Energy-Efficient Joint Estimation in Sensor Networks: Analog vs. Digital

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by  Shuguang Cui

ENERGY-EFFICIENTJOINTESTIMATION IN SENSOR NETWORKS:
ANALOGVS. DIGITAL
ShuguangCui1, Jin-JunXiao2, Andrea J.Goldsmith1, Zhi-QuanLuo2, and H. VincentPoor3
5 1WirelessSystemLab,DepartmentofElectricalEngineering,StanfordUniversity.
0 2DepartmentofElectricalandComputerEngineering,UniversityofMinnesota.
0 3DepartmentofElectricalEngineering,PrincetonUniversity.
2
ABSTRACT thesensornoisepdf. Recently,universaldecentralizedestimation
n
schemes(DES)withouttheknowledgeofnoisedistributionhave
a
Sensornetworksinwhichenergyisalimitedresourcesothat
J beenproposedin[7]and[8].In[7],theauthorconsideredtheuni-
energyconsumptionmustbeminimizedfortheintendedapplica- versalDESinahomogeneoussensornetworkwheresensorshave
1 tionareconsidered.Inthiscontext,anenergy-efficientmethodfor
observations ofthesamequality, whilein[8],theuniversalDES
2
the joint estimation of an unknown analog source under a given
inaninhomogeneoussensingenvironmentwasconsidered.These

] distortionconstraintisproposed. Theapproachispurelyanalog, proposed DESs require each sensor to send to the fusion center
T in which each sensor simply amplifies and forwards the noise- a short discretemessage withlength decided bythe local Signal
corrupted analog observation to the fusion center for joint esti-
I toNoiseRatio(SNR),whiletheperformanceisguaranteedtobe
s. mation. Thetotaltransmissionpoweracrossallthesensornodes withinaconstantfactorofthatachievedbytheBestLinearUnbi-
isminimizedwhilesatisfyingadistortionrequirementonthejoint
c asedEstimator(BLUE).Anassumptionintheseproposedschemes
[ estimate. Theenergy efficiency of thisanalog approach is com- isthatthechannelsbetweensensorsandthefusioncenterareper-
paredwithpreviouslyproposeddigitalapproacheswithandwith- fect,andallmessagesarereceivedbythefusioncenterwithoutany
1 outcoding. Itisshowninoursimulationthattheanalogapproach
distortion. However,duetopowerlimitationsandchannelnoise,
v
is more energy-efficient than the digital system without coding,
thesignalsentbyeachindividualsensortothefusioncenterwill
0
and in some cases outperforms the digital system with optimal
5 becorrupted. Therefore, thegoal ofthetransmissionsystemde-
coding.
0 signforthejointestimationproblemistominimizetheeffectof
1 thechannelcorruptionwhileconsumingtheminimumamountof
0 1. INTRODUCTION powerateachnode.
5
Historically, if the sensor observation is in analog form, we
0 A typical Wireless Sensor Network (WSN), as shown in Fig. 1,
have twomain options totransmit the observation fromthesen-
/ consistsof afusion center and alargenumber of geographically
s sorstothefusioncenter:analogordigitalcommunication. Forthe
c distributedsensors. Thesensorstypicallyhavelimitedenergyre- analog approach, wekeep theobservation signal analogandfur-
v: sourcesandcommunicationcapability.Eachsensorinthenetwork theruseanalogmodulationschemestotransmitthesignal,which
makesanobservationofthetargetofinterest,generatesalocalsig-
i isalsocalledanamplify-and-forwardapproach. Inthedigitalap-
X nal(eitheranalogordigital),andthensendsittothefusioncenter proach,wedigitizetheobservationintobits,possiblyapplychan-
wherethereceivedsensorsignalsarecombinedtoproduceafinal
r nel coding, then use digital modulation schemes to transmit the
a estimateoftheobservedsignal. Sensornetworksofthistypeare data. Itiswellknown([4],[5])thatforasingleGaussiansource
well-suited for situation awareness applications such as environ-
withanAWGNchannel,theamplify-and-forwardapproachisop-
mentalmonitoringandsmartfactoryinstrumentation.
timal.However,foranarbitrarysourcewithmultipleobservations
andmultipletransmissionchannels, itisunclear whichapproach
willleadtoasmallerpower consumption whilemeetingthedis-
Fusion Center
tortionrequirementsatthefusioncenter.
Another severe challenge facing sensor networks isthe hard
energyconstraint.Sinceeachsensorisequippedwithonlyasmall-
Fig.1.Sensornetworkwithafusioncenter.
sizebattery,forwhichreplacementisveryexpensiveifnotimpos-
Decentralizedestimationhasbeenstudiedfirstinthecontext sible, thepower consumption must beminimizedtoincrease the
of distributed control [1], distributed tracking [2], and most re- networklifetime[6].Therefore,energyefficiencycanbeusedasa
centlyinwirelesssensornetworks[3]. Amongthesestudies,itis performancecriteriontoevaluatedifferentsensornetworkdesigns
usuallyassumedthatthejointdistributionofthesensor observa- underthesamedistortionrequirement. In[9],adigitalsystemis
tionsisknown. Inpracticalsystems,theprobabilitydensityfunc- proposedtominimizethetotalpowerconsumptionforthejointes-
tion (pdf) of the observation noise is hard to characterize, espe- timationproblem,whereeachsensorquantizestheanalogobserva-
ciallyforalargescalesensornetwork.Thismotivatesustodevise tionintodigitalbitsandtransmitsthebitstothefusioncenterwith
signalprocessingalgorithmsthatdonotrequiretheknowledgeof uncodedMQAM.Thenumberofquantizationbitsforeachsensor
isoptimizedwiththetargetof minimizing thetotaltransmission
ThisresearchissupportedinpartbyfundsfromNationalSemicon-
poweracrossallthesensornodestoachieveagivendistortion. In
ductorandToyotaCorporation,bytheNaturalSciencesandEngineering
thispaper,weproposeananalogcounterpartforthesamesystem
Research Council of Canada, Grant No. OPG0090391, by the National
ScienceFoundation,GrantNo.DMS-0312416,bytheOfficeofNavalRe- and compare the energy efficiency between the analog approach
searchunderGrantN00014-03-1-0102. andthedigitalone. Asin[9],weassumethattheobservedsignal
isanalogandbounded,thefusioncenterdeploysthebestunbiased Thereceivedsignalvectorateachtimeinstanceisgivenby
linearestimator, theobservation noiseisuncorrelated acrossdif-
y=hθ+v, (5)
ferent sensors, and only the variance of the observation noise is
known. where
Our paper is organized as follows. Section II discusses the y = [y1,y2,··· ,yK]†,
problemformulation. SectionIIIcomparestheenergyefficiency h = [√α1g1,√α2g1, ,√αKgK]†,
···
nbuetmweereincatlheexaamnaplloegs.aSpepcrtoioacnhIVansdumthmeadriigeistaoluarpcpornocalcuhsiovnias.some v = [√α1g1n1+nc1,··· ,√αKgKnk+ncK]†,
and meanstranspose.
†
2. MINIMUMPOWERANALOGINFORMATION Accordingto[10],thebestlinearunbiasedestimator(BLUE)
COLLECTION forθisgivenby
WeassumethatthereareK sensorsandtheobservationxk(t)at θˆ = [h†R−1h]−1h†R−1y
sensorkisrepresentedasarandomsignalθ(t)corruptedwiththe
observationnoisenk(t):xk(t)=θ(t)+nk(t).Eachsensortrans- = K αkgk −1 K √αkgkyk , (6)
mfroitmstthheesxigkn(at)l’xs,kk(t=)to1,thef,uKsio.nWceenftuerrthwerhearsesuθm(te)tihsaetsntikm(ta)teids Xk=1σk2αkgk+ξk2! Xk=1σk2αkgk+ξk2
ofunknownstatisticsand·t·h·eamplitudeofx(t)isboundedwithin wherethenoisevariancematrixRisadiagonalmatrixwithRkk =
[ W,W], which isdefined by thesensing range of each sensor. σk2αkgk+ξk2withσk2thevarianceofthesensorobservationnoise
F−orsimplicityweassumeW = 1,butouranalysiscanbeeasily nk(t),k = 1,··· ,K. Thechannel noisevarianceξk2 isdefined
extendedtoanyvalues. Wealsoassumethatthenetworkissyn- bythenoisepowerspectraldensityandthebandwidthB.
chronized,whichmaybeenabledbyutilizingbeaconsignalsina Themeansquarederrorofthisestimatorisgivenas[10]
separatecontrolchannel. Var[θˆ] = [h†R−1h]−1
We assume that xk(t) is also band-limited and the informa- K −1
tion is contained within the frequency range [ B/2,B/2]. We = αkgk . (7)
ceonntsreidceerivaenr[a1n1a]l.ogThSeintrgalnesSmiditete-Bdasnigdn(aSlSisBg)ivsyesn−tebmy withacoher- Xk=1σk2αkgk+ξk2!
AccordingtoEq.(2),thetransmitpowerfornodekisbounded
yt(t)=2√αcos(ωct)x(t)+2√αsin(ωct)xˆ(t), (1) by 4W2αk. Therefore, the minimum power analog information
collectionproblemcanbecastas
K
forwhichtheaveragetransmissionpoweris
min W2α
k
P =4αPx 4αW2 (2) Xk=1
≤ K −1
α g
wofhxer(et)4.αisthetransmitterpowergainandPx isthepeakpower s.t. Xk=1σk2αkgkkk+ξk2! ≤D0
Thereceivedsignalatthefusioncenterisgivenby α 0, k=1, ,K
k
≥ ···
yr(t)=2√α√gcos(ωct)x(t)+2√α√gsin(ωct)xˆ(t)+nc(t) where D0 is the distortion target. However, this problem is not
(3) convexovertheαk’s.
wheregisthechannelpowergain,xˆ(t)istheHilberttransformof Letusdefine
x(t),andnc(t)isthechannelAWGN.Hence,attheoutputofthe r = αkgk = 1 .
coherentdetectorthesignalis k σk2αkgk+ξk2 σ2+ ξk2
k gkαk
1 B 1 B
y(t)=√α√gx(t)+ nI(t)cos(π t)+ nQ(t)sin(π t), Thentheaboveoptimizationproblemisequivalentto
2 c 2 2 c 2 K
(4) min W2α
k
where nIc(t)+jnQc (t) is the complex envelope of nc(t). After Xk=1
passingy(t)throughalow-passfilterandsamplingthebaseband
K
1
signalatasamplingrateB,wecanobtainanequivalentdiscrete- s.t. r
k
timesystem. Sincewe have K such sensors, theoverall system ≥ D0
isshowninFig.2. TheK transmitterssharethechannelviaFre- Xk=1 1
qspueecntcryalDefifivcisiieonncyMausltthipeleTiAmcecDesisvi(sFioDnMMAu)l,tiwplheiAchccheassst(hTeDsMamAe) rk = σk2+ gkξkα2k, αk ≥0, ∀k,
thatisusedin[9].
whereweseethatthevariableα canbecompletelyreplacedbya
k
functionofr . Therefore,theproblemcanbetransformedintoa
n k
1 2√α1 √g1 nc1nc2ncK problemwithvariables{r1,r2,...,rK}shownasfollows:
θ n2 2√α2 √g2 y1,··· ,yK min XkK=1Wg2kξk2 (cid:18)1−rrkkσk2(cid:19)
nK 2√αK √gK s.t. XkK=1rk ≥ D10; 0≤rk < σ1k2, ∀k (8)
Fig.2.AmplifyandForward
whichisconvexoverr .Theupperlimitonr inthesecondcon- conditions,thepowerisscaledbythefactor 1 ,whichmeansthat
straintisduetothefactkthatr =1/ σ2+ ξkk2 and ξk2 0. morepowerisusedtotransmitthesignalsfσrok2mthesensorswith
k k gkαk gkαk ≥
NowwesolveEq.(8).ItsLagra(cid:16)ngianGisgi(cid:17)venas betterobservationquality.
Wenowsolvetheoptimizationproblemforsomespecificex-
G(L,λ0) = kXK=1Wg2kξk2 (cid:18)1−rrkkσk2(cid:19) aadnmkdpisGletsh.e=Wtreana3ssm0suidmsBseioitsnhtadhtiesthtgaeaniccnheaaftnrndoem=lps1oenwmseo.rrAgkasitniongt[hk9e]=,fuwsdGei3ko.0g5nencweehnreaterteer
1 K σ2uni0form−lywithintherange[0.01,0.08]. WetakeB=10KHz
+λ r (9) k
0 D0 −Xk=1 k! saonrds,ξFk2i=g.3−(9a0)sdhBomw,skth=er1el,a·t·iv·e,pKow.Ferorsaavninexgasmcopmlepwariethd1w0i0thsethne-
for0 r 1 , k,whichleadstothefollowingKarush-Kuhn- uniformtransmissionstrategywhereallthesensorsusethesame
≤ k ≤ σK2 ∀ transmissionpowertoachievethegivendistortiontarget.Therel-
Tucker(KKT)conditions[12]:
ative power savings is plotted as a function of R = √Var(d),
W2ξ2 1 E(d)
k λ =0, k thedistancedeviationnormalizedbythemeandistance. Foreach
gk (1−rkσk2)2 − 0 ∀ valueofR,weaveragetherelativepowersavingsover100random
K runswhereineachrunthed ’sarerandomlygeneratedaccording
1 k
rk =0 tothegivenR.Asexpected,alargervariationofdistanceandcor-
Xk=1 − D0 respondingchannelqualityleadstoahigherpowersavingswhen
thisvariationisexploitedwithoptimalpowerallocation.
for 0 r 1 , k. Without loss of generality, we rank the
≤ k ≤ σK2 ∀ InFig.3(b),thenumberofactivesensorsoverRisshownfor
channelqualitysuchthat ξ12 ξ22 ... ξK2 wherewecallthe an example with10 sensors. Weseethat when thetransmission
g1 ≤ g2 ≤ ≤ gK
quantity gk thechannelSNR,andwedefine distancesfordifferentsensorsspanawiderangeofvalues(i.e.,R
ξk2 is large), more sensors can be shut off to save energy, since the
remainingsensorshaveverygoodchannels.
ξM2 A(M)
f(M)= gM , for1 M K, (10)
qB(M) ≤ ≤ %)
aUwtuannshkdiieeqnrLugKeeetAt1huue(n=sMlsefiaKs)msn.d=feT(KtmMehXMce1=hn)1sntu<iqhqceσuh1mg2ξeKtmm2shfKoaarTtsaanficlnd(loKnB[19d1(]≤i,)MtiwoξM<k2n)es=c1g≤a+inmavXMnKe=sdhλ1,of0iσwn(1m=2Kwth1(cid:16)−hait+WctBDhh1A1(i0cKs()aK.1Ks≥1)e)1w(cid:17)1i2es. Power saving over uniform transmission (in 1111267890111122.....067890155555.35 0(.4a) N0.4o5rm0a.5lize0d.55 dis0t.6anc0e.65 dev0.7iatio0.n75 (R0).8 0.85 Number of active sensors 1067891.....0578955550.3 0(.4b) N0.5orm0a.6lize0d.7 dis0t.8anc0e.9 dev1iatio1.n1 (R1).2 1.3
ropt= 1 1 W gk , k (11) Fig.3. (a)Powersavingsofoptimalpowerallocationvs. uniform
k σk2  − q√λ0 ∀ power;(b)Numberofactivesensorsversusdistancedeviation
where(x)+equals0whenx<0,andotherwiseisequaltox. Inourmodelweminimizethepowersum P ,i.e.,theL1-
Hence,bydefinition,wehave norm of the transmission power vector P = (Pk1,kP2,...,PK).
If the channel gainand thevariance of theobPservation noise for
ξ2 ropt
αopt = k k eachsensorareergodicallytime-varyingonablock-by-blockba-
k gk1−σk2rkopt sis, minimizingtheL1-normofPineachtimeblock minimizes
= ξk2 gkη 1 k=1, ,K , (12) E{ kPk}withE{}theexpectationoperation. Inotherwords,
gkσk2 (cid:18)rξk2 0− (cid:19) ··· 1 it1mPaximiEze0sthweiathveEragtehneobdaettleirfyeteimneer,gwyhaivcahiliasbdleefitoneedacahssensor
K k E{Pk} 0
andTαhke=ref0oroet,hethrweiosep,tiwmhaelrepoηw0e=r aAll(oKca1t)io/nB(sKtra1t)e.gy is divided p(wroePveadssubmyethtehafatcEtt0haistth1esameEfo0rallthesenEso0rs). T.hHisocwaenvebre,
intotwosteps.Inthefirststep,athresholdforkisobtainedaccord- K k E{Pk} ≥ E{K1 kPk}
whenthechannelisstaticandthevarianceoftheobservationnoise
ingtoEq.(10). ForchannelswithSNRworsethanthisthreshold, istime-invariant,minimizinPgtheL1-normmayPleadsomeindivid-
thecorrespondingsensorisshutoffandnopoweriswasted. For
ual sensors to consume toomuch power and die out quickly. In
the remaining active sensors, power should be assigned accord- thiscaseminimizingtheL∞-norm,i.e.,minimizingthemaximum
ing to Eq. (12). From Eq. (12) we see that when the channel is
oftheindividualpowervalues,isthemostfairforallsensors,but
fairlygood, i.e., gkη 1,wehaveαopt ξk2 η0,which thetotalpowerconsumptioncanbehigh. Asin[9],wecanmake
ξk2 0 ≫ k ∝ gkσk2 a compromise to minimize the L2-norm of P. In this way, we
meanstheoptimaqlsolutionisinverselyproportionqaltothesquare
canpenalizethelargetermsinthepowervectorwhilestillkeep-
rootofthechannelSNR.When gξkk2η0 isclosetoone,theopti- ingthetotalpowerconsumptionreasonablylow.FortheL2-norm
malsolutionmaynolongerhaveqsuchproperties. Forallchannel minimization,theproblemformulationbecomes
K W4ξ4 r 2 101
min k k AAnnaalloogg:: 2100 sseennssoorrss
s.t. XkK=1r gk2 1(cid:18)1; −0rkσkr2(cid:19)< 1 , k, (13) wer vector 100 DDiiggiittaall:: ufunncdoadmede nMtaQl AlimMi tw witihth 1 100 s oernthsoogrsonal channels
wsehcitciohnw, ewceacnosmolpvaeXkre=u1stihnekgp≥inowteDeri0rorefpfiociine≤tnmcyketohfodtσhske2[1a2n]a.∀loIngtahnedntehxet 2L−norm of the po 10−1
digital approaches previously discussed, where we minimize the
L2-normofP. WeusetheL2-normsinceitsimplifiesthecom- 10−22.5 3 3.5 D0 4 4.5 x 10−35
parison,butthecomparisoncanbemadeforanypowernorms.
Fig.4.ComparisonoftheL2-normofthepowervector
3. ANALOGVS.DIGITAL 5. REFERENCES
In order to transmit the observed analog signal with frequency [1] D.A.CastanonandD.Teneketzis,“DistributedEstimationAl-
range [ B/2,B/2] to the fusion center, each sensor in the dig- gorithmsforNonlinear Systems,”IEEETransactionsonAu-
italsys−temproposed in[9]firstsamplesthesignal atasampling tomaticControl,Vol.AC-30,pp.418–425,1985.
rateB, then quantizes each sample intobk bits, and finallyuses [2] A.S. Willsky, M. Bello, D.A. Castanon, B.C. Levy, and G.
uncoded MQAM to transmit the bk bits with a symbol rate B Verghese, “CombiningandUpdatingofLocalEstimatesand
and constellation size M = 2bk. Therefore, the total transmis- RegionalMapsAlongSetsofOne-dimensionalTracks,”IEEE
sionbandwidthinthepassbandisapproximatelyequaltoKBfor TransactionsonAutomaticControl,Vol.AC-27,pp.799–813,
the digital system where TDMA is used for the multiple access. 1982.
FortheSSBschemeusedinouranalogsystem,eachsensoronly
[3] H.C.Papadopoulos,G.W.Wornell,andA.V.Oppenheim,“Se-
occupies B/2 in the passband such that the total bandwidth re-
quential Signal Encoding from Noisy Measurements Using
quirementisKB/2whenFDMAisusedforthemultipleaccess.
QuantizerswithDynamicBiasControl,”IEEETrans.onIn-
Therefore,undertheassumptionoforthogonalchannelusage,the
formationTheory,Vol.47,pp.978–1002,2001.
analogsystemcansupportdoublethenumberofsensorsthatthe
digitalsystemsupportsinthesameamountofbandwidth. Thisis [4] T.J.Goblick,“TheoreticalLimitationsontheTransmissionof
mainlycausedbythefactthatthedigitalapproachproposedin[9] DatafromAnalogSources,”IEEETrans.Inform.Theory,vol.
forces the transmission symbol rate to be equal to the sampling IT-11,pp.558567,Oct.1965.
rateinthesourcecodingpart.
[5] M. Gastpar, B. Rimoldi and M. Vetterli, “To Code, or Not
Thepowerefficiencycomparisonbetweentheanalogapproach ToCode: LossySource-channel Communication Revisited,”
andthedigitalapproachisshowninFig.4, wherewedeploy 10 IEEETrans.on Information Theory, Vol. 49pp. 1147-1158,
sensors for the digital system and we plot the power curves for May2003.
boththecasewhereuncodedMQAMisused(thedottedline)and
[6] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-constrained
the case where single-user capacity-achieving channel codes are
Modulation Optimization,” to appear at IEEE Transac-
appliedtoeachtransmitter,whichgivesthefundamentalShannon
tions on Wireless Communications, 2003. Also available at
limit(thedashedline)forthedigitalsystemwithorthogonalchan-
http://wsl.stanford.edu/Publications.html.
nelusage.Fromthefigureweseethattheanalogapproachwith10
nodes,whichhavethesameobservationqualityandtransmission [7] Z.-Q. Luo, “Universal Decentralized Estimation in a Band-
distancesasinthedigitalsystem,ismoreenergyefficientthanthe width Constrained Sensor Network,” submitted to IEEE
digitalapproach withuncodedMQAM,butnotnecessarilymore Transactions on Information Theory. Also available at
energyefficientthanthefundamentallimitcurve. However,since http://www.ece.umn.edu/users/luozq/.
wecansupport20nodesintheanalogsystemwiththesameband- [8] Z.-Q. Luo, J.-J., Xiao, “Decentralized Estimation in an In-
widthrequirement,withthe20analognodeswecanachieveabet- homogeneous Sensing Environment,” at International Sypo-
terpowerefficiency(thesolidline)thanthedigitalsystemwith10 siumonInformationTheory,Chicago,2004.Alsoavailableat
nodesunder theoptimalsingle-userchannel coding. Thisistrue http://www.ece.umn.edu/users/luozq/.
even when the extra ten nodes in the analog system have worse
[9] J.Xiao,S.Cui,Z.Q.Luo,andA.J.Goldsmith,“JointEstima-
observationqualityandlargertransmissiondistancesthanthefirst
tioninSensorNetworksunderEnergyConstraint,”toappear
tennodes.
attheIEEEfirstconferenceonSensorandAdHocCommuni-
4. CONCLUSIONS cationsandNetworks,SantaClara,CA,October,2004.
[10] J.M.Mendel,LessonsinEstimationTheoryforSignalPro-
Inthispaper, wehaveshownthatforabounded sourcewithun-
cessing,Communications,andControl,PrenticeHall,Engle-
knownstatisticsandafusioncenterequippedwiththebestlinear
woodCliffs,NJ,1995.
unbiased estimator, we can minimize the total power consump-
tionacrossallthesensornodesunderacertaindistortionrequire- [11] S.Haykin,CommunicationSystems,3rdedition,JohnWiley
ment.Theinformationcollectioncanbeimplementedwithanana- &Sons,NewYork,1994.
logapproach,whichmaybemoreenergyefficientthanthedigital [12] S.Boyd,L.Vandenberghe,ConvexOptimization,Cambridge
approachwhenonlyorthogonal multipleaccessschemessuchas Univ.Press,Cambridge,U.K.,2003.
TDMAandFDMAareused.