A robust non-coherent distributed source 2-d angle of arrival estimation method under L-shaped array

Abstract

The application discloses a robust non-coherent distributed source two-dimensional direction of arrival estimation method under an L-shaped array, which establishes a generalized array flow type receiving model under the L-shaped array, the L-shaped array is composed of two uniform linear arrays respectively located on the x-axis and the z-axis of the xoz plane; the data output by the uniform linear arrays on the x-axis and the z-axis is sampled at N moments; statistical average operation and conjugate symmetry strategy are used to construct aperture expansion array mutual covariance vectors corresponding to the uniform linear arrays on the x-axis and the z-axis; according to the two aperture expansion array mutual covariance vectors, all direction angle estimation values with the x-axis as a reference and all direction angle estimation values with the z-axis as a reference are obtained, and then pairing is performed to obtain all two-dimensional direction of arrival estimation values; the method is not only suitable for uniform and non-uniform noise scenes, but also does not depend on prior information of the number of signal sources, and the practicability and flexibility of the algorithm are improved.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, and in particular relates to a robust two-dimensional angle of arrival estimation method for incoherent distributed sources under an L-shaped array. Background Technology

[0002] In cellular wireless communication systems, Angle of Arrival (AOA) estimation plays a crucial role, as it is key to base station downlink beamforming to ensure the performance of the cellular wireless communication system. While existing AOA estimation methods have yielded some excellent results, the vast majority are based on point source models with line-of-sight single-path propagation. However, in real-world cellular wireless communication systems, wireless signals often propagate along multiple paths in fast-fading channels. In such cases, incoherent distributed sources are better suited to characterize the actual scenario. However, due to the virtual expansion of the steering matrix of the incoherent distributed source array, existing AOA estimation algorithms based on point source models cannot be directly applied, posing a significant challenge to high-performance AOA estimation.

[0003] In recent years, with the deepening of theoretical research and the increasing demand for angle-of-arrival (AOA) estimation of incoherent distributed sources, some AOA estimation techniques based on incoherent distributed source models have been proposed. Representative algorithms include the weighted subspace algorithm published by M. Bengtsson et al. from the Royal Institute of Technology in Sweden in the journal IEEE Trans. Signal Processing; the rotation-invariant subspace algorithm published by S. Shahbazpanahi et al. from Ontario University of Technology and Y. Liu et al. from Harbin Engineering University in the journals IEEE Trans. Signal Processing and IEEE Signal Processing Letters; the beam compression algorithm published by Z. Zheng et al. from the University of Electronic Science and Technology of China in the journal IEEE Trans. Veh. Technology; and the reduced-rank subspace algorithm published by H. Chen et al. from Ningbo University and F. Gao et al. from Tsinghua University in the journals IEEE Sensors Journal, IEEE Trans. Aerosp. Electron. Systems, and IEEE Trans. Signal Processing, etc.

[0004] The aforementioned algorithms are all based on a uniform white noise model and require prior information about the number of sources to distinguish between the signal and noise subspaces. However, in practical applications, due to the non-ideals of hardware and receiving channels, the noise may be unknown or non-uniform, and the number of sources may also be unknown. Under these circumstances, the performance of existing algorithms deteriorates sharply or even fails. Therefore, providing a two-dimensional angle-of-arrival estimation method for incoherent distributed sources that overcomes the shortcomings of existing algorithms in practical applications has significant research value and practical application value. Summary of the Invention

[0005] This invention aims to address the limitations of existing technologies in estimating the two-dimensional angle of arrival (Angle of Arrival) of incoherent distributed sources, particularly the dependence on prior information about the number of signal sources in uniform noise environments. This invention provides a robust two-dimensional Angle of Arrival (Angle of Arrival) estimation method for incoherent distributed sources using an L-shaped array. This method is applicable not only to both uniform and non-uniform noise scenarios but also does not depend on prior information about the number of signal sources, thus improving the algorithm's practicality and flexibility.

[0006] The technical solution adopted by this invention to solve the above-mentioned technical problems is as follows: a robust two-dimensional angle of arrival estimation method for incoherent distributed sources under an L-shaped array, characterized by the following steps:

[0007] Step 1: Establish a generalized array manifold receiving model under the L-shaped array: Set up K far-field narrowband uncorrelated incoherent distributed sources incident on an L-shaped array composed of two uniform linear arrays. The two uniform linear arrays are located on the x-axis and z-axis of the xoz plane, respectively. The number of array elements is M, M>K, and the spacing between array elements is 1 / 2 of the carrier wavelength. The two uniform linear arrays have no shared array elements. The array element on the x-axis that is closest to the origin of the xoz plane is taken as the first array element on the x-axis, and the array element on the z-axis that is closest to the origin of the xoz plane is taken as the first array element on the z-axis.

[0008] Step 2: In the generalized array manifold receiving model, sample the data output by the uniform linear array on the x-axis at N time points; similarly, sample the data output by the uniform linear array on the z-axis at N time points.

[0009] Step 3: Using statistical averaging, perform cross-correlation calculations on the N time-series sampled data of the uniform linear array output on the x-axis and the N time-series sampled data of the output of the first element on the z-axis to obtain the cross-covariance vector r. xz Similarly, using statistical averaging, cross-correlation is performed on the sampled data at N time points of the uniform linear array output on the z-axis and the sampled data at N time points of the output of the first element on the x-axis to obtain the cross-covariance vector r. zx ;

[0010] Step 4: Construct r using a conjugate symmetry strategy. xz Aperture extension array cross-covariance vector r e Similarly, using a conjugate symmetry strategy, r is constructed. zx Aperture extension array cross-covariance vector r f ;

[0011] Step 5: If the number of signal sources K is known prior, then according to r e and r f And using a closed-form solution algorithm based on rotation-invariant subspace technology, we obtain K orientation angle estimates with the x-axis as the reference and K orientation angle estimates with the z-axis as the reference.

[0012] If the number of signal sources K is unknown, then according to r e and r f The sparse total least squares algorithm is used to obtain K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis.

[0013] Step 6: Pair the K azimuth angle estimates based on the x-axis and the K azimuth angle estimates based on the z-axis to obtain K pairs of two-dimensional angle of arrival estimates.

[0014] In step 2, the sampled data x(t) at time t of the uniform linear array output on the x-axis is approximately expressed as: x(t)≈A(θ)c0(t)+A′(θ)c1(t)+n x (t), the sampled data z(t) at time t of the uniform linear array output on the z-axis is approximately expressed as: z(t)≈A(β)c0(t)+A′(β)c2(t)+n z (t); where t = 1, ..., N, θ represents the direction angle variable of the signal emitted by the incoherent distributed source with the x-axis as the reference, β represents the direction angle variable of the signal emitted by the incoherent distributed source with the z-axis as the reference, and A(θ) represents the steering matrix of the uniform linear array on the x-axis, A(θ) = [a(θ1), ..., a(θ2)]. K )],a(θ k Let be the k-th column vector of A(θ), and let a(θ) be the column vector of A(θ). k ) represents θ k The relevant steering vector, θ k A(β) represents the direction angle of the signal emitted by the k-th incoherent distributed source relative to the x-axis, k = 1, ..., K, and A(β) represents the steering matrix of the uniform linear array on the z-axis, A(β) = [a(β1), ..., a(β2)]. K )],a(β k Let a(β) be the k-th column vector of A(β). k ) represents β k The relevant steering vector, βk The direction angle (θ) of the signal emitted by the k-th incoherent distributed source relative to the z-axis is given. k ,β k () represents the two-dimensional angle of arrival of the k-th incoherent source. This indicates that for a(θ) k The variable θ in ) k The new vector obtained by differentiation Indicates that for a(β) k The variable β in ) k The new vector obtained by differentiation, c0(t), represents the received signal vector on the L-shaped array after the signals emitted by all incoherent distributed sources at time t have propagated through their respective transmission paths, c0(t) = [c 1，0 (t), ..., c K，0 (t)] T c k，0 (t) is the k-th element in c0(t), c k，0 (t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k The received signal on the L-shaped array after propagation along the transmission path s k γ(t) represents the signal emitted by the k-th incoherent distributed source at time t. k，l (t) represents the complex path gain of the signal emitted by the k-th incoherent distributed source at time t propagating along the l-th transmission path, L k Let represent the number of transmission paths for the k-th incoherent distributed source, with the superscript "T" indicating transpose operation. c1(t) represents the received signal vector on a uniform linear array on the x-axis after propagation of the signals emitted by all incoherent distributed sources through their respective transmission paths at time t, and after angular expansion at their respective x-axis-based azimuth angles. c1(t) = [c 1，1 (t), ..., c K，1 (t)] T c k，1 (t) is the k-th element in c1(t), c k，1 (t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k Propagate along a transmission path, and at θ k The received signal on a uniform linear array along the x-axis after angular expansion in the directional direction. This represents the deviation between the x-axis-based azimuth angle of the signal emitted by the k-th incoherent distributed source at time t when propagating along the l-th transmission path and when propagating along the line-of-sight path. The mean is 0 and the variance is less than 3 degrees. These are random variables following a uniform or Gaussian distribution. c2(t) represents the received signal vector on a uniform linear array on the z-axis after propagation through all transmission paths of all incoherent sources at time t, and after angular expansion along their respective z-axis-based azimuth angles. c2(t) = [c 1，2 (t), ..., c K，2 (t)] T c k，2 (t) is the k-th element in c2(t), c k，2 (t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k Propagation along a transmission path, and in β k The received signal on a uniform linear array along the z-axis after angular expansion in the directional direction. This represents the deviation between the z-axis-based azimuth angle of the signal emitted by the k-th incoherent distributed source at time t when propagating along the l-th transmission path and when propagating along the line-of-sight path. The mean is 0, the variance is less than 3 degrees, and they are random variables following a uniform or Gaussian distribution. x (t) represents the Gaussian noise vector output by the uniform linear array on the x-axis at time t, where n z (t) represents the Gaussian noise vector output by the uniform linear array on the z-axis at time t, where n x (t) and n z (t) are unrelated to each other.

[0015] In step 3 Where z1(t) represents the sampled data of the output data of the first array element on the z-axis at time t, x1(t) represents the sampled data of the output data of the first array element on the x-axis at time t, the superscript "H" indicates the conjugate transpose operation, p represents the power vector of the received signals on the L-shaped array after the signals emitted by all incoherent distributed sources at N times have propagated through their respective transmission paths, p = [p1, ..., p K ] T p k Let p be the k-th element. k This indicates that the signal emitted by the k-th incoherent distributed source at N times is transmitted through L. k The power of the received signal on the L-shaped array after propagation along the transmission path. The superscript "*" indicates conjugate operation.

[0016] In step 4 Where, r e and r fAll of them have a dimension of (2M-1)×1, and J represents a permutation matrix in which the elements on the antidiagonal are 1 and the other elements are 0. Indicates by r zx The subvector formed by the second to the Mth elements, Indicates by r zx The subvector consisting of the second to the Mth elements.

[0017] In step 5, according to r e and r f The specific process of obtaining K estimated orientation angles based on the x-axis and K estimated orientation angles based on the z-axis using a closed-form solution algorithm based on rotation-invariant subspace technology is as follows:

[0018] Step 5.1a: r e Divide r into M sub-vectors with overlapping elements. e The m-th subvector obtained by partitioning Represented as Similarly, r f Divide r into M sub-vectors with overlapping elements. f The m-th subvector obtained by partitioning Where m = 1, ..., M, Indicates by r e The subvector consisting of the m-th element to the (M+m-1)-th element. Indicates by r f The subvector consisting of the m-th element to the (M+m-1)-th element;

[0019] Step 5.2a: Based on r e Construct r from all the sub-vectors obtained by partitioning xz The relevant M×M dimension matrix Similarly, according to r f Construct r from all the sub-vectors obtained by partitioning zx The relevant M×M dimension matrix Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k ;

[0020] Step 5.3a: For Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. e Then U e Divided into two sub-matrices and Similarly, for Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. f Then U f Divided into two sub-matrices and Among them, U e and U f The dimensions of all are M×K. Indicated by U e The submatrix formed by rows 1 to M-1, Indicated by U e The submatrix formed by rows 2 to M, Indicated by U f The submatrix formed by rows 1 to M-1, Indicated by U f The submatrix formed by rows 2 to M;

[0021] Step 5.4a: Calculation Then for H e Eigenvalue decomposition yields K eigenvalues ​​u1, ..., u2. K Then through the formula θ is calculated from k'∈[1,K]. k' The estimated value of k'∈[1,K] k'∈[1,K]; where the superscript u represents the pseudo-inverse operation of a matrix. k' Indicates H e The k'th eigenvalue obtained by eigenvalue decomposition, ∠u k' Indicates the relationship between u k' Phase take operation, θ k' This represents the k'th direction angle relative to the x-axis. Represents θ k' The estimated value;

[0022] Similarly, calculation Then for H f Eigenvalue decomposition is performed to obtain K eigenvalues ​​v1, ..., v2. K Then through the formula β is calculated from k'∈[1,K]. k' The estimated value of k'∈[1,K] Among them, v k' Indicates H f The k'th eigenvalue obtained by eigenvalue decomposition, ∠v k' Indicates to v k' Phase take operation, β k'This represents the k'-th direction angle relative to the z-axis. Indicates β k' The estimated value.

[0023] In step 5, according to r e and r f The specific process of obtaining K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis using the sparse total least squares algorithm is as follows:

[0024] Step 5.1b: Transfer r e Sparse representation is r e =(Ψ(θ)+E θ )p Nθ +e θ , will r f Sparse representation is r f =(Ψ(β)+E β )p Nβ +e β ; where Ψ(θ) represents the matrix [JA * (θ), A (2:M) (θ)] T sparse representation of A (2:M) (θ) represents the submatrix formed by the 2nd to Mth rows of A(θ), and Ψ(β) represents the matrix [JA * (β), A (2:M) (β)] T sparse representation of A (2:M) (β) represents the submatrix formed by rows 2 to M of A(β), E θ E represents the θ-related bias matrix resulting from the application of the generalized array manifold. β p represents the β-related bias matrix resulting from the application of the generalized array manifold. Nθ and p Nβ All are K-sparse vectors, p Nθ and p Nβ Both p contain non-zero elements and zero elements. Nθ The index of the k'-th non-zero element in the array is θ1, ..., θ2. K One of them, p Nβ The index of the k'-th non-zero element in the array is β1, ..., β K One of them, k'∈[1,K], e θ Indicates r e The deviation vector between the estimated value and the true value, e β Indicates r f The vector of deviations between the estimated and true values;

[0025] Step 5.2b: Construct the θ-dependent sparse global least squares optimization problem P1, described as: P1: Similarly, construct a β-related sparse total least squares optimization problem P2, described as: P2: Where ||||2 represents finding the l2 norm of a vector, |||| F Let ||||1|| denote the Frobenius norm of a matrix, ||||1|| denotes the l1 norm of a vector, and λ represents the regularization parameter, where λ > 0. p Nθ The estimated value, E represents θ The estimated value, p Nβ The estimated value, E represents β The estimated value;

[0026] Step 5.3b: Based on the θ-dependent sparse global least squares optimization problem P1 and the β-dependent sparse global least squares optimization problem P2, the alternating descent suboptimal algorithm is used to complete the optimization of P through alternating iterations. Nθ and p Nβ The estimation process is as follows:

[0027] Step 5.3.1b: Let iter represent the iteration number, and initialize iter to 1; let iter... max Indicates the maximum number of iterations;

[0028] Step 5.3.2b: Given and When calculating E in the iter-th iteration θ The estimated value and E β The estimated value Where, when iter=1 For p Nθ initial estimate For p Nβ initial estimate When iter > 1 In the (iter-1)th iteration, p Nθ The estimated value In the (iter-1)th iteration, p Nβ The estimated value;

[0029] Step 5.3.3b: Given and When calculating p in the iter-th iteration Nθ The estimated value and p Nβ The estimated value The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution.

[0030] Step 5.3.4b: Determine if iter is equal to iter max If so, the alternating iteration process ends, and the result will be... and Corresponding to p Nθ The final estimate and p Nβ The final estimated value; otherwise, let iter = iter + 1, and return to step 5.3.2b to continue execution; where "=" in iter = iter + 1 is the assignment symbol;

[0031] Step 5.4b: By finding p Nθ The index values ​​of the non-zero elements in the final estimate determine θ1, ..., θ2. K The estimated value p Nθ The index of the k'-th non-zero element in the final estimate is . θ represents the k'th direction angle relative to the x-axis. k' The estimated value; similarly, by finding p Nβ The index values ​​of the non-zero elements in the final estimate determine β1, ..., β2. K The estimated value p Nβ The index of the k'-th non-zero element in the final estimate is . This represents the k'th direction angle β relative to the z-axis. k' The estimated value.

[0032] In step 5.3.2b, The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution.

[0033] The specific process of step 6 is as follows:

[0034] Step 6.1: Calculate the cross-covariance matrix R of the L-shaped array. xz , Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k ;

[0035] Step 6.2: For R xz Perform column-wise vectorization to obtain vector r. v ;

[0036] Step 6.3: Based on r v A greedy algorithm is used to perform unambiguous pairing of K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis, as follows:

[0037] Step 6.3.1: Construct the complete basis matrix Ξ.

[0038]

[0039] ;in, Indicates the Kronecker product;

[0040] Step 6.3.2: Calculate |Ξ H r v | This yields K K×1 dimension output results. Where || represents the absolute value operation. This represents the output of the k'-th K×1 dimension;

[0041] Step 6.3.3: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] The maximum value corresponding to k' and Complete the pairing process, that is, if The largest, then and These are a pair of two-dimensional angle of arrival estimates for the same incoherent distributed source.

[0042] Compared with the prior art, the advantages of the present invention are as follows:

[0043] (1) By using two cross-covariance vectors to suppress the influence of noise and the expansion angle, and by using a conjugate symmetry strategy to expand the virtual array aperture, the method of the present invention is not only well applicable to both uniform and non-uniform noise, but also has good robustness and high estimation accuracy.

[0044] (2) Two-dimensional angle of arrival is independently estimated by using the closed solution algorithm based on rotation invariant subspace technology and the sparse total least squares algorithm respectively. Then, the parameter pairing strategy is adopted so that the method of the present invention can achieve unambiguous estimation of two-dimensional angle of arrival under the conditions of known and unknown number of signal sources. It has strong practicality, small constraints, and good flexibility. Attached Figure Description

[0045] Figure 1 This is a schematic diagram illustrating the implementation process of the method of the present invention;

[0046] Figure 2 This is a schematic diagram of the generalized array manifold receiving model under the L-shaped array in the method of the present invention;

[0047] Figure 3 The figures show the simulation results of the method of this invention, as well as existing rotation-invariant subspace algorithms and reduced-rank subspace algorithms, under the condition of known number of signal sources and uniform white noise.

[0048] Figure 4 The figures show simulation results of the method of this invention, as well as existing rotation-invariant subspace algorithms and reduced-rank subspace algorithms, under conditions of unknown number of signal sources and non-uniform noise.

[0049] Figure 5 The figures show simulation results of the method of this invention, as well as existing rotation-invariant subspace algorithms and reduced-rank subspace algorithms, under conditions of unknown number of signal sources, non-uniform noise, and different angular expansion variances. Detailed Implementation

[0050] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can implement it based on the description. The scope of protection of the present invention is not limited to the specific embodiments described herein.

[0051] This invention relates to a robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array, the implementation process of which is illustrated in the following diagram. Figure 1 As shown, the steps are as follows:

[0052] Step 1: Establish a generalized array manifold reception model under an L-shaped array: such as Figure 2As shown, K far-field narrowband uncorrelated incoherent sources are incident on an L-shaped array composed of two uniform linear arrays, where K ≥ 1. In this embodiment, K = 2. The two uniform linear arrays are located on the x-axis and z-axis of the xoz plane, respectively, and both have M elements, where M > K. In this embodiment, M = 10. The element spacing is half the carrier wavelength. The two uniform linear arrays have no shared elements. The element in the uniform linear array on the x-axis that is closest to the origin of the xoz plane is designated as the first element on the x-axis, and the element in the uniform linear array on the z-axis that is closest to the origin of the xoz plane is designated as the first element on the z-axis. Figure 2 In the diagram, 1, ..., M represent the nth array element on the x-axis or z-axis. Figure 2 In this diagram, the first element on the x-axis is labeled 1, and its distance from the origin of the xoz plane is half the carrier wavelength. The first element on the z-axis is also labeled 1, and its distance from the origin of the xoz plane is 0. However, in practice, the first element on the x-axis can also be positioned at the origin of the xoZ plane, and the first element on the z-axis can be positioned at half the carrier wavelength from the origin of the xoz plane.

[0053] Step 2: On the generalized array manifold receiving model, sample the data output by the uniform linear array on the x-axis at N time points; similarly, sample the data output by the uniform linear array on the z-axis at N time points.

[0054] Further specifying, in step 2, the sampled data x(t) at time t of the uniform linear array output on the x-axis is approximately expressed as: x(t)≈A(θ)c0(t)+A′(θ)c1(t)+n x (t), the sampled data z(t) at time t of the uniform linear array output on the z-axis is approximately expressed as: z(t)≈A(β)c0(t)+A′(β)c2(t)+n z (t); where N > 1, t = 1, ..., N, θ represents the direction angle variable of the signal emitted by the incoherent distributed source with the x-axis as the reference, β represents the direction angle variable of the signal emitted by the incoherent distributed source with the z-axis as the reference, A(θ) represents the steering matrix of the uniform linear array on the x-axis, A(θ) = [a(θ1), ..., a(θ2)] K )],a(θ k Let be the k-th column vector of A(θ), and let a(θ) be the column vector of A(θ). k ) represents θ k The relevant steering vector, θ k This represents the direction angle of the signal emitted by the k-th incoherent distributed source with the x-axis as the reference, k = 1, ..., K, and a(θ1) is the first column vector of A(θ).K Let be the Kth column vector of A(θ), and let A(β) represent the guiding matrix of the uniform linear array on the z-axis. A(β) = [a(β1), ..., a(β2)]. K )],a(β k Let a(β) be the k-th column vector of A(β). k ) represents β k The relevant steering vector, β k This represents the direction angle of the signal emitted by the k-th incoherent distributed source with the z-axis as the reference, and a(β1) is the first column vector of A(β). K ) is the Kth column vector of A(β), (θ) k ,β k () represents the two-dimensional angle of arrival of the k-th incoherent source. This indicates that for a(θ) k The variable θ in ) k The new vector obtained by differentiation Indicates that for a(β) k The variable β in ) k The new vector obtained by differentiation, c0(t), represents the received signal vector on the L-shaped array after the signals emitted by all incoherent distributed sources at time t have propagated through their respective transmission paths, c0(t) = [c 1，0 (t), ..., c K，0 (t)] T c k，0 (t) is the k-th element in c0(t), c k，0 (t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k The received signal on the L-shaped array after propagation along the transmission path s k γ(t) represents the signal emitted by the k-th incoherent distributed source at time t. k，l (t) represents the complex path gain of the signal emitted by the k-th incoherent distributed source at time t propagating along the l-th transmission path, L k Let represent the number of transmission paths for the k-th incoherent distributed source, with the superscript "T" indicating transpose operation. c1(t) represents the received signal vector on a uniform linear array on the x-axis after propagation of the signals emitted by all incoherent distributed sources through their respective transmission paths at time t, and after angular expansion at their respective x-axis-based azimuth angles. c1(t) = [c 1，1 (t), ..., c K，1 (t)] T c k，1 (t) is the k-th element in c1(t), c k，1(t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k Propagate along a transmission path, and at θ k The received signal on a uniform linear array along the x-axis after angular expansion in the directional direction. This represents the deviation between the x-axis-based azimuth angle of the signal emitted by the k-th incoherent distributed source at time t when propagating along the l-th transmission path and when propagating along the line-of-sight path. The mean is 0 and the variance is less than 3 degrees. These are random variables following a uniform or Gaussian distribution. c2(t) represents the received signal vector on a uniform linear array on the z-axis after propagation through all transmission paths of all incoherent sources at time t, and after angular expansion along their respective z-axis-based azimuth angles. c2(t) = [c 1，2 (t), ..., c K，2 (t)] T c k，2 (t) is the k-th element in c2(t), c k，2 (t) represents the signal emitted by the k-th incoherent distributed source at time t, after passing through L. k Propagation along a transmission path, and in β k The received signal on a uniform linear array along the z-axis after angular expansion in the directional direction. This represents the deviation between the z-axis-based azimuth angle of the signal emitted by the k-th incoherent distributed source at time t when propagating along the l-th transmission path and when propagating along the line-of-sight path. The mean is 0, the variance is less than 3 degrees, and they are random variables following a uniform or Gaussian distribution. x (t) represents the Gaussian noise vector output by the uniform linear array on the x-axis at time t, where n z (t) represents the Gaussian noise vector output by the uniform linear array on the z-axis at time t, where n x (t) and n z (t) are unrelated to each other.

[0055] Step 3: Using statistical averaging, perform cross-correlation calculations on the N time-series sampled data of the uniform linear array output on the x-axis and the N time-series sampled data of the output of the first element on the z-axis to obtain the cross-covariance vector r. xz Similarly, using statistical averaging, cross-correlation is performed on the sampled data at N time points of the uniform linear array output on the z-axis and the sampled data at N time points of the output of the first element on the x-axis to obtain the cross-covariance vector r. zx .

[0056] In this embodiment, Where z1(t) represents the sampled data of the output data of the first array element on the z-axis at time t, x1(t) represents the sampled data of the output data of the first array element on the x-axis at time t, the superscript "H" indicates the conjugate transpose operation, p represents the power vector of the received signals on the L-shaped array after the signals emitted by all incoherent distributed sources at N times have propagated through their respective transmission paths, p = [p1, ..., p K ] T p k Let p be the k-th element. k This indicates that the signal emitted by the k-th incoherent distributed source at N times is transmitted through L. k The power of the received signal on the L-shaped array after propagation along the transmission path. The superscript "*" indicates conjugate operation.

[0057] Step 4: Construct r using a conjugate symmetry strategy. xz Aperture extension array cross-covariance vector r e Similarly, using a conjugate symmetry strategy, r is constructed. zx Aperture extension array cross-covariance vector r f .

[0058] In this embodiment, Where, r e and r f All of them have a dimension of (2M-1)×1, and J represents a permutation matrix in which the elements on the antidiagonal are 1 and the other elements are 0. Indicates by r xz The subvector formed by the second to the Mth elements, Indicates by r zx The subvector consisting of the second to the Mth elements.

[0059] Step 5: If the number of signal sources K is known prior, then according to r e and r f Furthermore, a closed-form solution algorithm based on rotation-invariant subspace technology is used to obtain K orientation angle estimates with the x-axis as the reference and K orientation angle estimates with the z-axis as the reference.

[0060] Further specifying, based on r e and r f The specific process of obtaining K estimated orientation angles based on the x-axis and K estimated orientation angles based on the z-axis using a closed-form solution algorithm based on rotation-invariant subspace technology is as follows:

[0061] Step 5.1a: r e Divide r into M sub-vectors with overlapping elements. e The m-th subvector obtained by partitioning Represented as Similarly, r f Divide r into M sub-vectors with overlapping elements. f The m-th subvector obtained by partitioning Where m = 1, ..., M, Indicates by r e The subvector consisting of the m-th element to the (M+m-1)-th element. Indicates by r f The sub-vector consisting of the m-th element to the M+m-1-th element.

[0062] Here, according to It can be known Indicates by r e The subvector formed by the first element to the Mth element, Indicates by r e The subvector formed by the second element to the (M+1)th element, and so on, Indicates by r e The subvectors formed by the Mth element to the 2M-1th element have M-1 identical elements in every two adjacent subvectors; according to It can be known Indicates by r f The subvector formed by the first element to the Mth element, Indicates by r f The subvector formed by the second element to the (M+1)th element, and so on, Indicates by r f The subvectors formed by the Mth element to the 2M-1th element have M-1 identical elements in every two adjacent subvectors.

[0063] Step 5.2a: Based on r e Construct r from all the sub-vectors obtained by partitioning xz The relevant M×M dimension matrix Similarly, according to r f Construct r from all the sub-vectors obtained by partitioning zx The relevant M×M dimension matrix Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k Λ c0 The remaining elements in the array are 0.

[0064] Step 5.3a: For Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. e Then U e Divided into two sub-matrices and Similarly, for Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. f Then U f Divided into two sub-matrices and Among them, U e and U f The dimensions of all are M×K. Indicated by U e The submatrix formed by rows 1 to M-1, Indicated by U e The submatrix formed by rows 2 to M, Indicated by U f The submatrix formed by rows 1 to M-1, Indicated by U f The submatrix formed by rows 2 to M.

[0065] Step 5.4a: Calculation Then for H e Eigenvalue decomposition yields K eigenvalues ​​u1, ..., u2. K Then through the formula θ is calculated from k'∈[1,K]. k' The estimated value of k'∈[1,K] k'∈[1,K]; where the superscript u represents the pseudo-inverse operation of a matrix. k' Indicates H e The k'th eigenvalue obtained by eigenvalue decomposition, ∠u k' Indicates the relationship between u k' Phase take operation, θ k' This represents the k'th direction angle relative to the x-axis. Represents θ k' The estimated value, but It is not necessarily the direction angle θ of the signal emitted by the k-th incoherent distributed source with the x-axis as the reference. k The estimated value.

[0066] Similarly, calculation Then for H f Eigenvalue decomposition is performed to obtain K eigenvalues ​​v1, ..., v2. K Then through the formula β is calculated from k'∈[1,K]. k' The estimated value of k'∈[1,K] k'∈[1,K]; where, v k' Indicates H f The k'th eigenvalue obtained by eigenvalue decomposition, ∠v k' Indicates to v k' Phase take operation, β k' This represents the k'-th direction angle relative to the z-axis. Indicates β k' The estimated value, but It is not necessarily the direction angle β of the signal emitted by the k-th incoherent distributed source with z-axis as the reference. k The estimated value.

[0067] If the number of signal sources K is unknown, then according to r e and r f The sparse total least squares algorithm is used to obtain K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis.

[0068] Further specifying, based on r e and r f The specific process of obtaining K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis using the sparse total least squares algorithm is as follows:

[0069] Step 5.1b: Transfer r e Sparse representation is r e =(Ψ(θ)+E θ )p Nθ +e θ , will r f Sparse representation is r f =(Ψ(β)+E β )p Nβ +e β ; where Ψ(θ) represents the matrix [JA * (θ), A (2:M) (θ)] T sparse representation of A (2:M) (θ) represents the submatrix formed by the 2nd to Mth rows of A(θ), and Ψ(β) represents the matrix [JA * (β), A (2:M) (β)] T sparse representation of A (2:M) (β) represents the submatrix formed by rows 2 to M of A(β), E θ E represents the θ-related bias matrix resulting from the application of the generalized array manifold. β p represents the β-related bias matrix resulting from the application of the generalized array manifold.Nθ and p Nβ All are K-sparse vectors, p Nθ and p Nβ Both p contain non-zero elements and zero elements. Nθ The index of the k'-th non-zero element in the array is θ1, ..., θ K One of them, p Nβ The index of the k'-th non-zero element in the array is β1, ..., β K One of them, k'∈[1,K], that is, p Nθ The positions of the non-zero elements are represented by θ1, ..., θ2. K Non-one-to-one angle of arrival information, p Nβ The positions of the non-zero elements in the equation represent the positions of β1, ..., β2. K Non-one-to-one correspondence of angle of arrival information, e θ Indicates r e The estimated value and the true value (i.e., when the sampled data at N times tend to infinity r) e The deviation vector between the estimated values, e β Indicates r f The estimated value and the true value (i.e., when the sampled data at N times tend to infinity r) f The deviation vector between the estimated values, e θ and e β All of these are bias vectors generated from calculations based on a finite number of samples.

[0070] Step 5.2b: Construct the θ-dependent sparse global least squares optimization problem P1, described as: P1: Similarly, construct a β-related sparse total least squares optimization problem P2, described as: P2: Where ||||2 represents finding the l2 norm of a vector, |||| F Let ||||1|| denote the Frobenius norm of the matrix, ||||1|| denotes the l1 norm of the vector, and λ represents the regularization parameter, where λ > 0. In this embodiment, λ = 2. p Nθ The estimated value, E represents θ The estimated value, p Nβ The estimated value, E represents β The estimated value.

[0071] Step 5.3b: Based on the θ-dependent sparse global least squares optimization problem P1 and the β-dependent sparse global least squares optimization problem P2, the alternating descent suboptimal algorithm is used to complete the optimization of P through alternating iterations. Nθ and p NβThe estimation process is as follows:

[0072] Step 5.3.1b: Let iter represent the iteration number, and initialize iter to 1; let iter... max This represents the maximum number of iterations; in this embodiment, iter is taken as... max =3.

[0073] Step 5.3.2b: Given and When calculating E in the iter-th iteration θ The estimated value and E β The estimated value Where, when iter=1 For p Nθ initial estimate For p Nβ initial estimate When iter > 1 In the (iter-1)th iteration, p Nθ The estimated value In the (iter-1)th iteration, p Nβ The estimated value.

[0074] Here, The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution.

[0075] Step 5.3.3b: Given and When calculating p in the iter-th iteration Nθ The estimated value and p Nβ The estimated value The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution.

[0076] Step 5.3.4b: Determine if iter is equal to iter max If so, the alternating iteration process ends, and the result will be... and Corresponding to p Nθ The final estimate and p Nβ The final estimated value; otherwise, let iter = iter + 1, and return to step 5.3.2b to continue execution; where "=" in iter = iter + 1 is the assignment symbol.

[0077] Step 5.4b: By finding p Nθ The indices of the non-zero elements in the final estimate determine θ1, ..., θ2. K The estimated value p Nθ The index of the k'-th non-zero element in the final estimate is . θ represents the k'th direction angle relative to the x-axis. k' The estimated value, but It is not necessarily the direction angle θ of the signal emitted by the k-th incoherent distributed source with the x-axis as the reference. k The estimated value; similarly, by finding p Nβ The indices of the non-zero elements in the final estimate determine β1, ..., β2. K The estimated value p Nβ The index of the k'-th non-zero element in the final estimate is . This represents the k'th direction angle β relative to the z-axis. k' The estimated value, but It is not necessarily the direction angle β of the signal emitted by the k-th incoherent distributed source with z-axis as the reference. k The estimated value.

[0078] Step 6: Pair the K orientation angle estimates relative to the x-axis with the K orientation angle estimates relative to the z-axis to obtain K pairs of two-dimensional angle of arrival estimates. If K = 1, pairing is not required, but in practical applications, K is generally greater than 1.

[0079] To further specify, the specific process of step 6 is as follows:

[0080] Step 6.1: Calculate the cross-covariance matrix R of the L-shaped array. xz , Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k .

[0081] Step 6.2: For R xz Perform column-wise vectorization to obtain vector r. v .

[0082] Step 6.3: Based on r v A greedy algorithm is used to perform unambiguous pairing of K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis, as follows:

[0083] Step 6.3.1: Construct the complete basis matrix Ξ.

[0084] ;

[0085] in, Indicates the Kronecker product. Replace a(θ) k θ in ) k get Replace a(β) k β in ) k get

[0086] Step 6.3.2: Calculate |Ξ H r v | This yields K K×1 dimension output results. Where || represents the absolute value operation. This represents the output of the k'th K×1 dimension.

[0087] Step 6.3.3: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] The maximum value corresponding to k' and Complete the pairing process, that is, if The largest, then and These are a pair of two-dimensional angle of arrival estimates for the same incoherent distributed source.

[0088] The performance of the method of the present invention will be analyzed through simulation experiments below.

[0089] The simulation was performed using MATLAB software. The two-dimensional angles of arrival of the two incoherent distributed sources were {θ1 = 79.6°, β1 = 98.8°} and {θ2 = 105.3°, β2 = 127.2°}, respectively. The signals emitted by both incoherent distributed sources propagated along L1 = L2 = 100 transmission paths, and the complex path gain γ 1，l (t), γ 2，l(t) all satisfy a Gaussian distribution with a mean of 0 and a variance of 1. The x-axis and z-axis of the L-shaped array are uniform linear arrays consisting of 10 array elements spaced at half-carrier wavelengths. The number of sampling snapshots in the simulation is 200. Except for simulation experiment 3, the variance of the random variables generated by the angle expansion is 1.5°.

[0090] Simulation Experiment 1: This experiment demonstrates the effectiveness of the proposed method for two-dimensional angle-of-arrival estimation under conditions of a known number of signal sources and uniform white noise, and measures it using the root mean square error (RMSE) of the two-dimensional angle-of-arrival estimation. The signal-to-noise ratio (SNR) was varied from -9 dB to 6 dB. The simulation results are as follows: Figure 3 As shown, it can be seen that the method of the present invention achieves significantly improved estimation performance compared with the rotation-invariant subspace algorithm proposed by S. Shahbazpanahi et al. of Ontario University of Technology, Canada, and also achieves certain performance improvements compared with the reduced-rank subspace algorithm proposed by F. Gao et al. of Tsinghua University. At the same time, since the method of the present invention is a closed-form solution algorithm under the condition that the number of signal sources is known, its computational complexity is much lower than that of the reduced-rank subspace algorithm.

[0091] Simulation Experiment 2: This experiment demonstrates the effectiveness of the proposed method for two-dimensional angle-of-arrival estimation under conditions of unknown signal source quantity and non-uniform noise. The signal-to-noise ratio changes from -9 dB to 6 dB, and the covariance matrix of the Gaussian noise vector output by the uniform linear array on the x-axis is Q. x =diag(2.0, 8.0, 1.5, 2.5, 7.0, 1.0, 5.5, 9.0, 1.0, 2.0), where Q is the covariance matrix of the Gaussian noise vector output by the uniform linear array on the z-axis. y =diag(2.3, 1.5, 2.0, 3.0, 5.9, 3.5, 9.0, 1.0, 3.0, 6.8), where diag() calculates the diagonal matrix. Simulation results are as follows. Figure 4 As shown, the method of the present invention can achieve significantly better estimation performance than the comparative method under the condition of signal-to-noise ratio < 0 dB, demonstrating the superior noise robustness of the method of the present invention.

[0092] Simulation Experiment 3: This experiment demonstrates the effectiveness of the proposed method for estimating the two-dimensional angle of arrival under conditions of unknown signal source quantity, non-uniform noise, and different angular spread variances. The signal-to-noise ratio is fixed at -3 dB, and the covariance matrix of the Gaussian noise vector output by the uniform linear array on the x-axis is Q. x =diag(2.0, 8.0, 1.5, 2.5, 7.0, 1.0, 5.5, 9.0, 1.0, 2.0), where Q is the covariance matrix of the Gaussian noise vector output by the uniform linear array on the z-axis. y=diag(2.3, 1.5, 2.0, 3.0, 5.9, 3.5, 9.0, 1.0, 3.0, 6.8). The variance of the random variable generated by the angular expansion changes from 0.3° to 1.8°, and the simulation results are as follows. Figure 5 As shown, it can be seen that as the variance of the random variable generated by the angle expansion gradually increases, the estimation performance of the method of the present invention does not degrade significantly, and the estimation effect is far better than the comparison method, which proves the good applicability of the method of the present invention under the conditions of unknown number of signal sources, non-uniform noise and different angle expansion variances.

Claims

1. A robust non-coherent distributed source 2-D angle of arrival estimation method under L-shaped array, characterized in that The steps are as follows: Step 1: Establish a generalized array manifold receiving model under the L-shaped array: Set up K far-field narrowband uncorrelated incoherent distributed sources incident on an L-shaped array composed of two uniform linear arrays. The two uniform linear arrays are located on the x-axis and z-axis of the xoz plane, respectively. The number of array elements is M, M>K, and the spacing between array elements is 1 / 2 of the carrier wavelength. The two uniform linear arrays have no shared array elements. The array element on the x-axis that is closest to the origin of the xoz plane is taken as the first array element on the x-axis, and the array element on the z-axis that is closest to the origin of the xoz plane is taken as the first array element on the z-axis. Step 2: In the generalized array manifold receiving model, sample the data output by the uniform linear array on the x-axis at N time points; similarly, sample the data output by the uniform linear array on the z-axis at N time points. Step 3: using statistical average operation, cross-correlation operation is performed on the sampling data of N time points of the data output by the uniform linear array on the x axis and the sampling data of N time points of the data output by the first array element on the z axis, to obtain a cross-covariance vector r xz ; similarly, using statistical average operation, cross-correlation operation is performed on the sampling data of N time points of the data output by the uniform linear array on the z axis and the sampling data of N time points of the data output by the first array element on the x axis, to obtain a cross-covariance vector r zx ; Step 4: Construct the aperture expansion array cross-covariance vector r xz of r e ; likewise, construct the aperture expansion array cross-covariance vector r zx of r f ; Step 5: If the number of signal sources K is known a priori, then the r e and r f and the closed-form algorithm based on the rotational invariance subspace technique is used to obtain K direction angle estimates with respect to the x-axis and K direction angle estimates with respect to the z-axis. If the number of signal sources K is unknown, then according to r e and r f , K direction angle estimates with the x-axis as the reference and K direction angle estimates with the z-axis as the reference are obtained using the sparse total least squares algorithm. Step 6: Pair the K azimuth angle estimates based on the x-axis and the K azimuth angle estimates based on the z-axis to obtain K pairs of two-dimensional angle of arrival estimates.

2. The robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 1, characterized in that... In step 2, the sampling data x(t) of the tth moment of the data output by the uniform linear array on the x-axis is approximately expressed as: x(t)≈A(θ)c0(t)+A'(θ)c1(t)+n x In step 2, the sampling data z(t) of the tth moment of the data output by the uniform linear array on the z-axis is approximately expressed as: z(t)≈A(β)c0(t)+A'(β)c2(t)+n z (t);wherein t=1,...,N, θ represents the directional angle variable of the signal emitted by the non-coherent distributed source with the x-axis as the reference, β represents the directional angle variable of the signal emitted by the non-coherent distributed source with the z-axis as the reference, A(θ) represents the steering matrix of the uniform linear array on the x-axis, A(θ)=[a(θ1),...,a(θ K )], a(θ k ) is the kth column vector of A(θ), a(θ k ) represents the steering vector related to θ k , θ k represents the directional angle of the kth non-coherent distributed source with the x-axis as the reference, k=1,...,K, A(β) represents the steering matrix of the uniform linear array on the z-axis, A(β)=[a(β1),...,a(β K )], a(β k ) is the kth column vector of A(β), a(β k ) represents the steering vector related to β k , β k represents the directional angle of the kth non-coherent distributed source with the z-axis as the reference, (θ k ,β k ) represents the two-dimensional angle of arrival of the kth non-coherent distributed source, represents a new vector obtained by taking the derivative of the variable θ k in a(θ k ), represents a new vector obtained by taking the derivative of the variable β k in a(β k ), c0(t) represents the received signal vector of the signal emitted by all non-coherent distributed sources at the tth moment after propagating through all transmission paths respectively on the L-shaped array, c0(t)=[c 1,0 (t),...,c K,0 (t)] T , c k,0 (t) is the kth element in c0(t), c k,0 (t) represents the received signal of the signal emitted by the kth non-coherent distributed source at the tth moment after propagating through L k transmission paths on the L-shaped array, s k (t) represents a signal emitted by the kth non-coherent distributed source at the tth time, represents a complex path gain of a signal emitted by the kth non-coherent distributed source at the tth time propagating along the lth transmission path, k represents the number of transmission paths of the kth non-coherent distributed source, the superscript "T" represents a transposition operation, and c1(t) represents a received signal vector on a uniform linear array in the x-axis direction after signals emitted by all non-coherent distributed sources at the tth time propagate through all transmission paths thereof and are angularly spread in respective direction angles with the x-axis as a reference, 1,1 K,1 T , c k,1 (t) represents a signal emitted by the kth non-coherent distributed source at the tth time propagating through L k,1 transmission paths and being angularly spread in the direction θ k , and c k (t) represents a received signal on a uniform linear array in the x-axis direction after the signal is angularly spread in the direction θ represents a deviation between a direction angle with the x-axis as a reference when a signal emitted by the kth non-coherent distributed source at the tth time propagates along the lth transmission path and a direction angle with the x-axis as a reference when the signal propagates along a line-of-sight path, has a mean value of 0 and a variance less than 3xπ / 180 radians, is a random variable subject to a uniform distribution or a Gaussian distribution, and c2(t) represents a received signal vector on a uniform linear array in the z-axis direction after signals emitted by all non-coherent distributed sources at the tth time propagate through all transmission paths thereof and are angularly spread in respective direction angles with the z-axis as a reference, 1,2 K,2 T k,2 , c k,2 (t) represents a signal emitted by the kth non-coherent distributed source at the tth time propagating through L k transmission paths and being angularly spread in the direction β k , and c x (t) represents a received signal on a uniform linear array in the z-axis direction after the signal is angularly spread in the direction β represents a deviation between a direction angle with the z-axis as a reference when a signal emitted by the kth non-coherent distributed source at the tth time propagates along the lth transmission path and a direction angle with the z-axis as a reference when the signal propagates along a line-of-sight path, has a mean value of 0 and a variance less than 3xπ / 180 radians, is a random variable subject to a uniform distribution or a Gaussian distribution, and n x (t) represents a Gaussian distribution noise vector output by a uniform linear array in the x-axis direction at the tth time​​​​​​ z (t) represents the Gaussian noise vector output by the uniform linear array on the z-axis at time t, where n x (t) and n z (t) are unrelated to each other.

3. The robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 2, characterized in that... In step 3 Where z1(t) represents the sampled data of the output data of the first array element on the z-axis at time t, x1(t) represents the sampled data of the output data of the first array element on the x-axis at time t, the superscript "H" indicates the conjugate transpose operation, p represents the power vector of the received signal on the L-shaped array after the signals emitted by all incoherent distributed sources at N times have propagated through their respective transmission paths, p = [p1,...,p K ] T p k Let p be the k-th element. k This indicates that the signal emitted by the k-th incoherent distributed source at N times is transmitted through L. k The power of the received signal on the L-shaped array after propagation along the transmission path. The superscript "*" indicates conjugate operation.

4. The robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 3, characterized in that... In step 4 Where, r e and r f All of them have a dimension of (2M-1)×1, and J represents a permutation matrix in which the elements on the antidiagonal are 1 and the other elements are 0. Indicates by r xz The subvector formed by the second to the Mth elements, Indicates by r zx The subvector consisting of the second to the Mth elements.

5. A robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 4, characterized in that... In step 5, according to r e and r f , the specific process of obtaining K direction angle estimates based on the x-axis and K direction angle estimates based on the z-axis by using a closed-form solution based on the rotation invariant subspace technique is as follows: Step 5.1a: r e Divide r into M sub-vectors with overlapping elements. e The m-th subvector obtained by partitioning Represented as Similarly, r f Divide r into M sub-vectors with overlapping elements. f The m-th subvector obtained by partitioning Where m = 1,...,M, Indicates by r e The subvector consisting of the m-th element to the (M+m-1)-th element. Indicates by r f The subvector consisting of the m-th element to the (M+m-1)-th element; Step 5.2a: Based on r e Construct r from all the sub-vectors obtained by partitioning xz The relevant M×M dimension matrix Similarly, according to r f Construct r from all the sub-vectors obtained by partitioning zx The relevant M×M dimension matrix Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k ; Step 5.3a: For Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. e Then U e Divided into two sub-matrices and Similarly, for Perform eigenvalue decomposition to obtain The eigenvectors corresponding to the K largest eigenvalues ​​are given, and the K eigenvectors form a matrix U. f Then U f Divided into two sub-matrices and Among them, U e and U f The dimensions of all are M×K. Indicated by U e The submatrix formed by rows 1 to M-1, Indicated by U e The submatrix formed by rows 2 to M, Indicated by U f The submatrix formed by rows 1 to M-1, Indicated by U f The submatrix formed by rows 2 to M; Step 5.4a: Calculation Then for H e Eigenvalue decomposition is performed to obtain K eigenvalues ​​u1,...,u K Then, θ is calculated using the formula cc. k' The estimated value of k'∈[1,K] Among them, superscript u represents the pseudo-inverse operation of a matrix. k' Indicates H e The k'th eigenvalue obtained by eigenvalue decomposition, ∠u k' Indicates the relationship between u k' Phase take operation, θ k' This represents the k'th direction angle relative to the x-axis. Represents θ k' The estimated value; Similarly, calculation Then for H f Eigenvalue decomposition is performed to obtain K eigenvalues ​​v1,...,v K Then through the formula β was calculated k' The estimated value of k'∈[1,K] Among them, v k' Indicates H f The k'th eigenvalue obtained by eigenvalue decomposition, ∠v k' Indicates to v k' Phase take operation, β k' This represents the k'-th direction angle relative to the z-axis. Indicates β k' The estimated value.

6. A robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 4, characterized in that... In step 5, according to r e and r f , and using the sparse total least squares algorithm, the specific process of obtaining K direction angle estimates based on the x-axis and K direction angle estimates based on the z-axis is as follows: Step 5.1b: Transfer r e Sparse representation is r e =(Ψ(θ)+E θ )p Nθ +e θ , will r f Sparse representation is r f =(Ψ(β)+E β )p Nβ +e β ; where Ψ(θ) represents the matrix [(JA * (θ)) T ,(A (2:M) (θ)) T ] T sparse representation of A (2:M) (θ) represents the submatrix formed by the 2nd to Mth rows of A(θ), and Ψ(β) represents the matrix [(JA * (β)) T ,(A (2:M) (β)) T ] T sparse representation of A (2:M) (β) represents the submatrix formed by rows 2 to M of A(β), E θ E represents the θ-related bias matrix resulting from the application of the generalized array manifold. β p represents the β-related bias matrix resulting from the application of the generalized array manifold. Nθ and p Nβ All are K-sparse vectors, p Nθ and p Nβ Both p contain non-zero elements and zero elements. Nθ The index of the k'-th non-zero element in the array is θ1,...,θ K One of them, p Nβ The index of the k'-th non-zero element in the array is β1,...,β K One of them, k'∈[1,K], e θ Indicates r e The deviation vector between the estimated value and the true value, e β Indicates r f The vector of deviations between the estimated and true values; Step 5.2b: Construct the θ-dependent sparse global least squares optimization problem P1, described as: Similarly, construct the β-related sparse total least squares optimization problem P2, described as follows: Where |||2 represents finding the l2 norm of a vector, ||| F Let ||||1|| denote the Frobenius norm of a matrix, |||||1|| denotes the l1 norm of a vector, and λ represents the regularization parameter, where λ > 0. p Nθ The estimated value, E represents θ The estimated value, p Nβ The estimated value, E represents β The estimated value; Step 5.3b: The estimates of p and p are obtained by alternating iteration using an alternating descent suboptimal algorithm based on the sparse total least squares optimization problem P1 related to 0 and the sparse total least squares optimization problem P2 related to p, in particular by solving the following two problems iteratively: Nθ and p Nβ in particular by solving the following two problems iteratively: Step 5.3.1b: Let iter represent the iteration number, with the initial value of iter being 1; let iter max represent the maximum number of iterations. Step 5.3.2b: Given and When calculating E in the iter-th iteration θ The estimated value and E β The estimated value Where, when iter=1 For p Nθ initial estimate For p Nβ initial estimate When iter > 1 In the (iter-1)th iteration, p Nθ The estimated value In the (iter-1)th iteration, p Nβ The estimated value; Step 5.3.3b: Given and When calculating p in the iter-th iteration Nθ The estimated value and p Nβ The estimated value The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The calculation process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. Step 5.3.4b: Determine if iter is equal to iter max If so, the alternating iteration process ends, and the result will be... and Corresponding to p Nθ The final estimate and p Nβ The final estimated value; otherwise, let iter = iter + 1, and return to step 5.3.2b to continue execution; where "=" in iter = iter + 1 is the assignment symbol; Step 5.4b: By finding p Nθ The indices of the non-zero elements in the final estimate determine θ1,...,θ K The estimated value p Nθ The index of the k'-th non-zero element in the final estimate is . θ represents the k'th direction angle relative to the x-axis. k' The estimated value; similarly, by finding p Nβ The indices of the non-zero elements in the final estimate determine β1,...,β K The estimated value p Nβ The index of the k'-th non-zero element in the final estimate is . This represents the k'th direction angle β relative to the z-axis. k' The estimated value.

7. A robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 6, characterized in that... In step 5.3.2b, The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution. The solution process is as follows: Construct The l2-l1 norm minimization optimization problem is described as follows: Then, by solving the problem using second-order cone programming, we can obtain the solution.

8. A robust two-dimensional angle-of-arrival estimation method for incoherent distributed sources under an L-shaped array according to claim 5 or 6, characterized in that... The specific process of step 6 is as follows: Step 6.1: Calculate the cross-covariance matrix R of the L-shaped array. xz , Among them, Λ c0 Λ represents the received signal power matrix with a diagonal structure. c0 The dimension is K×K, Λ c0 The element in the k-th row and k-th column of the array is p. k ; Step 6.2: R xz is column vectorized to get vector r v ; Step 6.3: Based on r v A greedy algorithm is used to perform unambiguous pairing of K orientation angle estimates based on the x-axis and K orientation angle estimates based on the z-axis, as follows: Step 6.3.1: Construct the complete basis matrix Ξ. in, Indicates the Kronecker product; Step 6.3.2: Calculate |Ξ H r v | This yields K K×1 dimension output results. Where || represents the absolute value operation. This represents the output of the k'-th K×1 dimension; Step 6.3.3: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] The maximum value corresponding to k' and Complete the pairing process, that is, if The largest, then and These are a pair of two-dimensional angle of arrival estimates for the same incoherent distributed source.

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