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

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Pr(no error) ≥ Pr

 |*np*|

Pr(error) ≤ 1 −

**4. Selection criteria with multiple users**

**4.1. ML and MMSE selection criteria**

MDist or ME user selection criterion as follows:

≃ exp

= *P* ∏ *p*=1 Pr |*np*|

<sup>2</sup> < <sup>|</sup>*rp*,*<sup>p</sup>* <sup>|</sup> 2 4 

> *Q* ∏ *p*=1

 − min *p*

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

To maximize the performance, if *M* = 1, the user who can have the minimum PEP is chosen for a given MIMO detector. In Section 3, a few user selection criteria are derived, depending on the types of actually employed MIMO detectors. Note that only one user is selected (i.e., *M* = 1) in Section 3. To extend the user selection criteria to the case of *M* > 1 here and in the consecutive sections, we consider the combinatorial and greedy user selection criteria.

For a given *M* > 1, the set of the users who can access the channel can be found using the

*λ*min **H**<sup>H</sup> K**H**K 

respectively. If the ML detector is employed, the MDist user selection criterion can be used to choose the *M* users who can have the lowest BER, whereas the ME criterion is used to

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

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

choose the *M* users with the highest worst SNR (i.e., max-min SNR).

1 − exp


4*N*<sup>0</sup>

 |*np*|

<sup>2</sup> <sup>&</sup>lt; <sup>|</sup>*rp*,1<sup>|</sup>

<sup>2</sup> is a chi-square random variable with 2 degrees of freedom (or an exponential

= 1 − exp

 |*rp*,*p*| 2

2 <sup>4</sup> , <sup>∀</sup>*<sup>p</sup>*

<sup>2</sup> <sup>&</sup>lt; <sup>|</sup>*rp*,*p*<sup>|</sup>

 −|*rp*,*<sup>p</sup>* <sup>|</sup> 2 4*N*<sup>0</sup> 

4*N*<sup>0</sup>

2 4

. (20)

. Thus, from (20), the

as *<sup>N</sup>*<sup>0</sup> → 0. (21)

S (**H**K) (22)

(23)

bounded as

Since |*np*|

can be selected.

or

random variable), we have Pr

probability of error can be given by

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

The MD criterion derived in Section 3, with *M* = 1 for the LR-based MMSE-SIC detection, can be extended to the case with *M* > 1 as follows:

$$\mathcal{K}\_{\text{MD}} = \arg\max\_{\mathcal{K}} \left\{ \min\_{q} \left| r\_{q,q}^{(\mathcal{K})} \right| \right\} \tag{24}$$

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

$$\mathcal{K}\_{\text{ME}} = \arg\max\_{\mathcal{K}} \lambda\_{\text{min}} \left( \mathbf{G}\_{\mathcal{K}}^{\text{H}} \mathbf{G}\_{\mathcal{K}} \right) . \tag{25}$$

The user selection based on (22), (23), (24), and (25) is called the combinatorial user selection, because the users have to be selected by combinatorial (or exhaustive) search.
