# Distributed Extended Kalman Filtering Based Techniques for 3-D UAV Jamming Localization
## 4 Proposed Distributed Extended Kalman Filter Based 3D Jammer's Localization Schemes
### Distributed Extended Kalman Filter (DEKF) for 3D Localization  
In this method, a distributed scenario is considered, where each node employs the standard EKF by using the information of the received power from the jammer at the nearby boundary node of the jamming region, as shown in Figure 3. This scheme is termed as Distributed Extended Kalman Filter (DEKF). In the DEKF approach, every node process the jamming power and estimate the jammer's location locally without the need to collaborate with another node. Thus, for the jammer's 3D localization task with the DEKF, the state vector will take the following form:  
\[\mathbf{x_{k}}=[x_{I},y_{I},z_{I},v_{x},v_{y},v_{z},a_{x},a_{y},a_{z}]^{T} \tag{18}\]  
The Jammer's motion can be described using the well-known Kinetic equation model given by:  
\[\mathbf{x_{k+1}}=\mathbf{x_{k}}+\mathbf{v}\Delta t+\frac{1}{2}\mathbf{a} \Delta t^{2} \tag{19}\]Thus, the Jacobian matrix \(\mathbf{A}_{k}\) can be expressed as:  
\[\mathbf{A_{k}}\stackrel{{\Delta}}{{=}}\frac{\partial\mathbf{f}}{ \partial\mathbf{x_{k}}}\Big{|}_{(\hat{x}_{j,k-1},\hat{y}_{j,k-1},\hat{z}_{j,k-1 })}= \begin{bmatrix}1&0&0&dt&0&0&\frac{dt^{2}}{2}&0&0\\ 0&1&0&0&dt&0&0&\frac{dt^{2}}{2}&0\\ 0&0&1&0&0&dt&0&\frac{dt^{2}}{2}\\ 0&0&0&1&0&0&dt&0\\ 0&0&0&0&1&0&0&dt\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\end{bmatrix}  \tag{20}\]  
Additionally, the covariance matrix of process noise \(\mathbf{Q_{k}}\) and the covariance matrix of measurement noise \(\mathbf{R_{k}}\) can be written, respectively, as:  
\[\mathbf{Q_{k}}= \begin{bmatrix}\frac{dt^{4}}{4}&0&0&\frac{dt^{3}}{3}&0&0&\frac {dt^{2}}{2}&0&0\\ 0&\frac{dt^{4}}{4}&0&0&\frac{dt^{3}}{3}&0&0&\frac{dt^{2}}{2}&0\\ 0&0&\frac{dt^{4}}{4}&0&0&\frac{dt^{3}}{3}&0&0&\frac{dt^{2}}{2}\\ \frac{dt^{3}}{3}&0&0&\frac{dt^{2}}{2}&0&0&dt&0\\ 0&\frac{dt^{3}}{3}&0&0&\frac{dt^{2}}{2}&0&0&dt\\ \frac{dt^{2}}{2}&0&0&dt&0&0&1&0&0\\ 0&\frac{dt^{2}}{2}&0&0&dt&0&0&1&0\\ 0&0&\frac{dt^{2}}{2}&0&0&dt&0&0&1\end{bmatrix}  \tag{21}\]  
\[\mathbf{R_{k}}=diag\left(\sigma_{v_{x}}^{2},\sigma_{v_{y}}^{2},\sigma_{v_{z}} ^{2},\sigma_{a_{x}}^{2},\sigma_{a_{y}}^{2},\sigma_{a_{z}}^{2},\sigma_{P_{|B}} ^{2}\right) \tag{22}\]  
Now, assume that every sensor node provides the measurements of jammer's velocity components \((v_{x},v_{y},v_{z})\), acceleration \((a_{x},a_{y},a_{z})\) components, and the power received, which are captured by boundary node denoted as \(P_{JB}\). Hence, the measurement vector can be set up as:  
\[\mathbf{z_{k}}=[v_{x},v_{y},v_{z},a_{x},a_{y},a_{z},P_{JB}]^{T} \tag{23}\]  
Thus, in this scenario, both the observation function \(\mathbf{h}\) and the Jacobian matrix \(\mathbf{H}_{k}\) can be described, respectively, as:  
\[\mathbf{h}=[v_{x},v_{y},v_{z},a_{x},a_{y},a_{z},P_{JB}]^{T} \tag{24}\]  
and  
\[\mathbf{H_{k}}\stackrel{{\Delta}}{{=}}\frac{\partial\mathbf{h}}{ \partial\mathbf{x_{k}}}\Big{|}_{(\hat{x}_{j,k-1},\hat{y}_{j,k-1},\hat{z}_{j,k-1 })}=\begin{bmatrix}0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0  
(\(d_{IB}\)) given in Equation (1). The distance of jammer from the boundary node can be evaluated while using the following expression:  
\[d_{IB}=\sqrt{(x_{B}-x_{I})^{2}+(y_{B}-y_{J})^{2}+(z_{B}-z_{I})^{2}} \tag{26}\]  
where \((x_{B},y_{B},z_{B})\) is the boundary node.