138 ieee transactions on antennas and propagation,...

138 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 1, JANUARY 2014

Adjustment of Beamwidth and Side-Lobe Levelof Large Phased-Arrays Using Particle Swarm

Optimization TechniqueSong-Han Yang and Jean-Fu Kiang

Abstract—A particle swarm optimization technique is applied tomaintain a constant beamwidth of a large phased-array when itsmajor lobe is pointing away from the broadside direction. The firstside-lobe is suppressed to a minimum possible level, at the expenseof raising the other side-lobes. Tapering of excitation amplitudeacross the aperture is also exercised to help suppressing the side-lobe level.

Index Terms— Design optimization, phased arrays.

I. INTRODUCTION

L ARGE phased arrays have been proposed in solar powerstations, astronomy observatories, radar systems, and so

on. Themajor concerns in these applications include high powerconcentration ratio (PCR) of the major lobe, low side-lobe level,avoiding blind spots, maximizing power transmission, and so on[1]–[4].In [5]–[7], large phased arrays with Gaussian distribution

of excitation have been proposed as space antennas in a solarpower system, to transmit more power within the major lobeand reduce the side-lobe level simultaneously.Side-lobe reduction has been an important issue in array opti-

mization problems [8]–[15]. The Newton-Raphson method hasbeen combined with a conjugate gradient method to reconstructthe radiation pattern of a hexagonal planar array when some el-ements fail [8]. Local optimization techniques like this one mayfail to find the global optimum solution. Evolutionary types ofcomputational technique have been developed to search for theglobal optimum, for example, genetic algorithm (GA) [9], par-ticle swarm optimization (PSO) [10], [11], differential evolutionstrategy (DES) [12], ant colony optimization (ACO) [13], sim-ulated annealing (SA) [14], to name a few.K. K. Yan et al. proposed a real-genetic (continuous ge-

netic) algorithm to reduce the side-lobe level of a 30-elementlinear array and another circular array of the same size [9], inwhich the chromosome contains the information of phase andmagnitude. M. M. Khodier et al. applied a PSO algorithm to

Manuscript received June 21, 2013; revised August 15, 2013; acceptedSeptember 30, 2013. Date of publication October 24, 2013; date of currentversion December 31, 2013. This work was supported in part by the NationalScience Council, Taiwan, under contract NSC 100-2221-E-002-232.The authors are with the Department of Electrical Engineering and the Grad-

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

optimize the spacings between elements of a 32-element un-evenly spaced linear array, to minimize the side-lobe level andto control the null locations [11]. S. K. Goudos et al. presenteda comprehensive learning PSO (CLPSO) method, one variantof PSO’s, to do the same job as that in [10]. R. Bhattacharya etal. proposed a position mutated hierarchical PSO (PM-HPSO)to reduce the sidelobe level of an unevenly spaced 108-ele-ment planar thinned array [16]. The PM-HPSO is a hybridof the hierarchical PSO with time-varying acceleration coeffi-cients (HPSO-TVAC) algorithm [17] and the mutation scheme.In [18], an invasive weed optimization (IWO) method is pro-posed, which is claimed better than the PSO in both accuracyand efficiency, when tested upon an unevenly spaced 108-ele-ment planar thinned array. B.-K. Yeo et al. tried to compensatethe radiation pattern of a 32-element linear array with GA,PSO and hybrid PSO-GA [19]–[21], under the condition thatsome elements have failed.In this work, an adaptive PSO method [22] is used to opti-

mize the phases of elements in a large planar phased-array. Asystematic parameter adaptation scheme and convergence im-provement scheme are also used to make the PSO more effi-cient and effective. The goal is to maintain the beamwidth ofthe major lobe when it is swept away from the broadside di-rection, while suppressing the side-lobes to a minimum pos-sible level. The radiation characteristics of a hexagonal arrayis briefly described in Section II; the scheme of amplitude ta-pering and phase adjustment is presented in Section III, illus-trated with two hexagonal arrays of different sizes. The resultsof three larger arrays are presented in Section IV, followed bythe conclusion.

II. ARRAY OF HEXAGONAL APERTURES

Fig. 1 depicts a hexagonal planar array composed of 1 141hexagonal elements, with elements along each side ofthe outer perimeter. The side length of each element is , andthe separation between two adjacent elements is . To calculatethe radiation into free space, each element aperture supports anequivalent magnetic surface current of . Withoutloss of generality, make the approximation on all ele-ment apertures, thus .The electric field in the far-field region can be expressed as

(1)

0018-926X © 2013 IEEE

YANG AND KIANG: ADJUSTMENT OF BEAMWIDTH AND SIDE-LOBE LEVEL OF LARGE PHASED-ARRAYS 139

Fig. 1. Configuration of a hexagonal planar array composed of 1 141 hexagonalelements, each side of the outer perimeter contains elements, is theside length of each element, is the separation between two adjacent elements.In this sketch, the steering axis and the quadrature axis overlap with the axisand the axis, respectively. The subscripts, and , are the layer indices ofelements, counted along the steering axis and the quadrature axis, respectively.

with

(2)

where is the number of elements, is thecenter coordinate of the th element aperture, with tangentialelectric field , and . The array pattern, ,takes the form

(3)

and the element pattern, , is derived as [5]

(4)

The total radiated power is equal to the sum of powers radi-ated 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 hexag-onal elements, as shown in Fig. 1, with , ,

, , where and are the magni-tude of excitation at the center and the outermost layer, respec-tively. The diameter, , of the array, measured from one cornerto its opposite, is 0.598 m.Before tapering the magnitude on the element apertures, these

elements are first labeled in layers: The element at the center ofthe array is labeled as layer (0), the elements right adjacent tolayer (0) are labeled as layer (1), the elements right outside oflayer are labeled as layer . The magnitude of theexcitation field in each element aperture of layer is taperedas [5]

(8)

Fig. 2(a) shows the radiation pattern when the major lobe issteered to point at and . Compared with thepattern with the major lobe pointing at and ,the beamwidth of the major lobe is increased along the steeringdirection.In order to maintain the major-lobe width along the steering

direction, different tapering rates are adopted in the steering di-rection and the quadrature direction, respectively, as

(9)

where the subscripts, and , are the layer indices of ele-ments, counted along the steering axis and the quadrature axis,respectively.Figs. 2(b) and 3 show the radiation patterns of the array with

and . Note that the distribution of excitationfield strength on the array surface becomes more uniform whenis increased. The null-to-null beamwidth of the major lobe


Fig. 2. Radiation patterns (in dB): (a) , (b) , , (c), plus phase adjustment; , , ,

, , .

along the steering axis is reduced from 17.35 to 15.54 , whilethat along the quadrature axis is reduced from 17.04 to 14.06 .However, the side-lobe level is significantly raised.In a phased array, adjusting element locations is equivalent

to adjusting the phase of elements, and a location displacementof 0.75 cm at is equivalent to a phase change of70.8 . In this work, the elements of a phased array are fixed inlocation, but their phases are allowed to change from to70.8 [24]. To reduce the optimization complexity, the phase isadjusted along the steering axis; namely, the phases of elementapertures in the same row parallel to the quadrature axis are keptthe same.An adaptive particle swarm optimization (APSO) method

proposed by Zhan et al. [22] is adopted to adjust the phases,which includes a stage of evolutionary state estimation andanother stage of gbest variation with elitist learning strategy.In order to maintain the original beamwidth of the major lobe,

Fig. 3. Directivity gain along (a) steering axis and (b) quadrature axis; :

, : , , : , plus phaseadjustment; other parameters are the same as in Fig. 2.

TABLE IPARAMETERS CHOSEN IN THE APSO ALGORITHM

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

(10)

where are the phases, to be optimized,of rows of element apertures, with each row parallel to thequadrature axis; is the directivity gain (in dB) of thefirst side-lobe, at the angular direction of ; BW (in de-grees) and (in degrees) are the implemented beamwidthand the desired beamwidth, respectively, of the array; andis an empirical weighting factor. The first term on the righthand side of (10) is used to constrain the side-lobe level, andthe second term on the right hand side of (10) is used to keepthe beamwidth of the major lobe close to . The parametersused in the simulation are listed in Table I.Fig. 4(a) shows that the fitness value of the group best par-

ticle converges after 630 iterations. Fig. 4(b) shows the associ-ated phase adjustments of all the 39 rows. The side-lobe levelis reduced to 19.33 dB below the major-lobe level, but theother side-lobes are raised to about the same level as the firstside-lobe, as shown in Figs. 2(c) and 3. The beamwidth of themajor lobe is reduced to 15.54 , the same as pointing in thebroadside direction; but the peak gain drops by about 1 dB be-cause some power originally radiated within the major lobe hasbeen redistributed to the side-lobes.


Fig. 4. (a) Fitness value of group best particle and (b) phase adjustment; otherparameters 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.

This paper focuses on the synthesis and optimization of arraypatterns, without considering the mutual coupling effects [25].Fig. 5(a) demonstrates a simplified example of two-elementarray. The field amplitudes on apertures and are fixedas and , respectively, which will be adjusted using theapproach in this work to achieve the desired array pattern.Fig. 5(b) shows a more sophisticated alternative. The ampli-

tudes of the dominant mode, and , in the feeding guidesare decoupled from each other, and will be adjusted using theapproach in this work. Beneath , with , 2, the totalfield is consisted of the incident field, , its reflection, plushigher-order modes of the feeding guide. Equivalent electricand magnetic surface currents can be assumed on , to ac-count for its radiated fields. The field above is contributedby the equivalent surface currents on both and , so is thefield above . By imposing the condition that the tangentialfields beneath and above are continuous, a set of inte-gral equations can be derived. Numerical methods, like methodof moments, can be applied to solve for the equivalent surface

Fig. 6. Radiation patterns (in dB): (a) , (b) , , (c), plus phase adjustment; , , ,

, , .

currents on both and , then the field pattern of this two-el-ement array can be obtained on these surface currents. Methodslike this are rigorous, yet computationally expensive.The case in Fig. 5(a) can be viewed as an approximation to

that in Fig. 5(b). The apertures mainly radiate in the forward di-rection, their coupling effects to other elements could be consid-ered as secondary, if no surface waves are excited. The mutualcoupling effects can be further reduced with proper separation,, between elements, as marked in Fig. 1.Bearing these constraints in mind, when the same tapering

rates are applied to ’s in both cases of Fig. 5(a) and Fig. 5(b),major characteristics of the array pattern, like beamwidth of themajor lobe and side-lobe level, are expected to be similar. Whenthe optimization algorithm is applied, the amplitudes, ’s, inboth cases are expected to be similar, too.


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 . When themajor lobe is steered to point at and , apply thetapering rates of and . The major-lobe widthalong the steering axis is reduced from 8.68 to 7.59 , whilethat along the quadrature axis is reduced from 8.3 to 6.86 , asshown in Figs. 6(b) and 7.The parameters chosen in the APSO algorithm are the same

as listed in Table I, except . Similar to the case with, the first side-lobe level can be reduced to about

20 dB below the peak gain after the tapering rates and thephases are adjusted, and the major-lobe width is close to ,as shown in Fig. 7. Similar to the previous case, the other side-lobes are raised to about the same level as the first side-lobe, asshown in Fig. 6(c). Also note that the peak gain drops by about1.24 dB.Both the PSO [15] and the APSO [22] methods have been ap-

plied to the cases with and , andtheir performance is summarized in Table II. The side-lobes arereduced to a similar level with both methods, while the APSOconverges faster than the PSO. The simulations are carried outon a personal computer with an i5-2400 3.10 GHz CPU, 8 GBRAM memory, and Windows 7 operating system. The compu-tation time of APSO over 10 000 iterations to simulate the casesof and is 1.2 and 2 hours, respectively.Next, larger arrays with , 4.784 m, and 9.568

m, respectively, are simulated. The common parameters are

TABLE IIPERFORMANCE COMPARISON BETWEEN PSO [15] AND APSO [22] METHODS

Fig. 8. Directivity gain along steering-axis near the pointing direction: (a), , (b) , , (c) ,; : , : , , : ,

plus phase adjustment; , , ,.

, , and .The array with contains 18 961 hexagonalelements in 159 rows, The array withcontains 76 321 hexagonal elements in 319 rows,The array with contains 306 241 hexagonalelements in 639 rows.The characteristics of radiation patterns of these three arrays

are summarized in Table III. For the array with ,the side-lobe level is reduced to 17.07 dB below the peak gainafter 150 iterations. For the array with , the side-lobe level is reduced to 17.36 dB below the peak gain after


TABLE IIICHARACTERISTICS OF RADIATION PATTERNS WITH , 4.784, 9.568 m

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

80 iterations. For the array with , the side-lobelevel is only reduced to 14.21 dB below the peak gain after 20iterations.In these simulations, the side-lobe level along the quadrature

axis remains almost the same as when the major-lobe points tothe broadside direction. The side-lobe level along the steering-axis, on the other hand, is raised when the major-lobe pointsaway from the broadside direction. Fig. 8 shows the directivitygain, along the steering-axis, of the arrays with ,4.784 m and 9.568 m, respectively. The level of high-order side-lobes increases, but remains lower than that of the first side-lobe.

V. CONCLUSION

A particle swarm optimization technique has been appliedto adjust the phase of excitation field on all the element aper-tures of a large planar array, so that the beamwidth of the majorlobe can be restrained from broadening when it is pointing awayfrom the broadside direction. The first side-lobe is suppressedto a minimum possible level, at the expense of raising the otherside-lobes. Large arrays with corner-to-corner diameter of 2.392m, 4.784 m and 9.568 m, respectively, have been simulated.Their side-lobe level along the steering axis can be reduced to17.07 dB, 17.36 dB and 14.21 dB, respectively, as com-

pared with their peak gain. Tapering of excitation amplitudeacross the aperture is also considered to help suppressing thefirst side-lobe.

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Song-Han Yang was born in Tainan, Taiwan. Hereceived the B.S. degree in electrical engineeringfrom National Sun Yat-Sen University, Kaohsiung,Taiwan, in June 2011 and the M.S. degree inelectrical engineering from the Graduate Instituteof Communication Engineering, National TaiwanUniversity, Taipei, Taiwan, in 2013.He is currently serving as a Second Lieutenant in

the ROC Army, Taiwan.

Jean-Fu Kiang received the Ph.D. degree in elec-trical engineering, from the Massachusetts Instituteof Technology, Cambridge, MA, USA, in 1989.He has been a professor of the Department of

Electrical Engineering and the Graduate Instituteof Communication Engineering, National TaiwanUniversity since 1999. He has been interested inelectromagnetic applications and system issues, in-cluding wave propagation in different environments,antennas and arrays, radar and navigation, RF andmicrowave systems.

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