International Journal of Antennas and Propagation

Volume 2018, Article ID 6309192, 7 pages

https://doi.org/10.1155/2018/6309192

## A Hybrid Method Based on the Iterative Fourier Transform and the Differential Evolution for Pattern Synthesis of Sparse Linear Arrays

School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723000, China

Correspondence should be addressed to Xin-Kuan Wang; nc.ude.tuns@naukxw

Received 18 April 2018; Accepted 12 June 2018; Published 5 July 2018

Academic Editor: Francesco D'Agostino

Copyright © 2018 Xin-Kuan Wang and Gui-Bao Wang. 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

A hybrid method that combines the iterative Fourier transform (IFT) and the algorithm of differential evolution (DE) is proposed to address the synthesis of low sidelobe sparse linear array (SLA) including many elements. Firstly, a thinned linear array (TLA) with the lattices spaced at half wavelength is obtained by the IFT. Then, for the elements of the TLA whose left or right spacing are greater than half wavelength, their placements are selected as the candidates which will be further optimized by the DE, as long as the interelement spacing is not less than half wavelength. Consequently, the convergence pressure of DE is greatly alleviated for the reason that the selected elements only accounts for a small part of the total. Therefore, the SLA with improved sidelobe performance can be obtained at relative low hardware cost. Several numerical instances confirmed the effectiveness of the proposed method.

#### 1. Introduction

In some antenna applications, we focus the antenna array with narrow beamwidth to increase the resolution while allow sacrificing some antenna gain, so a certain percentage of the elements can be removed from a periodic, fully populated array. This is commonly known as thinned array (TA). For the advantages in reducing weight, power, and cost, TA had been used in satellite communications, ground-based radars, radio astronomy, and so on [1]. Over the past few decades, a number of tools, such as the genetic algorithm (GA) [2–4], the ant colony optimization (ACO) [5], the particle swarm optimization (PSO) [6], the biogeography-based optimization (BBO) [7], IFT [8–10], and the almost difference set- (ADS-) based approach [11], had been employed for TA synthesizing. In particular, the IFT [9] and ADS [11] had been confirmed very effective for thinning large planar arrays.

Another kind of structure analogous to TA, which has the elements arbitrarily located within the antenna aperture and thereby provides more degree of freedom to achieve sidelobe suppression, is called sparse array (SA). Unfortunately, the synthesis for SA refers to a question of multiple constraints including the minimum element spacing, the antenna aperture, and the number of elements. Therefore, the use of stochastic optimization algorithms such as the simulated annealing [12], PSO [13], DE [14, 15], and GA [16] is not a computationally efficient way for the synthesis of the SA including many elements. During the past few years, the deterministic approaches, such as the IFT [8–10], the methodology by means of density taper [17], had been proposed. Most of this kind of methods require a priori estimate for the array performance as well as a predefined number of elements within a given aperture size; therefore, they cannot guarantee the array fitting desired power patterns with maximally reduced number of elements. Recently, a noniterative method for linear array based on the matrix pencil method (MPM) [18] is proposed and confirmed to be effective and computationally efficient in synthesizing the massively sparse arrays. More notably, the Bayesian compressive sensing (BCS) had been employed [19] to address the same issue. The reports in [19, 20] shown that the BCS-based approach allows the design of linear arrangement fitting desired power patterns with at least 35% elements reduced comparing to the regular array configuration. Furthermore, in order to synthesize the asymmetrical shaped beams, an advanced term of the BCS is applied in [21], which could deal with the SLA matching arbitrary reference patterns.

In this manuscript, the synthesis of uniform-amplitude sparse linear array with many elements is considered. Naturally, the use of the stochastic method, typically as DE, always involves a large set of parameters and thereby requires high computational resources. To ease the impact, we propose a hybrid method that combines the IFT and DE (IFT-DE) to address the issue. The method is based on DE with the individual parameters constituted by the element locations. Differently, unlike the conventional DE in which all the individual parameters are randomly initialized, for the assumption that the element spacing is not less than ( denotes the wavelength), most element locations of this method are predetermined by a uniform excited TLA with the lattices spaced at that obtained by the IFT, the first step of the IFT-DE. Consequently, just a limited number of individual parameters that remain for DE are to be optimized. This will effectively narrow the solution space and thereby efficiently accelerate the convergence speed of DE.

The manuscript is mainly organized as follows. Section 2 describes the formulation of the proposed method. Section 3 provides a set of numerical examples to validate the effectiveness of the IFT-DE. Finally, the conclusion of this work is summarized in Section 4.

#### 2. Formulation of the Proposed Method

Consider a linear array with isotropic elements. The array factor can be expressed as follows [8]: where represents the th element excitation, and all the excitations are labeled as the collection . is the wavelength, and is the element spacing. , where denotes the angle measured from the broadside of the array.

The IFT in the proposed method is the same as that used in [10], which is a slight modification form to that presented in [8]. Therefore, the adaptively changed filling factor can be written as where refers to the step size when changes, is the iteration number with representing its maxima. When reaches its target value , can be written as

Accordingly, a single trial of the IFT-DE can be divided into three steps, as are in turn described below.

*Step 1. *Perform the IFT to get a thinned linear array [8–10].
(1)Randomly initialize the element excitations over the lattices with probability of 0.9 equal to one and the rest equal to zero.(2)Compute the array factor from through points IFFT with .(3)Force the values in the sidelobe region of to match the predefined value that is obtained by trial and error.(4)Compute from through points FFT.(5)Truncate the samples of to samples that coincide with the total number of lattices.(6)For , set the samples which have higher amplitudes to be equal to 1, while others equal to 0.(7)Decrease with step size of so that in each iteration cycle, just one element is removed for asymmetrical array while in the opposite, two elements located in symmetrical positions are removed.(8)Repeat steps (2)–(7) until arrives at , then the iteration is terminated and a thinned array with filling factor of is obtained.

*Step 2. *Select the element locations that need to be optimized.

For the above thinned array, select the elements whose left or right spacing are greater than , and label their initial locations as the set , where refers to the maximum number. Figure 1 illustrates a thinned array including six elements with the lattices spaced at . The symbols of solid circle indicate the activated elements, while the symbols “×” describe the other way (Figure 1). It can be seen that the elements numbered in 3rd, 4th, 5th, and 6th, whose initial locations are in turn recorded by , , , and , are deemed to meet the aforementioned selection rule.

*Step 3. *The optimization for the selected element locations using DE.
(1)Determine the lower and upper bounds for locations of the selected elements.Assume that a total of element locations need to be optimized with the initial value, and the lower and upper bounds for are, respectively, indicated by and . For the constraint that the minimum element spacing is not less than , the values of and are set to be determined as follows:
It depicts that the maximum range of the adjacent elements moving in the opposite direction is set equal, and the minimum distance after the movement is . To illustrate the above restriction, still in Figure 1 as an example, the location ranges (described by the double arrows) for the elements numbered in 3rd, 4th, 5th, and 6th are given by the following set of inequalities
(2)Perform DE.A conventional DE similar to that presented in [22] is used to find the SLA with minimum peak sidelobe level (PSLL), where the mutation strategy of DE/rand/1, as well as the operator of binomial crossover, is adopted. Once the overall steps are finished, the best fitness value so far is retained, and the method will proceed to the next trial until it reaches the maxima.