network coding for two-way relaying networks

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4476 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 9, NOVEMBER 2010

Network Coding for Two-Way Relaying NetworksOver Rayleigh Fading Channels

Wei Li, Jie Li, Senior Member, IEEE , and Pingyi Fan, Senior Member, IEEE

Abstract —Wireless network coding is a useful technology thatcan increase the total throughput of wireless networks. There are,however, few works focusing on wireless network coding overfading channels, which is an important characteristic of manyreal wireless channels and may result in performance degradation.To investigate the fading channel’s impact on network coding,based on the constant transmission power scheme and the channelinversion-transmission scheme, we analyze network throughputover Rayleigh fading channels. It is shown that, when the dif-ference between average channel gains over two broadcastingchannels is very large, the throughput of network coding greatlydecreases, and the advantage of network coding almost disap-

pears. To address this issue and to maximize the throughputof network coding over fading channels, we formulate the fad-ing compensation for network coding as optimization problemsand present the optimal transmission data rate and transmissionpower level of the relay node. Furthermore, to consider the real-ization problem of network coding over fading channels, includingunbalanced trafc load and asynchronization of packet arrivals,we present two opportunistic optimal network coding (OONC)schemes. Performance evaluation has shown that the proposedopportunistic schemes perform well in various scenarios.

Index Terms —Network coding, optimal data rate, Rayleighfading channels.

I. INTRODUCTION

W IRELESS networks are widely employed worldwide,such as cellular networks, wireless local area networks,

wireless sensor networks, etc. Due to the open-air-interfacenature of wireless channels, wireless networks face many chal-lenges. First, since channel resources may be shared by thewhole network, effective channel resource allocation becomesa critical problem. Second, the radio spectrum available forwireless services is extremely scarce while demands for theseservices are rapidly growing. Spectral efciency is therefore of

Manuscript received November 3, 2009; revised March 21, 2010 andMay 24, 2010; accepted May 31, 2010. Date of publication June 28, 2010;date of current version November 12, 2010. This work was supported in partby the National Natural Science Foundation of China/Research Grant CouncilJoint Research Scheme 60831160524, by the Tsinghua University InitiativeScientic Research Program, by the Grand-in-Aid for Scientic Research of theJapan Society for Promotion of Science, and by the open research fund of theNational Mobile Communications Research Laboratory, Southeast University,China. The review of this paper was coordinated by Prof. B. Hamdaoui.

W. Li is with the Department of Electronic Engineering, Tsinghua Univer-sity, Beijing 100084, China (e-mail: [email protected]).

J. Li is with the Graduate School of Systems and Information Engineer-ing, University of Tsukuba, Tsukuba Science City 305-8573, Japan (e-mail:[email protected]).

P. Fan is with the Department of Electronic Engineering, Tsinghua Uni-versity, Beijing 100084, China, and also with the National Mobile Commu-nications Research Laboratory, Southeast University, Nanjing 210096, China(e-mail: [email protected]).

Digital Object Identier 10.1109/TVT.2010.2053947

primary concern in the design of future wireless communicationsystems [1]. Third, energy saving is more important in wirelessnetworks than in wireline networks. Most wireless terminalsuse batteries, and their energy is limited. To prolong the net-work lifetime, saving energy is necessary. Fourth, wireless linksare subject to severe multipath fading due to randomly delayedreection, scattering, and diffraction. Fading leads to severedegradation in the signal-to-noise ratios (SNRs) of wirelesschannels. Hence, fading compensation is typically required.

Recently, network coding has attracted much attention in

wireless communications. Network coding was rst proposedby Ahlswede et al. [2] in 2000. Its principle is that the relaynode can receive packets from all the input links, encode theminto one packet rst, and then forward the encoded packet. It hasbeen proven that, in wireline networks, by employing network coding, a source node can send packets at the theoretical upperbound of transmission rate, i.e., the max-ow of a network.Li et al. [3] showed that linear network coding is sufcientto achieve the capacity in theory, and this work has greatlysimplied the process of seeking a feasible code. Subsequently,much work has been conducted in developing both theoreti-cal frameworks and engineering practices of network coding.Koetter and Medard [4] proposed an algebraic approach tonetwork coding on its solvability in multicast networks, bywhich the problems boil down to a set of matrix equations.Performance evaluation of network coding in terms of trafcdelay over some given practical graphs has been presented bysimulations [5]. Many algorithms for constructing linear codesover nite elds were given by Jaggi et al. [6]. Ho et al. [7],[8] investigated the randomized construction of multicast codes.Chou et al. [9] presented a distributed scheme for practicalnetwork coding.

Network coding can also be employed in wireless networks(e.g., [10]–[12]). Due to the broadcast nature of wireless chan-nels, wireless networks exhibit signicant data redundancy.

That is, at each hop, the transmitter delivers the same packetto multiple nodes within its radio range. Network coding canbe exploited to reduce redundancy and compress data, whichcan increase the overall network throughput, improve spectralefciency, and reduce energy consumption. Some typical work on wireless network coding is given as follows: Zhang et al.[13] investigated the multicast routing problem based on net-work coding and put forward a practical algorithm to obtainthe max-ow multicast routes in wireless ad-hoc networks.Katti et al. [14] proposed a protocol called COPE (which isa wireless network coding protocol), applied COPE in a real20-node wireless network, and evaluated the performance. Thetest results indicated that network coding increases network

0018-9545/$26.00 © 2010 IEEE



LI et al. : NETWORK CODING FOR TWO-WAY RELAYING NETWORKS OVER RAYLEIGH FADING CHANNELS 4477

Fig. 1. Example of the traditional store-and-forward scheme and network coding scheme. (a) Store-and-forward scheme. (b) Network coding scheme.

throughput. Liu and Xue [15] analyzed the rate region, i.e., thesum rate of a two-way relaying network with network codingover additive white Gaussian noise (AWGN) channels.

Previous work on wireless network coding, however, waslargely based on the condition that the wireless links are reliableand stable, which is similar to wireline networks. As a matterof fact, the wireless medium is fundamentally different fromits wireline counterpart. For wireless medium, signal fading isvery common due to the time-varying channel characteristic,which results in either a higher bit error rate (BER) or a larger

required transmit power. Previous designs of wireless network coding rarely took this characteristic of the wireless mediuminto account. We note that fading over a single link has beenwell studied in [16]; however, fading compensation based onnetwork coding is still an open problem.

In this paper, we will investigate network coding over fadingchannels. We will focus on two problems: First, we will inves-tigate the impact of channel fading on the network performancethrough theoretical analysis. Second, we will investigate thesolution of related problems by developing some methods,e.g., adjusting the transmission power level of the relay nodeaccording to channel-state information.

We rst analyze the performance of network coding in atwo-way relaying network over Rayleigh fading channels. Ina two-way relaying network, two source nodes S 1 and S 2 wantto transmit packets to each other, as shown in Fig. 1. In thisnetwork, signal fading greatly degrades the performance of wireless network coding. If network coding is employed, therelay node X will simultaneously transmit encoded packets toboth S 1 and S 2 . However, because the channel conditions of the two links could vary over time, it is not easy to guaranteethat the packets sent by X can perfectly be received by S 1 andS 2 , which degrades the throughput of wireless network coding.In particular, when the difference between the average channelgains of the two channels is very large, network throughputgreatly decreases, and the advantages of network coding almostdisappear.

To deal with this problem, a reasonable method is to adjust the transmission power level and transmission data rateof the relay node according to channel state. Recently, wehave analyzed the performance of wireless network codingover Rayleigh fading channels based on the constant transmis-sion power scheme and obtained its optimal transmission data

rate [17]. In this paper, we extend our previous work and moredetailedly analyze the outage capacity of network coding, basedon two typical transmission schemes, i.e., the constant trans-mission power scheme and the channel-inversion transmissionscheme. To increase the network throughput, we formulate thefading compensation for network coding over Rayleigh fadingchannels as two optimization problems and then obtain theoptimal transmission data rate and power level of the relaynode. Numerical and simulation results show that network throughput increases by employing the proposed optimal trans-mission data rate and power level. Furthermore, based on theperformance analysis, we consider selecting a relay node andobtain the optimal assignment location of the relay node, whichcan further improve the system performance.

In addition, we will also consider the problem of applyingnetwork coding in a practical wireless network in the fadingenvironment. In most previous work on network coding, globalsynchronization of packet arrivals is assumed throughout anetwork. However, in practical wireless networks, strict syn-chronization is difcult to satisfy. Asynchronization of packetarrivals may degrade network performance [19]. To deal withthe problem of asynchronization and unbalanced trafc load,we propose two novel opportunistic optimal network coding(OONC) schemes. Simulation results indicates that our devel-oped OONC schemes perform well in various scenarios.

The rest of this paper is organized as follows: In Section II,we outline the channel and network models. Then, we analyzethe optimal transmission data rate based on network codingover Rayleigh fading channels in Section III. In Section IV,we show some numerical results and analyze the selection of relay nodes. The OONC schemes are proposed in Section V.In Section VI, we evaluate the performance of the proposedschemes by simulations. Finally, we summarize our main re-sults and give concluding remarks in Section VII.

II. PRELIMINARIES AND SYSTEM DESCRIPTION

Considering a two-way relaying network, as shown in Fig. 1,there are two source nodes S 1 and S 2 , and a relay node X .Nodes S 1 and S 2 want to transmit packets to each other. It isassumed that all the three nodes in the wireless network sharea common frequency band. To reduce the wireless interference,more than one node is not allowed to transmit packets at thesame time. Otherwise, a collision will occur, and receiverscannot correctly decode the signal. All the nodes are operatingin the half-duplex mode, i.e., a node cannot simultaneouslytransmit and receive packets. Note that, although the two-wayrelaying network is simple, it is a fundamental building block of a wireless network, and operations of the relay node in a two-way relaying network are similar to that in a generic wirelessnetwork. In this paper, based on a two-way relaying network,we will investigate the impact of channel fading on the network




coding performance. The obtained results can be extended ingeneric wireless networks.

A. Channel Model

As a reasonable fading model for radio signals in heavily

built-up urban environments, Rayleigh fading is popularly em-ployed [20]. Over Rayleigh fading channels, the received powerP c follows an exponential distribution. Its probability densityfunction (pdf) is given by

f P c (ϕ) =1ϕ

· e− ϕ/ϕ (1)

where ϕ denotes the average received power [20].Let g(t), g(t) ≥ 0, denote the instantaneous channel power

gain of an arbitrary channel. In fact, channel gain g(t) rep-resents the power attenuation coefcient over the channel. Itis independent of the channel input, and its value is equal to

the ratio of the received power level to the transmitted powerlevel. In this paper, let gi (t) denote the channel gain of link (X, S i ), i = 1 , 2. Over Rayleigh fading channels, channel gaing(t) follows an exponential distribution, and its expected valueis denoted as g.

In addition, because the transmission power of a transmitteris limited, let P s denote the average transmit signal power.To simplify notations, let γ (t) = ( P s g(t)/N 0W ) denote thenormalized fading gain at the receiver, where N 0 denotes thepower spectral density of additive Gaussian white noise, andW denotes the received signal bandwidth. Assuming that allthe channels are with AWGN, then N 0 is a constant, andP s / (N 0W ) is a constant. The distribution of g(t) determinesthe distribution of γ (t), and vice versa . Consequently, γ (t)follows an exponential distribution, and its pdf is given by

f γ (γ ) =1γ

· e− γ/γ (2)

where γ is the expected value of γ (t). For simplicity, γ (t) isdenoted as γ without confusions.

In this paper, block fading is assumed. Block fading is amodel to treat time-selective fading in a tractable manner. Insuch a model, the channel gain over a fading channel is constantwhen each packet is being transmitted but independently variesfrom one packet to its next one [16].

Over a Rayleigh fading channel, the value of the channelgain g(t) is a variable. The receiver cannot correctly decode allreceived signals due to low SNR. In this paper, we use the out-age probability to characterize the probability of unsuccessfuldecoding. That is, if the received SNR falls below the minimumrequired SNR for the current data rate, the transmission fails.In this case, the system is said in outage [16]. The outageprobability is

P out = Pr( γ < γ 0) (3)

where γ 0 is the minimum required SNR for the current trans-mission data rate. That is to say, in this paper, we assume that,if the value of the received SNR is larger than a thresholdfor the current data rate, the receiver can correctly decode the

received signal; otherwise, the received signal cannot correctlybe decoded, and the transmission fails. In fact, it is reasonable,because Tse et al. showed that outage probability is a tightupper bound of the bit error probability in wireless fadingchannels [21].

Then, the average data rate received correctly is

C (P out ) = (1 − P out ) · R (4)

where R is the maximum data rate corresponding to outageprobability P out . In fact, the outage probability characterizesthe probability of data loss or, equivalently, the probability of deep fading.

The capacity with outage probability P out is correctly de-ned as the maximum data rate received, as given by

C out = maxP out

(1 − P out ) · R. (5)

Its basic premise is that, by allowing the system to lose somedata in the event of deep fading, a higher data rate can bemaintained than the case in which all data must correctlybe received, regardless of the fading state, like the case forShannon capacity.

B. Network Coding

Network coding is a useful technology that can effectivelyincrease the throughput of wireless networks. The procedure of network coding is shown in Fig. 1(b). In the two-way relayingnetwork, nodes S 1 and S 2 want to send packets to each other.At stage 1, node S 1 sends a packet a to relay node X . At

stage 2, node S 2 sends a packet b to relay node X . At stage 3,relay node X encodes packets a and b into a new packet a ⊕ band broadcasts the encoded packet a ⊕ b to both node S 1 andS 2 . Stage 3 is called coding stage , and the relay node X is calledcoding node . Then, node S 1 can obtain packet b by decoding thepacket a ⊕ b with packet a . Node S 2 can receive packet a in thesame way. This procedure requires three transmissions (stages)if there is no data loss.

As a comparison, a traditional relay protocol, referring to thestore-and-forward scheme, requires four transmissions (stages).The procedure of the store-and-forward scheme is shown inFig. 1(a). At stage 1, node S 1 sends a packet a to relay node X .At stage 2, relay node X forwards packet a to node S 2 . Atstage 3, node S 2 sends a packet b to relay node X . At stage 4,the relay node X forwards packet b to node S 1 . This procedurerequires four transmissions (stages) if there is no data loss.

Thus, compared with the store-and-forward scheme, by em-ploying network coding, the throughput can indeed increase by33%. Consequently, spectral efciency is improved, and energyconsumption for each packet may be reduced.

III. OPTIMAL TRANSMISSION -S CHEME ANALYSIS

As we observe, the performance analysis in Section II-Bon wireless network coding were based on reliable and stablewireless links, i.e., the packet sent by the relay node canperfectly be received by all the receivers. However, this is not




always true. For wireless links, signal fading is very common,usually resulting in a high BER, which degrades the perfor-mance of wireless network coding. For example, in Fig. 1(b),only when both S 1 and S 2 receive the packets successfullysent by X are the transmission successful for network codingscheme and the advantage of network coding apparent. If only

one or none of S 1 and S 2 correctly receives the packet sentby node X , it is somewhat similar to the store-and-forwardscheme. Unfortunately, due to the fading characteristic of wire-less channels, the channel gains of links (X, S 1) and (X, S 2)are variable, and the channels are not always reliable. Thatis, it is not easy to guarantee that the packets sent by X canperfectly be received by S 1 and S 2; thus, the performance of network coding is degraded. In particular, when the averagechannel gains of the two broadcasting channels are not equal,the throughput of the two receivers is not equal, which results ingreat throughput decrease in network-coding throughput. Theadvantage of network coding is reduced.

Thus, improvement of the performance of network coding infading environments is still an open problem. In the following,based on two typical transmission schemes, i.e., the constanttransmission-power scheme and the channel-inversion scheme,we investigate this problem and analyze how to adjust thetransmission power level and transmission data rate of the relaynode to increase network throughput.

A. Constant Transmission Power Scheme

In this part, we consider the case in which both the trans-mitter and receiver know the distribution of channel gain g(t),but the value of g(t) is only known at the receiver at time t .Equivalently, the instantaneous received SNR is only known atthe receiver. As assumed in Section II, the outage probabilityis adopted to characterize the probability of unsuccessful de-coding, i.e., only if the received SNR is larger than a thresholdfor the data rate can the received signal correctly be decoded.Since the transmitter has no knowledge of the instantaneouschannel gain, it is difcult to adjust the transmission powerand data rate according to the variety of g(t) to guaranteethat the received SNR is larger than the threshold. Therefore,a reasonable scheme is to x a transmission data rate and apower level independent of the instantaneous channel gain. Thisscheme refers to the CP (constant transmission power) scheme.

Here, the value of transmission data rate R is a design parameterbased on the outage probability. The objective is to nd theoptimal value of data rate R , at which the outage capacityachieves, i.e., the total throughput is maximized.

For the CP scheme, if the store-and-forward scheme is em-ployed, i.e., if there are only one transmitter and one receiver ineach transmission, as shown in Fig. 1(a), the transmitter xesa threshold of SNR, i.e., a minimum received SNR γ 0 , andtransmits at data rate R = W log2(1 + γ 0). Then

γ 0 = 2 R/W − 1. (6)

A packet is assumed to be correctly received when the instan-taneous received SNR is greater than or equal to γ 0 . When

the received SNR is smaller than γ 0 , the receiver declares anoutage. The probability of outage is

P out = Pr( γ < γ 0) =

γ 0

0

f γ (γ )dγ = 1 − e− γ 0 /γ . (7)

Consequently, the average data rate correctly received is

C (R) = (1 − P out ) · R = R · e− γ 0 /γ . (8)

Then, the optimization problem can be formulated as

Maximize : C (R) = R · e− γ 0 /γ

Subject to : R > 0

γ 0 = 2 R/W − 1. (9)

Here, (9) is called the single-link constant power (SLCP) outagecapacity problem.

If network coding is employed, the problem becomes morecomplicated. As shown in Fig. 1(b), at stage 1 or stage 2, thereare only one transmitter and one receiver in each transmission,which is similar to the store-and-forward scheme. The transmit-ter can adjust its transmit power level and data rate according tothe SLCP scheme (9) at stages 1 and 2. However, at the codingstage (stage 3), the relay node X broadcasts encoded packets toboth S 1 and S 2 , which is different from the store-and-forwardscheme. Thus, in the following part of this section, we onlyconsider the coding stage and analyze the outage capacity at thecoding stage. That is, the outage capacity for network codingin this paper is dened as the broadcasting outage capacity of the two relay channels over which the relay node broadcastsencoded packets.

At the coding stage, the outage probability of node S i is

P out ,i = 1 − e− γ 0 /γ i (10)

where γ 0 denotes the threshold SNR corresponding to data rateR at the coding stage, and γ i denotes the value of the averagefading gain of link (X, S i ), i = 1 , 2.

At the coding stage, there are four cases when the relay nodebroadcasts an encoded packet.

1) Both nodes S 1 and S 2 correctly receive the packet. Thetotal throughput is 2R .

2) Only node S 1 correctly receives the packet. The totalthroughput is R .3) Only node S 2 correctly receives the packet. The total

throughput is R .4) Neither node S 1 nor node S 2 correctly receives the

packet. The total throughput is 0.Therefore, the average throughput at stage 3 can be given by

C (R) = 2 R · (1− P out ,1)(1 − P out ,2)+ R · (1− P out ,1)P out ,2

+ R · P out ,1(1− P out ,2)

= R(e− γ 0 /γ 1 + e− γ 0 /γ 2 ). (11)

Comparing (8) and (11), there is only one term in (8), whichmeans that only one channel is considered in each transmission




for the store-and-forward scheme. However, there are two termsin (11), which means that two channels are considered at thecoding stage for the network-coding scheme.

Then, the objective of nding the optimal value of data rateR to maximize the total throughput can be formulated as thefollowing optimization problem:

Maximize : C (R) = R(e− γ 0 /γ 1 + e− γ 0 /γ 2 )

Subject to : R > 0

γ 0 = 2 R/W − 1. (12)

Problem (12) is called the network-coding constant power (NCCP) outage capacity problem. In fact, the solution of problem (12) can be obtained by computer search. In this case,we can obtain the optimal data rate Ropt ,nccp and the optimalSNR threshold γ o, nccp . The NCCP outage capacity can beexpressed as

C out ,nccp = Ropt ,nccp · (e− γ o, nccp /γ 1

+ e− γ o, nccp /γ 2

). (13)

B. Channel-Inversion Scheme

In this section, we consider the case in which both thetransmitter and the receiver can obtain the value of channelgain g(t) by some feedback protocols. In this case, the trans-mitter can adjust the transmission power to ensure that thereceived packets can correctly be decoded as the value of channel gain varies. Then, a suboptimal transmitter adaptationscheme, called the “channel-inversion (CI)” scheme, can beadopted [18]. For the channel inversion scheme, the transmitter

adapts the transmission power to maintain a constant receivedpower level, i.e., it inverts the channel fading. The channelthen appears similarly as a time-invariant AWGN channel. Thetransmitter can send packets at a constant data rate

R = W log2(1 + σ) (14)

where σ denotes the required constant received SNR.In a wireless network, with the traditional store-and-forward

scheme [i.e., there are only one transmitter and one receiverfor each transmission, as shown in Fig. 1(a)], the CI scheme isformulated as follows: The transmission power is given by

P (γ ) =σγ P s (15)

where P s is the maximal average transmission power. Tokeep the transmission power constraint, σ is selected as σ =(1/ E (1/γ )) , where E (1/γ ) denotes the expected value of 1/γ .The channel capacity with channel inversion can be given by

C = W log2 1 +1

E (1/γ ). (16)

On the other hand, for Rayleigh fading channels, the channelinversion cannot be realized due to very deep fading, i.e.,g(t) = 0 [18]. To deal with this problem, the node is allowednot to transmit packets in particularly deep fading states, refer-ring to outage states. In this case, a higher constant data rate

can be maintained. Such a scheme is called the “truncated CI”scheme [18]. It only compensates for fading above a certaincutoff fading gain γ 0 . That is, the transmission power is formu-lated as

P (γ ) =σγ · P s , γ ≥ γ 00, γ < γ

0

(17)

where γ 0 is based on the outage probability P out =Pr( γ <γ 0).Since the maximal average transmission power is P s , the valueof σ can be given by

1σ

=∞

γ 0

1γ

f γ (γ )dγ. (18)

The outage capacity with a given cutoff fading gain γ 0 isgiven by

C (γ 0) = W log2(1 + σ)Pr( γ ≥ γ 0). (19)

The corresponding problem of maximizing the total throughputis formulated as

Maximize : C (γ 0) = W log2(1 + σ)Pr( γ ≥ γ 0)Subject to : γ 0 > 0

1σ

=∞

γ 0

1γ

f γ (γ )dγ. (20)

The problem (20) is called the single-link channel-inversion(SLCI) outage capacity problem.

Although the truncated CI scheme based on the store-and-forward scheme has been well studied in [16], the problembecomes more complicated if network coding is employed,which is investigated in this paper. As shown in Fig. 1(b),stages 1 and 2 for network coding are similar to the store-and-forward scheme, and the transmitter can adjust its power leveland transmission data rate according to the SLCI problem (20)at stage 1 or stage 2. However, the coding stage (stage 3) isdifferent from the store-and-forward scheme, at which the relaynode X broadcasts encoded packets to both S 1 and S 2 . Thus,in the following part of this section, we only consider stage 3(coding stage). That is, the outage capacity for network codingin this paper is dened as the broadcasting outage capacity of

the two relay channels over which the relay node transmitsencoded packets.At the coding stage, the relay node needs to broadcast packets

to both S 1 and S 2 . Since the truncated CI scheme is adopted,only the links whose instantaneous fading gains are larger thana cutoff fading gain γ 0 will be considered. The correspondingtransmission power is

P (γ 1 , γ 2) =⎧⎪⎪⎨⎪⎪⎩

σmin {γ 1 ,γ 2 } P s , γ 1 ≥ γ 0 , γ 2 ≥ γ 0σγ 1

P s , γ 1 ≥ γ 0 , γ 2 < γ 0σγ 2

P s , γ 1 < γ 0 , γ 2 ≥ γ 00, γ 1 < γ 0 , γ 2 < γ 0

(21)

where γ 1 and γ 2 denote the instantaneous fading gains of links(X, S 1) and (X, S 2), respectively. As the preceding equation




shows, there are four cases when the relay node broadcastspackets to nodes S 1 and S 2 .

1) If the fading gains of both links (X, S 1) and (X, S 2) arelarger than the cutoff fading gain γ 0 , the relay node needsto transmit the encoded packets to both S 1 and S 2 . Thetotal throughput is 2R .

2) Only the fading gain of link (X, S 1) is larger than γ 0 .The relay node transmits packets to node S 1 . The totalthroughput is R .

3) Only the fading gain of link (X, S 2) is larger than γ 0 .The relay node transmits packets to node S 2 . The totalthroughput is R .

4) Neither link (X, S 1) nor link (X, S 2) is good enough.The relay node will not send packets. The total through-put is 0.

It needs to note that, for the CI scheme, if X determines totransmit a packet to a node, it will adjust the transmission powerto guarantee that the received SNR is larger than a threshold, so

that the receiver can correctly decode the received packet. Then,the outage capacity with a cutoff fading gain γ 0 can be given by

C (γ 0) = 2 R · Pr( γ 1 ≥ γ 0 , γ 2 ≥ γ 0)+ R · Pr( γ 1 ≥ γ 0 , γ 2 <γ 0)

+ R · Pr( γ 1 <γ 0 , γ 2 ≥ γ 0)

= R(e− γ 0 /γ 1 + e− γ 0 /γ 2 )

= W log2(1+ σ)(e− γ 0 /γ 1 + e− γ 0 /γ 2 ) (22)

where γ 1 and γ 2 denote the average fading gains of links(X, S 1) and (X, S 2), respectively.

In addition, the maximal average transmission power is P s ;hence, the value of constant received SNR σ should satisfy thefollowing constraint:

P (γ 1 , γ 2)f γ 1 (γ 1)f γ 2 (γ 2)dγ 1dγ 2 ≤ P s (23)

where f γ 1 (γ 1) and f γ 2 (γ 2) denote the pdf of γ 1 and γ 2 ,respectively. By using (2) and (21), (23) can be rewritten as

∞

γ 0

γ 2

γ 0

σγ 1

f γ 1 (γ 1)f γ 2 (γ 2)dγ 1dγ 2

+∞

γ 0

γ 1

γ 0

σ

γ 2f γ 1 (γ 1)f γ 2 (γ 2)dγ 2dγ 1

+∞

γ 0

γ 0

0

σγ 1

f γ 1 (γ 1)f γ 2 (γ 2)dγ 2dγ 2

+∞

γ 0

γ 0

0

σγ 2

f γ 1 (γ 1)f γ 2 (γ 2)dγ 1dγ 1 ≤ 1. (24)

For simplicity, let E 1(z) denote the exponential integral function of order 1, with the form

E 1(z) =

∞

z

e− t

t dt, z ≥ 0. (25)

Substituting (25) into (24), then σ must satisfy the followingcondition:

1σ

≥1γ 1

E 1γ 0γ 1

−∞

γ 0

e− γ 2 /γ 2

γ 21γ 1

E 1γ 2γ 1

dγ 2

+ 1γ 2

E 1 γ 0γ 2

−∞

γ 0

e− γ 1 /γ 1

γ 11γ 2

E 1 γ 1γ 2

dγ 1 . (26)

In this part, our objective is to choose the best transmissionpower level to maximize the total throughput. Thus, we havethe following optimization problem:

Maximize :C (γ 0)= W log2(1+ σ)(e− γ 0 /γ 1 + e− γ 0 /γ 2 )

Subject to :γ 0 > 0

1σ ≥

1γ 1 E 1

γ 0γ 1 −

∞

γ 0

e− γ 2 /γ 2

γ 21γ 1 E 1

γ 2γ 1 dγ 2

+1γ 2

E 1γ 0γ 2

−∞

γ 0

e− γ 1 /γ 1

γ 11γ 2

E 1γ 1γ 2

dγ 1 . (27)

The problem (27) is called the network-coding channel-inversion (NCCI) outage capacity problem. Its near-optimalsolution can be obtained by computer search.

By solving the problem (27), the optimal value of cutoff fading gain γ o, ncci and optimal data rate Ropt ,ncci can beobtained. The NCCI outage capacity can be expressed as

C out ,ncci = Ropt ,ncci (e− γ 0 ,ncci /γ 1 + e− γ 0 ,ncci /γ 2 ). (28)

Accordingly, the corresponding received SNR σopt ,ncci can alsobe obtained. Using (21), the optimal transmission power levelis obtained as well.

IV. NUMERICAL RESULTS AND ANALYSIS

A. Numerical Results

Letting the signal bandwidth W = 1 MHz, we will presentsome numerical results. Fig. 2 shows the outage capacity for

NCCP problem (12) and NCCI problem (27) with differentvalues of average fading gain. To evaluate the effect of thechannel gains of two broadcasting channels on the network coding outage capacity, let the outage capacity of a singlelink be the performance reference, which is also shown inFig. 2. Here, the outage capacity for network coding is thebroadcasting outage capacity of the two broadcasting channelsover which the relay node broadcasts the encoded packets. Thesingle-link outage capacities are the SLCP outage capacity (9)and SLCI outage capacity (20) of link (X, S 1).

In this gure, the outage capacity with the CI scheme is largerthan the corresponding outage capacity with the CP scheme.This is because, compared with the CP scheme, the transmitterin the CI scheme can obtain more channel information, i.e., thevalue of channel gain. The transmitter in the CI scheme can




Fig. 2. Average fading gain versus outage capacity.

adjust its transmission power level according to channel status.Hence, the outage capacity of the CI scheme is larger than thatof the CP scheme.

In addition, when γ 1 , which is the average fading gain of link (X, S 1), is xed, the NCCP and NCCI outage capacitywill increase as the value of γ 2 increases. Compared with theperformance reference, i.e., the single-link outage capacity, if the average fading gains of the two links are equal, the NCCPoutage capacity is exactly twice as much as the SLCP outagecapacity, and the NCCI outage capacity is a little smaller thanthe doubled SLCI outage capacity, which means that bothlink (X, S 1) and link (X, S 2) are sufciently utilized, and the

advantage of network coding is very apparent. However, if γ 1 γ 2 , the NCCP outage capacity is approximately equalto the SLCP outage capacity, and the NCCI outage capacity isapproximately equal to the SLCI outage capacity, which meansthat the advantage of network coding is reduced. The reasonis that, if γ 1 is much larger than γ 2 , the throughput of nodeS 1 will be much greater than node S 2 . Consequently, the totalthroughput is approximately equal to node S 1’s throughput. Ina word, the advantage of network coding cannot be exploitedapparently in the scenario that γ 1 γ 2 .

The preceding results give us a hint that suitable selection of relay node is an important issue to network performance. Alongthis line, we will consider how to select a relay node in the nextsection.

B. Selection of Relay Nodes

In Section III, we have analyzed how to choose the optimaltransmission power level to maximize the throughput, if therelay node is chosen. However, in some scenarios, there areseveral nodes that can be served as relay nodes in a wirelessnetwork, and only one relay node is enough in the network. Inthis case, selecting a better relay node to further improve thenetwork performance is also an important issue.

To the NCCP problem, we analyze how to select a relay nodein the aspects of capacity. Assuming that the average channelgain of link (X, S i ) is gi = β · d− α

i , where β is a constant

Fig. 3. Nodes’ locations.

coefcient, di is the length of link (X, S i ), i = 1 , 2, and α isthe path-loss (attenuation) factor usually satisfying 2 ≤ α ≤ 4,we will analyze the relationship between the NCCP outagecapacity and the location of the relay node.

Theorem 1: Assuming that the average channel gain of link (X, S i ) is gi = β · d− α

i , i = 1 , 2, the NCCP outage capacity ismaximized, if nodes X , S 1 , and S 2 are on a straight line and

d1 = d2 , where di denotes the distance between the relay nodeX and receiver S i .

Proof: As shown in Fig. 3, there are two candidate codingnodes X and X .

First, it is better to select a relay node on the straight line of two source nodes, because shorter distances between the nodesmean larger average channel gains.

Second, we analyze how the location of X affects the outagecapacity. Because gi = β · d− α

i , i = 1 , 2, and d2 = d − d1 , (11)can be written as

C (γ 0) = R e− γ 0 / (δd− α1 ) + e− γ 0 / (δ(d− d1 )− α ) (29)

where δ is a constant. By solving

∂C (γ 0)∂d1

= 0 (30)

we get d1 = d/ 2. In this case, d1 = d2 = d/ 2.In summary, the NCCP outage capacity is maximized, if the

relay node X , S 1 , and S 2 are on a straight line, and d1 = d2 .For the NCCI problem, a similar conclusion can also be

obtained. The detailed analysis to the NCCI problem is omittedhere.

V. OPPORTUNISTIC OPTIMAL NETWORK CODING SCHEME

To apply network coding in a practical network, there arestill some problems to be solved, e.g., the synchronization issueand the unbalanced trafc load. In most previous work onnetwork coding, global synchronization is assumed throughoutthe network. However, as some researchers have already no-ticed, global synchronization is actually too strict in practicalnetworks, and the absence of synchronization can greatly affectthe performance of network coding in wireline networks [5],[9], [19]. As a matter of fact, synchronization is more criticalin wireless fading channels than in wireline channels. Forexample, as shown in Fig. 1(b), at stage 3, if network codingis employed, the following condition that there are at least onepacket sent to S 1 and one sent to S 2 in node X ’s buffer should




be satised. Due to high BER and relatively high collisionprobability, the preceding condition may sometimes not besatised in wireless fading scenarios. In this case, node X stopsthe transmission, receives packets again, and waits for the nexttransmission, resulting in a waste of time slots. Unbalancedtrafc load can result in a similar problem.

To mitigate the impact of unbalanced trafc load andasychronization in wireless fading channels, we proposetwo OONC schemes, i.e., the OONC-CP scheme and theOONC-CI scheme, which combine the network coding andstore-and-forward scheme. Compared with the existing oppor-tunistic network coding schemes, the advantage of the proposedOONC schemes is that physical-layer information is consideredin OONC schemes. In existing schemes, e.g., in [15] and[19], the coding node selects the network-coding scheme orthe store-and-forward scheme just according to medium-accesslayer information, i.e., the buffer status. However, in this paper,our emphasis is fading channels; thus, in the proposed OONCschemes, the coding node decides which action to take, i.e.,network coding or forwarding, based on not only buffer statusbut on also channel state as well. In addition, the transmissionscheme of the physical layer is considered in OONC schemes,i.e., how to adjust the transmission data rate and transmissionpower level of relay nodes according to channel state, while itis not considered in existing schemes.

In the following, we will explain the OONC schemes. Basedon the CP scheme, the OONC-CP scheme is described asfollows:

For the relay node X , its procedure is given as follows:

Step 1) Node X receives packets and checks the buffer. If there are packets in its buffer, it begins to contendfor the channel. Go to step 2).

Step 2) If the relay node X has occupied the channel, itchecks its buffer. If there are some packets sent to S 1and some packets sent to S 2 , go to step 3); otherwise,go to step 4).

Step 3) The NCCP scheme is employed. Node X choosesone packet sent to S 1 and one packet sent to S 2 , en-codes them to one encoded packet, and then broad-casts the encoded packet. The transmission data rateis determined by (12). Go to step 5).

Step 4) The Store-and-forward scheme is employed. NodeX selects a packet and sends it to its destination.

The transmission data rate is determined by (9). Goto step 5).

Step 5) Prepare to receive packets. Go to step 1).

In addition, based on the CI scheme, the OONC-CI schemeis described as follows:

For the relay node X , its procedure is given as follows:

Step 1) Node X receives packets and checks the buffer. If there are packets in its buffer, it begins to contendfor the channel. Go to step 2).

Step 2) If the relay node X has occupied the channel, itchecks its buffer. If there are some packets sent to S 1and some packets sent to S 2 , go to step 3); otherwise,go to step 4).

Step 3) The NCCI scheme is employed. Node X checkschannel state information, selects receivers, and ad- justs its transmission power according to (21). Then,node X chooses one packet sent to S 1 and onepacket sent to S 2 , encodes them to one encodedpacket, and then transmits the encoded packet. The

optimal cutoff fading gain and transmission data rateare determined by (27). Go to step 5).Step 4) The Store-and-forward scheme is employed.

Node X selects a packet and sends it to its des-tination. The cutoff fading gain and transmissiondata rate are determined by (20). Go to step 5).

Step 5) Prepare to receive packets. Go to step 1).

The principle of the OONC schemes is that relay node Xneeds to make two decisions to determine how to transmit datapackets. First, X checks its buffer. If there are packets sent totwonodes S 1 and S 2 , respectively, network coding is employed;otherwise, the store-and-forward scheme is employed. Second,the relay node selects receivers and determines transmissionpower level according to channel-state information. In fact,this is also the second selection between network coding andforwarding. Because if the relay node decides to transmitpackets to only one receiver in the second decision accordingto channel-state information, the nal action of the relay nodewill be forwarding in spite of which action is selected in therst decision. For example, at step 3) of the OONC-CI scheme,if the NCCI scheme is employed, node X needs to check thechannel state. As (21) shows, only if the instantaneous fadinggains of both links (X, S 1) and (X, S 2) are larger than thecutoff fading gain γ 0 is network coding indeed employed, i.e.,

node X transmits encoded packets to both S 1 and S 2 . In othercases, node X transmits packets to one node or does not trans-mit packets, which is similar to the store-and-forward scheme.In addition, to maximize the total throughput, the transmissiondata rate and power level are determined by (9), (12), (20),and (27), respectively.

Some more details of the OONC-CP scheme and OONC-CIscheme are also given as follows:

First, in wireless networks, if two nodes are close to eachother and simultaneously transmit packets, a collision occurs,and it may result in the failure of decoding. In the proposedschemes, to avoid packet collisions, it requires that, when a

node wants to send a packet, it needs to contend for the channelrst. Only if a node, e.g., node X, has occupied the channel canit transmit packets, which means that other nodes within thetransmission range will keep quiet when node X is transmittingpackets. In this case, collisions are avoided. Then, the problemis how to contend for the channel. In this paper, the CSMA/CAprotocol can be employed to deal with this problem in a wirelessnetwork [23]. In the CSMA/CA protocol, a node requiring tosend data initiates the process by sending a request-to-send(RTS) packet. Then, the destination node replies with a clear-to-send (CTS) packet. After receiving the CTS packet, thesource node can begin to transmit data. If the source node doesnot receive the CTS packet due to collisions or low SNR, itretransmits an RTS packet. Any other node that receives theRTS or CTS packet but is neither source node nor destination




Fig. 4. Example of a large wireless network.

node should refrain from sending data for a given duration. Inthis case, when the source node is transmitting data packets,other nodes within the ranges of the source node and destinationnode will keep silent. Collisions are avoided.

Second, for the OONC-CI scheme, it requires that the relaynode X can obtain perfect CSI, i.e., the value of gi (t), i = 1 , 2,which is the channel power gain of link (X, S i ). Assuming thatthe value of gi (t), i = 1 , 2, varies not very fast, node X can ob-

tain the value of channel gain by exchanging RTS/CTS packetswith destination nodes in the proposed schemes. Because theCSMA/CA protocol is employed, if node X wants to send adata packet to a destination node, e.g., node S 1 , X should sendan RTS packet to S 1 and wait to receive a CTS packet. WhenS 1 replies with a CTS packet, it can add the value of transmittedpower level in the CTS packet. Then, by receiving the CTSpacket, X can estimate the received power level and obtain thetransmitted power level from the CTS packet. By comparing thetwo values, X obtains the value of instantaneous channel gaing1(t). As slow fading is assumed in this paper, after receivingthe CTS packet, X can transmit a data packet to S 1 accordingto the obtained value of

g1(t). This procedure can be performed

for each data transmission.Third, in the proposed schemes, relay node X can take two

actions, i.e., forwarding or network coding. To indicate whataction node X has taken, we can add 1 bit in the packetoverhead, which is denoted as “ACTIN.” ACTIN = 1 meansthat network coding is employed, and ACTIN = 0 meansthat forwarding is employed. Then, when node S 1 or S 2 re-ceives a packet, it decodes the packet and checks the value of the bit “ACTIN.” If ACTIN = 1 , a transmission nishes. If ACTIN = 1 , the receiver needs to recover the original packetaccording to network coding protocol.

Fourth, in the proposed schemes, if the received signal is notcorrectly decoded, the transmitter needs to retransmit the datapacket. In particular, for the OONC-CP scheme, the transmitter

cannot adjust the transmission power level according to thechannel state; thus, perhaps many packets cannot correctly bedecoded due to low SNR. Thus, retransmission is particularlyrequired if the OONC-CP scheme is employed.

In addition, the OONC schemes cannot only be applied ina two-way relaying network but can be expanded to a genericwireless network as well. In a large wireless network, if twonodes transmit packets to each other, every relay node on the

transmission path needs to relay packets from the two sourcenodes. Then, every relay node on the transmission path canbe treated as the relay node X in a two-way relaying network,and the transmission path can be separated as several two-wayrelaying networks. In this case, the proposed schemes can beapplied at the relay nodes. For example, there are seven nodesin Fig. 4. Node 1 wants to send packets a , c, e, etc., to node 7,and node 7 wants to send packets b, d, f , h , etc., to node 1.Fig. 4 shows the transmission procedure. In this gure, thewhole network can be separated as several two-way relayingnetworks. For instance, one of two-way relaying networks iscomposed of nodes 5, 6, and 7, in which node 6 is the relaynode. The proposed schemes can be employed at node 6. Atsteps 2) and 6), node 6 can transmit packets. By checking itsbuffer, only at step 6) can node 6 employ the network codingscheme. At step 2), node 6 just forwards packets. Node 6 alsoneeds to adjust its transmission data rate and power level basedon the proposed schemes.

VI. PERFORMANCE EVALUATION

In this section, we evaluate the performance of the pro-posed schemes by simulations. The performance comparisonbetween the proposed schemes and the existing store-and-forward scheme is also conducted. In the simulations, weconsider a two-way relaying network over Rayleigh fadingchannels, as shown in Fig. 1. In the network, there is only one




Fig. 5. Total throughput in simplied scenarios for the NCCP scheme.

Fig. 6. Total throughput in simplied scenarios for the NCCI scheme.

shared frequency band, and the bandwidth is 1.0 MHz. If notspecically given, the packet length is 500 bits, and the heavytrafc load is assumed, which means that S 1 and S 2 alwayshave packets to send. In addition, every result is based on tensimulations. The condential intervals are marked in gures,

and the condential level is 95%.

A. NCCP and NCCI Outage Capacity

First, we validate the NCCP outage capacity (12) and NCCIoutage capacity (27) by simulations. In this part, we considera simplied scenario. That is, we omit the procedure that S 1or S 2 transmits packets to the relay node X , and let X alwaysbroadcast encoded packets to S 1 and S 2 , i.e., only stage 3 inFig. 1(b) is considered. There are 10 000 transmissions in everysimulation.

Fig. 5 shows the average throughput under the simpliedscenarios for the NCCP scheme. Fig. 6 shows the averagethroughput under the simplied scenarios for the NCCI scheme.Compared with Fig. 2, the simulation results match the theo-

Fig. 7. Total throughput at different data rates for the OONC-CP scheme.

retical results very well. If packets are not transmitted at theoptimal data rate, i.e., Ropt ,nccp or Ropt ,ncci , the throughputwill decrease, which indicates that our proposed data ratesRopt ,nccp and Ropt ,ncci are the optimal data rates for the NCCPscheme and the NCCI scheme, respectively. In addition, boththe average fading gains of links (X, S 1) and (X, S 2) will affectthe throughput. As the value of γ 1 or γ 2 decreases, the totalthroughput will decrease.

B. Performance of OONC Schemes

In this part, we evaluate the performance of the OONC

schemes by simulations. We consider an actual two-way re-laying network. It is a distributed network, and there arecontentions and collisions among nodes. To deal with them,the CSMA/CA protocol is employed in our simulations. Someparameters are illustrated.

1) Transmitting a RTS/CTS/ACK packet takes 50 μs.2) The minimum time interval of two transmissions is 50 μs.3) The shortest and the longest periods of the sensing

channel for each transmission are 100 and 250 μs,respectively.

4) The simulation period is 120 s.As a comparison, the performances of the classical network-

coding scheme (store-and-forward is not exploited) and store-and-forward scheme are also presented.First, we show the simulation results of the CP schemes.Fig. 7 shows the total throughput with different values of the

average fading gains and different data rates for the OONC-CPscheme. In this gure, the total throughput at the optimal datarate Ropt ,nccp is larger than that at nonoptimal data rate. Figs. 5and 7 validate that Ropt ,nccp is the optimal data rate under bothsimplied and actual situations.

In Fig. 8, we compare the throughput of the OONC-CP, net-work coding (NCCP), and store-and-forward (SLCP) schemesunder balanced trafc load and unbalanced trafc load. Here,balanced trafc load means that the average trafc sent byS 1 and S 2 are equal. Unbalanced trafc load means that theaverage trafc sent by S 2 is equal to half of the average trafc




Fig. 8. Total throughput under balanced load and unbalanced load for theCP schemes (γ 2 = γ 1 ) .

Fig. 9. Effect of packet length on throughput for the CP schemes whenγ 2 = γ 1 (CPL: constant packet length; VPL: variable packet length).

sent by S 1 . Under balanced load, the throughput of the OONC-CP or NCCP scheme is almost twice as much as the throughputof the SLCP scheme. Under unbalanced load, the throughput of all schemes decreases, but the throughput of OONC-CP is much

larger than the throughput of the NCCP or SLCP scheme. Thereason is that, for the network coding scheme, only after packetsfrom both S 1 and S 2 arrive can the relay node X transmitencoded packets. Under unbalanced load, the packet from thelink with larger trafc load needs to wait for the packet fromthe link with smaller trafc load, which results in a waste of time. The store-and-forward scheme can mitigate the effect of unbalanced load, but its throughput is smaller than the network coding scheme. The OONC-CP scheme takes the advantagesof both the network coding and store-and-forward schemes.Fig. 8 indicates that the OONC scheme performs well underboth balanced load and unbalanced load.

We investigate the effect of packet length on the network throughput to the CP scheme shown in Fig. 9. In the simula-tions, CPL means that the packet length is 500 bits, and VPL

Fig. 10. Total throughput at different data rates for the OONC-CI scheme.

means that the packet length follows a uniform distribution andthat the average value is 500 bits. We observe that the network throughput is not sensitive to the conditions of constant packetlength or variable packet length. That is to say, the OONC issuitable for various situations.

Then, the simulation results of the CI schemes are shown asfollows:

Fig. 10 shows the total throughput as a function of differentvalues of the average fading gains and different data rates forthe OONC-CI scheme. In this gure, the total throughput atoptimal data rate is larger than the total throughput at nonop-timal data rate. Figs. 6 and 10 validate that Ropt ,ncci is theoptimal data rate under both simplied and actual situations. Inaddition, compared with Fig. 7, the throughput of the OONC-CI scheme is larger than that of the OONC-CP scheme. This isbecause, compared with the OONC-CP scheme, the transmitterin the OONC-CI scheme can obtain the value of instantaneouschannel gain, which can be used to adjust transmission powerlevel.

In Fig. 11, we compare the throughput of the OONC-CI, net-work coding (NCCI), and store-and-forward (SLCI) schemesunder balanced trafc load and unbalanced trafc load. Similarto Fig. 8, the OONC-CI scheme performs well under both thebalanced load and the unbalanced load.

Fig. 12 shows the effect of packet length on network through-

put to the CI schemes. In this gure, the network throughputis not sensitive to the conditions of constant packet length orvariable packet length. The OONC-CI is suitable for varioussituations.

VII. CONCLUSION

In this paper, we have investigated the performance of wire-less network coding over Rayleigh fading channels. We havefound that signal fading greatly degrades the performance of wireless network coding. In particular, when the differencebetween the average channel gains of the two relay channelsis very large, the advantage of network coding almost disap-pears. To increase the throughput for wireless network codingover Rayleigh fading channels, we have formulated the fading




Fig. 11. Total throughput for balanced load and unbalanced load for theCI schemes (γ 2 = γ 1 ) .

Fig. 12. Effect of packet length on throughput for the CI schemes whenγ 2 = γ 1 (CPL: constant packet length; VPL: variable packet length).

compensation as two optimization problems, i.e., NCCP andNCCI problems. By solving the optimization problems, theoptimal transmission data rate and optimal transmission powerlevel are presented. The simulations and numerical results have

shown that, if the relay node transmits packets at the optimaltransmission data rate and optimal transmission power, thenetwork throughput will increase. We have also consideredthe effect of the location of the relay node on the network throughput over fading channels and proven that, when therelay node and two source nodes are in a straight line and therelay node is exactly at the middle of the two source nodes,the network throughput can be maximized.

In addition, to consider the realization problem of network coding including asynchronization and unbalanced trafc loadin fading channels, two OONC schemes, i.e., the OONC-CPand OONC-CI schemes, have been designed, which combinethe network coding and store-and-forward schemes. Perfor-mance evaluation has shown that the OONC schemes performwell under various situations.

REFERENCES

[1] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading chan-nels under different adaptive transmission and diversity-combining tech-niques,” IEEE Trans. Veh. Technol. , vol. 48, no. 4, pp. 1165–1181,Jul. 1999.

[2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network informationow,” IEEE Trans. Inf. Theory , vol. 46, no. 4, pp. 1204–1216, Jul. 2000.

[3] S-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inf. Theory , vol. 49, no. 2, pp. 371–381, Feb. 2003.

[4] R. Koetter and M. Medard, “An algebraic approach to network coding,” IEEE/ACM Trans. Netw. , vol. 11, no. 5, pp. 782–795, Oct. 2003.

[5] T. Noguchi, T. Matsuda, and M. Yamamoto, “Performance evaluation of new multicast architecture with network coding,” IEICE Trans. Commun. ,vol. E86-B, no. 6, pp. 1788–1795, Jun. 2003.

[6] S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain, andL. Tolhuizen, “Polynomial time algorithms for multicast network codeconstruction,” IEEE Trans. Inf. Theory , vol. 51, no. 6, pp. 1973–1982,Jun. 2005.

[7] T. Ho, M. Medard, M. Effros, and D. R. Karger, “On randomized network coding,” in Proc. 41st Allerton Annu. Conf. Commun. Control Comput. ,Monticello, IL, 2003.

[8] T. Ho, M. Medard, R. Koetter, D. R. Karger, M. Effros, J. Shi, andB. Leong, “A random linear network coding approach to multicast,” IEEE Trans. Inf. Theory , vol. 52, no. 10, pp. 4413–4430, Oct. 2006.

[9] P. A. Chou, Y. Wu, and K. Jain, “Practical network coding,” in Proc. Allerton Conf. Commun. Control Comput. , Monticello, IL, 2003.[10] J. Zhang, K. B. Letaief, P. Fan, and K. Cai, “Network coding based signal

recovery for efcient scheduling in wireless networks,” IEEE Trans. Veh.Technol. , vol. 58, no. 3, pp. 1572–1582, Mar. 2009.

[11] D. S. Lun, M. Medard, and R. Koetter, “Efcient operation of wirelesspacket networks using network coding,” in Proc. IWCT , 2005.

[12] N. Dong, T. Tuan, N. Thinh, and B. Bose, “Wireless broadcast usingnetwork coding,” IEEE Trans. Veh. Technol. , vol. 58, no. 2, pp. 914–925,Feb. 2009.

[13] J. Zhang, P. Fan, andK. B. Letaief, “Network codingfor efcient multicastrouting in wireless ad-hoc networks,” IEEE Trans. Commun. , vol. 56,no. 4, pp. 598–607, Apr. 2008.

[14] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,“XORs in the air: Practical wireless network coding,” IEEE/ACM Trans. Netw. , vol. 16, no. 3, pp. 497–510, Jun. 2008.

[15] C.-H. Liuand F. Xue, “Network codingfor two-way relaying: Rate region,

sum rate and opportunistic scheduling,” in Proc. IEEE ICC , May 2008,pp. 1044–1049.

[16] A. J. Goldsmith, Wireless Communications . New York: CambridgeUniv. Press, 2005.

[17] W. Li, J. Li, and P. Fan, “Optimal data rate and opportunistic scheme onnetwork coding over Rayleigh fading channels,” in Proc. 5th Int. Conf. Mobile Ad-hoc Sens. Netw. , 2009, pp. 257–264.

[18] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with chan-nel side information,” IEEE Trans. Inf. Theory , vol. 43, no. 6, pp. 1986–1992, Nov. 1997.

[19] Y. Ma, W. Li, P. Fan, K. B. Letaief, and X. Liu, “On the characteristics of queueing and scheduling at encoding nodes for network coding,” Int. J.Commun. Syst. , vol. 22, no. 6, pp. 755–772, Jun. 2009.

[20] J. G. Proakis, Digital Communications , 3rd ed. Singapore:McGraw-Hill, 1995.

[21] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamentaltradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory , vol. 49,no. 5, pp. 1073–1096, May 2003.

[22] M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions .Deutsch, Germany: Verlag Harri Deutsch, 1984.

[23] IEEE Standard for Information Technology—Telecommunications and Information Exchange Between Systems—Local and Metropolitan Area Networks—Specic Requirements—Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specications , IEEEStd. 802.11, 1997.

[24] A. Argyriou, “Cross-layer and cooperative opportunistic network codingin wireless ad hoc networks,” IEEE Trans. Veh. Technol. , vol. 59, no. 2,pp. 803–812, Feb. 2010.

[25] B. K. Chalise and A. Czylwik, “Exact outage probability analysis for amultiuser MIMO wireless communication system with space-time block coding,” IEEE Trans. Veh. Technol. , vol. 57, no. 3, pp. 1502–1512,May 2008.

[26] X. W. Cui, Q. T. Zhang, and Z. M. Feng, “Outage probability for max-

imal ratio combining of arbitrarily correlated faded signals corrupted bymultiple Rayleigh interferers,” IEEE Trans. Veh. Technol. , vol. 55, no. 1,pp. 383–386, Jan. 2008.




Wei Li received the B.S. degree, in 2004, fromTsinghua University, Beijing, China, where he iscurrently working toward the Ph.D. degree with theDepartment of Electronic Engineering.

From January 2006 to March 2006, he vis-ited NICT Japan as a Researcher. From November2008 to October 2009, he visited the University of Tsukuba, Tsukuba, Japan, as a Visiting Foreign Re-

search Fellow. His research interests include network information theory, network coding, and wirelessad hoc networks.

Jie Li (SM’04) is currently a Professor with theGraduate School of Systems and Information En-gineering, University of Tsukuba, Tsukuba ScienceCity, Japan. He has served on the editorial boardsof the IPSJ Journal , Wiley Wireless Communica-tions and Mobile Computing , etc. His research in-terests are mobile distributed multimedia computingand networking, operating systems, network secu-rity, and modeling and performance evaluation of information systems.

Prof. Li is a Senior Member of the Associationfor Computing Machinery and a Member of Information Processing Societyof Japan (IPSJ). He has served as secretary for the Study Group on SystemEvaluation of IPSJ and on the editorial board of the IEEE T RANSACTIONSON VEHICULAR TECHNOLOGY . He has also served on the Steering Com-mittees of the SIG of System EVAluation (EVA) of IPSJ, the SIG of Data-base System of IPSJ, and the SIG of MoBiLe computing and ubiquitouscommunications of IPSJ. He is also on the program committees for sev-eral international conferences, such as the IEEE ICDCS, IEEE INFOCOM,IEEE GLOBECOM, and IEEE MASS.

Pingyi Fan (M’04–SM’09) received the B.S. degreefrom Hebei University, Hebei, China, in 1985, theM.S. degree from Nankai University, Tianjin, China,in 1990, and the Ph.D. degree from Tsinghua Univer-sity, Beijing, China, in 1994.

In 2002, he was promoted to Full Professor withTsinghua University. From August 1997 to March1998, he visited Hong Kong University of Science

and Technology, Kowloon, Hong Kong, as a Re-search Associate. From May 1998 to October 1999,he visited the University of Delaware, Newark, as a

Research Fellow. In March 2005, he visited NICT Japan as a Visiting Professor.From 2005 to 2009, he visited Hong Kong University of Science and Technol-ogy several times. He is the founding editor-in-chief of the International Jour-nal of Wireless Communications and Networking . In addition, he is currentlyserving as Editor for the Wiley Journal of Wireless Communication and MobileComputing , the Inderscience International Journal of Ad Hoc and UbiquitousComputing , and the Inderscience International Journal of Autonomous and Adaptive Communications Systems . From 2007 to 2009, he served as Editor forthe IEEE T RANSACTIONS ON WIRELESS COMMUNICATIONS . His researchinterests include beyond third-generation technology in wireless communi-cations such as multiple-input multiple-output, orthogonal frequency-divisionmultiplexing, multicarrier code-division multiple access, space time coding,low-density parity-check design, network coding, network information theory,and cross-layer design.

Dr. Fan is an overseas member of the Institute of Electronics, Informa-tion, and Communication Engineers. He has organized many internationalconferences, including the 2010 IEEE International Conference on WirelessCommunications, Networking and Information Security as Technical ProgramCommittee (TPC) Co-Chair; the IEEE International Communications Confer-ence as TPC Member; and Globecom from 2007 to 2010. He is also a Reviewerof more than 22 international journals, including 14 IEEE journals and threeEuropean Association for Signal Processing journals.

network coding for two-way relaying networks

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