**2. System model**

2 Recent Trends in Multiuser MIMO Communications

throughput based on greedy selection schemes.

with LR-based detectors.

approaches.

complexity compared with other approaches.

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

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

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

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

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

MIMO detector that is *actually* employed in a multiuser MIMO system.

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

#### **2.1. Multiuser MIMO system**

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,

define the set of the selected users' indexes as K = *k*(1), *k*(2), ..., *k*(*M*) . Then, over a slot duration, the received signal at the BS is given by

$$\mathbf{Y}\_{\mathcal{K}} = \mathbf{H}\_{\mathcal{K}} \mathbf{S}\_{\mathcal{K}} + \mathbf{N}\_{\prime} \tag{1}$$

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 (CSCG) random vector with **E n***l***n**<sup>H</sup> *l* = *N*0**I**, where **n***<sup>l</sup>* denotes the *l*th column of **N**. Note that **H**K = **H***k*(1) , ..., **H***k*(*M*) and that **S**K = **S***k*(1) , ..., **S***k*(*M*) .

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 selection methods.


#### **2.2. MIMO detection**

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.

#### *2.2.1. ML and linear detection*

For the sake of convenience, we omit the index set K. The ML detection is given by

$$\mathfrak{S}\_{\text{ml}} = \arg\min\_{\mathbf{s}\in\mathcal{S}^{M^p}} \left\lVert \mathbf{y} - \mathbf{H}\mathbf{s} \right\rVert^2,\tag{2}$$

where the complexity grows exponentially with *MP*.

Alternatively, an estimate of **s** can be obtained by a linear transformation as follows:

$$
\hat{\mathbf{s}} = \mathbf{W} \mathbf{y}\_{\prime} \tag{3}
$$

10.5772/57128

http://dx.doi.org/10.5772/57130

T. This

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<sup>T</sup> and **<sup>n</sup>**ex = [**n**<sup>T</sup> <sup>−</sup> <sup>√</sup>*c***s**T]

Lattice Reduction-Based User Selection in Multiuser MIMO Systems

Note that the size of **H**ex is the same as that of **G**ex which is 2*N* × *MP*. Let the QR factorization of **G**ex be **G**ex = **QR**, where **Q** is a matrix whose column vectors are

results in **y**ex = **H**ex**s** + **n**ex. Then, the LR-based SIC detection can be carried out with the

will use **n** to denote **n**¯. Note that **n** also includes the self-interference as mentioned in [7]. The SIC detection can be carried out with (7). The elements of the last row, the *MP*th layer, are detected first. Then, their contributions in the second last row are canceled and the signals of the (*MP* − 1)th row are detected. This operation is repeated up to the first row.

In this section, we derive user selection criteria depending on the type of actually employed MIMO detector, where a single user is selected to transmit signals to a BS at a time. Suppose

For detection method, the ML detector and two suboptimal detectors will be considered: one is the linear detector and the other is the SIC detector. As for the two suboptimal detectors,

Assuming that user *k* is selected, we omit the user index *k* for the sake of simplicity. To derive the selection criterion, we can consider the pairwise error probability (PEP). Suppose

> � � � 2 ≤ � � �**<sup>y</sup>** <sup>−</sup> **Hs**(1)

<sup>2</sup>*<sup>π</sup> <sup>e</sup><sup>z</sup>*2/2d*<sup>z</sup>* and **<sup>∆</sup>** <sup>=</sup> **<sup>s</sup>**(1) <sup>−</sup> **<sup>s</sup>**(2). Then, the following upper bound can be

�**Hd**¯ �<sup>2</sup> 2*N*<sup>0</sup>

that **s**(1) is transmitted, while **s**(2) is erroneously detected. Then, the PEP is given by

<sup>=</sup> Pr �� � �**<sup>y</sup>** <sup>−</sup> **Hs**(2)

= Q � �**H∆**� 2*N*<sup>0</sup>

**<sup>s</sup>**(1) <sup>→</sup> **<sup>s</sup>**(2)

� ≤ Q �

**Q**H**y**ex = **Q**H**G**ex**U**ex**s** + **Q**H**n**ex = **Rc** + **n**¯, (7)

ex**n**ex. Since the statistical properties of **n**¯ and **n** are the same, we

**Y***<sup>k</sup>* = **H***k***S***<sup>k</sup>* + **N**, (8)

� � � 2 �

, (9)

, (10)

orthonormal and **R** is upper triangular. Let **y**ex = [**y**<sup>T</sup> **0**]

that user *k* is chosen, the system model in (1) is simplified as

following signal:

**3.1. ML detector**

where <sup>Q</sup>(*x*) = � <sup>∞</sup>

obtained as

where **c** = **U**ex**s** and **n**¯ = **Q**<sup>H</sup>

**3. Single user selection criteria**

the LR is applied for better performance [6][7].

*P* �

*<sup>x</sup>* <sup>√</sup> 1 **<sup>s</sup>**(1) <sup>→</sup> **<sup>s</sup>**(2)

*P* � �

where **W** is a linear filter that is given by **W** = **H**H**H** + *c***I** <sup>−</sup><sup>1</sup> **<sup>H</sup>**. If *<sup>c</sup>* <sup>=</sup> 0, the linear detector corresponds to the zero-forcing (ZF) detector, while the minimum mean square error (MMSE) detector is obtained if *c* = *N*0/*Es*. Here, *Es* is the symbol energy and it is assumed that **E ss**<sup>H</sup> = *Es***I**.

To improve the performance of the detector, the LR is performed in the LR-based detection. A complex valued matrix can be converted into a real valued matrix for the LR as in [7]. Alternatively, the LR can be directly performed with a complex valued matrix as in [6], [8]. For convenience, in this chapter, we assume that the LR is performed with complex valued matrices.

For a given channel matrix **H**, the LR basis can be found as follows:

$$\mathbf{H} = \mathbf{G} \mathbf{U}\_{\prime} \tag{4}$$

where **U** is an (complex) integer unimodular matrix and **G** is a matrix whose column vectors are nearly orthogonal. The received signal can be rewritten as

$$\mathbf{y} = \mathbf{H}\mathbf{s} + \mathbf{n} = \mathbf{G}\mathbf{c} + \mathbf{n},\tag{5}$$

Under the MMSE criteria, the linear filter of LR-based MMSE linear detector is given by **W** = **<sup>G</sup>**H**<sup>G</sup>** + *<sup>N</sup>*<sup>0</sup> *Es* **<sup>U</sup>**H**<sup>U</sup>** −<sup>1</sup> **G**H.

#### *2.2.2. SIC detection*

An SIC detector is not a linear detector due to its cancellation operation. In [7], the LR-based SIC detectors are proposed. To generalize the LR-based SIC detector, define the extended channel matrix as **<sup>H</sup>**ex = [**H**<sup>T</sup> <sup>√</sup>*c***I**] T. The LR basis can be found as

$$\mathbf{H}\_{\rm ex} = \mathbf{G}\_{\rm ex} \mathbf{U}\_{\rm ex} \tag{6}$$

where **U**ex is a complex integer unimodular matrix and **G**ex is a matrix whose column vectors are nearly orthogonal. If the LR basis is not used, **U**ex = **I** (i.e., **G**ex = **H**ex).

Note that the size of **H**ex is the same as that of **G**ex which is 2*N* × *MP*. Let the QR factorization of **G**ex be **G**ex = **QR**, where **Q** is a matrix whose column vectors are orthonormal and **R** is upper triangular. Let **y**ex = [**y**<sup>T</sup> **0**] <sup>T</sup> and **<sup>n</sup>**ex = [**n**<sup>T</sup> <sup>−</sup> <sup>√</sup>*c***s**T] T. This results in **y**ex = **H**ex**s** + **n**ex. Then, the LR-based SIC detection can be carried out with the following signal:

$$\mathbf{Q}^{\rm H}\mathbf{y}\_{\rm ex} = \mathbf{Q}^{\rm H}\mathbf{G}\_{\rm ex}\mathbf{U}\_{\rm ex}\mathbf{s} + \mathbf{Q}^{\rm H}\mathbf{n}\_{\rm ex} = \mathbf{R}\mathbf{c} + \mathbf{\bar{n}},\tag{7}$$

where **c** = **U**ex**s** and **n**¯ = **Q**<sup>H</sup> ex**n**ex. Since the statistical properties of **n**¯ and **n** are the same, we will use **n** to denote **n**¯. Note that **n** also includes the self-interference as mentioned in [7].

The SIC detection can be carried out with (7). The elements of the last row, the *MP*th layer, are detected first. Then, their contributions in the second last row are canceled and the signals of the (*MP* − 1)th row are detected. This operation is repeated up to the first row.
