**2.1. Multi-cell MISO RBF**

In this case, *NR* = 1 and the system model in (1) reduces to

$$y\_k^{(c)} = \mathbf{h}\_k^{(c,c)} \sum\_{m=1}^{M\_c} \Phi\_m^{(c)} s\_m^{(c)} + \sum\_{l=1,\ l \neq c}^{\mathbb{C}} \sqrt{\gamma\_{l,c}} \mathbf{h}\_k^{(l,c)} \sum\_{m=1}^{M\_l} \Phi\_m^{(l)} s\_m^{(l)} + z\_k^{(c)},\tag{2}$$

31

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

, and uses them to

*Mc <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* =

+ *σ*<sup>2</sup> *I*, (6)

*<sup>k</sup>* [*φ*(*l*) 1 ,

*m*

*<sup>k</sup>* <sup>=</sup> *<sup>H</sup>*(*c*,*c*)

*<sup>k</sup>*,*<sup>m</sup>* . (7)

*Mc <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* =

*<sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* , then

*H*

], and *<sup>H</sup>***˜** (*l*,*c*)

*<sup>h</sup>***˜**(*c*,*c*)

*<sup>k</sup>*,*<sup>m</sup>* ||. The rationale is to maximize the power

**2.2. Multi-cell MIMO RBF schemes**

and RBF-AS, respectively, as follows:

a) The *<sup>c</sup>*-th BS generates *Mc* orthonormal beams, *<sup>φ</sup>*(*c*)

*<sup>H</sup>***˜** (*c*,*c*) *k*,−*m <sup>H</sup>***˜** (*c*,*c*) *k*,−*m H* +

*<sup>k</sup>* [*φ*(*c*)

SINR(MMSE,*c*)

*<sup>k</sup>*,*<sup>m</sup>* <sup>=</sup> *PT*

b2) MF: For the *m*-th beam, user *k* in the *c*-th cell does the following steps:

*Mc <sup>h</sup>***˜**(*c*,*c*) *k*,*m*

i. Estimate the effective channel through training with the *<sup>c</sup>*-th BS: *PT*

*<sup>k</sup>*,*<sup>m</sup>* / ||*h***˜**(*c*,*c*)

received from the desired beam. The receive signal now is

iii. Apply the MMSE receive beamformer, i.e., *t*

(*c*) *<sup>k</sup>*,*<sup>m</sup>* <sup>=</sup> *<sup>h</sup>***˜**(*c*,*c*)

*<sup>k</sup>*,−*<sup>m</sup>* <sup>=</sup> *<sup>H</sup>*(*c*,*c*)

BS is assumed to be distributed equally over *Mc* beams.

1. Training phase:

 *PT Mc <sup>H</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *<sup>m</sup>* .

BSs:

*<sup>W</sup>*(*c*) *<sup>k</sup>* <sup>=</sup> *PT Mc*

in which *<sup>H</sup>***˜** (*c*,*c*)

··· , *<sup>φ</sup>*(*l*) *Ml* ].

to the BS

 *PT Mc <sup>H</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *<sup>m</sup>* .

ii. Apply the MF, i.e., *t*

With multiple antennas, the MSs can apply receive diversity techniques to enhance the performance. In this chapter, we consider the three receiver designs, i.e., minimum-mean-square-error (MMSE), match-filter (MF), and antenna selection (AS), respectively. The necessity of employing such suboptimal schemes arises when MSs are constrained by the time-delay criterion or the complexity of the MMSE receiver. We formally define the multi-cell RBF with MMSE, MF, and AS receiver, denoted as RBF-MMSE, RBF-MF,

broadcast the training signals to all users in all cells. The total power of the *c*-th

ii. Estimate the interference-plus-noise covariance matrix through training with all

*C* ∑ *l*=1,*l*�=*c*

*<sup>m</sup>*−1, *<sup>φ</sup>*(*c*)

calculate and feedback the SINR corresponding to the *<sup>m</sup>*-th transmit beam *<sup>φ</sup>*(*c*)

*PTγl*,*<sup>c</sup> Ml*

*<sup>m</sup>*+1, ··· , *<sup>φ</sup>*(*c*)

(*c*) *<sup>k</sup>*,*<sup>m</sup>* =

*<sup>H</sup>*

*<sup>W</sup>*(*c*) *k* <sup>−</sup><sup>1</sup>

*<sup>H</sup>***˜** (*l*,*c*) *k <sup>H</sup>***˜** (*l*,*c*) *k*

*Mc*

 *PT Mc <sup>W</sup>*(*c*) *k* <sup>−</sup><sup>1</sup>

b1) MMSE: For the *m*-th beam, user *k* in the *c*-th cell does the following steps:

<sup>1</sup> , ··· , *<sup>φ</sup>*(*c*)

i. Estimate the effective channel through training with the *<sup>c</sup>*-th BS: *PT*

<sup>1</sup> , ··· ,*φ*(*c*)

*Mc*

Random Beamforming in Multi – User MIMO Systems

We consider a multi-cell RBF scheme, in which all BSs in different cells are assumed to be able to implement the conventional single-cell RBF [20] in a time synchronized manner, which is described as follows:


$$\begin{split} \text{SINR}^{(c)}\_{k,m} &= \frac{\frac{P\_T}{M\_c} \left| \boldsymbol{\hbar}^{(c,c)}\_{k} \boldsymbol{\Phi}^{(c)}\_{m} \right|^2}{\sigma^2 + \frac{P\_T}{M\_c} \sum\_{i=1, i \neq m}^{M\_c} \left| \boldsymbol{\hbar}^{(c,c)}\_{k} \boldsymbol{\Phi}^{(c)}\_{i} \right|^2 + \sum\_{l=1, l \neq c}^{C} \gamma\_{l,c} \frac{P\_T}{M\_l} \sum\_{i=1}^{M\_l} \left| \boldsymbol{\hbar}^{(l,c)}\_{k} \boldsymbol{\Phi}^{(l)}\_{i} \right|^2} \\ &= \frac{\eta\_c \left| \boldsymbol{\hbar}^{(c,c)}\_{k} \boldsymbol{\Phi}^{(c)}\_{m} \right|^2}{1 + \eta\_c \sum\_{i=1, i \neq m}^{M\_c} \left| \boldsymbol{\hbar}^{(c,c)}\_{k} \boldsymbol{\Phi}^{(c)}\_{i} \right|^2 + \sum\_{l=1, l \neq c}^{C} \mu\_{l,c} \sum\_{i=1}^{M\_l} \left| \boldsymbol{\hbar}^{(l,c)}\_{k} \boldsymbol{\Phi}^{(l)}\_{i} \right|^2} \end{split} \tag{3}$$

where *m* = 1, ··· , *Mc*.

• Opportunistic scheduling: The *c*-th BS schedules transmission to a set of *Mc* users for each time by assignning its *m*-th beam to the user with the highest SINR, i.e.,

$$k\_m^{(c)} = \arg\max\_{k \in \{1, \cdots, K\_\ell\}} \text{SINR}\_{k,m}^{(c)}.\tag{4}$$

Then, the achievable average sum-rate in bits-per-second-per-Hz (bps/Hz) of the *c*-th cell is given by

$$R\_{\rm RBF}^{(c)} = \mathbb{E}\left[\sum\_{m=1}^{M\_{\rm c}} \log\_2\left(1 + \text{SINR}\_{k\_m^{(c)}, m}^{(c)}\right)\right] = M\_{\rm c} \mathbb{E}\left[\log\_2\left(1 + \text{SINR}\_{k\_1^{(c)}, 1}^{(c)}\right)\right].\tag{5}$$

Note that a different feedback scheme for single-cell RBF systems is considered in [20]. Sharif *et. al.* assumes that each user can only sends its maximum SINR, i.e., max1≤*m*≤*<sup>M</sup>* SINR*k*,*m*, along with the index *m* in which the SINR is maximized. The objective for introducing this scheme is to get a fair comparison with OBF. In this case, (5) is only a close approximation of the *c*-th sum rate by ignoring the small probability that one user may be assigned more than one beams for transmission and diminishes to zero as *Kc* → ∞ [20]. In this chapter, the modified feedback scheme is considered since the focus is the rate performance under multi-user diversity, spatial receive diversity, and interference-limited effects.

#### **2.2. Multi-cell MIMO RBF schemes**

With multiple antennas, the MSs can apply receive diversity techniques to enhance the performance. In this chapter, we consider the three receiver designs, i.e., minimum-mean-square-error (MMSE), match-filter (MF), and antenna selection (AS), respectively. The necessity of employing such suboptimal schemes arises when MSs are constrained by the time-delay criterion or the complexity of the MMSE receiver. We formally define the multi-cell RBF with MMSE, MF, and AS receiver, denoted as RBF-MMSE, RBF-MF, and RBF-AS, respectively, as follows:

1. Training phase:

4 Recent Trends in Multiuser MIMO Communications

*Mc* ∑ *m*=1

*<sup>φ</sup>*(*c*) *<sup>m</sup> s* (*c*) *<sup>m</sup>* +

each BS is assumed to be distributed equally over *Mc* beams.

together with the corresponding beam index back to the BS.

<sup>=</sup> *<sup>η</sup><sup>c</sup>*

*Mc* ∑ *i*=1,*i*�=*m*

> *k* (*c*)

<sup>1</sup> + SINR(*c*) *k* (*c*) *<sup>m</sup>* ,*m*

multi-user diversity, spatial receive diversity, and interference-limited effects.

*Mc* ∑ *i*=1,*i*�=*m*

> *<sup>h</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *i* 2 +

 *<sup>h</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *i* 2 +

*σ*<sup>2</sup> + *PT Mc*

1 + *η<sup>c</sup>*

• In the training phase, the *<sup>c</sup>*-th BS generates *Mc* orthonormal beams, *<sup>φ</sup>*(*c*)

*C* ∑ *l*=1, *l*�=*c*

We consider a multi-cell RBF scheme, in which all BSs in different cells are assumed to be able to implement the conventional single-cell RBF [20] in a time synchronized manner, which is

uses them to broadcast the training signals to all users in the *c*-th cell. The total power of

• Feedback scheme: Each user in the *c*-th cell measures the signal-to-interference-plus-noise ratio (SINR) value for each of *Mc* beams (shown in (3) below), and feeds this scalar value

> *PT Mc <sup>h</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *m* 2

 *<sup>h</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *m* 2

• Opportunistic scheduling: The *c*-th BS schedules transmission to a set of *Mc* users for

Then, the achievable average sum-rate in bits-per-second-per-Hz (bps/Hz) of the *c*-th cell is

Note that a different feedback scheme for single-cell RBF systems is considered in [20]. Sharif *et. al.* assumes that each user can only sends its maximum SINR, i.e., max1≤*m*≤*<sup>M</sup>* SINR*k*,*m*, along with the index *m* in which the SINR is maximized. The objective for introducing this scheme is to get a fair comparison with OBF. In this case, (5) is only a close approximation of the *c*-th sum rate by ignoring the small probability that one user may be assigned more than one beams for transmission and diminishes to zero as *Kc* → ∞ [20]. In this chapter, the modified feedback scheme is considered since the focus is the rate performance under

= *Mc***E**

each time by assignning its *m*-th beam to the user with the highest SINR, i.e.,

*<sup>m</sup>* <sup>=</sup> arg max *<sup>k</sup>*∈{1,··· ,*Kc*}

*C* ∑ *l*=1,*l*�=*c*

> *µl*,*c Ml* ∑ *i*=1 *<sup>h</sup>*(*l*,*c*) *<sup>k</sup> <sup>φ</sup>*(*l*) *i* 2

SINR(*c*)

 log2 

*C* ∑ *l*=1,*l*�=*c* *γl*,*c PT Ml*

*Ml* ∑ *i*=1 *<sup>h</sup>*(*l*,*c*) *<sup>k</sup> <sup>φ</sup>*(*l*) *i* 2

*<sup>k</sup>*,*m*. (4)

<sup>1</sup> + SINR(*c*) *k* (*c*) <sup>1</sup> ,1 

<sup>√</sup>*γl*,*ch*(*l*,*c*) *k*

*Ml* ∑ *m*=1

*<sup>φ</sup>*(*l*) *<sup>m</sup> s* (*l*) *<sup>m</sup>* + *z* (*c*)

*<sup>k</sup>* , (2)

<sup>1</sup> , ··· ,*φ*(*c*)

*Mc* , and

, (3)

. (5)

*y* (*c*) *<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*(*c*,*c*) *k*

SINR(*c*) *<sup>k</sup>*,*<sup>m</sup>* =

where *m* = 1, ··· , *Mc*.

*<sup>R</sup>*(*c*) RBF <sup>=</sup> **<sup>E</sup>**  *Mc* ∑ *m*=1

log2 

given by

described as follows:

	- i. Estimate the effective channel through training with the *<sup>c</sup>*-th BS: *PT Mc <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* = *PT Mc <sup>H</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *<sup>m</sup>* .
	- ii. Estimate the interference-plus-noise covariance matrix through training with all BSs:

$$\mathbf{W}\_{k}^{(c)} = \frac{\mathbf{P}\_{\rm T}}{M\_{\rm c}} \tilde{\mathbf{H}}\_{k,-m}^{(c,c)} \left(\tilde{\mathbf{H}}\_{k,-m}^{(c,c)}\right)^{H} + \sum\_{l=1, l \neq c}^{\mathbb{C}} \frac{\mathbf{P}\_{\rm T} \gamma\_{l,c}}{M\_{l}} \tilde{\mathbf{H}}\_{k}^{(l,c)} \left(\tilde{\mathbf{H}}\_{k}^{(l,c)}\right)^{H} + \sigma^{2} \mathbf{I},\tag{6}$$

$$\begin{array}{l}\text{in which }\tilde{\mathbf{H}}\_{k,-m}^{(c,c)} = \mathbf{H}\_{k}^{(c,c)} \text{ [}\boldsymbol{\Phi}\_{1}^{(c)}\text{ }\cdots \text{ }\boldsymbol{\Phi}\_{m-1}^{(c)}\text{ }\boldsymbol{\Phi}\_{m+1}^{(c)}\text{ }\cdots \text{ }\boldsymbol{\Phi}\_{M\_{\varepsilon}}^{(c)}\text{], and }\tilde{\mathbf{H}}\_{k}^{(l,c)} = \mathbf{H}\_{k}^{(c,c)}\text{ [}\boldsymbol{\Phi}\_{1}^{(l)}\text{] }\cdots \text{ }\boldsymbol{\Phi}\_{M\_{\varepsilon}}^{(M\_{\varepsilon})}\text{ [}\boldsymbol{\Phi}\_{1}^{(l)}\text{] }\cdots \text{ }\boldsymbol{\Phi}\_{M\_{\varepsilon}}^{(l)}\text{].}\\\text{If }\cdots \text{, }\boldsymbol{\Phi}\_{M\_{l}}^{(l)}\text{].}\end{array}$$

iii. Apply the MMSE receive beamformer, i.e., *t* (*c*) *<sup>k</sup>*,*<sup>m</sup>* = *PT Mc <sup>W</sup>*(*c*) *k* <sup>−</sup><sup>1</sup> *<sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* , then calculate and feedback the SINR corresponding to the *<sup>m</sup>*-th transmit beam *<sup>φ</sup>*(*c*) *m* to the BS

$$\text{SINR}^{(\text{MMSE},c)}\_{k,m} = \frac{P\_T}{M\_c} \left(\tilde{\mathbf{h}}^{(c,c)}\_{k,m}\right)^H \left(\mathbf{W}^{(c)}\_k\right)^{-1} \tilde{\mathbf{h}}^{(c,c)}\_{k,m}.\tag{7}$$

	- i. Estimate the effective channel through training with the *<sup>c</sup>*-th BS: *PT Mc <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* = *PT Mc <sup>H</sup>*(*c*,*c*) *<sup>k</sup> <sup>φ</sup>*(*c*) *<sup>m</sup>* .
	- ii. Apply the MF, i.e., *t* (*c*) *<sup>k</sup>*,*<sup>m</sup>* <sup>=</sup> *<sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* / ||*h***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* ||. The rationale is to maximize the power received from the desired beam. The receive signal now is

$$\begin{split} r\_{k,m}^{(c)} = \left(\mathbf{t}\_{k,m}^{(c)}\right)^{H} \mathbf{y}\_{k}^{(c)} = \sqrt{\frac{P\_{T}}{M\_{\text{c}}}} \left(\mathbf{t}\_{k,m}^{(c)}\right)^{H} \tilde{\mathbf{f}}\_{k,m}^{(c,c)} \mathbf{s}\_{m}^{(c)} + \sqrt{\frac{P\_{T}}{M\_{\text{c}}}} \left(\mathbf{t}\_{k,m}^{(c)}\right)^{H} \mathbf{f}\_{k,-m}^{(c,c)} \mathbf{s}\_{-m}^{(c)} \\ + \sum\_{l=1, l\neq c} \sqrt{\frac{P\_{T}\gamma\_{l,c}}{M\_{l}}} \left(\mathbf{t}\_{k,m}^{(c)}\right)^{H} \mathbf{f}\_{k}^{(l,c)} \mathbf{s}\_{(l)} + \left(\mathbf{t}\_{k,m}^{(c)}\right)^{H} \mathbf{z}\_{k}^{(c)}, \end{split} \tag{8}$$

where *s* (*c*) <sup>−</sup>*<sup>m</sup>* = [*s* (*c*) <sup>1</sup> , ··· , *<sup>s</sup>* (*c*) *<sup>m</sup>*−1, *s* (*c*) *<sup>m</sup>*+1, ··· , *<sup>s</sup>* (*c*) *Mc* ] and *s* (*l*) *<sup>m</sup>* = [*s* (*l*) <sup>1</sup> , ··· , *<sup>s</sup>* (*c*) *Ml* ]. Through training with all BSs, user *k* in the *c*-th cell estimates the interferences' power in (8), which can be expressed equivalently as *<sup>h</sup>***˜**(*c*,*c*) *k*,*m <sup>H</sup> <sup>W</sup>*(*c*) *k <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* , in which *<sup>W</sup>*(*c*) *<sup>k</sup>* is defined as in (6). Each user then calculates and feedbacks the SINR corresponding to the *<sup>m</sup>*-th transmit beam *<sup>φ</sup>*(*c*) *<sup>m</sup>* to the BS, which can be expressed equivalently as follows

$$\text{SINR}\_{k,m}^{\text{(MF,c)}} = \frac{\frac{P\_T}{\tilde{M}\_c} ||\tilde{\hbar}\_{k,m}^{\text{(c,c)}}||^4}{\left(\tilde{\hbar}\_{k,m}^{\text{(c,c)}}\right)^H \left(\mathbf{W}\_k^{\text{(c)}}\right) \tilde{\hbar}\_{k,m}^{\text{(c,c)}}} \tag{9}$$

10.5772/57131

33

(Rx,*c*) *<sup>m</sup>* .

<sup>1</sup> ,1 . (14)

<sup>2</sup> . (15)

. (17)

SINR*k*,1 . (16)

*<sup>k</sup>*,*m*, which corresponds to

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

Random Beamforming in Multi – User MIMO Systems

*<sup>k</sup>*,*<sup>m</sup>* , (13)

<sup>1</sup> + SINR(Rx,*c*) *k* (Rx,*c*)

For each *<sup>m</sup>*-th beam, the *<sup>k</sup>*-th user selects antenna *<sup>n</sup>*(*c*)

*k*

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

log2 

**3. Literature survey on single-cell OBF/RBF**

cell index *c* is dropped for brevity. (3) and (5) hence reduce to

*R*RBF = *M***E**

SINR*k*,*m*, ∀*k*, *<sup>m</sup>* can be expressed as [20]

SINR*k*,*<sup>m</sup>* =

 log2 

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

(*s* + 1) *M* 

*<sup>k</sup>*,*<sup>m</sup>* back to the BS. 2. Scheduling phase: The *c*-th BS assigns the *m*-th beam to the user with the highest SINR

(Rx,*c*) *<sup>m</sup>* <sup>=</sup> arg max *<sup>k</sup>*∈{1,··· ,*Kc*}

3. Transmitting phase: The *c*-th BS uses the *m*-th beam to transmit data to user *k*

<sup>1</sup> + SINR(Rx,*c*) *k* (Rx,*c*) *<sup>m</sup>* ,*<sup>m</sup>*

The achievable sum-rate in bits per second per Hz (bps/Hz) of the *c*-th RBF-Rx cell can be

**Remark 2.1.** *The RBF-AS scheme consists of two selection processes: antenna selection at the MSs each with NR antennas and opportunistic scheduling at the BSs with Kc's users. The rate performance of RBF-AS is therefore equal to that of the (MISO-BC) RBF with NRKc single-antenna users in the c-th cell. The RBF-AS scheme is introduced here to provide a complete study and rigorous comparison.*

Since its introduction in the landmark paper [19], opportunistic communication has developed to a broad area with various constituent topics. In this section, we aim to present a succinct overview on the key developments of OBF/RBF, summarizing some of the most important results contributed to the field. Note that in the literature, virtually all the works consider the single-cell case. It is only quite recent that the rate performance of the multi-cell RBF and ad-hoc IC with user scheduling is explored in [24] [22] [23]. We therefore limit our survey to the single-cell OBF/RBF. In this section, the channel model (2) has *C* = 1 and the

> *PT <sup>M</sup>* <sup>|</sup>*hkφm*<sup>|</sup>

*<sup>M</sup>* <sup>∑</sup>*<sup>M</sup>*

1 + max *k*∈{1,··· ,*K*}

*M* − 1 +

*s* + 1 *η* 

The probability density function (PDF) and cumulative density function (CDF) of *S* :=

*σ*<sup>2</sup> + *PT*

2

*<sup>i</sup>*=1,*i*�=*<sup>m</sup>* <sup>|</sup>*hkφi*<sup>|</sup>

= *Mc***E**

 log2 

SINR(Rx,*c*)

*<sup>k</sup>*,*<sup>m</sup>* , and feeds SINR(AS,*c*)

SINR(AS,*c*)

expressed as

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

RBF-Rx <sup>=</sup> **<sup>E</sup>**

 *Mc* ∑ *m*=1

Note that the *<sup>k</sup>*-th user in the *<sup>c</sup>*-th cell knows *<sup>h</sup>***˜**(*c*,*c*) *k*,*m <sup>H</sup> <sup>W</sup>*(*c*) *k <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup>* but not *<sup>W</sup>*(*c*) *k* . b3) AS: The received signal at the *n*-th receive antenna of user *k* in the *c*-th cell is:

$$\mathbf{y}\_{k,n}^{(c)} = \mathbf{h}\_{k,n}^{(c,c)} \sum\_{m=1}^{M\_{\mathcal{L}}} \Phi\_m^{(c)} \mathbf{s}\_m^{(c)} + \sum\_{\substack{l=1,\ l \neq c}}^{C} \sqrt{\gamma\_{l,c}} \mathbf{h}\_{k,n}^{(l,c)} \sum\_{m=1}^{M\_l} \Phi\_m^{(l)} \mathbf{s}\_m^{(l)} + z\_{k,n'}^{(c)} \tag{10}$$

where *y* (*c*) *<sup>k</sup>*,*<sup>n</sup>* and *z* (*c*) *<sup>k</sup>*,*<sup>n</sup>* are the *<sup>n</sup>*-th element of *<sup>y</sup>*(*c*) *<sup>k</sup>* and *z* (*c*) *<sup>k</sup>* , respectively, and *<sup>h</sup>*(*l*,*c*) *<sup>k</sup>*,*<sup>n</sup>* <sup>∈</sup> **<sup>C</sup>**1×*Ml* is the *<sup>n</sup>*-th row of *<sup>H</sup>*(*l*,*c*) *<sup>k</sup>* , *<sup>n</sup>* ∈ {1, . . . , *NR*}, *<sup>l</sup>*, *<sup>c</sup>* ∈ {1, . . . , *<sup>C</sup>*}. For the *<sup>m</sup>*-th beam and *n*-th antenna, user *k* estimates the received power of the signal and the interferences through training with the *c*-th BS and all BSs, respectively. The following SINR value is then calculated:

$$\text{SINR}\_{k,n,m}^{(\text{AS},c)} = \frac{\frac{P\_T}{M\_c} \left| h\_{k,n}^{(c,c)} \phi\_m^{(c)} \right|^2}{\sigma^2 + \frac{P\_T}{M\_c} \sum\_{i=1, i \neq m}^{M\_c} \left| h\_{k,n}^{(c,c)} \phi\_i^{(c)} \right|^2 + \sum\_{l=1, l \neq c}^{C} \gamma\_{l,c} \frac{P\_T}{M\_l} \sum\_{i=1}^{M\_l} \left| h\_{k,n}^{(l,c)} \phi\_i^{(l)} \right|^2} . \tag{11}$$

Denote

$$\text{SINR}^{(\text{AS},c)}\_{k,m} := \max\_{n \in \{1, \dots, N\_R\}} \text{SINR}^{(\text{AS},c)}\_{k,n,m}.\tag{12}$$

For each *<sup>m</sup>*-th beam, the *<sup>k</sup>*-th user selects antenna *<sup>n</sup>*(*c*) *<sup>k</sup>*,*m*, which corresponds to SINR(AS,*c*) *<sup>k</sup>*,*<sup>m</sup>* , and feeds SINR(AS,*c*) *<sup>k</sup>*,*<sup>m</sup>* back to the BS.

2. Scheduling phase: The *c*-th BS assigns the *m*-th beam to the user with the highest SINR

$$k\_{m}^{(\text{Rx},c)} = \arg\max\_{k \in \{1, \dots, K\_{\ell}\}} \text{SINR}\_{k,m}^{(\text{Rx},c)}\,\text{}\,\tag{13}$$

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

6 Recent Trends in Multiuser MIMO Communications

+ ∑ *l*=1,*l*�=*c*

(*c*) *<sup>m</sup>*−1, *s* (*c*) *<sup>m</sup>*+1, ··· , *<sup>s</sup>*

SINR(MF,*c*) *<sup>k</sup>*,*<sup>m</sup>* =

Note that the *k*-th user in the *c*-th cell knows

*Mc* ∑ *m*=1

*<sup>φ</sup>*(*c*) *<sup>m</sup> s* (*c*) *<sup>m</sup>* +

*<sup>k</sup>*,*<sup>n</sup>* are the *<sup>n</sup>*-th element of *<sup>y</sup>*(*c*)

*Mc* ∑ *i*=1,*i*�=*m*

SINR(AS,*c*)

 *<sup>h</sup>*(*c*,*c*) *<sup>k</sup>*,*<sup>n</sup> <sup>φ</sup>*(*c*) *i* 2 +

 *PTγl*,*<sup>c</sup> Ml*

 *t* (*c*) *k*,*m H <sup>H</sup>***˜** (*l*,*c*) *<sup>k</sup> s*(*l*) +

(*c*) *Mc*

training with all BSs, user *k* in the *c*-th cell estimates the interferences' power in (8),

defined as in (6). Each user then calculates and feedbacks the SINR corresponding

*PT Mc*

*<sup>H</sup>*

 *<sup>h</sup>***˜**(*c*,*c*) *k*,*m*

<sup>√</sup>*γl*,*ch*(*l*,*c*) *k*,*n*

(*c*)

*<sup>k</sup>* , *<sup>n</sup>* ∈ {1, . . . , *NR*}, *<sup>l</sup>*, *<sup>c</sup>* ∈ {1, . . . , *<sup>C</sup>*}. For the *<sup>m</sup>*-th beam and

*C* ∑ *l*=1,*l*�=*c*

*γl*,*c PT Ml*

SINR(AS,*c*)

*<sup>k</sup>* and *z*

 *<sup>h</sup>***˜**(*c*,*c*) *k*,*m*

*C* ∑ *l*=1, *l*�=*c*

*n*-th antenna, user *k* estimates the received power of the signal and the interferences through training with the *c*-th BS and all BSs, respectively. The following SINR value

> *PT Mc <sup>h</sup>*(*c*,*c*) *<sup>k</sup>*,*<sup>n</sup> <sup>φ</sup>*(*c*) *m* 2

*<sup>k</sup>*,*<sup>m</sup>* :<sup>=</sup> max *<sup>n</sup>*∈{1,··· ,*NR*}

b3) AS: The received signal at the *n*-th receive antenna of user *k* in the *c*-th cell is:


> *<sup>W</sup>*(*c*) *k <sup>h</sup>***˜**(*c*,*c*) *k*,*m*

> > *<sup>H</sup>*

*Ml* ∑ *m*=1

*<sup>W</sup>*(*c*) *k <sup>h</sup>***˜**(*c*,*c*)

*<sup>φ</sup>*(*l*) *<sup>m</sup> s* (*l*) *<sup>m</sup>* + *z* (*c*)

*<sup>k</sup>* , respectively, and *<sup>h</sup>*(*l*,*c*)

*Ml* ∑ *i*=1 *<sup>h</sup>*(*l*,*c*) *<sup>k</sup>*,*<sup>n</sup> <sup>φ</sup>*(*l*) *i* 2

*<sup>k</sup>*,*n*,*<sup>m</sup>* . (12)

 *<sup>h</sup>***˜**(*c*,*c*) *k*,*m*

] and *s*

*<sup>H</sup>*

(*l*) *<sup>m</sup>* = [*s*

*<sup>W</sup>*(*c*) *k <sup>h</sup>***˜**(*c*,*c*)

*<sup>m</sup>* to the BS, which can be expressed equivalently as

 *PT Mc t* (*c*) *k*,*m H*

*<sup>H</sup>***˜** (*c*,*c*) *<sup>k</sup>*,−*ms* (*c*) −*m*

> (*c*) *Ml*

*<sup>k</sup>*,*<sup>m</sup>* , in which *<sup>W</sup>*(*c*)

, (9)

*<sup>k</sup>*,*<sup>m</sup>* but not *<sup>W</sup>*(*c*)

*<sup>k</sup>*,*n*, (10)

*<sup>k</sup>*,*<sup>n</sup>* <sup>∈</sup> **<sup>C</sup>**1×*Ml*

. (11)

]. Through

*<sup>k</sup>* is

*k* .

 *t* (*c*) *k*,*m H z* (*c*) *<sup>k</sup>* , (8)

(*l*) <sup>1</sup> , ··· , *<sup>s</sup>*

*r* (*c*) *<sup>k</sup>*,*<sup>m</sup>* = *t* (*c*) *k*,*m H <sup>y</sup>*(*c*) *<sup>k</sup>* = *PT Mc t* (*c*) *k*,*m H <sup>h</sup>***˜**(*c*,*c*) *<sup>k</sup>*,*<sup>m</sup> s* (*c*) *<sup>m</sup>* +

where *s*

follows

(*c*) <sup>−</sup>*<sup>m</sup>* = [*s*

> *y* (*c*) *<sup>k</sup>*,*<sup>n</sup>* <sup>=</sup> *<sup>h</sup>*(*c*,*c*) *k*,*n*

is the *<sup>n</sup>*-th row of *<sup>H</sup>*(*l*,*c*)

(*c*)

*σ*<sup>2</sup> + *PT Mc*

where *y*

Denote

(*c*) *<sup>k</sup>*,*<sup>n</sup>* and *z*

is then calculated:

SINR(AS,*c*) *<sup>k</sup>*,*n*,*<sup>m</sup>* = (*c*) <sup>1</sup> , ··· , *<sup>s</sup>*

to the *<sup>m</sup>*-th transmit beam *<sup>φ</sup>*(*c*)

which can be expressed equivalently as

3. Transmitting phase: The *c*-th BS uses the *m*-th beam to transmit data to user *k* (Rx,*c*) *<sup>m</sup>* .

The achievable sum-rate in bits per second per Hz (bps/Hz) of the *c*-th RBF-Rx cell can be expressed as

$$R\_{\rm RBF-Rx}^{(c)} = \mathbb{E}\left[\sum\_{m=1}^{M\_c} \log\_2\left(1 + \text{SINR}\_{k\_m^{(\rm Rx,c)},m}^{(\rm Rx,c)}\right)\right] = M\_c \mathbb{E}\left[\log\_2\left(1 + \text{SINR}\_{k\_1^{(\rm Rx,c)},1}^{(\rm Rx,c)}\right)\right].\tag{14}$$

**Remark 2.1.** *The RBF-AS scheme consists of two selection processes: antenna selection at the MSs each with NR antennas and opportunistic scheduling at the BSs with Kc's users. The rate performance of RBF-AS is therefore equal to that of the (MISO-BC) RBF with NRKc single-antenna users in the c-th cell. The RBF-AS scheme is introduced here to provide a complete study and rigorous comparison.*

#### **3. Literature survey on single-cell OBF/RBF**

Since its introduction in the landmark paper [19], opportunistic communication has developed to a broad area with various constituent topics. In this section, we aim to present a succinct overview on the key developments of OBF/RBF, summarizing some of the most important results contributed to the field. Note that in the literature, virtually all the works consider the single-cell case. It is only quite recent that the rate performance of the multi-cell RBF and ad-hoc IC with user scheduling is explored in [24] [22] [23]. We therefore limit our survey to the single-cell OBF/RBF. In this section, the channel model (2) has *C* = 1 and the cell index *c* is dropped for brevity. (3) and (5) hence reduce to

$$\text{SINR}\_{k,m} = \frac{\frac{P\_T}{M} \left| \hbar\_k \Phi\_m \right|^2}{\sigma^2 + \frac{P\_T}{M} \sum\_{i=1, i \neq m}^M \left| \hbar\_k \Phi\_i \right|^2}. \tag{15}$$

$$R\_{\rm RBF} = M \mathbb{E}\left\{ \log\_2 \left( 1 + \max\_{k \in \{1, \dots, K\}} \text{SINR}\_{k, 1} \right) \right\}. \tag{16}$$

The probability density function (PDF) and cumulative density function (CDF) of *S* := SINR*k*,*m*, ∀*k*, *<sup>m</sup>* can be expressed as [20]

$$f\_S(s) = \frac{e^{-s/\eta}}{\left(s+1\right)^M} \left(M - 1 + \frac{s+1}{\eta}\right). \tag{17}$$

$$F\_S(s) = 1 - \frac{e^{-s/\eta}}{\left(s+1\right)^{M-1}}.\tag{18}$$

*f*2,*m*(*x*) =

**3.2. Asymptotic analysis**

and large-system analyses.

*3.2.1. Large number of users*

 <sup>∞</sup> 0

 (*M*−*m*)*<sup>w</sup> m*−1

*Finally, the function fwm*,*βm*:*M*,*zm* (*w*, *<sup>β</sup>*, *<sup>z</sup>*) *is given in [26, (28)]*

× *M*−*m* ∑ *i*=0

 1 *η*

+ *z* + *w*

(*L* − 1)!(*l* − 1)!*β*¯*<sup>M</sup>*

*M* − *m i*

*where* U(*x*) *is the unit step function and β*¯ = 1 *due to the Rayleigh fading channel model.*

*m*−1

 (−1)*<sup>i</sup>*

*β* > 0, *w* > (*m* − 1)*β*, *z* < (*M* − 1)*β*, (25)

In addition, loose approximations for (5) are presented in [27] and [28]. We note that (19) involves only the exponential integral function, which is more efficiently computable than the Gaussian hyper-geometric functions in [27] [28] and the (exact) expression in Lemma 3.2. However, the sum-rate approximations in [27] and [28] can directly lead to some asymptotic results, e.g., the sum-rate scaling law *<sup>M</sup>* log2 log *<sup>K</sup>* as *<sup>K</sup>* <sup>→</sup> <sup>∞</sup>, while (19) and (20) do not.

The accurate expression for the achievable sum rate, albeit important, is too complicated. To reveal more insights on the performance of single-cell RBF, asymptotic analyses have been considered in other studies. There are two main approaches, namely, large-number-of-users

The conventional asymptotic investigation of OBF/RBF is to consider the number of users approaches infinity for a given finite SNR. Based on *extreme value theory* [29], one of the most

**Theorem 3.1.** *([20, Theorem 1]) For fixed M* ≤ *NT and PT, the single-cell RBF sum rate grows*

*RRBF*

In [21], Sharif *et. al.* show that *NT* log2 log *K* is also the rate scaling law of the optimal DPC scheme assuming perfect CSI as the number of users goes to infinity, for any given

<sup>1</sup> Strictly speaking, Theorem 3.1 only states for the single-cell RBF with the original feedback scheme in [20]. However, it is easy to see that the same result applies when the modified feedback scheme in Section 2.1 is considered.

important results in opportunistic communication is proved in [20]1.

lim *K*→∞

*double-logarithmically with respect to the number of users, i.e.,*

*fwm*,*βm*:*M*,*zm*

 *w*, *x* 1 *ρ*

exp

−*w*+*β*+*<sup>z</sup> β*¯ 

(*<sup>z</sup>* <sup>−</sup> *<sup>i</sup>β*)*M*−*m*−1U(*<sup>z</sup>* <sup>−</sup> *<sup>i</sup>β*),

*<sup>M</sup>* log2 log *<sup>K</sup>* <sup>=</sup> 1. (26)

(*m* − 2)!(*M* − *m* − 1)!

+ *z* + *w*

 , *z* 

Random Beamforming in Multi – User MIMO Systems

U(1 − (*m* − 1)*β*)×

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

0

*fwm*,*βm*:*M*,*zm* (*w*, *<sup>β</sup>*, *<sup>z</sup>*) = *<sup>M</sup>*![*<sup>w</sup>* <sup>−</sup> (*<sup>m</sup>* <sup>−</sup> <sup>1</sup>)*β*]

10.5772/57131

35

*dzdw*. (24)

where *η* = *PT*/(*Mσ*2) is the SNR per beam.

#### **3.1. Achievable rate**

With the feedback scheme stated in Section 2.1, the closed-form expression for the sum rate *R*RBF is given in the following lemma.

**Lemma 3.1.** *([22, Lemma 3.1], see also [55]) The average sum rate of the single-cell RBF is given by*

$$R\_{\rm RBF} = \frac{M}{\log 2} \sum\_{n=1}^{K} (-1)^n \binom{K}{n} \left[ \left( -\frac{n}{\eta} \right)^{n(M-1)} \frac{\varepsilon^{n/\eta} \operatorname{Ei}(-n/\eta)}{(n(M-1))!} \right.$$

$$-\sum\_{m=1}^{n(M-1)} \left( -\frac{n}{\eta} \right)^{m-1} \frac{(n(M-1) - m)!}{(n(M-1))!} \Big|\_{} \prime \tag{19}$$

*where Ei*(*x*) = *<sup>x</sup>* <sup>−</sup><sup>∞</sup> *<sup>e</sup><sup>t</sup> <sup>t</sup> dt is the exponential integral function.*

Assuming the feedback scheme in [20], (19) becomes a close approximation of the exact sum rate, especially when the number of users *K* is large. The exact expression of the sum rate is derived in [25], which involves a numerical integral with the SINR CDF of the first beam:

**Lemma 3.2.** *([25]) Assuming the feedback scheme in [20], the average sum rate of the single-cell RBF is given by*

$$R\_{\rm RBF}^{\prime} = M \int\_0^\infty \log\_2(1+\mathfrak{x}) f\_{\rm SINR}^{\prime}(\mathfrak{x}) d\mathfrak{x} \,\tag{20}$$

*in which*

$$f\_{\text{SINR}\_{k,1}}'(\mathbf{x}) = \sum\_{k=1}^{K} \frac{1}{M} \left(1 - \frac{1}{M}\right)^{k-1} \frac{K!}{(K-k)!(k-1)!} \left(F\_1(\mathbf{x})\right)^{K-k} \left[1 - F\_1(\mathbf{x})\right]^{k-1} f\_1(\mathbf{x})$$

$$+ \left(1 - \frac{1}{M}\right)^K \frac{1}{M-1} \sum\_{m=2}^{M} f\_{2,m}(\mathbf{x}),\tag{21}$$

$$F\_1(\mathbf{x}) = \int\_0^\mathbf{x} f\_1(t)dt,\tag{22}$$

$$f\_1(\mathbf{x}) = \sum\_{m=0} M - 1 \frac{(-1)^m M!}{(M - m - 1)! m!} \exp\left(-\frac{(1 + m)\mathbf{x}}{\eta (m\mathbf{x} - 1)}\right) \times$$

$$\times \frac{(M + 1/\eta - 1 + (1/\eta - mM + m)\mathbf{x})(1 - m\mathbf{x})^{M - 3}}{(1 + \mathbf{x})^M},\qquad(23)$$

$$f\_{\mathfrak{D},m}(\mathbf{x}) = \int\_0^\infty \int\_0^{\frac{(M-m)\mathfrak{D}}{m-1}} \left(\frac{1}{\eta} + z + w\right) f\_{\mathfrak{D}\_m \mathfrak{G}\_{\mathfrak{W}:M,\mathfrak{Z}\_m}} \left(w, \mathbf{x}\left(\frac{1}{\rho} + z + w\right), z\right) dz dw. \tag{24}$$

*Finally, the function fwm*,*βm*:*M*,*zm* (*w*, *<sup>β</sup>*, *<sup>z</sup>*) *is given in [26, (28)]*

$$f\_{\overline{w}\_m \beta\_{wM} z\_m}(w, \beta, z) = \frac{M! [w - (m - 1)\beta]^{m - 1}}{(L - 1)! (l - 1)! \tilde{\beta}^M} \frac{\exp\left(-\frac{w + \beta + z}{\beta}\right)}{(m - 2)! (M - m - 1)!} l \ell(1 - (m - 1)\beta) \times l \tag{25}$$

$$\times \sum\_{i = 0}^{M - m} \binom{M - m}{i} (-1)^i (z - i\beta)^{M - m - 1} l \ell(z - i\beta),$$

$$\beta > 0, w > (m - 1)\beta, z < (M - 1)\beta,\tag{25}$$

*where* U(*x*) *is the unit step function and β*¯ = 1 *due to the Rayleigh fading channel model.*

In addition, loose approximations for (5) are presented in [27] and [28]. We note that (19) involves only the exponential integral function, which is more efficiently computable than the Gaussian hyper-geometric functions in [27] [28] and the (exact) expression in Lemma 3.2. However, the sum-rate approximations in [27] and [28] can directly lead to some asymptotic results, e.g., the sum-rate scaling law *<sup>M</sup>* log2 log *<sup>K</sup>* as *<sup>K</sup>* <sup>→</sup> <sup>∞</sup>, while (19) and (20) do not.

## **3.2. Asymptotic analysis**

8 Recent Trends in Multiuser MIMO Communications

where *η* = *PT*/(*Mσ*2) is the SNR per beam.

*R*RBF is given in the following lemma.

*K* ∑ *n*=1

(−1)*<sup>n</sup>*

*R*′

*RBF* = *M*

*<sup>M</sup>* <sup>−</sup> <sup>1</sup> (−1)*mM*!

(*M* − *m* − 1)!*m*!

*K n*  −*n η*

*<sup>t</sup> dt is the exponential integral function.*

 <sup>∞</sup> 0

*<sup>k</sup>*−<sup>1</sup> *<sup>K</sup>*!

+ <sup>1</sup> <sup>−</sup> <sup>1</sup> *M*

*F*1(*x*) =

**3.1. Achievable rate**

*RRBF* <sup>=</sup> *<sup>M</sup>*

*where Ei*(*x*) = *<sup>x</sup>*

*is given by*

*in which*

*f* ′

*SINRk*1,1 (*x*) =

*f*1(*x*) = ∑

*m*=0

*K* ∑ *k*=1

1 *M* <sup>1</sup> <sup>−</sup> <sup>1</sup> *M*

log 2

<sup>−</sup><sup>∞</sup> *<sup>e</sup><sup>t</sup>*

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

With the feedback scheme stated in Section 2.1, the closed-form expression for the sum rate

**Lemma 3.1.** *([22, Lemma 3.1], see also [55]) The average sum rate of the single-cell RBF is given by*

−

Assuming the feedback scheme in [20], (19) becomes a close approximation of the exact sum rate, especially when the number of users *K* is large. The exact expression of the sum rate is derived in [25], which involves a numerical integral with the SINR CDF of the first beam: **Lemma 3.2.** *([25]) Assuming the feedback scheme in [20], the average sum rate of the single-cell RBF*

log2(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*)*<sup>f</sup>* ′

(*K* − *k*)!(*k* − 1)!

 *<sup>x</sup>* 0

exp  *<sup>K</sup>* 1 *M* − 1

− (<sup>1</sup> + *<sup>m</sup>*)*<sup>x</sup> η*(*mx* − 1)

<sup>×</sup> (*<sup>M</sup>* <sup>+</sup> 1/*<sup>η</sup>* <sup>−</sup> <sup>1</sup> + (1/*<sup>η</sup>* <sup>−</sup> *mM* <sup>+</sup> *<sup>m</sup>*)*x*)(<sup>1</sup> <sup>−</sup> *mx*)*M*−<sup>3</sup>

(*F*1(*x*))

*M* ∑ *m*=2

> ×

*n*(*M*−1) ∑ *m*=1

*<sup>n</sup>*(*M*−1) *<sup>e</sup>n*/*ηEi*(−*n*/*η*)

 −*n η*

(*n*(*M* − 1))!

(*s* + 1)

*<sup>M</sup>*−<sup>1</sup> . (18)

*<sup>m</sup>*−<sup>1</sup> (*n*(*<sup>M</sup>* <sup>−</sup> <sup>1</sup>) <sup>−</sup> *<sup>m</sup>*)! (*n*(*M* − 1))!

*SINRk*1,1 (*x*)*dx*, (20)

*<sup>K</sup>*−*<sup>k</sup>* [<sup>1</sup> <sup>−</sup> *<sup>F</sup>*1(*x*)]

*f*1(*t*)*dt*, (22)

(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*)*<sup>M</sup>* , (23)

*<sup>k</sup>*−<sup>1</sup> *<sup>f</sup>*1(*x*)

*f*2,*m*(*x*), (21)

, (19)

The accurate expression for the achievable sum rate, albeit important, is too complicated. To reveal more insights on the performance of single-cell RBF, asymptotic analyses have been considered in other studies. There are two main approaches, namely, large-number-of-users and large-system analyses.

#### *3.2.1. Large number of users*

The conventional asymptotic investigation of OBF/RBF is to consider the number of users approaches infinity for a given finite SNR. Based on *extreme value theory* [29], one of the most important results in opportunistic communication is proved in [20]1.

**Theorem 3.1.** *([20, Theorem 1]) For fixed M* ≤ *NT and PT, the single-cell RBF sum rate grows double-logarithmically with respect to the number of users, i.e.,*

$$\lim\_{K \to \infty} \frac{R\_{RBF}}{M \log\_2 \log K} = 1.\tag{26}$$

In [21], Sharif *et. al.* show that *NT* log2 log *K* is also the rate scaling law of the optimal DPC scheme assuming perfect CSI as the number of users goes to infinity, for any given

<sup>1</sup> Strictly speaking, Theorem 3.1 only states for the single-cell RBF with the original feedback scheme in [20]. However, it is easy to see that the same result applies when the modified feedback scheme in Section 2.1 is considered.

user's SNR. Essentially, the result implies that the intra-cell interference in a single-cell RBF system can be completely eliminated when the number of users is sufficiently large, and an "interference-free" MU broadcast system is attainable. This important result therefore establishes the optimality of single-cell RBF and motivates further studies on opportunistic communication. Various MIMO-BC transmission schemes with different assumptions on the fading model, feedback scheme, user scheduling, etc., can be shown achieving *NT* log2 log *K* as *K* → ∞, suggesting that it is a very universal rate scaling law (see, e.g., [30] [31] [28] [32] [33] [34]). Finally, it is worth noting that the RBF sum rate grows only *logarithmically* with *K*, i.e., *RRBF <sup>M</sup> <sup>M</sup>*−<sup>1</sup> log2 *<sup>K</sup>* <sup>→</sup> <sup>1</sup> as *<sup>K</sup>* <sup>→</sup> <sup>∞</sup>, when the background noise is ignored [30] [33]. More discussions on the large-number-of-users analysis will be given in Section 4.1.2 and 4.2.

10.5772/57131

37

<sup>1</sup> *<sup>v</sup>*2|2, (27)

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

Random Beamforming in Multi – User MIMO Systems

*<sup>m</sup>*=<sup>1</sup> is a solution of the GLPP. However,

*<sup>m</sup>*=<sup>1</sup> is a nontrivial GLPP. The idea of using more than

*<sup>d</sup>*(*v*1, *<sup>v</sup>*2) <sup>=</sup> *sin*(*v*1, *<sup>v</sup>*2) =

designing beamforming codebook for space-time codes [42] [43].

antennas, i.e., *<sup>M</sup>* <sup>≤</sup> *NT*, any orthonormal set {*φm*}*<sup>M</sup>*

the fairness of the system, finding {*φm*}*<sup>M</sup>*

multi-cell systems is introduced in [51].

**4.1. Multi-cell MISO RBF**

**4. Multi-cell RBF: Finite-SNR analysis**

is drawn.

**3.5. Other studies**

which is known as the chordal distance. The problem of finding the packing of *M* unit-norm vectors in **<sup>C</sup>***NT*×<sup>1</sup> that has the maximum minimum distance between any pair of them is called the *Grassmannian line packing problem* (GLPP). The GLPP appears in the problem of

Given that the number of transmit beams is less than or equal to the number of transmit

assuming that the BS sends *M* > *NT* beams to serve more users simultaneously and improve

*NT* beams is first proposed in [44] with *M* = *NT* + 1, and further studied in [45]. Zorba *et. al.* argue that the scaling law *M* log2 log *K* is still true for *M* = *NT* + 1 case [44], while the results in [45] imply that non-orthogonal beamforming matrix induces an interference-limited effect on the sum rate, and the multi-user diversity vanishes. Since both studies are based on approximated derivations, more rigorous investigations are necessary before any conclusion

Beam selection and beam power control algorithms for single-cell RBF are proposed in [46] [47] [33] [48]. The objective is to improve the rate performance especially when the number of users is not so large. The idea of employing a codebook of predetermined orthonormal beamforming matrices is introduced in [30] [31] [27]. While [30] [31] investigate RBF when quantized, normalized channel vectors are fed back to the BS, [49] studies the codebook design problem and the rate performance assuming that opportunistic selection is also performed on the codebook. These problems are related to Section 3.3 and the GLPP in Section 3.4. Fairness scheduling problem is studied in [19], in which the "*proportional fair scheduling* (PFS)" scheme is proposed. It is not surprising that most of the later developments approach PFS from a network layer's perspective. Notably, the convergence of PFS algorithm for many-user cases under general network conditions is proved in [50], and a global PFS for

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

For the single-cell RBF case, the SINR distributions given in (17) and (18) are obtained in [20].

sum-rate scaling law as the number of users per cell tends to infinity.

The following lemma establishes similar results for the multi-cell case

1 − |*v<sup>H</sup>*

#### *3.2.2. Large system*

The large-system analysis is a well-known and widely-accepted method to investigate the performance of communication systems. A recent application to single-cell RBF is introduced in [35], in which a general MIMO-BC setup with MMSE receiver and different fading models is considered. Assuming the numbers of transmit/receive antennas and data beams to approach infinity at the same time with fixed ratios for any given finite SNR, Couillet *et. al.* obtain "almost closed-form" numerical solutions which provide deterministic approximations for various performance criteria. Note that although these results are derived under the large-system assumption, Monte-Carlo simulations demonstrate that they can be applied to study small-dimensional systems with modest errors. However, [35] does not consider opportunistic scheduling, which is one of the main features of RBF.
