## 3 Imaging Algorithm Based on Modified Chirp Fourier Transform
### Accurate Rotation Ratio Estimation Based on Minimum Entropy  
Whether or not the rotation ratio estimation \(\widetilde{K}_{\Omega\omega}\) is accurate directly affects the final imaging results. For this reason, we took the estimated value obtained by DCFT as the initial value, and then used the minimum Shannon entropy criterion of the spectral envelope for accurate estimation. The basis for this decision is that when the estimated value \(\widetilde{K}_{\Omega\omega}\) is closer to the true rotation ratio \(K_{\Omega\omega}\), the smaller the spectral broadening of each scattering point sub-echo, the higher the spectral envelope sharpening degree, and the smaller the Shannon entropy of the envelope. The relationship between envelope waveform entropy and the rotation ratio is shown in Figure 4. It can be seen from the figure that the change in image entropy is a convex function, and the optimal rotation ratio is obtained when the function takes the minimum value. Therefore, the solution of the rotation ratio can be regarded as the optimal solution problem, with the minimum entropy function as the objective function and \(\widetilde{K}_{\Omega\omega}\) as the variable. At present, there are many kinds of optimization methods available in machine learning and deep learning [33; 34; 35]. The simplest and most commonly used method is the gradient descent method [20; 36], so we used this method to accurately estimate the rotation ratio. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(\sigma_{i}\) & \(f_{i}\) & \(k_{i}\) & **Estimated Value** & \(f_{i}\) & \(k_{i}\) \\ \hline \(i=1\) & \(1\) & \(40\) & \(320\) & \(i=1\) & \(40\) & \(320\) \\ \(i=2\) & \(1\) & \(75\) & \(30\) & \(i=2\) & \(75\) & \(30\) \\ \(i=3\) & \(1\) & \(-60\) & \(240\) & \(i=3\) & \(-60\) & \(240\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values and estimated values of signals. Figure 3: Discrete chirp Fourier transform (DCFT) of three component LFM signals. Suppose that the rotation ratio is \(K_{\Omega\omega}^{l}\) and the \(i\)th range cell spectral envelope obtained by DCFT in the form of Equation (16) is \(|\widetilde{s_{i}}(m)|_{l}\) at the \(l\) iteration. Then, the entropy of the ISAR image (objective function) is:  
\[H(l)=-\sum\limits_{i}\sum\limits_{m=1}^{M}p(i,m)_{l}\ln[p(i,m)_{l}] \tag{25}\]  
\[p(i,m)_{l}=\frac{|\widetilde{s}_{i}(m)|_{l}^{2}}{\sum\limits_{i}\sum\limits_{ m=1}^{M}|\widetilde{s}_{i}(m)|_{l}^{2}} \tag{26}\]  
where \(M\) represents the total number of azimuth sampling points. The relationship between the objective function and the rotation ratio is:  
\[\widetilde{K}^{l+1}_{\Omega\omega}=\widetilde{K}^{l}_{\Omega\omega}+\Delta K \tag{27}\]  
\[\Delta K=-\eta\frac{\partial H(l)}{\partial\widetilde{K}^{l}_{\Omega\omega}} \tag{28}\]  
where \(\eta\) is the search step size and \(\widetilde{K}^{0}_{\Omega\omega}\) is the rough estimate obtained by the DCFT method. Suppose \(\varepsilon\) is the termination condition, where the iteration terminates when \(|\Delta K|<\varepsilon\). If \(K_{\Omega\omega}^{l}\) is more accurate, a better azimuth image is obtained by the DCFT transform, a smaller entropy value is calculated by Equation (25), and the value in the graph is lower. The final estimate of the rotation ratio is:  
\[K_{\Omega\omega}^{l}=\underset{l}{\text{argmin}}[H(l)] \tag{29}\]  
After obtaining an accurate rotation ratio, azimuth signal focusing imaging can be accomplished by using MCFT transform once. Summarizing the above analysis, we obtained a block diagram of the ISAR Imaging algorithm of maneuvering targets based on MCFT, as shown in the Figure 5. The specific steps are:  
1. Input the radar echo signal;
2. Generate reference signal by measured reference range;
3. Perform dechirping processing for the echo signal and reference signal;
4. Range compression and translational compensation for the signal after dechirping;
5. Coarse rotation ratio estimation based on DCFT in Section 3.2;
6.