**3. Single user selection criteria**

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

$$\mathbf{Y}\_k = \mathbf{H}\_k \mathbf{S}\_k + \mathbf{N} \tag{8}$$

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

#### **3.1. ML detector**

4 Recent Trends in Multiuser MIMO Communications

where the complexity grows exponentially with *MP*.

where **W** is a linear filter that is given by **W** =

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

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

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

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

where **U** is an (complex) integer unimodular matrix and **G** is a matrix whose column vectors

Under the MMSE criteria, the linear filter of LR-based MMSE linear detector is given by

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

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).

T. The LR basis can be found as

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

are nearly orthogonal. The received signal can be rewritten as

**H**H**H** + *c***I**

**<sup>s</sup>**ˆml <sup>=</sup> arg min **<sup>s</sup>**∈S *MP* �**<sup>y</sup>** <sup>−</sup> **Hs**�<sup>2</sup> , (2)

**s**ˆ = **Wy**, (3)

**H** = **GU**, (4)

**y** = **Hs** + **n** = **Gc** + **n**, (5)

**H**ex = **G**ex**U**ex, (6)

<sup>−</sup><sup>1</sup> **<sup>H</sup>**. If *<sup>c</sup>* <sup>=</sup> 0, the linear detector

*2.2.1. ML and linear detection*

**E**

matrices.

**W** = 

**<sup>G</sup>**H**<sup>G</sup>** + *<sup>N</sup>*<sup>0</sup>

*2.2.2. SIC detection*

*Es* **<sup>U</sup>**H**<sup>U</sup>**

channel matrix as **<sup>H</sup>**ex = [**H**<sup>T</sup> <sup>√</sup>*c***I**]

−<sup>1</sup> **G**H.

**ss**<sup>H</sup> = *Es***I**.

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

$$P\left(\mathbf{s}\_{(1)} \rightarrow \mathbf{s}\_{(2)}\right) = \Pr\left(\left\|\mathbf{y} - \mathbf{H}\mathbf{s}\_{(2)}\right\|^2 \leq \left\|\mathbf{y} - \mathbf{H}\mathbf{s}\_{(1)}\right\|^2\right)$$

$$= \mathcal{Q}\left(\sqrt{\frac{\left\|\mathbf{H}\mathbf{A}\right\|}{2N\_0}}\right). \tag{9}$$

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

$$P\left(\mathbf{s}\_{(1)} \to \mathbf{s}\_{(2)}\right) \le \mathcal{Q}\left(\sqrt{\frac{\|\mathbf{H}\bar{\mathbf{d}}\|^2}{2\mathbf{N}\_0}}\right),\tag{10}$$

where

$$\bar{\mathbf{d}} = \arg\min\_{\mathbf{d}\in\mathcal{D}, \mathbf{d}\neq\mathbf{0}} ||\mathbf{H}\mathbf{d}||^2. \tag{11}$$

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, (16)

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, (17)

*<sup>p</sup>*,*<sup>p</sup>* denote the (*p*, *p*)th element of **R**

. (18)

*<sup>λ</sup>*min(**H**H**H**).

165

because <sup>Q</sup>(·) is a decreasing function and **<sup>∆</sup>**H(**H**H**H**)−1**<sup>∆</sup>** <sup>≤</sup> *<sup>λ</sup>*max(**H**H**H**)−1�**∆**�<sup>2</sup> <sup>=</sup> �**∆**�<sup>2</sup>

It is important to note that this ME criterion is valid for the LR-based linear detectors [6], [7].

*<sup>λ</sup>*min �

where **G***<sup>k</sup>* is the reduced basis from **H***k*. This ME criterion is the same as that in (14) except

As the LR is performed, the column vectors of **G**ex would be nearly orthogonal. In other words, the upper off-diagonal elements of **R** would be small. Thus the SIC detection performance would mainly depend on the diagonal elements of **R**. For convenience, let

from the *k*th user's channel **H***k*. Then, ignoring the interference terms (as they are canceled when the detection of the lower layers is successfully carried out with no error), the SNR of

> � min *p* � � � *r* (*k*) *p*,*p* � � � �

The MD criterion is also closely related to the minimum error probability criterion when the

Let *np* denote the *p*th element of **n**. Then, the LR-based SIC detection at the *P*th layer does

**G**<sup>H</sup> *<sup>k</sup>* **G***<sup>k</sup>* �

**G**H**G**�

2*N*<sup>0</sup>

(*k*)

*<sup>N</sup>*<sup>0</sup> . From this, the selection criterion can be given by

**x** = **Rc** + **n**. (19)

<sup>4</sup> . Thus, the LR-based SIC detection would have no

<sup>4</sup> , for all *p*. The probability of no error can be lower

�**∆U**�<sup>2</sup>

. From (16), the selection criterion becomes

Lattice Reduction-Based User Selection in Multiuser MIMO Systems

Therefore, the ME criterion in (14) can be used for the user selection criterion.

� ≤ Q � *λ*min �

�

*<sup>k</sup>*<sup>∗</sup> <sup>=</sup> arg max*<sup>k</sup>*

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

Let **c**(*i*) = **Us**(*i*), *i* = 1, 2. Then, from (15), the PEP is bounded as

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

�

that the channel matrix **H***<sup>k</sup>* is replaced by its reduced one **G***k*.

*<sup>c</sup>* = 0 (this is the case when *<sup>N</sup>*<sup>0</sup> → 0 or high SNR). Let *<sup>r</sup>*

*<sup>p</sup>* <sup>=</sup> <sup>|</sup>*<sup>r</sup>* (*k*) *p*,*p* | 2

SNR is high. For convenience, let **x** = **Q**H**y**. Then, (18) is rewritten as

<sup>2</sup> < <sup>|</sup>*rP*,*<sup>P</sup>*<sup>|</sup> 2

<sup>2</sup> < <sup>|</sup>*rp*,*<sup>p</sup>* <sup>|</sup> 2

*<sup>k</sup>*<sup>∗</sup> <sup>=</sup> arg max*<sup>k</sup>*

This selection criterion is referred to as the max-min diagonal term (MD) criterion.

*P* �

where **<sup>∆</sup><sup>U</sup>** <sup>=</sup> **<sup>c</sup>**(1) <sup>−</sup> **<sup>c</sup>**(2) <sup>=</sup> **<sup>U</sup>**

**3.3. SIC detectors**

not have error if <sup>|</sup>*nP*<sup>|</sup>

<sup>|</sup>*rP*,*<sup>P</sup>*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>

error across all the layers if |*np*|

<sup>2</sup> or <sup>|</sup>*nP*<sup>|</sup>

the *<sup>p</sup>*th layer of **<sup>H</sup>***<sup>k</sup>* becomes *<sup>γ</sup>*(*k*)

Here, D = � **<sup>d</sup>** = **<sup>s</sup>** − **<sup>s</sup>**′ � � � **<sup>s</sup>**, **<sup>s</sup>**′ ∈ S*<sup>P</sup>* � ⊂ **Z***<sup>P</sup>* + *j***Z***P*. For convenience, denote by *S*(**H**) the length of the shortest non-zero vector of the lattice generated by **H**. Then, we can see that *S*(**H**) = �**Hd**¯ �. From (10), if the ML detector is employed, the user selection criterion to minimize the error probability becomes

$$k^\* = \arg\max\_k S(\mathbf{H}\_k). \tag{12}$$

Throughout this chapter, the user selection criterion in (12) is referred to as the max-min distance (MDist) criterion as *S*(**H**) is the minimum distance of the lattice generated by **H**.

The problem to find a non-zero shortest vector in a lattice is called the shortest vector problem (SVP) and known to be NP-hard. For an approximation, the LLL algorithm in [4], which has a polynomial time complexity, can be used.

Another approximation can be considered by relaxing the constraint on **∆**. We have

$$\|\|\mathbf{H}\Delta\|\|^2 = \Delta^\mathbf{H}\mathbf{H}^\mathbf{H}\mathbf{H}\Delta \ge \|\|\Delta\|^2 \lambda\_{\text{min}}\left(\mathbf{H}^\mathbf{H}\mathbf{H}\right),\tag{13}$$

where *λ*min(**A**) stands for the minimum eigenvalue of **A**. This shows that the selection criterion can be based on the minimum eigenvalue of the channel matrix, i.e.,

$$k^\* = \arg\max\_k \lambda\_{\min} \left( \mathbf{H}\_k^H \mathbf{H}\_k \right). \tag{14}$$

Thus, each user can feed back its minimum eigenvalue of the channel matrix and the user who has the maximum *λ*min � **H**H *<sup>k</sup>* **<sup>H</sup>***<sup>k</sup>* � can be selected to access the channel. This selection criterion is referred to as the max-min eigenvalue (ME) criterion throughout this chapter.

#### **3.2. Linear detectors**

As the SNR increases, we have *c* → 0 (in this case, the MMSE detector becomes the ZF detector) and the PEP has the following upper bound:

$$\begin{split}P\left(\mathbf{s}\_{(1)}\rightarrow\mathbf{s}\_{(2)}\right) &= \mathcal{Q}\left(\frac{\|\mathbf{A}\|^2}{\sqrt{2N\_0\mathbf{A}^\mathbf{H}(\mathbf{H}^\mathbf{H}\mathbf{H})^{-1}\mathbf{A}}}\right) \\ &\leq \mathcal{Q}\left(\sqrt{\frac{\lambda\_{\text{min}}(\mathbf{H}^\mathbf{H}\mathbf{H})}{2N\_0}\|\mathbf{A}\|^2}\right),\end{split}\tag{15}$$

because <sup>Q</sup>(·) is a decreasing function and **<sup>∆</sup>**H(**H**H**H**)−1**<sup>∆</sup>** <sup>≤</sup> *<sup>λ</sup>*max(**H**H**H**)−1�**∆**�<sup>2</sup> <sup>=</sup> �**∆**�<sup>2</sup> *<sup>λ</sup>*min(**H**H**H**). Therefore, the ME criterion in (14) can be used for the user selection criterion.

It is important to note that this ME criterion is valid for the LR-based linear detectors [6], [7]. Let **c**(*i*) = **Us**(*i*), *i* = 1, 2. Then, from (15), the PEP is bounded as

$$P\left(\mathbf{s}\_{(1)} \to \mathbf{s}\_{(2)}\right) \le \mathcal{Q}\left(\sqrt{\frac{\lambda\_{\text{min}}\left(\mathbf{G}^{\text{H}}\mathbf{G}\right) \|\mathbf{A}\_{\text{U}}\|^2}{2N\_0}}\right) \tag{16}$$

where **<sup>∆</sup><sup>U</sup>** <sup>=</sup> **<sup>c</sup>**(1) <sup>−</sup> **<sup>c</sup>**(2) <sup>=</sup> **<sup>U</sup>** � **<sup>s</sup>**(1) <sup>−</sup> **<sup>s</sup>**(2) � . From (16), the selection criterion becomes

$$k^\* = \arg\max\_k \lambda\_{\min} \left( \mathbf{G}\_k^H \mathbf{G}\_k \right),\tag{17}$$

where **G***<sup>k</sup>* is the reduced basis from **H***k*. This ME criterion is the same as that in (14) except that the channel matrix **H***<sup>k</sup>* is replaced by its reduced one **G***k*.

#### **3.3. SIC detectors**

6 Recent Trends in Multiuser MIMO Communications

**<sup>d</sup>**¯ <sup>=</sup> arg min **<sup>d</sup>**∈D,**d**�=**<sup>0</sup>**

*<sup>k</sup>*<sup>∗</sup> <sup>=</sup> arg max*<sup>k</sup>*

Another approximation can be considered by relaxing the constraint on **∆**. We have

�**H∆**�<sup>2</sup> = **<sup>∆</sup>**H**H**H**H<sup>∆</sup>** ≥ �**∆**�2*λ*min

criterion can be based on the minimum eigenvalue of the channel matrix, i.e.,

*<sup>k</sup>*<sup>∗</sup> <sup>=</sup> arg max*<sup>k</sup>*

� **H**H *<sup>k</sup>* **<sup>H</sup>***<sup>k</sup>*

detector) and the PEP has the following upper bound:

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

� = Q �

> ≤ Q �

*P* �

of the shortest non-zero vector of the lattice generated by **H**. Then, we can see that *S*(**H**) = �**Hd**¯ �. From (10), if the ML detector is employed, the user selection criterion to minimize the

Throughout this chapter, the user selection criterion in (12) is referred to as the max-min distance (MDist) criterion as *S*(**H**) is the minimum distance of the lattice generated by **H**.

The problem to find a non-zero shortest vector in a lattice is called the shortest vector problem (SVP) and known to be NP-hard. For an approximation, the LLL algorithm in [4], which has

where *λ*min(**A**) stands for the minimum eigenvalue of **A**. This shows that the selection

*λ*min � **H**H *<sup>k</sup>* **<sup>H</sup>***<sup>k</sup>* �

Thus, each user can feed back its minimum eigenvalue of the channel matrix and the user

As the SNR increases, we have *c* → 0 (in this case, the MMSE detector becomes the ZF

�**∆**�2 �2*N*0**∆**H(**H**H**H**)−1**<sup>∆</sup>**

> *λ*min(**H**H**H**) 2*N*<sup>0</sup>

criterion is referred to as the max-min eigenvalue (ME) criterion throughout this chapter.

�**Hd**�2. (11)

*S*(**H***k*). (12)

, (13)

. (14)

, (15)

⊂ **Z***<sup>P</sup>* + *j***Z***P*. For convenience, denote by *S*(**H**) the length

� **H**H**H** �

� can be selected to access the channel. This selection

�

�**∆**�2 

where

Here, D =

�

error probability becomes

who has the maximum *λ*min

**3.2. Linear detectors**

**<sup>d</sup>** = **<sup>s</sup>** − **<sup>s</sup>**′

� � �

a polynomial time complexity, can be used.

**<sup>s</sup>**, **<sup>s</sup>**′ ∈ S*<sup>P</sup>*

�

As the LR is performed, the column vectors of **G**ex would be nearly orthogonal. In other words, the upper off-diagonal elements of **R** would be small. Thus the SIC detection performance would mainly depend on the diagonal elements of **R**. For convenience, let *<sup>c</sup>* = 0 (this is the case when *<sup>N</sup>*<sup>0</sup> → 0 or high SNR). Let *<sup>r</sup>* (*k*) *<sup>p</sup>*,*<sup>p</sup>* denote the (*p*, *p*)th element of **R** from the *k*th user's channel **H***k*. Then, ignoring the interference terms (as they are canceled when the detection of the lower layers is successfully carried out with no error), the SNR of the *<sup>p</sup>*th layer of **<sup>H</sup>***<sup>k</sup>* becomes *<sup>γ</sup>*(*k*) *<sup>p</sup>* <sup>=</sup> <sup>|</sup>*<sup>r</sup>* (*k*) *p*,*p* | 2 *<sup>N</sup>*<sup>0</sup> . From this, the selection criterion can be given by

$$k^\* = \arg\max\_k \left\{ \min\_p \left| r\_{p,p}^{(k)} \right| \right\}. \tag{18}$$

This selection criterion is referred to as the max-min diagonal term (MD) criterion.

The MD criterion is also closely related to the minimum error probability criterion when the SNR is high. For convenience, let **x** = **Q**H**y**. Then, (18) is rewritten as

$$\mathbf{x} = \mathbf{R}\mathbf{c} + \mathbf{n}.\tag{19}$$

Let *np* denote the *p*th element of **n**. Then, the LR-based SIC detection at the *P*th layer does not have error if <sup>|</sup>*nP*<sup>|</sup> <sup>|</sup>*rP*,*<sup>P</sup>*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> or <sup>|</sup>*nP*<sup>|</sup> <sup>2</sup> < <sup>|</sup>*rP*,*<sup>P</sup>*<sup>|</sup> 2 <sup>4</sup> . Thus, the LR-based SIC detection would have no error across all the layers if |*np*| <sup>2</sup> < <sup>|</sup>*rp*,*<sup>p</sup>* <sup>|</sup> 2 <sup>4</sup> , for all *p*. The probability of no error can be lower bounded as

$$\Pr(\text{no error}) \ge \Pr\left( |n\_p|^2 < \frac{|r\_{p,1}|^2}{4}, \forall p \right)$$

$$= \prod\_{p=1}^{P} \Pr\left( |n\_p|^2 < \frac{|r\_{p,p}|^2}{4} \right). \tag{20}$$

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Lattice Reduction-Based User Selection in Multiuser MIMO Systems

(24)

. (25)

*<sup>i</sup>*=<sup>0</sup> (*<sup>K</sup>* <sup>−</sup> *<sup>i</sup>*) possible user index sets.

, (26)

**4.2. LR-based MMSE and MMSE-SIC selection criteria**

compared to *P* in the case of *M* = 1.

can be extended to the case with *M* > 1 as follows:

In this subsection, the user selection criteria with LR-based detectors in Section 3 are extended to the case of *M* > 1, where the number of transmit layers are extended to *MP*,

The MD criterion derived in Section 3, with *M* = 1 for the LR-based MMSE-SIC detection,

 min *q r* (K) *q*,*q* 

*<sup>λ</sup>*min

The user selection based on (22), (23), (24), and (25) is called the combinatorial user selection,

The computational complexity of the user selection under the criteria derived in Section 4 grows rapidly with *M* or *K* as they are all combinatorial optimization problems. Thus, it is desirable to derive low complexity approaches for the user selection. In this section, we propose low complexity greedy approaches for the user selection. Note that we focus on the greedy user selection with a LR-based MIMO detector only as its performance is comparable to that of the ML detector and, more importantly, we can derive a computationally efficient

The user selection approaches in Section 4 have the complexity that becomes prohibitively

For each user index set, an LR of an *N* × *MP* complex channel matrix is to be performed. For example, when *K* = 10, *M* = *N* = 4 and *P* = 1, 10 × 9 × 8 × 7 = 5040 LRs of 4 × 4

To reduce the computational complexity in the user selection, we consider a greedy approach when a LR-based MIMO detector is employed. The resulting approach is called the LR-based greedy (LRG) user selection, which is of course suboptimal. The LRG user selection

1. Let *m* = 1 and K¯ = {1, . . . , *K*}. In order to select the first user, we can use any criterion.

**G**<sup>H</sup> *<sup>k</sup>* **G***<sup>k</sup>* 

*<sup>k</sup>*(1) <sup>=</sup> arg max *<sup>k</sup>*∈K¯ *<sup>λ</sup>*min

**G**<sup>H</sup> K**G**K 

<sup>K</sup>MD <sup>=</sup> arg max <sup>K</sup>

and the ME criterion for the LR-based MMSE detection can also be modified as

<sup>K</sup>ME <sup>=</sup> arg max <sup>K</sup>

because the users have to be selected by combinatorial (or exhaustive) search.

**5. LR-based greedy user selection using an updating method**

LR updating method in conjunction with greedy user selection.

high as *<sup>M</sup>* or *<sup>K</sup>* increases, because there are *<sup>U</sup>* <sup>=</sup> <sup>∏</sup>*M*−<sup>1</sup>

complex-valued channel matrices should be carried out.

For example, if the ME criterion is used, we have

**5.1. LR-based greedy user selection**

algorithm is summarized as follows:

Since |*np*| <sup>2</sup> is a chi-square random variable with 2 degrees of freedom (or an exponential random variable), we have Pr |*np*| <sup>2</sup> < <sup>|</sup>*rp*,*<sup>p</sup>* <sup>|</sup> 2 4 = <sup>1</sup> − exp −|*rp*,*<sup>p</sup>* <sup>|</sup> 2 4*N*<sup>0</sup> . Thus, from (20), the probability of error can be given by

$$\Pr(\text{error}) \le 1 - \prod\_{p=1}^{Q} \left( 1 - \exp\left(\frac{|r\_{p,p}|^2}{4N\_0}\right) \right)$$

$$\simeq \exp\left(-\min\_p \frac{|r\_{p,p}|^2}{4N\_0}\right) \text{ as } N\_0 \to 0. \tag{21}$$

Therefore, to minimize the probability of error, the user who has the maximum min*<sup>p</sup>* |*rp*,*p*| can be selected.
