## 3 Joint Power Control and Resource Allocation
### Power Allocation with Power Control Consideration  
One degenerated optimization work is proposing a power allocation scheme that considers power control, i.e., deriving the optimal power control and power allocation parameters when \(t\) and \(\rho\) are fixed. Equations (9) and (17) show that as \(P_{1}\) or \(P_{2}\) increases, the achievable upper bound of the objection function \(\min(R_{1},R_{2})\) in the MA phase increases monotonically. As \(P_{R}\) increases, \(\min(R_{1},R_{2})\) in the BC phase increases monotonically. Thus, the system achieves optimal transmission performance with the maximum attainable values of \(P_{1}\), \(P_{2}\) and \(P_{R}\) expressed as\[P_{1} =\min\biggl{(}P_{1,\max},\frac{\alpha Q}{|I_{1}|^{2}}\biggr{)} \tag{21}\] \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\] (22) \[P_{R} =\min\biggl{(}\eta\rho(\gamma_{\Sigma}+\sigma_{u,r}^{2})\frac{t}{ 1-t^{\prime}},\frac{Q}{|I_{r}|^{2}}\biggr{)} \tag{23}\]  
Equations (21)-(23) show that \(P_{1}\), \(P_{2}\) and \(P_{R}\) depend on \(\alpha(0<\alpha<1)\). When the sources have no transmit power limits, \(P_{1}\), \(P_{2}\) and \(P_{R}\) can be rewritten as  
\[P_{1}= \alpha Q/|I_{1}|^{2} \tag{24}\] \[P_{2}= (1-\alpha)Q/|I_{2}|^{2}\] (25) \[P_{R}= \min\biggl{(}\eta\rho\biggl{(}\biggl{(}\frac{|I_{1}|^{2}\alpha}{| I_{1}|^{2}}+\frac{|I_{2}|^{2}(1-\alpha)}{l_{2}}^{2}\biggr{)}Q+\] (26) \[\sigma_{u,r}^{2}\biggr{)}\frac{t}{1-t^{\prime}},\frac{Q}{|I_{r}| ^{2}}\biggr{)}\]  
By substituting Equations (24)-(26) into OP1, the optimization problem reduces to a one-dimensional optimization problem. Define \(R_{\min}=\min(R_{1},R_{2})\). This one-dimensional optimization problem can be expressed as  
\[\text{OP2}: \max R_{\min}\] s.t. \[\text{C3}^{\prime}: R_{\min}\leq t\cdot\min(C(Y_{ir}),C(Y_{MA})/2)),i=1,2 \tag{27}\] \[\text{C4}^{\prime}: R_{\min}\leq(1-t)\cdot C(Y_{ri}),i=1,2\]  
where  
\[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]0 \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]1 (29) \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]2 (30) \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]3  
and  
\[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]4 \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]5 (33) \[P_{2} =\min\biggl{(}P_{2,\max},\frac{(1-\alpha)Q}{|I_{2}|^{2}}\biggr{)}\]6  
Once the upper-bound of \(\text{C3}^{\prime}\) and \(\text{C4}^{\prime}\) as well as the intersection of \(\text{C3}^{\prime}\) and \(\text{C4}^{\prime}\) are determined, the optimal value of OP2 can be obtained by \(R_{\min}=\min(R_{1},R_{2})\). Thus, the first step is to find \(\alpha\), which maximizes \(t\cdot\min(C(Y_{ir}),C(Y_{MA})/2)\) and \((1-t)\cdot C(Y_{ri})\). Through some mathematical analysis, OP2 can be solved in the following two cases by comparing the two terms of \(P_{R}\) expressed in Equation (32). 3. When \(H_{1}<H_{2}\) In this case, \(C(\mathrm{Y}_{MA})\) and \(C(\mathrm{Y}_{ri})\) become monotonically decreasing functions. By analyzing the relationship of the intersection, one can obtain that \[\alpha_{3}^{\prime}=\left\{ \begin{aligned} &\min(\alpha_{0},\alpha_{1r,2r}), \quad\text{when }\alpha_{\min,1}\geq\alpha_{\max,2}\\ &\min(\alpha_{0},\alpha_{\min,1}),\quad\text{when }\alpha_{\min,1}< \alpha_{\max,2}\end{aligned} \right.\] (41) where \(\alpha_{\min,1}=\min(\alpha_{1r,r1},\alpha_{1r,2},\alpha_{1r,ma})\), \(\alpha_{\max,2}=\max(\alpha_{2r,r1},\alpha_{2r,2},\alpha_{2r,ma})\). The above analysis leads to the optimal power control parameter, which satisfies OP2 under Case 2 as  
\[\alpha_{c2}^{*}= \begin{cases}\alpha_{1}^{\prime},\;\;H_{1}=H_{2}\\ \alpha_{2}^{\prime},\;\;H_{1}>H2\\ \alpha_{3}^{\prime},\;\;H_{1}<H_{2}.\end{cases}  \tag{42}\]  
Combining the optimal power control parameters of Case 1 and Case 2, we have the following optimal power control solution:  
\[\alpha^{*}=