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Multi-pair two-way massive MIMO relaying with zero forcing: Energyefficiency and power scaling laws
Qian, J., Masouros, C., & Matthaiou, M. (2019). Multi-pair two-way massive MIMO relaying with zero forcing:Energy efficiency and power scaling laws. IEEE Transactions on Communications.https://doi.org/10.1109/TCOMM.2019.2959329
Published in:IEEE Transactions on Communications
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Download date:16. Mar. 2021
1
Multi-Pair Two-Way Massive MIMO Relayingwith Zero Forcing: Energy
Efficiency and Power Scaling LawsJun Qian, Student Member, IEEE, Christos Masouros, Senior Member, IEEE,
and Michail Matthaiou, Senior Member, IEEE
Abstract—In this paper, we study a multi-pair two-wayhalf-duplex decode-and-forward (DF) massive multiple-inputmultiple-output (MIMO) relaying system, in which multiplesingle-antenna user pairs can exchange information through amassive MIMO relay. For low-complexity transmission, zero-forcing reception/zero-forcing transmission (ZFR/ZFT) is em-ployed at the relay. First, we analytically study the large-scaleapproximations of the sum spectral efficiency (SE). Furthermore,we focus on three specific power scaling laws to study the trade-off between the transmit powers of each pilot symbol, each userand the relay, and also focus on how the transmit powers scalewith the number of relay antennas, M, to maintain a finiteSE performance. Additionally, we consider a practical powerconsumption model to investigate the energy efficiency (EE),and illustrate the impact of M and the interplay between thepower scaling laws and the EE performance. Finally, we considerthe system fairness via maximizing the minimum achievable SEamong all user pairs.
Index Terms—Decode-and-forward relaying, half-duplex, mas-sive MIMO, Max-min fairness, power scaling laws, spectralefficiency and energy efficiency, zero-forcing.
I. Introduction
MASSIVE multiple-input multiple-output (MIMO) hasbecome a key technology for the next-generation wire-
less communications with the potential of achieving highersystem capacity and data rate demands via simultaneouslyserving a significant number of users [1]–[3]. This is possiblesince large array gain and spatial multiplexing gain can beprovided [4], [5], leading to a huge improvement in the spectralefficiency (SE) and energy efficiency (EE) [6]. Generally,precoding is a commonly used technique in massive MIMO toensure downlink transmission and optimize link performance[7]–[9].
In a multi-pair relaying system, where users can exchangeinformation via a shared relay, the application of massiveMIMO techniques has attracted great attention due to thepotential of improving the network capacity, cellular coverage,system throughput, and enhancing the service quality for cell
J. Qian and C. Masouros are with the Department of Electronic andElectrical Engineering, University College London, London WC1E 7JE, U.K.(email: [email protected]; [email protected]).
M. Matthaiou is with the Institute of Electronics, Communications andInformation Technology (ECIT), Queen’s University Belfast, Belfast BT39DT, U.K. (email: [email protected]).
This work was supported by the Engineering and Physical SciencesResearch Council (EPSRC) project EP/M014150/1. The work of M. Matthaiouwas supported by EPSRC, UK, under grant EP/P000673/1.
edge users [10], [11]. Moreover, by deploying a large numberof antennas at the relay, the spatial diversity can be amplifiedwhile boosting the achievable performance [1]. Initially, one-way relaying systems were studied for multi-pair massiveMIMO relaying. For amplify-and-forward (AF) protocol, thepower control problem was studied in [12]; in addition, fordecode-and-forward (DF) protocol, the comparison of theachievable SE with different linear processing methods hasbeen studied in [11], while [13] has investigated the outageperformance of one-way DF relaying. However, one-wayrelaying might incur SE loss [14], [15]. In order to reduce theSE loss, two-way relaying is considered to improve the SE andextend the communication range while enabling bidirectionalcommunication [16]–[18]. Theoretically, in two-way relayingsystems, user pairs can exchange information via a sharedrelay in only two time slots, and the required time is muchshorter than that in one-way relaying system [19], [20]. Tothis end, multi-pair two-way relaying with massive arrays hasbeen widely studied, where more than one pair of users canbe served to exchange information [16], [21].
Normally, the AF protocol is investigated in most studiesof multi-pair two-way massive MIMO relaying, while the DFprotocol is typically overlooked. However, the AF relayingmight suffer from noise amplification [22]. In this case,DF two-way relaying is proposed as it can achieve betterperformance than AF relaying at low signal-to-noise ratios(SNRs) without noise propagation at the relay [22], [23]. Also,we recall that DF two-way relaying has the ability to performseparate precoding and power allocation on each relaying com-munication direction, at the cost of higher complexity [24].In some previous studies with two-way relaying, full-duplexhas been adopted [18], [20]. However full-duplex operationmay not be practically feasible due to the huge intensitydifference in near/far field of the transmitted/received signals.In this case, half-duplex operation, in which the relay transmitsand receives in orthogonal frequency or time resources, haspractical relevance and is considered in this paper [23], [25].
In the light of above, we study a multi-pair two-way half-duplex DF relaying system with zero-forcing (ZF) processingand imperfect CSI [22], [26]. This paper extends the work of[22], where only maximum ratio processing (MRC/MRT) isconsidered. A detailed analysis of the sum SE is presentedand we characterize a practical power consumption model toanalyze the EE performance of the proposed relaying system.In addition, power scaling scenarios which can improve the
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EE while maintaining the desired SE for large number ofrelay antennas [4] are studied in detail. Specifically, the maincontributions of this paper can be summarized as following• With a general multi-pair massive MIMO two-way relay-
ing system employing the DF protocol, we present a newlarge-scale approximation of the SE with ZF processingand imperfect CSI when the number of relay antennasapproaches infinity. To the best of our knowledge, noother prior work with this specific relaying system hasderived similar expressions due to the difficulty in ma-nipulating matrix inverses, which inherently kick in ZFtype of analysis.
• We characterize a practical power consumption modelderived from the relevant models in [27], [28]. It isutilized to analyze the EE performance of the proposedmulti-pair two-way relaying system.
• We investigate three power scaling laws inspired from[16], [22]. Our study illustrates that there exists a trade-off between the transmit powers of each user, each pilotsymbol and the relay; namely, the same SE, even the sameEE can be achieved with different configurations of thepower-scaling parameters. This provides great flexibilityin practical system design and forms a roadmap to selectthe optimal parameters to maximize the EE performancein particular scenarios.
• Motivated by the Max-Min fairness studies in [17], [29],we formulate an optimization problem to maximize theminimum achievable SE among all user pairs with im-perfect CSI in order to improve the sum SE and achievefairness across all user pairs. The complexity analysisof the proposed optimization problem has also beeninvestigated.
The structure of the paper is organized as follows: SectionII introduces the multi-pair two-way half-duplex DF relayingsystem model with ZF processing and imperfect CSI. Sec-tion III presents a large-scale approximation of the SE andcharacterizes the EE and its corresponding power consumptionmodel. Section IV demonstrates the power-scaling laws withdifferent parameter configurations in form of the asymptoticSE, while Section V illustrates the study of Max-Min fairness.Our numerical results are depicted in Section VI. Finally,Section VII concludes this paper.
Notation: We use HT , HH , H∗ and H−1 to represent thetranspose, conjugate-transpose, conjugate and the inverse ofmatrix H, respectively. Moreover, IM stands for an M × Midentity matrix. In addition, | · |, || · || and || · ||F denotesthe Absolute value, Euclidean norm and Frobenius norm,respectively. Then, CN (0,Σ) represents circularly symmetriccomplex Gaussian distribution with zero mean and covarianceΣ. Finally, E{·} is the expectation operator and diag(·) showsthe diagonal elements of a matrix.
II. SystemModel
As shown in Fig. 1, we investigate a multi-pair two-wayhalf-duplex DF relaying system, in which K pairs of single-antenna users, defined as TA,i and TB,i, i = 1, ...,K, exchangeinformation via a shared relay TR with M antennas, generally,
Fig. 1: Multi-pair two-way DF relaying system.
M � K. Moreover, we assume that there are no direct trans-mission links between user pairs. Normally, it is assumed thatmassive MIMO system operates in a time-division duplexing(TDD) mode [14], [30]. To this end, we assume that theproposed system is modeled as uncorrelated Rayleigh fading,works under TDD protocol and channel reciprocity holds [31],[32]. The uplink and downlink channels between TX,i, X = A, Band TR are denoted as hXR,i ∼ CN
(0, βXR,iIM
)and hT
XR,i,i = 1, ...,K, respectively, while βAR,i and βBR,i represent thelarge-scale fading parameters which are considered to beconstant in this paper for simplicity. Additionally, the channelmatrix can be formed as HXR =
[hXR,1, ...,hXR,K
]∈ CM×K ,
X = A, B.For the proposed relaying system, the data transmission
process can be divided into two phases with equal time slots.Generally, this two-phase protocol can be named as MultipleAccess Broadcast (MABC) protocol [23]. In the first MultipleAccess Channel (MAC) phase, all users transmit their signalsto the relay simultaneously. Therefore, the received signal atthe relay can be expressed as [28]
yr =
K∑i=1
(√pA,ihAR,ixAR,i +
√pB,ihBR,ixBR,i
)+ nR, (1)
where xXR,i is the Gaussian signal transmitted by the i-thuser TX,i with zero mean and unit power, pX,i is the averagetransmit power of TX,i, X = A, B. nR is the vector of additivewhite Gaussian noise (AWGN) at the relay whose elementsare independent and identically distributed (i.i.d) satisfyingCN (0, 1). For low-complexity transmission, linear processingis applied at the relay. Thus, the transformed signal can begiven by
zr = FMACyr , (2)
with FMAC ∈ C2K×M , the linear receiver matrix in the MAC
phase.In the second Broadcast Channel (BC) phase, the relay
first decodes the received information and then re-encodesand broadcasts it to users [22]. The linear precoding matrixFBC ∈ C
M×2K in the BC phase is applied to obtain the transmitsignal of the relay as
yt = ρDFFBCx, (3)
where x =[xT
A , xTB
]Trepresents the decoded signal and ρDF is
the normalization coefficient determined by the average relay
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power constraint E{||yt ||
2}
= pr. Therefore, the received signalsat TX,i, X = A, B can be given by
zX,i = hTXR,iyt + nX,i, (4)
with the standard AWGN at TX,i, nX,i ∼ CN(0, 1) , X = A, B.
A. Linear Processing
Generally, the inter-pair interference and inter-user interfer-ence can be eliminated by linear processing in massive MIMOsystems [4], [33]. In this paper, the basic linear processingscheme, ZF processing is applied at the relay to achieve low-complexity transmission. Thus, the linear processing matricesFMAC ∈ C
2K×M and FBC ∈ CM×2K for the proposed system
defined above can be given by [10], [34]
FMAC =
([HAR, HBR
]H [HAR, HBR
])−1[HAR, HBR
]H, (5)
FBC =[HBR, HAR
]∗([HBR, HAR
]T [HBR, HAR
]∗)−1, (6)
respectively. In (5)-(6) above, HXR are the estimated channels,X = A, B. To simplify the mathematical expressions in thefollowing, we assume that FAR
MAC ∈ CK×M , FBR
MAC ∈ CK×M
represent the first K rows and the rest K rows of FMAC ,respectively. Meanwhile, FRB
BC ∈ CM×K , FRA
BC ∈ CM×K stand
for the first K columns of and the rest K columns of FBC ,respectively.
B. Channel Estimation
In massive MIMO systems, it is important to considerimperfect CSI for realistic scenarios [11]. In TDD systems, thestandard way to estimate channels at the relay is to transmitpilots [27], [35]. In this case, among the coherence intervalwith length τc (in symbols), τp symbols are applied as pilotsymbols for channel estimation [22]. Generally, we assumethat all pilot sequences are mutually orthogonal and τp ≥ 2Kis required. Moreover, we assume that the minimum meansquare error (MMSE) estimator is employed at the relay toestimate channels [11], [27], [36]. Therefore, we can have thechannel estimates as
hXR,i = hXR,i + eXR,i, (7)
where hXR,i and eXR,i are the i-th columns of the estimatedmatrix HXR and estimation error matrix EXR, respectively,while HXR and EXR are statistically independent, X = A, B.pp represents the transmit power of each pilot symbol usedfor channel estimation, the elements in hXR,i and eXR,i areGaussian random variables with zero mean and varianceσ2
XR,i =τp ppβ
2XR,i
1+τp ppβXR,i, σ2
XR,i =βXR,i
1+τp ppβXR,i, X = A, B, respectively
[11].
III. Performance Analysis
A. Spectral Efficiency
In this subsection, we focus on the SE performance of theproposed half-duplex DF two-way relaying system. Generally,the large-scale approximation of the SE can be derived whenM → ∞.
1) Exact Expressions: In the MAC phase, according to (1)-(2), the transformed signal at the relay determined by the i-thuser pair can be expressed as
zr,i = zAr,i + zB
r,i, (8)
where zXr,i can be obtained by
zXr,i =
√pX,i
(FAR
MAC,i + FBRMAC,i
)hXR,ixX,i︸ ︷︷ ︸
desired signal
+√
pX,i
(FAR
MAC,i + FBRMAC,i
)eXR,ixX,i︸ ︷︷ ︸
estimation error
+∑j,i
√pX, j
(FAR
MAC,i + FBRMAC,i
)hXR, jxX, j︸ ︷︷ ︸
inter-user interference
+ FXRMAC,inR︸ ︷︷ ︸
noise
,
(9)and zr = zA
r + zBr ∈ C
K×1, with zXr ∈ C
K×1, X = A, B. With theassistance of (8)-(9), when we take the i-th pair of users intoconsideration, the estimation error, inter-user interference andcompound noise in zr,i can be given by
Ai = pA,i
(∣∣∣FARMAC,ieAR,i
∣∣∣2 +∣∣∣FBR
MAC,ieAR,i
∣∣∣2)+ pB,i
(∣∣∣FARMAC,ieBR,i
∣∣∣2 +∣∣∣FBR
MAC,ieBR,i
∣∣∣2) , (10)
Bi =∑j,i
pA, j
(∣∣∣FARMAC,ihAR, j
∣∣∣2 +∣∣∣FBR
MAC,ihAR, j
∣∣∣2)+
∑j,i
pB, j
(∣∣∣FARMAC,ihBR, j
∣∣∣2 +∣∣∣FBR
MAC,ihBR, j
∣∣∣2), (11)
Ci =∣∣∣∣∣∣FAR
MAC,i
∣∣∣∣∣∣2 +∣∣∣∣∣∣FBR
MAC,i
∣∣∣∣∣∣2, (12)
respectively. With the expressions of desired signals in (9) forzA
r,i and zBr,i, the SE of the specified user TX,i to the relay with
SINRXR,i, X = A, B, can be expressed as
RXR,i =τc − τp
2τcE{log2(1 + SINRXR,i)
}=τc − τp
2τcE{log2(1 +
pX,i
(∣∣∣FARMAC,ihXR,i
∣∣∣2 +∣∣∣FBR
MAC,ihXR,i
∣∣∣2)Ai + Bi + Ci
)}.
(13)Additionally, the standard lower capacity bound associatedwith the worst-case uncorrelated additive noise is consideredin this paper [22], [37]; therefore, the achievable SE of thei-th user pair in the MAC phase is given by Eqn. (14) on thetop of next page.
In the BC phase, via applying FBC to generate the relay’stransmit signal, the received signal at TX,i can be calculatedby (4). Take zA,i as an example in Eqn. (15) on the top of nextpage, while zB,i can be obtained by replacing the subscripts“AR”, “BR” in the channel vectors and corresponding vectors,the subscripts “RA”, “RB” in linear precoding vectors, and“A”, “B” in signal and noise terms with the subscripts “BR”,“AR”, the subscripts “RB”, “RA”, and “B”, “A” in zA,i,respectively. To this end, we can obtain the SE of the relay tothe i-th user TX,i, X = A, B by Eqn. (16) on the top of nextpage.
4
R1,i =τc − τp
2τc×E
log2
1 +
pA,i
(∣∣∣FARMAC,ihAR,i
∣∣∣2 +∣∣∣FBR
MAC,ihAR,i
∣∣∣2) + pB,i
(∣∣∣FARMAC,ihBR,i
∣∣∣2 +∣∣∣FBR
MAC,ihBR,i
∣∣∣2)Ai + Bi + Ci
. (14)
zA,i = ρDF hTAR,iF
RABC,ixB,i︸ ︷︷ ︸
desired signal
+ ρDFeTAR,iF
RABC,ixB,i︸ ︷︷ ︸
estimation error
+ ρDF
K∑j=1
hTAR,iF
RBBC,ixA,i + ρDF
∑j,i
hTAR,iF
RABC, jxB, j︸ ︷︷ ︸
inter-user interference
+ nA,i︸︷︷︸noise
,(15)
RRX,i =τc − τp
2τcE{log2(1 + SINRRX,i)
}=τc − τp
2τcE
{log2(1 +
∣∣∣hTXR,iF
RXBC,i
∣∣∣2∣∣∣eTXR,iF
RXBC,i
∣∣∣2 +K∑
j=1
(∣∣∣hTXR,iF
RABC, j
∣∣∣2 +∣∣∣hT
XR,iFRBBC, j
∣∣∣2) − ∣∣∣hTXR,iF
RXBC,i
∣∣∣2 + 1ρ2
DF
)}.
(16)
Meanwhile, the achievable SE of the i-th user pair in theBC phase is defined as the sum of the end-to-end SE fromTA,i to TB,i and from TB,i to TA,i [22], [28],
R2,i = min(RAR,i,RRB,i
)+ min
(RBR,i,RRA,i
). (17)
Therefore, the sum SE of the multi-pair two-way DF relay-ing system can be expressed as
R =
K∑i=1
Ri =
K∑i=1
min(R1,i,R2,i
), (18)
where Ri is the achievable SE of the i-th user pair for theproposed system determined by the minimum SE in MACand BC phases [15], [24].
2) Approximations: Practically, the large-scale approxima-tion of the SE for the i-th user pair studied in the followingcan be derived when the relay employs a great number ofantennas, i.e., M → ∞.
Lemma 1: When M → ∞, the inner product of any twocolumns in the estimated channel matrix HXR can be definedas [10], [38]
1M· hH
XR,ihXR, j →
{σ2
XR,i, i = j0, i , j
. (19)
Proof: Please see Appendix A.With an increasing number of relay antennas, the channel
vectors within HXR become asymptotically mutually orthogo-nal. As such, HH
XRHXR can be assumed to approach a diagonalmatrix [39]. Thus, according to Lemma 1, we can obtain
1M· HH
XRHXR → diag{σ2
XR,1, σ2XR,2, ..., σ
2XR,K
}, (20)
(HH
XRHXR
)−1→ diag
1M · σ2
XR,1
,1
M · σ2XR,2
, ...,1
M · σ2XR,K
,(21)
X = A, B. Therefore, the linear processing matrices FMAC andFBC can be simplified when M → ∞ as follows
FMAC →
(HH
ARHAR
)−1HH
AR(HH
BRHBR
)−1HH
BR
, (22)
FBC →
[H∗BR
(HT
BRH∗BR
)−1, H∗AR
(HT
ARH∗AR
)−1], (23)
respectively, while the normalization coefficient ρDF definedin Section II can be given by
ρDF =
√√ pr
E{∣∣∣∣∣∣FBC
∣∣∣∣∣∣2F
} =
√√√√√ M · prK∑
i=1
(1
σ2AR,i
+ 1σ2
BR,i
) . (24)
Corollary 1: With the DF protocol and the properties of ZFprocessing [40], when M → ∞, the large-scale approximationsassociated with Ri (Ri − Ri → 0) can be given by
R =
K∑i=1
Ri =
K∑i=1
min(R1,i, R2,i
). (25)
The approximations of the achievable SE in MAC and BCphases, and the SE from the user pair/relay to the relay/userpair can be expressed as
R1,i =τc − τp
2τc×
log2
1 +M(pA,i + pB,i)(
1σ2
AR,i+ 1
σ2BR,i
) [ K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
] ,(26)
R2,i = min(RAR,i, RRB,i
)+ min
(RBR,i, RRA,i
),
(27)
RAR,i =τc − τp
2τc×
log2
1 +MpA,i(
1σ2
AR,i+ 1
σ2BR,i
) [ K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
] ,
(28)
RRA,i =τc − τp
2τc× log2
1 +M · pr
(prσ2AR,i + 1)
K∑j=1
(1
σ2AR, j
+ 1σ2
BR, j
).(29)
RBR,i and RRB,i can be obtained by replacing the transmitpowers pA,i, pB,i, and the subscripts ”AR”, ”BR” with the
5
transmit powers pB,i, pA,i, and the subscripts ”BR”, ”AR” inRAR,i and RRA,i, respectively.Proof: Please see Appendix B.
B. Energy Efficiency
Generally, the EE is defined as the ratio of the sum SE tothe total power consumption of the proposed system and canbe given by [4], [27], [41],
ε =R
Ptotal, (30)
where R denotes the sum SE defined in (18), Ptotal representsthe total power consumption. In a practical system, the totalpower consumption consists of the transmitted signal power,the powers of operating static circuits and the RF componentsin each RF chain. Normally, each antenna is connected to oneRF chain [27]. Therefore, the power consumption model forthe users and the relay can be defined as [28],
Ptot,i =1
2τc
[ (τc − τp)pu + τp pp
ζi+ τc · PRF,i
]=
12τc
[ (τc − τp)pu + τp pp
ζi
]+
12
PRF,i
(31)
Ptot,r =1
2τc
[τc pr
ζr+ τc · PRF,r
]=
12
( pr
ζr+ PRF,r
), (32)
respectively. Note that Ptot,i represents the total power at i-th user and Ptot,r indicates the total power at the relay; ζr
and ζi denote the power amplifier efficiency for the relayand i-th user, respectively. The power consumption of the RFcomponents for single-antenna users and the relay with Mantennas can be defined as
PRF,i = PDAC,i + Pmix,i + P f ilt,i + Psyn,i, (33)
PRF,r = M(PDAC,r + Pmix,r + P f ilt,r) + Psyn,r, (34)
respectively. Psyn is the power consumption of the frequencysynthesizer, PDAC , Pmix and P f ilt are the power consumedby the digital-to-analog converters (DACs), signal mixers andfilters in the RF chain respectively [27]. As a result, the totalpower consumption Ptotal for the system can be re-expressedas
Ptotal = 2K · Ptot,i + Ptot,r + Pstatic, (35)
where Pstatic is the power of all the static circuits [4]. Tosimplify the power consumption model in the simulation, weassume that ζi = ζr = ζ, PDAC,i = PDAC,r = PDAC , Pmix,i =
Pmix,r = Pmix, P f ilt,i = P f ilt,r = P f ilt and Psyn,i = Psyn,r = Psyn
for i = 1, 2, ...,K.
IV. Power Scaling Laws
In this section, we investigate how the power-scaling lawsaffect achievable SE; and, in particular, how power reductionswith M maintain a desired SE. In the following, we considerthree power-scaling cases: a) only the transmit power ofeach pilot symbol is scaled; b) the transmit powers of datatransmission at each user and the relay are scaled; c) alltransmit powers are scaled, to demonstrate the interplay amongthe transmit power of each pilot symbol pp, the transmit power
of each user pu and the relay pr. For simplicity, we assumethat pA,i = pB,i = pu, i = 1, ...,K. We define that R1,i, R2,i, Ri,RXR,i and RRX,i, X = A, B, are asymptotic expressions of theachievable SE; additionally, without loss of generality in thefollowing, we define that
R =
K∑i=1
Ri =
K∑i=1
min(R1,i, R2,i
), (36)
R2,i = min(RAR,i, RRB,i
)+ min
(RBR,i, RRA,i
). (37)
1) Case A: Only the transmit power of the pilot symbolis scaled by M with pp =
Ep
Mγ , where Ep is a constant andγ > 0. This case is said to achieve power savings in the channeltraining stage.
Corollary 2: For pp =Ep
Mγ , with fixed pu, pr, Ep and γ > 0,when M → ∞, we can present the asymptotic results as
Ri −min(R1,i, R2,i
)→ 0. (38)
with
R1,i =τc − τp
2τclog2
1 +2 τpEp
Mγ−1(1
β2AR,i
+ 1β2
BR,i
) ( K∑j=1
(βAR, j + βBR, j
)+ 1
pu
),
(39)
RAR,i =τc − τp
2τclog2
1 +
τpEp
Mγ−1(1
β2AR,i
+ 1β2
BR,i
) ( K∑j=1
(βAR, j + βBR, j
)+ 1
pu
),
(40)
RRA,i =τc − τp
2τclog2
1 +
τpEp
Mγ−1 pr
K∑j=1
(prβAR,i + 1
) ( 1β2
AR, j+ 1
β2BR, j
),
(41)
and RBR,i and RRB,i can be obtained by replacing the subscripts”AR”, ”BR” in RAR,i and RRA,i with the subscripts ”BR”, ”AR”,respectively.
We can observe that Case A depends on the choice of γto scale the transmit power of each pilot symbol. From (39)-(41), we can know that when we reduce pp aggressively withγ > 1, Ri approaches zero. In contrast, when 0 < γ < 1, Ri
grows unboundedly. Additionally, when γ = 1, Ri convergesto a non-zero limit.
2) Case B: The transmit power of each pilot symbol pp isfixed, while other transmit powers are scaled with pu = Eu
Mα ,pr = Er
Mβ , where α ≥ 0 and β ≥ 0, and Eu, Er are constants. Inthis case, the potential power savings in data transmission arestudied.
Corollary 3: For pu = EuMα , pr = Er
Mβ , with fixed pp, Eu, Er
and α ≥ 0, β ≥ 0, when M → ∞, we can obtain
Ri −min(R1,i, R2,i
)→ 0. (42)
6
with
R1,i =τc − τp
2τclog2
1 +2 × Eu
Mα−1(1
σ2AR,i
+ 1σ2
BR,i
), (43)
RAR,i =τc − τp
2τclog2
1 +
EuMα−1(
1σ2
AR,i+ 1
σ2BR,i
), (44)
RRA,i =τc − τp
2τclog2
1 +
ErMβ−1
K∑j=1
(1
σ2AR, j
+ 1σ2
BR, j
). (45)
Similarly, RBR,i and RRB,i can be obtained by replacing thesubscripts ”AR”, ”BR” in RAR,i and RRA,i with the subscripts”BR”, ”AR”, respectively.
This case investigates that when both pu and pr are scaledwith M when M → ∞, the effects of estimation error andinter-user interference eliminate; thus, only the noise at usersand the relay remains to cause imperfection. When we cutdown pu and pr aggressively, namely, 1) α > 1, and β ≥ 0, 2)α ≥ 0, and β > 1, 3) α > 1, and β > 1, Ri reduces to zero. Onthe other hand, when we reduce both pu and pr moderately,which is, 0 ≤ α < 1 and 0 ≤ β < 1, Ri grows unboundedly.
Furthermore, for a specific scenario where both the transmitpowers of the relay and of each user are scaled with the samespeed α = β = 1, Ri converges to a non-zero limit,
R1,i =τc − τp
2τclog2
1 +2Eu(
1σ2
AR,i+ 1
σ2BR,i
), (46)
RAR,i = RBR,i =τc − τp
2τclog2
1 +Eu(
1σ2
AR,i+ 1
σ2BR,i
), (47)
RRA,i = RRB,i =τc − τp
2τclog2
1 +Er
K∑j=1
(1
σ2AR, j
+ 1σ2
BR, j
). (48)
We can see that the non-zero limit increases with respect toEu and Er, while decreasing with respect to the number of userpairs K. Also, if we apply 0 ≤ β < α = 1, the approximationof the sum SE is determined by the SE performance in theMAC phase, which means that R1,i given by (46) determinesRi when M → ∞. On the other hand, when 0 ≤ α < β = 1,the determination of SE appears in the BC phase; thus, Ri isdetermined by RRA,i and RRB,i given by (48).
3) Case C: This is a general case where all transmit powersare scaled, pp =
Ep
Mγ , pu = EuMα and pr = Er
Mβ , with γ ≥ 0, α ≥ 0and β ≥ 0, Ep, Eu and Er are constants.
Corollary 4: For pp =Ep
Mγ , pu = EuMα , pr = Er
Mβ with fixed Ep,Eu, Er and γ ≥ 0, α ≥ 0, β ≥ 0, when M → ∞, we can obtain
Ri −min(R1,i, R2,i
)→ 0. (49)
with
R1,i =τc − τp
2τclog2
1 +2 × τpEpEu
Mα+γ−1(1
β2AR,i
+ 1β2
BR,i
), (50)
RAR,i =τc − τp
2τclog2
1 +
τpEpEu
Mα+γ−1(1
β2AR,i
+ 1β2
BR,i
), (51)
RRA,i =τc − τp
2τclog2
1 +
τpEpEr
Mβ+γ−1
K∑j=1
(1
β2AR, j
+ 1β2
BR, j
). (52)
Similarly, RBR,i and RRB,i can be obtained by replacing thesubscripts ”AR”, ”BR” with the subscripts ”BR”, ”AR” inRAR,i and RRA,i, respectively.
As expected, the sum SE depends on the choice of α, β andγ. Additionally, α + γ determines the SE in the MAC phase,while β+γ determines the SE in the BC phase. When α = β >0 and α+ γ = 1, the trade-off between the transmit powers ofeach user/the relay and of each pilot symbol is displayed. Inthis case, if we reduce the transmit power of each pilot symbolaggressively, the channel estimate is corrupted and in orderto compensate this imperfection, the transmit power of eachuser/the relay should be increased. The limit of the asymptoticSE under this case can be given by
R1,i =τc − τp
2τclog2
1 +2τpEpEu(1
β2AR,i
+ 1β2
BR,i
), (53)
RAR,i = RBR,i =τc − τp
2τclog2
1 +τpEpEu(1
β2AR,i
+ 1β2
BR,i
), (54)
RRA,i = RRB,i =τc − τp
2τclog2
1 +τpEpEr
K∑j=1
(1
β2AR, j
+ 1β2
BR, j
). (55)
Moreover, we can see that when α > β ≥ 0 and α + γ = 1,the limit of Ri is determined by R1,i according to (53), whichmeans that we can improve the sum SE by increasing Ep andEu. In addition, the sum SE of the proposed system is anincreasing function of K based on (36). Meanwhile, when 0 ≤α < β and β+γ = 1, the limit of Ri is determined by RRA,i andRRB,i according to (55) which displays the trade-off betweenthe transmit powers of each pilot symbol and of the relay.
V. Max-Min Fairness AnalysisWith the assistance of the above mentioned SE analysis, and
as a further step forward from the power scaling laws, in thissection, our objective is to harness SE fairness among the userpairs. We achieve this purpose by maximizing the minimumachievable SE among all the user pairs; therefore, providingmax-min fairness [17], [29].
7
A. Spectral Efficiency Fairness
For analytical simplicity, the large-scale approximation inCorollary 1 is employed and we assume that the pilot powerpp is determined in advance. Moreover, we define that pA =
[pA,1, ..., pA,K]T , and pB = [pB,1, ..., pB,K]T . In this case, theoptimization problem can be formulated as
maxpA,pB,pr
mini∈1,...,K
Ri (56a)
subject to0 ≤ pr ≤ Pmax
r , 0 ≤ pA,i ≤ Pmaxu , 0 ≤ pB,i ≤ Pmax
u ,∀i (56b)
12
K∑i=1
(τc − τp
) (pA,i + pB,i
)τcζu
+pr
ζc
+ Po ≤ Pmax (56c)
Here, Pmax is the total power constraint, Pmaxu and Pmax
r are themaximum powers of each user and the relay, respectively and
Po =12
[2Kτp pp
τcζu+(2K+M)(PDAC +Pmix +P f ilt)+(2K+1)Psyn +
Pstatic
]is determined in Section III. According to Corollary 1,
we can rewrite the optimization problem (56) by introducingthe auxiliary variables t, t1, t2 as follows
maxpA,pB,pr ,t,t1,t2
t (57a)
s.t. (56b), (56c) (57b)t1 + t2 ≥ t (57c)
τc − τp
2τclog2
1 +
Mσ2AR,iσ
2BR,i
σ2AR,i+σ
2BR,i
(pA,i + pB,i
)K∑
j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
≥ t,∀i
(57d)
τc − τp
2τclog2
1 +
Mσ2AR,iσ
2BR,i
σ2AR,i+σ
2BR,i
pA,i
K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
≥ t1,∀i
(57e)
τc − τp
2τclog2
1 +pr M(
prσ2BR,i + 1
) K∑j=1
(1
σ2AR, j
+ 1σ2
AR, j
) ≥ t1,∀i
(57f)
τc − τp
2τclog2
1 +
Mσ2AR,iσ
2BR,i
σ2AR,i+σ
2BR,i
pB,i
K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
≥ t2,∀i
(57g)
τc − τp
2τclog2
1 +pr M(
prσ2AR,i + 1
) K∑j=1
(1
σ2AR, j
+ 1σ2
AR, j
) ≥ t2,∀i
(57h)
According to the property of logarithm function log( ab ) =
log(a) − log(b), we can observe that the SE of the proposedsystem can be defined as a difference of two concave functions
[42], we can rewrite the constraints with Ri = f (pA,i, pB,i, pr)−h(pA,i, pB,i, pr), where the specific mathematical expressionsfor TX,i, i = 1, ...,K and X = A, B can be given by
f1(pA,i, pB,i, pr
)=τc − τp
2τc×
log2
Mσ2AR,iσ
2BR,i
σ2AR,i + σ2
BR,i
(pA,i + pB,i
)+
K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
,(58)
fXR(pA,i, pB,i, pr
)=τc − τp
2τc×
log2
Mσ2AR,iσ
2BR,i
σ2AR,i + σ2
BR,i
pX,i +
K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
,(59)
fRX(pA,i, pB,i, pr
)=τc − τp
2τc×
log2
pr M +(prσ
2XR,i + 1
) K∑j=1
1σ2
AR, j
+1
σ2BR, j
, (60)
hXR(pA,i, pB,i, pr
)=τc − τp
2τclog2
K∑j=1
(pA, jσ
2AR, j + pB, jσ
2BR, j
)+ 1
,(61)
hRX(pA,i, pB,i, pr
)=τc − τp
2τclog2
(prσ2XR,i + 1
) K∑j=1
1σ2
AR, j
+1
σ2BR, j
.
(62)
Based on Corollary 1, it is clear that for R1,i,h1
(pA,i, pB,i, pr
)= hXR
(pA,i, pB,i, pr
), X = A, B. The specific
functions f (pA,i, pB,i, pr) and h(pA,i, pB,i, pr) defined above arejointly concave with respect to pA,i, pB,i, pr, i = 1, ...,K [42],[43], and the relaxed problem can be reformulated as
maxpA,pB,pr ,t,t1,t2
t (63a)
s.t. (56b), (56c), (57c) (63b)f1
(pA,i, pB,i, pr
)− h1
(pA,i, pB,i, pr
)≥ t,∀i (63c)
fAR(pA,i, pB,i, pr
)− hAR
(pA,i, pB,i, pr
)≥ t1,∀i (63d)
fRB(pA,i, pB,i, pr
)− hRB
(pA,i, pB,i, pr
)≥ t1,∀i (63e)
fBR(pA,i, pB,i, pr
)− hBR
(pA,i, pB,i, pr
)≥ t2,∀i (63f)
fRA(pA,i, pB,i, pr
)− hRA
(pA,i, pB,i, pr
)≥ t2,∀i (63g)
We can find that the difficulty in solving (63) lies in thecomponent hXR
(pA,i, pB,i, pr
)and hRX
(pA,i, pB,i, pr
), X = A, B.
Therefore, the value of(pA,i, pB,i, pr
)at k-th iteration is
supposed to be(p(k)
A,i, p(k)B,i, p(k)
r
). Since hXR
(pA,i, pB,i, pr
)and
hRX(pA,i, pB,i, pr
), X = A, B are concave and differentiable on
the considered domain, we can easily find the affine function asa Taylor first order approximation near
(p(k)
A,i, p(k)B,i, p(k)
r
)shown
in Eqn. (64)-(65) on the top of next page [42], [43]. Toupdate the objective in the (k + 1)-th iteration we replacehXR
(pA,i, pB,i, pr
)and hRX
(pA,i, pB,i, pr
), X = A, B by their
affine functions on the top of next page, respectively. There-fore, the optimization problem (63) at iteration (k + 1)-th with
8
h(k)XR
(pA,i, pB,i, pr
)=τc − τp
2τc×
log2
K∑j=1
(p(k)
A, jσ2AR, j + p(k)
B, jσ2BR, j
)+ 1
+
K∑j=1
(pA, j − p(k)
A, j
)σ2
AR, j +(pB, j − p(k)
B, j
)σ2
BR, j
ln 2(
K∑j=1
(p(k)
A, jσ2AR, j + p(k)
B, jσ2BR, j
)+ 1
), (64)
h(k)RX
(pA,i, pB,i, pr
)=τc − τp
2τc×
log2
(p(k)r σ2
XR,i + 1) K∑
j=1
1σ2
AR, j
+1
σ2BR, j
+
(pr − p(k)
r
)σ2
XR,i
K∑j=1
(1
σ2AR, j
+ 1σ2
BR, j
)ln 2
(p(k)
r σ2XR,i + 1
) K∑j=1
(1
σ2AR, j
+ 1σ2
BR, j
). (65)
convex constraints can be reformulated as
maxpA,pB,pr ,t,t1,t2
t (66a)
s.t. (56b), (56c), (57c) (66b)
f1(pA,i, pB,i, pr
)− h(k)
1(pA,i, pB,i, pr
)≥ t,∀i (66c)
fAR(pA,i, pB,i, pr
)− h(k)
AR(pA,i, pB,i, pr
)≥ t1,∀i (66d)
fRB(pA,i, pB,i, pr
)− h(k)
RB(pA,i, pB,i, pr
)≥ t1,∀i (66e)
fBR(pA,i, pB,i, pr
)− h(k)
BR(pA,i, pB,i, pr
)≥ t2,∀i (66f)
fRA(pA,i, pB,i, pr
)− h(k)
RA(pA,i, pB,i, pr
)≥ t2,∀i (66g)
To this end, via solving the optimization problem (66), thelower bound of the SE for each user pair can be obtained.Overall, the optimal solutions can be obtained by existingoptimization tool (CVX) and the iterative procedure of Max-Min fairness analysis can be summarized in Algorithm 1.
Algorithm 1 General iterative algorithm
Initialization: Set the iteration index k = 0, define a toleranceε > 0 and initial values for p(0)
A,i, p(0)B,i and p(0)
r , i = 1, ...,K.Repeat:
1: Solve optimization problem (76) and obtain the solutionsp(k)
A,i, p(k)B,i and p(k)
r , i = 1, ...K.2: Update (p(0)
A,i, p(0)B,i p(0)
r )=(p(k)A,i, p(k)
B,i p(k)r ), i = 1, ...K.
3: Set k = k + 1.Until:
4: |t(k) − t(k−1)| < ε or k ≥ L (maximum iteration number).Output: p∗A,i, p∗B,i and p∗r , i = 1, ...,K as the solutions.
B. Complexity Analysis
Recall that in the optimization problem (66), logarithmicfunctions are deployed in the constraints; therefore, the suc-cessive approximation method which constructs polynomialapproximations for all logarithmic terms is employed in CVXwhen solving this optimization problem [44]. To the bestof our knowledge, the exact complexity of the successiveapproximation method in CVX has not been determined in theliterature. Accordingly, we consider studying the complexityof the optimization problem with polynomial-approximationconstraints to obtain the complexity lower bound of Algorithm1.
Here, we make use of the fact that the optimization problemcan be considered as a quadratically constrained quadraticprogram (QCQP) in epigraph form [45] after constructingpolynomial approximations in CVX. With the assistance ofprevious studies [46], [47], the quadratic constraints can berewritten as linear matrix inequalities (LMIs). Note that linearconstraints can also be considered as LMI constraints. SinceAlgorithm 1 is an iterative process, solving the optimizationproblem by the interior-point method via CVX, as per themethodology in [48], we can similarly determine the lowerbound of the complexity of Algorithm 1 via the followingtwo parts:
1) Iteration Complexity: In Algorithm 1, with the giventolerance ε > 0, the number of required iterations to achievethe ε-optimal solution can be given by
Citer =
√√√Lnum∑j=1
k j · ln(1ε
) =√
6K2 + 4K + 3 · ln(1ε
), (67)
where Lnum is the number of LMI constraints and k j representsthe size of the j-th constraint, j = 1, ..., Lnum.
2) Per-Iteration Complexity: For each iteration, a searchdirection is generated by solving a system of n linear equationsin n unknowns with n = O(K2) [48]. To this end, thecomputation cost per iteration can be obtained by
Cper = n ·∑Lnum
j=1 k3j + n2 ·
∑Lnumj=1 k2
j + n3
= n · (3 + 2K + 8K3 + 24K4) + n2 · (3 + 2K + 4K2 + 12K3)+ n3.
(68)Hence, the lower bound of the total complexity Ctotal of
Algorithm 1 can be calculated by combining these two parts,
Ctotal = Citer ×Cper
=√
6K2 + 3K + 4 · ln( 1ε) · n · [(3 + 2K + +8K3 + 24K4)
+ n · (3 + 2K + 4K2 + 12K3) + n2].(69)
VI. Numerical Results
We now present simulation results to verify the abovestudies. Unless specifically noted, the following parameters areemployed in the simulation. We consider an LTE frame with[27] and we assume a coherence time τc = 196 (symbols) andthe length of the pilot sequences is τp = 2K, the minimumrequirement. For simplicity, we assume that the large-scale
9
fading parameters are βAR,i = βBR,i = 1 and each user hasthe same transmit power pA,i = pB,i = pu, i = 1, ...,K. For theproposed power consumption model, we assume that ζ = 0.38,PDAC = 7.8 mW, Pmix = 15.2 mW, P f ilt = 10 mW, Psyn = 25mW and Pstatic = 2 W.
A. Validation of Analytical Expressions
50 100 150
p
0
10
20
30
40
50
Su
m S
pe
ctr
al E
ffic
ien
cy (
bit/s
/Hz)
Exact pp=10 dB
Approx.
Exact pp=0 dB
Approx.
Exact pp=-5 dB
Approx.
Exact pp=-10 dB
Approx.
Exact pp=-10 dB, p
u=p
p=-5 dB
Exact, AF relaying
Exact pp=-20 dB, p
u=p
p=-5 dB
Exact, AF relaying
Fig. 2: Impact of τp for M = 400, K = 10, pu = 5 dB and pr = 10dB.
Fig. 2 shows the sum SE vs the length of pilot sequenceτp (in symbols). Note that the “Approx.” (Approximations)curves are obtained via applying Corollary 1, and the “Exact”(Exact results) curves are generated by (10)-(18). We canobserve that the large-scale approximations closely match theexact results and the sum SE can be maximized with anoptimal τ∗p at low pp. In contrast, the sum SE is a decreasingfunction of τp at moderate and high pp. To this end, inorder to achieve better sum SE performance, τp = 2K, theminimum requirement, is deployed in the channel estimationphase. Moreover, AF relaying with ZF processing studiedin [49]–[51] has been considered here as a benchmark to
further illustrate the performance of the proposed DF relayingsystem. It can be observed that when transmit powers are smallresulting in lower SINRs, the performance of DF relayingcan outperform that of AF relaying and an optimal τ∗p canbe obtained to maximize the sum SE with specific transmitpower configurations. Since in our following simulations, weconsider a more general power configuration, where SINRs arenot small enough for DF relaying to outperform AF relaying;therefore, we only focus on the performance of the proposedDF relaying in the following numerical results.
B. Power Scaling Laws
1) Case A: It can be easily observed in Fig. 3 (a) that thepower scaling law breaks down when γ = 0 and the sumSE grows unboundedly. For Case A, the curves named “Asy”(Asymptotic results) are presented according to Corollary 2.When 0 < γ < 1, the sum SE is an increasing function of M.In contrast, when γ = 1, the sum SE progressively reaches anon-zero limit, and when γ > 1, the sum SE approaches zerogradually. Moreover, the sum SE is a decreasing function ofγ; since with larger γ, the system would experience a lowerchannel estimation accuracy, which results in worse systemperformance.
Fig. 3 (b) verifies the impact of M on the EE when differentK and γ are applied. It is clearly shown that the sum SEsaturates when M is large, while Ptotal increases linearly withM, and accordingly the EE peaks at a certain value of M.In this case, an optimal M∗ can be selected to maximize theEE, especially when 0 ≤ γ < 1. Moreover, the EE decreasesmore significantly with a smaller K when M is large, and wecan observe that larger K can introduce a larger optimal γ∗ toobtain the maximum EE while achieving power savings.
2) Case B: Fig. 4 (a) investigates how the transmit powersof each user pu = Eu/Mα and the relay pr = Er/Mβ affectthe achievable SE. For Case B, the curves named “Asy”(Asymptotic results) are generated by Corollary 3. When α = 1and/or β = 1, the SE saturates to a non-zero limit. When α > 1,β > 1, the sum SE gradually reduces to zero. On the otherhand, when we cut down the transmit powers moderately, the
0 500 1000 1500 2000 2500 3000
Number of antennas
0
10
20
30
40
50
60
Su
m S
pe
ctr
al E
ffic
ien
cy (
bit/s
/Hz)
Exact =0.0
Approx.
Asy.
Exact =0.5
Approx.
Asy.
Exact =1
Approx.
Asy
Exact =1.5
Approx.
Asy
0 500 1000 1500 2000 2500 3000
Number of antennas
0
0.1
0.2
0.3
0.4
0.5
Energ
y E
ffic
iency (
bits/J
/Hz)
Approx. =0.0
Approx. =0.3
Approx. =0.6
Approx. =1
Approx. =0.0
Approx. =0.3
Approx. =0.6
Approx. =1
Solid Lines: K=10, Exact Results
Dashed Lines: K=30, Exact Results
Fig. 3: (a) Sum SE and (b) EE v.s. number of relay antennas M for K = 10, pu = 5 dB, pr = 10 dB and pp = Ep/Mγ with Ep = 5 dB.
10
0 500 1000 1500 2000 2500 3000
Number of antennas
0
5
10
15
20
25
30
35
Su
m S
pe
ctr
al E
ffic
ien
cy (
bit/s
/Hz)
Exact =0.5, =0.4
Approx.
Asy.
Exact =0.6, =0.8
Approx.
Asy.
Exact =1, =0.5
Approx.
Asy
Exact =0.3, =1
Approx.
Asy
Exact =1.3, =1.1
Approx.
Asy
0 500 1000 1500 2000 2500 3000
Number of antennas
0
0.2
0.4
0.6
0.8
1
1.2
En
erg
y E
ffic
ien
cy (
bits/J
/Hz)
Exact =0, =0
Approx.
Exact. =0.2, =0.3
Approx.
Exact. =0.3, =0.4
Approx.
Exact =0.5, =0.7
Approx.
Exact. =1, =0.5
Approx.
Exact. =0.5, =1
Approx.
Fig. 4: (a) Sum SE and (b) EE v.s. number of relay antennas for K = 10, pp = 5 dB, pu = Eu/Mα with Eu = 5 dB and pr = Er/Mβ withEr = 10 dB.
0 500 1000 1500 2000 2500 3000
Number of antennas
0
5
10
15
20
25
30
35
Sum
Spectr
al E
ffic
iency (
bit/s
/Hz)
Exact =0.5, =0.5, =0.5
Approx.
Asy.
Exact =1, =0.5, =0.7
Approx.
Asy
Exact =0.6, =0.8, =0.5
Approx.
Asy.
Exact =0.8, =1.1, =1.1Approx.
Asy
0 500 1000 1500 2000 2500 3000
Number of antennas
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
En
erg
y E
ffic
ien
cy (
bits/J
/Hz)
Exact =0, =0, =0
Approx.
Exact =0.2, =0.3, =0.4
Approx.
Exact =0.5, =0.5, =0.5
Approx.
Exact =0.6, =0.8, =0.5
Approx.
Exact =1, =0.5, =0.8
Approx.
Fig. 5: (a) Sum SE and (b) EE v.s. number of relay antennas for K = 10, pp = Ep/Mγ, pu = Eu/Mα with Ep = Eu = 5 dB and pr = Er/Mβ
with Er = 10 dB.
sum SE grows unboundedly. The channel estimation accuracykeeps stable and the transmission phase plays an importantrole in the SE performance.
The impact of M on the EE with different α and β isinvestigated in Fig. 4 (b). We can see that when 0 < α < 1and 0 < β < 1, the EE performance is better than that withoutpower scaling law and an optimal M∗ can be obtained tomaximize the EE. Therefore, the moderate power scaling in thetransmission phase can help to optimize the EE performance.In contrast, when the transmit powers are reduced aggressively,the EE is a decreasing function with respect to M. Withregards to this, by considering the trade-off between scalingparameters, appropriate values of α and β could be selectedto optimize the EE performance.
3) Case C: Fig. 5 (a) verifies the trade-off between thetransmit powers of each user, the relay and the pilot symbol.For Case C, the “Asy” (Asymptotic results) curves are obtainedvia Corollary 4. For the aggressive power-scaling scenario, thesum SE progressively converges to zero, as predicted. More-over, with the moderate power-scaling parameters, 0 < γ < 1,0 < α < 1 and 0 < β < 1, the sum SE increases with respectto M.
Fig. 5 (b) illustrates the impact of the number of relayantennas on the EE. It is clearly shown that the EE risesand then descends with respect to M while applying moderatepower-scaling parameters; thus, we can obtain the optimal M∗
to maximize the EE, e.g., with γ = 0.2, α = 0.3 and β = 0.4,the maximum EE around 1.25 bits/J/Hz can be obtained when
TABLE I: Average Run Time (in seconds) for three scenarios of Algorithm 1
K 2 4 6 8 10M = 500 Algorithm 1 9.318 10.021 12.745 13.136 14.721
Pmax = 23dB Algorithm 1 with equal power allocation 6.485 9.548 13.233 15.936 16.110pp = 5dB Uniform power allocation 4.100 5.529 7.504 8.997 11.298
11
-20 -15 -10 -5 0 5 10 15p
p (dB)
0
0.5
1
1.5
2
2.5
3
Min
SE
Requirem
ent (b
its/s
/Hz)
Algorithm 1 M=100
Algorithm 1 with equal user power
Uniform power allocation
Algorithm 1 M=200
Algorithm 1 with equal user power
Uniform power allocation
200 400 600 800 1000 1200 1400Number of relay antennas
0
0.5
1
1.5
2
2.5
3
3.5
4
Min
Achie
vable
SE
(bits/s
/Hz)
Algorithm 1 Pmax
=23dBAlgorithm 1 with equal user powerUniform power allocation
Algorithm 1 Pmax
=17dBAlgorithm 1 with equal user powerUniform power allocation
Fig. 6: Min achievable SE v.s. (a) pp with Pmax = Pmaxr = 23 dB and (b) number of relay antennas with pp = 5 dB, Pmax
r = Pmax for K = 10and Pmax
u = 10 dB.
M∗ ≈ 500. On the other hand, when we cut down the transmitpowers aggressively, the EE approaches to zero straightfor-wardly. Therefore, it is of crucial importance to determinethe scaling parameters to optimize the EE performance in aspecific power-limited scenario.
C. Max-Min Fairness
We consider three optimization scenarios with the minimumachievable SE among all user pairs: 1) Algorithm 1; 2)Algorithm 1 with equal user power, i.e. pA,i = pB,i = pu,i = 1, ...,K; 3) Uniform power allocation, i.e. pA,i = pB,i = pu,
i = 1, ...,K, 2K pu = pr, 12
(K∑
i=1
(τc−τp)(pA,i+pB,i)τcζu
+prζc
)+Po = Pmax.
For a more practical comparison, all users’ large-scale fadingparameters are different and can be generated via βk =
õ
Dυk,
where µ is the large-scale fading coefficient, Dk is the distancebetween the k-th user and the relay, υ is the path-loss exponent[52]. Following our benchmark work in [22], we considerβAR = [0.3188, 0.4242, 0.5079, 0.0855, 0.2625, 0.8010, 0.0292,0.9289, 0.7303, 0.4886], and βBR = [0.5785, 0.2373, 0.4588,0.9631, 0.5468, 0.5211, 0.2316, 0.4889, 0.6241, 0.6791].
Fig. 6 (a) shows the minimum achievable SE versus pp. Itcan be observed that the minimum achievable SE achieved viaAlgorithm 1 outperforms the other two scenarios, Algorithm1 with equal user power and uniform power allocation, espe-cially when pp is large enough. Moreover, larger number ofrelay antennas can help to increase the minimum achievableSE with the same total power constraint Pmax.
Fig. 6 (b) shows the minimum achievable SE with increasingnumber of relay antennas M. Similarly, a higher minimumachievable SE can be achieved by Algorithm 1 comparedwith the other two power allocation scenarios. The minimumachievable SE is an increasing function of M, especially witha larger total power constraint.
In Table I on the bottom of previous page, we display theaverage run time (in seconds) of three optimization scenariosdefined above with a given tolerance ε = 10−5. We can observethat, the running times for all three scenarios are increasing
with respect to K. Then, uniform power allocation has thesmallest number of constraints and, therefore, the running timefor this scenario is the shortest.
VII. Conclusion
This paper has studied the sum SE and EE performance ofa multi-pair two-way half-duplex DF relaying system with ZFprocessing and imperfect CSI. Note that this setup extendsconsiderably a stream of recent papers on massive MIMOrelaying by leveraging tools of Wishart matrix theory. In par-ticular, a large-scale approximation of the achievable SE wasdeduced. Meanwhile, a practical power consumption modelwas characterized to study the EE performance. Furthermore,in view of approximations, three specific power scaling lawswere investigated to present how the transmit powers of eachpilot symbol, each user and the relay can be scaled to improvethe system performance. These results have their own addingvalue as they translate mathematical formulations into systemdesign guidelines for power savings. Finally, a formulatedoptimization problem was studied to optimize the minimumachievable SE among all user pairs. Our numerical resultsdemonstrated emphatically that the proposed system with ZFprocessing is able to enhance the EE while preserving the SEperformance with moderate system configurations. Moreover,the simulation results of the optimization problem demon-strated that our proposed max-min fairness scheme can achievehigher minimum achievable SE among user pairs comparedwith the benchmark schemes where equal user power anduniform power allocation are applied. In our future work,we will consider the application of multi-pair two-way DFrelaying in TDD correlated massive MIMO systems as thesubsequent subject.
Appendix AProof of Lemma 1
In this appendix, we provide the calculations of Lemma 1.With the assumption that all estimated channels with hXR,i ∼
CN(0, σ2
XR,iIM
)and hXR, j ∼ CN
(0, σ2
XR, jIM
)are mutually
12
independent when i , j, i, j = 1, ...,K. When M → ∞, wecan haveIf i = j,
1M
hHXR,ihXR,i =
1M
∣∣∣hXR,i
∣∣∣2 =1M· Mσ2
XR,i = σ2XR,i, (70)
If i , j,
1M
hHXR,ihXR, j = 0. (71)
With the computation of (70)-(71), we can obtain Lemma 1 in(19).
Appendix BDerivation for approximations of the sum SEs
In this appendix, we present the detailed derivation for R1,iand RRX,i, while RXR,i can be obtained in a straightforward way.At first, some useful results widely used in the calculation aregiven in Lemma 2.
Lemma 2: Assume that hi ∼ CN(0, σ2
i IM
)and h j ∼
CN(0, σ2
jIM
)are mutually independent when i , j, i, j =
1, ...,K. Therefore, we have
∣∣∣hHi h j
∣∣∣2M2 →
σ4i , i = j
1Mσ2
i σ2j , i , j
, (72)
With the assistance of (20)-(23), Lemma 1 and Lemma 2,we derive the calculation of the corresponding approximationsin the following. First, we focus on R1,i, RXR,i, X = A, B inthe MAC phase, consisting of four terms defined above. WhenM → ∞, we can have1) Desired signal power of TX,i, X = A, B,
pX,i
(∣∣∣FARMAC,ihXR,i
∣∣∣2 +∣∣∣FBR
MAC,ihXR,i
∣∣∣2)→
pA,i
∣∣∣FARMAC,ihAR,i
∣∣∣2pB,i
∣∣∣FBRMAC,ihBR,i
∣∣∣2 →
pA,i
∣∣∣ 1Mσ2
AR,ihH
AR,ihAR,i
∣∣∣2pB,i
∣∣∣ 1Mσ2
BR,ihH
BR,ihBR,i
∣∣∣2→
{pA,i, X = ApB,i, X = B , (73)
2) Estimation Error Ai,
Ai → pA,i
∣∣∣ 1Mσ2
AR,i
hHAR,ieAR,i
∣∣∣2 +∣∣∣ 1Mσ2
BR,i
hHBR,ieAR,i
∣∣∣2+ pB,i
∣∣∣ 1Mσ2
AR,i
hHAR,ieBR,i
∣∣∣2 +∣∣∣ 1Mσ2
BR,i
hHBR,ieBR,i
∣∣∣2→
(pA,iσ
2AR,i + pB,iσ
2BR,i
)M
1σ2
AR,i
+1
σ2BR,i
, (74)
3) Inter-user Interference Bi,
Bi =∑j,i
pA, j
(∣∣∣FARMAC,i
(hAR, j + eAR, j
) ∣∣∣2 +∣∣∣FBR
MAC,i
(hAR, j + eAR, j
) ∣∣∣2)+
∑j,i
pB, j
(∣∣∣FARMAC,i
(hBR, j + eBR, j
) ∣∣∣2 +∣∣∣FBR
MAC,i
(hBR, j + eBR, j
) ∣∣∣2)→
∑j,i
pA, j
(∣∣∣FARMAC,ieAR, j
∣∣∣2 +∣∣∣FBR
MAC,ieAR, j
∣∣∣2)+
∑j,i
pB, j
(∣∣∣FARMAC,ieBR, j
∣∣∣2 +∣∣∣FBR
MAC,ieBR, j
∣∣∣2)→
∑j,i
pA, j
∣∣∣ 1Mσ2
AR,i
hHAR,ieAR, j
∣∣∣2 +∣∣∣ 1Mσ2
BR,i
hHBR,ieAR, j
∣∣∣2+
∑j,i
pB, j
∣∣∣ 1Mσ2
AR,i
hHAR,ieBR, j
∣∣∣2 +∣∣∣ 1Mσ2
BR,i
hHBR,ieBR, j
∣∣∣2→
1M
1σ2
AR,i
+1
σ2BR,i
∑j,i
(pA, jσAR, j + pB, jσBR, j
), (75)
4) Noise Ci,
Ci →∣∣∣∣∣∣ 1
Mσ2AR,i
hHAR,i
∣∣∣∣∣∣2 +∣∣∣∣∣∣ 1
Mσ2BR,i
hHBR,i
∣∣∣∣∣∣2 → 1M
1σ2
AR,i
+1
σ2BR,i
.(76)
Substituting (73)-(76) into (13)-(14), we can obtain R1,i, RXR,i,X = A, B in (26), (28).
Then, we focus on RRX,i in the BC phase. Similarly, thecorresponding terms in RRX,i, X = A, B can be computed asfollowing when M → ∞,
1) Normalization coefficient,
ρDF =
√√ pr
E{∣∣∣∣∣∣FBC
∣∣∣∣∣∣2} =
√√√√√ pr
E{
M∑i=1
2K∑j=1
∣∣∣FBC (i, j)∣∣∣2}
→
√√√√ prK∑
i=1
(1
Mσ2AR,i
+ 1Mσ2
BR,i
) . (77)
2) Desired signal,
∣∣∣hTXR,iF
RXBC,i
∣∣∣2 → ∣∣∣hTXR,i
1Mσ2
XR,i
h∗XR,i
∣∣∣2 → 1, (78)
3) Estimation error,
∣∣∣eTXR,iF
RXBC,i
∣∣∣2 → ∣∣∣eTXR,i
1Mσ2
XR,i
h∗XR,i
∣∣∣2 → σ2XR,i
Mσ2XR,i
, (79)
13
4) Inter-user interference,K∑
j=1
(∣∣∣hTXR,iF
RABC, j
∣∣∣2 +∣∣∣hT
XR,iFRBBC, j
∣∣∣2) − ∣∣∣hTXR,iF
RXBC,i
∣∣∣2=
K∑j=1
(∣∣∣ (hTXR,i + eT
XR,i
)FRA
BC, j
∣∣∣2 +∣∣∣ (hT
XR,i + eTXR,i
)FRB
BC, j
∣∣∣2)−
∣∣∣ (hTXR,i + eT
XR,i
)FRX
BC,i
∣∣∣2→
K∑
j=1
∣∣∣eTAR,iF
RBBC, j
∣∣∣2 +∑j,i
∣∣∣eTAR,iF
RABC, j
∣∣∣2K∑
j=1
∣∣∣eTBR,iF
RABC, j
∣∣∣2 +∑j,i
∣∣∣eTBR,iF
RBBC, j
∣∣∣2
→
K∑
j=1
∣∣∣eTAR,i
1Mσ2
BR, jh∗BR, j
∣∣∣2 +∑j,i
∣∣∣eTAR,i
1Mσ2
AR, jh∗AR, j
∣∣∣2K∑
j=1
∣∣∣eTBR,i
1Mσ2
AR, jh∗AR, j
∣∣∣2 +∑j,i
∣∣∣eTBR,i
1Mσ2
BR, jh∗BR, j
∣∣∣2
→
K∑
j=1
σ2AR,i
Mσ2BR, j
+∑j,i
σ2AR,i
Mσ2AR, j, X = A
K∑j=1
σ2BR,i
Mσ2AR, j
+∑j,i
σ2BR,i
Mσ2BR, j, X = B
. (80)
By applying (77)-(80) to (16), we can obtain RRX,i, X = A, B,in (29). With other additional computations, we complete theproof of the SE approximations shown in Corollary 1.
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Jun Qian (S’18) received the bachelor’s degreefrom Beihang University, China, in 2015, and theMSc. degree (Hons.) from University College Lon-don (UCL), U.K., in 2016, where she is currentlypursuing the PhD. degree with the Information andCommunications Engineering research group, De-partment of Electronic and Electrical Engineering.Her research interests include massive MIMO com-munications, wireless relaying systems and perfor-mance analysis. She was a recipient of the UCL’sMSc Telecommunications 2015/16 Best Overall Re-
search Project Prize.
Christos Masouros (M’ 06-SM’ 14) received theDiploma degree in Electrical and Computer Engi-neering from the University of Patras, Greece, in2004, and MSc by research and PhD in Electricaland Electronic Engineering from the University ofManchester, UK in 2006 and 2009 respectively. In2008 he was a research intern at Philips ResearchLabs, UK. Between 2009-2010 he was a ResearchAssociate in the University of Manchester and be-tween 2010-2012 a Research Fellow in Queen’sUniversity Belfast. In 2012 he joined University
College London as a Lecturer. He has held a Royal Academy of EngineeringResearch Fellowship between 2011-2016.
He is currently a Full Professor in the Information and CommunicationsEngineering research group, Dept. Electrical and Electronic Engineering,University College London. His research interests lie in the field of wire-less communications and signal processing with particular focus on GreenCommunications, Large Scale Antenna Systems, Cognitive Radio, interferencemitigation techniques for MIMO and multicarrier communications. He wasthe recipient of the Best Paper Awards in the IEEE GlobeCom 2015 and IEEEWCNC 2019 conferences, and has been recognised as an Exemplary Editorfor the IEEE Communications Letters, and as an Exemplary Reviewer for theIEEE Transactions on Communications. He is an Editor for IEEE Transactionson Communications, and IEEE Transactions on Wireless Communications.He has been an Associate Editor for IEEE Communications Letters, and aGuest Editor for IEEE Journal on Selected Topics in Signal Processing issues“Exploiting Interference towards Energy Efficient and Secure Wireless Com-munications” and “Hybrid Analog / Digital Signal Processing for Hardware-Efficient Large Scale Antenna Arrays”. He is currently an elected memberof the EURASIP SAT Committee on Signal Processing for Communicationsand Networking.
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 workingas a Postdoctoral Research Associate. He is currently a Reader (equivalentto Associate Professor) in Multiple-Antenna Systems at Queen’s UniversityBelfast, U.K. after holding an Assistant Professor position at Chalmers Uni-versity of Technology, Sweden. His research interests span signal processingfor wireless communications, massive MIMO systems, hardware-constrainedcommunications, mm-wave systems and deep learning for communications.
Dr. Matthaiou and his coauthors received the IEEE Communications Society(ComSoc) Leonard G. Abraham Prize in 2017. He was awarded the prestigious2018/2019 Royal Academy of Engineering/The Leverhulme Trust SeniorResearch Fellowship and recently received the 2019 EURASIP Early CareerAward. His team was also the Grand Winner of the 2019 Mobile WorldCongress Challenge. He was the recipient of the 2011 IEEE ComSoc BestYoung Researcher Award for the Europe, Middle East and Africa Region and aco-recipient of the 2006 IEEE Communications Chapter Project Prize for thebest M.Sc. dissertation in the area of communications. He has co-authoredpapers that received best paper awards at the 2018 IEEE WCSP and 2014IEEE ICC and was an Exemplary Reviewer for IEEE Communications Lettersfor 2010. In 2014, he received the Research Fund for International YoungScientists from the National Natural Science Foundation of China. He iscurrently the Editor-in-Chief of Elsevier Physical Communication. In the past,he was an Associate Editor for the IEEE Transactions on Communications,Associate Editor/Senior Editor for IEEE Communications Letters and wasthe Lead Guest Editor of the special issue on “Large-scale multiple antennawireless systems” of the IEEE Journal on Selected Areas in Communications.