# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 2 Multi-Channel Beam-Steering SAR System
### Spectrum Characteristics  
The azimuth spectrum of multi-channel BS-SAR has some particular characters due to steering beam and multi-channel sampling. The raw data support in the azimuth time-frequency domain (TFD) is outlined in Figure 3 for both multi-channel sliding spotlight SAR and multi-channel TOPSAR.  
Figure 2: Multi-channel sliding spotlight SAR and multi-channel TOPSAR. (**a**) is the multi-channel sliding spotlight and (**b**) is the multi-channel TOPSAR. Three different colors (green, red and yellow) in the figure represent three different channels.  
In Figure 3, three points (P1, P2 and P3) located on the left, center, and right of the azimuth are highlighted. The colored thin lines in this figure represent Doppler histories with negative slope, \(f_{r}\). The Doppler history of P2 is shifted upwards because of the constant Doppler centroid. The Doppler centroid of P1 is large in multi-channel sliding spotlight SAR because it experiences forward-looking squint illumination by the beam. By contrast, P3 is illuminated by backward-looking squint and has a smaller Doppler centroid. Consequently, the beam-steering results in a parallelogram TDF support which slants towards the lower right in multi-channel sliding spotlight SAR and towards the upper right in multi-channel TOPSAR. The steering bandwidth, \(B_{s}\), which occurs in multi-channel BS-SAR is then given by  
\[B_{s}=\left|k_{\omega}\right|\cdot T_{w}, \tag{4}\]  
where \(k_{\omega}\) is slope of the azimuth-varying Doppler centroid (or known as derotation rate) as [3]  
\[k_{\omega}=\frac{\partial\left(\frac{2v}{\lambda}\cos\left(\theta_{dc}\left(t \right)\right)\right)}{\partial t}\approx-\frac{2v}{\lambda}\sin\theta\omega_{ \theta}\approx-\frac{2v^{2}\text{sin}^{2}\theta}{\lambda r_{s}} \tag{5}\]  
where \(\theta_{dc}(t)\) is the instantaneous aspect angle.  
Due to the beam steering, the illumination time of any point target is expanded or shrunk by the factor \(1/\alpha(r)\), which results in their instantaneous bandwidth, \(B_{a}\), being weighted by the same factor. Figure 3 shows that the instantaneous bandwidth is larger than the PRF but less than \(N\cdot PRF\). The total bandwidth, including the instantaneous bandwidth \(B_{a}\) and the steering bandwidth \(B_{s}\) is then given by  
\[B_{t}=B_{a}+B_{s}=B_{3 dB}/\alpha\left(r\right)+\left|k_{\omega}\right|\cdot T _{w}, \tag{6}\]  
where \(B_{3 dB}\) is the bandwidth corresponding to the 3 dB antenna beam width.  
It can be seen that the total bandwidth, \(B_{t}\), is very large in the case of beam steering while the value \(N\cdot PRF\) is usually only a little larger than \(B_{3 dB}\). This means the bandwidth would span several PRFs, resulting in spectrum aliasing. Moreover, the operating PRF in multi-channel SAR systems becomes several (\(\sim\)\(N\)) times less than in single-channel SAR, so spectrum aliasing is very serious and needs to be removed.  
Figure 3: Time-frequency domain (TFD) diagram of multi-channel BS-SAR. (**a**) is for the multi-channel sliding spotlight and (**b**) is for the multi-channel TOPSAR. 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 the time and Doppler domain, respectively. The gradient color bars (blue, grey and gold) in the background represent the total PRF that multi-channel BS-SAR can achieve with \(N\) channels (here \(N=3\)). The purple area is the whole TFD of the multi-channel BS-SAR system and the shaded parts on both sides are the TFDs of edge points without full illumination.