## 4 Simulation Results  
In this section, simulation-based estimation errors are compared with analytically-derived estimation errors to validate the derived expressions for the analytically-derived estimation errors. The simulation condition is tabulated in Table 1.  
The results for the true azimuth and true elevation of \((\theta^{(0)}=120^{\circ},\psi^{(0)}=0^{\circ})\), \((\theta^{(0)}=120^{\circ},\psi^{(0)}=20^{\circ})\), \((\theta^{(0)}=120^{\circ},\psi^{(0)}=50^{\circ})\), and \((\theta^{(0)}=120^{\circ},\psi^{(0)}=80^{\circ})\) are illustrated in Figures 1(a),b and 1(b), respectively. Remember that, from (26), the analytic azimuth estimation error is always zero. The elevation estimation error with \"Analytic error (first order approx)\" and \"Analytic error (second order approx)\" are obtained from (34) and (35), respectively. For the elevation estimation error with \"Simulation error\", the exhaustive evaluation of (2), for given \(\theta^{(0)}\) and \(\psi^{(0)}\), with respect to two variables of \(\theta\) and \(\psi\) is performed, and \(\theta\) and \(\psi\) maximizing the cost function in (2) are chosen to be the estimate of the azimuth and the estimate of the elevation, respectively. The difference between the estimate of the azimuth and the true azimuth is defined as the simulation-based azimuth estimation error. The simulation-based elevation estimation error is similarly defined. In Figure 2, the \(y\)-axis represents the azimuth and elevation error and the \(x\)-axis represents the carrier frequency of the signal source. Note that the frequency of the array manifold is equal to 1 GHz, whose wavelength is denoted by \(\lambda\)  
Under the approximations of \(\cos(\Delta\psi)\approx 1\), \(\sin(\Delta)\approx\Delta\psi\), from (3), \(\Delta\psi\) for \(\psi^{(0)}=0^{\circ}\) is zero, which is why the analytic elevation error with the legend \"Analytic error (first order approx)\" is not shown in the lower plot of Figure 1(a). Remember that \(\Delta\psi\) denotes analytic elevation error. The results with the legend \"Analytic error (second order approx)\" show an excellent agreement with the simulation-based results in Figure 1(a), which validates our derivation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of sensors & 5 \\ Array architecture & UCA \\ Azimuth and elevation search step & 0.25\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation conditions. Figure 2: Analytic and simulated azimuth and elevation error with respect to the carrier frequency of the signal source for \(\psi^{(0)}=0^{\circ}\) and \(\psi^{(0)}=20^{\circ}\). As shown in the lower plot of Figure 2b, discrepancy between analytic results with the legend \"Analytic error (first order approx)\" in Figure 2b and the simulation error with the legend \"Simulation error\" occurs for an incident signal of \(\theta^{(0)}=120^{\circ}\), \(\psi^{(0)}=20^{\circ}\), and the discrepancy increases with the increase of \(|\Delta\psi|\), which is the absolute value of the simulation-based elevation estimation error. The simulation-based error is obtained from the cost function of an interferometer algorithm without any approximation. On the other hand, to get the analytic error, the approximations of \(\cos(\Delta\psi)\approx 1,\sin(\Delta\psi)\approx\Delta\psi\) are applied. Therefore, for the discrepancy between the simulated error and the analytically-derived error to be small, the approximation error in the Taylor series expansion should be small. In the approximation of \(\cos(\Delta\psi)\approx 1\), the absolute value of the error is bounded by \(\frac{\Delta\psi^{2}}{2}\), implying that the approximation error is not small enough to be negligible for large \(\Delta\psi\). As shown in the results with the legend \"Simulation error\" in the lower plot of Figure 2b, the largest value of the absolute value of \(\Delta\psi\), for the small true elevation angle of \(\psi^{(0)}=20^{\circ}\), is approximately \(10^{\circ}\) for the carrier frequency of 1.05 GHz.