## 4 Simulation Experiment
### Simulation Results  
In the simulation experiment, an aircraft scattering point model with 74 points was adopted, as shown in Figure 6. The simulation radar parameters were as follows: the transmitting signal wavelength was 0.05 m, the signal carrier frequency was 6 GHz, the signal bandwidth was 400 MHz, the pulse repetition frequency (PRF) was 200 Hz, the pulse width was 25.6 ms, the number of range sampling points was 256, the number of transmitting pulses was 256, the sampling frequency was 10 MHz, and the imaging time was 1.28 s. The target rotated at a uniform acceleration, with a rotating angular velocity of 0.05 rad/s and a rotating angular acceleration of 0.05 rad/s. The rotation ratio was \(K_{\Omega\omega}=1\). In order to express this clearly, radar parameters and target motion parameters are listed in Table 2.  
Figure 5: ISAR Imaging algorithm for maneuvering targets based on discrete chirp Fourier transform. Figure 6: Scattering point model for aircraft targets. Figure 7 shows the result of imaging the target directly using the RD algorithm, in which keystone transform is used to correct the cross-range unit migration [16]. It can be seen from the result that, for a maneuvering target, the RD algorithm causes serious defocusing in the azimuth direction, and the farther the scattering point from the reference center, the more serious the defocusing. DCFT was used to estimate the peak values of the 112 th, 122 th, 128 th, and 133 th range cells; the results are shown in Table 3. Figure 8 shows the MCDFT distribution diagram at the selected range cell. In the simulation, Gaussian noise was added, and the signal-to-noise ratio (SNR) was \(SNR=5\) dB. By averaging the four estimated rotational coefficients, we obtained the average value of \(\hat{K}_{\Omega\omega}=0.960\). This value is very close to the true value, \(K_{\Omega\omega}=1\), which shows that the parameter estimation method based on DCFT is effective. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline wavelength (m) & 0.05 & pulse number & 256 \\ carrier frequency (Ghz) & 6 & sampling frequency (MHz) & 10 \\ signal bandwidth (MHz) & 400 & processing time (s) & 1.28 \\ pulse width (ms) & 25.6 & Target angular velocity (rad/s) & 0.05 \\ PRF (Hz) & 200 & Target angular acceleration (rad/s\({}^{2}\)) & 0.05 \\ range sampling number & 256 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 7: Range Doppler (RD) imaging results. Figure 9 shows the minimum waveform entropy function iteration process based on gradient descent. After 30 iterations, we determined the minimum entropy, and the optimal rotation ratio was found. Figure 10 is the MCFT imaging result based on the optimal rotation ratio proposed in this paper. The result shows that the scattering points are clearly visible. The algorithm achieved a better focusing effect and eliminated the azimuth defocusing phenomenon. Figure 11 is the SPWVD-based time-frequency distribution diagram of the 113 th range cell, and Figure 12 is the corresponding Doppler spectrum before and after the MCFT focus process. It can be seen from Figure 11a that the range cell is a multi-component LFM signal (actually a three-component signal) before the azimuth focusing. From the corresponding azimuth-compressed spectrum (Figure 12a), the azimuth produces Doppler diffusion, and the close scattering point Doppler component is smeared; thus, the focusing effect was not achieved. It can be seen from Figure 11b that  
Figure 8: MCDFT distribution diagram at the (**a**) 113 th and (**b**) the 122 th range cell. Figure 10: The MCFT imaging result. Figure 9: Image entropy vs. iterations. after the MCFT azimuth focusing process, the range cell was adjusted to a multi-component single carrier frequency signal. From the corresponding azimuth Doppler spectrum (Figure 12b), the azimuth Doppler diffusion disappeared; all three targets can be separated in the Doppler domain.