on of a PIFA usi ng GA
In
the
design
process,
electric
and
magnetic
fields
have
to
be
analysed
in
order
to
evaluate
the
performance
of
the
antenna
.
Various
techniques
exist
for
the
analysis
of
electromagnetic
fields
and
microwave
propagation
.
To
gain
a
better

detailed
understanding
of
electromagnetic
interaction
and
fields,
numerical
simulation
techniques
are
favoured
against
approximate
analysis
methodologies
.
Empirical
methods
require
much
time
and
money
while
a
simple
model
is
more
flexible
and
easy
to
implement
.
To
account
for
the
electromagnetic
propagation
in
space,
a
variety
of
three

dimensional
full

wave
methods
are
available
.
Modelling Techniques
7
Desi gn & Opti mi sati on of a PIFA usi ng GA
A simple virtual model can be more flexible and much cheaper.
Finite
Element
Method
FEM
Transmission
Line Matrix
TLM
Finite
Difference
Time Domain
FDTD
PIFA Modelling
Finite

Difference
Time
Domain
(FDTD)
is
a
popular
and
among
the
most
widely
used
electromagnetic
numerical
modelling
technique
.
It
is
based
on
the
Finite

Difference
Method
(FDM),
developed
by
A
.
Thom
in
the
1920
s
.
8
Desi gn & Opti mi sati on of a PIFA usi ng GA
FDTD Space
•
FDTD starts by
di screti sing
a 3D
space i nto
rectangul ar cel l s,
whi ch are cal l ed
Yee Latti ce.
•
To represent the
di screte space i nto
a hi gh

l evel
programmi ng
l anguage, arrays
must be used.
•
3D Space and Cel l
si ze have to be
defi ned.
Absorbi ng Boundary
Condi ti ons
•
To sol ve for
unbounded
boundari es i n a
fi ni te computati on
space, an auxi l iary
boundary
condi ti on must be
i ntroduced to
effecti vel y absorb
al l
el ectromagneti c
energy i mpi ngi ng
on these
boundari es.
Source Exci tati on
•
Physi cal source
model s need to be
i ntroduced i n the
system to exci te
the fi el ds for
accurate ful l wave
anal ysis.
FDTD Eval uati on
•
The Vol tage
Standi ng Wave
Rati o (VSWR) i s
the key to
obtai ni ng the
bandwi dth of the
PIFA and thus, the
key to achi eve the
objecti ve of thi s
project.
•
VSWR i s
cal cul ated for
several
frequenci es i n the
2GHz band,
rangi ng from
1.9GHz to 2.5GHz.
FDTD Implementation
FDTD Implementation
The
Yee
lattice
is
specially
designed
to
solve
vector
electromagnetic
field
problems
on
a
rectilinear
grid
.
The
grid
is
assumed
to
be
uniformly
spaced,
with
each
cell
having
edge
lengths
∆x
,
∆
y
and
∆
z
.
Fig
.
2
shows
the
positions
of
fields
within
a
Yee
cell
.
Every
E
component
is
surrounded
by
four
circulating
H
components
.
Likewise,
every
H
component
is
surrounded
by
four
circulating
E
components
.
In
this
way,
the
curl
operations
in
Maxwell’s
equations
can
be
performed
efficiently
.
Equations
below
are
called
the
FDTD
field
advance
equations
or
the
Yee
field
advance
equations
Fig 2.
An FDTD cell or Yee cell showing the positions of
electric and magnetic field components
FDTD Space
9
Desi gn & Opti mi sati on of a PIFA usi ng GA
The
solution
space
is
normally
infinite
since
some
problems
require
that
one
or
more
of
the
boundaries
to
be
unbounded
.
For
practical,
purposes,
in
order
to
implement
FDTD,
the
spatial
domain
must
be
limited
in
size
because
it
is
impossible
for
any
computer
to
store
all
fields
in
the
entire
solution
space
if
the
spatial
domain
is
unbounded
.
Various
absorbing
boundary
conditions
(ABC)
have
been
used
for
truncating
the
FDTD
mesh
in
this
project
.
Absorbing Boundary Conditions
10
Desi gn & Opti mi sati on of a PIFA usi ng GA
One of the most
popul ar ABCs was
devel oped by
Mur
[15], based on the
Enqui st

Majda
formul ati on [6]. It
uses the
el ectromagneti c wave
equati on to esti mate
the magni tude and
propagati on di recti on
of the fi el ds near the
outer boundary and
cal cul ates the fi el ds
al ong thi s boundary.
Mur’s
Thi s techni que
i nterpol ates the fi el ds
i n space and ti me,
usi ng a Newton
backward

di fference
pol ynomi al. The
numeri cal predi cti on
of
Li ao’s
ABC i s usually
one order of
magni tude better
than that of the
second

order
Mur’s
.
Liao’s
The Hi gdon Boundary
Operator i s very
advantageous si nce i t
i nvol ves normal
deri vati ve onl y. It
produces hi gher l evel s
of absorpti on over
mul ti pl e angl es, and
has the same degree
of accuracy as the
second

order
Mur
wi th added fl exi bi l ity
of broadeni ng the
absorpti on band
Higdon
More recentl y,
Berenger
i nvented a
more sophi sti cated
ABC, cal l ed the
Perfectl y Matched
Layer (PML)
techni que. Thi s
techni que arti fi ci ally
creates a non

physi cal
absorbi ng medi um
(PML medi um)
adjacent to the outer
boundary of the FDTD
space
PML
FDTD Implementation
To
excite
the
PIFA
with
a
wide
range
of
frequencies,
a
Gaussian
pulse
implemented
as
soft
source
is
used
as
the
excitation
source
.
This
excitation
is
given
by
the
equation
:
where
ω
is
2
π
f
and
f
is the frequency of the pulse
t
is
[(
N
)
–
t
o
] and
N
is the number of time steps
∆
t
is the time step
t
o
is the time at which the pulse reaches the peak value of 1.
τ
controls the width of the pulse
The
Gaussian
excitation
has
some
variable
parameters
which
should
be
adjusted
to
fit
in
the
situation
where
the
excitation
is
being
used
.
Fig
.
3
illustrates
the
excitation
pulse
which
is
used
to
feed
the
antenna
Source Excitation
11
Desi gn & Opti mi sati on of a PIFA usi ng GA
FDTD Implementation
Fig 3. Excitation Gaussian Pulse
E vs. N
FDTD Implementation
The
Voltage
Standing
Wave
Ratio
(VSWR)
is
the
key
to
obtaining
the
bandwidth
of
the
PIFA
and
thus,
the
key
to
achieve
the
objective
of
this
project
.
In
order
to
obtain
the
VSWR,
the
input
impedance
of
the
PIFA
has
first
to
be
determined
.
Using
the
input
impedance,
a
scattering
parameter,
S
11
which
is
the
reflection
coefficient,
can
be
evaluated
and
consequently
the
VSWR
is
calculated
as
VSWR
is
calculated
for
several
frequencies
in
the
2
GHz
band,
ranging
from
1
.
9
GHz
to
2
.
5
GHz
.
A
graph
of
VSWR
against
frequencies
can
be
plotted
to
observe
the
parabolic
shape
of
the
curve
.
The
performance
of
the
antenna
is
then
evaluated
by
determining
the
bandwidth
from
the
range
of
frequencies
where
the
VSWR
is
less
than
2
(Fig
.
4
)
.
Fig 4. Graph of VSWR vs. Frequency
Performance Evaluation
12
Desi gn & Opti mi sati on of a PIFA usi ng GA
GA Optimisation
GA
is
a
very
powerful
search
and
optimisation
tool
which
works
differently
compared
to
classical
search
and
optimisation
methods
.
GA
is
nowadays
being
increasingly
applied
to
various
optimising
problems
owing
to
its
wide
applicability,
ease
of
use
and
global
perspective
.
As
the
name
suggests,
genetic
algorithms
borrow
its
working
principle
from
natural
genetics
.
Genetic
algorithms
(
GAs
)
are
stochastic
global
search
and
optimisation
methods
that
mimic
the
metaphor
of
natural
biological
evolution
.
GAs
operate
on
a
population
of
potential
solutions
applying
the
principle
of
survival
of
the
fittest
to
produce
successively
better
approximations
to
a
solution
.
At
each
generation
of
a
GA,
a
new
set
of
approximations
is
created
by
the
process
of
selecting
individuals
according
to
their
level
of
fitness
in
the
problem
domain
and
reproducing
them
using
operators
borrowed
from
natural
genetics
.
This
process
leads
to
the
evolution
of
populations
of
individuals
that
are
better
suited
to
their
environment
than
the
individuals
from
which
they
were
created,
just
as
in
natural
adaptation
.
Genetic Algorithms Concept
13
Desi gn & Opti mi sati on of a PIFA usi ng GA
GA Optimisation
Genetic
Algorithms
is
applied
to
the
whole
FDTD
process
which
acts
as
the
main
component
for
the
fitness
evaluation
.
GA
begins
its
search
with
a
random
set
of
solutions,
analyses
the
solutions
and
selects
the
best
ones
to
afterwards
converge
to
the
optimal
solution,
which
will
result
to
the
best
bandwidth
performance
.
The
working
principle
of
GAs
is
very
different
from
that
of
most
of
classical
optimisation
techniques
.
GA
is
an
iterative
optimisation
procedure
.
Instead
of
working
with
a
single
solution
in
each
iteration,
a
GA
works
with
a
number
of
solutions,
known
as
a
population,
in
each
iteration
.
A
flowchart
of
the
working
principle
of
a
simple
GA
is
shown
in
Fig
.
5
.
Fig 5. Working principles of a simple GA process
14
Desi gn & Opti mi sati on of a PIFA usi ng GA
Begin
Initialise population
Crossover
Mutation
Reproduction
Evaluation
Assign fitness
Stop
Gen = Gen + 1
Gen = 0
Yes
No
Condition
satisfied?
Working principles
GA Optimisation
In
this
project,
the
set
of
solutions
was
first
coded
in
binary
string
structures
and
Binary

Coded
GA
was
used
for
this
purpose
.
Then
Real

Coded
GA
was
used
for
improvement
in
convergence
and
precision
to
the
optimal
solution
.
The
GA
was
then
modified
to
a
hybrid
version
using
Clustering
technique
.
GA optimisation techniques
15
Desi gn & Opti mi sati on of a PIFA usi ng GA
In the Bi nary

Coded GA
(BCGA), the basi c bl ock
of the geneti c
al gori thm i s the
chromosome.
Each chromosome i s
composed of genes
descri bed as a bi nary
sequence of zeros and
ones.
Each gene i s associated
wi th a parameter to be
opti mi zed.
BCGA
Real coded GA (RCGA)
represents parameters
wi thout coding, which
makes representation of
the solutions very cl ose to
the natural formulation of
many problems.
Real

world optimization
probl ems often i nvolve a
number of characteristics,
whi ch make them difficult
to sol ve up to a required
l evel of satisfaction
RCGA
GA can sometimes get
stuck on sub optimal
sol ution wi thout any
progress to the real
opti mal solution.
One of the possible
sol ution to this problem is
to mai ntain a population
si ze as l arge as possible.
However, maintaining
l arge population i nvolve
hi gh cost to evaluate each
i ndividual. Therefore to
reduce the cost of
eval uation and accelerate
the convergence the
Hybri d Cl ustering GA i s
applied i n this work
Clustering
GA Optimisation
GA experimentation
16
Desi gn & Opti mi sati on of a PIFA usi ng GA
•
The population strings are represented as shown in Fig. 6 below:
•
where X
i
is a binary digit (0 or 1) and
i
taking values from 1 to 5. As illustrated in Fig.
3, the first 2 bits represent the parameter
f
x
, the next 2 bits the parameter
f
z
and the
last bit the height h of the radiating plate.
•
Each string is decoded, mapped and evaluated. The evaluation process involves the
FDTD method mentioned earlier.
Binary

Coded
GA
•
After experimenting the BCGA, RCGA was experimented to compare the convergence
and precision of the optimization process
•
In RCGA, decision variables are used directly to form chromosome

like structure.
Chromosome represents a solution and population is a collection of such solutions.
The operators modify the population of the solution to create new one.
•
For implementing the RCGA in order to solve problems developed in this model, the
following basic components are considered: Parameters of GA, Representation of
chromosomes, Initialization of chromosomes, Evaluation of fitness function, Selection
process, Genetic operators like crossover and mutation .
Real

Coded GA
f
x
f
z
h
X
1
X
2
X
3
X
4
X
5
Fig. 6. Population string known as chromosome
GA Optimisation
GA experimentation
17
Desi gn & Opti mi sati on of a PIFA usi ng GA
•
Clustering is a simple method of grouping the population into several small groups,
called as clusters. Fig. 7 illustrates the concept behind the conventional GA and the
modified clustered GA.
•
The algorithm evaluates only one representative for each cluster. The fitness of other
individuals are estimated from the representatives’ fitness. Using this method, large
population can be maintained with relatively less evaluation cost. One of the
important factors to take into consideration for clustering is the similarity measure.
This is commonly achieved using distance measures such as Euclidean distance, City
block distance and
Minkowski
distance .
•
There exist other clustering techniques namely the Hierarchical clustering, Overlapping
clustering and
Partitional
clustering. A hybrid GA with clustering based on the k

means
algorithms [11] from
Partional
clustering had been used in the presented work
because of its applicability and flexibility of specifying the number of clusters required
Clustering
GA
Fig. 7. Conventional GA vs. Clustered GA
Simulation
In
this
project,
MATLAB
has
been
opted
for
the
simulation
owing
to
its
distinct
advantages
over
other
programming
language
for
scientific
purposes
.
MATLAB
proved
to
be
suitable
for
the
simulation
although
the
processing
time
is
a
little
more
than
in
C
or
C++
.
MATLAB
facilitated
the
plotting
of
three

dimensional
graphs
and
debugging
of
the
program
is
done
easily
The
computer
program
is
written
according
to
the
FDTD
algorithm
by
following
all
the
conditions
necessary
for
convergence
of
solutions
.
To
be
more
flexible,
the
parameters,
such
as
the
solution
space,
frequency
of
excitation,
number
of
time
steps
and
others
defined
at
the
beginning
of
the
computer
program
may
be
modified
at
will
without
affecting
the
running
of
the
simulation
.
A
series
of
tests
were
carried
out
throughout
the
work
to
check
whether
the
implementation
of
the
FDTD
was
good
enough
to
evaluate
the
performance
of
the
PIFA
.
These
tests
were
carried
out
using
different
boundary
conditions,
different
excitation
pulses
and
different
computational
space
size
.
18
Desi gn & Opti mi sati on of a PIFA usi ng GA
Hi gh peaks at the
boundari es
refl ecti on of the waves at
the boundary causes
stati onary waves resul ti ng
to smal l standing waves
i nsi de the FDTD space
Wi thout
.
Good absorpti on of the
fi el ds
Not enough si gni fi cant
attenuati on
Hi gdon
.
Good absorpti on
Does not requi re
knowl edge fi el ds adjacent
to the cel l s
Does not perform wel l at
l ow frequency & Large
computati onal ti me
Dispersed
.
Provi des
refl ecti onl ess
boundary over broad
spectrum
Achi eves mi ni mal i n
computati onal cost and
memory requi rement
Mur’s
Simulation was carried out initially on different absorbing boundary conditions (Higdon,
Dispersed, Mur’s) as well as without any absorbing boundary condition.
Following are the simulation results
:
Absorbing Boundary Condition Simulation
19
Desi gn & Opti mi sati on of a PIFA usi ng GA
Simulation
Simulation
The
FDTD
mesh
size
has
to
be
defined
large
enough
for
the
waves
to
propagate
smoothly
.
A
very
large
mesh
size
would
obviously
give
better
approximation
of
the
fields
propagation
since
the
reflection
from
the
boundaries
would
be
very
far
from
the
source
(if
the
source
is
located
in
the
vicinity
of
the
centre
of
the
FDTD
space)
.
However,
a
very
large
mesh
size
would
automatically
increase
the
simulation
time
considerably
.
The
ground
plate
and
the
radiating
plate
are
assumed
to
be
infinitely
thin
perfect
conductors
and
their
conductivity
has
been
set
to
infinity
in
the
FDTD
model,
that
is,
they
have
been
considered
as
PEC
walls
in
the
FDTD
algorithm
.
In
this
work,
the
FDTD
mesh
size
was
set
to
approximately
20
cells
away,
in
all
direction,
from
the
PIFA
to
be
modelled
.
Thus,
within
90
time
steps,
the
fields
may
propagate
with
a
minimum
of
reflection
from
the
boundaries
and
the
simulation
took
approximately
24
hrs
to
display
a
single
value
of
the
VSWR
on
a
Pentium
4
,
1
.
86
GHz
computer
and
took
more
than
3
days
on
a
slightly
less
powerful
machine
Fig 8. FDTD Mesh Size
FDTD mesh size
20
Desi gn & Opti mi sati on of a PIFA usi ng GA
Results
The
PIFA
was
excited
using
a
Gaussian
waveform
of
frequency
ranging
from
1
.
9
GHz
to
2
.
5
GHz
and
the
boundary
condition
used
was
the
Mur’s
second
order
ABC
.
The
figures
show
the
top
and
side
views
of
the
PIFA
which
the
FDTD
algorithm
evaluated
.
The
feeding
point,
that
is,
the
source
location
can
be
varied
by
adjusting
the
parameters
f
x
and
f
z
.
The
height
of
the
radiating
plate
from
the
ground
plate
may
be
varied
by
changing
the
value
of
the
parameter
‘h’
.
The
variation
of
the
height
is
quite
small
(approximately
2
mm)
since
the
idea
of
the
project
is
to
maximise
the
bandwidth
of
the
PIFA
while
keeping
the
overall
dimensions
constant
.
Fig 9. Top and Side views of PIFA to be modelled
PIFA Modelled
21
Desi gn & Opti mi sati on of a PIFA usi ng GA
Results
The
frequency
range
of
interest
is
from
1
.
9
GHz
to
2
.
5
GHz
and
graphs
of
the
VSWR
against
the
frequencies
were
plotted
in
order
to
calculate
the
bandwidth
of
the
PIFA
.
It
is
noteworthy
that
the
smaller
is
the
frequency
interval
for
simulation,
the
smoother
is
the
graph
.
Owing
to
very
large
simulation
time
for
a
single
value
of
VSWR,
the
frequency
interval
was
taken
as
0
.
1
GHz
to
obtain
the
corresponding
value
of
VSWR
.
the
bandwidth
obtained
is
approximately
420
MHz
.
Fig 10. Graph of VSWR
vs
Frequency
Frequency Range
22
Desi gn & Opti mi sati on of a PIFA usi ng GA
Fig 11. E

field Propagation
The Bi nary Coded GA has proved to
be a very good opti mi si ng tool and i f
used properl y, i t may serve to sol ve
vari ous probl ems of search and
opti mi sati on.
Cl assical opti mi sing methods woul d
take much l onger ti me to fi nd the
opti mal sol uti on as compared to the
Bi nary GA method.
However, the opti mi sati on does not
al ways converge to the opti mal
sol uti on and someti mes get di verted
to some other sub

opti mal sol uti on.
BCGA
The Real Coded GA has been chosen
as the al ternati ve for Bi nary Coded
GA. The most i mportant feature of
the RCGA has been observed to be
the capaci ty to expl oi t l ocal
conti nui ti es.
Owi ng to the user of real
parameters i n RCGA, l arge domai ns
for the vari ables can be used as
opposed to BCGA i mpl ementati ons
where i ncreasi ng the domai n woul d
decrease the preci si on.
RCGA
The Hybri d GA by cl usteri ng, on the
other hand, has shown to converge
faster to the opti mal sol uti on.
Popul ati on si ze coul d be i ncrease
wi thout affecti ng the performance
of the opti mi sati on usi ng the Hybri d
GA by cl usteri ng.
The convergence l ooks si mi l ar to the
RCGA but performance i s much
better because of the FDTD
eval uati on of onl y the
representati ves of each cl uster.
Cl ustering
GA Outcome
23
Desi gn & Opti mi sati on of a PIFA usi ng GA
Results
Future Work
•
During
the
FDTD
simulation
process,
it
has
been
observed
that
processor
was
highly
overloaded
owing
to
the
large
computational
space
and
complexity
of
the
field
calculation
.
Consequently,
the
processing
of
the
FDTD
was
very
bulky
and
consumed
a
considerable
amount
of
processing
time
throughout
the
whole
optimisation
system
.
One
of
the
future
works
would
involve
improving
the
FDTD
process
through
code
and
logic
optimisation
.
Another
approach
would
imply
approximating
the
FDTD
process
through
other
simplified
models
achieving
the
same
results
.
FDTD Improvement
•
In
this
work,
we
have
formulated
and
solved
a
bandwidth
problem
for
the
PIFA
antenna
.
However,
in
Binary

Coded
GA
or
Real

Coded
GA,
a
difficulty
regarding
the
boundaries
of
the
decision
variables
was
encountered
.
The
optimisation
sometimes
converge
wrongly
or
take
very
long
to
converge
to
the
optimal
solution
.
As
part
of
the
future
work,
the
GA
has
to
be
analysed
thoroughly
and
improvements
identified
.
Few
of
the
approaches
would
be
:
Redefining
combination
of
operators
and
Hybrid
merging
of
GA
techniques
.
GA Improvement
24
Desi gn & Opti mi sati on of a PIFA usi ng GA
Thank you…
Desi gn & Opti mi sati on of a PIFA usi ng GA
25
Main
References
Pinho
,
P
.
T
.
,
Pereira,
J
.
R
.
,
"Design
of
a
PIFA
antenna
using
FDTD
and
Genetic
Algorithms",
Proc
IEEE
AP

S/URSI
International
Symp
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,
Boston,
United
States,
Vol
.
4
,
pp
.
700

703
,
July,
2001
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Rashid
A
.
Bhatti
,
Mingoo
Choi
,
JangHwan
Choi
,
and
Seong
Ook
Park,
“Design
and
Evaluation
of
a
PIFA
Array
for
MIMO

Enabled
Portable
Wireless
Communication
Devices”,
IEEE
Antenna
and
Propagation
Symposium
2008
,
San
Diego,
America,
July
5

12
,
2008
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Y
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Gao
,
X
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Chen,
Zhinong
Ying,
and
C
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Parini
,
“Design
and
performance
investigation
of
a
dual

element
PIFA
array
at
2
.
5
GHz
for
MIMO
terminals”,
IEEE
Transactions
on
Antennas
and
Propagation,
vol
.
55
,
no
.
12
,
2007
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K
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Deb
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“Optimization
for
engineering
design
:
Algorithms
and
examples”,
Prentice

Hall,
Delhi,
1995
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Gedney
and
Maloney,
“Finite
Difference
Time
Domain
modeling
and
applications”,
FDTD
Short
Course,
Mar
.
1997
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D
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Y
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Su,
D
.

M
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Fu,
and
D
.
Yu,
"Genetic
Algorithms
and
Method
of
Moments
for
the
Design
of
Pifas
",
Progress
In
Electromagnetics
Research
Letters,
Vol
.
1
,
9

18
,
2008
.
Maulik
U
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and
Bandyopadhyay
S
.
,
“Genetic
algorithm

based
clustering
technique”,
Journal
of
Pattern
Recognition
Society,
1999
.
Seront
,
G
.
and
Bersini
,
H
.
,
"A
new
GA

local
search
hybrid
for
continuous
optimization
based
on
multi
level
single
linkage
clustering,"
Proc
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of
GECCO

2000
,
pp
.
90
~
95
,
2000
.
Thanks
to
the
Tertiary
Education
Commission
(TEC)
of
Mauritius
for
sponsoring
my
post
graduate
research
work
at
the
University
of
Mauritius
.
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