researcharticle optimization of sparse concentric ring...

Research ArticleOptimization of Sparse Concentric Ring Arrays for Low Sidelobe

Kesong Chen Yafei Li and Jiajia Shi

School of Information and Communication Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China

Correspondence should be addressed to Yafei Li li1367356163com

Received 22 January 2019 Accepted 30 April 2019 Published 2 June 2019

Academic Editor Xiulong Bao

Copyright copy 2019 Kesong Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To lower the peak sidelobe level (PSLL) of sparse concentric ring arrays a method with multiple design constraints that embed afunction model into modified real genetic algorithm (MGA) and select the grid ring radii as optimization individual to synthesizesparse concentric ring arrays is proposed The multiple constraints include the array aperture the minimum element spacingand the number of elements The proposed method dynamically calculates the ratio of element on each ring and it has a fasterconvergence rate than other algorithmsTheMGAuses real number to code the optimization variable and it reduces the complexityof coding and improves the search efficiency Finally the results demonstrate the accuracy and effectiveness of the algorithm

1 Introduction

Antenna array plays an important role in technology includ-ing mobile communication radar system satellite andmedical treatment For example in [1] a concentric ringarray antenna is designed for reconfigurable isoflux patternwhich uses 61 elements and the frequency is 28 GHzThe antenna arrays can reduce the volume occupation andcomplex feeding and heat dissipation systems in a satelliteantenna system In [2] a sparse concentric rings array withisoflux coverage to earth surface is designed for low earthorbit (LEO) satellite system which optimizes the angularseparation between the antenna elements and the amplitudeexcitation across the array to generate the characteristics ofthe desired isoflux radiation The proposed antenna arrayscan generate a wide isoflux pattern with a good accuracy withrespect to the isofluxmask for LEO satellites In [3] a repeaterantenna is designed for wireless body area networks (WBAN)applications which can obtain the biological informationby implanting the device to human-body for treatmentdiagnosis and potential medical applications

The PSLL is an important criterion to evaluate the perfor-mance of the antenna array Thus with the multiple designconstraints including the number of elements the apertureof array and the minimum spacing of element it is animportant research topic to reduce the PSLL of antenna arrayin recent years [4] Study on array synthesis had been carried

out for many years and produced many classical methodsThese methodologies include Woodward Lawson methodand optimization calculation method [5 6] which can beused to improve pattern performance and Taylor synthesismethod [7] which can be applied to obtain a given patternWith the development of array antenna the scale of arraysbecomes larger and the structure of arrays becomes complexthe traditional method of array synthesis is no longer appli-cable So a lot of intelligence methods were introduced tosynthesize array antenna including genetic algorithm (GA)simulated annealing algorithm (SAA) differential evolutionalgorithm (DEA) and particle swarm optimization (PSO)[8ndash11] There are a few other topics about circular antennaarrays such as the design of nonuniform circular antennaarrays for low sidelobe [12 13] the comparison of algorithmsabout the multiobject design of circular antenna arrays [14]the design of electronically steerable linear arrays [15] andthe synthesis of sparse circular antenna arrays for acquiringdesired radiation beam pattern [16]This article selects GA tosynthesize the sparse concentric ring arrays for low PSLL In[17] GA was proposed In [8] GA was applied to optimizethinned array antenna and acquired efficient result In [18]a modified real genetic algorithm (MGA) was proposed tosynthesize the sparse concentric ring arrays

Due to the difference of element sparsity degree oneach auxiliary ring [19] a function model that presents the

HindawiInternational Journal of Antennas and PropagationVolume 2019 Article ID 1485075 8 pageshttpsdoiorg10115520191485075

2 International Journal of Antennas and Propagation

o

element

( )

m

(rn m)

x

z

y

Figure 1 Diagram of concentric ring array

relationship between ring radius and the degree of sparsity isproposed in this paper which is embedded into the process ofMGA to synthesize the sparse concentric ring arrays for lowPSLLThe proposed method dynamically calculates the ratioof element on each ring so it has a faster convergence ratethan other algorithms Besides theMGA uses real number tocode the optimization variable and it reduces the complexityof coding and improves the search efficiency Comparing theresult to that of other algorithms the proposed method canreduce PSLL efficiently and has a faster convergence

2 The Optimization Model of SparseConcentric Ring Array

In a planar array generate a point randomly the point isselected as centre point and then generate several circles withdifferent radius and place some elements on each ring theresulting planar array is a concentric ring array [20] Thediagram of concentric ring array is shown in Figure 1

Place an element in the centre of the concentric ring arraythe radiation pattern can be described as

119865 (u V) = 1 + 119867sum119899=1

120596119899119873nsum119898=1

exp [119895119896119903119899 (cos120593119898119906 + sin120593119898V)] (1)

where

119873119899 = number of elements in ring 119899119867 = number of rings119896 = 2120587120582120582 = wavelength120596119899 = element weights for ring 119899119903119899 = radius for ring 119899(119903119899 cos120593119898 119903119899 sin120593119898) = element location119906 = sin 120579 cos120593 V = sin 120579 sin120593120593119898 = 2120587(119898 minus 1)119873119899

For simplicity we assume that array antenna meets the idealconditions including the element that is isotropic and theelement that has uniform excitation amplitude and phaseshift The weight in the same ring is equal and the elementsin each ring are uniform distribution The direction of mainbeam of the array points to the normal direction of the arrayand the array pattern can be described as

119865 (u V) = 1 + 119873minus1sumi=1120596i exp [119895119896 (119903119894 cos120593119894119906 + 119903119894 sin120593119894V)] (2)

where119873 is the sum of element number 119903119894 is the radius of 119894thelement and 120593119894 is the angle of 119894th element

If the sparse concentric ring array has multiple con-straints including the number of rings 119867 the maximumarray aperture 119871 the minimum element spacing 119889119888 and thenumber of elements119873 the optimization goal is to obtain theminimum PSLL and the optimal mathematical model can bedescribed as

min 119875119878119871119871 (r0 1199031 119903119899 119903119867)119873119899 = lfloor2120587119903119899119889119899 rfloor 0 lt 119889119888 lt 119889119899 119899 = 1 2 119867

119867sum119899=0

119873119899 = 119873(3)

where

119875119878119871119871 (1199030 1199031 119903119899 119903119867) = max10038161003816100381610038161003816100381610038161003816119865 (119906 V)119865119865max

10038161003816100381610038161003816100381610038161003816 (4)

where 119865119865119898119886119909 is the maximum value of array pattern 119906 Vare the regions excluding the main beam 119873119899 is the elementnumber of 119899th ringsDue to the fact that the elements numberof each ring must be an integer the value of 119873119899 must berounded up or down to an integer To meet the requirementof minimum element spacing 119889119899 ge 119889119888 the digits to the rightof the decimal point are discarded

3 The Algorithm of Optimization

The flowchart of function model algorithm using MGA isshown in Figure 2

31 Function Model There is a conclusion in [19] in thesparse concentric ring array when the ring is away from thearray centre the more sparsity the element is the lower thePSLL is Thus we can propose a function model to presentthe relation between ring radius and the degree of sparsityWe select the data of [21] as fitting data shown in Table 1

In Table 1 119873119899 is the actual element number of 119899th ringsand 119873119899119898119886119909 is the maximum element number 119899th ring canplace

119873nmax = lfloor2120587119903119899 (120582)119889c rfloor (5)

119875119899 is element retention rate of 119899th ring

119875119899 = 119873119899119873nmax(6)

International Journal of Antennas and Propagation 3

Table 1 The fitting data element number (119873) ring radii (119903119899(120582)) actual element number in the rings (119873119899) maximum element number(119873119899119898119886119909) and element retention rate (119875119899)method PSLL (dB) 119873 119899 1 2 3 4 5 6

Opt 119903119899amp119873119899 [16] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

119873119899119898119886119909 9 17 26 37 47 59119875119899 99 1616 2526 3137 2647 3359

Table 2 The function model fits the data in Table 1

Function model Model parametera b c 1198772

Linear model 119891(119909) = 119886(sin(119909 minus 120587)) + 119887((119909 minus 10)2) + 119888 -01969 000228 06543 09602Gauss model 119891(119909) = 119886 sdot exp(minus((119909 minus 119887)119888)2) 1015 0716 4705 09168Sine model 119891(119909) = 119886 sdot sin(119887 sdot 119909 + 119888) 1054 02152 1633 09018Power model 119891(119909) = 119886 sdot 119909119887 1013 -02878 06812Rational model 119891(119909) = 119886(119909 + 119887) 5468 4315 07541Polynomial model 119891(119909) = 119886 sdot 119909 + 119887 -01331 1166 08791

Set Up the Initial Population

Is met end criteria

Yes

No

Start

Function Model

Calculate Fitness Value with Function Model

Selection

Crossover and Mutation

Output the Optimal Result

End

Figure 2 The flowchart of function model algorithm using MGA

Different function model is used to fit the data of Table 1 andthe result is shown in Table 2

Plot the fitting data and function model in Figure 3In the fitting function model the 1198772 is the determination

coefficient of fitting degree and the larger the 1198772 is thehigher the fitting degree is In Table 2 the linear model has

line1line2

line3

line4line5

line6

line1 Linear function modelline2 Gauss function modelline3 Sine function modelline4 Power function modelline5 Rational function modelline6 Polynomial function modelFitting data

315 25 3505 451 52 4rn()

04

05

06

07

08

09

1

11

12P n

Figure 3 Function model diagram

the maximum 1198772 so the linear model is selected as fittingfunction model

119865 (119883) = 119886 (sin (119883 minus 120587)) + 119887 ((119883 minus 10)2) + 119888 (7)

119883 is the radius of circular the value of 119865(119883) is the retentionrate of ring with radius 119883 119886 119887 119888 are the coefficients of thefunction model and they will be different with the variety ofthe number of elements and the size of the rings The actualelement number of rings with radii 119903119899 can be calculated asfollows

119873119899 = lfloor119865 (119903119899) sdot (2120587119903119899119889c )rfloor (8)


32 The Process of MGA Standard genetic algorithm uses01 binary code for individual and it has a low efficiencyMGA uses real number code for individual directly and ithas a high freedom of search and efficiency The main stepsof MGA include generating initial population calculatingfitness value selection crossover and mutation

321 Set Up the Initial Population We select ring radii 119877 =[1199030 1199031 1199032 119903119899 119903119867] as optimization individual In orderto meet the requirement of minimum element space 119889119888 anincreasing function is used to generate the ring radius asfollows

119903119899 = 119903119899minus1 + 119889119888 + 119886 sdot 119899 119886 gt 01199030 = 0119903H = 119871

2(9)

Generate individual repeatedly to make up the population 119875

119875 = [11987711198772 119877119896] (10)

119896 is the individual number of the population 119875

322 Calculate Fitness Value with Function Model Bring119877 into equation (5) so the maximum element number ofeach ring can be calculated Equation (7) can calculate theretention rate of each ring thus the actual element number ineach ring can be acquired Because the element in each ringis uniform distribution the angle of 119898th element on the 119899thring can be calculated as follows

120593119898 = 2120587 (119898 minus 1)119873119899 1 lt 119898 lt 119873119899 (11)

The polar coordinate representation of the individual is asfollows

119877119888 =

[[[[[[[[[[[[[

1199031 + 0119895 1199031 + 2120587119895 21198731 sdot sdot sdot 1199031 + 21205871198951198731 minus 11198731sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot119903119899 + 0119895 119903119899 + 2120587119895 2119873119899 sdot sdot sdot 119903119899 + 2120587119895119873119899 minus 1119873119899sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot119903119867 + 0119895 119903119867 + 2120587119895 2

119873119867 sdot sdot sdot 119903119867 + 2120587119895119873119867 minus 1119873119867

]]]]]]]]]]]]]

(12)

Substitute 119877119888 into equation (4) so the fitness value can becalculated

323 Judgement of TerminationCondition If the terminationconditions (the number of iterations reaches the limit or thePSLL meets the requirement) are met output the optimalindividual and theminimumPSLL of the current populationend Otherwise continue

324 Selection Operation Sort the individual according tofitness value

119875119904 = 119904119900119903119905 [11987711198772 119877119896] = [11987711990411198771199042 119877119904119896] (13)

Select the excellent individual by mean of truncated selectionrate 119901119905

119875119864 = [11987711990411198771199042 119877119904(lfloor119896sdot119901trfloor)] (14)

Replicate 119875119864 to make up new population 119875119899119890119908 as geneticoperator

119875119899119890119908 = [119875119864119875119864 ] = [11987711990411198771199042 119877119904(lfloor119896sdot119901trfloor)11987711990411198771199042 119877119904(lfloor119896sdot119901trfloor) 119877119904(119896minuslfloor119896lfloor119896sdot119901trfloorrfloorsdotlfloor119896sdot119901trfloor)]

(15)

The number of individuals in 119875119899119890119908 is equal to 119896325 Crossover and Mutation In order to meet the require-ment of minimum element spacing 119889119888 we need to set someconstraints in process of crossover and mutation

Crossover If the 119899th genes of 119894th and 119895th individuals areselected as crossover gene

The precrossover genetic operator is

119877119894 = [1199031198940 1199031198941 1199031198942 119903119894119899 119903119894119867]119877119895 = [1199031198950 1199031198951 1199031198952 119903119895119899 119903119895119867] (16)

The postcrossover genetic operator is

1198771198941015840 = [1199031198940 1199031198941 1199031198942 119903119895119899 119903119894119867]1198771198951015840 = [1199031198950 1199031198951 1199031198952 119903119894119899 119903119895119867]

(17)

where

119903119895(119899minus1) + 119889119888 le 119903119894119899 le 119903119895(119899+1) minus 119889119888 119899 = 1 2 119867 minus 1119903119894(119899minus1) + 119889119888 le 119903119895119899 le 119903119894(119899+1) minus 119889119888 119899 = 1 2 119867 minus 1 (18)

Mutation If the 119898th gene of 119897th individuals is selected asmutation gene

The premutation genetic operator is

119877119897 = [1199031198970 1199031198971 1199031198972 119903119897119898 119903119897119867] (19)

The postmutation genetic operator is

1198771198971015840 = [1199031198970 1199031198971 1199031198972 119903119897119898119906 119903119897119867] (20)

where

119903119897(119898minus1) + 119889119888 lt 119903119897119898119906 lt 119903119897(119898+1) minus 119889119888 119898 = 1 2 119867 minus 1 (21)

The individual after performing the crossover and mutationforms next generation population 119875g

119875119892 = [1198771101584011987721015840 1198771198961015840] (22)

4 Simulation

In order to verify the efficiency of proposedmethod two sim-ulations were performed We set some common parameters


Table 3 Comparing the optimal result of 119873 = 201 and 119867 = 7 with [21 22] Function model algorithm using MGA (FMAMGA) elementnumber (119873) ring radii (119903119899(120582)) and element number in the rings (119873119899)method PSLL (dB) 119873 119899 1 2 3 4 5 6 7

HGA [21] -2294 201 119903119899(120582) 10 159 214 288 366 498 --119873119899 12 19 26 36 45 62 --

Opt 119903119899 by MGA [22] -2545 201 119903119899(120582) 080 138 188 243 318 391 498119873119899 10 17 22 27 33 40 51

MGAFMA -2621 201 119903119899(120582) 072 122 172 239 318 393 498119873119899 9 15 21 28 35 41 51

Table 4 Comparing the optimal result of 119873 = 142 and 119867 = 6 with [21 22] Function model algorithm using MGA (FMAMGA) elementnumber (119873) ring radii (119903119899(120582)) and element number in the rings (119873119899)method PSLL (dB) 119873 119899 1 2 3 4 5 6

Opt 119903119899amp119873119899 [21] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

Opt 119903119899 by MGA [22] -2807 142 119903119899(120582) 076 136 209 292 377 470119873119899 9 16 26 30 27 33

FMAMGA -2819 142 119903119899(120582) 084 137 210 295 378 470119873119899 10 17 25 29 29 31

for two simulations such as selecting equation (4) as objectivefunction and the minimum element spacing 119889119888 = 1205822 theradiation pattern in the 119906-region (minus1 le 119906 le 1) and V-region(minus1 le V le 1) is sampled 1024 points The basic parametersof MGA are set as follows the number of genetic generationsis 200 the individual number of populations is 100 truncaterate is 50 crossover rate is 20 mutation rate is 2 andelitism is also employed

41 Optimizing Sparse Concentric Ring Arrays with the Num-ber of Elements N = 201 We set the same parameter valuesas [21 22] that is the number of elements 119873 = 201 and theaperture 119871 = 996120582 If ring number 119867 = 6 the arrays arealmost full and the element spacing is equal to nearly119889119888 thereis no room for optimization thus let the number of rings119867 =7 Set the coefficient of functionmodel 119886= -00072 119887= 00036and 119888 = 06973 Five independent experiments were carriedout in this simulation and convergence curve takes theaverage of 5 independent experimentsThe radiation patternthe element configuration diagram and the cut surface of 119906= 0 V = 0 are from the optimal result of 5 experiments Theelement configuration of the best sparse concentric ring arrayis shown in Figure 4 Figure 5 shows the radiation patternof the optimal result The cut surface of 119906 = 0 and V = 0 isshown in Figure 6 The PSLL of optimal result is -2621 dBand it is lower 327 dB than [21] and lower 076 dB than [22]Figure 7 shows the convergence characteristics of functionmodel algorithm using MGA and the proposed method hasfast convergence characteristic Contrast with that of [21 22]is shown in Table 3

42 Optimizing Sparse Concentric Ring Arrays with the Num-ber of Elements N= 142 According to [21 22] set the numberof elements 119873 = 142 the aperture 119871 = 94120582 and the number

minus5 0 5x()

y(

)

Element number201element o

minus5

0

5

Figure 4 The diagram of element configuration

of rings 119867 = 6 Set the coefficient of function model 119886 = -01969 119887 = 000228 and 119888 = 06543 Five independent exper-iments were carried out in this simulation and convergencecurve takes the average of 5 independent experiments Theradiation pattern the element configuration diagram and thecut surface of 119906 = 0 V = 0 are from the optimal result of5 experiments The element configuration of the best sparseconcentric ring array is shown in Figure 8 Figure 9 showsthe radiation pattern of the optimal result The cut surface ofradiation pattern when 119906 = 0 and V = 0 is shown in Figure 10The PSLL of optimal result is -2819 dB and it is lower037 dB than [21] and lower 012 dB than [22] Figure 11 showsthe convergence characteristics function model algorithmusing MGA and the proposed method has fast convergencecharacteristic Contrast with that of [21 22] is shown inTable 4


0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00

minus1minus1=sinsin u=sincos

Figure 5 Radiation pattern of the optimal result

sampling number 1024PSLL -2621 dB

u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)

uv0minus05 05 1minus1

Figure 6 The cut of radiation pattern 119906 = 0 V = 0

The single trial5 trials averaged

minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)

50 100 150 2000Generation

Figure 7 The convergence characteristics

minus6 minus4 minus2 0 2 4 6minus5

0

5Element number142element o

x()

y(

)


5 Conclusion

In this paper a function model that presents the relationshipbetween ring radius and the degree of sparsity is proposedwhich is embedded into the process ofMGA to synthesize thesparse concentric ring arrays for low PSLL Comparing theoptimal result with other literatures proves that the proposedmethod is an efficient way to reduce the PSLL and has afaster convergence rate It is reasonable and efficient to selectthe element number in the rings according to the functionmodel

Data Availability

The data of this paper can be accessed for the public the datahave been uploaded to github repository and all the datapresent in the research are available The chart docx dataused to support the findings of this study the result of figureand the program of this research have been deposited to the


0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)

minus05 0 05 1minus1uv



minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)



github repository these data can be accessed through httpsgithubcom1367356Optimization-of-sparse-concentric-ring-arrays-for-low-sidelobe

Conflicts of Interest

Kesong Chen Yafei Li and Jiajia Shi declare that there are noconflicts of interest regarding the publication of this paper

Acknowledgments

The research is funded by joint fund for civil aviation(U1233103)

References

[1] A R Maldonado M A Panduro C Del Rio Bocio andA L Mendez ldquoDesign of concentric ring antenna array fora reconfigurable isoflux patternrdquo Journal of ElectromagneticWaves and Applications vol 27 no 12 pp 1483ndash1495 2013

[2] M Ibarra M A Panduro A G Andrade and A ReynaldquoDesign of sparse concentric rings array for LEO satellitesrdquoJournal of Electromagnetic Waves and Applications vol 29 no15 pp 1983ndash2001 2015

[3] T Jinpil K Kyeol K Sunwoo et al ldquoDual-band on-bodyrepeater antenna for in-on-on WBAN applicationsrdquo Interna-tional Journal of Antennas and Propagation vol 2013 Article ID107251 12 pages 2013

[4] W-J Zhang L Li and F Li ldquoReducing the number of elementsin linear and planar antenna arrays with sparseness constrainedoptimizationrdquo IEEE Transactions on Antennas and Propagationvol 59 no 8 pp 3106ndash3111 2011

[5] H Steyskal ldquoThe Woodward-Lawson method - To bury or notto buryrdquo IEEE Antennas and Propagation Society Newslettervol 31 no 1 pp 35-36 1989

[6] P Minvielle E Tantar A-A Tantar and P Berisset ldquoSparseantenna array optimization with the cross-entropy methodrdquoIEEE Transactions on Antennas and Propagation vol 59 no 8pp 2862ndash2871 2011

[7] Q-Q He and B-Z Wang ldquoDesign of microstrip array antennaby using active element pattern technique combining withtaylor synthesis methodrdquo Progress in Electromagnetics Researchvol 80 pp 63ndash76 2008

[8] R L Haupt ldquoThinned arrays using genetic algorithmsrdquo IEEETransactions on Antennas and Propagation vol 42 no 7 pp993ndash999 1994

[9] A Trucco and V Murino ldquoStochastic optimization of linearsparse arraysrdquo IEEE Journal of Oceanic Engineering vol 24 no3 pp 291ndash299 1999

[10] D G Kurup M Himdi and A Rydberg ldquoSynthesis of uniformamplitude unequally spaced antenna arrays using the differen-tial evolution algorithmrdquo IEEE Transactions on Antennas andPropagation vol 51 no 9 pp 2210ndash2217 2003

[11] J Robinson andY Rahmat-Samii ldquoParticle swarmoptimizationin electromagneticsrdquo IEEE Transactions on Antennas and Prop-agation vol 52 no 2 pp 397ndash407 2004

[12] M A Panduro A L Mendez R Dominguez and G RomeroldquoDesign of non-uniform circular antenna arrays for side lobereduction using the method of genetic algorithmsrdquo AEU -International Journal of Electronics and Communications vol60 no 10 pp 713ndash717 2006


[13] M A Panduro and C A Brizuela ldquoEvolutionary multi-objective design of non-uniform circular phased arraysrdquo COM-PELThe International Journal for Computation and Mathemat-ics in Electrical and Electronic Engineering vol 27 no 2 pp 551ndash566 2008

[14] M A Panduro C A Brizuela J Garza S Hinojosa and AReyna ldquoA comparison of NSGA-II DEMO and EM-MOPSOfor the multi-objective design of concentric rings antennaarraysrdquo Journal of Electromagnetic Waves and Applications vol27 no 9 pp 1100ndash1113 2013

[15] M A Panduro C A Brizuela and D H Covarrubias ldquoDesignof electronically steerable linear arrays with evolutionary algo-rithmsrdquo Applied Soft Computing vol 8 no 1 pp 46ndash54 2008

[16] L A Garza L F Yepes D H Covarrubias M A Alonso andM A Panduro ldquoSynthesis of sparse circular antenna arraysapplying a tapering technique over reconstructed continuouscurrent distributionrdquo IETMicrowaves AntennasampPropagationvol 10 no 3 pp 347ndash352 2016

[17] J H Holland Adaptation in Natural and Artificial System MITPress 1992

[18] K Chen H Chen L Wang et al ldquoModified real GA for thesynthesis of sparse planar circular arraysrdquo IEEE Antennas andWireless Propagation Letters vol 15 p 1 2016

[19] R Das ldquoConcentric ring arrayrdquo IEEE Transactions on Antennasand Propagation vol 14 no 3 pp 398ndash400 1966

[20] R L Haupt ldquoThinned concentric ring arraysrdquo in Proceedings ofthe Antennas and Propagation Society International Symposium2008

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

[22] C Kesong Z Yongyun and N I Xiaolong ldquoOptimummethodof grid ring radii of sparse concentric rings arrayrdquo ChineseJournal of Radio Science 2016

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Submit your manuscripts atwwwhindawicom


o

element

( )

m

(rn m)

x

z

y

Figure 1 Diagram of concentric ring array

relationship between ring radius and the degree of sparsity isproposed in this paper which is embedded into the process ofMGA to synthesize the sparse concentric ring arrays for lowPSLLThe proposed method dynamically calculates the ratioof element on each ring so it has a faster convergence ratethan other algorithms Besides theMGA uses real number tocode the optimization variable and it reduces the complexityof coding and improves the search efficiency Comparing theresult to that of other algorithms the proposed method canreduce PSLL efficiently and has a faster convergence

2 The Optimization Model of SparseConcentric Ring Array

In a planar array generate a point randomly the point isselected as centre point and then generate several circles withdifferent radius and place some elements on each ring theresulting planar array is a concentric ring array [20] Thediagram of concentric ring array is shown in Figure 1

Place an element in the centre of the concentric ring arraythe radiation pattern can be described as

119865 (u V) = 1 + 119867sum119899=1

120596119899119873nsum119898=1

exp [119895119896119903119899 (cos120593119898119906 + sin120593119898V)] (1)

where

119873119899 = number of elements in ring 119899119867 = number of rings119896 = 2120587120582120582 = wavelength120596119899 = element weights for ring 119899119903119899 = radius for ring 119899(119903119899 cos120593119898 119903119899 sin120593119898) = element location119906 = sin 120579 cos120593 V = sin 120579 sin120593120593119898 = 2120587(119898 minus 1)119873119899

For simplicity we assume that array antenna meets the idealconditions including the element that is isotropic and theelement that has uniform excitation amplitude and phaseshift The weight in the same ring is equal and the elementsin each ring are uniform distribution The direction of mainbeam of the array points to the normal direction of the arrayand the array pattern can be described as

119865 (u V) = 1 + 119873minus1sumi=1120596i exp [119895119896 (119903119894 cos120593119894119906 + 119903119894 sin120593119894V)] (2)

where119873 is the sum of element number 119903119894 is the radius of 119894thelement and 120593119894 is the angle of 119894th element

If the sparse concentric ring array has multiple con-straints including the number of rings 119867 the maximumarray aperture 119871 the minimum element spacing 119889119888 and thenumber of elements119873 the optimization goal is to obtain theminimum PSLL and the optimal mathematical model can bedescribed as

min 119875119878119871119871 (r0 1199031 119903119899 119903119867)119873119899 = lfloor2120587119903119899119889119899 rfloor 0 lt 119889119888 lt 119889119899 119899 = 1 2 119867

119867sum119899=0

119873119899 = 119873(3)

where

119875119878119871119871 (1199030 1199031 119903119899 119903119867) = max10038161003816100381610038161003816100381610038161003816119865 (119906 V)119865119865max

10038161003816100381610038161003816100381610038161003816 (4)

where 119865119865119898119886119909 is the maximum value of array pattern 119906 Vare the regions excluding the main beam 119873119899 is the elementnumber of 119899th ringsDue to the fact that the elements numberof each ring must be an integer the value of 119873119899 must berounded up or down to an integer To meet the requirementof minimum element spacing 119889119899 ge 119889119888 the digits to the rightof the decimal point are discarded

3 The Algorithm of Optimization

The flowchart of function model algorithm using MGA isshown in Figure 2

31 Function Model There is a conclusion in [19] in thesparse concentric ring array when the ring is away from thearray centre the more sparsity the element is the lower thePSLL is Thus we can propose a function model to presentthe relation between ring radius and the degree of sparsityWe select the data of [21] as fitting data shown in Table 1

In Table 1 119873119899 is the actual element number of 119899th ringsand 119873119899119898119886119909 is the maximum element number 119899th ring canplace

119873nmax = lfloor2120587119903119899 (120582)119889c rfloor (5)

119875119899 is element retention rate of 119899th ring

119875119899 = 119873119899119873nmax(6)



Opt 119903119899amp119873119899 [16] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

119873119899119898119886119909 9 17 26 37 47 59119875119899 99 1616 2526 3137 2647 3359





Is met end criteria

Yes

No

Start

Function Model


Selection



End





line1line2

line3

line4line5

line6


315 25 3505 451 52 4rn()

04

05

06

07

08

09

1

11

12P n



119865 (119883) = 119886 (sin (119883 minus 120587)) + 119887 ((119883 minus 10)2) + 119888 (7)






119903119899 = 119903119899minus1 + 119889119888 + 119886 sdot 119899 119886 gt 01199030 = 0119903H = 119871

2(9)


119875 = [11987711198772 119877119896] (10)



120593119898 = 2120587 (119898 minus 1)119873119899 1 lt 119898 lt 119873119899 (11)


119877119888 =

[[[[[[[[[[[[[



]]]]]]]]]]]]]

(12)




119875119904 = 119904119900119903119905 [11987711198772 119877119896] = [11987711990411198771199042 119877119904119896] (13)





(15)




119877119894 = [1199031198940 1199031198941 1199031198942 119903119894119899 119903119894119867]119877119895 = [1199031198950 1199031198951 1199031198952 119903119895119899 119903119895119867] (16)


1198771198941015840 = [1199031198940 1199031198941 1199031198942 119903119895119899 119903119894119867]1198771198951015840 = [1199031198950 1199031198951 1199031198952 119903119894119899 119903119895119867]

(17)

where




119877119897 = [1199031198970 1199031198971 1199031198972 119903119897119898 119903119897119867] (19)


1198771198971015840 = [1199031198970 1199031198971 1199031198972 119903119897119898119906 119903119897119867] (20)

where



119875119892 = [1198771101584011987721015840 1198771198961015840] (22)

4 Simulation




HGA [21] -2294 201 119903119899(120582) 10 159 214 288 366 498 --119873119899 12 19 26 36 45 62 --

Opt 119903119899 by MGA [22] -2545 201 119903119899(120582) 080 138 188 243 318 391 498119873119899 10 17 22 27 33 40 51

MGAFMA -2621 201 119903119899(120582) 072 122 172 239 318 393 498119873119899 9 15 21 28 35 41 51


Opt 119903119899amp119873119899 [21] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

Opt 119903119899 by MGA [22] -2807 142 119903119899(120582) 076 136 209 292 377 470119873119899 9 16 26 30 27 33

FMAMGA -2819 142 119903119899(120582) 084 137 210 295 378 470119873119899 10 17 25 29 29 31




minus5 0 5x()

y(

)


minus5

0

5




0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00




u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)




0


x()

y(

)


5 Conclusion


Data Availability



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Opt 119903119899amp119873119899 [16] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

119873119899119898119886119909 9 17 26 37 47 59119875119899 99 1616 2526 3137 2647 3359





Is met end criteria

Yes

No

Start

Function Model


Selection



End





line1line2

line3

line4line5

line6


315 25 3505 451 52 4rn()

04

05

06

07

08

09

1

11

12P n



119865 (119883) = 119886 (sin (119883 minus 120587)) + 119887 ((119883 minus 10)2) + 119888 (7)






119903119899 = 119903119899minus1 + 119889119888 + 119886 sdot 119899 119886 gt 01199030 = 0119903H = 119871

2(9)


119875 = [11987711198772 119877119896] (10)



120593119898 = 2120587 (119898 minus 1)119873119899 1 lt 119898 lt 119873119899 (11)


119877119888 =

[[[[[[[[[[[[[



]]]]]]]]]]]]]

(12)




119875119904 = 119904119900119903119905 [11987711198772 119877119896] = [11987711990411198771199042 119877119904119896] (13)





(15)




119877119894 = [1199031198940 1199031198941 1199031198942 119903119894119899 119903119894119867]119877119895 = [1199031198950 1199031198951 1199031198952 119903119895119899 119903119895119867] (16)


1198771198941015840 = [1199031198940 1199031198941 1199031198942 119903119895119899 119903119894119867]1198771198951015840 = [1199031198950 1199031198951 1199031198952 119903119894119899 119903119895119867]

(17)

where




119877119897 = [1199031198970 1199031198971 1199031198972 119903119897119898 119903119897119867] (19)


1198771198971015840 = [1199031198970 1199031198971 1199031198972 119903119897119898119906 119903119897119867] (20)

where



119875119892 = [1198771101584011987721015840 1198771198961015840] (22)

4 Simulation




HGA [21] -2294 201 119903119899(120582) 10 159 214 288 366 498 --119873119899 12 19 26 36 45 62 --

Opt 119903119899 by MGA [22] -2545 201 119903119899(120582) 080 138 188 243 318 391 498119873119899 10 17 22 27 33 40 51

MGAFMA -2621 201 119903119899(120582) 072 122 172 239 318 393 498119873119899 9 15 21 28 35 41 51


Opt 119903119899amp119873119899 [21] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

Opt 119903119899 by MGA [22] -2807 142 119903119899(120582) 076 136 209 292 377 470119873119899 9 16 26 30 27 33

FMAMGA -2819 142 119903119899(120582) 084 137 210 295 378 470119873119899 10 17 25 29 29 31




minus5 0 5x()

y(

)


minus5

0

5




0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00




u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)




0


x()

y(

)


5 Conclusion


Data Availability



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119903119899 = 119903119899minus1 + 119889119888 + 119886 sdot 119899 119886 gt 01199030 = 0119903H = 119871

2(9)


119875 = [11987711198772 119877119896] (10)



120593119898 = 2120587 (119898 minus 1)119873119899 1 lt 119898 lt 119873119899 (11)


119877119888 =

[[[[[[[[[[[[[



]]]]]]]]]]]]]

(12)




119875119904 = 119904119900119903119905 [11987711198772 119877119896] = [11987711990411198771199042 119877119904119896] (13)





(15)




119877119894 = [1199031198940 1199031198941 1199031198942 119903119894119899 119903119894119867]119877119895 = [1199031198950 1199031198951 1199031198952 119903119895119899 119903119895119867] (16)


1198771198941015840 = [1199031198940 1199031198941 1199031198942 119903119895119899 119903119894119867]1198771198951015840 = [1199031198950 1199031198951 1199031198952 119903119894119899 119903119895119867]

(17)

where




119877119897 = [1199031198970 1199031198971 1199031198972 119903119897119898 119903119897119867] (19)


1198771198971015840 = [1199031198970 1199031198971 1199031198972 119903119897119898119906 119903119897119867] (20)

where



119875119892 = [1198771101584011987721015840 1198771198961015840] (22)

4 Simulation




HGA [21] -2294 201 119903119899(120582) 10 159 214 288 366 498 --119873119899 12 19 26 36 45 62 --

Opt 119903119899 by MGA [22] -2545 201 119903119899(120582) 080 138 188 243 318 391 498119873119899 10 17 22 27 33 40 51

MGAFMA -2621 201 119903119899(120582) 072 122 172 239 318 393 498119873119899 9 15 21 28 35 41 51


Opt 119903119899amp119873119899 [21] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

Opt 119903119899 by MGA [22] -2807 142 119903119899(120582) 076 136 209 292 377 470119873119899 9 16 26 30 27 33

FMAMGA -2819 142 119903119899(120582) 084 137 210 295 378 470119873119899 10 17 25 29 29 31




minus5 0 5x()

y(

)


minus5

0

5




0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00




u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)




0


x()

y(

)


5 Conclusion


Data Availability



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




HGA [21] -2294 201 119903119899(120582) 10 159 214 288 366 498 --119873119899 12 19 26 36 45 62 --

Opt 119903119899 by MGA [22] -2545 201 119903119899(120582) 080 138 188 243 318 391 498119873119899 10 17 22 27 33 40 51

MGAFMA -2621 201 119903119899(120582) 072 122 172 239 318 393 498119873119899 9 15 21 28 35 41 51


Opt 119903119899amp119873119899 [21] -2782 142 119903119899(120582) 076 136 209 299 378 470119873119899 9 17 25 31 26 33

Opt 119903119899 by MGA [22] -2807 142 119903119899(120582) 076 136 209 292 377 470119873119899 9 16 26 30 27 33

FMAMGA -2819 142 119903119899(120582) 084 137 210 295 378 470119873119899 10 17 25 29 29 31




minus5 0 5x()

y(

)


minus5

0

5




0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00




u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)




0


x()

y(

)


5 Conclusion


Data Availability



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

minus10

minus20

minus30

minus40Ra

diat

ion

patte

rn (d

B)

11

00




u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus265

minus26

minus255

minus25

minus245

minus24

minus235

minus23

minus225

minus22

minus215

PSLL

(dB)




0


x()

y(

)


5 Conclusion


Data Availability



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

minus10

minus20

minus30

minus40

Radi

atio

n pa

ttern

(dB)

11

00

minus1minus1=sinsin

u=sincos



u=0v=0

minus40

minus30

minus20

minus10

0

Radi

atio

n pa

ttern

(dB)




minus29

minus28

minus27

minus26

minus25

minus24

minus23

minus22

minus21

minus20

PSLL

(dB)






Acknowledgments


References


























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


researcharticle optimization of sparse concentric ring...

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