Performance analysis of collaborative spatio-temporal processing for wireless sensor networ

PerformanceAnalysisofCollaborativeSpatio-TemporalProcessingforWirelessSensor

Networks

‡C.Fischione,†A.Bonivento,†A.Sangiovanni-Vincentelli,§F.Santucci,‡K.H.Johansson

‡KTH,†U.C.Berkeley,§UniversityofL’Aquila

e-mail:‡{carlofi,kallej}@s3.kth.se,†{alvise,alberto}@eecs.berkeley.edu,§santucci@ing.univaq.it

Abstract—Spatio-Temporalprocessingisacontroltechniquetoincreasethequalityofthereceivedsignalsinwirelessnet-works.Outageeventshaveastronginfluencenotonlyontheperformanceofthephysicallayer,butalsoonrouting,MAC,andapplicationlayer.Inthispaper,weproposeanoutagebasedperformanceanalysisofcollaborativeSTPforWSNs.Afteranaccuratecharacterizationofthewirelesschannel,wederivetheoutagestatisticsasfunctionoftheSTPcoefficients,andinvestigatetheeffectsofSTPontheprobability,averagedurationandrateoftheoutageevents.Furthermore,weshowthatapropercontrolpolicyoftheSTPcoefficientscanbederivedaccordingtotherequirementsfromtheapplicationsandWSNscommunicationlayers.

IndexTerms:WirelessSensorNetworks(WSNs),Spatio-Temporalprocessing(STP),Fading,Outage,LevelCrossingAnalysis.

I.INTRODUCTION

Significantperformanceimprovementscanbeachievedifnodesinwirelessnetworkscooperate.Collaborativedistrib-utedspatio-temporalprocessing(STP)issuchatechnology,whichinthecontextofcellularradiosystemandad-hocwirelessnetworkshasbeenanareaofintenseresearch(seee.g.[1]).Forwirelesssensornetworks(WSNs),however,constraintsonpower,memory,informationprocessing,etc.,maketheextensionoftraditionalSTPtechniquestoWSNsdifficult.STPtechniquesforWSNsshouldalsotakeintoaccountcross-layermechanisms,whicharecrucialformanysensingandcontrolapplications[2],[3].

Somerecentcontributionscanbefoundinliterature.In[4]theauthorsinvestigatethepossibilityofshapingtheradiationdiagramusingarandomdistributionofnodes.Specifically,theauthorsshowthatusingrandomlyplacednodes,itisstillpossibletoobtaingoodradiationdiagram.However,theanalysisisbasedonanumberofidealassumptions,particularlyasimplemodelofthewirelesschannel.In[5]and[6],aninterestingframeworkisaddressed,wherenodesaregroupedinclustersandcollaboratetoreceiveandtrans-mitinformationusingadistributedalgorithmforthephasesynchronization.Moreover,theauthorsproposetheuseofanSTPtechniquebasedonMaximumRatioCombiningscheme,wherethenodesaresupposedtoperformanaccuratechannelestimation.In[7],amaximumlikelihoodapproachisproposedforthecommunicationbetweenasourceandadestinationwithmultiplerelays,andtheoptionsofrelayandforward,anddecodeandforwardareinvestigated.Aframeworkforperformanceanalysisinacellularradiosystem,butinterestingalsofromaWSNsperspective,canbefoundin[8],where

WorkdoneintheframeworkoftheHYCONNetworkofExellence,contractnumberFP6-IST-511368.

aMaximumRatioCombiningspatialdiversityschemewithcompositeRayleigh-Lognormalchannelmodelisstudied.Outageseventscanhavestronginfluenceontheperfor-manceofupperlayersintheprotocolstack,e.g.,[9],[10],[11].Despitetheirimportance,outageseventsforWSNshavebeenconsideredintheliterature[12]onlyrecently.Infact,itiswellknownthatoutagestatisticsareimportantperformancemeasuresusefulfortheoptimizationoftheMAC,time-slotduration,packetlength,etc.OutagestatisticsarealsoimportantfromtheperspectiveofapplicationsutilizingtheWSNs.AnimportantexampleiswhenaWSNisusedtogatherinformationforreal-timecontrolofaplant.Thestabilityoftheclosed-loopcontrolsystemmayrequireoutagesrateslowerthanacertainfrequencyrelatedtothesamplingfrequencyofthesensorsandactuatorsaswellasthedynamicsoftheplant.ThemaincontributionofthispaperisaninvestigationofSTPperformanceforWSNsintermsoftheoutagestatistics.WeprovideanaccurateframeworkfortheabstractionofthepropertiesofthephysicallayerwithcollaborativeSTP,takingintoaccountalltherelevantparametersthatmayinfluencetheupperlayerprotocolsandapplications.Afteraccuratemodellingofthephysicallayer,westudytheSTPperformancewithrespecttotheoutageevents.Specifically,theoutageprobability,theoutagerateandaverageoutagedurationareexplicitlyderivedasafunctionoftheSTPcoefficients.WeaimatdefininganoptimalcontroloftheSTPparameters,whichshouldbescalablewithrespecttoapplication,MAC,routingandpropagationconditionsofthephysicallayer.Ourapproachdiffersfrom[5]and[6],becausewefocusontheoptimizationofoutageperformanceinsteadofoptimalshapingtheequivalentradiationdiagramoftherandomsensorarray.Forreal-timeapplications,theoutageprobabilityisamorereasonablemeasurethanthebiterrorprobabilityanalysisaddressedin[7].Moreover,noneofthesecontributions,aswellas[8],takesintoaccountthepresenceofacorrelationstructureamongwirelesslinks,thatmayhavestrongeffectsontheperformance.

Therestofthepaperisorganizedasfollows:afteradescriptionofthesystemmodelandthewirelesschannelforWSNswithcollaborativeSTPinSectionII,inSectionIIItheoutagestatisticsarederived;inSectionIVasketchofpossibleoutagebasedSTPisoutlined;inSectionVnumericalexamplesarediscussed,and,finally,inSectionVIconclusionsandfuturedevelopmentsaregiven.

II.SYSTEMMODEL

ConsiderthescenarioinFig.1.Wehaveamovingsensornode,calledMS,thatbroadcastsasignalassociatedtoarealtimeapplication.ThereisaclusterofNsensornodesthatreceivethebroadcastedsignalandretransmitittoaSensor

Collectorinformation(SC).whereassociatedForexample,toantheinvertedMScouldpendulumtransmitasthestatusthecontrollerthecommunicationattheSCbetweentheactuatorattheMSin[2],andwhereWeassumethattherelayinghappenssensorthroughnodestheareWSN.

partofaWSNsourceseveraltechniquecoding.otherTherefore,techniquesSTPcanmaybebeseenimplemented,acomplementaryase.g.signalThereisrunningnodirectoverlinkthebetweenrelayingthesensorMSnodes.

andtheweighthasreceivedthetoreceivedpassthroughsignalstheSC,thetorelayingnodesthatcooperativelychannels,bytheSC.Thesensorimprovenodesthetransmitqualityusingofthedifferentsignaltimesignalsslotsfororexamplecodes.theyWeassumecouldusethattwothedifferentretransmissionfrequencies,frequencyThehappenslinksbetweenwithnotheappreciableMSanddelay.

oftherelayingnodesareandfixedfastfading.flat,timeWeassumevariantandtherelayaffectedbypathloss,shadowdescribingandinsensornodesandtheSCathevisibility.linksbetweenConsequently,thechannelcoefficientstogoodIndustrial,theoneapproximation.consideredinThisthem[5],setandupareareisassumedrepresentativebasicallyknownequivalentwithofthearecommonlyScientific,employedandinMedicalWSNs(ISM)[13].

transceivers,whichMoving Sensing NodeRelay Sensor NodesSensor CollectorFig.1.SystemoverviewoftheWirelessSensorNetwork.

fromTheautomatictheSCvariousreceivessimultaneouslytheretransmittedsignalsofsignalaBPSKwaysensornodes,andthereforeperformsinisexpressedmodulation,thesumofasfollows:

thethecomplexsignals.envelopeUndertheoftheassumptionreceivedy(t)=

󰀈Nwi[hi(t)s(t)+ni(t)]+n(t),(1)

i=1

wherecoefficients,wi,i=1,...,Narethecomplexinthewandforthesakeofnotationvaluessimplicity,oftheweweighting

includeiwirelessSC;alsohtheknownchannelgainsbetweenthenodesandnoiselinki(t),betweeni=1...,theN,isMSthechannelcoefficientoftheGaussiancomponentatthenodeiandsensori;ni(t)istheσ2noise(AWGN),withpower,andspectralisassumeddensityasgivenwhitebypresenti;finally,σ2atthen(receivert)includesofthetheSC,interferenceshavingpowerandspectralAWGNdensitynoisearen.Notethattheweightingcoefficientsw=[w1,w2,...,wshadowingThesetchannelbytheSTPalgorithm.

N]amultiplicativeandcoefficientfastmodelfading.hasfollowsWei(t)expressiscomprehensivesuchacoefficientofpath-loss,usinghi(t)=gi(t))takesintoaccountpath󰀁

[14]:

Li(t),(2)whereLi(tlossandshadowing,while

gi(t)=|gi(t)|ejφi(t)isthefastfading.Acommonassumption

2

forprocess,gi(t)thenisto|gbe(t)|aiscomplexRayleighzerodistributed.meanGaussianMoreover,randomiletusdefineprocess,pi(t)󰀁|gi(functionofwithpparametert)|2,then,)aregivenσpi(t)isanexponentialdistributedaspi.inThetheaveragefollowingand[14]:

correlationi(tE{pi(t)}󰀁mp2

i=2σpi

,(3)E{pi(t)pi(t−τ)}󰀁rp4

i(τ)=8σpiJ0

(2πfmτ),(4)

whereFurthermore,J0(·)isstatisticallywetheassumezeroorderthatBesselgfunctionofthefirstkind.

i(t)andgj(t)Theshadowingindependent.

,withi=j,arecomponentLi(t)hasalog-normaldistri-butionprocessLi(t)=eBi(t),whereBi(t)componenthavingisa2

GaussiantheexhibitsaveragemBiandvarianceσBi

.Theshadowingrandom

ofin[15]:

thepropagationautocorrelationscenario.acorrelationandWestructurethatisdependentoncross-correlationresortheretothefunctionsgeneralderivedmodel2

CτB(τ)=σ2−1τ2B

iBi

i

e,

(5)

1τ2

CστBBB

iBj(τ)=iσBjAcos[ϑij(t)+B]e

−2i

,(6)

where(measuredτBifrominisseconds),thede-correlationϑconstantoftheshadowing

ij(t)isFig.theMSandthenodei,andthetheangleMSandbetweenthenodethejlinks(seeThe2),environment.valuesandforAandConsequently,AandBareBtwoareconstantssuchthatA+B≤1.thedependentaverageandontheautocorrelation

propagationFig.between2.Thethecorrelationlinktowardstructurethenodeofjtheandchannelthenodegainsi.

isfunctionoftheanglefunctionofLi(t)canbeexpressedasfollows:

E{Li(t)}=emBi+21

σB2

i,

(7)

E{Li(t)Lj(t−τ)}=e

mBi+mBj+12σ2

B2i

+σBj

+2CBiBj(τ)

,andAfteramongkeepingacoherentinmindreceivertheindependencematchedtoofthe(8)

thetransmittedRayleighsignal+differentnodes,weexpresstheSignaltoInterferencefading[8]):

NoiseRatio(SINR)correspondingto(1)asfollows([14],SINR(t)=Ewhere󰀇sNi󰀇N

i=1|wi|2ri(t)

,(9)=1

|wi|2σi+σn|Esistheenergyofthetransmittedsignal,whileri(tassumedh(t)|2.Notethat(9)dependsontimesincethefading)=iismeaningtobetimevariant.Also,notethat|wi|2hasthefallsTheprobabilitybelowsignalofrelayingofaminimum(1)ispower.

saidtheoutagequalitytobeeventsthresholdinoutageisdefinedγ.ifasInthefollows:

particular,SINR(9)thePout=P(SINR(t)<γ).

(10)

III.STPOUTAGESTATISTICS

trivial.DerivingthestatisticsoftheSINR(9)isincombinationHowever,canofRayleigh-LognormalsincetheSINRisexpressedgeneralasalinearnonwell-knownrelyuponexistingresults[10],[16].randomWeprocesses,resorttowethemethodfor(seeextension[16]and[17]oftheforWilkinsonacomprehensiveMomentanalysismatchingandthethefollowingaccuracyapproximation:

oftheapproach).Specifically,weconsiderSINR(t)∼=eZ(t),

(11)

whereandZ(t)firstcovarianceisaandsecondCGaussianrandomprocesshavingaveragemZmomentsZ(τ),thatofarethederivedasafunctionofthemZ=ln󰀅SINR:

Mm2

1

M2(0)󰀆,(12)

mC=ln󰀅Mm2(τ)

Z(τ)M󰀆.m2(13)1where:

Mm1󰀁E{SINR(t)},

(14)Mm2(τ)

󰀁

E{SINR(t)SINR(t−τ)}.

(15)

expressedTheSINRasfollows:

indB,SINRdB,isaGaussianprocess,andis

SINRdB(t)=βZ(t)

(16)

withasβ=maverageandln10βmcovariancefunctionexpressed,)=β2Crespectively,SINRdB=ZandCSINRdB(τZ(τ),withforwardTheSINR/10.

outagederivationdBGaussianofdistributionallowsforastraight-Specifically,rateFoutageprobabilityPout(w),averageforageneraltheyout(wreferencecan),andbeoutageaveragedurationDout(w).onexpressedLevelasfollows(seee.g.[14]P=√1

2π󰀉CrossingTheory):

∞

out(w)e−y22dy,(17)uFw)=12π󰀂λ2−u2

out(λe2dy,(18)

0

Dout(w)=

Pout(w)

F),

(19)

out(wwhere

u=m󰀁SINRdBC−γdB=βmZ−γdB(20)SINRdB(0)

βσ,

Zλ0=CSINRdB(0),(21)λ−

d2C2=SINRdB(τ)dτ2,(22)

and(15),γdB=10logγ.IntheAppendix,theexpressionsfurtherThe(21)log-normaland(22)areof(14),approximationprovided.

fortheSINRallowsthat,probabilitywhenapproximatedistributionuisnegative,thestatisticsfunctioncanbeoftheoutagesintervals,to[10]describedwithparameterwithaRayleighϕ(w)=

3

u2imatedλ2/4λwith0,and,anexpressionhence,theaveragesimpleroutagewithrespectdurationto(19):

isapprox-Doutapprox(w)=󰀂

π2ϕ(w).(23)

Notemayproblems.

leadthatinto(23)remarkabletherearesimplificationsnotintegralfunctionstosolveofoptimizationw,andthisIV.SKETCHOFOUTAGEBASEDSTP

layerTheoutagestatistics(17)-(19)and(23),togetherproblems,constraints,canbeemployedtoformulateoptimizationwithcross-w.Thesolutionwheretheofthevariablesoptimizationaretheproblemweightingiscoefficientsderivedattheofprobability,theSC,fadingwhereitisnotrequiredthedeterministicknowledgeordernodemoments.butcoefficientsonlytheestimationforthecomputationofthefirstofandthesecondoutagefilters.withbyThee.g.Suchestimationscanbeperformedbyeachestimationsimplerunningoftheaverageorrecursivelow-passoptimization.thenodesquencySincearetransmittedfadingthechanneltoparameterstheparametersSCandusedperformedinthepropagation,correspondingsametheweightstoshouldthecoherencebeupdatedtimevarywithafre-atofleastthewirelessenergyAntimeexamplescale.

withtheoutageconsumptionofoutageofthebasedSTPistheminimizationofthevalueapplications,ofprobabilitythethresholdisnotnodesundertheconstraintthatthecouldlowerthanaminimumthreshold.Thewirelesssystemasforbesetbytherequirementsofthe[1].

dataorvoiceservicesinthirdgenerationtransmittedOncetheintoweightsthenodes.arecomputedThistransmissionattheSC,mighttheybehaveexpensivetobewilltermsofenergyconsumption,delay,etc.Ourfutureworksolutions.

includetheinvestigationofdistributedandsuboptimalV.NUMERICALEXAMPLES

eralIngenerality,frameworkthissection,arenumericalreportedexamplesanddiscussed.obtainedWithwithnoourlossgen-devisethehowwethepresentweightingsomecoefficientsspecificcasesmaybethattunedaretousefulofsatisfytotheWerequired50◦anglesconsiderconstraints.

ϑasystemscenariowithN=3sensors,wherei,jamongthecorrelationand115◦.Withnolossnodesofgenerality,arerandomlywesetchosentheshadowingbetweenoutdoorfrequencychannels.constantsTheAMS=isBassumed=0.5,tothatareassociatedtoTherefore,off.4GHz,andhavingtransmitaspeedwithof1acarrierc=2WenoisesassumethethatmaximumthepowerDopplerfrequencyisf.5m/s.m=10Hz.theoneachnodeisσspectraldensityoftheAWGN1=...=σN=σn=standardsymbolbe2

=deviationenergytonoiseratioEs/NN0,and0issetto10dB.The1.0,i=1of..3the.TheRayleighSINRcomponentsthresholdisaresettoassumedγto

dBσ.pSeei

Tab.IfortheshadowingdB=3EachThecurvesinFig.3-6areplottedsetting.

asfunctionofd1=|w1|2.legendcurvevalueofof0.theisreferredtothevalued2=|w2|2specifiedinthe01figure,whiled3=|w3|2issettotheconstantTab.InbeI,Fig.where3,thefortheoutageanycase.

parametersprobabilityisreportedforthecase1inavaluestrongestuniform,ofdaveragewiththeoftheshadowingareassumedtocomponent.exceptionofLookingthenodeFig.23,thatforexperienceseachd2,a1thatensuresaminimumoftheoutageprobability

Caseparameternode1node2node3mB1.02.01.01σiB1.01.01.0τii0.010.010.01mB1.01.01.02σiB1.02.01.0τii

0.010.010.013mBi,σB1.01.01.0τii0.0010.010.01TABLEI

PARAMETERSETTINGSFORTHENUMERICALEXAMPLES

canofbefound.Moreover,theminimumisthesameforpossibledwithpowers,whiletosetthetheexceptionthevalueoutageofofd0.05.Therefore,itcouldallcases2,2=beprobabilitytheweightingisstillcoefficientsminimized.

withlowinInFig.4,theoutageprobabilityisreportedforthecase2standardTab.I,wherethenode2isassumedtoexperiencealargerforprobabilityFig.3deviationarestillvalid.oftheMoreover,shadowing.Theobservationsmadehigherishigher,andthiscanbetheexplainedminimumoftheoutageeventsstandardlastlonger.

deviationoftheshadowingimplyobservingthatoutagethatFig.Noteaslargerthe3andthattherangevariationoftheoutageprobabilityinchannelFig.4,parameters.isdependentSevereonthechannelSTPcoefficientsconditionsasleadwelltoareIncomponentreportedFig.variation5andoftheoutages.

for6,thetheaverageoutagedurationandoutageratedelay.theLookingofthetonodecasetheresults,13experiencesofTab.I,wheretheshadowingasthereaislargeranincreasede-correlationloweroutagedecreasesoutageraterate.increases,Bythecontrary,whilehighertheaveragevaluesoutageofdofd1,2durationleadtoisofreachedasforbothdd1andd2increase,andacommonminimum1>0.25.Therefore,atrade-offinrequirementstheweightingmappedcoefficientsontooutagecouldberatesdevisedandduration.

accordingthechoicetothe10−1.4ytilibaborP egatuO10−1.5d2=0.05d2=0.09d2=0.13d2=0.17d2=0.2100.10.20.3d10.40.50.60.7Fig.the3.caseOutage1ofprobabilityTab.I.

asfunctionoftheSTPcoefficientsdi=|wi|2

forVI.CONCLUSIONSANDFUTUREDEVELOPMENTS

collaborativeInthispaper,STPanforinvestigationWSNswasofproposed.theoutageTheperformanceapproachofis

4

d2=0.05d2=0.09d2=0.13d2=0.17yd2=0.21tiliba10−1.4borP egatuO10−1.510−1.600.10.20.30.40.50.60.7d1Fig.4.OutageofprobabilityTab.I.

asfunctionoftheSTPcoefficientsdi=|wi|2forthecase2140120100) 1 − s( et80aR egatu60d2=0.05Od2=0.0940d2=0.13d2=0.1720d2=0.21000.10.20.30.40.50.60.7d1Fig.5.OutagerateasfunctionoftheSTPcoefficientsdi=|wi|2case3ofTab.I.

forthebasedasontheadoptionofagoodabstractionofphysicallayeranOurfunctionnecessitiesoptimizationmethodofthecanSTPpolicybeweightingcoefficients.

ofactuallytheusedforthederivationofexample,totheofweightingtheuppercoefficientslayersSTPofalgorithmtailoredtothecouldtheprotocolstack.Fordoensuremovingnotaffectanaveragethecontroloutageofdurationandanbeoutagetunedinrateorderthatfordesiredselectingsensor.theAnothersufficientexampleanapplicationnumberistheusemonitoredofourapproachbytheredundantqualitytheonesand,ofthethus,receivedofnodesthatensureasavingsignal,energy.whileturningoffthesleepingextensionanddisciplineofouralgorithmsmodeltoWeplantoinvestigateforincludethesensorjointoptimumSTP-sensingextensionnodesofcouldourbeapproachinteresting.

toWSNswithseveralnodes.Finally,movingVII.APPENDIX

asThefollows:

firstandsecondordermomentsof(9)canbederivedMm1=CA

󰀈Nd2mBi+21σB2

i2σpi

ei

,(24)

i=1

3x 10−3d2=0.052.5d2=0.09)sd2=0.13( noit2ad2=0.17ruD ed2=0.21gatu1.5O egarevA10.5000.10.20.30.40.50.60.7d1Fig.6.AverageOutage3ofTab.durationI.

asfunctionoftheSTPcoefficientsdi|w=i|2forthecaseMm2(τ)=

CA2

didj2σ2

pi

2σpjeBi+i=1󰀈

N,i=j󰀈N2mj=1

mBj

12[σ2B2i+σBj+2CBiBj(τ)]

+

CA2

󰀈

Nd4iσ8piJ0(2πfmτ)rνi(τ)

i=1

e

2mBi+σ2

B

i

+CBi(τ),(25)

whereCA=Es/(󰀇N

i=1diσi+σn)anddi=|wi2isderivedasfollows:

|2.λλ2

=−β

2d2

CZ(τ)dτ2(26)

d2Mm2(τ)

)

=

dτ2Mm2dτM−󰀃dMm2(τ󰀄2

,(27)

m22

where,areobtained:

afteralgebraicmanipulations,thefollowingexpression

dMm2(τ)󰀈N󰀈NdCBdτ

=

Gi,j(τ)

iBj(τ)

i=1j=1,j=i

dτ

+

󰀈NHi=1

󰀈Nτ)

dCBi(τ)

i,j(τ)+Ii(i=1

dτ

,(28)where

Gi,j(τ)

=

CAdidj2σ2

p2mBii2σpje+mBj

12[σ2B2i+σBj+2CBiBj(τ)]

,(29)Hi(τ)=CAd4iσ8p󰀄2mBi+σBiJ0(2πfmτ)e

2

i+CBi(τ),(30)

Ii(τ)=CAd4iσ8p2mBi+σB+CiJ0(2πfmτ)e

2

iBi(τ).(31)

5

andd2Mm2(τ)dGi,j(τ)dCBiBj(τ)

dτ2

=

󰀈NNi

j=1󰀈,i=j

dτdτ+

󰀈NGd2CBi,j(τ)iBj(τ)i

j=1󰀈N,i=j

dτ2

+

󰀈NdHi(τ)+i

dτ

󰀈NdIi(τ)dCBi(τ)i

dτdτ+

󰀈NId2Ci

Bi(τ)

i

dτ2,

(32)

withdGi,j(τ)

dCBiBj(τ)dτ=Gi,j(τ)dτ,(33)

dHi(τ)dτ

=

CAd4iσ8pJ0󰀄󰀄(2πfmτ)e

2mBi+σ2

B+CBi(τ)ii+Hi(τ)

dCBi(τ)

dτ

,(34)and

dIi(τ)

dτ=H(τ)+IdCi(τ)Bi(τ)i.(35)

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