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A Com parative Study Of Corrugated Horn Design ByEvolutionary TechniquesuAhmad HoorfarVillanova Univesrsity, Philadelphia, PA

Vahraz JamnejadJet Propulsion LaboratoxyCalifornia Institute of TechnologyPasadena, CA 91109vahraz.jamnejad@jpl.nasa.gov8 18-3542674

Ab s ~ u c t- orrugated horn antennas are frequently used asthe feed elements in ground-based reflector antennas forsatellite and deep space communications. A particularapplication is the multi-frequency feed horns for thereflector antennas of JPLMASA Deep Space Network(DSN). In this application, it is desirable to design a homthat has a nearly perfect circularly symmetric pattern (i.e.,identical E- and H-plane patterns), with zero or low cross-polarization, and at the same time achieves a specifiedbeamwidth and a low return loss at the design fresuencyrange. A parametric study of the corrugated horns, however,shows that the objective function relating the pattern shape,beamwidth and retun loss is a nonlinear function of thecorrugation dimensions and has many local optima. As aresult one has to resort to global optimization techniques,such as Evolutionaxy Algorithms (EA) or GeneticAlgorithms (GA), for a successful design of these antennas.Here an evolutionary programming (EP) algorithm is used tooptimize the pattem of a corrugated circular horn subject tovarious constraints on return loss, antenna beamwidth,pattem circularity, and low cross-polarization. EP algorithmswith different mutation operators including Gaussian,Cauchy and hybrid are applied and the results are compared.Also a hybrid EP-GA algorithm isused for comparison. TheEP algorithm is inherently suited for padlelization on aMassively Parallel Computer (MPP), which increases thecomputation speed by orders of magnitude. Examples ofdesign synthesis for a few corrugated horns are presented.The results show excellent and efficient optimiition of thedesired hom parametem.

TABLE FCONTENTS1. INTRODUCTION2. APPLICATIONOF E'TOHORN DESIGN3. NUMERICALRESULTS4. CONCLUSIONS

1. INTRODUCTIONCorrugated hom antennas are frequently used as the feed

elements in ground-based reflector antennas for satelliteanddeep space communications. In particular, for the latterapplication, it is desirable to design a hom that has a nearlyperfect circularly symmetric pattem (i.e., identical E- and H-plane patterns), with zero or low cross-polarization, and atthe same time achieves a specified beamwidth and a lowretum loss at the design frequency range. A parametric studyof the corrugated horns, however, shows hat the objectivefunction relating the pattem shape, beamwidth and S11 is anonlinear function of the corrugation dimensions and hasmany local optima.As a result one has to resort to a globaloptimization technique such as Evolutionary Algorithms fora successful design of these antennas.Using these techniques, extremely fast and efficient designand synthesis procedures can be implemented, sometimesresulting in highly unusual desigus which would otherwisebe extremely difficult, if not impossible, to achieve. Amongthe difficulties encountered in non-automated design byanalysis only are that a single input parametersimultaneously affects several output characteristics(pleiotropy) while a ;ingle output characteristic isdetermined by the simultaneous interaction of many inputparametem (POlygeny).Corrugated hom antennas are fkequently used as the feedelements in ground-based eflector antennas for satellite anddeep space communications. In particular, for the latterapplication, it is desirable to design a hom that has a nearlyperfect circularly symmetric pattern (i.e., identical E- and H-plane patterns), with zero or low cross-polarization, and atthe same time achieves a specified beamwidth and a lowretum loss at the design frequencies. A parametric study ofthe corrugated horns, however, shows that the objectivefunction relating the pattern shape, beamwidth and inputretum loss is a nonlinear function of the corrugationdimensions and has many local optima.As a result one has toresort to a global optimization technique such asEvolutionary Algorithms for a successful design of theseantennas.Evolutionary algorithms M e r substantially from more

' -7803-7651-X/03/317.00 2003 IEEE2 IEEEAC paper #1340, Updated Jan.09,2003

mailto:vahraz.jamnejad@jpl.nasa.govmailto:vahraz.jamnejad@jpl.nasa.gov

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traditional search and optimization methods. The mostsignificant differencesare:i) They search a population of points in parallel, not a singlepoint.ii) They do not require derivative information or otherauxiliary knowledge; only the objective function andcorresponding fitness levels influence the directions ofsearch.iii) They use probabilistic transition rules, not deterministiciv) They are generally more straightforward o apply.v) They can provide a number of potential solutions to agiven problem. The final choice is left to the user. Thus, incases where the particular problem does not have oneindividual solution, as in the case of multi-objectiveoptimization and scheduling problems, the evolutionaryalgorithm is ideally suited for identifying these alternativesolutions simultaneously. And finally,vi) Due to their inherently parallel nature, evolutionaxyalgorithms can be developed to speed up the computation byhamessing the power of massively parallel processingcomputers.The Evolutiom Algorithms (EAs) are in general multi-agent stochastic search methods that rely on a set of variationoperators to generate new offspring population. A selectionscheme is then used to probabilistically advance bettersolutions to the next generation and eliminate less-fitsolution accordingto the objective function being optimized.Among the three paradigms of EAs, namely, GeneticAlgorithms (GA), Evolutionary Programming (EP) andEvolution Strategies (ES), GA and EP have beensuccessfully applied to the design and optimization ofvarious antenna and microwave structures [1,2,3,4].

ones.

The variation operator used in GA is a combination ofcrossover and mutation with the former being the mainmechanism of change. The selection of the crossover andmutation probabilities is rather arbitrary and they are notadapted during evolution. EP, however, models theevolution at the species level, thus its variation operator isentirely based on mutation where adaptive andor self-adaptive techniques exist for adapting the parameters ofmutation operator during the evolution process. Mutation-based reproduction process in EP, coupled with the fact thatunlike the conventional GA, EP works directly withcontinuous parameters, provides a versatile tool in design ofthe problem specific variation operators for mul t i -me te rantenna optimization, and easy integration with availableapriori knowledge about the problem.In this work we have used an EP algorithm with a Gaussianmutation operatorto optimizepattem of a corrugated circularhorn subjectto various constraints on return loss and antennabeamwidth. A software code, based on generalized scatteringmatrix and mode-matching technique [5 ] , which has beendeveloped and modified at JPL over many years, is used forthe analysis of the corrugated hom and the calculation of the

radiated far field h m the hom. Examples on designsynthesis of a 45 section corrugated hom, with a total of 90optimization parameters, are presented.The nitial design forthe horn, as shown in Figure 1, is generated using variousguidelines in the literature [6,7] and by implementing asimple MATLAB program.

2. APPLICATIONF EP TOHORN DESIGNFor optimization purposes the N-section corrugated hom inFigure 1 is mathematically represented as a vector of lengthn = 2 N

-X = [rl ,rz....r , ;d , , d2 ....,d , ]'

where ri and di are radius and length of the i-th corrugatedsegment, respectively. For theminimization of the differencebetween theE- and the H-plane co-polar patterns, subject tothe constraints on the beamwidth and the overall return loss,we construct the fitness functionas,

[ if %sE(e b , o")al= 0 otherwise (3)

otherwise

in which E(@ # is the nonnalized co-polar pattem in dl3obtained by the application of the mode matching codealready mentioned.. Me in the fmt term is the number ofelevation angles, 8, sampled in the interval [0, 8 4 , nwhich we requhe a near circularly symmetric pattern. Thesecond and third terms in (2) penalize all the solutions thatviolate the constraint, q2 I (eb,oo) q l , on the normalizedgain value at e,,, while the last term penalizes those thatviolate the constraint, s,, S , , on the return loss. Theweighting factors wl, w2 and p are selected such that toprioritize the influence of various constraints on the fitnessfunction.

To optimize vector 5 in (l), we apply a continuousm e t e r meta-EP. The process consists of five basic steps:initialization, fitness evaluation, mutation, tournament andselection [8,9]. In particular, an initial population of pindividuals is formed through a uniform random or a biaseddistribution. We consider each individual to be a pair of real-valued vectors, 6 ,$ Vi E$?,...p} whereS% 113 2 ) . . . .-qh)] and 6 are the n-dimensional solutionand its corresponding strategy parameter (variance) vectors,

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respectively. 5 ' s are initialized based on the user-specifiedsearch domains, Ti E x,,,h, X-}", which may be imposed atthis stage. A meta-EP algorithm is now implemented bygenerating a single offspring (F', . 7 from each parent( F i , $ according to:

X i 'o ' ) = x i l i )+J;;i;SN,(0, 1N(0,l) rNt0,l)l (4)q i U ) = qio')efor j = 0,1,2,....n, where x(i) and vu) and are the jthcomponents of the solution vector and the variance vector,respectively. N(0,l) denotes a one-dimensional randomvariable with a Gaussian distribution of mean zero andstandard deviation one. Nj(0,l) indicates that the randomvariable is generated anew for each value of j. The scalefactors T and T' are commonly set to pJ'nde)',respectively, where n is the dimension of the search spacemechanism for generating new offspring population. Thedetails on the other steps of the meta-EP process can befound in [4,8,9].

3. NUMERICALESULTSAs an example, we have optimized a 45-segment corrugatedhorn. The geometry of the initial structure, beforeoptimization, is shown in Figure 1. The population size andthe number of opponents in the tournament selection of EPwere set to p = 10 and q = 4, respectively. To optimize theXvector in (l), the radii and lengths of the sections wererandomly initialized in the search domains 0.95ra < ri

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0I I --

I - pR(m

I I I I I I10 m m m mPolar argle indegrees

Figure 3. Comparison of horn patterns in two orthogonalplanes before &after optimization, Case I.

P o k argle ndegreesFigure 4. Pattern difference in the two planesbefore & after optimization, Case I.

Figure 7. Input S l l as a function of frequency for thehorn before and after optimization at 9.5GHz,Case I.

after

Figure 5 . Input SI1 as a function of fresuency for thehorn before and after optimization at 8GHz, Case 1.

F i W 8 . Comparison of horn patterns in two orthogonalplanes before &after optimization at 9.5 GHz,Case 11.

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ACKNOWLEDGMENTThe research described in this paper was carried out at the JetPropulsion Laboratory, Califomia Institute of Technology,under a contract with the National Aeronautics and SpaceAdministration. The authors appreciate the support andhelpll suggestions of Dr. Fa~zin Manshadi of JPLthroughout the course of this research.

Fmquency inGHzFigure 9. Input S1 1 as a fmction of frequency for the hombefore &after optimization at 9.5 GHZ, Case 11.

26 ,

Figure 10. Geometry of corrugated hom after optimization,Case 11.4. CONCLUSIONS

In this paper we have shown that the evolutionaryprogramming techniques can be successfully applied to theproblem of corrugated hom design. Many additional resultswill be shown at the conference. The next step already beingworked on, is the parallelization of the optimization programon a massively parallel computer such asSGIOrigin2000 atJPL. The present work was performed on the JPL CRAYSVl and takes many hours of computation, which can besubstantially reduced by parallelization. The parallel codewill be applied to the design multi-frequency X/X/Ka feedhoms of the JPLMASA Deep Space Network (DSN)reflector antennas.

REFERENCESD.S.Weile and E. Michielssen, Genetic algorithmoptimization applied to electmmagnetics: A review,IEEE Trans.Antennas Propagat., March 1997.Electromagnetic Optmization by Genetic Algorithms,edited by Y. Ramat-Samii and E. Michielssen, JohnWiley & Sons, 1999.A. Hoorfar andK. Chellapilla,Gain Optimization of amulti-layer printed dipole array usingevolutionaryprogramming, IEEE AP-S Int. Symp. ,Atlanta, June1998.A. Hoorfar,Mutation-based evolutionary algorithmsand their applications to optimization of antennas inlayered media, IEEE AP-S International Symposium,Orlando, FL, July 1999.Graeme L. James, Analysis and Design of E 1 1 toHE11 Corrugated Cylindrical Waveguide ModeConverters, IEEE Trans. Microwave Theory Tech., vol.MTT-29, no. 10, pp. 1059-1066,October 1981.P. J. B. Clarricoats and A. D. Olver, Corrugated hornsfor microwave antennas,Peter Peregriitus Ltd, 1984.T. A. Milligan, Modem Antenna Design, McGraw-Hill,Inc. 1985.D. B. Fogel, Evolutionary Computation: Toward a NewPhilosophy of Machine Intelligence, Piscataway, NJ:IEEE Press, 1994.T. Bkk, Evolutionary Algorithms in Theory andPractice, Oxford Univ. Press, 19%.

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Ahmad Hmrfar is an associate professor of electricalengineering at Villanova University, PA. He received M.S.and Ph.D. degrees in electrical engineering f b m theUniversity of Colorado at Boulder, in 1978 and 1984,respectively. From 1984-1986 he was a post-doctoralresearch associate in the Electromagnetics Laboratoryat theUniversity of Colorado. In 1986 he became a researchfaculty in the NSF center fo r MicrowaveMillimeter-wavesComputer-Aided Design (MIMICAD) n Boulder. Since 1988he has been with Villanova University where he is now anassociate professor of electrical engineering, programdirector of the ECE graduate program and director of theVillanova's Antenna Research Laboratory. His presentresearch interests include electromagnetic @Id theory,microwave and millimeter-wave antennas and circuits,numerical metho&, and evolutionary computationaltechniques. He was the chair of the joint Antennas andPropagation/'icrowave Theory and Techniques ( A P / m )Ch qter of the IEEE Philadelphia Section from 1993 to1995, and was the General Chairman of the Tweljth andThirteenth IEEE Benjamin Franklin Symposiums inMicrowave and Antenna Technologv held in 1994 and 1995,respectively. He is a senior member of IEEE and a memberof the International Union of Radio Science (URSJ.Vahraz Jamnejad is a principal engineerat the Jet Propulsion Laboratory,Califomia Institute of Technology. Hereceived his MS. nd Ph.D. in electricalengineeringfbm the University of Illinoisat Urbana-Champaign, specializing inelectromagneticsand antenms. At JPL, hehas been engaged in research andsoflware and hardware development invarious areas of spacecra8 antenna technology and satellitecommunication systems. Among other things, he has beeninvolved in the study, design, and development of groundand spacecraj antennas fo r fi ture generations of LandMobile Satellite Systems at L band, Personal Access SatelliteSystems at UKa band as well as eed arrays and reflectorsfor fi ture planetary misswns. His latest work oncommunication satellite systems involved the development ofground mobile antennas or x/Ka band mobile terminal, o ruse with ACTS satellite system. In the past few years, he hasbeen active in research in parallel computationalelectromagnetics as well as in developing antennas forMARS sample return orbiter. More recently he has studiedthe applicability of large arrays of small aperture reflectorantennas for the NASA Deep Space Network, and ispresently active in the design of aprototype array for thisqplication. Over the years, he has received many USpatents and NASA cert3cates of recognition.