## 3 The Proposed Method
### Construction of DCCF  
In this paper, DCCF is constructed in the range-frequency domain and the azimuth-time domain to reduce envelope shift and the order of phase response in the slow-time domain. Thereby, the accuracy of range migration correction is improved and the estimation of target motion parameters is simplified. Different from the method in reference [19], in this paper, the DCCF is constructed after RCMC, which reduces the coupling degree between the peak position of the DCCF and the target range migration. The obtained cross-correlation function has higher gain, which ensures the performance of parameter estimation. After the processing in Section 3.1, although the third and sixth items in Equation (16) still have the signal envelope shift caused by the residual range-azimuth coupling, the impact is small. The energy of the moving target is basically corrected to within a range cell. For Equation (13), the DCCF constructed in the range-frequency domain and azimuth-time domain is as follows [10; 25]\[S_{4}(f_{r},t_{m})=S_{3}(f_{r},t_{m})S_{3}^{*}(f_{r},t_{m}-t_{0}) \tag{19}\] \[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]  
where \(t_{0}\) denotes the fixed delay given by \(T_{s}/4\)[10; 25]. By combining Equations (13) and (19), the range cell migration factor concerning \(a_{3}\) in \(S_{4}(f_{r},t_{m})\) can be obtained. It can be written as  
\[H_{0}(f_{r},t_{m})=\exp\left\{j\frac{4\pi}{\lambda}\sqrt{\frac{f_{c}}{f_{c}+f_{ r}}}a_{3}\bigg{(}t_{0}^{3}+3t_{0}t_{m}^{2}-3t_{0}^{2}t_{m}\bigg{)}\right\} \tag{20}\]  
The range cell migration caused by \(a_{3}\) will be compensated in Equation (30). Substituting Equation (16) into Equation (19), we can obtain the last factor of Equation (19) as  
\[H_{1}(f_{r},t_{m},t_{0}) =\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\} \tag{21}\] \[\approx\exp\bigg{\{}-j\frac{4\pi f_{c}}{c}\Big{(}a^{\prime}_{1}t _{0}-a_{2}t_{0}^{2}+a_{3}t_{0}^{3}\Big{)}\bigg{\}}\] \[\quad\times\exp\bigg{\{}-j\frac{4\pi f_{c}}{c}\Big{(}2a_{2}t_{0}- 3a_{3}t_{0}^{2}\Big{)}t_{m}\bigg{\}}\ \times\exp\bigg{\{}-j\frac{12\pi f_{c}}{c}a_{3}t_{0}t_{m}^{2}\bigg{\}}\] \[\quad\times\exp\bigg{\{}-j\frac{2\pi f_{r}}{c}\Big{(}a^{\prime}_{ 1}t_{0}-a_{3}t_{0}^{3}\Big{)}\bigg{\}}\times\exp\bigg{\{}j\frac{6\pi f_{r}}{c} a_{3}\Big{(}t_{0}t_{m}^{2}-t_{0}^{2}t_{m}\Big{)}\bigg{\}}\]  
The fifth term in \(H_{1}(f_{r},t_{m},t_{0})\) still has range-azimuth coupling, which corresponds to the residual migration after range migration correction. Substituting Equation (21) into (19), and performing the IFFT in the range dimension, we can obtain  
\[S_{4}(f_{r},t_{m}) \tag{22}\] \[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})sinc\left[B\bigg{(}t_{r}- \frac{a^{\prime}_{1}t_{0}-a_{3}t_{0}^{3}}{c}+\frac{3a_{3}\big{(}t_{0}t_{m}^{2} -t_{0}^{2}t_{m}\big{)}}{c}\bigg{)}\right]\] \[\quad\times\exp\bigg{\{}-j\frac{4\pi f_{c}}{c}\Big{[}\Big{(}2a_{2 }t_{0}-3a_{3}t_{0}^{2}\Big{)}t_{m}+3a_{3}t_{0}t_{m}^{2}\Big{]}\bigg{\}}\]  
The residual range migration \(\Delta R=\big{|}3a_{3}\big{(}t_{0}t_{m}^{2}-t_{0}^{2}t_{m}\big{)}\big{|}\) is generally less than a range cell and can be ignored (we will use Equation (30) to compensate for the approximation error here). Therefore, the two-dimensional time domain cross-correlation function can be obtained as  
\[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]0 \[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]1 \[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]2 \[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]3  
where  
\[=|A|^{2}w_{a}(t_{m})w_{a}(t_{m}-t_{0})rect^{2}\bigg{(}\frac{f_{r}}{B} \bigg{)}\exp\{-j\Delta\Phi_{1}(f_{r},t_{m})\}\]4 \[=\Bigg{\{}