## 2 Proposed Method  
### Proposed Beam Weight Generation Mehtod  
Unlike the other beamforming systems, the beam weight in the proposed method is generated using a ZC sequence with CID information of the BS. The ZC sequence is especially suitable for beam weight generation in mmWave cellular systems with a beamformer because it allows an MS to \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Meaning** & **Notation** & **Meaning** \\ \hline \(N_{Tx}\) & Number of Tx antennas & \(n_{0}^{\gamma}\) & Amount of cyclic shift of \(v\)-th symbol for beam \\  & & & weight vector with root index \(\gamma\) \\ \(B\) & Number of subarrays & \(G_{p}^{\gamma}\) & Amount of cyclic shift of \(b\)-th subarray for \\ \(N_{SA}\) & Number of antenna elements in a subarray & \(d_{h,p}^{\gamma}\) & Complex constant determined by (\(\gamma\), \(b\), \(\gamma\)) \\ \(n_{Tx}\) & Tx antenna element index & \(\mathbf{m}_{i}\) & DFT-based beam weight vector for \(i\)-th Rx beam \\ \(n_{SA}\) & Antenna element index in a subarray & \(\mathbf{z}_{p}^{c,Rx}\) & AoA vector for \(p\)-th path in a cell with CID \(c\) \\ \(v\) & Symbol index in beam training period & \(\mathbf{z}_{p}^{c,Tx}\) & AoD vector for \(p\)-th path in a cell BS with CID \(c\) \\ \(\gamma\) & Root index of ZC sequence & \(\mathbf{H}_{p}\) & Channel matrix for \(p\)-th path \\ \(b\) & Subarray index & \(n_{p,i}^{\gamma}\) & Beam gain of \(i\)-th Rx beam of MS for \(p\)-th path \\ \(c\) & Cell ID (CID) & \(\theta_{p}^{c}\) & AoA of \(p\)-th path in a cell with CID \(c\) \\ \(i\) & Rx beam index & \(\theta_{p}^{c}\) & AoD of \(p\)-th path in a cell with CID \(c\) \\ \(p\) & Path index of channel & \(h_{p}\) & Channel coefficient of \(p\)-th path \\ \(k\) & Sample index of FFT output & \(\mathbf{w}_{p}\) & AWGN vector in \(v\)-th training symbol period \\ \(z_{n}^{\gamma}\) & \(n\)-th element of ZC sequence with a root & \(w_{i,p}\) & Noise signal received at \(i\)-th Rx beam in \(v\)-th \\  & index \(\gamma\) & & training symbol period \\  & Beam weight vector of BS with CID \(c\) in \(v\)-th & & Time-domain signal received from BS with CID \\ \(\mathbf{P}_{c}^{c}\) & symbol period & \(r_{x,j}^{c}\) & \(c\) at \(i\)-th Rx beam in MS \\ \(\mathbf{P}_{h,p}^{c}\) & Beam weight vector of \(b\)-th subarray of BS & \(\mathbf{\gamma}_{i,n}^{c}\) & IDFT of signal received from BS with CID \(c\) at \\  & with CID \(c\) in \(v\)-th symbol period & \(r_{i,n}^{c}\) & \(i\)-th Rx beam in MS \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation table. distinguish neighboring BSs and multiple beams generated by the BSs. Although the ZC sequence has not been used for beam weight generation previously, it is often used for the design of synchronization signal and pilot signal owing to its good correlation property and low peak-to-average power ratio (PAPR) [18; 19]. In the proposed method, the CID of the BS is mapped to the root index of the ZC sequence. It is assumed that a BS has a uniform linear array (ULA) with \(N_{Tx}\) antenna elements, which is divided into \(B\) subarrays (\(N_{SA}=N_{Tx}/B\) antenna elements in a subarray). The beam weights of the \(n_{Tx}\)-th antenna element in the \(v\)-th symbol of the beam training period are generated by the ZC sequence with a root index \(\gamma\) as follows [19; 20]:  
\[ \begin{split}[\mathbf{p}_{v}^{c}]_{n_{Tx}}=[\mathbf{p}_{b,p}^{ c}]_{n_{SA}}=d_{b,p}^{\gamma^{\prime}}[\mathbf{z}_{m^{\gamma}_{v}+C^{\gamma}_{ b}}^{\gamma^{\prime}}]_{nSA}=d_{b,p}^{\gamma^{\prime}}z_{m^{\gamma}_{v}+C^{ \gamma}_{b}+n_{SA}}^{\gamma^{\prime}}\\ =d_{b,p}^{\gamma^{\prime}}e^{-j\gamma(m^{\gamma}_{v}+C^{\gamma}_{ b}+n_{SA})(m^{\gamma}_{v}+C^{\gamma}_{b}+n_{SA})+1)/N_{SA}}\\ =d_{b,p}^{\gamma^{\prime}}e^{-j\gamma n_{SA}(n_{SA}+1)/N_{SA}}e^{-j2 \gamma(m^{\gamma}_{v}+C^{\gamma}_{b})/N_{SA}}e^{-j\gamma(m^{\gamma}_{v}+C^{ \gamma}_{b})(m^{\gamma}_{v}+C^{\gamma}_{b}+1)/N_{SA}}\end{split}  \tag{1}\]  
where \(z_{n}^{\gamma^{\prime}}=e^{-j\gamma n(n+1)/N_{SA}}\).