A Proposed Method for Synthesizing the Radiation Pattern of Linear
Antenna Arrays
AMIN AL KA’BI
Australian College of Kuwait, KUWAIT
Abstract – The design of antenna arrays is one of the most challenging optimization problems in recent research interests.
In this research work a new method of optimization proposed. This method called “Characteristics Evolution
Optimization” is based on parallel processing of streams of binary digits, and hence it can perform well in parallel
processing digital systems. In this article, a 16 - element linear antenna array has been taken into consideration, and the
performance of the proposed technique for synthesizing the radiation pattern of the array has been investigated and
compared with other existing techniques, such as DE (Differential Evolution), IWO (Invasive Weed Optimization), and
PSO (Particle Swarm Optimization). Various variants of Invasive Weed Optimization have been investigated as well. It
has been observed that the proposed method (Characteristics Evolution optimization) outperforms the other optimization
techniques significantly in different aspects.
Keywords: Pattern Synthesis, Linear Array, Antenna Arrays Characteristics Evolution Optimization, Radiation patterns
Received: August 12, 2021. Revised: March 16, 2022. Accepted: April 18, 2022. Published: May 19, 2022.
.
1. Introduction
Antenna Arrays play an important role in detecting and
processing signals arriving from different directions. The
role of antenna array synthesis is to determine the physical
layout of the array, and the amplitude and phase excitation
that produces a radiation pattern that is closest to the
desired radiation pattern. The shape of the desired pattern
can vary widely depending upon the application. Some
applications require a low sidelobe level, while other
applications require an interference reduction using null
control. However, the global synthesis of antenna arrays
that generate a desired radiation pattern are a highly non-
linear optimization problem, and hence analytical methods
are not applicable anymore. For this purpose, several
optimization techniques have been developed to suite non-
linear optimization problems. Many methods are bio-
inspired. These methods have proven to be highly
successful. Some of these are Genetic Algorithm (GA),
and Particle Swarm Optimization (PSO). In GA, a sample
of possible solutions is assumed, then mutation, crossover,
and selection are employed based on the concept of
survival of the fittest solution. Particle swarm optimization
(PSO) is a computational method in which optimization is
done by trying to improve a candidate solution problem at
each iteration with respect to a given measure of quality.
It is a population-based method. Here the population of
candidate solutions are known as particles. The position
and velocity of each particle are updated by a fitness
function. The objective of PSO is to find a solution for a
constrained minimization problem based on a particular
cost function.
In this research work a new method of optimizing the
synthesis of antenna array radiation/sensitivity patterns is
introduced. This method/algorithm is called
“Characteristics Evolution Optimization”. Firstly, the
linear array design synthesis problem is explained, and
hence, the new method is introduced with the solution of
the optimization problem, and compared to other
optimization methods such as Invasive Weed
Optimization (IWO), Particle Swarm Optimization (PSO),
and Differential Evolution Algorithm (DE).
2. Formulation Of the Design
Problem
To synthesize the radiation pattern of the linear
antenna array, the overall gain of the array as a function of
is required, i.e., . However, the class of this
function is large, as it includes large number of sum and
difference pattern components.
2. 1 Sum and Difference Patterns
Many applications of linear arrays involve the need to
produce sum and difference patterns such that the main
beam of the sum pattern points at , the twin main beams
of the difference pattern straddle , and both patterns
should exhibit a symmetrical sidelobe structure.
Figure 1 illustrates a linear antenna array with 2N
equally spaced elements, where the distance between the
elements can be adjusted to get the overall desired
radiation/sensitivity pattern of the array. Thus, the array
factor can be written as
(1)
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
111
Volume 21, 2022
For the sum pattern where , the above equation
can be expressed as
(2)
Figure 1. Linear Antenna Array with equally spaced 2N
elements.
For the difference pattern where , the array
factor can be expressed as
(3)
An array with 2N+1 elements is not suitable for the
generation of a difference pattern, due to the presence of
the central element. However, it can be used to produce a
sum pattern, and hence the pattern can be expressed as
(4)
3. Description Of the Proposed
Method
The algorithm called “Characteristics Evolution
Optimization CEO” is used to synthesize the radiation
pattern of the array as binary representations of the array
gains in various directions.
The main concept of this algorithm is based on the
tendency of less evolved organisms to adopt the most
significant characteristics of the more highly evolved
organisms, and simultaneously, modify their
characteristics accordingly. This process of adoption and
modification leads to continuous evolution, and hence
several diverse groups are formed with significant
differences amongst them.
In Characteristics Evolution Optimization algorithm
CEO, several groups with significant differences,
specifications, and characteristics are formed. These
groups are left to evolve independently for a specified
period of time. As a result, when one of the groups is found
to be more successful than other groups, the remaining
groups start merging with the successful group.
Eventually, the merged group evolves to obtain a higher
success. The proposed algorithm (CEO) tries to adopt this
procedure to obtain the optimum solution.
The step wise explanation of the algorithm is yet to be
explained.
3.1 Initialization
The radiation pattern of a linear array with 2N elements
shown in Figure 1 is considered. To initialize the pattern
synthesis, a population size of NP is considered, with each
particle being initialized in a N-dimensional space. The
particles are initialized with numbers ranging from 0 to R,
where R is the predefined range.
Each particle consists of N numbers to be converted
into their binary forms called “parts”. The bit length used
to represent the numbers can be defined by the user, where
larger bit lengths provide better accuracy. Decimal
numbers can be represented in binary forms by shifting the
decimal point to the right to appropriate steps, so that the
number on the left of the decimal point can be represented
in the allocated bit length. For example, to represent 2.765
in binary system using five bits would become 11011.
Once the particles are finalized, their fitness function is
calculated according to the optimization function at hand.
The particles are then arranged according to their fitness
values.
3.2 Segregation into Groups
The entire population is equally segregated into groups,
then the particles are arranged according to their fitness
values within their respective groups.
3.3 Adoption of Characteristics
In every group, the group members try to adopt the
characteristics of their corresponding leader, i.e. the
member with the best fitness value. In binary
representation, each bit is considered as a characteristic.
The importance of the characteristics increases from right
to left and the importance of the characteristics decreases
from left to right as shown in Figure 2.
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
112
Volume 21, 2022
Figure 2. A particle contains N parts. Each part consists
of several bits or characteristics.
Every particle has N parts, and each part has “Bit
Length” number to represent the number of
characteristics, as shown in Figure 3.
Figure 3. Particle representation. Every particle has N
parts, each part has “Bit Length” characteristics, with
values of 1 or 0.
The adoption process is illustrated in Figure 4. During
this process, every characteristic in any part is assigned a
particular adoption probability number that determines the
probability that a particular characteristic will be adopted.
During the adoption process in a group, every particle
adopts the characteristics of the best particle of that group,
according to the assigned adoption probability number.
Thus, there can be “Bit Length” number of adoption
probability numbers. For example, if the third
characteristic of the second part of the best particle in a
particular group is 1, then the chance of this 1 getting
transmitted to the third characteristic of second part of
some other particle of that same group is given by third
adoption probability number.
Figure 4. Adoption process. Every characteristic in every
part tries to adopt the characteristics from the
corresponding characteristic position and part of other
particle.
The adoption probability number is assigned to the
groups in a descending order from left to right, i.e., from
the most important characteristic to the least important
characteristic. The most important characteristics are
assigned the highest adoption probability numbers to make
sure that the other particles move closer to the best particle.
On the other hand, the less important characteristics
are assigned less adoption numbers, since we do not want
other particles to rapidly come at exactly the same position
as that of the best particle in the N-dimensional space. It is
required that other particles search wider space. Thus, as
the particles try to imitate the main characteristics of the
best particle of that group, they have the freedom to span
the nearby space using the less important characteristics.
3.4 Evolution of Characteristics
As the particles try to adopt the characteristics of the
best particle, they themselves try to evolve and search the
entire space by changing their own characteristics. Every
characteristic is given an evolution probability number
which determines the probability of the characteristics to
get altered, i.e., from 1 to 0 or from 0 to 1.
The evolution probability number is assigned to the
groups in an ascending order from left to right, i.e., from
the most important characteristics to the least important
characteristics. This is done to avoid extreme deviation
from their positions while preserving their rights to explore
the region around them. Thus, each particle evolves itself
individually, according to its evolution probability
number.
3.5 Competitive Selection
For a particular group, there are now three sets of
particle positions that have been formed. The first set
consists of the original positions, the second set is formed
after particles adopt the characteristics of the best particle
of the group, and the third set is formed after the particles
evolve their own characteristics.
Thus, for every particle, there are three positions
available in N-dimensional space, which enables
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
113
Volume 21, 2022
competitive selection of the best position for each particle
in the space.
However, there is another way of competitive selection,
that is: to arrange all the particle positions of a particular
group in accordance to their fitness values, and then to
select the best 1/3rd of the available positions. But this
method faces the problem of premature convergence,
which rises due to the fact that all the particles close to the
current best position would be given a preference.
3.6 Merging of the Groups
All groups continue to evolve independently for a
specified period of time. For the purpose of merging the
groups, a point of confidence is defined, and the best
fitness values of all the groups are recorded. Whenever any
group crosses the point of confidence, all the groups are
merged together into one single group. This directs all the
resources to search the space around the position of the
leading group, and all particles search their surrounding
spaces according to the previously mentioned rules.
4. Simulation Results
The proposed method (CEO) is used to synthesize the
radiation pattern of the linear antenna, and the results are
compared to other methods, such as Differential Evolution
algorithm (DE), Invasive Weed Optimization (IWO),
variants of IWO, and Particle Swarm Optimization (PSO).
The parametric setup used of the proposed algorithm is as
follows: “Bit Length” ranges from 40 to 50, initial range
of spread of the particles ranges from 0 to 10, and the
number of groups is set to 4, and each group is assumed to
have 50 particles. The population size after recombination
of the group is assumed to be 75, and the point of
confidence is set to 5. Hence, the four groups merge
together when the error goes below 5. The adoption
probability number of the characteristics is assumed to
vary from 40% for the leftmost, i.e., most important
characteristics to 20% for the rightmost characteristics.
The evolution probability numbers of the characteristics
starting from leftmost bit have values: 1%, 2%, 3%, 4%,
10% for the next 4 bits or characteristic, 20% for the next
6 characteristics, and 40% for the remaining bits or
characteristics. For the purpose of competitive selection,
all the particle positions of a particular group are arranged
according to their fitness values, and the best population
size NP is selected. The flow chart of Characteristics
Evolution Optimization algorithm is depicted in Figure 5.
The design problem statement is to synthesize the
radiation pattern of a linear array with 16 elements. The
maximum sidelobe level (desired_min_SL) is required to
be at -30dB. The function used to determine the error value
is abs(max_SL-desired_max_SL), where abs is the
absolute value function, and the desired maximum
sidelobe level (desired_max_SL) is -30dB. The angle
scanned for the sidelobes ranges from 0o to 77o and from
103o to 180o. Figure 6 illustrates the synthesized radiation
pattern of 16-elements linear array with -30dB sidelobe
level.
The parametric setup used for PSO, DE, IWO [9] and
its variants is as follows: For the Differential Evolution
algorithm (DE), the crossover constant is set to 0.5, and the
mutation factor is set to 0.2. The Number of populations is
assumed to be 400. For Particle Swarm Optimization
(PSO), w_max and w_min parameters are considered as
0.9 and 0.4, respectively, and
For the
Invasive Weed Optimization (IWO), the number of agents
is assumed to be 10 times the dimension. S_initial = 1,
S_final=0.00000001, the Maximum number of seeds is set
to have a value of 5, and the maximum number of
populations is assumed to be 20 times the dimension.
The variants of the IWO used in this example are
briefly described as follows: Modified IWO [1] uses a
|cos(iter)| term in calculating the standard deviation, to
allow for the fast convergence of the weeds present in
location of the global optimum solution without having to
wait for the standard deviation to decrease with iterations.
MIWO [2] uses a modified formula for calculating the
standard deviation, which is based not only on the iteration
but also on the fitness value of corresponding weed. So,
the standard deviation is different for every weed. This
gives opportunity to the far away weeds to get closer to the
global optimum solution, and prevent the close weeds to
get trapped. DIWO [2] merges the MIWO with the
differential evolution. It adopts the concept of mutation
and crossover from DE algorithm and applies it to MIWO
[2].
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
114
Volume 21, 2022
Figure 5. Flow chart of Characteristics Evolution
Optimization.
For the purpose of plotting the error graphs and finding
the beam widths, every algorithm is run 20 times. The
graphs shown in Figure 7 and Figure 8 are the average of
the 20 values. The lines in the graphs are artificially
smoothened for better presentation.
The performance of the algorithm vastly depends upon
the point of confidence, which determines the condition of
merging of the groups. The larger the point is, the sooner
groups will merge. However, this may cause the particles
to get stuck in a local optimum solution. The error plots for
different values of point of confidence are illustrated in
Figure 7 and Figure 8.
The performance also depends on the available “Bit
Length” of a given part. The larger the “Bit Length” is, the
better the performance will be.
However, it takes more time for the process to
converge, due to the difficulty involved in training a longer
sequence of bits. To achieve an acceptable performance,
the “Bit Length” is set initially to a small value, to allow
faster convergence, and at latter stages the “Bit Length” is
increased.
Figure 7 illustrates the error graphs for the radiation
pattern synthesis of 16-element linear array using different
algorithms. It can be seen that the proposed algorithm
exhibits quit better performance compared to other
algorithms.
Figure 6. Synthesized radiation pattern of 16-elements
linear array with -30dB sidelobe level.
Figure 7. Error plots for pattern synthesis of 16-element
array using different algorithms. Error has been
expressed in dB with respect to 15.
On the other hand, Figure 8 shows the error graphs for
pattern synthesis of the array using the proposed algorithm
for different points of confidence. It can be seen that better
performance can be achieved by increasing the point of
confidence.
Table 1. Listing of the average error and beamwidth
obtained by each algorithm after approximately 17000
functional evaluations.
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
115
Volume 21, 2022
Figure 8. Error plots for pattern synthesis of 16-element
array using the proposed algorithm for different points of
confidence.
5. Conclusions
In this paper a novel method of optimization
(Characteristics Evolution Optimization) is proposed. This
method employs the binary representation of numbers to
synthesize the radiation pattern of linear array antennas.
Since the method works in a parallel fashion, and only with
binary numbers, the method can provide significant
performance in parallel processing environment. In this
research work, the proposed method is used to synthesize
the radiation pattern of a 16-element antenna array. The
simulation results show that the proposed algorithm
exhibits significant performance compared to other
algorithms, such as PSO, DE, IWO and variants of IWO.
In its present form, the proposed method encounters
some difficulties under certain conditions. However, it is
expected that with further research the method can perform
very well in most of the optimization problems.
References
[1] Siddharth Pal, Swagatam Das, Aniruddha Basak, Ajith
Abraham, Ivan Zelinka, “Concentric Circular Antenna Array
Synthesis Using a Differential Invasive Weed Optimization
Algorithm”, IEEE 2010 International conference on soft
computing and pattern recognition.
[2] Aniruddha Basak, Ajith Abraham, Siddharth Pal, Vaclav
Snasel, “A modified invasive weed optimization algorithm
for the time-modulated linear antenna array synthesis”,
IEEE 2010
[3] Constantine A. Balanis and Panayiotis I. Ioannides,
Introduction to Smart Antennas, 1st edition: Morgan &
Claypool, 2007
[4] Iqroop Kaur, Sanjeev Kumar, “Linear Antenna Array
Optimization Techniques”, International journal of
engineering trends and technologies, 2011
[5] Herve Lebret, Stephen Boyd, “Antenna array pattern
synthesis via convex optimization”, IEEE transactions on
signal processing, 1997
[6] Siddharth Pal, Aniruddha Basak, Ajith Abraham, “Linear
antenna array synthesis using invasive weed optimization
algorithm”, 2009 International conference on soft computing
and pattern recognition.
[7] Ritwik Giri, Ariba Chowdhury, Arnob Ghosh, Swagatam
Das, Ajith Abraham, Vaclav Snasel, “A modified invasive
weed optimization algorithm for training of feed-forward
neural networks”, IEEE 2010
[8] Mohamadreza Ahmadi, Hamed Mojallali, “Chaotic invasive
weed optimization algorithm with applications to parameter
estimation of chaotic systems”, 2012 Elsevier
[9] Shaya Karimkashi, Ahmed A. Kishk, “Invasive weed
optimization and it’s features in electromagnetic”, IEEE
transactions in antennas and propagation, 2010
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.15
Amin Al Kabi
E-ISSN: 2224-2864
116
Volume 21, 2022
