Massive MIMO have drawn considerable attention as they enable significant capacity and coverage improvement in wireless cellular network. However, pilot contamination is a great challenge in massive MIMO systems. Under this circumstance, cooperation and three-dimensional (3D) MIMO are emerging technologies to eliminate the pilot contamination and to enhance the performance relative to the traditional interference-limited implementations. Motivated by this, we investigate the achievable sum rate performance of MIMO systems in the uplink employing cooperative base station (BS) and 3D MIMO systems. In our model, we consider the effects of both large-scale and small-scale fading, as well as the spatial correlation and indoor-to-outdoor high-rise propagation environment. In particular, we investigate the cooperative communication model based on 3D MIMO and propose a closed-form lower bound on the sum rate. Utilizing this bound, we pursue a “large-system” analysis and provide the asymptotic expression when the number of antennas at the BS grows large, and when the numbers of antennas at transceiver grow large with a fixed ratio. We demonstrate that the lower bound is very tight and becomes exact in the massive MIMO system limits. Finally, under the sum rate maximization condition, we derive the optimal number of UTs to be served.
Multiple-input multiple-output (MIMO) technology can provide a remarkable increase in data rate and reliability compared to the single-antenna system [
Currently, most of researches on MIMO systems are based on traditional two-dimensional (2D) MIMO channel model, which only involves the horizontal dimension while ignoring the effect of tilt angle in the vertical dimension. However, antenna tilt angle has a large impact on the performance of cooperative communication system as shown in [
BS cooperation is often referred to as multicell processing in which the BSs jointly decode the message based on the received signals (also known as coordinated multipoint (CoMP)). The capacity enhancement due to the BS cooperation has been extensively studied and has been shown to grow linearly with the number of BS receive antennas [
In this paper, we focus on the cooperating BSs which are interconnected through ideal links (e.g., optical fiber or high speed cable) to a central processor (CP), which has perfect channel state information (CSI). In detail, we investigate the sum rate performance of the 3D massive MIMO systems over BS cooperation in composite fading channels considering the effect of spatial correlation at the transmit side and I2O wireless propagation environment. In this scenario, we assume that the cell-edge UTs are uniformly distributed in a building with several floors and cell-center UTs are uniformly distributed in other areas of the cell. We note that the spatial nonstationary property and the near-field effect are the main properties for the realistic massive MIMO channel model which are interesting topics for additional research [ A cellular 3D MIMO uplink channel model is introduced, accommodating I2O high-rise propagation environments, Rayleigh lognormal fading model, distance dependent loss, and Kronecker correlated antennas. We derive a closed-form lower bound on the sum rate performance of the 3D MIMO ZF receivers. This bound does not involve complicated functions, and it can be computed fast and efficiently. With the help of the proposed lower bound, we analyze the asymptotic lower bound for 3D massive MIMO system under the cases that the average and total powers are fixed, respectively. The proposed bound applies to any finite number of antennas and remains relatively tight across the entire SNR and tilt angle ranges. Exploiting the results of (2), we also give a closed-form approximated solution for the number of UTs,
The rest of this paper is organized as follows. Section
In the following, we consider
System model of coordinated multicell 3D MIMO network (
Note that all UTs are classified into two categories: (i) cell-edge UTs, of which the index value is smaller than or equal to
The radiation pattern for the 3D antenna elements of the BS array follows the model proposed by 3GPP in [
Schematic illustration of 3D MIMO in single cell.
Further, we can obtain the composite antenna pattern in numerical value as follows:
It is noteworthy that adjusting antenna mechanical tilt angle requires a site visiting, which makes the adjustment process more expensive and time consuming. Therefore, all works in this paper assume a fixed mechanical tilt angle and only the electrical tilt angular variation is investigated.
Under the condition of BS cooperation, the aforementioned model in Section
The channel matrix
The elements of the diagonal matrix
We hereafter investigate the sum rate performance of the 3D MIMO ZF receivers under BS cooperation. Under this consideration, we assume that BSs have perfect CSI which can be obtained by channel reciprocity in TDD and feedback in FDD. The processing of the ZF receiving is formulized as
Clearly, the main challenge in analytically evaluating (
The main interesting point of this paper is to optimize
In the following, we derive a novel closed-form lower bound on the sum rate of 3D MIMO system under ZF receivers based on the result of [
The lower bound of the achievable sum rate performance of ZF receivers overcorrelated RLN 3D MIMO channels is given as follows:
Starting from (
For the
For shadowing fading, we use the fundamental property of a lognormal RV
By substituting (
For the uncorrelated RLN fading (
By substituting
It is easily seen that the sum rate monotonically grows with the mean of the lognormal shadowing, the antenna gain of both horizontal and vertical antenna radiation, and the number of BS antennas. However larger transceiver distance reduces the sum rate due to the increased path loss. Finally, we can observe that increasing the number of UTs or UT antennas do not always improve the system performance.
In this section, we investigate two issues: (i) the asymptotic sum rate for large transceiver antennas and (ii) the optimal number of UTs to achieve maximized sum rate.
By deploying large antenna arrays on BS, we can save the transceiver powers both UTs and BSs when the number of antennas grows large, while maintaining a given desired quality of service (QoS). In the following, we derive the asymptotic achievable sum rates for 3D massive MIMO system.
For 3D MIMO fading channels, the number of receive antennas tends to be unlimited with keeping a fixed
The proof follows by taking
For
The above corollary reveals that, when the number of receive antennas grows into infinity, the effects of small-scale fading and noise disappear. Furthermore, we can see that the asymptotic sum rate grows logarithmically with
For the 3D MIMO channels, as the numbers of BS and UT antennas grow to infinity with a fixed ratio
The proof follows, by substituting (
Clearly, the asymptotic lower bound increases linearly with the number of UTs,
For the 3D MIMO channels, as the number of receive antennas increases infinitely with
Substituting (
For the 3D MIMO channels, with the number of receive antennas growing to infinite with fixed
The proof is similar to Corollary
The Corollaries
In this section, the optimal number of UTs to maximize the system performance is investigated. From the former discussion, when the number of the BS antennas
For simplicity, we consider a simple scenario that UTs are uniformly distributed on a circle of radius
Consider the problem of finding the optimal number of UTs
To solve this problem, we begin with the following theorem which returns the approximated optimal solutions of (
Under the condition of the assumptions above, the optimal number of UTs
For the assumptions above and
By using (
To maximize (
The inequality (
For
Using the similar method to the case of
The objective function is concave with respect to
Denoting
Noticing the definition of
As anticipated, the sum rate of 3D MIMO systems increases with the number of BS antennas. Also, we can conclude that the optimal number of UT does not always increase with the number of the transmit antennas. Therefore, we can achieve the maximum sum rate with the ZF receivers from the optimal solution of (
In this section, we provide some numerical results to validate the accuracy of our analysis in a sectorized cell scenario. The multicell cellular system with
Value of parameter in simulation.
Parameter | Details | Value |
---|---|---|
|
Number of cells | 3 |
|
Radius of sector | 50 m |
|
Radius of building | 10 m |
|
Height of BS | 30 m |
|
Height of UT | 1.5 m |
|
Height of floor | 5 m |
|
Shadowing mean | 4 dB |
|
Shadowing standard deviation | 4 dB |
|
Path-loss exponent | 4 |
|
Number of floors | 3 |
|
Wall penetration loss | 0.01 (−20 dB) |
|
Inside loss | 0.5 |
|
Azimuth front-to-back ratio | 30 dB |
|
Horizontal HPBW | 70° |
|
Vertical HPBW | 7° |
|
Side lobe attenuation | 30 dB |
|
azimuth angles | 0°, 120°, −120° |
|
UT edge-cell to center-cell ratio | 1/3 |
|
Mechanical tilt angle | 0° |
We first investigate the tightness of the lower bound against the SNR in Figure
Simulate sum rate and lower bound against the SNR
Clearly, the bound remains sufficiently tight across the entire SNR range for all correlation coefficients. At high SNRs, the sum rate and the bounds increase approximately linearly with the SNRs. Moreover, the effect of correlation on the sum rate performance can be neglected at low SNR. From Figure
The lower bound versus the BS tilt angle is shown in Figure
Simulate sum rate and lower bound versus tilt angle (
In the following, the sum rate performance of the 3D massive MIMO system against the number of receive antennas for fixed
Analytical sum rate and lower bound versus the number of BS antennas (
Analytical sum rate and lower bound versus the number of BS antennas (
Finally, the optimal number of UTs
Analytical sum rate and lower bound versus the number of UTs (
From Figure
This paper investigates the sum rate of the 3D massive MIMO system over BS cooperation and BS antenna tilt angle. More importantly, we derived a novel and simple closed form for the lower bound of the sum rate. Based on the lower bound, we analyzed in detail the promising technology of the massive MIMO systems. In parallel, we also investigate the impacts of large-scale (log-normal shadowing fading, 3D antenna gain, distance path-loss, and I2O propagation loss) and small-scale fading, as well as the spatial correlation at the transmit side considering the cell-edge UTs and cell-center UTs. Besides, an expression is derived to obtain the approximate optimal number of UTs that maximizes the sum rate. This result has an important significance for the optimum UT configuration in practice. Although this model is simple, it can lead to easily following the derivations and quite insightful results on the benefits of MU-MIMO in a 3D MIMO and massive MIMO situations.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National 863 Project (no. 2014AA01A705), the Creative Research Groups of China (61121001), and National Science and Technology Major Project (no. 2013ZX03003009).