## 3 Estimation of Motion Parameters Based on the CCAE Method
### Performance Analysis  
The SNR of the wideband signal is defined as in [23; 31; 32]. If the mean of the \(\omega_{0}(t,t_{m})\) in Equation (2) is zero and the variance is \(\sigma^{2}\), then the SNR of \(s_{ur}(t,t_{m})\) can be written as \(SNR_{ur}=A^{2}/\sigma^{2}\), where \(A\) is the signal amplitude. Because of the stretching operation and analog-to-digital conversion, the bandwidth of the noise will decrease. If the sampling frequency is \(f_{s}\), then the noise bandwidth of the stretched signal decreases by the ratio \(B/f_{s}\). Consequently, the SNR of the stretched signal, i.e., Equation (5), can be written as:  
\[ \begin{split} SNR_{st}&=\frac{A^{2}}{\frac{f_{s}}{ B}\sigma^{2}}\\ &=SNR_{ur}\frac{B}{f_{s}}.\end{split}  \tag{25}\]  
Figure 3: The flowchart of the proposed cross-correlation of adjacent echoes (CCAE) method. In Equation (10), the first two terms contain the signal, and the third one is regarded as the noise term, which is composed of three terms, i.e., \(\omega_{1}(t,t_{m})\overline{s_{st,s}(t,t_{m+1})}\), \(\overline{\omega_{1}(t,t_{m+1})s_{st,s}(t,t_{m})}\) and \(\omega_{1}(t,t_{m})\overline{\omega_{1}(t,t_{m+1})}\). It is clear that \(\omega_{1}(t,t_{m})\overline{s_{st,s}(t,t_{m+1})}\) and \(\overline{\omega_{1}(t,t_{m+1})s_{st,s}(t,t_{m})}\) satisfy the Gaussian distribution, and \(\omega_{1}(t,t_{m})\overline{\omega_{1}(t,t_{m+1})}\) satisfy double Gaussian distribution [33]. There is no exact PDF for this distribution so far. However, this distribution can be approximately regarded as a Gaussian distribution in most cases [33; 34]. Therefore, the noise term in Equation (10) is approximate to the Gaussian noise. In addition, the means of these three terms are zero, and the variances of them are \(A^{2}\sigma^{2}f_{s}/B\), \(A^{2}\sigma^{2}f_{s}/B\) and \(\sigma^{4}f_{s}^{2}/B^{2}\), respectively. Hence, the SNR of \(s_{ac}(t,t_{m})\) is:  
\[ \begin{split} SNR_{ac}&=\frac{A^{4}}{\frac{f_{s}}{ B}(2A^{2}\sigma^{2}+\frac{f_{s}}{B}\sigma^{4})}\\ &=\frac{SNR_{st}}{2+1/SNR_{st}}.\end{split}  \tag{26}\]  
The velocity of the target is obtained by estimating the main peak position of the sinc function in Equation (13). This procedure can be considered similar to that of the frequency estimation of a complex sinusoidal signal \(s_{ac}(t,t_{m})\). For a complex sinusoidal signal with unknown amplitude, phase and frequency [35], the RMSE of frequency estimation can be expressed as:  
\[RMSE_{f}=\sqrt{\frac{6}{4\pi^{2}SNR\Delta^{2}N(N^{2}-1)}}, \tag{27}\]  
where \(\Delta\) is the sampling interval, \(N\) the number of samples and \(SNR\) the SNR of the sinusoidal signal [35]. According to the above definition, the RMSE of velocity of the proposed method is:  
\[ \begin{split} RMSE_{v}&=\frac{c}{2\gamma T_{pr}} \sqrt{\frac{6}{4\pi^{2}SNR_{ac}\Delta^{2}N(N^{2}-1)}}\\ &=\frac{c}{2\gamma T_{pr}}\sqrt{\frac{6(2 Bf_{s}+f_{s}^{2}/SNR_{ur })}{4\pi^{2}B^{2}SNR_{ur}\Delta^{2}N(N^{2}-1)}}.\end{split}  \tag{28}\]  
When \(N\) is large, the equation can be written as:  
\[ \begin{split} RMSE_{v}&\approx\frac{c}{2\gamma T_{pr }T}\sqrt{\frac{6(2 Bf_{s}+f_{s}^{2}/SNR_{ur})}{4\pi^{2}B^{2}SNR_{ur}N}}\\ &=\frac{c}{2 BT_{pr}}\sqrt{\frac{6(2 Bf_{s}+f_{s}^{2}/SNR_{ur})}{4 \pi^{2}B^{2}SNR_{ur}N}}.\end{split}  \tag{29}\]  
It is seen from Equation (29) that the estimation performance is affected by the PRI, the SNR of the UR signal and the bandwidth \(B\). When the acceleration is taken into consideration, the acceleration is obtained by subtracting two i.i.d variables \(\hat{f}_{m}\) and \(\hat{f}_{m+1}\) in Equation (23). The RMSE of the acceleration can be obtained according to Equation (23):  
\[ \begin{split} RMSE_{a}&=\frac{\sqrt{2}c}{2\gamma T _{pr}^{2}}\sqrt{\frac{6(2 Bf_{s}+f_{s}^{2}/SNR_{ur})}{4\pi^{2}(B^{2}SNR_{ur}) \Delta^{2}N(N^{2}-1)}}\\ &\approx\frac{\sqrt{2}c}{2 BT_{pr}^{2}}\sqrt{\frac{6(2 Bf_{s}+f_{s} ^{2}/SNR_{ur})}{4\pi^{2}B^{2}SNR_{ur}N}},\end{split}