Y\_[r,i]{}&=&\_[t=1]{}\^[3]{}h\_[tr]{}X\_[t,i]{}+Z\_[r,i]{}\[eqn:Gaussian\_model\]

where $h_{tr}$, for $t,r\in\{1,2,3\}$, are known channel gains. $X_{t,i}$ is the input signal with average power constraint:

[rcl]{} \_[i=1]{}\^n(x\_[t,i]{})\^2P\_t\[eqn:power\_cons\]

for $t\in\{1,2,3\}$. $Z_{r,i},\: r\in\{1,2,3\}$ is an independent and identically distributed (i.i.d) zero mean Gaussian noise component with unit power, i.e., $Z_{r,i}\sim{\mathcal{N}}(0,1)$.

![Gaussian three-user C-IFC.[]{data-label="fig:Gauschannelmodel"}](fig_gaus.eps){width="9cm"}

Now, we extend the results of Theorem \[thm:cap\_set2\], i.e., ${\mathcal{C}}_2$ and *Set2*, to the Gaussian case. The strong interference conditions of *Set2*, i.e., (\[eqn:set1\_I\]), (\[eqn:set2\_II\])-(\[eqn:set2\_IV\]) for the above Gaussian model, respectively, become ($Set_G$):

[rcl]{} h\_[33]{}\^2&&{h\_[31]{}\^2,h\_[32]{}\^2}\[eqn:set1\_I\_Gaus\]\
h\_[11]{}\^2P\_1+h\_[31]{}\^2P\_3(1-\_2\^2)+2h\_[11]{}h\_[31]{}\_1&&{A\_[12]{},h\_[13]{}\^2P\_1+h\_[33]{}\^2P\_3(1-\_2\^2)+2h\_[13]{}h\_[33]{}\_1}\
&&\[eqn:set1\_II\_Gaus\]\
h\_[22]{}\^2P\_2+h\_[32]{}\^2P\_3(1-\_1\^2)+2h\_[22]{}h\_[32]{}\_2&&{A\_[21]{},h\_[23]{}\^2P\_1+h\_[33]{}\^2P\_3(1-\_1\^2)+2h\_[23]{}h\_[33]{}\_2}\
&&\[eqn:set1\_III\_Gaus\]\
{A\_[12]{}B\_[12]{}}&&{A\_[21]{}B\_[21]{}}\[eqn:set1\_IV\_Gaus\]

where $-1\leq\rho_u\leq 1$ is the correlation coefficient between $X_u$ and $X_3$, i.e., $E(X_uX_3)=\rho_u\sqrt{P_uP_3}$ for $u\in\{1,2\}$, and $A_{ij}$ and $B_{ij}$ are defined as,

[rcl]{} A\_[ij]{}&=&\
B\_[ij]{}&=&

for $i,j\in \{1,2\}$.

\[thm:Gaus\_cap\_set1\] For the Gaussian three-user C-IFC, satisfying conditions (\[eqn:set1\_I\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]), the capacity region is given by

[rl]{} \_1\^G =\_[-1\_1,\_21:\_1\^2+\_2\^21 ]{} {&(R\_1,R\_2,R\_3): R\_1,R\_2,R\_3 0\
&R\_3(h\_[33]{}\^2P\_3(1-\_1\^2-\_2\^2))\[eqn:Gaus\_Cap1\_I\]\
&R\_[1]{}+R\_[3]{}(h\_[11]{}\^2P\_1+h\_[31]{}\^2P\_3(1-\_2\^2)+2h\_[11]{}h\_[31]{}\_1)\[eqn:Gaus\_Cap1\_II\]\
&R\_[2]{}+R\_[3]{}(h\_[22]{}\^2P\_2+h\_[32]{}\^2P\_3(1-\_1\^2)+2h\_[22]{}h\_[32]{}\_2)}\[eqn:Gaus\_Cap1\_III\]

where to simplify notation we define $$\label{eqn:theta}
\theta(x)\doteq \frac{1}{2}\log(1+x).$$

\[remark:Gaus\_cap\_set1\_I\] Condition (\[eqn:set1\_I\_Gaus\]) implies that Tx3 causes strong interference at Rx1 and Rx2. This fact enables Rx1 and Rx2 to decode $m_3$. Moreover, due to the first terms in the RHS of (\[eqn:set1\_II\_Gaus\]) and (\[eqn:set1\_III\_Gaus\]), $m_1$ and $m_2$ can be decoded at Rx2 and Rx1, respectively. Also, (\[eqn:set1\_II\_Gaus\])-(\[eqn:set1\_IV\_Gaus\]) provides strong interference conditions at Rx3, 