## 3 Joint Power Allocation for Co2P JDT under a Non-Unity TCPP
At this point, we introduce a much simpler CTP power expression for use in practice. The solid lines in Figure 2a-c show \(P_{2}^{*}\) versus \(\bar{\gamma}_{2}\)[dB], with \(\bar{\gamma}_{1}\)[dB] as a parameter, when \(P_{S}\) = 2, 4, and 8 W, respectively. In this study, we obtained the solid lines in Figure 2a-c from Equation (6) by using the bisection method [14]. From these solid lines, we see that in the condition \(\bar{\gamma}_{1}\geq\bar{\gamma}_{2}\), \(P_{2}^{*}\) is almost quadratic with respect to \(\bar{\gamma}_{2}\)[dB] over the corresponding Co2P JDT region (i.e., the region from the boundary point dividing the Co2P JDT mode and the non-JDT mode, \(\bar{\gamma}_{2}^{bd}\) to the balanced point of \(P_{2}^{*}=P_{1}^{*}\) at \(\bar{\gamma}_{2}=\bar{\gamma}_{1}\)). Therefore, we use an LQA CTP power expression for \(P_{2}^{*}\) versus \(\bar{\gamma}_{2}\)[dB], which is a quadratic line connecting the respective optimum CTP2 powers at the boundary and balanced points, while satisfying the slope at the balanced point. To derive the LQA CTP power expression in the condition \(\tilde{\gamma}_{1}\geq\tilde{\gamma}_{2}\), we first derive the boundary point. Given \(\tilde{\gamma}_{1}\), we obtain Equation (7) after substituting the boundary point condition (i.e., \(P_{2}^{*}=0\)) into Equation (6). \[ \begin{split}&-\frac{1}{P_{\mathrm{S}}\tilde{\gamma}_{1}}\left\{ \lim_{P_{2}^{*}\to 0}\gamma_{2}^{bd}e^{\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}}E_{1} \left(\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}\right)-\lim_{P_{2}^{*}\to 0} \frac{1}{P_{2}^{*}}e^{\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}}E_{1}\left( \frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}\right)\right.\\ &+\left(\tilde{\gamma}_{1}-\frac{1}{P_{\mathrm{S}}}\right)e^{ \frac{1}{P_{\mathrm{S}}\tilde{\gamma}_{1}}}E_{1}\left(\frac{1}{P_{\mathrm{S}} \tilde{\gamma}_{1}}\right)+\gamma_{2}^{bd}+\gamma_{1}\right\}\\ &\left.+\frac{\tilde{\gamma}_{2}^{bd}+\tilde{\gamma}_{1}}{\left( P_{\mathrm{S}}\tilde{\gamma}_{1}\right)^{2}}\left(P_{\mathrm{S}}\tilde{ \gamma}_{1}e^{\frac{1}{P_{\mathrm{S}}\tilde{\gamma}_{1}}}E_{1}\left(\frac{1}{ P_{\mathrm{S}}\tilde{\gamma}_{1}}\right)-\lim_{P_{2}^{*}\to 0}P_{2}^{*}\tilde{\gamma}_{2}^{bd}e^{ \frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}}E_{1}\left(\frac{1}{P_{2}^{*} \tilde{\gamma}_{2}^{bd}}\right)\right\}=0.\end{split}  \tag{7}\]  
Invoking the L'Hopital's rule, we obtain the following three limiting terms as  
\[\lim_{P_{2}^{*}\to 0}\gamma_{2}^{bd}e^{\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{ bd}}}E_{1}\left(\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}\right)=0, \tag{8a}\] \[\lim_{P_{2}^{*}\to 0}\frac{1}{P_{2}^{*}}e^{\frac{1}{P_{2}^{*}\tilde{ \gamma}_{2}^{bd}}}E_{1}\left(\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{bd}}\right) =\tilde{\gamma}_{2}^{bd},\] (8b) \[\lim_{P_{2}^{*}\to 0}P_{2}^{*}\tilde{\gamma}_{2}^{bd}e^{\frac{1}{P_{2}^{ *}\tilde{\gamma}_{2}^{bd}}}E_{1}\left(\frac{1}{P_{2}^{*}\tilde{\gamma}_{2}^{ bd}}\right)=0. \tag{8c}\]  
Substituting the three limiting terms in Equation (8) into Equation (7) and rearranging the terms, we obtain the boundary point as  
\[\tilde{\gamma}_{2}^{bd}=\frac{\tilde{\gamma}_{1}}{e^{\frac{1}{P_{\mathrm{S}} \tilde{\gamma}_{1}}}E_{1}\left(\frac{1}{P_{\mathrm{S}}\tilde{\gamma}_{1}} \right)}-\frac{1}{P_{\mathrm{S}}}. \tag{9}\]  
Next, we derive the slope of \(P_{2}^{*}\) with respect to \(\tilde{\gamma}_{2}\)[dB] at the balanced point. For ease of derivation, we set \(\alpha_{2}=\ln(\tilde{\gamma}_{2})\). By differentiating Equation (6) with respect to \(\alpha_{2}\), after some manipulations, we obtain Equation (10). \[\frac{dP_{2}^{*}}{d\alpha_{2}}=\frac{N_{T^{\prime}}}{D_{T^{\prime}}}, \tag{10}\]  
with  
\[