Analysis of Uniform Circular Arrays for Adaptive Beamforming Applications Using Particle Swarm Optimization Algorithm

Analysis of Uniform Circular Arrays for Adaptive Beamforming Applications Using Particle Swarm Optimization Algorithm Korany R. Mahmoud, 1,2 Mohamed I. Eladawy, 1 Rajeev Bansal, 2 Saber H. Zainud-Deen, 3 Sabry M. M. Ibrahem 1 1 Department of Electronics, Communications, and Computers, Faculty of Engineering, Helwan University, Cairo, Egypt 2 Department of Electrical and Computer Engineering, University of Connecticut, CT 06269 3 Department of Electronics and Electrical Communications Engineering, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt Received 11 January 2007; accepted 17 March 2007 ABSTRACT: In this article, the particle swarm optimization algorithm is used to calculate the complex excitations, amplitudes and phases, of the adaptive circular array elements. To illustrate the performance of this method for steering a signal in the desired direction and imposing nulls in the direction of interfering signals by controlling the complex excitation of each array element, two types of arrays are considered. A uniform circular array (UCA) and a planar uniform circular array (PUCA) with 16 elements of half-wave dipoles are examined. Also, the performance of an adaptive array using 3-bit amplitude and 4-bit phase shifters are studied. In our analysis, the method of moments is used to estimate the response of the dipole UCAs in a mutual coupling environment. VC 2007 Wiley Periodicals, Inc. Int J RF and Microwave CAE 18: 42 52, 2008. Keywords: smart antennas; adaptive beamforming; method of moments; mutual coupling; uniform circular arrays; particle swarm optimization algorithm; adaptive array I. INTRODUCTION Smart antenna systems have been widely considered to provide interference reduction and improve the capacity, data rates, and performance of wireless mobile communication [1, 2]. Smart antenna arrays with adaptive beamforming capability are very effective in the suppression of interference and multipath signals. The techniques of placing nulls in the antenna patterns to suppress interference and maximizing their Correspondence to: K. R. Mahmoud; e-mail: korany@engr. uconn.edu DOI 10.1002/mmce.20265 Published online 26 September 2007 in Wiley InterScience (www.interscience.wiley.com). gain in the direction of desired signal have received considerable attention in the past [3] and are still of great interest using evolutionary algorithms such as genetic algorithms (GA) [4, 5]. It is recognized that PSO algorithm is a practical and powerful optimization tool for a variety of electromagnetic and antenna design problems [6 10]. Compared to other evolutionary algorithms such as GA and simulated annealing, the PSO algorithm is much easier to understand and implement and requires minimum mathematical processing. A comparison between PSO and GA for the synthesis of amplitude-only, phase-only, and complex weighting for a specified far-field sidelobe envelope has been demonstrated in [11]. It indicated that the PSO algorithm is competitive with GA. For VC 2007 Wiley Periodicals, Inc. 42

UCA for Adaptive Beamforming Using PSO 43 amplitude-only synthesis, PSO performs better early on, but can be outperformed by the GA at a higher number of iterations. For phase-only synthesis, both algorithms have comparable performance, while GA slightly outperforms PSO for complex synthesis. In recent years, various versions of the PSO algorithm have been successfully used in linear [12, 13] and circular antenna array synthesis problems [14, 15]. The PSO is applied to an 18-element planar uniform circular array (PUCA) using phase-only control strategy for a synthesized beam pattern [14]. Also, the crossed PSO algorithm is applied to synthesize a PUCA of 19 elements with center-fed elements using a complex weight control strategy [15]. Many of the attempts on antenna array synthesis assume that the elements of the array are represented by isotropic point sensors isolated from one another [10, 12 14, 16, 17] or the element pattern can be modeled as a cosine function [9, 11, 15]. However, in practice, the elements of antenna arrays have finite physical dimensions and specific radiation characteristics. Since most of the beamforming algorithms ignore the effects of mutual coupling, especially in closely spaced antenna elements, the predicted system performances may not be accurate. Therefore, to evaluate accurately the resulting system performance of practical antenna arrays, the electromagnetic influence among the elements must be carefully considered. More recently, much attention has been paid to the effects of mutual coupling [18 20], and there have been studies integrating the GA with the method of moments (MoMs) [21]. In this paper, for adaptive arrays using space division multiple access, the optimal radiation pattern design of smart antennas is developed based on the particle swarm optimization (PSO) technique. Two types of arrays are considered in free space, the first one is a uniform circular array (UCA) and the second is a PUCA [16]. The array consists of center-fed halfwave dipoles. The dipoles are identical and oriented perpendicular to the plane of the array. The goal is to maximize the beam of the radiation pattern towards the intended user or Signal of Interest (SOI) and ideally obtain nulls in the directions of interfering signals or Signal not of Interest (SNOI). The performance of designed arrays is assessed using a full EM analysis based on the MoMs [22]. In this analysis, the mutual coupling effects between the array elements are fully taken into account. The methods of beam pattern synthesis based on controlling the complex weights (the amplitude and phase), the excitation amplitudes only, the phase values only, and the element position only have been extensively considered in the literature [3, 11, 23]. The most important method is based on controlling the complex weights since this technique utilizes fully the degrees of freedom for the solution space. Furthermore, the sidelobe level and the main beam characteristics can be controlled directly. On the other hand, it is also the most expensive approach considering the cost of both phase shifters and variable attenuators for all elements. Furthermore, when the number of elements in the antenna array increases, the computational time to find the values of element amplitudes and phases will also increase. This allows a trade-off between the quality of the constrained pattern and the complexity of the electronic control. PSO is used to adjust the weights of phase shift and amplitude of the excitation of each element of the array for beam synthesis. In this paper the PSO algorithm program was implemented using MAT- LAB-software version 7.0.4 and linked to a FOR- TRAN code program to simulate the antenna arrays using Microsoft Developer Studio 97. The paper is organized as follows. In Section II, a brief introduction to the PSO algorithm is presented. In Section III, the objective function is described. In Section IV, the antenna array design is explained. Numerical results will be discussed in Section V. Finally, Section VI presents the conclusions. II. PSO ALGORITHM PSO has attracted a lot of attention since its introduction in 1995 by Kennedy and Eberhart [24]. Many researchers have worked on improving PSO performance in various ways and developed many interesting variants. A new inertia weight parameter was incorporated into the original PSO algorithms by Shi and Eberhart [25]. In PSO, each solution is a point in the search space and may be regarded as a particle in the algorithm. In general, each particle flies through the D-dimensional problem space by learning from the best experiences of all the particles. Therefore, the particles have a tendency to fly towards better search area over the course of search process. In a new version of the PSO algorithm [26], each particle is attracted towards the best previous positions visited by its neighbors. In this case, we take into account two neighbors on each side. For an D-dimensional problem, the position of the ith particle is represented as X i 5 (x i1, x i2,..., x id ). Each row of the position matrix represents a possible solution to the optimization problem. The rate of the position change (velocity) for particle i is represented as V i 5 (v i1, v i2,..., v id ). To update the velocity matrix at each iteration k,

44 Mahmoud et al. Figure 1. algorithm. every particle should know its personal best and the global best position vectors in addition to the neighbor best position vector. The personal best position vector defines the position at which each particle attained its best fitness value up to the present iteration. The personal best position of the ith particle is represented as Pbest i 5 (pbest i1, pbest i2,..., pbest id ). The global best position vector defines the position in the solution space at which the best fitness value was achieved by all particles, and is defined by Gbest 5 (gbest 1, gbest 2,..., gbest D ). The best neighbor position vector discovered by the whole population is represented as Nbest 5 (nbest 1, nbest 2,..., nbest D ). The particles are manipulated according to the following equations: v kþ1 id Flowchart showing the main steps of the PSO ¼ x vk id þ c 1 rand 1 ðpbest id x k id Þ þ c 2 rand 2 ðgbest D x k id Þ þ c 3 rand 3 ðnbest D x k id Þ x kþ1 id ¼ xk id þ vkþ1 id ð1þ Dt ð2þ where c 1, c 2, and c 3 are the acceleration constants, which represent the weighting of stochastic acceleration terms that pull each particle towards pbest, gbest, and nbest positions. rand 1, rand 2, and rand 3 are three random numbers in the range [0, 1], x is the inertia weight introduced to balance between the global and local search abilities, Dt is taken as a unit time step. In our PSO algorithm, the terminology of soft and hard boundary conditions is applied to describe the way in which particles are enforced to stay inside the desired domain of interest [27]. The soft boundary conditions rely on a velocity clipping technique to prevent particles from explosion, where if v id exceeds a positive constant value specified by the user, the velocity of that dimension is assigned to be sign(v id )VD max, i.e., the velocity along each dimension is clamped to a maximum magnitude VD max. This is done to help keep the swarm under control. The maximum velocity was set to the upper limit of the dynamic range of the search (VD max 5 XD max ). The hard boundary condition is any boundary that uses a position-clipping criterion, where if x id exceeds XD max, then it is assigned to be XD max, and also if x id falls below XD min, then it is assigned to be Xmin D. In this case, time-varying maximum velocity is applied, where v id is changed linearly from VD max to 0.1VD max, because as the particles approach the optimal result it is preferred to have them move with lower velocities [28]. The concept of time-varying acceleration coefficients c 1 and c 2 is taken into account, in addition to the time-varying inertia weight factor, to effectively control the global search and convergence to the global best solution. An improved optimum solution was observed when changing c 1 from 2.5 to 0.5, changing c 2 from 0.5 to 2.5, and fixing c 3 to 1.0, over the full range of the search [29]. Shi and Eberhart [25] introduced a linearly decreasing inertia weight to the PSO. The weighting function, x, can be calculated from the following equation: TABLE I. Lengths, Spacings, and Performance Comparison for Optimized Yagi-Uda Antenna for Both Gain and Input Impedance PSO Optimized (Ref. [8]) GA Optimized (Ref. [31]) Element Length/2 Spacing Length/2 Spacing Driver 0.2431k 0.225k Reflector 0.2350k 0.2746k 0.239k 0.182k Director 1 0.2263k 0.1543k 0.224k 0.152k Director 2 0.2186k 0.2811k 0.217k 0.229k Director 3 0.2147k 0.3369k 0.211k 0.435k Director 4 0.2140k 0.3727k 0.220k 0.272k Gain (dbi) 12.627 12.58 Z in (X) 50.26 1 j0.085 49.64 2 j5.08

UCA for Adaptive Beamforming Using PSO 45 Figure 2. (a) Antenna array geometry of a UCA. (b) Antenna array geometry of a PUCA. x ¼ x max x max x min iter 0 max 3 iter ð3þ where, x max is the initial weight (0.9), x min is the final weight (0.4), iter 0 max is 0.75 of the maximum iteration number, and iter is the current iteration number. The value of x is fixed to x min after iter 0 max. The PSO algorithm used in this paper is very close to the crossed PSO algorithm used in [15], but in our algorithm a circular neighborhood topology [30] is used in addition to utilizing time-varying maximum velocity which decreases linearly from VD max to 0.1VD max over the full range of the search. Also, some parameters like c 1, c 2, and Dt are adjusted to obtain the best results, besides using soft and hard boundary conditions. Now it is necessary to see how an individual is represented in a PSO algorithm. For example, if we have (D/2) antenna elements, each element excitation has two variables (amplitude and phase). So for our array beamforming synthesis problem, the D-dimensional position vector is mapped to (D/2) amplitude and (D/2) phase weights. In the beginning, a population of I particles is generated with random positions (values) in the range of the solution space. Then a random velocity is assigned to each particle. As in all evolutionary computation techniques, there must be some function or method to evaluate the goodness of a position. The fitness function must take the position and return a single number representing the value of that position. By obtaining the fitness values, the Pbest, Gbest, and Nbest positions will be known. Then the new particle velocity (the amount of change in the particle s position) can be calculated by Eq. (1). Now it is simple to move each particle to its next location (position) using Eq. (2). After this process is carried out for each particle in the swarm, the process is repeated until the number of iteration is finished. Figure 1 shows a flowchart diagram of the main steps of the PSO algorithm used in this paper. It should be noted that the array geometry does not affect the performance of the PSO algorithm. As the number of array elements, the level of quantization of the amplitude and phase, and the boundaries for each variable increase, the PSO algorithm requires more time for convergence. III. OBJECTIVE FUNCTION The objective function provides the interface between the physical problem and the optimization algorithm. Figure 3. Radiation pattern comparison between our GA, PSO result and Ioannides and Balanis RLS result for isotropic PUCA.

46 Mahmoud et al. Objectiv function ¼ X N a i¼1 i Gðu i Þ X M b j¼1 j Gðu j Þ ð4þ Figure 4. Radiation pattern comparison between UCA and PUCA with respect to u (y 5 908) for half-wave dipole elements. In general, this could be antenna gain, weight, peak cross-polarization, or some kind of weighted sum of all these factors. So the quality of an antenna beamforming is expressed mathematically by an objective function. The following objective function rewards the antenna array for maximizing the output power toward the desired signal at u i and minimizing the total output power in the direction of the interfering signals at u j. where G is the antenna array gain and the constants a i and b j are the weights that control the contribution from each term to the overall objective function. The constant N represents the number of desired users and M represents the number of interferers. In our analysis, we take N 5 1 with two values for M: in the first case, M 5 2 and, in the other case, M 5 3. Next, the weights a i and b j are considered to be (a 1 5 20), (b 1 5 1), (b 2 5 1), and (b 3 5 1) to assign a higher priority to maximizing the output power toward the desired signal while minimizing the total output power in the direction of the interfering signals. IV. ANTENNA ARRAY DESIGN To illustrate the capabilities of the PSO algorithm as an optimization technique in antenna design, the lengths and spacings of several Yagi-Uda antennas Figure 5. (a) Amplitude and (b) phase excitation for each dipole element of the UCA. Figure 6. (a) Amplitude and (b) phase excitation for each dipole element of the PUCA.

UCA for Adaptive Beamforming Using PSO 47 TABLE II. and PUCA Directivity Comparison Between UCA Directivity (db) y 5 908 y 5 1008 y 5 1208 y 5 1308 UCA 12.27 11.88 11.14 9.66 PUCA 12.44 11.69 10.24 10.53 Figure 7. Radiation patterns for UCA with respect to u in different planes of y. optimized for various performance characteristics have been studied before [8]. Table I illustrates a comparison between the results obtained using PSO algorithm and the results proposed in Ref. [31] for an optimized six-element Yagi-Uda array for both gain and a matched input impedance (Z in )of50x. Better results have been obtained using PSO than other optimization techniques such as the GA [8]. Now, two different shapes of circular arrays using half-wave dipole elements are presented and compared with each other. An obvious advantage results from the symmetry of the UCA structure. Since a UCA does not have edge elements, directional patterns synthesized with a UCA can be electronically rotated in the plane of the array without a significant change of the beam shape [32, 33]. The first array considered is a UCA. The geometry of the array is shown in Figure 2a. This geometry consists of 16 elements uniformly distributed with a ring radius r 5 (8/2p)k. The second array is a PUCA. The geometry consists of two concentric circular arrays with uniformly distributed antenna elements as shown in Figure 2b. The first circular ring of radius r 1 5 (3/2p)k consists of six elements and the second circular ring of radius r 2 5 (5/2p)k consists of 10 elements. The antenna elements in both cases consist of vertical (z-directed) half-wave dipole elements equally spaced in the x y plane along a circular ring, where the distance between adjacent elements is d c 5 0.5k and the dipole wire radius is a 5 0.003369k. V. NUMERICAL RESULTS As an example of adaptive beamforming with a UCA and a PUCA, we considered the desired user at u 5 1808 while the other two users are at u 5 608 and u 5 2408, which are considered as interferers. Figure 3 shows a comparison between the resulting beam pattern for the PUCA when employing GA and the PSO algorithms for isotropic elements and the results obtained via recursive least squares (RLS) technique in Ref. [16]. Both GA and PSO algorithms are employed with a population size of 100 and 200 iterations. A stochastic sampling for selection is used for GA with a probability of crossover (P c 5 0.8), and probability of mutation (P m 5 0.001). All the unknown parameters were discretized using eight bits. It required about 25 s for both PSO and GA on a 32-bit Dell Precision Workstation 690 (Dual Core Intel (tm) Xeon (tm) Processor 5050 at 3.0 GHz) to Figure 8. Radiation patterns for PUCA with respect to u in different planes of y. Figure 9. Radiation patterns comparison for UCA with discrete and precise feeding.

48 Mahmoud et al. Figure 10. (a) Amplitude and (b) phase excitation for each element of the UCA. Figure 11. Radiation patterns comparison for PUCA fed with discrete and precise feeding. get the result. The PSO algorithm shows better performance than GA and RLS in directing the maximum towards the direction of the SOI while placing deeper nulls towards the angles of SNOIs. Therefore, we will focus on applying the PSO algorithm for beamformig the antenna array. Next, the more practical half-wave dipole elements (with mutual coupling) will be used for pattern synthesis. Figure 4 illustrates the resulting beam pattern for a UCA compared with the beam pattern of a PUCA. Figures 5 and 6 show the required amplitude and phase excitations of each element of the UCA and PUCA corresponding to the desired beam pattern of Figure 4. A first observation from these plots is that a good performance is obtained from PSO even when the mutual coupling between elements is fully taken into account. Also, it is noted that the PUCA achieves deeper nulls towards the angles of interfering signals compared with the UCA. The PSO algorithm with a swarm size of 100 and 200 iterations in this case required about 14.5 min to get the result. Figures 7 and 8 show the radiation patterns for the UCA and PUCA with respect to u in different planes of y (908, 1008, 1208, 1308), respectively. These figures indicate that a good behavior from the two types of arrays can be obtained in different planes. Note that, as Figure 12. (a) Amplitude and (b) phase excitation for each element of the PUCA.

UCA for Adaptive Beamforming Using PSO 49 Figure 13. Radiation patterns comparison for UCA with discrete and precise feeding. the elevation angle increases, the beamwidth increases and the gain decreases as depicted in Table II. From a practical view point, it would be difficult to feed the antenna array elements with the precise values of amplitudes and phases obtained, so we let the feeding excitation to have discrete values, where the amplitude is changed from 1.0 to 3.0 with step of 0.5, and the phase will change from 2p to p with step of p/4. In this case, the resulting beam pattern has been optimized by the PSO algorithm with a swarm size of only 50 and 200 iterations, which required about 7.25 min. Figure 9 shows the radiation pattern for the case of feeding each element of the UCA with stepped values of amplitudes and phases compared with the case of feeding with the original precise values. Figure 10 shows the required amplitude and phase excitations of each element to obtain this acceptable beam pattern. Also Figure 11 shows the required amplitude and phase excitations of each element of the PUCA to obtain the desired beam pattern of Figure 12. These results indicate that acceptable deep nulls can be obtained using 3-bit amplitude and 4-bit phase shifters; at the same time we can still obtain a high gain, 11.9 db for the UCA, and 12.3 db for the PUCA. Quantized steps of p/4 and 0.5 for phase and amplitude are used to compromise between the hardware cost and the beam pattern synthesis accuracy. Clearly if the quantized step size is decreased, the accuracy will increase but, on the other hand, the required convergence time will increase. To further illustrate the application of the technique, another case of adaptive beamforming with a UCA and a PUCA is considered, assuming one desired user and three interferers. The SOI impinges from u 1 5 1808 while the other three not of interest signals SNOIs are directed from u 2 5 308, u 3 5 1208, and u 4 5 3008. Figures 13 and 14 show a comparison between the desired radiation patterns for a UCA and a PUCA, respectively, in case of stepped and precise feeding. Table III shows the directivity for each case and the required amplitude and phase excitations of each element to obtain the beam patterns in Figures 13 and 14. As a comparison between a UCA and a PUCA, the UCA is shown to have the narrowest main beamwidth and, therefore, the best spatial resolution. This is due to the fact that the UCA has much larger overall size (the separation between nearly diametrically opposite elements is larger) than in a PUCA. However, for practical applications with size limitations, the dimension of the UCA can be challenging for actual implementation where the UCA area is 5.093k 2 while the area of the PUCA is only 1.989k 2. VI. CONCLUSIONS In this article, adaptive arrays of smart antennas such as a UCA and a PUCA are considered. The array consisted of center-fed half-wave dipoles and the mutual coupling effect between the array elements is fully taken into account. By integrating the PSO algorithm with the MoMs, the amplitudes and phases of the antennas are calculated. Good results are obtained in different planes for both UCA and PUCA. Also, acceptable gain and deep nulls are obtained using 3- bit amplitude and 4-bit phase shifters compared with precise values. The comparison between UCA and PUCA shows a trade-off between them in null depth, beamwidth, and overall size. Finally, it is anticipated that the above method can be applied to different types of adaptive arrays with different shapes. Figure 14. Radiation patterns comparison for PUCA with discrete and precise feeding.

50 Mahmoud et al. TABLE III. Amplitude and Phase Excitation for Each Element of the UCA and PUCA for Exact and Discrete Feeding UCA PUCA Element No. Exact Discrete Exact Discrete 1 3.00 ff 287.958 3.0 ff 08 2.92 ff 273.858 3.0 ff 2458 2 2.92 ff 265.448 3.0 ff 458 2.89 ff 235.428 2.5 ff 1358 3 3.00 ff 83.398 3.0 ff 1808 3.00 ff 21808 3.0 ff 2908 4 2.96 ff 2115.188 3.0 ff 2458 1.00 ff 1808 3.0 ff 458 5 2.16 ff 52.648 1.5 ff 1808 1.55 ff 1808 1.5 ff 458 6 2.37 ff 2104.448 3.0 ff 2458 1.94 ff 25.418 3.0 ff 1808 7 2.07 ff 215.768 3.0 ff 458 3.00 ff 127.468 3.0 ff 2908 8 1.14 ff 176.698 1.5 ff 1808 2.87 ff 1808 3.0 ff 08 9 3.00 ff 64.008 2.5 ff 21808 2.84 ff 270.118 3.0 ff 1808 10 1.54 ff 160.398 2.5 ff 21808 3.00 ff 133.258 3.0 ff 2458 11 2.98 ff 13.358 3.0 ff 458 2.56 ff 233.838 2.5 ff 1808 12 2.95 ff 179.938 3.0 ff 2908 1.20 ff 2142.048 3.0 ff 1358 13 2.85 ff 27.808 1.0 ff 1808 3.00 ff 13.138 3.0 ff 21808 14 3.00 ff 179.568 3.0 ff 2458 3.0 ff 150.608 3.0 ff 2458 15 2.99 ff 53.708 3.0 ff 1808 2.12 ff 220.608 3.0 ff 1808 16 3.00 ff 261.928 3.0 ff 458 2.33 ff 167.798 3.0 ff 08 Directivity (db) 12.92 12.10 11.87 10.71 REFERENCES 1. P.H. Lehne and M. Pettersen, An overview of smart antenna technology for mobile communications systems, IEEE Commun Surveys Tutorials 2 (1999), 2 13. 2. M. Chryssomallis, Smart antennas, IEEE Antennas Propag Mag 42 (2000), 129 136. 3. H. Steyskal, Simple method for pattern nulling by phase perturbation, IEEE Trans Antennas Propag 31 (1983), 163 166. 4. R.L. Haupt, Phase-only adaptive nulling with a genetic algorithm, IEEE Trans Antennas Propag 45 (1997), 1009 1015. 5. P.K. Varlamos and C.N. Capsalis, Electronic beam steering using switched parasitic smart antenna arrays, Prog Electromagn Res 36 (2002), 101 119. 6. D. Gies and Y. Rahmat-Samii, Particle swarm optimization for reconfigurable phase-differentiated array design, Microwave Opt Technol Lett 38 (2003), 168 175. 7. J. Robinson and Y. Rahmat-Samii, Particle swarm optimization in electromagnetics, IEEE Trans Antennas Propag 52 (2004), 397 407. 8. S.H. Zainud-Deen, K.R. Mahmoud, M. El-Adawy, and S.M.M. Ibrahem, Design of Yagi-Uda antenna and electromagnetically coupled curl antenna using particle swarm optimization algorithm, 22nd National Radio Science Conference (NRSC 2005), Cairo, Egypt, March 15 17, 2005. 9. T.B. Chen, Y.B. Chen, Y.C. Jiao, and F.S. Zhang, Synthesis of antenna array using particle swarm optimization, Microwave Conference Proceedings (APMC 2005, Asia-Pacific Conference Proceedings), Vol. 3, Suzhou, China, Dec 3-4, 2005, 4 pp. 10. M. Donelli, R. Azaeo, F.G.B. Natale, and A. Massa, An innovative computational approach based on a particle swarm strategy for adaptive phased-arrays control, IEEE Trans Antennas Propag 54 (2006), 888 898. 11. D.W. Boeringer and D.H. Werner, Particle swarm optimization versus genetic algorithms for phased array synthesis, IEEE Trans Antennas Propag 52 (2004), 771 779. 12. M.M. Khdier and C.G. Christodoulou, Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization, IEEE Trans Antennas Propag 53 (2005), 2674 2679. 13. M.H. Bataineh and J.I. Ababneh, Synthesis of aperiodic linear phased antenna arrays using particle swarm optimization, Electromagnetics 26 (2006), 531 541. 14. M. Benedetti, R. Azaro, D. Franceschini, and A. Massa, PSO-based real-time control of planar uniform circular arrays, IEEE Antennas Wireless Propag Lett 5 (2006), 545 548. 15. T.B. Chen, Y.L. Dong, Y.C. Jiao, and F.S. Zhang, Synthesis of circular antenna array using crossed particle swarm optimization algorithm, J Electromagn Waves Appl 20 (2006), 1785 1795. 16. P. Ioannides and C.A. Balanis, Uniform circular and rectangular arrays for adaptive beamforming applications, IEEE Antennas Wireless Propag Lett 4 (2005), 351 354. 17. M. Mouhamadou and P. Vaudon, Smart antenna array patterns synthesis: null steering and multi-user beam-

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52 Mahmoud et al. Saber H. Zainud-Deen was born in Menouf, Egypt, in November 1955. He received the B.Sc., M.Sc., and Ph.D. degrees from Menoufia University in 1978, 1982, and 1988, respectively, all in Electrical Engineering. From December 1978 to October 1982, he was an Instructor at Menoufia University. From November 1982 to October 1986, he was an Assistant Lecturer at Menoufia University. From March 1986 to 1988, he was a Research Assistant at Birmingham University, UK. From 1988 to 1993, he was a Lecturer in Menoufia University. From 1993 to 2000, he was a member of Technical Staff at Saudi Telecom Company, KSA. From 2000 to 2002, he was a Professor in Menoufia University. From 2002 to 2005, he was a Head of the Electrical and Communication Engineering Department at Menoufia University. Currently, he is a Professor in the Electrical and Communication Engineering Department at Menoufia University. His research interests are in wireless communications, smart antennas, and computational electromagnetics. Dr. Zainud-Deen is a member of IEEE. Sabry M. M. Ibrahem received the B.S. degree in Electrical Engineering from the High Industrial Institute, Cairo, in 1969, and the M.S. and D.Sc. degrees in Electrophysics from the George Washington University, Washington, DC, in 1978 and 1982, respectively. He is currently an Associate Professor of Electrical Engineering at Helwan University, Faculty of Engineering, Communication Department. His research interests include the areas of phased array antennas, antenna array design using genetic algorithms and other optimization techniques.