# On the Performance of LDPC-Coded Massive MIMO Schemes with Power-Ordered NOMA Techniques
## 2 System Characterization  
This article considers both conventional and cooperative NOMA associated with multi-layer MIMO. The number of \(T\) transmitting antennas of the multi-layer MIMO corresponds to the number of parallel flows of data (an increase of the symbols rate), while the number of \(R\) receiving antennas corresponds to the level of diversity. Multi-layer MIMO requires the number of \(R\) receiving antennas to be equal to or higher than the number of \(T\) transmitting antennas (see Figure 2). The Quadrature Phase Shift Keying (QPSK) modulation is assumed, associated with SC-FDE signals.  
The \(n\)th transmitted block, of \(N\) data symbols, sent by the \(t\)th antenna is denoted as \(x_{n}^{(t)}\), while the received block by the \(r\)th antenna is denoted as \(y_{n}^{(r)}\)[13]. The mapping between the time domain signal and the frequency domain signal for the transmitted block is defined as \(\left\{x_{n}^{(t)};n=0,1,\ldots,N-1\right\}=IDFT\left\{X_{k}^{(t)};k=0,1, \ldots,N-1\right\}\), i.e., by performing the discrete Fourier transform (DFT) of the time-domain block. Similar mapping for the received block is defined as \(\left\{y_{n}^{(r)};k=0,1,\ldots,N-1\right\}=IDFT\left\{Y_{k}^{(r)};k=0,1, \ldots,N-1\right\}\).  
After removing the cyclic prefix and assuming a cyclic prefix longer than the overall channel impulse response of each channel, the received frequency-domain signal becomes,  
\[\mathbf{Y}_{k}=\left[Y_{k}^{(1)},\ldots,Y_{k}^{(R)}\right]^{T}=\mathbf{H}_{k} \mathbf{X}_{k}+\mathbf{N}_{k} \tag{1}\]  
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, with \((r,t)th\) element \(\mathbf{H}_{k}^{(tr)}\), and where \(H_{k}\) denotes the channel frequency response for the \(k\)th subcarrier (which is assumed invariant during the transmission of a given block), and where the mapping between the time domain and frequency domain is defined by \(\left\{H_{k};k=0,1,\ldots,N-1\right\}=DFT\left\{h_{n};n=0,1,\ldots,N-1\right\}\). Moreover, \(N_{k}\) is the frequency-domain block channel noise for that subcarrier.  
Assuming a non-iterative receiver, the frequency domain estimated data symbols \(\mathbf{\tilde{X}}_{k}=\left[\widetilde{X}_{k}^{(1)},\ldots,\widetilde{X}_{k }^{(R)}\right]^{T}\) becomes:  
\[\mathbf{\tilde{X}}_{k}=\mathbf{B}_{k}\mathbf{Y}_{k} \tag{2}\]  
where \(\mathbf{B}_{k}\) is defined in [14] as:  
* \(\mathbf{B}_{k}=\left(\mathbf{H}_{k}^{H}\mathbf{H}_{k}\right)^{-1}\mathbf{H}_ {k}^{H}\) for ZF receiver;
* \(\mathbf{B}_{k}=\mathbf{H}_{k}^{H}\) for the MRC receiver.  
The ZF receiver is a linear algorithm that applies the pseudo-inverse of the frequency response of the channel for each frequency component of the channel. The ZF is very  
Figure 2: Block diagram of m-MIMO system with SC-FDE signals.  
efficient in removing the intersymbol interference but has the disadvantage of presenting noise enhancement when utilized in post-processing; therefore, it tends to degrade the performance for average to high levels of noise. Moreover, due to the high level of computation that results from the calculation of the pseudo-inverse of the channel, it is very complex, which also represents a high level of battery consumption. The MRC tends to mitigate these limitations due to its simplicity but generates 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. The iterative receiver implements the following function [14]:  
\[\widetilde{\mathbf{X}}_{k}=\mathbf{B}_{k}^{H}\mathbf{Y}_{k}-\mathbf{C}_{k} \widetilde{\mathbf{X}}_{k} \tag{3}\]  
The interference cancellation matrix \(\mathbf{C}_{k}\) can be computed as [14]:  
\[\mathbf{C}_{k}=\mathbf{A}_{k}^{H}\mathbf{H}_{k}-\mathbf{I} \tag{4}\]  
where \(\mathbf{I}\) is an \(R\times R\) identity matrix.