School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China
Copyright © 2016 Chuang Han and Ling 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.
A Feedback Particle Swarm Optimization (FPSO) with a family of fitness functions is proposed to minimize sidelobe level (SLL) and control null. In order to search in a large initial space and converge fast in local space to a refined solution, a FPSO with nonlinear inertia weight algorithm is developed, which is determined by a subtriplicate function with feedback taken from the fitness of the best previous position. The optimized objectives in the fitness function can obtain an accurate null level independently. The directly constrained SLL range reveals the capability to reduce SLL. Considering both element positions and complex weight coefficients, a low-level SLL, accurate null at specific directions, and constrained main beam are achieved. Numerical examples using a uniform linear array of isotropic elements are simulated, which demonstrate the effectiveness of the proposed array pattern synthesis approach.
There are many synthesis methods for array pattern of minimum sidelobe level (SLL) and null control [1–3]. These proposed methods attempt to find the best solution for sensors position distribution or complex weight coefficients of linear array. Compared with equally spaced arrays, unequally spaced arrays with optimally spaced sensors have advantages including their capability of achieving higher spatial resolutions or lower sidelobe, or we can use fewer sensors to meet similar pattern specifications by carefully designing the locations of array sensors [4–6]. By changing the amplitudes and phases of the array elements’ complex weights without any physical changes in the array, the method becomes suitable for adaptive processing applications in which the array pattern is dynamically adapted to the environments [7–10].
Although many studies have been published on array pattern synthesis, further research is still needed for the problems described below.
Firstly, only sensor positions are considered for minimum SLL and null control in [1, 5]. In , one example discussed the geometry of a 28-element array designed for SLL suppression in the region and prescribed nulls at three groups of symmetric direction. It shows that a broad null −50 dB deep is easily achieved by Particle Swarm Optimization (PSO). For the same condition, PM4HPSO-TVAC achieves a performance for synthesizing unequally spaced linear array with low SLL and null control. However, the constrained SLL is not an accurate null level but lower than one default value by considering sensor positions.
Secondly, many techniques proposed in the literature adjust the amplitude and phase of the array element to achieve a low SLL, if desired, nulls at specific directions with constrained main beam [7–10]. Phase only control widely used in phased arrays to provide beam scanning is inexpensive to produce and, also, is more likely to minimize excitation errors and preserve coherence . Only excitation amplitudes of each element are taken as optimization parameters using Taylor distribution and PSO in . The single optimized parameter may limit the effectiveness of SLL. In , numerical examples are used to demonstrate the effectiveness of achieving an accurate null level by changing the complex weights. Three nulls of −80 dB with a minimum average pattern value of 0.28 are achieved for an equally spaced linear array of 10 elements. However, the null level has a certain deviation with decreasing the number of elements even though the fitness has a large weight and the SLL has no obvious advantages by minimizing the pattern average level. It is likely to achieve the truly accurate null level and minimum SLL when both the weights and the positions are considered simultaneously with suitable fitness.
Thirdly, a simulated annealing (SA) based method was proposed in  to design an asymmetric array by optimizing both the sensor positions over a grid space and array complex weight coefficients. It does not simultaneously optimize all the parameters but perturbs the weight coefficient and position of each sensor in turn. It is possible that sparse arrays with continuously spaced sensors could have a high degree of freedom in lowering the SLL . In , an evolutionary method based on backtracking search optimization algorithm (BSA) is proposed for linear antenna array pattern synthesis with prescribed nulls at interference directions. Pattern nulling is obtained by controlling the amplitude, position, and phase of the antenna array elements, but the null depth is not constrained with a given value. An accurate null level with minimum SLL will be obtained by both considering position and complex weights in this paper.
The challenge of determining optimum parameter values simultaneously stems from the nonlinear and nonconvex dependency of the array factor to the weights and the sensor positions . In , the authors revealed that the synthesis of nonuniform array elements’ positions, excitations, and phases is a complicated nonlinear problem which contains a number of decision variables. The performance of the employed optimization scheme is an important factor in the success of a pattern synthesis method, in terms of solution quality, computational load, and stability. PSO has received a lot of attention due to its simplicity of implementation and its capability to escape from the traps of local optima [5, 14].
A lot of research is being carried out to address the main limitation of PSO, which is its tendency to converge prematurely at local optima [15, 16]. Modifications in the velocity calculation of PSO algorithm are proposed in . The authors proposed that personal best influence and initial velocity values have an important influence on the search procedure. In , the authors revealed that one of the aspects of PSO’s capability to find the global optima mainly depends upon its capability to explore the search space. The initial higher value of inertia weight applied to the last velocity improves the exploration of search space and its lower value towards the end of search helps to attain faster convergence .
Without loss of generality, the elements of the antenna array in this paper are isotropic radiators. However, the method can be generalized for different types of antennas, such as microstrip antenna and horn antenna. In , the authors presented a novel patch antenna design with high directivity in the broadside direction by using genetic algorithms (GA). This type of antenna has more advantage for the optimization of array pattern synthesis. Implementing the proposed method to other types of antennas is investigating by the authors. Progress will be shown in a separate paper.
In this paper, both element positions and complex weight coefficients are optimized for minimum SLL and accurate null level at specific directions with constrained main beam. Numerical examples are used to demonstrate the effectiveness of the proposed array pattern synthesis approach and the novelty of this paper will be described as follows:(i)The optimized objectives in the fitness function can obtain an accurate null level. The directly constrained SLL range reveals the capability to reduce SLL. In , the null level has no an accurate level, even though there is a big weighting factor for null control in the fitness. The objective of minimizing the pattern average value has influence on the null control and SLL minimization.(ii)Considering both element positions and complex weight coefficients, a low-level SLL, accurate null at specific directions, and constrained main beam are achieved.(iii)Feedback Particle Swarm Optimization (FPSO) is proposed for a large initial search space and fast convergence in local space with refined solution.
The remaining part of this paper is organized as follows. In Section 2, the array synthesis formulations for unequally linear array and PSO with initial conditions are briefly discussed. The FPSO method and different fitness functions are proposed and discussed in Section 3. Section 4 describes numerical examples and shows the comparative performance of the presented technique. Concluding remarks are given in Section 5.
2. Method on Array Synthesis Using PSO
Consider a uniformly planar array of isotropic elements with a spacing of as shown in Figure 1. The position of each sensor along the -axis can be written as . We just consider the array factor, which can be expressed as whereand (measured from -axis) is the pitch angle of radiation for the transmit array and the incidence of the plane wave for the receive array. is the wavelength and is the weight coefficient of the th sensor. Since is complex, it can be expressed as where and are the amplitude and phase of , respectively. Consequently, the array factor can be expressed as
Figure 1: Geometry of the -elements’ unequally linear array placed along the -axis.
Particle Swarm Optimization (PSO), also known as swarm intelligence, is a robust stochastic evolutionary computation technique based on the movement of intelligent swarms. The position and the velocity relationship after the th iteration between any two individuals are obtained by the following updating formula:where is a parameter called the inertia weight and the acceleration factors and are positive constants that control the relative impact of the personal (local) and common (global) knowledge on the movement of each particle with values ranging from 1.5 to 2.05. The terms and are independent, uniformly distributed random variables in the range of 0 to 1, and is the best previous position of while is the best overall position achieved by a particle within the entire population.
The initial swarm is formed by using a uniform array with an intersensor spacing of for each sensor and the complex weight coefficients of steering vectors where . Besides the design of the array pattern with element positions and complex weight coefficients discussed in , an extensive study has been made on the performance of the PSO under different conditions in the following sections, where several configurations and weights constraints were considered. The performance of these designs will be compared with those reported in  in terms of the null level, the SLL, and the convergence speed. Without loss of generality, we set the parameters of (3) to
To have the same range of complex weights used in [4, 7], we set the amplitude between 0 and 2 and the phase between and for every sensor. Here, we adopt the constraints of intersensor spacing between and which has a smaller search range than that of .
3. FPSO and Different Fitness
In , it was noted that the main limitation of PSO is its tendency to converge prematurely at local optima, and in , the iteration numbers to converge or the number of the fitness function evaluations was identified as an investigative topic.
In this paper, the solution includes amplitudes and phases constrained independently or with element positions simultaneously. A case study reveals that the choice of mutation probability and mutation step size has a strong influence on the convergence behavior of the swarm. For more mutation probability, there should be a large random velocity or large weighting coefficient for the velocity update before entering the global optimization range. To enhance the search of the precise solution that affects the index of pattern, a small value of velocity and the weighting coefficient are important for the absolute best value.
The concept of linearly decreasing inertia weight applied to Particle Swarm Optimization (LPSO) and the method of parameter strategy for PSO were proposed using the overshoot and the peak time of a transition in . In , dynamic inertia weight PSO (DIW-PSO) was proposed where the inertia weight for every particle is dynamically updated based on the feedback taken from the fitness of the best previous position found by the particle.
For a large initial search space and fast convergence in local space with refined solution, we propose a nonlinear inertia weight decided by subtriplicate function with feedback taken from the fitness of the best previous position. The feedback function with the inertia weight can be expressed as
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