# Angle-of-Arrival Estimation Using Difference Beams in Localized Hybrid Arrays
## 3 Proposed AOA Estimation Approach
### Estimation of \(Nu\)  
We apply Equation (4) to the estimation of \(Nu\), which is then used to suppress \(e^{jmNu}\) of \(s_{m}(t)\) in (1) followed by the estimation of \(u\). Substituting Equation (4) into Equation (2), we have  
\[P_{m}(u,t)=\sum_{n=0}^{N/2-1}e^{jn(u+a_{m}(t))}\Big{(}1-e^{i\frac{N}{2}(u+a_{ m}(t))}\Big{)}=-2 je^{i(N-1)\omega_{m}(u,t)}\frac{\sin^{2}\Big{(}\frac{N\omega_{ m}(u,t)}{2}\Big{)}}{\sin(\omega_{m}(u,t))}, \tag{5}\]  
Figure 2: An example of normalized synthesized patterns of difference beams and sum beams.  
where \(\omega_{m}(u,t)=(u+\omega_{m}(t))/2\). For convenience of illustration, we consider the first \(K\) subarrays even though the results apply to the remaining \(M-K\) subarrays. Therefore, \(\omega_{m}(u,t)\) is simplified as \(\omega_{m}(u,t)=\frac{u}{2}-\pi\big{(}\frac{m}{K}+\frac{t}{L}\big{)}\).  
Substituting Equation (5) into Equation (3), \(\rho_{m}(t)\) can be given by  
\[\rho_{m}(t)=|\tilde{s}(t)|^{2}G_{m}(u,t)e^{jNu}+z_{m}(t), \tag{6}\]  
where  
\[G_{m}(u,t)=4e^{\frac{-(N-1)\pi}{K}}\frac{\sin^{2}\Big{(}\frac{N\omega_{m}(u,t) }{2}\Big{)}\sin^{2}\Big{(}\frac{N\omega_{m+1}(u,t)}{2}\Big{)}}{\sin(\omega_{m} (u,t))\sin(\omega_{m+1}(u,t))}.\]  
As specified in Lemma 1[5], there exists a unique integer \(m^{\prime}\in[0,K-1]\) satisfying \(\sin(\omega_{m^{\prime}}(u,t))\sin(\omega_{m^{\prime}+1}(u,t))<0\). Given \(m^{\prime}\), we have  
\[4\sin^{2}\Big{(}\frac{N\omega_{m}(u,t)}{2}\Big{)}\sin^{2}\Big{(} \frac{N\omega_{m+1}(u,t)}{2}\Big{)}\] \[= 1-\cos(N\omega_{m}(u,t))-\cos(N\omega_{m+1}(u,t))+\cos(N\omega_ {m}(u,t))\cos(N\omega_{m+1}(u,t))\] \[= 1-2\cos\!\left(N\omega_{m^{\prime}}(u,t)+\frac{[2(m^{\prime}-m)- 1]Q\pi}{2}\right)\cos\!\left(\frac{Q\pi}{2}\right)+(-1)^{[2(m^{\prime}-m)-1]Q} \cos^{2}(N\omega_{m^{\prime}}(u,t)), \tag{7}\]  
since \(\omega_{m}(u,t)=\omega_{m^{\prime}}(u,t)+\frac{(m^{\prime}-m)\pi}{K}\). Considering two cases of \(Q\), i.e., even and odd, we have  
\[G_{m}(u,t)=\begin{cases}\frac{e^{j}\frac{e^{j}}{K}[1-\cos(N\omega_{m^{\prime} }(u,t))]^{2}}{\sin(\omega_{m}(u,t))\sin(\omega_{m+1}(u,t))},&\text{$Q$ is even}\\ \frac{-e^{j}\frac{e}{K}\sin^{2}(N\omega_{m^{\prime}}(u,t))}{\sin(\omega_{m}(u, t))\sin(\omega_{m+1}(u,t))}.&\text{$Q$ is odd}\end{cases} \tag{8}\]  
Furthermore, as stated in Lemma 2[5] that \(|\sin(\omega_{m^{\prime}}(u,t))\sin(\omega_{m^{\prime}+1}(u,t))|<|\sin(\omega _{m}(u,t))\sin(\omega_{m+1}(u,t))|\), \(\forall m\
eq m^{\prime}\), only \(G_{m^{\prime}}(u,t)\) with the largest amplitude has the opposite sign of the remaining since the numerator of \(G_{m}(u,t)\) does not change with \(m\). As a result, when the SNR is not very low, \(m^{\prime}\) can be determined by \(\rho_{m}(t)\) with the largest amplitude, i.e., \(m^{\prime}=\underset{m=0:K-1}{\operatorname{argmax}}\{|\rho_{m}(t)|\}\). Given \(m^{\prime}\), the signs of \(\rho_{m}(t)\) (\(m=0,...,K-1\)) can be aligned following \(\tilde{\rho}_{m}(t)=(-1)^{Q+1\{m=m^{\prime}\}}\rho_{m}(t)\) for \(\widehat{Nu}\), where \(\mathbb{1}\{\cdot\}\) is the indicator function. Note that, from the expression of \(\alpha_{m}(t)\), we have \(\rho_{m}(t)=\rho\mod\{m,K\}(t)\) (\(m=0,...,M-2\)) while ignoring \(z_{m}(t)\) in Equation (6), and hence the signs of \(\rho_{m}(t)\) can be further calibrated following Step 7 in Algorithm 1. As shown in Step 9 of Algorithm 1, \(\tilde{\rho}_{m}(t)\) across all subarrays and symbols can be coherently combined to improve the accuracy of \(\widehat{Nu}\).