# Cell Selection Technique for Millimeter-Wave Cellular Systems with Hybrid Beamforming
## 3 Proposed Cell Selection Technique  
### Cell and Beam Synchronization Signal (CBSS)  
In order to select a serving cell, we need to estimate channels (AoAs and AoDs) for all candidate cells in the initialization stage. In the proposed technique, CBSSs are simultaneously transmitted from all Tx beams of candidate BSs to reduce the processing time required for channel estimation. Here, it is assumed that the synchronization and CID detection are performed by the conventional primary synchronization signal (PSS) and secondary synchronization signal (SSS), used in LTE systems [25]. In addition, it is assumed that one additional OFDM symbol carries the proposed CBSS, next to the SSS. The CBSS needs to carry the CID, as well as the BID, in order to be detected in the multi-cellenvironment. If the CBSS carries only BID information, it will be impossible to determine which cell had transmitted the BID. Therefore, a large number of different sequences (CID \(\times\) BID) need to be generated. In order to transmit the information in one OFDM symbol, the BID is mapped in association with the CID in the CBSS. The proposed CBSS can be viewed as a combination of a Chu sequence and a polyphase sequence [26]. In the proposed CBSS, a new sequence is generated by mapping the CID to the root index of the Chu sequence and the BID to the index of the polyphase sequence. Figure 2 shows the generation process of the CBSS. The CBSS is defined by the product of a prime-length Chu sequence \(X^{c}(k)\) and a polyphase sequence \(P^{b}(k)\) in the frequency domain as follows:  
\[ \begin{array}{ll}X^{c,b}(k)=&X^{c}(k)P^{b}(k)\\ X^{c}(k)=&e^{\frac{|m^{2}k(k+1)}{N_{k}}},\ P^{b}(k)=e^{\frac{-2\pi nk}{N_{k}}}, \ k=0,1,\ldots,N_{z}-1\end{array}  \tag{6}\]  
where, \(c\in\mathcal{N}_{\text{C}}\), \(b\in\mathcal{N}_{B},r^{c}\in\mathcal{N}_{\text{R}}\), \(\mathcal{N}_{\text{C}}=\left\{ \begin{array}{cccc}0&1&\ldots&N_{\text{C}}-1 \end{array} \right\}\), \(\mathcal{N}_{B}=\left\{ \begin{array}{cccc}0&1&\ldots&N_{B}-1\end{array} \right\}\), and \(\mathcal{N}_{R}=\left\{ \begin{array}{cccc}1&2&\ldots&N_{R}\end{array} \right\}\) denote the CID, BID, [PERSON]'s root index mapped with CID \(c\), set of CIDs, set of BIDs, and set of all possible [PERSON]'s root indices, respectively. \(N_{R}\) and \(N_{z}\) denote the number of all possible [PERSON]'s root index and sequence length, respectively. In the polyphase sequence of CBSS, a parameter \(L\) is introduced to avoid the incorrect detection of cell ID when a symbol timing offset (STO) exists. Next, the properties of the CBSS are analyzed. The relationship between the frequency domain and time domain of the CBSS is given by:  
\[ \begin{array}{ll}x^{c,b}(n)&=\frac{1}{N}\sum\limits_{k=0}^{N_{z}-1}X^{c,b}( k)e^{\frac{2\pi nk}{N_{z}}}\\ &=\frac{1}{N}\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|m^{c}}{N_{z}}(k(k+1)-2 nkL(r^{ c})^{-1}+2 kn(r^{c})^{-1})}\\ &=e^{\frac{|m^{c}}{N_{z}}((r^{c})^{-1}(n-bL)^{2}+(n-bL))}\times\underset{N}{ \overset{N}{\overset{N}{\underset{k=0}}}}X\Big{(}k+(r^{c})^{-1}(n-bL)\Big{)}\\ \therefore x^{c,b}(n)&=x^{c}(0)X^{c}\left(\left(r^{c}\right)^{-1}(n-bL)\right) \end{array}  \tag{7}\]  
Figure 2: The generation process of CBSS. where \(x^{c}(0)=\frac{1}{\sqrt{N_{z}}}\Big{\langle}\frac{\chi r^{c}}{N_{z}}\Big{\rangle} \frac{1-\frac{J^{N_{z}}}{1-j}}{1-j}e^{\frac{-2\pi}{N_{z}}r^{c}\chi\gamma^{2}}\), \(\chi=\frac{N_{z}+1}{2},\ \gamma=\frac{N_{z}-1}{2}\). The autocorrelation function of the CBSS in the frequency domain is given by:  
\[ \begin{array}{ll}R^{c,b}(l)&=\sum\limits_{k=0}^{N_{z}-1}X^{c,b}(k)\Big{[}X^{ c,b}(k+l)\Big{]}^{*}\\ &=\sum\limits_{k=0}^{N_{z}-1}e^{\frac{|\pi}{N_{z}}(r^{c}I(l+1)-2 Lbl)}e^{\frac{ 2\pi}{N_{z}}(r^{c}kl)}\\ &=N_{z}e^{\frac{|\pi}{N_{z}}(r^{c}I(l+1)-2 Lbl)}\delta(\text{mod}(l,N_{z}))\end{array}  \tag{8}\]  
where \(\delta(\cdot)\), \(\left(r^{c}\right)^{-1}\), and \(\left\langle\chi r^{c}/N_{z}\right\rangle\) denote the Kronecker delta function, the multiplicative inverse of \(r^{c}\), and the Legendre symbol with a value of \(-1\), \(0\), or \(1\), respectively.