Download pdf - Multi-Relay Turbo-Coded Cooperative Diversity …hamouda/Jules_TVT...4458 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.62,NO.9,NOVEMBER2013 Multi-Relay Turbo-Coded Cooperative Diversity

4458 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013

Multi-Relay Turbo-Coded Cooperative DiversityNetworks Over Nakagami-m Fading Channels

Jules Merlin Moualeu, Student Member, IEEE, Walaa Hamouda, Senior Member, IEEE,HongJun Xu, Member, IEEE, and Fambirai Takawira, Member, IEEE

Abstract—This paper studies a multi-relay adaptive turbo-codedcooperative diversity scheme over Nakagami-m fading chan-nels. In the underlying scheme, after receiving a fraction ofthe codeword determined by a tradeoff parameter denoted byf , all the relays decode, and only the reliable relays forwardthe rest of the codeword to the destination. We assume binaryphase-shift keying modulation and analyze the scheme understudy in terms of bit error rate (BER) and outage probability.Union bounds on the BER are derived using the transfer functionbounding and the limit-before-average techniques. Furthermore,we examine the asymptotic behavior of the system in the high-signal-to-noise-ratio (SNR) regime using the derived pairwise er-ror probability (PEP). We also derive a closed-form expression forthe outage probability of the system and its high-SNR approxi-mation to evaluate the achievable diversity order. Simulations areprovided to assess the accuracy of our analytical work.

Index Terms—Coded cooperation, diversity order, error rate,fading channels, outage probability, turbo codes.

I. INTRODUCTION

U SER cooperation techniques (see [1] and [2]) allowsingle-antenna mobile terminals to share their physical re-

sources by creating a virtual antenna array. There are two maincooperative diversity strategies: amplify-and-forward (AF) anddecode-and-forward (DF). In the AF mode, the relay terminal(or cooperating user) amplifies the signal received from thesource terminal (partner) before forwarding the signal to thedestination terminal. On the other hand, in the DF mode,the relay terminal decodes and regenerates the received source’sinformation for retransmission to the destination (see, e.g.,

Manuscript received October 5, 2012; revised January 23, 2013 andMarch 15, 2013; accepted May 2, 2013. Date of publication May 31, 2013; dateof current version November 6, 2013. This work was supported by TELKOMSouth Africa through the Centre of Excellence Program. The material inthis paper was presented in part at the IEEE Wireless Communications andNetworking Conference, Shanghai, China, April 7–10, 2013. The review of thispaper was coordinated by Prof. C.-X. Wang.

J. M. Moualeu was with the School of Electrical, Electronic and ComputerEngineering, University of KwaZulu-Natal, Durban 4041, South Africa. Heis now with the Department of Electrical and Information Engineering, Uni-versity of the Witwatersrand, Johannesburg 2050, South Africa (e-mail: [email protected]).

W. Hamouda is with the Department of Electrical and Computer Engi-neering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail:[email protected]).

H. Xu is with the School of Electrical, Electronic and Computer Engi-neering, University of KwaZulu-Natal, Durban 4041, South Africa (e-mail:[email protected]).

F. Takawira is with the Department of Electrical and Information Engineer-ing, University of the Witwatersrand, Johannesburg 2050, South Africa (e-mail:[email protected]).

Digital Object Identifier 10.1109/TVT.2013.2265329

[3] and [4]). In [4], cooperative diversity with multiple re-lays is investigated wherein the authors show that reliabilityand high data rate can be achieved. In a different configu-ration, An and Hamouda [5] proposed a reduced complex-ity multiple-input multiple-output (MIMO) concatenated code,which can also be implemented in cooperative communications.

Recently, a novel framework that integrates channel codingwith cooperative diversity and termed coded cooperation wasintroduced in [6]. Hunter et al. [6] proposed an efficient cod-ing scheme by using rate-compatible punctured convolutionalcodes for the users. Similar to [6], in [7], a frame error rate(FER) is derived for the proposed scheme, and it is shown thatfull diversity can be achieved. Turbo codes have shown to be anatural fit in user cooperation [8]. Janani et al. [8] investigatedthe use of turbo codes in user cooperation in the context oforiginal coded cooperation and space–time cooperation. Theauthors in [9] derived some analytical bounds on the outageprobability for a single-relay convolutional-coded cooperativenetwork over Rayleigh fading channels, but no closed-formexpression was obtained. In [10], a convolutional-based dis-tributed scheme with a code design similar to [7] is proposed,but in a multi-relay scenario and evaluated their proposedscheme both in terms of bit error and outage probabilities. In asimilar work to [9], Elfituri et al. [10] found some approximatebounds on the bit error rate (BER) and outage probability for amulti-relay scenario.

The works on coded cooperation mentioned previously areconsidered over Rayleigh fading channels. A Rayleigh fad-ing model is practically unrealistic since it does not includethe statistical characteristics of a land mobile system, thecomplex indoor environments, and ionospheric radio links. Amodel that gives the best fit of these characteristics is theNakagami-m model, as it represents a generalized distributionwhere various fading environments (severe, light, or no fading)can be modeled. Many works have studied the uncoded coop-erative diversity schemes over Nakagami-m distribution (e.g.,in [11]–[17]). In [11], Ikki and Ahmed proposed a multi-relayscenario for uncoded DF scheme over independent and non-identically distributed (i.n.i.d.) Nakagami-m flat-fading chan-nels. Exact and approximate closed-form expressions of the biterror rate, the outage probability, and the channel capacity areobtained. In [12], the symbol error rate (SER) of the uncodedDF with multiple dual-hop relays over Nakagami-m fadingchannels is examined. Closed-form expressions for the SERand the outage probability were derived from the work in[13]–[15] for uncoded DF cooperative networks. The outageprobability performance for uplink cooperative code-division

0018-9545 © 2013 IEEE

MOUALEU et al.: MULTI-RELAY TURBO-CODED COOPERATIVE DIVERSITY NETWORKS 4459

multiple-access (CDMA) systems subject to Nakagami-m fad-ing using adaptive DF and AF schemes is investigated in [16]and [17], respectively. Under the same fading environment butwith a different setup, [18] studied the error rate performanceof a MIMO CDMA system.

Few works have focused on channel coding schemes withuser cooperation, i.e., coded cooperation, over Nakagami-mfading channels [19], [20]. For instance, in [20], the outageprobability of a transmission protocol identical to [6] (con-volutional coding) with two users under Nakagami-m fadingchannels is analyzed. However, no closed-form expression forthe exact outage probability was derived since the integral partin the outage probability expression is not easy to evaluate.Motivated by these observations, we consider a distributedturbo-coded system in a multi-relay cooperative network sub-jected to slow Nakagami-m fading distribution. To the best ofour knowledge, no work has been investigated on the integra-tion of turbo codes in multi-relay cooperative networks overNakagami-m fading channels. To provide better error rates andoutage probability, multi-relay adaptive cooperative schemesare preferred over the best relay selection scenarios, whichprovide better spectral efficiency [21] over their counterparts.Focus on the spectral efficiency is beyond the scope of thispaper. Moreover, there are very few research works in the liter-ature on multiple adaptive relays in coded cooperation (in thoseinstances, general fading is not even considered), and no closed-form expression of the true end-to-end outage probability isprovided. The schemes that are closely related to this paperare the schemes investigated in [7], [22], and [23]. In [7], theauthors considered a distributed turbo-coded system in a single-relay scenario over Rayleigh fading channels, where the sourceand the relay use the same puncturing pattern to transmit to thedestination. The work in [22] is an extension of [8] to a multi-relay scenario subjected to Rayleigh fading. However, our codedesign is dissimilar to [22], as well as the transmission protocol.Moreover, no analytical work on the outage probability isgiven, and our proposed analytical methodology on the BERis different from [22] as their analysis considered a fast-fadingscenario. Recently, Haghighat and Hamouda [23] proposed anovel signal-processing scheme emanating from [24], in whicha single relay listens to the source for a certain amount time andtransmits for the remainder of time using tradeoff parameter f .This parameter determines the amount of time that the relayshould listen and transmit. However, the complexity of thelatter transmission scheme is an issue since it resembles aturbo code embedded in another turbo code, hence requiringa complex decoding process. Furthermore, the authors in [23]only considered a single-relay scenario over Rayleigh fadingchannels. Our proposed scheme uses a less complex systemfrom the encoding and decoding point of view, in comparisonwith [23].

We consider a distributed turbo-coded cooperative systemsubject to a Nakagami-m fading distribution. Moreover, wederive the union bounds on the probability of bit errors forboth independent and identically distributed (i.i.d.) and i.n.i.d.Nakagami-m slow fading channels by using the transfer func-tion bounds and the limit-before-average technique that yieldtight bounds. Furthermore, an analytical closed-form expres-

sion of the outage probability is derived for an arbitrary numberof relays and different fading indices. This is done by evaluatingthe difficult integral part of the outage probability expression.We also provide approximations of both the pairwise errorprobability (PEP) and outage probability for the high-SNRregime and show that full diversity in the number of cooperatingrelays and fading parameters is achieved. The proposed systemis identical to a cooperative system with a tradeoff parameterf = 0 defined in [23]. Finally, we study through computersimulations the effects of f on the outage probability for variousvalues of f ranging from [0, 1].

The remainder of this paper is organized as follows. InSection II, we describe the system model. In Section III, weanalyze the upper bounds on the bit error probability followedby the outage probability analysis of the proposed scheme inSection IV. In Section V, a discussion on the extension toa multihop multibranch system is presented. The simulationand numerical results are presented in Section VI. Finally,conclusions are drawn in Section VII.

Notation: CN (0, σ2) denotes the circular symmetric com-plex Gaussian distribution with zero mean and variance σ2. Forrandom variable (RV) γij , E〈γij〉 = γ̄ij is the expected valueof γij .

II. SYSTEM MODEL

We consider a multi-relay dual-hop turbo-coded cooperativesystem over Nakagami-m fading channels. In what follows, wedenote the source, the nth relay and the destination nodes bys, rn, d, respectively, where n ∈ {1, . . . , L}, and L representsthe number of cooperating relays. All nodes are equippedwith a single antenna, and the relays operate in half-duplexmode, i.e., they cannot receive and transmit simultaneously.We assume that the channels are modeled as quasi-static, i.e.,the fading coefficients are constant for the entire duration ofa transmission frame (codeword) but vary independently fromone frame to another. We also assume that all receivers haveperfect knowledge of the channel statistics of the transmitter-to-receiver link. The cooperative transmission is divided intotwo phases, and the proposed system works as follows.

In the first phase, the source encodes a message of length Kby a turbo code of rate Rc = 1/3 and starts broadcasting thegenerated N -bit codeword. In our model, we consider a turbocode consisting of two parallel concatenated convolutionalcodes separated by an interleaver. We assume the destinationlistens to the entire codeword, whereas the relays only listento a fraction of the entire codeword. This idea stems from[24] in which predetermined phase duration for broadcast orcooperation commonly used in the literature is not employedin an effort to solve the half-duplex constraint. In [24], thelistening time denoted by f is determined by the relay basedon the channel conditions, i.e., the relay decides when to listenand when to transmit based on its received signal. Here, forsimplicity, this fraction of the entire codeword received by allthe relays is assumed to be equal (hence, f is identical for allthe relays) and is given by [23]

N ′ = K + (N −K)f, (1)


Fig. 1. Proposed distributed turbo-coded cooperative system with L relays.

where 0 ≤ f ≤ 1, and denotes the parameter that determinesthe amount of time that the relays listen to the source. Withoutloss of generality, we use f = 0 in this paper since it corre-sponds to our proposed scheme where the relays only receivethe noisy systematic bits. In practice, this can be done throughpuncturing by deleting the parity bits. Hence, N ′ = K andsimply means that all the relays receive a noisy version of thesystematic bits that can be done through puncturing prior todecoding. The received codeword at the destination and relaynodes are given by

ysd(1 : N) =√Pshsdxs(1 : N) + zsd(1 : N), (2)

ysrn(1 : K) =√

Pshsrnxs(1 : K) + zsrn(1 : K), (3)

where xs(1 :N)={xs(1), . . . , xs(K), xs(K+1), . . . , xs(N)},with xs(1 : K) representing the systematic bits and xs(K + 1 :N) representing the parity bits; hsd and hsrn are the fadingcoefficients for the s−d and s−rn links, respectively, which areobtained from a Nakagami-m distribution with E〈|hsd|2〉 = 1and E〈|hsrn |2〉 = 1 and parameters msd and msrn ; Ps is thetransmitted signal power at the source for the s−d and s−rnlinks; and zsd(1 : N) and zsrn(1 : K) represent the i.i.d. addi-tive white Gaussian noise (AWGN) modeled as CN (0, N0/2).All the relays are equipped with a turbo iterative decoder toestimate the punctured bits from the source node and an errordetection scheme via a cyclic redundancy check code.1

During the second phase, the decodable or reliable relays,i.e., the relays that have correctly decoded the source message,regenerate the source message and forward the punctured bits(parity bits) to the destination through orthogonal (in time)multiple-access channels. The nth received signal at the des-tination is given by

yrnd(K+1 : N)=√

Prhrndx̂s(K+1 : N)+zrnd(K+1 : N),(4)

1This is already included in the information message and, therefore, does notrepresent an additional overhead.

where x̂s(K + 1 : N) denotes the estimated parity bits; hrnd isthe fading coefficient for the rn−d links, with E〈|hrnd|2〉 = 1and parameter mrnd; Pr is the transmitted power at the relaysfor the rn−d links; and zrnd(K + 1 : N) denotes the i.i.d.AWGN with zero mean and variance N0/2 per dimension.The parity bits from the source and the decodable relays areoptimally combined by maximal-ratio combining (MRC), andthe output of the MRC is sent to a turbo decoder. Fig. 1shows the different stages of the proposed scheme during bothphases. It is worth mentioning that decoding at the destinationonly takes place after the second-hop transmission. In the casewhere no relay can successfully decode the source message,no transmission occurs in the second phase as all the relayskeep silent, and the destination starts decoding the first time-slot transmission message from the source.

Let us define the instantaneous received SNR betweenusers i and j by

γij = |hij |2γ̄ij , (5)

where |hij |2 is a gamma-distributed RV. The probability densityfunction (pdf) of γij is given by

p(γij) =m

mij

ij γmij−1ij

Γ(mij)γ̄mij

ij

exp

(−mijγij

γ̄ij

), (6)

where mij denotes the fading parameter of the i−j link andΓ(·) is the gamma function defined in [25, Eq. (8.130.1)].

III. UNION BOUNDS ON THE BIT ERROR RATE

Here, we derive the union bounds on the BER of the proposedscheme for L relays over Nakagami-m fading. In our analysis,we consider the binary phase-shift keying (BPSK) modulation.

A. PEP

The conditional PEP for a coded system is defined asthe probability of selecting an erroneous codeword x̃ = (x̃1,x̃2, . . . , x̃N ) when the codeword x = (x1, x2, . . . , xN ) is trans-mitted. As a baseline, we consider a single-link transmission.


Hence, the PEP conditioned on the instantaneous SNR γ ={γ(1), γ(2), . . . , γ(N)} is given by [26]

P (x → x̃|γ) = Q

⎛⎝√

2∑i∈ξ

γ(i)

⎞⎠ , (7)

where Q(x) denotes the Gaussian Q-function, γ(i) is theinstantaneous received SNR for code bit i, ξ is the set of allcode bits i for which x �= x̃, and the selection of the codewordx̃ over x is called an error event.2

In the proposed scheme, the number of relays that forward tothe destination during the second phase (those relays that havesuccessfully decoded the source message) varies from 0 to L.Hence, we denote Θ as the set of indices of cooperating relaysgiven by

Θ = {k1, k2, . . . , kM} ⊂ {1, 2, . . . , L}, (8)

where M = |Θ| denotes the cardinality of Θ.Assuming that γsd, γsr1 , . . . , γsrL , γr1d, . . . , γrLd are statis-

tically independent, and using (7), the exact expression of theend-to-end conditional PEP is given by

P (d|γsd, γsr1 , . . . , γsrL , γr1d, . . . , γrLd)

= Q(√

2dγsd)L∏

j=1

Q(√

2d1γsrj

)+

L∑M=1

(L

M

)

×

⎛⎝∏

i�∈ΘQ(√

2d1γsri)⎞⎠(∏

i∈Θ1 −Q

(√2d1γsri

))

×(Q

(√2dγsd +

1M

∑i∈Θ

d2γrid

)), (9)

2For the purpose of error analysis, we restrict ourselves to a chosen all-zerotransmitted codeword without loss of generality. Therefore, the PEP is not afunction of x̃ and x but only depends on d and can be given by P (d|γ), whered denotes the error-event Hamming weight.

where d = d1 + d2 denotes the error-event Hamming weight,with d1 and d2 representing the number of bits in the Hammingweight d that are sent through the source-to-relay channels andthe relay-to-destination channels, respectively. Moreover, thefirst term in (9) represents the case where all the relays and thedestination select the erroneous codeword x̃ over x and can beobtained using (7), whereas the second term means that someor all the reliable relays lead to an error event at the destination.

The alternative representation of the Gaussian Q-functionwas proposed by Craig [27] and can be written as

Q(x) =1π

π2∫

0

exp

(− x2

2 sin2 θ

)dθ. (10)

Using Craig’s formula (10), we can rewrite (9) as follows:

P (d|γsd, γsr1 , . . . , γsrL , γr1d, . . . , γsrL)

=

⎛⎜⎝ 1π

π2∫

0

exp

(− dγsd

sin2 θ

)dθ

⎞⎟⎠

×

⎛⎜⎝ L∏

j=1

⎡⎢⎣ 1π

π2∫

0

exp

(−d1γsrjsin2 θj

)dθj

⎤⎥⎦⎞⎟⎠

+

L∑M=1

(L

M

)⎛⎜⎝∏

i�∈Θ

1π

π2∫

0

exp

(−d1γsrjsin2 θi

)dθi

⎞⎟⎠

×

⎛⎜⎝∏

i∈Θ

⎡⎢⎣1 − 1

π

π2∫

0

exp

(−d1γsrisin2 θi

)dθi

⎤⎥⎦⎞⎟⎠

×

⎛⎜⎝ 1π

π2∫

0

exp

(−dγsd +

1M

∑i∈Θ d2γrid

sin2 θi

)dθi

⎞⎟⎠ . (11)

The unconditional PEP can be obtained by averaging (11)over the fading distribution and is given by (12), shown atthe bottom of the page, where p(γsd), p(γsri), and p(γrid)

P (d) =

⎛⎜⎝ 1π

π2∫

0

∞∫0

exp

(− dγsd

sin2 θ

)p(γsd)dγsddθ

⎞⎟⎠⎛⎜⎝ L∏

j=1

⎡⎢⎣ 1π

π2∫

0

∞∫0

exp

(−d1γsrjsin2 θj

)× p(γsrj )dγsrjdθj

⎤⎥⎦⎞⎟⎠

+

L∑M=1

(L

M

)⎛⎜⎝∏

i�∈Θ

1π

π2∫

0

∞∫0

exp

(−d1γsrjsin2 θi

)p(γsri)dγsridθi

⎞⎟⎠

×

⎛⎜⎝∏

i∈Θ

⎡⎢⎣1 − 1

π

π2∫

0

∞∫0

exp

(−d1γsrisin2 θi

p(γsri)dγsridθi

)⎤⎥⎦⎞⎟⎠

×

⎛⎜⎝ 1π

π2∫

0

∞∫0

∞∫0

exp

(−dγsd +

1M

∑i∈Θ d2γrid

sin2 θi

)p(γsd)p(γrid)dγsddγriddθi

⎞⎟⎠ , (12)


denote the pdf of RVs γsd, γsri , and γrid respectively, as givenby (6) with the subscripts changed to the corresponding links.To evaluate (12), we use a moment-generating-function-basedapproach with the following expression [26]:

∞∫0

exp(−sγij)p(γij)dγij =

(1 +

sγijmij

)−mij

, (13)

and after identifying (13) in (12) and performing some manip-ulations, (12) can be given by

P (d)=

⎛⎜⎝ 1π

π2∫

0

(1+

dγ̄sd

msd sin2 θ

)−msd

dθ

⎞⎟⎠

×L∏

j=1

⎛⎜⎝ 1π

π2∫

0

(1+

d1γ̄srjmsrj sin

2 θj

)−msrj

dθj

⎞⎟⎠

+

L∑M=1

(L

M

)⎛⎜⎝∏i�∈Θ

1π

π2∫

0

(1 +

d1γ̄srimsri sin

2 θi

)−msri

dθi

⎞⎟⎠

×

⎛⎜⎝∏

i∈Θ1 − 1

π

π2∫

0

(1 +

d1γ̄srimsri sin

2 θi

)−msri

dθi

⎞⎟⎠

× 1π

π2∫

0

(1 +

dγ̄sd

msd sin2 θ

)−msd

×∏i∈Θ

(1 +

d2γ̄rid

Mmrid sin2 θ

)−mrid

dθ. (14)

With the help of the work in [26] and [28], a closed-formexpression of the unconditional PEP expression in (14) forpositive integers msd, msrn , and mrnd can be obtained as in(15), shown at the bottom of the page, where 2F1(a, b; c;x) isthe Gauss hypergeometric function defined in [25, Eq. (9.111)],Λkj , Bl, Cl, and Il are given, respectively, by

Λjk =

dmrjd

−k

dsmrjd

−k

∏Ln=1n �=j

(1

1+d2γ̄rndmrndM s

)(mrjd − k

)!(

d2γ̄rjd

mrjd

)mrd−k, (16)

Bl =Al(

msd+k−1l

) , (17a)

Cl =

n=0∑k−1

(kn

)(msd+k−1

n

)Al, (17b)

P (d) =

⎡⎢⎣ (dγ̄sd/msd)

12

2√π(

1 + dγ̄sd

msd

)msd+12

·Γ(msd +

12

)Γ(msd + 1)

· 2F1

(1,msd +

12;msd + 1;

msd

msd + dγ̄sd

)⎤⎥⎦

×L∏

j=1

⎡⎢⎢⎣

(d1γ̄srj/msrj

) 12

2√π(

1 +d1γ̄srj

msrj

)msrj+ 1

2

·Γ(msrj +

12

)Γ(msrj + 1)

· 2F1

(1,msrj +

12;msrj + 1;

msrj

msrj + d1γ̄srj

)⎤⎥⎥⎦+

L∑M=1

(L

M

){ L∏j=M+1

[ (d1γ̄srj/msrj

) 12

2√π(

1 +d1γ̄srj

msrj

)msrj+ 1

2

·Γ(msrj +

12

)Γ(msrj + 1)

× 2F1

(1,msrj +

12;msrj + 1;

msrj

msrj + d1γ̄srj

)]}

×{

1 −M∏j=1

[ (d1γ̄srj/msrj

) 12

2√π(

1 +d1γ̄srj

msrj

)msrj+ 1

2

×Γ(msrj +

12

)Γ(msrj + 1)

· 2F1

(1,msrj +

12;msrj + 1;

msrj

msrj + d1γ̄srj

)

×L∑

j=1

mrjd∑

k=1

Λkj ×

(dγ̄sdmrjd

M

d2γ̄rjd

)k−1

2(

1 − dγ̄sdmrjdM

d2γ̄rjdmsd

)msd+k−1

×(

k−1∑l=0

(d2γ̄rjdmsd

dγ̄sdMmrjd− 1

)l

BlIl

(d2γ̄rjd

Mmrjd

)

−(dγ̄sdmrjdM

d2γ̄rjdmsd

)msd−1∑l=0

(1 −

dγ̄sdmrjdM

d2γ̄rjdmsd

)l

ClIl

(dγ̄sdmsd

))]}, (15)


Al =(−1)k−1+l

(k−1l

)(k − 1)!

k∏n=1

n �=l+1

(msd−k+n), (17c)

Il(x) = 1 −√

x

1 + x

(1+

l∑n=1

(2n− 1)!!n!2n(1 + x)n

). (17d)

Although the expression in (15) yields an exact closed-formexpression for the PEP and, more importantly, its numericalevaluation is not computationally prohibitive, it does not offeran insight about the effects of the different system parameters.In what follows, we examine the asymptotic performance ofthe PEP as SNR → ∞. Let γ̄sd = γ̄sr1 = · · · = γ̄srL = γ̄r1d =· · · = γ̄rLd = γ̄ → ∞. Then, (14) will reduce to

P (d) ≈ 1π

π2∫

0

(dγ̄)−msd

(d2Lγ̄

)−∑L

j=1mrjd

dθ. (18)

After integration and some manipulations, the approximate PEPexpression in (18) can be further expressed as

P (d) ≈ 12d−msd

(d2L

)−∑L

j=1mrjd

γ̄−(msd+

∑L

j=1mrjd

).

(19)

It is important to note that the PEP expression in (19) showsthat the diversity order of the system is msd +

∑Lj=1 mrjd.

B. Transfer Function Bounds

The weight distribution is essential in deriving the unionbounds on the error rates of a given code. However, for a turbocode with a given interleaver, obtaining the weight distributionis exceedingly difficult, rendering it impractical. This is dueto the fact that the redundant bits generated by the secondinterleaver are dependent on the input word and the manner inwhich the bits are interlaced. A viable solution to this approachis an exhaustive search over all possible cases, but for longinterleavers, it proves to be computationally intensive. In [29]and [30], the average weight distribution for turbo codes fora given length using the transfer function bounds technique isproposed. However, the computation method in [30] allows formore accurate numerical results. Therefore, in this paper, weuse the method proposed in [30].

Following the method in [30], a state transition matrix canbe obtained using the state transition for the code fragment(1, 17/13) in octal form and is given by

A(J, I,D)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

J JID 0 0 0 0 0 00 0 JID J 0 0 0 00 0 0 0 JD JI 0 00 0 0 0 0 0 JI JD

JID J 0 0 0 0 0 00 0 J JID 0 0 0 00 0 0 0 JI JD 0 00 0 0 0 0 0 JD JI

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(20)

where JjIiDd is a monomial with j being always equal to 1,and i and d are input and output dependent and take on the valueof 0 or 1. The corresponding transfer function is given by

T (J, I,D) =∑j≥0

∑i≥0

∑d≥0

JjIiDdt(j, i, d). (21)

Using the technique in [31] and after some manipulations,the recursion formula of t(j, i, d) can be obtained with the useof Matlab but is omitted here due to space limitations.

C. BER

The average BER can be upper bounded as [32]

P b ≤1K

∞∑d=df

a(d)P (d), (22)

where df represents the free distance, P (d) is the PEP, and a(d)is the number of error events with distance d given by

a(d) =K∑i=1

K∑d1=1

K∑d2=1︸︷︷︸

d=i+d1+d2

i

K

(i

K

)p(d1|i)p(d2|i), (23)

with

p(d1/2|i) =t(K, i, d1/2)(

Ki

) . (24)

Using (15) in (22) yields very loose bounds since many termson the right-hand side of (22) contribute to the sum over the en-tire SNR regime. For this reason, we revert to the limit-before-averaging technique [32] that uses the conditional union upperbound presented in (9) and results in much tighter bounds, asshown in Section VI. The average bit error probability becomes

P b≤∞∫0

· · ·∞∫0︸︷︷︸

L−fold

⎡⎣min

⎛⎝1

2,

1K

∞∑d=df

a(d)×P (d|γsd · · · γrLd)

⎞⎠⎤⎦

× p(γsd) · · · p (γrLd) dγsd · · · dγrLd. (25)

Note that the order of summation and integration cannot beinterchanged due to the minimization. Hence, the L-fold inte-gration will be numerically solved, which requires an intensivecomputation. However, an analysis on the computational com-plexity is beyond the scope of this paper.

IV. OUTAGE PROBABILITY

An important performance metric that best describes slowlyfading channels is the outage probability denoted by Pout. Inthe proposed scheme, the source broadcasts the entire codewordto the destination with an overall code rate Rc = 1/3, whereasthe cooperating relays received the source message with rateR1 = Rc/α, where α represents the cooperation ratio and isgiven as

α =K + f(N −K)

N. (26)


It should be pointed out that this analysis can also apply tovarious overall code rates Rc. Rc = 1/3 is used throughout thispaper because of the turbo code employed in this paper. Duringthe second phase, the decoding relays transmit the parity bitsonly with a transmission rate R2 = Rc/(1 − α). In the sequel,we derive a closed-form expression for the outage probability.

We first consider a direct transmission between source iand destination j as a baseline. The instantaneous capacity forslowly fading channels can be expressed as C(γij) = log2(1 +γij) [33]. The link is said to be in outage if the instanta-neous capacity falls below threshold Rc, which denotes thecode rate, and the corresponding outage event is C(γij) < Rc.Hence, the outage probability for a Nakagami-m fading can beevaluated as

Pout = Pr {C(γij) < Rc} =

2Rc−1∫0

p(γij)dγij

= 1 −Γ(m, (2Rc−1)m

γ̄ij

)Γ(m)

=γ(m, (2Rc−1)m

γ̄ij

)Γ(m)

(27)

where Pr{B} is the probability of event B, and γ(a, x) andΓ(a, x) denote the lower and upper incomplete gamma func-tions, respectively, as defined in [25, Eq. (8.350.1/2)].

A. Outage Analysis of the Proposed Scheme

In this scheme, the transmission for the codeword is doneover two phases. In the first phase, the source broadcasts theentire codeword to the destination with code rate Rc, whereasthe relays listen only to a part of the codeword with a code rateequivalent to R1 = Rc/α.

When relay rn does not successfully decodes the sourcemessage, it is said to be in outage, which is equivalent to theevent

C (γsrn) = log2 (1 + γsrn) < Rc/α. (28)

In the second phase, the decodable relays forward the paritybits to the destination. The transmission over the two phasescan be seen as time sharing between two independent channelswhere the second channel (rn−d) uses a fraction of the totaltime. Hence, the corresponding outage event involving a relayrn−d is given by

C (γsd, γrnd) = log2(1+γsd)+β(1−α) log2 (1+γrnd) < Rc

(29)

where β = 0 when there is no retransmission in the secondphase, and β = 1 otherwise.

The end-to-end outage probability at the destination giventhe two transmission phases is

Pout = Pr{γsd < 2Rc − 1

}·

L∏j=1

Pr{γsrj < 2Rc/α − 1

}

+

L∑M=1

(L

M

)

×

⎡⎣ L∏j=M+1

Pr{γsrj < 2Rc/α − 1

}

×M∏j=1

Pr{γsrj > 2Rc/α − 1

}

×Pr

⎧⎨⎩(1 + γsd)

α

⎛⎝1 +

M∑j=1

γrjd

⎞⎠1−α

< 2Rc

⎫⎬⎭⎤⎦

(30)

where the first term of the summation denotes the case whereall the relays are in outage and the second term corresponds tothe case where some or all relays may fully decode the sourcemessage (not in outage).

The expression in (30) can be rewritten using the first equa-tion in (27) in its integral form as

Pout=

⎛⎜⎝

2Rc−1∫0

p(γsd)dγsd

⎞⎟⎠⎛⎜⎝ L∏

j=1

⎡⎢⎣

2Rc/α−1∫0

p(γsrj

)dγsrj

⎤⎥⎦⎞⎟⎠

+

L∑M=1

(L

M

)⎛⎜⎝ L∏j=M+1

⎡⎢⎣

2Rc/α−1∫0

p(γsrj

)dγsrj

⎤⎥⎦⎞⎟⎠

×

⎧⎪⎨⎪⎩⎛⎜⎝ M∏

j=1

⎡⎢⎣1 −

2Rc/α−1∫0

p(γsrj

)dγsrj

⎤⎥⎦⎞⎟⎠

×∫

· · ·∫

︸︷︷︸A

p(γsd) ·M∏j=1

p(γrjd

)dγrjddγsd

⎫⎪⎪⎬⎪⎪⎭, (31)

where A corresponds to the region on integration given by

A =

{(γsd, γrjd

)|γsd ≥ 0, γrjd ≥ 0, 1 ≤ j ≤ M

(1 + γsd)

⎛⎝1 +

M∑j=1

γrjd

⎞⎠1−α}

. (32)

Given the constraints in (32), the region of integration can beexplicitly expressed as

A =

{γsd < 2Rc − 1 = a

γrjd < 2Rc/(1−α)

(1+γsd)1/(1−α) − 1 = b.(33)

After performing some integrations and using the explicitregion of integration as shown in (33), a more compact form


of (31) is given as

Pout =

⎡⎣γ

(msd,

msd

γ̄sd(2Rc − 1)

)Γ(msd)

⎤⎦

×

⎡⎣ L∏j=1

γ(msrj ,

msrj

γ̄srj(2Rc/α − 1)

)Γ(msrj )

⎤⎦

+

L∑M=1

(L

M

)⎡⎣ L∏j=M+1

γ(msrj ,

msrj


)Γ(msrj )

⎤⎦

×

⎧⎨⎩⎡⎣ M∏j=1

Γ(msrj ,

msrj


)Γ(msrj )

⎤⎦ M∏

j=1

1Γ(mrjd)

×2Rc−1∫0

mmsd

sd γmsd−1sd

Γ(msd)γ̄sdexp

(−msdγsd

γ̄sd

)

×γ

(mrjd,

mrjd

γ̄rjd

(2Rc/(1−α)

(1+γsd)1/(1−α)−1

))dγsd

}.

(34)

The outage probability expression in (34) is not in its closedform due to the existence of the integral expression. Hence, aclosed-form expression for the outage probability in (34) can beobtained by solving the following integral:

I(μ, ν, β) =

2Rc−1∫0

mmsd

sd γμsd

Γ(msd)γ̄sdexp(−νγsd)

×γ

(mrjd,

mrjd

γ̄rjd

(2Rcβ

(1 + γsd)β− 1

))dγsd, (35)

where μ = msd − 1, ν = (msd/γ̄sd), and β = (1/1 − α).The integral in (35) can be evaluated as shown in the

Appendix. Hence, the closed-form expression for the outageprobability can be derived as follows:

Pout =

⎡⎣γ

(msd,

msd

γ̄sd(2Rc − 1)

)Γ(msd)

⎤⎦

×

⎡⎣ L∏j=1

γ(msrj ,

msrj


)Γ(msrj )

⎤⎦+

L∑M=1

(L

M

)

×

⎡⎣ L∏j=M+1

γ(msrj ,

msrj


)Γ(msrj )

⎤⎦

×

⎧⎨⎩⎡⎣ M∏j=1

Γ(msrj ,

msrj


)Γ(msrj )

⎤⎦ M∏

j=1

1Γ(mrjd)

×[

msd

Γ(msd)γ̄sd

M∏j=1

((msd

γ̄sd

)−msd

×γ

(msd,

msd

γ̄sd(2Rc − 1)

),

− exp

(mrjd

γ̄rjd+ 1

))

×mrjd

−1∑n=0

n∑m=0

(−1)n+m

n!

(n

m

)(mrjd

γ̄rjd

)n

2Rcm1−α

×∞∑i=0

i∑l=0

(−1)i

i!

(i

l

)2

Rc(i−l)1−α

(msd

γ̄sd

)l

×(mrjd

γ̄rjd

)i−l(2Rc − 1)msd+l

msd + l

× 2F1

(msd + l,

i− l

1 − α,msd + l − 1; 1 − 2Rc

)].

(36)

B. Asymptotic Performance

To investigate the asymptotic behavior of (34), we assumeall the average SNRs to be sufficiently large, i.e., γ̄sd = γ̄srj =γ̄rjd = γ̄ → ∞. This condition will reduce (34) to

Pout ≈a∫

0

mmsd

sd γmsd−1sd

Γ(msd)γ̄exp

(−msdγsd

γ̄

)

×

⎡⎣ M∏j=1

γ(mrjd,

mrjd

γ̄ b)

Γ(mrjd)

⎤⎦ dγsd. (37)

Using γ(a, x) = xaa−11 F1 (a, 1 + a; −x) given in [25,

Eq. (8.351.2)], the expression in (37) can be rewritten as

Pout ≈a∫

0

mmsd

sd γmsd−1sd

Γ(msd)γ̄exp

(−msdγsd

γ̄

)

×

⎡⎣ M∏j=1

1Γ(mrjd)

mmrjd

−1

rjdγ̄−mrjdbmrjd

× 1F1

(mrjd, 1 +mrjd;−

mrjdb

γ̄

)⎤⎦ dγsd, (38)

where 1F1(a, b;x) is the confluent hypergeometric functiondefined in [25, Eq. (9.210.1)].

As γ̄→∞, the following approximations limx→∞ exp(−1/x) ≈ 1 and 1F1(a, b; 0) ≈ 1 can be used. After some algebraicmanipulations, the outage probability can be approximated asin (39), shown at the bottom of the next page, where Λ canbe numerically integrated and does not depend on γ̄. Theexpression in (39) confirms that the achievable diversity orderis msd +

∑Lj=1 mrjd, which translates to full diversity in the

number of relays and fading parameters.The derived PEP and outage probability expressions can

benefit practical system designs where Monte Carlo simulationsare very lengthy and limited, i.e., in the high-SNR region topredict the system performance or in a vast network (highnumber of cooperating relays are deployed). In this paper,


Fig. 2. Multihop multibranch cooperative diversity system.

it is noted that the relay nodes do not listen to the entirecodeword (systematic + parity bits), which can be implementedin practice by adjusting the puncturing window.

V. EXTENSION TO MULTIHOP MULTIBRANCH CODED

COOPERATIVE DIVERSITY

The system under study can be extended to a more generalscenario with multiple relays and multiple hops. In the sequel,a brief discussion of such a system will be given. A systemwith L+ 1 branches can be considered, i.e., {B0, B1, . . . , BL},where B0 corresponds to the direct transmission path betweenthe source and the destination. The remaining L branches{B1, . . . , BL} consist of NL relays. This scenario is shown inFig. 2, where we assume without loss of generality that eachbranch is composed of the same number of hops. The relaysin each branch forward to the next relay in the same hop ordestination, provided that they have successfully decoded thepackets transmitted by the preceding relays. Otherwise, theunsuccessful relays remain silent, and the corresponding branchis not considered at the destination. Furthermore, to avoidinterrelay interference, time-division multiple access (TDMA)can be used. At the destination, MRC is employed to combinethe signals arriving from reliable branches.

VI. NUMERICAL RESULTS

In what follows, we present the numerical results for bit errorand outage probabilities of the proposed transmission scheme.In our simulations, we consider an eight-state (1, 17/13) turbocode in octal form with overall code rate Rc = 1/3 and framesize K = 128. Simulation results are also presented to assessthe accuracy of the analytical results. We assume that allthe long-term average SNRs are equivalent, i.e., γ̄sd = γ̄sr1 =· · · = γ̄srL = γ̄r1d = · · · γ̄rLd = γ̄, and the fading coefficientsare fixed for the duration of an entire frame and vary indepen-dently from one frame to another.

Fig. 3 shows a comparison between the simulated BER andthe corresponding union bounds of the proposed scheme for

Fig. 3. BER comparison of (solid line) simulated BER and bounds (dashedline) of distributed turbo-coded cooperation over Nakagami-m fading channels.L = 1, and msd = msr = mrd = m.

Fig. 4. BER comparison of (solid line) simulated BER and (dashed line)bounds of distributed turbo-coded cooperation over Nakagami-m fading chan-nels. L = 2, and msd = msri = mrid = m, where i ∈ {1, 2}.

a single-relay scenario and i.i.d. fading parameters. It can benoted that the analytical BER confirms the simulated resultsfor the three cases presented here, i.e., m = 0.65, m = 1, andm = 1.5. Furthermore, the results show that diversity in thenumber of relays and fading parameters is achieved. In addition,we note that our analysis presented here yields tight bounds(less than 1 dB) for the different fading parameters.

Fig. 4 shows the BER performance comparison of the unionbounds on the BER and Monte Carlo simulations for L = 2

Pout ≈ γ̄−(msd+

∑L

j=1mrjd

)L∏

j=1

1Γ(mrjd)

a∫0

mmsd

sd γmsd−1sd

Γ(msd)m

mrjd−1

rjd

(2Rc/(1−α)

(1 + γsd)1/(1−α)− 1

)mrjd

dγsd

︸︷︷︸Λ

, (39)


Fig. 5. BER depending on various relay locations (x1, x2). x1 and x2 denotethe distances between the source and relays 1 and 2, respectively. msd =msrj = mrjd = 0.65, where j ∈ {1, 2}.

and different fading parameters as in Fig. 3. All the averagesubchannels SNRs are assumed to be identical. It can benoted that the union bounds on the BER are in agreementwith the simulated BER for different fading environments. Theanalytical methodology presented here validates our simulatedBER since it also yields tight bounds. We also note that fulldiversity order is achieved and is dependent on the number ofcooperating relays and fading indices.

The system model described earlier, i.e., E〈|hij |2〉 = 1where i ∈ {s, rn} and j ∈ {rn, d}, is unrealistic from a prac-tical point of view. Generally speaking, a more practicalscenario considers the effects of path loss. For simplicity, a linetopology (where all nodes are located on a line) is considered.In this scenario, E〈|hsd|2〉 = 1, E〈|hsrn |2〉 = (dsd/dsrn)

η , andE〈|hrnd|2〉 = (dsd/drnd)

η , where dsd, dsrn , and drnd are thedistances between the source and the destination, the sourceand the relays, the relays and the destination, respectively. dsdis normalized to 1 for simplicity, i.e., dsrn = l, drnd = 1 − l,and η is the path-loss coefficient of the wireless channel.Fig. 5 shows the BER performance depending on various relaypositions l = {0.2, 0.5, 0.7} for the proposed scheme. It can benoticed that the location of the relays relative to the source orthe destination greatly affects the system performance. We setη = 3 without loss of generality, and we consider L = 2 andthe different cases where the relays are closer to the sourceor the destination and at midpoint between the source and thedestination for m = 0.65. We denote (x1, x2) as the locationsof the relay nodes 1 and 2 relative to the source, respectively. Itcan be observed that, as one or both of the relays move closerto the destination, the BER performance degrades due to thepoor source-to-relay link as it becomes less reliable and, hence,becomes more prone to errors. It is also worth noticing that allthe curves in this figure have the same slope (i.e., same diversityorder), regardless of the location of the relays, and only SNRloss is incurred due to path loss.

Fig. 6 shows the outage probability for various numbersof relays L = {0, 1, 2, 3} and msd = mr1d = · · · = mrLd = 1,

Fig. 6. Outage probability for cooperative turbo-coded system with differentnumbers of relays L over Nakagami-m fading channels. msd = msrj =

mrjd = 1, where j ∈ {1, 2, 3}.

Fig. 7. Outage probability for cooperative turbo-coded system with differentm values and L = 2.

which is equivalent to Rayleigh fading channels, where L = 0corresponds to the non-cooperative case. We note a substantialgain as the number of cooperating relays increases. Moreover, itcan be also noted that our analytical results are quite accurate.This can be also shown in Fig. 7, where we set the numberof relays L = 2 and msd = mrjd �= msrj . We also observe theeffects of the fading parameters on the system performance aslarger m improves the outage probability. Full diversity in thenumber of relays and/or corresponding fading parameters fromrelays to destination and source to destination is achieved, aspredicted from our analysis.

In Fig. 8, we present the outage probability for non-identicalsource–destination and relay–destination Nakagami-m fadingchannels. This scenario is more practical since the links ina cooperative system are not necessarily similar. We showboth the theoretical results and simulation results. It is evidentthat our theoretical results are in good agreement with theMonte Carlo simulations.


Fig. 8. Outage probability for cooperative turbo-coded system with non-identical s−d and r−d Nakagami-m fading channels and L = 3.

Fig. 9. Outage probability versus tradeoff parameter f with identicalNakagami-m fading channels, L = 2, and i ∈ {1, 2}.

Since the cooperation ratio α is a function of f , it is possibleto see how the tradeoff parameter f varies with the outageprobability. In Fig. 9, the effect of the tradeoff parameter fon the outage probability is provided through computer simula-tions. Fig. 9 shows the outage probability with different valuesof f when all the average SNRs are set to 10 dB with i.i.d.Nakagami-m channels and L = 2. In this figure, we note thatthe value of f that minimizes the outage probability is aboutf = 0.3. However, we noted that the optimal tradeoff parameterf slightly varies with the number of cooperating relays. It canbe noted that as f → 1, which has no practical meaning, theoutage probability increases. This can be explained as, whenf = 1, the decoding relays dedicate little time to retransmissionas they spend all their time listening to the source (the entirecodeword is received). In this case, the system is similar toa non-cooperative case; hence, the performance deteriorates.Fig. 10 corroborates with Fig. 9 that shows f = 0.3 as theoptimal value. Moreover, it can be observed in Fig. 10 thatf = 1, which is similar to a non-cooperative scenario.

Fig. 10. Outage probability for various values of f with identicalNakagami-m fading channels, m = 1.5, and L = 2.

VII. CONCLUSION

We considered a distributed turbo-coded cooperation withmultiple relays over Nakagami-m fading channels. To avoiderror propagation, we used an adaptive scheme in which onlythe reliable relays forward to the destination during the secondphase. Both the analytical expressions of the bit error and out-age probabilities were derived. For the BER, we used the limit-before-average and transfer bounding techniques to providetight bounds. The analytical work of the proposed scheme interms of both bit error probability and outage probability waspresented and proven to be accurate. Finally, using the derivedBER and outage probability, we examined the asymptotic per-formance of the system where we showed that full diversity inthe number of cooperating relays and/or fading parameters canbe achieved.

APPENDIX

We first use the alternative representation of the lower incom-plete gamma function [25, Eq. (8.352.4)] given by

γ

(mrjd,

mrjd

γ̄rjd

(2Rcβ

(1 + γsd)β− 1

))= (mrjd − 1)!

×[

1 − exp

(mrjd

γ̄rjd

(1 − 2Rcβ

(1 + γsd)β

))

×mrjd

−1∑n=0

(mrjd/γ̄rjd)n

n!

(2Rcβ

(1 + γsd)β− 1

)n].

(40)

Using (40) in (35) and after some manipulations, I(μ, ν, β) canbe written as in (41), shown at the top of the next page. I1 canbe easily obtained from [25, Eq. (3.351.1)] and given by

I1 = ν−msdγ(msd, ν(1 − 2Rc)

). (42)


I(μ, ν, β) =

M∏j=1

mmsd

sd

Γ(msd)γ̄sd

⎡⎢⎢⎢⎢⎢⎣

2Rc−1∫0

γμsd exp(−νγsd)dγsd

︸︷︷︸I1

−2Rc−1∫0

γμsd exp(−νγsd) exp

⎛⎝mrjd

γ̄rjd−

2Rcβmrjd

γ̄rjd

(1 + γsd)β

⎞⎠mrjd

−1∑n=0

mnrjd

n!γ̄nrjd

(2Rcβ

(1 + γsd)β− 1

)n

︸︷︷︸I2

⎤⎥⎥⎥⎥⎥⎦ (41)

To obtain I2, we expand its form in (41), apply the binomialexpansion (1 − x)k =

∑kj=0

(kj

)(−1)jxj , and perform some

algebraic manipulations that yield

I2=exp

(mrjd

γ̄rjd+1

)mrjd−1∑

n=0

n∑m=0

(−1)n+mmnrjd

n!γ̄nrjd

(n

m

)2Rcβm

×2Rc−1∫0

γμsd exp

⎛⎝−νγsd −

2Rcβmrjd

γ̄rjd

(1 + γsd)β

⎞⎠ 1

(1 + γsd)βdγsd.

(43)

Using exp(−x) =∑∞

k=0((−1)k/k!)xk, applying the bino-mial expansion (g + h)k =

∑kj=0

(kj

)gk−jhj , and after some

manipulations, (43) is given by

I2=exp

(mrjd

γ̄rjd+1

)mrjd−1∑

n=0

n∑m=0

(−1)n+mmnrjd

n!γ̄nrjd

(n

m

)2Rcβm

×∞∑

k=0

k∑l=0

(−1)k

k!

(k

l

)2Rcβ(k−l)

(mrjd

γ̄rjd

)k−l(msd

γ̄sd

)l

×2Rc−1∫0

γμ+lsd

(1 + γsd)β(k−l)dγsd. (44)

Using [25, Eq. (3.194.1)] in (44) and the fact that 2F1(a, b;c;x) = 2F1(b, a; c;x), I2 can be obtained as

I2=exp

(mrjd

γ̄rjd+1

)mrjd−1∑

n=0

n∑m=0

(−1)n+mmnrjd

n!γ̄nrjd

(n

m

)2Rcβm

×∞∑

k=0

k∑l=0

(−1)k

k!

(k

l

)2Rcβ(k−l)

(msd

γ̄sd

)l

×(mrjd

γ̄rjd

)k−l(2Rc − 1)msd+l

msd + l

× 2F1(msd + l, β(j − l);msd + l − 1; 1 − 2Rc), (45)

where 2F1(b, a; c;x) denotes the Gauss hypergeometric func-tion defined in [25, Eq. (9.111)].

Using (45), (42), and (41) in (34) yields the closed-formexpression for the outage probability shown in (36).

ACKNOWLEDGMENT

The authors would like to thank the many helpful commentsand suggestions of the anonymous reviewers and the Editor ofthis journal.

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Jules Merlin Moualeu (S’06) was born inCameroon. He received the M.Sc.Eng. and Ph.D.degrees in electronic engineering from the Universityof KwaZulu-Natal, Durban, South Africa, in 2008and 2013, respectively.

From January 2011 to July 2011, he was a Vis-iting Scholar with Concordia University, Montreal,QC, Canada, under the Canadian CommonwealthScholarship Program (CCSP) offered by the ForeignAffairs and International Trade Canada (DFAIT).He is currently with the Department of Electrical

and Information Engineering, University of the Witwatersrand, Johannesburg,South Africa, as a Postdoctoral Fellow. His current research interests includedigital and wireless cooperative communications (physical layer), coding the-ory, space–time coding, and multiple-input multiple-output systems.

Walaa Hamouda (SM’06) received the M.A.Sc. andPh.D. degrees in electrical and computer engineeringfrom Queen’s University, Kingston, ON, Canada, in1998 and 2002, respectively.

Since July 2002, he has been with the Departmentof Electrical and Computer Engineering, ConcordiaUniversity, Montreal, QC, Canada, where he is cur-rently an Associate Professor. Since June 2006, hehas been the Concordia University Research Chairof Communications and Networking. His currentresearch interests include multiple-input multiple-

output space–time processing, cooperative communications, wireless networks,multiuser communications, cross-layer design, and source and channel coding.

Dr. Hamouda served or is serving as the Technical Cochair of the WirelessNetworks Symposium; the 2012 Global Communications Conference; theAd hoc, Sensor, and Mesh Networking Symposium of the 2010 InternationalCommunications Conference (ICC); and the 25th Queen’s Biennial Symposiumon Communications. He also served as the Track Cochair of the Radio AccessTechniques of the 2006 IEEE Vehicular Technology Conference (IEEE VTCFall 2006) and of the Transmission Techniques of the IEEE VTC Fall 2012.From September 2005 to November 2008, he was the Chair of the IEEEMontreal Chapter for Communications and Information Theory. He serves asan Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECH-NOLOGY, the IEEE COMMUNICATIONS LETTERS, and IET Wireless SensorSystems. He has received numerous awards, including the Best Paper Awardfrom ICC 2009 and the IEEE Canada Certificate of Appreciation in 2007and 2008.

HongJun Xu (M’07) received the B.Sc. degree fromthe University of Guilin Technology, Guilin, China,in 1984; the M.Sc. degree from the Institute of Tele-control and Telemeasure, Shi Jian Zhuang, China,in 1989; and the Ph.D. degree from the BeijingUniversity of Aeronautics and Astronautics, Beijing,China, in 1995.

From 1997 to 2000, he was a Postdoctoral Fellowwith the University of Natal, Durban, South Africa,and Inha University, Incheon, Korea. He is currentlya Professor with the School of Engineering, Univer-

sity of KwaZulu-Natal, Durban. He is the author of more than 20 journal papers.His research interests include digital and wireless communications and digitalsystems.

Fambirai Takawira (M’96) received the B.Sc.degree in electrical and electronic engineering(first-class honors) from Manchester University,Manchester, U.K., in 1981 and the Ph.D. degree fromCambridge University, Cambridge, U.K., in 1984.

In February 2012, he joined the University ofthe Witwatersrand, Johannesburg, South Africa, af-ter 19 years with the University of KwaZulu-Natal(UKZN), Durban, South Africa. At UKZN, he heldvarious academic positions, including Head of theSchool of Electrical, Electronic, and Computer En-

gineering, and just before his departure, he was the Dean of the Faculty ofEngineering. He has also held appointments with the University of Zimbabwe,Harare, Zimbabwe; the University of California San Diego, San Diego, CA,USA; British Telecom Research Laboratories; and National University ofSingapore, Singapore. His research interests include wireless communicationsystems and networks.

Dr. Takawira is a past Editor for the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS. He has served on several conference organizing commit-tees. He is currently serving as the Communications Society Director of theEurope, Middle East, and Africa regions for the 2012–2013 term.