## 2 Passive FSR Geometry  
The forward scattering phenomenon [16; 17] is schematically depicted in Figure 1.  
Figure 1: A typical passive forward scattering radar (FSR) geometry. T\({}_{\mathrm{x}}\)—transmitter, R\({}_{\mathrm{x}}\)—receiver, TGT—target, \(L\)—baseline, \(R_{1}\)—range from the transmitter to the target, \(R_{2}\)—range from the receiver to the target, \(D\)—range from the receiver to the crossing point, \(\beta\)—bistatic angle, \(\alpha\)—angle between the bistatic bisector and the velocity vector. In bistatic radars using an electromagnetic wave of a length \(\lambda\), the Doppler shift \(f_{d}\) produced by the target moving at the velocity \(V\) can be expressed as  
\[f_{d}=\frac{2V}{\lambda}\cos\left(\alpha\right)\cos\left(\beta/2\right), \tag{1}\]  
where \(\beta\) is the bistatic angle and \(\alpha\) is the angle between the bistatic bisector and the movement vector (see Figure 1). For such a spatial configuration where the target crosses the baseline (when the range information is lost and the Doppler frequency tends to 0 Hz) \(\beta\approx 180^{\circ}\), thus \(f_{d}\approx 0\) Hz, however, the target can be observed in a range \(\beta\in\left(180^{\circ}-\Delta,180^{\circ}+\Delta\right)\), where \(\Delta\) is a small angle, which delivers more information about the target trajectory. In the vicinity of \(f_{d}\approx 0\) Hz the signal impinging the receiving antenna can be defined as [20]  
\[\phi(t)=\frac{2\pi}{\lambda}\left[R_{1}(t)+R_{2}(t)-L\right]. \tag{2}\]  
Assuming that the target moves along a linear trajectory at a constant velocity \(v\) and the velocity vector (composed of \(v_{x}\) and \(v_{y}\) components corresponding respectively to the x- and y-axis) creates an angle with respect to the \(x\)-axis that is normal to the baseline \(\alpha=\tan^{-1}\left(\frac{v_{y}}{v_{x}}\right)\). If the target crosses the baseline at the point \(D\) from the receiver (see Figure 1) then \(x(t)=v_{x}t\) and \(y(t)=D+v_{y}t\) which leads to  
\[R_{1}(t)=\sqrt{x(t)^{2}+\left(L-y(t)\right)^{2}} \tag{3}\]  
and  
\[R_{2}(t)=\sqrt{x(t)^{2}+y(t)^{2}}. \tag{4}\]  
Apart from the Doppler history, the target forward radar cross section (FRCS) has a contribution to the signal reaching the receiver in the passive FSR system. Namely, a rectangular target of horizontal \(l_{h}\) and vertical \(l_{v}\) dimension such that \(l_{h}>>\lambda\) and \(l_{v}>>\lambda\), is given by [21]  
\[\sigma(t)=\frac{L^{2}}{R_{1}(t)R_{2}(t)}\frac{\cos\left(\theta_{T}(t)-\alpha \right)+\cos\left(\theta_{R}(t)+\alpha\right)}{2}\Pi\left(\frac{l_{h}}{\lambda }\left(\sin\left(\theta_{T}(t)-\alpha\right)+\sin\left(\theta_{R}(t)+\alpha \right)\right)\right) \tag{5}\]  
where the target aspect angle with respect to transmitter is \(\theta_{T}=\tan^{-1}\left(\frac{x(t)}{L-y(t)}\right)\) and with respect to receiver is \(\theta_{R}=\tan^{-1}\left(\frac{x(t)}{y(t)}\right)\) and \(\Pi\) is the function such that \(\Pi(x)=\frac{\sin(x)}{x}\). Then, the signal impinging the FSR receiving antenna can be written as  
\[s(t)=-\sigma(t)\sin\left(\phi(t)\right). \tag{6}\]  
Approximating Equation (6) by the third order Taylor polynomial around the crossing point \(t=t_{0}\), Equation (2) becomes  
\[\phi(t)\approx\pi\epsilon(t-t_{0})^{2}+\pi\zeta(t-t_{0})^{3}, \tag{7}\]  
where  
\[\epsilon=\frac{v_{x}^{2}}{\lambda}\left[\frac{1}{L-D}+\frac{1}{D}\right], \tag{8}\]  
and  
Figure 1:0  
and the latter equation goes to 0 for \(v_{y}=0\) or \(D=L/2\). Finally, the signal phase can be approximated as follows  
Figure 1:1where \(v_{x}\) denotes the velocity component which is perpendicular to the baseline \(L\), \(D\) expresses the range from the receiver to the crossing point, and \(t_{0}\) is the particular moment the target crosses the baseline. In fact, Equation (10) can be considered as a quadratic phase function resulting in the linear frequency-modulated signal. The modulation factor (also known as a chirp rate, frequency rate, frequency slope, etc.)