## 4 Precoding Matrix Design Based on Type II 5G-NR for MU-MIMO  
In this section, the design of the precoding matrices based on the Type II codebook is described. The main idea is to support communication to multiple UEs that share the same PRBs in MU-MIMO systems. In this case, a more sophisticated precoding design is required to cancel the interference in the spatial domain between the UEs at the cost of higher feedback overhead. In Release 15, only 2 data streams per UE are supported. However, enhanced precoding strategies are proposed in Release 16 to provide a finer frequency granularity supporting up to 4 layers and reducing signal overhead. Type II is only designed for single-panel antenna arrays with the configurations defined in Table 5.2.2.2.1-2 [25]. The main characteristic of Type II is that it applies a linear combination of \(L\) orthogonal beams per layer instead of just selecting the strongest one as in Type I. This strategy provides a more accurate representation of the dominant eigenvectors of the channel to obtain a precoding matrix that avoids interference between the UEs. Similar to Type I, this design can be represented as a two-step procedure for the \(k\)th user and \(d\)th layer denoted by \(\left[\mathbf{W}_{n}^{[k]}\right]_{d}=\mathbf{W}_{1}^{[kd]}\mathbf{w}_{2n}^{[ kd]}\). However, in the case of Type II codebook, the wideband matrix \(\mathbf{W}_{1}^{[kd]}\in\mathbb{C}^{N_{t}\times 2L}\) represents an orthogonal basis composed of \(L\) beams per polarization and layer, where \(L\in\{2,3,4\}\). Then, a linear combination of the chosen beams is performed by using \(\mathbf{w}_{2n}^{[kd]}\in\mathbb{C}^{2L\times 1}\) which is comprised of wideband and subband complex coefficients. With this design, the desired signal is coherently combined at the receiver side, while the interference signal from co-located UEs is canceled improving the SINR [2;30]. The wideband matrix for the \(k\)th user and \(d\)th layer is obtained as  
\[\mathbf{W}_{1}^{[kd]}= \begin{bmatrix}\mathbf{B}^{[kd]}&\mathbf{0}\\ \mathbf{0}&\mathbf{B}^{[kd]}\end{bmatrix} , \tag{14}\]  
where \(\mathbf{B}^{[kd]}=\left[\mathbf{b}_{0}^{[kd]},...,\mathbf{b}_{L-1}^{[kd]} \right]\in\mathbb{C}^{N_{1}N_{2}\times L}\) is composed of \(L\) beams obtained from a pre-defined DFT-codebook. Note that \(\mathbf{b}_{l}^{[kd]}\), \(\forall l=\{0,1,...,L-1\}\), is defined in (9) for the \(d\)th layer and \(k\)th UE based on a given horizontal (\(l_{d}^{[k]}\)) and vertical (\(m_{d}^{[k]}\)) indexes. The beams in \(\mathbf{B}^{[kd]}\) form an orthogonal basis to avoid inter-layer interference. Similar to Type I, the orthogonality can be obtained in the horizontal, vertical, or both dimensions at the same time. The PMI index set to obtain the wideband and subband matrices is defined as \(\mathbf{i}_{1}=\left[i_{11},i_{12},i_{13},i_{14}\right]\), and \(\mathbf{i}_{2}^{n}=\left[i_{21}^{n},i_{22}^{n}\right]\), respectively. The dimension of each subindex is given by \(i_{11}\in\mathbb{R}^{2\times 1}\), \(i_{12}\in\mathbb{R}^{1\times 1}\), \(i_{13}\in\mathbb{R}^{N_{u}\times 1}\), \(i_{14}\in\mathbb{R}^{2 LN_{u}\times 1}\), \(i_{21}^{n}\in\mathbb{R}^{2 LN_{u}\times 1}\), and \(i_{22}^{n}\in\mathbb{R}^{2 LN_{u}\times 1}\). Note that a higher overhead is required than in Type I. The computation of the wideband matrix \(\mathbf{W}_{1}\) is performed according to the \(\mathbf{i}_{1}\) index vector. The first problem of this design arises due to the long feedback required to identify the \(L\) beams for each layer and polarization. The main solution to reduce overhead is to first select \(L\) DFT orthogonal beams based only on the information reported in the \(i_{12}\in\{0,1,...,\binom{N_{1}N_{2}}{L}-1\}\) subindex. The algorithm proposed in section 5.2.2.3 of TS 38-214 [25] selects the \(L\) beams based on a binomial coefficients computation. Then, to increase the granularity, these DFT beams are oversampled in the horizontal and vertical dimensions by the subindex \(\mathbf{i}_{11}\).