An Optimization Framework for XOR-Assisted Cooperative Relaying in Cellular Networks

Transcription

1 n Optimization Framework for XOR-ssisted Cooperative Reaying in Ceuar Networks Hong Xu, Student Member, IEEE, Baochun Li, Senior Member, IEEE bstract This work seeks to address two questions in cooperative OFDM networks: First, how network coding based cooperative diversity can be expoited effectivey when overhearing is not readiy avaiabe. Second, how to reaize various forms of gains avaiabe, incuding muti-user diversity, cooperative diversity, and network coding. The main contribution of this paper is an unifying network utiity maximization framework that jointy considers reay assignment, reay strategy seection, channe assignment and power aocation. We formuate the optimization probem both with and without XOR-CD, a simpe XOR-assisted cooperative diversity scheme. We show that the optimization of physica ayer resource aocation with XOR-CD is equivaent to a weighted 3-set packing probem, which is NP-compete, and can be efficienty soved with provaby the best approximation factor. Without XOR-CD, the probem reduces to a weighted bipartite matching probem which can be optimay soved. Index Terms Cooperative communication, reays, network coding, resource aocation, OFDM, ceuar networks. 1 INTRODUCTION Network coding, a technique to aow coding capabiity in exchange for network capacity gain, has been utiized to improve performance in wireess networks in genera [2], [3]. In the context of cooperative diversity [4] [6], network coding has been everaged at the reay to mix packets from different cooperative sessions, provided that the reay overhears and successfuy decodes mutipe transmissions and these transmissions share a common destination [7] [9]. In this paper, we investigate the use of network coding in cooperative diversity from a new perspective of muti-channe networks. We assume the context of OFDM [10] based ceuar networks, which impose unique chaenges since overhearing is no onger naturay avaiabe as in previous work. Users cannot hear each other uness tuned to the same channe. Coding opportunities are therefore to be carefuy invented and engineered, rather than opportunisticay harvested. Moreover, network coding entais that the broadcast rate is confined to the worst rate among a inks invoved, aggravating the task of finding profitabe coding opportunities. In ight of these chaenges, we propose a simpe XOR-assisted cooperative diversity scheme caed XOR-CD. It expoits coding opportunities on bidirectiona traffic on the upink and downink of a mo- This work has been presented in part at IEEE INFOCOM, Rio de Janeiro, Brazi, pri 2009 [1]. The authors are with The Edward S. Rogers Sr. Department of Eectrica and Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4, Canada. Emai: {henryxu, bi}@eecg.toronto.edu. Hong Xu wi join the Department of Computer Science, City University of Hong Kong as an ssistant Professor starting from ugust bie station (MS). Bi-directiona traffic is profoundy avaiabe in ceuar networks, providing abundant network coding opportunities. Fig. 1 iustrates an exampe to show the basic idea of XOR-assisted cooperative diversity (XOR-CD). Bi-directiona traffic exists between MS and the base station (BS). The reay station (RS) performs cooperative reaying using orthogona channes. XOR network coding can be used here to mix packets and B at the RS and muticast a re-encoded packet ( B) using ony one subchanne. ssume that channe coding and moduation are inear, ( B) = B. The MS and BS can sti receive the intended information by XORing the coded packet with one that is known a priori to itsef. Therefore, cooperative diversity can sti be capitaized. subchanne 1 subchanne 2 subchanne 3 subchanne 4 B' RS B MS BS (a) Traditiona cooperative diversity scheme with subchanne assignment ' B (+B)' RS (+B)' B MS BS (b) XOR-assisted cooperative diversity scheme with subchanne assignment Fig. 1. The motivating scenario for XOR-CD in OFDM networks. ( ) represents moduation and channe coding. With XOR-CD, ony 3 subchannes are needed instead of 4 for conventiona DF. The benefits of XOR-CD are intuitive. In the idea case where channes are symmetric, and BS-RS and MS-RS channe quaities are identica, XOR-CD achieves the same transmission rate for both cooperative sessions invoved, with a saving of one subchanne and the power of one transmission compared B

2 to the conventiona Decode-and-Forward. The saved subchanne and power can be used to accommodate more cooperative sessions, thereby further improving the network throughput. The question then becomes how to effectivey reap the promising gains of XOR-CD in OFDM networks. Three kinds of gains can be expoited here: (i) mutiuser diversity gain: for a given data/reay subchanne, different MS experience independent fading, aowing us to assign a subchanne to the MS with the argest channe gain; (ii) cooperative diversity gain: The RS heps the intended receiver to combat fading and improve SNR through cooperative reaying; (iii) network coding gain: bi-directiona traffic is amenabe to network coding which is utiized at RS to make reaying more resource efficient, increasing the network capacity. Our main contribution in this paper is a unifying network utiity maximization framework (NUM) to tacke the above probem. It jointy considers the foowing dimensions of resource aocation: reay assignment, reay-strategy seection, and subchanne assignment for both MS and RS in a ce, which is referred to as the RSS-XOR probem. Through dua decouping, we show that the cross-ayer probem can be decouped into two subprobems: an appication ayer rate adaptation probem that is trivia to sove, and a physica ayer resource aocation probem which is much more difficut. Specificay, we prove that the RSS-XOR physica ayer resource aocation probem is NP-compete by transforming it into a weighted 3-set packing probem. We propose a poynomia-time agorithm to sove it with the best known constant approximation factor, based on an agorithm for the weighted independent set probem. We aso formuate the optimization probem with ony conventiona Decode-and-Forward cooperative diversity, which is referred to as the NO- XOR probem. Using the same decouping technique, we design an efficient agorithm that optimay soves the NO-XOR physica ayer resource aocation probem as a weighted bipartite matching probem. Finay, we extend to consider reay power aocation among cooperative sessions for both RSS-XOR and NO-XOR, and propose subgradient-based agorithms to sove them in the dua domain. The remainder of this paper is structured as foows. Sec. 2 summarizes reated work, and Sec. 3 introduces our system modes. In Sec. 4 we formay present our NUM framework, the RSS-XOR optimization probem and its counterpart NO-XOR probem, and extend both modes for power aocation. In Sec. 5, we present efficient agorithms to sove the difficut physica ayer resource aocation probems. We conduct extensive simuations to verify the effectiveness of agorithms in Sec. 6. Finay we give concuding remarks in Sec RELTED WORK This paper buids upon prior work on cooperative diversity, whose roots can be traced back to the reay channe mode studied in [11]. The popuarity of cooperative diversity is owed to [4] [6] where different reay strategies are deveoped. Recent research aims to expoit distributed antennas on neighboring nodes in the network, and has resuted in many protocos at both the physica ayer [12], [13] and the network ayer [14]. We aso buid on work on network coding introduced in [15]. It is shown in [2], [16] [19] that network coding combined with routing and scheduing can greaty improve throughput in wireess mutihop networks. simiar concusion is made for the information exchange paradigm in which two nodes exchange data via a reay [3], a scenario simiar to ours. However, the distinction is cear: in these prior work, cooperative diversity is not everaged as a mechanism to combat fading, since the two nodes cannot directy communicate without the reay. Recenty there are studies that incorporate network coding into cooperative communication. [8] is arguaby the first work that studies the diversity gain of network coding. daptive network coded cooperation, the idea of which is to match network-on-graph with code-on-graph to construct efficient network codes accounting for changing topoogy and ossy nature of wireess networks, is studied in [20]. In [9], a network coding based cooperative diversity scheme is proposed. The focus of these work is on the anaysis of diversity-mutipexing tradeoff of network coded cooperative diversity. They rey on the overhearing assumption with the singe shared channe mode, whie XOR-CD assumes a muti-channe setting and focuses on the resource aocation probem to reaize the promise of network coding. In this regard our work aso differs from studies of joint network and channe coding in two-way reay channe [21], [22]. Resource aocation in cooperative networks has been extensivey studied in previous work as we [23] [26]. [23] considers power aocation for a simpe triange network with one pair of source-destination and one reay. [24] considers muti-hop ad hoc networks and proposes a framework that jointy considers routing, reay seection and power aocation. [25], [26] considers OFDM based ceuar networks and are most reated to our work. [25] studies channe assignment and power aocation for muti-hop OFDM networks. [26] proposes soutions for joint optimization of channe assignment, reay strategy seection and power aocation in OFDM ceuar networks based on conventiona mpify-and-forward and Decode-and-Forward. Different from these efforts (summarized in Tabe 1), this work represents an eary attempt to study cooperative reaying in muti-channe networks with the use of network coding. We propose a nove di-

3 TBLE 1 Reated studies to this paper. Coding Channe Reay Power diversity assignment strategy aocation This paper [8], [9], [20] [23] [24] [25] [26] versity scheme with XOR that improves the resource efficiency of reaying and thereby boosts throughput performance. More importanty, we present a crossayer optimization framework to address the resource aocation probem for the cooperative network. Our framework jointy considers network coding, channe assignment, reay strategy seection and power aocation, which has not yet been discussed to our knowedge. Finay, our conference version [1] focuses on soving the physica ayer resource aocation probem with a specific utiity function. In this work, we adopt a network utiity maximization framework that generaizes to encompass many possibe choices of utiity functions at the upper ayer. We aso show that the upper ayer rate aocation probem and the ower ayer resource aocation probem can be decouped and soved independenty through a subgradient method, which is not present in [1]. 3 SYSTEM MODELS In this section, we introduce the underying system modes for our optimization framework. 3.1 Network Modes We consider a singe-ce OFDM network. The BS is communicating with each MS with bi-directiona traffic. The system operates in FDD mode, meaning that the upink and downink of a MS are assigned orthogona sets of subchannes. sma number of RS are empoyed in the ce to provide cooperative diversity. They may hep some MS for transmissions on their data subchannes, using reay subchannes from a reay channe poo orthogona to the data channe poo. One reay subchanne is used to support ony one data subchanne of a MS in conventiona cooperative diversity (CD). In the case of XOR-CD, one reay subchanne is used to support two data subchannes, one on the upink and one on the downink, as we iustrated in Sec. 1. We further assume that the BS and MS have infinite backog of traffic. It is then the case that cooperative transmissions progress concurrenty with direct data transmissions, which we sha discuss in more detai in Sec Decode-and-Forward (DF) is used as the conventiona CD scheme. ζ ψ Ω Φ L M() c i c r r s σ,c P p r,c i,c r p r,c i,c j,c r s TBLE 2 Key notations used in this paper. 3.2 Channe Modes set of data subchannes set of reay subchannes set of mobie stations set of reay stations set of inks MS corresponding to ink data subchanne c i ζ reay subchanne c r ψ reay station r Φ mobie station s Ω channe gain to noise ratio on ink channe c fixed power budget for direct transmission conventiona CD power aocation on ink XOR-CD power aocation on mobie station s We mode the wireess fading environment by arge scae path oss and shadowing, aong with sma scae frequency-seective Rayeigh fading. Fading between different subchannes are independent. We assume the network operates in a sow fading environment, so that channe estimation is possibe and fu channe side-information (CSI) is avaiabe, which makes the optimization feasibe. In a practica system, channe estimation is generay done at the receiver end and fed back to the base station, which then soves the optimization and informs a MS and RS the channe assignment, power eves, and reay strategies. Such assumptions about the fading environment are commony used as in [23], [25], [26]. so note that when the environment has fast fading components, optimization may be done in a statistica sense. In practica systems, it is usuay not feasibe to assign arbitrary subchannes for the upink and downink transmissions due to sef-interference. Usuay separate chunks of frequency bands are aocated in FDD mode. Such a practica constraint does not contradict our channe mode, however, as we can view any interference-free chunk of frequency bands as a subchanne, which is the basic unit of channe aocation in our probem. This constraint imits the fexibiity of channe aocation, and negativey impacts the throughput performance of cooperative and direct transmissions. Nevertheess, it does not affect the reative performance improvement of XOR-CD over comparative schemes, since the same constraint exists for a types of transmissions. n equa amount of power P is aocated for both direct and reay transmissions across a data and reay subchannes. In the extended modes with reay power aocation, however, RS can adjust the power eve for each of the reay subchannes they use in order to confine themseves to their power budget. 3.3 Notations Denote ζ, ψ, Ω and Φ as the set of data subchannes, reay subchannes, MS, and RS, respectivey. s Ω

4 denotes a MS and r Φ denotes a RS r, respectivey. L denotes a directed ink from the source S() to the destination D() where L is the set of inks. Each ink, being an upink or downink, has a corresponding MS s such that S() = s or D() = s. Let M() = s denote this reationship betweenands. Each ink can operate in one and ony one of three modes, namey the direct transmission mode, conventiona CD mode and XOR-CD mode, depending on the choice of reay strategy. Define the function R(c i,) as the achievabe rate of direct transmission on ink when it is assigned with subchanne c i. For conventiona CD, R(c i,c r,r,p r,ci,cr, ) is the achievabe rate function of, when RS r is assigned to be the reay for transmission on data subchanne c i, with aocated power p r,ci,cr on reay subchanne c r. For XOR-CD, R(c i,c j,c r,r,p r,ci,cj,cr s, s) denotes the achievabe rate function ifr is the reay ofsfor its upink transmission on c i and downink transmission on c j, with aocated power ps r,ci,cj,cr on reay subchanne c r. To assist the understanding of the anaysis, we summarize the key notations used throughout the paper in Tabe n Information Theoretic naysis We first provide an information theoretica anaysis in order to derive the rate functions for three transmission modes, especiay XOR-CD. The setup, incuding the compex channe gains on different inks, is shown in Fig. 2. Noises are modeed as i.i.d. circuary symmetric compex Gaussian noises CN(0,N 0 W) Direct Transmission For direct transmission of say ink B, the achievabe rate is found using the we-known formua (in b/s/hz): ( R(B,c 1 ) = og 2 1+ P h B,c 1 2 ), (3.1) ΓN 0 W where Γ is the gap to capacity and P denotes the direct transmission power. For notationa convenience we denote hb,c 1 2 ΓN as σ 0W B,c 1, where σ represents the channe gain-to-noise ratio. Then the rate function can be simpy expressed as: R(B,c 1 ) = og 2 (1+P σ B,c1 ). (3.2) Conventiona CD For the DF reay transmission for the traffic to B, assuming is a MS and B is the BS, first R 0 attempts to decode s message. If decoding is successfu, R 0 transmits to B with power p R0,c1,c4 B using reay subchanne c 4 as depicted in Fig. 2. Therefore, the maximum rate for this mode can be found to be R(c 1,c 4,R 0,p R0,c1,c4 B,B) = min{og 2 (1+P σ R0,c 1 ), og 2 (1+P σ B,c1 +p R0,c1,c4 B σ R0B,c 4 )}. (3.3) Compared to the resut in [6], ours does not have 1 2 before the expression. The reason is that a two-sot impementation is assumed in [6] with a shared channe, whereas in our OFDM-based muti-channe system, reay transmissions progress concurrenty with direct transmissions on orthogona channes. t any time sot, the concurrent reay transmission carries the message it received from the direct transmission in the previous time sot. For a reasonaby ong time period, say hundreds of time sots, the one-time-sot ead time can be safey ignored. R0 h R0,c 3 h R0B,c 4 h BR0,c 2 h R0,c 1 h B,c1 h B,c2 (a) Conventiona DF B h R0,c 3 h BR0,c 2 h R0,c 1 h B,c1 R0 h B,c2 (b) XOR-CD h R0B,c 3 Fig. 2. The channe modes for Decode-and-Forward and XOR-CD, where h,c denotes channe gain of ink when it is assigned subchanne c. Inspecting the rate function, we can see that increasing the reay power wi first increase the rate, but not any more after reaching a threshod, since the reay cannot deiver more information than what it can decode. Thus, the threshod vaue of the reay power is such that R(c 1,c 4,R 0, p R0,c1,c4 B,B) = og 2 (1+P σ R0,c 1 ) = og 2 (1+P σ B,c1 + p R0,c1,c4 B σ R0B,c 4 ), (3.4) which gives us p R0,c1,c4 B = σ R 0,c 1 σ B,c1 σ R0B,c 4 P. (3.5) Simiar anaysis can be carried out for R(c 2,c 3,R 0,p R0,c2,c3 B, B) and the threshod power p R0,c2,c3 B. These cacuations wi be used ater for the power contro probem in Sec XOR-CD The reay transmissions from R 0 to a MS and the BS B are done by performing XOR over the two messages and muticasting using a singe reay subchanne c 3. Therefore, the rate of this mode, for each of the two inks invoved, can be shown to be R(c 1,c 2,c 3,R 0,p R0,c1,c2,c3,) = min{og 2 (1+P σ R0,c 1 ),og 2 (1+P σ BR0,c 2 ), og 2 (1+P σ B,c1 +p R0,c1,c2,c3 σ R0B,c 3 ), og 2 (1+P σ B,c2 +p R0,c1,c2,c3 σ R0,c 3 )}, (3.6) The first two terms in (3.6) represent the maximum rate at which the reay can reiaby decode the source B

5 messages from both and B, whie the ast two terms represent the maximum rate at which and B can reiaby decode their intended message given repeated transmissions from R 0 s muticast, respectivey. Note that the upink and downink fows have the same rate given by (3.6). gain the threshod vaue of reay power at R 0 is such that R(c 1,c 2,c 3,R 0, p R0,c1,c2,c3,) = min{og 2 (1+P σ R0,c 1 ),og 2 (1+P σ BR0,c 2 )}. The detaied expression for p R0,c1,c2,c3 can then be derived. 4 N OPTIMIZTION FRMEWORK We present our optimization framework in this section. fter introducing the network utiity maximization framework, we first present RSS-XOR, our formuation for the joint channe assignment, reay assignment, and reay strategy seection probem with XOR-CD. We then provide another formuation with conventiona diversity schemes ony, i.e. the NO-XOR probem. We extend both formuations by considering reay power aocation. Finay we demonstrate that both probems under the NUM framework can be soved in the dua domain as a cross-ayer optimization probem. 4.1 The Network Utiity Maximization Framework We adopt a network utiity maximization framework where each data stream of a particuar ink has a utiity function, and the overa objective is to maximize the tota utiity of the network. The network utiity maximization framework (NUM) is originated from the semina work of Key [27], and has been extensivey appied to cross-ayer design probems in wireess networking [28]. utiity function is a concave and increasing function of the ink throughput that refects a MS s satisfaction. Depending on the appication the traffic is serving (e.g. voice, data), the utiity function can take on different shapes. Denote the throughput of ink as d, then utiity function can be denoted as U (d ). The objective of the optimization can be expressed as: max d U (d ) (4.1) L 4.2 The RSS-XOR Probem Our goa is to optimize the strategies of assigning appropriate reay subchannes to RS and data subchannes to MS, and pairing RS to the data subchannes of MS with different choices of reay strategies, in order to maximize the aggregated utiity. We now present the optimization constraints that refect these considerations in the foowing. For both upink and downink, traffic fas into three casses corresponding to the three transmission modes, namey direct traffic, conventiona CD traffic, and XOR-CD traffic. Introduce three0 1 decision variabes x ci, yr,ci,cr, and zs r,ci,cj,cr. x ci indicates whether inkon data subchanne c i is performing direct transmission. indicates whether ink is operating in conventiona CD mode with RS r and data-reay subchanne pair (c i,c r ). Each MS may be assigned mutipe such channe pairs depending on the instantaneous channe condition.zs r,ci,cj,cr indicates whether MS s is assigned with RS r and reay subchanne c r for its upink on data subchanne c i and downink on c j for XOR-CD. Since an equa amount of power P is used for each direct and reay transmission, throughput of ink can be characterized as foows: d = c i ζr(c i,)x ci + + c i,c j ζ,c r ψ,r Φ c i ζ,c r ψ,r Φ R ( c i,c r,r,p, ) R(c i,c j,c r,r,p,s)z r,ci,cj,cr s, where s = M(), L. (4.2) Reca that each data subchanne can ony be assigned to one ink which operates in one of the three modes. Therefore, L ( x ci + c r ψ,r Φ ) + +z r,cj,ci,cr s s Ω,r Φ,c j ζ,c r ψ ( z r,ci,cj,cr s ) 1, c i ζ, (4.3) where the first term accounts for the possibiity that c i is assigned for direct and conventiona CD modes, and the second term accounts for the possibiity of XOR-CD. Notice that this constraint aso impicity takes into consideration that each ink can ony operate in one of the three modes. Simiary, each reay subchanne can be assigned to ony one cooperative session, be it conventiona CD session or XOR-CD session. L r Φc i ζ + s Ω r Φc i ζ c j ζ z r,ci,cj,cr s 1, c r ψ. (4.4) Consequenty, the RSS-XOR probem becomes an integer program, with the objective (4.1) subject to constraints (4.2), (4.3), and (4.4). For ease of presentation, we use x,y and z to represent a the x ci s, s, and z r,ci,cj,cr s RSS-XOR: max x,y,z s as optimizing variabes. U (d ) L s.t. (4.2),(4.3), and (4.4). (4.5)

6 4.3 The NO-XOR Probem We aso provide the optimization formuation under NUM with ony conventiona cooperative diversity, i.e., the NO-XOR probem, which is studied as a baseine comparison. It can be readiy formuated in a simiar way as the RSS-XOR probem, with zs r,ci,cj,cr equa to zero for any c i,c j ζ, c r ψ, s Ω, r Φ. Formay, NO-XOR: max x,y U (d ) L s.t. d = c i ζr(c i,)x ci + c i ζ x ci + L L L r Φc i ζ 4.4 Power ocation R ( c i,c r,r,p, ), c r ψ r Φ r Φc r ψ 1, c i ζ, 1, c r ψ. (4.6) We can extend the two modes by incorporating an additiona constraint that each RS has a imited power budget. RS then has to aocate the right amount of power across a the cooperative sessions it supports in order to maximize the tota utiity. Mathematicay, the throughput constraints of both probems are updated by using R(c i,c r,r,p r,c,cr,) to repace R(c i,c r,r,p,) in (4.2), and using R(c i,c j,c r,r,p r,ci,cj,cr s, s) to repace R(c i,c j,c r,r,p,s) in (4.6). The constraint that the tota power of RS cannot exceed its budget can be expressed as foows for the RSS-XOR probem: L c i ζ c r ψ p r,ci,cr + s Ω c i ζ c j ζ c r ψ p r,ci,cj,cr s P r, r. (4.7) where P r denotes the power budget of RS r. RSS-XOR with power aocation can be formuated by adding constraint (4.7) into the origina formuation. For NO-XOR, the power constraint is simpy: L c i ζ c r ψ p r,ci,cr P r, r Φ. (4.8) The power aocation version of NO-XOR is simiary formuated by adding constraint (4.8) into (4.6). because if two sets of throughput using two different channe-rs-ink assignments and reay strategies are achievabe individuay, their inear combination is aso achievabe by a frequency-division mutipexing of the two sets of strategies. This idea for non-convex probems of muti-carrier systems is discussed in [29]. In particuar, using the duaity theory of [29], the foowing is true: Proposition 1: The RSS-XOR and NO-XOR probems, with the discrete seection of channes, RS and reay strategies, have zero duaity gap in the imit as the number of OFDM subchannes goes to infinity. detaied proof can be constructed aong the same ine of argument as in [29]. This proposition aows us to sove non-convex probems in their dua domain. though it requires number of channes to go to infinity, in reaity the duaity gap is very cose to zero as ong as number of channes is arge [26]. With this proposition, we show that the RSS-XOR and NO-XOR probems can be decouped into an appication ayer rate adaption probem and a physica ayer resource aocation probem, and be soved by soving these two probems separatey. Our technique is reminiscent of that in [26]. We focus on the basic RSS-XOR probem, whie the technique can be easiy appied to the NO-XOR probem and their power aocation extensions as we. First, introduce a new variabe t = [t 1,...,t,...,t L ], and rewrite the RSS-XOR probem as foows: max t,x,y,z U (t ) L s.t. d t, L, (4.2),(4.3), and (4.4). (4.9) Because U is an increasing function, when the objective of (4.9) is maximized, t must be equa to d. Thus (4.5) and (4.9) must have the same soution. The key step to decompose the probem is to reax the new constraint d t. The Lagrangian becomes L(λ,t,x,y,z) = L (U (t )+λ (d t )). (4.10) where λ being a dua variabe corresponding to ink. Observe the dua function { max L(λ,t,x,y,z) g(λ) = t,x,y,z (4.11) s.t. (4.2),(4.3), and (4.4). 4.5 Cross-ayer Optimization in the Dua Domain Both RSS-XOR and NO-XOR are non-convex probems because of the integer constraints x, y and z. Duaity gap for non-convex probems is non-zero in genera. However, in an OFDM system with many narrow subchannes, the optima soutions of RSS- XOR and NO-XOR are aways convex functions of P, now consists of two sets of variabes: appication ayer variabe t, and physica ayer variabes x,y,z. It can be readiy separated into two maximization subprobems, one rate adaptation probem in the appication ayer, g app (λ) = max t (U (t ) λ t ), (4.12) L

7 and a resource aocation probem in the physica ayer, max λ d x,y,z g phy (λ) = L (4.13) s.t. (4.2),(4.3), and (4.4). We can see that the optimization formuation provides a ayered approach to the network utiity maximization probem. The use of dua variabe λ contros the interaction between the ayers. It can be interpreted as a price signa that coordinates the throughput suppy and demand reationship between the physica and appication ayer. The physica ayer attempts to maximize the tota revenue given the perink rate price λ. higher vaue of λ attracts the physica ayer to aocate more resources to. The appication ayer, on the other hand, tries to maximize the tota net utiity, given the per-ink rate cost λ. higher vaue of λ causes the appication ayer to reduce its demand for throughput. Finay, since the network utiity maximization probem (4.9) has zero duaity gap, it can be soved by minimizing the dua objective: min g(λ) s.t. λ 0. (4.14) One way to sove the dua probem is to use a subgradient method that updates λ iterativey as shown in gorithm 1. gorithm 1 Subgradient method for soving (4.9). 1. Initiaize λ (0). 2. Given λ (k), sove the appication ayer and physica ayer subprobems (4.12) and (4.13), respectivey. Obtain the optima vaues t, x, y and z, and thus d. 3. Perform a subgradient update for λ, where ν (k) foows a diminishing step size rue: λ (k+1) r = [ λ (k) + (ν (k)) T (t d )] + 4. Return to step 2 unti convergence. Foowing a diminishing step size rue for choosing ν (k), the subgradient method above is guaranteed to converge to the optima dua variabes [30]. Carefu readers may be concerned with the sow convergence of the subgradient updates, especiay when the probem scaes up. Computationa experiences suggest that the compexity of subgradient updates is poynomia in the dimension of the dua probem, which is L for g(λ) [26]. Soving the appication ayer subprobem is straightforward. It can be readiy seen that the objective of (4.12) is maximized by maximizing each term in the summation separatey. U (t ) λ t is concave since by assumption U is a concave function of t. Thus, the optima throughput demand in the appication ayer t can be found by simpy taking the derivative of U (t ) λ t with respect to t and setting it to zero. The BS searches for t for each ink foowing this procedure. Many different choices of utiity functions are possibe. Since we assume infinite backog of data, we use the foowing utiity function definition for every ink throughout the rest of the paper: U (t ) = nt (4.15) This corresponds to the we-known proportiona fairness utiity mode [27]. Its merits incude the abiity to strike a good baance between throughput and fairness, and robustness with respect to changes in topoogy and power constraints [28]. We et each ink shares the same utiity function definition here for ease of iustration. The optimization however does not depend on this assumption to work. With this simpe utiity function, t can be readiy found as foows: t = 1 λ. (4.16) 5 SOLUTION LGORITHMS FOR THE PHYS- ICL LYER RESOURCE LLOCTION PROB- LEM We demonstrated that our network utiity maximization probems can be soved in their dua domain by soving the appication ayer and physica ayer subprobems separatey. We aso showed that the appication ayer subprobem is easy to sove. In this section, we tacke the more difficut physica ayer subprobem. The physica ayer resource aocation probem is essentiay an integer program. Conventiona approaches, such as branch and bound [31], are computationay expensive. Our soution agorithms need to be run frequenty at each scheduing epoch, making the task of deriving efficient heuristic agorithms imperative. Here we design efficient agorithms that sove the resource aocation for both the RSS-XOR and NO-XOR probems. Specificay, we first prove that RSS-XOR resource aocation is NP-compete and can be soved in poynomia-time with an approximation ratio of 1.5 using our agorithm. We then show that NO-XOR resource aocation can be optimay soved by transforming to weighted bipartite matching. Finay we design a subgradient agorithm to sove power aocation of the two probems in the dua domain. 5.1 Set Packing gorithm for RSS-XOR Resource ocation Soving the seemingy prohibitive RSS-XOR resource aocation probem (4.13) hinges on transforming to a

8 weighted set packing probem. We first estabish the equivaence and prove the hardness of the probem. We then propose our agorithm with a constant approximation factor. Proposition 2: The RSS-XOR resource aocation probem is equivaent to a maximum weighted 3-set packing probem, and is NP-compete. Proof: Construct a coection of channe sets C from a base setζ ψ as shown in Fig. 3. There are three kinds of channe sets, representing three transmission modes respectivey. (c i ), where c i ζ represents a the avaiabe channe sets for the direct transmission mode. (c i,c r ) where c i ζ,c r ψ corresponds to a the avaiabe data-reay channe combinations for the conventiona CD mode, with data subchanne c i and reay subchanne c r. The third kind, (c i,c j,c r ) corresponds to a the channe sets for the XOR-CD mode with data subchanne pair (c i,c j ) and reay subchanne c r, where c i,c j ζ,c r ψ. Sets intersect if they share at east one common eement, and are otherwise said to be disjoint. Each set has a corresponding weight, denoting the maximum objective vaue found across a possibe assignments of this channe set to different combinations of RS and inks. Specificay, w (ci) = max λ R(c i,), (5.1) w (ci,c r) = max λ R(c i,c r,r,p,).,r (5.2) For set (c i,c j,c r ), its weight is found over a possibe assignments of this set to combinations of RS and upink-downink of a MS, since it can ony be assigned to one MS. Formay, w (ci,c j,c r) = max s,r :s=m() λ R(c i,c j,c r,r,p,s). (5.3) w (ci,c j,c r) essentiay sums up upink and downink rates of s since one XOR-CD session incorporates two cooperative transmissions. The optimization (4.13) is to find the optima strategy to choose the transmission mode and assign RS and channes to each ink in order to maximize the aggregated utiity. The maximization is done over a inks. Equivaenty, we can interpret it as to find the optima strategy to seect disjoint channe combinations and assign RS and inks to them so as to maximize the objective. This is simpy a change of the order of summation in the objective of (4.13), when we substitute (4.2) into it. In this aternative interpretation, the maximization is done over a possibe channe sets by matching them to the best possibe inks and RS without dupicate use of channes. The soution found must exhaust a subchannes since we can aways improve the tota weight by adding sets corresponding to unassigned data and reay subchannes. The number of eements in a set is at most 3, therefore the probem is equivaent to weighted 3-set packing [32], which is NP-compete. the sets and their corresponding set weight are recorded in a tabe T assign. We see that, for sets (c i ), the size of weight search space is L ; for sets (c i,c r ) and (c i,c j,c r ), the search space size is Φ L. Thus, the weight construction process is of poynomia time compexity, given the number of three kinds of sets are aso poynomias of ζ and ψ. To propose a good approximation agorithm with reasonabe time compexity, first we construct an intersection graph G C of the set system C with the set of vertices V C and the set of undirected edges E C as shown in Fig. 3. Weighted set packing then can be generaized as a weighted independent set probem, the objective of which is to find a maximum weighted subset of mutuay non-adjacent vertices in G C [33]. The size of sets is at most 3, therefore G C is 3-caw free 1. The best known approximation for the weighted independent set probem in a caw-free graph is proposed in [33] and then acknowedged in [32], which we extend to form our agorithm. First we introduce a greedy agorithm, caed Greedy that prepares the groundwork. Define N(K,L) to be the set of vertices in L that intersect with vertices in K, i.e. N(K,L) = {u L : v K such that {u,v} E or u = v}. Greedy is a natura heuristic that repeatedy picks the heaviest vertex from among the remaining vertices and eiminate it and the adjacent vertices as shown in gorithm 2. gorithm 2 Greedy. 1. S 2. whie V C N(S,V C ) do 3. choose u V C N(S,V C ) with the maximum weight 4. S S {u} 5. end whie gorithm 3 pproximation agorithm for soving (4.13). 1. Construct the coection of weighted sets C and transform them into the weighted undirected graph G C. 2. Obtain a maxima independent set S using Greedy. 3. whie caw c such that T c improves w 2 (S) do 4. S S N(T c,s) T c 5. end whie 6. ssign channes to RS and inks by searching the entries in T assign corresponding to the sets present in S. natura thought to improve on the maxima independent set found by Greedy is to do oca search and repace a set with its caw with arger weight, which motivates our approximation agorithm summarized 1. Here a d-caw c is an induced subgraph that consists of an independent set T c of d nodes.

9 Data subchannes 1 2 Reay subchannes 1 2 Set system C Direct (1) (2) } transmission (1,1) (1,2) Conventiona (2,1) (2,2) } CD (1,2,1)(1,2,2) XOR-CD (2,1,1) (2,1,2) Intersection graph G C Fig. 3. Channe set construction and transformation into an intersection graph with 2 data subchannes and 2 reay subchannes. Vertices in G C correspond to sets in C. Edges are added between vertices whose corresponding sets intersect. as in gorithm 3. From [33], oca improvements on the square of tota weights sove the weighted independent set probem with a constant approximation factor of 1.5, which is the best resut known so far [32]. Therefore we have the foowing: Proposition 3: gorithm 3 provides at east 2 3 of the optimum of RSS-XOR physica ayer resource aocation probem (4.13). This is the best performance guarantee one can have uness a better agorithm can be found for the weighted independent set probem. 5.2 Matching gorithm for NO-XOR Resource ocation We now consider the NO-XOR physica ayer subprobem. Using the same technique presented in Sec. 4.5, the probem can be shown in the foowing form: max x,y λ d L s.t. d = c i ζr(c i,)x ci + R ( c i,c r,r,p, ), optimay soves the probem. c i ζ c r ψ r Φ x ci + Graph 1, c i ζ, L L r Φc r ψ 1, c r ψ. (5.4) 1 r Φ 2 L c i ζ Surprisingy, we find that it can be optimay soved in poynomia time. Specificay, Proposition 4: The NO-XOR physica ayer resource aocation probem is equivaent to weighted bipartite matching over a data and reay subchannes, and thus can be soved optimay. Proof: Construct a bipartite graph = (V 1 V 2,E) where V 1 and V 2 correspond to the set of data subchannes ζ and reay subchannes ψ respectivey, as shown in Fig. 4. We patch void vertices to V 2 to make V 2 = V 1 = ζ. The edge set E corresponds to ζ 2 edges connecting a possibe pairs of channes in two vertex sets. Each edge (k, j) carries three attributes, (w kj, kj,r kj ), where w kj = max λ R(k,j,r,P,),,r ( kj,r kj ) = argmax λ R(k,j,r,P,). (5.5),r For edges connecting data subchannes to void reay subchannes that we patched, the edge weights become (w kj, kj,0) where kj is the ink providing maximum objective vaue if data subchanne k is used. This essentiay captures the maximum objective vaue given by direct transmission. Observe that is bipartite, we can see the NO-XOR resource aocation probem (5.4) is equivaent to finding the maximum weighted bipartite matching on. The second attribute of an edge (k,j) in the maximum matching represents the ink assigned with this datareay subchanne pair (k, j), whie the third attribute dictates the transmission mode or the corresponding RS. 0 in the third attribute simpy means the ink shoud work in direct transmission mode. Hence, the maximum matching found represents the comprehensive assignment of RS, data and reay subchannes, as we as the transmission strategy decision, therefore. k. (w kj, kj,r kj ) 2. j V 1 V 2 Fig. 4. The graphica mode to show the equivaence of NO-XOR resource aocation probem (5.4) and weighted bipartite matching. Dotted vertices are void vertices patched. Not a inks are shown here. Severa good poynomia-time agorithms exist for soving the bipartite matching probem, of which the 1.

10 Hungarian agorithm [34] is a popuar choice. Since the graph construction is O( ζ 2 L ψ ), the entire agorithm is poynomia time. 5.3 Power ocation gorithm Finay, we turn our focus to the power aocation probem. Reca that the power imited versions of RSS-XOR and NO-XOR are proposed in Sec Consider the power imited RSS-XOR probem. Readiy we can see that it can be decouped into the same appication ayer subprobem as (4.12), and a physica ayer resource aocation of the foowing form, which incudes now an additiona power constraint (4.7): max x,y,z λ d L s.t. (4.2),(4.3),(4.4), and (4.7) (5.6) Therefore gorithm 1 can be used to sove it, provided that we can sove the physica ayer subprobem (5.6). In this section, we deveop a dua method to sove (5.6) efficienty. Introduce an Lagrangian mutipier vector µ to the power constraint (4.7) and the dua probem becomes where g(µ) =max L min µ 0 λ d + r ψ g(µ) (5.7) µ r ( P r p r,c,cr,c,c r ps r,ci,cj,cr s,c i,c j,c r s.t. (4.2) (4.4). (5.8) Since each MS s corresponds to two inks, we can equay spit power used for XOR-CD ps r,ci,cj,cr to these two inks without vioating the power constraint. Mathematicay, we et p r,ci,cj,cr = { 1 2 pr,ci,cj,cr s if s = M(); 0 otherwise. ) (5.9) The objective function of (5.8) can then be written as max (λ d µ r p r,c,cr ) µ r p r,ci,cj,cr r,c,c r r,c i,c j,c r + µ r P r. (5.10) r r µ rp r is constant in (5.10) since µ is given for each instance of g(µ). So soving the optimization (5.8) is equivaent to soving the foowing: max (λ d µ r p r,c,cr ) µ r p r,ci,cj,cr r,c,c r r,c i,c j,c r s.t. (4.2) (4.4). (5.11) Compared with the origina RSS-XOR physica ayer subprobem (4.13), the ony difference is the objective function which now incudes the cost of power. µ can be interpreted as a pricing variabe vector for reay power. (5.11) can be thought of as maximizing the tota throughput revenue minus the tota cost of reay power used, given the current prices of power at RS. This is easiy decomposed into maximization over every possibe set of data and/or reay channes. Therefore, it can be soved using the gorithm 3 in Sec. 5.1, with the weights being the maximum throughput revenue discounted by power cost instead of the maximum revenue ony. For ease of exporation, we dictate that the reay power for each cooperative session is set to the threshod vaue as derived in Sec and Then, w (ci,c r) = max,r w (ci,c j,c r) = max s,r λ R(c i,c r,r,p r,ci,cr :s=m(),) µ r p r,ci,cr, λ R(c i,c j,c r,r,p r,ci,cj,cr s,s) µ r p r,ci,cj,cr s. fter soving (5.11), the dua probem (5.7) can be readiy soved via a subgradient method which repeatedy updates the power prices according to the demand and suppy reationship at RS to reguate the power consumption. To summarize, the agorithm for soving the power imited RSS-XOR probem works as shown in gorithm 4. Notice that the subgradient gorithm 4 gorithm for soving (5.6). µ (0) 1. Initiaize µ (0). 2. Givenµ µ (k), sove the maximization probem (5.11) using gorithm 3. Obtain the soution vaues ˆp r,c,cr and ˆp r,ci,cj,cr, and the maxima independent set Ŝ. 3. Perform a subgradient update for µ, where ν (k) foows a diminishing step size rue: [ µ (k+1) r = µ (k) r ν r (P (k) r ˆp r,c,cr ) ] + ˆp r,ci,cj,cr,c,c r,c i,c j,c r 4. Return to step 2 unti convergence. 5. ssign channes to RS and inks by searching the entries in T assign corresponding to the sets present in Ŝ found in step 4. agorithm is suitabe for distributed impementation across RS. Each RS is abe to verify its power consumption, and update its own reay power price autonomousy according to ν (k) informed by BS. The updated prices can be transmitted to the BS with a negigibe amount of overhead. The dua method for power imited NO-XOR probem can be deveoped in a simiar way. 6 PERFORMNCE EVLUTION We dedicate this section to evauating the performance of our agorithms using simuations. Reca

11 that we use the proportiona fairness utiity function U (d ) = nd for each ink as mentioned in Sec Simuation Setup P N 0W We first introduce the simuation setup. The key of our experiment settings is to derive the achievabe data rate of a subchanne when it is aocated to a particuar MS, which requires computing the SNR vaue. We rey on the wireess channe simuator caed Chsim [35] to generate reaistic SNR resuts under different channe and mobiity modes, and adopt empirica parameters expained beow to mode the wireess fading environment. The subchanne bandwidth is set to be khz. Data subchannes are centered at 5 GHz, whie reay subchannes are centered at 2.5 GHz. The channe gain between two nodes at each subchanne can be decomposed into a sma-scae Rayeigh fading component and a arge-scae og norma shadowing with standard deviation of 5.8 and path oss exponent of 4. The inherent frequency seective property is characterized by an exponentia power deay profie with deay spread 15 µs. The time seective nature is captured by Dopper spread, which depends on the MS s speed. Throughout the simuation the MS are moving with speed uniformy distributed from 1 to 5 m/s according to random waypoint mode with zerosecond pause period. Without oss of generaity, the gap to capacity Γ is set to 1, meaning that for a given instantaneous channe gain, the physica ayer codewords adaptivey operate at the instantaneous achievabe rate of the reay protoco. This impies that an idea adaptive moduation and coding scheme (MC) is impemented. Γ can aso be set to a vaue arger than 1 to refect the gap between physica ayer impementation and the theoretica resut. The power constraint for each transmission is such that =23 db. This corresponds to a medium SNR environment. Such an experimenta setup is commony used in reated studies [25], [26]. 6.2 Performance of XOR-CD We first evauate the performance of XOR-CD using gorithm 1 with gorithm 3. We compare it to conventiona CD using gorithm 1 with the Hungarian agorithm [34]. We focus on the scenario where 10 MS are randomy ocated in a ce with a 100-meter radius. We set the number of data subchannes to be 100, and that of reay subchannes to be 30. For fairness 130 subchannes are used for the simuation of direct transmission. 1 RS is depoyed in the ce. Fig. 5 pots the network throughput for one second, with a samping period of 5 ms. The optimization therefore is done for 200 times. We can ceary see that XOR-CD outperforms conventiona DF cooperative diversity by around 30%. This is as expected, because XOR-CD conserves reay channes that can be utiized Throughput (Mbps) Conventiona CD Direct transmission Sampe index Fig. 5. Throughput comparison of XOR-CD against conventiona CD and direct transmission. Power efficiency Conventiona CD Sampe index Fig. 6. Power efficiency comparison of XOR-CD against conventiona CD. to support more cooperative sessions. To further iustrate its superiority in this aspect, we study XOR- CD s reaying resource efficiency. We evauate power efficiency defined as the ratio of throughput improvement over direct transmission and the amount of reay power used. Since we assume a fixed power for each reay subchanne without power aocation, this amounts to throughput improvement (%) power efficiency = number of reay subchannes. s seen in Fig. 6, XOR-CD s power efficiency is significanty better than that of conventiona CD. Finay, we notice that the conventiona diversity scheme aone provides over 20% improvement compared with simpe direct transmission. This diversity gain is simiar to the network coding gain, which further confirms the advantage of XOR-CD to doube the diversity gain without significant costs. 6.3 Effects of Reaying Resources Next, we study the effects of reaying resources on the performance of XOR-CD. We focus on two kinds of resources: RS and reay subchannes. Fig. 7 and Fig. 8 show the resuts, respectivey. Intuitivey, more RS provide better chances for MS to find a nearby

12 RS with better reay channe quaities. More reay subchannes aso enabe more cooperative sessions to take pace. Ceary, these two factors contribute to Throughput (Mb/s) Conventiona CD Number of RS (a) Throughput comparison. Power efficiency Conventiona CD Number of RS (b) Power efficiency comparison. Fig. 7. Effects of the number of RS. Number of reay subchannes is 20, number of data channes is 100, number of MS is 10, and ce radius is set to be 100m. Throughput (Mb/s) Conventiona CD Number of reay channes (a) Throughput comparison. Power efficiency Conventiona CD Number of reay channes (b) Power efficiency comparison. Fig. 8. Effects of the number of reay channes. Number of RS is 3, number of data channes is 100, number of MS is 10, and ce radius is set to be 100m. the increasing throughput refected in Fig. 7(a) and Fig. 8(a). XOR-CD consistenty maintains a 20% 30% gain over conventiona CD. Interestingy, when it comes to the power efficiency as defined in Sec. 6.2, we observe a cear discrepancy between the two kinds of resources. When we increase the number of RS, the tota amount of reay power used is unchanged since the tota number of reay subchannes, and thus the tota number of cooperative sessions, is unchanged. Thus, the power efficiency is improved as shown in Fig. 7(b). However, when we increase the number of reay subchannes, the tota amount of reay power used is proportionay increased, and efficiency is actuay decreasing in Fig. 8(b). This is because the optimization aways tries to expoit the argest performance gains first, which eads to the diminishing margina gain of using more reay subchannes and thus reay power. This observation suggests that we coud use a sma amount of reaying resources to obtain a reasonaby satisfactory improvement. 6.4 Effects of Path Loss Intuitivey, path oss increases when we increase the ce radius, and reaying becomes more beneficia for improving the throughput. This intuition is confirmed in Fig. 9, which pots the tota throughput improvement of XOR-CD over conventiona CD with different vaues of ce radius. We observe that as ce radius increases from 100m to 300m, the throughput improvement aso increases from around 40% to around 70%. We have conducted simuations with conventiona CD and observed the same trend. This observation suggests that reaying in genera is more hepfu for networks that are imited by path oss. Improvement (%) radius 100m radius 200m radius 300m Sampe index Fig. 9. Effects of path oss for 100 data subchannes, 30 reay subchannes, 10 MS and 3 RS. Empirica CDF Convergence and Duaity Gap 0.2 RSS XOR NO XOR Fig. 10. Empirica CDF of convergence iterations with 100 data subchannes and 30 reay subchannes. It is necessary to examine the convergence speed of our agorithms based on the subgradient methods. Fig. 10 pots the empirica CDF of convergence iterations of gorithm 1 for both RSS-XOR and NO-XOR probems. We observe that it takes on average around 180 iterations to sove the RSS-XOR probem, and around 130 iterations to sove the NO-XOR probem. No more than 220 and 157 iterations are needed for soving the RSS-XOR and NO-XOR probems, respectivey. We beieve that such a computation burden is affordabe for typica base stations with everincreasing computationa power. We aso examine the duaity gap of our probem as a resut of the finite number of OFDM subchannes. Since the RSS-XOR probem (4.5) is an NP-compete integer program with a non-inear objective function, we estimate its prima optimum by using the CVX sover [36] to sove (4.5). We pot the performance gap between CVX and our subgradient method with gorithm 3 that provides the dua optimum. The number of data subchannes is fixed at 100, and that of reay subchannes is set to 30, 60, and 90. Tabe. 3 shows the estimated duaity gap. We observe the gap is indeed sma and decreasing as the number of channes increases. This demonstrates the vaidity of soving the optimization in the dua domain. Note that CVX is significanty sower than our agorithms. 6.6 Performance of Power ocation Finay, we evauate the performance of our power aocation agorithms. We impement the subgradient

13 TBLE 3 Estimated duaity gap. # of reay subchannes Performance gap 5.46% 4.80% 4.24% based gorithm 4 for the RSS-XOR probem, as we as its counterpart for the NO-XOR probem. We enforce a uniform power constraint across RS. Each RS has reay power N P, where P is the power used for one direct transmission as in Sec We vary N from 10 to 100 and obtain Fig. 11. We observe from Fig. 11(a) that XOR-CD is ess sensitive to power constraints as refected by the margina improvement compared to conventiona CD. This is because XOR- CD utiizes power more efficienty, resuting in a ower power demand at RS. Therefore, pumping more power does not improve its performance substantiay. Throughput (Mbps) Conventiona CD Per RS power (P) Power efficiency Conventiona CD Per RS power (P) (a) Throughput comparison. (b) Power efficiency comparison. Fig. 11. Effects of power constraints for a 100m ce with 100 data subchannes, 30 reay subchannes, 10 MS, 3 RS. We aso evauate the effect of power aocation on reay power efficiency, which now is cacuated by dividing the throughput improvement (in %) by N, the per RS reay power. The resut is shown in Fig. 11(b). We observe that, expectedy, reay power efficiency is decreasing due to the inteigent power aocation that aways maximizes the throughput gain under a fixed power budget. XOR-CD consistenty provides better power efficiency up to the point that reay power is abundant for the conventiona CD to achieve the same throughput gain (the two ines in Fig. 11(b) converge as N increases). TBLE 4 Throughput vaues of different reay power profies. Reay power profie: Throughput Improvement RS1 RS2 RS3 (Mbps) (%) 10P 10P 10P P 9P 6P P 6P 9P P 7P 5P P 5P 7P The convergence speed of our power contro agorithms is much sower since they invove a nested subgradient update oop. In our simuations, we find that gorithm 4 with gorithm 1 for the power contro version of the RSS-XOR probem takes on average r P r = P R. Soving the around 1500 iterations to sove, which may not be feasibe for practica use. Given that power aocation yieds margina improvement for both XOR-CD and conventiona CD as discussed above, in most cases it is sufficient to adopt the simpe uniform power aocation across mobie stations. We finay study non-uniform power constraints at RS. For the same configuration and network topoogy with 3 RS, we fix the tota power constraint to be 30P and vary individua RS s power constraint. Tabe 4 summarizes the resuts of different reay power profies. Observe that aocating more power to RS1 has a positive effect on the average throughput, whie adjusting the constraints of RS2 and RS3 does not. The reason is that in our simuation RS are randomy ocated inside the ce. RS1 is ocated cosest to the BS, providing a much better reay channe for cooperative transmissions. ocating more power to RS1 boosts its reaying capacity and improves throughput. The resut aso suggests that power aocation at RS needs to be ocation-adaptive to best utiize resources. It is possibe to consider the optimization of reay power budget in our framework. We ony need to change the reay power budget P r from a constant to a new variabe, and add a new tota power constraint across a the P r : reay power budget optimization is technicay more invoved. Subgradient methods can sti be used by reaxing the additiona constraint, but the convergence of the agorithm wi be negativey affected. Since the performance gain of reay power budget aocation is margina as shown in Tabe 3, we do not to discuss this issue in detai here. 7 CONCLUSION This work represents an eary attempt to study network coding assisted cooperative diversity in mutichanne ceuar networks. We presented XOR-CD, a simpe cooperative diversity scheme with XOR in OFDM networks. s our main contribution, we proposed a unifying optimization framework based on network utiity maximization to expoit muti-user diversity, cooperative diversity and network coding jointy. We estabished the hardness of the decouped resource aocation probem in the physica ayer, and proposed efficient approximation and optima agorithms. Simuation resuts demonstrated that network coding has the potentia to significanty improve throughput of OFDM networks. REFERENCES [1] H. Xu and B. Li, XOR-assisted cooperative diversity in OFDM wireess networks: Optimization framework and approximation agorithms, in Proc. IEEE INFOCOM, [2] S. Chachuski, M. Jennings, S. Katti, and D. Katabi, Trading structure for randomness in wireess opportunistic routing, in Proc. of CM SIGCOMM, 2007.

14 [3] Y. Wu, P.. Chou, and S.-Y. Kung, Information exchange in wireess networks with network coding and physica-ayer broadcast, in Proc. of CISS, [4]. Sendonaris, E. Erkip, and B. azhang, User cooperation diversity-part I: System description, IEEE Trans. Commun., vo. 51, no. 11, pp , November [5], User cooperation diversity-part II: Impementation aspects and performance anaysis, IEEE Trans. Commun., vo. 51, no. 11, pp , November [6] J. N. Laneman, D. N. C. Tse, and G. W. Worne, Cooperative diversity in wireess networks: Efficient protocos and outage behavior, IEEE Trans. Inf. Theory, vo. 50, no. 12, pp , December [7] J. N. Laneman, Network coding gain of cooperative diversity, in Proc. of IEEE MILCOM, [8] Y. Chen, S. Kishore, and J. T. Li, Wireess diversity through network coding, in Proc. of IEEE WCNC, [9] C. Peng, Q. Zhang, M. Zhao, and Y. Yao, On the performance anaysis of network-coded cooperation in wireess networks, in Proc. of IEEE INFOCOM, [10] IEEE Standard, TM : ir interface for fixed wireess access systems, [11] T. M. Cover and.. E. Gama, Capacity theorems for the reay channe, IEEE Trans. Inf. Theory, vo. 25, no. 5, pp , September [12] D. Gunduz and E. Erkip, Opportunistic cooperation by dynamic resource aocation, IEEE Trans. Wireess Commun., vo. 6, no. 4, pp , pri [13] S. Savazzi and U. Spagnoini, Energy aware power aocation strategies for mutihop cooperative transmission schemes, IEEE J. Se. reas Commun., vo. 25, no. 2, pp , February [14]. Khandani, J. bounadi, E. Modiano, and L. Zheng, Cooperative routing in static wireess networks, IEEE Trans. Commun., vo. 55, no. 11, pp , November [15] R. hswede, N. Cai, S.-Y. Li, and R. Yeung, Network information fow, IEEE Trans. Inf. Theory, vo. 46, no. 4, pp , Juy [16] S. Katti, H. Rahu, W. Hu, D. Katabi, M. Medard, and J. Crowcroft, XORs in the air: Practica wireess network coding, in Proc. of CM SIGCOMM, Juy [17] P. Chaporkar and. Proutiere, daptive network coding and scheduing for maximizing throughput in wireess networks, in Proc. CM MobiCom, [18]. Khreishah, C. chun Wang, and N. Shroff, Cross-ayer optimization for wireess mutihop networks with pairwise intersession network coding, IEEE J. Se. reas Commun., vo. 27, no. 5, pp , June [19] N. Jones, B. Shrader, and E. Modiano, Optima routing and scheduing for a simpe network coding scheme, in Proc. IEEE INFOCOM, [20] X. Bao and J. Li, daptive network coded cooperation (NCC) for wireess reay networks: Matching code-on-graph with network-on-graph. IEEE Trans. Wireess Commun., vo. 7, no. 2, pp , February [21] C. Haus, F. Schreckenbach, I. Oikonomidis, and G. Bauch, Iterative channe and network decoding on a tanner graph, in Proc. 43rd erton Conference on Communications, Contro, and Computing, [22] S. Yang and R. Koetter, Network coding over a noisy reay: beief propagation approach, in Proc. IEEE Int. Symposium on Information Theory (ISIT), [23] J. Tang and X. Zhang, Cross-ayer resource aocation over wireess reay networks for quaity of service provisioning, IEEE J. Se. reas Commun., vo. 25, no. 4, pp , May [24] L. Le and E. Hossain, Cross-ayer optimization frameworks for mutihop wireess networks using cooperative diversity, IEEE Trans. Wireess Commun., vo. 7, no. 7, pp , Juy [25] S.-J. Kim, X. Wang, and M. Madihian, Optima resource aocation in muti-hop ofdma wireess networks with cooperative reay, IEEE Trans. Wireess Commun., vo. 7, no. 5, pp , May [26] T. C.-Y. Ng and W. Yu, Joint optimization of reay strategies and resource aocations in cooperative ceuar networks, IEEE J. Se. reas Commun., vo. 25, no. 2, pp , February [27] F. P. Key,. Mauoo, and D. Tan, Rate contro for communication networks: Shadow prices, proportiona fairness and stabiity, Journa of Operations Research Society, vo. 49, no. 3, pp , [28] G. Song and Y. G. Li, Cross-ayer optimization for ofdm wireess network - Part I and part II, IEEE Trans. Wireess Commun., vo. 4, no. 2, March [29] W. Yu and R. Lui, Dua methods for nonconvex spectrum optimization of muticarrier systems, IEEE Trans. Commun., vo. 54, no. 7, pp , Juy [30] S. Boyd and. Mutapcic, Subgradient methods, Lecture notes of EE364b, Stanford University, Winter Quarter subgrad method notes.pdf. [31] G. L. Nemhauser and L.. Wosey, Integer and Combinatoria Optimization. Wiey-Interscience, [32] B. Chandra and M. M. Hadórsson, Greedy oca improvement and weighted set packing approximation, J. gorithms, vo. 39, no. 2, pp , May [33] P. Berman, d/2 approximation for maximum weight independent set in d-caw free graphs, Nordic J. Comput., vo. 7, no. 3, pp , [34] C. Papadimitriou and K. Steigitz, Combinatoria Optimization: gorithms and Compexity. Prentice Ha, [35] S. Vaentin, Chsim a wireess channe simuator for omnet++, fachgebiete/research-group-computer-networks/projects/ chsim.htm, [36] CVX: Matab software for discipined convex programming, Hong Xu received his B.Engr. degree from the Department of Information Engineering, The Chinese University of Hong Kong, in 2007 and the M..Sc. and Ph.D. degrees from the Department of Eectrica and Computer Engineering, University of Toronto in 2010 and He wi join the Department of Computer Science, City University of Hong Kong as an ssistant Professor starting from ugust His research interests incude coud computing, datacenters, and computer networking and systems. He is a student member of CM and IEEE. Baochun Li received the B.Engr. degree from the Department of Computer Science and Technoogy, Tsinghua University, China, in 1995 and the M.S. and Ph.D. degrees from the Department of Computer Science, University of Iinois at Urbana-Champaign, Urbana, in 1997 and Since 2000, he has been with the Department of Eectrica and Computer Engineering at the University of Toronto, where he is currenty a Professor. He hods the Norte Networks Junior Chair in Network rchitecture and Services from October 2003 to June 2005, and the Be Canada Endowed Chair in Computer Engineering since ugust His research interests incude arge-scae distributed systems, coud computing, peer-to-peer networks, appications of network coding, and wireess networks. Dr. Li was the recipient of the IEEE Communications Society Leonard G. braham ward in the Fied of Communications Systems in In 2009, he was a recipient of the Mutimedia Communications Best Paper ward from the IEEE Communications Society, and a recipient of the University of Toronto McLean ward. He is a member of CM and a senior member of IEEE.