## 2 System Model
### The First Scenario  
The system has two hops from source to IRS and from IRS to destinations. These links are demonstrated in the system model, i.e., \(h_{S_{1},n}\), \(h_{1}D_{1},n\), \(h_{S_{1},n}\) and \(h_{I_{2}D_{1},n}\), along with their channel gains, denoted as \(h_{S_{1},n}=d_{S_{1}}^{-\alpha/2}\bar{h}_{S_{1},n}e^{-j\phi_{1n}}\), \(h_{I_{1}D_{1},n}=d_{I_{1}D_{1}}^{-\alpha/2}\bar{h}_{I_{1}D_{1},n}e^{-j\delta_{ I_{1}}^{(D_{1})}}\), \(h_{S_{1},n}=d_{S_{1}}^{-\alpha/2}\bar{h}_{S_{1},n}e^{-j\phi_{2n}}\) and \(h_{I_{2}D_{1},n}=d_{I_{2}D_{1}}^{-\alpha/2}\bar{h}_{I_{2}D_{1}}^{-\alpha/2} \bar{h}_{I_{2}D_{1},n}e^{-j\delta_{2n}^{(D_{1})}}\)[35]; the links \(S-I_{1}\), \(I_{1}-D_{1}\), \(S-I_{2}\) and \(I_{2}-D_{1}\) are associated with distances \(d_{S_{1}}\), \(d_{I_{1}D_{1}}\), \(d_{S_{1}}\) and \(d_{I_{2}D_{1}}\), respectively. Here, the path-loss coefficient is \(\alpha\), while \(\phi_{1n}\) and \(\phi_{2n}\) and \(\delta_{1n}^{(D_{1})}\) and \(\delta_{2n}^{(D_{1})}\) are the phases of the channel gains. \(\bar{h}_{S_{1},n}\), \(\bar{h}_{I_{1}D_{1},n}\), \(\bar{h}_{S_{1},n}\) and \(\bar{h}_{I_{2}D_{1},n}\) are the amplitudes of the channel gains, and they are adopted with a Rayleigh distribution. In the scope of our paper, it is reasonable to assume that the IRS achieves the full knowledge of the channels \(h_{S_{1},n}\), \(h_{I_{1}D_{1},n}\), \(h_{S_{1},n}\) and \(h_{I_{2}D_{1},n}\). The overall received signal at user \(D_{1}\) is given as [36]  
\[y_{D_{1}}^{(1)}=\Bigg{(}\sum_{n=1}^{N_{1}}h_{S_{1},n}h_{I_{1}D_{1},n}\eta_{1n }e^{i\theta_{1n}^{(D_{1})}}+\sum_{n=1}^{N_{2}}h_{S_{1},n}h_{I_{2}D_{1},n}\eta_ {2n}e^{i\theta_{2n}^{(D_{1})}}\Bigg{)}\big{(}\sqrt{P\chi_{1}}x_{1}+\sqrt{P \chi_{2}}x_{2}+\tau_{S}\big{)}+\tau_{D_{2}}+n_{D_{1}}, \tag{2}\]  
where \(\theta_{1n}^{(D_{1})}\) is the adjustable phase applied by the \(n\)th reflecting element of \(S-I_{1}-D_{1}\), \(\theta_{2n}^{(D_{1})}\) stands for the adjustable phase applied by the \(n\)th reflecting element of \(S-I_{2}-D_{1}\), \(\eta_{1n}\) is the reflection coefficient of \(I_{1}\) with \(\eta_{1n}\in(0,1]\), and \(\eta_{2n}\) represents the reflection coefficients of \(I_{2}\) with \(\eta_{2n}\in(0,1]\). The hardware impairment at the \(D_{1}\) is denoted as [32]  
\[\tau_{D_{21}}\sim CN\Bigg{(}0,Y_{D_{1}}^{2}P\Bigg{|}\sum_{n=1}^{N_{1}}h_{S_{1 },n}h_{I_{1}D_{1},n}\eta_{1n}e^{i\theta_{1n}^{(D_{1})}}+\sum_{n=1}^{N_{2}}h_{S _{1},n}h_{I_{2}D_{1},n}\eta_{2n}e^{i\theta_{2n}^{(D_{1})}}\Bigg{|}^{2}\Bigg{)}. \tag{3}\]  
The received signal to noise plus distortion ratio (SNDR) at the destination \(D_{1}\) is defined as  
\[\gamma_{D_{1}}^{(1)}=\frac{\gamma_{\chi 1}\Bigg{|}A_{1}^{(1)}e^{ \frac{j\big{(}\theta_{1n}^{(D_{1})}-\phi_{1n}-\theta_{1n}^{(D_{1})}\big{)}}{+A_ {2}^{(1)}e^{j\big{(}\theta_{2n}^{(D_{1})}-\phi_{2n}-\theta_{2n}^{(D_{1})}\big{)} }}\Bigg{|}^{2}}}{\Bigg{|}A_{1}^{(1)}e^{\frac{j\big{(}\theta_{1n}^{(D_{1})}- \phi_{1n}-\theta_{1n}^{(D_{1})}\big{)}}{+A_{2}^{(1)}e^{j\big{(}\theta_{2n}^{(D_ {1})}-\phi_{2n}-\theta_{2n}^{(D_{1})}\big{)}}}\Bigg{|}^{2}+\big{(}Y_{S}^{2}+Y_{ D_{1}}^{2}\big{)}\gamma\Bigg{|}A_{1}^{(1)}e^{\frac{j\big{(}\theta_{1n}^{(D_{1})}- \phi_{1n}-\theta_{1n}^{(D_{1})}\big{)}}{+A_{2}^{(1)}e^{j\big{(}\theta_{2n}^{(D_ {1})}-\phi_{2n}-\theta_{2n}^{(D_{1})}\big{)}}}\Bigg{|}^{2}+A_{2}^{(1)}e^{j \big{(}\theta_{2n}^{(D_{1})}-\phi_{2n}-\theta_{2n}^{(D_{1})}\big{)}}\Bigg{|}^{ 2}_{+1}}, \tag{4}\]  
where \(\gamma=\frac{P}{\sigma_{n}^{2}}\) is the average signal-to-noise ratio (SNR) at the source, \(A_{1}^{(1)}=d_{S_{1}}^{-\alpha/2}d_{I_{1}D_{1}}^{-\alpha/2}\)  
\(\sum_{n=1}^{N_{1}}\bar{h}_{S_{1},n}\bar{h}_{I_{1}D_{1},n}\eta_{1n}\), \(A_{2}^{(1)}=d_{S_{2}}^{-\alpha/2}d_{I_{2}D_{1}}^{-\alpha/2}\sum_{n=1}^{N_{2}} \bar{h}_{S_{1},n}\bar{h}_{I_{2}D_{1},n}\eta_{2n}\), \(\eta_{1n}=\eta,\forall_{1n}\) and \(\eta_{2n}=\eta,\forall_{2n}\). It is noted that from (4) that the highest value of \(\gamma_{D_{1}}^{(1)}\) can be obtained by disregarding the channel phases.