A coherent source adaptive beamforming method based on coprime array interpolation

Abstract

This invention discloses a coherent source adaptive beamforming method based on coprime array interpolation, belonging to the field of radar anti-jamming technology. It comprises two stages: a first stage of decoherence covariance matrix reconstruction and a second stage of adaptive beamformer construction. In the first stage, by constructing a sensor position model of the coprime array, a mathematical model of signal reception, descriptions of signal and interference, and the definition of the steering vector, the received signal is interpolated. The method utilizes the properties of decoherence by taking a diagonal matrix and the Vandermonde decomposition of the Taplez matrix to achieve effective separation of coherent source interference from the target signal. In the second stage, by reconstructing the interference and noise covariance matrices, an adaptive beamformer is designed, ultimately achieving high-precision detection and extraction of target signals in complex environments.

Description

Technical Field

[0001] This invention belongs to the field of radar anti-jamming technology, specifically relating to a coherent source adaptive beamforming method based on coprime array interpolation. Background Technology

[0002] Adaptive beamforming is a widely used technique in array signal processing, with applications in radar, sonar, wireless communications, and medical fields. It dynamically adjusts weights using received data to isolate the signal of interest and minimize interference and noise. Minimum variance distortion-free response (MRR) is a powerful adaptive beamforming technique that relies on precise knowledge of the desired signal and array configuration, providing high resolution and interference immunity. However, its performance degrades significantly due to mismatches in the steering vector of the signal of interest caused by non-ideal factors such as viewing direction error and wavefront distortion. Furthermore, performance drops sharply if the received data contains information about the SOI (Signal of Interest), which is unavoidable in real-world scenarios. Enhancing the robustness of adaptive beamformers has been a focus of extensive research over the past few decades.

[0003] In coherent source adaptive beamforming, multiple signal sources may generate the same or similar signals during transmission, which is particularly common in fields such as radar, sonar, and wireless communication. The signal similarity introduced by coherent sources makes it difficult for traditional beamforming methods to distinguish these signals, thus affecting the accuracy and performance of signal processing. Therefore, research on coherent source adaptive beamforming technology is particularly important. It can improve the system's resolution and anti-interference capabilities in environments with significant multipath effects or dense signal sources, ensuring reliable signal reception and processing. This is of great significance for improving the target recognition capabilities of radar systems, the underwater target detection capabilities of sonar systems, and the signal transmission quality of wireless communication systems. Therefore, research on adaptive beamforming technology in coherent source environments can not only overcome the limitations of traditional methods but also further expand its application scope in complex signal environments.

[0004] Firstly, adaptive beamforming algorithms cannot effectively address the problem of coherent interference sources: (1) Diagonal loading method: By adding constants to the diagonal of the covariance matrix, the stability and robustness of the beamforming algorithm are improved. However, in the environment of coherent sources, the loaded diagonal terms may not be sufficient to separate coherent signals, leading to performance degradation. This is because diagonal loading cannot effectively distinguish multiple coherent signals, thus failing to completely eliminate interference between coherent sources. (2) Eigenvalue decomposition method: Utilizing the eigenvalue decomposition information of the covariance matrix to enhance beamforming performance and suppress interference. In the face of coherent sources, eigenvalue decomposition may fail because coherent sources make the eigenvalue spectrum of the covariance matrix difficult to distinguish, making it impossible for the algorithm to accurately extract and separate the feature information of coherent signals. (3) Uncertainty set-based method: Considering array errors and uncertainties in the beamforming process, the robustness of the algorithm is improved by optimizing the uncertainty set. However, for coherent sources, error and uncertainty models are difficult to accurately describe signal characteristics because the high correlation between coherent signals leads to the failure of the uncertainty model, thus affecting the beamforming effect. (4) Covariance Matrix Reconstruction Method: Reconstructing the covariance matrix reduces estimation errors and noise, thus enhancing beamforming accuracy. However, coherent sources can make covariance matrix reconstruction inaccurate because coherent signals cause strong correlations in the matrix, making it difficult to effectively separate and suppress coherent signals during reconstruction. Faced with the challenge of coherent sources, existing adaptive beamforming methods may encounter significant performance degradation in practical applications. To overcome these problems, further research and development of novel beamforming techniques capable of effectively handling coherent signals are needed.

[0005] This invention presents a coherent source adaptive beamforming method based on coprime array interpolation, applicable to interference source scenarios. This method utilizes decoherence by taking a diagonal matrix and the Vandermonde decomposition property of the Taplez matrix to effectively separate coherent source interference from the target signal. Furthermore, it improves the robustness of the method by reconstructing the covariance matrices of interference and noise. This method effectively solves the problem of target signal energy leakage caused by coherent source interference and has high practical value for applications such as wireless signal reception. Summary of the Invention

[0006] In order to solve the technical problem of energy leakage in the reception of target signals caused by interference from coherent signal sources in practical scenarios, this invention aims to provide a coherent source adaptive beamforming method based on coprime array interpolation.

[0007] To solve the technical problem, the technical solution of the present invention is as follows:

[0008] An adaptive beamforming method for coherent sources based on coprime array interpolation, the method comprising:

[0009] The first stage is the reconstruction of the coherent covariance matrix, and the second stage is the construction of the adaptive beamformer.

[0010] The first stage involves constructing a sensor position model for a coprime array, a mathematical model for signal reception, a description of signals and interference, and a definition of the steering vector. This is followed by signal interpolation, utilizing the properties of decoherence of the diagonal matrix and the Vandermonde decomposition of the Taplez matrix to achieve effective separation of coherent source interference from the target signal. The second stage involves reconstructing the interference and noise covariance matrices and designing an adaptive beamformer, ultimately achieving high-precision detection and extraction of target signals in complex environments.

[0011] Furthermore, the first stage includes:

[0012] Received signal interpolation: Using the sensor positions of the coprime array, a virtual array is constructed through interpolation and an observation vector model is established to obtain the observation vector of the virtual ULA;

[0013] Derivation of the decoherence covariance matrix: The covariance matrix is ​​calculated using the observation vectors obtained through interpolation. The covariance matrix is ​​then reconstructed through matrix operations, ultimately constructing a new matrix that decoheres the interference signal.

[0014] Taplez-type matrix construction: The reconstructed covariance matrix is ​​expressed by constructing the Hermitian Taplez matrix, and further processing is used to obtain a more accurate model of the disturbance;

[0015] Atomic norm recovery of covariance matrix: Using atomic norm theory, the atomic l0 norm is relaxed to the atomic l1 norm. Through optimization, the required covariance matrix is ​​reconstructed and recovered, and finally a pseudo covariance matrix is ​​obtained for subsequent beamforming.

[0016] Furthermore, the second phase includes:

[0017] Orthogonal projection matrix construction: Construct a class covariance matrix and invert it to obtain the projection matrix C, which is used to eliminate the information of the desired signal while retaining interference and noise;

[0018] INCM matrix reconstruction: The covariance matrix is ​​processed using the projection matrix to reconstruct the INCM matrix, which preserves interference and noise information;

[0019] Calculating the weight vector: Finally, the weight vector is calculated based on the MVDR criterion, and the optimal weight vector is solved using the reconstructed INCM matrix to achieve adaptive beamforming of the target signal.

[0020] Furthermore, the sensor position model of the coprime array, the mathematical model of signal reception, the description of signals and interference, and the definition of the steering vector specifically include:

[0021] The sensor positions of the coprime array are:

[0022]

[0023] Where M and N are coprime numbers, the array aperture is (2M-1)Nd, d=λ / 2, and λ is the carrier wavelength;

[0024] There is one narrowband SOI and L far-field narrowband interferences coherent with the SOI; the DOAs of the signal and interferences originate from different directions θ0, θ1, ..., θ L The observation vector of the array is represented as:

[0025]

[0026] Where t is the time measure of the signal, s0(t), s l (t) represent the waveforms of the signal and the interference, respectively, where s l (t)=γ l s0(t), It is a non-zero complex number. It is independent and identically distributed (iid) additive white Gaussian noise. It is a steering vector related to the direction of the incoming wave;

[0027] The turning vector of the coprime array is:

[0028]

[0029] in,

[0030] Furthermore, the received signal interpolation specifically includes:

[0031] The augmented array construct interpolated array manifold is represented as:

[0032]

[0033] in, This represents the elements remaining in set U after removing elements from set S;

[0034] The observation vector of the virtual ULA is:

[0035]

[0036] Where γ0=1, These represent the guidance matrix and noise vector of the virtual ULA, respectively.

[0037] Furthermore, the decoherence covariance matrix derivation and the Taplez-type matrix construction specifically include:

[0038] Derivation of the decoherence covariance matrix:

[0039] Real ULA received signal Assume:

[0040]

[0041] The steering vector of a real ULA is defined as:

[0042]

[0043] The covariance matrix is:

[0044]

[0045] Among them, the vector γ is the reason why the part of the covariance matrix with respect to SOI and interference has a reduced rank;

[0046] Consider the turning vector of a coprime array Interpolation leads to The existence of a value of 0 allows for the use of... The calculated covariance matrix contains all zero columns; construct a selection matrix B, for... After removing all zeros from the column, we get:

[0047]

[0048] in, It is the covariance matrix after filtering out the target signal through orthogonal space. e i It is a column vector, with all elements being 0 except for the i-th element, which is 1.

[0049] Rephrased as:

[0050]

[0051] in:

[0052]

[0053] At this point, all prior information about the signal and interference is contained within... In Chinese, it is expressed as R S ,for:

[0054]

[0055] By constructing a new matrix to solve the coherence, we can obtain:

[0056]

[0057] The presence of coherent source interference caused ZZ H If it is not a diagonal matrix, retain its diagonal part to obtain a new matrix:

[0058]

[0059] in,

[0060] Construction of Taplez-type matrices:

[0061] Represented by Hermitian-Taiplz matrix Rewritten as:

[0062]

[0063] The Taplez matrix is ​​represented as:

[0064]

[0065] Furthermore, the construction of the atomic norm recovery covariance matrix specifically includes:

[0066] According to atomic norm theory, by selecting an off-grid atom set, the steering vectors of SOI and coherent interference in the received signal can be separated. The atom set is defined as follows:

[0067]

[0068] The l0 norm is very effective in finding the optimal sparse feature terms; therefore, the atomic l0 norm is used as a constraint, expressed as:

[0069]

[0070] In the l0-norm constraint problem, the convex hull of the l1-norm completely contains the convex hull of the l0-norm, meaning the l1-norm is the tightest convex relaxation of the l0-norm. Relaxing the atomic l0-norm to the atomic l1-norm is as follows:

[0071]

[0072] calculate The SCM is:

[0073]

[0074] Based on the above derivation and the prior conditions of the Hermitian-Taplitz matrix, the optimization problem is used to obtain the required pseudo-covariance matrix R. S The description is as follows:

[0075]

[0076] Among them, Π is used to distinguish RS Non-zero and known elements and zero and unknown elements binary matrix;

[0077] Introducing the ANM problem, we get:

[0078]

[0079] Where μ is a regularization parameter;

[0080] Minimizing the atomic norm can be equivalent to the following SDP problem:

[0081]

[0082] in, It is a Hermitian matrix.

[0083] Furthermore, the construction of the orthogonal projection matrix specifically includes:

[0084] Construct a covariance matrix as follows:

[0085]

[0086] Where I is the identity matrix, and the value of α affects the orthogonality between the constructed matrix and the desired signal, and should be a large value;

[0087] C is represented in the form of eigenvalues ​​and eigenvectors, where the eigenvalues ​​are μ. i Arranged in descending order:

[0088]

[0089] μ is the characteristic value arranged in descending order, and μ1 = M, μ i =1. ξ i μ and eigenvectors are corresponding eigenvalues;

[0090] The inverse of the constructed class covariance matrix C is represented as:

[0091]

[0092] based on If the condition is satisfied and α >> 1, then C is approximately:

[0093]

[0094] C -1 As the projection matrix B, it is:

[0095]

[0096] Furthermore, the INCM matrix reconstruction specifically includes:

[0097] The covariance matrix after projection matrix processing, where SOI information is eliminated and interference and noise information are retained, is as follows:

[0098]

[0099] The INCM matrix can be reconstructed from the corresponding eigenvalues ​​and eigenvectors as follows:

[0100]

[0101] in, for The smallest eigenvalue.

[0102] Furthermore, the calculation of the weight vector specifically includes:

[0103] The weight vector based on the MVDR criterion is:

[0104]

[0105] Compared with the prior art, the advantages of the present invention are as follows:

[0106] This invention proposes an adaptive beamforming method based on coprime array interpolation and interference-noise covariance matrix reconstruction, applicable to coherent interference signal sources and complex noise conditions. It can accurately eliminate coherent interference and achieve a better output signal-to-interference-plus-noise ratio (SNR). It is suitable for interference source scenarios. This method utilizes diagonal matrix decoherence and the van der Mund decomposition property of the Taplez matrix to effectively separate coherent source interference from the target signal. Simultaneously, the robustness of the method is improved through interference-noise covariance matrix reconstruction. This method effectively solves the problem of target signal energy leakage caused by coherent source interference and has high practical value for applications such as wireless signal reception. Attached Figure Description

[0107] Figure 1 Flowchart of the method described in this invention;

[0108] Figure 2 DOA estimation after interpolation of coprime arrays;

[0109] Figure 3 DOA estimation in orthogonal projection space;

[0110] Figure 4 Beam pattern reconstructed based on interference plus noise covariance matrix. Detailed Implementation

[0111] The specific implementation of the present invention is described below with reference to embodiments:

[0112] It should be noted that the structures, proportions, sizes, etc. shown in this specification are only used to complement the content disclosed in the specification for those skilled in the art to understand and read, and are not intended to limit the conditions under which the present invention can be implemented. Any modifications to the structure, changes in the proportions, or adjustments to the size, without affecting the effects and objectives that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0113] Furthermore, the terms such as "upper," "lower," "left," "right," "middle," and "one" used in this specification are merely for clarity of description and are not intended to limit the scope of the invention. Any changes or adjustments to their relative relationships, without substantially altering the technical content, should also be considered within the scope of the invention.

[0114] Example 1:

[0115] like Figure 1 As shown, the coherent source adaptive beamforming method based on coprime array interpolation described in this invention comprises two stages:

[0116] Phase 1: Reconstructing the coherence covariance matrix

[0117] The sensor positions of the coprime array are:

[0118]

[0119] Where M and N are coprime numbers, the array aperture is (2M-1)Nd, d=λ / 2, and λ is the carrier wavelength.

[0120] There is one narrowband SOI and L far-field narrowband interferences coherent with the SOI. The DOAs of the signal and interferences originate from different directions θ0, θ1, ..., θ2. L The observation vector of the array can be represented as:

[0121]

[0122] Where t is the time measure of the signal. s0(t), s l (t) represent the waveforms of the signal and the interference, respectively, where s l (t)=γ l s0(t), It is a non-zero complex number. It is independent and identically distributed (iid) additive white Gaussian noise. It is the steering vector related to the direction of the incoming wave.

[0123] The turning vector of the coprime array is:

[0124]

[0125] in,

[0126] Step 1: Interpolation of Received Signal

[0127] The augmented array construct interpolation array manifold can be represented as:

[0128]

[0129] in, This represents the elements remaining in set U after removing elements from set S.

[0130] The observation vector of the virtual ULA is:

[0131]

[0132] Where γ0=1, These represent the guidance matrix and noise vector of the virtual ULA, respectively.

[0133] Step 2: Derivation of the Decorative Covariance Matrix

[0134] Real ULA received signal Assume:

[0135]

[0136] The steering vector of a real ULA is defined as:

[0137]

[0138] The covariance matrix is:

[0139]

[0140] The vector γ is the reason why the part of the covariance matrix related to SOI and interference has a reduced rank.

[0141] Due to the turning vector of the coprime array Interpolation leads to The existence of a value of 0 allows for the use of... The calculated covariance matrix contains columns all zeros. Construct a selection matrix B, for... After removing all zeros from the column, we get:

[0142]

[0143] in, e i It is a column vector, where all elements are 0 except for the i-th element, which is 1.

[0144] Rephrased as:

[0145]

[0146] in,

[0147]

[0148] At this point, all prior information about the signal and interference is contained within... In Chinese, it is expressed as R S ,for:

[0149]

[0150] By constructing a new matrix to solve the coherence, we can obtain:

[0151]

[0152] The presence of coherent source interference caused ZZ H If it is not a diagonal matrix, retain its diagonal part to obtain a new matrix:

[0153]

[0154] in,

[0155] Step 3, Construction of Taplez-type matrices

[0156] Represented by Hermitian-Taiplz matrix Rewritten as

[0157]

[0158] The Taplez matrix is ​​represented as:

[0159]

[0160] Step 4: Atomic norm recovery of covariance matrix

[0161] According to atomic norm theory, the separation of SOI and coherent interference steering vectors in the received signal is achieved by selecting an off-grid atom set. The atom set is defined as follows:

[0162]

[0163] The l0 norm is very effective in finding the optimal sparse feature terms; therefore, the atomic l0 norm is used as a constraint, expressed as:

[0164]

[0165] In the l0-norm constraint problem, the convex hull of the l1-norm can completely contain the convex hull of the l0-norm; that is, the l1-norm is the tightest convex relaxation of the l0-norm. Relaxing the atomic l0-norm to the atomic l1-norm is as follows:

[0166]

[0167] calculate The SCM is:

[0168]

[0169] Based on the above derivation and the prior conditions of the Hermitian-Taplitz matrix, the optimization problem established can yield the required pseudo-covariance matrix R. S The description is as follows:

[0170]

[0171] Among them, Π is used to distinguish R S Non-zero and known elements and zero and unknown elements Binary matrix.

[0172] Introducing the ANM problem, we get:

[0173]

[0174] Here, μ is a regularization parameter.

[0175] Minimizing the atomic norm can be equivalent to the following SDP problem:

[0176]

[0177] in, It is a Hermitian matrix.

[0178] During the construction of the adaptive beamformer, the pseudo-covariance matrix R... S This provides the signal characteristic matrix after coherent interference is eliminated, forming the basis for calculating the orthogonal projection matrix and the INCM matrix. These matrices are then used to design the weight vectors of the beamformer, enabling it to accurately enhance the desired signal even in the presence of coherent interference. Figure 2 As shown, this is the DOA estimation after coprime array interpolation, which can accurately estimate the DOA of the target signal and the three decoherent coherent interference signals.

[0179] Phase Two: Adaptive Beamform Construction

[0180] Step 1: Constructing the orthogonal projection matrix

[0181] Construct a covariance matrix as follows:

[0182]

[0183] The value of α affects the orthogonality between the constructed matrix and the desired signal, and should be a relatively large value.

[0184] C can be represented in the form of eigenvalues ​​and eigenvectors, where the eigenvalues ​​are μ. i Arranged in descending order:

[0185]

[0186] μ is the characteristic value arranged in descending order, and μ1 = M, μ i =1. ξ i μ and μ are the corresponding eigenvectors and eigenvalues.

[0187] The inverse of the constructed class covariance matrix C can be represented as:

[0188]

[0189] based on If α >> 1, then C can be approximated as:

[0190]

[0191] C -1 It can be used as the projection matrix B, which is:

[0192]

[0193] Step 2, INCM matrix reconstruction

[0194] The covariance matrix after projection matrix processing, where SOI information is eliminated and interference and noise information are retained, is as follows:

[0195]

[0196] The INCM matrix can be reconstructed from the corresponding eigenvalues ​​and eigenvectors as follows:

[0197]

[0198] in, for The smallest eigenvalue. For example... Figure 3 As shown, this is the DOA estimation in orthogonal projection space, which can accurately obtain the DOA of the three interference signals after filtering out the target signal.

[0199] Step 3: Calculate the weight vector

[0200] The weight vector based on the MVDR criterion is:

[0201]

[0202] Example 2:

[0203] The interference suppression effect of this invention on coherent signal sources is demonstrated through actual tests under multiple interference conditions:

[0204] Operating parameters: Coprime array M=3, N=5, 11 array elements in total, element spacing set to half wavelength. Target arrival direction 0°, coherent source arrival directions -30°, 30°, 60°, SNR=20dB, INR received by each array element=30dB, snapshot number 100.

[0205] Specific procedures:

[0206] (1) Construct a coprime array model, M=3, N=5,0,3,5,6,9,10,12,15,20,25;

[0207] (2) Construct a regularization matrix B based on the actual sensor positions in the coprime array;

[0208] (3) Define the atomic set

[0209] (4) Constructing the atomic norm optimization problem;

[0210] (5) Solving the optimization problem yields the matrix.

[0211] (6) Construct the orthogonal projection matrix of the SOI direction;

[0212] (7) Processing with orthogonal projection matrix

[0213] (8) Reconstruct INCM as R INCM ;

[0214] (9) Calculate the weight function using the MVDR criterion;

[0215] Performance Comparison: Compared to traditional adaptive beamforming algorithms, this invention is suitable for coherent interference signal sources and complex noise conditions; it can accurately eliminate coherent interference and obtain a better output signal-to-interference-plus-noise ratio. For example... Figure 4 As shown, the power pattern obtained by this invention can effectively receive target signals and perform anti-interference processing on the directional region of interference signals, with a main-sidelobe ratio of 13.27 dB.

[0216] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0217] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0218] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0219] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0220] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

[0221] Many other changes and modifications can be made without departing from the concept and scope of this invention. It should be understood that this invention is not limited to the specific embodiments, and the scope of this invention is defined by the appended claims.

Claims

1. A method for adaptive beamforming of coherent sources based on coprime array interpolation, characterized in that, The method includes: The first stage is the reconstruction of the coherent covariance matrix, and the second stage is the construction of the adaptive beamformer. The first stage involves constructing a sensor position model for a coprime array, a mathematical model for signal reception, a description of signals and interference, and a definition of steering vectors. This is followed by signal interpolation, utilizing the properties of decoherence of the diagonal matrix and the Vandermonde decomposition of the Taplez matrix to achieve effective separation of coherent source interference from the target signal. The second stage involves reconstructing the interference and noise covariance matrices and designing an adaptive beamformer, ultimately achieving high-precision detection and extraction of target signals in complex environments. The first stage includes: Received signal interpolation: Using the sensor positions of the coprime array, a virtual array is constructed through interpolation and an observation vector model is established to obtain the observation vector of the virtual ULA; Derivation of the decoherence covariance matrix: The covariance matrix is ​​calculated using the observation vectors obtained through interpolation. The covariance matrix is ​​then reconstructed through matrix operations, ultimately constructing a new matrix that decoheres the interference signal. Taplez-type matrix construction: The reconstructed covariance matrix is ​​expressed by constructing the Hermitian Taplez matrix, and further processing is used to obtain a more accurate model of the disturbance; Atomic norm recovery of covariance matrix: Using atomic norm theory, the atomic l0 norm is relaxed to the atomic l1 norm. Through optimization, the required covariance matrix is ​​reconstructed and recovered, and finally a pseudo covariance matrix is ​​obtained for subsequent beamforming. The second phase includes: Orthogonal projection matrix construction: Construct a class covariance matrix and invert it to obtain the projection matrix C, which is used to eliminate the information of the desired signal while retaining interference and noise; INCM matrix reconstruction: The covariance matrix is ​​processed using the projection matrix to reconstruct the INCM matrix, which preserves interference and noise information; Calculating the weight vector: Finally, the weight vector is calculated based on the MVDR criterion, and the optimal weight vector is solved using the reconstructed INCM matrix to achieve adaptive beamforming of the target signal.

2. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The sensor position model, signal reception mathematical model, signal and interference description, and steering vector definition of the coprime array specifically include: The sensor positions of the coprime array are: Where M and N are coprime numbers, the array aperture is (2M-1)Nd, d=λ / 2, and λ is the carrier wavelength; There is one narrowband SOI and L far-field narrowband interferences coherent with the SOI; the DOAs of the signal and interferences originate from different directions θ0, θ1, ..., θ L The observation vector of the array is represented as: Where t is the time measure of the signal, s0(t), s l (t) represent the waveforms of the signal and the interference, respectively, where s l (t)=γ l s0(t), It is a non-zero complex number. It is independent and identically distributed (iid) additive white Gaussian noise. It is a steering vector related to the direction of the incoming wave; The turning vector of the coprime array is: in, 3. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The received signal interpolation specifically includes: The augmented array construct interpolated array manifold is represented as: in, M represents the elements remaining in set U after removing elements from set S; M and N are coprime numbers. The observation vector of the virtual ULA is: Where γ0=1, and These represent the guidance matrix and noise vector of the virtual ULA, respectively.

4. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The decoherence covariance matrix derivation and the construction of Taplez-type matrices specifically include: Derivation of the decoherence covariance matrix: Real ULA received signal Assume: The steering vector of a real ULA is defined as: The covariance matrix is: Where s0(t) is the waveform of the signal; the vector γ is the reason for the reduced rank of the SOI and interference parts of the covariance matrix; Consider the turning vector of a coprime array Interpolation leads to The existence of a value of 0 allows for the use of... The calculated covariance matrix contains all zero columns; construct a selection matrix B, for... After removing all zeros from the column, we get: in, It is the covariance matrix after filtering out the target signal through orthogonal space. e i It is a column vector, with all elements being 0 except for the i-th element, which is 1. Rephrased as: in: At this point, all prior information about the signal and interference is contained within... In Chinese, it is expressed as R S ,for: By constructing a new matrix to solve the coherence, we can obtain: Considering the presence of coherent source interference, ZZ H If it is not a diagonal matrix, retain its diagonal part to obtain a new matrix: in, Construction of Taplez-type matrices: Represented by Hermitian-Taiplz matrix Rewritten as: The Taplez matrix is ​​represented as:

5. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The construction of the atomic norm recovery covariance matrix specifically includes: According to atomic norm theory, by selecting an off-grid atom set, the steering vectors of SOI and coherent interference in the received signal can be separated. The atom set is defined as follows: The l0 norm is very effective in finding the optimal sparse feature terms; therefore, the atomic l0 norm is used as a constraint, expressed as: In the l0-norm constraint problem, the convex hull of the l1-norm completely contains the convex hull of the l0-norm, meaning the l1-norm is the tightest convex relaxation of the l0-norm. Relaxing the atomic l0-norm to the atomic l1-norm is as follows: calculate The SCM is: The optimization problem based on the prior conditions of the Hermitian-Pulitz matrix yields the required pseudo-covariance matrix R. S The description is as follows: Among them, Π is used to distinguish R S Non-zero and known elements and zero and unknown elements binary matrix; Introducing the ANM problem, we get: Where μ is a regularization parameter; Minimizing the atomic norm is equivalent to the following SDP problem: in, It is a Hermitian matrix.

6. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The construction of the orthogonal projection matrix specifically includes: Construct a covariance matrix as follows: Where I is the identity matrix, and the value of α affects the orthogonality between the constructed matrix and the desired signal, so a larger value is taken. C is represented in the form of eigenvalues ​​and eigenvectors, where the eigenvalues ​​are μ. i Arranged in descending order: μ is the characteristic value arranged in descending order, and μ1 = M, μ i =1, ξ i μ and eigenvectors are corresponding eigenvalues; The inverse of the constructed class covariance matrix C is represented as: based on If the condition is satisfied and α>>1, then C is approximately: C -1 As the projection matrix B, it is:

7. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 1, characterized in that, The INCM matrix reconstruction specifically includes: The covariance matrix after projection matrix processing, where SOI information is eliminated and interference and noise information are retained, is as follows: Representation: Taplez matrix; Reconstruct the INCM matrix R from the corresponding eigenvalues ​​and eigenvectors INCM for: in, for The smallest eigenvalue.

8. The coherent source adaptive beamforming method based on coprime array interpolation according to claim 7, characterized in that, The calculation of the weight vector specifically includes: The weight vector based on the MVDR criterion is:

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