# A Method of Detections' Fusion for GNSS Anti-Spoofing
## 2 The Proposed Method of Spoofing Detections' Fusion
### Valuation of the Belief Function for GNSS Spoofing Detections  
Belief functions are the key to the detections' fusion. They represent the reliability of the corresponding measurement or features of the signal. The belief function is calculated based on the basic probability assignment (BPA) in DST. As previously mentioned, BPA is equivalent to the belief function in the case of binary detections. However, there is no general way to get BPA or the belief function for a detection result. This section proposes a method to valuate the belief function and provides a brief discussion about two spoofing detections. Most of the spoofing detections are equivalent to hypothesis testings. They can be described as:  
\[\left\{ \begin{array}{ll}H_{0}:&T(x)\leq\gamma\\ H_{1}:&T(x)>\gamma\end{array} \right. \tag{6}\]  
where \(H_{0}\) and \(H_{1}\) are hypotheses that represent the absence and presence of a spoofing attack, and they are equivalent to \(A\) and \(C\) of Equation (1), respectively. \(T(x)\) is the test statistic of \(x\), which is a measurement of the signal. \(\gamma\) is the threshold of detection. Here, \(x\) could be any signal features and measurements, such as the absolute power or carrier-to-noise ratio (\(C/N_{0}\)) of the received signal, Doppler shift, outputs of the correlator, pseudorange measurement, etc. The detection of Equation (6) makes a hard decision to decide whether the signal is authentic or not. It is easy to implement, but it loses some information and increases the error decision. \(T(x)\) is a quantization of the anomaly of \(x\), and it also can be considered as a soft decision of the authenticity of signal. The belief function for the detection results of detector \(D_{i}\) can be defined as:  
\[\left\{ \begin{array}{ll}m_{i}(A)=f(T_{i},\gamma_{i})\\ m_{i}(C)=1-f(T_{i},\gamma_{i})\end{array} \right. \tag{7}\]  
where \(m_{i}(\cdot)\) is the belief function, \(T_{i}\) and \(\gamma_{i}\) represent the test statistic and threshold of \(D_{i}\), respectively, and \(f\) is the function that maps \(T_{i}\) and \(\gamma_{i}\) to a real number. The mapping is subject to:  
\[\left\{ \begin{array}{ll}f(T_{i},\gamma_{i})\in\left[\begin{array}{ll}0,&1 \end{array}\right]\\ m_{i}(A)>m_{i}(C)\;\;\mbox{when}\;\;T_{i}<\gamma_{i}\\ m_{i}(A)<m_{i}(C)\;\;\mbox{when}\;\;T_{i}>\gamma_{i}\\ m_{i}(A)=m_{i}(C)\;\;\mbox{when}\;\;T_{i}=\gamma_{i}\end{array} \right. \tag{8}\]  
This is in keeping with the decision rule of Equation (6). However, unlike Equation (6), \(m_{i}(\cdot)\) of Equation (7) provides a soft decision of the authenticity of the signal. Compared with the hard decision, it retains more information of \(x\). More than that, Equation (7) makes it possible to fuse the information from multiple detections to reduce the probability of error decisions. There are many forms of \(f\) to meet Equation (8). The \(p\)-value of hypothesis testing would have more proper to reflect the reliability of the results than the test statistic and threshold. However, the calculation of the tail probability for each obtained statistics in the method of the \(p\)-value is too complicated. Herein, two definitions of \(f\) are proposed for simplicity in the application. In what follows, \(T_{i}\) and \(\gamma_{i}\) are assumed as non-negative, and this is possible in a proper way. One definition of \(f\) can be given as:  
\[f_{1}(T_{i},\gamma_{i})=\left(\frac{1}{2}\right)^{T_{i}/\gamma_{i}} \tag{9}\]\(f_{1}(T_{i},\gamma_{i})\) is a decreasing function of \(T_{i}/\gamma_{i}\), and it is easy to implement. However, \(f_{1}(T_{i},\gamma_{i})\) descends slowly even if the anomaly is obvious enough. Besides, \(f_{1}(T_{i},\gamma_{i})\) is in a half closed interval \([\,0,\,1\,]\) because the base number of the power function is less than one. Another definition of \(f\) is given as:  
\[f_{2}(T_{i},\gamma_{i})=\left\{ \begin{array}{ll}1-\dfrac{T_{i}}{2\gamma_{i}} &T_{i}<2\gamma_{i}\\ 0&T_{i}\geq 2\gamma_{i}\end{array} \right.