CN114114187A - ADMM network direction finding method based on depth expansion under grid mismatch condition - Google Patents

ADMM network direction finding method based on depth expansion under grid mismatch condition Download PDF

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CN114114187A
CN114114187A CN202111384362.1A CN202111384362A CN114114187A CN 114114187 A CN114114187 A CN 114114187A CN 202111384362 A CN202111384362 A CN 202111384362A CN 114114187 A CN114114187 A CN 114114187A
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quantization error
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CN114114187B (en
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户盼鹤
苏晓龙
潘嘉蒙
刘振
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National University of Defense Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/417—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section involving the use of neural networks
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/14—Systems for determining direction or deviation from predetermined direction
    • G01S3/143—Systems for determining direction or deviation from predetermined direction by vectorial combination of signals derived from differently oriented antennae
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
    • G01S3/74—Multi-channel systems specially adapted for direction-finding, i.e. having a single antenna system capable of giving simultaneous indications of the directions of different signals
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/78—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using electromagnetic waves other than radio waves
    • G01S3/782—Systems for determining direction or deviation from predetermined direction
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

Abstract

The invention belongs to the technical field of array signal processing and machine learning, and particularly relates to an ADMM network direction finding method based on depth expansion under the condition of grid mismatch, which comprises the following steps: firstly, carrying out vectorization real number processing on a covariance matrix of array receiving data; then, calculating the first derivatives of the original redundant dictionary and the original redundant dictionary; then, constructing a spatial spectrum estimation layer in the depth expansion ADMM network; secondly, constructing a quantization error estimation layer in the deep-expansion ADMM network; thirdly, training the deeply expanded ADMM network; and finally, calculating the incident signal angle by using the output of the depth expansion ADMM network. The invention has the beneficial effects that: 1. compared with the model driven ADMM method, the method has high convergence rate; 2. compared with the ADMM network which is deeply expanded on the grid, the direction finding precision of the method is high.

Description

ADMM network direction finding method based on depth expansion under grid mismatch condition
Technical Field
The invention belongs to the technical field of array signal processing and machine learning, and particularly relates to an Alternating direction multiplier (ADMM) network direction finding method based on depth expansion under the condition of grid mismatch.
Background
The target direction finding has important application in the fields of radar, sonar and the like. When the sparse representation method is adopted for direction finding, when the target angle does not belong to the elements of the over-complete set, namely under the condition of redundant dictionary grid mismatch, a certain error exists between the estimated angle value and the real angle value. The deep expansion network converts the iterative process of the traditional method into a neural network cascade form, the parameters of the network have certain mathematical interpretability, the generalization capability is improved, and meanwhile, the convergence speed can be accelerated through training the network.
The reference "Real-Valued Sparse Learning for Off-Grid Direction-of-arrival (DOA) Estimation in Ocean dynamics [ J ]" (IEEE Journal of organic Engineering,2021,46(1): pages 172-182) converts the array received data to the Real number domain, and solves the Direction finding problem under the condition of Grid mismatch by adopting a Sparse Bayesian Learning method. Compared with a direction finding method based on a model on a grid, the method has high direction finding precision, but the calculation complexity is relatively large.
The direction finding problem under sparse representation is realized by the ADMM method in reference of Fast Compressive Sensing DOA Estimation Via ADMM solution [ J ] (2017IEEE International Conference on Information and Automation, pages 53 to 57, DOI:10.1109/ICInfA.2017.8078882), and compared with the method for solving the sparse representation problem by a CVX tool box, the method has smaller operation amount, but has relatively lower convergence speed compared with the method of depth expansion.
In reference to the document "Compressed LISTA expanding Toeplitz Structure [ J ]" (2019IEEE Radar Conference, pages 1 to 6, DOI:10.1109/radar.2019.8835826) an Iterative Shrinkage Threshold Algorithm (ISTA) is developed into a convolutional neural network in a Toeplitz matrix form, compared with the conventional deep learning orientation-based method, the network has certain mathematical interpretability of parameters and improved generalization capability, but compared with the ADMM method, the method needs to convert a redundant dictionary into an orthogonal matrix.
Disclosure of Invention
Aiming at the technical problems, the invention provides an ADMM network direction finding method based on depth expansion under the condition of grid mismatch.
The technical scheme of the invention is as follows: an ADMM network direction finding method based on depth expansion under the condition of grid mismatch comprises the following steps:
firstly, carrying out vectorization real number processing on the covariance matrix R of array receiving data.
The known antenna array is arranged in two-stage nesting, where x (N) represents the nth snapshot data received by the array, and N is 1, 2. The covariance matrix R of the array received data is calculated using the following equation:
Figure BDA0003362628050000021
wherein (·)HRepresenting a conjugate transpose operation.
Vectorizing real number processing is carried out on the covariance matrix R to obtain an array covariance vector
Figure BDA0003362628050000022
Figure BDA0003362628050000023
Wherein
Figure BDA0003362628050000024
The operation of taking the real part of a complex number is shown,
Figure BDA0003362628050000025
representing the imaginary operation of taking complex numbers, vec (·) represents arranging the covariance matrix into vectors in columns.
Second, calculate the original redundant dictionary
Figure BDA0003362628050000026
And first derivative of original redundant dictionary
Figure BDA0003362628050000027
Calculating an original redundant dictionary by
Figure BDA0003362628050000028
Figure BDA0003362628050000031
Wherein
Figure BDA0003362628050000032
a(θq)＝[a1(θq) a2(θq) … am(θq) … aM(θq)]T
(·)*Representing a conjugate operation (·)TWhich represents the operation of transposition by means of a transposition operation,
Figure BDA0003362628050000033
denotes Kronecker product, Q1, 2qRepresents the Q-th element of the overcomplete angle set, Q represents the number of the overcomplete angle sets under the original redundant dictionary, a (theta)q) Denotes thetaqThe direction vector of (a) is,
Figure BDA0003362628050000034
denotes a (theta)q) M is 1,2, M represents the number of array elements, ximThe mth array element position is shown, and λ represents the wavelength.
Calculating the first derivative of the redundant dictionary by
Figure BDA0003362628050000035
Figure BDA0003362628050000036
Wherein
Figure BDA0003362628050000037
Figure BDA0003362628050000038
And thirdly, constructing a spatial spectrum estimation layer in the depth expansion ADMM network.
The spatial spectrum estimation layer has L layers in total, and the I-th layer input of the spatial spectrum estimation is a grid mismatch redundant dictionary
Figure BDA0003362628050000039
And the array covariance vector generated in the first step
Figure BDA00033626280500000310
Figure BDA00033626280500000310
Figure BDA00033626280500000310
1,2, L, wherein B(l-1)The quantization error matrix representing the output of layer l-1 (calculated in the fourth step).
Spatial spectrum z output by I layer of spatial spectrum estimation layer(l)Can be calculated by the following formula:
Figure BDA0003362628050000041
u(l)＝Sρ(z(l)+η(l-1))
η(l)＝η(l-1)+z(l)-u(l)
wherein u is(l)Spatial spectrum z representing the output of the l-th layer(l)Corresponding optimization vector, η(l)Spatial spectrum z representing the output of the l-th layer(l)The corresponding dual vector is then used to generate the dual vector,
Figure BDA0003362628050000042
(·)-1representing an inversion operation, Sρ(. -) represents the soft threshold function:
Sρ(z(l)+η(l-1))＝sgn(z(l)+η(l-1))⊙max(|z(l)+η(l-1)|-ρ,0)
ρ represents a penalty parameter, sgn (·) represents a sign function, an | represents a Hadamard product, and | represents an absolute value operation. The input to layer 1 of the spatial spectrum estimation layer is as follows:
Figure BDA0003362628050000043
u(0)＝Sρ(z(0))
η(0)＝z(0)-u(0)
and fourthly, constructing a quantization error estimation layer in the depth expansion ADMM network.
The quantization error estimation layer has L layers in total, and the I-th layer input of the quantization error estimation is a quantization error covariance vector
Figure BDA0003362628050000044
And the first derivative of the redundant dictionary generated in the second step
Figure BDA0003362628050000045
Quantization error vector g output from the l-th layer of the quantization error estimation layer(l)Can be calculated by the following formula:
Figure BDA0003362628050000046
ζ(l)＝Sg(g(l)+τ(l-1))
τ(l)＝τ(l-1)+g(l)-ζ(l)
wherein ζ(l)Quantization error vector g representing the output of the l-th layer(l)Corresponding optimization vector, τ(l)Representing output of l-th layerQuantization error vector g(l)The corresponding dual vector.
Required quantization error matrix B in layer l-1 calculation of the third step(l-1)The elements on the diagonal can be calculated by:
Figure BDA0003362628050000051
wherein B is(l-1)The elements other than the diagonal lines are 0,
Figure BDA0003362628050000052
representing a quantization error matrix B(l-1)The q-th element on the diagonal line,
Figure BDA0003362628050000053
representing the output g of layer l-1 of the quantization error estimate(l-1)The q-th element of (a),
Figure BDA0003362628050000054
representing the l-1 layer output z of the spatial spectrum estimate(l-1)The qth element of (1).
The input of the 1 st layer of the quantization error estimation layer is B(0)＝0，ζ (0)0 and τ(0)＝0。
And fifthly, training the deep-expanded ADMM network.
In the training of the deep-developed ADMM network, parameters are updated by using a Back Propagation (BP) and Stochastic Gradient Descent (SGD) optimizer in PyTorch, and a loss function is calculated by the following formula
Figure BDA0003362628050000055
Wherein
Figure BDA0003362628050000058
Represents the square of 2-norm, | ·| non-woven phosphor1Denotes the 1-norm, ω1And ω2To representA regularization parameter.
And sixthly, calculating the angle of the incident signal by using the output of the depth expansion ADMM network.
After the constructed deep-expansion ADMM network is trained, vectorizing real number processing is carried out on the covariance matrix R of the array receiving data and the covariance matrix R is input into the deep-expansion ADMM network, and the spatial spectrum z output by the L-th layer of the spatial spectrum estimation layer is utilized(L)Has an angle theta corresponding to the kth spectral peakkAnd quantization error vector g outputted from Lth layer of quantization error estimation layer(L)The k-th peak is bkAngle of the kth incident signal
Figure BDA0003362628050000056
Can be calculated as
Figure BDA0003362628050000057
Where K is 1,2, …, K denotes the spatial spectrum z output from the L-th layer of the spatial spectrum estimation layer(L)I.e. the number of incident signals.
Compared with the prior art, the invention has the beneficial effects that:
1. compared with the model driven ADMM method, the method has high convergence rate;
2. compared with the ADMM network which is deeply expanded on the grid, the direction finding precision of the method is high.
Drawings
Fig. 1 is a schematic flow chart of an ADMM network direction finding method based on depth expansion under a lattice mismatch condition provided by the present invention;
FIG. 2 is a schematic diagram of a deep-developed ADMM network according to the present invention;
FIG. 3 is a schematic diagram of a scene using a two-level nested array for direction finding;
FIG. 4 is a spatial spectrum obtained using the present invention;
FIG. 5 illustrates quantization errors obtained using the present invention;
FIG. 6 shows mean square errors obtained by different methods for different network layers;
FIG. 7 shows the root mean square error obtained by different methods at different signal-to-noise ratios.
Detailed Description
The invention is further illustrated with reference to the figures and examples.
As shown in fig. 1, the method for depth-based deployment of ADMM network direction finding under lattice mismatch condition includes the following steps: firstly, carrying out vectorization real number processing on a covariance matrix of array receiving data; then, calculating the first derivatives of the original redundant dictionary and the original redundant dictionary; then, constructing a spatial spectrum estimation layer in the depth expansion ADMM network; secondly, constructing a quantization error estimation layer in the deep-expansion ADMM network; thirdly, training the deeply expanded ADMM network; and finally, calculating the incident signal angle by using the output of the depth expansion ADMM network.
In the fifth step, an embodiment of training the deep-developed ADMM network is as follows:
generating an over-complete angle set at an interval of-60 degrees and 60 degrees by taking 1 degree as an interval, randomly selecting two elements in the set as incident signal angles to generate array receiving data, carrying out vectorization real number processing on a covariance matrix of the array receiving data, and generating 14400 training samples as a training set. In training of a deeply-extended ADMM network, the epoch is set to 300, the mini-batch is set to 32, the parameters are updated using a Back Propagation (BP) and Stochastic Gradient Descent (SGD) optimizer in PyTorch, and the loss function is calculated by the following formula
Figure BDA0003362628050000071
Wherein
Figure BDA0003362628050000072
Represents the square of 2-norm, | ·| non-woven phosphor1Denotes the 1-norm, ω1And ω2The regularization parameters are represented.
As shown in fig. 2, a dashed box in the depth expansion ADMM network represents a spatial spectrum estimation layer, a dashed box represents a quantization error estimation layer, and each of the spatial spectrum estimation layer and the quantization error estimation layer has L layers.
In order to verify the positioning performance of the invention on the near-field source, three simulation experiments are used for explanation.
As shown in fig. 3, in the simulation experiment, a two-stage nested linear array is composed of 8 array elements, wherein a filled circle represents a 1 st-stage sub-array, and an empty circle represents a 2 nd-stage sub-array. The number of layers L of the depth-expanded ADMM network is set to 30.
The simulation experiment I is used for verifying the direction finding effectiveness of the invention under the condition of grid mismatch, and the angles of two incident signals in the experiment are respectively-10.95 degrees and 2.98 degrees. Fig. 4 is a spatial spectrum obtained using the present invention, and it can be seen that the two peaks correspond to the nearest-to-11 ° and 3 ° angles, respectively, of the incident signal. Fig. 5 shows quantization errors obtained by the present invention, and it can be seen that two peaks respectively correspond to off-grid values of an incident signal, which indicates that the present invention can realize accurate direction finding under the condition of grid mismatch.
The simulation experiment is used for verifying the convergence rate of the invention, and the angles of two incident signals in the experiment are-10.95 degrees and 2.98 degrees respectively. Fig. 6 shows the mean square error obtained by using different methods, where the abscissa is the number of network layers, the ordinate is the mean square error, the solid line represents the mean square error of the present invention at different number of layers, and the dotted line represents the mean square error of the model-driven ADMM method at different number of layers.
The simulation experiment is used for verifying the parameter estimation accuracy of the invention, and the angles of two incident signals in the experiment are respectively-10.95 degrees and 2.98 degrees. Fig. 7 shows the root mean square error obtained by using different methods, where the abscissa is the signal-to-noise ratio, the ordinate is the root mean square error, the solid line with o indicates the root mean square error of the present invention under different signal-to-noise ratios, and the dotted line with Δ indicates the root mean square error of the deeply expanded ADMM network on the grid under different signal-to-noise ratios.
The above description is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may occur to those skilled in the art without departing from the principle of the invention, and are considered to be within the scope of the invention.

Claims (3)

1. A depth-expansion-based ADMM network direction finding method under a grid mismatch condition is characterized by comprising the following steps:
firstly, carrying out vectorization real number processing on the covariance matrix R of array receiving data
The known antenna arrays are arranged in a two-stage nested manner, wherein x (N) is used for representing nth snapshot data received by the arrays, and N is 1, 2. The covariance matrix R of the array received data is calculated using the following equation:
Figure FDA0003362628040000011
wherein (·)HRepresenting a conjugate transpose operation;
vectorizing real number processing is carried out on the covariance matrix R to obtain an array covariance vector
Figure FDA0003362628040000012
Figure FDA0003362628040000013
Wherein
Figure FDA0003362628040000014
The operation of taking the real part of a complex number is shown,
Figure FDA0003362628040000015
representing the imaginary part operation of taking complex numbers, vec (·) representing arranging the covariance matrix into vectors in columns;
second, calculate the original redundant dictionary
Figure FDA0003362628040000016
And first derivative of original redundant dictionary
Figure FDA0003362628040000017
Calculating an original redundant dictionary by
Figure FDA0003362628040000018
Figure FDA0003362628040000019
Wherein
Figure FDA00033626280400000110
a(θq)＝[a1(θq) a2(θq) … am(θq) … aM(θq)]T
(·)*Representing a conjugate operation (·)TWhich represents the operation of transposition by means of a transposition operation,
Figure FDA00033626280400000111
denotes Kronecker product, Q1, 2qRepresents the Q-th element of the overcomplete angle set, Q represents the number of the overcomplete angle sets under the original redundant dictionary, a (theta)q) Denotes thetaqThe direction vector of (a) is,
Figure FDA00033626280400000112
denotes a (theta)q) M is 1,2, M represents the number of array elements, ximThe position of the mth array element is shown, and lambda represents the wavelength;
by means of the following formulaFirst derivative of computationally redundant dictionary
Figure FDA0003362628040000021
Figure FDA0003362628040000022
Wherein
Figure FDA0003362628040000023
Figure FDA0003362628040000024
Thirdly, constructing a spatial spectrum estimation layer in the deep-expansion ADMM network
The spatial spectrum estimation layer has L layers in total, and the I-th layer input of the spatial spectrum estimation is a grid mismatch redundant dictionary
Figure FDA0003362628040000025
And the array covariance vector generated in the first step
Figure FDA0003362628040000027
1,2, L, wherein B(l-1)A quantization error matrix representing the l-1 layer output;
spatial spectrum z output by I layer of spatial spectrum estimation layer(l)Can be calculated by the following formula:
Figure FDA0003362628040000028
u(l)＝Sρ(z(l)+η(l-1))
η(l)＝η(l-1)+z(l)-u(l)
wherein u is(l)Spatial spectrum z representing the output of the l-th layer(l)Corresponding optimization vector, η(l)Spatial spectrum z representing the output of the l-th layer(l)The corresponding dual vector is then used to generate the dual vector,
Figure FDA0003362628040000026
(·)-1representing an inversion operation, Sρ(. -) represents the soft threshold function:
Sρ(z(l)+η(l-1))＝sgn(z(l)+η(l-1))⊙max(|z(l)+η(l-1)|-ρ,0)
ρ represents a penalty parameter, sgn (·) represents a sign function, an | represents a Hadamard product, and | represents an absolute value operation;
fourthly, constructing a quantization error estimation layer in the depth expansion ADMM network
The quantization error estimation layer has L layers in total, and the I-th layer input of the quantization error estimation is a quantization error covariance vector
Figure FDA0003362628040000031
And the first derivative of the redundant dictionary generated in the second step
Figure FDA0003362628040000032
Quantization error vector g output from the l-th layer of the quantization error estimation layer(l)Can be calculated by the following formula:
Figure FDA0003362628040000033
ζ(l)＝Sg(g(l)+τ(l-1))
τ(l)＝τ(l-1)+g(l)-ζ(l)
wherein ζ(l)Quantization error vector g representing the output of the l-th layer(l)Corresponding optimization vector, τ(l)Quantization error vector g representing the output of the l-th layer(l)A corresponding dual vector;
required quantization error matrix B in layer l-1 calculation of the third step(l-1)The elements on the diagonal can be calculated by:
Figure FDA0003362628040000034
wherein B is(l-1)The elements other than the diagonal lines are 0,
Figure FDA0003362628040000035
representing a quantization error matrix B(l-1)The q-th element on the diagonal line,
Figure FDA0003362628040000036
representing the output g of layer l-1 of the quantization error estimate(l-1)The q-th element of (a),
Figure FDA0003362628040000037
representing the l-1 layer output z of the spatial spectrum estimate(l-1)The q element of (1);
fifthly, training the deep-developed ADMM network
In the training of the deeply-expanded ADMM network, the parameters are updated using a back-propagation and stochastic gradient descent optimizer in PyTorch, and the loss function is calculated by the following formula
Figure FDA0003362628040000038
Wherein
Figure FDA0003362628040000039
Represents the square of 2-norm, | ·| non-woven phosphor1Denotes the 1-norm, ω1And ω2Representing a regularization parameter;
sixthly, calculating the angle of the incident signal by using the output of the depth expansion ADMM network
Constructed deep-developed ADMM network passes throughAfter training, vectorizing real number processing is carried out on the covariance matrix R of array receiving data, the vectorized real number processing is input into a depth expansion ADMM network, and a spatial spectrum z output by an L-th layer of a spatial spectrum estimation layer is utilized(L)Has an angle theta corresponding to the kth spectral peakkAnd quantization error vector g outputted from Lth layer of quantization error estimation layer(L)The k-th peak is bkAngle of the kth incident signal
Figure FDA0003362628040000041
Can be calculated as
Figure FDA0003362628040000042
Where K is 1,2, …, K denotes the spatial spectrum z output from the L-th layer of the spatial spectrum estimation layer(L)I.e. the number of incident signals.
2. An ADMM network direction finding method based on depth expansion under the grid mismatch condition as claimed in claim 1, wherein: in the third step, the input of the layer 1 of the spatial spectrum estimation layer is as follows:
Figure FDA0003362628040000043
u(0)＝Sρ(z(0))
η(0)＝z(0)-u(0)。
3. an ADMM network direction finding method based on depth expansion under the grid mismatch condition as claimed in claim 1, wherein: in the fourth step, the input of the 1 st layer of the quantization error estimation layer is B(0)＝0，ζ(0)0 and τ(0)＝0。
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