## 2 RIS-Assisted Communication System Model  
Figure 1 shows the conceptual idea of a typical RIS-based communication system. We have considered the downlink communication for a multi-antenna base station (BS) to \(K\) single-user equipment (UE). In the BS, there are \(M\) uniform planar array antennas and the RIS has \(N\) number of reflective elements. The signal received by the \(k\)th UE is a combined signal from BS to RIS and from RIS to UE. Thus, the receiver signal via RIS for the \(k\)th UE can be represented as follows [32]:  
\[y_{k}=\mathbf{h}_{r,k}^{H}\mathbf{\Psi}\mathbf{H}_{b}x+n_{k}, \tag{1}\]  
where \(\mathbf{H}_{b}\in\mathbb{C}^{N\times M}\) is the channel from BS to RIS, \(\mathbf{h}_{r,k}\in\mathbb{C}^{1\times N}\) is the channel from RIS to the \(k\)th user, \(x\) is the transmitted symbol, and \(n_{k}\) is the additive white Gaussain noise (AWGN) with \(n_{k}\)\(\mathcal{CN}(0,\sigma_{k}^{2})\). \(\mathbf{\Psi}=\)diag\((\mathbf{c})\in\mathbb{C}^{N\times N}\) is the diagonal matrix presenting the phase-shift values of the reflecting elements of RIS with \((\mathbf{c})=[\alpha_{1}e^{i\theta_{1}},\alpha_{2}e^{i\theta_{2}},...,\alpha_ {N}e^{i\theta_{N}}]\), where \(\alpha\in[0,1]\) is the amplitude of the signal and \(\theta_{n}\in[0,2\pi]\) is the phase-shift value. In this paper, the constant amplitude coefficient (\(\alpha_{N}=1\)) is assumed [33]. Next, the entire channel from the BS to the \(k\)th UE through RIS is represented by \(\mathbf{h}_{r,k}^{H}\mathbf{\Psi}\mathbf{H}_{b}\in\mathbb{C}^{1\times M}\). Specifically, it is worth to note that the matrix \(\mathbf{\Psi}=\)diag\((\mathbf{c})\) is the diagonal matrix. Then, the aforementioned entire channel matrix \(\mathbf{h}_{r,k}^{H}\mathbf{\Psi}\mathbf{H}_{b}\) can be rewritten as follows [32]:  
\[\mathbf{h}_{2,k}^{H}\mathbf{\Psi}\mathbf{H}_{b}= \mathbf{h}_{2,n}^{H}\text{diag}(\mathbf{c})\mathbf{H}_{b}=\mathbf{ c}^{T}\text{diag}(\mathbf{h}_{2,k}^{H})\mathbf{H}_{b}\] \[\mathbf{H}_{k}= \text{diag}(\mathbf{h}_{2,k}^{H})\mathbf{H}_{b}.\]  
For the mmWave communication, BS-RIS channel can be expressed as follows [34]:  
\[\mathbf{H}_{b}=\sqrt{\frac{NM}{P_{b}}}\sum_{p=1}^{P_{b}}\delta_{p}\mathbf{a}_{ r}(\theta_{p},\gamma_{p}),\,\mathbf{a}_{t}(\phi_{p}), \tag{2}\]  
where \(\delta_{p}\) is the complex gain with the associated \(p\)th path, \(\theta_{p}(\gamma_{p})\) is azimuth (angle) of angle of arrival (AoA), \(\psi_{p}\) is the angle of departure (AoD), \(\mathbf{a}_{r}(\theta_{p},\gamma_{p})\) is the steering vector associated with receiver, and \(\mathbf{a}_{t}(\phi_{p})\) is the steering vector associated with transmitter. For a typical \(M=M_{1}\times M_{2}\) UPA, the array response vector is written as follows:  
\[\mathbf{a}(\theta,\gamma)=\frac{1}{\sqrt{M}}[e^{-j2\pi d\sin(\theta)\cos( \gamma)\mathbf{m}_{1}/\lambda}]\otimes[e^{-j2\pi d\sin(\gamma)\mathbf{m}_{2}/ \lambda}], \tag{3}\]where \(\mathbf{m_{1}}=[0,1,\ldots,M_{1}-1]\), \(\mathbf{m_{2}}=[0,1,\ldots,M_{2}-1]\), \(\lambda\) is the wavelength, \(d\) is the antenna spacing with \(d=\lambda/2\). The channel between RIS and UE can be expressed as follows:  
\[\mathbf{h_{r,k}^{H}}=\sqrt{\frac{N}{P_{r,k}}}\sum_{p_{2}=1}^{P_{r,k}}\delta_{p_ {2}}\mathbf{a}(\theta_{p_{2}},\gamma_{p_{2}}), \tag{4}\]  
where \(P_{r,k}\) is the number of paths between RIS and UE, \(\delta_{p_{2}}\) is the complex gain including path loss, and \(\theta_{p_{2}}\), \((\gamma_{p_{2}})\) is the azimuth (elevation) angle at RIS for the \(p_{2}\)th path.