International Journal of Antennas and Propagation

Volume 2019, Article ID 1064103, 13 pages

https://doi.org/10.1155/2019/1064103

## Multiobjective Synthesis of Linear Arrays by Using an Improved Genetic Algorithm

School of Electronic Science and Engineering, Jilin University, 130012 Changchun, China

Correspondence should be addressed to Bo Yang; moc.621@llaobgnay

Received 18 July 2018; Revised 5 September 2018; Accepted 4 April 2019; Published 22 July 2019

Academic Editor: Shiwen Yang

Copyright © 2019 Bo Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, an improved genetic algorithm with dynamic weight vector (IGA-DWV) is proposed for the pattern synthesis of a linear array. To maintain the diversity of the selected solution in each generation, the objective function space is divided by the dynamic weight vector, which is uniformly distributed on the Pareto front (PF). The individuals closer to the dynamic weight vector can be chosen to the new population. Binary- and real-coded genetic algorithms (GAs) with a mapping method are implemented for different optimization problems. To reduce the computation complexity, the repeat calculation of the fitness function in each generation is replaced by a precomputed discrete cosine transform matrix. By transforming the array pattern synthesis into a multiobjective optimization problem, the conflict among the side lobe level (SLL), directivity, and nulls can be efficiently addressed. The proposed method is compared with real number particle swarm optimization (RNPSO) and quantized particle swarm optimization (QPSO) as applied in the pattern synthesis of a linear thinned array and a digital phased array. The numerical examples show that IGA-DWV can achieve a high performance with a lower SLL and more accurate nulls.

#### 1. Introduction

Linear array pattern synthesis with multiobjective optimization has been a hot research topic in recent years [1–9]. The common objectives of such approach are the side lobe level (SLL), directivity, main beam width, and null depth, which may be contradictory to each other. To simplify the feed network of the array, discrete adjustment, instead of continuous adjustment, of the excitation amplitude and phase of the array elements can be carried out. The simple constructions can apply a thinned array by controlling the switch of the elements [10–14] and phase array with the use of an -bit digital phase shifter [15–17] or some other device.

Many optimization methods construct objective functions into one function, which is defined as aggregating function [1, 2]. The optimization method based on a backtracking search optimization algorithm (BSA) is used to synthesize array pattern with the prescribed nulls and the low SLL [2]. In [10], a suitable mapping method is used, and a modified particle swarm optimization (PSO) is found to be capable of effectively addressing the discrete optimization problems of linear array pattern synthesis. This strategy has been applied to thinned linear array pattern synthesis with a minimum SLL. A modified PSO algorithm called quantized PSO (QPSO) has been described for the synthesis of antenna array patterns by using digital phase shifters. The fitness function includes a weighted sum of the main beam width value, the sum of the null depths at the interference signal directions, and the SLL value [15]. This optimization method by aggregating function may get the result that looks good, but the parameters of coefficients in the fitness function are difficult to decide.

In [4], the authors showed that the linear antenna array design could be modeled as a multiobjective optimization problem (MOP). Evolutionary multiobjective optimization methods, such as NSGA-II, DEMO, SPEA-2, and EM-MOPSO, have been developed for the synthesis of linear antenna arrays and concentric ring antenna arrays, with the main beam width and SLL [1, 4–7] as objectives. NSGA-II [9] is a popular and efficient multiobjective genetic algorithm that has been used in several engineering design problems [18, 19]. For instance, NSGA-II has been extensively applied in the synthesis of antenna arrays [20–23] and is considered as one of the best evolutionary optimizers for multiobjective problems [7, 24]. What can be focused, the simulation results of optimum solutions distributed in the Pareto front (PF) should keep diversity for every fitness. The classic NSGA-II was applied for 40 elements thinned linear array [22], but the values of the null depth were only -32.7 and -22 dB with low SLL. Some hybrid algorithms, such as memetic generalized differential evolution (MGDE3) [20] and iterative fast Fourier transform (IFFT) with a judge factor introduced into the NSGA-II [23], have been used to try to improve the convergence and solution diversity.

Thinned arrays have been widely used in array pattern synthesis due to the excellent performance [11, 12, 14, 25, 26]. In [10], a suitable mapping method is used, and the modified PSO is found to be capable of effectively addressing the discrete optimization problems of linear array pattern synthesis. This strategy has been applied to thinned linear array pattern synthesis with a minimum SLL. In that study, a rounding strategy and real number PSO (RNPSO) are combined to enable PSO to solve 0-1 discrete optimization problems, integer optimization problems, and mixed optimization problems. In the mapping process, a round-down function is applied instead of a round function to guarantee that each integer value can be equally selected [10].

Phased arrays are considered to be the best solution in beam scanning and widely used in engineering [7, 15–17, 20]. With the advance in technology, digital phase shifters are now widely used in phased arrays to provide beam scanning and interference suppression [24]. A modified PSO algorithm called quantized PSO (QPSO) has been described for the synthesis of antenna array patterns with the use of digital phase shifters [15]. The solution space of the QPSO algorithm is restricted to the finite quantized integer values of the array phase coefficients. The QPSO searches for an optimal solution within the available quantized values of the digital phase shifters to minimize the fitness function, which includes the SLL value and interference suppression, while keeping the main beam unchanged. The fitness function includes a weighted sum of the mean beam width value, the sum of the null depths at the interference signal directions, and the SLL value. The QPSO searches for an optimal solution within the available quantized (discrete) values of the phase shifters that minimize the specified fitness function.

A genetic algorithm (GA) is generally believed to be suitable for discrete optimization problems because it uses a discrete coding method and deals directly with discrete variables. In the present study, an improved genetic algorithm with dynamic weight vector (IGA-DWV) is proposed for array pattern synthesis. The objectives of the linear array are synchronously optimized for different fitness functions. The innovations and the effectiveness obtained in this work are described below:

A multiobjective optimization problem for array pattern synthesis: A multiobjective evolutionary algorithm, instead of a single objective obtained by aggregating functions, is applied to linear array pattern synthesis. The SLL, directivity, and prescribed null positions are the optimized objectives for different arrays.

The application of the dynamic weight vector to address the diversity of solutions: To maintain the diversity of the selected solution in each generation, the solutions closer to the dynamic weight vector uniformly distributed on the PF can be chosen to the new population.

A lower SLL and a more accurate null position: A lower SLL and better null depth and width are obtained by the multiobjective optimization method and the appropriate coded method. The results show a higher performance than those in the existing research literature.

The remainder of this paper is organized as follows. In Section 2, the improved multiobjective genetic algorithm with dynamic weight vector is proposed. In Section 3, the multiobjective optimization problems of array pattern synthesis are presented. Section 4 simulates numerical examples and describes the comparative performance of the proposed technique. Finally, the conclusions are given in Section 5.

#### 2. Improved Multiobjective Genetic Algorithm with Dynamic Weight Vector

The multiobjective optimization problem can be expressed aswhere is the decision vector, and is the feasible region in the decision space. Here, is the number of objectives, and is the size of the decision vector.

Very often, because the objectives in (1) contradict each other, no point in minimizes all the objectives simultaneously; instead, the objectives have to be balanced. The best trade-offs among the objectives can be defined in terms of Pareto optimality.

Considering a minimization problem for each objective and , is said to dominate (written as ) if and only if for every , and for at least one . We can obtain a solution , the Pareto optimal to (1), if we cannot find a solution such that . Then, is called a Pareto optimal (objective) vector. In other words, any improvement in a Pareto optimal solution in one objective must lead to the deterioration of at least one other objective. The set of all the Pareto optimal solutions is called the Pareto set (PS), and the set of all the Pareto optimal objective vectors is the PF [27]. The multiobjective optimization algorithm aims to find the optimum solutions approximating the actual PF.

In this study, IGA-DWV is proposed for array pattern synthesis. Figure 1 shows the procedure. The key technology focuses on the coding for different problems and the selection operator.