## 3 Channel Estimation in Different Frequency Bands
The cascade channel \(\mathbf{G}\) can be expressed as:  
\[\mathbf{G}=\mathbf{H}\mathrm{diag}(\mathbf{h}_{k}) \tag{2}\]  
Figure 1: Frequency bandwidth and wavelength of wireless radio waves. Stacking \(T\) time slots of Equation (1) and combining Equation (2), by assuming \(x_{k,t}=1\) the received signal \(\mathbf{Y}_{k}=[\mathbf{y}_{k,1},\mathbf{y}_{k,2},\cdots,\mathbf{y}_{k,T}]\in\mathbb{C}^{M\times T}\) can be expressed as:  
\[\mathbf{Y}_{k}=\mathbf{G}\mathbf{\phi}+\mathbf{N}_{k} \tag{3}\]  
where \(\mathbf{\phi}=[\mathbf{\phi}_{1},\mathbf{\phi}_{2},\cdots,\mathbf{\phi}_{T}]\in\mathbb{C}^{N \times T}\) and \(\mathbf{N}_{k}=[\mathbf{n}_{k,1},\mathbf{n}_{k,2},\cdots,\mathbf{n}_{k,T}]\in\mathbb{C}^{M \times T}\). With the finite scattering properties of mmWave, it can be represented using the Saleh-Valenzuela (SV) model [22; 23; 24; 25]. There is a one-to-one relationship between spatial frequency and physical angle on the UPA side. Therefore, for simplicity of illustration, we will refer to the independent variable of the array steering vector as the angle or spatial frequency. The channel matrix \(\mathbf{H}\) between the RIS and the BS can be written as:  
\[\mathbf{H}=\sum_{l=1}^{L}\alpha_{l}\mathbf{a}_{N}(\psi_{l},v_{l})\mathbf{a}_{M}^{\rm H}( \omega_{l},\mu_{l}) \tag{4}\]  
where \(L\) represents the propagation scattering path between the BS and the RIS; \(\alpha_{l}\), \((\psi_{l},v_{l})\), and \((\omega_{l},\mu_{l})\) are the complex gain, angle of arrival (AoA), and angle of departure (AoD) of the \(l\)th path, respectively. The channel matrix \(\mathbf{h}_{k}\) between user \(k\) and the RIS can be written as:  
\[\mathbf{h}_{k}=\sum_{j=1}^{J_{k}}\beta_{kj}\mathbf{a}_{N}(\varphi_{k,j},\theta_{kj}), \quad\forall k\in K \tag{5}\]  
where \(J_{k}\) represents the propagation scattering path between the \(k\) users and the RIS; \(\beta_{kj}\) and \((\varphi_{k,j},\theta_{kj})\) are the complex gain and AoD of the \(j\)th path to the \(k\)th user, respectively. A steering vector of the \(P=P_{1}\times P_{2}\) UPA is \(\mathbf{a}_{P}(z,x)\in\mathbb{C}^{P\times 1}\), which can be expressed as [22; 23; 24]:  
\[\mathbf{a}_{P}(z,x)=\mathbf{a}_{P_{1}}(z)\otimes\mathbf{a}_{P_{2}}(x) \tag{6}\]  
where \(\mathbf{a}_{P_{1}}(z)\) and \(\mathbf{a}_{P_{2}}(z)\) can be expressed as Equations (7) and (8), respectively:  
\[\mathbf{a}_{P_{1}}(z)=[1,\mathrm{e}^{-\mathrm{j}2\pi z},\cdots,\mathrm{e}^{- \mathrm{j}2\pi(P_{1}-1)z}]^{\mathrm{T}} \tag{7}\]  
\[\mathbf{a}_{P_{2}}(x)=[1,\mathrm{e}^{-\mathrm{j}2\pi x},\cdots,\mathrm{e}^{- \mathrm{j}2\pi(P_{2}-1)x}]^{\mathrm{T}} \tag{8}\]  
where the variables \(z\) and \(x\) can be considered as the corresponding equivalent spatial frequencies with respect to the \(z\) axis and \(x\) axis of the UPA, respectively. The \(\mathbf{a}_{P_{1}}(z)\) and \(\mathbf{a}_{P_{2}}(x)\) are the two uniform linear arrays (ULAs) on the vertical (\(z\)-axis) and horizontal (\(x\)-axis) directions, respectively. The UPA is obtained by performing a Kronecker product calculation from two ULAs. The \(\rho\in[-90^{\circ},\ 90^{\circ}]\) and \(\xi\in[-180^{\circ},\ 180^{\circ}]\) are expressed as  
Figure 2: A RIS-aided multi-user MIMO system consisting of a BS with \(M\) antennas, a RIS with \(N\) reflective elements, and \(K\) single-antenna users, green elements in RIS are passive, yellow elements in RIS are active: (**a**) Cascade channel estimation and (**b**) separate channel estimation. the signal elevation and azimuth angles of the UPA, respectively. The relationship between the spatial frequency pair \((z,x)\) and the physical angle pair \((\rho,\xi)\) can be expressed as [23]:  
\[\mathbf{G}=\mathbf{H}\mathrm{diag}(\mathbf{h}_{k}) \tag{2}\]0  
where \(\lambda\) is the carrier wavelength, and \(d\) is the component spacing. We assume a one-to-one correspondence between physical angle and spatial frequency on the UPA side; then, it is necessary to satisfy \(d\leq\lambda/2\). The following evaluation metrics are commonly used in RIS channel estimation.