PATTERN SYNTHESIS OF NON-UNIFORM AMPLITUDE EQUALLY SPACED MICROSTRIP ARRAY ANTENNA USING GA, PSO AND DE ALGORITHMS

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International Journal of Advanced Research in Engineering and Technology
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Volume 7, Issue 2, March-April 2016, pp. 132–147, Article ID: IJARET_07_02_013
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___________________________________________________________________________
PATTERN SYNTHESIS OF NON-UNIFORM
AMPLITUDE EQUALLY SPACED
MICROSTRIP ARRAY ANTENNA USING
GA, PSO AND DE ALGORITHMS
K. Karuna Kumari
Department of ECE, GITAM University, Visakhapatnam, A.P., India
Prof. P. V. Sridevi
Department of ECE, Andhra University, Visakhapatnam, A.P., India
ABSTACT
In the recent years the development in communication systems requires the
development of low cost, minimal weight, low profile antennas that are
capable of maintaining high performance over a wide spectrum of
frequencies. This technological trend has focused much effort into the design
of a micro strip patch antennas. Nowadays Evolutionary Computation has its
growth to extent. Generally electromagnetic optimization problems generally
involve a large number of parameters. Synthesis of non-uniform linear
antenna arrays is one of the most important electromagnetic optimization
problems of the current interest. This paper is mainly focus on the reduction in
side lobe level of Microstrip linear arrays by using evolutionary
programming-real valued genetic algorithm (EP-GA), particle swarm
optimization algorithm(PSO) and Differential Evolution Algorithm(DE) . The
simulation results of non uniform linear arrays in terms of SLL is calculated.
The Best-Weights for 20 elements array are listed .The Radiation patterns are
presented for different number of elements. All the simulated results are
obtained by using MATLAB software.
Key words: Microstrip antenna Linear array, Real valued-Genetic Algorithm
(GA), Particle Swarm Optimization (PSO) algorithm, Differential Evolution
(DE) algorithm, Side-lobe Level.
Cite this Article: K. Karuna Kumari and Prof. P. V. Sridevi. Pattern Synthesis
of Non-Uniform Amplitude equally Spaced Microstrip Array Antenna Using
GA, PSO and DE Algorithms. International Journal of Advanced Research in
Engineering and Technology, 7(2), 2016, pp. 142–147.
http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=7&IType=2

Pattern Synthesis of Non-Uniform Amplitude equally Spaced Microstrip Array Antenna
Using GA, PSO and DE Algorithms
1. INTRODUCTION
Low side lobe antennas are becoming an increasingly important component of high
performance electronic systems, particularly those operating in heavy clutter and
jamming environments. Practically, the radiation pattern of antenna array has not only
having the main beam but also side lobes .Most of the power is confined in main
beam which provides the coverage in desired portion. Some of the power is also
remained in sidelobe levels that are nothing but wastage of transmitting power [16]. If
the sidelobe levels are very high, large amount of transmitting power is wastage. For
efficient use of transmitting power and desired radiation characteristics can be
achieved by using this five following control methods: 1) The geometrical
configuration of the overall array 2) The relative displacement between the elements
3) The excitation amplitude of the individual elements 4) The excitation phase of
individual elements 5) The relative pattern of the individual elements .Various
analytical and numerical methods have been used to optimize the side lobe levels
relative to the main beam by using above controls.
Several new optimization techniques have been emerged in the past two decades,
that mimic biological evolution, or the way biological entities communicate in nature.
Some of these algorithms have been used successively in many electromagnetic and
antenna problems include Genetic Algorithms (GA) [1], Particle Swarm Optimization
(PSO) [2], and the method of Differential Evolution (DE) Algorithm [3]. In the
electromagnetism research area, mainly the antenna synthesis, the use of these
optimizers is widely and clearly appreciated. To date, different GA, PSO and DE
algorithms have been successfully applied to different problems including antenna
design and the array synthesis.
Many efforts have also been devoted to compare these algorithms to each other.
Typically, such comparisons have been based on artificial numerical benchmark
problems. The goal of many studies was to verify that one algorithm outperformed
another on a given set of problems. In general, it has been possible to improve a given
standard method within a restricted set of benchmark problems by making minor
modifications to it.Recently, Genetic Algorithms (GA) ,particle swarm optimization
(PSO) and differential evolution (DE) have been introduced and particularly PSO has
received increased interest from the EC community. These techniques have shown
great promise in several real-world applications [4]-[7]. In this paper, we investigated
the performance of GA, DE, PSO for the optimization of SLL.
In the following part of the paper, the theory and formulation of the optimization
problem will be explained. After words, the details of the proposed ensemble based
Genetic Algorithms (GA), Particle Swarm Optimization (PSO) and Differential
Evolution (DE) algorithms will be given. Finally, the application of algorithms in the
optimization of SLL will be demonstrated for 20 element antenna array. The
corresponding results of three algorithms will be compared.
2. MICRO STRIP PATCH ANTENNA
Microstrip antennas are used not only as single element but are very popular in arrays.
Arrays are very versatile and are used to synthesize a required pattern that cannot be
achieved with a single element. Arrays increase the directivity, and perform various
other functions which would be difficult with any one single element.
For the microstrip antenna, the x-y plane (π/2, 0  π /2, 3 π /2  2 π) is the
principal E-plane. For this plane, the expression for the radiated fields is[8-9]

K. Karuna Kumari and Prof. P. V. Sridevi
  











 


 cos
2
sin
cos
2
cos
2
sin
0
0
0
Lk
hk
hk
E
2. DESIGN OF LINEAR ARRAY ANTENNAS
Consider a Linear array of 2N symmetric array elements uniformly spaced of distance
d between the elements in x-z plane. The array factor for the linear antenna array is
shown below [10].
Where k is the wave number, k=2ᴨ/λ, θ is the angle measured from the axis of the
array, θo is the steering angle, [ , ] is the excitation of the current for the
element on either side of the array midpoint to be controlled and d is the spacing
between elements. By the array factor, the fitness function is obtained. Here, the
fitness function is used to obtain the optimum weights and to achieve maximum SLL.
Assuming the linear array is symmetric about its center of the array.
Mathematically, the array factor of a 2N element linear array is given by [10]
 )coscos()5.0(cos2 01

  
dnkeIAF nJN
n N



2
k
Where k is the wave number
Fig 1: Geometry of Linear Antenna Array
θ is the angle measured from the axis of the array, θo is the steering
angle,[ , ] is the excitation of the current for the element on either side of the
array midpoint to be controlled and d is the spacing between elements. The fitness
function associated with this array is the maximum SLL of its associated far-field
pattern to be minimized. The general form of the fitness function is given by
)(
))(log20( 10


AFMax
AFMax
Fitness 
Where |AF(θ)|=|AF(θo)| 0 , θ
d

3. THE GENETIC ALGORITHM (GA)
The Genetic Algorithm (GA) is an optimization and global search technique based on
the mechanics of natural selection and natural genetics. A Genetic Algorithm allows a
population composed of many individuals to involve under specified selection rules to
a state that minimizes the cost function. This optimization algorithm is more powerful
for problems with more number of variables and local minima. GA is very efficient in
exploring the entire search space or the solution space, which is large and complex.
The Genetic algorithm is implemented using computer simulation [9]. Genetic
Algorithm may be represented as shown in Fig.3.
In computer algorithm, a chromosome is an array of genes, a number of
chromosomes make up one population. The chromosomes are generated randomly in
the selected space. Each chromosome has an associated fitness function, assigning a
relative merit to that chromosome. The algorithm begins with a large list of random
chromosomes.
Figure 2 Genetic Algorithm for optimization of side lobe level
yes
yes
No
No
New Chromosomes= parents+ new offspring
Select best SLL
Stop
SLL≤ SLL_ Max
SLL≤SLL_Max
Cross over and mutation to generate new
chromosome as child
Evaluate SLL for each chromosome
Chromosomes having best SLL as parent
Create initial population

Fitness functions are evaluated for each chromosome. The chromosomes are
ranked from the best-fit to the least-fit, according to their respective fitness functions.
Unacceptable chromosomes are discarded, leaving a superior species-subset of an
original list, which is the process of selection. Genes that survive become parents, by
crossing over some of their genetic material to produce two new offspring. The
parents reproduce enough to offset the discarded chromosomes. Thus, the total
number of chromosomes remains constant after every iteration. Mutations cause small
random changes in a chromosome. Fitness functions are evaluated for the offspring
and mutated chromosome, and the process is repeated. The algorithm stops after a set
number of iterations, or when an acceptable solution is obtained.
In the genetic algorithm, initial chromosomes are combination of random
chromosome and amplitude excitations of linear array instead of all random
chromosomes. The halves of the chromosome are discarded and new half of
chromosomes are generated from parents chromosome which are best fits to fitness
function. The cost function is the maximum side lobe level for the antenna pattern.
4. PRACTICAL SWARM OPTIMIZATION ALGORITHM
Practical swarm optimization (PSO) is a population-based self-adaptive search
optimization technique first introduced by Kennedy and Eberhart [11]. The PSO
method is becoming very popular due to its simplicity of implementation and ability
to quickly converge to a reasonably good solution. In the practical swarm algorithm,
the trajectory of each individual in the search space is adjusted by dynamically
altering the velocity of each particle, according to its own flying particles in the D-
dimensional search space can be represented as and
respectively.

Fig 3.PSO Algorithm for optimization of side lobe level
DE ALGORITHM
To update the velocity matrix at each iteration K, every particle should now its
personal best and global best position vectors in addition to the neighbor best position.
The personal best position vector defines the position at which each particle attained
its best fitness value up to the present iteration. The personal best position of the
particle is represented as the global best
position vector defines the position in the solution space at which the best fitness
value was achieved by all particles, and is defined by
[12]
Initialize particles with random position&
Velocity Vectors
Calculate fitness values for each particle
Is current fitness value
better than pBest
Assign current fitness as new pBest Keep previous pBest
Use each particle’s velocity value to update
its data values
Calculate velocity for each particle
Assign best particle’s p Best value to g Best
Target or maximum
epochs reached
NO
YES
NO
YES
sss
Ss
ss
St
ar
t
END
Start

The particles are manipulated according to the following equations:
(2)
(3)
In 1995, Price and Storn commenced the Differential Evolution (DE) algorithm
[10] which is based upon differential mutation operator. Practically there are many
problems with different types of objective functions such as non-linear, noisy, flat,
non-differentiable, non-continuous, and multi-dimensional or have many local
minima which are difficult to solve analytically. DE is a robust statistical method for
cost function minimization which does not make use of a single nominal parameter
vector but instead it uses a population of equally important vectors and is very
advantageous to find the most approximate solution to any type of problems.
Differential evolution is a simple, efficient and robust evolutionary algorithm, and
is usually mark as DE/x/y/z, where x denotes how the differential mutation base is
chosen , y denotes the number of vectors differences added to the base vector and z
indicates the crossover method. It has been successfully applied to the array synthesis
problems, electromagnetic inverse problems and many other scientific and
engineering problems. Although it has been reported that differential evolution
performs better than many other algorithms, it is still a dream for differential
evolution users to have a strategy perfectly balancing exploration and exploitation, or
equivalently, reliability and efficiency.
It has been well known that the critical idea behind the success of Differential
Evolution is the creative invention of differential mutation [13-15]. Different
differential mutation strategies balance exploration and exploitation differently. For
example, DE/Best/*/* generally converges faster due to the guidance by the best
individual but may be locally trapped because of loss of directivity, while
DE/Rand/*/* gains directivity at the cost of efficiency. In order to simultaneously,
provide both diversity and guidance so that exploration and exploitation can be more
efficiently balanced. Synthesis capability, reliability and efficiency of DE/Rand/*/*
are tested. The simulation results show that DE/Rand/*/* is able to achieve lower
peak side lobe levels and converge reliably and efficiently.
In DE algorithm the objective function is sampled by a set of initial points which
are chosen randomly from the entire search space. Then in the next step the algorithm
adds the weighted difference between the two randomly selected population vectors to
the third random population vector to generate a new parameter vector. This process
of generating the new parameter vector is called mutation. Now this parameter vector
is further mixed with the predefined parameters to produce the trial vector and this
process is called crossover. Then in the last step, called selection in which trial vector
is replaced by the target vector if the trial vector reduces the values of the cost
function then that obtained due to the target vector.
To realize the algorithm let the problem is function of D number of independent
parameters. In this work for N element array, the number of independent parameters
is 2N in which the first N parameters are the normalized amplitude coefficient of the
N array elements and remaining N parameters are to represent the static phase of each
element. Hence, if NP be the population size then the parameter vectors are
represented as

The target vector compared with the trial vector and the minimum value is
admitted to the next generation. The above steps are continued until the predefined
number of generations reached or the desired value of the cost function is obtained.
The entire process of DE for solving problem is shown in flowchart in Fig 4.
Fig4: DE Algorithm for optimization of SLL
C
R
O
S
S
O
V
E
R
S
E
L
E
C
T
I
O
N
M
U
T
A
T
I
O
N
Yes
Yes
No
Yes
No
No
Start Set G=0 and randomly initiate
Compute
i=1
Save the result and stop
+F( )
f (
G
f(
i=i+1
i=NP
G=G+1

6. RESUITS AND DISCUSSIONS
In this paper, the Non-Uniform linear antenna array is synthesized using global
optimizing techniques called GA, PSO and DE algorithms. The Non-Uniform array is
symmetric array of 2N elements towards the center of the array. The array having
progressive phase and uniform spacing between the elements is 0.5λ.The radiation
pattern of linear antenna array with N=20 elements at scan angles and deg are
given for GA, PSO and DE. In this paper, the array factor is the objective function for
optimizing the fitness /cost function called SLL. By using this evaluation techniques
the est weights are obtained for linear antenna arrays to reduce the SLL.
Figure 5 Radiation pattern of 20 element microstrip array antenna using GA
Figure 6 Cost function graph for 500 iterations
-1.5 -1 -0.5 0 0.5 1 1.5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Theta in (Radians)
NormalizedAmplitude(dB)
Optimized Radiation pattern of microstrip array using GA
0 50 100 150 200 250 300 350 400 450 500
-24
-23
-22
-21
-20
-19
-18
-17
-16
-15
No.of generation
costfunction

-1.5 -1 -0.5 0 0.5 1 1.5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Theta in (Radians)
Optimized Radiation pattern of microstrip array using PSO
0 50 100 150 200 250 300 350 400 450 500
-21
-20.5
-20
-19.5
-19
-18.5
-18
No.generation
Costfunction

-1.5 -1 -0.5 0 0.5 1 1.5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Theta in (Radians)
Optimized Radiation pattern of microstrip array using DE
0 50 100 150 200 250 300 350 400 450 500
-34
-32
-30
-28
-26
-24
-22
-20
-18
-16
No.of generations
costfunction

Figure 11 Radiation Pattern of Linear antenna array of 20 elements using GA, PSO and DE at
θ= (deg)
Figure 13 Radiation Pattern of Linear antenna array of 20 elements using GA, PSO and DE at
θ= (deg)
-1.5 -1 -0.5 0 0.5 1 1.5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Theta in (Radians)
Optimized Radiation pattern of microstrip array using GA,PSO,DE
GA
PSO
DE
-1.5 -1 -0.5 0 0.5 1 1.5
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Theta in Radians
Optimized Radiation pattern of microstrip array using GA,PSO,DE
GA
PSO
DE

Table 1 Best weights for GA, PSO and DE Algorithms with 200 iterations
Table 2 SLL for GA, PSO and DE Algorithms with 200 iterations
S.No
Optimized weights
using
GA(N=20)
Optimized weights
using
PSO (N=20)
Optimized weights
using
DE (N=20)
1 0.12379 1 0.1795
2 0.27239 1 0.2314
3 0.58451 0.71775 0.2507
4 0.36378 -0 0.2329
5 0.29836 1 0.3088
6 0.50807 1 0.3644
7 0.40648 -0 0.3992
8 0.6828 -0 0.6065
9 0.63655 1 0.5111
10 0.63238 0.27016 0.6688
11 0.66836 1 0.7699
12 0.50739 1 0.7218
13 0.86033 0.33163 0.7597
14 0.58251 1 0.7547
15 0.94851 1 1.0864
16 0.91995 1 0.8410
17 0.98516 1 0.9401
18 0.71477 1 1.0661
19 0.986 1 0.9943
20 0.92949 1 1.0953
GENETIC ALGORITHM
PARTICLE SWARM
ALGORITHM
DIFFERENTIAL
EVOLUTION
ALGORITHM
No.of
element
s
Side lobe
level(dB)
Converging
time (sec)
Side lobe
level(dB)
Converging
time (sec)
Side lobe
level(dB)
Converging
time (sec)
10 -27.99 37.62441 -27.808 29.252831 -38.8 14.43
20 -23.97 65.20003 -20.745 51.378521 -32.58 23.99
30 -22.70 93.89461 -20.726 74.199701 -28.12 32.49
40 -21.06 123.4181 -20.708 97.361576 -27.76 42.35
50 20.733 154.0718 -20.658 121.027607 -27.35 51.16
60 -20.645 187.3938 -20.574 144.979070 -27.06 59.89
70 -20.681 216.5338 -20.550 170.463793 -27.168 71.30
80 -20.68 248.7792 -20.393 195.657447 -25.198 83.39
90 -20.6 281.5972 -20.307 220.838639 -24.514 97.67
100 -20.572 281.5972 -20.214 247.091233 -23.625 114.02

Figure 14 Maximum Side lobe level with the number of elements
Figure 15 No. of elements verse Elapse time
10 20 30 40 50 60 70 80 90 100
-40
-38
-36
-34
-32
-30
-28
-26
-24
-22
-20
No.of Elements(N)
NormalisedAmplitude(dB)
No.of elements verse Normalised Amplitude(dB)
GA
PSO
DE
10 20 30 40 50 60 70 80 90 100
0
50
100
150
200
250
300
350
No.of Elements(N)
ElapseTime(seconds)
No.of elements verse Elapse Time(seconds)
GA
PSO
DE

In three algorithms N=20 elements are used. The radiation pattern of Linear
antenna array of 20 elements using GA, PSO algorithm and DE algorithm at
θ= (deg) and at θ= (deg) are represented. The radiation pattern of Linear antenna
array of 20 elements using GA, PSO and DE algorithms at θ= (deg) ,the SLL is -
23.97 dB , -20.45dB and -32.58 dB obtained respectively. The cost functions of
GA,PSO and DE algorithms of 200 iterations are represented. All the three algorithms
produced best results than normal linear antenna array having -13.6dB and
comparatively DE Algorithm produces maximum SLL than PSO and GA. All the
simulation results are obtained using MATLAB software
7. CONCLUSION
In this paper the optimization of 20 elements linear arrays in terms of side level using
genetic algorithm ,particle swarm optimization algorithm ,Differential Evaluations are
given. The side level for GA is -23.97 dB, PSO is - -20.45 dB and for DE is -32.58 for
20 element arrays. All three optimization techniques gives the good results for given
problem, however the elapsed time or converging time for genetic algorithm is more
when compared with PSO algorithm and DE algorithm. The number of generations
/iterations required to get best fitness value for DE is more compared with GA and
PSO.
REFERENCES
[1] Rahmat-Samii, Y., and Michielssen, E.: Electromagnetic Optimization by
Genetic Algorithms. (New York: Wiley-Interscience, 1999)
[2] Robinson, J., and Rahmat-Samii, Y.: Particle swarm optimization in
electromagnetics. IEEE Transactions on Antennas and Propagation, 2004, 52,
pp. 397–407.
[3] Storn, R. and K. Price, Differential evolution—A simple and efficient
heuristic for global optimization over continuous spaces, Journal of Global
Optimization, 11(4), pp. 341–359, 1997.
[4] M. M. Khodier and C. G. Christodoulou, Linear array geometry synthesis
with minimum side lobe level and null control using particle swarm
optimization, IEEE Transactions on Antennas and Propagation, 53(8),
August 2005.
[5] S. Yang, Y. B. Gan, and A. Qing, Sideband suppression in time modulated
linear arrays by the differential evolution algorithm, IEEE Antennas Wireless
Propagat. Lett., 1, pp. 173–175, 2002.
[6] D. Gies and Y. Rahmat-Samii, Particle swarm optimization for
reconfigurable phased-differentiated array design, Microw. Opt. Tech. Lett.,
38(3), pp. 168–175, Aug. 2003.
[7] Kurup, D., Himdi, M., and Rydberg, A.: Synthesis of uniform amplitude
unequally spaced antenna arrays using the differential algorithm, IEEE Trans.
Antennas Propagation, 2003, 51, pp. 2210–2217.
[8] K. Karuna Kumari, Dr.P.V.Sridevi Performance Evaluation of Circular
Microstrip Patch Array with Difference Dielectric Substrate Materials,
International Journal of Electronics and communication Engineering &
Technology 4(1) 2013,pp236–249
[9] K.Karuna Kumari, Dr.P.V.Sridevi, Side lobe level optimization of circular
Microstrip array antenna using Genetic Algorithm, IJECIERD 4(1) January-
Feb-2014pp 9–18

[10] F. Soltankarimi1, J. Nourinia and Ch. Ghobadi, Side Lobe Level
Optimization in Phased Array Antennas Using Genetic Algorithm, ISSST
A2004, Sydney, Australia, 30 Aug, 2 Sep 2004.
[11] J. Kennedy, R. Eberhart, Particle Swarm Optimization, IEEE Int. Conf.
Neural Networks (Perth, Australia), 4, 1995, pp.1942–1948.
[12] J.Robinson, Y.Rahmat-Samii, ParticleSwarm Optimization (PSO) in
Electromagnetics, IEEE Trans. Antennas Propagation, 52(2), February 2004,
pp. 397-407.
[13] S. Das, P. N. Suganthan, Differential Evolution: A survey of the State of- the-
Art, IEEE Transactions on Evolutionary Computation, 15(1), pp. 4–31, 2011
[14] Y. Wang, Z. Cai, and Q. Zhang, Differential evolution with composite trial
vector generation strategies and control parameters, IEEE Transactions on
Evolutionary Computation, 15(1), pp. 55–66, 2011.
[15] J.Robinson, Y.Rahmat-Samii, ParticleSwarm Optimization (PSO) in
Electromagnetics, IEEE Trans. Antennas Propagation, 52(2), February 2004,
pp. 397–407.
[16] Balanis, C.: Antenna Theory-Analysis and Design. (2nd Edition New York:
Wiley, 1997)
[17] Dr. Nagraj Kulkarni and Dr. S. N. Mulgi. A Novel Design and Development
of Superimposed Square Slot Two Element Rectangular Microstrip Array
Antenna for Wide Bandwidth and Intensified Gain. International Journal of
Advanced Research in Engineering and Technology, 5(1), 2014, pp. 187–
194.
[18] Gangadhar P Maddani, Sameena N Mahagavin, Shivasharanappa N Mulgi.
Design and Development of Microstrip Array Antenna for Wide Dual Band
Operation. International Journal of Electronics and communication
Engineering & Technology, 1(1), 2010, pp. 107–116.
[19] Gangadhar P Maddani, Sameena N Mahagavin, Shivasharanappa N Mulgi,
Rectangular Microstrip Array Antennas For Wide Triple Band Operation.
International Journal of Electronics and communication Engineering &
Technology, 1(1), 2010, pp. 53–61.
[20] James Blondin, Particle Swarm Optimization: A Tutorial September4, 2009.

PATTERN SYNTHESIS OF NON-UNIFORM AMPLITUDE EQUALLY SPACED MICROSTRIP ARRAY ANTENNA USING GA, PSO AND DE ALGORITHMS

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PATTERN SYNTHESIS OF NON-UNIFORM AMPLITUDE EQUALLY SPACED MICROSTRIP ARRAY ANTENNA USING GA, PSO AND DE ALGORITHMS