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Zhang and Zhang EURASIP Journal on Wireless Communications and
Networking (2017) 2017:183
DOI 10.1186/s13638-017-0968-2
RESEARCH Open Access
Thinning of antenna array via adaptive
memetic particle swarm optimization
Xiu Zhang and Xin Zhang*
Abstract
Massive multiple input multiple output antenna array is crucial for the fifth generation wireless communication.
Proper antenna array design can reduce interference among different signals and generate desirable beamforming.
Sparse antenna array is able to form narrower beam with lower sidelobe than equally spaced antenna array given the
same number of array elements. However, determining the position of elements is non-deterministic polynomial-time
hard. To effectively solve such problem, this paper proposes adaptive memetic particle swarm optimization (AMPSO)
algorithm. The algorithm adaptively tunes algorithmic parameters of particle swarm optimization (PSO). Moreover,
crossover operator is added to enhance local exploiting search information of PSO. Sparse antenna array design is
modeled as a minimization by thinning method. It is then tackled by the proposed algorithm. In terms of peak
sidelobe level, the AMPSO algorithm shows good performance compared with PSO and genetic algorithm.
Keywords: Sparse antenna array, Thinning array, Particle swarm optimization, Parameter control, Memetic computing
1 Introduction
Massive multiple input multiple output antenna array
is crucial for the fifth generation wireless communi-
cation [1]. No matter in cognitive radio networks, ad
hoc networks, or radar networks [2–5], antenna play
an important role to ensure data transmission and high
Quality of Services (QoS) under certain communication
requirements [6–9]. The design of antenna arrays is well
known to be a hard nonlinear programming problem [10].
Recently, many researchers attempt to create efficient and
effective optimization algorithms to solve the design of
antenna arrays. Stochastic algorithms are popularly used
because traditional optimization methods are not suitable
due to the unavailable of gradient information [11–13].
Stochastic algorithms also show good performance for
such design problems [14, 15].
Stochastic algorithms consist of two classes: evolu-
tionary algorithms (EAs) and swarm intelligence (SI)
algorithms. Informally, EAs contain genetic algorithm
(GA) [16], evolutionary strategies [17], and differential
evolution [18]. These algorithms simulate the evolu-
tion of genetic process of livings. They usually contain
*Correspondence: ecemark@mail.tjnu.edu.cn
Tianjin Key Laboratory of Wireless Mobile Communications and Power
Transmission, Tianjin Normal University, Tianjin, China
both mutation and crossover operators. SI algorithms
contain particle swarm optimization (PSO) [19], artifi-
cial bee colony (ABC) [20], neighborhood search opti-
mization [21], etc. SI algorithms emulate the social
behaviors of swarms or particles. They usually do not
contain crossover operator. Although EAs and SI are
created based on different nature, their positive com-
binations are able to result effective algorithms [22].
Inthepast,GA,PSO,andABChavebeenusedin
antenna designs [15, 23, 24]. Recently, covariance
matrix adaptation evolutionary strategy [25] and differ-
ential evolution [26] are applied to synthesize antenna
array patterns.
A large number of iterations is needed for standard
PSO algorithm to obtain a satisfactory solution [27].
This is not acceptable for users as antenna simula-
tion often takes a long time. In this paper, standard
PSO is modified by adding a crossover operator and
a parameter adaptation method. The idea is reason-
able as memetic with crossover operator can enlarge
and promote algorithm’s search; also parameter con-
trol is an effective way to make the algorithm adapt
to different design problems and saves the fine-tuning
efforts of users. The new algorithm is named as adap-
tive memetic particle swarm optimization (AMPSO),
which is applied to tackle antenna design. Numerical
© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and
reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the
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Zhang and Zhang EURASIP Journal on Wireless Communications and Networking (2017) 2017:183 Page 2 of 7
experiment is conducted studying the effect of adding
crossover and parameter adaptation. The results are dis-
cussed based on different metrics and compared with
other algorithms.
In the following, Section 2 introduces the antenna array
design considered in this paper. Section 3 gives stan-
dard PSO and the proposed AMPSO algorithm. Section 4
presents numerical simulation with discussions. Section 5
concludes the paper.
2 Sparse antenna array and related works
Sparse antenna array design can be classified to two cat-
egories. One refers to that array elements can be placed
anywhere between the aperture of antenna array. The
other is that thinning the distance to grids and posi-
tions in grids are equally spaced. In the first case, ele-
ments can be arbitrarily placed, hence the positions of
elements are continuous variables. In the second case,
elements could be placed on grids, hence its search space
is discrete and finite. However, when the number of array
elements increases up to tens or hundreds, search space of
such combinatorial optimization problem exponentially
increases, which brings huge trouble for conventional
optimization algorithms.
Given a sparse linear antenna array, as shown in Fig. 1,
the number of array elements is N(labels above zaxis),
the space between grids is d,andthenumberofgrid
positions along zaxis is M(labels below zaxis). Hence,
the aperture of this linear antenna array is (M−1)d.
To keep the same aperture, two array elements have to
be placed on the first and the last positions of grids.
In this model, the number of variables is N−2and
there are candidate M−2 positions to be placed. Sup-
pose all array elements have no order or priorities,
they have the same incentive amplitude and phase dis-
tribution. Thus, the total number of combinations of
elements is:
C(N−2)
(M−2)=(M−2)!
(N−2)!(M−N)!.(1)
The thinned linear antenna array can be modeled as
a minimization problem, whose objective is to mini-
mize peak sidelobe level (PSLL) and maximize main lobe
peak level. In this way, interference among elements
can be greatly avoided, and the data transmission can
Fig. 1 Flow chart of sparse linear antenna array
be realized in high array gain. The far field pattern is
expressed as:
E(u)=
M
i=1
δiIiejφiejkuxielpat(i),(2)
where δidenotes whether the ith grid is placed an element
(1 for yes and 0 for no); u=cos θ−cos θ0,Iiis the exci-
tation amplitude of element iand φiis the phase position
of element i;xidenotes the element position, its distance
from the first element is (i−1)d;θ0is the direction of main
beam, and θis the sweeping direction of linear antenna
array; elpat(i)is the direction pattern of element i.Sup-
pose all elements have the same excitation amplitude and
the same phase, then (6) is simplified to:
E(u)=
M
i=1
δiejkuxi.(3)
Thus, the optimization model of this type array is:
min f=−
|E(u)|
|MP|
s.t.1≤|u|≤ λC0
Nd
δi∈{0, 1},i=1, 2, ...,M
,(4)
where MP is mainlobe peak level, C0is a correction
parameter as the first zero point may shift in nonuniform
array. The constraint in (4) is to constrain the value of uso
that mainlobe field is removed in computing E(u).
Previously, many researches have reported in literature.
Wang et al. considered nonuniform antenna array design
for millimeter wave situations [28]. Zhao et al. studied
resource allocation problem in MIMO systems such as
time allocation [29] and power allocation [30].
3 Optimization algorithms
This section depicts standard PSO and the proposed
AMPSO algorithm.
3.1 Standard particle swarm optimization algorithm
The main procedures of standard PSO is expressed in
Fig. 2. It can be seen that the main loop of PSO is the
updating of velocities and positions of particles. The num-
berofparticlesisdenotedasNp, which also means the
size of population. As in initial population step, PSO
starts by a set of randomly created particles in given
search space. Usually, the evaluation of particles involves
the interaction with problem. In this paper, the problem
is nonuniform antenna design, which should be preset
ahead of executing algorithm. Then, the main generation
of standard PSO is executed. Compared with other swarm
intelligence approaches, like ABC, dragonfly algorithm
and biogeography-based optimization, the procedures of
PSO is more concise, though it contains more algorithmic
parameters than others.
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Zhang and Zhang EURASIP Journal on Wireless Communications and Networking (2017) 2017:183 Page 3 of 7
Fig. 2 Flow chart of standard particle swarm optimization algorithm
Let Ddenote problem dimension, updating velocity and
position of particle iis realized by the following equation:
vij(t+1)=wvij (t)+c1r1(pij −xij(t)) +c2r1(pgj −xij (t)) ,
(5)
xij(t+1)=xij (t)+vij(t+1),(6)
where subscript j∈[1,D] refers to the jth dimension; t
refers to the tth generation; r1and r2are random numbers
between 0 and 1. piand pgare respectively personal best
and global best positions that particle iwalked through. In
this equation, w,c1,andc2are three algorithmic parame-
ters of PSO. Their setting for good performance of algo-
rithm depends on the properties of practical problem. In
other words, they are sensitive to problem types.
Recently, many researches have been published about
parameter control of PSO. Inertia weight was adapted in
a stability-based manner, which could improve the perfor-
mance of PSO [31]. Khan et al. improved PSO by a random
mutation mechanism and a dynamic inertia weight adap-
tive method, which could facilitate algorithm convergence
[32]. Jin et al. improved PSO through enlarging explorative
ability of PSO to design a permanent magnet synchronous
machine [33].
3.2 Adaptive memetic PSO algorithm
As designated by no free lunch theory [34], the perfor-
mance of an optimization algorithm could be improved
by incorporating prior knowledge or properly hybrid with
other search operators or optimization algorithms. This
section will present a hybrid algorithm, as mentioned in
introduction, the modification contains two parts. The
first one is parameter control of PSO algorithm. The
second is hybrid with crossover operator.
The main procedures of the proposed algorithm is
given in Fig. 3. In this figure, crossover operator step
and parameter updating step are highlighted in bold. It
is well known that there are many crossover methods
reported in literature. Commonly used operators are uni-
form crossover, binomial crossover, exponential crossover,
arithmetic crossover. In this paper, arithmetic crossover
is used as its solid theoretic basis. Note that nonuni-
form antenna problem in this paper is thinned as a binary
optimization problem, hence arithmetic crossover is dis-
cretized to 0 if the value is less than 0.5; otherwise
set to 1.
The updating of parameters is based on the follow-
ing thought. In the first stage, algorithm should focus
on exploring search space and probe all candidate pos-
sible field. In the last stage, algorithm should focus on
exploiting the fields with possible global optima found
by the algorithm. Clearly, an intermediate stage should
exist during the evolutionary process of algorithm, which
may cost longer search time than the first and last stages.
This is because the locating of global optima is cru-
cial, and locating or trapping in local optimum is not
wanted. In this paper, the parameters w,c1,andc2are
computed by:
w(t+1)=wmax −(wmax −wmin)×1
1+exp(6−12t),
(7)
c1(t+1)=c1max−(c1max−c1min)×1
1+exp(12t−6),
(8)
c2(t+1)=c2max−(c2max−c2min)×1
1+exp(12t−6),
(9)
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Zhang and Zhang EURASIP Journal on Wireless Communications and Networking (2017) 2017:183 Page 4 of 7
Fig. 3 Flow chart of adaptive memetic particle swarm optimization
algorithm
where wmin and wmax are the maximum and minimum
value of parameter w; similarly, the minimum and max-
imum value of c1andc2arec1min,c1max and c2min,
c2max.
The adaptation of wis shown in Fig. 4; the adaptation of
c1andc2 is given in Fig. 5. wmax is 0.9 and wmin is 0.4. In
literature, a linear decrease of wfrom wmax to wmin is used.
Fig. 4 Parameter adaptation for w
While the adaptation of win this paper is alike sigmoid
function. As to c1andc2, both are adapted by the same
equation.
The PSO algorithm combined with crossover opera-
tor and parameter adaptation is abbreviated as AMPSO.
Compared with standard PSO, the AMPSO algorithm
needs more operations to perform two additional steps.
Forarithmeticcrossover,thecomputingofvelocity
requires 2D addition and 2D multiplication operations;
the computing of position also requires 2D addition and
2D multiplication operations. Thus, in each generation,
AMPSO needs 8D computer operations than standard
PSO.
Fig. 5 Parameter adaptation for c1andc2
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Zhang and Zhang EURASIP Journal on Wireless Communications and Networking (2017) 2017:183 Page 5 of 7
4 Numerical experiment
The AMPSO algorithm deals with nonuniform antenna
array design problem in this section. The numerical
results are also reported and discussed.
4.1 Experimental setting
Electromagnetic phenomena of antenna array systems
can be described using Maxwell’s electromagnetic field
equations, thus numerical methods of electromagnetic
field computation, such as finite element method (FEM),
method of moments, Monte Carlo method, and finite vol-
ume method, are in a better position for engineers to
appreciate and analyze the performance antenna array
systems. FEM is used in this paper to simulate antenna
array system.
Simulation configuration is described as follows.
Nonuniform antenna array design are thinned and then
used to study how AMPSO performs in handling binary
optimization design problems. The number of elements
is 25 (N=25), and the number of grids is 101 (D=
M=101). The interval between elements is 0.5λand the
aperture is 50λ.
StandardPSOandtheproposedAMPSOareapplied
to tackle the above design example. In addition, since
twomethodsareusedtoimprovePSO,itisnecessaryto
identify if both could shed positive effect on PSO algo-
rithm. PSO memetic with arithmetic crossover is denoted
as MPSO. The configurations of the test algorithms are
shown in Table 1. The AMPSO algorithm contains less
parameters than PSO. As Dis very large, Np is set to 100
based on our empirical experience.
As to termination condition, the maximum number of
function evaluations (MFE) is set to 5e4 for all algorithms,
i.e., MFE=50,000. Each algorithm is independently run 25
times to gain an average performance. The simulation is
implemented in Matlab, and executed on a personal com-
puter with 4-core 3.4 GHz CPU and 4 GB of memory. This
could provide a fair comparison environment for the test
algorithms.
4.2 Simulation results
The optimal objective function values attained by each
algorithm is shown in Table 2. Over 25 independent runs,
min denotes the minimum values; similarly, med, max and
mean are respectively the median, maximum and mean
values; std denotes standard deviation of values. Table 2
Table 1 Configuration of the PSO, MPSO, and AMPSO algorithms
Algorithm Parameters
PSO Np =100, w=0.8, c1=c2=2
MPSO Np =100, w=0.8, c1=c2=2, Cr =0.8
AMPSO Np =100, Cr =0.8
Table 2 Optimal fvalues found by the test algorithms for
antenna design
Algorithm min med max mean std
PSO −10.2747 −9.7138 −9.3686 −9.7785 0.2624
MPSO −11.0568 −10.5446 −10.1880 −10.5823 0.2897
AMPSO −11.3935 −10.7617 −10.4727 −10.8028 0.2619
contains the five metrics of results found by all algorithms.
It is observed from Table 2 that MPSO outperforms PSO
in terms of min, med, max, and mean; while the std
of results of MPSO is slightly greater than that of PSO.
This means that memetic with crossover may cause large
variance and increase unstable factor of PSO algorithm;
however, the results becomes more stable with parameter
adaptive method. Compared with PSO, AMPSO attains
better results in all five metrics. Moreover, AMPSO also
outperforms MPSO in all five metrics. This means that
both crossover operator and parameter adaptation are
useful for improve the performance of PSO. Both methods
can have positive effect on the algorithm.
The optimal result for thinning nonuniform antenna
array by the AMPSO algorithm is shown in Fig. 6. This
figure gives the far field pattern plot of Azimuth angle in
degree versus PSLL in dB. It can be seen that the differ-
ence between mainlobe and sidelobe is more than 11 dB,
which shows a high gain and satisfies the requirements
of such design. In addition, sidelobe near 80◦and 100◦
are depressed below to 20 dB. This is useful for reducing
signal interference between different array elements.
Figure 7 presents the convergence graphs of the best
results for the PSO, MPSO, and AMPSO algorithms. It
shows the best run of each algorithm over 25 repeated
trials. As the curves after 10,000 function evaluations
Fig. 6 The optimal result for thinning nonuniform antenna array
using AMPSO
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Zhang and Zhang EURASIP Journal on Wireless Communications and Networking (2017) 2017:183 Page 6 of 7
Fig. 7 Convergence graphs of the best results for PSO, MPSO, and
AMPSO algorithms
(FEs) are flat, hence, the xaxes are shown in log scale.
For solving real world design applications, this means
that less budget should be set by users. After 1000 FEs,
the gap between algorithms becomes larger, which means
that the proposed method improves the performance of
PSO. Moreover, both the addition of crossover operator
and parameter adaptation enhance the algorithm’s per-
formance. Equipped with both methods, the algorithm
attains the best solution. Thus, the proposed method
explores more than standard PSO and MPSO, and finally
locates better solution.
5Conclusions
This paper proposes to add crossover operator and
parameter adaptation to standard PSO algorithm for tack-
ling the design of nonuniform antenna array. The design
of antenna array is thinned to a finite candidate slots.
The increase of slot numbers brings enormous difficul-
ties to optimization algorithms. Hence, it is necessary to
study more efficient algorithms for such design problems.
This paper takes arithmetic crossover and uses rounding
method to discretized real values to binary values. More-
over, a novel parameter adaptation method is proposed.
Different from traditional linear decrease of parameter
values, parameters are adapted alike to sigmoid function.
In this way, sensitive parameters are adapted and could be
applied to a good many of problem types. Simulated on
an example with D=101, the proposed AMPSO algo-
rithm presents good performance in both convergence
process and solution quality. Moreover, both the use of
crossover and parameter adaptation are shown to be able
to positively affect the algorithm.
In antenna array model (4), far field pattern is consid-
ered, which is the main metric in evaluating the perfor-
mance of antenna. The larger the difference between main
lobe beam and sidelobe beam, the higher the transmission
efficiency is. However, the bandwidth of main lobe beam
is not included in this model. Large bandwidth antenna is
more reliable in receiving signals in complex terrain and
severe weather. This issue will be studied in the future.
Acknowledgements
This research was supported in part by the National Science Foundation of
China (Project Nos. 61601329, 61603275) and the Applied Basic Research
Program of Tianjin (Project Nos. 15JCYBJC52300, 15JCYBJC51500).
Authors’ contributions
XZ proposes the modified PSO method and writes most of this paper. XZ is in
charge of numerical simulation and proofreading of the paper. Both authors
read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
Received: 5 August 2017 Accepted: 20 October 2017
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