Research Article Design of Fully Digital Controlled Shaped Beam Synthesis Using Differential Evolution Algorithm
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1 Antennas and Propagation Volume 3, Article ID 7368, 9 pages Research Article Design of Fully Digital Controlled Shaped Beam Synthesis Using Differential Evolution Algorithm D. Mandal, A. Chatterjee, and A. K. Bhattacharjee 3 Department of Electronics and Communication Engineering, Bengal College of Engineering and Technology, West-Bengal, Durgapur-73, India DepartmentofElectronicsandCommunicationEngineering,Dr.B.C.RoyEngineeringCollege,West-Bengal,Durgapur-736,India 3 Department of Electronics and Communication Engineering, National Institute of Technology Durgapur, West-Bengal, Durgapur-739, India Correspondence should be addressed to D. Mandal; deb.mandal@gmail.com Received August 3; Revised 3 October 3; Accepted 3 October 3 Academic Editor: Dau-Chyrh Chang Copyright 3 D. Mandal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A shaped beam synthesis from a concentric ring array has been presented. Two different cases are considered. In the first case, a flat-top beam pattern and, in the second case, a cosec beam pattern have been generated. In both the cases, it has been ensured that the obtained beam patterns are not restricted in any single predefined φ-cut of the patterns; rather, their characteristics are retaining within a range of predefined φ-planes with some minor variations. The desired beam pattern under each of the individual case is obtained by finding optimum discrete excitations of the array elements. The optimum 4-bit amplitudes generated by four-bit digital attenuators and 5-bit phases generated by 5-bit digital phase shifters are computed using Differential Evolution () Algorithm. To illustrate the effectiveness of, the two beam patterns with specified characteristics are computed from the same array using Particle Swarm Optimization () algorithm and Genetic algorithm () by finding out optimum discrete excitations among the elements. Results clearly show the superiority of over and to handle the presented problem.. Introduction In mobile, satellite, and radar communication the important parameters are signal quality, system coverage, spectral efficiency, and so forth. To achieve these, efficient antenna design is the primary requirement. Antenna array synthesis is required to find radiation patterns from different array geometry and make these obtain patterns closer to their desired patterns either by varying its elements amplitude and phase or by reconfiguration of the array geometry. In variousapplicationsshapedbeamsareoftenrequiredbut major problems faced by shaped beams are high side lobe level and ripple. To minimize high sidelobe and ripple, an efficient evolutionary optimization algorithm has been chosen which is able to find out 4-bit optimum discrete elements amplitude and 5-bit optimum discrete phases of the array elements to achieve the desired shaped beam. Several approaches reported in the literature for generating shaped beams [ 6] are as follows. Azevedo proposed a technique based on FFT to generate shaped beams of cosec and flat-top pattern from a linear array antenna through the control of nonuniformly samples of the array factor, both in amplitude and phase [3]. Lei et al. generate low sidelobe cosecant square-shaped beam from linear antenna array by finding optimum amplitudes and phases of the array elements using the modified least square method [4]. Chatterjee et al. [5]proposedatechniquebased on finding optimum phases of the array elements modifying which over existing zero phases generates two different patterns. The search has been carried out using Gravitational Search Algorithm (GSA) [5]. In [6], a new technique has been proposed by Chatterjee et al. for obtaining dual beam pair where the states ( on/off ) of the array elements are modified using Firefly Algorithm (FA) for generating two different beam pairs. The method greatly simplified the design of the feed network [6]. The paper presents shaped beam synthesis of two ring concentric array of isotropic elements. Two different cases
2 Antennas and Propagation.75.5 Start Set G=and randomly initialize X i,g Compute X best,g Y [in terms of λ] X [in terms of λ] Figure : Concentric ring array of isotropic antennas in X-Y plane. Selection Crossover Mutation G=G+ i= G>G max No Yes V i,g =X best,g +F (X r,g X r,g ) u ji,g = { ji,g x ji,g if rand(,) CR otherwise f(u i,g ) <f(x i,g ) Yes f(u i,g )=f(x i,g+ ) Save the result and stop No X i,g =X i,g+ i=i+ D(θ, φ) db Flat-top (Case I) cosec (Case II) φ ( ) degree plane Figure : Desired patterns under two different design cases. have been considered. In the first case a flat-top beam is generated from the presented array by finding optimum 4- bit amplitudes and 5-bit phases, and in the second case a cosec pattern is generated from the same array by finding out another optimum 4-bit amplitudes and 5-bit phases of the elements. In both the cases the optimum discrete excitations are computed in such a manner that the obtained patterns are retaining their desired characteristics within a range of predefined φ-planes with some minor variations. Theoptimumdiscreteamplitudesandphasesforthetwo different cases are computed using Differential Evolution () algorithm [7 ]. To illustrate the effectiveness of, the shaped beams under two different cases are computed separately following the above procedure from the same array configuration using Particle Swarm Optimization () Yes i=np Figure 3: Flow chart of Differential Evolution () algorithm. [6, 9, ] algorithm and Genetic Algorithm () [ 6]. Results clearly show the effectiveness of over and for both the design cases presented in this problem.. Problem Formulation A concentric ring array of isotropic elements is considered. The far field pattern of the array shown in Figure can be written as [6] AF (θ, φ) = M N m m= n= No I mn e j[kr m sin θ cos(φ φ mn )+α mn ], () where M is the number of concentric rings; N m is the number of isotropic elements in mth ring; I mn is the excitation amplitude of mnth element; r m is the N m d m /π radius of the mth ring; d m is the inter element arc spacing of mth circle; k=π/λ,representswavenumber;λ is the wave length; θ, φ is the polar and azimuth angle; φ mn =nπ/n m is the angular location of the mnth element with n N m ;andα mn is the phase excitation of mnth element. The fitness function for the shaped beam pattern is defined as follows: F(ρ)=k (peaksll d max θ A {AFρ db (θ, φ)}) +k Δ, () where Δ is defined as Δ= AFρ db (θ ripple,φ) D(θ ripple,φ) db. (3) θ ripple In ()and(3), φ ( ) plane.
3 Antennas and Propagation (a) (c) Figure 4: Optimized flat-top patterns from the concentric ring array: (a) for φ=degree plan,e (b) for φ=5degree plane, (c) for φ= degree plane. (b) φ=5degree plane Figure 5: Optimized flat-top patterns from the concentric ring array for φ=5degree plane. In (), ρ is the unknown parameter set responsible for the desired beam pattern for the two different cases. ρ is defined as follows: ρ={i mn,α mn }, m M, n N m. (4) peaksll d is the desired value of peak SLL for the two different cases. A is the sidelobe region for the shaped beams of flattop and cosec patterns. θ ripple in () representstherangeof all θ, for the ripple parameter. The range of θ ripple for the flat-top beam is defined as to +5,and,forthecosec beam, the range of θ ripple is defined as to 3. D(θ, φ) db represent desired patterns under two different design cases for φ ( ) plane, shown in Figure. k and k are the weighting factors to give relative importance in each term of (). The values of k and k are chosen as one. For synthesis of shaped beam pattern under two different design cases, the fitness function has to be minimized separately under each of the cases by computing optimum set of discrete 4-bit I mn and optimum set of discrete 5-bit α mn among the array elements. The first part of () is proposed to reduce the peak SLL of the optimized beam pattern and second part is proposed to obtain the desired shape of the beam pattern while minimizing the value of F(ρ). In (), max θ A {AF ρ db (θ, φ)} represents the obtained maximum sidelobe level of the beam pattern within the prespecified range of φ-plane for the unknown parameter set ρ, which has to be determined. The term (peaksll d max θ A {AF ρ db (θ, φ)}) represents the deviation between the desired and obtained values of the peak SLL fortheparametersetρ. Thesecondpartof(), which is responsible for obtaining the desired shape of the beam pattern, is defined in (3). In (3), AF ρ db (θ ripple,φ) represents computed values of ripple parameter under different predefined φ-cut of the beam pattern for the parameter set ρ, and D(θ ripple,φ) db are the desired patterns shown in Figure. Theterm AF ρ db (θ ripple,φ) D(θ ripple,φ) db in (3) represent the deviations between these two parameters and finally their summation is computed to determine the total error in ripple under all predefined φ-cuts within the entire range of θ ripple fortheparametersetρ. The minimization of F(ρ) minimizes the two different terms associated with (), and hence the obtained
4 4 Antennas and Propagation (a) (b) (c) Figure 6: Optimized cosec patterns from the concentric ring array: (a) for φ=degree plane, (b) for φ=5degree plane, (c) for φ= degree plane φ=5degree plane Figure 7: Optimized cosec patterns from the concentric ring array for φ=5degree plane. beam-pattern approaches towards its desired one for thecomputedparametersetρ. Inthismanner,thefitness function of () serves the purpose of generating desired shaped beams while minimizing it individually under each of the design cases using,, and. 3. Algorithm Overviews and Parametric Setup 3.. Overview of Differential Evolution Algorithm. Differential Evolution () algorithm was introduced by Storn and Price. Similar to [ 6], is also based on population. It is a stochastic optimization method used to minimize an objective function. It has an advantage to find out true global minima and fast convergence using a few control parameters [7 ]. algorithm generates a population of NP in D dimensional search space called individuals. The individual of population in generation G can be written as X i,g = {x i,g,x i,g,...,x Di,G }, i =,,...,NP.Theinitial population covered the entire search space. At a generation G = the initial value of the jth parameter defined as follows: x ji, = rand(, ) (x up j x low j )+x low j,where i =,,...,NP, j =,,...,D.rand(, ) is uniformly distributed random variable within the range (, ); x low j and x up j are lower and upper bounds of jth parameter. Three steps mutation, crossover and selection can be described as follows. Mutation Operation. performs mutation operation to generate a mutant vector V i,g = {V i,g, V i,g,...,v Di,G } for each target vector X i,g. In this work, the strategy used is /best//bin which is defined as follows [7 ]: V i,g = X best,g +F (X r,g X r,g) r,r [,NP]and r =r =i. F is a real and constant factor, satisfying F [,]and X best,g is the vector which has best fitness at Gth generation. Crossover Operation. In this operation trial vector U i,g = {u i,g,u i,g,...,u Di,G } is generated from the target vector X i,g
5 Antennas and Propagation 5 Table : Desired and obtained results for Case I of the design problem. Specific φ cut Design parameters φ=degree plane Desired... Obtained Deviation (Δ) db Desired... Obtained φ=5degree plane Desired... Obtained Deviation (Δ) db Desired... Obtained φ=degree plane Desired... Obtained Deviation (Δ) db Desired... Obtained and mutant vector V i,g. The crossover strategy is defined follows: u ji,g ={ V ji,g x ji,g if rand (, ) CR otherwise. Crossover factor CR is const in the range of (, ). The value of CR is taken as.. Selection. The operation performs comparison between the objective function values at each trial vector f(u i,g ) and target vector f(x i,g ).Thevectorwhichhassmallerfitness function value remains in the next generation. Selection operation can be expressed as X i,g+ ={ U i,g if f(u i,g )<f(x i,g ) (6) X i,g otherwise. These three steps are repeated generation by generation until it reaches to its termination condition. Return the best vector in the current population (X best,g ) as the solution of the optimization problem. The flow chart of Differential Evolution is given in Figure Details of Parametric Setup. The individuals of the population for,, and are considered as (5) X=[I I I K α α α K ]. (7) The limits of the variables are defined as follows: I m, for m=,,...,k, (8) π α m π for m=,,...,k. The value of K in this problem becomes 3 and the search space dimension becomes 6. Basedontheguidelineprovidedin[7 ], the population size, scale factor (F), and crossover rate (CR) of are chosen as 5,.8, and.. The scheme used is: /best//bin and the maximum iteration number is chosen as 3. Swarm size in is taken as 5 and the initial population is chosen randomly. The values of C and C are chosen as [6, 9]. Time-varying inertia weight (w) is considered as decreasing linearly from.9 to.4. The maximum allowable velocity for each of the particle on dth dimension is considered as.9r d [6, 9], where r d is the difference between the maximum and minimum possible values of decision variables on dth dimension. The termination condition is chosen as a maximum iteration of 3. Population size in is taken as 5 and two-point crossover is chosen. Crossover probability and mutation probability are taken as.8 and., Roulette Wheel Selection is considered for the proposed problem, and the termination condition is chosen as a maximum iteration of 3. Other parametric setups of are taken from guidelines given in [ 6]. 4. Simulation Results A two ring concentric array of total 3 isotropic elements has beenconsidered.thenumberofelementsineachringofthe array is taken m,wherem is the ring number. The interelement spacing is considered as.5λ;thatis,d=.5λ, and the ring radii are computed as.795λ and.59λ. The presented results in this section are the best set of results obtained from different runs of each of the algorithm for each individual case. The design specifications of flat-top beam patterns computed separately using,, and and their corresponding obtained results in different φ-planes are shown in Table. FromTable, it can be seen that the obtained values of the design parameters using are better than and for the flat-top beam pattern. Three different φ-cuts of the obtained flat-top beams, computed individually using,, and are shown in Figure 4. InFigure4, the presented φ-cuts are degree, 5 degrees, and degrees. It can be ensured from Figure 4 that theobtainedflat-topbeamsarenotrestrictedinanysingle φ-plane within the prespecified range of φ.to observe the flat-top beam patterns from a different φ-plane which is outside its prespecified range ( φ ), a 5- degree φ-cut of the optimized beam patterns is presented in
6 6 Antennas and Propagation Algorithm ber Ring num- Table : Computed 4-bit amplitudes and 5-bit phases for Case I of the design problem. Excitation Elements Phase Amp Phase Amp Phase Amp Phase Amp Phase Amp Phase Amp
7 Antennas and Propagation 7 Table3:DesiredandobtainedresultsforCaseIIofthedesignproblem. Specific φ cut Design parameters φ=degree plane Desired... Obtained Deviation (Δ) Desired... Obtained φ=5degree plane Desired... Obtained Deviation (Δ) Desired... Obtained φ=degree plane Desired... Obtained Deviation (Δ) Desired... Obtained Best fitness Best fitness Generations Generations (a) (b) Figure 8: Convergence characteristics of,, and : (a) for Case I of the design problem (b) and for Case II of the design problem. Figure 5. FromFigure5, it can be seen that the patterns are deviated from its desired flat-top pattern of Figure 3,because they are not optimized for φ=5degree plane. The optimum 4-bit amplitudes and 5-bit phases of the array elements for the flat-top beams computed individually using,, and are shown in Table. The design specification for cosec beam patterns (Case II) and their corresponding obtained results in three different φ-planesareshownintable3. FromTable3, it canbeobservedthattheperformanceof,intermsof computing the design parameters for cosec beam pattern, is better than and. Figure 6 shows three different φ-cuts of the obtained cosec beam patterns, computed individually using,, and. In Figure 6, the presented φ-cuts of the optimized cosec beam patterns are degree, 5 degrees, and degrees. Figure 7 shows a 5-degree φ-cut of the optimized cosec beam patterns which clearly shows deviation from its desired cosec pattern of Figure 3, because they are not optimized for φ=5degree plane. The optimum 4-bit amplitudes and 5-bit phases of the array elements for the cosec beams computed individually using,, and are shown in Table 4. The comparative performance of,, and for the two different cases of design problem is shown in Table 5. Lowest mean fitness value of over and proves itself best performing algorithm over and. The convergence characteristics of the three algorithms for the two different cases of the presented problem are shown in Figure 8. FromFigure8(a), itcanbenoticed that the convergence of is better than and in terms of minimizing the fitness function of () for the flattop beam pattern. Similarly, Figure 8(b) clearly shows better convergence of over and in terms of minimizing the fitness function of ()forthecosec beam pattern. Table 6 shows the P values obtained through Wilcoxon s ranksumtest[, 7] between,, and for two different cases of design considerations. All the P values are less than.5 (5% significant level) which is a strong proof against null hypothesis indicating that better final fitness value obtained by the best algorithm is statistically significant and has not occurred by chance. 5. Conclusions Synthesis of shaped beam patterns from a concentric ring array antenna using Differential Evolution algorithm has been presented. For synthesis of shaped beam patterns, constrained side lobe and ripple are contemporarily taken into
8 8 Antennas and Propagation Algorithm ber Ring num- Table 4: Computed 4-bit amplitudes and 5-bit phases for Case II of the design problem. Excitation Elements Phase Amp Phase Amp Phase Amp Phase Amp Phase Amp Phase Amp
9 Antennas and Propagation 9 Table 5: Comparative performance of,, and. Different cases Algorithm Best fitness (out of ) Worst Mean Standard deviation Case I Case II Table 6: P values for Wilcoxon s two sided rank sum test. Different cases Comparison pair P-value / 7.473e 8 Case I /.9e 7 / 4.549e 6 / 6.786e 8 Case II / e 8 / e 7 account by minimizing properly formulated fitness function using Evolutionary Algorithm based procedure. Presented method is capable of producing beam patterns, which retains their desired characteristics within a range of predefined φplanes with some minor variations. Results clearly indicate a good agreement between the obtained and desired one. The presented method incorporates 4-bit amplitudes of the array elements, which ensure that the dynamic range ratio (DRR) remains within the limit of 6 which is helpful for reliable design of the feed network. Discrete excitations also reduced the number of attenuators and the phase shifters and hence are capable of reducing the cost and complexity of the system. Thecomparativeperformanceof,,andclearly showsthesuperiorityofoverandintermsof finding optimum solutions for the presented problem. The qualityofthesolutionsproducedindividuallyusing,, and for the two different cases of design considerations is analysed statistically and the superiority of is proven over and for the proposed problem. References [] C. A. Balanis, AntennaTheory,AnalysisandDesign, John Willy & Sons, New York, NY, USA, nd edition, 997. [] R. S. Elliott, Antenna Theory and Design, JohnWiley&Sons, New York, NY, USA, Revised edition, 3. [3] J. A. R. Azevedo, Shaped beam pattern synthesis with nonuniform sample phases, Progress in Electromagnetics Research B,vol.5,pp.77 9,8. [4] J. Lei, G. Fu, L. Yang, and D.-M. Fu, Wide band linear printed antenna array with low sidelobe cosecant square-shaped beam pattern, Progress in Electromagnetics Research C, vol.5,pp. 33 4,. [5] A.Chatterjee,G.K.Mahanti,andP.R.S.Mahapatra, Design of fully digital controlled reconfigurable dual-beam concentric ring array antenna using gravitational search algorithm, Progress in Electromagnetics Research C,vol.8,pp.59 7,. [6] A. Chatterjee, G. K. Mahanti, and A. Chatterjee, Design of a fully digital controlled reconfigurable switched beam concentric ring array antenna using firefly and particle swarm optimization algorithm, Progress in Electromagnetics Research B,vol.36,pp.3 3,. [7] R. Storn and K. Price, Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, vol., no. 4, pp , 997. [8] K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution A Practical Approach to Global Optimization. Natural Computing, Springer, New York, NY, USA, 5. [9] M.A.Panduro,C.A.Brizuela,L.I.Balderas,andD.A.Acosta, A comparison of genetic algorithms, particle swarm optimization and the differential evolution method for the design of scannable circular antenna arrays, Progress in Electromagnetics Research B,vol.3,pp.7 86,9. [] S. Das, A. Abraham, U. K. Chakraborty, and A. Konar, Differential evolution using a neighborhood-based mutation operator, IEEE Transactions on Evolutionary Computation, vol. 3, no. 3, pp , 9. [] J. Guo and J. Li, Pattern synthesis of conformal array antenna in the presence of platform using differential evolution algorithm, IEEE Transactions on Antennas and Propagation, vol.57,no.9, pp. 65 6, 9. [] R. L. Haupt, Introduction to genetic algorithms for electromagnetics, IEEE Antennas and Propagation Magazine, vol. 37, no., pp. 7 5, 995. [3]K.F.Man,K.S.Tang,andS.Kwong, Geneticalgorithms: concepts and applications, IEEE Transactions on Industrial Electronics,vol.43,no.5,pp ,996. [4] J. M. Johnson and Y. Rahmat-Samii, Genetic algorithms in engineering electromagnetics, IEEE Antennas and Propagation Magazine,vol.39,no.4,pp.7,997. [5] D. Marcano and F. Durán, Synthesis of antenna arrays using genetic algorithms, IEEE Antennas and Propagation Magazine, vol.4,no.3,pp.,. [6] M. A. Panduro, A. L. Mendez, R. Dominguez, and G. Romero, Design of non-uniform circular antenna arrays for side lobe reduction using the method of genetic algorithms, International Journal of Electronics and Communications, vol.6,no., pp , 6. [7] M. Hollander and D. A. Wolfe, Nonparametric Statistical Methods, John Wiley & Sons, New York, NY, USA, nd edition, 999.
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