## 2 Proposed Method
### CID and AoD Estimation Method  
This section describes the method of estimating CID and AoD, using the signals received at an MS in a mmWave cellular system. As discussed in Section 2, the received signal in Equation (3) becomes the summation of the DFT of the ZC sequences with the root index corresponding to CID and the cyclic shift corresponding to subarray index \(b\) when \(\phi_{p}^{c}\) is 0. When \(\phi_{p}^{c}\) is not 0, the received signal is additionally cyclic-shifted by the amount \(d\cos\phi_{p}^{c}\). Therefore, if the inverse discrete Fourier transform (IDFT) operation is applied to the received signal, the result becomes the summation of ZC sequences multiplied by a linear phase rotation with a slope corresponding to the subarray index \(b\) and \(d\cos\phi_{p}^{c}\)  
Figure 1: An example of Tx beamformers at two neighboring BSs when the proposed beam weight generation method is used. caused by AoD. After the IDFT operation, the IDFT of the signal received from the BS with CID \(c\) at the \(i\)-th Rx beam in the MS is given as follows:  
\[ \begin{array}{ll}\widetilde{r}_{i,n}^{c}&=\sum\limits_{v=0}^{N_{SA}-1}r_{v,i}^ {c}e^{-j2\pi vn/N_{Tx}}\\ &=\sum\limits_{b=0}^{B-1}h_{0}\eta_{0,c}^{c}e^{j2\pi nd\cos\phi_{F}^{c}}e^{-j2 \pi bn/N_{Tx}}z_{n}^{\gamma}+\omega_{i,n}^{c}+\widetilde{w}_{i,n},\,n=0,\ldots,N_{SA}-1\end{array}  \tag{4}\]  
where \(\omega_{i,n}^{c}=\sum\limits_{b=0}^{B-1}\sum\limits_{p=1}^{N_{P}-1}h_{p}\eta_ {p,c}^{c}e^{j2\pi nd\cos\phi_{F}^{c}}e^{-j2\pi bn/N_{Tx}}z_{n}^{\gamma}\). Here, \(h_{0}\) and \(\omega\) are the channel coefficient of the dominant path, and the remaining terms except the dominant path in the received signal. If the ZC sequence is descrambled after the IDFT operation, only the phase rotation term remains in the received signal. Then, the correlation between the IDFT of the received signal in Equation (4) and the ZC sequence used for beam weight generation in Equation (1) is performed using the \(N_{F}\)-point FFT as follows:  
\[A_{ik}^{c,\widetilde{\gamma}}=\sum\limits_{n=0}^{N_{F}-1}\widetilde{r}_{i,n}^ {c}(\widetilde{z}_{n}^{\gamma})^{*}e^{-j2\pi nk/N_{F}} \tag{5}\]  
where \(\widetilde{r}_{i,0\leq n<N_{Tx}}^{c}=\widetilde{r}_{i,n}^{c}\), \(\widetilde{r}_{i,N_{Tx}\leq n<N_{F}}^{c}=0\), \(\widetilde{z}_{0\leq n<N_{Tx}}^{\widetilde{\gamma}}=z_{n}^{\widetilde{\gamma }}\), and \(\widetilde{z}_{N_{Tx}\leq n<N_{F}}^{\widetilde{\gamma}}=0\). Here, \(k\) denotes the sample index of FFT output. When \(\widetilde{\gamma}\) is \(\gamma\), Equation (5) is given by  
\[ \begin{array}{ll}A_{ik}^{c,\gamma}&=\sum\limits_{b=0}^{B-1}h_{0}\eta_{0,b}^ {c}\sum\limits_{n=0}^{N_{Tx}-1}e^{j2\pi n(d\cos\phi_{0}^{c}+b/N_{Tx}-k/N_{F})} +\sum\limits_{n=0}^{N_{F}-1}(\omega_{i,n}^{c}+\widetilde{w}_{n})(\widetilde{z }_{n}^{\gamma})^{*}e^{-j2\pi nk/N_{F}}\\ &=\sum\limits_{b=0}^{B-1}h_{0}\eta_{0,b}^{c}A_{b,k}^{c,\gamma}+\sum\limits_{n=0 }^{N_{F}-1}(\omega_{i,n}^{c}+\widetilde{w}_{n})(\widetilde{z}_{n}^{\gamma})^{ *}e^{-j2\pi nk/N_{F}}\end{array}  \tag{6}\]  
where \(A_{b,k}^{c,\gamma}=e^{j\pi(N_{Tx}-1)\Theta_{b,k}^{c}}\sin(\pi N_{Tx}\Theta_{ b,k}^{\gamma})/\sin(\pi\Theta_{b,k}^{\gamma})\), \(\Theta_{b,k}^{\gamma}=\Theta_{k}^{\gamma}+b/N_{Tx}\), and \(\Theta_{k}^{\gamma}=d\cos\phi^{c}-k/N_{Tx}\). From \(A_{b,k}^{c,\gamma}\) the AoD of the dominant path can be estimated. First, the value of \(k\) maximizing the power of \(A_{b,k}^{c,\gamma}\) in Equation (6) is calculated by  
\[k_{b}^{\max}=\Re((N_{F}d\cos\phi^{c}+bN_{F}/N_{Tx})_{\varphi_{k}N_{F}}) \tag{7}\]  
where \(\Re\) denotes the round operation. Note that \(k_{b+1}^{\max}\) has an offset of \(N_{F}/N_{Tx}\) against \(k_{b}^{\max}\) in Equation (7) and \(A_{b+1,b_{b+1}}^{c,\gamma}\) has a phase offset of \(e^{j2\pi nd\cos\phi^{c}bN_{SA}}\) against \(A_{b,k_{b}}^{c,\gamma}\) in Equation (6).