# Cell Selection Technique for Millimeter-Wave Cellular Systems with Hybrid Beamforming
## 3 Proposed Cell Selection Technique
### Channel Estimation and Cell Selection  
Using the CBSSs, channels of candidate cells are estimated in the initialization stage. The channel estimation begins with the correlation between the received signal in Equation (1) and the known CBSSs. The correlation value between the signal received from the \(i\)-th Rx beam at the MS and reference CBSS is given by:  
\[ \begin{array}{llll}Z_{i}^{\overline{c},\overline{b}}&=&\sum\limits_{k=0}^{N _{z}-1}Y_{i}(k)\left[X^{\overline{c},\overline{b}}(k)\right]^{*}\\ &=&\sum\limits_{k=0}^{N_{z}-1}\sum\limits_{c=0}^{N_{z}-1}\sum\limits_{b=0}^{N _{z}-1}\mathbf{q}_{i}^{\prime}\mathbf{H}^{c}(k)\mathbf{p}^{c,b}X^{c,b}(k)\left[ X^{\overline{c},\overline{b}}(k)\right]^{*}+\sum\limits_{k=0}^{N_{z}-1}\mathbf{q}_{i} ^{\prime}\mathbf{W}(k)\left[X^{\overline{c},\overline{b}}(k)\right]^{*}\\ &=&\mathbf{q}_{i}^{\prime}\left(\sum\limits_{k=0}^{N_{z}-1}\mathbf{H}^{ \overline{c}}(k)X^{\overline{c},\overline{b}}(k)\left[X^{\overline{c}, \overline{b}}(k)\right]^{*}\right)\mathbf{p}^{\overline{c},\overline{b}}\\ &\quad+\sum\limits_{b=0,b,b\
eq\overline{b}}^{N_{z}-1}\mathbf{q}_{i}^{\prime} \left(\sum\limits_{k=0}^{N_{z}-1}\mathbf{H}^{\overline{c}}(k)X^{\overline{c}, b}(k)\left[X^{\overline{c},\overline{b}}(k)\right]^{*}\right)\mathbf{p}^{\overline{c},b}\\ &\quad+\sum\limits_{c=0,c\
eq\overline{c}}^{N_{z}-1}\sum\limits_{b=0}^{N_{z}-1} \mathbf{q}_{i}^{\prime}\left(\sum\limits_{k=0}^{N_{z}-1}\mathbf{H}^{c}(k)X^{ c,b}(k)\left[X^{\overline{c},\overline{b}}(k)\right]^{*}\right)\mathbf{p}^{c,b}+\sum \limits_{k=0}^{N_{z}-1}\mathbf{q}_{i}^{\prime}\mathbf{W}(k)\left[X^{\overline{c },\overline{b}}(k)\right]^{*}\end{array}  \tag{15}\]  
where \(X^{\overline{c},\overline{b}}(k)\) denote the reference CBSS with the reference CID \(\overline{c}\) and BID \(\overline{b}\). The correlation in Equation (15) should be performed for \(N_{\text{C}}\) cells and \(N_{B}\) Tx beams. Here, the vector \(\mathbf{q}_{i}\) is constructed to form an \(i\)-th Rx beam in the direction of \(m\)-th cluster, thereby receiving the signals only from the \(m\)-th cluster. If the \(N_{L}\) propagation rays contributed by the \(m\)th cluster are received within the sampling interval (\(T_{\text{S}}\)), the channel \(\mathbf{H}^{c}(k)\) with Tx and Rx beamforming can be written as \(\mathbf{H}^{\overline{c}}\). For example, when only two cells (cell 0 and cell 1) are present, the correlation in Equation (15) is given by:  
\[Z_{i}^{0,\overline{b}}=N_{z}\left(\mathbf{q}_{i}^{\prime}\mathbf{H}^{0}\mathbf{ p}^{0,\overline{b}}\right)+0+\sqrt{N_{z}}\sum\limits_{b=0}^{N_{z}-1}\left( \mathbf{q}_{i}^{\prime}\mathbf{H}^{1}\mathbf{p}^{1,b}\right) \tag{16}\]where the reference CID, \(\overline{c}\), is set to 0 and the noise effect is ignored. From the first two terms in Equations (15) and (16), it can be observed that there is no interference from other beams in the same cell. That is, the correlation value is high (\(N_{z}\)) when BIDs are the same, and the correlation value is zero when other BIDs are used. The third term in Equations (15) and (16) represents the interference term (\(\sqrt{N_{z}}\)) from beams in the other cells. In a vector form, \(\mathbf{q}_{i}\mathbf{H}^{\overline{c}}\mathbf{p}^{\overline{c},b}\) can also be written as follows:  
\[ \begin{array}{ll}\mathbf{q}_{i}^{\prime}\mathbf{H}^{\overline{c}}\mathbf{p} ^{\overline{c},b}&=\left(\left(\mathbf{p}^{\overline{c},b}\right)^{T}\otimes \mathbf{q}_{i}^{\prime}\right)\text{vec}(\mathbf{H}^{\overline{c}})\\ &=\left(\left(\mathbf{p}^{\overline{c},b}\right)^{T}\otimes\mathbf{q}_{i}^{ \prime}\right)(\mathbf{A}_{TXD}^{*}\otimes\mathbf{A}_{RXD})\mathbf{\beta}^{ \overline{c}}=\mathbf{A}^{\overline{c}}\mathbf{\beta}^{\overline{c}}\end{array}  \tag{17}\]  
where \(\mathbf{\beta}^{\overline{c}}\) is unknown path gains of \(N_{M}N_{L}\) propagation paths. The array response matrices (unitary discrete Fourier transform [DFT] matrices) at the BS and MS are defined for ULAs and UPAs, as follows:  
\[