PRFI suppression method based on Hankel structure and truncated nuclear norm regularization

Abstract

The invention discloses a PRFI suppression method based on a Hankel structure and truncation nuclear norm regularization. The method comprises the following steps: acquiring an SAR echo signal interfered by a PRFI signal; the SAR echo signals are constructed into a Hankel matrix, and the Hankel matrix comprises a low-rank PRFI signal interference component matrix and a sparse SAR useful signal component matrix; constructing a low-rank sparse decomposition model based on a Hankel matrix, wherein an objective function of the low-rank sparse decomposition model comprises a truncation nuclear norm regular term used for constraining a PRFI signal component and a sparse regular term used for constraining an SAR useful signal component; performing iterative solution on the low-rank sparse decomposition model by adopting an alternating direction multiplier method to obtain a PRFI signal interference component matrix and an SAR useful signal component matrix after separation; and carrying out inverse transformation on the SAR useful signal component matrix to obtain an SAR useful signal from which the PRFI signal is removed. According to the method, the loss of the SAR useful signal can be reduced while the PRFI signal interference is efficiently suppressed, so that the fidelity of the SAR useful signal is improved.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a PRFI suppression method based on Hankel structure and truncated kernel norm regularization. Background Technology

[0002] Synthetic Aperture Radar (SAR), as an active microwave remote sensing device, plays a vital role in surveying, reconnaissance, and disaster monitoring. However, the increasingly complex electromagnetic environment makes it frequently susceptible to various forms of radio frequency interference (RFI). Among these, pulse repetition frequency interval (PRFI), due to its high power and wide bandwidth characteristics, poses a particularly serious threat to SAR imaging quality.

[0003] Currently, existing techniques for suppressing PRFI can be mainly divided into the following categories: The first category is notch filtering methods, such as frequency-domain notch filtering (FNF) in the frequency domain and time-domain notch filtering (TNF) in the time domain. Although these methods are simple to implement, they can cause data loss in the spectrum or time domain, leading to loss of useful signals and the introduction of artifacts, especially when the PRFI intensity is high and its proportion is high, resulting in a significant decrease in image quality. The second category is semi-parametric methods based on sparse representation or low-rank matrix factorization. These methods decompose the interfered echo signal into low-rank interference components and sparse useful signal components, and achieve interference separation by optimizing the model. Among these, the closest existing technology is based on Robust Principal Component Analysis (RPCA) and its variants. This method uses the nuclear norm (the sum of all singular values) as a convex relaxation of the rank function and uses the L1 norm to constrain the sparse components.

[0004] However, the inventors discovered that the closest prior art has at least the following drawbacks: the nuclear norm, as a convex relaxation of the rank function, will indiscriminately minimize all singular values ​​in the matrix, which can easily lead to "over-penalization" of larger singular values. This results in a large error between the obtained low-rank interference components and the actual interference components, thereby causing a decrease in interference suppression performance and affecting the accuracy of interference suppression and the fidelity of the useful signal.

[0005] Therefore, how to reduce the error between low-rank interference components and the real interference components, thereby improving the accuracy of interference suppression and the fidelity of useful signals, is a technical problem that urgently needs to be solved. Summary of the Invention

[0006] To address the problem of reducing the error between low-rank interference components and the true interference components, thereby improving the accuracy of interference suppression and the fidelity of the useful signal, this invention provides a PRFI suppression method based on Hankel structure and truncated kernel norm regularization. The technical problem to be solved by this invention is achieved through the following technical solution: This invention provides a PRFI suppression method based on Hankel structure and truncated nuclear norm regularization, comprising: Acquire SAR echo signals that are interfered with by PRFI signals; The SAR echo signal is constructed into a Hankel matrix, which includes a low-rank PRFI signal interference component matrix and a sparse SAR useful signal component matrix. A low-rank sparse decomposition model is constructed based on the Hankel matrix. The objective function of the low-rank sparse decomposition model includes a truncated kernel norm regularization term for constraining PRFI signal components and a sparse regularization term for constraining SAR useful signal components. The truncated kernel norm is defined as the smallest singular value among the singular values ​​of the PRFI signal interference component matrix. Q The sum of singular values, Q To truncate parameters; The low-rank sparse decomposition model was solved iteratively by using the alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix. The SAR useful signal component matrix is ​​inversely transformed to obtain the SAR useful signal after removing the PRFI signal.

[0007] In one embodiment of the present invention, constructing a Hankel matrix from the SAR echo signal includes: The SAR echo signal is constructed into a Hankel matrix in the range direction using a pulse-by-pulse method.

[0008] In one embodiment of the present invention, a low-rank sparse decomposition model is constructed based on the Hankel matrix, including: Based on the low-rank characteristics of PRFI signals and the sparsity characteristics of useful SAR signals, a low-rank sparse decomposition model is constructed. The objective function of the low-rank sparse decomposition model is expressed as follows:

[0009] in, To simultaneously and Perform a minimize operation. The PRFI signal interference component matrix, For nuclear norm, Given the SAR useful signal component matrix, constraints are applied. and The sum is the Hankel matrix. , for Norm, for The first Q columns of the left singular vector matrix are truncated submatrices. for transpose, To constrain It is an orthogonal matrix. for The first Q columns of the right singular vector matrix are truncated submatrices. for transpose, To constrain It is an orthogonal matrix. For matrix traces, This is the regularization parameter.

[0010] In one embodiment of the present invention, the low-rank sparse decomposition model is iteratively solved using the alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix, including: First, the iterative solution of the low-rank sparse decomposition model is transformed into solving the following optimization problem:

[0011] Secondly, build The augmented Lagrange function, its expression is:

[0012] in, For Lagrange multiplier matrices, For penalty parameters, This is an inner product operation. It is the Frobenius norm; Finally, the augmented Lagrangian function is solved iteratively using the alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix.

[0013] In one embodiment of the present invention, the augmented Lagrangian function is iteratively solved using the alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix, including: Step (1), initialize the Lagrange multiplier matrix Penalty parameters The total number of iterations K, and the PRFI signal interference component matrix R and SAR useful signal component matrix to be solved. K is a positive integer; Step (2), for the k-th iteration, k < K, update the PRFI signal interference component matrix. At that time, the SAR useful signal component matrix is ​​fixed. Lagrange multiplier matrix and penalty parameters The PRFI signal interference component matrix is ​​updated using the singular value shrinkage operator. The updated PRFI signal interference component matrix is ​​obtained. ; Step (3) involves updating the SAR useful signal component matrix. At the same time, the PRFI signal interference component matrix is ​​updated regularly. Lagrange multiplier matrix and penalty parameters The useful signal component matrix of SAR is updated by a soft threshold shrinkage operator. The updated SAR useful signal component matrix is ​​obtained. ; Step (4) involves updating the Lagrange multiplier matrix. At the same time, the PRFI signal interference component matrix is ​​updated regularly. Updated SAR useful signal component matrix and penalty parameters And through the formula Update the Lagrange multiplier matrix The updated Lagrange multiplier matrix is ​​obtained. ; Step (5), in the penalty parameter At the same time, the PRFI signal interference component matrix is ​​updated regularly. Updated SAR useful signal component matrix and the updated Lagrange multiplier matrix And through the formula Update penalty parameters The updated penalty parameters are obtained. ,in, For hyperparameters, For penalty parameters The upper limit; Step (6): Determine if the iteration termination condition is met. If the iteration termination condition is met, stop the iteration and output the updated PRFI signal interference component matrix. and the updated SAR useful signal component matrix , to be used as the PRFI signal interference component matrix and the SAR useful signal component matrix, otherwise, let k=k+1 and return to step (2).

[0014] In one embodiment of the present invention, the updated PRFI signal interference component matrix is ​​calculated. The expression is:

[0015] in, It is a singular value shrinkage operator.

[0016] In one embodiment of the present invention, the updated SAR useful signal component matrix is ​​calculated. The expression is:

[0017] in, This is a soft threshold shrinkage operator.

[0018] In one embodiment of the present invention, the iteration termination condition is that the iteration number k is greater than the total iteration number K, or the difference between the solutions obtained from two consecutive iterations is less than a preset tolerance. .

[0019] In one embodiment of the present invention, an inverse transformation is performed on the SAR useful signal component matrix to obtain the SAR useful signal after removing the PRFI signal, including: The SAR useful signal component matrix is ​​subjected to anti-diagonal averaging to restore the SAR useful signal component matrix to a one-dimensional time-series signal, so as to obtain the SAR useful signal after removing the PRFI signal.

[0020] Another aspect of the present invention provides a storage medium storing a computer program for performing the steps of the PRFI suppression method based on Hankel structure and truncated nuclear norm regularization as described in any of the above embodiments.

[0021] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention constructs the SAR echo signal as a Hankel matrix, making the PRFI signal and the SAR useful signal exhibit a more significant difference in rank space. This improves the separability of the two signals and the ability to extract interference, reducing the error between low-rank PRFI signal interference components and sparse SAR useful signal interference components, thereby improving the accuracy of interference suppression and the fidelity of the useful signal. Secondly, conventional kernel norms uniformly compress all singular values, which may lead to erroneous attenuation of useful information. This invention, however, truncates the kernel norm regularization term and constrains only smaller singular values, preserving the principal component structure and more accurately retaining the SAR useful signal. This significantly reduces the loss of the SAR useful signal while efficiently suppressing interference, thus improving the fidelity of the SAR useful signal.

[0022] Furthermore, the method of this invention does not rely on the setting of prior parameters such as interference frequency and location, thus eliminating the dependence on filter parameters and notch filter frequencies found in traditional methods, and exhibiting good adaptability. This invention is applicable not only to simulation data but also demonstrates superior interference suppression performance in measured spaceborne SAR data, possessing strong engineering practical value and potential for widespread application.

[0023] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0024] Figure 1 This is a flowchart of a PRFI suppression method based on Hankel structure and truncated nuclear norm regularization provided by an embodiment of the present invention; Figure 2 This is a schematic diagram illustrating the low-rank property of a PRFI signal of a Hankel matrix provided in an embodiment of the present invention; Figure 3 This is a schematic diagram comparing the suppression results simulated under different suppression methods, provided by an embodiment of the present invention; Figure 4 This is a schematic diagram showing the comparison of actual inhibition results under different inhibition methods provided by an embodiment of the present invention. Detailed Implementation

[0025] To further illustrate the technical means and effects adopted by the present invention to achieve the intended purpose, the following describes in detail, with reference to the accompanying drawings and specific embodiments, a PRFI suppression method based on Hankel structure and truncated kernel norm regularization proposed according to the present invention.

[0026] The foregoing and other technical contents, features, and effects of the present invention will be clearly presented in the following detailed description of specific embodiments in conjunction with the accompanying drawings. Through the description of the specific embodiments, a more in-depth and concrete understanding can be gained of the technical means and effects adopted by the present invention to achieve its intended purpose. However, the accompanying drawings are for reference and illustration only and are not intended to limit the technical solutions of the present invention.

[0027] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations are intended to cover non-exclusive inclusion, such that an article or apparatus comprising a list of elements includes not only those elements but also other elements not expressly listed. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the article or apparatus that includes said element.

[0028] This invention addresses the problem of reducing the error between low-rank interference components and the true interference components, thereby improving the accuracy of interference suppression and the fidelity of the useful signal. It proposes a PRFI suppression method based on Hankel structure and truncated kernel norm regularization. Please refer to [link to relevant documentation]. Figure 1 The method includes the following steps: Step 1: Acquire the Synthetic Aperture Radar (SAR) echo signal that is interfered with by Pulse Repetition Frequency Interval (PRFI) signal.

[0029] The SAR echo signal is the original signal actually received by the SAR system. This signal is a superposition of three components, and its expression is: (1.1) in, , , These represent the SAR useful signal, PRFI signal, and noise signal, respectively. and These represent fast time (distance direction) and slow time (azimuth direction), respectively.

[0030] It should be noted that the useful signal of SAR refers to the echo signal from ground targets that the SAR system expects to receive.

[0031] Step 2: Construct the SAR echo signal into a Hankel matrix, which includes a low-rank PRFI signal interference component matrix and a sparse SAR useful signal component matrix.

[0032] PRFI signal Typically, PRFI signals can be modeled in two forms: a narrowband version, which is the superposition of multiple sinusoidal signals; and a wideband version, which is the superposition of multiple frequency-modulated signals. The envelope of these signals is usually a rectangular window and often accompanied by strong amplitude modulation. Previous studies have shown that Radio Frequency Interference (RFI) signals exhibit low-rank characteristics in both the frequency and time-frequency domains, but these characteristics may not apply to PRFI signals because they are intermittent, time-varying, and discontinuous in the spectrum. However, this invention finds that the low-rank characteristics of PRFI signals can be preserved using a Hankel matrix. In other words, SAR echo signals interfered with by PRFI signals can be constructed into a Hankel structure using time-delay embedding theory. For example, Figure 2 A schematic diagram illustrating the low-rank property of a PRFI signal with a Hankel structure is shown.

[0033] In this embodiment of the invention, the SAR echo signal is constructed into the Hankel matrix in the range direction in a pulse-by-pulse manner.

[0034] Specifically, first, the above equation (1.1) is discretized as follows: (1.2) in, (1.3) in, For discretized SAR echo signals, For discretized SAR echo signals, For discretized PRFI signals, For discrete noise signals, ~ For the SAR echo signal at the first sampling time ~ the second sampling time The value at each sampling time. ~ For the useful SAR signal at the first sampling time ~ the second The value at each sampling time. ~ For the PRFI signal at the first sampling time ~ the second sampling time The value at each sampling time. ~ For the noise signal at the first sampling time ~ the second sampling time The value at each sampling time. It represents the number of sampling points in the distance direction.

[0035] Then, all the data in equation (1.3) above are constructed into the following Hankel matrix form: (1.4) in, This represents an operator that converts a vector into a Hankel matrix. The SAR useful signal component matrix, The PRFI signal interference component matrix, This is the noise signal component matrix.

[0036] Specifically, the Hankel matrix representation of the SAR echo signal is as follows: (1.5) in, L It is the Hankel matrix. The dimension of the Hankel matrix ,in , This is the number of sampling points in the distance direction. The other three components ( The Hankel matrix of ) The representation is the same as the principle described above, and will not be repeated here.

[0037] Step 3: Construct a low-rank sparse decomposition model based on the Hankel matrix. The objective function of the low-rank sparse decomposition model includes a truncated nuclear norm (TNN) regularization term to constrain the PRFI signal components and a sparse regularization term to constrain the SAR useful signal components. The truncated nuclear norm is defined as the smallest singular value among the singular values ​​of the PRFI signal interference component matrix. Q The sum of singular values, Q This is for truncating parameters.

[0038] Specifically, based on a semi-parametric method using sparse SAR useful signals and low-rank RFI signals, PRFI signals can also be separated from SAR echo signals using a low-rank matrix with a Hankel structure. This is influenced by the traditional LRR model... Inspired by the regularization term protecting sparse SAR signals, the problem can be modeled as follows: (1.6) in, Represents the rank function. express Norm refers to the number of non-zero elements in a matrix. It is a regularization parameter, and it constrains... and The sum is the Hankel matrix. .

[0039] Due to the rank function and Norms are discrete and nonconvex; therefore, equation (1.6) is an NP-hard problem. A common solution is robust principal component analysis (RPCA), which uses the nuclear norm and Norm with respect to rank function and The norm is relaxed. Therefore, equation (1.6) can be updated to: (1.7) in, The nuclear norm is the sum of all singular values ​​of a matrix. express Norm, which is the sum of the absolute values ​​of all elements in a matrix, also constrains... and The sum is the Hankel matrix. .

[0040] Because the definition of the nuclear norm uses a summation operation on all singular values, all singular values ​​of the matrix are treated equally. This can lead to over-penalization of larger singular values ​​when solving for rank minimization in equation (1.7). This problem introduces errors into LRR, thus affecting interference suppression performance.

[0041] To address the error introduced by traditional low-rank representations, this invention introduces a truncated nuclear norm, TNN, to replace the traditional nuclear norm. Therefore, equation (1.7) can be updated as follows: (1.8) in, The truncation norm is defined as the smallest singular value among the singular values ​​of the PRFI signal interference component matrix R. Q Summing the singular values, and also constraining... and The sum is the Hankel matrix. .

[0042] Equation (1.8) performs better than Equation (1.7) because it introduces the TNN regularization term, which minimizes only some of the smaller singular values, reducing the error caused by minimizing all singular values ​​using the nuclear norm, and thus has a stronger rank approximation capability.

[0043] However, like the rank function, TNN is non-convex. Let the singular value decomposition of the PRFI signal interference component matrix R be... The left singular vector matrix , ~ For the first to the m-th left singular vectors, a diagonal matrix Right singular vector matrix , ~ Let be the first to the m-th right singular vectors.

[0044] U and V are truncated into A and B according to the truncation parameter, as follows: (1.9) in, for The first Q columns of the left singular vector matrix are truncated submatrices. for The first Q columns of the right singular vector matrix are truncated submatrices. Indicates the transpose operation. Q This is the truncation parameter, meaning only the first Q singular values ​​are truncated.

[0045] Furthermore, the TNN term can be reformulated as: (1.10) in, This represents the i-th singular value of the PRFI signal interference component matrix R. for Norm, for transpose, To constrain It is an orthogonal matrix. for transpose, To constrain It is an orthogonal matrix. For matrix The trace, also constrained and The sum is the Hankel matrix. .

[0046] Finally, according to equation (1.10), equation (1.8) can be modified to obtain the objective function of the low-rank sparse decomposition model.

[0047] Specifically, the objective function of the low-rank sparse decomposition model is expressed as follows: (1.11) in, To simultaneously and Perform the minimization operation, with the same constraints. and The sum is the Hankel matrix. .

[0048] Step 4: The low-rank sparse decomposition model is solved iteratively using the Alternating Direction Method of Multipliers (ADMM) to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix.

[0049] In this invention, the solution to equation (1.11) includes a two-stage approach: Phase 1: By performing singular value decomposition on a fixed Hankel matrix X, the following steps are calculated: and Second stage: In fixed and In the case of updating the PRFI signal interference component matrix R and the SAR useful signal component matrix S, we need to solve the following optimization problem: 1.12) Next, the augmented Lagrangian function of equation (1.12) is constructed, and its expression is as follows: (1.13) in, For Lagrange multiplier matrices, For penalty parameters, This is an inner product operation. It is the Frobenius norm.

[0050] Finally, the ADMM method is used to iteratively solve the augmented Lagrangian function by fixing other variables and alternately updating one variable, so as to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix.

[0051] Furthermore, for the k-th iteration, the subproblems that need to be solved are as follows: (1.14) In this embodiment of the invention, the augmented Lagrangian function is iteratively solved using the alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and SAR useful signal component matrix, specifically including: Step (1), initialize the Lagrange multiplier matrix Penalty parameters The total number of iterations K, and the PRFI signal interference component matrix R and SAR useful signal component matrix to be solved. K is a positive integer.

[0052] Step (2), for the k-th iteration, k < K, update the PRFI signal interference component matrix. At that time, the SAR useful signal component matrix is ​​fixed. Lagrange multiplier matrix and penalty parameters The PRFI signal interference component matrix is ​​updated using the singular value shrinkage operator. The updated PRFI signal interference component matrix is ​​obtained. .

[0053] Specifically, The update can be achieved by solving the following subproblems: (1.15) Here, equation (1.15) can be solved using the singular value contraction operator, and the solution can be expressed as: (1.16) in, It is a singular value shrinkage operator.

[0054] Step (3) involves updating the SAR useful signal component matrix. At the same time, the PRFI signal interference component matrix is ​​updated regularly. Lagrange multiplier matrix and penalty parameters The useful signal component matrix of SAR is updated by a soft threshold shrinkage operator. The updated SAR useful signal component matrix is ​​obtained. .

[0055] Specifically, The update can be achieved by solving the following subproblems: (1.17) Here, equation (1.16) can be solved using the soft threshold shrinkage operator, and the solution can be expressed as: (1.18) in, The soft threshold shrinkage operator is defined as follows: , It is a symbolic function.

[0056] Step (4) involves updating the Lagrange multiplier matrix. At the same time, the PRFI signal interference component matrix is ​​updated regularly. Updated SAR useful signal component matrix and penalty parameters And through the formula Update the Lagrange multiplier matrix The updated Lagrange multiplier matrix is ​​obtained. .

[0057] Step (5), in the penalty parameter At the same time, the PRFI signal interference component matrix is ​​updated regularly. Updated SAR useful signal component matrix and the updated Lagrange multiplier matrix And through the formula Update penalty parameters The updated penalty parameters are obtained. ,in, For hyperparameters, For penalty parameters The upper limit.

[0058] Step (6): Determine if the iteration termination condition is met. If the iteration termination condition is met, stop the iteration and output the updated PRFI signal interference component matrix. and the updated SAR useful signal component matrix , to be used as the PRFI signal interference component matrix R and the SAR useful signal component matrix S, otherwise, let k=k+1 and return to step (2) to execute repeatedly.

[0059] The iteration termination condition is that the number of iterations k is greater than the total number of iterations K, or the difference between the solutions obtained from two consecutive iterations is less than the preset tolerance. , specifically or .

[0060] Step 5: Perform an inverse transformation on the SAR useful signal component matrix to obtain the SAR useful signal after removing the PRFI signal.

[0061] Specifically, performing anti-diagonal averaging on the SAR useful signal component matrix can also be understood as performing a signal reconstruction operation on the SAR useful signal component matrix, restoring the SAR useful signal component matrix to a one-dimensional time-series signal, so as to obtain the SAR useful signal after removing the PRFI signal.

[0062] In this embodiment of the invention, the anti-diagonal averaging operation on the SAR useful signal component matrix can be expressed as:

[0063] in, To remove the PRFI signal from the useful SAR signal, To preserve the first row of the SAR useful signal component matrix S, a colon (:) indicates all columns in that row. This represents the extraction of a submatrix from the SAR useful signal component matrix S, specifically from row M+1 to row M+L-1 and column M. To horizontally concatenate the "first row" with the "transposed submatrix", the final SAR useful signal after removing the PRFI signal is obtained. Its dimension is 1 row, A complex matrix of columns.

[0064] In summary, this invention constructs the SAR echo signal as a Hankel matrix, making the PRFI signal and the SAR useful signal exhibit a more significant difference in rank space. This improves the separability of the two signals and the ability to extract interference, reducing the error between the low-rank PRFI signal interference components and the sparse SAR useful signal interference components, thereby improving the accuracy of interference suppression and the fidelity of the useful signal. Secondly, conventional kernel norms uniformly compress all singular values, which may lead to erroneous attenuation of useful information. This invention, however, truncates the kernel norm regularization term and constrains only smaller singular values, preserving the principal component structure and more accurately retaining the SAR useful signal. This significantly reduces the loss of the SAR useful signal while efficiently suppressing interference, thus improving the fidelity of the SAR useful signal.

[0065] Furthermore, the method of this invention does not rely on the setting of prior parameters such as interference frequency and location, thus eliminating the dependence on filter parameters and notch filter frequencies found in traditional methods, and exhibiting good adaptability. This invention is applicable not only to simulation data but also demonstrates superior interference suppression performance in measured spaceborne SAR data, possessing strong engineering practical value and potential for widespread application.

[0066] To verify the technical effects of the method provided in the embodiments of the present invention, an experimental comparison will be conducted next.

[0067] 1. Simulation Experiment For comparison, four traditional methods were selected: FNF, TNF, Eigen-Subspace Projection (ESP), and RPCA. Furthermore, this invention employs three evaluation metrics to quantify suppression performance: Peak Sidelobe Ratio (PSLR), Integrated Sidelobe Ratio (ISLR), and Impulse Response Width (IRW). In the experiments, the regularization parameter... Multiplication Based on experience, it is set as follows:

[0068] in, It is the number of sampling points in the distance direction. It represents the number of sampling points in the azimuth direction.

[0069] In the simulation experiment, this invention simulated an X-band airborne SAR system for acquiring point targets. An interference source (ground radar) was placed in the center of the scene, which intermittently transmitted PRFI to the SAR. The main simulation parameters are listed in Table 1 below: Table 1

[0070] Without loss of generality, the PRFI signal is set as a linear frequency modulated signal with a bandwidth of 30MHz. Figure 3 The impulse response functions under different suppression methods are shown, where (a) is the point target affected by PRFI interference; (b) is the suppression result of the FNF method; (c) is the suppression result of the TNF method; (d) is the suppression result of the ESP method; (e) is the suppression result of the RPCA method; and (f) is the suppression result of the method proposed in this invention. Table 2 lists the evaluation indicators of the five suppression methods.

[0071] Table 2

[0072] from Figure 3 Analysis reveals that, compared to point targets affected by PRFI signal interference, the FNF method suppresses most PRFI, but its sidelobes exhibit abnormal behavior. The ESP and RPCA methods retain some PRFI artifacts because the large bandwidth of PRFI makes it difficult for them to accurately identify the eigenvalues ​​or singularities corresponding to PRFI. Superficially, the TNF method's suppression performance is comparable to the method proposed in this invention, but quantitatively, the method of this invention is closer to the ideal level in both PSLR and ISLR metrics. It is noteworthy that, as shown in Table 2, all five methods maintain target resolution while suppressing PRFI.

[0073] 2. Actual measurement verification The simulation data above verifies the effectiveness and superiority of this invention in suppressing PRFI signal interference. To further verify the superiority of this method, actual measurement data is used for further illustration. This invention utilizes actual SAR data acquired by the European Space Agency's Sentinel-1A satellite, such as... Figure 4 As shown. The imaging area of ​​this data is located in Copenhagen, Denmark, and the data was acquired on February 27, 2020.

[0074] The bandwidth of the SAR signal is 56 MHz, while the bandwidth of the PRFI signal is approximately 10 MHz. Because the PRFI signal fails to align with the matched filter during imaging, it manifests as bright stripe artifacts in the image, such as... Figure 4 As shown in (a). Figure 4 The regions of interest (ROIs) marked in the figure illustrate the differences in suppression performance between FNF, TNF, ESP, RPCA, and the proposed method. (a) shows a SAR image interfered with by PRFI signals; (b) shows the suppression result of the FNF method; (c) shows the suppression result of the TNF method; (d) shows the suppression result of the ESP method; (e) shows the suppression result of the RPCA method; (f) shows the suppression result of the proposed method; (g) shows the ROIs marked with red boxes in (b)-(f); and (h) shows the ROIs marked with green boxes in (c) and (f). Specifically, [the figure would be inserted here]. Figure 4 The ROIs marked in red boxes in (b) to (f) are magnified to obtain Figure 4 (g) It is clear that the FNF, ESP, and RPCA methods are not suitable for suppressing wideband PRFI. These methods not only preserve some PRFI artifacts but also cause a loss of detail in some areas (deteriorated scene), destroying scene details, which is unacceptable for image interpretation. Figure 4 The ROIs marked with green boxes in (c) and (f) show that while the TNF method can completely eliminate PRFI, the signal-to-noise ratio (SNR) of the image is reduced (Deteriorated SNR) due to gaps in the pulses. In contrast, the method proposed in this invention preserves scene detail while suppressing PRFI, and its overall performance is superior to the other four methods.

[0075] In summary, whether under ideal simulation environments or complex experimental conditions, the PRFI interference suppression method based on Hankel structure and TNN proposed in this invention exhibits stronger suppression capabilities, better signal protection, and higher robustness, verifying its effectiveness and superiority in practical applications.

[0076] In the several embodiments provided by this invention, it should be understood that the apparatus and methods disclosed in this invention can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative. For example, the division of modules is merely a logical functional division, and in actual implementation, there may be other division methods. For example, multiple modules or components may be combined or integrated into another system, or some features may be ignored or not executed.

[0077] Furthermore, the functional modules in the various embodiments of the present invention can be integrated into one processing module, or each module can exist physically separately, or two or more modules can be integrated into one module. The integrated module can be implemented in hardware or in the form of hardware plus software functional modules.

[0078] Another embodiment of the present invention provides a storage medium storing a computer program for performing the steps of the PRFI suppression method based on Hankel structure and truncated kernel norm regularization described in the above embodiments.

[0079] Another aspect of the present invention provides an electronic device, including a memory and a processor. The memory stores a computer program, and when the processor invokes the computer program in the memory, it implements the steps of the PRFI suppression method based on the Hankel structure and truncated kernel norm regularization as described in the above embodiments. Specifically, the integrated modules implemented as software functional modules can be stored in a computer-readable storage medium. These software functional modules, stored in a storage medium, include several instructions to cause an electronic device (which may be a personal computer, server, or network device, etc.) or processor to execute some steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0080] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A PRFI suppression method based on Hankel structure and truncated kernel norm regularization, characterized in that, The method comprises the following steps: acquiring a SAR echo signal interfered by a PRFI signal; constructing the SAR echo signal into a Hankel matrix, the Hankel matrix comprising a low-rank PRFI signal interference component matrix and a sparse SAR useful signal component matrix; A low-rank sparse decomposition model is constructed based on the Hankel matrix, and a target function of the low-rank sparse decomposition model includes a truncated nuclear norm regular term for constraining a PRFI signal component and a sparse regular term for constraining a SAR useful signal component, the truncated nuclear norm is defined as a sum of the smallest first several singular values in singular values of the PRFI signal interference component matrix, Q is a truncated parameter, Q ​ solving the low-rank sparse decomposition model by using an alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and the SAR useful signal component matrix; performing inverse transformation on the SAR useful signal component matrix to obtain a SAR useful signal after removing the PRFI signal.

2. The Hankel structured and truncated kernel norm regularized PRFI suppression method according to claim 1, wherein, The step of constructing the SAR echo signal into a Hankel matrix comprises the following steps: constructing the SAR echo signal into the Hankel matrix in a pulse-by-pulse manner in the range direction.

3. The Hankel structured and truncated kernel norm regularization based PRFI suppression method of claim 1, wherein, The step of constructing a low-rank sparse decomposition model based on the Hankel matrix comprises the following steps: constructing the low-rank sparse decomposition model based on the low-rank characteristic of the PRFI signal and the sparse characteristic of the SAR useful signal, and the expression of the objective function of the low-rank sparse decomposition model is as follows: wherein is minimized subject to and , is the PRFI signal interference component matrix, is the nuclear norm, is the SAR useful signal component matrix, subject to and is a Hankel matrix , is norm, is a truncated submatrix of the first Q columns of the left-singular vector matrix of is the transpose of is subject to is an orthogonal matrix, is a truncated submatrix of the first Q columns of the right-singular vector matrix of is the transpose of is subject to is an orthogonal matrix, is the trace of the matrix , is a regularization parameter.

4. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method of claim 3, wherein, The step of solving the low-rank sparse decomposition model by using an alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and the SAR useful signal component matrix comprises the following steps: firstly, converting the step of solving the low-rank sparse decomposition model into solving the following optimization problem: Second, construct the augmented Lagrangian function, whose expression is: wherein, is a Lagrange multiplier matrix, is a penalty parameter, is an inner product operation, is a Frobenius norm; finally, solving the augmented Lagrange function by using an alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and the SAR useful signal component matrix.

5. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method of claim 4, wherein, The step of solving the augmented Lagrange function by using an alternating direction multiplier method to obtain the separated PRFI signal interference component matrix and the SAR useful signal component matrix comprises the following steps: Step (1), initializing a Lagrange multiplier matrix , a penalty parameter , a total iteration number K, and a PRFI signal interference component matrix R and a SAR useful signal component matrix , K is a positive integer; Step (2), for the kth iteration, k < K, update the PRFI signal-plus-interference component matrix , given the fixed SAR useful signal component matrix , the Lagrange multiplier matrix , and the penalty parameter , by the singular value shrinkage operator , to obtain the updated PRFI signal-plus-interference component matrix ; Step (3), updating the SAR signal component matrix , by fixing the updated PRFI signal interference component matrix , the Lagrange multiplier matrix , and the penalty parameter , updating the SAR signal component matrix by a soft-threshold shrinkage operator, to obtain the updated SAR signal component matrix ; Step (4) involves updating the Lagrange multiplier matrix. At the same time, the PRFI signal interference component matrix is ​​updated regularly. Updated SAR useful signal component matrix and penalty parameters And through the formula Update the Lagrange multiplier matrix The updated Lagrange multiplier matrix is ​​obtained. ; Step (5), updating the penalization parameter , the updated PRFI signal interference component matrix , the updated SAR useful signal component matrix , and the updated Lagrange multiplier matrix , and updating the penalization parameter by the formula , to obtain the updated penalization parameter , where is a hyperparameter, is an upper limit value of the penalization parameter . Step (6, judging whether the iteration termination condition is satisfied, if the iteration termination condition is satisfied, stopping iteration and outputting the updated PRFI signal interference component matrix and the updated SAR useful signal component matrix as the PRFI signal interference component matrix and the SAR useful signal component matrix, otherwise, letting k=k+1 and returning to step (2).

6. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method according to claim 5, wherein, Computing an updated PRFI signal interference component matrix The expression for the PRFI is: wherein is a singular value shrinkage operator.

7. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method of claim 5, wherein, Computing an updated SAR useful signal component matrix The expression for the SAR is: wherein, is a soft threshold shrinkage operator.

8. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method of claim 5, wherein, The iteration termination condition is that the iteration number k is greater than the total iteration number K, or the difference between solutions obtained by two adjacent iterations is less than a preset tolerance .

9. The Hankel-based structured and truncated kernel norm regularized PRFI suppression method of claim 1, wherein, The step of performing inverse transformation on the SAR useful signal component matrix to obtain a SAR useful signal after removing the PRFI signal comprises the following steps: performing inverse diagonal average operation on the SAR useful signal component matrix to restore the SAR useful signal component matrix into a one-dimensional time sequence signal to obtain a SAR useful signal after removing the PRFI signal.

10. A storage medium having stored therein a computer program, characterized in that, The computer program is used for executing the steps of the PRFI suppression method based on Hankel structure and truncated nuclear norm regularization in any one of claims 1 to 9.

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