## 4 Multi-User MIMO System Model  
Given the scenario described in Section 2, we consider the downlink communication between a single trisectorized BS and \(K\) users, and we focus on the precoding and combining scheme as depicted in Figure 5 of an OFDM Multi-User Multi-Layer MIMO system over wideband mmWave channels with \(Q\) subcarriers. The BS is equipped with \(N_{t}\) transmitting antennas per sector and each UT with \(N_{r}\) receiving antenna. Firstly, each of the \(K\) UTs placed inside the cell is assigned to one of the three sectors, the BS then transmits \(L\) data streams simultaneously to each UT, with \(L\leq N_{r}\ll N_{t}\) and \(L\,K\leq N_{l}\). Let \(\mathbf{s}_{k}(q)\) denote the data symbol vector to be transmitted to the \(k\)-th user at the subcarrier \(q\) and \(\mathbb{E}\left[\mathbf{s}_{k}(q)\,\mathbf{s}_{k}^{\mathbf{\mathrm{f}}}(q)\right] =1/L\,\mathbf{I}\), therefore, at the BS, \(L\,K\) data-symbols \(\mathbf{s}[(q)\ =\ [\mathbf{s}_{1}^{T}(q)\ \mathbf{s}_{2}^{T}(q)\ \cdots\ \mathbf{s}_{K}^{T}(q)]^{T}\) are precoded with a subcarrier-dependent precoder \(\mathbf{W}(q)=[\mathbf{W}_{1}(q)\ \mathbf{W}_{2}(q)\ \ldots\ \mathbf{W}_{K}(q)]\) of size \(N_{t}\times L\,K\), where \(\mathbf{W}_{k}(q)\) represents the precoder designed for the \(k\)-th user. A sum power constraint is imposed on the total transmission power \(P_{tot}\), i.e., \(\|\mathbf{W}(q)\|\leq P_{tot}/Q\). The data symbols are then transformed to the time domain through a \(N_{FFT}\)-point Inverse Fast-Fourier Transform (IFFT). A cyclic prefix (CP) of length \(D\) is then added to the data-symbols. The total transmitted signal at subcarrier \(q\) can be written as:  
\[\mathbf{x}(q)=\mathbf{W}(q)\,\mathbf{s}(q) \tag{65}\]  
We assume Time Division Duplexing (TDD) with channel reciprocity and ideal channel state information (CSI) at the BS, i.e., the BS can obtain long-term  
Figure 5: Multi-User Multi-Layer MIMO scheme.  
CSI for each user. Moreover, users communicate in the same time slot or resource and the BS:  
* resorts only to Space Division Multiple Access (SDMA) through precoding techniques;
* is able to process all \(KL\) users' signals, therefore we impose no limitation on the number of RF chains, i.e., the BS is able to employ at least \(KL\) parallel precoders.  
In our simulation, we generate the \(N_{r}\times N_{t}\times N_{taps}\) channel tensor \(\mathbf{\mathcal{H}}_{s,k}\) from sector \(s\) of the BS to user \(k\) in time domain as described in Section 3, we perform a \(N_{FFT}\)-point FFT along the 3 rd dimension for each RX-TX antenna pair, and we obtain a \(N_{r}\times N_{t}\times N_{FFT}\) frequency response channel tensor \(\mathbf{\tilde{\mathcal{H}}}_{s,k}\), where each \(N_{r}\times N_{t}\) matrix \(\mathbf{\tilde{\mathcal{H}}}_{s,k}(q)\) represents the flat MIMO channel of the \(q\)-th subband.  
After sector selection, the baseband equivalent receiving signal of the \(k\)-th user at subcarrier \(q\) can be written as follows:  
\[\mathbf{y}_{k}(q)=\mathbf{\tilde{\mathcal{H}}}_{k}(q)\,\mathbf{x}(q)=\mathbf{\tilde{ \mathcal{H}}}_{k}(q)\sum_{i=1}^{K}\mathbf{W}_{i}(q)\,\mathbf{s}_{i}(q)+\mathbf{ n}_{k}(q) \tag{66}\]  
where \(\mathbf{n}_{k}(q)\) is a length \(N_{r}\) noise vector whose entries are independent and identically distributed according to \(\mathcal{CN}(0,\sigma_{k}^{2})\) and are spatially white, i.e., \(\mathbb{E}\big{[}\mathbf{n}_{k}(q)\,\mathbf{n}_{k}^{H}(q)\big{]}=\sigma_{k}^{ 2}\mathbf{\tilde{\mathcal{I}}}\), moreover \(\sigma_{k}^{2}=k_{B}\,T\,F\,\Delta,\ \forall k\), where \(k_{B}\) is the Boltzmann constant, \(T\) the temperature, \(F\) the noise figure, and \(\Delta\) the size of the subband or subcarrier spacing (SCS).