## 5 Performance and Simulations of Instantaneous Frequency Rate Tracking Approach
### Detailed Estimation Algorithm  
In this section, the full algorithm of the instantaneous frequency rate tracking approach is described in detail. First, we examine the search range of the IFR given the complex sampling frequency \(F_{s}\). CPF for a signal without noise is computed by Equation (31):  
\[ \begin{split} CPF(n,\Omega)&=\sum_{m=0}^{(N-1)/2} \left[s(n+m)T_{s}\right]\times[s(n-m)T_{s}]exp(-j\Omega m^{2})\\ &=\exp\left[2\pi j\times 2\left(\phi_{0}+f\times nT_{s}+\frac{1}{2} \dot{f}n^{2}T_{s}^{2}\right)\right]\times\sum_{m=0}^{(N-1)/2}exp\left[j\left(2 \pi\frac{\dot{f}}{F_{s}^{2}}-\Omega\right)m^{2}\right],\end{split}  \tag{31}\]  
The above equation is maximized when:  
\[2\pi\dot{f}/F_{s}^{2}-\Omega=0, \tag{32}\]  
Thus, when the search range of digital IFR \(\Omega\) is \([-2\pi/F_{s},2\pi/F_{s}]\), the estimation range of \(\dot{f}\) is:  
\[-F_{s}\leq\dot{f}\leq F_{s}, \tag{33}\]  
The block diagram of the IFR tracking approach is shown in Figure 14.  
Figure 14: The block diagram of the instantaneous frequency rate (IFR) tracking approach. Figure 13: Mean squared error (MSE) performance of cubic phase function (CPF) and sequential cubic phase function (SCPF) vs. Cramer–Rao Lower Bound (CRLB): (**a**) \(N=201\); (**b**) \(N=401\). The detailed description of the algorithm is as follows:  
Step (1): Waveform generation  
At label 1 in Figure 14, generate the second-order phase model from the previous estimation parameters using Equation (34):  
\[\hat{\phi}_{local}=2\pi\times\left(\hat{\phi}_{0}+\hat{f}\times[t]+\frac{1}{2} \times\hat{f}\times[t]^{2}\right), \tag{34}\]  
where \(\hat{\phi}_{0}\), \(\hat{f}\), and \(\hat{\hat{f}}\) are the estimated phase, instantaneous Doppler frequency, and instantaneous Doppler rate in the previous computation iteration, respectively, and \([t]\) is the corresponding time instant sequence shown in Equation (35):  
\[[t]=[0,\frac{1}{Fs},\frac{2}{Fs},\cdots\frac{K}{Fs}], \tag{35}\]  
and \(K\) is the number of the total coherent integration samples. The estimated waveform could be computed by Equation (36):  
\[\hat{s}_{local}=exp[j\times\hat{\phi}_{local}], \tag{36}\]  
Note that the number of samples in the sequence \(\hat{s}_{local}\) is \(K+1\). Step (2): Frequency cancellation  
Construct the data sequence used for current estimation by combining \(K+1\) samples of \(\hat{s}_{local}\) and \(K\) samples of \(r[n]\), where \(r[n]\) come from the raw signal samples. As shown in Figure 12, there are two subsequences which are dot multiplied with each other to cancel the frequency term. One sequence is called the forward direction subsequence \(spos[n]\), and the other is called the backward direction subsequence \(sneg[n]\), which are defined in Equations (37) and (38), respectively:  
\[spos[n]:\{\hat{s}_{local}[K],r[1],r[2],\cdots,r[K]\}, \tag{37}\]  
\[sneg[n]:\{\hat{s}_{local}[K],\hat{s}_{local}[K-1],\hat{s}_{local}[K-2],\cdots, \hat{s}_{local}[0]\}, \tag{38}\]  
Thus, the correlation sequence could be obtained by Equation (39):  
\[-F_{s}\leq\dot{f}\leq F_{s}, \tag{33}\]0  
Notice that at this step, the data rate is still equal to the complex sampling rate of the raw signal sample. As the sampling rate of the raw signal sample is usually several hundred kHz, the search range for IFR is also at this order of magnitude. The prediction analysis of EDL dynamics in Section 3 shows that the maximum Doppler rate is no larger than 3.4 kHz/s; thus, the data rate could be reduced. Step (3): Down-sampling  
Down-sampling for the sequence \(scor[n]\) is realized by coherent summation. After down-sampling, the reduced rate complex data sequence is obtained, which is shown in Equations (40) and (41):  
\[-F_{s}\leq\dot{f}\leq F_{s}, \tag{33}\]1  
\[-F_{s}\leq\dot{f}\leq F_{s}, \tag{33}\]2where \(\text{Re}\{\cdot\}\) returns the real part of a complex number, and \(\text{Im}\{\cdot\}\) returns the imaginary part. \(M\) is defined by Equation (42):  
\[-F_{s}\leq\dot{f}\leq F_{s}, \tag{33}\]3  
and \(f_{s}\) is the data rate of the sequence \(sds[m]\).