# Cell Selection Technique for Millimeter-Wave Cellular Systems with Hybrid Beamforming
## 3 Proposed Cell Selection Technique
### Channel Estimation and Cell Selection
\begin{array}{ll}\mathbf{A}_{TXD}=\left\{\begin{array}{ll}\left[\begin{array} []{cccc}\mathbf{a}_{TX}^{\text{ULA}}(\phi_{1})&\cdots&\mathbf{a}_{TX}^{\text{ULA }}(\phi_{G_{TX}})\end{array}\right]&\text{for ULA}\\ \mathbf{a}_{TX}^{\text{UPA}}(\phi_{1},\theta_{1})&\cdots&\mathbf{a}_{TX}^{ \text{UPA}}\left(\phi_{G_{TX}^{\overline{c}}},\theta_{1}\right)&\cdots&\text{ for UPA}\\ \mathbf{a}_{TX}^{\text{UPA}}\left(\phi_{1},\theta_{G_{TX}^{\overline{c}}} \right)&\cdots&\mathbf{a}_{TX}^{\text{UPA}}\left(\phi_{G_{TX}^{\overline{c}}}, \theta_{G_{TX}^{\overline{c}}}\right)\end{array}\right]&\text{for ULA}\\ \mathbf{A}_{RXD}=\left\{\begin{array}{ll}\left[\begin{array}{cccc}\mathbf{a }_{RX}^{\text{ULA}}(\phi_{1})&\cdots&\mathbf{a}_{RX}^{\text{ULA}}(\phi_{G_{ RX}})\end{array}\right]&\text{for ULA}\\ \mathbf{a}_{RX}^{\text{UPA}}(\alpha_{1},\varphi_{1})&\cdots&\mathbf{a}_{RX}^{ \text{UPA}}\left(\alpha_{G_{RX}^{\overline{c}}},\varphi_{1}\right)&\cdots& \text{for UPA}\\ \mathbf{a}_{RX}^{\text{UPA}}\left(\alpha_{1},\varphi_{G_{RX}^{\overline{c}}} \right)&\cdots&\mathbf{a}_{RX}^{\text{UPA}}\left(\alpha_{G_{RX}^{\overline{c} }},\varphi_{G_{RX}^{\overline{c}}}\right)\end{array}\right]&\end{array} \right.\end{array} \tag{18}\]  
where \(\mathbf{A}_{TXD}\in\mathbb{C}^{N_{TX}\times G_{TX}}\) and \(\mathbf{A}_{RXD}\in\mathbb{C}^{N_{RX}\times G_{RX}}\) are defined for ULA. Here, \(G_{TX}\) and \(G_{RX}\) denote uniform fine grid points in \([-\pi/2,\pi/2)\) at the BS and MS, respectively. Also, \(\mathbf{A}_{TXD}\in\mathbb{C}^{N_{TX}\times(G_{TX}^{\overline{c}}\times G_{ TX}^{\overline{c}})}\) and \(\mathbf{A}_{RXD}\in\mathbb{C}^{N_{RX}\times(G_{RX}^{\overline{c}}\times G_{ RX}^{\overline{c}})}\) are defined for the UPA. Here, \(G_{TX}^{\overline{c}}(G_{TX}^{\overline{c}})\) and \(G_{RX}^{\overline{c}}\left(G_{RX}^{\overline{c}}\right)\) denote uniform fine grid points in the azimuth (elevation) domain with \([-\pi/2,\pi/2)\) (\([0,\pi/1)\)) at the BS and MS, respectively. In Equation (17), \(\mathbf{A}^{\overline{c}}\) is an \(N_{B}N_{I}\times G_{TX}G_{RX}\) and \(N_{B}N_{I}\times\left(G_{RX}^{\overline{c}}\times G_{RX}^{\overline{c}} \times G_{TX}^{\overline{c}}\right)\) matrix for ULA and UPA, respectively. In Equation (2), it is assumed that azimuth (elevation) AoA and AoD, \(\mathbf{a}_{TI}^{\overline{c}}\left(\mathbf{\varphi}_{TI}^{\overline{c}}\right)\) and \(\mathbf{\varphi}_{TI}^{\overline{c}}\left(\mathbf{\theta}_{TI}^{\overline{c}}\right)\), are taken from the above-mentioned uniform fine grids. Next, all the \(N_{B}N_{I}\) correlation values are stacked in a vector form as follows:  
\[\mathbf{Z}^{\overline{c}}=\left[ \begin{array}{cccc}\mathbf{Z}^{\overline{c}, 0}&\mathbf{Z}^{\overline{c},1}&\cdots&\mathbf{Z}^{\overline{c},N_{b}-1}\end{array} \right] \tag{19}\]  
\[\mathbf{Z}^{\overline{c},b}=\left[ \begin{array}{cccc}\mathbf{Z}^{\overline{c },b}_{0}&Z^{\overline{c},b}_{1}&\cdots&Z^{\overline{c},b}_{N_{i}-1}\end{array} \right] \tag{20}\]  
Equations (17) and (19) are used to obtain AoAs and AoDs of channel paths. The orthogonal matching pursuit (OMP) algorithm is used to obtain these values because of its simplicity and fast implementation [23; 28]. Stagewise OMP (StOMP) or generalized OMP (gOMP) algorithms can also be considered for estimation of AoAs and AoDs of channel paths [29; 30].