## 7 Simulation Results
### Simulation Results  
The simulation conditions are: the radius of the antenna plate is 200 mm; the frequency range of incident signal is 1-15 GHz. In each experiment, only one incident signal is considered. The azimuth and elevation of the incident signal are \((45^{\circ},60^{\circ})\) and \((45^{\circ},80^{\circ})\) respectively. Figure 10: Antenna placement. (**a**) 2D array of the planar spiral antenna; (**b**) 3D array of LPDA conformal antenna. Experiment 1 Comparison of two algorithms for solving ambiguous probability with different frequency. The incident angles of elevations are \(\beta=80^{\circ}\) and \(\beta=60^{\circ}\) respectively, and the SNR is 14 dB. The average number of phase difference is \(N=20\), and 200 Monte-Carlo experiments are performed. The results of solving ambiguous probability are shown in Figure 11. It can be seen from Figure 11a that the incident signals are from 1 GHz to 15 GHz. The probability of solving ambiguous problem of both the 3D array and the 2D array is more than 86%. The high probability of solving ambiguity verifies that the scheme proposed in this paper can achieve no ambiguous direction finding in ultra-wideband. As shown in Figure 11, the probability of solving ambiguity is decreasing with the frequency increasing. When we select actual value from the ambiguous values, the number of the ambiguous values at high frequency is enormous. The procedure gets more complicated and difficult which causes the behavior described above. It can be seen more intuitively in Equation (11). If the wavelength \(\lambda\) in the right of the equation becomes shorter, obviously the \(k\) in the left of the equation gets bigger and the number of ambiguous values increases. Comparing Figure 11a with Figure 11b, it can be seen that the probability of solving ambiguity at \(\beta=60^{\circ}\) is less than that at \(\beta=80^{\circ}\). This is due to the fact that the incident signal is relatively further with the antenna boresight at \(\beta=60^{\circ}\). The wave path difference between the antenna elements increases, which causes the ambiguous values of phase difference to increase. Thus the probability of solving ambiguity decreases at \(\beta=60^{\circ}\). Experiment 2 The comparison of root mean square error (RMSE) of two algorithms with respect to different frequencies and SNR. The elevation angles of the incident signals are \(\beta=80^{\circ}\) and \(\beta=60^{\circ}\), respectively, and the frequencies range from 1 GHz to 15 GHz. The average number of phase difference is \(N=40\), and the results of 200 Monte-Carlo experiments are shown in Figure 12 and Figure 13. Figure 12 depicts the RMSE with 2D array, and Figure 13 depicts the RMSE with 3D array. the RMSE of 3D array is slightly larger than that of 2D array at the same SNR. The result is consistent with the analysis of direction finding error as shown in Equation (28). The datatips in the contour plot reflect the viewpoint as mentioned above. This is because the virtual elements of 3D array use the addition and subtraction operations of the phase difference repeatedly. An example is shown in Equation (20). In order to clarify this issue, we assume that \(\Delta\phi_{FE}\) represents the measure error of \(\phi_{\mathrm{FE}}\), and \(\Delta\phi_{MF}\) represents the measure error of \(\phi_{MF}\). The error \(\Delta\phi_{GN}\) is equal to \(\Delta\phi_{FE}\) minus \(\Delta\phi_{MF}:\phi_{\mathrm{GN}}=\phi_{\mathrm{FE}}-\phi_{\mathrm{MF}}\). Obviously, the virtual baseline method can affect the direction finding accuracy. It can be seen that RMSE at \(\beta=60^{\circ}\) is larger than RMSE at \(\beta=80^{\circ}\) with the reason being that the incident signal direction is relatively further when the antenna boresight is at \(\beta=60^{\circ}\). The RMSE is affected by so many ambiguous values. Figure 12: Comparison of RMSE of 2D array with respect to different frequency and SNR. **(a)**\(\beta=80^{\circ}\); **(b)**\(\beta=60^{\circ}\). Figure 13: Comparison of RMSE of 3D array with respect to different frequency and SNR.