# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 2 Multi-Channel Beam-Steering SAR System
### Pre-Processing Approach  
The combination of filtering reconstruction and derotation is commonly adopted to tackle spectrum aliasing in multi-channel BS-SAR because of its high efficiency and effectiveness. Derotation is essentially a convolution operation, which can be implemented by complex function multiplication and FFT, but the aliasing caused by under-sampling will remain. In the pre-processing algorithm, the FFT can be replaced by filter reconstruction, which can overcome the remaining aliasing and simultaneously produce a uniformly sampled signal. The first phase function of pre-processing in a multi-channel BS-SAR system is given by  
\[h_{1}^{(i)}\left(t\right)=\exp\left\{-j\pi k_{\omega}\left(t+\frac{d_{i}}{2v} \right)^{2}-j\pi f_{d}\left(t+\frac{d_{i}}{2v}\right)+j\frac{\pi f_{r}d_{i}^{2} }{4v^{2}}\right\}\,, \tag{7}\]  
where \(f_{d}\) is the CDC. After multiplying by the first phase function \(h_{1}^{(i)}\left(t\right)\), the steering bandwidth \(B_{s}\) is eliminated and the CDC is compensated as shown in Figure 4a. Expanding the range \(R^{(i)}\left(t;r,t_{0}\right)\) by a second order Taylor expansion and ignoring the constant term, the signal expression of \(i\)-th channel after multiplying by the first phase function \(h_{1}^{(i)}\left(t\right)\) is  
\[s_{a}^{(i)}\left(t;r,t_{0}\right)=u_{a}\left(t+d_{i}/(2v);r,t_{0}\right)\,, \tag{8}\]  
where \(u_{a}^{(i)}\left(t;r,t_{0}\right)\) is the reference single-channel BS-SAR signal after first step of derotation, given by  
\[u_{a}\left(t;r,t_{0}\right)=rect\left[\frac{\alpha\left(r\right)vt-vt_{0}}{X_ {f}\left(r\right)}\right]\cdot rect\left[\frac{t}{T_{w}}\right]\cdot rect \left[\frac{vt_{0}}{X_{s}}\right]\exp\left\{j\pi f_{r}\left(t-t_{0}\right)^{ 2}-k_{\omega}\left(t-t_{0}\right)^{2}\right\}. \tag{9}\]  
As shown in Figure 4a, the bandwidth of \(s_{a}^{(i)}\left(t;r,t_{0}\right)\) is still larger than the PRF due to the under-sampling in each channel. Transforming (8) into the Doppler domain by performing FFT operation:  
Figure 4: TFD plots of multi-channel BS-SAR after multiplying by the first phase function. (**a**) is without DRR error and (**b**) is with DRR error. Three points are represented in red, yellow and green. The time and bandwidth ranges are shown at the bottom and left. \(A\left(t\right)\) and \(A\left(f\right)\) represent signal amplitudes in time domain and Doppler domain, respectively. The gradient color (blue, grey and gold) in background represent the total PRF with \(N\) channels (here \(N=3\)). The purple area is the whole TFD and the shaded parts on both sides are the TEDs of some edge points without full illumination. \[S_{a}^{\left(i\right)}\left(f;r\right)=\frac{1}{N}\sum\limits_{k=1}^{N}U_{a} \left(f+\left(k-1\right)\cdot f_{prf};r\right)e^{i2\pi\left(f+\left(k-1\right) \cdot f_{prf}\right)\frac{d_{i}}{2\upsilon}}, \tag{10}\] \[with-\frac{N\cdot f_{prf}}{2}+f_{d}<f<-\frac{\left(N-2\right) \cdot f_{prf}}{2}+f_{d}\]  
where \(U_{a}\left(f+\left(k-1\right)f_{prf};r\right)\) is the \(k\)-th sub-band of the spectrum of \(u_{a}\left(t;r,t_{0}\right)\). The transfer matrix of multi-channel BS-SAR after the derotation can be described by [7; 12]  
\[\mathbf{H}\left(f\right)=\frac{1}{N}\left( \begin{array}{ccc}e^{i2\pi\left(f +0\cdot f_{prf}\right)\frac{d_{0}}{2\upsilon}}&\cdots&e^{j2\pi\left(f+\left(N- 1\right)\cdot f_{prf}\right)\frac{d_{0}}{2\upsilon}}\\ \vdots&\ddots&\vdots\\ e^{j2\pi\left(f+0\cdot f_{prf}\right)\frac{d_{M}}{2\upsilon}}&\cdots&e^{j2\pi \left(f+\left(N-1\right)\cdot f_{prf}\right)\frac{d_{M}}{2\upsilon}}\end{array}  \right).