# Distributed Extended Kalman Filtering Based Techniques for 3-D UAV Jamming Localization
## 4 Proposed Distributed Extended Kalman Filter Based 3D Jammer's Localization Schemes  
The Extended Kalman Filtering is an extension of Linear Kalman Filter (LKF), which is capable to take into account the nonlinearity of the system model. More specifically, the EKF employs the first-order linearization of the nonlinear system in a recursive fashion to find the estimates current mean and the covariance of the state vector [30]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(B\) & Boundary node \\ \(B_{i}\) & Total boundary nodes \\ \(N\) & Unaffected node \\ \(N_{E}\) & Edge node \\ \(J\) & Jammer UAV \\ \(N_{l}\) & Jammed node \\ \((x_{l},y_{l},z_{l})\) & Jammer UAV position \\ \((x_{B},y_{B},z_{B})\) & Boundary node position \\ \((x_{N},y_{N},z_{N})\) & Node location \\ \(p_{lt}\) & Jammer’s transmission power \\ \(p_{t}\) & Node’s transmission power \\ \(RSS\) & Jammer received signal strength \\ \(RSS\) & Received signal strength \\ \(P_{JB}^{i}\) & Jamming power captured by boundary node \(i=1,2,\ \ldots,\ n\) \\ \(P_{IE}^{i}\) & Jamming power estimated at edge node \(i=1,2,\ \ldots,\ n\) \\ \(P_{EB}^{i}\) & Jamming power estimated at edge by boundary node \(i=1,2,\ \ldots,\ n\) \\ \(L_{i}^{i}\) & Link quality for link \(i=1,2,\ \ldots,\ n\) \\ \(SNR\) & Signal to noise ratio \\ \(\gamma\) & System threshold value \\ \(\beta\) & Distance ratio \\ \(d_{IE}\) & Distance from jammer to the edge node \\ \(d_{EB}\) & Distance from edge node to the boundary node \\ \(d_{JB}\) & Distance from jammer to the boundary node \\ \(n\) & Environmental factor \\ \(\mathbf{a}\) & Acceleration \\ \(\mathbf{x_{k}}\) & Current state \\ \(\mathbf{x_{k-1}}\) & Previous state \\ \(\mathbf{w_{k}}\) & Process noise \\ \(\mathbf{z_{k}}\) & Measurement vector \\ \(\mathbf{h_{k}}\) & Nonlinear function \\ \(\mathbf{v_{k}}\) & Measurement noise \\ \(\mathbf{Q}\) & Covariance matrix for prediction \\ \(\mathbf{R}\) & Covariance matrix for observation \\ \(\mathbf{H_{k}}\) & Jacobian matrix \\ \(\mathbf{K_{k}}\) & Kalman gain \\ \(\mathbf{A}\) & Transition matrix \\ \(\mathbf{v}\) & Velocity \\ \(X_{\sigma}\) & Gaussian noise \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of symbols. Consider the following non-linear state transition model:  
\[\mathbf{x_{k}}=\mathbf{f}(\mathbf{x_{k-1}},\mathbf{u_{k}})+\mathbf{w_{k}} \tag{10}\]  
where \(\mathbf{x_{k}}\) represents the state vector, \(\mathbf{u_{k}}\) is the input vector, \(\mathbf{f}\) represents the non-linear state transition function, and \(\mathbf{w_{k}}\) is a zero mean Gaussian process noise with a covariance matrix \(\mathbf{Q}\), i.e., \(\mathbf{w_{k}}\sim\mathbf{N}(\mathbf{0},\mathbf{Q})\). Now, the measurement vector \(\mathbf{z_{k}}\) can be expressed as:  
\[\mathbf{z_{k}}=\mathbf{h}(\mathbf{x_{k}},\mathbf{u_{k}})+\mathbf{v_{k}} \tag{11}\]  
where \(\mathbf{v_{k}}\) is a zero mean Gaussian measurement noise with a covariance matrix \(\mathbf{R}\), i.e., \(\mathbf{v_{k}}\sim\mathbf{N}(\mathbf{0},\mathbf{R})\). Here, \(\mathbf{h}\) represents non-linear function for the measurement process. The EKF estimates the current position in two main steps, prediction and update state [31].