## 4 Channel Modeling
### Indoor Physical Channel Model
#### 4.1.1 BS-RIS Channel  
Let us consider that \(M\) scatterers/reflectors are available between BS and RIS. These \(M\) scatterers are grouped into \(C\) clusters, each having \(S_{c}\) sub-rays for \(c=1,2,\ldots,C\), i.e., \(M=\sum_{c=1}^{C}S_{c}\). A cluster generally consists of sub-rays with similar spatial and/or temporal characteristics. The BS-RIS channel vector of size \(N\times N_{T}\) is defined as follows [25]:  
\[\mathbf{H}=\gamma\sum_{c=1}^{C}\sum_{s=1}^{S_{C}}\beta_{c,s}\sqrt{G_{e}\left( \theta_{c,s}^{BS-RIS}\right)L_{c,s}^{BS-RIS}}\;\mathbf{a}\left(\phi_{c,s}^{ BS-RIS},\theta_{c,s}^{BS-RIS}\right)\mathbf{a}^{T}\left(\phi_{c,s}^{BS},\theta_{c,s}^{ BS}\right)+\mathbf{H}_{\text{LoS}} \tag{12}\]  
where \(\gamma=\sqrt{\frac{1}{\sum_{c=1}^{C}S_{c}}}\) is the normalization factor, and \(\beta_{c,s}\) is the complex Gaussian distributed small-scale path gain corresponding to \(s^{th}\) sub-ray in \(c^{th}\) cluster. \(G_{e}\left(\theta_{c,s}^{BS-RIS}\right)\) is the RIS element gain and \(L_{c,s}^{BS-RIS}\) is the large-scale path attenuation associated with the \(s^{th}\) sub-ray in the \(c^{th}\) cluster. \(\mathbf{a}(\phi_{c,s}^{BS-RIS},\theta_{c,s}^{BS-RIS})\) and \(\mathbf{a}\left(\phi_{c,s}^{BS},\theta_{c,s}^{BS}\right)\) are the array response vectors for the elevation arrival angle \(\theta_{c,s}^{BS-RIS},\theta_{c,s}^{BS}\) and azimuth arrival angle \(\phi_{c,s}^{BS-RIS},\phi_{c,s}^{BS}\), respectively. The array response vectors are calculated by following the steps described in [25; 58]. The RIS element gain \(G_{e}\left(\theta_{c,s}^{BS-RIS}\right)\) is modeled as follows [25; 59]:  
\[G_{e}\left(\theta_{c,s}^{BS-RIS}\right)=2(2q+1)\text{cos}\left(\theta_{c,s}^{ BS-RIS}\right)^{2q},-\frac{\pi}{2}<\theta_{c,s}^{RIS}<\frac{\pi}{2} \tag{13}\]  
where  
\[q=0.25G_{e}(0)-0.5,G_{e}(0)=2\pi \tag{14}\]  
The Poisson distribution is commonly used to model the number of clusters in a mmWave communication \(C~{}\sim\mathcal{P}\left(\lambda_{p}\right)\). Here, \(\lambda_{p}\) is the variance, which depends on the operating frequency [60]. \[\lambda_{p}=\left\{ \begin{array}{ll}1.8\,,&f=28~{}GHz\\ 1.9\,,&f=73~{}GHz\end{array} \right. \tag{15}\]  
The cluster size is assessed to ensure a minimum of one cluster exists in the propagation environment. \[C=\text{max}\big{\{}1,\mathcal{P}\left(\lambda_{p}\right)\big{\}} \tag{16}\]  
In a given cluster, \(\phi_{c,s}^{BS-RIS}\) and \(\phi_{c,s}^{BS}\) follow a Laplacian distribution with mean \(\mathcal{U}\big{[}-\frac{\pi}{2},\frac{\pi}{2}\big{]}\). Similarly, the \(\theta_{c,s}^{BS-RIS}\) and \(\theta_{c,s}^{BS}\) are assumed to be conditionally Laplacian with a mean of \(\mathcal{U}\big{[}-\frac{\pi}{2},\frac{\pi}{2}\big{]}\). The path loss \(L_{c,s}^{BS-RIS}\) for \((c,s)^{th}\) path can be expressed as follows [25; 60]:  
\[L_{c,s}^{BS-RIS}=-20\text{log}_{10}\bigg{(}\frac{4\pi}{\lambda}\bigg{)}-10n \bigg{(}1+z\bigg{(}\frac{f-f_{0}}{f_{0}}\bigg{)}\bigg{)}\text{log}_{10}d_{c,s} -X_{\sigma} \tag{17}\]  
This model applies to real-time environments such as UMi and InH. Here, \(d_{c,s}\) is the propagation length of the \((c,s)^{th}\) path, which is measured according to the steps followed in [61], \(n\) is path loss exponent, and \(z\) is the system parameter, whereas \(f_{o}\) is the fixed reference frequency. 5G InH--indoor office is considered for indoor channel modeling, and the corresponding parameters such as \(n,~{}z,~{}\sigma\) are tabulated in Table 2[25; 61].