GNSS receiver array anti-interference method combining multivariate variational mode decomposition and dictionary learning

Abstract

The invention belongs to the communication technology, and particularly relates to a GNSS receiver array anti-interference method combining multivariate variational mode decomposition and dictionary learning, which comprises the following steps: obtaining each intrinsic mode component after an array receiving signal is subjected to multivariate variational mode decomposition, and reconstructing a noise-free signal after effective mode selection; and carrying out dictionary learning by using a noise-free signal, reconstructing an interference signal according to a trained dictionary, and subtracting the reconstructed interference signal from an array receiving signal to realize suppression of the interference signal. The method can effectively relieve the influence of noise components on signals, and guarantees the interference recognition precision of a subsequent algorithm. And the adopted dictionary learning algorithm can effectively identify interference signals of known types.

Description

Technical Field

[0001] The invention belongs to communication technology and relates to satellite navigation, and in particular to a GNSS receiver array anti-interference method combining multivariate variational mode decomposition and dictionary learning. Background Art

[0002] The Global Navigation Satellite System (GNSS) provides users with positioning, navigation, and timing (PNT) services, and plays a huge role in the military and civilian fields. However, since navigation satellites are in high orbits, the transmitted signals have low power when they reach the receiver, and are easily interfered by radio frequency signals in complex electromagnetic environments such as cities. When satellite signals cannot be accurately captured, the positioning error will expand to several meters or even hundreds of meters. In order to ensure that the receiver can reliably process satellite navigation signals, it is necessary to enhance the receiver's anti-interference ability.

[0003] However, the prior art still has the following problems:

[0004] 1. The existing methods can improve the GNSS signal reception capability in interference scenarios, but when multiple interference signals exist at the same time, its anti-interference performance will be reduced;

[0005] 2. Most existing studies focus on scenarios with a single interference signal. The anti-interference algorithms proposed in existing studies cannot effectively suppress interference signals in scenarios where multiple types of interference signals coexist. Summary of the invention

[0006] In view of the problem that in scenarios with various types of interference signals, traditional overcomplete dictionaries cannot accurately reconstruct signals due to the fixedness of their parameters, the present invention proposes a GNSS receiver array anti-interference method that combines multivariate variational mode decomposition and dictionary learning, including: obtaining each inherent mode component of the array received signal after multivariate variational mode decomposition, and reconstructing a noise-free signal after effective mode selection; using the noise-free signal to perform dictionary learning, reconstructing the interference signal according to the trained dictionary, and subtracting the reconstructed interference signal from the array received signal to suppress the interference signal.

[0007] Furthermore, obtaining each intrinsic modal component after the array receiving signal is subjected to multivariate variational modal decomposition includes the following steps:

[0008] Extracting predefined K multivariate modulated oscillations from input data x(t) comprising N data channels;

[0009] Each multivariate modulated oscillation is represented in analytical form using the Hilbert transform;

[0010] A cost function is established with the goal of minimizing the sum of the bandwidths of all modes and being able to reconstruct the input signal completely or in a least squares sense by superposition of the modes;

[0011] An optimization problem with the goal of minimizing the cost function is established, and the intrinsic mode components of the mixed interference signal are obtained by solving the optimization problem.

[0012] Furthermore, the process of screening out the signal components of the effective modes includes:

[0013] The Wassertein distance between each intrinsic modal component and the original signal is used to represent the similarity between each intrinsic modal component and the original signal;

[0014] The residual between a modal signal and its adjacent next modal signal on the same data channel is calculated, and the modal signal at the point where the residual is maximum and its previous signal are used for signal reconstruction.

[0015] Furthermore, the process of using the noise-free signal to learn the dictionary and reconstructing the interference signal according to the trained dictionary includes:

[0016] 100. Select pure interference signals as training samples for the dictionary in this paper, and extract K atoms from the training samples to construct the initial dictionary D 0 , set the initial number of iterations of the algorithm to q = 1 and the maximum number of iterations to Q;

[0017] 101. When updating the qth atom, the residual between the original signal and the dictionary atoms other than the current updated atom is calculated, and then the residual and the qth atom are reconstructed into a new optimization problem, and the non-zero columns of the sparse coefficients corresponding to the qth atom in the residual are screened out to form a new residual matrix;

[0018] 102. Perform singular value decomposition on the new residual matrix, use the first row of the left unitary matrix obtained by the singular value decomposition as the value of the qth atom, and use the product of the first row and first column of the diagonal matrix obtained by the singular value decomposition and the right unitary matrix as the value of the non-zero row of the sparse coefficient corresponding to the qth atom;

[0019] 103. Determine whether the maximum number of iterations has been reached. If not, set q=q+1 and return to step 101. If the maximum number of iterations has been reached, end the process.

[0020] In view of the lack of anti-interference methods for array antennas in GNSS scenarios, the present invention proposes an anti-interference method. First, the method considers that the received signal contains noise components, which will inevitably cause errors in the process of atomic update during dictionary learning. The multivariate variational mode decomposition algorithm is introduced to eliminate the influence of interference noise components by selecting the optimal mode; secondly, for the problem that the traditional overcomplete dictionary cannot accurately reconstruct the signal due to the fixedness of its parameters, the dictionary learning theory is introduced into array signal processing, and the signal reconstruction accuracy is improved by updating the dictionary atoms. In addition, the simulation results of the method of the present invention show that the proposed scheme can effectively alleviate the influence of noise components on the signal and ensure the interference recognition accuracy of the subsequent algorithm; and the adopted dictionary learning algorithm can effectively identify interference signals of known types. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 This is a principle block diagram of the anti-interference technology based on antenna array;

[0022] Figure 2 The present invention is used for multivariate variational mode decomposition and effective mode selection process;

[0023] Figure 3 It is the WD distance diagram between each component of the present invention and the original signal;

[0024] Figure 4 is a noise component elimination error diagram of the present invention;

[0025] Figure 5 The figure shows the comparison of NMSE of different algorithms according to the present invention as they change with INR. DETAILED DESCRIPTION

[0026] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0027] The present invention proposes a GNSS receiver array anti-interference method combining multivariate variational mode decomposition and dictionary learning, such as Figure 2 As shown, after multivariate variational mode decomposition of the N array receiving signal, the effective mode is selected to reconstruct the information to obtain a noise-free signal, which specifically includes: after the array receiving signal is subjected to multivariate variational mode decomposition, each inherent mode component is obtained, and after the effective mode is selected, the noise-free signal is reconstructed; the noise-free signal is used for dictionary learning, the interference signal is reconstructed according to the trained dictionary, and the array receiving signal is subtracted from the reconstructed interference signal to achieve suppression of the interference signal.

[0028] like Figure 1 As shown, the present invention considers an anti-interference system framework based on an antenna array, in which an antenna array with a specific shape is equipped at the front end of the satellite navigation receiver for receiving signals, and a suitable anti-interference processing structure is configured at the back end to realize interference detection and suppression and other processing.

[0029] Assume that there are K satellite navigation signals and L interference signals. All satellite navigation signals at time t are expressed as s(t) = [s1(t),…,s K (t)] T , all interference signals at time t are expressed as j(t)=[j1(t),…,j L (t)] T Since the present invention mainly studies the anti-interference algorithm before despreading, and does not involve the influence of the signal modulation method on the receiving anti-interference performance, for the convenience of analysis, direct sequence spread spectrum communication is used as an example for modeling and analysis. The navigation signal of the i-th satellite at the receiver is:

[0030]

[0031] Where Pi is the signal power, C i (t) is the spread spectrum code of the ith satellite at time t, D i (t) is the navigation message of the i-th satellite at time t; f i is the frequency of the i-th satellite, the spectrum includes the Doppler shift; φ i is the carrier phase of the i-th satellite.

[0032] A single interference method may not necessarily produce the expected interference effect. In the actual propagation environment, there are often complex interference signals. This invention will explore the composite interference signal composed of three types of interference: continuous wave interference, frequency modulation interference, and narrowband noise interference, namely:

[0033]

[0034] Among them, j l (t) represents the interference signal of type l.

[0035] The received signal of the N-element antenna array can be expressed as:

[0036] x(t)=A s s(t)+A j j(t)+n(t) (3)

[0037] in, is the received signal vector, each column of the received signal vector xn(t) corresponds to a signal received by a receiving channel, n(t) = [n1(t),…,n N (t)] Tis the receiver thermal noise vector, and denotes the array steering vectors of satellite signals and interference signals respectively, where a(ψ k ) is the steering vector of the kth signal, defined as:

[0038]

[0039] in, represents the phase shift caused by the distance between the current receiving antenna and the reference antenna, λ represents the wavelength of the received signal, and ψ k Represents the unit vector pointing to the direction of the signal incident in the coordinate system, z n is the position vector of the nth array element.

[0040] Since the strength of satellite navigation signals is usually very weak and the satellite signal energy is low when it reaches the ground receiver, for example, the signal-to-noise ratio of the GPS coarse acquisition code spread spectrum signal is about -20dB, which is very susceptible to radio frequency interference signals. The navigation signal is considered to be hidden by the thermal noise of the receiver; the reasons are as follows: from the perspective of signal power, the power density of the satellite navigation signal reaching the receiver after a long distance transmission is lower than the thermal noise power density of the receiver, and the satellite navigation signal uses spread spectrum modulation technology, which is significantly different from the interference signal. Therefore, it is considered that the noise is composed of the satellite navigation signal s(t) and white noise n(t). When there is no need to consider information such as the direction of the satellite signal, the GNSS anti-interference array receiving signal based on the antenna array can be simplified as:

[0041] x(t)=A j j(t)+n(t) (5)

[0042] Among them, n(t) contains satellite navigation signals and thermal noise signals.

[0043] Since the received signal contains noise components, the influence of noise components on matching pursuit cannot be ignored. And the fixed dictionary discretization indicates that if the actual received interference signal parameters do not completely match the dictionary atoms, the accuracy of matching pursuit will be reduced, and there will be inevitable errors. Most of the existing studies focus on the matching pursuit algorithm of the signal, but ignore the importance of dictionary atoms for sparse decomposition. In order to reduce the influence of the noise component of the received signal on the dictionary learning algorithm and improve the detection performance of the interference signal, the present invention proposes a dictionary learning algorithm combined with multivariate variational mode decomposition. By selecting the modal components with high correlation with the received signal to reconstruct the signal, the influence of the noise component on the dictionary learning process is removed, and then the initial dictionary is updated using the dictionary learning algorithm to obtain the atom that best matches the received interference signal, thereby detecting and suppressing the interference signal.

[0044] As a signal decomposition method, Multivariate Variational Mode Decomposition (VMD) determines the frequency center and bandwidth of each component by iteratively searching for the optimal solution of the variational model, and adaptively decomposes the original signal into a finite number of Intrinsic Mode Functions (IMF). The original signal can be obtained by superimposing all components. Multivariate Variational Mode Decomposition (MVMD), as a multivariate extension of VMD, is suitable for processing multi-channel and multi-dimensional signals. Specifically, K predefined multivariate modulated oscillations are extracted from the input data x(t) consisting of N data channels, that is:

[0045]

[0046] Among them, u k (t)=[u1(t),u2(t),…,u N (t)] T Represents the kth mode of the original signal, where each element satisfies the following form:

[0047] u k (t) = a k (t)cos(φ k (t)) (7)

[0048] Among them, a(t) is the amplitude function and φ(t) is the phase shift function.

[0049] From the above formula, we know that the original signal can be expressed as a superposition of the decomposed natural modes. By selecting the effective modes that contain more original signals and discarding the noise modes, the effect of signal denoising can be achieved. First, the bandwidth and center frequency of each natural mode are updated by constructing an optimization problem, and then the updated modes are selected to obtain the effective modes. The specific process is as follows:

[0050] The Hilbert transform constructs an analytical signal from a real signal, and then describes the phase of the analytical signal in time. This representation can be useful when analyzing time-varying amplitudes and instantaneous frequencies; therefore, the analytical signal is constructed with the help of the modal function obtained by the Hilbert transform, and a regular set of amplitudes and phases is defined. By using the Hilbert transform, u(t) is expressed in analytical form:

[0051]

[0052] where the Hilbert transform of u(t) is given by:

[0053]

[0054] The mode selection based on the multivariate model requires minimizing all modes The sum of the bandwidths of the input signals can be completely or in the least squares (LS) sense by superposition of modes. To achieve the above goal, we first need to convert u k (t) is expressed as the analytical signal form described in equation (8) u k The bandwidth of (t) can be obtained by harmonic shifting The L2-norm of the gradient function is used to estimate, so the cost function can be expressed as

[0055]

[0056] Since in the harmonic mixing the entire vector Both use a single frequency component ω k , the desired multivariate modulated oscillation has a single common frequency component ω in all channels k , so (10) can be further expressed as:

[0057]

[0058] in, represents the analytical modulation signal of the nth channel and the kth mode. The MVMD optimization problem is constructed as follows:

[0059]

[0060] Then construct the Lagrangian function to transform the constrained optimization problem into an unconstrained form:

[0061]

[0062] The penalty factor α is introduced to improve the fidelity of the reconstructed signal, and another term with the Lagrange multiplier λ is used to ensure that the constraints are strictly satisfied. For the solution of the above equation, the Alternating Direction Method of Multipliers (ADMM) can be used. ADMM decomposes the large global problem into multiple smaller and easier to solve local sub-problems through the decomposition-coordination process, and obtains the solution of the large global problem by coordinating the solutions of the sub-problems. Its modal Center frequency The iterative expression of the Lagrangian operator λ is:

[0063]

[0064] The update of each modal component in the frequency domain avoids the insufficiency of modal decomposition in the time domain and improves the accuracy of signal decomposition. Therefore, the problem is solved by Fourier transform to the frequency domain, and we can get

[0065]

[0066] Similarly, the update of the center frequency is expressed as

[0067]

[0068] Solve to get the updated center frequency

[0069]

[0070]

[0071] It can be seen from the above steps that the bandwidth and center frequency of each modal component are continuously updated until the convergence condition is reached to obtain the inherent modal component of the mixed interference signal.

[0072] Perform inverse Fourier transform on the spectrum of each IMF component finally obtained after iteration to obtain the time domain signal of each IMF. The specific expression is:

[0073]

[0074] The purpose of MVMD is to extract meaningful and interpretable components from the signal, which is the interference signal in this article. In the Nth channel, the Wassertein distance between each IMF component obtained by MVMD and the original signal is calculated to represent the similarity between each modal component and the original signal. The higher the similarity, the more original information is retained and the closer it is to the effective mode. The similarity function can be defined as:

[0075] L n (k)=W[pdf(u k,n (t)),pdf(x n (t))],k=1,…,K,n=1,…,N (20)

[0076] Among them, u k,n (t) represents the kth mode in the nth channel, x n (t) represents the received signal of the nth channel, L n (k) represents the WD distance between the selected mode and the received signal, and W[A,B] is the distance between the probability density functions (PDF) of A and B, which is expressed as:

[0077]

[0078] Among them, γ∈Γ(u k,n (t),x n (t)) indicates that γ belongs to u k,n (t) and x n (t), d(u k,n (t),x n (t)) is u k,n (t) and x n (t), p is the spatial geometric distance between them, and p is the norm index.

[0079] Calculate the residual based on the distance obtained by the similarity function:

[0080] θ n (k) = L n (k)-L n (k-1),(k=2,3,…,K) (22)

[0081] q = argmax[θ n (k)] (23)

[0082] Among them, θ n (k) is the difference of the similarity function of the nth channel, and q is the position of the modal component with the largest distance increment. For the effective mode, is the noise mode. Discarding the noise mode can reduce the dimension and reduce the computational complexity of subsequent sparse dictionary learning.

[0083] When there are interference signals of known signal types in the receiving environment, the received signal is decomposed by multivariate variational mode and the effective mode is selected, and the reconstructed signal can be approximately represented as a noise-free signal; when the signal selected by the effective mode is used for dictionary learning, the influence of the noise component on the dictionary update can be significantly reduced, and the matching accuracy can be improved. The coefficient matrix is ​​obtained by using the OMP algorithm, and then it is brought into the dictionary learning as the initial coefficient matrix, and the dictionary is updated.

[0084] The reconstructed signal x obtained after MVMD and effective mode selection n After M-point digital processing, the received signal vector of the nth channel is:

[0085]

[0086] where x n (m) is

[0087]

[0088] Where m is the sampling time, Respectively represent the firstcwi Continuous wave interference, the first fmi FM interference, the first nni narrowband noise interference, η(m) represents the residual clutter noise signal in the signal after multivariate mode decomposition and effective mode selection in the previous section. The interference signal expression is:

[0089]

[0090]

[0091]

[0092] Among them, m and T s are the sampling time and sampling period respectively, and f is the carrier frequency. Where P represents the power of the interference signal, θ represents the initial phase of the interference signal, represents the modulation rate of each FM signal, The variance of each narrowband noise signal is σ ξ A zero-mean stationary random process, is the FM index of each signal.

[0093] make:

[0094]

[0095] η=[η(0),η(1),…,η(M-1)] T (30)

[0096] Then equation (24) can be expressed in the form of a matrix vector

[0097] x n =j+η (31)

[0098] Where j represents the interference signal and η represents the residual noise.

[0099] The parameter set of the overcomplete dictionary should include the waveform style information of the interference signal. The interference signal model includes parameters such as carrier frequency, modulation frequency, modulation index, phase, etc. The atoms in the overcomplete atom library should have corresponding parameter space, and the atomic model can be directly set to d k =[d k (0)d k (1)…d k (M-1)] T , where d k (m) is

[0100]

[0101] Among them, d krepresents the kth atom, and then the overcomplete dictionary can be constructed as The corresponding sparse coefficient vector is Where M<K, and the sparsity is P.

[0102] The nth channel received signal vector is expressed as:

[0103] x n =Dγ n +η (33)

[0104] The pure interference signal is selected as the training sample of the dictionary in this paper, and atoms of the form (30) are extracted from the training sample to construct the initial dictionary Set the initial number of iterations of the algorithm to q=1 and the maximum number of iterations to Q.

[0105] Construct the following optimization problem

[0106]

[0107] Where U is the given sparsity threshold.

[0108] Now update the k'th atom in dictionary D. n Decomposed into the sum of the 1-rank matrix and the proposed k'th term, that is,

[0109]

[0110] Fix the other atoms, can be rewritten as:

[0111]

[0112] in, Residual. However, it is not possible to directly k' Solve directly may not be sparse. Therefore, it is necessary to k' Corresponding to The non-zero columns of The optimization problem is now described as minimizing Then the singular value decomposition (SVD) method is used to This breaks down into:

[0113]

[0114] In the update phase, take the first column of U as the new atom d k' , take the first row and first column of ΣΣ(1,1) and the first row of the right singular matrix V TThe product of (1,·) is used as the new sparse coefficient, that is,

[0115]

[0116] Repeat the above steps iteratively until the convergence condition is met.

[0117] Based on the trained dictionary D c and the sparse coefficient vector We can reconstruct the interference signal more accurately

[0118]

[0119] Subtracting the interference signal reconstructed here from the initial received signal can effectively suppress the interference signal.

[0120] Figure 3 It shows that after the signal is decomposed by multivariate variational mode, 8 IMF components are obtained and the distance increment after WD with the received signal is calculated respectively. The figure shows that the distance of the 4th mode is the critical distance. Therefore, the modal components of task IMF4 and later are noise modes, and the components of IMF3 and earlier are valid modes. Therefore, the first three modal components are selected to reconstruct the noise-free signal.

[0121] Figure 4 A curve chart showing the variation of NMSE with INR between the reconstructed signal and the original noise-free signal after effective mode selection is shown. The simulation results show that as the power of the interference signal increases, it becomes more reliable to extract useful parameters from the interference signal, and the performance of eliminating noise components also improves accordingly, which fully demonstrates the effectiveness of the selection of effective modes for removing noise components.

[0122] Figure 5 The interference signal detection errors of the traditional OMP algorithm, K-SVD algorithm and the MVMD-K-SVD algorithm proposed in this paper are shown. As can be seen from the figure, with the increase of the interference-to-noise power ratio, the detection error shows a downward trend; the reason is that with the increase of the power of the interference signal, the sparsely represented signal estimation accuracy is higher, and the dictionary atoms can more accurately match the parameters of the interference signal when updating, and find the optimal solution. In addition, the traditional K-SVD algorithm is used to directly update the dictionary atoms of the received signal. Due to the presence of noise components, the signal is affected, resulting in the reconstructed signal error of the optimal atom in the dictionary and the noisy signal matching compared with the noise-free signal. This also verifies the effectiveness of the algorithm in this paper.

[0123] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A GNSS receiver array anti-interference method based on multivariate variational mode decomposition and dictionary learning, characterized in that: include: The array receiving signal is decomposed by multivariate variational mode to obtain each inherent mode component, and after effective mode selection, the noise-free signal is reconstructed; The noise-free signal is used for dictionary learning, and the interference signal is reconstructed according to the trained dictionary. The reconstructed interference signal is subtracted from the array received signal to achieve interference signal suppression.

2. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 1, characterized in that: The array received signal is decomposed by multivariate variational mode to obtain each inherent mode component, which includes the following steps: Extracting predefined K multivariate modulated oscillations from input data x(t) comprising N data channels; Each multivariate modulated oscillation is represented in analytical form using the Hilbert transform; A cost function is established with the goal of minimizing the sum of the bandwidths of all modes and being able to reconstruct the input signal completely or in a least squares sense by superposition of the modes; An optimization problem with the goal of minimizing the cost function is established, and the intrinsic mode components of the mixed interference signal are obtained by solving the optimization problem.

3. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 1, characterized in that: Establish an optimization problem with the goal of minimizing the cost function, expressed as: Optimization problem: Constraints: Among them, u k,n represents the nth data channel and the kth modal signal of the original signal, u k,n (t) represents the nth data channel and the kth modal signal of the original signal at time t, ω k is the common frequency component of the kth modal signal, K is the number of modes, N is the number of data channels, represents the partial derivative operation with respect to time, represents the analytical modulation signal of the nth channel and the kth mode, ||·||2 represents the L2 norm of the obtained harmonic signal; x n (t) represents the input data of the nth data channel at time t.

4. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 4 is characterized in that: When solving the optimization problem, the Lagrangian function is constructed to convert the constrained optimization problem into an unconstrained form, which is expressed as: Among them, λ n represents the Lagrange multiplier when solving the nth channel; α is the penalty factor; <,> represents the inner product operation.

5. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 4, characterized in that: The multiplier alternating direction method iteratively solves the analytical modulation signal, center frequency and Lagrange multiplier of the nth channel and the kth mode through the decomposition and coordination process, that is: in, represents the analytical modulation signal of the nth channel and the kth mode obtained by the m+1th iteration; represents the Fourier transform of the nth channel signal; represents the Fourier transform of the ith mode of the nth channel; represents the Lagrange multiplier of the part ω>0 after Fourier transform; ω is the frequency of the signal after Fourier transform, ω k is the center frequency of the kth mode; represents the center frequency obtained by the m+1th iteration; represents the Lagrange multiplier obtained by the m+1th iteration; τ is the update parameter; To solve the mode of the nth channel and kth mode obtained.

6. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 1, characterized in that: The process of screening out the signal components of the effective mode includes: The Wassertein distance between each intrinsic modal component and the original signal is used to represent the similarity between each intrinsic modal component and the original signal; The residual between a modal signal and its adjacent next modal signal on the same data channel is calculated, and the modal signal at the point where the residual is maximum and its previous signal are used for signal reconstruction.

7. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 6, characterized in that: The similarity between the intrinsic modal component and the original signal is expressed as: L n (k)=W[pdf(u k,n (t)),pdf(x n (t))],k=1,…,K,n=1,…,N Among them, L n (k) represents the similarity between the nth data channel of the original signal, the kth modal signal and the original signal; W[A,B] is the distance between the probability density functions of A and B; pdf(·) represents the calculated probability density function; Γ(u k,n (t),x n (t)) represents all k,n (t) and x n (t) is the joint distribution set of marginal distributions; d(u k,n (t),x n (t)) is the nth data channel of the original signal and the kth modal signal u k,n (t) and the original signal x n (t) The spatial geometric distance between them, p is the norm index; Represents the distribution u k,n (t) Move to distribution x n (t) The required probability mass.

8. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 1, characterized in that: The process of using the noise-free signal for dictionary learning and reconstructing the interference signal based on the trained dictionary includes:

100. Select pure interference signals as training samples for the dictionary in this paper, and extract K atoms from the training samples to construct the initial dictionary D 0 , set the initial number of iterations of the algorithm to q = 1 and the maximum number of iterations to Q; 101. When updating the qth atom, the residual between the original signal and the dictionary atoms other than the current updated atom is calculated, and then the residual and the qth atom are reconstructed into a new optimization problem, and the non-zero columns of the sparse coefficients corresponding to the qth atom in the residual are screened out to form a new residual matrix; 102. Perform singular value decomposition on the new residual matrix, use the first row of the left unitary matrix obtained by the singular value decomposition as the value of the qth atom, and use the product of the first row and first column of the diagonal matrix obtained by the singular value decomposition and the right unitary matrix as the value of the non-zero row of the sparse coefficient corresponding to the qth atom; 103. Determine whether the maximum number of iterations has been reached. If not, set q=q+1 and return to step 101. If the maximum number of iterations has been reached, end the process.

9. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 8, characterized in that: The residual between the original signal and the remaining dictionary atoms is expressed as: Among them, E k' represents the residual between the original signal and the dictionary atoms except the k'th atom; d k represents the kth atom in the dictionary; x n Represents the input data of the nth data channel, Represents the sparse coefficient corresponding to the nth data channel and the kth atom.

10. The GNSS receiver array anti-interference method of joint multivariate variational mode decomposition and dictionary learning according to claim 8, characterized in that: The residual between the original signal and the remaining dictionary atoms is reconstructed with the qth atom into a new optimization problem, which is expressed as: Among them, E k' represents the residual between the original signal and the dictionary atoms except the k'th atom, d k' represents the k'th atom in the dictionary; represents the sparse coefficient corresponding to the nth data channel and the k'th atom; ||·|| F It means to find the Frobenius norm.

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