# A Method of Detections' Fusion for GNSS Anti-Spoofing
## 2 The Proposed Method of Spoofing Detections' Fusion
### The Model of Detections' Fusion  
Most of the spoofing detections focus on the anomalies of signal features or measurements. Generally, each detector aims at only one feature, and the result is evidence of the presence of the spoofing attack. However, this evidence would be inadequate and not absolute reliable in some cases. The combination of the results from multiple detectors is an approach to deal with this problem. A method based on DST to fuse multiple detectors is proposed in the following. According to DST, a complete frame of discernment in spoofing detection should contain all possible mutually exclusive declared propositions, such as the noise, authentic signal, counterfeit signal and the mixture of them. They indicate the identity of the signal. However, this complete frame of discernment sets up obstacles for implementation. The main problem is the evaluation of BPA. Thus, it is necessary to simplify this frame for application. In what follows, the noise, authentic and counterfeit signal are denoted as \"\(N\)\", \"\(A\)\" and \"\(C\)\", respectively, and the frame of discernment is denoted as \(\Theta\). In practice, noise is universal because the environment temperature cannot be absolute zero. The acquisition module of the GNSS receiver would determine the absence of satellite signals if there is only noise. Thus, \(N\) is redundant in \(\Theta\). In what follows, the object is limited to the satellite signal that has been processed by the receiver, regardless of its authenticity. In a broader sense, \(C\) does not only represent the spoofing signal, it also means the abnormal change of the satellite signal. Besides, \(A\) and \(C\) are exclusive for a signal. If authentic and counterfeit signals are mixed together in a processing channel and cannot be separated by the GNSS receiver, they should be marked as \"abnormal\", and the identified result should be \(C\). Thus, it is unnecessary to reserve a position for the mixture \"\(AC\)\" in \(\Theta\). Therefore, \(\Theta\) could be simplified as follows:  
\[\Theta=\{A,C\} \tag{1}\]  
That is, there are only two possible results of spoofing detector, \"authentic\" or \"counterfeit\"; the more generalized description is \"no spoofing\" or \"spoofing\". This simplification of \(\Theta\) facilitates the valuation of BPA and the belief functions, it also simplifies the combination rule. According to DST, for a possible result of one detector, the BPA is equivalent to the belief function because there are only two elements in \(\Theta\). Herein, the belief function is denoted as \(m_{i}(\cdot)\), which corresponds to detector \(D_{i}\). \(m_{i}(\cdot)\) follows:  
\[\left\{ \begin{array}{c}m_{i}(\phi)=0\\ m_{i}(A)+m_{i}(C)=1\end{array} \right. \tag{2}\]  
where \(\phi\) represents an empty set. \(m_{i}(\cdot)\) implies the believable degree of the evidence from detector \(D_{i}\). For example, \(D_{i}\) tends to determine the signal as \"\(A\)\" if \(m_{i}(A)>m_{i}(C)\); otherwise, it tends to determine the signal as \"\(C\)\". The conflict degree of different evidence can be defined as follows:  
\[\kappa=1-\Big{(}\prod_{i=1}^{N}m_{i}(A)+\prod_{i=1}^{N}m_{i}(C)\Big{)} \tag{3}\]  
The combination result is reliable if all evidence is coincident, while it is unreliable when the conflict degree is high. \(\kappa=1\) means that some evidence is completely opposite, such as \(m_{1}(A)=1\) and \(m_{2}(A)=0\). Due to the inference of the combination rule of DST [13] in the case of binary detection, the combined \(m(\cdot)\) are given as follows:\[\left\{ \begin{array}{l}m(A)=\dfrac{\prod\limits_{i=1}^{N}m_{i}(A)}{ \prod\limits_{i=1}^{N}m_{i}(A)+\prod\limits_{i=1}^{N}m_{i}(C)}\\ m(C)=\dfrac{\prod\limits_{i=1}^{N}m_{i}(C)}{\prod\limits_{i=1}^{N}m_{i}(A)+ \prod\limits_{i=1}^{N}m_{i}(C)}\end{array} \right. \tag{4}\]  
where \(N\) is the number of spoofing detectors. In the case that \(\kappa=1\), there is \(\prod m_{i}(A)+\prod m_{i}(C)=0\); \(m(\cdot)\) cannot be obtained according to Equation (4).