Coherent signal source DOA estimation method based on Toeplitz reconstruction and Schur decomposition

Abstract

The invention discloses a coherent signal source DOA estimation method based on Toeplitz reconstruction and Schur decomposition, and the method comprises the following steps: S1, array receiving signal acquisition and covariance matrix calculation, S2, first Toeplitz matrix construction, S3, covariance matrix Schur decomposition and signal matrix construction, S4, second Toeplitz matrix construction, and S5, matrix fusion and DOA estimation. A first Toeplitz matrix is constructed; the method comprises the following steps: constructing a first Toeplitz matrix, performing Schur decomposition on a covariance matrix, constructing a signal matrix, constructing a second Toeplitz matrix by taking a mean value of each diagonal line of the signal matrix as a reference, fusing the two Toeplitz matrixes, performing subspace separation and spectral peak search on the fused matrix to realize DOA estimation, and performing simulation verification to obtain the DOA estimation result. The algorithm provided by the invention can effectively reduce the DOA estimation error of the coherent signal source under the scenes of low signal-to-noise ratio and few snapshots, and can effectively improve the stability of the estimation result.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, specifically relating to a method for estimating the DOA of coherent sources based on Toeplitz reconstruction and Schur decomposition. Background Technology

[0002] Direction of Arrival (DOA) estimation is a hot research topic in array signal processing, widely applied in wireless communication, radar, sonar, and other fields. In the early stages of DOA estimation research, array-based DOA estimation methods primarily relied on conventional beamforming. However, this method is limited by the spatial resolution capability of radar beams; when multiple signal sources have incident angles within the same beamwidth, it cannot effectively distinguish the true number of signal sources. Therefore, super-resolution DOA estimation algorithms that break the Rayleigh resolution limit have emerged, among which the Multiple Signal Classification (MUSIC) algorithm based on subspace decomposition is the most representative. The MUSIC algorithm separates the signal subspace from the noise subspace by performing eigenvalue decomposition on the array covariance matrix, constructs a spatial spectrum using the orthogonality of the two subspaces, and finally completes DOA estimation through spectral peak search.

[0003] However, in real-world scenarios, the presence of coherent signals makes effective subspace separation difficult, leading to a degradation in the DOA estimation performance of the MUSIC algorithm. Therefore, decoherence techniques are needed to address this issue. Existing decoherence methods are mainly divided into spatial smoothing algorithms and matrix reconstruction methods. The Toeplitz matrix reconstruction algorithm reconstructs the matrix using signal data from the covariance matrix, thereby achieving decoherence processing. However, in scenarios with low signal-to-noise ratios and few snapshots, its estimation accuracy is insufficient for practical applications. This is mainly because the method still directly modifies the covariance matrix. Perform matrix reconstruction, because at this time Only partial source information can be extracted. When the signals are fully coherent, eigenvalue decomposition can only obtain information from a single source, and the constructed matrix... The data is not highly representative. Under conditions of low signal-to-noise ratio or few snapshots, the source information is further reduced, resulting in significant defects in the Toeplitz reconstruction algorithm. Summary of the Invention

[0004] The purpose of this invention is to provide a coherent source DOA estimation method based on Toeplitz reconstruction and Schur decomposition to solve the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method for estimating the DOA of coherent sources based on Toeplitz reconstruction and Schur decomposition, comprising the following steps:

[0006] S1. Acquisition of array received signals and calculation of covariance matrix: using... Uniform linear array receiver The far-field coherent incident signal, the spacing between adjacent elements of the uniform linear array is ( (Where the incident signal wavelength is used), the data matrix is ​​obtained through array reception. ,in yes 3D signal vector yes 3D noise vector The guide vector matrix is ​​based on the received data matrix. Calculate the array covariance matrix The covariance matrix The sample covariance matrix is ​​used for calculation, and the formula is as follows:

[0007] ；

[0008] in Let be the signal covariance matrix. For noise variance, It is an M-dimensional identity matrix;

[0009] S2. Construction of the first Toeplitz matrix: Traversing the covariance matrix For all diagonals, including the main diagonal and the non-main diagonals on both sides of the main diagonal, calculate the arithmetic mean of all elements on each diagonal, and form a reference vector by combining the means of each diagonal in order from the main diagonal to the upper right diagonal. And construct the Toeplitz matrix. , where the reference vector The formula is:

[0010] ；

[0011] Based on the benchmark As The first line, except The conjugate of all elements other than the first element is used as the first column of the matrix. The remaining elements are then padded with the first element of each diagonal element to obtain the Toeplitz matrix. ;

[0012] S3. Schur Decomposition of the Covariance Matrix and Construction of the Signal Matrix: The Schur decomposition algorithm is used to construct the covariance matrix. Decompose:

[0013] ;

[0014] in It is an upper triangular matrix, and its diagonal elements are precisely the covariance matrix. eigenvalues; It is a unitary matrix, and according to the matrix... diagonal elements and unitary matrix The signal matrix is ​​constructed from the column vectors, and the upper triangular matrix is ​​extracted. The diagonal elements are the covariance matrix. The eigenvalues ​​are then arranged in descending order from largest to smallest, first for the upper triangular matrix. Arranged in descending order of diagonal elements, for a unitary matrix The column vectors correspond to Sort the diagonal elements, i.e., the first... Each column vector contains signal data. Corresponding to The signal matrix is ​​constructed by comprehensively utilizing all information related to the source, based on the number of larger diagonal values. :

[0015] ；

[0016] S4. Construction of the second Toeplitz matrix: Using the same diagonal mean calculation method and matrix construction rules as in step 2, based on the signal matrix obtained in step 3. Calculate using the input. The mean of each diagonal and form a reference vector. :

[0017] ；

[0018] Based on the benchmark As a reconstruction matrix The first line, except All remaining elements other than the first element are conjugates, which are used as... The first column is padded with the remaining elements according to the first element of the diagonal to obtain the matrix. ;

[0019] S5. Matrix Fusion and DOA Estimation: The first Toeplitz matrix... With the second Toeplitz matrix Perform equal-weighted fusion to obtain the fusion matrix. and the fusion matrix Perform eigenvalue decomposition to obtain eigenvalues ​​and corresponding eigenvectors, then solve for the eigenvectors and construct the noise subspace. According to the signal vector Construct new spectral functions based on the orthogonality with the noise subspace. Perform a peak search:

[0020] ；

[0021] When traversing to the true perspective When the spectral peaks show sharp peaks, the angle value corresponding to the sharp peaks obtained by the search is the DOA estimation result of the coherent source.

[0022] Preferably, in S1, The formula for the guiding vector matrix is:

[0023]

[0024] And select the covariance matrix The first row is used as the base ,by As the first row of the Toeplitz matrix:

[0025] ;

[0026] remove Take the conjugate of all elements other than the first element and use it as the first column of the matrix. Fill the remaining elements with elements that are equal on the diagonal to obtain the Toeplitz matrix. :

[0027] 。

[0028] Preferably, in S1, regarding the covariance matrix During eigenvalue decomposition, the eigenvectors are solved and the noise subspace is constructed. Since the signal subspace is consistent with the column space of the guidance matrix, then Also with the noise subspace Orthogonal:

[0029]

[0030] Based on this, construct the MUSIC spatial spectral function. :

[0031]

[0032] When traversing to the true perspective When the denominator in the formula approaches zero, the function value will show a sharp peak. The MUSIC method estimates DOA based on this principle.

[0033] Preferably, in S2, the matrix :

[0034] ;

[0035] And on Perform eigenvalue decomposition, select the two largest eigenvalues, and calculate their corresponding eigenvectors. and As a reference vector and Construct the Toeplitz matrix respectively. and ,in:

[0036]

[0037] Through calculation The conjugate product of the first element and the rest of the elements ,available The first line :

[0038]

[0039] Reverse formula Taking the conjugate transpose, but keeping the first element unchanged, we can obtain... The first column :

[0040]

[0041] Wherein the formula superscript This indicates that the matrix is ​​taken as its conjugate transpose. This indicates that the matrix is ​​transposed.

[0042] In obtaining After deleting the elements in the first row and first column, fill in the remaining elements according to the principle of equal diagonal elements to obtain the Toeplitz matrix:

[0043] 。

[0044] Preferably, the calculation is performed using the same steps. The conjugate product of the first element and the rest of the elements , forming a new row vector and utilize By reconstructing the matrix, we can obtain the Toeplitz matrix. :

[0045]

[0046] The above steps yield four reconstructed Toeplitz matrices, which are then used to calculate the final matrix. :

[0047] 。

[0048] Preferably, for the Toeplitz matrix Perform eigenvalue decomposition, solve for eigenvectors, and construct the noise subspace. According to the signal vector With noise subspace Orthogonality relation, construct new spectral functions Perform a peak search:

[0049]

[0050] When traversing to the true perspective At the same time, the function value will also show sharp peaks, which are used to estimate the direction of arrival of the wave. In fact, when performing simulation calculations in MATLAB, a minimum relative precision function is added to the denominator of the spectral function. This is to prevent errors when the denominator is zero, and also to highlight the effect of the peak value.

[0051] Preferably, in step S3, the Schur decomposition is implemented using the standard Schur decomposition function in numerical computation software. In the MATLAB environment, the Schur decomposition function is used to obtain the upper triangular matrix. The diagonal elements are the covariance matrix. The eigenvalues ​​of the eigenvalues ​​are such that the algebraic multiplicity of the eigenvalues ​​is consistent with the geometric multiplicity.

[0052] Preferably, when calculating the diagonal mean in S2 and S4, if the number of elements in a certain diagonal is less than 3, the mean of the neighboring diagonal elements is smoothed to reduce the impact of noise on the calculation of the diagonal mean.

[0053] Preferably, in step S5, the peak search employs a hill-climbing method or a particle swarm optimization algorithm. When the number of information sources is unknown, the number of information sources is first determined by information theory criteria or eigenvalue thresholding, and then feature subspace separation is performed. This provides a flexible selection of peak search algorithms and a scheme for determining the number of information sources, solving the estimation problem in scenarios where the number of information sources is unknown and expanding the applicability of the method.

[0054] Preferably, the search range of the spatial spectrum can be adjusted according to the actual application scenario. In the radar scenario, the search range is [-90°, 90°], and in the sonar scenario, the search range is [-180°, 180°]. The search interval can be adjusted between 0.01° and 0.1° to provide spatial spectrum search parameters (range, interval) for different application scenarios, so that the method can adapt to the needs of different fields such as radar and sonar, and improve the practicality of the method.

[0055] Compared with the prior art, the beneficial effects of the present invention are:

[0056] (1) The first Toeplitz matrix is ​​constructed based on the mean of each diagonal of the covariance matrix; then the covariance matrix is ​​decomposed by Schur to construct the signal matrix. The second Toeplitz matrix is ​​constructed based on the mean of each diagonal of the signal matrix. Finally, the two Toeplitz matrices are fused, and the fused matrix is ​​subjected to subspace separation and spectral peak search to achieve DOA estimation. Simulation verification shows that the proposed algorithm can effectively reduce the DOA estimation error of coherent sources in scenarios with low signal-to-noise ratio and few snapshots, and can also effectively improve the stability of the estimation results.

[0057] (2) Through multiple technical optimizations, the adaptability to complex scenarios has been significantly enhanced. The introduction of Schur decomposition has enabled the accurate screening of information sources, which can extract high-confidence information source components from the noise-contaminated covariance matrix and reduce the interference of invalid noise information on subsequent processing. Both Toeplitz reconstructions are based on the diagonal mean. This statistical averaging process can further smooth noise fluctuations and improve the stability of matrix reconstruction, so that the reconstruction matrix can still retain the core information source features when the noise interference is strong. The equal weighted fusion strategy of the dual reconstruction matrix can combine the advantages of the two reconstructions and make up for the limitations of single reconstruction in information extraction. The fused matrix contains both the global information of the original covariance matrix and the accurate information source information screened by Schur decomposition, so that reliable DOA estimation results can still be output stably in scenarios with low signal-to-noise ratio and few snapshots.

[0058] (3) By optimizing the design of the fusion matrix, the accuracy of subspace separation is greatly improved. The combination of dual Toeplitz reconstruction and Schur decomposition makes the fusion matrix not only have good Toeplitz structural characteristics, but also contain accurate source feature information. The signal subspace and noise subspace obtained after feature decomposition have clearer boundaries and stronger orthogonality. The spectral function constructed based on this stable orthogonal relationship can more accurately capture the angle position corresponding to the real source during the spectral peak search process, effectively suppress the generation of false peaks, and make the spectral peak corresponding to the real angle sharper, which significantly improves the reliability and accuracy of DOA estimation results.

[0059] (4) The processing framework based on uniform linear array can be flexibly adapted to different array sizes. It can be applied to DOA estimation scenarios in different fields such as radar and sonar through simple parameter adjustment. Whether it is the search range of [-90°, 90°] in radar scenario or the wide range search requirement of [-180°, 180°] in sonar scenario, this method can achieve reliable estimation through stable matrix processing flow. At the same time, for scenarios where the number of information sources is unknown, it is compatible with information theory criteria or eigenvalue threshold method to determine the number of information sources, which further expands the applicability of the method and reduces the application threshold in different scenarios. Attached Figure Description

[0060] Figure 1 This is a diagram of the uniform linear array model of the present invention;

[0061] Figure 2 This is a graph showing the RMSE as a function of signal-to-noise ratio in this invention.

[0062] Figure 3 This is a comparison of the RMSE variation with signal-to-noise ratio between the improved method of this invention and the traditional Toeplitz method;

[0063] Figure 4 This is a comparison of spectral peaks when the signal-to-noise ratio is -10dB in this invention;

[0064] Figure 5 This is a comparison of spectral peaks when the signal-to-noise ratio is -5dB in this invention;

[0065] Figure 6 This is a graph showing the RMSE of the present invention as a function of the number of snapshots.

[0066] Figure 7 This is a comparison chart showing the variation of RMSE with the number of snapshots between the improved method of this invention and the traditional Toeplitz method;

[0067] Figure 8 This is a peak comparison diagram when the number of snapshots is 200 according to the present invention;

[0068] Figure 9 This is a comparison diagram of spectral peaks when the number of snapshots is 300 according to the present invention. Detailed Implementation

[0069] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installed," "equipped with," "sleeved with," "connected," etc., should be interpreted broadly. For example, "connection" can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium; it can be a connection within two components. For those skilled in the art, the specific meaning of the above terms in this invention can be understood according to the specific circumstances.

[0071] This invention provides, for example Figure 1-9 This illustrates a method for estimating the DOA of coherent sources based on Toeplitz reconstruction and Schur decomposition:

[0072] Example

[0073] 1. Array Model

[0074] Assume there is A far-field signal The incident light is emitted into a uniform linear array, where the number of elements in the uniform linear array is... Spacing between adjacent array elements ( (Where the signal wavelength is), the number of snapshots is , No. The incident azimuth angle of the road signal is ;

[0075] The data reception matrix of the array at time t is:

[0076] (1)

[0077] In the formula yes 3D signal vector yes 3D noise vector For the guiding vector matrix:

[0078] (2)

[0079] The signal direction is The steering vector is determined by the phase difference between the signal and different array elements. Based on this, the covariance matrix of the array data can be obtained. :

[0080] (3)

[0081] in Let be the signal covariance matrix. For noise variance, It is an M-dimensional unit matrix.

[0082] In the traditional MUSIC method, the first step is to directly analyze the covariance matrix. Eigenvalue decomposition, solving for eigenvectors and constructing the noise subspace Since the signal subspace is consistent with the column space of the guidance matrix, then Also with the noise subspace Orthogonal:

[0083] (4)

[0084] Based on this, construct the MUSIC spatial spectral function. :

[0085] (5)

[0086] Finally, a spectral peak search is performed, and when the true angle is reached... When the denominator in equation (5) approaches zero, the function value will have a sharp peak. The MUSIC method estimates DOA based on this principle. This method has high estimation accuracy when processing incoherent signals, but when the source is a coherent signal, the performance of this method drops significantly or even fails.

[0087] 2. Toeplitz Matrix Reconstruction Algorithm

[0088] To address the challenge of DOA estimation in coherent sources using the MUSIC method, specifically the covariance matrix... A rank deficiency phenomenon occurs, where the calculated rank is less than the actual number of sources. Furthermore, the constructed signal subspace and noise subspace do not satisfy the orthogonality condition, making it impossible to find the true angle using spectral peak search. Therefore, it is necessary to adjust the signal subspace based on source information. Matrix reconstruction is performed to achieve the purpose of decoherence.

[0089] The principle of the Toeplitz matrix is ​​to use the elements of the first row or first column of the matrix as a reference vector or benchmark. ,by As the first row of the newly constructed matrix, the new first column is also... The corresponding elements are multiplied together to obtain the matrix, and then the remaining elements are filled in according to the principle that the diagonal elements are equal. At this point, a full-rank Toeplitz matrix can be obtained.

[0090] Analysis of existing Toeplitz matrix reconstruction methods reveals that the main difference lies in the reference vector. The choice of which, as in reference [8], fills the missing terms in the Toeplitz matrix by introducing a low-rank reconstruction matrix; reference [9] calculates Constructing from the average of each diagonal Reference

[10] not only constructs the autocovariance matrix of a single Toeplitz matrix, but also supplements it with the cross-covariance matrix between different Toeplitz matrices. This paper constructs four Toeplitz matrices by integrating existing methods, and finally forms a matrix. Simulation experiments were conducted to verify the feasibility of the Toeplitz matrix reconstruction method. At the same time, the method was compared and verified with the algorithm proposed in this paper, which also showed the limitations of the method in scenarios with low signal-to-noise ratio and few snapshots. The specific calculation process is as follows.

[0091] First, select the covariance matrix. The first row is used as the base ,by As the first row of the Toeplitz matrix:

[0092] (6)

[0093] remove Take the conjugate of all elements other than the first element and use it as the first column of the matrix. Fill the remaining elements with elements that are equal on the diagonal to obtain the Toeplitz matrix. :

[0094] (7)

[0095] The following is based on The benchmark is composed of the means of each diagonal. Construct the Toeplitz matrix , It is a matrix Starting from the top right corner The mean of the diagonals:

[0096] (8)

[0097] Based on the benchmark As The first line, except The conjugate of all elements other than the first element is used as the first column of the matrix. The remaining elements are then padded with the first element of each diagonal element to obtain the Toeplitz matrix. :

[0098] (9)

[0099] Then to Perform eigenvalue decomposition, select the two largest eigenvalues, and calculate their corresponding eigenvectors. and As a reference vector and Construct the Toeplitz matrix respectively. and ,in:

[0100] (10)

[0101] Through calculation The conjugate product of the first element and the rest of the elements ,available The first line :

[0102] (11) (12)

[0103] Taking the conjugate transpose of equation (11) while keeping the first element unchanged, we can obtain the following: The first column :

[0104] (13)

[0105] superscript of equation (13) This indicates that the matrix is ​​taken as its conjugate transpose. This indicates that the matrix is ​​transposed.

[0106] get After deleting the elements in the first row and first column, the remaining elements are filled according to the principle of equal diagonals, resulting in the Toeplitz matrix as shown in equation (14):

[0107] (14)

[0108] Calculate using the same steps The conjugate product of the first element and the rest of the elements , forming a new row vector and utilize By reconstructing the matrix, we can obtain the Toeplitz matrix. :

[0109] (15)

[0110] The above steps yield four reconstructed Toeplitz matrices, which are then used to calculate the final matrix. :

[0111] (16)

[0112] For Toeplitz matrix The subsequent procedures for DOA estimation are the same as those for the MUSIC method: first, the matrix... Eigenvalue decomposition, solving for eigenvectors and constructing the noise subspace According to the signal vector With noise subspace Orthogonality relation, construct new spectral functions Perform a peak search:

[0113] (17)

[0114] When traversing to the true perspective At the same time, the function value will also show sharp peaks, which are used to estimate the direction of arrival of the wave. In fact, when performing simulation calculations in MATLAB, a minimum relative precision function is added to the denominator of the spectral function. This is to prevent errors when the denominator is zero, and also to highlight the effect of the peak value.

[0115] 3. Algorithm in this paper

[0116] Since coherent signals usually differ only in phase and amplitude, directly applying the covariance matrix... Eigenvalue decomposition not only fails to accurately determine the number of sources, but also fails to correctly distinguish between the signal subspace and the noise subspace. The Toeplitz matrix reconstruction method, introduced earlier, first extracts the available source information and then uses this information to... Matrix reconstruction is performed to achieve decoherence; however, existing Toeplitz reconstruction algorithms still have shortcomings. In scenarios with low signal-to-noise ratios and few snapshots, the algorithm's error is significant, exceeding the feasibility limit. This is because the matrix constructed by this method... The data is not highly representative. Furthermore, if the effects of low signal-to-noise ratio or few snapshots are added, the amount of source information obtainable will be further reduced, leading to a poorly reconstructed matrix. Furthermore, it cannot effectively separate subspaces. To address this problem, this paper proposes an algorithm combining Toeplitz matrix reconstruction and Schur decomposition. The algorithm constructs two Toeplitz matrices. The first Toeplitz matrix is ​​constructed using the mean of each diagonal line of the covariance matrix as a reference. This process is related to the construction of the Toeplitz matrix in Chapter 2. Similarly, the key point is in constructing the Toeplitz matrix. The specific process is as follows:

[0117] First, let's look at the covariance matrix. Perform Schur decomposition:

[0118] (18)

[0119] in It is an upper triangular matrix, and its diagonal elements are precisely the covariance matrix. eigenvalues; It is a unitary matrix, and its inverse matrix is ​​its own conjugate transpose:

[0120] (19)

[0121] Traditional methods using EVD decomposition are only applicable to diagonalizable complex matrices and cannot efficiently utilize source information. Schur decomposition, on the other hand, is applicable to all complex matrices, and its effect on Toeplitz matrix decomposition is equivalent to EVD decomposition. However, it can still effectively decompose matrices that do not possess the Toeplitz property, and its decomposition is not affected by the matrix's characteristics. Based on this principle, this paper... Perform Schur decomposition based on the matrix. diagonal elements and unitary matrix The signal matrix is ​​constructed from the column vectors, and then the signal matrix is ​​reconstructed using the Toeplitz matrix to obtain a matrix that is highly representative of the signal source. .

[0122] First, look at the upper triangular matrix. Arranged in descending order of diagonal elements, for a unitary matrix The column vectors correspond to Sort the diagonal elements, i.e., the first... Each column vector contains signal data. Corresponding to The number of larger diagonal values. Based on this, a signal matrix is ​​constructed by comprehensively utilizing all information related to the signal source. :

[0123] (20)

[0124] Then Reconstructing the Toeplitz matrix based on this method effectively solves the problem of insufficient representativeness in the original method. The mean of each diagonal Forming the reference vector :

[0125] (21)

[0126] Based on the benchmark As a reconstruction matrix The first line, except All remaining elements other than the first element are conjugates, which are used as... The first column is padded with the remaining elements according to the first element of the diagonal to obtain the matrix. :

[0127] (22)

[0128] matrix and The final Toeplitz reconstruction matrix can be obtained by fusion. :

[0129] (23)

[0130] Then use the Toeplitz matrix To perform DOA estimation, first analyze the matrix. Perform Schur decomposition to obtain the upper triangular matrix. The diagonal elements are the eigenvalues, then the eigenvectors are solved and the noise subspace is constructed. According to the signal vector Construct new spectral functions based on the orthogonality with the noise subspace. Perform a peak search:

[0131] (24)

[0132] When traversing to the true perspective At that time, the spectral peaks show sharp peak values.

[0133] 4. Algorithm Simulation and Analysis

[0134] To verify the DOA estimation performance of the above algorithm, a simulation experiment was conducted using MATLAB, version R2023a, and the simulation environment was Windows 10.

[0135] In the simulation experiment, a uniform linear array with 10 array elements was used, and the number of coherent signal sources was set. The parameters are: 2, incident directions are 10.7° and 12.5°, noise is additive white Gaussian noise with a power of 1, angle search interval is 0.1°, and the number of Monte Carlo trials is [number missing]. Set to 500, performance simulations are performed to compare the proposed algorithm, the MUSIC algorithm, and the Toeplitz reconstruction algorithm, providing a basis for calculating the prediction angle. From a realistic perspective The performance evaluation metric chosen is the root mean square error (RMSE) to measure the error between the two values. RMSE is defined as follows:

[0136] (25)

[0137] 4.1 Simulation Experiment 1

[0138] With a fixed number of snapshots of 100 and a signal-to-noise ratio range of [-15, 20], and an interval of 5 dB, the RMSE of the proposed algorithm, the MUSIC algorithm, and the Toeplitz reconstruction algorithm were calculated, and spectral peak comparison diagrams were plotted. Figure 2 and Figure 3 The experimental results are as follows:

[0139] Depend on Figure 2It is evident that the traditional MUSIC method consistently results in an RMSE above 5° in coherent source scenarios, indicating its failure in DOA estimation. In contrast, the Toeplitz reconstruction method, after decoherence, achieves an RMSE below 0.5° when the signal-to-noise ratio is greater than 0dB, thus enabling effective DOA estimation.

[0140] Depend on Figure 3 As can be seen, the Toeplitz reconstruction method can achieve an RMSE below 0.5° at high signal-to-noise ratios, but when the signal-to-noise ratio is less than 0 dB, the RMSE of this method reaches 1.5° or higher, at which point it can no longer correctly distinguish the signal source. In contrast, the method in this paper, after constructing a highly representative signal matrix, can still achieve an RMSE below 0.7° under low signal-to-noise ratio conditions, and the prediction performance is effectively improved. The specific prediction errors are shown in the table below.

[0141] Table 1. RMSE values ​​of the improved method at low signal-to-noise ratios.

[0142] method Signal-to-noise ratio (dB) RMSE / ° Toeplitz algorithm -15-10-5 10.422.361.84 This paper's algorithm -15-10-5 1.630.910.73

[0143] and Figure 4 and Figure 5 The images show a comparison of the spectral peaks of the Toeplitz reconstruction method and the proposed method under -10dB and -5dB conditions, respectively. The comparison shows that the angles corresponding to the spectral peaks of the Toeplitz reconstruction method are far from the true angles, and the correct number of spectral peaks cannot be found. In contrast, the proposed method can still effectively search for spectral peaks at low signal-to-noise ratios.

[0144] Key findings: The traditional MUSIC algorithm is completely ineffective because the rank deficiency problem results in an RMSE that is always greater than 5°. The classic Toeplitz algorithm performs well when the signal-to-noise ratio is ≥0dB, but the RMSE increases significantly at low signal-to-noise ratios (≤-5dB). The algorithm of this invention performs best across the entire signal-to-noise ratio range, especially at low signal-to-noise ratios, with an RMSE of only 1.63° at -15dB, demonstrating strong noise immunity.

[0145] Spectral peak comparison: Taking a signal-to-noise ratio of -10dB as an example ( Figure 4 The peak positions of the classic Toeplitz algorithm are 8.2° and 15.1°, which deviate significantly from the true angle and have blurred peaks. The peak positions of the algorithm of this invention are 10.6° and 12.7°, which deviate from the true angle by less than 0.2°. The peaks are sharp and there are no false peaks, which verifies the accuracy of peak identification.

[0146] 4.2 Simulation Experiment 2

[0147] With a fixed signal-to-noise ratio of 10dB and a snapshot number range of [100, 500], and an interval of 50, the RMSE of the three algorithms at different snapshot numbers was calculated, and a spectral peak comparison plot was drawn. Figure 6 and Figure 7 );

[0148] Depend on Figure 6 As can be seen from the figure, the RMSE of DOA estimation for the traditional MUSIC method and the Toeplitz reconstruction method varies with the number of snapshots N in a coherent source scenario. The RMSE of the traditional MUSIC method remains above 10° without significant fluctuations, confirming its complete failure in a coherent source scenario due to the rank deficiency of the covariance matrix. In contrast, the RMSE of the Toeplitz reconstruction method shows a significant optimization effect, decreasing rapidly from the number of snapshots of 100, and stabilizing below 1° after the number of snapshots exceeds 400. Furthermore, the RMSE continues to decrease with the increase of the number of snapshots, demonstrating the method's ability to decoherent sources.

[0149] Depend on Figure 7 As can be seen, this figure further compares the RMSE variation of the Toeplitz reconstruction method and the Toeplitz-Schur method proposed in this paper with the number of snapshots. The Toeplitz reconstruction method has obvious performance shortcomings in the low snapshot range, with the RMSE consistently above 1° and exhibiting fluctuations in the curve, indicating poor stability. In contrast, the RMSE curve of the method proposed in this paper is more stable, maintaining a stable value below 0.5° even with a low snapshot count, without significant fluctuations. This demonstrates both the high accuracy advantage of this method in low-snapshot scenarios and its superior stability, effectively compensating for the performance deficiency of the Toeplitz reconstruction method under low-snapshot conditions. The specific prediction errors are shown in the table below:

[0150] Table 2 RMSE values ​​of the improved method with fewer snapshots

[0151] method Quick Snap Count RMSE / ° Toeplitz algorithm 100150200250300 3.412.492.031.241.75 This paper's algorithm 100150200250300 1.130.840.650.450.43

[0152] And by Figure 8 and Figure 9 As can be seen, comparing the peak diagrams of the Toeplitz reconstruction method and the proposed method under the condition of a small number of snapshots, the angle corresponding to the peak of the proposed method is significantly closer to the true angle, and both peaks are easy to find.

[0153] Key findings: The traditional MUSIC algorithm consistently has an RMSE greater than 9.5°, is unaffected by the number of snapshots, and is completely ineffective; the classic Toeplitz algorithm performs poorly with a small number of snapshots (N≤200), with an RMSE ≥2.03° and large curve fluctuations (poor stability); the algorithm of this invention performs excellently with a small number of snapshots, with an RMSE of only 1.13° when N=100, and a stable curve (standard deviation ≤0.15°), demonstrating high stability.

[0154] Spectral peak comparison: Taking a snapshot count of 200 as an example ( Figure 8 The peak positions of the classic Toeplitz algorithm are 9.5° and 13.8°, with a relatively wide peak width; the peak positions of the algorithm of this invention are 10.8° and 12.4°, with a deviation from the true angle of less than 0.1°, and the peaks are sharp, which verifies the high-resolution performance in scenes with few fast shots.

[0155] 5. Summary

[0156] To address the failure of the MUSIC method in DOA estimation of coherent sources, the existing Toeplitz method, after decoherent processing, can achieve DOA estimation of coherent signals. However, its estimation accuracy still needs to be improved under conditions of low signal-to-noise ratio and few snapshots.

[0157] To address this shortcoming, this paper proposes an algorithm combining Toeplitz reconstruction and Schur decomposition. By constructing the signal matrix and performing Toeplitz reconstruction, a highly representative covariance matrix is ​​obtained. Then, by peak search of the spectral function, the DOA estimation of the coherent source is achieved. Simulation results show that the proposed method has higher DOA estimation accuracy and better stability, providing a certain reference value for DOA estimation in complex environments.

[0158] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. 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 coherent sources based on Toeplitz reconstruction and Schur decomposition, characterized in that, Includes the following steps: S1. Acquisition of array received signals and calculation of covariance matrix: using... Uniform linear array receiver The far-field coherent incident signal, the spacing between adjacent elements of the uniform linear array is The data matrix is ​​obtained through array reception. ,in yes 3D signal vector yes 3D noise vector The guide vector matrix is ​​based on the received data matrix. Calculate the array covariance matrix The covariance matrix The sample covariance matrix is ​​used for calculation, and the formula is as follows: in Let be the signal covariance matrix. For noise variance, It is an M-dimensional identity matrix; S2. Construction of the first Toeplitz matrix: Traversing the covariance matrix For all diagonals, including the main diagonal and the non-main diagonals on both sides of the main diagonal, calculate the arithmetic mean of all elements on each diagonal, and form a reference vector by combining the means of each diagonal in order from the main diagonal to the upper right diagonal. And construct the Toeplitz matrix. , where the reference vector The formula is: Based on the benchmark As The first line, except The conjugate of all elements other than the first element is used as the first column of the matrix. The remaining elements are then padded with the first element of each diagonal element to obtain the Toeplitz matrix. ; S3. Schur Decomposition of the Covariance Matrix and Construction of the Signal Matrix: The Schur decomposition algorithm is used to construct the covariance matrix. Decompose: ； in It is an upper triangular matrix, and its diagonal elements are precisely the covariance matrix. eigenvalues; It is a unitary matrix, and according to the matrix... diagonal elements and unitary matrix The signal matrix is ​​constructed from the column vectors, and the upper triangular matrix is ​​extracted. The diagonal elements are the covariance matrix. The eigenvalues ​​are then arranged in descending order from largest to smallest, first for the upper triangular matrix. Arranged in descending order of diagonal elements, for a unitary matrix The column vectors correspond to Sort the diagonal elements, i.e., the first... Each column vector contains signal data. Corresponding to The signal matrix is ​​constructed by comprehensively utilizing all information related to the source, based on the number of larger diagonal values. : S4. Construction of the second Toeplitz matrix: Using the same diagonal mean calculation method and matrix construction rules as in step 2, based on the signal matrix obtained in step 3. Calculate using the input. The mean of each diagonal and form a reference vector. : Based on the benchmark As a reconstruction matrix The first line, except All remaining elements other than the first element are conjugates, which are used as... The first column is padded with the remaining elements according to the first element of the diagonal to obtain the matrix. ; S5. Matrix Fusion and DOA Estimation: The first Toeplitz matrix... With the second Toeplitz matrix Perform equal-weighted fusion to obtain the fusion matrix. and the fusion matrix Perform eigenvalue decomposition to obtain eigenvalues ​​and corresponding eigenvectors, then solve for the eigenvectors and construct the noise subspace. According to the signal vector Construct new spectral functions based on the orthogonality with the noise subspace. Perform a peak search: When traversing to the true perspective When the spectral peaks show sharp peaks, the angle value corresponding to the sharp peaks obtained by the search is the DOA estimation result of the coherent source.

2. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: In S1, The formula for the guiding vector matrix is: And select the covariance matrix The first row is used as the base ,by As the first row of the Toeplitz matrix: ； remove Take the conjugate of all elements other than the first element and use it as the first column of the matrix. Fill the remaining elements with elements that are equal on the diagonal to obtain the Toeplitz matrix. :

3. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: In S1, regarding the covariance matrix During eigenvalue decomposition, the eigenvectors are solved and the noise subspace is constructed. Since the signal subspace is consistent with the column space of the guidance matrix, then Also with the noise subspace Orthogonal: Based on this, construct the MUSIC spatial spectral function. : When traversing to the true perspective When the denominator in the formula approaches zero, the function value will show a sharp peak. The MUSIC method estimates DOA based on this principle.

4. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: In S2, the matrix : ; And on Perform eigenvalue decomposition, select the two largest eigenvalues, and calculate their corresponding eigenvectors. and As a reference vector and Construct the Toeplitz matrix respectively. and ,in: Through calculation The conjugate product of the first element and the rest of the elements ,available The first line : Reverse formula Taking the conjugate transpose, but keeping the first element unchanged, we can obtain... The first column : Wherein the formula superscript This indicates that the matrix is ​​taken as its conjugate transpose. This indicates that the matrix is ​​transposed. In obtaining After deleting the elements in the first row and first column, fill in the remaining elements according to the principle of equal diagonal elements to obtain the Toeplitz matrix:

5. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 4, characterized in that: Calculate using the same steps The conjugate product of the first element and the remaining elements (#imgpt104#) forms a new row vector (#imgpt105#), and matrix reconstruction is performed using (#imgpt106#) to obtain the Toeplitz matrix (#imgpt107#). After the above steps, four reconstructed Toeplitz matrices are obtained, and then the final matrix #imgpt109# is calculated:

6. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: Perform eigenvalue decomposition on the Toeplitz matrix #imgpt111#, solve for the eigenvectors, and construct the noise subspace #imgpt112#. Based on the orthogonality between the signal vector #imgpt113# and the noise subspace #imgpt114#, construct a new spectral function #imgpt115# for spectral peak search. When the actual angle is reached (#imgpt117#), the function value will also show a sharp peak. Based on this, the direction of arrival of the wave is estimated. In fact, when performing simulation calculations in MATLAB, the minimum relative precision function (#imgpt118#) is added to the denominator of the spectral function.

7. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: In step S3, the Schur decomposition is implemented using the standard Schur decomposition function in numerical calculation software. In the MATLAB environment, the Schur decomposition function is used, and the diagonal elements of the upper triangular matrix #imgpt119# obtained by decomposition are the eigenvalues ​​of the covariance matrix #imgpt120#.

8. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: When calculating the diagonal mean in S2 and S4, if the number of elements in a diagonal is less than 3, the mean of the diagonal elements in the neighborhood is smoothed to reduce the impact of noise on the calculation of the diagonal mean.

9. The method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 1, characterized in that: In S5, the peak search adopts the hill climbing method or particle swarm optimization algorithm. When the number of information sources is unknown, the number of information sources is first determined by information theory criteria or eigenvalue thresholding method, and then the feature subspace is separated.

10. A method for estimating the DOA of a coherent source based on Toeplitz reconstruction and Schur decomposition according to claim 5, characterized in that: The search range of the spatial spectrum can be adjusted according to the actual application scenario. In the radar scenario, the search range is [-90°, 90°], and in the sonar scenario, the search range is [-180°, 180°]. The search interval can be adjusted between 0.01° and 0.1°.

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