**1. Introduction**

In wireless communications, the spectral efficiency can be improved by exploiting the space domain when antenna arrays are used. In particular, space-division multiple access (SDMA) [1–3] can be adopted with various beamforming techniques. If both the transmitter and the receiver are equipped with multiple antennas, the resulting channel becomes a multiple-input-multiple-output (MIMO) channel, which can provide a rich spatial diversity gain. In MIMO systems, it is often desirable to use the maximum likelihood (ML) detection to jointly detect received signals for optimal performance and full receive diversity. However, since the complexity of the ML detection exponentially grows with the number of transmit antennas, the ML detection approach becomes impractical for high-dimensional detection problems. To derive low-complexity suboptimal MIMO detectors, various approaches based on the properties of lattice are considered. For example, using the Lenstra-Lenstra-Lovasz (LLL) algorithm in [4], the lattice reduction (LR)-based ´ low-complexity detectors are proposed in [5–8], which can provide a full receive diversity gain with a near-ML performance. The basic idea of the LR-based MIMO detection is to generate a nearly orthogonal basis for a given channel matrix to mitigate the effect of (multiple antenna) interference.

Due to users' different locations and channel conditions, it is possible to exploit another diversity gain in a multiuser system, where the throughput can be maximized by choosing the user of the strongest channel gain at a time. The resulting diversity gain is called the multiuser diversity gain [9]. Multiuser systems can be extended to the case of MIMO systems [10], where the multiuser MIMO user selection plays a key role in increasing the throughput of downlink channels [11]. It is noteworthy that, by viewing the multiuser MIMO system as virtual antennas in a single-user MIMO system, various antenna selection techniques can be applied to user selection [12, 13]. A mutual information-based criterion is proposed in [12] to

terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Li et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Q. Li, Y. Li, L. Bai, and J. Choi , licensee InTech. This is an open access chapter distributed under the

select the antenna subset that maximizes the mutual information. In [13], a geometry-based criterion is developed with an LR-based linear detector to minimum the error probability. In general, user selection problems are combinatorial problems, and the complexity required to solve the problems could be prohibitively high for a large multiuser MIMO system. Thus, low-complexity suboptimal selection strategies are considered in [14–21], at the expense of degraded performance. In [14–17], a single antenna is selected at a time to maximize the throughput based on greedy selection schemes.

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**2. System model**

MIMO detection techniques.

**2.1. Multiuser MIMO system**

(CSCG) random vector with **E**

, ..., **H***k*(*M*)

the *A*-dimensional Cartesian product of S.

and will be denoted by **y**K, **s**K, and **n**, respectively.

 **H***k*(1)

**2.2. MIMO detection**

selection methods.

that **H**K =

define the set of the selected users' indexes as K =

duration, the received signal at the BS is given by

In this section, we introduce the model of multiuser MIMO system together with several

Consider the multiuser MIMO system with *K* users in uplink channels, where each user is equipped with *P* transmit antennas, and the base station (BS) is equipped with *N* receive antennas. Each user has an *N* × *P* channel matrix and a *P* × *L* signal matrix, which are denoted by **<sup>H</sup>***<sup>k</sup>* and **<sup>S</sup>***k*, respectively, where *<sup>k</sup>* ∈ {1, 2, ..., *<sup>K</sup>*}. Here, *<sup>L</sup>* is the number of symbols transmitted by a user. It is assumed that all the users share a common uplink channel and that *M* users can access the channel at a time, where *M* = ⌊*N*/*P*⌋. The channel is assumed to be a quasi-static block fading channel, with its channel matrix not varying over a time slot duration of *L* symbols. Here, a set of the *M* users who can access the channel could be updated for every time slot interval. Note that this selection problem can also be regarded as that with virtual antennas in a single-user MIMO system, where *MP* antennas are selected out of *KP* available antennas. Let *k*(*m*) be the *m*th selected user's index. For convenience,

where **H**K, **S**K, and **N** are the *N* × *MP* composite channel matrix, the *MP* × *L* transmitted signal matrix, and the *N* × *L* background noise matrix, respectively. We assume that each column vector of **N** is an independent zero-mean circularly symmetric complex Gaussian

Throughout this chapter, we assume that the channel state information (CSI) is perfectly known at the receiver. Furthermore, the following assumptions are used to derive user

A1) The elements of **S**<sup>K</sup> have a common signal alphabet, denoted by S, and S ⊂ **Z** + *j***Z**, where **<sup>Z</sup>** denotes the set of integer numbers and *<sup>j</sup>* <sup>=</sup> √−1. Furthermore, let <sup>S</sup> *<sup>A</sup>* represent

A2) The transmitted signals are uncoded. This implies that the user selection criteria in this chapter are based on uncoded bit error rate (BER). For uncoded signals, we can assume *L* = 1 (Note that this assumption is used to simplify the derivation of user selection criteria, while the length of slot can be any number). Thus, **Y**K, **S**K, and **N** are vectors

MIMO detection plays an important role in MIMO receivers. Within this chapter, several well known MIMO detectors including the ML detector, linear detectors, and successive

interference cancellation (SIC) detectors, together with LR are considered.

 **S***k*(1)

**n***l***n**<sup>H</sup> *l*

and that **S**K =

, ..., **S***k*(*M*)

 .

*k*(1), *k*(2), ..., *k*(*M*)

Lattice Reduction-Based User Selection in Multiuser MIMO Systems

**Y**K = **H**K**S**K + **N**, (1)

= *N*0**I**, where **n***<sup>l</sup>* denotes the *l*th column of **N**. Note

. Then, over a slot

Although the achievable rate or related signal-to-noise ratio (SNR) can be used for the user selection criterion, it would be more practical to use a certain performance measure that is directly related to the performance of the actual detector or decoder employed. Therefore, it is desirable to derive a user selection criterion that can maximize the performance of the MIMO detector that is *actually* employed in a multiuser MIMO system.

In this chapter, for the user selection in uplink channels of a cellular system, where a single user is selected to transmit signals to a base station (BS) at a time, the error probability is used for the user selection criterion to choose the user with the smallest error probability for given MIMO detectors. Various user selection criteria will be derived with the ML detector, LR-based detectors and other low-complexity suboptimal detectors. It will be shown that a near-optimal performance with a full diversity gain (i.e., multiuser diversity and multiple antenna diversity) can be achieved using the proposed user selection criteria in this chapter with LR-based detectors.

Based on the single user selection criteria derived, we will extend them to support multiple users at a time. This extension of the user selection (i.e., multiple user selection) is not straightforward, because the multiple-user selection problem becomes a combinatorial problem. If an exhaustive search is used for multiple user selection when an LR-based MIMO detector is employed, LR needs to be performed for all the possible channel matrices composited by a group of subchannel matrices of the selected users. Unfortunately, this results in a high computational complexity, because the number of user combinations is large. Therefore, we will propose a greedy user selection algorithm to reduce the computational complexity at the expense of degraded performance when LR-based detectors are used. Moreover, to further reduce the computational complexity, an iterative LR updating algorithm will be investigated. Based on a theoretical analysis in this chapter, we can show that, with the combinatorial user selection, the LR-based detection can achieve the same diversity as the ML detector. Through simulations, we will compare the performance obtained by our selection criteria (i.e., combinatorial and greedy ones) to other existing approaches.

With the LR-based detection employed, simulation results will confirm that our combinatorial user selection can provide the best performance, whereas the performance of the greedy user selection scheme could approach that of the combinatorial approach as the correlation between possible composite channel matrices decreases. It will also be shown that our greedy user selection provides a better performance and a significantly reduced complexity compared with other approaches.
