Multi-source DOA high-resolution estimation method and system based on cascade neural network

Abstract

The invention discloses a cascade neural network-based multi-source DOA high-resolution estimation method and system, and relates to the technical field of array signal processing, in the method, firstly, an interpretable spatial domain filtering network is designed based on a single-layer FNN, so that a target spatial domain is divided into a plurality of sub-spatial domains; and processing the output data of each sub-airspace by using a plurality of two-dimensional CNNs to obtain a multi-source DOA estimation result under the condition that the number of sources is unknown. In order to improve the DOA estimation precision, a pre-training and joint fine tuning network training method is also adopted, independent pre-training is performed on the interpretable SF-Net and each two-dimensional CNN, and end-to-end joint training is performed on the cascaded neural network on the basis, so that final network parameters are obtained, and multi-source DOA estimation is performed. According to the method, the network interpretability can be improved, the scale of a training data set is reduced, the network training difficulty is reduced, and better DOA estimation performance can be obtained under the condition of multiple information sources.

Description

Technical Field

[0001] The present invention relates to the technical field of array signal processing, and in particular to a multi-source DOA high-resolution estimation method and system based on a cascade neural network. Background Art

[0002] Direction of Arrival (DOA) estimation is an important research direction in array signal processing and is widely used in radar, communication, navigation and other fields. Over the past few decades, scholars have proposed a series of DOA estimation algorithms, including subspace decomposition algorithms, subspace fitting algorithms and sparse recovery algorithms. Among them, subspace decomposition algorithms represented by the Multiple Signal Classification (MUSIC) algorithm and the rotation-invariant subspace algorithm can achieve super-resolution DOA estimation, but their performance is poor under low signal-to-noise ratio conditions; subspace fitting algorithms represented by the maximum likelihood algorithm and the weighted subspace fitting algorithm can achieve good DOA estimation performance under low signal-to-noise ratio conditions, but their dependence on initial values ​​and computational complexity are high; sparse recovery algorithms represented by the L1 norm minimization algorithm and the orthogonal matching pursuit (OMP) algorithm have high estimation accuracy, but there are problems such as high computational complexity or the need for prior information on signal sparsity, which limits the application of such algorithms in practice.

[0003] In recent years, deep learning technology represented by deep neural network (DNN) has developed rapidly, providing new ideas for DOA estimation. Domestic and foreign scholars have successively applied autoencoders and fully connected networks (FNN) to DOA estimation, and obtained good estimation results under single-source conditions, but the performance is poor under multi-source conditions. In order to solve the problem of multi-source DOA estimation, scholars proposed a method of first dividing the target airspace into multiple sub-airspaces, and then performing DOA estimation on each sub-airspace separately. For example, some scholars use multiple autoencoders to divide the airspace, and each autoencoder is cascaded with a classifier composed of multiple layers of FNN to achieve multi-source DOA estimation. However, the performance of the spatial filtering network (SF-Net) composed of autoencoders is somewhat different from that of the ideal spatial filter, and there is a problem of poor interpretability. In addition, the network parameters of multi-layer FNN are relatively large, and the training difficulty is relatively high. In response to the problems of FNN, some scholars used a two-dimensional convolutional neural network (CNN) to process the array receiving data, and proposed different data set construction methods and network training methods for the cases where the number of sources is known and the number of sources is unknown, respectively, with higher multi-source DOA estimation performance. However, for the case where the number of sources is unknown, when the number of sources increases, the scale of the training data set constructed by this method will grow exponentially, the memory usage will increase, and the difficulty of network training will increase. Therefore, the application of this method in practice is subject to certain limitations.

[0004] Therefore, there is an urgent need for a DOA estimation scheme that can be used under multi-source conditions to solve the problems of poor interpretability, high training difficulty, and poor multi-source estimation performance in the above-mentioned existing technologies. Summary of the invention

[0005] The purpose of the present invention is to provide a multi-source DOA high-resolution estimation method and system based on a cascade neural network, which can improve the network interpretability, reduce the size of the training data set, improve the network training efficiency, and obtain better DOA estimation performance under multi-source conditions.

[0006] To achieve the above object, the present invention provides the following solutions:

[0007] The multi-source DOA high-resolution estimation method based on cascade neural network includes the following steps:

[0008] S1. Array receiving data preprocessing: performing covariance matrix estimation on the array receiving data obtained under multi-snap sampling conditions, obtaining the covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix, thereby obtaining preprocessed array receiving data;

[0009] S2, interpretable spatial domain filtering: design Q spatial domain filters based on a single-layer fully connected network, construct an interpretable spatial domain filtering network, initialize and train the interpretable spatial domain filtering network, use the trained interpretable spatial domain filtering network to divide the pre-processed array received data into Q sub-spatial domains, and perform matrixing and three-channeling processing respectively to obtain three-channel real-valued matrices after the Q spatial domain filtering;

[0010] S3, CNN spatial spectrum estimation: Design Q two-dimensional CNN estimators, construct DOA estimators, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize them, and output the CNN spatial spectrum estimation result;

[0011] S4, joint fine-tuning: the interpretable spatial domain filter network and the DOA estimator are combined to form a cascade neural network, and the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator are obtained by end-to-end joint fine-tuning training of the cascade neural network to form an optimal cascade neural network;

[0012] S5. Utilize the optimal cascade neural network to perform DOA estimation.

[0013] Further, in S1, array received data preprocessing: performing covariance matrix estimation on the array received data obtained under multi-snap sampling conditions, obtaining the covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix, and obtaining preprocessed array received data, specifically includes the following steps:

[0014] S101, perform covariance matrix estimation on the array received data obtained under the multi-snap sampling condition to obtain the covariance matrix:

[0015]

[0016] In the formula, (·) H represents conjugate transpose, Y represents the array receiving data matrix, and T represents the number of snapshots;

[0017] S102, the covariance matrix Perform vectorization processing, separate the real part and the imaginary part of the result, reconstruct it into a real vector, and obtain the preprocessed array receiving data

[0018]

[0019] Where vec(·) represents the vectorization operation on the matrix, Re represents the real part, and Im represents the imaginary part.

[0020] Furthermore, in S2, Q spatial domain filters are designed based on a single-layer fully connected network, an interpretable spatial domain filter network is constructed, and the interpretable spatial domain filter network is initialized and trained, specifically including the following steps:

[0021] S201, construct Q weight matrices that do not contain activation functions and biases A single-layer fully connected network is used as the spatial domain filter, q = 1, 2, ..., Q, Q is the number of sub-spatial domains divided;

[0022] S202, connecting Q spatial domain filters in parallel to obtain an interpretable spatial domain filter network;

[0023] S203, constructing a training data set for an interpretable spatial domain filtering network:

[0024]

[0025] In the formula, and They represent the input data set and label data set of the interpretable spatial domain filter network, P represents the number of grids divided in each sub-spatial domain, and J I is the number of repetitions, J S is the signal-to-noise ratio, For the first pqj I -j S Input data, L q,p is the pqth label data;

[0026] S204, the weight matrix W of each single-layer fully connected network in the interpretable spatial domain filtering network q To initialize:

[0027]

[0028] In the formula, (·) * represents conjugation, represents the Kronecker product, (·) T represents matrix transpose, H represents conjugate transpose; g q By solving the following equation:

[0029]

[0030] In the formula, I N×Nrepresents the unit matrix of size N×N, where N represents the number of grids into which the target airspace is divided. represents the qth sub-space domain, Represents the array manifold matrix corresponding to the P grids of the qth sub-space domain, express The pth grid The corresponding steering vector, p = 1, 2, ..., P, P represents The corresponding number of grid divisions is express The complement of , Θ is the target space, express The sth grid The corresponding steering vector, s = 1, 2, ..., S, S represents The corresponding number of grid divisions, δ represents the stopband attenuation rate;

[0031] S205, using the training data set, adopting the mean square error function as the network loss function, and obtaining the network parameters after the interpretable spatial domain filtering network training through back propagation algorithm learning

[0032]

[0033] In the formula, Indicates is the network parameter, with the first pqj I -j S Training data is the interpretable spatial domain filter network output of the input, J SF =PQJ I J S is the size of the training dataset.

[0034] Furthermore, in S2, the preprocessed array received data is divided into Q sub-spatial domains using a trained interpretable spatial domain filtering network, and matrixing and three-channeling are performed respectively to obtain three-channel real-valued matrices after Q spatial domain filtering, which specifically includes the following steps:

[0035] S2.1. Use Q trained single-layer fully connected networks to filter the pre-processed array received data and divide the data into Q sub-spatial domains:

[0036]

[0037] In the formula, r q represents the output data of the qth trained single-layer fully connected network, Represents the network parameters of the qth single-layer fully connected network obtained after training;

[0038] S2.2. The output of each single-layer fully connected network is processed into a complex number, a matrix, and a three-channel, and Q spatial domain filtered three-channel real-valued matrices X are obtained. q :

[0039]

[0040] In the formula, comp(r q )=r q(1:M×M) +jr q(M×M+1:2M×M) Indicates that r q Reconstructed into complex numbers, mat(·) represents matrix operation on vectors, and ∠(·) represents angle operation.

[0041] Furthermore, in S3, Q two-dimensional CNN estimators are designed, a DOA estimator is constructed, and each CNN estimator is initialized and trained, specifically including the following steps:

[0042] S301, design Q two-dimensional CNN networks including an input layer, a two-dimensional convolutional layer, a batch normalization layer, a fully connected layer and an output layer as two-dimensional CNN estimators corresponding to the Q sub-spatial domains;

[0043] S302, connecting Q two-dimensional CNN estimators in parallel to obtain a DOA estimator;

[0044] S303, construct a training data set for the DOA estimator:

[0045]

[0046] In the formula, and Represent the input dataset and label dataset of the qth CNN estimator respectively; is the number of combinations, J I is the number of repetitions, J S is the signal-to-noise ratio, K max Indicates the maximum number of signal sources. For Kj I -j S -j K training data, For Kj I -j S -j K Label data;

[0047] S304, initializing the network parameters of each CNN estimator in the DOA estimator:

[0048] The Glorot normalization method is used to initialize the convolution kernels of each convolutional layer and the weights of each fully connected layer, the bias vector is initialized to 0, the scale factor of each batch normalization layer is initialized to 1, and the translation factor is initialized to 0;

[0049] S305: Using the training data set, adopting the mean square error function as the network loss function, and learning the trained network parameters of the DOA estimator through the back propagation algorithm

[0050] In the formula, Indicates that the Kj K -j I -j S Training data As input, Ω q is the qth CNN estimator output of the network parameters, is the size of the training dataset.

[0051] Further, in S3, the three-channel real-valued matrix after filtering in each spatial domain is processed using the trained DOA estimator to obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize it, and output the CNN spatial spectrum estimation result, which specifically includes the following steps:

[0052] S3.1. Use Q trained CNN estimators to process the three-channel real-valued matrix after filtering in each spatial domain, and obtain the spatial spectrum estimation result P of each sub-spatial domain. q :

[0053]

[0054] In the formula, Indicates that the Xth q For input, is the qth trained CNN estimator output for the network parameters;

[0055] S3.2. Synthesize the spatial spectrum estimation results of all sub-spatial domains and output the CNN spatial spectrum estimation result P:

[0056]

[0057] Furthermore, in S4, the interpretable spatial domain filter network and the DOA estimator are combined to form a cascade neural network, and the cascade neural network is end-to-end jointly fine-tuned to obtain the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator, which specifically includes the following steps:

[0058] S401, using the output of the interpretable spatial domain filter network as the input of the DOA estimator, cascading them to obtain a cascade neural network;

[0059] S402, constructing a training data set for the cascade neural network:

[0060]

[0061] In the formula, and They represent the input dataset and label dataset of the cascade neural network respectively, and J I is the number of repetitions, J S is the signal-to-noise ratio, J α is the number of signal angle combinations; is the maximum number of signal sources, For Kj I -j S -j α training data, For Kj I -j S -j α Label data;

[0062] S403, initializing the network parameters of the cascade neural network:

[0063] The initialization network parameters of the cascade neural network are obtained according to the trained interpretable spatial domain filter network parameters and the trained DOA estimator network parameters:

[0064]

[0065] In the formula, Γ represents the network parameters of the cascade neural network;

[0066] S404, performing end-to-end joint fine-tuning training on the initialized cascade neural network to obtain the optimal network parameters Γ of the interpretable spatial domain filter network and the DOA estimator opt = {W opt ,Ω opt}:

[0067]

[0068] In the formula, Indicates that the Kj I -j S -j α Training data is the input and the output of the cascade neural network with Γ as the network parameter, is the size of the training dataset.

[0069] Further, in S5, using the optimal cascade neural network to perform DOA estimation specifically includes:

[0070] Based on the network parameters, the cascade neural network is used to process the T snapshot array data D actually received. real Processing is performed to obtain the DOA estimation result:

[0071]

[0072] In the formula,

[0073] The present invention also provides a multi-source DOA high-resolution estimation system based on a cascade neural network, which is used to execute the multi-source DOA high-resolution estimation method based on a cascade neural network, comprising:

[0074] The array receiving data preprocessing module is used to estimate the covariance matrix of the array receiving data obtained under the multi-snap sampling condition, obtain the covariance matrix and vectorize and separate the real part and the imaginary part of the covariance matrix to obtain the preprocessed array receiving data;

[0075] An interpretable spatial domain filtering module is used to design Q spatial domain filters based on a single-layer fully connected network, construct an interpretable spatial domain filtering network, initialize and train the interpretable spatial domain filtering network, and use the trained interpretable spatial domain filtering network to divide the pre-processed array received data into Q sub-spatial domains, and perform matrixing and three-channeling processing respectively to obtain three-channel real-valued matrices after Q spatial domain filtering;

[0076] The CNN spatial spectrum estimation module is used to design Q two-dimensional CNN estimators, construct a DOA estimator, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain, synthesize it, and output the CNN spatial spectrum estimation result;

[0077] A joint fine-tuning module is used to combine the interpretable spatial domain filter network and the DOA estimator to form a cascade neural network, and to obtain the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator by performing end-to-end joint fine-tuning training on the cascade neural network to form an optimal cascade neural network;

[0078] A high-resolution DOA estimation module is used to perform DOA estimation using the optimal cascade neural network.

[0079] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects: the multi-source DOA high-resolution estimation method and system based on cascade neural network provided by the present invention, the method constructs an interpretable spatial domain filtering network (SF-Net) based on a single-layer fully connected network, and performs sub-spatial domain division on the array received data; uses multiple convolutional neural networks to process the output of the spatial domain filtering network to realize subspace spectrum estimation; finally, performs subspace spectrum synthesis to obtain the multi-source DOA estimation result under the condition of unknown source number. Simulation results show that under different signal-to-noise ratios and source numbers, the method can obtain better DOA estimation performance than the traditional MUSIC algorithm. Compared with the typical deep neural network estimation algorithm, the present invention can greatly improve the network interpretability, reduce the scale of training data sets, reduce the difficulty of network training, and obtain better DOA estimation results under multi-source conditions. The present invention solves the problems of poor interpretability, high training difficulty, and poor multi-source estimation performance of DOA estimation algorithms based on deep neural networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0081] Figure 1 It is a structural diagram of the multi-source DOA estimation method based on cascade neural network of the present invention;

[0082] Figure 2 A network structure diagram of a CNN estimator according to an embodiment of the present invention;

[0083] Figure 3 : is the spatial domain response of different filters of SF-Net according to an embodiment of the present invention;

[0084] Figure 4 Output response of multi-source SF-Net according to an embodiment of the present invention;

[0085] Figure 5 1 is a comparison diagram of multi-source DOA estimation results using different algorithms in an embodiment of the present invention, wherein (a)-(d) represent the DOA estimation results of different algorithms corresponding to 2 sources, 3 sources, 6 sources, and 12 sources, respectively;

[0086] Figure 61 and 2 are the DOA estimation results of two sources of the multi-source DOA estimation algorithm based on the cascade neural network in an embodiment of the present invention, wherein (a) represents the estimation result corresponding to the conditions of signal-to-noise ratio SNR=10dB and angle interval Δ=5°; (b) represents the estimation result corresponding to the conditions of signal-to-noise ratio SNR=10dB and angle interval Δ=1°; (c) represents the estimation result corresponding to the conditions of signal-to-noise ratio SNR=-10dB and angle interval Δ=5°; (d) represents the estimation result corresponding to the conditions of signal-to-noise ratio SNR=-10dB and angle interval Δ=1°. DETAILED DESCRIPTION

[0087] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0088] The purpose of the present invention is to provide a multi-source DOA high-resolution estimation method and system based on a cascade neural network, which can improve the interpretability of the network, reduce the size of the training data set, reduce the difficulty of network training, and obtain better DOA estimation performance under multi-source conditions.

[0089] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0090] Example 1

[0091] like Figure 1 As shown, the multi-source DOA high-resolution estimation method based on a cascade neural network provided in an embodiment of the present invention includes the following steps:

[0092] S1. Array receiving data preprocessing: performing covariance matrix estimation on the array receiving data obtained under multi-snap sampling conditions, obtaining the covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix, thereby obtaining preprocessed array receiving data;

[0093] S2, interpretable spatial domain filtering: based on a single-layer fully connected network (FNN), Q spatial domain filters are designed, an interpretable spatial domain filter network (SF-Net) is constructed, the interpretable spatial domain filter network is initialized and trained, and the pre-processed array received data is divided into Q sub-spatial domains using the trained interpretable spatial domain filter network, and matrix and three-channel processing are performed respectively to obtain three-channel real-valued matrices after Q spatial domain filtering;

[0094] S3, spatial spectrum estimation: design Q two-dimensional CNN estimators, construct DOA estimators, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize them, and output the CNN spatial spectrum estimation result;

[0095] S4, joint fine-tuning: The interpretable spatial domain filter network and the DOA estimator are combined to form a cascade neural network (hereinafter referred to as SFCNN), and the optimal interpretable spatial domain filter network and DOA estimator network parameters are obtained by end-to-end joint fine-tuning training of the cascade neural network to form the optimal cascade neural network;

[0096] S5. Utilize the optimal cascade neural network to perform DOA estimation.

[0097] 1-1. About DOA estimation signal model:

[0098] Assume that there are K far-field narrowband signals at time t When incident on a uniform linear array consisting of M isotropic array elements, the array receives data as follows:

[0099] y(t)=A(θ)s(t)+n(t) (1)

[0100] In the formula, y(t)=[y1(t),y2(t),L,y M (t)] T ∈C M×1 represents a complex array of size M×1 receiving data vector, A(θ)=[a(θ1),a(θ2),...,a(θ K )]∈C M×K represents a complex array manifold matrix of size M×K, represents the kth incident signal complex steering vector of size M×1, represents the angle set of the incident signal, d represents the array element spacing, λ represents the signal wavelength, s(t)=[s1(t),s2(t),...,s k (t)] T ∈C K×1 represents the complex incident signal vector of size K×1, n(t)=[n1(t),n2(t),L,n M (t)] T ∈C M×1 represents the complex noise vector, (·) T Indicates transpose.

[0101] Under the condition of T snapshot sampling, the array receives data as follows:

[0102] Y=A(θ)S+N (2)

[0103] In the formula, Y=[y(t1),y(t2),L,y(t T )]∈C M×T represents the complex received data matrix of size M×T, S=[s(t1),s(t2),L,s(t T )]∈C K×T represents the complex incident signal matrix of size K×T, T represents the number of snapshots, N=[n(t1),n(t2),L,n(t T )]∈C M×T represents the complex noise matrix of size M×T.

[0104] Assume that the spatial angle range of possible incident signals is θ = [θ Start ,θ Final ], the target airspace Θ can be divided into N grids On this basis, it is assumed that the signal incident angles are all on the grid, that is, Then the array receives data approximately as follows:

[0105] Y=A(θ * )S * +N (3)

[0106] In the formula, represents a complex array manifold matrix of size M×N, S * =[s * (t1),L,s * (t T )]∈C N×T , N represents the number of grids into which the target airspace is divided.

[0107] Based on formula (3), the angle estimation of the incident signal can be obtained by using methods such as MUSIC, OMP or DNN, that is, the spatial spectrum P∈C N×1 Among them, the DNN-based DOA estimation method constructs a data set to train FNN or CNN, so that it can fit the complex nonlinear relationship between the array received data Y and the spatial spectrum P, which is expressed as:

[0108]

[0109] In the formula, Represents a nonlinear transformation operation. This embodiment aims to construct a new deep neural network to obtain a nonlinear transformation with better performance for multi-source DOA estimation.

[0110] 1-2. In this embodiment, in S1, array received data preprocessing: performing covariance matrix estimation on the array received data obtained under multi-snap sampling conditions, obtaining a covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix to obtain preprocessed array received data, specifically including:

[0111] Assume that the incident signal and noise are uncorrelated, and the noise has a mean of 0 and a variance of The complex Gaussian distribution of the array can be obtained according to formula (1) as follows:

[0112] R = E{y(t)y H (t)}=A(θ)E{s(t)s(t) H} H (θ)+R N (5)

[0113] In the formula, E{·} represents expectation, represents the noise covariance matrix, I M×M represents the identity matrix of size M×M;

[0114] In practice, the covariance matrix R is unknown, and it can be estimated based on T snapshot data according to formula (2), expressed as:

[0115]

[0116] In the formula, (·) H represents conjugate transpose;

[0117] To facilitate subsequent processing, the covariance matrix Perform vectorization processing, separate the real part and the imaginary part of the result, and reconstruct it into a real vector, which is expressed as:

[0118]

[0119] Where vec(·) means vectorizing the matrix, Re means taking the real part, and Im means taking the imaginary part.

[0120] 1-3. In this embodiment, in S2, Q spatial domain filters are designed based on a single-layer FNN, an interpretable SF-Net is constructed, and the interpretable SF-Net is initialized and trained, specifically including:

[0121] The existing DNN-based DOA estimation method uses autoencoders or CNNs to design spatial filtering networks, which can filter out signals outside the corresponding sub-spatial domains in the input data to a certain extent and perform sub-spatial domain division. However, the performance of the existing spatial filtering network is different from that of the ideal spatial filter, and there are problems such as high difficulty in network training and poor interpretability. Based on the principle of matrix spatial filtering, the present invention designs a spatial filter (hereinafter referred to as Interpretable SF-Net) that is simple to train and interpretable based on a single-layer FNN.

[0122] The function of spatial filtering is to allow the signal in the spatial domain of interest to pass without distortion, while suppressing the signal from the spatial domain of no interest. Let the qth matrix filter be G q ∈C M×M (corresponding to the qth sub-spatial domain), then based on formula (1), the filtered array received data can be obtained:

[0123]

[0124] In the formula, q = 1, 2, ..., Q, Q represents the number of sub-space domains divided, represents the filtered complex noise vector.

[0125] In order to make the array received data after filtering contain only the data corresponding to the qth sub-spatial domain, G q The following constraints should be met:

[0126]

[0127] In the formula, represents the qth sub-space domain, express is the complement of , Θ is the target space.

[0128] right and After discretization, the following optimization problem can be solved based on the stopband constrained passband minimum mean square criterion to obtain the optimal G q The estimate of is expressed as:

[0129]

[0130] In the formula, P represents The corresponding number of grid divisions is express The pth grid The corresponding steering vector, S represents The corresponding number of grid divisions, δ represents the stopband attenuation rate, express The pth grid The corresponding steering vector, express The sth grid The corresponding steering vector.

[0131] From equations (5) and (8), we can get the data covariance matrix after the qth matrix filter, which is expressed as:

[0132]

[0133] According to the properties of the Kronecker product, we can vectorize both ends of equation (11) to obtain:

[0134]

[0135] In the formula, represents the Kronecker product.

[0136] By performing real-imaginary separation on the vector and matrix in equation (12), we can obtain:

[0137] r q =W q r (13)

[0138] In the formula,

[0139] It can be found that formula (13) is consistent with “taking r as input and r as q is the output” and does not include activation function and bias, and the weight matrix is ​​W q Therefore, the present invention constructs Q single-layer FNNs as spatial domain filters, and obtains a weight matrix W with better filtering performance based on data training and learning. q On this basis, Q single-layer FNNs are connected in parallel to form SF-Net, which divides the array receiving data into Q sub-space domains, with input r and output

[0140] Given the number of array elements M, the array element spacing d, the wavelength λ, the number of sub-space domains Q, and the range of each sub-space domain The number of grids in each sub-spatial domain P, the signal-to-noise ratio variation range [SNR low ,SNR high ], number of signal-to-noise ratios J S , number of snapshots T and number of repetitions J I , the steps to construct the training dataset of interpretable SF-Net are as follows:

[0141] ① Select the qth sub-space and the pth grid point to determine the signal incident angle

[0142] ② Generate T snapshot array receiving data s q,p ∈C 1×T , calculate the signal steering vector Get array receiving data The signal amplitude is set to 1, s q,p Each element in is independent of each other and obeys a complex Gaussian distribution with a mean of 0 and a variance of 1. N is a complex Gaussian white noise signal, ρ n is the noise amplitude determined by the signal-to-noise ratio;

[0143] ③According to formula (6) and formula (7), estimate the covariance matrix And vectorize and separate the real and imaginary parts to obtain As input data for interpretable SF-Net;

[0144] ④According to Calculate the ideal signal covariance matrix, vectorize it and separate the real and imaginary parts to obtain r q,p , as the label data of interpretable SF-Net, is expressed as:

[0145]

[0146] ⑤Repeat steps ②-④J I times, obtained and

[0147] ⑥Change the signal-to-noise ratio and repeat steps ②-⑤ until the set signal-to-noise ratio range is traversed and the and

[0148] ⑦ Change p and q, repeat steps ②-⑥ until all grids and sub-spaces are traversed to obtain and Construct the input dataset and label dataset of interpretable SF-Net, expressed as:

[0149]

[0150] in, and Represent the input dataset and label dataset respectively, For the first pqj I -j S Input data, L q,p is the pqth label data;

[0151] It should be noted that: ① The scale of the constructed SF-Net dataset is J SF =J I J SQP;②The label dataset is provided by J I J S Group repeated data sets Composition; ③ According to the linear properties of a single-layer FNN without bias vector and activation function, the required spatial filtering performance can be obtained by training SF-Net using a single source array with an amplitude of 1 to receive data, without the need to use multiple source arrays with different amplitudes to receive data.

[0152] The network parameters of the interpretable SF-Net are the weight matrices of Q single-layer CNNs. The existing method can be used to solve equation (10) to obtain G q The weight matrix Wq of each single-layer CNN is initialized by estimating the weight matrix Wq of each single-layer CNN. The specific method is as follows:

[0153] make in(·) * represents conjugate, then the optimization function in equation (10) can be expressed as:

[0154]

[0155] In the formula, I N×N represents the identity matrix of size N×N, Represents the array manifold matrix corresponding to the P grids of the qth sub-spatial domain.

[0156] Similarly, the constraint condition of formula (10) can be expressed as:

[0157]

[0158] Based on equations (16) and (17), the following second-order cone programming problem can be obtained:

[0159]

[0160] The interior point method and other methods can be used to quickly solve equation (18) to obtain g q Therefore, the weight matrix of the qth single-layer CNN in SF-Net can be initialized as:

[0161]

[0162] In the formula,

[0163] After initialization, the training data set constructed based on formula (15) is used, and the mean square error function is used as the network loss function. The network parameters after interpretable SF-Net training are obtained through back propagation algorithm learning.

[0164]

[0165] In the formula, Indicates is the network parameter, with the first pqj I -j S Training data is the interpretable SF-Net output of the input, J SF =PQJ I J S is the size of the training dataset.

[0166] 1-4. In this embodiment, in S2, the pre-processed array received data is divided into Q sub-spatial domains using the trained interpretable SF-Net, and matrix and three-channel processing are performed respectively to obtain Q spatial domain filtered three-channel real-valued matrices, specifically including:

[0167] Use Q trained single-layer FNNs to filter the preprocessed array received data and divide the data into Q sub-space domains:

[0168]

[0169] In the formula, r q represents the output data of the qth trained single-layer FNN, Represents the qth single-layer FNN network parameters obtained after training.

[0170] On this basis, the output of each single-layer FNN is processed into complex numbers, matrices and three channels to obtain Q spatial domain filtered three-channel real-valued matrices X q :

[0171]

[0172] In the formula, comp(r q )=r q(1:M×M) +jr q(M×M+1:2M×M) Indicates that r q Reconstructed into complex numbers, mat(·) represents matrix operation on vectors, and ∠(·) represents angle operation.

[0173] 1-5. In this embodiment, in S3, Q two-dimensional CNN estimators are designed, a DOA estimator is constructed, and each CNN estimator is initialized and trained, specifically including:

[0174] Since CNN has certain advantages over other networks in terms of learning ability and number of parameters, the embodiment of the present invention designs a two-dimensional CNN estimator for the output of each spatial domain filter to obtain spatial spectrum estimation results in different sub-spatial domains. The specific structure is as follows: Figure 2 shown.

[0175] Each 2D CNN estimator includes 9 layers of networks, namely, input layer, 2D convolution layer C1, 2D convolution layer C2, 2D convolution layer C3, 2D convolution layer C4, fully connected layer F1, fully connected layer F2, fully connected layer F3 and output layer, where the input of the qth 2D CNN estimator is the qth three-channel real-valued matrix X q , the output is the qth subspace spectrum estimation, and the activation functions of the convolutional layer and the fully connected layer are both ReLU functions. In order to improve the convergence speed and stability of the network, a batch normalization (BN) layer is introduced after each convolutional layer. In addition, in order to avoid overfitting, each fully connected layer is processed using the data loss (Dropout) technology.

[0176] The C1 layer is used to extract features from the input data:

[0177]

[0178] In the formula, * represents convolution, represents the output of layer C1, Represented by the scale factor γ1 q and translation factor β1 q is the batch normalization operation for learning parameters, Represents the first convolution kernel, with dimensions of υ1×υ1×n0×n1, where υ1 represents the length and width of the convolution kernel, n1 represents the number of convolution kernels, and n0=3 represents the number of channels of the input data. represents a bias vector of dimension n1;

[0179] The C2, C3, and C4 layers are used to map data features to high-dimensional space:

[0180]

[0181] In the formula, Represents C i The output of the layer, Represents the i-th convolution kernel, with dimension υ i ×υ i ×n i-1 ×n i ,i=2,3,4,υ i Represents the length and width of the convolution kernel, n i Indicates the number of convolution kernels, y i-1 represents the output of the i-1th layer, Indicates dimension n i The bias vector of

[0182] The FC5, FC6 and FC7 layers are used to integrate feature information:

[0183]

[0184] In the formula, Indicates FC i Output of layer, i=5,6,7,P i Represents a random vector used for Dropout processing, whose elements follow the Bernoulli distribution, and represents the weight and bias vector of the i-th layer, and e represents the Hadamard product;

[0185] The output layer consists of a fully connected layer with P neurons and a Sigmoid layer. The output of the FC7 layer is mapped to the interval [0,1] to obtain the network output O q :

[0186]

[0187] In the formula, and Represent the weight and bias vector of the output layer respectively, Sigmoid(y)=1 / (1+e -y ).

[0188] By connecting Q CNN estimators in parallel, we can get the DOA estimator. Given the number of array elements M, the array element spacing d, the wavelength λ, the number of sub-space domains Q, and the range of each sub-space domain The number of grids in each sub-spatial domain P and the maximum number of signal sources in each sub-spatial domain K max , signal-to-noise ratio variation range [SNR low ,SNR high ], number of signal-to-noise ratios J S and the number of repetitions J I , the steps for constructing the training dataset for the qth CNN estimator are as follows:

[0189] ① Select the number of information sources K, where 0≤K≤K max ;

[0190] ② Calculate the array manifold matrix in is the incident signal angle set,

[0191] ③Calculate the data covariance matrix in is the incident signal amplitude set, ρ n is the noise amplitude determined by the signal-to-noise ratio;

[0192] ④ For R q Perform three-channel processing to obtain a three-channel real-valued matrix X q,K , as the input data of the qth CNN estimator, Xq,K (:,:,1)=Re(R q ), X q,K (:,:,2)=Im(R q ), X q,K (:,:,3)=∠(R q );

[0193] ⑤Build label data L q,K , expressed as:

[0194]

[0195] In the formula, Represents the angle corresponding to the pth grid in the qth sub-space;

[0196] ⑥Repeat steps ③-⑤J I times, obtained and

[0197] ⑦Change the signal-to-noise ratio and repeat steps ③-⑥ until the set signal-to-noise ratio range is traversed and the and

[0198] ⑧Change the incident signal angle set and repeat steps ②-⑦ until all possible angle combinations are traversed to obtain and in Indicates the number of combinations;

[0199] ⑨Change the number of information sources and repeat steps ②-⑧ until the maximum number of information sources in the sub-spatial domain is reached, and the input data set and label data set of the qth CNN estimator are obtained, which can be expressed as:

[0200]

[0201] In the formula, and Represents the input dataset and label dataset of the qth CNN estimator respectively For Kj I -j S -j K training data, For Kj I -j S -j K Label data;

[0202] It should be noted that: ① The size of a single CNN estimator dataset is ② The number of signal sources K = 0 corresponds to the case of no incident signal, that is, the input data only contains noise, and the label data is an all-zero vector; ③ Q CNN estimators construct data sets separately according to the same method.

[0203] The network parameters of the qth CNN estimator are The Glorot normalization method is used to normalize the convolution kernels of each convolution layer. and the weights of each fully connected layer Initialize and set the bias vector Initialize to 0, and set the Initialized to 1, Initialized to 0.

[0204] After initialization, the training data set constructed based on formula (28) can be used, and the mean square error function can be used as the network loss function to obtain the trained network parameters of the DOA estimator through back propagation algorithm.

[0205]

[0206] In the formula, Indicates that the Kj K -j I -j S Training data As input, Ω q is the output of the qth CNN estimator of the network parameters.

[0207] 1-6. In this embodiment, in S3, the three-channel real-valued matrix after filtering in each spatial domain is processed using the trained DOA estimator to obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize it, and output the CNN spatial spectrum estimation result, which specifically includes:

[0208] Use Q trained CNN estimators to process the three-channel real-valued matrix after filtering in each spatial domain to obtain the spatial spectrum estimation result P of each sub-spatial domain q :

[0209]

[0210] In the formula, Indicates that the Xth q For input, is the qth trained CNN estimator output of the network parameters.

[0211] The spatial spectrum estimation results of all sub-spatial domains are synthesized and the CNN spatial spectrum estimation result P is output:

[0212]

[0213] 1-7. In this embodiment, in S4, the interpretable SF-Net and the DOA estimator are combined to form a cascade neural network, and the cascade neural network is end-to-end jointly fine-tuned to obtain the optimal network parameters of the interpretable SF-Net and the DOA estimator, specifically including:

[0214] The output of the interpretable SF-Net is used as the input of the DOA estimator, and they are cascaded to obtain a cascaded neural network (hereafter referred to as SFCNN). Given the number of array elements M, the array element spacing d, the wavelength λ, and the target spatial angle range Θ = [θ Start ,θ Final ], number of grids N, number of snapshots T, signal-to-noise ratio range [SNR low ,SNR high ], number of signal-to-noise ratios J S , Maximum number of sources satisfy Number of signal angle combinations J α and the number of repetitions J I , the steps for constructing the SFCNN training dataset are as follows:

[0215] ① Select the number of information sources K, where

[0216] ② Calculate the array manifold matrix in is the incident signal angle set,

[0217] ③ Generate T snapshot data S and obtain array received data Y K =A(θ * )diag{ρ *}S+ρ n N, as the input data of SFCNN, where each element in S is independent of each other and obeys a complex Gaussian distribution with a mean of 0 and a variance of 1. is the incident signal amplitude set, N is the complex Gaussian white noise signal, ρ n is the noise amplitude determined by the signal-to-noise ratio;

[0218] ④According to formula (6) and formula (7), estimate the covariance matrix And vectorize and separate the real and imaginary parts to obtain

[0219] ⑤Build the label data L of SFCNN K , expressed as:

[0220]

[0221] In the formula, Indicates the angle corresponding to the nth grid point;

[0222] ⑥Repeat steps ②-⑤J I times, obtained and

[0223] ⑦Change the signal-to-noise ratio and repeat steps ②-⑥ until the set signal-to-noise ratio range is traversed and the and

[0224] ⑧Change the incident signal angle set and repeat steps ②-⑦ times, obtained and

[0225] ⑨Change the number of signal sources and repeat steps ②-⑧ until the maximum number of signal sources is reached. Construct the input dataset and label dataset of SFCNN, expressed as:

[0226]

[0227] It should be noted that: ① The size of the constructed SFCNN training dataset is ② Since each CNN estimator and SF-Net have been trained to obtain better network parameters, for a given number of sources, only a small number of angle combinations (i.e. ) Construct input data and label data to jointly fine-tune the network parameters of SFCNN to adapt it to the target spatial domain DOA estimation without traversing all possible angle combinations.

[0228] The network parameters of SFCNN can be expressed as Γ = {W, Ω}. The trained interpretable SF-Net and each CNN estimator have obtained good network parameters. Therefore, in the joint fine-tuning stage, the initialization of SFCNN can be directly determined by the trained SF-Net and CNN, that is, On this basis, SFCNN can be end-to-end jointly fine-tuned based on the training dataset constructed by formula (33) to obtain the optimal SF-Net and DOA estimator network parameters Γ opt = {W opt ,Ω opt}:

[0229] In the formula, Indicates that the Kj I -j S -j α Training data is the input and the output of the cascade neural network SFCNN with Γ as the network parameter.

[0230] 1-8. In this embodiment, in S5, DOA estimation is performed using the optimal cascade neural network, specifically including:

[0231] Based on the network parameters, a cascade neural network is used to process the T snapshot array data D actually received. real Processing is performed to obtain the DOA estimation result:

[0232]

[0233] In the formula,

[0234] Example 2

[0235] The embodiment of the present invention is a simulation experiment of the method described in Example 1.

[0236] In this embodiment, the DOA estimation performance of the method described in Example 1 is verified by simulation, and compared with the CNN method and the traditional MUSIC algorithm. All simulations are implemented based on MATLAB 2021a, the platform is Lenovo P920 graphics workstation, and the simulation parameters are: number of array elements M = 16, array element spacing d = λ / 2, target airspace range Θ = [-60°, 60°], number of grids N = 120, number of sub-airspaces Q = 6, number of sub-airspace grids P = 20, maximum number of signal sources per sub-airspace K max =2, Maximum number of signal sources in target airspace Number of snapshots T = 400, number of repetitions J I =100, SNR range [-10,10]dB, SNR number J S =5, number of angle combinations J α = 100. The method described in Example 1 has the same parameters as the CNN method, wherein the dimension of the convolution kernel of the C1 layer is υ1×υ1×n1=3×3×256, the step size is 2, and the dimension of the convolution kernel of the C2 to C4 layers is υ i ×υ i ×n i =2×2×256(i=2,3,4), the step size is 1, the number of neurons in layers FC5 to FC7 are 4096, 2048 and 1024 respectively, and the probability of Dropout is 0.2.

[0237] 2-1. About spatial filtering performance:

[0238] This example verifies the spatial filtering performance of SF-Net after training:

[0239] First, when the signal incident angle is Under the condition of , this embodiment calculates the spatial domain responses of Q different filters according to the following formula:

[0240]

[0241] In the formula, r n represents the SF-Net input data obtained by vectorizing and separating the real and imaginary parts of the ideal cosignal variance matrix, I q =[2(q-1)M 2 +1,2(q-1)M 2 +1,...,2qM 2 ] T represents the index corresponding to the qth filter, Indicated by r n As input, with the optimal parameter Γ of SFCNN opt W opt is the SF-Net output of the network parameters. Traversing all possible discrete angles in the target airspace, the airspace response of SF-Net can be obtained as follows: Figure 3 It can be seen that the Q filters of SF-Net can maintain a relatively constant response in the spatial domain of interest and obtain a response close to 0 in the spatial domain of no interest, indicating that SF-Net has good spatial domain filtering performance after training and can perform sub-spatial domain division on the array received data.

[0242] Assume that the number of signal sources is K = 8, the incident angle set is {-50°, -35°, -25°, -10°, 10°, 25°, 35°, 50°}, and generate T snapshot array receiving data Y K , and according to formula (6) and formula (7) to obtain the input data of SF-Net Then we get the output of SF-Net use Replace u in equation (36) n , the output responses of different filters of SF-Net can be obtained as follows Figure 4 As shown in the figure, it can be seen that the Q filters in SF-Net can effectively retain the incident signal in the corresponding sub-space domain and suppress the incident signal outside the space domain, which provides conditions for the subsequent use of CNN to estimate the subspace spectrum.

[0243] 2-2. About DOA estimation performance:

[0244] This example verifies the DOA estimation performance of the method described in Example 1 and its advantages over the CNN method and the MUSIC algorithm:

[0245] First, assuming the signal-to-noise ratio (SNR) = 0 dB, given four different numbers of signal sources (K = 2, 3, 6, 12), randomly generating incident signal angle sets, and using different methods to process the array received data to obtain the DOA estimation results as shown in the figure. Figure 5It should be noted that when the number of signal sources K is greater than 2, the training data set required by the CNN method is too large and exceeds the upper limit of the platform memory. Figure 5 (bd) only gives the DOA estimation results of the method described in Example 1 and the MUSIC algorithm. It can be seen that: under different signal source numbers, the method described in Example 1 can obtain better DOA estimation performance than the MUSIC algorithm; compared with the CNN method, the method described in Example 1 can still obtain accurate DOA estimation results when the number of signal sources is large (K = 3, 6, 12).

[0246] Next, assume that there are two incident signals in the target airspace, with an angle interval Δ of 5° or 1°. Figure 6 The DOA estimation results of the method described in Example 1 are given under the conditions of SNR equal to -10dB and 10dB and the incident signal angle traversing the entire target airspace (step size is 1°). It can be seen that: under different SNR and angle interval conditions, the method described in Example 1 can accurately estimate the angle of the two sources and has good resolution.

[0247] Example 3

[0248] The embodiment of the present invention provides a multi-source DOA high-resolution estimation system based on a cascade neural network, which is used to execute the multi-source DOA high-resolution estimation method based on a cascade neural network described in Example 1, including:

[0249] The array receiving data preprocessing module is used to estimate the covariance matrix of the array receiving data obtained under the multi-snap sampling condition, obtain the covariance matrix and vectorize and separate the real part and the imaginary part of the covariance matrix to obtain the preprocessed array receiving data;

[0250] An interpretable spatial domain filtering module is used to design Q spatial domain filters based on a single-layer FNN, construct an interpretable spatial domain filtering network, initialize and train the interpretable spatial domain filtering network, and use the trained interpretable spatial domain filtering network to divide the pre-processed array received data into Q sub-spatial domains, and perform matrixing and three-channeling processing respectively to obtain three-channel real-valued matrices after Q spatial domain filtering;

[0251] The CNN spatial spectrum estimation module is used to design Q two-dimensional CNN estimators, construct a DOA estimator, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain, synthesize it, and output the CNN spatial spectrum estimation result;

[0252] A joint fine-tuning module is used to combine the interpretable spatial domain filter network and the DOA estimator to form a cascade neural network, and to obtain the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator by performing end-to-end joint fine-tuning training on the cascade neural network to form an optimal cascade neural network;

[0253] A high-resolution DOA estimation module is used to perform DOA estimation using the optimal cascade neural network.

[0254] In summary, the multi-source DOA high-resolution estimation method and system based on cascade neural network provided by the present invention constructs an interpretable spatial domain filtering network based on matrix spatial domain filtering theory, and uses convolutional neural network to process the output signal of the filtering network to realize the nonlinear conversion of array received data to high-resolution spatial spectrum. Simulation results show that under different signal-to-noise ratios and number of sources, the method of the present invention can obtain better DOA estimation performance than the traditional MUSIC algorithm; compared with the existing typical CNN estimation algorithm, the method of the present invention can greatly reduce the scale of the training data set and reduce the difficulty of network training, and obtain better DOA estimation results under different number of sources.

[0255] The remaining technical features in this embodiment can be flexibly selected by those skilled in the art according to actual conditions to meet different specific practical needs. However, it is obvious to those skilled in the art that it is not necessary to adopt these specific details to implement the present invention. In other examples, in order to avoid confusing the present invention, the well-known components, structures or parts are not specifically described, which are all within the technical protection scope defined by the technical solution claimed for protection in the claims of the present invention.

[0256] Modifications and changes made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the scope of protection of the claims attached to the present invention. In the above description, a large number of specific details are set forth in order to provide a thorough understanding of the present invention. However, it is obvious to those of ordinary skill in the art that these specific details are not necessary to practice the present invention. In other examples, in order to avoid confusing the present invention, well-known technologies, such as specific construction details, operating conditions and other technical conditions, are not specifically described.

[0257] The principles and implementation methods of the present invention are described in this article using specific examples. The description of the above embodiments is only used to help understand the method and core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A multi-source DOA high-resolution estimation method based on cascade neural network, characterized in that: The following steps are involved: S1. Array receiving data preprocessing: performing covariance matrix estimation on the array receiving data obtained under multi-snap sampling conditions, obtaining the covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix, thereby obtaining preprocessed array receiving data; S2, interpretable spatial domain filtering: design Q spatial domain filters based on a single-layer fully connected network, construct an interpretable spatial domain filtering network, initialize and train the interpretable spatial domain filtering network, use the trained interpretable spatial domain filtering network to divide the pre-processed array received data into Q sub-spatial domains, and perform matrixing and three-channeling processing respectively to obtain three-channel real-valued matrices after the Q spatial domain filtering; S3, CNN spatial spectrum estimation: Design Q two-dimensional CNN estimators, construct DOA estimators, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize them, and output the CNN spatial spectrum estimation result; S4, joint fine-tuning: the interpretable spatial domain filter network and the DOA estimator are combined to form a cascade neural network, and the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator are obtained by end-to-end joint fine-tuning training of the cascade neural network to form an optimal cascade neural network; S5. Utilize the optimal cascade neural network to perform DOA estimation.

2. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S1, array received data preprocessing: performing covariance matrix estimation on the array received data obtained under multi-snap sampling conditions, obtaining the covariance matrix and performing vectorization and real-imaginary part separation on the covariance matrix, and obtaining the preprocessed array received data, specifically includes the following steps: S101, perform covariance matrix estimation on the array received data obtained under the multi-snap sampling condition to obtain the covariance matrix: In the formula, (·) H represents conjugate transpose, Y represents the array receiving data matrix, and T represents the number of snapshots; S102, the covariance matrix Perform vectorization processing, separate the real part and the imaginary part of the result, reconstruct it into a real vector, and obtain the preprocessed array receiving data Where vec(·) represents the vectorization operation on the matrix, Re represents the real part, and Im represents the imaginary part.

3. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S2, Q spatial domain filters are designed based on a single-layer fully connected network, an interpretable spatial domain filter network is constructed, and the interpretable spatial domain filter network is initialized and trained, specifically including the following steps: S201, construct Q weight matrices that do not contain activation functions and biases A single-layer fully connected network is used as the spatial domain filter, q = 1, 2, ..., Q, Q is the number of sub-spatial domains divided; S202, connecting Q spatial domain filters in parallel to obtain an interpretable spatial domain filter network; S203, constructing a training data set for an interpretable spatial domain filtering network: In the formula, and They represent the input data set and label data set of the interpretable spatial domain filter network, P represents the number of grids divided in each sub-spatial domain, and J I is the number of repetitions, J S is the signal-to-noise ratio, For the first pqj I -j S Input data, L q,p is the pqth label data; S204, the weight matrix W of each single-layer fully connected network in the interpretable spatial domain filtering network q To initialize: In the formula, (·) * represents conjugation, represents the Kronecker product, (·) T represents matrix transpose, H represents conjugate transpose; g q By solving the following equation: In the formula, I N×N represents the unit matrix of size N×N, where N represents the number of grids into which the target airspace is divided. represents the qth sub-space domain, Represents the array manifold matrix corresponding to the P grids of the qth sub-space domain, express The pth grid The corresponding steering vector, p = 1, 2, ..., P, P represents The corresponding number of grid divisions is express The complement of , Θ is the target space, express The sth grid The corresponding steering vector, s = 1, 2, ..., S, S represents The corresponding number of grid divisions, δ represents the stopband attenuation rate; S205, using the training data set, adopting the mean square error function as the network loss function, and obtaining the network parameters after the interpretable spatial domain filtering network training through back propagation algorithm learning In the formula, Indicates is the network parameter, with the first pqj I -j S Training data is the interpretable spatial domain filter network output of the input, J SF =PQJ I J S is the size of the training dataset.

4. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S2, the pre-processed array received data is divided into Q sub-spatial domains using a trained interpretable spatial domain filtering network, and matrixing and three-channeling are performed respectively to obtain three-channel real-valued matrices after Q spatial domain filtering, which specifically includes the following steps: S2.

1. Use Q trained single-layer fully connected networks to filter the pre-processed array received data and divide the data into Q sub-spatial domains: In the formula, r q represents the output data of the qth trained single-layer fully connected network, Represents the network parameters of the qth single-layer fully connected network obtained after training; S2.

2. The output of each single-layer fully connected network is processed into a complex number, a matrix, and a three-channel, and Q spatial domain filtered three-channel real-valued matrices X are obtained. q : In the formula, comp(r q )=r q(1:M×M) +jr q(M×M+1:2M×M) Indicates that r q Reconstructed into complex numbers, mat(·) represents matrix operation on vectors, and ∠(·) represents angle operation.

5. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S3, Q two-dimensional CNN estimators are designed, a DOA estimator is constructed, and each CNN estimator is initialized and trained, which specifically includes the following steps: S301, design Q two-dimensional CNN networks including an input layer, a two-dimensional convolutional layer, a batch normalization layer, a fully connected layer and an output layer as two-dimensional CNN estimators corresponding to the Q sub-spatial domains; S302, connecting Q two-dimensional CNN estimators in parallel to obtain a DOA estimator; S303, construct a training data set for the DOA estimator: In the formula, and Represent the input dataset and label dataset of the qth CNN estimator respectively; is the number of combinations, J I is the number of repetitions, J S is the signal-to-noise ratio, K max Indicates the maximum number of sources. For Kj I -j S -j K training data, For Kj I -j S -j K Label data; S304, initializing the network parameters of each CNN estimator in the DOA estimator: The Glorot normalization method is used to initialize the convolution kernels of each convolutional layer and the weights of each fully connected layer, the bias vector is initialized to 0, the scale factor of each batch normalization layer is initialized to 1, and the translation factor is initialized to 0; S305: Using the training data set, adopting the mean square error function as the network loss function, and learning the trained network parameters of the DOA estimator through the back propagation algorithm In the formula, Indicates that the Kj K -j I -j S Training data As input, Ω q is the qth CNN estimator output of the network parameters, is the size of the training dataset.

6. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S3, the three-channel real-valued matrix after filtering in each spatial domain is processed using the trained DOA estimator to obtain the spatial spectrum estimation result of each sub-spatial domain and synthesize them to output the CNN spatial spectrum estimation result, which specifically includes the following steps: S3.

1. Use Q trained CNN estimators to process the three-channel real-valued matrix after filtering in each spatial domain, and obtain the spatial spectrum estimation result P of each sub-spatial domain. q : In the formula, Indicates that the Xth q For input, is the qth trained CNN estimator output for the network parameters; S3.

2. Synthesize the spatial spectrum estimation results of all sub-spatial domains and output the CNN spatial spectrum estimation result P:

7. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S4, the interpretable spatial domain filter network and the DOA estimator are combined to form a cascade neural network, and the cascade neural network is end-to-end jointly fine-tuned to obtain the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator, which specifically includes the following steps: S401, using the output of the interpretable spatial domain filter network as the input of the DOA estimator, cascading them to obtain a cascade neural network; S402, constructing a training data set for the cascade neural network: In the formula, and They represent the input dataset and label dataset of the cascade neural network respectively, and J I is the number of repetitions, J S is the signal-to-noise ratio, J α is the number of signal angle combinations; is the maximum number of signal sources, For Kj I -j S -j α training data, For Kj I -j S -j α Label data; S403, initializing the network parameters of the cascade neural network: The initialization network parameters of the cascade neural network are obtained according to the trained interpretable spatial domain filter network parameters and the trained DOA estimator network parameters: In the formula, Γ represents the network parameters of the cascade neural network; S404, performing end-to-end joint fine-tuning training on the initialized cascade neural network to obtain the optimal network parameters Γ of the interpretable spatial domain filter network and the DOA estimator opt = {W opt ,Ω opt }: In the formula, Indicates that the Kj I -j S -j α Training data is the input and the output of the cascade neural network with Γ as the network parameter, is the size of the training dataset.

8. The multi-source DOA high-resolution estimation method based on cascade neural network according to claim 1 is characterized in that: In S5, using the optimal cascade neural network to perform DOA estimation specifically includes: Based on the network parameters, the cascade neural network is used to process the T snapshot array data D actually received. real Processing is performed to obtain the DOA estimation result: In the formula, 9. A multi-source DOA high-resolution estimation system based on a cascade neural network, used to execute the multi-source DOA high-resolution estimation method based on a cascade neural network according to any one of claims 1 to 8, characterized in that: include: The array receiving data preprocessing module is used to estimate the covariance matrix of the array receiving data obtained under the multi-snap sampling condition, obtain the covariance matrix and vectorize and separate the real part and the imaginary part of the covariance matrix to obtain the preprocessed array receiving data; An interpretable spatial domain filtering module is used to design Q spatial domain filters based on a single-layer fully connected network, construct an interpretable spatial domain filtering network, initialize and train the interpretable spatial domain filtering network, and use the trained interpretable spatial domain filtering network to divide the pre-processed array received data into Q sub-spatial domains, and perform matrixing and three-channeling processing respectively to obtain three-channel real-valued matrices after Q spatial domain filtering; The CNN spatial spectrum estimation module is used to design Q two-dimensional CNN estimators, construct a DOA estimator, initialize and train each CNN estimator, use the trained DOA estimator to process the three-channel real-valued matrix after filtering in each spatial domain, obtain the spatial spectrum estimation result of each sub-spatial domain, synthesize it, and output the CNN spatial spectrum estimation result; A joint fine-tuning module is used to combine the interpretable spatial domain filter network and the DOA estimator to form a cascade neural network, and to obtain the optimal network parameters of the interpretable spatial domain filter network and the DOA estimator by performing end-to-end joint fine-tuning training on the cascade neural network to form an optimal cascade neural network; A high-resolution DOA estimation module is used to perform DOA estimation using the optimal cascade neural network.

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