## 4 Simulations and Real Data Processing
### Comparison with the ACCF Method  
In this subsection, we compare the performance of the proposed CCAE method with the ACCF method [24; 25] at different SNRs, which varies from \(-30\) dB to \(2\) dB. One thousand Monte Carlo simulations are also carried out for each SNR condition. The other simulation parameters are listed in Table 3. In this experiment, the stretched signals and UR signals are both used in the proposed CCAE method, and the UR signals are used in the ACCF method, which can only be applied to the UR signals. Figure 4: The spectrum of the cross-correlation result. Figure 5: The performances of the proposed CCAE method with different parameters. We first evaluate the performance of the case, in which only velocity is taken into account. The velocity of the target is set to 100 m/s. Two pulse echoes are used only for velocity estimation. In Figure 6, it is seen that the estimated RMSEs of CCAE method on UR signals are similar to that of ACCF on UR signals, and the performance of CCAE on stretched signals is the best. As mentioned in Section 3.4, the noise term \(\omega_{1}(t,t_{m})\overline{\omega_{1}(t,t_{m+1})}\) is approximately regarded as a Gaussian noise, hence there is a gap between the theoretical and simulation results, which can be seen from Figure 6. However, on average, the RMSE gap is about 0.0543 dB for UR signals and 0.0179 dB for stretched signals. This shows that the noise approximation is reasonable. Furthermore, it is noted that the RMSEs of the CCAE deteriorate largely when the SNR is low. In low SNR cases, the amplitude of the self-term in the correlation result may be smaller than the noise amplitude. This may cause the failure of the peak search, so that the RMSEs deteriorate. In the parameter estimation problem, a threshold SNR means that the performance may deteriorate largely when the input SNR is lower than the threshold value. The threshold SNR is lower, the performance is better. In this experiment, the threshold SNR of CCAE/ACCF on UR signals is about \(-16\) dB, higher than that of CCAE on stretched signals (about \(-26\) dB). Generally, the theoretical RMSE is derived with the assumption that the spectrum peak can be successfully found. Therefore, there is no threshold SNR in the theoretical RMSE. We further evaluate the performances of the two methods with a second-order motion model. The velocity of the target is 100 m/s, and the acceleration is 10 m/s\({}^{2}\). Three pulses are used for velocity and acceleration estimation, the other parameters being the same as those of the previous experiments. The estimated acceleration results are shown in Figure 7, from which we can see that the RMSE performance of CCAE on UR signals is better than that of ACCF on UR signals. It can also be seen that the threshold SNR of CCAE on UR signals is smaller than that of ACCF on UR signals, i.e., \(-16\) dB versus \(-8\) dB. As has already been analyzed in Section 3.4, SNR decreases after cross-correlation. To estimate the acceleration, the conjugate multiplication has to be used twice in the ACCF method, but only once in the CCAE method. Therefore, the performance loss is more in the ACCF method than in the CCAE method. In addition, the performance of CCAE on stretched signals is better than that of CCAE on UR signals, due to the higher SNR of stretched signals. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Center Frequency (GHz)** & **Bandwidth (GHz)** & **Pulse Width (\%)** & **Sampling Frequency (MHz)** & **PRI (ms)** \\ \hline
9 & 1 & 100 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters of the comparison with the adjacent cross-correlation function (ACCF) method. Figure 6: RMSE comparison of velocity estimation by using CCAE on stretched signals, CCAE on uncompressed received (UR) signals and adjacent cross-correlation function (ACCF) on UR signals (without acceleration).