Xiangdong Jia＊, Mangang Xie, Meng Zhou, Hongbo Zhu, Longxiang Yang

1 College of Computer Science and Engineering, Northwest Normal University, Lanzhou 730070, China

2 Wireless Communication Key Lab of Jiangsu Province, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

* The corresponding author, email: jiaxd@nwnu.edu.cn

I. INTRODUCTION

Currently, the exponential growth of wireless data services driven by mobile Internet and smart devices has triggered the investigation of the fifth generation (5G) cellular wireless networks [1]-[2]. It has been predicted that,around 2020, the new 5G mobile networks are expected to be deployed [3]-[4]. The advanced 5G infrastructure will not only be a sheer evolution of the current network generations but,more significantly, a revolution in the information and communication technology (ICT)field [5].

On the one hand, over the past two decades, multiple-input multiple-output (MIMO)technique has been widely studied and ap-plied to many wireless standards such as the fourth generation cellular networks and IEEE 801.11n wireless local arear networks(LANs) since it can significantly improve the capacity and reliability of wireless systems. The MIMO concept is one of the most successful ones in recent years. However, in traditional MIMO systems the BS is equipped with only a small amount of antennas. For example, the LTE standard allows for up to eight antenna ports at BS. Basically, the more antennas the transmitter/receiver is equipped with, and the more degree of freedom that the propagation channel can provided, the better the performance in term of data rate or link reliability. Therefore, to further improve the performance of MIMO systems and to satisfy the ever-increasing capacity demands in 5G,recently multi-user massive MIMO or largescale MIMO has attracted a lot of attention in academy and industry and has been regarded as a promising enabler for the capacity increase of the future wireless networks [6]-[8].Multi-user massive MIMO was first proposed in multi-cellular noncooperative network [6].In massive MIMO systems, the base station(BS) is equipped with orders of magnitude more antennas, e.g., 100 or more, and simultaneously serves a multiplicity of channel nodes in the same radio channel, while users in each cell only have one antenna. As such, the array gain is expected to grow unboundly with the number of antennas at BS so that the multi-user interference, channel noise, and small-scale fading for any given number of users and any given powers of the interference users can be eliminated. This is due to the fact that when the number of antennas grows unlimited large,the random channel rectors between the users and the BS tend to pairwise orthogonal [9].

The authors consider a circularly distributed BS antenna array and obtain the asymptotic uplink rate of an arbitrary user by considering the asymptotic case where the number of antennas of BSs tends to infinity.

On the other hand, due to the inter-cell interference the traditional centralized antenna systems have low spectrum and energy efficiencies, especially at the cell edge. To overcome theses issues arising from centralized multiple antenna systems, recently the distributed antenna system (DAS) has also been gained great attention [10]. The idea of DAS was originally proposed to cover the dead spots in indoor wireless communications, and then implemented in cellular systems to improve cell coverage [11]. Different from in the traditional centralized multi-antenna systems such as centralized MIMO, in the DAS the antenna arrays of base station (BS) are separately distributed in a cell and connected to a baseband processing unit (BPU) via high-speed backbone links, for example, radio-over-fiber or microwave repeater. DAS is a promising method to mitigate the cell edge problem [12].Due to its promising features, the concept of DAS is developed for the current version of LTE-A.

Not surprisingly, incorporating DAS into massive MIMO is a strong candidate for the development of future energy-spectrum efficiency 5G wireless cellular networks. As a result, the novel scheme, distributed massive MIMO (DM-MIMO), not only inherits the advantages of both DAS and massive MIMO, but also sheds new light on higher performance. For example, one of the challenges with massive MIMO is the viability of mounting a large number of antennas on a BS. However, the distributed schemes can overcome this issue since in the scheme a large number of antennas are distributed over a large area and are connected to a central unit through backhaul links. On the contrary, the MIMO theory motivates a series of information-theoretical studies on the performance characterization of DASs. By regarding the channels between the user and multiple distributed antennas as a generalized MIMO channels, the powerful techniques developed for MIMO systems can be applied to DASs to identify the promising performance of DASs.Currently, to enhance the energy and spectrum efficiency of 5G systems, the DM-MIMO systems have been investigated widely [13]-[19]. By considering the limited backhaul link capacity, in [13] authors have pointed out that in distributed massive MIMO the backhaul link capacity becomes a bottleneck, and have obtained the maximum uplink sum rate for MD-MIMO systems. In [14], for a single-cell system with circularly distributed BS antennas and single-antenna users, the achievable uplink rate was obtained. The work [15] also focused on the single-cell DM-MIMO systems. Authors have investigated the uplink capacity when the number of users and the number of BS antennas both approach infinity with a fixed ratio. In [16], authors considered a distributed large-scale MIMO system consisting of multiple users and one base station with several distributed antenna sets. Based on random matrix theorem (RMT), authors have proposed a deterministic equivalent of ergodic sum rate and an algorithm for evaluating the capacity-achieving input covariance matrices.The obtained deterministic equivalent results are easy to compute and shown to be accurate for realistic system dimensions. In addition,they are shown to be invariant to several types of fading distribution. The reference [17] also focused on a single-cell DM-MIMO system.In this paper, authors have evaluated the performance loss caused by small-scale fading,but also the large-scale fading and path loss.Additionally, authors also accounted for the effect of spatial correlation at transmit side. Specially, authors have considered the classical lognormal model and proposed closed-form upper and lower bounds on the achievable sum rate. By using the second-order statistics in DM-MIMO systems, the problem of interference control has been investigated in [18].The work [19] also considered the single-cell DM-MIMO BS with grid antenna array layout and investigated the energy efficiency maximization problem. A low complexity channel-gain-based user cluster method has been proposed with the goal of improving energy efficiency. In [20], by accounting for real environmental parameters and antenna characteristics, a novel comprehensive channel model was proposed for DM-MIMO systems. Then,based on the proposed channel model, authors investigated the achievable spectral efficiency.

While the works [13]-[20] focused on the single-cell DM-MIMO systems, it is seen that only limited works considered the multi-cell ones. However, it is well known that, today and in the future, the cellular networks are the dominating mobile communication network structure, which consists of multiple cells so that the spectrum efficiency is improved. For such practical mobile communication cellular networks, when the BSs are equipped with distributed massive MIMO antenna array, the following problems should be considered carefully. All of the first, we should evaluate the effect of the interference from adjacent cells on systems such as in traditional centralized multi-cell systems [23]-[27]. In practice, due to the different distance from the interference users to the distributed antennas, it is challenging to investigate the effect of the interference from the adjacent cell on the performance of the users in central cell. Secondly, in the existing works [21]-[22] only the downlink transmission has been considered. For the achievable uplink rate, it has not been achieved so far. References [13]-[19] have evaluated the performance of distributed massive MIMO systems, but these references only focused on single-cell schemes. Finally, as a simplified scheme, the circularly distributed antenna array is a candidate. However, there is no work focusing on the effect of the radius of circular antenna array on system performance. The optimal radius of circular antenna arrays relies on the users in reference cell, but also the ones in interference cells. This is different from the single-cell systems. In single-cell systems, it is easy to achieve the optimal radius, but in multi-cell ones it is challenging to achieve it.

Therefore, inspired by the above literature review, in this paper we deal with the achievable uplink rate of the users in reference cell for DM-MIMO systems by considering the interference from the adjacent cells. Specially,we first model a distributed massive MIMO system as well as the corresponding assumptions. Compared with centralized MIMO systems, the distributed MIMO systems provide macro-diversity and have enhanced network coverage and capacity due to their open and flexible infrastructure. This is also different from the systems with distributed BSs, in which several BSs with multiple antennas cooperate with each other to jointly transmit or receive the information, and each BS has its own power constraint. Secondly, the DM-MIMO systems with arbitrary BS antenna topology would be investigated as well as the achievable uplink rate. Thirdly, to achieve the asymptotic performance we consider a circular antenna layout, which has a good balance among the tractability of theoretical analysis,practicality, and performance. Finally, the numerical results are presented to high the insight. In numerical analysis section, we also present the average asymptotic rate of a user by assuming that the users are randomly and uniformly located in each cell. The average rate does not depend on the locations of users.

Throughout this paper, we use boldface upper and lower letters to denote matrices and vectors. The notations (·)*, (·)T, and (·)Hstand for the operations of conjugation, transposition, and conjugate transposition, respectively.We also useTr(·), andE(·) to represent Euclidean norm, trace, and expectation. We usedenoting almost sure convergence.denotes the set ofcomplex value matrices. Finally, we usedenoting a circularly symmetric complex Gaussian vectorzwith zero mean and covariance matrixrepresents anidentity matrix.The Frobenius norm of a vector or matrix is represented as

Fig. 1 Multi-cell Multi-user distributed massive MIMO cellular systems

II. SYSTEM MODEL AND ASSUMPTIONS

Similar to [21], in Fig.1 we consider a multicell multi-user distributed massive MIMO cellular system consistingLhexagonal cells.Each cell consists of one BS equipped withantennas which are separately distributed in a cell and connected to a baseband processing unit via high-speed backbone links,andNsingle-antenna mobile user equipments(UEs). The numberMof antennas is assumed to be very large,e.g., a few hundreds. For simplicity, we assume that the distributed BS antennas have ideal cooperation with each other. As shown in Fig.1, we assume that the cell 1 is the reference cell or central cell (the corresponding BS called reference BS), and the otherL-1 cells are the interference cells.Each cell has the same area with cell radiusR. We also assume thatd0denotes the closest distance from the mobile user to BS’s antenna,

We focus on the uplink transmission of users in central cell (cell 1). On uplink links,it is assumed that all users simultaneously transmit data streams to their BSs in the same time-frequency resource. In the same cell,all users have the same transmit powerbut for different cells the transmit powers are maybe different,i.e.,Letdenote themassive MIMO channel matrix between theNusers in thel-th cell andMBS antennas in thei-th cell.Therefore, the receivedsignal vectorat the BS of the reference cell (cell 1) can be formulated as

whereDminlis the distance between then-th user in thel-th cell and them-th antenna of thei-th BS,vis the path-loss exponent with typical values ranging from 2 to 6,i.e.,

In this paper, the simple ZF linear receiver is employed at receiver. Letbe the ZFlinear receiver matrix at the central cell BS, which is given by

III. ACHIEVABLE UPLINK RATE OF A USER FOR ARBITRARY ANTENNA TOPOLOGY

Based on the multi-cell multi-user MD-MIMO system model presented in above section, in this section we first investigate the achievable uplink rate for arbitrary antenna topology for BS. To make this paper self-contained and also improve the readability of this paper, we first present some important results [28], which will be frequently involved in the sequent derivations.

3.1 Some important results

Lemma 1: Letwithbe a series of random matrices generated by the probability spacesuch that, forwithuniformly onN. Letwithbe random vector of i.i.d entries with zero mean, varianceand eighth-order moment of orderindependent ofThen,

almost surely.

Lemma 2:Letbe as in Lemma 1 andbe random, mutually independent with standard i.i.d entries of zero mean, variance 1/N, and eighth-order moment of orderindependent ofThen,

almost surely.

3.2 Uplink rate analysis

With above Lemmas, we investigate the achievable uplink rate of an arbitrary user for the DM-MIMO systems with arbitrary BS antenna topology. To this end, we first substitutealong with (1) into (5), having the output signal vector at ZF receivers given by

Although (10) presents the general expression for the achievable ergodic rateof an arbitrary user in central cell, it is seen that the derivation for the closed-form expression ofis challenging. To overcome this problem,by using the convexity ofand Jensen’s inequality, we have the following lower bound of the achievable rate, given by

where, with Jensen’s inequality, we define consideringwe have the result

This yields that

Therefore, substituting (14) along with (15)into (12),is expressed as

Combining (17) and (18) leads to

Therefore, combining (21), (20), (16), (12),and (11), we have the Theorem 1 that presents the uplink rate for then-th user in the central cell.

Theorem 1: Assume that the BS of central cell has the perfect CSI and the transmit power of each user in thel-th cell is scaled withMaccording towhereis fixed.Then, the achievable uplink rate of then-th user in the central cell is given approximately by

IV. ACHIEVABLE ASYMPTOTIC RATE FOR CIRCULARLY DISTRIBUTED ANTENNAS

Although Theorem 1 presents the general expression of the achievable uplink rate of an arbitrary user for distributed massive MIMO systems with arbitrary antenna topology,the expression ofgiven by (22) depends strongly on the position of users and the one of antennas. Besides this, the derivations also depend on the number of BSs’ antennas.This yields that the performance evaluation based on the derivation (22) is intractable in distributed massive MIMO systems when the number of distributed antennas is very large.In practice, as discussed in existing works, for massive MIMO systems the asymptotic performance is more favorable. Therefore, in this section we derive the asymptotic performance by considering the number of BS antennasMgrowing to infinity.

Due to the lack of effective mathematical method for arbitrary antenna topology, here we only consider the evenly circularly distributed antennas. The circular antenna layout is more practical due to its approximate balance among the tractability of theoretical analysis and performance evaluation.

The equation (23) indicates that we should achievewhich is the distance between then-th user in thel-th cell and them-th antenna of central BS. To this end, we employ Fig.2 to investigate the asymptotic uplink rate.

With equation (25) we consider the asymptotic scenario. Obviously, when the numberMof antennas at BS goes to infinity,equation (25) can be written as

Using the identity given by [31] leads to the asymptotic result of(26) is achieved

Theorem 2: When the BSs of massive MIMO systems are equipped with evenly circularly distributed antennas having antenna radius , the achievable asymptotic uplink rate of then-th user in central cell at distancefrom the cell center is given by

Fig. 2 Graphic illustration of the distance for circularly distributed antenna with radius r

where we define

By employing Legendre function, a compact expression foris presented in Theorem 2. Due to the fact that we have the identitywhereis the Gaussian hypergeometric function [30], it is very easy to obtain the evaluation of the achievable asymptotic uplink ratewith the assistance of some scientific computation software such as MATLAB. At the same time, with Theorem 2 we can obtain the average asymptotic uplink rate of a user in reference cell by taking the integral in term ofandrespectively, which is given by Theorem 3.

Theorem 3:For the multi-cell multi-user DM-MIMO systems, when the evenly circularly distributed BS antenna arrays are employed and the assumption that all users are randomly and uniformly located in each cell is considered, the average asymptotic uplink rateof a user can be formulated as

In (31), the random variables andsatisfying the closest distance constraint from the users to BS’ antenna,andare the probability density functions (PDFs) of the random variablesandrespectively.The PDFis given by

Fig. 3 Achievable asymptotic rate of a user versus the distancesand

With Theorem 3, it is easy to achieve the spectrum and energy efficiencies of the DM-MIMO systems.

V. NUMERICAL RESULTS

Based on the above analysis, in this section we present the numerical results to show the performance of the distributed massive MIMO systems as well as the impact of system parameters on the achievable uplink rate. During the analysis, a 7-cell hexagonal system layout is employed. We take the cell radiusR=1500m,the minimum distance between a BS antenna and a userIn each cell, all users have the same transmit powerwhereis fixed.The normalization within transmit power does not affect the behavior of performance curves but only affects the position of the curves on theEu-axis.

Note that, in Fig.3 we only present the numerical result in terms of the asymptotic uplink rate of a specific user based on Theorem 2. For simulations, it could be achieved by combing (25) and (22) so that the asymptotic performance can be simulated under the case where the number of antennas is large enough.However, observing (25), the simulation depends on the parameterwhich make it is difficult to perform simulations.

In Fig.3, we only present the asymptotic rate of an arbitrarily located user in central cell, which depends greatly on the locations of the users in central cell and interference calls.Obviously, in Fig.3 it is difficult to achieve the average experience of user service as well as a clear insight for the effect of system parameters on the achievable uplink rate. With Theorem 3, the average uplink rate can be simulated. Therefore, in Fig.4 we investigate the achievable average uplink rate of a user in central cell as well as the effect of system parameters. Fig.4 (a) presents the average uplink rate of a user versus the radiusof circular antenna array. At the same time, for comparison in Fig.4 (a) the average rate of a user for single sell layout scenario is also presented. Fig.4 (a)clearly shows that the average rate of a user in multi-cell scenario is inferior to the one of a user in single-cell scenario. This is due to the fact that the interference from the adjacent cells imposes a great effect on the achievable average rate. Additionally, it is seen easily that the radius of circular antenna arrays is of significance for enhancing the users’ average rate. Specially, when the radius of circular antenna arrays is small relatively, the increase ofwould contribute to the improvement of the achievable average uplink rate. However, when the radius of circular antenna array arrives at an optimal value, the circularly distributed massive MIMO systems achieve the optimal average uplink rate. Then, the average rate would decrease with the increase of.At the same time, the achievable uplink rate of a user for the centralized massive MIMO scenario can be evaluated with the radiusrof circular antenna arrays tending to zero. Obviously, the distributed massive MIMO systems greatly outperform the centralized massive MIMO ones. Fig.4 (b) is the average rate versus the transmit power of users under different path loss exponentv. We see that the path loss exponent has an outstanding effect on the average rate.

VI. CONCLUSIONS

Fig. 4 Effect of system parameters on average rate

The multi-cell multi-user distributed massive MIMO systems are investigated in terms of the achievable ergodic uplink rate. We first focus on the arbitrary BS antenna topology scenario and obtain the corresponding achievable uplink rate of an arbitrary user. The derivations show that in this scenario the uplink rate depends on the users’ access distance to each distributed antenna unit and the number of BSs’ antennas. Then, to obtain a clearer insight, we investigate the asymptotic uplink rate by considering the circularly and evenly distributed antenna arrays. It is achieved that the asymptotic uplink rate only depends on the distance from users’ position to the center of reference cell, and free of the number of BSs’antennas. The presented numerical results indicate that the distributed massive MIMO systems outperform greatly the centralized ones. The interference from the adjacent cells imposes great impact on system performance.At the same time, in numerical analysis section we present the average asymptotic uplink rate of a user by assuming the users randomly and uniformly located in a cell. From the numerical results we can obtain the optimal location of the radius of circularly distributed BS antenna arrays, at which the systems would obtain the maximum average uplink rate.

ACKNOWLEDGEMENT

The authors would like to thank the editors and the anonymous reviewers for their constructive comments and suggestions, which helped to improve the quality of this paper. This work was supported by the Natural Science Foundation of China under Grant 61261015 and 61561043, the 973 project 2013CB329104, the Natural Science Foundation of China under Grant 61372124, 61363059, and 61302100,the projects BK2011027, the Natural Science Foundation of Gansu Province for Distinguished Young Scholars (1308RJDA007),and by the Foundation Research Funds for the University of Gansu Province: ‘Massive MIMO channels modeling and estimation over millimeter wave band for 5G’.

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