A method for AoA estimation based on structured matrix embedding and restoration technology

Abstract

The present invention proposes an AoA estimation method based on structured matrix embedding and recovery technology. First, commercial WiFi equipment is used to collect channel state information (CSI) to obtain a CSI matrix. Then, the CSI matrix is ​​projected and a Hankel-Toeplitz matrix is ​​constructed, and the augmented Lagrangian function is used to transform the constrained problem into an unconstrained problem. Secondly, the alternating direction method of multipliers (ADMM) is used to restore the noise-suppressed CSI matrix. Finally, the restored CSI matrix is ​​used to estimate the AoA of the direct path through a matrix beam algorithm. The method designed by the present invention improves the robustness of the system and the estimation accuracy of AoA while greatly reducing the channel noise and without increasing additional channel bandwidth.

Description

Technical Field

[0001] The present invention belongs to the technical field of realizing perception by using commercial WiFi equipment, and specifically is a method for estimating the angle of arrival (AoA) based on structured matrix embedding and recovery technology, which realizes the AoA estimation of array antenna by processing channel state information (CSI). Background Art

[0002] With the rapid development of modern science and technology, indoor positioning is of great significance for security, navigation, material management, etc. Traditional positioning methods such as GPS (Global Positioning System), infrared sensor-based positioning methods and visual positioning methods perform well in outdoor environments. However, in complex indoor environments, the GPS signal strength is weak and cannot meet the needs of high-precision positioning. Moreover, infrared sensors and visual positioning methods are limited by the visual angle, especially in large buildings such as shopping malls, airports, and hospitals. They can only monitor targets within a limited angle, and their performance is limited. The wireless sensing positioning method based on WiFi makes up for the limitations of traditional sensors, has good performance in both line-of-sight and non-line-of-sight environments, and does not need to worry about privacy leakage. Thanks to the Multiple-Input Multiple-Output (MIMO) technology and Orthogonal Frequency Division Multiplexing (OFDM) technology, the wireless sensing positioning method based on WiFi has the advantages of high positioning resolution and anti-multipath interference. In addition, commercial WiFi devices are widely deployed in places such as office buildings, shopping malls, and homes, with a wide coverage and low cost. Therefore, using commercial WiFi devices for indoor positioning has gradually become a research hotspot.

[0003] In the field of wireless technology, fingerprint positioning solutions based on deep learning usually require detailed signal collection and marking of the entire environment in the initial deployment phase, which usually involves a lot of manpower and time investment, and its positioning accuracy is easily affected by environmental changes. However, the positioning method based on multidimensional parameter estimation does not need to collect a large amount of data in advance. It not only meets the real-time requirements, but also uses a variety of parameters to reduce the dependence of the positioning system on environmental changes, thereby improving the stability and adaptability of the system. In the early days, researchers used the received signal strength indicator (RSSI) information of WiFi to measure the distance between the receiver and the signal source. However, RSSI is greatly affected by obstacles, multipath effects and interference in the environment, and the positioning accuracy is relatively low. Subsequently, because CSI can perceive channel changes in the time domain, frequency domain and spatial domain more finely, and it contains the channel amplitude and phase information of multiple subcarriers on multiple links, it has the characteristics of being able to distinguish different paths. Therefore, researchers achieved high-precision indoor positioning by extracting CSI for signal processing.

[0004] However, in the case of using commercial WiFi devices, the existing methods have fewer antennas and lower bandwidth, which makes the signal susceptible to noise interference in a complex and multi-obstacle indoor environment, resulting in a decrease in positioning accuracy. To solve the above problems, the present invention provides an AoA estimation method based on structured matrix embedding and recovery technology, which constructs a Hankel-Toeplitz matrix, uses an augmented Lagrangian function to transform a constrained problem into an unconstrained problem, and then applies the alternating direction method of multipliers (ADMM) to restore the CSI matrix after noise suppression, and then uses a matrix bundle algorithm to estimate AoA. The present invention improves the robustness of the system and the estimation accuracy of AoA while greatly reducing channel noise and without increasing additional channel bandwidth. Summary of the invention

[0005] The purpose of the present invention is to provide an AoA estimation method based on structured matrix embedding and recovery technology in a WiFi sensor system, which can improve the robustness of the system and the estimation accuracy of AoA while greatly reducing channel noise and without increasing additional channel bandwidth.

[0006] The AoA estimation method based on structured matrix embedding and recovery technology described in the present invention specifically includes the following steps:

[0007] Step 1: In the WiFi sensor system, the transmitting device uses OFDM technology to modulate the RF signal onto multiple subcarriers, and then transmits it to the receiving device through the wireless channel to achieve parallel transmission and improve the bandwidth utilization. Consider a uniform linear array in space, and the distance between array elements is d. Since the signal propagates in the wireless channel with multipath effect, the lth (1≤l≤L) path reaches the uniform linear array, and the azimuth and flight time of the signal are θ respectively. l and τ l , the center frequency is f c , the subcarrier spacing is Δf, the packet time interval is Δt, and the speed of light is c. The channel frequency response (CFR) of a single packet on the kth (1≤k≤K) subcarrier at the i-th (1≤i≤I) antenna at the receiving end is expressed as:

[0008]

[0009] Among them, α l represents the complex attenuation of the lth path, σ i,k represents the channel noise.

[0010] Using the CSI information of the WiFi signal obtained by the receiving end, the CSI data matrix of a single received data packet can be written as:

[0011]

[0012] Among them, H i Represents the CSI data of the i-th (1≤i≤I) antenna.

[0013] Step 2: Establish an optimization model. The specific steps are as follows:

[0014] First, in order to construct a structured matrix, set the parameters M, N, m, and n as follows:

[0015]

[0016] If M or N is an even number, a 0 vector matching the matrix dimension is used as the data of the Mth row or Nth column of the CSI data matrix H, expressed as Where Ω represents the set of rows and columns of the matrix to be projected, that is, Ω = {row, col}, is the projection of the matrix on the subspace Ω, expressed as follows:

[0017] (1) If row and col are both odd numbers, then

[0018] (2) If row is an odd number and col is an even number, a zero vector with a dimension of row×1 is added to the right side of the matrix, and the matrix dimension is extended to row×(col+1);

[0019] (3) If row is an even number and col is an odd number, a 0 vector with a dimension of 1×col is added to the bottom of the matrix, and the matrix dimension is extended to (row+1)×col;

[0020] (4) If row and col are both even numbers, first add a 0 vector with a dimension of row×1 to the right side of the matrix, and then add a 0 vector with a dimension of 1×(col+1) to the bottom side of the matrix, and the matrix dimension is expanded to (row+1)×(col+1).

[0021] Then, to apply structured matrix embedding and recovery techniques, the rank-constrained optimization problem is written as:

[0022]

[0023] in, and represents the optimization variable, z={h,t} represents the set of variables to be optimized, represents the set of Hermitian positive semidefinite matrices whose rank is not greater than L, is the constructed Hankel-Toeplitz matrix, is the Hankel-Toeplitz operator, which means that the matrices in the set z are transformed according to arrangement, is the Hankel operator, is the Toeplitz operator,

[0024] and Respectively and The conjugation of.

[0025] Among them, the two-dimensional Hankel matrix It is expressed as:

[0026]

[0027] in, is an n×n Hankel matrix consisting of the jth row in the optimization variable h and is expressed as:

[0028]

[0029] Among them, h is the optimization variable, expressed as:

[0030]

[0031] Two-dimensional Toeplitz matrix It is expressed as:

[0032]

[0033] in, is an n×n Toeplitz matrix (except The matrix is ​​non-Hermitian), which is composed of the j-th row in the optimization variable t:

[0034]

[0035] Among them, t is the optimization variable, expressed as:

[0036]

[0037] Step 3: To facilitate the application of the ADMM algorithm, the rank-constrained optimization problem in step 2 is converted into an unconstrained optimization problem. The specific steps are as follows:

[0038] First, we introduce the auxiliary matrix This enables ADMM to solve the following rank-constrained problems:

[0039]

[0040] in, represents the objective function to be optimized. Moreover, represents the indicator function, when hour, when hour,

[0041] Then, the Hermitian Lagrange multiplier is introduced Use the augmented Lagrangian function to transform the constrained problem into an unconstrained problem:

[0042]

[0043] in, represents the inner product of X and Y, tr(·) represents the trace of the matrix, X H represents the conjugate transpose of X, and ρ is the penalty parameter. The form is the same, construct the auxiliary matrix Hermitian Lagrange multipliers

[0044] Step 4: Using the optimized variables in step 3, the ADMM algorithm is used to iterate and recover the original CSI data matrix from the channel noise. ADMM iteratively updates G. Λ, the specific steps are as follows:

[0045] First, initialize the auxiliary matrix G, the Hermitian Lagrange multiplier Λ and the optimization variable t to zero matrices, and the optimization variable h = H Ω , the number of iterations is represented by b, the maximum number of iterations is set to 3000, and the penalty parameter ρ is fixed to 1.

[0046] Then, the auxiliary matrix G b+1 Can be exist The projection on is obtained, and its update formula can be written as:

[0047]

[0048] Among them, projection After the Hermitian matrix is ​​subjected to truncated eigendecomposition, the remaining eigenvalues ​​are set to zero except for the largest L positive eigenvalues ​​(if the number of positive eigenvalues ​​is less than L, just set the other eigenvalues ​​to zero), and then the Hermitian matrix is ​​reconstructed.

[0049] Next, let the matrix W b =G b +ρ -1 Λ b , and can be written as

[0050]

[0051] For convenience of representation, the following arrangement is used to form W 1 , W 2 and W 3 The elements that make up C 1 , C 2 and C 3 , with dimensions n×Mn:

[0052]

[0053] Since the optimization variables h and t are separable, the derivative of the objective function with respect to {h, t} is equal to zero, and the update formula is obtained:

[0054]

[0055]

[0056] Among them, I is the N×N unit matrix, C 1,j , C2,j and C 3,j Respectively represent C 1 , C 2 and C 3 The j-th block matrix in , (·) -1 represents the inverse of the matrix, It means mapping an m×m matrix X to a 1×M-dimensional vector. The mapping rule is X i,j represents the element in the i-th row and j-th column of the matrix X, It means mapping an m×m Hermitian matrix X to a 1×m dimensional vector. The mapping rule is According to step 2, h b+1 and t b+1 structure

[0057] Again, the Hermitian Lagrange multiplier Λ b+1 The update formula is:

[0058]

[0059] Finally, when the original residual is calculated and the dual residual When the value of is small enough, the iterative process is judged to have converged, and the ADMM iteration is terminated. The termination conditions are as follows:

[0060]

[0061] Among them, set the absolute error ε abs =10 -4 , relative error ε rel =10 -5 , the original residual and the dual residual It is expressed as the following formula:

[0062]

[0063] Step 5: Iterate the obtained data in step 4 AoA and ToF are calculated using a two-dimensional matrix bundle algorithm. The specific steps are as follows:

[0064] First, construct the matrix It is expressed as:

[0065]

[0066] in, is the transformation matrix, expressed as:

[0067]

[0068] Perform singular value decomposition on the augmented matrix X:

[0069]

[0070] Among them, U s and U n denote the signal subspace and the noise subspace respectively, Σ s Contains the L largest singular values.

[0071] Then, the matrix bundle equation is constructed to solve the eigenvalues ​​related to AoA:

[0072] U s2 -λU s1 =0

[0073] Among them, U s1 and U s2 Respectively by U s Delete the last n rows and the first n rows to get. To obtain the relationship between AoA and eigenvalue, from the matrix bundle equation represents the Moore-Penrose pseudoinverse, and Ψ is obtained by eigenvalue decomposition μ The characteristic value of in,

[0074] Secondly, to obtain the relationship between ToF and eigenvalue, the transformation matrix Q is used to obtain:

[0075]

[0076] Where q(i) represents a column vector of dimension mn×1, where the element at the i-th position is 1 and the elements at the other positions are 0. Construct a matrix bundle equation to solve the eigenvalues ​​related to ToF:

[0077] U s2 -λU s1 =0

[0078] Among them, U sp1 and U sp2 Respectively by U s Delete the last n rows and the first n rows to get. To obtain the relationship between ToF and eigenvalue, the matrix bundle equation is obtained By eigenvalue decomposition, we can get Ψ ν The characteristic value of in,

[0079] Finally, the arrival angle of the lth path is and flight time The solution is:

[0080]

[0081] Step 6: Pair the arrival angle and flight time obtained in step 5. The specific steps are as follows:

[0082] First, μ and ν have the same eigenvector P:

[0083]

[0084] in, For μ The eigenvalue matrix of For ν The eigenvalue matrix of .

[0085] Then, ν The eigenvalues ​​of are used to calculate the multipath ToF, and the estimated ToFs are arranged in ascending order to obtain Ψ ν The eigenvectors of P, that is, the column vectors of P, are arranged in the order of ToF to form a matrix P′. The matrix A′ can be obtained as follows:

[0086] A′=(P′) -1 Ψ μ P′

[0087] Finally, the angle of arrival is inverted Get the correct pairing order and take the AoA corresponding to the minimum ToF as the arrival angle of the direct diameter.

[0088] Beneficial Effects

[0089] The present invention proposes an AoA estimation method based on structured matrix embedding and recovery technology. First, commercial WiFi equipment is used to collect CSI and obtain the CSI matrix. Then, the CSI matrix is ​​projected and the Hankel-Toeplitz matrix is ​​constructed. The noise-suppressed CSI matrix is ​​restored using ADMM. Finally, the AoA of the direct path is estimated using the matrix bundle algorithm using the restored CSI matrix.

[0090] The method designed by the present invention improves the robustness of the system and the estimation accuracy of AoA while greatly reducing channel noise and without increasing additional channel bandwidth. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 AoA estimation method based on structured matrix embedding and recovery technology of the present invention is a block diagram Specific implementation plan

[0092] The purpose of the present invention is to provide an AoA estimation method based on structured matrix embedding and recovery technology in a WiFi sensor system, which can improve the robustness of the system and the estimation accuracy of AoA while greatly reducing channel noise and without increasing additional channel bandwidth.

[0093] The AoA estimation method based on structured matrix embedding and recovery technology described in the present invention specifically includes the following steps:

[0094] Step 1: In the WiFi sensor system, the transmitting device uses OFDM technology to modulate the RF signal onto multiple subcarriers, and then transmits it to the receiving device through the wireless channel to achieve parallel transmission and improve the bandwidth utilization. Consider a uniform linear array in space, and the distance between array elements is d. Since the signal propagates in the wireless channel with multipath effect, the lth (1≤l≤L) path reaches the uniform linear array, and the azimuth and flight time of the signal are θ respectively. l and τ l , the center frequency is f c , the subcarrier spacing is Δf, the packet time interval is Δt, and the speed of light is c. The channel frequency response (CFR) of a single packet on the kth (1≤k≤K) subcarrier at the i-th (1≤i≤I) antenna at the receiving end is expressed as:

[0095]

[0096] Among them, α l represents the complex attenuation of the lth path, σ i,k represents the channel noise.

[0097] Using the CSI information of the WiFi signal obtained by the receiving end, the CSI data matrix of a single received data packet can be written as:

[0098]

[0099] Among them, H i Represents the CSI data of the i-th (1≤i≤I) antenna.

[0100] Step 2: Establish an optimization model. The specific steps are as follows:

[0101] First, in order to construct a structured matrix, set the parameters M, N, m, and n as follows:

[0102]

[0103] If M or N is an even number, a 0 vector matching the matrix dimension is used as the data of the Mth row or Nth column of the CSI data matrix H, expressed as Where Ω represents the set of rows and columns of the matrix to be projected, that is, Ω = {row, col}, is the projection of the matrix on the subspace Ω, expressed as follows:

[0104] (1) If row and col are both odd numbers, then

[0105] (2) If row is an odd number and col is an even number, a zero vector with a dimension of row×1 is added to the right side of the matrix, and the matrix dimension is extended to row×(col+1);

[0106] (3) If row is an even number and col is an odd number, a 0 vector with a dimension of 1×col is added to the bottom of the matrix, and the matrix dimension is extended to (row+1)×col;

[0107] (4) If row and col are both even numbers, first add a 0 vector with a dimension of row×1 to the right side of the matrix, and then add a 0 vector with a dimension of 1×(col+1) to the bottom side of the matrix, and the matrix dimension is expanded to (row+1)×(col+1).

[0108] Then, to apply structured matrix embedding and recovery techniques, the rank-constrained optimization problem is written as:

[0109]

[0110] in, and represents the optimization variable, z={h,t} represents the set of variables to be optimized, represents the set of Hermitian positive semidefinite matrices whose rank is not greater than L, is the constructed Hankel-Toeplitz matrix, is the Hankel-Toeplitz operator, which means that the matrices in the set z are transformed according to arrangement, is the Hankel operator, is the Toeplitz operator, and Respectively and The conjugation of.

[0111] Among them, the two-dimensional Hankel matrix It is expressed as:

[0112]

[0113] in, is an n×n Hankel matrix consisting of the jth row in the optimization variable h and is expressed as:

[0114]

[0115] Among them, h is the optimization variable, expressed as:

[0116]

[0117] Two-dimensional Toeplitz matrix It is expressed as:

[0118]

[0119] in, is an n×n Toeplitz matrix (except The matrix is ​​non-Hermitian), which is composed of the j-th row in the optimization variable t:

[0120]

[0121] Among them, t is the optimization variable, expressed as:

[0122]

[0123] Step 3: To facilitate the application of the ADMM algorithm, the rank-constrained optimization problem in step 2 is converted into an unconstrained optimization problem. The specific steps are as follows:

[0124] First, we introduce the auxiliary matrix This enables ADMM to solve the following rank-constrained problems:

[0125]

[0126] in, represents the objective function to be optimized. Moreover, represents the indicator function, when hour, when hour,

[0127] Then, the Hermitian Lagrange multiplier is introduced Use the augmented Lagrangian function to transform the constrained problem into an unconstrained problem:

[0128]

[0129] in, represents the inner product of X and Y, tr(·) represents the trace of the matrix, X H represents the conjugate transpose of X, and ρ is the penalty parameter. The form is the same, construct the auxiliary matrix Hermitian Lagrange multipliers

[0130]

[0131] Step 4: Using the optimized variables in step 3, the ADMM algorithm is used to iterate and recover the original CSI data matrix from the channel noise. ADMM iteratively updates G. Λ, the specific steps are as follows:

[0132] First, initialize the auxiliary matrix G, the Hermitian Lagrange multiplier Λ and the optimization variable t to zero matrices, and the optimization variable h = H Ω , the number of iterations is represented by b, the maximum number of iterations is set to 3000, and the penalty parameter ρ is fixed to 1.

[0133] Then, the auxiliary matrix G b+1 Can be exist The projection on is obtained, and its update formula can be written as:

[0134]

[0135] Among them, projection After the Hermitian matrix is ​​subjected to truncated eigendecomposition, the remaining eigenvalues ​​are set to zero except for the largest L positive eigenvalues ​​(if the number of positive eigenvalues ​​is less than L, just set the other eigenvalues ​​to zero), and then the Hermitian matrix is ​​reconstructed.

[0136] Next, let the matrix W b =G b +ρ -1 Λ b , and can be written as

[0137]

[0138] For convenience of representation, the following arrangement is used to form W 1 , W 2 and W 3 The elements that make up C 1 , C 2 and C 3 , with dimensions n×Mn:

[0139]

[0140] Since the optimization variables h and t are separable, the derivative of the objective function with respect to {h, t} is equal to zero, and the update formula is obtained:

[0141]

[0142]

[0143] Among them, I is the N×N unit matrix, C 1,j , C 2,j and C 3,j Respectively represent C 1 , C 2 and C 3 The j-th block matrix in , (·) -1 represents the inverse of the matrix, It means mapping an m×m matrix X to a 1×M-dimensional vector. The mapping rule is X i,j represents the element in the i-th row and j-th column of the matrix X, It means mapping an m×m Hermitian matrix X to a 1×m dimensional vector. The mapping rule is According to step 2, h b +1 and t b+1 structure

[0144] Again, the Hermitian Lagrange multiplier Λ b+1 The update formula is:

[0145]

[0146] Finally, when the original residual is calculated and the dual residual When the value of is small enough, the iterative process is judged to have converged, and the ADMM iteration is terminated. The termination conditions are as follows:

[0147]

[0148] Among them, set the absolute error ε abs =10 -4 , relative error ε rel =10 -5 , the original residual and the dual residual It is expressed as the following formula:

[0149]

[0150] Step 5: Iterate the obtained data in step 4 AoA and ToF are calculated using a two-dimensional matrix bundle algorithm. The specific steps are as follows:

[0151] First, construct the matrix It is expressed as:

[0152]

[0153] in, is the transformation matrix, expressed as:

[0154]

[0155] Perform singular value decomposition on the augmented matrix X:

[0156]

[0157] Among them, U s and U n denote the signal subspace and the noise subspace respectively, Σ s Contains the L largest singular values.

[0158] Then, the matrix bundle equation is constructed to solve the eigenvalues ​​related to AoA:

[0159] U s2 -λU s1 =0

[0160] Among them, U s1 and U s2 Respectively by U s Delete the last n rows and the first n rows to get. To obtain the relationship between AoA and eigenvalue, from the matrix bundle equation represents the Moore-Penrose pseudoinverse, and Ψ is obtained by eigenvalue decomposition μ The characteristic value of in,

[0161] Secondly, to obtain the relationship between ToF and eigenvalue, the transformation matrix Q is used to obtain:

[0162]

[0163] Where q(i) represents a column vector of dimension mn×1, where the element at the i-th position is 1 and the elements at the other positions are 0. Construct a matrix bundle equation to solve the eigenvalues ​​related to ToF:

[0164] U s2 -λU s1 =0

[0165] Among them, U sp1 and U sp2 Respectively by U s Delete the last n rows and the first n rows to get. To obtain the relationship between ToF and eigenvalue, the matrix bundle equation is obtained By eigenvalue decomposition, we can get Ψ ν The characteristic value of in,

[0166] Finally, the arrival angle of the lth path is and flight time The solution is:

[0167]

[0168] Step 6: Pair the arrival angle and flight time obtained in step 5. The specific steps are as follows:

[0169] First, μ and ν have the same eigenvector P:

[0170]

[0171] in, For μ The eigenvalue matrix of For ν The eigenvalue matrix of .

[0172] Then, ν The eigenvalues ​​of are used to calculate the multipath ToF, and the estimated ToFs are arranged in ascending order to obtain Ψ ν The eigenvectors of P, that is, the column vectors of P, are arranged in the order of ToF to form a matrix P′. The matrix A′ can be obtained as follows:

[0173] A′=(P′) -1 Ψ μ P′

[0174] Finally, the angle of arrival is inverted Get the correct pairing order and take the AoA corresponding to the minimum ToF as the arrival angle of the direct diameter.

Claims

1. An AoA estimation method based on structured matrix embedding and recovery technology, comprising the following steps: Step 1: Consider a uniform linear array in space, where the distance between array elements is d; since the signal propagates in the wireless channel with multipath effects, the lth path reaches the uniform linear array, where 1≤l≤L; the azimuth angle and flight time of the signal are θ respectively l and τ l , the center frequency is f c , the subcarrier spacing is Δf, the packet time interval is Δt, and the speed of light is c; the i-th antenna at the receiving end, on the k-th subcarrier, where 1≤i≤I and 1≤k≤K; the channel frequency response of a single data packet is expressed as: Among them, α l represents the complex attenuation of the lth path, σ i,k represents the channel noise, j represents the imaginary number symbol; Using the receiving end to obtain the CSI information of the WiFi signal, the CSI data matrix of a single received data packet is written as: Among them, η i represents the CSI data of the i-th antenna; Step 2: Establish an optimization model. The specific steps are as follows: First, set the parameters M, N, m, n as follows: If M or N is an even number, the 0 vector matching the matrix dimension is used as the data of the Mth row or Nth column of the CSI data matrix η, expressed as Where Ω represents the set of rows and columns of the matrix to be projected, that is, Ω = {row, col}, is the projection of the matrix on the subspace Ω; Then, to apply structured matrix embedding and recovery techniques, the rank-constrained optimization problem is written as: in, and represents the optimization variable, z = {h, ​​t} represents the set of variables to be optimized, ||·|| F represents the F-norm, represents the set of Hermitian positive semidefinite matrices whose rank is not greater than L, is the constructed Hankel-Toeplitz matrix, is the Hankel-Toeplitz operator, which means that the matrices in the set z are transformed according to arrangement, is the Hankel operator, is the Toeplitz operator, and Respectively and conjugation of; Step 3: To facilitate the application of the ADMM algorithm, the rank-constrained optimization problem in step 2 is converted into an unconstrained optimization problem. The specific steps are as follows: First, we introduce the auxiliary matrix This enables ADMM to solve the following rank-constrained problems: in, represents the objective function to be optimized; and, represents the indicator function, when hour, when hour, Then, the Hermitian Lagrange multiplier is introduced Using the augmented Lagrangian function Convert the constrained problem into an unconstrained problem: in, represents the inner product of X and Y, tr(·) represents the trace of the matrix, X H represents the conjugate transpose of X, ρ is the penalty parameter; and The form is the same, construct the auxiliary matrix Hermitian Lagrange multipliers Step 4: Using the optimization variables in step 3, use the ADMM algorithm to iterate. ADMM iteratively updates G. Λ, the specific steps are as follows: First, initialize the auxiliary matrix G, the Hermitian Lagrange multiplier Λ and the optimization variable t to zero matrices, and the optimization variable h = η Ω , the number of iterations is represented by b, the maximum number of iterations is set to 3000, and the penalty parameter ρ is fixed to 1; Then, the auxiliary matrix G b+1 Depend on exist The projection on is obtained; Next, let the matrix W b =G b +ρ -1 Λ b , written as Arrange the elements that make up W1, W2, and W3 as follows to form C1, C2, and C3, with dimensions n×Mn: The derivative of the objective function with respect to {h, t} is equal to zero, and the update formula is obtained: Among them, I′ is the N×N unit matrix, C 1,g , C 2,g and C 3,g denote the g-th block matrix in C1, C2 and C3 respectively, (·) -1 represents the inverse of the matrix, It means mapping an m×m matrix X to a 1×M-dimensional vector. The mapping rule is X i′,g represents the element in the i′th row and gth column of the matrix X, It means mapping an m×m Hermitian matrix X to a 1×m dimensional vector. The mapping rule is According to step 2, h b+1 and t b+1 structure Again, the Hermitian Lagrange multiplier Λ b+1 The update formula is: Finally, when the original residual is calculated and the dual residual When the value of is small enough, the iterative process is judged to have converged, and the ADMM iteration is terminated; Step 5: Iterate the obtained data in step 4 AoA and ToF are calculated using a two-dimensional matrix bundle algorithm; the specific steps are as follows: First, construct the matrix It is expressed as: in, is the transformation matrix, expressed as: Perform singular value decomposition on the augmented matrix X: D.UΣV H JU s Σ s V s H +U n Σ n V n H Among them, U s and U n denote the signal subspace and the noise subspace respectively, Σ s Contains the L largest singular values; Then, the matrix bundle equation is constructed to solve the eigenvalue λ1 related to AoA: U s2 -λ1U s1 =0 Among them, U s1 and U s2 Respectively by U s Delete the last n rows and the first n rows to get; To obtain the relationship between AoA and eigenvalue, the matrix bundle equation represents the Moore-Penrose pseudo-inverse. The eigenvalue of Ψ1 obtained by eigenvalue decomposition is in, And 1≤l≤L; Secondly, to obtain the relationship between ToF and eigenvalue, the transformation matrix Q is used to obtain: Among them, q(i″) represents a column vector with a dimension of mn×1, the element at the i″th position is 1, and the elements at the remaining positions are 0; the matrix bundle equation is constructed to solve the eigenvalue λ2 related to ToF: IN sp2 -λ2U sp1 =0 Among them, U sp1 and U sp2 Respectively by U s Delete the last n rows and the first n rows to get; To obtain the relationship between ToF and eigenvalue, the matrix bundle equation is obtained The eigenvalue of Ψ2 is obtained by eigenvalue decomposition. in, Finally, the arrival angle of the lth path is and flight time The solution is: Step 6: Pair the arrival angle and flight time obtained in step 5. The specific steps are as follows: First, Ψ1 and Ψ2 have the same eigenvector P: in, is the eigenvalue matrix of Ψ1, is the eigenvalue matrix of Ψ2; Then, the eigenvalue of Ψ2 is used to calculate the ToF of the multipath, and the estimated ToF is arranged in order from small to large, and the result is The eigenvectors of Ψ2, i.e., the column vectors of P, are arranged in the order of ToF to form the matrix P′. The matrix A′ is obtained by the following formula: A′=(P′) -1 Ψ1P′ Finally, the angle of arrival is inverted Get the correct pairing order and take the AoA corresponding to the minimum ToF as the arrival angle of the direct diameter.

2. A method for ADMM applied to structured matrix embedding and restoration techniques, comprising the following steps: Step 1: Establish an optimization model. The specific steps are as follows: First, in order to construct a structured matrix, set the parameters M, N, m, and n as follows: If M or N is an even number, the 0 vector matching the matrix dimension is used as the data of the Mth row or Nth column of the CSI data matrix η, expressed as Where Ω represents the set of rows and columns of the matrix to be projected, that is, Ω = {row, col}, is the projection of the matrix on the subspace Ω, expressed as follows: (1) If row and col are both odd numbers, then (2) If row is an odd number and col is an even number, a zero vector with a dimension of row×1 is added to the right side of the matrix, and the matrix dimension is extended to row×(col+1); (3) If row is an even number and col is an odd number, a 0 vector with a dimension of 1×col is added to the bottom of the matrix, and the matrix dimension is extended to (row+1)×col; (4) If row and col are both even numbers, first add a 0 vector with a dimension of row×1 to the right side of the matrix, and then add a 0 vector with a dimension of 1×(col+1) to the bottom side of the matrix, and the matrix dimension is expanded to (row+1)×(col+1); Then, to apply structured matrix embedding and recovery techniques, the rank-constrained optimization problem is written as: in, and represents the optimization variable, z = {h, ​​t} represents the set of variables to be optimized, ||·|| F represents the F-norm, represents the set of Hermitian positive semidefinite matrices whose rank is not greater than L, is the constructed Hankel-Toeplitz matrix, is the Hankel-Toeplitz operator, which means that the matrices in the set z are transformed according to arrangement, is the Hankel operator, is the Toeplitz operator, and Respectively and conjugation of; Among them, the two-dimensional Hankel matrix is the Hankel operator, expressed as: in, is an n×n Hankel matrix with 1≤g≤M, consisting of the g-th row in the optimization variable h, expressed as: Among them, h is the optimization variable, expressed as: Two-dimensional Toeplitz matrix is the Toeplitz operator, expressed as: in, is an n×n Toeplitz matrix with 1≤g≤M, and except The outer matrices are non-Hermitian matrices, consisting of the g-th row in the optimization variable t: Among them, t is the optimization variable, expressed as: Step 2: To facilitate the application of the ADMM algorithm, the rank-constrained optimization problem in step 1 is converted into an unconstrained optimization problem. The specific steps are as follows: First, we introduce the auxiliary matrix This enables ADMM to solve the following rank-constrained problems: in, represents the objective function to be optimized; and, represents the indicator function, when hour, when hour, Then, the Hermitian Lagrange multiplier is introduced Use the augmented Lagrangian function to transform the constrained problem into an unconstrained problem: in, represents the inner product of X and Y, tr(·) represents the trace of the matrix, X H represents the conjugate transpose of X, ρ is the penalty parameter; and The form is the same, construct the auxiliary matrix Hermitian Lagrange multipliers Step 3: Using the optimized variables in step 2, the ADMM algorithm is used to iterate and recover the original CSI data matrix from the channel noise; ADMM iteratively updates G, Λ, the specific steps are as follows: First, initialize the auxiliary matrix G, the Hermitian Lagrange multiplier Λ and the optimization variable t to zero matrices, and the optimization variable h = η Ω , the number of iterations is represented by b, the maximum number of iterations is set to 3000, and the penalty parameter ρ is fixed to 1; Then, the auxiliary matrix G b+1 Depend on exist The projection on is obtained, and its update formula is written as: Among them, projection After the Hermitian matrix is ​​subjected to truncated eigendecomposition, the remaining eigenvalues ​​are set to zero except for the largest L positive eigenvalues. If the number of positive eigenvalues ​​is less than L, just set the other eigenvalues ​​to zero and then reconstruct the Hermitian matrix. Next, let the matrix W b =G b +ρ -1 Λ b , written as For convenience of representation, the elements that make up W1, W2, and W3 are arranged as follows to form C1, C2, and C3, with dimensions n×Mn: Since the optimization variables h and t are separable, the derivative of the objective function with respect to {h, t} is equal to zero, and the update formula is obtained: Among them, I′ is the N×N unit matrix, C 1,g , C 2,g and C 3,g denote the g-th block matrix in C1, C2 and C3 respectively, (·) -1 represents the inverse of the matrix, It means mapping an m×m matrix X to a 1×M-dimensional vector. The mapping rule is X i′,g represents the element in the i′th row and gth column of the matrix X, It means mapping an m×m Hermitian matrix X to a 1×m dimensional vector. The mapping rule is According to step 2, h b+1 and t b+1 structure Again, the Hermitian Lagrange multiplier Λ b+1 The update formula is: Finally, when the original residual is calculated and the dual residual When the value of is small enough, the iterative process is judged to have converged, and the ADMM iteration is terminated. The termination conditions are as follows: Among them, set the absolute error ε abs =10 -4 , relative error ε rel =10 -5 , the original residual and the dual residual It is expressed as the following formula:

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