## 3 Joint Power Allocation for Co2P JDT under a Non-Unity TCPP
\tag{16}\]  
Then, we can obtain the CTP1 power to be actually allocated to CTP1, \(\hat{P}_{1}^{*}\) as \(\hat{P}_{1}^{*}=P_{S}-\hat{P}_{2}^{*}\). Furthermore, using \(\beta^{\prime}\left(\tilde{\gamma}_{1}\right)\), we obtain the LLA CTP2 power expression similar to Equation (12) of Ref. [8] as  
\[P_{2}^{l*}=\beta^{\prime}\left(\tilde{\gamma}_{1}\right)\left(\ln\tilde{\gamma }_{2}-\ln\tilde{\gamma}_{1}\right)+0.5P_{S}. \tag{17}\]  
In the condition \(\tilde{\gamma}_{1}\geq\tilde{\gamma}_{2}\), we calculate the CTP2 power to be actually allocated to CTP2, \(\hat{P}_{2}^{*}\) by using the LLA CTP2 power expressions of (17) as  
\[\hat{P}_{2}^{*}=\left\{ \begin{array}{ll}0,&\text{if }P_{2}^{l*}<0,\\ P_{2}^{l*},&\text{if }P_{2}^{l*}\geq 0.\end{array} \right. \tag{18}\]  
Then, we again can obtain the CTP1 power to be actually allocated to CTP1, \(\hat{P}_{1}^{*}\) as \(\hat{P}_{1}^{*}=P_{S}-\hat{P}_{2}^{*}\). In conclusion, we can decide whether Co2P JDT is performed or not by using Equation (16) or Equation (18). In the condition \(\tilde{\gamma}_{1}\geq\tilde{\gamma}_{2}\), Co2P JDT is performed only if \(\hat{P}_{2}^{*}>0\); single-point transmission by CTP1 is performed if \(\hat{P}_{2}^{*}=0\). In the condition \(\tilde{\gamma}_{1}<\tilde{\gamma}_{2}\), we can obtain Equation (19) of the LQA CTP power expression for \(P_{1}^{*}\) versus \(\tilde{\gamma}_{1}[\text{dB}]\) instead of \(P_{2}^{*}\) versus \(\tilde{\gamma}_{2}[\text{dB}]\) by exchanging the roles of CTP1 and CTP2, that is, \(P_{2}^{q*}\to P_{1}^{q*}\), \(\tilde{\gamma}_{2}[\text{dB}]\rightarrow\tilde{\gamma}_{1}[\text{dB}]\), \(\tilde{\gamma}_{1}[\text{dB}]\rightarrow\tilde{\gamma}_{2}[\text{dB}]\), \(\tilde{\gamma}_{2}^{bd}[\text{dB}]\rightarrow\tilde{\gamma}_{1}^{bd}[\text{ dB}]\), and \(\hat{\beta}\left(\tilde{\gamma}_{1}\right)\rightarrow\hat{\beta}\left(\tilde{ \gamma}_{2}\right)\) in (15). \[P_{1}^{q*} = \left\{-\frac{\hat{\beta}\left(\bar{\gamma}_{2}\right)}{\bar{\gamma }_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]}-\frac{0.5P_{S}}{\left(\bar{ \gamma}_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]\right)^{2}}\right\} \bar{\gamma}_{1}[\text{dB}]^{2}\] \[+\left\{\hat{\beta}\left(\bar{\gamma}_{2}\right)+\frac{2\hat{ \beta}\left(\bar{\gamma}_{2}\right)\bar{\gamma}_{2}[\text{dB}]}{\bar{\gamma} _{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]}+\frac{P_{S}\bar{\gamma}_{2 }[\text{dB}]}{\left(\bar{\gamma}_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{ dB}]\right)^{2}}\right\}\bar{\gamma}_{1}[\text{dB}]\] \[+\left\{\frac{\hat{\beta}\left(\bar{\gamma}_{2}\right)}{\bar{ \gamma}_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]}+\frac{0.5P_{S}}{ \left(\bar{\gamma}_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]\right)^{2 }}\right\}\bar{\gamma}_{1}^{bd}[\text{dB}]^{2}\] \[-\left\{\hat{\beta}\left(\bar{\gamma}_{2}\right)+\frac{2\hat{ \beta}\left(\bar{\gamma}_{2}\right)\bar{\gamma}_{2}[\text{dB}]}{\bar{\gamma} _{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{dB}]}+\frac{P_{S}\bar{\gamma}_{2 }[\text{dB}]}{\left(\bar{\gamma}_{1}^{bd}[\text{dB}]-\bar{\gamma}_{2}[\text{ dB}]\right)^{2}}\right\}\bar{\gamma}_{1}^{bd}[\text{dB}].\]  
We also obtain the LLA CTP power expression for \(P_{1}^{*}\) versus \(\bar{\gamma}_{1}[\text{dB}]\) as follows:  
\[P_{1}^{l*}=\beta^{\prime}\left(\bar{\gamma}_{2}\right)\left(\ln\bar{\gamma}_ {1}-\ln\bar{\gamma}_{2}\right)+0.5P_{S}. \tag{20}\]  
In Figure 2a-c, the dotted and dashed lines, respectively, indicate the CTP2 powers when the LQA and LLA CTP power expressions are used. From Figure 2a-c, we see that the CTP2 power using the LQA CTP power expression is closer to the corresponding optimum CTP power over a wider Co2P JDT region around the balanced point than that using the LLA CTP power expression.