2362 ieee transactions on wireless communications, …hamouda/jules_twt... · 2015. 1. 29. · 2362...

2362 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014

Relay Selection for Coded Cooperative Networkswith Outdated CSI over

Nakagami-m Fading ChannelsJules Merlin Moualeu, Member, IEEE, Walaa Hamouda, Senior Member, IEEE,

and Fambirai Takawira, Member, IEEE

Abstract—In this paper, we consider relay selection for turbocoded cooperative networks subject to Nakagami-m fading whenthe channel state information (CSI) is known at the receiver but isnot necessarily ideal. This non-ideality may be due to a feedbackdelay caused by the difference between the instantaneous CSIduring the transmission and the CSI at the time of relay selection,resulting in outdated CSI phenomena. The impact of the outdatedCSI on the proposed scheme is well investigated. A closed-formexpression for the exact outage probability is derived as wellas its asymptotic expression in the high signal-to-noise (SNR)regime. Moreover, upper bounds on the bit-error rate (BER) arepresented and a study of the diversity order reveals that for idealCSI, full diversity in the number of relays and fading parametersm is achieved as opposed to outdated CSI where the achievablediversity is equivalent to the diversity of a coded cooperativenetwork with a single relay.

Index Terms—Coded cooperation, diversity order, error rate,selection relaying, turbo codes.

I. INTRODUCTION

SELECTION relaying is a solution to the inefficient uti-lization of channel resources since it only requires two

orthogonal channels regardless of the number of relays. Relayselection schemes in single-hop wireless relaying networkshave extensively been investigated (See [1] – [5] and thereferences therein). Among these selection techniques, oppor-tunistic relay selection (ORS) [1] is a viable strategy fromthe implementation point of view, due to its low complexity.In time-varying channels, implementing relay selection maycause frequent relay switchings which can be detrimental tothe overall system performance. In [6], the authors used analternative to ORS to study the rate at which the switchingof a selected relay occurs in practice. Moreover, frequentrelay switchings may cause synchronization issues due tothe repeated initializations of the system each time a relayis selected. Hence, this leads to an increase in implemen-tation complexity and poor system performance. So far, the

Manuscript received November 1, 2013; revised January 12, 2014; acceptedMarch 4, 2014. The associate editor coordinating the review of this paper andapproving it for publication was L. Song.

The work of J. M. Moualeu and F. Takawira was supported in part by theCentre of Excellence in Telecommunications Access and Services (CeTAS).The work of W. Hamouda was supported in part by the Natural Sciences andEngineering Research Council of Canada (NSERC) grant 00861.

J. M. Moualeu and F. Takawira are with the Department of Electricaland Information Engineering, University of the Witwatersrand, Private Bag3, Wits 2050, Johannesburg, South Africa (e-mail: {jules.moualeu, fambi-rai.takawira}@wits.ac.za).

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

Digital Object Identifier 10.1109/TWC.2014.040214.132027

relay selection schemes presented, have dealt with uncodedcooperative relaying networks. In coded cooperation [7], [8],relay selection is seldom studied in the literature. For examplein [9], [10], Elfituri et al. studied an antenna/relay selectionin coded cooperation using convolutional codes. In general,CSI is used to accomplish relay selection and is assumed tobe known at the receiver.

The aforementioned studies consider an ideal CSI for relayselection which is impractical in real scenarios. From apractical point of view, the CSI at the time of transmissionmay be outdated due to a delayed feedback since the relayselection and data transmission instants can differ due tochannel conditions. In recent works, the effect of outdatedCSI on relay selection has been investigated (for e.g., [11 –[17]). In [11], the authors analyzed the behavior when CSIis subject to delay in the feedback channel for a decode-and-forward (DF)-based protocol over Rayleigh fading channels.In [12], the authors investigated the impact of outdated CSIfor relay selection in AF cooperative relaying under bothpartial and opportunistic relay selection schemes. Chen etal. [13] proposed a novel multiple relay selection (MRS)scheme to combat the severe diversity loss. Both amplify-and-forward (AF) and DF are considered in which the Nbest relays are opportunistically selected under outdated CSI.In [14], partial relay selection (PRS) and best relay selectionwith delayed CSI are studied for AF wireless relaying. Thework in [15] investigated the impact of channel estimationerrors and feedback delay in DF with relay selection andSuraweera et al. [16] analyzed the effect of outdated CSIon the performance of AF with the kth worst partial relayselection. In [18], the authors considered an AF cooperativesystem with direct transmission and analyzed the Shannoncapacity of the proposed scheme under outdated CSI. Theperformance analysis of the fixed-gain AF relay systems withinterference and thermal noise at the relay and destination wasstudied in [19]. However, in the above-mentioned works theimpact of outdated CSI on AF/DF cooperative networks hasbeen investigated for uncoded systems. So far, no work in theliterature considers the effects of outdated CSI on coded co-operative relaying networks. Moreover, the impact of outdatedCSI for cooperative relaying systems have been investigatedin fewer works under more general fading scenarios such asthe Nakagami-m fading (See [20] – [22]). However, thesestudies were undertaken for uncoded systems. In [20], theauthors considered an uncoded DF cooperative system andderived the exact closed-form expression of the outage prob-ability as a function of the correlation between the estimated

1536-1276/14$31.00 c© 2014 IEEE

MOUALEU et al.: RELAY SELECTION FOR CODED COOPERATIVE NETWORKS WITH OUTDATED CSI OVER NAKAGAMI-M FADING CHANNELS 2363

and actual channel values. The work in [21] evaluated thetransmission quality of a channel under Nakagami-m fadingwith both fading statistics and outdated CSI. Ferdinand etal.[22] investigated the uncoded AF with PRS over Nakagami-m fading channels with a feedback delay for both variable-gain and fixed-gain relay. However, [20] and [21] only studiedthe outage probability and no BER analysis was investigated.In [23], Wang et al. considered a multi-hop system in orderto address the issue of multi-user interference and designefficient transmission. In their work, they studied the sumdegrees of freedom (DoF) of a system in which a K-antennasource intends to communicate with K destination nodesthrough multiple layers each consisting of K full-duplexsingle-antenna relays with delayed channel state informationat transmitters (CSIT). It was shown that by treating themulti-hop multiple-input multiple-output (MIMO) broadcastnetwork as an entity, better maximum multiplexing gain couldbe achieved over the cascade approach with individual single-hop. However, outdated CSIT can be detrimental to the DoFgain. In [24], it was shown that by properly designing aninterference alignment (IA) scheme, the maximum multi-plexing gain achieving the sum DoF of a K-user MIMObroadcast channel could be attained with outdated CSI. In [25],the authors proposed a novel IA scheme at the relays thatcombines perfect delayed CSIT and imperfect instantaneousCSIT to achieve higher sum DoF for a two-hop multiple-input single output (MISO) system. In [26], the authorsinvestigated the design of efficient network codes for multi-user multi-relay wireless networks with slow fading channels.A non-binary network code construction based on maximumdistance separable (MDS) codes was proposed in order toachieve maximum diversity order for an arbitrary number ofsources and relays. Xiao and Skoglund [27] proposed a multi-user cooperative wireless networking system based on linearnetwork codes. The use of diversity network codes (DNCs)was also proposed in order to exploit in an efficient mannerthe diversity available by time-varying fading and cooperation.Eid et al. [28] studied a multi-relay coded cooperation forasynchronous direct-sequence code-division multiple-access(DS-CDMA) systems over slow fading channels. They showedthat by suppressing multi-user interference at the cooperativeend, the full benefits of coded cooperation can be achieved.

To the best of our knowledge, the study of relay selectionin coded cooperation taking into account outdated CSI overNakagami-m fading channels is not presently available in theliterature. To fill this void, we investigate the impact of out-dated CSI for relay selection in turbo-coded cooperation overNakagami-m fading channels. We consider a turbo coded sys-tem with a transmission scheme different from the traditionalturbo coded system. We derive a closed-form solution of theoutage probability for the scheme under study over Nakagami-m fading channels with any fading figure. Furthermore, a high-SNR approximation of the exact outage probability is derivedto evaluate the diversity order and is shown to be dependenton the correlation factor ρ. The system performance of theproposed system is also investigated through the pairwiseerror probability (PEP) analysis. A closed-form expressionof the PEP for all values of the fading parameter m (integerand non-integer) is provided and its asymptotic expression athigh SNR is examined to assess the diversity order of the

Fig. 1. Cooperative system model.

scheme for ideal and delayed CSI. We then provide a BERperformance analysis obtained by using the transfer functionbounds method and the limit-before-average-technique.

The remainder of the paper is organized as follows. InSection II, we describe the system model. In Section III,we analyze the outage probability of the proposed schemefollowed by the diversity analysis in Section IV. In SectionV, the BER analysis is presented. This includes the derivationof the closed-form expression of the PEP and its bahaviour athigh SNR. In Section VI, the diversity-multiplexing tradeoff(DMT) of the underlying scheme is discussed. Numericalresults are presented in Section VII. Finally conclusions aredrawn in Section VIII.

II. SYSTEM MODEL

We consider the cooperative relaying system illustrated inFig. 1, which consists of a single source node s communi-cating with a single destination node d with the help of Lrelay nodes denoted by ri, i = 1, · · · , L. The relays operatein half-duplex mode, i.e., they cannot receive and transmitsimultaneously. All nodes are equipped with a single transmitand/or receive antenna. It is also assumed that the fading co-efficients in the source-to-relay (s–r) and relay-to-destination(r–d) links are independent and identically distributed (i.i.d.),hence the subscript i denoting the relay number can be leftout in the subsequent analysis.

The cooperative transmission takes place in two phases.During the first phase, the source encodes a K-bit message bya turbo code of rate Rc and broadcasts the generated codewordof length N to the relays and destination. We assume that thedestination listens to the entire codeword whereas the relaysonly listen to a fraction of the entire codeword 1.

We consider that the relays only listen to a noisy versionof the systematic bits before attempting to decode. The re-ceived signals at the destination and the ith relay are givenrespectively by

ysd(1 : N) =√Pshsdx(1 : N) + nsd(1 : N), (1)

ysri(1 : K) =√Pshsrix(1 : K) + nsri(1 : K), (2)

where ri represents the ith relay, the vector x(1 : N)={x(1),· · · , x(K), x(K+1), · · · , x(N)} is binary phase shift keying

1In [29], an approach in which pre-defined phase durations for broadcastand cooperation commonly presented in the literature is proposed. In thisapproach, the time allocated for the relay to listen could vary depending onthe relay received channel. Hence, in practical scenarios, it is possible thatrelays only listen to a fraction of the codeword whereas the destination listensto the entire codeword.


(BPSK) modulated, with x(1 : K) denoting the systematic bitsand x(K+1 : N) representing the parity bits, hsd and hsri arethe Nakagami-m fading coefficients for the source-destinationand source-relays links respectively, with unit variance, Ps isthe transmit power at the source for the s−d and s− ri links,nsd(1 : N) and nsri(1 : K) represent the i.i.d. additive whiteGaussian noise (AWGN) modeled as CN (0, N0/2).

All the relays employ a turbo iterative decoder to estimatethe information sent by the source and a cyclic redundancycheck (CRC) code for error detection. In the second phase,the relay with the highest relay-destination instantaneous SNRbelonging to the decoding set (the set of relays that havesuccessfully decoded the source message), is selected toforward the parity bits to the destination. The received signalat the destination is given by

yrkd(K+1 : N) =√Prhrkdx(K+1 : N)+nrkd(K+1 : N),

(3)where the vector x(K + 1 : N) denotes the estimated paritybits, hrkd is the fading coefficient for the best relay-destinationlink and is subjected to a Nakagami-m distribution with unitvariance, Pr is the transmit power at the best relay node,nrkd(K + 1 : N) ∼ CN (0, N0/2) and

k = arg maxj∈D(s)

{γrjd}, (4)

where D(s) represents the decoding set and γrjd is theinstantaneous SNR at the selection instant, which can bedifferent from the actual SNR γrjd (used for retransmission)due to the time delay in the feedback channel. By definition,γrjd = |hrjd|2γ, where hrjd is an outdated version of hrd.

Both hrjd and hrjd are jointly Gaussian RV, and accordingto [11], hrjd conditioned on hrjd follows a Gaussian distri-bution given by

hrjd|hrjd ∼ CN(ρhrjd,

√1− ρ2

), (5)

where ρ is the correlation factor between hrjd and hrjd,modeled according to Jakes’ autocorrelation given by [30]

ρ = J0(2πfd,rjdTd

), (6)

with J0 denoting the zeroth order Bessel function of the firstkind, fd,rjd is the maximum Doppler frequency on the rj − dlink, and Td is the time difference between the actual channelvalue and its estimate.

III. OUTAGE PROBABILITY ANALYSIS

Outage probability represents an important performancemetric in wireless communications and is defined as theprobability that an instantaneous capacity C falls below atarget rate Rc. It can be mathematically formulated as

Pout = Pr{C(γ) < Rc} = Pr{log2(1 + γ) < Rc

}=

∫ 2Rc−1

0

p(γ)dγ, (7)

where γ and Pr{x} denote the instantaneous SNR and theprobability of x respectively and p(x) denotes the probabilitydensity function (PDF) of x.

In the proposed scheme, the destination listens to the entirecodeword transmitted by the source with a code Rc. The

relays listen only to the systematic bits sent by the sourcewith a code rate R1 = Rc

α and only the best relay forwardto the destination with a code rate R2 = Rc

1−α , whereα = K/N represents the cooperation ratio. The end-to-endoutage probability can be represented by

Pout = Pr{γsd < 2Rc − 1

}(Pr{γsr < 2Rc/α − 1

})L

+

L∑ϑ=1

(L

ϑ

)(Pr{γsr < 2Rc/α − 1

})L−ϑ

×(

Pr{γsr > 2Rc/α − 1

})ϑ

× Pr{(

1 + γsd)(1 + γrd

)1−α< 2R

}, (8)

where ϑ = |Θ| with Θ representing the set of indices of coop-erating relays given by Θ = {j1, j2, · · · , jϑ} ⊂ {1, 2, · · · , L}and |x| denotes the the cardinality of x. The first part of (8)represents D(s) = ∅ (with ∅ denoting the empty set), i.e.,all the relays in network are unreliable, whereas the secondexpression of (8) represents the case where at least one relayis reliable.

Following (7), the expression in (8) can be rewritten as

Pout =

(∫ 2Rc−1

0

p(γsd)dγsd

)(∫ 2Rc/α−1

0

p(γsr)dγsr

)L

+L∑

ϑ=1

(L

ϑ

)(∫ 2Rc/α−1

0

p(γsr)dγsr

)L−ϑ

×{(

1−∫ 2Rc/α−1

0

p(γsr)dγsr

)ϑ

×∫ a

0

∫ b

0︸︷︷︸A

p(γsd)p(γrd)dγrddγsd

},

(9)

where p(γsd) and p(γsr) denote the PDF of the RVs γsd andγsr, respectively. For Nakagami-m fading channels, p(γij) =m

mijij γmij−1

Γ(mij)γmij exp

(−mijγij

γij

)where {i, j} ∈ {s, (r, d)}, and A

corresponds to the region of integration given by

A{(γsd, γrd)|γsd ≥ 0, γrd ≥ 0, (1+γsd) (1 + γrd)

1−α< Rc

}.

(10)Considering (10), we can rewrite the constraint A as γrkd =

f(γsd)

given by

γrd <2Rc/(1−α)(

1 + γsd)1/(1−α)

− 1 � b. (11)

Since γrd > 0, we can easily obtain

γsd < 2Rc − 1 � a. (12)

However, the PDF of γrd, i.e., pγrkd(γrd) is not straighforward

since it involves relay selection based on outdated CSI andcan be obtained using the cumulative density function (CDF)derived in [20] by using the fact p(x) = ∂F(x)/∂x


pγrkd(γrd) = ϑ

∞∑n=0

ρnmmrd+nrd γmrd+n−1

rd γ−(mrd+n)

n!(1− ρ)mrd+2nΓ(mrd)Γ(mrd + n)

× exp

(− mrdγrd(1− ρ)γ

) ϑ−1∑j=0

(ϑ− 1

j

)(−1)j

× χ(mrd + n, ρ, j),

(13)

where

χ(mrd + n, ρ, j) =Γ(j + 1)(

11−ρ

+ j)mrd+n

×∑

τ0=τ1=···=τmrd−1=0

τ0+τ1+···+τmrd−1=j

mrd−1∏i=0

(1

i!(

11−ρ

+j)i

)τi

Γ(τi + 1)

⎞⎟⎟⎟⎟⎠

×(mrd + n− 1 +

mrd−1∑i=0

iτi

)!

(14)

Substituting (13) in (9), the resulting outage probabilityexpression Pout also contains a double integral as in (9), thatis not easy to evaluate. In what follows, we evaluate the doubleintegral expression (contained in the new outage probabilityexpression after substituting (13) in (9)) denoted by Id whichcan be written as

Id = ϑ

(∫ a

0

∫ b

0

mmsd

sd γmsd−1

Γ(msd)γmsdexp

(−msdγsd

γsd

)

×∞∑

n=0

ρnmmrd+nrd γmrd+n−1

rd

n!(1− ρ)mrd+2nΓ(mrd)Γ(mrd + n)γmrd+n

× γmrd+n−1rd exp

(− mrdγrd(1− ρ)γrd

)dγrddγsd

)

×ϑ−1∑j=0

(−1)jχ(mrd + n, ρ, j).

(15)

After some manipulations and evaluating the inner integralusing [31, Eq. 3.351.1], we obtain the following expressionshown at the top of the next page.

In order to evaluate the integral in (16), the alternative formof the lower incomplete Gamma function given in [31, Eq.8.352.4] by γ(a, x) = (a−1)!

(1− exp(−x)

∑n−1m=0

xm

m!

)can

be used and the expression in equation (16) can be rewrittenas shown in equation (17) on the next page.

The expression in (17) can be expanded using the binomialexpansion (1 − y)m =

∑mk=0

(mk

)(−1)kyk, to give (in (18)

as shown on the next page) where we define ν = mrd

(1−ρ)γrd

and β = 11−α . In (18), the inner integral can be evaluated

using [31, Eq. 3.351.1], whereas the outer integral can berearranged as shown in (19).Examining the exponential function in (19), one can rewriteit as shown in (20).After substituting (20) in (19) and performing some manipu-lations, (19) can further be expressed as in (21).Finally, using [31, Eq. 3.194.1] in (21) and substituting in (18),the outage probaility is given by (22), where 2F1 (w, x; y; z) is

the Gauss Hypergeometric function defined in [31, Eq. 9.111].The derived outage probability in (22) includes a series form(infinite series) that is not practical for numerical computation.Through some numerical simulations, it is noted that thederived expression converges after a finite number of terms.

IV. DIVERSITY ANALYSIS

In order to find useful insights into the diversity order doof our scheme, an easy to analyze expression of the outageprobability provided in (22) is needed. In what follows, anexpression of the asymptotic approximation of the outageprobability is derived. We consider that γsd = γsr = γrd =γ → ∞ and without loss of generality, we assume thatmsrj = msr, mrjd = mrd since the fading in all therespective links is i.i.d.

At high SNR, the outage probability in (8) can be reducedto

Pout ≈L∑

ϑ=1

Pr{(

1 + γsd)(1 + γrd

)1−α< 2R

}. (23)

Corollary 1: The diversity gain of the coded cooperativesystem with outdated CSI over Nakagami-m fading is givenby

do =

{msd +mrd, if ρ < 1msd + Lmrd, if ρ = 1

(24)

Proof: The proof is provided in the Appendix.

V. UPPER BOUNDS ON THE BIT ERROR PROBABILITY

In this section, we derive upper bounds on the BER for theproposed scheme under outdated CSI. To this end, we firstevaluate an exact expression for the unconditional PEP usedto derive the BER expression of a coded system. It is worthnoting that it is possible to get some insight of the systemthrough the PEP analysis. In addition, we obtain asymptotican expression of the derived PEP at high SNR to determinethe diversity order of the scheme under consideration.

A. Pairwise Error Probability

Assuming slow fading channel and perfect knowledge ofthe CSI at the receivers with the effects of delayed feedback,the end-to-end conditional PEP is given by

P (dH |γsd, γsr, γrkd) = Q(√

2dHγsd

)(Q(√

2d1γsr

))L+

L∑ϑ=1

(L

ϑ

)(Q(√

2d1γsr

))L−ϑ(1−Q

(√2d1γsr

))ϑ

×Q(√

2dHγsd + 2d2γrd

), (25)

where Q(•) denotes the Gaussian Q-function, dH = d1 + d2is the Hamming distance between the transmitted codeword cand the erroneous codeword c, d1 is the Hamming distancecorresponding to the frame in the s − r links and d2 is theHamming distance corresponding to the frame in the rk − dlink. Moreover, similar to (8), the first term in (25) representsthe case where all relays are unreliable, whereas the secondterm indicates that at least one relay is reliable.


Id =

∞∑n=0

ϑρnmmrd+nrd msd

n!(1− ρ)mrd+2nΓ(mrd)Γ(mrd + n)γmrd+nΓ(msd)γsd

(mrd

(1− ρ)γrd

)−mrd−n

×(∫ 2Rc−1

0

γ

(mrd + n,

2Rc/(1−α)mrd

(1− ρ)(1 + γsd)1/(1−α)γrd

− mrd

(1− ρ)γrd

)γmsd−1sd

× exp

(−msdγsd

γsd

)dγsd

)ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j).

(16)

Id =

∞∑n=0

ϑρnmmrd+nrd msd


(mrd

(1 − ρ)γrd

)−mrd−n

×(∫ 2Rc−1

0

(mrd + n− 1)!

[1− exp

(mrd

(1 − ρ)γrd

(1− 2Rc/(1−α)

(1 + γsd)1/(1−α)

))mrd+n−1∑m=0

1

m!

×(

mrd

(1− ρ)γrd

)m(2Rc/(1−α)

(1 + γsd)1/(1−α)

)m]

dγsd

)ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j).

(17)

Id =∞∑

n=0

ϑρnmmrd+nrd msd


(mrd

(1 − ρ)γrd

)−mrd−n[∫ 2Rc−1

0

Γ(mrd + n)

× γmsd−1sd exp

(−msdγsd

γsd

)dγsd −

∫ 2Rc−1

0

Γ(mrd + n)γmsd−1sd exp

(ν

(1− 2Rcβ

(1 + γsd)β

))

×mrd+n−1∑

k=0

νk

k!

k∑l=0

(k

l

)(−1)l−k 2Rclβ

(1 + γsd)lβ

dγsd

]ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j),

(18)

Id =

∞∑n=0

ϑρnmmrd+nrd msd


(mrd

(1− ρ)γrd

)−mrd−n[Γ(mrd + n)

×(msd

γsd

)−msd

γ

(msd,

msd

γsd

(2Rc − 1

))− Γ(mrd + n) exp(ν)

mrd+n−1∑k=0

k∑l=0

νk

k!

(k

l

)(−1)l−k2Rclβ

×∫ 2Rc−1

0

γmsd−1sd

(1 + γsd)lβ

exp

(−msdγsd

γsd

− 2Rcβ

(1 + γsd)β

)dγsd

]ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j),

(19)

exp

(−msdγsd

γsd

− 2Rcβ

(1 + γsd)β

)=

∞∑r=0

(−1)r

r!

(msdγsdγsd

+2Rcβ

(1 + γsd)β

)r

=

∞∑r=0

r∑s=0

2Rcβs(−1)r

r!

(r

s

)(msd

γsd

)r−sγr−ssd

(1 + γsd)βs

.

(20)

Using the alternative representation of the Gaussian Q-function [32] in (25) and averaging over the fading distri-bution, and after some manipulations with the aid of themoment generating function (MGF) defined in [33], yields aclosed-form expression of the end-to-end unconditional PEPfor integer values of the fading figure m. To derive an exactexpression for the average PEP for both integer and non-

integer values of fading parameter m, we use an accurateand simple approximate expression given by the sum of twoexponential functions, in which erfc(x) ≈ 1

6 exp(−x2

)+

12 exp

(− 43x

2)

[34]. Therefore, the resulting Gaussian Q-function is given by

Q(x) =1

12exp

(−x2

2

)+

1

4exp

(−2

3x2

). (26)


Id =

∞∑n=0

ϑρnmmrd+nrd msd


(mrd

(1− ρ)γrd

)−mrd−n[Γ(mrd + n)

×(msd

γsd

)−msd

γ

(msd,

msd

γsd

(2Rc − 1

))− Γ(mrd + n) exp(ν)

mrd+n−1∑k=0

k∑l=0

νk

k!

(k

l

)(−1)l−k2Rclβ

×∞∑r=0

r∑s=0

2Rcβs(−1)r

r!

(r

s

)(msd

γsd

)r−s ∫ 2Rc−1

0

γmsd+r−s−1sd

(1 + γsd)β(l+s)

dγsd

]ϑ−1∑j=0

(ϑ− 1

j

)(−1)j

× χ(mrd + n, ρ, j).

(21)

Pout =1

Γ(msd)ΓL(msr)

[γ

(msd,

msd

γsd

(2Rc − 1

))][γ

(msr,

msr

γsr

(2Rc/α − 1

))]L

+

L∑ϑ=1

(L

ϑ

)1

ΓL(msr)

[γ

(msr,

msr

γsr

(2Rc/α − 1

))]L−ϑ[γ

(msrj ,

msrj

γsrj

(2Rc/α − 1

))]ϑ

×∞∑

n=0

ϑρnmmrd+nrd msd

n!(1 − ρ)mrd+2nΓ(mrd)Γ(msd)γrdγsd

(mrd

(1− ρ)γrd

)−mrd−n

×[(

msd

γsd

)−msd

γ

(msd,

msd

γsd

(2Rc − 1

))− exp

(mrd

(1− ρ)γrd

)

×mrd+n−1∑

i=0

i∑l=0

∞∑r=0

r∑s=0

M−1∑j=0

1

i!

(mrd

(1− ρ)γrd

)i(i

l

)(−1)l2Rc/(1−α) (−1)r

r!

(r

s

)(msd

γsd

)r−s

× 2Rcs1−α

(2Rc − 1

)msd−r−s

msd − r − s2F1

(l + s

1− α,msd − r − s;msd − r − s+ 1; 1− 2Rc

)]

×(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j),

(22)

P (dH) =

(∫ ∞

0

[ 1

12exp (−dHγsd) +

1

4exp

(−4

3dHγsd

)]p(γsd)dγsd

)(∫ ∞

0

[ 1

12exp (−d1γsr)

+1

4exp

(−4

3d1γsr

)]p(γsr)dγsr

)L

+

L∑ϑ=1

(L

ϑ

)(∫ ∞

0

[ 1

12exp (−d1γsr)

+1

4exp

(−4

3d1γsr

)]p(γsr)dγsr

)L−ϑ(1−

∫ ∞

0

[ 1

12exp (−d1γsr) +

1

4exp

(−4

3d1γsr

)]

× p(γsr)dγsr

)ϑ ∫ ∞

0

∫ ∞

0

[ 1

12exp (−dHγsd − d2γrd) +

1

4exp

(−4

3(dHγsd + d2γrd)

)]p(γsd)

× p(γrd)dγsddγrd.

(27)

Using (26) in (25) and averaging over the fading distribu-tion, the average PEP is given by (27).

Substituting (13), p(γsd) and p(γsr) given in Section III,in (27), and after performing some manipulations, the exactexpression of the average PEP is given by (28) where thederived closed-form expression in (28) holds for all valuesof msd, msr and mrd. Furthermore, the average PEP derivedin (28) contains some infinite series that are not suited fornumerical computation for practical SNRs. The convergence

of power series is noted through numerical simulations aftera finite number of terms.

In what follows, we derive the asymptotic expression ofthe average PEP in the high SNR regime. For this analysis,it is worth rewriting the average PEP expression using theaforementioned Craig’s formula as (29).

Using the following assumption γsd = γsr = γrd = γ →


P (dH) =mmsd

sd mLmsrsr

γmsd

sd γLmsrsr

(1

12

(dH +

msd

γsd

)−msd

+1

4

(4dH3

+msd

γsd

)−msd)(

1

12

(d1 +

msr

γsr

)−msr

+1

4

(4d13

+msr

γsr

)−msr)L

+

L∑ϑ=1

(L

ϑ

)(1

12

(d1 +

msr

γsr

)−msr

+1

4

(4d13

+msr

γsr

)−msr)L−ϑ

×(

1

12

(d1 +

msr

γsr

)−msr

+1

4

(4d13

+msr

γsr

)−msr)ϑ ∞∑

n=0

ρnmmrd+nrd

n! (1− ρ)mrd+2n

γmrd+nrd

×ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ(mrd + n, ρ, j)

mmsd

sd

γmsd

sd

(1

12

(dH +

msd

γsd

)−msd(d2 +

mrd

(1− ρ) γrd

)−mrd−n

+1

4

(4dH3

+msd

γsd

)−msr(4d23

+mrd

(1− ρ) γrd

)−mrd−n)

(28)

P (dH) =

(1

π

∫ π2

0

∫ ∞

0

exp

(−dHγsd

sin2 θ

)p(γsd)dγsddθ

)(1

π

∫ π2

0

∫ ∞

0

exp

(−d1γsr

sin2 θ

)p(γsr)dγsrdθ

)L

+

L∑ϑ=1

(L

ϑ

)(1

π

∫ π2

0

∫ ∞

0

exp

(−d1γsr

sin2 θ

)p(γsr)dγsrdθ

)L−ϑ(1− 1

π

∫ π2

0

∫ ∞

0

exp

(−d1γsr

sin2 θ

)

× p(γsr)dγsrdθ

)ϑ(1

π

∫ π2

0

∫ ∞

0

∫ ∞

0

exp

(−dHγsd + d2γrd

sin2 θ

)p(γsd)p(γrd)dγsddγrddθ

).

(29)

∞, the expression in (29) can be reduced to

P (dH)γ→∞≈ 1

π

∫ π2

0

∫ ∞

0

∫ ∞

0

exp

(−dHγsd + d2γrd

sin2 θ

)× p(γsd)p(γrd)dγsddγrddθ.

(30)

After some algebraic manipulations and using the approxima-

tions(1 +

dH γij

mij

)−mij ≈(

dH γij

mij

)−mij

, it is easy to obtain

P (dH)γ→∞≈ 2mrd

(1− ρ)mrd Γ(mrd)γmsd+mrd

(dHmsd

)−msd

×(d2 +

mrd

(1− ρ) γ

)−mrd ϑ−1∑j=0

(ϑ− 1

j

)× (−1)jχ(mrd + n, ρ, j).

(31)

Case 1: 0 ≤ ρ < 1In (31), the following approximation can be made(d2 +

mrd

(1−ρ)γ

)−mrd ≈ d−mrd2 . In what follows, the resulting

expression corresponds to the high-SNR PEP for ρ < 1 andis given by

P (dH)γ→∞≈ Gc1 γ

−(msd+mrd), (32)

where Gc1 is given by

Gc1 =2mrdm

−msd

sd

(1− ρ)rd

dmsd

H dmrd2 Γ(mrd)

ϑ−1∑j=0

(ϑ− 1

j

)× (−1)jχ(mrd + n, ρ, j)

(33)

Case 2: ρ = 1Substituting ρ = 1 in (31) cannot hold since (1− ρ) γyields an indeterminate value. Using the alternative expressionfor p(γrd) as expressed in (47) and after performing someintegrations, the expression in (30) can further be expressedas

P (dH)γ→∞≈ L

2

(dHmsd

)−msd

γ−msd

∫ ∞

0

m2msd

rd

ξmrdΓ2(mrd)γmrd

× γmrd−1rd exp

(−d2γrd − mrdγrd

ξ

) L−1∑j=1

(L− 1

j

)

×( −1

Γ(mrd)

)j ∫ ∞

0

xmrd exp

(−mrdx

ξ

)× Γj

(mrd,

mrdx

γ

)dxdγrd,

(34)

where ξ = (1− ρ) γ. Following the steps similar to those inthe Appendix for the case ρ = 1, and with some additionalmanipulations the asymptotic PEP is given by

P (dH)γ→∞≈ mmsd

sd mmrd+L−1rd L

2dmsd

H Γ(msd)︸︷︷︸Gc2

γ−(msd+Lmrd), (35)

where Gc2 is also a constant.

B. Bit Error Probability

In the sequel, we evaluate the performance of the pro-posed scheme in terms of BER using the limit-before-average


method and the transfer bounds technique. The upper boundon the average BER is given by [36]

Pb ≤ 1

K

∞∑dH=df

a(dH)P (dH), (36)

where df denotes the free Hamming distance and a(dH) is thesum of bit errors for error event of distance dH and can beobtained using the transfer function bounds method as in [37]

a (dH) =

K∑i=1

K∑d1=1

K∑d2=1︸︷︷︸

dH=i+d1+d2

i

K

(i

K

)p (d1|i) p (d2|i) , (37)

where p(d1/2|i

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

(Ki

), with t(K, i, d1/2) ob-

tained from the transfer function of the given turbo codeT (J, I,D) =

∑j≥0

∑i≥0

∑d≥0 J

jIiDdt (j, i, d) by a recur-sive method using MATLAB, JjIiDd is a monomial with jequal to 1, i and d are input and output dependent and takeon the value 0 or 1. However, using (36) yields loose boundson the BER. A remedy is to use the limit-before-averagetechnique [38] that yields much tighter bounds. Hence, theupper bounds on the BER can be expressed as

Pb ≤∫ ∞

0

∫ ∞

0

∫ ∞

0

min

(1

2,1

K

∞∑d=df

a(dH)P (dH |γsd, γsr, γrd)

× p(γsd)p(γsr)p(γrd)dγsddγsrdγrd

).

(38)

A closed-form expression of (38) is difficult to obtain, sincethe summation and integration are not interchangeable due tothe minimization expression. Therefore, we resort to numericalmethods to find the solution to (38).

VI. DIVERSITY-MULTIPLEXING TRADEOFF

The diversity-multiplexing tradeoff (DMT) is a fundamentaltradeoff in the design of a diversity-achieving wireless com-munication system. The diversity and multiplexing gains arerespectively defined as [35]

do.= lim

γ→∞− log (Pout(γ))

log (γ), (39)

r.= lim

γ→∞C(γ)

log (γ), (40)

where C(γ) is the data rate a given scheme can support givenby C(γ) = 1− Pout(γ) and γ is the average SNR.

In order to derive the DMT of the underlying scheme athigh SNR, it is essential to determine the behavior of themultiplexing gain r = C(γ) log γ at high SNR. It can benoted that at high SNR, Pout(γ) tends to zero, hence r → 1.Using corollary 1 and (40), it is easy to obtain the DMT.

Theorem 2: The diversity-multiplexing gain of the pro-posed scheme is

do(r) =

{(msd +mrd)(1 − r), if ρ < 1(msd + Lmrd)(1− r), if ρ = 1

(41)

Fig. 2 depicts the DMT tradeoff of the proposed schemefor perfect and outdated CSI. It can be seen that the DMT is

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

Multiplexing gain, r

Div

ersi

ty g

ain,

do(r

)

Outdated CSI, L=3,5Perfect CSI, L=3Perfect CSI, L=5

Fig. 2. Diversity-multiplexing tradeoff of the turbo coded system schemefor outdated and perfect CSI. msd = 0.65, mrid = 1.5, i = 1, · · · , L.

invariant regardless of the number of relays for outdated CSI,whereas in the case of perfect CSI, the DMT is a function ofthe number of relays. Furthermore, in both cases, the DMT isa function of the fading parameters msd and mrd.

VII. NUMERICAL RESULTS

In this section, we present the numerical results for theoutage probability of the underlying transmission scheme. Inall simulations, the message length is K = 128 bits. Weconsider a turbo code with code rate Rc = 1/3, generatorpolynomial (1, 17/13) in octal form and without loss ofgenerality assume that all the average SNRs are equal andmsrj = msr, mrjd = mrd.

Fig. 3 shows a performance comparison of the simulatedand exact expression of the outage probability for ρ < 1(ρ = {0.1, 0.5, 0.8, 0.99}). It is clear that the exact outageprobability derived is in good agreement with the simulatedone. As noted, the achievable diversity order for outdated CSIis the same and independent of ρ and the number of availablerelays as can be confirmed by the same slope. Moreover, thisdiversity order is equal to the one of an adaptive turbo-codedDF with a single relay. This can be intuitively explained bynoting that even for the most outdated CSI, the selected relayis reliable (source-to-relay link is good) which translates to noerror propagation at the relay.

In Fig. 4, the performance of the outage probability forL = 4 and ρ = 0.5 is presented for different fading figures m.A comparison of the simulated and exact outage probabilitycan help verify the accuracy of the latter.

In Fig. 5, we show the outage probability versus thecorrelation factor ρ for both γ = 10–15dB. For L = 1, fulldiversity is always achieved and the outage probability is nota function of the correlation factor as the former does not varywith the correlation factor ρ. However, for L > 1, it can beseen that the outage probability is a function of ρ and slightlyvaries for ρ < 1.

In Fig. 6, similar to Fig. 5 we present the outage probabilityperformance as a function of the correlation factor ρ. It can benoted that for ρ �= 1, the performance improves slowly as ρincreases. For ρ = 1, there is a sudden change in slope which


−10 −5 0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Out

age

Pro

babi

lity

ρ=0.1ρ=0.5ρ=0.8ρ=0.99ρ=1

Fig. 3. Simulated (symbols) versus exact (solid lines) outage probability forρ = 0.1, 0.5, 0.8, 0.99, 1 and L = 4. msd = 0.65 and mrd = 0.85.

−10 −5 0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Out

age

Pro

babi

lity

m

sr=1, m=0.65

msr

=1, m=0.85

msr

=1, m=1

msr

=1, m=1.5

Fig. 4. Simulated (symbols) versus exact (solid lines) outage probability forρ = 0.5 and L = 4. msd = mrd = m.

can be attributed to the fact that full diversity is achieved atthat instance.

We also consider the effects of pathloss which representsa practical scenario. For simplicity, a line topology is consid-ered, i.e., the relay nodes are situated on the same line betweenthe source and the destination, with the distance between s andd normalized to 1. Furthermore, the distance between s andri, and ri and d is denoted as dsr and drd respectively, wheredsr = d and drd = 1 − d. In this scenario, the variancesE〈h2

sd〉 = 1, E〈h2sr〉 = 1/dη and E〈h2

rd〉 = 1/(1− d)η are as-sumed where η is the pathloss coefficient. For our simulations,without loss of generality, we consider d = 0.3 and η = 3 inFig. 7. For both cases, {msd = 0.65,msr = 1,mrd = 0.85}and {msd = 0.85,msr = 1,mrd = 1.5}, it can be seen thatthe diveristy order do = msd +mrd is achieved as predictedfrom our analysis for outdated CSI.

In Figs. 8 and 9, various levels of imperfect CSI areconsidered where one can note that full diversity in the numberof available relays L and fading figure m can be achievedwith ideal CSI. However, for non-ideal CSI, the achievable

0 0.2 0.4 0.6 0.8 110

−7

10−6

10−5

10−4

10−3

10−2

10−1

Correlation factor, ρ

Out

age

Pro

babi

lity

L=1L=3L=4

10dB

15dB

Fig. 5. Outage probability versus correlation factor ρ for various SNRs andnumber of relays. msdi = msri = mrid = 1, i = 1 · · ·L.

0 0.2 0.4 0.6 0.8 110

−7

10−6

10−5

10−4

10−3

10−2

Correlation factor, ρ

Out

age

Pro

babi

lity

msd

msr

i

=mrid=0.65

msd

msr

i

=mrid=0.85

msd

=msr

i

=mrid=1

Fig. 6. Outage probability versus correlation factor ρ for different m valuesand L = 5. msdi = msri = mrid, i = 1 · · ·L.

diversity order is identical to a single relay scenario. Moreover,both Figs. 8 and 9 show a comparison between the unionbounds on the BER and the simulated BER. As seen, thebounds on the BER are in good agreement with the simulationsfor different values of ρ. This validates the accuracy of ouranalytical framework presented in this work.

VIII. CONCLUSION

The effect of outdated CSI on turbo-coded cooperation withrelay selection subject to Nakagami-m fading was investigated.A closed-form expression for the outage probability and itsasymptotic expression at high SNR are derived. It was notedthat the outage probability performance is dependent on thelevel of CSI imperfection, with full diversity achieved for idealCSI only. In addition, a closed-form expression of the PEP isderived and the diversity analysis of the PEP corroborates withthe conclusions extracted from the outage probability study.

APPENDIX

We derive the high-SNR expression of the outage probabil-ity for turbo-coded cooperation with outdated CSI. Two casesarise: ρ < 1 and ρ = 1.


−10 −5 0 5 10 15 20 2510

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Out

age

Pro

babi

lity

ρ=0.1ρ=0.4ρ=0.7

msd

=0.65m

sr=1

mrd

=0.85

msd

=0.85m

sr=1

mrd

=1.5

Fig. 7. Outage probability performance for ρ = 0.1, 0.4, 0.7. E〈h2sd〉 = 1,

E〈h2sr〉 = 1/0.33 and E〈h2

rd〉 = 1/0.73.

0 5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Bit

Err

or R

ate

ρ=0.1ρ=0.7ρ=0.9ρ=1

Fig. 8. BER of the proposed scheme versus average SNR for variouscorrelation values ρ. L = 3 and msd = 0.65, msr = 1, mrd = 0.85.Solid lines: simulations, dashed lines: bounds.

Case 1: 0 ≤ ρ < 1It is easy to rewrite (23) as in (15) and assuming that γsd =γsr = γrd = γ → ∞,

Pout

γ→∞≈L∑

ϑ=1

∞∑n=0

ϑρnmmrd+nrd γ−(mrd+n)

n! (1− ρ)mrd+2n

Γ(mrd)Γ(mrd + n)

×(∫ a

0

mmsd

sd γmsd−1

Γ(msd)γmsdexp

(−msdγsd

γ

)

×∫ b

0

γmrd+n−1 exp

(− mrdγrd(1− ρ) γ

)dγrddγsd

)

×ϑ−1∑j=0

(ϑ− 1

j

)(−1)jχ (mrd + n, ρ, j) .

(42)

To solve the inner integral, we can use [31, Eq. 3.351.1] andafter some manipulations,

0 5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Average SNR (dB)

Bit

Err

or R

ate

ρ=0.7ρ=0.9ρ=1

Fig. 9. BER of the proposed scheme versus average SNR for variouscorrelation values ρ. L = 3 and msd = 1.5, msr = 1, mrd = 0.85.Solid lines: simulations, dashed lines: bounds.

Pout

γ→∞≈L∑

ϑ=1

∞∑n=0

ϑρnmmrd+nrd mmsd

sd γ−(msd+mrd+n)

n! (1− ρ)mrd+2n Γ(msd)Γ(mrd)Γ(mrd + n)

×(∫ a

0

γmsd−1sd exp

(−msdγsd

γ

)(mrd

(1− ρ) γ

)−mrd−n

× γ

(mrd + n,

mrdb

(1− ρ) γ

)dγsd

)ϑ−1∑j=0

(ϑ− 1

j

)

× (−1)jχ (mrd + n, ρ, j) .(43)

It can easily be noticed that at high SNR: (a) the maximumof the outage probability in (43) occurs when ϑ = L. (b)The dominant term in (43) is n = 0 of the infinite series.Following these observations and using the identity γ(a, x) =xa

1F1 (a, 1 + a;−x) in [31, Eq. 8.352.1],

Poutγ→∞≈ Lmmsd

sd mmrd

rd

(1− ρ)mrd Γ(msd)Γ2(mrd)γmsd+mrd

×(∫ a

0

γmsd−1sd exp

(−msdγsd

γ

)

× 1F1

(mrd, 1 +mrd;− mrdb

(1− ρ) γ

)dγsd

)

×L−1∑j=0

(L− 1

j

)(−1)jχ (mrd, ρ, j) ,

(44)

where 1F1 (x, y; z) denotes the confluent Hypergeometricfunction defined in [31, Eq. 9.210.1]. As γ → ∞, thehypergeometric function in (44) reduces to 1. Hence,

Pout

γ→∞≈ G′c1 γ

−(msd+mrd), (45)


where G′c1 is a constant value and is given by

G′c1 =

L−1∑j=0

(L− 1

j

)(−1)jχ (mrd, ρ, j)

×∫ 2Rc−1

0

γmsd−1sd

(2Rc/(1−α)

(1 + γsd)1/(1−α)

− 1

)mrd

dγsd,

(46)

where the integral part can be computed numerically yieldinga constant value.

Case 2: ρ = 1In this case, the observations (a) and (b) made for ρ < 1are still valid. Furthermore, a less simplified expression ofthe CDF in [20] and p(x) = ∂F(x)/∂x are used, and theasymptotic outage probability (i.e., γsd = γsr = γrd = γ →∞ is assumed) can be given by

Poutγ→∞≈ mmsd

sd m2mrd

rd

(1− ρ)mrd Γ(msd)Γ2(mrd)γmsd+2mrd

×(∫ a

0

γmsd−1sd exp

(−msdγsd

γ

)∫ b

0

γmrd−1rd

× exp

(− mrdγrd(1− ρ) γ

)dγrddγsd

)L−1∑j=0

(L− 1

j

)

× (−1)j

Γj(mrd)

∫ ∞

0

xmrd−1 exp

(− mrdx

(1− ρ) γ

)

× Γj

(mrd,

mrdx

γ

)dx.

(47)

The expression in (47) can further be simplifiedusing [31, Eq. 3.351.1]. Substituting Γj

(m, mx

γ

)≈

Γj(m) exp(−mjx

γ

)[31, Eq. 3.352.7] for high values of γ,

and after some algebraic manipulations, the expression in(47) can be rewritten as

Pout

γ→∞≈ mmsd

sd mmrd

rd

Γ(msd)Γ2(mrd)γmsd+2mrd

(∫ a

0

γmsd−1sd

× exp

(−msdγsd

γ

)γ

(mrd,

mrdb

(1− ρ) γ

)dγsd

)

×∫ ∞

0

xmrd−1 exp

(− mrdx

(1− ρ) γ

) L−1∑j=0

(L− 1

j

)

× (−1)j exp

(−mrdjx

γ

)dx.

(48)

The Binomial expansion (1 − y)m =∑m

j=0

(mj

)(−1)jyj

can be identified in (48). Moreover, at high values of y,exp

(− 1

y

)≈ 1− 1

ym . Hence (48) can be expressed as

Pout

γ→∞≈ mmsd

sd mLrd

Γ(msd)Γ2(mrd)γmsd+Lmrd

(∫ a

0

γmsd−1sd

× γ

(mrd,

mrdb

γ

)exp

(−msdγsd

γ

)dγsd

)

×∫ ∞

0

xmrd+L−2 exp

(− mrdx

(1− ρ) γ

)dx.

(49)

From (49), at high γ, δ ≈ 0 and hence γ(m, 1δ ) ≈ Γ(m)and exp

(− 1δ

) ≈ 0. After some manipulations, the asymptoticoutage probability can be expressed as

Pout

γ→∞≈ G′c2 γ

msd+Lmrd , (50)

G′c2 =

Kmmsd

sd mLrd

Γ(msd)Γ(mrd)

(2Rc − 1

)msd, (51)

with K =∫∞0

exp(−mrdx

δ

)dx being a constant.

By adopting the diversity order definition in [35], i.e.,d

.= limγ→∞ − log (Pout) / log (γ), the diversity order for

both cases (ρ < 1 and ρ = 1) can be obtained.

ACKNOWLEDGMENT

The authors would like to acknowledge the helpful com-ments and suggestions of the anonymous reviewers. We alsothank the Editor of this Journal.

REFERENCES

[1] A. Bletsas, H. Shin, and M. Z. Win, “Cooperative communications withoutage-optimal opportunistic relaying,” IEEE Trans. Wireless Commun.,vol. 6, no. 9, pp. 3450-3460, Sept. 2007.

[2] A. Bletsas, H. Shin, and M. Z. Win, “Outage optimality of opportunisticamplify-and-forward relaying,” IEEE Commun. Lett., vol. 11, no. 3, pp.261-263, Mar. 2007.

[3] I. Krikidis, J. S. Thompson, S. McLaughlin, and N. Goertz, “Amplify-and-forward with partial relay selection,” IEEE Commun. Lett., vol. 12,no. 3, pp. 235-237, Apr. 2008.

[4] M. Seyfi, S. Muhaidat, J. Liang, and M. Uysal, “Relay selection in dual-hop vehicular networks,” IEEE Signal Process. Lett., vol. 18, no. 2, pp.134-137, Feb. 2011.

[5] H. A. Saleh and W. Hamouda, “Cross-layer based transmit antennaselection for decision-feedback detection in correlated Ricean MIMOchannels,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 1677-1682,Apr. 2009.

[6] D. S. Michalopoulos, A. S. Lioumpas, G. K. Karagiannidis, and R.Schober, “Selective cooperative relaying over time-varying channels,”IEEE Trans. Commun., vol. 58, no. 8, pp. 2402-2412, July 2010.

[7] T. Hunter and A. Nosratinia, “Diversity through coded cooperation,”IEEE Trans. Wireless Commun., vol. 5, no. 2, pp. 283-289, Feb. 2006.

[8] M. Janani, A. Hedayat, T. E.Hunter, and A. Nosratinia, “Coded cooper-ation in wireless communications: space-time transmission and iterativedecoding,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 362-371,Feb. 2004.

[9] M. Elfituri, A. Ghrayeb, and W. Hamouda, “Antenna/Relay selection forcoded cooperative networks with AF relaying,” IEEE Trans. Commun.,vol. 57, no. 9, pp. 2580-2584, Sept. 2009.

[10] M. Elfituri, A. Ghrayeb, and W. Hamouda, “Antenna/Relay selection forcoded wireless cooperative networks,” in Proc. 2008 IEEE InternationalConference on Communications.

[11] J. L. Vicario, A. Bel, J. A. Lopez-Salcedo, and G. Seco, “Opportunisticrelay selection with outdated CSI: outage probability and diversityanalysis,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 2872-2876,June 2009.

[12] M. Soysa, H. Suraweera, C. Tellambura, and H. K. Garg, “Partial andopportunistic relay selection with outdated channel estimates,” IEEETrans. Commun., vol. 60, no. 3, pp. 840-850, Mar. 2012.

[13] M. Chen, T. C.-K. Liu, and X. Dong, “Opportunistic multiple relayselection with outdated channel state information,” IEEE Trans. Veh.Technol., vol. 61, no. 3, pp. 1333-1345, Mar. 2012.

[14] D. Michalopoulos, H. A. Suraweera, G. Karagiannidis, and R. Schober,“Amplify-and-forward relay selection with outdated channel estimated,”IEEE Trans. Commun., vol. 60, no. 5, pp.1278-1290, May 2012.

[15] M. Seyfi, S. Muhaidat and J. Liang, “Performance analysis of relayselection with feedback delay and channel estimation errors,” IEEESignal Process. Lett., vol. 18, no. 1, pp. 67-70, Jan. 2011.

[16] H. A. Suraweera, M. Soysa, C. Tellambura, and H. K. Garg, “Perfor-mance analysis of partial relay selection with feedback delay,” IEEESignal Process. Lett., vol. 17, pp. 531-534, June 2010.

[17] R. Mo, Y. H. Chew and C. Yuen, “Information rate and relay precoderdesign for amplify-and-forward MIMO relay networks with imperfectchannel state information,” IEEE Trans. Veh. Technol., vol. 61, no. 9,pp. 3958-3968, Nov. 2012.


[18] M. Torabi and D. Haccoun, “Capacity analysis of opportunistic relay-ing in cooperative systems with outdated channel information,” IEEECommun. Lett., vol. 14, no. 12, pp. 1137-1139, Dec. 2012.

[19] H. A. Suwaweera, D. S. Michalopoulos, and C. Yuen, “Performanceanalysis of fixed-gain AF relay system with interference in Nakagami-mfading channels,” IEEE Trans. Veh. Technol., vol. 61, no. 3, pp. 1457-1463, Mar. 2012.

[20] D. S. Michalopoulos, N. D. Chatzidiamantis, R. Schober, and G. K.Karagiannidis, “Relay selection with outdated channel estimates inNakagami-m fading,” in Proc. 2011 IEEE International Conference onCommunications.

[21] Y. Li, Q. Yin, L. Sun, H. Chen, and H. Wang, “A channel quality metricin opportunistic selection with outdated CSI over Nakagami-m fadingchannels,” IEEE Trans. Veh. Technol., vol. 61, no. 3, pp. 1427-1432,Mar. 2012.

[22] N. S. Ferdinand, N. Rajatheva, and M. Latva-aho, “Effects of feedbackdelay in partial relay selection over Nakagami-m fading channels,” IEEETrans. Veh. Technol., vol. 61, no. 4, pp. 1620-1634, May 2012.

[23] Z. Wang, M. Xiao, C. Wang, and M. Skoglund, “Degrees of freedomof multi-hop MIMO broadcast networks with delayed CSIT,” IEEEWireless Commun. Lett., vol. 2, no. 2 , pp. 207-210, Apr. 2013.

[24] M. A. Maddah-Ali and D. Tse, “Completely stale transmitter channelstate information is still very useful,” IEEE Trans. Inf. Theory, vol. 58,no. 7, p. 4418-4431, July 2012.

[25] Z. Wang, M. Xiao, C. Wang, and M. Skoglund, “On the degrees offreedom of two-hop MISO broadcast networks with mixed CSIT,” inProc. 2013 IEEE Global Communications Conference.

[26] M. Xiao, J. Kliewer, and M. Skoglund, “Design of network codes formultiple-user multiple-relay wireless networks,” IEEE Trans. Commun.,vol. 60, no. 12, pp.. 3755-3766, Dec. 2012.

[27] M. Xiao and M. Skoglund, “Multiple-user cooperative communicationsbased on linear network coding,” IEEE Trans. Commun., vol. 58, no.12, pp. 3345-3351, Dec. 2010.

[28] A. Eid, W. Hamouda, and I. Dayoub, “Performance of multi-relay codedcooperative diversity in asynchronous code-division multiple-access overfading channels”, IET Commun., vol. 5, no. 5, pp. 683-692, 2011.

[29] P. Mitran, H. Ochiai, and V. Tarokh, “Space-time diversity enhancementsusing collaborative communications,” IEEE Trans. Inf. Theory, vol. 51,no. 6, pp. 2041-2057, June 2005.

[30] W. C. Jakes, Microwave Mobile Communication. Wiley & Sons, 1974.[31] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products, 3rd ed. New York: Academic, 2007.[32] J. W. Craig, “A new, simple and exact result for calculating the

probability of error for two-dimensional signal constellations,” in Proc.1991 IEEE Military Communications Conference, pp. 571-575.

[33] M. K. Simon and M.–S. Alouini, Digital Communications over FadingChannels: A Unified Approach to Performance Analysis. New York:John Wiley and Sons, 2000.

[34] M. Chiani, D. Dardari, and M. K. Simon, “New exponential boundsand approximations for the computation of error probability in fadingchannels,” IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 840-845,July 2003.

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

[36] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.[37] D. Divsalar, S. Dolinar, and F. Pollara, “Transfer function bounds on the

performance of turbo codes,” TDA Progress Report 42-122, pp. 44-55,Aug. 1995.

[38] E. Malkamaki and H. Leib, “Evaluating the performance of convolu-tional codes over block fadings” IEEE Trans. Inf. Therory, vol. 45, no.5, pp. 1643-1646, July 1999.

Jules Merlin Moualeu (S’06–M’14) was born inCameroon. He received the MSc.Eng. and Ph.D.degrees in electronic engineering from the Univer-sity of KwaZulu-Natal, Durban, South Africa, in2008 and 2013, respectively. From January 2011 toJuly 2011, he was a Visiting Scholar at ConcordiaUniversity in Montreal, Canada, under the CanadianCommonwealth Scholarship Program (CCSP) of-fered by the Foreign Affairs and International TradeCanada (DFAIT).

He is with the Department of Electrical andInformation Engineering, University of the Witwatersrand in Johannesburg,South Africa, as a Postdoctoral Fellow. His current research interests are inthe general area of digital and wireless cooperative communications (physicallayer) and include coding theory, space-time coding, MIMO systems andcognitive radio networks.

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 a Professor. Since June 2006, he has beenConcordia University Research Chair in Commu-nications and Networking. His current research in-terests include multiple-input multiple-output space-

time processing, cooperative communications, wireless networks, multiusercommunications, cross-layer design, and source and channel coding.

Dr. Hamouda served as the Technical Co-chair of the Wireless NetworksSymposium, 2012 Global Communications Conference, the Ad hoc, Sensor,and Mesh Networking Symposium of the 2010 International CommunicationsConference (ICC), and the 25th Queen’s Biennial Symposium on Communica-tions. He also served as the Track Cochair of the Radio Access Techniques ofthe 2006 IEEE Vehicular Technology Conference (IEEE VTC Fall 2006) andthe Transmission Techniques of the IEEE VTC-Fall 2012. From September2005 to November 2008, he was the Chair of the IEEE Montreal Chapterin Communications and Information Theory. He has received numerousawards, including the Best Paper Award of the ICC 2009 and the IEEECanada Certificate of Appreciation in 2007 and 2008. He served as anassociate editor of the IEEE COMMUNICATIONS LETTERS and currentlyserves as an associate editor for the IEEE TRANSACTIONS ON VEHICULARTECHNOLOGY, the IEEE TRANSACTIONS ON SIGNAL PROCESSING and IETWireless Sensor Systems.

Fambirai Takawira (M’96) received the B.Sc. de-gree in electrical and electronic engineering (first-class honors) from Manchester University, Manch-ester, U.K., in 1981, and the Ph.D. degree fromCambridge University, Cambridge, U.K, in 1984.He joined the University of the Witwatersrand inJohannesburg, South Africa in February 2012 af-ter 19 years at the University of KwaZulu-Natal(UKZN). At UKZN he held various academic po-sitions including that of Head of the School ofElectrical, Electronic and Computer Engineering and

just before his departure he was Dean of the Faculty of Engineering. He hasalso held appointments at the University of Zimbabwe, the University ofCalifornia San Diego, British Telecom Research Laboratories, and NationalUniversity of Singapore. His research interests are in wireless communicationsystems and networks.

Dr. Takawira is a past editor of the journal IEEE TRANSACTIONS ONWIRELESS COMMUNICATIONS. He has served on several conference orga-nizing committees. He served as the Communications Society (COMSOC)Director for Europe, Middle East and Africa region for the 2012-2013 term.

2362 ieee transactions on wireless communications, …hamouda/jules_twt... · 2015. 1. 29. · 2362...

Documents