3. INTERMEDIATED GNSS SPOOFING DETECTOR BASED ON ANGLE OF
3.4 A Linear Regression Model of the Phase Double Differences to Improve the
3.4.4 Performance assessment with in-lab GNSS signals
In order to test the performance of the linear regression algorithm, we have generated an ensemble of spoofed and authentic GNSS signals in static conditions, using a commercial GNSS signal generator. The dataset is 2 hours long and the received 𝐶𝐶/𝑁𝑁0 is set to 42 dBHz for all the signals. A snapshot of the computed fDDs at the output of a pair of commercial receivers is plotted in Figure 3.45 for the first 400 s, where the three genuine signals (namely, PRN 5, PRN 16 and PRN 25) are visible, while the others are counterfeit, i.e., they are generated as coming from the same source. The estimated straight lines resulting from the linear regression algorithm presented in section pricewise linear model are also shown.
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Figure 3.45 Time series of the fractional DD measurements computed from a GNSS dataset, including both authentic and spoofed signals
After setting a normalized value of detection threshold 𝜆𝜆 = 6, the standard D3 algorithm gave the sequences of decisions shown in Figure 3.46 (for the first 400 s only, for the sake of readability). Events of missed-detection and false alarms sometimes happen. In the same condition, the LR-D3 algorithm provided the decisions reported in Figure 3.47: we can appreciate as in this case no events of false alarms or missed-detection happen, thus proving the robustness of this method.
However, the LR-D3 technique is more sophisticated than the D3 approach and which requires initialization time. The measured performance along 2 hours of signal simulation gave the result reported in Table 3.13.
Table 3.13 Comparison of detection performance for 2 hours of signal simulation: LR-D3 and standard D3 algorithms
LR-D3 (𝜆𝜆= 6)
Standard D3 (𝜆𝜆= 6)
Estimated missed-detection probability, 𝑃𝑃𝑀𝑀𝐷𝐷
< 0.02 % 0.12 %
Estimated false-alarm probability, 𝑃𝑃𝐹𝐹𝐹𝐹 < 0.02 % 3 %
Complexity O(n3) O(n)
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Figure 3.46 Decisions produced by the standard D3 algorithm
A detailed analysis of the behaviour of the LR-D3 algorithm applied to the GNSS measurements of the considered simulation is presented in the following Figure 3.47.
Figure 3.48 (a) shows the time series of the estimates of the slope 𝑎𝑎 of the fDDs for two signals in the simulation, namely PRN 7 ∈ 𝒮𝒮 and PRN 25 ∈ 𝒜𝒜, taken as examples. These two signals should be recognized as “separate” for all the time, therefore, in case of detection of events 𝐴𝐴7−25 or 𝐵𝐵7−25, they are false alarms.
Although stable convergence of the slope estimates is obtained in about 90 s, they are clearly separable in less than 10 s. In the successive time intervals, this convergence time is nearly 0. The estimated pairwise false alarm probability is presented in Figure 3.45 (b) as a function of λ. The same analysis is shown in Figure 3.45 (c) and Figure 3.45 (d) for the intercept estimates. The pairwise false alarm rate is 0 for any value of λ that is meaningful for the slope events (i.e., the intercept false alarm rate increases from 0 for values of normalized threshold three orders of magnitude bigger than for slope rates).
Performance assessment as a function of 𝑪𝑪/𝑵𝑵𝟎𝟎
The performance analysis of the LR-D3 algorithm is conducted in this paragraph as a function of the received 𝐶𝐶/𝑁𝑁0, generating three datasets at:
• dataset 1: 𝐶𝐶/𝑁𝑁0 = 39 dBHz;
• dataset 2: 𝐶𝐶/𝑁𝑁0 = 42 dBHz;
• dataset 3: 𝐶𝐶/𝑁𝑁0 = 45 dBHz;
The overall missed-detection and false alarm rates are shown in Figure 3.49, where it is possible to observe that the estimated missed-detection probability, 𝑃𝑃𝑀𝑀𝐷𝐷 is substantially independent from 𝐶𝐶/𝑁𝑁0 for any value of the detection threshold.
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Figure 3.47 Decisions produced by the LR-D3 algorithm
(a) (b)
(c) (d)
Figure 3.48 Examples of slope estimates (a) and intercept estimates (c), and associated pairwise false alarm rates for events A7-25(b) and B7-25(d). Here PRN
7∈ S and PRN 25∈ A
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Table 3.14 Detection performance as a function of C/N0
𝐶𝐶/𝑁𝑁0 = 39 dBHz
(𝜆𝜆 = 0.07) 𝐶𝐶/𝑁𝑁0 = 42 dBHz
(𝜆𝜆 = 0.07) 𝐶𝐶/𝑁𝑁0 = 45 dBHz (𝜆𝜆 = 0.07) Estimated missed-
detection probability, 𝑃𝑃𝑀𝑀𝐷𝐷
2.4% 0.3% 0.3%
Estimated false-alarm
probability, 𝑃𝑃𝐹𝐹𝐹𝐹 < 0.02% < 0.02% < 0.02%
Figure 3.49 Measured missed-detection rate and false alarm rate, evaluated on three data collections at different C/N0 (dataset 1: 39 dBHz; dataset 2: 42 dBHz,
dataset 3: 45 dBHz) as a function of the detection threshold λ
(1) Performance assessment as a function of antenna distance
Another test has been conducted to assess the robustness of the method with respect to the distance between the two antennas. The method is theoretically independent from the antenna distance, provided that the phase measurement noise is correctly modelled, so as to properly set the threshold in Figure 3.50. Figure 3.50 shows the estimated pairwise missed-detection probabilities as a function of the detection threshold 𝜆𝜆, for two signals coming from the same direction (𝐻𝐻0), both with 𝐶𝐶/𝑁𝑁0 = 50 dBHz. The distance between the antennas is indicated as a function of the carrier wavelength.
(a) (b)
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Figure 3.50 Measured pairwise missed-detection rate for the detection events Aij
and Bij evaluated on three data collections at different distance of two antennas In these Figures, the estimation tends to suffer from a smaller number of available measurements with respect to the previous simulations, nonetheless it can be confirmed that the antenna distance is not a relevant factor for the detection performance. The worse performance in missed-detection rate for the intercept at two wavelengths of distance is likely due to an effect of unmodelled clock bias between the two receivers, which resulted not correctly compensated for that test.