List of Symbols

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LH R EE

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User Pairing and Network Performance Optimization in Cooperative Wireless Network Coding

by

©Talha Rasheed

Athesis submitted tothe Schoo lof GraduateStudies in partialfulfilment oftherequirements for thedegree of

Masterof ComputerEngineering

Facultyof EngineeringandAppliedScience Memorial UniversityofNewfoundland

November2011

St.John's Newfoundland

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Abstract

In today'swireless networks,diversity is regarded as an efficient and established means to combat multi path fading.Moreover,user cooperation has emerged lately as an elegant technique to achieve spatial diversity over wireless channels,where the installation of multiple antennas on handheld,battery-powered, mobile terminals is often impractical. Recently,the application of network coding in cooperative wireless networks has gained increasing interest withits potentialto further boost the network performance, such as in terms of the achievable throughput. With network coding,the relaying nodes are allowed to linearly combine packets from multiple source nodes,and then forward the combined packets for better resource utilization.

We propose mutual user pairing in amulti-user infrastructure-based network-coded cooperative wireless networkto realize network coding, in the absence of dedicated relay nodes.We propose an optimal user pairing algorithm,and tailor it to maximize the network capacity. Next,we develop heuristic pairing algorithms which approach the optimal performance at a reduced complexity.Performance analysis is conducted in terms of the average capacity per user,average outage probability per user, and user-fairness.

For energy-constrainednetwork-coded cooperative networks,we subsequently address the problem of transmissionpower minimization.A joint optimization problem is formulated and solved to findthe pairingwhich maximizes the network capacity, and minimizes the transmission power, such that certain performance constraints in terms of the average capacity per user or average outage probability per user are satisfied.

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Acknow ledge ments

I am graciously thankful to mysupervisors,Dr.MohamedAhmed and Dr.Octavia Dobre.Without their continued guidance and support,this would not have beenpossible.

Iam alsothankfulto the Schoolof Graduate Studiesand the FacultyofEngineering for their support throughoutmy master'sprogram.Special thanks and regards to my office mates at the Computer Engineering ResearchLab;I will always cherish the memories.

Finally,I would like to thank myfriends and family.

TalhaRasheed

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List of Figures

I.I .A typical cooperativewirelessnetwork.

1.2.Butterflynetwork.

1.3.Two sources SI and S2communicatingwith thehelpof relaynodeR.8 1.4. Atypical wireless network with two sources transmittin gto a common destinationwith theassistance of acommon relaynode.

1.5. Systemmodel ofa wirelessnetwork. 12

1.6.Acooperative network withncommunicationpairs and m relays. IS 1.7Two sources transmittin gto a common destination;therelay overhears the 17 19 transmission s.

1.8 Cooperative wirelessnetwork.

2.1 Systemmodel underconsideration.Dotted andsolid lines represent source-and network-coded packettransmissionsrespectively. 24 2.2Packets transmitted by the pairednodesi andjin the two phases.In caseofinter-user transmissionfailure , an individualpacketistransm itted bytherelayingnodein the networkcod ingphase.

3.1.Thesystem model.Dotted andsolid linesrepresentsource- and network-coded-packettransmissions,respect ively .

25

36 3.2.Apotential matching in theweighted,undirected graph;the edges drawn

with thick linesarepartof the matchin g. 38

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3.3.The solid lines showthe edgesformin ga matching 3.4Acycle ofinner andoutervertices

40 40 5.1.Average capacity per user versusthe number of paired usersin thecell for

the proposedpairin g algorithms 54

5.2.Averageoutageprobabilit yper userversusthe numberof pairedusersin the cell for the max-max and random pairing algorithms. 55 5.3.Per-userthroughputJain's fairness index versus thenumber ofpairedusers in the cell for the proposedpairingalgorithms. 56 5.4. Optimal(minimum) powerallocation per user versus the number of paired usersin the cell,to meettheconstraint on maximum average outageprobability peruser.

5.5. Average capacityperuser versus thenumberof pairedusersin the cell. Theconstraintisin terms ofthe maximum average outage probability per user.

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57 5.6.Per-userthroughpu tJain's fairness index versus thenumber ofpaired usersin thecell.The constraint isin term s ofthemaximumaverageoutage probabilit yper user.

5.7. Minimumaveragetransmission powerper user versus thenumber of pairin gusers.Target capacityperuser=9.36 bps/Hz.

5.8.Average outage probability per userversusthenumberofpairingusers.

Targetcapacity per user=9.36 bps/Hz.

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5.9.Per-userthroughputJain' sfaime ssindexversusthe number of pairingusers.

Target capacityper user=9.36bps/H z.

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List of Symbols

k Relay assignment indexin two-wayrelay channels

e Exhaustiveset containing all possiblerela y assignment permut ations N_R Number of relaynodes

Nu«,., Number ofusersto bepairedinan infrastru cture-basednetwork Receivedinformation symbolsatdestinationnode Transmittedsource inform ation symbols

Channel coeffici entintegrating theeffectofpath-lo ss and Rayleigh fading Additive white Gaussiannoiseat thereceiver

No Power spectral densityofn C,um Sum networkcapacity

P Transmit power

r Averagesignal-to-noiseratio r Instantaneou s signal-to-noise ratio

R Packet code rate in case ofpoint-to-pointtransmission

Rate allocationfactorbetweendirecttransmission and networkcoding phase

P Outageprobability

n Exhaustiveset containing all possibleuser-pairin gs in network-coded network

~ User-pairingset in network-codednetwork dj General,undirectedweighted graph V Vert icesindjrepresenting the userstobepaired

E Setofedgesindj

Weights assigned toedges,E o(.{ Matchingindj

<Pout Averageoutage probabilit yfunction

<Pcap Average network capacityfunction

Tolerance for the bisection optimization

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List of Abbreviations

MIMO Multiple-Input Multiple-Output

S Source node

D Destination node

R Relay node

MRC Maximal Ratio Combining EGC Equal-Gain Combining SC Selection Combining TDMA Time-Division Multiple Access

XOR Exclusive-OR

SNR Signal-to-Noise Ratio ORA Optimal Relay Assignment CSI Channel State Information SER Symbol Error rate

BS Base Station

BER Bit Error Rate PER Packet Error Rate

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Chapter 1 Introduction

In modem wirelesscommunication network s,thereis aconsistently growing demandfor higher datarates,improved service quality,better coveragearea,andshorter processingtimes. The impedimentsto achievingthese goa ls are primarilythelimited available channelbandwidthand the dynamicnature ofthe wirelesschannels.Inaddition, wirelesschannelsare unpredictable,owingtotheeffects of smalland large scalefading [I]. The smallscale fading,usually simply termedas f adingis often themostdetrimental.

In awireless medium,multiplecopiesofthetransm itted signal,result ing from the randomscattering of the electromagneticwave from the surroundingobje ctsarriveat the receiver.Thesecopies arrive at the receiverhavingundergonedifferent channels,and thus arrive withdifferent gains,phase shifts,and delays.Themultiple copiesinterfereat the receiverand can add in a constructive or destructive fashion,whichresultsin the amplificationor the attenuation of thereceived signal. Incaseof attenuatio n, the signalis said tohaveundergonefading.Thismayresultin theunsucce ssfulrecepti on ofthe transmitted signal, as the receivermay not be able to distinguishthe received signalfrom thermalnoise[2]-[3].

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1.1Diversity in WirelessNetworks

Inwirelesscommunicationsystems,diversityisregarded as anefficientand established meansto combat the smallscalefading.Itisthetechniqu e by which multiple copies ofthetransmitt ed signalcan bereceived overindepe nde ntlyfadedchannelsatthe receiver andcombined.In case oneor morecop ies ofthe signalareaffected by severe fading,thereceivercan still detectthe signal fromthe othercopies.Thetermdiversity gain isusedtoquanti fythenumber ofindependently fadedcopiesof the transmitted signal at the receiver.In practice,independent channels canbe achieved primarilyin three physicaldomains:time, frequency,andspace. Diversitycould also be achievedinother form s suchas space-time diversity andcooperative diversity[4].

Time diversitycould be achieved bytransmittin gthe same signalmultiple times, in differenttime slots. Thesetime slotsshould be separatedatleastby thecoherence time ofthe channelsuch that it ismadesure thatthe channe lsat thesetime slotsare independ ent. The drawback of time diversityisthe decre aseddata rate and increased latency.Frequency diversitycanbe achieved bytransmitt ingmultiple copiesof thesame signal in different frequency bands. Thefrequencyseparationshould be enough to guarantee channel independence.However,more spectrum isrequir edtoachieve frequency diversity. Finally,space diversityis achieved by sendingand/o r receivingthe signal over multipleantennas,separated wellenough, such thatthechannels are independent. Spatial diversityon the other handneithercausesincreasedlatency,nor decreasesthebandwidth efficiency,and thereforehas attrac tedextensiveinterestfrom industry and researchcommunity inrecentyears.Communicationsystemsemploying

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multiple transmit and/or receive antennas are called Multiple-Input Multiple-Output (MIMO) systems.Itis important to situate the multiple transmit and/or receive antennas sufficiently far apart (usually more than half a wavelength) such that the fading over the channels between any pair of transmit and receive antennas is statistically independent.

Although the gains associatedwiththe use of multiple antennas in MIMO systems, such as improved channel capacity,higher throughput,better error performance,and energy efficiency,are very well established,there are certain limitations associated with their practical deployment. For instance, installing multiple antennas can often be impractical owing to the additional resource overhead,such as in terms of space for installing multiple antennas, or power.This is particularly true for mobile terminals,and these limitations on the installation of multiple antennas make the achievement of transmit diversity (from the end-user'sperspective) impractical.

To overcome these drawbacks,distributed nodes in a wireless network can cooperate and intelligently share their antennas to form the so-called virtual antenna arrays. This form of user cooperation has emerged lately as an elegant technique to achieve spatial diversity over wireless channels,such as in the form of cooperative diversity,which exploits the broadcasting nature of the wireless medium [5]. The notion itself stems from the classical relaying model with intelligent antenna sharing and signal combining at the receiver to realize spatial diversity.In cooperative transmission,users can utilize their time, frequency, and/or other resources to share their antennas to form virtual antenna arrays and emulate the operation of a MIMO system. Besides retaining the benefits innate toMIMO systems, cooperative diversity brings about few more,such as

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Fig.I.I.Atypicalcooperative wireless network.

improved energy efficiency, and has been widely shown to achieve remarkable performance gains in wirelessnetworks [4],[6].

1.2 Overview of Cooperative TransmissionProtocols

Fig.1.1shows a typical cooperative transmission network which consists of a source node (8) transmitting to a destination node (D) with the assistance of a relay node (R).Thecooperative transmission consistsof two phases . During thefirst phase,the sourcenode transmitsitsmessage to the destination (D). Dueto the broadcasting nature of the wireless medium,this message is overheardat the relay node (R). In the second phase, therelay node thenforwards the overheardpacket (after necessary processing) to the destination overanorthogo nalchannel.Thedestination then combines thetwo copies of the same packet received from the sourceand the relayover thetwo phases using any of the combining techniquessuch as Maximum-Ratio Combining (MRC), Equal-Gain

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Combining (EGC),or Selection Combining(SC) . This way,spatial diversityis achieved, asthetwo copiesof thesame packetare receivedoverpotentiallyuncorrelatedchannels.

The protocolsfor cooperativetransmissioncan bebroadly catego rizedon the basis of a number ofoptions. These could betherelaying strategy,relayingbehaviourincase of a decoding failure,and the type of coding employedin thesecond phase.For instance, some of the common relaying strategies are[4]:

• Amp lify-and-Forwa rd:In thistype ofrelaying strategy, therelaynode simply amplifies the receivedmessagefromthesource andforwards it tothedestination.

Amplify-and-Forward achievesthefull diversity gain. However,thedisadvantage ofthisprotocolisthatthe forwardedmessageisa noisy versionoftheoriginal message,asthe noiseaddedat therelaynode is alsoamplified.

• Decode-and-Forward:With Decode-and-Forward relaying ,therelaynodefirst decodesthe messagereceived from the source, re-encod esit,and forwardsthe source messageto thedestination.Decode-and-F orwardperform sbetter in case of goodsource-relay channel s,i.e., when the outage probabilit y over the source-relay link islow,whereasAmplify-and-Forw ardperformsbetter when thesource-relay channels are ofpoor qualit y.

• Compress -and-Forward: In thisprotocol,the relaying nodedigitizes and compressesthemessagereceivedfrom thesource in orderto decreasethe redundancy.The compressedmessageisthen re-encodedand forwardedtothe destination.The destinationthen combines thepackets from the sourceand relay.

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Some other relayingstrategiesinclude demodulate-and-forward and quantize-and- forward.Moreover ,the relaying protocols can also be static and adapti ve[4].Instatic protocolsthe relay node wouldalwaysforward thesource's packet,irrespective of whether it wasreceived successfullyor not. On the other hand ,protocols could alsobe adaptive,such that the relay forwardsthe source's messageonly ifit decodedthemessage correctly to avoiderror propagation.

1.3Introdu ctiontoNetwo rkCoding

Network coding was firstintroduced in [7] for wireline networks.The central notion behind network codingisto allowthe networknodestocombine the information packetsfrom multiple sourcesbeforetransmission,insteadof simply relaying/forwardin g them asin classicalrouting. Ineffect, the intermediate nodes in thenetwork betweenthe source and destination (such asrelaysand routers)canperform coding ofthepacketsto achievethe multicast capacityof the networkgraph. This is demonstratedinFig. 1.2 which showsa classic "butterfly"network.Itis assumed that the source S wantsto multicasttwo bitsa and b to two sinksDIand D2simultaneou sly,witheach link having a capacityof I bps.With traditional routing, each of the intermediate nodeswillsimply forward a copyof the packet they receive.Theshaded node can forwarda or b.This will make it impossibletoachieve the multicastcapacity of2 bps.However,with network coding,the intermediate relay node(which is shaded) can perform coding,which is a bit- wiseXOR operation,ona and b and multicastover the two outgoing links.Thisway,D I receivesa and a+b,and can recoverbasb=a+(a+b). In the same manner,D2receives b anda+b and canhence recovera.BothDIand D2 therefore receive at2 bps,

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s

J\ A w

1 ; :' ^[b

~O

01 02

s

A

1.;:1

^a

_O~

^a-b ^b

01 02

(a) (b)

Fig.1.2.Butterflynetwork [I].

and thusachievethe multicast capacity.

The utility of network coding in multicastwireline networks wasfirst demonstratedin[I].Ever since,itisextended to various wireless applications [I].Infact, wirelesspacket networks tendto benaturall y suited fornetwork coding owing tothe special characteri sticsof the wirelesslinks,such astheirbro adcastin gnatureand unreliability,forwhich network codingitself isa natural solution. Moreover,combined with thefact that protocoldesignfor wirelesscommunicationismuch moreflexiblethan for thewireline case,network coding seems an ideal meanstoachieve remarkable perform ance gains in wirelessnetwork s.

Owingto thesimplicityand the potential ofnetworkcodin g,thewireless communication researchcommunity has expended signific antinterest and effortto utilize it in a varietyofapplications in wirelessnetworks.Theserange from opportunistic routing in meshnetworks to distributed storage insensor networks[8].Networkcoding for wirelessnetworks isessentially acodingstrategyfor thedecode-and-forw ard

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Fig.1.3.Two sources81and82 communicatingwiththehelpofrelay nodeR.

cooperativetransmissionprotocol. With network coding, the relay node, after decodin g,is allowed toperformfurtherprocessingofthe source'spacket before forwardingit to thedestination. The applicationofnetwork codingin cooperative wireless networks hasrecently gained increasing interest[9],with itspotentialto significantly boostthenetwork throughp utand performance. A typicalexample of network coding inwireless networks isdepictedin Fig.1.3.The networkconsistsof two sources8Iand82swapping theirpackets withthe help ofthe relay nodeR,over orthogonalchannels.AssumingTime DivisionMultiple Access (TDMA),8I transmitsits packetfirst,followed by SZinthefirst phase. Meanwhile,the relaynodeRoverhearsboth these transmissions, and combinesthe two packets,for instance using the bit-wiseXOR operation,andthen broadcaststhe combinedpacketin the second phasewhich helpsboth source nodes 8Iand82 to achieve diversity gain.

Another networkcodin g scenar io ispresentedinFig.1.4,where thenetwork consistsof two sources8Iand82,transmittingto a commondestination(D) withthe help of therelaynode (R).Thesources8 I and 82 send their respective information packets to thedestinationnode (D) overorthogonalchannels duringthe first phase.Thesepackets

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S2

Fig. 104.A typical wireless network with two sources transmitting to a common destination with the assistanceof a common relaynode.

are also overheard at the relay node (R).The relay decodes the two information packets, and can subsequentlycombine the two packets,for instance using the bit-wise XOR operation. It thenforwards thecombined packetin the second phasewhich helpsboth sources S I and S2 to achieve diversitygain. Assumin gTDMA,a total of three time slots are required with network coding,whereasin caseof traditional routin g,the numberof requiredtime slots are four to achieve a diversityorderoftwo for bothnodes.

Thisdirectly results in a 25 percent throughput improvement.

The application of networkcodingtowireless network spromisesto changemany aspectsof networking.In effect,networkcodingdeviatesfrom theclassical networkin g approach where wireless networks are treated as physicalmeans of datatransportation, allowing for data manipulation within thenetwork.The application of network codingin wirelessnetworks has been studied ina variety of settings,including thecases of (a)two sources transmitting to a common destination[10]-[13],asis depicted in Fig.IA. This case

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is an important buildingblockfor numerous manifestations of wireless communication, suchas the infrastructure-basedcellularnetworks,

(b)multi-cast [14]-[1 5], where network coding is employedattheintermediary nodes in the networkto improve thethroughputfor information dissemination,and

(c) for two-way relay channels [16]-[19], for instance in ad hoc networks, where the intermediary nodes in the network serve as relays by forwarding the networkcoded packets forthe source-des tinatio npairs.

1.4RelaySelection in Cooperative WirelessNetworks

The design criterionwhich greatly impacts the performance of cooperative networks,both without and with networkcoding is the proper relay selection [16]. As user cooperation and intelligent relay selection can significantly boost the network throughput withanten nashari ng,an improperly selectedrelay can however deteriorate the system performance.

1.4.1Literature Reviewof Relay SelectionSchemes inCooper ative Networks

Directed by the significanceofrelay selection in cooperative networks,the problemof relay selection/assignmentis receiving extensive interest from the research community.The array of proposed solutions fall mainly into two categories:

infrastructure-orientedprotoco lswhich usually compriseof optimalsolutions(often based on exhaustive searches), and sub-optima l implementation-oriented

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heuristicsolutions.In this section, we survey some of the most conspicuous and representative publications in this area from the literature.

The authors in [20] address the issueof joint optimization of relay selectionand power allocation to maximize the average network capacity.They first proposean optimal solution for the joint optimization problem. However,to alleviate thecomplexity, theyseparate the joint optimization problem into the sub-problem of single bestrelay selection with uniform power distribution between thesource and relaynodes,and then optimal power allocation for the chosen source-relay pair.A so called"semi-distributed"

algorithm is then proposed for a network environment with multiple source-destination pairs where each relay node individually decides on its suitability to act as a relay, and the final decision is made by the central entity.Ithas beenshown that thesub-optimal algorithm with reduced computational complexity can provide comparable performance to that of the optimal scheme,which is based on exhaustive search. The authorsconsider the system model as shown in Fig.1.5[20].

The network consists of multiple source and dedicated relay nodes,and asingle destination node.The relays are assumed to operate in the Amplify-and-Forward mode.

For finding the optimal solution for a single source,the set of feasible relay nodes(i.e., the ones which can provide better capacityperformance than direct transmission)are searched for,and the one which maximizes(1.1)is selected as relay,

(1.1)

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0)

o o

0)

0 o o

o

Fig.I.5 .System model of a wireless network [20].

whereSNR is the Signal-to-Noise ratio at the transmitter,h_yandhJdare the channel coefficients from the source to relayj,and relayjto destination respectively,andNRis the number of relays. The channel coefficients integrate the multi path fading and the propagation path-loss.If none of the potential relay nodes offer an increased capacity over direct transmission, i.e.,if the set of feasible relay nodesisempty,thesource node goes with direct transmission.The authors then find an optimal solution for power allocation to further improve the performance after relay selection.

Following this optimal solution,the authors propose a semi-distributed relay selection scheme for a network environment which comprises multiple source and relay nodes,under the assumption of equal power allocation between a pair of source and relay.

The algorithm isdivided into two steps:feasibleset generation,and relay node allocation.

In the first phase, the nodes transmit hand-shaking packets before actual data transmission

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to allowthe relaynodesto estimate the channel gains from the sourceand destination nodes. All rela ynodescanhencedecide on theirfeasibilit y (this happensinadistributed fashion),and report their respectiveindices tothe destination .The destinationcan then performthe relaynodeallocationfrom thefeasible set byrandomlypicking arelaynode and assignin g it to one of the source nodes. Thissub-optim al scheme with lesscomputationalcomplexityisdemon stratedto achievenear-optimalperformance.

Theauthors of [21]proposethe so-called OptimalRelay Assignment(ORA) algorithmfor a network environment with multiple source and relaynodes.The objective isto maximizethe minimum capacity among the pairsofsource and destinat ionnodes.

The notablefeaturesof thisalgorithm are(i)guarantee of optimalit y, (ii) polynomial time complexity,and (iii)final capacityof every source-destination pair ismorethan that achievable with direct transmission.In the proposed scheme,a source-destination pair is assignedat mostonerelay,anda singlerelay nodecan assist at most onesource- destinationpair. Afteran initial "random" relaynode assignment ,thesolutionis adjusted in eachiteration toachievea greatervalueofthe objective function (the minimum capacity amongallsource-destination pairs).In particular,the source nodewith the lowestcapacityis identified and a better relay nodefor it is searched. However,incase the "better"relay ispre-assignedto some other source,another relay for that othersource nodeis searched for,andso on. Hencewithina single iteration,there are twopossibilities:

(i) a better solution(i.e.,a higher value of the objective function) isfound,and the algorithmmoveson to thenext iteration,or(ii)abetter solution could not be found,and the algorithm terminat es.The algorithm is shown to run inapolynom ialtime;also,itis arguedthat in caseof a non-optimalsolution, thealgorithm would keepon iterating,and

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would term inate onlyin case theassignmentsolution is optimal.The optima lityof the algorithmis alsoforma lly proven.

In [22],theauthorsconsiderrelay selectioninamultiple-accessnetwork witha single base station to extend the coverage area usingcooperation.Theauthorsderivethe optima lrelaylocationsbased ontwo cases,i.e.,ifthe destinationuses packetsfrom the relay as wellas thesourceMRCfor detection, or only thepacket from the relay node.In the formercase,the optimal(norma lized,wr.t.to the distance betweensourceand destination)relaylocation (along theline joining the sourceand destination) fromthe destinationis shown tobe

(1.2)

wherepis thepath loss exponent. In casep~2,an interestingobservation isthat the optimal relaylocationiscloser tothe source node.In the case of no-MRCat the receiver, the optimal relaypositionis shown tobeatthe mid-pointbetween thesourceand destination alongthe linejoiningthe sourceanddestination.Theautho rs thenpropose a simpledistributedalgorithm-nearestneighbour routing,in which the relay nearestto the sourcenode canbe selectedas thehelper. Thoug hfarfrom optimal,itis very easy to implementinadistributed fashion.

1.4.2 Literature Review of Relay Selection Schemes in Cooperative Networks employing Network Coding

Networkcoding has recently been studiedextensivelyfor cooperativewireless networks as thecombiningofdata at intermediaterelaynodes can furtherimprove the

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Fig.I.6.A cooperative network withncommunicationpairs and m relays[23]-[24]

network throughput aswell asrobustness.In particular,thetwo-way relaychannel model hasreceived themost interest asit couldbe regardedasthebasicbuildingmodule of many wireless networks.Relay selection in networkcodingenvironm entsisparticularly interestingasmorethan one source nodeshavetobeinvolvedin the relay selection processasopposedt ojustoneinconventionalcooper ativenetworks.In this section,someof themostrepresentative schemes from theliteratur e addressing relay selection/ass ignment in cooperativenetwork swith networkcodingare surveyed.

In [23]theauthorsconsiderthesystem modelas shown inFig. 1.6.Thenumber of relaynodes is assumedto be greateror equal to thenumber of communicating pairs,and the direct link between the pairsis ignored.Moreover,onlyasingle relayisassigned to everypair.For ease of comprehension,it is assumedthatone of the nodesin the communicatingpair isthe Source (S) and the otherone isthedestin ation (D). In thefirst timeslot,the nodeS transmit sitspacketwhich isreceivedand decod ed at the selected

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relay.Similarl y,in thesecond timeslot,node0 transm itsitspacketand it isreceived and decoded at therelaynode .TherelaythenXORsthetwopacket s and broadcaststhe network coded packet which isthen heardbyboth Sand 0 (therebysaving onetimeslot compared with traditionalrelayingusingTDMA for instance). The authorsthenpropose an optimal anda sub-optimalscheme forbest-relay selection.They considerthechannel coefficients over the two links,i.e.,thesource-relay and relay-destination,and assume thatthe weaker of the two coefficients will dominatethe end-to-end performance. The proposed optimal relayassignment criterion is such thattheminimum channelcoefficient over thetwolinks ismaximized . For the optimalsolution,all possible assignment permut ations are considered (whichareP:'

IN

R ,wherePreprese nts permut ations, in caseofNRrelays and m pairs).If 0 denotesthesetconsistingofall possible permut ations, the indexof the optimalassignment,k· ,is given by

(1.3)

whereIhIk.ministheweakestsource-relay or relay-destination sub-channel.Theauthors then propose a sub-optimalscheme by exploiting thecorrel ationwithin the elementsof set 0.Theset 0 ispartitionedinto

P :' IN

R'smallersubsets. The subsets containing correlatedelements are not searched for,hence reducing thenumber ofpermut ations over which the search is run.

In [17],theauthors propose analog networkcodingusin gdifferentialmodulation over two-w ayrelay channels,suchthat the ChannelState Information (CSI) isnot requiredto be known at the source,destination ,or the relaynodes, and istherefore

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S2

Fig.l.7.Two sources transmitting to a common destination; the relay overhears the transmissions [II].

estimated.Only a single pair of sources is considered in the model with multiple intermediary relay nodes. An optimal relay selection criterion is proposed; the relay which minimizes the estimated sum SymbolError Rate(SER) of thetwosources is selected, according to

where SER\.k(hi,*,h2,k )andSER2,kare the estimated Symbol Error Ratesfor Source I and Source 2, respectively,for relayk,hl,kis the channel coefficient from Source I to relayk,andh₂,kis the channel coefficient from Source 2 to relayk.

The best-relay selection is carried out by only one source; hence the decision making node has to calculatethe SER for the other source node.The authors then propose a simple sub-optimal relay selection scheme,in which the relay which minimizesthe maximum estimated SER of the two sources is selected,i.e.,

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Thesub-optimal min-max scheme is demonstratedto perform verycloseto the optimal solution,especially as thenumber of availablerelaynodesincreases.

Amultiple-access scenario asdepictedinFig.1.7isconsideredin [II].Thetwo sources transmit their respective packets to the basestation (BS) in thefirst phase,which compri sestwo timeslots.Thesepackets arealsooverheardat theintermed iatenodes.In thesecond phase (i.e.,the networkcoding phase),theselected rela ycombin esthe decoded packetsfrom the sources in the first phaseand relaysthe networkcodedpacket totheBS.A single transmi ssion from therelaythushelpsbothsources to achieve diversity gain.For relay selection, theauthors propose aratherunappealin g solutionof exhaustivesearchfor the best relay (in termsof maximi zation of thesum capacityofthe twonodes).Thisscheme is infeasiblefornetwork environme ntswhich usually comprise multiple relaynodes;development of implementation- oriented solutions is anextremely interesting and worth-whilearea for futureinvestigation .

In the workson cooperativewirelessnetworkcodin g surveye din thissection,and within othersfrom the literature,the relays are assumedto be dedicated,i.e.,theytransmit nothin gfor themsel ves when relaying.In practicethistranslatestothe fact thatthe relayingnodecannottransm itforitsel f while it ishelping another user. Apossibili ty is for the network providerto deploystand-alone dedicatednodesto actasrelays.In effect, the assumptionof dedicated relaynodesplaces additional constraints onwireless terminals,or necessitatesadditionalinfrastructurefrom theserviceproviderto support the network.

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Fig. 1.8.Cooperativewirelessnetwork.

Moreover,in the caseofmultiple-accessnetworks(i.e., thecase ofmultiple sources transmittingto a common destination , such asa base station in [25]),trulymulti- userenvironments arenotconsidered. The number ofsources in the networkislimited to two,and the issueof scalabilityto real-worldmultiusernetworksis not addressed.

Moreover,theassumption of thepresenceof dedicated relaysin thenetwork is maintained .

1.5 ThesisMotivation andContributions

In perspective of theoutlinedlimitationsof related works,we are motivatedto addressthe problem of partner selection (pairing)inatrulymulti-user environment, where usersemploynetworkcodingtotransmittoacommon destination (e.g. a base station inacellularenvironment ),in the absence of dedicatedrelaynodes.This is an important

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communicationscenario, and to the best of our knowledge, the problem ofmutual user pairing in such multi-user environments has not been addressed previously in the literature.Inthe absence of dedicated relay nodes, and as shown in Fig.1.8,users are considered tomutually pair among themselves to realize network coding. The pairing should be performed to optimize certain system performance metrics, such as network capacity,outage probability, and/or fairness. Nodes constituting a pair periodically swap the roles of source and relay for the mutual benefit of achieving diversitygain.

Our objectives are:

(a) to address the problem of mutual user pairing in a multiuser environment,such as to optimize certain system performance parameters,and

(b) in conjunction with the user pairing schemes, to address the transmission power optimization,with constraints on certain network performance metrics.

Themajor contributions of this thesis are summarized as follows:

I.We formulate and solve an optimization problem to obtain the user pairing which optimizes system performance metrics. We tailor our algorithm to maximize the network capacity,but this can also be used to optimize the outage probability,user-fairness,or other performance metrics.

2. The optimality of the algorithm is verified; however, to address the computational complexity,we then propose implementation-oriented heuristic user pairing algorithms. The heuristic schemes are designed to approach the optimal performance at a significantly reduced complexity. We propose

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algorithms which address averagenetwork capacity,average outageprobability, and user-fairness.The performanceofthe optimal and heuri stic algorithmsis investi gatedthrough exten siv e simulatio ns.

3.Oncetheproblem of userpairingis solved,wenext addresstheissue ofpower minimization,and solvea joint optimizationproblem.We perform userpairing to maximi zethe total networkcapacity,and minimizethetransmi ssionpower per user ,such that certain networkperformanc econstr aint , suchasin term s of the average capacityor averageoutageprobability,is satisfied.

List of Publications:

Our work,duringthe course of this thesishasresult edin the followingpublication s:

T.Rasheed,M.H. Ahmed,and O.A.Dobre,"User-Pairing forCapacity Maximizati on in CooperativeWirelessNetwork Codin g,"submitted toIEEEICC 2012.

T. Rasheed,M.H.Ahmed,O.A.Dobre,and M.Saad,"Optimal User-Pairingin Cooperati veWirelessNetwork Coding withConstrain edPower Minimization,"accept ed toIEEERWS 2012.

T.Rasheed,Y. P.Chen,O.A.Dobre, and M. H.Ahmed,"Medium Access Control in Wireless Sensor Networks:Contemporary DesignIssues and FutureResearch Directions,"inProc.IEEE NECEC2010.

T. Rasheed,M.H.Ahmed,and O.A.Dobre,"CooperativeCommunica tion for Cogniti veRadioNetworks,"in Proc.IEEE NECEC 2010.

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T.Rasheed,M.H.Ahmed,and O.A.Dobre,"Relay Selection Schemesfor Cooperative Commun icatio nand Network Coding:A Survey," in Proc.IEEE NECEC 2010.

1.6 Organi zationof the Thesis

The rest of the thesisisorgani zedas follows.In Chapter 2 we layoutthesystem model,and then computethe capacityand outage probabilityfor the network-cod ed cooperation under consideration.Chapter 3 describe sthe pairing algorithmsto realize network coding.We propose variousoptimaland heuristicpairingschemeswhichaddress network performance parameters,such ascapacity,outage probability,and user-fairn ess.

InChapter 4,we perform power minimization ,and solve the joint optimizationproblem to minimizethe transmission power,while meetingcertain constraint s on thenetwork performance.Performance analysisof the proposedalgorithmsisconductedin Chapter 5, with extensivesimulations. Scenariosare highlightedas to when certain (pairingand joint/constrainedoptimiza tion)algorithms are preferable over others. Chapter 6 summarizes the findings of thisthesis,outlines the main conclusion s,and finallypresents recommendationsfor possiblefutureresearch direction s.

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Chapter 2 Capacity and Outage Probability Analysis of Network-Coded Cooperation

In this chapter,we outline the system andsignal model for the network-coded cooperation. We subsequentlyperform thecapacity and outage analysis of the network- coded cooperation by presenting the capacity and outage probabilityexpressions.For sufficiently large packet length,the outage probability demonstrates a lower bound on the packet error rate [26].Throughout the analysis,we assume perfectly orthogonal channels, exhibiting quasi-static (i.e.block)Rayleigh fading,and half-duplex transmissions.

Section 2.I outlines thesystem and the signal model. The network-coded cooperation scenariounderconsiderationis presentedin Section 2.2.Subsequently,the capacity and outage probability analysis is performed in Section 2.3.

2. 1SystemMode l

The system model of the network coded cooperation considered in this work is shown in Fig.2.1.We considera singlecell with an even number of users(NIL"'..)'Nodes areuniformly and randomlydistributedover the entire cell and are assumed to be equippedwith singleantennas. We assume no dedicated relay nodes inthe cell.

Usersstrategicallypair among themselves,and periodically swap the roles of the source

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Fig.2. l .System modelunder consideration.Dotted andsolid linesrepresent source-and network-codedpackettransm issionsrespectively.

andrelay torealize netwo rkcoding,and achievespatialdiversity.Nodes constitutinga pair firstbroadcasttheirrespective packetsto the base station,and also overhear each other'stransmissions.Incase ofa successful detection of the partner'spacket,a network- coded packetis subseq uent lytransmittedby the overhearingnode,which helps both nodes in the pair to achievediversity gain.

Thereceived signalattherelay or destination nodes is given by

y[m]=h[m]x[m]+n[m] (2.1)

wherex[m]isthetransmitt ed signal,h[m]isthe channelcoefficientwhichintegratesthe effectofpathloss and frequency non-selectiveRayleigh fading,andm is thetime index. The termn[m]isthe zero-meanadditivewhiteGaussiannoise (AWGN) with

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_ ......j)jn·ctTruml11i"illl1l·hu,...---._."IctwllrkClldil1\:'·hu,c_

I . •

,from l1odc;

I

sJ frol11l1odc.i [S;lf}SJfrom lll' dCi

I

Sj$si from nodcj

I

Fig.2.2.Packets transmitted by the paired nodes i andjin the two phases.Incaseof inter-user transmission failure,an individual packet is transmitted by the relaying node in

the network coding phase.

power spectral density(No),capturing the effect ofthermal noise at the receiver.

We model the inter-user and user-destination channels asnon-ideal(i.e. noisywith Rayleigh fading). Thus, a node constituting a pair sometimes may not be able to detect the packet of its partner,and as a result,it may not always forward the network-coded packet to help its partner. The network-coded packet transmission and detection of a pair of nodes follow the model proposed in [27]. The communication with the common destination (such as a base station or access point) is performed over two phases,and each phase consists of two orthogonal channels (we assume Time Division Multiple Access (TDMA)in this work).This model is depictedin Fig.2.2,where it is assumed that nodes i and jconstitute a pair,where i,jE{I,...,Nu.,m}'and i*- j.The node i transmits its packet to the base station in the first time slot during the first phase,i.e.,the direct transmission phase, while nodejoverhears.Subsequently ,nodejtransmits its packet in the second time slot while nodeioverhears. This is followed by the second,orthe network coding phase of transmissionI.Now,if node ihad decoded its partner' spacket in the previous phase, it would combine it with its ownpacket, and send the network coded packet to the base station.Otherwise,node i would send an additional packet foritself.

IThe terms"first phase"and "direct transmission"phase,and"second phase"and

"network coding phase"are used interchangeably in the context.

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Meanwhile,nodejdoesthe samein the secondtimeslotofthe second phase.Atthe base station, thetwo independentlyfadednetworkcodedpackets areco mbined using anyofthe well-knowncombining techniques, such as Selection Combining (SC),Equal-Gain Combining(EGC),ormaximratio combining(MRC) [3].Thispacket isthenjointl y decoded with thepacketsreceived in thefirst phase torecoverthe information bits. A maximumdiversity orde roftwo foreach user can thereforebe achieve d.Thisconcludes thetwophases of communicationwith thebase station.

The energy allocation isnon-equalbutsymmetric(with respect tothetwophases), i.e.,individualnodeswithin the pairmayusedifferenttransm issionpowersina single phase,but thetransm issionpowerof aparticularnodeis equalin thetwophases.Cyclic redund ancychecks areassumed todetectdecodin gerrors at thereceivingnodes.

Moreover,incorporatinganadditional flagbitinthe packetstransm ittedinthesecond phasehelpsthebase station determ inethe successof inter-usertransmissions,and hence thenatureof the packetsreceivedin the second phase.

Noteworth yisthefactthatwe assume no dedicat edrelays in the cell,astherelay nodes also transm itforthemselves when relaying.Moreover,since userstransmit over orthogonalchannels,there isno same-cell interference.Allchannels,i.e.inter-user and source-des tination,areassumed tobe spatiallyindependent,frequency flatRayleigh fading, with pureAWGN.Weassumeblock fading,such that allchannels remain constantduring the two phases.The signal modelfor thetwo-phasenetworkcoded cooperation scenario isformallypresentednext.

2.1.1 Signal Model

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In the first phase ,the source nodeitransmit sL/2 symbo ls,andthereforethe time indexm=I,...,L/2. Forthe source -to-des tination transmi ssion ,the symbols receiv ed atthedestinationsare givenby

(2.2)

wheresJm]are thetransmitt ed sourceinformatio nsym bols,nD[m]isthe AWGN noise at thereceiver,and thechannelcoefficient(h,,n[m])captures the effectofpathloss and frequencynon-selective Rayleighfading.We assumeperfect channel state informationat all receivers,i.e.,thechannelcoe fficientsare perfectl yestimated ,and thatperfect synchronization existsbetweennodes which per form coherent detecti on . Thechannel coefficie nt is assumed tobe constantoverthetwophases (includi ng2Lsymbo ls),and the depend en cy ofhon timem ishenceforthdropped. The received symbo lsatnodej are

(2.3)

wherenj[m]isthe AWGN noise at node j,andh,)sthe coefficient ofthe channel from nodeitonodej.Similarly,form=^L/2+I,...,L,nodej (nowassumingtherole of source)sends itspackettothe base station, which is overheard byi.The received symbols atDandiaregive n respecti vely as

(2.4) and

(2.5)

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whe res)m]aresymbo ls transm ittedbynodej,n,[m]isthenoise at node i, and hJ./Jand hi., are the coefficientsof the channel between jandD,andjandi, respectively.In the second phase of transmi ssion, i and j transmit for m=L+I,...,3L/2 and m=3L/2+I,...,2L,respectively. The received symbo lsatD fromi andjare given respectivelyby

(2.6)

and

where'ED'denotesthebit-wise XORoperator.

Incasethepartn erdoesnot decode thesource'spacket, ittransmits additional symbolsforitse lf duringthe secon dphaseoftransmission .

2.2Capa cityand Outage Analysis oftheNetwork CodedCoopera tion

Inwirelesscommunica tion, thedynam ic and time-var yin gnature ofthe fading channels makesthedesign ofcommun ication systems extremelychallenging.Anefficient meansto combat the effectsoftime-varying fading over wirelesschannels is throughthe use of spatia l diversity.In this workwe consider networ k-coded cooperationasa cooperative transmi ssionapproach toreali ze spatial diversity.We conside r mutualuser pairing, whe re users strateg ically pair,andswap theroles ofsourceand relaytoreal ize network codingandachievespat ial diversity.Therelay nodes are not dedica ted, i.e.,they transmi tfortheirpartner,as wellas forthemselves whenrelaying.

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Theinter-source andsource -des tination channel capaci tiesfor nodesiandjare functionsofthe corres pondingchannelcoeffic ients,and,the refo re they arerandom variab les.Moreove r,anoutageoveralink is definedas theeventof throughputfalling below atargetinform at ion rate.Weusetheoutageprobabilit y ata certain rate as ametr ic of thepacket error rate(PER) fortheblock-ba sedtransmissionsunderconsideration[28].

The inter-so urce channel s are modeled asnon-ideal (due tonoise andfading),and successful decod ing at therelayisnot guaranteed.Thistranslates to thefact thattherelay forwards anetwork codedpacket in the second phaseonlyif it decodeditspartner's packet correc tly. Otherwise,it transmit sits own packet only. Hence,the average throughput ofthepairdepend s on the successofinter-sourcetransm issions, whichmust first bedetermin ed.

2.2.1 Direct Transmission Phase

In thedirecttransmi ssionphase,nodesiand jsequentiallybroadcasttheir respectivepack ets,cont ainin gk inform ationbits,tothebase stationandalso overhear eachother's transmis sion s.The inter-source information theoreticchann el capacityfor nodeiisC,.}=log2(1+r,)[bits/sec/Hz], wherer.,=/h,.}1

2P,/Noisthe instantaneous

SNRofthe inter-so urce link, with

p,

asthetransmitpower. Anoutageoccurswhenever Ci.}<2R,where Risthepacket inform ation rate incase of the point -to-point transmi ssion . For Raylei ghfadin g,the outage probabilit yfornodeiis givenas[27]

P,.}⁼1_exp( _

2;,.~

^I),

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(2.8)

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wheref_i.)is the averageSNR of the inter-sourcelink. The outageprobabilityfor node jcansimilarly be calculatedby replacingfi•

ibyf_i.iin(2.8).

2.2.2 Netw or kCod ing Phase

The successof inter-source packettransmissionscan leadtothe following four distinctcases[27] :

Case A: When both nodesiandj forming a pair decode each other'spackets ,theyboth transmitthe network-coded packet in thesecondphase,which results in a full cooperation scenario,for that pair.

Case B: When noneof the two nodesdecodeeach other'spacket,they sendadditional packetsfor themselvesin thesecond phase,and thesystem returnsto anon-cooperative scenario, for that pair of packets.

Case C: When onlynodej decodes i,and not vice-versa,onlynodejtransmit sthe network-coded packet in thesecond phase (which helps both nodes),whereasnode irepeats its own packet.

CaseD: When onlyidecodesj'spacket, and notvice-versa, onlynodeitransmitsthe network-codedpacket in the second phase (whichhelps both nodes),whereasnode

jrepeats its own packet.

We considermaximumratiocombinin g (MRC) at the destination,which forms thecombined packetbythe weightedsum of the receivedpacketsoverthe twophases.To

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determin ethe channelcapacityand outageprobabilityforthefourpossibl e cases,parts of thepackets fro m nodesiandjwhichare usedfordecodin g at thedestin ation should be identified.In this subsection ,weperform thecapacityandoutage analysisfor thefou r possibl e casesfornodeionly. A similarapproach holdsfornodej.The underlying assumption isthat nodesiandj constitute apair , and mutuallycooperatetorealize network coding.The algorithms for userpairingin a multiuser environmentwill formall y bepresented in the followin gchapter.

Case A: Both nodesiandjcomprisingthepairdecodeeachother'spacketsin thedirecttransmi ssionphase.Each nodetransmitsthenetwork coded packet (s,E9Sj)in the networkcoding phase.For decoding,a packet[s;,(s,E9Sj)']of len gthNisform ed, where theprime denoted the MRC.Asthispacketcontain s 2k inform ation bits,its code

rateis

*

⁼^{2R. The}^two^parts^of^this^packetare essentially received overparallel channels whosecapacitiesadd together.The outageeventfor nodeiis[27]

whereais thefraction of timeallocatedto thefirstphase.From theperspecti ve of capacity, the effect ofMRC at thereceiverisreflectedbythe additionof the tworeceived SNR s (as in the second termin2.9).Theoutage probabilit y of the event in(2.9)is approximated as(the derivationisshown in Appendix A)

(2.10)

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Thisrepresents the outageprobabilitygiven the occurrenceof CaseA.The probabilityof occurrenceof Case A is given by the product ofprobabilities of successful decodin g at nodesiandj whichcan becomput edfrom (2.8).Definingthe overall outageprobabilityasP"D,Awhere'A'indicatesthe case,we get

P,,lJ,A=(I- P"j).(I-~")·p,,D· ^(2.11)

CaseB:Neither ofthe two nodesiandjconstituting the pair decode each other's packets. Eachsource nodetransm its additio nalpackets foritself.Atthedestination,a packet[sp Sj]is formedwhose coderate isR.Theoutageevent in this case is[27]

wherethe twoterms in(2.12)comefrom thecontributionsto the totalcapacity from the two phases,respectively.Following the same approach as in Case A,the outage probability is approxi matedas

(2.13)

Cas eC:O nlyjcan correctly decode fs packet,but not vice versa.In thiscase, nodej helpsi,butitransm its for itself duringthe networkcoding phase.The (2 -a)Nandcode rate of2R/(2-a).Theoutageeventfor nodeiinthis case is inform ation symbolsofiare decoded from thepacket[s" (s;EBSj)'s;]with length of

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Following the same approach as in Case A,the outage probabilityiscomputedas

(2.15)

CaseD:Onlynodeicancorrectlydecodenodej'spacket and not viceversa.In this caseihelpsj,butjtransmitsfor itselfduring the network codingphase. To decode Psinformationsymbols,a packet[Si,( Siffis))]of code rate2Risformed at the destination.

The outageeventfor thiscase is

and following the same approachasCase A,theoutage probabilityis approximatedas

_ _ _ [2

^2R

- I]

P,.D.D"'P,.j'(l-~./) ,-r- .

I,D

(2.17)

The total outageprobabilityis thesum of theoutage probabilitiesfor the four cases,i.e.

2.3Conclus ion

p,

⁼P'.D.A+P,.D.B+P"D.C+P,.D,D ^(2.18)

InthisChapter,we presentedthe signal and system modelfor thenetwork -coded cooperation under consideration.We presented and capacityand outage probability analysisfor a pair of nodes,consideringnon-idealinter-userchannel s.In the next chapter,

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we address the challenging problem of the mutual pairingofusersin themulti-user cellularenvironment. More specifically,we proposeand present optimal and heuristic user-pairingstrategies to address various network performance metrics,such as average capacity,avera geoutage probability, and user-fairne ss.

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Chapter 3 User Pairing in Network-Coded Cooperative Wireless Networks

3.1. Mut ua lUser Pairin gto Realize NetworkCodi ng

We address the problem of themutual pairing of users,or partner selectionina multi-usernetwork-codedcooperative wireless network,to achievespatial diversity.As outlinedinChapter2,users,havingdatato transmit ,mutuallypair among themselvesto realizenetworkcoding,whiletransmittingtoacommondestination.This could be an accesspoint ina wirelesslocal areanetwork or a base stationina cellular environment.

Twonodes constitutingapairperiodically swapthe rolesofsourceandrelayforthe mutualbenefitof achievingdiversity gain. Hence,onlyusers withdata to transmit participateinto cooperation,and idleusersare notengaged.This system modelis depictedinFig.3.1.

Transmission toacommondestination ina wireless network is animportant communicationscenario,and to thebestof our knowledge,theproblemof mutualuser pairingin suchmulti-userenvironments hasnot been addressed previouslyin the literature.

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Fig.3.1.The system model. Dotted and solidlines representsource-and network- coded- packettransmissions,respectively.

3.2User Pairing to Optimize System Performance

As shown in Fig.3.1,users strategically pair among themselvesto realize spatial diversity .Forthisnetwork-coded cooperationscenario under consideration,theuser pairingstrategydirectlyimpactstheoverall networkperformance. Moreo ver,theuser pairing can be performedto optimizecertain network performance metrics,such as maximizingthetotal network capacity, minimizing the outage probability,and/or maximizing theper-userthroughputfairness.

In this chapter,we firstformulate andsolvean optimization problem(using the maximum weightedmatchingalgorithm)to obtain the userpairingwhichyields the

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maximumachievab le total networkthro ughput. In orderto facilitatethepairing process, wesubseq uently propose implementation-orientedheuristic algorithmswhich approach the optimal performance atareduced computational complexity.In particular, we proposemax-max pairing to maximizethe network capacity ata significa ntly reduced complexity.Moreover,max-min pairingalgo rithm isproposed to minimize theoutage probability,withavery low complexity.

3.2.1Optimal User Pairing~'to Maximize Netw or kCa pacity

We formulateandsolve the problemof determining the optimal user-pairing~' which maximizesthe total network capacity.Wehave theset of all possiblepairing sets

Il,such that everyset~

En

isthe pairing containingN",m/2 disjointuserpairs.Each

pairing 1'!? is therefore a symmetric mapping of elements from the set XE{I, 2,...,N"",..}to thesetJYE{I,2 ,...,N",er..}'with the restrictionofan element fromX not beingmapped to thesame element inJY.The goal isto find the optimal pairin g1'!?' that maximizes the totalnetworkcapacity given by:

Therefore ,

C,um=LiCi'

~'=arg

w::

^C,um^(~).

(3.1)

(3.2)

Atfirst glance,thislookslike the problemof maximum weighted matching(i.e., pairing)in bipartite graphs,and any ofthe assignment algorithms,such asthe well-known Hungarian algorithm [29],seems asacandidate solution. However ,asit wasobserved,a

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Fig.3.2. A potentialmatchingin the weighted, undirected graph; the edges drawn with thicklines are part ofthe matching.

weight matrix W, with zeros on the main diagonal and symmetric entries,

twi,

=[W]l.1=C"D+C l,IJ'whereC,.DandC}.Dare the source-destination channel

capac itiesforiandj, respectivel y, and[Wl,.}and [W]}.Idescribetheweight ofthe assignmentofnodeitoj, and nodej toi,respectively(whereiandj constitutea potent ialpair),did not alwaysleadto a symmetricassignment.To find the optimal solution,we therefore model thisproblem asmaximumweighted matchin gingeneral graphs.

Weconstructaweightedundirected graphc9=(V,E),where the verticesVare the usersto bepaired, connectedbytheset ofedgesE.Furthermor e,1V1=Nu,mand 1£1=Nu<e"(Nu,m-1)/2(asthegraph isfully connected),where1.ldenotes the cardinality ofthe set. Eachedge(i,j)has anassociatedweightw,,}=C"D+Cl.D.The goalis to find thematching (i.e., pairing)with themaximumtotal weight.Thismaximum weighted

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matching coversall the verticesin the graph,andeachvertexis connectedonly to a single edge.Moreover,eachedgein the graphconnects twodistinct vertices.One such potential matching for a weighted graph with four nodesisshown inFig.3.2.Itisnoteworthythat the edge with themaximumweightmay not be apart ofthemaximum weighted matching.

When thenumberof usersto be pairedislarge,theproblem of finding the optimal pairing (i.e., thematching with the maximumtotalweight) isclearly far from trivial, whereasanexhaustivesearchisprohibitively expensive.To solve thispairingproblem, we useJack Edmond's maximum weightedmatchingalgorithmfor generalgraphs,which is describ edin [30].In the following,wepresent a succinct description ofthe algorithm,and the readeris referred to[30]formoredetails.

The ideaistostart with an emptypairing,and then,during eachstage, tofindan augmenting pathin thegraph whichyields the maximum increaseinweight.The blossoms methodisusedforfinding theaugmentingpathsin the graph.To explain this problem of maximumweighted matching ingeneral graphs,we clarify some terms from graph theory. Amatchingin agraphis a setof edges, such thatnotwo edgessharea commonvertex.A sample matchin ginanon-full y connectedgraph, consisting of8 verticesisshown in Fig.3.3.Furthermore, a vertex in the graph withrespectto a matchin g 0{{isfreeifn one oftheedgesin thematchin gareincidenton thatvertex.Analternating pathinthe graphwith respecttothematchin gelfis such thatitsedges alternately belong to thematching olf,and notto the matchin gott .Moreover,anaugmentingpath is an alternating pathbetweenfree vertices.

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The matchin gelfisnotmaximu mmatchingif andonly ifthereis an augmenting path withrespect toelf.We searchfor the augmenting paths in the graph byperforming

. . . '

8 0

0

8

Fig.3.3. Thesolid lines showtheedgesformin g amatch ing.

--- ---

Fig. 3.4.A cycle ofinner andoutervertices.

abreadth-fir st searchstartingfrom free vertices. We callanedge in thematching as 'solid'andanedge notin thematching at'dotted'. To search forthe augmenting path froma free vertex,webuildatree of alternating paths. Theroot,as wellas allthevertices whichare at anevendistancefromthe rootare called'innervertices' .If we run into a freeinner vertex,thenanaugmentingpathto thatvertex can be constructed.

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Thestepofbuildingthe treeisbased onscanningan outervertex,v.Each solid edge (v, w),wherewisnot alreadyin thetree,is added tothetree.Vertex wisdesignated an'inner' ,and the solidedge(w,x),which isunique, and isincidentwithwis added to thetree,andxislabeled as 'outer' .

Duringthe processof scanningtheoutervertexv,if we encounteran edge(v,w), inwhichwis outer,wethen form a cycle as inFig.3.4.In this case,we contractthe cycleto forma super-ve rtex,calledablossom,andcontinueso on.Moreover, if we encounterafree vertex, thenanaugmenting path can beconstructed from theroottothat vertex.We show thiswith anexample.Consider the followinggraph:

0- · · ··0-0·· · · ·cp

0-CP- 6 53

6· · · · ·0

Startingwithabreadth-first searchfrom vertex1,we see cycle 5-10-9 in thefollowing graph.

0---- -0-0----0-C): _~

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A blossomis formed by shrinking vertices 5, 10,and 9,and thesearch iscontinued.

0···0-0···· ·0-e. ~

We thenshrink(5, 10,9),6,8 into a singlevertex.

o I 0 I 0 I

0 ···0-0··· ··~·0

Wehence find an augmenting path in the shrunk graph.By unshrinking,the following augmneting pathin the original graph can be found.

8 ···· ·0-0···0-0· ···0-0· ····0-0· ····0

We start with an empty pairing,and during each stage find an augmentingpath in the graph whichleads to the maximumincrease in weight. The algorithmsolves the pairing problem inO(N³)time,and avoidsthe need for an exhaustive search. Moreover,if the number of usersto be paired is large,the set of userscan besplit into randomlychosen smallergroups to reduce the complexityof the algorithm,while howevercompromising the performance.

3.3Heuri stic User PairingAlgor ithms- Appr oaching Optimal Perf orm an ce In this section, we proposecomputationallysimpler heuristicuser-pairin g schemes to simplify the pairing process.In particular,we proposemax-maxpairingto maximize

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thetotalnetworkcapacit y.More over ,max-min pairin gisproposedtominimizethe average outage probability.

3.3. 1Max-max pairing

This algo rithm pair suserswith the obj ective ofapproachingthe optima l capacity at amuch reducedcomputational com plexity.A weig ht matrixWwithzeroson main diagon al,andsymme tric entries[Wl,J=[W1J"=C"D+Cl ,Dis established, whereiand j are potential pairs.TheO(N³)algorithm isformallypresented in the following : a)Initialize an empty pairing~,

b)Selec t thelargestelement fromW,for instanc e[Wl,l'and form thepairby augmenting~withiandj,

c) Update W byremo vin gthe rowsandcolumns correspond ingto thepairform edin(b), d) Continue from(b) until~iscomplete and all nodeshave been paired ,

Max-max pairinghasthe same big 0 complexityasthe optimalpairing,which depict sthat itsca lessim ilarly tothechangesin inputsize,asthe optima l pairin g, However,max-m axpairing is significa ntly comput ation all y simpler than the optimal pai ring, asit requir es simpler computation s.Thisis also refle ctedin the averagesimulation timeswhich arereferredto in Chapter5.

3.3.2Max-min pairing

Thisheuri sticalgorithm isdesignedtoaddressthesystem outageprobabilit y.We startwith theweakestuser (in terms of the SNRtotheBS)in thecell and pair itwith the

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user having the strongest of the weaker of source-relay and relay-dest inationlinks,since theoutageperformance is alwaysdeterminedby the weaker of thetwo links [31],and continueso onforother users. Thealgorithm has complexityofO(N²) ,and is formally presented as follows:

a) Initialize an empty pairing~,

b) Select a nodeiwiththe lowestYi.Dand pair it with} with max[min(Yi,J' Yj,lJ)], c) Augmentthe pairing~with the pair formed in (b), and updatetheset of eligible nodes.

d) Continuefrom(b) until~is completeandallnodes havebeenpaired.

Apparent ly,max-minpairing is computationallyefficient because it involves cheap computations.This is also reflected by the simulation times as stated in Chapter 5.

3.3.3Random pairing

Pairingusers randomly isthe most straight-forwardstrategy,and is the simplest to implementinpractice. From the set of eligible users,two randomlychosen nodes are paired.~is augmented,theset of eligibleusers is updated,and the algorithm repeats until all users have been paired. Althoughrandom selection is not an effective way of pairing, we include it here for comparison purposes.

3.4 Conclusion

In thischapter,weconsidered the problemof mutual userpairing in network-coded cooperative networks.We proposed an optimal pairing algorithm,and tailored it to maximize the network capacity.We subsequently proposed computationally simpler

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heuristic pairing algorithms. In particular,we proposed the max-max pairing with the objective of maximizing the network capacity.Moreover, we proposed the max-min pairing to minimize the outage probability.

The performance analysis of the proposed optimal and heuristic algorithms is presented in Chapter 5,wherethese are compared in terms of average capacity,average outage probability,and user-fairness. The suitability of these algorithms,in view of varying system performance requirements is also discussed.

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Chapter 4 Power Minimization: Joint & Constrained Optimization

In energy-constrained wirelessnetworks,the designof energyefficient protocols is imperative.For the network-codedcooperationscenario under consideration ,wehave emphasizedthat the gains associatedwith cooperation and network codingarethe improvedthroughput and outageperformance ,brought about bythe achieved spatial diversity.However,for energy constrainedwireless networks such as sensor and cellular networks,where minimizing the energy consumption is one of the objectives,these performance gains canbe traded-offwithenergy savings,and can therefore resultin significantly improved battery lifetimes.

In this chapter, we consider power minimization ,and solvea joint optimization problem.In the joint optimization problem,we perform user pairing to maximizethe total network capacity,and minimize the transmission power per user, such that certain network performance constraintin terms ofthe average outage probabilityper user,or the average capacity peruseris satisfied.We use the maximum weighted matching algorithm (asdescribedin Chapter 3, Section 3.2.1)to obtainthe optimal userpairing which leadsto the maximum total networkcapacity.Subsequently,we use the bisectionoptimization

46

List of Symbols

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Contents

List of Figures

List of Symbols

List of Abbreviations

Chapter 1 Introduction

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Chapter 4

Power Minimization: Joint & Constrained Optimization

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