# IRS-Enabled Ultra-Low-Power Wireless Sensor Networks: Scheduling and Transmission Schemes
## 2 System Model  
In this section, we describe the channel models used in the IRS-equipped WSN system. To clarify the mathematical symbols used, we list them in Table 1. The abbreviations used throughout the paper are listed in Table 2.  
Consider the system shown in Figure 2, with one single antenna user, a BS equipped with \(M\) antennas, and \(L\) IRS sensor nodes, hereinafter denoted as ISNs. In every ISN, a sensor unit or node is attached to an IRS that modulates the information from the sensor onto a carrier reference signal sent from the BS to the user. Each IRS is equipped with \(N\) IRS elements. The BS and all IRSs have no shared control links to assist in optimized transmission quality. Assuming a uniform linear array (ULA) at the BS and uniform planar array (UPA) at the IRSs, the known (due to the static nature of the BS and deployed sensors, \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(M\) & Number of BS antennas \\ \(N\) & Number of IRS elements \\ \(L\) & Number of ISNs \\ \(\mathbf{H}_{1J}\in\mathbb{C}^{M\times N}\) & LoS BS-IRS \(l\) channel \\ \(\mathbf{h}_{2J}\in\mathbb{C}^{N}\) & IRS \(L\)-user channel \\ \(\mathbf{h}_{1}(t)\in\mathbb{C}^{M}\) & End-to-end channel \\ \(\beta_{1}^{l}\) & Path loss factor for \(\mathbf{H}_{1J}\) \\ \(\beta_{2}^{l}\) & Path loss factor for \(\mathbf{h}_{2J}\) \\ \(\beta^{l}\) & Path loss factor for \(\mathbf{h}_{l}(t)\) \\ \(\mathbf{a}_{l}\in\mathbb{C}^{M}\) & BS array response vector \\ \(\mathbf{b}_{l}\in\mathbb{C}^{N}\) & IRS \(l\) array response vector \\ \(\mathbf{\Theta}_{l}(t)\) & Phase shift diagonal matrix for IRS \(l\) \\ \(\mathbf{v}_{l}(t)\) & Phase shift vector for IRS \(l\) \\ \(\mathbf{\theta}_{l,n}(t)\) & Phase shift for the \(n^{th}\) element of IRS \(l\) \\ \(\mathbf{G}_{l}=\mathbf{H}_{1J}\text{diag}(\mathbf{h}_{2J})\in\mathbb{C}^{M \times N}\) & Reformulation of the channels \\ \(\mathbf{x}_{l}\) & Reference known signal \\ \(\mathbf{f}_{l}\in\mathbb{C}^{M}\) & Beamforming vector designated for ISN \(l\) \\ \(P_{l}\) & Reference signal power \\ \(\gamma_{l}\) & Signal to noise ratio \\ \(X_{l}(t)\) & BPSK signal \\ \(p_{e}\) & Cross-coverage probability \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Important notations for system model. and because we can place the ISNs in the LoS of the BS, the BS knows \(\mathbf{H}_{1,l}\forall l\)) LoS BS-IRS \(l\) channel matrix \(\mathbf{H}_{1,l}\) can be written as  
\[\mathbf{H}_{1,l}=\sqrt{\beta_{1}^{l}\mathbf{a}_{l}\mathbf{b}_{l}^{H}}. \tag{1}\]  
Here, \(\beta_{1}^{l}\) is the path loss factor for the BS-IRS \(l\) channel and \(\mathbf{a}_{l}\in\mathbb{C}^{M}\) is the array response vector for the BS, while \(\mathbf{b}_{l}\in\mathbb{C}^{N}\) is the array response vector for the IRS \(l\) and \(\mathbf{H}_{1,l}\in\mathbb{C}^{M\times N}\). As we assume a ULA at the BS, the steering vector \(\mathbf{a}_{l}\) can be written as  
\[\mathbf{a}_{l}=[1,e^{-\beta kd_{BS}\cos(\phi_{l})},\dots,e^{-\beta kd_{BS}(M-1) \cos(\phi_{l})}]^{T}, \tag{2}\]  
where \(d_{BS}\) is the inter-antenna spacing, \(k=2\pi/\lambda_{c}\) is the wave number, \(\lambda_{c}\) is the carrier wavelength, and \(\phi_{l}\) is the elevation angle-of-departure (AoD) from the BS to IRS \(l\)[13]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Acronym** & **Definition** \\ \hline IRS & Intelligent reflecting surface \\ ISN & IRS sensor node \\ WSN & Wireless sensor network \\ BS & Base station \\ RR & Round-robin \\ PF & Proportional fairness \\ OS & Opportunistic scheduling \\ RF & Radio frequency \\ IoT & Internet of Things \\ SWIPT & Simultaneous wireless information and power transfer \\ LoS & Line of Sight \\ UPA & Uniform planar array \\ ULA & Uniform linear array \\ AoA & Angle of arrival \\ AoD & Angle of departure \\ SNR & Signal to noise ratio \\ CSI & Channel state information \\ BPSK & Binary phase shift keying \\ LDPC & Low-density parity check \\ BER & Bit error rate \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of abbreviations.