Enhanced Spatial Spoofing Detection With and Without Direction of Arrival Estimation

Spoofing, the transmission of false global navigation satellite system signals, threatens receivers. The first step in dealing with spoofers is to detect the attack. Once a spoofing attack is detected, spatial detection methods using an array of antennas can additionally mitigate spoofing through spatial nulling. This article presents a spatial spoofing detection approach using the estimated antenna steering vector (ASV) from direction of arrival (DOA) estimation. Furthermore, the method is compared with blind detection and post-DOA estimation methods. Simulation results show that the approach is comparable to the current state-of-the-art if the same a priori information is used. However, superior performance is achieved if the attitude of the antenna array is known. The presented approach represents an alternative method for spatial spoofing detection that leverages the DOA estimates but remains in the ASV domain. Therefore, it benefits from noise modeling and ambiguity resolution.


I. INTRODUCTION
Spoofing is the falsified transmission of global navigation satellite system (GNSS) signals [1], [2], and it misleads a GNSS receiver to an erroneous position, velocity, and time (PVT) solution.Spoofing attacks are increasing [3], indicating that it is a significant threat to the reliability and security of GNSS receivers and needs to be countered.Many spoofing detection techniques exist [2], [4], [5], [6], but many of them are transient detectors: they monitor the tracking and PVT changes over time.Furthermore, such detectors have limited mitigation capabilities.
Fig. 1 shows such a single-transmitter attack.The authentic GNSS signals are distributed over the sky, but all the spoofing signals originate from a single direction.Consequently, an authentic GNSS will show a sky plot with distributed satellites (bottom left in blue), but a spoofed GNSS sky plot will receive signals clustered from one direction (bottom right in red).The distributed versus clustering concept forms the basis of several spatial spoofing detection techniques [7], [8], [9], [10], [11], [12], [13].
Distributed multitransmitter spoofing attacks are currently difficult to execute and highly unlikely [14], [15], making single-transmitter attacks more viable.It results in spoofing signals originating from the same direction.Therefore, spatial detection techniques using an array of antennas are viable to determine the direction of the spoofing signals and to mitigate them through spatial null steering [8], [16] or to locate the spoofer and physically remove it.It demonstrates a clear advantage to spoofing handling for spatial processing methods.
Current spatial detection methods are either blind and use no a priori information [11], [12], [13], resulting in low performance, or relying entirely on the final direction of arrival (DOA) estimation outputs [7], [8], [9], [10], limiting the performance to the DOA estimator.This article presents a spatial method to detect spoofing signals that bridge these two approaches: it utilizes the a priori information from the PVT solution and satellite ephemeris but detects using the antenna steering vector (ASV).The results show comparable performance to the current state-of-the-art uniformly most powerful invariant (UMPI) detector [10], emphasizing excellent performance.However, if the attitude of the GNSS receiver (i.e., orientation in space) is perfectly known (an unfair comparison), it is superior to the current state-ofthe-art.The performance improvement is attributed to the correct Gaussian noise modeling (other detectors make a Gaussian approximation), and the ASV-based comparison intrinsically corrects ambiguities in the DOA estimation.
The rest of this article is organized as follows.Section II provides context to multiantenna processing in GNSS.Section III describes the signal models and gives background to DOA estimation, and Section IV presents current state-of-the-art spatial spoofing detection methods.Section V introduces the new spatial detection method, and Section VI shows the results and discusses the performance of the detectors.Finally, Section VII concludes this article.

II. SPATIAL PROCESSING IN GNSS
Within the context of GNSS, there are four spatial architectural approaches for spatially detecting and mitigating spoofing signals.
First, multiple distributed receivers collaborate: A popular approach is to use differential GNSS to compare the pseudorange measurements between receivers [17], [18].This approach improves when the carrier phase is considered.Furthermore, it may also facilitate spoofing localization [19].Alternatively, collaborative receiver autonomous integrity monitoring (RAIM) with multiple receivers detects and excludes potential spoofed signals [20].Collaborative approaches are simple to implement, as the spoofing detection is done after the GNSS signal processing.However, they require multiple distributed receivers with communication links, resulting in an overall complex deployment architecture.
Second, GNSS satellites transmit right-hand circular polarized signals.The polarization is altered by the environment (i.e., multipath reflections) and the received antenna.Therefore, the polarization at reception indicates spoofing [21] or facilitates DOA estimation for spoofing detection [22].Analyzing the polarization requires a more complex receiver, but a single compact antenna can be used [23].
Third, moving [24], [25], [26] or rotating antennas [27], [28] creates a synthetic aperture that facilitates DOA estimation.Such approaches only need a single receiver with low hardware requirements but require the receiver to move.Mechanical wear and tear and the capability of the receiver to move limit this approach.
Finally, colocated antennas facilitate phased array processing [29], [30].Such systems require multiple antennas, making them more expensive and larger.However, they allow both signal DOA estimation and remove the spoofing signals, resulting in superior mitigation capabilities.This article focuses on phased array processing, allowing for static, stand-alone detection.Therefore, it applies to fix installed reference and monitoring systems.

III. SIGNAL MODELS AND PREPROCESSING
This section presents the precorrelation and postcorrelation signal models and briefly introduces DOA estimation.

A. Signal Model for an Antenna Array
The signal model for an N-sized antenna array that receives L signals is defined as follows [31]: where x(t ) ∈ C N×1 is the received complex signal vector per antenna element, s(t ) ∈ C 1×L is the complex signal vector from each satellite vehicle (SV), n(t ) ∈ C N×1 is a complex additive white Gaussian noise (AWGN) vector, and A ∈ C N×L is a Vandermonde matrix [32] of the complex steering vectors mapping each satellite vehicle (SV) signal to an antenna element.The steering matrix composes of the individual ASV where a l ∈ C N×1 is the ASV of the lth SV signal.The ASV is a function of the antenna position in three-dimensional (3-D) space d n ∈ R 3×1 and the wave vector of the satellite signal Finally, the wave vector of the satellite signal k l relates to the spatial vector u l ∈ R 3×1 in the direction of the satellite where λ c is the carrier wavelength of the signal.As a practical GNSS example, λ c ≈ 19 cm for GPS L1.

B. Postcorrelation Signal Model
GNSS systems use code-division multiple access to transmit multiple signals on the same center frequency.Through correlation with the correct pseudo-random noise sequence r l (t ), a signal can be isolated in the code domain (i.e., despreading the signal) [29], also known as postcorrelation processing where y l [m] ∈ C N×1 is the postcorrelation signal for the mth correlation epoch, s l (t ) ∈ C 1×1 is the received complex signal from the lth SV, and n(t ) ∈ C N×1 is the altered noise vector.Postcorrelation allows the signals to be isolated before DOA estimation, i.e., reducing L signals to a effectively signal L = 1.This isolation improves performance and benefits from the noise suppression properties of the integration gain.
The GNSS tracking channels are already locked onto the satellite signal in postcorrelation processing [29].Therefore, the carrier Doppler and code delay are already compensated.Furthermore, the correlator outputs of each antenna channel are jointly used to steer the delay locked loop, frequency locked loop, and phase locked loop, for enhanced signal tracking performance [33].

C. DOA Estimation
Several DOA estimation algorithms exist [31].However, only multiple signal classification (MUSIC) is considered for this article [34], as it has good performance and is popular [31].
First, the autocovariance matrix (ACM) Ryy,l ∈ C N×N of the received signal is determined by accumulating over M snapshots The auto-covariance matrix (ACM) is a Hermitian matrix and can be separated into signal and noise subspaces through eigenvalue decomposition where V s ∈ C N×L contains the signal eigenvectors related to the largest L eigenvalues as composed in the diagonal matrix s ∈ C L ×L , and L ) .A significant limitation of the multiple signal classification (MUSIC) algorithm is that the number of signals L should be known for ideal operation and noise subspace separation [31].However, in the case of postcorrelation processing, it may be assumed that L = 1 if no multipath reflections or additional spoofing signals are present (see the hypotheses in Section IV).MUSIC is a Pisarenko estimator, and theoretically, the correct ASV (i.e., a l ) will be orthogonal to the noise subspace V ω .Therefore, the correct ASV can be estimated with âl = arg max where âl is the estimated ASV. MUSIC requires a processing-intensive parametric sweep, but some methods to limit the search exist [31].The selection of the parametric sweep directly impacts DOA performance (e.g., if it misses the largest peak) and complexity (i.e., how many values are tested).The spatial vector u l determines the final DOA.DOA accuracy is impacted by antenna coupling effects, calibration inaccuracies, and ambiguities.Several solutions to these exist [31].In the context of this article, ambiguities in the DOA are at the forefront, as they result in biased errors.These biases are significant limitations for several detection methods.

IV. SPATIAL DETECTION METHODS
The section presents several state-of-the-art spatial detection methods.The methods use different information for detection.Fig. 2 shows a flow diagram of the different methods and their place in the processing chain.Note that some methods (e.g., attitude sum of squares of error (SSE) detection) are later in the chain and require more processing and a priori information than others (e.g., blind detection).
The generalized spoofing hypothesis test is set up as follows.Note that a mixture of authentic and spoofed signals is not considered in this definition.In a tin can attack [12] or if the received power of the spoofing signals are significantly higher than the authentic ones, these hypotheses are valid.Later in the article, a mixed constellation will also be considered.

1) Null hypothesis
A detection metric x is compared with a threshold λ, and the outcome determines if the null hypothesis H 0 or the alternative hypothesis H 1 are accepted In this case, the probability density function (PDF) of the alternative hypothesis H 1 is assumed on the right of the PDF of the null hypothesis H 0 .However, it may also be the other way, wherein the detection logic is inverted.The probability of (correct) detection P D is defined as the probability of correctly selecting the alternative hypothesis H 1 , given that the signal is indeed spoofed [35] Conversely, the probability of false alarm P FA is defined as the probability of falsely selecting the alternative hypothesis H 1 , given that the signal is authentic H 0 In detector design, the probability of detection P D should be maximized, and the probability of false alarm P FA should be minimized.A detector that maximally separates the PDF of the detection metric for the two hypotheses is superior.One method to analyze detector performance considers the receiver operating characteristic (ROC) for a specific scenario [35].The ROC plots the probability of false alarm P FA versus the probability of detection P D .Any point on the line represents the statistics for a single threshold λ.Therefore, the ROC shows the potential for all possible threshold λ settings.Finally, the area under the curve (AUC) of a ROC curve provides a summarized performance of a detector.An AUC = 1 is the perfect detector with no errors, and an AUC = 0.5 is a poor detector as it is randomly guessing results.A value between AUC = 0.5 and AUC = 1 is an adequately designed detector, and a value below AUC = 0.5 indicates a detection logic issue or an inappropriate detector design.Section VI evaluates the detector performances with the receiver operating characteristic (ROC) curves and the AUC for the different detectors.

A. Joint Spoofing Detection and Attitude Estimation
The attitude of the antenna array is the orientation offset between the local coordinate frame relative to the global coordinate frame of the antenna array.Matching the received DOA measurement to the expected direction facilitates the attitude estimation.If the signals are spoofed, then the attitude estimation results in significant estimation errors, facilitating spoofing detection [7], [8], [9].
The measured DOA estimates for satellites U ∈ R 3×L are a rotated version of the correct direction U ∈ R 3×L U = R(φ)U + N (12) where R(φ) is a rotation matrix, N ∈ R 3×L is a noise matrix, and U combines the spatial vectors u l for each SV The ideal spatial vectors u l are determined by the receiver position and the satellite ephemeris.The rotation matrix R(φ) rotates the constellation with the roll, pitch, and yaw angles in the Euler vector φ.
A least squares (LS) approach solves for the rotation matrix based on this minimization constraint [8] After the attitude is estimated (i.e., estimated Euler angles φ), the sum of squares of error (SSE) of the estimation can be determined as follows: where R UU is the covariance matrix of the measurements.
This method is simple and uses the pre-existing infrastructure of a postcorrelation multiantenna GNSS receiver, which makes it practical.However, it is suboptimal as the Gaussian noise model implied in (12) and ( 14) is invalid: the spatial vectors are unit vectors resulting in rotations, not additions.This mapping to the unit sphere is demonstrated in Fig. 3, where the Gaussian error is projected onto a manifold on the spatial unit sphere.Nevertheless, the method works adequately and is well documented in the literature [7], [8], [9].
A Monte Carlo simulation of 10 6 runs demonstrates the error shaping in a practical scenario.First, a unit vector u in a random direction is generated, and a random noise vector n is added to it 18) Second, the error vector is normalized to have unitary magnitude Third, the effective error due to the spherical manifold is determined e = e x , e y , e z = u e − u. (21) Finally, the power spectral density (PSD) of the mean noise n and the mean error ē is plotted for comparison in Fig.
The figure shows that the noise model (solid blue line) n has the expected Gaussian curve, but the error on the sphere (dotted orange line) ē has a more pointy form.The kurtosis of the noise model is κ n = 2.994, which is close to the theoretical value of 3, but the kurtosis of the error is κ e = 3.591.It indicates that the error on the sphere is leptokurtic, resulting in fatter tails in the distribution.The overall error is smaller than the noise |ē| ≤ |n| because radial errors are cut off due to the normalization process [see (20)].This effect is also reflected in the standard deviation reducing from σ n = 0.01733 to σ e = 0.01414.Even though an error reduction is desired, it does not meet the Gaussian error assumption required for the LS approach.
Fig. 5 shows the PDF for the angular error (i.e., the angle between u e and u) of the Monte Carlo simulation.The figure demonstrates that it follows a Rayleigh distribution [35].Even though the angular errors are small (99.9% are below 3.7 • , and the mean angular error is 1.24 • ).It emphasizes that even with low noise, the resultant PDF of Fig. 4 does not approximate to be Gaussian.

B. Arc Detection
The arc-based detector is relative and a uniformly most powerful invariant (UMPI) detector [10].The detector uses the great circle arcs δ(l, p) between two spatial vectors u l and u p (the spatial vectors are unit vectors) Therefore, it is a relative detector and does not require the attitude of the antenna array to be estimated or corrected.
There are N c combinations Next, all N c combination of arcs are added to a single vector The arc-based UMPI detector assumes that the arcs are Gaussian distributed or Gaussian overbounded at least [10].If the signals are authentic, then the mean is the expected great circle arcs δ as determined by the PVT solution and the satellite ephemeris.However, if the signals are spoofed, then it is assumed that all signals are spatially correlated, and the great circle arcs have zero mean.It can be summarized as follows: where R δδ is the covariance matrix between the circle arcs, and N (μ, σ 2 ) represents a system of Gaussian distributions with mean μ and variance σ 2 .There are correlations between the arcs that need to be included in the covariance matrix, and it is strongly advised to consult the original article of Rothmaier et al. [10] for a detailed description.The arc-based UMPI detector is a Neyman-Pearson (NP) test [35], also known as a likelihood ratio test.Furthermore, to simplify the calculations, the logarithm of the function is taken.The final detector ARC is derived as [10] follows: The spoofing hypothesis test is framed as follows: Like the attitude SSE detector, this detector requires the expected satellite vectors and the related covariance to operate.Therefore, the same a priori information is needed, making these detectors a viable comparison.A limitation of both approaches is that noise is assumed Gaussian, which it cannot be as follows.
1) The error is constrained to the orientation or mapping of the spatial unit vector and is projected on a manifold (see Fig. 3).
2) The DOA estimation error of an antenna array depends strongly on the array geometry (i.e., d n ) and the wave vector (i.e., k l ), resulting in different errors' magnitudes and PDF as projected to the array manifold [36].
3) The DOA estimation may have ambiguous results (especially if the array is linear or planar) [37].
However, the Gaussian assumption is adequate for most cases.An alternative approach detects based on prerecorded data with empirically determined PDF [10], but it results in additional lookup tables limiting quick detection.Furthermore, it requires considerable calibration, limiting general use.

C. Blind Detection
All the previous methods require a priori information of the antenna array (i.e., antenna position d n ) and the correct constellation (i.e., u l ).However, blind methods can still facilitate spoofing detection without this information [13].Blind detection is interesting because it allows spoofing even if the antenna array is unknown, uncalibrated, or tampered.
A simple blind method consists of comparing the angle θ b (l, p) between two ASVs As there are N c combinations between all the satellites, a simple method to combine them is to take the mean squared value between them The mean squared angle θ b between all ASVs is used as the spoofing detection metric.If all of the satellites are spoofed, then this value should tend to zero (small spatial angles), but if the constellation is authentic, it should tend to a larger value.Therefore, the hypothesis test is framed as follows: The approach requires no a priori information.Hence, it has less leverage than the attitude SSE and ARC detectors; consequently, inferior performance is expected.However, the tradeoff between performance and a priori information can be considered in practical systems.Furthermore, this approach is independent of the underlying error PDF of the measurements, increasing robustness.

V. STEERING VECTOR-BASED SPOOFING DETECTION
A limitation of attitude SSE and ARC methods is that both rely entirely on the DOA outputs (i.e., u l ) and discard the ASV (i.e., a l ).Furthermore, both also assume Gaussian noise, which is not wholly valid.In contrast, the blind method only uses the ASV (i.e., a l ) and discards the a priori information of the expected constellation (i.e., u l ), resulting in inferior detection results.This section presents an ASV detector that considers the DOA estimates.
If the entire constellation is spoofed, then all the ASVs are approximately the same where a can be approximated as the mean ASV Therefore, the received signals are considered complex Gaussian variables The correlation measurements υ, the theoretical ASV α, the estimated ASV α, the mean ASV α, and the covariance matrix R υυ need reformatting before detection Using the same NP approach to the arc-based detector and the appropriate PDF, the detector can be determined as follows: The real operator is required for the normalization as the data are complex valued [35].This detector uses the raw data from the postcorrelation (i.e., υ), the mean estimated ASV from the DOA (i.e., α), the theoretically expected ASV (i.e., α), and the ACM from DOA estimation (i.e., R −1 υυ ).Furthermore, the measurement errors are Gaussian as the detector is done in the ASV domain and not over unit vectors, making the additive white Gaussian noise (AWGN) assumption valid.Furthermore, as the detector is based on the ASV, it circumvents the DOA ambiguities.
However, this detector has two significant limitations making it highly impractical.First, the correlated vector υ should be correctly scaled for every SV; otherwise, the detection performance significantly degrades.Even if each correlated vector y l is normalized, channel gain mismatches may still significantly affect the performance [38].A more straightforward method is to replace the measurement array with the ASV estimated by the DOA and use it as a detector It solves the calibration issue and benefits from the Pisarenko estimation.However, it projects the noise onto a unit sphere shaping it.As a result, it degrades the detector performance as the assumptions in ( 38) and ( 38) is no longer valid.It means that this detector is now making the same fallacy as the attitude SSE and the ARC detectors, but it solves the calibration issue.
The second issue is that the correct ASV is used for this detector, i.e., the α already incorporates the attitude corrected values.Consequently, the detector is clairvoyant as all information are already available, making it unfair compared with the other detectors.
The attitude estimator of Section IV-A is used to get a unit vector projected to the antenna arrays orientation ǔl to make the detector practical The projected unit vector ǔl determines the expected ASV for the antenna array Finally, the projected vectors from the attitude estimation can be transformed and used for detection The benefit of this approach is that it uses the same information as the attitude SSE and ARC detectors, but it uses the ASV domain for detection that could improve detection performance for larger antenna arrays.One limitation of this method is that it is linked to the attitude estimation, resulting in degradation if the attitude estimation fails.
The final detector is not as ideal as the original derivation in (48).However, further innovations may result in functional yet improved performance and future research topics.

VI. RESULTS
Two Monte Carlo simulations of 10 5 runs evaluate the detection methods.In the first simulation, a completely spoofed scenario is compared to a completely authentic one.In the second simulation, only a partially spoofed scenario is considered, i.e., the spoofer only has a capability to generate half of the satellites.
In both Monte Carlo simulations, a four-, five-, six-, and seven-element (N ∈ {4, 5, 6, 7}) uniform circular array with a half-wavelength radius with omnidirectional antenna elements is simulated with no coupling effects.Furthermore, the antenna array is simulated with an attitude with ±10 • roll, ±10 • pitch, and ±180 • yaw uniformly distributed errors.A constellation consisting of six SV is randomly generated with an elevation mask of 10 • .All SVs have the same carrier-to-noise density ratio (C/N 0 ) in a measurement and are generated for 22 and 25 dBHz (with higher C/N 0 s, the detection is almost perfect and requires significantly more Monte Carlo runs to validate accurately, making it impractical).The signals are processed with 1 ms integration, and M = 10 epochs of data are processed.Furthermore, perfect correlation is assumed, i.e., no code or carrier offset in correlation.The DOA algorithm is a standard MUSIC method but with two parametric sweep stages.The first stage has 1 • resolution and searches over all positive elevation angles.The second stage searches ±2 • around the result of the first stage with 0.1 • resolution.This two-stage approach requires less processing with precise results.With four elements [see Fig. 7(a)], all detectors have poor results.This is caused by the high noise level and the antenna array ambiguities.If larger antenna arrays are used, the performance significantly improves for all detectors.It emphasizes the limitation of the four-element antenna array ambiguities.

A. Completely Spoofed Scenario
With five elements, the ASV detector with the theoretically known reference ASV ["ASV-T" with AUC = 0.954, see (49)] has the best ROC of all detectors.However, this is not a fair comparison, as it is a clairvoyant detector that uses the theoretically perfect ASV as a reference.Therefore, it is expected to have superior performance.However, if the estimated ASV after attitude determination is used as a reference ["ASV-E" with AUC = 0.743, see (54)], then the comparison is valid to the other detectors.Furthermore, the ASV detector has improved results to the arc-based UMPI detector ("Arc," with AUC = 0.667).The SSE detector ("SSE" with AUC = 0.578) has the worst performance, confirming the observations of Rothmaier et al. [10].Finally, blind detection ("Blind" with AUC = 0.684) has decent performance.As the blind has competitive performance, the DOA estimation often fails with low CN 0 , highlighting the limitation of DOA dependence.
The performance of all methods improves with six and seven elements.In a general trend, it shows that "ASV-T" ≥ "ASV-E" ≥ "Blind."This trend is in line with the expectations.The SSE detector performs well with a low probability of false alarm P FA (typically if < 0.15) but is soon overtaken by other detectors.In all scenarios, it has the lowest AUC.The "Arc" detector also performs well with a low probability of false alarm P FA , where it outperforms the "ASV-E" and "Blind" detectors, which indicates good operation as long as DOA is mostly successful with no ambiguity issues.Furthermore, the "Arc" detector has AUC values superior to "Blind" but inferior to "ASV-E."Finally, it must be highlighted that the "Arc" line is unsmooth, which indicates that the PDF calculation for the ROC curve is suboptimal.Therefore, it affects the results to a limited degree.Fig. 7 shows the ROC curves for 25 dBHz CN 0 .With four antenna elements, most approaches but the blind method have poor results, further highlighting issues with antenna array ambiguities.With the larger antenna array, excellent performance with all detectors is achieved.Generally, it is observed that the AUCs are "ASV-T" ≥ "ASV-E" ≥ "Arc" ≥ "SSE" ≥ "Blind."The "ASV-T" detector is clairvoyant and it is expected to have the best results.Conversely, the "Blind" detector uses the least information and is expected to have the worse results.The other three methods ("ASV-E," "Arc," and "SSE") use the DOA results and are in the midfield between the two extremes.Of these three, "SSE" has the poorest results as it is limited by the attitude estimation, making it prone to further estimation errors."ASV-E" and "Arc" have good performance, even though they are still bounded by the Gaussian noise assumption.As the "ASV-E" additionally uses the information from the ASV, it has better ambiguity resolution capabilities than the "Arc" detector.It emphasizes that the robustness of the DOA should be considered when developing a detector.
The results indicate that DOA and attitude estimation are often the limiting factors.In the ideal case, where the estimation errors are Gaussian, methods that rely on them work well.However, if the estimation process fails, e.g., due to ambiguities or outliers, then methods that rely on them quickly degrade.It should be mentioned that altering the DOA estimation (e.g., changing the parameter testing space or granularity or using more snapshots M) significantly impacts the performance of the spoofing detectors.Therefore, the robust design of the estimators is crucial.

B. Partially Spoofed Scenario
Fig. 8 shows the results for a partially spoofed constellation.For these constellations, half of the satellites are authentic, and the other half are spoofed.
None of the methods can distinguish the authentic and partial constellations from each other for the four-element antenna array.In the other scenarios, the "Blind" and "Arc" are barely above the random line, indicating that these methods struggle with partial constellations.The "SSE" method performs well for low probability of false alarm but then dips below the random line for high probability of false alarm.Finally, the "ASV-T" and "ASV-E" perform well for five to seven antenna elements (AUCs between 0.645 and 0.874).However, it is significantly lower than for a completely spoofed scenario (compare Fig. 6).It shows that the newly presented methods are superior for partial constellations.
These results show that partially spoofed scenarios are more challenging to detect but possible.Iterative receiver autonomous integrity monitoring (RAIM)-like approaches [7], [20] to identify a subset of spoofed scenarios were not considered in this article but could improve the partial constellation detection.

VII. CONCLUSION
This article introduces a spatial spoofing detection method that uses the estimated ASV from the DOA estimation process for an antenna array.Results show that the new method has comparable performance to the current state-of-the-art detectors, illustrating a valuable alternative for spoofing detection.It demonstrates that the degradation caused by attitude estimation is countered by using the ASV and the correct error modeling.However, if the attitude is known, such as in a fixed-installed reference station, the performance significantly improves.
A more extensive systematic evaluation of the method, including a theoretical foundation, is proposed for future research.Several extensions, such as evaluating the effect of constellation size, are possible.Live tests with a real antenna array are proposed to validate the results.Furthermore, other real-world effects, such as multipath, impede spoofing detection.

Fig. 1 .
Fig. 1.Spoofing scenario description with the geometry of a single spoofing transmitter versus a constellation of global navigation satellite system (GNSS) satellites.The resulting authentic sky plot (left) and a spoofed sky plot (right) are shown at the bottom.

Fig. 2 .
Fig. 2. Flow diagram of different detectors (blue) with relation to the signal preprocessing (red), i.e., correlation and DOA estimation, and detector processing (green), i.e., attitude estimation and arc calculation.
H 0 : Only authentic signals and noise are received.2) Alternative hypothesis H 1 : Only spoofed signals and noise are received.

Fig. 6
Fig. 6 compares the detector ROC curves.The legend displays the AUC for each ROC curve.With four elements [see Fig.7(a)], all detectors have poor results.This is caused by the high noise level and the antenna array ambiguities.If larger antenna arrays are used, the performance significantly improves for all detectors.It emphasizes the limitation of the four-element antenna array ambiguities.With five elements, the ASV detector with the theoretically known reference ASV ["ASV-T" with AUC = 0.954, see (49)] has the best ROC of all detectors.However, this is not a fair comparison, as it is a clairvoyant detector that uses the theoretically perfect ASV as a reference.Therefore, it is expected to have superior performance.However, if the estimated ASV after attitude

Fig. 8 .
Fig. 8. ROC on a linear scale for 22 dBHz for different antenna array sizes for a partial constellation.(a) Four elements.(b) Five elements.(c) Six elements.(d) Seven elements.