# Passive Bistatic Ground-Based Synthetic Aperture Radar: Concept, System, and Experiment Results
## 5 Discussions  
### Artifact Suppression  
As shown in Figures 10 and 15, strong artifacts exist in the range direction, which are generated by the waveform structure of the used satellite digital TV signal. More specifically, the amplitude-filtered spectrum and the frequency gap between two adjacent TV channels, as shown in Figure 3, make the range compression nonoptimal. To reduce the influence of the amplitude-filtered spectrum,  
Figure 19: SAR image overlaid with the aerial view provided by Google maps. Note that the two strong targets at about 60 m and 65 m are located at a parking lot. Figure 18: SAR image of natural and man-made targets: The right subfigure is the zoomed version of the left subfigure with specific focuses on the fence, light pole, tree, and house. as mentioned previously, the flat spectrum can be extracted for range compression. For each TV channel, according to Equation (9), the extraction operation can be expressed as  
\[s_{W}(f)=W(f-f_{c})s(f)\simeq W(f-f_{c})A_{0}\exp[-j2\pi f(R_{1}+R_{2})] \tag{36}\]  
where \(W(f)\) is a window function in the frequency domain, given by  
\[W(f)= \begin{cases}1&\left|f\right|\leq f_{s}(1-\alpha)\\ 0&\text{otherwise}\end{cases}  \tag{37}\]  
With respect to the frequency gaps, based on the known spectrum in Equation (36) and some assumed signal models, the gapped signal components can be estimated, i.e., the frequency gaps can be filled. For instance, spatially variant apodization (SVA) based [30], compressive sensing based [49], and low-rank matrix completion based [37] methods have been used for frequency gap filling in different PBR applications. In the following, the Super-SVA-based method [37] is introduced. Super-SVA [50] is based on SVA that is an adaptive amplitude windowing method with the capability to effectively suppress the sidelobes of Fourier-transform without broadening the mainlobe [51]. According to this property, Super-SVA extrapolates the spectrum of a signal by iteratively using SVA and the inverse weighting operations. In our case, at each antenna position, the Super-SVA-based method can be applied to each TV channel to expand its bandwidth. Mathematically, at each antenna position, for the \(n\)-th (\(n=1\), \(2\),..., \(12\)) TV channel, according to Equation (36), the windowed frequency domain signal can be expressed as  
\[s_{W}^{n}(f)=W(f-f_{c}^{n})A_{0}\exp[-j2\pi f(R_{1}+R_{2})] \tag{38}\]  
With the SVA weighting function \(\mathbf{W}_{SVA}(\mathbf{f})\) in the frequency domain [37], the Fourier-transform of the \(n\)-th TV channel is given by  
\[\mathcal{X}_{SVA}^{n}=\mathbf{F}_{n}^{H}[s_{W}^{n}(\mathbf{f})\odot W_{SVA}(\mathbf{f})] \tag{39}\]  
where \(\mathbf{F}_{n}\) is the Fourier-transform matrix for the \(n\)-th TV channel. Then, by performing the inverse Fourier-transform, a band-unlimited signal can be obtained as  
\[s_{SVA}^{n}(\mathbf{f})=\mathbf{F}_{n}\mathcal{X}_{SVA}^{n} \tag{40}\]  
Defining an inverse weighting function in the frequency domain as \(\mathbf{W}_{inv}(\mathbf{f})\), which is the inverse Fourier-transform of the mainlobe of a _sinc_ function in the time domain, the spectrum extrapolated signal can be obtained as [37]  
\[s_{Super-SVA}^{n}(\mathbf{f})=s_{SVA}^{n}(\mathbf{f})/W_{inv}(\mathbf{f}) \tag{41}\]  
which has a wider bandwidth than the original signal in Equation (38). However, its bandwidth cannot be infinite because the inverse process should be truncated to avoid the problem of singularities. After inverse weighting and truncating, the center portion of the extrapolated signal in Equation (41) should be replaced with the original signal in Equation (38) to improve the accuracy, giving the first-time gap filling result. This procedure should be repeated for several times to fill the entire frequency gap between two TV channels. Since two adjacent TV channels have the same gapped frequencies, a simple average process can be conducted to make the reconstruction more accurate.