## 2 System Model
Based on this signal, the channel is estimated at each UE and a CSI is fedback, including the PMI. With this information, the gNB designs the precoding matrix. Finally, the precoded data symbols are transmitted to each UE avoiding interference. where \(\mathbf{H}_{j}^{[k]}\in\mathbb{C}^{N_{r}\times N_{t}}\) represents the channel matrix at the \(j\)th subcarrier between the gNB and the \(k\)th UE. Furthermore, \(\mathbf{n}_{j}^{[k]}\in\mathbb{C}^{N_{r}\times\mathrm{cl}}\) is the circularly symmetric additive white Gaussian noise vector (AWGN) at the \(k\)th UE \(\left(\mathbf{n}_{j}^{[k]}\sim\mathcal{CN}\left(0,\sigma_{j}^{2[k]}\mathbf{I}_ {N_{r}}\right)\right)\). The first term in (1) represents the desired signal while the second term is the interference produced by the multiplexed UEs that share the same PRBs. The SE at the \(j\)th subcarrier and \(k\)th UE is computed as  
\[R_{j}^{[k]}=\sum_{d=1}^{N_{s}}\log_{2}\left(1+SINR_{j}^{[kd]}\right), \tag{2}\]  
where \(SINR_{j}^{[kd]}\) denotes the SINR at the \(j\)th subcarrier, \(k\)th UE, and \(d\)th data stream, \(\forall d\in\{1,2,...,N_{s}\}\). For a given precoding and combining matrix, the SINR can be written as [40]  
\[SINR_{j}^{[kd]}=\frac{\left|\left[\mathbf{U}_{j}^{[k]}\right]^{H}\mathbf{H}_{ j}^{[k]}\left[\mathbf{W}_{n}^{[k]}\right]_{d}\right|^{2}}{\left[\mathbf{U}_{j}^{ [k]}\right]_{d}^{H}\mathbf{Q}_{j}^{[kd]}\left[\mathbf{U}_{j}^{[k]}\right]_{d} },\forall j\in\mathcal{A}_{n}. \tag{3}\]  
It must be pointed out that the denominator in (3) includes the interference terms and also the noise variance, expressed through the covariance matrix \(\mathbf{Q}_{j}^{[kd]}\in\mathbb{C}^{N_{r}\times N_{r}}\). This is detailed in the following equation where the interference plus noise covariance matrix \(\mathbf{Q}_{j}^{[kd]}\) is defined as  
\[\mathbf{Q}_{j}^{[kd]}=\sum_{l=1}^{K}\mathbf{H}_{j}^{[k]}\mathbf{W}_{n}^{[l]} \left(\mathbf{W}_{n}^{[l]}\right)^{H}\left(\mathbf{H}_{j}^{[l]}\right)^{H}- \mathbf{H}_{j}^{[k]}\left[\mathbf{W}_{n}^{[k]}\right]_{d}\left[\mathbf{W}_{n} ^{[k]}\right]_{d}^{H}\left(\mathbf{H}_{j}^{[k]}\right)^{H}+\sigma_{j}^{2[k]} \mathbf{I}_{N_{r}}. \tag{4}\]  
The major goal of this paper is to evaluate the SE bounds achieved by the precoding matrices designed within the 5G-NR specifications. However, the standard does not specify the combining filter design at the receiver side for downlink data transmission since it is vendor specific based on their own research. In this work, the MMSE combining method is applied to maximize the SINR based on the precoding matrix previously computed at the gNB side [3; 40]. With this strategy, the highest SE is attained. Nevertheless, the design of the combining matrix is limited by the knowledge that the UE can acquire from the channel. It must be noted that in a multi-user scenario, each UE is only able to estimate its channel to the gNB supported by the CSI-RS. However, a full representation of the channels with the remaining users can not be obtained. Therefore, according to this restriction, the combining matrix is designed as  
\[\left[\mathbf{U}_{j}^{[k]}\right]_{d}=\frac{\left(\tilde{\mathbf{Q}}_{j}^{[kd ]}\right)^{-1}\mathbf{H}_{j}^{[k]}\left[\mathbf{W}_{n}^{[k]}\right]_{d}}{ \left\|\left(\tilde{\mathbf{Q}}_{j}^{[kd]}\right)^{-1}\mathbf{H}_{j}^{[k]} \left[\mathbf{W}_{n}^{[k]}\right]_{d}\right\|},\forall j,\forall k,\forall d, \tag{5}\]  
where the covariance matrix \(\tilde{\mathbf{Q}}_{j}^{[kd]}\) only considers the channel between the gNB and the \(k\)th UE as follows  
\[\tilde{\mathbf{Q}}_{j}^{[kd]}=\mathbf{H}_{j}^{[k]}\mathbf{W}_{n}^{[k]}\left( \mathbf{W}_{n}^{[k]}\right)^{H}\left(\mathbf{H}_{j}^{[k]}\right)^{H}-\mathbf{ H}_{j}^{[k]}\left[\mathbf{W}_{n}^{[k]}\right]_{d}\left[\mathbf{W}_{n}^{[k]} \right]_{d}^{H}\left(\mathbf{H}_{j}^{[k]}\right)^{H}+\sigma_{j}^{2[k]}\mathbf{I }_{N_{r}}. \tag{6}\]  
The subsequent sections describe the design of the precoding matrix according to the 5G-NR standard for SU-MIMO and MU-MIMO systems.