**4. Linear beamforming**

2

*i ki n*

å <sup>+</sup>

The performance of a MIMO system is measured by a global objective function of the indi‐ vidual data rates or SINRs *U* (*r*1, ⋯, *r <sup>K</sup>* ). From the system perspective it is desirable to optimize *U* (∙ ) instead of the individual rates *ri* ∀*i* ∈ *S* since the latter are coupled by the transmit powers and the beamforming weights in (2). Thus the performance depends on how efficiently the resources are allocated to each user and how effectively the interference from other users is mitigated. In this chapter we optimize the global utility function modeled as the sum rate maximization problem using BF subject to global power constraints. For the case

*k k <sup>F</sup> <sup>k</sup>*

where ∙ *<sup>F</sup>* denotes the Frobenius norm, *β<sup>k</sup>* is a priority weight associated to user *k* defined a priory by upper layers of the communications system to take into account QoS, fairness, or another system constraint. Finding a solution of (3) is a complex problem due to the nature of the optimum **P** and **W** and each solution depends on the system requirements expressed by the weights *βk* [14]. The computation of optimal beamforming weights **w***<sup>k</sup>* involves SINR balancing [11] and since the weights do not have a closed-form, iterative computational demanding algorithms have been proposed to determine them [6], [15]. Indeed, problem (3)

Let Ω={1, ⋯, *K*} be the set of all competing users where *K* is larger than the number of available antennas at the BS, i.e., |Ω|= *K* ≥ *Nt*, where |Ω| denotes the cardinality of the set Ω. In order to exploit the optimization dimension provided by MUD, it is necessary to select a set of users *S* whose channel characteristics maximize the sum rate when they transmit simultaneously in the same radio resource. Such characteristics are defined by the type of beamforming scheme, the power constraints, the SNR regime, and the deployment character‐ istics (*Nt* and *K*). The sum rate maximization with user selection optimization problem can be

å £

**h w**

SINR *k kk k*

=

and the instantaneous achievable data rate of user *k* is *rk* =log2 (1 + SINR*<sup>k</sup>* ).

where *K* ≤ *Nt* the general optimization problem is given by

*BF*

, <sup>1</sup> R = max s.t. *K*

**W P**

is NP-hard even when all priority weights *βk* are equal [16].

=

b

**3. Problem formulation**

28 Contemporary Issues in Wireless Communications

**3.1. Multiuser scenario**

defined as:

*p p*

*i k*

¹

2 2

{ <sup>2</sup> 1/2

*r P*

s

**h w** (2)

**WP** (3)

In this section we describe the structure of two sub-optimal linear beamforming schemes and the optimal power allocation for each one of them. It is assumed perfect CSI at the transmitter which can be attained through time-division duplex (TDD) scheme assuming channel reciprocity [6]. Notice that the weight vectors multiply the intended symbols in (1) which can be seen as a form of precoding, henceforward we use the terms beamforming and precoding interchangeably.

#### **4.1. Zero Forcing Beamforming**

In Zero Forcing Beamforming (ZFBF), the channel matrix **H** at the transmitter is processed so that orthogonal channels between the transmitter and the receiver are created, defining a set of parallel subchannels [5]. The Moore-Penrose pseudo inverse of **H**(*S*) is given by [17]:

$$
\tilde{\mathbf{W}}(\mathbf{S}) = \mathbf{H}(\mathbf{S})^H (\mathbf{H}(\mathbf{S}) \mathbf{H}(\mathbf{S})^H)^{-1} \tag{5}
$$

and the ZFBF matrix is given by the normalized column vectors of (5) as **<sup>W</sup>**(*S*)= **<sup>w</sup>**¯1 / **<sup>w</sup>**¯1 , <sup>⋯</sup>, **<sup>w</sup>**¯|*<sup>S</sup>* <sup>|</sup> / **<sup>w</sup>**¯|*<sup>S</sup>* <sup>|</sup> . Under ZFBF scheme the sum rate maximizing power allocation is given by the water-filling algorithm and according to [5] the information rate achieved with optimum **P** in (3) is given by :

$$\mathcal{R}^{ZFF}\{\mathbf{H}(\mathbf{S})\} = \sum\_{i=1}^{|\mathbf{S}|} \left(\log(\mu b\_i)\right)^{+} \tag{6}$$

where *bi* =( **<sup>H</sup>**(*S*)**H**(*S*)<sup>H</sup> *ii* -1)-1 is the effective channel gain of the *i*th user and its allocated power is *pi* =(*μbi* - 1)+, the water level *μ* is chosen to satisfy ∑ *i*∈*S* (*μ* - *bi* -1)+ =*P* and (*x*)<sup>+</sup> =max {*x*, 0}. If all users in *S* are allocated with nonzero power, the water level has the compact form [10], [18]: *<sup>μ</sup>* <sup>=</sup> <sup>1</sup> <sup>|</sup>*<sup>S</sup>* <sup>|</sup> ( **<sup>W</sup>**¯(*S*) *<sup>F</sup>* <sup>2</sup> <sup>+</sup> *<sup>P</sup>*) <sup>=</sup> <sup>1</sup> <sup>|</sup>*<sup>S</sup>* <sup>|</sup> (*P* + ∑ *i*∈*S bi* -1).

### **4.2. Zero forcing dirty paper**

Suboptimal throughput maximization in Gaussian BC channels has been proposed in several works [5], [18], [19] based on the QR-type decomposition of the channel matrix **H**(*S*)=**L**(*S*)**Q**(*S*) obtained by applying Gram-Schmidt orthogonalization (GSO) [20]. **L**(*S*) is a lower triangular matrix and **Q**(*S*) has orthonormal rows. The beamforming matrix given by **W**(*S*)=**Q**(*S*)<sup>H</sup> generates a set of interference channels *yi* =*l ii pi si* + ∑ *j*<*i l ij p <sup>j</sup> sj* + *ni* , *i* =1, ⋯, *k* while no infor‐ mation is sent to users *k* + 1, ⋯, *K*. In order to eliminate the interference component *Ii* = ∑ *j*<*i l ij p <sup>j</sup> sj* of the *i*th user, the signals *p <sup>j</sup> sj* for *i* =1, ⋯, *k* are obtained by successive dirty paper encoding, where *Ii* is non-causally known. This precoding scheme was proposed in [5] and the authors showed that the precoding matrix forces to zero the interference caused by users *j* >*i* on each user *i*, therefore this scheme is called zero-forcing dirty-paper (ZFDP) coding. The information rate achieved with optimum **P** in (3) under the ZFDP scheme is given by [5]:

$$\mathcal{R}^{ZFDP}(\mathbf{H(S)}) = \sum\_{i=1}^{|\mathbf{S}|} \left( \log(\mu d\_i) \right)^{+} \tag{7}$$

where *di* =|*l ii*|2 is the squared absolute value of *l ii* and *μ* is the solution to the water-filling equation ∑ *i*∈*S* (*μ* - *di* -1)+ <sup>=</sup>*P*, which defines the *i*th power as *pi* =(*μdi* - 1)+.
