IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 5, MAY 2013 2865

Arrays of Concentric Rings of Elements: Synthesis of PencilBeams With and Without Allowance for Nonexcitation

Blockage

R. Eirey-Pérez, J. A. Rodríguez-González, F. J. Ares-Pena, andG. Franceschetti

Abstract—Pencil beamsmay be synthesized for arrays of concentric ringsof radiating elements by fitting the field pattern to that of a Taylor patternfor a circular aperture of the same radius. Nonexcitation blockage sup-presses inner rings. Here we examine whether this needs to be taken intoaccount in the synthesis procedure.

Index Terms—Annular arrays, antenna arrays, antenna radiation pat-tern synthesis.

I. INTRODUCTION

A problem that can arise in antenna engineering is the need to locatethe antenna around a central non-radiating zone, or window, a situationreferred to by Milligan as “nonexcitation blockage” [1]. If the windowand the outer contour of the antenna are both circular, the result is anannular aperture. The main effects of the central window are to raiseside lobe levels, reduce gain and beamwidth, and shift nulls. The firstto minimize the elevation of side lobe levels in synthesizing radiationpatterns for annular apertures was Ludwig, who achieved a pencil beamwith low wide-angle side lobes by means of an excitation distributionthat tapered to zero at the edge of the blocking structure [2]. Soon after-wards, Sachidananda and Ramakrishna employed a simplex algorithmto optimize the excitation of this kind of aperture for both sum and dif-ference monopulse modes [3].Although radiation patterns for annular arrays can doubtless be syn-

thesized by appropriate adaptation of existing techniques for full cir-cular arrays (see, for example, [4], or the indirect approaches via con-tinuous aperture distributions of [5] and [6]), we know of no publishedaccount of any such adaptation. This silence poses the question ofwhether adaptation is absolutely necessary, or whether, on the contrary,little is lost by simply suppressing the central part of a circular arraydesigned to radiate the desired pattern.Here, we first present a simple direct, deterministic synthesis tech-

nique for arrays of concentric rings of elements required to produce agiven array pattern. We then report the results of applying this methodto the generation of Taylor-like pencil beams [7], [8] when there isnonexcitation blockage, comparing the performance of blocked arraysdesigned taking blockage into account with that of blocked arrays de-signed ignoring blockage.

Manuscript received May 24, 2012; revised November 20, 2012; acceptedJanuary 17, 2013. Date of publication January 25, 2013; date of current versionMay 01, 2013. This work was supported by the Spanish Ministry of Educationand Science under Project TEC2008-04485 and by the Xunta de Galicia underProject 09TIC006206PR.R. Eirey-Pérez, J. A. Rodríguez-González, and F. J. Ares-Pena are with the

Radiating Systems Group, Department of Applied Physics, Faculty of Physics,University of Santiago de Compostela, 15782 Santiago de Compostela, Spain(e-mail: raquel.eirey@usc.es; ja.rodriguez@usc.es; francisco.ares@usc.es).G. Franceschetti is with the University Federico II of Napoli, Italy (e-mail:

gfrance@unina.it).Color versions of one or more of the figures in this communication are avail-

able online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2013.2242832

II. METHOD

We consider arrays with quadrantal symmetry, consisting of concen-tric rings of elements. The rings are spaced a distance apart, andthe elements of each ring are equally spaced upon it. The -th ringfrom the centre of the full circular array (i.e. an array lacking a centralwindow) has elements and radius . The posi-tion of the -th element of its first quadrant is given by

In keeping with the symmetry of the desired pencil beam, all the ele-ments on ring have the same excitation .If central blockage involves the suppression of the innermost rings

of an -ring full circular array, the field generated by theresulting annular array is given by

(1)

where as usual is the wavenumber and and are polar and az-imuthal angles. Since the pattern is approximately -symmetric (thesuppression of the inner rings having eliminated the source of most de-viation from -symmetry), (1) may be approximated by

(2)

For an array of the above kind, with inner rings removed, a radi-ation pattern similar to a circular Taylor pattern of given and max-imum side lobe level [7], [8] may be synthesized by first samplingthe Taylor pattern with respect to the polar angle , and then obtainingthe by fitting (2) to the samples by least squares [9]. This proce-dure takes central nonexcitation blockage into account and is referredto in the following as the “blockage-respecting” procedure. In what fol-lows, we compare its results with those obtained by the corresponding“blockage-ignoring” procedure, i.e. when the are obtained by fitting(2) for the full circular array and then ignoring (equivalently,setting to zero) the excitations of the inner rings. The variables com-pared (calculated in each case for the blocked array, i.e. after settingthe excitations of the inner rings to zero) are the dynamic range ratio

, maximum side lobe level , and directivity .We present results for an array of 20 rings with separation .

From each Taylor pattern used as template, 130 equispaced sampleswere taken over the range . Only real excitations wereconsidered, i.e. only arrays with all their elements in phase with eachother.Pattern synthesis calculations were performed using MATLAB

R2009b on a desktop PC with a Core i7 processor running at 3.2 GHz.

III. NUMERICAL RESULTS

Figs. 1 and 2 show plots of , and against thenumber of missing inner rings when starting from Taylor patterns with

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2866 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 5, MAY 2013

Fig. 1. Influence of the number of blocked inner rings, , on side lobe level, di-rectivity and dynamic range ratio when starting from Taylor patterns with nom-inal and (P), 5 (Q), 7 (R). Circles , blockage-re-specting procedure; triangles , blockage-ignoring procedure.

nominal s of 25 dB (Fig. 1) or 30 dB (Fig. 2) and values of3, 5 or 7.In general, regardless of the starting Taylor pattern, and regardless

of whether blockage was or was not taken into account in the synthesisprocedure, increasing the blockage over the range increased

sigmoidally to between 12 and 11 dB. Blockage-respectingsyntheses achieved levels some 3–5 dB better than blockage-ignoringsyntheses over the linear part of the sigmoid.Over the same blockage range , the directivity achieved

by blockage-respecting synthesis fell linearly to about 2 dB below thatof the unblocked array, 35.0–35.5 dB. If blockage was ignored duringsynthesis, directivity was essentially unaffected by removal of up to 5or 6 rings, but then fell linearly to about 1 dB below that of the un-blocked array.In the unblocked array, the dynamic range ratio fell

as increased and as the nominal of the starting Taylor patternrose. When synthesis ignores blockage, blockage only affects the

Fig. 2. Influence of the number of blocked inner rings, , on side lobe level, di-rectivity and dynamic range ratio when starting from Taylor patterns with nom-inal and (P), 5 (Q) or 7 (R). Circles , blockage-re-specting procedure; triangles , blockage-ignoring procedure.

dynamic range ratio if the extreme excitations of the unblockedarray lie in the blocked rings. was in fact unaffected orvery slightly reduced when synthesis ignored blockage. By contrast,when synthesis took blockage into account, showeda marked rise and fall as increased, though this trend bore asuperimposed oscillation and the peak decreased relativeto the corresponding blockage-ignoring curve, and shifted to highervalues, as increased and as the nominal of the starting Taylorpattern rose.The above findings are borne out by Fig. 3, which shows plots

of , and against for starting Taylor patternsof with nominal s of 20, 25 and 30 dB. Theseplots additionally illustrate the slight fall in directivity that occurs asnominal Taylor is lowered when blockage is taken into accountin the synthesis.Computation times were short, 0.1–0.5 s, regardless of whether

blockage was respected or ignored in the synthesis procedure.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 5, MAY 2013 2867

Fig. 3. Influence of the number of blocked inner rings, , on side lobe level,directivity and dynamic range ratio when starting from Taylor patterns with

and nominal (P), 25 dB (Q) or 30 dB (R).Circles , blockage-respecting procedure; triangles , blockage-ignoringprocedure.

IV. DISCUSSION AND CONCLUSIONS

For antenna arrays with concentric ring geometry, the radiationpattern fitted by least squares to a Taylor pattern for a circular apertureof the same radius is similar to the starting Taylor pattern as regardsmaximum side lobe level and directivity. Progressive omission ofinner rings (nonexcitation blockage) markedly raises the maximumside lobe level and slightly reduces the directivity. Taking blockageinto account during the synthesis procedure results in slightly lessdirectivity and a somewhat lower maximum side lobe level than ifblockage is ignored and the pattern is synthesized without omissionof any rings of elements. The dynamic range ratio falls very slightlywith increasing blockage if blockage is ignored during synthesis,and peaks at intermediate blockage values (omission of 5–7 elementrings, i.e. the inner 7–15% of the aperture), when blockage is takeninto account.

It is surprising that directivity and dynamic range ratio were notonly not significantly deteriorated by ignoring blockage but wereactually improved. Prompted by an anonymous reviewer, we won-dered whether this apparent anomaly might also occur with othersynthesis techniques. As a step towards elucidation of this issue,we compared the results of respecting and ignoring blockage in thecase of a technique consisting in first sampling the continuous Taylordistribution and then subjecting the resulting discrete distribution tostochastic optimization. The blockage-respecting variant of this tech-nique consisted in sampling the Taylor distribution onto the complete(unblocked) array, zeroing the blocked rings of elements, and thenoptimizing the remaining excitations by simulated annealing (SA).The blockage-ignoring variant consisted in sampling the Taylor dis-tribution onto the complete array, optimizing this set of excitationsby SA, and then zeroing the inner rings. In both cases, the SAcost function included terms for , dynamic range ratio and di-rectivity, and the weights of these terms were the same in both cases.The differences between the results of the two variants were qual-itatively the same as for the method described above in section II:the blockage-respecting procedure afforded better but worsedirectivity and dynamic range ratio. These results were not signifi-cantly altered by increasing the number of equispaced elements inthe rings so as to maintain an approximately constant element densitythroughout the array.It is concluded that, for arrays with the geometry considered in this

communication, taking blockage into account when synthesizing apencil beam by the method of Section II is only worth while if theresulting improvement in maximum side lobe level compensates forthe accompanying loss of directivity and increase in dynamic rangeratio (a case in point might be a radar application in which tightcontrol of side lobe level is required to avoid jamming). Otherwise, itis preferable to synthesize the pattern for the full array (with no ringsomitted), even though the central rings will not in fact exist. It remainsto be seen to what extent this conclusion may be generalizable toannular arrays that have other element arrangements or to synthesismethods other than those considered here.

REFERENCES

[1] T. A.Milligan, Modern Antenna Design, 2nd ed. Hoboken, NJ, USA:Wiley, 2005, pp. 208–211.

[2] A. C. Ludwig, “Low sidelobe aperture distributions for blocked andunblocked circular apertures,” IEEE Trans. Antennas Propag., vol.AP-30, no. 5, pp. 933–946, 1982.

[3] M. Sachidananda and S. Ramakrishna, “Constrained optimization ofmonopulse circular aperture distribution in the presence of blockage,”IEEE Trans. Antennas Propag., vol. AP-31, no. 2, pp. 286–293,1983.

[4] M. Vicente-Lozano, F. Ares-Pena, and E. Moreno, “Pencil-beam pat-tern synthesis with a uniformly excited multi-ring planar antenna,”IEEE Antennas Propag. Mag., vol. 42, no. 6, pp. 70–74, 2000.

[5] O. M. Bucci, T. Isernia, and A. F. Morabito, “Optimal synthesis ofdirectivity constrained pencil beams by means of circularly symmetricaperture fields,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp.1386–1389, 2009.

[6] O. M. Bucci, T. Isernia, and A. F. Morabito, “A deterministic approachto the synthesis of pencil beams through planar thinned arrays,” Prog.Electromagn. Res., vol. 101, pp. 217–230, 2010.

[7] T. T. Taylor, “Design of circular apertures for narrow beamwidth andlow sidelobes,” IRE Trans. Antennas Propag., vol. AP-8, pp. 17–22,1960.

[8] R. S. Elliott, Antenna Theory andDesign. Hoboken, NJ, USA:Wiley,2003, pp. 213–218, Revised Edition.

[9] R. Eirey-Pérez, M. Álvarez-Folgueiras, J. A. Rodríguez-González, andF. Ares-Pena, “Arbitrary footprints from arrays with concentric ringgeometry and low dynamic range ratio,” J. Electromagn. Waves Ap-plicat., vol. 24, no. 13, pp. 1795–1806, 2010.