A MODIFIED DIFFERENTIAL EVOLUTION ALORITHM IN SPARSE LINEAR ANTENNA ARRAY SYNTHESIS Kaml Dmller Department of Electrcal-Electroncs Engneerng rne Amercan Unversty North Cyprus, Mersn TURKEY kdmller@gau.edu.tr Al Haydar Department of Computer Engneerng rne Amercan Unversty North Cyprus, Mersn TURKEY ahaydar@gau.edu.tr ABSTRACT A modfcaton on the classcal Dfferental Evoluton (DE) algorthm, based on randomzaton of the mutaton scale factor, s proposed n the lnear antenna array synthess. An example of postonphase synthess of unequally spaced lnear antenna array wth the mnmum peak sdelobe level s presented and compared wth some publshed results. The comparson clearly ndcates that the proposed modfcaton outperforms the exstng results n the recent lterature obtaned by varants of DE algorthm. Keywords: Dfferental Evoluton, Antenna Array, Sdelobe Suppresson. INTRODUCTION There has been an ncreasng focus on antenna array desgn due to the ncrease both n sze and n the varety of wreless applcatons. One of the man objectves of array desgn s beam shapng of the radaton pattern by adjustng parameters such as geometrcal confguraton, relatve dsplacements, exctaton ampltudes, exctaton phases and relatve patterns of dentcal elements (Balans, 1982). In some desgn problems, such as when a very hgh drectvty s requred n the selected drecton, analytcal technques cannot gve adequate results. Thus, t s requred to embed some numercal optmzaton technques nto the desgn process. In ths paper, a smple and effectve modfcaton on the classcal DE algorthm s proposed and s appled for the poston-phase synthess of the lnear antenna arrays. The obtaned results are compared wth the results of some varants of the DE algorthm obtaned from the recent lterature (Kurup et al., 2003), (Ln & Qng, 2010) and (oudos et al., 2011). CONTEXT AND REVIEW OF LITERATURE The Dfferental Evoluton s one of the evolutonary algorthms that extensvely appled for syntheszng of antenna arrays and some other electromagnetc problems (Rocca et al., 2011). Dfferental evoluton (DE) algorthm was proposed by Prce and Storn n 1995 (Storn & Prce, 1997), (Prce et al., 2005). It s an effectve, robust and smple global optmzaton algorthm whch has a few control parameters. The DE algorthm s used n antenna array synthess problems extensvely. A classcal DE algorthm s used n syntheszng unequally spaced antenna arrays n (Kurup et al., 2003). Dynamc Dfferental 25
Evoluton (DDE) algorthm (Qng, 2006), whch dffers from the DE algorthm by the strategy used for updatng ndvduals, s appled to antenna synthess problems n (Ln & Qng, 2010). The effect of angular resolutons whch are used to span the space n optmzaton has also been nvestgated n (Ln & Qng, 2010). A modfed verson of the DDE algorthm, whch s called Modfed Dfferental Evoluton Strategy (MDES), s also appled to some antenna synthess problems (Chen et al., 2008). Another varant of the DE algorthm, whch adaptvely adjusts the control parameters, s Self Adaptve Dfferental Evoluton (SADE) algorthm (Brest et al., 2006). The SADE algorthm s also appled to some antenna and mcrowave desgn problems (oudos et al., 2011). METHOD Sdelobe Suppresson of Antenna Arrays We consder an array of 2N sotropc antennas, whch are placed symmetrcally along the x-axs, as shown n Fg. 1. The radaton pattern of the gven array s symmetrc wth respect to the x-axs. The array factor AF s a functon of the angle θ, whch represents the angular separaton from y-axs and t can be wrtten as follows (Balans, 1982), (Ln & Qng, 2010), AF( x N x, I, φ,θ) = I exp 2π sn(θ) + φ (1) = N λ where λ s the wavelength and three vectors, x, I and φ, contan the postons, exctaton currents and exctaton phases of the antenna elements. The array factor s a functon of only the angle θ for a syntheszed antenna array n whch x, I and φ are determned. y θ -x N -x N-1 -x 3 -x 2 -x 1 x 1 x 2 x 3 x N-1 x N x Fgure 1. eometry of the 2N element symmetrc lnear antenna array along the x-axs The peak sdelobe level (PSLL) of the antenna array s defned as (Ln & Qng, 2010) AF(x, I,φ,θ) PSLL( x, I, φ) = max (2) θ S AF(x, I,φ,0) where S s the space spanned by the angle θ excludng the predefned man lobe wth the center at θ=0. The objectve functon s selected to mnmze PSLL, snce the focus of array synthess of our work s to mnmze the peak sdelobe levels of the unequally spaced antenna arrays wth unform ampltude exctaton. We consder the poston-phase synthess that the exctaton currents are the same for all elements (.e. I- =I=1, for =1, 2, 3,, N) and elements are assumed to be symmetrc (.e. x- =x, and φ-=φ for =1, 2, 3,, N).Then, N couples of real numbers (x, φ ; =1, 2, 3,, N), where x s the poston and φ [0,π/2] s the phase of the th element are formng the soluton space. 26
Dfferental Evoluton Algorthm The DE algorthm s a stochastc, parallel drect search method. It can be brefly stated as follows: Intalzaton: Intalze the populaton of sze P n N dmensons. Mutaton In each generaton, each N-dmensonal soluton (parent) vector z, =1, 2 P s mutated to obtan the perturbed (mutant) vector v + that s produced by v + 1 = zr + F.(z r z 1 2 r3 ) (3) where r 1, r 2, r 3 are three mutually dfferent ntegers, whch are randomly chosen from the set {1,2,, P}. They are also dfferent from the value. Mutaton scale parameter (F) s a real constant number whch has a control on the ampltude of the dfference (zr zr ). 2 3 Crossover The crossover operator s manly appled n order to ncrease the dversty of the mutant vector. In ths step, the parent vector 1 vector y +. Selecton z together wth the perturbed vector In the selecton phase, the ftness of the tral vector y + v + are recombned to obtan the tral s calculated and t s compared wth the ftness of the parent vector z. If the ftness of the tral vector s better than ts parent vector, then the tral vector replaces the parent vector n order to advance to the next generaton. Otherwse, the parent vector s kept n the next generaton wthout any change. Proposed Modfcaton on DE Algorthm In the modfed DE algorthm, the only change s performed on the selecton of F value. Instead of selectng the F as a real and constant value, t s selected randomly by generatng a unform random number Fj [0, 1], where j=1, 2,, N and =1, 2,, P. Ths random selecton of F enables the algorthm to delve the regon more deeply hence not stackng to local solutons. NUMERICAL RESULTS The modfed DE algorthm s appled n the poston-phase syntheszng of a 32-element lnear array. The populaton sze of the modfed DE algorthm s set to 60. Crossover rate s set to 0.95. We run the program for 30 tmes for each problem and the best soluton sets are presented and compared wth publshed results. Table 1. Comparson of the Best PSLL s obtaned by Varants of DE Algorthms n db Phase-Poston Recalculated Reported Ths Paper -23.6508 DE [3] -23.2181-23.34 DDE [7] -23.5726-23.45 SADE [10] -23.3462-23.59 MDES [8] -23.1259-23.45 27
For each run, 3000 teratons are carred out. We set the desred beamwdth to 6.3. The angular resoluton s set to 0.2 whch s also suggested n (Ln & Qng, 2010). The mnmum and the maxmum dstances between any adjacent elements are set to 0.5λ and λ, respectvely. In order to compare our results wth the ones that are publshed n lterature on the same bass, all PSLLs are recalculated wth an angular resoluton of 0.0018 (=π/100000) by usng the postons and phases gven each reference. The recalculated and the reported best PSLLs are gven n Table I. It should be noted that even the average PSLLs for the 30 runs wth the modfed DE algorthm, whch s calculated as -23.6218 db, s better than all the reported results. The standard devaton of PSLLs of 30 runs s calculated as 0,037659 db, whch clearly shows how robust the modfed DE s for the analyzed problem. The best soluton set, whch s obtaned by the proposed DE algorthm, s gven n Table II. Table 2. The Best Soluton Set Obtaned by the Proposed DE Algorthm No Poston Phase (x /λ) (Degrees) 1 0.250 41.84 2 0.750 42.13 3 1.250 45.47 4 1.750 42.92 5 2.250 47.29 6 2.754 44.25 7 3.314 44.11 8 3.864 42.96 9 4.509 42.52 10 5.098 42.64 11 5.751 44.06 12 6.560 37.85 13 7.429 40.05 14 8.268 72.41 15 9.242 12.60 16 10.114 40.72 The frst three sdelobes of the best synthess of the proposed DE algorthm are compared wth the best soluton sets of DDE (Ln & Qng, 2010) and SADE (oudos et al., 2011) algorthms n Fg. 2. It can be observed that the obtaned soluton set by usng the modfed DE algorthm outperforms the best soluton sets, whch were presented n (Ln & Qng, 2010) and (oudos et al., 2011). DISCUSSION AND CONCLUSION The proposed modfcaton on the DE algorthm ams to construct a deeper search n the regon of nterest. The experments that we have conducted have shown us that ths modfcaton of the DE algorthm does not only mprove the soluton qualty, but also ncreases the robustness of the algorthm when t s appled to the selected antenna synthess problems. Moreover, we observe that the randomzaton of the mutaton scale factor s very effectve n the problems that have many local mnmums n the regon of nterest. 28
Fgure 2. The frst three sdelobes of poston phase synthess of 32-element lnear array obtaned by usng dfferent DE algorthms. REFERENCES Balans, C. A. (1982). Antenna Theory. New York: John Wley. Brest, J., rener, S., Boskovc, B., Mernk, M., & Zumer, V. (2006). Self-adaptng control parameters n dfferental evoluton: A comparatve study on numercal benchmark problems. IEEE Trans. Evol. Comput., 10, 646 657. Chen, Y., Yang, S., & Ne, Z. (2008). The applcaton of a modfed dfferental evoluton strategy to some array pattern synthess problems. IEEE Antennas and Wreless Propag. Lett, 56 (7), 1919-1927. oudos, S. K., Sakavara, K., Samaras, T., Vafads, E. E., & Sahalos, J. N. (2011). Self-adaptve dfferental evoluton appled to real-valued antenna and mcrowave desgn problems. IEEE Trans. Antennas Propag., 59 (4), 1286-1298. Kurup, D.., Hmd, M., & Rydberg, A. (2003). Synthess of unform ampltude unequally spaced antenna arrays usng the dfferental evoluton algorthm. IEEE Trans. Antennas Propag, 51 (9), 2210-2217. Ln, C., & Qng, A. (2010). Synthess of unequally spaced antenna arrays by usng dfferental evoluton. IEEE Trans. Antennas Propag., 58 (8), 2553-2561. Prce, K. V., Storn, R. M., & Lampnen, J. A. (2005). Dfferental evoluton: a practcal approach to global optmzaton. Berln: Sprnger. Qng, A. (2006). Dynamc dfferental evoluton strategy and applcatons n electromagnetc nverse scatterng problems. IEEE Trans. eosc. Remote Sensng, 44 (1), 116 125. Rocca, P., Massa, A., & Olver,. (2011).,. Olver and A. Massa, Dfferental Evoluton as Appled to Electromagnetcs,, vol. 53, no. 1, 38-49, February 2011. IEEE Antennas and Propagaton Magazne, 53 (1), 38-49. Storn, R., & Prce, K. (1997). Dfferental evoluton-a smple and effcent heurstc for global optmzaton over contnuous spaces. J. lobal Optmzaton, 341-359. 29