# Performance Analysis of Massive MIMO for Cell-Boundary Users

## Abstract

In this paper, we consider massive multiple-input multiple-output (MIMO) systems for both downlink and uplink scenarios, where three radio units (RUs) connected via one digital unit (DU) support multiple user equipments (UEs) at the cell-boundary through the same radio resource, i.e., the same time-frequency slot. For downlink transmitter options, the study considers zero-forcing (ZF) and maximum ratio transmission (MRT), while for uplink receiver options it considers ZF and maximum ratio combining (MRC). For the sum rate of each of these, we derive simple closed-form formulas. In the simple but practically relevant case where uniform power is allocated to all downlink data streams, we observe that, for the downlink, vector normalization is better for ZF while matrix normalization is better for MRT. For a given antenna and user configuration, we also derive analytically the signal-to-noise-ratio (SNR) level below which MRC should be used instead of ZF. Numerical simulations confirm our analytical results.

Minor Revision

## 1Introduction

Multiple-input multiple-output (MIMO) wireless communication techniques have evolved from single-user MIMO (SU-MIMO) to multi-user MIMO (MU-MIMO) systems [1]. To approach the capacity of the MIMO broadcast channel, [2] proposed simple zero-forcing (ZF)-based linear algorithms, where the transmitter and the receivers are equipped with multiple antennas. The authors in [4] intensively investigated the optimality of the linear matched type-combining filter and they assumed an infinite number of antennas at the receiver. [4] proved that a simple linear beamforming (coordinated beamforming in the paper) asymptotically approaches the sum capacity achieved by dirty paper coding (DPC).

Recently, massive MIMO (a.k.a. large-scale MIMO) has been proposed to further maximize network capacity and to conserve energy [5]. The authors in [6] proposed massive MIMO systems that use simple linear algorithms such as maximum ratio combining (MRC) for the uplink and maximum ratio transmission (MRT) for the downlink. To further increase sum rate performances, several network MIMO algorithms with multiple receive antennas have been proposed [8]. These systems, however, assume that the network supports a maximum of three users through a relatively small number of transmit antennas.^{1}

Massive MIMO systems in multi-cell environments have also been studied in [5]. Multi-cell massive MIMO is prone to some critical issues. These include pilot contamination, which becomes, in time division duplex (TDD) systems, the main capacity-limiting factor, especially when MRT is used. Joint spatial division and multiplexing is proposed in [12] to employ frequency division duplex (FDD) systems. The authors in [13] proposed a pilot alignment algorithm for a cognitive massive MIMO system. The authors in [7] investigated downlink performance with MRT and ZF precoder for a massive MIMO system.^{2}

The assumption of an infinite number of antennas at the BS for a finite number of users somehow trivializes many problems (e.g., under this limit MRC/MRT have the same performance as ZF). A more meaningful system scaling is considered in [10], where the number of antennas per BS and the number of users both go to infinity with a fixed ratio. In this case, the results of the infinite number of BS antennas per user can be recovered by letting this ratio become large. This more refined analysis, however, illuminates all the system performance regimes. For example, in [16] the “massive MIMO” regime is defined as the regime where pilot contamination dominates with respect to multi-user interference; it is observed that this regime occurs only when the number of BS antennas per user is impractically large. These conclusions are also reached, independently and in parallel, in [10]. In particular, [10] considers a multi-cell architecture formed by small clusters of cooperating BSs. The authors proposed a system where the users are partitioned into homogeneous classes and the downlink MIMO precoding scheme is optimized for each class. Then, a scheduler optimally allocates the time-frequency transmission resource across the different user classes, yielding an inherent multi-mode multi-cell massive MIMO system. One of the main findings of [10] is that it is convenient to serve users at the cell center in a single-cell mode, while users at the cell edge should be served by small cooperative clusters formed by the closest three neighboring cell/sectors.

This paper, motivated by the results in [10], focuses on the cell-edge user performance-the system bottleneck for both the uplink and the downlink. The massive MIMO system under consideration consists of multiple radio units (RUs) connected to one another by optical fibers, and further connected to a centralized digital unit (DU), as illustrated in Figs. ?(a) and ?. Through the optical fibers, each RU can share data messages and channel state information. With respect to all three neighboring BSs forming a cluster, the cell-edge users have symmetric and spatially isotropic channel statistics. Hence, the system is equivalent to, as shown in Fig. (b), a single-cell massive MIMO system with cell-edge users that are located in the low signal-to-noise-ratio (SNR) regime (for the high SNR regime, we provide some results to provide a detailed analysis, such as for determining which normalization method is better for MRT). As in the prior work referenced above, we consider the performance of linear precoding/filtering schemes such as ZF and MRT for the downlink and ZF and MRC for the uplink. In particular, we consider two possible normalizations of the precoding filters for the downlink, referred to as vector and, respectively matrix normalization. The main contributions of this paper are as follows:

Tighter ergodic achievable sum rate of ZF and MRT/MRC: Tighter ergodic achievable sum rate of ZF and MRT/MRC: We provide a methodology of simple but quite accurate approximations while considering the property ofnear deterministc, which is defined in Section . This approximation method is valid for massive MIMO systems at low/high SNR. The effective channel matrix tends to an identity matrix when MRT/MRC are assumed with perfect CSI in massive MIMO systems. This means that the random channels becomenear deterministicdue to the property of the law of large numbers. If the number of user is large, the sum of the interference powers cannot becomenear deterministic, which means the sum of those still have randomness and do not converge to zero. The researchers in [5] concluded that with perfect CSI at the BS and a large , the performance of a MU-MIMO system with a transmit power per user scaling with is equal to the performance of a single-input-single-output system. Assuming a large number of users, the conclusion in [5] does not hold because the sum of the interference powers does not converge to zero. From this aspect, considering the condition ofnear deterministicand the large number of users is mathematically significant in analyzing the performance of massive MIMO systems. We investigate whether the signal-to-interference-plus-noise-ratio (SINR) term in the ergodic sum rate is able to approximatenear deterministicsuch as the form of the expectation when goes to infinity in the low or high SNR regime. From this approximation of the SINR term, we derive simple approximations for the ergodic achievable sum rate of ZF and MRT/MRC at the low and high SNR regimes considering two simple power normalization methods at the downlink.The derived approximations are accurate and far simpler to evaluate than the

asymptotic expressionsgiven in [10], which were obtained through asymptotic random matrix theory [17] and usually given in terms of the solution of multiple coupled fixed-point equations. Thanks to their simplicity, the proposed approximations are suitable to analyze the low and high SNR regimes. From the derived approximations, we investigate a suitable normalization method of MRT and transceiver mode selection algorithms. In their asymptotic expressions, the authors in [7] did not consider normalization methods; thus the expressions were unable to classify which normalization method was better. Numerical results demonstrate the tightness of our analysis.^{3}

Downlink precoding normalization methods: We compare matrix and vector normalization for downlink precoding, in the simple and practical case of uniform power allocation over all downlink streams. It is well known that these normalizations, with optimal (waterfilling) power allocation and ZF precoding yield identical results [18]. However, with practical suboptimal power allocation and in the finite antenna regime these normalizations are generally not equivalent. Most prior work on multi-user MIMO paid scant attention to this issue. For example, the authors in [19], considered matrix normalization and those in [20] considered vector normalization in [20] with neither making any mention of why different normalizations were used; [19] is part I of their papers dealing with linear precoding (ZF/MMSE), and [20] is part II focusing on non-linear precoding (vector perturbation). Thus, the comparison between vector and matrix normalization remains an open problem. To solve this issue, we find a suitable normalization method for MRT precoding by using the proposed approximations and a suitable normalization method for ZF precoding by using the arithmetic-geometric inequality.

Transceiver mode selection algorithms: We propose two transceiver mode selection algorithms from transmit power and the number of active users perspectives. In [21], it is concluded that ZF is better for cell center users, i.e., high SNR, and MRT is better for cell-boundary users, i.e., low SNR, in a downlink system. However, [21] did not consider transceiver mode selection as a function of SNR (i.e., of the transmit power, for a given pathloss law and cell geometry) for the same class of edge users. In this paper, we explain how much transmit power and/or number of active users are needed for ZF to provide a better sum rate than MRT (for downlink) or MRC (for uplink). In particular, we find the optimal MIMO mode selection scheme in terms of closed-form thresholds of the transmit power, where the thresholds depend on the number of edge users.

Note that since our system model is simplified by making assumptions, our analytical results are more accurate than the work given demonstrated in [10]. Furthermore, our-closed form expressions are based on the assumption of infinite , and when goes to infinity the approximation is quite accurate. Even when is finite, though still a large number, the approximations are quite accurate. In prior work on massive MIMO with linear receivers/precoders, reserchers have provided the simple lower bounds of the sum rate performance [5], or the complex closed-form expressions of that [16], which are less accurate than our analysis. We anticipate our contributions to yield a wide range of insights for related studies, such as those on performance analysis on MIMO, power normalization methods, and the trade-off between ZF and MRT/MRC.

This paper is organized as follows. In Section 2, we introduce the considered system model and problem statement with respect to precoding normalization methods and beamforming techniques. In Section 3, we introduce some mathematical motivations and preliminaries useful for analysis. In Section 4, we analyze i) the ergodic performance of ZF and MRT precoding, ii) which precoding normalization method is better for each precoder, and iii) the ergodic performance for cell-boundary users with the best normalization method. In Section , we provide an approximation of the achievable ergodic uplink sum rate. In Section 6, we propose transceiver mode selection algorithms with i) a power threshold and ii) the number-of-users cross point of ZF- and MRT-precoding techniques. Numerical results are shown in Section 7. Section 8 presents our conclusions and future work.

## 2System Model: Massive MIMO

Consider a massive MIMO system as shown in Figs. and . One DU (cloud BS) controls three RUs and users. Each RU is connected with one another by optical fibers. Fig. (a) shows that the cloud BS provides a massive MIMO environment to cell-edge users under the assumption that RU has a relatively small number of antennas (more practical in the recent antenna configuration; we will use this system model in Section 7). Fig. (b) illustrates the equivalent model of Fig. (a) considered as single-cell massive MIMO systems with cell edge users. We assume that the cloud BS has antennas and each user equipment (UE) is equipped with one antenna. In this paper, we do not consider pilot contamination and assume perfect CSI at the RU. We also assume that the channel is flat fading and the elements of a channel matrix are modeled as independent complex Gaussian random variables with zero mean and unit variance. The channel between the cloud BS (one DU and three RUs) and the -th user is denoted by an row vector (). A channel matrix between the cloud BS and all UEs consists of channel vectors . Let denote the column vector of transmit precoding and represent the transmit symbol for the -th UE at downlink. Similarly, let denote the column vector of receive combining filter for the -th UE at uplink. Also, let be the additive white Gaussian noise vector. Then, the received signal at the -th UE is expressed by

where, denotes the total network transmit power across three RUs. Also, the received signal for the -th UE at the cloud BS is expressed by

where, and denote the transmit power per each user and the transmit symbol of the -th user at uplink, respectively.

### 2.1Downlink

Eq. (Equation 1) contains the desired signal, interference, and noise terms. To eliminate the interference term, we use the following precoding:

where is a precoding matrix consisting of each column vector .

To satisfy the power constraint, we consider two methods, i.e., vector/matrix normalizations. The normalized transmit beamforming vectors (columns of a precoding matrix) with vector/matrix normalizations are given as and , respectively. Note that vector normalization imposes equal power per downlink stream, while matrix normalization yields streams with different power. In this paper, to simplify, we do not consider a power optimization that could yield a complexity problem in very large array antenna systems.

#### ZF/MRT with vector normalization

The received signal at the -th UE can be expressed as follows:

#### ZF/MRT with matrix normalization

Similarly, we can rewrite the received signal with matrix normalization as such:

### 2.2Uplink

Similar to the downlink system, to eliminate the interference term, and to maximize the SNR in (Equation 2), we use the following combining filter at the RUs:

where is a combining filter matrix consisting of each column vector . Here, we do not consider normalization since it does not change SNR values in the uplink scenario.

### 2.3Summary of the Main Results

We first summarize the main results based on our analyses. These results are mathematically motivated from some random matrix theorems, shown in Section 3.

#### Asymptotic downlink sum rate

In Section 4, we derive the simple and tight approximations of the ergodic achievable sum rate of ZF and MRT. We also find that ZF with vector normalization is better than ZF with matrix normalization while MRT with matrix normalization is better than MRT with vector normalization. The proposed ergodic achievable sum rates of ZF with vector normalization at low SNR and MRT with matrix normalization at low SNR are given by

#### Asymptotic uplink sum rate

We also evaluate the approximations of the ergodic achievable sum rate of ZF and MRC in the uplink case. The proposed ergodic achievable sum rates of ZF at low SNR and MRC at low SNR are given by

#### Transceiver mode selection algorithm

In Section 6, we propose two transceiver mode selection algorithms from i) a transmit power and ii) the number of active users perspectives. We explain how much transmit power and/or the number of active users are needed for ZF to provide a better sum rate than MRT/MRC. The thresholds are given in *Lemma*s ?- ?.

## 3Mathematical Motivations and Preliminaries

In this section, we introduce some mathematical motivations and preliminaries to evaluate an asymptotic analysis for network massive MIMO, which will be used in Sections Section 4 and Section 5.

### 3.1Achievable Rate Bound

In this paper, to maximize the achievable sum rate of downlink/uplink systems, we evaluate the closed forms of each system’s performance. The achievable rates are bounded as follows:

by using *Jensen’s Inequality* of convex and concave functions where , , and represent signal power, interference power, and noise power, respectively. Note that we use these bounds only for ZF cases and show that our results, based on the bounds, are more accurate than prior work [5], [7], [16].

### 3.2Expectation and Variance of Random Vectors

### 3.3Effective Channel

Generally, a transceiver uses MRT or MRC in massive MIMO, which means that the effective channel of the desired signal becomes one and the interference signal becomes zero as the number of antennas () goes to infinity, as illustrated:

### 3.4Signal and Interference Power

Note that the terms and do not converge to and zero, respectively, as goes to infinity since their variance does not go to zero, which means that they still have randomness.

### 3.5Chebyshev’s Inequality

Let be a random variable with variance , and be scalars, and be a random variable with variance , respectively. If is not very large in comparison with , but , then is *near deterministic*, i.e., . For fixed , with high probability. We use () to approximate a SINR term in ergodic sum rate expressions.

As an example, to analyze the rate in the high SNR regime, if converges to zero, then converges to .

### 3.6Ergodic Achievable Sum Rate of Massive MIMO Systems at the low/high SNR regime

### 3.7Arithmetic-Geometric Inequality

This is a simple application of *Jensen’s Inequality*.

## 4Asymptotic Downlink Sum Rate for Cell-boundary Users

In this section, we derive the achievable rate bounds, and show which normalization method is suitable for ZF- and MRT-type precoding at the downlink. We assume that the cell-boundary users are in the low SNR regime () and that the cell center users are in the high SNR regime (); we also assume that the desired power term or the interference power term becomes *near deterministic* in the low/high SNR regime. Based on our analytical results, we will also show which precoding technique is desired for cell-boundary users.

### 4.1Ergodic Performance

#### Achievable rate bounds for ZF precoding

i) Lower bounds: The lower bound of the ergodic sum rate for the ZF precoding is well known, as follows [7]:

using the property of Wishart matrices [17].

ii) Vector normalization-upper bound: From (Equation 3), we can derive the SINR of the upper bound of vector normalization in the ZF case, as given by

where results from the diversity order of ZF , as shown in [23]. From (Equation 5), the upper bound of vector normalization in the ZF case can be represented as

iii) Matrix normalization-upper bound: From (Equation 4), the SINR of the upper bound of matrix normalization in the ZF case can be expressed as

From (Equation 6), the first upper bound (with an expectation form) of matrix normalization in the ZF case can be represented as

By using *Lemma* ?, (Equation 7) can further be expressed as

So the second upper bound (without an expectation form) of matrix normalization in the ZF case can be given by

#### Achievable rate for MRT precoding

i) Vector normalization-low SNR regime: From (Equation 3), we can derive the ergodic achievable sum rate of vector normalization in low SNR as follows:

ii) Vector normalization-high SNR regime: Similarly, we can get the ergodic achievable sum rate of vector normalization in high SNR as follows:

iii) Matrix normalization-low/high SNR regime: From (Equation 4), we can evaluate the ergodic achievable sum rate of matrix normalization in low/high SNR by using the following formation:

### 4.2Comparison between Vector and Matrix Normalizations

#### Performance comparison of ZF

To find which normalization technique is better in ZF, we let in *Lemma* ? directly.

where and . From (), since the desired power and the interference power are one and zero respectively, the power normalization per user only affects the performance of ZF. From the perspective of *Jesen’s Inequality*, the sum rate with the different power allocation per user is the upper bound of the sum rate with the same power allocation per user at the instant channel and arbitrary SNR. We can conclude that, in the ZF case, vector normalization is always better than matrix normalization.

#### Performance comparison of MRT

From (Equation 9)-(Equation 11), a comparison of the ergodic achievable sum rate is given by

at low and high SNR, respectively. It is well known that approach methods that maximize the desired signal are better than those that mitigate the interference power at low SNR. The effective desired channel gain can be maximized with MRT. The desired power per user scales down in proportion to its power in the vector normalization while the desired power per user scales down in the same proportion over all users. This means that the gains of better channels with vector normalization tend to scale down more than the gains of better channels with matrix normalization. This yields the difference of the desired power term of () and () at low SNR. Therefore, we confirm that, for MRT precoding, matrix normalization is always better than vector normalization at low SNR. We conclude, however, that there is marginal performance gap between vector normalization and matrix normalization at high SNR.

### 4.3Ergodic Achievable Sum Rate for Cell-boundary Users with the Best Normalization Method

As explained in Section , we conclude that the suitable normalization methods are vector normalization for ZF and matrix normalization for MRT. We assume that the transmit power () is small for cell-boundary users (low SNR regime). Using the property of ZF precoding, the ergodic achievable sum rate of ZF is represented as where () results from ( ?) with . Eq. ( ?) indicates that the achievable sum rate of ZF precoding can approach its upper bounds at low SNR by ( ?). Thus, the ergodic achievable sum rate of ZF with vector normalization at low SNR is given by From (Equation 11), we find the ergodic achievable sum rate of matrix normalization in low SNR:

## 5Asymptotic Uplink Sum Rate for Cell-Boundary Users

We have focused on a downlink scenario with a sum power constraint. In this section, we investigate an uplink case, where each user has its own power constraint. From (Equation 2), the ergodic achievable sum rate for the uplink, , is

From (Equation 14), we can derive the ergodic achievable uplink sum rate of MRC, , as follows:

We approximate the ergodic achievable sum rate of MRC as follows:

i) High SNR regime:

ii) Low SNR regime:

Similar to ZF precoding, the ergodic sum rate for ZF for uplink at low SNR, , is

## 6Transceiver Mode Selection

### 6.1Algorithm

In this section, we propose two transceiver mode selection algorithms from i) transmit power and ii) the number of active users perspectives. To provide a mathematically simple solution, we first introduce *Lemma* ? and *Lemma* use a power threshold as follows:

Lemma

? helps the RUs select one of the precoders, i.e., ZF or MRT, with respect to the transmit power of the cloud BS. Also, the power policy of the cloud BS could be adjusted by the power threshold that is a function of and . Therefore, the RUs could find a suitable precoding mode according to the user’s location. Similarly, *Lemma* ? could be applied to the uplink case.

The proposed power threshold would be affected by a specific number of users, so a power cross point, that refers to , exists where MRT or MRC is always better for any number of active users. Since *Lemma*s and are monotonic increasing functions of (), and have minimum values at . These points become and .

Now we investigate , which is a transceiver mode selection threshold when the transmit power at the transceiver is larger than and when the number of active users varies.

Lemma

s ?- ? provide a proper solution for the low SNR regime like a cell-boundary. For example, if the users have very low SNR, which means or is always lower than or , the cloud BS should use MRT or MRC to increase a sum rate. Also, the cloud BS should use MRT or MRC for users having transmit power larger than (especially in the low SNR regime) when the number of active users is larger than .

### 6.2Performance Comparison for a Large Number of Users Case

We derive the ergodic sum rate of ZF with vector normalization at low SNR (or upper bound of ZF with vector normalization) when

This is equal to the result in [19]. From this result, we are able to gather insights into user scheduling. We also derive the ergodic sum rate of MRT with any normalization and the ergodic sum rate of MRC when In the special case of MRC, if the transmit power per user is scaling down with from the sum power constraint , i.e., , then From () and (), we conclude that the performance of MRT and MRC is bounded by when , even SNR goes to infinity. This differs from the result in [5] () because the authors in [5] did not consider the case of a large number of users.

Next, at in downlink, we check the difference of the gradient between the rates of ZF and MRT. If the gradient of the rate of ZF is larger than that of MRT, the rate of ZF with vector normalization is larger than that of MRT when . In the other case, the rate of MRT with matrix normalization is larger than that of ZF when . The difference of the gradient between the rates of ZF and MRT is expressed as

where denotes the gradient of the curve at . Similarly, is the gradient of the curve at . In general, cell-boundary users have relatively low SNR and, as we assumed, the cloud BS has large-scale antennas, meaning is much larger than . Therefore, if exists, (Equation 19) is always positive. We also confirm this through numerical comparisons as shown in Fig. ?. From this observation, we realize that MRT precoding is suitable for cell-boundary users if the number of active users is larger than .

## 7Numerical results

For numerical comparisons, we assume that each RU has eight transmit antennas; thus the cloud BS has a total of 24 antennas. *Note that any number of antennas can be used and this constraint is not really related to our system.* This assumption is based on the parameters of 3GPP LTE-advanced; Release 10 supports eight Node B antennas [24].

Fig. (a) shows the achievable sum rate of ZF for downlink at low SNR. We compare the simulation results with their theoretical upper bound. As mentioned in Section 4, the ergodic achievable sum rate of ZF with vector normalization approaches its upper bound at low SNR while that of ZF with matrix normalization approaches its first upper bound. Note that the *first upper bound* is obtained by Monte Carlo’s simulation because is unknown. Fig. (b) also describes the achievable sum rate of MRT with ( dB) total SNR. Our achievable sum rate result is almost the same as the numerical results for vector normalization. There is, however, a gap between the simulation results and the proposed achievable sum rate for matrix normalization. This is because the proposed achievable sum rate form is accurate when SNR is lower than . Our analysis is more accurate than the closed forms in [10], which are the same as the lower bound of ZF. In addition, our expression of ergodic sum rate is more accurate than the closed forms in [10] that are given by and these could be the lower bounds of our expression. From this comparison, we could confirm (as was also shown in Section 4) that ZF with vector normalization is better. In contrast, MRT with matrix normalization is better at getting an improved sum rate performance at low SNR.

Figs. (a) and (b) show that the results from (Equation 16) and (Equation 17) are approximately the same as the ergodic achievable uplink sum rate of MRC where and , respectively. Note the large gap between the ergodic achievable uplink sum rate and the lower bound of MRC with finite shown in [5]. The legend, *Simulation*, indicates the ergodic achievable uplink sum rate of MRC (Equation 15) while the proposed analysis in Figs. (a) and (b) indicates the approximation shown in (Equation 16) and (Equation 17), respectively. Note that the lower bound in [5] is given by and the closed form in [16] is given by In addition, the sum rate approaches to when at high SNR, and the sum rate approaches to when at low SNR. This shows that the results of () and () could hold even is finite.

In Fig. (a), we illustrate the achievable sum rate of MRT precoding and ZF precoding at downlink. Fig. (b) illustrates the achievable sum rate of MRC and ZF at uplink ( = dB and dB are calculated by *Lemma*s ? and ? with and at downlink and uplink, respectively). Note that a cross point of MRT (or MRC) curve and ZF curve is the power threshold. It is well known that MRT/MRC performs better than ZF at low SNR. This is the same in the massive MIMO systems, and we can easily find a borderline between MRT/MRC region and ZF region with the proposed simple equation. Figs. (a) and (b) show that MRT/MRC is always better than ZF at very low SNR regardless of . Used for simulations were dB at downlink and dB at uplink, and . This result verifies *Lemma* ?. We summarize also our conclusions in Tables ?, ?, and ?.

In Fig. (a), we also compare the achievable sum rates of ZF precoding with MRT precoding when the total transmit SNR is dB. It shows that ZF with vector normalization is better while MRT with matrix normalization is better for achieving higher sum rates. Fig. (b) illustrates the achievable sum rates of ZF- and MRT-precoding with dB transmit SNR. The result is similar to that found in Fig. (a). As mentioned in Section 6, in the low SNR regime, using MRT precoding is generally better when the number of active users is larger than . Also, we realize through Figs. (a) and (b) that as SNR decreases, shifts to the left. This means that the cloud BS could determine a precoding by at low SNR.

The performance of MRT/MRC increases as increases while the performance of ZF decreases as increases. This is because the ergodic sum rate of ZF at goes to a very small constant at downlink, as shown in (Equation 18). Similarly, at uplink, the ergodic sum rate of ZF at could also close to a very small value since its lower bound is zero.

Figs. (a) and (b) show that the achievable sum rate of MRT/MRC at low SNR. To predict the tightness of our analysis for infinite , we compare the simulation result of our analysis with that of the closed form in [16] and the lower bound in [7] and [14] in the downlink scenario with a large number of antennas at the BS (up to ). Similarly, we compare the simulation result of our analysis with that of the closed form in [16] and the lower bound in [5] in the uplink scenario. Both numerical results of MRT/MRC show that, when goes to infinity, our analyses are quite tighter than the work given in [5]. Note that the closed form in [16] is very tight when but as increases not as tight as our analysis in the low SNR regime and the uplink scenario. The computation complexity is the same as the lower bounds in [5] but the accuracy is better than the closed form in [16]. Therefore, we conclude that the proposed ergodic sum rate is tighter and simpler than the previous work.

Precoding normalization technique | |
---|---|

Vector normalization Matrix normalization | |

Matrix normalization Vector normalization | |

Downlink | ||

Uplink | ||

Precoding technique | ||
---|---|---|

Cell-center | Large | Zero-forcing |

Cell-boundary | Small | MRT/MRC |

## 8Conclusions

In this paper, we proposed massive MIMO systems supporting multiple cell-boundary UEs. For precoding designs, we first derived the achievable sum rate bounds of zero-forcing (ZF) and the approximation of the ergodic achievable sum rate of maximum ratio transmission (MRT) with vector/matrix normalization. Through analytical and numerical results, we confirmed that vector normalization is better for ZF and that matrix normalization is better for MRT. We also investigated the optimal mode-switching point as functions of the power threshold and the number of active users in a network. According to the mathematical and numerical results the BS can select a transceiver mode to increase the sum rate for both downlink and uplink scenarios. We anticipate our analysis providing insights for related studies, such as those on performance analysis of MIMO, power normalization methods, and the trade-off between ZF and MRT/MRC. In future work, we will consider more practical scenarios including limited cooperation among RUs and cooperation delay.

### @.1Proof of Lemma 1

Let be the -th element of . Since is an i.i.d. complex Gaussian random variable with zero mean and unit variance, i.e., , is a Gamma random variable with unit shape parameter and unit scale parameter, i.e., from the relationship between Rayleigh distribution and Gamma distribution. Therefore, we can say that is an exponential random variable with unit parameter (), i.e., by the property of Gamma distribution and exponential distribution. Since the -th moment of the exponential random variable is , we can obtain , , and .

#### Proof of Lemma 1. 3)

The expectation of is given by where . The notation denotes a permutation operator. The variance of is also derived by The variance terms and the covariance terms in ( ?) are calculated by

where .

#### Proof of Lemma 1. 4)

The expectation of is

where . The variance of is also expressed as where