Super-resolution spatial spectrum estimation method based on optimal structured virtual domain tensor filling of coprime planar array

Abstract

The application discloses an ultra-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling, mainly solves the problem that patch missing elements in a virtual domain tensor are difficult to be effectively filled and spatial spectrum resolution performance is limited, and the implementation steps are as follows: tensor signal modeling of a coprime planar array; derivation of an augmented virtual planar array based on inter-correlation tensor dimension merging; construction of a virtual domain tensor based on mirror expansion of a non-continuous virtual planar array; virtual domain sub-tensor superposition transformation reconstruction of the virtual domain tensor; optimal structured virtual domain tensor optimization based on virtual domain sub-tensor dimensions; structured virtual domain tensor filling based on an alternating direction multiplier method; and ultra-resolution spatial spectrum estimation by decomposing the filled structured virtual domain tensor. The application realizes optimal filling of a coprime planar array virtual domain tensor, fully utilizes all non-continuous virtual domain statistical information of the coprime planar array for ultra-resolution spatial spectrum estimation, and can be used for target positioning.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of array signal processing, and particularly relates to a spatial spectrum estimation technology based on sparse array tensor signal statistical processing, in particular to a super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling, which can be used for target positioning. BACKGROUND

[0002] As a technology for describing the spatial energy distribution of array signals, spatial spectrum estimation is widely used in radar, communication, geological exploration and other fields. At present, the increasing complexity of application scenarios continuously improves the performance requirements such as accuracy and resolution of spatial spectrum estimation. Compared with the traditional uniform array, the coprime array, as a typical sparse array architecture with systematic structure, has the advantages of large aperture and high resolution, which lays the foundation for the breakthrough of spatial spectrum estimation performance. In the coprime planar array scene, since the received signal covers the three-dimensional spatial characteristics, modeling and analysis of the signal through tensor can preserve the original structure of the multi-dimensional signal of the coprime planar array, so as to exploit the multi-dimensional signal characteristics. Based on the tensor second-order statistics of the coprime planar array, the augmented multi-dimensional non-continuous virtual array is derived, and the continuous part is extracted for virtual domain tensor processing, which can realize the spatial spectrum estimation of Nyquist matching. However, such processing method discards a large number of non-continuous virtual array elements, which causes serious loss of virtual domain statistical information, and limits the performance of spatial spectrum estimation such as accuracy and resolution.

[0003] In the field of image restoration, the low-rank tensor filling technology can fill the missing elements randomly distributed in the image tensor. However, the missing elements in the equivalent virtual domain tensor derived from the coprime planar array do not meet the premise of random distribution; therefore, the traditional low-rank filling technology cannot effectively fill the virtual domain tensor. Therefore, how to fill the virtual domain tensor with patch missing elements so as to fully utilize the non-continuous virtual domain statistical information of the coprime planar array and comprehensively improve the performance of spatial spectrum estimation is a technical problem to be solved but full of challenges. SUMMARY

[0004] The present application aims to solve the problem that the patch missing elements in the virtual domain tensor cannot be effectively filled and the performance of spatial spectrum resolution is limited by the existing method, and proposes a super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling, which provides a feasible idea and effective solution for realizing the super-resolution coprime planar array spatial spectrum estimation by constructing the optimal structured virtual domain tensor to effectively fill the dispersed missing elements in the virtual domain tensor.

[0005] The application aims to realize the technical scheme of a super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling.

[0006] (1) The receiving end uses 4M x M y +N x N y -1 physical antenna elements, and is structured according to the structure of a coprime planar array; wherein M x , M x , M y , N y are a pair of coprime integers; the coprime planar array is decomposed into two sparse uniform sub-arrays and wherein contains 2M x ×2M y antenna elements, and the element spacing in the x-axis direction and the y-axis direction is N x d and N y d respectively, contains N x ×N y antenna elements, and the element spacing in the x-axis direction and the y-axis direction is M x d and M y d respectively, and the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, i.e. d = λ / 2;

[0007] Suppose there are K far-field narrowband non-correlated signal sources from directions, θ k and are the azimuth angle and the elevation angle of the kth incident signal source respectively, k = 1, 2, …, K, and the T sampling snapshot signals of the sparse uniform sub-array are represented by a three-dimensional tensor as follows:

[0008]

[0009] wherein s k = [s k,1 , s k,2 , …, s k,T ] T is the multi-snapshot sampling signal waveform corresponding to the kth incident signal source, [·] T represents the transposition operation, represents the vector outer product, is a noise tensor independent of each signal source, and are respectively The steering vectors in x and y directions corresponding to the signal source with DOA are denoted as

[0010]

[0011]

[0012] where and denote the sparse uniform subplanar array The actual positions of the physical antenna elements in x and y directions, and

[0013] The received signal of the sparse uniform subplanar array is denoted by another three-dimensional tensor

[0014]

[0015] where is a noise tensor independent of each signal source, and are the steering vectors in x and y directions corresponding to the signal source with DOA The steering vectors in x and y directions corresponding to the signal source with DOA are denoted as

[0016]

[0017]

[0018] where and denote the sparse uniform subplanar array The actual positions of the physical antenna elements in x and y directions, and

[0019] (2) The second-order cross-correlation tensor is obtained by calculating the cross-correlation statistics of tensors

[0020]

[0021] where denotes the power of the kth incident signal source, denotes the cross-correlation noise tensor, <·,·> r denotes the tensor contraction operation of two tensors along the rth dimension, E[·] denotes the mathematical expectation operation, (·) * denotes the conjugate operation; define two dimension sets​​ and By dimension merging of the cross-correlation tensor , a virtual domain signal

[0022]

[0023] where, and are equivalent to a non-continuous virtual surface array with the steering vectors in x-axis and y-axis corresponding to the signal source with the DOA , respectively. denotes the Kronecker product; the size of the non-continuous virtual surface array is where the whole rows and columns contain holes.

[0024] (3) Constructing the non-continuous virtual surface array The virtual surface array with respect to the mirror image of the coordinate axis and superimposing and in the third dimension into a three-dimensional non-continuous virtual cubic array with the size of Here, and rearrange the elements in the conjugate transpose signal of the virtual domain signal to correspond to the positions of the virtual array elements in , to obtain the virtual domain signal corresponding to the virtual surface array Superimpose and in the third dimension to obtain the virtual domain tensor corresponding to the non-continuous virtual cubic array is expressed as:

[0025]

[0026] where, and are the steering vectors in x-axis and y-axis of the non-continuous virtual cubic array corresponding to the signal source with the DOA , and and correspond to the elements in x-axis and y-axis directions of being set to zero at the hole positions,

[0027]

[0028] denotes the mirror transformation factor vector corresponding to and ; due to the non-continuous virtual plane array contains holes in the whole row and the whole column, the non-continuous virtual cube array is obtained by superimposing and its mirror part contains missing elements, i.e. holes, in the whole patch, the corresponding virtual domain tensor contains zero elements in the whole patch;

[0029] (4) A virtual domain sub-tensor x of the virtual domain tensor is obtained by cutting a translation window with size P y × P x × 2; contains elements with indices (1: P y -1), (1: P x -1), (1:2) in three dimensions respectively; then, the translation window is translated by one element along the x-axis and the y-axis direction respectively, and is divided into L y × L x virtual domain sub-tensors, denoted as s x = 1, 2, …, L y , s y = 1, 2, …, L x ; the size of the translation window ranges from:

[0030]

[0031]

[0032] and L y , L x , P y , P y satisfy the following relationship:

[0033]

[0034]

[0035] The virtual domain sub-tensors with the same s y index subscript are superimposed in the fourth dimension to obtain L x virtual domain sub-tensors with size P y × P x × 2 × L​​x four-dimensional tensors; further, superimposing the L y four-dimensional tensors in a fifth dimension, a five-dimensional virtual domain tensor This five-dimensional virtual domain tensor covers the spatial angle information in the x-axis and y-axis directions, the spatial mirror transformation information, and the spatial translation information in the x-axis and y-axis directions; the dimension set is defined as Then, the virtual domain tensor transformation of dimension merging is performed on to obtain a three-dimensional structured virtual domain tensor

[0036]

[0037] The three dimensions of the structured virtual domain tensor respectively represent the spatial angle information, the spatial translation information, and the spatial mirror transformation information; thus, the missing elements in the virtual domain tensor are randomly distributed to the three spatial dimensions covered by the structured virtual domain tensor

[0038] (5) Since the dispersion degree and the proportion of zero elements in the structured virtual domain tensor are closely related to the effect of tensor filling, in order to ensure that the dispersion degree of zero elements in the structured virtual domain tensor is maximum and the proportion is minimum, the dimension size of the virtual domain sub-tensor needs to be optimized, that is, the value of (P x , P y ) is optimized and selected, so as to obtain the optimal structured virtual domain tensor. The specific process is as follows: according to each value combination (P x , P y ), the sum of the Euclidean distances between all zero elements in the structured virtual domain tensor corresponding to the value combination is calculated:

[0039]

[0040] Wherein, Ω represents the position index set of zero elements in , and and represent the coordinates of any two positions in the set Ω. Here, represents the total number of zero elements in ; the dispersion degree of zero elements in the structured virtual domain tensor is determined by the parameter ψ; correspondingly, the proportion of zero elements in the structured virtual domain tensor is represented as:

[0041]

[0042] Taking into account both maximizing the dispersion of zero elements in the structured virtual field tensor and minimizing the proportion of zero elements. The dimension optimization problem of virtual domain subtensors is expressed as:

[0043]

[0044]

[0045]

[0046] Traversing P x and P y Range of values and All values ​​within, each group (P) x ,P y Each value corresponds to a value of the objective function. Select a set (P) corresponding to the maximum value of the objective function x ,P y The value of ) is the optimal virtual domain subtensor. Dimension size;

[0047] (6) Design a structured virtual domain tensor filling optimization problem based on the alternating direction multiplier method:

[0048]

[0049]

[0050]

[0051] Among them, optimization variables It is the filled structured virtual domain tensor, corresponding to a virtual uniform cubic array. express The matrix expanded along the b-th dimension, α b The nuclear norm weighting constant must satisfy α1 + α2 + α3 = 1, ||·|| * Represents the nuclear norm number, in order to ensure The three matrix nuclear norms It can be optimized independently, and this problem introduces... Three auxiliary tensors express The set of position indices of non-zero elements Indicates that the tensor is in Mapping on, Represent the zero tensor; introduce dual variables The Lagrangian function of the above optimization problem is expressed as:

[0052]

[0053] where ρ > 0 is a compensation factor, [·x·] denotes the tensor inner product, and ‖·‖ F denotes the Frobenius norm; the target variable is solved iteratively by minimizing the Lagrangian function to obtain the filled structured virtual domain tensor

[0054] (7) The filled structured virtual domain tensor Theoretically, it is modeled as:

[0055]

[0056] where, is the spatial factor of

[0057]

[0058]

[0059] denote the virtual uniform cubic array the steering vectors along the x-axis and y-axis directions,

[0060]

[0061]

[0062] are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of windowing the virtual domain sub-tensor, respectively; the filled structured virtual domain tensor is decomposed by canonical polyadic to obtain the estimated values of the three factor vectors p(μ k ,ν k ), q(μ k ,ν k ) and c(μ k ,ν k ), denoted as and The structured virtual domain tensor signal subspace is constructed

[0063]

[0064] where orth(·) denotes the matrix orthogonalization operation; the noise subspace is denoted by V s ​​Obtain:

[0065]

[0066] where I denotes the identity matrix, (·) H denotes the conjugate transpose operation;

[0067] Iterate the two-dimensional direction of arrival θ and respectively, the azimuth and elevation angles in the range of [-90°, 90°] and [0°, 180°] are iterated, and the corresponding parameters and construct the corresponding virtual uniform cubic array steering vector is expressed as:

[0068]

[0069] get the spatial spectrum corresponding to the two-dimensional direction of arrival :

[0070]

[0071] Further, the coprime surface array structure described in step (1) is specifically described as follows: a pair of sparse uniform sub-surface arrays and are constructed in the plane coordinate system xoy contains 2M x ×2M y antenna elements, the element spacing in the x-axis direction and the y-axis direction is N x d and N y d, respectively, and the position coordinates in xoy are {(N x dm x ,N y dm y ), m x = 0, 1,..., 2M x -1, m y = 0, 1,..., 2M y -1}; contains N x ×N y antenna elements, the element spacing in the x-axis direction and the y-axis direction is M x d and M y d, respectively, and the position coordinates in xoy are {(M x dn x ,M y dn y ), n x ​=0,1,...,N x -1,n y =0,1,...,N y -1};M x N x and M y N y They are a pair of coprime integers; and Subarrays are combined by overlapping array elements at coordinate system position (0,0) to obtain an actual array containing 4M elements. x M y +N x N y -1 coprime surface array of physical antenna elements.

[0072] Furthermore, the derivation of the cross-correlation tensor described in step (2) is, in practice, By estimating tensors and The cross-correlation statistics are obtained, i.e., the sampled cross-correlation tensor.

[0073]

[0074] Furthermore, in step (6), by minimizing the Lagrange function Iterative solution of the objective variable In the (η+1)th iteration, and Updated to:

[0075]

[0076]

[0077]

[0078] Target variable The closed-form solution is:

[0079]

[0080]

[0081] in, Representation matrix Threshold singular value decomposition operation, min(X1,X2) represents the singular values ​​of X, U X V X Let fold represent the left and right singular matrices of X. (b) [·] indicates tensor expansion. (b)diag(c) represents a diagonal matrix with the elements in vector c as diagonal elements, max(·) represents a maximum value operation, and min(·) represents a minimum value operation.

[0082] Compared with the prior art, the present application has the following advantages:

[0083] (1) The present application designs an optimal reconstruction criterion for a virtual domain tensor, and maximizes the dispersion degree of missing elements in the virtual domain tensor to construct a structured virtual domain tensor, thereby laying a foundation for effectively filling the virtual domain tensor with missing elements in a patch.

[0084] (2) The present application proposes a structured virtual domain tensor filling method based on an alternating direction multiplier method, and fully utilizes all the discontinuous virtual domain statistical information of a coprime surface array, thereby realizing super-resolution spatial spectrum estimation for the coprime surface array under the condition of Nyquist matching. BRIEF DESCRIPTION OF DRAWINGS

[0085] Figure 1 is a general flowchart of the present application.

[0086] Figure 2 is a schematic diagram of a coprime surface array constructed by the present application.

[0087] Figure 3 is a schematic diagram of a discontinuous virtual cubic array constructed by the present application.

[0088] Figure 4 is a schematic diagram of a virtual domain sub-tensor interception process designed by the present application.

[0089] Figure 5 is a spatial spectrum estimation effect diagram of the method proposed by the present application. DETAILED DESCRIPTION

[0090] The technical solutions of the present application will be further described in detail below with reference to the accompanying drawings.

[0091] In order to solve the problems that the existing method cannot effectively fill the missing elements in a virtual domain tensor in a patch and the spatial spectrum resolution performance is limited, the present application proposes a super-resolution coprime surface array spatial spectrum estimation method based on optimal structured virtual domain tensor filling, which effectively fills the missing elements in a virtual domain tensor in a patch through an optimal reconstruction criterion for a virtual domain tensor and an alternating direction multiplier method, so as to realize super-resolution coprime surface array spatial spectrum estimation. Figure 1 The implementation steps of the present application are as follows:

[0092] Step 1: Tensor signal modeling of a coprime surface array. At the receiving end, a 4M x M y +N x N y-1 physical antenna elements to construct a coprime surface array, such as Figure 2 A pair of sparse uniform sub-surface arrays and are constructed in the plane coordinate system xoy, where contains 2M x ×2M y antenna elements, the element spacing in the x-axis direction and the y-axis direction are N x d and N y d respectively, whose position coordinates in xoy are {(N x dm x ,N y dm y ), m x =0,1,...,2M x -1, m y =0,1,...,2M y -1}; contains N x ×N y antenna elements, the element spacing in the x-axis direction and the y-axis direction are M x d and M y d respectively, whose position coordinates in xoy are {(M x dn x ,M y dn y ), n x =0,1,...,N x -1, n y =0,1,...,N y -1};M x , N x and M y , N y are a pair of coprime integers respectively; the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, i.e. d = λ / 2; and and are combined into sub-arrays in a way that the elements at the position (0, 0) of the coordinate system overlap, obtaining a coprime surface array actually containing 4M x M y +N x N y -1 physical antenna elements.

[0093] Suppose there are K far-field narrowband non-correlated signal sources from directions, after superimposing the T sampling snapshot signals of the sparse uniform sub-surface array in the third dimension, a three-dimensional tensor signal can be obtained, which can be modeled as:

[0094]

[0095] Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T To represent the multi-shot sampled signal waveform corresponding to the k-th incident signal source, [·] T This indicates the transpose operation. Represents the vector outer product. For noise tensors that are independent of each signal source, and They are respectively The guiding vectors in the x and y axes correspond to the direction of incoming wave. The signal source is represented as:

[0096]

[0097]

[0098] in, and Representing sparse uniform sub-arrays The actual positions of the physical antenna elements in the x-axis and y-axis directions, and Similarly, sparse uniform subarrays The T sampled snapshot signals can be used with another three-dimensional tensor express:

[0099] in, For noise tensors that are independent of each signal source, and They are respectively The guiding vectors in the x and y axes correspond to the direction of incoming wave. The signal source is represented as:

[0100]

[0101]

[0102] in, and Representing sparse uniform sub-arrays The actual positions of the physical antenna elements in the x-axis and y-axis directions, and

[0103] Step 2: Derive the augmented virtual matrix based on cross-correlation tensor dimension merging. This is done by calculating the tensor signal. and the cross-correlation statistics of the tensor signals

[0104]

[0105] where, denotes the power of the kth incident signal source, denotes the cross-correlation noise tensor, r denotes the tensor product operation of two tensors along the rth dimension, E[·] denotes the operation of taking mathematical expectation, (·) * denotes the conjugate operation. In practice, the cross-correlation statistics of the tensor signals and are estimated, i.e., the sampled cross-correlation tensors

[0106]

[0107] By merging the dimensions of the cross-correlation tensors that represent the spatial information of the same direction, the steering vectors corresponding to the two sparse uniform sub-arrays form a difference set in the exponential term, and thus a two-dimensional augmented virtual array is constructed. Specifically, since the 1st and 3rd dimensions of the cross-correlation tensor represent the spatial information of the x-axis direction, and the 2nd and 4th dimensions represent the spatial information of the y-axis direction, two dimension sets of the cross-correlation tensor are merged to obtain a virtual domain signal

[0108]

[0109] where, and are equivalent to a non-continuous virtual array with steering vectors in the x-axis and y-axis, respectively, corresponding to the signal source with the DOA , and denotes the Kronecker product. The size of the non-continuous virtual array is which contains holes in the whole rows and columns, Here, in order to simplify the derivation process, the cross-correlation noise tensor is omitted in the theoretical modeling step with respect to ; however, in practice, since the sampled cross-correlation tensor is used to replace the theoretical cross-correlation tensor is still covered in the virtual domain signal statistical processing process;

[0110] Step 3: Constructing the virtual domain tensor based on the mirror image expansion of the non-continuous virtual surface array Virtual surface array with respect to the mirror image of the coordinate axis And And Stacking in the third dimension into a three-dimensional non-continuous virtual cubic array with a size of As shown in Here, Figure 3 And Reordering the elements in the conjugate transpose signal of the virtual domain signal to correspond to the positions of each virtual array element in , so as to obtain the virtual domain signal corresponding to the non-continuous virtual surface array Stacking and in the third dimension to obtain the virtual domain tensor corresponding to the non-continuous virtual cubic array is expressed as:

[0111]

[0112] Wherein, and are the guide vectors of the non-continuous virtual cubic array in the x-axis and y-axis directions, corresponding to the signal source with a wave direction of , and and correspond to the elements of the hole positions in the x-axis and y-axis directions in , respectively,

[0113]

[0114] indicate the mirror image transformation factor vectors corresponding to and ; since the non-continuous virtual surface array contains whole rows and whole columns of holes, the non-continuous virtual cubic array obtained by stacking and its mirror image part contains elements (holes) missing in patches, so the corresponding virtual domain tensor contains zero elements in patches;

[0115] ​Step 4: Reconstruct the virtual domain tensor through the superposition transformation of virtual domain subtensors. Since coprime arrays do not satisfy the Nyquist sampling theorem, in order to achieve Nyquist-matched signal processing on a virtual uniform cubic array, it is necessary to reconstruct the virtual domain tensor. The missing elements in the array are filled to correspond to a virtual uniform cubic array. However, existing tensor imputation techniques based on the low-rank criterion rely on the randomized distribution of missing elements in the tensor, and therefore cannot imput virtual domain tensors with large areas of missing elements. Effective filling is required. Therefore, it is necessary to reconstruct the virtual domain tensor. To disperse its clustered missing elements, the specific process is as follows: design a system of size P x ×P y A translation window of ×2 is used to extract the virtual domain tensor. A virtual domain subtensor It includes The indices in the three dimensions are (1:P) x -1), (1:P y -1), (1:2) elements; then, by translating the translation window one element at a time along the x-axis and y-axis, we can then... Divided into L x ×L y A virtual domain subtensor, such as Figure 4 As shown, it is represented as s x =1,2,…,L x s y =1,2,…,L y The range of values ​​for the size of the translation window is:

[0116]

[0117]

[0118] And L x L y P x P y The following relationship exists between them:

[0119]

[0120]

[0121] Will have the same s y Virtual domain subtensor with index subscript Superimposing in the fourth dimension yields L. y Each dimension is P x ×Py ×2×L x The four-dimensional tensor; further, this L y The superposition of four-dimensional tensors in the fifth dimension yields a five-dimensional virtual domain tensor. This five-dimensional virtual domain tensor It covers spatial angle information along the x and y axes, spatial mirror transformation information, and spatial translation information along the x and y axes; Merging is performed along the first and second dimensions representing spatial angular information, and simultaneously along the fourth and fifth dimensions representing spatial translation information, while retaining the third dimension representing spatial mirror transformation information to construct a structured virtual domain tensor. Specifically, the dimension set is defined. Then for A virtual domain tensor transformation involving dimension merging yields a three-dimensional structured virtual domain tensor.

[0122]

[0123] The three dimensions represent spatial angular information, spatial translation information, and spatial mirror transformation information, respectively. Therefore, the virtual domain tensor... Large swathes of missing elements are randomly distributed into the structured virtual domain tensor. The three spatial dimensions it covers;

[0124] Step 5: Obtain the optimal structured virtual domain tensor based on virtual domain subtensor dimension optimization. During the reconstruction of the virtual domain tensor, the size of the translation window, i.e., the virtual domain subtensor... Dimension size (P) x ,P y This will affect the structured virtual domain tensor. The dispersion and proportion of zero elements are crucial factors, and these two indicators are closely related to the effectiveness of tensor filling. To ensure... The zero element has the highest dispersion and the smallest proportion, so the dimension of the virtual domain subtensor needs to be optimized, that is, the dimension of (P) needs to be optimized. x ,P y The optimal structured virtual domain tensor is obtained by optimizing the selection of the value of ) based on each combination of values ​​(P). x ,P y ), calculate the corresponding structured virtual domain tensor The sum of the Euclidean distances between all pairwise zero elements in the set:

[0125]

[0126] Where Ω represents The set of indexes of the zero element and Let represent the coordinates of any two positions within set Ω, where . express The total number of zero elements. Structured virtual field tensor. The dispersion of zero elements is determined by the parameter ψ; correspondingly, the structured virtual domain tensor The proportion of zero elements in the equation can be expressed as:

[0127]

[0128] Taking into account both maximizing the dispersion of zero elements in the structured virtual field tensor and minimizing its proportion of zero elements. The dimension optimization problem of virtual domain subtensors can be expressed as:

[0129]

[0130]

[0131]

[0132] Traversing P x and P y Range of values and All values ​​within, each group (P) x ,P y Each value corresponds to a value of the objective function. Select a set (P) corresponding to the maximum value of the objective function x ,P y The value of ) is the optimal virtual domain subtensor. Dimension size;

[0133] Step 6: Filling Structured Virtual Domain Tensors Based on Alternating Direction Method of Multipliers. Design an optimization problem for filling structured virtual domain tensors based on the Alternating Direction Method of Multipliers (ADMM):

[0134]

[0135]

[0136]

[0137] Among them, optimization variables It is the filled structured virtual domain tensor, corresponding to a virtual uniform cubic array. express The matrix expanded along the b-th dimension, α b The nuclear norm weighting constant must satisfy α1 + α2 + α3 = 1, ||·|| * Represents the nuclear norm number, in order to ensure The three matrix nuclear norms It can be optimized independently, and this problem introduces... Three auxiliary tensors express The set of position indices of non-zero elements Indicates that the tensor is in Mapping on, Represent the zero tensor; introduce dual variables. The Lagrangian function of the above optimization problem can then be expressed as:

[0138]

[0139] Where ρ>0 represents the compensation factor, [·×·] represents the tensor inner product, and ‖·‖ F Represents the Frobenius norm. The objective variable is solved iteratively by minimizing the Lagrange function. In the (η+1)th iteration, and Updated to:

[0140]

[0141]

[0142]

[0143] Target variable The closed-form solution is:

[0144]

[0145]

[0146] in, Representation matrix Threshold singular value decomposition operation, min(X1,X2) represents the singular values ​​of X, U X V X Let fold represent the left and right singular matrices of X. (b) [·] indicates tensor expansion. (b)diag(c) denotes a diagonal matrix with the elements in vector c as the diagonal elements, max(·) denotes the max operation, and min(·) denotes the min operation. Through the iteration of the above alternating direction multiplier method, the filled structured virtual domain tensor

[0147] Step 7: Decompose the filled structured virtual domain tensor to realize super-resolution spatial spectrum estimation. The filled structured virtual domain tensor can be theoretically modeled as:

[0148]

[0149] wherein, is the spatial factor of

[0150]

[0151]

[0152] respectively represent the virtual uniform cubic array steering vectors along the x-axis and y-axis directions,

[0153]

[0154]

[0155] respectively are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of intercepting the virtual domain sub-tensor by the translation window. The filled structured virtual domain tensor is subjected to canonical polyadic decomposition, and the estimated values of the three factor vectors p(mu k , nu k ), q(mu k , nu k ) and c(mu k , nu k ) are represented as and The structured virtual domain tensor signal subspace

[0156]

[0157] wherein orth(·) represents the matrix orthogonalization operation; the noise subspace is represented by The V s can be obtained as:

[0158] ​​

[0159] Where I represents the identity matrix, (·) H This indicates the conjugate transpose operation.

[0160] Traversing the two-dimensional wave direction of arrival Calculate the corresponding parameters And construct the corresponding virtual uniform cubic array The guiding vector Represented as:

[0161]

[0162] Here, θ∈[-90°,90°], Obtain the corresponding two-dimensional direction of arrival. Spatial spectrum for:

[0163]

[0164] The effects of the present invention will be further described below with reference to simulation examples.

[0165] Simulation Example: A coprime array is used to receive the incident signal, and its parameters are selected as M. x =2,M y =3, N x =3, N y =4, meaning the coprime matrix of the architecture contains a total of 4M. x M y +N x N y -1 = 35 physical array elements. Assume there are two narrowband incident signals with incident azimuth and elevation angles of [35°, 20°] and [45.5°, 40.5°], respectively. According to the virtual domain subtensor dimension optimization problem proposed in this invention, the optimal virtual domain subtensor dimension is obtained as 7 × 14 × 2, corresponding to the optimal structured virtual domain tensor. Its dimensions are 56×238×2. The nuclear norm weight constant is taken

[0166] Simulation experiments were conducted using 300 sampling snapshots under the condition of SNR=0dB. The normalized spatial spectrum estimation results corresponding to the method proposed in this invention are as follows: Figure 5 As shown, the x-axis and y-axis represent the azimuth and elevation angles of the incident signal source, respectively. It can be seen that the method proposed in this invention can form precise spectral peaks at the directions of arrival corresponding to the two incident signal sources, demonstrating the excellent performance of the proposed spatial spectrum estimation method in terms of accuracy and resolution.

[0167] The above merely describes preferred embodiments of the present application, and the present application is not limited to the above. Any person skilled in the art, without departing from the technical scope of the present application, can make many possible changes and modifications to the technical solutions of the present application, or modify equivalent embodiments with equivalent changes. Therefore, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present application, without departing from the technical scope of the present application, still falls within the protection scope of the technical solutions of the present application.

Claims

1. A super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling, characterized in that, Comprising the following steps: (1) the receiving end uses 4M x M y +N x N y -1 physical antenna elements, according to the structure of the coprime surface array; wherein M x , N x and M y , N y are a pair of coprime integers; the coprime surface array is decomposed into two sparse and uniform sub-surface arrays and wherein contains 2M x ×2M y antenna elements, and the element spacing in the x-axis direction and the y-axis direction is N x d and N y d respectively, contains N x ×N y antenna elements, and the element spacing in the x-axis direction and the y-axis direction is M x d and M y d respectively, and the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, i.e. d = λ / 2; Assume there are K far-field narrowband uncorrelated signal sources from directions, θk k and are the azimuth and elevation angles of the k-th incident signal source, respectively, k = 1, 2, …, K. The T sampled snapshot signals of a sparse uniform subaperture array are represented by a three-dimensional tensor as follows: Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T To represent the multi-shot sampled signal waveform corresponding to the k-th incident signal source, [·] T This indicates the transpose operation, and ° indicates the vector cross product. For noise tensors that are independent of each signal source, and They are respectively The guiding vectors in the x and y axes correspond to the direction of incoming wave. The signal source is represented as: wherein, and respectively represent sparse uniform sub-arrays actual positions of physical antenna elements in the x-axis and y-axis directions, and Sparse uniform sub-aperture array of received signals with another three-dimensional tensor representation: wherein is the noise tensor independent of each signal source, and respectively the steering vector in the x-axis and y-axis directions, corresponding to the signal source with the DOA is expressed as: wherein, and respectively represent sparse uniform sub-arrays actual positions of the physical antenna elements in the x-axis and y-axis directions, and (2) The cross-correlation tensor and is obtained by taking the cross-correlation statistics of the tensors wherein, represents the power of the kth incident signal source, represents the cross-correlation noise tensor, <·,·> r represents the tensor contraction operation of two tensors along the rth dimension, E[·] represents the taking mathematical expectation operation, (·) * represents the conjugate operation; defines two dimension sets and by dimension merging the cross-correlation tensor a virtual domain signal wherein, and respectively, is equivalent to a non-continuous virtual planar array The steering vectors in x and y axes, corresponding to the signal sources with DOA , denotes the Kronecker product; the non-continuous virtual planar array has a size of which contains the holes in the whole rows and columns, (3) Constructing a non-continuous virtual surface array Regarding the mirror image of the coordinate axis And And Superimposed in the third dimension into a three-dimensional non-continuous virtual cubic array with a size of Here, And The conjugate transpose signal of the virtual domain signal The elements in the signal are rearranged to correspond to the positions of each virtual array element in The virtual domain signal corresponding to the virtual surface array is Superimposed in the third dimension And The virtual domain tensor corresponding to the non-continuous virtual cubic array is represented as:​ in, and These are non-continuous virtual cubic arrays. The guiding vectors on the x and y axes correspond to the direction of incoming waves. The signal source, and and Corresponding to Set the elements representing the hole positions along the x-axis and y-axis to zero. denote the mirror transformation factor vectors corresponding to and Due to the non-continuous virtual plane array containing holes in the whole row and the whole column, the virtual domain tensor and its mirror part superimposed non-continuous virtual cubic array containing holes in the whole row and the whole column, the virtual domain tensor contains holes in the whole row and the whole column. (4) Through a space of size P x ×P y A translation window of ×2 is used to extract the virtual domain tensor. The first virtual domain subtensor It includes The indices in the three dimensions are (1:P) x -1), (1:P y -1), (1:2) elements; then, translate the translation window one element at a time along the x-axis and y-axis respectively, and Divided into L x ×L y A virtual domain subtensor, denoted as s x =1,2,…,L x s y =1,2,…,L y The range of values ​​for the size of the translation window is: and L x , L y , P x , P y satisfy the following relationship: The virtual domain tensors with the same s y Index subscript virtual domain sub-tensors Superimpose in the fourth dimension to obtain L y four-dimensional tensors with P x × P y × 2 × L x dimensions; further, superimpose the L y four-dimensional tensors in the fifth dimension to obtain a five-dimensional virtual domain tensor This five-dimensional virtual domain tensor covers the spatial angle information in the x-axis and y-axis directions, the spatial mirror transformation information, and the spatial translation information in the x-axis and y-axis directions; define a dimension set Then, perform virtual domain tensor transformation on by dimension merging to obtain a three-dimensional structured virtual domain tensor The three dimensions of the structured virtual domain tensor respectively represent spatial angular information, spatial translation information, and spatial mirror transformation information; thus, the missing elements in the virtual domain tensor are randomly distributed over the three spatial dimensions covered by the structured virtual domain tensor (5) Due to the structured virtual domain tensor The dispersion and proportion of zero elements are closely related to the effect of tensor filling. In order to ensure... The zero element has the highest dispersion and the smallest proportion, so the dimension of the virtual domain subtensor needs to be optimized, that is, the dimension of (P) needs to be optimized. x ,P y The optimal structured virtual domain tensor is obtained by optimizing the selection of the value of ) , specifically by: selecting the optimal structured virtual domain tensor based on each combination of values ​​(P) x ,P y ), calculate the corresponding structured virtual domain tensor The sum of the Euclidean distances between all pairwise zero elements in the set: where Ω denotes the set of position indices of zero elements in Ω, and denotes the coordinates of any two positions in Ω, where z1, z2 = 1, 2, …, denotes the total number of zero elements in Ω; the structured virtual domain tensor the dispersion of zero elements in Ω is determined by the parameter ψ; correspondingly, the structured virtual domain tensor the proportion of zero elements in the structured virtual domain tensor is expressed as: The dispersion degree of zero elements in the structured virtual domain tensor is maximized and the proportion of zero elements is minimized The dimension optimization problem of the virtual domain sub-tensor is represented as: P x and P y value range and Each group of (P x , P y ) values corresponds to a target function value Select a group of (P x , P y ) values corresponding to the maximum value of the target function, which is the optimized virtual domain sub-tensor dimension size (6) Design a structured virtual domain tensor filling optimization problem based on the alternating direction multiplier method: Among them, optimization variables It is the filled structured virtual domain tensor, corresponding to a virtual uniform cubic array. express The matrix expanded along the b-th dimension, α b The nuclear norm weight constant must satisfy... ‖·‖ * Represents the nuclear norm number, in order to ensure The three matrix nuclear norms It can be optimized independently, and this problem introduces... Three auxiliary tensors express The set of position indices of non-zero elements Indicates that the tensor is in Mapping on, Represent the zero tensor; introduce dual variables If b = 1, 2, 3, then the Lagrangian function of the above optimization problem can be expressed as: where p > 0 denotes a compensation factor, [ · x · ] denotes the tensor inner product, and || · || F denotes the Frobenius norm F denotes the Frobenius norm; the objective variable is solved iteratively by minimizing the Lagrangian function obtaining the filled structured virtual domain tensor (7) structured virtual domain tensor after padding Theoretically modeled as: wherein is a spatial factor, respectively represent a virtual uniform cubic array guiding vectors along the x-axis and y-axis directions, are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of translational windowing and intercepting the virtual domain sub-tensors, respectively; and performing canonical polyadic decomposition to obtain estimated values of three factor vectors p(μ k ,ν k ), q(μ k ,v k ) and c(μ k ,v k ), denoted as and constructing a structured virtual domain tensor signal subspace where orth(·) denotes the matrix orthogonalization operation; and denotes the noise subspace, by V s is obtained. where I denotes the identity matrix, (·) H denotes the conjugate transpose operation; Traversing two-dimensional boresight directions θ and are azimuth and elevation angles, respectively, that are traversed in the value ranges [-90°, 90°] and [0°, 180°], respectively, the corresponding parameters are calculated and a corresponding virtual uniform cubic array is constructed with steering vector is expressed as: a spatial spectrum of the corresponding two-dimensional direction of arrival is given by: is given by:

2. The super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling according to claim 1, characterized in that, The coprime planar array structure described in step (1) is specifically described as follows: a pair of sparse and uniform sub-planar arrays are constructed on a planar coordinate system xoy and wherein contains 2M x ×2M y antenna elements, the element spacing in the x-axis direction and the y-axis direction are N x d and N y d respectively, and the position coordinates on the xoy are {(N x dm x ,N y dm y ), m x =0, 1,..., 2M x -1, m y =0, 1,..., 2M y -1}; contains N x ×N y antenna elements, the element spacing in the x-axis direction and the y-axis direction are M x d and M y d respectively, and the position coordinates on the xoy are {(M x dn x ,M y dn y ), n x =0, 1,..., N x -1, n y =0, 1,..., N y -1};M x , N x and M y , N y are a pair of coprime integers respectively; and and are combined into a subarray in a manner that the elements overlap at the position (0, 0) of the coordinate system, thereby obtaining a coprime planar array actually containing 4M x M y +N x N y -1 physical antenna elements.

3. The super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling according to claim 1, characterized in that, The cross-correlation tensor derivation described in step (2) in practice, By estimating the cross-correlation statistics of the tensor and one obtains, i.e. uses the sampled cross-correlation tensor instead of the cross-correlation tensor 4. The super-resolution coprime planar array spatial spectrum estimation method based on optimal structured virtual domain tensor filling according to claim 1, characterized in that, In step (6), the Lagrangian function The target variable is solved iteratively At the η+1 iteration, and is updated to: Target variable The closed-form solution is: wherein, denotes a threshold singular value decomposition operation of a matrix min(X1,X2) denotes the singular values of X, U X denotes the left singular matrix of X, V X denotes the right singular matrix of X, fold (b) [·] denotes the tensor expansion [·] (b) denotes the inverse operation of, diag(c) denotes a diagonal matrix with the elements of the vector c as diagonal elements, max(·) denotes the maximum operation, min(·) denotes the minimum operation.​

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