# Pattern Synthesis of Linear Antenna Arrays Using Enhanced Flower Pollination Algorithm

Authors: Urvinder Singh; Rohit Salgotra
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

In this paper, a new variant of flower pollination algorithm (FPA), namely, enhanced flower pollination algorithm (EFPA), has been proposed for the pattern synthesis of nonuniform linear antenna arrays (LAA). The proposed algorithm uses the concept of Cauchy mutation in global pollination and enhanced local search to improve the exploration and exploitation tendencies of FPA. It also uses dynamic switching to control the rate of exploration and exploitation. The algorithm is tested on standard benchmark problems and has been compared statistically with state of the art to prove its worthiness. LAA design is a tricky and difficult electromagnetic problem. Hence to check the efficacy of the proposed algorithm it has been used for synthesis of four different LAA with different sizes. Experimental results show that EFPA algorithm provides enhanced performance in terms of side lobe suppression and null control compared to FPA and other popular algorithms.


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## Body

## 1. Introduction

Antenna arrays find their application in number of wireless applications such as radar, sonar, mobile, TV, and satellite. Antenna arrays are favored as these have high directivity, reduce power consumption, increase spectral efficiency and also have beam steering capability [1]. The design of antenna arrays with desired radiation pattern has found great interest in the literature. Antenna arrays design is intricate and nonlinear problem. Hence number of optimization techniques such as genetic algorithm (GA), differential algorithm (DE), particle swarm optimization (PSO), biogeography based optimization (BBO) and many others [2–21] have been used to synthesize these. It is required that antenna arrays radiate in desired directions so that these do no add to electromagnetic pollution. This can be achieved if the energy is maximum in the main lobe and minimum in side lobes. Moreover, to avoid interference from undesired directions and to circumvent jamming, null placement in radiation pattern has also gained importance. So, overall the main issues in designing the antenna arrays are reducing the side lobe level (SLL) and placing nulls in the desired directions of radiation pattern.Linear antenna arrays (LAA) consist of number of antenna elements arranged in a straight line. LAA have become very popular because of their simple geometry and applications. LAA design has been investigated by numerous researchers using several optimization algorithms in the past. Recioui has used PSO for SLL reduction of LAA [2]. Khodier and Christodoulou have synthesized LAA for maximum SLL reduction and null placement. They have designed three arrays for different sizes and showed that PSO gives better results than uniform and quadrature programming method (QPM) [3]. Rattan et al. have applied GA [7] for the optimization of LAA. Their results were better than those obtained using PSO [3]. DE has been employed by Lin et al. to design unequally spaced LAA [8]. They have discussed about the impact of angular resolution on the final results. Dib et al. have compared the performance of self-adaptive DE (SADE) and Tagachi’s method for optimization of LAA [9]. Cengiz and Tokat have used GA, memetic algorithm (MA) and tabu search (TS) to optimize three different LAA [10]. Singh et al. [11] have applied BBO to synthesize uniform and nonuniform LAA and found that BBO is better than PSO and other methods for the design of arrays. BBO has also been used by Sharaqa and Dib to design LAA [12]. Ant colony optimization has been used by Rajo-lglesias and Quevedo-Teruel for the synthesis of LAA for SLL reduction and null placement [13]. A multiobjective approach using multiobjective DE (MOEA/D-DE) has been employed by Pal et al. to optimize LAA [14]. The authors have noted that MOEA/D-DE gives better trade-off curves between null placement and SLL. LAA synthesis using fitness adaptive differential evolution algorithm (fiADE) has been done by Chowdhury et al. [15]. Guney and Onay have utilized harmony search algorithm (HSA) for SLL minimization and null placement in different directions of radiation pattern of LAA [16]. A new algorithm based on the breeding strategy of cuckoos known as cuckoo optimization algorithm (COA) has been employed to optimize three different nonuniform LAA [17]. The authors have compared the results with the popular algorithms like DE, PSO, firefly algorithm (FA) and found that the results provided by COA in terms of SLL reduction and null placement are better than competitive algorithms. Moreover, convergence curves of different algorithms are compared and it is concluded that COA provides faster convergence than the other methods. Guney and Durmus have used back scattering algorithm (BSA) for the pattern nulling of LAA [18]. Khodier has used cuckoo search (CS) to optimize antenna arrays [19]. An enhanced version of PSO named as comprehensive learning PSO (CLPSO) has been used to design three different LAA [20]. Recently, a novel algorithm which mimics the behaviour of flowers known as flower pollination algorithm (FPA) has been used to synthesize LAA [21]. Though number of algorithms have been proposed for synthesis of LAA, these suffer from certain shortcomings like getting stuck in local minima, slow convergence speeds and require precise parameter tuning. So, in this work, the authors propose an enhanced version of FPA known as enhanced FPA (EFPA). The newly proposed algorithm uses the concept of Cauchy distribution to follow large steps in global pollination, enhanced local search and dynamic switch probability to control the rate of local and global pollination. It has better exploration and exploitation capability and is also less likely to stuck in local minima.The rest of the paper is organized as follows: Section2 gives details about the basic FPA algorithm, Section 3 proposes a new EFPA algorithm, Section 4 gives result and discussion, and Section 5 provides an extensive conclusion.


## 2. Flower Pollination Algorithm

Flowers are the most fascinating plant species and have been dominating earth from the cretaceous period, partly from about 125 million years. Almost 80% of plant species are flowering and it has been made possible by the process of pollination [22]. Pollination in flowers, refer to transfer of pollen grains from one flower to another via pollinators such as wind, diffusion, bees, birds, bats and other animals [23]. If the pollination process is through insects, it is called biotic pollination and if it takes place via wind or diffusion, it is called abiotic pollination. Overall, 200,000 varieties of flower pollinators exist in nature. Pollinators such as honey bee has been found to show some specific phenomenon known as flower constancy. This phenomenon helps them to visit only specific flower species and by pass others, hence maximizing the transfer of pollens from a particular plant species. This helps pollinators in finding better food sources and minimize the exploration or learning cost [24]. Pollination can further be classified based upon the pollens of same or different flowers. If a pollinator transfers pollens from one flower to another, it is called cross-pollination while if it does for the same species, it is self-pollination.Based upon the phenomenon of pollination, flower constancy and behaviour of pollinators, flower pollination algorithm (FPA) was proposed by Yang [25]. He proposed four sets of rules for governing this algorithm as follows:(1)
For long distances, the pollination used is biotic cross-pollination; the process is called global pollination and is performed via Levy flights.(2)
Abiotic self-pollination process forms the basis of local pollination.(3)
The reproduction probability of different flowers involved in pollination is considered as flower constancy.(4)
A switch probabilityp∈[0,1] is defined for controlling local and global pollination.For simplicity, a single flower producing only one pollen gamete is considered. This means a single solution for problem under test is equivalent to a pollen gamete or a flower. From the rules above, two key features of global and local pollination have been used to formulate FPA.The first step in FPA is global pollination, where pollinators such as insects carry pollen to long distances. This process helps in pollination of the best fit flower or solution so far and is represented asR∗. Rule 1 with a combination of flower constancy is represented as(1)xit+1=xit+LλR∗-xit,where xit is the ith solution of the problem in the tth iteration, L is the step size, and Levy flight is generally used to mimic this phenomenon [25]. Levy flight is drawn from a Levy distribution as(2)L~λΓλsin⁡πλ/2⁡π1s1+λ,s≫s0>0,where Γ(λ) is the standard gamma function with a step size of s>0.In the second step, Rule2 is used along with flower constancy to represent local pollination as(3)xit+1=xit+ϵxjt-xkt,where xjt and xkt are pollen gametes of same plants but different flowers and ϵ is uniformly distributed in [0,1]. This phase mimics glower constancy at a local scale or in a limited search space.So, we can say that global and local pollination carry out pollination activities at large and small scale, respectively. In practice, we use a switching probability based on Rule4 to define the extent of local and global pollination. In the next section, a new enhanced version of flower pollination algorithm is proposed.


## 3. Enhanced Flower Pollination Algorithm

In the recent past, a large number of researchers have focused on enhancing the basic capabilities of FPA. The algorithm due to its linear nature, make it suitable for deeper analysis. But it has been proved qualitatively and quantitatively in [26] that the FPA algorithm has a very limited scope for optimization problems at hand. Also, the performance of FPA has not been analyzed to a deeper level and the algorithm is still to prove its worthiness for becoming a state-of-the-art algorithm. Keeping in view the above analysis, a new version of FPA namely EFPA has been proposed. The newly proposed EFPA aims at providing three different modifications to the basic FPA. These include Cauchy based global pollination, enhanced local pollination based upon experience of current best flower pollinator as well as local flowers in proximity, and, thirdly, using dynamic switching probabilities.(I) Cauchy Based Global Pollination. In the global pollination phase, instead of using a standard Levy distribution, a Cauchy based operator C(δ) is used. This operator is basically a Cauchy random variable with distribution given by(4)δ=12+1πarctanxg.The Cauchy density function is given by (5)fCauchy0,gx=1πgg2+x2.The general equation of global pollination becomes (6)xit+1=xit+CδR∗-xit,where g is a scale parameter and its value is generally set to 1. The use of Cauchy operator allows for larger mutation by searching the search space at a faster pace and further accounts for avoiding premature convergence.(II) Enhanced Local Pollination.The second modification is added in the local pollination phase. Here based upon the experience of local and current best pollinators, the position of new pollinators is updated. Further, if the fitness of new position is greater than the old one, each pollinator updates its position with respect to the previous one. The general equation is given by(7)xit+1=xit+aR∗-xit+bxjt-xkt,whereaandb are uniformly distributed random numbers in the range of [0,1]. Also, the solutions xjt and xkt, corresponds to jth and kth flower pollinator in the group with j≠k. This phase enhances the local search capabilities of FPA algorithm.(III) Dynamic Switch Probability. In FPA, local and global pollinations are controlled by switching probability p∈[0,1]. For a standard state-of-the-art algorithm, it is imperative to follow more global search at the start and as the algorithm progresses more intensive local search is followed. Using this basic concept, the value of switching probability is selected dynamically. The switch probability is updated by following a general formula as(8)p=p-maxiter-tmaxiter×0.1.The above general equation decreases the value ofp linearly with iterations and hence adds to intensive global search at the beginning and local search towards the end. Here maxiter corresponds to the maximum number of iterations andt is the current iteration. The pseudocode for the EFPA is given in Pseudocode 1.Pseudocode 1:Pseudocode of EFPA.
Start: Initializen random flowers Define switch probability,p∈[0,1] Defined-dimensional objective function fro problem under test Find current best solutionR∗ from the initial population while  iterations < maximum number of iterations for  i=1:n if  rand < p Draw ad-dimensional Cauchy step vector obeying Cauchy distribution Global Pollination:xit+1=xit+C(δ)(R∗-xit) else Local Pollination:xit+1=xit+aR∗-xit+b(xjt-xkt) end if evaluatexit+1 ifxit+1  is better than xit,   update end if Updatep by:  p=p-maxiter-tmaxiter×(0.1) end for Update the final best solution. end while end


## 4. Results and Discussion

### 4.1. Results of Benchmark Functions

The performance of EFPA is evaluated by applying it to two set of optimization problems. One set includes test functions and other is the real-world application of antenna array design. The algorithm is also compared with other states of the art to prove its competitiveness. The simulations are performed on Intel Core i3 personal computer with Windows 10 operating system using MATLAB version 7.10.0 (R2010a). EFPA is tested on seven well-known benchmark problems [27] as shown in Table 1 with search range and optimal solution. For comparison purposes, artificial bee colony (ABC) [28], DE [29], bat algorithm (BA) [30] and FPA [25] are used. The maximum number of function evaluations for each algorithm is set to 30 × 1000, with 30 as the population size and 1000 as the maximum number of iterations. The parameter settings of all algorithms are given in Table 2. All the algorithms are run 50 times and results in terms of best, worst, mean and standard deviation are reported. Since because of the stochastic nature of algorithm, Wilcoxon rank-sum test has also been performed to significantly prove the better performance of EFPA. The performance in terms of time-complexity has also been shown in Table 5 to prove its competitiveness. Time-complexity is basically taken for each of the algorithms in the benchmark problems and it has been calculated for one run for each algorithm.Table 1
Description of test functions.
Test problems Objective function Search range Optimum value D Sphere function f 1 x =  ∑ i = 1 D x i 2 [−100, 100] 0 10 Weierstrass function f 2 x =  ∑ i = 1 D ∑ k = 0 k max ⁡ a k cos ⁡ 2 π b k x i + 0.5 - D  ∑ k = 0 k max ⁡ a k cos ⁡ ( 2 π b k · 0.5 ) ;  w h e r e  a = 0.5 ,   b = 3 ,   k max ⁡ = 20 [−0.5, 0.5] 0 30 Schaffer function f 3 x = 1 n - 1 s i · sin ⁡ 50.0 s i 1 / 5 + 1 2  s i = x i 2 + x i + 1 2 [−100, 100] 0 30 Griewank f 4 x = 1 4000  ∑ i = 1 N x i 2 -  ∏ i = 1 N cos ⁡ x i i + 1 [−600, 600] 0 30 Penalized 1 function f 5 ( x ) = π n 10 sin 2 ⁡ π y 1 +  ∑ i = 1 n - 1 y i - 1 2 1 + 10 sin 2 ⁡ π y i + 1 + y n - 1 2  +  ∑ i = 1 n u x i , 10,100,4 [−50, 50] 0 30 y i = 1 + x i + 1 4 ;   u ( x i ,  a ,  k ,  m ) = k x i - a m x i > a 0 - a < x i < a k - x i - a m x i < - a Ackley function f 6 x = - 20 exp ⁡ ⁡ - 0.2 1 D ∑ i = 1 D x i 2 - e x p  1 D ∑ i = 1 D cos ⁡ 2 π x i + 20 + e [−100, 100] 0 30 Rosenbrock function f 7 =  ∑ i = 1 N - 1 100 x i + 1 - x i 2 2 + x i - 1 2 [−100, 100] 0 30Table 2
Parameter settings of various algorithms.
Algorithm Parameters Values DE F 1.5 CR 0.8 Population size 30 Maximum iterations 1000 Stopping criteria Max iteration  ABC Colony size 30 Number of food sources Colony size/2 Limit 100 Maximum cycles 1000 Stopping criteria Max iteration  FPA Population size 30 Probability switch 0.6 Maximum iterations 1000 Stopping criteria Max iteration  BA Population size 50 Loudness 0.5 Pulse rate 0.5 Maximum number of iterations 200 Stopping criteria Max iteration  EFPA Population size 30 Probability switch Dynamic and linearly decreasing Maximum iterations 1000 Stopping criteria Max iterationThe comparison of comparison for various test functions is given in Table3. The best and worst values give the most optimal and least optimal solution among the 50 solutions from the maximum number of runs. Mean gives the average values of results and is used when the best and worst values do not give comparable results. But mean may give same results for widely different results. So, standard deviation is also used to calculate results. For functions f1, f4, f5, f6, and f7 it can be seen that EFPA performs much better than other algorithms except for f2 where ABC gives better results. For function f3 ABC and EFPA gives same results. Further statistical analysis from the Table 4 verify the results. In Table 5, time-complexity for each algorithm is shown. Here time-complexity show that EFPA takes comparatively less time for solving the benchmark problem for most of the benchmark functions in comparison to FPA, BA and ABC. For DE, only four functions give better time-complexity while, for the rest three, again EFPA is better. Overall, we can say that EFPA is superior to ABC, DE, FPA, and BA. In the next subsection, EFPA is applied for synthesis of LAA.Table 3
Comparison of results for various test functions.
Objective function Algorithm Best Worst Mean Standard deviation f 1 x DE 4.43 E + 03 2.97 E + 04 1.54 E + 04 6.22 E + 03 ABC 2.74 E + 03 3.67 E + 04 2.74 E + 04 5.93 E + 05 BA 8.08 E + 03 3.54 E + 04 1.96 E + 04 7.01 E + 03 FPA 1.00 E + 02 1.00 E + 03 3.61 E + 02 1.66 E + 02 EFPA 3.35 E - 15 4.72 E - 07 1.82 E - 08 6.85 E - 08  f 2 x DE 1.64 E + 01 3.82 E + 01 2.54 E + 01 4.51 E + 00 ABC 2.53 E + 00 5.90 E + 00 3.90 E + 00 8.40 E - 01 BA 2.51 E + 01 4.30 E + 01 3.36 E + 01 4.24 E + 00 FPA 3.83 E + 01 5.72 E + 01 5.12 E + 01 4.64 E + 00 EFPA 1.34 E + 01 2.66 E + 01 2.00 E + 01 3.10 E + 00  f 3 x DE 7.95 E - 05 2.29 E - 01 4.64 E - 02 5.37 E - 02 ABC 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00 BA 3.06 E - 14 3.47 E - 01 8.95 E - 02 9.22 E - 02 FPA 0.00 E - 00 1.98 E - 12 1.03 E - 13 3.39 E - 13 EFPA 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00  f 4 x DE 4.89 E + 01 3.68 E + 02 1.54 E + 02 7.19 E + 01 ABC 1.06 E + 00 1.76 E + 00 1.25 E + 00 1.73 E - 01 BA 2.94 E + 00 1.18 E + 01 5.77 E + 00 1.63 E + 00 FPA 5.71 E - 01 1.04 E + 00 9.25 E - 01 9.76 E - 02 EFPA 0.00 E - 00 1.45 E - 01 2.90 E - 03 2.06 E - 02  f 5 x DE 1.78 E + 06 6.92 E + 08 2.24 E + 08 2.27 E + 08 ABC 3.49 E - 02 2.77 E + 07 1.07 E + 08 8.91 E + 07 BA 3.40 E + 05 5.55 E + 07 1.58 E + 07 1.53 E + 07 FPA 2.81 E + 00 1.11 E + 01 6.81 E + 00 1.87 E + 00 EFPA 1.25 E + 00 3.1 E + 01 8.93 E + 00 6.40 E + 00  f 6 x DE 2.00 E + 01 2.00 E + 01 2.00 E + 01 0.00 E + 00 ABC 2.00 E + 01 2.02 E + 01 2.00 E + 01 5.30 E - 03 BA 1.97 E + 01 2.00 E + 01 1.99 E + 01 2.92 E - 02 FPA 2.05 E + 01 2.11 E + 01 2.10 E + 01 8.37 E - 02 EFPA 7.29 E + 00 1.33 E + 01 9.95 E + 00 1.29 E + 00  f 7 x DE 4.21 E + 08 3.78 E + 10 8.63 E + 09 1.00 E + 10 ABC 2.18 E + 04 3.61 E + 05 2.96 E + 04 5.39 E + 04 BA 8.62 E + 08 1.29 E + 10 3.07 E + 09 2.10 E + 09 FPA 1.48 E + 05 6.49 E + 06 2.30 E + 06 1.72 E + 06 EFPA 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00Table 4
p-test values of various algorithms.
Objective function BA FPA ABC DE EFPA f 1 x 7.06 E - 18 7.06 E - 18 7.06 E - 18 7.06 E - 18 NA f 2 x 7.06 E - 18 7.06 E - 18 NA 3.25 E - 09 7.06 E - 18 f 3 x 3.31 E - 20 6.89 E - 17 NA 3.31 E - 20 NA f 4 x 4.73 E - 20 4.73 E - 20 4.73 E - 20 4.73 E - 20 NA f 5 x 7.06 E - 18 7.70 E - 02 4.21 E - 04 7.06 E - 18 NA f 6 x 7.06 E - 18 7.06 E - 18 7.06 E - 18 3.31 E - 20 NA f 7 x 3.31 E - 20 1.06 E - 17 3.31 E - 20 3.31 E - 20 NATable 5
Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds).
Function DE ABC BA FPA EFPA f 1 x 1.284464 2.527247 1.190053 2.328157 1.965927 f 2 x 15.98023 15.62962 19.02841 15.22211 14.80168 f 3 x 1.185347 2.470673 1.047533 2.183295 1.900515 f 4 x 3.022755 5.167423 4.165333 2.183153 2.030502 f 5 x 2.848487 5.988428 2.974728 3.755994 3.498411 f 6 x 1.667705 3.186360 2.261759 2.346810 2.383599 f 7 x 1.693011 3.265792 2.087382 3.384049 2.999886


### 4.2. Results of LAA

A symmetrical LAA consists of2N elements withN elements on both sides of the origin and arranged in a straight line as shown in Figure 1. The nth element is excited with a current of amplitude In and phase φn. The position of nth element is represented by xn. The array factor (AF) of such an antenna is given by [1](9)AF∅=2∑n=1NIncos⁡kxncos⁡∅+φn.Here k is the wave number and is given by 2π/λ. The symmetric nature of the array reduces the computational cost as onlyN elements are needed to be optimized. Moreover, these symmetric arrays give symmetric radiation pattern which is desired for number of applications.Figure 1
Geometry of a linear antenna array.The AF for LAA in (9) is a function of excitation currents (both In and φn) and positions xn. If current excitations are held uniform (i.e., In=1 and φn = 0°), then the LAA is function of xn only and array is called nonuniform LAA. Hence for nonuniform LAA, AF reduces to (10)AFϕ=2∑n=1Ncos⁡kxncos⁡ϕ⁡.The design of nonuniform LAA has gained importance in the past and several researchers have synthesized it using different optimization techniques [3, 7, 9, 11–21]. The array factor of a nonuniform array is aperiodic by virtue of nonuniform spacing between the elements. This aperiodic nature of arrays helps in obtaining lower SLL with lesser number of elements for a given antenna size. Furthermore, the uniform element excitations help in reducing the feed network cost and complexity. However, the relationship between the array factor and element spacing is nonconvex and nonlinear. So, this makes the design of nonuniform arrays difficult.For directing antenna arrays energy in a specific direction, it is desired that its radiation pattern should have low side lobes. Moreover, nulls are also required in the direction of interference. The desired radiation pattern can be achieved by suitably adjusting the positions of antenna elements for nonuniform LAA. The adjustment of positions is a tricky and nonlinear problem. So, optimization algorithms are well suited for this type of problems. For applying optimization to antenna problem, it is imperative to define a fitness function which is given as(11)fit=SLL+α∗max⁡0,BW-BWd-1+α∗∑k=1Kmax⁡0,AFdBϕk-NudB. In the above fitness function, SLL is the side lobe level, AFdB is the array factor in decibels, BW and BWd are the calculated and desired beamwidth of the main lobe, ϕk gives the direction of the kth null, NudB is the required null depth in dB, K is the number of the required nulls, and α is a very large number. The fitness function given in (11) uses a penalty method which penalizes any antenna design which does not meet the required constraints of beamwidth and nulls (if desired). This is achieved by taking value α to be very large. The BW here is obtained computationally from the radiation pattern data.


#### 4.2.1. Linear Antenna Optimization

This subsection presents the application of EFPA to optimization of nonuniform LAA. In the first example, a 12-element nonuniform antenna is taken for optimization. The goal of optimization is to minimize the SLL in the regionϕ∈ [98°,180°]. Because of symmetry there are only six element positions which are to be optimized. The number of generations for EFPA are taken as 1000. The population size taken as 20 in order to make fair comparison between the results of FPA [21] and EFPA. The total number of function evaluations are 20 ∗ 1000 = 2000 for both the FPA [21] and EFPA. As the aim of the optimization is to suppress the SLL only, so the value of α is taken as 0 in the fitness equation of (11). The EFPA algorithm is run for 30 times and best results are listed in this work. The optimum positions obtained using EFPA are shown in Table 6. The performance of EFPA is contrasted against other popular algorithms in terms of SLL in Table 7. The minimum SLL obtained by EFPA is −21.07 dB and is better than GA [10], FiADE [15], TS [10], MA [10], FPA [21] and PSO [15]. The comparison of convergence curve between EFPA and FPA is shown in Figure 2 which clearly shows faster convergence of EFPA than the FPA [21]. The FPA converged in about 700 iterations whereas EFPA took only 225 iterations to converge to the final solution. This is due to enhanced exploration due to Cauchy operator and better exploitation in EFPA. The radiation pattern of EFPA antenna array is shown in Figure 3. For comparison, the radiation patterns of GA [10], FiADE [15], and PSO [15] are also shown in Figure 3.Table 6
Geometry of 12-element LAA (with respect toλ/2) obtained using EFPA.
Element 1 2 3 4 5 6 Position 0.3765 0.9734 1.7376 2.5051 3.4827 4.6744Table 7
Performance of different algorithms for 12-element LAA.
Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPA SLL (dB) −18.96 −19.10 −18.77 −17.90 −20.764 −18.40 −21.07Figure 2
Convergence plots of 12-element LAA.Figure 3
Radiation plot of 12-element EFPA optimized LAA.In the next example, a 22-element nonuniform LAA is optimized for reducing SLL in the regionϕ∈[98°, 180°] and also imposing a null in the direction of 99°. For achieving the required pattern, the value of NudB is set to −60 dB. It is established fact that as SLL is decreased the width of the main lobe in radiation pattern increases. So, for this design, the desired beamwidth BWd is taken as 18°. The value of α is taken as 106 in fitness function. The optimum positions are shown in Table 8. The results are compared in Table 9 which clearly indicate better performance of EFPA as against BSA [18], MOEA/DE [14], HSA [16], PSO [15], GA [10], MA [10], TS [10], and FPA [21]. The SLL of EFPA antenna array is −26.31 dB which is much lower than that obtained using other algorithms. Furthermore, the null in the direction of 99° is −75.58 dB which is less than the desired value of −60 dB and deeper than MOEA/DE [14], GA [7], PSO [3] and MA [10]. In order to compare the performance of EFPA and FPA in terms of convergence speed, FPA is also run for the same objective with same population size. The convergence graphs of EFPA and FPA are plotted in the Figure 4. The convergence characteristics clearly substantiates the superior performance of EPFA in terms of faster convergence speed. The radiation plots of EFPA, HSA [16], BSA [18], and MOEA/DE [14] optimized arrays are shown in Figure 5.Table 8
Geometry of 22-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 Position 0.4937 0.7965 1.2153 2.1313 2.5003 3.1327 3.9654 4.5877 5.5435 6.6391 8.0603Table 9
Performance of different algorithms for 22-element LAA.
Algorithm BSA [18] TS [10] MOEA/D-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPA SLL (dB) −23.54 −17.17 −20.93 −23.28 −20.68 −15.73 −18.11 −23.81 −26.31  Null (dB) (99°) −104.61 −67.94 −69.64 −103.3 −49.94 −54.29 −73.92 −101.71 −75.58Figure 4
Convergence plots of 22-element LAA.Figure 5
Radiation plot of 22-element EFPA optimized LAA.In order to show the capability of EFPA in synthesizing the radiation pattern with multiple nulls, a 28-element nonuniform LAA is optimized for SLL reduction in the regionϕ∈[100°, 180°] and having nulls in the directions of 120°, 122.5°, and 125°. The desired beamwidth is set to the value of 8.35° with tolerance of ±1°. The optimum geometry obtained using EFPA is shown in Table 10. The comparison of convergence curves between EFPA and BBO [17], CS [17], DE [17], FA [17], and COA [17] is given in Figure 6 which shows that EFPA reaches the optimum solution in less number of iterations. The comparison of results in terms of SLL and null values achieved is given in Table 11 which again clearly indicates the superior performance of EFPA as against other methods. The SLL of EFPA is −22.90 dB which is lower by 1 dB, 9.24 dB, 9.3 dB, 1.04 dB, 8.26 dB, 9.24 dB, 1.27 dB, and 0.30 dB than BSA [18], QPM [3], COA [17], ACO [13], GA [7], CLPSO [20], and FPA [21], respectively. The null values are also equal to or less than −60 dB. Figure 7 shows the comparison of QPM [3], HSA [16], BSA [18], and EFPA optimized array radiation plots.Table 10
Geometry of 28-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Position 0.5880 1.2142 2.5291 2.8291 4.1205 4.9374 6.1006 6.9646 8.1374 9.5186 10.7399 12.4399 14.1399 15.8399Table 11
Performance of different algorithms for 28-element LAA.
Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −21.90 −13.24 −13.6 −21.86 −14.64 −13.60 −21.63 −22.60 −22.90 Null (dB)(120°) −82.49 −48.49 −52.74 −60.08 −52.74 −59.25 −60.04 −78.45 −60 Null (dB)(122.5°) −93.59 −48.35 −51.66 −60.05 −59.20 −75.53 −60.01 −96.51 −60 Null (dB)(125°) −80.49 −89.3 −61.46 −60.10 −43.58 −62.52 −60.00 −82.58 −67.66Figure 6
Convergence plots of 28-element LAA.Figure 7
Radiation plot of 28-element EFPA optimized LAA.The last design example illustrates the use of EFPA to optimize the element positions for minimization of SLL and null placement for 32-element LAA. The SLL is to be minimized in the regionϕ∈ [93°, 180°] and null is to be placed in the direction of 99°. The required beamwidth BWd is 7.1°. The plot of convergence for EFPA is shown in Figure 8. Also for the sake of comparison, the convergence plots of BBO [17], CS [17], DE [17], FA [17], and COA [17] are shown in the same figure. The element positions for EFPA optimized array are shown in Table 12. Table 13 compares the performance of different optimization methods for design of 32-element LAA. The SLL of proposed EFPA antenna array is −23.73 dB which is least in Table 13. The reduction is considerable as compared to BSA [18], QPM [3], PSO [3], HSA [16], and GA [7] methods. (Figure 9).Table 12
Geometry of 32-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Position 0.5268 1.1101 2.2759 2.9840 3.8883 4.5802 5.7817 6.6474 7.7093 8.8370 9.7490 11.0885 12.6565 14.3043 16.0043 17.7043Table 13
Performance of different algorithms for 32-element LAA.
Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −20.50 −18.80 −19.51 −17.73 −22.83 −16.24 −22.75 −23.10 −23.73 Null (dB)(99°) −107.50 −62.20 −88.08 −34.74 −62.63 −62.94 −60 −130.60 −60.00Figure 8
Convergence plots of 32-element LAA.Figure 9
Radiation plot of 32-element EFPA optimized LAA.


## 4.1. Results of Benchmark Functions

The performance of EFPA is evaluated by applying it to two set of optimization problems. One set includes test functions and other is the real-world application of antenna array design. The algorithm is also compared with other states of the art to prove its competitiveness. The simulations are performed on Intel Core i3 personal computer with Windows 10 operating system using MATLAB version 7.10.0 (R2010a). EFPA is tested on seven well-known benchmark problems [27] as shown in Table 1 with search range and optimal solution. For comparison purposes, artificial bee colony (ABC) [28], DE [29], bat algorithm (BA) [30] and FPA [25] are used. The maximum number of function evaluations for each algorithm is set to 30 × 1000, with 30 as the population size and 1000 as the maximum number of iterations. The parameter settings of all algorithms are given in Table 2. All the algorithms are run 50 times and results in terms of best, worst, mean and standard deviation are reported. Since because of the stochastic nature of algorithm, Wilcoxon rank-sum test has also been performed to significantly prove the better performance of EFPA. The performance in terms of time-complexity has also been shown in Table 5 to prove its competitiveness. Time-complexity is basically taken for each of the algorithms in the benchmark problems and it has been calculated for one run for each algorithm.Table 1
Description of test functions.
Test problems Objective function Search range Optimum value D Sphere function f 1 x =  ∑ i = 1 D x i 2 [−100, 100] 0 10 Weierstrass function f 2 x =  ∑ i = 1 D ∑ k = 0 k max ⁡ a k cos ⁡ 2 π b k x i + 0.5 - D  ∑ k = 0 k max ⁡ a k cos ⁡ ( 2 π b k · 0.5 ) ;  w h e r e  a = 0.5 ,   b = 3 ,   k max ⁡ = 20 [−0.5, 0.5] 0 30 Schaffer function f 3 x = 1 n - 1 s i · sin ⁡ 50.0 s i 1 / 5 + 1 2  s i = x i 2 + x i + 1 2 [−100, 100] 0 30 Griewank f 4 x = 1 4000  ∑ i = 1 N x i 2 -  ∏ i = 1 N cos ⁡ x i i + 1 [−600, 600] 0 30 Penalized 1 function f 5 ( x ) = π n 10 sin 2 ⁡ π y 1 +  ∑ i = 1 n - 1 y i - 1 2 1 + 10 sin 2 ⁡ π y i + 1 + y n - 1 2  +  ∑ i = 1 n u x i , 10,100,4 [−50, 50] 0 30 y i = 1 + x i + 1 4 ;   u ( x i ,  a ,  k ,  m ) = k x i - a m x i > a 0 - a < x i < a k - x i - a m x i < - a Ackley function f 6 x = - 20 exp ⁡ ⁡ - 0.2 1 D ∑ i = 1 D x i 2 - e x p  1 D ∑ i = 1 D cos ⁡ 2 π x i + 20 + e [−100, 100] 0 30 Rosenbrock function f 7 =  ∑ i = 1 N - 1 100 x i + 1 - x i 2 2 + x i - 1 2 [−100, 100] 0 30Table 2
Parameter settings of various algorithms.
Algorithm Parameters Values DE F 1.5 CR 0.8 Population size 30 Maximum iterations 1000 Stopping criteria Max iteration  ABC Colony size 30 Number of food sources Colony size/2 Limit 100 Maximum cycles 1000 Stopping criteria Max iteration  FPA Population size 30 Probability switch 0.6 Maximum iterations 1000 Stopping criteria Max iteration  BA Population size 50 Loudness 0.5 Pulse rate 0.5 Maximum number of iterations 200 Stopping criteria Max iteration  EFPA Population size 30 Probability switch Dynamic and linearly decreasing Maximum iterations 1000 Stopping criteria Max iterationThe comparison of comparison for various test functions is given in Table3. The best and worst values give the most optimal and least optimal solution among the 50 solutions from the maximum number of runs. Mean gives the average values of results and is used when the best and worst values do not give comparable results. But mean may give same results for widely different results. So, standard deviation is also used to calculate results. For functions f1, f4, f5, f6, and f7 it can be seen that EFPA performs much better than other algorithms except for f2 where ABC gives better results. For function f3 ABC and EFPA gives same results. Further statistical analysis from the Table 4 verify the results. In Table 5, time-complexity for each algorithm is shown. Here time-complexity show that EFPA takes comparatively less time for solving the benchmark problem for most of the benchmark functions in comparison to FPA, BA and ABC. For DE, only four functions give better time-complexity while, for the rest three, again EFPA is better. Overall, we can say that EFPA is superior to ABC, DE, FPA, and BA. In the next subsection, EFPA is applied for synthesis of LAA.Table 3
Comparison of results for various test functions.
Objective function Algorithm Best Worst Mean Standard deviation f 1 x DE 4.43 E + 03 2.97 E + 04 1.54 E + 04 6.22 E + 03 ABC 2.74 E + 03 3.67 E + 04 2.74 E + 04 5.93 E + 05 BA 8.08 E + 03 3.54 E + 04 1.96 E + 04 7.01 E + 03 FPA 1.00 E + 02 1.00 E + 03 3.61 E + 02 1.66 E + 02 EFPA 3.35 E - 15 4.72 E - 07 1.82 E - 08 6.85 E - 08  f 2 x DE 1.64 E + 01 3.82 E + 01 2.54 E + 01 4.51 E + 00 ABC 2.53 E + 00 5.90 E + 00 3.90 E + 00 8.40 E - 01 BA 2.51 E + 01 4.30 E + 01 3.36 E + 01 4.24 E + 00 FPA 3.83 E + 01 5.72 E + 01 5.12 E + 01 4.64 E + 00 EFPA 1.34 E + 01 2.66 E + 01 2.00 E + 01 3.10 E + 00  f 3 x DE 7.95 E - 05 2.29 E - 01 4.64 E - 02 5.37 E - 02 ABC 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00 BA 3.06 E - 14 3.47 E - 01 8.95 E - 02 9.22 E - 02 FPA 0.00 E - 00 1.98 E - 12 1.03 E - 13 3.39 E - 13 EFPA 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00  f 4 x DE 4.89 E + 01 3.68 E + 02 1.54 E + 02 7.19 E + 01 ABC 1.06 E + 00 1.76 E + 00 1.25 E + 00 1.73 E - 01 BA 2.94 E + 00 1.18 E + 01 5.77 E + 00 1.63 E + 00 FPA 5.71 E - 01 1.04 E + 00 9.25 E - 01 9.76 E - 02 EFPA 0.00 E - 00 1.45 E - 01 2.90 E - 03 2.06 E - 02  f 5 x DE 1.78 E + 06 6.92 E + 08 2.24 E + 08 2.27 E + 08 ABC 3.49 E - 02 2.77 E + 07 1.07 E + 08 8.91 E + 07 BA 3.40 E + 05 5.55 E + 07 1.58 E + 07 1.53 E + 07 FPA 2.81 E + 00 1.11 E + 01 6.81 E + 00 1.87 E + 00 EFPA 1.25 E + 00 3.1 E + 01 8.93 E + 00 6.40 E + 00  f 6 x DE 2.00 E + 01 2.00 E + 01 2.00 E + 01 0.00 E + 00 ABC 2.00 E + 01 2.02 E + 01 2.00 E + 01 5.30 E - 03 BA 1.97 E + 01 2.00 E + 01 1.99 E + 01 2.92 E - 02 FPA 2.05 E + 01 2.11 E + 01 2.10 E + 01 8.37 E - 02 EFPA 7.29 E + 00 1.33 E + 01 9.95 E + 00 1.29 E + 00  f 7 x DE 4.21 E + 08 3.78 E + 10 8.63 E + 09 1.00 E + 10 ABC 2.18 E + 04 3.61 E + 05 2.96 E + 04 5.39 E + 04 BA 8.62 E + 08 1.29 E + 10 3.07 E + 09 2.10 E + 09 FPA 1.48 E + 05 6.49 E + 06 2.30 E + 06 1.72 E + 06 EFPA 0.00 E - 00 0.00 E - 00 0.00 E - 00 0.00 E - 00Table 4
p-test values of various algorithms.
Objective function BA FPA ABC DE EFPA f 1 x 7.06 E - 18 7.06 E - 18 7.06 E - 18 7.06 E - 18 NA f 2 x 7.06 E - 18 7.06 E - 18 NA 3.25 E - 09 7.06 E - 18 f 3 x 3.31 E - 20 6.89 E - 17 NA 3.31 E - 20 NA f 4 x 4.73 E - 20 4.73 E - 20 4.73 E - 20 4.73 E - 20 NA f 5 x 7.06 E - 18 7.70 E - 02 4.21 E - 04 7.06 E - 18 NA f 6 x 7.06 E - 18 7.06 E - 18 7.06 E - 18 3.31 E - 20 NA f 7 x 3.31 E - 20 1.06 E - 17 3.31 E - 20 3.31 E - 20 NATable 5
Time-complexity comparison for various algorithms (time taken by each algorithm for single run in seconds).
Function DE ABC BA FPA EFPA f 1 x 1.284464 2.527247 1.190053 2.328157 1.965927 f 2 x 15.98023 15.62962 19.02841 15.22211 14.80168 f 3 x 1.185347 2.470673 1.047533 2.183295 1.900515 f 4 x 3.022755 5.167423 4.165333 2.183153 2.030502 f 5 x 2.848487 5.988428 2.974728 3.755994 3.498411 f 6 x 1.667705 3.186360 2.261759 2.346810 2.383599 f 7 x 1.693011 3.265792 2.087382 3.384049 2.999886


## 4.2. Results of LAA

A symmetrical LAA consists of2N elements withN elements on both sides of the origin and arranged in a straight line as shown in Figure 1. The nth element is excited with a current of amplitude In and phase φn. The position of nth element is represented by xn. The array factor (AF) of such an antenna is given by [1](9)AF∅=2∑n=1NIncos⁡kxncos⁡∅+φn.Here k is the wave number and is given by 2π/λ. The symmetric nature of the array reduces the computational cost as onlyN elements are needed to be optimized. Moreover, these symmetric arrays give symmetric radiation pattern which is desired for number of applications.Figure 1
Geometry of a linear antenna array.The AF for LAA in (9) is a function of excitation currents (both In and φn) and positions xn. If current excitations are held uniform (i.e., In=1 and φn = 0°), then the LAA is function of xn only and array is called nonuniform LAA. Hence for nonuniform LAA, AF reduces to (10)AFϕ=2∑n=1Ncos⁡kxncos⁡ϕ⁡.The design of nonuniform LAA has gained importance in the past and several researchers have synthesized it using different optimization techniques [3, 7, 9, 11–21]. The array factor of a nonuniform array is aperiodic by virtue of nonuniform spacing between the elements. This aperiodic nature of arrays helps in obtaining lower SLL with lesser number of elements for a given antenna size. Furthermore, the uniform element excitations help in reducing the feed network cost and complexity. However, the relationship between the array factor and element spacing is nonconvex and nonlinear. So, this makes the design of nonuniform arrays difficult.For directing antenna arrays energy in a specific direction, it is desired that its radiation pattern should have low side lobes. Moreover, nulls are also required in the direction of interference. The desired radiation pattern can be achieved by suitably adjusting the positions of antenna elements for nonuniform LAA. The adjustment of positions is a tricky and nonlinear problem. So, optimization algorithms are well suited for this type of problems. For applying optimization to antenna problem, it is imperative to define a fitness function which is given as(11)fit=SLL+α∗max⁡0,BW-BWd-1+α∗∑k=1Kmax⁡0,AFdBϕk-NudB. In the above fitness function, SLL is the side lobe level, AFdB is the array factor in decibels, BW and BWd are the calculated and desired beamwidth of the main lobe, ϕk gives the direction of the kth null, NudB is the required null depth in dB, K is the number of the required nulls, and α is a very large number. The fitness function given in (11) uses a penalty method which penalizes any antenna design which does not meet the required constraints of beamwidth and nulls (if desired). This is achieved by taking value α to be very large. The BW here is obtained computationally from the radiation pattern data.


### 4.2.1. Linear Antenna Optimization

This subsection presents the application of EFPA to optimization of nonuniform LAA. In the first example, a 12-element nonuniform antenna is taken for optimization. The goal of optimization is to minimize the SLL in the regionϕ∈ [98°,180°]. Because of symmetry there are only six element positions which are to be optimized. The number of generations for EFPA are taken as 1000. The population size taken as 20 in order to make fair comparison between the results of FPA [21] and EFPA. The total number of function evaluations are 20 ∗ 1000 = 2000 for both the FPA [21] and EFPA. As the aim of the optimization is to suppress the SLL only, so the value of α is taken as 0 in the fitness equation of (11). The EFPA algorithm is run for 30 times and best results are listed in this work. The optimum positions obtained using EFPA are shown in Table 6. The performance of EFPA is contrasted against other popular algorithms in terms of SLL in Table 7. The minimum SLL obtained by EFPA is −21.07 dB and is better than GA [10], FiADE [15], TS [10], MA [10], FPA [21] and PSO [15]. The comparison of convergence curve between EFPA and FPA is shown in Figure 2 which clearly shows faster convergence of EFPA than the FPA [21]. The FPA converged in about 700 iterations whereas EFPA took only 225 iterations to converge to the final solution. This is due to enhanced exploration due to Cauchy operator and better exploitation in EFPA. The radiation pattern of EFPA antenna array is shown in Figure 3. For comparison, the radiation patterns of GA [10], FiADE [15], and PSO [15] are also shown in Figure 3.Table 6
Geometry of 12-element LAA (with respect toλ/2) obtained using EFPA.
Element 1 2 3 4 5 6 Position 0.3765 0.9734 1.7376 2.5051 3.4827 4.6744Table 7
Performance of different algorithms for 12-element LAA.
Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPA SLL (dB) −18.96 −19.10 −18.77 −17.90 −20.764 −18.40 −21.07Figure 2
Convergence plots of 12-element LAA.Figure 3
Radiation plot of 12-element EFPA optimized LAA.In the next example, a 22-element nonuniform LAA is optimized for reducing SLL in the regionϕ∈[98°, 180°] and also imposing a null in the direction of 99°. For achieving the required pattern, the value of NudB is set to −60 dB. It is established fact that as SLL is decreased the width of the main lobe in radiation pattern increases. So, for this design, the desired beamwidth BWd is taken as 18°. The value of α is taken as 106 in fitness function. The optimum positions are shown in Table 8. The results are compared in Table 9 which clearly indicate better performance of EFPA as against BSA [18], MOEA/DE [14], HSA [16], PSO [15], GA [10], MA [10], TS [10], and FPA [21]. The SLL of EFPA antenna array is −26.31 dB which is much lower than that obtained using other algorithms. Furthermore, the null in the direction of 99° is −75.58 dB which is less than the desired value of −60 dB and deeper than MOEA/DE [14], GA [7], PSO [3] and MA [10]. In order to compare the performance of EFPA and FPA in terms of convergence speed, FPA is also run for the same objective with same population size. The convergence graphs of EFPA and FPA are plotted in the Figure 4. The convergence characteristics clearly substantiates the superior performance of EPFA in terms of faster convergence speed. The radiation plots of EFPA, HSA [16], BSA [18], and MOEA/DE [14] optimized arrays are shown in Figure 5.Table 8
Geometry of 22-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 Position 0.4937 0.7965 1.2153 2.1313 2.5003 3.1327 3.9654 4.5877 5.5435 6.6391 8.0603Table 9
Performance of different algorithms for 22-element LAA.
Algorithm BSA [18] TS [10] MOEA/D-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPA SLL (dB) −23.54 −17.17 −20.93 −23.28 −20.68 −15.73 −18.11 −23.81 −26.31  Null (dB) (99°) −104.61 −67.94 −69.64 −103.3 −49.94 −54.29 −73.92 −101.71 −75.58Figure 4
Convergence plots of 22-element LAA.Figure 5
Radiation plot of 22-element EFPA optimized LAA.In order to show the capability of EFPA in synthesizing the radiation pattern with multiple nulls, a 28-element nonuniform LAA is optimized for SLL reduction in the regionϕ∈[100°, 180°] and having nulls in the directions of 120°, 122.5°, and 125°. The desired beamwidth is set to the value of 8.35° with tolerance of ±1°. The optimum geometry obtained using EFPA is shown in Table 10. The comparison of convergence curves between EFPA and BBO [17], CS [17], DE [17], FA [17], and COA [17] is given in Figure 6 which shows that EFPA reaches the optimum solution in less number of iterations. The comparison of results in terms of SLL and null values achieved is given in Table 11 which again clearly indicates the superior performance of EFPA as against other methods. The SLL of EFPA is −22.90 dB which is lower by 1 dB, 9.24 dB, 9.3 dB, 1.04 dB, 8.26 dB, 9.24 dB, 1.27 dB, and 0.30 dB than BSA [18], QPM [3], COA [17], ACO [13], GA [7], CLPSO [20], and FPA [21], respectively. The null values are also equal to or less than −60 dB. Figure 7 shows the comparison of QPM [3], HSA [16], BSA [18], and EFPA optimized array radiation plots.Table 10
Geometry of 28-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Position 0.5880 1.2142 2.5291 2.8291 4.1205 4.9374 6.1006 6.9646 8.1374 9.5186 10.7399 12.4399 14.1399 15.8399Table 11
Performance of different algorithms for 28-element LAA.
Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −21.90 −13.24 −13.6 −21.86 −14.64 −13.60 −21.63 −22.60 −22.90 Null (dB)(120°) −82.49 −48.49 −52.74 −60.08 −52.74 −59.25 −60.04 −78.45 −60 Null (dB)(122.5°) −93.59 −48.35 −51.66 −60.05 −59.20 −75.53 −60.01 −96.51 −60 Null (dB)(125°) −80.49 −89.3 −61.46 −60.10 −43.58 −62.52 −60.00 −82.58 −67.66Figure 6
Convergence plots of 28-element LAA.Figure 7
Radiation plot of 28-element EFPA optimized LAA.The last design example illustrates the use of EFPA to optimize the element positions for minimization of SLL and null placement for 32-element LAA. The SLL is to be minimized in the regionϕ∈ [93°, 180°] and null is to be placed in the direction of 99°. The required beamwidth BWd is 7.1°. The plot of convergence for EFPA is shown in Figure 8. Also for the sake of comparison, the convergence plots of BBO [17], CS [17], DE [17], FA [17], and COA [17] are shown in the same figure. The element positions for EFPA optimized array are shown in Table 12. Table 13 compares the performance of different optimization methods for design of 32-element LAA. The SLL of proposed EFPA antenna array is −23.73 dB which is least in Table 13. The reduction is considerable as compared to BSA [18], QPM [3], PSO [3], HSA [16], and GA [7] methods. (Figure 9).Table 12
Geometry of 32-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Position 0.5268 1.1101 2.2759 2.9840 3.8883 4.5802 5.7817 6.6474 7.7093 8.8370 9.7490 11.0885 12.6565 14.3043 16.0043 17.7043Table 13
Performance of different algorithms for 32-element LAA.
Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −20.50 −18.80 −19.51 −17.73 −22.83 −16.24 −22.75 −23.10 −23.73 Null (dB)(99°) −107.50 −62.20 −88.08 −34.74 −62.63 −62.94 −60 −130.60 −60.00Figure 8
Convergence plots of 32-element LAA.Figure 9
Radiation plot of 32-element EFPA optimized LAA.


## 4.2.1. Linear Antenna Optimization

This subsection presents the application of EFPA to optimization of nonuniform LAA. In the first example, a 12-element nonuniform antenna is taken for optimization. The goal of optimization is to minimize the SLL in the regionϕ∈ [98°,180°]. Because of symmetry there are only six element positions which are to be optimized. The number of generations for EFPA are taken as 1000. The population size taken as 20 in order to make fair comparison between the results of FPA [21] and EFPA. The total number of function evaluations are 20 ∗ 1000 = 2000 for both the FPA [21] and EFPA. As the aim of the optimization is to suppress the SLL only, so the value of α is taken as 0 in the fitness equation of (11). The EFPA algorithm is run for 30 times and best results are listed in this work. The optimum positions obtained using EFPA are shown in Table 6. The performance of EFPA is contrasted against other popular algorithms in terms of SLL in Table 7. The minimum SLL obtained by EFPA is −21.07 dB and is better than GA [10], FiADE [15], TS [10], MA [10], FPA [21] and PSO [15]. The comparison of convergence curve between EFPA and FPA is shown in Figure 2 which clearly shows faster convergence of EFPA than the FPA [21]. The FPA converged in about 700 iterations whereas EFPA took only 225 iterations to converge to the final solution. This is due to enhanced exploration due to Cauchy operator and better exploitation in EFPA. The radiation pattern of EFPA antenna array is shown in Figure 3. For comparison, the radiation patterns of GA [10], FiADE [15], and PSO [15] are also shown in Figure 3.Table 6
Geometry of 12-element LAA (with respect toλ/2) obtained using EFPA.
Element 1 2 3 4 5 6 Position 0.3765 0.9734 1.7376 2.5051 3.4827 4.6744Table 7
Performance of different algorithms for 12-element LAA.
Algorithm FiADE [15] MA [10] GA [10] PSO [15] FPA [21] TS [10] EFPA SLL (dB) −18.96 −19.10 −18.77 −17.90 −20.764 −18.40 −21.07Figure 2
Convergence plots of 12-element LAA.Figure 3
Radiation plot of 12-element EFPA optimized LAA.In the next example, a 22-element nonuniform LAA is optimized for reducing SLL in the regionϕ∈[98°, 180°] and also imposing a null in the direction of 99°. For achieving the required pattern, the value of NudB is set to −60 dB. It is established fact that as SLL is decreased the width of the main lobe in radiation pattern increases. So, for this design, the desired beamwidth BWd is taken as 18°. The value of α is taken as 106 in fitness function. The optimum positions are shown in Table 8. The results are compared in Table 9 which clearly indicate better performance of EFPA as against BSA [18], MOEA/DE [14], HSA [16], PSO [15], GA [10], MA [10], TS [10], and FPA [21]. The SLL of EFPA antenna array is −26.31 dB which is much lower than that obtained using other algorithms. Furthermore, the null in the direction of 99° is −75.58 dB which is less than the desired value of −60 dB and deeper than MOEA/DE [14], GA [7], PSO [3] and MA [10]. In order to compare the performance of EFPA and FPA in terms of convergence speed, FPA is also run for the same objective with same population size. The convergence graphs of EFPA and FPA are plotted in the Figure 4. The convergence characteristics clearly substantiates the superior performance of EPFA in terms of faster convergence speed. The radiation plots of EFPA, HSA [16], BSA [18], and MOEA/DE [14] optimized arrays are shown in Figure 5.Table 8
Geometry of 22-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 Position 0.4937 0.7965 1.2153 2.1313 2.5003 3.1327 3.9654 4.5877 5.5435 6.6391 8.0603Table 9
Performance of different algorithms for 22-element LAA.
Algorithm BSA [18] TS [10] MOEA/D-DE [14] HSA [16] PSO [15] GA [10] MA [10] FPA [21] EFPA SLL (dB) −23.54 −17.17 −20.93 −23.28 −20.68 −15.73 −18.11 −23.81 −26.31  Null (dB) (99°) −104.61 −67.94 −69.64 −103.3 −49.94 −54.29 −73.92 −101.71 −75.58Figure 4
Convergence plots of 22-element LAA.Figure 5
Radiation plot of 22-element EFPA optimized LAA.In order to show the capability of EFPA in synthesizing the radiation pattern with multiple nulls, a 28-element nonuniform LAA is optimized for SLL reduction in the regionϕ∈[100°, 180°] and having nulls in the directions of 120°, 122.5°, and 125°. The desired beamwidth is set to the value of 8.35° with tolerance of ±1°. The optimum geometry obtained using EFPA is shown in Table 10. The comparison of convergence curves between EFPA and BBO [17], CS [17], DE [17], FA [17], and COA [17] is given in Figure 6 which shows that EFPA reaches the optimum solution in less number of iterations. The comparison of results in terms of SLL and null values achieved is given in Table 11 which again clearly indicates the superior performance of EFPA as against other methods. The SLL of EFPA is −22.90 dB which is lower by 1 dB, 9.24 dB, 9.3 dB, 1.04 dB, 8.26 dB, 9.24 dB, 1.27 dB, and 0.30 dB than BSA [18], QPM [3], COA [17], ACO [13], GA [7], CLPSO [20], and FPA [21], respectively. The null values are also equal to or less than −60 dB. Figure 7 shows the comparison of QPM [3], HSA [16], BSA [18], and EFPA optimized array radiation plots.Table 10
Geometry of 28-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Position 0.5880 1.2142 2.5291 2.8291 4.1205 4.9374 6.1006 6.9646 8.1374 9.5186 10.7399 12.4399 14.1399 15.8399Table 11
Performance of different algorithms for 28-element LAA.
Algorithm BSA [18] QPM [3] PSO [3] COA [17] ACO [13] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −21.90 −13.24 −13.6 −21.86 −14.64 −13.60 −21.63 −22.60 −22.90 Null (dB)(120°) −82.49 −48.49 −52.74 −60.08 −52.74 −59.25 −60.04 −78.45 −60 Null (dB)(122.5°) −93.59 −48.35 −51.66 −60.05 −59.20 −75.53 −60.01 −96.51 −60 Null (dB)(125°) −80.49 −89.3 −61.46 −60.10 −43.58 −62.52 −60.00 −82.58 −67.66Figure 6
Convergence plots of 28-element LAA.Figure 7
Radiation plot of 28-element EFPA optimized LAA.The last design example illustrates the use of EFPA to optimize the element positions for minimization of SLL and null placement for 32-element LAA. The SLL is to be minimized in the regionϕ∈ [93°, 180°] and null is to be placed in the direction of 99°. The required beamwidth BWd is 7.1°. The plot of convergence for EFPA is shown in Figure 8. Also for the sake of comparison, the convergence plots of BBO [17], CS [17], DE [17], FA [17], and COA [17] are shown in the same figure. The element positions for EFPA optimized array are shown in Table 12. Table 13 compares the performance of different optimization methods for design of 32-element LAA. The SLL of proposed EFPA antenna array is −23.73 dB which is least in Table 13. The reduction is considerable as compared to BSA [18], QPM [3], PSO [3], HSA [16], and GA [7] methods. (Figure 9).Table 12
Geometry of 32-element (with respect toλ/2) LAA obtained using EFPA.
Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Position 0.5268 1.1101 2.2759 2.9840 3.8883 4.5802 5.7817 6.6474 7.7093 8.8370 9.7490 11.0885 12.6565 14.3043 16.0043 17.7043Table 13
Performance of different algorithms for 32-element LAA.
Algorithm BSA [18] PSO [3] HSA [16] QPM [3] CS [19] GA [7] CLPSO [20] FPA [21] EFPA SLL (dB) −20.50 −18.80 −19.51 −17.73 −22.83 −16.24 −22.75 −23.10 −23.73 Null (dB)(99°) −107.50 −62.20 −88.08 −34.74 −62.63 −62.94 −60 −130.60 −60.00Figure 8
Convergence plots of 32-element LAA.Figure 9
Radiation plot of 32-element EFPA optimized LAA.


## 5. Conclusions

FPA is a novel nature inspired algorithm which mimics the pollination of flowers. But it has some weaknesses like poor exploitation capability and slow convergence speed. So, to overcome these problems an improved version of FPA called EFPA has been proposed in this paper. Three modifications in the basic FPA have been proposed in the enhanced version. The modifications in EFPA help in providing better search capability and help it to escape local minima. The proposed algorithm has been tested on seven benchmark functions. The results show that EFPA is able to find global optima of most of the benchmark functions. Moreover, the EFPA is utilized for pattern synthesis of nonuniform LAA. Four different antenna arrays of different sizes and different design constraints have been taken. When EFPA is used for SLL minimization only, it reaches the optimum result in less number of iterations than FPA. For multiobjective design with SLL suppression, null, and beamwidth control, EFPA obtains better performance than the results of well-known algorithms reported in literature. The results indicate the potential ability of EFPA to be used for optimization for other antenna designs.



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*Source: 7158752-2017-02-20.xml*