International Journal of Antennas and Propagation

Volume 2015, Article ID 295012, 10 pages

http://dx.doi.org/10.1155/2015/295012

## Shaped Beam Pattern Synthesis of Antenna Arrays Using Composite Differential Evolution with Eigenvector-Based Crossover Operator

Radiocommunications Laboratory, Section of Applied and Environmental Physics, Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Received 6 May 2015; Revised 5 July 2015; Accepted 12 July 2015

Academic Editor: Kerim Guney

Copyright © 2015 Sotirios K. Goudos. 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

This paper addresses the problem of designing shaped beam patterns with arbitrary arrays subject to constraints. The constraints could include the sidelobe level suppression in specified angular intervals, the mainlobe halfpower beamwidth, and the predefined number of elements. In this paper, we propose a new Differential Evolution algorithm, which combines Composite DE with an eigenvector-based crossover operator (CODE-EIG). This operator utilizes eigenvectors of covariance matrix of individual solutions, which makes the crossover rotationally invariant. We apply this novel design method to shaped beam pattern synthesis for linear and conformal arrays. We compare this algorithm with other popular algorithms and DE variants. The results show CODE-EIG outperforms the other DE algorithms in terms of statistical results and convergence speed.

#### 1. Introduction

Array synthesis is a classic and challenging optimization problem, which has been extensively studied using several analytical or stochastic methods [1–4]. Common optimization goals in array synthesis are the sidelobe level suppression and the matching of the main lobe to a desired pattern. Thus, the optimization problem is usually to find a set of element excitations and/or positions that closely match a desired pattern. The desired pattern shape can vary widely depending on the application. The shaped beam pattern synthesis problem has received wide attention over the years, and several methods or techniques have been reported in the literature [5–11]. Among others these methods include Tabu search [5], ant colony optimization [11], and linear programming [7]. Additionally, evolutionary algorithms (EAs) like Genetic Algorithms (GAs), Particle Swarm Optimization (PSO), and Differential Evolution (DE) [12], which is a population-based stochastic global optimization algorithm, have been applied to a variety of array design problems [13–21]. Several DE variants or strategies exist. One of the DE advantages is that very few control parameters have to be adjusted in each algorithm run.

Composite DE (CODE) [22] is an adaptive DE variant, which combines three different trial vector generation strategies with three preset control parameter settings. The above combination is performed in a random way in order to generate trial vectors. The main advantage of CODE is that it has a simple structure and thus it is very easy to be implemented in any programming language. In this way significant computational costs spent on searching using a trial-and-error procedure can be avoided. The DE performance depends on control parameters, mutation strategies, and crossover operators. Most of the DE strategies or variants use the binomial crossover operator, which has been found to produce better results than the exponential crossover operator [23]. The authors in [24] propose an alternative crossover operator, namely, the eigenvector-based crossover. This operator utilizes the eigenvector information of the covariance matrix of the population to rotate the coordinate system. Additionally, this crossover operator can be applied to any DE variant. In this paper, we incorporate this scheme to the CODE algorithm and therefore introduce a new algorithm, the Composite DE with Eigenvector-Based Crossover (CODE-EIG).

In this paper, we apply CODE-EIG to the shaped beam pattern synthesis problem. More specifically, we utilize the CODE-EIG algorithm for designing different linear arrays cases and a conformal array case. Additionally, in order to evaluate the new algorithm, we compare the algorithm’s performance with other DE strategies, PSO, and GAs.

This paper is organized as follows: A brief description of the CODE and the CODE-EIG algorithm is given in Section 2. Section 3 describes the problem formulation and the objective functions and presents the numerical results for different array design cases. Finally, the conclusion is given in Section 4.

#### 2. The Composite Differential Evolution (CODE) Algorithm

A population in DE consists of NP vectors , , where is the generation number. The population is initialized randomly from a uniform distribution. Each -dimensional vector represents a possible solution, which is expressed as

The population is initialized as follows:where and are -dimensional vectors of the lower and upper bounds, respectively, and is a uniformly distributed random number within .

The initial population evolves in each generation with the use of three operators: mutation, crossover, and selection. Depending on the form of these operators, several DE variants or strategies exist in the literature [12, 25]. The choice of the best DE strategy depends on problem type [23]. In CODE the following three strategies are used for trial vector generation. These include DE/rand/1bin, DE/rand-to-best/2/bin, DE/rand/2/bin, and DE/current-to-rand/1 [26]. In these strategies, a mutant vector for each target vector is computed by DE/rand/1/bin DE/rand/2/bin DE/current-to-rand/1where , , , , and are randomly chosen indices from the population, which are different from index , is a mutation control parameter, and is a randomly generated number from a uniform distribution within the interval . The basic idea of CODE [22] is to generate a trial vector for each of the three above-mentioned strategies. The control parameters settings are randomly chosen. Therefore, CODE maintains a strategy candidate pool, consisting of the three strategies given in (3a), (3b), and (3c). A parameter candidate pool is also used with the settings given below:These are strategies and settings that have been widely used. Therefore, their properties have been extensively studied. The authors in [22] have modified the DE/rand/2/bin strategy by using a random number from the interval instead of the first mutation control parameter. The implementation of the CODE is quite simple. The best trial vector of the three generated is chosen. Then it is compared with the old vector and it replaces it if its fitness function value is smaller.

After mutation in each generation for every vector of the population, the binary crossover operator is applied to generate a trial vector whose coordinates are given bywhere , is a number from a uniform random distribution within the interval , a randomly chosen index from , and CR the crossover constant from the interval .

In CODE, the crossover operator is not applied for the DE/current-to-rand/1 strategy. CODE uses a greedy selection operator, which for minimization problems is defined bywhere , are the fitness values of the trial and the old vector, respectively. Therefore, the newly found trial vector replaces the old vector only when it produces a lower objective function value than the old one. Otherwise, the old vector remains in the next generation. The stopping criterion for the CODE or DE in general is usually the generation number or the number of objective function evaluations.

##### 2.1. CODE with Eigenvector-Based Crossover Operator (CODE-EIG)

The authors in [24] propose an eigenvector-based crossover operator, which utilizes eigenvectors of covariance matrix of individual solutions. Thus, the crossover is rotationally invariant. To avoid losing diversity of population, the offspring can be stochastically born from the parents with either the standard coordinate system or the rotated coordinate system. The authors in [24] also introduce a new parameter to control the probability of selecting one of the coordinate systems. They have shown that this scheme can increase the population diversity and prevent premature convergence. Additionally, another significant advantage of this operator is that it can be applied to any crossover strategy with minimal changes. Therefore, it can enhance any existing DE variant. The main idea is to exchange the information between the target vector and the mutant vector in the eigenvector basis instead of the natural basis. We consider the same population of NP -dimensional vectors defined in the previous section. The covariance between th and th dimension of the population in the th generation is given by [24]where , are the mean values of the variables in the th and th dimension, respectively.

To compute the eigenvector basis we need to factorize the covariance matrix into a canonical formwhere is the square matrix () whose th column is the eigenvector of and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues. The factorization of a matrix into a canonical form is called eigendecomposition. The authors in [24] use Jacobi’s method [27] for eigendecomposition. When the eigenvector basis is found, the th target vectors can be expressed by ; the th mutant vectors can be expressed by . Then, a predefined crossover operator, such as binomial crossover, will exchange some of the elements of the mutant vector with some of the elements of its target vector to form a trial vector. The trial vector is then given by [24]where is the conjugate transpose of the eigenvector basis and is a crossover operator on two vectors and , where is a new control parameter introduced in [24] which is called eigenvector ratio between 0 and 1 that determines the ratio of the eigenvector-based crossover operator and the other crossover operator. The main feature of this approach is that no matter how the crossover operator exchanges the elements in the eigenvector basis, the crossover behavior will become rotationally invariant in the natural basis.

Therefore, to create the CODE-EIG variant we use (3a), (3b), and (3c) to generate the mutant vector and (9) to create the trial vector. Additionally, this new variant requires the setting of the eigenvector ratio . It must be noted that when then only the eigenvector crossover operator is used, while when then only the binary crossover operator is used.

The CODE-EIG algorithm is outlined below:(1)Initialize the CODE-EIG parameters. That is, set the maximum number of generations , the population size number NP, and the eigenvector ratio .(2)Initialize a uniformly distributed random population of NP individuals. Set .(3)Evaluate objective function value for every vector of the population.(4)For each vector of the population, generate three mutant vectors , , , each with one of the three strategies given in (3a), (3b), and (3c) and with a control parameter setting randomly selected from the parameter candidate pool.(5)Apply the eigenvector-based crossover operator according to (9) and create three trial vectors , , .(6)Evaluate objective function values for the trial vectors.(7)Select the best of the trial vectors.(8)Apply the selection operator according to (6).(9)Keep the best solution so far.(10)If the maximum number of generations is reached, then exit; otherwise increment generation number and go to step 4 for the next generation.The CODE-EIG flowchart is depicted in Figure 1.