# Optimized Combination of Local Beams for Wireless Sensor Networks
## 2 System Model  
We consider a WSN, operating in mmWave spectrum band, with a cluster having \(I\) sensor nodes \(\mathcal{C}=\{c_{1},\cdots,c_{I}\}\) and \(A\) (\(A\ll I\)) APs \(\mathcal{A}=\{a_{1},\cdots,a_{A}\}\) which are in directions of \(\varphi_{1},\cdots,\varphi_{A}\), respectively. The sets of \(\mathcal{C}\) and \(\mathcal{A}\) can be used as the identifications (IDs) in the system. Under the assumption that the two terminals are coplanar on the \(x-y\) plane, the \(i\)th node is located at the coordinates of \(\left(r_{i},\psi_{i}\right)=\left(\sqrt{x_{i}^{2}+y_{i}^{2}},tan^{-1}\left( \frac{y_{i}}{x_{i}}\right)\right)\), and the \(j\)th AP is a Euclidean distance \(\rho_{i}\left(\varphi_{j}\right)=\sqrt{r_{i}^{2}+r_{A}^{2}-2r_{A}r_{i}cos\left( \varphi_{j}-\psi_{i}\right)}\approx r_{A}-r_{i}cos\left(\varphi_{j}-\psi_{i}\right)\) away from the corresponding node when \(r_{i}\ll r_{A}\). Due to the severe path loss within the cluster, the nodes need beamforming structures for the optimal directional link by considering operations of neighbor nodes. However, because they suffer from the limited battery-powered capacity, complex beamforming structures having multiple RF chains, (e.g., digital and hybrid schemes [34]), have disadvantages in terms of long-term operations of WSN. Thus, we apply a simple analog beamforming structure to each node, which is composed of a ULA with \(N\)\(d\)-wavelength spaced isotropic elements, \(N\) phase shifters, and a single RF chain. When the nodes are at the origin and the ULAs are parallel to the \(x\)-axis, their array factor can be simply defined as  
\[\mathcal{AF}\left(\phi\right)=\sum\limits_{n=0}^{N-1}e^{jknd[\cos\phi-\cos \theta]} \tag{1}\]  
where \(\phi\in[\ -\pi,\pi\ )\) is the azimuth angle, \(k=2\pi/\lambda\) is the propagation constant, \(\lambda\) is the wavelength at the operating frequency, and \(\vartheta\in[\ -\pi,\pi\ )\) is the steering angle of the ULA. However, in most cases, the positions of the nodes and the broadside directions of the ULAs are arbitrary. Therefore, we modify (1) to  
\[\mathcal{AF}_{i}\left(\phi\ |\ \theta_{i}\right)=\sum\limits_{n=0}^{N-1}e^{-jk \left\{\rho_{i}(\phi)-nd[\cos(\hat{\varphi}_{i}(\phi)-\Omega_{i})-\cos\theta_ {i}]\right\}} \tag{2}\]  
with the following assumptions: (1) the leftmost element of the ULA is placed on \(\left(r_{i},\psi_{i}\right)\) following the random distribution; (2) \(\left(r_{i},\psi_{i}\right)\) is the origin of the local \(u-v\) plane, where the \(u\) and \(v\) axes are parallel to \(x\) and \(y\) axes, respectively; (3) the included angle \(\Omega_{i}\) between the \(u\)-axis and the line of the elements follows the uniform distribution of \(\mathcal{U}\left[-\pi,\pi\ \right)\) due to the irregular surface where the nodes are mounted. In addition, \(\mathcal{AF}_{i}\left(\cdot\right)\) is the array factor of the \(i\)th node; \(\hat{\varphi}_{i}\left(\phi\right)=cos^{-1}\left(\hat{\mathbf{a}}_{i,i}\cdot \hat{\mathbf{u}}\right)\) is the local azimuth angle with respect to the \(u\)-axis, which is simply converged to \(\phi\) under the far-field condition; \(\theta_{i}\) is the steering angle of \(i\)th node; \(\hat{\mathbf{a}}_{i,j}\) is the unit radial vector from \(\left(r_{i}cos\psi_{i},r_{i}sin\psi_{i}\right)\) to \(\left(r_{A}cos\phi,r_{A}sin\phi\right)\); and \(\hat{\mathbf{u}}=\left[1\ \ \ 0\right]\) is the unit vector in the direction of the \(u\)-axis. To combine the beams coherently in the direction of \(\phi=\varphi_{j}\), \(\mathfrak{S}\) starts sharing the data signal and the synchronization bits [35] with the \(I_{c}\) collaborative nodes inside its maximum communication radius \(R\), denoted as the \(\mathcal{D}=\{d_{1},\cdots,d_{I_{c}}\}\).