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: [email protected]
AbstractβIn 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 Termsβantenna 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 π equis-paced isotropic elements with interelement spacing π. Theseselements are distributed on a circle of radius π in the π₯ β π¦plane. π is azimuth angle starting forward x axis. π is pitch
R
Ο
ΞΈ
Fig. 1. Illustration of uniform circular array antenna
angle starting forward z axis.The azimuth angle of ππ‘βelementππ is given by the following equation:
ππ = 2ππ
π, π = 1, β β β π (1)
The radius of the circle is :
π =π
2π ππ( ππ )
(2)
or the the distance between tow consecutive elements are π =π/2,
π =π
4π ππ( ππ )
(3)
The radiation pattern of the circular array, shown in figure 1,can be described by its array factor given by:
π΄πΉ (π, π) =
πβπ=1
πΌπππ(ππ π ππππππ (πβππ)+πΌπ)) (4)
with π is the number of the uniform circular array elements,π = 2π
π is the free space wavenumber,π is the wavelength,π β [βπ/2 π/2], π β [0 π] and πΌπ,πΌπ are amplitude andphase of each element, respectively.
πΌπ = βππ π πππ0πππ (π0 β ππ) (5)
(π0, π0) 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 βππ0,π0(πΌ)
subject to πππ,ππ(πΌ) = πΏππ with
{π = 1, . . . ,ππ
π = 1, . . . , ππ
πππ,ππ(πΌ) β€ πΏππ with
{π = ππ + 1, . . . ,ππ = ππ + 1, . . . , π
β2 β€ πΌππ β€ 2 with
{π = 1, . . . ,ππ = 1, . . . , π
(6)Where (π0, π0) is the direction of the main lobe, (ππ, ππ)
is the equality constraints matrix size and (ππ, ππ) is theinequality constraints matrix size.
ππ,π(πΌ) = β£πβ
π=1
πβπ=1
πππππ(π(πβ1)ππ₯π’π₯+π(πβ1)ππ¦π’π¦+πΌππ)β£2
(7)is the matrix of the objective function, πππ,ππ
(8) is thematrix of equality constraints and πππ,ππ
(9) is the matrix ofinequality constraints.
πππ,ππ=
β‘β’β’β’β£
ππ1,π1ππ2,π1
. . . ππππ ,π1
ππ1,π2ππ2,π2
. . . ππππ ,π2
.... . .
...ππ1,πππ
ππ2,πππ. . . ππππ ,πππ
β€β₯β₯β₯β¦ (8)
πππ,ππ=
β‘β’β’β’β£
ππππ+1,πππ+1ππππ+2,πππ+1
. . . πππ,πππ+1
ππππ+1,πππ+2ππππ+2,πππ+2
. . . πππ,πππ+2
.... . .
...ππππ+1,ππ
ππππ+2,ππ. . . πππ,ππ
β€β₯β₯β₯β¦
(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 ββππ0,π0(πΌππ)π π+ 1
2ππππππ
subject to βπππ,ππ (πΌππ)π π+ πππ,ππ (πΌππ) = πΏππ
{π = 1, . . . ,ππ
π = 1, . . . , ππ
βπππ,ππ(πΌππ)π π+ πππ,ππ(πΌππ) β€ πΏππ
{π = ππ + 1, . . . ,ππ = ππ + 1, . . . , π
π β βπβπ
(10)
where πππ is the BFGS (Broyden, Fletcher, Goldfarb, andShanno) approximation of the Hessian β2
πΌπΏ(πΌππ, πππ).The lagrangian function πΏ is defined as
πΏ(πΌ, π) = βππ0,π0(πΌ) +
πβπ=1
πβπ=1
πππ(πππ,ππ(πΌ)β πΏππ) (11)
whereπ =
(π1, π2 . . . ππ
)(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
πΌπ+1 = πΌπ + ππππ (13)
where,πΌπ is the current point, ππ is a search direction vector,which is used to form a new iterate πΌπ+1, ππ β]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 0.5π. The computed element ( amplitude) currentexcitations of figures 2 , 3, 4 and 5 are given in Table I.
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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
β80 β60 β40 β20 0 20 40 60 80β70
β60
β50
β40
β30
β20
β10
0
Ο(deg)
Am
plitu
e(dB
)
Optimisation with constraint
ΞΈ =β30Β°
Fig. 2. Circular antenna array pattern with main beam at (π = β30β, π =β30β) for π = 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 (π = β30β, π = β30β).The maximum side lobe level is β15ππ΅. In figure 3, the
pattern with one null imposed at (π = 30β, π = β30β). Themaximum side lobe level and the null depth are β15ππ΅ andβ70ππ΅, respectively. Figure 4 shows the radiation patternwith tow imposed nulls at (π = β50β, π = β30β) and(π = 60β, π = β30β).
β80 β60 β40 β20 0 20 40 60 80β70
β60
β50
β40
β30
β20
β10
0
Ο(deg)A
mpl
itue(
dB)
Optimisation with constraint
ΞΈ =β30Β°
Fig. 3. Radiation pattern for π = 20 using the SQP results with main beamat (π = β30β, π = β30β) and one null at (π = 30β, π = β30β).
β80 β60 β40 β20 0 20 40 60 80β70
β60
β50
β40
β30
β20
β10
0
Ο(deg)
Am
plitu
e(dB
)
Optimisation with constraint
ΞΈ =β30Β°
Fig. 4. Radiation pattern for π = 20 using the SQP results with main beamat (π = β30β, π = β30β) and tow nulls at (π = β50β, π = β30β) and(π = 60β, π = β30β).
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(π = β30β, π = β30β) and one null at (π = β4β, π = β30β)
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and broad null (Ξπ = 6β) centered at 37β. Figure 6 presentsradiation pattern with main beam at (π = β30β, π = β30β)and tow nulls at (π = β3β, π = β30β), (π = 40β, π = β30β)and broad null (Ξπ = 7β) centered at 54β .
β80 β60 β40 β20 0 20 40 60 80β70
β60
β50
β40
β30
β20
β10
0
Ο(deg)
Am
plitu
e(dB
)
Optimisation with constraint
ΞΈ =β30Β°
Fig. 5. Radiation pattern for π = 20 using the SQP results with main beamat (π = β30β, π = β30β) and one null at (π = β4β, π = β30β) andbroad null (Ξπ = 6β) centered at 37β.
β80 β60 β40 β20 0 20 40 60 80β70
β60
β50
β40
β30
β20
β10
0
Ο(deg)
Am
plitu
e(dB
)
Optimisation with constraint
ΞΈ =β30Β°
Fig. 6. Radiation pattern for π = 20 using the SQP results with mainbeam at (π = β30β, π = β30β) and tow nulls at (π = β3β, π = β30β),(π = 40β, π = β30β) and broad null (Ξπ = 7β) centered at 54β .
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.
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