Antenna Pattern Control of Planar Arrays for Long

8/11/2019 Antenna Pattern Control of Planar Arrays for Long

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Antenna pattern control of planar arrays for longdistance communications

J.A.R. Azevedo

Engineering and Mathematics Department, University of Madeira, Campus da Penteada, 9000-390 Funchal, Portugal

Received 15 October 2007; received in revised form 31 March 2008; accepted 31 March 2008

Abstract

As an alternative to standard antennas and due to the advent of low-cost digital processors and materials, adaptive antenna arrays areincreasingly considered for applications in long distance communications. The pattern beam control potentialities provide the perfor-mance improvement of the communication systems. In this context, this work presents a technique that permits to synthesize the antennapattern of a planar array. Considering that the relationship between the array factor and the array excitations for the far field region is aFourier transform in the appropriate variables, the non-uniform sampling procedure is extended for planar arrays. These arrays permit afurther control of the antenna pattern since two dimensions of space are used. The technique considers a number of array factor pointsequal to the array size, which is used to impose appropriated values in the antenna pattern. The produced array factor passes throughthose points with the desired values, which allows the synthesis of antenna patterns with appropriated characteristics. The theory is pre-sented in order to develop the necessary direct expressions to calculate the array excitation. The array factor can be obtained using theinverse Fourier transform. As application examples, with this technique it is possible to define the level of each sidelobe of the antennapattern, to control the main beam and to impose nulls in prescribed directions of the pattern.2008 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Planar antenna arrays; Pattern control; Non-uniform sampling; Pencil beam patterns

1. Introduction

A typical antenna used for long distance communica-tions, such as satellite earth stations, is the parabolic reflec-tor antenna. Large gains are produced to overcome thefade effects with an adequate diameter of the dish. In recent

times, small-aperture earth terminals are increasingly imple-mented using antenna arrays (Saunders, 2004). Communica-tions through the ionosphere can be significantly affected byits presence. For instance, ionospheric scintillation imposesa major problem to satellite systems (Tulunay and Bradley,2004). The mitigation of ionospheric effects is demanded. Inthis context, adaptive systems have great potentialities,where an adaptive antenna is a very important component

due to the capability of controlling the radiation pattern.The concept of adaptive antenna system corresponds tothe processing of the individual element signals in order toreact to the environment, steering its beam toward a desiredsignal while steering a null toward an undesired one (Krausand Marhefka, 2002). Therefore, a huge and expensive dish

antenna can be replaced by an array of small, low-cost, mass-produced antenna units for similar applications.In many communication systems it is desirable for the

antenna to have a narrow beam with very low sidelobe levels.A typical large parabolic reflector may have significantsidelobes when compared with the obtained from a phasedarray. Controlling the amplitude and phase of each arrayelement it has resulted in array designs with levels down50 dB or more. On the other hand, GPS satellites use ashapedbeam to equalize signal strength acrossthe entire cov-erage area.

0273-1177/$36.00 2008 COSPAR. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.asr.2008.03.032

E-mail address: [email protected]

www.elsevier.com/locate/asr

Available online at www.sciencedirect.com

Advances in Space Research 43 (2009) 16031610
mailto:[email protected]:[email protected]


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A linear array allows beam steering in one dimension. Aplanar array has two dimensions of control, permitting anarrow pencil beam to be produced and the control ofthe shape in both directions. The relationship betweenthe array factor and the array excitations for the far fieldregion is a Fourier transform. Taking into account this

relation, a technique was developed for linear arrays basedon non-uniform sampling of the array factor (Azevedo andCasimiro, 2005). The ability to increase the energy in somedirections and create nulls in the pattern in other directionswas implemented by the control of certain points of thearray factor. In this work, the non-uniform sampling pro-cedure is extended for planar arrays. The technique consid-ers a number of array factor samples equal to the arraysize, which is used to impose values in appropriated posi-tions of the pattern.

Several authors have considered methods to deal withplanar arrays.Taylor (1960)presented a technique to pro-duce low sidelobes for circular continuous apertures. The

array factor is given by the Hankel transform of the sourcedistribution. The method ofTseng and Cheng (1968)alsopermits the control of the sidelobe level, synthesizingChebyshev patterns with planar discrete arrays. The arrayfactor is defined by the trigonometric form of the Fourierseries. Both techniques provide limited control, since onlypermit a uniform sidelobe structure. The method presentedin this work permits the control of the individual sidelobelevels. Therefore, not only Chebyshev patterns can be syn-thesized but also different regions of the sidelobe structurecan have different levels permitting the attenuation of inter-ferences that could influence the system performance. The

work ofBucci et al. (2002)provides different sidelobe levelsbut utilizes minimization techniques in the synthesis prob-lem. The method proposed here is very simple since it con-trols directly the sidelobe levels through the samples of thearray factor.

2. Non-uniform sampling technique for planar arrays

Considering a planar array in the x an ydirections, thearray factor is the inverse Fourier transform of the sourcedistribution, cx;y,

Fbx;by Z

1

1

Z 1

1

cx;yejbxxbyydxdy 2p2F1cx;y 1

where F means the Fourier transform and the bar underthe variable indicates the temporal Fourier transform.The variables bx and by are given by

bx b sinh cos/

byb sinh sin/2

with b = 2p/k, k the wavelength, h the angle between thezdirection and the point of the far field and / is the angle

between the x direction and the projection of the far field

point in the xy plane. The visible region for the radiationis a window defined by b2x b

2y6 b

2.Many authors consider Eq. (1) only for continuous

source distributions. For discrete arrays the array factoris usually defined by a Fourier series with the involvedvariables different from the continuous, as it can be veri-

fied inBucci et al. (2002). To generalize the procedure andjustify the used variables, Eq.(1)will be applied to deter-mine the relation between the array factor and the dis-crete source distribution. This procedure can be foundin Casimiro and Azevedo (2005) where a linear discretesource distribution may also be represented by a continu-ous function with the appropriate use of Dirac impulses.This permits to unify the analysis and synthesis of contin-uous and discrete arrays. Therefore, for a discrete arrayEq. (1) becomes

Fbx;by

Z

1

1Z

1

1 cx;

yejbxxbyy

dxdy

Z 11

Z 11

X1n1

X1m1

cxn;ymdxxn;yymejbxxbyydxdy

X1n1

X1m1

cxn;ymejbxxnbyym 3

Given the array factor, the array excitation is obtainedthrough the direct Fourier transform,

cx;y 1

2p2

Z 11

Z 11

Fbx;byejbxxbyydbxdby

12p2

FFbx; by

4

The far field of a planar array consisting ofMNele-ments positioned in a rectangular lattice of the x, y planewith inter-element spacingdx and dy, respectively, is givenby

FTfeFbx;by

Fbx;by

PN12

nN12

PM12

mM12cxn;yme

jnbxdxmbydy N;Modd

PN21

nN2

PM2 1

mM2cxn;yme

j 2n12 bxdx2m1

2 bydy N;Meven

8>:

5

where fe is the element pattern assuming that all elementshave the same embedded element pattern. In general, thisassumption applies for large arrays where all elementsexperience the same environment. In this work, it will beassumed that the array elements have omni-directionalradiating characteristics, which means fe= 1.

For faster calculation of the array factor, the Fast Fou-rier Transform (FFT) can be used. This is very importantwhen the number of array elements is very high as it hap-pens on planar arrays. Although the array factor be a con-tinuous function, the FFT can also be applied to the

calculation of the array excitation, if is considered an

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appropriate number of sampling points in order to mini-mize the aliasing effect.

Applying to(5) a changing of variables to both expres-sions, the array factor have the same form,

Fbx;by ejbx

N12 dxejby

N12 dyX

N1

n0

XM1m0

cxn;ymejnbxdxmbydy 6

This result is in appropriated form to apply the FFT, sincemost algorithms have the summation beginning inn= m= 0.

For the control of the array factor it can be imposedsome desired points and values of this function. For sim-plicity let us consider dx= dy= d and M=N. I f i t i sintended that the array factor pass through the (bxi,byk)

position with the Fbxi;

byk value, then

XN1n0

XN1m0

cxn;ymejnbxidmbykd Fbxi;byke

jbxiN1

2 dejbykN1

2 d 7

If a sufficient number of equations were used, the arrayexcitation is obtained by solving the system in order to

c(xn,ym). This can be accomplished in an easier way usingmatrix analysis. A possibility consists of defining the fol-lowing matrix products,

Therefore, the array excitation is

C X1:F:Y1 9

Although this procedure permits to obtain the array exci-tation, only N+ Ndifferent points of the array factor can beimposed. To deal with N N sample points, the systemdefined in Eq. (7) should be solved using a matrix ofN2 N2,

In this case, the array excitation is characterized through a

F X:C:Y

X

1 ejbx0d ej2bx0d . . . ejN1bx0d



.

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1 ejbxN1d ej2bxN1d . . . ejN1bxN1d

26666666664

37777777775

Y

1 1 1 . . . 1

ejby0d ejby1d ejby2d . . . ejbyN1d

ej2by0d ej2by1d ej2by2d . . . ej2byN1d

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ejN1by0d ejN1by1d ejN1b2d . . . ejN1byN1d

2666666664

3777777775

F

Fbx0;

by0ejN1

2 b

x0b

y0d

Fbx0;

by1ejN1

2 b

x0b

y1d

. . .

Fbx0;

byN1ejN1

2 b

x0b

yN1d

Fbx1;by0ejN1

2 bx1by0d Fbx1;by1ejN1

2 bx1by1d. . . Fbx1;byN1e

jN12 bx1byN1nd

Fbx1;by0ejN1

2 bx2by0d Fbx2;by1ejN1

2 bx2by1d. . . Fbx2;byN1e

jN12 bx2byN1d

.

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.

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FbxN1;by0ejN1

2 bxN1by0d FbxN1;by1ejN1

2 bxN1by1d. . . FbxN1;byN1e

jN12 bxN1byN1d

266666666664

377777777775

C

cx0;y0 cx0;y1 cx0;y2 . . . cx0;yN1

cx1;y0 cx1;y1 cx1;y2 . . . cx1;yN1

cx2;y0 cx2;y1 cx2;y2 . . . cx2;yN1

.

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cxN1;y0 cxN1;y1 cxN1;y2 . . . cxN1;y0N1

26666666664

37777777775

8

J.A.R. Azevedo/ Advances in Space Research 43 (2009) 16031610 1605


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vector. The same has been done for the non-uniform samplesof the array factor. The proceduredefines a system wherethenumber of equations is equal to the number of unknowns.Therefore, the technique that permits to impose N Nval-ues of the array factor using NNarray elements is appliedusing Eq.(10)and the array excitation can be calculated by

C A1:F 11

Through Eq.(3), the inverse Fourier transform of this ar-ray excitation gives the array factor.

3. Results for the pencil beam pattern

The objective is to obtain a pencil beam pattern whosesidelobe level (SLL) can be specified. The desired directiv-ity for the antenna is defined with an appropriated numberof array elements and the beam direction is controlled withthe phase of the array excitation.

Although the previously section presents a procedurethat obtains an array factor that passes through some spe-cific points, this is not enough to control the sidelobe struc-ture of the beam pattern. It is also necessary to determinethe appropriated positions of those points, in order to pro-duce a pattern with controlled SLL. The uniform planararray can be considered as the initial approximation, whosearray factor corresponds to the Sinc function in the bxandby variables. Then, NNsamples of the array factor aremoved to the sidelobe peak positions and the intended val-ues for those samples are attributed. A new approximationfor the array factor is calculated as it was previouslyreferred. If necessary other corrections of the sidelobe

structure can be made.

To demonstrate the application of the presented tech-nique, an array with N= 9 is considered, consisting of 81elements. The distance between elements is d= 0.5k. Theinitial N N samples are in positions bxn= np/ Nd,bym=mp/Nd, n, m =(N1)/2,. . .,0, 1,. . .,(N1)/2.The sample values are null except the central one, whichtakes the value Fbxn;bym 1, n =m= 0. With these sam-ple positions and values the A matrix and the F vector arecalculated. From Eq.(11) the corresponding array excita-tion is determined. Applying Eq. (3), the approximatedarray factor is calculated. Fig. 1(a) shows the result inthe bx and by variables andFig. 1(b) presents the sidelobestructure. As it can be observed, the sidelobes are very highwith a value around13 dB below the main lobe level. Forfurther comparisons, the directivity calculated by theequation

D0

4pFh;/j j2maxR

2p0

Rp2

0 Fh;/j j2 sinhdhd/ 12

gives 23.6 dB. It was considered only half of the spaceabove the array and the result was obtained numerically.

Let us suppose that it is intended to define the maximumsidelobe level as SLL =30 dB. FromFig. 1, it is verifiedthat neither the sidelobes in the line bx= 0 and by= 0respect this condition nor the four sidelobes aroundbx,by= 2/k. Thus, to produce a pattern with the specifiedSLL, the nearest sample of each sidelobe is moved for thepeak position and it is imposed the desired value with theappropriated sign. This sign could be the same as the one

produced by the uniform array. However, the shape of

F A:C

A

1 ejby0d ej2by0d . . . ejN1by0d ejbx0d ejbx0by0d . . . ejN1bx0by0d

1 ejby1d ej2by1d . . . ejN1by1d ejbx1d ejbx1by1d . . . ejN1bx1by1d

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1 ejbyN1d ej2bN1d . . . ejN1byN1d ejbxN1d ejbxN1byN1d . . . ejN1bxN1byN1d

266664

377775

C

cx0;y0

cx0;y1

cx0;y2

.

.

.

cx0;yN1

cx1;y0

cx1;y1

.

.

.

cx1;yN1

.

.

.

cxN1;y0N1

266666666666666666666666664

377777777777777777777777775

F

Fbx0;by0ejbx0by0

N12 d

Fbx1;by1ejbx1by1

N12 d

Fbx2;by2ejbx2by2

N12 d

.

.

.

FbxN1;

byN1ejbxN1byN1

N12 d

26666666664

37777777775

10

http://-/?-http://-/?-


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the main lobe is more circular if all lobes around have the

same sign. Considering the latter, Fig. 2shows the resultsafter this correction of the sidelobe levels. The highest side-lobe level is 22.5 dB, which means almost 10 dB belowthe previous result. The directivity is 22.9 dB, correspond-ing to 0.7 dB below in relation to the uniform array. Ifanother correction is made by moving the previous samplesfor the new peaks, the approximated array factor is the onerepresented in Fig. 3. Now, all sidelobes are around30 dB. The directivity is 22.5 dB, being 1.1 dB below inrelation to the uniform array. The corresponding sourcedistribution is represented in Fig. 4. The four corner ele-ments have excitations at least 35 dB below the excitationin the centre of the array. Therefore, the corner elementscan be removed without affecting the directivity. Thismakes the antenna cheaper and lighter. In fact the squarearray ofFig. 3behaves like a circular array with equal ringsidelobes. For comparison it was calculated the array fac-tor of a circular array with 77 elements operating with uni-form illumination. The directivity of this array is 23.9,which means 1.4 dB higher than the produced by the pre-sented technique. However, the circular array has a maxi-mum sidelobe level of 18 dB against the 30 dBproduced by the proposed technique.

More appropriated to visualize the radiation in thespace, Fig. 5 presents the three-dimensional patterns for

the uniform array and for the approximated array factor

Fig. 1. Array factor for the uniform planar array: (a) in bx and byvariables; (b) two-dimensional pattern.

Fig. 2. Array factor for the planar array after one correction of thesidelobe levels: (a) in bx and by variables; (b) two-dimensional pattern.

Fig. 3. Array factor forSLL=30 dB: (a) inbxand byvariables; (b) two-

dimensional pattern.



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with SLL=30 dB. It is clear the increase in performanceof the second pattern related to the sidelobe level structure.

If it is intended to synthesize a planar array with side-lobe levels of 40 dB below the main lobe, practically all

sidelobes produced by the uniform array factor must belowered. Therefore, the positions of all peaks are deter-mined and the desired values are imposed. The sign ofthe sidelobes can be the same as the one produced by thearray factor of the uniform array. In fact, either makingthis attribution or changing the sign of all sidelobes, exceptthose in the positionsbx,by= 0, the difference in directivity

is only 0.06 dB. Considering the first case, four correctionsin the sidelobe structure have permitted to obtain the resultofFig. 6. The directivity is 21.6 dB and the source distribu-tion is represented inFig. 7. In contrast to linear arrays, asidelobe of a planar array can be a maximum in one direc-tion but not on the other one (relative peaks). Thus, in thealgorithm for correction of the sidelobe levels, the arrayfactor samples are first imposed in the absolute peaks. Ifexist, the others samples are placed in the relative peaks

-2-1.5

-1-0.5

00.5

1

1.52

-2

-1

0

1

2

0

0.005

0.01

0.015

0.02

0.025

0.03

Fig. 4. Source distribution for the array factor ofFig. 3.

Fig. 5. Comparison between patterns: (a) uniform array pattern; (b)

pattern obtained by the presented technique. Fig. 6. Array factor for SLL=40 dB.



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but with the care that from iteration to iteration the sam-ples must be around the previous positions, providing theconvergence of the method.

Another important application of the technique is tocreate nulls in prescribed directions. As an example, letus suppose that it is intended to impose a null in h= 40,/= 120 on the previous pattern of SLL=40 dB. Inorder to create a wider null in this direction, three nullsare imposed around the desired one: h= 39, /= 120;

h= 39, /= 119; and h= 39, /= 121. The nulls arecreated moving the samples of the array factor aroundthe null for these positions. With the new samples theapproximated arrays factor is determined. Since the side-lobe level structure may be changed, a correction of thesidelobe peaks is performed as it was previously presented.For the example, two corrections produce the desired pat-tern and the directivity is the same as the initial pattern.

Fig. 8 shows the pattern contour and the position wherethe null was created.

4. Conclusions

Two parameters are important for antennas design inorder to minimize the fading effects produced by the iono-sphere. The first one is the directivity, which multiplied by

the antenna efficiency specifies the gain of the antenna. Thesecond one is the sidelobe level structure referred to themain lobe to minimize the potential sources of interference.An interesting case is the pencil beam pattern, where themain lobe is pointed to the desired direction of radiation.The appropriated number of array elements permits to syn-thesize a narrow beam for the desired directivity. In thispaper, it was presented a technique that controls the side-lobe levels of the array factor for planar patterns in a sim-ilar manner as done for linear arrays. This case revealed tobe more difficult to control the diagram due to the numberof sidelobes spread on a plane and its characteristics. Foran array with a specified number of elements, the array

excitation is calculated imposing the same number of sam-ples of the array factor. These points are used to controlthe shape of the pattern in a non-uniform samplingprocedure.

References

Azevedo, J.A., Casimiro, A.M. Antenna arrays synthesis using aninterpolation-FFT technique, in: Eleventh International Symposiumon Antenna Technology and Applied Electromagnetics, Saint-Malo,France, pp. 200201, June, 2005.

Bucci, O.M., Caccavale, L., Isernia, T. Optimal far-field focusing ofuniformly spaced arrays subject to arbitrary upper bounds in nontarget

directions. IEEE Trans. Anten. Propag. 50, 15391544, 2002.

-2

-1

0

1

2

-2

-1

0

1

2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Fig. 7. Source distribution for the array factor ofFig. 5.

Fig. 8. Contour pattern and null in h = 40, / = 120.



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Casimiro, A.M., Azevedo, J.A. New techniques that unify the procedureof the analysis and synthesis of antenna arrays. J. Electromagn. WavesAppl. 19, 18811896, 2005.

Kraus, J.D., Marhefka, R.J. Antennas for All Applications, third edMcGraw-Hill, New York, 2002.

Saunders, S.R. Antennas and Propagation for Wireless CommunicationSystems. John Wiley & Sons, Chichester, 2004.

Taylor, T.T. Design of circular apertures for narrow beamwidth and lowsidelobes. IRE Trans. Anten. Propag. AP-8, 1722, 1960.

Tseng, F.I., Cheng, D.K. Optimum scannable planar arrays with aninvariant sidelobe level. Proc. IEEE 56, 17711778, 1968.

Tulunay, Y.K., Bradley, P. The impact of space weather on communi-cation. Ann. Geophys. (Suppl.) 47, 929944, 2004.


Antenna Pattern Control of Planar Arrays for Long

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