# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## Appendix A
The ranges of \(t^{(1)}\) and \(t^{(2)}\) when \(t_{0}=-\frac{X_{s}}{2v}\) and \(t_{0}=\frac{X_{s}}{2v}\) are derived as  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[-\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,-\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{31}\]  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]  
\[t^{(2)}\big{|}_{t_{0}=-X_{s}/(2v)}\in\left[-\frac{r_{s}^{\epsilon}}{r}\frac{ X_{s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,-\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{ r}-1\right|\right], \tag{33}\]  
\[t^{(2)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{r_{s}^{\epsilon}}{r}\frac{X_{ s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r} -1\right|\right]. \tag{34}\]  
Then \(t^{1}\) and \(t^{2}\) can also be expressed as  
\[t^{(1)}=\left.t^{(1)}\right|_{t_{0}=-X_{s}/(2v)}+\frac{\delta}{(1+\delta)\, \alpha\,(r)}\left(\frac{X_{s}}{2v}+t_{0}\right), \tag{35}\]  
\[t^{(1)}=\left.t^{(1)}\right|_{t_{0}=X_{s}/(2v)}-\frac{\delta}{(1+\delta)\, \alpha\,(r)}\left(\frac{X_{s}}{2v}-t_{0}\right), \tag{36}\]  
\[t^{(2)}=\left.t^{(2)}\right|_{t_{0}=-X_{s}/(2v)}+\frac{r_{s}^{\epsilon}}{r} \left(\frac{X_{s}}{2v}+t_{0}\right), \tag{37}\]  
\[t^{(2)}=\left.t^{(2)}\right|_{t_{0}=X_{s}/(2v)}-\frac{r_{s}^{\epsilon}}{r} \left(\frac{X_{s}}{2v}-t_{0}\right). \tag{38}\]  
First taking the multi-channel sliding spotlight mode, in which \(r_{s}>r>0\), substitute (36) and (38) into \(t_{l}^{(1)}-t_{l}^{(2)}\), then \[t_{l}^{(1)}-t_{l}^{(2)} = t_{l}^{(1)}\Big{|}_{t_{0}=X_{s}/\left(2v\right)}-\frac{\delta}{\left(1 +\delta\right)\alpha\left(r\right)}\left(\frac{X_{s}}{2v}-t_{0}\right)-\left.t_{ l}^{(2)}\Big{|}_{t_{0}=X_{s}/\left(2v\right)}+\frac{r_{s}^{e}}{r}\left(\frac{X_{s}}{2v}-t_{0} \right)\right.\] \[= \left.\Delta t_{l}\left(t_{0}\right)\right|_{t_{0}=X_{s}/\left(2 v\right)}+\left(\frac{r_{s}^{e}}{r}-\frac{\delta}{\left(1+\delta\right)\alpha \left(r\right)}\right)\left(\frac{X_{s}}{2v}-t_{0}\right)\]  
where  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]0 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]1 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]2 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]3 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]4 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]5  
Notice that we have used (22) and the following equation  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]6  
Therefore,  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]7 \[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]8  
Next, substitute (10) and (11) into \(\Delta t_{r}(t_{0})\) as  
\[t^{(1)}\big{|}_{t_{0}=X_{s}/(2v)}\in\left[\frac{\delta}{(1+\delta)\,\alpha\, (r)}\frac{X_{s}}{2v}-\frac{1}{2}\frac{X_{\alpha\theta}\,(r)}{(1+\delta)\,v} \left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|,\frac{\delta}{(1+ \delta)\,\alpha\,(r)}\frac{X_{s}}{2v}+\frac{1}{2}\frac{X_{\alpha\theta}\,(r)} {(1+\delta)\,v}\left|\frac{r_{s}}{r}-\frac{\delta}{\alpha\,(r)}\right|\right], \tag{32}\]9 \[t^{(2)}\big{|}_{t_{0}=-X_{s}/(2v)}\in\left[-\frac{r_{s}^{\epsilon}}{r}\frac{ X_{s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,-\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{ r}-1\right|\right], \tag{33}\]0  
where  
\[t^{(2)}\big{|}_{t_{0}=-X_{s}/(2v)}\in\left[-\frac{r_{s}^{\epsilon}}{r}\frac{ X_{s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,-\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{ r}-1\right|\right], \tag{33}\]1 \[t^{(2)}\big{|}_{t_{0}=-X_{s}/(2v)}\in\left[-\frac{r_{s}^{\epsilon}}{r}\frac{ X_{s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,-\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{ r}-1\right|\right], \tag{33}\]2 \[t^{(2)}\big{|}_{t_{0}=-X_{s}/(2v)}\in\left[-\frac{r_{s}^{\epsilon}}{r}\frac{ X_{s}}{2v}-\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{r}-1\right|,-\frac{r_{s}^{ \epsilon}}{r}\frac{X_{s}}{2v}+\frac{1}{2}T_{w}\left|\frac{r_{s}^{\epsilon}}{ r}-1\right|\right], \tag{33}\]3  
as similar to (10).