## 3 System Model
### Framework for UEs Under isIC and HWDs  
This section elaborates on the system model for a multi-user scenario, considering the iSIC and HWIs. The SNR achieved during the direct transmission and cooperative relaying of the IOS-aided C-HNOMA system is described. The received signal at UE during direct transmission is given by the following:  
\[y_{d-u}^{r,f}=\mathbf{h_{d}}\Theta^{r,f}\mathbf{g_{d-u}}+w_{u}+n_{u} \tag{2}\]  
where \(\left[\mathbf{h_{d}}\right]_{L\times 1}\) is the channel between gNB and IOS. \(\mathbf{h_{d}^{d}}=\left[\alpha_{1}e^{-j\theta_{1}},\alpha_{2}e^{-j\theta_{2} },\;\ldots,\;\alpha_{L}e^{-j\theta_{L}}\right]\), \(\alpha_{l}e^{-j\theta_{l}}\sim\mathcal{CN}(0,1),l=1,2\ldots L\), \(\mathbf{h_{d}}\sim\mathcal{CN}(0,1)\). The IOS phase shift matrix is \(\left[\Theta^{r,f}\right]_{L\times L}=diag\left(\chi_{1}^{r,f}e^{j\theta_{1}^ {r,f}},\;\chi_{2}^{r,f}e^{j\theta_{2}^{r,f}},...,\chi_{L}^{r,f}e^{j\theta_{L}^ {r,f}}\right)\), \(\theta_{l}\;e(0,2\pi]\). \(\chi_{l}^{r,f}e\left(\chi_{l}^{r},\chi_{l}^{f}\right)\), \(\chi_{l}^{r}e(0,1)\). The IOS operates in three modes: energy splitting (ES), mode switching (MS), and time splitting (TS) [14; 15]. In this context, the MS protocol is preferred due to its simpler hardware complexity compared to the ES protocol. The entire time frame is utilized for reflection/refraction in the MS protocol, leading to more efficient spectrum use compared to the TS protocol [31]. \(\chi_{l}^{r}=0\) for the UE under the refracting side of the IOS and \(\chi_{l}^{r}=1\) for the UE under the reflecting side of the IOS. Similarly, \(\chi_{l}^{f}e(0,1)\), \(\chi_{l}^{f}=1\) for the UE under the refractive side of the IOS and \(\chi_{l}^{f}=0\) for the UE under the reflecting side of the IOS. \(n_{u}\sim\mathcal{CN}\big{(}0,\sigma_{u}^{2}\big{)}\) is the noise component added at the UE. It is assumed that \(\sigma_{u}^{2}\) is common for all the UE, where \(\sigma_{1}^{2}=\sigma_{2}^{2}=...=\sigma_{M}^{2}=\sigma_{u}^{2}\). \(w_{u}\) is the hardware noise term \(w_{u}e(w_{s},\;w_{w})\). \(\left[\mathbf{g_{d-u}}\right]_{L\times 1}\) is the channel between the IOS and UE, which is given as follows:  
\[\mathbf{g_{d-u}}=\left[ \begin{array}{c}\beta_{1}e^{-j\varphi_{1}},\;\beta_{2 }e^{-j\varphi_{2}},\;\ldots,\beta_{L}e^{-j\varphi_{L}}\end{array} \right]^{H}\]  
, where \(\beta_{l}e^{-j\varphi_{l}}\sim\mathcal{CN}\big{(}0,\delta_{u}^{2}\big{)}\). The phase compensation produced via the IOS is \(\theta_{l}=-\psi_{l}-\varphi_{l}\). The cumulative channel between gNB and the UE is \(H_{u}=\mathbf{h_{d}}\Theta^{r,f}\mathbf{g_{d-u}}\). The SE achieved by the SUE during direct transmission is given by the following:  
\[\Gamma_{d-s,p}^{i}=\frac{1}{M}\text{log}_{2}\left(1+\frac{G_{s}\rho\mu_{si}}{ \varepsilon G_{s}\rho\mu_{wi}+G_{s}\rho\left[k_{g}^{2}+k_{g}^{2}\right]+1}\right) \tag{3}\]  
where  
\[G_{s}=\left\{ \begin{array}{cl}\left|H_{2i-1}\right|^{2}&;p=NN-FF\\ \left(\frac{1-\left(-1^{i}\right)}{2}\right)\left|H_{2i-1}\right|^{2}+\left( \frac{1+\left(-1^{i}\right)}{2}\right)\left|H_{2i-2}\right|^{2}&;p=OO-EE\\ \left|H_{2i-1}\right|^{2}&;p=SW-SW&;i=1,2,\ldots,\;M/2\\ \left|H_{i}\right|^{2}&;p=SW\\ \left|H_{r_{1}}\right|^{2}&;p=RP\end{array} \right. \tag{4}\]  
\(\rho=\frac{p}{\sigma_{u}^{2}}\) is the instantaneous SNR. The SE achieved by the WUE during direct transmission is given by the following:\[\Gamma_{d-w,p}^{i}=\frac{1}{M}\text{log}_{2}\left(1+\frac{G_{w}\rho\mu_{wi}}{G_{w }\rho\mu_{si}+G_{w}\rho\left[k_{g}^{2}+k_{w}^{2}\right]+1}\right) \tag{5}\]  
where  
\[G_{w}=\left\{ \begin{array}{cl}\left|H_{2i}\right|^{2}&;p=NN-FF\\ \left(\frac{1-\left(-1\right)}{2}\right)\left|H_{2i+1}\right|^{2}+\left(\frac {1+\left(-1\right)^{i}}{2}\right)\left|H_{2i}\right|^{2}&;p=OO-EE\\ \left|H_{2i}\right|^{2}&;p=SW-SW\\ \left|H_{M+1-i}\right|^{2}&;p=SW\\ \left|H_{r_{2}}\right|^{2}&;p=RP\end{array} \right. ;i=1,2,\ldots,\ M/2 \tag{6}\]  
Here, \(\left|H_{u}\right|^{2}\) is the cumulative channel gain of the \(u\)th UE, where \(\left|H_{u}\right|^{2}\epsilon\{G_{s},G_{w}\}\).