**5.3. Optimality of multi-cell RBF**

So far, we have characterized the achievable DoF region for the multi-cell RBF schemes that require only partial CSI at the transmitter. One important question that remains unaddressed yet is how the multi-cell RBF performs as compared to the optimal transmission scheme (e.g., IA) for the multi-cell downlink system with the full transmitter CSI, in terms of achievable DoF region. In this subsection, we attempt to partially answer this question by focusing on the high-SNR/DoF regime. Note that we only consider the multi-cell MIMO RBF schemes. Discussions for the MISO RBF can be drawn from either the RBF-MMSE, RBF-MF, or RBF-AS by setting *NR* = 1.

## *5.3.1. Single-cell case*

24 Recent Trends in Multiuser MIMO Communications

numerical *R*RBF−MMSE (1)

> *,m)* log2 ρ

> *,m)* log2 ρ

*<sup>d</sup>*RBF−MMSE,1*(*α*<sup>1</sup>*

numerical *R*RBF−MF (1)

numerical *R*RBF−AS (1)

We denote *<sup>d</sup>*RBF-Rx(*α*, *<sup>m</sup>*) <sup>=</sup> *<sup>d</sup>*RBF-Rx,1(*α*1, *<sup>m</sup>*) ,··· , *<sup>d</sup>*RBF-Rx,*C*(*αC*, *<sup>m</sup>*)

of receive diversity to multi-user diversity, therefore, is significant.

*<sup>d</sup>*RBF−MF/AS,1*(*α*<sup>1</sup>*

0

*5.2.2. DoF region characterization*

is given in the following theorem

*system is given by*

spatial receive diversity, and interference.

D*RBF-Rx*(*α*) = *conv*

*<sup>γ</sup>*2,1 = −1dB, and *<sup>K</sup>*<sup>1</sup> = ⌊*ρα*<sup>1</sup> ⌋.

5

10

15

20

Sum−rate [bps/Hz]

25

30

35

40

2 4 6 8 10 12 14 16 18

log2 ρ

**Figure 7.** Comparison of the numerical sum-rate and the DoF scaling law, with *C* = 2, [*M*1, *M*2]=[4, 2], *NR* = 3, *α*<sup>1</sup> = 1,

of characterizing the achievable sum-rate for RBF under the effects of multi-user diversity,

given in Lemma 5.3 or 5.4. The characterization of the DoF region for the multi-cell RBF-Rx

**Theorem 5.3.** *Assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* ), *<sup>c</sup>* <sup>=</sup> 1, . . . , *C, the achievable DoF region of a C-cell RBF-Rx*

*where conv denotes the convex hull operation and "Rx" stands for either MMSE, MF, or AS as usual.*

Fig. 8 depicts the DoF region of a two-cell system employing either the RBF-MMSE or RBF-MF/AS. We assume that *NT* = 4, *NR* = 2. The region's boundaries for RBF-MMSE and RBF-MF/AS are denoted by solid and dashed lines, respectively. When *α*<sup>1</sup> and *α*<sup>2</sup> are small, the DoF region is greatly expanded in the case of the RBF-MMSE. In fact, compared to the RBF-MF, RBF-AS, and MISO RBF, an *exponentially* less number of users in each cell is required to achieve a certain DoF region in the RBF-MMSE (see Lemma 5.3). The assistance

We see that receive diversity is indeed beneficial for RBF systems. However, there exists a tradeoff between the rate/DoF performance and the complexity/delay time. The option thus

*<sup>d</sup>RBF-Rx*(*α*, *<sup>m</sup>*), *Mc* ∈ {0, ··· , *NT*}, *<sup>c</sup>* = 1, ··· , *<sup>C</sup>*

*T*

, with *d*RBF-Rx,*c*(*αc*, *m*)

, (50)

First, we consider the single-cell case to draw some useful insights. It is well known that the maximum sum-rate DoF for a single-cell MIMO-BC with *NT* transmit antennas and *<sup>K</sup>* ≥ *NT* users each with *NR* receive antennas under independent channels is *NT* [1] [55], which is achievable by the DPC scheme or even simple linear precoding schemes. However, it is not immediately clear whether such a result still holds for the case of *<sup>K</sup>* <sup>=</sup> <sup>Θ</sup>(*ρα*) <sup>≫</sup> *NT* with *α* > 0, since in this case *NT* may be only a lower bound on the maximum DoF. We thus have the following proposition.

53

http://dx.doi.org/10.5772/57131

. (51)

*5.3.2. Multi-cell case*

that DRBF-Rx(*α*) ⊆ DMIMO(*α*) ⊆ DUB(*α*).

*bound for a C-cell downlink system is given by*

*i.e.,* D*RBF-Rx*(*α*) ≡ DUB(*α*)*, if*

*large per-cell user densities.*

**6. Summary**

<sup>D</sup>UB(*α*) =

*• RBF-MMSE: <sup>α</sup><sup>c</sup>* ≥ *CNT* − *NR,* ∀*<sup>c</sup>* ∈ {1, ··· , *<sup>C</sup>*}*. • RBF-MF/AS: <sup>α</sup><sup>c</sup>* ≥ *CNT* − <sup>1</sup>*,* ∀*<sup>c</sup>* ∈ {1, ··· , *<sup>C</sup>*}*.*

numbers of per-cell users are not so large.

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

**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*

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,*

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

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

In this chapter, we have introduced the OBF/RBF and the current developments in the literature. First of all, we have given an overview for the single-cell case, summarizing some of the most important contributions so far. Furthermore, we have reviewed the recent investigations on the rate performance of multi-cell RBF systems in both finite- and high-SNR regimes. These results are useful for the optimal design of multi-cell RBF in practical cellular systems. In particular, it is revealed that collaboration among the BSs in assigning their respective numbers of data beams based on different per-cell user densities is essential to achieve the optimal throughput tradeoffs among different cells. Moreover, the results show that spatial receive diversity is also a significant factor to be considered, noting that there

<sup>+</sup> : *dc* <sup>≤</sup> *NT*, *<sup>c</sup>* <sup>=</sup> 1, ··· , *<sup>C</sup>*

Random Beamforming in Multi – User MIMO Systems

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

(*d*1, *<sup>d</sup>*2, ··· *dC*) ∈ **<sup>R</sup>***<sup>C</sup>*

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

**Proposition 5.1.** *([24, Proposition 4.1]) Assuming K* = Θ(*ρα*) *with α* > 0*, the maximum sum-rate DoF of a single-cell MIMO-BC with NT transmit antennas is d*∗(*α*) = *NT.*

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 is given in the following proposition

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


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 number of users.

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.
