Optimization of Radiation Pattern for Circular Antenna Array using Genetic Algorithm
and Particle Swarm Optimization with Combined Objective Function
TinhNguyen Dinh1
-
(Faculty of Radio-Electronic Engineering, Le Quy Don Technical University, Hanoi, Vietnam
tinhnd_k31@lqdtu.edu.vn)
Copyright © The Institute of Electronics and Information Engineers(IEIE)
Keywords
Circular antenna array, Phased array antenna, Particle swarm optimization, Genetic algorithm, Amplitude distribution
1. Introduction
Circular antenna array (CAA) is capable of 360$^{\circ}$ coverage by an omnidirectional
beam, multi-beam, or a narrow beam that could scan 360$^{\circ}$ [1]. Therefore, the CAAs are used in radar, sonar, navigation systems, and base station
antennas in information systems. The omnidirectional beam of CAAs (or cylindrical
arrays) is often used in the transmit mode of sonars [2]. To generate the omnidirectional radiation pattern, the elements on the circle are
excited in phase with uniform amplitude distribution [2,3]. With this transmit mode producing umbrella-shaped beam patterns or sector beam patterns,
the reduction of side-lobe level (SLL) in receive mode by a narrow beam is crucial
to suppress scattering noise from side-lobe directions. As a result, the study focuses
on decreasing SLL and increasing directivity in narrow beam generation mode for CAAs.
With the complex configuration, the reduction of SLL for CAA is more challenging than
for linear antenna arrays. In [4,5], CAA is transformed into a virtual linear array. Then, based on the calculation for
this virtual linear array, the optimal amplitude distribution to mitigate SLL and
generate the desired beamwidth for CAA is determined. This method requires multiple
conversion steps, and the equally spaced elements with a distance being less than
half a wavelength. Besides, this method is limited by the accuracy of the transformation,
and could not produce both a desired SLL and the narrowest main beam (the maximum
directivity) [6]. Another method in [7] is to build the mathematical expression describing the radiation pattern, then analyze
simulation results to determine the optimal amplitude distribution reducing SLL to
less than the required value and to generate the narrowest half-power beamwidth (HPBW).
This method also requires the distance between two successive elements on the circle
to be less than half a wavelength.
Thanks to the simplicity, ease of implementation, and convergence ability, the genetic
algorithm (GA) and particle swarm algorithm (PSO) are widely applied for optimizing
the configuration and parameters of some antenna arrays [8-10]. Therefore, GA and PSO are also used to determine the amplitude distribution for
CAA (or cylindrical array) to reduce SLL as shown in the literature [11-16]. However, these studies have not simultaneously decreased SLL to less than the required
value and increased the directivity to a maximum value. Some studies [13-16] have ignored the element pattern when analyzing the array pattern and determining
the optimal amplitude distribution. Unlike the linear antenna array, the radiation
pattern of CAA cannot be calculated by multiplying the array factor with the element
pattern, it is necessary to consider the element pattern in the summation when analyzing
the radiation pattern of CAA.
With the construction of a combined objective function for GA and PSO, the paper determines
the amplitude distribution to optimize the radiation pattern for CAA by decreasing
SLL to less than the required value and increasing directivity when considering the
element patterns. Optimization of the array pattern is carried out in both cases of
uniform CAA and non-uniform CAA with the distances between two adjacent elements on
a circle being more than half a wavelength. The simulation results demonstrate the
effectiveness of the proposed solution when determining the amplitude distribution
that reduces the SLL to less than -20 dB and maximizes directivity in some cases.
This paper is organized as follows. Section 2 describes the radiation pattern of CAA.
Section 3 constructs the combined objective function to optimize the radiation pattern
for CAA using GA and PSO. Section 4 presents simulation results. Finally, Section
5 concludes the paper.
2. Radiation Pattern of CAA
Consider a CAA as shown in Fig. 1 with N elements arranged at arbitrary intervals on the circle corresponding to the
coordinates A$_{1}$, A$_{2}$, ..., A$_{\mathrm{N}}$. In this Figure, l$_{1}$, l$_{2}$,
${\ldots}$, l$_{N-1}$, and l$_{N}$ are distances along the circular from A$_{1}$ to
A$_{2}$, A$_{2}$ to A$_{3}$, ${\ldots}$, A$_{\mathrm{N-1}}$ to A$_{\mathrm{N}}$, and
A$_{\mathrm{N}}$ to A$_{1}$, respectively. As a result, the radius of the circle R
and the distances l$_{1}$, l$_{2}$, ${\ldots}$, l$_{N}$ are related to each other
by the expression:
Set l$_{0}$ = 0, the coordinate of the n$^{th}$ element is ($R\cos \theta _{n}$, $R\sin
\theta _{n}$, 0), and $\theta _{n}$ is given by:
Choose the coordinate origin O(0,0,0) as the reference point (or the phase reference).
Consider any direction in space defined by the directive unit vector . The path-length difference between points A$_{\mathbf{n}}$ and O directed to P is
determined as follows [2,17]:
In the receiver mode, the active angle intercepted by the active elements on the circle
is usually chosen to be less than or equal to 180$^{\circ}$. Assuming the number of
active elements is M (M ${\leq}$ N), the amplitude distribution and phase distribution
of the array with the M element are (a$_{1}$, a$_{2}$, ${\ldots}$, a$_{M}$) and (${\psi}$$_{1}$,
${\psi}$$_{2}$, ${\ldots}$, ${\psi}$$_{M}$), respectively. The array factor when considering
the elements in the array as omnidirectional antennas is determined by
where j is the imaginary unit which is defined as$j^{2}=-1$, $k=\frac{2\pi }{\lambda
}$ is the wave number, and$\lambda $ is the wavelength.
To steer the main beam to the direction with angular coordinates ($\theta _{0}$, $\varphi
_{0}$), the excitation phase for elements in the CAA must satisfy the following conditions
Substituting the excitation phase in (5) into Eq. (4), the array factor when steering the main beam to the direction$(\theta _{0}$, $\varphi
_{0})$ is determined as
When taking into account the radiation pattern, the element patterns of the m$^{th}$
element in the azimuth direction, and elevation direction can be chosen as follows
[1,7,18]:
Therefore, the array pattern considering the element patterns is expressed as
Since the CAA is placed in the x-y plane, the radiation pattern synthesis for CAA
is usually interested in the azimuth direction (${\varphi}$ = 0). With the suppression
of the radiation pattern in the elevation plane, the array pattern in the azimuthal
plane is given by
In the case of uniform CAA, when only the array pattern in the azimuth plane is considered,
formulas (5) and (9) become:
Fig. 1. Geometry Model of CAA.
3. Combined Objective Function to Optimize the Radiation Pattern for CAA Using GA
and PSO
3.1 Constructing a Combined Objective Function to Optimize both SLL and Directivity
for CAA
When only considering the radiation pattern in the azimuth plane, the maximum value
of SLL is determined according to the function $F(\theta )$as the following
where $\theta _{nu1}$ and $\theta _{nu2}$ are the angles at the first nulls to the
left and right relative to the main beam, respectively.
When steering the main beam to the middle of the active arc, the nulls in (12) are calculated as
HPBW of CAA in case all elements in the circle are exited the same amplitude is approximately
[1]
The HPBW is mainly determined according to the elements in the half of the circle.
With amplitude distributions reducing SLL, the HPBW is widened compared with the uniform
distribution. Therefore, when choosing the number of elements blocking an angle $\theta
_{active}$ ($\theta _{active}\leq 180^{\circ}$) with an amplitude distribution that
differed from the uniform distribution, the HPBW is determined by
where ${\alpha}$ represents the beam extension due to the use of distributions other
than the uniform distribution and the reduction of active element number. In this
study, ${\alpha}$ is chosen equal to 1.85 corresponding to a beam broadening of about
85\% compared to the uniform distribution for elements in the half of the circle to
reduce the SLL of radiation patterns in CAAs.
Therefore, the objective function for mitigating SLL to less than SLL$_{0}$ and maximizing
directivity D (in decibels) can be built as the following expression
where $\left\lceil x\right\rceil $ and $\left\lfloor x\right\rfloor $are the round
functions toward integers of arbitrary real number x, which are determined by$\left\lceil
x\right\rceil =\min \left\{p\in Z,\hspace{0pt}\,p\geq x\right\}$, and $\left\lfloor
x\right\rfloor =\max \left\{q\in Z,\hspace{0pt}\,q\leq x\right\}$. $sign\left(x\right)$
is the sign function defined as follows
When suppressing the radiation pattern in the elevation plane, the directivity D in
the azimuth plane is given by
Eq. (16) mathematically describes the objective function to optimize multiple objectives including
reducing SLL to a required value and increasing the directivity. Optimization according
to the objective function (16) includes two stages: in stage 1, when SLL$_{max}$ ${\leq}$ SLL$_{0}$, the second
term is zero, and optimization is performed until SLL$_{max}$ = SLL$_{0}$. In the
second stage, when SLL$_{max}$ ${\geq}$ SLL$_{0}$, the first term is zero, and optimization
is implemented to increase D to D$_{max}$ (equivalent to a decrease in the HPBW).
With the optimization of these two stages, it is possible to apply nature-inspired
algorithms such as GA and PSO to determine the amplitude distribution that decreases
SLL smaller than SLL$_{0}$ and increases D to the maximum value.
3.2 GA with Combined Objective Function to Determine Optimal Amplitude for CAA
The GA uses many different operators in the optimization process including encoding
schemes, crossover, mutation, and selection [19]. A string of M chromosomes is used to represent the M amplitude values of the corresponding
elements in the CAA, which are real values in the interval [0,1]. The first generation of the population is initialized with the K populations. These
individuals in the next generation are produced by crossover, mutation, and selection
based on fitness function computation. This generation evaluation is continued until
the objective function's desired fitness level is achieved. With the objective function
built in section 3.1, the optimization is completed when the objective function converges
to a value.
3.3 PSO with Combined Objective Function to Determine Optimal Amplitude for CAA
Consider a hyperspace consisting of M dimensions corresponding to the M amplitude
values of the CAA. With K particles, each particle moves to its best position ($p_{besti}^{t}$)
and the global best position ($g_{best}^{t}$ ) in the swarm. The optimal solution
being equivalent to the position of the particle is achieved when the objective function
F$_{ob}$ reaches its smallest value. The velocity vectors v and position vectors x
of the i$^{th}$ particles (1 ${\leq}$ i ${\leq}$ K}) in the M-dimensional hyperspace
at the t+1 iteration are determined as the following expressions [20,21]
where $\omega $ is the inertia weight, r$_{1}$ and r$_{2}$ are random vectors uniformly
distributed in the range [0,1]$^{M}$, c$_{1}$ and c$_{2}$ are the acceleration coefficients.
4. Simulation Results
4.1 Simulation for Uniform CAA
To illustrate the effectiveness of the GA and PSO with the combined objective function
for uniform CAA, the study considers a uniform CAA with 30 elements, the distance
between two consecutive elements on the circle is 0.6${\lambda}$, the active arc length
is 108$^{\circ}$ equivalently to the number of elements selected as 10. With the selection
of the blocking angle of 108$^{\circ}$ corresponding to the first 10 elements, the
main beam is at 54$^{\circ}$ when steering to the middle of the active arc. With 30
elements, the number of main beams in the middle of the arc is 30 according to the
initial position and 30 displacements of active elements. These main beams are 12$^{\circ}$
apart. Suppose it is necessary to determine the amplitude distribution so that the
SLL is less than or equal to SLL$_{0}$ = -20 dB.
With GA, the population size is selected to be 100, the number of generations (iterations)
is 5000, the crossover probability is 1, and the mutation probability is 0.02. With
PSO, the acceleration coefficients in Eqs. (19) and (20) are determined as follows c$_{1}$ = c$_{2}$ = 2, and the inertia coefficient is decreased
linearly from 0.9 to 0.4 equivalent to from the beginning to the last iteration [22]. The numbers of particles and iterations are chosen as 100 and 5000, respectively.
To evaluate the convergence of the algorithms, the programs to determine the amplitude
distribution using GA and PSO are run 3 times, and then the run with the smallest
objective function is selected. With this method, the minimum objective functions
using the GA and PSO in MATLAB 2015A are expressed in Fig. 2.
Fig. 2 illustrates that the objective functions obtained by the GA and PSO reach -33.87
dB and -33.79 dB, respectively. These objective functions converge well when the number
of iterations is 5000. The normalized amplitude distributions derived from the GA
and PSO with the combined objective function are shown in Table 1.
To demonstrate the effectiveness of the amplitude distributions for uniform CAA generated
from the proposed GA and PSO, Fig. 3 depicts radiation patterns obtained from the proposed solution, from the amplitude
distribution using PSO with the objective function in [16], and from conventional amplitude distributions such as Dolph-Chebyshev window, Kaiser
window, and raised cosine-squared weighting. The input parameters and number of runs
for PSO in [16] are also chosen as those for the proposed PSO. To ensure SLL less than or equal to
-20 dB, the Dolph-Chebyshev distribution has a side-lobe attenuation (SLA) of -42
dB, the Kaiser distribution has the coefficient ${\beta}$ = 4.18, and the raised cosine-squared
distribution has ${\Delta}$ = 0.161 [7].
In Fig. 3, the beam patterns derived from the proposed GA, proposed PSO, PSO in [16], the Dolph-Chebyshev distribution (SLA = -42 dB), Kaiser distribution (${\beta}$
= 4.18), and raised cosine-squared distribution (${\Delta}$ = 0.161) are denoted by
solid red curves, dashed blue curves, dashdot black curves, dotted brown curves, curves
with magenta triangles, and curves with green circles, respectively. The parameters
of the radiation patterns in Fig. 3 are described in detail in Table 2.
Table 2 shows that with the same input parameters, the proposed GA and proposed PSO produced
an SLL less than or equal to -20 dB, while PSO in [16] could not produce an SLL smaller than the required value of -20 dB. The directivity
obtained from the proposed GA reaches the largest value of 13.87 dB, then raised cosine-squared
distribution with ${\Delta}$ = 0.161 attains a value of 13.83 dB. The proposed PSO
and PSO in [16] together produce a directivity of 13.79 dB. The two conventional distributions including
Dolph-Chebyshev (}SLA = -42 dB) and Kaiser (${\beta}$ = 4.18) generate the lowest
directivities with values of 13.78 dB and 13.69 dB, respectively. The HPBW obtained
from the proposed GA and proposed PSO achieves the two narrowest values of 13.16$^{\circ}$
and 13.31$^{\circ}$, respectively. The remaining others produce HPBW smaller than
these two values. Therefore, the amplitude distributions from the proposed GA and
proposed PSO could provide radiation patterns with SLL less than or equal to the required
value of -20 dB, largest directivity, and smallest HPBW.}
Fig. 2. Combined Objective Function of GA and PSO.
Fig. 3. Radiation patterns of uniform CAA with amplitude distributions from solutions.
Table 1. Normalized amplitude distributions for the uniform CAA from the proposed GA and proposed PSO.
|
Index
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
GA
|
0.1411
|
0.3450
|
0.4141
|
0.7017
|
0.9447
|
1.0000
|
0.7563
|
0.4426
|
0.2940
|
0.1752
|
|
PSO
|
0.1621
|
0.3529
|
0.4544
|
0.8414
|
1.0000
|
1.0000
|
1.0000
|
0.4048
|
0.3650
|
0.1270
|
Table 2. The parameters of the radiation patterns in the uniform CAA.
|
Parameters
|
SLL (dB)
|
D (dB)
|
HPBW (°)
|
|
Proposed GA
|
-20.00
|
13.87
|
13.16
|
|
Proposed PSO
|
-20.00
|
13.79
|
13.31
|
|
PSO in [16]
|
-18.94
|
13.79
|
13.33
|
|
Dolph-Chebyshev (SLA = -42 dB)
|
-20.00
|
13.78
|
13.48
|
|
Kaiser (β = 4.18)
|
-20.00
|
13.69
|
13.80
|
|
raised cosine-squared (Δ = 0.161)
|
-20.00
|
13.83
|
13.33
|
4.2 Simulation for Non-uniform CAA
Non-uniform CAAs could reduce grating lobes when steering the main beams in different
directions [23]. In addition, when manufacturing errors are large, a uniform CAA changes to a non-uniform
CAA. To highlight the effectiveness of the proposed GA and proposed PSO for non-uniform
CAA, the section investigates a non-uniform CAA with 30 elements, the distances between
two successive elements on the circle are values in the range [0.55${\lambda}$, 0.65${\lambda}$].
The values l$_{1}$, l$_{2}$, ..., l$_{N}$ could be generated using normally distributed
pseudorandom numbers in [0.55${\lambda}$, 0.65${\lambda}$] by MATLAB software. An
example of a set of 30 values is [0.6021, 0.5587, 0.5649, 0.6291, 0.5697, 0.6032,
0.6007, 0.6327, 0.6259, 0.6426, 0.5879, 0.6409, 0.5608, 0.6430, 0.6114, 0.6167, 0.6118,
0.5895, 0.5881, 0.5919, 0.5773, 0.5517, 0.6010, 0.6001, 0.5602, 0.5873, 0.6140, 0.5740,
0.6099, 0.6117] ${\lambda}$. With the choice of the first one-third of the circle,
the active angle is 107.9872$^{\circ}$. As a result, the main beam is steered to the
middle of the active arc corresponding to the steering angle of 53.9936$^{\circ}$.
The programs determining optimal amplitude distributions using the proposed GA and
proposed PSO on MATLAB are also run 3 times with the same input parameter in section
4.1. The minimum objective functions of the runs using the proposed GA and proposed
PSO are illustrated in Fig. 4.
From Fig. 4, the objective functions derived by the proposed GA and proposed PSO for non-uniform
CAA together achieve -33.94 dB. These objective functions also converge well when
the number of iterations is 5000 as in the case of uniform CAA. The normalized amplitude
distributions obtained from the proposed GA and proposed PSO for the non-uniform CAA
are shown in Table 3.
To facilitate a comparison between the cases, this section also uses the parameters
for PSO in [16] similar to section 4.1. With this requirement of the SLL being less than or approximately
equal to -20 dB, the Dolph-Chebyshev distribution, the Kaiser distribution, and the
raised cosine-squared distribution have SLA = -56.2 dB, ${\beta}$ = 5.09, ${\Delta}$
= 0.088, respectively. Radiation patterns obtained from the amplitude distributions
are expressed in Fig. 5.
In Fig. 5, the solid red curves, dashed blue curves, dashdot black curves, dotted brown curves,
curves with magenta triangles, and curves with green circles also depict the beam
patterns derived from the proposed GA, proposed PSO, PSO in [16], the Dolph-Chebyshev distribution (SLA = -56.2 dB), Kaiser distribution (${\beta}$
= 5.09), and raised cosine-squared distribution (${\Delta}$ = 0.088), respectively.
From Fig. 5, the parameters of the radiation patterns are presented in Table 4.}
Table 4 shows that the PSO in [16] produces the radiation pattern with the smallest SLL being -20.16 dB. The Dolph-Chebyshev
distribution and Kaiser distribution could not generate the radiation pattern with
the SLL being less than or equal to -20 dB despite the change of distribution parameters.
With PSO in [16], the amplitude distribution provides the radiation pattern with the larger directivity
and the smaller HPBW compared with three conventional distributions including Dolph-Chebyshev,
Kaiser, and raised cosine-squared. It can be seen that with non-uniform CAA, the distributions
from the proposed solution improve parameters including the SLL and the directivity
significantly compared to the conventional distributions such as the Dolph-Chebyshev,
Kaiser, and raised cosine-squared.
Even though it produces a value less than -20 dB, the SLL generated from PSO in [16] could not be fully controlled at the required value. While the amplitude distributions
from the proposed GA and proposed PSO generate a larger directivity than that from
the PSO in [16] with a together approximate value of 13.94 dB. The HPBWs obtained by the proposed
GA and proposed PSO are 12.82$^{\circ}$ and 12.78$^{\circ}$, respectively, which are
smaller than that derived by PSO in [16]. With a nearly equal SLL, the proposed GA and proposed PSO produce significantly
smaller HPBW than PSO [16]. Especially, in comparison with conventional distributions such as Dolph-Chebyshev,
Kaiser, and raised cosine-squared, the distributions from the proposed GA and proposed
PSO enhance the parameters of the radiation patterns more considerably. The improvements
obtained from the proposed solution in Table 4 are due to the proposed solution optimizing multiple objectives to both reduce SLL
to less than -20 dB and increase the directivity (or reduce HPBW). These results in
Sections 4.1 and 4.2 show that, in the case of non-uniform CAA, the effectiveness
in improving the parameters of radiation patterns by the proposed GA and proposed
PSO is much increased compared with the case of uniform CAA. With the ability to optimize
many parameters, the objective function constructed in this study is not only applicable
to GA, PSO, and CAA but it can also be applied to other optimization algorithms as
well as other antenna arrays. These issues will be studied in more detail in the future.
Fig. 4. Combined Objective Function of GA and PSO for non-uniform CAA.
Fig. 5. Radiation patterns of non-uniform CAA with amplitude distributions from solutions.
Table 3. Normalized amplitude distributions for the non-uniform CAA from the proposed GA and proposed PSO.
|
Index
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
GA
|
0.2672
|
0.4421
|
0.5110
|
0.9358
|
0.9729
|
1.0000
|
0.7433
|
0.4816
|
0.3320
|
0.1377
|
|
PSO
|
0.2951
|
0.4610
|
0.4834
|
1.0000
|
1.0000
|
1.0000
|
0.7624
|
0.4519
|
0.3482
|
0.1473
|
Table 4. The parameters of the radiation patterns in the non-uniform CAA.
|
Parameters
|
SLL (dB)
|
D (dB)
|
HPBW (°)
|
|
Proposed GA
|
-20.00
|
13.94
|
12.82
|
|
Proposed PSO
|
-20.00
|
13.94
|
12.78
|
|
PSO in [16]
|
-20.16
|
13.73
|
13.57
|
|
Dolph-Chebyshev (SLA = -56.2 dB)
|
-19.97
|
13.43
|
14.78
|
|
Kaiser (β = 5.09)
|
-19.88
|
13.38
|
14.97
|
|
raised cosine-squared (Δ = 0.088)
|
-20.00
|
13.51
|
14.46
|
5. Conclusion
The paper proposed a solution determining the amplitude distribution to optimize the
radiation pattern for CAA using GA and PSO with a combined objective function. The
amplitude distributions from the proposed solution produced the radiation pattern
with the SLL being less than or equal to -20 dB, maximum directivity, and smallest
HPBW compared to conventional solutions when the distances between two adjacent elements
on a circle were more than half a wavelength and the radiation patterns of each element
were taken into account. The proposed solution was effective in both cases of uniform
CAA and non-uniform CAA. In addition, the combined objective function from this study
could be applied to other antenna arrays to generate the radiation pattern with SLL
smaller than the required value and maximum directivity.
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Nguyen Dinh Tinh received his B.E. in Electronics-Telecommunications, the M.E.,
and a Ph.D. degree in Radar Navigation Engineering from Le Quy Don Technical University,
Vietnam, in 2008, 2012, and 2023, respectively. He is now a lecturer at Le Quy Don
Technical University, Hanoi, Vietnam. His research interests include antennas, microwave
circuit design, signal processing, machine learning, synthetic aperture sonar, and
sonar engineering.