# A Method of Detections' Fusion for GNSS Anti-Spoofing
## 2 The Proposed Method of Spoofing Detections' Fusion
### Performance of Detections' Fusion  
A spoofing detection can be fused with others by the method as mentioned previously, as long as it could be depicted by a hypothesis testing as Equation (6). There are two types of errors in hypothesis testings. One is false alarm and the other is missed detection. The former is the incorrect rejection of the true hypothesis \(H_{0}\), while the latter is the failure to reject the false hypothesis \(H_{1}\). The probabilities of these two errors can be described as follows:  
\[\left\{ \begin{array}{ll}P_{fa}=P(H_{1}\big{|}H_{0})\\ P_{md}=P(H_{0}\big{|}H_{1})\end{array} \right. \tag{12}\]  
where \(P_{fa}\) is the false alarm probability and \(P_{md}\) is the missed detection probability. They are contradictory. For a given sample set, \(P_{md}\) increases with the reduction of \(P_{fa}\), and vice versa. \(P_{fa}\) and \(P_{md}\) cannot be reduced simultaneously unless the sample size increases, but this is infeasible especially in practice. Detection aiming at data set for a long time cannot reflect the real-time changes of signals. This limits the performance of single spoofing detection. However, it is different in case that multiple detections are fused, because the false alarms would not happen simultaneously in different detectors. This means that false alarms of fused detection would be much less than those of each detector. Herein, a brief derivation of \(P_{fa}\) is given in the case that only two detections are fused. The probabilities of \(m_{i}(\cdot)\) under hypotheses \(H_{0}\) and \(H_{1}\) are denoted as \(P_{0}^{(i)}\) and \(P_{1}^{(i)}\), respectively. The false alarm probability of detector \(D_{i}\) is denoted as \(P_{fa}^{(i)}\), and the corresponding missed detection probability is denoted as \(P_{md}^{(i)}\). In the case that only two detections are fused, \(m(C)>0.5\) is equivalent to \(m_{1}(C)+m_{2}(C)>1\), and \(m(A)\geq 0.5\) is equivalent to \(m_{1}(A)+m_{2}(A)\geq 1\). The derivation is provided in Appendix A. Based on Equations (4) and (5), \(P_{fa}\) of fused detections can be given as follows:  
\[ \begin{array}{rcl}P_{fa}&=&P\bigg{(}m(A)<0.5\Big{|}H_{0}\bigg{)}\ =\ P \bigg{(}m(C)>0.5\Big{|}H_{0}\bigg{)}\\ &=&P_{0}^{(1)}\bigg{(}m_{1}(C)>0.5\bigg{)}\cdot P_{0}^{(2)}\bigg{(}m_{2}(C)>1- m_{1}(C)\Big{|}m_{1}(C)>0.5\bigg{)}\\ &&+\ P_{0}^{(2)}\bigg{(}m_{2}(C)>0.5\bigg{)}\cdot P_{0}^{(1)}\bigg{(}m_{1}(C)>1- m_{2}(C)\Big{|}m_{2}(C)>0.5\bigg{)}\\ &&-\ P_{0}^{(1)}\bigg{(}m_{1}(C)>0.5\bigg{)}\cdot P_{0}^{(2)}\bigg{(}m_{2}(C)>0.5\bigg{)}\\ &<&P_{fa}^{(1)}+P_{fa}^{(2)}-P_{fa}^{(1)}\cdot P_{fa}^{(2)}\end{array}  \tag{13}\]  
where \(0.5\) is the threshold of \(m(\cdot)\) to determine the authenticity of the signal. \(P_{fa}^{(1)}+P_{fa}^{(2)}-P_{fa}^{(1)}\cdot P_{fa}^{(2)}\) is the false alarm probability of two detectors if their results are not fused together. In such a case, each false alarm would be mistaken for a spoofing attack no matter in which detector it happens. This can be explained as Figure 3.  
The shadow areas in Figure 3a represent the region of false alarms under the hypothesis of \(H_{0}\) in the case that two detections are not fused, while in Figure 3b, the shaded region is lessened in the case that two detectors are fused together. This is the graphical interpretation of the inequality (13). Meanwhile, the dark shaded area in Figure 3a, denoted as \(R_{1}\), represents the region in which false  
Figure 3: Regions of the false alarm in the case that detections are fused or not fused under hypothesis \(H_{0}\). (**a**) Two detections are not fused; (**b**) two detections are fused. alarms happen simultaneously in both \(D_{1}\) and \(D_{2}\). Denoting the shaded area in Figure 3b as \(R_{2}\), \(R_{1}\) is a subset of \(R_{2}\), i.e., \(R_{1}\subset R_{2}\). Thus, there is:  
\[P_{fa}^{(1)}\cdot P_{fa}^{(2)}<P_{fa} \tag{14}\]  
The above inequality can be deduced as follows:  
\[