## 3 Estimation of Motion Parameters Based on the CCAE Method
### Proposed CCAE Algorithm  
At the beginning, we apply the conjugate multiplication to both UR signals and stretched signals. Then, it is seen that the conjugate multiplication result of the UR signals is similar to that of the stretched signals. The UR signal at slow time \(t_{m}\) is multiplied with the conjugate of the adjacent UR signal at slow time \(t_{m+1}\):  
\[s_{ac}^{ur}\left(t,t_{m}\right) =s_{ur}\left(t,t_{m}\right)\cdot\text{conj}\left(s_{ur}\left(t,t_ {m+1}\right)\right) \tag{6}\] \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]  
where:  
\[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\] \[\quad\times\text{rect}\left(\frac{t-\tau_{p}\left(t_{m+1}\right) }{T}\right)\] \[\quad\times e^{\beta\tau_{T}\left(\tau_{p}\left(t_{m+1}\right)- \tau_{p}\left(t_{m}\right)\right)t}\] \[\quad\times e^{\beta\tau_{T}\left(\tau_{p}\left(t_{m+1}\right)- \tau_{p}\left(t_{m}\right)\right)}\] \[\quad\times e^{\beta\tau_{T}\left(\tau_{p}^{2}\left(t_{m}\right) -\tau_{p}^{2}\left(t_{m+1}\right)\right)},\]  
Figure 2: The “stretching” idea of the proposed cross-correlation of adjacent echoes (CCAE) method. \[s_{cr}\left(t,t_{m}\right)= \sum_{q=0}^{P-1}\sum_{p=0,p\
eq q}^{P-1}A_{q}A_{p}\,\text{rect} \left(\frac{t-\tau_{q}\left(t_{m}\right)}{T}\right) \tag{8}\] \[\times\text{rect}\left(\frac{t-\tau_{p}\left(t_{m+1}\right)}{T}\right)\] \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]0 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]1 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]2 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]3 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]4 (9) \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]5  
where the superscript \(\overline{x}\) denotes the conjugate of \(x\) and the superscript \(ur\) indicates that the multiplication is performed on the UR signals. In Equation (6), the first term, i.e., the self-term \(s_{se}\left(t,t_{m}\right)\), is the product of adjacent echoes from the same scatterer, the second term, i.e., the cross-term \(s_{cr}\left(t,t_{m}\right)\), is the product of adjacent echoes from different scatterers and the third term \(\omega_{2}^{\mu r}(t,t_{m})\) is the noise after conjugate multiplication. Similarly, when the conjugate multiplication is performed on the stretched signals, we can obtain:  
\[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]6 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]7  
where:  
\[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]8 \[=s_{se}\left(t,t_{m}\right)+s_{cr}\left(t,t_{m}\right)+\omega_{2 }^{ur}(t,t_{m}),\]9 \[s_{se}\left(t,t_{m}\right) =\sum_{p=0}^{p-1}A_{p}^{2}\text{rect}\left(\frac{t-\tau_{p}\left( t_{m}\right)}{T}\right) \tag{7}\]0  
is the noise added to the stretched signal after conjugate multiplication. The first two terms of the right-hand side of Equation (10) are the same in Equation (6), where the exponential term \(e^{j2\pi\gamma\left(\tau_{p}\left(t_{m+1}\right)-\tau_{p}\left(t_{m}\right) \right)t}\) contains the information of the distance difference between the slow time \(t_{m}\) and \(t_{m+1}\). The difference lies in that the data rate of the stretched signal is lower than that of the UR signal, and the noise bandwidth is smaller, resulting in the SNR of the stretched signal being higher than that of the UR signal. Consequently, the stretched signal is adopted in the following derivation. In Equations (7) and (8), there are two rectangular window functions. The lengths of the two functions are equal, while their positions are different. In other words, the nonzero parts of the two window functions do not completely overlap. This will cause loss in signal energy.