## 4 Simulation Results
Considering the worst case of \(\Delta\psi=10^{\circ}\), the approximation \(\cos(\Delta\psi)\approx 1\) induces an error, which can be expressed as the absolute value of \(1-\cos(10^{\circ})\). Since, in deriving the results with the legend \"Analytic error (first order approx)\", \(\cos(\Delta\psi)\approx 1\) is used, the approximation of \(\cos(\Delta\psi)\approx 1\) induces an error of the absolute value of \(1-\cos(10^{\circ})\) for \(\Delta\psi=10^{\circ}\), which is why the largest discrepancy between the results with the legend \"Simulation error\" and the results with the legend \"Analytic error (first order approx)\" occurs at the carrier frequency of 1.05 GHz. The discrepancy between the results with the legend \"Simulation error\" and the results with the legend \"Analytic error (first order approx)\" at the other carrier frequencies is smaller than the discrepancy at the carrier frequency of 1.05 GHz: from the simulation-based results in the lower plot of Figure 2b, the absolute value of the elevation estimation error, \(\Delta\psi\), is greatest at the carrier frequency of 1.05 GHz, implying that the largest discrepancy should occur at that frequency. Remember that, in the approximation of \(\cos(\Delta\psi)\approx 1\), the approximation error increases with the increase of the absolute value of \(\Delta\psi\). To reduce the approximation error in \(\cos(\Delta\psi)\approx 1\) for large \(\Delta\psi\), \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\) is adopted, and the associated results with the legend \"Analytic error (second order approx)\" are illustrated in the lower plot of Figure 2b. Since the approximation error in \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\) is much smaller than that in \(\cos(\Delta\psi)\approx 1\) for large \(\Delta\psi\), the agreement between the results with the legend \"Analytic error (second order approx)\" and the simulated results is much better than the agreement between the results with the legend \"Analytic error (first order approx)\" and the simulated results. This observation is consistent with the fact that the approximation \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\) is quite accurate for the worst case of \(\Delta\psi=10^{\circ}\) in comparison with the approximation \(\cos(\Delta\psi)\approx 1\). Note that the superiority of the approximation \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\) over the approximation \(\cos(\Delta\psi)\approx 1\) is clear when the absolute value of the elevation estimation error, \(\Delta\psi\), is large. The agreement between the analytically-derived results and the simulation-based results in the lower plot of Figure 2b can be improved by reducing the approximation error. This can be achieved by adopting the second order Taylor series, not the first order Taylor series expansion, in getting the analytic error. In the second order approximation of \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), the absolute value of the error is bounded by \(\frac{\Delta\psi^{4}}{4}\). In the lower plot of Figure 2b, the results for the incident signal of azimuth = 120\({}^{\circ}\), elevation = 10\({}^{\circ}\) are illustrated, where the analytic error with the legend \"Analytic error (second order approx)\" is obtained from the second order Taylor approximation, not the first order Taylor approximation. The analytic results with the legend \"Analytic error (second order approx)\" are closer to the simulation-based results than the analytic results with the legend \"Analytic error (first order approx)\", illustrating the improvement by adopting \(\cos(\Delta\psi)\approx 1-\frac{\Delta\psi^{2}}{2}\), rather than \(\cos(\Delta\psi)\approx 1\). It is clear that the agreement between the analytic results with the legend \"Analytic error (second order approx)\" and simulation-based results is better than the agreement between the analytic results with the legend \"Analytic error (First order approx)\" in the lower plots of Figures 2b and 3a,b.