CN116719061A - GNSS induced deception detection method based on RMS sliding envelope and SVM - Google Patents

Abstract

The invention discloses a GNSS induced deception detection method based on an RMS sliding envelope and an SVM, which comprises the steps of firstly capturing and tracking a visible satellite from a GNSS intermediate frequency signal by using a software receiver, and taking IQ data, a carrier-to-noise ratio C/N0 and Doppler frequency shift of an ELP tracking loop as original characteristics; then setting window length, calculating extreme points of the original features, and interpolating the maximum points and the minimum points by using a cubic spline interpolation method to obtain upper and lower envelope curves of the original features; calculating the sliding root mean square of the envelope curve as a new processed characteristic; after normalizing the new features, performing feature dimension reduction by using principal component analysis; and finally, dividing the features into a training set and a testing set, and inputting a testing sample after training a support vector machine to obtain the most classified result. The invention reduces the data redundancy and the operation time, is easy to realize, has low complexity, has low requirements on the receiver, greatly reduces the equipment cost of the receiver, and has wider application scenes.

Description

GNSS induced deception detection method based on RMS sliding envelope and SVM

Technical Field

The invention relates to the technical field of satellite navigation interference detection, in particular to a GNSS induced fraud detection method based on an RMS sliding envelope and an SVM.

Background

Global satellite positioning systems (GNSS) have provided location, velocity and time (PVT) information for residential, commercial and military users since 1994, and each major nation has been developing their own GNSS systems, the beidou BDS of china, the GPS of the united states, the Galileo of europe, and the GLONASS of russia and known as the four large GNSS systems of the world today, many components of critical infrastructure and widely used applications have begun to rely on the continued availability of PVT information after the global positioning system has entered the commercial and civilian markets, and will have significant impact or even destroyed the global industry once the global navigation satellite system suddenly shuts down or is disturbed to provide erroneous time-location information.

The ever-increasing complexity of the electromagnetic environment and the dramatic changes in international form present serious challenges for satellite navigation security applications, in civilian GNSS systems, navigation signals are very weak in power over long distances and are susceptible to fraud because the signal structure is public. With the development of scientific technology, the development of integrated electronics technology, sensor technology and radio technology, the deception jamming becomes easier to realize, the cost is lower, the operation is more flexible, and the navigation deception must be detected, weakened or eliminated. The detection of fraud is critical, and the satellite navigation fraud can be further reduced or eliminated only if it is detected correctly, or the location of the source of fraud can be further determined to be removed.

In recent years, a plurality of solutions are provided for GNSS spoofing detection at home and abroad, including various methods such as absolute power monitoring, relative power monitoring, correlation peak detection method, signal angle of arrival detection method, time consistency detection, autonomous integrity detection (RAIM), navigation and time information fusion detection. The absolute power monitoring deception jamming detection technology is to set a threshold value for the power of a signal received by a receiver, and when the power of the received signal exceeds the set threshold value, the signal can be confirmed to be deception, but the method can cause false alarm in judgment due to factors such as antenna type, antenna gesture, multipath influence and the like. The principle of signal angle of arrival monitoring is based on the assumption that the rogue interferer is a single antenna interferer, the direction angle of the transmitted rogue signal at the receiver antenna is the same. By monitoring the angle of arrival, detection of the angle of arrival of the received satellite navigation signal can be performed using multiple antennas of the receiver, and whether the angle of arrival is the same can be determined to identify rogue interference. The basic principle of time consistency monitoring is as follows: most of the information carried by the spoofed signal and the time of the real signal cannot be synchronized, and the spoofing device can bring a certain doppler shift to the spoofed receiver, so that the shift is monitored by a local high stability and high accuracy clock, and the time difference between the observed time and the satellite signal ephemeris time is compared, and the method is mainly aimed at generating spoofing interference. The basic principle of autonomous integrity monitoring (RAIM) of a receiver is that a sampling detection method is adopted to realize deception detection on received satellite signals, the receiver determines that at least 4 satellite signals are needed for self position, then 4 satellite signals are randomly extracted for multiple times from all received signals, the position of the receiver is calculated, and whether deception interference exists in the signals is judged through multiple times of comparison of the calculated positions.

The method and the technology are difficult to popularize because the additional hardware equipment is arranged to obtain good detection performance, but the economic cost or the implementation difficulty is increased, and the deceptive interference detection technology based on information calculation is difficult to popularize because the algorithm is used for judging after the data analysis processing of the receiver is acquired, and is a research focus of the current deceptive interference detection technology without changing a satellite navigation signal system and installing additional equipment.

Disclosure of Invention

Based on the technical problems in the background art, the invention provides a GNSS induced fraud detection method based on an RMS sliding envelope and an SVM.

The invention provides a GNSS induced deception detection method based on an RMS sliding envelope and an SVM, which comprises the following steps:

step 1: capturing visible satellites from GNSS intermediate frequency signals received by a radio frequency front end by using a GNSS software receiver, tracking the captured visible satellites, calculating SQM characteristics and carrier-to-noise ratio C/N0 based on IQ branch outputs of advanced, instant and lagged loops in a tracking stage, and simultaneously estimating Doppler frequency shift, wherein the calculated 5 characteristics are used as original characteristics;

step 2: after calculating sliding RMS envelope for the 5 features in the step 1, using mean variance normalized features and principal component analysis to reduce dimensions, and taking the processed data as training data of new features;

step 3: training the SVM classifier through the training data obtained in the step 2, and obtaining a trained SVM model after training is finished; judging whether a deception signal exists or not through the trained SVM model; comparing the predicted label with the real label, calculating various evaluation indexes, and adjusting parameters to optimize the performance of the model;

step 4: and (3) detecting signals received by the GNSS receiver through the trained SVM model in the step (3), and completing GNSS deception jamming detection based on the RMS envelope and the SVM in signal capturing and tracking.

The detection steps of the invention are as follows:

(1) Processing and storing GNSS intermediate frequency signals received by the radio frequency front end;

(2) Capturing visible satellites therefrom using a GNSS software receiver;

(3) Tracking and calculating SQM characteristics of the j-th visible satellite: including Ratio, delta and ELP:

(3.1) calculating the in-phase and quadrature branch outputs of the lead, immediate, and lag loops of the tracking loop, denoted as I _E 、I _P 、I _L And Q _E 、Q _P 、Q _L ：

(3.2) calculating the SQM feature of the j-th satellite:

wherein I is _E 、I _P 、I _L In-phase components, Q, of the outputs of the leading, immediate, lagging code correlation branches, respectively _E 、Q _P 、Q _L The quadrature components of the outputs of the early, immediate, and late code correlation branches, respectively.

(4) Estimating carrier-to-noise ratio C/N0 of the j-th satellite:

(4.1) calculating an average normalized value of wideband power and narrowband power from the in-phase and quadrature branch outputs calculated in step (3.1):

where K is the number of correlators, T is the bit duration of the navigation data, and M is the number of correlation integrator outputs.

(4.2) C/N0 can be estimated from the average normalized value of the wideband power and the narrowband power:

(5) Estimating the Doppler shift of the j-th satellite:

wherein f _s For the carrier frequency of satellite transmissions, V _s The tangential velocity of the satellite is C, the propagation velocity of the signal is C, and A is the angle between the vector diameter of the user to the satellite and the tangential velocity vector of the user to the satellite.

(6) Calculating the RMS sliding envelope of the original features of the j-th satellite obtained in the steps (3) - (5), including Ratio, delta, ELP, C/N0 and f _d ：

(6.1) obtaining the maximum points of all the original features, and marking as I _m ；

(6.2) pair I _m Interpolation is carried out on the maximum value point set in the model number to obtain an upper envelope curve;

(6.3) obtaining the minimum value points of all the original features, and marking as I _n ；

(6.4) pair I _n Interpolation is carried out on the minimum value point set in the model to obtain a lower envelope curve;

(6.5) the length w of the sliding window and the sliding step μ;

(6.6) calculating the sliding RMS of the envelope curves in steps (6.2) and (6.4). The input discrete sequence X (k) sliding RMS calculation formula is:

(7) Repeating steps (3) - (6) until the sliding RMS envelope of all captured satellites in view of step (2) is calculated;

(8) Taking all the processed data in the step (7) as new characteristics, and preprocessing:

(8.1) normalized preprocessing of data using mean variance:

(8.2) feature selection of the normalized data using principal component analysis to reduce data dimensionality while ensuring that data information is not lost, and checking the quality of the combined features. The re-projection can be expressed as follows:

wherein a is _ij Is constant, n is projection spaceDimension b of (b) _ij Is one of iterative processesGroup constant, iterating until the error between the reprojection feature and the original feature is minimum, PC _i Is a linear combination of the original features:

(9) Parameters of the SVM are set and the data set is divided.

(9.1) the support vector machine constructs a hyperplane for classification by mapping the raw training data into a multidimensional space, the hyperplane being constructed to divide the different classes. In the SVM parameter setting, C is a penalty coefficient, which is understood as a weight for adjusting the preference of two indexes (interval size, classification accuracy) in the optimization direction, that is, the higher C is, the more intolerant the error is, the easier the fitting is, the smaller C is, the easier the fitting is, the bigger or smaller C is, and the generalization capability is poor. Setting C to 0.98;

(9.2) selecting an RBF function as kernel;

(9.3) after gamma is selected as a kernel, a parameter carried by the function implicitly determines the distribution of the data mapped to a new feature space, the larger the gamma is, the smaller the support vector is, the smaller the gamma value is, the more support vectors are, the number of the support vectors influences the training and predicting speed, and the gamma is set to be 0.2;

(9.4) when the data set was divided, 70% was used as training data, and the remaining 30% was used as test data.

(10) Training and evaluating the model.

(10.1) training the SVM on the training set after setting the parameters and dividing the data set according to step (9);

(10.2) subsequently evaluating the model on the test set. After inputting the test sample, the SVM outputs a predictive label of 1 or-1, where 1 represents the hypothesis H ₁ I.e., the detected sample is a spoofed GSNN signal; -1 represents hypothesis H ₀ I.e. the detected sample is a normal signal and is not attacked by GNSS spoofing.

Compared with the traditional SQM detection method, the invention reduces the redundancy of data by extracting the upper envelope and the lower envelope, so that the data distribution of the deception signal and the real signal is more discrete, the detection rate is greatly improved, fewer characteristics and less calculation time are used, and better detection performance is achieved.

Drawings

FIG. 1 is a flow chart of an example of an implementation of the present invention;

FIG. 2 is a schematic diagram of a model of satellite navigation system induced fraud and detection;

FIG. 3 is a diagram of correlation peak search results of a navigation satellite software receiver based on an FFT algorithm acquisition process;

FIG. 4 is a schematic diagram of the frame of the tracking loop and IQ data acquisition;

FIG. 5 is a diagram showing the true ELP signal and the rogue ELP signal and the data distribution for the visible satellite GPS-03;

FIG. 6 is a graph of the true ELP signal and the rogue ELP signal of GPS-03 and the data distribution for different window lengths;

FIG. 7 is a graph of ROC of GNSS fraud detection at different window lengths;

FIG. 8 is a graph of F1 score and time-consuming statistics of GNSS fraud detection at different window lengths;

fig. 9 is a diagram of confusion matrices detected under different spoofing scenarios.

Detailed Description

The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments.

Referring to fig. 1-9, a GNSS induced fraud detection method based on an RMS sliding envelope and an SVM includes the steps of:

1. processing and storing GNSS intermediate frequency signals received by the radio frequency front end, capturing visible satellites by using a GNSS software receiver, and storing result data;

the step 1 specifically comprises the following steps:

1.1GNSS signals are transmitted in space and are incident on GNSS receiving antennas of users, the amplitudes are greatly weakened after attenuation and thermal noise interference, the frequencies are too high, and the radio frequency front end utilizes the combination of an amplifier, a down converter, a band-pass filter and a self-oscillator, so that the signals have higher amplitudes, lower frequencies and smaller bandwidths. The front end output of satellite number k is:

wherein, P, τ,respectively representing power, time delay and carrier phase, T _s Is the sampling interval, C, D is the time nT respectively _s Corresponding spreading code sequence and navigation data sequence, ζ (nT _s ) Is zero mean and variance sigma ² K is the PRN number of the satellite, f is the intermediate frequency to which the front end down-converts the carrier frequency;

since the signal structure of the spoofing interference is generated by imitating the real signal of the GPS, the spoofing signal received by the receiver is:

wherein the superscript a, s respectively indicates a true signal and a spoofed signal, and when the receiver is spoofed, the received signal is a combination of signals from a plurality of satellites:

wherein, gamma ^a And gamma ^s PRN sets of true and spoofed signals, respectively;

the cross-correlation function of the local code and the received hybrid satellite signal can be obtained by carrier stripping and coherent integration for 1ms, expressed as:

R(τ)＝R ^a (τ)+R ^s (τ)+ζ′t,τ)

wherein R is ^a (. Cndot.) represents the cross-correlation function of the real satellite signal with the local code; r is R ^s (-) represents the cross-correlation function of the spoofed signal with the local code; ζ' (t, τ) represents the cross-correlation result between the filtered gaussian white noise and other satellite signals and the local code;

the 1.2 receiver must know which satellites are visible to the user, so the satellite signal needs to be captured, the capturing phase mainly focuses on frequency and code phase, maximum value searching is carried out on correlation peaks of signal power, when the maximum value exceeds a set threshold value, the satellites are captured with corresponding frequency and phase and keep tracking, the block diagram of a tracking loop is shown in fig. 4, and in-phase and quadrature components of the output of leading, immediate and lagging code correlation branches of the tracking process can be expressed as:

where d is the correlator spacing, τ is the time delay, θ ₀ For initial phase, θ ₁ Is a phase delay;

2. the original features are calculated. The original features comprise CN0, doppler frequency shift and SQM features calculated according to the IQ branch output of the ELP tracking loop;

the step 2 specifically comprises the following steps:

2.1 calculating carrier-to-noise ratio, carrier-to-noise density ratio (C/N0) estimation is based on the ratio of the wideband power of the signal to its narrowband power for measuring the quality of the obtained GNSS signal:

wherein T is the bit duration of the navigation data, M is the number of outputs of the correlation integrator, mu _NP Is broadband power and narrowband powerThe average normalized value of the ratio is calculated as:

wherein: k is the number of correlators;

2.2SQM characteristic calculation, the general statistical index for SQM monitoring is Ratio, delta, ELP, and the specific formula is:

wherein I is _E 、I _P 、I _L In-phase components, Q, of the outputs of the leading, immediate, lagging code correlation branches, respectively _E 、Q _P 、Q _L The quadrature components of the outputs of the early, immediate, and late code correlation branches, respectively.

3. Calculating extreme points of original features, setting X (t) as a discrete sequence, and meeting the following conditions:

X(t)∈[x(t ₁ ),x(t ₂ ),...,x(t _n )]＝[x ₁ ,x ₂ ,…,x _n ],

t∈[t(1),t(2),…,t(n)]＝[t ₁ ,t ₂ ,…,t _n ]

x (t) has M maxima and N minima, corresponding sequence subscripts (I) _m ,I _n ) Between (T) _m ,T _n ) And the function value (U, V) is noted as:

I _m ＝[I _m (1),I _m (2),…,I _m (M)]

I _n ＝[I _n (1),I _n (2),…,I _n (N)]

4. interpolating the extreme points by using a cubic spline interpolation method, and calculating the sliding RMS of the characteristic envelope curve;

the step 4 specifically comprises the following steps:

4.1 interpolation function y Using cubic spline _i ＝f(x _i ) For each segment interval t _i ,t _i+1 ]Interpolation is carried out to obtain all minimum value points I excluding the end points _n The lower envelope curve of the interpolation section formed by the piecewise interpolation curves is a lower envelope curve excluding the left and right end points, and the upper envelope curve is obtained for all maximum point differences in the same way;

4.2 setting the window length, which determines the number of sample points of the sliding window, has an important influence on the smoothness and data distribution of the subsequent whole envelope. The window length is set according to the sample length.

4.3 calculating the sliding RMS of the envelope, wherein the calculation formula is as follows:

where w is the window length and μ is the sliding step size.

Fig. 5 shows the raw data and sliding RMS envelope data of the SQM feature ELP of the navigation satellite GPS-03.

5. Data preprocessing and creating a data set.

The step 5 mainly comprises the following steps:

5.1 data normalization, taking the sliding RMS envelope, the features were normalized using the following formula:

wherein, the liquid crystal display device comprises a liquid crystal display device,representing the expectations of the ith eigenvalue of the kth acquisition satellite, ±>Representing the standard deviation of the ith eigenvalue of the kth acquisition satellite.

5.2 feature dimension reduction, principal Component Analysis (PCA) is a technique that reduces the dimension of a dataset. The basic method is from the original spaceTo another space->The projection may be represented by the following equation:

wherein a is _ij Constant, PC _i Is the linear combination of original features, n is the projection spaceIs a dimension of (c).

Principal component analysis uses re-projection to check the quality of the combined features, which can be expressed as follows:

wherein b _ij As a set of constants in the iterative process, iterating continuously until the error between the re-projection characteristic and the original characteristic is minimum;

5.3 data set partitioning for training and evaluation of SVM classifier, 70% of the data set was used for training and 30% was used for evaluation of trained classifier.

6. Training and testing the SVM classifier.

The step 6 mainly comprises the following steps:

6.1 training an SVM classifier, constructing a hyperplane for classification by mapping the original training data into a multidimensional space, the hyperplane being constructed to divide different classes, the hyperplane with the largest distance between support vectors being set as the hyperplane ultimately used for classification, the data point closest to the hyperplane being called the "support vector", so that, in order to be able to classify with the smallest classification error, our goal is to maximize the distance between the edge data point and the hyperplane, in order to find the optimal hyperplane, the hyperplane can be defined as:

w ^T x+b＝0

wherein w is ^T Is the normal vector of the hyperplane, contains weights of different data points, X is the sample point defining the hyperplane, and b is the bias constant.

The dataset is represented by the vector z=z ₁ ,z ₂ ,…,z _m And corresponding label y _i ∈[-1，1]Composition, wherein y _i =1 is positive sample, y _i = -1 is a negative sample, further a decision boundary is defined, which should correctly divide all points into:

in making all z=z ₁ ,z ₂ ,…,z _m In the hyperplane of data separation, only one hyperplane capable of enabling the separation margin of two types of samples to reach the optimal hyperplane exists, and the optimal solution can be found through optimization:

6.2 in the SVM parameter setting, C is a punishment coefficient, and is understood as a weight for adjusting the preference of two indexes (interval size and classification accuracy) in the optimization direction, namely, the higher C is, the less tolerant of errors, the easier the overfitting is, the smaller C is, the easier the underfitting is, the oversized or undersized C is, and the generalization capability is poor. Setting C to 0.98;

6.3 selecting RBF function as kernel;

6.4 gamma is a parameter of the RBF function after the RBF function is selected as kernel, and the distribution of the data mapped to a new feature space is implicitly determined, wherein the larger the gamma is, the smaller the gamma value is, and the more the support vectors are. The number of support vectors affects the speed of training and prediction. Gamma is set to 0.2.

7. And evaluating the SVM classifier.

7.1 after training the SVM classifier, evaluating on a test sample set, and determining an evaluation index first. The evaluated indexes include area AUC (Area Under Curve) under ROC (Receiver operating characteristic curve) curve, accuracy, precision, recall, and F1 score. Wherein AUC is defined as the area enclosed by the coordinate axis under the ROC curve, the numerical value of the area is not more than 1, and the closer the AUC is to 1.0, the higher the authenticity of the detection method is; when the value is lower than 0.5, the authenticity is the lowest, and the application value is not provided; the accuracy is the proportion of the number of correctly classified samples to the total number of samples, the overall accuracy is represented, and the samples fail when unbalanced; the precision is the ratio of the number of correctly classified positive samples to the number of all classified positive samples, and only the precision of the positive samples is represented; the recall ratio is a ratio of the number of positive samples to all positive samples, which is correctly predicted, and represents the probability that fraud is not missed; f1 score measures both recall and precision. The calculation mode of the index is as follows:

7.2 input test samples, model output predictive labels, SVM output predictive labels 1 or-1, where 1 represents hypothesis H ₁ I.e., the detected sample is a spoofed GSNN signal; -1 represents hypothesis H ₀ I.e. the detected sample is a normal signal and is not attacked by GNSS spoofing.

And 7.3, comparing the predicted label y_pred in the step with the real label y_true to obtain the number of samples of correct classification and error classification, namely TP, TN, FP, FN, and substituting the number of samples into the formula in the step (7.1) to solve various indexes.

And 7.4, evaluating each index in the steps, adjusting parameter settings in the steps (6.2) - (6.4), and re-implementing the steps (7.1) - (7.3) until the model is trained to the optimal state.

8. Real-time samples were tested. After the trained model is determined to achieve better performance on the test set, the test is conducted on the sample received in real time, so that GNSS deception jamming is detected in real time.

The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.

Claims (8)

1. The GNSS induced spoofing detection method based on the RMS sliding envelope and the SVM is characterized by comprising the following steps:

step 1: capturing visible satellites from GNSS intermediate frequency signals received by a radio frequency front end by using a GNSS software receiver, tracking the captured visible satellites, calculating SQM characteristics and carrier-to-noise ratio C/N0 based on IQ branch outputs of advanced, instant and lagged loops in a tracking stage, and simultaneously estimating Doppler frequency shift, wherein the calculated 5 characteristics are used as original characteristics;

step 2: after calculating sliding RMS envelope for the 5 features in the step 1, using mean variance normalized features and principal component analysis to reduce dimensions, and taking the processed data as training data of new features;

step 3: training the SVM classifier through the training data obtained in the step 2, and obtaining a trained SVM model after training is finished; judging whether a deception signal exists or not through the trained SVM model; comparing the predicted label with the real label, calculating various evaluation indexes, and adjusting parameters to optimize the performance of the model;

step 4: and (3) detecting signals received by the GNSS receiver through the trained SVM model in the step (3), and completing GNSS deception jamming detection based on the RMS envelope and the SVM in signal capturing and tracking.

2. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 1, wherein in step 2, the RMS sliding envelope is calculated by: finding extreme points of all original features, wherein a maximum point set is recorded as I _m The minimum value point set is marked as I _n The maximum point represents the maximum boundary of the original feature change in a certain interval, and the minimum value represents the minimum boundary of the original feature change in a certain period.

3. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 1, wherein in step 2, the RMS sliding envelope is calculated by: interpolation is carried out on the subareas of the extreme point set, and the upper envelope curve and the lower envelope curve of the original characteristics can be obtained after the complete set is traversed.

4. A GNSS induced fraud detection method based on an RMS sliding envelope and an SVM according to claim 1, wherein in the step 2, the RMS sliding envelope X has a sliding RMS value:

where w is the length of the sliding window, μ is the sliding step size, and γ (i) is the root mean square of the ith window.

5. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 4, wherein the length w of the sliding window determines the flatness of the envelope and redundancy of the data, and the length of the sliding window is not more than 20% of the length of the original feature samples.

6. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 1, wherein in the step 1, the extracted feature parameters F= [ Ratio, delta, ELP, C/N0, F _D ]The calculation mode is as follows:

wherein I is _E 、I _P 、I _L In-phase components, Q, of the outputs of the leading, immediate, lagging code correlation branches, respectively _E 、Q _P 、Q _L Orthogonal components output by the lead, immediate and lag code correlation branches respectively; k is the number of correlators, T is the bit duration of navigation data, and M is the number of output of the correlation integrator; f (f) _s For the carrier frequency of satellite transmissions, V _s The tangential velocity of the satellite is C, the propagation velocity of the signal is C, and A is the angle between the vector diameter of the user to the satellite and the tangential velocity vector of the user to the satellite.

7. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 1, wherein the method comprises the steps ofIn the step 3, if there is a spoofing signal, the SVM outputs a predictive label of 1, otherwise, outputs a predictive label of-1, and corresponds to the binary hypothesis H respectively ₀ And H ₁ 。

8. The method for detecting GNSS induced fraud based on RMS sliding envelope and SVM according to claim 7, wherein the binary hypothesis is a hypothesis of presence or absence of GNSS satellite fraud expressed by the following formula:

H ₀ :

H ₁ :

wherein T is the detection statistic, ζ is the channel noise, the superscript a and s represent the true signal and the spoofing signal, respectively, and the subscript k represents the kth satellite.