National Communications Research Laboratory, Southeast University, Nanjing 210096, China

* The corresponding author, email: jinshi@seu.edu.cn

I. INTRODUCTION

In recent years, massive multiuser MIMO is one of the promising candidates for future generation wireless cellular due to it provides higher spectral and energy efficiency [1] [2].Most importantly, authors in [3] has theoretically demonstrated that when the base station(BS) is deployed large scale antenna array(e.g., hundreds of antennas), the transmitter/receiver adopt the MRT precoding/maximum ratio combing (MRC) beamforming techniques, huge potential of MIMO system can be aroused.

Numerous works have studied the achievable spectral efficiency (SE) with different precoding or detection schemes under assumption that the channel model is a general Rayleigh fading. For example, the work of [2] obtained the lower bounds for the uplink achievable rate under perfect or imperfect channel state information (CSI) for Rayleigh fading channel. Authors in [4] extended these results for Ricean channels with an arbitrary-rank deterministic component. For the downlink transmission,the work in [5] analyzed the total achievable rate of downlink massive MIMO systems with MRT and zero force (ZF) precoding, but did not give the exact expression. In [6], authors performed the total achievable SE of massive MIMO systems with MRT and ZF precoding.However, those simplified formulas for the total achievable SE were presented in the high and low-signal-to-noise ratios (SNR) regimes.From these previous works, no works was to establish a comprehensive analysis on the achievable ergodic SE and EE with MRT pre-coding and an arbitrary finite antenna number.

We aim to analyze the downlink achievable SE and EE of systems, where the BS is deployed arbitrary finite antenna number. An exact analytical result for the achievable SE by considering MRT precoding is obtained.Furthermore, our theoretical results showcase that the increasing number of BS antennas can boost the achievable SE of system, whilst the achievable SE tends to a saturated rate in the large SNR case. Furthermore, an important observation is that the increasing number of users is beneficial for the achievable EE and one can find out an optimal antenna number to maximize the EE of multiuser MIMO system.

The authors obtained an exact expression for the achievable SE,which is very tight for the entire SNR regime with arbitrary finite antenna number.The authors further studied the EE of system by considering a practical energy consumption model in this paper.

II. SYSTEM MODEL

Considering the downlink communication of multiuser MIMO, in which one BS is deployedtransmit antennas and located in the center of a circle cell,users are randomly distributed within a circle cell (i.e.Whentransmit antennas serves simultaneously withusers, for userk, the received signal can be modeled as

Note that the expectation of the achievable SE is carried over all Rayleigh fading channel realizations, which makes sure the achievable SE to be ergodic. Therefore, the total achievable ergodic SE of the multiuser MIMO system is given by

Herein, we assume that the BS has perfect CSI which can potentially be obtained, e.g.,through exploiting uplink channel feedback in frequency division duplexing systems or channel reciprocity in time division duplex systems. A basic MRT processing is adopted by the BS, i.e., we choose. Compared with other linear precoding technologies, for example ZF and linear minimum mean square error (LMMSE), MRT precoding has low computational complexity and also it does not involve matrix inverse calculations.Moreover, MRT precoding is the most attractive technology for massive multiuser MIMO systems since the different channel links among users become orthogonal to each other when the antenna number at the BS goes to infinity.

III. THE PERFORMANCE ANALYSIS

We obtain a tractable expression of the achievable ergodic SE in order to further analyze the EE of MIMO system. Based on these analytical results, we examine the impact of several system physical insights, such as SNR,the number of BS antennas and users, on the achievable ergodic SE and EE of system.

3.1 Achievable ergodic SE

Theorem 1:For independent and identically distributed (i.i.d.) Rayleigh fast fading channel under perfect CSI at the BS, an accurate closed-form for the achievable ergodic SE with MRT precoding is given as

Proof:According to the result of (2), for userk, the achievable ergodic SE can be rewritten as

which can be further evaluated to

which can be factorized as

According to the above results, we know thatis a sum ofindependent not necessarily identically distributed gamma, where the shape parameter equals toand oneand scale parameter equals to one.With the help of the results in [8], the p.d.f. ofis given by

With a similar method, we know thatis a sum ofindependent and identically distributed gamma, where both of the shape parameter and the scale parameter equal to onerespectively. The p.d.f. ofis given by

To this end, we have obtained the p.d.f.s ofand. And then we will exploit the p.d.f.s in (9) and (10) to evaluateandterms,which can be expressed, respectively, as

and

With the help of the following identity in[9]

We obtain

and

Substituting (14) and (15) into (6), and some basic manipulations, derives the analytical result.

Theorem1reveals thatin (4) depends on the antenna number, user number and SNR.It is worth noting that the analytical expression in (4) involves a special function of exponential integral, we cannot directly observe the influence of these physical parameters on the achievable ergodic SE of the system. Next, we present a special case of the achievable ergodic SE asapproaches infinity.

Corollary 1:When the SNR at the BS goes to infinite, the achievable ergodic SE in (4)reduces to

Proof:Before the proof, we start by factorizing (4) as

The proof starts from the formula in [11,Eq. (8.352.5)], which states that

It is interesting to observe fromCorollary1 that as the SNR grows without bound,becomes a constant, whose value depends on both the number of BS antennas and users.We know that fixed the number of users and under the high SNR regime, the achievable ergodic SE increases withsince this makes channel vectors asymptotically orthogonal. On the other hand, with fixed antenna number,decreases with the users number increase indicating more degradation due to interference from more users. This is due to users mutual interference becomes significant.

Corollary 2:For the special case ofbut with fixedand SNR, the downlink achievable ergodic SE becomes a logarithmic function withwhich is reduced to

Proof: Before the proof, we recall the result in (17). By applying the results in [12], we obtain the approximation expression

By substituting this result into (17) and performing by some basic algebraic manipulations, we draw the conclusion.

FromCorollary2, it is interesting to notice that the achievable ergodic SE is a monotonically increasing logarithmic function of.This observation implies that massive antenna arrays deployed at the BS plays an important role in multiuser MIMO systems and contribute substantially towards improving the achievable SE, which verifies the intuitive result in [3].

3.2 Achievable EE

The EE is a significant performance criterion of communication systems. The achievable EE is defined by the proportion of the total achievable SE to the total power consumption,which can be established as

According to an energy consumption model in [10], the total power consumption can be categorized as follows: (a) radio frequency(RF) power at the BSwhich has a reciprocal relationship with the drain efficiency of power amplifier, (b) internal non-RF power expended by each antenna(c) the power required by the circuit components of each user(d) the power consumption contributing to operating precoding(e) the basic power consumption at BS. Thus, the total power consumption of system is given by

Corollary 3:The achievable EE function is strictly increases withand is a quasi-concave withMoreover, a unique global optimalalways exists.

Proof:Substituting (3) and (24) into (23),and dividing the denominator and numerator by, we have

From (25), we observe that the evaluation of the denominator almost remains unchanged due to the number of items stays the same.However, the evaluation of the numerator deceases withincreases, which leads tofunction is strictly increases with. Similarly,dividing the denominator and numerator bywe have

Fig. 1 Simulated and analytical results for total achievable SE versus SNR for and 256

Fig. 2 Simulated and analytical results for total achievable SE versus for different SNR cases

Corollary3 reveals important trends of the EE onand. We have an explicit guideline that the EE of multiuser MIMO system increases withand existing an optimumguarantees the EE maximization.

IV. NUMERICAL SIMULATION

We provide Monte-Carlo simulation to validate our derived analytical results in this section. In our simulation, the available channel bandwidthB=200 kHz, transmission powerWatt and power amplifier efficiency= 1/0.39, power consumption of each antennaWatt, power consumption of each userWatt, basic power consumption=20 Watt. All simulation results were obtained by averaging over 100,000 independent channel realizations.

Fig. 1 shows the analytical results for the total achievable SE and the numerical simulation results various SNR. We can see that the curves of theoretical results and Monte-Carlo simulation have a perfect match with the entire SNR case, which efficiency validates the analytical results inTheorem1. Furthermore, the total achievable SE tends to a saturated constant value in the larger SNR case, which is validated the analytical analysis inCorollary1. For the reason that as the SNR increase,interference of MIMO system from inter-user will also grows. In addition, we compare the total achievable SE for different numbers of BS antennas. As expected, massive antennas MIMO system is desirable for the reason that the BS equipped with large scale antenna can provide enough antenna array gain.

Fig. 2 depicts the analytical results for the total achievable SE and the numerical simulation results versus the antenna number. We observe that the total achievable SE increases the number of BS antennas, which implies that a large scale antenna number significant benefits the total achievable SE. We compare the total achievable SE for different number of users case, i.e.= 5 and 10, respectively.We find that the increasing speed of the total achievable SE with a larger number of users is much faster than for smaller numbers, which implies that the achievable SE of system can be improved via serving more users.

Fig.3 depicts the total achievable EE versusIt can be seen that for given number of users, the achievable EE first increases and then decreases whenvaries from 10 to 350. As expected, the curves are in agreement with the analytical result inCorollary3, which implies that there exists a rational antenna number to achieve an optimal EE of MIMO system.Moreover, we observe that for a great number of users, the optimal antenna number required the BS also becomes huge for the reason that it is necessary to use massive array antennas to compensate for the increased user interference.In addition, we compare the total achievable EE for different user numbers. As expected,fixed antenna number, the total achievable EE increases as the number of users. This is because more users benefit the total achievable SE under a larger number of BS antennas at the BS.

V. CONCLUSIONS

In this paper, we have studied the achievable SE and EE of downlink multiuser MIMO system with MRT precoding. We have obtained an exact expression for the achievable SE,which is very tight for the entire SNR regime with arbitrary finite antenna number. According to this analytical result, we further studied the EE of system by considering a practical energy consumption model. Our results indicated that the total achievable SE tends to a saturated rate in the large SNR case, while it increases logarithmically with the antenna number. Furthermore, analytical results also indicated that the achievable EE increases with user number and there is an optimal antenna number to maximize the EE of systems.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China under Grants 61531011 and 61450110445, the International Science and Technology Cooperation Program of China under Grant 2014DFT10300 and China Scholarship Council.

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