# Optimization of Intelligent-Reflecting-Surface-Enabled Covert Communication in Multi-Sensor Systems
## 3 System Model  
### Communication Scenario and Equivalent Channel  
We examine a discrete-time additive white Gaussian noise channel model with complex values, as depicted in Figure 1. The central processing unit (CPU) broadcasts \(n\) complex-valued symbols \(\mathbf{S_{R}}=\{S_{R,j}\}_{i=1}^{n}\) to \(N\) sensors, and each sensor, denoted as the \(k\)-th sensor, transmits feedback information to the CPU in \(n\) complex-valued symbols \(\mathbf{S_{F,k}}=\{S_{F,k,j}\}_{i=1}^{n}\). To optimize the transmission rate, both the broadcast and feedback signals follow Gaussian distributions, where \(S_{R,i}\sim\mathcal{CN}(0,2\sigma_{R}^{2})\) and \(S_{F,k,i}\sim\mathcal{CN}(0,2\sigma_{F,k}^{2})\). The transmitter, represented by the intelligent reflecting surface (IRS) or Alice, aims to transmit \(n\) complex-valued symbols \(\mathbf{c}=\{c_{i}\}_{i=1}^{n}\) to reach the target sensor, [PERSON]. Meanwhile, the detector, [PERSON], endeavors to detect any access attempts. All transmission and feedback activities occur within a single time slot. Consequently, the \(i\)-th symbol received by [PERSON] and [PERSON] can be expressed as:  
\[\widehat{Y}_{B,i} =h_{RB}S_{R,j}+h_{RA}h_{AB}S_{R,i}c_{i}+\sum_{k=1}^{N}h_{FB,k}S_{ F,k,i}+\widehat{N}_{B,i}, \tag{1}\] \[\widehat{Y}_{W,i} =h_{RW}S_{R,i}+h_{RA}h_{AW}S_{R,i}c_{i}+\sum_{k=1}^{N}h_{FW,k}S_{ F,k,i}+\widehat{N}_{W,i}, \tag{2}\]  
where \(\widehat{N}B,i\) and \(\widehat{N}W,i\) represent independent and identically distributed (i.i.d.) Gaussian noise components with variances \(2\widehat{\sigma}B^{2}\) and \(2\widehat{\sigma}W^{2}\), respectively. Specifically, \(\widehat{N}B,i\sim\mathcal{CN}(0,2\widehat{\sigma}B^{2})\) and \(\widehat{N}W,i\sim\mathcal{CN}(0,2\widehat{\sigma}W^{2})\). The channel coefficients from the central processing unit (CPU) to [PERSON], [PERSON], and [PERSON] are denoted by \(hRA\), \(hRB\), and \(h_{RW}\), respectively. Additionally, the channel coefficients from [PERSON] to [PERSON] and from [PERSON] to [PERSON] are represented by \(h_{AB}\) and \(h_{AW}\), respectively. The channel coefficients from the \(k\)-th sensor to [PERSON] and [PERSON] are denoted by \(h_{FB,k}\) and \(h_{FW,k}\), respectively. For simplicity, we assume that the CPU broadcasts \(n\) symbols with a phase angle of zero, and all channel coefficients are universally known. Given the multitude of sensors, [PERSON] and [PERSON] face challenges in demodulating all signals. Consequently, the feedback signals from the sensors are treated as interference. Considering the composition of feedback signals from \(N\) sensors, the channel can be equivalently expressed as:  
\[Y_{B,i} =h_{RB}S_{R,i}+h_{RA}h_{AB}S_{R,i}c_{i}+N_{B,i}, \tag{3}\] \[Y_{W,i} =h_{BW}S_{R,i}+h_{RA}h_{AW}S_{R,i}c_{i}+N_{W,i}, \tag{4}\]  
where \(N_{B,i}\sim\mathcal{CN}(0,2\sigma_{B}^{2})\) and \(N_{W,i}\sim\mathcal{CN}(0,2\sigma_{W}^{2})\) with \(\sigma_{B}^{2}=\mathbb{E}(|\sum_{k=1}^{N}h_{FB,k}S_{F,k,i}|^{2})+\widehat{ \sigma}_{B}^{2}\) and \(\sigma_{W}^{2}=\mathbb{E}(|\sum_{k=1}^{N}h_{FW,k}S_{F,k,i}|^{2})+\widehat{ \sigma}_{B}^{2}\).  
Based on the scenario and actual situation considerations, we assume that [PERSON] and [PERSON] can recover the broadcast information \(\mathbf{S}_{R}\) and perform a subtraction operation on their observations to improve transmission performance and detection performance. Then, the \(i\)-th symbol received by [PERSON] and [PERSON] can be equivalently rewritten as  
\[Y_{B,i} =h_{RA}h_{AB}S_{R,i}c_{i}+N_{B,i} \tag{5}\] \[Y_{W,i} =h_{RA}h_{AW}S_{R,i}c_{i}+N_{W,i}. \tag{6}\]  
In the following context, our problem will be analyzed on equivalent channels (5) and (6).