- Research
- Open Access
Nonuniform antenna array design by parallelizing three-parent crossover genetic algorithm
- Xin Zhang^{1, 2},
- Qinglian Zhang^{1, 2} and
- Xiu Zhang^{1, 2}Email author
https://doi.org/10.1186/s13638-017-0895-2
© The Author(s) 2017
Received: 3 April 2017
Accepted: 29 May 2017
Published: 12 June 2017
Abstract
Antenna plays a very important role in wireless communication. Array elements laid out nonuniformly could achieve better frequency reliability and lower sidelobe level than uniform spaced elements. Designing desirable nonuniform antenna array requires the tuning of distances between each element, excitation, amplitude, and so on. Such design problem can be solved by genetic algorithm. Based on a recently genetic algorithm modification, this paper attempts to invent a parallel framework to enhance the efficiency of genetic algorithm. The parallel framework increases exploitation search of genetic algorithm. The effectiveness of the proposed algorithm is firstly demonstrated on toy problems with different kinds of problem complexity. The proposed algorithm is then verified on nonuniform antenna array design. Compared with genetic algorithm without parallelization, the novel algorithm attains better design solution and saves a lot computation time.
Keywords
- Nonuniform antenna array
- Genetic algorithm
- Parallel computing
- Antenna design
- Numerical optimization
1 Introduction
In radar sensor networks, antenna arrays are quite often used to increase the diversity gain and reduce the probability of missed detection and false alarm [1, 2]. How to deploy the antenna arrays is a science and art. Antenna arrays could be deployed in uniform or nonuniform to achieve the best performance in target detection [3–5].
Since Hertz and Marconi invented the first antenna, it has becoming more and more important in social life, and now, it is indeed an indispensable part of our daily life [3]. Existing in a three-dimensional world that composed of the scope of beam, three-dimensional radian, square angle, and solid angle, antenna can be a transducer or a transverter between the guided wave and free space wave, which can be described by basic electrical paraments such as input impedance, radiation resistance, gain, efficiency, directional diagram, polarization mode, and beam width. It can be seen that antenna is nowadays a crucial factor in wireless communication, though design-efficient and reliable antenna is a difficult problem as such design is a simulation-based time-consuming task [6]. Parallel execution of antenna design is able to greatly save simulation time [7].
Genetic algorithm (GA) was proposed by J. Holland in the 1970s. It is characteristic of imitating biological evolution mechanism in nature. The paradigm of GA is based on Darwin’s evolution theory and Mendel’s heredity theory. It could automatically acquire and accumulate the acknowledgement of the search space in the exploring process and reasonably control the search process toward optimal solution [8]. GA generally begins with an initial population, according to the fitness function, to evaluate the fitness of all individuals, and then, it chooses excellent individuals, intersects, and mutates to introduce new individuals to population. Finally, GA is able to make its population to evolve until met the given accuracy or reached the largest genetic algebra [9].
Since the 1990s, GA has become popular in solving both combinational problems and continuous optimization problems [10, 11]. Wind farm micro-siting is the decision problem for determining the optimal placement of wind turbines in consideration of the wake effect. An enhanced genetic algorithm (EGA), which is customized to the properties of wind farm dimensions, was proposed to solve such problem. The experimental results show that the EGA can obtain the decision both effectively and efficiently as compared to other metaheuristic approaches [12]. A better match to the field data was reached with the optimization parameters set with the genetic algorithm. In order to check to what extent the model replicated reality, model validation was also addressed. Results showed that a genetic algorithm is usefully applicable in the calibration process of the microscopic traffic simulation model [13]. Parallel GA strategy has been applied to optimize reservoir operation [14]. Alansi et al. implement GA in field-programmable gate array for space division multiple access techniques [15].
Besides GA, other evolutionary algorithms have been applied in real-world problems [16–19]. Min et al. considered pulse compression and sidelobe suppression in [20]. Yan et al. studied parallel algorithm in cognitive radar network [21]. Dhaliwal and Pattnaik compared several evolutionary algorithms on the design of fractal antenna [22]. Zhang et al. applied artificial bee colony algorithm to solve antenna design [23].
As a famous evolutionary algorithm, GA shows good performance in dealing with antenna design problem. This paper attempts to propose a parallel framework based on a three-parent crossover GA. This framework plans to enhance exploitation of GA so that solutions could be refined to a better extent than original algorithm in the later evolutionary stage. The goodness of the proposed framework is studied on both toy problems and nonuniform antenna design problem.
In the following, Section 2 introduces the design of nonuniform antenna array and related works. Section 3 describes three-parent crossover GA and the proposed parallel framework. Section 4 reports numerical simulation results and discussions. Conclusion is made in Section 5.
2 Nonuniform antenna design and related works
Relatively, though in the early 1960s nonuniform array has begun to be studied, its synthesis remains to be solved [24]. Usually, the synthesis is considered in a given number of arrays and array response, determining the array element position and incentive distribution. Among them, under a given number of elements and a given shape of antenna, the most important subject is how to design array element space and incentive of phase distribution reasonably so as to make lowest the array’s peak sidelobe level. On the one hand, array element response is the complex exponential function of its position; hence, the synthesis problem of array element is a nonlinear optimization problem; on the other hand, considering engineering application, the array element space must satisfy certain constraint (e.g., no less than a given value) to reduce the mutual coupling among elements. In addition, the ratio of the largest and the smallest array element incentive (i.e., current taper ratio (CRT)) need to be close to 1, which will make the antenna transmission power as large as possible.
where N is the number of array elements, d _{ i } denotes the position of the ith element, L is the array aperture with d _{1}=0 and d _{ N }=L. Function f is the peak sidelobe level (PSLL), and the objective is to minimize PSLL.
where k=2π/λ,u= cosθ− cosθ _{0} with θ is the sweeping angle from the direction of array, and θ _{0} is the position targeted by the main beam.
The binary particle swarm optimization in conjunction with the time-domain discrete Green’s function method was proposed for designing of printed ultra wideband (UWB) antennas [4]. A topology optimization method based on the method of moments for configuration design of planar metallic antenna was proposed in [25]. Bhatia et al. proposed a novel design of modified printed circular monopole antenna with defective ground structure for wideband applications. The basic design of antenna was comprised of a partial rectangular ground plane. The antenna design was further modified by employing a triangular notch in the ground plane to enhance the impedance bandwidth [26].
3 The used algorithms
This section describes three-parent crossover GA and our parallel framework.
3.1 Three-parent crossover genetic algorithm
Three-parent crossover GA (GA-TPC) was proposed by Elsayed and his colleagues [27]. A diversity operator is used in this algorithm to keep good population diversity. The algorithm attains good performance in competition problems on real parameter optimization. By taking this version, it probably outperforms old GA versions in dealing with antenna design problems.
The structure of GA-TPC consists of four steps: initial population, tournament selection, three-parent crossover, and diversity operator. The later three steps constitute main generation of this algorithm.
where x _{ i } is the ith solution (1 ≤ i ≤ N _{ p } where N _{ p } is the population size), r _{ i } denotes a vector of real numbers randomly generated between 0 and 1, and x ^{max} and x ^{min} are the boundary vectors of search space. For most practical problems, feasible ranges of decision variables are easy to set; in case feasible initial solutions are hard to create, feasibility detection should be added to the algorithm.
Tournament selection. Parent selection is realized by tournament selection method with tournament size tc=2 and with elitism 1, which means the best individual always keeps in the selection pool.
where v _{ i } are candidate offsprings, r _{ i } (i=1,2,3) are generated according to (N(0,σ)).
where u _{ i },i=1,2,3 are offsprings after current generation. Survivor selection of GA-TPC is alike to other GA versions. A new population is build by choosing the best N _{ p } individuals from offsprings and archive pool.
3.2 The proposed parallel framework
where x _{best} is the best so far individual produced in coarse search stage, r _{ i } (i∈[1,N _{ p }]) are random vectors generated according to (N(0,1)), \(\mathbf {x}_{p}^{\text {max}}\) and \(\mathbf {x}_{p}^{\text {min}}\) are respectively the vector of maximal and minimal values of all dimensions in the final population of coarse search stage. The physical meaning of this operation is to create a population of individuals normally distributed around the best so far individual. The standard deviation of normal distribution relies on the range of decision variables obtained in the first stage. Besides good initialization, algorithmic parameters are set causing the algorithm performing more exploitative search in the fine search stage.
In terms of algorithmic parameter, large N _{ p } assigns high explorative ability of GA, whereas small N _{ p } causes GA more exploitative ability. Moreover, GA with large N _{ p } converges slower than it with small N _{ p }. The convergence rate of GA is also related with crossover rate, though three-parent crossover relies on normal distribution and does not have crossover rate parameter. As to probabilities in diversity operator, p _{ i } (i=1,2,3) are set to 0.01, 0.1, and 0.1, respectively. They are fixed based on our experience and the author’s suggestion.
4 Numerical simulation
In this section, the proposed parallel framework is studied and compared with GA without parallel.
4.1 Experimental setting
Configuration of GA-TPC, parallel GA-TPC, and parallel GA-MR algorithms
Algorithm | Parameters |
---|---|
GA-TPC | N _{ p }=90, σ=0.5 |
Parallel GA-TPC | \(N_{p}^{1}=60\), σ ^{1}=0.5, \(N_{p}^{2}=30\), σ ^{2}=1.0 |
Parallel GA-MR | N _{ p }=320, N _{ g }=8, P _{ g }=80, R _{ m }=0.5, P _{ c }=1000 |
Toy functions: 5 unimodal functions and 5 multimodal functions
Number | Function | Property |
---|---|---|
f _{1} | Spherical function | Unimodal, separable, scalable |
f _{2} | Axis parallel hyper-ellipsoid, weighted sphere | Unimodal, separable, scalable |
f _{3} | Perm function | Unimodal, nonseparable, scalable |
f _{4} | High-conditioned elliptic function | Unimodal, separable, scalable |
f _{5} | Schwefel’s problem 1.2 | Unimodal, nonseparable, scalable |
f _{6} | Rosenbrock’s function | Multimodal, nonseparable, scalable |
f _{7} | Rastrigin’s function | Multimodal, separable, scalable |
f _{8} | Griewank function | Multimodal, nonseparable, scalable |
f _{9} | Ackley’s function | Multimodal, nonseparable, scalable |
f _{10} | Zakharov function | Multimodal, nonseparable, scalable |
In the study of toy functions and antenna design, the maximal number of objective function evaluations (MFE) is fixed at 5000D. For each test case, an algorithm is independently run 25 times to reach an average performance of the algorithm. For parallel GA-TPC, computational resources are equally split to stages 1 and 2. That is MFE = 2500D for stage 1 and MFE = 2500D for stage 2. It can be seen from the parallel framework that communication between multiple cores happens when creating initial population at stage 2. Because computational resource is equally split and the same algorithm structure is used, it is expected that multiple cores would finish running nearly the same time so that wasteful waiting of the synchronization of multiple cores is reduced to minimum. Thus, good parallel efficiency could be reached in the proposed parallel framework. Both algorithms and problems are programmed in Matlab. The simulation is performed on a personal computer with four-core 3.40 GHz CPU and 4 GB random access memory.
4.2 Results on toy functions
Optimal function values found by GA-TPC and parallel GA-TPC over 25 independent runs
Algorithm | GA-TPC | Parallel GA-TPC | U test | ||
---|---|---|---|---|---|
f(·) | med | std | med | std | p-value |
f _{1} | 6.56E −04 | 2.64E + 00 | 0.00 | 0.00 | 4.58E −08 |
f _{2} | 6.07E −08 | 6.38E −03 | 0.00 | 0.00 | 1.10E −06 |
f _{3} | 0.00 | 0.00 | 0.00 | 0.00 | N/A ^{a} |
f _{4} | 6.17E −08 | 5.19E −04 | 0.00 | 0.00 | 1.90E −05 |
f _{5} | 1.16E + 12 | 9.85E + 13 | 1.49E + 12 | 1.68E + 13 | 0.4377 |
f _{6} | 1.04E + 01 | 1.16E + 02 | 6.43E + 00 | 1.88E + 01 | 0.0052 |
f _{7} | 3.98E + 00 | 1.68E + 00 | 4.97E + 00 | 3.22E + 00 | 0.0708 |
f _{8} | 1.28E −01 | 1.30E −01 | 9.07E −02 | 5.13E −02 | 0.0478 |
f _{9} | 0.00 | 0.00 | 0.00 | 0.00 | N/A ^{a} |
f _{10} | 4.39E −02 | 7.88E −01 | 0.00E + 00 | 2.31E −01 | 6.34E −08 |
Results found by parallel GA-MR and parallel GA-TPC over 25 independent runs
Algorithm | Parallel GA-MR | Parallel GA-TPC | U test | ||
---|---|---|---|---|---|
f(·) | med | std | med | std | p value |
f _{1} | 0.00 | 0.00 | 0.00 | 0.00 | N/A ^{a} |
f _{2} | 4.92E −05 | 1.43E −07 | 0.00 | 0.00 | 8.66E −05 |
f _{3} | 0.00 | 0.00 | 0.00 | 0.00 | N/A ^{a} |
f _{4} | 5.60E −08 | 8.93E −05 | 0.00 | 0.00 | 5.25E −05 |
f _{5} | 9.90E + 14 | 9.54E + 13 | 1.49E + 12 | 1.68E + 13 | 0.3077 |
f _{6} | 7.29E + 00 | 1.96E + 01 | 6.43E + 00 | 1.88E + 01 | 0.2072 |
f _{7} | 4.05E + 00 | 2.86E + 00 | 4.97E + 00 | 3.22E + 00 | 0.0644 |
f _{8} | 2.08E −01 | 1.36E −01 | 9.07E −02 | 5.13E −02 | 0.0130 |
f _{9} | 0.00 | 0.00 | 0.00 | 0.00 | N/A ^{a} |
f _{10} | 2.02E −06 | 5.47E −01 | 0.00E + 00 | 2.31E −01 | 0.0379 |
4.3 Results on antenna design
Results found by GA-TPC and parallel GA-TPC over 25 independent runs for antenna design
Algorithm | GA-TPC | Parallel GA-TPC | ||||
---|---|---|---|---|---|---|
Metric | min | med | max | min | med | max |
f | −20.88 | −20.57 | −19.43 | −21.36 | −21.02 | −20.78 |
d _{1} | 0 | 0 | 0 | 0 | 0 | 0 |
d _{2} | 0.72 | 0.67 | 1.65 | 0.99 | 0.77 | 0.78 |
d _{3} | 1.56 | 1.52 | 2.31 | 1.74 | 1.55 | 1.56 |
d _{4} | 2.21 | 2.22 | 2.92 | 2.59 | 2.26 | 2.24 |
d _{5} | 2.73 | 2.81 | 3.47 | 3.21 | 2.75 | 2.75 |
d _{6} | 3.31 | 3.39 | 4.03 | 3.71 | 3.28 | 3.29 |
d _{7} | 3.77 | 3.82 | 4.51 | 4.30 | 3.76 | 3.81 |
d _{8} | 4.23 | 4.34 | 5.03 | 4.73 | 4.30 | 4.37 |
d _{9} | 4.73 | 4.89 | 5.44 | 5.22 | 4.81 | 4.69 |
d _{10} | 5.25 | 5.31 | 5.97 | 5.61 | 5.24 | 5.24 |
d _{11} | 5.62 | 5.77 | 6.45 | 6.11 | 5.65 | 5.75 |
d _{12} | 6.25 | 6.30 | 6.98 | 6.57 | 6.25 | 6.28 |
d _{13} | 6.75 | 6.92 | 7.47 | 7.09 | 6.74 | 6.75 |
d _{14} | 7.33 | 7.34 | 7.97 | 7.49 | 7.31 | 7.31 |
d _{15} | 8.07 | 8.10 | 8.52 | 8.16 | 8.09 | 8.14 |
d _{16} | 8.89 | 8.87 | 9.03 | 9.02 | 8.84 | 8.88 |
d _{17} | 9.74 | 9.74 | 9.74 | 9.74 | 9.74 | 9.74 |
By conducting U test on 25 trials on this problem, parallel GA-TPC significantly outperforms GA-TPC with significance level α=0.05. In terms of running time, the design simulation of GA-TPC costs 5295.2 (s) over 25 trials, while the design simulation of parallel GA-TPC takes 2891.5 (s). The time ratio on this problem is 1.83, which is alike to the ratio on toy functions. Thus, it is reasonable to conclude that the proposed parallel framework can effectively and efficiently improve the performance of GA-TPC algorithm.
5 Conclusions
Recently, many researchers or users apply genetic algorithm (GA) to design antennas. As antenna design is a difficult and a simulation-based time-consuming task, GA is a useful tool to deal with such problem. This paper focuses on improving the efficiency of GA based on the idea of parallel computing. Different existing approaches, our research is based on a state-of-the-art version of GA, denoted as GA-TPC. Moreover, in the proposed parallel framework, algorithm evolution is equally split to two stages. In stage 1, algorithm search is guided toward exploration through the use of large population size and relatively large standard deviation (std) in normal distribution. In stage 2, algorithm search is guided toward exploitation through using small population size and large std. Moreover, initial individuals of stage 2 are created around the best individual returned in stage 1. The distribution of individuals in stage 2 also relies on the variance of individuals in the last population of stage 1.
The proposed framework is instantiated on GA-TPC, denoted as parallel GA-TPC. Tested on both toy problems and antenna design problem, parallel GA-TPC finds better solutions than GA-TPC and a state-of-the-art parallel modification of GA (i.e., parallel GA-MR). Moreover, parallel GA-TPC reaches smaller std of function values of final solutions than GA-TPC and parallel GA-MR. This means that the proposed framework is effective and stable. Furthermore, computational time of parallel GA-TPC is saved about 80% compared with the time of GA-TPC under the same simulation environment.
The proposed framework shows good performance in antenna design, though it has an overhead that the extension of the framework to more cores is not easy. The difficulty of multiple cores extension attributes to algorithmic parameter setting. Dividing algorithm evolution to two stages is intuitively feasible as GA has exploration and exploitation. Using more cores means to split evolution to more stages which has to be meaningful and effective. This direction will be studied in the future.
Declarations
Acknowledgements
This research was supported in part by the National Science Foundation of China (Project No. 61603275, 61601329) and the Doctoral Fund Project of Tianjin Normal University (Project No. 043-135202XB1602).
Authors’ contributions
XZ proposes the parallel framework based on GA-TPC and writes most of this paper except that XZ writes the numerical simulation section. QZ is in charge of the computer simulation of the algorithm and the collection of the results. XZ organizes the whole paper as well as the proofreading. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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