**4.1. Multi-cell MISO RBF**

In this subsection, we review the recent results on the achievable sum rate of the multi-cell MISO RBF scheme under finite-SNR regime [22]. In particular, we first consider an extension of Lemma 3.1 to the multi-cell case subject to the ICI, and furthermore show the asymptotic sum-rate scaling law as the number of users per cell tends to infinity.

For the single-cell RBF case, the SINR distributions given in (17) and (18) are obtained in [20]. The following lemma establishes similar results for the multi-cell case

**Lemma 4.1.** *([22, Lemma 3.2], see also Section 4.2) In the multi-cell MISO RBF, the PDF and CDF of the SINR S* := *SINR*(*c*) *<sup>k</sup>*,*m,* <sup>∀</sup>*k*, *m, have closed-form expressions given by*

$$f\_S^{(c)}(s) = \frac{e^{-s/\eta\_c}}{(s+1)^{M\_c-1} \prod\_{l=1, l \neq c}^{\mathbb{C}} \left(\frac{\mu\_{l,c}}{\eta\_c}s + 1\right)^{M\_l}} \left[\frac{1}{\eta\_c} + \frac{M\_c - 1}{s+1} + \sum\_{l=1, l \neq c}^{\mathbb{C}} \frac{M\_l}{s + \frac{\eta\_c}{\mu\_{l,c}}}\right],\tag{28}$$

$$F\_{\mathcal{S}}^{(c)}(s) = 1 - \frac{e^{-s/\eta\_{\mathcal{L}}}}{(s+1)^{M\_{\mathcal{L}}-1} \prod\_{l=1, l \neq c}^{\mathcal{C}} \left(\frac{\mu\_{l,\mathcal{L}}}{\eta\_{\mathcal{C}}}s + 1\right)^{M\_{l}}} \,. \tag{29}$$

10.5772/57131

39

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

numerical analytical

Random Beamforming in Multi – User MIMO Systems

  � � � � � � � *x*=−1

  � � � � � � �

*x*=−*ηc*/*µl*,*<sup>c</sup>*

, (32)

. (33)

5 10 15 20 25 30 35 40

Number of users

 

∏*<sup>C</sup> l*�=*c* � *<sup>x</sup>* <sup>+</sup> *<sup>η</sup> µl* �*nMl*

1

*t*�=*l*,*c* � *<sup>x</sup>* <sup>+</sup> *<sup>η</sup> µt* �*nMt*

1

*<sup>d</sup>n*(*Mc*−1)−*p*+<sup>1</sup> *dxn*(*Mc*−1)−*p*+<sup>1</sup>

(*x* + 1)*n*(*Mc*−1)+<sup>1</sup> ∏*<sup>C</sup>*

If the original feedback scheme in [20] is employed, (30) becomes a close approximation of the *c*-th sum rate. Note that no other results than Theorem 4.1 are available in the literature

Fig. 1 shows the analytical and numerical results on the RBF sum rate as a function of the number of users for both single-cell and two-cell systems. In the single-cell case, *M* = *NT* = 3, *η* = 20 dB, while in the two-cell case, *K*<sup>1</sup> = *K*2, *M*<sup>1</sup> = *M*<sup>2</sup> = *NT* = 3, *η*<sup>1</sup> = *η*<sup>2</sup> = 20 dB, *µ*2,1 = 6 dB, and *µ*1,2 = 10 dB. It is observed that the sum-rate expressions in (19) and (30) are very accurate. Thus, it is possible to use Theorem 4.1 to compute all the sum-rate tradeoffs among different cells in a multi-cell RBF system, which leads to the achievable rate region. However, such a characterization requires intensive computations, and does not provide any

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

4

*and thus given by [52, (2.102)]:*

*An*,*c*,*<sup>p</sup>* <sup>=</sup> <sup>1</sup>

(*nMl* − *<sup>q</sup>*)!

for the multi-cell RBF sum rate.

*4.1.2. Asymptotic sum rate as Kc* → ∞

*An*,*l*,*<sup>q</sup>* <sup>=</sup> <sup>1</sup>

useful insights.

6

8

single−cell

**Figure 1.** Comparison of the analytical and numerical results on the RBF sum rate.

(*n*(*Mc* − 1) − *p* + 1)!

*<sup>d</sup>nMl*−*<sup>q</sup> dxnMl*−*<sup>q</sup>*  

two−cell

10

Sum Rate (bps/Hz)

12

14

16

#### *4.1.1. Sum rate with finite K*

With Lemma 4.1, Lemma 3.1 is readily generalized to the multi-cell case in the following theorem.

**Theorem 4.1.** *([22, Theorem 3.1]) The total sum rate of a C-cell MISO RBF system equals to* ∑*<sup>C</sup> <sup>c</sup>*=<sup>1</sup> *<sup>R</sup>*(*c*) *RBF, in which the individual sum rate of the c-th cell, R*(*c*) *RBF, is given by*

$$\begin{split} R\_{\mathrm{RBF}}^{(c)} &= \frac{M\_{\mathrm{L}}}{\log 2} \sum\_{n=1}^{K\_{\mathrm{c}}} (-1)^{n} \binom{\mathrm{K}\_{\mathrm{c}}}{n} \prod\_{l=1, l \neq c}^{\mathrm{C}} \left( \frac{\eta\_{\mathrm{c}}}{\mu\_{l,c}} \right)^{n M\_{\mathrm{l}}} \times \\ & \left\{ \sum\_{p=1}^{n(M\_{\mathrm{L}}-1)+1} \frac{A\_{\mathrm{R},c,p}}{(p-1)!} \left[ e^{\frac{\mu}{\eta\_{\mathrm{c}}}} \left( -\frac{n}{\eta\_{\mathrm{c}}} \right)^{p-1} Ei \left( -\frac{n}{\eta\_{\mathrm{c}}} \right) - \sum\_{m=1}^{p-1} \left( -\frac{n}{\eta\_{\mathrm{c}}} \right)^{m-1} (p-1-m)! \right] \right. \\ & \left. + \sum\_{l=1, l \neq c}^{\mathrm{C}} \sum\_{q=1}^{nM\_{l}} \frac{A\_{\mathrm{u},l,q}}{(q-1)!} \left[ e^{\frac{\mu}{\eta\_{\mathrm{c}}}} \left( -\frac{n}{\eta\_{\mathrm{c}}} \right)^{q-1} Ei \left( -\frac{n}{\mu\_{l,c}} \right) \right. \\ & \left. - \sum\_{m=1}^{q-1} \left( -\frac{n}{\eta\_{\mathrm{c}}} \right)^{m-1} \left( \frac{\mu\_{l,c}}{\eta\_{\mathrm{c}}} \right)^{q-m} (q-1-m)! \right] \right\}, \quad (30) \end{split}$$

*where An*,*c*,*p's and An*,*l*,*q's are the coefficients from the following partial fractional decomposition:*

$$\frac{1}{n(\mathbf{x}+1)^{n(M\_{\ell}-1)+1} \prod\_{l \neq \pm \mathcal{L}}^{\mathbb{C}} \left(\mathbf{x} + \frac{\eta\_{\mathcal{L}}}{\mu\_{l,\mathcal{L}}}\right)^{nM\_{l}}} = \sum\_{p=1}^{n(M\_{\ell}-1)+1} \frac{A\_{\mathrm{n},\mathcal{L},p}}{(\mathbf{x}+1)^{p}} + \sum\_{l=1, l \neq \pm \mathcal{L}}^{\mathbb{C}} \sum\_{q=1}^{nM\_{l}} \frac{A\_{\mathrm{n},l,q}}{\left(\mathbf{x} + \frac{\eta\_{\mathcal{L}}}{\mu\_{l,\mathcal{L}}}\right)^{q}},\tag{31}$$

**Figure 1.** Comparison of the analytical and numerical results on the RBF sum rate.

*and thus given by [52, (2.102)]:*

12 Recent Trends in Multiuser MIMO Communications

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

*<sup>F</sup>*(*c*)

*Mc*−1

*C* ∏ *l*=1,*l*�=*c*

*RBF, in which the individual sum rate of the c-th cell, R*(*c*)

 *<sup>C</sup>* ∏ *l*=1,*l*�=*c*  *η<sup>c</sup> µl*,*c*

> *<sup>p</sup>*−<sup>1</sup> *Ei* − *n ηc* − *p*−1 ∑ *m*=1

*<sup>q</sup>*−<sup>1</sup> *Ei* − *n µl*,*c* 

> − *n ηc*

*n*(*Mc*−1)+1 ∑ *p*=1

*where An*,*c*,*p's and An*,*l*,*q's are the coefficients from the following partial fractional decomposition:*

− *q*−1 ∑ *m*=1

=

 *µl*,*<sup>c</sup> ηc s* + 1 *Ml*

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

*Mc*−1

With Lemma 4.1, Lemma 3.1 is readily generalized to the multi-cell case in the following

**Theorem 4.1.** *([22, Theorem 3.1]) The total sum rate of a C-cell MISO RBF system equals to*

*nMl* ×

(*s* + 1)

*of the SINR S* := *SINR*(*c*)

*4.1.1. Sum rate with finite K*

theorem.

*<sup>R</sup>*(*c*)

*RBF* <sup>=</sup> *Mc* log 2

+

*n*(*Mc*−1)+<sup>1</sup> ∑ *p*=1

> *C* ∑ *l*=1,*l*�=*c*

(*x* + 1)*n*(*Mc*−1)+<sup>1</sup> ∏*<sup>C</sup>*

*Kc* ∑ *n*=1

> *nMl* ∑ *q*=1

> > 1

*l*�=*c <sup>x</sup>* <sup>+</sup> *<sup>η</sup><sup>c</sup> µl*,*c nMl*

(−1)*<sup>n</sup>*

*An*,*c*,*p* (*p* − 1)!

> *An*,*l*,*<sup>q</sup>* (*q* − 1)!

*Kc n*

> *e n ηc* − *n ηc*

 *e n µl*,*c* − *n ηc*

∑*<sup>C</sup> <sup>c</sup>*=<sup>1</sup> *<sup>R</sup>*(*c*)

*f* (*c*)

**Lemma 4.1.** *([22, Lemma 3.2], see also Section 4.2) In the multi-cell MISO RBF, the PDF and CDF*

 1 *ηc* +

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

*Mc* − 1 *<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup>

*RBF, is given by*

 − *n ηc*

*<sup>m</sup>*−<sup>1</sup> *<sup>µ</sup>l*,*<sup>c</sup>*

*An*,*c*,*p* (*<sup>x</sup>* <sup>+</sup> <sup>1</sup>)*<sup>p</sup>* <sup>+</sup>

*ηc*

*<sup>q</sup>*−*<sup>m</sup>*

*C* ∑ *l*=1,*l*�=*c* *nMl* ∑ *q*=1

 *<sup>x</sup>* <sup>+</sup> *<sup>η</sup><sup>c</sup> µl*,*c*

*An*,*l*,*<sup>q</sup>*

*<sup>m</sup>*−<sup>1</sup>

(*p* − 1 − *m*)!

(*q* − 1 − *m*)!

, (30)

*<sup>q</sup>* , (31)

*C* ∑ *l*=1,*l*�=*c*

*Ml <sup>s</sup>* <sup>+</sup> *<sup>η</sup><sup>c</sup> µl*,*c*

. (29)

, (28)

*<sup>k</sup>*,*m,* <sup>∀</sup>*k*, *m, have closed-form expressions given by*

$$A\_{n,\mathcal{L},p} = \frac{1}{(n(M\_{\mathcal{L}}-1)-p+1)!} \frac{d^{n(M\_{\mathcal{L}}-1)-p+1}}{dx^{n(M\_{\mathcal{L}}-1)-p+1}} \left[ \frac{1}{\prod\_{l \neq c}^{\mathbb{C}} \left(x + \frac{\eta}{\mu\_l}\right)^{nM\_l}} \right] \Big|\_{x=-1} \tag{32}$$

$$A\_{n,l,q} = \frac{1}{(nM\_l - q)!} \frac{d^{nM\_l - q}}{dx^{nM\_l - q}} \left[ \frac{1}{(x+1)^{n(M\_l - 1) + 1} \prod\_{l \neq l,c}^{C} \left( x + \frac{\eta}{\mu\_l} \right)^{nM\_l}} \right] \bigg|\_{x = -\eta\_c/\mu\_{l,c}}.\tag{33}$$

If the original feedback scheme in [20] is employed, (30) becomes a close approximation of the *c*-th sum rate. Note that no other results than Theorem 4.1 are available in the literature for the multi-cell RBF sum rate.

Fig. 1 shows the analytical and numerical results on the RBF sum rate as a function of the number of users for both single-cell and two-cell systems. In the single-cell case, *M* = *NT* = 3, *η* = 20 dB, while in the two-cell case, *K*<sup>1</sup> = *K*2, *M*<sup>1</sup> = *M*<sup>2</sup> = *NT* = 3, *η*<sup>1</sup> = *η*<sup>2</sup> = 20 dB, *µ*2,1 = 6 dB, and *µ*1,2 = 10 dB. It is observed that the sum-rate expressions in (19) and (30) are very accurate. Thus, it is possible to use Theorem 4.1 to compute all the sum-rate tradeoffs among different cells in a multi-cell RBF system, which leads to the achievable rate region. However, such a characterization requires intensive computations, and does not provide any useful insights.
