## 2 System Model
### The Second Scenario (the Benchmark)  
The overall received signal at the user \(D_{1}\) is given as [33; 36]  
\[ \begin{array}{l}y_{D_{1}}^{(2)}=\left(\sum\limits_{n=1}^{N_{1}}h_{S_{1},n}h_ {I_{1}D_{2},n}\eta_{1n}e^{i\theta_{1n}^{(D_{1})}}+\sum\limits_{n=1}^{N_{2}}h_{S _{1},n}h_{I_{1}D_{2},n}\eta_{2n}e^{i\theta_{2n}^{(D_{1})}}+\sum\limits_{n=1}^{ N_{3}}h_{S_{1},n}h_{I_{3}D_{2},n}\eta_{3n}e^{i\theta_{3n}^{(D_{1})}}\right)(\sqrt{P \chi_{1}}x_{1}+\sqrt{P\chi_{2}}x_{2}+\tau_{5})\\ +\tau_{D_{2}}+n_{D_{1}},\end{array}  \tag{13}\]  
where \(\tau_{D_{31}}\sim CN\left(0,Y_{D_{1}}^{2}P\left|\sum\limits_{n=1}^{N_{1}}h_{S _{1},n}h_{I_{1}D_{1},n}\eta_{1n}e^{i\theta_{1n}^{(D_{1})}}\right.\right.+\sum \limits_{n=1}^{N_{2}}h_{S_{1},n}h_{I_{2}D_{1},n}\eta_{2n}e^{i\theta_{2n}^{(D_{ 1})}}\)  
\(+\sum\limits_{n=1}^{N_{3}}h_{S_{1},n}h_{I_{3}D_{2},n}\eta_{3n}e^{i\theta_{3n}^{ (D_{2})}}\left|\vphantom{\sum\limits_{n=1}^{N_{3}}h_{S_{1},n}h_{I_{3}D_{2},n} \eta_{3n}e^{i\theta_{3n}^{(D_{2})}}}\right|^{2}\Biggr{)}\). In addition, \(h_{S_{1},n}\) and \(h_{I_{3}D_{1},n}\) are the channel gains with \(h_{S_{1},n}=d_{S_{1}}^{-\alpha/2}\bar{h}_{S_{1},n}e^{-i\phi_{3n}}\), \(h_{I_{3}D_{1},n}=d_{S_{1}}^{-\alpha/2}\bar{h}_{I_{3}D_{1},n}e^{-i\phi_{3n}^{(D _{1})}}\)[35], where \(d_{S_{1}}\) and \(d_{I_{3}D_{1}}\) are the distances for the \(S-I_{3}\) link, \(I_{3}-D_{1}\) link, respectively, and \(\phi_{3n}\) and \(\delta_{3n}^{(D_{1})}\) are the phases of the channel gains. \(\bar{h}_{S_{1},n}\) and \(\bar{h}_{I_{3}D_{1},n}\) following a Rayleigh distribution are the amplitudes of the channel gains. Moreover, we assume that the IRS has full knowledge of the channel phases of \(h_{S_{1},n}\), \(h_{I_{3}D_{1},n}\). Similar to (4), the received SNDR at the destination \(D_{1}\) is defined as  
\[ \begin{array}{l}\gamma_{D_{1}}^{(2)}=\dfrac{\gamma\chi_{1}\left|A_{1}^{(1)}+ A_{2}^{(1)}+A_{3}^{(1)}e^{\left\{\theta_{3n}^{(D_{1})}-\phi_{3n}-\delta_{3n}^{(D_{1})} \right\}}\right|^{2}}\\ \gamma\chi_{2}\left|A_{1}^{(1)}+A_{2}^{(1)}+A_{3}^{(1)}e^{i\left(\theta_{3n}^{ (D_{1})}-\phi_{3n}-\delta_{3n}^{(D_{1})}\right)}\right|^{2}+\left(Y_{5}^{2}+Y_ {D_{1}}^{2}\right)\gamma\left|A_{1}^{(1)}+A_{2}^{(1)}+A_{3}^{(1)}e^{i\left( \theta_{3n}^{(D_{1})}-\phi_{3n}-\delta_{3n}^{(D_{1})}\right)}\right|^{2}+1 \end{array} ,\end{array} \tag{14}\]  
\[ \begin{array}{l}\text{where }A_{3}^{(1)}=d_{S_{1}}^{-\alpha/2}d_{S_{1}}^{- \alpha/2}\sum\limits_{n=1}^{N_{3}}\bar{h}_{S_{1},n}\bar{h}_{I_{3}D_{1},n}\eta_ {3n}\eta_{3n}=\eta,\forall_{3n}.\end{array} \]  
In (14), to obtain the maximum value of \(\gamma_{D_{1}}^{(2)}\), we need to eliminate the channel phases. Similar to the method in [35], by setting the phases \(\theta_{3n}^{(D_{1})}=\phi_{3n}+\delta_{3n}^{(D_{1})}\) for \(3n=1,\ldots,N_{3}\), the maximal \(\gamma_{D_{1}}^{(2)}\) can be written as  
\[ \begin{array}{l}\gamma_{D_{1}}^{(2)}=\dfrac{\gamma\chi_{1}\left|A_{1}^{(1)}+ A_{2}^{(1)}+A_{3}^{(1)}\right|^{2}}{\gamma\chi_{2}\left|A_{1}^{(1)}+A_{2}^{(1)}+A_{3}^{(1)} \right|^{2}+\left(Y_{5}^{2}+Y_{D_{1}}^{2}\right)\gamma\left|A_{1}^{(1)}+A_{2}^ {(1)}+A_{3}^{(1)}\right|^{2}+1}.\end{array}  \tag{15}\]  
The overall received signal at the user \(D_{2}\) is given as [33; 36]  
\[ \begin{array}{l}y_{D_{2}}^{(2)}=\left(\sum\limits_{n=1}^{N_{1}}h_{S_{1},n}h_ {I_{1}D_{2},n}\eta_{1n}e^{i\theta_{1n}^{(D_{2})}}+\sum\limits_{n=1}^{N_{2}}h_{S _{1},n}h_{I_{1}D_{2},n}\eta_{2n}e^{i\theta_{2n}^{(D_{2})}}+\sum\limits_{n=1}^{N_{ 1}}h_{S_{1},n}h_{I_{3}D_{2},n}\eta_{3n}e^{i\theta_{3n}^{(D_{2})}}\right)(\sqrt{P \chi_{1}}x_{1}+\sqrt{P\chi_{2}}x_{2}+\tau_{5})\\ +\tau_{D_{2}}+n_{D_{2}},\end{array}  \tag{16}\]  
\[ \begin{array}{l}\text{where }\tau_{D_{32}}\sim CN\left(0,Y_{D_{2}}^{2}P\left| \sum\limits_{n=1}^{N_{1}}h_{S_{1},n}h_{I_{1}D_{2},n}\eta_{1n}e^{i\theta_{1n}^{(D _{2})}}+\sum\limits_{n=1}^{N_{2}}h_{S_{1},n}h_{I_{2}D_{2},n}\eta_{2n}e^{i\theta_ {2n}^{(D_{2})}}\right.\\ +\left.\sum\limits_{n=1}^{N_{3}}h_{S_{1},n}h_{I_{3}D_{2},n}\eta_{3n}e^{i\theta_ {3n}^{(D_{2})}}\right|^{2}\right)\cdot\theta_{3n}^{(D_{2})}\end{array}