# Improved 2D Ground Target Tracking in GPS-Based Passive Radar Scenarios
## 4 Target Tracking  
PR measurements are applied directly to a Cartesian tracking stage based on the Kalman filter [22], to obtain the target location. The application of the Kalman filter to target tracking is common in PR [23; 24; 25]. The core of the tracker is a recurrent prediction-correction process that is usually performed by stochastic filters. These filters use a recurrent prediction/correction process to estimate the state of the target taking into consideration a set of noisy measurements and a dynamic model that represents the target movement.  
The performance of the Kalman filter is described with Expressions (18) and (19), which depend on the following elements: \(\mathbf{x_{k}}\) is the state vector, \(\mathbf{\hat{x_{k}}}\) and \(\mathbf{\hat{x_{k}}}\) are the prior and posterior estimations of the state vector, \(\mathbf{P_{k}}^{-}\) and \(\mathbf{P_{k}}\) are the prior and posterior estimations of the state covariance matrix, \(\mathbf{F}\) and \(\mathbf{H}\) are the transition and measurement matrices, \(\mathbf{Q}\) and \(\mathbf{R}\) are the process and measurement noises covariance matrices, \(\mathbf{K_{k}}\) is the Kalman gain which corrects the vector prior estimation with the process innovation vector \(\mathbf{y_{k}}=\mathbf{z_{k}}-\mathbf{H}\mathbf{\hat{x_{k}}}^{-}\), where \(\mathbf{z_{k}}\) is the noisy measurement. Expression (18) corresponds to the prediction phase, where an estimate of the current state variables is obtained:  
\[\mathbf{\hat{x_{k}}}^{-} =\mathbf{F}\mathbf{\hat{x_{k-1}}} \tag{18}\] \[\mathbf{P}_{k}^{-} =\mathbf{F}\mathbf{P}_{k-1}\mathbf{F}^{T}+\mathbf{Q}\]  
These estimates are updated using a weighted average of the next measurements, once observed.  
\[\mathbf{K_{k}} =\mathbf{P}_{k}^{-}\mathbf{H}^{T}(\mathbf{H}\mathbf{P}_{k}^{-} \mathbf{H}^{T}+\mathbf{R})^{-1} \tag{19}\] \[\mathbf{\hat{x_{k}}} =\mathbf{\hat{x_{k}}}^{-}+\mathbf{K_{k}}\mathbf{y_{k}}\] \[\mathbf{P_{k}} =(1-\mathbf{K_{k}}\mathbf{H})\mathbf{P}_{k}^{-}\]  
In order to use a Kalman filter for tracking moving targets, a dynamic model of the target motion must be designed. Since the intended targets are ground vehicles, motion in a plane with X and Y coordinates is considered, and a state vector is defined whose elements are the position, velocity, and acceleration of the target in both dimensions (\(\mathbf{x^{T}}=[x~{}v_{x}~{}a_{x}~{}y~{}v_{y}~{}a_{y}]\), where \(T\) is the transpose operation). A quasi-constant acceleration model in both x and y dimensions is selected to describe the expected dynamics of the targets, the acceleration derivative being modelled as a white noise process with power spectral density \(q\)[26]. Therefore, the state function and state noise covariance matrices can be defined as \(\mathbf{F}=diag(\mathbf{F_{1}})\) and \(\mathbf{Q}=diag(\mathbf{Q_{1}})\), where \(\mathbf{F_{1}}\) and \(\mathbf{Q1}\) are the matrixes presented in (20).  
\[\mathbf{F_{1}}=\begin{pmatrix}1&T&T^{2}\\ 0&1&T\\ 0&0&1\end{pmatrix},~{}~{}\mathbf{Q_{1}}=q\begin{pmatrix}T^{5}/20&T^{4}/8&T^{3 }/6\\ T^{4}/8&T^{3}/3&T^{2}/2\\ T^{3}/6&T^{2}/2&T\end{pmatrix} \tag{20}\]  
Due to the linearity of the selected state function, the filter estimation stage is implemented with the standard Kalman filter equations (18). On the contrary, the geometrical considerations of the GPS-based PR presented in Section 2.1 give rise to non-linear relationships between the measurement vector \(\mathbf{z_{k}}=[m,p,\phi]^{T}\) and the local Cartesian plane where the state vector \(\mathbf{x_{k}}\) is defined. For this reason, the suboptimal extended Kalman filter approach is selected to produce the filter correction stage expressed in (21), tackling the non-linearities of the measurement function \(h\) by a local linearisation [27].  
\[\mathbf{K_{k}} =\mathbf{P}_{k}^{-}\mathbf{H}^{T}(\mathbf{H}\mathbf{P}_{k}^{-} \mathbf{H}^{T}+\mathbf{V}_{k}\mathbf{R}\mathbf{V}_{k}^{T})^{-1}\] (21) \[\mathbf{x_{k}} =\mathbf{x_{k}^{-}}+\mathbf{K_{k}}(\mathbf{z_{k}}-h(\mathbf{ \hat{x_{k}}}^{-},0)\] \[\mathbf{P_{k}} =(1-\mathbf{K_{k}}\mathbf