## 3 Beamformer Design
### Design Principle  
Following the SIC rule, the decoding condition at the \(\ell\)-th cooperative UT is \(\gamma_{k}^{\ell}\geq\gamma_{k}^{*},\ k=1,\ldots,J\). Here, \(\gamma_{k}^{*}=2^{\tau_{k}}-1\) is the threshold for decoding the \(k\)-th message signal and \(\tau_{k}\) is the data rate of \(x_{k}\). On the other hand, at the \(j\)-th non-cooperative UT, the SIC decoding condition is given as  
\[\gamma_{k}^{j}\geq\gamma_{k}^{*},\ k=1,\ldots,j. \tag{6}\]  
Collecting the sum of signal strength terms at the \(k\)-th UT, we define \(\alpha_{k}\) in (7) with the minimum operator.  
\[\alpha_{k}=\left\{\begin{array}{l}\min[|\mathbf{H}_{1}\mathbf{w}_{A}\|^{2}, \ldots,|\mathbf{H}_{L}\mathbf{w}_{A}\|^{2}+\sum_{m=1}^{L-1}\|\mathbf{G}_{L,m} \mathbf{w}_{T,m}\|^{2},\min_{m=k,\ldots,J}\{|\mathbf{h}_{m}\mathbf{w}_{A}|^{2} +\sum_{\ell=1}^{L}|\mathbf{g}_{m,\ell}\mathbf{w}_{T,\ell}|^{2}\}],\\ L<k\leq J,\\ \min[|\mathbf{H}_{1}\mathbf{w}_{A}\|^{2},\ldots,|\mathbf{H}_{L}\mathbf{w}_{A} \|^{2}+\sum_{m=1}^{L-1}\|\mathbf{G}_{L,m}\mathbf{w}_{T,m}\|^{2},\min_{m=L+1, \ldots,J}\{|\mathbf{h}_{m}\mathbf{w}_{A}|^{2}+\sum_{\ell=1}^{L}|\mathbf{g}_{m, \ell}\mathbf{w}_{T,\ell}|^{2}\}],\\ 1\leq k\leq L.\end{array}\right. \tag{7}\]  
Proposition 1 reveals the design principle of \(\mathbf{w}_{A}\) and \(\mathbf{w}_{T,1},\ldots,\mathbf{w}_{T,L}\) for the cooperative NOMA system.  
Proposition 1: _To maximize the total rates \(\sum_{k=1}^{J}r_{k}^{*}\), the \(L+1\) beamformer vectors \(\mathbf{w}_{A}\) and \(\mathbf{w}_{T,1},\ldots,\mathbf{w}_{T,L}\) should be designed such that \(\alpha_{k},\ k=1,\ldots,J\) should be maximized by equalizing all the terms in the minimum operation so that \(\alpha_{1}=\alpha_{2}=\cdots=\alpha_{J}\)._  
Proof: The condition in (6) can be rearranged as  
\[\min_{m=k+1,\ldots,J}\gamma_{k}^{m}\geq\gamma_{k}^{*},\ k=1,\ldots,J. \tag{8}\]  
Together with the condition at the cooperative UTs, we have for \(x_{k}(n)\) to be decoded at the cooperative UTs and the UTs that need to decode it  
\[\begin{array}{l}\min_{m=1,2,\ldots,J}\gamma_{k}^{m}\geq\gamma_{k}^{*},\ k \leq L,\\ \min_{m=1,\ldots,L,k+1,\ldots,J}\gamma_{k}^{m}\geq\gamma_{k}^{*},\ L<k\leq J.\end{array} \tag{9}\]  
Note that the SINR expressions in (3) and (4) indicate that \(\gamma_{k}^{m},\ (m>L)\) is a monotonic function of \(|\mathbf{h}_{m}\mathbf{w}_{A}|^{2}+\sum_{\ell=1}^{L}|\mathbf{g}_{m,\ell} \mathbf{w}_{T,\ell}|^{2}\) and \(\gamma_{k^{\prime}}^{\ell}(\ell\leq L)\) is a monotonic function of \(\|\mathbf{H}_{l}\mathbf{w}_{A}\|^{2}+\sum_{m=1}^{\ell-1}\|\mathbf{G}_{\ell,m} \mathbf{w}_{T,m}\|^{2}\). The definitions of \(\alpha_{k}\)'s in (7) allow us to re-write (9) concisely as  
\[\frac{\alpha_{k}P_{k}}{\alpha_{k}\sum_{m=k+1}^{J-1}P_{m}+1}\geq\gamma_{k}^{*},\ k=1,\ldots,J. \tag{10}\]  
Therefore, the rate \(r_{k}^{*}\) on the right-hand side of (10) is maximized by maximizing \(\alpha_{k}\); hence, maximizing the total rates \(\sum_{k=1}^{J}r_{k}^{*}\) corresponds to maximizing \(\alpha_{k},\ k=1,\ldots,J\). Here, from the \(k=1\) case to the \(k=L\) case, produce the minimum of \(\alpha_{k}\) as \(\alpha_{1}=\ldots=\alpha_{L}\)by including all the gain terms of UTs in the minimum operation. Without any further constraint on the beamformer, the optimum beamforming condition of maximizing the minimum of two variables is equaling the two variables. Therefore, applying this fact from \(\alpha_{J}\) to \(\alpha_{1}\) one-by-one assures that the last statement of the Proposition holds.  
Defining \(\beta_{k}=\min_{m=k,\ldots,J}\{|\mathbf{h}_{m}\mathbf{w}_{A}|^{2}\}\), the design principle of \(\mathbf{w}_{A}\) for the non-cooperative NOMA system appeared in Reference [12] can be easily re-derived as \(|\mathbf{h}_{1}\mathbf{w}_{A}|^{2}=\cdots=|\mathbf{h}_{J}\mathbf{w}_{A}|^{2}\) and \(\beta_{1}=\cdots=\beta_{J}\) equivalently. From the principle in Proposition 1, we may contrive a scheme to maximize the rates (equivalently \(r_{k}^{*}\)) with a fixed power or to minimize the power with fixed rates. We follow the latter approach in the next subsection.