## 5 Performance and Simulations of Instantaneous Frequency Rate Tracking Approach  
### Basics of Cubic Phase Function and Squaring Loss Issue  
The discrete-time cubic phase function (CPF) is defined by Equation (25) [17]:  
\[CP(n,\Omega)=\sum_{m=0}^{K}r(n+m)r(n-m)e^{-j\Omega m^{2}}, \tag{25}\]  
where \(r(n)\) is the digitized raw signal samples, \(m\) is the lag time, \(K\) is an even number, and \(\Omega\) is the frequency rate bin. The basic idea of the CPF is simple: folding up the raw data and making a cross-product followed by a summation results in the energy of the CPF concentrated along with the frequency rate law of the signal. Thus, the maximization of Equation (25) leads to a direct estimation of the instantaneous frequency rate (IFR) estimation. As proved in the Appendix A, the ML estimation of frequency rate for a linear chirp signal is shown in Equation (26); thus, the ML estimation of IFR could be obtained by Equation (27). According to the asymptotic properties of ML, the statistics to estimate IFR are asymptotically unbiased, and asymptotically attains the Cramer-Rao Lower Bound (CRLB). \[\hat{\hat{f}}=\underset{\hat{f}}{\operatorname{argmax}}\bigg{(}\sum_{i=-N/2}^{ (N/2)-1}\bigg{[}\tilde{r}_{i}\times e^{-2\pi(\frac{j}{2}\hat{t}_{i}^{2})}\bigg{]} \bigg{)}^{2}, \tag{26}\]  
\[\hat{\hat{f}}_{IFR}=\underset{\Omega}{\operatorname{argmax}}|\sum_{m=0}^{K}r (n+m)r(n-m)e^{-j\Omega m^{2}}\big{|}^{2}, \tag{27}\]  
The main factor that limits the performance of the CPF estimator is the operation of cross-production. Cross-production helps cancel out the frequency term, leaving only the IFR term, but it also brings in the influence of squaring loss caused by the cross-production of thermal noise, which degrades the IFR tracking performance at a low SNR region. The squaring loss could be computed by Equation (28), and the numerical value of squaring loss (SL) for each input SNR is shown in Figure 11, which shows that the SL becomes significant when the input SNR decreases:  
\[SL=1+\frac{1}{SNR_{in}}, \tag{28}\]  
For the Mars EDL case, the Doppler drifts linearly with time when the spacecraft is approaching Mars, as shown in the first 100 s in Figure 3. The Doppler dynamics is small, and the CNR of the received signal would be moderate or high. So, the Doppler phase of the received signal could be precisely reconstructed, which could be used as a priori Doppler phase information. We propose a sequential CPF (SCPF) estimator, which uses this information to eliminate the squaring loss effect in order to enhance the IFR tracking performance at a low SNR. The SCPF statistics is computed by Equation (29). As shown in the equation, to cancel the frequency term, \(K\) samples of \(r(n+m)\) still come from the raw signal samples, while \(K+1\) samples of \(\hat{s}(n-m)\) are generated from the previous parameter estimations. This process is illustrated in Figure 12:  
\[\widehat{CP}(n,\Omega)=\left|\sum_{m=0}^{K}r(n+m)\hat{s}(n-m)e^{-j\Omega m^{2} }\right|^{2}, \tag{29}\]In each computation iteration of this sequential estimation manner, the IFR is first estimated using Equation (29), followed by the demodulation from the raw signal samples. Then, the frequency and initial phase are separately estimated using a ML estimator [23]. Finally, a second-order phase model is generated using the estimated IFR, frequency, and the initial phase, and it is then used in the next computation iteration. To evaluate the estimation performance of the proposed SCPF, the mean squared error (MSE) of the estimation at different SNRs is compared with the standard CPF, as well as CRLB. The CRLB of IFR in the digital domain is given by Equation (30) [20]:  
\[\mathrm{var}(a_{2})\geq\frac{90}{N^{5}SNR}, \tag{30}\]  
where \(a_{2}=\pi T_{s}^{2}\hat{f}\), and \(T_{s}\) is the complex sampling interval. It could be seen that the signal sample number \(N\) greatly influences the variance of IFR.