**2. System model**

The principle behind this optimum coding technique is that the BS knows the interference for a given user and can pre-subtract it before transmission, which yields the capacity of an interference free channel. However, DPC is a nonlinear process that requires successive encoding and decoding whose complexity is prohibitive in practical systems and other suboptimal techniques are preferred instead. In the literature, DPC has been interpreted as *beamforming* (BF) [5] which is a SDMA scheme where data streams of different users are encoded independently and multiplied by weight vectors in order to mitigate mutual inter‐ ference. Although BF is a suboptimal multiuser transmission scheme, several works (e.g., [3], [5]) have shown that it can achieve a large portion of the DPC rate and its performance is closeto-optimal for large *Nt* and *K*. Nevertheless, the optimization of the downlink BF weight vectors is a non-convex problem [6]. The specific selection of the weight vector of a given user may affect the performance of other users, i.e., the achievable signal-to-interference-plus-noise ratio (SINR) of one user is coupled with the other users weight vectors and transmit powers [6]. Since the weight vectors must be optimized jointly, optimum BF is a complicated task and suboptimal weights given by Zero-Forcing (ZF) based methods can be used [5]. The joint optimization of the beamforming weights, the power allocation, and the set of links that are scheduled under SDMA mode is performed for a given global objective function. In BC system literature a meaningful figure of merit is the total sum rate since it quantifies the total data flow for a specific BF scheme and the open problems are power allocation and user selection. If the set of scheduled users meets that *Nt* ≥ *K Nr*, power allocation can be given either by Lagrangian methods (convex optimization [7]) or by equal power allocation (close-to-optimal for the high SNR regime [8]–[12]). However, when the set of users is larger than the number of spatial resources at the BS, i.e., *Nt* < *K Nr*, user selection is required. The selection of the optimum set of users that maximizes the sum rate for a given BF scheme with optimum power allocation is a mixed non-convex problem. Indeed, in systems where users must be allocated in different radio resources (slots or sub-channels) finding the optimum subsets of users under SDMA mode for each radio resource is a NP-complete problem whose optimal solution can only be found via exhaustive search (ES) [13]. Recent research works have proposed a number of feasible heuristics algorithms that find a suboptimal yet acceptable solution to the sum rate maximization problem with SDMA communication. The literature of MU-MIMO has been focused on ZF-based BF schemes due to their tractability and the fact that some properties of the wireless channels can be used to estimate the reliability of joint transmission for a given set of users. The main objective of joint scheduling and BF is to make better decisions at the media access control (MAC) layer by exploiting information from the physical (PHY) layer. The literature reviewed in this chapter addresses the scheduling (user selection) process in the MAC layer using PHY layer information without considering constraints from upper layers. Our aim is to provide a comprehensive overview of algorithms proposed over the last years regarding joint user selection and SDMA schemes that solve the sum rate maximization

problem in MU-MISO BC systems.

26 Contemporary Issues in Wireless Communications

Consider a single-cell with a single base station equipped with *Nt* antennas, and a set *S* of *K* single-antenna users (*Nr* =1) illustrated in Fig. 1. We assume perfect CSI at the BS and the channel coefficients are modeled as independent random variables, with a zero-mean circu‐ larly symmetric complex Gaussian distribution (Rayleigh fading). The signal received by the *k*th user is given by:

$$\mathbf{y}\_{\mathbf{k}} = \sqrt{\mathbf{p}\_{\mathbf{k}}} \mathbf{h}\_{\mathbf{k}} \mathbf{w}\_{\mathbf{k}} \mathbf{s}\_{\mathbf{k}} + \sum\_{i \neq \mathbf{k}}^{\mathbf{K}} \sqrt{\mathbf{p}\_{i}} \mathbf{h}\_{\mathbf{k}} \mathbf{w}\_{i} \mathbf{s}\_{i} + \mathbf{n}\_{\mathbf{k}} = \mathbf{h}\_{\mathbf{k}} \times \mathbf{n}\_{\mathbf{k}} \tag{1}$$

where *sk* is the intended symbol for user *k*, **x**∈ℂ*Nt*×1 is the transmitted signal vector from the base station antennas, **h**k∈ℂ1×*<sup>N</sup> <sup>t</sup>* is the channel vector to the user *k*. Each user ignores the modulation and coding of other users, i.e., it is assumed single-user detection where each user treats the signals intended for other users as interference. *nk* ∽(0, *σ<sup>n</sup>* 2 ) is the additive zero mean white Gaussian noise with variance *σ<sup>n</sup>* 2 . The entries of the block fading channel **H**= **h**<sup>1</sup> *<sup>T</sup>* , <sup>⋯</sup>, **<sup>h</sup>***<sup>K</sup> <sup>T</sup> <sup>T</sup>* are normalized so that they have unitary variance, and the transmitted signal **x**=∑*<sup>k</sup>* =1 *<sup>K</sup> pk***w***<sup>k</sup> sk* has an average power constraint **<sup>x</sup>***<sup>H</sup>* **<sup>x</sup>** <sup>≤</sup>*P* where <sup>∙</sup> is the expectation operation. Since the noise has unit variance, *P* represents the SNR.

**Figure 1.** Multiuser MISO Broadcast System

For linear spatial processing at the transmitter, the BF matrix can be defined as **W** = **w**1, ⋯, **w***<sup>K</sup>* , the symbol vector as **s**= s1, ⋯, s*K* and **P**=*diag* p1, ⋯, p*<sup>K</sup>* is the matrix whose main diagonal contains the powers. The SINR of the *k*th user is given by:

$$\text{SINR}\_k = \frac{p\_k \left| \mathbf{h}\_k \mathbf{w}\_k \right|^2}{\sum\_{i \neq k} p\_i \left| \mathbf{h}\_k \mathbf{w}\_i \right|^2 + \sigma\_n^2} \tag{2}$$

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