Thinning of antenna array via adaptive memetic particle swarm optimization
© The Author(s) 2017
Received: 5 August 2017
Accepted: 20 October 2017
Published: 6 November 2017
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.
Massive multiple input multiple output antenna array is crucial for the fifth generation wireless communication . 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 . 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: evolutionary algorithms (EAs) and swarm intelligence (SI) algorithms. Informally, EAs contain genetic algorithm (GA) , evolutionary strategies , and differential evolution . These algorithms simulate the evolution of genetic process of livings. They usually contain both mutation and crossover operators. SI algorithms contain particle swarm optimization (PSO) , artificial bee colony (ABC) , neighborhood search optimization , 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 combinations are able to result effective algorithms . In the past, GA, PSO, and ABC have been used in antenna designs [15, 23, 24]. Recently, covariance matrix adaptation evolutionary strategy  and differential evolution  are applied to synthesize antenna array patterns.
A large number of iterations is needed for standard PSO algorithm to obtain a satisfactory solution . This is not acceptable for users as antenna simulation 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 reasonable as memetic with crossover operator can enlarge and promote algorithm’s search; also parameter control 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 adaptive memetic particle swarm optimization (AMPSO), which is applied to tackle antenna design. Numerical experiment is conducted studying the effect of adding crossover and parameter adaptation. The results are discussed 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 standard 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 categories. 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 positions in grids are equally spaced. In the first case, elements 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.
where MP is mainlobe peak level, C 0 is a correction parameter as the first zero point may shift in nonuniform array. The constraint in (4) is to constrain the value of u so 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 . Zhao et al. studied resource allocation problem in MIMO systems such as time allocation  and power allocation .
3 Optimization algorithms
This section depicts standard PSO and the proposed AMPSO algorithm.
3.1 Standard particle swarm optimization algorithm
where subscript j∈[1,D] refers to the jth dimension; t refers to the tth generation; r 1 and r 2 are random numbers between 0 and 1. p i and p g are respectively personal best and global best positions that particle i walked through. In this equation, w, c 1, and c 2 are three algorithmic parameters of PSO. Their setting for good performance of algorithm 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 performance of PSO . Khan et al. improved PSO by a random mutation mechanism and a dynamic inertia weight adaptive method, which could facilitate algorithm convergence . Jin et al. improved PSO through enlarging explorative ability of PSO to design a permanent magnet synchronous machine .
3.2 Adaptive memetic PSO algorithm
As designated by no free lunch theory , the performance 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.
where w min and w max are the maximum and minimum value of parameter w; similarly, the minimum and maximum value of c1 and c2 are c1min, c1max and c2min, c2max.
The PSO algorithm combined with crossover operator and parameter adaptation is abbreviated as AMPSO. Compared with standard PSO, the AMPSO algorithm needs more operations to perform two additional steps. For arithmetic crossover, the computing of velocity 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.
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 volume 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 0.5λ and the aperture is 50λ.
Configuration of the PSO, MPSO, and AMPSO algorithms
Np=100, w=0.8, c1=c2=2
Np=100, w=0.8, c1=c2=2, Cr=0.8
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 computer 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
Optimal f values found by the test algorithms for antenna design
This paper proposes to add crossover operator and parameter adaptation to standard PSO algorithm for tackling 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 difficulties 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. Moreover, 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 algorithm 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 considered, which is the main metric in evaluating the performance 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.
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).
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.
The authors declare that they have no competing interests.
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