# Advanced Hybrid Beamforming Technique in MU-MIMO Systems
## 4 Advanced Hybrid Beamforming System  
As mentioned in Section 2, FCS hybrid beamforming which each RF chain is connected to all antennas achieves full array gain and obtains high average sum rate. To reduce the complexity of hardware system, PCS hybrid beamforming system with the same number of antennas connected to each RF chain is proposed. Therefore, this paper proposes the beam search algorithm to increase the performance at the PCS hybrid beamforming system based on codebook. Also, it is assumed in this paper that the transmitter knows the angle of departure (AOD) of the channels and perfect channel state information (CSI) is available. Figure 7 shows the PCS hybrid beamforming system and the feedback process of azimuth and elevation angles for the multi-paths. At the analog precoder, the azimuth and elevation angles are determined according to the number of \(B\)-bits used by the transmitter and the beam is formed according to the \(B\)-bits. The \(k\)-th user transmits the information for the received azimuth and elevation angles of multi-paths to the trasmitter. By using the azimuth and elevation angles of multi-paths proportional to the number of \(B\)-bits, \(B^{2}\) beam patterns are formed. If 1-bit beamformer is used at the transmitter, the user sends the information of the azimuth and elevation angles for the multi-paths. Among them, information for the azimuth and elevation angle of the path with the highest power is used. In addition, only one beam pattern is formed. If 2-bit beamformer is used at the transmitter, information for the azimuth and elevation angle of the paths with the highest and the second power is used. In addition, four beam patterns are formed by the combination of azimuth and elevation angles. Table 1 shows the example of the azimuth and elevation angles for the multi-paths when transmitter uses the 3-bit. Index 1 means the highest power path and index 2 means the next highest power path. Index 3 is the path with the high power after index 2. Therefore, Table 2 shows several beam patterns that can be combined by using azimuth and elevation angles corresponding to the multi-paths of Table 1. \begin{table}
\begin{tabular}{c|c|c} \hline
**Multi-Path Index** & **Elevation Angle (\(\theta\))** & **Azimuth Angle (\(\phi\))** \\ \hline
1 & \(\frac{1}{6}\pi\) & \(-\frac{1}{3}\pi\) \\
2 & \(\frac{\sqrt{2}}{2}\pi\) & \(\frac{1}{3}\pi\) \\
3 & \(\frac{1}{3}\pi\) & \(\frac{\sqrt{2}}{2}\pi\) \\ \hline Range of angle & \([0,\,\pi]\) & \([-\frac{1}{3}\pi,\,\frac{1}{2}\pi]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Example of the azimuth and elevation angles for the multi-paths. Figure 7: The PCS hybrid beamforming system including the feedback process. The azimuth and elevation angles among the formed beams are estimated according to RSS as follows,  
\[\left(\hat{\phi}_{k},\,\hat{\theta}_{k}\right)=\arg\max\left|\mathbf{H}_{k} \mathbf{V}_{RF}\mathbf{V}_{D_{k}}\right|^{2}, \tag{9}\]  
where \(\hat{\phi}_{k}\) denotes an optimal elevation angle to be transmitted to the \(k\)-th user corresponding to the RSS. \(\hat{\theta}_{k}\) denotes an optimal azimuth angle to be transmitted to the \(k\)-th user corresponding to the RSS. According to the RSS, a weight matrix based on the azimuth and elevation angles is formed in the analog precoder and the data stream is transmitted to the user through the antenna array. For PCS hybrid beamforming system, the equation of sum rate for the \(k\)-th user can be expressed as follows,  
\[R_{k}=\log_{2}\left(1+\frac{\left|\mathbf{H}_{k}\mathbf{V}_{RF}\mathbf{V}_{D_{ k}}\right|^{2}}{\sigma^{2}+\sum\limits_{l\
eq k}\left|\mathbf{H}_{k}\mathbf{V}_{ RF}\mathbf{V}_{D_{l}}\right|^{2}}\right), \tag{10}\]  
where \(R_{k}\) denotes the achievable rate of the \(k\)-th user and \(\sum\limits_{l\
eq k}\left|\mathbf{H}_{k}\mathbf{V}_{RF}\mathbf{V}_{D_{l}} \right|^{2}\) denotes an interference signal power to the \(k\)-th user by another received RF chain.