2662 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 8, AUGUST 2005

Efficiency-Constrained Particle Swarm Optimizationof a Modified Bernstein Polynomial for Conformal

Array Excitation Amplitude SynthesisDaniel W. Boeringer and Douglas H. Werner, Fellow, IEEE

Abstract—As various enabling technologies advance, conformalphased arrays are finding more numerous applications. Because aconformal array is curved, new far field pattern behaviors emergeand many of the traditional linear and planar phased array syn-thesis methods are not valid. This paper starts by reviewing theequations for the far field of a curved phased array, and providesa generalized definition of aperture efficiency appropriate for con-formal arrays. A modified Bernstein polynomial, defined with justfive parameters, is introduced which provides a flexible method tospecify a variety of smooth unimodal amplitude distributions thatare shown to give good sidelobe levels and aperture efficiencies.By using particle swarm optimization of the modified Bernsteinpolynomial parameters constrained to provide a specified apertureefficiency, a family of aperture distributions and correspondingfar field patterns is produced that allows aperture efficiency to betraded for sidelobe level.

Index Terms—Antenna arrays, antenna radiation patterns, con-formal antennas, optimization methods.

I. INTRODUCTION

FOR conformal phased arrays, generally both the amplitudeand phase of the array elements must be adjusted in orderto maintain low sidelobes as the array is scanned. While it isnatural to choose the phase so it focuses the beam in the de-sired scan direction, the choice of amplitude weights for lowsidelobes and good aperture efficiency is not so obvious. Tradi-tional linear phased array analysis usually assumes that the arrayelements are collinear, equally spaced, and identical, which pro-vide many simplifications that facilitate analytical solutions andinsights into array performance and behavior [1]. When an arrayis curved, the noncollinear element arrangement obstructs manycommon mathematical and computational simplifications, sothat some of these simplifications no longer apply and the lineararray methods may not extrapolate to curved arrays. In partic-ular, since the patterns of individual elements point in differentdirections and not all elements contribute equally in the beampointing direction [2], the total far field cannot be written as theproduct of an array factor with an element pattern. This alsomotivates the attenuation of obscured elements as a functionof scan angle, which does not occur in the linear case. Despite

Manuscript received November 22, 2004; revised February 28, 2005.D. W. Boeringer was with the Department of Electrical Engineering, The

Pennsylvania State University, University Park, PA 16802 USA. He is now withNorthrop Grumman Corporation, Baltimore, MD 21203 USA.

D. H. Werner is with the Department of Electrical Engineering, The Pennsyl-vania State University, University Park, PA 16802 USA (e-mail: dhw@psu.edu).

Digital Object Identifier 10.1109/TAP.2005.851783

these challenges, the antenna community is increasingly inter-ested in the advantages promised by conformal antennas arrays,which include the increased field of view enabled by an an-tenna that is wrapped around a surface, as well as the weightand volume savings associated with more seamless integrationon many platforms.

One of the simplest and most intuitive approaches to con-formal array pattern synthesis is to orient a conventional lowsidelobe amplitude distribution such as Taylor weights along aplane perpendicular to the desired scan direction and sampleit at the element locations projected to this plane [3]–[6], pos-sibly while accounting for mutual coupling or element patterneffects. Although this technique can provide low sidelobes, theresulting distribution may not have the best aperture efficiencyfor the obtained sidelobe level. If the exact desired far field pat-tern of the curved array is known, least squares methods canbe used to provide the closest corresponding weights [7]–[9].Unfortunately, the precise far field pattern is not usually knowna priori nor is it desirable to try to guess it. Typically one doesnot care exactly how the sidelobes are arranged, only that theyare below some level, and to impose an arbitrary pattern isunlikely to give a result that is optimal in other respects suchas aperture efficiency.

A “do not exceed” sidelobe threshold is a difficult nonlinearconstraint not generally handled by analytical methods, con-sequently several numerical techniques have been applied tothe synthesis problem for curved phased arrays. Simulated an-nealing has been applied to the synthesis of csc [2] patterns [10],and pencil beam patterns have been synthesized using nonlinearprogramming [11], alternating projections [12], and a general-ized projection algorithm [13]. In each case, however, the op-timized patterns are obtained without regard for aperture effi-ciency. A common shortcoming of many numerical methods isthat the optimized weights may oscillate in amplitude across thearray. This may theoretically give good sidelobes when mutualcoupling is neglected, but may not be realizable in practice be-cause of mutual coupling effects that tend to smooth out theeffective amplitude distribution. A much better approach is toconstrain the amplitude distribution to be smooth and unimodal,such as the modified Bernstein polynomial introduced in thispaper. It allows the amplitude distribution to shift in the direc-tion of scan, and using an expression for aperture efficiency gen-eralized to include conformal arrays, optimization of this mod-ified Bernstein polynomial provides a way to trade aperture ef-ficiency for low sidelobes.

0018-926X/$20.00 © 2005 IEEE

BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2663

Fig. 1. Geometry and parameters of elements in a curved array. The conformalarray radiates from the convex side.

II. CURVED PHASED ARRAY ANALYSIS

This section reviews equations for curved array far field pat-terns and aperture efficiency, which reduce to familiar conven-tional equations when the array is flat. An optimization criterionthat promotes main beam quality in addition to low sidelobes isintroduced.

A. Far Field Antenna Patterns for a Curved Array

Consider elements distributed on a two-dimensional planarcurve, with locations and oriented normal to the curvewith direction cosines . Assuming that the radiating el-ements are perfectly matched and radiate only from the convexside of the curved array, the element gain pattern of the

th element can be written as

(1)

where is the far field angle measured from the -axis, isthe effective area of the radiating element, and is the wave-length of array operation. The geometry and associated param-eters for the curved array are shown in Fig. 1. For a linear arrayalong the -axis, and , and this reduces to thefamiliar characteristic which is proportional to the el-ement’s projected area with the far field angle and is oftencited as a theoretical bound on the pattern of an array apertureelement [14], [15], [83]. The more general expression above for

merely reorients this characteristic to point inthe direction that the element is aimed.

The far field pattern of an array of of these elementscan be written [16], [17] as

(2)

where and are the amplitude and phase weights of the thelement. In general for a conformal antenna, the element patterncannot be factored outside the summation because the elementorientation varies from element to element. This expression re-duces to the familiar result for the linear phased array along the

axis when and all the are the same [18]. Inthis paper the phase weights will always be chosen to makea phase front in the direction of the chosen scan angle , as

(3)

This deterministic choice of element phase weights coheresall the elements at and guarantees the specified scan angleindependent of the amplitude weights or sidelobe optimizationcost function.

B. Aperture Efficiency for a Curved Array

The generalized aperture efficiency for a conformal antennaused here follows the treatment of Hessel and Sureau [19],[84]. Starting with the realized gain function for perfectlymatched elements

(4)

when all the are phased in the direction of the maximumgain array weights can be written as

(5)

where the denominator normalizes the total array power tounity. Under these conditions the maximum possible gain

can be written as

(6)

For a planar array with identical elements is thesame for each element, and so the maximum gain array weightscorrespond to uniform illumination, as expected for a planararray. For a conformal array, the maximum gain array weightsput the most power into elements that point more toward thedesired scan angle . Taken to the extreme, elements on ageneral conformal array that happen not to point at all in the de-sired scan direction have which implies thatthe associated element weights ; in other words, ele-ments that do not look in the desired scan direction are turnedoff.

2664 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 8, AUGUST 2005

Now the generalized aperture efficiency for a set of ampli-tude weights can be defined relative to the maximum realizedgain condition as

(7)

where the element weights are normalized for unity power as

(8)

For a planar array

(9)

and so this expression for the aperture efficiency simplifies tothe conventional expression of aperture efficiency for a lineararray as

(10)

If the amplitude weights are not normalized, then this ex-pression for planar array aperture efficiency can be modified toa perhaps more familiar form [20]

(11)

The application of Taylor amplitude weights [21] providesbetter sidelobes on linear arrays, but of course this comes atthe expense of decreased antenna efficiency. When all(uniform illumination) for a flat array, , and whentapering such as Taylor weights are applied. Taylor weights arecommonly used for flat phased arrays because they generallyprovide the best aperture efficiency for a given sidelobe level[22], [23], but these lower sidelobe levels do come at the cost ofreduced antenna efficiency.

C. Sidelobe and Main Beam Optimization Criteria

In this paper, one goal will be the optimization of sidelobelevel, by minimizing a cost function related to the sidelobe level.Optimizers are well known for giving “what was asked for in-stead of what was really wanted,” so it is important to define

Fig. 2. Far field pattern family illustrating the difficulties with the traditionaldefinition of sidelobe level definition. (a) An artificial family of possible far fieldantenna patterns. (b) Subset of the antenna patterns with sidelobes technicallybelow �30 dB.

for the optimizer exactly what is meant by sidelobes and side-lobe level. The artificial family of far field antenna patterns inFig. 2 illustrates the potential problem in optimizing far fieldantenna sidelobes. If a peak sidelobe is simply identified as themaximum far field level outside the nulls or dips surroundingthe main beam, then many of the patterns shown in Fig. 2 tech-nically have good sidelobes, although they are clearly of poorquality because of the main beam undulations or shoulders. Ifsuch a simple definition of sidelobe level is used as an optimizercost function then the optimizer could legitimately return resultswith these undesirable shoulders.

To address this problem an additional measure will be placedon the main beam quality. The second derivative of the mainbeam becomes positive at the onset of any main beam undula-tion or shoulders, i.e.,

(12)

The sidelobe level at which this occurs will be called themain beam shoulder level, and the optimizer will treat this asa sidelobe. In this paper, the second derivatives for this mainbeam criterion are found numerically from the calculated pat-tern points. The cost function will be the worst (highest) of thetraditional peak sidelobe level or this main beam shoulder level.Linear array patterns resulting from classical weights like uni-form illumination or Taylor weights have no main beam shoul-ders, in which case this cost function reduces to the traditional

BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2665

Fig. 3. Sidelobe and main beam shoulder levels for the patterns of Fig. 2. The largest of peak sidelobe level and main beam shoulder level is an appropriate costfunction for far field pattern characterization.

peak sidelobe criterion. Fig. 3 illustrates these two potential cri-teria for the artificial antenna patterns of Fig. 2. The combinedpeak sidelobe level and main beam shoulder level criterion cor-rectly penalizes the outermost cases with undulating main beamshoulders.

III. A MODIFIED BERNSTEIN POLYNOMIAL FORARRAY SYNTHESIS

This section introduces a modified Bernstein polynomial forphased array excitation synthesis. The modified Bernstein poly-nomial provides a flexible method for specifying smooth uni-modal functions, which are typically expected and desired forlinear and conformal array aperture amplitude weights. It is de-fined with five parameters , , , , and , on the in-terval as

(13)

for , , , , and. Previous work by the authors [24] proposed a four

parameter version of this modified Bernstein polynomial where.

The complicated-looking ratio at the front of this expressionis merely a normalization factor providing a maximum value ofunity at the single peak which occurs when as shown inthe top of Fig. 4. This facilitates a shift of the excitation max-imum in the direction of scan providing the appropriate attenua-tion of obscured elements. As illustrated in the middle of Fig. 4,

and specify the left and right endpoints andrespectively. The bottom of Fig. 4 shows that increasingsharpens the left side of the peak of , while increasing

Fig. 4. Top: The parameter A controls the unimodal peak location. Middle:The parametersC andC set the height of the left and right endpoints. Bottom:N and N control the sharpness of the two sides of the unimodal peak.

2666 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 8, AUGUST 2005

sharpens the right side. When compared to the conventional thorder Bernstein polynomial [25] of order given by

(14)

it can be seen that the modified Bernstein polynomial reducesto the conventional Bernstein polynomial when

(15)

where , , and , and theleading factor in the large parentheses is merely a constant thatproperly scales the peak value [26].

By its definition the modified Bernstein polynomialcannot have any oscillations, which is a common drawback forthe output of many optimization and curve fitting routines. Ifa highly oscillatory distribution were to be implemented in aphased array, mutual coupling between elements would tend toaverage out the weights, usually changing the effective aperturedistribution and distorting the resulting far field pattern. Theinherently smooth modified Bernstein polynomial avoids thisproblem completely.

IV. PARTICLE SWARM OPTIMIZATION

This section introduces the particle swarm optimizer, and de-scribes its implementation for the optimization of array am-plitude weights for curved antenna pattern synthesis using themodified Bernstein polynomial.

A. Particle Swarm Optimization Overview

Particle swarm optimization is a recently conceived optimiza-tion technique that possesses several highly desirable qualities,including the important property that the basic algorithm is veryeasy to understand and implement. It resembles other evolu-tionary optimization methods such as genetic algorithms, butrequires less bookkeeping and generally fewer lines of code.Particle swarm optimization developed from simulations of syn-chronous bird flocking and fish schooling, when the investi-gators realized that their algorithms possessed an optimizingcharacteristic [27]–[29]. Consider the simultaneous optimiza-tion of variables. A collection or swarm of particles is de-fined, where each particle is assigned a random position in the

-dimensional problem space so that each particle’s positioncorresponds to a candidate solution to the optimization problem.The solutions corresponding to each of these particle positions isscored to obtain a scalar cost based on how well each solves theproblem. Governed by both deterministic and stochastic updaterules, these particles then fly to new positions in the -dimen-sional problem space, which are then mapped to the problemspace and scored. As the particles explore the problem hyper-space, each particle remembers the best position that it has everfound (i.e., its own personal best or more precisely, its localbest). Each particle also knows the best position found by anyparticle in the swarm, called the global best. As the optimiza-tion proceeds, particles are pulled toward these known solutionswith linear spring-like attraction forces. Overshoot and under-shoot combined with stochastic adjustment explore the -di-

mensional problem hyperspace, eventually settling down neara good solution. This process can be modeled as a dynamicalsystem, although the behavior is extraordinarily complex evenwhen extremely simplified update rules are applied to just asingle particle [30]–[32]. This new optimization technique isvery promising, and researchers are just beginning to apply itto electromagnetic problems [33]–[42].

As a new method of evolutionary optimization, particleswarm optimization is often compared to older methods suchas genetic algorithms. Good performance can generally beobtained by both methods. Since the evaluation of the costfunction tends to dominate the overall computation budgetfor electromagnetic optimization, the computational overheadrequirements of both optimization algorithms are usuallynegligible. The two methods traverse the problem hyperspacedifferently, which suggests that there may be optimizationscenarios better suited to one method versus the other. Therecould well be some electromagnetic optimization problems thathave a multidimensional cost function topography for whichthe dynamics of the particle swarm method is quite well suited.A few studies have been performed comparing particle swarmoptimization to genetic algorithms [38], [39], [43], [44], evo-lutionary optimization [45], [46], and ant colony optimization[47]. Some research groups are interested in combining particleswarm techniques with the breeding and selection concepts ofgenetic algorithms [48]–[50]. Many variants of the original par-ticle swarm optimizer have been proposed, such as optimizingmultiple subswarms simultaneously [51]–[53], adding negativeentropy to stir up the particles [54], sharing discoveries among alocal neighborhood of particles [55]–[58], time-varying searchgoals [59], using particle swarms to direct the mutations inevolutionary programming [60], dispersing clustered particlesto increase diversity [61], and using fuzzy logic rules to adap-tively update algorithm parameters [62]. Such variants mayalso find productive applications in electromagnetics. Manyresearchers have had success in hybridizing various algorithmsto obtain better performance than any single algorithm alone;the particle swarm optimizer provides a new ingredient for themix. Particle swarm optimizers have also been proposed forinteger programming [63] and binary optimization [64], [65].A key characteristic of the particle swarm optimizer is that thebasic algorithm itself is highly robust yet remarkably simple toimplement. While possessing similar capabilities as the geneticalgorithm, the much simpler implementation and reduced book-keeping of the particle swarm is appealing. The simpler thealgorithm, the more people can take advantage of it! Althoughrelatively new to the antennas and propagation community,particle swarm optimization appears to have good possibilitiesfor electromagnetic optimization, and there are many excitingpotential applications that have yet to be explored.

B. Particle Swarm Optimization for Curved Array Synthesis

The application of particle swarm optimization to phasedarray synthesis is very recent [24], [37]–[39], [42]. Considerthe particle swarm optimization of the modified Bernsteinpolynomial parameters to obtain good sidelobe performance.The algorithm starts by initializing a group of particles (50in this paper), with random positions in a five-dimensional

BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2667

TABLE IVARIABLES USED IN PARTICLE SWARM OPTIMIZATION UPDATE EQUATIONS

hyperspace, constrained between zero and one in each dimen-sion. The five dimensions of the particle correspond to the fiveparameters of the modified Bernstein polynomial. For eachparticle a random velocity is also initialized, with values in eachof the five dimensions between 1 and 1. As described later,each particle’s five position numbers are mapped to , , ,

and , and the corresponding far field pattern sidelobesare scored for each particle. After all these particles are scored,the best performer is identified as the initial global best.

Now the particles are flown through the problem hyperspacefor a specified number of iterations, using the following sto-chastic velocity and position update rules. For each particlein turn, the first step is to update the velocity separately alongeach dimension according to the velocity update rule

(16)

where the various quantities are discussed in detail below andare summarized in Table I.

Three components typically contribute to this new velocity.The first term is sometimes referred to as “inertia,” “mo-mentum,” or “habit,” which is just proportional to the oldvelocity and is the tendency of the particle to continue in thesame direction it has been traveling. This component is scaledby the constant [66], which keeps the particles in motion anddiscourages premature convergence. Some researchers find ad-vantage in allowing this multiplier to take random values [67],decrease during the optimization to encourage local searchingat the end of the optimization process [68]–[72], randomly varyduring the optimization [73], or be set to zero altogether [74].For this study is taken to be a constant 0.4 throughout theoptimization process, which is found to give good results onvarious problems studied by the authors.

The second term of the velocity update equation is a linear at-traction toward the best position ever found by the given particle(often called local best), scaled by the product of a fixed con-stant and a random number between zero and one. Adifferent random number is used for each dimension of each par-ticle on each iteration. This component is variously referred to as“memory,” “self-knowledge,” “nostalgia,” or “remembrance.”

The third term of the velocity update equation is a linear at-traction toward the best position found by any particle (oftencalled global best), scaled by the product of a fixed constant

and another random number between zero and one,again chosen anew for each dimension of each particle on eachiteration. This component is variously referred to as “cooper-ation,” “social knowledge,” “group knowledge,” or “shared in-formation.” Following common practice in the literature [75],

for the cases considered in this paper. Theseparadigms allow particles to profit both from their own discov-eries as well as the discoveries of the swarm as a whole, mixinglocal and global information uniquely for each particle on eachiteration. The further a particle is from one of these previouslyfound best locations, the more strongly the particle is pulled inthat direction.

The algorithm limits the resulting velocity to a maximumroot mean square value [76], [77] by the rule

(17)

where held constant at 0.3 throughout the optimiza-tion is found to give good results on various problems studiedby the authors. Although many researchers clip the velocity tosome limit along each dimension separately, the rootmean square limit used here seems to work quite well, perhapsbecause it preserves the direction of the updated velocity.Like the inertia weight , large values of orencourage global searching while small values encouragelocal searching [78]. Some researchers find an advantage indecreasing the velocity limit during the optimization process,to localize the search at the end of the optimization.

Next, the new position of the particle is calculated by addingthe new velocity vector to the old particle position vector as

(18)

where a unit time step is assumed. If any dimension of the newposition vector is less than zero or more than one, it is clippedto stay within this range. This new position is then mapped tothe modified Bernstein polynomials parameters and the new re-sulting far field pattern is scored for peak sidelobe and mainbeam shoulder level. If this position has the best score (lowestsidelobes and shoulders) that this particle has found so far, thenit is retained as the local best memory for this particle. If in ad-dition, this position has the best score of any particle so far, thenit is further retained as the global best for the entire swarm. Thefinal array distribution is taken as that generated by the globalbest scoring particle after a specified number of iterations isreached. For this paper 30 iterations (30 iterations 50 par-ticles = 1,500 cost function evaluations) are used.

The current implementation of the particle swarm optimizeruses asynchronous updates [79], where the global best is up-dated after each particle, rather than waiting until all the parti-cles have been scored on a given iteration. This makes the mostcurrent global best information known to all particles as soon asit becomes available. This seems to enhance performance oversynchronous updates, although asynchronous updating is not asconducive to parallel processing as synchronous updates. The

2668 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 8, AUGUST 2005

Fig. 5. Implementation of update equations for particle swarm optimization.

operation of the particle swarm optimizer is summarized withthe psuedocode listed in Fig. 5.

V. RESULTS

A curved array is described using a simple Bézier curve, and ascanned example using conventional Taylor weights is shown togive poor sidelobe and main beam characteristics. By using par-ticle swarm optimization of the modified Bernstein polynomialparameters with a convenient constraint for constant apertureefficiency, a family of aperture distributions and correspondingfar field patterns is produced that allows aperture efficiency tobe traded for sidelobe level while maintaining good main beamquality.

A. Geometry of a Curved Array

Use of a Bézier curve to describe the array curvature providesa more general case than the typical assumption of circular ge-ometry. The parametric equation that represents a simple arraycurvature is given by a one-dimensional Bézier curve with

(19)

and

(20)

where and are in wavelengths, and are thepreviously described conventional Bernstein polynomials. Thearray to be analyzed in this paper contains 30 elements spaced0.5 wavelengths apart along this curve, as shown in Fig. 6(a).For this curved array scanned 30 , Fig. 6 shows that a conven-tional 30 dB Taylor weighting only gives 16 dB sidelobes and13.2 dB shoulders, with generalized efficiency .

Fig. 6. Taylor weights do not lower the peak sidelobes on a curved array.(a) 30 element curved phased array scanned 30 . (b) 30 dB Taylor weights.(c) Resulting far field pattern with�16 dB peak sidelobes and a�13.2 dB peakmain beam shoulder.

Fig. 7. Particle swarm performance improves as the solution evolves. The dotsabove each iteration denote the sidelobe levels corresponding to each of the 50particles’ positions. The line marks the global best performance.

B. Optimization With Constant Aperture Efficiency

In order for the particle swarm optimizer to find the best side-lobe level for a specified aperture efficiency, it is desirable thatthe optimizer only return candidate solutions with that aper-ture efficiency. Fortunately, a simple adjustment of the modifiedBernstein polynomial facilitates this. The five-dimensional can-didate particle positions are always clipped between zero andone, i.e., for the th particle, . These positioncoordinates are first mapped into initial values for the modifiedBernstein polynomial as

(21)

BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2669

TABLE IIMAPPING OF THE GLOBAL BEST PARTICLE’S POSITION TO THE MODIFIED BERNSTEIN POLYNOMIAL GOVERNING THE AMPLITUDE WEIGHTS. THE PARTICLEPOSITION VALUES (LEFT) ARE MAPPED TO THE INITIAL VALUES OF THE MODIFIED BERNSTEIN POLYNOMIAL (CENTER). THE VALUE x = 1:3294 ADJUSTSTHE APERTURE EFFICIENCY TO THE SPECIFIED VALUE � = 0:822 TO OBTAIN THE FINAL MODIFIED BERNSTEIN POLYNOMIAL PARAMETER VALUES (RIGHT)

Since the peak of the distribution shifts in the direction ofscan, is constrained here to be greater than or equal to 0.5,while and can initially cover nearly the entire valid range0 to 1. Observation of the parameter values taken during priorexhaustive empirical trials suggests the initial range of 3–10 for

and .Next, to obtain the desired aperture efficiency, a single value

is found to modify these initial values as

(22)

where is chosen to provide the specified aperture efficiency .The parameter remains unchanged. Decreasing increasesthe aperture efficiency, by broadening the peaks and raising theendpoints; in particular, forces the distribution to be uni-form. Conversely, increasing lowers the aperture efficiency.At every cost function evaluation, a golden section search [80] isused to find the value of that provides the desired aperture effi-ciency. Thus, the particle swarm optimizer only searches param-eter values in a finite range while constraining the efficiency ateach step in a very simple manner, making the optimization fo-cused and efficient. Other possible polynomial expansions suchas a Taylor series do not generally have inherently limited co-efficients, which slows convergence since the range over whichan optimizer must search is then unlimited.

This algorithm is now implemented for the curved array givenabove, to optimize the modified Bernstein polynomial parame-ters to obtain the best sidelobe and shoulder level given an arbi-trary desired efficiency and scan direction of 30 .The performance of the particle swarm during the optimiza-tion process as measured by the peak sidelobe or main beamshoulder level cost function is illustrated in Fig. 7 for 30 it-erations. The best particle’s final position and correspondingmapping to the modified Bernstein polynomial parameters withadjustment for is given in Table II. The resulting opti-mized amplitude distribution and corresponding antenna patternis shown in Fig. 8. Note that the optimized amplitude weights areshifted in the direction of scan, consistent with the intuitive ex-pectation implied by the generalized aperture efficiency that ele-ments which do not point in the desired scan direction should beattenuated. The choice of provides sidelobes around

30.5 dB.

Fig. 8. Optimized amplitude distribution and corresponding antenna pattern.(a) 30 element curved phased array scanned 30 . (b) Modified Bernsteinpolynomial weights. (c) Resulting far field pattern with �30.5 dB peaksidelobes.

C. Independent Efficiency-Constrained Optimization ofSidelobe Performance

Using particle swarm optimization with the aperture effi-ciency constraint, the sidelobe and main beam shoulder levelsare now independently optimized for a selection of efficiencyvalues, at a 30 scan angle. Using the optimization parametersin the previous section, the modified Bernstein polynomialparameters are independently optimized to obtain the best side-lobe level for 13 equally spaced values of aperture efficiency

, requiring patternevaluations for the cost function. In each case the optimizationcost function is the combined peak sidelobe level and mainbeam shoulder level criterion described above. The optimizedparameters of the modified Bernstein polynomial as a functionof aperture efficiency and peak sidelobe level are shown inFig. 9 and Fig. 10, and the resulting amplitude distributionsand corresponding far field antenna patterns are qualitativelyillustrated in Fig. 11. Note how both the amplitude distributionsand the antenna patterns transition smoothly from one to the

2670 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 8, AUGUST 2005

Fig. 9. Optimized A, C , and C vary with efficiency and optimized sidelobes.

Fig. 10. Optimized N and N vary with efficiency and optimized sidelobes.

Fig. 11. Optimized amplitude distributions and corresponding antennapatterns. (a) Amplitude distributions transitioning smoothly from � = 0:76 infront to � = 1:0 in back. (b) Corresponding antenna patterns also transitionsmoothly.

next as the efficiency varies. Because this array is curved, theillumination function for is not uniform.

One way to arrive at intermediate amplitude distributions thatare not specifically optimized is to interpolate between the op-timized values of the modified Bernstein polynomial parame-ters. Since the values of the optimized parameters are knownas a function of aperture efficiency, then given a desired aper-ture efficiency , the corresponding five parameters ofthe modified Bernstein polynomial for good sidelobe perfor-mance can be interpolated from the optimized values of Fig. 9and Fig. 10. Some examples of the peak sidelobe performancefor linear parameter interpolation are shown in Fig. 12 and thestricter peak sidelobe and main beam shoulder performance isshown in Fig. 13. The peak sidelobes characteristic interpo-lates almost perfectly, while some main beam distortion is pos-sible at a few interpolated efficiency values in Fig. 13. Althoughthe second derivative test for main beam distortion indicates a

BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2671

Fig. 12. Interpolated parameters provide cases between optimized points.

Fig. 13. Interpolated parameters can raise main beam shoulders slightly.

possible main beam problem, the actual far field patterns forthe suspicious cases reveal that the actual distortion is negli-gible. It is likely that the adjacent optimized patterns are close todisplaying some variation in the second derivative, which thenmanifests itself in the interpolation process. If desired, even thisslight distortion can be removed by optimizing these specificcases.

VI. CONCLUSION AND FUTURE WORK

This paper illustrates a new approach to curved phased arraysynthesis using the particle swarm optimization of a modifiedBernstein polynomial. In addition to the usual peak sidelobecriterion, a useful main beam criterion is developed that helpsensure good main beam quality in the optimized results. Themodified Bernstein polynomial is defined by just five parame-ters for simplicity and fast optimization, and provides smoothunimodal amplitude distributions that are demonstrated to givegood sidelobe performance and aperture efficiency. Recently,genetic algorithms have been used to synthesize low sidelobeweighting distributions while maximizing the aperture effi-ciency [81], [82]. The results of this paper go a step further forthe more general conformal phased array case, constraininga particle swarm optimizer to provide only solutions with thespecified aperture efficiency.

When the optimization is repeated for a selection of efficiencyvalues, the optimized amplitude distributions and the resultingantenna patterns transition smoothly from one to the next asthe efficiency varies. These optimized results can be interpo-

lated to obtain amplitude distributions with intermediate valuesof aperture efficiency and peak sidelobe level. There are manyadditional exciting avenues to explore for future work. Opti-mizing the modified Bernstein polynomial as a function of ad-ditional variables such as scan angle, curvature, and elementspacing could provide valuable insights into curved array phe-nomenology. Another possible line of investigation for futurework is a multimodal formulation of the modified Bernsteinpolynomial, which could accommodate undulating array curva-tures with more than one hump or extrema.

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BOERINGER AND WERNER: EFFICIENCY-CONSTRAINED PARTICLE SWARM OPTIMIZATION 2673

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Daniel W. Boeringer received the B.S.E.E. and the B.S. degrees in computerscience in 1990 and the M.S.E.E. degree in 1994 from the University of NorthCarolina, Charlotte, and the Ph.D. degree in electrical engineering from ThePennsylvania State University, University Park, under a corporate fellowshipfrom Northrop Grumman Corporation, Baltimore, MD.

He is an Advisory Engineer with Northrop Grumman Electronic Systems,Baltimore, where he designs antenna systems for various platforms and appli-cations. His interests include phased-array architecture and optimization.

Douglas H. Werner (S’81–M’89–SM’94–F’05) re-ceived the B.S., M.S., and Ph.D. degrees in electricalengineering and the M.A. degree in mathematicsfrom The Pennsylvania State University (PennState), University Park, in 1983, 1985, 1989, and1986, respectively.

He is a Professor in the Department of ElectricalEngineering, Penn State. He is a Member of theCommunications and Space Sciences Lab (CSSL)and is affiliated with the Electromagnetic Com-munication Research Lab. He is the Director of

the Computational Electromagnetics and Antennas Research Lab (CEARL)http://labs.ee.psu.edu/labs/dwernergroup/. He is also a Senior Scientist inthe Electromagnetics and Environmental Effects Department of the AppliedResearch Laboratory at Penn State. He has published numerous technicalpapers and proceedings articles and is the author of nine book chapters. He isan Editor of Frontiers in Electromagnetics (Piscataway, NJ: IEEE Press, 2000).He also contributed a chapter for Electromagnetic Optimization by GeneticAlgorithms (New York: Wiley Interscience, 1999) as well as Soft Computing inCommunications (New York: Springer, 2004). He is a former Associate Editorof Radio Science. His research interests include theoretical and computationalelectromagnetics with applications to antenna theory and design, microwaves,wireless and personal communication systems, electromagnetic wave interac-tions with complex media, meta-materials, fractal and knot electrodynamics,neural networks, genetic algorithms and particle swarm optimization.

Dr. Werner is a Member of Eta Kappa Nu, Tau Beta Pi, Sigma Xi, theAmerican Geophysical Union (AGU), the International Union of RadioScience (URSI) Commissions B and G, and the Applied ComputationalElectromagnetics Society (ACES). He was presented with the 1993 AppliedComputational Electromagnetics Society (ACES) Best Paper Award and wasalso the recipient of a 1993 URSI Young Scientist Award. In 1994, he receivedThe Pennsylvania State University Applied Research Laboratory OutstandingPublication Award. He was the recipient of a College of Engineering PSESOutstanding Research Award and Outstanding Teaching Award, in March 2000and March 2002, respectively, and was recently presented with an IEEE CentralPennsylvania Section Millennium Medal. He has also received several Lettersof Commendation from Penn State’s Department of Electrical Engineering forOutstanding Teaching and Research. He is an Editor of the IEEE Antennas andPropagation Magazine.

tocEfficiency-Constrained Particle Swarm Optimization of a ModifiedDaniel W. Boeringer and Douglas H. Werner, Fellow, IEEEI. I NTRODUCTION

Fig.€1. Geometry and parameters of elements in a curved array. TII. C URVED P HASED A RRAY A NALYSISA. Far Field Antenna Patterns for a Curved ArrayB. Aperture Efficiency for a Curved ArrayC. Sidelobe and Main Beam Optimization Criteria

Fig.€2. Far field pattern family illustrating the difficulties wFig.€3. Sidelobe and main beam shoulder levels for the patterns III. A M ODIFIED B ERNSTEIN P OLYNOMIAL FOR A RRAY S YNTHESIS

Fig.€4. Top: The parameter $A$ controls the unimodal peak locatiIV. P ARTICLE S WARM O PTIMIZATIONA. Particle Swarm Optimization OverviewB. Particle Swarm Optimization for Curved Array Synthesis

TABLE€I V ARIABLES U SED IN P ARTICLE S WARM O PTIMIZATION U PDFig.€5. Implementation of update equations for particle swarm opV. R ESULTSA. Geometry of a Curved Array

Fig.€6. Taylor weights do not lower the peak sidelobes on a curvFig.€7. Particle swarm performance improves as the solution evolB. Optimization With Constant Aperture Efficiency

TABLE€II M APPING OF THE G LOBAL B EST P ARTICLE ' S P OSITION Fig.€8. Optimized amplitude distribution and corresponding antenC. Independent Efficiency-Constrained Optimization of Sidelobe P

Fig. 9. Optimized $A$, $C_{0}$, and $C_{1}$ vary with efficiencyFig. 10. Optimized $N_{0}$ and $N_{1}$ vary with efficiency and Fig.€11. Optimized amplitude distributions and corresponding antFig.€12. Interpolated parameters provide cases between optimizedFig.€13. Interpolated parameters can raise main beam shoulders sVI. C ONCLUSION AND F UTURE W ORKR. C. Hansen, Significant Phased Array Papers . Dedham, MA: ArteW. H. Kumar, Preface to the special issue on conformal arrays, II. Chiba, K. Hariu, S. Sato, and S. Mano, A projection method prK.-I. Haryu, I. Chiba, S. Mano, and T. Katagi, Null points adjusN. Kojima, K.-I. Hariu, and I. Chiba, Low sidelobe pattern synthA. Ludwig, Curved array pattern synthesis, in 1985 IEEE AntennasJ. Zheng, Pattern synthesis of cylindrical phased array by usingP. N. Fletcher and M. Dean, Least squares pattern synthesis for G. Mazzarella and G. Panariello, Pattern synthesis of conformal J. A. Ferreira and F. Ares, Pattern synthesis of conformal arrayY.-C. Jiao, W.-Y. Wei, L.-W. Huang, and H.-S. Wu, A new low-sideH. Steyskal, Pattern synthesis for a conformal wing array, in 20O. M. Bucci, A. Capozzoli, and G. D'Elia, Power pattern synthesiW. Wasylkiwskyj, Element pattern bounds in uniform phased array,H. J. Stalzer, J. Shmoys, and A. Hessel, Element pattern of dualE. A. Wolff, Antenna Analysis . New York: Wiley, 1966.A. Tennant, A. F. Fray, D. B. Adamson, and M. W. Shelley, Beam sT. C. Cheston and J. Frank, Phased array radar antennas, in RadaA. Hessel and J.-C. Sureau, On the realized gain of arrays, IEEER. J. Mailloux, Phased Array Antenna Handbook . Boston, MA: ArteT. T. Taylor, Design of line-source antennas for narrow beamwidtR. C. Hansen, Phased Array Antennas . New York: Wiley, 1998.D. W. Boeringer and D. H. Werner, Particle swarm optimization ofL. Piegl and W. Tiller, The NURBS Book . Berlin: Springer, 1997.E. W. Weisstein . Bernstein Polynomial [Online] http://mathworldR. Eberhart and J. Kennedy, A new optimizer using particle swarmJ. Kennedy and R. Eberhart, Particle swarm optimization, in 1995J. Kennedy and R. C. Eberhart, Swarm Intelligence . San FranciscJ. Kennedy, The behavior of particles, in Proc. 7th Ann. Conf. EM. Clerc and J. Kennedy, The particle swarm explosion, stabilityE. Ozcan and C. K. Mohan, Particle swarm optimization: surfing tG. Ciuprina, D. Ioan, and I. Munteanu, Use of intelligent-particB. Brandstatter and U. Baumgartner, Particle swarm optimization J. Robinson, S. Sinton, and Y. Rahmat-Samii, Particle swarm, genJ. Robinson and Y. Rahmat-Samii, Particle swarm optimization in D. Gies and Y. Rahmat-Samii, Reconfigurable array design using pD. W. Boeringer and D. H. Werner, A comparison of particle swarmD. Gies and Y. Rahmat-Samii, Particle swarm optimization (PSO) fL. Matekovits, M. Mussetta, P. Pirinoli, S. Selleri, and R. E. ZD. Gies and Y. Rahmat-Samii, Vector evaluated particle swarm optR. J. W. Hodgson, Particle swarm optimization applied to the atoR. C. Eberhart and Y. Shi, Comparison between genetic algorithmsP. J. Angeline, Evolutionary optimization versus particle swarm V. Miranda and N. Fonseca, EPSO best-of-two-worlds meta-heuristiL. Schoofs and B. Naudts, Swarm intelligence on the binary constM. Løvbjerg, T. K. Rasmussen, and T. Krink, Hybrid particle swarP. J. Angeline, Using selection to improve particle swarm optimiS. Naka, T. Genji, T. Yura, and Y. Fukuyama, Practical distributF. van den Bergh and A. P. Engelbrecht, Effects of swarm size onY. Shi and R. A. Krohling, Co-evolutionary particle swarm optimiF. van den Bergh and A. P. Engelbrecht, Training product unit neX.-F. Xie, W.-J. Zhang, and Z.-L. Yang, Dissipative particle swaJ. Kennedy and R. Mendes, Population structure and particle swarT. Krink, J. S. Vesterstrom, and J. Riget, Particle swarm optimiR. Mendes, P. Cortez, M. Rocha, and J. Neves, Particle swarms foJ. Kennedy, Small worlds and mega-minds: effects of neighborhoodB. Al-kazemi and C. K. Mohan, Multi-phase generalization of the C. Wei, Z. He, Y. Zhang, and W. Pei, Swarm directions embedded iM. Løvbjerg and T. Krink, Extending particle swarm optimizers wiY. Shi and R. C. Eberhart, Fuzzy adaptive particle swarm optimizE. C. Laskari, K. E. Parsopoulos, and M. N. Vrahatis, Particle sJ. Kennedy and R. C. Eberhart, A discrete binary version of the J. Kennedy and W. M. Spears, Matching algorithms to problems: anY. Shi and R. Eberhart, A modified particle swarm optimizer, in X. Hu and R. C. Eberhart, Adaptive particle swarm optimization: P. N. Suganthan, Particle swarm optimizer with neighborhood operR. C. Eberhart and Y. Shi, Comparing inertia weights and constriE. C. Laskari, K. E. Parsopoulos, and M. N. Vrahatis, Particle sA. A. Abido, Particle swarm optimization for multimachine power Y. Shi and R. C. Eberhart, Empirical study of particle swarm optA. I. El-Gallad, M. El-Hawary, A. A. 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