## 3 Joint Power Allocation for Co2P JDT under a Non-Unity TCPP
To obtain the LQA CTP2 power expression, we express the quadratic equation for \(P_{2}^{*}\) versus \(\tilde{\gamma}_{2}\)[dB] as \(P_{2}^{d*}=A\tilde{\gamma}_{2}\)[dB]\({}^{2}+B\tilde{\gamma}_{2}\)[dB]\(+C\). This quadratic equation must meet the following three conditions: \(P_{2}^{d*}=0\) at \(\tilde{\gamma}_{2}=\tilde{\gamma}_{2}^{bd}\) (i.e., the boundary point condition), \(P_{2}^{d*}=0.5P_{S}\) at \(\tilde{\gamma}_{2}=\tilde{\gamma}_{1}\) (i.e., the balanced point condition), and the slope at the balanced point \(\hat{\beta}\left(\tilde{\gamma}_{1}\right)\). Using these three conditions, we obtain three simultaneous equations with respect to \(A\), \(B\), and \(C\):  
\[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]0\[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]1 \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]2  
Solving these three simultaneous equations, we obtain Equation (15) of the LQA CTP power expression for \(P_{2}^{*}\) versus \(\tilde{\gamma}_{2}[\text{dB}]\). \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]3 \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]4 \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]5 \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]6 \[N_{\beta^{\prime}}= \left\{\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{ P_{S}^{3}}+\left(\frac{dP_{2}^{*}}{d\alpha_{2}}\right)^{2}\frac{64}{P_{S}^{4} \tilde{\gamma}_{1}}+\frac{dP_{2}^{*}}{d\alpha_{2}}\frac{16}{P_{S}^{2}}+\frac{ dP_{2}^{*}}{d\alpha_{2}}\frac{32}{P_{S}^{3}\tilde{\gamma}_{1}}+\frac{4}{P_{S}^{2} \tilde{\gamma}_{1}}\right\}\]7  
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 LQA CTP2 power expression of Equation (15) as  
\[\hat{P}_{2}^{*}=\left\{ \begin{array}{ll}0,&\text{if }P_{2}^{q*}<0,\\ P_{2}^{q*},&\text{if }P_{2}^{\tilde{q}*}\geq 0.\end{array} \right.