of relays $R_r$ and $R_t$, there exist $P_{S}$ and $P_{R_t}$ such that $\mathbb{I}(R_t R_r) =0$, $\text{SNR}_{R_t D} \geq \gamma_0$ and $\text{SINR}_{S R_r} \geq \gamma_0$. The minimum power levels $P_{S}^*$ and $P_{R_t}^*$ are achieved when $\text{SNR}_{R_t D} = \text{SINR}_{S R_r} = \gamma_0$, and are given by

\[eq:P\_optimal\] $$\begin{aligned}
P_{S}^* &= \frac{ \gamma_0 n_{R_r}}{g_{S R_r}} \; , \label{eq:P_S_optimal} \\
P_{R_t}^* &=\max \left\{ \frac{\gamma_0 n_D}{g_{R_t D}} , \frac{n_{R_r}\gamma_0 (\gamma_0 + 1)}{g_{R_t R_r}} \right\} \; \label{eq:P_Rt_optimal}.\end{aligned}$$

For IC to take place, according to , we have $$\begin{aligned}
\label{eq:index2}
{g_{R_t R_r} P_{R_t}} \geq \gamma_0({g_{S R_r}P_{S} +  n_{R_r}})\end{aligned}$$ Given that $P_{R_t}$ is chosen such that is fulfilled, then becomes $$\frac{g_{S R_r}P_{S}}{n_{R_r}} \geq  \gamma_0\; ,$$ and since $P_S$ decreases monotonically with $P_{R_t}$ (see equation  for $\mathbb{I}(R_t R_r) =0$), while it requires a minimum value such that equation  is fulfilled, the minimum power of $S$ is given with equality; i.e., $$\begin{aligned}
P_{S}^* = \frac{ \gamma_0 n_{R_r}}{g_{S R_r}} \; .\end{aligned}$$ Substituting into we have ${g_{R_t R_r} P_{R_t}} \geq n_{R_r}\gamma_0 (\gamma_0 + 1)$. Hence, the minimum $P_{R_t}$ is given by $$\begin{aligned}
P_{R_t}^* =\max \left\{ \frac{\gamma_0 n_D}{g_{R_t D}} , \frac{n_{R_r}\gamma_0 (\gamma_0 + 1)}{g_{R_t R_r}} \right\} \; .\end{aligned}$$

Proposition \[prop:1\] provides the minimum power levels of $S$ and $R_t$, provided that their 