*5.3.2. Multi-cell case*

26 Recent Trends in Multiuser MIMO Communications

numerical *R*DPC numerical *R*RBF−MMSE

 *NT* log2 ρ

is given in the following proposition

*scheme is DoF-optimal if*

number of users.

*• RBF-MMSE: <sup>α</sup>* ≥ *NT* − *NR. • RBF-MF/AS: <sup>α</sup>* ≥ *NT* − <sup>1</sup>*.*

Sum−rate [bps/Hz]

2 3 4 5 6 7 8 9 10

**Proposition 5.1.** *([24, Proposition 4.1]) Assuming K* = Θ(*ρα*) *with α* > 0*, the maximum sum-rate*

Proposition 5.1 confirms that the maximum DoF of the MIMO-BC is still *NT*, even with the asymptotically large number of users that scales with SNR, i.e., multi-user diversity does not yield any increment of "interference-free" DoF. The optimality of the single-cell RBF schemes

**Proposition 5.2.** *([24, Proposition 4.4]) Assuming K* = Θ(*ρα*)*, the single-cell MIMO RBF-Rx*

From the above proposition, it follows that the single-cell MIMO RBF schemes achieve the maximum DoF with *M* = *NT* if the number of users is sufficiently large, thanks to the multi-user diversity effect that completely eliminates the intra-cell interference with a large

As an example, we compare the numerical sum-rates and the DoF scaling law in Fig. 9, in which the DPC and RBF-MMSE are employed, and *<sup>α</sup>* ≥ *NT* − *NR* . We consider two single-cell systems with the following parameters: (1) *<sup>M</sup>* <sup>=</sup> *NT* = 3, *NR* = 2, *<sup>α</sup>* = 1, *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋; and (2) *<sup>M</sup>* <sup>=</sup> *NT* = 4, *NR* = 3, *<sup>α</sup>* = 1.2, *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋. The rates and scaling law of system 1 and 2 are denoted as the solid and dash lines, respectively. Note that in both cases, the DPC and RBF-MMSE sum-rates follow the (same) DoF scaling law quite closely. This example clearly

demonstrates the optimality of the RBF given a large per-cell number of users.

**Figure 9.** Comparison of the numerical DPC and RBF-MMSE sum-rates, and the DoF scaling law with *α* ≥ *NT* − *NR*.

*DoF of a single-cell MIMO-BC with NT transmit antennas is d*∗(*α*) = *NT.*

log2 ρ For the convenience of analysis, we use DUB(*α*) to denote an upper bound on the DoF region defined in (40), for a given *α* in the multi-cell case. Clearly, under Assumption 1, it follows that DRBF-Rx(*α*) ⊆ DMIMO(*α*) ⊆ DUB(*α*).

The following proposition establishes the DoF region upper bound DUB(*α*).

**Proposition 5.3.** *([24, Proposition 4.3]) Given Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* )*, c* <sup>=</sup> 1, ··· , *C, a DoF region upper bound for a C-cell downlink system is given by*

$$\mathcal{D}\_{\text{UB}}(\mathfrak{a}) = \left\{ (d\_1, d\_2, \dots, d\_{\mathbb{C}}) \in \mathbb{R}\_+^{\mathbb{C}} : d\_{\mathbb{C}} \le \mathcal{N}\_{\text{T}}, \mathcal{c} = 1, \dots, \mathcal{C} \right\}. \tag{51}$$

The optimality of the multi-cell MIMO RBF-Rx schemes is shown in the following proposition

**Proposition 5.4.** *([24, Proposition 4.3]) Given Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* )*, c* <sup>=</sup> 1, ··· , *C, the multi-cell MIMO RBF-Rx scheme achieves the "interference-free" DoF region of a C-cell downlink system, i.e.,* D*RBF-Rx*(*α*) ≡ DUB(*α*)*, if*


*As a consequence, we also have* D*RBF-Rx*(*α*) ≡ D*MIMO*(*α*)*, i.e., the RBF-Rx scheme is optimal given large per-cell user densities.*

The above proposition implies that the multi-cell MIMO RBF schemes are indeed DoF-optimal when the numbers of users in all cells are sufficiently large. Due to the overwhelming multi-user diversity gain, RBF compensates the lack of full CSI at transmitters without any compromise of DoF degradation. However, it is important to point out that such a result should not undermine the benefits of having the more complete CSI at transmitters in practical multi-cell systems, where more sophisticated precoding schemes than RBF such as IA-based ones can be applied to achieve substantial throughput gains, especially when the numbers of per-cell users are not so large.
