# Intelligent Reflecting Surface-Assisted Wireless Communication Using RNNs: Comprehensive Insights
## 3 System Model
### IRS System Structure  
The IRS-assisted communication model consists of a transmitter, a receiver, and IRS-reflecting elements. In this study, we consider IRS-assisted communication as shown in Figure 4. We consider each user, \(I\), to have a single antenna, and the serving base station (BS) has a uniform linear array of \(L\) antennas. The number of IRS elements is considered as \(N\). The received signal for the \(i\)-th (\(i=1,2,3,...,I\)) user from the IRS can be expressed as follows:  
\[y_{i}=\mathbf{H}_{z,i}^{H}\Phi\mathbf{H}\mathbf{x}+n_{i}, \tag{1}\]  
where \(y_{i}\) is the received signal at the \(i\)-th user, \(\mathbf{x}\in\mathbb{C}^{L\times 1}\) is the transmitted signal, \(\mathbf{H}\in\mathbb{C}^{N\times L}\) is the channel matrix from the BS to IRS, \(\mathbf{H}_{z,i}^{H}\) is the channel from the IRS to the \(i\)-th user, and \(n_{i}\sim\mathcal{CN}(0,\sigma^{2})\) is the additive white Gaussian noise (AWGN) at the \(i\)-th user. Again, \(\Phi=diag(\mathbf{d})\in{}^{N\times N}\) represents a diagonal matrix that contains phase shift values of reflective elements of the IRS, where \(\mathbf{d}=[\beta_{1}e^{i\phi_{1}},\beta_{2}e^{i\phi_{2}},...,\beta_{N}e^{i \phi_{N}}]^{T}\in\mathbb{C}^{N\times 1}\), and \(\phi_{N}\in[0,2\pi]\) and \(\beta_{N}\in[0,1]\) denote the phase coefficient and amplitude of the \(i\)-th reflective element, respectively. For the experiment in this study, we consider \(\beta_{N}=1\). We also consider that the direct link between BS and the user exists and is expressed as \(\mathbf{H}_{d}\in\mathbb{C}^{L\times 1}\). The \(\Phi\) is considered as a diagonal matrix, i.e., \(\Phi=diag(\mathbf{d})\). The channel from the BS to the user via the IRS can be transformed as follows:  
\[\mathbf{H}_{z,i}^{H}\Phi\mathbf{H}=\mathbf{H}_{z,i}^{H}diag(\mathbf{d}) \mathbf{H}=\mathbf{d}^{T}diag(\mathbf{H}_{z,i}^{H})\mathbf{H}, \tag{2}\]  
\[\mathbf{H}_{c}=\mathbf{H}_{z,i}^{H}\Phi\mathbf{H}, \tag{3}\]  
where \(\mathbf{H}_{c}\) is the cascaded channel for the \(i\)-th user and \(\mathbf{H}_{c}\in^{N\times L}\), which relies on the downlink channel state information. We can write the total channel using both the cascaded channel and direct channel as follows:  
\[\mathbf{y}_{ti}=(\mathbf{H}_{c}+\mathbf{H}_{d})\mathbf{x}+\mathbf{n}_{i}, \tag{4}\]  
where the direct communication link is \(\mathbf{H}_{d}\) from the BS to the \(i\)-th user. The three-dimensional (3D) Saleh-Valenzuela channel model is adapted for millimeter-wave (mmWave) propagation [28]. This method is a statistical channel model in a multipath propagation scenario. The channel model for mmWave propagation can be expressed as follows [10]:  
\[\mathbf{H}_{mm}=\sqrt{\frac{N}{T}}\sum_{t=1}^{T}\eta_{t}\mathbf{a}(\rho_{t}^{H},\phi_{t}^{H}), \tag{5}\]  
where \(\mathbf{H}\) is the channel vector, the \(t\)-th path complex gain is represented as \(\eta_{t}\), and the total number of paths is \(T\). The elevation angle of departure and azimuth angle of departure for transmission signals are \(\phi_{t}^{H}\) and \(\rho_{t}^{H}\), respectively. The array response vector is expressed as \(\mathbf{a}(\rho_{t}^{H},\phi_{t}^{H})\). We can express the array response vector as follows:  
\[\mathbf{a}(\rho,\phi)=\frac{1}{\sqrt{N}}[e^{-\beta\pi d\sin(\rho)\cos(\phi) \mathbf{n}_{1}/\lambda}]\otimes[e^{-\beta\pi d\sin(\phi)\mathbf{n}_{2}/ \lambda}], \tag{6}\]  
where \(\lambda\) is the carrier wavelength, \(d\) is the antenna spacing, \(\mathbf{n}_{1}=[0,1,...,N_{1}-1]\), \(\mathbf{n}_{2}=[0,1,....,N_{2}-1]\), and (\(N=N_{1}\times N_{2}\)). For proper communication, the antenna spacing satisfies the condition of \(d=\lambda/2\).