## 7 Simulation Results
### Simulation Results
**(a)**\(\beta=80^{\circ}\); **(b)**\(\beta=60^{\circ}\). Experiment 3 The RMSE comparison of two algorithms with respect to different frequencies and azimuths. The elevation angle of incident signal is \(\beta=80^{\circ}\), and the frequencies range from 1 GHz to 15 GHz. The SNR is 20 dB, and the average number of phase difference is \(N=40\). The result of 200 Monte-Carlo experiments is shown in Figure 14. It can be seen from Figure 14 that the RMSE of the two arrays at low frequency is larger than that at high frequency. The higher the frequency is, the lower the RMSE is, which is consistent with the analysis of errors in Section 6. The RMSE of 3D array is slightly larger than planar array at the same azimuth which is caused by the fact that virtual elements of 3D array use the addition and subtraction operations of the phase difference repeatedly. Experiment 4 The RMSE comparison of two algorithms with respect to different elevation angles. The azimuth angle of incident signal is \(\alpha=45^{\circ}\), and the frequency is 6 GHz. The elevation angles range from \(45^{\circ}\) to \(90^{\circ}  
Experiment 5 The RMSE and average CPU time comparison of two algorithms with MUSIC. The azimuth and elevation of incident signal is \((45^{\circ},80^{\circ})\) and the frequency is 8 GHz. The average number of phase difference \(N\) is 40, and the results of 200 Monte-Carlo experiments are shown in Figure 16. It can be seen that the RMSE of 3D array is slightly larger than that of 2D array. However, the MUSIC algorithm performs the best among all methods. This is because the MUSIC algorithm uses a spectrum peak searching technique with high accuracy. The CPU time comparison of the three algorithms is shown in Table 1. The PC with Intel Core 3.4 GHZ, 16 GB RAM is used for simulation. The SNR is 20 dB, and the frequency of incident signal is 8 GHz. 100, 200 and 500 Monte-Carlo experiments are considered respectively. It can be seen that the algorithm of 2D array is the fastest and the MUSIC algorithm is the slowest. The algorithm of 3D array is in the middle of the two algorithms mentioned above. It is caused by the fact that the algorithm of 3D array spends a lot of time on the matrix inversion. However, the computation of the spectrum peak searching is tremendous in comparison with the two algorithms mentioned above. The power of the direction finding system is assumed to be \(P_{t}\). For 100 independent Monte-Carlo experiments, the energy consumption of 2D array, 3D array and MUSIC algorithm are \(0.54P_{t}\), \(4.53P_{t}\) and \(20.06P_{t}\), respectively. If the energy of the direction finding system is fixed, then the working time of the proposed algorithm of 2D and 3D is much longer than that of MUSIC. Thus, the proposed algorithm outperforms than MUSIC, which has a low energy consumption. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **100** & **200** & **500** \\ \hline
2D array & 0.54 s & 0.94 s & 2.22 s \\
3D array & 4.53 s & 9.05 s & 22.29 s \\ MUSIC & 20.06 s & 41.79 s & 108.10 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The consuming time of three algorithms. Figure 16: Comparison of RMSE with different elevation.