**3. Problem formulation**

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

$$\mathbf{R}^{\mathrm{BF}} = \max\_{\mathbf{W}, \mathbf{P}} \sum\_{k=1}^{K} \beta\_k r\_k \qquad \text{s.t.} \ \left\| \mathbf{W} \mathbf{P}^{1/2} \right\|\_{F}^{2} \le P \tag{3}$$

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

#### **3.1. Multiuser scenario**

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 defined as:

User Selection and Precoding Techniques for Rate Maximization in Broadcast MISO Systems http://dx.doi.org/10.5772/58937 29

$$\mathfrak{R}(\mathbf{S}) = \max\_{\mathbf{S} \in \Omega : \|\mathbf{S}\| \le N\_t} R^{\mathrm{BF}}(\mathbf{H}(\mathbf{S})) \tag{4}$$

where **H**(*S*) is the row-reduced channel matrix containing only the channels of the subset of selected users *S* and *R BF* (**H**(*S*)) is the achievable sum rate for such set. User scheduling is a real time operation whose computational complexity and implementation efficiency affect the performance of upper-layers. Moreover, finding the optimal solution of (4) requires an exhaustive search over a search space of size *L* =∑*i*=1 *Nt K* ! / (*i* !(*K* - *i*)!), which is the number of all ordered combinations of users.

Since the computation of the optimal solution of the sum rate maximization problem implies the joint optimization over **P**, **W**, and *S*, the original problem (4) can be relaxed by taking one or more of the following actions: 1) by using beamforming weights with a defined structure; 2) based on linear beamforming the power allocation can be performed either by Lagrangian methods or by equal power allocation; and 3) based on the structure of the linear beamforming it is possible to design user selection algorithms that exploit information contained in **H** and at the same time, to achieve a good trade-off between performance and complexity. In the following sections we will study systems with equally prioritized users (*β<sup>k</sup>* =1 ∀*k* ∈Ω) and the characteristics of state-of-the-art user selection algorithms that find suboptimal yet acceptable solutions to the non-convex and combinatorial problem (4).
