Uniform Circular Phased Arrays synthesis usingSQP Algorithm

A. HammamiR.Ghayoula

and A. GharsallahUnite de recherche :

Circuits et systmes electroniques HFFaculte des Sciences de Tunis,

Campus Universitaire Tunis EL-manar,2092,

TunisieEmail: hammami.a.fst@gmail.com

Abstract—In this paper, a uniform circular array antenna withsynthesis technique based on Sequential Quadratic Programming(SQP) algorithm is discussed. The SQP algorithm is a robustmethod for the optimization with inequality constraints. SQPalgorithm has been used for steering the main lobe and reducingthe side lobe level for the radiation pattern by controlling onlythe amplitude of the array elements. In order to illustratethe performance of the proposed method, several examples ofuniform circular antennas array patterns are performed.

Index Terms—antenna arrays, circular arrays, optimizationmethods,sequential Quadratic Programming

I. INTRODUCTION

Antennas arrays become important in communication ap-plications like sonar, radar and communications. Indeed, theyreduce the electromagnetic environment pollution by minimiz-ing the side lobe level and enhance spectral efficiency.Among the different types of antenna arrays,circular antennaarray has become very popular in cellular mobile applicationsand wireless communications [1]-[5].In last decade, several methods have been developed forantenna array pattern synthesis [6]-[16]. In this paper, we pro-pose the use of the Sequential Quadratic Programming (SQP)algorithm to determine an optimum set of amplitude excitationcoefficients for circular arrays that provide a radiation patternwith maximum side lobe level reduction.

The rest of the paper is arranged as follows: the uniformcircular antenna arrays is presented in section II. Section IIIshows optimization of uniform circular phased arrays using thesequential quadratic programming (SQP) algorithm. SectionIV demonstrate the simulation results, and finally, section VImakes conclusions.

II. UNIFORM CIRCULAR ANTENNA ARRAYS

Consider a circular antenna array composed of 𝑁 equis-paced isotropic elements with interelement spacing 𝑑. Theseselements are distributed on a circle of radius 𝑅 in the 𝑥 − 𝑦plane. 𝜑 is azimuth angle starting forward x axis. 𝜃 is pitch

R

φ

θ

Fig. 1. Illustration of uniform circular array antenna

angle starting forward z axis.The azimuth angle of 𝑛𝑡ℎelement𝜑𝑛 is given by the following equation:

𝜑𝑖 = 2𝜋𝑖

𝑁, 𝑖 = 1, ⋅ ⋅ ⋅𝑁 (1)

The radius of the circle is :

𝑅 =𝑑

2𝑠𝑖𝑛( 𝜋𝑁 )

(2)

or the the distance between tow consecutive elements are 𝑑 =𝜆/2,

𝑅 =𝜆

4𝑠𝑖𝑛( 𝜋𝑁 )

(3)

The radiation pattern of the circular array, shown in figure 1,can be described by its array factor given by:

𝐴𝐹 (𝜃, 𝜑) =

𝑁∑𝑛=1

𝐼𝑛𝑒𝑗(𝑘𝑅𝑠𝑖𝑛𝜃𝑐𝑜𝑠(𝜑−𝜑𝑛)+𝛼𝑛)) (4)

with 𝑁 is the number of the uniform circular array elements,𝑘 = 2𝜋

𝜆 is the free space wavenumber,𝜆 is the wavelength,𝜃 ∈ [−𝜋/2 𝜋/2], 𝜑 ∈ [0 𝜋] and 𝐼𝑛,𝛼𝑛 are amplitude andphase of each element, respectively.

𝛼𝑛 = −𝑘𝑅𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0 − 𝜑𝑛) (5)

(𝜃0, 𝜑0) is the direction of maximum radiation.

III. OPTIMIZATION OF UNIFORM CIRCULAR PHASED

ARRAYS USING A SQP ALGORITHM

Our objective is to steer the main beam in the direction ofdesired signal and to reduce or to suppress interfering signalsfrom prescribed directions while receiving desired signal froma chosen direction by amplitude only control. The form ofoptimization problem is expressed in mathematical terms as:

minimise −𝑓𝜃0,𝜑0(𝐼)

subject to 𝑓𝜃𝑖,𝜑𝑗(𝐼) = 𝛿𝑖𝑗 with

{𝑖 = 1, . . . ,𝑚𝑒

𝑗 = 1, . . . , 𝑛𝑒

𝑓𝜃𝑘,𝜑𝑙(𝐼) ≤ 𝛿𝑘𝑙 with

{𝑘 = 𝑚𝑒 + 1, . . . ,𝑚𝑙 = 𝑛𝑒 + 1, . . . , 𝑛

−2 ≤ 𝐼𝑚𝑛 ≤ 2 with

{𝑚 = 1, . . . ,𝑀𝑛 = 1, . . . , 𝑁

(6)Where (𝜃0, 𝜑0) is the direction of the main lobe, (𝑚𝑒, 𝑛𝑒)

is the equality constraints matrix size and (𝑚𝑖, 𝑛𝑖) is theinequality constraints matrix size.

𝑓𝜃,𝜑(𝛼) = ∣𝑀∑

𝑚=1

𝑁∑𝑛=1

𝑎𝑚𝑛𝑒𝑗(𝑘(𝑚−1)𝑑𝑥𝑢𝑥+𝑘(𝑛−1)𝑑𝑦𝑢𝑦+𝛼𝑚𝑛)∣2

(7)is the matrix of the objective function, 𝑓𝜃𝑖,𝜑𝑗

(8) is thematrix of equality constraints and 𝑓𝜃𝑘,𝜑𝑙

(9) is the matrix ofinequality constraints.

𝑓𝜃𝑖,𝜑𝑗=

⎡⎢⎢⎢⎣

𝑓𝜃1,𝜑1𝑓𝜃2,𝜑1

. . . 𝑓𝜃𝑚𝑒 ,𝜑1

𝑓𝜃1,𝜑2𝑓𝜃2,𝜑2

. . . 𝑓𝜃𝑚𝑒 ,𝜑2

.... . .

...𝑓𝜃1,𝜑𝑛𝑒

𝑓𝜃2,𝜑𝑛𝑒. . . 𝑓𝜃𝑚𝑒 ,𝜑𝑛𝑒

⎤⎥⎥⎥⎦ (8)

𝑓𝜃𝑘,𝜑𝑙=

⎡⎢⎢⎢⎣

𝑓𝜃𝑚𝑒+1,𝜑𝑛𝑒+1𝑓𝜃𝑚𝑒+2,𝜑𝑛𝑒+1

. . . 𝑓𝜃𝑚,𝜑𝑛𝑒+1

𝑓𝜃𝑚𝑒+1,𝜑𝑛𝑒+2𝑓𝜃𝑚𝑒+2,𝜑𝑛𝑒+2

. . . 𝑓𝜃𝑚,𝜑𝑛𝑒+2

.... . .

...𝑓𝜃𝑚𝑒+1,𝜑𝑛

𝑓𝜃𝑚𝑒+2,𝜑𝑛. . . 𝑓𝜃𝑚,𝜑𝑛

⎤⎥⎥⎥⎦

(9)

Recently, the Sequential Quadratic Programming (SQP)method has been successfully applied to solve the problem(6) [6]-[7]. SQP algorithm is the most widely used algorithmto solve nonlinear optimization problem with constraints. Thisalgorithm for constrained optimization is an iteration-typemethod. Its main idea is based on linearisations of theconstraints and a quadratic model of the objective function.

The quadratic programming problem solved at each iterationof SQP can be defined as

minimise −∇𝑓𝜃0,𝜑0(𝛼𝑘𝑙)𝑇 𝑑+ 1

2𝑑𝑇𝑀𝑘𝑙𝑑

subject to ∇𝑓𝜃𝑖,𝜑𝑗 (𝛼𝑘𝑙)𝑇 𝑑+ 𝑓𝜃𝑖,𝜑𝑗 (𝛼𝑘𝑙) = 𝛿𝑖𝑗

{𝑖 = 1, . . . ,𝑚𝑒

𝑗 = 1, . . . , 𝑛𝑒

∇𝑓𝜃𝑘,𝜑𝑙(𝛼𝑘𝑙)𝑇 𝑑+ 𝑓𝜃𝑘,𝜑𝑙(𝛼𝑘𝑙) ≤ 𝛿𝑘𝑙

{𝑘 = 𝑚𝑒 + 1, . . . ,𝑚𝑙 = 𝑛𝑒 + 1, . . . , 𝑛

𝑑 ∈ ℝ𝑀∗𝑁

(10)

where 𝑀𝑘𝑙 is the BFGS (Broyden, Fletcher, Goldfarb, andShanno) approximation of the Hessian ∇2

𝛼𝐿(𝛼𝑘𝑙, 𝜆𝑘𝑙).The lagrangian function 𝐿 is defined as

𝐿(𝛼, 𝜆) = −𝑓𝜃0,𝜑0(𝛼) +

𝑚∑𝑘=1

𝑛∑𝑙=1

𝜆𝑘𝑙(𝑓𝜃𝑘,𝜑𝑙(𝛼)− 𝛿𝑘𝑙) (11)

where𝜆 =

(𝜆1, 𝜆2 . . . 𝜆𝑛

)(12)

is the matrix of the Lagrange multiplier.To solve the QP sub-problem, described on equation (11), SQPalgorithm generate a sequence of points approximating to thesolution by the procedure

𝛼𝑘+1 = 𝛼𝑘 + 𝜇𝑝𝑑𝑘 (13)

where,𝛼𝑘 is the current point, 𝑑𝑘 is a search direction vector,which is used to form a new iterate 𝛼𝑘+1, 𝜇𝑝 ∈]0, 1] is asuitable step length parameter.

IV. SIMULATIONS AND RESULTS

In this section, numerical results for the optimizationproblem mentioned above are presented. Theses resultsillustrate the performance of the proposed method , sequentialQuadratic Programming (SQP) Algorithm, by controllingthe amplitude excitation of each array element in order tosteer the main beam in desired direction, and impose single,multiple and/or broad nulls in directions of interfering signal.

To demonstrate the validity of the proposed approach, weapply the amplitude result found to feed a circular arraywith twenty antenna elements. The spacing between tow arrayelements is 0.5𝜆. The computed element ( amplitude) currentexcitations of figures 2 , 3, 4 and 5 are given in Table I.

2

TABLE ICOMPUTED ELEMENT AMPLITUDE ARRAY WEIGHT FOR FIGURES 2-6

Synthesized excitations

Fig 2 3 4 5 6

n Amplitude excitation

1 0.7968 0.4483 0.4223 0.4058 0.1848

2 0.5110 0.7082 0.6261 1.1945 0.1707

3 0.0081 0.5544 0.5635 0.2886 1.4157

4 0.9253 0.4666 0.4623 0.5385 0.1229

5 -0.8512 -0.1067 -0.0923 -0.0249 0.6759

6 1.4684 0.6038 0.6100 0.6211 -0.5942

7 1.0551 0.7863 0.7747 0.7635 0.0724

8 2.0000 2.0000 2.0000 2.0000 1.4687

9 2.0000 2.0000 1.9062 2.0000 1.3207

10 1.5785 1.4210 1.4746 0.8484 -0.3664

11 0.7951 0.4482 0.4222 0.3258 0.2285

12 0.5124 0.7082 0.6261 0.8197 0.2935

13 0.0100 0.5544 0.5634 0.2405 1.2627

14 0.9234 0.4666 0.4622 0.9973 -0.0148

15 -0.8502 -0.1068 -0.0923 -0.2671 0.7144

16 1.4681 0.6038 0.6100 0.7970 -0.3778

17 1.0562 0.7862 0.7746 0.6064 0.1567

18 2.0000 2.0000 2.0000 2.0000 1.8206

19 2.0000 2.0000 1.9063 2.0000 1.4922

20 1.5759 1.4209 1.4745 1.4956 -0.2645

−80 −60 −40 −20 0 20 40 60 80−70

−60

−50

−40

−30

−20

−10

0

φ(deg)

Am

plitu

e(dB

)

Optimisation with constraint

θ =−30°

Fig. 2. Circular antenna array pattern with main beam at (𝜑 = −30∘, 𝜃 =−30∘) for 𝑁 = 20 using the SQP results

The radiation pattern plot in figures 2-6 obtained by usingsequential quadratic algorithm SQP demostrates the ability ofthis method to impose nulls and wide sector in direction ofinterfering signal by controlling only the amplitude excitationof each array elements.Figure 2 shows the radiation pattern obtained by controllingthe amplitude with main baim at (𝜑 = −30∘, 𝜃 = −30∘).The maximum side lobe level is −15𝑑𝐵. In figure 3, the

pattern with one null imposed at (𝜑 = 30∘, 𝜃 = −30∘). Themaximum side lobe level and the null depth are −15𝑑𝐵 and−70𝑑𝐵, respectively. Figure 4 shows the radiation patternwith tow imposed nulls at (𝜑 = −50∘, 𝜃 = −30∘) and(𝜑 = 60∘, 𝜃 = −30∘).

−80 −60 −40 −20 0 20 40 60 80−70

−60

−50

−40

−30

−20

−10

0

φ(deg)A

mpl

itue(

dB)

Optimisation with constraint

θ =−30°

Fig. 3. Radiation pattern for 𝑁 = 20 using the SQP results with main beamat (𝜑 = −30∘, 𝜃 = −30∘) and one null at (𝜑 = 30∘, 𝜃 = −30∘).

−80 −60 −40 −20 0 20 40 60 80−70

−60

−50

−40

−30

−20

−10

0

φ(deg)

Am

plitu

e(dB

)

Optimisation with constraint

θ =−30°

Fig. 4. Radiation pattern for 𝑁 = 20 using the SQP results with main beamat (𝜑 = −30∘, 𝜃 = −30∘) and tow nulls at (𝜑 = −50∘, 𝜃 = −30∘) and(𝜑 = 60∘, 𝜃 = −30∘).

The radiation pattern shown in figure 5 and 6 demonstrates theability of the synthesis method based on sequential quadraticalgorithm SQP algorithm to prescribe a wide band interferencesignal. Figure 5 shows adiation pattern with main beam at(𝜑 = −30∘, 𝜃 = −30∘) and one null at (𝜑 = −4∘, 𝜃 = −30∘)

3

and broad null (Δ𝜑 = 6∘) centered at 37∘. Figure 6 presentsradiation pattern with main beam at (𝜑 = −30∘, 𝜃 = −30∘)and tow nulls at (𝜑 = −3∘, 𝜃 = −30∘), (𝜑 = 40∘, 𝜃 = −30∘)and broad null (Δ𝜑 = 7∘) centered at 54∘ .

−80 −60 −40 −20 0 20 40 60 80−70

−60

−50

−40

−30

−20

−10

0

φ(deg)

Am

plitu

e(dB

)

Optimisation with constraint

θ =−30°

Fig. 5. Radiation pattern for 𝑁 = 20 using the SQP results with main beamat (𝜑 = −30∘, 𝜃 = −30∘) and one null at (𝜑 = −4∘, 𝜃 = −30∘) andbroad null (Δ𝜑 = 6∘) centered at 37∘.

−80 −60 −40 −20 0 20 40 60 80−70

−60

−50

−40

−30

−20

−10

0

φ(deg)

Am

plitu

e(dB

)

Optimisation with constraint

θ =−30°

Fig. 6. Radiation pattern for 𝑁 = 20 using the SQP results with mainbeam at (𝜑 = −30∘, 𝜃 = −30∘) and tow nulls at (𝜑 = −3∘, 𝜃 = −30∘),(𝜑 = 40∘, 𝜃 = −30∘) and broad null (Δ𝜑 = 7∘) centered at 54∘ .

V. CONCLUSION

In this paper, a method for circular array antenna patternsynthesis based on the sequantial quadratic algorithm SQPby controlling only the amplitude has been presented. SQP

method is the most powerful and the most widely used to solvenolinear optimization problem. SQP algorithm transforms thenolinear optimization problem to sequence of a quadraticsubproblem which is obtained by linearizing the constraintsand approximating the Lagrangian function.Simulation results show that the SQP algorithm is efficient forsynthesizing a circular array antenna pattern by the amplitude-only control.

REFERENCES

[1] Mohammad Shihab, Y. Najjar, N. Dib, M. Khodier, Design of non-uniformcircular antenna arrays using particle swarm optimization, Journal ofELECTRICAL ENGINEERING, VOL. 59, NO. 4, 2008, 216-220 3rd ed.Harlow, England: Addison-Wesley, 1999.

[2] ZAINUD-DEEN, S.MADY, E.AWADALLA, K.HARS-HER, H. Con-trolled radiation pattern of circular antenna array, IEEE Antennas andPropagation Symp. (2006), 3399-3402.

[3] PESIK, L.PAUL, D.RAILTON, C.HILTON, G. BEACH, M. : techniquefor modelling eight element circular an- tenna array Electronics Letters42 No. 14 (July 2006), 787-788.

[4] PANDURO, M. Mendez, A. Dominguez, R. ROMERO, G. Design of non-uniform circular antenna arrays for side lobe reduction using the methodof genetic algorithms, Int. J.Electron. Commun. (AEU) 60 (2006), 713-717.

[5] MAHLER, W. LANDSTORFER, F. Design and optimisation of anantenna array for WiMAX base stations, IEEE/ACES InternationalConference on Wireless Communications and Applied ComputationalElectromagnetics (2005), 1006-1009.

[6] M. Mouhamadou, P. Vaudon, “Complex Weight Control of Array PatternNulling,” International Journal of RF and Microwave Computer-AidedEngineering, pp. 304–310, 2007.

[7] M. Mouhamadou, P. Vaudon, “Smart Antenna a Array Patterns Synthesis:Null Steering and multi-user Bealforming by Phase control,” Progress InElectromagnetics Research, Vol. 60, pp. 95–106, 2006.

[8] R.L. Haupt, “Phase-only adaptive nulling with a genetic algorithm,” IEEETrans Antennas Propag 45 , pp. 1009-1015, 1997.

[9] K. Guney and A. Akdagli, “Null steering of linear antenna arrays usinga modified tabu search algorithm,” Prog Electromagn Res , 33 , pp. 167-182, 2001.

[10] N. Karaboga, K. Guaney, and A. Akdagli, “Null steering of linearantenna arrays with use of modified touring ant colony optimizationalgorithm,” Wiley Periodicals; Int J RF Microwave Computer-Aided Eng, 12,pp. 375-383, 2002.

[11] R.Ghayoula, N.Fadlallah, A.Gharsallah, M.Rammal,“ Phase-Only Adap-tive Nulling with Neural Networks for Antenna Array Synthesis,” IETMicrow. Antennas Propag., Vol. 3, pp. 154–163, 2009.

[12] R.Ghayoula, N.Fadlallah, A.Gharsallah, M.Rammal, “Neural NetworkSynthesis Beamforming Model for One-Dimensional Antenna Arrays ,”The Mediterranean Journal of Electronics and Communications, ISSN:1744-2400, Vol. 4, No. 1, pp.126–131, 2008.

[13] D.I. Abu-Al-Nadi and M.J. Mismar, “Genetically evolved phase-aggregation technique for linear arrays control,” Prog Electromagn Res,43,pp. 287-304, 2003.

[14] S. Yang, Y.B. Gan, and A. Qing, “Antenna-array pattern nulling using adifferential evolution algorithm,” Wiley Periodicals; Int J RF MicrowaveComputer- Aided Eng, 14 ,pp. 57-63, 2004.

[15] T.H. Ismail, D.I. Abu-AL-Nadi, and M.J. Mismar, “Phase-only controlfor antenna pattern synthesis of linear arrays using LevenbergMarquartalgorithm,” Electromagnetics ,24 , pp. 555-564, 2004.

[16] M.M. Khodier and C.G. Christodoulou, “Linear array geometry synthe-sis with minimum sidelobe level and null control using particle swarmoptimization,” IEEE Trans Antennas Propag ,53 ,pp. 2674-2679, 2005.

4