# Cell Selection Technique for Millimeter-Wave Cellular Systems with Hybrid Beamforming
## 3 Proposed Cell Selection Technique
### Channel Estimation and Cell Selection
Using the vector \(\mathbf{a}^{\overline{c}}\) obtained by OMP, comprised of indices of AoAs and AoDs for all \(N_{M}N_{L}\) propagation paths, multipath gains \(\mathbf{\beta}^{\overline{c}}\) can be estimated as follows:  
\[\mathbf{\widetilde{\beta}}^{\overline{c}}=\left(\mathbf{A}^{\overline{c}^{ \prime}}(\mathbf{a}^{\overline{c}})\mathbf{A}^{\overline{c}}(\mathbf{a}^{ \overline{c}})\right)^{-1}\mathbf{A}^{\overline{c}^{\prime}}(\mathbf{a}^{ \overline{c}})\mathbf{Z}^{\overline{c}} \tag{21}\]  
Then, the estimated channel matrix can be constructed for each candidate cell as follows:  
\[\mathbf{\widetilde{H}}^{\overline{c}}=\mathbf{A}_{RXD}diag\left(\mathbf{ \widetilde{\beta}}^{\overline{c}}\right)\mathbf{A}^{\prime}{}_{TXD} \tag{22}\]  
Finally, a serving cell should be selected in the initialization stage, taking into consideration the hybrid beamforming structure, because a hybrid beamforming system will be used in the data transmission stage. Here, we select the serving cell that maximizes channel capacity in the hybrid beamforming system. The unconstrained optimal precoder \(\mathbf{F}_{\mathrm{opt}}^{\overline{r}}\) for hybrid beamformer is obtained by performing a singular value decomposition (SVD) of the estimated channel in Equation (22) as \(\widetilde{\mathbf{H}}^{\overline{r}}=\mathbf{U}^{\overline{r}}\mathbf{\Sigma} ^{\overline{r}}\mathbf{V}^{\overline{c}*}\)[13]. Here, \(\mathbf{U}^{\overline{r}}\) is an \(N_{RX}\times N_{RX}\) unitary matrix, \(\mathbf{\Sigma}^{\overline{r}}\) is an \(N_{RX}\times N_{TX}\) diagonal matrix, and \(\mathbf{V}^{\overline{r}}\) is an \(N_{TX}\times N_{TX}\) unitary matrix. The matrix \(\mathbf{V}^{\overline{r}}\) can be expressed as:  
\[\mathbf{V}^{\overline{r}}=\left[ \begin{array}{cccc}\mathbf{V}_{1}^{\overline {r}}&\mathbf{V}_{2}^{\overline{r}}&\ldots&\mathbf{V}_{N_{TX}}^{\overline{r}} \end{array} \right] \tag{23}\]  
where \(\mathbf{V}_{N_{TX}}^{\overline{r}}=\left[ \begin{array}{cccc}V_{N_{TX},1}^{ \overline{r}}&V_{N_{TX},2}^{\overline{r}}&\ldots&V_{N_{TX},N_{TX}}^{\overline{ r}}\end{array} \right]^{T}\in\mathbb{C}^{N_{TX}\times 1}\). The optimal unconstrained precoder for \(\widetilde{\mathbf{H}}^{\overline{r}}\) is given by \(\mathbf{F}_{\mathrm{opt}}^{\overline{r}}=\mathbf{V}_{1}^{\overline{r}}\). In general, the channel capacity of a MIMO system is expressed in terms of the channel matrix \(\widetilde{\mathbf{H}}^{\overline{r}}\) and precoder \(\mathbf{F}_{\mathrm{opt}}^{\overline{r}}\), as follows [31]:  
\[R^{\overline{r}}=\log_{2}\left|\mathbf{I}_{N_{s}}+\frac{\mathrm{SNR}}{N_{s}} \widetilde{\mathbf{H}}^{\overline{r}}\mathbf{F}_{\mathrm{opt}}^{\overline{r} }\times\left(\mathbf{F}_{\mathrm{opt}}^{\overline{r}}\widetilde{\mathbf{H}}^ {\overline{r}}\right)^{\prime}\right| \tag{24}\]  
When \(N_{RF}\) Tx chains are available in the hybrid beamformer at the BS, Equation (24) can be rewritten as:  
\[R^{\overline{r}}=\log_{2}\left|\mathbf{I}_{N_{s}}+\frac{\mathrm{SNR}}{N_{s}} \mathbf{\Sigma}_{N_{s}}^{\overline{r}}\mathbf{V}_{N_{s}}^{\overline{r}} \mathbf{F}_{\mathrm{opt}}^{\overline{r}}\times\left(\mathbf{\Sigma}_{N_{s}}^ {\overline{r}}\mathbf{V}_{N_{s}}^{\overline{r}}\mathbf{F}_{\mathrm{opt}}^{ \overline{r}}\right)^{\prime}\right| \tag{25}\]  
where \(\mathbf{\Sigma}_{N_{s}}^{\overline{r}}=\left[ \begin{array}{cccc}\Sigma_{1}^ {\overline{r}}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\Sigma_{N_{RF}}^{\overline{r}}\end{array} \right]\) and \(\mathbf{V}_{N_{k}}^{\overline{r}}=\left[ \begin{array}{cccc}\mathbf{V}_{1}^ {\overline{r}}&\cdots&\mathbf{V}_{N_{RF}}^{\overline{r}}\end{array} \right]\),\(N_{s}=N_{RF},N_{RF}\in N_{TX}\) and \(N_{RF}\leq N_{TX}\). By repeating the same procedure for all candidate cells, the cell providing the maximum channel capacity is selected as the optimal serving cell as follows:  
\[\mathrm{Serving\;Cell}(\mathcal{\mathcal{\mathcal{\mathcal{\mathcal{\mathcal{ \mathcal{\mathcal{\mathcal{\mathcal{\mathcal{\mathcal{\mathcal{\mathcal{ \mathcal{\mathcal{ \mathcal{ \mathcal{    \mathcal{      }}}}}}}}}}}}}}}}}}}}}}}  
and UPA.