# IRS-Enabled Ultra-Low-Power Wireless Sensor Networks: Scheduling and Transmission Schemes
## 2 System Model
Figure 2: IRS-equipped sensor nodes (ISN)s in a WSN with \(L\) nodes. The solid lines represent downlink transmission, while uplinks are shown by the dashed lines. The IRS is envisioned to be a planar array of \(N=N_{x}N_{z}\) elements, where \(N_{x}\) denotes the number of horizontally placed elements and \(N_{z}\) is the number of vertically placed elements. The array response vector \(\mathbf{b}_{l}\) for a UPA at IRS \(l\) is expressed as  
\[\mathbf{b}_{l}=\mathbf{b}_{l,x}\otimes\mathbf{b}_{l,z} \tag{3}\]  
Here, \(\otimes\) denotes the Kronecker product and  
\[\mathbf{b}_{l,x} =[1,e^{-jkl_{IRS}\sin(\varphi_{l})\cos(\phi_{l})},\ldots,e^{-jkl_ {IRS}(N_{x}-1)\sin(\varphi_{l})\cos(\phi_{l})}]^{T}, \tag{4}\] \[\mathbf{b}_{l,z} =[1,e^{-jkl_{IRS}\cos(\varphi_{l})},\ldots,e^{-jkl_{IRS}(N_{z}-1) \cos(\varphi_{l})}]^{T}, \tag{5}\]  
where \(d_{IRS}\) is the inter-element spacing and \(\phi_{l}(\varphi_{l})\) denote the azimuth (elevation) angle-of-arrival AoA of the path from BS to IRS \(l\)[14]. We define the IRS \(l\) to the user channel as \(\mathbf{h}_{2,l}\in\mathbb{C}^{N}\), where \(\mathbf{h}_{2,l}\sim\mathcal{CN}(0,\beta_{2}^{l}\mathbf{I}_{N})\), \(\beta_{2}^{l}\) is the path loss factor and \(\mathbf{I}_{N}\) is the \(N\times N\) identity matrix. Assuming the direct link between the BS and user is obstructed due to high penetration losses, the end-to-end channel from the BS to the user through IRS \(l\) is provided by  
\[\mathbf{h}_{l}(t)=\mathbf{H}_{1,l}\mathbf{\Theta}_{l}(t)\mathbf{h}_{2,l}= \mathbf{G}_{l}\mathbf{v}_{l}(t)\sim\mathcal{CN}(0,\beta_{l}\mathbf{H}_{1,l} \mathbf{H}_{1,l}^{H}), \tag{6}\]  
where \(\beta_{l}=\beta_{l}^{l}\beta_{2}^{l}\) is the total path loss factor for the cascaded IRS channel. Because \(\mathbf{\Theta}_{l}(t)\) is a diagonal matrix, we can rewrite \(\mathbf{H}_{1,l}\mathbf{\Theta}_{l}(t)\mathbf{h}_{2,l}\) as \(\mathbf{G}_{l}\mathbf{v}_{l}(t)\), in which \(\mathbf{G}_{l}=\mathbf{H}_{1,l}\text{diag}(\mathbf{h}_{2,l})\in\mathbb{C}^{M \times N}\). Each IRS element \(n\) has has a phase denoted by \(\theta_{l,n}(t)\). The phase shift matrix \(\mathbf{\Theta}_{l}(t)\in\mathbb{C}^{N\times N}\) is diagonal with elements containing the phase shifts of IRS \(l\), and is defined as \(\mathbf{\Theta}_{l}(t)=\text{diag}(\mathbf{v}_{l}(t))\) and \(\mathbf{v}_{l}(t)=[e^{j\theta_{1,1}(t)},\ldots,e^{j\theta_{l,N}(t)}]^{T}\), where the \(\text{diag}(\cdot)\) operator transforms a vector into a diagonal matrix. Note that we parameterize \(\mathbf{\Theta}_{l}\) with a time variable \((t)\); this is because, as we shall see, \(\mathbf{\Theta}_{l}\) changes as a function of time in order to impart modulation onto the signal reflected from the IRS. The slow-varying channel \(\mathbf{h}_{2,l}\) that remains constant for time \(\tau_{\text{C}}\) is called the coherence time, and corresponds to a number of symbols \(T_{\text{S}}\) such that \(\tau_{\text{C}}=T_{\text{S}}\tau_{\text{S}}\) and \(\tau_{\text{S}}\) is the transmission time for one symbol. We call a coherence interval \(\tau_{\text{C}}\) a frame which is indexed by a frame index \(t_{f}\). Furthermore, the ISNs are blindly operated, i.e., they do not take channel variations into account when reconfiguring their respective phase shift matrices. Next, we describe and study a transmission scheme for the IRS-assisted WSN.