## 2 RIS-Assisted System and Channel Model
### System Model  
Figure 2 depicts a RIS communication system where \(I\) user equipment (UE) contains a single antenna, and the base station (BS) contains \(M\) uniform linear array (ULA) antennas. The total number of RIS is taken to be uniform planner array (UPA) \(N\) elements. For the \(i\)-th UE \((i=1,2,...,I)\), the received signal through RIS can be stated as follows [6]:  
\[\mathbf{y_{i}}=\mathbf{H}_{2,i}^{H}\mathbf{\Omega}\mathbf{H}_{br}\mathbf{x}+ \mathbf{n_{i}}, \tag{1}\]  
where \(\mathbf{x}\in\mathbb{C}^{M\times 1}\) represents the precoded transmit signal with \(Q\) subcarriers in each OFDM symbol. Inverse discrete Fourier transform (IDFT) is used to process the OFDM data symbols after they are transformed to the time domain. Then, the CP data are inserted as a guard interval. Discrete Fourier transform (DFT) is carried out at the UE after the CP data are removed. Thereafter, \(\mathbf{y_{i}}\) is the signal that is received at the \(i\)-th UE. \(\mathbf{H}_{br}\in\mathbb{C}^{N\times M}\) is the channel matrix from BS to RIS, the channel matrix between RIS and the \(i\)-th UE is \(\mathbf{H}_{2,i}\in\mathbb{C}^{N\times 1}\), and at the \(i\)-th user, the additive white Gaussian noise is represented \(n_{i}\sim\mathcal{CN}(0,\sigma^{2})\). The phase shift values of RIS elements are represented by \(\mathbf{\Omega}=\text{diag}(\mathbf{\Psi})\in\mathbb{C}^{N\times N}\), which is the diagonal matrix of \(\mathbf{\Omega}\). Each element can be characterized as \(\mathbf{\Psi}=[\gamma_{1}e^{j\xi_{1}},\gamma_{2}e^{j\xi_{2}},\dots,\gamma_{N} e^{j\xi_{N}}]^{T}\in\mathbb{C}^{N\times 1}\), where the phase shift coefficient and the amplitude for the \(n\)-th reflective elements are \(\xi_{n}\in[0,2\pi]\) and \(\gamma_{n}\in[0,1]\). Consequently, the symbol \(\mathbf{\Omega}\) can be written as follows:  
\[\mathbf{\Omega}=\begin{bmatrix}\gamma_{1}e^{j\xi_{1}}&\dots&\dots&\dots\\ \dots&\gamma_{2}e^{j\xi_{2}}&\dots&\dots\\ \vdots&\vdots&\dots&\vdots\\ \dots&\dots&\dots&\gamma_{N}e^{j\xi_{N}}\end{bmatrix}. \tag{2}\]  
The constant amplitude coefficient is assumed to be \(\xi_{n}=1\) for convenience in calculations [9]. In this study, the direct communication channel \(\mathbf{H}_{d}\in\mathbb{C}^{M\times 1}\) between BS to UEs is also considered. The BS to the \(i\)-th UEs channel \(\mathbf{H}_{2,i}^{H}\mathbf{\Omega}\mathbf{H}_{br}\in\mathbb{C}^{M\times 1}\) via RIS is assumed. In particular, it is important to remember that the matrix \(\mathbf{\Omega}=\text{diag}(\mathbf{\Psi})\) is a diagonal matrix. Thereafter, the previous whole channel matrix \(\mathbf{H}_{2,i}^{H}\mathbf{\Omega}\mathbf{H}_{br}\) can be reformed as follows:\[\mathbf{H}_{2,i}^{H}\mathbf{\Omega H}_{br}= \mathbf{H}_{2,i}^{H}\mathrm{diag}(\Psi)\mathbf{H}_{br}=\mathbf{V}^ {\mathbf{T}}\mathrm{diag}(\mathbf{H}_{2,i}^{H})\mathbf{H}_{br}, \tag{3}\] \[\mathbf{H}_{cas}= \mathbf{H}_{2,i}^{H}\mathbf{\Omega H}_{br}, \tag{4}\]  
where the cascaded channel for the \(i\)-th UEs is \(\mathbf{H}_{cas}\in\mathbb{C}^{N\times M}\) and only relies on the downlink CSI [46]. Therefore, the total received signal \(\mathbf{y_{rit}}\) at the \(i\)-th UEs via the cascaded channel and the direct communication link can be written as follows [47]:  
\[\mathbf{y_{rit}}=\mathbf{H}_{2,i}^{H}\mathbf{\Omega H}_{br}\mathbf{x}+\mathbf{H}_ {d}\mathbf{x}+\mathbf{n_{i}}, \tag{5}\]  
where \(\mathbf{H}_{d}\) is the direct communication link from BS to the \(i\)-th UEs.