# 2D-Unitary ESPRIT Based Multi-Target Joint Range and Velocity Estimation Algorithm for FMCW Radar
## 4 Simulations and Analysis
### Resolution Comparison in the Multi-Target Scenario  
In this simulation, firstly, two targets with the same initial velocity and close range are, respectively, set as \(R_{0}=[11.9,12]\) m and \(v_{0}=[5.8,5.8]\) m/s to compare the range resolution of various algorithms. Secondly, two targets with the same initial range and close velocity are, respectively, set as \(R_{0}=[11,11]\) m and \(v_{0}=[5.6,5.8]\) m/s to compare the velocity resolution of various algorithms.  
Figures 4 and 5, respectively, show the scatter plots of the range resolution and velocity resolution of various algorithms when \(SNR=10\) dB for 10 independent Monte Carlo experiments. As can be seen, the proposed algorithm can distinguish two targets, while the other algorithms cannot. Therefore, the resolution ability of the proposed algorithm is higher than the other algorithms in this case.  
Figure 6 shows the RMSE values of various algorithms versus SNR for range estimation of two targets with closer range. As can be seen from this figure, the RMSE of the proposed algorithm decreases as the SNR increases, which is less than the MMEMP algorithm. The 2D-FFT algorithm and the 2D-CZT algorithm fail to distinguish two targets with range difference 0.1 m due to the fence effect, and thus RMSE values of these algorithms are determined by the proximity of the estimated frequencies to the frequency points. Figure 6b shows that the 2D-Root MUSIC algorithm has the largest performance failure due to the estimated false targets.  
Figure 7 shows the RMSE values of various algorithms versus SNR for velocity estimation of two targets with similar velocities. As can be seen from this figure, the RMSE values of the proposed algorithm reach \(10^{-2}\) order when SNR \(>4\) dB, and the RMSE values of the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Preset \(R_{0}\)/m and \(v_{0}\)/(m/s)**} & \multicolumn{5}{c}{**RMSE of \(R_{0}\)/m and \(v_{0}\)/(m/s)**} \\ \cline{2-6}  & **2D-FFT** & **2D-CZT** & **2D-Root MUSIC** & **MMEMP** & **Proposed Algorithm** \\ \hline \(R_{0}=22\) m & 0.322100 & 0.004764 & 0.015790 & 0.001526 & 0.001526 \\ \(v_{0}=2\) m/s & 0.437500 & 0.016009 & 0.022567 & 0.002876 & 0.002872 \\ \hline \(R_{0}=90\) m & 0.298346 & 0.008075 & 0.008887 & 0.001655 & 0.001656 \\ \(v_{0}=20\) m/s & 0.312500 & 0.024992 & 0.021406 & 0.003242 & 0.003240 \\ \hline \(R_{0}=32\) m & 0.143093 & 0.005348 & 0.010997 & 0.000681 & 0.000678 \\ \(v_{0}=-5\) m/s & 0.312500 & 0.040518 & 0.058606 & 0.002764 & 0.002762 \\ \hline \(R_{0}=78\) m & 0.258566 & 0.006998 & 0.015488 & 0.001759 & 0.001761 \\ \(v_{0}=-13\) m/s & 0.500000 & 0.020319 & 0.022531 & 0.002142 & 0.002136 \\ \hline \hline \end{tabular}
\end{table}
Table 2: RMSE of the range and velocity estimates when the SNR is 10 dB.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Preset \(R_{0}\)/m and \(v_{0}\)/(m/s)**} & \multicolumn{5}{c}{**RMSE of \(R_{0}\)/m and \(v_{0}\)/(m/s)**} \\ \cline{2-6}  & **2D-FFT** & **2D-CZT** & **2D-Root MUSIC** & **MMEMP** & **Proposed Algorithm** \\ \hline \(R_{0}=22\) m & 0.322100 & 0.026561 & 0.685941 & 0.018371 & 0.018137 \\ \(v_{0}=2\) m/s & 0.437500 & 0.233469 & 3.55283 & 0.043890 & 0.043179 \\ \hline \(R_{0}=90\) m & 0.334004 & 0.049870 & 0.753553 & 0.015121 & 0.015405 \\ \(v_{0}=20\) m/s & 0.312500 & 0.203403 & 18.63478 & 0.019307 & 0.019125 \\ \hline \(R_{0}=32\) m & 0.143093 & 0.066198 & 1.16336 & 0.013487 & 0.013138 \\ \(v_{0}=-5\) m/s & 0.312500 & 0.437058 & 12.319505 & 0.008981 & 0.008957 \\ \hline \(R_{0}=78\) m & 0.258566 & 0.053769 & 0.468652 & 0.018281 & 0.017876 \\ \(v_{0}=-13\) m/s & 0.500000 & 0.558930 & 15.504426 & 0.028887 & 0.028395 \\ \hline \hline \end{tabular}
\end{table}
Table 3: RMSE of the range and velocity estimates when the SNR is \(-10\) dB.