## 3 Beamforming Optimization with AN
### Beamforming Optimization with AN in the MISO SWIPT CR-NOMA Network  
The SU-Tx power for transmitting to SUs and EH users is minimized in the proposed network by satisfying the QoS requirements of each user and in accordance with the decoding order of the SIC indicated in (4), as follows:  
\[\min_{\{\textbf{v}_{l},\textbf{w}_{m},\beta_{l}\},\textbf{f}} \sum_{l=1}^{L}\left\|\textbf{v}_{l}\right\|^{2}+\sum_{m=1}^{M} \left\|\textbf{w}_{m}\right\|^{2}\] s.t. \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\] \[\sum_{l^{\prime}=1}^{L}\left|\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\right|^{2}+\sum_{m=1}^{M}\left|\textbf{g}_{l}^{H}\textbf{w}_{m}\right| ^{2}+\left|\textbf{g}_{l}^{H}\textbf{f}\right|^{2}+\sigma_{l}^{2}\geq\frac{ \psi_{l}}{(1-\theta_{l})},\,\forall_{l}\] (11c) \[\sum_{l=1}^{L}\left|\textbf{e}_{m}^{H}\textbf{v}_{l}\right|^{2}+ \sum_{m^{\prime}=1}^{M}\left|\textbf{e}_{m}^{H}\textbf{w}_{m^{\prime}}\right| ^{2}+\left|\textbf{e}_{m}^{H}\textbf{f}\right|^{2}+\sigma_{\rm{EH},m}^{2}\geq \xi_{m},\,\forall_{m}\] (11d) \[\sum_{l=1}^{L}\left\|\textbf{v}_{l}\right\|^{2}+\sum_{m=1}^{M} \left\|\textbf{w}_{m}\right\|^{2}+\left\|\textbf{f}\right\|^{2}=P^{\rm{max}},\] (11e) \[\sum_{l=1}^{L}\left|\textbf{h}_{k}^{H}\textbf{v}_{l}\right|^{2}+ \sum_{m=1}^{M}\left|\textbf{h}_{k}^{H}\textbf{w}_{m}\right|^{2}+\left|\textbf{ h}_{k}^{H}\textbf{f}\right|^{2}\leq\phi_{\rm{PU},k},\,\forall_{k}\] (11f) \[1>\theta_{l}>0,\,\forall l\] (11g) \[\left|\textbf{g}_{l}^{H}\textbf{v}_{L}\right|^{2}\geq\left| \textbf{g}_{l}^{H}\textbf{v}_{L-1}\right|^{2}\geq...\geq\left|\textbf{g}_{l}^{ H}\textbf{v}_{1}\right|^{2},\,\forall l, \tag{11h}\]  
where \(\gamma_{i}\) is the minimum targeted SINR for the \(i\)-th message, \(i\in\mathbb{Q}\), \(\psi_{l}\) is the minimum harvested energy required by the \(l\)-th SU, \(\xi_{m}\) is the minimum harvested energy required by the \(m\)-th EH user, \(p^{\rm{max}}\) denotes the maximum available power at the SU-Tx, and \(\phi_{\rm{PU},k}\) denotes the maximum permissible interference with the k-th PU. Constraint (11b) is to guarantee a minimum SINR for the messages from \(s_{l}^{\rm{SU}}\) to \(s_{L}^{\rm{SU}}\) at the \(l\)-th SU, while constraints (11c) and (11d) represent the minimum harvested energy required by the SUs and the EH users, respectively. Constraint (11e) indicates the maximum available power, \(p^{\rm{max}}\) that can be consumed by the SUs, the EH users, and AN. Constraint (11f) indicates that the transmit power for the SUs, the EH users, and for AN must not exceed the maximum permissible interference with the \(k\)-th PU, given by \(\phi_{\rm{PU},k}\). Problem (11) is not convex or challenging to solve due to the coupling between the PS ratios, \(\theta_{l}\), and the precoding vectors, \(\textbf{v}_{l}\), \(\textbf{w}_{l}\), and \(\textbf{f}\), in constraint (11b). Therefore, problem (11) cannot be solved directly. We derive an optimal solution via SDR [38] for problem (11). Define \(\textbf{V}_{l}=\textbf{v}_{l}\textbf{v}_{l}^{H}\), \(\textbf{W}_{m}=\textbf{w}_{m}\textbf{w}_{m}^{H}\), \(\textbf{F}=\textbf{f}\textbf{f}^{H}\), \(\textbf{G}_{l}=\textbf{g}_{l}\textbf{g}_{l}^{H}\), \(\textbf{E}_{m}=\textbf{e}_{m}\textbf{e}_{m}^{H}\), and \(\textbf{H}_{k}=\textbf{h}_{k}\textbf{h}_{k}^{H}\), \(\forall_{l}\), \(\forall_{m}\), \(\forall_{k}\).