## 3 Beamforming Optimization with AN
### Beamforming Optimization with AN in the MISO SWIPT CR-NOMA Network
\[\left(\sum_{l^{\prime}=1}^{i-1}\operatorname{[PERSON]}\left(\mathbf{G} _{l}\mathbf{V}_{l^{\prime}}\right)+\sum_{m=1}^{M}\operatorname{[PERSON]}\left( \mathbf{G}_{l}\mathbf{W}_{m}\right)+\operatorname{[PERSON]}\left(\mathbf{G}_{l} \mathbf{F}\right)+\sigma_{l}^{2}+\frac{\sigma_{cn,l}^{2}}{\theta_{l}}\right) \gamma_{i}\] \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]0 \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]1 (12c) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]2 (12d) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]3 (12e) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]4 (12f) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]5 (12g) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]6 (12h) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]7 (12i) \[\frac{\theta_{l}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l}\big{|}^{ 2}}{\theta_{l}\left(\sum_{l=1}^{i-1}\big{|}\textbf{g}_{l}^{H}\textbf{v}_{l^{ \prime}}\big{|}^{2}+\sum_{m=1}^{M}\big{|}\textbf{g}_{l}^{H}\textbf{w}_{m} \big{|}^{2}+\big{|}\textbf{g}_{l}^{H}\textbf{f}\big{|}^{2}+\sigma_{l}^{2} \right)+\sigma_{cn,l}^{2}}\geq\gamma_{i},\,\forall l,\,i\in\mathbb{Q},\,l\leq i \tag{11b}\]8  
Problem (12) is still non-convex due to the rank-one constraints. Then, we can drop the rank-one constraints to get an SDP problem denoted as (12)-SDR, which is convex and can be solved by using the CVX toolbox of MATLAB.