# Passive Bistatic Ground-Based Synthetic Aperture Radar: Concept, System, and Experiment Results
## 3 Signal Processing
### Imaging and Displacement Estimation
Moreover, for the reference antenna with \((x_{p},y_{ref})\) denoting its \(p\)-th antenna position, based on the plane wave approximation and Taylor series expansion, it can be derived that  
\[R_{1}^{p}(x_{0},y_{0})=R_{1,2}(x_{0},y_{0})-R_{1,1}^{p}\simeq y_{0}-y_{ref} \tag{23}\]  
The maximal approximation error in Equation (23) will be increased for the target with a larger azimuth position \(x_{0}\), as shown in Figure 13, where the simulation parameters are: \(y_{T}=-36\),000 km and \(x_{p}\) is from \(-0.6\) m to \(0.6\) m with a step of \(5\) mm. It can be seen that, with the considered local area imaging, the approximation error is always smaller than \(0.15\) mm (corresponding to a phase error of \(\pi/80\)). Therefore, the plane wave approximation is used for PB-GB-SAR imaging to simplify the imaging process, as, in such a case, the TV satellite position is not necessary to be known in advance. Then, the complex amplitude of the point at \((x_{0},y_{0})\) can be estimated by BPA as  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]0  
To effectively calculate Equation (24), the FFT of zero-padded \(s_{p}(f)\) with the interpolation process or the type-II nonuniform FFT (NUFFT) [48] can be used. In this paper, the latter method is adopted, which can be expressed as  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]1  
However, when a fine grid size is selected for imaging, the computational complexity of Equation (25) is high. In the following, similar to GB-SAR, RMA is formulated for PB-GB-SAR. At first, the frequency domain received signal from the whole observation scene at the \(p\)-th antenna position is expressed as  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]2  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]3 Using the stability of the peak amplitudes of reference signal autocorrelation function to determine the antenna moving direction: (**left**) experiment setup and (**right**) calculation results. \[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]4 Approximation error for targets at different positions. The right subfigure is for the targets with \(y_{0}=100\). where \(\sigma(x,y)\) is the reflection coefficient of the target at \((x,y)\) and \(\Omega\) denotes the observation scene. Then, by Fourier-transform, with respect to \(x_{p}\), we can get  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]5  
where  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]6  
Based on the principle of stationary phase (PSP), \(S_{1}(k_{x},k)\) can be approximated by  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]7  
By substituting Equation (29) into Equation (27), we have  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]8  
Conventionally, to use FFT to increase the computing speed, the interpolation process is used to form a signal matrix uniformly sampled in the \(kx\)-\(ky\) domain, which can be expressed as  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]9  
where the arrow stands for interpolation. It should be noted that, different from GB-SAR with \(k_{y}=\sqrt{4k^{2}-k_{x}^{2}}\), for PB-GB-SAR, we have \(k_{y}=\sqrt{k^{2}-k_{x}^{2}}+k\).