## 2 System Model and Problem Formulation
Meanwhile, the cascade channels S-RIS, RIS-D, and RIS-R, expressed by \(H_{SI}=\left[h_{SI_{1}},\,\ldots,\,h_{SI_{N}}\right]^{T}\in\mathbb{C}^{N \times 1}\), \(H_{ID}=\left[h_{ID_{1}},\,\ldots,\,h_{ID_{N}}\right]^{T}\in\mathbb{C}^{N\times 1}\), and \(H_{IR}=\left[h_{IR_{1}},\,\ldots,\,h_{IR_{N}}\right]^{T}\in\mathbb{C}^{N\times 1}\), represent other reflective paths from the RIS. A generalized expression for all channels is given as Equation (1):  
\[b=\beta_{b}\,e^{i\,\phi_{b}} \tag{1}\]  
where \(b\in\ B=\{h_{SD},\ h_{SR},\ h_{DR},\ H_{SI_{n}},\ H_{ID_{n}},\ H_{IR_{n}}\},\ n\in N\), and \(\phi_{b}\in\left[-\pi,\ \pi\right]\) represents the phase of \(b\). In Equation (1), \(\beta_{b}\) represents the magnitude of \(b\), which is modeled with  
Figure 1: RIS-assisted batteryless IoT network setup. a Rayleigh distribution. In power-splitting mode, passive IoT devices perform energy harvesting and data transmission simultaneously, with each process utilizing a portion of the incoming RF power on the device [36]. Let \(P_{D}\) denote the received power at the IoT device, which reflects \(\beta P_{D}\) for communication and absorbs \((1-\beta)P_{D}\) for energy harvesting, where \(\beta\in~{}(0,\,1)\) is the device's reflection coefficient. We utilize constant envelope modulation, specifically \(M\)-ary phase-shift keying (M-PSK), to ensure that \(\beta\) remains fixed. In this work, we apply a linear energy harvesting model, where the harvested power at the IoT device is expressed as Equation (2):  
\[P_{hrv}=\delta(1-\beta)P_{D} \tag{2}\]  
where \(\delta\), with values between 0 and 1, represents the efficiency of power conversion. Typical values for \(\delta\) range from 0.2 to 0.6. Although practical energy harvesting circuits have non-linear characteristics and a certain activation threshold, the linear model is sufficiently accurate within specific operating ranges. Non-linear models are also commonly used, as detailed in [37; 38], but for the sake of brevity, these details are omitted. Our solution can easily extend to these more complex models as well. Let the source transmit a modulated carrier signal \(\sqrt{P}\)\(s\), where \(P\) represents the transmission power, and the signal \(s\) is normalized such that \(\mathbb{E}\Big{\{}\lvert s\rvert^{2}\Big{\}}=1\). The IoT device receives the signal through the direct channel \(h_{SD}\) and the cascade channels provided by the RIS, denoted as \(H_{ID}\). The signal received by the IoT device can be expressed as Equation (3):  
\[y_{D}=\sqrt{P}\ \Big{(}h_{SD}+H_{ID}^{T}\Psi H_{SI}\Big{)}s \tag{3}\]  
where \(\Psi=\text{diag}(\psi_{1},\psi_{2},\dots,\psi_{n})\in\mathbb{C}^{\text{N}\times \text{N}}\) is a diagonal matrix capturing the reflection characteristics (both attenuation and phase shifts) of the RIS elements. Here, \(\psi_{n}=\eta_{n}\text{e}^{i\phi_{n}}\), where \(\eta_{n}\) is the magnitude of attenuation, and \(\phi_{n}\in[-\pi,\ \pi]\) represents the phase shift of the \(n\)-th reflector, with \(n\in N\). According to our model, the IoT device needs to harvest enough energy to power its operations. With the linear energy harvesting model, as shown in Equation (2), the harvested power at the IoT device must meet the following energy constraint (Equation (4)):  
\[\delta(1-\beta)P_{D}\geq P_{b} \tag{4}\]  
where \(P_{D}=P\big{\lvert}h_{SD}+H_{ID}^{T}\Psi H_{SI}\big{\rvert}^{2}\) is the total power received by the IoT device, and \(P_{b}\) represents the minimum activation power threshold required for the energy harvester at the IoT device. The IoT device harvests energy from the incoming signal and modulates it using its own data \(q\), where \(q\) is a normalized symbol chosen from a multi-level modulation scheme (M-ary), satisfying \(\mathbb{E}\Big{\{}\lvert q\rvert^{2}\Big{\}}=1\), before transmitting it to the IoT receiver.