## 2 Materials and Methods
### System Model
#### 2.1.2 Stage of \(IRS_{m}-BS\)  
It was assumed that the stage from \(IRS_{m}\) reflected the signal to \(BS\), which is referred to as the stage of \(IRS_{m}-BS\), and a signal propagation schematic of this phase is shown in Figure 3.  
Figure 3: Signal propagation schematic of the \(IRS_{m}-BS\) stage. Figure 2: Signal propagation schematic of the \(US-IRS_{m}\) stage. As shown in Figure 3, the channel from the \(k\)th reflection element of the \(m\)th IRS (\(IRS_{m}\)) to the \(w\)th antenna of \(BS(AN_{w})\) is denoted as \(h2_{m,k,w}\). Its unit is dB, and the specific expression is shown in Equation (3). \[h2_{m,k,w}=\sum\limits_{c_{m,k,w}=1}^{C_{m,k,w}}\sqrt{\beta_{2m}}\cdot\alpha 2c_{m,k,w}\cdot e^{j2\pi\frac{d}{\lambda}[(k-1)\cos{( \varphi_{m,k,w})}+(w-1)\cos{(\theta_{m,k,w})}]} \tag{3}\]  
where \(C_{m,k,w}\) is the total number of paths from the \(k\)th reflecting element of \(IRS_{m}\) to the \(w\)th antenna, \(\beta 2_{m}\) is the large-scale decay factor from \(IRS_{m}\) to \(BS\), \(\alpha 2c_{m,k,w}\) is the path fading coefficient of the \(c_{m,k,w}\)th path, \(\lambda\) is the signal wavelength, \(\theta_{c_{m,k,w}}\) is the arrival angle of the path \(c_{m,k,w}\), and \(\varphi_{c_{m,k,w}}\) is the launching angle of the path \(c_{m,k,w}\); they are all in a range of \([0,\pi]\). We assume that the total channel in the stage of \(IRS_{m}-BS\) is \(\mathbf{H2}_{m}\in\mathbb{C}^{W\times K}\), where \(\mathbb{C}^{W\times K}\) denotes the complex matrix of \(W\) rows and \(K\) columns, as expressed in Equation (4). The detailed expression of Equation (4) is shown in (A1) of Appendix A. \[\mathbf{H2}_{m}= \begin{bmatrix}h2_{m,1,1}&\cdots&h2_{m,K,1}\\ \vdots&\ddots&\vdots\\ h2_{m,1,W}&\cdots&h2_{m,K,W}\end{bmatrix}  \tag{4}\]  
Therefore, the channel through which the signal emitted by \(US\) arrived at the \(w\)th antenna of \(BS\) via the \(k\)th reflector element of \(IRS_{m}\) is denoted as \(h_{m,k,w}\). Its unit is dB, and the specific expression is shown in Equation (5). \[ \begin{split}& h_{m,k,w}\\ &=h1_{m,k}\cdot\Xi_{m,k}\cdot h2_{m,k,w}\\ &=\sqrt{\beta_{m}}\sum\limits_{l_{m,k}=1}^{L_{m,k}}\alpha 1_{l_{m,k}}\sum\limits_{c_{m,k,w}=1}^{C_{m,k,w}}\alpha 2c_{m,k,w} \eta_{m,k}e^{\frac{\varphi_{m,k,w}+j2\pi\frac{d}{\lambda}[(k-1)[\cos{(\varphi _{c_{m,k,w}})}-\cos{(\vartheta_{l_{m,k}})}]}{+(w-1)\cos{(\theta_{c_{m,k,w}})}},\end{split}  \tag{5}\]  
where \(\beta_{m}\) is the large-scale fading coefficient from \(US\) through \(IRS_{m}\) to \(BS\), and \(\Xi_{m,k}\) is the reflection coefficient of the \(k\)th reflection element of \(IRS_{m}\), as expressed in Equation (6). \[\Xi_{m,k}=\eta_{m,k}e^{j\omega_{m,k}}, \tag{6}\]  
where \(\eta_{m,k}\) denotes the reflection amplitude of the \(k\)th reflection element of \(IRS_{m}\). In this study, it was assumed that \(\eta_{m,k}=1\), so \(\omega_{m,k}\in[0,2\pi]\) denotes the reflection phase of the \(k\)th reflection element of \(IRS_{m}\). Thus, the total channel of \(IRS_{m}\)-assisted \(US\) communicating with \(BS\) can be expressed as \(\mathbf{H}_{m}\in\mathbb{C}^{W\times 1}\), where \(\mathbb{C}^{W\times 1}\) denotes the complex matrix of \(W\) rows and one column. This is expressed as Equation (7). The detailed expression of Equation (7) is shown in (A2) of Appendix A. \[\mathbf{H}_{m}=\mathbf{H2}_{m}diag(\Xi_{m})\mathbf{H1}_{m}= \begin{bmatrix} \sum\limits_{k=1}^{K}h_{m,k,1}&\cdots&\sum\limits_{k=1}^{K}h_{m,k,W}\end{bmatrix} ^{T} \tag{7}\]  
where \(diag(\Xi_{m})\) represents the reflection coefficient matrix of \(IRS_{m}\), expressed as Equation (8),  
\[diag(\Xi_{m})= \begin{bmatrix}\eta_{m,1}e^{j\omega_{m,1}}&0&0\\ 0&\ddots&0\\ 0&0&\eta_{m,K}e^{j\omega_{m,K}}\end{bmatrix} . \tag{8}\]  
In addition,  
\[diag(\Xi_{m})\mathbf{H1}_{m}=diag(\mathbf{H1}_{m})\Xi_{m}, \tag{9}\]such that \(\mathbf{Y}_{m}=\mathbf{H2}_{m}diag(\mathbf{H1}_{m})\). Let \(\mathbf{Y}_{m}\in\mathbb{C}^{W\times K}\) be the cascade channel of \(US-IRS_{m}-BS\). Consequently, Equation (7) can be rewritten as Equation (10).