Direction of Arrival Estimation Method for L-type Coprime Arrays with Split Subarrays Based on Fourth-order Sampled Covariance Tensor Denoising

Abstract

The present invention discloses a method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising, which mainly solves the problem of signal structure damage and high-order virtual domain statistics being interfered by noise terms in the existing method. The implementation steps are: constructing an L-type coprime array with linear sub-arrays split; modeling the received signal of the L-type coprime array and deriving its second-order cross-correlation matrix; deriving a fourth-order covariance tensor based on the cross-correlation matrix; realizing fourth-order sampling covariance tensor denoising based on kernel tensor thresholding processing; deriving a fourth-order virtual domain signal based on the denoised sampling covariance tensor; constructing a denoised structured virtual domain tensor; and obtaining the direction of arrival estimation result by decomposing the structured virtual domain tensor. The present invention makes full use of the high-order tensor statistical distribution characteristics of the constructed sub-array-split L-type coprime array, realizes high-precision two-dimensional direction of arrival estimation by denoising virtual domain tensor signal processing, and 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 high-order virtual domain statistics of a multidimensional sparse array, and specifically is a sub-array split L-type coprime array direction of arrival estimation method based on fourth-order sampling covariance tensor denoising, 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, high resolution and high degree of freedom. It can break through the limitation of Nyquist sampling rate and improve the comprehensive performance of DOA estimation. In order to achieve DOA estimation matching Nyquist sampling rate in the coprime array scenario, the common practice is to derive the coprime array received signal into a high-order statistical model, and realize DOA estimation based on virtual domain signal processing by constructing an augmented virtual uniform array. However, the existing methods usually model the received signal as a vector and derive the virtual domain signal by vectorizing the covariance matrix of the received signal. In the scenario of deploying a multi-dimensional coprime array, since the received signal covers multi-dimensional spatiotemporal information, the vectorized signal processing method loses the original structural information of the coprime array received signal. As a multi-dimensional data type, tensor can be used to represent complex electromagnetic information and retain the original structure of the received signal, so it is gradually applied to the field of array signal processing. However, the existing tensor signal processing method is only effective under the premise of matching Nyquist sampling rate, and has not yet involved the high-order statistical analysis of coprime array sparse signals and their virtual domain extension.

[0003] As an important tool for extracting multidimensional signal features, tensor decomposition is highly sensitive to noise. Traditional virtual domain derivation methods based on high-order signal statistics often introduce complex noise terms, which brings great challenges to the virtual domain expansion of coprime arrays based on tensor models. On the one hand, traditional methods derive augmented virtual domains based on the autocorrelation statistics of received signals, and the noise power introduced by noise autocorrelation will interfere with the tensor statistics processing; on the other hand, traditional methods estimate high-order sampling covariance statistics based on statistical calculations of sampled signals, which introduces high-order sampling noise, thus seriously affecting the decomposition processing of high-order covariance tensors. Therefore, how to overcome the interference of noise power and high-order sampling noise at the same time in the scenario of multi-dimensional coprime arrays, perform denoised virtual domain tensor derivation, and achieve high-precision two-dimensional direction of arrival estimation based on denoised virtual domain tensor processing 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 damaged multidimensional sparse array received signal structure and interference of high-order virtual domain statistics by noise terms in existing methods, and to propose a sub-array split L-type coprime array direction of arrival estimation method based on fourth-order sampling covariance tensor denoising, which provides a feasible idea and effective solution for achieving high-precision two-dimensional direction of arrival estimation through high-order tensor statistics denoising processing.

[0005] The object of the present invention is achieved through the following technical scheme: a method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising, the method comprising the following steps:

[0006] (1) Use at the receiving end physical antenna array elements, constructing an L-shaped mutually prime array with linear sub-arrays; the L-shaped mutually prime array consists of two mutually prime linear arrays located on the x-axis and the y-axis Composed of two mutually prime linear arrays and The first array element is arranged from the position 1 on the x-axis and y-axis respectively; the mutually prime linear array Included array elements, among which, and are a pair of relatively prime integers, |·| represents the cardinality of a set; and represents the position of each element of the L-type coprime array on the x-axis and y-axis, where The unit interval d is taken as half the wavelength of the incident narrowband signal;

[0007] (2) Assume that there are K The far-field narrow-band incoherent signal source in the direction of the L-shaped coprime linear array is formed. The received signal is modeled 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 is the number of sampling snapshots, o represents the vector outer product, is the noise that is independent of each signal source, for The steering vector corresponds to the direction of the incoming wave. The signal source is expressed as:

[0010]

[0011] in, [·]T represents the transpose operation; by finding and The cross-correlation statistics of

[0012]

[0013] in, represents the power of the kth incident signal source, E{·} represents the mathematical expectation operation, (·) H represents the conjugate transpose operation, (·) * represents the conjugation operation;

[0014] (3) Calculate the second-order cross-correlation matrix The autocorrelation of

[0015]

[0016] In practice, it can be obtained by sampling the covariance tensor of order 4 Approximate substitution, that is:

[0017]

[0018] in,

[0019]

[0020] is the fourth-order sampling noise tensor; The The elements are represented as It follows an approximate complex Gaussian distribution, and its approximate variance It is expressed as:

[0021]

[0022] where λ 1 ,λ 2 and λ 3 Represents the three sub-variance terms and The combined weight of represents the noise power;

[0023] (4) For the fourth-order sampling covariance tensor Perform a higher-order singular value decomposition:

[0024]

[0025] in, Represents the kernel tensor, which contains the The projection of the signal and noise components in and Indicates the corresponding Singular matrices in four dimensions; Perform thresholding, that is, The elements less than or equal to the noise threshold ∈ are set to zero, while the elements greater than the noise threshold ∈ are retained, thereby obtaining the kernel tensor after thresholding The elements in are represented as:

[0026]

[0027] in, express No. elements, the noise threshold ∈ is:

[0028]

[0029] Furthermore, using the thresholded kernel tensor Multiply by four singular matrices Y (1) , Y (2) , Y (3) and Y (4) , get the denoised sampling covariance tensor It is expressed as:

[0030]

[0031] (5) Define dimension sets By sampling the covariance tensor for denoising Perform dimension-merged tensor transformation to obtain a fourth-order virtual domain signal

[0032]

[0033] in, and By forming difference arrays on the exponential terms, augmented virtual linear arrays on the x-axis and y-axis are constructed. represents the Kronecker product; Corresponding to a two-dimensional non-continuous virtual cross array A virtual uniform cross array is included in and are virtual uniform linear arrays on the x-axis and y-axis respectively; and The positions of the virtual array elements are expressed as and in and From a non-contiguous virtual cross array Virtual domain signal Extract the virtual uniform cross array The elements of each virtual array element position in are obtained The corresponding fourth-order virtual domain signal

[0034] (6) From and Extract subarrays separately as a pan window; then, the pan windows and By translating one virtual array element interval d along the negative semi-axis direction of the x-axis and the y-axis, we can get J x Virtual uniform linear subarray and J y Virtual uniform linear subarray j x =1,2,...,J x , j y =1,2,...,J y , The virtual uniform subarray The corresponding virtual domain signal can be expressed as Fixed y Index, Superimposing in the third dimension, we get J y A three-dimensional virtual domain tensor, then, this J y The three-dimensional virtual domain tensors are superimposed in the fourth dimension to obtain a four-dimensional denoised structured virtual domain tensor It is expressed as:

[0035]

[0036] in,

[0037]

[0038]

[0039] They are and The guidance vector,

[0040]

[0041]

[0042] are the translation factors along the x-axis and y-axis respectively;

[0043] (7) Denoising the structured virtual domain tensor through canonical polyadic decomposition (CPD) Perform tensor decomposition and obtain The estimated values ​​of each spatial factor, namely from Extract parameters from and And according to {μ 1 (k), μ 2 (k)} and two-dimensional direction of arrival The relationship between the two-dimensional wave direction of arrival is obtained The closed form solution of .

[0044] Furthermore, the L-type coprime array structure of the linear subarrays in step (1) is specifically described as follows: It consists of a pair of sparse uniform linear subarrays, the two sparse uniform linear subarrays contain and antenna array elements, and the array element spacing is and The two sparse linear uniform subarrays are combined in such a way that the first array elements overlap to obtain Coprime linear array

[0045] Furthermore, the fourth-order sampling noise tensor described in step (3) Will and The The elements are represented as and but The The elements are represented as:

[0046]

[0047] because and They obey approximate complex Gaussian distribution, namely:

[0048]

[0049]

[0050]

[0051] therefore

[0052]

[0053]

[0054]

[0055]

[0056]

[0057] Thus, it is concluded It also obeys an approximate complex Gaussian distribution, and its approximate variance It is expressed as:

[0058]

[0059] Furthermore, the fourth-order virtual domain signal derivation described in step (5) corresponds to a virtual uniform cross array Virtual domain signal It can be expressed as:

[0060]

[0061] in,

[0062]

[0063]

[0064] Respectively and The guidance vector of .

[0065] Furthermore, the two-dimensional direction of arrival estimation process described in step (7) is Extract parameters from and

[0066]

[0067]

[0068] Among them, ∠(·) represents the complex number argument operation; according to {μ 1 (k), μ 2 (k)} and two-dimensional direction of arrival The relationship between and Get a 2D DOA estimate The closed-form solution of is:

[0069]

[0070]

[0071] Further, in step (7), according to the uniqueness condition of CPD, The following conditions must be met to undertake CPD:

[0072]

[0073] where κ(·) represents the Kruskal rank of the matrix, and for The factor matrix of and Substituting into the CPD uniqueness condition inequality, we get in represents the rounding up operation; therefore, the maximum number of targets that can be estimated by the proposed method is

[0074] Compared with the prior art, the present invention has the following advantages:

[0075] (1) The present invention obtains the second-order signal statistics after removing the noise power interference by performing cross-correlation calculation on the received signals of the L-type coprime array with sub-array separation, and further expands the fourth-order covariance tensor based on this, realizing the derivation of the virtual domain tensor;

[0076] (2) Based on the statistical characteristics analysis of the fourth-order sampling covariance tensor of mutually prime arrays, the present invention designs a fourth-order sampling covariance tensor denoising method based on kernel tensor threshold filtering, which lays a foundation for suppressing sampling noise interference and constructing a denoised virtual domain tensor;

[0077] (3) The present invention proposes a structured superposition mechanism for denoising virtual domain signals, and performs tensor decomposition and angle information extraction on the constructed denoising structured virtual domain tensor, thereby achieving accurate two-dimensional direction of arrival estimation under underdetermined conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 It is the overall flow chart of the present invention.

[0079] Figure 2It is a structural schematic diagram of the sub-array split L-type coprime array proposed in the present invention.

[0080] Figure 3 It is a schematic diagram of a virtual uniform cross array and its virtual uniform subarray constructed by the present invention.

[0081] Figure 4 This is the result diagram of the two-dimensional underdetermined direction of arrival estimation using the traditional Tensor MUSIC method.

[0082] Figure 5 It is a diagram of the two-dimensional underdetermined direction of arrival estimation result of the method proposed in the present invention. DETAILED DESCRIPTION

[0083] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0084] In order to solve the problems of signal structure damage and high-order virtual domain statistics being interfered by noise terms in the existing methods, the present invention proposes a method for DOA estimation of sub-array split L-type coprime array based on fourth-order sampling covariance tensor denoising. By deriving the high-order tensor statistics of the sub-array split L-type coprime array, a denoising technique for the sampling covariance tensor is designed, and high-precision two-dimensional DOA estimation is achieved based on denoised virtual domain tensor signal processing. Figure 1 , the implementation steps of the present invention are as follows:

[0085] Step 1: Construct an L-shaped coprime array with linear subarrays. Use Physical antenna array elements construct an L-shaped coprime array with linear sub-arrays, such as Figure 2 As shown: Construct a mutually prime linear array on the x-axis and y-axis respectively Included antenna array elements, where and are a pair of relatively prime integers, |·| represents the cardinality of the set; these two relatively prime linear arrays and The first array element of is arranged from the position 1 on the x-axis and y-axis respectively. Therefore, the two coprime linear arrays that form the L-type coprime array and Do not overlap; use and represents the position of each element of the L-type coprime array on the x-axis and y-axis, where The unit interval d is taken as half of the wavelength of the incident narrowband signal; the two separated mutually prime linear arrays constituting the L-type mutually prime array Each of them consists of a pair of sparse uniform linear subarrays, and the two sparse uniform linear subarrays contain and Antenna array elements, The array element spacing is and And combine them in the way that the first array elements overlap to obtain Coprime linear array

[0086] Step 2: Modeling of the received signal of the L-type coprime array and derivation of its second-order cross-correlation matrix. Assume that there are K signals from The far-field narrow-band incoherent signal source in the direction of the L-shaped mutually prime array will form two mutually prime linear arrays. and The received signal is modeled as:

[0087]

[0088] 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 is the number of sampling snapshots, represents the vector outer product, is the noise that is independent of each signal source, for The steering vector corresponds to the direction of the incoming wave. The signal source is expressed as:

[0089]

[0090] in, Represents the transpose operation; by finding coprime linear arrays and Sampling signal and The cross-correlation statistics of

[0091]

[0092] in, represents the power of the kth incident signal source, E{·} represents the mathematical expectation operation, (·) H represents the conjugate transpose operation, (·) * Represents the conjugate operation; by performing cross-correlation calculation on the received signal, the noise The noise power term introduced by the autocorrelation calculation is in represents the noise power and I represents the identity matrix.

[0093] Step 3: Derive the fourth-order covariance tensor based on the cross-correlation matrix. In order to achieve the augmented virtual array derivation, the fourth-order statistics of the L-type coprime array are further derived based on the second-order cross-correlation statistics. Specifically, by Find its autocorrelation and get the fourth-order covariance tensor

[0094]

[0095]

[0096] In practice, it can be done by estimating the received signal and The fourth-order statistics are obtained, that is, the fourth-order sampling covariance tensor

[0097]

[0098] in,

[0099]

[0100] is the fourth-order sampling noise tensor. and The The elements are represented as and but The The elements can be represented as:

[0101]

[0102] in, because and They obey approximate complex Gaussian distribution, namely:

[0103]

[0104]

[0105]

[0106] therefore

[0107]

[0108]

[0109]

[0110]

[0111]

[0112] Thus, it is concluded It also obeys an approximate complex Gaussian distribution, and its approximate variance It is expressed as:

[0113]

[0114] where λ 1 ,λ 2 and λ 3 Represents the three subvariances and The combined weight of .

[0115] Step 4: Implement fourth-order sampling covariance tensor denoising based on kernel tensor thresholding. Perform a high-order singular value decomposition on it:

[0116]

[0117] in, Represents the kernel tensor, which contains the The projection of the signal and noise components in and Indicates the corresponding Singular matrices in four dimensions; Perform thresholding, that is, The elements less than or equal to the noise threshold ∈ are set to zero, and the elements greater than the noise threshold ∈ are retained, so as to obtain the kernel tensor after thresholding The elements in are represented as:

[0118]

[0119] in, express No. elements, the noise threshold ∈ is:

[0120]

[0121] Furthermore, using the thresholded kernel tensor Multiply by four singular matrices Y (1) , Y (2) , Y (3) and Y(4) , get the denoised sampling covariance tensor It is expressed as:

[0122]

[0123] Step 5: Derive the fourth-order virtual domain signal based on the denoised sample covariance tensor. By combining the denoised sample covariance tensor The dimension of spatial information in the same direction can be represented by two coprime linear arrays and The conjugate steering vector and A difference array is formed on the exponential term, thereby constructing an augmented virtual linear array on the x-axis and y-axis respectively, corresponding to a two-dimensional non-continuous virtual cross array Specifically, the denoised sample covariance tensor The first and third dimensions represent the spatial information in the x-axis direction, and the second and fourth dimensions represent the spatial information in the y-axis direction; for this purpose, the dimension set is defined By sampling the covariance tensor for denoising Perform a tensor transformation with dimension merging to obtain a non-contiguous virtual cross array The fourth-order virtual domain signal

[0124]

[0125] in, and By forming difference arrays on the exponential terms, augmented virtual linear arrays on the x-axis and y-axis are constructed. represents the Kronecker product. A virtual uniform cross array is included The structure is as Figure 3 As shown, and are virtual uniform linear arrays corresponding to the x-axis and y-axis respectively. and The positions of the virtual array elements are and in and

[0126] From a non-contiguous virtual cross array Virtual domain signal Extract the virtual uniform cross array The elements of each virtual array element position in are obtained The corresponding virtual domain signal Modeled as:

[0127]

[0128] in,

[0129]

[0130]

[0131] Respectively and The guidance vector of .

[0132] Step 6: Construct the denoised structured virtual domain tensor. Considering the virtual uniform cross array Two virtual uniform linear arrays and Symmetric about the x=1 and y=1 axes, respectively. and Extract subarrays separately as a pan window; then, the pan windows and By translating one virtual array element interval d along the negative semi-axis direction of the x-axis and the y-axis, we can get J x Virtual uniform linear subarray and J y Virtual uniform linear subarray like Figure 3 As shown, here, j x =1,2,...,J x , j y =1,2,...,J y , The virtual uniform subarray The corresponding virtual domain signal can be expressed as Virtual domain signals with adjacent index subscripts and There is a one-step translation relationship between them in the y-axis direction. Similarly, and There is a one-step translation relationship in the x-axis direction. Therefore, these virtual domain signals are stacked into a structured virtual domain tensor. Specifically, fixed j y Index subscript, Superimposing in the third dimension, we get J y A three-dimensional virtual domain tensor, then, this J yThe three-dimensional virtual domain tensors are superimposed in the fourth dimension to obtain a denoised structured virtual domain tensor It is expressed as:

[0133]

[0134] in,

[0135]

[0136]

[0137] They are and The guidance vector,

[0138]

[0139]

[0140] are the translation factors along the x-axis and y-axis respectively.

[0141] Step 7: Obtain the DOA estimation result by decomposing the structured virtual domain tensor. Through canonical polyadic decomposition (CPD), we can get The estimated values ​​of each spatial factor, namely from Extract parameters from and

[0142]

[0143]

[0144] Among them, ∠(·) represents the complex number angle operation. Finally, according to the parameter {μ 1 (k), μ 2 (k)} and two-dimensional direction of arrival The relationship between and Get a 2D DOA estimate The closed-form solution of is:

[0145]

[0146]

[0147] According to the uniqueness condition of CPD, for the tensor The following conditions must be met to undertake CPD:

[0148]

[0149] Where K(·) represents the Kruskal rank of the matrix, and for The factor matrix of and Substituting into the CPD uniqueness condition inequality, we get in represents the rounding up operation; therefore, the maximum number of targets that can be estimated by the proposed method is

[0150] The effects of the present invention are further described below in conjunction with a simulation example.

[0151] Simulation example: Using a sub-array split L-type coprime array to receive the incident signal, the parameters are selected as follows: That is, the L-type coprime array of the architecture contains antenna array elements. Assuming there are 22 incident narrowband signals, the two-dimensional parameter μ of the direction of arrival is 1 (k) and μ 2 (k) are uniformly distributed on [-0.97, 0.97]. Subvariance combination weight λ 1 =1,λ 2 =0.25,λ 3 =1. The proposed method for DOA estimation of L-type coprime arrays with sub-array separation based on fourth-order sampling covariance tensor denoising is compared with the traditional Tensor Multiple Signal Classification (Tensor MUSIC) method. Under the conditions of signal-to-noise ratio SNR=-5dB and sampling snapshot number T=500, the two-dimensional DOA estimation performance of the above methods under underdetermined conditions is as follows: Figure 4 and Figure 5 shown.

[0152] It can be seen that under underdetermined conditions, the proposed method can accurately estimate the two-dimensional direction of arrival of all sources, while the Tensor MUSIC method cannot effectively estimate the two-dimensional direction of arrival of all sources. Compared with the traditional Tensor MUSIC method, the proposed method achieves accurate estimation of the two-dimensional direction of arrival by constructing a denoised virtual domain tensor while suppressing noise power and sampling high-order noise interference, and has a more superior direction of arrival estimation performance under underdetermined conditions.

[0153] In summary, the present invention explores the statistical distribution characteristics of the high-order sampling covariance tensor by constructing the relationship between the multidimensional virtual domain of the L-type coprime array and the denoised high-order tensor statistics, and designs a denoising processing method for the high-order sampling covariance tensor based on this; furthermore, a structured spatial segmentation and superposition mechanism of the denoised high-order virtual domain signal is established to construct a denoised structured virtual domain tensor, and by performing tensor decomposition on it, the accurate estimation of the two-dimensional wave arrival direction is achieved, and its closed-form solution is given.

[0154] 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 method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising, characterized in that: The following steps are involved: (1) Use at the receiving end physical antenna array elements, constructing an L-shaped mutually prime array with linear sub-arrays; the L-shaped mutually prime array consists of two mutually prime linear arrays located on the x-axis and the y-axis Composed of two mutually prime linear arrays and The first array element is arranged from the position 1 on the x-axis and y-axis respectively; the mutually prime linear array Included array elements, among which, and are a pair of relatively prime integers, |·| represents the cardinality of a set; and represents the position of each element of the L-type coprime array on the x-axis and y-axis, where The unit interval d is taken as half the wavelength of the incident narrowband signal; (2) Assume that there are K The far-field narrow-band incoherent signal source in the direction of the L-shaped coprime linear array is formed. The received signal is modeled 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 is the number of sampling snapshots, represents the vector outer product, is the noise that is independent of each signal source, for The steering vector corresponds to the direction of the incoming wave. The signal source is expressed as: in, [·] T Represents a transpose operation; by asking and The cross-correlation statistics of in, represents the power of the kth incident signal source, E{·} represents the mathematical expectation operation, (·) H represents the conjugate transpose operation, (·) * represents the conjugation operation; (3) Calculate the second-order cross-correlation matrix The autocorrelation of In practice, the covariance tensor is sampled by fourth order Approximate substitution, that is: in, is the fourth-order sampling noise tensor; The The elements are represented as It follows an approximate complex Gaussian distribution, and its approximate variance It is expressed as: where λ1, λ2 and λ3 represent the three sub-variance terms and The combined weight of represents the noise power; (4) For the fourth-order sampling covariance tensor Perform a higher-order singular value decomposition: in, Represents the kernel tensor, which contains the The projection of the signal and noise components in and Indicates the corresponding Singular matrices in four dimensions; Perform thresholding, that is, The elements less than or equal to the noise threshold ∈ are set to zero, while the elements greater than the noise threshold ∈ are retained, thereby obtaining the kernel tensor after thresholding The elements in are represented as: in, express No. elements, the noise threshold ∈ is: Using the thresholded kernel tensor Multiply by four singular matrices Y (1) , Y (2) , Y (3) and Y (4) , get the denoised sampling covariance tensor It is expressed as: (5) Define dimension sets By sampling the covariance tensor for denoising Perform dimension-merged tensor transformation to obtain a fourth-order virtual domain signal in, and By forming difference arrays on the exponential terms, augmented virtual linear arrays on the x-axis and y-axis are constructed. represents the Kronecker product; Corresponding to a two-dimensional non-continuous virtual cross array A virtual uniform cross array is included in and are virtual uniform linear arrays on the x-axis and y-axis respectively; and The positions of the virtual array elements are expressed as and in and From a non-contiguous virtual cross array Virtual domain signal Extract the virtual uniform cross array The elements of each virtual array element position in are obtained The corresponding fourth-order virtual domain signal (6) From and Extract subarrays separately as a pan window; then, the pan windows and By translating one virtual array element interval d along the negative semi-axis direction of the x-axis and the y-axis, we can get J x Virtual uniform linear subarray and J y Virtual uniform linear subarray j x =1,2,...,J x , j y =1,2,...,J y , The virtual uniform subarray The corresponding virtual domain signal can be expressed as Fixed y Index, Superimposing in the third dimension, we get J y A three-dimensional virtual domain tensor, then, this J y The three-dimensional virtual domain tensors are superimposed in the fourth dimension to obtain a four-dimensional denoised structured virtual domain tensor It is expressed as: in, They are and The guidance vector, are the translation factors along the x-axis and y-axis respectively; (7) Denoising the structured virtual domain tensor through CPD Perform tensor decomposition and obtain The estimated values ​​of each spatial factor, namely from Extract parameters from and And according to {μ1(k),μ2(k)} and the two-dimensional wave arrival direction The relationship between the two-dimensional wave direction of arrival is obtained The closed form solution of .

2. The method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, characterized in that: The L-type coprime array structure of the linear subarrays in step (1) is specifically described as follows: It consists of a pair of sparse uniform linear subarrays, the two sparse uniform linear subarrays contain and antenna array elements, and the array element spacing is and The two sparse linear uniform subarrays are combined in such a way that the first array elements overlap to obtain Coprime linear array 3. The method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, characterized in that: The fourth-order sampling noise tensor described in step (3) Will and The The elements are represented as and but The The elements are represented as: because and They obey approximate complex Gaussian distribution, namely therefore Thus, it is concluded It also obeys an approximate complex Gaussian distribution, and its approximate variance It is expressed as:

4. The method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, characterized in that: The fourth-order virtual domain signal derivation described in step (5) corresponds to a virtual uniform cross array Virtual domain signal It can be expressed as: in, Respectively and The guidance vector of .

5. The method for estimating the direction of arrival of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, characterized in that: The two-dimensional direction of arrival estimation process described in step (7) is as follows: Extract parameters from and Among them, ∠(·) represents the complex number argument operation; according to {μ1(k), μ2(k)} and the two-dimensional wave arrival direction The relationship between and Get a 2D DOA estimate The closed-form solution of is:

6. The method for DOA estimation of a sub-array-split L-type coprime array based on fourth-order sampling covariance tensor denoising according to claim 1, characterized in that: In step (7), according to the uniqueness condition of CPD, The following conditions must be met to undertake CPD: Where K(·) represents the Kruskal rank of the matrix, and for The factor matrix of and Substituting into the CPD uniqueness condition inequality, we get in represents the rounding up operation; therefore, the maximum number of targets that can be estimated by this method is

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