Synthesis of Sparse Arrays Based On CIGA (Convex Improved Genetic Algorithm)

In this paper, a novel hybrid algorithm on beam pattern synthesis of sparse arrays is proposed,which aims at minimizing the peak sidelobe level (PSLL). Sparse arrays can provide higher spatial resolution and relatively lower sidelobe than general arrays, but it is necessary to solve the multi-constraint problem of nonconvex nonlinear. Thus, we propose a Convex Improved Genetic Algorithm (CIGA) that can adjust the positions and the excitation coefficients of arrays to achieve the minimum PSLL. The CIGA is an effective two-step approach to the synthesis of sparse array. Firstly, Improved Genetic Algorithm is proposed, which is suitable for beam pattern synthesis of sparse arrays. The Improved Genetic Algorithm is adopted to adjust the positions of arrays to achieve the local optimum PSLL, and then convex optimization method is used to calculate the excitation coefficients in expectation of reaching the minimum PSLL. Simulation results show that the PSLL obtained by CIGA is about 5dB better than the published methods in sparse linear arrays and prove that the CIGA is superior to the published methods.


I. INTRODUCTION
Nowadays, antenna arrays can be widely applied to a system including radar, radio astronomy and satellite communication system. Synthesis of antenna arrays has been researched for several decades.
Different from synthesis of single-antenna, synthesis of antenna arrays can be considered as a multiconstraint optimization problem, which requires more advanced optimization methods [1]. In many practical applications, antenna arrays often require low sidelobe. For example, the PSLL of uniform Substrate Integrated Waveguide slot antenna arrays is close to 13.60 -dB, but in common radar system, the lower PSLL is often required [2]. Therefore, it is necessary to research an effective method to depress the PSLL of antenna arrays.
Analyzing the expression of the array factor, we find that the element positions and amplitude and phase of the excitation coefficients are the main factors affecting the radiation pattern of antenna arrays. Generally, there are two categories of array layouts for unequally spaced arrays: Thinned Arrays, where selecting the appropriate array elements from the equally spaced arrays [3]; sparse arrays, where the array elements can be arranged arbitrarily within the array aperture [4]. It is clear that the former reduce the degree of freedom of the optimization process, and cannot get the minimum Synthesis of Sparse Arrays Based On CIGA (Convex Improved Genetic Algorithm) PSLL, so the former is limited in scope of application. The latter can make the PSLL as low as possible, but algorithm is relatively complex [5].
In the past few years, many synthesis methods of antenna arrays have been developed. The methods of antenna arrays synthesis based on stochastic optimization algorithm are proved effectively, such as genetic algorithm (GA) [6,7], differential evolution (DE) [8], particle swarm optimization (PSO) [9], invasive weed optimization [10] and cuckoo search algorithm [11], dynamic parameters differential evolution algorithm (DPDE) [12], improved chicken swarm optimization [13]. Such methods are effective for optimizing amplitude and phase of the excitation coefficients of the equally spaced arrays or optimizing the element positions with the uniform excitation coefficients, but it cannot achieve the truly minimum PSLL when both the excitation coefficients and the element positions are considered. An effective method has been proposed in [14], which can consider both the excitation coefficients and the element positions simultaneously, but only the intelligent optimization algorithm is used, which leads to the result without good stability. Some other synthesis techniques are much more efficient than the stochastic optimization algorithms, such as compressive sensing based synthesis approaches in [15] and [16], bayesian compressive sensing based synthesis approaches in [17] and [18], the matrix pencil methods in [19] and [20], two step approach (TSA) [21]. However, these methods reduce the degree of freedom of antenna arrays, leading to a local optimal solution eventually. Recently, a better arrays synthesis method can be achieved while improving the computational efficiency, utilizing a global optimization algorithm that applies the density tapper technology [22].
In this paper, inspired by the idea that presented in [22], the novel convex improved genetic algorithm (CIGA) is proposed for the non-uniform excitation coefficients arrays that the array elements can be arranged arbitrarily within the array aperture. Firstly, the uniform excitation coefficients are set under the condition of unchanged aperture of the array to obtain the optimal array layout. Then, this problem is transformed into a convex optimization problem that can be processed by Interior Point Method, Gradient Descent Method, or the like. Finally, the minimum PSLL can be obtained and experimental results verify the effectiveness and outperformance of the proposed method.

A. Array factor
In the following discussion, we will consider the case of planar arrays because linear arrays can be regarded as a special planar array. For example, there is an array of N radiation elements and the array factor AF of planar arrays can be mathematically formulated as follows [23]: where ( , ) n n x y is the coordinate of the nth array element, n w is a complex variable that represents the excitation coefficient of the nth array element. = 2 b p l is the free space wavenumber, l is the wavelength.  The radiation pattern of planar arrays can be expressed as [24]: where 1 W and 2 W represent the main lobe region and sidelobe region, respectively.

B. Formulation description, NP-Hardness and convex optimization 1) Problem description
The purpose of this paper is to find an optimal set of the positions and amplitude and phase of the excitation coefficients, so that the PSLL is as low as possible. Thus, this optimization problem can be described as: where ( , ) q f represents the visible area,

2) NP-Hardness and convex optimization
To prove that this optimization problem is a NP-hard, we first simplify the continuous optimization problem to the discrete optimization problem. Take a 10-element linear arrays as an example and set

III. PROBLEM SOLUTION
The proposed algorithm is divided into two steps, the details will be described in this section.
Step A. optimal positions Under the condition of the uniform excitation coefficients, the problem is transformed into a single objective optimization problem for finding the optimal positions. Since the problem of finding the optimal positions is non-linear and non-convex, an intelligent optimization algorithm is usually used. In this paper, the optimal array element positions are solved by the IGA. The optimization model can be expressed as: The process of the IGA is as follows: 2) Selection: The traditional roulette method is adopted, and the possibility of the children reservation are determined by size of the individual fitness. The fitness of the ith is i f , the probability of selection SP can be expressed as: 1 1, 2, , 3) Crossover: Firstly, the population is divided into a set of odd number of individuals and a set of even number of individuals. The integers in [1, N] are randomly selected as the intersection points, and then the crossover is carried out according to the crossover probability, and the respective partial genes are exchanged at the intersection point of the paired individuals and forming a pair of new individuals. To achieve a better optimizing performance than the traditional GA, we introduce adaptive crossover probability c P : where ave f and max f represent the average fitness and the maximum fitness, respectively. In this way, c P can change dynamically during optimization process. In order to preserve the characteristics of genetic algorithm and prevent it from becoming the general stochastic search algorithm, the constants 1 k and 1 x are introduced. According to experience, 1 k is 1.9 and 1 x is 0.5.

4) Mutation:
The mutation probability m P of an individual is a parameter as important as the crossover probability c P . A smaller fitness leads to a higher mutation probability. m P is computed by max 2 2 arcsin( ) 1 2 where the constants 2 k and 2 x are introduced to prevent the maximum mutation probability m P from exceeding 0.1, according to experience, we set 2 0.18 k = and 2 =0.4 x .
Step B. optimal excitation coefficients After the optimal positions of the array elements are obtained, the problem can be transformed into a convex optimization problem, and the optimal excitation coefficients are obtained by using the traditional convex optimization methods (Interior Point Method, Newton-Decline Method and so on).
The optimization model can be expressed as:

A. Linear array antennas scenario
We consider a N-element linear sparse arrays, shown in Fig. 2. Fig. 2. N-element linear sparse arrays.
To explain the effectiveness of the algorithm proposed in the step A, we compare the performance of the IGA with that of GA. We consider a linear sparse arrays with =98.5 L l and 152 N = . Table Ⅰ summarizes the comparative results between GA and the IGA, after 50 independent runs, the worstcase PSLL, the best-case PSLL and the average PSLL obtained by the IGA are better than that of GA.
In addition, the CPU running time obtained by the IGA are shorter than that of GA. Fig. 3 shows the convergence during the optimization process obtained by GA and the IGA.
It can be seen that the accuracy, the convergence and the CPU running time can be effectively improved by using IGA. Note that the IGA is the first step of CIGA, the IGA is used to achieve the optimum array geometry that must meet the minimum spacing constraint. Once the optimum array geometry has been defined, we could calculate their excitation coefficients by means of convex programming, such as the CVX and SeDumi.  Fig. 3. Convergence during the optimization process obtained by GA and IGA.
In order to illustrate the superiority of the proposed algorithm in this paper, the CIGA is compared with Dynamic parameters differential evolution algorithm (DPDE) [12], Two Step Approach (TSA) [21] and Improved Genetic Algorithm [14]. The minimum elements spacing c d is 0.5l , which is the same as that of the reference algorithms. The element coordinate vector represents the minimum elements spacing vector and v represents the positions information of the array elements that need to be optimized.
Example A: In order to make comparison with the literature, the CIGA is set the same array parameters as provided in [12], =98.5 L l and 152 N = . Fig. 4 (a) shows the normalized radiation pattern of the best solution obtained by the CIGA. Fig. 4 (b) depicts the PSLL after that the proposed method runs 50 times. The best PSLL is 29.02 -dB and the mean PSLL is 28.55 -dB, as Table Ⅱ shows, which is about 5dB better than the PSLL obtained by these algorithms mentioned in [12].   Table Ⅳ, Table Ⅴ. The CIGA has achieved a best PSLL of 23.14 -dB for the 17-element arrays and 24.76 -dB for the 37-element arrays, as Table Ⅲ depicts, which are better than those algorithms mentioned by [21]. The pattern of 17-element arrays and 37-element arrays are plotted in Fig. 5 (a) and Fig. 5 (b).   Example C: In this example, we consider a linear sparse arrays with 25 L l = and =37 N and the selected parameters are the same as those in [14]. Fig. 6 (a) and Fig. 6 (b) display the optimal positions and amplitude of the normalized excitation coefficients. As exhibited in Fig. 7, the best PSLL of the CIGA is 23.68 -dB, which is about 1.2 dB better than the best result of the method in [14].

B. Planar array antennas scenario
We consider a sparse planar array with 100 radiation elements in the range of 10 10 l ĺ .The optimal array layout obtained by the CIGA is shown in Fig. 8. Fig. 9 (a) clearly shows the radiation pattern of the planar antenna arrays obtained by the CIGA, for a more intuitive view of the PSLL of the array pattern, power mask is drawn at the PSLL. It is clearly shows that the CIGA has achieved the best PSLL of 13.30 -dB, which is better than that of improved chicken swarm optimization [13]. Fig. 9 (b) shows the top view of radiation pattern of the planar antenna arrays. In some special applications, more antenna elements are often required to obtain the desired the radiation pattern. Thus, planar antenna arrays with 400 radiation elements are applied to validate the performance of the proposed CIGA for the large-scale arrays. In Fig. 10 (a), the results show that the PSLL is 14.65 -dB, which is lower than that of the improved chicken swarm optimization proposed in [13]. Fig. 10 (b) shows the top view of radiation pattern of the planar antenna arrays. Obviously, the energy is more concentrated than that of the 100 elements planar antenna arrays. V. CONCLUSION To solve the nonconvex nonlinear problem in the synthesis of sparse arrays, a novel hybrid algorithm is proposed in this paper. The proposed algorithm has the advantages of the IGA and the convex optimization method. The PSLL obtained by the CIGA can get 20% reduction of that of the swarm intelligent optimization algorithms [12]- [14], [21] in sparse linear arrays. In sparse planar arrays, the PSLL obtained by the CIGA can get 7% reduction of these methods mentioned in [13]. In addition, it can be seen from the results of multiple independent runs that the proposed algorithm has good stability. Whether a linear arrays with a small number of elements or a large-scale planar arrays, the proposed CIGA provides better performance than the other published methods.