## 2 System Model
The vector \(\mathbf{n}_{\ell}(n)\) denotes an \(N\times 1\) additive noise vector at the \(\ell\)-th cooperative UT antennas with \(\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})\) distribution, while \(n_{j}(n)\) denotes the additive noise at the  
Figure 1: The cooperative non-orthogonal multiple access (NOMA) system with \(L\) cooperating user terminals (UTs) among \(J\) NOMA UTs. The cooperating UTs use \(L\) separate time slots for the cooperation phases. \(j\)-th UT antenna with \(\mathcal{CN}(0,1)\) distribution. Then, the received signal at the \(j\)-th UT at the time slot \(n+1\) can be written as  
\[ \begin{array}{ll}\mathbf{y}_{j}(n+1)&=\mathbf{G}_{j,1}\mathbf{w}_{T,1}x(n)+ \mathbf{n}_{j}(n+1),\;j=2,\ldots,L\\ y_{j}(n+1)&=\mathbf{g}_{j,1}\mathbf{w}_{T,1}x(n)+\eta_{j}(n+1),\;j=L+1,\ldots,J,\end{array}  \tag{2}\]  
where \(\mathbf{n}_{j}(n+1)\) and \(\eta_{j}(n+1)\) are the additive noise signals at the \(j\)-th UT with \(\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N})\) and \(\mathcal{CN}(0,\sigma^{2})\) distributions, respectively. If \(L\geq 2\), the 2-nd cooperative UT again starts decoding the messages from the observations in (1) and \(\mathbf{w}_{R,2}^{H}\mathbf{y}_{2}(n+1)\) according to the SIC manner. Then, it re-encodes the \(J\) messages and transmits the message \(x(n)\) in the time slot \(n+2\). Using \(j\) previous observations, similar processes are repeated until the \(L\)-th cooperative UT, which requires \(L\) time slots and the cooperation ends after the \((n+L)\)-th time slot. It may look unnecessary for the \(\ell\)-th cooperative UT to retransmit all the \(J\) messages including the message signal destined to itself after it decodes the \(J\) messages. However, in NOMA with SIC decoding, each UT needs to decode all the message signals with indices preceding the index of the UT's own signal; thus, it is reasonable for the cooperative UTs to repeat all the message signals to help the decoding at the UTs. The \(j\)-th (\(j\geq L+1\)) UT has \(L+1\) observations from the time slots from \(n\) in (1) to \(n+L\) so that it can decode the messages \(x_{1}(n),\ldots,x_{j}(n)\) in SIC manner. Assuming the maximum ratio combining (MRC) of those observations, the signal-to-interference-plus-noise ratio (SINR) for the signal \(x_{k}(n),\;k\leq j\) is as follows:  
\[\gamma_{k}^{j}=\frac{\left[|\mathbf{h}_{j}\mathbf{w}_{A}|^{2}+\sum_{\ell=1}^{ L}|\mathbf{g}_{j,\ell}\mathbf{w}_{T,\ell}|^{2}\right]P_{k}}{\left[|\mathbf{h}_{j} \mathbf{w}_{A}|^{2}+\sum_{\ell=1}^{L}|\mathbf{g}_{j,\ell}\mathbf{w}_{T,\ell}| ^{2}\right]\sum_{m=k+1}^{J}P_{m}+\sigma^{2}}. \tag{3}\]  
On the other hand, at the \(\ell\)-th cooperative UT, there are \(\ell\) observations for the decoding of \(x_{k}(n)\). Assuming MRC with \(\mathbf{w}_{R,\ell}=(\mathbf{H}_{\ell}\mathbf{w}_{A})^{H}/\|\mathbf{H}_{\ell }\mathbf{w}_{A}\|\) at slot \(n\) and \(\mathbf{w}_{R,\ell}=(\mathbf{G}_{\ell,m}\mathbf{w}_{T,m})^{H}/\|\mathbf{G}_{ \ell,m}\mathbf{w}_{T,m}\|\) at slot \(n+m,\;(m\leq L)\), the SINR for \(x_{k}(n)\) at the cooperative UT is given as follows  
\[\gamma_{k}^{\ell}=\frac{[\|\mathbf{H}_{\ell}\mathbf{w}_{A}\|^{2}+\sum_{m=1}^{ \ell-1}\|\mathbf{G}_{\ell,m}\mathbf{w}_{T,m}\|^{2}]P_{k}}{[\|\mathbf{H}_{\ell }\mathbf{w}_{A}\|^{2}+\sum_{m=1}^{\ell-1}\|\mathbf{G}_{\ell,m}\mathbf{w}_{T,m} \|^{2}]\sum_{m=k+1}^{J}P_{m}+\sigma^{2}}. \tag{4}\]