# Blind Spoofing GNSS Constellation Detection Using a Multi-Antenna Snapshot Receiver +
## 5 Experimental Setup and Methodology  
Two experimental setups were used: the first one used a six-element UCA. The array was connected to a six-channel receiver which consisted of two synchronized three-channel Flexiband front-ends [50], synchronized with a 10 MHz clock and triggered simultaneously. These front-ends have a maximum sample-rate of 81 MHz at 8 bit I/Q; however, for this experiment, a common sampling rate of 10.125 MHz was used. One minute of data has been recorded and then evaluated in post-processing. A picture of the system is shown in Figure 5. The recording was made under open-sky conditions with the purpose to determine the probability of false alarm for the developed detectors. The second setup used a single antenna connected to each channel of the receiver via a splitter. For this recording, all signals will have similar and static array steering vectors. Therefore, this was considered a rudimentary spoofing simulation, without the need for an anechoic chamber or a live setup. With this recording, the probability of detection for the detectors was evaluated. Only Global Positioning System (GPS) L1 C/A signals were considered in this experiment. A total of 15,000 snapshots were taken and evaluated for each test. Snapshots of length 2, 3, 6 and 10 ms were extracted from the recordings. In the acquisition, coherent integration of 1 ms was selected, and 1, 2, 5 and 10 epochs were added incoherently, respectively. The integration time significantly influenced the performance of acquisition and the accuracy of the estimation for the beamforming steering vector. Acquisition was done on each snapshot, followed by the estimation of the beamforming steering vector for each acquired satellite. Lastly, the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\) was calculated from the array steering vectors (see Section 3). In the analysis of the data the following detection metrics were evaluated:  
* the steering vector angle between a pair of satellites, i.e., the creation of the detection matrix \(\mathbf{D}\) (see Section 3),
* the fraction of detections in the detection matrix \(\mathbf{D}\) (see Section 4.1),
* the Eigenvalue maximum-to-mean ratio of the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\) (see Section 4.2),
* the number of clusters of the Louvain algorithm based on the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\) (see Section 4.3),
* the community size maximum-to-mean ratio for the Louvain algorithm based on the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\)(see Section 4.3), and
* the modularity \(Q\) from Louvain clustering based on the normalized correlation matrix \(\mathbf{C}_{\mathrm{norm}}\)(see Section 4.3). For each detection metric, the probability density function (PDF) \(p(x)\) was calculated from the recorded data and compared. This allowed for the statistics of the metrics to be evaluated. To generate the PDF, a histogram from every detection metric was first calculated. The histogram was then  
Figure 5: Photo of the measurement system. ©IEEE. Reprinted, with permission, from [22]. normalized according to the bin-widths to obtain the PDF. Since a histogram is limited by a finite resolution, it is a discontinuous approximation of the true PDF. This is shown by the \"step\" like function in some of the plots. The histogram bin-sizes were changed depending on the metric used. From the PDF the mean \(\mu\) and standard deviation \(\sigma\) can also be determined [51]:  
\[\mu=\mathrm{E}\left[X\right]=\int_{-\infty}^{\infty}xp(x)dx\,, \tag{37}\]  
\[\sigma=\sqrt{\mathrm{E}\left[X^{2}\right]-\mathrm{E}\left[X\right]^{2}}=\sqrt{ \int_{-\infty}^{\infty}x^{2}p(x)dx-\mu^{2}}\,, \tag{38}\]  
where \(\mathrm{E}\left[\cdot\right]\) is the expected function for the random variable \(X\). Both the mean \(\mu\) and standard deviation \(\sigma\) are shown in the figures to allow for comparison of the detectors. This supports the analysis by summarizing the probability densities.