Multipair Two-Way Half-Duplex DF Relaying with Massive Arrays andImperfect CSI

Kong, C., Zhong, C., Matthaiou, M., Björnson, E., & Zhang, Z. (2018). Multipair Two-Way Half-Duplex DFRelaying with Massive Arrays and Imperfect CSI. IEEE Transactions on Wireless Communications. DOI:10.1109/TWC.2018.2808952

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Download date:14. May. 2018

1

Multipair Two-Way Half-Duplex DF Relaying withMassive Arrays and Imperfect CSI

Chuili Kong, Student Member, IEEE, Caijun Zhong,Senior Member, IEEE, Michail Matthaiou,Senior Member,IEEE, Emil Bjornson,Senior Member, IEEE, and Zhaoyang Zhang,Member, IEEE

Abstract—This paper considers a two-way half-duplex decode-and-forward relaying system where multiple pairs of single-antenna users exchange information via a multiple-antennarelay.Assuming that the channel knowledge is non-ideal and the relayemploys maximum ratio processing, we derive a large-scaleapproximation of the sum spectral efficiency (SE) that is tightwhen the number of relay antennas,M , becomes very large.Furthermore, we study how the transmit power scales withMto maintain a desired SE. In particular, three special power-scaling cases are discussed and the corresponding asymptotic SEis deduced with clear insights. Our elegant power-scaling lawsreveal a tradeoff between the transmit powers of the user/relayand pilot symbol. Finally, we formulate a power allocationproblem in terms of maximizing the sum SE and obtain alocal optimum by solving a sequence of geometric programmingproblems.

Index Terms—Decode-and-forward, geometric programming,massive MIMO, power-scaling law, two-way relaying.

I. I NTRODUCTION

Massive multiple-input multiple-output (MIMO) is one ofthe key technologies for the next-generation wireless commu-nications due to its distinctive features and advantages overconventional MIMO systems. Some key benefits include: 1)very narrow beams and little inter-user interference due tothe asymptotic channel orthogonality; 2) low computationalcomplexity since linear signal processing is asymptoticallyoptimal; 3) transmit power can be made extremely low, since itscales down inversely proportional to the number of antennaswhen perfect channel state information (CSI) is available.Inaddition to the use of massive MIMO in cellular networks,

Manuscript received August 14, 2017, revised December 19, 2017, acceptedFebruary 12, 2018. The work was supported in part by the National Scienceand Technology Major Project of China 2017ZX03001002–003,the NationalNatural Science Foundation of China under Grant 61671406, U1709219,61725104, and the Zhejiang Provincial Natural Science Foundation of China(LR15F010001), and in part by the Open Research Fund of StateKeyLaboratory of Integrated Services Networks under Grant ISN19-05. Theassociate editor coordinating the review of this paper and approving it forpublication was B. Clerckx.(Corresponding author: Caijun Zhong.)

C. Kong, C. Zhong and Z. Zhang are with the Institute of Informationand Communication Engineering and also with the Zhejiang ProvincialKey Laboratory of Information Processing, Communication and Networking,Zhejiang University, Hangzhou 310027, China, and also withthe StateKey Laboratory of Integrated Services Networks, Xidian University, Xi’an710126, China (e-mail: kcldut@163.com; caijunzhong@zju.edu.cn; sun-rise.heaven@gmail.com).

M. Matthaiou is with the Institute of Electronics, Communications andInformation Technology (ECIT), Queen’s University Belfast, Belfast, BT39DT, U.K. (email: m.matthaiou@qub.ac.uk).

E. Bjornson is with the Department of Electrical Engineering(ISY), Linkoping University, Linkoping, SE-581 83, Sweden (email:emil.bjornson@liu.se).

the technique has been analyzed in other applications, suchas cognitive radio, heterogeneous networks, energy harvestingunits, and relaying systems, etc. [1]–[4].

The massive MIMO technique also finds an importantapplication in the multipair relaying system, where multiplepairs of users simultaneously establish communication linkswith the aid of a shared relay with large antenna arrays.The resulting multipair massive MIMO relaying system hasattracted substantial attention from both academia and industrybecause of its ability to enhance the coverage and service qual-ity for cell edge users [5]–[8]. The initial works on multipairmassive MIMO relaying mainly focus on the one-way relayingsystems. For one-way multipair massive MIMO amplify-and-forward (AF) relaying systems, the work [9] studied the max-min user selection problem and the power allocation issue.Later in [10], taking into account the semi-blind gain controland relay oscillation, the authors proposed a low-complexitypower control scheme. In contrast, for the decode-and-forward(DF) protocol, [5] compared the achievable spectral efficiency(SE) of two linear processing techniques, i.e., zero-forcing(ZF) and maximum-ratio (MR) in Rayleigh fading channels.Then, [11] extended the analysis to the Ricean fading case. Theenergy efficiency was optimized in [12]. Yet, such one-waymechanism incurs a 50% SE loss. To reduce this loss, two-way relaying becomes a potential solution [13]–[16], wherethetwo communicating nodes execute bidirectional simultaneousdata transmission.

Recently, there has been intensive research on two-waymassive MIMO relaying. For example, the power-scaling lawsof MR and/or ZF processing methods are characterized forhalf-duplex [17], [18] and full-duplex [19], [20] schemes,respectively. The work [21] considered the maximization oftheenergy efficiency of the system subject to maximum power andminimum SE constraints. Nevertheless, one major limitationof the above works is that perfect CSI is assumed. Sinceobtaining perfect CSI is very challenging in the context ofmassive MIMO, it is important to look into the realisticscenario with imperfect CSI. By employing the minimummean-square-error (MMSE) estimation at the relay, [22], [23]studied the impact of pilot power on the SE, [24] maximizesthe energy efficiency using the max-min approach, while [25]adopted the composite channel estimation method to reducethe pilot overhead by half and demonstrated that this approachoutperforms the individual channel estimation scheme in [22],[23] when the coherence interval is smaller. However, all theaforementioned two-way massive MIMO relaying works focuson the AF protocol, and the DF case is largely overlooked.

2

Unlike AF relaying, DF relaying does not suffer from theproblem of noise amplification. Thus, DF two-way relayingmay achieve better performance than AF two-way relay, espe-cially at low signal-to-noise ratios (SNRs) [26]. Besides this,DF two-way relaying has the flexibility of performing separatepower allocation/precoding for relaying the communicationon each direction. Therefore, it is of great interest to studythe performance of two-way massive MIMO relaying systemsadopting the DF protocol.

Motivated by this, in the current work, we consider amultipair two-way DF relaying system, taking into accountchannel estimation errors, and present a comprehensive anal-ysis of the achievable SE and power-scaling law of MRprocessing. Specifically, the main contributions of this paperare summarized as follows:

• We propose a general multipair massive MIMO two-way relaying system employing the DF protocol, andpresent a large-scale approximation of the SE underthe imperfect CSI when the number of relay antennasapproaches infinity.

• We characterize new power-scaling laws, which general-ize the results presented in [5], [17], [18]. It turns outthat there exists a trade-off between the transmit powersof each user, pilot symbol and the relay; in other words,the same SE can be achieved with different combinationof power-scaling parameters, which offers great flexibilityin the design of practical systems.

• To improve the sum SE, we study the power allocationproblem subject to a sum power constraint. Local op-timum solutions are obtained by solving a sequence ofgeometric programming (GP) problems. Our numericalresults suggest that the proposed power allocation strategysignificantly improves the sum SE.

The remainder of the paper is organized as follows: SectionII introduces the multipair two-way half-duplex DF relayingsystem model. Section III presents a large-scale approximationof the SE, with imperfect CSI, while Section IV studies thepower-scaling laws of different system configurations. Thepower allocation problem is discussed in Section V. Thenumerical results are verified in Section VI. Finally, SectionVII provides some concluding remarks.

Notation: We use bold upper case letters to denote matrices,bold lower case letters to denote vectors and lower case lettersto denote scalars. Moreover,(·)H , (·)∗, (·)T , and(·)−1 repre-sent the conjugate transpose operator, the conjugate operator,the transpose operator, and the matrix inverse, respectively.Also, || · || is the Euclidian norm,|| · ||F denotes the Frobeniusnorm, and|·| is the absolute value. In addition,x ∼ CN (0,Σ)denotes a circularly symmetric complex Gaussian randomvector x with covariance matrixΣ, while Ik is the identitymatrix of sizek. Finally, the statistical expectation operator isrepresented byE{·}, the variance operator is Var{·}, and thenotation

a.s.→ means almost sure convergence.

II. SYSTEM MODEL

Consider a multipair two-way relaying system, whereNpairs of single-antenna users, denoted as TA,i and TB,i, i =

1, . . . , N , exchange information with each other, under theassistance of a shared relay TR equipped withM antennas,shown in Fig. 1. We assume that the direct links betweenTA,i and TB,i do not exist due to shadowing. Also, the relayoperates in the half-duplex mode, i.e., it cannot transmit andreceive simultaneously.

M,1

TA

,TA i

,TA N

,1TB

,TB i

,TB N

TR

1

T

ARg

1ARg

,

T

AR ig

,AR ig

,

T

AR Ng

,AR Ng

1

T

RBg

1RBg

,

T

RB ig

,RB ig

,

T

RB Ng

,RB Ng

Fig. 1: Illustration of the multipair two-way relaying system.

It is assumed that the system works under a time divisionduplex (TDD) protocol and channel reciprocity holds. As such,the uplink and downlink channels between TA,i and TR can bedenoted asgAR,i ∼ CN (0, βAR,iIM ) andgT

AR,i, respectively.Similarly, the channels between TB,i and TR are denotedas gRB,i ∼ CN (0, βRB,iIM ) and gT

RB,i, i = 1, . . . , N ,respectively. This model is known as uncorrelated Rayleighfading, andβAR,i and βRB,i model the large-scale effect,which are assumed to be constant over many coherenceintervals and known a priori. For notational convenience, thechannel vectors can be collected together in a matrix formas GAR , [gAR,1, . . . ,gAR,N ] ∈ CM×N and GRB ,

[gRB,1, . . . ,gRB,N ] ∈ CM×N .For the considered multipair two-way relaying system, the

information transmission process consists of two separatephases. In the first phase, i.e., multiple-access channel (MAC)phase, theN user pairs TA,i and TB,i simultaneously transmittheir respective signals to TR. Thus, the received signal at TR

is given by

yr =N∑

i=1

(√pA,igAR,ixA,i +

√pB,igRB,ixB,i

)+ nR, (1)

wherexA,i andxB,i are Gaussian signals with zero mean andunit power transmitted by thei-th user pair,pA,i and pB,i

are the average transmit power of TA,i and TB,i, respectively,andnR is a vector of additive white Gaussian noise (AWGN)at TR, whose elements are identically and independently dis-tributed (i.i.d.)CN (0, 1). Note that to keep the notation cleanand without loss of generality, we take the noise variance tobe1 in this paper. To reduce the complexity, single-user lineardetection is considered at the relay. As such, the transformedsignal after linear processing can be expressed as

rDF = WTyr, (2)

whereWT ∈ C2N×M is the linear receiver matrix, which willbe specified shortly.

In the second phase, i.e., broadcasting (BC) phase, the relayfirst decodes the received information, and then re-encodes

3

and broadcasts it to the users. A linear precoding matrixJ ∈ CM×2N is applied to the decoded signalx. As such,the transmit signal of TR is given by

yDFt = ρDFJx, (3)

where x =[xTA,x

TB

]Twith xA = [xA,1, . . . , xA,N ]T and

xB = [xB,1, . . . , xB,N ]T , andρDF is the normalization coeffi-cient, which is determined by the average power constraint atthe relay, i.e.,E

{||yDF

t ||2}= pr. Hence, the signals received

at TX,i (whereX ∈ {A,B}) can be expressed as

zDFX,i = gT

XR,iyDFt + nX,i, (4)

wherenX,i ∼ CN (0, 1) represents the AWGN at TX,i. Pleasenote that, to simplify notation, we introducegBR,i which isdefined asgBR,i , gRB,i due to the channel reciprocity.

A. Channel Estimation

Since we consider a block fading system, the channelsGAR

andGRB are not known and need to be estimated at the relayin every coherence interval. The typical way of doing this inTDD systems is to transmit pilots [27]. To this end, duringeach coherence interval of lengthτc (in symbols),τp symbolsare used for channel training. In this case, TA,i and TB,i

simultaneously transmit mutually orthogonal pilot sequencesto TR. Thus, the received pilot matrix at TR is

Yp =√τpppGARΦ

TA +

√τpppGRBΦ

TB +Np, (5)

wherepp is the transmit power of each pilot symbol,Np isAWGN matrix including i.i.d.CN (0, 1) elements, while thei-th columns ofΦA ∈ Cτp×N andΦB ∈ Cτp×N are the pilotsequences transmitted from TA,i and TB,i, respectively. Sinceall pilot sequences are assumed to be mutually orthogonal,τp ≥ 2N is required, and we have thatΦT

AΦ∗A = IN ,

ΦTBΦ

∗B = IN , andΦT

AΦ∗B = 0N .

As in [5], [28], we assume that TR uses the MMSEestimator to estimateGAR andGRB. As such, we have

gAR,i = gAR,i + eAR,i, (6)

gRB,i = gRB,i + eRB,i, (7)

wheregAR,i, gRB,i, eAR,i, andeRB,i are thei-th columns ofthe estimated matricesGAR, GRB, and the estimation errormatricesEAR andERB, respectively, which are mutually in-dependent. The elements ofgAR,i, eAR,i are Gaussian randomvariables with zero mean, varianceσ2

AR,i and σ2AR,i, respec-

tively, whereσ2AR,i ,

τpppβ2AR,i

1+τpppβAR,iand σ2

AR,i ,βAR,i

1+τpppβAR,i.

Similarly, the elements ofgRB,i, and eRB,i are complexGaussian random variables with zero mean, varianceσ2

RB,i

and σ2RB,i, respectively, whereσ2

RB,i ,τpppβ

2RB,i

1+τpppβRB,iand

σ2RB,i ,

βRB,i

1+τpppβRB,i.

B. Linear Processing Matrices

To keep the complexity costs at the relay to reasonablelevels, a simple linear processing scheme is used at the relay.

With the MR method1, the processing matrixWT ∈ C2N×M

andJ ∈ CM×2N are given by

WT =[

GAR, GRB

]H

, (8)

J =[

GRB , GAR

]∗

, (9)

respectively, whileρDF is given by

ρDF =

√pr

E {||J||2F}=

√√√√√

pr

MN∑

n=1

(

σ2AR,n + σ2

RB,n

) . (10)

III. SPECTRAL EFFICIENCY

In this section, we investigate the SE (in bit/s/Hz) of thetwo-way half-duplex DF relaying system. In particular, a large-scale approximation of the SE is deduced whenM → ∞.

In the MAC phase, a linear processing matrixWT is appliedto the received signals prior to signal detection, hence, thepost-processing signals at the relay are given by

rDF = (11)

GHAR

(N∑

i=1

(√pA,igAR,ixA,i +

√pB,igRB,ixB,i

)+ nR

)

GHRB

(N∑

i=1

(√pA,igAR,ixA,i +

√pB,igRB,ixB,i

)+ nR

)

,

where the topN elements ofrDF stand for the signals fromTA,i (i = 1, . . . , N ), while the bottomN elements ofrDF

represent the signals from TB,i (i = 1, . . . , N ). Without lossof generality, we focus only on thei-th pair of users, i.e., TA,i

and TB,i, which is given by (12). Since the relay has imperfectCSI, it treats the channel estimates as the true channels todecode the signals. To this end, using a standard lower capacitybound based on the worst-case uncorrelated additive noise [30]yields the sum achievable SE of thei-th user pair in the MACphase:

RDF1,i =

τc − τp2τc

× (13)

E

log2

1 +ADF

i +BDFi

E

{(CDF

i +DDFi + EDF

i

)|GAR, GRB

}

,

where the inner and outer expectations are taken over the

1Note that MR is a very attractive linear processing technique in the contextof massive MIMO systems due to its low complexity. Most importantly, itcan be implemented in a distributed manner [27], [29]. Leveraging on theproperties of long Gaussian vectors, the extensions to the ZF and MMSEprocessing can be easily made by using the same technique as in the MRmethod. In addition, to simplify the notations, we have assumed that theMAC and BC phases take place in the same coherence interval. However,this is not necessary.

4

rDFi = rDF

i + rDFN+i =

√pA,i

(gHAR,igAR,i + gH

RB,igAR,i

)xA,i +

√pB,i

(gHAR,igRB,i + gH

RB,igRB,i

)xB,i

︸ ︷︷ ︸

desired signal

(12)

+√pA,i

(gHAR,ieAR,i + gH

RB,ieAR,i

)xA,i +

√pB,i

(gHAR,ieRB,i + gH

RB,ieRB,i

)xB,i

︸ ︷︷ ︸

estimation error

+∑

j 6=i

(√pA,j

(gHAR,igAR,j + gH

RB,igAR,j

)xA,j +

√pB,j

(gHAR,igRB,j + gH

RB,igRB,j

)xB,j

)

︸ ︷︷ ︸

inter-user interference

+(gHAR,i + gH

RB,i

)nR

︸ ︷︷ ︸

compound noise

.

estimation errors and channel estimates, respectively, and

ADFi = pA,i

(|gH

AR,igAR,i|2 + |gHRB,igAR,i|2

), (14)

BDFi = pB,i

(|gH

AR,igRB,i|2 + |gHRB,igRB,i|2

), (15)

CDFi = pA,i

(|gH

AR,ieAR,i|2 + |gHRB,ieAR,i|2

)(16)

+ pB,i

(|gH

AR,ieRB,i|2 + |gHRB,ieRB,i|2

),

DDFi =

∑

j 6=i

pA,j

(|gH

AR,igAR,j|2 + |gHRB,igAR,j|2

)(17)

+∑

j 6=i

pB,j

(|gH

AR,igRB,j|2 + |gHRB,igRB,j|2

),

EDFi = ||gAR,i||2 + ||gRB,i||2. (18)

In addition, the SE of the TX,i → TR (X ∈ {A,B}) linkcan be obtained as

RDFXR,i =

τc − τp2τc

× (19)

E

log2

1 +XDF

i

E

{(CDF

i +DDFi + EDF

i

)|GAR, GRB

}

.

In the BC phase, the relay broadcasts to all users using theMR principle; hence the received signal at TX,i is given by

zDFX,i = ρDF

N∑

j=1

(gTXR,ig

∗RB,jxA,j + gT

XR,ig∗AR,jxB,j

)+ nX,i.

(20)

As in [5], [25], [30], we consider the realistic case where theusers do not have any instantaneous CSI and instead only havestatistical CSI since the acquisition of instantaneous CSIat theuser is extremely costly. Note that this type of lower boundshas already been proved to be very good due to the channelhardening [31]2. Therefore, after performing the partial self-interference cancellation according to the statistical knowledgeof the channel gains, the post-processing signals at TA,i can

2Note that there are two typical ways to obtain the instantaneous CSI, i.e.,downlink training and feedback. However, both methods incur huge overheads,hence they are not scalable in the massive MIMO ecosystem.

be re-expressed as

zDFA,i = zDF

A,i − ρDFE{gTAR,ig

∗RB,i

}xA,i (21)

= ρDFE{gTAR,ig

∗AR,i

}xB,i

︸ ︷︷ ︸

desired signal

+ ρDF(gTAR,ig

∗AR,i − E

{gTAR,ig

∗AR,i

})xB,i

︸ ︷︷ ︸

gain uncertainty

+ ρDF(gTAR,ig

∗RB,i − E

{gTAR,ig

∗RB,i

})xA,i

︸ ︷︷ ︸

residual self-interference

+ ρDF

∑

j 6=i

(gTAR,ig

∗RB,jxA,j + gT

AR,ig∗AR,jxB,j

)

︸ ︷︷ ︸

inter-user interference

+ nA,i︸︷︷︸

noise

,

and the post-processing signals at TB,i is obtained by replacingA andB in (21) with B andA.

Therefore, the SE of the TR → TX,i link is expressed as

RDFRX,i =

τc − τp2τc

log2(1 + SINRDF

RX,i

), (22)

where SINRDFRX,i is given by (23) (on the top of the next page).

Now, according to [32]–[36], the sum SE of thei-th userpair over both MAC and BC phases for the considered two-way DF relaying can be expressed as

RDFi = min

(RDF

1,i, RDF2,i

), (24)

whereRDF2,i is the sum SE of thei-th user pair in the BC phase,

which is given by the sum of the end-to-end SE from TA,i toTB,i (i.e., min

(RDF

AR,i, RDFRB,i

)) and the end-to-end SE from

TB,i to TA,i (i.e., min(RDF

BR,i, RDFRA,i

)),

RDF2,i = min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

). (25)

Thus, the sum SE of the multipair two-way DF relayingsystem is

RDF =

N∑

i=1

RDFi . (26)

When TR employs a very large antenna array, i.e.,M → ∞,a large-scale approximation of the SE of thei-th user pair ispresented in the following theorem.

Theorem 1:With the DF protocol, as the number of relayantennas grows to infinity, then we haveRDF

i − RDFi

M→∞−→ 0,whereRDF

i is given by

RDFi , min

(

RDF1,i, R

DF2,i

)

, (27)

5

SINRDFRX,i =

|E{gTXR,ig

∗XR,i

}|2

Var{

gTXR,ig

∗XR,i

}

+ Var{

gTAR,ig

∗RB,i

}

+∑

j 6=i

(

E

{

|gTXR,ig

∗RB,j|2

}

+ E

{

|gTXR,ig

∗AR,j|2

})

+ 1ρ2

DF

. (23)

where

RDF1,i ,

τc − τp2τc

log2

1 +tA,i + tB,i

(

σ2AR,i + σ2

RB,i

)

qi

, (28)

RDF2,i , min

(

RDFAR,i, R

DFRB,i

)

+ min(

RDFBR,i, R

DFRA,i

)

, (29)

with

RDFAR,i ,

τc − τp2τc

log2

1 +tA,i

(

σ2AR,i + σ2

RB,i

)

qi

, (30)

RDFRA,i ,

τc − τp2τc

log2

1 +

prMσ4AR,i

(prβAR,i + 1)N∑

j=1

(

σ2AR,j + σ2

RB,j

)

,

(31)

tA,i , pA,i

(Mσ4

AR,i + σ2AR,iσ

2RB,i

), (32)

tB,i , pB,i

(Mσ4

RB,i + σ2AR,iσ

2RB,i

), (33)

qi , pA,iσ2AR,i + pB,iσ

2RB,i (34)

+∑

j 6=i

(pA,jβAR,j + pB,jβRB,j) + 1,

and RDFBR,i and RDF

RB,i are obtained by replacing the transmitpowerspA,i, pB,i, and the subscripts “AR”, “RB” with thetransmit powerspB,i, pA,i, and the subscripts “RB”, “AR” inRDF

AR,i and RDFRA,i, respectively.

Proof: See Appendix A.Theorem 1 provides a large-scale approximation of thei-

th user pair’s SE. More specifically,RDF1,i, R

DFAR,i, and RDF

BR,i

are computed by utilizing Lemma 1 in Appendix A, whileRDF

RA,i and RDFRB,i are the exact expressions forRDF

RA,i andRDF

RB,i. We can see that all the signal-to-interference-plus-noiseratio (SINR) of RDF

1,i, RDFAR,i, R

DFBR,i, R

DFRA,i, and RDF

RB,i canbe expressed in the form ofax+b

cx+d, wherea > 0, b ≥ 0, c > 0,

d > 0, and x denotes the transmit power, i.e.,pA,i, pB,i,or pr. Since ax+b

cx+dincreases withx and converges toa

cas

x → ∞, we conclude that 1)RDFi is an increasing function

of pA,i, pB,i, and pr; and 2) whenpA,i → ∞, pB,i → ∞,and/orpr → ∞, RDF

i converges to a non-zero limit, due tostrong inter-user interference. Moreover, we observe thatRDF

i

increases with the number of relay antennasM , indicating thestrong advantage of employing massive antenna arrays at therelay, while decreases with the number of user pairsN , whichis expected since larger number of users increases the amountof inter-user interference.

IV. POWER-SCALING LAWS

In this section, we pursue a detailed investigation of thepower-scaling laws; that is, how the powers can be reduced

with M while retaining a desired SE. In the case wherethe users have different power levels from the beginning, thedifferences can be absorbed into theσ2

AR,i and σ2RB,i terms

without loss of generality. Thus, we assume that all the usershave the same transmit power, i.e.,pA,i = pB,i = pu. Then,we can characterize the interplay between the relay’s transmitpower pr, the user’s transmit powerpu, and the transmitpower of each pilot symbolpp, asM grows to infinity. Moreprecisely, we consider three different scenarios:

• Scenario A: Fixedpu andpr, while pp =Ep

Mγ with γ > 0,andEp being a constant. Such a scenario represents thepotential of power saving in the channel training stage.

• Scenario B: Fixedpp, while pu = Eu

Mα , pr = Er

Mβ , withα ≥ 0 andβ ≥ 0, andEu, Er are constants. Hence, thechannel estimation accuracy remains unchanged, and theobjective is to study the potential power savings in thedata transmission stage.

• Scenario C: This is the most general case wherepu =Eu

Mα , pr = Er

Mβ , andpp =Ep

Mγ , with α ≥ 0, β ≥ 0, andγ > 0, Eu, Er, andEp are constants.

1) Scenario A:We present the following power-scaling lawfor Scenario A.

Theorem 2:With the DF protocol, for fixedpu, pr andEp,whenpp =

Ep

Mγ with γ > 0, asM → ∞, we have

RDFi − min

(RDF

1,i, RDF2,i

) M→∞−→ 0, (35)

where

RDF1,i ,

τc − τp2τc

× (36)

log2

1 +pu

τpEp

Mγ−1

(β4AR,i + β4

RB,i

)

(

β2AR,i + β2

RB,i

)(

puN∑

j=1

(βAR,j + βRB,j) + 1

)

,

RDF2,i , min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

), (37)

6

with

RDFAR,i ,

τc − τp2τc

× (38)

log2

1 +pu

τpEp

Mγ−1β4AR,i

(

β2AR,i + β2

RB,i

)(

puN∑

j=1

(βAR,j + βRB,j) + 1

)

,

RDFRA,i ,

τc − τp2τc

× (39)

log2

1 +

prτpEp

Mγ−1β4AR,i

(prβAR,i + 1)N∑

j=1

(

β2AR,j + β2

RB,j

)

,

andRDFBR,i andRDF

RB,i are obtained by replacing the subscripts“AR”, “RB” in RDF

AR,i and RDFRA,i with the subscripts “RB”,

“AR”, respectively.As can be seen, the large-scale approximation of the SE

RDFi in Scenario A depends on the choice ofγ. When we cut

down pp too much, i.e.,γ > 1, RDFi converges to zero. On

the other hand, when0 < γ < 1, RDFi grows unboundedly.

Finally, whenγ = 1, RDFi converges to a non-zero limit.

2) Scenario B:Next, we turn our attention to Scenario Band present the following result.

Theorem 3:With the DF protocol, for fixedpp, Eu, andEr,whenpu = Eu

Mα , pr = Er

Mβ , with α ≥ 0, β ≥ 0, asM → ∞,we have

RDFi − min

(RDF

1,i, RDF2,i

) M→∞−→ 0, (40)

where

RDF1,i =

τc − τp2τc

log2

(

1 +Eu

Mα−1

σ4AR,i + σ4

RB,i

σ2AR,i + σ2

RB,i

)

, (41)

RDF2,i = min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

), (42)

with

RDFAR,i =

τc − τp2τc

log2

(

1 +Eu

Mα−1

σ4AR,i

σ2AR,i + σ2

RB,i

)

, (43)

RDFRA,i =

τc − τp2τc

log2

1 +

Er

Mβ−1

σ4AR,i

N∑

j=1

(

σ2AR,j + σ2

RB,j

)

,

(44)

andRDFBR,i andRDF

RB,i are obtained by replacing the subscripts“AR”, “RB” in RDF

AR,i and RDFRA,i with the subscripts “RB”,

“AR”, respectively.Theorem 3 indicates that when both the transmit power of

each userpu and the transmit power of the relaypr are scaleddown inversely proportional toM (asM → ∞), the effectsof estimation error, residual self-interference, and inter-userinterference vanish, and the only remaining impairment comesfrom the noise at users and the relay. Moreover, when eachuser’s transmit power is sufficiently large, i.e.,Eu → ∞, thelarge-scale approximation ofRDF

i is determined only byRDFRA,i

and RDFRB,i, suggesting that the bottleneck of SE appears in

the BC phase. In contrast, when the relay’s transmit powerbecomes large, i.e.,Er → ∞, the large-scale approximationof RDF

i is determined only byRDF1,i, R

DFAR,i, R

DFBR,i, indicating

that the bottleneck of SE occurs in the MAC phase.Also, when we cut down the transmit powers of the relay

and/or of each user too much, namely, 1)α > 1, andβ ≥ 0,2) α ≥ 0, andβ > 1, 3) α > 1, andβ > 1, RDF

i convergesto zero. On the contrary, when we cut down both the transmitpowers of the relay and of each user moderately, i.e.,0 ≤α < 1 and0 ≤ β < 1, RDF

i grows unboundedly. So the mostimportant task is how to select the parametersα and β tomakeRDF

i converge to a non-zero finite limit. We discuss thisin the following corollaries.

Corollary 1: With the DF protocol, for fixedpp, Eu, andEr, whenα = β = 1, namely,pu = Eu

M, pr = Er

M, asM →

∞, the SE of thei-th user pair has the limit

RDFi → min

(RDF

1,i, RDF2,i

), (45)

where

RDF1,i =

τc − τp2τc

log2

(

1 +Eu

(σ4AR,i + σ4

RB,i

)

σ2AR,i + σ2

RB,i

)

, (46)

RDF2,i = min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

), (47)

with

RDFAR,i =

τc − τp2τc

log2

(

1 +Euσ

4AR,i

σ2AR,i + σ2

RB,i

)

, (48)

RDFRA,i =

τc − τp2τc

log2

1 +

Erσ4AR,i

N∑

j=1

(

σ2AR,j + σ2

RB,j

)

, (49)

andRDFBR,i andRDF

RB,i are obtained by replacing the subscripts“AR”, “RB” in RDF

AR,i and RDFRA,i with the subscripts “RB”,

“AR”, respectively.Corollary 1 reveals that when both the transmit powers of

the relay and of each user are scaled down with the samespeed, i.e.,1/M , RDF

i converges to a non-zero limit. Moreover,this non-zero limit is an increasing function with respect to Eu

andEr , while a decreasing function with respect to the numberof user pairsN .

Corollary 2: With the DF protocol, for fixedpp, Eu, andEr, whenα = 1 and0 ≤ β < 1, namely,pu = Eu

M, pr = Er

Mβ ,asM → ∞, the SE of thei-th user pair has the limit

RDFi → τc − τp

2τclog2

(

1 +Eu

(σ4AR,i + σ4

RB,i

)

σ2AR,i + σ2

RB,i

)

. (50)

Corollary 2 suggests that when we cut down the transmitpower of each user too much, i.e.,0 ≤ β < α = 1, thelarge-scale approximation of the SE is determined by theperformance in the MAC phase, i.e.,RDF

1,i, which depends onlyonEu, and is independent ofEr. This result is expected sincewhen the transmit power of each user is much less than thetransmit power of the relay, the bottleneck of SE occurs in theMAC phase. On the other hand, when the transmit power of

7

the relay is cut down more compared with that of each user,i.e., 0 ≤ α < β = 1, the bottleneck of SE appears in the BCphase, thusRDF

i is determined byRDFRA,i andRDF

RB,i as shownin the following corollary.

Corollary 3: With the DF protocol, for fixedpp, Eu, andEr, when0 ≤ α < 1 andβ = 1, namely,pu = Eu

Mα , pr = Er

M,

asM → ∞, the SE of thei-th user pair has the limit

RDFi → τc − τp

2τclog2

1 +

Erσ4AR,i

N∑

j=1

(

σ2AR,j + σ2

RB,j

)

(51)

+τc − τp2τc

log2

1 +

Erσ4RB,i

N∑

j=1

(

σ2AR,j + σ2

RB,j

)

.

3) Scenario C:Finally, a corresponding power-scaling lawfor Scenario C is obtained as follows.

Theorem 4:With the DF protocol, for fixedEu, Er, andEp, whenpu = Eu

Mα , pr = Er

Mβ , andpp =Ep

Mγ , with α ≥ 0,β ≥ 0, andγ > 0, asM → ∞, we have

RDFi − min

(RDF

1,i, RDF2,i

) M→∞−→ 0, (52)

where

RDF1,i =

τc − τp2τc

log2

(

1 +τpEuEp

Mα+γ−1

β4AR,i + β4

RB,i

β2AR,i + β2

RB,i

)

, (53)

RDF2,i = min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

), (54)

with

RDFAR,i =

τc − τp2τc

log2

(

1 +

τpEuEp

Mα+γ−1β4AR,i

β2AR,i + β2

RB,i

)

, (55)

RDFRA,i =

τc − τp2τc

log2

1 +

τpErEp

Mβ+γ−1 β4AR,i

N∑

j=1

(

β2AR,j + β2

RB,j

)

, (56)

andRDFBR,i andRDF

RB,i are obtained by replacing the subscripts“AR”, “RB” in RDF

AR,i and RDFRA,i with the subscripts “RB”,

“AR”, respectively.

As expected, the large-scale approximation of the SERDFi

depends on the relationship betweenα, β, andγ. Moreover,the termα + γ determines the SE in the MAC phase, whileβ+γ determines the SE in the BC phase, as elaborated in thefollowing corollaries.

Corollary 4: With the DF protocol, for fixedEu, Er, andEp, whenα = β > 0 andα + γ = 1, namely,pu = Eu

Mα ,pr = Er

Mβ , andpp =Ep

Mγ , with γ > 0, asM → ∞, the SE ofthe i-th user pair has the limit

RDFi → min

(RDF

1,i, RDF2,i

), (57)

where

RDF1,i =

τc − τp2τc

log2

(

1 +τpEuEp

(β4AR,i + β4

RB,i

)

β2AR,i + β2

RB,i

)

,

(58)

RDF2,i = min

(RDF

AR,i, RDFRB,i

)+ min

(RDF

BR,i, RDFRA,i

), (59)

with

RDFAR,i =

τc − τp2τc

log2

(

1 +τpEuEpβ

4AR,i

β2AR,i + β2

RB,i

)

, (60)

RDFRA,i =

τc − τp2τc

log2

1 +

τpErEpβ4AR,i

N∑

j=1

(

β2AR,j + β2

RB,j

)

, (61)

andRDFBR,i andRDF

RB,i are obtained by replacing the subscripts“AR”, “RB” in RDF

AR,i and RDFRA,i with the subscripts “RB”,

“AR”, respectively.

Corollary 4 presents a trade-off between the transmit powersof each pilot symbol and of each user/the relay. In otherwords, if we cut down the transmit power of each pilotsymbol too much, which causes poor channel estimationaccuracy, the transmit power of each user/the relay should beincreased to compensate this imperfection and maintain thesame asymptotic SE.

Corollary 5: With the DF protocol, for fixedEu, Er, andEp, whenα > β ≥ 0 and α + γ = 1, namely,pu = Eu

Mα ,pr = Er

Mβ , andpp =Ep

Mγ , with γ > 0, asM → ∞, the SE ofthe i-th user pair has the limit

RDFi → τc − τp

2τclog2

(

1 +τpEuEp

(β4AR,i + β4

RB,i

)

β2AR,i + β2

RB,i

)

.

(62)

From Corollary 5, we can see that the limit ofRDFi is an

increasing function with respect toEu andEp, indicating thatwe can boost the SE by increasing the transmit power of eachuser and of each pilot symbol. In addition, the limit ofRDF

i isindependent ofN , indicating that the sum SE of the systemis an increasing function with respect toN .

Corollary 6: With the DF protocol, for fixedEu, Er, andEp, when 0 ≤ α < β and β + γ = 1, namely,pu = Eu

Mα ,pr = Er

Mβ , andpp =Ep

Mγ , with γ > 0, asM → ∞, the SE ofthe i-th user pair has the limit

RDFi → τc − τp

2τclog2

1 +

τpErEpβ4AR,i

N∑

j=1

(

β2AR,j + β2

RB,j

)

(63)

+τc − τp2τc

log2

1 +

τpErEpβ4RB,i

N∑

j=1

(

β2AR,j + β2

RB,j

)

.

Corollary 6 provides the trade-off between the transmitpowers of the relay and of each pilot symbol.

8

V. POWER ALLOCATION

Power allocation is an effective means to enhance the sumSE of the system. In this section, we assume that the designfor channel training stage is done in advance, i.e., the pilotpower pp is determined. We are interested in designing apower allocation algorithm in the data transmission stagemaximizing the sum SE subject to a total power constraint,

i.e.,N∑

i=1

(pA,i + pB,i) + pr ≤ P . For notational simplicity,

we defineN , {1, . . . , N}, pA , [pA,1, . . . , pA,N ]T , and

pB , [pB,1, . . . , pB,N ]T .

For analytical tractability, we use the large-scale approx-imation (27) in Theorem 1 instead of the exact expression(24). Hence, the power allocation optimization problem isformulated as

maximizepA,pB,pr

N∑

i=1

RDFi (64)

subject toN∑

i=1

(pA,i + pB,i) + pr ≤ P

pA ≥ 0,pB ≥ 0, pr ≥ 0

RDFi ≥ Rmin, i ∈ N

where Rmin is the minimum SE requirement of thei-thuser pair. Sincelog(·) is an increasing function, (64) can beequivalently reformulated asPDF

1 :

minimizepA,pB,pr

γDFi

,γDFA,i

,γDFB,i

N∏

i=1

(1 + γDF

i

)−1

subject to γDFi ≤ aDF

i pA,i + bDFi pB,i

N∑

j=1

(cDFi,jpA,i + dDF

i,jpB,i

)+ 1

, i ∈ N

γDFi ≤ γDF

A,i + γDFB,i + γDF

A,iγDFB,i, i ∈ N

γDFA,i ≤ min

{aDFi pA,i

gi,

preDFi pr + fDF

i

}

, i ∈ N

γDFB,i ≤ min

{

bDFi pB,i

gi,

pr

eDFi pr + fDF

i

}

, i ∈ N

N∑

i=1

(pA,i + pB,i) + pr ≤ P

pA ≥ 0,pB ≥ 0, pr ≥ 0

(γDFi

)−1(

22τcRminτc−τp − 1

)

≤ 1, i ∈ N

where γDF ,[γDF1 , . . . , γDF

N

]T, γDF

A ,[γDFA,1, . . . , γ

DFA,N

]T,

γDFB ,

[γDFB,1, . . . , γ

DFB,N

]T, gi =

N∑

j=1

(cDFi,jpA,i + dDF

i,jpB,i

)+ 1,

aDFi =

Mσ4AR,i+σ2

AR,iσ2RB,i

σ2AR,i

+σ2RB,i

, bDFi =

Mσ4RB,i+σ2

AR,iσ2RB,i

σ2AR,i

+σ2RB,i

,

cDFi,j =

{

σ2AR,i, j = i,

βAR,j , j 6= i,,

eDFi =

βRB,i

Mσ4RB,i

N∑

j=1

(σ2AR,j + σ2

RB,j

), fDF

i =

1Mσ4

RB,i

N∑

j=1

(σ2AR,j + σ2

RB,j

), anddDF

i,j , eDFi , fDF

i are obtained

by replacing the subscripts “AR”, “RB” with “RB”, “AR” incDFi,j , e

DFi , fDF

i , respectively.The above problemPDF

1 is identified as a comple-mentary geometric programming (CGP) problem, whichis nonconvex. Also,γDF

i , γDFA,i, and γDF

B,i are consideredas the SINR of min

(RDF

1,i, RDF2,i

), min

(RDF

AR,i, RDFRB,i

), and

min(RDF

BR,i, RDFRA,i

), respectively. In addition, we have re-

placed the equality “=” with “ ≤” in the first four constraintsof problem PDF

1 ; however, this does not change or relaxthe original problem (64), since the objective function isdecreasing withγDF

i . Therefore, we can guarantee that thesefour constraints must be active at any optimal solution ofPDF

1 .Although the CGP problem is nonconvex, we can obtain

its local optimum solution by jointly solving a sequenceof convex GP problems; this is a technique that has beenwidely used in the resource allocation literature, such as [5],[22], [37]–[40]. Next, we are dedicated to transformingPDF

1

into a standard GP problem that can be solved efficientlywith standard optimization tools such as CVX or ggplab.Since the objective function1 + γDF

i can be approximated

by a monomial functionωDFi

(γDFi

)µDFi , whereµDF

i =γDFi

1+γDFi

and ωDFi =

(γDFi

)−µDFi(1 + γDF

i

)(which are obtained by

guaranteeing that both the values and the first gradients ofthe approximations and of the original functions are equalat the same specific points, i.e., by solving the equations

ωDFi

(γDFi

)µDFi = 1 + γDF

i and ωDFi µDF

i

(γDFi

)µDFi −1

= 1), themain challenge to convertPDF

1 into a GP problem is totransform the first two inequality constraints into the formof posynomials. According to [40], [41], the geometric meanis no larger than the arithmetic mean for any set of positivenumbers; thus we have

aDFi pA,i + bDF

i pB,i ≥(

aDFi pA,i

νDFA,i

)νDFA,i(

bDFi pB,i

νDFB,i

)νDFB,i

, (65)

where νDFA,i =

aDFi pA,i

aDFi pA,i+bDF

i pB,i, νDF

B,i =bDFi pB,i

aDFi pA,i+bDF

i pB,i, and

pA,i, pB,i are the initialization values.As a result, the first inequality constraint inPDF

1 can beapproximated by [40]

γDFi ≤

(aDFi pA,i

νDFA,i

)νDFA,i(

bDFi pB,i

νDFB,i

)νDFB,i

N∑

j=1

(cDFi,jpA,i + dDF

i,jpB,i

)+ 1

, i ∈ N . (66)

Now, we focus on the approximation of the second inequal-ity constraint. Following the idea proposed in [37, Lemma1], we use a monomial functiong(x, y) = ηxλ1yλ2 toapproximatef(x, y) = x + y + xy near an arbitrary pointx, y > 0. To make the approximation accurate, we need toensure that the following equations hold and then obtain itssolution:

x+ y + xy = ηxλ1 yλ2

1 + y = ηλ1xλ1−1yλ2

1 + x = ηλ2xλ1 yλ2−1

⇒

λ1 = x(1+y)x+y+xy

,

λ2 = y(1+x)x+y+xy

,

η = (x+ y + xy) x−λ1 y−λ2 .

9

To this end, the second inequality constraint inPDF1 can be

approximated by

γDFi ≤ ηDF

i

(γDFA,i

)λDFA,i(γDFB,i

)λDFB,i , i ∈ N , (67)

whereηDFi =

(γDFA,i + γDF

B,i + γDFA,iγ

DFB,i

) (γDFA,i

)−λDFA,i(γDFB,i

)−λDFB,i ,

λDFA,i =

γDFA,i(1+γDF

B,i)γDFA,i

+γDFB,i

+γDFA,i

γDFB,i

,

λDFB,i =

γDFB,i(1+γDF

A,i)γDFA,i

+γDFB,i

+γDFA,i

γDFB,i

, andγDFA,i, γ

DFB,i are the initialization

values.We now outline the steps to solve the original problemPDF

1

in Algorithm 1.Note that we have removedωDF

i in the objective function,since they do not affect the optimization problem. Also, fiveextra inequalities as trust region constraints are included,which limit how much the variables are allowed to differfrom the current guessγDF

i , γDFA,i, and γDF

B,i. The limit of anyconvergent sequence generated by Algorithm 1 is a Karush-Kuhn-Tucker point, and the detailed proof can be found in[42]. The result holds provided Slater’s constraint qualificationcondition holds. The parameterθ > 1 controls the desiredaccuracy. More precisely, whenθ is close to 1 it provides goodaccuracy for the monomial approximation but with slowerconvergence speed, and vice versa ifθ is large. As discussedin [41], θ = 1.1 offers a good tradeoff between the accuracyand convergence speed.

Algorithm 1 focuses on the case where each user transmitswith a different power, and yields a local optimum of theoriginal problemPDF

1 by solving a sequence of GPs. Now, weturn our attention to the scenario where all the users transmitwith the same power, i.e.,pA,i = pB,i = pu and Rmin isvery low such that the constraintRDF

i ≥ Rmin, i ∈ N can beneglected; hence, the problem (64) reduces to the followingspecial case:

PDF3 : maximize

pu,pr

N∑

i=1

RDFi

subject to 2Npu + pr ≤ P, pu ≥ 0, pr ≥ 0.

Theorem 5:PDF3 is a convex optimization problem.

Proof: See Appendix B.Since the optimization problemPDF

3 is convex, the optimalsolutionspDF,opt

u ∈(0, P

2N

]andpDF,opt

r ∈ (0, P ] maximizing thesum SE can be obtained efficiently by adopting some standardtechniques, such as the bisection method with respect toP ,due to the convexity of the optimization problemPDF

3 .

VI. N UMERICAL RESULTS

We now present numerical results to validate the aboveanalytical results. Unless otherwise specified, the followingset of parameters are used in simulation. The length of thecoherence interval isτc = 196 (symbols), chosen by theLTE standard. The different large-scale fading parametersare arbitrarily generated byβAR,i = zi (rAR,i/r0)

α andβRB,i = zi (rRB,i/r0)

α, where zi is a log-normal randomvariable with standard deviation8 dB, rAR,i and rRB,i arethe locations of TAR,i and TRB,i from the relay,α = 3.8is the path loss exponent, andr0 denotes the guard interval

Algorithm 1 Successive approximation algorithm forPDF1

1) Initialization. Define a toleranceǫ and parameterθ. Setk =1, the initial values ofγDF

i , γDFA,i andγDF

B,i are chosen accordingto the SINR in Theorem 1. Also, we setpA,i = pB,i =

P4N ,

2) Iteration k. Compute µDFi =

γDFi

1+γDFi

,

νDFA,i =

aDFi pA,i

aDFi pA,i+bDF

i pB,i, νDF

B,i =bDFi pB,i

aDFi pA,i+bDF

i pB,i,

ηDFi =

(γDFA,i + γDF

B,i + γDFA,iγ

DFB,i

) (γDFA,i

)−λDFA,i(γDFB,i

)−λDFB,i ,

λDFA,i =

γDFA,i(1+γDF

B,i)γDFA,i

+γDFB,i

+γDFA,i

γDFB,i

, λDFB,i =

γDFB,i(1+γDF

A,i)γDFA,i

+γDFB,i

+γDFA,i

γDFB,i

. Then,

solve the following GP problemPDF2 :

minimizepA,pB,pr

γDFi

,γDFA,i

,γDFB,i

N∏

i=1

(γDFi

)−µDFi

subject to θ−1pDFA,i ≤ pDF

A,i ≤ θpDFA,i, i ∈ N

θ−1pDFB,i ≤ pDF

B,i ≤ θpDFB,i, i ∈ N

θ−1γDFi ≤ γDF

i ≤ θγDFi , i ∈ N

θ−1γDFA,i ≤ γDF

A,i ≤ θγDFA,i, i ∈ N

θ−1γDFB,i ≤ γDF

B,i ≤ θγDFB,i, i ∈ N

γDFi (uA,i)

−νDFA,i

(

bDFi pB,i

νDFB,i

)−νDFB,i

gi ≤ 1, i ∈ N

γDFi

(ηDFi

)−1 (γDFA,i

)−λDFA,i(γDFB,i

)−λDFB,i ≤ 1, i ∈ N

γDFA,i

(aDFi

)−1p−1A,igi ≤ 1, i ∈ N

γDFB,i

(bDFi

)−1p−1B,igi ≤ 1, i ∈ N

γDFA,ip

−1r

(eDFi pr + fDF

i

)≤ 1, i ∈ N

γDFB,ip

−1r

(

eDFi pr + fDF

i

)

≤ 1, i ∈ NN∑

i=1

(pA,i + pB,i) + pr ≤ P

pA ≥ 0,pB ≥ 0, pr ≥ 0

(γDFi

)−1(

22τcRminτc−τp − 1

)

≤ 1, i ∈ N

whereuA,i =aDFi pA,i

νDFA,i

.

Denote the optimal solutions byp(k),DFA,i , p

(k),DFB,i , γ

(k),DFi ,

γ(k),DFA,i , γ(k),DF

B,i , i ∈ N .

3) Stopping criterion. If maxi |p(k),DFA,i − pDF

A,i| < ǫ and/or

maxi |p(k),DFB,i − pDF

B,i| < ǫ and/ormaxi |γ(k),DFi − γDF

i | < ǫ

and/ormaxi |γ(k),DFA,i −γDF

A,i| < ǫ and/ormaxi |γ(k),DFB,i −γDF

B,i| <ǫ, stop; otherwise, go to step 4).4) Update initial values. Set pDF

A,i = p(k),DFA,i , pDF

B,i = p(k),DFB,i ,

γDFi = γ

(k),DFi , γDF

A,i = γ(k),DFA,i , γDF

B,i = γ(k),DFB,i , andk = k+1.

Go to step 2).

which specifies the nearest distance between the users andthe relay. The relay is located at the center of a cell witha radius of1000 meters andr0 = 100 meters. We chooseN = 5, βAR = [0.2688, 0.0368, 0.00025, 0.1398, 0.0047], andβRB = [0.0003, 0.00025, 0.0050, 0.0794, 0.0001]. The length

10

of the pilot sequences isτp = 2N which is the minimumrequirement. We assume that each user has the same transmitpower, i.e.,pA,i = pB,i = pu.

A. Validation of analytical expressions and evaluating theimpact ofτp

We assume thatpp = pu, and that the total transmit powerof theN user pairs is equal to the transmit power of the relay,i.e., pr = 2Npu.

Fig. 2(a) shows the sum SE versus the transmit power ofeach userpu for different number of relay antennas. Note thatthe “Approximations” curves are obtained by using (27), andthe “Numerical results” curves are generated according to (24)by averaging over104 independent channel realizations. Ascan be readily observed, the large-scale approximations arevery accurate, especially for large antenna arrays. Moreover,we can see that increasing the number of relay antennassignificantly yields higher SE, as expected.

Fig. 2(b) studies the impact ofτp on the sum SE. Wecan see that at moderate and lowpp, there is an optimalτp maximizing the sum SE. Also, the optimal length of pilotsequence decreases with the transmit power of pilot sequencepp. In contrast, at highpp, i.e., pp = 10 dB, the sum SE is adecreasing function with respect toτp, which means that theoptimal τp is equal to2N .

B. Power-scaling laws

In this subsection, we provide numerical simulation resultsto verify the power-scaling laws presented in the previoussubsections, and investigate the potential for power savingwhen employing large number of antennas at the relay. Sinceour goal is to show the general power saving behavior and itis unnecessary to pay much attention to one particular user’slocation, here we setβAR,i = βRB,i = 1. Note that thecurves labeled as “Approximations” are obtained accordingto Theorems 1.

1) Scenario A: Fig. 3 verifies the analytical results forScenario A. The curves labelled as “Scenario A”, are plottedaccording to Theorem 2. It can be readily observed that inthe largeM regime, the asymptotic curves converge to theexact curves, demonstrating the accuracy of the asymptoticanalysis. In addition, whenγ > 1, i.e., γ = 2, the SEgradually approaches zero. In contrast, when0 < γ < 1, i.e.,γ = 0.8, the SE of both schemes grows unbounded. Finally,when γ = 1, the SE converges to a non-zero limit for bothschemes.

2) Scenario B:Fig. 4 investigates how the scaling of thetransmit power of each userpu = Eu

Mα and the transmit powerof the relaypr = Er

Mβ affects the achievable SE. Note thatthe curves labelled as “Scenario B” are generated by usingTheorem 3, while the curves labelled as “Scenario B-Case X”with X ∈ {I, II , III } are plotted according to Corollaries 1–3,respectively.

Fig. 4(a) studies three different cases according to the valuesof α andβ. In agreement with Corollaries 1–3, the sum SEsaturates in the asymptotical largeM regime for all the three

−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

pu (dB)

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

Numerical results: M = 200Approximations: M = 200Numerical results: M = 50Approximations: M = 50

(a) Validation of analytical expressions

20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

10

τp

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

p

p = 10 dB

pp = −10 dB

pp = −15 dB

pp = −20 dB

(b) Impact ofτp

Fig. 2: Validation of analytical expressions and impact ofτpfor N = 5, pp = pu andpr = 2Npu.

500 1000 1500 2000 2500 3000

6

8

10

12

14

Number of relay antennas M

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

Scenario AApproximations

500 1000 1500 2000 2500 30000

5

10

Number of relay antennas M

Scenario AApproximations

γ = 1

γ = 2

γ = 0.8

Fig. 3: Sum SE versus the number of relay antennasM forN = 5, pu = 10 dB, pr = 20 dB, andpp = Ep/M

γ withEp = 10 dB.

cases. As readily observed, Case I and Case II achieve thesame performance due to the setting ofEr = 2NEu.

Fig. 4(b) illustrates the other two extreme scenarios wherethe transmit power down-scaling is either too aggressive ortoomoderate. For the former scenario, three different cases arestudied, i.e.,α > 1, β ≥ 0, α ≥ 0, β > 1, andα > 1, β > 1.

11

500 1000 1500 2000 2500 30006

7

8

9

Number of relay antennas M

Scenario B−Case IApproximations

500 1000 1500 2000 2500 30006

7

8

9

Number of relay antennas M

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

Scenario B−Case IIApproximations

500 1000 1500 2000 2500 3000

10

15

20

Number of relay antennas M

Scenario B−Case IIIApproximationsα = 0.4, β = 1

α = 1, β = 0.2

α = 1, β = 1

(a) Case I II, and III

500 1000 1500 2000 2500 30000

5

10

Number of relay antennas M

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

Scenario BApproximations

500 1000 1500 2000 2500 3000

10

15

20

Number of relay antennas M

Scenario BApproximationsα = 0.9, β = 0.7

α = 0.6, β = 0.8

α = 0.6, β = 1.2

α = 1.5, β = 1.2

α = 1.3, β = 0.8

(b) Zero limit and unbounded

Fig. 4: Sum SE versus the number of relay antennasM forN = 5, pp = 10 dB, pu = Eu/M

α with Eu = 10 dB, andpr = Er/M

β with Eu = 20 dB.

As expected, when the number of relay antennas increases,the sum SE gradually reduces to zero. However, the speedof reduction varies significantly depending on the scalingparameters. The larger the scaling parameters, the faster thedecay SE. On the other hand, when we cut down the transmitpowers of each user and of the relay moderately, the sum SEgrows unboundedly.

3) Scenario C:Fig. 5 demonstrates the tradeoff between theuser/relay power and the pilot symbol power. For illustrationpurposes, two extreme scenarios where the transmit powerdown-scaling is either too aggressive or too moderate areconsidered. For the former scenario, two sets of curves aredrawn according toα = 1.3, β = 1.1, γ = 0.5 andα = 0.8,β = 0.6, γ = 1, which satisfyα+ γ = 1.8 andβ + γ = 1.6.When the number of relay antennas grows large, the sum SEof all system configurations smoothly converges to zero, aspredicted. Moreover, the gaps between the two sets of curvesreduce withM and eventually vanish. This indicates that aslong asα + γ and β + γ are the same, the asymptotic sumSE remains unchanged. Now, let us focus on the two curvesassociated withN = 5. Interestingly, we see that the curveassociated withγ = 0.5 yields better sum SE in the finiteantenna regime, despite the fact that the user or relay poweris

over-reduced compared to theγ = 1 case, which suggests thatit is of crucial importance to improve the channel estimationaccuracy. The same behavior appears for the unbounded SEscenario whereα+ γ = 0.6 andβ + γ = 0.7.

500 1000 1500 2000 2500 30000

1

2

3

Number of relay antennas M

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

α = 1.3, β = 1.1, γ = 0.5α = 0.8, β = 0.6, γ = 1

500 1000 1500 2000 2500 30000

10

20

30

Number of relay antennas M

α = 0.5, β = 0.6, γ = 0.1α = 0.4, β = 0.5, γ = 0.2

N = 2

N = 5

N = 5

N = 10

Fig. 5: Sum SE versus the number of relay antennasM forpu = Eu/M

α with Eu = 10 dB, pr = Er/Mβ with

Er = 15 dB, andpp = Ep/Mγ with Ep = 0 dB.

C. Power allocation

Fig. 6(a) illustrates the impact of the optimal power al-location scheme on the sum SE when all users’ large-scalefading are different. The optimal power allocation curvesare generated by Algorithm 1. As a benchmark scheme forcomparison, we also plot the sum SE with uniform powerallocation, i.e., the relay transmit power equals to the sumuser transmit power. As can be observed, the optimal powerallocation policy respectively provides79.11% and 82.67%SE enhancement forpp = 10 dB and pp = 20 dB whenM = 300, indicating that high channel estimation accuracyslightly boosts the power allocation efficiency. Moreover,byfocusing on the case where every user has the same transmitpower, i.e,pA,i = pB,i = pu, we can see that the optimaluser transmit power is a decreasing function with respect tothe number of user pairsN , for a given power budgetP = 10dB.

Fig. 6(b) further examines the impact of key system pa-rameters such asM , andpp on the optimal power allocationscheme. As can be seen, with fixed number of user pairsN = 5, the optimal user transmit powerpDF,opt

u with M = 100is larger than that withM = 50, suggesting that we shouldincrease the transmit power of each user when the number ofrelay antennas is large. In addition, when the pilot trainingpower increases, i.e., frompp = −20 dB to pp = 0 dB, theoptimal user transmit powerpDF,opt

u also increases, indicatingthat when the channel estimation accuracy is improved, weneed to use a higher transmit power for each user.

VII. C ONCLUSION

The paper studied the sum SE of a multipair two-way DFrelaying system with the MR processing by taking realisticCSI assumption into account. In particular, a closed-form

12

100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

Number of relay antennas M

Optimal power allocation: p

p = 10 dB

Uniform power allocation: pp = 10 dB

Optimal power allocation: pp = 20 dB

Uniform power allocation: pp = 20 dB

−10 −8 −6 −4 −2 00

2

4

6

8

pu (dB)

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

N = 10N = 5

−0.33.3

(a) Benefits of optimal power allocation and impact ofN

−10 −8 −6 −4 −2 00

5

10

pu (dB)

M = 100M = 50

−10 −8 −6 −4 −2 00

2

4

6

8

pu (dB)

Sum

spe

ctra

l effi

cien

cy (

bit/s

/Hz)

p

p = 0 dB

pp = −20 dB

−0.2

−0.3

−0.3

−0.4

(b) Impact ofM andpp

Fig. 6: Sum SE for a given power budgetP = 10 dB.

large-scale approximation of the SE was obtained that istight when the number of relay antennas is large. Basedon the approximation, the system’s power-scaling laws werecharacterized, which demonstrated that the transmit powersof the users, relay, and pilot symbol can be substantiallyreduced to maintain the desired SE. In addition, it was revealedthat there exists a tradeoff between the user/relay transmitpower and pilot symbol power, which provides great flexibilityfor the design of practical systems. Finally, the transmitpowers of each user and the relay were jointly optimized interms of maximizing the sum SE, which substantially enhancethe performance compared to the uniform power allocationscheme. For the special case where all users have the sametransmit power, it turns out that the users should decrease theirtransmit power when the number of user pairs becomes large.On the other hand, the users should increase their transmitpower when the number of relay antennas increases or whenthe channel estimation accuracy improves.

APPENDIX APROOF OFTHEOREM 1

Here we only present the detailed derivation forRDF1,i and

RDFRA,i, sinceRDF

AR,i and RDFBR,i can be obtained in a straight-

forward way, whileRDFRB,i can be derived in the same fashion

as RDFRA,i.

First, we focus on (13), which consists of five terms: 1)desired signal power of TB,i A

DFi ; 2) desired signal power of

TA,i BDFi ; 3) estimation errorCDF

i ; 4) inter-user interferenceDDF

i ; 5) compound noiseEDFi . For each of these five terms, we

will subsequently derive a deterministic equivalent expression.Before proceeding, we first review some useful results,

which are given in the following lemma.Lemma 1:Let x ∼ CN (0, σ2

xIM ) andy ∼ CN (0, σ2yIM ).

Assume thatx andy are mutually independent. Then, we have

1

Mx†x

a.s.→ σ2x, M → ∞, (68)

1

Mx†y

a.s.→ 0, M → ∞, (69)

1

M2|x†y|2 − 1

Mσ2xσ

2y

a.s.→ 0, M → ∞. (70)

Now, we compute the five terms one by one.1) Deterministic equivalent forADF

i : we have 1M2A

DFi =

pA,i

M2

(|gH

AR,igAR,i|2 + |gHRB,igAR,i|2

).

Then, by invoking Lemma 1, we have

1

M2ADF

i − pA,i

(

σ4AR,i +

1

Mσ2AR,iσ

2RB,i

)

a.s.→ 0. (71)

2) Deterministic equivalents forBDFi , CDF

i , DDFi , andEDF

i :Similarly, we obtain

1

M2BDF

i − pB,i

(

σ4RB,i +

1

Mσ2AR,iσ

2RB,i

)

a.s.→ 0, (72)

1

M2CDF

i − (73)

1

M

(σ2AR,i + σ2

RB,i

) (pA,iσ

2AR,i + pB,iσ

2RB,i

) a.s.→ 0,

1

M2DDF

i − (74)

1

M

∑

j 6=i

(pA,jβAR,j + pB,jβRB,j)(σ2AR,i + σ2

RB,i

) a.s.→ 0,

1

M2EDF

i − 1

M

(σ2AR,i + σ2

RB,i

) a.s.→ 0. (75)

Substituting (71), (72), (73), (74), and (75) into (13) and(19), and after some algebraic manipulations, we obtainRDF

1,i,RDF

AR,i, RDFBR,i.

Now, we turn our attention to deriveRDFRA,i.

1) ComputeE{gTAR,ig

∗AR,i

}:

E

{gTAR,ig

∗AR,i

}= E

{||gAR,i||2

}+ E

{eTAR,ig

∗AR,i

}(76)

= Mσ2AR,i.

2) Compute Var{gTAR,ig

∗AR,i

}:

Var{gTAR,ig

∗AR,i

}(77)

= E

{||gAR,i||4

}+ E

{|eTAR,ig

∗AR,i|2

}−M2σ4

AR,i

= Mσ2AR,iβAR,i.

3) Compute Var{gTAR,ig

∗RB,i

}:

Var{gTAR,ig

∗RB,i

}(78)

= E

{|gT

AR,ig∗RB,i|2

}− |E

{gTAR,ig

∗RB,i

}|2 = Mσ2

RB,iβAR,i.

4) Compute∑

j 6=i

(E

{|gT

AR,ig∗RB,j |2

}+ E

{|gT

AR,ig∗AR,j |2

}):

13

For j 6= i, we obtain

E

{|gT

AR,ig∗RB,j |2

}= Mσ2

RB,jβAR,i, (79)

E

{|gT

AR,ig∗AR,j |2

}= Mσ2

AR,jβAR,i. (80)

Thus, we have∑

j 6=i

(E

{|gT

AR,ig∗RB,j |2

}+ E

{|gT

AR,ig∗AR,j |2

})(81)

= MβAR,i

∑

j 6=i

(σ2AR,j + σ2

RB,j

).

Combining (76), (77), (78), and (81) completes the proof.

APPENDIX BPROOF OFTHEOREM 5

Using the same argument as in the proof of Theorem 5, itcan be proved that the objective function inPDF

3 is maximizedwhen2Npu + pr = P .

Before studying the properties ofRDFi , we first present the

following useful lemma.

Lemma 2:The functionsg1(x) = log2

(

1 + a1xb1x+c1

)

and

g2(x) = log2

(

1 + a2(d2−x)b2(d2−x)+c2

)

are all strictly concave withrespect tox whena1, b1, c1, a2, b2, c2, d2 > 0.

Now, focusing onRDFi and substituting2Npu + pr = P

into RDFi , it is easy to show thatRDF

1,i, RDFAR,i, andRDF

BR,i canbe reformulated asg1(pu), while RDF

RA,i, and RDFRB,i can be

reformulated asg2(pu), hence,RDF1,i, RDF

AR,i, RDFBR,i, RDF

RA,i,and RDF

RB,i are all concave functions with respect topu.Due to the convexity preservation property of point-

wise maximum and nonnegative weighted sums operations[43], RDF

2,i = min(

RDFAR,i, R

DFRB,i

)

+ min(

RDFBR,i, R

DFRA,i

)

is also a concave function with respect topu. Therefore,min

(

RDF1,i, R

DF2,i

)

is a concave function, which completes theproof.

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Chuili Kong (S’15) received the B.S. degree in elec-tronics and information engineering from the DalianUniversity of Technology, Dalian, China, in 2013.She is currently working towards the Ph.D. degree ininformation and communication engineering at theZhejiang University. From October 2016 to October2017, she was a visiting scholar at University ofCalifornia, Irvine. She was an exemplary reviewerfor IEEE TRANSACTIONS ONWIRELESSCOMMU-NICATIONS in 2017. Her research interests includemassive MIMO systems and relaying communica-

tions.

Caijun Zhong (S’07-M’10-SM’14) received theB.S. degree in Information Engineering from theXi’an Jiaotong University, Xi’an, China, in 2004,and the M.S. degree in Information Security in 2006,Ph.D. degree in Telecommunications in 2010, bothfrom University College London, London, UnitedKingdom. From September 2009 to September 2011,he was a research fellow at the Institute for Electron-ics, Communications and Information Technologies(ECIT), Queens University Belfast, Belfast, UK.Since September 2011, he has been with Zhejiang

University, Hangzhou, China, where he is currently an associate professor.His research interests include massive MIMO systems, full-duplex communi-cations, wireless power transfer and physical layer security.

Dr. Zhong is an Editor of the IEEE TRANSACTIONS ONWIRELESSCOM-MUNICATIONS, IEEE COMMUNICATIONS LETTERS, EURASIP JOURNAL

ON WIRELESS COMMUNICATIONS AND NETWORKING, and JOURNAL OFCOMMUNICATIONS AND NETWORKS. He is the recipient of the 2013 IEEEComSoc Asia-Pacific Outstanding Young Researcher Award. Heand hiscoauthors has been awarded a Best Paper Award at the WCSP 2013. He wasan Exemplary Reviewer for IEEE TRANSACTIONS ON COMMUNICATIONS

in 2014.

Michail Matthaiou (S’05–M’08–SM’13) was bornin Thessaloniki, Greece in 1981. He obtained theDiploma degree (5 years) in Electrical and Com-puter Engineering from the Aristotle University ofThessaloniki, Greece in 2004. He then received theM.Sc. (with distinction) in Communication Systemsand Signal Processing from the University of Bristol,U.K. and Ph.D. degrees from the University ofEdinburgh, U.K. in 2005 and 2008, respectively.From September 2008 through May 2010, he waswith the Institute for Circuit Theory and Signal

Processing, Munich University of Technology (TUM), Germany working as aPostdoctoral Research Associate. He is currently a Reader (equivalent to As-sociate Professor) in Multiple-Antenna Systems at Queen’sUniversity Belfast,U.K. after holding an Assistant Professor position at Chalmers University ofTechnology, Sweden. His research interests span signal processing for wirelesscommunications, massive MIMO, hardware-constrained communications, andperformance analysis of fading channels.

Dr. Matthaiou and his coauthors received the 2017 IEEE CommunicationsSociety Leonard G. Abraham Prize. He was the recipient of the2011 IEEEComSoc Best Young Researcher Award for the Europe, Middle East andAfrica Region and a co-recipient of the 2006 IEEE Communications ChapterProject Prize for the best M.Sc. dissertation in the area of communications.He was co-recipient of the Best Paper Award at the 2014 IEEE InternationalConference on Communications (ICC) and was an Exemplary Reviewerfor IEEE COMMUNICATIONS LETTERS for 2010. In 2014, he received theResearch Fund for International Young Scientists from the National NaturalScience Foundation of China. In the past, he was an AssociateEditor forthe IEEE TRANSACTIONS ON COMMUNICATIONS, Associate Editor/SeniorEditor for IEEE COMMUNICATIONS LETTERSand was the Lead Guest Editorof the special issue on “Large-scale multiple antenna wireless systems” ofthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He wasa co-chair of the Wireless Communications Symposium (WCS) at IEEEGLOBECOM 2016.

Emil Bj ornson (S’07, M’12, SM’17) received theM.S. degree in Engineering Mathematics from LundUniversity, Sweden, in 2007. He received the Ph.D.degree in Telecommunications from KTH Royal In-stitute of Technology, Sweden, in 2011. From 2012to mid 2014, he was a joint postdoc at the Alcatel-Lucent Chair on Flexible Radio, SUPELEC, France,and at KTH. He joined Linkoping University, Swe-den, in 2014 and is currently Associate Professor andDocent at the Division of Communication Systems.

He performs research on multi-antenna commu-nications, Massive MIMO, radio resource allocation, energy-efficient com-munications, and network design. He is on the editorial board of the IEEETRANSACTIONS ONCOMMUNICATIONS and the IEEE TRANSACTIONS ON

GREEN COMMUNICATIONS AND NETWORKING. He is the first author ofthe textbooksMassive MIMO Networks: Spectral, Energy, and HardwareEfficiency(2017) andOptimal Resource Allocation in Coordinated Multi-CellSystems(2013). He is dedicated to reproducible research and has made a largeamount of simulation code publicly available.

Dr. Bjornson has performed MIMO research for more than ten years andhas filed more than ten related patent applications. He received the 2016 BestPhD Award from EURASIP, the 2015 Ingvar Carlsson Award, and the 2014Outstanding Young Researcher Award from IEEE ComSoc EMEA. He hasco-authored papers that received best paper awards at WCSP 2017, IEEEICC 2015, IEEE WCNC 2014, IEEE SAM 2014, IEEE CAMSAP 2011, andWCSP 2009.

15

Zhaoyang Zhang(M’00) received his Ph.D. degreein communication and information systems fromZhejiang University, Hangzhou, China, in 1998. S-ince then he has been with the College of Informa-tion Science and Electronic Engineering, ZhejiangUniversity, where he became a full professor in2005. He has a wide variety of research interestsincluding information theory and coding theory, sig-nal processing for communications and in networks,computation-and-communication theoretic analysis,etc. He was a co-recipient of four international

conference Best Paper / Best Student Paper Awards. He is currently servingas Editor for IEEE Transactions on Communications, IET Communicationsand some other international journals. He served as GeneralChair, TPCCo-Chair or Symposium Co-Chair for many international conferences andworkshops like ChinaCOM 2008, ICUFN 2011/2012/2013, WCSP 2013,Globecom 2014 Wireless Communications Symposium, and VTC-Fall 2017Workshop HMWC, etc.