A method for estimating the DOA of coherent signals based on tensor decomposition

Abstract

This invention presents a method for estimating the Direction of Ability (DOA) of coherent signals based on tensor decomposition, belonging to the field of signal processing technology. The method includes the following steps: Step S1, dividing a two-dimensional surface array into forward and backward subarrays to construct a third-order tensor model; Step S2, performing parallel factorization on the third-order tensor model to estimate the array manifold matrix corresponding to the first subarray; Step S3, using the estimation result of the array manifold matrix corresponding to the first subarray, performing two-dimensional DOA estimation. This invention can estimate the two-dimensional DOA of coherent signals without rank recovery even under low signal-to-noise ratio and small snapshot conditions, and it achieves high accuracy and good robustness to noise.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, and particularly relates to a method for estimating the DOA of coherent signals based on tensor decomposition. Background Technology

[0002] Due to the influence of complex electromagnetic environments, multipath propagation, and mutual coupling between array elements, the signals received by array elements are often correlated. However, the received covariance matrix of correlated signals is usually rank-deficient, making it impossible to correctly divide the signal subspace and noise subspace. Therefore, the impact on traditional subspace algorithms such as Music and Esprit is particularly significant. To solve the problem of rank deficiency in the signal covariance matrix, many algorithms have been developed to recover the rank of the correlated signal covariance matrix, and then estimate the DOA of the signal from the recovered full-rank covariance matrix, such as forward and backward spatial smoothing. However, traditional forward and backward spatial smoothing algorithms have high signal-to-noise ratio requirements. At low signal-to-noise ratios, the accuracy of the algorithm is severely reduced, and its robustness to noise is low. Summary of the Invention

[0003] The purpose of this invention is to provide a coherent signal DOA estimation method based on tensor decomposition. This method can estimate the two-dimensional DOA of coherent signals without the need for covariance matrix rank recovery under conditions of low signal-to-noise ratio and small snapshot size. Moreover, the two-dimensional DOA estimation has high accuracy and good robustness to noise.

[0004] To achieve the above objectives, the present invention employs the following technical solution:

[0005] A method for estimating the DOA of a coherent signal based on tensor decomposition, the method comprising the following steps:

[0006] Step S1: Divide the two-dimensional surface array into forward and backward subarrays to construct a third-order tensor model;

[0007] Step S2: Perform parallel factorization on the third-order tensor model to estimate the array manifold matrix corresponding to the first subarray;

[0008] Step S3: Use the estimation result of the array flow matrix corresponding to the first subarray to perform two-dimensional DOA estimation.

[0009] Furthermore, the specific implementation process of step S1 includes:

[0010] Step S11: Determine the first subarray in the two-dimensional array, and move it along the forward and backward directions of the two-dimensional array to divide the two-dimensional array into multiple subarrays.

[0011] In this subarray, each subarray has the same number of rows and columns, and any two adjacent subarrays partially overlap, with the overlapping parts having the same number of array elements.

[0012] Step S12: Obtain the coherent signal data received by each subarray to determine the covariance data of each subarray;

[0013] Step S13: Arrange the covariance data of the multiple subarrays into a third-order tensor model by forward slicing.

[0014] Furthermore, in step S2, the specific process of parallel factorization includes:

[0015] Step S21: Expand the third-order tensor model modulo 1, modulo 2 and modulo 3 respectively to determine the modulo 1 expansion matrix, modulo 2 expansion matrix and modulo 3 expansion matrix;

[0016] Step S22: Using the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix, perform trilinear least squares solution on the three factor matrices of the third-order tensor model to determine the estimated result of the array manifold matrix corresponding to the first subarray.

[0017] Furthermore, in step S21, the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix are respectively:

[0018] Z1=(A * ☉B)A T +N1;

[0019] Z2=(B☉A)(A * ) T +N2;

[0020] Z3=(A☉A * B T +N3;

[0021] Where Z1, Z2, and Z3 are the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix, respectively; A, A * B and A are the first factor matrix, the second factor matrix, and the third factor matrix, respectively; A T 、(A * ) T and B T N1, N2, and N3 are the transpose matrices corresponding to the first factor matrix, the second factor matrix, and the third factor matrix, respectively; N1, N2, and N3 are the first noise matrix, the second noise matrix, and the third noise matrix corresponding to the modulo 1, modulo 2, and modulo 3 expansions, respectively.

[0022] Furthermore, in step S22, the specific process of solving the trilinear least squares problem includes:

[0023] Step S221: Perform Tuker decomposition and compression on the third-order tensor to obtain the kernel tensor;

[0024] Step S222: Perform parallel factorization on the kernel tensor to obtain the initial matrices of the first factor matrix, the second factor matrix, and the third factor matrix;

[0025] Step S223: Set the initial value of the iteration number k to 1;

[0026] Step S224: Using the initial matrices and corresponding expansion matrices of two of the factor matrices, estimate the remaining factor matrix to obtain the estimated values ​​of the first factor matrix, the second factor matrix, and the third factor matrix.

[0027] Step S225: Determine whether the iteration number k is less than the threshold. If yes, increment the iteration number k by 1 and assign it to k, and use the estimated values ​​of the three factor matrices as the initial three factor matrices, then return to step S224. If no, use the estimated value of the first factor matrix as the estimated value of the array flow matrix of the first subarray.

[0028] Furthermore, in step S224, when the remaining factor matrix is ​​the first factor matrix, the estimated result of the transpose of the first factor matrix is:

[0029]

[0030] in, This is the estimate of the transpose of the first factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0031] Furthermore, in step S224, when the remaining factor matrix is ​​the second factor matrix, the estimated result of the transpose of the second factor matrix is:

[0032]

[0033] in, This is the estimate of the transpose of the second factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0034] Furthermore, in step S224, when the remaining factor matrix is ​​the third factor matrix, the estimated result of the transpose of the third factor matrix is:

[0035]

[0036] in, This is the estimate of the transpose of the third factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0037] Furthermore, in step S3, the specific process of the two-dimensional DOA estimation includes:

[0038] Step S31: Calculate the product matrix of the array manifold matrix corresponding to the estimated first subarray and its corresponding conjugate transpose matrix;

[0039] Step S32: Using the product matrix, perform direction finding estimation to obtain the DOA estimation results for each coherent signal.

[0040] In summary, the technical solution of the present invention has the following technical effects:

[0041] This invention constructs a third-order tensor model by dividing a two-dimensional surface array into forward and backward subarrays. It then estimates the array manifold matrix corresponding to the first subarray using the parallel factorization of the third-order tensor model. Finally, it performs two-dimensional DOA estimation by utilizing the rotation invariance of the estimated array manifold matrix corresponding to the first subarray. This achieves high-precision DOA estimation without rank recovery, even under low signal-to-noise ratio and small snapshot conditions, with high estimation accuracy and robustness to noise. Attached Figure Description

[0042] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0043] Figure 1 This is a schematic diagram of the DOA estimation method for coherent signals based on tensor decomposition according to the present invention.

[0044] Figure 2 This is a schematic diagram of the forward and backward movement of a one-dimensional subarray.

[0045] Figure 3 (a) and (b) in the figure are respectively schematic diagrams of scatter plots for DOA estimation of one-dimensional coherent signals and root mean square error curves for DOA estimation of one-dimensional coherent signals by the method of the present invention and the traditional method;

[0046] Figure 4 In the figures, (a) and (b) are respectively a scatter plot of DOA estimation for two-dimensional coherent signals and a schematic diagram of the root mean square error curve of DOA estimation for two-dimensional coherent signals using the method of the present invention and the conventional method. Detailed Implementation

[0047] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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.

[0048] This embodiment presents a method for estimating the DOA of coherent signals based on tensor decomposition, referencing... Figure 1 The coherent signal DOA estimation method includes the following steps:

[0049] Step S1: Divide the two-dimensional surface array into forward and backward subarrays to construct a third-order tensor model.

[0050] This embodiment employs a spatial smoothing algorithm to partition a two-dimensional planar array into forward and backward subarrays, and stacks the covariance data of each subarray into a third-order tensor. The specific implementation process includes:

[0051] Step S11: Determine the first subarray in the two-dimensional array, and move it along the forward and backward directions of the two-dimensional array to divide the two-dimensional array into multiple subarrays.

[0052] This embodiment first determines the first subarray 1, and then follows... Figure 2 The forward and backward movement methods shown result in M ​​forward subarrays and M backward subarrays. Each subarray has the same number of rows a (antenna elements) and columns b (antenna elements), and any two adjacent subarrays partially overlap, with the same number of elements in the overlapping part. Both a and b are greater than or equal to the number of incident signals, and the total number of subarrays is greater than or equal to the number of incident signals.

[0053] Step S12: Obtain the coherent signal data received by each subarray to determine the covariance data of each subarray;

[0054] Step S13: Arrange the covariance data of the multiple subarrays into a third-order tensor model by forward slicing.

[0055] Step S2: Perform parallel factorization on the third-order tensor model to estimate the array manifold matrix corresponding to the first subarray.

[0056] This embodiment utilizes the construction method of this third-order tensor to achieve the uniqueness of parallel factorization, and finally obtains the array flow matrix corresponding to the first subarray. The specific process of parallel factorization includes:

[0057] Step S21: Expand the third-order tensor model modulo 1, modulo 2 and modulo 3 respectively to determine the modulo 1 expansion matrix, modulo 2 expansion matrix and modulo 3 expansion matrix;

[0058] In this embodiment, the modulo 1 expansion matrix, the modulo 2 expansion matrix, and the modulo 3 expansion matrix are as follows:

[0059] Z1=(A * ☉B)A T +N1;

[0060] Z2=(B☉A)(A * ) T +N2;

[0061] Z3=(A☉A * B T +N3;

[0062] Where Z1, Z2, and Z3 are the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix, respectively; A, A * A and B are the first factor matrix, the second factor matrix, and the third factor matrix, respectively (ideally, A and A' are the first factor matrix, the second factor matrix, and the third factor matrix). * (where they are conjugate matrices); A T 、(A * ) T and B T N1, N2, and N3 are the transpose matrices corresponding to the first factor matrix, the second factor matrix, and the third factor matrix, respectively; N1, N2, and N3 are the first noise matrix, the second noise matrix, and the third noise matrix corresponding to the modulo 1, modulo 2, and modulo 3 expansions, respectively.

[0063] Step S22: Using the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix, perform trilinear least squares solution on the three factor matrices of the third-order tensor model to determine the estimated result of the array manifold matrix corresponding to the first subarray.

[0064] In this embodiment, the three factor matrices are the first factor matrix, the second factor matrix, and the third factor matrix. The trilinear least squares solution in this embodiment fixes all factor matrices except for one, and iteratively solves for each factor matrix. For example, if A is fixed... * And B, use the following formula to calculate A:

[0065]

[0066] Among them, ||·|| F This represents the Frobenius norm.

[0067] If A and B are fixed, A can be calculated using the following formula. * :

[0068]

[0069] Among them, ||·|| F This represents the Frobenius norm.

[0070] If A and A are fixed * The following formula is used to calculate B:

[0071]

[0072] Among them, ||·|| F This represents the Frobenius norm.

[0073] In summary, the specific process of solving the trilinear least squares problem in this embodiment includes:

[0074] Step S221: Perform Tuker decomposition and compression on the third-order tensor to obtain the kernel tensor;

[0075] Step S222: Perform parallel factorization on the kernel tensor to obtain the initial matrices of the first factor matrix, the second factor matrix, and the third factor matrix;

[0076] Step S223: Set the initial value of the iteration number k to 1;

[0077] Step S224: Using the initial matrices and corresponding expansion matrices of two of the factor matrices, estimate the remaining factor matrix to obtain the estimated values ​​of the first factor matrix, the second factor matrix, and the third factor matrix.

[0078] In this embodiment, when the remaining factor matrix is ​​the first factor matrix, the estimation result of the transpose of the first factor matrix is:

[0079]

[0080] in, This is the estimate of the transpose of the first factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0081] When the remaining factor matrix is ​​the second factor matrix, the estimated result of the transpose of the second factor matrix is:

[0082]

[0083] in, This is the estimate of the transpose of the second factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0084] When the remaining factor matrix is ​​the third factor matrix, the estimated result of the transpose of the third factor matrix is:

[0085]

[0086] in, This is the estimate of the transpose of the third factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

[0087] Step S225: Determine whether the iteration number k is less than the threshold. If yes, increment the iteration number k by 1 and assign it to k, and use the estimated values ​​of the three factor matrices as the initial three factor matrices, then return to step S224. If no, use the estimated value of the first factor matrix as the estimated value of the array flow matrix of the first subarray.

[0088] Step S3: Use the estimation result of the array flow matrix corresponding to the first subarray to perform two-dimensional DOA estimation.

[0089] In this embodiment, the array flow matrix corresponding to the first subarray has a Vandermonde structure. Utilizing the rotation invariance of the array flow matrix corresponding to the first subarray, the ESPRIT algorithm is employed for two-dimensional DOA estimation. The specific process of two-dimensional DOA estimation includes:

[0090] Step S31: Calculate the product matrix of the array manifold matrix corresponding to the estimated first subarray and its corresponding conjugate transpose matrix.

[0091] The product matrix in this embodiment is:

[0092]

[0093] Where R is the product matrix; This is the estimated result of the array flow matrix A corresponding to the first subarray; It is the conjugate transpose of the estimated array manifold matrix A corresponding to the first subarray.

[0094] Step S32: Using the product matrix, perform direction finding estimation to obtain the DOA estimation results for each coherent signal.

[0095] In this embodiment, the DOA estimation result for each coherent signal is as follows:

[0096]

[0097] in, The estimated DOA of the k-th coherent signal; θ k and This represents the phase information (i.e., elevation and azimuth angles) of the k-th coherent signal in the product matrix R.

[0098] from Figure 3 Coherent signal DOA estimation of a one-dimensional array and Figure 4 Monte Carlo simulation results of DOA estimation of coherent signals in a two-dimensional array show that the method of the present invention can accurately estimate coherent signals. Under low signal-to-noise ratio conditions, the performance of DOA estimation, whether one-dimensional or two-dimensional, is better than that of traditional forward and backward spatial smoothing, and the DOA estimation accuracy is high.

[0099] This embodiment constructs a third-order tensor model by dividing a two-dimensional surface array into forward and backward subarrays. Using the parallel factorization of the third-order tensor model, the array manifold matrix corresponding to the first subarray is estimated. Through the rotation invariance of the estimated array manifold matrix corresponding to the first subarray, two-dimensional DOA estimation is performed. This achieves high-precision DOA estimation without rank recovery, even under low signal-to-noise ratio and small snapshot conditions, with high estimation accuracy and high robustness to noise.

[0100] Please note that the technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, not all possible combinations of the technical features in the above embodiments have been described. However, as long as the combination of these technical features does not contradict each other, it should be considered within the scope of this specification. The above embodiments only illustrate several implementation methods of this application, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be pointed out that for those skilled in the art, several modifications and improvements can be made without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method for estimating the DOA of a coherent signal based on tensor decomposition, characterized in that, The coherent signal DOA estimation method includes the following steps: Step S1: Divide the two-dimensional surface array into forward and backward subarrays to construct a third-order tensor model; Step S2: Perform parallel factorization on the third-order tensor model to estimate the array manifold matrix corresponding to the first subarray; In step S2, the specific process of parallel factorization includes: Step S21: Expand the third-order tensor model modulo 1, modulo 2 and modulo 3 respectively to determine the modulo 1 expansion matrix, modulo 2 expansion matrix and modulo 3 expansion matrix; Step S22: Using the modulo 1 expansion matrix, modulo 2 expansion matrix and modulo 3 expansion matrix, perform trilinear least squares solution on the three factor matrices of the third-order tensor model to determine the estimated result of the array manifold matrix corresponding to the first subarray. In step S22, the specific process of solving the trilinear least squares problem includes: Step S221: Perform Tuker decomposition and compression on the third-order tensor to obtain the kernel tensor; Step S222: Perform parallel factorization on the kernel tensor to obtain the initial matrices of the first factor matrix, the second factor matrix, and the third factor matrix; Step S223: Set the initial value of the iteration number k to 1; Step S224: Using the initial matrices and corresponding expansion matrices of two of the factor matrices, estimate the remaining factor matrix to obtain the estimated values ​​of the first factor matrix, the second factor matrix, and the third factor matrix. Step S225: Determine whether the iteration number k is less than the threshold. If so, increment the iteration number k by 1 and assign it to k, and use the estimated values ​​of the three factor matrices as the initial three factor matrices, then return to step S224. If not, use the estimated value of the first factor matrix as the estimated value of the array flow matrix of the first subarray. Step S3: Using the estimation result of the array flow matrix corresponding to the first subarray, perform two-dimensional DOA estimation; In step S3, the specific process of the two-dimensional DOA estimation includes: Step S31: Calculate the product matrix of the array manifold matrix corresponding to the estimated first subarray and its corresponding conjugate transpose matrix; Step S32: Using the product matrix, perform direction finding estimation to obtain the DOA estimation results for each coherent signal.

2. The coherent signal DOA estimation method according to claim 1, characterized in that, The specific implementation process of step S1 includes: Step S11: Determine the first subarray in the two-dimensional array, and move it along the forward and backward directions of the two-dimensional array to divide the two-dimensional array into multiple subarrays. In this subarray, each subarray has the same number of rows and columns, and any two adjacent subarrays partially overlap, with the overlapping parts having the same number of array elements. Step S12: Obtain the coherent signal data received by each subarray to determine the covariance data of each subarray; Step S13: Arrange the covariance data of the multiple subarrays into a third-order tensor model by forward slicing.

3. The coherent signal DOA estimation method according to claim 2, characterized in that, In step S21, the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix are respectively: Z1=(A * ⊙B)A T +N1; Z2=(B☉A)(A * ) T +N2; Z3=(A☉A * B T +N3; Where Z1, Z2, and Z3 are the modulo 1 expansion matrix, modulo 2 expansion matrix, and modulo 3 expansion matrix, respectively; A, A * B and A are the first factor matrix, the second factor matrix, and the third factor matrix, respectively; A T 、(A * ) T and B T N1, N2, and N3 are the transpose matrices corresponding to the first factor matrix, the second factor matrix, and the third factor matrix, respectively; N1, N2, and N3 are the first noise matrix, the second noise matrix, and the third noise matrix corresponding to the modulo 1, modulo 2, and modulo 3 expansions, respectively.

4. The coherent signal DOA estimation method according to claim 3, characterized in that, In step S224, when the remaining factor matrix is ​​the first factor matrix, the estimated result of the transpose of the first factor matrix is: in, This is the estimate of the transpose of the first factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

5. The coherent signal DOA estimation method according to claim 4, characterized in that, In step S224, when the remaining factor matrix is ​​the second factor matrix, the estimated result of the transpose of the second factor matrix is: in, This is the estimate of the transpose of the second factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

6. The coherent signal DOA estimation method according to claim 4, characterized in that, In step S224, when the remaining factor matrix is ​​the third factor matrix, the estimated result of the transpose of the third factor matrix is: in, This is the estimate of the transpose of the third factor matrix; superscript. This indicates the pseudo-inverse operation on the matrix; ⊙ represents the KhatriRao product operation.

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