Performance evaluation of robust D 3 implementations

3. INTERMEDIATED GNSS SPOOFING DETECTOR BASED ON ANGLE OF

3.3 Performance Analysis of the Dispersion of Double Differences Algorithm to

3.3.2 Performance evaluation of robust D 3 implementations

In order to increase the robustness of the detection against the noise effect, in the previous section proposed the averaging of the fractional DDs along short time windows before taking the decision. Indicating with 𝜂𝜂 the number of measurements within such averaging window and assuming independent noise samples along the time, the noise variance of the averaged measurements is evidently reduced by a factor 𝜂𝜂:

𝜎𝜎�𝑘𝑘2 =𝜎𝜎𝑘𝑘2

𝜂𝜂 ,∀𝑘𝑘 ∈ (𝒮𝒮 ∪ 𝒜𝒜) (3.44)

This operation has the following positive effects on the overall performance:

If a new detection threshold 𝜉𝜉̅𝑗𝑗𝑘𝑘2 =�𝜎𝜎�𝑗𝑗2+𝜎𝜎�𝑘𝑘2�𝜉𝜉2 =𝜉𝜉𝑗𝑗𝑘𝑘2 /𝜂𝜂 is used in (3.18), then the pairwise 𝑃𝑃𝑚𝑚𝑑𝑑 and overall 𝑃𝑃𝑀𝑀𝐷𝐷 remain the same, while the 𝑃𝑃𝑓𝑓𝑓𝑓 in Figure 3.28 is reduced by a rigid shift right-wise of the curves, corresponding to a contraction of the abscissa axis by a factor 1/𝜂𝜂;

If we maintain the threshold 𝜉𝜉𝑗𝑗𝑘𝑘2 =�𝜎𝜎𝑗𝑗2+ 𝜎𝜎𝑘𝑘2�𝜉𝜉2 in (3.18), then the pairwise 𝑃𝑃𝑚𝑚𝑑𝑑 in Figure 3.25 and overall 𝑃𝑃𝑀𝑀𝐷𝐷 in Figure 3.30 are reduced by a rigid shift left-wise of the curves, corresponding to an expansion of the abscissa axis by a factor 𝜂𝜂; on the other hand, the 𝑃𝑃𝑓𝑓𝑓𝑓 remains unaltered.

An example of this second case is reported in Figure 3.32, where it is possible to appreciate how the averaging technique remarkably reduces the probability of missed-detection. A drawback of this method is that the averaging correlates the series of decisions along time, so that independent decisions can be taken just every 𝜂𝜂 measurements.

A different method to increase the robustness of the detection algorithm is to use a second antenna baseline (i.e. to add a third antenna to the system) to run another instance of the 2-baselines detection algorithm: A spoofing detection is declared if and only if both baselines have detected the same counterfeit signals at the same instant.

On the basis of this rule, the set of probabilities reported in Table 3.2 describe the expected performance of this method, where we use the subscripts 𝑏𝑏1,𝑏𝑏2 to indicate the two baselines separately and the superscript (2𝑏𝑏) to indicate a quantity referred to the algorithm that employs two baselines jointly. The product of probabilities decreases both 𝑃𝑃𝐷𝐷(2𝑏𝑏) and 𝑃𝑃𝐹𝐹𝐹𝐹(2𝑏𝑏) with respect to their counterpart along the single baseline, while consequently increases 𝑃𝑃𝑀𝑀𝐷𝐷(2𝑏𝑏).

66

Figure 3.32 Estimated PMD for the D3 algorithm with averaged fractional DDs, under the H0 condition and for different averaging window lengths η

The benefit of this method can be appreciated by comparing the ROC curves associated to this method to the ones associated to the single baseline method, as shown in Figure 3.33 where the continuous line indicates 2-baselines method and the dotted line indicates 1-baseline method, for several values of λ. The interesting observations are that:

For any value of 𝑃𝑃𝑀𝑀𝐷𝐷(2𝑏𝑏) =𝑃𝑃𝑀𝑀𝐷𝐷, the false alarm of the 2-baselines method is lower:

𝑃𝑃𝐹𝐹𝐹𝐹(2𝑏𝑏) <𝑃𝑃𝐹𝐹𝐹𝐹 for the same value of 𝜆𝜆.

For any given value of 𝜆𝜆, the performance set (𝑃𝑃𝑀𝑀𝐷𝐷∗ ,𝑃𝑃𝐹𝐹𝐹𝐹∗ ) is always 𝑃𝑃𝑋𝑋𝑋𝑋(2𝑏𝑏)<𝑃𝑃𝑋𝑋𝑋𝑋, where ′𝑋𝑋𝑋𝑋′ is either ′𝑀𝑀𝐷𝐷′ or ′𝐹𝐹𝐴𝐴′.

The 2-baselines method avoids the introduction of temporal correlation among close decisions and does not increase the minimum number of detectable signals, but comes at the cost of one additional antenna and one additional receiver in the setup.

The above analysis could be extended to more than two baselines as done in[59], in order to further reduce the probability of false alarm without affecting the minimum number of detectable signals, but this is considered straightforward and not pursued in this work.

67

Finally, an interesting evolution of this method has been presented in the next section, which exploits the short-term linearity of the DD measurements to move the detection rule from the difference of fractional measurements, as in (3.18), to the parameters of the linear model of the DDs; the new method, based on a linear regression of the measured fractional DDs (LR-D3), improves the sensitivity and the minimum number of detectable single-source signals.

Table 3.2 Statistical performance of the D3 algorithm with two baselines Correct detection (𝐻𝐻0)

Event: 𝐷𝐷𝑖𝑖𝑗𝑗,𝑏𝑏1∩ 𝐷𝐷𝑗𝑗𝑘𝑘,𝑏𝑏1∩ 𝐷𝐷𝑖𝑖𝑗𝑗,𝑏𝑏2∩ 𝐷𝐷𝑗𝑗𝑘𝑘,𝑏𝑏2

Probability: 𝑃𝑃𝐷𝐷(2𝑏𝑏) =𝑃𝑃𝐷𝐷,𝑏𝑏1⋅ 𝑃𝑃𝐷𝐷,𝑏𝑏2 Missed-detection (𝐻𝐻0)

Probability: 𝑃𝑃𝑀𝑀𝐷𝐷(2𝑏𝑏) = 1− 𝑃𝑃𝐷𝐷(2𝑏𝑏) False alarm (𝐻𝐻1)

Event: 𝐴𝐴𝑖𝑖𝑗𝑗,𝑏𝑏1∩ 𝐴𝐴𝑗𝑗𝑘𝑘,𝑏𝑏1∩ 𝐴𝐴𝑖𝑖𝑗𝑗,𝑏𝑏2∩ 𝐴𝐴𝑗𝑗𝑘𝑘,𝑏𝑏2

Probability: 𝑃𝑃𝐹𝐹𝐹𝐹(2𝑏𝑏) =𝑃𝑃𝐹𝐹𝐹𝐹,𝑏𝑏1⋅ 𝑃𝑃𝐹𝐹𝐹𝐹,𝑏𝑏2 where events are defined as:

𝐷𝐷𝑗𝑗𝑘𝑘,𝑏𝑏𝑐𝑐: �Λ𝐷𝐷3(𝑗𝑗,𝑘𝑘) <𝜉𝜉𝑗𝑗𝑘𝑘2 �ℎ0� along the baseline 𝑏𝑏𝑛𝑛 𝐴𝐴𝑗𝑗𝑘𝑘,𝑏𝑏𝑐𝑐: �Λ𝐷𝐷3(𝑗𝑗,𝑘𝑘) < 𝜉𝜉𝑗𝑗𝑘𝑘2 �ℎ1� along the baseline 𝑏𝑏𝑛𝑛

68

Figure 3.33 Comparison of ROC curves for the D3 spoofing detection algorithm with 1 and 2 baselines, for several non-centrality parameters λ

Ambiguities

The single-baseline, single-epoch setup analysed in Sub-Section 3.3.1 presents some intrinsic ambiguity, shortly mentioned above. First, the ambiguity produced by the rotational symmetry of the angular measurement cos(𝛼𝛼𝑖𝑖), cos(𝛼𝛼0) around the baseline axis, as discussed in Sub-Section 3.3.1. It is possible to argue that the risk of finding an authentic satellite on an ambiguous direction increases with the sine of 𝛼𝛼𝑖𝑖, towards the antenna boresight, as the circumference of ambiguity is maximum. This situation can be resolved with a second no-colinear baseline, whose region of ambiguity is maximally disjoint from the first baseline (e.g., orthogonal baselines).

The second cause of ambiguity is the reduction of the DDs to their fractional parts.

Since the condition under test is in fact 𝐷𝐷

𝜆𝜆𝐶𝐶�cos(𝛼𝛼𝑖𝑖)−cos(𝛼𝛼0)�= 0, then it is ambiguous for any 𝐷𝐷>𝜆𝜆2𝐶𝐶. The region of unambiguous measurements is the range

𝐷𝐷

𝜆𝜆𝐶𝐶��cos(𝛼𝛼𝑖𝑖)−cos(𝛼𝛼0)�� < 1, where the cosines difference is bounded in [−2,2]. With some algebra the angular range for which the fractional DDs are not ambiguous becomes

2�sin�𝛼𝛼𝑖𝑖 +𝛼𝛼0

2 �sin�𝛼𝛼𝑖𝑖− 𝛼𝛼0

2 ��<λC 𝐷𝐷

(3.45)

10-6 10-4 10-2 100

H

0: Probability of missed detection: P MD(2b) , P MD 10-8

10-6 10-4 10-2 100

H 1: Probability of false alarm: P FA (2b), P FA

Receiver Operating Curves with 2 baselines, parameterized as per

0.01 4 16 36 64 100 0.01 4 16 36 64 100 =

2 baselines

1 baseline

69

Using this formulation, it is possible to prove that the dimension of the range of unambiguous sums of measurements (𝛼𝛼𝑖𝑖+𝛼𝛼0) is inversely proportional to the antenna distance and to the angular difference with respect to the reference signal direction |𝛼𝛼𝑖𝑖− 𝛼𝛼0|.