# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 2 Multi-Channel Beam-Steering SAR System
### Signal Mode  
A multi-channel BS-SAR system transmits a linear frequency modulated signal and each sub-channel receives the backscattered echoes. Since parameter errors in azimuth are the focus of this paper, for simplicity only the azimuth signal is considered.  
In multi-channel BS-SAR, antenna steering is combined with multi-channel technology. In Figure 2, \(T_{w}\) is the overall illumination time, \(X_{s}\) represents the fully illuminated swath in azimuth, and \(X_{f}(r)\) represents the azimuth antenna footprint length. In addition, there are two parameters specific to multi-channel BS-SAR. One is \(\omega_{\theta}\), the beam-steering rate, which is usually constant. Since the effective velocity [30] of the satellite is also approximately invariant, the beam is assumed to focus on a virtual rotation point (VRP). The beam rotates clockwise around the VRP in multi-channel sliding spotlight SAR and counterclockwise in multi-channel TOPSAR. Another new parameter, \(r_{s}\), is the distance from the SAR sensor to the VRP at the central illumination time, while \(r\) is the distance from the SAR sensor to the target at zero-Doppler time. \(\omega_{\theta}\) and \(r_{s}\) are positive in multi-channel sliding spotlight mode and negative in multi-channel TOPS mode.  
Figure 1: The combined modes with multi-channel technology and beam-steering technology. Three different colors (green, red, and yellow) in the figure represent three different channels.  
We define a factor \(\alpha(r)\), which multiplies the azimuth resolution in multi-channel BS-SAR due to the steering beam, as  
\[\alpha\left(r\right)=\frac{r_{s}-r}{r_{s}}. \tag{1}\]  
It can be seen that \(\alpha\left(r\right)<1\) in multi-channel sliding spotlight SAR and \(\alpha\left(r\right)>1\) in multi-channel TOPSAR. So, the azimuth resolution in multi-channel sliding spotlight SAR is finer than in multi-channel strip-map SAR, while coarser in multi-channel TOPSAR. Reference [15] provides another definition of \(\alpha\left(r\right)\), which is essentially the same.  
Assuming that there are \(N\) channels and the azimuth antenna pattern is rectangular with 3 dB beamwidth, the azimuth signal from a point target located at \(\left(r,vt_{0}\right)\) acquired by \(i\)-th receiver can be expressed as  
\[ss_{a}^{\left(i\right)}\left(t;r,t_{0}\right)=rect\left[\frac{\alpha\left(r \right)\left(vt+d_{i}/2\right)-vt_{0}}{X_{f}\left(r\right)}\right]\cdot rect \left[\frac{vt+d_{i}/2}{vT_{w}}\right]\cdot rect\left[\frac{vt_{0}}{X_{s}} \right]\exp\left\{-j\frac{4\pi}{\lambda}R^{\left(i\right)}\left(t;r,t_{0} \right)\right\}. \tag{2}\]  
In Equation (2), \(t\) is the slow time in azimuth, \(rect\left[\bullet\right]\) represents the rectangular function, \(d_{i}\) represents the distance between the \(i\)-th channel and the reference channel, which usually is at the center of the antenna, \(v\) is the effective velocity of the SAR sensor, and \(\lambda\) is the wavelength. \(R^{\left(i\right)}\left(t;r,t_{0}\right)\) is the instantaneous distance between the \(i\)-th equivalent phase center of the sensor and an arbitrary target, given by [17]  
\[R^{\left(i\right)}\left(t;r,t_{0}\right)=\frac{1}{2}R\left(t-t_{0};r\right)+ \frac{1}{2}R\left(t+\frac{d_{i}}{v}-t_{0};r\right)\approx R\left(t+\frac{d_{ i}}{2v}-t_{0};r\right)+\frac{f_{r}d_{i}^{2}}{16v^{2}}, \tag{3}\]  
where \(R\left(t;r\right)=\sqrt{r^{2}+\left(vt\right)^{2}-2 vt\cos\theta}\), in which \(\theta\) is the aspect angle depending on \(r\), and \(f_{r}\) is Doppler rate. It can be seen that the signal of the \(i\)-th channel can be seen as a replica of the reference signal-channel BS-SAR impulse response with a delay.