**6.1. Diversity Gain Analysis from Error Probability**

Through the following diversity gain analysis, we can see the impact of each MIMO detector on the performance of multiuser systems.

### *6.1.1. Diversity Gain of Combinatorial User Selection with ML and MMSE Detectors*

Using the pairwise error probability (PEP), we can find the diversity order from multiple receive antennas as well as multiple user selection.

**Theorem 6.1.** *The average PEP of the ML detector with the M selected users under the MDist user selection criterion, denoted by P*ml *<sup>e</sup> , is upper-bounded as*

$$P\_{\varepsilon}^{\text{ml}} \le c\_1 \left(\frac{\|\sigma\_h^2 \mathbf{d}\|^2}{N\_0}\right)^{-N\left\lfloor\frac{K}{M}\right\rfloor} + o\left(\left(\frac{\|\sigma\_h^2 \mathbf{d}\|^2}{N\_0}\right)^{-N\left\lfloor\frac{K}{M}\right\rfloor + 1}\right),\tag{31}$$

*where c*<sup>1</sup> <sup>&</sup>gt; <sup>0</sup> *is constant, and* **<sup>d</sup>** <sup>=</sup> **<sup>s</sup>**(1) <sup>−</sup> **<sup>s</sup>**(2) *(here,* **<sup>s</sup>**(*i*) ∈ S *MP and* **<sup>s</sup>**(1) �<sup>=</sup> **<sup>s</sup>**(2)*).*

*Proof.* See Section 8.1.

14 Recent Trends in Multiuser MIMO Communications

(*iP*)3*N* log(*iP*)

*h* .

*selection criterion, denoted by P*ml

on the performance of multiuser systems.

*P* ml *<sup>e</sup>* <sup>≤</sup> *<sup>c</sup>*<sup>1</sup>

receive antennas as well as multiple user selection.

 �*σ*<sup>2</sup> *h***d**�<sup>2</sup> *N*<sup>0</sup>

when large *K* and *M* are considered.

*<sup>i</sup>*=1(*<sup>K</sup>* <sup>−</sup> *<sup>i</sup>* <sup>+</sup> <sup>1</sup>)*<sup>O</sup>*

with variance *σ*<sup>2</sup>

(*iP*)3*N* log(*iP*)

**6. Diversity Analysis and Numerical Results**

**6.1. Diversity Gain Analysis from Error Probability**

<sup>∑</sup>*M*−<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *<sup>O</sup>*

∑*<sup>M</sup>*

denotes the case of *K* = 10, *N* = 8, (*M*, *P*)=(4, 2), respectively.

**Average value of** *η* Number of columns in **H**K 2 3 4 5 6 7 8 Sum

Note that the superscript <sup>1</sup> denotes the case of *K* = 10, *N* = 8, (*M*, *P*)=(8, 1) and the superscript <sup>2</sup>

possible cases of (*M*, *P*)=(8, 1) and (*M*, *P*)=(4, 2). Based on these results, we can observe that the complexity is significantly reduced if UBLR is employed. We also note that with the LRG, the complexity for the case of (*M*, *P*)=(8, 1) is higher than that of (*M*, *P*)=(4, 2) as expected (a large *M* implies a higher complexity). We can also show

In this section, we consider the diversity gain of the combinatorial user selection approaches with various detectors, such as the ML, MMSE, and LR-based SIC detectors. We derive lower bounds on the diversity gain of them. Since the diversity gain analysis of the proposed greedy user selection approach is difficult, we rely on simulations, from which we can show that our proposed LRG/UBLRG user selection approach has a similar diversity gain and comparable performance to the combinatorial one. Throughout this section, we assume that the elements of the channel matrix **H**K are independent zero-mean CSCG random variables

Through the following diversity gain analysis, we can see the impact of each MIMO detector

Using the pairwise error probability (PEP), we can find the diversity order from multiple

**Theorem 6.1.** *The average PEP of the ML detector with the M selected users under the MDist user*

 �*σ*<sup>2</sup> *h***d**�<sup>2</sup> *N*<sup>0</sup>

<sup>−</sup>*N*⌊ *<sup>K</sup> <sup>M</sup>* ⌋+<sup>1</sup> 

*6.1.1. Diversity Gain of Combinatorial User Selection with ML and MMSE Detectors*

*<sup>e</sup> , is upper-bounded as*

<sup>−</sup>*N*⌊ *<sup>K</sup> M* ⌋ + *o*

*where c*<sup>1</sup> <sup>&</sup>gt; <sup>0</sup> *is constant, and* **<sup>d</sup>** <sup>=</sup> **<sup>s</sup>**(1) <sup>−</sup> **<sup>s</sup>**(2) *(here,* **<sup>s</sup>**(*i*) ∈ S *MP and* **<sup>s</sup>**(1) �<sup>=</sup> **<sup>s</sup>**(2)*).*

. Compared to the complexity of LRG which is upper-bounded as

, the UBLRG scheme has a lower complexity, especially

(*MP*)3*N* log(*MP*)

, (31)

 +

**Table 2.** The average value of *η* in the LRG and UBLRG user selection with the CLLL based MMSE-SIC detector is used.

that the complexity of UBLRG is upper-bounded as (*K* − *M* + 1)*O*

LRG<sup>1</sup> 0.2909 0.9029 1.8022 3.0633 4.7711 7.2925 12.1228 30.2457 UBLRG1 0.2904 0.5851 0.8940 1.2708 1.7653 2.5620 4.7728 12.1404 LRG<sup>2</sup> 0.2926 n/a 1.7977 n/a 4.7663 n/a 12.0856 18.9422 UBLRG2 0.2879 n/a 1.4952 n/a 3.0191 n/a 7.3761 12.1783

This theorem shows that a full receive diversity gain of *N* together with a partial multiuser diversity gain of at least ⌊ *<sup>K</sup> <sup>M</sup>* ⌋ can be achieved by the ML detector under the MDist user selection criterion. This result is derived under the fact that there are at least ⌊ *<sup>K</sup> <sup>M</sup>* ⌋ statistically independent alternative combinations of the composite channel matrix **H**<sup>K</sup> for *M* users. Hence, this result is a lower bound on the diversity gain. In fact, there are more combinations for **H**K, which are not independent, that can increase the multiuser diversity gain. By simulations, we will further demonstrate the impact of the combinations of *M* selected users that are not independent.

**Theorem 6.2.** *The average PEP of the MMSE detector with the selected M users under the ME user selection criterion, denoted by P*mmse *<sup>e</sup> , is upper-bounded as*

$$P\_{\varepsilon}^{\text{mass}} \le c\_2 \left(\frac{\sigma\_h^2 \|\mathbf{d}\|^2}{N\_0}\right)^{-(N-P+1)\lfloor\frac{K}{M}\rfloor} + o\left(\left(\frac{\sigma\_h^2 \|\mathbf{d}\|^2}{N\_0}\right)^{-(N-P+1)\lfloor\frac{K}{M}\rfloor+1}\right),\tag{32}$$

*where c*<sup>2</sup> > 0 *is constant.*

*Proof.* See Section 8.2.

This theorem shows that for the MMSE detector, the ME user selection criterion may not be able to exploit a full receive diversity.

#### *6.1.2. Diversity Gain of Combinatorial User Selection with LR-based Detector*

**Theorem 6.3.** *The average PEP of the LR-based SIC detector with the selected M users under the MD user selection criterion, denoted by P*lr *<sup>e</sup> , is upper-bounded as*

$$P\_e^{\mathbb{I}^\*} \le c\_3 \left(\frac{||\sigma\_h^2 \mathbf{d}||^2}{N\_0}\right)^{-N\left\lfloor\frac{K}{M}\right\rfloor} + o\left(\left(\frac{||\sigma\_h^2 \mathbf{d}||^2}{N\_0}\right)^{-N\left\lfloor\frac{K}{M}\right\rfloor+1}\right),\tag{33}$$

*where c*<sup>3</sup> > 0 *is constant.*

*Proof.* See Section 8.3.

This theorem shows that a full receive diversity gain of *N* together with the same partial multiuser diversity gain, ⌊ *<sup>K</sup> <sup>M</sup>* ⌋, as with the ML detector, can be achieved by the LR-based detector under the MD user selection criterion. From these results, we can see that the LR-based detector is as good as the ML detector with respect to the diversity gains.
