## 2 Proposed Method
### Proposed Beam Weight Generation Mehtod
Here, \(z_{n}^{\gamma}\), \(\mathbf{p}_{v}^{c}\) and \(\mathbf{p}_{b,v}^{c}\) are the \(n\)-th element of the ZC sequence with a root index \(\gamma\), beam weight vector with \(N_{Tx}\), \(b\)-th subarray beam weight vector with \(N_{SA}\) elements, respectively. Also, \(m^{\gamma}_{v}\) and \(G^{\gamma}_{b}\) are the amount of cyclic shift of the \(v\)-th symbol and \(b\)-th subarray, respectively. The parameters, \(c=\gamma-1\), \(b=\lfloor n_{Tx}/B\rfloor\), \(n_{SA}=(n_{Tx})_{\varphi_{N_{SA}}}\), and \(z_{n}^{\gamma}\) are the CID, subarray index, antenna element index in a subarray, and \(n\)-th element of a ZC sequence with a root index \(\gamma\) and length \(N_{SA}\), respectively. \(\mathbf{z}_{m^{\gamma}_{v}+G^{\gamma}_{b}}^{\gamma^{\prime}}\) denotes a vector composed of the ZC sequence with the root index \(\gamma\) and cyclic shift \(m^{\gamma}_{v}+G^{\gamma}_{b}\). Also, \(\%\) and \(\left[.\right]_{n}\) denote the modulo operation and \(n\)-th element in a vector. In addition, \(d_{b,p}^{\gamma}\) and \(v\) denote a complex constant determined by parameters (\(\gamma\), \(b\), \(v\)) and the index of Tx beamforming weight vector of BS, respectively. The values of \(c\), \(b\), and \(v\) are ranged from 0 to \((N_{SA}-2)\), from 0 to \((B-1)\), and from 0 to \((N_{SA}-1)\), respectively. The Tx beams are generated by increasing the index \(v\) in Equation (1) from 0 to \((N_{SA}-1)\). This procedure is repeated until one round of an Rx beam sweep is completed. If \((\gamma m^{\gamma}_{v})_{\varpi_{N_{SA}}}\), \((\gamma G^{\gamma}_{b})_{\varphi_{N_{SA}}}\), and \(d_{b,p}^{\gamma^{\prime}}\) are given by \(v\), \(b\), and \(\left(\left[\mathbf{z}_{0}^{\gamma^{\prime}}\right]_{m^{\gamma}_{v}+G^{\gamma} _{b}}\right)^{*}\) respectively, \(\left[\mathbf{p}_{b,v}^{c}\right]_{n_{SA}}\) can be expressed as  
\[\left[\mathbf{p}_{b,v}^{c}\right]_{n_{SA}}=z_{n_{SA}}^{\gamma^{\prime}}e^{-j2 \pi\eta n_{SA}/N_{SA}}e^{-j2\pi\eta n_{SA}/N_{SA}} \tag{2}\]  
Note that the beampatterns generated by the subarray beam weight vector in Equation (1) are determined by the parameters (\(\gamma\), \(b\), \(v\), \(N_{SA}\)). The signal received at the \(i\)-th Rx beam of the MS from BS with CID \(c\) in the \(v\)-th period can be represented by [5]  
\[ \begin{split} r_{v,i}^{c}&=(\mathbf{m}_{i})^{H}\sum\limits_{ b=0}^{B-1}\sum\limits_{p=0}^{N_{p}-1}\mathbf{H}_{p}\mathbf{p}_{b,v}^{c}x_{v}^{c}+( \mathbf{m}_{i})^{H}\mathbf{w}_{v}\\ &=\sum\limits_{b=0}^{B-1}\sum\limits_{p=0}^{N_{p}-1}h_{p}\Big{\{}( \mathbf{m}_{i})^{H}\mathbf{a}_{p}^{c,Rx}\Big{\}}\Big{\{}(\mathbf{a}_{p}^{c,Tx })^{H}\mathbf{p}_{b,p}^{c}\Big{\}}x_{v}^{c}+(\mathbf{m}_{i})^{H}\mathbf{w}_{v} \\ &=\sum\limits_{b=0}^{B-1}\sum\limits_{p=0}^{N_{p}-1}h_{p}\Big{\{}( \mathbf{m}_{i})^{H}\mathbf{a}_{p}^{c,Rx}\Big{\}}\sum\limits_{n=0}^{N_{p}-1}e^ {j2\pi mrd\cos\phi_{p}^{c}}\big{[}\mathbf{p}_{b,v}^{c}\big{]}_{n}x_{v}^{c}+w_{v,i }\\ &=\sum\limits_{b=0}^{B-1}\sum\limits_{p=0}^{N_{p}-1}h_{p}\gamma_{p,j}^{N_{ Tx}-1}z_{n_{Tx}}^{\gamma^{\prime}=c+1}e^{-j2\pi(b+v-N_{Tx}d\cos\phi_{p}^{c})n_{Tx}/N_{Tx}}x_{v}^{c}+w_{v,i }\\ &=\sum\limits_{b=0}^{B-1}\sum\limits_{p=0}^{N_{p}-1}h_{p}\gamma_{p,j}^{c} \gamma_{b,v}^{c}+w_{v,i}\end{split}  \tag{3}\]where \(\mathbf{H}_{p}=\sum\limits_{p=0}^{N_{p}-1}h_{p}\mathbf{a}_{p}^{c,Rx}(\mathbf{a}_{p} ^{cTx})^{H}\), \(w_{v,i}=(\mathbf{m}_{i})^{H}\mathbf{w}_{v}\), \(\eta_{p,i}^{c}=(\mathbf{m}_{i})^{H}\mathbf{a}_{p}^{c,Rx}\), \(\left[\mathbf{a}_{p}^{c,Rx}\right]_{n}=e^{2\pi mud\cos\phi_{p}^{c}}\), \(\left[\mathbf{a}_{p}^{c,Tx}\right]_{n}=e^{2\pi mud\cos\phi_{p}^{c}}\), and \(y_{b,p}^{c}\)=\(\sum\limits_{n_{Tx}=0}^{N_{T}}2\mathbf{a}_{n_{Tx}}^{j=c-1}e^{-j\pi(h+v-N_{Tx}d\cos \phi_{p}^{c})\pi_{Tx}/N_{Tx}}\). Here, \(\mathbf{m}_{i}\) is the DFT-based beam weight vector for the \(i\)-th Rx beam. Also. \(\mathbf{a}_{p}^{c,Rx}\) and \(\mathbf{a}_{p}^{c,Tx}\) are AoA and AoD vectors for the \(p\)-th path in a cell with CID \(c\). \(\mathbf{H}_{p}\) and \(\eta_{p,i}^{c}\) are channel matrix and Rx beam gain for the \(p\)-th path, respectively.