Investigation and correction of error in impedance tube using intelligenttechniques Technology Journal of Applied Researchand

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Availableonlineatwww.sciencedirect.com

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology14(2016)405–414

Original

Investigation and correction of error in impedance tube using intelligent techniques

J. Niresh

^a,∗

, S. Neelakrishnan

^a

, S. Subha Rani

^b

aDepartmentofAutomobileEngineering,PSGCollegeofTechnology,Coimbatore641004,India

bDepartmentofElectronics&CommunicationEngineering,PSGCollegeofTechnology,Coimbatore641004,India Received8March2016;accepted28September2016

Availableonline29November2016

Abstract

Errorsariseinthemeasurementofsoundabsorptioncoefficientusingimpedancetubeduetovariousfactors.Minimizingtheerrorsrequireaddi- tionalhardwareorpropercalibrationofcertaincomponents.Thispaperproposesanewintelligenterrorcorrectionmechanismusingmathematical modellingandsoftcomputingparadigms.Alowcostimpedancetubeisdesigned,developedanditsperformanceiscomparedwithacommercially availablestandardtube.Aparticleswarmoptimizationandneuralnetworkbasedsystemisdevelopedtoreducetherandomandsystematicerrors inthedevelopedimpedancetube.Theproposedsystemistestedusingvariousporousandnon-porousfunctionaltextilematerialsandtheresults arevalidated.AsignificantreductioninerrorisobtainedatallfrequencyrangeswithPSObasedpredictionmethod.

Keywords:Neuralnetwork;PSO;Impedancetube;Soundabsorptioncoefficient;Errorcorrection

1. Introduction

Measurementofsoundabsorptioncoefficientisconsideredto beimportantindeterminingtheacousticpropertiesofmaterials consideredforuseinnoisecontrol.Soundabsorptioncoefficient isaquantitythatrepresentsthepercentageofsoundabsorbedby amaterial.Thisismeasuredusingthephenomenonofreflection ofsoundwaves. Soundwavesaregeneratedwithinamedium andtransmittedtowardsthetestsample.Bymeasuringtheinci- dentandreflected waves, reflection coefficientandhencethe acousticimpedancecanbecalculated.Thetwostandardmeth- ods for determining the absorption coefficient are Standing WaveRatio (SWR) (Iso, 1996) methodandtransferfunction method(InternationalOrganizationforStandardization,2001).

Thestandingwavepatternanditspressuremeasurementareused intheformer,whereasinthelatter,transferfunctionisused.A

∗Correspondingauthor.

E-mailaddress:[email protected](J.Niresh).

PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.

methodtomeasuresoundabsorptioncoefficientof acoustical materialsusedaswallorceilingtreatmentsisproposedin(Iso, 2003),providingresultsatrandomincidence.Anoptimization methodbasedonflowresistivitytoimprovereproducibilityis proposedin(Jeong&Chang,2015).Anewtechniquetomeasure soundabsorptioncoefficientusingsoundintensityprobesisdis- cussedin(Bonfiglio,Prodi,Pompoli,&Farina,2006).Another methodthatusesechoimpulsetechniquetomeasureacoustic impedance is elaborated in (Garai & Pompoli, 2001), which requires further post processing of results to obtain accurate measurements.Theacousticimpedanceisgenerallycalculated overawide rangeof frequenciesandthesestandardmethods introduceerrorsinthemeasurementsetup.Variousprocedures tomitigatetheseerrorshavebeendealtintheliterature.

Biasandrandomerrorsoccurwhilefindingthetransferfunc- tionbetweenthemicrophonepositions.In(Pilon,Panneton,&

Sgard,2004),the effectofairgapinstandingwavetubewas discussedwithexperimentalresults.Theimpactofbarometric pressure, sample mounting and error in apparatus was stud- ied and an analysis was presented in(Tinianov & Babineau, 2006).Themeasurementisalsoaffectedbydispersion,which

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Atwo microphone threecalibrationmethodwas proposed in(Gibiat&Laloe,1990).Inthismethod,impedanceofthree known devices isconnectedsubsequently andthe impedance oftheunknowndeviceiscalculatedbyusingthethreeknown impedanceandthetransferfunctionobtainedfromtheunknown device.

A multiple microphone method was proposed to measure pressureatmorethanthreepositions.Theacousticimpedance iscalculatedfromthetransferfunctionofafewcombinations ofmicrophones.Animprovedmethodwasproposedin(Cho&

Nelson,2002)using least squarecurvefittingandoptimizing the response of all microphone positions. Here again, multi- plemicrophoneswereemployedtodeterminevarioustransfer functionvaluesandchoosingthebest.

Alltheabovemethodsoferrorcorrectioninvolveaconstruc- tional change in the impedancetube or at least a calibration mechanism, whichis notstandardized. To overcome the dif- ficulty,anovelintelligencebasederrorcorrectionmechanism is developed based on soft computing techniques. In this paper, error reduction inimpedance tubeis consideredas an optimization problem. Numerous soft computing techniques suchas FireflyAlgorithm(NasiriandMeybodi, 2012),Simu- latedAnnealing(Mantawy,Abdel-Magid,&Selim,1998),Ant colonyoptimization(Xiao,Zhou,&Zhang,2004),ArtificialBee colony(Karaboga&Basturk,2007),BatandArtificialimmune system(AIS)(Taha, Mustapha,& Chen,2013),have already beenappliedtodifferentoptimizationproblem.Neuralnetworks havebeenusedinvariousapplicationareasforerrorcorrection andproved to be efficient. In (Ren, Xu, Sun, &Yue, 2011), thermalerrorcorrectioniscarriedoutusingbackpropagation neural networkand roughsets in CNCmachine. Errormini- mizationinradiooccultation electrondensityretrieval (Pham

& Juang, 2015),wind speed forecast (Zjavka, 2015), colour correction(Zhuo,Zhang,Dong,Zhao,&Peng,2014)andcoor- dinateboringmachine(Yangetal.,2014)withneuralnetworks havebeenaddressedintheliterature.PSOiscombinedwitha timedifferenceof arrivalalgorithmtoreduceerrorinfinding emitterlocationforvariousapplications(Cakir,Kaya,Yazgan, Cakir,&Tugcu,2014).Calibrationofthreeaxismagnetometer isdoneusingParticleSwarmOptimization(PSO)anditsvari- ant(Wu,Wu,Hu,&Wu,2013)byreducingthesensorerrors.

Variousotherapplicationsincludetoolpositioning(Mahapatra

&Devi,2013),predictionofgeometricerrorsinrobotics(Alici,

Jagielski,AhmetS¸ekercio˘glu,&Shirinzadeh,2006).Tomini- mizeerrorinimpedancetube,softcomputingtechniqueshave notbeenappliedyetintheliterature.Neuralnetworkhasbeen usedtomodeldifferentmaterialstoestimatethesoundabsorp- tioncoefficient(Gardner,O’Leary,Hansen,&Sun,2003).PSO (Kennedy&Eberhart,1995)andANN(Sarle,1994)havebeen choseninthispaper,asthesetwoalgorithmshavealreadybeen usedsuccessfullyintheliteratureasdiscussedearlier.Theerrors arisinginthemeasurementofacousticimpedanceduetovarious reasonsismodelledmathematicallyfromsampledataavailable through experiments. Neural network and PSO are designed to reduce the errors,andtheir results are comparedwith the standardtube.

2. Impedancetube

Totestthesoundabsorptioncoefficient,atransferfunction based impedance tubehas been developed. One microphone methodisemployedbyusingasinglemicrophoneattwoloca- tion successively, thereby avoiding phase mismatch between twodifferentmicrophones(Allard,1993).Thetheoryoftrans- ferfunctionmethodhasalreadybeendiscussedextensivelyin theliterature(Chung&Blaser,1980a,1980b).Thedeveloped impedancetubealongwiththeentiresetupisshowninFigure1.

2.1. Methodology

In theproposedmethodology,asinusoidalsoundsourceis usedthatgeneratesplanewavesinthetube.Thesoundpressure ismeasuredattwolocationsincloseproximitytothesample.

Thecomplexacoustictransferfunctionofthetwomicrophones signalisdeterminedandthenusedfor calculatingthe normal – incidence complex reflectionfactor(r),normal – incidence absorptioncoefficient(α)andimpedanceratioofthetestmate- rial.Itisfoundthatthefrequencyrangedependsonthediameter ofthetubeandthedistancebetweenthemicrophonepositions.

The normal – incidence complex reflection factor can be calculatedusingtheformula

r= r e^j∅r = H12−HI

HR−H12e^2jk⁰^x¹ (1)

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Fig.2.Impedancetubeschematic.

wherex1,distancebetweensampleandthefarthermicrophone location;Φr,phaseangleofnormalincidencereflectionfactor;

H12,transferfunctionfrommicrophoneonetotwo,definedby complexratiop2/p1=S12/S21.HR&HIarethetransferfunction fortheincidentwaveandtransferfunctionforreflectedwave.

Thesoundabsorptioncoefficientcanbecalculatedas:

α=1−|r|²=1−r²1−r²i (2)

2.2. Tubefabricationfortransfer–functionmethod

Theimpedancetubeismadefromarigid,smooth,non-porous walls.Toavoidvibrationresonanceintheworkingfrequency range,thewallsofthetubearemadethickandheavy.Thetube ismadefrombrassandtheholesfor microphonesaredrilled totherequiredtoleranceof±0.2mm.Tosealthemicrophone hermetically,rubbersealsareinsertedattheholeswheremicro- phonesareplaced.Arigidplungerisfittedattheendofthetube tobackthesamplematerial.Therigidplungerismadeofmild metalwithathicknessof23.8mm.Amembraneloudspeaker isconnectedattheoppositeendofthetube.Thesurfaceofthe loudspeakermembranecoverstwo-thirdsofthecross-sectional areaoftheimpedancetube.Loudspeakeriscontainedinaninsu- latingboxinordertoavoidairborneflankingtransmissiontothe microphonesasshowninFigure2.

2.3. Tubecalculationfortransferfunctionmethod

Forsoundabsorptioncoefficientmeasurement,thegenerated soundsignalshallbeaplanewave.Therefore,thetubehastobe longenoughtoensurethedevelopmentofplanesoundwaves betweensoundsourceandthesample.ThestandardISO10534- 2recommendsatubelengthofatleastthricethediameter,as non-planewaveswilldisappearatthatdistance.Alengthofat least 10–15 times the diameters ispreferred (Seybert, 2010).

The upper workingfrequencyis chosen insuch a wayas to avoidtheoccurrenceofnon-planewavemodepropagationandto

assureaccuratephasedetection(RYU,n.d.).Theupperlimiting frequencyfucanbecalculatedfromthefollowingcondition.

D<0.58λu (3)

Theinnerdiameter oftubefor thisdesignis d=104.8mm which gives the upper liming frequency of fu=1.8kHz. The limitingfrequencydependsonthedistancebetweenthemicro- phones,s0,asdepictedinFigure2,whichshouldexceed5%of thewavelengthcorresponding tothelowerfrequencyinterest.

Therefore,s0determinesthelowerlimitingfrequencyflusing thefollowingcondition.

s0>0.05λl (4)

Additionally,thefollowingconditionhastobesatisfied.

f_us0<0.45c0 (5)

Fortheselecteddistances0=30cm,thelowerlimitingfre- quencyisfl=58Hz.Alargerspacingbetweenthemicrophones hasbeenchoseninordertoenhancetheaccuracyofmeasure- ment(Suhanek,Jambrosic,&Domitrovic,2008).Thespacing betweenthesoundsourceandmicrophonexaccordingtoISO 10534-2shouldbe

x>3d>314.4mm (6)

For thisstudy, the spacing between the soundsource and microphone istaken as x=720mm.The distancex2 between the test sample andthe microphone nearest toit depends on thetypeofsample.Tomeetthehighlyasymmetricalnatureof sample,thevalueischosenasbelow.

x2=2d=209.6mm (7)

Finally,thetotaltubelengthisgivenby

l=x2+x+s0=1420mm (8)

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800 0.615 0.666 8.12 0.143 0.144 0.70 0.359 0.382 6.41

1000 0.562 0.581 3.36 0.177 0.181 2.26 0.456 0.459 0.66

1250 0.526 0.531 0.88 0.224 0.228 1.79 0.579 0.582 0.52

1600 0.473 0.481 1.50 0.297 0.306 3.03 0.732 0.739 0.96

Averageerror 8.41 4.05 3.29

3. Systematicandrandomerrorsinmeasurement

Errorsinanimpedancetubecanbeclassifiedintointrinsic andextrinsicerrorsbasedonthesourceof theerror.Intrinsic errorsariseduetotheinternalpartsoftheimpedancetubeinclud- ing microphone error, specimen mounting error, loudspeaker distortion andductresonance. Extrinsic errorsare duetothe instrumentsusedtoaidthemeasurementofacousticimpedance likesignalsource error,powersupplyerroranderrorsdueto electroniccircuitry. The influenceof measurement apparatus, relatedinstruments,andthesamplematerial’sbehaviourcauses systematic errors in the readings. Systematic errors include errorsintrinsictoimpedancetubeoperation,errorsintrinsicto testspecimenmountinganderrorsintrinsictothemeasurement instrumentation. It may be due to the attenuation within the tube, probesandstructure-bornevibration.Errors inthe electroniccircuitry alsoadds to thesesystematic errors.Random errorsarealsointroducedinthereadingduetoenvironmental andoperationalconditions.

Thermalconductionintheacousticboundarylayernearthe tubewallattenuatesthewavemotionanddecreasesthestanding waveratio.Backingcavityalsoplaysaroleinthemeasurement ofimpedance.Theimpedancemeasuredincludestheinteraction oftestspecimenaswellascavityimpedance.Systematicerrors canbeavoidedbyadjustingthecavitydepthforzeroreactance.

Thepressureprobehasafinitesizeandhenceintroducesasys- tematicerrorduetodeviationfromtheRayleighendcorrection valueof0.85rp.Itisthusrequiredtodeterminetheendcorrec- tionbyexperimentalmethodbeforemeasurement.Mechanical vibrationsofthetubealsointroducesystematicerrors,especially atlowvaluesofpressure.

4. Errordependencies

Theinaccuraciesinatubeareduetoanumberoffactors.

Butsomefactorsmayberepetitive.Someoftheerrorscanbe

correctedusingcalibrationwhileothererrorsmayvarywiththe operatingconditionsoftheimpedancetube.Theseerrorsmay require dynamiccalibrationandcanbedifficulttocorrect.IN thispaper,inordertoimprovetheaccuracyoftheresults,the dependencyofaccuracyonmeasurableparametersisanalyzed.

Itisobservedthatthequantityoferrorvarieswiththefrequency oftheinputsignal.Theimpedancetubeproducessimilarmea- surementsforamaterialatagivenfrequencyoftheinputsignal.

Hence the inaccuraciesinthe measurements are attributedto thenatureofthetube,electroniccircuitandothercomponents ofthesystem.Theaccuratemeasurementisafunctionofmea- suredvaluefromtheimpedancetubeanderrorfactorasinEq.

(9).

Error=α−αcor (9)

whereαcoristhedesiredaccuratemeasurementascompared withB&Kimpedancetubeαisthemeasurementfromastandard impedancetubewhichisacceptedtobeaccurate.Severaltest materialsarecollectedandtheirsoundabsorptioncoefficients ascalculatedfromstandardB&Kimpedancetubeandthedevel- opedimpedancetubeandfewsamplesareshowninTable1.

HereAisthe measurementfromthe developedimpedance tubeandBisthemeasurementfromstandardB&Kimpedance tube.Itisobservedfromthetablethattheerrorvarieswithfre- quencyoftheinputsoundwaveandalsothenatureofmaterial, asdifferentmaterialsshoweddifferentaverageerror.

Sinceitisobservedthattheerrorproducedbythesystemhas somedependenciesonthefrequencyoftheinputsoundwave andtheabsorptioncoefficientofthematerialweformulatethe errorfunctionasinEq.(10)

Error=function(f,α)+c (10)

wherecistheconstanterrorinthesystem.Theconstanterrors inthesystemcanbeeradicatedwithcalibration.Buterrormay alsohavelinearandpolynomialdependencieswithfrequency andalso the actualsoundabsorptioncoefficient, αcor.Hence

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Fig.3.PSOmovement.

Eq.(10)canbewrittenas

Error=c+d1f+d2f²+d3f³+···+e1αcor

+e2αcor²+··· (11)

IgnoringthehigherordersinEq.(11),

Error=c+d1f +d2f²+e1αcor+e2αcor² (12) Eq.(12)isthemathematicalmodel forerrorinthesystem which is composed of a constant error, linear and quadratic dependencieswithfrequencyandabsorptioncoefficientmea- surementsfromtheimpedancetube.Ourobjectiveistominimize theerrorinmeasurements.Whilemakingimprovementsinthe hardware quality is costly, solving Eq. (12) can gives us an alternateapproachwhichdoesnotinvolveanycost.

5. Softcomputingbasederrorcorrectionsystem 5.1. PSObasedsystem

Thesolutiontoacomplexequation withmanynumbersof unknownscanbesolvedbyanumberofmethods.Inthispaper, ameta-heuristicmethodandneuralnetworkmethodareusedto solvetheequation.ParticleSwarmOptimizationisanintelligent algorithmwhichmimicsthebehaviourofbirdsflockingtowards asourceoffood.Thealgorithmconsistsofentitiescalled“parti- cles”whicharedefinedbytheirco-ordinatesinthesearchspace.

Inourproblem,thesearchspaceisoffivedimensionsandeach positionprovidesadifferentvalueonsubstitutingintotheobjec- tivefunction.Meta-heuristicmethodconsiderserrormodelling througherrorminimizationoptimization,whileneuralnetworks approachtrains theneuronstomimicthestandardimpedance tube.ParticleSwarmOptimizationischosenbecauseofitscapa- bilitiesinsolvingcomplexnon-linearsystemsandisusedfor correctingerrorsindifferentapplications.InPSO,theparticles moveaccordingtotheirvelocitiestodifferentpositionsinthe searchspace.Duetotheirmovement,thepositionco-ordinates arealteredandthefitnessvalueofeachparticlechanges.This happens inan iterative process.These velocities are updated (Fig.3)in everyiteration accordingto the mathematicalEq.

(13).

Vnew=Vold∗w+c1∗rand∗(LB−Xold)

+c2∗rand∗(GB−Xold) (13)

whereVnewisthe velocityofthe particleinthe currentitera- tion,Voldisthevelocityoftheparticleinthepreviousiteration, c1,c2 andwaretunableparameters,LBislocalbestsolution oftheparticleandGBistheglobalbestsolutionoftheentire swarm.Thelocalbestisthebestsolutionobtainedbyanindivid- ualparticlethroughouttheiterations.Theglobalbestisthebest solutionobtainedyetbythealgorithm.Fromtheaboveequation itisobservedthattheparticlesmovetowardsbettersolutionsin aswarmingnature.Astheparticlesmoveinthesearchspace, newsolutionsareobtained.Foreverynewsolution,thefitness valueofthesolutionisfoundinordertodetermineifthenew solutionisbetterorworsecomparedtotheprevioussolutions.

ThefitnessfunctionisderivedfromEq.(12)as Min function(c,d1,d2,e1,e2)

=Min

_N

i=1

Error(i)−c+d1fi+d2fi2+e1αi+e2αi2

(14) whereNisthetotalnumberofsampledataandiisthesample number.Alsotheinputvariablesareconstrainedinthepenta- dimensionalsearchspaceasgiveninEqs.(15)–(19).

cmin>c>cmax (15)

d1min>d1>d1max (16)

d2min>d2>d2max (17)

e1min>e1>e1max (18)

e2min>e2>e2max (19)

Attheendofalliterations,theglobalbestsolutionisobtained which is used for compensating the error. The steps for the processareasfollows

STEP1:Thefollowingparametersareinitializedforopti- mization

(i) Searchspacedimension (ii) Numberofparticles

(iii) Maximumandminimumlimits (iv) Maximumnumberofiterations

STEP2:Theparticlesaredistributedrandomlywithinthe searchspace

STEP3:Thefitnessvaluefor alltheparticlesarefound usingthefitnessfunction

STEP4:Thelocalbestandglobalbestvaluesarefound fromtheinitialsolution

STEP5:Ineachiteration,

(i) ThevelocityoftheparticlesisupdatedusingEq.(13) (ii) Thenewpositionsoftheparticlesisfound

(iii) The newpositionsare checked for adherenceto the constraints

(iv) Thefitness valueof the newposition isfoundusing fitnessfunctionasinEq.(14)

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(vi) STEP6:Aftercompletionofalltheiterations,theglobal bestsolutionisobtainedandisconsideredforerrorcom- pensationmodel.

PSOhasseveralsimulationparameterswhichwilldefinethe movementoftheparticlesthroughthesearchspace.Hencethe idealparametersforoptimizationshouldbefoundbyrepeated optimizationwithdifferentsimulationparameters.

5.2. Neuralnetworkbasederrorcorrector

ArtificialNeuralnetworks(ANN)consistofgroupsofenti- tiescalledneuronswhichmimictheoperationsofabiological neuron. These artificial neurons are interconnected and they possesstheabilitytolearnandrecognizepatterns.Theiractiv- ityissimilartothehumanbrain.AtrainedANNcanrecognize patternsthat aretoocomplex forhumans andothercomputer techniques.The trainedANNcanprovideresultsto“whatif”

scenarios and predict results when sufficient information is providedfor analysis.In thebrain, knowledge isacquired by formingnewinterconnectionsbetweentheneurons.Everynew memorytriggersanewconnectioninthebrain. The stronger theconnection,thebetterthememory.Usuallytheconnection strengthisimprovedbyrepeatedtraining.Inthesameway,the artificialneuronsareinterconnectedinacomplexnetworkand thenetworkisalteredwithnewexperiences.ANNconsistsof numberoflayersofindividualneurons.Theneuronsinasingle layercommunicatewiththeneuronsintheneighbouringlayers.

Thenumberoflayersalsodeterminestheextentoflearningof thenetwork.Morelayerswillfacilitatemorerobustnetworkbut itslowsdownthelearningprocess.Newlearning’sarestored as weighting factorin the connections. Theseweightingfac- torsarecorrectedthroughatechniquecalledbackpropagation whichisafeedbackmechanism.Thepatternrecognitionability iftheANNdifferswiththetypeofneuralnetworkandfeedback mechanism.Severalclassesofneuralnetworksareconsidered forourproblem.

5.2.1. Feedforwardnetwork(FFN)

ThisisthesimplesttypeofANNwheretheinformationflows inonlyonedirection,andtherearenotanyloopsinthenetwork, asshowninFigure4.Thefirstlayeriscalledastheinputlayer becauseitreceivestheinputvaluesandthelastlayeriscalledthe

outputlayerbecauseitprovidestheresultsoftheneuralnetwork.

Thelayersinthemiddlearecalledhiddenlayers,becausethey representthe internalstates ofthenetworkandareconcealed fromtheinputsandoutputs.Eachneuronconnectionisinitial- izedbyarandomweightwhichaffectthedataflowingacross theneurons.Theweightsareadjustedinafeedbackmechanism in an iterative process called back propagation. The iterative processcontinuesuntilsufficientaccuracyisreached.

5.2.2. Radialbasisfunctionnetwork(RBFN)

Radial BasisFunction networksaresimilartoFFNexpect thattheyhavearadialbasisfunctionforactivation.Activation functiondeterminestheoutputoftheneuron,andisdetermined byaradialbasisfunction.Theoutputofthenetworkisacombi- nationofparametersoftheneuronandtheinputtotheneurons.

Eachneuronhasaprototypevectorwhichiscomparedwiththe inputoftheneuron.Theradialdistancebetweenthetwodeter- minestheoutputoftheneuron.Thevaluedropsexponentially asthedistanceincreases.

5.2.3. ProbabilisticNeuralNetwork(PNN)

AProbabilisticNeuralNetworkisafeedforwardneuralnet- workwhichhasastatisticalalgorithmcalledkerneldiscriminant analysisandhasanadditionallayercalledsummationlayerin additiontothepreviousthreelayers(Fig.5).PNNshowsfaster learningwhencomparedwithstandardFFNandhasinherently parallelstructure.

Themeasurementfromthedesignedimpedancetubeandfre- quencyarefedastheinputstothenetwork.Theperformance oftheneuralnetworkisdependentonthetotalnumberofinput datasets.Thedatasetsarecollectedbyrepeatedexperimenton theimpedancetubewithdifferentmaterials.

5.3. Testingandvalidation

The simulation parameters chosen for PSO are given in Table2.

Since,themethodisastochasticmethod,thereisboundto beminordeviationintheresultsandruntime.Hencetheopti- mizationisrepeatedformanytimeswithdifferentparameterset andthebestvalueoutoftheseoptimizationsisconsidered.The

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Table2

SimulationparametersofPSO.

S.no Parameter Value

1 Numberofparticles 50

2 Numberofiterations 100

3 c1 1.5

4 c2 2.5

5 w 0.3

Table3

PerformanceofPSOwithinputdataset.

c1=1.5 c1=1 c1=3 c2=2.5 c2=3 c2=1

BaseMSE 0.009405 0.009405 0.009405

PSOmodelledMSE 0.008449 0.008449 0.008451

%Improvement 10.16 10.16 10.14

Meanruntime(s) 4.375 4.410 4.432

Table4

PSOoptimizedsolutionset.

Parameter Value×10⁻⁸

c −0.1202

d1 −0.5415

d2 −0.3512

e1 −0.2530

e2 −0.3218

9.6×10^–3 9.4

9.2

9

8.8

Mean square error value

8.6

8.40 10 20 30 40 50 60

No of iterations

70 80 90 100

Fig.6.ConvergencecharacteristicsofPSO.

errorvaluesafteroptimizationwithdifferentvaluesofc1,c2for inputdatasetisgivenbelowinTable3.

Theabovetableshowsthattheparameterchangesonlyhave averyminorchangeintheperformanceofthealgorithm.This showstherobustnessofthealgorithminsolvingthisproblem.

Theoptimizedvaluesoftheerrorcompensatingparametersare giveninTable4.

Figure 6 shows the convergence characteristics of PSO attaininganearlyconvergenceanditseffectivenessinsolving theproblem.Thestandarddeviationfromthe bestsolutionis almostnonewhichshowsthatPSO reachesglobalminimain almostallthetrials.

Table5

TestdatasetforvalidationwithPSOmodelledsystem.

F(Hz) Soundabsorptioncoefficient

Resinfelt–3000g/m²and45mm

B A ErrorA(%) C ErrorC(%)

80 0.048 0.045 6.25 0.0480 6.67

100 0.049 0.046 6.12 0.0490 6.52

125 0.063 0.059 6.35 0.0629 6.61

160 0.093 0.091 2.15 0.0929 2.09

200 0.125 0.122 2.40 0.1249 2.38

250 0.167 0.164 1.80 0.1668 1.71

315 0.229 0.222 3.06 0.2286 2.97

400 0.302 0.299 0.99 0.3014 0.80

500 0.391 0.387 1.02 0.3901 0.80

630 0.509 0.505 0.79 0.5076 0.51

800 0.635 0.632 0.47 0.6327 1.11

1000 0.775 0.762 1.68 0.7715 1.25

1250 0.889 0.885 0.45 0.8835 0.17

1600 0.965 0.961 0.41 0.9560 0.52

Averageerror 2.42 2.34

×10^–3 4

3.5

2.5

1.5

0.5 3

2

1

0 10

Mean error value

20

Number of intermediate values

30 40

Fig.7.Performanceofneuralnetworkwithvaryingnumberofdatasets.

ThesolutionsetgiveninTable4isusedtocorrecttheerror inthesystemtoacertainextent.Inthiswaytheoverallaccu- racyoftheimpedancetubeisimproved.Theresultswithatest dataisgiveninTable5.HereCisthe PSOaided errorcom- pensatedmeasurement.CiscalculatedusingEqs.(9)and(12).

FromTable5itisclearthattheaverageerrorhasdroppedby nearly37%.Thisshowstheeffectivenessoferrorcompensation techniquewithameta-heuristicapproach.

Totestusingneuralmethod,theinputdatasetisexpandedby makinglargenumberofmeasurementsatintermediatefrequen- ciestoobtainsufficienttrainingoftheneuralnetwork.Figure7 showstheeffectivenessofneuralnetworkmethodinerrorcom- pensationfordifferentnumberofinputdataset.

Neuralperformanceimproveswithincreasedinputdataset.

Theneuralnetworkistrainedwiththeexpandeddatasetandthe trainingparametersarechosenasinTable6.

Withtheabovetrainingparameters,theperformanceoftrain- ing,validationandtestingwereobservedas0.168972,0.11176 and0.006083respectively.Figure8showstheerrorhistogram

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Instances 300 200 100 0

–0.06065 –0.05577 –0.05088 –0.04112 –0.03623 –0.03135 –0.02646 –0.02158 –0.01669 –0.01181 –0.00692 –0.00204 –0.002844 –0.007728 0.01261 0.0175 0.02238 0.02727 0.03215–0.046

Errors

Fig.8.Errorhistogramoftrainedneuralnetwork.

Best validation performance is 0.011176 at epoch 76 10³

10²

10¹

10⁰

10^–1

10^–2

10^–3 0

Sum squared error (sse)

10 20 30 40 50

82 epochs

60 70 80

Train Validation Test Best

Fig.9.Performanceofneuralnetworktraining.

Fig.10.Neuralnetworkvalidation.

andFigure9showstheperformanceoftheneuralnetworktrain- ingwhileFigure10displaysthevalidationofneuralnetwork.

Withtheexpandedinputdataset,theneuralnetworksmod- ellederrorcompensationproducedanaverageerrorof0.0013 improvingtheaccuracyoftheimpedancetubeby69.7%.When the trainedneural networkwasused tocompensatetheerrors oftheimpedancetubewithatestdataset,theresultwasworse comparedtothe systemwithouterrorcompensation.Thetest dataistestedwiththethreetypesofneuralnetworkaidederror compensatorandtheresultsareshowninTable7.FromTable7 it canbeinferred thatthe systemwithRBFN neuralnetwork hastheleasterrorratewhencomparedtootherneuralnetworks.

However,neuralnetworkhasnotprovidedbettererrorcorrec- tionforthesampleunderstudy.Itisclearthattheaverageerror hasincreasednearlytwice.Henceneuralnetworksrequirelotof inputdatasetsforeffectiveoperation,whichiseffortconsuming.

A comparison of PSO anddifferent neural network modelling reveals that PSO provides gooderror compensation at

Table7

Testdatawithneuralnetworksmodellederrorcompensation.

Resinfelt–3000g/m²and45mm

B A ErrorA FFN ErrorFFN RBFN ErrorRBFN PNN ErrorPNN

80 0.048 0.045 6.25 0.0416 7.56 0.0482 0.42 0.0487 1.46

100 0.049 0.046 6.12 0.0435 5.43 0.0545 9.18 0.0550 12.2

125 0.063 0.059 6.35 0.0539 8.64 0.0586 6.98 0.0585 7.14

160 0.093 0.091 2.15 0.0765 15.9 0.0826 11.1 0.0815 12.3

200 0.125 0.122 2.40 0.1035 15.1 0.1081 13.5 0.1065 14.8

250 0.167 0.164 1.80 0.1421 13.3 0.1758 5.27 0.1792 7.31

315 0.229 0.222 3.06 0.2039 8.15 0.2165 5.46 0.2189 4.41

400 0.302 0.299 0.99 0.2855 4.52 0.2940 2.65 0.2519 16.5

500 0.391 0.387 1.02 0.3896 0.67 0.3284 1.79 0.3165 19.0

630 0.509 0.505 0.79 0.5291 4.77 0.4999 1.79 0.5121 0.61

800 0.635 0.632 0.47 0.6723 6.38 0.6151 3.13 0.6230 1.89

1000 0.775 0.762 1.68 0.8200 7.61 0.7544 2.66 0.7488 3.38

1250 0.889 0.885 0.45 0.9121 3.06 0.8870 0.22 0.8991 1.14

1600 0.965 0.961 0.41 0.9274 3.50 0.9942 3.03 0.9630 0.21

Averageerror 2.42 7.43 4.81 7.33

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Fig.11.Comparativeanalysis.

Table8

Outputwithdifferenttestsamples.

Glasswool–800g/m²and40mm Softfelt–40mm+hardfelt–5mm Resinfelt–2000g/m²and30mm+2mmHL Errorwithout

compensator

Errorwith PSO

Errorwith RBFN

Errorwithout compensator

Errorwith PSO

Errorwith RBFN

Errorwithout compensator

Errorwith PSO

Errorwith RBFN

80 04 4.00 9.05 0.3 0.29 0.66 0.3 0.30 0.08

100 04 4.00 7.10 0.8 0.79 0.13 0.8 0.80 0.16

125 03 2.90 2.40 0.2 0.19 0.58 0.2 0.19 0.98

160 04 3.90 1.80 0.2 0.19 0.57 0.2 0.19 2.14

200 03 2.90 6.30 0.4 0.38 0.31 0.4 0.39 2.47

250 01 0.80 9.80 5.0 4.97 2.70 5.0 4.98 2.47

315 03 2.70 9.20 4.0 3.96 1.18 4.0 3.96 2.57

400 02 1.40 6.60 0.5 0.44 1.84 0.5 0.44 0.27

500 02 1.10 1.00 1.0 0.91 1.38 1.0 0.91 0.41

630 13 11.6 13.6 0.9 0.76 0.20 0.9 0.76 0.75

800 03 0.70 0.20 3.9 4.12 2.74 3.9 4.13 1.98

1000 03 0.50 7.70 2.0 1.64 1.80 2.0 1.65 1.32

1250 02 3.50 3.40 4.2 4.74 1.55 4.2 4.75 5.83

1600 04 5.00 54.2 5.8 4.90 3.23 5.8 4.90 2.49

Averageerror 3.6 3.2 9.5 2 1.94 1.35 2.08 2.02 2.71

Table9

Performanceofthealgorithms.

S.no. Method MSE

1 Meansquareerrorbeforemodelling 0.004286

2 MeansquareerrorafterPSO-modelling 0.003498 3 MeansquareerrorafterFFN-modelling 0.020267 4 MeansquareerrorafterRBFN-modelling 0.016585 5 MeansquareerrorafterPNN-modelling 0.019413

allfrequencies,whileneuralnetworkperformsonlyatthemid frequencyrange.Atotherfrequencies,theerrorishigherthan theoriginaldataasevidentfromFigure11.Similarexperiments wereconductedonothermaterialsandtheaccuracyoftheoutput increasedbyasignificantnumber,atallfrequencies(Table8).In thetable,HLindicatescotton/linenblendakaHalbleinen.With neuralnetworkmodel,theerrorisreducedforsoftfelt,whereas increasedforothermaterials,andhenceisnotareliablesystem.

PSOmodelledsystemprovidesbetteraccuracyforallsamples andhenceisrobustandreliable.

Similartestswereconductedwith20samplesandtheaverage meansquareerror(MSE)obtainedforeachmethodisgivenin Table9.

6. Conclusion

A mathematicalmodel based error correctionsystem was developed withan intelligent error compensator tominimize errorsinaimpedancetube.PSOandthreetypesofneuralnet- workmodelsnamely,feedforward,RBFNandPNNwereused totestthesystemwithdifferenttestsamples.PSOfaredwellin termsofreductioninerroratallfrequencies,whileneuralnet- workbasedsystemsreducedtheerroronlyatlimitedfrequency ranges.Thisisattributedtothelargetraining datasetrequired forneuralnetwork.Ofthethreetypesofneuralnetworkmodels, RBFNprovidedbettererrorcorrectioncomparedtoothertwo

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