UNIVERSITY OF TRENTO A VERSATILE ENHANCED GENETIC ALGORITHM FOR PLANAR ARRAY DESIGN

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1 UNIVERSITY OF TRENTO DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY Povo Trento (Itly), Vi Sommrive 14 A VERSATILE ENHANCED GENETIC ALGORITHM FOR PLANAR ARRAY DESIGN Mssimo Donelli, Slvtore Corsi, Frncesco De Ntle, Dvide Frnceschini, nd Andre Mss August 2004 Technicl Report DIT

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3 A Verstile Enhnced Genetic Algorithm for Plnr Arry Design Mssimo Donelli*, Slvtore Corsi**, Frncesco De Ntle*, Dvide Frnceschini*, nd Andre Mss* Deprtment of Informtion nd Communiction Technology University of Trento Vi Sommrive, Trento, ITALY E-mil: Deprtment of Electronics University of Pvi Vi Ferrt, Pvi, ITALY

4 A Verstile Enhnced Genetic Algorithm for Plnr Arry Design Mssimo Donelli, Slvtore Corsi, Frncesco De Ntle, Dvide Frnceschini, nd Andre Mss Indexing terms: Plnr rry, Antenn Arry Synthesis, Genetic Algorithms. Abstrct - In order to synthesize plnr, sprse, nd periodic rrys, numericl procedure bsed on n enhnced genetic lgorithm is proposed. The method mximizes suitbly defined single-objective fitness function itertively cting on the sttes nd the weights of the elements of the rry. Such cost function is relted to the shpe of the desired bem pttern, to the number of ctive elements nd to others user-defined rry-pttern constrints. To preliminrily ssess the effectiveness of the pproch, selected numericl experiments re performed. The obtined results seem to confirm its fesibility. Moreover, given the heterogeneity of the test benchmrks, the verstility is pointed out s key-feture of the implemented methodology. 2

5 I. INTRODUCTION The synthesis of the bem pttern of two-dimensionl rrys is imed t defining the optiml configurtion of positions nd weights of the rry elements to fulfill severl userdefined constrints (e.g., the reduction of the side-lobes pek (SLP) under fixed threshold, nrrow min lobe, reduced number of rry elements, smll dimension of the rry, etc ). An nlyticl solution to such problem is not vilble nd numericl techniques re generlly used. In [1] method bsed on the liner-progrmming theory hs been pplied to reduce the level of the side-lobes of thinned rry when the number of elements nd the min-lobe width re fixed. Such n pproch turned out to be computtionlly expensive in deling with lrge rrys becuse of n excessive increse in the memory requirement nd computtionl lod. To overcome this drwbck, simulted nneling (SA) procedure hs been proposed in [2] nd pplied to synthesize weighted rry. However, since single-gent globl optimiztion procedures re chrcterized by reduced convergence rte, improved nd more effective (in terms of convergence rte of the itertive process) methodologies re required. In such frmework, this pper presents multiple-gent optimiztion technique imed t fully exploiting the key-fetures of genetic-bsed methodologies (GA) [3] in deling with two-dimensionl rrys [4]. The computtionl efficiency of the presented method hs been improved with suitbly defined hybridiztion nd through the definition of grdient-bsed optimizer. It should be pointed out tht it is out of the min scope of such reserch work to focus on the lgorithmic issues (hybrid codings nd grdient-bsed optimizers re commonplce in GA usge). Nevertheless, these spects re deeply investigted in order to define verstile pproch, being ble to successfully del with lrge clss of plnr structures nd synthesis problems with vrious constrints (voiding the customiztion of the method to single kind of two-dimensionl rrys). The mnuscript is orgnized s follows. In Section II, the mthemticl formultion of the pproch is presented. To ssess the fesibility nd the verstility of the pproch, Section III shows the results of selected/representtive numericl experiments. Finlly, some conclusions nd guidelines for future developments re drwn (Sect. IV). II. SYNTHESIS PROCEDURE MATHEMATICAL FORMULATION Generlly speking, the problem of the synthesis of two-dimensionl rry is chrcterized by different nd conflicting requirements to be stisfied. It is n exmple of 3

6 multi-objective problem. Unlike the strtegy described in [5], the proposed pproch does not consider multi-objective optimiztion but it ims t reducing the multi-objective problem into clssicl single-objective one through n d-hoc combintion of severl terms relted to the physicl prmeters of the rry. Let us consider typicl set of requirements to be stisfied: Minimum discrepncy between the synthesized nd reference bem pttern, ( u v) Nrrow min lobe; Low level of the side-lobes; Uniform level of the side-lobes; Reduced number of ctive rry elements, N. The single-objective function is defined s liner combintion of these constrints p d, ; f ( X, W) = 1 k f (1) ( X, W) + k N + k f ( X, W) + k ( X, W) 1 BP 2 3 SLP 4 fsll where the unknown prmeters to be optimized re the sttes X = x,..., x ] nd the weights [ 1 N W = w,..., w ] of the rry elements, which re ssumed to be locted in N positions of n [ 1 N λ 2 eqully spced two-dimensionl grid ( λ being the free-spce wvelength). The terms of the cost function re defined s follows f f BP SLL ( X, W) = ( p( u,v) p ( u,v) ) f SLP (u,v) S ( X, W) = mx{ p( u, )} d v R R u v 1 1 dudv ( X, W) = mx{ p( u,v) } v { p( u, )} v R R u v 1 1 R R u v 1 1 where u = sinα - sinα 0 nd v = sinβ - sinβ 0 tke into ccount the direction of rrivl of the impinging signl (defined by the ngulr coordintes (α,β)), nd the steering direction 0 = (α 0,β 0 ) (Fig. 1). R is rel vlue llowing the min lobe to be excluded from the clcultion of the side-lobes level nd S defines the rnge for which p( u v) pd ( u, v), >, p(u,v) being the normlized bem pttern; k 1, k 2, k 3 nd k 4 re normlizing coefficients, chosen empiriclly. 4

7 The solution of the rising problem is obtined through the mximiztion of (1). Towrds this end, n enhnced GA (EGA) hs been used. GAs re optimiztion lgorithms bsed on the Drwinin theory of evolution [6]. Stndrd GAs [7], [8] consider popultion of P tril solutions coded in binry strings clled chromosomes nd rnked ccording to the vlue of suitble objective function (clled fitness function). The best solutions re selected nd undergo crossover nd muttion opertors to generte new popultion. The itertive procedure is stopped when fixed threshold for the fitness function ( f mximum number of genertions hs been reched ( i = I EGA η EGA ) or, i being the itertion number). In the following, the most relevnt fetures (compred to stndrd implementtions) of the enhnced GA-bsed procedure will be detiled. A. CHROMOSOME REPRESENTATION To ccurtely represent the unknown prmeters, hybrid coding hs been used. The sttes nd the weights of the rry elements hve been represented with boolen nd rel prmeters, respectively. The chromosome Φ is hybrid-coded string Φ = x n = { xn, n = 1,..., N; wn, n = 1,..., N} = { Φ m ; m = 1,...,2N} { 0,1} w { W W } n min, mx (2) where W min nd W mx define the rnge of vrition of the weight coefficients. B. GENETIC OPERATORS Concerning the genetic opertors, becuse of the chromosome representtion, suitble mixed rel-boolen crossover [9] hs been defined. Two selected (ccording to roulette wheel schem [10]) chromosomes, (1) Φ nd rules re used ccording to the gene under test. If Φ (2) Φ, re superimposed nd different crossover m = w, then n (1) ' (1) (2) [ Φ m ] = rφ m + ( 1 r) Φ m (2) ' (2) (1) [ Φ ] = rφ + ( 1 r) Φ m m m (3) r being rndom vlue between 0 nd 1. If mintined. Otherwise, the stte of the sensor is chosen rndomly. Φ m = x, the stte of the n-th sensor is n 5

8 The muttion opertes with different elements deth nd birth probbilities ( p d nd p b, respectively) nd performs rndom perturbtion of the weight coefficients in the rnge { W min,w mx }. The muttion nd the crossover opertors re pplied with probbility p m nd p c, respectively. In order to increse the convergence rte of the itertive process, grdient-serch opertor is defined. It is pplied when fixed threshold for the cost function hs been chieved ( f ( Φ i ) ηocg, ( p) Φ = rg min min ( ( Φ ) f i j = 1,.., i p= 1,..., P ). The sensor sttes re frozen nd the weight coefficients re modified to optimize the bem pttern shpe. More in detil, strting from the optiml tril solution reched until now, Φ = Φ, k 0, the following steps re itertively performed: (). ( Φ ) nd ( ) f SLP i, k (b). If fslp( i k ) > ηslp f SLL i, k Φ re evluted; i, k i = Φ, then Φ i, k + 1 = Φi, k +αkdk where α k is chosen ccording to the stndrd Polk-Ribière conjugte-grdient method [11] being [ d ] [ f ( Φ )] if m [ 1, ] SLP i, k N m k =. (4) m 0 otherwise Otherwise, the tril solution is updted by considering new serch direction given by [ d ] [ f ( Φ )] if m [ 1, ] SLL i, k N = m k. (5) m 0 otherwise (c). If k = or fixed threshold for the SLP hs been ttined ( fslp ( Φ i, k ) ηconv K mx ), then the itertive procedure stops nd the best chieved tril solution is ssumed s the finl prmetric description of the synthesized rry. A flowchrt of the EGA-bsed procedure is shown in Figure 2. III. NUMERICAL RESULTS To ssess the effectiveness nd the verstility of the proposed rry synthesis method, lrge number of numericl tests, relted to different rry geometries nd vrious constrints, hs been performed. Moreover, some reference test cses hve been considered nd the obtined results re compred with those reported in the relted literture. In order to give some quntittive informtion on the method performnce, let us define set of representtive quntities. Concerning the geometric dimension of the rry, let us 6

9 indicte s occuption domin D, the minimum squre re where the rry elements lie. Moreover, let us define the rry sprseness coefficient (hereinfter indicted with ρ S ) equl to the rtio between the occuption domin nd the normlized number of ctive rry elements to quntify the lloction density of the rry elements ρ S D = 2 N λ N 2 (6) N being the number of ctive rry elements. The first numericl experiment dels with the test cse considered in [1], [2] nd it is relted to plnr rry. The vlues of the EGA prmeters, chosen ccording to the guidelines in [8], re: P = 160, I EGA = 400, p c = (crossover probbility), nd ξ M = 0.01 (muttion probbility). Moreover, fter n exhustive clibrtion process, the vlues of the thresholds turned out to be: η = 0. 1, η = , nd η = As fr s the fitness OCG conv coefficients re concerned, the following vlues (heuristiclly-defined) hve been considered: k = 0.12, k = 0. 12, k 3 = 0., nd k = 4 4. Figure 3 shows the bem pttern of the synthesized rry (owing to the symmetry properties of p(u,v), only the vlues in the rnge u [-1,1] nd v [0,1] re shown). The rry is mde up of N = 64 ctive elements. The pek of the side-lobes level turns out to be equl to db nd the min lobe width (ML) is u -6dB = v -6dB = These vlues indicte tht such solution is closer to the optiml one [1] thn tht shown in [2]. In [2] the sme min lobe width hs been chieved but with lrger number of rry elements ( N = 67 ) nd higher side-lobes level ( SLP = 24. 3dB ). Moreover, the EGA-bsed procedure llows reduction in the vlue of the sprseness coefficient s consequence of the ccumultion of the rry elements shown in Fig. 4. EGA ρ S is equl to 40 while the rrys synthesized in [1] nd [2] re chrcterized by ρ S = 62 nd = 61. 4, respectively. In prticulr, the relevnce of ρ S such result is further confirmed by the rndom rrys theory, which estimtes n verge side-lobe level equl to 18 db for 64 rndom-plced element rry [12]. Becuse of the sttisticl nture of the optimiztion pproch, ech test cse hs been solved by running severl times the EGA-bsed procedure to ssess its relibility. As confirmed from the sttistics reported in Tb. I, the pproch shows good stbility. The verge vlues of the 7

10 chrcteristic prmeters re very close to those of the optiml rry with smll stndrd devitions. Finlly, Fig. 5 shows the behvior of the pek of the side-lobes level versus the number of ctive elements. For completeness, the difference between the bem pttern of N = 62 ctive elements ( ML = nd SLP = db ) nd the optiml rry is given in Fig. 6, thus evidencing n increse in the min lobe width ( ML = versus ML = ). Concerning the second test cse, it dels with the optimiztion of N = rry [13]. The synthesis process is imed t chieving fixed level of the side-lobes ( 20 db ) long the xes u = 0 ( SLP ) nd v = 0 ( SLP ), by minimizing the number of ctive rry v u elements. The EGA method is ble to synthesize thinned rry of N = 79 ctive elements with the following bem-pttern chrcteristics: SLP u = db, SLP v = db, u 6 db = 0.123, v 6 db = positively compres with tht obtined in [13] ( ρ S = 108 ).. The rry sprseness coefficient is equl to 79, which To confirm the verstility of the pproch, let us consider the reduction of the side-lobes level by preserving the min-lobe width s well. The EGA succeeds in synthesizing N = 90 - element rry with reduction of the SLP of bout 3.50 db ( = 90). Such result clerly ρ S points out the flexibility of the proposed method in successfully hndling the trde-off between the side-lobe pek nd the number of ctive elements. Tb. II summrizes the bem pttern chrcteristics of the synthesized rrys nd Figs. 7-8 show the bem pttern behvior on the reference xes nd the rry lyouts, respectively. For completeness, to give n ide of the reproducibility of the synthesis results, sttisticl evlution of the configurtions obtined fter mny EGA executions is provided in Tb. I. IV. CONCLUSIONS A verstile method for the synthesis of sprse plnr rrys hs been proposed. The method llows the specifiction of multiple constrints relted to the min bem width, the side-lobes level, the loction nd the number of ctive rry elements. Towrds this purpose, the originl multiple-objective problem hs been recst in the optimiztion of singleobjective cost function. Such mximiztion hs been crried out by itertively thinning nd weighting the rry elements with synthesis strtegy defined by n enhnced hybrid-coded genetic lgorithm. The effectiveness, relibility nd flexibility of the pproch hve been 8

11 ssessed through selected test cses. The obtined results re fvorbly compred with those reported in the relted literture nd chieved employing deterministic nd/or stochstic methodologies. Moreover, thnks to the introduction of the hybridiztion nd of the grdientbsed optimizer, the EGA-bsed procedure llows reduction of bout 15 % in terms of required execution time. Future developments will be imed t deling with other geometries nd constrints s well s t considering time-vrying environment where vrious interferences occur. Such sitution generlly ppers in rel communictions nd requires fesible nd verstile procedure, being ble to perform rel-time rry synthesis by dptively tuning ntenn chrcteristics nd prmeters. Becuse of the verstility of the EGA, it seems good cndidte to fce this problem. 9

12 REFERENCES [1] S. Holm, B. Elgetun, G. Dhl, Properties of the bempttern of weight- nd lyoutoptimized sprse rrys, IEEE Trns. Ultrsonics, Ferroelectrics, nd Frequency Control, vol. 44, pp , [2] A. Trucco, Thinning nd weighting of lrge plnr rrys by simulted nneling, IEEE Trns. Ultrsonics, Ferroelectrics, nd Frequency Control, vol. 46, pp , [3] D. E. Goldberg, Genetic Algorithms in Serch, Optimiztion nd Mchine Lerning. Addison-Wesley, Reding, MA, [4] F. Ares-Pen, Appliction of genetic lgorithms nd simulted nneling to some ntenn problems, in Electromgnetic Optimiztion by Genetic Algorithms, eds. Y. Rhmt-Smii nd E. Michielssen, Wiley & Sons, pp , [5] D. S. Weile nd E. Michielssen, Integer-coded preto genetic lgorithm design of constrined ntenn rrys, Electronic Lett., vol. 32, pp , [6] J. H. Hollnd. Adpttion in Nturl nd Artificil Systems. University of Michign Press, Ann Arbor, [7] D. S. Weile nd E. Michielssen, Genetic lgorithm optimiztion pplied to electromgnetics: A review, IEEE Trns. Antenns Propgt., vol. 45, pp , [8] Y. Rhmt-Smii nd E. Michielssen, Electromgnetic Optimiztion by Genetic Algorithms. Wiley & Sons, New York, [9] S. Corsi, A. Mss, nd M. Pstorino, A computtionl technique bsed on relcoded genetic lgorithm for microwve imging purposes, IEEE Trns. Geoscience nd Remote Sensing, vol. 38, pp , [10] L. Dvis, Hndbook of Genetic Algorithms. Vn Nostrnd Reinhold, [11] E. Polk, Computtionl Methods. Acdemic Press, New York, [12] Steinberg, Principles of Aperture nd Arry System Design. New York: Wiley, [13] R. L. Hupt, Thinned rrys using genetic lgorithms, IEEE Trns. Geoscience Antenns Propgt., vol. 42, pp ,

13 FIGURE CAPTIONS Figure 1. Plnr rry geometry nd nottions. Figure 2. Flowchrt of the Enhnced Genetic Algorithm procedure (EGA) Figure 3. N = element plnr rry. Bem pttern of the N = 64 -element synthesized rry ( SLP = db, u v ). 6 db = 6 db = Figure 4. N = element plnr rry. Arry lyouts defined with () liner progrmming method [1] ( ρ S = 62 ), (b) simulted nneling pproch [2] ( ρ S = ), nd (c) EGA-bsed method ( ρ S = 40). Figure 5. N = element plnr rry. EGA-bsed method. Side-lobe pek ( SLP ) s function of the number of ctive rry elements ( N ) for severl optimized rrys. Figure 6. N = element plnr rry. EGA-bsed method. Difference between the bem pttern of the N = 62 -element rry (u -6dB = 0.340) nd of the N = 64- element rry (u -6dB = 0.327). Figure 7. N = element plnr rry. Bem power ptterns long the u nd v xes. Figure 8. N = element plnr rry. Arry lyouts defined with () GA-bsed method [13] ( N = 108, ρ S = 108 ), (b) EGA-bsed method ( N = 79, = 79 ), nd (c) EGA-bsed method ( N = 90, ρ S = 90 ). ρ S 11

14 TABLE CAPTIONS Tble I. Sttistics of the rry prmeters fter mny runs of the EGA-bsed procedure (the superscript * indictes the verge vlue between u -6dB nd v -6dB ) Tble II. N = element plnr rry. Comprison between the rry prmeters of the solution obtined with the GA-bsed pproch [13] nd with the EGA-bsed method. 12

15 z z 0 α 0 β 0 y 0 y x 0 x Fig. 1 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 13

16 START i = 1 i = i + 1 l = 1 l < P? yes no yes f l ( Φ i ) η? OCG no yes p c p? no k = 0 Φ = k Φ i Crossover Reproduction no f ( Φ k ) η? SLP yes Perform one GS SLP Perform one GS SLL d itertion ( Φ ) = k f SLP k d itertion ( Φ ) = k f SLL k no p m p? yes k = k+1 Muttion yes f ( Φ k ) η? EGA Evlute fitness l f ( Φ i ) STOP yes i = I EGA? no no l = l + 1 Fig. 2 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 14

17 Fig. 3 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 15

18 6 y [lmd] x [lmd] y [lmd] x [lmd] y [lmd] x [lmd] Fig. 4 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 16

19 Best EGA results EGA best compromise [2] SLP [db] Number of ctive elements (N ) Fig. 5 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 17

20 Fig. 6 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 18

21 bem power pttern [db] u bem power pttern [db] EGA () EGA (b) [13] v Fig. 7 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 19

22 5 4 y [lmd] x [lmd] 4 y [lmd] x [lmd] 4 y [lmd] x [lmd] Fig. 8 M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 20

23 Arry Geometry N vg σ N SLP vg σ SLP MLW -6dB ρ D,vg weighted (*) (*) 0.65 (*) (*) weighted (*) Tb. I M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 21

24 Synthesis Method N SLP u SLP v u -6dB v -6dB SLP u,v Best in [13] db db db Proposed Approch () db db db Proposed Approch (b) db db db Tb. II M. Donelli et l., A Verstile Enhnced Genetic Algorithm for. 22