# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 3 Estimation Methods for DRR, CDC Error, and Phase Inconsistency Errors
### Estimation for Phase Inconsistency Errors  
Phase inconsistency error estimation is important for fine spectrum reconstruction and imaging in multi-channel SAR systems. The existing estimation methods for multi-channel strip-map SAR are mainly based on the MUSIC method [21]. This kind of methods are all based on the frequency consistency of all signals from different azimuth positions. In multi-channel strip-map SAR, all the signals from different targets will overlapped in Doppler domain as in Figure 7a. It can be seen that there are two ambiguous spectra in the grey parts, which can be used to construct the covariance matrix of Doppler spectrum. Then the signal subspace and the noise subspace can be extracted and are used to calibrate the phase inconsistency errors. However, in the multi-channel BS-SAR (Figure 7b), the Doppler spectrum of targets at different azimuth position is not overlapped because of the azimuth-varying Doppler centroid. Therefore, it is impossible to determine the ambiguity number in a certain Doppler gate and there is not appropriate spectrum that MUSIC method can employ. References [27; 28] propose estimation methods for multi-channel strip-map SAR based on the spectrum energy distribution, namely AWLS and DSO. The principle of these methods is that the spectrum energy in the effective bandwidth will leak out due to the phase inconsistency errors. Then some merit indicators can be used to calibrate the phase inconsistency errors. However, only part of spectrum is considered in both method which restricts their robustness to CDC error and their feasibilities to multi-channel BS-SAR was not investigated. In this section, the principle of optimization is extended into the multi-channel BS-SAR based on the expression of signal after pre-processing. The smaller the phase inconsistency errors, the smoother and more concentrated the spectrum. Figure 7: The Doppler frequency aliasing in multi-channel strip-map SAR (**a**) and multi-channel BS-SAR (**b**). The three colors represent three targets in azimuth. \(f\) and \(f_{a}\) denote the baseband frequency and physical frequency, respectively. Image Entropy is then adopted to assess how well the spectrum is concentrated after pre-processing. The advantage of using image entropy is that it can simultaneously evaluate the spectrum inside and outside the effective bandwidth, which can achieve better performance that AWSL and DSO. Another benefit of using the whole spectrum is that the CDC error can be ignored while the AWLS and DSO methods require knowledge of the CDC. Using \(m\) and \(l\) to represent the azimuth and range pixel indices respectively, rewrite Equation (12) in discrete form with phase inconsistency errors:  
\[\mathbf{\tilde{S}}_{a}\left(m,l\right)=\Gamma\mathbf{H}\left(m\right)\mathbf{ U}_{a}\left(m,l\right), \tag{38}\]  
where \(\mathbf{\tilde{S}}_{a}\left(m,l\right)\) is the discrete form of \(\mathbf{S}_{a}\left(f,r\right)\) with phase inconsistency errors, given as  
\[\mathbf{\tilde{S}}_{a}\left(m,l\right)=\left[\mathbf{\tilde{S}}_{a}^{\left(0 \right)}\left(m,l\right),\mathbf{\tilde{S}}_{a}^{\left(1\right)}\left(m,l \right),\cdots,\mathbf{\tilde{S}}_{a}^{\left(N-1\right)}\left(m,l\right)\right] ^{T} \tag{39}\]  
and \(\Gamma\) is the phase inconsistency error matrix:  
\[\Gamma=diag\left\{\mathbf{g}\right\}=diag\left\{\mathrm{e}^{\mathrm{i}\tilde{ S}_{0}},\mathrm{e}^{\mathrm{i}\tilde{S}_{1}},\cdots,\mathrm{e}^{\mathrm{i}\tilde{S}_{N-1}} \right\}. \tag{40}\]  
The unambiguous spectrum \(\mathbf{U}_{a}\left(m,l\right)\) can then be estimated by  
\[\mathbf{U}_{a}\left(m,l\right)=\mathbf{P}\left(m\right)\Gamma^{-1}\mathbf{ \tilde{S}}_{a}\left(m,l\right), \tag{41}\]  
where \(\mathbf{P}\left(m\right)\) is the discrete form of \(\mathbf{P}\left(f\right)\) in (15).