# Feasibility Study of Passive Bistatic Radar Based on Phased Array Radar Signals
## 4 Coherent Integration Method Based on Radon-NUFFT
### Signal Model  
Suppose the bistatic velocity of the target is \(v_{0}\), the initial distance between the target and transmitter is \(R_{t0}\), and the initial distance between the target and receiver is \(R_{r0}\); then, the bistatic range sum of the target can be expressed as  
\[R\left(t_{m},\hat{t}\right)=R_{t0}+R_{r0}-v_{0}\left(t_{m}+\hat{t}\right)=R_{0 }-v_{0}\left(t_{m}+\hat{t}\right) \tag{9}\]  
where \(R_{0}=R_{t0}+R_{r0}\) and \(R_{0}\) is the initial range sum.  
Then the time delay between the target echo and the transmitted signal is \(\Delta t=R\left(t_{m},\hat{t}\right)/c\). As the transmitted signal is noted as Equation (3), the echo signal from the target can be represented as  
\[s_{r}\left(t_{m},\hat{t}\right)=\sigma_{r}A\,\text{Rect}\left(\frac{\hat{t}- \Delta t}{\tau_{m}}\right)\exp\left[j\pi\frac{B}{\tau_{m}}\left(\hat{t}-\Delta t \right)^{2}\right]\exp\left[j2\pi f_{c}\left(t_{m}+\hat{t}-\Delta t\right)\right] \tag{10}\]  
where \(\sigma_{r}\) is the scattering coefficient of target and the signals to be analyzed have the same CF noted as \(f_{c}\) and BW noted as \(B\).  
The baseband signal of the target echo after down-conversion can be given as  
\[s_{b}\left(t_{m},\hat{t}\right)=\sigma_{r}A\,\text{Rect}\left(\frac{\hat{t}- \Delta t}{\tau_{m}}\right)\exp\left[j\pi\frac{B}{\tau_{m}}\left(\hat{t}-\Delta t \right)^{2}\right]\exp\left(-j2\pi f_{c}\Delta t\right) \tag{11}\]After the reconstruction of reference signal, pulse compression is performed and the result can be represented as  
\[\begin{split}& s_{PC}\left(t_{m},\hat{t}\right)=\sigma_{r}A\sqrt{B \tau_{m}}\,\text{sinc}\left[\pi\left(B+f_{d}\right)\left(\hat{t}-\frac{2\left( R_{0}-v_{0}t_{m}\right)}{c}+\frac{f_{d}}{\mu_{m}}\right)\right]\exp\left(-j\pi \frac{f_{d}^{2}}{\mu_{m}}\right)\\ &\cdot\exp\left[-j4\pi\frac{\left(f_{c}-f_{d}\right)\left(R_{0}- v_{0}t_{m}\right)}{c}\right]\exp\left[j\pi f_{d}\left(\hat{t}-\frac{2\left(R_{0}-v_{0}t_{ m}\right)}{c}+\frac{f_{d}}{\mu_{m}}\right)\right]\end{split} \tag{12}\]  
where \(f_{d}=2v_{0}f_{c}/c\) denotes the Doppler frequency and \(\mu_{m}=B/\tau_{m}\) is the chirp rate. Based on the assumption that the radar signal is narrowband signal and \(v_{0}\ll c\), the above formula can be simplified as follows:  
\[s_{PC}\left(t_{m},\hat{t}\right)=A_{m}\,\text{sinc}\left[\pi B\left(\hat{t}- \left(\frac{2\left(R_{0}-v_{0}t_{m}\right)}{c}-\frac{f_{d}}{\mu_{m}}\right) \right)\right]\exp\left(j2\pi f_{d}t_{m}\right) \tag{13}\]  
where \(A_{m}\) is the complex amplitude of the signal.  
It can be observed from Equation (13) that the amplitude of the signal is proportional to \(\sqrt{B\tau_{m}}\), since the signal pulse width is agile, the amplitudes of different pulses will be different. Also, the signal envelope varies with \(\left[2\left(R_{0}-v_{0}t_{m}\right)/c-f_{d}/\mu_{m}\right]\), which can be divided into two terms for analysis. The former term \(\left[2\left(R_{0}-v_{0}t_{m}\right)/c\right]\) is the RCM caused by target motion; the latter term \(f_{d}/\mu_{m}\) is caused by the range-Doppler coupling effect. Since the slow time is nonuniform, which will cause nonuniform RCM and since \(\mu_{m}\) is different for different pulses, it is necessary to compensate the RCM and range-Doppler coupling of different pulses accordingly. In addition, the phase of the signal with respect to the slow time is \(\exp\left(j2\pi f_{d}t_{m}\right)\), so the nonuniform slow time will lead to the nonuniform phase fluctuations among different pulses and traditional Fourier transform among slow time dimension cannot be applied.