## Appendix A Analysis of the Angular Dependence of the Signal Acquisition Performance  
Given the angular selectivity of a vector sensor, the transmitter position (transmitter angle) changes the power distribution among the channels of the vector sensor receiver [4]. To understand how the transmitter position affects the signal acquisition performance studied in this paper, suppose the transmitter is located at the angle \(\overline{\varphi}\) in the \(x-y\) plane. Using the angular von Mises PDF [21] for the signal angle-of-arrival \(\varphi\) around the mean \(\overline{\varphi}\)  
\[f(\varphi)=\frac{\exp(\
u\cos(\varphi-\overline{\varphi}))}{2\pi I_{0}(\
u)},\;\
u\geq 0,\;-\pi<\overline{\varphi}<\pi, \tag{19}\]  
we can obtain the closed-form expressions for the \(x\) and \(y\) channel powers that we need in this study. In the above PDF, \(\
u\) controls the width of the PDF and \(I_{\ell}(. )\) is the \(\ell\)-th order modified Bessel function of the first kind. An example of this PDF for \(\overline{\varphi}=30^{\circ}\) and \(\
u=20\) is shown in Figure 19. The von Mises PDF is widely used for the characterization and analysis of angular variables, and includes or approximates several other angular PDFs [22]; moreover, given its mathematical form, it can result in useful closed-form formulas for channel correlation functions and channel powers of the vector sensor, as demonstrated below. Let \(h_{p}(x_{0},y_{0})\) represent the acoustic pressure at a point \((x_{0},y_{0})\). By definition, the spatial correlation function of \(h_{p}\) is given by \(c_{h_{p}}(\Delta x,\Delta y)=E[h_{p}(x_{0},y_{0})h_{p}^{*}(x_{0}+\Delta x,y_{0}+ \Delta y)]\). Upon writing \(h_{p}(x_{0},y_{0})\) and \(h_{p}(x_{0}+\Delta x,y_{0}+\Delta y)\) as superpositions of plane waves received at the points \((x_{0},y_{0})\) and \((x_{0}+\Delta x,y_{0}+\Delta y)\) in the two-dimensional plane, respectively, it can be shown that \(c_{h_{p}}(\Delta x,\Delta y)=\eta_{p}^{2}E_{\varphi}[\exp\{j\kappa\sqrt{\left( \Delta x\right)^{2}+\left(\Delta y\right)^{2}}\cos(\varphi-\varphi_{0})\}]\)[23], where \(\eta_{p}^{2}=E[|h_{p}|^{2}]\) is the power of \(h_{p}\), \(\varphi_{0}=\tan^{-1}(\Delta y/\Delta x)\), \(\kappa=2\pi/\lambda\) and \(\lambda\) is the wavelength. By substituting \(f(\varphi)\) of (A1) and computing the mathematical expectation with respect to \(\varphi\) using [24], we obtain the following spatial correlation function  
\[c_{h_{p}}(\Delta x,\Delta y)=\eta_{p}^{2}\frac{J_{0}\left(j\sqrt{v^{2}-\kappa^{ 2}(\left(\Delta x\right)^{2}+\left(\Delta y\right)^{2})+j2\
u\kappa(\Delta x \cos\overline{\varphi}+\Delta y\sin\overline{\varphi})}\right)}{J_{0}(j\
u)}\] (A2)  
where \(J_{\ell}(0)\) is the \(\ell\)-th order Bessel function of the first kind and \(I_{\ell}(\zeta)=j^{-\ell}J_{\ell}(j\zeta)\). Since the acoustic particle velocity \(x\) and \(y\) components are the spatial gradients of the acoustic pressure along the \(x\) and \(y\) axes, that is, \(h_{x}=\partial h_{p}/\partial x\) and \(h_{y}=\partial h_{p}/\partial y\), taking certain derivatives of \(c_{h_{p}}(\Delta x,\Delta y)\) in (A2), according  
\[E[h_{p}h_{y}^{*}] = (jx)^{-1}\partial c_{h_{p}}(\Delta x=0,\Delta y)/\partial\Delta y \big{|}_{\Delta y=0}\,,\] (A7) \[= -j\eta_{p}^{2}\frac{I_{1}(y)}{I_{0}(y)}\sin\overline{\varphi}.\]  
Using Equations (A3) and (A4) and this property of Bessel functions \(\zeta J_{2}(\zeta)=2J_{1}(\zeta)-\zeta J_{0}(\zeta)\)[24], it can be shown that \(\eta_{x}^{2}+\eta_{y}^{2}=\eta_{p}^{2}\). Since the sum of the \(x\) and \(y\) particle velocity channel powers is constant, it means that as \(\overline{\varphi}\), which specifies the transmitter position, increases counterclockwise with respect to the positive \(x\) axis, the \(x\)-channel power \(\eta_{x}^{2}\) decreases, whereas the \(y\)-channel power \(\eta_{y}^{2}\) increases. To see how this affects the signal acquisition performance of the ring vector sensor receiver, we produce \(P_{A}\) versus \(P_{FA}\) performance curves via simulations, as explained below.