## 3 Beamforming Optimization with AN
### Beamforming Optimization with an in the MISO SWIPT CR-ZF Network  
In this section, we include a comparative scheme based on ZF to solve the power minimization problem in the cognitive MISO SWIPT system. The ZF scheme allows the nullification of the interference signals from other users, which is considered to be a common approach used to simplify the system design [35; 40]. At the \(l\)-th SU user, we design the ZF scheme to null the interference from other SU's messages and from the energy signals. At the \(k\)-th PU user, the ZF scheme nulls the interference from the SU's messages and from the energy signals. However, to increase the harvested energy at the EH users, we allow interference signals at the EH users. Then, we define\(\mathbf{A}_{I}=[\mathbf{g}_{1},...,\mathbf{g}_{I-1},\mathbf{g}_{1+1},...,\mathbf{ g}_{L},\mathbf{h}_{1},...,\mathbf{h}_{K}]\in\mathbb{C}^{N\times(L+K-1)}\) and \(\mathbf{B}_{m}=[\mathbf{g}_{1},...,\mathbf{g}_{L},\mathbf{h}_{1},...,\mathbf{ h}_{K}]\in\mathbb{C}^{N\times(L+K)}\). Therefore, we denote the beamforming vectors as follows:  
\[\mathbf{\tilde{v}}_{I}=\frac{\mathbf{\Omega}_{AJ}\mathbf{\Omega}_{AJ}^{H} \mathbf{g}_{I}}{\left\|\mathbf{\Omega}_{AJ}\mathbf{\Omega}_{AJ}^{H}\mathbf{g} _{I}\right\|}, \tag{17}\]  
\[\mathbf{v}_{I}=\sqrt{p_{\text{SU}J}}\mathbf{\tilde{v}}_{I}, \tag{18}\]  
\[\mathbf{\tilde{w}}_{m}=\frac{\mathbf{\Omega}_{B,m}\mathbf{\Omega}_{B,m}^{H} \mathbf{e}_{m}}{\left\|\mathbf{\Omega}_{B,m}\mathbf{\Omega}_{B,m}^{H}\mathbf{ e}_{m}\right\|}, \tag{19}\]  
\[\mathbf{w}_{m}=\sqrt{p_{\text{EH},m}}\mathbf{\tilde{w}}_{m}, \tag{20}\]  
where \(p_{\text{SU}J}\) and \(p_{\text{EH},m}\) are the power variables to be optimized, \(\mathbf{\Omega}_{AJ}\in\mathbb{C}^{N\times(N-(L+K-1))}\) is the orthogonal basis of the null space of \(\mathbf{A}_{I}\) and \(\mathbf{\Omega}_{B,m}\in\mathbb{C}^{N\times(N-(L+K))}\) is the orthogonal basis of the null space of \(\mathbf{B}_{m}\). Based on the ZF scheme, the transmit power minimization problem (11) can be transformed into the following problem:  
\[\min_{\{p_{\text{SU}J},p_{\text{EH},m},\theta_{I}\},\mathbf{f}} \sum_{l=1}^{L}p_{\text{SU}J}+\sum_{m=1}^{M}p_{\text{EH},m}\] (21a) s.t. \[\frac{\theta_{l}p_{\text{SU}J}\left|\mathbf{g}_{l}^{H}\mathbf{ \tilde{v}}_{I}\right|^{2}}{\theta_{l}\left(\left|\mathbf{g}_{l}^{H}\mathbf{f} \right|^{2}+\sigma_{l}^{2}\right)+\sigma_{cn,l}^{2}}\geq\gamma_{l},\,\forall l \tag{21b}\] \[p_{\text{SU}J}\left|\mathbf{g}_{l}^{H}\mathbf{\tilde{v}}_{I} \right|^{2}+\left|\mathbf{g}_{l}^{H}\mathbf{f}\right|^{2}+\sigma_{l}^{2}\geq \frac{\psi_{l}}{(1-\theta_{l})},\,\forall_{l}\] (21c) \[\sum_{l=1}^{L}p_{\text{SU}J}\left|\mathbf{e}_{m}^{H}\mathbf{ \tilde{v}}_{I}\right|^{2}+\sum_{m^{\prime}=1}^{M}p_{\text{EH},m^{\prime}} \left|\mathbf{e}_{m}^{H}\mathbf{\tilde{w}}_{m^{\prime}}\right|^{2}+\left| \mathbf{e}_{m}^{H}\mathbf{f}\right|^{2}+\sigma_{\text{EH},m}^{2}\geq\xi_{m}, \,\forall_{m}\] (21d) \[\sum_{l=1}^{L}p_{\text{SU}J}+\sum_{m=1}^{M}p_{\text{EH},m}+\left\| \mathbf{f}\right\|^{2}=p^{\text{max}},\] (21e) \[\left|\mathbf{h}_{k}^{H}\mathbf{f}\right|^{2}-\phi_{\text{PU},k} \leq 0,\,\forall_{k}\] (21f) \[\mathbf{v}_{I}=\sqrt{p_{\text{SU}J}}\mathbf{\tilde{v}}_{I}, \tag{18}\]0  
Next, we solve the problem (21) by using the SDR technique. We denote \(\mathbf{F}=\mathbf{f}\mathbf{f}^{H}\), \(\mathbf{G}_{l}=\mathbf{g}_{l}\mathbf{g}_{l}^{H}\), \(\mathbf{E}_{m}=\mathbf{e}_{m}\mathbf{e}_{m}^{H}\), \(\mathbf{H}_{k}=\mathbf{h}_{k}\mathbf{h}_{k}^{H}\), \(\left\|\mathbf{f}\right\|^{2}=\text{Tr}\left(\mathbf{F}\right)\), \(\left|\mathbf{g}_{l}^{H}\mathbf{f}\right|^{2}=\text{Tr}\left(\mathbf{G}_{l} \mathbf{F}\right)\), \(\left|\mathbf{e}_{m}^{H}\mathbf{f}\right|^{2}=\text{Tr}\left(\mathbf{E}_{m} \mathbf{F}\right)\) and \(\left|\mathbf{h}_{k}^{H}\mathbf{f}\right|^{2}=\text{Tr}\left(\mathbf{H}_{k} \mathbf{F}\right)\). \[\mathbf{v}_{I}=\sqrt{p_{\text{SU}J}}\mathbf{\tilde{v}}_{I}, \tag{18}\]1 (22a) s.t.