## 2 System Model
### Propagation and Formulations
#### 2.2.1 Odd Transmission Block
The PA factors for devices \(D_{i}\) for \(i=\{N,\ldots,1\}\) are, respectively, denoted by \(\alpha_{i}\), constrained to \(\alpha_{1}<\ldots<\alpha_{N}\), \(\alpha_{1}+\ldots+\alpha_{N}=1\), and given by  
\[\alpha_{i}=i\Bigg{/}\sum_{n=1}^{N}n. \tag{7}\]  
Note that Expression (7) is derived from the feature studied [7]. However, the devices in [7] were ordered where \(D_{1}\) is the farthest device. However, the system model shown Figure 1 indicates that device \(D_{N}\) has the farthest distance from the coupled relays. In term of NOMA theory, the farthest device must be allocated the biggest PA factor. As a result, the PA factor \(\alpha_{N}\) for device \(D_{N}\) is the largest value among the PA factors, whereas device \(D_{1}\) is the nearest distance from the coupled relays. Therefore, device \(D_{1}\) is allocated the smallest PA factor. SIC is another feature of NOMA theories which is implemented at the user. The user therefore implements SIC to detect messages in the received signal. In [7; 12], the authors investigated NOMA networks with a random number \(N\) of users. The users repeated the SIC phases until their own messages were successfully detected in the received signal. It is important to note that \(N\) devices exist in our model (Figure 1), and, therefore, \(N\) SIC phases at the relay \(R_{1}\). After selecting the received signal as given in (5), in the first SIC phase, relay \(R_{1}\) detects the message \(x_{N}\) of device \(D_{N}\) as a result of the constraint of the PA factors \(\alpha_{N}>\ldots>\alpha_{1}\). In the second SIC phase, relay \(R_{1}\) detects the message \(x_{N-1}\) of device \(D_{N-1}\) after removing the \(x_{N}\) symbol from the received signals. The relay \(R_{1}\) repeats SIC until it successfully detects the last symbol \(x_{1}\). However, the present study examines a massive MIMO underlay cooperative NOMA network in contrast to the schemes presented in [29; 31], where the author studied cooperative Single-Input-Single-Output (SISO)-NOMA schemes. Fortunately, the authors of [23; 24] also investigated a MIMO-NOMA network with TAS and obtained Signal-to-Interference-plus-Noise Ratios (SINRs), where the devices detected information by applying SIC. To optimize system performance, our study considers the massive MIMO technique in combination with a TAS protocol, where relay \(R_{1}\) selects the best received signal \(\max\Biggl{\{}\left|\mathbf{Y}_{R_{1}}^{\left(T_{1}^{(odd)}\right)}\right| \Biggr{\}}\) for SIC. In the first SIC phase, relay \(R_{1}\) decodes the \(x_{N}\) symbol from the best received signal \(\max\Biggl{\{}\left|\mathbf{Y}_{R_{1}}^{\left(T_{1}^{(odd)}\right)}\right| \Biggr{\}}\) by treating the data symbols \(x_{j}=\{x_{1},\ldots,x_{N-1}\}\)and AWGN \(n_{R_{1}}\) as interference. The SINR is therefore obtained when relay \(R_{1}\) decodes the \(x_{i}\) symbol, as follows:  
\[\max_{[A_{0}\times A_{1}]}\{\gamma_{ \begin{subarray}{c}T_{1}^{(odd )}\\ I=\{N,\ldots,1\}\end{subarray} }^{( \begin{subarray}{c}T_{1}^{(odd)}\\ \gamma_{R_{1}-x_{i}}\\ I=\{N,\ldots,1\}\end{subarray} }\right\} =\frac{(1-\lambda)\max_{[A_{0}\times A_{1}]}\Bigl{\{}\left|\mathbf{ H_{0}}\right|^{2}\Bigr{\}}\alpha_{i}\rho_{S}}{(1-\lambda)\max_{[A_{0} \times A_{1}]}\Bigl{\{}\left|\mathbf{H_{0}}\right|^{2}\Bigr{\}}\rho_{S}\sum \limits_{j=1}^{i-1}\alpha_{j}+1},\text{for }i>1, \tag{8}\] \[=(1-\lambda)\max_{[A_{0}\times A_{1}]}\Bigl{\{}\left|\mathbf{H_{0} }\right|^{2}\Bigr{\}}\alpha_{i}\rho_{S},\qquad\qquad\text{for }i=n=1, \tag{9}\]  
where \(\rho_{S}\) is the transmission Signal-to-Noise Ratio (SNR) and \(\rho_{0}=P_{S}/N_{0}\). Maximization of the instantaneous bit rate threshold is achieved at relay \(R_{1}\) when relay \(R_{1}\) decodes the message \(x_{i}\) for \(i=\{N,\ldots,1\}\) as follows:  
\[\max_{[A_{0}\times A_{1}]}\{\mathbf{R}_{ \begin{subarray}{c}T_{1}^{(odd)}\\ I=\{N,\ldots,1\}\end{subarray} }^{( \begin{subarray}{c}T_{1}^{(odd)}\\ I=\{N,\ldots,1\}\end{subarray} }\}=\frac{1}{2}\text{log}_{2}\left(1+\max_{[A_{0 }\times A_{1}]}\left\{\gamma_{