# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 3 Estimation Methods for DRR, CDC Error, and Phase Inconsistency Errors
### Analysis of DRR Error
#### 3.1.2 Scaling Effect  
The duration of \(\left\{t\right\}\) of an arbitrary point target is still constant but is increased or decreased by a factor given by:  
\[\beta\left(\delta\right)=\frac{1}{\left(1+\delta\right)}\left(1-\frac{\delta} {\alpha\left(r\right)}\frac{r}{r_{s}}\right)=\frac{1}{\left(1+\delta\right)} \left(1-\frac{\delta}{\alpha\left(r\right)}\left(1-\alpha\left(r\right)\right) \right)=1-\frac{\delta}{\left(1+\delta\right)\alpha\left(r\right)}. \tag{28}\]  
Since equivalent PRF (see in (20)) is a function of \(k_{\omega}\), the number of samples of \(\left\{t\right\}\) for an arbitrary point target is also increased or decreased:  
\[\frac{N_{t}^{e}}{N_{t}}=\frac{f_{prf2}^{e}\beta\left(\delta\right)}{f_{prf2}}= \left(1+\delta\right)\beta\left(\delta\right)=1+\delta-\frac{\delta}{\alpha \left(r\right)}, \tag{29}\]  
where \(N_{t}^{e}\) and \(N_{t}\) are the numbers of samples of a target with and without DRR error, respectively. Here \(f_{prf2}^{e}\) is the equivalent PRF after pre-processing with DRR error. When \(\delta>0\), \(N_{t}\) is reduced for multi-channel sliding spotlight SAR and increased for multi-channel TOPSAR, while the opposite occurs when \(\delta<0\). Hence we refer to this as a'scaling effect'. Although \(N_{t}\) will reduce in multi-channel sliding spotlight SAR with \(\delta>0\) and in multi-channel TOPSAR with \(\delta<0\), the number of samples for all targets illustrated will increase because the'symmetrical shift' will separate the energy along azimuth (see Appendix B).  
Taking the multi-channel TOPS mode as an example, a simple simulation is developed using the parameters listed in Table 1. Figure 5 shows the results after pre-processing. Three points are located at different range and azimuth positions in the scenario in Figure 5a. With DRR error (\(\delta=0.03\)), signal shift occurs after pre-processing in Figure 5b, whereas the time ranges \(\left\{t\right\}\) of three point targets are aligned after pre-processing without DRR error in Figure 5d. The number of samples clearly increase for DRR error in Figure 5c.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Image Parameters** & **Multi-Channel TOPS** & **Multi-Channel Sliding Spotlight** \\ \hline Wavelength & 0.03 m & 0.03 m \\ Elevation angle & 30.0\({}^{\circ}\) & 30.0\({}^{\circ}\) \\ Orbit height & 630 km & 630 km \\ Eccentricity & 0.0011 & 0.0011 \\ Orbit inclination angle & 98\({}^{\circ}\) & 98\({}^{\circ}\) \\ Pulse width & 10 \(\mu\)s & 10 \(\mu\)s \\ PRF & 1700 Hz & 1700 Hz \\ Channel number & 6 & 6 \\ Signal bandwidth & 80 MHz & 500 MHz \\ Sampling rate & 96 MHz & 600 MHz \\ Swath & 100 km \(\times\) 100 km (A \(\times\) R) & 20 km \(\times\) 20 km (A \(\times\) R) \\ Antenna length & 9 m & 9 m \\ Azimuth resolution & 5 m & 0.5 m \\ Aspect angle & 90\({}^{\circ}\) & 90\({}^{\circ}\) \\ Effective velocity & 7275.8 m/s & 7275.8 m/s \\ \(r_{s}\) & \(-\)202, 942.8 m & 1,323,920.6 m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.