## 2 Proposed Method
### CID and AoD Estimation Method
Using this relationship, the correlation values for the estimation of CID and AoD are obtained by combining \(A_{b,k}^{c,\gamma}\) as follows:  
\[\widetilde{A}_{i,k}^{c,\widetilde{\gamma}}=\frac{1}{B}\sum\limits_{b=0}^{B-1}A_ {i,b_{b}}^{c,\widetilde{\gamma}}e^{-j2\pi d\cos\widetilde{\phi}_{b}bN_{SA}} \tag{8}\]  
where \(\widetilde{\phi}_{k}= \begin{cases}\cos^{-1}(2k/(N_{F})),&0\leq k<N_{F}/2\\ \cos^{-1}(-2(N_{F}-k)/(N_{F})),&N_{F}/2\leq k<N_{F}\end{cases} k=0\
eg N_{F}-1, \quad\text{and}\quad k_{b}=(k+bN_{F}N_{0}/N_{Tx})_{\varphi_{k}N_{F}}\). In Equation (8), the peak value is located at \(k\) corresponding to the AoD of the path with the maximum received signal for each Rx beam. Using the combined correlation value, the AoD are estimated as follows:  
\[\hat{\phi}^{\hat{\gamma}}= \begin{cases}\cos^{-1}(2\hat{k}^{\hat{\gamma}}/(N_{F} )),&0\leq\hat{k}^{\hat{\gamma}}<N_{F}/2\\ \cos^{-1}(-2(N_{F}-\hat{k}^{\hat{\gamma}})/(N_{F})),&N_{F}/2\leq\hat{k}^{\hat{ \gamma}}<N_{F}\end{cases}  \tag{9}\]where \(\{i_{\max},\hat{\gamma},\hat{\xi}_{\gamma}\}=\operatorname*{argmax}_{[i,\hat{\gamma},k]}\lvert\overline{A}_{i,\hat{k}}^{c,\hat{\gamma}}\rvert^{2}\). Here, \(i_{\max}\) means the Rx beam index with the maximum received power. The CID corresponding to the estimated AoD is \(\hat{c}=\hat{\gamma}-1\). Figure 2 summarizes the procedure of CID and AoD estimation using the signals received from neighboring BSs with beam weights generated by the proposed method. Although the beamformer generated by the proposed beam weight generation method has a random beam pattern, the Signal-to-Noise Ratio (SNR) gain can be obtained by the correlation operation in Equation (5). The SNR after the correlation operation in Equation (5) can be obtained using the signal and noise terms in \(\mathbf{r}_{b}^{c}\). Here, \(\mathbf{r}_{b}^{c}\) denotes a received signal vector \([r_{0,i}^{c}\ldots\ r_{N_{SA-1},i}^{c}]^{T}\). When the channel has only LoS path and both \(B\) and \(\eta_{0,i}^{c}\) is 1, the signal component of \(\mathbf{r}_{b}^{c=\gamma-1}\) is given by  
\[\mathbf{y}_{0}^{\gamma-1}=[y_{0,0}^{\gamma-1}\ y_{0,1}^{\gamma-1}\ldots\ y_{0,N_{Tx}-1}^{\gamma-1}]^{T}=\mathbf{F}^{\gamma-1}\mathbf{M}\mathbf{z}_{0}^{ \gamma}x_{0}^{c}=\mathbf{F}^{\gamma-1}\overline{\mathbf{z}}_{0}^{\gamma}x_{0 }^{c} \tag{10}\]  
where \([\mathbf{M}]_{k}^{n}=\mathrm{e}^{-\beta\pi kn/N_{Tx}}\), \([\overline{\mathbf{z}}_{0}^{\gamma}]_{k}=z_{k}^{-\gamma-1}e^{-\beta\pi\frac{( \gamma-1)(1+N_{Tx})}{2}k/N_{Tx}}/N_{Tx}\sqrt{N_{Tx}}\bigg{(}\frac{\alpha^{+} \gamma}{N_{Tx}}\bigg{)}\frac{1+\beta^{N_{Tx}}}{1+j}e^{-\frac{i\beta\pi^{-} \alpha^{+}}{N_{Tx}}}\)[20], \(\alpha^{\pm}=(N_{Tx}\pm 1)/2\), \((\gamma_{-1}\gamma)_{\varphi_{\text{N}}N_{Tx}}=1\). Here, \(\gamma_{-1}\) denotes a modulo inverse of \(\gamma\). When \(N\) is a prime number, \((a/N_{Tx})\) becomes a Legendre symbol. The term \((a/N_{Tx})\) becomes \(\pm 1\) when \(\alpha\) is not 0. Also, \([\mathbf{F}^{\phi^{\gamma-1}}]_{k}^{\delta}\) is the element in the \(k\)-th row and \(\delta\)-th column of \(\mathbf{F}^{\phi^{\gamma-1}}\), which is a circular matrix and a function of AoD. \(\mathbf{F}^{\phi^{\gamma-1}}\) is expressed as follows:  
\[[\mathbf{F}^{\phi^{\gamma-1}}]_{k}^{\delta}=e^{j\pi(\mathbf{G}_{k}^{\gamma})( N_{Tx}-1)}\sin(\pi nN_{Tx}\mathbf{G}_{k}^{\gamma})/\sin(\pi n\mathbf{G}_{k}^{ \gamma})/N_{Tx} \tag{11}\]  
where \((\mathbf{F}^{\phi^{\gamma-1}})^{H}\mathbf{F}^{\phi^{\gamma-1}}=\mathbf{I}\). When the Tx signal \(x_{v}^{c}\) is a constant \(1/\sqrt{N_{Tx}}\), the correlation value of \((\mathbf{F}^{\phi^{\gamma-1}}\overline{\mathbf{z}}_{0}^{\gamma})^{H}\mathbf{ y}_{0}^{\gamma-1}\) becomes \((N_{Tx})^{2}\) and its power becomes \((N_{Tx})^{3}\). Also, when \(\mathbf{w}\) is a noise vector \([w_{0,i}\ldots w_{N_{SA-1},i}]^{T}\), the average power of \((\mathbf{F}^{\phi^{\gamma-1}}\overline{\mathbf{z}}_{0}^{\gamma})^{H}\mathbf{w}\) becomes \(\sigma^{2}(N_{Tx})^{2}\).