Massive MIMO-Enabled Full-Duplex Cellular Networks

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCOMM.2017.2731768, IEEE Transactions on Communications

IEEE TRANSACTIONS ON COMMUNICATIONS Massive MIMO-Enabled Full-Duplex Cellular Networks Arman Shojaeifard, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Marco Di Renzo, Senior Member, IEEE, Gan Zheng, Senior Member, IEEE, Khairi Ashour Hamdi, Senior Member, IEEE, Jie Tang, Member, IEEE

Abstract—We provide a theoretical framework for the study outperform the state-of-the-art cellular standards jointly in of massive multiple-input multiple-output (MIMO)-enabled full- terms of spectral efficiency (SE) and energy efficiency (EE). duplex (FD) cellular networks in which the residual self- Moreover, under increasingly scarce spectrum, the transceiving interference (SI) channels follow the Rician distribution and other channels are Rayleigh distributed. In order to facilitate of information over the same radio-frequency (RF) resources, bi-directional wireless functionality, we adopt (i) in the downlink i.e. full-duplex (FD) mode [5], has become a topic of interest (DL), a linear zero-forcing with self-interference-nulling (ZF- for 5G and beyond [6], [7]. In theory, FD technology can SIN) precoding scheme at the FD base stations (BSs), and (ii) in double the achievable sum-rate of half-duplex (HD) radios, the uplink (UL), a self-interference-aware (SIA) fractional power where orthogonal RF partitioning is typically employed to control mechanism at the FD mobile terminals (MTs). Linear ZF receivers are further utilized for signal detection in the UL. The avoid the over-powering self-interference (SI). In practice, results indicate that the UL rate bottleneck in the FD baseline however, the respective FD over HD SE gain predominantly single-input single-output (SISO) system can be overcome via depends on the SI cancellation capability. exploiting massive MIMO. On the other hand, the findings may Recently, there has been major breakthroughs in SI can- be viewed as a reality-check, since we show that, under state-of- cellation using any combination of (i) spatial/angular isolation the-art system parameters, the spectral efficiency (SE) gain of FD massive MIMO over its half-duplex (HD) counterpart is largely and (ii) subtraction in digital/analog domains [8]. On the other limited by the cross-mode interference (CI) between the DL and hand, FD, beyond point-to-point, remains in its infancy. In the UL. In point of fact, the anticipated two-fold increase in SE particular, the introduction of cross-mode interference (CI) is shown to be only achievable when the number of antennas between the downlink (DL) and the uplink (UL), in addition tends to be infinitely large. to the SI, significantly increases the complexity for large Index Terms—Full-duplex, cellular network, massive MIMO, scale FD cellular setups. Many relevant works have emerged self-interference, cross-mode interference, uplink power control, recently, including FD for small-cell (SC) [9]–[11], relay Rician fading channel, stochastic geometry theory. (RL) [12]–[14], cloud radio access network (CRAN) [15], and heterogeneous cellular network (HCN) [16]–[18]. A general I.INTRODUCTION consensus from early results is that the FD over HD SE The fifth-generation mobile network (5G) is expected to gain mostly arises in the DL and that the UL is the main roll out from 2018 onwards as a remedy for tackling the performance bottleneck. For example, the authors in [19] have existing capacity crunch [3]. A key 5G technology is massive shown that bi-directional cellular systems with baseline single- multiple-input multiple-output (MIMO), or large scale an- input single-output (SISO), achieve double the DL rate at the tenna system (LSAS), where the base stations (BSs) equipped cost of more than a thousand-fold reduction in the UL rate. A with hundreds of antennas simultaneously communicate with potential strategy for tackling this limitation is to exploit the multiple mobile terminals (MTs) [4]. Massive MIMO, via large degrees of freedom (DoF) in massive MIMO for better spatial-multiplexing and directing power intently, can greatly resilience against SI and CI [20], [21].

A. Shojaeifard and K.-K Wong are with the Communications and Information Systems Group, Department of Electronic and Electrical Engineering, University College A. Related Works London, London WC1E 7JE, United Kingdom. (e-mail: [email protected]; kai- [email protected]). M. Di Renzo is with the Laboratoire des Signaux et Systemes,` CNRS, CentraleSupelec,´ In [22], the authors considered a FD BS with large scale Univ Paris Sud, Universite´ Paris-Saclay, 3 rue Joliot Curie, Plateau du Moulon, 91192, antenna array serving multiple HD single-antenna MTs, and Gif-sur-Yvette, France. (e-mail: [email protected]). G. Zheng is with the Wolfson School of Mechanical, Electrical, and Manufacturing proposed a linear extended zero-forcing (ZF) precoder to Engineering, Loughborough University, Loughborough LE11 3TU, United Kingdom. (e- suppress SI at the receiving antennas, subject to perfect mail: [email protected]). K. A. Hamdi is with the Microwave and Communication Systems Group, School of channel state information (CSI). The sum-rate in FD mode Electrical and Electronic Engineering, University of Manchester, Manchester M13 9PL, was almost doubled versus that in the HD case, and the United Kingdom. (e-mail: [email protected]). J. Tang is with the School of Electronic and Information Engineering, South China optimal ratio of transmit over receive antennas was found University of Technology, Guangzhou, and with the State Key Laboratory of Integrated to be approximately 3 [22]. The extension to a FD BS with Services Networks, Xidian University, China. (e-mail: [email protected]). This work was supported by the United Kingdom Engineering and Physical Sciences large scale antenna array serving HD multi-antenna MTs using Research Council (EPSRC) under Grants EP/N008219/1 and EP/N007840/1, and by the linear block diagonalization (BD) beamforming was provided National Natural Science Foundation of China under Grant 61601186. Parts of this work were presented at the IEEE GLOBECOM 2016, Washington, D.C., in [23]. With asymptotically large antenna array size, the United States [1], and at the IEEE ICC 2017, Paris, France [2]. optimal ratio of transmit over receive antennas was shown

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to converge to the ratio of DL over UL streams. In [24], We further reduce the computational complexity in certain under a setup similar to that in [22], the FD sum-rates with special cases. In particular, we adopt the proposed framework different spatial isolation and SI subtraction methods were to study the FD versus HD cellular network performance. compared. On the other hand, non-linear transceiver design With baseline SISO, we derive a tight bounded closed-form for maximizing the sum-rate in FD multi-user MIMO was function of the FD over HD SE gain and show its optimal studied in [25] and [26]. The works highlighted above have point occurs in the ratio of the BS and MT transmit powers considered a single-cell network. Intuitively, the introduction being equal to one. With massive MIMO, we derive single- of SI and CI has huge implications in a FD multi-cell multi- integral expressions for the FD and HD SEs and hence utilize user MIMO paradigm necessitating rigorous investigation. multivariate non-linear curve fitting to develop a closed-form Several studies of FD multi-cell MIMO cellular networks approximation of the corresponding FD over HD SE gain as have also been recently reported. In [27], the DL and UL a function of the number of antennas and users. ergodic sum-rates in a deterministic FD multi-cell multi-user The validity of the proposed theoretical framework is con- MIMO setup were characterized using conjugate-beamforming firmed using MC simulations. Our findings highlight that (CB). In addition, the throughput performance in FD dis- the key characteristics of massive MIMO, in terms of high tributed MIMO systems with BS cooperation (i.e., FD network transmit/receive array gain and lower BS/MT transmit power, MIMO) was investigated in [28]. In particular, the authors allow for achieving significant performance gains over other utilized spatial interference-alignment (IA) for tackling the CI FD multi-cell setups. On the other hand, the SE gain of FD bottleneck, and characterized the FD over HD multiplexing over HD massive MIMO cellular network, under state-of-the- gain in closed-form. Moreover, in [29], success probability art system parameters, is shown to be predominantly limited expressions in cellular systems with FD relaying BSs under by the CI between the DL and the UL. In point of fact, different beamforming and interference-cancellation strategies the corresponding sum-rate gain is shown to increase only were characterized using stochastic geometry theory. On the logarithmically in the antenna array size, with the anticipated other hand, the design and analysis of randomly-deployed FD two-fold increase in SE only achieved as the number of cellular networks with directional antennas was provided in antennas tends to be infinitely large. [30]. In particular, the authors derived analytical expressions for the DL and UL coverage probabilities under passive SI suppression and fixed power control. C. Organization The remainder of this paper is organized as follows. The massive MIMO-enabled FD cellular network is described in B. Contributions Section II. The proposed UL power control mechanism is Motivated by the above, in this work, we provide a the- introduced in Section III. The SE analysis is provided in oretical stochastic geometry-based framework for the study Section IV, followed by a comparison of FD over HD SE of FD massive MIMO cellular networks using the Poisson gain in Section V. Numerical results are provided in Section point process (PPP)-based abstraction model of BSs and MTs. VI, and finally, conclusions are drawn in Section VII. We jointly consider the DL and UL of a FD multi-cell multi- user massive MIMO system where the BSs and MTs are transceiving over the same RF resources. We adopt the Rician D. Notation distribution for the residual SI fading channels, thus captur- We use the following notation in this work. X is a matrix ing performance under generalized SI cancellation capability. with (i, j)-th entry {X}i,j; x is a vector with k-th element All other fading channels are modeled using the Rayleigh {x}k; T , †, and + are the transpose, Hermitian, and pseudo- distribution. In the DL, we devise a linear zero-forcing with inverse operations; Ex{.} is the expectation; Fx(.) is the self-interference-nulling (ZF-SIN) precoder to jointly suppress cumulative density function (c.d.f.); Px(.) is the probability residual SI and multi-user interference. On the other hand, density function (p.d.f.); P(x) is the probability; Mx(.) is we propose a self-interference-aware (SIA) fractional power the m.g.f.; |x| is the modulus; kxk and kXk are the Euclidean control mechanism at the FD MTs to keep SI below a certain and Frobenius norms; I(.) is the identity matrix; Null(.) is a threshold. We derive the distributions, and in certain cases nullspace; H(.) is the Heaviside step function; δ(.) is the Delta moments of the proposed UL transmit power mechanism. function; CN (µ, ν2) is the complex Gaussian distribution with Moreover, in the UL, for signal detection at the FD massive mean µ and variance ν2; Γ(.) and Γ(., .) are the Gamma and MIMO BSs, linear ZF receivers are utilized for suppressing incomplete (upper) Gamma functions; Γ(κ, θ) is the Gamma multi-user interference. distribution with shape parameter κ and scale parameter θ; The signals distributions are derived under the linear pro- Ln(.) is the Laguerre polynomial; Ei(.), S (.), and C (.) are cesses described above. We characterize the DL and UL SEs the exponential, Sine, and Cosine integral functions; erfi(.) using a moment-generating-function (m.g.f.)-based approach is the imaginary error function; Qm(., .) is the Marcum and derive the signals conditional statistics in closed-form. Q-function; 2F1(., .; .; .), 2F˜1(., .; .; .), 0F1 (; .; .), 0F˜1 (; .; .), The proposed framework can be readily used to calculate the pFq(.; .; .), pF˜q(.; .; .) are the Gauss, Regularized Gauss, con- SEs via three-fold integrals, versus (i) the manifold integrals fluent, Regularized confluent, generalized, and Regularized m,n . involved in the direct capacity evaluation approach, and (ii) the generalized hypergeometric functions; and Gp,q ( . | (.)) is the highly resource-intensive Monte-Carlo (MC) simulations [31]. Meijer-G function, respectively.

SHOJAEIFARD et al.: MASSIVE MIMO-ENABLED FULL-DUPLEX CELLULAR NETWORKS

II.SYSTEM DESCRIPTION path-loss model with exponent β (> 2). Note that CSI in time- A. Network Topology division duplex (TDD)-based massive MIMO systems can be acquired based on channel reciprocity through UL training. In this work, we consider a FD cellular setup with BSs and In this work, we assume sophisticated channel estimation MTs respectively deployed on the two-dimensional Euclidean algorithms with sufﬁcient training information are used to Φ(d) Φ(u) space according to independent stationary PPPs and obtain perfect CSI [37]. The reader is referred to [38] for λ(d) λ(u) l k with spatial densities and . Let and denote the the impact of imperfect CSI on the performance of multi-cell l k locations of the -th BS and the -th MT, respectively. Their multi-user massive MIMO systems. respective Euclidean distance is therefore dl,k = kl − kk. The BSs are assumed to be equipped with N transmit and N t r C. Beamforming Design receive antennas (Nt + Nr RF chains in total), respectively. The MTs are in turn assumed to be equipped with single Next, we discuss the linear precoding and decoding strate- transmit/receive antennas (two RF chains in total). Each FD gies for the massive MIMO-enabled FD cellular network under BS, using linear beamforming, is considered to simultaneously consideration. T T U×Nt serve U FD MTs in the DL and UL per resource block Let Gl = [gl,k]1≤k≤U ∈ C denote the combined DL channels from the l-th BS to its U MTs. We use s = [32]. We assume the condition U ≤ min(Nt,Nr) holds, thus, l T U×1 2 scheduling is not required here. [sl,k]1≤k≤U ∈ C , E |sl,k| = 1, to denote the DL By invoking the Slivnyak’s theorem [33], the DL analysis complex symbol vector from the l-th BS to its U MTs. Here, is carried out for an arbitrary MT o assumed to be located at we consider the case where each BS equally allocates its (d) the center. Here, we consider a cellular association strategy total transmit power p among its U MTs. The normalized Nt×U where the reference MT is served by a massive MIMO BS precoding matrix at the l-th BS is Vl = [vl,k]1≤k≤U ∈ C , 2 b which provides the greatest received signal power. For E kvl.kk = 1. Hence, the DL received signal is given by (u) (u) homogeneous cellular deployments, this is equivalent to the (1), as shown at the top of the next page, where Φl (⊂ Φ ) cellular association strategy based on the closest transmitter- is the set of scheduled MTs in the cell of BS l, p(u) is the k,l −β (d) k-th scheduled MT transmit power for sending s to BS l, receiver distance, i.e., b = arg max dl,o , ∀l ∈ Φ [34]. k,l The UL analysis, on the other hand, is carried out for the and ηo is the complex additive white Gaussian noise (AWGN) 2 reference MT o signal at its serving massive MIMO BS with mean zero and variance σd, respectively. Nr ×U b. The corresponding transceiver distance p.d.f. is given by Next, let Hl = [hk,l]1≤k≤U ∈ C represent the (d) (d) 2 compound UL channel matrix at the l-BS with respect to Pdb,o (r) = 2πλ r exp −πλ r . Note that the alternative decoupling approach for cellular association [35] results in the its U scheduled MTs. The normalized decoding matrix at T T U×Nr loss of channel reciprocity in massive MIMO systems. the l-th BS is denoted using Wl = [wk,l]1≤k≤U ∈ C , 2 It is important to note that the set of scheduled MTs is E kwk,lk = 1. The corresponding post-processing UL strictly not an independent process as a result of the inherent signal is given by (2), as shown at the top of the next page, Nr ×1 spatial dependencies arising from (i) the cellular association where ηb ∈ C is the AWGN vector with mean zero and 2 strategy, and (ii) the constraint of each BS serving multi- covariance matrix σuINr . ple MTs per resource block. For the sake of mathematical Proposition 1. In the DL, we adopt a linear ZF-SIN precoder tractability, in the same spirit as in [36], we invoke the where the transmit antenna array (conditioned on Nt ≥ Nr + following assumption. U) is utilized to jointly suppress SI and multi-user interference Assumption 1. The set of scheduled MTs, conditioned on at the receiving antennas. This is achieved at the BS l by the spatial constraints imposed by the cellular association setting the columns of Vl equal to the normalized columns of ˆ+ ˆ† ˆ ˆ† −1 Nt×U ˆ strategy and the number of MTs being served by each BS Gl = Gl (GlGl ) = [gˆl,k]1≤k≤U ∈ C where Gl = † † −1 per resource block, is modeled as an independent stationary Gl(INt − Gl,l(Gl,lGl,l) Gl,l). PPP with density λ(u). Proof: See Appendix A. Note that the proposed null-steering precoder differs from B. Channel Model the extended ZF scheme in [22] where ‘all-zero’ streams Let g ∈ C1×Nt , G ∈ CNr ×Nt , and G ∈ CNr ×Nt are sent for the purpose of suppressing SI, i.e., in [22] l,k l,j l,l sl the (normalized) transmit signal vector Vl · with denote the channel from the l-BS to the k-th MT, the channel 0Nr ×1 Nt×(U+Nr ) from the l-th BS to the j-th BS, and the residual SI channel Vl ∈ C is set equal to the normalized columns ˆ+ ˆ† ˆ ˆ† −1 Nt×(U+Nr ) Nr ×1 of G = G (GlG ) = [gˆl,k]1≤j≤U+N ∈ C at the l-th BS, respectively. Moreover, hk,l ∈ C , hk,i, l hl i l r Gˆ = Gl and hk,k are respectively the channel from the k-th MT to where l Gl,l . the l-th BS, the channel from the k-th MT to the i-th MT, Proposition 2. In the UL, a linear ZF decoder, eliminating and the residual SI channel at the k-th MT. The residual SI multi-user interference, is employed with the normalized rows (for brevity, hereafter, refered to as SI) channels are subject † † of H+ = (H H )−1H = [hˆT ]T ∈ CU×Nr set as the to Rician fading with independent and identically-distributed l l l l k,l 1≤k≤U row vectors of W , at the BS l. (i.i.d.) elements drawn from CN (µ, υ2). All other channels are l modeled using Rayleigh fading with i.i.d. elements drawn from In the subsequent parts of this section, we characterize the CN (0, 1). Here, we use the unbounded distance-dependent DL and UL signal-to-interference-plus-noise ratios (SINRs).

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q (d) β q (d) β q (d) β p − 2 p − 2 X p X − 2 yd = U db,o gb,ovb,osb,o + U db,o gb,o vb,ksb,k + U dl,o gl,oVlsl | {z } k∈Φ(u)\{o} l∈Φ(d)\{b} d (intended signal) b | {z } | {z } dd (multi-user interference) d,d (inter-cell interference) q β q X (u) − 2 (u) + pk,l dk,o hk,osk,l + po,b ho,oso,b + ηo (1) (u) |{z} k∈Φ ,l∈Φ(d)\{o,b} | {z } noise l si,d (self-interference) | {z } u,d (cross-mode interference)

q β q β q β (u) − 2 T X (u) − 2 T X (u) − 2 T yu = po,b do,b wo,bho,bso,b + pk,b dk,b wo,bhk,bsk,b + pk,l dk,b wo,bhk,bsk,l | {z } k∈Φ(u)\{o} k∈Φ(u),l∈Φ(d)\{b} u (intended signal) b l | {z } | {z } uu (multi-user interference) u,u (inter-cell interference)

q (d) β q (d) p X − 2 T p T T + U dl,b wo,bGl,bVlsl + U wo,bGb,bVbsb + wo,bηb (2) l∈Φ(d)\{b} | {z } | {z } si,u (self-interference) noise | {z } d,u (cross-mode interference)

D. Downlink SINR suppressing SI at the BS side results in a loss of Nr (number of receive antennas) DoF in the DL antenna array gain. The DL received SINR at the reference MT is given by

Xd The Rayleigh fading model applies to cases without line- γd = 2 (3) Id,d + Iu,d + Isi,d + σd of-sight (LOS), e.g., with afar transceiver distances. In FD p(d) −β p(d) P −β setups, however, the nodes transmit and receive antennas are where X = d G , I = (d) d G , d U b,o b,o d,d U l∈Φ \{b} l,o l,o co-located. Hence, the Rician fading model, which takes into P (u) −β (u) Iu,d = (u) (d) pk,l dk,oHk,o, Isi,d = po,b Ho,o, k∈Φl ,l∈Φ \{o,b} account the different LOS and scattered paths, can be invoked 2 2 2 Gb,o , |gb,ovb,o| , Gl,o , kgl,oVlk , Hk,o , |hk,o| , and to capture performance under generalized SI cancellation 2 gˆb,o Ho,o |ho,o| . The ZF-SIN precoding vector vb,o = capability [8]. , kgˆb,ok is selected in the direction of the projection of gb,0 on Null([gb,k]1≤k≤U,k6=0, Gb,b). The nullspace spanned by the Assumption 3. The SI channel power gain at the reference MT SI and multi-user interference is Dd , Nt − Nr − U + 1 o is a non-central Chi-squared random variable with Rician dimensional. For analytical tractability, we assume that the q KΩ factor K and fading attenuation Ω, such that µ , outer-cell precoding matrices have independent column vectors K+1 q Ω [36], [39]. As a result, the channel power gain from each and ν , K+1 . The corresponding p.d.f. and m.g.f. are interfering BS in the DL is interpreted as the aggregation respectively given by of multiple separate beams from the projection of the cross- 1 + K (1 + K)h g PHo,o (h) = exp − K + link channel vector l,o onto the one-dimensional precoding Ω Ω vectors vl,k. The scheduled MTs, on the other hand, transmit r ! K(1 + K)h using single-antennas (in all directions). × I 2 (6) 0 Ω Assumption 2. The channel power gains at the reference MT o, from the intended BS b, interfering BS l, and scheduled and MT k, are respectively Gb,o ∼ Γ(Dd, 1), Gl,o ≈ Γ(U, 1), and 1 + K zKΩ M (z) = exp − . (7) Hk,o ∼ Γ(1, 1). Note that the p.d.f. and m.g.f. expressions of a Ho,o 1 + K + zΩ 1 + K + zΩ Gamma-distributed random variable G(.) with shape parameter κ and scale parameter θ can be respectively expressed It should be noted that the modiﬁed Bessel function of the as ﬁrst kind can be expressed through hypergeometric functions, gκ−1 g ˜ χ 2 P (g) = exp − (4) e.g., here, I0 (χ) = 0F1 ; 1; 2 . G(.) θκΓ(κ) θ and Remark 2. The SI channel power gain at the reference MT 1 o can be approximated using Gamma moment matching as MG (z) = κ . (5) 2 (.) (1 + θz) µ2+ν2 2µ2+ν2 ν2 Ho,o ≈ Γ(κ, θ), where κ , (2µ2+ν2)ν2 and θ , µ2+ν2 Remark 1. Utilizing the linear ZF-SIN precoder for spatially [40].

SHOJAEIFARD et al.: MASSIVE MIMO-ENABLED FULL-DUPLEX CELLULAR NETWORKS

E. Uplink SINR In this work, we propose an LTE-compliant SIA fractional Next, we express the UL received SINR from the reference power control mechanism where the MTs adjust their transmit MT at its serving BS as power based on the distance-dependent path-loss, SI, and maximum available transmit power. The motivation behind X u this is in line with the notion that the MTs with limited γu = 2 (8) Iu,u + Id,u + σu number of antennas rely on digital/analog domain interference (u) −β P (u) cancellation strategies, and hence may experience severe SI where Xu = po,b do,b Ho,b, Iu,u = k∈Φ(u),l∈Φ(d)\{b} pk,l l and CI affecting the DL operation. −β p(d) P −β dk,b Hk,b, Id,u = U l∈Φ(d)\{b} dl,b Gl,b, Ho,b , Specifically, in this work, an arbitrary scheduled MT k T 2 T 2 T 2 |wo,bho,b| , Hk,b , |wo,bhk,b| , and Gl,b , kwo,bGl,bVlk . transmits to its serving BS l using ˆ T −2 † −1 Note kho,bk = {(Hb Hb) }o,o ∼ Erlang(Du, 1) where (u) ψβ −1 (u) pk,l = min p0dk,l ,ISIHk,k, p (9) Du , Nr − U + 1. In turn, Wb is selected independently from hk,b and Gl,b. We recall the assumption that Vl has (u) where p0, ψ (∈ (0, 1]), ISI, and p are respectively the independent column vectors. normalized power density, compensation factor, tolerable SI Assumption 4. The channel power gains at the reference BS level, and maximum transmit power at the MT [36]. The value b, from the intended MT o, interfering MT k, and interfering of ISI can be set as the difference in the noise floor power from BS l are respectively modeled using Ho,b ∼ Γ(Du, 1), Hk,b ∼ the gain of the MT SI cancellation capability. The distribution Γ (1, 1), and Gl,b ≈ Γ(U, 1). of the transmit power in this setup can be developed. It is important to note that it is certainly feasible to apply Lemma 1. The c.d.f. and p.d.f. of the transmit power of a other linear precoding schemes such as CB in FD massive typical MT under the SIA fractional power control mechanism, (u) ψβ −1 (u) MIMO systems [27]. In such cases, the SI channel power pk,l = min p0dk,l ,ISIHk,k, p , are respectively given by T 2 gain, e.g., at the reference BS b, Gb,b kwo,bGb,bVbk , (10) and (11), as shown at the top of the next page, where , 2 ψβ needs to be characterized. In [40], the distribution of the (d) p ISI ΞI(p) = πλ and ΞII(p) = . SI with arbitrary linear beamforming design over FD multi- p0 pΩ user MIMO Rician fading channels was approximated using Proof: See Appendix B. Gamma moment matching. In particular, for FD multi-user Remark 3. The proposed LTE-compliant SIA fractional power massive MIMO systems, we obtain the following result. control mechanism can be viewed as a generalization of the existing approaches for UL power control including total Corollary 1. With arbitrary linear precoders in the DL (u) (such as CB and ZF), the SI channel power gain at the (without ISI) and truncated (without ISI and p ) fractional reference massive MIMO BS b can be approximated using power control schemes. Gamma moment matching as Gb,b ≈ Γ(κ, θ), where κ = The computation of SE can be greatly simplified with a 2 U µ2+ν2 4 2 2 4 and θ = (U+2)µ +2µ ν +ν [40]. non-direct methodology requiring only the moments of the (U+2)µ4+2µ2ν2+ν4 µ2+ν2 signals involved [43]. On the other hand, the p.d.f. expression In the case of imperfect knowledge of SI channel, one has in (11), is a highly non-linear piecewise function. Hence, to account for the impact of estimation error. For example, the exact moments of the proposed SIA fractional power ˆ Nr ×Nt let Gb,b ∈ C denote the minimum mean-square error control mechanism required for the calculation of SE cannot (MMSE) estimate of Gb,b. Consider the estimation error given be derived in closed-form. Next, we develop results for the ˆ by ξb,b , Gb,b−Gb,b with elements drawn independently from moments of the SIA power control in certain special cases CN ($, ς2). Hence, we can characterize the corresponding where the corresponding p.d.f. has a more tractable form. average SI as in the following result. It should be noted that a Meijer-G function can be readily Corollary 2. With the MMSE estimation process above, under calculated using common software for numerical computation. arbitrary linear beamforming design, the average SI channel A Meijer-G function can also be expressed in terms of a power gain at the reference massive MIMO BS b in the UL hypergeometric function based on the results from [44]. n o T ˆ 2 PU T 2 is given by E{kwo,bGb,bVbk } = E k=1 |wo,bξb,bvb,k| Corollary 3. The p.d.f. of the transmit power of a typical = U($2 + ς2) [40]. MT under the SIA fractional power control mechanism can be simplified in certain special cases. III.SELF-INTERFERENCE-AWARE POWER CONTROL For K = 0 (Rayleigh SI channel), p0 → +∞ (no path-loss compensation), and I → +∞ (no constraint on the SI), we In long-term-evolution (LTE) standards, UL fractional SI respectively obtain (12), (13), and (14), as shown at the top power control is defined to account for the path-loss effect of the next page. [41]. Recently, interference-aware fractional power control has been proposed to ensure that the power adjustment intended Lemma 2. The [-th positive moment of the transmit power of a for path-loss compensation does not cause undesired interfer- typical MT under the SIA fractional power control mechanism ence to neighboring nodes [42]. Intuitively, power control has admits a closed-form expression in certain special cases. Let perhaps an even more essential role to play in FD cellular (d) ˆ πλ ˆ ISI ΞI = 2 and ΞII = . setups. This topic remains somewhat unexplored however. βψ Ω p0

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√ p (u) (u) F (u) (p) = 1 − exp (−ΞI(p)) 1 − Q1 2K, 2(1 + K)ΞII(p) 1 − H p − p + H p − p (10) pk,l

 √ q  (u) (u) (u) (u)  exp −ΞI p 1 − Q1 2K, 2(1 + K)ΞII p δ p − p p ≥ p   "  exp (−Ξ (p)) 2Ξ (p) √  I I 1 − Q 2K, p2(1 + K)Ξ (p) + (1 + K)Ξ (p) P (u) (p) = 1 II II p p ψβ (11) k,l   #   ˜ (u)  × exp (− (K + (1 + K)ΞII(p))) 0F1 (; 1; K(1 + K)ΞII(p)) p < p 

 (u) (u) (u) (u)  exp −ΞI p 1 − exp −ΞII p δ p − p p ≥ p  P (u) (p) = ! p exp (−Ξ (p)) 2Ξ (p) (12) k,l  I I (1 − exp (−Ξ (p))) + Ξ (p) exp (−Ξ (p)) p < p(u)  p ψβ II II II

 √ q (u) (u) (u)  1 − Q1 2K, 2(1 + K)ΞII p δ p − p p ≥ p Pp(u) (p) = (13) k,l  2 ˜ (u) (1 + K)ΞII p exp (− (K + (1 + K)ΞII(p))) 0F1 (; 1; K(1 + K)ΞII(p)) p < p

 1 − exp −Ξ p(u) δ p − p(u) p ≥ p(u)  I P (u) (p) = p 2Ξ (p) (14) k,l  I exp (−Ξ (p)) p < p(u)  ψβp I

ˆ[ ˆ2 ˆ ! ˆ pˆ ˆ2 ˆ !! (u)[ Ξ 3,0 Ξ ΞII ΞI ΞII 3,0 Ξ ΞII (2[)! p = √II G −[+1,0, 1 I − G −[− 1 ,0, 1 I + (15) E k,l π 0,3 2 4 2 0,3 2 2 4 ˆ2[ ΞI

For p(u) → +∞ (no constraint on the maximum transmit IV. SPECTRAL EFFICIENCY ANALYSIS power), K = 0 (Rayleigh SI channel), ψ = 1 (compensation The notion of doubling the SE by going from HD to FD factor), and β = 4 (path-loss exponent), we can obtain (15), has been a key driving force behind the surge of interest in as shown at the top of this page. this topic. This is however only a theoretical upper-bound For p → +∞ (no path-loss compensation) and p(u) → 0 concerning point-to-point links under perfect SI cancellation +∞ (no constraint on the maximum transmit power), capability. In multi-cell environments, many factors come into [ play, such as increased interference complexity and intensity, p(u) = (1 + K)[Ξˆ[ Γ(1 − [) F ([; 1; −K). (16) E k,l II 1 1 which requires rigorous investigation prior to the potential Further, for K = 0 (Rayleigh SI channel), adoption of FD technology in 5G and beyond. To facilitate performance analysis and optimization, we [ (u) ˆ[ provide a framework for the computation of the DL and UL E p = ΞIIΓ(1 − [). (17) k,l SEs in the massive MIMO-enabled FD cellular network. We utilize a m.g.f.-based methodology, which avoids the need for For p(u) → +∞) (no constraint on the maximum transmit the direct computation of the SINR p.d.f. by requiring only power) and I → +∞ (no constraint on the SI), SI the m.g.f.s of the different signals involved [45], [46]. [ ψβ[ (u) Γ( 2 +1) E pk,l = ψβ[ . (18) Ξˆ 2 I A. Downlink Spectral Efﬁciency Further, for ψ = 1 (compensation factor), and β = 4 (path- We proceed by providing an explicit expression for the loss exponent), calculation of the SE in the DL. (u)[ Γ (1 + 2[) E p = . (19) Theorem 1. The DL SE in the FD massive MIMO cellular k,l Ξˆ2[ I network is given by (20), as shown at the top of the next page. Proof: See Appendix C. Proof: The result follows directly from [45, Lemma 1].

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Z +∞ Z +∞ Z +∞ Sd,f = E {log2 (1 + γd)} = log2(e) MIsi,d|p(z)MIu,d|p(z) 1 − MXd|r(z) 0 0 0 2 exp −zσd × M (z) P (u) (p) P (r) dz dr dp Id,d|r p db,o (20) z k,l

" (d) −U 2 (d) −U M (z) = exp − πλ(d) r2 z p r−β + 1 − 1 + Γ U + z p Id,d|r U β U

 2  #! U+ 2 Γ 1 − β ! p(d) β β Uβ+2 ˜ 2 2 r ×  z − Ur 2F1 U + 1, U + ; U + + 1; − (d)  (22) U Γ(U) β β p z U

2 2 2 M (z) = exp −πUλ(d) (zp) β Γ 1 − Γ 1 + (23) Iu,d|p β β

Z +∞ Z +∞ Z +∞ Su,f = E {log2 (1 + γu)} = log2(e) MId,u (z) 1 − MXu|p,r(z) MIu,u|p,r(z) 0 0 0 2 exp −zσu × Pp(u) (p) Pdb,o (r) dz dr dp (25) z k,l

" β β !#! 2 2 2 2r 2 2 r M (z) = exp −πUλ(d) (zp) β Γ 1 − Γ 1 + −r2 1− F 1, 1 + ; 2 + ; − Iu,u|p,r β β zp (β + 2) 2 1 β β zp (27)

 2 2  2 Γ 1 − Γ U + (d) p(d) β β β MId,u (z) = exp −πλ z U 2  (28) U β Γ(U)

Next, we provide explicit expressions for the conditional Theorem 2. The UL SE in the FD massive MIMO cellular m.g.f.s of the different DL intended and interfering signals. network is given by (25), as shown at the top of this page. Lemma 3. The conditional m.g.f.s of the different DL signals Proof: The result follows directly from [45, Lemma 1]. in the FD massive MIMO cellular network are given by −D The conditional m.g.f.s of the different UL intended and p(d) −β d MXd|r(z) = 1 + z U r , (21) interfering signals required for SE calculation can be derived in closed-form as in the following lemma. (22), (23), as shown at the top of this page, and 1 + K zpKΩ Lemma 4. The conditional m.g.f.s of the different UL signals M (z) = exp − . (24) Isi,d|p 1 + K + zpΩ 1 + K + zpΩ in the FD massive MIMO cellular network are given by Proof: See Appendix D. −β−Du MXu|p,r(z) = 1 + zpr , (26) Remark 4. The CI at the reference MT in (23), is from (27), and (28), as shown at the top of this page. the aggregation of interference from all other scheduled MTs which are approximated using an independent PPP with Proof: See Appendix D. spatial density Uλ(d) based on Assumption 1. Remark 5. The UL inter-cell interference in (27) is derived (d) B. Uplink Spectral Efﬁciency using a spatially-thinned PPP with spatial density Uλ and circular exclusion region with radius r based on the spatial Similarly, we provide a systematic approach for the calcu- constraints from Assumption 1 [36], [38]. lation of the SE in the UL.

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C. Special Cases ence in the UL for β = 4 (path-loss exponent) is given by √ r2 In certain special cases, the conditional m.g.f.s of the signals M (z) = exp −πUλ(d) zp arccot √ . (34) Iu,u|p,r zp can be simpliﬁed further as highlighted in what follows. Corollary 7. The CI conditional m.g.f. in the UL for U = 1 Corollary 4. The conditional m.g.f. of the inter-cell interfer- (single-user) and β = 4 (path-loss exponent) is given by ence in the DL for U = 1 (single-user), and β = 4 (path-loss π2 p exponent) is given by M (z) = exp − λ(d) zp(d) . (35) Id,u 2 2 !! (d)p (d) r MI |r(z) = exp −πλ zp arccot . The SEs in (20) and (25) require three-fold integral com- d,d p (d) zp putations - versus the manifold integrals involved in the direct (29) p.d.f.-based approach. It is possible to further reduce the Corollary 5. The conditional m.g.f. of the CI in the DL with computational complexity in certain special cases with UL β = 4 (path-loss exponent) is given by power control. Note that in the case of baseline SISO, we take into account the SI at the FD BS side. π2 √ M (z) = exp − Uλ(d) zp . (30) Iu,d|p 2 Lemma 5. For Nt,Nr, U = 1 (baseline SISO), K = 0 2 2 (Rayleigh SI channel), σd, σu = 0 (interference-limited re- The above can be further simpliﬁed under special cases of UL (d) gion), and β = 4 (path-loss exponent), the SEs (ω = p power control. p for DL and ω = p for UL) are reduced to double-integral (u) p(d) For p → +∞ (no constraint on the maximum transmit expressions in (36), as shown at the bottom of this page. power), K = 0 (Rayleigh SI channel), ψ = 1 (compensation (d) Further, for SI = 0 (perfect SI subtraction), the SEs (ω = p factor), and β = 4 (path-loss exponent), we can obtain (31), p for DL and ω = p for UL) are given by (37), as shown at as shown at the bottom of this page. p(d) the bottom of this page. Further, for p0 → +∞ (no path-loss compensation), (u) For p0 → +∞ (no path-loss compensation) and p → 2! 1 π2Uλ(d) +∞ (no constraint on the maximum transmit power), the SE 3,0 1 ˆ MI (z) = √ G 0, 2 ,1 zΞII . (32) expressions in (37) can be reduced to single-fold integrals in u,d π 0,3 4 (38) and (39), as shown at at the top of the next page. (u) On the other hand, for ISI → +∞ (no constraint on the SI), For ISI → +∞ (no constraint on the SI) and p → +∞ 1 (no constraint on the maximum transmit power), the SE MIu,d (z) = √ . (33) expressions in (37) can be reduced to single-fold integrals in 1 + π zp 2 0 (40) and (41), as shown at the top of the next page. Corollary 6. The conditional m.g.f. of the inter-cell interfer- Proof: See Appendix E.

ˆ !2! 1 1 3,0 UΞI π √ M (z) = + √ G 0, 1 ,1 Ξˆ 1 + zp Iu,d π √ 0,3 2 II 0 1 + 2 zp0 π 2 2 ˜ p˜ ˆ !2! UΞI ΞII 3,0 UΞI π √ − √ G − 1 ,0, 1 Ξˆ 1 + zp 0,3 2 2 II 0 (31) 2 π 2 2

" Z +∞ Z +∞ 2πλ(d) √ πλ(d) π √ S = log (e) √ ω sin √ + ω (s + arccot(s)) {.},f 2 2 0 0 Ω (1 + s ) Ω 2 πλ(d) π √ πλ(d) π √ × C √ + ω (s + arccot(s)) + cos √ + ω (s + arccot(s)) Ω 2 Ω 2 # π πλ(d) π √ √ (u) × − S + ω (s + arccot(s)) Pp (p) ds dp (36) 2 Ω 2 k,l

Z +∞ Z +∞ 2 S{.},f = log2(e) Pp(u) (p) ds dp (37) 0 0 2 √π k,l (1 + s ) 2 ω + s + arccot(s)

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" √ s Z +∞ 2 π π I S = log (e) 1 − SI d,f 2 2 (d) 0 (1 + s )(s + arccot(s)) 2 (s + arccot(s)) p Ω ! π 2 I π 2 I − SI exp − SI 2 (s + arccot(s)) p(d)Ω 2 (s + arccot(s)) p(d)Ω ! s !!# π 2 I π I × Ei SI − πerﬁ SI ds (38) 2 (s + arccot(s)) p(d)Ω 2 (s + arccot(s)) p(d)Ω

s " s Z +∞ 4 I √ 2 (s + arccot(s)) I 2 (s + arccot(s))2 S = log (e) SI π + SI exp − u,f 2 2 (d) (d) 0 π (1 + s ) p Ω π p Ω π ! ! s !!# I 2 (s + arccot(s))2 I 2 (s + arccot(s)) I × SI Ei SI − πerﬁ SI ds (39) p(d)Ω π p(d)Ω π p(d)Ω

s  s   s  Z +∞ (d) (d) (d) (d) −4λ p (d) p (d) p Sd,f = log2(e) 2 exp 2λ (s + arccot(s))  Ei −2λ (s + arccot(s))  ds (40) 0 1 + s p0 p0 p0

" s Z +∞ 2 π2λ(d) p(d) Su,f = log2(e) 2 1 + 0 (1 + s )(s + arccot(s)) 2 (s + arccot(s)) p0  s   s  # π2λ(d) p(d) π2λ(d) p(d) × exp   Ei −  ds (41) 2 (s + arccot(s)) p0 2 (s + arccot(s)) p0

V. FULL-DUPLEXVERSUS HALF-DUPLEX then derive results for the multi-user massive MIMO setup. Network design insights are accordingly drawn. The SE expressions developed facilitate performance analysis and optimization for generalized FD cellular deployments. A. Baseline SISO At the same time, the proposed framework can serve as a benchmark tool for comparing the performance of FD over HD Here, we compare the FD versus HD performance for the systems. Although an explicit expression for the correspond- baseline SISO case. ing gain cannot be obtained due to the highly complex SE Lemma 6. The FD and HD SEs for N t,N r, U = 1 (baseline expressions involving multiple improper integrals, we proceed SISO), p(u) (ﬁxed MT transmit power), SI = 0 (perfect SI 2 2 by providing results in certain special cases. subtraction), σd, σu = 0 (interference-limited region), and In what follows, Sd,h and Su,h are respectively used to β = 4 (path-loss exponent) are respectively given by (42) denote the per-user SEs in the DL and the UL of a HD cellular and (43), as shown at the bottom of this page. Further, network. To facilitate performance comparison between the bounded closed-form expressions of the FD and HD SEs are FD and HD systems, we consider the SE over two time slots, respectively given by (44) and (45), where (46), as shown at i.e., Sf = 2(Sd,f + Su,f ) for FD, and Sh = Sd,h + Su,h for the top of the next page. HD, respectively. We ﬁrst study the baseline SISO case and Proof: See Appendix F.

q p(u) q p(d) +∞ 2π Z 1+s2 p(d) + p(u) + 8(s + arccot(s)) Sf = log (e) ds (42) 2 q (u) q (d) 0 π p π p 2 p(d) + s + arccot(s) 2 p(u) + s + arccot(s)

Z +∞ S 4 h = log2(e) 2 ds (43) 0 (1 + s )(s + arccot(s))

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     2 (d) 2 S˜ 8 8 π p π − 8 f ≤ 2 log2(e) Ψ  − 1 + Ψ  − 1 , < < (44)   q p(u)   q p(d)  π − 8 p(u) π π 1 + p(d) π 1 + p(u)

8 S˜ ≤ 2 log (e)Ψ − 1 (45) h 2 π

√ √ 1 π √ √ (1 + α) (5 − α) arccot ( α) + α log 4 (1 + α) − 2 (5 − 4 α + α) (1 − 2 α − α) Ψ(α) = √ (46) α 2 1+α (25 − 6α + α )

8 8 Ψ √ − 1 + Ψ − 1 2 2 S˜ π(1+ x) π 1+ √1 π π − 8 maximize f = x subject to: < x < (47) (d) S˜ 8 x= p h Ψ π − 1 π − 8 π p(u)

Z +∞ 8 S = log (e) ds (49) f 2 2 π 0 (1 + s ) s + arccot(s) + 2

Remark 6. The HD SE for the special case described in Note that with equivalent DL and UL transmit powers, the Lemma 6 is independent of the system parameters. The exact FD baseline SISO exact SE expression from (42) reduces to and bounded HD baseline SISO SEs over two time slots (in (49), as shown at the top of this page. nat/s/Hz) are approximately 3 and 2.72, respectively. On the Remark 7. Base on the result from Lemma 7, the highest other hand, the exact and bounded FD baseline SISO SEs exact and bounded SE percentage gains of FD over HD for for this special case are functions of the BS and MT transmit baseline SISO are respectively ∼9% and ∼13% (these values powers only. also correspond to the individual SE gains in the DL and the UL). We can infer that without advanced techniques for Based on the above remark, we next study the problem of tackling the CI, even under perfect SI subtraction, the FD finding the optimal fixed transmit powers which result in the baseline SISO system achieves only modest improvements in largest FD over HD SE gain for the case of baseline SISO. (d) SE over its HD counterpart. The SE function in HD mode, Sh in (43), is affine in p > 0 and p(u) > 0. On the other hand, it can be shown that the SE B. Massive MIMO function in FD mode, Sf in (42), is strictly quasi-concave in p(d) > 0 and p(u) > 0. This guarantees the existence of Next, we analyze the FD versus HD SE for massive MIMO- a unique maximum solution under a positive transmit power enabled cellular networks. S S region. h and f , however, involve improper integrals. Thus, Lemma 8. The FD and HD SEs with massive MIMO, ZF an exact closed-form solution cannot be obtained here. Hence, beamforming, Up (BS transmit power), p (MT transmit power), we utilize the bounded SE expressions developed in (44) and 2 2 SI = 0 (perfect SI subtraction), σd, σu = 0 (interference- (45). limited region), and β = 4 (path-loss exponent) are respectively given by (50) and (51), as shown at the top of the Lemma 7. Consider the bounded FD and HD SEs for next page. For U = 1 (single-user), these expressions can N t,N r, U = 1 (baseline SISO), p(u) (MT transmit power), 2 2 be respectively reduced to (52) and (53), as shown at the top SI = 0 (perfect SI subtraction), σd, σu = 0 (interference- of the next page. limited region), and β = 4 (path-loss exponent) from Lemma 6. Proof: The proof follows from a similar approach to that We can formulate the optimization problem in (47), as shown provided in Appendix F. at the top of this page, for the highest FD over HD SE gain. The solution to this problem is x∗ = 1, resulting in The SE expressions for the case of multi-user massive !∗ MIMO derived in Lemma 8 involve single-fold integrals. Next, S˜ 2Ψ 4 − 1 f = π . (48) we employ multivariate non-linear curve fitting in order to S˜ 8 h Ψ π − 1 provide a closed-form approximation for the FD versus HD massive MIMO SE gain as a function of the number of Proof: See Appendix G. antennas and users.

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U−Nr +∞ 2 1 Z 4 1 + s 1 − 1 + s2 Sf = log (e) 2 2 Γ U+ 1 0 1 + s p π ( 2 ) s + U arccot(s) + U Γ(U) 2 1 U−Nt ! 1 + s 1 − 1 + s2 + ds (50) −U √ Γ U+ 1 s 1 2U+1 1 ˜ 1 3 2 ( 2 ) π U 1 + s2 − s Γ U + 2 2F1 U + 2 , U + 1; U + 2 ; −s + π UΓ(U) + 2

2 1 U−Nr Z +∞ 1 + s 1 − 1 + 2 S 2 s h = log2(e) 2 0 1 + s s + U arccot(s)

2 1 U−Nt ! 1 + s 1 − 1 + s2 + ds (51) −U √ Γ U+ 1 1 2U+1 1 ˜ 1 3 2 ( 2 ) s 1 + s2 − Us Γ U + 2 2F1 U + 2 , U + 1; U + 2 ; −s + π Γ(U)

Z +∞ −Nr −Nt ! S 4 1 1 f = log2(e) π 2 − 1 + 2 − 1 + 2 ds (52) 0 s + arccot(s) + 2 s s

Z +∞ −Nr −Nt ! S 2 1 1 h = log2(e) 2 − 1 + 2 − 1 + 2 ds (53) 0 s + arccot(s) s s

Corollary 8. The FD over HD massive MIMO SE gain examining the validity of the proposed analytical framework. (d) 4 with ZF beamforming, N (number of transmit and receive The BS deployment density is considered to be λ = π antennas), Up (BS transmit power), p (MT transmit power), per unit area (km×km). The total system bandwidth is taken 2 2 SI = 0 (perfect SI subtraction), σd, σu = 0 (interference- to be B = 20 MHz. The noise powers in the DL and the 2 limited region), and β = 4 (path-loss exponent) can be UL are calculated using σ = −170 + 10 log10 (B) + Nf 1 approximated using multivariate non-linear curve ﬁtting as (dBm), where Nf is the noise ﬁgure [49]. Note that the results

4 ! from the MC simulations are obtained considering (i) the S 9 U 25 f ≈ 2 − . (54) scheduled MTs follow from a PPP (Assumption 1), (ii) non- S 1 h 10 N 10 central Chi-squared distribution for the SI channel power gains Remark 8. The FD over HD massive MIMO SE gain loga- (Assumption 3), and (iii) Gamma distribution for the other rithmically (i) increases in the number of antennas, and (ii) channels power gains (Assumptions 2 and 4). decreases in the number of users. Furthermore, the anticipated 1) Different FD Cellular Setups: We compare the DL and two-fold increase in SE of massive MIMO-enabled FD versus UL SE performance of different FD cellular networks, namely, HD cellular networks is only achieved as the number of massive MIMO (with ZF-SIN precoder and ZF decoder) antennas tends to be infinitely large with orders of magnitude and baseline SISO in Fig. 1. In order to facilitate a fair smaller number of users (N → +∞, N U). comparison whilst capturing the inherent characteristics of It is important to note that the choice of FD over HD the two systems, we consider (i) per-user SEs, although each operation is primarily to enhance SE, which inherently results massive MIMO BS is considered to be serving several MTs per in reduced EE [47]. Some potential solutions for alleviating resource block, (ii) greater BS transmit power in the baseline this issue include modified scheduling and power control SISO case, and (iii) equivalent residual SI channel statistics. mechanisms [48]. A rigorous assessment of the EE in FD The results confirm prior findings that the UL rate is the main massive MIMO cellular networks is postponed to future work. performance bottleneck in FD cellular systems with baseline SISO. Here, the UL performance is severely limited due to the impact of CI, especially with the large disparity in the VI.NUMERICAL RESULTS BS and MT transmit power levels, as well as the imperfect SI In this section, we provide several numerical examples in subtraction. Furthermore, it can be observed that significant SE order to assess the performance of the FD massive MIMO gains in the DL and (especially) in the UL can be achieved cellular network under different settings of system parameters. by exploiting the large scale antenna array with linear ZF- MC simulations are accordingly provided for the purpose of SIN precoding and linear ZF decoding. By increasing the 1The properties related to the goodness of fit of the closed-form approxi- antenna array size, further improvements can be realized from mation in (54) are R2 ≈ 0.999 and estimated variance of ≈ 1.17 × 10−3. (i) greater transmit/receive antenna array gains, and (ii) the

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3.5 25 FD Massive MIMO FD Baseline SISO 3 MC DL UL 2.5 0

2 Efﬁciency [b/s/Hz]

1.5 Efﬁciency Gain [%]

−25 Spectral 1 Spectral Nr = 200 Nr = 100 0.5 Nr = 50 Nr = 5 MC 0 −50 2 2.5 3 3.5 4 4.5 5 0 100 200 300 400 500

Path-Loss Exponent BS Transmit Antenna Array Size

(d) 4 2 (d) 4 2 (d) Fig. 1. System parameters are: λ = π BSs/km , massive MIMO (Nt = Fig. 3. System parameters are: λ = π BSs/km , U = 5, p = 30 dBm, (d) (u) 80, Nr = 20, U = 8, p = 30 dBm), baseline SISO (Nt = 1, Nr = 1, p = 23 dBm, p0 = −80 dBm, B = 20 MHz, Nf = 10 dB, β = 4, (d) (u) U = 1, p = 43 dBm), p = 23 dBm, p0 = −80 dBm, B = 20 MHz, ψ = 1, Ω = −90 dB, K = 2. Nf = 10 dB, ψ = 1, Ω = −80 dB, K = 1.

80 2) FD versus HD Massive MIMO: Next, we investigate the performance of the FD massive MIMO cellular network with 60 respect to its HD counterpart over a wide range of MT SI channel attenuation in Fig. 2. It can be seen that any potential 40 improvements from the FD operation occurs for MT SI channel attenuation well below −80 dB. This trend indicates that 20 unless advanced interference mitigation solutions are available at the MTs, the FD technology should be strictly used at the BS 0 side (serving HD DL and UL MTs). Nevertheless, even with nearly perfect SI cancellation capability at the reference FD −20 MT, with single-user MIMO (U = 1), the maximum SE gain Efﬁciency Gain [%] −40 of the FD massive MIMO cellular network over its analogous HD variant is 47% (resulting from 73% and 1% increases in

Spectral −60 DL + UL the DL and UL SEs, respectively). It should be noted that DL this negligible change in the UL SE comes as a result of the −80 UL overwhelming CI from the BSs transmitting in the DL in FD MC mode. In addition, it can be seen from Fig. 2 that by increasing −100 U = 10 the number of MTs (whilst using the same total transmit power U = 1 at the BS side), the FD over HD UL SE gain may be improved, −120 −120 −110 −100 −90 −80 −70 −60 however, this comes at the cost of reduced relative DL SE gain due to the additional CI from the MTs transmitting in the UL MT SI Channel Attenuation [dB] in FD mode.

(d) 4 2 3) Transmit/Receive Antenna Array Size: The impact of the Fig. 2. System parameters are: λ = π BSs/km , Nt = 350, Nr = 50, (d) (u) different number of transmit and receive antennas on the SE p = 30 dBm, p = 23 dBm, p0 = −80 dBm, B = 20 MHz, Nf = 10 dB, β = 4, ψ = 1, K = 1. gain of the massive MIMO-enabled FD system over its HD counterpart is illustrated in Fig. 3. It can be observed that the corresponding performance gain increases in the number of potential to linearly reduce the transmit powers of the BSs transmit antennas due to the reduced impact of the DL array and MTs without degrading the received signal-to-noise ratio gain loss from the ZF-SIN precoding scheme and improved (SNR). It should be noted that the MC results conﬁrm the resilience against interference. The optimal ratio of transmit validity of the proposed analytical framework. over receive antennas, on the other hand, depends on the

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7 4

3.5 6

5 2.5

p(u) = 0.2 W 4 2 p(u) = 0.05 W Efﬁciency [b/s/Hz] MC Efﬁciency [b/s/Hz] DL DL + UL 1.5 UL 3 DL MC Spectral Spectral UL SIA 1 Fractional 2 Fixed 0.5

1 0 0 5 10 15 20 −120 −100 −80 −60 −40 −20 0

BS Transmit Power [W] MT SI Channel Attenuation [dB]

Fig. 4. System parameters are: λ(d) = 4 BSs/km2, N = 150, N = 50, (d) 4 2 π t r Fig. 5. System parameters are: λ = π BSs/km , Nt = 150, Nr = 50, U = 4, p0 = −80 dBm, B = 20 MHz, Nf = 10 dB, β = 4, ψ = 1, U = 4 p(d) = 30 p(u) = 23 p = −80 ISI = 25 , dBm, dBm, 0 dBm, σ2 dB, Ω = −100 dB, K = 0. (u) pf = 23 dBm, B = 20 MHz, Nf = 10 dB, β = 4, ψ = 1, K = 0.

particular system settings. A rigorous study of this aspect is case of fixed as well as fractional power allocation strategies left to future work. It is important to note that the results means that the DL performance significantly suffers in the from Fig. 3 reaffirm our theoretical findings in Remark 8, case of large MT SI channel attenuation. The proposed scheme that the FD over HD massive MIMO SE gain increases only can therefore serve as an important safe-guard mechanism for logarithmically in the antenna array size. ensuring that a certain maximum SI level is not exceeded. 4) Transmit Power Budget: It has been shown in [4] that the transmit energy can be linearly conserved in the VII.SUMMARY number of antennas. This is a contributing factor in tackling the FD UL rate bottleneck through massive MIMO, as was We provided a theoretical framework using tools from demonstrated in Fig. 1. We depict the impact of different stochastic geometry and point processes for the study of BS transmit power and MT maximum transmit power in Fig. massive MIMO-enabled FD cellular networks. The DL and 4. Intuitively, increasing p(d), or p(u), respectively improves UL SEs with linear ZF-SIN precoding, SIA fractional power the corresponding SE in the DL, or the UL; the relative gain control, and linear ZF receiver were characterized over Ri- however decreases for larger power budgets. Further, the DL cian SI and Rayleigh intended and other-interference fading and UL SEs are conflicting functions in the BS/MT transit channels. The results highlighted the promising potential of powers. In general, we observe that a large difference in the massive MIMO towards unlocking the UL rate bottleneck in DL/UL power levels is deteriorating to the overall performance FD communication systems. On the other hand, the results in FD mode. Due to the large disparity in the DL/MT SEs, demonstrated that the anticipated two-fold increase in SE of however, the direct optimization of the FD sum-rate places the the massive MIMO-enabled FD cellular network overs its HD focus on the DL, see Fig. 4. It is therefore important for any variant is only achievable in the asymptotic antenna region. respective optimization to strike a balance (e.g., by applying weights) between the DL and UL SEs. This is however beyond APPENDIX A the scope of this work. To jointly suppress the (residual) SI and multi-user inter- 5) Uplink Power Control: The results presented so far were ference in the DL, considering Gl, Hl, and Gl,l are full-rank based on the conventional UL fractional power control mech- and estimated without error, the conditions (i) WlGl,lVl = anism defined in the existing LTE standards for HD cases. 0U×U , and (ii) gl,kvl,j = 0, ∀j 6= k must be satisfied. Next, we study the performance of the massive MIMO-enabled The singular value decomposition (SVD) of Gl,l is given † FD cellular network with different fixed (at maximum power), h (1) (0)i by Ul,lΣl,l V V where Ul,l holds the left singular conventional, and proposed SIA fractional power control pro- l,l l,l V(1) tocols under different SI channel attenuation in Fig. 5. It can vectors, Σl,l is the matrix of singular values, and l,l and V(0) be observed that the lack of self-interference-awareness in the l,l are the right singular matrices of the non-zero and zero

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singular values, respectively [50]. For SI nulling, the precoder Hence, with some basic algebraic manipulations, we arrive at is designed such that its column vectors are in the nullspace of Lemma 1. V(0) † † −1 Gl,l (i.e., Vl = l,l ) using INt −Gl,l(Gl,lGl,l) Gl,l. At the same time, to suppress multi-user interference in the DL, we APPENDIX C design the proposed ZF-SIN precoder Vl using the pseudo- Consider the case where the p.d.f. of the UL transmit power ˆ inverse of the projection channel matrix Gl = Gl(INt − is given by † † −1 Gl,l(Gl,lGl,l) Gl,l). r (d) p Pp(u) (p) = exp −πλ APPENDIX B k,l p0 The proposed SIA fractional power control mechanism c.d.f. " (d) r # 2πλ p ISI ISI ISI can be expressed as × 1 − exp − + 2 exp − . 4p p0 pΩ p Ω pΩ (u) (C.1) F (u) (p) = 1 − H p − p pk,l By utilizing the integral identities (where n > 0 and [ > n) (u) (u) × P pˆk ≤ p + H p − p (B.1) +∞ 1−n−[ Z Γ n (u) ψβ −1 exp (−xn) x[−n dx = (C.2) where pˆ = min p0d ,ISIH . Hence, we can write k k,l k,k 0 n and F (u) (p) = F ψβ (p)+F −1 (p)−F ψβ (p) F −1 (p) pˆ p d ISIH p d ISIH k 0 k,l k,k 0 k,l k,k ! 1 ! 3,0 ψβ 1 1 p I G n−[−1,0, 2 = d ≤ + H ≥ SI Z +∞ 0,3 4 P k,l P k,k √ 1 [−n p0 p exp − x − x dx = √ , x π 1 ! 0 ψβ p ISI (C.3) − P dk,l ≤ P Hk,k ≥ . (B.2) p0 p we can arrive at (15). By considering the transmitter-receiver distance distribution, Next, consider the following transmit power p.d.f. 1 ! (1 + K)ISI (1 + K)ISI ψβ p Pp(u) (p) = 2 exp − K + P dk,l ≤ k,l p Ω pΩ p0 ˜ K(1 + K)ISI 1 × 0F1 ; 1; . (C.4) p ψβ pΩ Z p0 = 2πλ(d)r exp −πλ(d)r2 dr 0 We derive the following integral identity (where [ < 1 and 2 ! ψβ n > [) (d) p = 1 − exp −πλ , (B.3) +∞ p0 Z 1 1 exp F˜ ; 1; p[−n dp p 0 1 p and the SI channel power gain non-central Chi-squared distri- 0 bution, = L[−n+1(1)Γ(n − [ − 1). (C.5) ˜ ISI Hence, by utilizing Lm(z) = 0F1 (−m; 1; z), we can arrive at P Hk,k ≥ p (16). It should be noted that with no path-loss compensation, the ﬁrst and beyond positive moments ([ ≥ 1) of p(u) are Z +∞ 1 + K (1 + K)h k,l = exp − K + inﬁnite. ISI Ω Ω p Finally, note the following unbounded transmit power p.d.f. r ! 2 2 ! K(1 + K)h (d) ψβ ψβ × I0 2 dh 2πλ p (d) p Ω Pp(u) (p) = exp −πλ . k,l ψβp p0 p0 s ! √ 2(1 + K)ISI (C.6) = Q1 2K, , (B.4) pΩ By utilizing the integral identity (C.2), we can obtain (18).

we can obtain APPENDIX D 2 ! ψβ Let Φ, λ, and denote the PPP, density, and exclusion (d) p E F (u) (p) = 1 − exp −πλ P −β pˆ region radius, respectively. Hence, I = Qxkxk k p0 x∈Φ where x and Q are respectably the location and channel s !! x √ 2(1 + K)I power gain of an arbitrary interferer with respect to a typical 1 − Q 2K, SI . (B.5) 1 pΩ receiver at the origin. We proceed as follows ( !) Through differentiating the above, we can derive the p.d.f. X −β MI (z) = E exp −z Qxkxk expression in (B.6), as shown at the top of the next page. x∈Φ

SHOJAEIFARD et al.: MASSIVE MIMO-ENABLED FULL-DUPLEX CELLULAR NETWORKS

2 ! 2 s !! ψβ (d) βψ √ (d) p 2πλ p 2(K + 1)ISI Ppˆ(u) (p) = exp −πλ 1 − Q1 2K, k p0 ψβp p0 pΩ ! (1 + K)I (1 + K)I K(1 + K)I + SI exp − K + SI F˜ ; 1; SI (B.6) p2Ω pΩ 0 1 pΩ

(i) n −β N o = lim EN EQx,x exp −zQxkxk and →+∞ D 1 (ii) 2 2 {exp (−αQ )} = , (D.6) = lim exp πλ D − E EQx x (1 + αV )U D→+∞ −β we can arrive at × EQx,x exp −zQxkxk − 1 " Z D 2 (iii) 2 E = lim exp 2πλ Q R MI (z) = exp − πλ − E + E x −β U D→+∞ E (zV E + 1) !! 2 2 −β Γ 1 − Γ U + Uβ+2 × exp −zQ kRk − 1 dR β β 2 Uβ E x + (zV ) β − Γ(U) Uβ + 2 (zV )U (iv) 2 2 −β #! = exp − πλ Q Γ 1 − − Γ 1 − , zQxE β E x β β 2 2 E × 2F1 U + 1,U + ; U + + 1; − . (D.7) 2 β β zV β 2 −β × (zQx) + E exp − zQxE − 1 (D.1) where (i) is written considering N conditional interferers are i.i.d. in a circular region of radius D (> E ) around APPENDIX E the center with the limit as D → +∞; (ii) is from char- acterizing N as a Poisson random variable with expected Consider the cases where the DL and UL SEs are respec- 2 2 value πλ D − E and hence utilizing the Poisson identity tively simpliﬁed to E ζN = exp (E {N } (ζ − 1)); (iii) is written using the Z +∞ Z +∞ Z +∞ 2πλ(d)rp(d) p.d.f. of the distance R (where E < R < D) S = log (e) d,f 2 (1 + zpΩ) r4 + zp(d) 2R 0 0 0 ! !! Pkxk(R) = 2 2 ; (D.2) p (d) D − E (d) π √ p (d) zp 2 exp −πλ zp + zp arctan 2 + r (iv) is obtained by invoking the integral identity (where α > 0 2 r

and β > 2) P (u) (p) dz dr dp (E.1) pk,l 2 β −β 2α and Ekxk exp −αkxk = β (D 2 − E 2) Z +∞ Z +∞ Z +∞ 2πλ(d)rp 2 α 2 α S = log (e) × Γ − , − Γ − , , (D.3) u,f 2 1 + zp(d)Ω) (r4 + zp) β D β β E β 0 0 0 √ π p √ zp and taking the limit as D → +∞. exp −πλ(d) zp(d) + zp arctan + r2 2 r2 To proceed, the distribution of the channel power gain Pp(u) (p) dz dr dp. (E.2) should be speciﬁed. Here, we consider the general case where k,l Qx ∼ Γ(U, V ), which can be used to capture a wide range z 4 Utilizing, in the order given, substitution with p(d) → u , of MIMO setups [46]. Hence, through utilizing the following conversion from Cartesian to polar coordinates with u → integral identities (where α > 0 and β > 2) R sin(T ) and r → R cos(T ) (with Jacobian R), the integral 2 identity 2 Γ U + β 2 β β EQx Qx = V , (D.4) Z +∞ Γ(U) R 2 4 exp −R dR 0 1 + R 2 1 π 2 Γ U − β + 1 = sin (1) C (1) + − cos (1) S (1) , (E.3) EQx Γ 1 − , αQx = 2 2 β (αV )U Γ(U + 1) √ and substitution with T → arctan s, we can arrive at (36). 2 1 × 2F1 U, U − + 1; U + 1; − , (D.5) The same process can be applied in the case of complete SI β αV removal but with the reduced integral identity (C.1) to obtain

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(37). Finally, we can respectively arrive at (38)-(39) using The joint HD DL/UL SE in the above is affine in x. It +∞ can be shown that the second derivative of the SE function Z √ 2 1 1 1 π d S S √ exp − dp = π + Ei(1) − erfi(1) in FD mode, dx2 (2 ( d,f + d,f )), is always positive for 0 p p + p p e e 2 π π−8 2 (E.4) π−8 < x < 1, always negative for 1 < x < π , and zero for x = 1. Consequently, L is convex in x and the and at (40)-(41) using objective function in (47) is strictly quasi-concave in x, with Z +∞ 1 √ a maximum point at x∗ = 1. √ exp (− p) dp = −2 exp(1)Ei(−1). (E.5) 0 p + p REFERENCES [1] A. Shojaeifard, K. K. Wong, M. D. Renzo, K. A. Hamdi, and J. Tang, “Design and analysis of full-duplex massive MIMO cellular networks,” APPENDIX F in Proc. IEEE Globecom (GC) Wkshps., Dec. 2016, pp. 1–6. [2] A. Shojaeifard, K. K. Wong, M. D. Renzo, G. Zheng, K. A. Hamdi, and By computing the special case of 2 (Sd,f + Su,f ) and J. 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