# Passive Bistatic Ground-Based Synthetic Aperture Radar: Concept, System, and Experiment Results
## 3 Signal Processing
### Imaging and Displacement Estimation  
As mentioned previously, since a high azimuth resolution is desired, the surveillance antenna is moved along a positioner to generate a synthetic aperture, while the reference antenna is stationary and its looking angle is adjusted to the satellite, as shown on the left of Figure 4. For the \(p\)-th position of the surveillance antenna (\(x_{p}\), \(y_{p}\)), the range compression result obtained by combing \(N\) TV channels is given by  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}[s_{surv}^{p}(\mathbf{f})s_{ref}^{ p}(\mathbf{f})^{*}] \tag{17}\]  
where \(\mathbf{F}\) is the Fourier-transform matrix. Since the targets within short ranges (i.e., 100 m) are considered, it is not necessary to calculate the time delays corresponding to the far range targets. Therefore, Equation (17) is modified to  
\[\mathbf{\chi}_{p}=\mathbf{F}^{H}s_{p}(\mathbf{f})=\mathbf{F}^{H}\mathbf{\sum}_{m=1}^{M}s_{p}^{m}( \mathbf{f})=\mathbf{F}^{H}\mathbf{\sum}_{m=1}^{M}s_{surv}^{p,m}(\mathbf{f})s_{ref}^{p,m}(\mathbf{f })^{*} \tag{18}\]  
where \(M=T_{\text{int}}/T_{\text{p}}\), with \(T_{\text{p}}\) being the processing time. By doing so, the computational complexity can be reduced for the following imaging methods. After range compression at each antenna position, azimuth compression can be conducted by the time domain BPA, which estimates the complex amplitude of the target at (\(x_{0}\), \(y_{0}\)) as  
\[\sigma_{BPA}(x_{0},y_{0})=\sum\
olimits_{p=1}^{P}\mathbf{\chi}_{p}[R_{1}(x_{0},y_ {0})+R_{2}^{p}(x_{0},y_{0})] \tag{19}\]  
where  
\[R_{1}(x_{0},y_{0})=R_{1,2}(x_{0},y_{0})-R_{1,1}=\sqrt{{x_{0}}^{2}+(y_{0}-y_{T} )^{2}}-(y_{ref}-y_{T}) \tag{20}\]  
denotes the difference between the satellite-to-target distance and the satellite-to-reference-antenna distance, and  
\[R_{2}^{p}(x_{0},y_{0})=\sqrt{(x_{0}-x_{p})^{2}+(y_{0}-y_{p})^{2}} \tag{21}\]  
denotes the distance between the target and the surveillance antenna. It should be noted that, to calculate Equation (20), the position of the TV satellite should be known in advance, and, to calculate Equation (21), the surveillance antenna position should be determined by its moving step \(\Delta x\) and the angle between its moving direction and the \(x\) axis, i.e., \(\varphi_{0}\) on the left of Figure 4.  
To simplify the calculation of Equations (20) and (21), it is desired that the angle \(\varphi_{0}\) is zero, i.e., the antenna moving direction should be parallel with the \(x\) axis. To this end, the reference antenna faced to the satellite is moved together with the surveillance antenna, as shown on the right of Figure 4.  
The peak amplitude stability of the autocorrelation function of the received reference signal can be checked to make sure that the antenna moving direction is always parallel with the \(x\) axis (the wavefront of the satellite digital TV signal). For example, Figure 12 shows that when the antenna moving direction is parallel with the wavefront, the normalized autocorrelation peak amplitude (the first measurement is acted as the reference) is stable along the synthetic aperture with a mean value of 0.9972 and a standard deviation of 0.0078. When the moving direction is inclined, the normalized amplitude will be changed along the synthetic aperture, so requiring for a correction. According to the geometry shown on the right of Figure 4, Equation (21) can be modified to  
\[R_{2}^{p}(x_{0},y_{0})=\sqrt{(x_{0}-x_{p})^{2}+y_{0}^{2}} \tag{22}\]  
where \(x_{p}=-L/2+(p-1)\Delta x\) is the \(p\)-th antenna position, which is only dependent on the moving step.