Multi-frequency distributed nested array DOA estimation method based on low-rank constraint

Abstract

The invention provides a multi-frequency distributed nested array DOA estimation method based on low-rank constraint, and the method comprises the steps: S1, building a receiving signal model of a distributed nested array, and obtaining a receiving signal based on the receiving signal model; s2, calculating a covariance matrix of the received signal, and performing vectorization processing on the covariance matrix to obtain a first virtual array corresponding to the distributed nested array; s3, extrapolating sub-arrays in the first virtual array to fill part of holes existing under the long baseline condition to obtain a second virtual array; s4, zero elements are inserted into the remaining hole positions to obtain a third virtual array, and the rank of the covariance matrix corresponding to the third virtual array is recovered; s5, establishing a matrix reconstruction model combining the low-rank constraint and the sparsity constraint, and solving to obtain a complemented covariance matrix; and S6, carrying out characteristic decomposition on the complemented covariance matrix to obtain a final DOA estimation value.

Description

Technical Field

[0001] This invention relates to the fields of radar and communication technology, and in particular to a method for estimating the direction of arrival (DOA) of a multi-frequency distributed nested array based on low-rank constraints. Background Technology

[0002] Distributed arrays, due to their advantage of equivalent large aperture, show broad application prospects in target detection, high-precision positioning, and tracking. Compared with conventional arrays, distributed arrays can expand the array aperture without increasing the number of array elements through the sparse arrangement of subarrays or array elements, thereby improving array gain and angle measurement accuracy. Direction of arrival (DOA) estimation, as an important research direction in array signal processing, has significant application value in radar, sonar, wireless communication, and navigation systems. Traditional DOA estimation methods are mainly based on centralized arrays. For example, super-resolution subspace algorithms such as Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) are mostly based on uniform arrays, and their DOA estimation degrees of freedom and angle estimation performance are limited by the number of array elements. Increasing the number of array elements leads to increased hardware complexity and cost.

[0003] As the spatial sampling theorem states, the synthetic radiation pattern of a distributed array contains grating lobes, which can affect target detection and parameter estimation. To address this, researchers have proposed several typical sparse arrays to achieve large apertures while avoiding array ambiguity, such as coprime arrays and nested arrays. In long-baseline distributed arrays, the large physical distance between subarrays leads to missing signals between them. This problem severely impacts the performance of long-baseline arrays, especially in direction estimation and signal reconstruction, where data completion becomes a challenge. Current DOA estimation methods for distributed arrays include virtual array interpolation algorithms, dual-scale deambiguation algorithms, and low-rank matrix reconstruction algorithms. Low-rank matrix reconstruction algorithms can better utilize virtual arrays to expand the aperture compared to other algorithms. One such algorithm generates a uniform linear array through coprime array interpolation and constructs a Toeplitz matrix, transforming the rank minimization problem into a nuclear norm minimization Toeplitz matrix reconstruction problem. This is then solved using convex optimization tools, achieving high-precision DOA estimation.

[0004] However, existing DOA estimation methods based on low-rank matrix reconstruction have the following problems: (1) Current DOA estimation methods based on low-rank matrix reconstruction algorithms are mainly focused on single sparse arrays, and their application in distributed array scenarios has not been fully explored. Moreover, most existing matrix completion methods use convex functions such as the nuclear norm as the optimal approximation of the rank function, lacking an accurate approximation of the matrix rank structure, which leads to increased signal reconstruction error, especially under low signal-to-noise ratio conditions, the performance drops significantly. (2) As the array aperture increases, the computational load of related convex optimization algorithms increases exponentially, which will bring a huge computational burden in practical applications. It is difficult to meet the real-time requirements, which restricts its engineering application.

[0005] Despite some progress in related research, traditional distributed array DOA estimation methods suffer from virtual array holes under long baseline conditions, leading to fuzzy angle estimation, limited resolution, and performance degradation under low signal-to-noise ratio and small snapshot number conditions. It is evident that current distributed array DOA estimation still faces significant challenges. Therefore, this invention proposes a high-precision, low-complexity DOA estimation method suitable for distributed nested arrays. Summary of the Invention

[0006] In view of this, the present invention provides a multi-frequency distributed nested array DOA estimation method based on low-rank constraints to alleviate the problems of large angle estimation error and high computational complexity in existing distributed array DOA estimation methods.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows:

[0008] In embodiments of this disclosure, a method for estimating the DOA of a multi-frequency distributed nested array based on low-rank constraints is provided, comprising: Step S1: establishing a received signal model of the distributed nested array and obtaining the received signal based on the received signal model; Step S2: calculating the covariance matrix of the received signal and vectorizing the covariance matrix to obtain a first virtual array corresponding to the distributed nested array; Step S3: extrapolating the subarrays in the first virtual array to fill in some holes existing under long baseline conditions to obtain a second virtual array; Step S4: inserting zero elements at the remaining hole positions to obtain a third virtual array and restoring the rank of the covariance matrix corresponding to the third virtual array; Step S5: establishing a matrix reconstruction model combining low-rank constraints and sparsity constraints, and solving to obtain the completed covariance matrix; Step S6: performing eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate.

[0009] According to embodiments of this disclosure, a distributed nested array includes multiple identical second-level nested subarrays arranged collinearly.

[0010] According to an embodiment of this disclosure, in step S3, the subarrays in the initial virtual array are extrapolated using multi-frequency information within the bandwidth range to expand the aperture.

[0011] According to an embodiment of this disclosure, in step S4, the rank of the covariance matrix corresponding to the third virtual array is recovered using the Toeplitz matrix reconstruction method.

[0012] According to an embodiment of this disclosure, in step S5, a matrix reconstruction model with joint low-rank constraints and sparsity constraints is established, and the completed covariance matrix is ​​obtained by using an iterative reweighted least squares algorithm.

[0013] According to an embodiment of this disclosure, in step S6, the MUSIC algorithm is used to perform eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate.

[0014] According to the embodiments of this disclosure, in step S3, since the signal energy and noise at different frequency points are different, the covariance matrix of the working frequency point and the supplementary frequency point is normalized when filling the hole.

[0015] According to an embodiment of this disclosure, in step S4, zero elements are inserted at the hole positions to form a continuous and uniform third virtual array, and the Hermite Toeplitz covariance matrix after the aperture is expanded is constructed using Hermite symmetry.

[0016] According to the embodiments of this disclosure, in step S5, the covariance matrix corresponding to the third virtual array is modeled as a linear combination of the covariance matrix under the dictionary matrix using the matrix reconstruction model, and the coefficient matrix of the linear combination has low rank. The optimization problem with dual constraints is constructed and solved by minimizing the approximate rank through the Schatten-p norm.

[0017] According to an embodiment of this disclosure, step S6 includes: performing eigenvalue decomposition on the completed covariance matrix and further dividing it into a received signal subspace and a noise subspace; using the orthogonality of the noise eigenvector and the signal eigenvector to obtain a spatial spectrum estimation function; and obtaining the estimated value of the direction of arrival by the peak value of the spatial spectrum.

[0018] This invention presents a low-rank constraint-based multi-frequency distributed nested array DOA estimation method. Based on a distributed array structure with nested subarrays, it constructs a larger virtual array aperture under the same number of array elements by optimizing the subarray arrangement, thereby improving spatial resolution. This invention utilizes multi-frequency information within the received bandwidth to extrapolate and fill virtual array holes. Different frequency signals are mapped to different virtual array element positions, thus extrapolating some missing element information of the virtual array and reducing the impact of holes on DOA estimation. A low-rank matrix reconstruction technique is used to construct the covariance matrix of the received signal of the virtual array. Zero elements are inserted at the extrapolated hole positions of the virtual array, and the Toeplitz matrix reconstruction method is used to restore the rank of the covariance matrix. A matrix reconstruction model combining low-rank and sparse constraints is constructed. The optimization problem with dual constraints is constructed by minimizing the approximate rank using the Schatten-p norm, improving the accuracy and noise resistance of DOA estimation. An efficient solution for the low-rank sparse constraint problem using iterative reweighted least squares is derived, reducing computational complexity and improving the real-time performance of the algorithm. After obtaining the covariance matrix of the received signal from the virtual array without missing data, the MUSIC algorithm is used for high-resolution DOA estimation. Attached Figure Description

[0019] The above and other objects, features and advantages of the present invention will become more apparent from the following description of embodiments of the invention with reference to the accompanying drawings, in which:

[0020] Figure 1 The flowchart of the multi-frequency distributed nested array DOA estimation method based on low-rank constraints proposed in this invention is illustrated in the schematic diagram.

[0021] Figure 2 This diagram schematically illustrates the comparison between the spatial spectrum estimation results obtained by the estimation method proposed in this invention and the actual DOA results.

[0022] Figure 3 The diagram illustrates the radiation patterns obtained based on different array states with the same number of array elements.

[0023] Figure 4 The diagram illustrates the variation of the root mean square error with the signal-to-noise ratio when estimating using only a single-frequency received signal and the DOA estimation method proposed in this invention.

[0024] Figure 5 The diagram illustrates the variation of the root mean square error with the number of snapshots when estimating using only a single-frequency received signal and the DOA estimation method proposed in this invention. Detailed Implementation

[0025] This invention provides a low-rank constraint-based method for DOA estimation of multi-frequency distributed nested arrays. By combining multi-frequency information and a low-rank matrix reconstruction algorithm, it fully utilizes the virtual array extension of the distributed nested array to achieve high-precision signal reconstruction and unambiguous angle estimation. Compared with existing distributed array DOA estimation methods, it achieves a larger virtual aperture and robustness to low signal-to-noise ratio and small snapshot number, while reducing computational complexity. First, the covariance matrix of the received data from the distributed nested array is vectorized, mapping the original array to a virtual array model. Then, multi-frequency information within the bandwidth is used to extrapolate the virtual array, reducing the hole effect caused by subarray spacing. For the remaining hole region, a low-rank reconstruction problem of the received signal covariance matrix is ​​constructed. A joint recovery model is built by combining sparsity, and the completed matrix is ​​obtained by solving iterative reweighted least squares. Finally, a subspace-based algorithm is applied to the completed covariance matrix for DOA estimation. Compared with other methods, the DOA estimation method of this invention can not only achieve high-precision and unambiguous angle estimation, but also meet the real-time requirements in practical applications, providing a reliable DOA estimation solution for large-scale distributed nested array systems.

[0026] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0027] In this embodiment of the invention, a method for estimating the DOA of a multi-frequency distributed nested array based on low-rank constraints is provided, such as... Figure 1 As shown, the DOA estimation method includes:

[0028] Step S1: Establish a received signal model for the distributed nested array, and obtain the received signal based on the received signal model;

[0029] Step S2: Calculate the covariance matrix of the received signal and vectorize the covariance matrix to obtain the first virtual array corresponding to the distributed nested array;

[0030] Step S3: Extrapolate the subarrays in the first virtual array to fill in some of the holes that exist under the long baseline condition, and obtain the second virtual array;

[0031] Step S4: Insert zero elements at the remaining hole positions to obtain a third virtual array, and restore the rank of the covariance matrix corresponding to the third virtual array;

[0032] Step S5: Establish a matrix reconstruction model with joint low-rank constraints and sparsity constraints, and solve for the completed covariance matrix; and

[0033] Step S6: Perform eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate.

[0034] According to an embodiment of the present invention, the distributed nested array includes multiple identical second-level nested subarrays arranged collinearly. For example, it includes I second-level nested subarrays, where I ≥ 2, and the number of array elements of the uniform linear array inside each nested subarray is... The spacing between array elements is The number of elements in the outer uniform linear array is The spacing between array elements is To achieve the maximum degrees of freedom with the same number of array elements, when the total number of array elements is even, in order to obtain the virtual uniform linear array to the greatest extent possible, The total number of array elements in the distributed nested array is: Assuming the baseline length between each subarray is D, and the bandwidth of the array's received signal is large enough to cover the information at the desired frequency, consider, for example, a distributed nested array consisting of two identical nested subarrays (I=2). The total number of array elements is 12. The baseline length of the two subarrays is D=28.

[0035] In operation S1, for multiple frequencies respectively Continuous wave signals (i=1…l) are emitted from K independent far-field narrowband targets. The received signal vector, scattered onto the distributed array subarray, is represented as:

[0036] ;

[0037] in, Frequency The reflection coefficient of the k-th signal, where t=1…T is the number of snapshots. This represents the additive white Gaussian noise vector. It is the steering vector, represented as:

[0038] ;

[0039] in, This indicates the wavelength corresponding to the frequency.

[0040] The received signal vector is low-pass filtered and then down-converted to baseband form to obtain:

[0041] ;

[0042] in, Represents the guidance matrix. ; Represents the signal matrix, .

[0043] When the array operates at a certain frequency At that time, the array receives the following signal:

[0044] ;

[0045] in, Indicates operating frequency The guiding matrix below, This represents the signal matrix at that frequency. This represents the noise matrix.

[0046] According to an embodiment of the present invention, in operation S2, assuming the signal is an uncorrelated signal, the array output is at the frequency point. The covariance matrix is:

[0047] ;

[0048] in, For the signal at frequency point The covariance matrix. For the k-th signal at frequency point Energy.

[0049] Furthermore, by vectorizing the received data covariance matrix, we can obtain:

[0050] ;

[0051] in, , Represents the identity matrix. This represents the Khatri-Rao product. For the received signal at the operating frequency, its covariance matrix, after vectorization, forms a virtual array. For the same distributed array, frequency points within the bandwidth range are selected. Then the array at the frequency point The received signal can be represented as:

[0052] ;

[0053] The array at the frequency point After vectorizing the covariance matrix of the received signal, the first virtual array corresponding to the distributed nested array is obtained, which can be represented as:

[0054] ;

[0055] in, Among them, when the frequency point This refers to the broadening of the first virtual array, when the frequency point It is a proportional reduction of the first virtual array.

[0056] According to an embodiment of the present invention, in step S3, the subarrays in the initial virtual array are extrapolated using multi-frequency information within the bandwidth range to expand the aperture. Since the signal energy and noise at different frequencies are different, the covariance matrix of the working frequency and the supplementary frequency is normalized when filling the holes.

[0057] Specifically, for a virtual array with nested subarrays, multi-frequency information can be extrapolated to fill some of the holes caused by the long baseline, thereby increasing the array aperture and obtaining a second virtual array. For example, for the nested array configuration with 12 elements above, the first and twelfth elements can be set to operate in dual-band. Since the signal energy and noise at different frequencies may be different, the array covariance matrices of both the operating and supplementary frequencies must be normalized during the supplementation process. The normalized covariance matrix can be expressed as:

[0058] ;

[0059] in, Represents the normalized matrix The (i,j)th element, In frequency point The number of array elements used to fill in the missing terms of the operating frequency point. Indicates the frequency of the i-th array element. The received signal, Indicates the frequency of the j-th array element. The conjugate received signal, This represents the covariance matrix of the received signal. Indicates frequency point The conjugate transpose of the received signal. Indicates frequency point The received signal, with normalized energy and noise, can be expressed as follows:

[0060] ;

[0061] ;

[0062] in, Frequency point The signal energy, Frequency point noise energy, It is the normalized signal energy. It is the normalized noise energy.

[0063] According to an embodiment of the present invention, in step S4, a third virtual array with zero elements and continuous uniformity is inserted at the remaining hole positions, and the rank of the covariance matrix corresponding to the third virtual array is recovered using the Toeplitz matrix reconstruction method.

[0064] Specifically, matrix reconstruction is performed using the low-rank characteristics of the received signal to improve the accuracy and robustness of aperture completion. Zero elements are inserted at the aperture positions of the second virtual array after multi-frequency information filling to form a continuous and uniform third virtual array. ,use The Hermite Toeplitz matrix after expanding the aperture using Hermite symmetry construction, where, To remove redundant elements and average the received signal of the third virtual array at the same virtual array element positions, let... Let M represent the number of elements in the third virtual array, then:

[0065] ;

[0066] The obtained covariance matrix It is composed of signal data from multiple different frequency points. The signal at each frequency point can be regarded as a signal received by the array from different locations, and the noise is sparse.

[0067] According to an embodiment of the present invention, in step S5, a matrix reconstruction model with joint low-rank constraints and sparsity constraints is established, and the completed covariance matrix is ​​obtained by using an iterative reweighted least squares algorithm.

[0068] The covariance matrix corresponding to the third virtual array is modeled as a linear combination of the covariance matrix under the dictionary matrix using the matrix reconstruction model. The coefficient matrix of the linear combination has low rank. The optimization problem with dual constraints is constructed and solved by minimizing the approximate rank using the Schatten-p norm.

[0069] Specifically, the covariance matrix of the third virtual array is modeled as a covariance matrix using a low-rank representation model. Linear combinations under a dictionary matrix, where the coefficient matrix of the linear combination has low rank. Using... The Schatten-p norm represents the low-rank constraint. Compared to the nuclear norm, the Schatten-p norm is closer to the rank function and can better characterize the matrix rank minimization problem. The optimization problem is represented as follows:

[0070] ;

[0071] Where Z is the linear combination coefficient matrix to be optimized, and E is the noise matrix to be optimized. This is the covariance matrix corresponding to the third virtual array. Describing the Schatten-p norm, express Norm, The problem is restated as follows, using regularization parameters:

[0072] ;

[0073] in, The function to be optimized.

[0074] If L, S, U, and V are set as temporary parameters for intermediate calculation steps to simplify formula derivation, the above formula can be further expressed as follows:

[0075] ;

[0076] in, Represents the trace of a matrix. It is a unit array. Indicates transpose. Represents the i-th column of the matrix. This represents the Frobenius norm of the matrix.

[0077] make ;

[0078] but ;

[0079] right Differentiating, we get: ,make ;

[0080] in, This represents the partial derivative.

[0081] right Differentiate each column.

[0082] ;

[0083] in, ;

[0084] make ,Right now ;

[0085] Further results were obtained:

[0086] ;

[0087] The above equation is the Sylvester equation, which can be solved using MATLAB's lyap function to obtain the completed covariance matrix.

[0088] It should be noted that for the constructed matrix reconstruction model with joint low-rank and sparse constraints, in addition to using the Schatten-p norm for rank minimization approximation, nuclear norm minimization, weighted nuclear norm minimization, and truncated nuclear norm minimization can also be used to complete the matrix.

[0089] According to an embodiment of the present invention, in step S6, the MUSIC algorithm is used to perform eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate. This includes:

[0090] The completed covariance matrix is ​​eigenvalued and further divided into a received signal subspace and a noise subspace. The spatial spectrum estimation function is obtained by utilizing the orthogonality of the noise eigenvector and the signal eigenvector. The estimated value of the direction of arrival can be obtained by the peak value of the spatial spectrum.

[0091] Specifically, The low-rank representation of the signal can be further analyzed using the MUSIC algorithm to obtain the DOA estimate. This is followed by eigenvalue decomposition, which further divides the signal into a signal subspace and a noise subspace, i.e.:

[0092] ;

[0093] in, and They are the signal subspace and the noise subspace, respectively. Given diagonal matrices, and utilizing the orthogonality of the noise eigenvectors and signal eigenvectors, the spatial spectrum estimation function is obtained:

[0094] ;

[0095] in, Indicates the guide vector. This indicates the conjugate transpose, and the direction of arrival can be estimated by the peak value of the spatial spectrum.

[0096] It should be noted that, in addition to the MUSIC algorithm, the ESPRIT algorithm, Root-MUSIC algorithm, and compressed sensing algorithms can also be used to estimate the spatial spectrum of the completed covariance matrix.

[0097] Furthermore, the spatial spectrum estimation results of the DOA estimation method proposed in this invention were simulated. Under the conditions of SNR=0dB and T=400, 11 uncorrelated targets uniformly distributed between -45 degrees and 45 degrees were selected and incident on a distributed array composed of two identical nested subarrays for DOA estimation. The DOA estimation method proposed in this invention ( Figure 2 A comparison between the estimated results and the actual DOA results (hereinafter referred to as this invention) Figure 2As shown. This invention has high estimation accuracy for multiple unrelated sources, and the spectral peaks are sharp.

[0098] Distributed nested arrays with expanded apertures, distributed nested arrays without padded apertures, and uniform linear arrays, all with a total of 12 array elements, were selected. The array radiation patterns under these different array configurations were plotted and compared. The results are as follows: Figure 3 As shown, compared with a uniform array with the same number of array elements, a distributed array exhibits a narrower main lobe width in its radiation pattern due to its larger array aperture.

[0099] Based on the above experiments, the performance of the DOA estimation method proposed in this invention under different signal-to-noise ratios and snapshot numbers was verified. First, with the snapshot number fixed at 100, five signal sources were uniformly distributed between 30° and -20° and incident on a uniform linear array. The performance of the DOA estimation method proposed in this invention was examined as the signal-to-noise ratio increased from -10dB to 20dB. Figure 4 The changing trend of the estimation performance of the method (hereinafter referred to as the present invention), and its comparison with the DOA estimation method based on single-frequency received signals (hereinafter referred to as the present invention). Figure 4 The results are as follows (referred to as single frequency in Chinese) are compared. Figure 4 As shown. Subsequently, with a fixed signal-to-noise ratio of 5dB, the number of snapshots was increased from 100 to 1000, and the DOA estimation method proposed in this invention was plotted. Figure 5 The present invention (hereinafter referred to as the present invention) estimates the performance curve as a function of the number of snapshots, and compares it with the DOA estimation method based on single-frequency received signals (The present invention is referred to as the present invention). Figure 5 The results are as follows (referred to as single frequency in Chinese) are compared. Figure 5 As shown in the figure. Simulation results show that, compared with the traditional DOA estimation method based on single-frequency array, the proposed multi-frequency distributed nested array DOA estimation method based on low-rank constraints exhibits better DOA estimation performance under different signal-to-noise ratios and snapshot numbers, verifying its effectiveness in improving estimation accuracy.

[0100] In summary, the low-rank constraint-based multi-frequency distributed nested array DOA estimation method of this invention, unlike DOA estimation methods using sparse arrays such as coprime arrays, utilizes a distributed nested array for DOA estimation. The distributed array employed has a longer baseline distance, constructing a larger virtual array aperture under the same number of array elements, thus improving spatial resolution. Simultaneously, it utilizes multiple frequency information within the bandwidth range to construct a virtual array for extrapolating the apertures of the distributed array's virtual array. Furthermore, it constructs a matrix reconstruction model combining low-rank and sparse constraints, and constructs a dual-constraint optimization problem by minimizing the approximate rank using the Schatten-p norm, improving the accuracy and noise resistance of DOA estimation. An efficient solution for the low-rank sparse constraint problem using iterative reweighted least squares is derived. Unlike existing techniques that use convex optimization, the proposed algorithm reduces computational complexity and improves real-time performance.

[0101] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. It should be noted that implementations not illustrated or described in the drawings or the main text of the specification are forms known to those skilled in the art and are not described in detail. Furthermore, the definitions of the elements and methods described above are not limited to the various specific structures, shapes, or methods mentioned in the embodiments, and those skilled in the art can easily modify or substitute them.

[0102] Furthermore, unless specifically described or required to occur in a specific order, the order of the above steps is not limited to those listed above and can be varied or rearranged according to the desired design. Moreover, the above embodiments can be used in combination with each other or with other embodiments based on design and reliability considerations; that is, technical features from different embodiments can be freely combined to form more embodiments.

[0103] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for estimating the DOA of a multi-frequency distributed nested array based on low-rank constraints, comprising: Step S1: Establish a received signal model for the distributed nested array, and obtain the received signal based on the received signal model; Step S2: Calculate the covariance matrix of the received signal and vectorize the covariance matrix to obtain the first virtual array corresponding to the distributed nested array; Step S3: Extrapolate the subarrays in the first virtual array to fill in some of the holes that exist under the long baseline condition, and obtain the second virtual array; Step S4: Insert zero elements at the remaining hole positions to obtain a third virtual array, and restore the rank of the covariance matrix corresponding to the third virtual array; Step S5: Establish a matrix reconstruction model with joint low-rank and sparsity constraints, and solve for the completed covariance matrix; and Step S6: Perform eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate.

2. The multi-frequency distributed nested array DOA estimation method according to claim 1, wherein the distributed nested array comprises multiple identical second-level nested subarrays arranged collinearly.

3. The DOA estimation method for multi-frequency distributed nested arrays according to claim 1, In step S3, the subarrays in the initial virtual array are extrapolated using multi-frequency information within the bandwidth range to expand the aperture.

4. In the multi-frequency distributed nested array DOA estimation method according to claim 1, in step S4, the Toeplitz matrix reconstruction method is used to recover the rank of the covariance matrix corresponding to the third virtual array.

5. In the multi-frequency distributed nested array DOA estimation method according to claim 1, in step S5, a matrix reconstruction model with joint low-rank constraints and sparsity constraints is established, and the completed covariance matrix is ​​obtained by using the iterative reweighted least squares algorithm.

6. In the multi-frequency distributed nested array DOA estimation method according to claim 1, in step S6, the MUSIC algorithm is used to perform eigenvalue decomposition on the completed covariance matrix to obtain the final DOA estimate.

7. In the multi-frequency distributed nested array DOA estimation method according to claim 3, in step S3, since the signal energy and noise at different frequency points are different, the covariance matrix of the working frequency point and the supplementary frequency point is normalized when performing hole filling.

8. In the multi-frequency distributed nested array DOA estimation method according to claim 4, in step S4, zero elements are inserted at the hole positions to form a continuous and uniform third virtual array, and the Hermite-Toeplitz covariance matrix after expanding the aperture is constructed using Hermite symmetry.

9. In the multi-frequency distributed nested array DOA estimation method according to claim 5, in step S5, the covariance matrix corresponding to the third virtual array is modeled as a linear combination of the covariance matrix under the dictionary matrix using the matrix reconstruction model, and the coefficient matrix of the linear combination has low rank. The optimization problem with dual constraints is constructed and solved by minimizing the approximate rank through the Schatten-p norm.

10. The multi-frequency distributed nested array DOA estimation method according to claim 6, step S6 includes: The completed covariance matrix is ​​eigenvalued and further divided into a received signal subspace and a noise subspace. By utilizing the orthogonality between noise eigenvectors and signal eigenvectors, the spatial spectrum estimation function is obtained: and The direction of arrival can be estimated by the peak value of the spatial spectrum.

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