Two-dimensional Direction of Arrival Estimation Method for Mutually Prime Arrays Based on Virtual Domain Tensor Filling

Abstract

The present invention discloses a two-dimensional direction of arrival estimation method for coprime arrays based on virtual domain tensor filling, which mainly solves the problems of multi-dimensional signal structure information loss and virtual domain statistics that cannot be fully utilized in existing methods. The implementation steps are: constructing coprime arrays; tensor modeling of coprime array received signals; constructing an augmented non-continuous virtual array based on coprime array cross-correlation tensor transformation; deriving a virtual domain tensor based on the mirror expansion of the non-continuous virtual array; dispersing its missing elements in pieces through virtual domain tensor reconstruction; filling virtual domain tensors based on tensor nuclear norm minimization; and decomposing the filled virtual domain tensor to obtain the direction of arrival estimation result. The present invention makes full use of all non-continuous virtual domain tensor statistics information corresponding to coprime arrays to achieve high-precision direction of arrival estimation, which can be used for target positioning.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and in particular to a statistical signal processing technology based on second-order high-dimensional statistics of a sparse array virtual domain, and specifically is a two-dimensional direction-of-arrival estimation method for a coprime array based on virtual domain tensor filling, which can be used for target positioning. Background Art

[0002] As a sparse array with a systematic structure, the coprime array has the advantages of large aperture and high resolution, and can break through the performance bottleneck of the traditional uniform array DOA estimation in terms of estimation performance and cost overhead. Since the sparsely arranged array elements of the coprime array do not meet the Nyquist sampling rate, in order to achieve Nyquist matching of DOA estimation, the common practice is to calculate the second-order statistics of the coprime array received signal to construct an augmented non-continuous virtual array, and extract the continuous part from it to achieve Nyquist matching processing based on the second-order equivalent signal in the virtual domain. Furthermore, in order to make full use of all non-continuous virtual array elements, the existing method fills the non-continuous virtual array to achieve improved DOA estimation performance. However, the above method usually represents the received signal as a vector and derives the virtual domain second-order equivalent signal by vectorizing the received signal covariance matrix; in the scenario of deploying a coprime array, since the coprime array received signal covers two-dimensional spatial information, this vectorized signal processing method destroys the original spatial information structure of the coprime array received signal, which will cause serious performance loss.

[0003] In order to preserve the structured information of multi-dimensional received signals, tensors, as a multi-dimensional data type, have begun to be applied in the field of array signal processing to characterize received signals covering multi-dimensional spatial information, and to perform feature analysis and effective information extraction, thereby achieving high-precision and high-resolution DOA estimation. However, when it comes to the statistical processing of virtual domain tensors of coprime arrays, the augmented multi-dimensional non-continuous virtual array will have whole pieces of holes, resulting in the existence of missing elements in the corresponding virtual domain tensor. Traditional tensor filling methods used in image restoration are based on the premise of the random distribution of missing elements in the tensor, so they cannot effectively fill the virtual domain tensor. Therefore, for the virtual domain tensor model of coprime arrays, how to effectively utilize all the non-continuous virtual domain tensor statistical information to achieve high-precision and high-resolution two-dimensional DOA estimation is still an urgent problem to be solved. Summary of the invention

[0004] The purpose of the present invention is to address the problems of multi-dimensional signal structured information loss and inability to fully utilize virtual domain statistics in existing methods, and to propose a two-dimensional direction of arrival estimation method for coprime arrays based on virtual domain tensor filling, which provides a feasible idea and effective solution for fully utilizing all non-continuous virtual domain tensor statistics information corresponding to coprime arrays to achieve high-precision and high-resolution two-dimensional direction of arrival estimation based on Nyquist matching.

[0005] The object of the present invention is achieved through the following technical solution: a method for estimating two-dimensional direction of arrival of mutually prime arrays based on virtual domain tensor filling, the method comprising the following steps:

[0006] (1) The receiving end uses 4M x M y +N x N y -1 physical antenna array element, which is constructed according to the structure of a mutually prime array; where M x 、N x and M y 、N y are a pair of coprime integers respectively; the coprime matrix is ​​decomposed into two sparse uniform sub-matrices and in Contains 2M x ×2M y There are antenna array elements, and the array element spacing in the x-axis direction and the y-axis direction is N x d and N y d, Contains N x ×N y The antenna array elements have an array element spacing of M in the x-axis direction and M in the y-axis direction. x d and M y d, the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, that is, d = λ / 2;

[0007] (2) Assume that there are K Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, respectively, k = 1, 2, ..., K, then the sparse uniform sub-array The T sample snapshot signal is represented by a three-dimensional tensor It is expressed as:

[0008]

[0009] Among them, s k =[s k,1 ,s k,2 ,…,s k,T ]T is the multi-snap sampling signal waveform corresponding to the kth incident signal source, [·] T represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as:

[0010]

[0011]

[0012] in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and

[0013] Sparse uniform sub-array The T sample snapshot signal is represented by another three-dimensional tensor It is expressed as:

[0014]

[0015] in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as:

[0016]

[0017]

[0018] in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and

[0019] By finding the three-dimensional tensor and The cross-correlation statistics of

[0020]

[0021] in, 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 mathematical expectation operation, (·) * represents the conjugate operation; the cross-correlation noise tensor Only at the (1,1,1,1)th position there is a value Elements of represents the noise power, while the elements at other positions are all 0;

[0022] (3) Define dimension sets By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a virtual domain signal

[0023]

[0024] in, and By forming difference arrays on the exponential terms, a two-dimensional augmented virtual array along the x-axis and y-axis is constructed. represents the Kronecker product; therefore, Corresponding to a size of Non-continuous virtual array The non-continuous virtual array It contains holes in entire rows and columns, i.e. missing elements;

[0025] (4) Constructing a non-continuous virtual array Virtual Array Mirrored on Coordinate Axis and will and In the third dimension, it is superimposed into a A three-dimensional non-contiguous virtual cube array here, and Correspondingly, the virtual domain signal The conjugate transposed signal Rearrange the elements in to correspond to The position of the virtual array element in the virtual array is obtained Virtual domain signal Will and Superposition is performed in the third dimension to obtain the corresponding non-continuous virtual cubic array Virtual domain tensor of It is expressed as:

[0026]

[0027] in, and Discontinuous virtual cube arrays The steering vectors on the x-axis and y-axis correspond to the incoming wave direction. The signal source is There are holes in and Corresponding to The elements of the hole positions in the x-axis and y-axis directions are set to zero.

[0028]

[0029] Indicates that it corresponds to and The mirror transformation factor vector; due to the non-continuous virtual array contains holes for entire rows and columns, and its mirror image Non-continuous virtual cube array obtained by superposition contains patches of holes, corresponding to non-continuous virtual cube arrays Virtual domain tensor of This includes the missing elements of the whole piece;

[0030] (5) Design a size P x ×P y ×2 translation window to select virtual domain tensor A sub-tensor of It includes The indexes in the three dimensions are (1:P x -1), (1:P y -1), (1:2); then, translate the window one element along the x-axis and y-axis respectively. Split into L x ×L y sub-tensors, represented as s x =1,2,…,L x ,s y =1,2,…,L y ; The value range of the translation window size is:

[0031]

[0032]

[0033] And L x , Ly , P x , P y The following relationship is satisfied:

[0034]

[0035]

[0036] will have the same y Index sub-tensor Superimposing in the fourth dimension, we get L y The dimension is P x ×P y ×2×L x ; further, this L y The four-dimensional tensors are superimposed 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, spatial mirror transformation information, and spatial translation information in the x-axis and y-axis directions; defines the dimension set Then through The dimensions of the virtual domain are merged to obtain a three-dimensional reconstructed virtual domain tensor

[0037]

[0038] The three dimensions of represent the spatial angle information, spatial translation information and spatial mirror transformation information respectively. Therefore, the original virtual domain tensor The missing elements in the patch are randomly distributed to In the three spatial dimensions covered;

[0039] (6) Design a virtual domain tensor filling optimization problem based on minimization of the tensor nuclear norm:

[0040]

[0041]

[0042] Among them, the optimization variables is the padded virtual domain tensor, corresponding to a virtual uniform cubic array ‖·‖ * represents the tensor nuclear norm, express The position index set of non-missing elements in , Represents a tensor in Mapping on; Since the nuclear norm is a convex function, the virtual domain tensor filling problem based on minimizing the tensor nuclear norm is a solvable convex optimization problem. Solving this convex optimization problem, we get

[0043] (7) Filled virtual domain tensor It is expressed as:

[0044]

[0045] in, for The spatial factor,

[0046]

[0047]

[0048] They represent virtual uniform cubic arrays The steering vectors along the x-axis and y-axis,

[0049]

[0050]

[0051] are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of translating the window to intercept the sub-tensor; for the filled virtual domain tensor Perform canonical polyadic decomposition to obtain three factor vectors p k ,q k and c k The estimated value of and Extract from it contained in and The angle parameter of the exponential term is used to obtain the two-dimensional direction of arrival estimation result.

[0052] Furthermore, the mutually prime array structure described in step (1) is specifically described as follows: construct a pair of sparse uniform sub-arrays on the plane coordinate system xoy and in Contains 2M x ×2M y There are antenna array elements, and the array element spacing in the x-axis direction and the y-axis direction is N x d and N y d, whose position coordinates on xoy are {(N x dm x ,N y dm y ),mx =0,1,...,2M x -1,m y =0,1,...,2M y -1}; Contains N x ×N y The antenna array elements have an array element spacing of M in the x-axis direction and M in the y-axis direction. x d and M y d, whose position coordinates on 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 relatively prime integers respectively; and The subarrays are combined in such a way that the array elements at the coordinate system (0,0) overlap, and the actual array containing 4M x M y +N x N y -1 coprime array of physical antenna elements.

[0053] Furthermore, the derivation of the cross-correlation tensor described in step (2) is, in practice, By calculating the tensor and The cross-correlation statistics of are approximated by the sampled cross-correlation tensor

[0054]

[0055] Furthermore, in step (7), the filled virtual domain tensor Perform canonical polyadic decomposition to obtain factor vector and Then the parameter and from and Extracted as:

[0056]

[0057]

[0058] Among them, ∠(·) represents the complex number argument operation, a (a) represents the ath element of a vector a; here, according to and The Kronecker structure, η 1 ∈[1,P x P y -1] and η 2 ∈[1,L x L y -1] respectively satisfy mod(η 1 ,Px)≠0 and mod(η 2 ,P y )≠0, and δ 1 ∈[1,P x P y -P x ],δ 2 ∈[1,L x L y -L x ], mod(·) represents the remainder operation; according to the parameter (μ k ,v k ) and the two-dimensional direction of arrival The relationship between , we can get the two-dimensional direction of arrival estimation The closed-form solution is:

[0059]

[0060]

[0061] Compared with the prior art, the present invention has the following advantages:

[0062] (1) The present invention derives an augmented non-continuous virtual array based on the cross-correlation tensor, and uses the mirror image expansion of the non-continuous virtual array to construct a three-dimensional non-continuous virtual cubic array and its corresponding virtual domain tensor, which fully retains the structural information of all non-continuous virtual domain statistics of the coprime array;

[0063] (2) The present invention proposes a virtual domain tensor filling mechanism for non-continuous virtual arrays. The missing elements are dispersed through virtual domain tensor reconstruction to meet the low-rank fillability of the virtual domain tensor, thereby effectively filling the virtual domain tensor and achieving high-precision and high-resolution two-dimensional wave direction of arrival estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 It is the overall flow chart of the present invention.

[0065] Figure 2 It is a schematic diagram of the coprime array structure constructed by the present invention.

[0066] Figure 3 It is a schematic diagram of the augmented non-continuous virtual array derived by the present invention.

[0067] Figure 4 It is a schematic diagram of a non-continuous virtual cubic array constructed by the present invention.

[0068] Figure 5 This is a comparison chart of the accuracy performance of the direction of arrival estimation of the method proposed in the present invention under different signal-to-noise ratio conditions.

[0069] Figure 6 This is a comparison chart of the accuracy performance of the direction of arrival estimation of the method proposed in the present invention under different sampling snapshot numbers. DETAILED DESCRIPTION

[0070] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0071] In order to solve the problems of multi-dimensional signal structure information loss and inability to fully utilize virtual domain statistics in existing methods, this paper proposes a two-dimensional direction of arrival estimation method for coprime arrays based on virtual domain tensor filling, which effectively fills the missing elements of the original virtual domain tensor to achieve Nyquist matching of coprime array two-dimensional direction of arrival estimation. Figure 1 , the implementation steps of the present invention are as follows:

[0072] Step 1: Construct a coprime array. Use 4M at the receiving end x M y +N x N y -1 physical antenna element to construct a coprime array, such as Figure 2 As shown: Construct a pair of sparse uniform sub-arrays on the plane coordinate system xoy and in Contains 2M x ×2M y There are antenna array elements, and the array element spacing in the x-axis direction and the y-axis direction is N x d and N y d, whose position coordinates on 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 The antenna array elements have an array element spacing of M in the x-axis direction and M in the y-axis direction. xd and M y d, whose position coordinates on 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 mutually prime integers respectively; the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, that is, d = λ / 2; and The subarrays are combined in such a way that the array elements at the coordinate system (0,0) overlap, and the actual array containing 4M x M y +N x N y -1 mutually prime array of physical antenna elements;

[0073] Step 2: Tensor modeling of the mutually prime array receiving signal. Assume that there are K signals from Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, k = 1, 2, ..., K, and the sparse uniform sub-arrays in the coprime arrays are After the T sampling snapshot signals are superimposed in the third dimension, a three-dimensional tensor signal can be obtained. Modeled as:

[0074]

[0075] Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T is the multi-snap sampling signal waveform corresponding to the kth incident signal source, [·] T represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as:

[0076]

[0077]

[0078] in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and

[0079] Similarly, the sparse uniform sub-array The received signal can be expressed as a three-dimensional tensor express:

[0080]

[0081] in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as:

[0082]

[0083]

[0084] in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and

[0085] By finding the three-dimensional tensor signal and The cross-correlation statistics of

[0086]

[0087] in, represents the power of the kth incident signal source, represents the four-dimensional cross-correlation noise tensor, <·,·> r represents the tensor contraction operation of two tensors along the rth dimension, E[·] represents the mathematical expectation operation, (·) * represents the conjugate operation. Here, the cross-correlation noise tensor Only at the (1,1,1,1)th position there is a value Elements of represents the noise power, and the elements at other positions are all 0. In practice, By calculating the tensor signal and The cross-correlation statistics of are approximated by the sampled cross-correlation tensor

[0088]

[0089] Step 3: Construct an augmented non-continuous virtual array based on the cross-correlation tensor transformation of the coprime arrays. Contains the corresponding two sparse uniform sub-arrays and By merging the spatial information The dimension of spatial information in the same direction can make the steering vectors corresponding to two sparse uniform sub-arrays form a difference array in the exponential term, thereby constructing a two-dimensional augmented virtual array. Specifically, the cross-correlation tensor The first and third dimensions (through the guidance vector and Represents) the spatial information in the x-axis direction, the 2nd and 4th dimensions (through the guidance vector and Represents) represents the spatial information in the y-axis direction; for this purpose, the dimension set is defined By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a virtual domain signal

[0090]

[0091] in, and Equivalent to a non-continuous virtual array The steering vectors on the x-axis and y-axis correspond to the incoming wave direction. The signal source, represents the Kronecker product. Non-continuous virtual array Size And contains holes in the entire row and column (i.e. missing elements), such as Figure 3 As shown, Here, in order to simplify the derivation process, the cross-correlation noise tensor In About is omitted in the theoretical modeling step; however, in practice, due to the use of the sampled cross-correlation tensor Alternative Theory Cross-Correlation Tensor It is still covered in the virtual domain signal statistical processing;

[0092] Step 4: Derive the virtual domain tensor based on the mirror extension of the non-continuous virtual array. Construct a non-continuous virtual array Virtual Array Mirrored on Coordinate Axis and will and In the third dimension, it is superimposed into a A three-dimensional non-contiguous virtual cube array like Figure 4 Here, and Correspondingly, the virtual domain signal The conjugate transposed signal Arrange the elements in to correspond to The position of the virtual array element in the virtual array can be obtained Virtual domain signal Will and Superposition is performed in the third dimension to obtain the corresponding non-continuous virtual cubic array Virtual domain tensor of It is expressed as:

[0093]

[0094] in, and Discontinuous virtual cube arrays The steering vectors on the x-axis and y-axis correspond to the incoming wave direction. signal source; due to There are missing elements (holes) in and Corresponding to The elements of the hole positions in the x-axis and y-axis directions are set to zero.

[0095]

[0096] Indicates the corresponding and The mirror transformation factor vector; due to the non-continuous virtual array contains holes for entire rows and columns, and its mirror image Non-continuous virtual cube array obtained by superposition contains patches of holes, corresponding to non-continuous virtual cube arrays Virtual domain tensor of This includes the missing elements of the whole piece;

[0097] Step 5: Distribute the missing elements through virtual domain tensor reconstruction. In order to construct a virtual uniform cubic array for Nyquist-matched signal processing, it is necessary to reconstruct the virtual domain tensor. Fill in the missing elements in the array, thus corresponding to a virtual uniform cubic array However, low-rank tensor filling technology is based on the premise of random distribution of missing elements in the tensor, and cannot be used for virtual domain tensors with large-scale missing elements. To do this, we reconstruct the virtual domain tensor Disperse the missing elements into pieces. The specific process is: by designing a size of P x ×P y ×2 translation window to select virtual domain tensor A sub-tensor of It includes The indexes in the three dimensions are (1:P x -1), (1:P y -1), (1:2); then, translate the translation window along the x-axis and y-axis by one element respectively, then Split into L x ×L y sub-tensors, represented as s x =1,2,…,L x ,s y =1,2,…,L y The range of the translation window size is:

[0098]

[0099]

[0100] And L x , L y , P x , P y The following relationship is satisfied:

[0101]

[0102]

[0103] will have the same y Index sub-tensor Superimposing in the fourth dimension, we get L y The dimension is P x ×P y ×2×L x ; further, this L tThe four-dimensional tensors are superimposed in the fifth dimension to obtain a five-dimensional virtual domain tensor. This five-dimensional virtual domain tensor It 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; Merge along the 1st and 2nd dimensions representing the spatial angle information, merge along the 4th and 5th dimensions representing the spatial translation information, and retain the 3rd dimension representing the spatial mirror transformation information. The specific operation is: define the dimension set Then through The dimensions of the 3D reconstructed virtual domain tensor can be obtained by merging

[0104]

[0105] The three dimensions of represent the spatial angle information, spatial translation information and spatial mirror transformation information respectively. Therefore, the virtual domain tensor The missing elements in the patch are randomly distributed to In the three spatial dimensions covered;

[0106] Step 6: Virtual domain tensor filling based on minimization of tensor nuclear norm. In order to reconstruct the virtual domain tensor To fill in, we design the following virtual domain tensor filling optimization problem based on minimization of the tensor nuclear norm:

[0107]

[0108]

[0109] Among them, the optimization variables is the padded virtual domain tensor, corresponding to a virtual uniform cubic array ‖·‖ * represents the tensor nuclear norm, express The position index set of non-missing elements in , Represents a tensor in Since the nuclear norm is a convex function, the virtual domain tensor filling problem based on minimizing the tensor nuclear norm is a solvable convex optimization problem. Solving this convex optimization problem, we can get

[0110] Step 7: Decompose the filled virtual domain tensor to obtain the DOA estimation result. It can be expressed as:

[0111]

[0112] in, for The spatial factor,

[0113]

[0114]

[0115] They represent virtual uniform cubic arrays The steering vectors along the x-axis and y-axis,

[0116]

[0117]

[0118] are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of translating the window to intercept the sub-tensor. Perform canonical polyadic decomposition to obtain three factor vectors p k ,q k and c k The estimated value of and Then the parameter and Available from and Extracted from:

[0119]

[0120]

[0121] Among them, ∠(· represents the complex number argument operation, a (a) represents the ath element of a vector a; here, according to and The Kronecker structure, η 1 ∈[1,P x P y -1] and η 2 ∈[1,L x L y -1] respectively satisfy mod(η 1 ,P x )≠0 and mod(η 2 ,P y )≠0, and δ 1 ∈[1,P x P y -P x ],δ 2 ∈[1,L x Ly -L x ], mod(·) represents the remainder operation. According to the parameter (μ k ,v k ) and the two-dimensional direction of arrival The relationship between , we can get the two-dimensional direction of arrival estimation The closed-form solution is:

[0122]

[0123]

[0124] The effects of the present invention are further described below in conjunction with a simulation example.

[0125] Simulation example: Using a coprime array to receive the incident signal, its parameters are selected as M x =2,M y =3, N x =3, N y =4, that is, the coprime array of the architecture contains 4M x M y +N x N y -1=35 physical array elements. The translation window size of the sub-tensor is 6×15×2. Assuming that there are two narrowband incident signals, the azimuth and elevation angles of the incident direction are [30.6°, 25.6°] and [40.5°, 50.5°] respectively. The two-dimensional wave arrival direction estimation method of the coprime array based on virtual domain tensor filling proposed in the present invention is compared with the traditional multiple signal classification (Multiple Signal Classification, MUSIC) method and tensor multiple signal classification (Tensor MUSIC) method which only uses the continuous part of the virtual domain. Under the condition of sampling snapshot number T=300, the performance comparison curve of the root mean square error (Root-mean-square Error, RMSE) changing with the signal-to-noise ratio SNR is plotted, as shown in FIG. Figure 5 As shown; under the condition of SNR = 0dB, the performance comparison curve of RMSE changing with the number of sampling snapshots T is plotted, as shown Figure 6 shown.

[0126] from Figure 5 and Figure 6From the comparison results, it can be seen that the method proposed in the present invention has performance advantages in the accuracy of DOA estimation, whether in different expected signal-to-noise ratio (SNR) scenarios or in different sampling snapshot numbers (T) scenarios. Compared with the traditional MUSIC method, the method proposed in the present invention fully utilizes the structured information of the coprime array receiving signal by constructing a virtual domain tensor, thereby having better DOA estimation performance; compared with the Tensor MUSIC method, the performance advantage of the method proposed in the present invention comes from the use of all non-continuous virtual domain statistical information through virtual domain tensor filling, while the Tensor MUSIC method only extracts the continuous part of the non-continuous virtual array for virtual domain signal processing, resulting in the loss of virtual domain statistical information.

[0127] In summary, the present invention realizes the random distribution of missing elements in pieces through virtual domain tensor reconstruction, and on this basis, designs a virtual domain tensor filling method based on minimization of the tensor nuclear norm, successfully utilizes all non-continuous virtual domain statistical information, and realizes high-precision two-dimensional wave arrival direction estimation of coprime arrays.

[0128] The above is only a preferred embodiment of the present invention. Although the present invention has been disclosed as a preferred embodiment, it is not intended to limit the present invention. Any technician familiar with the art can make many possible changes and modifications to the technical solution of the present invention by using the above disclosed methods and technical contents without departing from the scope of the technical solution of the present invention, or modify it into an equivalent embodiment of equivalent changes. Therefore, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solution of the present invention still falls within the scope of protection of the technical solution of the present invention.

Claims

1. A two-dimensional direction of arrival estimation method for mutually prime arrays based on virtual domain tensor filling, characterized in that: The following steps are involved: (1) The receiving end uses 4M x M y +N x N y -1 physical antenna array element, which is constructed according to the structure of a mutually prime array; where M x 、N x and M y 、N y are a pair of coprime integers respectively; the coprime matrix is ​​decomposed into two sparse uniform sub-matrices and in Contains 2M x ×2M y There are antenna array elements, and the array element spacing in the x-axis direction and the y-axis direction is N x d and N y d, Contains N x ×N y The antenna array elements have an array element spacing of M in the x-axis direction and M in the y-axis direction. x d and M y d, the unit interval d is taken as half of the wavelength λ of the incident narrowband signal, that is, d = λ / 2; (2) Assume that there are K Far-field narrowband uncorrelated signal source in the direction, θ k and are the azimuth and elevation angles of the kth incident signal source, respectively, k = 1, 2, ..., K, then the sparse uniform sub-array The T sample snapshot signal is represented by a three-dimensional tensor It is expressed as: Among them, s k =[s k，1 ,s k，2 ,…,s k，T ] T is the multi-snap sampling signal waveform corresponding to the kth incident signal source, [·] T represents the transpose operation, represents the vector outer product, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as: in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and Sparse uniform sub-array The T sample snapshot signal is represented by another three-dimensional tensor It is expressed as: in, is the noise tensor independent of each signal source, and They are The steering vectors in the x-axis and y-axis directions correspond to the incoming wave directions. The signal source is expressed as: in, and Represent the sparse uniform sub-arrays The actual position of the physical antenna elements in the x- and y-axis directions, and By finding the three-dimensional tensor and The cross-correlation statistics of in, 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 mathematical expectation operation, (·) * represents the conjugate operation; the cross-correlation noise tensor Only at the (1, 1, 1, 1)th position there exists a value Elements of represents the noise power, while the elements at other positions are all 0; (3) Define dimension sets By cross-correlation tensor Perform dimension-merged tensor transformation to obtain a virtual domain signal in, and By forming difference arrays on the exponential terms, a two-dimensional augmented virtual array along the x-axis and y-axis is constructed. represents the Kronecker product; therefore, Corresponding to a size of Non-continuous virtual array The non-continuous virtual array It contains holes in entire rows and columns, i.e. missing elements; (4) Constructing a non-continuous virtual array Virtual Array Mirrored on Coordinate Axis and will and In the third dimension, it is superimposed into a A three-dimensional non-contiguous virtual cube array here, and Correspondingly, the virtual domain signal The conjugate transposed signal Rearrange the elements in to correspond to The position of the virtual array element in the virtual array is obtained Virtual domain signal Will and Superposition is performed in the third dimension to obtain the corresponding non-continuous virtual cubic array Virtual domain tensor of It is expressed as: in, and Non-contiguous virtual cube arrays The steering vectors on the x-axis and y-axis correspond to the incoming wave direction. The signal source is There are holes in and Corresponding to The elements of the hole positions in the x-axis and y-axis directions are set to zero. Indicates that it corresponds to and The mirror transformation factor vector; due to the non-continuous virtual array contains holes for entire rows and columns, and its mirror image Non-continuous virtual cube array obtained by superposition contains patches of holes, corresponding to non-continuous virtual cube arrays Virtual domain tensor of This includes the missing elements of the whole piece; (5) Design a size P x ×P y ×2 translation window to select virtual domain tensor A sub-tensor of Included The indexes in the three dimensions are (1∶P x -1), (1∶P y -1), (1:2); then, the translation window is translated one element along the x-axis and y-axis directions respectively, Split into L x ×L y sub-tensors, represented as The value range of the translation window size is: And L x , L y , P x , P y The following relationship is satisfied: will have the same y Index sub-tensor Superimposing in the fourth dimension, we get L y The dimension is P x ×P y ×2×L x ; further, this L y The four-dimensional tensors are superimposed 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, spatial mirror transformation information, and spatial translation information in the x-axis and y-axis directions; defines the dimension set Then through The dimensions of the virtual domain are merged to obtain a three-dimensional reconstructed virtual domain tensor The three dimensions of represent the spatial angle information, spatial translation information and spatial mirror transformation information respectively. Therefore, the original virtual domain tensor The missing elements in the patch are randomly distributed to In the three spatial dimensions covered; (6) Design a virtual domain tensor filling optimization problem based on minimization of the tensor nuclear norm: Among them, the optimization variables is the padded virtual domain tensor, corresponding to a virtual uniform cubic array ||·|| * represents the tensor nuclear norm, express The position index set of non-missing elements in , Represents a tensor in Mapping on; Since the nuclear norm is a convex function, the virtual domain tensor filling problem based on minimizing the tensor nuclear norm is a solvable convex optimization problem. Solving this convex optimization problem, we get (7) Filled virtual domain tensor It is expressed as: in, for The spatial factor, They represent virtual uniform cubic arrays The steering vectors along the x-axis and y-axis, are the spatial translation factor vectors corresponding to the x-axis and y-axis directions in the process of translating the window to intercept the sub-tensor; for the filled virtual domain tensor Perform canonical polyadic decomposition to obtain three factor vectors p k ,q k and c k The estimated value of and Extract from it contained in and The angle parameter of the exponential term is used to obtain the two-dimensional direction of arrival estimation result.

2. The method for estimating two-dimensional direction of arrival of mutually prime arrays based on virtual domain tensor filling according to claim 1 is characterized in that: The mutually prime array structure described in step (1) is specifically described as: constructing a pair of sparse uniform sub-arrays on the plane coordinate system xoy and in Contains 2M x ×2M y There are antenna array elements, and the array element spacing in the x-axis direction and the y-axis direction is N x d and N y d, whose position coordinates on 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 The antenna array elements have an array element spacing of M in the x-axis direction and M in the y-axis direction. x d and M y d, whose position coordinates on xoy are {(M x dn x , M y dn y ), nx=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 mutually prime integers; and Subarrays are combined in such a way that the array elements at the coordinate system (0, 0) overlap, and the actual array containing 4M x M y +N x N y -1 coprime array of physical antenna elements.

3. The method for estimating two-dimensional direction of arrival of mutually prime arrays based on virtual domain tensor filling according to claim 1, characterized in that: The derivation of the cross-correlation tensor described in step (2) is, in practice, By calculating the tensor and The cross-correlation statistics of are approximated by the sampled cross-correlation tensor 4. The method for estimating two-dimensional direction of arrival of mutually prime arrays based on virtual domain tensor filling according to claim 1, characterized in that: In step (7), the filled virtual domain tensor Perform canonical polyadic decomposition to obtain factor vector and Then the parameter and from and Extracted as: Among them, ∠(·) represents the complex number argument operation, a (a) represents the ath element of a vector a; here, according to and The Kronecker structure, η1∈[1,P x P y -1] and η2∈[1, L x L y -1] respectively satisfy mod(η1, P x )≠0 and mod(η2,P y )≠0, and δ1∈[1,P x P y -P x ],δ2∈[1,L x L y -L x ], mod(·) represents the remainder operation; according to the parameter (μ k , v k ) and the two-dimensional direction of arrival The relationship between , we can get the two-dimensional direction of arrival estimation The closed-form solution is:

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