# Distributed Extended Kalman Filtering Based Techniques for 3-D UAV Jamming Localization
## 3 Concept of Distance Ratio (\(\beta\))  
In this section, we described the concept of distance ratio which was introduced in [29]. Moreover, we extended the distance ratio concept to three-dimensional space \((x,y,z)\). This concept will be later utilized in order to develop the EKF algorithm in Section 4. Note that, Table 1 defines the main parameters. The distance ratio concept is based on signal to noise ratio at the edge node \(N_{E}\), jamming power received by boundary node \(B\) and Received Signal Strength (RSS) from \(B\), as shown in Figure 4. Here, \(d_{JB}\) and \(d_{JE}\) are the distances from the jammer to the boundary node and from the jammer to the edge node, respectively. Thus, we can express \(d_{JB}\) and \(d_{JE}\), respectively, as:  
\[d_{JB}=\sqrt{(x_{B}-x_{J})^{2}+(y_{B}-y_{J})^{2}+(z_{B}-z_{J})^{2}} \tag{1}\]  
Figure 3: Nodes classification during jamming attack.  
Figure 2: Representation of the link quality and JRSS while jammer is moving. The solid arrow describes the jamming signal and the dotted red line is the JRSS received by the base station. The link between the nodes depicted by the dot green line and the power received by the boundary node from its neighbor represented as a solid green line.  
(1)  
The SNR decreases when the jammer moves towards the target node until it drops to below an acceptable threshold value. This results in a jammed node and communication channel being completely blocked. Therefore, the node has SNR approximately equal to the system threshold value (\(SNR\approx\gamma\)) located on the jamming edge. We use this observation in order to estimate the unknown distances between the jammer and edge node. The jamming power received by the boundary node is following the Log-distance shadowing model where it an extension of the Friis equation and it is proportional inverse to the distance, as follows:  
\[P_{JB}=P_{t}+k-10n\log_{10}d_{JB}+X_{\sigma} \tag{3}\]  
where \(P_{JB}\) is the jamming power received at distance \(d\), and \(P_{t}\) is the jammer's transmission power, the path loss exponent \(n\), depending on the environment and it is variant from physical environment to other. In this paper we assumed it equal to 2 for free space environment, or Line of Sight (LoS). The Gaussian noise with zero mean denoted by \(X_{\sigma}\). \(k\) is constant and depend on the antenna characteristics. \(d_{JB}\) is the distance between jammer and the boundary node.  
Similarly, the jamming power received by the edge node (represented by \(P_{JE}\)) can be expressed as:  
\[P_{JE}=P_{t}+k-10n\log_{10}d_{JE}+X_{\sigma} \tag{4}\]  
Next, we define the distance ratio (\(\beta\)) as the ratio of \(d_{JE}\) to \(d_{JB}\), that is:  
\[\beta=\frac{d_{JE}}{d_{JB}} \tag{5}\]  
Now, using the geometry that is shown in Figure 4, we conclude that the distances \(d_{JE}\), \(d_{JB}\), and \(d_{EB}\) are related as:  
\[d_{JB}=d_{JE}+d_{EB} \tag{6}\]  
Thus, we can also relate \(d_{EB}\) and \(d_{JB}\) while using the following:  
\[(1-\beta)=\frac{d_{EB}}{d_{JB}} \tag{7}\]  
As a result, we can relate the power terms \(P_{JB}\) and \(P_{JB}\) with the aid of distance ratio to obtain:  
\[P_{JE}=P_{JB}-10n\log_{10}(\beta) \tag{8}\]  
Figure 4: Illustration of distance ratio. The two base stations positioned outside the jamming range wherever the green circle depicts the edge node and placed at the jamming region.  
Similarly, it can be shown that \(P_{EB}\) and \(P_{JB}\) can be related as:  
\[P_{EB}=P_{JB}-10n\log_{10}\left(1-\beta\right) \tag{9}\]  
Later, in Section 3, we will explain how the above equations can help us in the development of a distributed Extended Kalman Filter (EKF) algorithm for the jammer's 3D location estimation.