## 5 Performance and Simulations of Instantaneous Frequency Rate Tracking Approach
### Selection of Processing Parameters  
There are two important processing parameters in each computation iteration of the IFR tracking algorithm. One parameter is the data rate after the down-sampling processing, and the other is the integration time of each iteration. As there is no incoherent integration operation in the algorithm, the total integration time of each iteration equals to the coherent integration time. For the data rate after down-sampling, the summation operation is after the frequency cancellation. This is similar to the processing manner in a digital phase locked loop (DPLL) [24]. In a DPLL, a complex mixing operation is carried out between the incoming raw signal samples and the local predicted signal waveform samples, followed by a summation of the dot multiplied samples, which are used to generate the difference phase estimation between the incoming raw signal and the local predicted signal. In the IFR tracking algorithm, the summation of the dot-multiplied samples is used to estimate the instantaneous frequency rate, in contrast to the difference phase in a DPLL. There are two factors to be considered to determine the data rate \(f_{s}\) after down-sampling. One factor is the sample SNR after down-sampling, which could be computed by Equation (51):  
\[SNR(\mathrm{dB})=CNR(\mathrm{dBHz})-10\times log_{10}(f_{s}), \tag{51}\]  
which indicates that \(f_{s}\) should be as small as possible to improve the sample SNR. The other factor is the search range for IFR, which is shown in Equation (33). Thus, the data rate could be selected by Equation (52), which maximizes the sample SNR and preserves the search range for IFR:  
\[\underset{f_{s}}{\mathrm{argmin}}\Big{\{}f_{s}\geq\Big{|}\dot{f}_{\mathrm{max }}\Big{|}\Big{\}}, \tag{52}\]  
where \(\Big{|}\dot{f}_{\mathrm{max}}\Big{|}\) is the maximum value of the Doppler rate. Considering that the maximum Doppler rate is around 3400 Hz/s, the data rate \(f_{s}\) is selected to be 4 kHz. For the integration time of each iteration, the main factor to be considered is the energy loss in the coherent summation due to a second-order Doppler rate. As there are two coherent summation operations in the IFR tracking approach, which are separately implemented to down-sample the samples after frequency cancellation and compute the SCPF statistics, the coherence loss function shown in Equation (53) is used again to evaluate the influence of the second Doppler rate:  
\[G=\frac{1}{N}\sum\
olimits_{i=-\frac{N}{2}}^{\frac{N}{2}-1}\left\{-2\pi j \times\left[\frac{1}{6}\ddot{f}\times(i\times T_{s})^{3}\right]\right\}\times \frac{1}{L}\sum\
olimits_{i=-\frac{1}{2}}^{\frac{1}{2}-1}\left\{-2\pi j\times \left[\frac{1}{6}\ddot{f}\times(i\times t_{s})^{3}\right]\right\}, \tag{53}\]  
\[T_{s}=\frac{1}{Fs},\;t_{s}=\frac{1}{f_{s}} \tag{54}\]  
where \(N\) is the number of raw signal samples that are added together to generate a new signal sample after down-sampling, \(L\) is the number of the signal samples that is used to compute the SCPF statistics, \(N\times L=K\), and \(K\) is the total number of raw signal samples that are processed in each computation iteration of the IFR tracking approach. Thus, the total coherent gain account for the influence of the second-order Doppler rate could be computed by Equation (55):  
\[CoherentGain=10\times log_{10}(K)+10\times log_{10}([|G|^{2}]), \tag{55}\]  
The whole EDL flight process consists of two independent parts: an aerobraking flight and parachute deployment. The characteristics of these two flight parts are quite different. In aerobraking flight, the Doppler and its derivatives' curves vary with time smoothly, and the maximum value of the second-order Doppler rate is no larger than 200 Hz/s\({}^{2}\). Meanwhile, the Doppler rate and second-order Doppler rate suffers an impulse dynamic when the parachute is deploying suddenly, and the maximum value of the second-order Doppler rate is no larger than 500 Hz/s\({}^{2}\).