## 4 Performance and Simulation of 2D Maximum Energy Search Approach
### Performance Analysis and Processing Parameter Influence Evaluation  
To make an evaluation and determine the estimation threshold of the 2D search approach, a simulation is implemented based on the EDL trajectory of MSL. Consider that the maximum Doppler rate is about 3400 Hz/s, and the second-order Doppler rate is about 440 Hz/s\({}^{2}\), the coherent integration time is set to 0.01 s, and the total incoherent integration time is set to 0.5 s for all of the EDL process parameter estimations. The detailed processing parameters are shown in Table 1.  
In the analysis shown in [PERSON], et al. [8], Doppler dynamics are varied with time, and different processing parameters are selected separately for the aerobraking flight phase and parachute deployment phase; this is a refinement processing of the raw data, and the performance could get better. In our analysis, we just use one set of the processing parameters to deal with the simulated data for the reason that the time instant when unfolding the parachute is not known in advance, as well as for analysis simplicity. Figure 7: Coherent loss due to residual Doppler rate for different coherent integration time: (**a**) 0.01 s (\(\Delta F=100\) Hz); (**b**) 0.1 s (\(\Delta F=10\) Hz). To evaluate the carrier detection performance, a simulation is made for different CNR based on the EDL trajectory of MSL. Five hundred trials are made for each CNR, and the probability of missed carrier acquisition in the simulation is counted and compared with the theoretical prediction computed from Equations (16)-(18), which is shown in Figure 8. The simulated results are well consistent with the performance prediction, and the detection SNR for the selected processing parameters is about 22.5 dB/Hz. The large deviation of the simulation result from the prediction value when CNR is 22 dB/Hz is due to the limited simulation trials. The influence of the coherent integration time and the incoherent integration time on the performance of carrier detection is different. The coherent integration is more efficient. As an illustration, Figure 9 is a performance comparison of two integration approaches: the blue dotted line is 0.01 s of coherent integration with 50 blocks of incoherent summation, which corresponds to a total 0.5 s integration, while the red solid line is 0.25 s of coherent integration only. It could be seen that the 0.25-s integration has a better detection performance, although the first approach has an integration time of 0.5 s. \begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameters** & **Values** \\ \hline Sampling frequency & 200 kHz \\ FFT resolution & 100 Hz \\ Number of cells/FFT & 2000 \\ Coherent integration time & 0.01 s \\ Incoherent integration time & 0.5 s \\ Number of averaged FFTs & 50 \\ Frequency search space & [\(-100\) kHz, 100 kHz] \\ Doppler rate search space & [0 Hz/s, 4000 Hz/s] \\ Doppler rate resolution & 100 Hz/s \\ Doppler rate cells & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for 2D search approach. FFT: fast Fourier transform. Figure 8: Comparisons of carrier acquisition probability between theoretical prediction and simulations. According to Equation (20), the main limitation of expanding the coherent integration is the existence of the large Doppler rate. Short coherent integration (corresponding to small FFT resolution, \(\Delta F\)) also leads to the inaccurate estimation of Doppler frequency, which in turn limits the carrier detection performance. Besides, as the complex sampling frequency is high (for example, 200 kHz), the sampling time is quite short (1/200 kHz = 5e\({}^{-6}\) s), and the time numerical contribution to residual phase due to residual Doppler frequency and residual Doppler rate is different in Equation (22). In mathematics:  
\[t_{i}\gg\frac{1}{2}t_{i}^{2}\Rightarrow\Delta ft_{i}\gg\frac{1}{2}\Delta ft_{i }^{2}\,, \tag{24}\]  
when the residual Doppler frequency is numerically equal to the residual Doppler rate, \(\Delta f=\Delta\dot{f}\).