# Cell Selection Technique for Millimeter-Wave Cellular Systems with Hybrid Beamforming
## 3 Proposed Cell Selection Technique
### Cell and Beam Synchronization Signal (CBSS)
As seen in Equation (7), the time domain version of the CBSS can be represented as a Chu sequence with a cyclic shift and its amplitude is given by \(\left|x^{c,b}(n)\right|=1/\sqrt{N_{z}}\) for all \(n\). From Equations (7) and (8), it can be seen that the time domain version of CBSS, \(x^{c,b}(n)\), has a constant amplitude and zero-autocorrelation except at the lag \(l\) where \(\text{mod}(l,N)=0\), thereby satisfying the constant-amplitude zero-autocorrelation (CAZAC) property. The CAZAC property holds if and only if Chu's root index and sequence length are co-prime. The intra-cell correlation (correlation between two beams \(b\) and \(\overline{b}\) in the same cell) and inter-cell correlation (correlation between a beam in one cell and a beam in the adjacent cell) are analyzed in an ideal condition, ignoring the channel and noise effects, as follows:  
\[ \begin{array}{ll}\text{Z}^{c}\overline{b}&=\sum\limits_{k=0}^{N_{z}-1}X^{c, b}(k)\Big{[}X^{c,\overline{b}}(k)\Big{]}^{*}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|\pi}{N_{z}}(r^{c}k(k+1)-2 bLk-r^{c}k(k+ 1)+2\overline{b}Lk)}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|\pi}{N_{z}}kL(\overline{b}-b)}=N_{z} \delta\Big{(}\text{mod}\Big{(}b,\overline{b}\Big{)}\Big{)}\end{array}  \tag{9}\]  
and  
\[ \begin{array}{ll}\text{Z}^{c}\overline{b}&=\sum\limits_{k=0}^{N_{z}-1}X^{c, b}(k)\Big{[}X^{\overline{c},\overline{b}}(k)\Big{]}^{*}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|\pi}{N_{z}}(r^{c}k(k+1)-2 bLk-r^{c}k(k+ 1)+2\overline{b}Lk)}\\ &=e^{\frac{-|\pi}{N_{z}}L(\overline{b}-b)((r^{c}-r^{c})^{-1}L(\overline{b}-b) +1)}\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|\pi}{N_{z}}k(k+1)(r^{c}-r^{c})}\\ &=e^{\frac{-|\pi}{N_{z}}L(\overline{b}-b)((r^{c}-r^{c})^{-1}L(\overline{b}-b) +1)}\sum\limits_{k=0}^{c\overline{b}}\end{array}  \tag{10}\]  
where  
\[\sum\limits_{k=0}^{c\overline{b}}=\sqrt{N_{z}}\left\{ \begin{array}{ll} \left\langle\frac{x|r^{c}-r^{c}|}{N_{z}}\right\rangle\frac{1-j^{N_{z}}}{1-j}e ^{\frac{-2\pi}{N_{z}}(r^{c}-r^{c})\chi\gamma^{2}},\ r^{c}>r^{\overline{c}}\\ \left\langle\frac{x|r^{c}-r^{c}|}{N_{z}}\right\rangle\frac{1+j^{N_{z}}}{1+j}e ^{\frac{-2\pi}{N_{z}}(r^{c}-r^{c})\chi\gamma^{2}},\ r^{c}<r^{\overline{c}}\end{array} \right. \tag{11}\]  
Equation (10) can be derived using the Gauss sum property [27]. From Equations (9) and (10), it can be observed that the intra-cell correlation has a peak amplitude \(N_{z}\) only when \(b=\overline{b}\) for \(b,\overline{b}\in\mathcal{N}_{B}\), and the amplitude of the inter-cell correlation is given by \(\sqrt{N_{z}}\). As can be seen in Equations (9) and (10), CBSSs with the same Chu's root indices are perfectly orthogonal, and CBSSs with different Chu's root indices are not orthogonal. However, the interference term caused by the beam transmitted from neighboring BSs is relatively small compared to the signal from the reference beam. For example, the ratio between \(\sqrt{N_{z}}\) and \(N_{z}\) is 3% when \(N_{z}=1021\). When an STO (\(\delta_{t}\)) is present, the intra-cell correlation is given by:  
\[ \begin{array}{ll}Z^{c,\overline{b}}&=\sum\limits_{k=0}^{N_{z}-1}X^{c,b}(k)e^ {\frac{-2 mk_{k}}{N_{z}}}\left[X^{c,\overline{b}}(k)\right]^{*}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{k\pi}{N_{z}}(r^{\prime}k(k+1)-2 bLk-r^{ \prime}k(k+1)+2\overline{b}Lk-2 bk_{\overline{t}})}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{k\pi}{N_{z}}k(L(\overline{b}-b)-\delta_{ t})}\\ &=N_{z}\delta\left(\text{mod}\left(b-\overline{b},\delta_{t}\right)\right) \end{array}  \tag{12}\]  
The correlation value in Equation (12) becomes high when the following condition is satisfied:  
\[L=\frac{\delta_{t}}{\Delta b},\ L>0 \tag{13}\]  
where \(\Delta b\) is defined as \(\overline{b}-b\). To avoid the condition in Equation (13), the parameter \(L\) needs to be adjusted and can be set to the following value:  
\[L>\delta_{t_{\text{max}}},\ L_{\text{max}}=\left\lfloor\frac{N_{z}}{N_{B}}\right\rfloor \tag{14}\]  
where \(\delta_{t_{\text{max}}}\) and \(L_{\text{max}}\) denote the maximum STO and maximum value of parameter \(L\), respectively.