International Journal of Antennas and Propagation

Volume 2016, Article ID 1956047, 7 pages

http://dx.doi.org/10.1155/2016/1956047

## Location Optimization for Square Array Antennas Using Differential Evolution Algorithm

Department of Electrical & Electronics Engineering, Faculty of Engineering, Gazi University, Maltepe, 06570 Ankara, Turkey

Received 28 March 2016; Accepted 24 April 2016

Academic Editor: Sotirios K. Goudos

Copyright © 2016 Erkan Afacan. 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 recent works thinned arrays giving minimum peak sidelobe levels for planar square antenna arrays are obtained using Hadamard difference sets. In the current work thinned array configurations giving lower peak sidelobe levels than those given in the literature are obtained for square arrays of , , , and elements. Differential evolution algorithm is used in the determination of the antenna locations.

#### 1. Introduction

Thinned antenna arrays offer advantages of cost, weight, and power consumption. If the array elements are located periodically then it is difficult to control sidelobe level (SLL) effectively. One way of overcoming this problem is to place array elements aperiodically. So, a planar antenna array can be placed on a two-dimensional grid with equal spacing. For thinning, some of the antenna elements from the array are removed or turned off, and the remaining active elements are fed with equal amplitude currents.

Since the synthesis problem for planar thinned arrays is complex and cannot be solved analytically, several global optimization algorithms are proposed in the literature such as simulated annealing [1], ant colony optimization [2], invasive weed optimization [3], genetic algorithm [4–7], particle swarm optimization [8, 9], differential evolution [10], and teaching learning based optimization algorithm [11].

In [1], simulated annealing is applied to thinning and weighting of planar arrays with element numbers as large as . Ant colony optimization is applied to a elements’ planar array in [2], and the SLLs in and planes are sought to be minimized. Invasive weed optimization which is inspired from the phenomenon of colonization of invasive weeds in nature is applied to a elements’ planar array in [3].

In [4] a elements’ planar antenna array is optimized by using genetic algorithm and a thinned array filled 54% with 108 elements is obtained. Since the problem geometry is symmetric optimization is realized for a elements’ antenna array in fact. A multiple agent optimization technique based on genetic algorithm is presented in [5]. Authors have reduced the multiobjective problem into a classical single objective one, so it has been possible to specify multiple constraints within the method. The proposed method is applied to and elements’ planar arrays. In [6] the same problem given in [4] is studied with modified real genetic algorithm and a thinned array filled 50% with 100 elements is obtained. A genetic algorithm based on orthogonal design is utilized to optimize the peak sidelobe level (PSLL) of a elements’ planar array in [7].

In [8] binary and real particle swarm optimizations are used to synthesize a elements’ planar array with two different cost functions. In [9] a feedback based binary particle swarm optimization is proposed. The algorithm is applied to , , and elements’ square arrays. The cost function is taken as a linear combination of PSLL and thinning percentage.

Differential evolution (DE) algorithm is a widely used algorithm in antenna array synthesis problems for linear [12–19], circular [20–22], planar [10, 23–25], and conformal arrays [26]. DE is also applied to Yagi-Uda antennas in [27, 28].

In [12], DE algorithm is applied to the synthesis of low sidelobe linear antenna arrays. Authors have shown that DE converges faster than real-coded genetic algorithm. DE algorithm is applied to the synthesis of uniform amplitude, unequally spaced linear arrays in [13]. Position-only, phase-only, and position-phase syntheses are considered and it is observed that position-phase synthesis gives better results than two others. The effect of element spacing and element number on SLL is investigated in detail. In [14], DE algorithm is used for linear array pattern synthesis with prescribed nulls at the direction of interferences while SLLs are kept below desired levels. Only the array element excitation amplitudes are controlled. A pattern synthesis method based on DE is proposed for uniform amplitude thinned linear phased arrays in [15]. The locations and phase excitations of the thinned array elements are optimized. Reduction of grating lobes and minimization of SLLs are aimed at while keeping the main beam pointed to the desired direction. In [16], time modulation is used for the design of unequally spaced linear arrays. DE algorithm is adopted to optimize the positions and switch-on times of elements in the array, and SLLs and sideband levels are tried to be suppressed with beamwidth constraint. A dynamic DE algorithm with the best of random differential mutation is used in the synthesis of unequally spaced linear arrays in [17]. Position-only and position-phase syntheses are considered and lowest PSLLs are investigated. In [18], DE with strategy adaptation is applied to the sparse linear array synthesis. The goal of the optimization is taken as to suppress SLL by finding the optimum element positions and to set main lobe to a desired beamwidth. In [19], composite DE with eigenvector base crossover is applied to shaped beam pattern synthesis problem for linear arrays and for a conformal cylindrical array.

In [20], a numerical approach based on DE algorithm is proposed for the pattern synthesis of concentric ring array antennas with the constrained first null beamwidth. It is shown that the method gives lower PSLLs with the optimization of ring spacing and element number in each ring. A modified form of DE called DE with global and local neighborhoods is used for designing a thinned concentric planar circular antenna array in [21]. Reducing SLL and increasing the number of switched-off elements are aimed at. It is shown that any desired SLL value can be obtained while maintaining the number of switched-off elements at a reasonable value. In [22], a synthesis method for shaping beam pattern of a concentric ring array antenna using DE and genetic algorithm is proposed. Authors have shown that both methods are capable of generating the desired shaped beams.

In [10], a Boolean differential evolution algorithm based on Boolean algebra is proposed. The method has only one control parameter which is the crossover rate. Authors have applied the method to and elements’ square arrays, and they have sought the minimum PSLL without the constraint of thinning percentage. The method is also applied to a elements’ planar array with a different cost function which is the sum of the maximum SLLs in and planes. In [23], DE algorithm and a binary-coded genetic algorithm are jointly applied to the minimization of the SLL in a square planar array. It is shown that the SLL of the planar array can be decreased without a large change in directivity. An FFT-based synthesis method for generating dual radiation pattern from a rectangular planar array is presented in [24]. Array elements phases are modified while amplitudes are commonly shared. The FFT method is incorporated with self-adaptive DE algorithm. Different beam pairs having different PSLLs are investigated. The results obtained from self-adaptive DE, DE, and particle swarm optimization are compared with each other. In [25], fractal-based planar antenna array optimization using DE is considered. Sierpinski carpet fractal array concept is utilized and PSLL reduction and element number reduction by finding optimum element positions and amplitudes are aimed at.

In [26], a synthesis method based on DE algorithm is presented for conformal antenna arrays in the presence of a platform. Element phases are determined by the scan angle and array geometry, and element amplitudes are optimized via DE so as to reduce SLLs to desired levels. In [27], DE is applied to the optimization of the geometric parameters of Yagi-Uda antennas. A hybrid DE algorithm is proposed which incorporates simplified quadratic interpolation in [28]. The method is applied to Yagi-Uda antennas and wideband patch antenna design.

The modified teaching learning based optimization algorithm proposed in [11] is applied to square array and elements’ planar array.

Difference sets [29–32] and hybrid methods [33–35] are also introduced for the optimization of planar arrays. In [32], the sidelobe control problem is directly tried to be solved by applying the properties of difference sets. In [29–31], Hadamard difference sets are used to obtain thinned arrays giving minimum SLLs for planar square antenna arrays.

A hybrid approach which combines genetic algorithm and cyclic difference sets is proposed in [34]. Hadamard difference sets based method given in [29] is combined with a binary particle swarm optimization in [33]. The proposed hybrid approach is applied to , , and square arrays. The authors have expressed the cost function as a weighted sum of the PSLL and thinning percentage. In [35], McFarland difference sets and a binary genetic algorithm are used for the synthesis of thinned rectangular arrays.

In [36] thinned array configurations giving lower PSLLs than given in [29] were obtained for square arrays of and elements.

In the current work thinned array configurations giving lower PSLLs than given in [29–31] are obtained for square arrays of , , , and elements. For the given square arrays, under the constraints of given number of array elements, the optimum thinned array configurations are tried to be found. The obtained PSLLs are also found to be lower than given in [36] for square arrays of and elements. Differential evolution algorithm is used in the determination of the antenna locations.

#### 2. The Geometry of the Problem

The planar antenna array under investigation is shown in Figure 1. The array factor for this planar antenna array can be written asHere, is the amplitude of the th array element current and it takes the values of 1 or 0. A value of 1 for means that the antenna element is turned on or active, and a value of 0 for means that this antenna element is turned off or passive. is the total number of antenna elements, and and denote the and coordinates of the th element in wavelengths, respectively. The distance between grid points is assumed to be , where is the wavelength. The array factor given in (1) can be also expressed as a two-dimensional sum, but one- dimensional indexing is found to be more efficient from the programming point of view.