# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 3 Estimation Methods for DRR, CDC Error, and Phase Inconsistency Errors
### Estimation of the CDC Error  
Since the Doppler spectrum of multi-channel SAR data is ambiguous, the CDC estimation method for single-channel SAR data cannot be used directly. Reference [20] proposes the SCCC method for a multi-channel SAR system, but this will fail when the spatial sampling of multi-channels is severely non-uniform, which restricts its application. In fact, the CDC can be determined based on the pre-processed signal. As noted in Section 2.3, the signal envelope after pre-processing is bell-shaped,  
Figure 5: Demonstration of the influence of DRR: (**a**) shows a scene with three target points located at different range and azimuth positions; (**b**) illustrates the symmetrical shift of the azimuth signal after pre-processing with DRR error; (**c**) shows the scaling effect of the azimuth signal after pre-processing with DRR error; (**d**) the result of pre-processing without DRR error.  
similarly to the azimuth antenna pattern. So, the methods in [30], such as power balance and peak detecting, can be used on the pre-processed signal to estimate the CDC. In this section, these methods are extended to be used in the estimation of CDC error.  
If there is a CDC error, \(\Delta f_{d}\), the signal after the first phase multiplication with \(h_{1}^{(i)}\left(t\right)\) will include \(\Delta f_{d}\). Due to the transformation of the pre-processing, the Doppler shift introduced by a CDC error can be seen as a time delay in the signal after pre-processing as  
\[s_{\mathrm{c}}^{\mathrm{c}}\left(t;r,t_{0}\right)=rect\left[\frac{t+\frac{ \Delta f_{d}}{\Delta\
u}-\frac{\delta}{\left(1+\delta\right)\alpha\left(r \right)}t_{0}}{\frac{X_{f}\left(r\right)}{v}\frac{r_{g}}{r}\frac{r_{g}}{\left(1 +\delta\right)}\left(1-\frac{\delta}{\alpha\left(r\right)}\frac{r}{r_{g}} \right)}\right]rect\left[\frac{vt_{0}}{X_{s}}\right]\exp\left\{j\pi\frac{k_{ \omega}f_{r}}{k_{\omega}^{\epsilon}-f_{r}}\left(t-t_{0}\right)^{2}\right\}. \tag{33}\]  
From (33), it can be seen that the time range \(\left\{t\right\}\) of a point target is shifted by the DRR error and CDC error. So, the CDC error can be integrated into the estimation of DRR error. The time delay of an arbitrary point target after pre-processing with DRR error and DC error is given by  
\[t_{s2}=\frac{\delta}{\left(1+\delta\right)}\frac{t_{0}}{\alpha\left(r\right)}- \frac{\Delta f_{d}}{k_{\omega}^{\epsilon}}. \tag{34}\]  
For two point targets located at (r,\(vt_{1}\)) and (r, \(vt_{2}\)), the time delays after processing are  
\[\left\{\begin{array}{l}t_{s2,1}=\frac{\delta}{\left(1+\delta\right)}\frac{t _{1}}{\alpha\left(r\right)}-\frac{\Delta f_{d}}{k_{\omega}^{\epsilon}}\\ t_{s2,2}=\frac{\delta}{\left(1+\delta\right)}\frac{t_{2}}{\alpha\left(r\right) }-\frac{\Delta f_{d}}{k_{\omega}}\end{array}\right. \tag{35}\]  
It is then easy to obtain the DRR error and CDC error by solving Equation (35). The DRR error can also be solved using (32), and \(\Delta f_{d}\) can be obtained from (35). If we split the raw data equally as in Section 3.2, \(t_{1}=-t_{2}\). Therefore, the CDC error can be also be obtained as  
Figure 6: Flowchart of the method to estimate DRR and CDC errors.  
\[\Delta f_{d}=-\frac{1}{2}k_{w}^{c}\left(t_{s2,1}+t_{s2,1}\right). \tag{36}\]  
The flowchart of the estimation of DRR error and CDC error is shown in Figure 6. To obtain sufficient precision, the estimation procedure can be iterated until the termination condition is met as  
\[\left|\Delta k_{w}^{(q)}\right|<\Delta k_{w,th},\quad\left|\Delta f_{d}^{(q)} \right|<\Delta f_{d,th}, \tag{37}\]  
where \(\Delta k_{w}^{(q)}\) and \(\Delta f_{d}^{(q)}\) are the estimated DRR and CDC error after the \(q\)-th iteration respectively and \(\Delta k_{w,th}\) and \(\Delta f_{d,th}\) denote their thresholds for the precision.