## 2 System Model
### MRC-ZFB/MRT-SC-ZFB with FD Scenario  
The secure MRC-ZFB/MRT-SC-ZFB protocol employs two phases and the beamforming vector \(\mathbf{w}_{ZF}\), \(\mathbf{w}_{ZF1}\) are designed for each phase. For MRC-ZFB phase, we put forward an optimal beamforming scheme which maximizes the achievable secrecy rate of MRC-ZFB CRNs based on FD operation. Relay can adopt \(N_{R}\) antennas gain for receiving the signal from the Alice by MRC operation, and Bob uses \(N_{B}\) antennas for transmitting jamming signals to eavesdropper Eve simultaneously by using ZFB operation. ZFB is employed to get rid of malicious jamming signals for meeting the constraints at PU. Now, defining the \(N_{B}\times(1+N_{R})\) channel matrix, i.e., \(\mathbf{H}_{BZ}=[\mathbf{h}_{BP},\mathbf{H}_{BR}]\), where \(\mathbf{h}_{BP}\), \(\mathbf{H}_{BR}\) are \(N_{B}\times 1\), \(N_{R}\times N_{B}\) channel link matrices Bob-PU and Bob-Relay, respectively, but the number of the antennas at Bob should be more than \(N_{R}+1\) (\(N_{B}>N_{R}+1\)). It is assumed that Relay employs a linear receiver \(\mathbf{w}_{r}\) with \(\left\|\mathbf{h}_{AR}\right\|\) = 1 for detecting signal. Additionally, MRC is employed at the Relay, denoted as \(\mathbf{w}_{r}=\mathbf{h}_{AR}/\left\|\mathbf{h}_{AR}\right\|\)[38]. The optimal expression of the weight vector \(\mathbf{w}_{ZF}\) is given by  
\[\begin{array}{l}\max_{\mathbf{w}_{ZF}}\left|\mathbf{h}_{BE1}^{\dagger} \mathbf{w}_{ZF}\right|\\ s.t.\left|\mathbf{h}_{BP}^{\dagger}\mathbf{w}_{ZF}\right|=0,\left|\mathbf{h}_ {AR}^{\dagger}\mathbf{H}_{BR}\mathbf{w}_{ZF}\right|=0\delta\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbfmathbfmathbfmathbfmathbfmathbfmathbfmathbf }}}}}}}}}}}}}}}}}\\ \end \  
where \(\mathrm{T}^{\perp}\!=\!\left(\mathbf{I}-\mathbf{H}_{BZ}\!\left(\mathbf{H}_{BZ} \mathbf{H}_{BZ}^{\dagger}\right)^{-1}\mathbf{H}_{BZ}^{\dagger}\right)\) is the correlation matrix with rank \(N_{B}-\left(N_{R}+1\right)\). Consequently, the instantaneous SNR at the Relay and the instantaneous signal-to-interference-plus-noise ratio (SINR) at the Eve are written as  
\[\gamma_{AR,2}\!=\!\frac{P_{S}}{\sigma_{R}^{2}}\|\mathbf{h}_{AR}\|^{2}, \tag{8}\]  
and  
\[\gamma_{AE,2}\!=\!\frac{P_{S}|\mathbf{h}_{AE}|^{2}}{P_{J}|\mathbf{h}_{BE1}^{ \dagger}\mathbf{w}_{ZF}|^{2}\!+\!\sigma_{E}^{2}}, \tag{9}\]  
where the interference power of Bob-Eve is denoted by \(P_{J}\). For the FD mechanism, when researchers investigate the performance limitations of information theory (such as capacity), the assumption that SI is not taken into consideration is widely used [33]. Therefore, for solving the closed-form expressions of SOP of FD operation at Bob, a detailed introduction will be presented in Section 3. For MRT-SC-ZFB with FD phase, the Relay uses the \(N_{R}\) antennas to transmit information to Bob. Then Bob selects the best \(i\)-th antenna to receive information from Relay and the remaining \(N_{B}-1\) antennas transmit the jamming signal to Eve simultaneously. Therefore, Relay adopts MRT operation to decode and retransmit signal, and Bob adopts SC-ZFB operation to receive signals simultaneously to transmit jamming signals. The aim of SC-ZFB operation is to maximize the received SNR at Bob while avoiding the leakage of security information to the Eve and the interference to PU. For the MRT-SC-ZFB phase, it requires \(N_{B}\geq 3\). Thus, the optimal expression of the optimal weight vector \(\mathbf{w}_{ZF1}\) is expressed as  
\[ \begin{array}{l}\max_{\mathbf{w}_{ZF1}}\left|\mathbf{h}_{BE2}^{\dagger} \mathbf{w}_{ZF1}\right|\\ s.t.\left|\mathbf{h}_{BP}^{\dagger}\mathbf{w}_{ZF1}\right|\boldsymbol{\xi}\! \left|\mathbf{w}_{ZF1}\right|_{F}=1,\end{array}  \tag{10}\]  
where \(\mathbf{h}_{BE2}\) denote the \(\left(N_{B}-1\right)\times 1\) channel vector between the remaining \(N_{B}-1\) antennas of [PERSON] and [PERSON]. \(\mathbf{h}_{BP}\) denote the \(\left(N_{B}-1\right)\times 1\) channel vector between the remaining \(N_{B}-1\) antennas of [PERSON] and [PERSON].