# Performance Analysis of Interferometer Algorithm under Phase Measurement Error
## 6 Numerical Results  
In this section, a computer simulation was conducted on the MATLAB R2017a version. The simulation environment is set as shown in Table 1. The phase error is assumed to be a Gaussian or a uniform random variables. We compared the MSE of the Monte Carlo simulation for various values of measurement noise variance. The \(x\)-axis of the Figure represents the standard deviation, and the \(y\)-axis represents the MSE value on a log scale. The simulation was performed by fixing the azimuth angle to 120 degrees and changing the elevation angle to 15\({}^{\circ}\), 45\({}^{\circ}\), and 75\({}^{\circ}\). The noise of the phase is assumed to be the zero-mean uniform random variables. Since the random variables are assumed to be uniformly distributed between \(-A\) degrees and \(A\) degrees, standard deviation of uniform random variables is \(\sigma=\sqrt{\frac{A^{2}}{3}}\). Since \(A\) values are 1\({}^{\circ}\), 5\({}^{\circ}\), 9\({}^{\circ}\), 13\({}^{\circ}\) and 17\({}^{\circ}\), standard deviations are 0.57\({}^{\circ}\), 2.88\({}^{\circ}\), 5.19\({}^{\circ}\), 7.5\({}^{\circ}\) and 9.81\({}^{\circ}\), respectively. Figures 2-4 show the simulation results for the Gaussian random error, and Figures 5-7 show the simulation results for the uniform random error. The elevation was set to 15\({}^{\circ}\), 45\({}^{\circ}\), and 75\({}^{\circ}\), and the azimuth was set to 120\({}^{\circ}\) in all cases. It is shown that the MSE of the azimuth is larger than that of the elevation for high true elevation incident signal. On the other hand, it is shown that the MSE of elevation is larger than that of the azimuth for low true elevation incident signal. In all cases, it can be confirmed that the analytic MSEs show agreements with the MSEs of the Monte Carlo simulation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of sensors & 5 \\ Array architecture & Uniform Circular Array \\ Monte-Carlo iteration & 1000 \\ Radius of array & 0.5\(\lambda\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation condition. Figure 3: Performance for Gaussian\(-\)distributed phase measurement error with \(\theta^{(0)}=120^{\circ}\) and \(\psi^{(0)}=45^{\circ}\). This figure shows the MSEs of azimuth and elevation with respect to phase error at a signal with medium incident elevation. It can be confirmed that the MSE of elevation is similar to that of the azimuth at medium elevation(uniform circular array). Figure 2: Performance for Gaussian\(-\)distributed phase measurement error with \(\theta^{(0)}=120^{\circ}\) and \(\psi^{(0)}=15^{\circ}\). This figure shows the mean square errors (MSEs) of azimuth and elevation with respect to phase error at a signal with low incident elevation. It can be confirmed that the MSE of elevation is larger than that of the azimuth at low elevation(uniform circular array). Figure 4: Performance for Gaussian\(-\)distributed phase measurement error with \(\theta^{(0)}=120^{\circ}\) and \(\psi^{(0)}=75^{\circ}\). This figure shows the MSEs of azimuth and elevation with respect to phase error at a signal with high incident elevation. It can be confirmed that the MSE of elevation is smaller than that of the azimuth at high elevation(uniform circular array). Figure 5: Performance for uniformly\(-\)distributed phase measurement error with \(\theta^{(0)}=120^{\circ}\) and \(\psi^{(0)}=15^{\circ}\). This figure shows the MSEs of azimuth and elevation with respect to phase error at a signal with low incident elevation(uniform circular array). It can be confirmed that the MSE of elevation is larger than that of the azimuth at low elevation(uniform circular array). From the upper figures of Figures 2-7, it is clear that azimuth MSE increases with the increase of the true elevation angle, \(\psi^{(0)}\). On the other hand, from the lower figures of Figures 2-7, it is clear that elevation MSE decreases with the increase of the true elevation angle, \(\psi^{(0)}\).