# Blind Spoofing GNSS Constellation Detection Using a Multi-Antenna Snapshot Receiver +
## 7 Discussion of Results  
As the integration time used for acquisition increases, the estimation of the array steering vector improves. The variance of the array steering vectors is consequently reduced, resulting in an improved representation of the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\). In the spoofed case, the correlation matrix tends to be a unit matrix \(\mathbf{I}\). Therefore, it is expected and observed in all results, that superior performance is achieved with longer integration times.  
Using a detector to estimate the spoofing of a pair of satellites resulted in sufficient spoofing detection. However, it only considers two satellites at a time and is sensitive to the spatial relationship between the two satellites. When a system of satellites is observed, better results can be obtained, as what is shown in the system of detectors.  
Three metrics have shown to be useful for detecting a spoofed scenario as all three of these have demonstrated perfect detection capabilities for the given scenario. Firstly, the fraction of detections is built upon the detection of a single pair of satellites. This method combines all detections, hence, it is robust against a single false detection through increased redundancy. This method is straightforward to implement but requires the design of two detectors. Secondly, an Eigenvalue decomposition-based detector was evaluated. This metric exploits the fact that similar satellites would result in a sparse signal sub-space of the Eigenvectors. A maximum-to-mean Eigenvalue ratio was used. This method is more computationally expensive than the fraction of detections but requires fewer calibration and detectors to be developed. Thirdly, a clustering approach based upon the Louvain algorithm was presented. This metric observes the similarity of the array steering vectors as edges between communities. This method has similar advantages and disadvantages to the Eigenvalue based metric. All other metrics presented have been found to have inadequate performance but are nonetheless interesting from a research point of view and for evaluation purposes.  
The results are compiled from a high number of observations including the use of different integration lengths. Therefore, a high degree of accuracy and confidence in the methods were achieved. However, only a single spoofed and a single real scenario has been evaluated. This is a significant  
Figure 14: CDF comparison of the modularity after clustering for a six-element array.  
limitation to the presented results, as the authors believe that the number of acquired satellites, the distribution of the satellites within the sky (i.e., are the satellites close to each other), the recording scenario including degradation effects (e.g., multi-path, shadowing, array coupling effects etc.), the received signal strength of the satellites (i.e., the carrier-to-noise density ratio (\(C/N_{0}\))), the antennas used, and the array geometry will have an impact on the results. As such, further studies including additional observations are required to improve the validity of the results.  
In the presented results, the complete constellation is either real or spoofed. However, in many spoofing attacks, a mixed constellation of real and spoofed signals is observed. This is not accounted for in the results but it is believed that Eigenvalue and clustering techniques could benefit in a mixed spoofing attack. For future research, it is proposed that mixed attacks are considered such that the spoofing detection metrics could be improved. Lastly, in a mixed spoofing scenario, a secondary goal could be to identify spoofed satellites and to remove these from the PVT solution. Such spoofing mitigation methods are theoretically possible using the metrics presented in this paper; however, the feasibility thereof still needs to be further investigated.