Abstract
A backtracking search optimization algorithm (BSA) is proposed for the synthesis of concentric circular antenna arrays (CCAAs) with the low sidelobe levels at a fixed beamwidth. Several numerical examples of CCAA patterns with the single, multiple, and broad nulls imposed at the directions of interference are also given to illustrate the performance and flexibility of the proposed algorithm. BSA is a relatively new population based evolutionary optimization algorithm. The numerical results show that the design of CCAA using BSA provides good sidelobe levels with a fixed beamwidth. The nulling patterns obtained by BSA are also very good. The quality of results obtained by the BSA is checked by comparing with that of several metaheuristic algorithms in the literature. BSA is also compared with differential search (DS) and bacterial foraging algorithms (BFA) in terms of iterative performances.
1. Introduction
It is wellknown that the planar arrays have better control over the radiation pattern than the linear arrays. Besides other planar arrays such as circular and rectangular arrays, concentric circular antenna array (CCAA) has also become favourable in the radar and wireless communication systems. CCAA is capable of scanning electronically its main beam in the plane of the array without significant degradation of the array pattern. CCAA also has a good flexibility in terms of array pattern synthesis both in narrowband and in broadband applications [1–18].
For the antenna array synthesis problems, traditional optimization techniques involve particular disadvantages, owing to the nonlinearity. Starting points of optimization process become important to obtain good solutions if the dimension of the solution space is increased. They have also tendency to convergence to local optimum solutions. Additionally, gradient search methods also need continuous and differentiable cost functions [1]. For this reason, in recent years, the metaheuristic techniques [1–3, 19–34] based on the genetic algorithm, ant colony optimization, particle swarm optimization (PSO), differential evolution, tabu search, clonal selection, bees algorithm, harmony search, plant growth simulation, seeker optimization, evolutionary programming (EP), and bacteria foraging optimization (BFO) have become more popular, and they have been used in solving antenna array pattern synthesis problems. In this paper, an alternative metaheuristic technique based on a backtracking search optimization algorithm (BSA) [35] is employed for the synthesis of CCAA with the low sidelobe levels at a fixed beamwidth. Furthermore, several numerical examples of CCAA patterns with the single, multiple, and broad nulls imposed at the directions of interference are also presented. Additionally, in order to examine the computational performance of BSA, the convergence curve of BSA is compared with those of differential search (DS) [36] and bacterial foraging algorithm (BFA) [37].
The main motivation for the development of BSA is to achieve a simpler and more effective search algorithm [35]. Unlike many optimization algorithms, BSA has a single control parameter. Moreover, the problem solving performance of BSA is not oversensitive to the initial value of this parameter. BSA has a simple structure that is effective, fast, and capable of solving multimodal problems and that enables it to easily adapt to different numerical optimization problems. In [35], BSA was statistically compared with PSO, artificial bee colony algorithm (ABC), covariance matrix adaptation evolution strategy (CMAES), adaptive differential evolution algorithm (JDE), comprehensive learning particle swarm optimizer (CLPSO), and selfadaptive differential evolution algorithm (SADE). 75 boundaryconstrained and three realworld benchmark problems were employed for the comparisons. In general, the simulations and comparisons in [35] showed that BSA can solve the benchmark problems more successfully than the compared algorithms. BSA was also used in [38] for circular antenna array design.
The rest of the paper is organized as follows. In Section 2, the general problem formulations for the synthesis of CCAA are given. A detailed explanation for the BSA is presented in Section 3. Numerical results are given in Section 4. Finally the conclusion of the paper is presented in Section 5.
2. Problem Formulation
Let us assume that a number of isotropic elements are placed in multiple concentric circular rings which differ in radius as illustrated in Figure 1. There are concentric circular rings in  plane and the th ring has a radius and the number of elements is , where . If the array elements are equally spaced within the ring, the azimuth angle from the axis for th element is given by Thus, the array factor of the concentric circular antenna array can be formulated as follows: where is the excitation amplitude of the th element in the th ring and is the excitation amplitude of the central element placed at the origin. In this paper, is fixed to 0 for all optimization examples in order to direct the peak of the main beam to the direction.
In this paper, we consider designing antenna arrays with the low maximum sidelobe level (MSL) at a fixed first null beamwidth (FNBW). Dynamic range ratio (DRR) is also considered in this paper since it is important for some practical considerations. Additionally, null steering on the sidelobe region is also studied in this paper. Thus, the following cost function is used: where , , , and are the weight factors. , , , and are the functions used for suppressing MSL, fixing FNBW, decreasing DRR, and locating nulls, respectively. The function can be given as follows: where and are the two angles at the first nulls on each side of the main beam. can be formulated as follows: where is the array factor in dB and is the desired MSL value. The function in (3) can be given by where and are the FNBW value obtained by BSA and the desired maximum FNBW, respectively. The function can be given by where and are the maximum and minimum amplitude values of the antenna array elements obtained by the BSA, respectively. is the desired maximum DRR value. In a similar manner, is given as follows: and are the obtained and desired maximum null depth level (NDL) values, respectively. In this way, the cost function can manage the MSL, FNBW, DRR, and NDL values.
3. Backtracking Search Optimization Algorithm (BSA)
BSA [35] is one of the new population based evolutionary algorithms. It is based on an iterative process which tries to minimize the objective function. BSA consists of five evolutionary mechanisms: initialization, selectionI, mutation, crossover, and selectionII. The general flowchart of BSA is illustrated in Figure 2.
3.1. Initialization
BSA initially scatters the population members in the solution space by using a uniform random distribution function: where and are the population size and the problem dimension, respectively. is the uniform distribution function and is the position of the th population member in the solution space. and in (9) are lower and upper bounds of solution space, respectively.
3.2. SelectionI
In the SelectionI stage, BSA generates the historical population utilized to determine the search direction. The following equation is used to initialize the historical population: The following equation gives a chance to BSA to redesign at the beginning of each iteration: where and are the uniform real numbers in the range of in order to decide if the is selected from previous generation. The following equation is used for shuffling the order of the population members: where the function is a random shuffling function.
3.3. Mutation
The mutant members of BSA are generated by using the following function: where is a real number to control the step size amplification of the search direction. BSA employed historical data to determine the search direction of the population members by taking into account values.
3.4. Crossover
The final form of the trialpopulation is generated by the crossover process of the BSA. The crossover process of the BSA is given in a pseudocode form in Algorithm 1. The crossover step includes two steps. The first strategy uses mixrate (Algorithm 1, lines 2–4). The second strategy allows only one randomly chosen individual to mutate in each trial (Algorithm 1, line 6). After the crossover process, the individuals may exceed the search space boundaries. In order to restrict the movement area of the individuals, the procedure given in Algorithm 2 is activated when needed. The restriction process provides a random regeneration in the valid search space for the individuals which exceed the boundaries.


3.5. SelectionII
In selectionII process of BSA, values are updated by a greedy selection strategy that uses which have better fitness than the corresponding values. Global minimum value is replaced by the best individual of when value is better than global minimum value. The detailed descriptions of BSA can be found in [35].
4. Numerical Results
In this paper, a CCAA which has three rings is synthesized by using BSA for all examples. The number of innermost, middle, and outermost rings has 4, 6, and 8 array elements, respectively. Additionally, CCAA with a central element is also considered. The main goal is to obtain array patterns having low MSL and narrow FNBW values. DRR is also taken into account for the practical considerations. In order to test the flexibility of the algorithm, single, double, triple, and wide nulls on the sidelobe region of the pattern are also located by using BSA. The distances between two elements in the innermost, middle, and outermost rings are 0.55λ, 0.606λ, and 0.75λ, respectively. The population sizes of BSA for the all examples in this section are set to 30. The maximum iteration number is selected as 5000. The special parameter of BSA mixrate is fixed to 1. Step size amplification is determined as , where . The optimization processes are carried on a computer which has 2.8 GHz i7 processor and 4 GB RAM. The software used for the optimization implementations of BSA is MATLAB.
To show the performance and flexibility of the proposed BSA on the CCAA pattern synthesis, comprehensive comparisons have been made in this paper. In these comparisons, the results of BSA are compared with the results of nine different algorithms from the literature [1–3]. These algorithms are EP [1], PSO [1], PSO with constriction factor and inertia weight approach (PSOCFIWA) [1], canonical PSO (CPSO) [2], real coded genetic algorithm (RGA) [2], craziness based PSO (CRPSO) [2], hybrid evolutionary programming (HEP) [2], BFO [3], and binary coded genetic algorithm (BGA) [3]. In order to properly compare the results of BSA with the results of the other nine algorithms [1–3], the desired MSL, FNBW, and DRR values are chosen to be similar to those of nine algorithms presented in [1–3].
The CCAA considered for the first group of examples does not have a central array element. BSA is employed to achieve a pattern which has low MSL and narrow FNBW values. The DRR values of the CCAAs are also considered in the cost function in the same optimization process. The array patterns obtained by BSA used in this paper are compared with the patterns of EP [1], PSO [1], PSOCFIWA [1], CPSO [2], RGA [2], CRPSO [2], HEP [2], BFO [3], and BGA [3] in Figures 3–11, respectively. The MSL, FNBW, and DRR values obtained by BSA and the other nine algorithms are given in Table 1. It is apparent from Figures 3–11 and Table 1 that MSL, FNBW, and DRR values of the patterns produced by using BSA are better than those of EP [1], PSO [1], PSOCFIWA [1], CPSO [2], RGA [2], CRPSO [2], HEP [2], BFO [3], and BGA [3]. The amplitude values calculated by BSA for the patterns presented in Figures 3–11 are given in Table 2.
Moreover, the convergence curve of BSA is compared with those of DS [36] and BFA [37] for the first example represented by Figure 3. The maximum iteration number and the population number for all three algorithms are 700 and 50, respectively. Whereas DS algorithm does not need any extra parameter definition, the algorithm parameters of BFA are set as follows: number of chemotactic steps is 175, length of swim is 4, number of reproduction steps is 4, number of eliminationdispersal events is 1, and length of run is 0.4. Figure 12 shows the convergence curves of BSA, DS, and BFA for the first example in the first group of examples. From Figure 12, it is clearly seen that BSA can converge the better cost function value at the end of the iteration than DS and BFA.
For the second group of examples, the CCAA with central array element is considered. Figures 13–17 show the patterns obtained by BSA. For a comparison, the patterns achieved by EP [1], PSOCFIWA [1], PSO [1], BFO [3], and BGA [3] are also included in these figures. The MSL, FNBW, and DRR values obtained by BSA are compared with those of EP [1], PSOCFIWA [1], PSO [1], BFO [3], and BGA [3] in Table 3. As it can be seen from Figures 13–17 and Table 3, all MSL and FNBW values achieved by BSA are better than those of the compared algorithms in [1] and [3]. Additionally, the DRR values calculated by BSA are better than those of EP [1], PSOCFIWA [1], BFO [3], and BGA [3]. On the other hand, the DRR value obtained by BSA used in this paper is the same with that of PSO [1]. The amplitude values calculated by BSA for the patterns presented in Figures 13–17 are given in Table 4.
In the last group of examples, four synthesis examples are studied in order to examine the capability of BSA for steering single, multiple, and wide nulls on the sidelobe region of CCAA pattern. The amplitude values of array elements are controlled by BSA. A CCAA pattern with a single null imposed at 70° is considered as the first pattern nulling example. Figure 18 shows the radiation pattern obtained by BSA. From Figure 18, it can be seen that NDL value at 70° is 106.9 dB and MSL value is −33 dB. For the second and third pattern nulling examples, double (70° and 108°) and triple (53°, 70°, and 109°) nulls are placed by using BSA on the sidelobe region of the CCAA pattern. Figures 19 and 20 show the patterns having double and triple nulls, respectively. For both examples, all NDL values are deeper than −85 dB and all MSL values are less than −30 dB. In the last example, a CCAA pattern with wide null at 75° with is synthesized by using BSA. Figure 21 illustrates the pattern achieved by BSA. From Figure 21, it can be seen that NDL value is −62 dB and MSL value is −30.1 dB. The amplitude values calculated by BSA for the patterns presented in Figures 18–21 are given in Table 5.
From Figures 3–11 and 13–17, it can be concluded that BSA is capable of synthesizing CCAA patterns with low MSL and narrow FNBW values. Besides, BSA can be used to achieve low DRR values within the same optimization process. Furthermore, according to Figures 18–21, it can be said that BSA has capacity to synthesize the CCAA patterns with low MSL values and single, double, triple, and wide nulls at the desired directions.
5. Conclusion
In this paper, BSA is used for the synthesis of CCAA patterns with the low sidelobe levels at a fixed beamwidth and patterns with the single, multiple, and broad nulls imposed at the directions of interference. It can be concluded that the design of CCAA using BSA provides good sidelobe levels and null depth levels with a fixed beamwidth. The comparisons made between the results obtained by BSA and several metaheuristic algorithms show that BSA is a very competitive algorithm for the synthesis of antenna array patterns. Additionally, the convergence curve comparison shows that BSA has better convergence characteristic than DS and BFA.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.