## 3 Joint Power Allocation for Co2P JDT under a Non-Unity TCPP  
We formulate the problem of JPA for Co2P JDT and Co3P JDT under a TCPP constraint, to maximize the ECC in Rayleigh fading channels. This problem can be expressed as  
\[\max_{P_{i}} \tilde{C}_{jdt},\] (3a) s.t. \[\sum_{i=1}^{M}P_{i}\leq P_{S}, \tag{3b}\] \[0\leq P_{i}\leq P_{S},\ i=1,\cdots,M, \tag{3c}\]  
where \(\tilde{C}_{jdt}\) denotes the ECC achieved by the CoMP JDT network; and \(M\) denotes the number of CTPs in the JDT cooperating set. It should be noted that \(\tilde{C}_{jdt}\) is the single-branch ergodic capacity in the case of non-cooperative transmission (i.e., non-JDT mode) and the ECC in the case of cooperative transmission (i.e., CoMP JDT mode). In Rayleigh fading channels, the instantaneous diversity-combined signal-to-noise ratio in Equation (1) is the sum of two independent exponential random variables \(P_{1}\gamma_{1}\) and \(P_{2}\gamma_{2}\). Hence, the probability density function by the Co2P JDT network can be expressed as  
\[f_{\tau_{co2}}\left(\gamma_{co2}\right)=\left\{ \begin{array}{ll}\epsilon \frac{-\frac{\gamma_{co2}}{P_{1}\gamma_{1}}-\frac{\gamma_{co2}}{P_{2}\gamma_{2 }}}{\bar{P}_{1}\gamma_{1}-P_{2}\gamma_{2}},&\text{if }P_{1}\bar{\gamma}_{1}\
eq P_{2}\bar{ \gamma}_{2},\\ \frac{\gamma_{co2}}{P_{1}^{2}\gamma_{1}^{2}}e^{-\frac{\gamma_{co2}}{P_{1}\gamma _{1}}},&\text{if }P_{1}\bar{\gamma}_{1}=P_{2}\bar{\gamma}_{2},\end{array} \right. \tag{4}\]  
Then, the ECC achieved by the Co2P JDT network is given by Equation (5) [8]  
\[C_{2,jdt}=\left\{ \begin{array}{ll}\frac{\log_{2}(e)}{P_{1}\bar{\gamma}_{1}- P_{2}\bar{\gamma}_{2}}\left\{P_{1}\bar{\gamma}_{1}e^{\frac{1}{P_{1}\bar{ \gamma}_{1}}}E_{1}\left(\frac{1}{P_{1}\bar{\gamma}_{1}}\right)-P_{2}\bar{\gamma }_{2}e^{\frac{1}{P_{2}\bar{\gamma}_{2}}}E_{1}\left(\frac{1}{P_{2}\bar{\gamma}_ {2}}\right)\right\},&\text{if }P_{1}\bar{\gamma}_{1}\
eq P_{2}\bar{\gamma}_{2},\\ \log_{2}(e)\left\{1-\left(\frac{1}{P_{1}\bar{\gamma}_{1}}-1\right)e^{\frac{1}{ P_{1}\bar{\gamma}_{1}}}E_{1}\left(\frac{1}{P_{1}\bar{\gamma}_{1}}\right)\right\},& \text{if }P_{1}\bar{\gamma}_{1}=P_{2}\bar{\gamma}_{2},\end{array} \right. \tag{5}\]  
where \(E_{1}(\cdot)\) denotes the first-order exponential integral function [13]. Figure 1: CoMP JDT networks: (a) Co2P case (b) Co3P case. To calculate the optimum CTP powers to maximize the ECC, substituting \(P_{1}=P_{S}-P_{2}\) into the top of Equation (5), differentiating the resulting ECC expression with respect to \(P_{2}\), and setting the derivative to zero, we obtain the following expression:  
\[\frac{\log_{2}\left(e\right)}{\left(P_{S}-P_{2}^{*}\right)\bar{ \gamma}_{1}-P_{2}^{*}\bar{\gamma}_{2}}\left\{\left(\bar{\gamma}_{2}-\frac{1}{P _{2}^{*}}\right)e^{\frac{1}{P_{2}^{*}\bar{\gamma}_{2}}}E_{1}\left(\frac{1}{P_ {2}^{*}\bar{\gamma}_{2}}\right)\right.\] \[\left.+\left(\bar{\gamma}_{1}-\frac{1}{P_{S}-P_{2}^{*}}\right)e^{ \frac{1}{\left(P_{S}-P_{2}^{*}\right)\bar{\gamma}_{1}}}E_{1}\left(\frac{1}{ \left(P_{S}-P_{2}^{*}\right)\bar{\gamma}_{1}}\right)+\bar{\gamma}_{1}+\bar{ \gamma}_{2}\right\} \tag{6}\] \[+\frac{\left(\bar{\gamma}_{1}+\bar{\gamma}_{2}\right)\log_{2} \left(e\right)}{\left\{\left(P_{S}-P_{2}^{*}\right)\bar{\gamma}_{1}-P_{2}^{* }\bar{\gamma}_{2}\right\}^{2}}\left\{P_{2}^{*}\bar{\gamma}_{2}e^{\frac{1}{P_ {2}^{*}\bar{\gamma}_{2}}}E_{1}\left(\frac{1}{P_{2}^{*}\bar{\gamma}_{2}}\right)\right.\] \[-\left.\left(P_{S}-P_{2}^{*}\right)\bar{\gamma}_{1}e^{\frac{1}{ \left(P_{S}-P_{2}^{*}\right)\bar{\gamma}_{1}}}E_{1}\left(\frac{1}{\left(P_{S} -P_{2}^{*}\right)\bar{\gamma}_{1}}\right)\right\}=0.\]  
Here, \(P_{1}^{*}\) and \(P_{2}^{*}\) denote the respective optimum CTP powers to be allocated to CTP1 and CTP2 for Co2P JDT. Because Equation (6) contains several complicated operations such as exponentials, exponential integrals, and their products, we can solve Equation (6) by only using numerical techniques. Thus, the optimum CTP powers for Co2P JDT might not be used in practice owing to the high complexity of implementation.