## 2 System and Signal Characterization  
This article considers Quadrature Phase Shift Keying (QPSK) modulation, MIMO and SC-FDE signals, as depicted in Figure 2. Since the considered MIMO is the multi-layer transmission, the number of receiving antennas \(R\) needs to be equal to or higher than the number of transmitting antennas \(T\). As the number of receiving antennas \(R\) increases, the diversity also increases, achieving better performances. It is worth noting that there are \(T\) parallel data streams being transmitted. Consequently, an increase of the number of transmitting antennas results in an increase of the symbols rate. The \(t\)th antenna has a block of \(N\) data symbols \(\left\{x_{n}^{(t)};n=0,1,\ldots,N-1\right\}=IDFT\)  
\(\left\{X_{k}^{(t)};k=0,1,\ldots,N-1\right\}\) to send. Appendix B contains a list of symbols that can be used for clarification. At the receiver side, the received block associated with the \(r\)th user is represented by \(\left\{y_{n}^{(r)};k=0,1,\ldots,N-1\right\}=IDFT\left\{Y_{k}^{(r)};k=0,1, \ldots,N-1\right\}\)[17]. As with other SC-FDE schemes, a cyclic prefix longer than the maximum overall channel impulse response is appended to each transmitted block and removed at the receiver. In this case, the corresponding frequency-domain block \(\left\{Y_{k}^{(r)};k=0,1,\ldots,N-1\right\}\) satisfies  
\[\mathbf{Y}_{k}=\left[Y_{k}^{(1)},\ldots,Y_{k}^{(R)}\right]^{T}=\mathbf{H}_{k }\mathbf{X}_{k}+\mathbf{N}_{k} \tag{1}\]  
Figure 2: Block diagram of m-MIMO system with SC-FDE signals. where \(\mathbf{X}_{k}=\left[X_{k}^{(1)},\ldots,X_{k}^{(T)}\right]^{T}\) and where \(\mathbf{H}_{k}\) denotes the \(T\times R\) channel matrix for the \(k\)th subcarrier (which is assumed invariant during the transmission of a given block), with \((r,t)th\) element \(\mathbf{H}_{k}^{(tr)}\) and with \(\left\{H_{k}^{(tr)};k=0,1,\ldots,N-1\right\}=DFT\left\{h_{n}^{(tr)};n=0,1, \ldots,N-1\right\}\). Moreover, \(N_{k}\) is the frequency-domain block channel noise for that subcarrier and \(\mathbf{N}_{k}=\left[N_{k}^{(1)},\ldots,N_{k}^{(R)}\right]^{T}\) denotes the \(1\times R\) channel matrix of the noise. As defined in [15], using a non-iterative receiver, the frequency domain estimated data symbols \(\widetilde{\mathbf{X}}_{k}=\left[\widetilde{X}_{k}^{(1)},\ldots,\widetilde{X }_{k}^{(R)}\right]^{T}\) comes:  
\[\widetilde{\mathbf{X}}_{k}=\mathbf{B}_{k}\mathbf{Y}_{k} \tag{2}\]  
where \(\mathbf{B}_{k}\) is post-processing matrix of the receiver defined in [15], separately, for the ZF and MRC, namely as \(\mathbf{B}_{k}=\left(\mathbf{H}_{k}^{H}\mathbf{H}_{k}\right)^{-1}\mathbf{H} _{k}^{H}\) for ZF and \(\mathbf{B}_{k}=\mathbf{H}_{k}^{H}\) for the MRC. It is known that the ZF requires a high level of computation, as it computes the pseudo-inverse of the channel matrix, for each frequency component. The MRC receiver mitigates such limitation, but it presents some level of residual interference generated in the decoding process for moderate values of \(T/R\), which can be mitigated by employing an iterative receiver. Considering \(R\gg 1\), which is the scenario of m-MIMO and assuming small correlation between the channels of different transmitting and receiving antennas, the elements outside the main diagonal of \(\mathbf{A}_{k}^{H}\mathbf{H}_{k}\) are much lower than the ones at its diagonal, where \(\left(i,i^{\prime}\right)^{th}\) elements of the matrix \(\mathbf{A}\) are defined for the MRC as [15]:  
\[\left[\mathbf{A}\right]_{i,i^{\prime}}=\left[\mathbf{H}\right]_{i,i^{\prime}}^ {H} \tag{3}\]  
For moderate values of \(T/R\), the level of interference can still be representative, which can be mitigated by implementing the iterative receiver [15]:  
\[\widetilde{\mathbf{X}}_{k}=\mathbf{B}_{k}^{H}\mathbf{Y}_{k}-\mathbf{C}_{k} \tilde{\mathbf{X}}_{k} \tag{4}\]  
where \(\widetilde{\mathbf{X}}_{k}=\left[\widetilde{X}_{k}^{(1)},\ldots,\widetilde{X }_{k}^{(R)}\right]^{T}\) stands for the frequency domain estimated data symbols.