# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 2 Multi-Channel Beam-Steering SAR System
### Pre-Processing Approach
\tag{11}\]  
Equation (10) can be then rewritten in matrix form as  
\[\mathbf{S}_{a}\left(f,r\right)=\mathbf{H}\left(f\right)\mathbf{U}_{a}\left(f, r\right), \tag{12}\]  
where  
\[\mathbf{S}_{a}\left(f,r\right)=\left[S_{a}{}^{\left(0\right)}\left(f,r\right),S_{a}{}^{\left(1\right)}\left(f,r\right),\cdots,S_{a}{}^{\left(N-1\right)} \left(f,r\right)\right]^{T}, \tag{13}\]  
and  
\[\mathbf{U}_{a}\left(f,r\right)=\left[U_{a}\left(f+0\cdot f_{prf};r\right),U_{ a}\left(f+1\cdot f_{prf};r\right),\cdots,U_{a}\left(f+\left(N-1\right) \cdot f_{prf};r\right)\right]^{T}\!\!\!. \tag{14}\]  
The reconstruction filter is then given by  
\[\mathbf{P}\left(f\right)=\mathbf{H}^{-1}\left(f\right)=[\mathbf{P}_{1}\left( f\right),\mathbf{P}_{2}\left(f\right),\cdots,\mathbf{P}_{N}\left(f\right)]^{T}, \tag{15}\]  
where \(\left[\bullet\right]^{-1}\) indicates the matrix inverse and \(\left[\bullet\right]^{T}\) denotes matrix transposition. Then the unambiguous Doppler spectrum \(\mathbf{U}_{a}\left(f,r\right)\) can be obtained after summing the weighted \(S_{a}^{\left(i\right)}\left(f;r\right)\) for all Doppler gates as  
\[\mathbf{U}_{a}\left(f,r\right)=\mathbf{P}\left(f\right)\mathbf{S}_{a}\left(f, r\right). \tag{16}\]  
Note \(\mathbf{P}\left(f\right)\) may fail to reconstruct the spectrum because the transfer matrix \(\mathbf{H}\left(f\right)\) is singular when the receivers' samplings coincide spatially. References [31; 32; 33] provide the modified reconstruction filters for this case. Using filter reconstruction in the standard derotation operation is superior to use of the FFT. Signals of \(N\) channels after the first phase multiplication are transformed into the Doppler domain and combined as a whole unambiguous spectrum by the reconstruction filtering, which plays a similar role to the FFT in the implementation of convolution. On the other hand, the signal produced by the reconstruction filter can also be regarded as being in the time domain [14] because the FFT in pre-processing is one of steps of derotation implement, which is essentially a convolution operation in time domain. The last phase multiplication in the pre-processing can then be performed as in single-channel BS-SAR, with a phase function given by  
\[h_{2}\left(t\right)=\exp\left\{-j\pi k_{\omega}t^{2}\right\}. \tag{17}\]  
The signal after the pre-processing can then be expressed as \[s_{b}\left(t;r,t_{d}\right)=rect\left[\frac{t}{\frac{X_{f}(r)}{v}\frac{r_{s}}{r}} \right]rect\left[\frac{t-\frac{r_{s}}{r}t_{0}}{T_{w}\left(\frac{r_{s}}{r}-1 \right)}\right]rect\left[\frac{vt_{0}}{X_{s}}\right]\exp\left\{j\pi\frac{2v^{2 }\text{cos}^{2}\theta}{\lambda\left(r_{s}-r\right)}(t-t_{0})^{2}\right\}. \tag{18}\]  
Please note that there are three envelope terms in (18) and \(vt_{0}\) in the third term represents the azimuth position of arbitrary targets. Whatever the value of \(t_{0}\), the first term is contained in the second term, as demonstrated in [34]. Therefore, the time range of signal \(s_{b}\left(t;r,t_{d}\right)\) is determined by the first term, and (18) can be simplified as  
\[s_{b}\left(t;r,t_{d}\right)=rect\left[\frac{t}{\frac{X_{f}(r)}{v}\frac{r_{s}}{ r}}\right]\exp\left\{j\pi\frac{2v^{2}\text{cos}^{2}\theta}{\lambda\left(r_{s}-r \right)}(t-t_{0})^{2}\right\}. \tag{19}\]  
This means the time ranges of the pre-processed signals from targets at different azimuths are independent of their azimuth positions and overlap in time. For simplicity, we use the symbol \(\left\{t\right\}\) to represents the time range of the pre-processed signal. Because only one FFT is employed in the pre-processing, the overlapped envelope is bell-shaped, and it is applicable to perform azimuth antenna pattern correction and signal weighting at this stage. After pre-processing, the equivalent PRF is given as [34]  
\[f_{prf2}=\frac{N_{a}\cdot\left|k_{\omega}\right|}{N\cdot f_{prf}}, \tag{20}\]  
where \(N_{a}\) is the total processing number in azimuth [34] after zero-padding. It should be selected to ensure \(f_{prf2}>B_{t}\).