In this paper a pattern synthesis method based on Evolutionary Algorithm is presented. A Flat-top beam pattern has been generated from a concentric ring array of isotropic elements by finding out the optimum set of elements amplitudes and phases using Differential Evolution algorithm. The said pattern is generated in three predefined azimuth planes instate of a single phi plane and also verified for a range of azimuth plane for the same optimum excitations. The main beam is steered to an elevation angle of 30 degree with lower peak SLL and ripple. Dynamic range ratio (DRR) is also being improved by eliminating the weakly excited array elements, which simplify the design complexity of feed networks

Bhattacharjee, A. K.

Kola, K. S.

Mandal, D.

Roy, V. P.

Tewary, J.

Advanced Electromagnetics (E-Journal)

ADVANCED ELECTROMAGNETICS, VOL. 5, NO. 3, DECEMBER 2016Synthesis of Steered Flat-top Beam Pattern Using Evolutionary AlgorithmDebasis Mandal1∗, Kalyan S. Kola 2, Jyotirmay Tewary 3,Ved P. Roy4, and Anup K. Bhattacharjee51,2,3,4Bengal College of Engineering and Technology, Durgapur, India5National Institute of Technology, Durgapur, India*corresponding author, E-mail: deb.mandal22@gmail.comAbstractIn this paper a pattern synthesis method based on Evolu-tionary Algorithm is presented. A Flat-top beam patternhas been generated from a concentric ring array of isotropicelements by finding out the optimum set of elements am-plitudes and phases using Differential Evolution algorithm.The said pattern is generated in three predefined azimuthplanes instate of a single ϕ plane and also verified for arange of azimuth plane for the same optimum excitations.The main beam is steered to an elevation angle of 30o withlower peak SLL and ripple. Dynamic range ratio (DRR) isalso being improved by eliminating the weakly excited ar-ray elements, which simplify the design complexity of feednetworks.1. IntroductionThe scanned shaped beam has a wide range of applicationsin radar and satellite based communications. These shapedbeams faced the problem of high side lobe and ripple. Theconcentric circular ring array antenna (CCRA) is very effi-cient to steer the beam patterns, which enables to increasethe capacity in directional communication systems. Vari-ous techniques have been reported in the literature as givenbelow [3–8].Azevedo proposed a technique to generate shaped beampatterns from a linear array antenna through the controlof nonuniformly samples of the array factor, both in am-plitude and phase based on Fast Fourier Transform (FFT)[3]. By minimizing the First Null Beam Width (FNBW)and peak side lobe level (SLL) the best beam directivityhas been obtained by Wooh and Shi to determine the opti-mum transducer parameters [4] and also investigated thatthe inter-element spacing and the number of elements playsan important role in determining transducer performances.Mandal et al. generates shaped beams from concentric ringarray antenna using discrete excitations in a range of pre-defined ϕ planes with zero elevation angle [5]. Reyna et al.described the design procedure of steerable concentric ringsarray for low side lobe level using Genetic Algorithm [6].A new steerable reflector which can electronically steer areflected beam over +/ − 40o in two dimensions for bothpolarizations have been proposed and implemented [7] bySievenpiper et al. A different approach proposed by Mandalet al. for generating shaped beams (Flat-top and cosec2)−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5X [in terms of λ]Y [in terms of  λ]Figure 1: Geometry of concentric ring array Antennahave been carried out using Differential Evolution Al-gorithm, which improves the dynamic range ratio by elimi-nating weekly excited array elements [8].2. Problem FormulationIn this paper the peak SLL and ripple of the steered Flat-topbeam pattern in three different planes of (θ0 = 30o, ϕ =0o) , (θ0 = 30o, ϕ = 15o) and (θ0 = 30o, ϕ = 30o) arereduced by finding out the optimum set of elements ampli-tudes and phases using Differential Evolution (DE) algo-rithm [9–12]. Scanning any beam from an array requiresprogressive phase gradient considered as 30o. Patterns arealso generated in three arbitrarily chosen azimuth plane us-ing same excitations, where the pattern retains their shapewith some minor variations. Weakly excited array elementsare removed (or turn off) from the optimized array withoutdistorting the obtained pattern to achieve better DynamicRange Ratio (DRR).A concentric ring array of isotropic elements shown inFig. 1 is considered. The far field pattern of the arrayscanned to a specified angle can be defined as [1, 2, 6].AF (θ, ϕ) =M∑m=1Nm∑n=1Imnejkrm[sinθcos(ϕ−ϕmn)−sinθ0cos(ϕ0−ϕmn)](1)where, M= number of concentric rings;Nm= number of isotropic elements in m-th ring;Imn excitation amplitude of mn-th element;rm radius of the m-th ring;dm inter element arc spacing of m-th circle;k = 2pi/λ, represents wave number;λ is the wave length;θ,ϕ Elevation and azimuth angle;ϕmn= 2npiλ is the angular location of the mn-th elementwith, 1 ≤ n ≤ Nm and(θ0, ϕ0) = steering angle.Table 1: Ring radius and number of elements per ringM 1 2 3 4Nm 5 10 15 20rm(λ) 0.3979 0.7958 1.1935 1.5915The fitness function of the Flat-top beam pattern for thisapproach is defined as:F (ρ) = k1{peakSLLd − maxθ∈A (AF ρdB(θ, ϕ))}2 + k2 ×4(2)where,4 is defined for Flat-top beam pattern as:4 =∑θi∈{−15o+θo to 150+θo}|AF ρdB(θi, ϕ)−D(θi, ϕ)|(3)In equation, 2 and 3 ϕ ∈ (0o to 30o) at θ = 30o plane.ρ is the unknown parameter set responsible for the de-sired beam patterns. Which is defined as follows:ρ = {Imn, αmn} ; 1 ≤ m ≤M and 1 ≤ n ≤ Nm(4)peakSLLd are the desired value of peak SLL for Flat-top beam pattern. A is the sidelobe region for the pattern.DdB (θ, φ) is the desired pattern shown in Fig. 2 at ϕ =0o, 15o and 30o plane. The range of θi for Flat-top beampattern is −15o + θo to 15o + θo .k1 and k2 are the weighting factors. The fitness functionhas to be minimized by finding out optimum set of ampli-tudes and phases using Differential Evolution (DE) Algo-rithm.3. Differential Evolution AlgorithmDifferential Evolution Algorithm (DE) is one of the bestpopulation based evolutionary Algorithm introduced byStorn and Price in the year of 2005, which is able to find out−90 −75 −60 −45 −30 −15 0 15 30 45 60 75 90−20−15−10−50θ in degrees|D(θ,ϕ)|ϕ ∈ 0o− 30oFigure 2: Desired scanned Flat-top beam patterntrue global minima with less control parameters and fasterconvergence. After a randomly initialization and generationin D dimensional search space three main steps involvedin DE Algorithm are [8–12] namely Mutation, Crossoverand Selection. In Mutation process mutant vector Vi,G isgenerated corresponding to the target vector Xi,G. Dur-ing crossover operation the trial vector Ui,G is generatedusing mutant vector Vi,G and target vector Xi,G. Finally,in Selection process the objective function values are com-pared with each trial vector and those produces the best fit-ness function retain for the next generation. These stepsare being repeated till the predefined value of generation,results the best solution of the objective function in the cur-rent population (Xbest,G).Pseudo code of Differential Evolution Algorithm (DE)1. Initialize G = 0 and rand (Xi,G)2. Compute Xbest,G until G > Gmax2.1 initialize i = 12.2 Vi,G = Xbest,G + F.(Xr1,G −Xr2,G)2.3 Uji,G = Vji,G, where rand(0, 1) 6 CRotherwise, xji,G2.4 check f(Ui,G < f(Xi,G if yesgo to 2.4.1 else 2.4.22.4.1 f(Ui,G) = f(Xi,G+1))2.4.2 Xi,G = Xi,G+12.5 check i = NP , if yes go to 2.5.1else go to 2.5.22.5.1 G = G+ 12.5.2 i = i+ 13. Loop back to step 2.2The population size, scale factor (F ), and crossoverrate (CR) of DE are chosen as 50, 0.8, and 0.2 re-spectively [8–12],. The DE scheme used in this paperis “DE/best/1/bin” with maximum iteration number of5000.87−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50 ϕ = 0 degree planeθ in degrees|AF| dB−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50ϕ = 15 degree planeθ in degrees|AF| dB−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50 ϕ = 30 degree planeθ in degrees|AF| dB  Desired flattop beam patternsObtained flattop beam patterns(c)(b)(a)Figure 3: Scanned Flat-top patterns after removing theweakly excited elements from the array (a) ϕ = 0o (b)ϕ = 15o (c) ϕ = 30o4. Simulation ResultsA concentric ring array of 50 isotropic elements with 0.5λinter-element spacing has been considered (Figure 1).Table 2: Desired and obtained result for scanned Flat-toppattern in predefined ϕ planesθo ϕ Design Flat-top Pattern(Degree) (Degree) Parameters Desired Obtained30oϕ = 0oPeak SLL (dB) -15.00 -14.26754 (dB) 0.00 9.8496ϕ = 15oPeak SLL (dB) -15.00 -13.59424 (dB) 0.00 9.5019ϕ = 30oPeak SLL (dB) -15.00 -13.154 (dB) 0.00 9.8783The number of elements present in each ring with theirrespective ring radius is given in Table 1. Figure 2 andFigure 3 depict the desired and obtained Flat-top beampattern in all predefined azimuth planes considering thescanning angle of 30o respectively.Figure 4: Optimum excitations (a) Amplitudes and (b)PhasesThe design specification of steered Flat-top beam pat-tern with its corresponding desire and obtained results aregiven in Table 2. From Table 2 it can be observed clearlythat the peak SLL for all three predefined azimuth planesare very much closer to its desired value.Optimum values of excitation for the array elementsobtained using the DE algorithm are presented in Table 3where the elements having amplitudes less than 0.04 con-sidered as ′off ′, by making impedance equal to the sourceimpedance of the fed elements resulting improvement inDRR level. The obtained values of normalized amplitudeshaving a range in between 0.0125 to 1 provide the DRRlevel 80. Whereas, by making the elements ′off ′ havingamplitudes less than 0.04, the range of amplitudes are nowbecome 0.059 to 1 results improved DRR level 16.949 with-out distorting the shape of the beam. Thus simplify the de-88Table 3: Amplitudes and Phases of the Array ElementsRing Normalized PhasesNo. Amplitudes in Degrees1. 0.603, 0.969, 0.992, 162.72, -180.00, 177.12,1.000, 0.860, -161.64, 178.20,2.1.000, 0, 1.000, 166.68, 0, 99.36,1.000, 1.000, 0, -176.04, 177.48, 0,0.141, 0.834, 1.000, 180.00, 180.00, -180.00,0.970, -180.00,3.1.000, 0, 0.524, -180.00, 0, 179.28,0, 1.000, 0, 0, 113.76, 0,0.321, 1.000, 0.997, 167.40, -87.84, 180.00,0, 1.000, 0, 0, 180.00, 0,0.229, 0.629, 0.371, -76.32, -180.00, 119.16,4.0.985 0.544 0.999 54.36 -45.00 12.600.354 0.999 0.258 54.00 -180.00 142.560.059 0.452 0.982 -115.56 12.96 -48.601.000 0.075 0.107 45.00 31.32 123.840.720 0.930 0 35.64 114.84 00.549 0.271 0.467 -168.84 3.24 37.800.262 0.063 -73.44 -154.08Figure 5: Optimized array after removing weekly excitedelements (green represents “on” and red “off”)sign complexity of feed networks (less number of attenua-tors and phase shifters are required). Figure 4(a) and Fig-ure 4(b) depicts element wise normalized amplitudes andphases respectively.After eliminating weakly excited array elements themodified array geometry is shown in Figure 5. The excita-tions are also being verified by considering three arbitrarilychosen azimuth planes (ϕ = 7.5o, 22.5oand 35o) presentedin Figure 6 where the steered beam pattern has been gener-ated with some minor variations with respect to its desiredparameters illustrated in Table 4.−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50 ϕ = 7.5 degree planeθ in degrees|AF| dB−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50ϕ = 22.5 degree planeθ in degrees|AF| dB−90−75−60−45−30−15 0 15 30 45 60 75 90−25−20−15−10−50ϕ = 35 degree planeθ in degrees|AF| dB  Desired flattop beam patternsObtained flattop beam patterns(a)(b)(c)Figure 6: Scanned Flat-top patterns in arbitrary planes (a)ϕ = 7.5o (b) ϕ = 22.5o (c) ϕ = 35o.Table 4: Obtained results for scanned Flat-top pattern inarbitrary ϕ planesθo ϕ Design Flat-top Pattern(Degree) (Degree) Parameters Obtainedθo = 30oϕ = 7.5oPeak SLL (dB) -16.29954 (dB) 11.3549ϕ = 22.5oPeak SLL -13.29574 (dB) 10.8354ϕ = 35oPeak SLL -11.464 (dB) 14.383789From Table 4 it can be observed that for an arbitraryangle of 7.5o which is in-between the range of predefinedplane 0o to 15o has a value of peak SLL −16.2995 dB andripple (4) 11.3549 dB where as for the arbitrary angle of22.5o the value of peak SLL and ripple (4) are −13.2957dB and 10.8354 dB respectively which is adequate to itsdesired value but for an arbitrary angle of 35o which is be-yond the pre defined range of azimuth plane (ϕ = 30o )the obtained peak SLL and ripple are far deviated from itsdesired value. The convergence curve of DE algorithm hasbeen given in Figure 7.0 1000 2000 3000 4000 50002004006008001000GenerationsBest fitness  DEFigure 7: Convergence curve of Differential Evolution Al-gorithm (DE).5. ConclusionsSynthesis of a Flat-top beam pattern of concentric ring arrayantenna for a scanning angle of 30o has been proposed andassessed using Differential Evolution algorithm. This pat-tern has been generated in three predefined azimuth planesinstead of any particular ϕ plane. Optimum excitationsused for generating the Flat-top beam pattern are also veri-fied for three arbitrarily chosen azimuth angles to illustratethat those excitations are useful not only in some prede-fined ϕ planes rather a range of azimuth plane. For syn-thesis of Flat-top beam patterns; side lobe and ripple areminimized by minimizing properly formulated fitness func-tion using evolutionary Algorithm. Dynamic Range Ratio(DRR) have also been improved by turning off the weaklyexcited array elements. The presented method can also beapplied to synthesize other array geometries.References[1] Balanis, C. A., Antenna Theory, Analysis and design,3rd ed. New York, USA: Jhon Willy & sons, 1997.[2] Elliott, R. S., Antenna Theory & Design, Revised Edi-tion. Wiley-IEEE Press, Dec, 2002.[3] Azevedo, J. A. R, Shaped beam pattern synthesis withnonuniform sample phases, Progress in Electromag-netics Research B, Vol. 05, 77–90, 2008.[4] Wooh, S. C, and Shi, Y, Optimum beam steering oflinear phased arrays, Wave Motion, Vol. 29,No. 03,245–265, 1999.[5] Mandal, D., Chatterjee, A. and Bhattacharjee, A. K.Design of Fully Digital Controlled Shaped Beam Syn-thesis Using Differential Evolution Algorithm, In-ternational Journal of Antennas and Propagation,Vol. 2013, 1–9, 2013.[6] Reyna, A., Panduro, M. A., Covarrubias, D. H., andMendez, A. Design of steerable concentric rings ar-ray for low side lobe level , Scientia Iranica, Vol. 19,No. 03, 727–732, 2012.[7] Sievenpiper, D. F., Schaffner, J. H, Song, H. 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Synthesis of Steered Flat-top Beam Pattern Using Evolutionary Algorithm

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