Abstract
New optimization-based methods have been developed to use measured direction-of-arrival (DoA) information in order to classify received global navigation satellite system signals into authenticated and spoofed sets and to augment that information with pseudorange information when DoA information alone is insufficient to achieve the needed classification. These methods are designed for a system that is being developed to mitigate spoofing and jamming by using signals from a controlled radiation pattern antenna. These new spoofing classification methods operate on DoA outputs from trackers of various signals. This paper presents a multi-hypothesis test that considers all possible hypotheses regarding the authenticated and spoofed sets of tracked signals. A combinatorial analysis is performed in which all possible authenticated-set/spoofed-set classifications are generated for a given set of tracked signals and the correct authenticated set is determined among the different combinations. Results from Monte Carlo simulations show that using a combined DoA and pseudorange method is suitable for determining the correct combinations.
1 INTRODUCTION
For global navigation satellite system (GNSS) receivers to be resilient, they must counteract two main threats: jamming signals and spoofed signals. One method for creating resilient GNSS receivers is to use a multi-element antenna array. This paper starts by analyzing a particular multi-element antenna array developed by Toyon, Inc., and later generalizes this analysis to any multi-element antenna. The Toyon antenna array is a controlled radiation pattern antenna (CRPA) system that contains seven patch antennas, each with a right-hand circular polarization (RHCP)-sensitive and a left-hand circular polarization (LHCP)-sensitive feed. The patches are oriented in a “bug eye” shape. An image of a candidate antenna, like the one considered in the present paper, is shown in Figure 1. In this paper, it is assumed that the direction-of-arrival (DoA)-dependent transfer functions are known for all antenna feeds and for their radiofrequency (RF) front-end electronics for both RHCP and LHCP incident signals. The transfer function calibrations used in the analysis in this paper have been measured in an anechoic chamber.
A multi-element antenna array of the type considered in this paper
Note that the methods presented in this paper apply to any CRPA for which calibrated transfer functions are available for each feed with known DoA dependence (Esswein & Psiaki, 2018). There is no need to use the particular Toyon CRPA to which some of these analyses apply, nor is there a need to use a CRPA that includes LHCP feeds. Of course, when using the methods presented herein, the performance of different CRPAs will vary with the properties of the array. In general, an array with more elements is likely to achieve better resilience to spoofing relative to an array with fewer elements.
The methods developed in this paper assume that each signal is acquired and tracked. The signal tracking algorithms must perform well and may take the form of the vector tracking method described by Esswein and Psiaki (2018). This technique is suitable for applications that require high reliability, such as for ship and aircraft navigation and for autonomous vehicles. This paper focuses on developing anti-spoofing techniques for a multi-element antenna array in the presence of spoofed and jamming signals. These anti-spoofing techniques use optimization-based methods that operate on measured DoA information in order to classify received GNSS signals into authenticated and spoofed sets. The spoofing classification method presented in this paper operates on the DoA outputs from trackers of various signals and uses the trackers’ computed estimation error covariances for DoA estimates. Similiar to other works that discuss DoA-based methods (Appel et al., 2016; Konovaltsev et al., 2013a, 2013b; Lo et al., 2018; Marcos et al., 2018; Meurer et al., 2012, 2016; Psiaki et al., 2013, 2014; Rothmaier et al., 2019, 2021b), with DoA sensitivity achieved through the use of multiple antennas or antenna motion, the present method relies on the known DoAs in reference coordinates and performs attitude determination as part of its calculations.
Previous work has explored combining DoA methods with pseudorange residuals. Esswein and Psiaki (2019) discussed a combinatorial classification method using a combination of DoA methods and pseudorange residuals. This work is an extension of that previous work. Rothmaier et al. (2021a) discussed a framework for combining multiple metrics into a single metric using the generalized likelihood ratio test. One example, analyzed in this paper, combines a DoA metric with a pseudorange residual metric.
This paper makes three main contributions to the subject of developing receivers that are resilient to GNSS spoofing. The first contribution is the development of a new DoA cost metric and associated multi-hypothesis test that uses the whole set of acquired signals to differentiate between spoofed and authentic signals. The second contribution is the incorporation of pseudoranges into the cost metric along with the DoA metric to further enhance differentiation between spoofed and authentic signals. The third contribution is an investigation into the effectiveness of the new algorithms against a multi-transmitter spoofer with geometric separation of the transmitters.
The remainder of this paper is organized as follows. Section 2 discusses the presented DoA-based algorithm for sorting signals into authenticated and spoofed groups. It details the attitude parameterization used to denote the DoA of a signal in antenna coordinates, the DoA combination method utilized for sorting into authenticated and spoofed signal sets, the use of signal polarization in the form of phasor estimates in order to differentiate combinations with similar costs, and a generalized combination method based on DoA terms and pseudoranges. Section 3 details the multiple simulations developed in this paper and their results. In Section 4, we consider a relaxation of the core assumptions made in this paper. This section discusses the assumptions made for the combination method presented herein, with a subsection for each relaxed assumption. In Section 4.1, we allow more than one spoofed signal to exist for any given pseudorandom noise (PRN) code. In Section 4.2, we relax an assumption concerning the DoAs of the spoofed signals, considering the possibility of multiple DoAs being distributed among the set of spoofed signals. Finally, a summary and conclusions are presented in Section 5.
2 METHODS FOR SORTING RECEIVED SIGNALS INTO AUTHENTICATED AND SPOOFED SUBSETS
The anti-spoofing methods developed in this paper are utilized downstream of signal acquisition and tracking. A spoofer will typically boost its signal power in order to wrest control of the receiver’s tracking loops from the authentic Global Positioning System (GPS) signal. Smart spoofers will attempt to keep the signal power of each spoofed signal only slightly higher than that of its corresponding authentic GPS signal. Such a strategy makes it difficult to distinguish an authentic signal from a spoofed signal based on the signal power alone. Therefore, the methods presented herein require that the victim receiver acquire and track all available PRN-code signals, including multiple signals for any given PRN code if more than one signal is detected during acquisition. This paper assumes that the tracking algorithm for each signal is a combined delay/phase/polarization/DoA-locked loop that is implemented via Kalman filter techniques, as we previously reported (Esswein & Psiaki, 2018). In that work (Esswein & Psiaki, 2018), the Kalman tracking filter relies on a knowledge of the CRPA calibration in order to form estimates of the DoA-based states. Other tracking techniques could be used; however, they may require moderate changes to the basic techniques developed here. The Kalman filter for each signal relies on models of the polarization- and DoA-dependent manifold of each antenna feed. Analysis of this system indicates that it can track signal DoAs to degree-level accuracy (Esswein & Psiaki, 2018). The Kalman filter state for the tracking loop of a given signal is defined as follows:
where τ is the spreading code start time for a particular PRN code period, xRI, xRQ, xLI, and xLQ are the real and imaginary components of phasors for the RHCP and LHCP signal components, ω is the carrier Doppler shift, α is the carrier Doppler shift rate, xDoA and yDoA constitute a parameterization of the signal DoA, and vxDoA and vyDoA are the “velocities” of the DoA parameters. The index n indicates the particular signal being tracked.
The methods presented herein for signal classification are applied to the outputs of tracking loops. These methods make two assumptions:
The maximum number of spoofed signals for a given PRN code is one.
All of the spoofed signals arrive from a single direction.
In Section 4, each of these assumptions will be relaxed.
2.1 DoA Parameterization
The direction of an incoming signal is defined relative to a body-fixed antenna coordinate system via the state-vector elements xDoA and yDoA. The unit direction vector that defines the DoA of a given signal can also be parameterized by an azimuth angle, ψ, and a co-elevation angle, θ, as shown in Figure 2. Figure 2 also shows the body-fixed antenna coordinate system axes for the +X, +Y, and +Z directions.
DoA of a signal in antenna coordinates
The (negative) DoA unit direction vector pointing from the antenna to the n-th satellite can be defined in body-fixed coordinates using the corresponding azimuth and co-elevation angles as follows:
The rightmost term in this equation involves rn, which denotes the known (negative) DoA unit direction vector for the same signal in reference coordinates, often Earth-centered/Earth-fixed coordinates. This term also involves the 3 × 3 matrix A(q), which is an orthonormal rotation matrix that transforms from reference coordinates to the antenna body-fixed coordinate frame. The rotation matrix is a function of the 4 × 1 attitude quaternion q, which is subject to the following normalization constraint:
The vector filter state defined in Equation (1) uses an alternate DoA parameterization instead of ψ and θ because there exists a singularity in the azimuth/co-elevation space at θ = 0 radians. To ensure that the attitude parameterization is nonsingular for the co-elevation range of 0 ≤ θ ≤ θmax, the signal tracking filter of Esswein and Psiaki (2018) uses an alternate representation of the attitude that is similar to the Gibbs parameterization reported by Wertz (1978). The representation of Esswein and Psiaki (2018) differs from the Gibbs parameterization in that it is two-dimensional rather than three-dimensional and is scaled up by a factor of two. The new representation allows θmax to have an upper bound of π radians. The relationship between the tracking filter attitude representation in terms of xDoA and yDoA and the foregoing azimuth/co-elevation representation is shown below:
The (negative) DoA vector can also be written in terms of the DoA parameterization of the tracking filter:
The DoA-based signal classification technique presented in this paper requires two additional vectors that are orthogonal to bn and to each other:
and:
where the vectors
and:
The vector triad [bn,(cn / ‖cn‖), (dn / ‖dn‖)] is orthonormal and right-handed. The vectors cn and dn are useful for defining a negative log-likelihood cost function, which is important for DoA-based signal classification. The usefulness of these newly defined vectors will be demonstrated in the next subsection.
2.2 DoA-Based Distinction Between Authentic and Spoofed Signals
The classification of signals into authenticated and spoofed sets uses a combinatorial analysis that performs an exhaustive search through 2NPRN combinations, where NPRN is the number of unique satellite PRNs that have been successfully acquired and tracked. This method amounts to a multi-hypothesis test that considers all possible hypotheses regarding the authenticated and spoofed sets of tracked signals. Each hypothesis assigns all instances of all tracked PRN codes to either the authenticated set or the spoofed set. This analysis assumes that the receiver acquires and tracks only one or two instances of each PRN code/satellite. If two instances are tracked, then each considered combination assigns one of these signals to the authenticated set and the other to the spoofed set. If only one instance is tracked, then each considered combination assigns the instance to either the authenticated set or the spoofed set. The authenticated and spoofed sets are different for each considered combination.
After the signals have been divided into the authenticated and spoofed sets of a given combination, a cost is assigned to that combination based on two optimization problems. The first optimization problem focuses on the combination’s set of authenticated signals. Here, the measured DoAs in antenna coordinates and the known DoAs in reference coordinates are used to define a negative log-likelihood cost function that equals half the sum of the squares of the normalized fit errors between the measured and reference DoAs after transformation of the latter into antenna body coordinates using A(q), as in Equation (2). The optimization finds the attitude quaternion q that minimizes this cost function. The second optimization problem finds the minimum of another negative log-likelihood cost function defined by the measured DoAs of the combination’s assumed spoofed signals in antenna coordinates. This cost function equals half the weighted sum of the squared differences between the measured DoAs of the spoofed signals in antenna coordinates and an estimate of the common DoA of all spoofed signals, also given in antenna coordinates. The best estimate of the common DoA of the spoofed signals is the DoA that minimizes this cost function. The cost assigned to the given combination equals the sum of the minimum cost values associated with the solutions of its two negative log-likelihood estimation problems. The combination of authenticated and spoofed sets that produces the lowest total of these two costs is deemed to correctly identify the authentic and spoofed signals. The use of minimized negative log-likelihood cost functions to define the test statistic implies that the test is not Neyman–Pearson-optimal. A Neyman–Pearson-optimal test would need a priori distributions for the unknown free parameters and would require integration over these parameters in order to marginalize them out of the distribution. It is generally difficult to obtain reasonable a priori distributions and to perform the needed integrals, whereas determining optimal estimates is typically easier. In this context, “sub-optimal” indicates that, for a given probability of false alarm and a given signal power, the the probability of missed detection will not be as low for this test statistic as a Neyman–Pearson-optimal test statistic. Although the result is strictly sub-optimal relative to Neyman–Pearson optimality, this approach usually produces a reasonably small probability of missed detection because its use of a minimized negative log-likelihood statistic gives a result that is only slightly different from the Neyman–Pearson integrals, as the integrated probability distribution is highest near the optimal estimates and, therefore, contributes the most to the required integrals.
To determine the total cost for a given combination, the requisite cost functions for the authenticated and spoofed signals must be defined. The signals from the authenticated set are used to define an attitude determination cost function, whereas all of the signals from the spoofed set are used to define a spoofer-direction cost function.
The attitude determination problem for the authenticated signals is developed from the following DoA measurement model for the n-th signal:
where νxDoAn is the error of the signal tracking filter in its xDoAn estimate and νyDoAn is the error in its yDoAn estimate. The expression on the extreme left-hand side of this equation is a first-order Taylor series approximation of the expression shown in the middle. This measurement model equation is slightly non-standard in that the error terms νxDoAn and νyDoAn appear on the same side of the equations as the “measurement” term bn. Although this constitutes a vector equation with three components, there are actually only two pieces of scalar information because the vectors on both sides of the equation are unit vectors (only to first order in νxDoAn and νyDoAn on the extreme left-hand side), which implies that the length component of the equation carries no information.
This measurement equation can be transformed into a pair of scalar equations via multiplication of both sides by
The simple form of the left-hand side of this equation results from the orthogonality of the vector triad
This takes the form of a standard measurement model in which the 2 × 1 “measurement” vector is the [0; 0] vector on the left-hand side of this equation.
A negative log-likelihood cost function for the three-axis attitude can then be defined as half the sum of the weighted squares of the measurement errors νxDoAn and νyDoAn from Equation (12) summed over all of the satellites in the presumed authenticated set. The cost function takes the following form:
where Nsat is the number of signals of interest in the authenticated set. The matrix Rxyn is a 2 × 2 square root information matrix for the estimates xDoAn and yDoAn from the tracking filter for the n-th signal. The associated 2 × 2 computed covariance matrix of the extended Kalman filter for the estimation errors in these two state-vector elements is
By finding the optimal q that minimizes the cost in Equation (13), one finds the best-fit direction cosine matrix for the measurement models in Equation (12). Although JAtt constitutes a cost function for attitude determination, its more important role in this analysis is that of evaluating each proposed authenticated set. If all of the signals in a given proposed authenticated set are indeed authentic GNSS signals, then each bn will correspond closely to its respective reference coordinate vector rn after transformation into antenna body coordinates via A(qopt), with qopt being the attitude quaternion estimate that minimizes JAtt (q). The optimal attitude quaternion, qopt, should produce the lowest value of JAtt (q) relative to all other combinations. The solution for qopt minimizes J Att (q) using a Gauss–Newton method specially tailored to meet the normalization constraint on the quaternion given in Equation (3). A reasonable first guess for q can be found by applying Davenport’s q-method to the data in
The spoofed-signal DoA cost function assumes that every spoofed signal comes from the same direction. Let us suppose that rspf is the unit direction vector in antenna body-fixed coordinates of the common (negative) DoA of all of the spoofed signals. Then, the following DoA measurement model applies to the n-th signal if it is part of the spoofed set for a given combination:
Note that Equation (14) and Equation (12) are similar, despite the fact that Equation (12) applies to the case for authenticated signals and Equation (14) applies to the case for spoofed signals. The difference is that the term A(q)rn with known rn and unknown q is replaced by the unknown vector rspf in this equation. The vector rspf is subject to the following normalization constraint:
A negative log-likelihood cost function for the spoofed set can be defined as half the sum of the weighted squares of the measurement errors from Equation (14) summed over all of the spoofed satellites for a given combination. This cost function then takes the following form:
where Nspf is the number of signals of interest in the spoofed set. For all scenarios, Nsat + Nspf = Nsoi, where Nsoi is the total number of signals of interest. The conventions adopted in Equations (13) and (16) assume that the Nsoi signals of interest are ordered so that the authenticated ones constitute the first Nsat signals and the spoofed ones constitute the last Nspf signals. For each different hypothesis regarding the authenticated and spoofed sets, an appropriate reordering of signals will be needed.
A singular-value decomposition can be used to find the optimal rspf that minimizes the cost function in Equation (16) subject to the unit normalization constraint in Equation (15). The singular-value calculation takes the following form:
The (2Nspf) × 3 matrix on the right-hand side of this equation is the input to the singular-value decomposition, and the three matrices on the left-hand side are the outputs. Uspf is a (2Nspf) × (2Nspf) orthonormal matrix. Σspf is a 3 × 3 diagonal matrix with non-negative singular values on its diagonal in decreasing order so that (Σspf)11 = σ1 ≥ (Σspf)22 = σ2 ≥ (Σspf)33 = σ3 ≥ 0. Vspf is a 3 × 3 orthonormal matrix. The optimal value of the spoofer DoA unit direction vector rspf is equal to the third column of Vspf. The minimum cost is as follows:
The final cost γi for each combination i is then the sum of the optimal values of these two cost functions:
The combination with the lowest γ is deemed to be the correct combination of authentic and spoofed signals.
2.3 Polarization-Based Distinction Between Authentic and Spoofed Signals
There can exist cases in which an authentic GNSS signal has a DoA close to the DoA of the spoofed signals. In this scenario, it can be difficult to differentiate some of the lowest-score combinations from each other. To solve this problem, RHCP and LHCP phasor estimates can be used to choose between the lowest-score combinations. GNSS signals are RHCP signals; therefore, it can be assumed that a signal with significant non-zero LHCP phasor amplitude is either a spoofed signal or a multipath signal. The most likely combination is the one whose authentic signals all have LHCP phasor amplitudes of nearly zero.
Figures 3 and 4 present quadrature phasor components vs. in-phase phasor components of the same PRN, with Figure 3 corresponding to the authentic RHCP signal and Figure 4 corresponding to a spoofed signal that includes both LHCP and RHCP components. The likely authenticity of the signal in Figure 3 is readily apparent because the LHCP phasor points (orange) are clustered near the origin, whereas the RHCP phasor points (blue) lie far from the origin. It is not unusual for the quadrature components of the RHCP phasor points to be small for a GNSS signal, as in Figure 3, if the receiver is using a phase-locked loop that tracks the RHCP phase. In contrast, Figure 4 shows LHCP and RHCP phasor points that are all remote from the origin. In fact, the orange cluster of LHCP phasor points is approximately the same distance from the origin as the blue cluster of RHCP phasor points. Thus, both signal components have approximately the same power, signifying that the signal is linearly polarized and is likely a spoofed signal. Therefore, if the LHCP phasor magnitudes are sufficiently large relative to the RHCP phasor magnitudes, then the signal should be deemed to originate from a spoofer.
Phasors of RHCP and LHCP signal components for an authentic GPS signal
Phasors of RHCP and LHCP signal components for a spoofed GPS signal
The use of DoAs and phasor estimates provides a solution for determining the correct authenticated and spoofed combinations. However, the efficacy of using polarization differentiation relies on a number of assumptions. The first assumption is that the spoofer antenna is not RHCP. A spoofed antenna that is RHCP would produce a signal with small LHCP phasors, similar to authentic GPS satellite antennas. A second assumption is that the RHCP and LHCP responses of the antenna elements are calibrated for all relevant elevation angles. This compensates for the reduced ability of a patch antenna to distinguish between polarizations at low elevations in the model. The non-planar antenna array modeled in this paper mitigates this lowered ability because at least one antenna will view the signal at a higher elevation. A third assumption is that the environmental noise is primarily due to thermal noise. The in-phase and quadrature plots show signals from an environment in which all noise present is due to thermal noise. Therefore, these plots represent ideal cases and do not reflect phenomena such as multipath, which can increase the LHCP phasors of a signal. Thus, although polarization differentiation is a useful tool to help differentiate the lowest DoA combinations with similar magnitude costs, it may not be a sufficiently reliable method.
2.4 Joint DoA- and Pseudorange-Based Distinction Between Authentic and Spoofed Signals
There is another possible strategy for deciding between multiple proposed authenticated/spoofed combinations with similarly low values of the DoA-based classification statistic γi from Equation (19). This strategy uses pseudorange-based cost functions for the authenticated and spoofed sets in each combination. The correct combination should have an authenticated set with a low weighted sum of squares of its pseudorange residuals after solving for its authenticated navigation solution. The cost function is then as follows:
where Pn is the measured pseudorange corresponding to the n-th signal of interest, σn is the standard deviation for the measured pseudorange, and
The signals in the spoofed set of a given combination can also be used to produce a navigation solution, the false solution that the spoofer wants the victim receiver to accept. There are two cases in which a spoofer would produce a non-consistent set of signals and, therefore, a poor false solution. In the first case, the spoofer does not create self-consistent signals. In the second case, the receiver is the victim of multiple spoofers, where each spoofer is consistent with itself, but not with the other spoofers. The authors have chosen not to consider either case for this work. The classification algorithm can also produce a pseudorange cost function for the spoofed set that minimizes the sum of squares of its pseudorange residuals. This cost function takes the following form:
The
The ephemerides used to compute the navigation solution for both cost functions are from the GPS navigation message decoded from the tracked signals. If the signal is authentic, then we obtain the authentic version of the transmitted ephemeris data. If the signal is spoofed, then we obtain a spoofed version of the transmitted ephemeris data.
Given the two optimized pseudorange cost values in Equations (20) and (21), the new multi-hypothesis test statistic that includes both the DoA and pseudorange costs is as follows:
Suppose that the spoofed signals all come from a common DoA, that the spoofed signals have been designed to produce a consistent navigation solution, and that the spoofed solution differs appreciably from the authenticated navigation solution. Then, the lowest value of this test statistic should be able to distinguish the combination that correctly splits its received signals into authenticated and spoofed sets. This correct splitting capability should be possible even if some of the authentic signal DoAs are very near the common DoA of all of the spoofed signals.
3 SIMULATION & RESULTS
Three separate simulation scenarios have been developed in order to demonstrate the different capabilities of the methods presented in the previous section. For the first simulation, a truth-model simulation has been used to synthesize a set of signals at the outputs of the CRPA’s RF front-ends, which each contain five authentic RHCP GPS coarse acquisition code signals, with each authentic signal having a different DoA. Along with the authentic signals, four spoofed signals transmitted from a single linearly polarized antenna and five jammers are added to the simulation. Each jammer has its own distinct DoA. The PRNs for the spoofed signals do not necessarily correspond to the PRNs for the authentic signals. Additional signal noise is modeled as wideband thermal noise with a flat power spectral density. The simulation includes the impact of DoA-dependent transfer function models of all of the antenna feeds on phase, amplitude, and PRN code distortion. The simulation models a 14-feed antenna system mounted on a vehicle, with the vehicle rotating at a rate of 20°/s. This simulation models the effects of anti-jam capabilities presented in our previous work (Esswein & Psiaki, 2018), which rely on blind pre-correlation nulling of jammer signals that uses a frequency-binned approach.
Figure 5 shows the results of all of the total costs, γ, for each of the 2NPRN combinations. Note that NPRN is the number of unique PRNs found amongst the tracked signals, not the number of tracked signals themselves, and can be less than the total number of tracked signals. In this case, NPRN equals 6, which implies that there are 64 combinations in total. This simulation uses the purely DoA-based hypothesis test statistic. Therefore, the total cost γi for each combination is computed using Equation (19). From Figure 5, it is clear that one of the combinations has a significantly lower cost than all of the others by more than three orders of magnitude. This combination also corresponds to the combination with all of the authentic GNSS signals in the authenticated set and all of the spoofed signals in the spoofed set. This case shows that this authentication method can be utilized as part of a resilient position, navigation, and timing system for a GNSS receiver in the presence of both spoofed and jammed signals.
Plot of the optimal-cost-based hypothesis test statistic for each combination
The first simulation shows that this method can successfully select the correct combination using estimates computed downstream from vector tracking. However, this scenario corresponds to only one set of authentic directions with one spoofed direction. Thus, performing a few more cases would not give a full picture as to how well the algorithm can determine the correct combination for any group of authentic and spoofed signal directions. However, performing many simulations like this first one would be too expensive because of the need to simulate high-bandwidth RF front-end outputs for each antenna feed of the CRPA.
Therefore, a second simulation that is much simpler has been developed. This simulation is sufficiently simple that a 1000-case set of Monte Carlo simulations can be run. The number of satellites tracked, for each Monte Carlo run, is determined by the number of satellites visible at the receiver location and the scenario time, with an elevation mask of 10°. Different locations will track a different number of satellites. The number of authentic signals can range from 5 to 12. For each simulated authentic signal, there will be a corresponding simulated spoofed signal. Although this simulation assumes the GPS constellation only, this analysis would also work for multiple GNSS constellations. Multiple constellations, however, would increase the number of satellites and would thus affect the computational workload and detection reliability. The scenario time is held constant for all Monte Carlo cases. This simulation generates authentic signal directions based on an actual receiver location and the ephemerides of an example GPS constellation and randomly generates one spoofed signal direction for all of the spoofed signals. A type of randomness is generated in the DoAs of the authentic signals by randomly generating different “truth” receiver locations for the 1000 different Monte Carlo cases. These signal directions are used to produce “truth” values for tracking filter outputs xDoAn and yDoAn for each of the authentic and spoofed signals. Random measurement noise values for νxDoAn and νyDoAn are generated using a random number generator and are added to the corresponding truth values of xDoAn and yDoAn in order to produce simulated noisy DoA measurements for each of the authentic and spoofed signals. Each simulated measurement noise vector [νxDoAn;νyDoAn] is sampled from a zero-mean Gaussian distribution with covariance equal to
Figure 6 presents a histogram of the ratio between two DoA-only hypothesis test statistics from Equation (19) selected from the 2NPRN combinations for each of the 1000 Monte Carlo runs. This ratio takes the form of
Histogram of the ratio between the true combination cost and the lowest false cost for the DoA-only case
To test the robustness of the algorithm against this challenge, the simulation was modified to no longer randomize the spoofed direction. Instead, the spoofed DoA was set to one of the authentic directions. Figure 7 displays a histogram of the test statistic ratios for this new test. One can see that many cases produce ratios greater than one. This result indicates that the DoA-only method cannot reliably determine the correct authenticated and spoofed signal sets if there is only a small separation between the DoA of any of the authentic signals and the common DoA of all of the spoofed signals.
Histogram of the ratio between the true combination cost and the lowest false cost for the DoA-only case with an authentic signal direction that aligns with the common direction of the spoofed signals
The same Monte Carlo cases that were used to generate Figures 6 and 7 were re-used to evaluate the method that also considers pseudorange terms in its multi-hypothesis test statistic, as in Equation (22). These simulated pseudorange measurements are determined by the ephemeris and a “true” user position for the authenticated signals and a “false” user position for the spoofed signals. The differences between the “true” and “false” positions have been set randomly and are on the order of 100 m.
Similar to Figures 6 and 7, Figures 8 and 9 present histograms of all of the cost ratios for each of the 1000 Monte Carlo runs with randomized spoofer DoAs (Figure 8) and authentic-signal-aligned spoofer DoAs (Figure 9), but Figures 8 and 9 are based on the DoA-plus-pseudorange test statistic. Figure 8 shows that inclusion of the pseudorange cost terms in the test statistic decreases the magnitude of the cost ratios. In contrast to the maximum ratio of 0.814 for the DoA-only case in Figure 6, the maximum ratio in Figure 8 is 0.237, notably smaller than the DoA-only value and, therefore, much further from a potential mis-identification of the authenticated and spoofed sets.
Histogram of the ratio between the true combination cost and the lowest false cost for the DoA-plus-pseudorange method when considering the 1000 Monte Carlo cases with randomized spoofer DoAs
Histogram of the ratio between the true combination cost and the lowest false cost for the DoA-plus-pseudorange metric with an authentic signal direction that aligns with the common direction of the spoofed signals
Figure 9 shows that by including pseudorange cost components in the metric, all of the correct combinations are chosen, even when the spoofer DoA aligns with one of the authentic signal DoAs, unlike the result obtained by using the DoA-only test statistic for the same cases, as shown in Figure 7, where many incorrect combinations are chosen. This result implies that the DoA-plus-pseudorange method achieves smaller ratios compared with the DoA-only method. It is important to note, however, that while all combinations are correctly identified in Figure 9, a number of ratios are still high, with one reaching approximately 0.99. These high values occur when the pseudorange and DoA for two signals of the same PRN are the same or nearly so. Having the same pseudorange for a single PRN does not mean that the true and spoofed user positions are the same. Rather, the two positions lie on a sphere of the same radius from the GPS satellite. The joint DoA-plus-pseudorange multi-hypothesis test is robust because it is aided by the pseudoranges when there is only a small angular separation between the direction of the spoofer and the direction of one or more authentic satellites, whereas it can rely on DoA separation when there is only a small position separation. The DoA-only method does not give accurate results when DoA diversity between the authentic signals and the spoofed signals is not maintained. Of course, it is possible that a particular authentic signal may have the same DoA and pseudorange as its spoofed counterpart. In this unlikely event, the DoA-plus-pseudorange method may also fail to identify the correct authenticated and spoofed signal sets.
4 RELAXATION OF SOME ASSUMPTIONS
The multi-hypothesis test statistics described in Section 2 rely on two assumptions in order to achieve the results shown in the previous section. However, it is possible to relax, or possibly eliminate, these assumptions by analyzing and/or modifying the multi-hypothesis test.
4.1 Generalized Combination Method
The first of the original assumptions permitted at most one instance of a spoofed signal for any given acquired and tracked PRN code. In this subsection, we allow the number of spoofed signals to be more than one per PRN code. Here, we assume that there are NPRN distinct acquired and tracked PRN codes. The m-th PRN code is assumed to have Dm distinct instances. The maximum number of distinct instances of all acquired and tracked PRN codes is Dmax, such that 1 ≤ Dm ≤ Dmax for all m = 1, …, NPRN and Dm = Dmax for at least one value of m. At most, one of the instances of the m-th PRN code is an authentic signal, and the other (Dm –1) signals are spoofed signals. It is possible that all Dm instances are spoofed signals.
The multi-hypothesis test for this scenario considers combinations with (Dmax + 1) sets of acquired and tracked PRN codes. The first set is the presumed authenticated set. The other Dmax sets are the presumed spoofed sets. Each combination assigns each acquired and tracked version of a given PRN code to one and only one of these sets. A total of (Dmax +1 − Dm) sets will have no instance of the m-th PRN code signal. The total number of unique combinations of such sets is as follows:
The Dmax! denominator in this formula accounts for the fact that the hypothesis test only counts a given combination of Dmax spoofed sets once without regard to its ordering in the list of spoofed sets.
The multi-hypothesis test statistic calculation for this scenario assumes that each of the Dmax spoofed sets has all of its signals arrive from a common DoA because they are all transmitted from the same antenna. The statistic also assumes that the signals of a given spoofed set produce PRN codes that are consistent with some spoofed navigation solution. Different spoofed sets are allowed to have different spoofer DoAs and different self-consistent spoofed navigation solutions.
The DoA-only multi-hypothesis test statistic calculation for a given combination proceeds as follows. The optimal value of J Att (q) is computed for the presumed authenticated set. For each of the Dmax spoofed sets, a unique optimal rspf that minimizes the corresponding value of the spoofer DoA cost function JSpf (rspf) from Equation (16) is computed. Note that the summation for each unique spoofed set in its corresponding JSpf (rspf) definition for a given combination is taken over a different set of acquired and tracked signals than for all of the other spoofed sets of that same combination. The multi-hypothesis test statistic for the i-th combination is then as follows:
where
The DoA-plus-pseudorange multi-hypothesis test statistic calculation for the i-th combination takes the following form:
where
It would be straightforward to develop a simulation that explores the merits of this proposed extension of these methods. Yet, no such simulation has been developed; therefore, no simulation results are presented for this extension.
4.2 Multiple-Direction Spoofer
In this paper, we assumed that the spoofer comes from only one direction and used the cost function in Equation (16) to determine how well the spoofed set fits to a single direction. A very sophisticated (and more expensive) spoofer might transmit its spoofed signals from multiple directions. Therefore, it is appropriate to characterize the robustness of both the DoA method as well as the DoA-plus-pseudorange method for a case with spoofer signals coming from multiple directions.
To measure authentic/spoofed classification robustness in the presence of multiple spoofed-signal DoAs, a new Monte Carlo simulation was performed, and its outputs were used to evaluate the robustness of the DoA-plus-pseudorange method. No evaluation of the original DoA-only method was performed, as it seems obvious that the DoA-only method will perform very poorly when spoofed signals arrive from multiple directions. A modified DoA-only method is evaluated for this challenging situation. In this simulation, the spoofed signals come from two different directions instead of one. The distance between the true and spoofed user positions is no longer randomized, but is set to a particular value. Figure 10 shows the number of incorrect combinations for the DoA-plus-pseudorange method out of 1000 total Monte Carlo cases for varying separations between the true and spoofed user positions. As the distance between the two positions increases, the number of incorrect combinations decreases until reaching zero at a separation of 100 km. This separation is unrealistically large, indicating that the DoA-plus-pseudorange method is not particularly robust to multi-directional spoofing attacks. This lack of robustness arises from the DoA cost function of the spoofed combination, which explicitly assumes that there is a single spoofing direction. An incorrect combination is chosen when an authentic direction is close to one of the spoofed directions, similar to the problem above, except that now the cost of the correct combination has a large spoofed cost component. The pseduorange cost functions must be able to offset this increased cost by strongly penalizing incorrect combinations. This pseudorange-based compensation is effective only when the authentic and spoofed positions are sufficiently far apart.
Dependence of the number of incorrect multi-hypothesis decisions for a multi-directional spoofer with varying position separation when using the DoA-plus-pseudorange method
Because this issue arises from the spoofed DoA cost function, a promising option is to de-weight this function in the overall test statistic. A modified version of the DoA-plus-peudorange test statistic from Equation (22) uses a positive de-weighting factor, β. This factor is chosen to be a value in the range of 0 < β < 1. With this factor, the test statistic from Equation (22) takes the following form:
Similarly, the DoA-only test statistic of Equation (19) can be transformed into the following form in order to de-weight the contribution of the spoofed DoA term:
A 1000-case Monte Carlo simulation was performed to evaluate the weighted DoA-only test statistic of Equation (27) and the weighted DoA-plus-pseudorange test statistic of Equation (26). The simulation uses a separation of 100 m between the true and spoofed positions. Figures 11 and 12 show histograms of the test statistic ratios for each Monte Carlo case for the weighted DoA-only metric of Equation (27) (Figure 11) and the weighted DoA-plus-pseudorange metric of Equation (26) (Figure 12). In both cases, the weighting value is set to β = 10−5. Both weighted test statistics perform better than the un-weighted DoA-plus-pseudorange test statistic, shown in Figure 10, in determining the correct combination. However, one case that uses the weighted DoA-only test statistic still chooses the incorrect combination, as indicated by one ratio in Figure 11 being larger than unity. The weighted DoA-plus-pseudorange test statistic identifies the correct combination for all cases. Its highest ratio is below 0.16 in Figure 12, indicating the robustness of this method. This result is significant because it demonstrates that a weighted version of the DoA-plus-pseudorange metric provides this method with the ability to correctly identify authentic and spoofed signals with some level of robustness to multi-directional spoofers. This advantage is obtained without increasing the total number of combinations that must be tested. The effects of using the same β parameter were tested using the same Monte Carlo cases used in Figures 6 and 8. Recall that in these cases, the spoofed signals all come from a common direction. This test was carried out to verify whether this new method can be utilized against single- and multi-direction spoofers without a priori knowledge regarding which type of spoofer is mounting the attack. The corresponding highest ratios for these runs, to three significant digits, were 0.927 and 0.250, respectively. These values represent an increase from the highest ratios in Figures 6 and 8. It is important to note, however, that both runs using this β parameter determined the correct authentic and spoofed sets. Moreover, the increase in ratios was significantly lower for the DoA-plus-pseudorange than the DoA-only test statistic.
Histogram of the ratio between the true combination metric and the lowest false metric for the weighted DoA-only metric for the situation in which spoofed signals arrive from two distinct directions
Histogram of the ratio between the true combination metric and the lowest false metric for the weighted DoA-plus-pseudorange metric for the situation in which spoofed signals arrive from two distinct directions
5 SUMMARY AND CONCLUSIONS
This paper has developed and performed simulations to assess two anti-spoofing multi-hypothesis tests for multi-element antenna arrays. The first method uses only DoA data to distinguish between authentic and spoofed signals. This method seeks an authenticated set of signals that has the diversity of DoAs expected based on satellite geometry and the receiver’s navigation solution. This method also seeks a spoofed set of signals that lack diversity in their DoAs. The second method utilizes the DoA data criteria while also considering the pseudorange data of the received signals. This second method also seeks self-consistent sets of measured pseudoranges for both the authenticated and spoofed sets based on their respective navigation solutions. The performance of these methods has been studied using a variety of criteria. The first criterion is based on the ability of the method to determine the correct combination of authenticated and spoofed signal sets regardless of the direction separation between the authentic GNSS signals and the spoofed signals. In this case, the combined DoA-plus-pseudorange approach succeeded in determining the correct combinations whereas the DoA-only method performed poorly if one (or more) authentic signal DoA was near the common DoA of the spoofed signals. Another criterion is based on the robustness of the method to edge cases, particularly the case of multiple spoofed signal directions. For the case in which the spoofer is sophisticated (and expensive) enough to split its false signals into two subsets such that it transmits from two distinct DoAs, the combined DoA-plus-pseudorange method was able to distinguish the correct authenticated and spoofed sets, given a sufficient position separation or by de-weighting the spoofed DoA component of the test metric. The DoA-only method of signal classification experiences a high failure rate when there are two spoofer directions, regardless of the scenario. For the case in which there are multiple spoofers for a given PRN, the DoA-plus-pseudorange method could be extended to search for the correct combination of authenticated signals and multiple sets of spoofed signals. Thus, the combined DoA-plus-pseudorange method provides a robust approach for distinguishing authentic GNSS signals from spoofed signals for applications that require high reliability.
HOW TO CITE THIS ARTICLE
Esswein, M. C., & Psiaki, M. L. (2025). Classification of authentic and spoofed GNSS signals using a calibrated antenna array. NAVIGATION, 72(1). https://doi.org/10.33012/navi.675
ACKNOWLEDGMENTS
This work was supported in part by the Toyon Research Corporation under subcontract number SC17-C023-1on Army Phase II STTR prime contract W56KGU-17-C-0013. The Toyon contract monitor was Kenan Ezal.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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