A high-precision estimation method for the angle of arrival

Abstract

The present invention discloses a high-precision estimation method for the angle of arrival, which successively includes the following steps: A: First, establish an off-grid DOA bilinear system model, and then factorize the off-grid DOA bilinear system model using the Bayesian formula; B: Estimate the sparse incident wave signals S ; C: Estimate the off-grid error parameter vector β ; D: According to the obtained sparse incident wave signals S and the off-grid error parameter vector β , perform high-precision calculation of the angle of arrival through the off-grid DOA bilinear system model obtained in step A. The present invention can solve the contradiction between complexity and robustness in off-grid angle of arrival estimation, effectively improve the calculation accuracy of the angle of arrival, and has obvious performance advantages in scenarios with fewer snapshots such as airborne radars.

Description

Technical Field

[0001] The present invention relates to the field of array signal processing, and in particular to a high-precision estimation method for the angle of arrival in array signal processing. Background Art

[0002] Estimating the direction of arrival (DOA) of multiple far-field signals using an antenna array is a fundamental problem in array signal processing and has been widely applied in fields such as radar, sonar, and earthquake prediction. For the DOA estimation problem, researchers at home and abroad have proposed various types of methods, such as subspace decomposition, support vector machines, matrix eigen-space decomposition, etc. Each method has its own advantages and disadvantages. With the emergence of compressive sensing technology, researchers have classified DOA estimation as a sparse recovery problem and proposed various estimation algorithms, such as the orthogonal matching pursuit algorithm. In general sparse estimation methods, the DOA range is divided into a uniform grid, assuming that the actual DOA is accurately located on a certain grid. Compared with methods such as subspace partitioning and eigen-space decomposition, compressive sensing-based methods usually can achieve better performance, especially when the number of snapshots is small. Sparse Bayesian learning is a class of classical sparse recovery algorithms. By adding a sparse guiding prior to variables, it can achieve better estimation performance with fewer observations. However, when the true incoming wave direction is not exactly located on the grid, the above scheme will lead to relatively serious estimation bias. On the one hand, a finer grid division will lead to a higher computational complexity, and on the other hand, it will make the column vectors of the sensing matrix have a higher correlation, resulting in a worse convergence of the estimation algorithm.

[0003] In recent years, multiple teams at home and abroad have proposed various sparse reconstruction algorithms based on off-grid signals, assuming that the true angle does not have to be limited to the grid and can calculate the error and correct the estimated value. For example, the root sparse Bayesian learning method (RSBL) can achieve good estimation performance in the linear array scenario. Due to its low complexity, the classical approximate message passing algorithm (AMP) has been widely used in the sparse Bayesian learning framework, but the ordinary AMP algorithm will cause divergence in the iterative process when encountering a relatively non-Gaussian distributed observation matrix. Summary of the Invention

[0004] The object of the present invention is to provide a high-precision estimation method for the angle of arrival, which can solve the contradiction between complexity and robustness in off-grid angle of arrival estimation, effectively improve the calculation accuracy of the angle of arrival, and has obvious performance advantages in scenarios with fewer snapshots such as airborne radar.

[0005] The present invention adopts the following technical solutions:

[0006] A high-precision estimation method for the angle of arrival successively includes the following steps:

[0007] A: First, establish the off-grid DOA bilinear system model Y = (A + ∑ n β n E n )S+W=Φ{β}S+W;

[0008] Among them, Φ{β}=A+Ediag(β)=A+∑ n β n E n , E n =[0,…,e(θ n ),…,0]∈C M×N , the off-grid DOA bilinear system model is a shorthand for the received signal vector Y = (A (θ) + E (θ) diag (β)) S + W; Y = (A (θ) + E (θ) diag (β)) S + W is y t =(A(θ)+E(θ)diag(β))s t +w t Multiple observation forms; suppose there is a uniform linear array composed of M non-directional array elements, and there are K narrow-band far-field incoming wave signals, y t represents the received signal vector at the t-th snapshot, A(θ)=[a(θ1),...,a(θ N )]∈C M×N , C M×K represents a complex-valued matrix with dimension M×K; assuming that the angle space θ of the incoming signal is divided into N grids, it can be expressed as a vector form θ=[θ1,…,θ N ], each grid size is Δ θ =π / N; matrix E(θ)=[e(θ1),…,e(θ N )] represents the error matrix, and the vector β=[β1,…β N ] T represents the grid error parameter vector, the grid error parameter n=1,…,N,s t ∈C N×1 represents the sparse incoming signal vector of length N; L is the number of snapshots, w t represents a Gaussian white noise vector with a mean of 0 and a variance of 1 / λ; the matrix Y = [y 1 ,y 2 ,…,y L ]、S=[s 1 ,s 2 ,…,s L ] and W = [w 1 ,w 2 ,…,w LThe set matrices representing the received signal vector, the incident wave signal vector, and the Gaussian white noise vector respectively, and the matrix S has the same sparse characteristics; A and E are the abbreviations of the array manifold matrix A(θ) and the error matrix E(θ) respectively; θ is the angular space of the incident wave signals;

[0009] Then, factorize the off-grid DOA bilinear system model using Bayes' formula;

[0010]

[0011] Among them, the sparse incident wave signal vector s t is defined as two identical equivalent vectors and both represent the sparse incident wave signal vector, and the mathematical expression is δ(·) is the delta function; represents that in the t-th snapshot, the likelihood function of the received signal vector y t is in the form of a complex Gaussian, and the likelihood function is expressed as where represents a complex Gaussian distribution with a mean of and a covariance matrix of λ -1 I, where I represents a diagonal matrix; the probability distribution represents the prior distribution of the vector , and γ represents the hyper-prior parameter vector; all incident wave signal vectors for t = 1:L have the same sparse characteristics, share the same hyper-prior parameter vector γ; the off-grid error parameter vector β follows a uniform distribution p(β) = U[-Δ θ / 2, Δ θ / 2], and the prior distribution of the noise variance is assumed to be p(λ) = 1 / λ;

[0012] B: Estimate the sparse incident wave signal S;

[0013] C: Estimate the off-grid error parameter vector β;

[0014] D: According to the obtained sparse incident wave signal S and the off-grid error parameter vector β, perform high-precision calculation of the angle of arrival through the off-grid DOA bilinear system model obtained in step A.

[0015] In the said step B, the approximate message passing algorithm is applied to estimate the sparse incident wave signal S.

[0016] In the said step C, the expectation maximization method is applied to estimate the off-grid error parameter vector β

[0017] In the said step A, the steps for establishing the off-grid DOA bilinear system model are as follows,

[0018] A1: Suppose there is a uniform linear array composed of M non-directional array elements, and there are K narrowband far-field incoming signals. Then the received vector at the t-th snapshot is expressed as

[0019]

[0020] where, y t ∈C M×1 is the received signal vector of all M array elements at the t-th snapshot. The symbol C M×1 represents a complex-valued vector of dimension M×1, is the array manifold matrix. The symbol C M×K represents a complex-valued matrix of dimension M×K, represents the true incoming signal vector at the t-th snapshot. C K×1 represents a complex-valued vector of dimension K×1, w t represents a Gaussian white noise vector with a mean of 0 and a variance of 1 / λ, and λ represents the noise precision;

[0021] A2: Decompose the array manifold matrix into vector form

[0022] where, the vector represents the steering vector of the k-th incoming signal, and the decomposition is expressed as j represents the imaginary unit, represents the true arrival angle of the k-th incoming signal, k = 1, …, K; the angle space θ of the incoming signals is divided into N grids, expressed in vector form as θ = [θ1, …, θ N , and the size of each grid is Δ θ = π / N; each angle θ n corresponds to a potential incoming signal source n = 1, …, N; the directions of the incoming signals are sparsely distributed in the angle domain, that is, N >> K, and a sparse incoming signal vector is formed

[0023] A3: Reconstruct the array manifold matrix as A(θ) = [a(θ1),..., a(θ N )] ∈ C M×N ;

[0024] A4: Introduce the off-grid model. Assume that the true arrival angle of the signal source does not exactly coincide with any grid, that is, assume there is an off-grid error parameter According to the off-grid model, the first-order Taylor expansion of the steering vector is:

[0025]

[0026] Among them, the off-grid error parameter β n is defined as vector represents the error vector, where represents the derivative of the steering vector with respect to the angle of arrival ; the subscript n k represents the grid closest to the true angle of arrival ; represents grid n k corresponding angle of arrival;

[0027] A5: According to the first-order Taylor expansion, rewrite the received signal vector at the t-th snapshot as:

[0028] y t =(A(θ)+E(θ)diag(β))s t +w t ;

[0029] where the matrix E(θ)=[e(θ1),…,e(θ N )] represents the error matrix, the vector β=[β1,…β N T represents the off-grid error parameter vector, s t ∈C N×1 represents the sparse incoming wave signal vector to be solved with length N; when there are L snapshots in the DOA estimation model,

[0030] A6: When there are L snapshots in the DOA estimation model, write the received signal vector at the t-th snapshot in step A5 in the form of multiple observations:

[0031] Y=(A(θ)+E(θ)diag(β))S+W;

[0032] where the matrices Y=[y 1 ,y 2 ,…,y L , S=[s 1 ,s 2 ,…,s L and W=[w 1 ,w 2 ,…,w L represent the set matrices of the received signal vector, the incoming wave signal vector, and the Gaussian white noise vector respectively. The matrix S has the same sparse characteristic, that is, the positions of non-zero elements in each column are the same, and the angle space θ of the wave signal is a fixed value;

[0033] ​Abbreviate the array manifold matrix \(A(\theta)\) and the error matrix \(E(\theta)\) as \(A\) and \(E\) respectively, and abbreviate the multiple observation form in step A6 as

[0034] \(Y=(A + E\mathrm{diag}(\beta))S+W\); (1)

[0035] A8: Since only the outlier error parameter vector \(\beta\) in the expression \((A + E\mathrm{diag}(\beta))\) in equation (1) is unknown, redefine it as

[0036] \(\varPhi\{\beta\}=A + E\mathrm{diag}(\beta)=A+\sum_{i = 1}^{N}\beta_{i}e_{i}\); (2) n \(\beta\) n \(E\) n ; (2)

[0037] where \(\beta\) n represents the outlier error parameter, and \(E\) n is defined as \(E\) n =\([0,\cdots,e(\theta_{i}),\cdots,0]\in\mathbb{C}^{M\times1}\); (2) n ),\cdots,0]\in\mathbb{C}^{M\times1}\); (2) M×N Transform the DOA estimation model in equation (1) into a bilinear form

[0038] \(Y=(A+\sum_{i = 1}^{N}\beta_{i}e_{i})S + W=\varPhi\{\beta\}S+W\) (3) n \(\beta\) n \(E\) n )S + W=\varPhi\{\beta\}S+W\) (3)

[0039] where \(S=[s_{1},\cdots,s_{L}]\) represents the \(L\) sparse incoming wave signal vectors to be solved; only when 1 ,\cdots,s L represents the \(L\) sparse incoming wave signal vectors to be solved; only when \(s_{i}\neq0\), \(\beta_{i}\) n takes non - zero values, and the vectors \(\beta\) and \(S\) have exactly the same sparse characteristics;

[0040] A9: According to the off - grid DOA bilinear system model shown in formula (3), using the constraint relationship between data, factorize the joint distribution function of all known and unknown variables in the system using Bayes' formula

[0041]

[0042] The steps of B are as follows:

[0043] B1: Initialize the outlier error parameter vector \(\beta\) and the intermediate variable as an all - zero vector of length \(N\), initialize the intermediate variable as 1, initialize the noise precision as \(\lambda = 1\), and initialize the hyper - prior parameter vector \(\gamma\) as an all - 1 vector of length \(N\); then go to step B2;

[0044] B2: Calculate the sparse wave signal vector The posterior probability is in Gaussian form where the mean and the variance are calculated respectively as

[0045]

[0046] Then go to step B3;

[0047] In the formula, the operation symbol <·> represents taking the average of a vector, I represents a diagonal matrix, and y represents the received signal vector;

[0048] B3: Using the mean and the variance obtained in step B2, as well as the intermediate variables and initialized in step B1, calculate the intermediate variables and which are respectively

[0049]

[0050] Then go to step B4;

[0051] B4: Using the hyper-prior parameter γ initialized in step B1 and the intermediate variables and obtained in step B3, calculate the posterior distribution of the intermediate variable to be in Gaussian form where the mean and the variance are calculated respectively as

[0052]

[0053] Then go to step B5;

[0054] B5: Using the mean and the variance of the intermediate variable obtained in step B4, calculate the hyper-prior parameter γ as

[0055]

[0056] Then go to step B6; where L represents the number of snapshots in DOA estimation;

[0057] B6: Using the mean and the variance of the intermediate variable obtained in step B4, as well as the intermediate variables and obtained in step B3, calculate the intermediate variables and

[0058]

[0059] Then enter step B7;

[0060] B7: Determine whether the set number of iterations is reached. If so, end the iteration; if not, return to step B2; finally, determine the sparse incident wave signal vector

[0061] Steps B2 to B6 described above are the iteration process, and the number of iterations is set to 15 times.

[0062] Step C described above includes the following specific steps:

[0063] C1: Initialize the off-grid error parameter β as a vector of all 0s with length N, substitute β into formula (2) to construct the matrix Φ{β}; then enter step C2;

[0064] C2: Use the variance obtained in step B2 For t = 1:L, calculate the intermediate variable matrix D as

[0065]

[0066] Then enter step C3;

[0067] C3: Use the sparse incident wave signal vector obtained in step B7 and E defined in formula (2) n , for n = 1, 2,..., N, respectively construct the matrix H ∈ C N×N and the vector α ∈ C N×1 as

[0068] α = (α1…α i …α N ) T ;

[0069] Among them, the elements in the matrix and vector H ij , i, j = 1, 2,..., N, α i , i = 1, 2,..., N,

[0070]

[0071]

[0072] i, j represent the element labels of the matrix H and the vector α, and the matrix E i , E j is the same as the matrix E in formula (2) nHave the same definition; then proceed to step C4;

[0073] C4: Obtain the estimated value of the out-of-grid error parameter vector β, β = H -1 α; then proceed to step C5;

[0074] C5: Determine whether the set number of iterations has been reached. If so, end the iteration; if not, return to step C2; finally, obtain the out-of-grid error parameter vector β.

[0075] As described above, steps C2 to C4 are the iterative process, and the number of iterations is set to 5 times.

[0076] The present invention converts the DOA estimation model into a bilinear problem and uses the bilinear VAMP algorithm to solve this problem. The present invention first converts the out-of-grid DOA estimation model into a bilinear problem and uses the bilinear VAMP algorithm to deduce and operate on the bilinear model; then establishes a DOA estimation simulation model to compare the proposed algorithm with the existing methods in the literature. The results show that the algorithm proposed by the present invention has obvious performance advantages in scenarios with fewer snapshots such as airborne radars. Brief Description of the Drawings

[0077] Figure 1 Is the flow schematic diagram of the present invention;

[0078] Figure 2 Is the comparison diagram of the estimated value and the true value in one Monte Carlo simulation of the present invention;

[0079] Figure 3 Is the change curve diagram of the present invention and the normalized mean square error with the number of snapshots;

[0080] Figure 4 Is the change curve diagram of the estimation performance with the signal-to-noise ratio when the number of snapshots L is 2 for the present invention and the existing algorithms;

[0081] Figure 5 Is the change curve diagram of the estimation performance with the signal-to-noise ratio when the number of snapshots L is 4 for the present invention and the existing algorithms;

[0082] Figure 6 Is the change curve diagram of the estimation performance with the signal-to-noise ratio when the number of snapshots L is 6 for the present invention and the existing algorithms;

[0083] Figure 7 Is the change curve diagram of the estimation performance with the signal-to-noise ratio when the number of snapshots L is 8 for the present invention and the existing algorithms. Detailed Embodiment

[0084] The following describes the present invention in detail with reference to the drawings and embodiments:

[0085] Such as Figure 1As shown in the figure, the high-precision estimation method for the angle of arrival of the present invention includes the following steps:

[0086] A: Establish an off-grid DOA bilinear system model Y = (A + Σ n β n E n )S + W = Φ{β}S + W, and use Bayes' formula for factorization;

[0087] A1: Suppose there is a uniform linear array composed of M non-directional array elements, and there are K narrowband far-field incoming signals. Then the received vector at the t-th snapshot is expressed as

[0088]

[0089] where, y t ∈C M×1 is the received signal vector of all M array elements at the t-th snapshot. The symbol C M×1 represents a complex-valued vector of dimension M×1, is the array manifold matrix. The symbol C M×K represents a complex-valued matrix of dimension M×K, represents the true incoming signal vector at the t-th snapshot. C K×1 represents a complex-valued vector of dimension K×1, w t represents a Gaussian white noise vector with a mean of 0 and a variance of 1 / λ, where λ represents the noise precision. A2: Decompose the array manifold matrix into vector form:

[0090]

[0091] where, the vector represents the steering vector of the k-th incoming signal and can be decomposed as j represents the imaginary unit, represents the true angle of arrival of the k-th incoming signal, k = 1,…,K; Since the goal of DOA estimation is to use the received vector y t to estimate the angles of arrival of all K incoming signals and amplitudes, therefore, in the present invention, the angle space θ of the incoming signals is divided into N grids, expressed in vector form as θ = [θ1,…,θ N , and the size of each grid is Δ θ = π / N. Assume that each angle θ n corresponds to a potential incoming signal source n = 1,…,N; Since the directions of the incoming signals are sparsely distributed in the angle domain, that is, N >> K, a sparse incoming signal vector is thus formed

[0092] A3: Reconstruct the array manifold matrix into A(θ) = [a(θ1),..., a(θ N )] ∈ C M×N ;

[0093] A4: Since the accuracy of DOA needs to be improved, the most direct method is to divide the angular space θ of the incoming wave signal into smaller grids, that is, increase the number of grids N. However, increasing the number of grids N will lead to an increase in complexity and make the correlation of the matrix A(θ) larger. And a matrix A(θ) with high correlation will lead to a deterioration in estimation performance, cause divergence of iterative algorithms such as message passing, and result in estimation failure. To avoid the above problems, an off-grid model is introduced in the present invention, that is, it is assumed that the true arrival angle of the signal source does not exactly coincide with any grid, that is, it is assumed that there is an off-grid error parameter According to the off-grid model, the steering vector is first-order Taylor expanded as:

[0094]

[0095] where the off-grid error parameter β n is defined as The vector represents the error vector, where represents the derivative of the steering vector with respect to the arrival angle ; the subscript n k represents the grid closest to the true arrival angle , represents the grid n k corresponding arrival angle.

[0096] A5: According to the first-order Taylor expansion, the received signal vector at the t-th snapshot is rewritten as:

[0097] y t = (A(θ) + E(θ)diag(β))s t + w t ;

[0098] where the matrix E(θ) = [e(θ1), …, e(θ N )] represents the error matrix, the vector β = [β1, … β N T represents the off-grid error parameter vector, s t ∈ C N×1 represents the sparse incoming wave signal vector of length N to be solved.

[0099] A6: When there are L snapshots in the DOA estimation model, the above formula is written in the form of multiple observations ​

[0100] Y = (A(θ) + E(θ)diag(β))S + W;

[0101] where the matrix Y = [y 1 , y 2 , …, y L , S = [s 1 , s 2 , …, s L , and W = [w 1 , w 2 , …, w L represent the set matrices of the received signal vector, the incident wave signal vector, and the Gaussian white noise vector, respectively. And the matrix S has the same sparse feature, that is, the positions of non-zero elements in each column are the same. The angular space θ of the incident wave signal in the above formula is a fixed value.

[0102] A7: Denote the array manifold matrix A(θ) and the error matrix E(θ) as A and E respectively, then the DOA estimation model is denoted as

[0103] Y = (A + Ediag(β))S + W; (1)

[0104] It can be observed that the unknown parameters in the above DOA estimation model are S and β. In the field of signal processing, the above DOA estimation model belongs to a bilinear model that simultaneously estimates two sets of independent parameters using a set of observations.

[0105] A8: Since only the outlier error parameter vector β is unknown in the expression (A + Ediag(β)) in formula (1), it can be redefined as

[0106] Φ{β} = A + Ediag(β) = A + ∑ n β n E n ; (2)

[0107] where β n represents the outlier error parameter, and E n is defined as E n = [0, …, e(θ n ), …, 0] ∈ C M×N , then the DOA estimation model in formula (1) is transformed into a bilinear form

[0108] Y = (A + Σ n β n E n )S + W = Φ{β}S + W (3)

[0109] where S = [s 1 ,..., s Ldenote L sparse incident wave signal vectors to be solved. According to the physical meaning of vector β in Equation (1), it can be known that only when β n takes non-zero values, so it can be determined that vector β and matrix S have exactly the same sparse characteristics, that is, the positions of non-zero elements are the same.

[0110] A9: According to the off-grid DOA bilinear system model shown in formula (3), using the constraint relationship between data, factorize the joint distribution function of all known and unknown variables in the system by using Bayes' formula.

[0111]

[0112] In the above formula, the sparse incident wave signal vector s t is defined as two identical equivalent vectors and All three are exactly equal, that is All represent the sparse incident wave signal vector, and the mathematical expression is where δ(·) is the delta function; represents the likelihood function of the received signal vector y t at the t-th snapshot. Since the noise follows a complex Gaussian distribution with variance 1 / λ, the likelihood function can be expressed in complex Gaussian form where represents a complex Gaussian distribution with mean and covariance matrix λ -1 I, where I represents a diagonal matrix; the probability distribution represents the prior distribution of vector , and γ represents the hyper-prior parameter vector. Since the present invention assumes that all incident wave signal vectors t = 1:L have common sparse characteristics, then share the same hyper-prior parameter vector γ. The off-grid error parameter vector β follows a uniform distribution p(β) = U[-Δ θ / 2, Δ θ / 2], and the prior distribution of the noise variance is assumed to be p(λ) = 1 / λ. According to the factorization of the bilinear model shown in formula (4), the bilinear approximation message passing and expectation maximization algorithms can be used to derive the algorithm for the bilinear model. The specific steps are divided into two parts: estimating the sparse incident wave signal S and the off-grid error parameter β.

[0113] B: Estimate the sparse incident wave signal S;

[0114] B1: Assume that the off-grid error parameter vector β is known, then the matrix Φ{β} is also known, making the bilinear model shown in formula (3) degenerate into a single linear model. When solving the single linear problem, apply the approximate message passing algorithm to the sparse incident wave signal vector Perform an estimation, and the specific estimation process is as follows:

[0115] Initialize the outlier error parameter vector β and the intermediate variables as a vector of all zeros with length N, and the intermediate variable is initialized to 1, the noise precision is initialized to λ = 1, and the hyperprior parameter vector γ is initialized to a vector of all 1s with length N; then proceed to step B2;

[0116] B2: Calculate the posterior probability of the sparse wave signal vector to be in Gaussian form where the mean and the variance are calculated respectively as

[0117]

[0118] Then proceed to step B3;

[0119] In the formula, the operation symbol <·> represents taking the average of a vector, I represents a diagonal matrix, and y represents the received signal vector. Using the average of the variance vector in the present invention can improve the robustness of the algorithm.

[0120] B3: Utilize the mean and the variance obtained in step B2, as well as the intermediate variables and initialized in step B1 to calculate the intermediate variables and which are respectively

[0121]

[0122] Then proceed to step B4;

[0123] B4: Utilize the hyperprior parameter γ initialized in step B1 and the intermediate variables and obtained in step B3 to calculate the posterior distribution of the intermediate variable to be in Gaussian form where the mean and the variance are calculated respectively as

[0124]

[0125] Then proceed to step B5;

[0126] B5: Utilize the mean and the variance of the intermediate variable obtained in step B4 to calculate the hyperprior parameter γ as

[0127]

[0128] Then enter step B6; where L represents the number of snapshots in DOA estimation.

[0129] B6: Use the intermediate variable mean and variance obtained in step B4, and the intermediate variable and obtained in step B3 to calculate the intermediate variables and

[0130]

[0131] Then enter step B7;

[0132] B7: Determine whether the set number of iterations is reached. If so, end the iteration; if not, return to step B2; finally determine the sparse incident wave signal vector

[0133] In this embodiment, steps B2 to B6 are the iterative process, and the number of iterations is set to 15 times.

[0134] C: Estimate the off-grid error parameter vector β;

[0135] The present invention estimates the off-grid error parameter vector β using the expectation maximization method, and the calculation steps are as follows:

[0136] C1: Initialize the off-grid error parameter β as a vector of all 0s with length N, substitute β into formula (2) to construct the matrix Φ{β}; then enter step C2;

[0137] C2: Use the variance obtained in step B2 to calculate the intermediate variable matrix D for t = 1:L as

[0138]

[0139] Then enter step C3;

[0140] C3: Use the sparse incident wave signal vector obtained in step B7 and E n defined in formula (2), n = 1, 2,..., N to construct the matrix H ∈ C N×N and the vector α ∈ C N×1 as

[0141] α = (α1…α i …α N ) T;

[0142] Among them, the elements H in the matrix and the vector ij , i, j = 1, 2,..., N, α i , i = 1, 2,..., N,

[0143]

[0144]

[0145] where i, j represent the element labels of the matrix H and the vector α. Here, the matrix E i , E j has the same definition as the matrix E in formula (2); then go to step C4; n ; then go to step C4;

[0146] C4: Obtain the estimated value of the out-of-grid error parameter vector β, β = H -1 α; then go to step C5;

[0147] C5: Determine whether the set number of iterations has been reached. If so, end the iteration; if not, return to step C2; finally, obtain the out-of-grid error parameter vector β.

[0148] In this embodiment, steps C2 to C4 are iterative processes, and the number of iterations is set to 5 times.

[0149] D: Utilize the sparse incoming wave signal vector S obtained in step B and the out-of-grid error parameter β obtained in step C, and combine with the DOA estimation model Y = (A + ∑ n β n E n )S + W = Φ{β}S + W established in step A to finally obtain a high-precision estimated value of the angle of arrival.

[0150] In terms of complexity, the present invention can be divided into two steps, B and C. Among them, step B2 requires matrix multiplication Φ T {β}Φ{β}, with a complexity of The remaining operations in step B are all in scalar form, with a complexity of where the symbol represents the order of complexity. Step C requires matrix inversion, with a complexity of Therefore, the overall complexity of the present invention can be written as

[0151] To illustrate the performance of the present invention, a simulation environment can be established for numerical simulation and compared with existing algorithms at home and abroad. The simulation environment selects a uniform linear array, the number of array elements is M = 30, then the number of virtual grids is N = 30, that is, the grid width is 6°, the number of snapshots is set to L = 2 - 12, and the threshold γ is setmax = 10 3 To measure the effectiveness of the present invention, in the simulation, the existing grid-based sparse Bayesian learning (OGSBL), root-finding SBL (RTSBL), and Gauss-Siedel root (GSROOT) are selected for comparison. In addition, the Cramer-Rao bound (CRB) is selected as a reference.

[0152] Figure 2 It is a comparison graph of the estimated value Estimated and the true value True Angle in a single Monte Carlo simulation. In the simulation, the true angles are selected as [-28.6, -18.6, 3.5, 15.6, 31.7]. Figure 2 In it, the abscissa is the angle and the ordinate is the absolute value of the estimated vector s2. In this simulation, the signal-to-noise ratio SNR = 30 dB is set. From Figure 2 it can be seen that the present invention can accurately capture the off-grid angles and accurate powers of 5 incoming wave signals, and the angle estimation results approach 0 at the angles without incoming wave signals.

[0153] Figure 3 It is a curve graph of the normalized mean square error (NMSE) of the present invention and existing algorithms varying with the number of snapshots. The numerical simulation results are the average values of 500 Monte Carlo simulations. In each simulation, the direction of arrival is based on [-30, -18, 6, 18, 30], and a uniform random angle offset of (-3, 3) is superimposed. Figure 3 In it, the signal-to-noise ratio SNR = 20 dB is set, and the number of incoming wave signals is K = 5. From the simulation results, it can be seen that when the number of snapshots is large, the performance of the present invention, i.e., Bad-VAMP in the figure, is relatively close to that of the existing RTSBL and GSROOT. However, as the number of snapshots decreases, the performance of the RTSBL and GSROOT methods deteriorates significantly, making the present invention show a large performance gain. From Figure 3 it can also be seen that due to not considering the grid deviation, the performance of the OGSBL algorithm is poor, thus proving the necessity of the off-grid estimation algorithm.

[0154] Figures 4 to 7 They are respectively curve graphs of the estimation performance varying with the signal-to-noise ratio when the number of snapshots L is 2, 4, 6, and 8. From Figures 4 to 7 it can be obtained the same conclusion as Figure 3 that is, the present invention has obvious performance advantages when the number of snapshots is small. To achieve the same estimation performance, the present invention requires fewer snapshots and is more suitable for fast time-varying scenarios such as airborne radars.

Claims

1. A high-precision estimation method for the angle of arrival, characterized in that: The following steps are included in sequence: A: First, establish the off-grid DOA bilinear system model Y = (A + ∑ n β n E n )S + W = Φ{β}S + W; where, Φ{β} = A + Ediag(β) = A + ∑ n β n E n ,E n = [0,…,e(θ n ),…,0] ∈ C M×N ,Φ{β} represents a matrix with parameter β, and C M×N represents a complex-valued matrix of dimension M×N. The off-grid DOA bilinear system model is a shorthand for the received signal vector Y = (A(θ) + E(θ)diag(β))S + W; Y = (A(θ) + E(θ)diag(β))S + W is the multiple observation form of y t = (A(θ) + E(θ)diag(β))s t + w t ; Suppose there is a uniform linear array composed of M non-directional array elements, and there are K narrowband far-field incoming wave signals. y t represents the received signal vector at the t-th snapshot. A(θ) = [a(θ1),...,a(θ N )] ∈ C M×N ; Suppose the angular space θ of the incoming wave signals is divided into N grids, expressed in vector form as θ = [θ1,…,θ N , and the size of each grid is Δ θ = π / N; The matrix E(θ) = [e(θ1),…,e(θ N )] represents the error matrix, and the vector β = [β1,…β N T represents the off-grid error parameter vector. The off-grid error parameter represents the true arrival angle of the k-th incoming wave signal, represents the arrival angle corresponding to grid n k , n = 1,…,N. The subscript n k represents the grid closest to the true arrival angle . s t ∈ C N×1 represents the sparse incoming wave signal vector of length N to be solved; L is the number of snapshots, and w t represents a Gaussian white noise vector with a mean of 0 and a variance of 1 / λ, where λ represents the noise precision; The matrices Y = [y 1 ,y 2 ,…,y L , S = [s 1 ,s 2 ,…,s L and W = [w 1 ,w 2 ,…,w L ​Denote the set matrices of the received signal vector, the incident wave signal vector, and the Gaussian white noise vector respectively. The matrix S has the same sparse characteristics; A and E are the abbreviations of the array manifold matrix A(θ) and the error matrix E(θ) respectively; θ is the angular space of the incident wave signal. Then the off-grid DOA bilinear system model is factorized using the Bayesian formula; Among them, the sparse incident wave signal vector s t is defined as two identical equivalent vectors and both represent the sparse incident wave signal vector, and the mathematical expression is δ(·) is the delta function; represents the likelihood function of the received signal vector y t at the t-th snapshot. The likelihood function is expressed in the form of complex Gaussian where represents a complex Gaussian distribution with a mean of and a covariance matrix of λ -1 I. I represents the diagonal matrix; the probability distribution represents the prior distribution of the vector . γ represents the hyper-prior parameter vector; all incident wave signal vectors have the common sparse characteristic, share the same hyper-prior parameter vector γ; the off-grid error parameter vector β follows a uniform distribution p(β) = U[-Δ θ / 2, Δ θ / 2], and the prior distribution of the noise variance is assumed to be p(λ) = 1 / λ; B: Estimate the sparse incoming signal S; C: Estimate the off-grid error parameter vector β; D: Based on the obtained sparse incoming wave signal S and the off-grid error parameter vector β, the off-grid DOA bilinear system model obtained in step A is used to perform high-precision calculation of the angle of arrival.

2. The high-precision estimation method of the angle of arrival according to claim 1, wherein: In the step B, the sparse incoming signal S is estimated by applying an approximate message passing algorithm.

3. The high-precision estimation method of the angle of arrival according to claim 1, characterized in that: In the step C, the off-grid error parameter vector β is estimated by using the expectation maximization method.

4. The high-precision estimation method of the angle of arrival according to claim 1, wherein: In step A, the steps for establishing the off-grid DOA bilinear system model are as follows: A1: Assume that there is a uniform linear array composed of M non-directional array elements and there are K narrow-band far-field incoming wave signals. Then the receiving vector at the t-th snapshot is expressed as where y t ∈ C M×1 is the received signal vector of all M array elements at the t-th snapshot, and the symbol C M×1 denotes a complex-valued vector of dimension M×1, is the array manifold matrix, and the symbol C M×K denotes a complex-valued matrix of dimension M×K, denotes the true incident signal vector at the t-th snapshot, C K×1 denotes a complex-valued vector of dimension K×1, w t denotes a Gaussian white noise vector with mean 0 and variance 1 / λ, where λ represents the noise precision; A2: Decompose the array manifold matrix into a vector form Among them, the vector represents the steering vector of the k-th incoming wave signal, and the decomposition is expressed as j represents the imaginary unit, represents the true arrival angle of the k-th incoming wave signal, k = 1, …, K; the angular space θ of the incoming wave signals is divided into N grids, expressed in vector form as θ = [θ1, …, θ N , and the size of each grid is Δ θ = π / N; each angle θ n corresponds to a potential incoming wave signal source The directions of the incoming wave signals are sparsely distributed in the angular domain, that is, N >> K, forming a sparse incoming wave signal vector A3: Reconstruct the array manifold matrix into A(θ) = [a(θ1),..., a(θ N )] ∈ C M×N ; A4: Introduce an off-grid model, assuming that the true angle of arrival of the signal source does not exactly coincide with any grid, that is, assuming the existence of an off-grid error parameter According to the off-grid model, the steering vector is first-order Taylor expanded as: Among them, the off-grid error parameter β n is defined as vector denotes the error vector, where denotes the derivative of the steering vector with respect to the angle of arrival ; the subscript n k denotes the grid closest to the true angle of arrival ; denotes the grid n k and the corresponding angle of arrival; A5: According to the first-order Taylor expansion, the received signal vector at the t-th snapshot is rewritten as: y t = (A(θ) + E(θ) diag(β)) s t + w t ; Among them, the matrix \(E(\theta)=[e(\theta_1),\ldots,e(\theta N )]\) represents the error matrix, and the vector \(\beta = [\beta_1,\ldots,\beta N \) T represents the out-of-grid error parameter vector, and \(s t \in\mathbb{C} N×1 represents the sparse incident wave signal vector of length \(N\) to be solved; A6: When there are L snapshots in the DOA estimation model, the received signal vector at the t-th snapshot in step A5 is written as a multiple observation form: Y=(A(θ)+E(θ)diag(β))S+W; Among them, the matrix Y = [y 1 , y 2 , …, y L , S = [s 1 , s 2 , …, s L , and W = [w 1 , w 2 , …, w L respectively represent the set matrices of the received signal vector, the incident wave signal vector, and the Gaussian white noise vector. The matrix S has the same sparse characteristics, that is, the positions of the non-zero elements in each column are the same, and the angular space θ of the wave signal is a fixed value; A7: The array flow matrix A(θ) and the error matrix E(θ) are abbreviated as A and E respectively, and the multiple observation form in step A6 is abbreviated as Y=(A+Ediag(β))S+W;(1) A8: Since only the off-grid error parameter vector β is unknown in the expression (A+Ediag(β)) in formula (1), it can be redefined as Φ{β} = A + Ediag(β) = A + ∑ n β n E n ; (2) where β n represents the off-grid error parameter, and E n is defined as En = [0, …, e(θn), …, 0] ∈ C M×N , and the DOA estimation model in equation (1) is transformed into a bilinear form Y = (A + ∑ n β n E n )S + W = Φ{β}S + W (3) where \(S = [s 1 ,\cdots,s L \) represents \(L\) sparse incoming wave signal vectors to be solved; only when , \(\beta n takes non-zero values, and the vector \(\beta\) and the matrix \(S\) have exactly the same sparse characteristics; A9: According to the off-grid DOA bilinear system model shown in formula (3), the constraint relationship between data is used to factorize the joint distribution function of all known and unknown variables in the system using the Bayesian formula.

5. The high-precision estimation method of the angle of arrival according to claim 1, wherein: The step B comprises the following specific steps: B1: Initialize the outlier error parameter vector β and intermediate variables as a vector of all zeros with length N, and initialize the intermediate variable to 1, initialize the noise precision to λ = 1, and initialize the hyper-prior parameter vector γ as a vector of all ones with length N; then proceed to step B2; B2: Calculate the posterior probability of the sparse wave signal vector to be Gaussian where the mean and the variance are calculated respectively as Then proceed to step B3; In the formula, the operator <·> represents averaging the vector, I represents the diagonal matrix, and y represents the received signal vector; B3: Use the mean obtained in step B2 and variance as well as the intermediate variables initialized in step B1 and Calculate the intermediate variables and respectively as Then proceed to step B4; B4: Using the hyper - prior parameter γ initialized in step B1 and the intermediate variable obtained in step B3 and calculate that the posterior distribution of the intermediate variable is in Gaussian form where the mean and the variance are calculated respectively as Then proceed to step B5; B5: Use the intermediate variable obtained in step B4 to calculate the mean value and variance and calculate the hyper-prior parameter γ as Then proceed to step B6; where L represents the number of snapshots in DOA estimation; B6: Use the intermediate variable obtained in step B4 mean value and variance and the intermediate variable obtained in step B3 and calculate the intermediate variables and Then proceed to step B7; B7: Determine whether the set number of iterations is reached. If so, end the iteration; if not, return to step B2; finally, determine the sparse incident wave signal vector 6. The high-precision estimation method of the angle of arrival according to claim 5, characterized in that: The steps B2 to B6 are an iterative process, and the number of iterations is set to 15 times.

7. The high-precision estimation method of the angle of arrival according to claim 5, characterized in that: The step C comprises the following specific steps: C1: Initialize the off-grid error parameter β to a vector of all zeros with a length of N, substitute β into formula (2) to construct the matrix Φ{β}; then proceed to step C2; C2: Use the variance obtained in step B2 t = 1:L Calculate the intermediate variable matrix D as Then proceed to step C3; C3: Utilize the sparse incoming wave signal vector obtained in step B7 and E defined in formula (2) n , n = 1, 2,..., N to construct matrix H ∈ C N×N and vector α ∈ C N×1 respectively as Among them, the elements H in the matrix and the vector if , i, f = 1, 2,..., N, α i , i = 1, 2,..., N, i and f represent the element labels of matrix H and vector α, matrix E i , E f is the same as matrix E in formula (2) n and has the same definition; then proceed to step C4; C4: Obtain the estimated value of the off-grid error parameter vector β, β = H -1 α; then proceed to step C5; C5: Determine whether the set number of iterations has been reached. If so, end the iteration; if not, return to step C2; and finally obtain the off-grid error parameter vector β.

8. The high-precision estimation method of the angle of arrival according to claim 7, characterized in that: The steps C2 to C4 are an iterative process, and the number of iterations is set to 5 times.

Images