Linear Array Pattern Synthesis Using Restriction in Search Space for Evolutionary Algorithms:

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1 205 IEEE 2nd International Conference on Recent Trends in Information Systems (ReTIS) Linear Array Pattern Synthesis Using Restriction in Search Space for Evolutionary Algorithms: A Comparative Study Archit Ghosh, Tamal Das, Soumyo Chatterjee Dept. of Electronics Communication Engineering Heritage Institute of Technology Kolkata-70007, India soumyo.chatterjee@heritageit.edu Sayan Chatterjee 2 Dept. of Electronics Telecommunication Engineering Jadavpur University Kolkata , India sayan34@gmail.com Abstract In this article, a comparative study between population based optimization methods with rom restricted search space definition applied in the pattern synthesis of linear antenna arrays is presented. Synthesis problem of reduced side lobe level narrow beamwidth is considered. The design objective further considers the optimization of excitation amplitude uniform inter element spacing using rom restricted search space definition by particle swarm optimization differential evolution methods. As examples simulation of elements have been considered. Effectiveness of the restriction in search space is proved through statistical parametric analysis. Further comparison with published work has been carried out to prove the superiority of restricted search Particle Swarm Optimization. Keywords Linear Array; Pattern Synthesis; Restriction in search space; Evolutionary Algorithms. I. INTRODUCTION Present day communication systems often requires low side lobe level (SLL) first null beamwidth (FNBW) to avoid degradation of total radiated power. Objective of achieving reduced SLL narrow beamwidth simultaneously are conflicting in nature as improvement in one results in degradation of other. Conventional analytical methods like Taylor one parameter method have the ability to achieve low SLL only []. This limitation led to the application of evolutionary algorithms in solving the conflicting optimization objectives of low SLL narrow beamwdith. In last two decade, several new population based evolutionary optimization methods have been developed by different researchers to solve the conflicting optimization problem of low SLL narrow beamwidth [2-9]. Motivation behind each algorithm is achievement of desired objectives with less computation time, low stard deviation small mean cost function value. According to open literature the most popular efficient algorithms in solving the aforementioned design objectives are Particle Swarm Optimization [2-4] Differential Evolution algorithms[5-6]. In these algorithms excitation amplitudes, inter element spacing, phase have been either considered individually or in combination as optimization parameters. Performance of PSO DE primarily depends on search space definition which conventionally is rom in nature. Consequently, for efficient optimization the designer must have some idea of probable solutions hence is a limitation. Recently a new concept of restriction in search space has been proposed which overcomes the limitation of rom search space definition. In [2] synthesis of 4, 6 elements linear array with reduced SLL level of -20 db has been reported using restriction in search space of PSO algorithm with excitation amplitude as optimization parameter. The restriction method is based on solution Chebyshev polynomial for linear array developed by Dolph. Similar method of restriction using Taylor One Parameter distribution is reported in [3-4], where in low SLL narrow beamwidth are considered as optimization both individually together. In [4], excitation amplitude is considered as optimization parameter. Results reveal that only excitation amplitude perturbation with restriction is not sufficient in achieving reduced SLL narrow beamwidth. In linear array, inter element spacing variation affects the FNBW value whereas variation in excitation amplitude affects the SLL value. Hence when simultaneous controls of both are required, it is desirable to consider both excitation amplitude inter element spacing as optimization parameters. In this work, synthesis problem of low or reduced SLL narrow beamwidth in linear array is addressed through design of elements broadside linear array of isotropic radiators, using PSO DE algorithms with without restriction in search space. Excitation amplitudes uniform inter-element spacing are considered as optimization parameters for each optimization algorithm. The restriction in search space is being developed by using Taylor One Parameter distribution newly developed design expression for Dolph Chebyshev linear array. Effectiveness of restriction in search is proved through statistical parametric comparison of simulation results. Comparison with other published work available in open literature proves the superiority of restriction in search space for PSO. II. PROBLEM FORMULATION In a conventional array, radiation pattern for an array of isotropic radiators is mathematically modeled in an expression called as array factor (AF). The normalized array factor /5/$ IEEE 92

2 expression for even numbered (2N) linear array is represented in (). N AF ( θ) = a n () i= 2i cos kd cos θ 2 If the linear array is odd numbered (2N+), then corresponding normalized array factor expression is as expressed in (2) [0]. N AF ( θ) = a n cos[ ( i ) kd cosθ] (2) i= In (2), d is the uniform inter-element spacing in wavelengths (), k is the wave number expressed as k=2/, a i is the excitation amplitude of i th element is the elevation angle. In order to realize the desired objectives two error functions related to SLL reduction narrow FNBW are formulated. The first error function (EF ) is defined as: AF( θ) max20log AF( θ) EF = SLL(dB) d (3) max SLL (db) d is the target SLL value the max function represents peak SLL value obtained at each iteration cycle. AF() max is the maximum array factor for a particular value of elevation angle (). For a broadside position, AF() max occurs at a of 90 the same has been adopted in the present work. The second error function (EF 2 ) is as defined in (4). EF ( FNBW) (FNBW c 2 = d ) (4) In (4), (FNBW) d is the desired first null beamwidth (FNBW) c is the calculated FNBW at each iteration cycle. By using weighted sum method [3], a single equation is formulated by combining two error functions with a priority of weights on multi objective. Thus the cost function (CF), is as expressed in (5). CF = EF + EF 2 (5) In (5), are the weighting factor, which in turn determines the influence or priority level of each error function for achieving objectives. At this point of discussion it seems that the whole optimization process depends on the value of weighting factors, which makes the process undesirably subjective. According to [4] [8], SLL reduction narrow beamwidth problem can be hled separately by varying excitation amplitude inter-element spacing respectively. Consequently, sensitivity of low SLL narrow beamwidth contributes more towards excitation amplitude perturbation variation in inter-element spacing respectively. Objectives of SLL reduction narrow beamwidth are proposed to be realized using error functions EF EF 2 respectively in the present paper. Excitation amplitude uniform inter-element spacing are considered as optimization parameter for both the error functions is perturbed simultaneously using PSO DE algorithm for achieving the minimum non-negative value. From the above discussion it is evident that the variation in excitation amplitude primarily affects the EF value whereas variation in uniform interelement spacing primarily affects the EF 2. Hence, keeping equal to along with these set of optimization parameter each objective can be easily achieved without any tradeoff which in turn eliminates the subjective nature of weighted sum method. III. EVOLUTIONARY ALGORITHMS AND LINEAR ARRAY SYNTHESIS Evolutionary algorithms uses mechanisms inspired by biological evolution, such as reproduction, mutation, recombination, selection. Cidate solutions to the optimization problem play the role of individuals in a population, the fitness function determines the quality of the solutions. Evolution of the population then takes place through repeated iterations[2-6],[8],[9]. In this study primarily restriction in search space of PSO DE are considered for comparative study. The two algorithms are explained briefly in the following sub-sections. A. Particle Swarm Optimization Algorithm The PSO is inspired by the behavior of bird flying or fish schooling; introduced by Kennedy Eberhart in 995 as a new population based algorithm []. In stard PSO (SPSO), a swarm consists of a set of particles each particles represents a potential solution of an optimization problem. Considering the ith particle of the swarm with N particles in a D dimensional search space, its position velocity at iteration t are denoted by X i (t) = (x i (t), x 2 i (t),,x D i (t)) V i (t) = (v i (t), v 2 i (t),,v D i (t)). The velocity of each particle is updated using (6). (6) where I =,2,..N N is the population size; d =, 2..D D is the dimension of search space ; r () is a uniformly distributed rom numbers in the interval [0,]; acceleration coefficients C C 2 are non negative acceleration coefficients. Pbest i (t) = (Pbest i (t),..,pbest D i (t)), called the personal best solution representing the best solution found by ith particle itself until iteration t; (Gbest (t),.., Gbest D (t)), called the global best solution, represents the global best solution found by all particles until iteration t. Parameter w is the inertia weight to balance the global local search abilities of particles in the search space, which is given by (7). (7) In (7), w max is the initial weight, w min is the final weight, t is the current iteration number T is the maximum iteration number. The updated particle position is represented in (8) under the condition x d min x d i (t+) x d max, where x d min x d max represent lower upper bounds of the dth variable respectively. 93

3 (8) B. Differential Evolution Algorithm In 995, Price Storn proposed a new floating point encoded evolutionary algorithm for global optimization named it DE [7] owing to a special kind of differential operator, which they invoked to create new offspring from parent chromosomes instead of classical crossover or mutation. Following section, outlines the brief description of classical DE algorithm. Like any other evolutionary algorithm, DE also starts with a population of N D-dimensional search variable vectors. Considering the ith particle of the swarm with N particles in a D dimensional search space, its position at iteration t are denoted by X i (t) = (x i (t), x 2 i (t),,x D i (t)). Now in each generation (or one iteration of the algorithm) to change each population member x i (t) (say), a Donor vector v i (t) is created. To create v i (t) for each ith member, three other parameter vectors (say the r, r2, r3th vectors) are chosen in a rom fashion from the current population. Next, a scalar number F scales the difference of any two of the three vectors the scaled difference is added to the third one the donor vector v i (t) is created. Mathematically the generation process is expressed in (9). (9) The mutation operator is as expressed in (0). (0) In (0), parameter c is represented as c=log0(f min /F max )/t. Next, to increase the potential diversity of the population a crossover scheme comes to play. The donor vector exchanges its components based on cross over rate (CR) creates a trial vector as expressed in (). () The cross over rate is defined as () where T is the total number of iterations. In this way for each trial vector x i (t) an offspring vector u i (t) is created. To keep the population size constant over subsequent generations, the next step of the algorithm calls for selection to determine which one of the target vector the trial vector will survive in the next generation, i.e., at time t = t +. DE actually involves the Darwinian principle of Survival of the fittest in its selection process which may be outlined as represented in (3) [7]. (3) C. Restriction in Search Space The concept of restriction in search space is first reported in [3],[4] for PSO. In this article the concept is extended to DE wherein for each particle X i (t) represents the possible excitation amplitude uniform inter-element spacing satisfying the desired objective. The upper lower bound for excitation amplitudes is defined using Taylor One parameter distribution in following two steps. For a particular array configuration (2N or 2N+) desired SLL, excitation amplitudes are calculated. From the excitation amplitude value the maximum amplitude value is assigned to upper bound of excitation amplitude minimum value is assigned to the lower bound. In order to define the upper boundary for inter-element spacing, the design expression for broadside Dolph Chebyshev array is used [6]. It must be noted that Dolph Chebyshev array has been considered as it gives the minimum possible beamwidth for a desired SLL given linear array configuration. The lower bound for uniform inter-element spacing is set to 0.5 to avoid mutual coupling. The main purposes of restriction are summarized as: To eliminate the initial romness of the search space defining an improved initial search space with higher probability of finding optimum solution. To improve the computation efficiency of evolutionary algorithms (PSO DE). In this study, restriction has been applied to both PSO DE to prove the effectiveness of restricted search. As such the restricted version of PSO DE are referred as Restricted Search PSO (RSPSO) Restricted Search DE (RSDE). IV. EXPERIMENTAL RESULTS In order to prove the effectiveness of restriction in search space on both PSO DE, elements linear array with SLL objective of -20 db FNBW of 4 degree 0 degree are considered. A. Experimental Setup Parametric setup of an algorithm depends upon the optimization problem it is supposed to hle [2]-[9]. Accordingly, parametric setup for all the algorithms for each design problem instantiations is set is summarized in Table I. In Table I, r i represents the difference between the maximum minimum possible values of the i-th optimization parameter. Total number of iterations for each optimization algorithm is kept at 00. The same parametric setup is also used for restricted search DE (RSDE) restricted search PSO (RSPO). Dimension D depends upon the number of elements considered. Consequently for even number of elements D= ( elements/2) + for odd numbered of elements it is D= (( elements+)/2) +. The initial search for restricted search space is defined for all the cidates same initial search space is used by both RSDE RSPSO. Same initialization method has been adopted for SPSO DE wherein same rom search definition is used. While solving problems using evolutionary computing algorithms (SPSO, DE) it is a common practice to 94

4 have multiple independent runs so that optimum solutions of higher quality can be obtained [3]. TABLE I. PARAMETRIC SETUP FOR DE AND SPSO ALGORITHM SPSO DE Parameter Value Parameter Value swarm size agents c (c max =2.5 c 2min =0.5) increased from 2.5 to 0.5 CR (CR max =0.95 CR min =0.2) increased from 0.2 to 0.95 c 2 (c max =2.5 c 2min =0.5) Inertia weight w (w max =0.9 w min = 0.4) v d,max decreased from 2.5 to 0.5 decreased from 0.9 to *r i F (F max =0.6 F min =0.) exponentially decreased from 0.6 to 0. In the present work, larger number of independent runs does not affects the optimum results significantly. In addition to this, large number of independent run unnecessarily increases the overall computation time. Hence, in the present work 25 independent runs are considered. B. Statiscal Analysis In order to underst the significance of data that are subject to rom variation statistical hypothesis test is required. In statistics, a result is statistically significant if it has been predicted as unlikely to have occurred by chance alone, according to a pre determined threshold probability called as significance level [5]. Present work involves data sets subjected to rom variation hence requires a statistical hypothesis test. One such non parametric statistical hypothesis test that has been used in the present work is Wilcoxon s rank sum test. In this test, two related samples, matched samples or repeated measurements on a single sample are compared to assess whether their population mean ranks differ [5]. The result of Wilcoxon s rank sum test is represented by p-value indicating the estimated probability of rejecting the null hypothesis of a study question when that hypothesis is true. The null hypothesis in present work is a hypothesis of no difference between set of minimum cost function values for different optimization objective, at each iteration obtained using conventional DE PSO their restricted counterparts (RSDE RSPSO). Typical value of significance level for the above mentioned test is 5%(0.05). As such if p-value is less than 0.05 then the null hypothesis is rejected indicating significant difference in the chosen set of data. The results of Wilcoxons ranked sum test for two design instances of elements linear array are summarized in Table II. From Table II it is observed that for each design instance, p value is less than 0.05 which in turn represents significant difference in the best set of minimum cost function data obtained by the contestant algorithms over 25 independent trials. TABLE II. WILCOXON S RANK SUM TEST DATA COMPARING RSPSO WITH ALL OTHER ALGORITHMS FOR THREE DESIGN INSTANCES elements Method Comparison p-value RSPSO /SPSO RSPSO /DE RSPSO /RSDE RSPSO /SPSO RSPSO /DE RSPSO /RSDE Further study of mean cost function (MCF) stard deviation (SD) over 25 independent trials is carried out. The results of the study are summarized in Table III from which it is observed that for all the design instances, MCF SD exhibits minimum value for RSPSO RSDE. It is also observed that SD MF for RSPSO is least in each design instances which indicates its robustness over other contestant algorithms. TABLE III. MEAN AND STANDARD DEVIATION OF COST FUNCTION OVER 25 INDEPENDENT TRIAL RUNS Statistical elements parameter SPSO RSPSO DE RSDE MCF SD MCF SD It is common practice in the evolutionary computing domain to measure the CPU execution time required to reach convergence. Table IV shows the CPU execution time taken by each algorithm over three design instances. TABLE IV. COMPARISON OF CPU EXECUTION TIME AND ITERATION VALUE AT CONVERGENCE OF THE ALGORITHMS Parameters CPU Execution times (sec.) Iteration value at Convergence Elements SPSO RSPSO DE RSDE As an example, convergence characteristics with four contestant algorithms for elements linear array are shown in Fig.. The best convergence graph is marked in solid black line which happens to be associated with RSPSO with convergence at 27 th iteration (inset graph) for elements linear array. 95

5 optimum uniform inter element spacing value is also least using RSPSO algorithm. As a result the aperture size of linear array realized using RSPSO is least compared to DE, IWO CPSO algorithms. For example, in case of elements linear array using RSPSO algorithm, optimum inter element spacing is whereas using SPSO, DE RSDE the values are , respectively. Consequently, it can be said that RSPSO algorithm exhibits a better optimized solution for achieving multi objective requirement of desired SLL FNBW. The corresponding excitation amplitude values are tabulated in Table VI. TABLE V. ANTENNA PARAMETERS AND THERE COMPARISON FOR THREE DESIGN INSTANCES Fig.. Convergence graph for elements linear array for SLL of -20 db FNBW of 0 degree. C. Parametric Analysis Comparative Study Fig. 2. represents the radiation pattern of elements linear array (corresponding to best of the 25 runs) obtained by solving four algorithms. The radiation parameters there values considered in the analysis are summarized in Table V. elements (SLL d=-20db FNBW d = 4 degree) (SLL d=-20db FNBW d = 0 degree) Algorithm Excitation amplitude ratio Optimum interelement spacing () RSPSO SPSO DE RSDE RSPSO SPSO DE RSDE TABLE VI. EXCITATION AMPLITUDE VALUES OBTAINED USING FOR CONTESTANT ALGORITHM eleme nts Algorithm Optimum excitation amplitude (a, a 2, a 3, a 4..a N) RSPSO 2.484,2.3003, 2.06,.4394,.5496,.284 SPSO , , , 4.33, 2.589, DE 4.80, , , 4.02, , RSDE , 2.585,.7564,.6784,.2238,.2869 Fig. 2. Radiation pattern of elements optimized linear array using four algorithms with the objective of desired SLL of -20 db desired FNBW of 0 degree. From Table V it is observed that for all the design instances, desired SLL FNBW have been achieved for four contestant algorithms. Further, the optimum excitation amplitude ratio exhibits minimum value for RSPSO algorithm for all the design instances considered. Due to minimization of excitation amplitude ratio in the RSPSO algorithm, the energy consumption in realizing the actual array is minimized resulting in simple feed network less area of occupancy. In continuation to ongoing analysis, it is observed that RSPSO SPSO DE RSDE.470, 2.386, , ,.7777, ,.243,.4082,.5976, , , , , , , , , , , , , , , , , , , , ,.9687, , , ,.07,.665,.7648,

6 TABLE VII. COMPARATIVE STUDY OF 20, 26 AND 30 ELEMENT BROADSIDE ARRAY WITH UNEQUAL EXCITATION AMPLITUDE AND UNIFORM INTER ELEMENT SPACING elements 20 (case-i) 26 (case-ii) 30 (case-iii) Algorithm MSLL FNBW HPBW Directivity Antenna Aperture () NI Stard Deviation Excitation Amplitude ratio RSPSO ABC [] FFA [] RSPSO ABC [] FFA [] RSPSO ABC [] FFA [] Table VII shows a comparative study between different design instances of linear array design realized using RSPSO algorithm, ABC FFA algorithm [9]. Number of particles (NP) is 20 number of iterations (T) is 200. Moreover desired SLL for case-i, case-ii case-iii has been considered to be db, db db respectively. Further, desired FNBW for the three cases are considered to be.4 degree, 8.8 degree 7.6 degree respectively. Robustness of the algorithms has been tested through stard deviation of 20 independent runs for each algorithm. From Table 8, it is observed that for all the design instances considered in this study, RSPSO algorithm is able to reach the desired objectives with minimum number of iteration cycle remarkably least stard deviation. Though there is slight increase in antenna aperture for linear array realized using RSPSO, an improvement in directivity has been observed for this case. From Table VII, it has been observed that for 20 elements linear array, antenna aperture increase form 9.5 to 9.85 directivity improves from 8.32 to Further, excitation amplitude ratio is also least in case of linear arrays realized using RSPSO algorithm, indicating a simpler feed network as reported in [2]. V. CONCLUSION In present work, synthesis of linear array with reduced SLL narrow beamwidth has been addressed using two evolutionary algorithms PSO DE with restriction in initial search space definition. Non uniform excitation amplitude uniform inter-element spacing is taken as optimization parameter. Effectiveness of restricted search space method using in both DE PSO has been realized through three design instances of elements linear array of isotropic radiators. Among the four contestant algorithms RSPSO outperforms other two algorithms. Hence it can be concluded that restriction is more effective with PSO than that of DE. Comparative study with published work further firmly establishes the effectiveness of the proposed method using RSPSO algorithm. As observed, for obtaining required SLL value of db FNBW of.4 degree for 20 elements linear array stard deviation obtained using RSPSO is 0.0 as compared to that of value obtained using ABC FFA algorithms respectively with slightly greater antenna aperture. References [] T.T.Taylor, Design of line source antennas for narrow beamwidth low sidelobes, IRE Transaction on Antennas Propagation, Vol. 7, pp. 955, pp. 6-28,. [2] S.Chatterjee, S. Chatterjee, D. R. 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