Radar-oriented order constraint Jacobi characteristic value self-ordering method and system

Abstract

The invention discloses a radar-oriented order constraint Jacobi eigenvalue self-sorting method and system, solves the problem of resource consumption caused by an additional sorting module in a Jacobi iteration eigenvalue decomposition post-processing flow in the prior art, realizes direct output of ordered eigenvalues and eigenvectors, and omits an independent sorting module. The method comprises the steps that a real symmetric matrix A obtained through radar baseband signal sampling and covariance estimation is processed to obtain a compressed array, and a feature vector matrix V is initialized; in the parallel iteration process, index pairs are generated according to a scheduling sequence, sub-matrixes are extracted, two types of asymmetric rotation operators are generated through a condition driving mechanism, and a feature vector matrix is synchronously updated to track feature vectors; global data replacement is carried out, and non-diagonal element energy judgment convergence is calculated; after iteration is ended, feature values arranged in a descending order can be directly extracted from the main diagonal of the final compressed array, and corresponding feature vectors are output and used for direction of arrival estimation.

Description

Technical Field

[0001] This invention relates to the field of radar signal processing technology, and in particular to a method and system for order-constrained Jacobian eigenvalue self-sorting for radar. Background Technology

[0002] In the field of radar signal processing, high-precision direction-of-arrival (DOA) estimation is one of the core tasks. Classical super-resolution DOA estimation algorithms such as MUSIC rely on accurate and fast eigenvalue decomposition (EVD) of the antenna array covariance matrix for performance. For modern radar systems such as phased array radars, it is necessary to process multiple snapshots of data within an extremely short time window. Therefore, EVD operations must be completed on the order of microseconds, which places stringent demands on the real-time performance of computing hardware.

[0003] In terms of hardware platform selection, FPGA has become the mainstream platform for implementing EVD modules in radar real-time signal processing systems due to its customizable parallel computing architecture, deterministic low processing latency, and efficient pipelined processing capabilities. In FPGA-based EVD implementations, the parallel Jacobi iterative method is widely adopted because of its regular computational structure and natural suitability for hardware parallelization. Existing common parallel Jacobi schemes typically include the following steps: dividing the symmetric matrix into multiple 2×2 submatrices for parallel processing; calculating the rotation operator and applying it to the submatrices to achieve diagonalization; iterating until the energy of the off-diagonal elements of the matrix converges below a threshold.

[0004] However, the existing technical solutions described above have an inherent drawback that severely restricts the real-time performance of radar systems: since the core objective of the Jacobi method is only diagonalization, its iterative process lacks a mechanism for sorting eigenvalues. Therefore, after convergence, the obtained eigenvalues ​​and corresponding eigenvectors are unordered in the output. In radar DOA estimation applications, it is necessary to distinguish the signal subspace from the noise subspace based on the magnitude of the eigenvalues, usually in descending order, and select the largest eigenvalues ​​and their corresponding eigenvectors for subsequent spatial spectrum estimation. Therefore, existing solutions must cascade a separate sorting module after the EVD calculation module. This sorting module needs to sort the eigenvalue sequence and simultaneously adjust the column order of the eigenvector matrix.

[0005] This post-ordering module brings significant problems: First, it requires additional FPGA logic resources, such as comparators, multiplexers, and complex state control logic; second, the ordering operation is serially dependent on the EVD iteration process, making pipeline optimization impossible and introducing non-negligible additional processing delays; finally, this leads to an increase in the resource and timing overhead of the entire EVD processing chain, becoming a bottleneck for improving the real-time performance of radar signal processing.

[0006] Therefore, it is necessary to eliminate the resource and latency overhead caused by independent sorting modules in order to meet the urgent need of radar systems for high real-time signal processing. Summary of the Invention

[0007] This invention provides a self-sorting method and system for order-constrained Jacobian eigenvalues ​​for radar, which solves the resource consumption problem caused by the additional sorting module in the post-processing flow of Jacobian iterative eigenvalue decomposition in the prior art. It enables the EVD module to directly output ordered eigenvalues ​​and corresponding eigenvectors, thereby omitting the independent sorting module.

[0008] In a first aspect, the present invention provides a self-ordering method for order-constrained Jacobian eigenvalues ​​for radar, the method comprising: S1, Receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compression array, and initialize the n-order eigenvector matrix V. S2, based on the preset scheduling sequence, generates multiple column index pairs that need to be processed in the current parallel iteration cycle, and extracts the diagonal submatrix from the radar compressed array according to the index of each index pair. And according to the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; S3, using the first type of rotation operator For the diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. S4. After completing the update of the current parallel iteration cycle, the data in the updated radar compressed array is globally replaced according to the preset scheduling rules to obtain the replaced radar compressed array. It is then determined whether the replaced radar compressed array meets the iteration termination condition. If not, the replacement radar compressed array and the updated feature vector matrix V are used as the basis to return to step S2 for the next round of iteration; if yes, the process proceeds to step S5. S5 extracts the main diagonal elements from the final radar compressed array to obtain n eigenvalues ​​arranged in descending order, and outputs the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

[0009] In conjunction with the first aspect, in one possible implementation, the data structure preprocessing of the radar signal target matrix A to obtain a radar compressed array includes: By utilizing the symmetry of the symmetric matrix, the upper triangular portion of the symmetric matrix is ​​extracted to obtain the upper triangular matrix; The upper triangular matrix is ​​stored in a one-dimensional array to obtain the radar compressed array.

[0010] In conjunction with the first aspect, in one possible implementation, the statement based on the diagonal submatrix... The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ,include: Calculate the principal rotation angle used to eliminate the off-diagonal elements of the diagonal submatrix. and the main rotation angle Convert to master rotation parameters ; Compare the diagonal submatrix The first diagonal element Second diagonal element Size; If the first diagonal element Greater than or equal to the second diagonal element Then the first type of rotation operator Second type of rotation operator All equal to the principal rotation parameters ; If the first diagonal element Less than the second diagonal element Then for the principal rotation parameters Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator .

[0011] In conjunction with the first aspect, in one possible implementation, the main rotation angle The calculation formula is expressed as: ; in, Represents the first diagonal element; Indicates the second diagonal element; Indicates the first off-diagonal element; Indicates the second off-diagonal element; This represents the arctangent function.

[0012] In conjunction with the first aspect, in one possible implementation, the main rotation parameters are... Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator ,include: The main rotation parameters Applying an order adjustment operator yields the correction matrix. ; According to the correction matrix For the main rotation parameters The first type of rotation operator is obtained. Second type of rotation operator .

[0013] In conjunction with the first aspect, in one possible implementation, the use of the first type of rotation operator... For the diagonal submatrix Perform a contract transformation to obtain the updated radar compressed array, including: Using the first type of rotation operator Construct rotation matrix According to the rotation matrix For the diagonal submatrix Perform an orthogonal rotation transformation to eliminate the diagonal submatrix. The off-diagonal elements in the matrix are used to update the diagonal submatrix. The diagonal elements in the matrix are used to obtain the updated diagonal submatrix. ; The updated diagonal matrix Write back to the radar compressed array to obtain the updated radar compressed array.

[0014] In conjunction with the first aspect, in one possible implementation, the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained, including: Based on the second type of rotation operator For the two columns in the feature vector matrix V that correspond to the current column index pair, perform a similarity transformation; The two transformed column vectors are used as the updated feature vector columns and written back to the feature vector matrix V to obtain the updated feature vector matrix V.

[0015] In conjunction with the first aspect, in one possible implementation, the step of globally permuting the data in the updated radar compressed array according to a preset scheduling rule to obtain a permuted radar compressed array includes: Based on the cyclic scheduling sequence, the element indices in the updated radar compressed array are remapped through a fixed permutation network to generate new index mapping relationships; Using the new index mapping relationship, the data in the updated radar compressed array is rearranged to obtain a permuted radar compressed array whose data distribution meets the scheduling requirements of the next parallel iteration cycle.

[0016] Secondly, the present invention provides a radar-oriented order-constrained Jacobian eigenvalue self-sorting system, the system comprising: The data preprocessing module is used to receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compressed array, and initialize the n-order eigenvector matrix V. The operator calculation module is used to generate multiple column index pairs that need to be processed in the current parallel iteration cycle according to a preset scheduling sequence, and extract the diagonal submatrix from the radar compressed array based on the index of each index pair. And according to the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; Parallel processing module for using the first type of rotation operator For the diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. The global permutation module is used to perform global permutation on the data in the updated radar compressed array according to the preset scheduling rules after completing the update of the current parallel iteration cycle, to obtain the permuted radar compressed array, and to determine whether the permuted radar compressed array meets the iteration termination condition. If not, it returns to the operator calculation module for the next round of iteration based on the permuted radar compressed array and the updated feature vector matrix V; if yes, it enters the output module. The output module is used to extract the main diagonal elements from the final radar compressed array, obtain n eigenvalues ​​arranged in descending order, and output the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

[0017] One or more technical solutions provided in this invention have at least the following technical effects or advantages: This invention receives baseband signals acquired by a radar array, samples and estimates the covariance of the baseband signals to obtain an n-order real symmetric radar signal target matrix A to be decomposed. The radar signal target matrix A undergoes data structure preprocessing to obtain a radar compressed array, and an n-order eigenvector matrix V is initialized. By converting the radar baseband signal into a real symmetric matrix through sampling and covariance estimation, the noise resistance and stability of signal processing are enhanced. The data structure preprocessing to generate the radar compressed array significantly reduces storage requirements and subsequent computational complexity, facilitating hardware implementation. Initializing the eigenvector matrix provides a foundation for subsequent iterations, ensuring the accuracy of eigenvalue decomposition. Based on a preset scheduling sequence, multiple column index pairs to be processed within the current parallel iteration cycle are generated. Diagonal submatrices are extracted from the radar compressed array according to the indices of each index pair. The first and second type rotation operators corresponding to the current parallel iteration cycle are calculated based on the diagonal submatrices. The preset scheduling sequence optimizes the order of parallel iterations, avoids processing conflicts, and maximizes computational efficiency. Generating column index pairs supports parallel processing of multiple submatrices, accelerating the iteration speed. Extracting diagonal submatrices focuses on local calculations, simplifying the operation. The calculated rotation operators provide a key tool for matrix diagonalization, effectively eliminating off-diagonal elements. The first type of rotation operator is used to perform a congruent transformation on the diagonal submatrix to obtain an updated radar compressed array; simultaneously, a similarity transformation is performed on the corresponding columns of the eigenvector matrix V according to the second type of rotation operator to obtain updated eigenvector columns, and thus an updated eigenvector matrix V. The congruent transformation updates the radar compressed array, gradually realizing matrix diagonalization while strictly maintaining matrix symmetry and eigenvalue invariance; the synchronous similarity transformation updates the eigenvector matrix, ensuring the accuracy of eigenvector tracking; the parallel update operation significantly improves the processing speed, making it very suitable for the needs of real-time radar signal processing. After completing the update of the current parallel iteration cycle, the data in the updated radar compressed array is globally permuted according to the preset scheduling rules to obtain the permuted radar compressed array. The energy of all off-diagonal elements is calculated based on the permuted radar compressed array. If the energy is not lower than a preset threshold, the process returns to step S2 for the next iteration, using the updated radar compressed array and the updated feature vector matrix V as the basis. Global permutation and data reorganization optimizes the data layout for the next iteration, helping to accelerate the convergence process. Calculating the energy of off-diagonal elements as a convergence criterion ensures accuracy while avoiding excessive iteration, effectively saving computational resources. The loop return mechanism enables continuous automatic optimization throughout the process until the preset convergence criterion is met. When the energy is lower than the preset threshold, the iteration terminates. The main diagonal elements are extracted from the final radar compressed array to obtain n feature values ​​arranged in descending order, and the final feature vector matrix V is output as the corresponding feature vector for radar signal direction-of-arrival estimation.The iterative termination condition based on the energy threshold ensures the best balance between the accuracy and computational efficiency of eigenvalue decomposition; the method of directly extracting eigenvalues ​​from the compressed array is highly efficient; the final output eigenvalues ​​and eigenvector matrix provide direct and reliable input for radar signal direction-of-arrival estimation, thereby improving the overall accuracy of radar signal source localization and the real-time performance of system processing. Attached Figure Description

[0018] Figure 1 A flowchart illustrating the steps of the order-constrained Jacobian eigenvalue self-sorting method for radar provided in this embodiment of the invention; Figure 2 A detailed flowchart illustrating the symmetric matrix eigenvalue decomposition system with embedded sorting function provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of "symmetric matrix compression" provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of a "Round Forward Sequence" provided in an embodiment of the present invention; Figure 5 This is a schematic diagram of the angles corresponding to the off-diagonal matrix provided in an embodiment of the present invention; Figure 6 This is a schematic diagram illustrating the flow of data in a parallel processing unit array, as provided in an embodiment of the present invention. Figure 7 The convergence curve of the sum of squares of off-diagonal elements provided in this embodiment of the invention; Figure 8 A heat map of the diagonalization process provided in an embodiment of the present invention; Figure 9 A heatmap of feature vector orthogonality provided in an embodiment of the present invention; Figure 10 This is an RTL view of the eigenvalue decomposition system provided in an embodiment of the present invention; Figure 11 This is a schematic diagram of the top-level architecture of the Jacobian eigenvalue decomposition hardware system provided in an embodiment of the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0020] In a first aspect, the present invention provides a self-sorting method for order-constrained Jacobian eigenvalues ​​for radar applications, see [link to relevant documentation]. Figure 1 The method includes the following steps S1 to S5.

[0021] S1, receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compressed array, and initialize the n-order eigenvector matrix V. Specifically, in step S1, the radar signal target matrix A undergoes data structure preprocessing to obtain a radar compressed array, including: (1) Using the symmetry of the symmetric matrix, extract the upper triangular part of the symmetric matrix to obtain the upper triangular matrix; (2) Store the upper triangular matrix into a one-dimensional array to obtain the radar compressed array.

[0022] For example, first receive from The raw baseband echo signal acquired by a uniform linear array radar antenna array. This signal can be characterized as the superposition of reflected echoes from multiple potential targets in space and noise. Let the signal vector received by the array in a certain snapshot be: (1) in, Indicates the first Each element at any moment The received complex baseband signal.

[0023] For digital processing, the system satisfies the Nyquist sampling theorem rate. Discretization sampling is performed to obtain a set of lengths of Quick shot data sequence .

[0024] Subsequently, to estimate the spatial statistical properties of the received signal, the system calculates the sample covariance matrix (SCM) of the sampled dataset. Its unbiased estimation expression is: (2) in, The sample mean vector This represents the conjugate transpose. In practical applications, if the signal has zero mean or has undergone past mean preprocessing, it can be simplified to: (3) This matrix The dimension is ,matrix No. Line number Column elements Characterized the first With the The covariance between the received signals of each array element contains the spatial orientation information of the target. Because the covariance operation satisfies... (in (indicating conjugation), therefore It is a Hermitian conjugate symmetric matrix. For real signals or systems that have undergone specific processing, It degenerates into a real symmetric matrix. In the scenario described in this invention, after necessary preprocessing (such as taking the real part or a specific transformation), the final n-order real symmetric radar signal target matrix A to be decomposed is obtained, that is, let... Or through equivalent transformations to make and ,in This is the number of array elements (i.e., the matrix order). This matrix... It will be used as input to subsequent advanced signal processing algorithms such as eigenvalue decomposition to estimate parameters such as direction of arrival (DOA).

[0025] When processing the covariance matrix in algorithms such as MUSIC and subspace projection, the matrix to be decomposed is usually a symmetric matrix. To optimize storage resource consumption and reduce the complexity of subsequent calculations, this invention first performs data structure preprocessing on the n-order real symmetric radar signal target matrix A to be decomposed. This is achieved by utilizing the symmetric properties of the matrix, i.e. Only its upper triangular portion (including the main diagonal) is extracted and stored in a one-dimensional linear storage space, such as... Figure 3 The compression process is shown below. This reduces the processing and storage of off-diagonal elements by nearly half. Simultaneously, the system initializes an n-order identity matrix V as the starting point for iterating over the eigenvector matrix.

[0026] S2, based on the preset scheduling sequence, generates multiple column index pairs that need to be processed in the current parallel iteration cycle, and extracts the diagonal submatrix from the radar compressed array according to the index of each index pair. And based on the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; Here, the diagonal submatrix Represented as: (4) in, Represents the first diagonal element; Indicates the second diagonal element; Indicates the first off-diagonal element; This represents the second off-diagonal element.

[0027] Specifically, in step S2, based on the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator This includes the following steps S21 to S25.

[0028] S21, Calculate the principal rotation angle used to eliminate off-diagonal elements of the diagonal submatrix. and the main rotation angle Convert to master rotation parameters ; Here, the main rotation angle The calculation formula is expressed as: (5) in, Represents the first diagonal element; Indicates the second diagonal element; Indicates the first off-diagonal element; Indicates the second off-diagonal element; Represents the arctangent function; due to symmetry, .

[0029] S22, Compare the diagonal submatrix The first diagonal element Second diagonal element Size; S23, if the first diagonal element Greater than or equal to the second diagonal element Then the first type of rotation operator Second type of rotation operator All equal to the principal rotation parameters ; S24, if the first diagonal element Less than the second diagonal element Then for the principal rotation parameter Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator ; Here, in step S24, the main rotation parameters are... Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator ,include: (1) Set the principal rotation parameters Applying an order adjustment operator yields the correction matrix. ; (2) Based on the correction matrix For the principal rotation parameters The first type of rotation operator is obtained. Second type of rotation operator .

[0030] For example, this invention employs a pulsating array structure to perform parallel Jacobian iterations. For instance... Figure 3 As shown, the n-order matrix is ​​divided into multiple 2x2 sub-matrix processing units (PEs), including Diagonal processing units and One off-diagonal processing unit.

[0031] In the In each iteration, for each diagonal processing unit specified by a cyclic scheduling sequence (such as a Round-Robin sequence), the 2x2 submatrix it processes can be expressed as Equation (4). Here, k is related to the size of the n-order real symmetric radar signal target matrix A. .

[0032] To eliminate off-diagonal elements A principal rotation angle needs to be calculated. Main rotation angle This is expressed as formula (5). From this, a set of principal rotation parameters can be obtained. ,in, , In the traditional Jacobi method, this set of parameters is directly used for all subsequent related rotation operations.

[0033] To achieve automatic descending order of eigenvalues ​​during iteration, this invention does not directly use the master rotation parameter. Instead, it introduces a condition-driven asymmetric rotation operator generation mechanism. This mechanism is based on the current first diagonal element. Second diagonal element The magnitude relationship of the principal rotation parameters Conditional adjustments are made to generate two sets of functionally separated, asymmetric rotation operators: the first type of rotation operator. Second-type rotation operator used for updating radar signal target matrix A Used for updating the eigenvector matrix V.

[0034] The specific generation logic is as follows: 1. Sorting condition judgment: If the element originally belongs to the lower triangle, compare the size of the diagonal elements.

[0035] 2. Operator generation: like This indicates that the current diagonal elements are in descending order and no intervention is needed. At this point, both types of operators are equal to the principal rotation parameter: (6) like This indicates that the current order does not meet the descending order requirement, and sorting intervention must be activated. At this point, by adjusting the master rotation parameters... Applying an order adjustment operator is equivalent to superimposing a 90-degree rotation on the original rotation, generating a new operator that can swap the positions of diagonal elements while eliminating off-diagonal elements. This operator can be derived from a correction matrix. express: (7) To maintain the orthogonality of the entire transformation, the update of the eigenvector matrix V must also employ the same correction; therefore: (8) This mechanism ensures that after each rotation, the elements on the diagonal always evolve towards a globally ordered goal. This decision-making process is also reflected in… Figure 2 In the process.

[0036] S3, using a first-type rotation operator diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. Specifically, in step S3, the first type of rotation operator is used. diagonal submatrix Perform contract transformation to obtain the updated radar compressed array, including the following steps S31 to S32.

[0037] S31, using a first-type rotation operator Construct rotation matrix According to the rotation matrix diagonal submatrix Perform orthogonal rotation transformations to eliminate diagonal submatrices. Find the off-diagonal elements in the matrix and update the diagonal submatrix. The diagonal elements in the matrix are used to obtain the updated diagonal submatrix. ; S32, update the diagonal submatrix Write back to the radar compressed array to obtain the updated radar compressed array.

[0038] Specifically, in step S3, according to the second type of rotation operator A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, which in turn yields the updated eigenvector matrix V, including: (1) Based on the second type of rotation operator For the two columns in the eigenvector matrix V that correspond to the current column index pair, perform a similarity transformation; (2) Take the two column vectors obtained after the transformation as the updated eigenvector columns, write them back to the eigenvector matrix V, and obtain the updated eigenvector matrix V.

[0039] For example, after generating two types of rotation operators, the system performs update operations on the radar signal target matrix A and the eigenvector matrix V in parallel.

[0040] Diagonal submatrix The update is accomplished by the following formula, where the rotation matrix is... By the first type of rotation operator constitute: (9) After this transformation, the new off-diagonal elements and Theoretically set to zero, in FPGA implementation it approaches zero due to quantization errors, while new diagonal elements... and satisfy The order.

[0041] Each off-diagonal processing unit receives a first-class rotation operator generated by the two diagonal processing units in its corresponding row and column, such as Figure 5 As shown. Suppose a non-diagonal submatrix is ​​subjected to a row rotation operator. Sum and column rotation operators The effect of this, and its update formula is: (10) To maintain the correspondence between eigenvectors and eigenvalues, the eigenvector matrix V must be updated synchronously with A. This update operation uses a type II rotation operator. For each diagonal processing unit, the index pairs involved... The p-th and q-th columns of matrix V will undergo rotation transformation: (11) in, This represents the column vector of matrix V. Since the column indices involved in all n / 2 diagonal processing units are different in a single parallel step, these n / 2 sets of column update operations can be executed in complete parallel.

[0042] S4. After completing the update of the current parallel iteration cycle, the data in the updated radar compressed array is globally replaced according to the preset scheduling rules to obtain the replaced radar compressed array. It is then determined whether the replaced radar compressed array meets the iteration termination condition. If not, the replacement radar compressed array and the updated feature vector matrix V are used as the basis to return to step S2 for the next round of iteration; if yes, the process proceeds to step S5. Specifically, in step S4, the data in the updated radar compressed array is globally replaced according to the preset scheduling rules to obtain the replaced radar compressed array, including the following S1 to S42.

[0043] S41, based on the cyclic scheduling sequence, the element indices in the updated radar compressed array are remapped through a fixed permutation network to generate new index mapping relationships; S42, using the new index mapping relationship, rearrange the data in the updated radar compressed array to obtain a permuted radar compressed array whose data distribution meets the scheduling requirements of the next parallel iteration cycle.

[0044] For example, after completing one parallel rotation, in order to process new off-diagonal elements in the next iteration, a global repositioning of the elements of the radar signal target matrix A is required. This repositioning follows a preset Round-Robin scheduling sequence, such as... Figure 4 The 8th order forward sequence is shown.

[0045] To facilitate implementation on FPGAs with fixed data paths, this rearrangement is accomplished using a fixed permutation network. For example... Figure 6 As shown, data flows systematically between processing units (PEs). For example, elements on the diagonal and in the first row are translated according to specific rules, while the positions of other elements are moved accordingly based on the position changes of the leading elements (diagonal and first row elements) in their respective rows and columns. For an 8-order matrix, this structured data flow ensures that after seven parallel rotations and rearrangements, all off-diagonal elements in the upper triangular region have been processed once, completing one full Jacobian iteration (Sweep).

[0046] After data rearrangement, return to step S2 to begin a new round of iterative calculation until the total energy of the off-diagonal elements converges to a sufficiently small threshold. During the iterative calculation, two conditions are preset: a maximum number of iterations and a convergence threshold. Iteration continues until the maximum number of iterations is reached. Even if the total energy of the off-diagonal elements has converged below the threshold during the iteration process, the iteration will not terminate prematurely but will continue to complete all preset iterations. If, after completing the maximum number of iterations, the total energy of the off-diagonal elements has still not converged below the threshold, the iteration will also terminate.

[0047] S5. When the energy is lower than the preset threshold, the iteration terminates. Extract the main diagonal elements from the final radar compressed array to obtain n eigenvalues ​​arranged in descending order, and output the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

[0048] For example, when the iteration termination condition is met, the main diagonal elements of the radar signal target matrix A stored in the processing unit array are the eigenvalues ​​arranged in descending order, and each column of the eigenvector matrix V is the corresponding eigenvector. Depending on subsequent application requirements, the upper triangular matrix stored in one-dimensional space can be restored to a complete n-order diagonal matrix form, completing the entire self-sorting eigenvalue decomposition process.

[0049] In a simulation experiment provided by this invention, MATLAB was used to verify the effectiveness of the proposed self-ordering Jacobian iterative matrix eigenvalue decomposition method. An 8th-order symmetric matrix was selected as the test object, and the maximum number of scans was set to 10. Key data during the iteration process were recorded, including the sum of squares of off-diagonal elements, the matrix transformation process, and the orthogonality of eigenvectors.

[0050] Figure 7 The convergence curve of the sum of squares of off-diagonal elements is shown, after normalization. As can be seen from the figure, the algorithm exhibits excellent convergence characteristics: initially, the normalized sum of squares of off-diagonal elements is 1.0; after the first scan, it rapidly decreases to approximately 0.25; after the second scan, it decreases to approximately 0.05; and after the third scan, it approaches 0. From the fourth scan onwards, the sum of squares of off-diagonal elements essentially converges to 0, indicating that the matrix has been sufficiently diagonalized. This result proves that the Jacobi iterative method has a fast convergence speed, typically requiring only 3-4 complete scans to achieve the accuracy requirements for engineering applications.

[0051] Figure 8 The evolution of matrix A during the first four scans is visually illustrated using a heatmap. Scan 0 (the initial state) shows that the matrix has obvious off-diagonal elements, with large positive values ​​(orange-red) on the diagonal. As the iteration progresses, scans 1 and 2 show that the magnitudes of the off-diagonal elements gradually decrease, and the color becomes lighter. By scan 3, the matrix has essentially become diagonalized, with the off-diagonal areas approaching white and values ​​close to 0, while the diagonal elements retain significant eigenvalue information. This visualization clearly verifies that the Jacobian rotation transformation can effectively eliminate off-diagonal elements, gradually transforming a symmetric matrix into a diagonal matrix.

[0052] The orthogonality of eigenvector matrices is an important criterion for the correctness of eigenvalue decomposition algorithms. Figure 9A heatmap of the eigenvector orthogonality VTV during the first four scans is shown. As can be seen from the graph, regardless of whether it's scan 0 (the initial identity matrix) or subsequent scans 1, 2, and 3, VTV maintains the standard identity matrix shape: diagonal elements are always 1 (dark orange), and off-diagonal elements are always 0 (close to white). This indicates that the eigenvector matrix strictly maintains orthogonality throughout the entire iteration process, verifying that the orthogonality property of the rotation transformation is well preserved, and demonstrating the algorithm's excellent numerical stability.

[0053] Simulation experiments verified the effectiveness of the proposed method from three key dimensions: convergence, diagonalization effect, and orthogonality. The algorithm achieves full diagonalization of the matrix within 3-4 scans, demonstrating fast convergence. During the iteration process, off-diagonal elements are effectively eliminated, while diagonal elements retain eigenvalue information. The eigenvector matrix strictly maintains orthogonality throughout the process. These results provide a reliable theoretical foundation for subsequent FPGA hardware implementation, proving the feasibility and effectiveness of the proposed method in practical applications.

[0054] like Figure 10 As shown, Jacobi iterative eigenvalue decomposition has been implemented in subspace projection signal processing projects. The hls_spai module integrates Jacobi iterative eigenvalue decomposition logic, driven by the global clock ap_clk and the reset signal ap_rst_n. This module receives main antenna sampling data via the AXI-Stream interface antenna_main_sample, calculates eigenvectors using Jacobi iteration and constructs the signal subspace, filters out interference through projection operations, and sends the clean signal downstream via the output interface signal_clear. It also supports flow control handshaking via the TREADY signal.

[0055] like Figure 11 As shown, the Jacobian iteration is globally controlled by a finite state machine (FSM). External raw data first enters the covariance matrix generation module, and the calculation result is stored in the data cache RAM. Subsequently, the core computing unit reads the cached data and, combined with the index provided by the sorting module, calculates the Jacobian twisting factor. This factor is then fed into a pipelined left and right multiplication module to update the matrix data, and simultaneously updates the eigenvector storage module. After iteration, the eigenvalue storage module outputs the results, which are then sorted in descending order by the sorting and selector module, finally outputting the target eigenvalues ​​and their corresponding eigenvectors.

[0056] Table 1: Comparison of resource consumption before and after optimization of the Jacobi iteration module in this embodiment of the invention

[0057] As shown in Table 1, before optimization, the algorithm mainly adopted a serial computing method. Although it consumed less hardware resources, it could not fully utilize the parallel characteristics of the FPGA, resulting in a processing latency of more than 30,000 cycles. After HLS optimization, although the logic resources and DSP units were increased by about 3 times (trading space for time), the overall processing latency was successfully reduced by about 78%, significantly improving the real-time processing capability of the system.

[0058] This invention, after calculating the principal rotation parameters for diagonalizing the 2x2 submatrix, does not apply them directly, but instead introduces a sorting condition judgment step. This step is based on the current diagonal elements ( and The order adjustment operator (i.e., an orthogonal correction transformation) is dynamically activated based on the size relationship of the diagonal elements. This operator is used to correct the main rotation parameters, thereby generating a new set of rotation operators that simultaneously embody the dual intentions of diagonalization and sorting. This method rewrites the rotation transformation rules conditionally based on the real-time state of the diagonal elements.

[0059] When the sorting condition determination step is triggered, this invention derives a pair of functionally heterogeneous, asymmetric rotation operators from a single master rotation parameter. Specifically, it generates a first type of rotation operator. It is specifically designed to perform rotation transformations on radar signal target matrix A, including its diagonal and off-diagonal sub-blocks, that can both diagonalize and swap diagonal elements; and simultaneously generate second-class rotation operators. This is specifically designed to perform orthogonal transformations on the eigenvector matrix V. This decouples a single rotation task into two clearly defined, path-separated operator update processes, ensuring that they remain mathematically compatible. and This is a key technical means to achieve "transformation equals sorting".

[0060] This invention deeply integrates the sorting requirement into each atomized rotation operation through the aforementioned order-constrained orthogonal transformation mechanism. This makes the order of diagonal elements a convergence invariant that is actively maintained and optimized during the iteration process. The final ordered state of eigenvalues ​​is no longer an additional processing result after algorithm convergence, but an inherent property achieved synchronously with the energy convergence process of off-diagonal elements. This technical solution, which achieves the native ordered arrangement of eigenvalues ​​in the iterative computation flow, fundamentally eliminates the dependence on independent sorting hardware modules and their related control logic.

[0061] This invention can accurately identify off-diagonal elements that logically correspond to the lower triangular region of the original matrix by parsing the row and column index relationships generated in a round-robin sequence at a specific iteration stage. The aforementioned sorting condition judgment and asymmetric operator derivation mechanism are only triggered when processing these elements. This design seamlessly integrates algorithmic sorting innovation with data structure-level storage optimization, achieving self-sorting functionality without sacrificing storage and computational efficiency, demonstrating a high degree of optimization in hardware and software co-design.

[0062] Secondly, the present invention provides a radar-oriented order-constrained Jacobian eigenvalue self-sorting system, the system comprising: The data preprocessing module is used to receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compressed array, and initialize the n-order eigenvector matrix V. The operator computation module is used to generate multiple column index pairs that need to be processed in the current parallel iteration cycle according to a preset scheduling sequence, and extract the diagonal submatrix from the radar compressed array based on the index of each index pair. And based on the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; Parallel processing module for using first-class rotation operators diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. The global permutation module is used to perform global permutation on the data in the updated radar compressed array according to the preset scheduling rules after the current parallel iteration cycle is updated, to obtain the permuted radar compressed array, and to determine whether the permuted radar compressed array meets the iteration termination condition. If not, it returns to the operator calculation module for the next round of iteration based on the permuted radar compressed array and the updated feature vector matrix V; if yes, it enters the output module. The output module is used to extract the main diagonal elements from the final radar compressed array, obtain n eigenvalues ​​arranged in descending order, and output the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

[0063] For example, consider a 4-element uniform linear array radar operating at 1 GHz. Two far-field signal sources are incident from 30° and 60° directions. After the array acquires the baseband signal, a 4th-order real symmetric target matrix A is obtained through sampling and covariance estimation, for example: A=[[4.0,1.0,0.5,0.2],[1.0,3.0,0.6,0.3],[0.5,0.6,2.0,0.4],[0.2,0.3,0.4,1.0]].

[0064] Data preprocessing module: Receives baseband signals, performs sampling and covariance estimation to obtain matrix A. The upper triangular elements of matrix A, including the diagonal, are stored as a radar compressed array: [4.0, 1.0, 0.5, 0.2, 3.0, 0.6, 0.3, 2.0, 0.4, 1.0]. The 4th-order eigenvector matrix V is initialized as the identity matrix.

[0065] Operator computation module: Based on the parallel scheduling sequence, the first cycle generates index pairs (1,2) and (3,4). It extracts the corresponding 2x2 diagonal submatrices from the compressed array; for example, for (1,2), it obtains the submatrix [[4.0,1.0],[1.0,3.0]]. It calculates the Jacobian rotation angle for each submatrix, generating the corresponding first and second type rotation operators.

[0066] Parallel processing module: The first type of rotation operator is used to perform a congruent transformation on each diagonal submatrix, updating the corresponding elements in the compressed array, for example, eliminating the off-diagonal elements at positions (1,2) and (3,4). Simultaneously, the second type of rotation operator is used to perform a similarity transformation on the corresponding columns of the eigenvector matrix V, updating the eigenvectors.

[0067] Global Permutation Module: After completing the current cycle update, the module performs a global permutation of the data in the compressed array according to the scheduling rules, rearranging the element order. It calculates the sum of squares of all off-diagonal elements as energy and compares it with a preset threshold, such as 1e-6. If the energy is not lower than the threshold, the next iteration begins with the updated array and matrix V.

[0068] Output module: The iteration terminates when the off-diagonal energy falls below a threshold. The main diagonal elements are extracted from the final compressed array to obtain eigenvalues ​​sorted in descending order, for example, λ1=5.0, λ2=3.0, λ3=2.0, λ4=1.0. The corresponding eigenvector matrix V is output for subsequent direction-of-arrival (DOA) estimation algorithms, such as the MUSIC algorithm, ultimately yielding signal DOA estimates of 30° and 60°.

[0069] The various embodiments described in this specification are presented in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. All or part of this invention can be used in numerous general-purpose or special-purpose computer system environments or configurations. Examples include: personal computers, server computers, handheld or portable devices, tablet devices, mobile communication terminals, multiprocessor systems, microprocessor-based systems, programmable electronic devices, network PCs, minicomputers, mainframe computers, and distributed computing environments including any of the above systems or devices, etc.

[0070] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the present invention.

Claims

1. A self-sorting method for order-constrained Jacobian eigenvalues ​​in radar systems, characterized in that, include: S1, Receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compression array, and initialize the n-order eigenvector matrix V. S2, based on the preset scheduling sequence, generates multiple column index pairs that need to be processed in the current parallel iteration cycle, and extracts the diagonal submatrix from the radar compressed array according to the index of each index pair. And according to the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; S3, using the first type of rotation operator For the diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. S4. After completing the update of the current parallel iteration cycle, the data in the updated radar compressed array is globally replaced according to the preset scheduling rules to obtain the replaced radar compressed array. It is then determined whether the replaced radar compressed array meets the iteration termination condition. If not, the replacement radar compressed array and the updated feature vector matrix V are used as the basis to return to step S2 for the next round of iteration; if yes, the process proceeds to step S5. S5 extracts the main diagonal elements from the final radar compressed array to obtain n eigenvalues ​​arranged in descending order, and outputs the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

2. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 1, characterized in that, The process of preprocessing the radar signal target matrix A to obtain a radar compressed array includes: By utilizing the symmetry of the symmetric matrix, the upper triangular portion of the symmetric matrix is ​​extracted to obtain the upper triangular matrix; The upper triangular matrix is ​​stored in a one-dimensional array to obtain the radar compressed array.

3. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 1, characterized in that, The diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ,include: Calculate the principal rotation angle used to eliminate the off-diagonal elements of the diagonal submatrix. and the main rotation angle Convert to master rotation parameters ; Compare the diagonal submatrix The first diagonal element Second diagonal element Size; If the first diagonal element Greater than or equal to the second diagonal element Then the first type of rotation operator Second type of rotation operator All equal to the principal rotation parameters ; If the first diagonal element Less than the second diagonal element Then for the principal rotation parameters Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator .

4. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 3, characterized in that, The main rotation angle The calculation formula is expressed as: ； in, Represents the first diagonal element; Indicates the second diagonal element; Indicates the first off-diagonal element; Indicates the second off-diagonal element; This represents the arctangent function.

5. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 3, characterized in that, The main rotation parameters Apply the order adjustment transformation and apply it according to the transformed principal rotation parameters. The first type of rotation operator is obtained. Second type of rotation operator ,include: The main rotation parameters Applying an order adjustment operator yields the correction matrix. ; According to the correction matrix For the main rotation parameters The first type of rotation operator is obtained. Second type of rotation operator .

6. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 1, characterized in that, The first type of rotation operator is used For the diagonal submatrix Perform a contract transformation to obtain the updated radar compressed array, including: Using the first type of rotation operator Construct rotation matrix According to the rotation matrix For the diagonal submatrix Perform an orthogonal rotation transformation to eliminate the diagonal submatrix. The off-diagonal elements in the matrix are used to update the diagonal submatrix. The diagonal elements in the matrix are used to obtain the updated diagonal submatrix. ; The updated diagonal matrix Write back to the radar compressed array to obtain the updated radar compressed array.

7. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 1, characterized in that, The second type of rotation operator A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained, including: Based on the second type of rotation operator For the two columns in the feature vector matrix V that correspond to the current column index pair, perform a similarity transformation; The two transformed column vectors are used as the updated feature vector columns and written back to the feature vector matrix V to obtain the updated feature vector matrix V.

8. The order-constrained Jacobian eigenvalue self-sorting method for radar according to claim 1, characterized in that, The step of globally replacing the data in the updated radar compressed array according to a preset scheduling rule to obtain a replaced radar compressed array includes: Based on the cyclic scheduling sequence, the element indices in the updated radar compressed array are remapped through a fixed permutation network to generate new index mapping relationships; Using the new index mapping relationship, the data in the updated radar compressed array is rearranged to obtain a permuted radar compressed array whose data distribution meets the scheduling requirements of the next parallel iteration cycle.

9. A self-ordering system for order-constrained Jacobian eigenvalues ​​for radar, characterized in that, include: The data preprocessing module is used to receive the baseband signal collected by the radar array, sample and estimate the covariance of the baseband signal to obtain the n-order real symmetric radar signal target matrix A to be decomposed, perform data structure preprocessing on the radar signal target matrix A to obtain the radar compressed array, and initialize the n-order eigenvector matrix V. The operator calculation module is used to generate multiple column index pairs that need to be processed in the current parallel iteration cycle according to a preset scheduling sequence, and extract the diagonal submatrix from the radar compressed array based on the index of each index pair. And according to the diagonal submatrix The calculation is performed to obtain the first type of rotation operator corresponding to the current parallel iteration cycle. Second type of rotation operator ; Parallel processing module for using the first type of rotation operator For the diagonal submatrix Perform a contractual transformation to obtain the updated radar compressed array; simultaneously, based on the second type of rotation operator... A similarity transformation is performed on the corresponding columns of the eigenvector matrix V to obtain the updated eigenvector columns, and then the updated eigenvector matrix V is obtained. The global permutation module is used to perform global permutation on the data in the updated radar compressed array according to the preset scheduling rules after completing the update of the current parallel iteration cycle, to obtain the permuted radar compressed array, and to determine whether the permuted radar compressed array meets the iteration termination condition. If not, it returns to the operator calculation module for the next round of iteration based on the permuted radar compressed array and the updated feature vector matrix V; if yes, it enters the output module. The output module is used to extract the main diagonal elements from the final radar compressed array, obtain n eigenvalues ​​arranged in descending order, and output the final eigenvector matrix V as the corresponding eigenvector for radar signal direction of arrival estimation.

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