# Low-Power Beam-Switching Technique for Power-Efficient Collaborative IoT Edge Devices
## 2 System Model and Motivation
When the far-field condition \((A\gg r_{m})\) is applied, the received signal at \(AP\) is given as  
\[y(\varphi|\,\mathbf{\Theta}_{M},\mathbf{\Omega}_{M})=z\sum_{m_{m}\in\mathbb{M }}\sqrt{P_{m}}\alpha_{m}\beta(\Theta_{i,m},\Omega_{m})+n, \tag{2}\]  
where \(\beta(\Theta_{i,m},\Omega_{m})\) is the radiation pattern with \(\Xi_{i,m}^{\prime}=\left[\frac{-\pi}{N_{i}}+\Theta_{i,m}+\Omega_{m},\frac{\pi }{N_{i}}+\Theta_{i,m}+\Omega_{m}\right]\), \(\Omega_{m}\sim U[\,-\,\pi,\pi\,)\) is the random deviation angle due to the irregular surface where the sensors are mounted, and \(\mathbf{\Omega}_{M}=\left[\Omega_{1},\cdots,\Omega_{M}\right]\in\left[-\pi, \pi\right]^{M}\) and \(\mathbf{\Theta}_{M}=\left[\Theta_{i,1},\cdots,\Theta_{i,M}\right]\in\left[- \pi,\pi\right]^{M}\). \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Symbols** & \(\mathbb{N}\) & \(\boldsymbol{n_{n}}\) & \(\boldsymbol{(r_{m},\psi_{n})}\) & \(\boldsymbol{(A,\varphi_{0})}\) \\ \hline Meaning & Set of \(n_{n}\) & Index of sensor & Coordinates of \(n_{n}\) & Coordinates of \(AP\) \\ \hline
**Symbols** & \(\hat{\mathcal{B}}(\Theta_{i,n})\) & \(N_{t}\) & \(\Theta_{i,n}\) & \(\hat{\varphi}_{n}(\varphi)\) \\ \hline Meaning & Bampattern of BSS & Amplitude \(\hat{\mathcal{B}}(\Theta_{i,n})\) & of \(\hat{\mathcal{B}}(\Theta_{i,n})\) & Steering angle of \(i\)-th beam & Local \(\hat{\text{\emph{Azimuth}}}\) angle \\ \hline
**Symbols** & \(\varphi\) & \(\Xi_{i,n}\) & \(\mathbb{S}\) & \(s^{k}\) \\ \hline Meaning & Global azimuth angle & Interval \(\hat{\mathcal{B}}(\Theta_{i,n})\) & of \(\hat{\mathcal{S}}\) of \(s^{k}\) & Index of active sensor \\ \hline
**Symbols** & \(z_{k}\) & \(\mathbb{M}^{k}\) & \(m_{m}^{k}\) & \(P_{m}\) \\ \hline Meaning & Data symbol & Set of \(m_{m}^{k}\) & Neighbor sensors of \(s^{k}\) & Transmission power \\ \hline
**Symbols** & \(\Gamma_{m}(\varphi_{0})\) & \(d_{m}(\varphi_{0})\) & \(\lambda\) & \(\alpha_{m}\) \\ \hline Meaning & Closed-loop initial phase & Distance between \(m_{m}\) and \(AP\) & Operating wavelength & Wireless channel \\ \hline
**Symbols** & \(\hat{\mathcal{B}}(\Theta_{i,m},\Omega_{m})\) & \(\Omega_{m}\) & \(\mathbf{\Omega}_{M}\) & \(\mathbf{\Theta}_{M}\) \\ \hline Meaning & Beampattern with \(\Omega_{m}\) & Random deviation angle & Set of \(\Omega_{m}\) & Set of \(\Theta_{i,m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the symbols in the system model. \([-\pi,\pi]^{M}\) represents the deviation and steering vector of \(\mathbb{M}\), respectively. Moreover, the received SNR is also given as  
\[F(\mathbf{\Theta}_{M})=\frac{\left|y(\varphi_{0}|\mathbf{\Theta}_{M},\mathbf{\Omega}_{M})-n \right|^{2}}{\sigma_{n}^{2}}. \tag{3}\]  
Figure 6a shows the simulated SNR with various values of \(N_{t}\). As observed, there are considerable gaps between the curves of \"ideal\" and \"practical\". Especially, with the increment of \(N_{t}\), the difference becomes more distinct (e.g., when \(M=12\), the differences at \(N_{t}=3\) and at \(N_{t}=12\) are 8.8 dB and 18.6 dB, respectively). This can be explained by the fact that when the higher \(N_{t}\) is equivalent to the higher directivity; this leads to the increased number of sensors which do not illuminate \(AP\). From the above results, it is recognized that an efficient algorithm for beam switching is required for the SNR enhancement in our scenario.