On Relay Assignment in Network-Coded Cooperative Systems

868 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 3, MARCH 2011

On Relay Assignment inNetwork-Coded Cooperative Systems

Xuehua Zhang, Ali Ghrayeb and Mazen Hasna

Abstract—We consider in this paper relay assignment forcooperative systems with multiple two-way relay channels. Thenodes corresponding to one two-way relay channel (henceforthreferred to as pair) communicate with each other through arelay. The relays use network coding to simultaneously transmitthe signals corresponding to the pairs they are assigned to. Wepropose two relay assignment schemes. One scheme considersall possible relay assignment permutations and selects the onethat yields the best performance, and the other one considersonly a subset of these permutations and selects the best one.The advantage of the latter is that it results in a significantreduction in computational complexity, in addition to makingthe analysis more tractable. We analyze the performance of theseschemes over asymmetric independent Rayleigh fading channels.We also consider semi-symmetric and symmetric channels asspecial cases. We derive closed-form expressions for the end-to-end bit error rate performance for all scenarios and showthat the full diversity order is achieved, which is the numberof available relays. We present several examples to verify thetheoretical results.

Index Terms—Cooperative networks, network coding, relayassignment, two-way relay channels.

I. INTRODUCTION

THE concept of network coding was first proposed byAhlswede, et. alidea is that the intermediate node lin-

early combines the received data instead of sending themdirectly. Various relaying protocols have been proposed forthe two-way relaying channels, whereby two nodes exchangeinformation between each other through one or more relays.The relay nodes normally operate either in the decode-and-forward (DF) or amplify-and-forward (AF) mode. All pro-posed protocols that use network coding can be classified intotwo types: Three-step schemes [2], [3], and two-step schemes[4]. In the former schemes, the relay receives and decodes thebits received from both nodes in the first two steps. In thethird step, the relay applies exclusive-or (XOR) to both bitsand broadcasts the resulting bit to both nodes. This resultsin saving one step in this way as compared to traditionalrelaying (without network coding). The authors in [5] and[6] study how to cope with the performance degradation dueto the asymmetry in the channel, i.e., when the relay is not

Manuscript received March 1, 2010; revised July 4, 2010, October 27, 2010,and November 24, 2010; accepted November 24, 2010. The associate editorcoordinating the review of this paper and approving it for publication wasX.-G. Xia.

This work was supported by Qatar National Research Fund (QNRF) GrantXP0034.

X. Zhang and A. Ghrayeb are with the Department of Electrical andComputer Engineering, Concordia University, Montreal, Quebec, Canada (e-mail: [email protected], [email protected]).

M. Hasna is with the Department of Electrical Engineering, Qatar Univer-sity, Doha, Qatar (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2010.011811.100302

equally far away from the nodes. In [5], a solution to thetime optimization of each phase is given, whereas in [6],hierarchical modulation at one source is employed. In thetwo-step schemes, the relay only decodes the sum of thesignals received and maps it to a corresponding zero or one.In [7], a two-step AF scheme named analog network codingis presented.

Another related direction of research in cooperative com-munications is selection diversity, which is a fundamentaldiversity technique that can be found in many works in theliterature. A relay selection scheme termed opportunistic relay-ing based on the end-to-end channel instantaneous quality isproposed in [8]. For this relay selection scheme, there are twomain relay selection methods: Proactive opportunistic relayingand reactive opportunistic relaying [9]. The main differencesbetween these two methods are as follows. For proactiveopportunistic relaying, the relaying selection is performedbefore transmission. But for reactive opportunistic relaying,the relay selection is performed after transmission and thisrelay must be one of the relays that successfully decoded thesignal received from the source. The performance of thesetwo schemes with or without direct path and AF relaying orDF relaying are studied in [9]-[15], and it is shown that thefull diversity order is achieved. In [14], the authors extendreactive opportunistic relaying to a network environment,which involves multiple simultaneous unidirectional transmit-ting source-destination pairs. Relay assignment for multiplepairs in relay networks is also considered in [16] and [17].

Recently, some efforts have been made to exploit selectiondiversity with network coding in cooperative networks [18],[19]. Relay selection for bidirectional DF relaying is firststudied in [18]. In [19], the authors study the performanceof two-way AF relaying with relay selection similar to theopportunistic relaying scheme proposed in [8]. Besides thesepapers, there are some other papers that study the performanceof network coding with relay selection for multiple source-destination pairs. In [20] and [21], the performance of theXOR-based network coding combined with relay selection formultiple source-destination pairs is analyzed. Only the bestrelay is chosen to help all pairs. The best relay is selectedbased on the end-to-end instantaneous channel gains, whichis similar to the proactive opportunistic relaying proposed in[9].

The authors in [22] propose another network coding schemefor relay selection with multiple source-destination pairs.Unlike [20] and [21], the scheme in [22] is based on selectingthe best relay from the relays that have successfully decodedthe message from all sources, which is similar to the reactiveopportunistic relaying scheme proposed in [9]. However, the

1536-1276/10$25.00 c⃝ 2010 IEEE

ZHANG et al.: ON RELAY ASSIGNMENT IN NETWORK-CODED COOPERATIVE SYSTEMS 869

schemes in [20] and [21] are somewhat impractical since thedestination is assumed to be able to have correctly received themessages sent by all the other sources. The scheme in [22]does not have this assumption where it transmits a randomlinear combination of the columns using an underlying spacetime block codes (STBC) matrix as a network coding schemeat the relay [23]. But as the number of pairs increases, thedetection complexity of this scheme becomes unacceptable.Also, we notice that the best relay is chosen from the relaysthat have successfully decoded the message from all thesources, which leads to a nonzero probability that there willbe no qualified relays to select from. So although both twoschemes are attractive due to their high throughput, they lackpracticality.

In light of the above, it is clear that two-way relaying withor without relay selection has been largely studied in manyaspects.1 However, to the best of our knowledge, no workhas been done to solve the following problem: How to assignrelays, in a network setting, to the network pairs in conjunctionwith network coding. This is addressed in this paper.2 Inparticular, we propose two relay assignment schemes. Oneis based on the entire set of relay assignment permutations,and another based on a subset of these permutations. Com-paring with the work of [14], there are two main differences.Firstly, the problem that we are trying to solve here is therelay assignment problem for two-way relaying with networkcoding, while the relay assignment problem in [14] is for one-way relaying without network coding. The schemes in [14] cannot be applied directly to solve the problem at hand. Secondly,although both of our relay assignment schemes are based onthe end-to-end channel qualities, they are based on proactiveopportunistic relaying, while the schemes in [14] are based onreactive opportunistic relaying. Since the relay assignment inour schemes is done before the actual transmission, only theselected relays need to decode the information of the pairs thatthey are helping, while for the schemes in [14], all the relayshave to try to decode all the information from all sources andtell the destinations whether they have successfully decoded ornot, which introduces a computational burden at the relays.3

We examine the performance of the proposed assignmentschemes on symmetric and asymmetric independent Rayleighfading channels. In the symmetric case, all subchannels havethe same average signal-to-noise ratio (SNR), whereas in theasymmetric case, the subchannels have different SNRs. Weanalyze the performance of these assignment schemes andderive a closed form expression for the end-to-end bit errorrate performance. We show that the maximum diversity orderis achieved, which is the number of relays. We also presentseveral examples through which we validate the theoreticalresults.

The remainder of the paper is organized as follows. Thesystem model is presented in Section II. In Section III, the

1As two-way relaying is a basic building block in most wireless networks,multiple two-way relay channels coexist in practical systems such as wirelesssensor network and vehicular ad-hoc networks (VANETs), to name a few.

2We remark that relay assignment for multiple-pair networks was consid-ered in [26], but without network coding.

3Some work had been done to remedy the impact of error propagationdue to imperfect detection at the relay nodes (see [24], [25] and referencestherein.)

S m

S 1

D m

D 1

R n

R 2

R 1

Fig. 1. A cooperative network with 𝑚 bidirectional communication pairsand 𝑛 relays.

proposed relay assignment schemes are presented. We analyzethe performance of these schemes assuming binary phaseshift keying (BPSK) in Section IV. The diversity order ofthe system, based on the suboptimal criterion, is derived inSection V. We present several numerical examples in SectionVI. Section VII concludes the paper.

II. SYSTEM MODEL

The system model considered in this paper is shown in Fig.1. As shown in the figure, the network consists of 𝑚 pairs (i.e.,𝑚 two-way relay channels) and 𝑛 relay nodes where 𝑛 ≥ 𝑚.Again, the latter assumption is justified because any node ina network can serve as a relay node. The nodes of each paircommunicate with each other through one relay node usingorthogonal subchannels. For simplicity, we assume that thereis no direct path between the source and destination nodes.However, the relay assignments schemes analyzed here canbe extended to the case when there is a direct path.

Each of the nodes is equipped with a single antenna andoperates in a half-duplex mode. We assume that all nodesuse BPSK signaling. In the first time slot, one of the nodesof each pair transmits its signal. That is, 𝑚 nodes transmitsimultaneously in the first time slot using frequency divisionmultiple access (FDMA) [27]. In the second time slot, theremaining 𝑚 nodes transmit simultaneously their signals tothe relays. We assume that a central controller in the networkhas the channel state information (CSI) of all the links. Weacknowledge that the complexity of this scheme grows withthe number of pairs in the network, which is a known problemfor such networks. The best 𝑚 relays are then assigned to the𝑚 pairs where one relay is intended to serve one pair. Eachrelay node decodes the two received signals (over the twotime slots), XORs the decoded signals, and broadcasts theresulting signal to all nodes. Note that the selected 𝑚 relayswill have to transmit using orthogonal channels (either in timeor frequency).

The network subchannels are assumed to experience inde-pendent slow and frequency non-selective Rayleigh fading.Let ℎ𝑆𝑖𝑅𝑗 , ℎ𝐷𝑖𝑅𝑗 , ℎ𝑅𝑗𝑆𝑖 and ℎ𝑅𝑗𝐷𝑖 (for 𝑖 = 1, . . . ,𝑚, 𝑗 =1, . . . , 𝑛) denote the fading coefficients between the 𝑖th


source−𝑗th relay, 𝑖th destination−𝑗th relay, 𝑗th relay−𝑖thsource and 𝑗th relay−𝑖th destination, respectively. We considerboth independent identically distributed (i.e., symmetric) andindependent but non-identically distributed (i.e., asymmetric)Rayleigh fading channels over all transmitting pairs and re-lays.4 To avoid ambiguity, we refer to one of the nodes ofeach pair as source (𝑆) and the other node as destination (𝐷),although it is a two-way communication between 𝑆 and 𝐷.

Let 𝑦𝑆𝑖𝑅𝑗 and 𝑦𝐷𝑖𝑅𝑗 denote the received signals at the 𝑗threlay from the 𝑖th source and 𝑖th destination (over the twotime slots), respectively. These signals can be expressed as

𝑦𝑆𝑖𝑅𝑗 =√𝜌𝑖,𝑗ℎ𝑆𝑖𝑅𝑗𝑥𝑆𝑖 + 𝑛𝑆𝑖𝑅𝑗 ,

and𝑦𝐷𝑖𝑅𝑗 =

√𝜌𝑖,𝑗ℎ𝐷𝑖𝑅𝑗𝑥𝐷𝑖 + 𝑛𝐷𝑖𝑅𝑗 ,

respectively, where 𝜌𝑖,𝑗 denotes the average SNR of the(𝑖, 𝑗)th subchannel, 𝑥𝑆𝑖 denotes the transmitted signal fromnode 𝑆𝑖, 𝑥𝐷𝑖 denotes the transmitted signal from node 𝐷𝑖,and 𝑛𝑆𝑖𝑅𝑗 and 𝑛𝐷𝑖𝑅𝑗 are additive white complex Gaussiannoise (AWGN) samples with zero mean and unit variance.The relay then uses maximum likelihood (ML) detection todetect the two signals (arriving from the pair nodes overtwo time slots). That is, �̂�𝑆𝑖𝑅𝑗 =sign(ℜ𝑒{𝑦𝑆𝑖𝑅𝑗ℎ

∗𝑆𝑖𝑅𝑗

}), and�̂�𝐷𝑖𝑅𝑗 =sign(ℜ𝑒{𝑦𝐷𝑖𝑅𝑗ℎ

∗𝐷𝑖𝑅𝑗

}).After the two signals are detected at the relay nodes, they

are XORed as 𝑥𝑅𝑗 = 𝑥𝑆𝑖𝑅𝑗 ⊕ �̂�𝐷𝑖𝑅𝑗 = −(�̂�𝑆𝑖𝑅𝑗 �̂�𝐷𝑖𝑅𝑗 ), andthe resulting signal is broadcasted to all nodes in the thirdtime slot. The signals received at the source and destinationnodes of each pair are expressed as

𝑦𝑅𝑗𝑆𝑖 =√𝜌𝑗,𝑖ℎ𝑅𝑗𝑆𝑖𝑥𝑅𝑗 + 𝑛𝑅𝑗𝑆𝑖 ,

and𝑦𝑅𝑗𝐷𝑖 =

√𝜌𝑗,𝑖ℎ𝑅𝑗𝐷𝑖𝑥𝑅𝑗 + 𝑛𝑅𝑗𝐷𝑖 ,

respectively. Consequently, the final decoded bitsat the source and destination of each pair areexpressed as �̂�𝑆𝑖 =sign(ℜ𝑒{−𝑥𝑆𝑖𝑦𝑅𝑗𝑆𝑖ℎ

∗𝑅𝑗𝑆𝑖

}), and�̂�𝐷𝑖 =sign(ℜ𝑒{−𝑥𝐷𝑖𝑦𝑅𝑗𝐷𝑖ℎ

∗𝑅𝑗𝐷𝑖

}), respectively.

III. RELAY ASSIGNMENT CRITERIA

In this section, we discuss two relay assignment criteria.The first criterion considers all possible relay permutations,whereas the second criterion limits the search over a subsetof the available permutations. Details of these criteria are inorder.

A. Full Set (optimal) Selection

The optimal assignment scheme depends on the sub-channel instantaneous SNR between the relay nodes andthe pair nodes, namely, 𝛾𝑆𝑖𝑅𝑗

, 𝛾𝑅𝑗𝐷𝑖, 𝛾𝐷𝑖𝑅𝑗

, and 𝛾𝑅𝑗𝑆𝑖

(for 𝑖 = 1, . . . ,𝑚, 𝑗 = 1, . . . , 𝑛) where 𝛾𝑆𝑖𝑅𝑗=

𝜌𝑖,𝑗∣∣ℎ𝑆𝑖𝑅𝑗

∣∣2. To simplify the notation, let 𝜸𝑖𝑗 denote the set

{𝛾𝑆𝑖𝑅𝑗, 𝛾𝑅𝑗𝐷𝑖

, 𝛾𝐷𝑖𝑅𝑗, 𝛾𝑅𝑗𝑆𝑖

}, which is the set of theinstantaneous SNRs between the 𝑗th relay and the nodes

4The assumption of having four independent subchannels for each pair canbe justified since FDMA is used for the uplink and downlink channels.

of the 𝑖th pair. The objective here is to optimize the end-to-end bit error rate performance. Since there are two hopsseparating the end nodes in each pair, the weaker link is theone that dominates the end-to-end bit error rate performance.This certainly applies to all pairs. Therefore, the optimalrelay assignment scheme is the one that results in the bestsubchannel among the weakest ones.

To elaborate, let Φ be the set containing all assignmentpermutations. Given that there are 𝑚 pairs and 𝑛 relays,the size of Φ is obviously 𝑃𝑛

𝑚 = 𝑛!(𝑛−𝑚)! . Each element

of Φ consists of all the fading coefficients (4𝑚 of themsince there are 𝑚 pairs and each pair has four subchannels)corresponding to that particular relay assignment. To simplifythe presentation, let 𝜙𝑘 denote the 𝑘th element of Φ for𝑘 = 1, 2, . . . , 𝑃𝑛

𝑚, and let 𝛾𝑘,min denote the smallest elementin 𝜙𝑘, i.e., the weakest subchannel. Accordingly, the optimalassignment, denoted by 𝜙𝑘∗ , has index 𝑘∗

𝑘∗ = argmax𝑘

{𝛾𝑘,min, 𝑘 = 1, 2, . . . , 𝑃𝑛

𝑚

}. (1)

To elaborate, consider the example in which 𝑚 = 3 and𝑛 = 4. Consequently, there are 24 possible relay assignmentpermutations, which are illustrated in Table I. In the table,the first entry of each row indicates the relay assigned to thefirst pair, the second is the relay assigned to the second pairand the third is the relay assigned to the third pair. To relateΦ to Table I, consider the first row of the table. The threeentries of this row correspond to 𝜸1

1, 𝜸22 and 𝜸3

3, respectively.Each of these elements represents four fading coefficients, asdefined before, thus there are 12 fading coefficients comprising𝜙1. Obviously, 𝛾1,min in (1) is the minimum of those 12coefficients. Once the minimum of each row is obtained, thepermutation corresponding to the largest is selected.

While this assignment scheme is optimal, it suffers fromhigh complexity because it is essentially brute-force. Forexample, when 𝑛 = 𝑚 = 10, there are 𝑃𝑛

𝑚 = 3628800permutations to search over. In addition, there is correlationbetween certain rows of Φ, which makes the performanceanalysis difficult. For instance, rows one and five are correlatedsince in both cases, 𝑅1 is assigned to the first pair. Suchcorrelation makes it extremely difficult to find a closed formexpression for the probability density function (pdf) of theresulting fading coefficients. These reasons motivate us toconsider a subset assignment scheme, described next, wherebythe permutations considered are not correlated.

B. Subset (suboptimal) Selection

The objective here is to divide the set Φ into subsets suchthat the correlation among the permutations within a subsetis eliminated. Then the best permutation within one subset isselected. The number of subsets is 𝑁 = 𝑃𝑛

𝑚/𝑛, which resultsin a significant reduction in complexity. For instance, when𝑛 = 𝑚 = 10, the number of permutations to consider reducesfrom 3628800 to 10. Another advantage of this selectioncriterion is the fact it becomes easier to obtain the pdf of thecoefficients corresponding to the selected permutation sincesuch permutations are mutually independent. The disadvantageis some degradation in performance, as we will demonstratelater.


TABLE IALL POSSIBLE RELAY ASSIGNMENT PERMUTATIONS FOR 𝑚 = 3 AND

𝑛 = 4.

pair 1 pair 2 pair 3

𝑅1 𝑅2 𝑅3

subset 1 𝑅2 𝑅3 𝑅4

𝑅3 𝑅4 𝑅1

𝑅4 𝑅1 𝑅2

𝑅1 𝑅3 𝑅2


𝑅3 𝑅1 𝑅4

𝑅4 𝑅2 𝑅1

𝑅1 𝑅2 𝑅4


𝑅3 𝑅4 𝑅2

𝑅4 𝑅1 𝑅3

𝑅1 𝑅4 𝑅2


𝑅3 𝑅2 𝑅4

𝑅4 𝑅3 𝑅1

𝑅1 𝑅3 𝑅4


𝑅3 𝑅1 𝑅2

𝑅4 𝑅2 𝑅3

𝑅1 𝑅4 𝑅3


𝑅3 𝑅2 𝑅1

𝑅4 𝑅3 𝑅2

Remark 1: The full set and subset assignment criteria areequivalent for the following cases: 𝑚 = 1, any 𝑛; and𝑚 = 𝑛 = 2. However, for most other cases, especially as𝑛 increases, there will be a performance degradation since thenumber of permutations to be searched over will be less.

In the following, we shall outline the steps that lead todividing Φ into the desired subsets. In the process, we willrefer to Table I to make things clear.

1) Construct 𝑁 subsets, each having 𝑛 permutations, i.e.,rows. There are 𝑚 entries in each row.

2) Fill the first entry of the first row of each subset withone relay, and this relay should be the same in all ofthese entries. Note that this relay could be any of thepossible relays, but once one is selected, it should bethe same. For instance, In Table I, this relay is 𝑅1.

3) Fill the remaining 𝑚− 1 entries of the first row of eachsubset with all other permutations of the remaining 𝑛−1relays. There are 𝑃𝑛−1

𝑚−1 of such permutations. Referringto Table I, there are six permutations.

4) Fill the remaining entries of each column of each subsetwith the remaining relays without repetition. For eachsubset, this can be accomplished by starting with the firstentry of each column and increase the index of the relayas you go down until all relays are used up. This processensures that a relay is used only once in any givencolumn and any given row, which is key to eliminatecorrelation among the permutations of a subset. Thisshould be clear from Table I.

IV. PERFORMANCE ANALYSIS

As mentioned above, due to the correlation between someof the elements of Φ, it is not easy to obtain a closed formexpression for the pdf of the coefficients corresponding tothe selected permutation. However, when such correlation isnot present, obtaining such pdf is straightforward. As such,although we can not analyze the optimal scheme, the analysisof the subset scheme gives a performance upper bound on thatof the optimal one. In this section, we analyze the end-to-endbit error rate performance with the subset assignment schemeover asymmetric Rayleigh fading channels. We also extendthis analysis to the semi-symmetric and symmetric cases. Wefirst derive a closed form expression for the bit error rate, andthen derive an upper bound on the bit error rate to demonstratethat the full diversity order is achieved (the latter is done inSection V). Throughout this section, we assume that all nodesuse BPSK.

A. The End-to-End Bit Error Rate Performance

There are two ways of having an error at the relay. Whenthe signal from the source to the relay is detected correctly,but the signal from the destination to the relay is detectedincorrectly. The second scenario is the opposite of the first.However, when both signals are in error or both are correct,the relay is not in error. Consequently, the bit error probabilityat the selected relay 𝑅 can be obtained as

𝑃𝑒,𝑅 = 𝑃𝑒,𝑆𝑖𝑅 (1− 𝑃𝑒,𝐷𝑖𝑅) + 𝑃𝑒,𝐷𝑖𝑅 (1− 𝑃𝑒,𝑆𝑖𝑅) , (2)

where 𝑃𝑒,𝑆𝑖𝑅 is the probability of making an error over the𝑆𝑖 −𝑅 link. The rest of the variables are similarly defined.

There are also two ways of making an error at either node ofeach pair. When the relayed bit is in error and this erroneousbit is received correctly by either node of each pair, and whenthe relayed bit is correct but is flipped during transmission. Inthis case, the bit error rate at node 𝑆𝑖 is expressed as

𝑃𝑒,𝑆𝑖 = (1− 𝑃𝑒,𝑅𝑆𝑖)𝑃𝑒,𝑅 + (1− 𝑃𝑒,𝑅)𝑃𝑒,𝑅𝑆𝑖 . (3)

Plugging (2) into (3), we have

𝑃𝑒,𝑆𝑖 = 𝑃𝑒,𝑆𝑖𝑅 (1− 𝑃𝑒,𝑅𝑆𝑖) (1− 𝑃𝑒,𝐷𝑖𝑅)

+𝑃𝑒,𝐷𝑖𝑅 (1− 𝑃𝑒,𝑅𝑆𝑖) (1− 𝑃𝑒,𝑆𝑖𝑅)

+𝑃𝑒,𝑅𝑆𝑖 − 𝑃𝑒,𝑆𝑖𝑅𝑃𝑒,𝑅𝑆𝑖 (1− 𝑃𝑒,𝐷𝑖𝑅)

−𝑃𝑒,𝑅𝑆𝑖𝑃𝑒,𝐷𝑖𝑅 (1− 𝑃𝑒,𝑆𝑖𝑅) . (4)

Similarly, we can express the error rate expression at node 𝐷𝑖

as

𝑃𝑒,𝐷𝑖 = 𝑃𝑒,𝑆𝑖𝑅 (1− 𝑃𝑒,𝑅𝐷𝑖 ) (1− 𝑃𝑒,𝐷𝑖𝑅)

+𝑃𝑒,𝐷𝑖𝑅 (1− 𝑃𝑒,𝑅𝐷𝑖 ) (1− 𝑃𝑒,𝑆𝑖𝑅)

+𝑃𝑒,𝑅𝐷𝑖 − 𝑃𝑒,𝑆𝑖𝑅𝑃𝑒,𝑅𝐷𝑖 (1− 𝑃𝑒,𝐷𝑖𝑅)

−𝑃𝑒,𝑅𝐷𝑖𝑃𝑒,𝐷𝑖𝑅 (1− 𝑃𝑒,𝑆𝑖𝑅) . (5)

In order to get the expression of 𝑃𝑒,𝑆𝑖 and 𝑃𝑒,𝐷𝑖 , we need toget expressions for 𝑃𝑒,𝑆𝑖𝑅, 𝑃𝑒,𝐷𝑖𝑅, 𝑃𝑒,𝑅𝑆𝑖 , and 𝑃𝑒,𝑅𝐷𝑖 .

In the following subsections, we consider the performanceof various scenarios. We first start with the case when thesubchannels are independent, non-identical Rayleigh fading(i.e., asymmetric). We then consider the semi-symmetric case


in which the 𝑆𝑖 − 𝑅 links are identical, the 𝐷𝑖 − 𝑅 linksare identical, the 𝑅 − 𝑆𝑖 links are identical, and the 𝑅 − 𝐷𝑖

links, but these four sets of subchannels are non-identical. Wealso consider the case when all channels are symmetric, i.e.,independent and identical. Clearly, the latter two cases arespecial cases of the asymmetric case.

B. Asymmetric Channels

We now find expressions for 𝑃𝑒,𝑆𝑖𝑅, 𝑃𝑒,𝐷𝑖𝑅, 𝑃𝑒,𝑅𝑆𝑖 , and𝑃𝑒,𝑅𝐷𝑖 so that we can get closed-form expressions for 𝑃𝑒,𝑆𝑖

and 𝑃𝑒,𝐷𝑖 defined by (4) and (5), respectively. To be able to doso, we need to find the pdf of the random variables involved,namely, 𝛾𝑆𝑖𝑅, 𝛾𝐷𝑖𝑅, 𝛾𝑅𝑆𝑖

and 𝛾𝑅𝐷𝑖. Although these variables

are independent and different (on average), their pdfs will besimilar where we can use one expression to express the fourpdfs. As such, to make the presentation simpler, we denotethese variables by 𝛾𝑖𝑘 for 𝑘 = 1, 2, 3, 4 where 𝛾𝑖1 = 𝛾𝑆𝑖𝑅,𝛾𝑖2 = 𝛾𝐷𝑖𝑅, 𝛾𝑖3 = 𝛾𝑅𝑆𝑖

, and 𝛾𝑖4 = 𝛾𝑅𝐷𝑖.

The pdf of 𝛾𝑖𝑘 can then be expressed as [21]

𝑓𝛾𝑖𝑘(𝛾) =

𝑛∑𝑗=1

𝜆𝑖𝑗,𝑘 (𝜆𝑗 − 𝜆𝑖𝑗,𝑘) 𝑒−𝜆𝑖𝑗,𝑘𝛾

⋅𝛾∫

0

⎡⎣𝑒−(𝜆𝑗−𝜆𝑖𝑗,𝑘)𝜃

∏𝑚 ∕=𝑗

(1− 𝑒−𝜆𝑚𝜃

)⎤⎦ 𝑑𝜃

+

𝑛∑𝑗=1

⎡⎣𝜆𝑖𝑗,𝑘𝑒

−𝜆𝑗𝛾∏𝑚 ∕=𝑗

(1− 𝑒−𝜆𝑚𝛾

)⎤⎦ , (6)

where 𝜆𝑖𝑗,1 ≜ 1𝛾𝑆𝑖𝑅𝑗

, 𝜆𝑖𝑗,2 ≜ 1𝛾𝐷𝑖𝑅𝑗

, 𝜆𝑖𝑗,3 ≜ 1𝛾𝑅𝑗𝑆𝑖

,

and 𝜆𝑖𝑗,4 ≜ 1𝛾𝑅𝑗𝐷𝑖

) (for 𝑖 = 1, . . . ,𝑚, 𝑗 = 1, . . . , 𝑛)

and 𝜆𝑗 ≜∑𝑚

𝑖=1 (𝜆𝑖𝑗,1 + 𝜆𝑖𝑗,2 + 𝜆𝑖𝑗,3 + 𝜆𝑖𝑗,4) . 𝛾𝑆𝑖𝑅𝑗=

𝜌𝑖,𝑗𝐸[∣∣ℎ𝑆𝑖𝑅𝑗

∣∣2] is the average SNR for the 𝑆𝑖 − 𝑅𝑗 link.The other terms are similarly defined.

After some manipulations and carrying out the integration,𝑓𝛾𝑖𝑘

(𝛾) can be simplified as

𝑓𝛾𝑖𝑘(𝛾) =

𝑛∑𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘 (𝜆𝑗 − 𝜆𝑖𝑗,𝑘)

𝜆𝑗 − 𝜆𝑖𝑗,𝑘 + 𝜏

⋅𝑒−𝜆𝑖𝑗,𝑘𝛾(1− 𝑒−(𝜆𝑗−𝜆𝑖𝑗,𝑘+𝜏)𝛾)

+𝑛∑

𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘𝑒−(𝜆𝑗+𝜏)𝛾 , (7)

where the set 𝑇 𝑛𝑗 is obtained by expanding the product∏

𝑚 ∕=𝑗(1 − 𝑒−𝜆𝑚𝛾) and then taking the logarithm of eachterm, and sign(𝜏 ) corresponds to the sign of each term inthe expansion [28]. That is,

𝑇 𝑛𝑗 =

⎧⎨⎩𝜏 :

𝑛∏𝑚=1𝑚 ∕=𝑗

(1− 𝑒−𝜆𝑚𝛾) =∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝑒−𝜏𝛾

⎫⎬⎭ . (8)

The moment generating function (MGF) of 𝛾𝑖𝑘 can be calcu-lated as [29]

𝑀𝛾𝑖𝑘(𝑠) =

∞∫0

𝑒−𝑠𝛾𝑓𝛾𝑖𝑘(𝛾) 𝑑𝛾. (9)

Plugging (7) into (9) and carrying out the integration, we have

𝑀𝛾𝑖𝑘(𝑠) =

𝑛∑𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘 (𝜆𝑗 − 𝜆𝑖𝑗,𝑘)


⋅(

1

𝜆𝑖𝑗,𝑘 + 𝑠− 1

𝜆𝑗 + 𝜏 + 𝑠

)

+

𝑛∑𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘

𝜆𝑗 + 𝜏 + 𝑠. (10)

The average bit error probability (for BPSK) is then givenby [29]

𝑃𝑒,𝑘 =1

𝜋

𝜋/2∫0

𝑀𝛾𝑖𝑘

(1

sin2 𝜃

)𝑑𝜃. (11)

Plugging (10) into (11) and after some manipulations, 𝑃𝑒,𝑖𝑘

can be expressed as

𝑃𝑒,𝑖𝑘 =

𝑛∑𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘(𝜆𝑗 − 𝜆𝑖𝑗,𝑘)


⋅(

1

𝜆𝑖𝑗,𝑘𝐼1

(1

𝜆𝑖𝑗,𝑘

)− 1

𝜆𝑗 + 𝜏𝐼1

(1

𝜆𝑗 + 𝜏

))

+

𝑛∑𝑗=1

∑𝜏∈𝑇𝑛

𝑗

sign(𝜏 )𝜆𝑖𝑗,𝑘

𝜆𝑗 + 𝜏𝐼1

(1

𝜆𝑗 + 𝜏

), (12)

where 𝐼1(𝑐) is given as [29]

𝐼1(𝑐) =1

𝜋

𝜋/2∫0

sin2 𝜃

sin2 𝜃 + 𝑐𝑑𝜃 =

1

2

(1−

√𝑐

1 + 𝑐

). (13)

Note that when 𝑘 = 1, 2, 3, 4, 𝑃𝑒,𝑖𝑘 corresponds to 𝑃𝑒,𝑆𝑖𝑅,𝑃𝑒,𝐷𝑖𝑅, 𝑃𝑒,𝑅𝑆𝑖 , and 𝑃𝑒,𝑅𝐷𝑖 respectively. Plugging these ex-pressions in (4), we obtain the end-to-end bit error rateexpression at node 𝑆𝑖. Similarly, we can get the bit error rateexpression at node 𝐷𝑖 using (5).

C. Semi-symmetric Channels

In this case, it is assumed that all subchannels between therelay nodes and 𝑆𝑖 for 𝑖 = 1, 2, . . . ,𝑚 have the same averageSNR; and all subchannels between the relay nodes and 𝐷𝑖

for 𝑖 = 1, 2, . . . ,𝑚 have the same average SNR. As such,the bit error rate performance at all 𝑆𝑖 for 𝑖 = 1, 2, . . . ,𝑚is the same. The same is true for all 𝐷𝑖 for 𝑖 = 1, 2, . . . ,𝑚.However, the performance at the nodes of the same pair canbe different.

We now define 𝜆1 ≜ 1𝛾𝑆𝑅

, 𝜆2 ≜ 1𝛾𝐷𝑅

, 𝜆3 ≜ 1𝛾𝑅𝑆

and

𝜆4 ≜ 1𝛾𝑅𝐷

. Also, let 𝛾𝑘 for 𝑘 = 1, 2, 3, 4 represent 𝛾𝑆𝑅,𝛾𝐷𝑅, 𝛾𝑅𝑆 and 𝛾𝑅𝐷, respectively. Note that we dropped theindices from these variables because the performance is thesame for all pairs (as explained before). Accordingly, (6) canbe rewritten as

𝑓𝛾𝑘(𝛾) = 𝑛𝜆𝑘 (𝑚𝜆− 𝜆𝑘) 𝑒

−𝜆𝑘𝛾

⋅𝛾∫

0

[𝑒−(𝑚𝜆−𝜆𝑘)𝜃(1− 𝑒−𝑚𝜆𝜃)𝑛−1

]𝑑𝜃

+𝑛𝜆𝑘𝑒−𝑚𝜆𝛾

(1− 𝑒−𝑚𝜆𝛾

)𝑛−1, (14)


where 𝜆 = 𝜆1+𝜆2+𝜆3+𝜆4. Applying binomial expansion to𝑓𝛾𝑘

(𝛾) and after some simple algebraic manipulations, (14)can be expressed as

𝑓𝛾𝑘(𝛾) =

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝜆𝑘𝑒

−𝜆𝑘𝛾(𝑚𝜆− 𝜆𝑘)

𝑚𝜆− 𝜆𝑘 +𝑚𝜆𝑗

⋅(1 − 𝑒−(𝑚𝜆−𝜆𝑘+𝑚𝜆𝑗)𝛾)

+𝑛𝜆𝑘

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑒−𝑚𝜆(𝑗+1)𝛾 . (15)

Plugging (15) into (9) and carrying out the integration, wehave

𝑀𝛾𝑘(𝑠) =

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝜆𝑘(𝑚𝜆− 𝜆𝑘)


⋅(

1

𝜆𝑘 + 𝑠− 1

𝑚𝜆(𝑗 + 1) + 𝑠

)

+

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝜆𝑘

𝑚𝜆(𝑗 + 1) + 𝑠. (16)

Plugging (16) into (11) and after some manipulations, the biterror rate can be expressed as

𝑃𝑒,𝑘 =

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝜆𝑘(𝑚𝜆− 𝜆𝑘)


⋅(

1

𝜆𝑘𝐼1

(1

𝜆𝑘

)− 1

𝑚𝜆(𝑗 + 1)𝐼1

(1

𝑚𝜆(𝑗 + 1)

))

+𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝜆𝑘

𝑚𝜆(𝑗 + 1)𝐼1(

1

𝑚𝜆(𝑗 + 1)),

(17)

where 𝑃𝑒,𝑘 for 𝑘 = 1, 2, 3, 4 correspond to 𝑃𝑒,𝑆𝑅, 𝑃𝑒,𝐷𝑅,𝑃𝑒,𝑅𝑆 , and 𝑃𝑒,𝑅𝐷, respectively. Then plugging the resultingexpressions into (4) and (5), we have the end to end bit errorprobability at node 𝑆 and node 𝐷, respectively.

D. Symmetric Channels

For independent, identical Rayleigh fading channels, theaverage SNRs for all channel is the same, i.e., 𝜆𝑘 definedjust after (6) are the same, and define 𝛽 ≜ 𝜆𝑘, implying that𝜆 = 4𝛽. Consequently, (17) simplifies to

𝑃𝑒 =

𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛𝛽(4𝑚− 1)

4𝑚− 1 + 4𝑚𝑗

⋅(1

𝛽𝐼1

(1

𝛽

)− 1

4𝑚𝛽(𝑗 + 1)𝐼1

(1

4𝑚𝛽(𝑗 + 1)

))

+𝑛−1∑𝑗=0

(−1)𝑗(𝑛− 1

𝑗

)𝑛

4𝑚(𝑗 + 1)𝐼1(

1

4𝑚𝛽(𝑗 + 1)).

(18)

In this case, the bit error rate at all nodes is the same,which can be obtained by plugging (18) into (4) or (5). Ifwe let 𝛽 = 1, that is, the channel gains are modeled as zeromean, unit variance complex Gaussian random variables, theresulting expression reduces to expression (10) derived in [31].

V. ACHIEVABLE DIVERSITY ORDER

Showing that the full diversity order is achieved with thesubset assignment criterion directly proves that the optimalassignment scheme achieves the full diversity as well. Whilethe final expressions derived above are exact, they do notreveal the diversity order of the system. In this section, wederive an upper bound on 𝑃𝑒 that proves that the maximumdiversity order, which is 𝑛, is achieved. In order to achieve thisgoal, we simply assume that all the channel gains are mod-eled as zero mean, unit variance complex Gaussian randomvariables, i.e., symmetric channels, but this does not changethe diversity order of the system over asymmetric channels,as demonstrated in [30].

With this assumption, the average bit error rate of all thelinks are the same. Thus, we simply use 𝑃𝑒,𝑆𝑅 to representthe bit error rate of all links. For 𝑃𝑒,𝑆𝑖 = 𝑃𝑒,𝐷𝑖 , we use 𝑃𝑒

to represent the end-to-end bit error rate. To this end, we canupper bound 𝑃𝑒 in (4) and (5) as

𝑃𝑒 = 2 (1− 𝑃𝑒,𝑆𝑅) (1− 𝑃𝑒,𝑆𝑅)𝑃𝑒,𝑆𝑅

+ [1− 2𝑃𝑒,𝑆𝑅 (1− 𝑃𝑒,𝑆𝑅)]𝑃𝑒,𝑆𝑅

≤ 3𝑃𝑒,𝑆𝑅. (19)

The last line is obtained by assuming that 0 ≤ 𝑃𝑒,𝑆𝑅 << 1,which is true in the practical SNR range. From (19), we knowthat to prove that 𝑃𝑒 achieves full diversity order, it is sufficientto prove that 𝑃𝑒,𝑆𝑅 achieves full diversity.

Note that 𝑃𝑒,𝑆𝑅 can be expressed as

𝑃𝑒,𝑆𝑅 =

∞∫0

𝑄(√

2𝜌ℎ)𝑓 (ℎ) 𝑑ℎ, (20)

where 𝑓(ℎ) is the pdf of ∣ℎ𝑆𝑅∣2 corresponding to the selectedrelay, which is given as [21]

𝑓(ℎ) = 𝑛 (4𝑚− 1) 𝑒−ℎ

ℎ∫0

𝑒−(4𝑚−1)𝜃(1− 𝑒−4𝑚𝜃

)𝑛−1𝑑𝜃

+𝑛𝑒−4𝑚ℎ(1 − 𝑒−4𝑚ℎ)𝑛−1. (21)

Using (20) and (21), we can express 𝑃𝑒,𝑆𝑅 as 𝑃𝑒,𝑆𝑅 = 𝑃𝑒1 +𝑃𝑒2 , where

𝑃𝑒1 =

∫ ∞

0

𝑄(√

2𝜌ℎ) [

𝑛𝑒−4𝑚ℎ(1− 𝑒−4𝑚ℎ)𝑛−1]𝑑ℎ, (22)

and

𝑃𝑒2 =

∞∫0

𝑄(√

2𝜌ℎ) (

𝑛 (4𝑚− 1) 𝑒−ℎ)

⋅⎛⎝ ℎ∫

0

𝑒−(4𝑚−1)𝜃(1− 𝑒−4𝑚𝜃

)𝑛−1𝑑𝜃

⎞⎠ 𝑑ℎ.(23)

We now examine the behavior of the expressions of 𝑃𝑒1 and𝑃𝑒2 in terms of the diversity order. Using Chernoff bound,we can upper bound 𝑃𝑒1 as shown in (24). Note that the lastexpression in (24) is actually the Beta function, given as [32]

𝐵 (𝑥, 𝑦) =

∫ 1

0

𝑡𝑥−1 (1− 𝑡)𝑦−1 𝑑𝑡 =Γ (𝑥) Γ (𝑦)

Γ (𝑥+ 𝑦),


𝑃𝑒1 ≤∞∫0

1

2

[𝑛𝑒−4𝑚ℎ(1 − 𝑒−4𝑚ℎ)𝑛−1

]𝑒−𝜌ℎ𝑑ℎ = − 𝑛

8𝑚

∞∫0

𝑒−𝜌ℎ(1 − 𝑒−4𝑚ℎ)𝑛−1𝑑𝑒−4𝑚ℎ =𝑛

8𝑚

1∫0

𝑡𝜌

4𝑚 (1− 𝑡)𝑛−1𝑑𝑡. (24)

for Re {𝑥} > 0 and Re {𝑦} > 0, and Γ (𝑥) is the Gammafunction. Armed with the above results, and when 𝜌 issufficiently large, we can express the last expression in (24)as

𝑃𝑒1 ≤ 𝑛

8𝑚𝐵

( 𝜌

4𝑚+ 1, 𝑛

)=

𝑛

8𝑚

Γ(

𝜌4𝑚 + 1

)Γ (𝑛)

Γ(

𝜌4𝑚 + 1 + 𝑛

)=

𝑛!

8𝑚

(𝑛∏

𝑖=1

( 𝜌

4𝑚+ 𝑖

))−1

= 𝑂(𝜌−𝑛), (25)

which suggests that the full diversity order is achieved.Similarly, using Chernoff bound, we can upper bound 𝑃𝑒2

as

𝑃𝑒2 ≤∞∫0

1

2𝑛 (4𝑚− 1) 𝑒−(𝜌+1)ℎ

⋅⎛⎝ ℎ∫

0

𝑒−(4𝑚−1)𝜃(1− 𝑒−4𝑚𝜃

)𝑛−1𝑑𝜃

⎞⎠ 𝑑ℎ

=𝑛 (4𝑚− 1)

2(1 + 𝜌)

∞∫0

𝑒−(4𝑚+𝜌)ℎ(1− 𝑒−4𝑚ℎ

)𝑛−1𝑑ℎ

=𝑛 (4𝑚− 1)

8𝑚(1 + 𝜌)

1∫0

𝑡𝜌

4𝑚 (1− 𝑡)𝑛−1

𝑑𝑡

=(4𝑚− 1)𝑛!

8𝑚(1 + 𝜌)

(𝑛∏

𝑖=1

( 𝜌

4𝑚+ 𝑖

))−1

= 𝑂(𝜌−(𝑛+1)

), (26)

which suggests that the diversity order of this term is 𝑛+ 1,which is one more than the maximum diversity order. Since theperformance is dominated by the term with the lower diversityorder, we conclude that the diversity order of the bit error ratewith the subset assignment criterion is indeed 𝑛, which isthe full diversity order. Consequently, the optimal assignmentcriterion achieves the full diversity order as well.

VI. SIMULATION RESULTS

We present in this section some simulation results to val-idate the bit error rate expressions derived in the previoussections. Specifically, we confirm that both assignment cri-teria achieve the full diversity order. We also examine thedegradation in performance due to using the subset assignmentcriterion as opposed to the optimal one. Throughout thissection, we assume that all nodes use BPSK.

In Fig. 2, we compare our subset scheme with two other ex-isting relay assignment schemes, which are random selection

5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

bit e

rror

rat

e

Random selectionm=2 n=3 proposed (suboptimal)m=3 n=3 proposed (suboptimal)m=2 n=3 sequential selectionm=3 n=3 sequential selection

Fig. 2. Performance of the suboptimal assignment scheme, random selectionand sequential selection (over symmetric channels).

and sequential selection over symmetric channels. In randomselection, relay assignment is randomly selected from the setΦ, which contains all the assignment permutations. It turns outthat the performance of this scheme is the same as the caseof one pair with one relay. In sequential selection, we firstpick one relay and one pair by finding the largest value of 𝜸𝑖

𝑗

in Φ, then remove this pair and this relay from Φ. The samething repeats until all pairs have their corresponding selectedrelays. As such, all pairs have the same opportunity to beserved by the best relay, the second best relay, etc., leadingto equivalent performances among all of them. Furthermore,since the performance is dominated by the case when the pairis assigned last in the process, the overall diversity order ofthis scheme is 𝑛−𝑚+1. That is, when the last pair is assigneda relay, only 𝑛−𝑚+ 1 relays are left for assignment, hencethe relays contribute only 𝑛 − 𝑚 + 1 to the diversity order.For example, as shown in the figure, the diversity order is onefor 𝑚 = 3, 𝑛 = 3 and two for 𝑚 = 2, 𝑛 = 3, whereas it isshown that our scheme achieves full diversity order, which is𝑛.

In Fig. 3, we present the bit error rate performance for thefollowing cases: 𝑚 = 𝑛 = 2; 𝑚 = 2, 𝑛 = 3; and 𝑚 = 𝑛 = 3over symmetric channels. In all cases, we consider both theoptimal and subset assignment criteria. For the 𝑚 = 𝑛 =2 case, as shown in the figure, the diversity is two and theperformance is the same since both criteria are equivalent (asnoted before). The diversity increases to three in the caseswhen 𝑛 = 3. In addition, we can see the degradation in SNRdue to the subset assignment scheme, which is a little over1 dB at bit error rate 1 × 10−5. Finally, we observe that theperformance of the optimal assignment criterion improves as𝑚 decreases from three to two, while the diversity order is


5 10 15 20 25 30 3510

−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

bit e

rror

rat

e

m=2 n=2m=2 n=3 optimalm=2 n=3 suboptimalm=3 n=3 optimalm=3 n=3 suboptimal

Fig. 3. Bit error rate performance comparison between the optimal andsuboptimal assignment schemes for various 𝑚 and 𝑛.

5 10 15 20 25 30 3510

−7

10−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

bit e

rror

rat

e

m=2 n=2m=2 n=3m=3 n=3m=2 n=4Theory

Fig. 4. Bit error rate performance (simulated and theoretical) of thesuboptimal assignment scheme.

the same in both cases. This is attributed to the fact that when𝑚 < 𝑛, the number of ways the relays can be assigned tothe 𝑚 pairs is more, giving rise to the possibility of finding abetter permutation.

In Fig. 4, we compare the simulated bit error rate to thetheoretical expression derived in Section IV for symmetricchannels. In our simulations, we set all channel variances toone. It is obvious that all the communication nodes have thesame average bit error probability. In particular, we considerthe cases: 𝑚 = 2, 3 and 𝑛 = 2, 3, 4. In all cases, weconsider the subset assignment scheme. As shown in thefigure, the performance improves as 𝑛 increases. Also, theperformance improves as 𝑚 decreases while fixing 𝑛 (similarto the observation in Fig. 3.) We can also see the perfect matchbetween theory and simulations, which validates the derivedbit error rate expression for symmetric channels.

In Fig. 5, we show the bit error performance of thesubset assignment scheme for asymmetric channels.

5 10 15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

ρ (dB)

bit e

rror

rat

e

n=2 D

1−−>S

1

n=2 S1−−>D

1

n=2 D2−−>S

2

n=2 S2−−>D

2

n=3 Di−−>S

i

n=3 Si−−>D

i

theory

Fig. 5. Bit error rate performance (simulated and theoretical) of thesuboptimal assignment scheme over asymmetric channels.

We consider two cases: 𝑚 = 2 and 𝑛 = 2, 3.To simplify the presentation, let 𝝊𝑖

𝑗 denote the set

{𝐸[∣∣ℎ𝑆𝑖𝑅𝑗

∣∣2] , 𝐸[∣∣ℎ𝑅𝑗𝑆𝑖

∣∣2] , 𝐸[∣∣ℎ𝐷𝑖𝑅𝑗

∣∣2] , 𝐸[∣∣ℎ𝑅𝑗𝐷𝑖

∣∣2]}where 𝐸

[∣∣ℎ𝑆𝑖𝑅𝑗

∣∣2] represents the variance of the channelcoefficient of the 𝑆𝑖 −𝑅𝑗 link. The other terms are similarlydefined. In the simulations, for the case 𝑚 = 2, 𝑛 = 2, werandomly set 𝝊1

1 =(1, 1

8 ,14 ,

12

), 𝝊2

2 =(

112 , 1, 1

3 ,17

),

𝝊12 =

(16 ,

114 ,

13 ,

12

), and 𝝊2

1 =(19 ,

15 ,

110 ,

19

). As such,

all the nodes have different average bit error rates, whichare displayed as four curves in the figure. It is shown fromthe curves that the full diversity order is achieved, whichis 𝑛 = 2 for all the nodes in the network. As for the case𝑚 = 2, 𝑛 = 3, we set 𝝊𝑖

𝑗 =(13 ,

14 ,

12 , 1

). In this case, since

the channels are semi-symmetric, the average bit error ratesare the same for the different pairs, hence there are only twocurves for this case. As we expect, the link 𝑆𝑖 −→ 𝐷𝑖 has abetter performance than 𝐷𝑖 −→ 𝑆𝑖 because the channel fromthe relay to 𝐷𝑖 is better than that from the relay to 𝑆𝑖. Inaddition, we observe that full diversity order is achieved forboth nodes, which is 𝑛 = 3. In addition, we can also observethe perfect match between theory and simulations, whichvalidates the bit error rate expression derived for asymmetricchannels.

VII. CONCLUSIONS

We addressed in this paper relay assignment schemes forcooperative systems with multiple two-way relay channelscooperating through relays that employ network coding. Weexplored two assignment criteria: optimal and subset overasymmetric, semi-symmetric and symmetric channels. Wederived closed-form expressions for the end-to-end bit errorrate performance and showed that the full diversity order isachieved in all cases.

Throughout the paper, we assumed that a single relay helpsup to one pair. This might pose some limitation from animplementation point of view. Instead, one might considerassigning one relay to multiple pairs at the same time. One


way to achieve this is to use higher order modulation schemesat the relay nodes where the bits comprising a symbol corre-spond to different pairs. The challenge that comes to mindimmediately, among others, would be to find the modulationsize to use and how to assign the pairs to the relays. Thiswill be tackled in future work. In addition, our proposed relayassignment schemes are centralized ones, which require signif-icant overheads for large networks, Therefore, an interestingremaining topic to be studied is to find efficient distributedrelay assignment schemes.

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Xuehua Zhang received her M.A.Sc degree inelectrical engineering from the Concordia Univer-sity, Montreal, QC, Canada in 2010. She is cur-rently working toward the Ph.D. degree in electricalengineering at Concordia University. Her currentresearch interests include wireless communications,cooperative communications and network coding.

Ali Ghrayeb (S’97-M’00-SM’06) received thePh.D. degree in electrical engineering from the Uni-versity of Arizona, Tucson, USA in 2000. He is cur-rently an Associate Professor with the Departmentof Electrical and Computer Engineering, ConcordiaUniversity, Montreal, QC, Canada.

He is a co-recipient of the IEEE Globecom 2010Best Paper Award. He holds a Concordia UniversityResearch Chair in Wireless Communications. He isthe coauthor of the book Coding for MIMO Com-munication Systems (Wiley, 2008). His research in-

terests include wireless and mobile communications, error correcting coding,MIMO systems, wireless cooperative networks, and cognitive radio systems.

Dr. Ghrayeb has instructed/co-instructed technical tutorials at several majorIEEE conferences. He serves as an Associate Editor of the IEEE TRANS-ACTIONS ON WIRELESS COMMUNICATIONS and IEEE TRANSACTIONS

ON SIGNAL PROCESSING. He served as an Associate Editor of IEEETRANSACTIONS ON VEHICULAR TECHNOLOGY and the Wiley WirelessCommunications and Mobile Computing Journal.

Mazen O. Hasna (S’94-M’03-SM’08) received thebachelor degree from Qatar University, Doha, Qatarin 1994, the master’s degree from the University ofSouthern California (USC), Los Angeles in 1998,and the Ph.D. degree from the University of Min-nesota, Twin Cities in 2003, all in electrical engi-neering. In 2003, Dr. Hasna joined the electricalengineering department at Qatar University as anassistant professor. Currently, he serves as the Deanof Engineering at Qatar University. Dr. Hasna’sresearch interests span the general area of digital

communication theory and its application to performance evaluation of wire-less communication systems over fading channels. Current specific researchinterests include cooperative communications, ad hoc networks, cognitiveradio, and network coding.

Documents

On Relay Assignment in Network-Coded Cooperative Systems