# Low-Power Beam-Switching Technique for Power-Efficient Collaborative IoT Edge Devices
## 2 System Model and Motivation  
The symbols, which we used in this proposed system models, are summarized in Table 1. As observed in Figure 4, uplink data transmission between a WSN having \(N\) sensors, \(\mathbb{N}=\{n_{1},\cdots,n_{N}\}\), and an AP, \(AP\), being far from \(\mathbb{N}\), is considered. Both the terminals are positioned in the \(x-y\) plane, and their polar coordinates are represented as \((r_{n},\psi_{n})\) and \((A,\varphi_{0})\), respectively. Each sensor is equipped with a BSS with an antenna array connected to a beam switching matrix and circuit, and this selects one of the directional beams for the transmission. It is also assumed that due to the power constraint of wireless  
Figure 3: The concept of the collaborative beamforming (black dots: sensors; red dot: access point (AP)). sensors and the limited number of antennas in BSSs, there are no direct uplinks between each sensor and \(AP\). As shown in Figure 5, the beampattern of the BSS is simply defined as  
\[\hat{\mathcal{B}}(\Theta_{i,n})= \begin{cases}N_{t}&if\quad\tilde{\phi}_{n}(\varphi )\in\Xi_{i,n}\\ \\ 0&if\quad\tilde{\phi}_{n}(\varphi)\
otin\Xi_{i,n},\end{cases}  \tag{1}\]  
where \(\Theta_{i,n}=\frac{\pi(2i-N_{t}-1)}{N_{t}}\) is the steering angle of the \(i\)th beam with \(i\in\{1,\cdots,N_{t}\}\), \(\tilde{\phi}_{n}(\varphi)\) is the local azimuth angle of \(n_{n}\), \(\varphi\in[\ -\pi,\pi)\) is the global azimuth angle, and \(\Xi_{i,n}=\left[\frac{-\pi}{N_{t}}+\Theta_{i,n},\frac{\pi}{N_{t}}+\Theta_{i,n}\right]\) is the azimuth interval with \(\Xi_{i,n}\cap\Xi_{i^{\prime},n}=\emptyset\) for \(i\
eq i^{\prime}\). Figure 4: The system model (**a**) without or (**b**) with the effect of irregular mounting surface. Figure 5: The beampatterns of the BSS with \(N_{t}=3\) and \(\Delta\Xi_{i,m}=2\pi/N_{t}\). Let us state that one of the active source sensors, \(s^{k}\in\mathbb{S}=\left\{s^{1},\cdots,s^{K}\right\}\), \(K\ll N\), needs to transmit the data symbol \(z^{k}\in\mathbb{C}\), being satisfied with \(E\left[z^{k}\right]=0\), \(E\left[z^{k}\right]^{2}=1\), and \(E\left[z^{k}\
eq z^{k^{\prime}}\right]=0\) for \(k\
eq k^{\prime}\), in the direction of \(AP\). To establish the direct transmission between \(s^{k}\) and \(AP\) without considering a multi-hop strategy, the scheme of the CB is introduced to coherently combine beams of \(\mathbb{M}^{k}\) at \((A,\varphi_{0})\), where \(\mathbb{M}^{k}=\left\{m_{1}^{k},\cdots,m_{M}^{k}\right\}\) is satisfied with the condition of \(\cup_{k=1}^{K}\mathbb{M}^{k}=\mathbb{N}\) and \(\mathbb{M}^{k}\cap\mathbb{M}^{k^{\prime}}=\emptyset\) for \(k\
eq k^{\prime}\), and its elements are within the coverage radius \(R\) of \(s^{k}\). From here, the superscript \(k\) is omitted to pursue concise representation. As the preparatory step of the CB, each sensor selects the beam that minimizes \(\left|\phi_{m}(\varphi_{0})-\Theta_{i,m}\right|\). Afterward, \(s\) shares \(z\) with \(\mathbb{M}\), and then \(m_{m}\) broadcasts the data signal \(z\sqrt{P_{m}}e^{-j\Gamma_{m}(\varphi_{0})}\) through its beam [41]. Here, \(P_{m}\) is the transmission power of \(m_{m}\), \(\Gamma_{m}(\varphi_{0})=-\frac{2\pi}{\sqrt{A}}d_{m}(\varphi_{0})\) is the closed loop initial phase for the carrier synchronization of \(\mathbb{M}\), \(d_{m}(\varphi_{0})=\sqrt{A^{2}+\left(r_{m}\right)^{2}-2 Ar_{m}\cos(\varphi_{0 }-\psi_{m})}\) is the Euclidean distance between \(m_{m}\) and \(AP\), and \(\lambda\) is the wavelength of the operating frequency. Let \(\alpha_{m}\sim\exp\left\{N\big{(}0,\sigma_{\alpha}^{2}\big{)}\right\}\) denote the time-invariant and log-normal wireless channel between \(m_{m}\) and \(AP\)[59; 60].