Adjustment of Beamwidth and Side-Lobe Level of Large Phased-Arrays Using Particle Swarm Optimization Technique Song-Han Yang and Jean-Fu Kiang

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138 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 1, JANUARY 2014 Adjustment of Beamwidth and Side-Lobe Level of Large Phased-Arrays Using Particle Swarm Optimization Technique Song-Han Yang and Jean-Fu Kiang

Abstract—A particle swarm optimization technique is applied to optimize the spacings between elements of a 32-element un- maintain a constant beamwidth of a large phased-array when its evenly spaced linear array, to minimize the side-lobe level and major lobe is pointing away from the broadside direction. The first to control the null locations [11]. S. K. Goudos et al. presented side-lobe is suppressed to a minimum possible level, at the expense of raising the other side-lobes. Tapering of excitation amplitude a comprehensive learning PSO (CLPSO) method, one variant acrosstheapertureisalsoexercised to help suppressing the side- of PSO’s, to do the same job as that in [10]. R. Bhattacharya et lobe level. al. proposed a position mutated hierarchical PSO (PM-HPSO) Index Terms— Design optimization, phased arrays. to reduce the sidelobe level of an unevenly spaced 108-element planar thinned array [16]. The PM-HPSO is a hybrid of the hierarchical PSO with time-varying acceleration coeffi- I. INTRODUCTION cients (HPSO-TVAC) algorithm [17] and the mutation scheme. In [18], an invasive weed optimization (IWO) method is pro- ARGE phased arrays have been proposed in solar power posed, which is claimed better than the PSO in both accuracy L stations, astronomy observatories, radar systems, and so and efficiency, when tested upon an unevenly spaced 108-ele- on. The major concerns in these applications include high power ment planar thinned array. B.-K. Yeo et al. tried to compensate concentration ratio (PCR) of the major lobe, low side-lobe level, the radiation pattern of a 32-element linear array with GA, avoiding blind spots, maximizing power transmission, and so on PSO and hybrid PSO-GA [19]–[21], under the condition that [1]–[4]. some elements have failed. In [5]–[7], large phased arrays with Gaussian distribution In this work, an adaptive PSO method [22] is used to opti- of excitation have been proposed as space antennas in a solar mize the phases of elements in a large planar phased-array. A power system, to transmit more power within the major lobe systematic parameter adaptation scheme and convergence im- and reduce the side-lobe level simultaneously. provement scheme are also used to make the PSO more effi- Side-lobe reduction has been an important issue in array opti- cient and effective. The goal is to maintain the beamwidth of mization problems [8]–[15]. The Newton-Raphson method has the major lobe when it is swept away from the broadside di- been combined with a conjugate gradient method to reconstruct rection, while suppressing the side-lobes to a minimum pos- the radiation pattern of a hexagonal planar array when some el- sible level. The radiation characteristics of a hexagonal array ements fail [8]. Local optimization techniques like this one may is briefly described in Section II; the scheme of amplitude ta- fail to find the global optimum solution. Evolutionary types of pering and phase adjustment is presented in Section III, illus- computational technique have been developed to search for the trated with two hexagonal arrays of different sizes. The results global optimum, for example, genetic algorithm (GA) [9], par- of three larger arrays are presented in Section IV, followed by ticle swarm optimization (PSO) [10], [11], differential evolution the conclusion. strategy (DES) [12], ant colony optimization (ACO) [13], simulated annealing (SA) [14], to name a few. K. K. Yan et al. proposed a real-genetic (continuous ge- II. ARRAY OF HEXAGONAL APERTURES netic) algorithm to reduce the side-lobe level of a 30-element Fig. 1 depicts a hexagonal planar array composed of 1 141 linear array and another circular array of the same size [9], in hexagonal elements, with elements along each side of which the chromosome contains the information of phase and the outer perimeter. The side length of each element is ,and magnitude. M. M. Khodier et al. applied a PSO algorithm to the separation between two adjacent elements is . To calculate the radiation into free space, each element aperture supports an Manuscript received June 21, 2013; revised August 15, 2013; accepted equivalent magnetic surface current of . Without September 30, 2013. Date of publication October 24, 2013; date of current version December 31, 2013. This work was supported in part by the National loss of generality, make the approximation on all ele- Science Council, Taiwan, under contract NSC 100-2221-E-002-232. ment apertures, thus . The authors are with the Department of Electrical Engineering and the Grad- The electric fieldinthefar-field region can be expressed as uate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan (e-mail: [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.2013.2287280 (1)

(4)

The total radiated power is equal to the sum of powers radiated out of all the element apertures [5], [23]

(5)

The directive gain of the array can thus be calculated as [5], [23]

(6)

(7)

III. AMPLITUDE TAPERING AND PHASE ADJUSTMENT Consider a hexagonal planar array composed of 1 141 hexagonal elements, as shown in Fig. 1, with , , , ,where and are the magnitude of excitation at the center and the outermost layer, respectively. The diameter, , of the array, measured from one corner Fig. 1. Configuration of a hexagonal planar array composed of 1 141 hexagonal elements, each side of the outer perimeter contains elements, is the to its opposite, is 0.598 m. side length of each element, is the separation between two adjacent elements. Before tapering the magnitude on the element apertures, these In this sketch, the steering axis and the quadrature axis overlap with the axis elements are first labeled in layers: The element at the center of and the axis, respectively. The subscripts, and , are the layer indices of elements, counted along the steering axis and the quadrature axis, respectively. the array is labeled as layer (0), the elements right adjacent to layer (0) are labeled as layer (1), the elements right outside of layer are labeled as layer . The magnitude of the with excitation field in each element aperture of layer is tapered as [5]

(8)

Fig. 2(a) shows the radiation pattern when the major lobe is steered to point at and . Compared with the pattern with the major lobe pointing at and , (2) the beamwidth of the major lobe is increased along the steering direction. where is the number of elements, is the In order to maintain the major-lobe width along the steering center coordinate of the th element aperture, with tangential direction, different tapering rates are adopted in the steering di- electric ﬁeld ,and . The array pattern, , rection and the quadrature direction, respectively, as takes the form (9) (3) where the subscripts, and , are the layer indices of elements, counted along the steering axis and the quadrature axis, and the element pattern, , is derived as [5] respectively. Figs.2(b)and3showtheradiationpatternsofthearraywith and . Note that the distribution of excitation ﬁeld strength on the array surface becomes more uniform when is increased. The null-to-null beamwidth of the major lobe 140 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 1, JANUARY 2014

Fig. 3. Directivity gain along (a) steering axis and (b) quadrature axis; : , : , , : , plus phase adjustment; other parameters are the same as in Fig. 2.

TABLE I PARAMETERS CHOSEN IN THE APSO ALGORITHM

while keeping all the side-lobe level below a threshold, we choose the following ﬁtness function:

(10)

Fig. 2. Radiation patterns (in dB): (a) ,(b) , ,(c) where are the phases, to be optimized, , plus phase adjustment; , , , , , . of rows of element apertures, with each row parallel to the quadrature axis; is the directivity gain (in dB) of the first side-lobe, at the angular direction of ;BW (in de- along the steering axis is reduced from 17.35 to 15.54 ,while grees) and (in degrees) are the implemented beamwidth that along the quadrature axis is reduced from 17.04 to 14.06 . and the desired beamwidth, respectively, of the array; and However, the side-lobe level is significantly raised. is an empirical weighting factor. The first term on the right In a phased array, adjusting element locations is equivalent hand side of (10) is used to constrain the side-lobe level, and to adjusting the phase of elements, and a location displacement the second term on the right hand side of (10) is used to keep of 0.75 cm at is equivalent to a phase change of the beamwidth of the major lobe close to . The parameters 70.8 . In this work, the elements of a phased array are fixed in used in the simulation are listed in Table I. location, but their phases are allowed to change from to Fig. 4(a) shows that the fitness value of the group best par- 70.8 [24]. To reduce the optimization complexity, the phase is ticle converges after 630 iterations. Fig. 4(b) shows the associ- adjusted along the steering axis; namely, the phases of element ated phase adjustments of all the 39 rows. The side-lobe level apertures in the same row parallel to the quadrature axis are kept is reduced to 19.33 dB below the major-lobe level, but the the same. other side-lobes are raised to about the same level as the first An adaptive particle swarm optimization (APSO) method side-lobe, as shown in Figs. 2(c) and 3. The beamwidth of the proposed by Zhan et al. [22] is adopted to adjust the phases, major lobe is reduced to 15.54 , the same as pointing in the which includes a stage of evolutionary state estimation and broadside direction; but the peak gain drops by about 1 dB be- another stage of gbest variation with elitist learning strategy. cause some power originally radiated within the major lobe has In order to maintain the original beamwidth of the major lobe, been redistributed to the side-lobes. YANG AND KIANG: ADJUSTMENT OF BEAMWIDTH AND SIDE-LOBE LEVEL OF LARGE PHASED-ARRAYS 141

Fig. 4. (a) Fitness value of group best particle and (b) phase adjustment; other parameters are the same as in Fig. 2.

Fig. 5. An array of two elements, (a) fixed field distribution on each aperture, (b) fixed incident amplitude in each feeding guide.

Fig. 6. Radiation patterns (in dB): (a) ,(b) , ,(c) This paper focuses on the synthesis and optimization of array , plus phase adjustment; , , , , , . patterns, without considering the mutual coupling effects [25]. Fig. 5(a) demonstrates a simplified example of two-element array. The field amplitudes on apertures and are fixed as and , respectively, which will be adjusted using the currents on both and ,thenthefield pattern of this two-el- approach in this work to achieve the desired array pattern. ement array can be obtained on these surface currents. Methods Fig. 5(b) shows a more sophisticated alternative. The ampli- like this are rigorous, yet computationally expensive. tudes of the dominant mode, and , in the feeding guides The case in Fig. 5(a) can be viewed as an approximation to are decoupled from each other, and will be adjusted using the that in Fig. 5(b). The apertures mainly radiate in the forward di- approach in this work. Beneath ,with ,2,thetotal rection, their coupling effects to other elements could be consid- field is consisted of the incident field, , its reflection, plus ered as secondary, if no surface waves are excited. The mutual higher-order modes of the feeding guide. Equivalent electric coupling effects can be further reduced with proper separation, and magnetic surface currents can be assumed on , to ac- , between elements, as marked in Fig. 1. count for its radiated fields. The field above is contributed Bearing these constraints in mind, when the same tapering by the equivalent surface currents on both and ,soisthe rates are applied to ’s in both cases of Fig. 5(a) and Fig. 5(b), field above . By imposing the condition that the tangential major characteristics of the array pattern, like beamwidth of the fields beneath and above are continuous, a set of inte- major lobe and side-lobe level, are expected to be similar. When gral equations can be derived. Numerical methods, like method the optimization algorithm is applied, the amplitudes, ’s, in of moments, can be applied to solve for the equivalent surface both cases are expected to be similar, too. 142 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 1, JANUARY 2014

TABLE II PERFORMANCE COMPARISON BETWEEN PSO [15] AND APSO [22] METHODS

Fig. 7. Directivity gain along steering-axis (a) and quadrature-axis (b); : , : , , : , plus phase adjustment; other parameters are the same as in Fig. 6.

IV. ARRAYS WITH LARGER SIZE Next, consider a larger hexagonal planar array with , which is composed of 4 681 hexagonal elements. The other parameters are , , , ,and . Fig. 6(a) shows the radiation pattern of the array, with the tapering rate of .Whenthe major lobe is steered to point at and , apply the tapering rates of and . The major-lobe width along the steering axis is reduced from 8.68 to 7.59 ,while that along the quadrature axis is reduced from 8.3 to 6.86 ,as showninFigs.6(b)and7. The parameters chosen in the APSO algorithm are the same as listed in Table I, except . Similar to the case with ,theﬁrst side-lobe level can be reduced to about 20 dB below the peak gain after the tapering rates and the Fig. 8. Directivity gain along steering-axis near the pointing direction: (a) , ,(b) , ,(c) , phases are adjusted, and the major-lobe width is close to , ; : , : , , : , as shown in Fig. 7. Similar to the previous case, the other side- plus phase adjustment; , , , lobes are raised to about the same level as the ﬁrst side-lobe, as . shown in Fig. 6(c). Also note that the peak gain drops by about 1.24 dB. Both the PSO [15] and the APSO [22] methods have been ap- , , and . plied to the cases with and ,and The array with contains 18 961 hexagonal their performance is summarized inTableII.Theside-lobesare elements in 159 rows, The array with reduced to a similar level with both methods, while the APSO contains 76 321 hexagonal elements in 319 rows, converges faster than the PSO. The simulations are carried out The array with contains 306 241 hexagonal on a personal computer with an i5-2400 3.10 GHz CPU, 8 GB elements in 639 rows. RAM memory, and Windows 7 operating system. The compu- The characteristics of radiation patterns of these three arrays tation time of APSO over 10 000 iterations to simulate the cases are summarized in Table III. For the array with , of and is 1.2 and 2 hours, respectively. the side-lobe level is reduced to 17.07 dB below the peak gain Next, larger arrays with , 4.784 m, and 9.568 after 150 iterations. For the array with ,theside- m, respectively, are simulated. The common parameters are lobe level is reduced to 17.36 dB below the peak gain after YANG AND KIANG: ADJUSTMENT OF BEAMWIDTH AND SIDE-LOBE LEVEL OF LARGE PHASED-ARRAYS 143

TABLE III CHARACTERISTICS OF RADIATION PATTERNS WITH , 4.784, 9.568 m

The number before (after) the slash sign is along the steering (quadrature) axis.

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[15] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in [25] S. K. Goudos, K. Siakavara, T. Samaras, E. E. Vafiadis, and J. N. Sa- electromagnetics,” IEEE Trans. Antennas Propagat., vol. 52, no. 2, pp. halos, “Sparse linear array synthesis with multiple constraints using 397–407, Feb. 2004. differential evolution with strategy adaptation,” IEEE Antennas Wire- [16] R. Bhattacharya, T. K. Bhattacharyya, and R. Garg, “Position mu- less Propagat. Lett., vol. 10, pp. 670–673, 2011. tated hierarchical particle swarm optimization and its application in synthesis of unequally spaced antenna arrays,” IEEE Trans. Antennas Propagat., vol. 60, no. 7, pp. 3174–3181, Jul. 2012. [17] A. Ratnaweera, S. K. Halgamuge, and H. C. Watson, “Self-organizing hierarchical particle swarm optimizer with time-varying acceleration Song-Han Yang was born in Tainan, Taiwan. He coefficients,” IEEE Trans. Evol. Comput., vol. 8, no. 3, pp. 240–255, received the B.S. degree in electrical engineering Jun. 2004. from National Sun Yat-Sen University, Kaohsiung, [18] S. Karimkashi and A. A. Kishk, “Invasive weed optimization and its Taiwan,inJune2011andtheM.S.degreein features in electromagnetics,” IEEE Trans. Antennas Propagat., vol. electrical engineering from the Graduate Institute 58, no. 4, pp. 1269–1278, Apr. 2010. of Communication Engineering, National Taiwan [19] B. K. Yeo and Y. Lu, “Array failure correction with a genetic algo- University, Taipei, Taiwan, in 2013. rithm,” IEEE Trans. Antennas Propagat., vol. 47, pp. 823–828, May He is currently serving as a Second Lieutenant in 1999. the ROC Army, Taiwan. [20] B. K. Yeo and Y. Lu, “Adaptive array digital beamforming using com- plex-coded particle swarm optimization-genetic algorithm,” in Proc. Asia-Pacific Microwave Conf., Dec. 2005, DOI: 10.1109/APMC.2005. 1606380. [21] B. K. Yeo and Y. Lu, “Fast array failure correction using improved particle swarm optimization,” in Proc. Asia-Pacific Microwave Conf., Jean-Fu Kiang received the Ph.D. degree in elec- Dec. 2009, pp. 1537–1540. trical engineering, from the Massachusetts Institute [22]Z.H.Zhan,J.Zhang,Y.Li,andH.S.H.Chung,“Adaptiveparticle of Technology, Cambridge, MA, USA, in 1989. swarm optimization,” IEEE Trans. Syst. Man Cyber., vol. 39, no. 6, He has been a professor of the Department of pp. 1362–1381, Dec. 2009. Electrical Engineering and the Graduate Institute [23] C. A. Balanis, Antenna Theory, 3rd ed. New York, NY, USA: Wiley, of Communication Engineering, National Taiwan 2005, pp. 64–69. University since 1999. He has been interested in [24] Y. B. Tian and J. Qian, “Improve the performance of a linear array electromagnetic applications and system issues, in- by changing the spaces among array elements in terms of genetic algo- cluding wave propagation in different environments, rithm,” IEEE Trans. Antennas Propagat., vol. 53, no. 7, pp. 2226–2230, antennas and arrays, radar and navigation, RF and Jul. 2005. microwave systems.