## 2 System Model
### IRS Based Communication  
In this section, we describe the IRS network architecture and other related consideration. Figure 1 shows a communication scenario with the IRS where the base station (BS) has \(M\) uniform planner array (UPA) antenna and each user (UE) has a single antenna. The number of IRS elements is considered as \(N\). The received signal via IRS for each UE can be expressed as follows [19]:  
\[\mathbf{y}_{i}=\mathbf{H}_{ui}^{H}\mathbf{\Psi}\mathbf{H}_{b}\mathbf{x}+\mathbf{n}, \tag{4}\]  
where \(\mathbf{y}_{i}\) is the received signal at the \(i\)-th UE, the transmitted signal is \(\mathbf{x}\in\mathbb{C}^{M\times 1}\), \(\mathbf{H}_{b}\in\mathbb{C}^{N\times M}\) is the channel matrix between BS and IRS, \(\mathbf{H}_{ui}\in\mathbb{C}^{1\times N}\) is the channel matrix from IRS to user \(i\), and \(n\sim\mathcal{CN}(0,\sigma^{2})\) is the additive white Gaussian noise at the \(i\)-th user. \(\mathbf{\Upsilon}\) is the diagonal matrix where \(\mathbf{\Upsilon}=diag(\mathbf{\xi})\in\mathbb{C}^{N\times N}\) represents the phase shift values of IRS elements. Each element can be defined as \(\mathbf{\xi}=[\eta_{1}e^{i\omega_{1}},\eta_{2}e^{i\omega_{2}},\ldots,\eta_{N}e^{ i\omega_{N}}]\in\mathbb{C}^{N\times 1}\), where \(\eta_{n}\in[0,1]\) denotes the amplitude and \(\omega_{n}\in[0,2\pi]\) is the phase shift coefficient of the \(n\)-th reflective element. Thus, \(\mathbf{\Upsilon}\) can be written as:  
\[\mathbf{\Upsilon}= \begin{bmatrix}\eta_{1}e^{i\omega_{1}}&\ldots&\ldots&\ldots\\ \cdots&\eta_{2}e^{i\omega_{2}}&\ldots&\ldots\\ \vdots&\vdots&\ldots&\vdots\\ \ldots&\ldots&\ldots&\eta_{N}e^{i\omega_{N}}\end{bmatrix} . \tag{5}\]  
For easy calculation, the constant amplitude coefficient is considered as \(\omega_{n}=1\). The total channel with direct communication link can be represented by [20]:  
\[\mathbf{y}_{it}=\mathbf{H}_{ui}^{H}\mathbf{\Psi}\mathbf{H}_{b}\mathbf{x}+\mathbf{H}_{d}\mathbf{x}+ \mathbf{n}, \tag{6}\]  
Figure 1: Intelligent reflecting surface in the multiuser MIMO communication system. where \(\mathbf{H}_{d}\in\mathbb{C}^{M\times 1}\) is the direct channel between BS and UE. The channel matrices \(\mathbf{H}_{ui}\), \(\mathbf{H}_{b}\), and \(\mathbf{H}_{d}\) follow the Rayleigh fading distribution, and each column \(k\) is modeled as:  
\[h_{ui(1,k)} =\kappa h_{ui(1,k)} \tag{7a}\] \[h_{b(1,k)} =\kappa\hat{h}_{b(1,k)}\] (7b) \[h_{d(1,k)} =\kappa\hat{h}_{d(1,k)}, \tag{7c}\]  
where \(\kappa\) is the path loss factor and \(\hat{h}_{ui(1,k)}\), \(\hat{h}_{b(1,k)}\), and \(\hat{h}_{d(1,k)}\) are \(\sim\mathcal{CN}(0,\sigma^{2})\). In addition, we consider the Saleh-Valenzuela channel model [8], which is applicable for a multipath propagation environment. The general theory of the Saleh-Valenzuela model for mmWave communication can be modeled as follows:  
\[\mathbf{h}=\sqrt{\frac{N}{L}}\sum_{l=0}^{L}\alpha_{l}\mathbf{a}(\gamma_{l}^{H },\phi_{l}^{H}), \tag{8}\]  
where \(\mathbf{h}\) is the channel vector, \(\alpha_{l}\) is the complex gain of the \(l\)-th path, \(L\) is the total number of paths, \(\gamma_{l}^{H}\) is the azimuth angle of departure, and \(\phi_{i}^{H}\) is the elevation angle of departure and \(\mathbf{a}(\gamma_{l}^{H},\phi_{l}^{H})\) is the array response vector. For a typical \(N_{1}\times N_{2}\) UPA, the array response vector written as follows [21]:  
\[\mathbf{a}(\gamma,\phi)=\frac{1}{\sqrt{N}}[e^{-j2\pi d\sin(\gamma)\cos(\phi) \mathbf{n_{1}}/\lambda}]\otimes[e^{-j2\pi d\sin(\phi)\mathbf{n_{2}}/\lambda}], \tag{9}\]  
where \(\mathbf{n_{1}}=[0,1,\dots,N_{1}-1]\) and \(\mathbf{n_{2}}=[0,1,\dots,N_{2}-1]\), \(\lambda\) is the carrier wavelength, and \(d\) is the antenna spacing fulfilling the condition \(d=\lambda/2\).