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Synthesis of cosecant linear antenna array pattern using a novel modified invasive weeds optimization.

I. INTRODUCTION

The recent developments in wireless telecommunications technologies require networks with enhanced characteristics like capacity and coverage. Whatever its complexity, a single antenna element is not able to fit with the requests of environment, but these limitations could be overcome by the synthesis of antenna arrays. In wireless mobile telecommunications, the most commonly used topology is the uniform linear array [1]. Moreover, in the past decades, many researches have been performed in the field of linear array synthesis. The goal is to fit with new requirements in radiation characteristics by altering current distribution (amplitude and/or phase). The principal requirements to be satisfied are minimum side lobe level (SLL) [2], nulls towards interferes [3], shaped beam [4]-[6].

The synthesis of antenna array that gives a desired pattern by controlling amplitude and phase is a nonlinear problem. Since the classical synthesis methods often fall in local optima, many stochastic methods have been used for solving the array design problems. Among the recent and robust algorithms which have been successfully employed, these methods include: genetic algorithm (GA) [7], ant colony optimization (ACO) [8], particle swarm optimization (PSO) [9], flocks of starling (FSO) [10], cat swarm optimization (CSO) [11], artificial bees colony (ABC) [12] and plant growth simulation algorithm (PGSA) [13].

On the other hand, in [14], Mehrabian and Lucas proposed a new algorithm based on the ecological behaviour of colonizing weeds, known as invasive weeds optimization (IWO). In the last few years, IWO became popular in the field of electromagnetic problems due to its robustness and flexibility [15].

In this paper, a new modified version of IWO, known as MIWO, is presented for the synthesis of linear array antenna to obtain an array pattern with shaped beam. All the steps in our modified IWO are the same in the original IWO except the reproduction process. The procedure of reproduction is based on a new dispersal technique as follow. For each iteration index, a random number is generated and compared to a fixed number called the probability of mutation (Pm). If the generated number is greater than Pm, the algorithm will replace the value of the calculated standard deviation SD of the current iteration by its initial value. This process of mutation allows a larger dispersion of seeds around the parent plant, thus avoiding getting stuck in local minima. To illustrate the performance of the proposed algorithm, MIWO is applied to synthesis of both pencil beam patterns and shaped beam patterns.

II. ANTENNA ARRAY PROBLEM FORMULATION

Let us assume that a 2A-element linear array is symmetric with respect to its center, placed along x axis as shown in Fig. 1. Mathematically, the Array Factor (AF) can be written as

AF([theta]) = 2[N.summation over (n=1)][a.sub.n] cos([2[pi]/[lambda]][x.sub.n] sin [theta] + [[phi].sub.n]), (1)

where [lambda] is the wavelength, [theta] is the measured angle from the axis of the array, [a.sub.n] and [[phi].sub.n] represent the amplitude and phase weights of the nth element on each side of the array, respectively, [x.sub.n] is the position of the nth element in the array.

In our study, by varying both amplitude and phase of each element in the array, the characteristics of the array factor will be controlled during the synthesis process. To overcome the effect of mutual coupling, an allowed distance of interelements is fixed to 0.5 [lambda].

In order to synthesize the array desired pattern, both in the cosecant region and in the predetermined side-lobe regions, a cost function for these requirements is given by [16]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [w.sub.1] and [w.sub.2] are the weights of the cost function, generally both equal to 1. ([w.sub.1] = [w.sub.2] = 1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

here [M.sub.1] = 1.122[csc(cos(95.8)) x csc(cos(99))],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

here [M.sub.2] = csc(cos(97[degrees])) x csc(cos(99[degrees])).

As it can be seen, [S.sub.max] allows a tolerance of 1 dB (1.122 in linear scale, as was extracted from the expression of [M.sub.1]) with [S.sub.min] between 95.8[degrees] and 120[degrees]. Here [S.sub.max] and [S.sub.min] represent the maximum and minimum shaping region, respectively. A desired pattern must be situated inside the shaping region. To achieve this goal, our MIWO method is used.

III. MODIFIED INVASIVE WEEDS OPTIMIZATION

IWO is a recent method, compared to genetic algorithm (GA) and other traditional stochastic methods, based on the colonizing behavior of the invasive weeds in the nature. Plants invade cropping field, occupy the free space around this field and grow to get new weeds, and so on. Starting from random positions of a non-uniform linear array, the optimum positions of the elements will be found by suppressing side lobes in both regions outside the main beam, with symmetric nulls in some directions. In the proposed MIWO, a mutation process in the calculation of SD is added; this allows a larger dispersion of seeds around the parent plant. Hence, new weeds will then grow randomly. Their number is related to the fitness value of their parents. The algorithm of MIWO method is organized as follows.

A. Initialization

First, a finite number of plants are randomly spread over search space (N dimensions). This initial population is denoted as POP = [[P.sub.1], [P.sub.2] ..., [P.sub.pop_ini]] where pop_ini is the number of generated plants. Each plant is considered as a proposed solution in the search space and termed as [P.sub.i] = [[I.sub.1], [I.sub.2], ..., [I.sub.N]], where [I.sub.n] represent both amplitude and phase of the nth element on each side of the array.

B. Evaluation

The fitness reflects an evaluation of how good the plant is. The optimal plant (vector solution) is the one which minimizes the cost function defined by the designer.

C. Reproduction

Depending on its fitness value, a plant can produce a number of seeds from 0 to a fixed number. The best plant in the colony will produce a maximum number of seeds, while the worst plant cannot produce any seed. Between these extremely plants fitness, the number of produced seeds is given by

Ns([P.sub.i]) = integer[Nbr], (5)

where

Nbr = [m.sub.s] + (Ms - [m.sub.s]/BC - WC)[C([P.sub.i]) - WC], (6)

where Ms and [m.sub.s] are the maximum and minimum numbers of generated seeds, respectively. BC and WC are the best and worst cost in the actual population, respectively. C(Pi) is the cost function of ith plant in the population and [Nbr] is a mathematical function that gives the greatest integer of Nbr.

Next, the produced seeds will be spread around its parent with a random distance using uniform distribution with the mean equal to 0 and a standard deviation SD that will decrease iteratively following the equation below

[SD.sub.iter] = ([itr.sub.max] - itr)/[([itr.sub.max]).sup.mod]([SD.sub.ini] - [SD.sub.fnl]) + [SD.sub.fnl], (7)

where, [itr.sub.max] is the maximum number of iterations and itr is the actual iteration index. [SD.sub.ini] and [SD.sub.fnl] present the initial and final values of standard deviation, respectively; mod is the modulation index, usually equal to 3.

D. Mutation

In IWO, many works investigate on the choice of standard deviation value, as the most important parameter in this optimization method. A good parameter choice is essential for a fast convergence of the process. Many works have been done to achieve its optimal value. For instance, in [17], each weed in the population has its own standard deviation value depending to its cost function in the actual population; whereas a periodical variation can be added to the standard deviation.

In the present study, the idea is inspired from the mutation process of the genetic algorithms (GA) [7]. First, a probability [P.sub.m] is fixed. If [P.sub.m] is lower than a randomly generated value in the range [0, 1], then the standard deviation value of the current iteration [SD.sub.iter] will be replaced by its initial value [SD.sub.ini], otherwise, the standard deviation [SD.sub.iter] will be calculated as indicated in (7). The steps of the proposed algorithm are as follows:

Step 1: A random number is generated.

Step 2: A fixed probability of mutation Pm is chosen.

Step 3: If the random number is greater than Pm then [SD.sub.iter] = [SD.sub.ini];

Else [SD.sub.iter] is calculated from (7).

Step 4: Repeat this algorithm for each iteration.

E. Limitation

The seeds will grow and become new weeds. These weeds will be added to the colony till a maximum number of plants pop_max is reached. In this case, a competitive exclusion will begin to keep only the best pop_maxth plants in the colony, and discard the rest.

F. Stop Criteria

Generally, a maximum number of iterations is taken as a stop criterion.

IV. NUMERICAL RESULTS

In order to assess the performance of the proposed MIWO for the synthesis of linear arrays, two examples have been chosen and presented. Each example has 2N number of elements. All algorithms have been run 50 times. The best results are compared. A comparison is presented between the proposed method and other instantiations in the literature such as ABC [12], PSO [9] and ACO [8]. In this study, the initial population is fixed to 4N and the maximum number of population is fixed to twice of initial population. The standard deviations [SD.sub.ini] and [SD.sub.fnl] are set to 0.05 and [10.sup.-7], respectively. The iteration number is fixed to 100 and [P.sub.m] is fixed to 0.5 and will be decreased iteratively.

In the first example, a shaped beam pattern synthesis of 2N = 24 elements has been examined by controlling the amplitude and phase of each element on each side of the array. The inter-element spacing is considered equal to 0.5[lambda]. In Fig. 2, the cosecant pattern is obtained using MIWO by determining both amplitude and phase of the array elements, when the optimal values are tabulated in Table I.

In this case, as in [16], a reduction of 20 % in the number of array elements is obtained in comparison with the same goal achieved for 30 elements using tabu search algorithm (TSA) [4].

In the same way, the problem dimension (N = 12) is reduced to the half when the symmetry in controlling parameters is proposed. The dynamic range ratio (DRR) of amplitude is acceptable (equal to 13.06), in comparison with the very high value (DRR = 171.65) found by [6] and 15.26 found by [4]. Fig. 3 shows the evolution of standard deviation values versus the number of iterations, when more than 32 mutations are shown. Causing to the iterative decrease of Pm, it can be seen that most mutations are found in the first half range of iterations.

V. COMPARATIVE STUDY

In order to evaluate the capabilities of the proposed method, a good instantiation of the design problem is solved using our MIWO, ACO [8], PSO [9], and CLPSO [18]. A non-uniform linear array of 2N = 10 elements is considered with uniform excitation ([a.sub.n] = 1 and [[phi].sub.n] = 0). By controlling the inter-elements spacing, an optimized array pattern should present a minimum SLL with a beam-width equal or less to that of the corresponding uniform array. The number of iterations is fixed to 1000, and the cost function is given by [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where S is the space spanned by the angle [theta] excluding the main beam, [BW.sub.c] and [BW.sub.d] denote the calculated and desired beamwidth, respectively and [XI] is a very large number.

The optimal positions are listed in Table II, when the corresponding radiation pattern is shown in Fig. 4. As it can be seen, the best design is obtained by MIWO when a minimum SLL is lower than--19.06 dB with a beamwidth close to that of the uniform array.

VI. CONCLUSIONS

In this paper, a new algorithm denoted as modified invasive weeds optimization (MIWO) is introduced for the synthesis of both cosecant array pattern and pencil beam pattern of linear array. MIWO has been compared to other methods when the different results show the robustness and the performance of the proposed method. Moreover, the use of a mutation in the standard deviation calculation allows the search space to be better explored avoiding falling in local minimum. In consequence, the convergence will be accelerated. Many other extensions in array design could be also easy for implementation. Because of the good performance of MIWO, it can be used as an alternative optimization method to solve many problems in antenna design.

http://dx.doi.org/10.5755/j01.eee.21.5.13332

Manuscript received May 14, 2015; accepted August 11, 2015.

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El Hadi Kenane (1, 2), Farid Djahli (1), Arres Bartil (1)

(1) LIS Laboratory, Institute of Electronics, University of Setif 1, 19000 Setif, Algeria

(2) LGE Laboratory, Department of Electronics Engineering, University of M'sila, Ichbilia St. 28000 M'sila, Algeria

kenaneh@yahoo.fr

TABLE I. OPTIMIZED ELEMENT EXCITATIONS (AMPLITUDE
AND PHASE) OBTAINED USING MIWO FOR THE ARRAY FACTOR
SKETCHED IN FIG. 1.

Element      Normalized     Static phase
number (n)    amplitude        weights
               weights     ([[phi].sub.n])
             ([a.sub.n])

1              0.9835          20.7204
2              0.8638          57.5860
3              0.5365          86.3196
4              0.3402          91.3742
5              0.3137          87.1511
6              0.2787         124.6700
7              0.2753         138.9213
8              0.2006         143.9706
9              0.1441         148.9622
10             0.1890         179.9825
11             0.0753         151.3751
12             0.1124         179.9522

TABLE II. OPTIMIZED ELEMENT POSITIONS OF 10 ELEMENT
LINEAR ARRAY USING MANY METHODS IN TERM OF
WAVELENGTH.

n   uniform    MIWO    ACO [8]   CLPSO [18]   PSO [9]

1   0.2500    0.2290   0.2500      0.2515     0.2515
2   0.7500    0.7352   0.5500      0.7110     0.5550
3   1.2500    1.2490   1.0500      1.2080     1.0650
4   1.7500    1.8973   1.5500      1.8350     1.5000
5   2.2500    2.6450   2.1500      2.5585     2.1100
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Author:Kenane, El Hadi; Djahli, Farid; Bartil, Arres
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Date:May 1, 2015
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