# Secure Communication for Uplink Cellular Networks Assisted with Full-Duplex Device-to-Device User
## 3 Problem Formulation and Analysis  
In this section, we propose a secure communication scheme that maximizes the achievable rate of the D2D link with the constraint of the secrecy rate of the cellular uplink. The optimization problem is solved by deriving the power allocation factors of CU and D1. First of all, the optimal power allocation factor of D1 is obtained according to the monotonicity of the objective function. Then after verifying that the solution is the global minimum point, the closed-form solution of the optimal power allocation factor of the CU is derived. The instantaneous SINR at D1 can be described as:  
\[\gamma_{D1}=\frac{\beta P_{C}\gamma_{C1}}{(1-\beta)P_{C}\gamma_{C1}+\kappa P_ {D}\gamma_{1}+1} \tag{6}\]  
where \(\gamma_{C1}\) and \(\gamma_{1}\) are channel-to-noise ratios (CNRs) of the link from CU to D1 and the self-interference channel, respectively, defined as:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]  
\[\gamma_{1}=\big{|}f_{1}\big{|}^{2}/\sigma^{2} \tag{8}\]  
The achievable rate of D1 is given by:  
\[R_{1}=W\log_{2}\left(1+\frac{\beta P_{C}\gamma_{C1}}{(1-\beta)P_{C}\gamma_{C1 }+\kappa P_{D}\gamma_{1}+1}\right) \tag{9}\]  
where the parameter \(W\) is the system bandwidth. D1 employs the FD AF protocol to assist the cellular uplink transmission. The jamming signal is assumed to be known in advance by the BS, so that the BS is able to cancel out the jamming signal from the received signal. Then the instantaneous SINR at BS is expressed as:  
\[\gamma_{B}=\frac{\alpha\beta P_{C}P_{D}\gamma_{C1}\gamma_{B1}}{\kappa\alpha P _{D}^{2}\gamma_{1}\gamma_{B1}+\alpha P_{D}\gamma_{B1}+P_{C}\gamma_{C1}+\kappa P _{D}\gamma_{1}+1} \tag{10}\]  
where \(\gamma_{B1}\) is CNRs of the link from D1 to BS, defined as:  
\[\gamma_{B1}=|h_{B1}|^{2}/\sigma^{2} \tag{11}\]The achievable rate of cellular uplink communication is given by:  
\[R_{C}=W\log_{2}\!\left(1+\frac{\alpha\beta P_{C}P_{D}\gamma_{C1}\gamma_{B1}}{ \kappa\alpha P_{D}^{2}\gamma_{1}\gamma_{B1}+aP_{D}\gamma_{B1}+P_{C}\gamma_{C1}+ \kappa P_{D}\gamma_{1}+1}\right) \tag{12}\]  
Due to the reuse of the cellular uplink spectrum, the cellular uplink will cause interference to D2. So the instantaneous SINR at D2 can be described as:  
\[\gamma_{D2}=\frac{(1-\alpha)P_{D}\gamma_{12}}{P_{C}\gamma_{C2}+\alpha P_{D} \gamma_{12}+1} \tag{13}\]  
where \(\gamma_{C2}\) and \(\gamma_{12}\) are the CNRs of the links from CU and D1 to D2, respectively, defined as:  
\[\gamma_{C2}=|h_{C2}|^{2}/\sigma^{2} \tag{14}\]  
\[\gamma_{12}=|h_{12}|^{2}/\sigma^{2} \tag{15}\]  
The achievable rate of D2D communication is given by:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]0  
For simplicity, \(R_{D}\) can be rewritten as:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]1  
where the parameter \(F(\alpha)\) is given by:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]2  
Therefore, the optimization function of the proposed scheme can be expressed as:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]3 \[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]4 (19b) \[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]5 (19c) \[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]6  
where \(R_{th}\) represents the secrecy rate threshold of the cellular uplink. Then, substituting (9) and (12) into the constraint (19a) with secrecy rate, as follows:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]7  
where the parameter \(\eta\) is given as:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]8  
According to (20), we can derive that:  
\[\gamma_{C1}=|h_{C1}|^{2}/\sigma^{2} \tag{7}\]9From (17), we can easily see that \(R_{D}\) is a monotonically increasing function with respect to \(F(\alpha)\), so we obtain the optimal object \(R_{D}\) (19) by solving the maximum value of \(F(\alpha)\).