Antenna array design by a contraction adaptive particle swarm optimization algorithm
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Abstract
Massive multiple input multiple output (MIMO) has drawn intensive attention of researchers for solving huge data transmissions in the fifth generation wireless communication. Antennas aligning in array bring in advantages for interference reduction of incoming wave signals and interference offset among antenna elements. The design of antenna array is studied in this paper. The design is turned into a maximization problem. A contraction adaptive particle swarm optimization (CAPSO) method is proposed to solve the problem. Different from previous methods, CAPSO is based on a contraction factor which limits the variation neighborhood during the solution searching process. The adaptive technique can tune the searching range of CAPSO. Simulations are reported comparing the CAPSO method with other methods. Convergence rate is analyzed through two toy functions. CAPSO shows a fast convergence property. Then it is used to solve antenna array design. The CAPSO method shows good performance for different numbers of array elements.
Keywords
Antenna array MIMO Particle swarm optimization Numerical optimizationAbbreviations
 5G
Fifth generation
 CAPSO
Contraction adaptive particle swarm optimization
 GA
Genetic algorithm
 IPSO
Improved particle swarm optimization
 MIMO
Multiple input multiple output
 NP
Number of particles
 PSO
Particle swarm optimization
 QoS
Quality of service
1 Introduction
Massive multiple input multiple output (MIMO) is a hot topic for providing good quality of service (QoS) in the fifth generation (5G) wireless mobile communication [1]. Not only data transmission efficiency but also transmission reliability and privacy security have to be considered to assure good QoS [2, 3]. Moreover, new communication technology is also welcome such as cooperative communication, heterogeneous network, and compressive sensor network [4, 5, 6]. Among different network structures, resource allocation, network efficiency, throughput, and other aspects need to be studied as soon as possible [7, 8].
Antenna is essential in physical layer communication to assure data transmission in 5G wireless communication [9]. Antenna array is even more useful due to high gain and reliability of arranging many elements [10]. Optimal antenna design methods can be classified to deterministic methods and heuristic methods. Deterministic methods often converge very quickly but requiring prior knowledge of geometric structure of antennas [11, 12, 13]. To solve difficult design models, relaxation techniques are generally utilized to reduce the complexity of models [14, 15, 16]. Parallel computing, matrix decomposition, and other methods are also utilized to speed up the convergence rate for deterministic methods [17, 18, 19, 20, 21].
Heuristic algorithms for antenna design include genetic algorithm (GA) [22], particle swarm optimization (PSO) [23], differential evolution [24, 25], and neighborhood field optimization [26]. Recently, improved heuristic algorithms have been used to design wireless communication network designs, power allocation, and antennas [27, 28].
PSO has been studied and applied to many different problems in the real world [29]. Since its creation, standard PSO has been improved from many aspects including parameter control, search equation, and neighborhood network structure. This paper attempts to tune algorithmic parameters based on contraction and accelerating factors.
The above is discussed for the purpose of the paper. There are two contributions in the paper. The first is the improved parameter adaptation based on contraction, which improves the form of the iterative formula for the PSO algorithm. Another one is the convergence rate analysis and application to antenna design with different numbers of antenna elements.
Section 3 briefly introduces the antenna design problem and related works. Section 4 introduces standard PSO and the proposed contraction adaptive particle swarm optimization (CAPSO) algorithms. Section 6 reports simulation results compared with other methods, and the paper is concluded in Section 7.
2 Method and experiment
The study of the paper is to effectively solve antenna array designs. The study is accomplished by two parts. First, antenna array designs are expressed as an optimization problem. The problem model contains the main factors in antenna array designs. Second, a contraction adaptive particle swarm optimization algorithm is used to solve the problem model. The efficiency of the algorithm should be analyzed to assure the problem model being effectively solved. Comparisons are made by taking three algorithms which are of the same type as the proposed CAPSO algorithm. Comparison results show that the CAPSO algorithm is more efficient than other algorithms. Then, it is used to solve antenna array designs. The experiment is based on numerical simulations. To verify the effectiveness of the proposed method, designs are modeled as easy to use and scalable type. The analysis shows that the proposed method is useful and effective to solve antenna array designs.
3 The design of antenna array and related works
For more than a century, especially after the Second World War, antenna theory, design, and application have been rapidly developed. Antenna array is an important type of antenna [30]. An antenna system consisting of two or more discrete antennas is called an antenna array. There are many kinds of antenna arrays, according to the arrangement of elements, wired array, and plane array [31, 32].
A linear array is an antenna array consisting of a plurality of units separated from each other and centered on a line [33]. A linear array is divided into uniform linear array and nonuniform linear array: uniform linear array means the equal distance between the adjacent antenna unit and the constant incentive phase difference between adjacent units, that is to say each unit is a linear array excited by the law of equality. Nonuniform linear array refers to that the distance between the adjacent antenna elements is not equal, and each element is a linear array inspired by the law of nonsynchronous progressive phase.
where k=2π/λ, λ is the wavelength, G is the number of array elements, and PM is the peak value of the main lobe of antenna array. θ_{0} is the incoming direction of the main beam. d_{c} is the minimum distance between two successive elements. I_{i} and φ_{i} are the amplitude of excitation and phase, respectively.
In model (1), d_{i} could be any real numbers such that d_{i}−d_{i−1}≥d_{c} and d_{i+1}−d_{i}≥d_{c}. Moreover, I_{i} and φ_{i} are also variables to be defined by users. These can be set as parameters in the antenna array system.
4 The CAPSO method
It can be seen from (2), φ, c_{1} and c_{2} are algorithmic parameters as well as NP.
The adaptive formula of φ is previously studied in [35].
In the CAPSO method, parameters φ, c_{1} and c_{2} are all adapted based on (6), (7), and (8). c_{1} and c_{2} are based on the iteration process of the method. Their values gradually decrease from \(c_{i}^{\text {max}}\) to \(c_{i}^{\text {min}}\) (i=1,2). Based on (8), φ also decreases along with iteration. Note that c_{1}+c_{2}≥4 is required to fulfill the root operation. Thus, the CAPSO method performs a large search step size in the former iteration process, while performing small search step size in the later iteration process. The procedures of CAPSO are identical to the philosophy of heuristic methods [36, 37].
5 Numerical experiment settings
The CAPSO method will be studied on toy functions and then applied to solve antenna array designs in this section.

\(f_{1}(\textbf {x}) = 100\left (x_{1}^{2}x_{2}^{2}\right)\left (1x_{1}^{2}\right)\);

\(f_{2}(\textbf {x}) = \left (4x_{1}^{2}2.1x_{1}^{4}+x_{1}^{6}/3+x_{1}x_{2} 4x_{2}^{2}+x_{2}^{4}\right)\);
Both functions are unconstrained maximization problems with two independent variables, which can take any real numbers. The CAPSO method is independently tested 25 times on each function with t^{max}=10,000.
In the simulation of antenna array designs, excitation amplitude I_{i} and phase φ_{i} for element i are set to constant. d_{i} between adjacent elements are independent variables and computed based on distance of adjacent elements. The length of array L is set to Nλ. Hence, with different numbers of elements G, L is automatically changed. The design problem is then scalable and can be tested with any number of array elements.
6 Results and discussion
Result comparison of the CAPSO method versus standard PSO, IPSO, and APSO methods
Method  f(·)  Mean  std  p 

PSO  f _{1}  1099.20  434.68  2.16E −7 
f _{2}  4033.60  32.00  7.31E −6  
IPSO  f _{1}  908.80  222.27  2.37E −7 
f _{2}  3072.00  448.85  1.15E −7  
APSO  f _{1}  939.20  210.12  2.36E −7 
f _{2}  3172.80  446.58  1.20E −7  
CAPSO  f _{1}  449.60  553.93  N/A 
f _{2}  2004.80  46.64  N/A 
Figure 3 shows a far field pattern associated with the optimal solution when G=10. It can be seen from the figure that the main lobe is about 10 dB better than the nearest side lobe. Moreover, the range of the main lobe is about 5° around θ_{0}=90°. Thus, the attained antenna array G=10 is a wideband case.
Figure 4 shows a far field pattern associated with the optimal solution when G=30. It can be seen from the figure that the main lobe is about 14 dB better than the nearest side lobe. Moreover, the range of main lobe is about 2° around θ_{0}=90°. Thus, the attained array G=30 is narrower than the array G=10.
Figure 5 shows a far field pattern associated with the optimal solution when G=50. It can be seen from the figure that the main lobe is about 16.5 dB better than the nearest side lobe. Moreover, the range of main lobe is about 2° around θ_{0}=90°. Thus, the attained array G=50 is similar as the array G=30, but is narrower than the array G=10.
From the above simulation, it can be seen that the gain of antenna array increases with the number of elements. However, the band width decreases with the number of elements from 10 to 30 array elements. The band width does not decrease with array elements from 30 to 50. Because the tested antenna array problem scales from 10 to 50 array elements, the results show that our method is suitable to such designs.
7 Conclusion
Radio waves are widely used in communication, broadcasting, target detection, navigation, and other fields. The transmission and reception of radio waves depend on antennas.
This paper focuses on how to optimize the positions of antenna array elements so as to maximize the performance of antenna in transmitting and receiving data. The far field pattern and side lobe level value are main metrics of the performance of antenna array. The proposed CAPSO method is based on contraction factor adaptation. Accelerating factors are also adapted.
A numerical experiment is tested on two toy functions for analyzing the convergence rate of the CAPSO method. Compared with three other methods, CAPSO can achieve good convergence performance. The simulation on antenna array shows that the CAPSO method is able to find good solution, though when the number of elements becomes large, the band width decreases. The band width does not decrease too much from 30 elements to 50 elements. Because the tested antenna array problem is scalable, the results show that our method is suitable to such designs.
Notes
Acknowledgements
Not applicable.
Funding
This research was supported in part by the National Natural Science Foundation of China (Project No. 61601329, 61603275, 61601328), the Tianjin Higher Education Creative Team Funds Program, and the Applied Basic Research Program of Tianjin (Project No. 18JCYBJC16100).
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Authors’ contributions
XZ proposes the algorithm framework based on PSO and writes most of this paper except that XZ writes the antenna array model section. DL is in charge of the computer simulation of the algorithm and collection of the results. YW organizes the whole paper as well as contributes useful discussion. All 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.
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