# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 4 Experiments and Discussion
### Performance Analysis of Estimation Methods  
To test the performance of proposed estimation methods, the Monte Carlo experiments with 100 times are carried on. In each experiment, the DRR error is set as 3%; the CDC error is set as 100 Hz and phase inconsistency errors are set random between \(\pm 30^{\circ}\). As illustrated in Section 3.5, the SNR after pre-processing in multi-channel BS-SAR system depends on the uniformity of the spatial sampling. Therefore, we here analyze the performance of estimation methods for DRR error, CDC error, and phase inconsistency errors against SNR and \(Fu\). Since the results of multi-channel sliding spotlight  
Figure 10: Azimuth signals of multi-channel TOPSAR after pre-processing: (**a**) shows the amplitude of signals with DRR error (dotted green line), CDC error (dashed blue line), phase inconsistency errors (dash-dot yellow line) and without error (red line); (**b**) shows the amplitude of the signals after antenna correction. SAR is similar with multi-channel TOPSAR, the performance is analyzed based on multi-channel TOPSAR for convenience. For the phase inconsistency errors, the root-mean-square error (RMSE) is defined as:  
\[\sigma=\sqrt{\frac{1}{N-1}\sum_{n=1}^{N}\left(\hat{\zeta}_{i}-\zeta_{i}\right)^{ 2}}, \tag{53}\]  
where \(\hat{\zeta}_{i}\) is the estimate of \(\zeta_{i}\). Figure 11: Imaging results of multi-channel TOPSAR without errors (**a**), with DRR error (**b**), CDC error (**c**), phase inconsistency errors (**d**–**h**) are corresponding azimuth profiles. Imaging results of multi-channel sliding spotlight SAR without errors (**i**), with DRR error (**j**), CDC error (**k**), phase inconsistency errors (**l**–**p**) are corresponding azimuth profiles. The performance of the DRR error estimation method against SNR and \(Fu\) is shown in Figure 12. The estimation errors for DRR decrease as SNR increases for both values of \(Fu\) used, but the estimation error for \(Fu\) = 0.9 is much larger than the figure for \(Fu\) = 1.05. This can be explained by the azimuth ambiguity signal rate. When \(Fu\) = 0.9, the operating PRF of the system is lower than in the uniform case. As a result, more ambiguous power is aliased into the processing bandwidth. The fluctuation at \(Fu\) = 1.1 when SNR = 0 dB in Figure 12b can be explained by the same. As the estimation precisions of DRR error can also be impacted by the azimuth ambiguity signal rate, expect the SNR. When \(Fu\) increases from 1 to 1.2, the SNR scaling factor (see (49)) will increase, which will degrade the \(SNR_{out}\) and result worse estimation precision. On the other hand, larger \(Fu\) means larger PRF and less ambiguity signal rate according the sampling theory. Then the less ambiguity will alleviate the negative tendency of estimation precision brought by worse \(Fu\). In the case here, the scaling factor domains the upward tendency of estimation precision from \(Fu\) = 1 to \(Fu\) = 1.1 and then the ambiguity signal rate becomes the dominated factor after \(Fu\) = 1.1. It also can be seen that this conflict between scaling factor and ambiguity signal rate is depended on the SNR. When SNR is large enough (e.g., 20 dB), the estimation error has no obvious correlation with the uniformity factor. The performance of the CDC error estimation methods against SNR and \(Fu\) is shown in Figure 13. In this experiment, we compare the SCCC method with the symmetric shift (SS) method proposed in this paper. The SS method achieves slightly better precision than SCCC for all SNR and is robust against the uniformity factor \(Fu\), while the SCCC method is sensitive to \(Fu\). This is because the SCCC method depends on the correlation between the adjacent pulses. As the \(Fu\) increases, the estimation error of SCCC method has the increasing tendency. The drop at \(Fu\) = 1.1 is only a bad case, which should be ignored.