## Appendix A Analysis of the Angular Dependence of the Signal Acquisition Performance
**Numerical Simulations:** We consider an isotropic noise field [4; 12] that renders the noise covariance matrix of \(\Sigma_{\mathbf{n}}=\text{diag}(\sigma_{x}^{2},\sigma_{y}^{2},\sigma_{p}^{2})\) for the three-channel vector sensor receiver, where \(\sigma_{x}^{2}=\sigma_{y}^{2}=\sigma_{p}^{2}/2\). Then, we generate 1000 complex normal realizations of the \(3N\times 1\) complex noise vector \(\mathbf{n}\) in (9) that includes \(N\) samples of the noise for each of the three channels of the vector sensor receiver. Then, using (12), we simulate 1000 values for the decision statistic \(\Lambda\) that, for a given \(P_{FA}\), allows the computation of the corresponding decision threshold \(\gamma\) such that \(P_{FA}=\text{Pr}(\Lambda>\gamma|H_{0})\) and \(H_{0}\) represents the noise-only scenario. To compute \(P_{A}\) for this \(\gamma\) according to \(P_{A}=\text{Pr}(\Lambda>\gamma|H_{1})\), where \(H_{1}\) represents the signal-in-noise scenario, we need to simulate 1000 values for the decision statistic \(\Lambda\) under \(H_{1}\) and using (12) as well. Therefore, 1000 realizations of the \(3N\times 1\) complex vector \(\mathbf{r}=\mathbf{Sh}+\mathbf{n}\) of signal plus noise in (3) are also generated, where \(\mathbf{r}\) includes \(N\) samples of the received LFM signal plus noise for each of the three channels of the vector sensor receiver. Here, we note that based on (6), the \(3N\times 3\) complex matrix \(\mathbf{S}\) is composed of \(N\) LFM signal samples, where the LFM signal and its parameters are given in Section 4.1. Moreover, to generate the realizations of \(\mathbf{r}\), we generate 1000 complex normal realizations of the three channels of the vector sensor receiver \(\mathbf{h}=[h_{x}h_{y}\ h_{p}]^{T}\) with the covariance matrix of \(\Sigma_{\mathbf{h}}=E[\mathbf{hh}^{\dagger}]\), for which the elements are given by Equations (A3)-(A7). By repeating the above simulation steps for the \(P_{FA}\) values shown in Figures 10 and 11, the associated \(P_{A}\) values are computed and used to graph the performance curves presented in these two figures. The considered numerical values for the parameters in the simulations are \(\sigma_{p}^{2}=0.1\), \(N=1000\), \(\
u=20\), and \(\overline{\varphi}=10^{\text{o}}\) or \(80^{\text{o}}\), which indicates that the transmitter is close to the \(x\) axis or the \(y\) axis, respectively. The simulated \(P_{A}\) versus \(P_{FA}\) curve for each individual channel of the ring vector sensor receiver is shown in Figure 10, when the transmitter is close to the \(x\) axis. We observe that the acquisition performance of the \(x\) channel is better than that of the \(y\) channel. This can be attributed to the higher SNR of the \(x\) channel, \(10\log_{10}(\eta_{x}^{2}/\sigma_{x}^{2})=12.7\) dB, compared to the SNR of the \(y\) channel, \(10\log_{10}(\eta_{y}^{2}/\sigma_{y}^{2})=2\) dB. The situation is reversed and the \(y\) channel performance becomes better than the \(x\) channel, as seen in Figure 11, when the transmitter is close to the \(y\) axis. This is because in this case, SNRs of the \(x\) and \(y\) channels are changed to 2 and 12.7 dB, respectively, due to the change in the transmitter position. Additionally, we observe that in both figures, the multichannel combining method performs better than each individual channel.