Large Deviations Analysis for the Detection of 2D Hidden Gauss-Markov Random Fields Using Sensor Networks

0.17 MB·English

by  Youngchul Sung

LARGEDEVIATIONSANALYSIS FORTHEDETECTION OF2D HIDDEN
GAUSS-MARKOVRANDOMFIELDS USINGSENSOR NETWORKS
YoungchulSung†, H. VincentPoorandHeejung Yu
ABSTRACT (GMRFs)usingnoisyobservations.InparticularweconsiderSen-
sorsijlocatedona2Dlattice .Ondenotingthe(noisy)measure-
8 Thedetectionofhiddentwo-dimensionalGauss-Markovrandomfieldsus- mentsofSensorijasY andIadoptingaNeyman-Pearsonformu-
0 ing sensor networks is considered. Under a conditional autoregressive lation,wecanmodeltheijdetectionproblemvianullandalternative
0 model, theerror exponent fortheNeyman-Pearson detector satisfying a hypothesesgivenby
fixedlevelconstraintisobtainedusingthelargedeviationsprinciple. For
2
asymmetricfirstorderautoregressivemodel,theerrorexponentisgiven H0:Yij =Wij,ij∈I vs. H1:Yij =Xij+Wij,ij∈I, (1)
n explicitly intermsoftheSNRandanedgedependencefactor(fieldcor-
Ja rsetrleantigotnh).isTsheeenbetohadviivoirdeofinthtoetewrroorreegxipoonnsednetpaesnadifnugncotniotnheofvaclourereolaftitohne (iw.ih.de.r)e{W(0ij,}σ2r)epnroeisseentwsiitnhdaepkennodwenntvaanrdianidceentσic2a,lalynddisXtriibjuteids
N { }
SNR.AthighSNR,uncorrelatedobservationsmaximizetheerrorexponent a stationary GMRF on the2D lattice independent of themea-
0 foragivenSNR,whereasthereisnon-zerooptimalcorrelationatlowSNR. surementnoise W . Thus,theobseIrvationsamplesforma2D
2 Basedontheerrorexponent,theenergyefficiency(definedastheratioof { ij}
hiddenGMRFunder .
thetotalinformationgatheredtothetotalenergyrequired)ofadhocsen- H1
] sornetworkfordetectionisexaminedfortwosensordeploymentmodels:
T aninfiniteareamodelandandinfinitedensitymodel. Forafixedsensor Yij
I density, theenergyefficiencydiminishes tozeroatrateO(area−1/2)as (i,j)
s. theareaisincreased.Ontheotherhand,non-zeroefficiencyispossiblefor Xij Sensorij
c increasingdensitydependingonthebehaviorofthephysicalcorrelationas Xij Yij
r
[ afunctionofthelinklength. Wij
IndexTerms-Neyman-Pearsondetection,errorexponent,GMRF r
1
v 1. INTRODUCTION Fig.1.Sensorsona2DLattice :HiddenMarkovStructure
Consider the design of a sensor network for the detection of a I
3
correlatedstochasticsignalinafixedarea. Manyquestionsarise
7 1.1. SummaryofResults
0 insuchadesign: How dothefieldcorrelation andmeasurement TheexacterrorprobabilityofthedetectionoftheNeyman-Pearson
3 signal-to-noise(SNR)affectthedetectionperformance? Whatis testisnotavailableinclosed-forminthegeneralcorrelatedcase,
. theoptimalsensordensity,i.e.,thenumberofnodesperunitarea? includingthehypotheses(1).Hence,weinvokethethelargedevi-
1 Whatistheinformationandenergytrade-offinsuchasensornet- ationsprincipleandusetheerrorexponentofthedetectionprob-
0 workwithadhocrouting?Toaddresstheseissues,severalstudies ability(or,moreconveniently,itscomplement,themissprobabil-
8 ity)asanalternativeperformancemeasure.Forafixedfalse-alarm
based on one-dimensional (1D) spatial signal models have been
0 conducted(see,e.g.,[1]and[2]). However,thereisanimportant level,themissprobabilityPM decaysexponentiallyasthesample
: size nincreases, and the error exponent isdefined asthe rateof
v difference between 1D signal models and actual spatial signals. exponentialdecay,i.e.,
i Suppose that we take observations from sensors placed equidis-
rX tmanaytlythaelonnbgeavliineweterdanassecstamlaipdleosvegrenaegriavteendabryeaa.Tohnee-odbimseernvsaitoionnasl K=∆nl→im∞−n1 logPM (2)
a processandtheresultsfromtimeseriesanalysiscouldbeapplied underthegivenconstraint(i.e.,thefalsealarmprobabilityPF
≤
toinvestigatetheirstatisticalproperties. However,thereisnoreal α).Theerrorexponentisagoodperformancecriterioninthelarge
notionof‘signalflow’ordependencedirectionalongthetransect sampleregimesinceitallowsthedesignertoestimatethenumber
asthereisinamoretraditionallyobtainedtimeseries. Forsam- of samples required for a given detection performance. Hence,
plesfromsensorsdeployed over atwo-dimensional (2D) area, it efficient design can be examined through the error exponent for
isnecessary toconsider thesignaldependence inalldirectionin largescalesensornetworks.
the plane, and as a consequence, answering the above questions Here, we adopt the conditional autoregression (CAR) model
becomesmoredifficult. for the signal, and derive a closed-form expression for the error
To address the above questions in a 2D setting, in this pa- exponent ofthemissprobability(whichisindependentofα)in
K
per,weconsiderthedetectionof2DGauss-Markovrandomfields thespectraldomain. Wedosobyexploitingthespectralstructure
oftheCARsignalandtherelationshipbetweentheeigenvaluesof
†Y.SungandH.YuarewiththeDept. ofElectricalEngineering,KoreaAd- the block circulant approximation toa block Toeplitzmatrixde-
vancedInstituteofScienceandTechnology(KAIST),Daejeon305-701,SouthKo-
scribingthe2Dcorrelationstructure.Inparticular,itisshownthat
rea. Email:[email protected]@stein.kaist.ac.kr. H.V.Pooriswiththe
theerrorexponentforthedetectionof2DhiddenGMRFisanex-
Dept. ofElectricalEngineering,PrincetonUniversity,Princeton,NJ08544. Email:
tensionofthatinthe1DcaseobtainedbySungetal.[3].Asinthe
[e-mail]BrainKorea21
1Dcase, itisshownthati.i.d. (and, thus, uncorrelated)observa-
Project,theSchoolofInformationTechnology,KAIST.TheworkofH.V.Poorwas
supportedinpartbytheU.S.NationalScienceFoundationunderGrantsANI-03- tionsmaximizetheerrorexponentforagivenSNRwhentheSNR
38807andCNS-06-25637. ishigh. Ontheotherhand,thereisanoptimalnon-zerodegreeof
correlationatlowSNR.Interestingly,itisseenthatthereisadis- and Cov(Xij,Xi′j′) =∆ E{XijXi′j′} = c(i−i′,j −j′) for
continuityintheoptimalcorrelationstrengthasafunctionofSNR. somefunctionc(, ).
· ·
Intheperfectlycorrelatedcase, theerrorexponent iszeroasex- Fora2DstationaryGMRF X ,thecovariance γ isdefined
ij ij
pected.FortheerrorexponentasafunctionofSNR,wewillshow { } { }
as
sthtraetntghtehearrtohrigexhpSoNneRn.tincreasesaslogSNRforagivencorrelation γij =E{Xi′j′Xi′+i,j′+j}=E{X00Xij}, (4)
whichdoesnotdependoni′orj′duetothestationarity. Further,
Weconsidertwoasymptoticregimesmodellingthesensorde-
thespectraldensityfunctionofazero-meanandstationaryGaus-
ploymentin2D:aninfiniteareamodelwithafixeddensityandan
infinitedensitymodelwithafixedarea. Applyingtheresults,we sianprocess{Xij}onI∞withcovarianceγij isdefinedas
obtaintheasymptoticbehavioroftheenergyefficiency,definedas 1
f(ω ,ω )= γ exp( ι(iω +jω )), (5)
theratioofthetotalinformationgatheredtotherequiredenergyto 1 2 4π2 ij − 1 2
obtaininformationfromtheareaforanadhocnetworkwithmini- ijX∈I∞
mumhoproutingtothefusioncenter. Fortheinfiniteareamodel, whereι= √ 1and(ω ,ω ) ( π,π]2. Notethatthisisa2D
theenergyefficiencydecaystozerowithrateO(area−1/2)aswe extensionoft−heconvent1iona2l 1∈Dd−iscrete-timeFouriertransform
increasethecoveragearea. Fortheinfinitedensitymodel,onthe (DTFT).
otherhand,anon-zeroefficiencyispossibleifthedecayrateofthe
Definition3(Theconditionalautoregression(CAR)) AGMRF
errorexponent (density)asafunctionofdensityisslowerthan canbespecifiedusingasetoffullconditionalnormaldistributions
K
O(density(1−δ)/2),whereδisthepropagationconstantδ 2. withmeanandprecision:
≥
1T.h2e. dReteelcattieodnWofoGrkauss-Markov processesinGaussiannoiseisa E{Xij|X−ij} = −θ100 i′j′∈XI∞6=00θi′j′Xi+i′,j+j′, (6)
classicalproblem. See[4]andreferencestherein. However,most
Prec{Xij|X−ij} = θ00>0, (7)
work in this area considers only 1D signals or time series. A
closed-form error exponent was obtained anditspropertieswere whereX−ij denotesthesetofallvariablesexceptXij.
investigatedfor 1D hiddenGauss-Markov random processes [3]. ItisshownthattheGMRFdefinedbytheCARmodel(6)-(7)is
azero-meanstationaryGaussianprocesson withthespectral
Largedeviationsanalyseswereusedtoexaminetheissuesofop- ∞
densityfunction[6] I
timalsensordensityandoptimalsamplingwereexaminedwitha
1DsAignnaelrrmorodexelpoinne[1n]tawnads[o2b].tainedforthedetectionof2DGM- f(ω1,ω2)= 4π12 ij∈I∞θijexp1(−ι(iω1+jω2)) (8)
RFsin[5],wherethesensorsarelocatedrandomlyandtheMarkov if P
graph is based on the nearest neighbor dependency enabling a
loop-free graph and further analysis. In this work, however, the |{θij 6=0}|<∞, θij =θ−i,−j, θ00>0, (9)
measurement noisewasnot captured. Our workherefocuses on {θij}issothatf(ω1,ω2)>0, ∀(ω1,ω2)∈(−π,π]2.(10)
theerrorexponentforthedetectionof2DhiddenGMRFona2D Weassumethatthe2Dstochasticsignalin(1)isgivenbyasta-
infinitelattice,whichallowsfortheconsiderationofmeasurement tionaryGMRFdefinedbytheCARmodel(6)-(7)and(9)-(10).
noise. InparticularweexaminetheaboveCARmodelandinves- Then,theobservationspectrumunderthetwohypotheses(1)are
tigateofthedetectionperformancewithrespect(w.r.t.)tovarious given,respectively,by
designparameterssuchascorrelationstrength,measurementSNR,
sensordensityandarea. Sy(ω ,ω )= σ2 and Sy(ω ,ω )= σ2 +f(ω ,ω ).
0 1 2 4π2 1 1 2 4π2 1 2
2. DATAMODEL
3. PERFORMANCEMEASURE:ERROREXPONENT
Definition1(GMRF[6]) ArandomvectorX=(X ,X , ,X )
1 2 n
Rn is a Gauss-Markov random field w.r.t. a labelled·g·r·aph In this section, we investigate the performance of the Neyman-
∈=(ν, )withmeanµandprecisionmatrixQ>0,ifitsprob- Pearson detector with level α (0,1) for a 2D CAR signal in
aGbilitydeEnsityfunctionisgivenby ∈
noisy observations. Weobtain theerror exponent inthespectral
1 domainforthisproblembyexploitingthespectralstructureofthe
p(X)=(2π)−n/2|Q|1/2exp − (X−µ)TQ(X−µ) , (3)
2 CARsignalandtherelationshipbetweentheeigenvaluesofblock
„ «
circulantandblockToeplitzmatricesrepresenting2Dcorrelation
andQ =0 l,m foralll=m.Here,νisthesetof
lm6 ⇐⇒{ }∈E 6 structure.
allnodes 1,2, ,n and isthesetofedgesconnectingpairs
{ ··· } E
ofnodes,whichrepresenttheconditionaldependencestructure. Theorem1(ErrorExponent) ConsiderNeyman-Pearsondetec-
tionbetweenthehypotheses(1)withthemodel(6)-(7)andwith
Notethatthemeanandtheprecisionmatrixfullycharacterize
level α (0,1). Assuming that conditions (9 and 10) hold, the
aGMRF.NotealsothatthecovariancematrixQ−1iscompletely errorexp∈onentofthemissprobabilityisindependent ofαandis
dense in general while the precision matrix Q has nonzero ele- givenby
mentsQ onlywhenthereisanedgebetweennodeslandmin
lm
theMarkovrandomfield.Hence,whenthegraphisnotfullycon- K = 1 π π 1logσ2+4π2f(ω1,ω2)
nected, the precision matrixis sparse. The 2D indexing scheme 4π2 Z−πZ−π„2 σ2
(i,j)canbeproperlyconvertedtoan1DschemetoapplyDefini- 1 σ2 1
tion1. Fromhereon,weusethe2Dindexingschemeforconve- + 2σ2+4π2f(ω1,ω2) − 2«dω1dω2, (11)
nience. 1 π π
Definition2(Stationarity) A 2D GMRF on 2D doubly infinite = 4π2 Z−πZ−πD(N(0,S0y(ω1,ω2))||N(0,S1y(ω1,ω2))dω1dω2,
lattice ∞ issaidtobestationaryifthemeanvectorisconstant whereD( )denotestheKullback-Leiblerdivergence.
I ·||·
Proof: isgivenbythealmost-surelimitoftheasymptotic Corollary1 TheerrorexponentfortheNeyman-Pearsondetector
Kullback-LeKiblerrateK=limn→∞n1 logpp01,,nn(yn)evaluatedunder forthehypotheses(1)withtheSFAR2Dsignalmodelisgivenby
p [7]. UsingthefactthatwehaveGaussiandistributionsunder
0,n 1 π π 1 SNR
bothhypotheses,wehave Ks = 4π2Z−πZ−π„2log 1+(2/π)K(4ζ)(1−2ζcosω1−2ζcosω2)!
K = nl→im∞n1„12logddeett((ΣΣ01,,nn)) + 12ynT(Σ−1,1n−Σ−0,1n)yn«, +121+(2/π)K(4ζ)(1−21ζSNcRosω1−2ζcosω2) −12«dω1dω2. (15)
ThenapproximatingtheblockToeplitzcorrelationmatrixwitha Note that the SNR and correlation are separated in (15), which
blockcirculantmatrixandapplyingthethe2DGrenander-Szego¨ enablesustoinvestigatetheeffectsofeachtermseparately.
theorem,weobtainthelimitofeachtermasfollows.
3.2. PropertiesoftheErrorExponent
1 1 π π Ks
n1logdet(Σ1,n) → 4π2 Z−πZ−πlog(σ2+4π2f(ω1,ω2))dω1dω2, fFuinrsctt,ioitnisofretahdeileydgseeednefpreonmdeCncoeroflalacrtyor1ζth(a0t Ksζisac1o/n4ti)nfuooruas
nlogdet(Σ0,n) → logσ2, givenSNR.Thevaluesof sattheextremeco≤rrelati≤onsaregiven
n1ynTΣ−1,1nyn → 4π12 Z−ππZ−ππ σ2+4π2σf2(ω1,ω2)dω1dω2, bie.xyi.pdne.occtteiandsge,at(nhi.daet.,Kζ(=i0s)g0i=)v,etnhπ2KebycaonrdollaKry(1re)du=ce∞st.oTStheeirne’fsolreem, minathaes
n1ynTΣ−0,1nyn → 1, 1 Ks 1 1
almostsurely.(cid:4) Ks= 2log(1+SNR)+2(1+SNR)−2 =D(N(0,σ2)||N(0,σ2+Ps)).
Thistheorem is a 2D extension of the error exponent of 1D Forthe perfectlycorrelated case (ζ = 1/4), on theother hand,
hidden Gauss-Markov model based on state-space structure ob- s = 0. In fact, in this case as well as in the i.i.d. case, the
K
tainedin[3]. Intuitively,theerrorexponent(11)canbeexplained two-dimensionality is irrelevant. The known result that PM
usingthefrequencybinningargument.Foreach2Dfrequencyseg- Θ(n−1/2)fortheperfectlycorrelatedcaseisapplicable. ∼
mentdω dω ,thespectraareflat,i.e.,thesignalsareindependent For intermediate values of correlation, we evaluate (15) for
1 2
and Stein’s lemma can be applied for the segment. The overall severaldifferentSNRvalues,asshowninFig. 2. Itisseenthatat
Kullback-Leiblerdivergenceisthesumofcontributionsfromeach
bin. 0.75 0.1
SNR = 10 dB SNR = 0 dB
0.7 0.095
3.1. SymmetricFirstOrderAutoregression 0.09
Tcaouortorienrlevagetrsieotisngsaiaotnendt(hSSeNFbAReR,hw)a,vedifoeursrcotrhfiebtrehdceobenyrsriotdhreerecxthopenodnsiyetmniotmnasestaricfufinrcsttioonrdoefr Error exponent, E00..0056..5556 Error exponent, E000...00000..6780055578
λ
E{Xij|X−ij} = κ(Xi+1,j+Xi−1,j+Xi,j+1+Xi,j−1), 0.45 0.00.5056
Prec{Xij|X−ij} = κ>0, 0.40 0.05 0.1 ζ 0.15 0.2 0.25 0.050 0.05 0.1 ζ 0.15 0.2 0.25
(a) (b)
where0 λ κ. (Thisisasufficientconditiontosatisfy(9)-
≤ ≤ 4
(1λ0)..)InNtohtiesmheordeetlh,atthθe0c0o=rreκlatainodnθis1,s0ym=mθe−t1ri,0cf=orθe0a,c1h=seθt0o,f−f1ou=r 00..003367 SNR = −3 dB 0.00.108159 SNR = −5 dB
− 0.035
inecnxoetinteghdnheisbt2iioooDrnniancolgafcstneahouwedsa1eelDsh.daeGvTpeaheuefnsodsSue-FrMnAncaeyRrikgomohnvbloyodrraeoinlnndgitsohnmaeodpspierrmesovcpiineloesutshys,eewstfahommiuceprhal(enph.ilanaHsgnetfahurree)l Error exponent, E0000....00000.3333012343 Error exponent, E00..0000..110067115578
0.016
directions.ThespectrumoftheSFARisgivenby 00..002289 0.0155
1 0.0270 0.05 0.1 ζ 0.15 0.2 0.25 0.0150 0.05 0.1 ζ 0.15 0.2 0.25
f(ω1,ω2)= 4π2κ(1 2ζcosω 2ζcosω ). (12) (c) (d)
1 2
− −
Wedefinetheedgedependencefactorζby
ζ =∆ λ, 0 ζ 1/4. (13) F(ci)gS.2N.RK=s-a3sdaBf,u(ndc)tiSoNnRof=ζ-:5(ad)BSNR=10dB,(b)SNR=0dB,
κ ≤ ≤
Notethatζ = 0corresponds tothei.i.d. casewhereasζ = 1/4
highSNR ismonotonicallydecreasingasζ increases. Hence,
s
correspondstotheperfectlycorrelatedcase.Hence,thecorrelation K
i.i.d.observationsgivethebesterrorperformanceforagivenvalue
strengthcanbecapturedinthissinglequantityζforSFARsignals.
ofSNRwhenSNRislarge,asinthe1Dcase[3]. Aswedecrease
ThepoweroftheSFARisobtainedusingtheinverseFouriertrans-
theSNR,itisobservedthatasecondmodegrowsnearζ = 1/4.
formviatherelation(5),andisgivenby
AswefurtherdecreasetheSNR,thevalueofζofthesecondmode
2K(4ζ) 1 shiftstoward1/4,andthevalueofthesecondmodeexceedsthat
Ps =γ00 = πκ , „0≤ζ ≤ 4«, (14) of the i.i.d. case. Hence, there isa discontinuity in the optimal
correlationasafunctionofSNRinthe2Dcaseevenifthemaximal
where K() isthe complete ellipticintegral of the firstkind [8]. itselfiscontinuous. Thisisnotthecasein1D.
TheSNRi·sgivenbySNR = Ps = 2K(4ζ). Usingeq. (11)and Ks Withregardto asafunctionofSNR,itisstraightforward
σ2 πκσ2 Ks
the SNR,weobtain the error exponent inthe SFARsignal case, toseethatitiscontinuousandincreasesattheratelogSNRathigh
denotedby andgiveninthefollowingcorollary. SNRforagivenvalueofζ.
s
K
4. ADHOCNETWORKING:INFORMATION-ENERGY Sincer isfixed, andE donotchangewithn,and(17)
n s link
TRADE-OFF follows.(cid:4) K
Theanalyticalresultsintheprevioussectioncanbeappliedtoan-
Theorem3(Infinitedensitymodel) Fortheinfinitedensitymodel,
swersomefundamentalquestionsinthedesignofsensornetworks
anon-zeroefficiencyispossibleifthedecayrateoftheerrorexpo-
fordetectionapplications.Weconsideraplanaradhocsensornet-
nent asafunctionofdensityisslowerthan
workwithminimumhoprouting. Tosimplifytheanalysis,weas- Ks
sumethat(2n+1)2sensorsarelocatedonthegrid[ n:1:n]2
− O density(1−δ)/2 . (19)
withspacingr ,asshowninFig.1,andafusioncenterislocated
n “ ”
at thecenter (0,0). Weassumethat the measurement Y isde-
ij
liveredtothefusioncenterusingtheminimumhoprouting,which Proof:Fortheinfinitedensitymodel,wehave
requiresahopcountof i + j.
| | | | r =Θ(n−1), rδ =Θ(n−δ), density=Θ(n2).
4.1. Physicalcorrelationmodel n n
Theactual physical correlationinthismodel can beobtained by From(18),wehaveη = (r )/n1−δ. If asafunctionofr
s n s n
solvingapropercontinuousindex2Dstochasticdifferentialequa- decaysslowerthann1−δ,KηdoesnotdiminKishtozero.(cid:4)
tion(SDE),e.g., Thenon-zeroefficiencyintheasymptoticregimedependson
∂ 2 ∂ 2 thedecayrateof sasafunctionofrn.Since s(ζ)isgiven,this
+ −ξ2 X(x,y)=u(x,y), dependsonthefuKnctionsf andginSection4.K1andthepropaga-
"„∂x« „∂y« # tionlossfactorδ.
whereu(x,y)istheprocessnoiseandξisaparameterdetermin-
5. CONCLUSIONS
ingthecorrelationstrengthofthefield. BysolvingaproperSDE,
the edge correlation factor ρ is given, as a function of the edge Wehaveconsidered thedetection of2D GMRFsfromnoisyob-
lengthrn,by servations. Wehaveadopted theCARmodel for thesignal, and
ρ=f(rn). have used a spectral domain approach to derive the error expo-
Typically,f()isapositiveandmonotonicallydecreasingfunction nentfortheNeyman-Pearsondetectorsatisfyingafixedlevelcon-
ofr . Furthe·r,wehaveamonotonemappingg :ρ ζ fromthe straint. Underthesymmetricfirstorderautoregressivemodel,we
n
edgecorrelationfactorρtotheedgedependence fa→ctorζ, which have obtained the error exponent explicitly in terms of the SNR
maps zeroand one tozero and 1/4, respectively. Thus, wehave andtheedgedependencefactor. Wehaveinvestigatedtheproper-
ζ = g(f(r )), and for given physical parameters (with a slight tiesoftheerrorexponentasafunctionofSNRandcorrelation.We
n
abuseofnotation), haveseenthatthebehavioroftheerrorexponentw.r.t.correlation
strength isdivided into two regions depending on SNR. At high
s(SNR,ζ)= s(SNR,g(f(rn)))= s(SNR,rn). SNR, i.i.d. (and, thus, uncorrelated) observations maximize the
K K K
error exponent for a given SNR, whereas there is non-zero opti-
WewillusetheargumentsSNRandζfor properlyifnecessary.
Ks malvalueofcorrelationatlowSNR.Further,ithasbeenseenthat
4.2. Energyefficiency thereisadiscontinuityfortheoptimalcorrelationasafunctionof
Wenow consider theenergy efficiencyof theadhoc sensor net- SNR.Basedontheerrorexponent, wehavealsoinvestigatedthe
workasthenetworksizegrows. Theenergyefficiencyη canbe energyefficiencyofadhocsensornetworkfordetectionapplica-
definedas tions.Forafixednodedensity,theenergyefficiencydecaystozero
η= totalgatheredinformationIt, (16) withrateO(area−1/2)asweincreasethearea.Ontheotherhand,
totalrequiredenergyEt non-zeroefficiencyispossiblewithincreasingdensitydepending
where I is given by the product of the number of sensors and onphysicalcorrelationstrengthasafunctionofthelinklength.
t
theinformation pereachsensor. Weconsidertwoasymptotic
s
regimes for theKincrease in network size: an infinite area model 6. REFERENCES
with fixed density and an infinitedensity model withfixed area.
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