## 3 The Proposed Method
### Parameter Estimation Based on SAC Algorithm and 2D-FFT  
In this paper, the DCCF calculation is performed after RCMC, which reduces the coupling degree between the cross-correlation peak position and the target. Therefore, the SAC algorithm is more suitable for estimating the azimuth modulation frequency of \(S_{5}(t_{m})\)[26]. Based on the above analysis, we know that \(\omega\) in Equation (24) is a constant. The amplitude factor of \(S_{5}(t_{m})\) has been neglected since it has no impact on the peak position of \(S_{SAC}(t_{m})\). Let us perform FFT on Equation (24), then move the center of the Doppler spectrum to zero frequency. According to the principle of stationary phase, the azimuth spectrum can be simplified as [27; 28]  
\[S_{5}(f_{m})=FFT[S_{5}(t_{m})]=\exp\biggl{\{}-j\pi\frac{f_{m}^{2}}{F_{2}} \biggr{\}},-\frac{B_{a}}{2}\leqslant f_{m}\leqslant\frac{B_{a}}{2} \tag{25}\]  
where \(f_{m}\) represents the Doppler frequency and \(B_{a}\) represents the Doppler bandwidth. The principle of the SAC algorithm is shown in Figure 3. It can be seen that the SAC algorithm can estimate the value of \(F_{2}\) without iteration. The Doppler spectrum of the signal is divided into left and right sub-bands, and then the cross-correlation function after sub-band translation is obtained. The cross-correlation peak position can be used to estimate the Doppler modulation frequency. The spectrum of \(S_{5}(f_{m})\) is divided into left and right parts. The left and right spectra obtained by shifting \(+\Delta f_{m}\) and \(-\Delta f_{m}\), respectively, can be expressed as [26]  
\[ \begin{cases}S_{\text{SL}}(f_{m})=\exp\left\{-j\pi\frac{(f_{m}-\Delta f_{m})^ {2}}{F_{2}}\right\},-\frac{B_{a}}{2}+\Delta f_{m}\leqslant f_{m}\leqslant \Delta f_{m}\\ S_{\text{SR}}(f_{m})=\exp\left\{-j\pi\frac{(f_{m}+\Delta f_{m})^{2}}{F_{2}} \right\},-\Delta f_{m}\leqslant f_{m}\leqslant\frac{B_{a}}{2}-\Delta f_{m}\end{cases}  \tag{26}\]  
where \(\Delta f_{m}\) is given by \(\frac{B_{a}}{4}\)[26]. A single-frequency signal can be obtained by Equation (27). \[S_{SAC}(f_{m})=S_{\mathrm{SL}}(f_{m})S_{SR}^{*}(f_{m})=\exp\left(j2\pi\frac{2 \Delta f_{m}}{F_{2}}f_{m}\right) \tag{27}\]  
We can obtain \(S_{SAC}(t_{m})\) by performing IFFT on \(S_{SAC}(f_{m})\). The peak of \(S_{SAC}(t_{m})\) will appear at \(\sigma_{tm}\). The estimated value of \(a_{3}\) can be obtained with Equation (28). \[\sigma_{tm}=\frac{2\Delta f_{m}}{F_{2}},\;a_{3}=-\frac{\Delta f_{m}\lambda}{ 3\sigma_{tm}t_{0}} \tag{28}\]  
It should be noted that when using Equation (24) to extract the slow time-domain chirp signal, the range-frequency domain and azimuth-time domain coupling term (the fifth term) in Equation (21) is ignored, resulting in a residual range migration. Therefore, the estimated value of \(a_{3}\) is used to construct a phase factor to compensate for the ignored range migration in Equation (24). According to Equation (20), the constructed phase compensation factor can be written as  
\[H_{2}(f_{r},t_{m})=\exp\left\{j\frac{4\pi}{\lambda}\sqrt{\frac{f_{c}}{f_{c}+f_ {r}}}\widehat{a}_{3}\left(t_{0}^{3}+3t_{0}t_{m}^{2}-3t_{0}^{2}t_{m}\right)\right\} \tag{29}\]  
Multiplying Equations (19) and (29), the compensated signal can be obtained as  
\[ \begin{split} S_{6}(f_{r},t_{m})&=S_{4}(f_{r},t_{m} )H_{2}(f_{r},t_{m})\\ &=w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\left(\frac{f_{r}}{B} \right)\;\exp\left\{-j2\pi F_{0}f_{r}\right\}\exp\left\{j2\pi F_{1}t_{m}\right\} \end{split}  \tag{30}\]  
The energy of \(S_{6}(f_{r},t_{m})\) can be accumulated by performing range IFFT and azimuth FFT on the compensated \(S_{6}(f_{r},t_{m})\). The range-time domain and azimuth-Doppler domain signal obtained can be written as [7]  
\[ \begin{split} S_{6}(t_{r},f_{m})&=FFT_{t_{m}}\left\{ IFFT_{f_{r}}[S_{6}(f_{r},t_{m})]\right\}\\ &=A\sigma_{0}(t_{r}-F_{0})\sigma_{0}(f_{m}-F_{1})\end{split}  \tag{31}\]  
where \(\delta_{0}(\cdot)\) denotes the Dirac delta function. The peak positions of Equation (31) correspond to \(F_{0}\) and \(F_{1}\), respectively.