## 4 Joint Power Allocation for Co3P JDT
\[\left\{{{P_{1}^{*}}^{2}}^{\prime}{\hat{\gamma}}_{1}-\frac{{P_{1}^{*}} ^{2}{\tilde{\gamma}}_{1}\left({P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}-{P_{2}^{* }}^{\prime}{\tilde{\gamma}}_{2}\right)}{{P_{1}^{*}}^{*}{\tilde{\gamma}}_{1}-{P _{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}}-\frac{{P_{1}^{*}}^{2}{\tilde{\gamma}}_ {1}\left({P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}-{\tilde{\gamma}}_{3}\right) }{{P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}-{P_{3}^{*}}^{\prime}{\tilde{\gamma }}_{3}}-{P_{1}^{*}}^{\prime}\right\}\] \[\cdot\frac{\log_{2}\left(e\right){\tilde{\gamma}}_{1}}{\left({P _{1}^{*}}{\tilde{\gamma}}_{1}-{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}\right) \left({P_{1}^{*}}{\tilde{\gamma}}_{1}-{P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{ 3}\right)}{{P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}-{P_{3}^{*}}^{\prime}{ \tilde{\gamma}}_{3}}\right)}^{\frac{1}{{P_{1}^{*}}{\tilde{\gamma}}_{1}}}{E_{ 1}\left(\frac{1}{{P_{1}^{*}}{\tilde{\gamma}}_{1}}\right)}\] \[+\left\{{P_{2}^{*}}^{2}{}^{\prime}{\tilde{\gamma}}_{2}-\frac{{P_ {2}^{*}}^{2}{\tilde{\gamma}}_{2}}{{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}-{P _{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}}\right)}{{P_{3}^{*}}^{\prime}{\tilde{ \gamma}}_{2}-{\tilde{P}_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}}-\frac{{P_{2}^{ *}}^{2}{\tilde{\gamma}}_{2}\left({P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}-{ \tilde{\gamma}}_{3}\right)}{{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}-{P_{3}^ {*}}^{\prime}{\tilde{\gamma}}_{3}}-{P_{2}^{*}}^{\prime}\right\}\] \[\cdot\frac{\log_{2}\left(e\right){\tilde{\gamma}}_{2}}{\left({P _{2}^{*}}{\tilde{\gamma}}_{2}-{P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}\right) \left({P_{2}^{*}}{\tilde{\gamma}}_{2}-{P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{ 3}\right)}{{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}}\] \[+\left\{{2P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{3}-\frac{{P_{3}^{ *}}^{2}{\tilde{\gamma}}_{3}\left({\tilde{\gamma}}_{3}-{\tilde{P}_{1}^{*}}^{ \prime}{\tilde{\gamma}}_{1}\right)}{{P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{3}-{ P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}}-\frac{{P_{3}^{*}}^{2}{\tilde{\gamma}}_{3} \left({\tilde{\gamma}}_{3}-{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2}\right)}{{P _{3}^{*}}^{\prime}{\tilde{\gamma}}_{3}-{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{ 2}}-1\right\} \tag{23}\] \[\cdot\frac{\log_{2}\left(e\right){\tilde{\gamma}}_{3}}{\left({P _{3}^{*}}{\tilde{\gamma}}_{3}-{P_{1}^{*}}^{\prime}{\tilde{\gamma}}_{1}\right) \left({P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{3}-{P_{2}^{*}}^{\prime}{\tilde{ \gamma}}_{2}\right)}{{P_{3}^{*}}^{\prime}{\tilde{\gamma}}_{3}}\right.\] \[+\frac{\log_{2}\left(e\right){P_{1}^{*}}^{\prime}{P_{2}^{*}}}{ \left({P_{1}^{*}}{\tilde{\gamma}}_{1}-{P_{2}^{*}}^{\prime}{\tilde{\gamma}}_{2} \right)\left({P_{1}^{*}}{\tilde{\gamma}}_{1}-{P_{3}^{*}}^{\prime}{\tilde{\gamma }}_{3}\right)}+\frac{\log_{2}\left(e\right){P_{2}^{*}}^{\prime}{P_{2}^{*}}{ \tilde{\gamma}}_{2}^{2}}{\left({P_{2}^{*}}{\tilde{\gamma}}_{2}-{P_{1}^{*}}^{ \prime}{\tilde{\gamma}}_{1}\right)\left({P_{2}^{*}}{\tilde{\gamma}}_{2}-{P_{3}^ {*}}^{\prime}{\tilde{\gamma}}_{3}\right)}\] \[+\frac{\log_{2}\left(e\right)P_{2}^{2}\bar{\gamma}_{2}^{2}}{\left( P_{2}\bar{\gamma}_{2}-P_{1}\bar{\gamma}_{1}\right)\left(P_{2}\bar{\gamma}_{2}-P_{3} \bar{\gamma}_{3}\right)}e^{\frac{1}{P_{2}\bar{\gamma}_{2}}}E_{1}\left(\frac{1} {P_{2}\bar{\gamma}_{2}}\right)\]0  
In Equation (23), \({P_{1}^{*}}\), \({P_{2}^{*}}\), and \({P_{3}^{*}}\) denote the respective optimum CTP powers to be allocated to CTP1, CTP2, and CTP3 for Co3P JDT; and \((\cdot)^{\prime}\) denotes \(d(\cdot)/dP_{3}^{*}\). At this point, we again introduce a simple CTP power expression for use in practice. The solid lines in Figure 3a-c show \({P_{3}^{*}}\) versus \({\tilde{\gamma}}_{3}\)[dB], with \({\tilde{\gamma}}_{1}\)[dB] and \({\tilde{\gamma}}_{2}\)[dB] as parameters, in the condition \({\tilde{\gamma}}_{1}\geq{\tilde{\gamma}}_{2}\geq{\tilde{\gamma}}_{3}\). In this study, the solid lines were obtained using Equation (23) by exhaustive search. From these solid lines, we see that \({P_{3}^{*}}\) is almost linear with respect to \({\tilde{\gamma}}_{3}\)[dB] over the corresponding Co3P JDT region.