Cat Swarm Optimization

CatSwarmOptimizationasAppliedtoTime-ModulatedConcentricCircularAntennaArray:AnalysisandComparisonWithOtherStochasticOptimizationMethods

GopiRam,DurbadalMandal,RajibKar,andS.P.Ghoshal

Abstract—Inthiscommunication,a9-ringtime-modulatedconcentriccircularantennaarray(TMCCAA)withisotropicelementshasbeenstud-iedbasedonanevolutionaryoptimizationalgorithmcalledcatswarmoptimization(CSO)forthereductionofsidelobelevel(SLL)andimprove-mentintheDirectivity,simultaneously.ThecomparativecasestudiesasCase-1andCase-2aremadewiththreecontrolparameterslikeinterele-mentspacinginrings,interringradii,andtheswitching“ON”timesofringswiththehelpofsamealgorithm.ExperimentalresultsshowaconsiderableSLLreductionwithrespecttotheuniformlyexcitedcase.ThenumericalresultsshowCase-2outperformsCase-1withrespecttoSLLandDirectivity.Apartfromthis,thepowersradiatedatthecenter/fundamentalfrequencyandthefirsttwosidebandfrequencies,anddynamicefficiencyhavebeencomputed.Ithasbeenobservedthatasthesidebandfrequencyincreases,boththepowersradiatedbyharmonicfrequenciesandsidebandlevels(SBLs)decrease.

IndexTerms—Catswarmoptimization(CSO),concentriccirculararrays,FNBW,radio-frequency(RF)switch,sidelobelevel(SLL),timemodulation.

evolutionaryoptimizationalgorithmstoelectromagneticshasenablednewsolutionandapplicationthatwasnotobtainablebyearliertechniques.

Differentevolutionaryalgorithms,geneticalgorithm(GA)[5]andparticleswarmoptimization(PSO)[6],[7],differentialevolution(DE)[8],[9]havebeenusedintheprocessoflowsidelobearraypatternsynthesisofTMCCAA.Inthiswork,forsimultaneousreductionofSLLaswellasimprovementinDirectivityoftheuniformlyexcitedTMCCAA[8],CSO[16],[17]isapplied.ThisCSOtechniquepro-ceedsbyoptimizingswitching“ON”timeweightofeveryringinthearray;alltheelementsintheringareuniformlyexcitedforthesame“ON”timeweightofthering.HighspeedandperiodicRFswitchisusedfortheswitchingpurpose.Duetotheaddi-tionaldegreeoffreedomastime,theSLLcanbefurtherreducedascomparedtothepublishedresultsin[5]and[8],whilekeep-inguniformamplitudeexcitations.Thestatisticalanalysisandt-test[18]havebeendonetoprovethesuperioroptimizationperformanceofCSO.

II.DESIGNEQUATIONS

Fig.1showsthegeneralconfigurationofCCAAwithPconcentriccircularrings,wherethepth(p=1,2,...,P)ringhasaradiusrpandthecorrespondingnumberofelementsintheringisNp.

Inthetimemodulation,eachringofCCAAiscontrolledbyahigh-speedperiodicRFswitchUp(t)[1].Suppose,thearrayoperatesattheoperatingfrequencyf0(Hz)andT0isthetimeperiodoftheoperatingfrequency.

Time-modulationperiodofthemodulatingswitchUp(t)isTprf;hence,time-modulationfrequencyfprf=1/Tprf.Generally,thetime-modulationfrequencyfprfismuchlowerthantheoperatingfre-quencyf0.ItmeansthatTprf>T0[1].Operatingfrequencyf0andtime-modulationfrequencyfprfareindependentofeachother.Now,instatedofexcitingeachringcontinuously,eachringisturned“ON”forthefixedtimedurationofτpwithapulserepetitionperiodTprfislyingintherangeτp≤Tprf>T0.DuetoRFswitchUp(t),theantennawillnotonlyradiateattheoperatingfrequency(f0)butalsowillradiateatdifferentsidebandharmonicsofmodulatingfrequency(fprf).SidebandharmonicsoftheradiationpatternarenotbecauseofoperatingfrequencyofthesignalbutduetotheperiodicswitchUp(t).Switch-ontimeofeachringofTMCCAAisτp(0≤τp≤Tprf).ThefarfieldofTMCCAAisobtainedasfollows:AF(θ,t)

⎫⎧

NpP⎬⎨󰀋󰀋j2πf0t

1+IpUp(t)exp[jkrpsinθcos(φ−φpi)]=e

⎭⎩p=1i=1

(1)

whereIp(Ip=1,throughoutthisstudy)istheexcitationamplitude

ofanelementonthepthcircularring;k=2π/λ;λbeingthesig-nalwavelength.θandφsymbolizethezenithanglefromthepositivez-axisandtheazimuthanglefromthepositivex-axistotheorthogonalprojectionoftheobservationpoint,respectively.Itmaybenotedthatiftheazimuthangleisassumedtobe0◦,i.e.,φ=00,(1)maybewrit-tenasanaperiodicfunctionofθwithaperiodofπradian.Theangleφpiistheelement-to-elementangularseparationmeasuredfromthepositivex-axis.Theelementsineachringareassumedtobeuniformlydistributed

󰀁󰀂i−1

(2)φpi=2π;p=1,...,P;i=1,...,Np.

Np

I.INTRODUCTION

ShanksandBickmore[1],Kummeretal.[2],andLewisandEvins[3]haveproposedtime-modulationtechniquebyprovidingradio-frequency(RF)switchingofantennaelementswith“time”asanextracontrolparametersoastoprovideanadditionaldegreeoffree-domtoimprovetheoverallradiationcharacteristicsoftime-modulatedantennaarrays(TMAs).

ManyrecentstudiesonTMAshavebeenfocusedontwomaintracks:1)thesynthesisofdesiredpatternatthecenterfrequencyandthejointminimizationofthelevelsoftheundesiredharmonicsgen-eratedbyON–OFFcommutationoftheswitches[4]–[9];and2)theexploitationofthesidebandradiationsforthedesignofmultifunc-tionalormultibeamantennaarrays[10]–[15].Dealingwiththefirstproblem,thepresentworkhasbeenproposedwiththeoptimizationofDirectivityandsidelobelevel(SLL)ofTMCCAAusinganovelevolutionaryalgorithmcatswarmoptimization(CSO)inthefieldofelectromagnetics.

TheprimarydisadvantageofTMAispresenceofsidebandsatharmonicfrequencies[1].Theseharmonicsaregenerallyunwantedastheywasteenergyandmaycauseinterferenceinotherpartsofradiospectrum[4]–[15].Toovercomethisdrawback,manyswitch-ingschemeshavebeenintroducedtosuppressthesesignalsbymeansofsuppressingthemaximumlevelsofsidebands.Theuseof

ManuscriptreceivedSeptember19,2014;revisedFebruary21,2015;acceptedJune09,2015.DateofpublicationJune11,2015;dateofcurrentversionSeptember01,2015.ThisworkwassupportedbytheDepartmentofScienceandTechnology,GovernmentofIndiaunderProjectSB/EMEQ-319/2013.

G.Ram,D.Mandal,andR.KararewiththeElectronicsandcommunicationDepartment,NationalInstituteofTechnology,Durgapur713209,India(e-mail:gopi203hardel@gmail.com).

S.P.GhoshaliswiththeElectricalEngineeringDepartment,NationalInstituteofTechnology,Durgapur713209,India(e-mail:spghoshalnitdgp@gmail.com).

Colorversionsofoneormoreofthefiguresinthiscommunicationareavailableonlineathttp://ieeexplore.ieee.org.

DigitalObjectIdentifier10.1109/TAP.2015.2444439

0018-926X©2015IEEE.Personaluseispermitted,butrepublication/redistributionrequiresIEEEpermission.

Seehttp://www.ieee.org/publications_standards/publications/rights/index.htmlformoreinformation.

IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.63,NO.9,SEPTEMBER201181

Fig.1.Concentriccircularantennaarray(CCAA).

Theperiodicswitch“ON–OFF”functionUp(t)intimedomainas

representedin(1)canbedecomposedintoFourierseriesinfrequencydomain,givenby

U󰀋∞p(t)=

ampej2πmfprft

(3)

m=−∞

where

amp=

Ipτp

sinc(mfprfτp)e−jπmfprfτpTprf

.(4)

Equation(4)showscurrentexcitationvalueforthemthmultiple

offprf.From(1),(3),and(4),thetotalfar-fieldarrayfactorcanbeexpandedintoFourierseriesandgivenby

⎧⎫

AF(θ,t)=󰀋∞⎨1+

󰀋P󰀋

Npa⎬mpexp[jkrpsinθcos(φ−φpi)]m=−∞

⎩p=1i=1

⎭

ej2π(f0+m.fprf)t.

(5)

Thefarfieldof(5)containsinfinitenumberofm.fprfcom-ponents,wherem=0,±1,±2,...,±∞.Them.fprfcomponent

arrayfactorisgivenby

AFm(θ,t)=ej2π(f0+m.fprf)t

⎧×⎨P⎩1+󰀋

󰀋Np⎫

ajkr⎬mpexp[psinθcos(φ−φpi)]p=1

i=1

⎭.(6)

From(6),onecanexpressthefollowingarrayfactorsAF0(θ,t),AF1(θ,t),andAF2(θ,t)fortheoperatingfrequency,thefirstandthesecondpositivesidebandfrequencies,respectively,whichcanbeusedtosynthesizethedesiredradiationpatternatf0,f0+fprf,andf0+2fprf.

Similartotime-modulatedlineararrays[4],theDirectivityofTMCCAAisgivenby

DTMCCAA=

󰀉∞|AF󰀊0(θ0,φ0)|2󰀊2ππ

(7)

41

π

m=−∞

00

|AFm(θ,φ)|2sinθdθdφwhereDTMCCAAistheDirectivityofTMCCAA;θ0denotesthe

directioninwhichAF(θ,φ)hasmaximumvalue;andAFm(θ,φ)denotestheradiationpatternatthemthordersidebandfrequencym.fprf.

ThepowerradiatedbyTMCCAA,Pm(m=0,±1,±2,...,±∞)isgivenby

2π

󰀌

π

P󰀋

∞󰀌m=

m(θ,φ)|2sinθdθdφ.

(8)

m=−∞,

00

|AFDynamicefficiency(ηd)isdefinedastheratioofpowerradiated

bythearrayatthecenterfrequency(m=0)andthesumofpowersradiatedbythearrayatthecenterfrequencyaswellasthesidebandharmonicfrequencies.

Themainobjectiveinthiscommunicationistooptimizethreedifferentcontrolparameters;thefirstoptimizingparameteristheswitchingtimesequenceτpofeachring.Thesecondoptimizingparameterisradiusofeachring(r,andfor1,ther2,...interring,rp).spacingThereshouldas0.5λbesomeconstraintsasr1≥0.5λ≤rp<λ.Thethirdoptimizingparameterisinterelementspacingineachring,andthisparameteralsofollowstheminimumconstraintof0.5λ≤dp<λ.AnadditionalseparationofΔpisaddedsothatrp=rp−1+λ/2+Δp;where

0≤Δp≤λ.

(9)

Theobjective/fitnessisexpressedasfollows:

fg(τP,rp,dp)=w11∗SLLg

fitnessmax(τP,rp,dp)|f0

+w22∗SBLgmax(τP,rp,dp)|f0+fprf+w33∗SBLgmax(τP,rp,dp)|f0+2fprf+w44∗(1/DTMCCA_max)

(10)

wheregdenotesthegthiterationcycle;SLLmax|fSLLatthecenterfrequency;SBL0isthe

maximum|max|f0+fprfandSBLmaxf0+2fprfarethemaximumsidebandlevels(SBLs)atthefirstandsecondsidebandfrequencies,respectively,andw11,w22,w33,w44aretheweightingfactorsofvarioustermstoemphasizethedifferentcontributionstothefitnessfunction.DTMCCA_maxisthemaximumDirectivityoftheTMCCAA.

III.NUMERICALRESULTS

ThissectiongivestheexperimentalresultsforvariousTMCCAAdesignsobtainedbyRGA,PSO,DE,andCSOalgorithms.Thiscom-municationadoptstheevolutionaryoptimizationalgorithmscalledCSO[16],[17].DirectivityiscalculatednumericallybySimpson’s1/3rule.Velocityofelectromagneticwaveinthefreespaceisgivenbyc=3.0×108m/sec;operatingfrequencyf0=3.0GHz;time-modulationfrequencyfprf=1/Tprf.Time-modulationperiodofthemodulatingswitchUi(t)isTprf=1μs,then,onecancalculatetime-modulationfrequencyfprf=1/Tprf.CurrentexcitationweightIpiskeptuniform(Ip=1).Fortheuniform9-ringCCAAof279ele-mentswithringradiusrp=p.λ/2andinterelementspacingineachringdp=λ/2.SLL,FNBW,andDirectivityare−17.4dB,14.76◦,and29.35dB,respectively.Tokeepdp≥λ/2andallowsufficientinterelementspacing,thedigitstotherightofthedecimalpointaredropped.TableIshowsoptimizedswitchingtimesequence,ringradii,andoptimalnonuniforminterelementspacingobtainedbyCSO.

A.Case-1:OptimizingOnlySwitchingTimeSequenceofEachRingTableIIshowsthatforCase-1,SLLandDirectivityobtainedbyCSOaremuchbetterthanthoseobtainedbyRGA,PSO,DE,uniformCCAA,andthoseobtainedin[8],withalittleincreaseinFNBW.

4182IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.63,NO.9,SEPTEMBER2015

TABLEI

OPTIMIZINGSWITCHINGTIMESEQUENCE,RINGRADII,ANDOPTIMALNONUNIFORMINTERELEMENTSPACINGBYCSO

TABLEII

OPTIMALSLL,SBL1,SBL2,FNBW,P0,P1,P2,DIRECTIVITY,DYNAMICEFFICIENCYOFTMCCAAOBTAINEDBYRGA,PSO,DE,

ANDCSOALGORITHMS

T-TEST

TABLEIII

VALUESANDP-VALUESOBTAINEDFORCOMPARISONOFCSO

WITHOTHERALGORITHMSFORCASE-2

Fig.2.Radiationpatterns(dB)obtainedbyCSOforCase1.

Fig.3.Radiationpatterns(dB)obtainedbyCSOforCase-2.

B.Case-2:OptimizingSwitchingTimeSequence,RingRadii,andOptimalNonuniformInterelementSpacing

TableIIshowsthatforCase-2,SLLandDirectivityobtainedbyCSOaremuchbetterthanthoseobtainedbyRGA,PSO,DE,andthoseoftheuniformCCAA,withalittleincreaseinFNBW.Moreover,Case-2outperformsCase-1withrespecttoSLLandDirectivity.Figs.2and3showtheradiationpatterns(dB)obtainedbyCSOforCase-1andCase-2,respectively.

Theanalysisofallthenumericalresultsofthetwocasestud-iesdepictedinTableIIshowsthatDirectivityandSLLobtainedbyRGA,PSO,DE,andCSOforCase-2aremuchbetterthanthoseofthesamealgorithmsforCase-1.CSOforCase-2outperformsRGA,

PSO,andDE,fortheCase-1andCase-2,uniformCCAA,andthoseobtainedin[5]and[8]withrespecttoreducedSLLandimprove-mentinDirectivity.TableIIalsoshowsthatastheharmonicfrequencyincreasestheSBLsdecrease,consequently,theassociatedpowersdecrease.ThesametableshowsthatthevaluesofSBL1andSBL2arehigherthanSLL,butthepowerassociatedwiththefundamentalfrequencyismuchhigherthanthecorrespondingpowersP1andP2associatedwithSBL1andSBL2,respectively.ThismeansthateventhoughTMCCAAhasadrawbackofharmonicradiations,powersradi-atedbythesidebandsareveryless.SoonecansaythatCSOforCase-2yieldsthebestoptimalresultswithrespecttoDirectivityandSLL,respectively,forTMCCAA.

Comparisonsofaccuraciesbasedont-test:

TableIIIshowst-testvaluesobtainedamongtheRGA,PSO,DE,andCSO.Whenthet-testvaluegiveninTableIIIishigherthan1.676,2.009,2.403,2.678,2.937,3.261(β=50),thereisasignificantdiffer-encebetweenthetwoalgorithms(CSOoverRGA,PSO,andDE)with95%,97.5%,99%,99.5%,99.75%,99.9%confidencelevels,respec-tively[18].Thus,fromstatisticalanalysis,itisclearthatCSOoffersmorerobustandpromisingresultsascomparedtootheralgorithms.TableIVshowstheexecutiontimes,meanfitnessfunctionvalues,stan-darddeviationsforthefitnessfunctionsappliedforCase-2,obtainedbyRGA,PSO,DE,andCSO.

Inthiscommunication,p-valueusesstudent’stcumulativedis-tributionfunction[18].TableIIIshowsthatallp-valuesobtainedforCSOwithrespecttotheotheralgorithmsaremuchlesserthan0.05,Then,onceagain,itisclearthatCSOoutperformsRGA,PSO,andDE.

IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.63,NO.9,SEPTEMBER201183

TABLEIV

EXECUTIONTIMES,MEANFITNESSFUNCTIONVALUE,STANDARDDEVIATIONFORTHEFITNESSFUNCTIONSAPPLIEDFORCASE-2,

OBTAINEDBYRGA,PSO,DE,ANDCSO

IV.CONCLUSION

Inthiscommunication,a9-ring,279-elementtime-modulatedcon-centriccircularantennaarray(TMCCAA)isconsideredforsimultane-ousreductionofSLLandimprovementofDirectivitybythealgorithmcalledCSO.Twocasestudieshavebeentaken,Case-1:optimizingonlyswitchingtimesequenceofeachring;andCase-2:optimiz-ingswitchingtimesequence,ringradii,andnonuniforminterele-mentspacing.Authors’contributionsinthiscommunicationareinmanyfolds:1)thenewapplicationofCSOalgorithmintheopti-maldesignofTMCCAAforsimultaneousoptimizationofSLLandDirectivity;2)simulationresultsshowthatCase-2outperformsCase-1withrespecttoSLLandDirectivity;3)CSOforCase-2outperformsRGA,PSO,andDEforboththeCase-1andCase-2,uniformCCAA,andthoseobtainedin[5]and[8]withrespecttoreducedSLLandimprovementinDirectivity;4)powerradiatedatthesecondsidebandfrequencyismuchlowerthanthepowerradiatedatthefirstsidebandfrequencyandthatatthecenterfrequency;and5)fromthesimula-tionresultsitisclearthatastheharmonicfrequencyincreases,powersandSBLsoftheradiatedharmonicfrequenciesbytheantennarapidlydecrease.Ascomparedtothemethodofnonuniformamplitudeexcita-tionweightingmethod,controllingtheswitch-ontimeofeachelementisthebettermethodwhichcanbemorepreciselyandmorerapidlyrealized.Forstatisticalanalysis,t-testhasalsobeendoneforthetestofstabilityofCSOoverRGA,PSO,andDEforCase-2.t-testresultsshowthatCase-2withCSOgivesthebestqualitynear-optimalsolu-tionsascomparedtoRGA,PSO,DE,andCSO.ThefinalinferencemaybedrawninfavorofCSO,whichprovestobethebestconsideringallaspectsofradiationofTMCCAA.Thus,thenumericalresultsmakeitclearthatfortheoptimaldesignofTMCCAA,optimizingalgorithmsandselectionofproperoptimizingvariablesrelatedtothearrayplayimportantroles.

ACKNOWLEDGMENT

TheauthorswouldliketothankSERB,DepartmentofScienceandTechnology,GovernmentofIndia.

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