Achievable Rate of Multi-Cell Downlink Massive
MIMO Systems with D2D Underly
Ashraf Al-Rimawi
∗Member, IEEE, Laith Ibrahim
†Member, IEEE, and Wessam Ajib
‡Senior member, IEEE
Department of Electrical and Computer Engineering∗, Birzeit University, PalestineDepartment of Technology and Applied Science†, Al-Quds Open University, Palestine Department of Computer Science ‡, University of Quebec in Montreal, Canada
Email: aalrimawi@birzeit.edu∗, librahim@qou.edu†, ajib.wessam@uqam.ca‡,
Abstract—In this paper, a new analytical framework model based on stochastic geometry for Device-To-Device (D2D) com-munication underlaying multi-cell massive Multi-Input-Multi-Output (MIMO) system is proposed. Assuming Maximum Ratio Transmission or Zero Forcing precoding scheme for cellular downlink transmission, the impact of RF mismatches and achiev-able rate of cellular user are analytically derived. The studied model assumes truncated Gaussian distribution to model RF mismatches, D2D interference, inter cell interference, and intra-cell interference. Accordingly, closed form expressions of lower-bound achievable data rate for cellular users is derived. More-over, asymptotic performance analysis under the assumption of large number of antennas has been performed. Simulation results are found to coinside with the theoritical results and validated our model.
Index Terms—Masive MIMO, D2D, Linear Precoding, Reci-procity Error.
I. INTRODUCTION
Future generations of wireless networks just as fifth generation (5G) and beyond are always aiming for higher spectrum efficiency, better link reliability and larger data rate [1]. The combination of massive input multiple-output (MIMO) technology and device-to-device (D2D) is one of the most promising approaches and a key factor in future wireless networks to achieve its targeted goals. D2D communication technology allows operators to be more flexible in terms of offloading data traffic from the core networks by enabling neighboring users to communicate with each other using direct links. Thus, it possess the ability of enhancing the spectrum efficiency and therefore improving the resource utilization.
On another hand, massive MIMO has been a hot topic for researchers recently [2], [3]. It refers to the deployment of very large antenna arrays in base stations in order to serve a given number of users with the same time and frequency block simultaneously. In such technology the inter-antenna interference can be eliminated and a high data rate can be achieved by using linear precoding techniques such as Maximum Ratio Transmission (MRT) and Zero Forcing (ZF) [3].
Radio frequency (RF) mismatches phenomena in Time Division Duplexing (TDD) based massive MIMO system is an important issue that should be taken into account [4]. In such systems, and since the actual values of the Uplink (UL) channel is used to estimate the Downlink (DL) channel
values, the Radio-Frequency (RF) mismatches cause a random deviation on the estimated DL channel values. The later is known as reciprocity error [5].
II. RELATEDWORKANDCONTRIBUTION
Recently, a considerable number of studies analyzed the performance of massive MIMO systems (with and without D2D) in different scenarios and conditions [6]–[8]. In [6], spectral and energy efficiency of D2D communication under-lying massive MIMO cellular network has been evaluated. Works in [7] investigate the coexistence between massive MIMO and D2D. In [8], a spectral efficiency analysis for DL massive MIMO systems with MRT precoding in a single cell scenario has been analyzed for perfect and imperfect channel state information (CSI).
Although the above works provide good results about the performance of linear precoding in different massive MIMO system setups, they did not consider the reciprocity error. However, the authors in [5] consider and analyze the impact of reciprocity error on the performance of linear precoding but they consider only a single cell.
In this work, considering the imperfection of channel esti-mation and channel reciprocity error, we analyze the system performance in terms of CUE achievable data rate in a multicell DL massive MIMO system with D2D underly and with linear precoding schemes. More specifically, the main contributions of this work are as follows:
• With imperfect channel reciprocity and imperfect CSI,
the achievable data rate of cellular users in the considered system is derived for ZF and MRT precoding schemes.
• Asymptotic analysis has been provided under the assump-tion of a large number of base staassump-tions (BS) antennas and high transmission power. Closed-form expression for the achievable data rate for ZF and MRT has been derived.
• A performance comparison of the two considered
pre-coding schemes has been performed for different system parameters and in the presence of channel reciprocity error and channel estimation error.
Notations : E (.) denotes the expectation operator, and var (.) is the mathematical variance. The operators (.)T
and (.)Hrepresent transpose and conjugate transpose, respectively.
Fig. 1: System Model
III. SYSTEMMODEL
As shown in Fig.1, we consider a downlink multi-cell massive MIMO systems consisting of B number of BSs. Each BS is equiped with an array of M antennas and serves K cellular users (CUEs) with a single antenna. We assume that the locations of BSs are fixed and the users are distributed uniformly in the cellular coverage area.
In addition, we consider a random number of D2D groups which are also distributed uniformly in the cellular coverage area. We assume in each group only one D2D link is active at the same time sharing the same radio resource block with the cellular network. The location of D2D transmitter is modeled
by a homogeneous Poisson point process (HPPP) Φd with
density λD and the D2D receiver is randomly located at a
fixed distance from its corresponding transmitter in the D2D coverage area. Furthermore, we consider that each BS uses md
degrees of freedom to cancel the interference from the nearest D2D interferes. Besides, the BSs and D2D transmitters are assumed to be at a constant power Pc and Pd, respectively.
On the other hand, we consider that the system operates in TDD mode with MRT and ZF linear precoding schemes. In our model, not only UL and DL channels are considered, but also the impact of the effective response of transmitter (Tx) and receiver (Rx) RF modules at the BS is taken into account.
Let H∈ CM ×K, H
btand Hbrdenote the UL channel, transmit
and receive RF matrices at the BS b, respectively. The M×M diagonal matrix of Hbtcan be expressed as
Hbt= diag (hbt,1, hbt,2, . . . , hbt,i, . . . , hbt,M) , (1)
where hbt,i (i = 1, 2, . . . , M) are RF gains characterized as follow
hbt,i= Abt,ieθbt,i. (2)
Here, we consider the amplitude of RF gain Abt,i and phase
θbt,iare independent random variables and distributed accord-ing to truncated Gaussian distribution as in [5], which is more generalized and realistic compared to uniform distribution
model in [9]. Similarly, M × M diagonal matrix Hbr can
be denoted as in (1). Therefore, the amplitude and phase
reciprocity errors of both Tx and Rx fronts can be modeled as Abt,i∼ NT αbt, 0, σ2 bt , Abt,i∈ [at, bt] (3) φbt,i∼ NT θbt, 0, σ2 θt , φbt,i∈ [θt,1, θt,2] (4) Abr,i∼ NT αbr, 0, σ2 br , Abr,i∈ [ar, br] (5) φbr,i∼ NT θbr, 0, σ2θt , φbr,i∈ [θr,1, θr,2] , (6)
respectively, where NT denotes truncated Gaussian distribu-tion.
We assume αbt, αbr, θbt and θbr remain constant within the considered coherence time of the channel. Let’s de-note E[ejθbt], E[ejθbr], E[A
bt,i], E[Abr,i], var (Abt,i) and var (Abr,i) by gt, gr, αt, αr, σt2, and σ2r, respectively.
In fact, the channel responses at both sides of Tx and Rx are not the same. Therefore, we define RF mismatches as the ratio between Hbtand Hbr, i.e,
Er= HbtH−1br = diag h bt,1 hbr,1, . . . hbt,i hbr,i, . . . hbt,M hbr,M . (7) In practice, various environment factors may cause the ele-ments of RF mismatches matrix in (7) to be different from each other.
By considering channel reciprocity within the channel coher-ence interval, the estimated DL channel can be expressed as [5]
ˆ
Hd = ˆHTu=1 − τ2HTHbr+ τ VT, (8)
where V is M × K dimensional matrix which is represent
the channel error. In this paper, we assume both H and V are independent, identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. The parameter τ ∈ [0, 1] is the estimation variance and it reflects the accuracy of the channel estimation.
By taking the effect of Hbt into consideration, the actual DL
channel Hd can be given by
Hd= HTHbt. (9)
Without loss of generality, the received signal at the kthCUE in cell b is given by yb,k = PcλhTHbtwb,ksb,k Desired Signal + K i=1,i=k PcλhTb,kHbtwb,isb,i Inter-User Interference + l=b K i=1 PcλhT l,kHbtwl,isl,i Intra-cell interference + j∈Φd Pd||xj||− α 2hb,jsb,j+ nb,j Inter-D2D interference , (10)
where wb,k represents the linear precoding vector for user k which is a function of the DL channel estimation ˆHd. We assume that the average transmit power for each CUE is the same. λ is the normalization parameter, such that
E[trλ2wwH] = 1. n is additive white Gaussian noise
IV. ACHIEVABLEERGODICRATE
In this section, we derive the achievable ergodic rate Rb,k
of the received signal at the kth CUE in cell b.
The achievable ergodic rate Rb,k can be expressed as Rb,k= E [log (1 + SINR)] = E log 1 + Ps PI+ σ2 k , (11)
where the desired signal power Psand the interference power PI can be expressed as Ps=PcλhTH btwb,ksb,k 2 , (12) and PI =| K i=1,i=k PcλhT b,kHbtwb,isb,i + l=b K i=1 PcλhT l,kHbtwl,isl,i + j∈Φd Pd||xj||− α 2hb,jsb,j+ nb,j|2, (13) respectively.
Similarly to [10], it is difficult to calculate the ergodic rate directly from (11). Therefore, we propose to use Jensen’s inequality to obtain a lower bound of the achievable rate [10], as: Rb,k ≥ log 1 + E 1 SINR −1 , (14)
where the expected value of signal-to-interference plus noise ratio (SINR) can be approximated by [11]
E P s PI+ σ2 k ≈ E [Ps] E 1 PI+ σ2 k . (15)
V. SIGNALMODELS OFLINEARPRECODERS
With large number of antennas at BS, the precoding scheme should balance the complexity and the performance of massive MIMO systems [2], [12].
A. Maximum Ratio Transmission MRT
MRT is a precoding scheme that maximizes the received signal-to-noise ratio (SNR) at the CUE, i.e, the MRT precod-ing employed by the BS can be expressed as [13]
Wmrt= ˆH H
d , (16)
The expected values of the desired signal power, and interfer-ence power for MRT scheme can be given by
E [Ps,mrt] =E PcλmrthTHbtwb,k,mrtsb,k 2 =PcAt K 1 − τ2A r((M − 1) AI + 2) + τ2 (1 − τ2) Ar+ τ2 , (17) and E [PI,mrt] =MPcλ2mrtAt BK − 1 + τ2A r+ (Bk − 1) τ2 +2 (πλD) α 2 P dΓ md+ 1 − α 2 (α − 2) Γ (md) , (18) respectively. Note that AI = α 2 tσr2 (α2 t+σt2)(α2r+σ2r)|gt| 2|g r|2 is the aggregated
reciprocity error factor, Ar= α2r+ σ2r, At=α2t+ σ2t, and
λmrt= 1
M K(1−τ2)Ar+τ2).
By substituting equations (17) and (18) into equation (15), the
closed-form expression of the output SINR for the kth CUE
can be obtained. B. Zero Forcing ZF
The ZF precoding matrix can be expressed as [14] Wzf = ˆH H d ˆ HdHˆ H d −1 , (19)
Similarly, the expected values of the desired signal power, and interference power for ZF can be given by
E [Ps,zf] =E PcλzfhTHbtwb,k,zfsb,k 2 =PcM − K K BI, (20) and E [PI,zf] =| K i=1,i=k PcλzfhTb,kHbtwb,isb,i + l=b K i=1 PcλzfhTl,kHbtwl,isl,i + j∈ΦD Pd||xj||− α 2hb,jsb,j+ nb,j|2 =Pc(At− BI)KB − 1K +2 (πλD) α 2 P dΓ md+ 1 −α 2 (α − 2) Γ (md) , (21) respectively, where BI = (1−τ2)AIAtAr (1−τ2)Ar+τ2 , and λzf = M −K K ((1 − τ2) Ar+ τ2).
By substituting equations (20) and (21) into equation (15), the closed-form expression of the output SINR in presence of ZF
precoding scheme for the kthCUE can be obtained.
VI. ASYMPTOTICSINR ANALYSIS
In this section, we focus on investigating the effect of imperfect channel reciprocity error on the performance of MRT and ZF precoding schemes. Assuming large number of
BSs antennas (M → ∞), and the number of users in cell b is
greater than one (K >> 1), closed form expressions of SINR for MRT and ZF can be approximated as
0 0.2 0.4 0.6 0.8 1 Estimation variance parameter ( ) -20 -15 -10 -5 0 5 10 SINR (dB) (a): MRT precoding AI=1 AI=0.7 AI=0.5 AI=0.3 AI=0.1 0 0.2 0.4 0.6 0.8 1 Estimation variance parameter ( ) -20 -15 -10 -5 0 5 10 15 20 25 30 SINR (dB) (b): ZF precoding AI=1 AI=0.7 AI=0.5 AI=0.3 AI=0.1
Fig. 2: SINR in dB with respect to the estimation variance parameter (τ ) and for different values of the aggregated reciprocity error factor (AI).
and ˜ SINRk,zf ≈M − KK PcBI Pc(At− BI) B + ID+ σ2k , (23) respectively. ID = 2(πλD) α 2PdΓ(md+1−α2)
(α−2)Γ(md) is the interference from D2D
communications in the desired cell.
As special cases, if we remove the channel estimation error from equations (22) and (23) by assuming that the estimation variance parameter τ equals zero, then we obtain:
˜ SINRk,mrt≈ PcAtMAI K (PcAtB + σ2 k+ ID) , (24) and ˜ SINRk,ZF ≈ Pc(M − K) AI KPc(1 − AI) B + σ2k+ ID . (25)
Based on equations (24) and (25), it can be noted that:
• The perfect channel reciprocity can be obtained when
AI = 1. In this case, we have the same results derived in [13].
• Higher SINR can be achieved when the level of the
aggregated reciprocity error factor AI increases. • The effect of noise and infinite D2D interference depends
primarily on the BS transmission power Pc. i.e., the noise
and D2D interference are completely vanished for high BS transmit power.
VII. NUMERICALRESULTS
In this section, we present the simulation and numerical results obtained to validate the theoretical analysis in sections V and VI. Here we consider, unless otherwise specified, the
available D2D transmit power equal to Pd = 20 dBm, the
number of mobile users is fixed to K = 10 users per cell, the number of BS antennas M = 300, the number of cells B = 5, the path loss exponent α = 3 and the transmit BS
power Pc = 15 dB. Furthermore, we consider equal power
-20 0 20 40 Pd (dB) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 SE ( bi t/se c /Hz) (a): MRT precoding B=1 (Analytical) B=1 (Simulation) B=2 (Analytical) B=2 (Simulation) B=4 (Analystical) B=4 (Simulation) -20 0 20 40 Pd (dB) 0 1 2 3 4 5 6 7 SE ( bi t/se c /Hz) (b): ZF precoding B=1 (Analytical) B=1 (Simulation) B=2 (Analytical) B=2 (Simulation) B=4 (Analystical) B=4 (Simulation)
Fig. 3: The achievable rate as a function of BS transmit power
(Pc) for different number of cells (B) and with imperfect
channel estimation with τ2=0.1.
0 50 100 150 200 M: Number of BS Antennas 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ac h iev ab le R a te ( bi t/se c /Hz) (a): MRT precoding K=10 K=15 K=25 0 50 100 150 200 M: Number of BS Antennas -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Ac h iev ab le R a te ( bi t/se c /Hz) (b): ZF precoding K=10 K=15 K=25
Fig. 4: The achievable rate for different number of user per cell (K) and with imperfect channel estimation with τ2=0.1.
allocation for K users and a small deviation of the amplitude errors i.e., Ar= At= 1.
Numerical results in Fig. 2 show the impact of the estimation variance parameter τ on the system performance for a different values of the aggregated reciprocity error factor AI. It can
be clearly shown that as τ approaches to 1, the system performance approaches to zero. This is because at this point the channel estimation is completely uncorrelated with the actual channel response. In contrast, as τ approaches to zero, the performance goes high due to the accuracy of the channel estimation. Moreover, at a specific value of τ , the system
performance goes up and down as AI changed. As AI goes
to 1, a perfect channel reciprocity is maintained and hence a
performance gain is obtained. While as AI goes to zero the
channel reciprocity error increases. Thus, a performance loss is obtained.
-10 0 10 20 Pc (dB) 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Ac h
ievable Rate (bit/sec/Hz)
(a): MRT precoding D=0.1 D=0.3 D=0.9 -10 0 10 20 Pc (dB) 0.5 1 1.5 2 2.5 3 3.5 4 Ac h
ievable Rate (bit/sec/Hz)
(b): ZF precoding
D=0.1 D=0.3 D=0.9
Fig. 5: The achievable rate as a function of BS transmit power (Pc) for different D2D densities (λD) and with imperfect channel estimation with τ2=0.1.
with respect to the transmit BS power on the achievable rate for MRT and ZF precodings. It can be observed that increasing B will deteriorate the achievable rate at high transmit power range. While, as the transmit power goes small, the achievable rate loss becomes smaller. Moreover, at high BS transmit power, the achievable rate becomes lower bounded by a constant ratio for both schemes which validates the asymptotic analysis in equations (24) and (25).
Fig. 4 shows the achievable rate versus the number of antennas M with different number of mobile users per cell K. In this scenario, we consider K is 10, 15 and 25 users respectively. Clearly, the achievable rate can be enhanced by increasing M , while deploying more users results in decreasing the rate. In Fig. 5 the achievable rate versus the BS transmit power for
different values of D2D density λD and for both precoding
schemes is depicted. It can be shown that at a certain D2D density and as the BS transmit power increase, the achievable rate also increases for both schemes. However, the effect of D2D density at a specific transmit power range can be regarded as two phase effect. At low BS transmit power range, the effect of increasing D2D density is high and results in system performance degradation. This is because the D2D interference is much higher than the BS transmit power leading to low SINR and hence a degradation in the achievable rate is observed. However, at high BS transmit power range, the D2D interference power is much lower than the BS transmit power. Therefore, the rate will not be greatly affected by λD
in this power region. This validates the asymptotic analysis in Section VI.
Furthermore, from Figs. 2, 3, 4 and 5 a performance compar-ison of the two considered precoding schemes is observed. It can be clearly seen that ZF achieves higher rate than MRT for all considered scenarios. Thus, we conclude that ZF performs better than MRT in a multi-cell downlink massive MIMO systems even though MRT is much simpler in practice especially for very high number of antennas.
VIII. CONCLUSIONS
In this paper, a lower bound on the achievable data rate of a multi-cell downlink massive MIMO system with MRT and ZF precoders and D2D communications has been derived. In addition, the asymptotic values of SINR at high BS transmit power and for a large number of BS antennas has been analyzed. Based on our analytical results, we find that the achievable data rates for both precoding schemes increase with increasing the number of BS antennas and its transmit power, and decrease with the number of cellular users and the number of cells. In addition, at high BS transmit power, we find that the impact of D2D interference and noise can be eliminated.
IX. ACKNOWLEDGMENT
This work was supported by Palestine Quebec Science Bridge (PQSB), Birzeit University and Zamala Fellowship program initiated by the Bank of Palestine (BoP) in partnership with taawon.
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