Multipair Two-Way DF Relaying with Cell-Free Massive MIMO
Anastasios K. Papazafeiropoulos, Pandelis Kourtessis, Symeon Chatzinotas, John M. Senior
11 Multipair Two-Way DF Relaying with Cell-Free Massive MIMO
Anastasios K. Papazafeiropoulos, Pandelis Kourtessis, Symeon Chatzinotas, John M. Senior,
We consider a two-way half-duplex decode-and-forward (DF) relaying system with multiple pairs of single-antenna users assistedby a cell-free (CF) massive multiple-input multiple-output (mMIMO) architecture with multiple-antenna access points (APs). Underthe practical constraint of imperfect channel state information (CSI), we derive the achievable sum spectral efficiency (SE) for afinite number of APs with maximum ratio (MR) linear processing for both reception and transmission in closed-form. Notably,the proposed CF mMIMO relaying architecture, exploiting the spatial diversity, and providing better coverage, outperforms theconventional collocated mMIMO deployment. Moreover, we shed light on the power-scaling laws maintaining a specific SE as thenumber of APs grows. A thorough examination of the interplay between the transmit powers per pilot symbol and user/APs takesplace, and useful conclusions are extracted. Finally, differently to the common approach for power control in CF mMIMO systems,we design a power allocation scheme maximizing the sum SE.
Index Terms —Two-way relaying, cell-free massive MIMO systems, decode-and-forward, power-scaling law, beyond 5G MIMO.
I. I
NTRODUCTION
Massive multiple-input multiple-output (mMIMO) systems,where a large number of antennas in both collocated anddistributed setups serves simultaneously a lower number ofusers, has become one of the key fifth-generation (5G) physical-layer technologies towards higher throughput and energyefficiency by means of simple linear signal processing [1], [2].Recently, a distributed mMIMO architecture under the nameof cell-free (CF) mMIMO, enjoying the benefits of networkMIMO, has emerged by providing higher coverage probabilityand exploiting the diversity against shadow fading [3]. Inparticular, a CF mMIMO system includes a large number ofsingle-antenna access points (APs) that is connected to a centralprocessing unit (CPU) and serves jointly all users by means ofcoherent joint transmission/reception. In this direction, in [4],APs were equipped with multiple antennas to increase the arrayand diversity gains. Generally, CF mMIMO systems outperformsmall cells (SCs) and collocated deployments but given thatthis is an emerging promising architecture at its infancy, itsstudy is limited [4]–[11]. For example, authors in [5] achievedbetter data rates by means of a user-centric approach, wherethe APs serve a group of users instead of all of them, while in[9], the realistic spatial randomness of the APs was taken intoaccount by means of a Poisson point process (PPP) to obtainthe coverage probability in CF mMIMO systems.Multipair relaying systems, where multiple pairs of userscommunicate simultaneously by means of a relay to enhancethe network coverage, have been improved further by theintroduction of the mMIMO characteristics, which enhance thespatial diversity and achieve an order of magnitude spectralefficiency (SE) improvement [12], [13]. This technique, knownas multipair mMIMO relaying, initially considered one-waytransmission by means of amplify-and-forward (AF) as wellas decode-and-forward (DF) protocols [12]–[15]. For instance,
A. Papazafeiropoulos is with the Communications and Intelligent SystemsResearch Group, University of Hertfordshire, Hatfield AL10 9AB, U. K., andwith SnT at the University of Luxembourg, Luxembourg. P. Kourtessis andJohn M. Senior are with the Communications and Intelligent Systems ResearchGroup, University of Hertfordshire, Hatfield AL10 9AB, U. K. S. Chatzinotasis with the SnT at the University of Luxembourg, Luxembourg. E-mails:[email protected], [email protected], [email protected]. in the case of DF relaying, the authors in [12] examined theachievable SE in Rayleigh fading channels for both maximum-ratio (MR) and zero-forcing (ZF) linear processing, while in[13], optimization of the energy efficiency was performed. Inreference to AF relaying, the power allocation and max-minuser selection was investigated in [14].Unfortunately, the one-way transmission strategy incurs anSE degradation by [13], [16]. To reduce this loss, thetwo-way mMIMO relaying, where bidirectional simultaneousdata transmission applies, has attracted significant attention[17]–[23]. Hence, its main advantage in comparison to one-wayrelaying is the reduction of the time required for informationexchange between the user pairs into just two time slots.For instance, [17] and [18] characterized the power-scalinglaws for half-duplex (HD) and full-duplex (FD) transmission,respectively. Still, first works assumed only perfect channel stateinformation (CSI), which is highly unrealistic since practicalsystems have the availability of only imperfect CSI. Thus, atraining phase with pilot transmission and minimum mean-square-error (MMSE) estimation at the relay was consideredin [19] and [20] for AF and DF protocols, respectively. Apartfrom evaluating the impact of imperfect CSI on the systemperformance, several other important questions arise. Forexample, [21] and [22] investigated the interplay among thetransmit powers of the pilot, user, and relay for DF and AF,respectively. Interestingly, distributed relaying for multipair two-way channels has been studied but only for AF and perfectCSI conditions [23], [24].
A. Motivation
This work relies on vital observations: i) Collocated mMIMOrelaying in [21] has not fully taken advantage of the benefitsof practical distributed mMIMO systems. In particular, CFmMIMO systems exploit the spatial diversity, provide bettercoverage since the APs are closer to the users, and suppress the a r X i v : . [ c s . I T ] F e b inter-AP interference due to the APs cooperation . ii) Two-wayrelaying is more advantageous than one-way transmission. iii)Given that AF undergoes noise amplification, DF may performbetter at a low signal-to-noise ratio (SNR) [25]. Thus, DFis a more attractive method for mMIMO relaying normallyoperating at the low power regime [1]. iv) DF is more flexiblethan AF in the case of two-way relaying since it can allowpower allocation in both directions [26]. v) Although someworks have studied FD for CF mMIMO systems [27]–[29],FD may not be advisable for CF mMIMO systems due to thelarge power level difference of the transmit/received signalsin the near/far-field while the APs might be quite close to theusers. vi) The existing literature on CF mMIMO relaying hasnot studied at all the insightful power-scaling laws. B. Contributions and Outcomes
The main contributions are summarized as follows. • Motivated by the above observations, we establish thetheoretical framework for a two-way CF mMIMO relayingsystem with imperfect CSI by employing the DF protocol,where a large number of distributed APs plays therole of a relay . Notably, the two-way design enhancesthe CF mMIMO relaying performance by achieving asignificant increase in the SE with comparison to one-way communication but also CF mMIMO is expectedto enhance two-way transmission due its accompanyingadvantages compared to collocated mMIMO systems. • Contrary to the existing work [21], which has studiedtwo-way mMIMO relaying with a large number ofcollocated antennas, we have accounted for the emergingCF mMIMO architecture at the position of the largecollocated deployment. Notably, although [21] proposesasymptotic approximations of the SE for a large numberof antennas in terms of deterministic equivalents, weconsider a finite number of APs, where the analysis ismore general while we also cover the scenario, where thenumber of APs grows to infinity . In addition, contrary to Some other differences between the two are architectures follow. First, inCF mMIMO systems, each AP obtains its local CSI. In this way, we achieveto alleviate the computational burden at the CPU. Moreover, CF mMIMOsystems achieve increased fairness with respect to the users since, in CF, it ismore possible to have an AP close to a user. Also, CF achieves lower latencysince the APs are closer to the users. Actually, their advantageous layout ismore attractive for mobile edge computing and caching. Notably, in [8], beingone of the references pointing to the differences between CF mMIMO andsingle-cell collocated mMIMO systems, the authors highlighted the differencesin the generation of the correlation matrices and the allocation of the pilots. This framework will be the ground to study other interesting topics onCF mMIMO systems and two-way relaying such as the impact of imperfectbackhaul links [6]. Among other differences between this and [21], we would like to highlightthat the channel model and the power constraints are different. In our work, thepower constraints concern each AP and the channel model describes the linkfrom each AP to a user. On the contrary, in [21], the power constraints concernthe single collocated relay, and the channel model describes the link from thecollocated relay to a user. Moreover, our expressions include summations overthe number of APs, corresponding to the contributions from different APs.Such summations do not appear in [21] and result in simplified equationswhile in this work, equations are more complicated and require differentmanipulations which are relevant to the CF mMIMO analysis. Also, in thiswork, the channel estimation takes place at each AP. On the contrary, in [21],the channel estimation is performed at the collocated relay. [23], which assumed the AF protocol and the unrealisticassumption of perfect CSI, we have considered imperfectCSI and the DF protocol, being more suitable for CFmMIMO systems. Also, differently to [27]–[29] whichconsidered FD transmission for CF mMIMO systems, wefocus on a totally different architecture being HD two-way transmission, and we investigate the problems ofpower-scaling laws and power allocation by maximizingthe sum SE. Notably, previous works on CF consideredonly the max-min fairness method to obtain the powercontrol coefficients. • We derive the achievable sum SE of a two-way CFmMIMO relaying network employing the DF protocolwith imperfect CSI and linear processing by means ofmaximum ratio combiner (MRC) for the uplink as well asmaximum ratio transmission (MRT) for the downlink inclosed form for a finite number of APs and we demonstratebetter performance over the collocated deployment with alarge number of antennas . To the best of our knowledge,no other prior work has obtained similar expressions underthe CF mMIMO relaying consideration. • We carry out an asymptotic analysis for this architecture toinvestigate the power-scaling laws maintaining a specificSE as the number of APs increases. Notably, this is theunique work on CF mMIMO relaying that obtains powerscaling laws, which are different compared to the scalinglaws in [21] since they include summations with respect tothe number of APs and the relevant variables correspond todifferent APs instead of a relay with collocated antennas.We observe a trade-off among the transmit power of eachpilot symbol, user, and relay, and we shed light on theimpact of the scaling parameters. • We formulate an optimization problem for CF mMIMOsystems, maximizing the sum SE by keeping constant thetotal transmit power, in order to obtain the necessary powercontrol coefficients while previous works on CF wererelied on a power control by using the max-min fairnessapproach. Numerical results show the improvement of thesum SE compared to uniform power allocation.
C. Paper Outline
The remainder of this paper is organized as follows. Sec-tion II presents the system model of a two-way CF mMIMOrelaying system with multiple antennas APs employing the DFprotocol. This section includes also the channel estimation anddata transmission phases. Section III provides the SE analysisfor a finite number of APs. Section IV presents the power-scaling laws under different power settings while Section Vaddresses the optimal power allocation. The numerical results The application of other linear techniques such as zero-forcing presentscertain trade-offs between complexity and performance. For instance, theydemand more backhaul, which might be prohibitive in the case of largedistributed networks (CF mMIMO systems). In particular, the results couldalso be extended by incorporating the more robust MMSE processing, assuggested recently in [8]. Given the MMSE intractability, the extension cantake place by simulations or by the deterministic equivalent analysis [9], [30],[31]. The significance of these observations suggests them to be an interestingtopic of future research. are discussed in Section VI, and Section VII concludes thepaper.
D. Notation
Vectors and matrices are denoted by boldface lower andupper case symbols, respectively. The symbols ( · ) T , ( · ) H ,and ( · ) ∗ express the transpose, Hermitian transpose, andconjugate operators, respectively. The expectation and varianceoperators are denoted by E [ · ] and var [ · ] , respectively. Also, b ∼ CN ( , Σ ) represents a circularly symmetric complexGaussian vector with zero mean and covariance matrix Σ .II. S YSTEM M ODEL
As illustrated in Fig. 1, we consider a CF mMIMO archi-tecture, where a set of APs playing the role of distributedrelays, assists the exchange of information in a multipairtwo-way relaying system. Specifically, we assume that aset W = { , . . . , W } of W = |W| communication userpairs, consisted of users T A ,i and T B ,i , i = 1 , . . . , W , isserved simultaneously by means of the set M = { , . . . , M } of M = |M| APs in the same time-frequency resources.Moreover, all APs connect to a CPU via perfect backhaullinks. Certain conditions such as severe shadowing do notallow the existence of direct links between the user pairs. Also,each AP is equipped with N antennas, and each user has asingle antenna. All the nodes, i.e., the user pairs and the APsare randomly distributed in a wide area and operate in the HDmode.The system operation takes place within a coherence intervalunder the time division duplex (TDD) protocol where channelreciprocity is met [2]. The data transmission phase includestwo stages, namely, the multiple-access channel (MAC) andbroadcasting (BC) stages. In the former, i.e., the MAC stage,all users transmit to the APs. In particular, the message, sentby user i , is decoded by means of joint processing at the CPUwhere all APs have sent their received signals. In a similarway, during the second stage, all APs transmit simultaneouslyto all user pairs.The channel model includes both small and large-scalefading. Especially, the large-scale fading describes the effectsof shadowing and path-loss. Also, the large-scaling fadingchanges slowly, i.e., it can be assumed constant for severalcoherence intervals while the small-scale fading stays staticduring the duration of a coherence interval but it changesfrom one interval to the next. In mathematical terms, thechannel vector between T A ,i and m th AP is given by h mi ∈ C N × ∼ CN ( , α A ,mi I N ) , known as uncorrelated Rayleighfading, where α A ,mi represents the large-scale fading of thecorresponding link [3]. Similarly, the channel between T B ,i and m th AP is denoted by g mi ∈ C N × ∼ CN ( , α B ,mi I N ) with α B ,mi being the large-scale fading of this link. A. Channel Estimation
Given that the propagation channels are piece-wise constantover a coherence interval, both h mi and g mi need to beestimated in every interval by means of pilot transmission. Hence, let τ c and τ p be the durations of the coherence intervaland the uplink training in symbols ( τ p < τ c ) [3]. Both T A ,i and T B ,i , i = 1 . . . , W , send simultaneously orthogonal pilotsequences ϕ A ,i ∈ C τ p × and ϕ B ,i ∈ C τ p × , respectively .Note that this mutual orthogonality requires τ p ≥ W , ϕ H A ,i ϕ B ,i = 0 , and ϕ H A ,i ϕ A ,j = ϕ H B ,i ϕ B ,j = 0 , ∀ i (cid:54) = j [12].In addition, we assume (cid:107) ϕ A ,i (cid:107) = (cid:107) ϕ B ,i (cid:107) = 1 . Thus, the m th AP receives Y p ,m = √ τ p p p W (cid:88) i =1 (cid:0) h mi ϕ H A ,i + g mi ϕ H B ,i (cid:1) + W p ,m , (1)where W p ,m is an N × τ p matrix describing the additive whiteGaussian noise (AWGN) and having i.i.d elements distributedas CN (0 , . Also, p p expresses the normalized transmit signal-to-noise ratio (SNR) of each pilot symbol.Following the approach in [12], we obtain the MMSEestimated channels for the i th pairbased on [33] as h mi = ˆ h mi + ˜ h mi (2) g mi = ˆ g mi + ˜ g mi , (3)where ˆ h mi ∼ CN ( , φ A ,mi I N ) and ˜ h mi ∼ CN ( , e A ,mi I N ) are the estimated and estimation error channel vectors, beingmutually independent with φ A ,mi = τ p p p α ,mi τ p p p α A ,mi and e A ,mi = α A ,mi τ p p p α A ,mi , respectively. Similar expressions hold for theestimated channel and estimation error vectors in (3), i.e., ˆ g mi ∼ CN ( , φ B ,mi I N ) and ˜ g mi ∼ CN ( , e B ,mi I N ) aremutually independent with φ B ,mi = τ p p p α ,mi τ p p p α B ,mi and e B ,mi = α B ,mi τ p p p α B ,mi , respectively. B. Data Transmission
The communication takes place in two phases, as describednext.
1) Phase I
It is known as a MAC phase, where all user pairs simultane-ously transmit their data to all relay nodes, being the APs. Inother words, the received signal by AP m from the W pairs T A ,i , T B ,i is given by y m = √ p u W (cid:88) i =1 (cid:0) √ η A ,i h mi q A ,i + √ η B ,i g mi q B ,i (cid:1) + n u ,m , (4)where q A ,i and q B ,i are the data by the i th user pair weightedby the power control coefficients η A ,i and η B ,i , respectively.Note that E {| q A ,i | } = E {| q B ,i | } = 1 while ≤ η A ,i ≤ and ≤ η B ,i ≤ . Also, p u denotes the normalized uplinkSNR and n u ,m ∼ CN ( , I N ) is the AWGN vector at AP m .By accounting for linear detection by means of the linearreceiver matrix W m = [ W m, A , W m, B ] ∈ C N × W , the m th Works on CF mMIMO systems usually assume non-orthogonal pilotsamong the users. However, popular works in this area exist that are based onthe assumption of orthogonal pilots e.g., [32]. In this work, which is the firstone studying the multipair two-way transmission with CF mMIMO relaying,we have assumed orthogonal pilots among users to enable comparison withexisting works on collocated mMIMO systems. Future studies on multipairtwo-way transmission could include in the design the use of non-orthogonalpilots.
Fig. 1:
A multipair two-way CF mMIMO relaying network with M multi-antenna APs and W user pairs. AP multiplies its received signal y m with the transpose of thelinear detector and obtains the post-processed signals as r m = (cid:20) W H m, A y m W H m, B y m (cid:21) . (5)Herein, we assume application of MRC due to its lowcomplexity. Also, it is suggested for implementation in adistributed fashion (locally at the APs), and it results insimplified expressions as suggested initially in [3]. Note thatthe top W rows of r m correspond to W signals from T A whilethe remaining W bottom rows stand for the signals from T B ,i ( i = 1 , . . . , W ) . Next, all APs sent their processed signals tothe CPU through perfect backhaul links and the CPU obtains r = M (cid:88) m =1 r m (6) = (cid:34)(cid:80) Mm =1 W H m, A y m (cid:80) Mm =1 W H m, B y m (cid:35) . (7)Now, we focus on the detection of the transmitted symbols ofthe i th user pair at the CPU. Specifically, we denote r m i = M (cid:88) m =1 ( W H m, A ) i y m (8) r m W + i = M (cid:88) m =1 ( W H m, B ) i y m (9)where r m i and r m W + i are the detected symbols from T A ,i and T B ,i , respectively. Note that ( X ) i denotes the i th row of X .Thus, the total detected symbols at the CPU from both T A ,i and T A ,i are given by ˜ r i = M (cid:88) m =1 r m i + r m W + i (10) = M (cid:88) m =1 (cid:16) ˆ h H mi + ˆ g H mi (cid:17) y m . (11)
2) Phase II
This phase includes encoding of the received informationand broadcasting it to all user pairs. Specifically, AP m applieslinear precoding matrices in terms of MRT given by ˆ G m and ˆ H m to transmit the signals q A and q B to T A ,i and T B ,i by using the power control coefficients η A ,m and η B ,m ,respectively. Thus, the transmit signal is written as x m = √ p d (cid:16) ˆ G ∗ m η / ,m q A + ˆ H ∗ m η / ,m q B (cid:17) , (12)where p d is the maximum normalized transmit powerwhile for notational convenience, we have denoted ˆ G m =[ˆ g m , . . . , ˆ g mW ] ∈ C N × W , ˆ H m = (cid:104) ˆ h m , . . . , ˆ h mW (cid:105) ∈ C N × W , η A ,m = diag ( η A ,m , . . . , η A ,mW ) , η B ,m = diag ( η B ,m , . . . , η B ,mW ) , q A = [ q A , , . . . , q A ,W ] T , and q B =[ q B , , . . . , q B ,W ] T . Note that the power control coefficients arechosen to satisfy the power constraint, E {(cid:107) x m (cid:107) } ≤ p d , whichgives N W (cid:88) i =1 ( η A ,mi φ B ,mi + η B ,mi φ A ,mi ) ≤ . (13)Also, we have a total power constraint for all APs acting asrelay nodes, i.e., E { (cid:80) Mm =1 (cid:107) x m (cid:107) } = p r , which results in p d = p r N (cid:80) Wi =1 (cid:80) Mm =1 ( η A ,mi φ B ,mi + η B ,mi φ A ,mi ) . (14)The received signal at T A ,i by all APs is written as z A ,i = M (cid:88) m =1 h T mi x m + n A ,i = √ p d M (cid:88) m =1 W (cid:88) j =1 h T mi (cid:16) η / ,mj ˆ g ∗ mj q A ,j + η / ,mj ˆ h ∗ mj q B ,j (cid:17) + n A ,i . Similarly, T B ,i receives by all APs z B ,i = √ p d M (cid:88) m =1 W (cid:88) i =1 g T mi (cid:16) η / ,mj ˆ g ∗ mj q A ,j + η / ,mj ˆ h ∗ mj q B ,j (cid:17) + n B ,i . (15) Note that n X ,i ∼ CN ( , I N ) is the additive noise at T X ,i ( X ∈ { A , B } ). III. SE A NALYSIS
This section presents the SE performance analysis of theDF two-way CF mMIMO with MRC/MRT linear processingby means of exact and closed-form expressions.
1) Phase I
The received signal, given by (11) can be written as in (16)at the top of next page after substituting the received signalgiven by (4). Also, we have used (2) and (3) since the m thAP has imperfect CSI and considers the estimated channelsas its true channels. Taking as a reference the i th user pair,we obtain its achievable SE in the MAC phase by means ofthe use-and-then-forget capacity bounding technique wherethe CPU uses only statistical knowledge of the channel whenperforming the detection and the unknown terms are treated asuncorrelated additive noise [2, Ch. 3], [34]. Note that, in thecase of mMIMO, this bound exploits channel hardening andbecomes tighter as the number of antennas increases. Reliedon this assumption, many works in CF mMIMO exploited thatchannel hardening appears in the case of a large number ofAPs. However, in [35], it was shown that, in general, channelhardening is not met in CF mMIMO systems with single-antenna APs, but it appears if multi-antenna APs are considered.Numerical results in Section VI, relied on this assumption,verify the tightness of this bound. Thus, the achievable SE isgiven by R MAC i = τ c − τ p τ c log (cid:0) γ MAC i (cid:1) , (17)where the corresponding SINR is given by γ MAC i = DS MAC A i + DS MAC B i EE MAC A ,i + EE MAC B ,i + IUI MAC i + N MAC i (18)with the various terms provided by DS MAC A i = η A ,i (cid:12)(cid:12)(cid:12) E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17)(cid:111)(cid:12)(cid:12)(cid:12) , (19) DS MAC B i = η B ,i (cid:12)(cid:12)(cid:12) E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi g mi + ˆ g H mi g mi (cid:17)(cid:111)(cid:12)(cid:12)(cid:12) , (20) EE MAC A ,i = η A ,i var (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17)(cid:111) , EE MAC B ,i = η B ,i var (cid:110) M (cid:88) m =1 √ η B ,i (cid:16) ˆ h H mi g mi + ˆ g H mi g mi (cid:17)(cid:111) , IUI
MAC i = W (cid:88) j (cid:54) = i η A ,j E (cid:110)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 ˆ h H mi h mj (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) M (cid:88) m =1 ˆ g H mi h mj (cid:12)(cid:12)(cid:12) (cid:111) + W (cid:88) j (cid:54) = i η B ,j E (cid:110)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 ˆ h H mi g mj (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) M (cid:88) m =1 ˆ g H mi g mj (cid:12)(cid:12)(cid:12) (cid:111) , (21) N MAC i = 1 p u E (cid:110) (cid:107) M (cid:88) m =1 ˆ h mi (cid:107) + (cid:107) M (cid:88) m =1 ˆ g mi (cid:107) (cid:111) . (22) Note that DS , EE , IUI ik , and N express the desired signal (DS)part, the estimation error (EE) part, the inter-user interference(IUI), and the thermal noise and can refer to both MAC andBC phases as well as users T A ,i and T B ,i . Moreover, theachievable SE of the link T X ,i where X ∈ { A , B } , is writtenas R MAC X ,i = τ c − τ p τ c log (cid:0) γ MAC X ,i (cid:1) (23)with signal-to-interference-plus-noise ratio (SINR) given by γ MAC X ,i = DS MAC X i EE MAC A ,i + IUI MAC i + N MAC i . (24)
2) Phase II
Since in practice, the users are not aware of the instantaneousCSI, we take advantage again of the channel hardening and theuse-and-then-forget bound from the massive MIMO (mMIMO)literature [34]. Thus, we assume that user T A ,i has knowledgeonly of its statistics and performs partial self-interferencecancellation to obtain ˆ z A ,i = z A ,i − √ p d E (cid:110) M (cid:88) m =1 √ η A ,mi h T mi ˆ g ∗ mi (cid:111) q A ,i (25) = √ p d E (cid:110) M (cid:88) m =1 √ η B ,mi h T mi ˆ h ∗ mi (cid:111) q B ,i + √ p d (cid:32) M (cid:88) m =1 √ η B ,mi h T mi ˆ h ∗ mi − E (cid:40) M (cid:88) m =1 √ η B ,mi h T mi ˆ h ∗ mi (cid:41)(cid:33) q B ,i + √ p d (cid:32) M (cid:88) m =1 √ η A ,mi h T mi ˆ g ∗ mi − E (cid:40) M (cid:88) m =1 √ η A ,mi h T mi ˆ g ∗ mi (cid:41)(cid:33) q A ,i + √ p d W (cid:88) j (cid:54) = i M (cid:88) m =1 (cid:16) √ η A ,mj h T mi ˆ g ∗ mj q A ,j + √ η B ,mj h T mi ˆ h ∗ mj q B ,j (cid:17) + n A ,i , (26)where the first term expresses the desired signal, the secondand third terms express the gain uncertainty, the fourth andfifth terms represent the residual self-interference, the sixthand seventh terms describe the inter-pair interference while thelast term denotes the noise. The achievable SE of T A ,i duringthe BC phase is obtained by R BC A ,i = τ c − τ p τ c log (cid:0) γ BC A ,i (cid:1) , (27)where γ BC A ,i is given by γ BC A ,i = DS BC A i BU BC A i + BU BC B i + (cid:80) j (cid:54) = i (cid:16) IUI BC A j + IUI BC B j (cid:17) + p d (28) ˜ r i = √ p u (cid:18) √ η A ,i E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17) (cid:111) q A ,i + √ η B ,i E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi g mi + g H mi ˆ g mi (cid:17) (cid:111) q B ,i (cid:124) (cid:123)(cid:122) (cid:125) desired signal + √ η A ,i M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17) q A ,i − √ η A ,i E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17) (cid:111) q A ,i (cid:124) (cid:123)(cid:122) (cid:125) estimation error, A + √ η B ,i M (cid:88) m =1 M (cid:88) m =1 (cid:16) ˆ h H mi g mi + ˆ g H mi g mi (cid:17) q B ,i − √ η B ,i E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi g mi + ˆ g H mi g mi (cid:17) (cid:111) q B ,i (cid:124) (cid:123)(cid:122) (cid:125) estimation error, B + W (cid:88) j (cid:54) = i M (cid:88) m =1 √ η A ,j (cid:16) ˆ h H mi h mj + ˆ g H mi h mj (cid:17) q A ,j + W (cid:88) j (cid:54) = i M (cid:88) m =1 √ η B ,j (cid:16) ˆ h H mi g mj + ˆ g H mi g mj (cid:17) q B ,j (cid:124) (cid:123)(cid:122) (cid:125) inter-user interference (cid:19) + M (cid:88) m =1 (cid:16) ˆ h H mi + ˆ g H mi (cid:17) n u ,m (cid:124) (cid:123)(cid:122) (cid:125) post-pocessed noise . (16)with DS BC A i = (cid:12)(cid:12)(cid:12) E (cid:110) M (cid:88) m =1 √ η B ,mi h T mi ˆ h ∗ mi (cid:111)(cid:12)(cid:12)(cid:12) , (29) BU BC A i = var (cid:110) M (cid:88) m =1 √ η B ,mi h T mi ˆ h ∗ mi (cid:111) , (30) BU BC B i = var (cid:110) M (cid:88) m =1 √ η A ,mi h T mi ˆ g ∗ mi (cid:111) , (31) IUI BC A i = E (cid:110)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 √ η A ,mj h T mi ˆ g ∗ mj (cid:12)(cid:12)(cid:12) (cid:111) , (32) IUI BC B i = E (cid:110)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 √ η B ,mj h T mi ˆ h ∗ mj (cid:12)(cid:12)(cid:12) (cid:111) . (33)Note that BU refers to the beamforming gain uncertainty (BU).Similarly, we obtain the achievable SE of T B ,i , R BC B ,i , afterobtaining the post-processed signal at user T B ,i , ˆ z B ,i , by meansof a similar expression to (26). Hence, the achievable SE of the i th pair T A ,i to T B ,i is given by min (cid:16) R MAC A ,i , R BC B ,i (cid:17) while theSE for the opposite direction is min (cid:16) R MAC B ,i , R BC A ,i (cid:17) with theindividual SEs obtained previously. As a result, the achievableSE of of the i th pair during BC is given by the sum R BC i = min (cid:0) R MAC A ,i , R BC B ,i (cid:1) + min (cid:0) R MAC B ,i , R BC A ,i (cid:1) . (34)According to [16], [26], the achievable SE of the i th pairover both phases is given by (17) and (34) as R i = min (cid:0) R MAC i , R BC i (cid:1) , (35)and the achievable sum SE of a multipair two-way CF mMIMOrelaying system is given by R = W (cid:88) i =1 R i . (36) Theorem 1:
The achievable sum SE of a multipair two-wayCF mMIMO relaying system with DF protocol and MRC/MRTlinear processing, for any finite M and W , is given by (36)including the SEs provided by (37)-(39) at the top of the nextpage with ¯X being the complement of X ∈ { A , B } . Proof:
See Appendix A.Notably, our procedure results in exact closed-form resultswhile other similar works rely on approximations. Also,regarding the dependence of the individual SEs with respectto the transmit power, we observe that they are interference-limited as expected [21]. Furthermore, we notice an increaseof the SEs with the number of APs by taking advantage of theCF mMIMO architecture, which is studied in depth below.IV. P
OWER E FFICIENCY
Herein, we present a detailed study concerning the achievablepower savings by letting the number of APs grow large, i.e., M → ∞ . These savings are known as power-scaling laws thatallow preserving a specific SE while reducing the transmitpowers.We assume that all the users have the same transmit power,i.e., no power control is considered. In particular, let η A ,i = η B ,i = 1 for the sake of simplicity, we are going to shed lighton the impact of the transmit power per user, pilot symbol, andrelay on the separate SEs in the large APs limit, i.e., M → ∞ . A. Scenario A: p p = E p M α , and fixed p r , p u This scenario concerns the study of the power efficiency inthe training phase.
Proposition 1:
For fixed p r , p u , and E p , when p p = E p M α R MAC i = τ c − τ p τ c log p u N (cid:18)(cid:16)(cid:80) Mm =1 √ η A ,i φ A ,mi (cid:17) + (cid:16)(cid:80) Mm =1 √ η B ,i φ B ,mi (cid:17) (cid:19)(cid:80) Mm =1 (cid:16)(cid:80) Wj =1 p u ( η A ,j α A ,mj + η B ,j α B ,mj ) + 1 (cid:17) ( φ A ,mi + φ B ,mi ) , (37) R MAC X ,i = τ c − τ p τ c log p u η A ,i N (cid:16)(cid:80) Mm =1 φ X ,mi (cid:17) (cid:80) Mm =1 (cid:16)(cid:80) Wj =1 p u ( η A ,j α A ,mj + η B ,j α B ,mj ) + 1 (cid:17) ( φ A ,mi + φ B ,mi ) , (38) R BC X ,i = τ c − τ p τ c log N p r (cid:16)(cid:80) Mm =1 √ η ¯X ,mi φ X ,mi (cid:17) (cid:80) Mm =1 (cid:80) Wj =1 ( p r α X ,mi + 1) ( η A ,mj φ B ,mj + η B ,mj φ A ,mj ) . (39)with α > and M → ∞ , we obtain γ MAC i = p u N E p M α (cid:18)(cid:16)(cid:80) Mm =1 α A ,mi (cid:17) + (cid:16)(cid:80) Mm =1 α B ,mi (cid:17) (cid:19) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj )+1 ( α A ,mi + α B ,mi ) , (40) γ MAC
X,i = p u N E p M α (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj )+1 ( α A ,mi + α B ,mi ) , (41) γ BC X ,i = N p r E p M α (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( p r α X ,mi + 1)( α B ,mj + α A ,mj ) . (42)Clearly, the choice of α defines the result. For example, in(40), the order of γ MAC i is O (cid:0) M − α (cid:1) , which implies that γ MAC i → , if α > . On the other hand, if < α < , γ MAC i → ∞ . However, when α = 1 , the corresponding SINRsresult in a finite limit. Similar comments holds for the SINRsgiven by (41) and 42. Note that the corresponding non-zerolimits are written as γ MAC i = p u N E p (cid:18)(cid:16)(cid:80) Mm =1 α A ,mi (cid:17) + (cid:16)(cid:80) Mm =1 α B ,mi (cid:17) (cid:19) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj )+1 ( α A ,mi + α B ,mi ) , (43) γ MAC
X,i = p u N E p (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj )+1 ( α A ,mi + α B ,mi ) , (44) γ BC X ,i = N p r E p (cid:16)(cid:80) Mm =1 α X ,mi (cid:17)(cid:80) Mm =1 (cid:80) Wj =1 ( p r α X ,mi + 1)( α B ,mj + α A ,mj ) . (45) B. Scenario B: p u = E u M β and p r = E r M γ , and fixed p p This strategy focuses on the power efficiency of solely thedata transmission phase.
Proposition 2:
For fixed p p , E u , and E r when p u = E u M β , p r = E r M γ with β ≥ , γ ≥ , and M → ∞ , we obtain γ MAC i = N E u M β (cid:32) M (cid:88) m =1 φ A ,mi (cid:33) + (cid:32) M (cid:88) m =1 φ B ,mi (cid:33) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , (46) γ MAC X ,i = N E u M β (cid:16)(cid:80) Mm =1 φ X ,mi (cid:17) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , (47) γ BC X ,i = N E r M γ (cid:16)(cid:80) Mm =1 φ X ,mi (cid:17) (cid:80) Mm =1 (cid:80) Wj =1 ( φ A ,mj + φ B ,mj ) . (48)It is straightforward to show that the order of both γ MAC i and γ MAC X ,i is O (cid:0) M − β (cid:1) while the order of γ BC X ,i is O (cid:0) M − γ (cid:1) .As before, the selection of the parameters β and γ affectsdirectly the corresponding SINRs. Especially, we observe that if ≤ β < and ≤ γ < , the SINRs grow unboundedly. Also,we notice that under certain circumstances, being equivalentto reducing further the transmit powers of each user or/andthe relay by means of β > or/and γ > , the sum SE of the i th pair R i tends to zero because R MAC i or/and R BC i tend tozero. Especially, R i → when one of the conditions 1) β ≥ and γ > , 2) β > and γ ≥ , 3) β > and γ > is met.As a result, in order to make γ MAC i and γ MAC
X,i converge to anon-zero limit, we should have β = 1 while when γ = 1 , γ BC X,i takes a finite value. These limits are given by γ MAC i = N E u (cid:18)(cid:16)(cid:80) Mm =1 φ A ,mi (cid:17) + (cid:16)(cid:80) Mm =1 φ B ,mi (cid:17) (cid:19)(cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , (49) γ MAC X ,i = N E u (cid:16)(cid:80) Mm =1 φ X ,mi (cid:17) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , (50) γ BC X ,i = N E r (cid:16)(cid:80) Mm =1 φ X ,mi (cid:17) (cid:80) Mm =1 (cid:80) Wj =1 ( φ A ,mj + φ B ,mj ) . (51)It is worthwhile to mention that Proposition 2 reveals thatin the large number of APs limit, the reduction of both thetransmit power of the APs and the users proportionally to M − cancels out the effects of inter-user interference, residual interference, and estimation error. The following corollariesshow how the sum SE R i changes by varying β and γ . Corollary 1:
When β = 1 and ≤ γ < , the SE of the i thuser pair as M → ∞ is written as R i = τ c − τ p τ c × log N E u (cid:32) M (cid:88) m =1 φ A ,mi (cid:33) + (cid:32) M (cid:88) m =1 φ B ,mi (cid:33) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) . (52)According to this corollary, R i is equal to R MAC i since R BC i = 0 , which means that the SE of the i th user pair dependsonly on Phase I (MAC phase). The explanation relies on thefact that since we have reduced the transmit power per usermuch less compared to the transmit power of the APs actingas relays, the MAC phase will present lower performance.Remarkably, in this case, R i does not depend on the numberof users. Also, this SE does not depend on E r , but it increaseswith E u . Corollary 2:
When ≤ β < and γ = 1 , the SE of the i thuser pair as M → ∞ is written as R i = τ c − τ p τ c log N E r (cid:16)(cid:80) Mm =1 φ A ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( φ A ,mj + φ B ,mj ) + τ c − τ p τ c log N E r (cid:16)(cid:80) Mm =1 φ B ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( φ A ,mj + φ B ,mj ) . (53)Corollary 2 denotes that the SE of the i th user pair appearsa bottleneck in the BC phase because the transmit power of theAPs during this phase has been cut down more than the transmitpower of each user. Herein, we notice that R i decreases withthe number of user pairs W while it increases with E r and itis independent of E u . Corollary 3:
When β = γ = 1 , the SE of the i th user pairas M → ∞ is written as R i = min (cid:0) R MAC i , R BC i (cid:1) , (54)where R MAC i = τ c − τ p τ c × log N E u (cid:32) M (cid:88) m =1 φ A ,mi (cid:33) + (cid:32) M (cid:88) m =1 φ B ,mi (cid:33) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) (55) R BC i = min (cid:0) R MAC A ,i , R BC B ,i (cid:1) + min (cid:0) R MAC B ,i , R BC A ,i (cid:1) , (56) with R MAC X ,i and R BC X ,i given by using (50) and (51), respectively.This corollary indicates that if we reduce the transmit powerof the APs and users simultaneously and equally to /M , boththe MAC and BC affect R i . C. Scenario C: p p = E p M α , p u = E u M β , and p r = E r M γ Such a scenario is the most general, where we can achievepower savings in both training and data transmission phases.
Proposition 3:
When p p = E p M α p u = E u M β , and p r = E r M γ with α ≥ , β ≥ , γ ≥ , and E p , E u , E r constants, as M → ∞ , we obtain γ MAC i = N E p E u M α + β (cid:32) M (cid:88) m =1 α A ,mi (cid:33) + (cid:32) M (cid:88) m =1 α B ,mi (cid:33) (cid:80) Mm =1 ( α A ,mi + α B ,mi ) , (57) γ MAC X ,i = N E p E u M α + β (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) (cid:80) Mm =1 ( α A ,mi + α B ,mi ) , (58) γ BC X ,i = N E p E r M α + γ (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) (cid:80) Mm =1 (cid:80) Wj =1 ( α A ,mj + α B ,mj ) . (59)Following the same procedure as before, we observe thatthe orders of the SINRs in the MAC and BC phases are O (cid:0) M − α − β (cid:1) and O (cid:0) M − α − γ (cid:1) , respectively. Hence, thecorresponding SINRs converge to non-zero limits only when α + β = 1 and α + γ = 1 . Otherwise, they can growunboundedly or tend to zero. These limits are given by γ MAC i = N E p E u (cid:32) M (cid:88) m =1 α A ,mi (cid:33) + (cid:32) M (cid:88) m =1 α B ,mi (cid:33) (cid:80) Mm =1 ( α A ,mi + α B ,mi ) , (60) γ MAC X ,i = N E p E u (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) (cid:80) Mm =1 ( α A ,mi + α B ,mi ) , (61) γ BC X ,i = N E p E r (cid:16)(cid:80) Mm =1 α X ,mi (cid:17) (cid:80) Mm =1 (cid:80) Wj =1 ( α A ,mj + α B ,mj ) . (62)The following corollaries present trade-offs between thetransmit powers of the pilot symbols and the APs and/or users. Corollary 4:
When α + β = 1 and β > γ ≥ , the SE ofthe i th user pair as M → ∞ is written as R i = τ c − τ p τ c × log N E p E u (cid:32) M (cid:88) m =1 α A ,mi (cid:33) + (cid:32) M (cid:88) m =1 α B ,mi (cid:33) (cid:80) Mm =1 ( α A ,mi + α B ,mi ) . (63)The inequality implies that α + γ < . Hence, γ BC X ,i → ,and the SE of the i th user pair is determined only by the MACPhase. Clearly, R i does not depend on the number of user pairs, which results in the increase with W of the sum SEgiven by (36). Corollary 5:
When α + γ = 1 and γ > β ≥ , the SE ofthe i th user pair as M → ∞ is written as R i = τ c − τ p τ c log N E p E r (cid:16)(cid:80) Mm =1 α A ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj ) + τ c − τ p τ c log N E p E r (cid:16)(cid:80) Mm =1 α B ,mi (cid:17) M (cid:88) m =1 W (cid:88) j =1 ( α A ,mj + α B ,mj ) . (64)Herein, the inequality suggests that α + β < , which meansthat γ MAC X ,i → , and only the BC phase defines the SE of the i th user pair. Corollary 6:
When α + β = 1 and β = γ ≥ , the SE ofthe i th user pair as M → ∞ is written as R i = min (cid:0) R MAC i , R BC i (cid:1) , (65)where R i = τ c − τ p τ c × log N E u (cid:32) M (cid:88) m =1 φ A ,mi (cid:33) + (cid:32) M (cid:88) m =1 φ B ,mi (cid:33) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) (66) R BC i = min (cid:0) R MAC A ,i , R BC B ,i (cid:1) + min (cid:0) R MAC B ,i , R BC A ,i (cid:1) , (67)with R MAC X ,i and R BC X ,i given in terms of (61) and (62), respec-tively.In other words, both conditions α + β = 1 and α + γ = 1 are fulfilled. By shedding further light on this corollary, weobserve an interplay appearing among the transmit powers.For example, a reduction of the pilot transmit power wouldresult in a degradation of the estimated channel that could bebalanced by an increase of the transmit power of the users/APsduring the transmission phase to preserve the performance withrespect to the SE. Remark 1:
In all corollaries above, the corresponding SEcould be boosted by increasing the involved E i , i = p , u , r . Forexample, Corollary 2 suggests that R i can be increased with E r by increasing the transmit power of the APs.V. P OWER A LLOCATION
Different from the previous section, where the transmitpowers of all users were assumed equal for the sake ofexposition of the scaling laws, in this section, we elaborateon the optimal power allocation among the users and the APsduring both MAC and BC phases, respectively. We assume thatthe power allocation takes place during the data transmissionphase while the power design of the training phase in terms of p p has previously being determined. In particular, we followthe procedure in [12], [21] and adapt it according to our systemarchitecture having M distributed APs as relay nodes.We focus on the maximization of the sum SE constrainedto a total power P , i.e., p u (cid:80) Wi =1 ( η A ,i + η B ,i ) + p r ≤ P . Inparticular, the formulation of the power allocation optimizationis described by max η A , η B ,p r W (cid:88) i =1 R i (68a) subject to p u W (cid:88) i =1 ( η A ,i + η B ,i ) + p r ≤ P η A ≥ , η B ≥ , p d ≥ , p r ≥ R i ≥ R min , i ∈ W where we have denoted η A = [ η A , , . . . , η A ,W ] T and η B =[ η B , , . . . , η B ,W ] T , while R min expresses the minimum SE ofthe i th pair. Given that the logarithm is an increasing function,the optimization, given by (68a), can be written as min ˜ η A , ˜ η B ,p r γ i ,γ A ,i ,γ B ,i W (cid:89) i =1 (1 + γ i ) − (69a) subject to γ i ≤ a ,i ˜ η A ,i + a ,i ˜ η B ,i (cid:80) Wj =1 ( a ,ij ˜ η A ,j + a ,ij ˜ η B ,j ) + 1 (69b) γ A ,i ≤ min { a ,i ˜ η A ,i c i , p r p r b B,i + c B,i } , i ∈ W (69c) γ B ,i ≤ min { a ,i ˜ η B ,i c i , p r p r b A,i + c A,i } , i ∈ W (69d) γ i ≤ γ A ,i + γ B ,i + γ A ,i γ B ,i , i ∈ W (69e) W (cid:88) i =1 (˜ η A ,i + ˜ η B ,i ) + p r ≤ P (69f) ˜ η A ≥ , ˜ η B ≥ , p d ≥ , p r ≥ (69g) γ − i (cid:18) τ c R min τ c − τ p − (cid:19) ≤ , i ∈ W (69h)where we have defined ˜ η A ,i = p u η A ,i , ˜ η B ,i = p u η B ,i , a ,i = N ( (cid:80) Mm =1 φ A ,mi ) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , a ,i = N ( (cid:80) Mm =1 φ B ,mi ) (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , a ,ij = (cid:80) Mm =1 α A ,mj (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , a ,ij = (cid:80) Mm =1 α B ,mj (cid:80) Mm =1 ( φ A ,mi + φ B ,mi ) , c i = (cid:80) Wj =1 ( a ,ij ˜ η A ,j + a ,ij ˜ η B ,j ) + 1 , b X,i = (cid:80) Mm =1 (cid:80) Wj =1 α X ,mi ( η A ,mj φ B ,mj + η B ,mj φ A ,mj ) N ( (cid:80) Mm =1 √ η ¯X ,mi φ X ,mi ) , and c X,i = (cid:80) Mm =1 (cid:80) Wj =1 ( η A ,mj φ B ,mj + η B ,mj φ A ,mj ) N ( (cid:80) Mm =1 √ η ¯X ,mi φ X ,mi ) . Moreover, γ i , γ A ,i , and γ B ,i correspond to the SINRs of R i , min (cid:16) R MAC A ,i , R BC B ,i (cid:17) and min (cid:16) R MAC B ,i , R BC A ,i (cid:17) , respectively.The latter optimization problem is nonconvex since it falls tothe category of complementary geometric programming (CGP).However, its solution can be obtained by solving a sequenceof convex GP problems [12], [21], [36], [37]. Initially, as in[37, Lem. 1], we approximate the objective function (1 + γ i ) by the monomial function δ i γ µ i i , where δ i = ( γ ◦ i ) − µ i (1 + γ ◦ i ) and µ i = γ ◦ i γ ◦ i +1 [12]. Next, we transform the two inequalities described by (69b) and (69e) into posynomials, in order toresult in a GP problem. Specifically, the first inequality iswritten as a ,i ˜ η A ,i + a ,i ˜ η B ,i ≥ (cid:18) a ,i ˜ η A ,i φ A ,i (cid:19) φ A ,i (cid:18) a ,i ˜ η B ,i φ B ,i (cid:19) φ B ,i , (70)where φ A ,i = a ,i ˜ η ◦ A ,i a ,i ˜ η ◦ A ,i + a ,i ˜ η ◦ B ,i , φ B ,i = a ,i ˜ η ◦ B ,i a ,i ˜ η ◦ A ,i + a ,i ˜ η ◦ B ,i with ˜ η ◦ A ,i and ˜ η ◦ B ,i denoting the initialization values. In (70), wehave applied a known property, expressing that, for any setof positive numbers, the geometric mean is no larger than thearithmetic mean [38]. Hence, substitution of (70) into (69b)gives γ i ≤ (cid:16) a ,i ˜ η A ,i q A ,i (cid:17) q A ,i (cid:16) a ,i ˜ η B ,i q B ,i (cid:17) q B ,i (cid:80) Wj =1 ( a ,ij ˜ η A ,j + a ,ij ˜ η B ,j ) + 1 , i ∈ W . (71)For the second inequality, given by (69e), we follow theprocedure in [21] to approximate f ( x, y ) = x + y + xy near anarbitrary point x ◦ , y ◦ > by means of the monomial function g ( x, y ) = ζx λ y λ in terms of ζ , λ , and λ . These parametersare obtained in [21] as λ = x ◦ (1+ y ◦ ) x ◦ + y ◦ + x ◦ y ◦ , λ = y ◦ (1+ x ◦ ) x ◦ + y ◦ + x ◦ y ◦ , ζ = ( x ◦ + y ◦ + x ◦ y ◦ ) ( x ◦ ) − λ ( y ◦ ) − λ . Thus, we have γ i ≤ ζ i γ λ A ,i A ,i γ λ B ,i B ,i , i ∈ W (72)where ζ i = (cid:0) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i (cid:1) (cid:0) γ ◦ A ,i (cid:1) − λ A ,i (cid:0) γ ◦ B ,i (cid:1) − λ B ,i , λ A ,i = γ ◦ A ,i ( γ ◦ B ,i ) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i , and λ B ,i = γ ◦ B ,i ( γ ◦ A ,i ) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i with γ ◦ A ,i , γ ◦ B ,i being the initialization values. The algorithm stepsare provided by Algorithm 1, where the parameter θ > definesthe desired accuracy as a trade-off with convergence speed.Especially, as θ approaches , we result in better accuracywhile the convergence speed is slow.VI. N UMERICAL R ESULTS
This section depicts the analytical results provided bymeans of Theorem 1 and Propositions 1-3 that illustrate theperformance of a multi-pair two-way CF mMIMO system. Forthe sake of comparison, we have accounted for a conventionaltwo-way collocated massive MIMO architecture employingDF as described by [21]. Also, our analytical results areaccompanied by Monte Carlo simulations by means of independent channel realizations, in order to verify them andshow their tightness. A. Simulation Setup
Unless otherwise stated, the following set of parameters isused during the simulations. In particular, we consider M =200 APs and W = 5 user pairs uniformly distributed in an areaof × . Each AP is equipped with N = 3 antennas. Notethat the area edges are wrapped around to avoid the boundaryeffects. Also, we assume that the coherence time and bandwidthare T c = 1 ms and B c = 200 kHz , respectively, which meansthat the coherence block consists of channel uses. Theorthogonality among pilots requires at least τ p = 2 W = 10 .Moreover, we assume that p p , p u , and p r correspond to thenormalized powers, obtained by dividing ¯ p p = ¯ p u , and ¯ p r by Algorithm 1
Successive approximation algorithm1.
Initialisation : Define the parameter θ and the tolerance (cid:15) .Set k = 1 , ˜ η A ,i = ˜ η B ,i = P W , p r = P while γ ◦ i , γ ◦ A ,i , and γ ◦ B ,i are chosen by means of Theorem 1.2. Iteration k : Evaluate µ i = γ ◦ i γ ◦ i +1 , φ A ,i = a ,i ˜ η ◦ A ,i a ,i ˜ η ◦ A ,i + a ,i ˜ η ◦ B ,i , φ B ,i = a ,i ˜ η ◦ B ,i a ,i ˜ η ◦ A ,i + a ,i ˜ η ◦ B ,i , ζ i = (cid:0) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i (cid:1) (cid:0) γ ◦ A ,i (cid:1) − λ A ,i (cid:0) γ ◦ B ,i (cid:1) − λ B ,i , λ A ,i = γ ◦ A ,i ( γ ◦ B ,i ) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i , λ B ,i = γ ◦ B ,i ( γ ◦ A ,i ) γ ◦ A ,i + γ ◦ B ,i + γ ◦ A ,i γ ◦ B ,i .Then, solve the GP problem: min ˜ η A , ˜ η B ,p r γ i ,γ A ,i ,γ B ,i W (cid:89) i =1 γ − µ i i (73a) subject to θ − ˜ η A ,i ≤ ˜ η A ,i ≤ θ ˜ η A ,i , i ∈ W (73b) θ − ˜ η B ,i ≤ ˜ η B ,i ≤ θ ˜ η B ,i , i ∈ W (73c) θ − γ ◦ i ≤ γ i ≤ θγ ◦ i , i ∈ W (73d) θ − γ ◦ A ,i ≤ γ A ,i ≤ θγ ◦ A ,i , i ∈ W (73e) θ − γ ◦ B ,i ≤ γ B ,i ≤ θγ ◦ B ,i , i ∈ W (73f) c i γ i (cid:18) a ,i ˜ η A ,i φ A ,i (cid:19) − φ A ,i (cid:18) a ,i ˜ η B ,i φ B ,i (cid:19) − φ B ,i ≤ , i ∈ W (73g) γ i ζ − i γ − λ A ,i A ,i γ − λ B ,i B ,i ≤ , i ∈ W (73h) γ A ,i c i ( a ,i ˜ η A ,i ) − ≤ i ∈ W (73i) γ B ,i c i ( a ,i ˜ η B ,i ) − ≤ i ∈ W (73j) γ A ,i p − ( p r b B,i + c B,i ) ≤ , i ∈ W (73k) γ B ,i p − ( p r b A,i + c A,i ) ≤ , i ∈ W (73l) W (cid:88) i =1 (˜ η A ,i + ˜ η B ,i ) + p r ≤ P (73m) ˜ η A ≥ , ˜ η B ≥ , p d ≥ , p r ≥ (73n) γ − i (cid:18) τ c R min τ c − τ p − (cid:19) ≤ , i ∈ W (73o)Let ˜ η (cid:63) A ,i , ˜ η (cid:63) B ,i , γ (cid:63)i , γ (cid:63) A ,i , γ (cid:63) B ,i , i ∈ W .3. If max i | ˜ η (cid:63) A ,i − ˜ η ◦ A ,i | < (cid:15) and/or max i | ˜ η (cid:63) B ,i − ˜ η ◦ B ,i | < (cid:15) and/or max i | γ (cid:63)i − γ ◦ i | < (cid:15) and/or max i | γ (cid:63) A ,i − γ ◦ A ,i | < (cid:15) and/or max i | γ (cid:63) B ,i − γ ◦ B ,i | < (cid:15) → Stop. Otherwise, go to step .4. Update initial values . Set k = k +1 , ˜ η ◦ A ,i = ˜ η (cid:63) A ,i , ˜ η ◦ B ,i = ˜ η (cid:63) B ,i , γ ◦ i = γ (cid:63)i , γ ◦ A ,i = γ (cid:63) A ,i , γ ◦ B ,i = γ (cid:63) B ,i , and go to step .the noise power N P = W c × κ B × T × N F . The variousparameters are found in Table I. Power control in terms ofmaximizing the sum SE is considered only in Subsection VI-D.Hence, without any power control, we assume that in theuplink, all users transmit with full power, i.e., η A ,i = η B ,i =1 , ∀ i . Similarly, in the downlink, all APs transmit with fullpower, which means η mi = (cid:16) N (cid:80) Wi =1 ( φ B ,mi + φ A ,mi ) (cid:17) − by satisfying (13). Also, the transmit powers during the MACand BC phases are assumed equal, i.e., W p u = p r .We take into account for [8, Remark 4], and thus, we consider TABLE I: Parameters Values for Numerical Results
Description Values
Number of APs M = 200 Number of Antennas/AP N = 3 Number user pairs W = 5 Carrier frequency f = 2 GHz Power per pilot symbol ¯ p p = 100 mW Uplink transmit power ¯ p u = 100 mW Path loss exponent α = 4 Communication bandwidth W c = 20 MHz Coherence bandwidth B c = 200 KHz Coherence time T c = 1 ms Duration of uplink training τ p = 10 samplesBoltzmann constant κ B = 1 . × − J / K Noise temperature T = 290 K Noise figure N F = 9 dB the 3GPP Urban Microcell model in [39, Table B.1.2.1-1] asa more appropriate benchmark for CF mMIMO systems thanthe established model presented initially in [3] because of twomain reasons: i) although CF mMIMO systems are more likelysuggested for shorter distances, the model in [3] assumes thatshadowing is met for users found further from an AP, andii) the COST-Hata model, used in [3] is suitable for macro-cellswith APs being at least far from the users and at least above the ground while the CF setting suggests APs foundat lower height and being very close to the users. Specifically,the large-scale fading coefficient, described by this mode for a carrier frequency, is given by α X ,mk [ dB ] = − . − . (cid:18) d X ,mk (cid:19) + F X ,mk , where d X ,mk expresses the distance between AP m and user k while F X ,mk ∼ CN (cid:0) , (cid:1) describes the shadow fading.In addition, the shadowing terms between different users areassumed to be correlated as E { F X ,mk F X ,ij } = 4 − δ X ,kj / only when m = i , where δ X ,kj is the distance between users k and j . B. Demonstration of basic properties
Initially, we assume that p p = p u as well as p r = 2 W p u ,which mean that users transmit equal power during the trainingand data transmission phases.Fig. 2 presents the sum SE versus p u with varying numberof APs along with Monte-Carlo simulations verifying theanalytical expressions since the lines almost coincide. Also,we notice that SE saturates at high SNR due to the inter-userinterference, as expected. Moreover, SE increases with theincreasing number of APs M .Fig 3(a) depicts the sum SE versus the number of APs M .Also, we have considered the scenario of genie receivers at theusers during the BC phase. In other words, we have assumedthat the corresponding receivers are aware of instantaneousCSI and not just its statistics. Since the gap between the twolines is small, the downlink channel hardens and no extratraining is required. In addition, for the sake of comparison,we have included the collocated scenario with a base stationat the center of the area being the relay node and having M N antennas, α X ,mk = α X k , η k = M η mk , ∀ m , and in general, all the parameters equal across the index m [27]. It can be seenthat the CF mMIMO relay setting outperforms the collocatedlayout because the diversity against path-loss and shadow fadingis exploited. Moreover, we have considered the conventionalorthogonal scheme where the transmission of each pair takesplace at different time slots or frequency bands. As the numberof APs increases, the two-way CF MIMO performs betterbecause the effect from the inter-user interference decreases.Thus, when M is low, the orthogonal scheme performs better,but the mMIMO system behaves better as M increases andchannels become orthogonal, which means a mitigation of theinterference. Compared to Fig. 3(a), which assumes W = 5 userpairs, Fig. 3(b) shows the sum SE in the case of W = 20 userpairs. In the latter figure, the outperformance of the CF mMIMOsetting over the collocated scenario is more pronounced becauseof its concomitant advantages. Hence, we observe that theperformance gap at M = 400 AP is and , when W = 5 and W = 20 , respectively. C. Power-scaling laws
Herein, we verify Propositions 1-3 providing the power-scaling laws, also denoted as asymptotic results that correspondto the scenarios A-C mentioned earlier. In addition, weelaborate on the resultant power savings with comparison tothe analytical exact results provided by Theorem 1.Fig. 2:
Sum SE per versus the uplink transmit power p p = p u for varyingnumber of APs M with validation by Monte-Carlo simulations ( N = 3 , W = 5 , and p r = 2 W p u ). Fig 4 sheds light into Scenario A by depicting the sumSE with respect to the number of APs M for varying scalingin terms of the parameter α . In general, we observe that theasymptotic results approach the exact curves as M → ∞ .When α = 0 . < α < , R i → ∞ , while if α = 1 , theasymptotic SE saturates and approaches the analytical result.The third group of curves corresponds to α = 1 . > . In suchcase, R i tends to zero.Figs. 5 and 6 illustrate the properties regarding the powersavings of Scenario B described by Proposition 2. Especially,in any of the cases i) β = 1 and < γ < , ii) < β < and γ = 1 , and iii) β = γ = 1 , we show in Fig. 5 that the (a)(b) Fig. 3:
Sum SE versus the number of APs M for different scenarios ( N = 3 , p p = p u , and p r = 2 W p u ), when (a) W = 5 and (b) W = 20 user pairs. asymptotic results converge to specific values and approachthe exact results in the large number of APs regime ( M → ∞ )as described by Corollaries 3, 4. In the upper set of plots ofFig. 6, we observe that when β or γ is greater than one, whichmeans that the transmit power of the MAC or BC phase iscut down too much, the sum SE approaches zero as M → ∞ .In fact, the larger the parameter, being greater than one, thefaster the decrease of the SE to zero. Furthermore, when boththe transmit powers of users and APs are reduced tolerablysuch that β < and γ < , the sum SE R i increases withoutbound as the lower set of lines of Fig. 6 reveals.Fig. 7 represents Scenario C describing the interplay betweenthe pilot symbol power and and the users/relay powers. Thesums α + β and α + γ determine the behavior of the sum SEin the large number of APs limit. Hence, if we set α = 1 . , β = 1 . , γ = 0 . and α = 0 . , β = 1 . , γ = 0 . , we observethat both lines converge to zero as M → ∞ since, in both cases,we have α + β = 2 . and α + γ = 1 . . Notably, the two linesconverge to each other (their gap decreases) as M increases, Fig. 4: Sum SE versus the number of APs M by means of asymptotic(Scenario A) and exact analysis (Theorem 1) for N = 3 , W = 5 , p p = E p /M α with E p = 10 dB . i.e., the asymptotic sum SE is the same because the two sumsare kept identical. Furthermore, the line with α = 0 . providesbetter SE for finite number of APs although the transmit relaypower is cut down more because the channel is estimatedwith higher quality. The middle set of lines demonstrates thatthe sum SE grows without bound when both α + β < and α + γ < simultaneously. The third subfigure illustrates that ifany of the two following conditions are satisfied, the sum SEapproaches a non-zero limit. Specifically, if i) α + β = 1 and β > γ ≥ or ii) α + γ = 1 and γ > β ≥ , then R i saturates.Fig. 5: Sum SE versus the number of APs M by means of asymptotic(Scenario B) and exact analysis (Theorem 1) for N = 3 , W = 5 , p u = E u /M β with E u = 10 dB , and p r = E r /M γ with E r = 10 dB (finitelimits). D. Power allocation
Fig. 8 shows the performance of the two-way CF mMIMOsystem with optimal power allocation using Algorithm 1 when M = 200 APs and with varying channel estimation accuracy Fig. 6:
Sum SE versus the number of APs M by means of asymptotic(Scenario B) and exact analysis (Theorem 1) for N = 3 , W = 5 , p u = E u /M β with E u = 10 dB , and p r = E r /M γ with E r = 10 dB (zero andunbounded limits). Fig. 7:
Sum SE versus the number of APs M by means of asymptotic(Scenario C) and exact analysis (Theorem 1) for N = 3 , W = 5 , p p = E p /M α with E p = 10 dB , p u = E u /M β with E u = 10 dB , and p r = E r /M γ with E r = 10 dB (zero, unbounded, and finite limits). by means of p p while θ = 1 . to achieve good accuracywith reasonable convergence time. Also, we assume that thetotal power budget is P = 10 dB . Moreover, we consideruniform power allocation by using Theorem 1 for the sake ofcomparison. Notably, the optimal power allocation performsbetter than uniform power allocation, and it results in animprovement of . and . when p p = 10 dB and p p = 15 dB , respectively. In other words, a better channelestimation barely affects the sum SE.VII. C ONCLUSION
This paper investigated the sum SE of a multipair two-wayHD relaying system assisted by a CF mMIMO architectureemploying MR processing and accounting for imperfect CSI.Contrary to the common collocated mMIMO layout, thedistributed CF mMIMO design achieves higher performance. Fig. 8:
Sum SE versus the number of APs M by means of uniform (Theorem1) and optimal (Algorithm 1) power allocations for p p = 10 dB (blue lines)and p p = 15 dB (red lines). Furthermore, power-scaling laws, achieving to scale the trans-mit powers of the users and APs while maintaining the desiredSE, were obtained. Also, the trade-offs regarding these laws wasexamined. Finally, we performed an optimal power allocationconcerning the transmit powers of the APs and users duringthe data transmission phase towards the improvement of theSE with comparison to uniform power allocation.A
PPENDIX AP ROOF OF T HEOREM γ MAC i . The desired signalsof T A ,i and T B ,i in (18) are written as E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi h mi + ˆ g H mi h mi (cid:17)(cid:111) = N M (cid:88) m =1 φ A ,mi , (74) E (cid:110) M (cid:88) m =1 (cid:16) ˆ h H mi g mi + ˆ g H mi g mi (cid:17)(cid:111) = N M (cid:88) m =1 φ B ,mi , (75)since h mi and g mi are independent. We continue with thederivations of EE MAC A ,i , EE MAC B ,i , IUI
MAC i , and N MAC i . Specifi-cally, we have EE MAC A ,i = η A ,i (cid:32) var (cid:110) M (cid:88) m =1 ˆ h H mi h mi (cid:111) + var (cid:110) M (cid:88) m =1 ˆ g H mi h mi (cid:111)(cid:33) , (76)where (76) is obtained because the variance of a sum ofindependent RVs is equal to the sum of the variances. The firstterm of (76) is obtained as var (cid:110) M (cid:88) m =1 ˆ h H mi h mi (cid:111) = M (cid:88) m =1 E (cid:110)(cid:12)(cid:12)(cid:12) ˆ h H mi h mi − E (cid:110) ˆ h H mi h mi (cid:12)(cid:12)(cid:12) (cid:111)(cid:111) (77) = M (cid:88) m =1 (cid:18) E (cid:110)(cid:12)(cid:12)(cid:12) ˆ h H mi h mi (cid:12)(cid:12)(cid:12) (cid:111) − (cid:12)(cid:12)(cid:12) E { ˆ h H mi ˆ h mi } (cid:12)(cid:12)(cid:12) (cid:19) = M (cid:88) m =1 (cid:18) E (cid:110)(cid:12)(cid:12)(cid:12) (cid:107) ˆ h mi (cid:107) + ˆ h H mi ˜ h mi (cid:12)(cid:12)(cid:12) (cid:111) − N φ ,mi (cid:19) (78) = M (cid:88) m =1 (cid:16) E (cid:110) (cid:107) ˆ h mi (cid:107) (cid:111) + E (cid:110) | ˆ h H mi ˜ h mi | (cid:111) − N φ ,mi (cid:17) (79) = N M (cid:88) m =1 (cid:0) ( N + 1) φ ,mi + φ A ,mi e A ,mi − N φ ,mi (cid:1) (80) = N M (cid:88) m =1 α A ,mi φ A ,mi , (81)where (77) follows again because the variance of a sum ofindependent RVs is equal to the sum of the variances. In(78), we have considered that ˜ h mi is independent of ˆ h mi andhas zero mean. The identity E {(cid:107) ˆ h mi (cid:107) } = N ( N + 1) φ ,mi has been used in 80, and (81) follows after some algebraicmanipulations since e A ,mi = α A ,mi − φ A ,mi . More easily, thesecond term of (76) is given by var (cid:110) M (cid:88) m =1 ˆ g H mi h mi (cid:111) = N M (cid:88) m =1 α A ,mi φ B ,mi . (82)Hence, EE MAC A ,i becomes my means of (81) and (82) EE MAC A ,i = N η A ,i M (cid:88) m =1 α A ,mi ( φ A ,mi + φ B ,mi ) . (83)In the same way, we derive EE MAC B ,i = N η B ,i M (cid:88) m =1 α B ,mi ( φ A ,mi + φ B ,mi ) . (84)The term, describing the inter-user interference, is obtained as IUI
MAC i = W (cid:88) j (cid:54) = i (cid:32) η A ,j M (cid:88) m =1 (cid:16) E {| ˆ h H mi h mj | } + E {| ˆ g H mi h mj | } (cid:17)(cid:33) + W (cid:88) j (cid:54) = i (cid:32) η B ,j M (cid:88) m =1 (cid:32) E {| ˆ h H mi g mj | } + E {| M (cid:88) m =1 ˆ g H mi g mj | } (cid:33)(cid:33) = W (cid:88) j (cid:54) = i (cid:32) η A ,j M (cid:88) m =1 ( φ A ,mi α A ,mj + φ B ,mi α B ,mj ) (cid:33) + W (cid:88) j (cid:54) = i (cid:32) η B ,j M (cid:88) m =1 ( φ A ,mi α B ,mj + φ B ,mi α A ,mi ) (cid:33) = N W (cid:88) j (cid:54) = i M (cid:88) m =1 ( η A ,j α A ,mj + η B ,j α B ,mj )( φ A ,mi + φ B ,mi ) , (85)where we have applied the property E {| X + Y | } = E {| X | } + E {| Y | } , holding between two independent random variables X and Y with E { X } = 0 . Next, the noise term becomes N MAC i = 1 p u M (cid:88) m =1 ( φ A ,mi + φ B ,mi ) . (86)Substitution of (74), (75),(83), (84), (85), and (86) into (18)and (23) provides R MAC i as well as R MAC X ,i for X ∈ { A , B } .The proof continues with the derivation of R BC A ,i by meansof the computation of (29)-(33). Specifically, regarding the desired signal, we have E (cid:110) M (cid:88) m =1 √ p d ,m η B ,mi h T mi ˆ h ∗ mi (cid:111) = E (cid:110) M (cid:88) m =1 √ p d ,m η B ,mi (cid:107) ˆ h mi (cid:107) (cid:111) + E (cid:110) M (cid:88) m =1 √ p d ,m η B ,mi ˜ h T mi ˆ h ∗ mi (cid:111) = N M (cid:88) m =1 √ p d ,m η B ,mi φ A ,mi , (87)where we have used the independence between ˆ h mi and ˜ h mi .Also, BU BC A i and BU BC B i are obtained by following the sameprocedure with (81). In particular, we have BU BC A i = N M (cid:88) m =1 η B ,mi α A ,mi φ A ,mi , (88) BU BC B i = N M (cid:88) m =1 η A ,mi α A ,mi φ B ,mi . (89)Next, regarding IUI BC A i , we have E (cid:110)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 η / ,mj h T mi ˆ g ∗ mj (cid:12)(cid:12)(cid:12) (cid:111) = M (cid:88) m =1 η A ,mj E {| h T mi ˆ g ∗ mj | } = N M (cid:88) m =1 η A ,mj α A ,mi φ B ,mj , (90)where in the first equation, we have used again that E {| X + Y | } = E {| X | } + E {| Y | } . The last term, IUI BC B i , is obtainedsimilarly as IUI BC B i = N M (cid:88) m =1 η B ,mj α A ,mi φ A ,mj . (91)By using (87)-(91) and (14), we obtain R BC A ,i and conclude theproof since R BC B ,i can be derived in the same fashion.R EFERENCES[1] E. Larsson et al. , “Massive MIMO for next generation wireless systems,”
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