# Improved 2D Ground Target Tracking in GPS-Based Passive Radar Scenarios
## 3 Ground Target 2D Localization Scheme
### Target Localization  
The measurement vector provides information of the target's bistatic range (\(m\)), and DoA (\(\phi\)), as input parameters to estimate the target's position in local Cartesian space. In addition, the target motion is expected to be parallel to the local Cartesian plane at a known altitude from the PR location, \(alt\).  
The measured bistatic range (\(m\)), defines a three-dimensional ellipsoid of revolution with the GPS satellite and the PR located at the foci. The equation of this ellipsoid is (13) in an auxiliary coordinate system, \(\{x_{aux},y_{aux},z_{aux}\}\), which is obtained from the local Cartesian space by a first rotation on the z-axis with an angle \(\phi_{rot}=\phi_{sat}+\pi\), followed by a second rotation on the y-axis according to the satellite's inclination, \(\theta_{sat}\). These two rotations place the position of the transmitter at \((-L,0,0)\) in the auxiliary coordinate system as shown in Figure 6.  
\[\frac{(x_{aux}+L/2)^{2}}{a^{2}}+\frac{y_{aux}^{2}}{b^{2}}+\frac{z_{aux}^{2}}{ c^{2}}=1 \tag{9}\]  
where \(a=(m+L)/2\) is the length of the major semi-axis and \(b=c=\sqrt{a^{2}-(L/2)^{2}}\) is the length of the minor semi-axis. The DoA of the target in the new coordinate system obtained after applying the first rotation is \(\phi^{\prime}=\phi-\phi_{rot}\), whereas the target altitude remains invariant. The target positions fulfil the following equations:  
\[\begin{split} x^{\prime}&=y^{\prime}/tan(\phi^{ \prime})\\ z^{\prime}&=alt\end{split} \tag{10}\]  
After applying the second rotation in the y-axis according to the inclination of the satellite, \(\theta_{sat}\), the target positions in the auxiliary coordinate system have to satisfy the following equations:  
\[\begin{split} x_{aux}&=x^{\prime}cos(\theta_{sat}) +z^{\prime}sin(\theta_{sat})=ey_{aux}+d\\ y_{aux}&=y^{\prime}\\ z_{aux}&=-x^{\prime}sin(\theta_{sat})+z^{\prime} cos(\theta_{sat})=fy_{aux}+g\end{split} \tag{11}\]  
with \(e=cos(\theta_{sat})/tan(\phi^{\prime})\), \(d=altsin(\theta_{sat})\), \(f=-sin(\theta_{sat})/tan(\phi^{\prime})\), and \(g=altcos(\theta_{sat})\). By substituting equations (11) in (9), and after several organisation operations, the second order equation shown in (12) is obtained.  
\[a^{\prime}y_{aux}^{2}+b^{\prime}y_{aux}+c^{\prime}=0 \tag{12}\]  
Figure 6: PR geometry: local Cartesian space (blue), auxiliary coordinate system (purple).  
where:  
\[\begin{split} a^{\prime}&=\frac{e^{2}}{a^{2}}+\frac{1}{b^{ 2}}+\frac{f^{2}}{c^{2}}\\ b^{\prime}&=2\bigg{(}\frac{(d+L/2)e}{a^{2}}+\frac{ fg}{c^{2}}\bigg{)}\\ c^{\prime}&=\frac{(d+L/2)^{2}}{a^{2}}+\frac{g^{2}}{c^{ 2}}-1\end{split} \tag{13}\]  
Solving (12) and substituting in (10), two possible target locations are obtained, after the first z-axis rotation. The ambiguous solution can be solved considering the target arrival angle \(\phi^{\prime}\). Finally, the target location in the local Cartesian space is obtained by inverting the first rotation of the z-axis with expressions in (14).  
\[\begin{split} x&=x^{\prime}cos(-\phi_{rot})-y^{ \prime}sin(-\phi_{rot})\\ y&=x^{\prime}sin(-\phi_{rot})+y^{\prime}cos(-\phi_ {rot})\\ z&=z^{\prime}=alt\end{split} \tag{14}\]