Sparse array direction of arrival estimation method based on cnn convolution kernel tensor decomposition

Abstract

The application discloses a sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition, mainly solves the problem of high network training cost and low efficiency of the existing sparse array direction of arrival estimation method based on a convolution neural network, and the implementation steps are as follows: constructing a two-dimensional nested coprime planar array; sparse tensor signal modeling and cross-correlation tensor derivation of the nested coprime planar array; constructing a virtual domain tensor input of the convolution neural network; CNN convolution kernel canonical polyadic decomposition and compressed convolution calculation; two-dimensional direction of arrival estimation based on compressed convolution neural network fast training. Through tensor decomposition of the high-dimensional CNN convolution kernel, the application designs a corresponding sparse tensor statistic compressed convolution calculation mode, accelerates the training process of the convolution neural network, realizes the two-dimensional direction of arrival estimation of the sparse array which meets the requirements of high performance and high efficiency, and can be used for target direction finding.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of array signal processing, and particularly relates to high-order statistical processing technology based on sparse tensor signals, and specifically relates to a sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition, which can be used for high-performance target direction finding. BACKGROUND

[0002] Compared with a traditional uniform array, a sparse array has the advantages of large aperture and high resolution, and can break through the performance bottleneck of uniform array direction of arrival estimation in estimation performance and cost overhead. A sparse array with a systematic structure derives an augmented virtual array based on second-order statistics of received signals, so as to realize direction of arrival estimation through virtual domain second-order equivalent signal processing. As a typical sparse array architecture, a nested array can derive a continuous virtual array, and therefore is widely used. However, a traditional virtual domain signal processing method represents received signals as vectors, and derives virtual domain second-order equivalent signals through a covariance matrix of the vectorized received signals. With the continuous expansion of the dimension of a sparse array in actual application, the received signals of a multi-dimensional sparse array cover multi-dimensional space-time information, and such a vectorized signal processing method will destroy the original structure of the multi-dimensional received signals, causing serious performance loss. In order to describe more complex spatial electromagnetic properties, a tensor as a multi-dimensional data type is widely used to represent received signals covering multi-dimensional space-time information. An existing sparse array tensor signal processing method derives a second-order virtual domain tensor corresponding to a sparse tensor signal, represents the second-order virtual domain tensor as a canonical polyadic (CP) model, and directly performs CP decomposition to realize direction of arrival estimation. However, such a method based on the CP model is often difficult to adapt to harsh signal propagation environments, and is difficult to meet the performance requirements of high precision and low computational complexity in scenes such as low signal-to-noise ratio and real-time requirement.

[0003] To cope with the above challenges, a convolutional neural network (CNN) extracts potential features from high-dimensional data based on a data-driven mode, and is regarded as a promising scheme for DOA estimation. The CNN can construct sparse tensor statistics as a multi-channel input, extract multi-dimensional signal features layer by layer based on convolution operations, and effectively resist noise disturbance and non-ideal signal deviation at the input end. However, in a traditional CNN, one network layer corresponds to a high-dimensional CNN convolution kernel, which contains a large number of parameters to be trained; as the number of network layers increases, the training efficiency of the CNN will be increasingly low. In a computing resource-limited scenario, it is often difficult to completely train the CNN while ensuring network performance. Therefore, it is a challenging technical problem to design a CNN with high performance and high training efficiency to realize sparse array DOA estimation that meets performance requirements. SUMMARY

[0004] The present application aims to solve the problems of high network training cost and low efficiency of existing sparse array DOA estimation methods based on a CNN, and provides a sparse array DOA estimation method based on CNN convolution kernel tensor decomposition.

[0005] The present application aims to solve the problems of high network training cost and low efficiency of existing sparse array DOA estimation methods based on a CNN, and provides a sparse array DOA estimation method based on CNN convolution kernel tensor decomposition.

[0006] (1) The receiving end uses M x M y +N x N y -1 physical antenna elements, which are arranged according to the structure of a two-dimensional nested coprime planar array; the nested coprime planar array is divided into a uniform planar subarray and a sparse planar subarray wherein contains M x ×M y antenna elements, and the element spacing d is half of the wavelength λ of the incident narrowband signal, i.e. d = λ / 2, contains N x ×N y antenna elements, and the spacings in the x-axis and y-axis directions are M x d and M y d, respectively;

[0007] (2) Assuming that there are K far-field narrowband uncorrelated signal sources from directions, θ k and Let azimuth and elevation be the azimuth and elevation angles of the k-th incident signal source, respectively, where k = 1, 2, ..., K. Then, consider the uniform planar subarray within the nested coprime planar array. The T snapshot samples are superimposed in the third dimension to obtain a three-dimensional tensor signal. The model is as follows:

[0008]

[0009] Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T Let [·] be the waveform vector of the multi-shot sampled signal corresponding to the k-th incident signal source. T This indicates the transpose operation. Represents the vector outer product. For a noise tensor that is independent of each signal source, a x (k) and a y (k) are respectively The guiding vectors in the x-axis and y-axis directions are represented as follows:

[0010]

[0011]

[0012] in, Similarly, sparse planar subarrays The received signal uses a three-dimensional tensor signal Represented as:

[0013]

[0014] in, b is a noise tensor that is independent of each signal source. x (k) and b y (k) are respectively The guiding vectors in the x-axis and y-axis directions are represented as follows:

[0015]

[0016]

[0017] By obtaining tensor signals and The cross-correlation statistics are used to obtain the second-order cross-correlation tensor.

[0018]

[0019] in, This represents the power of the k-th incident signal source. Represents the four-dimensional cross-correlation noise tensor, <·,·> r Let E· denote the tensor shrinking operation along the r-th dimension, and let E· denote the expected value operation. * Indicates the conjugate operation;

[0020] (3) Define the dimension set By merging cross-correlation tensors The dimension representing spatial information in the same direction is obtained as follows:

[0021]

[0022] in, and By forming a difference array over the exponent terms, a set of size M is constructed. x N x ×M y N y Virtual uniform planar array Indicates the Kronecker product. For noise terms; arrange the elements in U to correspond to The position of the virtual array element is used to obtain the second-order equivalent signal of the virtual domain:

[0023]

[0024] in, and They are virtual uniform planar arrays Guiding vectors along the x-axis and y-axis directions, For the corresponding noise term; extract the imaginary and real parts of the second-order equivalent signal Q in the virtual domain, and superimpose them in the third dimension to obtain a three-dimensional virtual domain tensor. As input to a convolutional neural network;

[0025] (4) Set the four-dimensional CNN convolution kernel corresponding to the first convolutional layer. Where (P1, Q1) represents the convolution size, H0 = 2 represents the number of input channels, and H1 represents the number of output channels of the first convolutional layer; for Canonical polyadic decomposition yields:

[0026]

[0027] in, For the kernel tensor, and They are respectively the corresponding A four-dimensional factor matrix, where R is the canonical polyadic rank, × r Represent the tensor-matrix product along the r-th dimension; use the kernel tensor and factor matrix to represent the virtual domain tensor. Compressed convolution is performed to obtain the output feature map of the first convolutional layer. I1 = M x N x -P1-2δ+1,J1=M y N y -Q1-2δ+1, where δ is the size of zero padding, the compressed convolution calculation process can be represented by three steps:

[0028]

[0029]

[0030]

[0031] in, for The (i0,j0,h0)th element in the array, where i0 = 1, 2, ..., M x N x j0 = 1, 2, ..., M y N y i1 = 1, 2, ..., I1, j1 = 1, 2, ..., J1, h0 = 1, 2, ..., H0, h1 = 1, 2, ..., H1, r0, r1, r, r′ = 1, 2, ..., R; Similarly, set another L-1 four-dimensional CNN convolutional kernels. The kernel tensors of CNN convolutional kernels are obtained through canonical polyadic decomposition. With factor matrix {V l,1 V l,2 V l,3 V l,4 The corresponding feature maps are then calculated and output layer by layer based on compressed convolution. Thus, a compressed convolutional neural network of depth L is constructed to realize two-dimensional direction of arrival estimation for sparse arrays;

[0032] (5) Using a parameter matrix Output feature map of the Lth convolutional layer After weighting, the two-dimensional direction-of-arrival estimation results of the output layer are obtained:

[0033]

[0034] Where vec(·) represents the vectorization operation, For real two-dimensional direction of arrival ; define the loss function of the output layer:

[0035]

[0036] where ‖·‖1 and ‖·‖2 represent 1-norm and 2-norm respectively, and η represents a conversion threshold calculated by a loss function; based on gradient descent of the loss function Θ, fast training of the compressed convolutional neural network is completed; and two-dimensional direction of arrival estimation in an actual application scenario is realized by using the trained complete compressed convolutional neural network.

[0037] Further, the nested coprime planar array structure in step (1) is specifically described as: a uniform planar subarray The position coordinates of the antenna array elements in the coordinate system are {(m x d,m y d)|m x =0,1,…,M x -1,m y =0,1,…,M y -1}; a sparse planar subarray The position coordinates of the antenna array elements in the coordinate system are {(n x M x d,n y M y d)|n x =0,1,…,N x -1,n y =0,1,…,N y -1}; {M x ,N x} and {M y ,N y} are a pair of coprime integers; and and are combined into a subarray in the manner of overlapping the position elements of the coordinate system origin, and since the arrangement of the elements of and satisfies the coprime number condition, the position elements of the remaining positions except the coordinate system origin do not overlap, and a nested coprime planar array containing M x M y +N x N y -1 antenna array elements is obtained.

[0038] Further, the second-order cross-correlation tensor is derived, and in practice, the sampling cross-correlation statistics of the three-dimensional tensor signals and are calculated to obtain an approximation, i.e., a sampling cross-correlation tensor

[0039]

[0040] Further, the compressed convolutional neural network accelerates the training process in step (5), calculates the gradient of the loss function Θ with respect to the convolutional layer kernel tensor and the gradient of the factor matrix {V l,1 ,V l,2 ,V l,3 ,V l,4} is calculated, and the trained parameters therein are updated by using gradient descent; the trained parameters in the virtual domain tensor {V {V l,1 ,V l,2 ,V l,3 ,V l,4} are iteratively updated by inputting several virtual domain tensors, and the fast training of the compressed convolutional neural network is completed; in actual application, the precise two-dimensional direction of arrival estimation result is obtained by performing the compressed convolution calculation of the trained complete compressed convolutional neural network

[0041] Compared with the prior art, the present application has the following advantages:

[0042] (1) The present application expands the sparse array signal sampling mode to the tensor space, establishes the derivation mechanism between the sparse tensor signal processing and the virtual domain tensor, and matches the virtual domain tensor to the input layer of the convolutional neural network, thereby providing the basis for improving the direction of arrival estimation precision performance;

[0043] (2) The present application uses the low-rank property of the high-dimensional CNN convolution kernel to perform tensor decomposition, greatly compresses the trained network parameters contained therein, thereby accelerating the training process of the constructed compressed convolutional neural network, reducing the training cost, and realizing the sparse array direction of arrival estimation which meets the requirements of high performance and low cost. BRIEF DESCRIPTION OF DRAWINGS

[0044] Figure 1 is the overall flow chart of the present application.

[0045] Figure 2 is the schematic diagram of the CNN convolution kernel canonical polyadic decomposition and compressed convolution calculation designed by the present application.

[0046] Figure 3 is the comparison diagram of the direction of arrival estimation precision performance of the method proposed by the present application under different signal-to-noise ratio conditions.

[0047] Figure 4 is the comparison diagram of the direction of arrival estimation precision performance of the method proposed by the present application under different sampling snapshot conditions. DETAILED DESCRIPTION

[0048] The technical solutions of the present application are further described in detail below with reference to the accompanying drawings.

[0049] In order to solve the problems of high network training cost and low efficiency existing in the sparse array direction of arrival estimation method based on the existing method of convolutional neural network, the present application proposes a sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition, which uses the low-rank property of high-dimensional CNN convolution kernel to decompose it to obtain low-dimensional projection factor, and then performs compressed convolution calculation on the sparse tensor statistics to extract multi-dimensional signal features layer by layer, realizing the sparse array direction of arrival estimation considering the requirements of high performance and low cost. Figure 1 The implementation steps of the present application are as follows:

[0050] Step 1: Construct a two-dimensional nested coprime planar array. The receiving end uses M x M y +N x N y -1 physical antenna elements, which are structured according to the structure of the two-dimensional nested coprime planar array. The nested coprime planar array is decomposed into a uniform planar subarray and a sparse planar subarray Wherein contains M x ×M y antenna elements, and the element spacing d is half of the wavelength λ of the incident narrowband signal, i.e. d = λ / 2, The position coordinates of the antenna elements in the coordinate system are {(m x d,m y d)|m x = 0,1,…,M x -1,m y = 0,1,…,M y -1}; contains N x ×N y antenna elements, and the element spacing in the x-axis direction and the y-axis direction is M x d and M y d, respectively. The position coordinates in the coordinate system are {(n x M x d,n y M y d)|n x = 0,1,…,N x -1,n y = 0,1,…,N y -1};{M x ,N x} and {M y ,N y} are a pair of coprime integers; and Subarrays are combined by overlapping array elements at the origin of the coordinate system. and The array elements are arranged to satisfy the condition of coprime numbers, therefore, the array elements at positions other than the origin of the coordinate system do not overlap, thus obtaining the actual array containing M. x M y +N x N y -1 nested coprime planar array of antenna elements;

[0051] Step 2: Sparse tensor signal modeling and cross-correlation tensor derivation for nested coprime planar arrays. Assume there are K signals from... Far-field narrowband uncorrelated signal source in the direction of θ k and Let azimuth and elevation be the azimuth and elevation angles of the k-th incident signal source, respectively, where k = 1, 2, ..., K. Then, consider the uniform planar subarray within the nested coprime planar array. The T snapshot samples are superimposed in the third dimension to obtain a three-dimensional tensor signal. The model is as follows:

[0052]

[0053] Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T Let [·] be the waveform vector of the multi-shot sampled signal corresponding to the k-th incident signal source. T This indicates the transpose operation. Represents the vector outer product. For a noise tensor that is independent of each signal source, a x (k) and a y (k) are respectively The guiding vectors in the x-axis and y-axis directions are represented as follows:

[0054]

[0055]

[0056] in, Similarly, sparse planar subarrays The received signal can be a three-dimensional tensor signal. Represented as:

[0057]

[0058] in, b x (k) and b y (k) are respectively The steering vectors in x-axis and y-axis directions are denoted as:

[0059]

[0060]

[0061] The second-order cross-correlation tensor is obtained by calculating the cross-correlation statistics of the three-dimensional tensor signals

[0062]

[0063] wherein, denotes the power of the kth incident signal source, denotes the four-dimensional cross-correlation noise tensor, <·,·> r denotes the tensor contraction operation of two tensors along the rth dimension, E[·] denotes the mathematical expectation operation, (·) * denotes the conjugate operation; in practice, The sampling cross-correlation tensor is approximated by calculating the sampling cross-correlation statistics of the tensor signals and

[0064]

[0065] Step 3: Construct the virtual domain tensor input of the convolutional neural network. Since the cross-correlation tensor contains the spatial information of the uniform planar subarray and the sparse planar subarray , by merging the dimensions representing the spatial information in the same direction in , the steering vectors corresponding to the same direction of the two subarrays form a difference set array in the exponential term, thereby constructing an augmented virtual array. Specifically, the 1st and 3rd dimensions of the cross-correlation tensor represent the spatial information in the x-axis direction, and the 2nd and 4th dimensions represent the spatial information in the y-axis direction; therefore, the dimension set is defined. By dimension merging on the cross-correlation tensor

[0066]

[0067] wherein, and by forming a difference set array in the exponential term, an augmented virtual array is constructed, which has a size of M x N​​x ×M y N y Virtual uniform planar array Indicates the Kronecker product. For the corresponding noise terms; arrange the elements in U to correspond to The position of the virtual array element is used to obtain the second-order equivalent signal of the virtual domain:

[0068]

[0069] in, and They are virtual uniform planar arrays Guiding vectors along the x-axis and y-axis directions. For the corresponding noise term; extract the imaginary and real parts of the second-order equivalent signal Q in the virtual domain, and superimpose them in the third dimension to obtain a three-dimensional virtual domain tensor. As input to a convolutional neural network;

[0070] Step 4: Canonical polyadic decomposition and compressed convolution calculation of CNN convolution kernels. Virtual domain tensors are extracted using a convolutional neural network. The signal features are used to estimate the direction of arrival. Specifically, a four-dimensional CNN convolution kernel corresponding to the first convolutional layer is set. Where (P1, Q1) represents the convolution size, H0 = 2 represents the number of input channels, and H1 represents the number of output channels of the first convolutional layer; to reduce the number of trainable parameters in the CNN convolutional kernel, such as Figure 2 As shown, its low-rank property is used to perform canonical polyadic decomposition, which is expressed as:

[0071]

[0072] in, For the kernel tensor, and They are respectively the corresponding A four-dimensional factor matrix, where R is the canonical polyadic rank, × r This represents the tensor-matrix product along the r-th dimension; thus, the number of trainable parameters contained in the kernel tensor and factor matrix obtained after CNN kernel decomposition is significantly compressed. Simultaneously, the kernel tensor and factor matrix are used to modify the virtual domain tensor. Compressed convolution is performed to obtain the output feature map of the first convolutional layer. I1 = M x N x -P1-2δ+1,J1=M y Ny -Q1-2δ+1, where δ is the size of zero padding, the compressed convolution calculation process can be represented by three steps:

[0073]

[0074]

[0075]

[0076] in, for The (i0,j0,h0)th element in the array, where i0 = 1, 2, ..., M x N x j0 = 1, 2, ..., M y N y i1 = 1, 2, ..., I1, j1 = 1, 2, ..., J1, h0 = 1, 2, ..., H0, h1 = 1, 2, ..., H1, r0, r1, r, r′ = 1, 2, ..., R; Similarly, construct another L-1 four-dimensional convolution kernels. The kernel tensors are obtained by canonical polyadic decomposition of CNN convolution kernels. With factor matrix {V l,1 V l,2 V l,3 V l,4} and outputs the corresponding feature maps layer by layer based on the compressed convolution calculation criterion. Thus, a compressed convolutional neural network of depth L is constructed to realize two-dimensional direction of arrival estimation;

[0077] Step 5: Two-dimensional direction-of-arrival estimation based on fast training of a compressed convolutional neural network. This utilizes a parameter matrix. Output feature map of the Lth convolutional layer After weighting, the two-dimensional direction-of-arrival estimation result of the output layer is obtained, which is expressed as:

[0078]

[0079] Where vec(·) represents the vectorization operation, For true two-dimensional wave direction of arrival Estimate the value; define the loss function for the output layer:

[0080]

[0081] Where ||·||1 and ||·||2 represent the 1-norm and 2-norm respectively, and η represents the transformation threshold for calculating the loss function, which is generally set to 1; the loss function Θ is calculated with respect to the kernel tensor of the convolutional layer. gradients of the factor matrices {V l,1 ,V l,2 ,V l,3 ,V l,4}, and updates the to-be-trained parameters therein using gradient descent; iteratively updates the to-be-trained parameters in the compressed convolutional neural network by inputting a plurality of virtual domain tensors {V l,1 ,V l,2 ,V l,3 ,V l,4}, and completes the fast training of the compressed convolutional neural network; in actual application, the precise two-dimensional direction-of-arrival estimation result is obtained by performing compressed convolution calculation of the trained complete compressed convolutional neural network

[0082] The effect of the application will be further described below in combination with a simulation example.

[0083] Simulation example: the received signal is received by a nested co-prime planar array, and the parameters are selected as M x =M y =4, N x =N y =5, that is, the nested co-prime planar array of the architecture contains M x M y +N x N y -1=40 physical array elements. The derived virtual uniform planar array has 20x20 virtual array elements, and the size of the corresponding virtual domain tensor input is 20x20x2. The training cost and efficiency of the compressed convolutional neural network based on convolution kernel tensor decomposition proposed in the application and the traditional uncompressed convolutional neural network are compared. The depth of the convolutional neural network is set to L=3, and the sizes of the convolution kernels are 5x5x2x16, 5x5x16x32 and 3x3x32x32 respectively. For the proposed compressed convolutional neural network, the canonical polyadic rank of the CNN convolution kernel tensor decomposition is {3, 3, 2}. Assuming that there are 2 incident signals, in the training phase, 19200 samples are generated to train the network, and the signal azimuth and elevation angle of each sample are randomly generated in the range of [15°, 65°], and the signal-to-noise ratio (SNR) is randomly selected from the set {-15dB, -10dB, -5dB, 0dB}. The experiment proves that the amount of to-be-trained parameters contained in the convolution kernel of the traditional convolutional neural network is 22816, and the floating point calculation amount consumed in training is 4.114x10 6 ; in comparison, the amount of to-be-trained parameters contained in the kernel tensor and the factor matrix of the proposed compressed convolutional neural network is compressed to 366, and the floating point calculation amount consumed in training is only 1.843x10 6 .

[0084] Furthermore, the estimation accuracy of the proposed method is compared with that of traditional convolutional neural network methods and canonical polyadic decomposition methods. Under the condition of 150 sampling snapshots T, performance comparison curves of root-mean-square error (RMSE) as a function of signal-to-noise ratio are plotted, as shown below. Figure 3 As shown; under the condition of signal-to-noise ratio (SNR) = -5dB, the performance comparison curve of RMSE as a function of sampling snapshot number T is plotted, as follows. Figure 4 As shown in the comparison results, compared with traditional uncompressed convolutional neural networks, the method proposed in this invention can achieve similar two-dimensional direction-of-arrival (DOA) estimation accuracy, while reducing training loss by 200 times. Compared with traditional methods based on canonical polyadic models, the method proposed in this invention effectively extracts deep multi-dimensional signal features through multi-layer compressed convolution calculations, maintaining better estimation accuracy and stability under conditions of low signal-to-noise ratio and low snapshot sampling number. In summary, this invention achieves sparse array two-dimensional DOA estimation that meets the requirements of low cost and high performance by performing tensor decomposition on the high-dimensional convolutional kernels of convolutional deep networks and designing corresponding compressed convolution calculation methods.

[0085] The above description is merely a preferred embodiment of the present invention. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make many possible variations and modifications to the technical solutions of the present invention using the methods and techniques disclosed above, or modify them into equivalent embodiments with equivalent changes, without departing from the scope of the technical solutions of the present invention. Therefore, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall still fall within the protection scope of the technical solutions of the present invention.

Claims

1. A sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition, characterized in that, Comprising the following steps: (1) The receiver uses M x M y +N x N y -1 physical antenna array elements are configured according to a two-dimensional nested coprime planar array structure; this nested coprime planar array is decomposed into a uniform planar subarray. and a sparse planar subarray in Includes M x ×M y There are 1 antenna array elements, and the element spacing d is half the wavelength λ of the incident narrowband signal, i.e., d = λ / 2. Contains N x ×N y There are 1 antenna array elements, with spacing M along the x-axis and y-axis respectively. x d and M y d; (2) Assume there are K far-field narrowband uncorrelated signal sources from θkand φkare the azimuth and elevation angles of the kth incident signal source, k = 1, 2, …, K, respectively. k The T snapshot signals of the uniform planar subarray in the nested coprime planar array are stacked in the third dimension to obtain a three-dimensional tensor signal Modeling as: ​​ Among them, s k =[s k,1 ,s k,2 ,…,s k,T ] T Let [·] be the waveform vector of the multi-shot sampled signal corresponding to the k-th incident signal source. T This indicates the transpose operation. Represents the vector outer product. For a noise tensor that is independent of each signal source, a x (k) and a y (k) are respectively The guiding vectors in the x-axis and y-axis directions are represented as follows: wherein, Similarly, the sparse planar sub-array The received signal is represented as a three-dimensional tensor signal is represented as: wherein is a noise tensor independent of each signal source, b x (k) and b y (k) are respectively Guiding vectors in the x-axis and y-axis directions are represented as: By taking the cross-correlation statistics of the tensor signals and the second-order cross-correlation tensor wherein, 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 taking mathematical expectation operation, (·) * represents the conjugate operation; (3) defining the set of dimensions by merging the cross-correlation tensors the dimensions representing the same directional spatial information in the middle table, we have: where and The difference set array is formed on the exponential term, and a virtual uniform planar array with size of M x N x ×M y N y is constructed denotes the Kronecker product, is the noise term; the elements in U are arranged to correspond to the positions of the virtual array elements in , and the virtual domain second-order equivalent signals are obtained: wherein, and are virtual uniform planar arrays steering vectors along the x and y axis directions, are the corresponding noise terms; the imaginary and real parts of the virtual domain second order equivalent signal Q are extracted, stacked in the third dimension, to obtain a three-dimensional virtual domain tensor as input to a convolutional neural network; (4) Set the four-dimensional CNN convolution kernel corresponding to the first convolution layer where (P1, Q1) represents the convolution size, H0=2 represents the number of input channels, and H1 represents the output channel number of the first convolution layer; and canonical polyadic decomposition is performed to obtain: in, For the kernel tensor, and They are respectively the corresponding A four-dimensional factor matrix, where R is the canonical polyadic rank, × r Represent the tensor-matrix product along the r-th dimension; use the kernel tensor and factor matrix to represent the virtual domain tensor. Compressed convolution is performed to obtain the output feature map of the first convolutional layer. I1 = M x N x -P1-2δ+1,J1=M y N y -Q1-2δ+1, where δ is the size of zero padding, the compressed convolution calculation process can be represented by three steps: wherein, is the (i0,j0,h0)th element in x N x , j0=1,2,…,M y N y , i1=1,2,…,I1, j1=1,2,…,J1, h0=1,2,…,H0, h1=1,2,…,H1, r0,r1,r,r'=1,2,…,R; similarly, set another L-1 four-dimensional CNN convolution kernels Obtain their corresponding kernel tensors by canonical polyadic decomposition of the CNN convolution kernels and factor matrices {V l,1 , V l,2 , V l,3 , V l,4} and calculate the corresponding feature maps layer by layer based on compressed convolution Thus, a compressed convolutional neural network with a depth of L is constructed for realizing sparse array two-dimensional direction of arrival estimation; (5) using a parameter matrix the output feature map of the Lth convolutional layer weighted to obtain a two-dimensional direction-of-arrival estimation result of the output layer where vec( ) denotes a vectorization operation, for real two-dimensional direction of arrival estimates; define the loss function for the output layer: Wherein ‖·‖1 and ‖·‖2 represent 1-norm and 2-norm respectively, and η represents a conversion threshold calculated by a loss function; based on gradient descent of the loss function Θ, fast training of the compressed convolutional neural network is completed; and using the trained compressed convolutional neural network, two-dimensional direction of arrival estimation is realized in an actual application scenario.

2. The sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition according to claim 1, characterized in that, The nested co-prime planar array structure described in step (1) is specifically described as follows: a uniform planar sub-array The position coordinates of the antenna elements in the coordinate system are {(m x d,m y d)|m x = 0, 1, …, M x -1,m y = 0, 1, …, M y -1} and a sparse planar sub-array The position coordinates of the antenna elements in the coordinate system are {(n x M x d,n y M y d)|n x = 0, 1, …, N x -1,n y = 0, 1, …, N y -1} respectively; {M x ,N x} and {M y ,N y} are a pair of co-prime integers respectively; and and The sub-arrays are combined in a manner that the elements at the origin position of the coordinate system are overlapped. Since the element arrangement of and satisfies the co-prime number condition, the elements at positions other than the origin position of the coordinate system are not overlapped, thereby obtaining a nested co-prime planar array actually containing M x M y +N x N y -1 antenna elements.

3. The sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition according to claim 1, characterized in that, The second-order cross-correlation tensor as described in step (2) The derivation, in practice, By computing the sampled cross-correlation statistics of the three-dimensional tensor signal and approximated by the sampled cross-correlation tensor 4. The sparse array direction of arrival estimation method based on CNN convolution kernel tensor decomposition according to claim 1, characterized in that, The compressed convolutional neural network accelerates the training process, and calculates the gradient of the loss function Θ with respect to the convolutional layer kernel tensor and the gradient of the factor matrix {V l,1 , V l,2 , V l,3 , V l,4} is calculated, and the to-be-trained parameters therein are updated by using gradient descent; the to-be-trained parameters in the virtual domain tensor {V , V l,1 , V l,2 , V l,3 , V l,4} are iteratively updated by inputting a plurality of virtual domain tensors, and the fast training of the compressed convolutional neural network is completed; in actual application, the accurate two-dimensional direction of arrival estimation result is obtained by performing compressed convolution calculation of the trained complete compressed convolutional neural network

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