## 4 Simulation Results
In the results with \"Simulation error\" in the lower plots of Figures 3a,b., the largest value of the absolute value of the elevation estimation error, \(\Delta\psi\), is approximately \(2^{\circ}\) and \(0.5^{\circ}\), respectively, both of which are much smaller than \(10^{\circ}\). Remember that the largest value of the absolute value of the simulation-based elevation estimation error in Figure 2b is approximately \(10^{\circ}\). The error bound of the absolute value of the error in the approximation \(\cos(\Delta\psi)\approx 1\) is given by \(\frac{\Delta\psi^{2}}{2}\), which gets smaller with the decrease of \(|\Delta\psi|\). Therefore, the approximation error due to the first order Taylor expansion in getting the analytic error gets smaller with the decrease of \(|\Delta\psi|\). The discrepancies between the simulated error and the analytically-derived error in the lower plot of Figure 3a is smaller than that in the lower plot of Figure 2b since simulation-based \(|\Delta\psi|\) in the lower plot of Figure 3a is smaller than that in the lower plot of Figure 2b, both for the results with \"Analytic error (first order approx)\" and for the results with \"Analytic error (second order approx)\". Similarly, both for the results with \"Analytic error (first order approx)\" and for the results with \"Analytic error (second order approx)\", the discrepancy between the simulated error and the analytically-derived error in the lower plot of Figure 3b is smaller than that in the lower plot of Figure 3a since simulation-based \(|\Delta\psi|\) in the lower plot of Figure 3b is smaller than that in the lower plot of Figure 3a. In Figures 4 and 5, the \(x\)-axis, the \(y\)-axis, and the \(z\)-axis represent the true azimuth angle of the incident signal, the true elevation angle of the incident signal, and the estimation error, respectively. The true azimuth angle of the incident signal increases from zero to 359 degrees in increments of \(5^{\circ}\). The true elevation angle of the incident signal increases from zero to 89 degrees in increments of \(5^{\circ}\). The frequency of the array manifold is 1 GHz, and the carrier frequency of the incident signal is 950 MHz. Figure 4 shows the simulation-based azimuth estimation error with respect to the true azimuth angle and the true elevation angle of the incident signal. Figure 5a illustrates the simulation-based elevation estimation error with respect to the true azimuth angle and the true elevation angle of the incident signal. Figure 5b shows the analytically-derived elevation error. It is shown that the error of the azimuth is always zero regardless of the true value of the azimuth and the elevation, and the error of the elevation depends only on the true elevation angle of an incident signal not on the true azimuth of an incident signal. Figures 6-10 show the simulation-based elevation error and analytical elevation error with respect to the true elevation. In Figure 6a,b, the frequencies of the incident signal are 950 MHz and 960 MHz, respectively. Figures 7a,b, 8a,b, 9a,b, and 10a,b illustrate the results for 970 MHz, 980 MHz, 990 MHz, 1010 MHz, 1020 MHz, 1030 MHz, 1040 MHz, and 1050 MHz, respectively. The analytically-derived elevation estimation errors show excellent agreement with the simulation-based elevation estimation error in Figures 6-10. For the evaluation of the analytically-derived elevation estimation error, the approximations of \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\) and \(\sin(\Delta\psi)\approx\Delta\psi\) are adopted. Figure 4: Simulation-based azimuth error (\(f_{RF}=\frac{c}{\lambda_{RF}}=950\) MHz). (**a**) Simulation error elevation with respect to the true azimuth and true elevation. (**b**) Simulation error with respect to the true azimuth. Figure 5: Analytic elevation error and simulation-based elevation error with respect to the true azimuth and the elevation of the incident signal (\(f_{RF}=\frac{c}{\lambda_{RF}}=950\) MHz).