# Passive Bistatic Ground-Based Synthetic Aperture Radar: Concept, System, and Experiment Results
## 3 Signal Processing
### Signal Model  
For the \(n\)-th channel with \(n=1\), 2,..., \(N\) (\(N=12\)), the satellite digital TV signal can be expressed as  
\[s_{0}^{n}(t)=\sum_{i=-\infty}^{\infty}A_{n}\exp(j\varphi_{i}^{n})h(t-iT_{s}) \exp(j2\pi f_{c}^{n}t) \tag{8}\]  
where \(t\) is time; \(A_{n}\) is the amplitude; \(h(t)\) is the impulse response of the root-raised-cosine (RRC) filter [47] with a duration of \(T_{s}\); \(f_{c}^{n}\) is the carrier frequency; and \(\varphi_{i}^{n}\) is the phase of the \(i\)-th symbol. For QPSK modulation, \(\varphi_{i}^{n}\) equals to \(\pi\)/4, 3\(\pi\)/4, 5\(\pi\)/4, or 7\(\pi\)/4. The frequency transfer function of the RRC filter can be expressed as [47]  
\[H(f)= \begin{cases}1&\big{|}f\big{|}\leq f_{s}(1-\alpha)\\ \sqrt{\frac{1}{2}+\frac{1}{2}\sin\frac{\pi}{2f_{s}}\Big{(}\frac{f_{s}-\big{|}f \big{)}}{\alpha}}&f_{s}(1-\alpha)\leq\big{|}f\big{|}\leq f_{s}(1+\alpha)\\ 0&\big{|}f\big{|}\geq f_{s}(1+\alpha)\end{cases}  \tag{9}\]  
where \(f_{s}=0.5\)/\(T_{s}\) is the Nyquist frequency and \(\alpha=0.35\) is the roll-off factor. Figure 11: (**Left**) phase and (**right**) amplitude difference of the cross-correlation peaks for 15 min. Based on Equations (8) and (9), the spectrum of the \(n\)-th TV channel with an integration time of \(T_{\text{int}}\) can be expressed as  
\[s_{0}^{n}(f)=\left(H(f-f_{c}^{n})\sum_{i=-\infty}^{\infty}A_{n}e^{ib^{n}_{i}t}e^{ -j2\pi(f-f_{c}^{n})IT_{s}}\right)\ast\left(T_{\text{int}}e^{-j\pi T_{\text{int}} (f-f_{c}^{n})}\sin c[T_{\text{int}}(f-f_{c}^{n})]\right) \tag{10}\]  
where * denotes the convolution process. It can be learned from Equation (10) that the spectrum of each TV channel is determined by the spectrum of RRC filter and a _sinc_ function. When a long integration time is used, the main lobe of the _sinc_ function is narrow and the spectrum of each TV channel is dominated by the spectrum of RRC filter. In other words, the spectrum is amplitude-filtered by Equation (9) in such a case. According to Figure 4, and with the amplitudes \(A_{n\!ef}\) and \(A_{s\!urv}\) that are independent of the frequency, the received reference signal and surveillance signal from all the TV channels can be expressed as  
\[s_{ref}(t)=\sum_{n=1}^{N}A_{n\!ef}s_{0}^{n}[t-R_{1,1}/c] \tag{11}\]  
and  
\[s_{s\!urv}(t)=\sum_{n=1}^{N}A_{s\!urv}s_{0}^{n}[t-\left(R_{1,2}+R_{2}\right)/c] \tag{12}\]  
Range compression can be conducted by the cross-correlation technique, which will be more easily computed in the frequency domain by fast Fourier-transform (FFT). The frequency-domain reference signal and surveillance signal are given by  
\[s_{ref}(f)=\sum_{n=1}^{N}A_{n\!ef}s_{0}^{n}(f)\exp(-j2\pi fR_{1,1}/c)=A_{n\!ef }s_{0}(f)\exp(-jkR_{1,1}) \tag{13}\]  
and  
\[s_{s\!urv}(f)=\sum_{n=1}^{N}A_{s\!urv}s_{0}^{n}(f)\exp[-j2\pi f(R_{1,2}+R_{2})/ c]=A_{s\!urv}s_{0}(f)\exp[-jk(R_{1,2}+R_{2})] \tag{14}\]  
where \(k=2\)\(\pi f/c\) denotes the wavenumber and \(s_{0}(f)\) is the spectrum of all TV channels. Then, the bistatic time delay of the target can be calculated by the inverse Fourier-transform of the following frequency domain signal. \[s(f)=s_{s\!urv}(f)s_{ref}^{*}(f)=A_{0}\Big{|}s_{0}(f)\Big{|}_{2}\exp[-jk(R_{1} +R_{2})] \tag{15}\]  
where \((\cdot)^{*}\) denotes complex conjugate, \(R_{1}=R_{1,2}-R_{1,1}\), and \(A_{0}=A_{s\!urv}A_{ref}^{*}\). For PBR, the inverse filtering (i.e., reciprocal filtering)-based method [29; 32] can be used to avoid the influence of the IO waveform, i.e., the influence of \(s_{0}(f)\) in Equation (15), resulting in  
\[s_{s\!urv}(f)=\frac{s_{s\!urv}(f)}{s_{n\!ef}(f)}=\frac{A_{s\!urv}}{A_{n\!ef}} \exp[-jk(R_{1}+R_{2})] \tag{16}\]  
Since the inverse filtering needs a pure reference signal and its mismatched filtering property will reduce the output SNR, Equation (16) can only be used in the conditions where a template reference signal can be obtained and the SNRs of targets are high enough.