2012 04 Arbitrary Footprints from Arrays with Concentric Ring Geometry and Low Dynamic Range Ratio.pdf

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Journal of Electromagnetic Wavesand ApplicationsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tewa20

Arbitrary Footprints from Arrays

with Concentric Ring Geometry

and Low Dynamic Range RatioR. Eirey-Pérez a , M. Álvarez-Folgueiras b , J. A.

Rodríguez-González c & F. Ares-Pena d

a Department of Applied Physics, Faculty of Physics,University of Santiago de Compostela, 15782 Santiago deCompostela, Spainb Department of Applied Physics, Faculty of Physics,University of Santiago de Compostela, 15782 Santiago deCompostela, Spainc Department of Applied Physics, Faculty of Physics,University of Santiago de Compostela, 15782 Santiago deCompostela, Spaind Department of Applied Physics, Faculty of Physics,University of Santiago de Compostela, 15782 Santiago deCompostela, Spain;, Email: [email protected] online: 03 Apr 2012.

To cite this article: R. Eirey-Pérez , M. Álvarez-Folgueiras , J. A. Rodríguez-González &F. Ares-Pena (2010) Arbitrary Footprints from Arrays with Concentric Ring Geometry andLow Dynamic Range Ratio, Journal of Electromagnetic Waves and Applications, 24:13,1795-1806

To link to this article: http://dx.doi.org/10.1163/156939310792486601

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J. of Electromagn. Waves and Appl., Vol. 24, 1795–1806, 2010

ARBITRARY FOOTPRINTS FROM ARRAYS WITHCONCENTRIC RING GEOMETRY AND LOW DYNAMICRANGE RATIO

R. Eirey-Pérez, M. Álvarez-FolgueirasJ. A. Rodŕıguez-González, and F. Ares-Pena

Department of Applied PhysicsFaculty of PhysicsUniversity of Santiago de Compostela15782 Santiago de Compostela, Spain

Abstract—A common method of antenna array synthesis uses least-squares approximation to fit the radiation pattern of the array to a setof samples of an ideal desired pattern F id. However, it has long beenknown that discontinuities in F id prevent close approximation, andthat sampling a target pattern F

tar that approximates F

id can afford

better results. Here we show, for the case of circular planar arrays,that an appropriate target pattern can be obtained by modification of a Taylor pattern for a circular aperture.

1. INTRODUCTION

In view of the physical unrealizability of ideal antenna radiationpatterns such as a perfectly flat-topped beam, designers sinceSchelkunoff [1] have sought to identify and implement the realizablepattern that, given various conditions or constraints, constitutes theleast-squares approximation to the ideal pattern. For a numberof situations there exist function-analytic solutions for the bestapproximation to a given field (see, for example, [2–4]). Themain problem faced by these methods is due to the least-squaressolution being an optimum in the mean, because of which it mayallow significant deviation from the ideal pattern in regions whereclose approximation is of particular interest [3]. Since electronic

computation facilities became widespread, this difficulty has beenReceived 11 May 2010, Accepted 8 July 2010, Scheduled 21 July 2010

Corresponding author: F. Ares-Pena ([email protected]).



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1796 Eirey-Pérez et al.

tackled by seeking the least-squares fit to a set of samples of the desiredpattern [5]; although the same problem recurs to some extent, it may

be handled by assigning different weights to different sample points [6].Unfortunately, it is not always easy to see how a given set of sampleweights will affect the solution.

An alternative approach to the waywardness of least-squaressolutions is to replace the ideal pattern F id with a discontinuity-free target pattern F tar that is close to the desired pattern, thoughnot necessarily in the least-squares sense [3]. Here we show, forthe case of planar arrays required to generate arbitrary footprintpatterns (i.e., patterns consisting of a flat-topped main beam witha contour of given shape, surrounded by low side lobes), thatan appropriate target pattern can be obtained by angle-dependenthomothetic transformation (ADHT) of a circular Taylor pattern thathas been shaped by the method of Elliott and Stern (ES) [7, 8]; in aprevious paper [9] we described the use of a target pattern of this kindfor Woodward-Lawson synthesis, which of course requires a rectangulararray. Once the target pattern has been obtained, it can be sampled,and the excitations of a given array can be optimized to obtain theleast-squares fit to the samples. Finally, array realizability can befacilitated by imposing a lower limit on element excitations [10].

The following papers recently published in PIER journals may beconsidered relevant [11–22].

2. METHOD

Given an ideal footprint F id that is defined by a contour C in thespace of spherical coordinates (θ, φ), and is to be achieved by an arrayof characteristic radius a, we proceed as follows.

Step 1. We begin by deforming C to a circle C around some

appropriate internal point. For simplicity, this point is assumed hereto be θ = 0, so that the deformation takes the form (θ, φ) → (θ, φ),where sin θ = h(φ)sin θ for some function h defined on [0, 2π).

Step 2. We next apply the ES method (Elliott and Stern 1988) toobtain a footprint of contour C , maximum ripple Rd and maximumside lobe level SLLd that could be generated by a circular aperture of radius a, i.e., for some appropriate M and p we find tn and sn suchthat the required circular footprint is given by

F ES(t) = f (t)

M

n=1

1−

t2

(tn+ jsn)2

1−

t2

(tn− jsn)2ε M + p

n=M +1

1−

t2

t2n

(1)

where t = (2a/λ)sin θ, ε is 1 if the field must be real and 0 otherwise,



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Concentric ring geometry and low dynamic range ratio 1797

and

f (t) = 2

J 1(πt)

πt

M + p+εM

n=1

1 −

t2

γ 21n−1

(2)

J 1 being the Bessel function of the first kind and order 1, and πγ 1n thenth zero of J 1. M (or 2M if the field is to be real) is the number of nulls of J 1(πt)/πt that must be filled to cover the area within C

, and p is the number of additional nulls that must be moved to achieve theside lobe specification or steepen rolloff. Although the determinationof tn and sn is an iterative local optimization process, it is neverthelessvery rapid.

Step 3. The pattern F ES obtained in Step 2, or a sample of itspoints (see Step 4 below), is then reverse-transformed into the targetpattern F tar by replacing sin θ

with h(φ)sin θ in the expression for t.Step 4. F tar is sampled at appropriate points. If the sampling

scheme is sufficiently dense as not to need to take the morphology of F tar into account explicitly, then in Step 3 F tar need only be calculatedat the sample points.

Step 5. The excitations I i of the array are optimized by least-squares to afford the pattern best fitting the F tar samples. Like thecalculation of F ES in Step 2, the least-squares calculation is much faster

than stochastic optimization methods.Steps 3 –5 , 3 –5 , etc. If ε is 0 in Equation (1), and F ES(t)

therefore complex, the power pattern F ES(t)∗F ES(t) corresponding to

the set {(tn, sn)} is also the pattern corresponding to any set {(tn, s

n)},where sn = ±sn [7]. In general this leads to a total of 2

M differentF tar and the same number of solutions {I i}. Therefore, from amongthese 2M solutions, one can choose the one with the best combinationof side lobe level, ripple and/or dynamic range ratio I max/I min, whereI max and I min are the maximum and minimum excitation amplitudes,

respectively.Step 6. If necessary, the element excitations I i can be retouched soas to reduce the dynamic range ratio. For example, thresholds T zero andT floor (T zero < T floor ) can be defined for the amplitude of the normalizedexcitation I i/I max, excitations I i such that |I i/I max| < T zero can be setto zero, and excitations I i such that T zero ≤ |I i/I max| < T floor can beamplified by |I max/I i|T floor . This procedure, which ensures a dynamicrange ratio of 1/T floor , is generally found to result in less patterndegradation than the suppression of all excitations with amplitudes lessthan T floor . Apart from reducing the dynamic range ratio, eliminatingelements also, of course, reduces the weight of the antenna.



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3. EXAMPLES

The method described above was applied to the examples that followusing Matlab (R2009b) on a desktop computer with an Intel Core i7processor running at 3.2 GHz. In all cases, computation time was about4 or 5 seconds.

3.1. Rectangular Footprint, Circular Array, Complex Field

We wish to generate a rectangular footprint of aspect ratio 2 : 1, centredon θ = 0 and defined in the first quadrant by

v = 0.12 0 ≤ u


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0 2 4 6 8 100

2

4

6

8

10

x

y

[λ]

[ λ ]

Figure 1. First quadrant of the array used in the complexfield example, before eliminationof weakly excited elements.

0 2 4 6 8 100

2

4

6

8

10

x

y

[λ]

[ λ ]

Figure 2. First quadrant of the array used in the complexfield example, after elimination of weakly excited elements.

of the antenna, the field function to be fitted to the F tar samples isgiven by

F (θ, φ) = 420

m=1

mn=1

I mn

cos(kxmn sin θ cos φ)cos(kymn sin θ sin φ) (6)

where I mn is the excitation of the nth element on the mth ring, and kis the wavenumber.

The pattern produced by Step5 has a maximum side lobe level of −22.34 dB and lowest ripple troughs of −2.60 dB in the footprint, andthe array excitation distribution has a dynamic range ratio of 471. If the excitations are retouched in Step6 using thresholds T zero = 1/16

and T floor = 1/14 to achieve a dynamic range ratio of 14, 92 elements(44%) are zeroed in each quadrant, the maximum side lobe level islowered to −22.59 dB, and the ripple troughs are raised to −2.51 dB.The resulting array is shown in Fig. 2, and the pattern it generatesin Figs. 3 and 4. Note that although this pattern underperforms asregards both side lobe level and ripple, the initial specifications couldbe met simply by increasing M and/or p in Step 2 so as to be ableto impose stricter specifications on F ES. The alternative solutionscorresponding to Steps 3–5 of the general method (see Section 2)

are no improvement.The pattern obtained above may be compared with the results of sampling the ideal pattern (a perfectly flat-topped rectangular columnrising 25 dB above a perfectly flat background), instead of F ES. Before



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Figure 3. Power pattern of the complex field generated by the arrayof Fig. 2 (projection on the (u, v) plane).

Figure 4. Power pattern of the complex field generated by the arrayof Fig. 2 (perspective view).



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Table 1. Values of tn and sn forF

ES in the complex field example.

n tn sn1 0.5967 0.52252 1.7837 0.52683 3.6420 04 4.3039 05 5.2119 0

Table 2. Values of tn and sn forF

ES in the real field example.

n tn sn1 1.0225 1.14242 3.0445 1.08293 5.4319 04 6.1570 05 7.1293 0

retouching, the least-squares fit to the ideal pattern has a side lobe levelas high as −17.80 dB and ripple troughs of −2.03 dB, and the dynamicrange ratio of the array is 2576. Retouching (with the same thresholdsas above) eliminates 161 elements per quadrant (77%), leaving an arraythat generates a pattern with side lobe levels of −13.79 dB and rippletroughs of −1.65 dB.

3.2. Rectangular Footprint, Circular Array, Real Field

To approximate the same power pattern as above while ensuring a real

field, a task that requires more nulls to be filled, we use a larger circulararray comprising 32 rings. The between-ring spacing is again λ/2, andthe same formulae as above are used for ρm, xmn and ymn (so the radiusof the outermost ring is 15.75λ). The values of M and p in Step 2,and the sampling scheme of Step4, are likewise the same as in thecomplex field case; the values of tn and sn determining F ES are listedin Table 2. Before retouching, the resulting pattern has a side lobelevel of −22.74 dB and ripple troughs of −1.37 dB, and the dynamicrange ratio of the array is 15,097. Retouching with T zero = 1/56 and

T floor = 1/54 eliminates 394 elements per quadrant (75%; see Fig. 5)and achieves a pattern with a maximum side lobe level of −22.56dBand ripple troughs of −1.56 dB (Figs. 6 and 7). As in the complex fieldcase, achievement of −25 dB side lobes and −1.0 dB ripple troughswould require stricter specifications for F ES.

3.3. Irregular Footprint, Circular Array, Complex Field

Figure 8 shows the outline of the desired footprint in our final example,in which we are allowed 12 rings of elements and once more aim for a

peak side lobe level of −25 dB and ripple troughs no deeper than −1dB.Fig. 8 also shows, superimposed on the desired footprint contour C ,the circular contour C ; since C has no analytic expression, neither has



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Figure 5. First quadrant of thearray used in the real field ex-ample, after elimination of weaklyexcited elements.

Figure 6. Power pattern of thereal field generated by the arrayof Fig. 5 (projection on the (u, v)plane).

Figure 7. Power pattern of the real field generated by the array of Fig. 5 (perspective view).



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Figure 8. An irregular desiredbeam contour C , and the circularcontour C into which it is trans-formed by ADHT in the last ex-ample of Section 3.

Figure 9. The array used in thelast example of Section 3, afterelimination of weakly excited el-ements.

Figure 10. Power pattern of the real field generated by the array of Fig. 9 (projection on the (u, v) plane).

the transform function h(φ), which must be obtained numerically. Weuse the same element placement scheme, M and p as in the previous



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Figure 11. Power pattern of the real field generated by the array of Fig. 9 (perspective view).

examples. Since there is no symmetry in this footprint, F tar must be

sampled over the whole hemisphere (we use 360 φ-cuts 1◦ apart, with25 equispaced samples in each θ-cut), and instead of using Equation (6)to reconstruct the field, we must use

F (θ, φ) =12

m=1

4mn=1

I mn

exp { jk (xmn sin θ cos φ + ymn sin θ sin φ)} (7)

This affords a solution with a dynamic range ratio of 120 thatgenerates a power pattern with a peak side lobe level of −18.95 dB andripple troughs of −1.38 dB. Retouching the excitation distribution with

T zero = 1/13 and T floor = 1/10 eliminates 97 elements (31%; Fig. 9),producing a power pattern with a peak side lobe level of −18.85dBand ripple troughs of −1.64 dB (Figs. 10 and 11).

4. CONCLUSION

Footprint patterns generated by antenna arrays must be evaluated interms of their quality parameters (side lobe level, ripple, etc.) ratherthan the fit of the pattern to an ideal flat-topped column. The quality

of least-squares fit for the ideal pattern is significantly poorer thanthat of least-squares fit for patterns constructed by angle-dependenthomothetic transformation of a circular footprint obtained by theElliott-Stern method for circular continuous apertures.



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ACKNOWLEDGMENT

This work was supported by the Spanish Ministry of Science andTechnology through project TEC2008-04485.

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2012 04 Arbitrary Footprints from Arrays with Concentric Ring Geometry and Low Dynamic Range Ratio.pdf