*4.1.2. Asymptotic sum rate as Kc* → ∞

An attempt to extend the famous scaling law *M* log2 log *K* to the multi-cell RBF case has been made in [53] based on an approximation of the SINR's PDF (which is applicable if

**Figure 2.** Comparison of the numerical sum rate and the sum-rate scaling law for RBF.

the SNR and INRs are all roughly equal), by showing that the same asymptotic sum-rate *Mc* log2 log *Kc* for each individual cell holds as the single-cell case. However, we note that with the exact SINR distributions in Lemma 4.1, a rigorous proof can be obtained, leading to the following proposition.

**Proposition 4.1.** *([22, Proposition 3.1]) For fixed Mc's and PT, c* = 1, ··· , *C, we have*

$$\lim\_{K\_{\mathcal{C}} \to \infty} \frac{R\_{RBF}^{(\mathcal{C})}}{M\_{\mathcal{C}} \log\_2 \log K\_{\mathcal{C}}} = 1. \tag{34}$$

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41

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*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*, the CDF of the random*

Random Beamforming in Multi – User MIMO Systems

*<sup>η</sup><sup>c</sup> <sup>s</sup>*)*Ml .*

, (35)

*ds<sup>j</sup>* , (36)

. (37)

. (38)

**4.2. Multi-cell MIMO RBF**

*variable S* := *SINR*(*MMSE*,*c*)

three cases.

*in which*

To study the achievable sum-rates of the RBF-Rx schemes, it is necessary to investigate the SINRs given in (7), (9), and (12). The following lemmas establish the SINR distributions in

> � <sup>∑</sup>*NR*−<sup>1</sup> *<sup>i</sup>*=<sup>0</sup> *<sup>ζ</sup>is<sup>i</sup>*

(<sup>1</sup> <sup>+</sup> *<sup>s</sup>*)*Mc*−<sup>1</sup> <sup>∏</sup><sup>∑</sup> *Mc*−<sup>1</sup>

*NR*−1 ∑ *k*=0

*k* ∑ *j*=0 �

*<sup>η</sup><sup>c</sup> <sup>s</sup>*)*Ml*

*<sup>l</sup>*=1,*l*�=*<sup>c</sup>* (<sup>1</sup> <sup>+</sup> *<sup>µ</sup>l*,*<sup>c</sup>*

*<sup>k</sup>*,*<sup>m</sup> can be expressed as*

*k*−*j c dj T*0

*<sup>η</sup><sup>c</sup> <sup>s</sup>*)*Ml*

*<sup>k</sup>*,*<sup>m</sup> can be expressed as*

*NR*

*<sup>l</sup>*=1,*l*�=*<sup>c</sup>* (<sup>1</sup> <sup>+</sup> *<sup>µ</sup>l*,*<sup>c</sup>*

(−1)*<sup>j</sup> sk*

(*k* − *j*)!*j*!*η*

*<sup>l</sup>*=1,*l*�=*<sup>c</sup>* (<sup>1</sup> <sup>+</sup> *<sup>µ</sup>l*,*<sup>c</sup>*

**Lemma 4.2.** *([24, Corollary 3.1], see also [54]) Given NR* ≤ <sup>∑</sup>*<sup>C</sup>*

*FS*(*s*) = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*s*/*η<sup>c</sup>*

−*s*/*η<sup>c</sup>*

*<sup>T</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> (1 + *s*)*Mc*−<sup>1</sup> ∏*<sup>C</sup>*

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*s*/*η<sup>c</sup>*

*Mc*−1

In Fig. 3, we present the per-cell SINR CDFs of the RBF-MMSE, RBF-MF, and RBF-AS with the following setup: *<sup>C</sup>* = 4, *<sup>η</sup>*<sup>1</sup> = 20 dB, *NR* = 3, *<sup>M</sup>*<sup>1</sup> = 3, [*µ*2,1, *<sup>µ</sup>*3,1, *<sup>µ</sup>*4,1] = [0, −3, 3] dB, and [*M*2, *M*3, *M*4] = [3, 2, 4]. The numerical results are obtained with Monte-Carlo simulations, while the analytical results are computed based on Lemma 4.2, 4.3, and 4.4. As a reference, we also present the numerical CDF of the MISO RBF with single-antenna users. It is observed that spatial receive diversity does help improving the SINR performance. However, only in the RBF-MMSE scheme that there exists a tremendous gain. Finally, we note that the SINR

With Lemma 4.2, 4.3, and 4.4, it is possible to extend Theorem 4.1 to multi-cell MIMO RBF systems. The results posses complicated expressions and, again, does not lead to useful

*C* ∏ *l*=1,*l*�=*c* � *µl*,*<sup>c</sup> ηc s* + 1 �*Ml*

(*s* + 1)

performance of the RBF-MF is *not* always better than that of the RBF-AS.

insights. We are more interested in the following proposition

*in which <sup>ζ</sup><sup>i</sup> is the coefficient of s<sup>i</sup> in the product* (<sup>1</sup> <sup>+</sup> *<sup>s</sup>*)*Mc*−<sup>1</sup> <sup>∏</sup><sup>∑</sup> *Mc*−<sup>1</sup>

**Lemma 4.3.** *([24, Theorem 3.2]) The CDF of S* := *SINR*(*MF*,*c*)

*FS*(*s*) = <sup>1</sup> − *<sup>e</sup>*

**Lemma 4.4.** *([24, Corollary 3.2]) The CDF of S* := *SINR*(*AS*,*c*)

*FS*(*s*) =

*<sup>k</sup>*,*<sup>m</sup> in (7) is*

Fig. 2 depicts both the numerical and theoretical asymptotic sum-rates for a single-cell RBF system, and the first cell of a two-cell RBF system. In the single-cell case, *η* = 10 dB, and *M* = *NT* = 4, while in the two-cell case, *M*<sup>1</sup> = *M*<sup>2</sup> = *NT* = 4, *η*<sup>1</sup> = 10 dB, and *µ*2,1 = 6 dB. We observe that the convergence to the sum-rate scaling law *Mc* log2 log *Kc* is rather slow in both cases. For example, even with *K* or *Kc* to be 105, the convergence is still not clear. In fact, Fig. 2 indicates that the sum-rate might follow a (single-) logarithmically scaling law. Furthermore, Proposition 4.1 implies that the sum-rate scaling law *Mc* log2 log *Kc* holds for any cell regardless of the ICI as *Kc* → ∞. As a consequence, this result implies that each cell should apply the maximum number of transmit beams, i.e., *Mc* = *NT*, ∀*c*, to maximize the per-cell throughput. Such a conclusion may be misleading in a practical multi-cell system with non-negligible ICI. The above two main drawbacks, namely, slow convergence and misleading conclusion, limit the usefulness of the conventional sum-rate scaling law *Mc* log2 log *Kc* for the multi-cell RBF.

#### **4.2. Multi-cell MIMO RBF**

14 Recent Trends in Multiuser MIMO Communications

single cell first cell scaling law Mc

**Figure 2.** Comparison of the numerical sum rate and the sum-rate scaling law for RBF.

the following proposition.

Sum Rate (bps/Hz)

Number of users

10<sup>1</sup> 10<sup>2</sup> 10<sup>3</sup> 10<sup>4</sup> 10<sup>5</sup>

the SNR and INRs are all roughly equal), by showing that the same asymptotic sum-rate *Mc* log2 log *Kc* for each individual cell holds as the single-cell case. However, we note that with the exact SINR distributions in Lemma 4.1, a rigorous proof can be obtained, leading to

> *<sup>R</sup>*(*c*) *RBF Mc* log2 log *Kc*

Fig. 2 depicts both the numerical and theoretical asymptotic sum-rates for a single-cell RBF system, and the first cell of a two-cell RBF system. In the single-cell case, *η* = 10 dB, and *M* = *NT* = 4, while in the two-cell case, *M*<sup>1</sup> = *M*<sup>2</sup> = *NT* = 4, *η*<sup>1</sup> = 10 dB, and *µ*2,1 = 6 dB. We observe that the convergence to the sum-rate scaling law *Mc* log2 log *Kc* is rather slow in both cases. For example, even with *K* or *Kc* to be 105, the convergence is still not clear. In fact, Fig. 2 indicates that the sum-rate might follow a (single-) logarithmically scaling law. Furthermore, Proposition 4.1 implies that the sum-rate scaling law *Mc* log2 log *Kc* holds for any cell regardless of the ICI as *Kc* → ∞. As a consequence, this result implies that each cell should apply the maximum number of transmit beams, i.e., *Mc* = *NT*, ∀*c*, to maximize the per-cell throughput. Such a conclusion may be misleading in a practical multi-cell system with non-negligible ICI. The above two main drawbacks, namely, slow convergence and misleading conclusion, limit the usefulness of the conventional sum-rate

= 1. (34)

**Proposition 4.1.** *([22, Proposition 3.1]) For fixed Mc's and PT, c* = 1, ··· , *C, we have*

lim *Kc*→∞

scaling law *Mc* log2 log *Kc* for the multi-cell RBF.

log2

log K <sup>c</sup>

To study the achievable sum-rates of the RBF-Rx schemes, it is necessary to investigate the SINRs given in (7), (9), and (12). The following lemmas establish the SINR distributions in three cases.

**Lemma 4.2.** *([24, Corollary 3.1], see also [54]) Given NR* ≤ <sup>∑</sup>*<sup>C</sup> <sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*, the CDF of the random variable S* := *SINR*(*MMSE*,*c*) *<sup>k</sup>*,*<sup>m</sup> in (7) is*

$$F\_{\mathbb{S}}(\mathbf{s}) = 1 - \frac{e^{-s/\eta\_{\mathbb{C}}} \left(\sum\_{i=0}^{N\_{\mathbb{R}}-1} \zeta\_{i} \mathbf{s}^{i}\right)}{(1+s)^{M\_{\mathbb{C}}-1} \prod\_{l=1, l \neq \boldsymbol{\mathcal{L}}}^{\sum M\_{\mathbb{C}}-1} (1 + \frac{\mu\_{l\boldsymbol{\mathcal{L}}}}{\eta\_{\boldsymbol{\mathcal{L}}}} \mathbf{s})^{M\_{\mathbb{I}}}},\tag{35}$$

*in which <sup>ζ</sup><sup>i</sup> is the coefficient of s<sup>i</sup> in the product* (<sup>1</sup> <sup>+</sup> *<sup>s</sup>*)*Mc*−<sup>1</sup> <sup>∏</sup><sup>∑</sup> *Mc*−<sup>1</sup> *<sup>l</sup>*=1,*l*�=*<sup>c</sup>* (<sup>1</sup> <sup>+</sup> *<sup>µ</sup>l*,*<sup>c</sup> <sup>η</sup><sup>c</sup> <sup>s</sup>*)*Ml .*

**Lemma 4.3.** *([24, Theorem 3.2]) The CDF of S* := *SINR*(*MF*,*c*) *<sup>k</sup>*,*<sup>m</sup> can be expressed as*

$$F\_{\mathbb{S}}(s) = 1 - e^{-s/\eta\_{\mathbb{C}}} \sum\_{k=0}^{N\_{\mathbb{R}}-1} \sum\_{j=0}^{k} \frac{(-1)^j s^k}{(k-j)! j! \eta\_{\mathbb{C}}^{k-j}} \frac{d^j T\_0}{ds^j},\tag{36}$$

*in which*

$$T\_0 = \frac{1}{(1+s)^{M\_\ell - 1} \prod\_{l=1, l \neq c}^{C} (1 + \frac{\mu\_{l\varepsilon}}{\eta\_{\varepsilon}}s)^{M\_l}}.\tag{37}$$

**Lemma 4.4.** *([24, Corollary 3.2]) The CDF of S* := *SINR*(*AS*,*c*) *<sup>k</sup>*,*<sup>m</sup> can be expressed as*

$$F\_{\mathcal{S}}(s) = \left(1 - \frac{e^{-s/\eta\_c}}{(s+1)^{M\_\mathcal{L}-1} \prod\_{l=1, l\neq c}^{\mathbb{C}} \left(\frac{\mu\_{l,c}}{\eta\_c}s+1\right)^{M\_l}}\right)^{N\_R}.\tag{38}$$

In Fig. 3, we present the per-cell SINR CDFs of the RBF-MMSE, RBF-MF, and RBF-AS with the following setup: *<sup>C</sup>* = 4, *<sup>η</sup>*<sup>1</sup> = 20 dB, *NR* = 3, *<sup>M</sup>*<sup>1</sup> = 3, [*µ*2,1, *<sup>µ</sup>*3,1, *<sup>µ</sup>*4,1] = [0, −3, 3] dB, and [*M*2, *M*3, *M*4] = [3, 2, 4]. The numerical results are obtained with Monte-Carlo simulations, while the analytical results are computed based on Lemma 4.2, 4.3, and 4.4. As a reference, we also present the numerical CDF of the MISO RBF with single-antenna users. It is observed that spatial receive diversity does help improving the SINR performance. However, only in the RBF-MMSE scheme that there exists a tremendous gain. Finally, we note that the SINR performance of the RBF-MF is *not* always better than that of the RBF-AS.

With Lemma 4.2, 4.3, and 4.4, it is possible to extend Theorem 4.1 to multi-cell MIMO RBF systems. The results posses complicated expressions and, again, does not lead to useful insights. We are more interested in the following proposition

**Figure 3.** Comparison of the numerical and analytical CDFs of the per-cell SINR.

**Proposition 4.2.** *([24, Proposition 4.2]) For fixed Mc, NR and PT, c* = 1, ··· , *C, we have*

$$\lim\_{K\_{\mathcal{C}} \to \infty} \frac{R\_{\mathrm{RBF-Rx}}^{(\mathcal{C})}}{M\_{\mathcal{C}} \log\_2 \log K\_{\mathcal{C}}} = 1,\tag{39}$$

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<sup>+</sup>; (40)

. (41)

log *<sup>ρ</sup>* <sup>→</sup> *<sup>α</sup>* with

 ,

Random Beamforming in Multi – User MIMO Systems

In this section, we introduce the high-SNR/DoF analysis and the studies on multi-cell MISO/MIMO RBF. We first specify the DoF region, which was defined in [9] and is restated

**Definition 5.1.** *(General DoF region) The DoF region of a C-cell MIMO-BC system is defined as*

<sup>+</sup> : <sup>∀</sup>(*ω*1, *<sup>ω</sup>*2, ··· , *<sup>ω</sup>C*) <sup>∈</sup> **<sup>R</sup>***<sup>C</sup>*

*C* ∑ *c*=1 *ωc <sup>R</sup>*(*c*) *sum* log2 *<sup>ρ</sup>*

*sum are the non-negative rate weight, the achievable DoF,*

+;

*C* ∑ *c*=1 *ωc <sup>R</sup>*(*c*) *RBF-Rx* log2 *<sup>ρ</sup>*

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

*C* ∑ *c*=1

*sum*, *<sup>R</sup>*(2)

*ωcdc* ≤ lim

*sum*, ··· , *<sup>R</sup>*(*C*) *sum*)*.*

<sup>+</sup> : <sup>∀</sup>(*ω*1, *<sup>ω</sup>*2, ··· , *<sup>ω</sup>C*) <sup>∈</sup> **<sup>R</sup>***<sup>C</sup>*

max *Mc*∈{0,··· ,*NT* }

*and the sum rate of the c-th cell, respectively; and the region* R *is the set of all the achievable sum-rate*

If the multi-cell MIMO RBF schemes are deployed, the achievable DoF region defined in (40)

**Definition 5.2.** *(DoF region with RBF) The DoF region of a C-cell MIMO RBF system is defined as*

*ωcdc* ≤ lim *ρ*→∞ 

Furthermore, the DoF regions are denoted as DMISO and DRBF, respectively, under a

Note that the DoF region is, in general, applicable for any number of users per cell, *Kc*. However, if it is assumed that all *Kc*'s are constant with *ρ* → ∞, it can be shown that the DoF regions for the multi-cell RBF schemes given in (41) will collapse to the null point, i.e., a zero DoF for all the cells, due to the intra-/inter-cell interference2. It thus follows that for the analytical tractability, the DoF region characterization for the multi-cell RBF requires that *Kc* increases in a certain order with the SNR, *ρ*. [24] [22] thus make the following assumption3: **Assumption 1.** *The number of users in each cell scales with <sup>ρ</sup> in the order of <sup>ρ</sup>α<sup>c</sup> , with <sup>α</sup><sup>c</sup>* <sup>≥</sup> <sup>0</sup>*, denoted by Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ), *<sup>c</sup>* <sup>=</sup> 1, . . . , *C, i.e., Kc*/*ρα<sup>c</sup>* <sup>→</sup> *<sup>δ</sup> as <sup>ρ</sup>* <sup>→</sup> <sup>∞</sup>*, with <sup>δ</sup> being a positive constant*

<sup>2</sup> A rigorous proof of this claim can be deduced from Theorem 5.2 and 5.3 later for the special case of *α<sup>c</sup>* = 0, ∀*c*, i.e.,

<sup>3</sup> In [23], Tajer *et. al.* consider a more general assumption on the number of ad-hoc SISO links *n*, i.e., log *<sup>n</sup>*

*<sup>ρ</sup>*→<sup>∞</sup> sup *R*∈R

below for convenience.

is reduced to

*independent of αc.*

*α* > 0.

all cells having a constant number of users.

D*RBF-Rx* =

D*MIMO* =

*where <sup>ρ</sup> is the per-cell SNR; <sup>ω</sup>c, dc, and R*(*c*)

*tuples for all the cells, denoted by <sup>R</sup>* = (*R*(1)

*in which "Rx" denotes either MMSE, MF or AS.*

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

*C* ∑ *c*=1

multi-cell MISO setup. Certainly, DRBF ⊆ DMISO and DRBF-Rx ⊆ DMIMO.

*where "Rx" denotes any receive diversity scheme at the users, e.g., MMSE, MF, or AS.*

Essentially, this proposition extends Proposition 4.1 to the multi-cell MIMO case. Note that a similar result for a single-cell setup is obtained in [20] [21]. The proposition implies that spatial receive diversity only provides *marginal* gain to the rate performance of multi-cell MIMO RBF systems. However, as shown in Section 4.1.2, the traditional asymptotic analysis might not accurately characterize the RBF sum-rate in the condition of a small number of users (even up to *K*, *Kc* = 105). This negative conclusion on the receive diversity, therefore, might be misleading. The benefit of receive diversity to multi-cell RBF systems will be discussed further in Section 5.2.

#### **5. Multi-cell RBF: High-SNR analysis and the degrees of freedom region**

Motivated by the major limitations of the finite-SNR analysis, [22] and [23] propose using the high-SNR/DoF analysis to study the rate performance of multi-cell/IC systems with opportunistic scheduling. In particular, the multi-cell MISO RBF and ad-hoc SISO IC are considered in [22] and [23], respectively. A further extension is [24], in which the multi-cell MIMO RBF schemes are investigated to reveal the benefit of receive diversity. It has been shown that the high-SNR/DoF analysis provides a succinct description of the interplay between the multi-user diversity, spatial receive diversity and spatial multiplexing gains achievable in RBF.

In this section, we introduce the high-SNR/DoF analysis and the studies on multi-cell MISO/MIMO RBF. We first specify the DoF region, which was defined in [9] and is restated below for convenience.

**Definition 5.1.** *(General DoF region) The DoF region of a C-cell MIMO-BC system is defined as*

$$\mathcal{D}\_{\text{MIMO}} = \left\{ (d\_1, d\_2, \dots, d\_{\mathbb{C}}) \in \mathbb{R}\_+^{\mathbb{C}} : \forall (\omega\_1, \omega\_2, \dots, \omega\_{\mathbb{C}}) \in \mathbb{R}\_+^{\mathbb{C}}; \right. \tag{40}$$

$$\sum\_{c=1}^{\mathbb{C}} \omega\_c d\_c \le \lim\_{\rho \to \infty} \sup\_{\mathbf{R} \in \mathcal{R}} \sum\_{c=1}^{\mathbb{C}} \omega\_c \frac{R\_{\text{sum}}^{(c)}}{\log\_2 \rho} \},$$

*where <sup>ρ</sup> is the per-cell SNR; <sup>ω</sup>c, dc, and R*(*c*) *sum are the non-negative rate weight, the achievable DoF, and the sum rate of the c-th cell, respectively; and the region* R *is the set of all the achievable sum-rate tuples for all the cells, denoted by <sup>R</sup>* = (*R*(1) *sum*, *<sup>R</sup>*(2) *sum*, ··· , *<sup>R</sup>*(*C*) *sum*)*.*

If the multi-cell MIMO RBF schemes are deployed, the achievable DoF region defined in (40) is reduced to

**Definition 5.2.** *(DoF region with RBF) The DoF region of a C-cell MIMO RBF system is defined as*

$$\mathcal{D}\_{\text{RBF-Rx}} = \left\{ (d\_1, d\_2, \dots, d\_{\text{C}}) \in \mathbb{R}\_+^{\mathbb{C}} : \forall (\omega\_1, \omega\_2, \dots, \omega\_{\text{C}}) \in \mathbb{R}\_+^{\mathbb{C}}; \right. $$

$$\sum\_{c=1}^{\mathbb{C}} \omega\_c d\_c \le \lim\_{\rho \to \infty} \left[ \max\_{M\_c \in \{0, \dots, N\_{\text{Tr}}\}} \sum\_{c=1}^{\mathbb{C}} \omega\_c \frac{R\_{\text{RBF-Rx}}^{(c)}}{\log\_2 \rho} \right] \}. \tag{41}$$

*in which "Rx" denotes either MMSE, MF or AS.*

16 Recent Trends in Multiuser MIMO Communications

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

discussed further in Section 5.2.

achievable in RBF.

**Figure 3.** Comparison of the numerical and analytical CDFs of the per-cell SINR.

 *F(1)*

*S (s)*

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>

RBF−MMSE, numerical RBF−MMSE, analytical RBF−MF, numerical RBF−MF, analytical RBF−AS, numerical RBF−AS, analytical RBF, numerical

= 1, (39)

 *s*

**Proposition 4.2.** *([24, Proposition 4.2]) For fixed Mc, NR and PT, c* = 1, ··· , *C, we have*

*<sup>R</sup>*(*c*) *RBF-Rx Mc* log2 log *Kc*

Essentially, this proposition extends Proposition 4.1 to the multi-cell MIMO case. Note that a similar result for a single-cell setup is obtained in [20] [21]. The proposition implies that spatial receive diversity only provides *marginal* gain to the rate performance of multi-cell MIMO RBF systems. However, as shown in Section 4.1.2, the traditional asymptotic analysis might not accurately characterize the RBF sum-rate in the condition of a small number of users (even up to *K*, *Kc* = 105). This negative conclusion on the receive diversity, therefore, might be misleading. The benefit of receive diversity to multi-cell RBF systems will be

**5. Multi-cell RBF: High-SNR analysis and the degrees of freedom region** Motivated by the major limitations of the finite-SNR analysis, [22] and [23] propose using the high-SNR/DoF analysis to study the rate performance of multi-cell/IC systems with opportunistic scheduling. In particular, the multi-cell MISO RBF and ad-hoc SISO IC are considered in [22] and [23], respectively. A further extension is [24], in which the multi-cell MIMO RBF schemes are investigated to reveal the benefit of receive diversity. It has been shown that the high-SNR/DoF analysis provides a succinct description of the interplay between the multi-user diversity, spatial receive diversity and spatial multiplexing gains

lim *Kc*→∞

*where "Rx" denotes any receive diversity scheme at the users, e.g., MMSE, MF, or AS.*

Furthermore, the DoF regions are denoted as DMISO and DRBF, respectively, under a multi-cell MISO setup. Certainly, DRBF ⊆ DMISO and DRBF-Rx ⊆ DMIMO.

Note that the DoF region is, in general, applicable for any number of users per cell, *Kc*. However, if it is assumed that all *Kc*'s are constant with *ρ* → ∞, it can be shown that the DoF regions for the multi-cell RBF schemes given in (41) will collapse to the null point, i.e., a zero DoF for all the cells, due to the intra-/inter-cell interference2. It thus follows that for the analytical tractability, the DoF region characterization for the multi-cell RBF requires that *Kc* increases in a certain order with the SNR, *ρ*. [24] [22] thus make the following assumption3:

**Assumption 1.** *The number of users in each cell scales with <sup>ρ</sup> in the order of <sup>ρ</sup>α<sup>c</sup> , with <sup>α</sup><sup>c</sup>* <sup>≥</sup> <sup>0</sup>*, denoted by Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ), *<sup>c</sup>* <sup>=</sup> 1, . . . , *C, i.e., Kc*/*ρα<sup>c</sup>* <sup>→</sup> *<sup>δ</sup> as <sup>ρ</sup>* <sup>→</sup> <sup>∞</sup>*, with <sup>δ</sup> being a positive constant independent of αc.*

<sup>2</sup> A rigorous proof of this claim can be deduced from Theorem 5.2 and 5.3 later for the special case of *α<sup>c</sup>* = 0, ∀*c*, i.e., all cells having a constant number of users.

<sup>3</sup> In [23], Tajer *et. al.* consider a more general assumption on the number of ad-hoc SISO links *n*, i.e., log *<sup>n</sup>* log *<sup>ρ</sup>* <sup>→</sup> *<sup>α</sup>* with *α* > 0.

Considering the number of per-cell users to scale polynomially with the SNR is general as well as convenient. The linear scaling law, i.e., *Kc* = *βcρ*, with constant *β<sup>c</sup>* > 0, is only a special case of *Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ) with *<sup>α</sup><sup>c</sup>* <sup>=</sup> 1; if *Kc* is a constant, then the corresponding *α<sup>c</sup>* is zero. As will be shown later in this section, Assumption 1 enables us to obtain an efficient as well as insightful characterization of the DoF region for the multi-cell RBF. Note that the DoF region under Assumption 1 can be considered as a generalization of the conventional DoF region analysis based on IA [9] for the case of finite number of users, to the case of asymptotically large number of users that scales with the SNR. For the notational convenience, we use DMISO(*α*), DMIMO(*α*), DRBF(*α*), and DRBF-Rx(*α*) to denote the achievable DoF regions corresponding to *Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ), *<sup>c</sup>* = 1,··· ,*C*, and *<sup>α</sup>* = [*α*1, ··· , *<sup>α</sup>C*] *T*.

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Random Beamforming in Multi – User MIMO Systems

(44)

(45)

*RBF*(1) = 2 from (45).

integer DoF in the literature with a finite *K*). Moreover, it is observed that for any given <sup>0</sup> < *<sup>α</sup>* < *NT* − 1, assigning more transmit beams by increasing *<sup>M</sup>* initially improves the sum-rate DoF if *M* ≤ *α* + 1; however, as *M* > *α* + 1, the DoF may not necessarily increase with *M* due to the more dominant inter-beam/intra-cell interference. Note that the term *M* −1 in the denominator of (43a) is exactly the number of interfering beams to one particular beam. Thus, Lemma 5.1 provides a succinct description of the interplay between the available multi-user diversity (specified by *α* with a larger *α* denoting a higher user density or the number of users in a cell), the level of the intra-cell interference (specified by *M* − 1), and the

It is also of great interest to obtain the maximum achievable DoF for a given *α*. The result is

**Theorem 5.1.** *([22, Theorem 4.1]) For the single-cell MISO RBF with NT transmit antennas and user density coefficient α, the maximum achievable DoF and the corresponding optimal number of*

*NT*, *<sup>α</sup>* > *NT* − 1.

*NT*, *<sup>α</sup>* > *NT* − 1.

In Fig. 4, we compare the numerical sum rate and the analytical scaling law in Lemma 5.1. It is observed that the newly obtained sum-rate scaling law, *<sup>R</sup>*RBF <sup>≈</sup> *<sup>d</sup>*RBF(*α*, *<sup>M</sup>*)log2 *<sup>ρ</sup>*, in the single-cell RBF case is very accurate, even for small values of SNR *ρ* and number of users *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋. Compared with Fig. 2 for the conventional scaling law, *<sup>R</sup>*RBF <sup>≈</sup> *<sup>M</sup>* log2 log *<sup>K</sup>*, a much faster convergence is observed here. The DoF approach thus provides a more efficient way of characterizing the achievable sum rate for single-cell RBF. Also observe that the sum rate for *M* = 2 is higher than that for *M* = 4. This is because with *NT* = 4 and *α* = 1 in this

Since many previous studies have observed that adjusting the number of beams according to the number of users in single-cell RBF can improve the achievable sum rate (see, e.g., [46] [33] [48]), our study here can be considered as a theoretical explanation for such an observation. Fig. 5 shows the maximum DoF and the corresponding optimal number of transmit beams versus the user density coefficient *α* with *NT* = 4 for single-cell RBF, according to Theorem 5.1. It is observed that to maximize the achievable sum rate, we should only transmit more data beams when the number of users increases beyond a certain threshold. It is

<sup>4</sup> The notations ⌊*α*⌋ and {*α*} denote the integer and fractional parts of a real number *α*, respectively.

⌊*α*⌋ + 1, *<sup>α</sup>* ≤ *NT* − 1, 1 ≥ {*α*}(⌊*α*⌋ + <sup>2</sup>),

⌊*α*⌋+<sup>1</sup> , *<sup>α</sup>* <sup>≤</sup> *NT* <sup>−</sup> 1, {*α*}(⌊*α*⌋ <sup>+</sup> <sup>2</sup>) <sup>&</sup>gt; 1,

⌊*α*⌋ + 1, *<sup>α</sup>* ≤ *NT* − 1, 1 ≥ {*α*}(⌊*α*⌋ + <sup>2</sup>), ⌊*α*⌋ + 2, *<sup>α</sup>* ≤ *NT* − 1, {*α*}(⌊*α*⌋ + <sup>2</sup>) > 1,

*RBF*(1) = 2 is *<sup>M</sup>*<sup>∗</sup>

*RBF*(*α*) = 4 with *<sup>M</sup>* <sup>=</sup> *NT* <sup>=</sup> 4 is attained when *<sup>α</sup>* <sup>≥</sup> <sup>3</sup>

achievable spatial multiplexing gain or DoF, *d*RBF(*α*, *M*).

 

*α*(⌊*α*⌋+2)

 

shown in the following theorem.

*d*∗ *RBF*(*α*) =

*M*∗

also observed that the maximum DoF *<sup>d</sup>*<sup>∗</sup>

*RBF*(3) = 4.

since *<sup>M</sup>*<sup>∗</sup>

*RBF*(*α*) =

example, the optimal number of beams to achieve *<sup>d</sup>*<sup>∗</sup>

*transmit beams are*<sup>4</sup>

#### **5.1. Multi-cell MISO RBF**

In this subsection, we review the MISO RBF study in [22]. It is revealed that the high-SNR analysis provides an efficient way of characterizing the achievable sum rate for RBF, even for small values of SNR and number of users. In single-cell RBF, the DoF result shows that adjusting the number of beams according to the number of users can improve the achievable sum rate, thus providing a theoretical explanation for observations in, e.g., [46] [33] [48]. Extending to the multi-cell case, it is furthermore unfolded 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.

#### *5.1.1. Single-cell case*

First, we consider the DoF for the achievable sum rate in the single-cell RBF case without the ICI. The cell index *c* in (2) thus is dropped for brevity. In the single-cell case, the DoF region collapses to a line, bounded by 0 and *<sup>d</sup>*<sup>∗</sup> RBF(*α*), where *<sup>d</sup>*<sup>∗</sup> RBF(*α*) <sup>≥</sup> 0 denotes the maximum DoF achievable for the RBF sum rate.

The achievable DoF for single-cell RBF is defined with a given pair of *α* and *M* as

$$d\_{\rm RBF}(\alpha, M) = \lim\_{\rho \to \infty} \frac{R\_{\rm RBF}}{\log\_2 \rho} = \lim\_{\eta \to \infty} \frac{R\_{\rm RBF}}{\log\_2 \eta} \tag{42}$$

since *<sup>η</sup>* = *<sup>ρ</sup>*/*M*. Thus, *<sup>d</sup>*<sup>∗</sup> RBF(*α*) = max *M*∈{1,··· ,*NT* } *<sup>d</sup>*RBF(*α*, *<sup>M</sup>*) for a given *<sup>α</sup>* ≥ 0. The DoF *dRBF*(*α*, *M*) is characterized in the following lemma.

**Lemma 5.1.** *([22, Lemma 4.1]) Assuming K* = Θ(*ρα*)*, the DoF of single-cell MISO RBF with <sup>M</sup>* ≤ *NT orthogonal transmit beams is given by*

$$d\_{\rm RBF}(\mathfrak{a}, M) = \left\{ \begin{array}{c} \frac{\mathfrak{a}M}{M-1}, & 0 \le \mathfrak{a} \le M-1, \\ \end{array} \right. \tag{43a}$$

$$\begin{array}{cc} \cdot & \cdot\\ \cdot & \cdot \end{array} \qquad \Big( \quad M\_{\prime} \qquad \qquad \qquad \mathfrak{a} > M - 1. \tag{43b}$$

With RBF and under the assumption *K* = Θ(*ρα*), it is interesting to observe from Lemma 5.1 that the achievable DoF can be a non-negative real number (as compared to the conventional integer DoF in the literature with a finite *K*). Moreover, it is observed that for any given <sup>0</sup> < *<sup>α</sup>* < *NT* − 1, assigning more transmit beams by increasing *<sup>M</sup>* initially improves the sum-rate DoF if *M* ≤ *α* + 1; however, as *M* > *α* + 1, the DoF may not necessarily increase with *M* due to the more dominant inter-beam/intra-cell interference. Note that the term *M* −1 in the denominator of (43a) is exactly the number of interfering beams to one particular beam. Thus, Lemma 5.1 provides a succinct description of the interplay between the available multi-user diversity (specified by *α* with a larger *α* denoting a higher user density or the number of users in a cell), the level of the intra-cell interference (specified by *M* − 1), and the achievable spatial multiplexing gain or DoF, *d*RBF(*α*, *M*).

18 Recent Trends in Multiuser MIMO Communications

**5.1. Multi-cell MISO RBF**

*5.1.1. Single-cell case*

collapses to a line, bounded by 0 and *<sup>d</sup>*<sup>∗</sup>

achievable for the RBF sum rate.

since *<sup>η</sup>* = *<sup>ρ</sup>*/*M*. Thus, *<sup>d</sup>*<sup>∗</sup>

Considering the number of per-cell users to scale polynomially with the SNR is general as well as convenient. The linear scaling law, i.e., *Kc* = *βcρ*, with constant *β<sup>c</sup>* > 0, is only a special case of *Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ) with *<sup>α</sup><sup>c</sup>* <sup>=</sup> 1; if *Kc* is a constant, then the corresponding *α<sup>c</sup>* is zero. As will be shown later in this section, Assumption 1 enables us to obtain an efficient as well as insightful characterization of the DoF region for the multi-cell RBF. Note that the DoF region under Assumption 1 can be considered as a generalization of the conventional DoF region analysis based on IA [9] for the case of finite number of users, to the case of asymptotically large number of users that scales with the SNR. For the notational convenience, we use DMISO(*α*), DMIMO(*α*), DRBF(*α*), and DRBF-Rx(*α*) to denote the achievable

In this subsection, we review the MISO RBF study in [22]. It is revealed that the high-SNR analysis provides an efficient way of characterizing the achievable sum rate for RBF, even for small values of SNR and number of users. In single-cell RBF, the DoF result shows that adjusting the number of beams according to the number of users can improve the achievable sum rate, thus providing a theoretical explanation for observations in, e.g., [46] [33] [48]. Extending to the multi-cell case, it is furthermore unfolded that collaboration among the BSs in assigning their respective numbers of data beams based on different per-cell user densities

First, we consider the DoF for the achievable sum rate in the single-cell RBF case without the ICI. The cell index *c* in (2) thus is dropped for brevity. In the single-cell case, the DoF region

RBF(*α*), where *<sup>d</sup>*<sup>∗</sup>

*R*RBF log2 *<sup>ρ</sup>* <sup>=</sup> lim

*η*→∞

*R*RBF

*T*.

RBF(*α*) <sup>≥</sup> 0 denotes the maximum DoF

log2 *<sup>η</sup>* (42)

*<sup>d</sup>*RBF(*α*, *<sup>M</sup>*) for a given *<sup>α</sup>* ≥ 0. The DoF

, 0 ≤ *α* ≤ *M* − 1*,* (43a)

*M*, *α* > *M* − 1*.* (43b)

DoF regions corresponding to *Kc* <sup>=</sup> <sup>Θ</sup>(*ρα<sup>c</sup>* ), *<sup>c</sup>* = 1,··· ,*C*, and *<sup>α</sup>* = [*α*1, ··· , *<sup>α</sup>C*]

is essential to achieve the optimal throughput tradeoffs among different cells.

The achievable DoF for single-cell RBF is defined with a given pair of *α* and *M* as

*ρ*→∞

*M*∈{1,··· ,*NT* }

 

**Lemma 5.1.** *([22, Lemma 4.1]) Assuming K* = Θ(*ρα*)*, the DoF of single-cell MISO RBF with*

*αM M* − 1

With RBF and under the assumption *K* = Θ(*ρα*), it is interesting to observe from Lemma 5.1 that the achievable DoF can be a non-negative real number (as compared to the conventional

*d*RBF(*α*, *M*) = lim

RBF(*α*) = max

*dRBF*(*α*, *M*) is characterized in the following lemma.

*dRBF*(*α*, *M*) =

*<sup>M</sup>* ≤ *NT orthogonal transmit beams is given by*

It is also of great interest to obtain the maximum achievable DoF for a given *α*. The result is shown in the following theorem.

**Theorem 5.1.** *([22, Theorem 4.1]) For the single-cell MISO RBF with NT transmit antennas and user density coefficient α, the maximum achievable DoF and the corresponding optimal number of transmit beams are*<sup>4</sup>

$$d\_{RBF}^{\*}(\mathfrak{a}) = \begin{cases} \lfloor \mathfrak{a} \rfloor + 1, & \mathfrak{a} \le \mathcal{N}\_{T} - 1, 1 \ge \{\mathfrak{a}\}(\lfloor \mathfrak{a} \rfloor + 2), \\ \frac{\mathfrak{a}(\lfloor \mathfrak{a} \rfloor + 2)}{\lfloor \mathfrak{a} \rfloor + 1}, & \mathfrak{a} \le \mathcal{N}\_{T} - 1, \{\mathfrak{a}\}(\lfloor \mathfrak{a} \rfloor + 2) > 1, \\ \mathcal{N}\_{T}, & \mathfrak{a} > \mathcal{N}\_{T} - 1. \end{cases} \tag{44}$$
 
$$M^{\*} = (\mathfrak{a})^{\*} + 1, \quad \mathfrak{a} \le \mathcal{N}\_{T} - 1, 1 \ge \{\mathfrak{a}\}(\lfloor \mathfrak{a} \rfloor + 2), \\ \Delta = \frac{1}{2} \mathcal{N}\_{T} - 1. \tag{45}$$

$$M\_{\rm RBF}^{\*}(\mathfrak{a}) = \begin{cases} \mathfrak{a}^{\*} \mathbb{I} & \text{if } \mathfrak{a} \subset \mathfrak{a}^{\*} \mathbb{I} \quad \mathfrak{a}^{\*} \mathbb{I} = \{\mathfrak{a}^{\*}\} \{\mathfrak{a}^{\*} \} \cdots \mathfrak{a}^{\*} \\ \lfloor \mathfrak{a} \rfloor + 2, & \mathfrak{a} \le N\_{\mathbb{T}} - 1, \{\mathfrak{a}\} (\lfloor \mathfrak{a} \rfloor + 2) > 1, \\ N\_{\mathbb{T}'} & \text{if } \mathfrak{a} > N\_{\mathbb{T}} - 1. \end{cases} \tag{45}$$

In Fig. 4, we compare the numerical sum rate and the analytical scaling law in Lemma 5.1. It is observed that the newly obtained sum-rate scaling law, *<sup>R</sup>*RBF <sup>≈</sup> *<sup>d</sup>*RBF(*α*, *<sup>M</sup>*)log2 *<sup>ρ</sup>*, in the single-cell RBF case is very accurate, even for small values of SNR *ρ* and number of users *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋. Compared with Fig. 2 for the conventional scaling law, *<sup>R</sup>*RBF <sup>≈</sup> *<sup>M</sup>* log2 log *<sup>K</sup>*, a much faster convergence is observed here. The DoF approach thus provides a more efficient way of characterizing the achievable sum rate for single-cell RBF. Also observe that the sum rate for *M* = 2 is higher than that for *M* = 4. This is because with *NT* = 4 and *α* = 1 in this example, the optimal number of beams to achieve *<sup>d</sup>*<sup>∗</sup> *RBF*(1) = 2 is *<sup>M</sup>*<sup>∗</sup> *RBF*(1) = 2 from (45). Since many previous studies have observed that adjusting the number of beams according to the number of users in single-cell RBF can improve the achievable sum rate (see, e.g., [46] [33] [48]), our study here can be considered as a theoretical explanation for such an observation.

Fig. 5 shows the maximum DoF and the corresponding optimal number of transmit beams versus the user density coefficient *α* with *NT* = 4 for single-cell RBF, according to Theorem 5.1. It is observed that to maximize the achievable sum rate, we should only transmit more data beams when the number of users increases beyond a certain threshold. It is also observed that the maximum DoF *<sup>d</sup>*<sup>∗</sup> *RBF*(*α*) = 4 with *<sup>M</sup>* <sup>=</sup> *NT* <sup>=</sup> 4 is attained when *<sup>α</sup>* <sup>≥</sup> <sup>3</sup> since *<sup>M</sup>*<sup>∗</sup> *RBF*(3) = 4.

<sup>4</sup> The notations ⌊*α*⌋ and {*α*} denote the integer and fractional parts of a real number *α*, respectively.

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 *dRBF \* (*α*)*

Random Beamforming in Multi – User MIMO Systems

 *MRBF \* (*α*)*

, (47)

*RBF* (*α*) with *NT* = <sup>4</sup>.

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

0 0.5 1 1.5 2 2.5 3 3.5 4

α

**Theorem 5.2.** *([22, Theorem 4.2]) Assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* ), *<sup>c</sup>* <sup>=</sup> 1, . . . , *C, the achievable DoF region*

This theorem implies that we can obtain the DoF region of multi-cell RBF DRBF(*α*) by taking the convex hull over all achievable DoF points *d*RBF(*α*, *m*) with all different values of *m*, i.e.,

In Fig. 6, we depict the DoF region of a two-cell RBF system with *NT* = 4, and for different user density coefficients *α*<sup>1</sup> and *α*2. The vertices of these regions can be obtained by setting appropriate numbers of beams 0 ≤ *<sup>M</sup>*<sup>1</sup> ≤ 4 and 0 ≤ *<sup>M</sup>*<sup>2</sup> ≤ 4, while time-sharing between these vertices yields the entire boundary. To achieve the maximum sum-DoF of both cells, it is observed that a rule of thumb is to transmit more beams in the cell with a higher user density, and when *α*<sup>1</sup> and *α*<sup>2</sup> are both small, even turn off the BS of the cell with the smaller user density. Since the maximum sum-DoF does not consider the throughput fairness, the DoF region clearly shows all the achievable sum-rate tradeoffs among different cells, by observing its (Pareto) boundary as shown in Fig. 6. It is also observed that switching the two BSs to be on/off alternately achieves the optimal DoF boundary when the numbers of users in both cells are small, but is strictly suboptimal when the user number becomes large (see

Furthermore, consider the case without any cooperation between these two BSs in assigning their numbers of transmit beams, i.e., both cells act selfishly by transmitting *Mc* = *NT* beams to intend to maximize their own DoF. The resulted DoF pairs *d*RBF([*α*1, *α*2], [4, 4]) for three

*RBF* (*α*) and optimal number of beams *<sup>M</sup>*<sup>∗</sup>

1

*of a C-cell RBF system is given by*

<sup>D</sup>*RBF*(*α*) = *conv*

*where conv denotes the convex hull operation.*

different BS beam number assignments.

the dashed line in Fig. 6).

**Figure 5.** The maximum DoF *d*<sup>∗</sup>

1.5

2

2.5

3

3.5

4

**Figure 4.** Comparison of the numerical sum rate and the scaling law *dRBF* (*α*, *<sup>M</sup>*)log2 *<sup>ρ</sup>*, with *NT* <sup>=</sup> <sup>4</sup>, *<sup>α</sup>* <sup>=</sup> <sup>1</sup>, and *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋.

#### *5.1.2. Multi-cell case*

We then extend the high-SNR analysis for the single-cell RBF to the more general multi-cell RBF subject to the ICI. For convenience, we denote the achievable sum-rate DoF of the *c*-th cell as *<sup>d</sup>*RBF,*c*(*αc*, *<sup>m</sup>*) = lim*ρ*→<sup>∞</sup> *<sup>R</sup>*(*c*) RBF log2 *<sup>ρ</sup>* , where *<sup>m</sup>* = [*M*1,··· ,*MC*] *<sup>T</sup>* is a given set of numbers of transmit beams at different BSs. We then state the next lemma on the achieve DoF of the *c*-th cell.

**Lemma 5.2.** *([22, Lemma 4.2]) In the multi-cell MISO RBF, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable DoF of the c-th cell dRBF*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

$$d\_{\rm RBF,c}(\mathfrak{a}\_{\mathfrak{c}},\mathfrak{m}) = \left\{ \begin{array}{c} \frac{\mathfrak{a}\_{\mathfrak{c}}M\_{\mathfrak{c}}}{\sum\_{l=1}^{\mathbb{C}}M\_{l} - 1}, & 0 \le \mathfrak{a}\_{\mathfrak{c}} \le \sum\_{l=1}^{\mathbb{C}}M\_{l} - 1, \\ \end{array} \right. \\ \tag{46a}$$

$$\begin{array}{ccccc}\hline \text{---} & \text{-} & \text{M}\_{\text{C}} & \text{-} & \text{-} & \text{M}\_{\text{C}} - 1. \\\hline \end{array} \qquad \qquad \qquad \begin{array}{ccccc}\text{-} & \text{-} & \text{-} & \text{M}\_{\text{I}} - 1. \\\hline \end{array} \tag{46b}$$

Similar to Lemma 5.1 for the single-cell case, Lemma 5.2 reveals the relationship among the multi-user diversity, the level of the interference, and the achievable DoF for multi-cell RBF. However, as compared to the single-cell case, there are not only *Mc* − 1 intra-cell interfering beams, but also ∑*<sup>C</sup> <sup>l</sup>*=1,*l*�=*<sup>c</sup> Ml* inter-cell interfering beams for any beam of the *c*-th cell in the multi-cell case, as observed from the denominator in (46a), which results in a decrease in the achievable DoF per cell.

Next, we obtain characterization of the DoF region defined in (41) for the multi-cell RBF with any given set of per-cell user density coefficients, denoted by *<sup>α</sup>* = [*α*1, ··· , *<sup>α</sup>C*] *T* in the following theorem; for convenience, we denote *<sup>d</sup>*RBF(*α*, *<sup>m</sup>*) <sup>=</sup> � *<sup>d</sup>*RBF,1(*α*1, *<sup>m</sup>*) ,··· , *dRBF*,*C*(*αC*, *m*) �*T* , with *d*RBF,*c*(*αc*, *m*) given in Lemma 5.2.

**Figure 5.** The maximum DoF *d*<sup>∗</sup> *RBF* (*α*) and optimal number of beams *<sup>M</sup>*<sup>∗</sup> *RBF* (*α*) with *NT* = <sup>4</sup>.

**Theorem 5.2.** *([22, Theorem 4.2]) 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 system is given by*

$$\mathcal{D}\_{\text{RBF}}(\mathfrak{a}) = \mathsf{conv}\left\{ \mathsf{d}\_{\text{RBF}}(\mathfrak{a}, \mathfrak{m})\_{\prime} M\_{\mathbb{C}} \in \{0, \cdot, \cdot, \, N\_{\mathbb{T}}\}, \, \mathfrak{c} = 1, \, \cdots, \, \mathbb{C} \right\}, \tag{47}$$

*where conv denotes the convex hull operation.*

20 Recent Trends in Multiuser MIMO Communications

10

cell as *<sup>d</sup>*RBF,*c*(*αc*, *<sup>m</sup>*) = lim*ρ*→<sup>∞</sup> *<sup>R</sup>*(*c*)

*5.1.2. Multi-cell case*

beams, but also ∑*<sup>C</sup>*

*dRBF*,*C*(*αC*, *m*)

achievable DoF per cell.

�*T*

cell.

15

20

Sum Rate (bps/Hz)

25

30

6 7 8 9 10 11 12 13 14 15 16

 *M=4* numerical *R*RBF

 *d(*α*,M)* log2

ρ

*<sup>T</sup>* is a given set of numbers of

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*,* (46a)

*T*

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*.* (46b)

 *M=2*

log2 ρ

**Figure 4.** Comparison of the numerical sum rate and the scaling law *dRBF* (*α*, *<sup>M</sup>*)log2 *<sup>ρ</sup>*, with *NT* <sup>=</sup> <sup>4</sup>, *<sup>α</sup>* <sup>=</sup> <sup>1</sup>, and *<sup>K</sup>* <sup>=</sup> ⌊*ρα*⌋.

We then extend the high-SNR analysis for the single-cell RBF to the more general multi-cell RBF subject to the ICI. For convenience, we denote the achievable sum-rate DoF of the *c*-th

log2 *<sup>ρ</sup>* , where *<sup>m</sup>* = [*M*1,··· ,*MC*]

transmit beams at different BSs. We then state the next lemma on the achieve DoF of the *c*-th

**Lemma 5.2.** *([22, Lemma 4.2]) In the multi-cell MISO RBF, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable*

*Mc*, *α<sup>c</sup>* > ∑*<sup>C</sup>*

Similar to Lemma 5.1 for the single-cell case, Lemma 5.2 reveals the relationship among the multi-user diversity, the level of the interference, and the achievable DoF for multi-cell RBF. However, as compared to the single-cell case, there are not only *Mc* − 1 intra-cell interfering

multi-cell case, as observed from the denominator in (46a), which results in a decrease in the

Next, we obtain characterization of the DoF region defined in (41) for the multi-cell RBF with any given set of per-cell user density coefficients, denoted by *<sup>α</sup>* = [*α*1, ··· , *<sup>α</sup>C*]

in the following theorem; for convenience, we denote *<sup>d</sup>*RBF(*α*, *<sup>m</sup>*) <sup>=</sup> � *<sup>d</sup>*RBF,1(*α*1, *<sup>m</sup>*) ,··· ,

, 0 ≤ *α<sup>c</sup>* ≤ ∑*<sup>C</sup>*

*<sup>l</sup>*=1,*l*�=*<sup>c</sup> Ml* inter-cell interfering beams for any beam of the *c*-th cell in the

*αcMc*

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>

RBF

 

∑*<sup>C</sup>*

, with *d*RBF,*c*(*αc*, *m*) given in Lemma 5.2.

*DoF of the c-th cell dRBF*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

*dRBF*,*c*(*αc*, *m*) =

This theorem implies that we can obtain the DoF region of multi-cell RBF DRBF(*α*) by taking the convex hull over all achievable DoF points *d*RBF(*α*, *m*) with all different values of *m*, i.e., different BS beam number assignments.

In Fig. 6, we depict the DoF region of a two-cell RBF system with *NT* = 4, and for different user density coefficients *α*<sup>1</sup> and *α*2. The vertices of these regions can be obtained by setting appropriate numbers of beams 0 ≤ *<sup>M</sup>*<sup>1</sup> ≤ 4 and 0 ≤ *<sup>M</sup>*<sup>2</sup> ≤ 4, while time-sharing between these vertices yields the entire boundary. To achieve the maximum sum-DoF of both cells, it is observed that a rule of thumb is to transmit more beams in the cell with a higher user density, and when *α*<sup>1</sup> and *α*<sup>2</sup> are both small, even turn off the BS of the cell with the smaller user density. Since the maximum sum-DoF does not consider the throughput fairness, the DoF region clearly shows all the achievable sum-rate tradeoffs among different cells, by observing its (Pareto) boundary as shown in Fig. 6. It is also observed that switching the two BSs to be on/off alternately achieves the optimal DoF boundary when the numbers of users in both cells are small, but is strictly suboptimal when the user number becomes large (see the dashed line in Fig. 6).

Furthermore, consider the case without any cooperation between these two BSs in assigning their numbers of transmit beams, i.e., both cells act selfishly by transmitting *Mc* = *NT* beams to intend to maximize their own DoF. The resulted DoF pairs *d*RBF([*α*1, *α*2], [4, 4]) for three

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*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> *NR,* (48a)

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*,* (49a)

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>*.* (49b)

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> *NR.* (48b)

schemes such as MF or AS. This reflects a tradeoff between the rate/DoF performance and

Similarly to the MISO case, the DoF of the *c*-th RBF-Rx cell is defined as *d*RBF-Rx,*c*(*αc*, *m*) =

**Lemma 5.3.** *([24, Lemma 4.1]) In the multi-cell RBF-MMSE, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable*

*αcMc*

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> *NR*

**Lemma 5.4.** *([24, Lemma 4.2]) In the multi-cell RBF-MF/AS, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable*

*αcMc*

*<sup>l</sup>*=<sup>1</sup> *Ml* <sup>−</sup> <sup>1</sup>

The DoF scaling laws *<sup>d</sup>*RBF-MMSE,*c*(*αc*, *<sup>m</sup>*)log2 *<sup>ρ</sup>* and *<sup>d</sup>*RBF-MF/AS,*c*(*αc*, *<sup>m</sup>*)log2 *<sup>ρ</sup>* are compared in Fig. 7. As a reference, we also show the sum-rates of the RBF-MMSE, RBF-MF, and RBF-AS. Again, a good match is observed between the scaling law and the numerical results,

We first remark that the MMSE receiver produces an *interference mitigation* effect to the DoF performance. From Lemma 5.2 and 5.3, it is observed that *NR* − 1 interferences can be effectively eliminated in the RBF-MMSE. In particular, with *NR* degrees of freedom at the receiver, the users can use one for receiving signal and *NR* − 1 for suppressing the interferences. Secondly, in terms of sum-rate, the gain captured by employing either RBF-MF or RBF-AS is only marginal comparing to the MISO RBF with single-antenna users, as shown in Lemma 5.4. The benefit of receive diversity, therefore, clearly depends on the availability of the interferences' CSI at the users. In the RBF-MMSE, this CSI is the interference-plus-noise

to the users in the RBF-MF/AS, it is expected that no significant gain is achieved by these

At this point, we observe a sharp contrast between the conclusions based on the high-SNR/DoF and large-number-of-users analysis. In the conventional analysis, Proposition 4.2 effectively implies that there is no significant rate gain for the multi-cell RBF with receive diversity scheme at the users, and also no loss with intra-/inter-cell interferences in the system as the number of users goes to infinity. Recalling the convergences of the two approaches, the high-SNR/DoF analysis, therefore, is a more efficient method

*Mc*, *α<sup>c</sup>* > ∑*<sup>C</sup>*

*Mc*, *α<sup>c</sup>* > ∑*<sup>C</sup>*

, 0 ≤ *α<sup>c</sup>* ≤ ∑*<sup>C</sup>*

, 0 ≤ *α<sup>c</sup>* ≤ ∑*<sup>C</sup>*

*<sup>k</sup>* . In fact, since there is no information about the interferences given

the complexity/delay time of RBF systems.

*dRBF-MMSE*,*c*(*αc*, *m*) =

*dRBF-MF/AS*,*c*(*αc*, *m*) =

even with small values of the SNR and number of users.

log2 *<sup>ρ</sup>* . We then obtain the DoF of the *<sup>c</sup>*-th cell.

*DoF of the c-th cell dRBF-MMSE*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

 

∑*<sup>C</sup>*

*DoF of the c-th cell dRBF-MF/AS*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

 

∑*<sup>C</sup>*

*5.2.1. DoF of the c-th cell*

RBF-Rx

covariance matrix *<sup>W</sup>*(*c*)

schemes comparing with the MISO RBF.

lim*ρ*→<sup>∞</sup> *<sup>R</sup>*(*c*)

**Figure 6.** DoF region of two-cell RBF system with *NT* = 4.

sets of *α* are shown in Fig. 6 as *P*1, *P*2, and *P*3, respectively. It is observed that the smaller the user densities are, the further the above non-cooperative multi-cell RBF scheme deviates from the Pareto boundary. In general, the optimal DoF tradeoffs or the boundary DoF pairs are achieved when both cells cooperatively assign their numbers of transmit beams based on their respective user densities, especially when the numbers of users in both cells are not sufficiently large. Since the information needed to determine the optimal operating DoF point is only the individual cell user density coefficients, the DoF region provides a very useful method to globally optimize the coordinated multi-cell RBF scheme in practical systems.

#### **5.2. Multi-cell MIMO RBF schemes: Benefit of spatial receive diversity**

*Spatial receive diversity* is another topic which is not fully understood even in a single-cell RBF setup. As discussed in Section 4.2, RBF with and without receive diversity schemes still achieve the same sum-rate scaling law, assuming that the number of users per cell goes to infinity for any given user's SNR. Based on the conventional asymptotic analysis, it is thus concluded that receive diversity only provides *marginal* gain to the rate performance.

The objective of this subsection is to review the recent results on the rate performance of multi-cell MIMO RBF systems under the high-SNR/DoF analysis [24]. Contrasting with the large-number-of-users analysis, it is found that the benefit of receive diversity is significant for the RBF scheme. In particular, receive diversity introduces an interference mitigation effect on the DoF performance. As a consequence, an substantial less number of users in each cell is required to achieve a given DoF region comparing to the case without receive diversity. However, the gain is observed only with MMSE receiver but not with suboptimal schemes such as MF or AS. This reflects a tradeoff between the rate/DoF performance and the complexity/delay time of RBF systems.

#### *5.2.1. DoF of the c-th cell*

22 Recent Trends in Multiuser MIMO Communications

0

0.5

1

 *P3*

1.5

2

2.5

 *d2*

**Figure 6.** DoF region of two-cell RBF system with *NT* = 4.

systems.

3

3.5

4

4.5

0 1 2 3 4

sets of *α* are shown in Fig. 6 as *P*1, *P*2, and *P*3, respectively. It is observed that the smaller the user densities are, the further the above non-cooperative multi-cell RBF scheme deviates from the Pareto boundary. In general, the optimal DoF tradeoffs or the boundary DoF pairs are achieved when both cells cooperatively assign their numbers of transmit beams based on their respective user densities, especially when the numbers of users in both cells are not sufficiently large. Since the information needed to determine the optimal operating DoF point is only the individual cell user density coefficients, the DoF region provides a very useful method to globally optimize the coordinated multi-cell RBF scheme in practical

**5.2. Multi-cell MIMO RBF schemes: Benefit of spatial receive diversity**

*Spatial receive diversity* is another topic which is not fully understood even in a single-cell RBF setup. As discussed in Section 4.2, RBF with and without receive diversity schemes still achieve the same sum-rate scaling law, assuming that the number of users per cell goes to infinity for any given user's SNR. Based on the conventional asymptotic analysis, it is thus concluded that receive diversity only provides *marginal* gain to the rate performance.

The objective of this subsection is to review the recent results on the rate performance of multi-cell MIMO RBF systems under the high-SNR/DoF analysis [24]. Contrasting with the large-number-of-users analysis, it is found that the benefit of receive diversity is significant for the RBF scheme. In particular, receive diversity introduces an interference mitigation effect on the DoF performance. As a consequence, an substantial less number of users in each cell is required to achieve a given DoF region comparing to the case without receive diversity. However, the gain is observed only with MMSE receiver but not with suboptimal

 *d1*

*=1.5*

 *P2*

α*1*

*=4.7*, <sup>α</sup>*<sup>2</sup>*

α*1 =7*, <sup>α</sup>*<sup>2</sup> =7*

*=3.7*

 *P1*

α*1 =1*, <sup>α</sup>*<sup>2</sup>* Similarly to the MISO case, the DoF of the *c*-th RBF-Rx cell is defined as *d*RBF-Rx,*c*(*αc*, *m*) = lim*ρ*→<sup>∞</sup> *<sup>R</sup>*(*c*) RBF-Rx log2 *<sup>ρ</sup>* . We then obtain the DoF of the *<sup>c</sup>*-th cell.

**Lemma 5.3.** *([24, Lemma 4.1]) In the multi-cell RBF-MMSE, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable DoF of the c-th cell dRBF-MMSE*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

$$d\_{\text{RBF-MMSE},\mathcal{L}}(\mathfrak{a}\_{\mathfrak{c}},\mathfrak{m}) = \left\{ \begin{array}{c} \mathfrak{a}\_{\mathfrak{c}}M\_{\mathbb{C}} \\ \sum\_{l=1}^{\mathbb{C}}M\_{l} - N\_{\mathbb{R}} \end{array} , \quad 0 \le \mathfrak{a}\_{\mathfrak{c}} \le \sum\_{l=1}^{\mathbb{C}}M\_{l} - N\_{\mathbb{R}'} \\ \tag{48a}$$

$$\mathfrak{a}\_{\mathfrak{c}} > \sum\_{l=1}^{\mathbb{C}}M\_{l} - N\_{\mathbb{R}} . \tag{48b}$$

**Lemma 5.4.** *([24, Lemma 4.2]) In the multi-cell RBF-MF/AS, assuming Kc* <sup>=</sup> <sup>Θ</sup> (*ρα<sup>c</sup>* )*, the achievable DoF of the c-th cell dRBF-MF/AS*,*c*(*αc*, *<sup>m</sup>*)*, c* ∈ {1, . . . , *<sup>C</sup>*}*, for a given <sup>m</sup> is*

$$d\_{\text{RBF-MF/AS},\mathcal{L}}(\mathfrak{a}\_{\mathfrak{c}},\mathfrak{m}) = \begin{cases} \frac{\mathfrak{a}\_{\mathfrak{c}}M\_{\mathfrak{c}}}{\sum\_{l=1}^{\mathbb{C}}M\_{l} - 1}, & 0 \le \mathfrak{a}\_{\mathfrak{c}} \le \sum\_{l=1}^{\mathbb{C}}M\_{l} - 1, \end{cases} \tag{49a}$$

$$\begin{array}{ccccc} & & \text{ $M\_{\odot}$ } & & \text{ $\mathbf{a}\_{\odot}$ } > \sum\_{l=1}^{\mathbb{C}} M\_{l} - 1. & & \text{ $\mathbf{a}\_{\odot}$ } & \text{ $M\_{\odot}$ } \\ \end{array}$$

The DoF scaling laws *<sup>d</sup>*RBF-MMSE,*c*(*αc*, *<sup>m</sup>*)log2 *<sup>ρ</sup>* and *<sup>d</sup>*RBF-MF/AS,*c*(*αc*, *<sup>m</sup>*)log2 *<sup>ρ</sup>* are compared in Fig. 7. As a reference, we also show the sum-rates of the RBF-MMSE, RBF-MF, and RBF-AS. Again, a good match is observed between the scaling law and the numerical results, even with small values of the SNR and number of users.

We first remark that the MMSE receiver produces an *interference mitigation* effect to the DoF performance. From Lemma 5.2 and 5.3, it is observed that *NR* − 1 interferences can be effectively eliminated in the RBF-MMSE. In particular, with *NR* degrees of freedom at the receiver, the users can use one for receiving signal and *NR* − 1 for suppressing the interferences. Secondly, in terms of sum-rate, the gain captured by employing either RBF-MF or RBF-AS is only marginal comparing to the MISO RBF with single-antenna users, as shown in Lemma 5.4. The benefit of receive diversity, therefore, clearly depends on the availability of the interferences' CSI at the users. In the RBF-MMSE, this CSI is the interference-plus-noise covariance matrix *<sup>W</sup>*(*c*) *<sup>k</sup>* . In fact, since there is no information about the interferences given to the users in the RBF-MF/AS, it is expected that no significant gain is achieved by these schemes comparing with the MISO RBF.

At this point, we observe a sharp contrast between the conclusions based on the high-SNR/DoF and large-number-of-users analysis. In the conventional analysis, Proposition 4.2 effectively implies that there is no significant rate gain for the multi-cell RBF with receive diversity scheme at the users, and also no loss with intra-/inter-cell interferences in the system as the number of users goes to infinity. Recalling the convergences of the two approaches, the high-SNR/DoF analysis, therefore, is a more efficient method

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0 1 2 3 4

depends on the communication system in consideration. The RBF-MMSE is a good choice if there are small numbers of users in the cells while the constrains on the complexity and delay time are slack. However, when the receivers are required to be simple and there are many users in the system, the RBF-MF, RBF-AS, or even MISO RBF, is more favourable.

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

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

 *d1*

α*1*

*=3.5*, <sup>α</sup>*<sup>2</sup>*

*=4.3*

α*1 =6*, <sup>α</sup>*<sup>2</sup> =6*

0

**Figure 8.** DoF regions of different RBF-MMSE and RBF-MF/AS systems.

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

by setting *NR* = 1.

*5.3.1. Single-cell case*

the following proposition.

0.5

1

1.5

2

2.5

 *d2*

3

α*1*

*=0.8*, <sup>α</sup>*<sup>2</sup>*

*=1*

3.5

4

4.5

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

of characterizing the achievable sum-rate for RBF under the effects of multi-user diversity, spatial receive diversity, and interference.

#### *5.2.2. DoF region characterization*

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>*) *T* , with *d*RBF-Rx,*c*(*αc*, *m*) given in Lemma 5.3 or 5.4. The characterization of the DoF region for the multi-cell RBF-Rx is given in the following theorem

**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 system is given by*

$$\mathcal{D}\_{\text{RBF-Rx}}(\mathfrak{a}) = \mathsf{comv}\left\{ \mathbf{d}\_{\text{RBF-Rx}}(\mathfrak{a}, \mathfrak{m})\_{\prime} \, \boldsymbol{M}\_{\mathbb{C}} \in \{ 0, \cdots, \, \mathrm{N}\_{\mathbb{T}} \}, \; \mathfrak{c} = 1, \cdots, \; \mathrm{C} \right\},\tag{50}$$

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

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

**Figure 8.** DoF regions of different RBF-MMSE and RBF-MF/AS systems.

depends on the communication system in consideration. The RBF-MMSE is a good choice if there are small numbers of users in the cells while the constrains on the complexity and delay time are slack. However, when the receivers are required to be simple and there are many users in the system, the RBF-MF, RBF-AS, or even MISO RBF, is more favourable.
