# IRS-Enabled Ultra-Low-Power Wireless Sensor Networks: Scheduling and Transmission Schemes
## 7 WSN Transmission Power and Range
From Figure 12, it can be observed that with certain values of \(N\), the battery lifetime of a single ISN is higher than that of a LoRa transmitter-equipped sensor node; it is worth noting that future developments in IRSs can be expected to make this gap between the ISN and a transmitter-equipped sensor node even larger. \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **Time Duration (s)** & **Power Consumption (mW)** \\ \hline Sensing unit & \(T_{sen}=0.025\) & 10.5 \\ Transmitter & \(T_{tr}=0.0065\) & 92.4 \\ Processor & \(T_{proc}=0.0335\) & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation Parameters. Next, in Figure 13, we focus on how the battery lifetime of an ISN decreases with increasing \(N\), as adding more elements linearly increases the power consumption of the ISN to operate and adjust the IRS phases and deplete the sensor's battery capacity, as in (50). Moreover, Figure 13 shows the proportional increase in range \(d\), which is the distance from the ISN to the user. To construct the relationship between range and number of elements \(N\), recall \(\beta_{2}^{l}=\frac{10^{-3}}{d^{3}}\) in Table 3; using (13), we can write  
\[\gamma_{l} =\frac{P_{l}|\mathbf{f}_{l}^{H}\mathbf{H}_{1,l}\mathbf{\Phi}_{l} \mathbf{h}_{2,l}|^{2}}{\sigma^{2}} \tag{52}\] \[=\beta_{2}^{l}\frac{P_{l}|\mathbf{f}_{l}^{H}\mathbf{H}_{1,l}\mathbf{ \Phi}_{l}\mathbf{z}_{2,l}|^{2}}{\sigma^{2}},\] (53) \[=\big{(}\frac{10^{-3}}{d^{3}}\big{)}\frac{P_{l}|\mathbf{f}_{l}^{H }\mathbf{H}_{1,l}\mathbf{\Phi}_{l}\mathbf{z}_{2,l}|^{2}}{\sigma^{2}}, \tag{54}\]  
where \(\mathbf{h}_{2,l}=\sqrt{\beta_{2}^{l}}\mathbf{z}_{2,l}\) and \(\mathbf{z}_{2,l}\sim\mathcal{CN}(0,\mathbf{I}_{N})\). In terms of \(d\), this becomes  
\[d=\bigg{(}\frac{10^{-3}P_{l}|\mathbf{f}_{l}^{H}\mathbf{H}_{1,l}\mathbf{\Phi}_{l} \mathbf{z}_{2,l}|^{2}}{\gamma_{l}\sigma^{2}}\bigg{)}^{1/3}, \tag{55}\]  
and we can fix \(\gamma_{l}=3\) dB; then, \(R_{l}\approx 0.8\) (19). It can be seen that increasing \(N\) expands the sensor's range. Moreover, increasing the transmitting power of the BS plays a significant role in extending the sensor range, as shown in Figure 13.  
Figure 12: Battery lifetime of a single sensor node with an IRS (with different element number \(N\)) and without an IRS vs. measurement period \(T_{m}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Characteristics** & **Value** \\ \hline Sensing unit voltage & \(V_{s}=2\) V \\ Transmitter and processor voltage & \(V_{b}=3.3\) V \\ Battery capacity & C = 950 mAh \\ Self-discharge current & \(l_{s}=7.5\times 10^{-3}\) mA \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Battery characteristics. In Figure 14, we simulate and compare the lifetime of a WSN equipped with IRSs vs. a typical WSN without IRSs. We assume fair transmission (such as with RR) for both scenarios and that all sensor nodes are aware of their turn in transmission, and use a measurement period \(T_{m}=30\) s. At any period, only one node is active; the other \((L-1)\) nodes are in sleep mode. The total battery energy in the system for \(L\) sensor nodes is \(E_{total}=LE_{battery}\). The average current consumption of the entire network \(I_{avg,net}\) can be derived as  
\[I_{avg,net}=\frac{1}{T_{m}}\big{(}I_{tr}T_{tr}+I_{sen}T_{sen}+I_{proc}T_{proc }+I_{s}T_{s}+(L-1)T_{s}I_{s}\big{)}, \tag{56}\]  
and the WSN lifetime \(L_{WSN}\) in hours can then be found as  
\[L_{WSN}=\frac{E_{total}}{I_{avg,net}}. \tag{57}\]  
Figure 14 shows that for certain values of \(N\), i.e., \((N=5,10,15)\), the lifetime of the WSN with IRSs is larger than the WSN without IRSs, which means that the performance is improved under IRSs with constraints on the number of elements in regard to energy consumption. Moreover, with an increasing number of sensor nodes, the lifetime of WSN increases logarithmically due to the discharge current in sleep mode. Note that this simulation assumes that each sensor is scheduled the same number of times, which happens in the scheduling schemes described in this paper for RR, PF, and OS.