CN107102291B - The relatively prime array Wave arrival direction estimating method of mesh freeization based on virtual array interpolation - Google Patents

Abstract

The invention discloses a method for estimating the direction of arrival of a gridless coprime array based on virtual array interpolation, which mainly solves the problem of information loss caused by the non-uniformity of virtual arrays in the prior art. The implementation steps are: constructing a coprime array at the receiving end; using the coprime array to receive the incident signal and modeling; calculating the equivalent virtual signal corresponding to the signal received by the coprime array; constructing an interpolation virtual array and modeling; constructing an interpolation virtual array Multi-sampling snapshot signal of the array and its sampling covariance matrix; construct the projection matrix and define the projection operation related to the projection matrix; design and solve the optimization problem based on the minimization of the kernel norm of the interpolated virtual array signal covariance matrix; according to The reconstructed interpolated virtual array covariance matrix is used for DOA estimation. The invention improves the degree of freedom and resolution of the direction of arrival estimation, and can be used for passive positioning and target detection.

Description

Direction of Arrival Estimation Method for Meshless Coprime Array Based on Virtual Array Interpolation

technical field

The invention belongs to the technical field of signal processing, in particular to the estimation of direction of arrival of radar signals, acoustic signals and electromagnetic signals, in particular to a method for estimation of direction of arrival of a gridless coprime array based on virtual array interpolation, which can be used for Passive localization and object detection.

Background technique

Direction-of-Arrival (DOA) estimation is an important branch in the field of array signal processing. It can effectively process the signal to realize DOA estimation of the signal, which has important application value in the fields of radar, sonar, voice, wireless communication and so on.

The degree of freedom of the DOA estimation method refers to the number of incident signal sources it can estimate. The existing DOA estimation methods usually use uniform linear arrays for signal reception and modeling, but the degree of freedom of the methods based on uniform linear arrays is limited by the actual number of antenna elements. Specifically, for a uniform linear array containing L antenna elements, the degree of freedom is L-1. Therefore, when the number of incident signal sources in a certain spatial range is greater than or equal to the number of antenna elements in the array, the existing methods using uniform linear arrays cannot perform effective DOA estimation.

The coprime array can increase the DOA estimation degree of freedom under the premise of a certain number of antenna elements, so it has received extensive attention from the academic community. As a typical manifestation of coprime sampling technology in the spatial domain, coprime array provides a systematic sparse array architecture solution, and can break through the bottleneck of limited degrees of freedom of traditional uniform linear arrays and achieve DOA estimation method. improvement. The existing DOA estimation methods based on coprime arrays mainly derive the coprime arrays into the virtual domain by using the properties of prime numbers, and form an equivalent virtual uniform linear array to receive signals to achieve DOA estimation. Since the number of virtual array elements contained in the virtual array is larger than the actual number of antenna array elements, the degree of freedom is effectively improved. However, since the virtual array derived from the coprime array is a non-uniform array, many existing signal processing methods based on uniform linear arrays cannot be directly applied to the virtual array to equivalently receive signals to achieve effective DOA estimation. A commonly used solution for DOA estimation methods using coprime arrays is to use only the continuous elements in the virtual array to form a virtual uniform linear array for DOA estimation, but this results in the loss of part of the original information and the performance of correlation estimation. decrease.

At the same time, many DOA estimation methods currently need to pre-set the spatial grid points of the assumed direction of arrival of the signal in the design process of the optimization problem. With the improvement of the accuracy requirements for direction of arrival estimation results, the pre-set spatial grid points for these DOA estimation methods will become denser and denser, which leads to a sharp increase in computational complexity. Not only that, but in practical situations, it is inevitable that the direction of arrival of some signals cannot completely fall on the preset grid points, resulting in inherent model mismatch errors.

SUMMARY OF THE INVENTION

The purpose of the present invention is to propose a method for estimating the direction of arrival of a gridless coprime array based on virtual array interpolation, which fully utilizes all the information provided by the non-uniform virtual array, and It ensures the estimation of direction of arrival without meshing, thereby improving the DOA estimation degree of freedom and resolution, and reducing the computational complexity of DOA estimation to a certain extent.

The object of the present invention is achieved through the following technical solutions: a method for estimating the direction of arrival of a gridless coprime array based on virtual array interpolation, comprising the following steps:

(1) The receiving end uses M+N-1 antennas, and is structured according to the co-prime array structure; where M and N are co-prime integers;

(2) Assuming that there are K far-field narrow-band incoherent signal sources from the directions of θ ₁ , θ ₂ ,..., θ _K , then the (M+N-1)×1-dimensional coprime array received signal x(t) can be constructed The model is:

Among them, _sk (t) is the signal waveform, n(t) is the noise component independent of each signal source, a(θ _k ) is the steering vector in the direction of θ _k , expressed as:

Among them, p _i d,i=1,2,...,M+N-1 represents the actual position of the i-th physical antenna element in the coprime array, and p ₁ =0; d is half of the incident narrowband signal wavelength λ , that is, d=λ/2, [ ] ^T represents the transpose operation. A total of T sampling snapshots are collected to obtain the sampling covariance matrix of the received signal of the coprime array

Here, ( ) ^H represents the conjugate transpose operation;

(3) Calculate the equivalent virtual signal corresponding to the received signal of the coprime array: the sampling covariance matrix of the received signal of the vectorized coprime array Obtain the virtual array equivalent received signal v:

in, is (M+N-1) ² ×K-dimensional virtual array steering matrix, contains the power of the K incident signal sources, is the noise power, i _v =vec( _IM+N-1 ). Here, vec( ) represents the vectorization operation, that is, stacking the columns in the matrix to form a new vector, ( ) ^* represents the conjugation operation, Represents the Kronecker product, and _IM+N-1 represents the (M+N-1)×(M+N-1) dimensional identity matrix. The position of each virtual array element in the virtual array corresponding to the vector v is

remove set The repeated virtual array elements at the corresponding positions of the repeated elements in , get a non-uniform virtual array Its corresponding equivalent virtual signal v _c can be obtained by selecting the element at the corresponding position in the vector v;

(4) Construct an interpolated virtual array and its received signal and model it: First, for a non-uniform virtual array Under the premise of keeping the original position of the virtual array element unchanged, insert several virtual array elements into the non-consecutive positions, so that the non-uniform virtual array Converted to a uniform virtual array with a spacing of d, the array aperture is the same as that of a coprime array, and the number of virtual array elements is increased The interpolated uniform virtual array contains a total of virtual array elements, where |·| represents the potential of the set, and the corresponding equivalent virtual signal v _I can be obtained by inserting 0 into the vector _vc , and the position of inserting 0 is the same as The position of the virtual array element inserted in corresponds to;

(5) Construct the interpolated virtual array multi-sampling snapshot signal and its sampling covariance matrix: cut into L _I contiguous subarrays of length L _I , where

Correspondingly, interpolate the virtual array The multi-sampled snapshot signal of can be obtained by intercepting the corresponding elements in the vector v _I , namely: v _I,l ,l=1,2,...,L _I consists of L _I +1-1th to 2L _I -1th elements in v _I. Next, the sampling covariance matrix R _v of _VI can be obtained as follows:

Wherein, <v _I > _i represents the equivalent received signal corresponding to the virtual array element whose position is id;

(6) Construct the projection matrix and define the projection operation: the dimension of the projection matrix P is the same as that of R _v . If an element in the matrix R _v is 0, the value of the element at the same position in the projection matrix P is also 0; otherwise, the projection matrix P The value of the element in the corresponding position is 1. definition It is the projection operation, in which the variable in parentheses is a matrix with the same dimension as P, and the projection operation is realized by multiplying each element in the variable matrix with the element at the corresponding position in the projection matrix P, and a matrix with the same dimension as the matrix P is obtained. ;

(7) Design and solve the optimization problem based on the minimization of the kernel norm of the interpolated virtual array signal covariance matrix. Using the interpolated virtual array covariance matrix R _v obtained in (5) as a reference value, find a Toeplitz matrix with the smallest nuclear norm as the covariance matrix of the interpolated virtual array signal, and the difference between it and R _v is required to be less than a certain Threshold, the following optimization problem with vector z as a variable can be constructed:

in, express The nuclear norm of , represents the Hermitian symmetric Toeplitz matrix with the vector z as the first column; ∈ is the threshold constant, which is used to constrain the reconstruction error of the covariance matrix; It is guaranteed that the reconstructed covariance matrix satisfies the condition of positive semi-definite; ‖·‖ _F represents the Frobenius norm. Solving the above convex optimization problem can get the optimal value Correspondingly, the reconstructed Toeplitz matrix is the interpolated virtual array covariance matrix;

(8) According to the reconstructed interpolated virtual array covariance matrix Do direction of arrival estimation.

Further, the coprime array structure described in step (1) can be specifically described as: first select a pair of coprime integers M, N; then, construct a pair of sparse uniform linear subarrays, wherein the first subarray contains M The antenna array elements with spacing Nd are located at 0,Nd,…,(M-1)Nd, and the second sub-array contains N antenna elements with spacing Md, whose positions are 0,Md,…,(N -1) Md; then, the two sub-arrays are combined in a manner that the first array element overlaps to obtain a non-uniform co-prime array structure that actually contains M+N-1 antenna elements.

Further, the sampling _covariance matrix R _v of VI constructed in step (5) can also be equivalently obtained by the following method:

Further, the convex optimization problem in step (7) can be transformed into the following optimization problem with vector z as a variable:

where μ is the regularization parameter used to trade off the matrix during minimization Reconstruction error and kernel norm of z.

Further, the DOA estimation in step (8) may adopt the following methods: multiple signal classification method, rotation invariant subspace method, multiple signal classification method for root finding, covariance matrix sparse reconstruction method, etc.

Further, in step 8, the direction of arrival is estimated by the multiple signal classification method, which is specifically: draw the virtual domain spatial spectrum P _MUSIC (θ):

where d(θ) is the L _I ×1-dimensional interpolation virtual array steering vector, corresponding to a segment of virtual uniform array from 0 to (L _I -1)d; _En is L _I ×(L _I -K ) dimension matrix, representing the interpolated virtual array covariance matrix θ is the assumed direction of arrival of the signal; find the peaks on the spatial spectrum P _MUSIC (θ) through spectral peak search, and arrange the response values corresponding to these peaks from large to small, and take the first K peaks The corresponding angular direction is the result of DOA estimation.

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

(1) The present invention introduces the idea of array interpolation in the equivalent virtual field of the coprime array, and makes full use of all the information provided by the virtual array. A uniform linear virtual array is constructed by interpolating virtual array elements in the non-uniform virtual array. While retaining all the information received by the original non-uniform virtual array, the constructed virtual domain signal model satisfies the Nyquist signal model. sampling law;

(2) The present invention designs the optimization problem based on the idea of minimizing the kernel norm of the covariance matrix of the interpolated virtual array signal, and does not need to pre-define the space grid points in the process of designing the optimization problem, and realizes the direction of arrival without grids estimation, while ensuring the resolution and computational efficiency of DOA estimation;

(3) The optimization problem based on the reconstruction of the interpolated virtual array covariance matrix proposed by the present invention ensures that the optimal solution result is a Hermitian symmetric Toeplitz matrix, so that the error between the optimal solution and the theoretical covariance matrix is smaller. Since the theoretical covariance matrix of the uniform linear array incoherent received signal satisfies the Toeplitz structure, the reconstruction of the covariance matrix using its Toeplitz characteristic as a prior constraint can make the difference between the reconstruction result and the real value smaller, thereby improving the DOA estimation performance.

Description of drawings

FIG. 1 is a block diagram showing the overall flow of the method of the present invention.

FIG. 2 is a schematic structural diagram of a pair of sparse uniform sub-arrays forming a coprime array in the present invention.

FIG. 3 is a schematic structural diagram of a coprime array in the present invention.

FIG. 4 is a schematic structural diagram of an interpolation virtual array in the present invention.

FIG. 5 is a schematic diagram of a method for dividing an interpolation virtual array in the present invention.

FIG. 6 is a schematic diagram of the spatial power spectrum used to reflect the degree of freedom performance of the method proposed in the present invention.

FIG. 7 is a schematic diagram of a normalized spatial spectrum used to reflect the resolution performance of the method proposed in the present invention.

Detailed ways

The technical solutions and effects of the present invention will be described in further detail below with reference to the accompanying drawings.

For the application of DOA estimation in practical systems, coprime arrays have attracted much attention because they can break through the limitation of the number of physical array elements on the degree of freedom through the calculation and statistical signal processing of equivalent virtual array signals. However, limited by the non-uniformity of virtual arrays, many methods currently choose to use continuous virtual array elements for DOA estimation, resulting in information loss. At the same time, many methods pre-set the spatial grid points of the assumed direction of arrival signal before performing DOA estimation, which causes inherent mismatch errors and conflicts between computational complexity and estimation accuracy. In order to make full use of all the information contained in the non-uniform virtual array and avoid the problem of limited estimation resolution caused by the predefined spatial grid points, the present invention provides a gridless interactive grid based virtual array interpolation. The method for estimating the direction of arrival of the mass array, referring to FIG. 1, the implementation steps of the present invention are as follows:

Step 1: use M+N-1 antenna array elements at the receiving end to construct a coprime array; first, select a set of coprime integers M and N; then, referring to Figure 2, construct a pair of sparse uniform linear sub-arrays, in which the first One sub-array contains M antenna elements with a spacing of Nd, and their positions are 0, Nd,..., (M-1)Nd; the second sub-array contains N antenna elements with a spacing of Md, and their positions are 0 ,Md,...,(N-1)Md; the unit spacing d is taken as half of the wavelength of the incident narrowband signal, that is, d=λ/2; then, the first antenna element of the two sub-arrays is regarded as the reference array element, refer to In Fig. 3, the reference array elements of the two sub-arrays are overlapped to realize the sub-array combination, and a non-uniform co-prime array structure actually containing M+N-1 antenna array elements is obtained.

Step 2: Use a coprime array to receive the signal and model it. Assuming that there are K far-field narrow-band incoherent signal sources from the directions of θ ₁ , θ ₂ ,..., θ _K , the non-uniform coprime array with the structure of step 1 is used to receive the incident signal, and the (M+N-1)×1 dimension is obtained. The coprime array receives the signal x(t), which can be modeled as:

Among them, _sk (t) is the signal waveform, n(t) is the noise component independent of each signal source, a(θ _k ) is the coprime array steering vector in the direction of θ _k , expressed as

Among them, p _i d,i=1,2,...,M+N-1 represents the actual position of the i-th physical antenna element in the coprime array, and p ₁ =0; d is half of the incident narrowband signal wavelength λ , that is, d=λ/2, [ ] ^T represents the transpose operation. A total of T sampling snapshots are collected to obtain the sampling covariance matrix of the received signal of the coprime array

where (·) ^H represents the conjugate transpose operation.

Step 3: Calculate the equivalent virtual signal corresponding to the signal received by the coprime array. Sampling Covariance Matrix of Received Signals of Vectorized Coprime Array Obtain the virtual array equivalent received signal v:

in, is (M+N-1) ² ×K-dimensional virtual array steering matrix, contains the power of the K incident signal sources, is the noise power, i _v =vec( _IM+N-1 ). Here, vec( ) represents the vectorization operation, that is, stacking the columns in the matrix to form a new vector, ( ) ^* represents the conjugation operation, Represents the Kronecker product, and _IM+N-1 represents the (M+N-1)×(M+N-1) dimensional identity matrix. The position of each virtual array element in the virtual array corresponding to the vector v is in

remove set The repeated virtual array elements at the corresponding positions of the repeated elements in , get a non-uniform virtual array Its corresponding equivalent virtual signal v _c can be obtained by selecting elements at corresponding positions in the vector v.

Step 4: Construct the interpolation virtual array and its received signal modeling. Referring to Figure 4, for a non-uniform virtual array Under the premise of keeping the original position of the virtual array element unchanged, insert several virtual array elements (as shown by the hollow circle in Fig. Converted to a uniform virtual array with a spacing of d, the array aperture is the same as that of a coprime array, and the number of virtual array elements is increased The interpolated virtual array contains a total of virtual array elements, where |·| represents the potential of the set. The equivalent virtual signal v _I corresponding to the interpolated virtual array can be obtained by filling the corresponding positions of the holes in the vector _vc with 0.

Step 5: Construct an interpolated virtual array multi-sampling snapshot signal and its sampling covariance matrix. Referring to Figure 5, the virtual array will be interpolated cut into L _I contiguous subarrays of length L _I , where

because The virtual array elements in are symmetrically distributed with zero positions, is always odd, so L _I is an integer. Correspondingly, interpolate the virtual array The multi-sampling snapshot signal can be obtained by intercepting the corresponding elements in the vector v _I , namely: V _I =[v _I,1 ,v _I,2 ,...,v _I,LI ], where v _I,l ,l= 1,2,...,L _I consists of L _I +1-1th to 2L _I -1th elements in v _I. Next, the sampling covariance matrix R _v of _VI can be obtained as follows:

Wherein, <v _I > _i represents the equivalent received signal corresponding to the virtual array element whose position is id. Since the virtual array elements in the interpolated virtual array are symmetrically distributed with respect to the null position, the equivalent virtual received signal on it is in a conjugate relationship with the null position, so the above sampling covariance matrix can also be equivalently obtained in the following way:

Step 6: Construct the projection matrix and define the projection operation. Since the covariance matrix R _v obtained in step 5 contains 0 inserted in step 4, all the elements on the diagonal line of its corresponding position are 0. According to such a structure, a projection matrix P with the same dimension as R _v is defined. If an element at a certain position in R _v is 0, the value of the element at the same position in the projection matrix P is also 0; otherwise, the corresponding element in the projection matrix P is 0. The element value of position is 1. definition It is the projection operation, where the variable in the parentheses is a matrix with the same dimension as P, and the projection operation is realized by multiplying each element of the variable matrix with the element at the corresponding position in the projection matrix P, and a matrix with the same dimension as the matrix P is obtained. .

Step 7: Design and solve the optimization problem based on the minimization of the kernel norm of the interpolated virtual array signal covariance matrix. Using the interpolated virtual array covariance matrix R _v obtained in step 5 as a reference value, find a Toeplitz matrix with the smallest nuclear norm as the covariance matrix of the interpolated virtual array signal, and the difference between it and R _v is required to be less than a certain threshold , the optimization problem with the vector z as a variable can be constructed as follows:

in, express The nuclear norm of , represents the Hermitian symmetric Toeplitz matrix with the vector z as the first column; ∈ is the threshold constant, which is used to constrain the reconstruction error of the covariance matrix; It is guaranteed that the reconstructed covariance matrix satisfies the condition of positive semi-definite; ‖·‖ _F represents the Frobenius norm. Solving the above convex optimization problem can get the optimal value The above convex optimization problem can be transformed into the following optimization problem with the vector z as a variable:

where μ is the regularization parameter used to trade off the matrix during minimization Reconstruction error and kernel norm of z. Solving the above convex optimization problem can get the optimal value Correspondingly, the reconstructed Toeplitz matrix is the interpolated virtual array covariance matrix.

Step 8: According to the reconstructed interpolated virtual array covariance matrix Do direction of arrival estimation. By introducing classical methods, such as multiple signal classification method, rotation invariant subspace method, root multiple signal classification method, covariance matrix sparse reconstruction method, etc., the reconstructed interpolated virtual array covariance matrix By performing the operation, the DOA estimation result can be obtained. Taking the multiple signal classification method as an example, draw the virtual domain spatial spectrum P _MUSIC (θ):

where d(θ) is the L _I ×1-dimensional interpolation virtual array steering vector, corresponding to a segment of virtual uniform array from 0 to (L _I -1)d; _En is L _I ×(L _I -K ) dimension matrix, representing the interpolated virtual array covariance matrix θ is the assumed direction of arrival of the signal; find the peaks on the spatial spectrum P _MUSIC through spectral peak search, and arrange the response values corresponding to these peaks from large to small, and take the corresponding values of the first K peaks. The angular direction is the result of DOA estimation.

On the one hand, the present invention introduces the idea of virtual array interpolation, and inserts virtual array elements on the basis of the derived original virtual array, thereby converting the original non-uniform virtual array into a virtual uniform array, while retaining the original non-uniform virtual array. All information avoids the mismatch of statistical signal processing models caused by the non-uniformity of the original virtual array and the information loss problem caused by the traditional method of intercepting the virtual uniform sub-array; on the other hand, the signal covariance matrix based on the virtual array is introduced. The idea of kernel norm minimization is used to design the optimization problem to reconstruct the covariance matrix of the interpolated virtual array, and realize the meshless DOA estimation on the virtual domain.

The effect of the present invention will be further described below in conjunction with a simulation example.

Simulation example 1: The co-prime array is used to receive the incident signal, and its parameters are selected as M=3, N=5, that is, the co-prime array of the architecture includes M+N-1=7 physical array elements. Assume that the number of incident narrowband signals is 9, and the incident direction is uniformly distributed in the spatial angle domain of -50° to 50°; the signal-to-noise ratio is set to 30dB, the number of sampling snapshots is T=500; the regularization parameter μ is set to 0.25.

The spatial power spectrum of the method for estimating the direction of arrival of a gridless coprime array based on virtual array interpolation proposed by the present invention is shown in FIG. 6 , where the vertical dotted line represents the actual direction of the incident signal source. It can be seen that the method proposed in the present invention can effectively distinguish the nine incident signal sources. For the traditional method using a uniform linear array, only 6 incident signals can be resolved by using 7 physical antenna elements. The above results show that the method proposed in the present invention achieves an increase in the degree of freedom.

Simulation example 2: The co-prime array is used to receive the incident signal, and its parameters are also selected as M=3, N=5, that is, the co-prime array of the architecture contains M+N-1=7 physical antenna elements in total; it is assumed that the incident narrowband signal The number is 2, and the incident direction is -0.5° to 0.5°. The rest of the parameter settings are the same as those of simulation example 1. It can be seen from the normalized spatial spectrum shown in FIG. 7 that the method proposed in the present invention can effectively distinguish the directions of arrival of the two short-range signals, which shows the good resolution performance of the method.

To sum up, the method proposed in the present invention makes full use of all the information on the non-uniform virtual array, and can realize the grid-free estimation of the incident signal when the number of signal sources is greater than or equal to the number of physical antennas, and the DOA is increased. Estimated degrees of freedom and resolution. In addition, compared with the traditional method using uniform linear array, the physical antenna array elements and radio frequency modules required in practical application of the method proposed by the present invention can also be correspondingly reduced, which reflects economy and high efficiency.

Claims (6)

1. A gridding-free co-prime array direction of arrival estimation method based on virtual array interpolation is characterized by comprising the following steps:

(1) the receiving end uses M + N-1 antennae and is constructed according to a co-prime array structure; wherein M and N are relatively prime integers;

(2) suppose there are K from θ₁，θ₂，…，θ_KA directional far-field narrow-band incoherent signal source, then the (M + N-1) × 1 dimensional co-prime array received signal x (t) can be modeled as:

wherein s is_k(t) is a signal waveform, n (t) is a noise component independent of each signal source, and a (theta)_k) Is theta_kThe steering vector of the direction is expressed as:

wherein p is_id, i-1, 2, …, M + N-1 denotes the actual position of the ith physical antenna element in the co-prime array, and p₁0; d is half the wavelength λ of the incident narrowband signal, i.e. d ═ λ/2,[·]^Trepresenting a transpose operation; collecting T sampling snapshots to obtain sampling covariance matrix of co-prime array received signal

Here, (.)^HRepresents a conjugate transpose operation;

(3) calculating equivalent virtual signals corresponding to the co-prime array receiving signals: sampling covariance matrix of vectorized co-prime array received signalObtaining a virtual array equivalent received signal v:

wherein,is (M + N-1)²A virtual array steering matrix of dimension xK,including the power of K incident signal sources,as the noise power, i_v＝vec(I_M+N-1) (ii) a Here, vec (-) denotes a vectorization operation, i.e., stacking columns in a matrix in order to form a new vector (-)^*It is meant a conjugate operation of the two,denotes the kronecker product, I_M+N-1Represents an (M + N-1) × (M + N-1) -dimensional identity matrix; the position of each virtual array element in the virtual array corresponding to the vector v is

Removing collectionsRepeating virtual array elements at the positions corresponding to the repeating elements to obtain a non-uniform virtual arrayIts corresponding equivalent virtual signal v_cCan be obtained by selecting elements at corresponding positions in the vector v;

(4) constructing an interpolation virtual array and a received signal thereof and modeling: first for non-uniform virtual arraysUnder the premise of keeping the original virtual array element position unchanged, a plurality of virtual array elements are inserted into discontinuous positions, so that the non-uniform virtual array is formedConverting into an interpolated uniform virtual array with a spacing of d, the same array aperture as the co-prime array, and an increased number of virtual array elementsThe interpolated uniform virtual array comprisesA virtual array element, where | represents the potential of the set, its corresponding equivalent virtual signal v_ICan be passed through vector v_cWhere 0 is inserted, the position of 0 is inserted andthe positions of the inserted virtual array elements are corresponding;

(5) constructing an interpolated uniform virtual array multi-sampling snapshot signal and a sampling covariance matrix thereof: will be provided withIs cut into L_IEach length is L_IIn a continuous sub-array of

Accordingly, the uniform virtual array S is interpolated_IThe multi-sampling snapshot signal can be obtained by intercepting a vector v_IThe corresponding elements in (1) are obtained, namely:v_I，l，l＝1，2，...，L_Iby v_IMiddle L_I+1-L to 2L_I-l elemental compositions; then, V_IIs sampled covariance matrix R_vCan be obtained by the following method:

wherein,the equivalent received signal corresponding to the virtual array element with the position id is represented;

(6) constructing a projection matrix and defining projection operations: dimension and R of projection matrix P_vSame if the matrix R_vIf a certain element in the projection matrix P is 0, the value of the element at the same position in the projection matrix P is also 0; otherwise, the element value of the corresponding position in the projection matrix P is 1; definition ofThe projection operation is realized by multiplying each element in the variable matrix and an element at a corresponding position in the projection matrix P one by one to obtain a matrix with the same dimension as the matrix P;

(7) designing an optimization problem based on interpolation virtual array signal covariance matrix kernel norm minimization and solving the optimization problem; using the interpolated virtual array covariance matrix R obtained in (5)_vAs a reference value, find a Toeplitz matrix with the minimum kernel norm as the covariance matrix of the interpolated virtual array signal, and require that it and R are equal_vIs less than a certain threshold, the following optimization problem with vector z as a variable can be constructed:

wherein,to representThe number of the nuclear norms of (c),representing a hermitian symmetric Toeplitz matrix with a vector z as a first column; e is a threshold constant and is used for constraining the reconstruction error of the covariance matrix;the reconstructed covariance matrix is ensured to meet the semi-positive definite condition; i | · | purple wind_FRepresents the Frobenius norm; solving the optimization problem can obtain the optimized valueAccordingly, the reconstructed Toeplitz matrixInterpolating a virtual array covariance matrix;

(8) interpolating virtual array covariance matrix from reconstructedAnd estimating the direction of arrival.

2. The method of estimating direction of arrival of latticeless co-prime array based on virtual array interpolation of claim 1, wherein: the coprime array structure in the step (1) can be specifically described as follows: firstly, selecting a pair of relatively prime integers M, N; then, constructing a pair of sparse uniform linear sub-arrays, wherein the first sub-array comprises M antenna array elements with the spacing Nd and the positions of the M antenna array elements are 0, Nd,. and (M-1) Nd, and the second sub-array comprises N antenna array elements with the spacing Md and the positions of the N antenna array elements are 0, Md,. and (N-1) Md; and then, performing sub-array combination on the two sub-arrays according to a mode that the first array element is overlapped to obtain a non-uniform co-prime array structure actually containing M + N-1 antenna array elements.

3. The method of estimating direction of arrival of latticeless co-prime array based on virtual array interpolation of claim 1, wherein: v constructed in step (5)_IIs sampled covariance matrix R_vCan also be obtained equivalently by the following method:

4. the method of estimating direction of arrival of latticeless co-prime array based on virtual array interpolation of claim 1, wherein: the convex optimization problem in step (7) can be converted into the following optimization problem with vector z as a variable:

where μ is a regularization parameter, for weighting the matrix during minimizationThe reconstruction error and the nuclear norm of z.

5. The method of estimating direction of arrival of latticeless co-prime array based on virtual array interpolation of claim 1, wherein: the direction of arrival estimation in step (8) may adopt one of the following methods: a multiple signal classification method, a rotation invariant subspace method, a root-finding multiple signal classification method, and a covariance matrix sparse reconstruction method.

6. The method of estimating direction of arrival of latticeless co-prime array based on virtual array interpolation of claim 1, wherein: in step (8), estimating the direction of arrival by a multiple signal classification method, specifically: drawing a virtual domain space spectrum P_MUSIC(θ)：

Wherein d (θ) is L_IInterpolating the virtual array steering vector in dimension x 1, corresponding to positions from 0 to (L)_I-1) a segment of a virtual uniform array of d; e_nIs L_I×(L_I-K) dimensional matrix representing a reconstructed interpolated virtual array covariance matrixThe noise subspace of (1); θ is the assumed direction of arrival of the signal; finding spatial spectra P by spectral peak search_MUSICAnd (theta) arranging the response values corresponding to the peak values from large to small, and taking the angle directions corresponding to the first K peak values, namely the estimation result of the direction of arrival.