# Machine-Learning-Based Optimization for Multiple-IRS-Aided Communication System
## 2 System Model  
### Network Model  
As shown in Figure 1, we consider a multiple-IRS-assisted downlink communication system, where multiple IRSs are deployed to aid transmission from the base station (BS) to the user equipment (UE). In this work, we focus on the single-cell scenario. Similar to [5], the interference from other cells is ignored. The consideration of multiple cells is left for future work. Let the set \(\mathcal{L}~{}=\{1,\ldots,L\}\) denote the multiple IRSs and the set \(\mathcal{K}=\{1,\ldots,K\}\) denote the multiple UEs, respectively. The BS is assumed to be equipped with \(M\) transmit antennas, while each UE is equipped with a single antenna. In addition, each IRS is equipped with \(N\) reflecting units that are configured via a smart controller that is wirelessly linked to the BS [29]. Note that these controllers can serve as either receivers for channel estimation or reflectors for relaying signals to UEs [30]. In this paper, similar to [5; 6; 7; 8; 9; 10; 11], all channels are considered to be quasi-static flat-fading, and full CSI is assumed. The CSI can be obtained by using the reduced length of the pilot sequence, such as in [2]. The consideration of imperfect CSI is left for our future work. Let \(\mathbf{h}_{dk}^{H}\in\mathbb{C}^{1\times M}\) denote the direct channel from the BS to the \(k\)-th UE, while \(\mathbf{G}_{l}\in\mathbb{C}^{N\times M}\) and \(\mathbf{h}_{rkl}^{H}\in\mathbb{C}^{1\times N}\) are the indirect channels from the BS to the \(l\)-th IRS, and from the \(l\)-th IRS to the \(k\)-th UE, respectively. At the transmitter side, the BS performs a linear precoding transmission with one beamforming vector assigned to each served UE. Thus, the signal transmitted by the BS is expressed as  
\[\mathbf{x}=\sum_{k=1}^{K}\mathbf{w}_{k}s_{k}, \tag{1}\]  
where \(\mathbf{w}_{k}\)\(\mathbb{C}^{M\times 1}\) denotes the transmit beamforming vector of the \(k\)-th UE with data symbol \(s_{k}\). It is assumed that \(s_{k}\)'s are modeled as random variables with zero mean and unit variance, i.e., \(\mathbb{E}[{|s_{k}|}^{2}]=1,\forall k\). Thus, the total consumed power transmitted at the BS is calculated as  
\[P_{T}=\sum_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}. \tag{2}\]  
At the reflecting side, each IRS receives the signals from the transmitting BS and reflects the tuned signals. Specifically, the incident signal, symbolized by \(\hat{x}_{n}\), \(\hat{x}_{n}\) at the \(n\)-th element of the \(l\)-th IRS, is multiplied by a constant-modulus reflection coefficient. Thus, the reflected signal of the \(n\)-th element at the \(l\)-th IRS can be expressed as  
\[\hat{y}_{l,n}=e^{i\theta_{l,n}}\hat{x}_{n},\forall l,n, \tag{3}\]  
where \(\theta_{l,n}\)\(\epsilon\)\([0,\ 2\pi)\) denotes the angle of the \(n\)-th unit at the \(l\)-th IRS. As such, the reflected signal at each \(l\)-th IRS is expressed as  
\[\hat{\mathbf{y}}_{l}=\mathbf{\Theta}_{l}\hat{x},\forall l, \tag{4}\]  
where \(\mathbf{\Theta}_{l}=diag(\ e^{i\theta_{l,1}},\ldots,\ e^{i\theta_{l,N}})\)\(\epsilon\)\(\mathbb{C}^{N\times N}\), \(\hat{\mathbf{y}}_{l}=[\hat{y}_{l,1},\ldots,\hat{y}_{l,N}]^{T}\), and \(\hat{\mathbf{x}}=[\hat{x}_{1},\ldots,\hat{x}_{N}]^{T}\). At the receive side, the \(k\)-th UE at the location of \((x_{k},y_{k},z_{k})\) receives signals from both the BS and the IRSs. Due to the high path loss, only the signals reflected by the IRSs are considered once. Hence, the total received signal at the \(k\)-th UE is given by  
\[y_{k}= \underbrace{\left(\sum_{l\in\mathcal{L}}\mathbf{h}_{\textit{rkl}}^{H} \mathbf{\Theta}_{l}\ G_{l}+\mathbf{h}_{\textit{dkk}}^{H}\right)\mathbf{w}_{k}s_{k}}_{ \text{Desired signal}}\ \ \ +\underbrace{\sum_{j\
eq k}\left(\sum_{l\in\mathcal{L}}\mathbf{h}_{\textit{rkl}}^{H} \mathbf{\Theta}_{l}\ G_{l}+\mathbf{h}_{\textit{dkk}}^{H}\right)\mathbf{w}_{j}s_{j}}_{ \text{Interference}}+\underbrace{n_{k}}_{\text{Noise}},\forall k\in\mathcal{K}, \tag{5}\]  
Figure 1: Architecture of IRS-aided system.