Array element failure fault-tolerant direction of arrival estimation method and device based on incomplete tensor reconstruction

Abstract

The invention provides an array element failure fault-tolerant arrival direction estimation method and device based on incomplete tensor reconstruction, and belongs to the technical field of array signal processing. The method comprises the following steps: carrying out array element working state detection on a received signal of an L-shaped sensor array so as to construct three-order incomplete signal tensors of two direction arrays; performing cross-correlation operation on the third-order incomplete signal tensor to obtain an incomplete cross-correlation tensor, performing virtual array expansion processing to obtain a fifth-order incomplete augmented tensor, and reshaping the fifth-order incomplete augmented tensor into a third-order incomplete target tensor; and generating a mask tensor corresponding to the third-order incomplete target tensor, and then constructing and solving an optimization problem to obtain a factor matrix decomposition result so as to extract an estimation result of the direction of arrival. According to the method, the array physical defect problem is converted into a data recovery task, the self-healing capability defined by software is realized, and the robustness and accuracy of DOA estimation in the array element random failure scene are remarkably improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of array signal processing, and particularly relates to a method and device for estimating direction of arrival (DOA) of an array element failure tolerance based on incomplete tensor reconstruction. BACKGROUND

[0002] The signal direction of arrival (DOA) estimation is a key technology in the fields of radar, sonar, wireless communication and radio astronomy, and aims to determine the angle of a far-field signal source relative to a sensor array. The L-shaped array is widely used due to its simple structure and the ability to provide two-dimensional angle information.

[0003] In recent years, DOA estimation algorithms based on tensor algebra, such as parallel factor analysis (PARAFAC), have attracted widespread attention due to their ability to automatically realize multi-dimensional parameter pairing and improve degrees of freedom. However, these advanced algorithms are based on a strict premise that the physical structure and response characteristics (array manifold) of the sensor array are accurately known.

[0004] In practical applications, sensor array elements may randomly fail due to hardware aging, physical damage or environmental influences. The failure of array elements can destroy the geometric integrity of the array, causing the array manifold to deviate from its ideal mathematical model (such as the Vandermonde structure of a uniform linear array). This structural damage is fatal to traditional tensor algorithms, causing a catastrophic decline in performance and making it impossible to accurately estimate the DOA of the signal.

[0005] Existing solutions mainly include two types: one is to repair the failed array elements through hardware maintenance or replacement, which is costly and not timely, and in some application scenarios (such as satellites and deep-sea probes) it is even impossible to implement; the other is to compensate for array defects through complex software calibration algorithms, but such algorithms usually have high computational complexity and poor performance when there are a large number of failed array elements, making it difficult to achieve the desired effect. Therefore, developing a DOA estimation algorithm that can maintain high precision and robustness in the case of array element failure has important theoretical significance and application value. SUMMARY

[0006] The present application aims to overcome the shortcomings of traditional DOA estimation algorithms in the prior art, which are highly dependent on the integrity of the array structure and have severely deteriorated performance when array elements fail. A method and device for estimating the direction of arrival (DOA) of an array element failure tolerance based on incomplete tensor reconstruction are proposed. By converting the physical array element failure problem into a mathematical tensor data recovery task, the present application gives the sensor array a software-defined "self-healing" capability, greatly improving the robustness and reliability of target detection and positioning in radar, sonar and wireless communication systems under harsh working conditions.

[0007] The embodiment of the first aspect of the present application provides a method for estimating the direction of arrival of a failed array element fault-tolerant incomplete tensor reconstruction, comprising:

[0008] Performing array element working state detection on the received signals of the L-shaped sensor array to construct a third-order incomplete signal tensor of two direction arrays;

[0009] Performing cross-correlation operation on the third-order incomplete signal tensor of the two direction arrays to obtain an incomplete cross-correlation tensor and performing virtual array expansion processing, and then obtaining a fifth-order incomplete augmented tensor;

[0010] Reshaping the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through a generalized tensorization operation;

[0011] Generating a mask tensor corresponding to the third-order incomplete target tensor;

[0012] Constructing a weighted parallel factor analysis decomposition optimization problem for the mask tensor and solving the problem to obtain a factor matrix decomposition result;

[0013] Based on the factor matrix decomposition result, an estimation result of the direction of arrival is obtained.

[0014] In one specific embodiment of the present application, the third-order incomplete signal tensor of two direction arrays is constructed, comprising:

[0015] 1) Performing array element working state detection on the received signals of the L-shaped sensor array to generate a binary mask vector of the array element;

[0016] Wherein, the L-shaped sensor array is composed of two mutually perpendicular uniform linear arrays each containing M array elements, and is used for receiving signals from K far-field narrowband signal sources, wherein the x-direction array and the z-direction array are respectively arranged along the x-axis and the z-axis;

[0017] The working state of each array element in the array is detected to obtain two binary mask vectors with a length of M and ;

[0018] Wherein, the vector corresponds to the M array elements on the x-axis, and is a vector containing M elements, wherein the i-th element records the working state of the i-th physical array element on the x-axis;

[0019] The vector corresponds to the M array elements on the z-axis, and is a vector containing M elements, wherein the i-th element records the working state of the i-th physical array element on the z-axis;

[0020] In the two vectors, the value 1 of an element indicates that the corresponding physical array element is normal; the value 0 indicates that the corresponding physical array element has failed;

[0021] 2) Based on the result of step 1), establish a third-order incomplete signal tensor of each direction array in the L-shaped sensor array;

[0022] wherein the actual received data matrix of length N shots obtained from the x-axis and z-axis arrays is subarray partitioned to construct a third-order incomplete signal tensor and ;

[0023] wherein the specific partitioning manner of the subarray partitioning is as follows:

[0024] 2-1) Determine the length L of the subarray, and the value of L satisfies , wherein K is the estimated maximum number of sources, and M is the total number of array elements; the number of subarrays ;

[0025] Then the subarray length of the x direction and the z direction is , and the number of subarrays of the x direction and the z direction is ;

[0026] 2-2) Based on the received data matrix of the x-axis array , the process of constructing a third-order incomplete signal tensor is as follows:

[0027] Construct a Hankel matrix: for each column of , a dimensional Hankel matrix is constructed, and the jth column of the matrix is composed of a data segment with a length of L from the jth element in the original signal vector ;

[0028] Stack to form a tensor: stack the N Hankel matrices corresponding to N shots along the third dimension, i.e., the shot dimension, wherein the nth Hankel matrix constitutes the nth slice of the third-order tensor , denoted as , wherein: represents all data in the dimension;

[0029] 2-3) Based on the received data matrix of the z-axis array , the process of constructing a third-order incomplete signal tensor is as follows:

[0030] Construct a Hankel matrix: for each column, i.e., the signal vector of the tth shot of , a A dimensional Hankel matrix, wherein the j-th column of the Hankel matrix is ​​formed by the original signal vector. It consists of a data segment of length L starting from the j-th element;

[0031] Stacking to form a tensor: Stack the N Hankel matrices corresponding to the N snapshots along the third dimension, i.e., the snapshot dimension, where the nth Hankel matrix forms a third-order tensor. The nth slice is denoted as .

[0032] In a specific embodiment of the present invention, the expression for calculating the incomplete cross-correlation tensor is as follows:

[0033]

[0034] in, For an incomplete fourth-order cross-correlation tensor; N is the total number of snapshots; and These represent two-dimensional data slices of the x-axis and z-axis signal tensors at the nth snapshot time; This represents the outer product operation. This indicates the conjugate operation.

[0035] In a specific embodiment of the present invention, obtaining the fifth-order incomplete augmented tensor includes:

[0036] 1) For the incomplete fourth-order cross-correlation tensor Perform a conjugate inversion operation, which consists of two actions: dimension flipping, i.e., flipping the tensor... All four dimensions are reversed; element-wise conjugate is taken, that is, the complex conjugate of each element in the reversed tensor is taken;

[0037] Thus, a result is obtained with A fourth-order tensor of the same size is denoted as... ;

[0038] 2) The original incomplete fourth-order cross-correlation tensor And obtained in step 1) By splicing along a completely new fifth dimension of length 2, we obtain a fifth-order incomplete augmented tensor, denoted as . .

[0039] In a specific embodiment of the present invention, the step of reshaping the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization includes:

[0040] 1) Slice and reassemble the fifth-order incomplete augmented tensor to obtain a new fifth-order incomplete intermediate tensor. The specific steps are as follows:

[0041] 1-1) From the original fifth-order tensor In the middle, index slicing is performed along the first dimension (x-axis) and the third dimension (z-axis) of the subarray to extract four overlapping sub-data blocks of the same size, where:

[0042] Sub-block 1: ;

[0043] Sub-block 2: ;

[0044] Sub-block 3: ;

[0045] Sub-block 4: ;

[0046] Where [a:b] represents the Python-style slice index, that is, from index a to b-1;

[0047] 1-2) The four sub-data blocks obtained in step 1-1) Stacking along a new fourth dimension yields a new fifth-order incomplete intermediate tensor, denoted as . ;

[0048] 2) For incomplete intermediate tensors Performing a generalized tensor quantization operation, denoted as Thus Remodeled into a third-order incomplete target tensor ;in:

[0049] Will The first and third dimensions are merged to form The new first dimension;

[0050] Will The second and fourth dimensions are merged to form The new second dimension;

[0051] The fifth dimension remains unchanged, becoming The new third dimension.

[0052] In a specific embodiment of the present invention, generating the mask tensor corresponding to the third-order incomplete target tensor includes:

[0053] 1) Create two binary vectors of length M, all of which are 1, as pseudo-signals when there is no signal input;

[0054] 2) Using real physical mask vectors and The two binary vectors obtained in step 1) are multiplied element by element to simulate the effects of array element failure;

[0055] 3) Using the two binary vectors processed in step 2) as input, repeat the process of subarray partitioning, cross-correlation calculation, data augmentation and dimension reshaping to obtain the corresponding third-order incomplete target tensor;

[0056] 4) Binarize the third-order incomplete target tensor obtained in step 3) to generate the final mask tensor. ;

[0057] Specifically, a third-order tensor with the same size as the third-order incomplete target tensor obtained in step 3) and all elements initialized to zero is created as the initial mask tensor. ; Take all non-zero elements in the third-order incomplete target tensor obtained in step 3) and initialize them. The value at the corresponding position is set to 1, indicating that the data is reliable; all zero elements in the third-order incomplete target tensor obtained in step 3) are initialized. A value of 0 at the corresponding position indicates that the data is unreliable; after traversing all elements of this third-order incomplete target tensor, the final mask tensor is obtained. .

[0058] In one specific embodiment of the present invention, the objective of the optimization problem is based on a third-order incomplete objective tensor. and mask tensor Solve for a set of optimal low-rank factor matrices. The objective function of the optimization problem is:

[0059]

[0060] in, For a third-order incomplete target tensor, Its lengths are its three dimensions; R is the factor matrix to be solved, and R is the preset number of signal sources; This represents the element-wise product of Hadamah; This represents the Frobenius norm.

[0061] In a specific embodiment of the present invention, obtaining the factor matrix decomposition result includes:

[0062] 1) Initially, let the iteration number t=0, and then perform a factoring operation on the factor matrix. Perform random initialization, denoted as follows: , and ;

[0063] 2) Let the current iteration round be the (t+1)th iteration;

[0064] 3) In the current iteration, update the factor matrix. :

[0065] Fixed factor matrix obtained in the previous round and ,calculate and Khatri-Rao product ;

[0066] For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the first dimension; where, for the i-th row, the update process is as follows:

[0067] Expand the matrix from the modulo-1 of the incomplete target tensor. Extract the i-th row vector. Expand the matrix from the modulus-1 of the mask tensor. Extract the mask vector of the i-th row. According to the mask vector Find the set of all position indices where the value is 1; from and Select the elements and rows corresponding to the index set to obtain the valid data vectors. And effective KR product matrix Updated by solving the following standard linear least squares problem. :

[0068]

[0069] The solution to this problem is:

[0070]

[0071] The equivalent expression for this solution process is:

[0072]

[0073] in, It is a diagonal matrix whose diagonal elements are composed of mask vectors. constitute;

[0074] Factor matrix After traversing all rows, the updated matrix is ​​obtained. ;

[0075] 4) In the current iteration, update the factor matrix. :

[0076] Fixed factor matrix and ,calculate and Khatri-Rao product ;

[0077] For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the second dimension; where, for the j-th row, the update expression is:

[0078]

[0079] in, and These are third-order incomplete target tensors and mask tensor After modulo-2 expansion, the row vector and diagonal weight matrix corresponding to the j-th row are obtained.

[0080] Factor matrix After traversing all rows, we obtain the updated matrix. ;

[0081] 5) In the current iteration, update the factor matrix. :

[0082] Fixed factor matrix and ,calculate and Khatri-Rao product:

[0083] ;

[0084] For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the third dimension;

[0085] For the k-th row, the update expression is:

[0086]

[0087] in, and These are third-order incomplete target tensors and mask tensor After modulo-3 expansion, the row vector and diagonal weight matrix corresponding to the k-th row are obtained.

[0088] Factor matrix After traversing all rows, we obtain the updated matrix. ;

[0089] 6) After each iteration is completed, the reconstruction error or DOA estimation error is calculated and compared with the corresponding error of the previous round to determine whether the convergence condition is met: if not, let t=t+1 and then return to step 3) to enter the next iteration; if the condition is met, the iteration ends and the factor matrix decomposition result is obtained.

[0090] 7) Repeat steps 1) to 6) multiple times, each time generating a new set of random initial values ​​and iterating. Select the decomposition result that minimizes the objective function value of the optimization problem among all attempts as the final factor matrix estimate.

[0091] In one specific embodiment of the present invention, obtaining the estimated direction of arrival includes:

[0092] 1) For the matrix The kth column Calculate the azimuth-related complex phase term :

[0093]

[0094] in, It is a column vector The second element, It is its first element;

[0095] Calculate the complex phase term related to pitch angle :

[0096]

[0097] in, It is a column vector The third element;

[0098] 2) Based on the results of step 1), calculate the estimated azimuth angle. :

[0099]

[0100] Calculate the estimated pitch angle :

[0101]

[0102] in, It is a standard mathematical function used to calculate the phase angle factor matrix of a complex number. ;

[0103] 3) Repeat steps 1) to 2) to finally obtain a set of automatically paired two-dimensional DOA estimation results for all R signal sources. .

[0104] A second aspect of the present invention provides a fault-tolerant arrival direction estimation device for incomplete tensor reconstruction of array element failures, comprising:

[0105] The third-order signal tensor construction module is used to detect the array element working status of the received signal of the L-shaped sensor array in order to construct the third-order incomplete signal tensor of the array in two directions.

[0106] The array expansion module is used to perform cross-correlation operation on the third-order incomplete signal tensors of the two directional arrays to obtain incomplete cross-correlation tensors and perform virtual array expansion processing to obtain a fifth-order incomplete augmented tensor.

[0107] The tensor reconstruction module is used to reshape the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization operations.

[0108] The mask tensor generation module is used to generate the mask tensor corresponding to the third-order incomplete target tensor;

[0109] The factor matrix decomposition module is used to construct and solve a weighted parallel factor analysis decomposition optimization problem on the mask tensor to obtain the factor matrix decomposition result.

[0110] The arrival direction estimation module is used to obtain the arrival direction estimation result based on the factor matrix decomposition result.

[0111] A third aspect of the present invention provides an electronic device comprising:

[0112] At least one processor; and a memory communicatively connected to said at least one processor;

[0113] The memory stores instructions that can be executed by the at least one processor, and the instructions are configured to perform the above-described method for estimating the arrival direction of array element failures in incomplete tensor reconstruction.

[0114] A fourth aspect of the present invention provides a computer-readable storage medium storing computer instructions for causing the computer to execute the above-described method for estimating the arrival direction of array element failures in incomplete tensor reconstruction.

[0115] The features and beneficial effects of this invention are as follows:

[0116] High robustness and "self-healing" capability: This invention equates physical failure to data loss and achieves software-defined "self-healing" through "data interpolation" using a tensor-complete algorithm. Even under conditions of random failure of a large number of array elements, it can still maintain high estimation accuracy, with a gradual performance degradation rather than catastrophic collapse.

[0117] High efficiency and one-step solution: The weighted alternating least squares algorithm used in this invention directly outputs a factor matrix containing DOA information without having to fill in the entire tensor before decomposition, thus avoiding additional computational overhead and making it suitable for scenarios with real-time requirements.

[0118] Low cost and high applicability: This invention provides a purely software solution for addressing array defects, avoiding expensive hardware maintenance costs. The method can be extended to defect arrays with different geometries, as long as their signals can be constructed as tensors with low-rank properties. Attached Figure Description

[0119] Figure 1 This is an overall flowchart of the incomplete tensor reconstruction array element failure fault-tolerant arrival direction estimation method according to an embodiment of the present invention.

[0120] Figure 2 This is a scatter plot of DOA estimation results under different array element failure ratios in a specific embodiment of the present invention.

[0121] Figure 3 This is a performance comparison curve of the root mean square error (RMSE) of DOA estimation as a function of signal-to-noise ratio (SNR) in a specific embodiment of the present invention. Detailed Implementation

[0122] This invention proposes a method and apparatus for estimating the arrival direction of array element failures in incomplete tensor reconstruction. The invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0123] A first aspect of this invention proposes a method for estimating the arrival direction of array element failures in incomplete tensor reconstruction, comprising:

[0124] The operating status of the array elements is detected by the received signal of the L-shaped sensor array in order to construct the third-order incomplete signal tensor of the two-directional array;

[0125] Cross-correlation is performed on the third-order incomplete signal tensors of the two directional arrays to obtain an incomplete cross-correlation tensor, and then virtual array expansion processing is performed to obtain a fifth-order incomplete augmented tensor.

[0126] The fifth-order incomplete augmented tensor is reshaped into a third-order incomplete target tensor through a generalized tensor quantization operation.

[0127] Generate the mask tensor corresponding to the third-order incomplete target tensor;

[0128] A weighted parallel factor analysis decomposition optimization problem is constructed and solved for the mask tensor to obtain the factor matrix decomposition results;

[0129] Based on the factor matrix decomposition results, the estimated direction of arrival is obtained.

[0130] The core idea of ​​the method described in this embodiment is to creatively transform the problem of physical defects in the array into a data recovery problem in a high-dimensional data space. This method does not attempt to repair the physical array manifold, but rather considers the signal acquired by a defective array as equivalent to the ideal data tensor generated by a perfect array having missing data at certain locations. Based on this, a series of mathematical transformations are performed on the observed signal to construct an incomplete target tensor with missing terms. Then, utilizing the inherent low-rank property of this tensor, a factor matrix embedding the DOA parameters is directly recovered using tensor completeness techniques, thereby achieving robust estimation of the DOA.

[0131] In a specific embodiment of the present invention, the overall process of the incomplete tensor reconstruction array element failure fault-tolerant arrival direction estimation method is as follows: Figure 1 As shown, it includes the following steps:

[0132] 1) Detect the element operating status of the received signal from the L-shaped sensor array to construct a third-order incomplete signal tensor for the two directional arrays; the specific steps are as follows:

[0133] 1-1) Detect the working status of the array elements in the received signal of the L-shaped sensor array and generate the binary mask vector of the array element.

[0134] In this embodiment, it is assumed that an L-shaped array consisting of two mutually perpendicular uniform linear arrays (ULAs), each containing M array elements, is used to receive signals from K far-field narrowband signal sources, wherein the x-direction array and the z-direction array are deployed along the x-axis and z-axis, respectively.

[0135] This embodiment first detects or acquires preset device status information for the operating status of each array element. This information is recorded in two binary mask vectors of length M. and middle.

[0136] Specifically:

[0137] vector Corresponding to the M elements on the x-axis, it is a vector containing M elements, where the i-th element... The working state of the i-th physical array element on the x-axis is recorded.

[0138] vector This corresponds to M elements on the z-axis. It is a vector containing M elements, where the i-th element... The working state of the i-th physical array element on the z-axis is recorded.

[0139] In these two vectors, a value of "1" indicates that the corresponding physical element is functioning normally and can output a valid signal; a value of "0" indicates that the corresponding physical element has failed and cannot output a valid signal. For example, if the third element on the x-axis fails, then the vector... The third element is 0, that is .

[0140] Furthermore, in a preferred embodiment of the present invention, state detection can be achieved through a data-driven online self-calibration method, which has the advantage of requiring no additional hardware testing equipment. The specific steps are as follows:

[0141] 1-1-1) Quick capture for calibration:

[0142] The L-shaped sensor array is used to acquire short snapshots of data containing environmental noise or non-specific signals when not targeting a specific target signal.

[0143] 1-1-2) Calculate the energy of a single channel:

[0144] For each physical array element, calculate the average signal energy or power of the receiving channel within the acquired snapshot.

[0145] 1-1-3) Statistical Analysis and Threshold Determination:

[0146] Calculate the median and standard deviation of the energy for all M channels. Based on statistical principles, the received energy of a normally functioning array element should fall within a reasonable distribution range. If the received energy of an array element is significantly lower than the statistical center of other array elements (e.g., lower than "median - 3 standard deviations"), or its energy value is lower than an absolute minimum energy threshold set based on the system noise level, then the array element can be considered to be in a failed state.

[0147] Using this energy detection method, the system can dynamically and in real-time evaluate the operating status of each array element and generate the aforementioned binary mask vector. and .

[0148] 1-2) Based on the results of step 1-1), establish the third-order incomplete signal tensor of each direction array in the L-shaped sensor array.

[0149] In this embodiment, the actual received data matrix (both N snapshots in length) obtained from the x-axis and z-axis arrays in step 1-1) is used. We divide the matrix into subarrays (in dimensionality) to construct a third-order incomplete signal tensor. and .

[0150] The specific method for dividing the subarray is as follows:

[0151] 1-2-1) Determine the length L of the subarray. To effectively utilize the rotation invariance of the array in subsequent processing and ensure the stability of the algorithm, the value of the subarray length L needs to satisfy... The condition is given by K, where K is the estimated maximum number of information sources and M is the total number of array elements. A preferred value is... (Round down).

[0152] After determining the subarray length L, the number of subarrays That is, it was determined to be In this embodiment, the subarray parameters in the x and z directions are set to be the same, i.e. and .

[0153] 1-2-2) Construct an incomplete signal tensor along the x-axis.

[0154] In this embodiment, the received data matrix is ​​based on the x-axis array. Construct a third-order incomplete signal tensor The process is as follows:

[0155] Constructing the Hankel matrix: For Each column (i.e., the signal vector of each snapshot) , build a A dimensional Hankel matrix. The j-th column of this matrix is ​​formed by the original signal vector. It consists of a data segment of length L starting from the j-th element.

[0156] Stacking to form a tensor: Stack the N Hankel matrices corresponding to the N snapshots along the third dimension (the snapshot dimension). This forms the nth Hankel matrix, which is then a third-order tensor. The nth "slice", that is , where: represents all data in this dimension.

[0157] 1-2-3) Construct an incomplete signal tensor along the z-axis.

[0158] In this embodiment, the received data matrix is ​​based on the z-axis array. Construct a third-order incomplete signal tensor The process is as follows:

[0159] Constructing the Hankel matrix: For Each column (i.e., the signal vector of each snapshot) , build a A dimensional Hankel matrix. The j-th column of this matrix is ​​formed by the original signal vector. It consists of a data segment of length L starting from the j-th element.

[0160] Stacking to form a tensor: Stack the N Hankel matrices corresponding to the N snapshots along the third dimension (the snapshot dimension). This forms the nth Hankel matrix, which is then a third-order tensor. The nth "slice", that is .

[0161] Through the above operations, the two-dimensional received data matrix and Successfully reshaped into a third-order tensor and Because the original data contains zero or error values ​​due to array element failures, this incomplete information is directly reflected in the constructed Hankel matrix and the final third-order tensor, making it an incomplete signal tensor.

[0162] 2) Based on the results of step 1), perform cross-correlation on the third-order incomplete signal tensors of the two directional arrays to obtain incomplete cross-correlation tensors, and then perform virtual array expansion processing to obtain fifth-order incomplete augmented tensors; the specific steps are as follows:

[0163] 2-1) Based on the results of step 1), calculate the incomplete cross-correlation tensor.

[0164] In this embodiment, the calculation process aims to estimate the second-order statistical properties of the signal by averaging the data from all snapshots, thereby eliminating the dependence on specific signal waveforms and random noise. The specific calculation expression is as follows:

[0165]

[0166] in, For an incomplete fourth-order cross-correlation tensor; N is the total number of snaps. n is the index of the snap, traversing from the first snap (n=1) to the last snap (n=N). and These represent two-dimensional data "slices" of the x-axis and z-axis signal tensors at the nth snapshot time. This represents the outer product operation. This indicates the conjugate operation. The symbol represents the element-wise summation of the outer product results calculated from all N snapshots. This means dividing the accumulated result by the total number of snapshots to perform a time average, which mathematically corresponds to the expectation operator. A practical estimate.

[0167] 2-2) Data augmentation is performed on the incomplete cross-correlation tensor obtained in step 2-1) to obtain a fifth-order incomplete augmented tensor.

[0168] Based on the incomplete fourth-order cross-correlation tensor obtained in step 2-1) This embodiment utilizes the inherent conjugate symmetry of a uniform linear array (ULA) to construct an augmented data tensor. The purpose of this operation is to artificially construct more information redundancy and specific symmetry structures in the data, which helps to extract angle parameters more stably and accurately in subsequent steps and effectively improves the degree of freedom of the algorithm.

[0169] The specific steps for data augmentation are as follows:

[0170] 2-2-1) Constructing the conjugate symmetric part:

[0171] For incomplete fourth-order cross-correlation tensors Perform a conjugate inversion operation. This operation consists of two actions: dimension flipping, i.e., flipping the tensor... All four dimensions are reversed; element-wise conjugate is taken, that is, the complex conjugate of each element in the reversed tensor is taken.

[0172] Through these two actions, a new one is obtained, which is... A fourth-order tensor of the same size is denoted as... .

[0173] 2-2-2) The splicing forms a fifth-order augmented tensor:

[0174] The original incomplete fourth-order cross-correlation tensor and the conjugate symmetric part constructed in step 2-2-1). It is stacked along a brand new fifth dimension with a length of 2.

[0175] The concatenated result is a fifth-order incomplete augmented tensor, denoted as . .

[0176] In this new fifth-order tensor, taking the first "slice" along the fifth dimension yields the original... Taking the second "slice" yields... .

[0177] Through this data augmentation step, the method described in this embodiment transforms the input incomplete fourth-order tensor into a fifth-order incomplete tensor with higher dimensions and richer structure. This provides the necessary input for subsequent generalized tensor quantization operations.

[0178] 3) The fifth-order incomplete augmented tensor obtained in step 2) is reshaped into a third-order incomplete target tensor through generalized tensor quantization.

[0179] In this embodiment, this step aims to transform the structurally complex high-dimensional tensor obtained in step 2) into a structurally clear third-order tensor that can be directly used for parallel factorization. This process includes two sub-steps: factor recombination and dimension merging, as detailed below:

[0180] 3-1) Slice and recombine the fifth-order incomplete augmented tensor obtained in step 2) to obtain a new fifth-order incomplete intermediate tensor.

[0181] In this embodiment, the fifth-order incomplete augmented tensor obtained in step 2) is... A delicate slicing and recombination operation is performed. The purpose of this operation is to extract data sub-blocks with specific phase translation relationships from the original tensor and explicitly encode these relationships into a new dimension, thereby constructing an intermediate tensor that can directly reflect two-dimensional rotation invariance.

[0182] This process is mathematically equivalent to constructing a fifth-order incomplete tensor. The specific operating steps are as follows:

[0183] 3-1-1) Define four sub-data blocks:

[0184] From the original fifth-order tensor In the middle, index slicing is performed along the first dimension (the internal dimension of the x-axis subarray) and the third dimension (the internal dimension of the z-axis subarray) to extract four overlapping sub-data blocks of the same size, where:

[0185] Sub-block 1: ;

[0186] Sub-block 2: ;

[0187] Sub-block 3: ;

[0188] Sub-block 4: ;

[0189] Here, [a:b] represents the Python-style slice index, i.e., from index a to b-1. These four sub-blocks correspond to the data obtained by performing "no translation", "x-direction translation", "z-direction translation", and "simultaneous x and z-direction translation" on the virtual two-dimensional array, respectively.

[0190] 3-1-2) Stacking to form intermediate tensors:

[0191] The four sub-data blocks obtained in step 3-1-1) Stack along a new fourth dimension. The result of the stacking is a new fifth-order incomplete intermediate tensor, denoted as . In this new tensor In this context, the fourth dimension has a length of 4, and the indices j=0,1,2,3 of this dimension correspond to the four sub-blocks in the original data. .

[0192] Through this slicing and recombination step, this embodiment transforms the translation invariance (i.e., phase relationship) hidden within the data into an explicit tensor dimension that can be processed later, paving the way for the final extraction of angular information.

[0193] 3-2) Perform dimension merging on the fifth-order incomplete intermediate tensor obtained in step 3-1).

[0194] In this embodiment, the incomplete intermediate tensor Perform generalized tensor quantization, i.e., dimension merging. This operation is denoted as... This is reshaped into the final third-order incomplete objective tensor. The specific meaning of this dimension merging operation is:

[0195] Will The first and third dimensions are merged to form The new first dimension.

[0196] Will The second and fourth dimensions are merged to form The new second dimension.

[0197] The fifth dimension remains unchanged, becoming The new third dimension.

[0198] Through this operation, this embodiment ultimately obtains a third-order incomplete target tensor, denoted as . The dimension of this tensor is This final construction It will be used as the direct input for the subsequent weighted tensor decomposition step.

[0199] 4) Based on the results of step 3), generate the corresponding mask tensor.

[0200] In this embodiment, the incomplete target tensor obtained in step 3) is precisely labeled. Which elements are reliable and which are unreliable due to array element failure needs to be determined by generating a [database name]. Same-dimensional binary mask tensor . specific generation The method is as follows:

[0201] 4-1) Create two binary vectors of length M, where all elements are "1", to serve as "pseudo-signals" when there is no signal input.

[0202] 4-2) Using the real physical mask vector obtained in step 1) and The two binary vectors obtained in step 4-1) are multiplied element by element to simulate the effects of array element failure.

[0203] 4-3) Using the two binary vectors processed in step 4-2) as input, repeat the calculation process from step 1-2) to step 3), including subarray partitioning, cross-correlation calculation, data augmentation and dimension reshaping, to obtain the corresponding third-order incomplete target tensor.

[0204] 4-4) Binarize the third-order incomplete target tensor obtained in step 4-3) to generate the final mask tensor. The specific steps are as follows:

[0205] Create a third-order tensor of the same size as the third-order incomplete target tensor obtained in step 4-3), with all elements initialized to zero, as the initial mask tensor. Specifically, a third-order tensor of the same size as the third-order incomplete target tensor obtained in step 3) is created, with all elements initialized to zero, and used as the initial mask tensor. ; Take all non-zero elements in the third-order incomplete target tensor obtained in step 3) and initialize them. The value at the corresponding position is set to 1, indicating that the data path at that position is reliable; all zero elements in the third-order incomplete target tensor obtained in step 3) are initialized... A value of 0 at the corresponding position indicates that the data path at that position is blocked due to array element failure, and the data is unreliable; the initial mask tensor After all elements in the mask are updated, the final mask tensor is obtained. .

[0206] 5) For the mask tensor obtained in step 4), construct and solve the weighted parallel factor analysis (PARAFAC) decomposition optimization problem to obtain the factor matrix decomposition results; the specific steps are as follows:

[0207] 5-1) Construct an optimization problem.

[0208] One of the core innovations of this embodiment is that it creatively transforms the complex physical fault tolerance problem into a well-defined DOA parameter estimation problem, which is constructed as a weighted parallel factorization optimization problem. The goal of this problem is to obtain the DOA parameter estimate from the known third-order incomplete objective tensor. and mask tensor A set of optimal low-rank factor matrices can be directly solved. .

[0209] Specifically, the objective function of this optimization problem is constructed as follows:

[0210]

[0211] in, For the aforementioned third-order incomplete target tensor, Its lengths are its three dimensions; R is the factor matrix to be solved, and R is the preset number of signal sources; This represents the element-wise product of Hadamah; This represents the Frobenius norm.

[0212] Solving this optimization problem involves finding an optimal set of factor matrices such that the reconstructed tensor is derived from the mask tensor. The marked known observation locations, and the observation data The weighted error is minimized. By solving this optimization problem, a factor matrix containing the DOA parameters can be directly obtained, realizing the software-defined array "self-healing" function.

[0213] 5-2) The alternating least squares algorithm is used to solve the weighted parallel factor analysis (PARAFAC) decomposition problem.

[0214] In this embodiment, the optimization problem constructed in step 5-1) is solved using an alternating least squares (ALS) algorithm with a multiple restart strategy. This algorithm solves the problem by analyzing the factor matrix. The algorithm iteratively updates the solution to find the optimal solution. In a single ALS iteration, the algorithm updates the solution sequentially in a fixed order during each iteration. Three matrices. The core of this update process is to transform the optimization problem of the third-order tensor into a series of weighted linear least squares subproblems that are easier to solve for matrices or vectors.

[0215] For a single ALS solution, the specific steps are as follows:

[0216] 5-2-1) Initialization of the factor matrix.

[0217] Initially, let the iteration number t=0, and then perform iterations on the three factor matrices. Assign initial values, denoted as follows: , and In this embodiment, random initialization is used. Specifically, the three matrices are filled with complex random numbers that follow a standard normal distribution.

[0218] 5-2-2) Let the current iteration round be the (t+1)th iteration;

[0219] 5-2-3) In the current iteration round, update the factor matrix. :

[0220] Fixed factor matrix obtained in the previous round and First, calculate their Khatri-Rao product. Then, for the factor matrix each line (in , For a third-order incomplete target tensor The length of the first dimension is updated independently. For the i-th row, the update process is as follows:

[0221] Expand the matrix from the modulo-1 of the incomplete target tensor. Extract the i-th row vector. Expand the matrix from the modulus-1 of the mask tensor. Extract the mask vector of the i-th row. Based on the mask vector Find the set of all indexes where the value is 1. and In this process, only the elements and rows corresponding to the above index set are selected to obtain the effective data vectors. And effective KR product matrix The update is achieved by solving the following standard linear least squares problem. :

[0222]

[0223] The solution to this problem is:

[0224]

[0225] The updated matrix can be obtained by traversing all rows i. .

[0226] It should be noted that the solution process for each row described above is sometimes expressed in a more compact theoretical form in mathematical literature:

[0227]

[0228] in, It is a diagonal matrix whose diagonal elements are composed of mask vectors. The formula is mathematically equivalent to the step-by-step solution method described above.

[0229] 5-2-4) In the current iteration round, update the factor matrix. :

[0230] Fix the factor matrix that was just updated in this round. And the factor matrix of the previous round First, calculate their Khatri-Rao product, but this time in a different order: Then, for the factor matrix each line (in , For a third-order incomplete target tensor The second dimension is updated independently. This process requires expanding the tensor modulo-2, which physically means using the second dimension of the tensor as the row of the new matrix. For the j-th row, the update process is the same as the previous update. The update is exactly the same for all rows, that is, by selecting the valid data related to that row and solving a standard linear least squares problem, the update expression is:

[0231]

[0232] in, and These are third-order incomplete target tensors and mask tensor After modulo-2 expansion, the row vector and diagonal weight matrix corresponding to row j are obtained. The updated matrix can be obtained by iterating through all rows j. .

[0233] 5-2-5) In the current iteration round, update the factor matrix. .

[0234] Fix the factor matrix that was just updated in this round. and First, calculate their Khatri-Rao product: Then, for the factor matrix each line (in , For a third-order incomplete target tensor The length of the third dimension is updated independently. This process requires expanding the tensor modulo-3. For the k-th row, the update formula is:

[0235]

[0236] in, and These are third-order incomplete target tensors and mask tensor After modulo-3 expansion, the row vectors and diagonal weight matrix corresponding to the k-th row are obtained. The updated matrix can be obtained by iterating through all rows k. .

[0237] 5-2-6) After each iteration is completed, calculate the reconstruction error or directly calculate the DOA estimation error, and compare it with the error of the previous iteration to determine whether the convergence condition is met. If the convergence condition is not met in the current iteration, let t = t + 1, and then return to step 5-2-3) to enter the next iteration. The next iteration will start from updating the matrix. Started, and updating At that time, the matrix obtained from the previous iteration (t+1th iteration) will be used. and .

[0238] In this embodiment, the iterative process terminates when any of the following conditions are met:

[0239] Error convergence: The rate of change of the reconstruction error calculated in two consecutive iterations is less than a preset convergence threshold. This threshold is required to be a sufficiently small positive number to balance computational accuracy and convergence speed. In this embodiment, a preferred value is... .

[0240] Maximum number of iterations reached: The number of iterations reaches a preset upper limit to prevent the algorithm from getting stuck in an infinite loop. This parameter must be a sufficiently large positive integer to ensure the algorithm has enough time to converge. In this embodiment, a preferred value is 200 iterations.

[0241] 5-2-7) To improve the stability of the algorithm and avoid getting trapped in local optima, this embodiment employs a multiple restart strategy: that is, repeating steps 5-2-1) to 5-2-6) multiple times, so that the entire ALS iteration process is executed independently multiple times (e.g., 15 times), each time starting the iteration attempt from a newly generated set of random initial values. Finally, the decomposition result corresponding to the attempt that minimizes the objective function value of the optimization problem among all attempts is selected as the final factor matrix estimate.

[0242] 6) Extract the arrival direction from the factor matrix obtained in step 5).

[0243] In this embodiment, after the ALS iterative algorithm in step 5) converges, its final output is a set of reconstructed factor matrices. All information regarding the two-dimensional direction of arrival (DOA) is encoded in the factor matrix. This step aims to decode the DOA angle estimates of all signal sources from this matrix.

[0244] From the matrix The kth column ( The specific steps for extracting the angle information of a single signal source are as follows:

[0245] 6-1) Extract the complex phase term.

[0246] This step utilizes the factor matrix. The specific column structure extracts the angle-related complex phase term through ratio operations between elements. This ratio operation utilizes the inherent scale invariance of parallel factorization, which can automatically eliminate arbitrary scale factors introduced during the decomposition process, thereby obtaining normalized phase information.

[0247] Calculate the complex phase term related to azimuth angle :

[0248]

[0249] in, It is a column vector The second element, It is its first element.

[0250] Calculate the complex phase term related to pitch angle :

[0251]

[0252] in, It is a column vector The third element.

[0253] 6-2) Based on the results of step 6-1), calculate the two-dimensional arrival direction.

[0254] This step recovers the actual angle value from the complex phase term obtained in the previous step using inverse trigonometric function operations.

[0255] Calculate the estimated azimuth angle :

[0256]

[0257] Calculate the estimated pitch angle :

[0258]

[0259] in, It is a standard mathematical function used to calculate the phase angle (in radians) of a complex number.

[0260] For factor matrix Repeat steps 6-1) and 6-2) for each column to obtain a set of automatically paired two-dimensional DOA estimation results for all R signal sources. .

[0261] The advantage of this method is that it directly obtains the DOA estimate from the tensor decomposition result, avoiding the parameter pairing problem and additional spectral peak search steps in traditional methods.

[0262] This embodiment transforms the physical element failure problem into a mathematical tensor completion problem, achieving robust DOA estimation even with up to 60% element failure. This method is theoretically sound, computationally efficient, and highly fault-tolerant, providing an effective solution for unavoidable hardware failures in practical applications.

[0263] Figure 2 This is a scatter plot of DOA estimation results under different element failure ratios in a specific embodiment of the present invention. In this embodiment, an L-shaped array composed of two mutually perpendicular 10-element uniform linear arrays is used, with the element spacing set to half the center wavelength of the signal. The received signal contains four far-field narrowband signal sources with true directions of arrival (DOA) of (58°, 66°), (67°, 130°), (76°, 86°), and (85°, 105°), respectively, and the number of snapshots is set to 500. The figure shows the scatter plot distribution of DOA estimates for four different array element failure levels under a fixed signal-to-noise ratio of 10 dB. Figure 2The first figure shows the case with a small amount of missing data (9.8% missing data). The TCDA estimation results are closely clustered around the actual DOA locations, with a root mean square error (RMSE) of 0.20°, close to the ideal 0.14° boundary. The second figure shows the scenario with moderate failure (29.6% missing data). The algorithm maintains excellent robustness, with an RMSE of only 0.22°. The third figure shows the performance under severe failure (39.3% missing data). Although the estimation variance increases slightly, the TCDA RMSE is still controlled within 0.33°, accurately distinguishing all signal sources. The fourth figure shows the extreme test results under extremely severe failure (59.4% missing data). Although traditional methods would completely fail under this condition, the method described in this embodiment can still provide usable estimation results, with an RMSE of 0.39°. The results show that the method described in this embodiment has excellent "self-healing" capabilities, with performance decreasing gradually with the degree of physical damage, effectively bridging the gap between catastrophic failure and perfect array performance.

[0264] Figure 3 This is a performance comparison curve of the root mean square error (RMSE) of DOA estimation as a function of signal-to-noise ratio (SNR) in a specific embodiment of the present invention, demonstrating the performance under different numbers of array element failures. This embodiment uses 1000 Monte Carlo experiments to statistically reduce the impact of random noise. Figure 3 In the figure, the horizontal axis represents the signal-to-noise ratio (SNR), ranging from -10 dB to 20 dB; the vertical axis represents the root mean square (RMS) error. Experiments were conducted with different scenarios ranging from 1 to 8 randomly failed array elements. All performance curves exhibited the expected behavior: the RMS error decreased with increasing SNR, and all curves were above the Cramer-Rhodes boundary (CRB). In the low SNR region, the performance curves for different numbers of failures clustered closely and approximated the curves of the ideal (no failure) case, indicating that additive noise is the dominant factor in the error. In the high SNR region, the curves were clearly separated and presented different "error bases," the level of which was proportional to the number of failures, indicating that the performance limit of the algorithm is determined by the amount of lost spatial information. The most important finding is that the TCDA algorithm exhibits a "smooth degradation" characteristic: its performance does not catastrophically collapse after a few failures, but rather decreases smoothly and predictably with increasing failure numbers. Notably, in the case of 1 to 3 failed array elements, the algorithm shows almost no performance loss compared to the ideal case. Even under severe conditions with eight failed array elements, the algorithm maintains robust performance, with the root mean square error remaining below 1 degree when the signal-to-noise ratio is above -2 dB. This predictable and smooth performance degradation confirms that the invention is a highly reliable and robust choice for varying noise conditions and unavoidable hardware failures.

[0265] To implement the above embodiments, a second aspect of the present invention proposes a fault-tolerant arrival direction estimation device for incomplete tensor reconstruction of array element failures, comprising:

[0266] The third-order signal tensor construction module is used to detect the array element working status of the received signal of the L-shaped sensor array in order to construct the third-order incomplete signal tensor of the array in two directions.

[0267] The array expansion module is used to perform cross-correlation operation on the third-order incomplete signal tensors of the two directional arrays to obtain incomplete cross-correlation tensors and perform virtual array expansion processing to obtain a fifth-order incomplete augmented tensor.

[0268] The tensor reconstruction module is used to reshape the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization operations.

[0269] The mask tensor generation module is used to generate the mask tensor corresponding to the third-order incomplete target tensor;

[0270] The factor matrix decomposition module is used to construct and solve a weighted parallel factor analysis decomposition optimization problem on the mask tensor to obtain the factor matrix decomposition result.

[0271] The arrival direction estimation module is used to obtain the arrival direction estimation result based on the factor matrix decomposition result.

[0272] It should be noted that the aforementioned explanation of an embodiment of a fault-tolerant arrival direction estimation method for array element failure based on incomplete tensor reconstruction also applies to a fault-tolerant arrival direction estimation device for array element failure based on incomplete tensor reconstruction in this embodiment, and will not be repeated here. According to an embodiment of the present invention, a fault-tolerant arrival direction estimation device for array element failure based on incomplete tensor reconstruction involves detecting the array element operating status of the received signal of an L-shaped sensor array to construct two third-order incomplete signal tensors for the directional arrays; performing cross-correlation operations on the three-order incomplete signal tensors of the two directional arrays to obtain an incomplete cross-correlation tensor and performing virtual array expansion processing to obtain a fifth-order incomplete augmented tensor; reshaping the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization; generating a mask tensor corresponding to the third-order incomplete target tensor; constructing and solving a weighted parallel factor analysis decomposition optimization problem on the mask tensor to obtain the factor matrix decomposition result; and obtaining the arrival direction estimation result based on the factor matrix decomposition result. This enables the sensor array to achieve a software-defined "self-healing" capability by transforming the physical array element failure problem into a mathematical tensor data recovery task, which greatly improves the robustness and reliability of radar, sonar and wireless communication systems in target detection and positioning under harsh conditions.

[0273] To implement the above embodiments, a third aspect of the present invention provides an electronic device, comprising:

[0274] At least one processor; and a memory communicatively connected to said at least one processor;

[0275] The memory stores instructions that can be executed by the at least one processor, and the instructions are configured to perform the above-described method for estimating the arrival direction of array element failures in incomplete tensor reconstruction.

[0276] To implement the above embodiments, a fourth aspect of the present invention provides a computer-readable storage medium storing computer instructions for causing the computer to execute the above-described method for estimating the arrival direction of array element failures in incomplete tensor reconstruction.

[0277] It should be noted that the computer-readable medium described in this disclosure can be a computer-readable signal medium or a computer-readable storage medium, or any combination thereof. A computer-readable storage medium can be, for example,—but not limited to—an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of a computer-readable storage medium may include, but are not limited to: an electrical connection having one or more wires, a portable computer disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination thereof. In this disclosure, a computer-readable storage medium can be any tangible medium containing or storing a program that can be used by or in connection with an instruction execution system, apparatus, or device. In this disclosure, a computer-readable signal medium can include a data signal propagated in baseband or as part of a carrier wave, carrying computer-readable program code. Such propagated data signals can take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A computer-readable signal medium can be any computer-readable medium other than a computer-readable storage medium, which can send, propagate, or transmit a program for use by or in connection with an instruction execution system, apparatus, or device. The program code contained on the computer-readable medium can be transmitted using any suitable medium, including but not limited to: wires, optical fibers, RF (radio frequency), etc., or any suitable combination thereof.

[0278] The aforementioned computer-readable medium may be included in the aforementioned electronic device; or it may exist independently and not assembled into the electronic device. The aforementioned computer-readable medium carries one or more programs, which, when executed by the electronic device, cause the electronic device to perform a method for estimating the fault-tolerant arrival direction of array element failures in an incomplete tensor reconstruction, as described in the above embodiment.

[0279] Computer program code for performing the operations of this disclosure can be written in one or more programming languages ​​or a combination thereof, including object-oriented programming languages ​​such as Java, Smalltalk, and C++, and conventional procedural programming languages ​​such as the "C" language or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network—including a local area network (LAN) or a wide area network (WAN)—or can be connected to an external computer (e.g., via the Internet using an Internet service provider).

[0280] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0281] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0282] Any process or method described in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or more executable instructions for implementing a particular logical function or process, and the scope of the preferred embodiments of this application includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the function involved, as will be understood by those skilled in the art to which embodiments of this application pertain.

[0283] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-included system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device. More specific examples (a non-exhaustive list) of computer-readable media include: an electrical connection having one or more wires (electronic device), a portable computer disk drive (magnetic device), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). In addition, computer-readable media can even be paper or other suitable media on which programs can be printed, because programs can be obtained electronically, for example, by optically scanning paper or other media, followed by editing, interpreting or otherwise processing as necessary, and then stored in computer memory.

[0284] It should be understood that various parts of this application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented using software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0285] Those skilled in the art will understand that all or part of the steps of the methods in the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, the program includes one or a combination of the steps of the method embodiments.

[0286] Furthermore, the functional units in the various embodiments of this application can be integrated into a processing module, or each unit can exist physically separately, or two or more units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium.

[0287] The storage medium mentioned above can be a read-only memory, a disk, or an optical disk, etc. Although embodiments of this application have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting this application. Those skilled in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of this application.

Claims

1. A method for estimating the arrival direction of array elements in incomplete tensor reconstruction with fault tolerance, characterized in that, include: The operating status of the array elements is detected by the received signal of the L-shaped sensor array in order to construct the third-order incomplete signal tensor of the two-directional array; Cross-correlation is performed on the third-order incomplete signal tensors of the two directional arrays to obtain an incomplete cross-correlation tensor, and then virtual array expansion processing is performed to obtain a fifth-order incomplete augmented tensor. The fifth-order incomplete augmented tensor is reshaped into a third-order incomplete target tensor through a generalized tensor quantization operation. Generate the mask tensor corresponding to the third-order incomplete target tensor; A weighted parallel factor analysis decomposition optimization problem is constructed and solved for the mask tensor to obtain the factor matrix decomposition results; Based on the factor matrix decomposition results, the estimated direction of arrival is obtained.

2. The method according to claim 1, characterized in that, The construction of the third-order incomplete signal tensor for the two directional arrays includes: 1) Detect the working status of the array elements in the received signal of the L-shaped sensor array and generate the binary mask vector of the array element; The L-shaped sensor array is composed of two mutually perpendicular uniform linear arrays, each containing M array elements, used to receive signals from K far-field narrowband signal sources, wherein the x-direction array and the z-direction array are deployed along the x-axis and z-axis, respectively. The operating state of each element in the array is detected, resulting in two binary mask vectors of length M. and ; Where, vector The M elements corresponding to the x-axis are represented by a vector containing M elements, where the i-th element is... The working state of the i-th physical array element on the x-axis is recorded; vector The M elements corresponding to the z-axis are represented by a vector containing M elements, where the i-th element is... The working state of the i-th physical array element on the z-axis is recorded; In these two vectors, a value of 1 indicates that the corresponding physical element is functioning normally; a value of 0 indicates that the corresponding physical element has failed. 2) Based on the results of step 1), establish the third-order incomplete signal tensor for each direction of the L-shaped sensor array; Specifically, the actual received data matrix of length N snapshots, obtained from the x-axis and z-axis arrays, is subdivided to construct a third-order incomplete signal tensor. and ; The specific method for dividing the subarray is as follows: 2-1) Determine the length L of the subarray, where the value of L satisfies Where K is the estimated maximum number of information sources, M is the total number of array elements, and the number of subarrays. ; Then the subarray lengths in the x and z directions are The number of subarrays in the x and z directions is ; 2-2) Received data matrix based on x-axis array Construct a third-order incomplete signal tensor The process is as follows: Constructing the Hankel matrix: For Each column , build a A dimensional Hankel matrix, the j-th column of which is formed by the original signal vector. It consists of a data segment of length L starting from the j-th element; Stacking to form a tensor: Stack the N Hankel matrices corresponding to the N snapshots along the third dimension, i.e., the snapshot dimension, where the nth Hankel matrix forms a third-order tensor. The nth slice is denoted as , where: represents all data in this dimension; 2-3) Received data matrix based on z-axis array Construct a third-order incomplete signal tensor The process is as follows: Constructing the Hankel matrix: For Each column, i.e., the signal vector of the t-th snapshot , build a A dimensional Hankel matrix, wherein the j-th column of the Hankel matrix is ​​formed by the original signal vector. It consists of a data segment of length L starting from the j-th element; Stacking to form a tensor: Stack the N Hankel matrices corresponding to the N snapshots along the third dimension, i.e., the snapshot dimension, where the nth Hankel matrix forms a third-order tensor. The nth slice is denoted as .

3. The method according to claim 2, characterized in that, The expression for calculating the incomplete cross-correlation tensor is as follows: in, For an incomplete fourth-order cross-correlation tensor; N is the total number of snapshots; and These represent two-dimensional data slices of the x-axis and z-axis signal tensors at the nth snapshot time; This represents the outer product operation. This indicates the conjugate operation.

4. The method according to claim 3, characterized in that, The obtained fifth-order incomplete augmented tensor includes: 1) For the incomplete fourth-order cross-correlation tensor Perform a conjugate inversion operation, which consists of two actions: dimension flipping, i.e., flipping the tensor... All four dimensions are reversed; element-wise conjugate is taken, that is, the complex conjugate of each element in the reversed tensor is taken; Thus, a result is obtained with A fourth-order tensor of the same size is denoted as... ; 2) The original incomplete fourth-order cross-correlation tensor And obtained in step 1) By splicing along a completely new fifth dimension of length 2, we obtain a fifth-order incomplete augmented tensor, denoted as . .

5. The method according to claim 4, characterized in that, The process of reshaping the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization includes: 1) Slice and reassemble the fifth-order incomplete augmented tensor to obtain a new fifth-order incomplete intermediate tensor. The specific steps are as follows: 1-1) From the original fifth-order tensor In the middle, index slicing is performed along the first dimension (x-axis) and the third dimension (z-axis) of the subarray to extract four overlapping sub-data blocks of the same size, where: Block 1: ; Sub-block 2: ; Block 3: ; Block 4: ; Where [a:b] represents the Python-style slice index, that is, from index a to b-1; 1-2) The four sub-data blocks obtained in step 1-1) Stacking along a new fourth dimension yields a new fifth-order incomplete intermediate tensor, denoted as . ; 2) For incomplete intermediate tensors Performing a generalized tensor quantization operation, denoted as Thus Remodeled into a third-order incomplete target tensor ;in: Will The first and third dimensions are merged to form The new first dimension; Will The second and fourth dimensions are merged to form The new second dimension; The fifth dimension remains unchanged, becoming The new third dimension.

6. The method according to claim 5, characterized in that, The generation of the mask tensor corresponding to the third-order incomplete target tensor includes: 1) Create two binary vectors of length M, all of which are 1, as pseudo-signals when there is no signal input; 2) Using real physical mask vectors and The two binary vectors obtained in step 1) are multiplied element by element to simulate the effects of array element failure; 3) Using the two binary vectors processed in step 2) as input, repeat the process of subarray partitioning, cross-correlation calculation, data augmentation and dimension reshaping to obtain the corresponding third-order incomplete target tensor; 4) Binarize the third-order incomplete target tensor obtained in step 3) to generate the final mask tensor. ; Specifically, a third-order tensor with the same size as the third-order incomplete target tensor obtained in step 3) and all elements initialized to zero is created as the initial mask tensor. ; Take all non-zero elements in the third-order incomplete target tensor obtained in step 3) and initialize them. The value at the corresponding position is set to 1, indicating that the data is reliable; all zero elements in the third-order incomplete target tensor obtained in step 3) are initialized. The value at the corresponding position is 0, indicating that the data is unreliable; the initial mask tensor After all elements in the mask are updated, the final mask tensor is obtained. .

7. The method according to claim 6, characterized in that, The objective of the optimization problem is to solve the problem based on a third-order incomplete objective tensor. and mask tensor Solve for a set of optimal low-rank factor matrices. The objective function of the optimization problem is: in, For a third-order incomplete target tensor, Its lengths are its three dimensions; R is the factor matrix to be solved, and R is the preset number of signal sources; This represents the element-wise product of Hadamah; This represents the Frobenius norm.

8. The method according to claim 7, characterized in that, The obtained factor matrix decomposition results include: 1) Initially, let the iteration number t=0, and then perform a factoring operation on the factor matrix. Perform random initialization, denoted as follows: , and ; 2) Let the current iteration round be the (t+1)th iteration; 3) In the current iteration, update the factor matrix. : Fixed factor matrix obtained in the previous round and ,calculate and Khatri-Rao product ; For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the first dimension; where, for the i-th row, the update process is as follows: Expand the matrix from the modulo-1 of the incomplete target tensor. Extract the i-th row vector. Expand the matrix from the modulus-1 of the mask tensor. Extract the mask vector of the i-th row. According to the mask vector Find the set of all position indices where the value is 1; from and Select the elements and rows corresponding to the index set to obtain the valid data vectors. And effective KR product matrix Updated by solving the following standard linear least squares problem. : The solution to this problem is: The equivalent expression for this solution process is: in, It is a diagonal matrix whose diagonal elements are composed of mask vectors. constitute; Factor matrix After traversing all rows, the updated matrix is ​​obtained. ; 4) In the current iteration, update the factor matrix. : Fixed factor matrix and ,calculate and Khatri-Rao product ; For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the second dimension; where, for the j-th row, the update expression is: in, and These are third-order incomplete target tensors and mask tensor After modulo-2 expansion, the row vector and diagonal weight matrix corresponding to the j-th row are obtained. Factor matrix After traversing all rows, we obtain the updated matrix. ; 5) In the current iteration, update the factor matrix. : Fixed factor matrix and ,calculate and Khatri-Rao product: ； For factor matrix each line Perform independent updates. , For a third-order incomplete target tensor The length of the third dimension; For the k-th row, the update expression is: in, and These are third-order incomplete target tensors and mask tensor After modulo-3 expansion, the row vector and diagonal weight matrix corresponding to the k-th row are obtained. Factor matrix After traversing all rows, we obtain the updated matrix. ; 6) After each iteration is completed, the reconstruction error or DOA estimation error is calculated and compared with the corresponding error of the previous round to determine whether the convergence condition is met: if not, let t=t+1 and then return to step 3) to enter the next iteration; if the condition is met, the iteration ends and the factor matrix decomposition result is obtained. 7) Repeat steps 1) to 6) multiple times, each time generating a new set of random initial values ​​and iterating. Select the decomposition result corresponding to the attempt that minimizes the objective function value of the optimization problem among all attempts as the final factor matrix estimate.

9. The method according to claim 8, characterized in that, The estimated result of the arrival direction includes: 1) For the matrix The kth column Calculate the azimuth-related complex phase term : in, It is a column vector The second element, It is its first element; Calculate the complex phase term related to pitch angle : in, It is a column vector The third element; 2) Based on the results of step 1), calculate the estimated azimuth angle. : Calculate the estimated pitch angle : in, It is a standard mathematical function used to calculate the phase angle factor matrix of a complex number. ; 3) Repeat steps 1) to 2) to finally obtain a set of automatically paired two-dimensional DOA estimation results for all R signal sources. .

10. A fault-tolerant arrival direction estimation device for incomplete tensor reconstruction of array elements, characterized in that, include: The third-order signal tensor construction module is used to detect the array element working status of the received signal of the L-shaped sensor array in order to construct the third-order incomplete signal tensor of the two-directional array. The array expansion module is used to perform cross-correlation operation on the third-order incomplete signal tensors of the two directional arrays to obtain incomplete cross-correlation tensors and perform virtual array expansion processing to obtain a fifth-order incomplete augmented tensor. The tensor reconstruction module is used to reshape the fifth-order incomplete augmented tensor into a third-order incomplete target tensor through generalized tensor quantization operations. The mask tensor generation module is used to generate the mask tensor corresponding to the third-order incomplete target tensor; The factor matrix decomposition module is used to construct and solve a weighted parallel factor analysis decomposition optimization problem on the mask tensor to obtain the factor matrix decomposition result. The arrival direction estimation module is used to obtain the arrival direction estimation result based on the factor matrix decomposition result.

Images