## 3 Precoding Matrix Design Based on Type I 5G-NR for SU-MIMO  
The design of the precoding matrix according to the Type I codebook for SU-MIMO is described in this section. This is a particular case of the system model described in Section 2 assuming \(K=1\). The main goal is to transmit the signal over the \(N_{s}\) strongest propagation paths of the channel matrix to the target UE. Inter-layer interference is avoided with the precoding matrix designed at the gNB based on the PMI reported from the UE. Table 1 defines the supported parameters for Type I. According to TS 38-214 [25], the gNB is configured with a maximum number of 32 virtual antenna ports for the downlink transmission. Without loss of generality, the virtual antenna ports are assumed to be equal to the physical transmit antennas in this paper. Then \(N_{t}=2N_{g}N_{1}N_{2}\), where \(N_{g}\) is the number of antenna panels, \(N_{1}\) and \(N_{2}\) are the dual-polarized (DP) antenna elements in the horizontal and vertical dimension, respectively, as illustrated in Figure 1. Note that the product by 2 indicates the use of DP antennas. A single or multi-panel antenna is supported where each panel can be designed following a uniform linear (ULA) or planar array (UPA). Up to \(N_{s}=8\) and \(N_{s}=4\) layers transmission is available for single-panel and multi-panel, respectively. The maximum number of layers \(N_{s}\) depends on the number of antenna ports such that \(N_{s}\leq\min(N_{t},N_{r})\). Since more users are commonly located in the azimuth plane, then a higher number of antennas is available in the horizontal dimension to provide a higher spatial resolution. A maximum number of 16 antenna elements in a single-panel array is supported with a configuration of \(N_{1}=16\), \(N_{2}=1\) for ULA or \(N1=8\), \(N_{2}=2\) for UPA [31]. The main objective in Type I is to select \(N_{s}\) precoding vectors from a predefined DFT-codebook set that corresponds to the strongest propagation paths. The precoding vector for the \(d\)th layer in the \(n\)th subband is designed as  
\[[\mathbf{W}_{n}]_{d}=\mathbf{W}_{1}^{[d]}\mathbf{w}_{2n}^{[d]}, \tag{7}\]  
where \(\mathbf{W}_{1}^{[d]}\in\mathbb{C}^{N_{t}\times 2}\) and \(\mathbf{w}_{2n}^{[d]}\in\mathbb{C}^{2\times 1}\) are related to wideband and subband parameters, respectively. Note that the index \(k\) is omitted in (7) since a single user is considered in this section. The wideband matrix \(\mathbf{W}_{1}^{[d]}\) is defined as follows  
\[\mathbf{W}_{1}^{[d]}= \begin{bmatrix}\mathbf{b}^{[d]}&\mathbf{0}\\ \mathbf{0}&\mathbf{b}^{[d]}\end{bmatrix} , \tag{8}\]  
where \(\mathbf{b}^{[d]}\in\mathbb{C}^{N_{1}N_{2}\times 1}\) indicates the direction of the strongest beam in the \(d\)th layer. This factor is independent of the frequency and polarization since it only depends on large-scale properties of the channel [31]. Note that since orthogonal polarization is assumed, then the same beam is transmitted over the two sets of DP antennas without causing interference. Consequently, \(\mathbf{W}_{1}^{[d]}\) has a diagonal block structure. For an UPA, the \(d\)th beam \(\mathbf{b}^{[d]}(l_{d},m_{d})\) is obtained from a DFT-codebook set as a function of the horizontal and vertical indexes \(l_{d}\) and \(m_{d}\), respectively, as follows [2; 30]  
\[\mathbf{b}^{[d]}(l_{d},m_{d})=\mathbf{w}_{ULA}(l_{d})\otimes\mathbf{w}_{ULA}( m_{d}), \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Maximum** & **Number of** & **Number of** & **Number of** & **Supported** \\  & **Number of** & **Antenna Ports** & **Antenna** & **Configurations of** \\  & **Layers** & & **Panels** & **(\(N_{1},N_{2}\)) and (\(O_{1},O_{2}\))** \\ \hline Type I & 8 & 2, 4, 8, 12, 16, & 1 & Table 5.2.2.2.1-2 \\ Single-Panel & & 24, 32 & & TS-38.214 \\ Type I & 4 & 8, 16, 32 & 2, 4 & Table 5.2.2.2.2-1 \\ Multi-panel & & & & TS-38.214 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Supported parameters by Type I downlink codebook-based transmission.