## 3 Beamformer Design
### Two Designs  
According to the design principles of the cooperative NOMA in Proposition 1, we may propose two feasible designs in this subsection. In addition, a simple modification of the designs results in a design for the non-cooperative NOMA system. Since Proposition 1 states that all the sums of channel gain terms reaching the UTs are required to be equalized for the optimal points, such condition can be met with SDP based approaches through the following steps. Note that the typical application of SDP is composed of linear matrix inequalities (LMI) with semi definite parameter matrices [28] so that we can build a parameter matrix \(\mathbf{W}\) from the beamforming vectors and then the optimality condition in Proposition 1 can be absorbed in the LMIs as follows. First, we define the \((M+LN)\times 1\) vector set of \(\tilde{\mathbf{h}}_{j}=[\mathbf{h}_{j}^{T},\mathbf{g}_{j}^{T},\ldots,\mathbf{ g}_{j,L}^{T}]^{T}\), \(j=L+1,\ldots,J\) and the \(N\times(M+(L-1)N)\) concatenated matrices of \(\mathcal{H}_{1}=[\mathbf{H}_{1},\mathbf{0}_{N},\ldots,\mathbf{0}_{N}], \mathcal{H}_{2}=[\mathbf{H}_{2},\mathbf{G}_{2,1},\mathbf{0}_{N},\ldots, \mathbf{0}_{N}],\ldots,\mathcal{H}_{L}=[\mathbf{H}_{L},\mathbf{G}_{L,1},\ldots,\mathbf{G}_{L,L-1}]\), where \(\mathbf{0}_{N}\) is the \(N\times N\) zero matrix. With another definition of the \((M+LN)\times 1\) concatenated beamformer vector \(\mathbf{w}=[\mathbf{w}_{A}^{T},\mathbf{w}_{T,1}^{T},\ldots,\mathbf{w}_{T,L}^{ T}]^{T}\), we may form the following SDP. \[ \begin{array}{c}\min_{\mathbf{W}_{\in}\mathbb{C}^{M+N}}\mathbf{Tr}[\mathbf{ W}]\\ \text{s.t.}\ \mathbf{Tr}[\mathbf{H}_{i}\mathbf{W}]\geq\gamma^{*},\ j=L+1,\ldots,J \\ \mathbf{Tr}[\mathcal{H}_{j}^{H}\mathcal{H}_{i}\mathcal{H}_{j}\mathbf{W}]\geq \gamma^{*},\ j=1,2,\ldots,L\\ \text{rank}\{\mathbf{W}\}=1.\end{array}  \tag{11}\]  
Note that the last rank constraint requires the rank reduction process as in Reference [28] before the solutions \(\mathbf{w}_{A}\) and \(\mathbf{w}_{T,\ell}\) are available. Here, we use the matrix definitions \(\mathbf{H}_{j}=\tilde{\mathbf{h}}_{j}\tilde{\mathbf{h}}_{j}^{H}\) and \(\mathbf{W}=\mathbf{w}\mathbf{w}^{H_{j}}\gamma^{*}\) is the threshold uniformly applied to all UT channels to control the total throughput \(\sum_{k=0}^{J-1}r_{k}^{*}\). The SDP in (11) has the same structure as the one in Reference [28] due to the similarity between the NOMA system and the multicast system. A defect of the problem in (11) is that we have no control over the balance of the AP power (\(\|\mathbf{w}_{A}\|^{2}\)) and the cooperative UT power (\(\|\mathbf{w}_{T,\ell}\|^{2}\)) in the solution of (\(\mathbf{w}\)). Since the cooperative UTs have power limitation in practice, we apply additional constraints on the cooperative UT power of the problem in (11) so that we can have better control on the division of the power. We modify (11) as follows. \[ \begin{array}{c}\min_{\mathbf{W}_{\in}\mathbb{C}^{M+N}}\mathbf{Tr}[\mathbf{W} ]\\ \text{s.t.}\ \mathbf{Tr}[\mathbf{E}_{\ell}\mathbf{W}\mathbf{E}_{\ell}]\leq P_{ \ell},\ \ell=1,\ldots,L\\ \mathbf{Tr}[\tilde{\mathbf{H}}_{j}\mathbf{W}]\geq\gamma^{*},\ j=L+1,\ldots,J \\ \mathbf{Tr}[\mathcal{H}_{j}^{H}\mathcal{H}_{j}\mathbf{W}]\geq\gamma^{*},\ j=1,2, \ldots,L\\ \text{rank}\{\mathbf{W}\}=1.\end{array}  \tag{12}\]  
Here, \(\mathbf{E}_{\ell}=\text{diag}[\mathbf{0}_{N\ell},\mathbf{1}_{N},\mathbf{0}_{ N(L=\ell)}]\) and \(P_{\ell}\) is the desired power level (\(P_{d,\ell}\)) of the \(\ell\)-th cooperative UT. Again, the rank reduction produces a \(\mathbf{w}_{A}\) that complies with \(\mathbf{w}_{T,\ell}\), \(\ell=1,\ldots,L\) having the desired cooperative UT power. Compared to the SDP in (11), the power control capability of (12) is earned at the expense of some performance loss since the feasible region of (12) is a subset of that in (11). Note that a slight modification of (11) leads to the SDP for the beamforming in the non-cooperative NOMA system as  
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