## 5 Performance and Simulations of Instantaneous Frequency Rate Tracking Approach
### Tracking Threshold Evaluation  
According to the working mechanism of the IFR tracking approach, there are mainly two factors that greatly influence the performance. One factor lies at label 2 in Figure 14, where the input raw signal samples are complex mixed with the waveform samples estimated in the last computation iteration. The frequency term is cancelled out, leaving only the frequency rate term to be estimated. If there is a mismatch of the instantaneous frequency between the input signal samples and the estimated waveform samples, the effect of residual frequency will arise, and the performance to estimate the IFR will be deteriorated. Another factor lies between labels 4 and 6. If the IFR is not estimated accurately enough, the estimation performance of frequency will also be influenced. To numerically evaluate the two factors on tracking threshold, take Equation (22) again to compute the effects of residual frequency on the estimation of IFR, and the effects of the residual frequency rate on the estimation of frequency, which are shown in Figure 16.  
Figure 15: The coherent gain vs. coherent integration time for different second-order Doppler rate. It could be seen from Figure 16 that the residual frequency greatly influences the estimation performance of IFR. Although it does not introduce any estimation bias, the residual frequency degrades the signal energy severely. Two Hz of residual frequency leads to 4 dB (or \(10\times log_{10}(0.4)\)) in SNR degradation. As a comparison, the frequency estimation is more tolerant with the residual frequency rate, which is also shown in this figure. The residual frequency rate will not lead to a estimation bias for frequency, and a residual frequency rate of 50 Hz/s leads to only 2.4 dB (or \(10\times log_{10}(0.6)\)) in SNR degradation. Thus, a rule of thumb for determining the tracking threshold for the IFR tracking approach is Equation (56):  
\[\Delta f\leq 1\;\mathrm{Hz}, \tag{56}\]  
To theoretically compute the SNR for tracking threshold, the mean square errors (MSE) for frequency and frequency rate are laid out. As shown by [PERSON], et al. [20], when the SNR is high, the CRLB for frequency and frequency rate estimation without outlier are as shown in Equations (57) and (58):  
\[\mathrm{var}\Big{(}\hat{f}|no\;outlier\Big{)}\geq\frac{1.5}{\pi^{2}}\frac{1}{ SNR\times T_{s}^{2}N^{3}}, \tag{57}\]  
\[\mathrm{var}\Big{(}\hat{\hat{f}}|no\;outlier\Big{)}\geq\frac{90}{\pi^{2}}\frac{ 1}{SNR\times T_{s}^{4}N^{5}}, \tag{58}\]  
where SNR is the sample signal-to-noise ratio, \(T_{s}\) is the complex sampling interval, and \(N\) is the number of samples for parameters estimation. When an outlier occurs, the estimation error variance could be found by assuming that each point within the search range has equal a priori probability of being chosen; thus, the frequency estimate may take on any value between \(-Fs/2\) and \(Fs/2\), and the frequency rate estimation may take on any value between \(-f_{s}\) and \(f_{s}\), and the corresponding variance could be computed by Equations (59) and (60):  
\[\mathrm{var}\Big{(}\hat{f}|outlier\Big{)}=\frac{1}{Fs}\int_{-Fs/2}^{Fs/2}x^{2} dx=\frac{Fs^{2}}{12}, \tag{59}\]  
\[\mathrm{var}\Big{(}\hat{\hat{f}}|outlier\Big{)}=\frac{1}{2 fs}\int_{-fs}^{fs}x^ {2}dx=\frac{fs^{2}}{3}, \tag{60}\]  
Figure 16: Effect of Doppler dynamic errors on the coherent integration: (**a**) Doppler frequency error; (**b**) Doppler rate error. According to the total probability formula, the lower bound for frequency estimation MSE is computed by Equation (61):  
\[ \begin{split}\mathrm{var}\Big{(}\hat{f}\Big{)}=\big{(}1-P_{acq1} \big{)}\times\mathrm{var}\Big{(}\hat{f}|outlier\Big{)}+P_{acq1}\times\mathrm{ var}\Big{(}\hat{f}|no\ outlier\Big{)}\geq\big{(}1-P_{acq1}\big{)}\times\frac{F_{2}^{2}}{12}+\\ P_{acq1}\times\frac{15}{n^{2}}\frac{1}{SNR\times T_{s}^{2}N^{3}}, \end{split}  \tag{61}\]  
where \(P_{acq1}\) is the probability of correct frequency estimation. Similarly, the lower bound for frequency rate estimation MSE is computed by Equation (62):  
\[