# Optimized Combination of Local Beams for Wireless Sensor Networks
## 3 Proposed Algorithm  
As the first step of designing the practical algorithm to suppress the sidelobes, we consider the channel model first. Because the mmWave channel is sparse, it can be geometrically modeled as [38; 39]  
\[ \begin{array}{rcl}h_{i,j}\left(\varphi_{j}\mid\vartheta_{i}\right)&=&\sqrt{ \frac{1}{K_{ij}^{\prime}}}\left[\sqrt{K_{ij}}\beta_{i,j}^{1}\mathcal{AF}_{i} \left(\varphi_{j}\mid\vartheta_{i}\right)+\sum\limits_{\ell=2}^{L_{ij}}\frac{ \beta_{i,j}^{\ell}}{\sqrt{L_{ij}}}\mathcal{AF}_{i}\left(\varphi_{i,j}^{\ell} \mid\vartheta_{i}\right)\right]\\ &=&\sqrt{\frac{1}{K_{ij}^{\prime}}}\left[\sqrt{K_{ij}}\beta_{i,j}^{1}\sum \limits_{n=0}^{N-1}e^{-jk\left\{\rho_{i}\left(\varphi_{j}\right)-nd\left[\cos \left(\varphi_{j}-\Omega_{i}\right)-\cos\vartheta_{i}\right]\right\}}\right.\\ &+&\left.\frac{1}{\sqrt{L_{ij}}}\sum\limits_{\ell=2}^{L_{ij}}\sum\limits_{n=0 }^{N-1}\beta_{i,j}^{\ell}e^{-jk\left\{\rho_{i}\left(\varphi_{i,j}^{\ell} \right)-nd\left[\cos\left(\varphi_{i,j}^{\ell}-\Omega_{i}\right)-\cos \vartheta_{i}\right]\right\}}\right].\end{array}  \tag{5}\]  
where \(K_{i,j}\) is the Ricean K-factor between the node \(d_{i}\) and the AP \(a_{j}\), \(K_{ij}^{\prime}=1+K_{i,j}\), \(\beta_{i,j}^{\ell}\sim\mathcal{CN}\left(0,1\right)\) is the complex Gaussian channel coefficient of the \(\ell\)th path, \(\varphi_{i,j}^{\ell}\sim\mathcal{U}\left[\ -\ \pi,\pi\ \right)\) is the angle of departure (AoD) of the the non-line-of-path (NLOS) paths, and \(L_{i,j}\sim\mathcal{U}\left[\ 2,4\right]\) is the integer number of the paths. Additionally, using (5), the combined signal at \(a_{j}\) is given as  
\[y_{j}\left(\boldsymbol{\vartheta},\boldsymbol{\Gamma}_{j}\right)=\sum\limits_ {i=1}^{L_{\mathrm{c}}}\sqrt{P_{i}}zh_{i,j}\left(\varphi_{j}\mid\vartheta_{i} \right)e^{jk\Gamma_{i,j}}+n, \tag{6}\]  
where \(\boldsymbol{\vartheta}=\left[\vartheta_{1},\cdots,\vartheta_{L_{\mathrm{c}}}\right]\) and \(\boldsymbol{\Gamma}_{j}=\left[\Gamma_{1,j},\cdots,\Gamma_{L_{i,j}}\right]\) are the vectors of the steering angles and the phase offsets, individually; \(P_{i}\) is the transmission power of the \(i\)th node; \(z\) is the data signal satisfied with \(E\left\{\left|z\right|^{2}\right\}=1\); and \(n\sim\mathcal{CN}\left(0,\sigma_{n}^{2}\right)\) is the additive white Gaussian noise (AWGN) observed at \(a_{j}\). Then, the INR for \(j\
eq 1\) are given as  
\[ \begin{array}{rcl}\xi_{j}\left(\boldsymbol{\vartheta},\boldsymbol{\Gamma}_{1 }\right)&=&\frac{\left|y_{j}\left(\boldsymbol{\vartheta}, \boldsymbol{\Gamma}_{1}\right)-n\right|^{2}}{\sigma_{n}^{2}}\\ &=&\frac{\left|\sum\limits_{i=1}^{L_{\mathrm{c}}}\sqrt{P_{i}}zh_{i,j}\left( \varphi_{j}\mid\vartheta_{i}\right)e^{jk\Gamma_{i,j}}\right|^{2}}{\sigma_{n}^ {2}},\end{array}  \tag{7}\]  
and the average INR is defined as  
\[ \begin{array}{rcl}\Xi\left(\boldsymbol{\vartheta},\boldsymbol{\Gamma}_{1 }\right)&=&\sum\limits_{j=2}^{A}\xi_{j}\left(\boldsymbol{\vartheta}, \boldsymbol{\Gamma}_{1}\right)/\left(A-1\right)\\ &=&\sum\limits_{j=2}^{A}\left|y_{j}\left(\boldsymbol{\vartheta},\boldsymbol{ \Gamma}_{1}\right)-n\right|^{2}/\left[\sigma_{n}^{2}\left(A-1\right)\right]\\ &=&\sum\limits_{j=2}^{A}\left|\sum\limits_{i=1}^{L_{\mathrm{c}}}\sqrt{P_{i}}zh_ {i,j}\left(\varphi_{j}\mid\vartheta_{i}\right)e^{jk\Gamma_{i,j}}\right|^{2}/ \left[\sigma_{n}^{2}\left(A-1\right)\right].\end{array}