Self-adaptive differential evolution applied to real-valued antenna and microwave design problems

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Self-Adaptive Differential Evolution Applied to Real-Valued Antenna and Microwave Design Problems Sotirios K. Goudos, Member, IEEE, Katherine Siakavara, Member, IEEE, Theodoros Samaras, Member, IEEE, Elias E. Vafiadis, Member, IEEE, and John N. Sahalos, Life Fellow, IEEE

Abstract—Particle swarm optimization (PSO) is an evolutionary algorithm based on the bird fly. Differential evolution (DE) is a vector population based stochastic optimization method. The fact that both algorithms can handle efficiently arbitrary optimization problems has made them popular for solving problems in electromagnetics. In this paper, we apply a design technique based on a self-adaptive DE (SADE) algorithm to real-valued antenna and microwave design problems. These include linear-array synthesis, patch-antenna design and microstrip filter design. The number of unknowns for the design problems varies from 6 to 60. We compare the self-adaptive DE strategy with popular PSO and DE variants. We evaluate the algorithms’ performance regarding statistical results and convergence speed. The results obtained for different problems show that the DE algorithms outperform the PSO variants in terms of finding best optima. Thus, our results show the advantages of the SADE strategy and the DE in general. However, these results are considered to be indicative and do not generally apply to all optimization problems in electromagnetics. Index Terms—Differential evolution (DE), evolutionary algorithms (EAs), linear array synthesis, microwave filter design, optimization methods, particle swarm optimization (PSO), patch antenna design.

I. INTRODUCTION

S

EVERAL evolutionary algorithms (EAs) have emerged in the past decade that mimic biological entities behavior and evolution In this paper we consider particle swarm optimization (PSO) [1] and Differential evolution (DE) [2], [3]. PSO [1] is an evolutionary algorithm based on the bird fly. It is an easy-to-implement algorithm. PSO has been used successfully in constrained and unconstrained electromagnetic design problems [4]–[24]. Differential evolution (DE) [2], [3] is a population-based stochastic global optimization algorithm. Several DE variants or strategies exist. An overview of both PSO and DE algorithms and the hybridizations of these algorithms with other soft computing tools can be found in [25]. The classical DE strategy has Manuscript received December 13, 2009; revised May 08, 2010; accepted August 28, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2011.2109678

been applied to microwave structures [26]–[28], antenna design [29]–[35], signal optimization [36] and microwave imaging applications [37]–[44]. DE produced better results than PSO on numerical benchmark problems with low and medium dimensionality (30 and 100 dimensions) [45]. However, on noisy test problems, DE was outperformed by PSO. In [46] a comparative study between DE and PSO variants is presented for the design of radar absorbing materials (RAM). The number of problem dimensions was 10 and DE outperformed the PSO variants in terms of convergence speed and best values found. The shape reconstruction of a perfectly conducting 2-D scatterer using DE and PSO is presented in [40], [44]. Also both algorithms have been applied to 1-D small-scale inverse scattering problems [43]. In these cases, DE outperformed PSO. In [47] a comparison between DE, PSO and Genetic algorithms (GAs) for circular array design is presented. DE and PSO showed similar performances and both of them had better performance compared to GAs. One of the DE advantages is that very few control parameters have to be adjusted in each algorithm run. However, the control parameters involved in DE are highly dependent on the optimization problem. Therefore, it is not always an easy task to tune these parameters. Recently a novel DE strategy has been applied to numerical benchmark problems that self-adapts the control parameters (SADE) [48]. SADE has been applied successfully to a microwave absorber design problem [49]. In this paper, SADE is compared with other algorithms. The comparison is performed on common real-valued antenna and microwave design problems. These problems include linear-array synthesis with sidelobe level suppression and null control in specified directions. In order to evaluate the algorithms’ performance combined with a numerical solver, we apply the algorithms to the design of a dual-band E-shaped patch-antenna and of a microstrip bandpass filter. As numerical solver we employ FEKO [50], a commercially available EM solver. We compare the SADE with two PSO variants and the classical DE/rand/1/bin strategy. The numerical results show the advantages of the SADE approach and the DE in general. However, these results cannot lead to the general conclusion that DE outperforms PSO in all optimization problems in electromagnetics. This paper is organized as follows: In Section II we describe the PSO and DE algorithms. We present the numerical results in Section III. Finally, we give the conclusion in Section IV.

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GOUDOS et al.: SADE APPLIED TO REAL-VALUED ANTENNA AND MICROWAVE DESIGN PROBLEMS

II. EVOLUTIONARY ALGORITHMS A. Initialization A population (or swarm) in both PSO and DE consists of vectors (or particles) , where is the generation number. The population is initialized randomly from a uniform distribution. Each -dimensional vector represents a possible solution, which is expressed as

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Clerc [51] suggested the use of a different velocity update rule, which introduced a parameter called constriction factor. The role of the constriction factor is to ensure convergence when all the particles have stopped their movement. The velocity update rule is then given by

(4) (5)

(1) The population is initialized as follows: (2) where and are -dimensional vectors of the lower is a uniformly and upper bounds respectively and . The stopping criterion distributed random number within for both PSO and DE is usually the generation number or the number of objective-function evaluations. B. Particle Swarm Optimization In PSO, the particles move in the search space, where each particle position is updated by two optimum values. The first one is the best solution (fitness) that has been achieved so far. This value is called . The other one is the global best value obtained so far by any particle in the swarm. This best value is . called and , the velocity update rule After finding the is an important factor in a PSO algorithm. The most commonly used algorithm defines that the velocity of each particle for every problem dimension is updated with the following:

(3) where sion,

is the particle velocity in the dimendenotes the current iteration and the previous, is the particle position in the nth dimension, , are uniformly distributed random numbers in (0,1), is a parameter known as the inertia weight and and are the learning factors. The parameter (inertia weight) is a constant between 0 and 1. This parameter represents the particle’s fly without any external influence. The higher the value of , or the closer it is to one, the more the particle stays unaffected from pbest and gbest. The parameter represents the influence of the particle memory on its best position, while the parameter represents the influence of the swarm best position. Therefore, in the Inertia Weight PSO (IWPSO) algorithm the parameters to be determined are: the swarm size (or population size), usually 100 or less, the cognitive learning factor and the social learning factor (usually both are set to equal to 2.0), the inertia weight and the maximum number of iterations. It is common practice to linearly decrease the inertia weight starting from 0.9 or 0.95 to 0.4.

where and . This PSO algorithm variant is known as constriction factor PSO (CFPSO). Boundary conditions in PSO play a key role as it is pointed out in [52], [53]. In this paper we have applied the reflective walls boundary conditions. C. Differential Evolution In DE algorithms, the initial population evolves in each generation with the use of three operators: mutation, crossover and selection. Depending on the form of these operators several DE variants or strategies exist in the literature [3], [54]. The choice of the best DE strategy depends on the problem type [55]. The most popular is the one known as DE/rand/1/bin strategy. In this strategy, a mutant vector for each target vector is computed by (6) and are randomly chosen indices from the where , population and is a mutation control parameter. After mutation, the crossover operator is applied to generate a trial vector whose coordinates are given by if if (7) , is a number from a uniform where , a randomly random distribution from the interval chosen index from and the crossover constant . DE uses a greedy selection operator, from the interval which for minimization problems is defined by if otherwise

(8)

, are the fitness values of the trial where and the old vector respectively. Therefore, the newly found trial replaces the old vector only when it produces vector a lower objective-function value than the old one. Otherwise, the old vector remains in the next generation. The stopping criterion for the DE is usually the generation number or the number of objective-function evaluations. D. Self-Adaptive Differential Evolution (SADE) Storn has suggested [3] to choose the differential evolution and from the intervals and control parameters

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, respectively and to set . The correct selection of these control parameter values is, frequently, a problemdependent task. Multiple algorithm runs are often required for fine-tuning the control parameters. In [48] a novel strategy is proposed for the self-adapting of DE control parameters. This strategy is based on DE/rand/1/bin scheme. Each vector is exvalues. The basic idea is based on tended with its own and the evolution of the control parameters. New vectors are found by using the improved values of the control parameters. These vectors are more likely to survive and produce offspring. In turn, the newly found vectors propagate the improved values of the control parameters to the next generation. Therefore, the control parameters are self-adjusted in every generation for each individual according to the following scheme: if otherwise if otherwise

(9)

where

are uniform random numbers , , are the lower and the upper limits of F set to 0.1 and 0.9, respectively and and represent the probabilities to adjust the control parameters. The authors in [48] have set both these probabilities to 0.1 after several trials. They have tested the SADE performance with several low dimension benchmarks. Their conclusion is that the self-adaptive strategy is better or at least comparable to the classical DE DE/rand/1/bin strategy considering the quality of the solutions found. Therefore, by using the self-adaptive strategy the user does not have to adjust the and parameters while the time complexity does not increase. More details about the Self-adaptive DE strategy can be found in [48]. III. NUMERICAL RESULTS It must be pointed out that several PSO and DE variants exist in the literature. In order to select, the best algorithm for every problem one has to consider the problem characteristics. For example, micro-PSO performs very well for microwave image reconstruction [56]. Another key issue is the selection of the algorithm control parameters, which is also in most cases problemdependent. In this paper, we compare SADE with common PSO and DE algorithms. The control parameters for these algorithms are those that commonly perform well regardless of the characteristics of the problem to be solved. We apply all algorithms to real-valued antenna and microwave design problems. These include linear-array synthesis, patch-antenna and microstrip filter design. The linear-array synthesis design cases are executed 50 times for each algorithm. In all other cases, the algorithms are run for 20 independent trials. The best results are compared. All algorithms are compiled using the same compiler (Borland C++ Builder 5.0) on a PC with Intel Core 2 Duo E8500 at 3.16 GHz with 4 GB RAM running Windows XP. The C source code for the DE algorithm was the one given by Storn in [57]. This code was modified to include the self-adaptive DE strategy.

Fig. 1. Geometry of a 2N-element linear array along the x-axis.

The best value, the worst value, the mean and the standard deviation of the last generation computed by each algorithm are to presented here. In the standard DE, is set to 0.5 and 0.9. In the PSO algorithms and are set equal to 2.05. For . The velocity is upCFPSO, these values result in dated asynchronously, which means that the global best position is updated the moment it is found. The stopping criterion for all algorithms is the generation number. For all examples the population size and the number of generations is set equal for all algorithms. The same initial conditions are used for all algorithms. A. Linear-Array Synthesis We consider a 2N-element uniformly excited linear array symmetrically placed along the x-axis (Fig. 1). The array factor in the x-z plane is expressed as (10) where is the wavelength, and are the position and the phase of the th element, respectively, and and are the corresponding vectors. Equation (11) expressed in dB is written as (11) where is the direction of the maximum. The optimization goal is the sidelobe level (SLL) suppression by finding the optimum element positions and phases. This problem is defined by the minimization of the objective function (12) where is the set of theta angles that are outside the angular range of the mainlobe. In order to find the maximum SLL the first zero has to be found first. This is accomplished by using a simple search algorithm that scans the AF values after the mainlobe maximum. A brief description of the search algorithm follows. We start from the mainlobe maximum value and increase . We calculate the difference of the the AF scan in steps of current AF value minus the previous one. If this difference is negative, the angle is increased again and the next AF value is calculated. On the other hand, when the difference becomes positive, which means that the AF scan passed the mainlobe, the first zero is found.

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TABLE I COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-PHASE SYNTHESIS

= 0:6

d

Fig. 3. Best Array patterns obtained by SADE for the 32-element array case for d : .

= 06

TABLE II COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-ONLY SYNTHESIS

d

=

Fig. 2. Convergence rate for the 32-element array case (a) position-only syn: . thesis and (b) position-phase synthesis for d

=06

For sidelobe level suppression when null control in specified directions is required, the objective function may be defined as (13) where is the number of the required null directions, are weight factors and is defined as if otherwise.

and

(14)

is the desired null level in dB and the direction of Here, null. the We examine different design options for uniformly excited arrays, which include position-only, phase-only and position-phase synthesis. For position-only and phase-only synthesis the number of unknowns is , while for position-phase synthesis this number is . We assume that for is the minimum distance between two all cases, where adjacent elements. is varied depending on The maximum allowed distance the design case as in [30]. A population of 100 vectors is selected for all algorithms. The total number of iterations is set to 2000. The first design case is that of a 32-element array. We consider first that and we apply all algorithms to position-only and position-phase synthesis. For position-only

synthesis, all algorithms converge to the same best value in all trials. The convergence rate of the average objective function for this design case is given in Fig. 2(a). It is obvious that CFPSO, DE and SADE converge faster than the classical IWPSO algorithm. For position-phase synthesis, the algorithms obtain different results. The convergence-rate plot for this case is shown in Fig. 2(b). The DE strategies and the CFPSO converge faster than the IWPSO. The classical DE strategy converges faster than the SADE strategy but both obtain similar results. Table I presents the results for all algorithms. The self-adaptive DE produces slightly better results than the classical DE and both are better than the PSO ones. The patterns of the best array designs obtained for both position-only and position-phase synthesis are shown in Fig. 3. SADE obtains an SLL of (instead of in [30]) for position-only synthesis. For position-phase synthesis the best SLL value obtained by SADE ( in [30]). is For the second design case we set . Table II shows the results for position-only synthesis. Both DE strategies clearly outperform the PSO algorithms. The DE strategies obtain smaller standard deviation values than the PSO ones. Table III shows the results for position-phase synthesis. SADE

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TABLE III COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-PHASE SYNTHESIS

d

=

Fig. 5. Best Array patterns obtained by SADE for the 32-element array case . for d

=

TABLE IV ELEMENT POSITIONS AND PHASES FOR THE 32-ELEMENT ARRAY

Fig. 4. Convergence rate for the 32-element array case (a) position-only syn. thesis and (b) position-phase synthesis for d

=

obtains the best objective-function value of all algorithms, while it outperforms the other algorithms in terms of mean and worst values. The convergence-rate plots for these cases are shown in Fig. 4(a) and (b). SADE, DE and CFPSO converge at a similar speed. The patterns of the best designs are shown in Fig. 5. The (instead of maximum SLL values are [30]) and (instead of [30]) for position-only and position-phase synthesis respectively. Table IV shows the position and phase values of the best designs obtained for all the 32-element array design cases. The next design case is that of a 60-element array. We apply the algorithms to phase-only and position-phase synthesis. We as in [30]. Table V presents the statistical reset sults of the objective-function values for both phase-only and position-phase synthesis. It is evident again that both DE strategies produce better results than the PSO ones. For phase-only

synthesis, the classical DE strategy obtains slightly better results than the SADE. However, SADE obtains a lower standard deviation value. For position-phase synthesis, SADE clearly outperforms the other algorithms. CFPSO obtains a better best value than the classical DE, but the classical DE produces better results than the CFPSO algorithm in terms of mean and standard deviation values. Fig. 6(a) and (b) show the convergence-rate plots. CFPSO, DE and SADE converge at similar speeds. Fig. 7 presents the patterns of the best designs obtained for both phase-only and position-phase design cases. The max(instead of imum SLL of the phase-only synthesis is in [30]) while for position-phase synthesis the SLL (instead of in [30]). Table VI value is presents the optimum phases and positions derived by the SADE strategy for phase-only and position-phase synthesis. It must be pointed out that the authors in [30] have not used the classical DE strategy. Instead, they have used a DE variant that generates two trial vectors for every vector of the population using different strategies. The values for the control parameters are not reported for all cases in [30]. For the 32-element

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Fig. 7. Best Array patterns obtained by SADE for the 60-element array case for d = 0:7.

TABLE VI ELEMENT POSITIONS AND PHASES FOR THE 60-ELEMENT ARRAY

Fig. 6. Convergence rate for the 60-element array case (a) phase-only synthesis and (b) position-phase synthesis for d 0:7.

=

TABLE V COMPARATIVE RESULTS FOR 60 ELEMENTS PHASE-ONLY AND POSITION-PHASE SYNTHESIS

case with the population size is set to 320 and the number of iterations is 300. This results in objective-function evaluations. In our results, we have objective-function evaluations. Thus, the number of objective-function evaluations is slightly higher in our case, which could explain the small differences with [30]. We also notice that in the design cases where only positions or phases are optimized the differences with [30] are small, ranging from 0.11 dB to 0.13 dB. This indicates that in such cases where the problem search space is small, it is less probable to obtain a significantly better solution using a number of objective-function evaluations similar to [30]. We obtain slightly

better improvements to the solutions of [30] in the 32-element position-phase synthesis design cases (0.18 dB to 0.25 dB). The better improvement of 2.21 dB is achieved for the 60-element position-phase synthesis. For this case, the search space becomes larger (60 unknowns) and it is therefore more probable to obtain better solutions. The final linear-array synthesis design case is a case of SLL suppression and null control in three directions. The desired null directions are at 30, 32.5 and 35 degrees. The desired null . The array consists of 28 elements as in level is set to

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TABLE VIII ELEMENT POSITIONS FOR THE 28-ELEMENT ARRAY WITH NULL CONTROL

Fig. 8. Convergence rate for the 28-element array case, position-only synthesis 0:7. for d

=

TABLE VII COMPARATIVE RESULTS FOR 28 ELEMENTS POSITION-ONLY SYNTHESIS WITH NULL CONTROL

Fig. 9. Best Array patterns obtained by SADE and DE for the 28-element array = 0:7. case for d

[8]. Table VII shows the results. It is evident that SADE obtains better results than the other algorithms. Up to numerical errors, the same values are obtained by the two DE strategies. The standard deviation values denoted as zero in Table VII are . The DE strategies converge at similar speeds, less than slightly faster than the CFPSO algorithm as shown in Fig. 8. IWPSO converges at slower speed, but the final objective-function value is close to those obtained by the other algorithms. The best patterns obtained by the classical DE and the Selfadaptive DE are shown in Fig. 9. The SLL obtained in [8] using and the null values are below . Both PSO is . The best DE strategies obtain nulls at a level lower than SLL values obtained by the classical DE and the Self-adaptive and respectively. The element DE are positions for these cases are presented in Table VIII.

Fig. 10. Geometry of a E-shaped patch antenna.

B. E-Shaped Patch-Antenna Design The use of microstrip patch antennas in wireless communication systems provides several advantages like low profile, low cost and ease of fabrication. E-shaped patch antennas extend the patch functionality and bandwidth. They are suitable for dual-band or wide-band designs. The design of such antennas requires the determination of the geometrical parameters of the antenna that satisfy the design requirements at the desired frequencies. PSO and DE have been in several occasions [58], [59] applied to E-shaped patch-antenna design. The geometry of an E-shaped patch antenna is given in Fig. 10. Two parallel slots are incorporated into the rectangular patch. A coaxial feed is used, which introduces two possible lengths to generate different resonant frequencies.

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TABLE IX COMPARATIVE RESULTS FOR THE E-SHAPED PATCH-ANTENNA DESIGN

The

E-shaped

patch has six design parameters: . These are the patch width , the patch length , the slot width , the slot length , the slot and the feed position . The ground-plane size position and a 5.5 mm thick air substrate is fixed at is used as in [59]. The design goal is to minimize the magnitude in two frequencies 5 GHz and 5.5 GHz, which implies the minimization of the objective function

Fig. 11. Convergence rate for the E-shaped patch-antenna design.

(15) where is the vector of the antenna geometry. In order to maintain the E-shape, additional restrictions apply to the design parameters. These are [58]

Fig. 12. Simulated S

(16) The computation of the objective function requires the use of a full-wave numerical method. The E-shaped patch antenna is modeled in FEKO. In order to integrate the in-house source code of the evolutionary algorithms with FEKO, a wrapper program is created. The implementation of a separate optimizer that calls external software to do the simulations for the evaluation of the objective function can also be found in [60]. A population of 10 vectors is selected for all algorithms. The total number of generations is set to 200. In this case execution time plays an important role. Therefore, fewer objective-function evaluations are selected than in the linear-array synthesis case. Table IX presents the statistical results of the objective-function values for each algorithm. Fig. 11 shows the convergence rate. SADE seems to perform better than the other algorithms. SADE obtains the best objective-function value, the highest mean value and the smallest standard deviation value. Both DE strategies perform better than the PSO algorithms. IWPSO obtains a better best value than CFPSO but CFPSO produces better mean and standard deviation values than IWPSO. SADE has a faster convergence speed than the other DE algorithm. The DE strategies converge faster than the PSO algorithms. The frequency response of the best designs found by the selfadaptive DE (Des 1) and the classical DE (Des 2) algorithms is given in Fig. 12. A finer mesh is used in FEKO model with 100 frequency points to provide a smooth response. The Des

curves for best designs found by SADE and DE.

TABLE X DESIGN PARAMETERS FOR THE BEST DESIGNS FOUND BY DE AND SADE (mm)

1 antenna has a bandwidth between 4.78 and 5.91 GHz with , while the Des 2 design has a similar bandwidth between 4.8 and 5.91 GHz. For both designs the values are lower than between 4.88 and 5.73 GHz. Table X has the design parameters obtained for both designs. The surface current distributions for designs Des 1 and Des 2 at 5 GHz and 5.5 GHz are shown in Fig. 13. C. Microstrip Bandpass Filter The final example presents the design of a bandpass microstrip filter (Fig. 14). A substrate with dielectric constant equal to 9 and 0.66 mm thickness is considered [61]. Again the filter is modeled in FEKO. In [62], [63] the space-mapping technique is used for filter design. This is accomplished in conjunction with FEKO. Such a filter design problem can be defined by two objectives subject to two constraint functions. in the passband The first objective is to maximize the frequency range. The second objective is to minimize the in the stopband frequency range. Additionally, constraints can

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TABLE XI DESIGN SPECIFICATIONS FOR THE BAND-PASS FILTER

Fig. 13. Simulated surface current distribution of the two best designs (a) Des 1 at 5 GHz, (b) Des 1 at 5.5 GHz, (c) Des 2 at 5 GHz, (d) Des 2 at 5.5 GHz.

Fig. 15. Convergence rate for the microstrip bandpass filter design case. TABLE XII COMPARATIVE RESULTS FOR THE BAND-PASS FILTER DESIGN CASE

Fig. 14. Bandpass microstrip filter geometry.

be set for levels in both the passband and the stopband frequency range. An effective way to combine the above objectives in one objective function is to use an exact penalty method [64]. Therefore, this design problem is defined by the minimization of the objective function

(17) where is the vector of filter geometry, and define the corresponding passband and the stopand define the minimum and band frequency ranges, maximum allowable values in the passband and stopband frequency ranges respectively and is a very large number. In a penalty method the feasible region is expanded, but a large cost or “penalty” is added to the original objective function for solutions that lie outside the original feasible region. Therefore is chosen large enough to ensure that solutions that do not fulfill constrains result in large fitness values. Table XI shows the filter design specifications. For each FEKO run, 17 frequency sweeps are taken in the frequency range 4.3–5.5 GHz. A population of 15 vectors is selected for

TABLE XIII DESIGN PARAMETERS FOR THE BAND-PASS FILTER DESIGN CASE (mm)

all algorithms. The total number of generations is set to 100. The same control parameters that are given in the previous sections are used for all algorithms. In the objective function was set equal to . The results for all algorithms are shown in Table XIII. It is evident that the SADE and the DE outperform the PSO algorithms. The DE obtains the best value but SADE obtains the highest mean and the smallest standard deviation values. The CFPSO performance is also close to that of the DE strategies. Fig. 15 presents the convergence rate. The DE strategies and CFPSO seem to converge at similar speeds. The frequency response of the best design obtained by SADE is depicted in Fig. 16. It is evident that in the frequency bands between 4.3 and 4.6 GHz and between 5.2 and 5.5 GHz the value lies below . Table XIII shows the design parameters for this filter.

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TABLE XIV COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-ONLY SYNTHESIS  (POPULATION SIZE=30) d

=

Fig. 16. Best filter design case frequency response found by SADE.

Fig. 18. Convergence rate for the 28-element array case, position-only syn0:7 and population size=30. thesis for d

=

TABLE XV COMPARATIVE RESULTS FOR 28 ELEMENTS POSITION-ONLY SYNTHESIS WITH NULL CONTROL (POPULATION SIZE=30)

Fig. 17. Convergence rate for the 32-element array case position-only synthesis  and population size = 30. for d

=

D. Variation of the Population Size We have chosen to run again two examples from the lineararray synthesis cases with a smaller population size. Our goal is to determine if and to what extent convergence rate and statistical results depend on the population size. The population size is set to 30 and all algorithms run for 50 independent trials. The first case is that of position-only synthesis for the 32-ele. The statistical results are shown in ment array with Table XIV while the convergence rate is depicted in Fig. 17. We observe that although the number of objective-function evalua, SADE obtains again the tions is reduced to same best value as with a population size of 100. Also both DE strategies obtain higher best values than those reported in [30]. The convergence-rate plot is quite similar to that of Fig. 4(a). The second case is that of the 28-element array with null control. Table XV shows the statistical results. DE and SADE produce similar results. The standard deviation value denoted as as in Table VII. Fig. 18 zero in Table XV is less than presents the convergence rate. It seems that CFPSO, DE and SADE converge at similar speeds as in Fig. 8. Given the results above, we conclude that a decreasing population size does not significantly modify the convergence rate. However, there is a slight deterioration in all statistical results.

IV. CONCLUSION The SADE strategy has been applied to common design problems in electromagnetics. The number of unknowns for the design problems varies from 6 to 60. The SADE results have been compared with other popular evolutionary algorithms. The DE strategies outperform the PSO algorithms in terms of best values found. However, these results are indicative and do not generally apply to all optimization problems in electromagnetics. Compared with the classical DE strategy the Self-adaptive DE obtains similar convergence rate results and produces in average the same or better results. The main advantage of SADE is the fact that it requires only the adjustment of two parameters: the population size and the number of iterations. CFPSO has produced better results than the IWPSO algorithm and converges at a similar speed with the DE strategies. Both PSO algorithms have presented a larger dispersion of values, which is due to their ability to escape from local optima. All algorithms can be combined with a numerical method. The SADE strategy can be easily applied to other microwave and antenna design problems and it can be used in conjunction with an EM solver software.

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This application of the SADE strategy to other design problems will be part of our future work. In addition, in our future work, we intend to explore the applicability of the SADE strategy to multi-objective DE algorithms. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their detailed and valuable comments. REFERENCES [1] J. Kennedy and R. Eberhart, “Particle swarm optimization,” presented at the IEEE Int. Conf. on Neural Networks, Piscataway, NJ, 1995. [2] R. Storn and K. Price, “Differential evolution-A Simple and efficient adaptive scheme for global optimization over continuous spaces,” 1995, Tech. Rep. TR-95-012. [3] R. Storn and K. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimiz., vol. 11, pp. 341–359, 1997. [4] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Antennas Propag., vol. 55, pp. 556–567, 2007. [5] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, pp. 397–407, 2004. [6] M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1761–1776, 2005. [7] S. M. Cui and D. S. Weile, “Application of a parallel particle swarm optimization scheme to the design of electomagnetic absorbers,” IEEE Trans. Antennas Propag., vol. 53, pp. 3616–3624, 2005. [8] M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2674–2679, 2005. [9] M. Benedetti, R. Azaro, and A. Massa, “Memory enhanced PSO-based optimization approach for smart antennas control in complex interference scenarios,” IEEE Trans. Antennas Propag., vol. 56, pp. 1939–1947, 2008. [10] S. Selleri, M. Mussetta, P. Pirinoli, R. E. Zich, and L. Matekovits, “Some insight over new variations of the particle swarm optimization method,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 235–238, 2006. [11] S. M. Mikki and A. A. Kishk, “Quantum particle swarm optimization for electromagnetics,” IEEE Trans. Antennas Propag., vol. 54, pp. 2764–2775, 2006. [12] Y. Najjar, Y. Moneer, and N. Dib, “Design of optimum gain pyramidal horn with improved formulas using particle swarm optimization,” Int. J. RF Microw. Comput.-Aided Engng., vol. 17, pp. 505–511, 2007. [13] P. J. Bevelacqua and C. A. Balanis, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 3442–3449, 2007. [14] J. Nanbo and Y. Rahmat-Samii, “Analysis and particle swarm optimization of correlator antenna arrays for radio astronomy applications,” IEEE Trans. Antennas Propag., vol. 56, pp. 1269–1279, 2008. [15] N. Dib and M. Khodier, “Design and optimization of multi-band wilkinson power divider,” Int. J. RF Microw. Comput.-Aided Engng., vol. 18, pp. 14–20, 2008. [16] S. K. Goudos, I. T. Rekanos, and J. N. Sahalos, “EMI reduction and ICs optimal arrangement inside high-speed networking equipment using particle swarm optimization,” IEEE Trans. Electromagn. Compat., vol. 50, pp. 586–596, 2008. [17] L. Lizzi, F. Viani, R. Azaro, and A. Massa, “A PSO-driven spline-based shaping approach for ultrawideband (UWB) antenna synthesis,” IEEE Trans. Antennas Propag., vol. 56, pp. 2613–2621, 2008. [18] F. Afshinmanesh, A. Marandi, and M. Shahabadi, “Design of a single-feed dual-band dual-polarized printed microstrip antenna using a boolean particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 56, pp. 1845–1852, 2008. [19] K. R. Mahmoud, M. I. Eladawy, R. Bansal, S. H. Zainud-Deen, and S. M. M. Ibrahem, “Analysis of uniform circular arrays for adaptive beamforming applications using particle swarm optimization algorithm,” Int. J. RF Microw. Comput.-Aided Engng., vol. 18, pp. 42–52, 2008.

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[43] A. Semnani, M. Kamyab, and I. T. Rekanos, “Reconstruction of one-dimensional dielectric scatterers using differential evolution and particle swarm optimization,” IEEE Geosci. Remote Sensing Lett., vol. 6, pp. 671–675, Oct. 2009. [44] I. T. Rekanos, “Conducting scatterer reconstruction using differential evolution and particle swarm optimization,” presented at the 23rd Annu. Rev. Progr. Appl. Computat. Electromagn. (ACES), Verona, Italy, 2007. [45] J. Vesterstrom and R. Thomsen, “A comparative study of differential evolution, particle swarm optimization and evolutionary algorithms on numerical benchmark problems,” presented at the Congress on Evolutionary Computation, Portland, OR, 2004. [46] S. K. Goudos, Z. D. Zaharis, K. B. Baltzis, C. S. Hilas, and J. N. Sahalos, “A comparative study of particle swarm optimization and differential evolution on radar absorbing materials design for EMC applications,” presented at the Int. Symp. on Electromagnetic Compatibility-EMC Europe, 2009. [47] M. A. Panduro, C. A. Brizuela, L. I. Balderas, and D. A. Acosta, “A comparison of genetic algorithms, particle swarm optimization and the differential evolution method for the design of scannable circular antenna arrays,” Progr. Electromagn. Res. B, pp. 171–186, 2009. [48] J. Brest, S. Greiner, B. Boskovic, M. Mernik, and V. Zumer, “Selfadapting control parameters in differential evolution: A comparative study on numerical benchmark problems,” IEEE Trans. Evol. Comput., vol. 10, pp. 646–657, 2006. [49] S. K. Goudos, “Design of microwave broadband absorbers using a selfadaptive differential evolution algorithm,” Int. J. RF Microw. Comput.Aided Eng., vol. 19, pp. 364–372, 2009. [50] User’s Manunal, FEKO Suite 5.2 FEKO, 2003 [Online]. Available: www.feko.info [51] M. Clerc, “The swarm and the queen: towards a deterministic and adaptive particle swarm optimization,” presented at the Congress on Evolutionary Computation, Washington, DC, 1999. [52] T. Huang and A. S. Mohan, “A hybrid boundary condition for robust particle swarm optimization,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 112–117, 2005. [53] S. Xu and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antennas Propag., vol. 55, pp. 760–765, Mar. 2007. [54] R. Storn, “Differential evolution research—trends and open questions,” Studies Comput. Intell., vol. 143, pp. 1–31, 2008. [55] E. Mezura-Montes, J. Velazquez-Reyes, and C. A. C. Coello, “A comparative study of differential evolution variants for global optimization,” presented at the Genetic and Evolutionary Computation Conf., Seattle, WA, 2006. [56] T. Huang and A. S. Mohan, “A microparticle swarm optimizer for the reconstruction of microwave images,” IEEE Trans. Antennas Propag., vol. 55, pp. 568–576, Mar. 2007. [57] R. Storn, Differential Evolution Homepage [Online]. Available: http:// www.icsi.berkeley.edu/~storn/code.html [58] N. Jin and Y. Rahmat-Samii, “Parallel particle swarm optimization and finite-difference time-domain (PSO/FDTD) algorithm for multiband and wide-band patch antenna designs,” IEEE Trans. Antennas Propag., vol. 53, pp. 3459–3468, 2005. [59] L. Zhang, Z. Cui, Y. C. Jiao, and F. S. Zhang, “Broadband patch antenna design using differential evolution algorithm,” Microw. Opt. Technol. Lett., vol. 51, pp. 1692–1695, 2009. [60] D. J. Bekers, S. Monni, S. M. van de Berg, A. M. van de Water, B. J. Morsink, C. Alboin, V. Ducros, M. Celikbas, J. Blanche, N. Fiscante, G. Gerini, J. P. Martinaud, M. Rochette, and G. H. C. van Werkhoven, “Optimization of phased arrays integrated with FSS and feeding elements based on parametric models,” presented at the 2nd Eur. Conf. on Antennas and Propagation, 2007. [61] A. Hennings, E. Semouchkina, A. Baker, and G. Semouchkin, “Design optimization and implementation of bandpass filters with normally fed microstrip resonators loaded by high-permittivity dielectric,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1253–1261, 2006. [62] S. Koziel and J. W. Bandler, “SMF: A user-friendly software engine for space-mapping-based engineering design optimization,” presented at the Signals, Systems and Electronics Int. Symp., 2007. [63] S. Koziel, J. W. Bandler, and K. Madsen, “A space-mapping framework for engineering Optimization; theory and implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3721–3730, 2006. [64] G. Di Pillo, “Exact Penalty Methods,” in Algorithms for Continuous Optimization: The State of the Art, E. Spedicato, Ed. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1994, pp. 1–45.

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Sotirios K. Goudos (S’00–M’05) was born in Thessaloniki, Greece, in 1968. He received the B.Sc. degree in physics, the M.Sc. degree in electronics, and the Ph.D. degree in physics from the Aristotle University of Thessaloniki, in 1991 1994, and 2001, respectively, and the Master in Information Systems from the University of Macedonia, Greece, in 2005. Since 1996, he has been working in the Telecommunications Center, Aristotle University of Thessaloniki. His research interests include antenna and microwave structures design, electromagnetic compatibility of communication systems, evolutionary computation algorithms and semantic web technologies. Dr. Goudos is a member of the Greek Physics Society and the Greek Computer Society.

Katherine Siakavara (M’04) received the B.Sc. degree in physics, the M.Sc. degree in electronics, and the Ph.D. degree, all from the Aristotle University of Thessaloniki, Greece, in 1977, 1979, and 1982, respectively. She is currently an Assistant Professor in the Department of Physics, University of Thessaloniki, Greece. She has authored or coauthored more than 60 papers in peer reviewed journals and international conferences. Her research interests include applied electromagnetics, antennas and microwaves.

Theodoros Samaras (S’93–A’97–M’02) received the Physics degree from Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1990, the M.Sc. degree in medical physics (with distinction) from the University of Surrey, Surrey, U.K., in 1991, and the Ph.D. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996. In 1998, he was with the Bioelectromagnetics (BIOEM)/Electromagnetic Compatibility (EMC) Group, Swiss Federal Institute of Technology, Zürich, Switzerland, where he was mainly involved with studying the temperature increase due to the absorption of electromagnetic energy in materials and the effect of heat diffusion in electromagnetic dosimetry. He subsequently joined the Hyperthermia Unit, Erasmus Medical Center, Rotterdam, The Netherlands, where he conducted research on treatment quality of superficial hyperthermia. In December 1999, he returned to the Aristotle University of Thessaloniki, where he is currently an Assistant Professor. His research interests include numerical techniques with applications in biomedical technology, EMC and telecommunications. Dr. Samaras was the recipient of a Marie Curie Fellowship awarded by the European Commission.

Elias E. Vafiadis (M’86) was born in Thessaloniki, Greece, in 1952. He received the B.Sc. degree in physics and the M.Sc. degree in electronics from the Aristotle University of Thessaloniki, in 1975 and 1979, respectively, and the Ph.D. degree in electrical engineering from the Democritus University of Thrace, Xanthi, Greece, in 1985. On February 1981, he joined the Electrical Engineering Department, Democritus University of Thrace, as a Research Associate. From 1986 to 1993, he served as a Lecturer and Assistant Professor in the Microwaves Laboratory, Xanthi, Greece. From 1993 to 2003, he was an Assistant Professor and, since 2003, he has been an Associate Professor in the Department of Physics, School of Science, Aristotle University of Thessalonik. His research interests include application of electromagnetic theory to waveguiding and radiating structures and CAD techniques for microwave circuits design. Dr. Vafiadis is a member of the Hellenic Physical Society.

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John N. Sahalos (M’75–SM’84–F’06–LF’10) received the B.Sc. and Ph.D. degrees in physics, in 1967 and 1974, respectively, and the Diploma (BCE+MCE) in Civil Engineering and the Professional Diploma of Postgraduate Studies in Electronic Physics, both in 1975, all from the Aristotle University of Thessaloniki, (AUTH), Greece. From 1971 to 1974, he was a Teaching Assistant and, from 1974 to 1976, he was an Instructor in the Department of Physics, AUTH. In 1976, he worked at the ElectroScience Laboratory, Ohio State University, Columbus, as a Postdoctoral University Fellow. From 1977 to 1986, he was a Professor in the Electrical Engineering Department, University of Thrace, Greece, and Director of the Microwaves Laboratory. Since 1986, he has been a Professor at the School of Science, AUTH, where he is the Director of postgraduate studies in electronic physics and the Director of the Radio-Communications Laboratory (RCL). During 1981-82, he was a Visiting Professor at the Department of Electrical and Computer Engineering, University of Colorado, Boulder. During 1989 to 1990, he was a Visiting Professor at the Technical University of Madrid, Spain. He is the author of three books (in Greek), six book chapters, and more than 350 articles published in the scientific literature. He also is the author of the book The Orthogonal Methods of Array Synthesis, Theory and the ORAMA Computer Tool (Wiley, 2006). His research interests are in the areas of antennas, high frequency techniques, communications, EMC/EMI, microwaves and biomedical engineering.

Dr. Sahalos is a Professional Engineer and a Consultant to industry. In 2002-04, he was on the Board of Directors of the OTE, the largest Telecommunications Company in Southeast Europe. He served as a Technical Advisor on several national and international committees, as well as, in several Mobile Communications Companies. Since 1992, he has been a member of Commissions A and E of URSI. Since 1998, he is the President of the Greek committees of URSI. He is the President of the section on Informatics, Telecommunications and Systems of the National Committee of Research and Technology. He is an honorary member of the Radio-electrology Society, a member of the Greek Physical Society, and a member of the Technical Chamber of Greece. He is the creator and leader of an EMC network with five laboratories (three from the academy and two from the industry). He has been honored with a special investigation fellowship from the Ministry of Education & Science, Spain. He also has been honored by several Institutes and Organizations. He graduated/mentored nearly 25 Ph.D. students and coauthored two best paper awards at international conferences. He received the 2009 IEEE-AP Society award for exceptional performance as a reviewer of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has been on the editorial board of three scientific journals. He was elected by department representatives of the Aristotle University of Thessaloniki as the Vice-Chairman of the Research Committee of AUTH for the period 2007-2010. Since June 2010, and for the next two years, he will be with the five member International Consulting Committee of the GRNET S.A.