## 4 Performance Analysis
### Simulation Results  
To verify the effectiveness of NC-Euler-WESPRIT DPD algorithm, we applied Monte Carlo experiments to evaluate the performance. Define root mean square error (RMSE) as  
\[RMSE=\frac{1}{Q}\sum_{q=1}^{Q}\sqrt{\frac{1}{Mon}\sum_{mn=1}^{Mon}\left[\left( \hat{x}_{q,mn}-x_{q}\right)^{2}+\left(g_{q,mn}-y_{q}\right)^{2}\right]}, \tag{73}\]  
where \(Mon\) represents the number of Monte Carlo experiments and \(Q\) stands for the number of targets. \(\left(\hat{x}_{q,mn},\hat{y}_{q,mn}\right)\) denotes the estimated position of the \(q\)th target in the \(mn\)th experiment, \(\left(x_{q},y_{q}\right)\) represents the real position of the \(q\)th target. See Table 2 for other simulation parameters. Moreover, to show the heteroscedasticity of the received SNR at each observation station [36], we define  
\[\sigma_{error}^{2}=\frac{1}{G}\sum_{g=1}^{G}\|\alpha_{g}-\bar{\alpha}_{g}\|^{2}, \tag{74}\]  
where  
\[\bar{\alpha}_{g}=\left[\alpha_{g,1},\alpha_{g,2}\cdots,\alpha_{g,Q}\right]^{ T}\in\mathbb{R}^{Q\times 1}, \tag{75}\]  
\[\bar{\alpha}_{g}=\frac{1}{Q}\sum_{q=1}^{Q}\alpha_{g,q} \tag{76}\]  
\[\bar{\alpha}_{g}=\left[\bar{\alpha}_{g},\bar{\alpha}_{g},\cdots,\bar{\alpha} _{g}\right]^{T}\in\mathbb{R}^{Q\times 1} \tag{77}\]  
Figure 2: The running time of different algorithms varying with search grids. Simulation 1: Discuss the performance of NC-Euler-WESPRIT algorithm with different array elements. Considering that five far-field targets impact six observation stations, NC phases and locations of targets are (20, 40, 50, 60, 70) and \([(-1900,-1900),(-1850,1900),(100,100),(1850,1900),(1900,-1900)]\), respectively. \(\sigma^{2}_{error}=10\), \(K=300\), \(SNR=30\) dB and the observation locations are \([(-800,-900),(-480,-700),(-160,-1000),(160,-800),(480,-1100),(800,-1200)]\). \(M=4\) in Figure 3 and \(M=8\) in Figure 4. The spectral peak diagrams show that NC-Euler-WESPRIT algorithm can successfully distinguish the targets even when the number of array elements at each observation station is less than that of targets. Besides, the estimation performance of NC-Euler-WESPRIT algorithm is getting better and better with the increase of the number of array elements. \begin{table}
\begin{tabular}{l l l} \hline
**Simulation Parameters** & **Value** & **Unit** \\ \hline \(SNR\)(signal to noise ratio) & 0\(\sim\)30 & dB \\ \(G\)(number of observation stations) & 6 & - \\ \(Q\)(number of targets) & 5 & - \\ \(\lambda\)(wavelength) & 1 & m \\ \(d\)(interval of array elements) & \(\lambda/2\) & m \\ \(M\)(number of antennas) & 6 & - \\ \(K\)(number of snapshots) & 10\(-\)300 & - \\ \(Mon\)(number of Monte Carlo) & 1000 & - \\ \(\sigma^{2}_{error}\)(heteroscedasticity) & 10 & - \\ \(\mathbf{p}\)(target locations) & \((-3000,-3000)\) (0,0) (3000,3000) & m \\  & \((-2500,3000)\) (2500,\(-2500)\) & \\ \(\mathbf{\varphi}\)(NC phase) & [10,40,30,50,60] & rad \\ \(\mathbf{u}\)(observation locations) & \((-6000,-9000)\) (\(-3600,-7000)\) (\(-1200,-10,000\)) & m \\  & \((1200,-8000)\) (3600,\(-11,0000)\) (6000,\(-12,000\)) & \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of simulation experiments. Figure 3: Cost function of NC-Euler-WESPRIT (\(M=4\)). Simulation 2: Discuss the performance of NC-Euler-WESPRIT algorithm with different snapshots. Considering that five far-field targets impact six observation stations. \(SNR=20\,\mathrm{dB}\), \(M=5\). \(K=50\) in Figure 5 and \(K=300\) in Figure 6. The other simulation parameters are the same as simulation 1. The contour figures indicate that NC-Euler-WESPRIT algorithm can successfully distinguish the targets with small snapshots. In addition, the sidelobe of the spectral peak becomes smaller with the increase of snapshots. Figure 4: Cost function of NC-Euler-WESPRIT (\(M=8\)). Figure 5: Contour diagram of NC-Euler-WESPRIT (\(K=50\)). Simulation 4: Discuss the RMSE performance of NC-Euler-WESPRIT algorithm with different \(\sigma_{error}^{2}\) varying with SNR. Considering that three far-field targets impact six observation stations.