Pareto Optimal Synthesis of the Linear Array Geometry for Minimum Sidelobe Level and Null Control During Beam Scanning

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1 Pareto Optimal Synthesis of the Linear Array Geometry for Minimum Sidelobe Level and Null Control During Beam Scanning F. Günes, F. Tokan Department of Electronics and Communication Engineering, Faculty of Electrics and Electronics, Yildiz Technical University, Yildiz 34349, Istanbul, Turkey Received 2 November 2009; accepted 20 March 2010 ABSTRACT: In this work, synthesis of the linear array geometry is put forward as a constrained vector optimization problem whose components are to meet the minimum sidelobe level (SLL) and control of the wide/narrow null placement during beam scanning whose range can vary from zero to the wide bands. As these synthesis objectives generally conflict with each other, nondominated solutions are acquired using the Nondominated Sorting Genetic Algorithm- II (NSGA-II) as a fast nondominated genetic sorting algorithm. Then, the Pareto frontiers are obtained using these trade-off solution sets between the maximum SLL, null control, and scanning range to provide a view of all design options. Thus, the pattern features resulted from these Pareto frontiers are valid for any chosen main beam direction within its full prescribed beam scanning range. The same Pareto optimal synthesis procedure can be applied to a thinned linear antenna array. A thinned linear antenna array is obtained by a simple genetic optimization by rounding the excitation amplitudes either to 1 or 0 to minimize the maximum of side lobe level (MSLL) during the beam scanning within the prescribed region as stated in the multiobjective function. Finally, some typical Pareto optimal radiation patterns of the scanning arrays are synthesized with only perturbating the positions of the array elements, and their full electromagnetic wave simulations are also completed to examine the resulted mutual coupling effects between the elements of the arrays. It can be concluded that the Pareto optimal synthesis procedure gives the scanning linear antenna arrays with successful radiation performance. VC 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE 20: , Keywords: linear array; sidelobe suppression; null control; beam scanning; constrained multiobjective optimization; pareto-optimal solutions; nondominated sorting genetic algorithm I. INTRODUCTION Mathematically, the synthesis of an antenna array is a highly nonlinear multiobjective optimization problem whose objectives are generally conflicting with each other. As far as the linear antenna array is concerned, among the main objectives, gain of the main beamwidth, minimum SLL, and narrow/wide null control during beam scanning can be considered. In literature heretofore, many synthesis methods have been developed by considering each objective individually for a chosen fixed main beam direction. In the works of [1 3], the methods are concerned with Correspondence to: F. Günes; mailto:gunes@yildiz.edu.tr DOI /mmce Published online 26 July 2010 in Wiley Online Library (wileyonlinelibrary.com). suppressing the SLL, while preserving the gain of the main beam. Other methods deal with only the interference suppression and null control [4 9]. However, typically in [10], sidelobe suppression and null control are taken together into account with the maximum radiation in broadside, using a novel stochastic algorithm, particle swarm optimization, where the algorithm combines these multiple objectives into a single function, commonly known as aggregating functions. On the other hand, in computational science, different strategies are used for solving this type of classical multiobjective optimization problem. On one hand, the decision making is reduced into a single objective, using weights for each objective, and scalar optimization is used to find the corresponding solution. Such an approach entails repetition of the procedure of assigning weights until a satisfactory solution is found. VC 2010 Wiley Periodicals, Inc. 557

2 558 Günes and Tokan Thus, the decision making is applied at the end of the optimization procedure. Another approach is to find the Pareto optimal frontier that contains all the Pareto optimal solutions [11 15]. A Pareto optimal solution is a nondominated solution which, by definition, is the best that can be achieved for one objective without disadvantaging at least one other objective. Recently, many works have been focused on the multiobjective evolutionary algorithms (MOEAs) that use nondominated sorting and sharing [11 15] to find Pareto optimal frontiers. The basic advantages of an evolutionary algorithm in multiobjective optimization are: (1) it can generate a population of efficient solutions (nondominated solutions) in a single run and (2) it eliminates the use of weighted parameters or aggregation functions. It has also become a preferred method for multiobjective optimization problems that are too complex for traditional techniques. Among the typical works on the electromagnetic synthesis that use the MOEAs, the works [16 19] can be considered that use the particle swarm, genetic and ant colony evolutionary algorithms. The goal of this work is to synthesize the linear antenna array geometry, i.e., to determine the physical layout of the array, using the Pareto optimal frontiers based on the required objectives for the radiation pattern. As is well known, beam scanning within the required range is one of the most significant properties expected from an array. During beam scanning, maintaining the MSLL at its minimum possible level and controlling nulls are inevitably demanded. In this work, these objectives are achieved as nondominated solutions only by the perturbations of the array elements positions in a linear geometry, while keeping a uniform excitation over the array aperture and with good quality directivity. Here, the NSGA-II [15, 16] is used in forming the Pareto optimal solutions as a fast nondominated genetic sorting algorithm. Then, the Pareto frontiers are obtained using the trade-off solution sets between the MSLLs, null control, and scanning range to provide a view of all design options. Thus, for a chosen synthesized linear antenna array geometry, without using any attenuation network in the feeding system, only the phase programming for the beam scanning, it would be possible keeping not only the MSLL at its minimum level, at the same time placing narrow/wide nulls in the required directions. To minimize mutual coupling effects between the elements, the minimum interelement spacing is obeyed in the array geometry synthesis process, and the full electromagnetic wave simulations of the worked example are also presented, which verify the synthesized radiation patterns. In this work, thinning process is also applied by a simple genetic optimization to find out the positions of the thinned elements for an optimal radiation pattern subject to the required objectives. In the next section, fundamentals of the Pareto optimization are given. The third section formulates the linear antenna array problem, whereas the forth section describes the Pareto genetic optimal synthesis of the nonuniform linear phased arrays. Typical worked examples are given in the fifth section, and the conclusions are listed in the sixth section. II. MULTIOBJECTIVE OPTIMIZATION AND PARETO OPTIMAL SOLUTION SETS Multiobjective optimization is the process of the simultaneously minimization or maximization of m objective functions ~f ð~xþ ¼ðf 1 ð~xþ; f 2 ð~xþ; ; f m ð~xþþ with respect to n decision variables ~x ¼ðx 1 ; x 2 ; x 3 ; :::; x n Þ subject to the given constraints in the decision space X. Thus, the multiple-objective function ~f : X ) Y evaluates the quality of the specific solution by assigning an objective vector ~y ¼ðy 1 ; y 2 ; y 3 ;:::; y m Þ in the Y objective space: Minimize=maximize; f i ð~xþ i ¼ 1; 2; ::::; m: ð1aþ subject to g i ð~xþ 0; i ¼ 1; 2; ::::; j; ð1bþ h i ð~xþ ¼0 i ¼ 1; 2; ::::; k; ð1cþ x il x i x iu i ¼ 1; 2; ::::; n: ð1dþ The constraints given by (1.b), (1.c), and (1.d) determines a feasible D region in the decision space x [ R n, and the multiple-objective function ~f maps this feasible region D into an objective function space y [ R n as shown in Figure 1 where n ¼ 3, m ¼ 2. If any of the objective functions f i ð~xþ; i ¼ 1:2; ::::; m are conflicting, no single objective vector dominates another objective vector. Instead, the outcome is a set of solutions, and the concept of nondomination or Pareto optimality must be used to characterize the objectives [14]. A nondominated solution is an optimal solution, if it is not dominated by another solution. Here, a domination relation must be defined: For the solution vectors ~x 1 and ~x 2, when the following conditions are met: (i) ~x 1 is at least as good as ~x 2 for all the objectives and (ii) ~x 1 is strictly better than ~x 2 for at least one objective, then ~x 1 is said to dominate ~x 2. In the case where ~x 1 and ~x 2 dominate other solution vectors but not each other, they are deemed mutually optimal solutions and referred to as Pareto optimal. The set of Pareto optimal solutions reflects the trade-off surfaces between Figure 1 Mapping between the decision and objective spaces and the Pareto optimal frontier for n ¼ 3, m ¼ 2. International Journal of RF and Microwave Computer-Aided Engineering/Vol. 20, No. 5, September 2010

3 Pareto Optimal Synthesis of the Linear Array 559 the different objectives. This set of Pareto optimal solutions is referred to as the Pareto frontiers (Fig. 1). In this work, the evolutionary multiobjective optimization procedure we used is NSGA-II proposed by Deb et al. [14, 15]. In the work of Deb et al. [14, 15], a fast nondominated sorting genetic algorithm is presented that alleviates all the main difficulties of the previous nondominated sorting multiobjective evolutionary algorithms. Specifically, O(MN 3 ) computational complexity, where M is the number of objectives and N is the population size, is overcome by a fast nondominated sorting approach. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best with respect to fitness and spread solutions. Simulation results on difficult test problems show that in most problems the NSGA-II is able to find much better spread of solutions and better convergence near the true Pareto optimal frontier compared to Paretoarchived evolution strategy and strength Pareto evolutionary algorithms. Moreover, the definition of dominance is modified to solve constrained multiobjective problems efficiently. Therefore, we chose this algorithm for its ease of implementation and its efficient computation of nondominated ranks within high degree of accuracy. The detailed theory together with literature can be found in [14, 15]. III. PROBLEM FORMULATION The far-field radiation pattern of a linear array at an arbitrary direction (h, /) when mutual coupling effects between the array elements neglected can be given using pattern multiplication as: FFðh; /Þ ¼EPðh; /Þ : AFðh; /Þ (1) where EP(h,/) represents the element pattern and AF(h,/) is the array factor. In this work, a linear phased array of 2N half-wave dipole antennas placed symmetrically along y-axis is considered as shown in Figure 2. Because of the symmetry of the array geometry, dependence of the array factor in the azimuth plane can be expressed as: AFð/Þ ¼2 XN n¼1 A n cos½kd n sin / þ b n Š (2) where k is the wavenumber and A n, b n and d n are the excitation amplitude, phase and location of the nth element, respectively. If it is assumed that the maximum radiation of the array is required to be oriented at an angle / 0 ( 90 / 0 90 ), the phase excitation b n of the n th element must be equal to [20]: kd n sin / þ b n j /¼/o ¼ 0 ) b n ¼ kd n sin / o (3) Thus, by controlling the progressive phase difference between the elements, the maximum radiation can be squinted in any required direction to form a scanning array. As in phased array technology, the scanning must Figure 2 be continuous, the system should be capable of continuously varying the progressive phase between the elements. In practice, this is accomplished electronically by the use of ferrite or diode phase shifters [20]. Otherwise, miniature or micromachined motors could also be placed under each element to achieve wide-angle beam scanning without the need of phase shifters [21]. If b n in (3) is placed in (2), the array factor can be obtained as follows: AF ¼ 2 XN n¼1 A n cos½kd n ðsin / sin / 0 ÞŠ (4) In this work, as the maximum levels of the sidelobe regions are desired to be minimized and narrow/wide nulls in any required locations are generated by means of perturbations in the elements positions, by introducing D n displacement to the n th element, eq. (4) becomes: AF ¼ 2 XN n¼1 A n cos½kðd n þ D n Þðsin / sin / 0 ÞŠ (5) Regarding the EP(h, /), half-wave dipoles have been considered. Half-wavelength dipole is one of the most commonly used antennas when used separately or in the array environment [22]. Because its radiation resistance is 73 ohms, which is very near the 75 ohm characteristic impedance of some transmission lines [20]. The far electrical field of a half-wave dipole placed along y-axis as oriented to the z-axis can be expressed as: E h ffi jg I oe jkr 2pr Linear antenna array geometry. cosð kl 2 cos hþ cosðkl 2 Þ sin h In (6), g represents the intrinsic impedance of the medium and l is the length of the dipole where I 0 ¼ constant. The far electric field of a half-wave dipole of the linear array in Figure 2 in the azimuth plane can be obtained by substituting h ¼ 90 in (6) as: E h ffi jg I oe jkr 2pr (6) (7) International Journal of RF and Microwave Computer-Aided Engineering DOI /mmce

4 560 Günes and Tokan Figure 3 (a) Normalized pattern of the 40-element thinned conventional array with the excitation pattern of [ ]in the symmetrical configuration when compared with the patterns of the 40-element conventional array and the full-wave simulation patterns (b) Convergence curve for obtaining thinned version of the conventional array. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] Thus, normalized radiation pattern of the linear phased array of half-wave dipole antennas becomes in the azimuth plane: FFð/Þ ¼ XN n¼1 A n cos½kðd n þ D n Þðsin / sin / 0 ÞŠ (8) Now the problem statement simply reduces to applying the Pareto Genetic algorithm (PGA) to (8) to find the perturbation amount D n S of the array elements that will result in an array beam with a narrow/wide null generations in the required directions and minimization of the maximum levels of sidelobe regions of the radiation pattern, while the main beam is scanning within the prescribed ranges. IV. PARETO GENETIC OPTIMAL SYNTHESIS OF THE LINEAR PHASED ANTENNA ARRAYS In this work, we are interested in the synthesis of the geometry of a linear antenna array with minimum SLL and narrow/wide nulls generation at the interference locations, while scanning the main beam between the required scan angles. In the worked examples, only a single null generation will be considered, the approach can be extended to the case of the multiple nulls. However, in this bi-objective optimization case, as the objectives are in conflict with each other, they must be minimized together without making each other worse. Thus, the problem statement can be arranged as follows: n Minimize f 1 ð ~ o DÞ ( " ¼ Minimize max maxffð/; ~ DÞ /¼/ o /BW 2 ; /¼ 90 o maxjffð/þj /¼90o /¼/ o þ / BW 2 n Minimize f 2 ð ~ o DÞ 8 < ¼ Minimize max maxffð/; ~ DÞ /¼/ nullu : /¼/ nulll /o ¼/ ou / o ¼/ ol /o ¼/ ou / o ¼/ ol ) ð9þ 9 = ; ð10þ subject to 0:1 jd n j (11) where / 0 and / BW are the direction of the maximum radiation and the corresponding main beamwidth, respectively; whereas / ol and / ou stand for the lower and upper boundaries of the scanning angles. Here, (9) and (10) are stated for suppression of the MSLL and null generation at the interference locations, while scanning the main beam between the required scan angles, respectively. In (9) and (10), the decision vector ~x is replaced with the n-dimensioned perturbation vector ~ D where n ¼ N/2, N is the element number. In both (9) and (10), the minimization operator is applied to the maximum values chosen between all the greatest values within the sidelobe and null regions during main beam scanning. Equation (11) gives the limitations of the perturbation amount to minimize the mutual coupling effects between the elements. International Journal of RF and Microwave Computer-Aided Engineering/Vol. 20, No. 5, September 2010

5 Pareto Optimal Synthesis of the Linear Array 561 Figure 4 Pareto frontiers of the null region within the interval 69 to 71 and minimum possible SLL of the optimized (a) conventional (b) thinned conventional arrays for the given scanning ranges. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] V. TYPICAL EXAMPLES FOR THE PARETO OPTIMAL SYNTHESIS OF LINEAR PHASED ARRAYS In this section, some typical examples resulting from the application of Pareto optimization using the enhanced NSGA II to the objectives stated by the (9), (10), and (11) are presented. In the worked examples, a uniformly excited linear antenna array is considered that has 40 halfwave dipole antennas with initially uniform interelement spacing of 0.5 k. Hereafter, this original array configuration is named as conventional array. The second antenna array to be considered is the thinned version of the conventional array, which is obtained by a simple genetic optimization by rounding the excitation amplitudes to either 1 or 0 to minimize the maximum SLL during the beam scanning within the prescribed region as stated in the objective function in (9). Thus, the excitation amplitudes of the thinned array are shown to be as [ ],andtheresulting radiation pattern is given compared as the conventional and the full electromagnetic wave (CST) simulation patterns in Figure 3a, which is obtained only after nine generations as seen from its convergence characteristic in Figure 3b. Hereafter, all the radiation patterns synthesized by pattern multiplication are verified by their corresponding full-wave simulations using half-wave dipoles at 2.6 GHz. In Figure 3a, it can be seen that 5 db reduction in the MSLL is achieved by turning off the appropriate six elements of the conventional array that results in a decrease of necessity for energy, cost, and complexity of the array. For all the antenna arrays, the decision vector consists of a displacement vector of the elements positions, and its dimension is equal to the half of the element number because of the symmetrical array geometry (Fig. 2). As the bi-objective case is considered by the eqs. (9) (11) as taking the scanning range as a free parameter, Pareto frontiers are formed in the objective plane of the maximum International Journal of RF and Microwave Computer-Aided Engineering DOI /mmce

6 562 Günes and Tokan TABLE I Levels of the Nonoptimized Cases in the Plane of the Maximum Levels of the Whole Side Lobe and the Null Region Within the Interval 698 to 718 Conventional Array Thinned Conventional Array MSLL (db) max[ff(69 to 71 )] MSLL max[ff(69 to 71 )] Main beam directed to Scanning between [( 5 )to5 ] Scanning between [( 10 )to10 ] * 22.11* Scanning between [( 20 )to20 ] 13.29* 30.43* Scanning between [( 30 )to30 ] Scanning between [( 40 )to40 ] Scanning between [( 50 )to50 ] levels of the radiation pattern within the required null and the whole side lobe regions, depending on the beam scanning range. Thus, Pareto optimal solutions given with the frontiers are a set of equally valid solutions at a fixed main beam scan range. It should be noted that the pattern features resulted from these Pareto frontiers are valid for any chosen main beam direction within its full prescribed beam scanning range. Hereafter, some typical radiation pattern examples will be given when compared with their full wave counterparts and the conventional patterns. Figures 4a and 4b show the Pareto frontiers of the optimized conventional and thinned conventional arrays, respectively, for the scanning ranges of [( 50 )to50 ], [( 40 )to40 ], [( 30 )to30 ], [( 20 )to20 ], [( 10 ) to 10 ], [( 5 )to5 ] and [0 ] for the case of a narrow null between 69 and 71 and minimum possible SLL. Besides, the maximum levels of the null and SLL regions for the nonoptimized cases of conventional and thinned conventional arrays for the chosen scanning ranges are highlighted in Figures 4a and 4b, respectively. Table I summarizes the maximum levels of null and SLL regions of conventional and thinned conventional arrays for all the scanning ranges of the nonoptimized cases. The isolated single points highlighted in the Pareto frontiers plane are shown in Table I with the stars (*) to verify the success of the Pareto optimization process. Thus, the Pareto frontiers of the trade-off solution sets resulted from NSGA-II in Figures 4a and 4b provide a view of all design options for the predetermined cases. In the next stage, a decision must be made for choosing the solution set to be synthesized depending on the application. As can be observed from Figures 4a and 4b, an increase in the scanning range follows a decrease in the reduction performance of the MSLL and maximum level within the null region of the linear array. The first example illustrates the synthesis of a 40-element linear array directed to 15 for the chosen point ( 13.7 db; 34 db) within the scanning range [( 20 ) 20 ], which corresponds to the Pareto optimized version of the highlighted original point in the Pareto frontier plane in Figure 4a. Thus, the regions of suppressed SLL are [( 90 )to12 ]and[18 to 90 ] with the maximum level of 13.7 db, and the prescribed narrow null region is [69 to 71 ] with the maximum level of 34 db. In addition, the radiation patterns of the conventional and CST simulation version of the Pareto optimized patterns are also shown in Figure 5. Half-power beam width is nearly constant at / HPBW ¼ for all the patterns and the first column in Table III gives the necessary displacements of the elements for the synthesized radiation pattern. The second example illustrates the synthesis of a 34-element linear array obtained by thinning the 40-element Figure 5 Normalized radiation patterns for the conventional array, optimized array, and CST simulation of the half-wave dipoles for the chosen Pareto optimal point in Fig. 4(a). Here, the main beam direction is chosen as 15. Thus, the regions of suppressed SLL are [( 90 ) to 12 ] and [18 to 90 ] with the maximum level of 13.7 db, and the prescribed narrow null region is [69 to 71 ] with the maximum level of 34 db. / HPBW for both the conventional and optimized arrays. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] International Journal of RF and Microwave Computer-Aided Engineering/Vol. 20, No. 5, September 2010

7 Pareto Optimal Synthesis of the Linear Array 563 Figure 6 Normalized radiation patterns for the thinned array, optimized thinned array, and CST simulation of the half-wave dipoles for the chosen Pareto optimal point in Fig. 4(b). Here, the main beam direction is chosen as 10. Thus, the regions of suppressed SLL are [( 90 )to7 ] and [13 to 90 ] with the maximum level of 18.6 db, and the prescribed narrow null region is [69 to 71 ] with the maximum level of 26 db. / HPBW for both the thinned and optimized arrays. Figure 7 Pareto frontiers of the prescribed broad null region within the interval 65 to 75 and minimum possible SLL of the optimized (a) conventional (b) thinned conventional arrays for the given scanning ranges. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] International Journal of RF and Microwave Computer-Aided Engineering DOI /mmce

8 564 Günes and Tokan TABLE II Levels of the Nonoptimized Cases in the Plane of the Maximum Levels of the Whole Side Lobe and the Null Region Within the Interval 65 degree to 75 degree Conventional Array Thinned Conventional Array MSLL (db) max[ff(65 to 75 )] (db) MSLL (db) max[ff(65 to 75 )] (db) Main beam directed to db Scanning between [( 5 )to5 ] db Scanning between [( 10 )to10 ] db Scanning between [( 20 )to20 ] db Scanning between [( 30 )to30 ] * 17.94* db Scanning between [( 40 )to40 ] 13.24* 24.68* db Scanning between [( 50 )to50 ] db conventional array, directed to 10 for the chosen point ( 18.6 db; 26 db) within the scanning range [( 10 )to 10 ] which corresponds to the Pareto optimized version of the highlighted original point in the Pareto frontier plane in Figure 4b. Thus, the amount of 0.66 db and 3.89 db reductions are achieved at the maximum levels of sidelobe ([( 90 )to7 ]and[13 to 90 ]) and null regions ([69 to 71 ]), respectively. The radiation patterns of the thinned conventional and CST simulation version of the Pareto-optimized patterns are also shown in Figure 6. Half-power beamwidth is again nearly constant at / HPBW for all the patterns and the second column of Table III gives the necessary displacements of the elements for the synthesized radiation pattern. Figures 7a and 7b shows the Pareto frontiers of the conventional and thinned conventional arrays, respectively, for the same scanning ranges, for the case of a broad null between 65 and 75, and Table II gives all the corresponding solution pairs of the nonoptimized cases, where the highlighted isolated single points in the Pareto frontiers planes are indicated by stars. For the broad nulling case, two work examples are exhibited. The third example shows the synthesis of a 40-element linear antenna array directed to 20 for the chosen point ( 13.3 db; 32.7 db) within the scanning range [( 40 ) 40 ], which corresponds to the Pareto optimized version of the highlighted original point in the Pareto frontier plane in Figure 7a. Thus, the regions of suppressed SLL are [( 90 )to16 ] and [24 to 90 ] with the maximum level of db, and the prescribed narrow null region is [65 to 75 ] with the maximum level of 32.7 db. In addition, the radiation patterns of the conventional and CST simulation version of the Pareto optimized patterns are also shown in Fig. 8. Half-power beam width is nearly constant at / HPBW ¼ for all the patterns and the third column in Table III gives the necessary displacements of the elements for the synthesized radiation pattern. The final work example illustrates the synthesis of the 34-element thinned conventional array directed to 25 for the chosen point ( 18 db; 20 db) within the scanning range [( 30 )to30 ] which corresponds to the optimized version of the highlighted original point in the Pareto frontier plane in Figure 7b. Thus, the amount of 0.06 db and 4.06 db reductions are achieved at the maximum levels of sidelobe ([( 90 )to21 ] and [29 to 90 ]) and null regions ([69 to 71 ]), respectively. The radiation patterns of the thinned and CST simulation version of the Pareto-optimized patterns are also shown in Figure 9. Half-power beamwidth is again nearly constant at / HPBW for all the patterns, and the fourth column in Table III gives the necessary displacements of the elements for the synthesized radiation pattern. Figure 8 Normalized radiation patterns for the conventional array, optimized array, and CST simulation of the half-wave dipoles for the chosen Pareto optimal point in Fig. 7(a). Here, the main beam direction is chosen as 20. Thus, the regions of suppressed SLL are [( 90 ) to 16 ] and [24 to 90 ] with the maximum level of db, and the prescribed broad null region is [65 to 75 ] with the maximum level of 32.7 db. / HPBW for both the conventional and optimized arrays. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] International Journal of RF and Microwave Computer-Aided Engineering/Vol. 20, No. 5, September 2010

9 Pareto Optimal Synthesis of the Linear Array 565 TABLE III Optimized Element Position Perturbations D n for Figs. 5, 6, 8 and 9 D n (in k) Element Number Figure 5 Figure 6 Figure 8 Figure VI. CONCLUSIONS The antenna synthesis method carried out in this work is comprised of the following stages: 1. Synthesis of the scanning linear antenna array pattern is put forward as a constrained vector optimization problem whose components are to meet the minimum sidelobe level, control of the wide/narrow null placement during beam scanning whose range can vary from zero to the wide bands. 2. In this work, these objectives are achieved as nondominated solutions only by perturbations in the array elements positions in a linear geometry, while keeping a uniform excitation over the array aperture and a good quality of directivity. 3. Here, the NSGA-II is used in forming the Pareto optimal solutions as a fast nondominated genetic sorting algorithm. 4. Then, the Pareto frontiers are obtained using the tradeoff solution sets between the MSLL, null control, and scanning range to provide a view of all design options. 5. The same Pareto optimal synthesis procedure can be applied to a thinned linear antenna array. A thinned linear antenna array version is obtained using a simple genetic optimization by rounding the excitation amplitudes to either 1 or 0 to minimize the MSLL during beam scanning within the prescribed region as stated in the multiobjective function given by (9). Finally, the proposed method is applied to the synthesis of either fully populated or thinned linear arrays using only Figure 9 Normalized radiation patterns for the thinned array, optimized array, and CST simulation of the half-wave dipoles for the chosen Pareto optimal point in Fig. 7(b). Here, the main beam direction is chosen as 25. Thus, the regions of suppressed SLL are [( 90 )to 22 ] and [28 to 90 ] with the maximum level of 18 db, and the prescribed broad null region is [65 to 75 ] with the maximum level of 22 db. / HPBW for both the thinned and optimized arrays. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] International Journal of RF and Microwave Computer-Aided Engineering DOI /mmce

10 566 Günes and Tokan perturbations in the elements positions, based on the requirements of the chosen Pareto optimal points. Furthermore, the full electromagnetic wave simulations of these antenna arrays are also completed, and thus, the mutual coupling effects are examined. It can be concluded that the proposed method is successful to meet the Pareto optimal requirements with only perturbations in the elements positions. REFERENCES 1. K.K. Yan and Y.L. Lu, Sidelobe reduction in array pattern synthesis using genetic algorithm, IEEE Trans Antennas Propag 45 (1997), P.J. Bevelacqua and C.A. Balanis, Minimum sidelobe levels for linear arrays, IEEE Trans Antennas Propag 55 (2007), D.A. Tonn and R. Bansal, Reduction of sidelobe levels in interrupted phased array antennas by means of a genetic algorithm, Int J RF Microwave Comput Aided Eng 17 (2007), M.H. Er, Linear antenna array pattern synthesis with prescribed broad nulls, IEEE Trans Antennas Propag 38 (1990), R.L. 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Volakis, Antenna engineering handbook, McGraw Hill, New york, BIOGRAPHIES Filiz Günes received her MSc degree in Electronic and Communication Engineering from the Istanbul Technical University in She attained her PhD degree in communication engineering from the Bradford University, London, in She worked as a research fellow at the same university between 1979 and 1983 with contracts from the European Space Agency and British Defence Minister in the areas of the propagation and electromagnetic compatibility. Since 1983, she has been with the Yildiz Technical University, where she is currently a Full Professor and chair of the Electronic and Communication Engineering Department. Her current research interests are in the areas of multivariable network theory, device modeling, computer-aided circuit design, microwave amplifiers, microwave filters, broadband matching circuits, monolithic microwave integrated circuits, and electromagnetic compatibility, antenna arrays. Fikret Tokan was born in Lüleburgaz, Turkey, on August 25, He received the BS degree in Electronics and Communications Engineering from the Yildiz Technical University in 2002 and MS degree from the Yildiz Technical University, Istanbul, in Communication Engineering in He has been currently working as a research assistant toward his PhD degree. His current research interests are electromagnetic waves, propagation, antenna arrays, scattering, and numerical methods. International Journal of RF and Microwave Computer-Aided Engineering/Vol. 20, No. 5, September 2010