## 2 System Model  
In this paper, we investigate downlink transmissions in a cognitive MISO SWIPT NOMA system with \(L\) SUs, and \(M\) EH receivers that coexist with \(K\) PUs via the underlay scheme, as shown in Figure 1. In the proposed underlying CR system, the secondary network can operate in the same frequency band as the primary network, provided that the interference towards the primary users is kept at a tolerable level. We also consider the presence of one malicious eavesdropper that attempts to intercept confidential information of the SUs. Furthermore, the SU-Tx is equipped with \(N\) antennas, while there are single antennas for the PU, SU, and EH receivers, and for the eavesdropper. Moreover, we consider the SUs to be hybrid users capable of decoding information and harvesting energy at the same time by employing a PS structure in the receiver, while energy is harvested by the EH receivers. Moreover, the SU-Tx transmits AN to cause interference with the eavesdropper while improving information secrecy. The SU-Tx is in charge of transmitting the signals towards the SU and the EH receivers, as well as, design the beamforming vectors for the multi-antenna scheme. Then, the signal transmitted from the SU-Tx can be expressed as:  
\[\mathbf{x}=\sum_{l=1}^{L}\mathbf{v}_{ls}^{\text{SU}}+\sum_{m=1}^{M}\mathbf{w}_ {m}s_{m}^{\text{EH}}+\mathbf{f}\mathbf{r}, \tag{1}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Abbreviation** & **Term** \\ \hline AN & Artificial noise \\ BS & Base station \\ CR & Cognitive radio \\ CSI & channel state information \\ DF & Decode-and-forward \\ ID & Information decoding \\ IoT & Internet of things \\ MISO & Multiple-input single-output \\ MIMO & Multiple-input multiple-output \\ NOMA & Non-orthogonal multiple-access \\ PA & Power allocation \\ PLS & Physical-layer security \\ PS & Power-splitting \\ PSO & Particle swarm optimization \\ SIC & Successive interference cancelation \\ SDP & semidefinite programming \\ SINR & Signal-to-Interference-plus-noise ratio \\ SSR & Secrecy sum-rate \\ QoS & Quality of service \\ TDMA & Time-division multiple-access \\ SDR & Semidefinite relaxation \\ SWIPT & Simultaneous wireless information and power transfer \\ ZF & Zero-forcing \\ SU-Tx & Secondary transmitter \\ SU\({}_{\text{l}}\) & \(l\)-th Secondary user \\ EH\({}_{\text{m}}\) & \(m\)-th EH user \\ \(PU_{k}\) & \(k\)-th Primary user \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main abbreviations. where \(\mathbf{v}_{l}\in\mathbb{C}^{N\times 1}\) represents the beamforming vector for the \(l\)-th SU with \(l\in\mathbb{Q}\triangleq\{1,2,..,L\}\), \(s_{l}^{\text{SU}}\) is the transmitted signal of the \(l\)-th SU with unit power \(\left|s_{l}^{\text{SU}}\right|^{2}=1\), \(\mathbf{w}_{m}\in\mathbb{C}^{N\times 1}\) denotes the beamforming vector for the \(m\)-th EH receivers with \(m=1,2,..,M\), \(s_{m}^{\text{EH}}\) is the energy signal of the \(m\)-th EH receiver with unit power \(\left|s_{m}^{\text{EH}}\right|^{2}=1\), \(\mathbf{f}\in\mathbb{C}^{N\times 1}\) corresponds to the beamforming vector for the AN, and \(r\) is a zero-mean Gaussian random variable with unit power \(\left|r\right|^{2}=1\), which is artificial jamming. Beamforming vector \(\mathbf{f}\) is used to prevent anything from affecting legal transmissions, and artificial jamming \(r\) is randomly generated to effectively disturb any eavesdropping [14]. Accordingly, the received signal at the \(l\)-th SU is written as  
\[y_{\text{SU},l}=\mathbf{g}_{l}^{H}\mathbf{v}_{l}s_{l}^{\text{SU}}+\mathbf{g}_{ l}^{H}\left(\sum_{l^{\prime}=1,l^{\prime}\
eq l}^{L}\mathbf{v}_{l^{\prime}}s_{ l^{\prime}}^{\text{SU}}+\sum_{m=1}^{M}\mathbf{w}_{m}s_{m}^{\text{EH}}+\mathbf{f}r \right)+n_{\text{SU},l},l\in\mathbb{Q}, \tag{2}\]  
where \(\mathbf{g}_{l}^{H}\in\mathbb{C}^{N\times 1}\) is the channel vector from the SU-Tx to the \(l\)-th SU, \(n_{\text{SU},l}\sim\mathcal{CN}\left(0,\sigma_{l}^{2}\right)\) is the additive white Gaussian noise (AWGN) at the \(l\)-th SU, with zero mean and variance denoted as \(\sigma^{2}\).