# Blind Spoofing GNSS Constellation Detection Using a Multi-Antenna Snapshot Receiver +
## 4 System of Detections
### Fraction of Detections  
The first method is to consider the number of detections in the upper triangle of the detection matrix \(\mathbf{D}\). The size of the detection matrix \(\mathbf{D}\) depends on the number of successfully acquired satellites \(N_{\text{s}}\). The total number of detection values \(N\) in the upper triangle is:  
\[N=\frac{N_{\text{s}}(N_{\text{s}}-1)}{2}\,. \tag{23}\]  
The detector observes the number of detections in this triangle. If at least \(M\) detections are made, then the system is selected as being spoofed. If it is assumed that all of the satellites are either completely spoofed or only real signals (i.e., no combined real and spoofed signals), the detection probabilities for the system can be determined to be:  
\[P_{D}^{(M)}=\sum_{m=M}^{N}\binom{N}{m}P\left(d|S\right)^{m}P\left(\bar{d}|S \right)^{N-m}\,, \tag{24}\]  
where \(\binom{N}{m}\) is the binomial coefficient, \(P\left(d|S\right)\) is the conditional probability that a value is detected \(d\), given that the signals are spoofed \(S\). The inverse is indicated by \(\bar{d}\) (i.e., it is not detected). Similarly, the probability of a missed detection \(P_{M}^{(M)}\), the probability of false alarm \(P_{FA}^{(M)}\), and the probability of a correct rejection \(P_{R}^{(M)}\) can also be determined:  
\[P_{M}^{(M)}=\sum_{m=0}^{M-1}\binom{N}{m}P\left(d|S\right)^{m}P\left(\bar{d}|S \right)^{N-m} \tag{25}\]  
\[P_{FA}^{(M)}=\sum_{m=M}^{N}\binom{N}{m}P\left(d|\bar{S}\right)^{m}P\left(\bar{ d}|\bar{S}\right)^{N-m} \tag{26}\]  
\[P_{R}^{(M)}=\sum_{m=0}^{M-1}\binom{N}{m}P\left(d|\bar{S}\right)^{m}P\left(\bar{ d}|\bar{S}\right)^{N-m} \tag{27}\]  
This approach improves overall performance as it uses each combination of the satellites to verify whether the complete constellation is spoofed or not. Thereby, false detections from spatially similar satellites can be efficiently suppressed. Further, by combining all of the detections the overall system improvement allows for increased tuning freedom within individual detection.  
One issue of this approach of defining a minimum number of detection, is that the total number of detectors \(N\) is inconsistent. The total number of detectors depends on the size of the matrix \(\mathbf{C}_{\text{norm}}\), which in turn is dependent on the number of satellites that were successfully acquired. To overcome this, a fraction of detections \(\frac{M}{N}\) is used. This is called an \(M\) of \(N\) detector [39], but also referred to as binary integration [40; 41] or coincidence detection [42; 43]. A fraction of detections \(\frac{M}{N}\) approach makes the system independent on the number of detectors to evaluate, as it is normalized. For example, if at least half of the values should be detected, then the system detector threshold is defined as \(\frac{M}{N}=\frac{1}{2}\). If too few detections are made then the null-hypothesis \(H_{0}\) of no spoofing is accepted. Otherwise, the alternative hypothesis \(H_{1}\) is accepted, where the signals have similar origins and are considered spoofed. This can be formally defined as:  
\[\lambda_{\text{frac}}=\frac{M}{N}\underset{H_{1}}{\overset{H_{0}}{\gtrless}} \frac{\sum_{n,m}D(n,m)-N_{\text{s}}}{2N}\,, \tag{28}\]where \(D(i,j)\) is the \(i\)-th row and the \(j\)-th column of the detection matrix \(\mathbf{D}\) (described in Section 3), and \(\lambda_{\text{frac}}\) is the detector threshold for the fraction of detections metric. It can further be simplified using Equation (23):  
\[\lambda_{\text{frac}}\overset{H_{0}}{\gtrless}\frac{\sum_{n,m}D(n,m)-N_{ \text{s}}}{N_{\text{s}}(N_{\text{s}}-1)}\,. \tag{29}\]  
The fraction of detection is a simple yet effective method of combining multiple detectors.