## 2 System Model and Evaluated Precoding Schemes
### Analog Beamforming Schemes  
As mentioned above, in mmW systems, the high number of antennas required to achieve a high quality link makes fully-digital implementations hardly feasible in practice, due to the high cost and power consumption of having as many RF chains as antennas. For that reason, less complex precoding schemes either in the analog (i.e., radio-frequency) domain or combining the processing in both digital baseband and analog domain are receiving increasing attention [11].  
In RF beamforming, phase shifts are applied to each antenna element of the array in order to steer the beam towards a certain direction. Figure 1 shows an example of analog beamforming (ABF) architecture, with \(P<<N_{t}\) RF chains and a set of \(N_{t}\) phase shifters applied to each RF chain. The outputs of the \(i\)-th phase shifters of every RF chain are combined to feed the \(i\)-th antenna element in the array. Using this architecture, \(P\) data streams can be simultaneously transmitted with full array gain and up to \(P\) different users can be spatially multiplexed. In this work, we consider a simple and practical method for mmW beamforming. Motivated by the fact that the channel at high frequencies is characterized by limited scattering, in particular for the UEs in line of sight (LoS) condition, we assume that (a) beamformers are designed only in the RF domain, and (b) the digital stage is optimized to schedule up to \(P\) UEs in each subframe, each UE being served by a specific RF beamformer. More sophisticated schemes could be used [10; 11], for instance, by optimizing the digital precoding weights to be used in the baseband domain. However, these schemes are expected to provide relevant gain only when most of the UEs have a highly frequency selective channel, as in scenarios where there are mostly UEs in non line of sight (NLoS) conditions. Therefore, in our analysis, we will intentionally compare a simple ABF scheme against fully DP. Future work will include a more thorough evaluation on how much of the performance gap between these two schemes can be filled by more advanced (but at the same time more complex) hybrid beamformers.  
The RF beamforming vector for the \(k\)-th UE, denoted by \(\mathbf{w}_{RF_{k}}\), is selected from a codebook based on the discrete Fourier transform (DFT) matrix, which provides a grid of beams such as in [10], useful also for channel limited feedback setups. Elements of the codebook are of this form:  
\[\mathbf{w}_{RF}^{(c)}=\frac{1}{\sqrt{N_{t}}}\left[1,e^{\frac{2\pi c}{C}},\dots,e^{\frac{2\pi c}{C}(N_{t}-1)}\right]^{T}, \tag{2}\]  
where \(c=0,1,\dots,C-1\) is the index of the codebook elements and \(C\) the number of vectors in the codebook. We assume \(C=N_{t}\), that is to say, maximum diversity order. The specific beamforming vector for UE \(k\) is chosen to maximize the sum rate over the whole allocated band \(\mathcal{B}\) as:  
\[\mathbf{w}_{RF_{k}}=\operatorname*{arg\,max}_{\mathbf{w}_{RF}^{(c)}}\sum_{b \in\mathcal{B}}\log_{2}\left(1+\frac{P_{t}\left|\mathbf{h}_{k,b}\mathbf{w}_{ RF}^{(c)}\right|^{2}}{N+I}\right), \tag{3}\]where \(P_{t}\) is the transmit power, and \(N\) and \(I\) are the noise and long term interference power at the receiver, respectively. Although many other codebook designs could be used, DFT-based precoding is chosen in this work to achieve a good tradeoff between complexity and performance.  
Note that the whole bandwidth is available at each RF chain and multiple UEs are multiplexed in the spatial domain. Equal power sharing among users is considered, meaning, for example, that, when \(P\) users are simultaneously multiplexed by a BS, the available power per RF chain must be scaled by \(1/P\) to fulfill a per-BS power restriction.