## 3 Joint Power Control and Resource Allocation
### Optimal JoTAPS Scheme  
This subsection derives the jointly optimum time allocation and power splitting ratio (JoTAPS) with a fixed transmit power. Firstly, we give an initial power control parameter \(\alpha=\hat{\alpha}\). Then, the source nodes can determine the transmit power according to Equations (21) and (22), which are \(P_{1}=\min(P_{1,\max},\frac{\delta Q}{|l_{1}|^{2}})\) for \(S_{1}\) and \(P_{2}=\min\Bigl{(}P_{2,\max},\frac{(1-\hat{\alpha})Q}{|l_{2}|^{2}}\Bigr{)}\) for \(S_{2}\). In this case, the previous optimization problem transforms into  
\[\text{OP3:}\ \ \max\ \min(R_{1},R_{2})\] (46) s.t. \[\text{C2:}\where  
\[C(\hat{\Upsilon}_{rl})=\log_{2}\left(1+\frac{|h_{i}|^{2}\eta\rho(\gamma_{\Sigma}+ \sigma_{u,r}^{2})\frac{t}{1-t}}{\sigma_{u,i}^{2}+\sigma_{i}^{2}}\right) \tag{49}\]  
Define \(R_{\min}=\min(R_{1},R_{2})\). The constraint of rate region C3 and C4 can be rewritten as  
\[C\hat{3}:R_{\min}\leq t\cdot g_{1}(t,\rho) \tag{50}\] \[C\hat{4}:R_{\min}\leq(1-t)\cdot g_{2}(t,\rho) \tag{51}\]  
where  
\[g_{1}(t,\rho)=\min(C(Y_{1r}),C(Y_{2r}),\frac{1}{2}C(Y_{max})) \tag{52}\] \[g_{2}(t,\rho)=\min(C(\hat{\Upsilon}_{r1}),C(\hat{\Upsilon}_{r2})) \tag{53}\]  
Substituting the rewritten constraint in Equations (48), (50) and (51) into OP3, we have  
\[\text{OP4: }\max_{t,\rho}\ R_{\min} \tag{54}\] \[\text{s.t. }\mathbb{C2},C\hat{3},C\hat{4},C5,C6\]  
Some analysis reveals that with respect to \(\rho\) (fix t) or with respect to \(t\) (fix \(\rho\)), OP4 is a convex optimization problem. Theorem 3.2: _Given \(\alpha=\hat{\alpha}\), OP4 is a convex optimization problem with respect to \(\rho\) (fix t) or with respect to \(t\) (fix \(\rho\))._  
Proof: To prove that OP4 is a convex optimization problem, we need to prove that the objective function and the constraint are both convex or affine functions. The objective function of OP4 is a constant, and when \(t\) and \(\rho\) are fixed, C2 is a linear function. Then, we derive the convex function properties of C3 and C4. When \(t\) is fixed, the first and second derivatives of \(C(Y_{a})\), \(a=\{1r,2r,MA\}\) in \(g_{1}(t,\rho)\) with respect to \(\rho\) are  
\[\frac{\partial C(Y_{a})}{\partial\rho}=-\frac{\gamma_{a}\sigma_{b }^{2}}{\ln 2}\cdot\frac{1}{(\bar{\rho}\sigma_{u}^{2}+\sigma_{b}^{2})(\bar{\rho}m_ {a}+\sigma_{b}^{2})} \tag{55}\] \[\frac{\partial^{2}C(Y_{a})}{\partial^{2}\rho}=-\frac{\gamma_{a} \sigma_{b}^{2}}{\ln 2}\cdot\frac{\sigma_{u}^{2}(\bar{\rho}\sigma_{u}^{2}+ \sigma_{b}^{2})+m_{a}(\bar{\rho}\sigma_{u}^{2}+\sigma_{b}^{2})}{(\bar{\rho} \sigma_{u}^{2}+\sigma_{b}^{2})(\bar{\rho}m_{a}+\sigma_{b}^{2})} \tag{56}\]  
where \(\bar{\rho}=1-\rho\), \(m_{a}=\sigma_{u}^{2}+\gamma_{a}\), \(\gamma_{a}=\{\gamma_{1},\gamma_{2},\gamma_{\Sigma}\}\) are three cases of \(a\in\{1,2,\Sigma\}\). Since \(t\) is fixed, \(\rho\in(0,1)\). Thus \(t\cdot\frac{\partial^{2}C(Y_{1r})}{\partial\rho^{2}}<0\), \(t\cdot\frac{\partial^{2}C(Y_{2r})}{\partial\rho^{2}}<0\), \(\frac{t}{2}\cdot\frac{\partial^{2}C(Y_{2r})}{\partial\rho^{2}}<0\). From the properties of convex functions, we conclude that if \(f(x)\) and \(g(x)\) are convex (or concave) functions, then \(\min(f(x),g(x))\) are convex (or concave) as well. Clearly \(g_{1}(t,\rho)\) is concave with respect to \(\rho\) when \(t\) is fixed; thus, C3 is concave with respect to \(\rho\). When \(t\) is fixed, the first and second derivatives of \(C(Y_{b}),b=\{r1,r2\}\) in \(g_{2}(t,\rho)\) with respect to \(\rho\) are  
\[\text{C2:}\where  
\[C(\hat{\Upsilon}_{rl})=\log_{2}\left(1+\frac{|h_{i}|^{2}\eta\rho(\gamma_{\Sigma}+ \sigma_{u,r}^{2})\frac{t}{1-t}}{\sigma_{u,i}^{2}+\sigma_{i}^{2}}\right) \tag{49}\]0 \[\text{C2:}\where  
\[C(\hat{\Upsilon}_{rl})=\log_{2}\left(1+\frac{|h_{i}|^{2}\eta\rho(\gamma_{\Sigma}+ \sigma_{u,r}^{2})\frac{t}{1-t}}{\sigma_{u,i}^{2}+\sigma_{i}^{2}}\right) \tag{49}\]1  
where \(m_{\Sigma=\sigma_{u}^{2}+\gamma_{\Sigma}}\), \(m_{i}=\sigma_{u}^{2}+\sigma_{i}^{2}\), \(i=\{1,2\}\). Since \(t\) is fixed, \(\rho\in(0,1)\), \(\frac{\partial^{2}C(Y_{ii})}{\partial\rho^{2}}<0\).