Digitally Controlled Steered Dual Beam Pattern Synthesis of a Rectangular Planar Array Antenna in a Range of Azimuth Plane Using Evolutionary Algorithms

This paper presents a pattern synthesis method to generate dual-beam patterns of a rectangular planar array of isotropic antennas in a particular scanning angle using Evolutionary Algorithms. The dual-beam patterns are cosec2 pattern and pencil beam pattern, and both the patterns are steered to an elevation angle of 20 degrees (θ = 20◦). Moreover, each pattern is synthesized in three azimuth planes (φ = 0◦, 5◦, and 10◦). The isotropic elements are uniformly spaced, and nonuniform excitations are applied to achieve the desired patterns. These patterns are obtained by applying the optimum set of common elements amplitude and phases for the cosecant-squared pattern only. The optimum 4-bit discrete amplitudes and 5-bit discrete phases are produced using Differential Evolutionary (DE) Algorithm, Genetic Algorithm (GA), Particle Swarm Optimization (PSO) Algorithm, and Firefly Algorithm (FA). These discrete excitations are helpful to reduce the Dynamic Range Ratio (DRR) and the design complexity of the feed networks. The excitations are also verified in a range of arbitrarily chosen azimuth planes. The patterns are generated in the same steering angle with minor variations of the desired parameters. The outcomes established the superiority of DE over PSO, GA, and FA and the effectiveness of the proposed method.


INTRODUCTION
The array antenna having scanning capability is very useful in wireless communication, mainly in radarrelated applications. However, generating the cosec 2 beam and pencil beam at a range of azimuth planes along with an elevation angle faced a high sidelobe with an enormous ripple problem. Various approaches reported in literature for generating array patterns and dual beam patterns are as follows [1][2][3][4][5][6][7][8][9][10][11].
Multiple beam patterns using simulated annealing algorithm [5] have been generated by Diaz et al. Lei et al. proposed and developed a process for generating a cosecant-squared beam pattern of a linear antenna array by incorporating optimum amplitudes and phases of the isotropic array elements by applying the modified least square method [6]. Both uniform and Gaussian distributions of common amplitudes are applied to generate shaped beam patterns (flattop and a cosecant-squared). Different sets of phases are also used for different beam patterns. For finding these excitations, the Woodward-Lawson technique [7] has been introduced by Durr et al. Chatterjee et al. proposed a method for generating dual beams using optimum sets of radial amplitudes and phases of a concentric ring array antenna. These excitations are achieved by using Gravitational Search Algorithm (GSA) [8]. Chatterjee et al. also proposed a Firefly Algorithm (FA) technique for generating a dual-beam pattern of a concentric ring antenna array. The applied elements are isotropic, and the excitations are used in the rings of the CCAA instate of the elements. The expected patterns are generated by applying specific states or combinations of switches [9]. A dual radiation pattern of the rectangular planar array has been generated at different azimuth planes with zero degrees elevation angle using the evolutionary algorithm by Mandal et al. [10]. A flattop and pencil beam is also generated using common amplitudes distribution and phases for shaped beams at zero degrees elevation by Mandal et al. in 2015 [11]. Kenane et al. proposed a method of synthesizing two different non-uniform antenna arrays Linear and Circular, using Dynamic Invasive Weeds Optimization (IWO), which helps to get deep nulls in the directions of interferences with low sidelobe levels [12].
In this paper, a cosec 2 pattern and a pencil beam pattern from a planar array [1][2][3][4] of 90 isotropic elements are obtained by finding out the optimum set of common elements amplitudes for both the patterns and a group of phases for cosec 2 shaped beam using Evolutionary Algorithms. Here 20 • progressive phase has also been applied to scan the beam patterns on that elevation angle. The patterns have been generated in three predefined azimuth planes using the excitations achieved by DE, GA, PSO, and FA. This approach also verifies that the patterns retain their desired parameters within a range of azimuth planes instead of a single φ plane. These are also proved by selecting some arbitrary φ planes and obtained similar patterns by applying the same excitations for each Evolutionary Algorithm with some minor variations. The received amplitudes (4-bit) and phases (5-bit) are both digitals for providing a lower Dynamic Range Ratio (DRR). These discrete excitations are used to shorten the design complexity of the feed network as DRR is less, so less number of attenuators and phase shifters are required. The comparative performance of these four Evolutionary Algorithms, Differential Evolution algorithms (DE), Genetic Algorithm, Particle Swarm optimization algorithms, and Firefly Algorithm (FA) are also analyzed.

PROBLEM FORMULATIONS
A planar array of 90 isotropic elements is considered. The far-field pattern of the array shown in Figure 1 can be written as [1][2][3][4]: I mn e j[kmdx(sin θ cos φ−sin θo cos φo)+kndy(sin θ sin φ−sin θo sin φo)+αmn] Here, I mn is the excitation amplitude of mn-th element; α mn denotes the phase excitation of mn-th element; M and N denote the number of isotropic elements in X and Y directions, respectively; d x and d y = 0.5λ represent the inter element spacing along with X and Y directions; θ and φ are polar and azimuth angle; θ o and φ o are beam pointing angle; k = 2π λ is the wavenumber. The fitness function for the dual-beam pattern (cosec 2 and Pencil Beam) is defined as: where ∆ is defined as: In Equations (2) and ( ρ is the unknown parameter set responsible for the desired beam patterns for this approach. ρ is defined as follows: peakSLL d 1 and peakSLL d 2 are the desired values of peak SLL for cosec 2 and pencil beam patterns. S 1 and S 2 are sidelobe regions for both patterns. D dB (θ, ϕ) is the desired pattern shown in Figure 2 at   Particle Swarm Optimization (PSO), and DE. This Optimization Algorithm DE [13][14][15][16][17][18][19] can find out true global minima with fewer control parameters and achieve convergence quickly for nonlinear problems. DE is an efficient and effective global optimizer that is population-based. DE algorithm has used a very powerful stochastic search technique. In DE, three major parameters like population size, scaling factor, and crossover rate are crucially responsible for algorithm's performance. The flowchart of DE is shown in Figure 3. To minimize a function f (X), where, where D is the search space dimension or number variables. If the population size is considered as N , the population matrix can be expressed as Here g is the generation and n = 1, 2, 3, . . . , N . Initial population is generated randomly between upper bound (x U i ) and lower bound (x U i ). After random initialization and generation in D dimensional search space, the three main steps involved in the DE Algorithm are Mutation, Recombination, and Selection.
In the Mutation process, three different vectors x g r1n , x g r2n , and x g r3n are selected from each parameter vector. Donor vector v g+1 n can be written as follows. v g+1 F is called Scale Factor, and the value is between 0 and 1. During the Recombination operation the trial vector u g+1 n,i is generated using target vector x g n,i and donor vector v g+1 n,i . Finally, in the Selection process, the objective function values are compared with each target vector x g n,i and trial vector u g+1 n,i . Those who produce the lowest function value, i.e., the best fitness function, are selected for the next generation. These steps are repeated until it reaches the predefined value of generation. The result, i.e., the best solution of the objective function in the current population, is expressed as X best,G .
For solving this optimization problem, the population size is considered as 50, the scale factor (F ) taken as 0.8, and the crossover rate (CR) of DE chosen as 0.2, whereas the scheme used in this nonlinear problem is DE/best/1/bin with maximum iteration number of 2000 [13][14][15][16][17][18][19].

An Overview of Particle Swarm Optimization Algorithm (PSO)
With the inspiration of social behavior of birds flock, fish school, etc., a population-based stochastic optimization technique is developed by Kennedy and Eberhart in 1995. The name of this optimization algorithm is Particle Swarm Optimization. It utilizes a population of potential solutions to probe the search space concurrently. The individuals are called the particles, and the population is called the swarm in PSO. In the search space, each particle is associated with a position and velocity. As the swarm is a set of N particles, it can be expressed as: s = {x 1 , x 2 , . . . , x n }, and the position and velocity of a particle in n-dimensional problem is defined as [20][21][22][23][24][25] The velocity and positions of the particles are swaps according to the iterations, and these swaps are controlled by the execution of the particles itself and that of the other particles. The algorithm can be summarized as follows: Step 1: Initialization The position and velocity of N number of particles are initialized randomly at iteration t using the following equations: for i = 1, 2, 3, . . . , N and j = 1, 2, 3, . . . , n (11) where x min and x max are the lower and upper bounds of the problem. Within the range [0, 1] rand is an uniform random variable.
Step 2: Fitness evolution at the current iteration Compute the fitness at the current generation for each particle in the swarm, i.e., evaluate f (x i ) for i = 1, 2, . . . , N .
Step 3: Compute pbest i and gbest At current iteration, pbest i is the best previous position of the i-th particle, and gbest is the global best position among all the particles present in the swarm using the following equations [20][21][22][23][24][25][26][27] Step 4: Update particles position and velocity Update the velocity and position at iteration t of the particles using the following equations for i = 1, 2, . . . , N and j = 1, 2, . . . , n. Two uniformly distributed random variables rand 1 and rand 2 have a range in between [0, 1]. c 1 and c 2 weighting factors are called cognitive and social parameter. w is known as the inertia weight. It is expressed as: where w min and w max are set to 0.4 and 0.9, and T is the maximum iteration, which restrict the velocity of the particles such that |v ij (t + 1)| ≤ v max , for i = 1, 2, . . . , N and j = 1, 2, . . . , n. This is termed as velocity clamping and is accomplished using the equation given bellow: Step 5 Repeat steps 2-5 until iterations reach their maximum limit. Retain the values of gbest as the final result. The swarm size of PSO is taken as 50 with the randomly chosen initial population. The values of C 1 and C 2 are considered as 2 [20][21][22][23][24][25][26][27]. The time-varying inertia weight (w) is chosen in the way where the merits are decreased linearly from 0.9 to 0.4. For every particle on the dth dimension the maximum admissible velocity is considered as 0.9r d , where r d is the deflection between the maximum and minimum possible utility of the decision variables on this dth dimension. At a maximum iteration of 2000, the termination condition is considered.

Parametric Setup of Genetic Algorithm (GA) and Firefly Algorithm (FA)
In GA, the Population size is considered 50, and the two-point crossover schemes are selected. The probabilities of Crossover and mutation are chosen as 0.08 and 0.01. In the Selection process, the Roulette Wheel scheme is thought about for this proposed problem, and the termination condition is chosen as a maximum iteration of 2000. Other parametric setups of the Genetic Algorithm are selected from the recommendation given in [28][29][30][31][32]. In FA number of fireflies 50 is taken, β 0 = 0.20, γ = 0.25, α = 1, and Search space dimension and Choice of initial population are considered as 48 and random, at a maximum iteration of 2000 [9].

RESULTS
A planar array of 90 uniformly placed isotropic elements has been considered. M number of elements are uniformly placed along X direction, and N number of elements are placed along Y direction. where M = 10 and N = 5 have been chosen. The inter element spacing is considered as 0.5λ, i.e., d x = 0.5λ and d y = 0.5λ shown in Figure 1.
The design parameters of the steered dual-beam patterns along with their corresponding obtained results are shown in Table 1. From Table 1, it has been seen that by applying DE the obtained values of the peak side lobe level for the cosec 2 beam pattern in three different predefined planes are −19.4309 dB, −19.2812 dB, and −19.3472 dB corresponding to their desired value of −20.00 dB. The parameter ripple (∆) is incorporated to measure the total deviation between the obtained (shown in Figure 4) and desired patterns (shown in Figure 2 Similarly, Table 1 shows the obtained and desired values of peak SLL and ripple ∆ using Genetic   Figure 4, Figure 5, Figure 6, and Figure 7 depict the obtained steered dual-beam patterns using DE, GA, PSO, and FA, respectively. The element wise amplitudes and phases of the array elements obtained using DE for generating the beam patterns are shown in Figure 8, and their corresponding values of normalized amplitudes and phases are written in Table 3. These normalized amplitudes have 2 4 , i.e., 16 different levels between 0.625 and 1. Similarly, the phases have 2 5 , i.e., 32 different levels between −180 • and +180 • . Figure 9 and Figure 10 show the 4-bit discrete amplitudes and 5-bit discrete phases using GA and PSO. The exact value of discrete 4-bit amplitudes and 5-bit phases are given in Table 4 and Table 5. In Figures 11, 12, 13, and 14, the beam patterns are generated in three arbitrarily chosen azimuth planes for the same excitations with some minor variation. In each figure, the first arbitrary azimuth angle is chosen as 2.7 degrees (0 • < 2.7 • < 5 • in between the predefined φ plane); the second is 7.8 degrees (within the predefined φ plane 5 • and 10 • ); and the third is 12.1 degrees (> 10 • beyond the predefined azimuth plane). Figure 4 shows the array pattern in predefined azimuth planes whereas Figure 11 shows the pattern in arbitrarily chosen azimuth plane, and in both the cases, the optimum excitations are generated using DE. The obtained cosec 2 beam patterns follow the desired beam pattern shown in Figure 2 within the coverage range of elevation angle (20 • to 50 • ).
Similarly, (Figure 5, Figure 12 ), (Figure 6, Figure 13 ) and (Figure 7, Figure 14 ) have nearly equal values of the design parameters obtained by using the optimum excitations produced by GA, PSO, and FA for predefined azimuth angle and arbitrary plane.
In Table 2, the values of design parameters are mentioned for the arbitrary angles for all four EAs. The obtained values of design parameters for arbitrarily selected φ planes and predefined φ planes are comparable. So for this array synthesis method, it is not essential to consider all the azimuth planes rather some predefined azimuth planes which ensure a range where the pattern retains the desired characteristic. Figure 15 shows the convergence curves of DE, GA, PSO, and FA. From these convergence curves it is clearly observed that DE is superior to GA, PSO, and FA as fitness value of DE is less than the others. The Computations have been done in MATLAB 2015a with core 2 duo processor, 3 GHz, 4 GB RAM. Table 6 below

CONCLUSION
A steered dual-beam planar array antenna has been synthesized in three different azimuth angles. Two different beam patterns (cosec 2 and a pencil beam) are scanned to a particular elevation angle and again synthesized in three predefined azimuth planes using three well known Evolutionary Algorithms. Each Evolutionary Algorithm generates optimum 4-bit discrete amplitudes and 5-bit discrete phases to achieve the desired parameters of the patterns. After selecting and achieving the similar pattern in arbitrary planes, the presented method shows its capability to produce the same beam pattern not only in the predefined azimuth planes rather a range of azimuth planes. The design parameters like peak side lobe level (peak SLL) and ripple ∆ are reduced by finding the optimum set of array excitations using DE, GA, and PSO Algorithm. The Dynamic Range Ratio (DRR) is also reduced by using these digital excitations, which results in the smaller number of attenuators and phase shifters in the feed network. This pattern synthesis method also ensures that the desired patterns retain their desired specifications with some minor variations in a range of azimuth plane on the scanning angle also rather in a particular predefined φ plane with zero elevation angle. An acceptable concurrence between the desired and obtained outcomes validated the proposal, and the introduced method can also be used to synthesize other array geometries.