A meshless block sparse angle estimation method based on element mutual coupling optimization

Abstract

The application provides a non-grid block sparse angle estimation method based on element mutual coupling optimization, and is characterized by the following steps: step one, establishing an array receiving signal model under unknown element mutual coupling interference; step two, constructing a block sparse representation model under unknown element mutual coupling interference; step three, establishing a sparse reconstruction model based on mixed kernel-l1 norm minimization under block sparsity; step four, deducing an equivalent convex optimization reconstruction structure based on semi-definiteness; step five, deducing a non-grid Lagrange dual solution form; and step six, solving by using a convex optimization algorithm to realize high-precision angle estimation. The non-grid block sparse reconstruction method of the application effectively suppresses the discrete grid error without calibrating the mutual coupling coefficient, and improves the angle estimation precision under unknown element mutual coupling.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of array signal processing, and specifically designs a gridless block sparse angle estimation method based on element mutual coupling optimization. BACKGROUND

[0002] Array signal processing utilizes multiple antenna elements to realize multi-point parallel sampling processing of space signal field, and obtains additional spatial degrees of freedom by virtue of the spatial characteristics of the array. Unlike general signal processing methods, array signal processing can effectively utilize spatial degrees of freedom for flexible beam control, and obtain higher signal gain and stronger signal anti-interference capability. After years of development, array signal processing technology has been successfully applied to the fields of astronomy observation, radar, sonar, communication system and biomedical engineering. Target angle direction estimation is one of the important tasks of array signal processing. After decades of development, angle estimation algorithms have developed multiple signal classification (MUSIC) algorithm, Capon algorithm, etc. In addition, influenced by the booming compressed sensing technology in the past decade, a batch of array angle estimation algorithms based on sparse reconstruction have emerged by utilizing the characteristics of target signals in space.

[0003] The premise of realizing high-precision angle estimation by the sparse reconstruction type angle estimation algorithm is to utilize the accurately known array flow pattern information to construct a high-precision sparse representation dictionary, which requires that the structure and response of the array system are accurately known. When the array system has unknown element mutual coupling interference, the sparse representation dictionary constructed on the basis of the array flow pattern cannot effectively represent the spatial target angle distribution, and therefore the angle estimation performance under sparse reconstruction will be poor. On the other hand, the discrete grid mismatch problem is one of the inherent problems of the sparse reconstruction type algorithm. The premise of realizing effective angle estimation under sparse reconstruction is that the real target angle is contained in the pre-divided spatial discrete grid points, and the densely divided discrete dictionary grid is helpful to improve the accuracy of sparse representation, but too dense grid points will cause too large correlation between sparse dictionary elements, which is not conducive to accurate sparse reconstruction, and will also lead to high dimension of sparse reconstruction and increase of algorithm solving complexity. In addition, the observation space of the target angle is continuous, and no matter how dense the discrete grid dictionary is, it cannot ensure that the real angle falls on the pre-divided grid points, resulting in the discrete grid mismatch problem, off-grid error and further decline of angle estimation accuracy. SUMMARY

[0004] To solve the above technical problems, the present application provides a gridless block sparse angle estimation method based on element mutual coupling optimization, characterized by comprising the following steps:

[0005] Step one, establish an array received signal model under unknown element mutual coupling interference;

[0006] Step 2: Construct a block sparse representation model under the mutual coupling interference of unknown array elements;

[0007] Step 3: Establish a sparse reconstruction model based on hybrid kernel-l1 norm minimization under block sparsity;

[0008] Step 4: Derive the equivalent semidefinite convex optimization reconstruction structure;

[0009] Step 5: Derive the meshless Lagrange dual solution form;

[0010] Step 6: Solve using convex optimization algorithms to achieve high-precision angle estimation.

[0011] Furthermore, step one includes: assuming a uniform linear array with N array elements, the antenna element positions are distributed as d = [d0, d1, ..., d2]. N-1 ] T It is not a general assumption that the first antenna is d0 = 0, then d n = (n-1)d, where d is set as half wavelength. Considering the mutual coupling interference between the first P antenna elements in this uniform linear array, the element mutual coupling interference matrix is ​​introduced as follows:

[0012]

[0013] in Let ρ be the mutual coupling coefficient on the i-th antenna element. i and These represent the amplitude and phase of the mutual coupling coefficient, respectively. If K far-field, uncorrelated targets illuminate the linear array, assuming the incoming wave directions are θ = [θ1, θ2, ..., θ...] K ] T , where θ k Let the direction of arrival of the k-th target be denoted as . Then, the baseband received signal of the target with element-to-element mutual coupling interference in the t-th snapshot can be expressed as:

[0014] in Let λ be the steering vector corresponding to the k-th target, λ represent the wavelength of the carrier wave, C be the mutual coupling matrix defined by equation (1), and A = [a(θ1),...,a(θ2)]. K [)] represents the array manifold matrix, s(t) represents the signal amplitude matrix, and n(t) is additive white Gaussian noise with a mean of 0 and a variance of .

[0015] Furthermore, step two includes: setting the target θ k The corresponding guide vector is simplified as follows:

[0016] a(θk )=[1,β(θ k ),…,β(θ k ) N-1 ] T (3)

[0017] in make The guiding vector representing the mutual coupling of array elements has the following parameterized structure:

[0018]

[0019] In the formula Ψ(θ) k )∈C N×(2P-1) It is a block diagonal matrix with the following structure:

[0020]

[0021] z(θ k c) is a (2P-1)×1 dimensional vector whose P-th element is 1, denoted as .

[0022] z(θ k c) = [μ1,…,μ P-1 ,1,α1,…,α P-1 ] T (6)

[0023] Where element μ i and α i The definition is as follows:

[0024]

[0025]

[0026] The constant term is denoted as:

[0027]

[0028] Therefore, the array manifold under mutual coupling of array elements has the following block matrix representation:

[0029]

[0030] in:

[0031]

[0032] Λ is a block diagonal matrix, denoted as:

[0033]

[0034] In the formula Given a (2P-1)×1 dimensional vector, the array receiving signal model under mutual coupling of array elements can be simplified as follows:

[0035] X = BΛS + N (13)

[0036] Discretize the continuous space to construct discrete grid angles, where S = [s(1) s(2) … s(T)], T is the number of snapshots, and Θ = {θ1, θ2, …, θ L}, θ l Let L be the angle value of the discrete grid, L be the number of discrete grid points, and L >> K. Substituting this discrete grid set into matrix BΛ, we obtain the block sparse representation model of the array received signal under mutual coupling interference:

[0037]

[0038] in Dictionary of block sparse representations:

[0039]

[0040] remember It is a block sparse matrix containing L sub-blocks Q l It has the following sparse form:

[0041]

[0042] Therefore, the sparse representation model of the array block under the mutual coupling of unknown array elements is as follows:

[0043] X = BQ + N (17).

[0044] Furthermore, step three includes: considering the block sparse matrix Q, there exists a matrix stacking form as follows:

[0045] Q = [Q1, Q2, ..., Q L ] T (18)

[0046] Therefore, submatrix Q l Let Q be a matrix with rank less than or equal to 1, where the submatrix is ​​0 when the rank is 0. Therefore, the block sparse reconstruction model for Q is as follows:

[0047]

[0048] Among them ||·|| F Let F denote the matrix norm, and ζ > 0 denote the regularization parameter. After convex relaxation, the matrix kernel norm minimization problem is the best convex relaxation of the matrix rank minimization problem. Therefore, equation (19) can be relaxed to the form of matrix kernel norm summation, i.e., the kernel-l1 norm problem:

[0049]

[0050] where || · || * denotes the nuclear norm of a matrix.

[0051] Further, the step four comprises: considering the l-th sub-block Q l , there exists the following singular value decomposition:

[0052] Q l = U l Σ l V l H (21)

[0053] where Σ l is a diagonal matrix consisting of singular values, U l and V l denote left and right singular value matrices respectively. Let P l = U l Π l, 1W l H , G l = W l Π l,2 V l H where Σ l = Π l,1 Π l,2 , W l is an r x r dimensional unitary matrix, i.e. W l H W l = I, then the matrix Q l can be written as:

[0054] Q l = U l Π l,1 W l H W l Π l,2 V l H = P l G l (22)

[0055] According to the definition of the nuclear norm, we have:

[0056] ||Q l || * = ||Σ l || * = ||Π l,1 Π l,2 || * ≤ ||Πl,1 || F ||Π l,2 || F (23)

[0057] =||P l || F ||G l || F

[0058] By Cauchy inequality theorem, the equality holds if and only if , and by the arithmetic mean-geometric mean inequality, we have

[0059]

[0060] Thus, the core norm of matrix Q l is equivalent to the solution of the following minimization problem:

[0061]

[0062] Based on this, the mixed core-l1 norm of block matrix Q can be rewritten as:

[0063]

[0064] where P = blkdiag(P1, …, P L ) is an MLxrL dimensional block diagonal matrix, G = [G1, …, G L ] T is an rLxT dimensional matrix stacked by G l . Thus, based on the core-l1 norm minimization problem, we can rewrite it as:

[0065]

[0066] denotes a set of MLxrL dimensional block diagonal matrices, and the atoms in the set are composed of L Mxr dimensional sub-matrices as the main diagonal. Fixing matrix P in the above equation and taking the partial derivative equal to zero, we can get the optimal solution expression of matrix G as:

[0067]

[0068] Substituting equation (28) into equation (27), we can simplify it to:

[0069]

[0070] Define the block diagonal matrix Γ = PP H , we know that Γ is an MLxML dimensional matrix and has a semi-definite structure, so we have denotes the set of positive semi-definite block-diagonal matrices with elements of dimension ML x ML and diagonal blocks of dimension M x M. Since λ > 0, the coefficients can be dropped without affecting the optimal result of the model, so the above equation can be equivalent to:

[0071]

[0072] The above equation has the following equivalent semi-definite optimization structure:

[0073]

[0074] where Z is an N x N Hermitian matrix.

[0075] Further, the step five includes: first considering the Lagrange dual model corresponding to equation (31) as follows:

[0076]

[0077] where Y0and Y1are dual operator matrices, Y0is an N x N semi-definite matrix, and Y1is an N x T general matrix. T and I M denote the T-dimensional and M-dimensional identity matrices, respectively, and assume that θ is an arbitrary angle in continuous space, and let β(θ) = e jπsinθ Consider the dictionary matrix B(θ) = Ψ(θ) at angle θ, and construct the extended steering vector as follows:

[0078]

[0079] Obviously, there is an N x NM selection matrix J that satisfies B(θ) = JΩ(θ), and let M(θ) = B H (θ)Υ0B(θ), then:

[0080] M(θ) = Ω H (θ)J H Υ0JΩ(θ) (34)

[0081] On the other hand, for the identity matrix I M there is the following equivalent structure:

[0082] I M = Ω H (θ)HΩ(θ) (35)

[0083] where H is an NM x NM matrix and satisfies:

[0084]

[0085] where blktr M{·} represents the trace operation of the block matrix. According to formula (34) and (35), the constraint term can be changed into:

[0086]

[0087] Therefore, the block sparse reconstruction model can be rewritten as the following meshless optimization model:

[0088]

[0089] Further, the step six comprises: solving formula (38) by using a convex optimization tool package CVX to obtain the optimal solution of the meshless optimization model The meshless high-precision angle estimation is obtained by solving the root of a polynomial:

[0090]

[0091] The meshless high-precision angle estimation is obtained by solving the root of a polynomial:

[0092] The present application utilizes the meshless block sparse reconstruction method to effectively suppress the discrete grid error without calibrating the mutual coupling coefficient, and improves the angle estimation precision under the condition of unknown array mutual coupling. BRIEF DESCRIPTION OF DRAWINGS

[0093] Figure 1 is a simplified array mutual coupling model schematic diagram of the present application;

[0094] Figure 2 is a whole structure framework diagram of the present application;

[0095] Figure 3 is an angle estimation performance comparison diagram of the present application;

[0096] Figure 4 is a comparison diagram of the root mean square error changing with the signal-to-noise ratio of the present application;

[0097] Figure 5 is a comparison diagram of the root mean square error changing with the number of shots of the present application;

[0098] Figure 6 is a comparison diagram of the root mean square error changing with different target angle intervals of the present application. DETAILED DESCRIPTION

[0099] The method provided by the present application is described in more detail below in combination with the drawings:

[0100] The application is a kind of uniform linear array under unknown array mutual coupling interference based on block sparse meshless angle estimation method. The application is aimed at the array sparse reconstruction problem under unknown array mutual coupling interference, and proposes a kind of block sparse meshless angle estimation method based on block sparse. The method realizes the embedding operation of mutual coupling coefficients by performing necessary parameterization transformation on the target steering vector under array mutual coupling, and then realizes the block sparse representation form without the influence of array mutual coupling. Then, the equivalent semi-definite sparse reconstruction model of the kernel-norm minimization is given, and the meshless block sparse angle estimation is derived by means of the Lagrange dual model. The method realizes high-precision meshless angle estimation without array mutual coupling calibration by means of block sparse reconstruction.

[0101] The target parameter estimation of the application mainly includes the following aspects:

[0102] 1. The target steering vector under array mutual coupling is parameterized and transformed, the block structure array flow matrix under the embedding of mutual coupling coefficients is constructed, and the complete dictionary of block sparse is divided based on the block array flow matrix

[0103] Figure 1 is a simplified array mutual coupling model schematic diagram of the application. As shown in Figure 1 The model used in the application is a uniform linear array composed of N antenna elements, and the element spacing is half wavelength. In the figure, c p represents the array mutual coupling coefficient, which is related to the distance related to mutual coupling.

[0104] The first parameterization change is performed on the target steering vector under array mutual coupling, and the array flow matrix B of block sparse structure is constructed. Let the target θ k The corresponding steering vector is:

[0105] a(θ k )=[1,β(θ k ),…,β(θ k ) N-1 ] T (40)

[0106] Wherein The steering vector under array mutual coupling After parameterization change, it can be rewritten as:

[0107]

[0108] In the formula, Ψ(θ k )∈C N×(2P-1) is a block diagonal matrix, and the structure is as follows,

[0109]

[0110] z(θ kc) is a (2P-1) x 1 vector with its Pth element being 1, denoted as

[0111] z(θ k ,c) = [μ1,..., μ P-1 ,1, α1,..., α P-1 ] T (43)

[0112] where elements μ i and α i are defined as follows,

[0113]

[0114]

[0115] is a constant term, denoted as:

[0116]

[0117] Thus, the block-sparse array flow matrix under element mutual coupling can be written as:

[0118]

[0119] The second one is to discretize the continuous space and divide it into discrete grid angles, where S = [s(1) s(2)... s(T)], T is the number of snapshots, Θ = {θ1, θ2,..., θ L}, θ l is the value of the divided discrete grid angle, and L is the number of discrete grid points. Thus, the block-sparse overcomplete dictionary is denoted as:

[0120]

[0121] 2. Construct a block-sparse representation model under element mutual coupling using a block-sparse dictionary, and construct a sparse reconstruction model based on a mixed kernel-l1 norm.

[0122] The first one is to construct a block-sparse representation model under element mutual coupling. Using the discrete grid point set Θ, a block-diagonal matrix can be constructed as follows:

[0123]

[0124] where is a (2P-1) x 1 vector. Thus, the array received signal can be denoted as follows:

[0125]

[0126] is a row-sparse matrix, where sl denotes is a vector and satisfies the following condition:

[0127]

[0128] denotes It can be seen that Q is a block-sparse matrix. Thus the block-sparse representation model under the mutual coupling of array elements can be denoted as:

[0129]

[0130] Secondly, a sparse reconstruction model based on the mixed nuclear-l1 norm is constructed using the block-sparse representation model. In order to solve the block-sparse matrix Q, an optimization model based on the mixed rank-l1 norm minimization can be used to solve it as follows:

[0131]

[0132] However, the rank operation of the matrix is an NP-hard problem, and an effective algorithm cannot be found to solve it, so it needs to be convexly relaxed. The nuclear norm of the matrix is the best convex relaxation of the rank of the matrix, so an optimization solving model based on the mixed nuclear-l1 norm minimization can be constructed as follows:

[0133]

[0134] 3. In order to realize fast optimization solving, an equivalent matrix change is used to construct a semi-positive sparse reconstruction model

[0135] Consider the lth sub-block Q l of Q, consider that it has the following singular value decomposition:

[0136] Q l = U l Σ l V l H (55)

[0137] In the formula, Σ l is a diagonal matrix composed of singular values. Let P l = U l Π l,1 W l H , G l = W l Π l,2 V l H , where Σ l = Π l,1 Π l,2 , W l is a r x r dimensional unitary matrix, i.e. W lH W l = I, then the matrix Q l can be written as:

[0138] Q l = U l Π l,1 W l H W l Π l,2 V l H = P l G l (56)

[0139] According to the definition of the nuclear norm, we have:

[0140] ||Q l || * = ||Σ l || * = ||Π l,1 Π l,2 || * ≤ ||Π l,1 || F ||Π l,2 || F (57)

[0141] = ||P l || F ||G l || F

[0142] By the Cauchy-Schwarz inequality theorem, the equality holds if and only if . Moreover, by the arithmetic mean-geometric mean inequality, we have

[0143]

[0144] Thus, the nuclear norm of the matrix Q l is equivalent to the solution of the following minimization problem:

[0145]

[0146] Further, the mixed nuclear-l1 norm computation of the block matrix Q has the following equivalent form:

[0147]

[0148] Here P = blkdiag(P1,..., P L ) is an ML x rL block diagonal matrix, G = [G1,..., G L ] T is a block matrix composed of Gl Stacked rL x T matrix. In which denotes a set of ML x rL block-diagonal matrices, and the atoms in the set are composed of L M x r sub-matrices as the main diagonal. Thus the original optimization problem (54) can be rewritten as:

[0149]

[0150] Taking the partial derivative of the above equation with respect to the fixed matrix P and setting it to zero, the optimal solution of matrix G can be obtained as:

[0151]

[0152] Substituting equation (62) into equation (61) can be simplified as:

[0153]

[0154] Define the block-diagonal matrix Γ = PP H , which is an ML x ML matrix and has a semi-definite structure, so we have denotes a set of semi-definite block-diagonal matrices, and the dimensions of the elements in the set are ML x ML, and the dimensions of the diagonal blocks are M x M. Since λ > 0, the coefficients in the equation can be omitted without affecting the optimal result of the model, so the above equation can be equivalent to:

[0155]

[0156] and it has the following semi-definite optimization structure:

[0157]

[0158] where Z is an N x N Hermitian matrix.

[0159] 4. Constructing a meshless sparse solution structure under the Lagrange dual

[0160] Consider the Lagrange dual model of equation (65):

[0161]

[0162] In which Υ0 and Υ1 are dual operator matrices, Υ0 is an N x N semi-definite matrix, and Υ1 is an N x N ordinary matrix. T and I M denote the T-dimensional and M-dimensional identity matrices, respectively. Construct the extended steering vector as follows:

[0163]

[0164] where denotes the Kronecker product of matrices. The block-structured array flow pattern has the following relationship with the steering vector in equation (67):

[0165] B(θ) = JΩ(θ) (68)

[0166] J is a selection matrix. Then the constraint term M(θ) = B H (θ)Υ0B(θ) can be rewritten as:

[0167] M(θ) = Ω H (θ)J H Υ0JΩ(θ) (69)

[0168] On the other hand, for the unit matrix I M there exists the following equivalent structure:

[0169] I M = Ω H (θ)HΩ(θ) (70)

[0170] where H is an NMxNM matrix and satisfies:

[0171]

[0172] where The matrix T n is a Toeplitz matrix, where the nth diagonal line element parallel to the main diagonal line is 1, and the rest of the elements are zero. When n = 0, it represents the main diagonal line of the matrix. Substituting equations (69) and (70) into the constraint term of equation (66) has

[0173]

[0174] Combined with equations (71) and (72), the meshless sparse reconstruction model under the Lagrange dual can be obtained:

[0175]

[0176] 5. The meshless sparse reconstruction model (73) is solved by using a convex optimization solving algorithm, and then the robust and high-precision target angle estimation is realized

[0177] High-precision angle estimation is realized by solving the roots of the following polynomial:

[0178]

[0179] Therefore, the gridless block sparse DOA estimation method based on mutual coupling optimization provided by the application mainly solves the problem of angle estimation precision decline and angle resolution loss caused by mutual coupling interference in actual array signal processing system. The application firstly performs necessary parameter transformation on the array steering vector with mutual coupling, and constructs an array signal processing model based on block sparse representation on the basis of the parameter transformation, and embeds the mutual coupling parameters into the to-be-measured sparse vector. In order to realize high-precision angle estimation, an equivalent semi-definite optimization structure based on mixed kernel-l1 norm minimization is derived, and then a gridless angle estimation is realized by using a dual optimization method, and high-precision DOA estimation results are obtained without mutual coupling calibration compensation. Compared with the prior art, the application has the following characteristics:

[0180] 1. The application embeds the mutual coupling coefficients into the to-be-measured sparse variable by using the block sparse structure without changing the sparsity of the signal, and high-precision angle estimation can be realized without mutual coupling calibration.

[0181] 2. The application derives a sparse reconstruction model based on mixed kernel-l1 norm minimization, which guarantees the convexity of the solution model and the global optimality of the sparse solution.

[0182] 3. The application derives a Lagrange dual model based on the mixed kernel-l1 norm minimization model, and provides a gridless angle estimation method, which effectively solves the spatial discrete grid mismatch problem in the sparse framework.

[0183] Figure 2 is the overall structural framework of the application. As shown in Figure 2 , the gridless block sparse angle estimation method based on mutual coupling optimization of the application includes the following steps:

[0184] Step 1, establishing an array received signal model under unknown mutual coupling interference

[0185] Suppose that the number of antenna elements of a uniform linear array is N, the antenna element position distribution is d=[d0,d1,...,d N-1 ] T , and the first antenna is not generally assumed to be d0=0, then d n =(n-1)d, where d is set to half a wavelength. Considering that there is mutual coupling interference between the first P antenna elements in the uniform linear array, the mutual coupling interference matrix is introduced as follows:

[0186]

[0187] where is the mutual coupling coefficient on the i-th antenna element, and i and These represent the amplitude and phase of the mutual coupling coefficient, respectively. If K far-field, uncorrelated targets illuminate the linear array, assuming the incoming wave directions are θ = [θ1, θ2, ..., θ...] K ] T , where θ k Let the direction of arrival of the k-th target be denoted as . Then, the baseband received signal of the target with element-to-element mutual coupling interference in the t-th snapshot can be expressed as:

[0188]

[0189] in Let λ be the steering vector corresponding to the k-th target, λ represent the carrier wavelength, C be the mutual coupling matrix defined by equation (1), and A = [a(θ1),...,a(θ2)]. K ] represents the array manifold matrix. s(t) represents the signal amplitude matrix. n(t) is additive white Gaussian noise with a mean of 0 and a variance of . Step 2: Construct a block sparse representation model under the mutual coupling interference of unknown array elements

[0190] target θ k The corresponding guide vector is simplified as follows:

[0191] a(θ k )=[1,β(θ k ),…,β(θ k ) N-1 ] T (77)

[0192] in make The guiding vector representing the mutual coupling of array elements has the following parameterized structure:

[0193]

[0194] In the formula Ψ(θ) k )∈C N×(2P-1) It is a block diagonal matrix with the following structure:

[0195]

[0196] z(θ k c) is a (2P-1)×1 dimensional vector whose P-th element is 1, denoted as .

[0197] z(θ k c) = [μ1,…,μ P-1 ,1,α1,…,α P-1 ] T (80)

[0198] Where element μ iand α i The definition is as follows:

[0199]

[0200]

[0201] The constant term is denoted as:

[0202]

[0203] Therefore, the array manifold under mutual coupling of array elements has the following block matrix representation:

[0204]

[0205] in:

[0206]

[0207] Λ is a block diagonal matrix, denoted as:

[0208]

[0209] In the formula It is a (2P-1)×1 dimensional vector. The array receiving signal model under mutual coupling of array elements can be simplified as follows:

[0210] X = BΛS + N (87)

[0211] Discretize the continuous space to construct a discrete grid with angles Θ = {θ1, θ2, ..., θ}. L Let L be the number of discrete grid points, and L >> K. Substituting this discretely partitioned grid set into matrix BΛ, we can obtain a block sparse representation model of the array received signal under mutual coupling interference:

[0212]

[0213] in Dictionary of block sparse representations:

[0214]

[0215] remember It is a block sparse matrix containing L sub-blocks Q l It has the following sparse form:

[0216]

[0217] Therefore, the sparse representation model of the array block under the mutual coupling of unknown array elements is as follows:

[0218] X = BQ + N (91)

[0219] Step three, establishing the sparse reconstruction model based on mixed kernel-l1 norm minimization under block sparsity

[0220] Considering the block sparse matrix Q, there is a matrix stacking form as follows:

[0221] Q = [Q1, Q2, …, Q L ] T (92)

[0222] Therefore, the sub-matrix Q l is a matrix with rank less than or equal to 1 (when the rank is 0, the sub-matrix is 0). Thus, the block sparse reconstruction model about Q is as follows:

[0223]

[0224] where ||·||F F represents the matrix F norm, and ζ > 0 represents a regularization parameter. However, the rank sum constraint is an NP-hard problem, and an effective solution algorithm cannot be found, so it is usually processed by convex relaxation. The kernel norm minimization problem of the matrix is the best convex relaxation of the matrix rank minimization problem, so formula (93) can be relaxed into the form of the sum of the kernel norms of the matrix, that is, the kernel-l1 norm problem:

[0225]

[0226] where ||·||* * represents the kernel norm of the matrix.

[0227] Step four, deriving an equivalent convex optimization reconstruction structure based on semi-definiteness

[0228] Considering the lth sub-block Q l , there is a singular value decomposition as follows:

[0229] Q l = U l Σ l V l H (95)

[0230] where Σ l is a diagonal matrix composed of singular values, U l and V l represent left and right singular matrices respectively. Let P l = U l Π l,1 W l H , G l = W l Π l,2 V lH where Σ l = Π l,1 Π l,2 , W l is an r x r dimensional unitary matrix, i.e. W l H W l = I, then the matrix Q l can be written as:

[0231] Q l = U l Π l,1 W l H W l Π l,2 V l H = P l G l (96)

[0232] According to the definition of the nuclear norm, we have:

[0233] ||Q l || * = ||Σ l || * = ||Π l,1 Π l,2 || * ≤ ||Π l,1 || F ||Π l,2 || F (97)

[0234] = ||P l || F ||G l || F

[0235] By the Cauchy-Schwarz inequality theorem, the equality holds if and only if , and by the arithmetic mean-geometric mean inequality, we have:

[0236]

[0237] Therefore, the nuclear norm of the matrix Q l is equivalent to the solution of the following minimization problem:

[0238]

[0239] Based on this, the mixed nuclear-l1 norm of the block matrix Q can be rewritten as:

[0240]

[0241] where P = blkdiag(P1,..., P L ) is an MLxrL block diagonal matrix, G = [G1,..., G L ] T is an rLxT matrix stacked by G l . Thus the kernel-l1 norm minimization problem can be rewritten as:

[0242]

[0243] denotes a set of MLxrL block diagonal matrices, and each atom in the set is composed of L Mxr sub-matrices as the main diagonal. Fixing matrix P in the above equation and taking the partial derivative equal to zero, the optimal solution of matrix G can be obtained as:

[0244]

[0245] Substituting equation (102) into equation (101), we can simplify it as:

[0246]

[0247] Define a block diagonal matrix Γ = PP H , we know that Γ is an MLxML matrix and has a semi-definite structure, so we have denotes a set of semi-definite block diagonal matrices, and the dimension of each element in the set is MLxML, and the dimension of the diagonal block is MxM. Since λ > 0, the coefficient in the equation can be omitted without affecting the optimal result of the model, so the above equation can be equivalent to:

[0248]

[0249] The above equation has the following equivalent semi-definite optimization structure:

[0250]

[0251] where Z is an N x N Hermitian matrix.

[0252] Step five, derive the meshless Lagrange dual solution form

[0253] First consider the Lagrange dual model corresponding to equation (105) as follows:

[0254]

[0255] where Υ0 and Υ1 are dual operator matrices, Υ0 is an N x N semi-definite matrix, and Υ1 is an N x T ordinary matrix. T and I MI and M are unit matrices of T and M dimensions, respectively. Assume that θ is an arbitrary angle in continuous space, and let β(θ) = e jπsinθ Consider the dictionary matrix B(θ) = Ψ(θ) at angle θ. The extended steering vector is constructed as follows:

[0256]

[0257] Obviously, there is an N x NM selection matrix J that satisfies B(θ) = JΩ(θ). Let M(θ) = B H (θ)Υ0B(θ), then:

[0258] M(θ) = Ω H (θ)J H Υ0JΩ(θ) (108)

[0259] On the other hand, for the unit matrix I M There is an equivalent structure as follows:

[0260] I M = Ω H (θ)HΩ(θ) (109)

[0261] In the formula, H is an NM x NM matrix and satisfies:

[0262]

[0263] Where blktr M {·} represents the trace operation of the block matrix. According to formula (108) and (109), the constraint term can be changed to:

[0264]

[0265] Therefore, the block sparse reconstruction model can be rewritten as the following meshless optimization model:

[0266]

[0267] Step six, use convex optimization algorithm to solve, realize high-precision angle estimation

[0268] Use the convex optimization tool CVX to solve formula (112) to obtain the optimal solution of the meshless optimization model Obtain the meshless high-precision angle estimation by solving the roots of the polynomial:

[0269]

[0270] The effectiveness of the present application is illustrated by the following examples:

[0271] (I) Simulation conditions and content ​

[0272] 1. Performance of angle estimation for uniform linear array in presence of unknown mutual coupling among array elements

[0273] Consider a uniform linear array (N=10) with 64 array elements, and the distance between adjacent elements is half wavelength. Assume that the first three array elements of the array have unknown mutual coupling (P=4), and the mutual coupling coefficients are c1=0.1547+0.1577j, c2=0.1167-0.1089j and c3=0.2457-0.2609j, respectively. Assume that there are two far-field non-coherent point sources, and the true angles of arrival are θ1=-15.3854° and θ2=7.9107°. In addition, assume that the noise term of the received signal is a zero-mean Gaussian white noise with variance and the active signal and the noise term are not correlated. In the setting of signal-to-noise ratio SNR=10dB and snapshot number T=8, the spatial angle -90°~90° is uniformly divided by a step of 1° to form an initial spatial grid.

[0274] 2. Relationship between root mean square error of target angle estimation in presence of unknown mutual coupling among array elements and signal-to-noise ratio

[0275] Consider a uniform linear array (N=10) with 64 array elements, and the distance between adjacent elements is half wavelength. Assume that the first three array elements of the array have unknown mutual coupling (P=4), and the mutual coupling coefficients are c1=0.1547+0.1577j, c2=0.1167-0.1089j and c3=0.2457-0.2609j, respectively. The sampling snapshot number is T=8. Assume that there are two far-field non-coherent point sources, and the directions of arrival are -15.3854° and 7.9107°, respectively. In addition, assume that the noise term of the received signal is a zero-mean Gaussian white noise with variance and the active signal and the noise term are not correlated. In the simulation, the initial division step of the spatial discrete grid is 1°. Here, the root mean square error (RMSE) of angle estimation is defined as where is the angle estimation value of the kth target in the ith Monte-Carlo simulation experiment; I is the number of Monte-Carlo simulation experiments; and K is the number of estimated targets. Here, 200 independent Monte-Carlo experiments are performed, the signal-to-noise ratio changes from -10dB to 20dB with an interval of 5dB.

[0276] 3. Relationship between root mean square error of target angle estimation in presence of unknown mutual coupling among array elements and snapshot number

[0277] Consider a uniform linear array (N=10) with 64 elements, the element spacing is half wavelength, assume that the first three antenna elements of the array exist unknown element mutual coupling effect (P=4), wherein the mutual coupling coefficients are c1=0.1547+0.1577j, c2=0.1167-0.1089j and c3=0.2457-0.2609j. The signal-to-noise ratio is SNR=5dB. Assume that there are two far-field non-coherent point source signals, and the directions of arrival are-15.3854° and 7.9107° respectively. In addition, assume that the noise term of the received signal is a zero-mean Gaussian white noise with a variance of and the active signal and the noise term are not related. In the simulation, the initial division step of the spatial discrete grid is 1°. Here the root mean square error (RMSE) of angle estimation is defined as is the angle estimation value of the kth target in the ith Monte-Carlo simulation experiment; I is the number of Monte-Carlo simulation experiments; K is the number of estimated targets. Here, 200 independent Monte-Carlo experiments are performed, and the sampling snapshot number varies from 10 to 310 with an interval of 50.

[0278] 4. The root mean square error of target angle estimation with unknown element mutual coupling interference changes with the angle interval

[0279] Consider a uniform linear array (N=10) with 64 elements, the element spacing is half wavelength, assume that the first three antenna elements of the array exist unknown element mutual coupling effect (P=4), wherein the mutual coupling coefficients are c1=0.1547+0.1577j, c2=0.1167-0.1089j and c3=0.2457-0.2609j. The signal-to-noise ratio of the received signal is set to 5dB, and the sampling snapshot number is 8. Assume that there are two far-field non-coherent signals, and the directions of arrival are-7.8197° and-7.8197°+δ, respectively, δ represents the target angle interval. In addition, assume that the noise term of the received signal is a zero-mean Gaussian white noise with a variance of and the active signal and the noise term are not related. Here the root mean square error (RMSE) of angle estimation is defined as is the angle estimation value of the kth target in the ith Monte-Carlo simulation experiment; I is the number of Monte-Carlo simulation experiments; K is the number of estimated targets. Here, 200 independent Monte-Carlo experiments are performed, and the angle interval δ varies from 0.5° to 20°, wherein the interval between the first two test points is 0.5°, and the subsequent interval is increased by 3° from 2°.

[0280] (II) Simulation results

[0281] ​​1. Performance of target angle estimation of uniform linear array under unknown element mutual coupling interference

[0282] Figure 3 The angle estimation performance comparison chart of the present application. Figure 3 The normalized spatial spectrum of the sparse Bayesian array calibration algorithm (SBAC algorithm), the improved l1-SVD algorithm (V-l1-SVD algorithm) and the meshless angle estimation algorithm proposed in the present application are respectively given in the figure. It can be seen from the figure that the spectrum peak of the meshless block sparse angle estimation algorithm based on element mutual coupling optimization proposed in the present application is closer to the real target angle, and the SBAC algorithm and the V-l1-SVD algorithm cannot obtain accurate spatial spectrum due to the existence of spatial discrete mesh mismatch problem. Figure 3 The superiority of the present application in target angle estimation can be intuitively proved.

[0283] 2. The root mean square error of target angle estimation of the present application under unknown element mutual coupling interference changes with signal-to-noise ratio

[0284] Figure 4 The root mean square error of the present application changes with the signal-to-noise ratio relationship curve comparison chart. Figure 4 The root mean square error of the present application and the sparse Bayesian array calibration algorithm (SBAC algorithm) and the improved l1-SVD algorithm (V-l1-SVD algorithm) under element mutual coupling target angle estimation changes with signal-to-noise ratio, and the Cramer-Rao lower bound (CRLB) of the system model under the existence of element mutual coupling interference problem is also drawn in the figure. It can be seen from the figure that the estimation performance of the present application is obviously better than that of SBAC, V-l1-SVD and other algorithms under the condition of low signal-to-noise ratio, and excellent performance is also shown under the condition of high signal-to-noise ratio, and it can be seen that the RMSE curve of the present application is closer to the Cramer-Rao lower bound (CRLB). It is proved that the meshless block sparse angle estimation algorithm based on element mutual coupling optimization proposed in the present application has excellent estimation performance.

[0285] 3. The root mean square error of target angle estimation of the present application under unknown element mutual coupling interference changes with the number of shots

[0286] Figure 5 The root mean square error of the present application changes with the number of shots relationship curve comparison chart. Figure 5The figure shows the relationship between the root mean square error of the target angle estimation of the present application, the sparse Bayesian array calibration algorithm (SBAC algorithm) and the improved l1-SVD algorithm (V-l1-SVD algorithm) under the influence of element mutual coupling and the change of the sampling snapshot number, and the Cramer-Rao lower bound (CRLB) of the system model under the influence of element mutual coupling error is also plotted in the figure. As can be seen from the figure, the angle estimation performance of the V-l1-SVD algorithm is quite poor, which is due to the fact that the algorithm truncates the effective aperture of the array in order to suppress the element mutual coupling problem, resulting in a loss of angle estimation performance. For the SBAC algorithm, although its estimation performance has been significantly improved compared with the V-l1-SVD algorithm, the angle estimation performance cannot be further improved due to the influence of the grid mismatch problem. On the other hand, as can be seen from the figure, the performance of the grid-free sparse angle estimation algorithm based on element mutual coupling optimization proposed in the present application is closer to the CRLB, showing a robust angle estimation performance.

[0287] 4, the relationship between the root mean square error of the target angle estimation of the present application and the change of the angle interval

[0288] Figure 6 is the curve comparison figure of the root mean square error of the present application with the change of different target angle intervals. Figure 6 The figure shows the relationship between the root mean square error of the target angle estimation of the present application, the sparse Bayesian array calibration algorithm (SBAC algorithm) and the improved l1-SVD algorithm (V-l1-SVD algorithm) under the influence of element mutual coupling and the change of the angle interval. From Figure 6 It can be seen that the SBAC and V-l1-SVD algorithms have low angle estimation performance and insufficient angle resolution under the condition of small angle interval due to the influence of the spatial discrete grid mismatch problem. In contrast, the grid-free sparse angle estimation algorithm based on element mutual coupling optimization proposed in the present application can effectively suppress the influence of element mutual coupling and off-grid error since it does not need to divide the discrete dictionary grid, and therefore achieves a high precision estimation performance under the condition of small angle interval. Therefore, it can be known that the angle resolution capability of the algorithm proposed in the present application is better.

[0289] It should be noted that the above is only a schematic description and elaboration of the present application, and those skilled in the art should understand that any modification and replacement of the present application belongs to the protection scope of the present application.

Claims

1. A gridless sparse angle estimation method based on array element mutual coupling optimization, characterized in that, Includes the following steps: Step 1: Establish an array received signal model under unknown mutual coupling interference of array elements; Step 2: Construct a block sparse representation model under the mutual coupling interference of unknown array elements; Step 3: Establish a sparse reconstruction model based on hybrid kernel-l1 norm minimization under block sparsity; Step 4: Derive the equivalent semidefinite convex optimization reconstruction structure; Step 5: Derive the meshless Lagrange dual solution form; Step 6: Solve using a convex optimization algorithm to achieve high-precision angle estimation. Step one includes: assuming a uniform linear array with N array elements, the antenna element positions are distributed as d = [d0, d1, ..., d2]. N-1 ] T It is not a general assumption that the first antenna is d0 = 0, then d n = (n-1)d, where d is set as half wavelength. Considering the mutual coupling interference between the first P antenna elements in this uniform linear array, the following element mutual coupling interference matrix is ​​introduced: in Let ρ be the mutual coupling coefficient on the i-th antenna element. i and These represent the amplitude and phase of the mutual coupling coefficient, respectively. If K far-field, uncorrelated targets illuminate the linear array, assuming the incoming wave directions are θ = [θ1, θ2, ..., θ...] K ] T , where θ k Let the direction of arrival of the k-th target be denoted as . Then, the baseband received signal of the target with element-to-element mutual coupling interference in the t-th snapshot is expressed as: in Let λ be the steering vector corresponding to the k-th target, λ represent the carrier wavelength, C be the mutual coupling matrix defined by equation (75), and A = [a(θ1),...,a(θ2)]. K [)] represents the array manifold matrix, s(t) represents the received signal amplitude matrix, and n(t) is additive white Gaussian noise with a mean of 0 and a variance of . Step two includes: setting the target θ k The corresponding guide vector is simplified as follows: a(θ k )=[1,β(θ k ),…,β(θ k ) N-1 ] T (3) in make The guide vector representing the mutual coupling of array elements has the following parameterized structure: In the formula Ψ(θ) k )∈C N×(2P-1) It is a block diagonal matrix with the following structure: z(θ k c) is a (2P-1)×1 dimensional vector whose P-th element is 1, denoted as . z(θ k ,c)=[μ1,…,μ P-1 ,1,a1,…,a P-1 ] T (6) Where element μ i and α i Defined as: The constant term is denoted as: Therefore, the array manifold under mutual coupling of array elements has the following block matrix representation: in: Λ is a block diagonal matrix, denoted as: In the formula Given a (2P-1)×1 dimensional vector, the array receiving signal model under mutual coupling of array elements simplifies to: X = BΛS + N (13) Discretely divide the continuous space into discrete grid angles, where S = [s(1) s(2)…s(T)], T is the number of snapshots, and Θ = {θ1,θ2,…,θ} L }, θ l Let L be the discrete grid angle value, L be the number of discrete grid points, and L >> K. Substituting this discrete grid set into matrix BΛ, we obtain the block sparse representation model of the array received signal under mutual coupling interference: in Dictionary of block sparse representations: remember It is a block sparse matrix containing L sub-blocks Q l It has the following sparse form: Therefore, the sparse representation model of the array block under the mutual coupling of unknown array elements is: X = BQ + N (17).

2. The meshless block sparse angle estimation method based on array element mutual coupling optimization according to claim 1, characterized in that, Step three includes: Considering the block sparse matrix Q, there exists the following matrix stacking form: Q=[Q1,Q2,…,Q L ] T (18) Therefore, submatrix Q l Let Q be a matrix with rank less than or equal to 1, where the submatrix is ​​0 when the rank is 0. Therefore, the block sparse reconstruction model for Q is: Among them ||·|| F Let F denote the matrix norm, and ζ > 0 denote the regularization parameter. After convex relaxation, the matrix kernel norm minimization problem is the best convex relaxation of the matrix rank minimization problem. Therefore, the relaxation of equation (19) is in the form of the following matrix kernel norm summation, that is, the kernel-l1 norm problem: In the formula ||·|| * The nuclear norm of a matrix is ​​denoted by .

3. The meshless sparse angle estimation method based on array element mutual coupling optimization according to claim 2, characterized in that, Step four includes: considering the l-th sub-block Q l It has the following singular value decomposition: Q l =U l S l V l H (21) In the formula Σ l U is a diagonal matrix composed of singular values. l and V l Let P represent the left and right singular matrices respectively. l =U l Π l,1 W l H G l =W l Π l,2 V l H , including Σ l =Π l,1 Π l,2 W l W is an r×r dimensional unitary matrix. l H W l =I, then matrix Q l Note: Q l =U l P l,1 W l H W l P l,2 V l H =P l G l (22) According to the definition of nuclear norm, we have: According to Cauchy's inequality theorem, if and only if When the equation holds true, the equality holds, and we can also know from the arithmetic square inequality: Therefore, matrix Q l The nuclear norm of is equivalent to solving the following minimization problem: Based on this, the calculation of the mixing kernel-l1 norm of the block matrix Q is rewritten as follows: In the formula, P = blkdiag(P1,…,P L G is an ML×rL dimensional block diagonal matrix, where G = [G1, ..., G2]. L ] T It is G l The stacked rL×T dimensional matrix can then be rewritten based on the kernel-l1 norm minimization problem as follows: Let G represent the set of ML×rL-dimensional block diagonal matrices, where the atoms in the set are composed of L M×r-dimensional submatrices forming the main diagonal. With the matrix P fixed, taking the partial derivative of the above equation and finding it equal to zero, we obtain the optimal solution expression for matrix G as follows: Substituting equation (28) into equation (27), we can simplify to: Define a block diagonal matrix Γ = PP H It can be seen that Γ is an ML×ML dimensional matrix with a positive semidefinite structure, therefore we have Let represent the set consisting of positive semi-definite block diagonal matrices, where the element dimension of the set is ML×ML and the dimension of the diagonal blocks is M×M. Since ζ>0, the coefficients in the equation are omitted without affecting the optimal result of the model. Therefore, the above equation is equivalent to: The above equation has the following equivalent positive semidefinite optimization structure: Where Z is an N×N dimensional Hermitian matrix.

4. The meshless sparse angle estimation method based on array element mutual coupling optimization according to claim 3, characterized in that, Step five includes: First, consider the Lagrange dual model corresponding to equation (31) as follows: In the formula, γ0 and γ1 are dual operator matrices, γ0 is an N×N positive semi-definite matrix, and γ1 is an N×T ordinary matrix. T and I M Let β(θ) = e^(-T) and M-dimensional identity matrices, respectively. Assume θ is an arbitrary angle in continuous space. jπsinθ Considering the dictionary matrix B(θ) = Ψ(θ) under angle θ, the extended guiding vector is constructed as follows: Clearly, there exists an N×NM dimensional choice matrix J satisfying B(θ)=JΩ(θ), denoted as M(θ=B H If (θ)γ0B(θ), then: M(θ)=Ω H (i)J H γ0JΩ(θ) (34) On the other hand, for the identity matrix I M The following equivalent structure exists: I M =Oh H (θ)HΩ(θ) (35) In the formula, H is an NM×NM dimensional matrix and satisfies: blktr M {·} represents the trace operation of the block matrix. According to equations (34) and (35), the constraint terms become: Therefore, the block sparse reconstruction model is rewritten as the following meshless optimization model:

5. The meshless block sparse angle estimation method based on array element mutual coupling optimization according to claim 4, characterized in that, Step six includes: solving equation (38) using the convex optimization toolkit CVX to obtain the optimal solution of the meshless optimization model. By solving the polynomial: The root is used to obtain a high-precision angle estimate without mesh.

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