Sparse array coherent signal source DOA estimation method based on Toeplitz matrix reconstruction

Abstract

The invention belongs to the technical field of direction of arrival, and relates to a sparse array coherent signal source DOA estimation method based on Toeplitz matrix reconstruction. Comprising the following steps: S1, assuming that a coherent signal source is incident to a sparse array, further constructing a sparse array observation signal model, and further solving a theoretical covariance matrix of the sparse array observation signal model; s2, performing solution operation on the theoretical covariance matrix in the S1; s2, on the basis of the S2, further constructing a Hermitian Toeplitz matrix through virtual array interpolation, and further realizing DOA estimation based on the virtual array interpolation; and S3, obtaining a virtual ULA based on the sparse array of the S1 by combining the S2, performing solution operation on the virtual ULA, and then constructing a Toeplitz matrix based on physical array interpolation so as to realize DOA estimation based on virtual array interpolation. According to the algorithm provided by the invention, the coherent signal source can be accurately estimated under the condition that the degree of freedom and the array aperture are not lost. Simulation experiment results show that the proposed algorithm is superior to the existing algorithm in the aspect of estimating the coherent source.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of direction of arrival, and relates to a sparse array coherent source DOA estimation method based on Toeplitz matrix reconstruction. BACKGROUND

[0002] Direction of Arrival (DOA) estimation, which determines the azimuth angle of the incident source by using statistical methods, has been a topic of considerable interest in sensor array processing for quite a long time. It has applications in target positioning, wireless communication, autonomous driving, unmanned aerial vehicle swarms, and many other fields. Among various algorithms for DOA estimation, the MUSIC algorithm and the ESPRIT algorithm based on subspace have become the most mainstream algorithms due to their superior resolution, accuracy, and efficiency. The premise assumption of these two algorithms is non-coherent sources, i.e., full-rank covariance matrix. However, in practical applications, coherent sources produced in complex environments such as multipath propagation in mobile communication, background reflection in radar detection, and signal leakage in hardware devices need to be considered. This high correlation between sources is enough to cause the failure of the MUSIC algorithm and the ESPRIT algorithm, because it will cause the source covariance matrix to become singular.

[0003] In order to overcome the adverse effects of coherence, many decorrelation algorithms have been proposed to restore the rank of the covariance matrix. However, most decorrelation algorithms are only compatible with uniform linear arrays, although uniform linear arrays play an irreplaceable role in avoiding spatial aliasing, they encounter considerable challenges in the trade-off between performance and cost. In the past fifteen years, great progress has been made in sparse array DOA estimation. However, in the array signal model, non-coherent sources are usually considered rather than coherent sources.

[0004] At present, the research on DOA estimation of coherent sources with sparse array is still in its infancy, and the number of available literatures is limited. The methods based on sparsity can effectively identify coherent sources. Taking active sparse array as an example, the l1 SVD algorithm is applied to the sum and difference joint array to estimate coherent sources, and the Least Absolute Shrinkage And Selection Operator (LASSO) algorithm is applied to the sum and difference joint array to estimate mixed non-coherent sources and coherent sources. The l1 SVD algorithm and the LASSO algorithm are essentially grid-based sparse reconstruction problems, and their accuracy depends on the accuracy of the predefined grid. Therefore, an overly dense grid may bring excessive computational burden, thus hindering real-time applications. Recently, several algorithms have been proposed from the perspective of no grid for DOA estimation of coherent sources with sparse array. Some scholars proposed a method for estimating coherent sources by using coprime array interpolation. This algorithm uses an interpolation array to construct a Toeplitz matrix, then solves the Nuclear Norm Minimization (NNM) problem to recover the interpolated Toeplitz matrix, and applies a subspace-based algorithm to the recovered Toeplitz matrix to analyze the angle of coherent sources. In addition, some scholars use the non-zero columns of the interpolated covariance matrix to recover the noise-free Toeplitz matrix, thereby constructing an Atomic Norm Minimization (ANM) problem to realize DOA estimation. This method is not sensitive to the phase difference between sources, and makes use of more degrees of freedom, thus having higher resolution. Some scholars proposed Ambiguity-Free SVD (AF-SVD) algorithm and Augmented Signal Subspace (ASS) algorithm. Both of them perform generalized eigenvalue decomposition on the covariance matrix to obtain the eigenvector corresponding to the signal subspace, and use it to construct a Hankel matrix to estimate coherent sources. The former uses the coprimality of subarrays to obtain a unique solution, while the latter maps the signal subspace of the sparse array to the signal subspace corresponding to the ULA with the same physical aperture, and obtains the DOA estimation by solving the NNM problem based on the Hankel matrix. However, the AF-SVD algorithm and the ASS algorithm cannot estimate more coherent sources than the number of sensors. SUMMARY

[0005] In view of the deficiencies of the prior art, the application further improves the estimation performance of coherent sources in the sparse array framework. Compared with the typical resolution method of the sparse array, the application quickly and accurately estimates the coherent sources without losing degrees of freedom. Specifically, first, the difference between the covariance matrix of incoherent sources and coherent sources is studied, and a new decorrelation strategy is proposed, that is, to separate the diagonal elements and off-diagonal elements in the covariance matrix of coherent sources. Although the information contained in the off-diagonal elements is deleted, the diagonal elements (i.e. source power) are still retained, so that we can extract a new matrix similar in structure to the covariance matrix of incoherent sources from the covariance matrix of coherent sources. Subsequently, array interpolation is performed in the virtual domain (physical domain) to activate the received information contained in the virtual (physical) sensors, so as to maximize the use of the degrees of freedom provided by the sparse array for DOA estimation. Simulation results show that the proposed algorithm is superior to existing algorithms in estimating coherent sources.

[0006] In order to achieve the above technical effects, the application proposes a sparse array coherent source DOA estimation method based on Toeplitz matrix reconstruction, comprising:

[0007] The sparse array coherent source DOA estimation method based on Toeplitz matrix reconstruction comprises:

[0008] S1: assuming that coherent sources are incident to the sparse array, constructing a sparse array observation signal model, and solving the theoretical covariance matrix of the sparse array observation signal model;

[0009] S2: performing decorrelation on the theoretical covariance matrix of S1;

[0010] S3: based on S2, further constructing a Hermitian Toeplitz matrix through virtual array interpolation, and then realizing DOA estimation based on virtual array interpolation;

[0011] S4: combining S3 and based on the sparse array of S1, obtaining an imaginary ULA, performing decorrelation on the imaginary ULA, then constructing a Toeplitz matrix based on physical array interpolation, and then realizing DOA estimation based on physical array interpolation, and finally realizing the sparse array coherent source DOA estimation based on Toeplitz matrix reconstruction.

[0012] Preferably, the sparse array observation signal model of the lth sampling snapshot constructed by S1 is:

[0013]

[0014] In the formula, s(l) represents a reference signal source, β=[β1,β2,…,β K ]​T Denotes the coherence coefficient vector, where β k θ is a non-zero complex constant. k p represents the angle of the k-th source. i d represents the position of the i-th element. The cardinality of a set is denoted by . No. The position of each array element; β k Let n represent the coherence coefficient of the k-th source. s (l) represents an additive white Gaussian noise vector independent of the signal source. Represents an array manifold matrix. This represents the source vector of the coherent signal.

[0015] Preferably, the theoretical covariance matrix in S1 for:

[0016]

[0017] In the formula, L represents the total number of sampling snapshots, l is the variable, and the l-th snapshot is (·). H This indicates the conjugate transpose.

[0018] Preferably, the solution for S2 is... for:

[0019]

[0020] in, Represents the source covariance matrix The estimated value, Indicates noise power. express The identity matrix.

[0021] Preferably, in S3, the correction for:

[0022]

[0023] Wherein, the angle estimate is defined as Based on correction The estimated angle is further obtained as follows This yields a DOA estimate based on virtual array interpolation.

[0024] Preferably, in S4, the correction for:

[0025]

[0026] In the formula, based on the correction The estimated angle is further obtained as follows Thus, a DOA estimation based on physical array interpolation is obtained.

[0027] Compared with the prior art, the present application has the beneficial effects that:

[0028] The present application studies the DOA estimation of coherent sources based on sparse array. By separating the diagonal elements and off-diagonal elements in the covariance matrix of the sources, decorrelation operation is realized. Then, from the perspective of virtual domain and physical domain, Toeplitz matrix reconstruction problem is established for DOA estimation. Compared with the existing coherent DOA estimation algorithm based on sparse array, the decorrelation algorithm proposed in this paper maintains the complete degree of freedom and array aperture, and has better performance. In particular, the proposed algorithm is also applicable to the underdetermined case where the number of coherent sources exceeds the actual number of sensors. Although the proposed algorithm has significant advantages in coherent source estimation, the RMSE performance tends to saturate and deviates from CRB at high SNR. Therefore, future work will focus on developing techniques that can improve the accuracy of sparse array coherent source DOA estimation at high SNR. BRIEF DESCRIPTION OF DRAWINGS

[0029] The accompanying drawings are included to provide a further understanding of the present application, and constitute a part of the specification, illustrate the present application together with the embodiments thereof, and explain the principles of the present application, and are provided to give the conveyer a thorough and complete understanding of the present application, and are included as a part of the specification.

[0030] In the drawings:

[0031] Figure 1 Comparison of covariance matrices for five sources: (a) incoherent sources, (b) coherent sources;

[0032] Figure 2 Comparison of spatial spectra of K=5 coherent sources uniformly distributed between-30° and 30°: (a) ULA+FBSS, (b) NNM, (c) ANM, (d) ASS, (e) l1-SVD, (f) LASSO, (g) TMR-VAI, (h) TMR-PAI, where SNR=0dB, the number of snapshots is L=100, the blue solid line represents the estimation result, and the red dashed line corresponds to the actual incident direction of the coherent source;

[0033] Figure 3 Comparison of spatial spectra of K=7 coherent sources uniformly distributed between-30° and 30°: (a) ULA+FBSS, (b) NNM, (c) ANM, (d) ASS, (e) l1-SVD, (f) LASSO, (g) TMR-VAI, (h) TMR-PAI, where SNR=0dB, the number of snapshots is L=100. The blue solid line represents the estimation result, and the red dashed line corresponds to the actual incident direction of the coherent source;

[0034] Figure 4 Spatial spectrum comparison of K=9 coherent sources uniformly distributed between -30° and 30°, where SNR=0dB and snapshot number L=100. The blue solid line represents the estimation result, while the red dashed line corresponds to the actual incident direction of the coherent source on the horizontal axis: (a) ANM, (b) LASSO, (c) TMR-VAI, (d) TMR-PAI;

[0035] Figure 5 Comparison of RMSE of K=5 coherent sources evenly distributed between -30° and 30°: (a) RMSE as a function of SNR, (b) RMSE as a function of snapshot number;

[0036] Figure 6 Resolution comparison of two neighboring coherent sources distributed at θ1=-1° and θ2=1°: (a) ULA+FBSS, (b) NNM, (c) ANM, (d) ASS, (e) l1-SVD, (f) LASSO, (g) TMR-VAI, (h) TMR-PAI, where SNR=0dB and the number of snapshots is L=100;

[0037] Figure 7 This is a flowchart of the method of the present invention. Detailed Implementation

[0038] The following is in conjunction with the appendix Figure 1 -Appendix Figure 7 The preferred embodiments of the present invention will be described herein. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0039] Example:

[0040] Methods for estimating the DOA of sparse array coherent sources based on Toeplitz matrix reconstruction include:

[0041] S1: Assuming a coherent source is incident on a sparse array, and further constructing a sparse array observation signal model, the theoretical covariance matrix of the sparse array observation signal model is then obtained; specifically including:

[0042] S1.1: Assume the direction is θ = {θ k K far-field narrowband coherent sources |k=1,2,…,K} are incident on a sparse array. θ k p represents the angle of the k-th source. i d represents the position of the i-th element. The cardinality of a set is denoted by . No. The positions of the array elements. Where d = λ / 2 is the unit element spacing and λ is the signal wavelength, then the sparse array observation signal at the l-th sampling snapshot time is... Modeled as:

[0043]

[0044] In the formula, s(l) represents the reference signal source, and β = [β1, β2, ..., β K ] T Denotes the coherence coefficient vector, where β k β is a non-zero complex constant. k Let n represent the coherence coefficient of the k-th source. s (l) represents an additive white Gaussian noise vector independent of the signal source. Represents an array manifold matrix;

[0045] Therefore, the k-th signal source is Then the coherent signal source vector This represents the sparse array. manifold matrix, and Indicates direction θ k The corresponding guide vector. Represents the signal vector Uncorrelated noise vector. Symbol (·) T This represents the transpose operator. It represents the imaginary unit.

[0046] S1.2: Based on S1.1, the theoretical covariance matrix of the sparse array observation signal is:

[0047]

[0048] Where k and q are variables, representing the k-th and q-th information sources, respectively, and θ q This represents the angle of the q-th information source. Indicates the array steering vector. express The identity matrix; the asterisk means conjugate;

[0049] S1.3: In formula (2) of S1.2:

[0050]

[0051] Represents the source covariance matrix. Indicates the power of the reference signal source. Let E[·] represent the noise power, and let I[·] represent the statistical expectation operator. T Let T represent the T×T dimensional identity matrix, denoted by (·).H and (·) * denote the conjugate transpose and conjugate operator respectively, and l denotes the lth snapshot;

[0052] S1.4: In practical signal processing, the theoretical covariance matrix is usually approximated by the sample covariance matrix, i.e.

[0053]

[0054] where L denotes the total number of sampling snapshots, and l is a variable denoting the lth snapshot.

[0055] S2: Perform the operation of solving the theoretical covariance matrix of S1; including:

[0056] By constructing difference joint arrays, sparse arrays can provide more degrees of freedom than the actual number of array elements for DOA estimation, but an important prerequisite is to assume that the sources are non-coherent, which has been studied and confirmed in a large number of papers on sparse array design and virtual array processing. Under the assumption of non-coherent sources, the source covariance matrix is recalculated as:

[0057]

[0058] where denotes the power of the kth source. diag(·) denotes the diagonalization operator.

[0059] It can be seen that the covariance matrix of non-coherent sources is obviously different in structure from that of coherent sources. Specifically, since the sources are not related, the off-diagonal elements of R s are all close to 0, and thus can be regarded as a diagonal matrix. In contrast, since the sources are highly correlated, the off-diagonal elements of R are all close to 1, and thus cannot be ignored. An example is given to illustrate this difference. The covariance matrices of 5 non-coherent sources and coherent sources are shown in Figure 1 (a) and Figure 1 (b), respectively. It can be seen that R cannot be regarded as a diagonal matrix like R s . In addition, the ranks of the two are rank(R s ) = 5 and Therefore, the correlation between sources also causes the rank loss of the covariance matrix, thereby causing various sparse array processing techniques to fail.

[0060] However, it is worth noting that although the correlation causes the structure of R to be complex, the diagonal elements still contain the key information of the source power. This indicates that it is also possible to accurately estimate coherent sources by only using the diagonal elements of R .

[0061] S2.1: To distinguish the covariance matrix of coherent sources The diagonal and off-diagonal elements in equation (2) are reformulated as follows:

[0062]

[0063] In the formula, It is composed of source power vector The diagonal matrix formed, and It is by A matrix composed of off-diagonal elements; since the source power is distributed in On the diagonal, therefore, the second term in equation (6) can be eliminated. To achieve the goal of decoherence.

[0064] S2.2: As can be seen from the above analysis, it is necessary to estimate the coherent signal source vector. To obtain Therefore, the least squares method is used to obtain the sparse array observation signal model shown in equation (1). The estimated value is:

[0065]

[0066] In the formula, Θ represents all possible directions of the information source, unlike... Corresponding to a given angle,

[0067] This corresponds to all angles within the search range; therefore, equation (7) is an underdetermined problem with potentially infinitely many solutions.

[0068] S2.3: In order to obtain a unique solution, (7) is transformed into the following constraint minimization problem:

[0069]

[0070] ω represents the Lagrange multiplier vector.

[0071] S2.4: The Lagrange form of equation (8) is:

[0072]

[0073] calculate about The derivative is

[0074]

[0075] ω denotes the Lagrange multiplier vector, let achievable The optimal solution is

[0076]

[0077] Then, substituting the constraint condition in equation (8) can obtain

[0078]

[0079] Finally, substituting ω in equation (11) can obtain

[0080]

[0081] Let Then, the estimation value of the source covariance matrix can be expressed as

[0082]

[0083] S2.5: According to the estimation result in equation (14), equation (6) can be re-expressed as:

[0084]

[0085] In the formula,

[0086] S2.6: Eliminate the second term in equation (15) After that, is updated to

[0087]

[0088] So far, the non-diagonal elements in the coherent source covariance matrix have been successfully eliminated.

[0089] S3: On the basis of S2, further construct the Hermitian Toeplitz matrix through virtual array interpolation, and then realize the DOA estimation based on virtual array interpolation; S3 includes:

[0090] S3.1: Vectorize equation (16) in step 2 to obtain the signal vector r:

[0091]

[0092] In the formula, represents the estimation value of p, vec(·) represents the vectorization operator, represents the Khatri-Rao product, represents the power of the kth source; p i and p j represent the element positions of the sparse array, which are the previous p i ​d and p j d, which is omitted here due to normalization;

[0093] S3.2: In equation (16) The corresponding element positions are given by the following difference joint array:

[0094]

[0095] S3.3: For the model in equation (17), the co-arrayMUSIC algorithm can be used to obtain the direction of the signal source. However, for sparse arrays with holes in the differential co-array, such as coprime arrays, it is necessary to use virtual array interpolation technology to construct an equivalent observed signal vector corresponding to the ULA. Assuming it contains... The shortest ULA is but It includes both virtual sensors and nominal sensors (i.e., holes). Since we cannot obtain prior information about the received signals corresponding to the nominal sensors, we initialize them to zero and construct the observation signal vector corresponding to the ULA by combining it with the signal vector r. as follows:

[0096]

[0097] In the formula, <·> i This represents the i-th element of the vector;

[0098] S3.4: Based on the symmetry of the difference joint array, it can be known that... Therefore, use Construct the following Hermitian Toeplitz matrix:

[0099]

[0100] In the formula,

[0101] S3.5: Based on matrix filling theory, construct the following optimized model for reconstructing the received signal corresponding to the nominal sensor:

[0102]

[0103] In the formula, V is a binary matrix used to distinguish between non-zero and zero elements in T, η1 represents the threshold of the fitting error, Toep(z) represents the Hermitian Toeplitz matrix with vector z as the first column, and the symbol ⊙ represents the Hadamard product, ||·|| F The Frobenius norm is represented by the symbol ≥, which indicates a matrix inequality, and rank(·) represents the rank of the matrix.

[0104] S3.6: The rank minimization problem in (21) is NP-hard, and it is convexly relaxed to the following problem by using the nuclear norm minimization:

[0105]

[0106] where μ1is a penalty parameter for balancing the fitting error and the nuclear norm term, and ||·||Frepresents the Frobenius norm. *

[0107] S3.7: Since (22) can be further converted to the following trace minimization problem:

[0108]

[0109] where Tr(·) denotes the trace operator of a matrix.

[0110] S3.7: Since the reconstructed Hermitian Toeplitz matrix Toep(z) corresponds to the observed signals of a ULA, the subspace-based algorithms such as MUSIC and ESPRIT can be used to obtain the angle estimation results. Define the angle estimation value as Since corresponds to all angles in the search range, then may not be accurate enough. By using θ (1) to re-correct as:

[0111]

[0112] where The expression of can be divided into the following two cases.

[0113] Case 1: Under the underdetermined condition (i.e. ), the solution is similar to (7), and we have

[0114]

[0115] Case 2: Under the overdetermined condition (i.e. ), can be obtained by solving the following objective function

[0116]

[0117] The derivative of with respect to is

[0118]

[0119] Since is a column full-rank matrix, then​​ The optimal solution of equation (27) is

[0120]

[0121] Thus, we have

[0122]

[0123] S3.8: Based on equation (24), repeat the steps of equations (15)-(23) to get the new estimated angle which is the DOA estimation based on virtual array interpolation.

[0124] The main steps of TMR-VAI coherent sources DOA estimation algorithm combined with decorrelation strategy are summarized in Algorithm 1.

[0125]

[0126]

[0127] S4: Combine S3 and get a virtual ULA based on the sparse array of S1, perform the operation of the virtual ULA, then construct the Toeplitz matrix based on physical array interpolation, and then realize the DOA estimation based on physical array interpolation. S4 includes:

[0128] S4.1: From the perspective of sensor deployment alone, the sparse array can be regarded as a special form of ULA, that is, by deploying virtual sensors at the positions where there are no sensors in , a virtual ULA is obtained based on the sparse array of S1, that is, a virtual ULA with the same array aperture as the sparse array is generated The observation signal model of the virtual ULA is constructed as:

[0129]

[0130] In equation (30), denotes the streamer matrix corresponding to the virtual ULA , and denotes the steering vector pointing to θ k , and denotes the noise vector irrelevant to the signal vector

[0131] From equation (1) and equation (30), we find that the observation model of the sparse array can be obtained by dimension reduction of the observation model of the virtual ULA, that is:

[0132] ​​

[0133] where, is a compressed reduced dimensionality matrix with binary elements "1" and "0". Specifically, the mth element in is equal to the nth element in , i.e., Ψ m,n = 1, while the rest of the elements are 0. For example, regarding the sparse array

[0134]

[0135] and its corresponding hypothetical ULA with the same physical aperture

[0136] the compressed reduced dimensionality matrix Ψ can be written as

[0137]

[0138] S4.2: Based on S4.1, the theoretical covariance matrix of the sparse array observation signal represented by equation (3) is re-expressed as:

[0139]

[0140] where, denotes the covariance matrix of the hypothetical ULA observation signal, and Ψ is the compressed reduced dimensionality matrix;

[0141] S4.3: Ψ H is a diagonal matrix, whose binary elements on the main diagonal represent the actual sensors ("1") and the hypothetical sensors ("0"), respectively, so that:

[0142]

[0143] S4.4: For the hypothetical ULA, the estimated value is:

[0144]

[0145] where,

[0146] S4.5: After removing the non-diagonal elements in , the covariance matrix of the hypothetical ULA observation signal is represented as:

[0147]

[0148] where, is the identity matrix of ;

[0149] S4.5: As can be seen from equation (39), The elements related to the virtual sensors are all zeros, while the other elements are one-to-one corresponding to the elements in Therefore, based on S4.4, the following optimization model is established to activate the virtual sensors to correspond to the received signals:

[0150]

[0151] where B is a binary matrix used to distinguish the non-zero elements corresponding to the actual sensors and the zero elements corresponding to the virtual sensors in η2 represents the fitting error threshold; Toep(y) represents the Hermitian Toeplitz matrix with vector y as the first column;

[0152] S4.6: The optimization model in equation (40) is further convexly relaxed into the following minimization problem about the trace:

[0153]

[0154] where μ2 is a penalty parameter used to balance the fitting error and the nuclear norm term;

[0155] S4.7: The angle estimation result can be obtained by using the subspace-based algorithm on the reconstructed Hermitian Toeplitz matrix Toep(y) To improve the estimation accuracy, the following correction can be further made

[0156]

[0157] where Finally, based on equation (31), the steps of equations (38)-(40) are repeated to obtain the new estimated angle The DOA estimation based on physical array interpolation is realized.

[0158] The main steps of the TMR-PAI coherent source DOA estimation algorithm combined with the decorrelation strategy are summarized in Algorithm 2.

[0159]

[0160]

[0161] Simulation experiment:

[0162] 1. In this section, the excellent performance of the proposed algorithm in coherent source DOA estimation will be verified through simulation experiments. In the following experiments, the classic coprime array and its corresponding ULA with the same physical array aperture are used to collect coherent sources. More specifically, two coprime numbers M = 4 and N = 5 are used to deploy the sensors located at​ The corresponding ULA consists of 17 sensors to ensure the consistency of the array aperture.

[0163] To comprehensively evaluate the performance of the TMR-VAI algorithm and the TMR-PAI algorithm, we compare them with several previously developed DOA estimation algorithms for coherent sources, including the FBSS algorithm (hereinafter referred to as ULA+FBSS, as the FBSS algorithm is only applicable to the ULA), the NNM algorithm, the ANM algorithm, the ASS algorithm, the l1-SVD algorithm, and the LASSO algorithm. It should be noted that, except for the ULA+FBSS algorithm, all the above-mentioned algorithms involving convex optimization problems are implemented with the CVX toolbox.

[0164] ULA+FBSS, all the above-mentioned algorithms involving convex optimization problems are implemented with the CVX toolbox.

[0165] 1.1 Computational time

[0166] In the first experiment, we evaluate the computational complexity of different test algorithms by comparing their average running times in 100 Monte Carlo experiments. All algorithms are executed on the same notebook computer using MATLAB R2020a software, which is equipped with an Intel(R) Core(TM) i7-10750H CPU and 8 GB RAM. It is assumed that there are K = 3 coherent sources distributed at θ1= -5°, θ2= 0°, and θ3= 5° in the simulated scenario, and the sensor array collects a total of L = 300 snapshots. The angle search range is limited to [-10°, 10°]. Table 1 records the running times of each algorithm at different search step sizes Δθ. Obviously, the ULA+FBSS algorithm has the shortest running time because it does not need to solve any complex optimization problem. The l1-SVD algorithm and the LASSO algorithm both need to traverse the predefined grid points to search for the target, resulting in relatively large computation times, and the computation time increases exponentially with the increase in grid density. When estimating coherent sources, the LASSO algorithm bears the greatest computational burden because it needs to estimate all the elements in the source covariance matrix. In contrast, the computation times of the remaining algorithms differ, but they are all within the same order of magnitude. The computation times of these algorithms are higher than that of the ULA+FBSS algorithm, but they are much lower than those of the l1-SVD algorithm and the LASSO algorithm. It should be noted that the computation times of the ANM algorithm and our proposed algorithms are slightly higher than those of the NNM algorithm and the ASS algorithm because they need to reconstruct higher-dimensional matrices. In addition, our proposed algorithms involve two optimization processes, and their computation times are approximately twice those of the ANM algorithm.

[0167] Table 1 Running times of different algorithms at different search step sizes (unit: seconds)

[0168]

[0169]

[0170] 1.2 Spatial spectrum

[0171] In the second experiment, we compare the estimation performance of different algorithms for different number of coherent sources. First, we consider that the sensor array observes K = 5 coherent sources with coherence coefficient vector and they are uniformly distributed in [-30°, 30°]. The signal-to-noise ratio (SNR) and the number of sampling snapshots are set to 0 dB and L = 100, respectively. The spatial spectrum obtained by searching the source directions in the range of -40° to 40° with a step of 0.5° is shown in Figure 2 It can be seen that the spectral peaks in each subfigure are around the actual incident angles, which means that all the algorithms have the ability to effectively estimate the five coherent sources.

[0172] However, Figure 2 (d) the spatial spectrum of the presented ASS algorithm is slightly inferior to other algorithms, with not only a relatively flat spectrum peak but also a relatively large deviation from the actual incident direction. This is because the ASS algorithm only relies on the eigenvector corresponding to the largest eigenvalue of the signal covariance matrix observed by the coprime array when constructing the augmented Hankel matrix. Although this strategy is simple, it inevitably discards a large amount of key information contained in other eigenvectors, thereby weakening the DOA estimation performance.

[0173] Then, we increase the number of coherent sources to 7, with the corresponding coherence coefficient vector being Meanwhile, we use the other simulation parameters in Figure 2 Figure 3 (a), Figure 3 (b), and Figure 3 (d) show that the ULA+FBSS algorithm, the NNM algorithm, and the ASS algorithm have a sharp decline in performance when trying to estimate 7 coherent sources. The fundamental reason for this is the inherent processing mechanism of each algorithm, i.e.

[0174] ​ULA+FBSS algorithm needs to go through spatial smoothing process, while NNM algorithm and ASS algorithm involve the construction of augmented Toeplitz matrix and augmented Hankel matrix, respectively. These steps make the actually available degrees of freedom greatly reduced to half of the physical array aperture. In the design example, whether a 17-element ULA or an 8-element coprime array is adopted, the actually available degrees of freedom of the three algorithms are only 9, which is far from enough to accurately identify 7 closely distributed coherent sources under the given signal-to-noise ratio and limited number of sampling snapshots. This phenomenon highlights the limitations of traditional algorithms in dealing with high-density coherent sources due to the loss of degrees of freedom.

[0175] It is worth noting that although the ANM algorithm can identify 7 coherent sources, the spatial spectrum characteristics are relatively poor, specifically, the peak response corresponding to some angles is low, which to some extent affects the accuracy and reliability of angle estimation. In contrast, Figure 3 (e), Figure 3 (f), Figure 3 (g) and Figure 3 (h) respectively present the l1-SVD algorithm, LASSO algorithm, TMR-VAI algorithm and TMR-PAI algorithm, all of which exhibit relatively sharp spectral peaks, which indicates that they can more effectively estimate all incident sources, thereby breaking through the performance bottleneck encountered by traditional algorithms due to the limited degrees of freedom. In addition, compared with the l1-SVD algorithm and LASSO algorithm based on grid division, the TMR-VAI algorithm and TMR-PAI algorithm proposed by us belong to the gridless DOA estimation algorithm, which can identify the sources in a shorter time.

[0176] Figure 3 Spatial spectrum comparison of K=7 coherent sources uniformly distributed between-30° and 30°, where SNR=0dB and the number of snapshots is L=100. The blue solid line represents the estimation result, and the red dashed line corresponds to the actual incident direction of the coherent source.

[0177] Finally, we compare the angle estimation performance of several algorithms under underdetermined conditions. We further increase the number of coherent sources to 9, and the corresponding coherence coefficient vector is The remaining simulation parameters are consistent with Figure 2 and Figure 3 From Figure 4 , it can be seen that the performance of each algorithm is significantly different. Figure 4 As shown in (a), the ANM algorithm has obvious spurious peaks in the spatial spectrum, and these larger peak responses are prone to cause angle estimation deviation, thereby reducing the accuracy of estimation. Figure 4(b) The problem of missing source estimation occurs in the spatial spectrum of the displayed LASSO algorithm, because the strong correlation between the sources makes them exhibit similar features in the sparse representation framework, thus confusing the recognition ability of the algorithm, resulting in the loss of some sources in the estimation process. Both the TMR-VAI algorithm and the TMR-PAI algorithm effectively identify 9 coherent sources, as shown in Figure 4 (c) and Figure 4 (d). This result not only illustrates the effectiveness of the source covariance matrix correction strategy in the virtual domain and the physical domain to resolve coherence, but also verifies the superiority of the TMR-VAI algorithm and the TMR-PAI algorithm in the task of coherent source DOA estimation under the underdetermined condition.

[0178] Figure 4 Comparison of the spatial spectrum of K = 9 coherent sources uniformly distributed between -30° and 30°, where SNR = 0 dB and the number of snapshots L = 100. The blue solid line represents the estimation result, and the red dashed line corresponds to the actual incident direction of the coherent source.

[0179] 1.3 Estimation accuracy

[0180] In the third experiment, we quantitatively analyze the accuracy of the compared algorithms for coherent source DOA estimation by calculating the root mean square error (RMSE). The RMSE is calculated as follows

[0181]

[0182] where Q represents the number of Monte Carlo experiments, and θ k (q) represents the estimated value of θ

[0183] Considering the simulation scenario in Figure 2 , we calculate the RMSE of each algorithm under 200 Monte Carlo experiments, and plot these RMSE points and the Cramér-Rao Bound (CRB) in Figure 5 . Figure 5 (a) The curve of RMSE versus signal-to-noise ratio is plotted, where the number of sampling snapshots is set to 100. It can be observed that as the signal-to-noise ratio increases, the estimation error of each algorithm gradually decreases, and when the signal-to-noise ratio is greater than -5 dB, the estimation performance tends to be saturated. The limitations of the NNM algorithm and the ASS algorithm in utilizing degrees of freedom result in larger estimation errors. Although the ANM algorithm and the proposed algorithm can theoretically utilize all degrees of freedom, the latter performs better in actual estimation, especially when the signal-to-noise ratio exceeds -10 dB, each

[0184] The RMSE points are lower than the ANM algorithm. The estimation accuracy of the l1-SVD algorithm and the LASSO algorithm cannot be comparable with our proposed algorithms due to the limitation of the search grid points.

[0185] We also notice that the TMR-VAI algorithm and the TMR-PAI algorithm have similar estimation performance, which reflects the consistency of the sparse array and its corresponding virtual ULA in reconstructing the Toeplitz matrix. On the other hand, Figure 5 (b) The RMSE curves plotted against the number of sampling snapshots have similar results, where the signal-to-noise ratio is set to -10 dB. It can be seen that the estimation error of each algorithm gradually decreases with the increase of the number of sampling snapshots, but the estimation error of the proposed TMR-VAI algorithm and the TMR-PAI algorithm is closest to the CRB, achieving the highest estimation accuracy among the compared algorithms, followed by the ANM algorithm, the l1-SVD algorithm and the LASSO algorithm, and finally the NNM algorithm and the ASS algorithm. These phenomena fully demonstrate that the proposed TMR-VAI algorithm and the TMR-PAI algorithm are superior to the previously proposed algorithms in estimating coherent sources.

[0186] 1.4 Resolution

[0187] In the fourth experiment, we compare the resolution capabilities of all test algorithms. Assume that two closely spaced coherent sources located at θ1= -1° and θ2= 1° impinge on the sensor array, and the coherence coefficient vector is The signal-to-noise ratio SNR = 10 dB, the number of sampling snapshots is L = 100, and the search step size is set to 0.1°. Figure 6 The energy distribution diagrams of the normalized spatial spectrum corresponding to each algorithm in 100 Monte Carlo experiments are plotted, where the brighter area reflects the larger energy of the corresponding element, and the horizontal coordinate corresponding to the maximum energy area corresponds to the DOA estimation result. From Figure 6 It can be seen that the ULA+FBSS algorithm, the NNM algorithm and the ASS algorithm only present a single bright column in the estimation result, which indicates that they cannot effectively resolve the two coherent sources. The reason is that these three algorithms only use 9 degrees of freedom in actual processing, and the loss of degrees of freedom reduces the resolution performance of DOA estimation. In contrast, the remaining algorithms all present two clearly distinguishable bright columns around the actual incident direction of the signal source, which indicates that they can effectively identify closely spaced coherent sources. Although Figure 6 (e) The estimation result of the l1-SVD algorithm shown is already quite close to the true value, but there are still some false peaks in the estimation result, which will have an adverse effect on the estimation accuracy to some extent. Since the ANM algorithm, the LASSO algorithm and our proposed TMR-VAI algorithm and TMR-PAI algorithm increase the available degrees of freedom to 17, the estimation result is closer to the true value, thus exhibiting stronger resolution capability.

[0188] Conclusions:

[0189] The present application studies the problem of DOA estimation for coherent sources based on sparse array. By separating the diagonal elements and off-diagonal elements in the covariance matrix of sources, decorrelation operation is realized. Then, from the perspective of virtual domain and physical domain, Toeplitz matrix reconstruction problem is established for DOA estimation. Compared with the existing coherent DOA estimation algorithms based on sparse array, the decorrelation algorithm proposed in this paper maintains the complete degrees of freedom and array aperture, and has better performance. In particular, the proposed algorithm is also applicable to the underdetermined case where the number of coherent sources exceeds the actual number of sensors. Although the proposed algorithm has significant advantages in coherent source estimation, the RMSE performance tends to saturate and deviates from the CRB at high SNR. Therefore, future work will focus on developing techniques that can improve the accuracy of sparse array DOA estimation for coherent sources at high SNR.

[0190] The above shows and describes the basic principles, main features and advantages of the present application. Those skilled in the art should understand that the present application is not limited to the above embodiments, and the above embodiments and descriptions in the specification are only to illustrate the principles of the present application. Without departing from the spirit and scope of the present application, various changes and improvements can be made to the present application, and these changes and improvements all fall within the scope of the claimed present application. The scope of protection of the present application is defined by the appended claims and their equivalents.

Claims

1. A method for estimating the DOA of sparse array coherent sources based on Toeplitz matrix reconstruction, characterized in that, include: S1: Assuming a coherent source is incident on a sparse array, construct a sparse array observation signal model and solve for the theoretical covariance matrix of the sparse array observation signal model. S2: Solve the theoretical covariance matrix of S1; S3: Based on S2, the Hermitian Toeplitz matrix is ​​further constructed through virtual array interpolation, thereby realizing DOA estimation based on virtual array interpolation; S4: Combining S3 and obtaining a hypothetical ULA based on the sparse array of S1, the hypothetical ULA is solved, and then a Toeplitz matrix based on physical array interpolation is constructed, thereby realizing DOA estimation based on physical array interpolation, and finally realizing DOA estimation of sparse array coherent source based on Toeplitz matrix reconstruction.

2. The DOA estimation method for sparse array coherent sources based on Toeplitz matrix reconstruction according to claim 1, characterized in that: The sparse array observation signal x at the l-th sampling snapshot time constructed by S1 s (l) The model is: In the formula, s(l) represents the reference signal source, and β = [β1, β2, ..., β K ] T Denotes the coherence coefficient vector, where β k θ is a non-zero complex constant. k This represents the angle of the k-th source. Let β represent the cardinality of the set. k Let represent the coherence coefficient of the k-th source. This represents an additive white Gaussian noise vector independent of the signal source. Represents an array manifold matrix. This represents the source vector of the coherent signal.

3. The DOA estimation method for sparse array coherent sources based on Toeplitz matrix reconstruction according to claim 2, characterized in that, The theoretical covariance matrix in S1 for: In the formula, L represents the total number of sampling snapshots, l is the variable, the l-th snapshot, and (·) H This indicates the conjugate transpose.

4. The DOA estimation method for sparse array coherent sources based on Toeplitz matrix reconstruction according to claim 3, characterized in that, S2 solution yields for: in, Represents the source covariance matrix The estimated value, Indicates noise power. express The identity matrix, diag(·) denotes the diagonalization operator.

5. The DOA estimation method for sparse array coherent sources based on Toeplitz matrix reconstruction according to claim 2, characterized in that, In S3, corrections for: Wherein, the angle estimate is defined as Based on correction The estimated angle is further obtained as follows This yields a DOA estimate based on virtual array interpolation.

6. The DOA estimation method for sparse array coherent sources based on Toeplitz matrix reconstruction according to claim 5, characterized in that: S4, corrected for: In the formula, This corresponds to the hypothetical ULA. The manifold matrix, based on the modified The estimated angle is further obtained as follows This yields a DOA estimate based on physical array interpolation.

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