Wireless Communications and Mobile Computing

Volume 2017 (2017), Article ID 5709367, 7 pages

https://doi.org/10.1155/2017/5709367

## Diversity-Multiplexing-Nulling Trade-Off Analysis of Multiuser MIMO System for Intercell Interference Coordination

^{1}IT and Mobile Communication Division, Samsung Electronics, Soowon, Republic of Korea^{2}School of Electrical Engineering, Korea University, Seoul, Republic of Korea

Correspondence should be addressed to Chung G. Kang

Received 16 September 2017; Accepted 6 November 2017; Published 10 December 2017

Academic Editor: Mostafa Zaman Chowdhury

Copyright © 2017 Jinwoo Kim and Chung G. Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A fundamental performance trade-off of multicell multiuser multiple-input multiple-output (MU-MIMO) systems is explored for achieving intercell and intracell interference-free conditions. In particular, we analyze the three-dimensional diversity-multiplexing-nulling trade-off (DMNT) among the diversity order (i.e., the slope of the error performance curve), multiplexing order (i.e., the number of users that are simultaneously served by MU-MIMO), and nulling order (i.e., the number of users with zero interference in a victim cell). This trade-off quantifies the performance of MU-MIMO in terms of its diversity and multiplexing order, while nulling the intercell interference toward the victim cell in the neighbor. First, we design a precoding matrix to mitigate both intercell and intracell interference for a linear precoding-based MU-MIMO system. Then, the trade-off relationship is obtained by analyzing the distribution of the signal-to-noise ratio (SNR) at the user terminals. Furthermore, we demonstrate how DMNT can be applied to estimate the long-term throughput for each mobile station, which allows for determining the optimal number of multiplexing order and throughput loss due to the interference nulling.

#### 1. Introduction

Multiuser MIMO (MU-MIMO) scheme, allowing a base station (BS) to communicate with multiple users simultaneously, provides an opportunity to boost the sum capacity through precoding, even when each user has only one antenna. For example, zero-forcing transmit beamforming (ZFBF) is one of the practical multiuser transmission strategies for MU-MIMO systems [1]. By designing one user’s beamforming vector to be orthogonal to other selected users’ channel vectors, ZFBF can completely eliminate the multiuser interference corresponding to intracell interference in cellular systems. Furthermore, using more transmit antenna can increase the number of users simultaneously served by MU-MIMO or enhance the error performance of each link between the BS and user.

Despite the theoretical attractiveness, the capacity gain promised by MIMO techniques has been shown to degrade severely in a multicell environment. To suppress the intercell interference, the authors in [2–5] investigated a coordinated beamforming scheme using multiple antennas at the BS. The achievable rate region of the MISO interference channel, in the case where the full channel information is shared among BSs, was derived in [2, 3], with instantaneous and statistical CSI, respectively. Distributed beamforming with a virtual SINR framework was proposed in [4]. The theoretical results in [2–4], however, are limited to only one user in the victim cell. The authors in [5] assumed that the interference experienced by multiple users in the victim cells is suppressed. Some studies on the interference mitigation in the cooperative beamforming for multiuser systems have been studied from the perspective of scheduling issues [6–9]. A low-complexity random beamforming method, which only requires sharing of user indices, has been suggested with analytic throughput expressions [6]. In [7], the authors provided a transmission beamforming scheme for interference nulling with user selection. Also, the reduced complexity algorithms for joint user selection in adaptive coordination scheme were designed in [8]. However, the unrealistic special homogeneous case, in which all users have the same average SNR, is assumed in [8]. In [9], we have considered a generalized intercell interference coordination problem and proposed a two-step coordination procedure to choose a cell-edge user and decide the coordination. However, the research in [6–9] was not extended to MU-MIMO, that is, only dealing with the multiple users in the serving cell.

In this paper, we analyze the three-dimensional DMNT among the diversity order (i.e., slope of error performance curve), multiplexing order (i.e., the number of users simultaneously served by MU-MIMO scheme), and nulling order (i.e., the number of other cell users subject to zero interference in a victim cell), while providing the victim cell with intercell interference nulling. We consider an interference-free environment in which the BS in each cell employs a precoding matrix with antennas, so as to null the intracell interference while mitigating the intercell interference. It is assumed that all users are equipped with a single antenna. Our contribution is to reveal the fundamental property of performance trade-off, given by , in multicell MU-MIMO subject to intercell and intracell interference-free conditions. To the best of our knowledge, there has never been any rigorous justification for this particular property in previous works. Note that the current diversity-multiplexing-nulling trade-off (DMNT) is quite different from the well-known diversity-multiplexing trade-off (DMT) that deals with the multiple antenna gain to be achieved simultaneously by any coding scheme (e.g., space-time coding) in the point-to-point MIMO system [10]. Meanwhile, we demonstrate that our DMNT can be applicable to estimating the long-term user throughput, which allows for determining the optimal multiplexing order and throughput loss due to interference nulling.

The rest of this paper is organized as follows. We present some preliminaries for our analysis in Section 2. Section 3 presents a system model and the precoding matrix design under consideration. Our analysis results for the diversity-multiplexing-nulling trade-off are given in Section 4. Section 5 demonstrates how DMNT can be applied to estimate the long-term throughput for each MS. Finally, concluding remarks are given in Section 6.

#### 2. Preliminaries

We first introduce some results from previous works, which are useful for our analysis.

*Definition 1. *Let be independent complex Gaussian random vectors with zero mean vector (i.e., ) and identity covariance matrix (i.e., ). If , where is the matrix, then is said to have a* complex Wishart distribution* with degrees of freedom [11], that is, .

By using vectors, we may define the matrix as The probability density function of for is given aswhere . If , that is, is defined as a random variable, then it has a Chi-squared distribution with degrees of freedom, that is, . This result immediately follows by substituting into the probability density function (pdf) of the Wishart distribution.

Lemma 2. *If and is partitioned aswhere is and the Schur complement of block is also a Wishart matrix with a distribution of . In [12], the Schur complement of block is given as *

*Proof. *See proof of Theorem in [13].

#### 3. Signal Model and Precoding Matrix Design

We consider MU-MIMO downlink systems in which the BS serves a set of selected mobile stations (MSs) simultaneously in a serving cell, while imparting interference to the MSs in victim cells, as illustrated in Figure 1. We assume that there are and MSs in the serving cell and victim cells, respectively. Let be a subset of indices for users that are intended for transmission by the BS . The user set is dynamically selected by a scheduler in the BS. At the serving cell, we design a wireless link equipped with transmit antennas at the BS and a single receive antenna at each MS. The MSs in the victim cells also employ a single receive antenna and do not perform any type of interference mitigation. Let us denote and as complex Gaussian channel vector and beamforming vector for the th MS, respectively. For a subset , we define and . The received signal at the th MS in the subset is represented bywhere and are the data symbol and the Additive White Gaussian Noise (AWGN) with variance of , respectively. We impose average power constraints, and . The received signals in (5) are rewritten by the aggregated received signal vector aswhere and . Let denote an channel vector from the serving BS to the th MS in the victim cell. Note that MSs in the victim cells are those subject to intercell interference. In our current system model in Figure 1, MSs in the virtual victim cell can be considered as those multiplexed with MSs in the serving cell while satisfying the intercell interference-free condition. For the MSs in the victim cell, the aggregated received signal vector is given bywhere , which can be known to the serving BS by a sounding signal [14]. Note that a desired signal of the victim MS is not represented in (7); that is, is just an intercell interference vector for the victim MSs, which would be controlled by the BS in the serving cell.