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Antenna Array Synthesis with Chebyshev-GeneticAlgorithm MethodA.HammamiR.Ghayoulaand A.GharsallahUnit´e de recherche:Circuits et systmes´e lectroniques HFFacult´e des Sciences de Tunis,Campus Universitaire Tunis EL-manar,2092,TunisieEmail:hammami.a.fst@Abstract—This paper presents the application of a Multiobjec-tive Optimisation Genetic Algorithm for the synthesis of linear antenna arrays.This method for antenna pattern synthesis can suppress multiple interferences by placing nulls at the directions of the interfering sources and placing the main beam in the direction of the desired signal by controlling the phase and the amplitude.To verify the performances of the method,several illustrative examples are provided to justify the proposed phase-amplitude perturbations approach based on genetic algorithms.Index Terms—Synthesis method,Genetic-algorithm,Chebyshev ,antenna array.I.I NTRODUCTIONThe antenna array pattern synthesis consist in steering nulls to the direction of interference and placing the main beam directed to the desired signal.In this last decade,several an-tenna pattern synthesis methods have been developed based on variety of popular methods[1]-[9].However,antenna pattern synthesis,generally,can be classified into three categories.The first category requires that the antenna possesses nulls in the desired directions,which is known as Schelkunoff method. The second category,the patterns are required to exhibit a desired distribution in the entire visible region.This is also known as beam as beam shaping,and can be achieved by using either Fourier Transform or Woodward-Lawson Methods [12].The last category requires that the patterns produced possess narrow beams and low side lobes.This can be achieved through the Taylor line-source,binomial method,SQP method [1][2],Dolph-Chebyshev method[13]and Genetic algorithm method[14]-[18].In this paper,the technique proposed is based on genetic algorithm method to synthesis steered beams with nulls in desired direction by controlling both the amplitude and phase. The paper is organized as follows;the radiation pattern formulation is presented in section II.The genetic algorithm method is presented in section III.Section IV shows the Antenna array synthesis by genetic algorithm method with the simulation results.Section V,andfinally,section VI makes conclusions.II.R ADIATION P ATTERN F ORMULATIONAn antenna array is a multiple radiating elements arranged in space and interconnected to produce a directional radiation pattern.The radiation pattern of an antenna array is determined by the number of antenna elements,the type of single elements used,their positions in space,and the amplitude and phase of the currents feed them[11].We consider a linear array (figure1)of N equispaced isotropic antenna elements with interelemnet spacingλ/2,(whereλis the signal wavelength). These elements are excited with current a i and with a gradient of phaseφi(i=1,2,...,N).Fig.1.Diagram of an amplitude and phase adaptive linear array. The array factor can be written asF(θ)=i=N∑i=1a i e j(kx i sinθ+φi)(1)Whereφi∈[φ1,φ1,...,φN]represents the phase excita-tion of the i tℎelement(the antenna in the beginning is taken as reference of phase:φ1=0),x i represents the position of the i tℎelement,k=2π/λis the wave number,θis the angle of incidence of desired signal,andλis the signal wavelength.The problem of interference suppression in antenna array is tofinding the amplitude and the phase for excitations in order to obtain a radiation pattern with the desired set of characteristics(minimum of radiation in direction of the interfering signals and the maximum of radiation indirection of the useful signal).The mathematical model of multi-objective problem is formulated as minimizing an objective function subject to set of constraints.This problem can be written asminimise−fθi (φ)subject to fθi (φ)≤δj with j=m e+1,...,m−2π≤φn≤2πwitℎn=1,...,N(2) Wherefθ(φ)=∣n=N∑n=1a n e j(kx n sinθ+φn)∣2(3)fθi =[fθ1,θ2,...,θme)]T is the vector of objective func-tions,m e is the numbers of the desired signal,θi,θj andδj are the i tℎdirections of the desired signals,the j tℎdirections of interfering signals,and the levels in the regions of the suppressed sectors respectively,and m is the number of the sampled angular direction.a i,with i=1,2,...,N,is a amplitude of each array element which isfind by-40dB Chebyshev.III.G ENETIC A LGORITHMThe genetic algorithms(AG)are algorithms of optimization. They are techniques inspired of the evolution of the natural species.This theory is based on the principle of evolution introduced by Charles Darwin,whose principle is to create by chance a”population”of solutions.The characteristics of each solution represent genes[10].Then,one evaluates each solution.The individuals best adapted will be subjected to a succession of operators.The others will be eliminated.Rising generation obtained overall will be adapted to the problem than the preceding one.This process is repeated until the birth of a better solution.The creation of a new population is established starting from the preceding one asfigure2shows it.1)Phase of selection:This operation makes it possible to adapt the individuals most adapted to the problem.2)Phase of crossing:This phase relates to the pairs of individuals resulting from the phase of selection and consists of the exchange of parts of chromosomes.This exchange allows the birth of the children obtained starting from the chromosomes of the parents.3)Phase of change:this third operator is introduced to mitigate the disappearance of information(bits)of the popu-lation.Its role consists in modifying by chance,with a certain probability,the value of a bit[3].IV.N UMERICAL R ESULTSTo demonstrate the validity of the proposed method for steering single,multiple and broad nulls with the imposed directions by controlling the element excitations,several com-puter simulation examples using a linear array with onehalfFig.2.Flow chart of Genetic algorithm.wave interelement spaced twenty isotropic elements were presented.A40−dB Chebysheff pattern for a20equispaced linear elements is used as the initial pattern,as shown in figure3.The shape of the amplitude distribution,determinedFig.3.The initial−40dB Tschebyscheff pattern.by Chebyshev method at−40dB,can be used to reduce the sidelobes of the radiation pattern.For a focused beam the amplitude distribution is always symmetric about the center of the array.The optimized excitation magnitudes elements 2Fig.4.The element excitation required to achieve the desired pattern(20 isotropic elements spaced)given with the Tschebyscheff method. algorithm is applied in order to determine the phases of radiating elements.These phases represent the”chromosome”that is in turn composed by”genes”.Figure5shows the chromosome construction.The genetic algorithm parameters which are used to determine these results are:population size 20,number of generations500,crossover probability0.8.geneφ1φ2φ3...φN−2φN−1φNGA cℎromosomeFig.5.Chromosome construction.Figure6shows the radiation pattern obtained by controlling both the amplitude and phase with one imposed null at −40∘.The maximum SLL and the null depth are−30dB and75dB,respectively.The radiation pattern plot infigure3-4 obtained by using genetic algorithm method demonstrates the aptitude of this technique to prescribe a wide band interference signal.Figure4shows the radiation pattern obtained by controlling both the amplitude and phase with a broad null Δθ=5centered at35.The desired nulls are deeper than 60dB.Thefigure5shows the ability of genetic algorithm method of creating null in the direction of interference at 60and a wide band interference signal centered at-45.The maximum side lobe level SLL and the null depth of thefigure 5are−28.5and−60dB,respectively.The computed element (amplitude and phase)current excitations offigures6,7and 8are given in TABLE I.V.C ONCLUSIONIn conclusion,we have proposed a method for antenna pattern synthesis based on the genetic algorithms(AG)to formTABLE IC OMPUTEDE LEMENT C OMPLEX ARRAY WEIGHTF OR F IGURES6,7AND8Synthesized excitationsFig567n Amplitude at-40dB phase excitation(degrees)10.118216.43169.977692.585420.1660-14.6007-22.1914121.229330.2641-44.6794-51.9815153.048540.3817-77.821-84.3627179.995550.5121-108.840-115.6460-141.953560.6461-139.582-145.6933-113.959970.7727180.000180.0000-81.818680.8803157.972151.1695-45.675990.9587126.735120.0326-17.621110 1.000094.70387.629311.742011 1.000064.47857.954643.7583120.958733.38725.582180.3406130.8803 1.7712-5.7812105.9068140.7727-29.950-35.7790136.4725150.6461-60.375-68.5887169.0420160.5121-92.081-98.9445180.0000170.3817-124.24-129.9865-128.8854180.2641-154.850-162.3380-98.3040190.1660173.879167.9294-66.0978200.1182142.841135.6758-31.0885 nulls at the directions of interfering signals by controlling the phase and the amplitude of each array element.The computer simulation results show that the amplitude-phase control,using the genetic algorithms(AG)and-40dB Chebyshev amplitude, is efficient for forming single and broad nulls imposed at the directions of interference and maximum main beam in the direction of the desired signal.3Fig.7.Pattern synthesis with a wide sectors imposed around−35.Fig.8.Pattern synthesis with null imposed at60and a wide sectors imposed around−45.A CKNOWLEDGMENTThe authors would like to thank...R EFERENCES[1]M.Mouhamadou,P.Vaudon,Complex Weight Control of Array 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