# Clutter Cancellation and Long Time Integration for GNSS-Based Passive Bistatic Radar
## 2 GNSS-Based Bistatic Radar System Geometry Signal Model  
The radar system in this paper consists of a satellite transmitter and stational receiver antenna on the ground. The receiver is equipped by two channels. One (referred to as reference channel) collects the direct path signal that is transmitted by illuminators as the reference signal to accomplish the cross-correction, and the other (surveillance channel) captures the target echoes from the area of interest. Figure 1 illustrates the geometric configuration of GNSS_based bistatic radar. O-XYZ defines a Cartesian coordinate system, where the radar receiver is located at the origin O. The transmitter is one GNSS satellite that is located at \(P_{TX}(x_{s};y_{s};z_{s})\) and moves at a velocity \(V_{s}\big{(}v_{sx^{\prime}};v_{sy^{\prime}};v_{sz}\big{)}\). The target is an airplane with a velocity and acceleration \(v_{t}\) and \(a_{t}\), respectively. Assuming that an air target within the receiver antenna for the whole aperture time \(T_{a}\). The instantaneous position of the air target can be expressed as  
\[P_{t}(t_{m})=P_{t0}+v_{t}t_{m}+\frac{1}{2}a_{t}t_{m}^{2}=[x_{0},y_{0},z_{0}]+ \big{[}v_{tx},v_{ty},v_{tz}\big{]}\cdot t_{m}+\frac{1}{2}\big{[}a_{tx},a_{ty},a _{tz}\big{]}\cdot t_{m}^{2} \tag{1}\]  
where \(t_{m}\in\left[-(\frac{T_{a}}{2}),(\frac{T_{a}}{2})\right]\) is the slow-time. Let \(R_{D}(t_{m})\), \(R_{T}(t_{m})\), and \(R_{R}(t_{m})\) denote, respectively, the baseline between the satellite and the receiver, the instantaneous range from satellite to target, and target to receiver at time \(t_{m}\) during the coherent processing interval (CPI). \(R(t_{m})\) is the instantaneous bistatic range, which is given by the range difference between the target echo and direct signal  
\[R(t_{m})=R_{T}(t_{m})+R_{R}(t_{m})-R_{D}(t_{m}) \tag{2}\]  
Additionally, the Doppler frequency can be expressed as  
\[f_{d}(t_{m})=\frac{1}{\lambda}\cdot\frac{dR(t_{m})}{dt_{m}} \tag{3}\]  
where \(\lambda\) is the signal wavelength. Because GNSS signals are continuous in time, they do have a period of \(T_{c}=1\) ms. The captured data can be formatted in a two-dimensional scheme by defining the equivalent pulse in signal processing. The fast time dimension is defined as the sample of each period of the ranging code, and the slow time dimension is the equivalent pulses for coherent processing, as shown in Figure 2.  
Figure 1: Geometric configuration of Global Navigation Satellite Systems_based (GNSS_based) bistatic radar. The received echo after the cross-correction can be modelled, as follows [18]  
\[s(r,t_{m})=AQ[r-R(t_{m})]\exp(-j\frac{2\pi R(t_{m})}{\lambda}) \tag{4}\]  
where \(A\) is the backward scattering coefficient of the target, \(Q(\cdot)\) is the cross-correction function between target echo and reference signal, and \(c\) is the speed of light. The Taylor series of the bistatic range \(R(t_{m})\) around the reference time is  
\[R(t_{m})\cong R_{0}+\lambda f_{dc}t_{m}+\frac{1}{2}\lambda f_{dmr}t_{m}^{2} \tag{5}\]  
where \(R_{0}\) is the initial range at the reference time and \(f_{dc}\) and \(f_{dmr}\) are the Doppler centroid (DR) and Doppler modulated rate (DMR) of the target. Generally, a second-order expansion is sufficient for describing the Doppler variation of the air target. Hence, the received signal in (4) can be rewritten as  
\[ \begin{array}{ll}s(r,t_{m})&=AQ\Big{[}r-(R_{0}+\lambda f_{dc}t_{m}+\frac{1} {2}\lambda f_{dmr}t_{m}^{2})\Big{]}\\ &\cdot\exp(-j\frac{2\pi}{\lambda}(R_{0}+\lambda f_{dc}t_{m}+\frac{1}{2} \lambda f_{dmr}t_{m}^{2}))\end{array}  \tag{6}\]  
It is clear that the signal power of the GNSS_based bistatic radar is extremely low, which limits the target detection distance of the radar system. Long time integration is needed in order to improve the detection performance. However, it can be seen from (6) that both the range and Doppler are time-varying during the aperture time. Proper strategies are needed to eliminate the range and Doppler migrations.