Article Non-Orthogonal Multiple Access for Ubiquitous Wireless Sensor Networks Asim Anwar 1

16 Pages·2017·2.39 MB·English

sensors
Article
Non-Orthogonal Multiple Access for Ubiquitous
Wireless Sensor Networks
AsimAnwar1,Boon-ChongSeet1,* ID andZhiguoDing2
1 DepartmentofElectricalandElectronicEngineering,AucklandUniversityofTechnology,Auckland1010,
NewZealand;[email protected]
2 SchoolofComputingandCommunications,LancasterUniversity,LancasterLA14YW,UK;
[email protected]
* Correspondence:[email protected];Tel.:[phone]
Received:31December2017;Accepted:7February2018;Published:8February2018
Abstract: Ubiquitous wireless sensor networks (UWSNs) have become a critical technology
for enabling smart cities and other ubiquitous monitoring applications. Their deployment,
however,canbeseriouslyhamperedbythespectrumavailabletothesheernumberofsensorsfor
communication. TosupportthecommunicationneedsofUWSNswithoutrequiringmorespectrum
resources,thepower-domainnon-orthogonalmultipleaccess(NOMA)techniqueoriginallyproposed
for5thGeneration(5G)cellularnetworksisinvestigatedforUWSNsforthefirsttimeinthispaper.
However, unlike 5G networks that operate in the licensed spectrum, UWSNs mostly operate in
unlicensedspectrumwheresensorsalsoexperiencecross-technologyinterferencesfromotherdevices
sharingthesamespectrum. Inthispaper,wemodeltheinterferencesfromvarioussourcesatthe
sensorsusingstochasticgeometryframework. Toevaluatetheperformance,wederiveatheorem
and present new closed form expression for the outage probability of the sensors in a downlink
scenario under interference limited environment. In addition, diversity analysis for the ordered
NOMAusersisperformed. Basedonthederivedoutageprobability,weevaluatetheaveragelink
throughputandenergyconsumptionefficiencyofNOMAagainstconventionalorthogonalmultiple
access(OMA)techniqueinUWSNs. Further,therequiredcomputationalcomplexityfortheNOMA
usersispresented.
Keywords: cross-technology interference; non-orthogonal multiple access; stochastic geometry;
unlicensedspectrum;ubiquitouswirelesssensornetworks
1. Introduction
Non-orthogonalmultipleaccess(NOMA)isemergingasastrongcandidateforadoptionasthe
multipleaccesstechnologytoenhancesystemcapacityfor5thGeneration(5G)cellularsystems[1,2].
In NOMA, the transmitted signals of multiple users are multiplexed in the power domain using
sametime,frequencyorcoderesource,anddemultiplexedbyapplyinganinterferencecancellation
techniqueatthereceiver[3].
Althoughoriginallyproposedforcellularsystems,NOMAexhibitsstrengthsthatweconsider
ashighlyrelevanttoaddressingthedeploymentchallengesofubiquitouswirelesssensornetworks
(UWSNs),i.e.,large-scalenetworksofwirelesssensorsdenselydeployedforubiquitousmonitoring
of physical environments. Specifically, for a given spectrum bandwidth, NOMA can enable more
simultaneousconnectionsthanexistingapproacheswithouttheoverheadsofcodingandspreadingto
facilitatetheseparationofusers’signalsatthereceiver[4]. Thisisparticularlyattractiveforsupporting
massiveconnectivitywithoutrequiringmorespectrumresourcesinUWSNs.
NOMA can be applicable to both uplink (sensors-to-sink) and downlink (sink-to-sensors)
communicationwherepowerfulsinknodescanperformtheequivalentroleofbasestations(BSs)for
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thetasksofusergroupingandtransmissionpowerallocation. InUWSN,however,thereisagreater
motivationandchallengetoapplyNOMAinthedownlink(DL).
Firstly,UWSNswithlargegeographiccoveragetypicallyuseshort-rangemulti-hopcommunication
to conserve energy [5]. The size of the routing table at each node increases with the number of
destinations. Unlike in uplink communication where the sink node is the main destination of
all sensors’ outgoing traffic, the routing table size for DL communication can grow prohibitively
with a massive number of sensors as destinations [6]. In this case, NOMA can offer a practical
solutionbyenablingdirectDLtransmissionsfromsinknodetomultiplesensorssimultaneously. DL
transmissions are initiated when sink node queries a specific sensor or group of sensors for some
information [7,8], or communicates information essential for their operations such as sleep-wake
schedules[9]. Further,unlike5G,mostUWSNsoperateinunlicensedspectrumwheresensorsalso
experiencecross-technology(CT)interferences[10]fromotherdevicessharingthesamespectrum.
The main novelty and contributions of this paper are as follows. For the first time, NOMA
is proposed as a spectrum efficient means of supporting massive connectivity in UWSNs, and its
performance in a DL scenario where sink transmits to a group of sensors using NOMA under
CT and other interferences is investigated using stochastic geometry [11]. The sensors, sinks and
CT nodes can reside randomly and independently of each other in a two-dimensional (2D) plane.
Hence, their spatial topologies can be modeled with three different homogeneous Poisson Point
Processes(PPPs). Wefurtherderivetheclosed-formexpressionfortheoutageprobabilityatprobe
receiver’s location and analyze its diversity order. We also present the average link throughput
and energy consumption efficiency analysis to gain better understanding of applying NOMA to
UWSN and benchmark the performance against conventional orthogonal multiple access (OMA).
Numericalresultsareshowntovalidatetheaccuracyoftheperformedanalysisaswellascomparethe
outage,throughputandenergyefficiencyperformancesbetweentheNOMAandOMAbasedUWSNs.
Moreover,acomputationalcomplexityanalysisisperformedtoevaluatethecomplexityrequiredby
SICunitsofsensorreceiverstodecodeNOMAmessagesignals.
The rest of the paper is organized as follows. Section 2 reviews the related works. Section 3
describes the network model underlying our study. Section 4 derives the outage probability and
diversityorderatthesensorreceiverunderinterferenceconstraints. Thenetworkperformancesin
terms of the average link throughput and energy consumption efficiency are then defined based
onthederivedoutageprobabilityinSection5. Next,wepresentthenumericalresultsinSection6,
andconcludethepaperinSection7.
2. RelatedWorks
Early studies have focused on the performance comparison between NOMA and orthogonal
frequencydivisionmultipleaccess(OFDMA).Throughsystemlevelsimulations,theauthorsin[12]
reported30–40%gaininoverallcellthroughputforNOMAoverOFDMAunderfrequency-selective
widebandschedulingandpowerallocation.
In[13],theauthorsanalyzedtheperformanceofDLNOMAwithrandomlydeployedusers. They
evaluated performance under two scenarios: i.e., when user’s quality of service (QoS) and user’s
channelconditiondetermineitsrate. Theresultsshowthatappropriaterateandpowerallocation
wouldresultinbetteroutageandergodicsumrateinNOMAthanOMA.
ThefairnessachievedbyDLNOMAisdiscussedin[14]. Theauthorsproposedoptimalpower
allocationcoefficientsbasedontheavailabilityofperfectaveragechannelstateinformation. Results
showedthatNOMAwithproposedpowerallocationmaintainshighfairnessandachievessuperior
outageperformance(userrate)comparedtotimedivisionmultipleaccess(TDMA)andNOMAwith
fixedpowerallocation.
Theauthorsin[15]consideredaNOMAemployedinanunderlaycognitiveradio(CR)network.
ThesecondaryBScommunicateswithsecondaryusers(SUs)usingNOMA.Theprimarytransmitters
(PTs)spatialtopologyismodeledbyahomogenousPPPintheprimarynetwork. TheSUreceptions
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areinterferedbyPTstransmissions. Stochasticgeometrytoolsarethenutilizedtoanalyzeinterference
andoutageattheSUreceiver. TheresultsdemonstratethatNOMAachievesloweroutageprobability
comparedtoconventionalOMA.
In[16],amulti-inputmulti-output(MIMO)NOMAframeworkbasedontheprincipleofsignal
alignmentisproposed. TheperformanceofMIMONOMAsystemisevaluatedunderfixedandCR
inspiredpowerallocationstrategies. UnlikeconventionalMIMOsystemsbasedonOFDMwherea
singlesub-channelisallocatedtooneuser, multipleusersinMIMONOMAsystemcanshareone
sub-channel. The results show that the proposed framework can improve reception reliability of
existingMIMONOMAsystems. Thesameauthorsfurtherproposedaprecodingdesignin[17]for
MIMONOMAinaDLscenariotosuppressinter-clusterinterferencewhereasintra-clusterinterference
isminimizedusingsuccessiveinterferencecancellation(SIC)technique. Theauthorsin[18]analyzed
theperformanceofMIMONOMAandMIMOOMAsystemswhenmultipleusersaregroupedinto
clusters. Further,theyproposedanadmissioncontrolschemetoadaptivelytrade-offbetweensum
rateandthenumberofusersthatcanbeadmittedintoacluster. TheresultsshowthatMIMONOMA
canoutperformMIMOOMAintermsofbothsumrateanduserfairness.
NOMAwithtopologicalinterferencemanagement(TIM)forsingleinputsingleoutput(SISO)
broadcastchannel(BC)isproposedin[19]. ThetotalKusersaredividedintoTgroups. TheBSapplies
DLNOMA,whichresultsininter-andintra-groupinterferences. Theauthorsappliedtheproposed
TIMandSICtominimizeinter-group,andintra-groupinterference,respectively. Resultsshowedthat
theproposedschemeachievessuperiorsumratecomparedtotraditionalTDMA.
In [20], the authors analyzed a multi-cell uplink NOMA cellular network using stochastic
geometry. ThelocationsofBSsandcellularusersaremodeledbyPoissonclusterprocess. Closed-form
expressionsfortheLaplacetransformoftheinterferenceatBSarederivedbyconsideringbothintra-
andinter-clusterinterferencesundervariousSICscenarios. NOMAisshowntooutperformOMAin
termsofaverageratecoverage.Theauthorsin[21]proposedauserschedulingschemebasedonwhich
aclosed-formexpressionforpowerallocationisderivedtomaximizeenergyefficiencyofdownlink
NOMAsystemsunderimperfectchannelstateinformation. Resultsshowthattheproposedscheme
canachievehigherenergyefficiencythanexistingschemesandOMA.Similarly,in[22],theauthors
presentedenergyefficientuserschedulingandpowerallocationschemesforNOMAwithnotonly
imperfectchannelstateinformationbutalsomutualcross-tierinterferenceinheterogeneousnetworks.
Theystudiedthetrade-offsbetweendatarateandenergyperformances,andreportedahighenergy
efficiency of NOMA in such networks. In [23], a joint subcarrier and power allocation method is
proposed for NOMA based amplify-and-forward relaying that can secure communications in the
presenceofeavesdroppersthroughcooperativejamming. Theproposedmethodshowsenhanced
NOMAenergyefficiencyandsecurityoverrandomresourceallocation. Morerecently,theauthors
in[24]consideredthreecognitiveNOMAarchitectures: underlay,overlayandCR-inspiredNOMA
architectures,andpresentedcooperativerelayingstrategiesformitigatinginter-andintra-network
interferencesinordertoimprovetheirreceptionreliability.
Todate,NOMAhasbeeninvestigatedonlyinthecontextofcellularnetworks. TheuseofNOMA
forUWSNhasnotbeenproposedinliterature,andthereportedperformancegainsofNOMAover
itscounterpartscannotbestraightforwardlyclaimedforUWSN.Thisisbecause,unlikeincellular
network,thesinknodeinaUWSNhasnocontroloveralltransmitterswithinitscoverage,including
sensorsandsinksofotherUWSNsunderdifferentadministrativedomainsorCTnodesthatshare
the same spectrum such as WiFi and Bluetooth devices. Hence, it is important to investigate the
performanceofaUWSNemployingNOMAunderinterference-limitedscenario.
3. NetworkModel
WeconsideraUWSNcomprisingofsensorandsinknodes,whicharerandomlydistributedin
an infinite 2D plane. Further, it is assumed that the CT nodes are also co-located with sensor and
sinknodes. TheseCTnodesarenotapartofUWSNbutareoperatinginthesamefrequencyband,
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andhencecauseinterferencetothereceptionofprobereceiver. Thespatialtopologyofthesensor,sink
sainndk CaTndtr CanTs mtraitntesrmnitotdere sniosdmeso dise mledodbeyletdhr beey hthormeeo gheonmouosgePnPoPuss, dPePnPost,e ddebnyotΞeSdE b,Ξy SKSaEnd, ΞCSKT wanitdh
inten switiieths iλnte,nλsitieas ndλ, ,res panecdt ively., respectively.
CT SE SK SECT SK CT
WWee ffooccuuss oonn aa DDLL ttrraannssmmiissssiioonn sscceennaarriioo wwhheerree aa ssiinnkk nnooddee ccoommmmuunniiccaatteess wwiitthh sseennssoorr nnooddeess
uussiinngg NNOOMMAA.. TTooa vavoiodida mambibgiugiutyit,yw, ewree freerfteor tthoe tsheen ssoernsroecre riveceersivaesrsu saesr su.sWeres.a Wlsoe raelfseor troeftehre tpor othbee
psirnokbne osdineka nsoadtee satst raa tnessmt tirttaenr,smwhititcehr,i wschoinchsi dise creodnstiodbeereldo ctaot ebde alotctahteecde antt etrheo fcaendtiesrc oAf aw ditihscr a(cid:1827)d iwusitRh.
TrahdeiuMs N(cid:1844).O TMheA (cid:1839)us eNrsOaMreAc ounsesirds earreed ctoonbseiduenreifdo rtom blye duinstifroibrumtleyd dinisstirdibeudtiesdc iAn,siadses hdoiswc n(cid:1827)i,n aFs isghuorwen1.
iFnu Frtihgeurr,eb 1y. pFruorbtheerre,c beyiv perr,owbee raelcweaivyesrm, weaen altwheayms- tmheNanO tMheA mu-stehr .NOMA user.

(a)

(b)
FFiigguurree 11.. (a()a )AA reraelaizliaztaiotino nofo fΞSE , ,ΞSK aanndd ΞCTt rtarnasnmsmitittetrerp prorocecsesseses;s;a anndd ((bb)) aann iilllluussttrraattiioonn ooff
SE SK CT
ssiinnkk--ttoo--sseennssoorrss ccoommmmuunniiccaattiioonn uussiinngg NNOOMMAA uunnddeerr iinntteerrffeerreennccee ooff ootthheerr ssiinnkk,, sseennssoorr aanndd CCTT nnooddeess..
All communication links in the network follow a composite Rayleigh fading and distance
All communication links in the network follow a composite Rayleigh fading and distance
dependent path-loss channel model. The channel between the m-th user and test transmitter is given
dependentpath-losschannelmodel. Thechannelbetweenthem-thuserandtesttransmitterisgiven
bbyy hhmm=hhˆˆmm((11+dmd)αm)1/−21, /w2,hwerhee rheˆmhˆ manadn dddmm rreepprreesseenntt tthhee RRaayylleeiigghh ffaaddiinngg cchhaannnneell ggaaiinn,, aanndd ddiissttaannccee
bbeettwweeeenn tthhee tteesstt trtraannssmmititteter raanndd mm-t-ht husuesre, rr,ersepsepceticvtievlyel, ya,nadn dα iiss tthhee ppaatthh lloossss eexxppoonneennt.t .WWe eusues ae
baobuonudnedde dpaptaht hlolsoss smmodoedle tlot oavavooidid isissusue eoof fssiningguulalarritiyty aatt ssmmaallll ddiissttaanncceess [[1111,,1133]].. WWiitthhoouutt lloossss ooff
ggeenneerraalliittyy,, wwee aassssuummeeN NOOMMAAu usesresr’sc’ hcahnannenlegl agianisnasr aeroer doredreedreads a|hs1 |2h1≤2 ......≤ |hhMM2|2. .CCoonnsseeqquueennttllyy,,
the power allocation coefficients under NOMA are sorted as  ... with M  1. The
1 M im1 i
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the power allocation coefficients under NOMA are sorted as β ≥ ... ≥ β with ∑M β = 1.
1 M i=m+1 i
ThetesttransmittersendsasuperimposedsignaltoallNOMAusersandthereceivedsignalatthe
m-thuserisgivenas:
M
∑(cid:112)
r = h β Ps +n , (1)
m m i i m
i=1
where P is the transmission power of test transmitter, s is the message signal of i-th sensor node,
i
andn istheadditivewhiteGaussiannoise(AWGN)withzeromeanandvarianceσ2.
m
ForeachNOMAuser,therearetwotypesofinterferencethatinterferethereceptionofdesired
signal. Firstistheintra-userinterferencethatispresentduetothesuperpositionofmultipleusersin
NOMA,andthesecondisduetothetransmissionofundesiredtransmittersinthenetwork. SICis
employedateachuserreceivertomitigatetheintra-userinterference. Theoptimaldecodingorderof
SICisinthesequenceofincreasingchannelgains. Therefore,m-thuserdecodesthemessagesignalsof
alljusers,j < m,beforedecodingitsownmessage,andtreatsthemessagesignalsoftheusersj > m
asnoise.
Forthesecondtype,weconsiderthattheinterferencelinksamongunwantedtransmittersand
probeNOMAreceiveraredominatedbypathloss. Assumetheprobereceiverislocatedatoriginof
thecoordinatesystemi.e.,d = (x ,0)withx (cid:54)=0,thenthetotalinterferenceatprobeuser’slocation
m 0 0
canbewrittenas, I = ∑w∈ΞSE(1+dαw)−1+∑x∈ΞSK\x0(1+dαx)−1+∑y∈ΞCT(1+dαy)−1,wheredw, dx
andd representthedistancesbetweenundesiredtransmittingsensors,sinks,CTnodesandtheprobe
y
receiver,respectively. Further,sensorsandsinksoperateonverylowpowerlevels,thustheNOMA
users may experience excessive interference from a nearby transmitter(s) leading to a situation of
completeoutage. Toavoidthis,weconsideraguardzoneofradiusd aroundeachNOMAuser,within
0
whichnounwantedtransmitterisallowedtotransmit[25].
AlistofcommonlyusedvariablesissummarizedinTable1.
Table1.Commonlyusedvariables.
Notation Description
α Pathlossexponent
βj Powerallocationcoefficientofj-thuser
(cid:118)j TargetSINRthresholdforj-thuser
Pm Outageprobabilityofthem-thuser
out
Φ(·,·;·) Confluenthyper-geometricfunction
Λm→j SINRatm-thusertodecodej-thusermessage
Λm SINRatm-thusertodecodeitsownmessage
κ Averageinterferencelevel
χ AveragesystemSNR
dm Distancebetweentesttransmitterandm-thuser
d Guardzoneradiusaroundeachsensorreceiver
0
hm Channelbetweenm-thNOMAuserandtesttransmitter
M TotalnumberofNOMAusers
L (·) LaguerrepolynomialofdegreeS
S
N,T Gaussian–Chebyshevparameters
P Transmitpoweroftesttransmitter
Pc Constantpowerconsumptionofcircuits
R RadiusofdiscA
TPm Averagelinkthroughputofuserm
ς Overallenergyconsumptionefficiency
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LetΛm→j representsthesignal-to-interference-and-noiseratio(SINR)atthem-thNOMAuserto
decodethemessagesignalofj-thuser,j < m,thenΛm→j canbecomputedasfollows:
|h |2β χ
Λm→j = |h |2χ∑Mm βj +κI+1, (2)
m i=j+1 i
whereχ(cid:44) P/σ2istheaveragesystemsignal-to-noiseratio(SNR),κ (cid:44) P /σ2istheaverageinterference
I
level,andP isthecommonmaximumtransmissionpoweravailabletosink,sensorandCTnodes.
I
Ifallthej < musersaredecodedandremovedsuccessfullybythem-thuserfromitsobservation
signalr ,thenSINRrequiredtodecodeitsownmessageisgivenby:
m
|h |2β χ
Λm→m = |h |2χ∑Mm mβ +κI+1. (3)
m i=m+1 i
4. OutageandDiversityAnalysis
4.1. OutageProbabilityAnalysis
Inthissection, wepresentanexactanalysisoftheoutageprobabilityform-thuserunderthe
considerednetworksetting.Let(cid:118) andR representthetargetSINR,andtherateforuserj,respectively,
j j
where 1 ≤ j ≤ M and (cid:118)j = 2Rj −1. To simplify notation, we define ∆m,j (cid:44) (cid:8)Λm→j < (cid:118)j(cid:9) as the
outageeventatm-thuserwhenitfailstodecodethemessageofj-thuser,1≤ j ≤ m. Consequently,
theoutageprobabilityform-thuser,denotedbyPm ,canbewrittenasfollows:
out
Pm =1−P (∆c ∩...∩∆c ), (4)
out r m,1 m,m
where∆c isthecomplementeventof∆ . Toproceedfurther,rewrite∆c as:
m,j m,j m,j
(cid:26) (cid:27)
∆cm,j = (cid:8)Λ(cid:110)m→j > (cid:118)j(cid:9) = |hm|2χ∑|hiMm=|j+2β1jβχi+κI+1 > (cid:118)(cid:111)j
= |h |2χ(β −(cid:118) ∑M β ) > (cid:118) (κI+1) (5)
m j j i=j+1 i j
(=a)(cid:110)|h |2 > τ(κI+1)(cid:111),
m j
where τ = (cid:118) [χ(β −(cid:118) ∑M β )]−1. Step (a) providesthefollowingessentialconditiontokeep
j j j j i=j+1 i
NOMAoperational:
M
∑
C1: β −(cid:118) β >0. (6)
j j i
i=j+1
whenconditionC1isviolated,them-thuserwillalwayssufferoutage,irrespectiveofthechannelSNR.
Further,bydefiningτ∗ =max{τ ,...,τ },Pm canbewrittenas:
m 1 m out
(cid:16) (cid:17)
Pm =1−Pr |h |2 > τ∗(κI+1) . (7)
out m m
To proceed forward, we notice that |h |2 are the ordered channel gains. Let F (·) and f (·)
m X X
denotethecumulativedistributionfunction(CDF),andtheprobabilitydensityfunction(PDF),ofX
(cid:12) (cid:12)2
respectively. Then,theorderedandunorderedchannel(cid:12)(cid:101)h(cid:12) havefollowingrelationships[26]:
(cid:12) (cid:12)
(cid:32) (cid:33)
M∑−m M−m (−1)q(cid:104) (cid:105)m+q
F (t) = µ F (t) . (8a)
|hm|2 m q m+q |(cid:101)h|2
q=0
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(cid:32) (cid:33)
f (t) = µ M∑−m M−m (−1)q(cid:104)F (t)(cid:105)m+q−1f (t), (8b)
|hm|2 m q |(cid:101)h|2 |(cid:101)h|2
q=0
whereµ = M![(M−m)!(m−1)!]−1.Tothisend,thefollowingtheorempresentstheexactexpression
m
forPm .
out
Theorem1. Theoutageprobabilityofthem-thNOMAusercanbederivedas:
Pomut = Θ1 ∑S Ψseυgs(cid:26)2∑K Re(cid:20)e−λπ[(e−ckd0−α−1)d20+ck∑tT=1ηte−utckd0−α]+iπΘgs(cid:21)
(cid:34) s=1 (cid:32) k=0 (cid:33)
M−m M−m
× µm q∑=0 q (−1)q∑nN=1ψn(Φ(δ,1+δ;−an(κgs+1))+εΦ(1+δ,2+δ;−an(κgs+1))) (9)
×(cid:16)1−e−τm∗(κgs+1)tnΦ(δ,1+δ;−an(κgs+1))(cid:17)m+q−1(cid:21)(cid:27),
where Ψs = wsegs, ws = S!(S+Γ1()S2+[L1S)+g1s(gs)]2, LS(·) is the Laguerre polynomial of degree S, gs are the
roots of L (·), c = υ + iπk, υ = ρ−log(ξ)/Θ, ρ is a real number, ξ is the desired relative accuracy,
S k 0 Θ 0√
Θ is a scaling parameter, i = −1, K is the number of terms used to invert the Laplace transform,
(cid:113)
λ = λ +λ +λ ,η = 1d2−αω 1−φ2u−δ,ω = π/T,u = (1+φ )/2,φ =cos((2t−1)π/2T),
SE SK CT t 2 0 t t t t t t t
ψn = ωn(cid:112)1−θn2τm∗(κz+1)e−τm∗(κz+1)tn, θn = cos((2n−1)π/2N), ωt = π/N, tn = (1+θn)/2,
a = τ∗t Rα,δ =2/α,ε = δ(1+δ)−1Rα,TandNarecomplexity-accuracytradeoffparameters,andΦ(·,·;·)
n m n
isaconfluenthyper-geometricfunction.
Proof1. SeeAppendixA.
4.2. DiversityAnalysis
Inthissubsection,wepresentthediversityanalysisfortheorderedNOMAusersinhighSNR
regime. Thediversityorderofthem-thuseroutageprobabilityisdefinedas:
logPm
D = −lim out. (10)
χ→∞ logχ
ToobtainD,wefirstnoticefromEquation(7)thattheasymptoticoutageprobabilityinhighSNR
P∞
regime,denotedas canbeexpressedas:
m
(cid:16) (cid:17)
P∞ =Pr |h |2 < t∗ , (11)
m m
where t∗ = τ(cid:101)mκIχ−1, τ(cid:101)m = max{τ1,...,τm} and τm = (cid:118)m[(βm−(cid:118)m∑iM=m+1βi)]−1. Next, when
χ → ∞ and t∗ →0,similartoEquation(8a),theCDFoftheorderedchannelisexpressedas:
(cid:32) (cid:33)
F∞ (t∗) = µ M∑−m M−m (−1)q(cid:104)F∞ (t∗)(cid:105)m+q (12)
|hm|2 m q=0 q m+q |(cid:101)h|2
WenoticeintheCDFoftheunorderedchannelinAppendixAthat Φ(δ,1+δ;−t∗Rα) →1 and
e−t∗ ≈1−t∗. Hence,F∞ (t∗)inEquation(12)canbeapproximatedas:
|(cid:101)h|2
F∞ (t∗) ≈ t∗ (13)
|(cid:101)h|2
SubstitutingEquation(13)intoEquation(12),F∞ (t∗)canbeexpressedas:
|hm|2
F|∞hm|2(t∗) ≈ ϑ(cid:16)τ(cid:101)mκIχ−1(cid:17)m+o(cid:104)(cid:16)τ(cid:101)mκIχ−1(cid:17)m(cid:105), (14)
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whereϑ = µ /m. BasedonEquation(14),P∞ in(11)isgivenas:
m m
P∞m ≈ χ1m(cid:90)0∞ϑ(τ(cid:101)mκz)mfI(z)dz (15)
(cid:124) (cid:123)(cid:122) (cid:125)
Ψ
ItcanbeobservedthatΨisaconstant. Hence,Equation(15)canbeexpressedasfollows:
P∞ ≈ Aχ−m+o(χ−m). (16)
m
Finally, substituting Equation (16) into Equation (10), the diversity order experienced by the
m-th user is found to be m. It can be observed from Equation (15) that interference in integral
Ψ is independent of χ and hence the factor χ−m dominates Ψ in Equation (15) under high
SNR. This indicates that the interference-limited NOMA based UWSN becomes equivalent to an
interference-freenetwork. Thisresultondiversityordercanbeinterpretedasfollows. First,them-th
useravailsexactlymchancestodecodeitsownmessage. Second,them-thuserhas(m-1)interferences
fromthehigherorderusersthatneedtobecancelledoutbyapplyingSIC.Hence,itobtainsadiversity
ofm.
5. ThroughputandEnergyConsumptionEfficiency
5.1. LinkThroughputEfficiency
Thelinkthroughputefficiencybetweenthem-thuserandthetesttransmitter,denotedbyTP
m
andmeasuredin[bits/s/Hz]isdefinedas[27]:
TP = (1−Pm )log (1+(cid:118) ). (17)
m out 2 m
Correspondingly,theaveragelinkthroughputefficiencyofthenetworkcanbefoundas:
M
∑
TP = ( TP )/M (18)
avg m
m=1
5.2. EnergyConsumptionEfficiency
NOMA is used by the sink node to communicate with the sensors. Since NOMA is typically
appliedontopofanunderlyingaccesstechnologytobetterreusethetransmissionresources, e.g.,
timeslots,frequencychannels,orspreadingcodes,anadditionalSICunitisrequiredatthesensorsto
decodethedesiredmessage. Itisthusimportanttoanalyzetheoverallenergyconsumptionefficiency
ςoftheNOMAbasedUWSN.Thetotalenergyconsumptionalongthesignalpath(communication
andcircuits)canbedecomposedintothreemaincomponents[28]: thepowerconsumedbypower
amplifiersP ,thepowerconsumedbySICunittoprocessinformationP,andthepowerconsumedby
a s
allothercircuitunits(filters,mixers,frequencysynthesizer,etc.) P. Hence,theoverallςinJoules/bit
c
canbeexpressedas:
ς = (P +MP +P)/R (19)
a c s b
whereR = B·TP isthebitrate,BisthechannelbandwidthinHz,P = νP,ν = ν /ν ,ν isthe
b avg a 2 1 1
drainefficiencyofthepoweramplifier,andν isthepeak-to-averageratio.
2
TheP isconsideredasaconstant,andP canberegardedastheaveragepowerconsumedby
c s
SICunitsofallscheduledsensorreceivers. TofindP,weneedtocomputethepowerconsumedby
s
m-thsensor,P ,toprocessN = |B·TP |bits. OurapproachistoexpressP intermsoftherequired
m b m m
computationalcomplexityC .
m
Sensors2018,18,516 9of16
To proceed forward, we first require a power consumption model for the sensor device.
ConsideringthesensorasCMOSdevice,thepowerconsumedduringcomputationinastaticCMOS
deviceisgivenas[29]:
P =C V2(cid:125) , (20)
m eff m m
(cid:125)
whereC ,V and aretheeffectiveswitchingcapacitance,supplyvoltage,andclockfrequencyof
eff m m
(cid:125)
m-thsensor,respectively. Further,V and aredirectlyrelatedas:
m m
(cid:125) = ∂V , (21)
m m
where∂ >0isadesignparameterthattakesavalueofO(B)[30]. Dynamicvoltagescaling(DVS)is
astandardtechniqueforconservingpowerinCMOSdevicesthroughdynamicallyadjuststheclock
frequencybyscalingthevoltageaccordingtoprocessingload[31]. ForagivenN ,V canbescaledto
b m
(cid:125)
match withC /s. BasedonEquations(20)and(21),P canbeexpressedas:
m m m
(cid:20)C (cid:21)3
P =C m (22)
m eff
∂
andC isgivenintermsofnumberoffloatingpointoperations(FLOPs)perbitdecision[32]as:
m
C = N [2N M+5M+8∑M−1m+(M−1)(5+2N )]
m b s m=1 s (23)
+2MN (M−1)+2MN +M+Mlog (M)
s b 2
where N isthenumberofsamplestorepresentonebit. Finally, basedonEquations(22)and(23),
s
substitutingP = M−1∑M P intoEquation(19)obtainstheoverallς.
s m=1 m
6. NumericalResults
This section presents results to verify the accuracy of outage probability, link throughput
and energy consumption analysis for a UWSN which uses NOMA for downlink (sink-to-source)
transmission. Wecalculatethecoefficients β accordingtothepowerallocationschemeproposed
m
forNOMAin[13],withβ = (M−m+1)/µ,whereµisselectedsuchthat∑M β =1. Inallthe
m m=1 m
simulations,weuseparametervaluesinTable2unlessotherwisestated.
Table2.Simulationparameters.
Parameter Description Value(s)
α Pathlossexponent 4
βm Powerallocationcoefficientofm-thuser {0.6,0.3,0.1}1,2,3
(cid:118)m SINRthresholdform-thuser {0.9,1.5,2}1,2,3
λSE IntensityofΞSE 10−3
λSK IntensityofΞSK 10−4
λCT IntensityofΞCT 10−3
κ Averageinterferencelevel 1dB
σ2 Noisepower −90dBm
χ AveragesystemSNR 10to50dB
K Fourierseriesterms 10
M TotalNOMAusers 3
N Gaussian–Chebyshevparameter 3
P Testtransmittertransmissionpower −10to30dBm
R Transmissionradiusoftesttransmitter 10m
S DegreeofLaguerrepolynomial 2
T Gaussian–Chebyshevparameter 5
Figure2showstheoutageprobabilityofeachm-thuserundertheimpactofdifferenttransmission
radius R ofthesinknode. Thesolidanddashedcurvesareanalyticalresultsobtainedbyplotting
Sensors 2018, 18, x FOR PEER REVIEW 10 of 16
K Fourier series terms 10
M Total NOMA users 3
N Gaussian–Chebyshev parameter 3
P Test transmitter transmission power −10 to 30 dBm
R Transmission radius of test transmitter 10 m
S Degree of Laguerre polynomial 2
T Gaussian–Chebyshev parameter 5
Sensors2018,18,516 10of16
SenFsiogrsu 2r0e1 82, 1s8h,o xw FOs Rth PeE oERu tRaEgVeI EpWro b ability of each m-th user under the impact of different transmiss1i0o nof 16
radius R of the sink node. The solid and dashed curves are analytical results obtained by
p(9lo),twtinhgil e(9s)i,m wKuh lialtei osnimreusluatlitFosonau rrereisseuhr lostwes raniereisn tse“hrom”wsan n idn ““+●””t aonvda l“i+d”a tteo ovuarliddearti1ev 0oa tuior ndse.rIitvcaatinonbse. oItb csearnv bede
othbasterinvcerde athsianMt gin Rcrreeassuinltgs iRnT horiegtsahule lNtrsOo iunM thaAigg euhpseerr orosbu atbagileit yprdoubeabtioliitny cdreuaes teod inpcarte3ha sloesds .pFatuhr tlhosesr., Fthuertohredre, rthede
oNrdOeMreAd NusOerMsNAh a uvseedrsif hfearveenG tdaoiufufsetsarigeanent–p oCeurhfteoabgrmye saphneecrvfe o,praamsraaemnaccehet,e hara ss eaacdhif hfearse na tdci3hff aenrennetl ccohnandniteiol nco.ndition.
P Test transmitter transmission power −10 to 30 dBm
R Transmission radius of test transmitter 10 m
S Degree of Laguerre polynomial 2
T Gaussian–Chebyshev parameter 5
Figure 2 shows the outage probability of each m-th user under the impact of different transmission
radius R of the sink node. The solid and dashed curves are analytical results obtained by
plotting (9), while simulation results are shown in “●” and “+” to validate our derivations. It can be
observed that increasing R results in higher outage probability due to increased path loss. Further, the
ordered NOMA users have different outage performance, as each has a different channel condition.

Figure 2. Impact of (cid:1844) on outage probability of each m-th user.
Figure2.ImpactofRonoutageprobabilityofeachm-thuser.
Figure 3 demonstrates the average outage probability under different interference levels ,
Figure3demonstratestheaverageoutageprobabilityunderdifferentinterferencelevelsκ,defined
defined immediately after Equation (2). Reducing  expectedly lowers the outage probability.
immediatelyafterEquation(2). Reducingκexpectedlylowerstheoutageprobability. Figure4further
Figure 4 further shows the outage probability of each m-th user under the impact of different path
showstheoutageprobabilityofeachm-thuserundertheimpactofdifferentpathlossα. Theresults
loss α. The results show that NOMA achieves lower outage probability than OMA for different
showthatNOMAachievesloweroutageprobabilitythanOMAfordifferentvaluesofα. Thelink
values of α. The link throughput efficiency of each m-th user with increasing transmit power of test
throughputefficiencyofeachm-thuserwithincreasingtransmitpoweroftesttransmitter(sinknode)
transmitter (sink node) are shown in Figure 5. NOMA achieves better throughput efficiency than
areshowninFigure5. NOMAachievesbetterthroughputefficiencythanOMA,asalltheordered
OMA, as all the ordered NOMA users have lower outage probability than their OMA counterparts.
NOMAusershaveloweroutageprobabilitythantheirOMAcounterparts.
Figure6showstheenergyconsumptionefficiencyςcomparisonbetweenNOMAandOMAasa
Figure 2. Impact of (cid:1844) on outage probability of each m-th user.
functionoftransmitpowerP.Theresultsareobtainedbyconsideringtwoandthreeusers. Thepower
allocatioFnigcuoreef fi3c ideenmtsoannsdtraStIeNsR thther easvheoraldgse foourtMag=e 3praorebagbivileitny inunTdaebrle d2i.ffFeorernMt i=nt2e,rtfheereynacree lcehvoeslesn ,
asdβemfi=ne{d0 .i8m,0m.2e}dmia=t1e,2ly,a nafdte(cid:118)r mE=qu{a0ti.o9n,1 .(52})m. =R1e,2druecsipnegc tive leyx.pFeucrttehdelry, Cloefwf e=rs2 t×he1 0o−u1t5aFgaer apdrsobfoarb7il0ity.
nmFiCgMurOe S4 tfeucrhtnhoerlo sghyo[w33s ]t,hceh aonuntaelgbe apnrdowbaidbtilhitBy o=f 2eaMchH mz,-tNhs u=se4r suanmdpelre tshteo irmeppraecste notf odnifefebriet,nat npdath
cirlcousist pαo. wTehrec orenssuulmts psthioonwP cth=at2 0NdOBMmA. O avcehriaelvl,eNs OloMwAer aocuhtiaevgees pbreotbtearbςilitthya nthOanM OAMduAe ftoora dloifwfeerrent
ouvtaaglueepsr oobf aαb.i lTithyew lihnikc hthrreosuulgtshpinulta erfgfeicriRenbciyn oEfq euaacthio nm(-1th9 )uasnedr wcointhse iqnucerenatlsyinhgi gthraenrsςm. Hit opwoewveerr ,otfh etest
ςotfrabnotshmsicttheerm (seisnakr encoodme)p aarrae bslheobweyno nind Faitgruarnes m5.i tNpOowMeAr oafc1h5iedvBems ,bwethteicrh thinrdoiucgahtepsutht eefifmicpieonrtcayn cthean
ofOopMtiAm,i zaisn agllt htheep oorwdeerreadn NdOraMteAal ulosceartsi ohnavine NloOwMerA oubtaasgeed pUroWbSaNbisli.ty than their OMA counterparts.

Figure 3. Impact of (cid:2018) on average outage probability. The arrow means that the bottom first line
(blue) is for k=1 dB, second line (red) is for k= 5 dB, third line (green) is for k= 10 dB, forth line
(purple) is for k= 20 dB, ordered by the direction of the arrow.

Figure 3. Impact of (cid:2018) on average outage probability. The arrow means that the bottom first line
Figure3.Impactofκonaverageoutageprobability.Thebottomfirstline(blue)isfork=1dB,second
(blue) is for k=1 dB, second line (red) is for k= 5 dB, third line (green) is for k= 10 dB, forth line
line(red)isfork=5dB,thirdline(green)isfork=10dB,andfourthline(purple)isfork=20dB,
ord(epruerdpbley) tish efodri kre=c 2ti0o ndBo,f othrdeearrerdo wby. the direction of the arrow.