# An Efficient Ground Moving Target Imaging Method for Airborne Circular Stripmap SAR
## 5 Experimental Results
### Validation of the Proposed Imaging Method  
In this subsection, numerical experiments are conducted to investigate the performance of the proposed imaging method. The parameters of CSSAR are given in Table 1. Five moving targets, named T1, T2, T3, T4, and T5, are simulated, and their parameters are presented in Table 3. The signal-to-noise ratio (SNR) is set to be \(-10\) dB. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**IRW Broadening**} & \
\begin{tabular}{c} **PSLR** \\ **(dB)** \\ \end{tabular} &
\begin{tabular}{c} **ISLR** \\ **(dB)** \\ \end{tabular} \\ \hline \multirow{2}{*}{Original range model} & Azimuth & 0.00\% & \(-13.23\) & \(-10.12\) \\ \cline{2-5}  & Range & 0.00\% & \(-13.26\) & \(-10.04\) \\ \hline \multirow{2}{*}{Proposed range model} & Azimuth & 0.00\% & \(-13.12\) & \(-9.86\) \\ \cline{2-5}  & Range & 0.00\% & \(-13.26\) & \(-10.04\) \\ \hline \multirow{2}{*}{Second-order [PERSON] approximated range model} & Azimuth & 2.03\% & \(-11.05\) & \(-9.06\) \\ \cline{2-5}  & Range & 0.00\% & \(-13.26\) & \(-10.04\) \\ \hline \multirow{2}{*}{Third-order [PERSON] approximated range model} & Azimuth & 1.35\% & \(-12.91\) & \(-9.25\) \\ \cline{2-5}  & Range & 0.00\% & \(-13.26\) & \(-10.04\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured quality parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\mathbf{v_{x}}\) **(m/s)** & \(\mathbf{v_{y}}\) **(m/s)** & \(\mathbf{a_{x}}\) **(m/s2)** & \(\mathbf{a_{y}}\) **(m/s2)** & \(\mathbf{r_{0}}\) **(km)** & \(\mathbf{\theta_{0}}\) **(rad)** \\ \hline
**T1** & \(-29\) & \(20\) & \(-0.5\) & \(0.3\) & \(15.8\) & \(0\) \\ \hline
**T2** & \(-5\) & \(10\) & \(0.1\) & \(0.2\) & \(15.9\) & \(0.01\) \\ \hline
**T3** & \(21\) & \(10\) & \(-0.4\) & \(-0.3\) & \(16.0\) & \(0\) \\ \hline
**T4** & \(5\) & \(-5\) & \(0.2\) & \(-0.5\) & \(16.1\) & \(-0.01\) \\ \hline
**T5** & \(-24\) & \(-20\) & \(0.5\) & \(0.4\) & \(16.2\) & \(0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the moving targets. In using the GA, first, the value ranges of the parameters to be searched and the encoding accuracies should be determined so that we can encode these parameters. In this paper, it is assumed that \(v_{x}\in[-30\) m/s, \(30\) m/s, \(v_{y}\in[-30\) m/s, \(30\) m/s], \(a_{x}\in[-1\) m/s\({}^{2}\), \(1\) m/s\({}^{2}\)], \(a_{y}\in[-1\) m/s\({}^{2}\), \(1\) m/s\({}^{2}\)]. Then, according to (3)-(9) and (12)-(14), the value ranges of \(v_{c}\), \(\alpha\), and \(\beta\) can be figured out and the results are: \(v_{c}\in[-297.31\) m/s, \(362.59\) m/s\(],\alpha\in[-7.75\times 10^{5}\) m\({}^{2}\)/s,  
Figure 5: Focusing results. (**a**) azimuth profile, (**b**) range profile. The parameters of the target are \(R_{c}=21\) km, \(v_{x}=3\) m/s, \(v_{y}=-12\) m/s, \(a_{x}=0.6\) m/s\({}^{2}\) and \(a_{y}=-0.3\) m/s\({}^{2}\). 7.75 \(\times\) 10\({}^{5}\) m\({}^{2}\)/s], \(\beta\)\(\in\)[\(-\)50.34 m/s, 50.34 m/s]. The encoding accuracies for the three parameters can be set according to the required focusing quality. To keep the azimuth impulse response width (IRW) broadening within 2%, the Doppler chirp rate error should be less than \(1/T_{a}^{2}\), where \(T_{a}\) is the synthetic aperture time [16]. The Doppler chirp rate is related to \(v_{e}\) and \(\alpha\), and their relationship is as follows  
\[K_{a}=-\frac{4}{\lambda}\bigg{(}\frac{v_{e}^{2}}{2R_{c}}-\frac{\alpha^{2}}{8R_ {c}^{3}}\bigg{)} \tag{30}\]  
In the experiments, the bound for the Doppler chirp rate error is set to be \(1/(3T_{a}^{2})\). Then, based on (30), the encoding accuracies for \(v_{e}\) and \(\alpha\) can be given by (31) and (32), respectively. \[\Delta v_{e}=\min_{v_{e}}\left\{-v_{e}+\sqrt{v_{e}^{2}+\frac{\lambda R_{c}}{12 T_{a}^{2}}}\right\} \tag{31}\]  
\[\Delta\alpha=\min_{\alpha}\left\{-\alpha+\sqrt{\alpha^{2}+\frac{\lambda R_{c} ^{3}}{3T_{a}^{2}}}\right\} \tag{32}\]  
The derivation for (31) and (32) are presented in Appendix A.