## 3 Estimation of Motion Parameters Based on the CCAE Method
### Proposed CCAE Algorithm
The length of the non-overlapping part is determined by the target's motion, i.e.,  
\[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\]1 \[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\]2 \[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\]3  
where \(\Delta R_{pq}(t_{m})=R_{p}(t_{m+1})-R_{q}(t_{m})\). In common radar applications, the ratio of the non-overlapping part's length to the length \(T\) of the transmitted signal (i.e., \(2\Delta R_{pq}(t_{m})/cT\)) is much less than one. For example, if \(T=1\) ms and \(T_{pr}=10\) ms, then the length of the target is \(50\) m and velocity \(6000\) m/s. Computation of the maximum value of \(\Delta R_{pq}(t_{m})\) by \(\Delta R_{pq}(t_{m})=50+6000\times 0.01=110\) m gives the ratio \(2\Delta R_{pq}(t_{m})/cT\approx 0.073\%\). Thus, the influence of the non-overlapping part can be considered negligible. Then, by substituting Equation (3) into Equation (10) and applying the FFT operation to Equation (10) for energy focusing, one gets:  
\[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\]4  
where \(S_{se}(f,t_{m})\), \(S_{cr}(f,t_{m})\), \(W_{2}(f,t_{m})\) are the FFT results of \(s_{se}(t,t_{m})\), \(s_{cr}(t,t_{m})\) and \(\omega_{2}(t,t_{m})\), respectively, i.e.,  
\[ \begin{split} S_{se}\left(f,t_{m}\right)&=\sum_{p=0 }^{P-1}A_{p}^{2}\text{sinc}\left[T\left(f-\frac{2\gamma\Delta R_{pp}(t_{m})}{c} \right)\right]\\ &\quad\quad\quad\quad\quad\quad\times e^{i\frac{4\pi}{\lambda_{c} }\Delta R_{pp}(t_{m})}e^{i\frac{4\pi}{c^{2}}\gamma(R_{p}^{2}(t_{m})-R_{p}^{2}( t_{m+1}))},\end{split}  \tag{14}\]  
\[ \begin{split} S_{cr}\left(f,t_{m}\right)&=\sum_{q=0 }^{P-1}\sum_{p=0,p\
eq q}^{P-1}\text{sinc}\left[T\left(f-\frac{2\gamma\Delta R _{pq}(t_{m})}{c}\right)\right]\\ &\quad\quad\quad\quad\quad\times A_{q}A_{p}e^{i\frac{4\pi}{\lambda _{c}}\Delta R_{pq}(t_{m})}\\ &\quad\quad\quad\quad\quad\times e^{i\frac{4\pi}{c^{2}}\gamma(R_ {p}^{2}(t_{m})-R_{q}^{2}(t_{m+1}))},\end{split}  \tag{15}\]  
where \(\lambda_{c}\) is the wavelength corresponding to the center frequency \(f_{c}\). In Equations (14) and (15), the exponential terms are constant phase terms, which have no impact on estimating the frequency of the spectrum, i.e., Equation (13). For simplicity, the sinc functions are extracted for further discussion in the following sections. Define:  
\[S_{si}\left(f,p,q\right)=\text{sinc}\left[T\left(f-\frac{2\gamma\Delta R_{pq}( t_{m})}{c}\right)\right]. \tag{16}\]  
Let it be assumed that the peak position of \(S_{si}(f,p,q)\) is \(\tilde{f}_{pq}\). When \(p=q\), Equation (16) is the same as the sinc function in Equation (14). Since all scatterers are assumed to locate along the line of sight, the distance difference between \(t_{m}\) and \(t_{m+1}\) is the same for every scatterer of the target. Namely, \(\Delta R_{pp}(t_{m})=\Delta R_{qq}(t_{m})\) and \(\tilde{f}_{pp}=\tilde{f}_{qq}\) for \(p\
eq q\). The energy of all scatterers will accumulate on one peak, which can be regarded as the main peak of the spectrum. When \(p\
eq q\), because of the differences in the combinations of \(p\) and \(q\), the distance difference \(\Delta R_{pq}(t_{m})\) will bring a number of frequency components \(\tilde{f}_{pq}\)\((p\
eq q)\) into the spectrum. The amplitudes of these spectral lines are much less than that of the main spectral line \(\tilde{f}_{pp}\). As seen from Equations (12) and (14), the main peak position of the spectrum is determined by the distance difference of the target between slow time \(t_{m}\) and \(t_{m+1}\). Therefore, the target parameters can be obtained by estimating the frequency of the main peak position. This can be implemented by using the frequency estimation methods, such as FFT and [PERSON]'s method [30]. The estimated frequency can be expressed as:  
\[