synthesis of uniformly excited concentric ring arrays by

Research ArticleSynthesis of Uniformly Excited Concentric Ring Arrays by theStrategy of Partial Density Tapering and the Algorithm ofDifferential Evolution

Xin-Kuan Wang Gui-Bao Wang Jianke Jia and Chao-Jun Huang

School of Physics and Telecommunication Engineering Shaanxi University of Technology Hanzhong 723001 China

Correspondence should be addressed to Xin-Kuan Wang wxkuansnuteducn

Received 21 July 2020 Revised 10 September 2020 Accepted 29 September 2020 Published 10 October 2020

Academic Editor Luciano Mescia

Copyright copy 2020 Xin-Kuan Wang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A new strategy of density tapering called the partial density tapering (PDT) accompanied with the algorithm of differentialevolution (DE) is proposed to suppress the peak sidelobe level (PSL) of uniform excited concentric ring arrays (UECRA) withisotropic elements rough performing the PDT a sound starting solution for DE can be generated en the ring filling factor(RFF) is introduced so that the optimization of the number of elements can be transformed into the optimization of RFFs withinthe tapered thresholds and thereby the real coding can be directly used with respect to the consideration of parallel encodingstrategy Finally the UECRA featuring improved PSL performance can be obtained by limited runs of conventional DE Severalnumerical instances for UECRA with aperture sizes ranging from small to large scale confirmed the outperformance of theproposed method

1 Introduction

For the capability of achieving rotationally symmetricalbeam pattern concentric ring array (CRA) had found theapplications in radar direction finding radio-astronomysatellite communications and so on [1ndash3] erefore thesynthesis for CRA had been a topic of considerable study bymany investigators [1ndash17] In particular most of the pre-vious efforts focused on the uniform excited (equal feed)CRA due to the simplified feeding network and maximizedefficiency of DC-to-RF power conversion as it is comparedto the use of amplitude tapering technique According to theliteratures we can retrieve mainly three types of methodswhich are developed to design concentric ring arrays(CRAs) e first is deterministic approaches (DAs) [3ndash9]which usually refer to emulating the continuous referenceaperture distribution where a proper rule of density tapering[4ndash7] is employed to determine the element locations In thepast few years several new forms of DAs are proposed todesign UECRA Angeletti et al [3] presented a method

where both locations and dimensions of the radiating ele-ments are jointly optimized to improve the aperture effi-ciency of UECRA In literature [8] Morabito and Nicolacidescribed a way to the optimal power synthesis of mask-constrained shaped beams In literature [9] by utilizing thecharacteristics of first kind Bessel functions for representingthe array factor Kedar presented a technique for synthesis ofwideband wide-scan UECRA with low sidelobe level In allthe DAs have a fast calculation speed which could be usedfor the synthesis of very large UECRA However this type ofapproaches cannot guarantee the methods which convergeto the global optima

e second type of method is the use of biologicallyinspired global optimization procedures such as the geneticalgorithm (GA) [10] and its improved forms [11ndash13] Haupt[10] earlier described the way to optimize the elementplacement in a UECRA to obtain the low PSL by using theGA en Chen et al [11] presented a modified real GA(MGA) which makes ring radii as the only optimal variablesto suppress the PSL of UECRA For the same issue Jiang

HindawiInternational Journal of Antennas and PropagationVolume 2020 Article ID 5495846 7 pageshttpsdoiorg10115520205495846

et al [12] proposed an improved integer genetic algorithm(IIGA) In literature [13] a multiobjective GA (MOGA) isdescribed and proved to be effective for designing the CRAwhere both the half-power-beamwidth (HPBW) and PSL ofthe CRAs at different scanning directions are optimized

e third type of approaches for designing UECRA is thehybrid algorithms (HAs) [2 14ndash17] By combining theadvantages of some traditional optimization strategies thedeterministic procedures the global optimization tools etcthe use of this type of methods may achieve global opti-mization at a relative low-computational burden Sometypical contributions are reviewed below Carlin et al [2]delivered a two-step methodology which determines thesparsest auxiliary layout of concentric current ringsmatching a user-defined reference pattern through aBayesian compressive sensing method and then yields theconcentric ring isophoric array by strategy of density ta-pering Furthermore through splitting the synthesis ofcircular array into a global optimization procedure on astrongly reduced search space and a convex programmingproblem Bucci and Pinchera [14] presented a hybrid ap-proach which is proved effective for the synthesis of high-directivity pencil beam patterns Other notable HAs such asthe method in [15] which combines a convex optimizationand a determined approach together the strategy [16] byusing weighting density tapering and GA the approach [17]integrating an improved discrete cuckoo search algorithm(IDCSA) and a cuckoo search-invasive weed algorithm(CSIWA) [17]

Of all the previous efforts the biological-based methodssuch as GA [10ndash12] and MOGA [13] have the advantages ofglobal optimization but are only suitable for dealing withsmall arrays Actually the reports for synthesizing UECRAof large scale by a limited runs of GA [10] are proved to betime consuming and thereby local minima is resulted eglobal optima may be obtained by many runs of the methodwhich means the needing of huge computational effortsAccording to the authorsrsquo opinion improving the quality ofthe initial solution is a reasonable way to speed up theconvergence speed of the stochastic algorithms In view ofthis point and with the purpose of suppressing the PSL ofUECRA with isotropic elements a new strategy of densitytapering named partial density tapering (PDT) is proposedin this paper e idea of PDT is derived from the techniqueof amplitude tapering typical as the solution when per-forming Taylor tapering to achieve an ultralow sidelobearray e amplitudes of element excitations show a tapereddistribution from the center of array to the outermost Inparticular for the elements located near the center of thearray the normalized excited amplitudes are almost equaland close to 1 For this reason we first divide the aperture ofUECRA into M concentric rings where a total of m0 rings(m0 is the nearest integer less than or equal to M2) close tothe center of the array are set to be equal-spaced and theinterspacings of remaining (M minus m0) rings are set to besequentially increased from the inside to the outside of thearray en we initialize the element distribution of m0inner rings to be fully populated while the ring filling factor(RFF defined as the ratio of the current number of elements

of a ring to the number of elements when the ring is fullypopulated) of the rest rings are constrained by a propertapered thresholds For this reason we name the strategy asPDT

After the implementation of PDT both the ring radii andRFF of the outer (M minus m0) rings are optimized by a con-ventional algorithm of DE thus we call the method PDT-DE Unlike the IIGA [12] where a parallel encoding strategyis adopted this method transforms the number of elementson a ring into the value of RFF between (01) and therebyreal coding can be directly used in the optimizationprocedure

e paper is organized as follows Section 2 describes theformulation of the proposed method Section 3 provides aset of numerical examples to validate the effectiveness of thePDT-DE Finally the conclusion of this work is summarizedin Section 4

2 Formulation of the Proposed Method

For a single element at the center the array factor of aUECRA with isotropic elements can be represented by

F(u v) 1 + 1113944M

m11113944

Nm

n1e

jkrm ucosϕmn+vsin ϕmn( ) (1)

where rm (m 1 2 M) is the radius of ring m thatconsists of Nm isotropic elements M denotes the totalnumber of rings and k is the wave number ϕmn depicts theangular location of element n in ring m which is expressedby ϕmn 2π(n minus 1)Nm u sin θ cos ϕ and v sin θ sinϕ θand ϕ are the elevation and azimuth angles in sphericalcoordinates

For a given ring we specify that the minimum arc lengthinterval between adjacent elements is 05λ (λ denotes thewavelength) en the maximum number of elements thatcan be filled in ring m (we mean that the ring m is fullypopulated) is given by

Nmaxm int

2πrm

05λ1113874 1113875 m 1 2 M (2)

where int(A) denotes the function that rounds the value ofAto the nearest integer less than or equal to A

Suppose a UECRA consisting of M rings is synthesizedusing the PDT-DE within a circular aperture of radius Rthus the detailed procedures of the proposed method aredepicted as follows

21 Aperture Division Divide the aperture into M con-centric rings where both R and M should conform to therelationship

isin R minus M middot d0

st εge 05λ d0 isin (05λ 06λ)1113896 (3)

where isin is the difference of the radii between aperture radiusand the outermost ring when we assume that all the rings arespaced at d0 For convenience rewrite d0 as

2 International Journal of Antennas and Propagation

d0 (05 + 01σ)λ (4)

where σ is a randomly selected value from interval (01)Furthermore we force m0 int(M2) and the radii of m0rings close to the center of the aperture can be expressed by

rm m middot d0 m 1 2 m0 (5)

For the rest (M minus m0) rings located away from the centerof the aperture the radii are specified by

rm0+q m0 + q( 1113857 middot d0 + Δdq

st 0ltΔdq ltisin q 1 2 M minus m0

⎧⎨

⎩ (6)

For convenience a vector is introduced to describe thecollection of Δdq q 1 2 M minus m0 as

Δd Δd1Δd2 ΔdMminusm01113966 1113967 (7)

Let Δrm rm minus rmminus1 which means the difference of radiibetween mth and (m minus 1)th rings then we have

Δrm d0 for 2lemlem0

Δrm gt d0 form0 ltmleM1113896 (8)

According to equations (6) and (7) and in order to meetthe requirement of density tapering the variables in vectorΔd need to be arranged in ascending order To determineΔd a new vector is defined as

τ τ1 τ2 τMminusm0 τ1+Mminusm0

τ2+Mminusm0 τL+Mminusm0

1113966 1113967

(9)

where L is an integer that is randomly selected from 0 to 5and τi (1le ileL + M minus m0) is the random number uni-formly distributed within (01) rough sorting τi1113864 1113865 inascending order we get a distinct vector

τprime τ1prime τ2prime τMminusm0prime τ1+Mminusm0prime τ2+Mminusm0

prime τL+Mminusm0prime1113966 1113967

(10)

Based on (3) and (10) we divide ϵ into (L + M minus m0)

unequal parts and the value of each part is described by

δi ϵ

1113936Mminusm0+Li1 τi

prime middot τiprime i 1 2 L + M minus m0 (11)

Let a new vector to record (11) be

δ δ1 δ2 δL δL+1 δL+Mminusm01113966 1113967 (12)

Previous derivation shows that δi (i 1 2 L+

M minus m0) increases sequentially with the increment of sub-scripts Furthermore we split δ into two vectors (denoted asδprime and δPrime) and record them as

δprime δ1 δ2 δL1113864 1113865 (13)

δPrime δL+1 δL+Mminusm01113966 1113967 (14)

Let Δd δPrime therefore Δdq δL+q (q 1 2 M

minusm0)

After the implementation of equations (3)ndash(14) all thering radii of UECRA are determined For rings from 1th tom0th the spacings are equal to d0 Differently for rings from(m0 + 1)th to Mth the ring intervals are greater than d0 andthey are gradually increased from the inside to the outside ofthe array

e use of vector δprime is to make a slight adjustment to theradius of outermost ring so that rM leR In particular whenL 0 δprime becomes to an empty set thus we have rM R orelse rM ltR e operation helps to meet the design targetsunder possible smallest aperture radius

22 Initialize the Element Distribution Firstly the ringsfrom 1th to m0th are fully populated and the number ofelements for each ring is determined by (2) e totalnumber of elements in the full structure is recorded as1113936

m0m1 Nmax

m For the rest rings the ring filling factors (RFFs)are expressed by

fq Nm0+q

Nmaxm0+q

times 100 q 1 2 M minus m0 (15)

where Nm0+q denotes the current number of elements in ringm0 + q and Nmax

m0+q is determined by (2) We set a new vectorto describe (15) as

fRFF f1 f2 f3 fMminusm01113966 1113967 (16)

where fq (q 1 2 M minus m0) are bounded by

fminq ltfq ltf

maxq q 1 2 M minus m0 (17)

For convenience the collections of fminq1113966 1113967 and fmax

q1113966 1113967

are respectively labeled as

fminRFF f

min1 f

min2 f

min3 f

minMminusm0

1113966 1113967 (18)

fmaxRFF f

max1 f

max2 f

max3 f

maxMminusm0

1113966 1113967 (19)

Both fmaxRFF and fmin

RFF constitute the upper and lower fillingthresholds for fRFF In order to meet the requirement ofdensity tapering the variables of the two vectors should begiven in descending order respectively In more detail weset fmin

q (q 1 2 M minus m0) to be arranged indescending order within the interval [025 08] and ac-cordingly fmax

q (q 1 2 M minus m0) are determined by

fmaxq f

minq + η1 whenf

minq ge 075

fmaxq f

minq + η2 otherwise

⎧⎪⎨

⎪⎩(20)

where we specify η1 isin [01 02] η2 isin [01 025] so that theRFFs for rings from (m0 + 1)th to Mth are less than 10 Notonly that the curves of RFF thresholds need to have arelatively gentle tapering to expand the solution space andenhance the capability of global convergence of the algo-rithm erefore a scope 005le η2 minus η1 le 01 is recom-mended However the parameters of η1 and η2 are notstrictly limited as long as the values ensure that the RFFs areless than 10 and both the RFF thresholds satisfies a propertapering distribution

International Journal of Antennas and Propagation 3

23 DE Optimizing e individual vector of the populationis constituted by σ τ and fRFF which can be described as

X σ τ fRFF1113864 1113865 (21)

with

0lt σ lt 1

0lt τi lt 1 i 1 2 L + M minus m0

fminq ltfq ltf

maxq q 1 2 M minus m0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(22)

Both σ and τ are used to adjust the ring radii of UECRAwhile fRFF determines the number of elements that can befilled in different rings Suppose τmax 1 1 1 andτmin 0 0 0 and both the two vectors have the samelength with τ Correspondingly the upper and lowerthresholds for X can be expressed by

Xmax 0 τmax fmaxRFF1113864 1113865

Xmin 1 τmin fminRFF1113966 1113967

⎧⎨

⎩ (23)

According to (23) more individuals of DE are generatedthrough the equation

X Xmin + Rand middot Xmax minus Xmin( 1113857 (24)

where Rand is a random vector that has the same length withX and the variables in Rand are the random numbersuniformly distributed within (01) e fitness function aredefined as

min PSL ζ(X) (25)

where ζ(X) denotes the UECRA that can be obtained withminimum PSL (the fitness value) After the whole pop-ulation is formed the individuals are optimized by a con-ventional DE and evaluated by the fitness function describedin (25) Simultaneously the elite strategy is adopted so thatthe UECRA featuring minimum PSL is retained

3 Numerical Examples

e proposed method is applied to the design cases similarto that taken from [10ndash12 16] where a single element isexisted at the center of UECRA e parameters involved in

the method including the population size the maximumnumber of iterations in each generation and crossoverprobability as well as the scaling factor are equal to 50 2009 and 08 respectively

In the first case a circular aperture with radius of 45λ isconsidered among which the array consists of 7 rings and142 elements According to the proposed method the innerthree rings are fully populated For the four outer rings werenumbered them from inside to the outside as 1 2 3and 4 eoretically the upper and lower thresholds forfilling factors of the four rings should be prespecified as aredescribed by (18)ndash(20) However since the total number ofelements is given in advance only the RFF thresholds of1sim3 rings need to be predetermined e number of el-ements in the 4 ring is equal to the difference between 141and the total number of elements included in the remainingsix rings We specify the lower and upper RFF thresholds as[08 065 05] and [10 09 075] respectively e obtainedarray after 50 runs of the PDT-DE has the PSL of minus2995 dBwhich is 213 dB lower than the value obtained by the GA[10] and 162 dB lower than the report obtained by the MGA[11] Not only that compared with the report in [10] theaperture radius is reduced from 47λ to 426λ and the di-rectivity (denoted as D) is lost by 07 dB

Similarly when we increase the total number of elementsin UECRA to 183 and the number of rings to 8 the PSL ofsynthesized array is equal to -2786 dB which shows a PSLreduction of 228 dB with a little sacrifice at directivity as it iscompared to the value demonstrated by GA [10] Describedin Figures 1 and 2 are the 3D pattern of previous two in-stances where N denotes the total number of elements Inaddition for the case when N 201 the best array after 50runs of the PDT-DE has a PSL equal to minus3034 dB eperformance of the array shows a PSL suppression of 121 dBcompared to the value obtained by the MGA [11] and a PSLsuppression of 613 dB compared to that demonstrated bythe IFT-DE [18] More detailed results including the ringradii and the number of elements in each ring are given inTable 1

In the fourth case we consider a circular array withaperture radius of 15λ and the total number of elements of1302en we set the maximum number of runs equal to 20to alleviate the computational burden e best fitness value

ndash5ndash10ndash15ndash20

Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05

ndash1 ndash1ndash05

0u

051

Figure 1 3D pattern of the UECRA when N 142


is obtained at the 7th run where the array has the PSL ofminus3463 dB as shown in Figure 3(a) e value is 167 dBlower than the report demonstrated by IIGA [12] with a

011 dB loss of directivity Correspondingly a reduction ofaperture radius about 036λ is resulted Figures 3(b) and 3(c)describe the 3D and u-cut pattern of the array respectively


Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


Table 1 Synthesis results obtained by PDT-DE for the cases N 142 183 and 201

R(λ) N PSL (dB) D (dB) m 1 2 3 4 5 6 7 8

426 142 minus2995 2819 rm(λ) 05 10 15 206 268 345 426 ―Nm 6 12 18 21 29 26 29 ―

45 183 minus2786 2839 rm(λ) 05 10 15 20 25 307 368 45Nm 6 12 18 25 25 28 25 43

50 201 minus3034 2930 rm(λ) 05 10 15 20 265 336 411 50Nm 6 12 18 25 33 35 31 40

ndash27

ndash28

ndash29

ndash30

ndash31

Fitn

ess v

alue

(dB)

ndash32

ndash33

0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash34

ndash351

ndash1 ndash08 ndash06 ndash04 ndash02 0u

(c)

(b)

02 04 06 08 1

3 5 7 9 11Number of runs

(a)

13 15 17 1920Nor

mal

ized

ampl

itude

(dB)

ndash10

ndash20

Nor

mal

ized

ampl

itude

(dB)

ndash30

ndash40

ndash50

0

105

0v ndash05


0 u05

1

undashcut pattern

Figure 3 (a) Convergence of fitness value (b) 3D pattern and (c) u-cut


Furthermore the method is extended for the synthesisof large-scale UECRA within an aperture radius of 20λamong which the array consist of 2256 elements and 29rings e best array by 20 runs of the PDT-DE has the PSLof minus3633 dB and directivity of 3954 dB which show about225 dB PSL reduction with only 012 dB loss of directivityas compared to the reports obtained by HS [16]

Figures 4 and 5 depict the 3D pattern and elementdistribution within the aperture of the UECRA respec-tively For the same array the ring radii and the numberof elements in each ring are tabulated in Table 2 Moresynthesis results by using different methods are comparedin Table 3 where we can conclude that the PDT-DE hasthe advantages in PSL suppression of UECRA with re-spect to other methods and the PSL performance of thearray become more impressive for large scale UECRA byusing the proposed method All the numerical resultsmentioned above are obtained by using a PC equippedwith an 8 GB RAM and the Intel I7-6700 processor thatoperates at 34 GHz

4 Conclusion

e proposed method is capable of achieving UECRA withlower PSL than the published reports and the loss of di-rectivity can be neglected Not only that it has the advantage


Nor

mal

ized

ampl

itude

(dB)

ndash25ndash30ndash35ndash40ndash45ndash50

0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050

x (λ)ndash5ndash10ndash15ndash20

Figure 5 Element distribution of the UECRA when N 2256

Table 2 Results of ring radii (rm) and number of elements (Nm)

for the UECRA including 2256 elements

m rm Nm m rm Nm m rm Nm

1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―

Table 3 Comparative results by using different methods

R(λ) N Methods PSL (dB) D (dB) 3dB beamwidth(Δu)

470142

GA [10] minus2782 2889 ―470 MGA [11] minus2833 ― ―426 PDT-DE minus2995 2819 012845 183 GA [10] minus2558 285 ―45 PDT-DE minus2786 2839 0115498

201MGA [11] minus2913 ― ―

50 IFT-DE [18] minus2421 ― ―50 PDT-DE minus3034 2930 010810 623 IIGA [12] minus3244 3475 0056910 PDT-DE minus3360 3433 00615 1302 IIGA [12] minus3296 3749 003881464 PDT-DE minus3463 3738 00420 2256 HS [16] minus3408 3966 0029620 PDT-DE minus3633 3954 003225 3465 IIGA [12] minus3364 4137 002362478 PDT-DE minus3558 4119 0024


of easy encoding with respect to IIGA erefore the PDT-DE provides an effective way for the synthesis of UECRAwith aperture sizes ranging from small to large scale Fur-thermore by some modifications the method can also beextended for the sidelobe suppression of sparse linear arrays

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported in part by the National NaturalScience Foundation of China under Grant nos 6160127261772398 and 61972239 Key Research and DevelopmentProgram Projects of Shaanxi Province under Grant no2019SF-257 Scientific Research Plan of Shaanxi ProvincialEducation Department under Grant no 19JK0175 andScientific Research Plan of Shaanxi University of Technologyunder Grant no SLGRCQD2009

References

[1] H EI-Kamchouchi ldquoUltra-low-sidelobe-level concentric-ring-array pattern synthesis using Bessel neural networksrdquo IEEEAntennas and Propagation Magazine vol 52 no 4 pp 102ndash105 2010

[2] M Carlin G Oliveri and A Massa ldquoHybrid BCS-deter-ministic approach for sparse concentric ring isophoricarraysrdquo IEEE Transactions on Antennas and Propagationvol 63 no 1 pp 378ndash383 2015

[3] P Angeletti G Toso and G Ruggerini ldquoArray antennas withjointly optimized elements positions and dimensions part IIplanar circular arraysrdquo IEEE Transactions on Antennas andPropagation vol 62 no 4 pp 1627ndash1639 2014

[4] R Willey ldquoSpace tapaering of linear and planar arraysrdquo IRETransactions on Antennas and Propagation vol 10 no 4pp 369ndash377 1962

[5] T A Milligan ldquoSpace-tapered circular (ring) arrayrdquo IEEEAntennas and Propagation Magazine vol 46 no 3 pp 70ndash732004

[6] O M Bucci and S Perna ldquoA deterministic two dimensionaldensity taper approach for fast design of uniform amplitudepencil beams arraysrdquo IEEE Transactions on Antennas andPropagation vol 59 no 8 pp 2852ndash2861 2011

[7] Y Jiang and S Zhang ldquoAn innovative strategy for synthesis ofuniformly weighted circular aperture antenna array based onthe weighting density methodrdquo IEEE Antennas and WirelessPropagation Letters vol 12 pp 725ndash728 2013

[8] A F Morabito and P G Nicolaci ldquoOptimal synthesis ofshaped beams through concentric ring isophoric sparse ar-raysrdquo IEEE Antennas andWireless Propagation Letters vol 16pp 979ndash982 2017

[9] A Kedar ldquoDeterministic synthesis of wide scanning sparseconcentric ring antenna arraysrdquo IEEE Transactions on An-tennas and Propagation vol 67 no 12 pp 7387ndash7395 2019

[10] R L Haupt ldquoOptimized element spacing for low sidelobeconcentric ring arraysrdquo IEEE Transactions on Antennas andPropagation vol 56 no 1 pp 266ndash268 2008

[11] K-S Chen Y-Y Zhu X-L Ni and H Chen ldquoLow sidelobesparse concentric ring arrays optimization using modifiedGArdquo International Journal of Antennas and Propagationvol 2015 pp 1ndash5 2015

[12] Y Jiang S Zhang Q Guo and M Li ldquoSynthesis of uniformlyexcited concentric ring arrays using the improved integerGArdquo IEEE Antennas andWireless Propagation Letters vol 15pp 1124ndash1127 2016

[13] D Bianchi S Genovesi and A Monorchio ldquoConstrainedpareto optimization of wide band and steerable concentricring arraysrdquo IEEE Transactions on Antennas and Propagationvol 60 no 7 pp 3195ndash3204 2012

[14] O M Bucci and D Pinchera ldquoA generalized hybrid approachfor the synthesis of uniform amplitude pencil beam ring-arraysrdquo IEEE Transactions on Antennas and Propagationvol 60 no 1 pp 174ndash183 2012

[15] X Zhao Q Yang and Y Zhang ldquoA hybrid method for theoptimal synthesis of 3-D patterns of sparse concentric ringarraysrdquo IEEE Transactions on Antennas and Propagationvol 64 no 2 pp 515ndash524 2016

[16] Y Jiang S Zhang Q Guo and X Luan ldquoA hybrid strategybased on weighting density and genetic algorithm for thesynthesis of uniformly weighted concentric ring arraysrdquo IEEEAntennas and Wireless Propagation Letters vol 16 pp 186ndash189 2017

[17] G Sun Y Liu Z Chen S Liang A Wang and Y ZhangldquoRadiation beam pattern synthesis of concentric circularantenna arrays using hybrid approach based on cuckoosearchrdquo IEEE Transactions on Antennas and Propagationvol 66 no 9 pp 4563ndash4576 2018

[18] L Wang X-K Wang G Wang and J-K Jia ldquoA two-stepmethod for the low-sidelobe synthesis of uniform amplitudeplanar sparse arraysrdquo Progress In Electromagnetics ResearchM vol 86 pp 153ndash162 2019


et al [12] proposed an improved integer genetic algorithm(IIGA) In literature [13] a multiobjective GA (MOGA) isdescribed and proved to be effective for designing the CRAwhere both the half-power-beamwidth (HPBW) and PSL ofthe CRAs at different scanning directions are optimized

e third type of approaches for designing UECRA is thehybrid algorithms (HAs) [2 14ndash17] By combining theadvantages of some traditional optimization strategies thedeterministic procedures the global optimization tools etcthe use of this type of methods may achieve global opti-mization at a relative low-computational burden Sometypical contributions are reviewed below Carlin et al [2]delivered a two-step methodology which determines thesparsest auxiliary layout of concentric current ringsmatching a user-defined reference pattern through aBayesian compressive sensing method and then yields theconcentric ring isophoric array by strategy of density ta-pering Furthermore through splitting the synthesis ofcircular array into a global optimization procedure on astrongly reduced search space and a convex programmingproblem Bucci and Pinchera [14] presented a hybrid ap-proach which is proved effective for the synthesis of high-directivity pencil beam patterns Other notable HAs such asthe method in [15] which combines a convex optimizationand a determined approach together the strategy [16] byusing weighting density tapering and GA the approach [17]integrating an improved discrete cuckoo search algorithm(IDCSA) and a cuckoo search-invasive weed algorithm(CSIWA) [17]

Of all the previous efforts the biological-based methodssuch as GA [10ndash12] and MOGA [13] have the advantages ofglobal optimization but are only suitable for dealing withsmall arrays Actually the reports for synthesizing UECRAof large scale by a limited runs of GA [10] are proved to betime consuming and thereby local minima is resulted eglobal optima may be obtained by many runs of the methodwhich means the needing of huge computational effortsAccording to the authorsrsquo opinion improving the quality ofthe initial solution is a reasonable way to speed up theconvergence speed of the stochastic algorithms In view ofthis point and with the purpose of suppressing the PSL ofUECRA with isotropic elements a new strategy of densitytapering named partial density tapering (PDT) is proposedin this paper e idea of PDT is derived from the techniqueof amplitude tapering typical as the solution when per-forming Taylor tapering to achieve an ultralow sidelobearray e amplitudes of element excitations show a tapereddistribution from the center of array to the outermost Inparticular for the elements located near the center of thearray the normalized excited amplitudes are almost equaland close to 1 For this reason we first divide the aperture ofUECRA into M concentric rings where a total of m0 rings(m0 is the nearest integer less than or equal to M2) close tothe center of the array are set to be equal-spaced and theinterspacings of remaining (M minus m0) rings are set to besequentially increased from the inside to the outside of thearray en we initialize the element distribution of m0inner rings to be fully populated while the ring filling factor(RFF defined as the ratio of the current number of elements

of a ring to the number of elements when the ring is fullypopulated) of the rest rings are constrained by a propertapered thresholds For this reason we name the strategy asPDT

After the implementation of PDT both the ring radii andRFF of the outer (M minus m0) rings are optimized by a con-ventional algorithm of DE thus we call the method PDT-DE Unlike the IIGA [12] where a parallel encoding strategyis adopted this method transforms the number of elementson a ring into the value of RFF between (01) and therebyreal coding can be directly used in the optimizationprocedure

e paper is organized as follows Section 2 describes theformulation of the proposed method Section 3 provides aset of numerical examples to validate the effectiveness of thePDT-DE Finally the conclusion of this work is summarizedin Section 4

2 Formulation of the Proposed Method

For a single element at the center the array factor of aUECRA with isotropic elements can be represented by

F(u v) 1 + 1113944M

m11113944

Nm

n1e

jkrm ucosϕmn+vsin ϕmn( ) (1)

where rm (m 1 2 M) is the radius of ring m thatconsists of Nm isotropic elements M denotes the totalnumber of rings and k is the wave number ϕmn depicts theangular location of element n in ring m which is expressedby ϕmn 2π(n minus 1)Nm u sin θ cos ϕ and v sin θ sinϕ θand ϕ are the elevation and azimuth angles in sphericalcoordinates

For a given ring we specify that the minimum arc lengthinterval between adjacent elements is 05λ (λ denotes thewavelength) en the maximum number of elements thatcan be filled in ring m (we mean that the ring m is fullypopulated) is given by

Nmaxm int

2πrm

05λ1113874 1113875 m 1 2 M (2)

where int(A) denotes the function that rounds the value ofAto the nearest integer less than or equal to A

Suppose a UECRA consisting of M rings is synthesizedusing the PDT-DE within a circular aperture of radius Rthus the detailed procedures of the proposed method aredepicted as follows

21 Aperture Division Divide the aperture into M con-centric rings where both R and M should conform to therelationship

isin R minus M middot d0

st εge 05λ d0 isin (05λ 06λ)1113896 (3)

where isin is the difference of the radii between aperture radiusand the outermost ring when we assume that all the rings arespaced at d0 For convenience rewrite d0 as


d0 (05 + 01σ)λ (4)






⎧⎨

⎩ (6)









1113966 1113967

(9)




(10)



δi ϵ







δprime δ1 δ2 δL1113864 1113865 (13)



minusm0)




m0m1 Nmax


fq Nm0+q

Nmaxm0+q






fminq ltfq ltf



q1113966 1113967


fminRFF f

min1 f

min2 f

min3 f

minMminusm0

1113966 1113967 (18)

fmaxRFF f

max1 f

max2 f

max3 f

maxMminusm0

1113966 1113967 (19)





fmaxq f

minq + η1 whenf

minq ge 075

fmaxq f


⎧⎪⎨

⎪⎩(20)




X σ τ fRFF1113864 1113865 (21)

with

0lt σ lt 1


fminq ltfq ltf


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(22)




⎧⎨

⎩ (23)




min PSL ζ(X) (25)









Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051






Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051



R(λ) N PSL (dB) D (dB) m 1 2 3 4 5 6 7 8

426 142 minus2995 2819 rm(λ) 05 10 15 206 268 345 426 ―Nm 6 12 18 21 29 26 29 ―

45 183 minus2786 2839 rm(λ) 05 10 15 20 25 307 368 45Nm 6 12 18 25 25 28 25 43

50 201 minus3034 2930 rm(λ) 05 10 15 20 265 336 411 50Nm 6 12 18 25 33 35 31 40

ndash27

ndash28

ndash29

ndash30

ndash31

Fitn

ess v

alue

(dB)

ndash32

ndash33

0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash34

ndash351


(c)

(b)

02 04 06 08 1


(a)

13 15 17 1920Nor

mal

ized

ampl

itude

(dB)

ndash10

ndash20

Nor

mal

ized

ampl

itude

(dB)

ndash30

ndash40

ndash50

0

105

0v ndash05


0 u05

1

undashcut pattern





4 Conclusion



Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050






1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―



470142


201MGA [11] minus2913 ― ―




Data Availability




Acknowledgments


References




















d0 (05 + 01σ)λ (4)






⎧⎨

⎩ (6)









1113966 1113967

(9)




(10)



δi ϵ







δprime δ1 δ2 δL1113864 1113865 (13)



minusm0)




m0m1 Nmax


fq Nm0+q

Nmaxm0+q






fminq ltfq ltf



q1113966 1113967


fminRFF f

min1 f

min2 f

min3 f

minMminusm0

1113966 1113967 (18)

fmaxRFF f

max1 f

max2 f

max3 f

maxMminusm0

1113966 1113967 (19)





fmaxq f

minq + η1 whenf

minq ge 075

fmaxq f


⎧⎪⎨

⎪⎩(20)




X σ τ fRFF1113864 1113865 (21)

with

0lt σ lt 1


fminq ltfq ltf


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(22)




⎧⎨

⎩ (23)




min PSL ζ(X) (25)









Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051






Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051



R(λ) N PSL (dB) D (dB) m 1 2 3 4 5 6 7 8

426 142 minus2995 2819 rm(λ) 05 10 15 206 268 345 426 ―Nm 6 12 18 21 29 26 29 ―

45 183 minus2786 2839 rm(λ) 05 10 15 20 25 307 368 45Nm 6 12 18 25 25 28 25 43

50 201 minus3034 2930 rm(λ) 05 10 15 20 265 336 411 50Nm 6 12 18 25 33 35 31 40

ndash27

ndash28

ndash29

ndash30

ndash31

Fitn

ess v

alue

(dB)

ndash32

ndash33

0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash34

ndash351


(c)

(b)

02 04 06 08 1


(a)

13 15 17 1920Nor

mal

ized

ampl

itude

(dB)

ndash10

ndash20

Nor

mal

ized

ampl

itude

(dB)

ndash30

ndash40

ndash50

0

105

0v ndash05


0 u05

1

undashcut pattern





4 Conclusion



Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050






1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―



470142


201MGA [11] minus2913 ― ―




Data Availability




Acknowledgments


References





















X σ τ fRFF1113864 1113865 (21)

with

0lt σ lt 1


fminq ltfq ltf


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(22)




⎧⎨

⎩ (23)




min PSL ζ(X) (25)









Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051






Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051



R(λ) N PSL (dB) D (dB) m 1 2 3 4 5 6 7 8

426 142 minus2995 2819 rm(λ) 05 10 15 206 268 345 426 ―Nm 6 12 18 21 29 26 29 ―

45 183 minus2786 2839 rm(λ) 05 10 15 20 25 307 368 45Nm 6 12 18 25 25 28 25 43

50 201 minus3034 2930 rm(λ) 05 10 15 20 265 336 411 50Nm 6 12 18 25 33 35 31 40

ndash27

ndash28

ndash29

ndash30

ndash31

Fitn

ess v

alue

(dB)

ndash32

ndash33

0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash34

ndash351


(c)

(b)

02 04 06 08 1


(a)

13 15 17 1920Nor

mal

ized

ampl

itude

(dB)

ndash10

ndash20

Nor

mal

ized

ampl

itude

(dB)

ndash30

ndash40

ndash50

0

105

0v ndash05


0 u05

1

undashcut pattern





4 Conclusion



Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050






1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―



470142


201MGA [11] minus2913 ― ―




Data Availability




Acknowledgments


References























Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051



R(λ) N PSL (dB) D (dB) m 1 2 3 4 5 6 7 8

426 142 minus2995 2819 rm(λ) 05 10 15 206 268 345 426 ―Nm 6 12 18 21 29 26 29 ―

45 183 minus2786 2839 rm(λ) 05 10 15 20 25 307 368 45Nm 6 12 18 25 25 28 25 43

50 201 minus3034 2930 rm(λ) 05 10 15 20 265 336 411 50Nm 6 12 18 25 33 35 31 40

ndash27

ndash28

ndash29

ndash30

ndash31

Fitn

ess v

alue

(dB)

ndash32

ndash33

0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash34

ndash351


(c)

(b)

02 04 06 08 1


(a)

13 15 17 1920Nor

mal

ized

ampl

itude

(dB)

ndash10

ndash20

Nor

mal

ized

ampl

itude

(dB)

ndash30

ndash40

ndash50

0

105

0v ndash05


0 u05

1

undashcut pattern





4 Conclusion



Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050






1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―



470142


201MGA [11] minus2913 ― ―




Data Availability




Acknowledgments


References






















4 Conclusion



Nor

mal

ized

ampl

itude

(dB)


0

105

0v ndash05


0u

051


20

15

10

5

0

ndash5

y (λ)

ndash10

ndash15

ndash2020151050






1 052 6 11 572 71 21 1146 962 104 13 12 624 78 22 1213 1083 156 19 13 676 84 23 1281 1064 208 26 14 728 91 24 1358 1125 260 32 15 782 87 25 1440 1176 312 39 16 839 89 26 1538 1117 364 45 17 897 101 27 1659 1218 416 52 18 956 105 28 1816 759 468 58 19 1019 98 29 200 16310 520 65 20 1081 87 ― ― ―



470142


201MGA [11] minus2913 ― ―




Data Availability




Acknowledgments


References





















Data Availability




Acknowledgments


References




















synthesis of uniformly excited concentric ring arrays by

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