A meshless parameter estimation method based on sparse array in non-uniform noise background

Abstract

The application discloses a non-uniform noise background-based sparse array gridless parameter estimation method, and the implementation steps are as follows: a sparse array structure is arranged at a receiving end; a sparse array receiving signal in a non-uniform noise background is modeled; a covariance matrix of the array receiving signal is calculated; receiving data of a virtual differential common array is calculated through vectorization and de-redundancy; a distribution to which a fitting error is subjected is derived, and a parameter estimation optimization model based on atomic norm minimization and avoiding a complex regularization parameter determination process is constructed; a semi-positive definite programming form of a primal problem and a dual problem is derived; and a wave direction estimation result is obtained through a MUSIC algorithm or polynomial root seeking. In addition, on the basis of fully utilizing a virtual differential common array of the sparse array to provide extended degrees of freedom and all information, the application restrains the influence of non-uniform noise by constraining the sparsity of data atoms to be recovered and selecting a regularization parameter from a chi-square distribution table according to a distribution to which a fitting error between the data to be recovered and actual receiving data not affected by noise is subjected, overcomes the grid mismatch problem caused by parameter domain discretization, avoids a complex process of determining the regularization parameter, and can obtain a higher-precision source direction estimation result.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, and relates to the determination of the direction of arrival of an incident signal source using a sparse array affected by non-uniform noise. Specifically, it is a meshless parameter estimation method based on a sparse array under non-uniform noise background. Background Technology

[0002] As an important branch of array signal processing, direction-of-arrival (DOA) estimation of incident sources has wide applications in many fields such as radar, sonar, and wireless communication. Among them, subspace algorithms such as MUSIC and ESPRIT can provide high-precision estimation results. However, subspace methods require the ideal assumption of uniform noise. When there is unknown environmental interference or non-uniform noise caused by different array element hardware conditions, the performance of subspace algorithms will degrade or even fail.

[0003] In addition, considering that sparse arrays can overcome the limitation of the number of physical array elements to handle the situation where the number of information sources is greater than the number of sensors (underdetermined) compared to uniform arrays, and can improve the accuracy of parameter estimation by utilizing extended virtual apertures, some algorithms based on sparse arrays and robust to non-uniform noise have been proposed. The scheme in reference [1] (He ZQ, Shi ZP, Huang L. Covariance sparsity-aware DOA estimation for nonuniform noise[J]. Digital Signal Processing, 2014, 28: 75-81.) estimates the azimuth of the source in space by removing the diagonal elements of the sparse array covariance matrix and then dividing the spatial angle into grids; reference [2] (Liu K, Zhang Y D. Coprime array-based DOA estimation in unknown nonuniform noise environment[J]. Digital Signal Processing, 2018, 79: 66-74.) recovers the noiseless sparse array covariance matrix by matrix reconstruction or matrix completion; reference [3] (Huang, Y., Z. Zheng and W. Wang, DOA Estimation Using Coprime Array in the Presence of Unknown Nonuniform Noise. Circuits, Systems, and Signal) Processing, 2022, 41(5): p.3000-3010.) Based on the recovery of the hole elements of the sparse matrix covariance matrix, the diagonal elements are recovered by linear prediction to suppress non-uniform noise. However, the above method restricts the estimated parameters to a pre-defined grid, which may lead to grid mismatch with the target, or require a complex process to determine the regularization parameter value to simultaneously constrain the sparsity of the target and the degree of fit with the received data.

[0004] Therefore, how to design a gridless method for estimating the azimuth of incident sources without requiring a complex process to determine regularization parameters when sparse arrays are affected by non-uniform noise is an important problem to be solved. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by proposing a meshless parameter estimation method based on the atomic norm for sparse arrays affected by non-uniform noise. This method constrains the sparsity of the received data from the virtual differential array in a continuous parameter space, fully utilizing all the information provided by the virtual array and avoiding the grid mismatch problem. Furthermore, it incorporates the distribution of the difference between the ideal noiseless received data to be recovered and the actual data, directly selecting regularization parameters from the chi-square distribution table, thus avoiding the complex process of determining regularization parameters.

[0006] This invention is achieved through the following technical solution, which includes the following steps:

[0007] (1) The receiving end uses a sparse array structure with M array elements, and the positions of the array elements are as follows: Where, d i It belongs to the set of positive integers, i = 1, ..., M, where λ is the signal wavelength;

[0008] (2) K narrowband far-field signals at angles Incident on this sparse array; against a non-uniform noise background, the received data of the array at time t is: Where A=[α(θ1),…,α(θ K [)] is the manifold matrix. Let be the guiding vector; where j is the imaginary unit, with the superscript [·]. T This represents the transpose operation; s(t) = [s1(t), ..., s K (t)] T Let n be the incident signal vector; n(t) be the non-uniform noise.

[0009] (3) Calculate the data covariance matrix using the sampled data at T time points: Where p k Let σ be the power of the k-th source. m 2 Let σ be the noise power on the m-th element. For non-uniform noise, σ is the noise power on different elements. m 2 Different from each other, indicated by superscript [·] H The conjugate transpose operation is represented by `diag{·}`, which represents a diagonal matrix constructed with the vector inside the parentheses as its diagonal elements.

[0010] (4) Vectorize the data covariance matrix to obtain: Where vec(·) represents the vectorization operation. ⊙ and These are the Kronecker product and the Kronecker product, respectively. * To represent the conjugate operation, e mThis represents the m-th column of an identity matrix of dimension M; The corresponding virtual array element positions are Integration The redundant elements in the data are used to obtain the virtual differential array received data. in It makes The set of all (m, n) that are true. yes A collection composed of unique elements. Represents a set The momentum, and Representing sets respectively The m-th and n-th elements and the set The i-th element, Γ 1(i，M(m-1)+n) Let Γ1 be the element in the i-th row and M(m-1)+n-th column;

[0011] (5) Use Indicates inclusion The smallest uniform linear array, constructed based on The set of atoms for recovering the noiseless received data y: Define the atomic norm of y Where inf{·} denotes the infimum, and conv(·) denotes the convex hull of the atomic set;

[0012] (6) Constructing the received data The correspondence between the data to be recovered and the data y: Where the dimension of Γ2 is Its element in the m-th row and n-th column is Represents a set The potential; C is obtained by having a dimension of d M +1 identity matrix, remove the first row and The row corresponding to the central hole element is obtained;

[0013] (7) Simultaneously constrain the sparsity of the atomic composition of y and y with To determine the degree of fit, an optimization problem model based on minimizing the atomic norm is constructed: Where ||·||2 represents the matrix L2 norm, and ε is a pre-set fitting error threshold;

[0014] (8) Derivation The distribution it follows: in, Γ′1 is obtained by removing the first row from Γ1. Describing the degrees of freedom as The asymptotic chi-square distribution; choosing η from the chi-square distribution table guarantees... It is true with a certain probability;

[0015] (9) Since the atomic norm constrains sparsity in infinite dimensions, combined with step (8), the form of the solvable optimization problem is derived: Where T(y) represents the Hermitian Toeplitz matrix constructed with y as the first column, tr(·) represents the trace of the matrix, and T(y)≥0 is used to constrain T(y) to be a positive semi-definite matrix;

[0016] (10) Derive the SDP form of the dual problem of the atomic norm minimization problem:

[0017]

[0018] Where Re(·) denotes the real part operation, This indicates the inverse operation.

[0019] (11) Use the MUSIC algorithm or polynomial root finding to determine the target orientation. estimate:

[0020] Where V is The noise subspace, which is composed of d M The eigenvectors are composed of +1-K smallest eigenvalues. The optimal solution is obtained by solving the optimization problem in step (9); or, in, The optimal solution is obtained by solving the optimization problem in step (10).

[0021] Furthermore, the function of matrix Γ1 mentioned in step (4) is to match the corresponding positions in the virtual difference matrix. The average of the elements is taken to achieve the purpose of removing redundancy.

[0022] Furthermore, the steps described in step (4) Corresponding to the positions of the array elements in the virtual differential co-array, this method does not require It is a holeless continuous difference array, and the array types applicable to this invention are not limited to common coprime arrays or nested arrays.

[0023] Furthermore, the atomic norm mentioned in step (5) It is the atomic l0 norm The convex approximation.

[0024] Furthermore, the matrix Γ2 mentioned in step (6) serves to receive data from the virtual differential co-array. Fill the hole locations with zeros; the function of matrix C is to extract... The data shared by y, due to The first element is contaminated by noise. The first row of elements in y is also deleted.

[0025] Furthermore, the objective function of the optimization problem described in step (9) is derived from the following relationship:

[0026] Furthermore, the dual problem of the atomic norm minimization problem in step (10) is obtained through Lagrange analysis, and since strong duality holds, solving this dual problem is equivalent to solving the original optimization problem in step (7).

[0027] Compared with the prior art, the present invention has the following advantages:

[0028] First, this invention uses a sparse array for direction of arrival estimation, which overcomes the disadvantage of using a uniform linear array where the degree of freedom is limited by the number of physical array elements, and makes full use of the degree of freedom and large aperture provided by the virtual differential array.

[0029] Second, this invention constrains the sparsity of virtual differential array received data in a continuous parameter space, avoiding the grid mismatch problem caused by parameter discretization, and realizes high-resolution gridless parameter estimation of the source azimuth under the influence of non-uniform noise on physical array elements.

[0030] Third, this invention derives the chi-square distribution of the error between the received data of the virtual array to be recovered and the actual received data, thereby achieving the purpose of selecting regularization parameters from the chi-square distribution table and avoiding the complex process of determining regularization parameters. Attached Figure Description

[0031] Figure 1 This is a flowchart of the overall process for a meshless parameter estimation method based on sparse arrays under non-uniform noise background.

[0032] Figure 2(a) shows the spatial estimation spectrum of the proposed method under the influence of non-uniform noise in the case of 12 incident sources.

[0033] Figure 2(b) shows the spatial estimation spectrum of the proposed method under the influence of non-uniform noise in the case of two closely spaced signal sources.

[0034] Figure 3 This is a comparison chart showing the mean square error of azimuth estimation as a function of signal-to-noise ratio for the proposed method in the overdetermined case compared with other sparse array non-uniform noise processing methods.

[0035] Figure 4 This is a comparison chart showing the mean square error of orientation estimation of the proposed method under overdetermined conditions with the number of snapshots, compared to other sparse array non-uniform noise processing methods.

[0036] Figure 5 This is a comparison chart showing the mean square error of azimuth estimation as a function of signal-to-noise ratio between the proposed method of this invention and other sparse array non-uniform noise processing methods under underdetermined conditions.

[0037] Figure 6 This is a comparison chart showing the mean square error of orientation estimation as a function of the number of snapshots for the proposed method under underdetermined conditions, compared with other sparse array non-uniform noise processing methods. Detailed Implementation

[0038] The technical solution and effects of the present invention will be further described below with reference to the accompanying drawings. The present invention provides a meshless parameter estimation method based on a sparse array under non-uniform noise background, referring to... Figure 1 The implementation steps of this invention are as follows:

[0039] Step 1: Arrange the M=6 array elements respectively in A sparse array for receiving signals is constructed at the location, where λ is the wavelength of the incident signal.

[0040] Step 2: K narrowband far-field signals are incident on the sparse array in the aforementioned non-uniform noise environment. The antenna receiver in the array samples the spatial target signal, and the array receives the data as x(t) = [x1(t), x2(t), ..., x6(t)]. T x i (t) represents the received data of the i-th array element, where i = 1, 2, ... 6.

[0041] Step 3: Calculate the covariance matrix of the array received data using T sampling snapshots:

[0042]

[0043] Step 4: Obtain the received data of the virtual differential array through vectorization and redundancy removal:

[0044] in It makes The set of all (m, n) that are true. yes A collection composed of unique elements. Represents a set The momentum, and Representing sets respectively The m-th and n-th elements and the set The i-th element, Γ 1(i，M(m-1)+n) Let Γ1 be the element in the i-th row and M(m-1)+n-th column.

[0045] Step 5: Calculate the variance matrix of the fitting error. Γ′1 is obtained by removing the first row from Γ1.

[0046] Step Six: Solve the following optimization problem using MATLAB's CVX toolbox: T(y)≥0, where the dimension of Γ2 is Its element in the m-th row and n-th column is C is achieved by having dimension d M +1 identity matrix, remove the first row and The row corresponding to the hole element is obtained, and η = 3.95 is directly selected from the chi-square distribution table.

[0047] Step 7: Solve the following SDP-type optimization problem using MATLAB's CVX toolbox:

[0048]

[0049] Step 8: Use the MUSIC algorithm or polynomial root finding to determine the target orientation. estimate: Where V is The noise subspace, which is composed of d M The eigenvectors are composed of +1-K smallest eigenvalues. The optimal solution is obtained by solving the optimization problem in step six; or... in, The optimal solution is obtained by solving the optimization problem in step seven. The effects of this invention will be further described below with simulation examples.

[0050] Simulation Example 1: The antenna array is a sparse array with M = 6 elements, located at positions [0 3 5 6 10 14]λ / 2, where λ is the signal wavelength. The array is affected by Gaussian non-uniform noise, with a noise mean of 0 on different elements. The corresponding noise variance matrix is ​​diag{11.2, 5.2, 10, 3.4, 11.2, 5.2, 10}. The snapshot number and signal-to-noise ratio (SNR) are set to 3000 and 30dB, respectively. The definition of SNR is: For signal power, This represents the noise power on the m-th array element. Figure 2(a) shows the spatial estimation spectrum obtained by the method proposed in this invention, assuming 12 equal-power far-field narrowband sources are incident on the sparse array at angles of [-65°, 63.7°]. Figure 2(b) shows the spatial estimation spectrum obtained by the method proposed in this invention, assuming two closely spaced sources are incident on the sparse array at angles of {-0.4°, 0.4°}. Simulation results show that the method proposed in this invention can suppress the influence of non-uniform noise, fully utilize the degrees of freedom and larger aperture provided by the virtual differential array, and can cope with underdetermined scenarios where the number of sources is much higher than the number of array elements, while also demonstrating high resolution.

[0051] Simulation Example 2: 500 Monte Carlo experiments were conducted. Assuming four equal-power far-field narrowband sources were incident on the sparse array at angles of [-45°, -17.3°, -10.4°, 38.1°], with a fixed snapshot number of 500, the root mean square error (RMSE) of parameter estimation as a function of SNR is shown in the curve. Figure 3 As shown; additionally, with SNR fixed at 5dB, the curve of RMSE changing with the number of snapshots is shown below. Figure 4 As shown in the figure, it can be concluded from the figure that the method proposed in this invention has the closest RMSE to CRB compared to other methods. This is because the method proposed in this invention not only overcomes the mesh mismatch problem while suppressing the influence of non-uniform noise, but also fully utilizes the degrees of freedom of sparse matrix expansion, avoiding the complex process of determining regularization parameters and the impact of inappropriate regularization parameters on estimation performance, thereby achieving the effect of improving estimation accuracy.

[0052] Simulation Example 3: This study investigates the variation of the mean square error (MSE) with signal-to-noise ratio (SNR) and snapshot number under underdetermined conditions. It assumes seven sources are incident on the aforementioned sparse array at angles of [-65°, -44.3°, -23.6°, -2.9°, 17.8°, 38.5°, 59.2°], keeping other parameters the same as in Simulation Example 2. Figure 5 Plot the RMSE as a function of SNR when the number of snapshots is fixed at 500. Figure 6 The figure shows the change of RMSE with the number of snapshots when the SNR is fixed at 15dB. As can be seen from the figure, the method proposed in this invention still has higher estimation accuracy than other methods, and there is no need to redetermine the regularization parameter. However, for the method in reference [2], the determination of its regularization parameter is related to the sparsity of the information source. Inappropriate regularization parameters lead to a serious reduction in algorithm performance, which indirectly confirms the superiority of the method proposed in this invention.

[0053] In summary, this invention presents a meshless parameter estimation method for sparse arrays affected by non-uniform noise. By constraining the sparsity of the noiseless received data to be recovered in the continuous parameter space using the atomic norm, and utilizing a regularization parameter selected from a chi-square distribution table based on the fitting error distribution between the noiseless data to be recovered and the actual received data unaffected by non-uniform noise, an optimization problem for parameter estimation is constructed, and the dual problem SDP form without eigenvalue decomposition is derived. This method fully utilizes the information and degrees of freedom provided by the virtual difference co-array of the sparse array, suppresses the influence of non-uniform noise, achieves source orientation estimation in the continuous parameter domain through the atomic norm, overcomes the problems caused by grid discretization, and eliminates the need for a complex process to determine the regularization parameter. This method exhibits superior performance in practical applications of direction-of-arrival estimation and target detection.

Claims

1. A sparse matrix based meshless parameter estimation method under non-uniform noise background, characterized in that, comprising the steps of: (1) The receiving end uses a sparse array with M elements, and the element position is where d i is a positive integer set, i = 1, …, M, and λ is the signal wavelength; (2) K narrowband far-field signals at angles Incident on this sparse array; against a non-uniform noise background, the received data of the array at time t is: Where A=[α(θ1),···,α(θ K [)] is the manifold matrix. Let be the guiding vector; where j is the imaginary unit, with the superscript [·]. T This represents the transpose operation; s(t) = [s1(t), ..., s K (t)] T Let n be the incident signal vector; n(t) represents non-uniform noise. (3) Calculate the data covariance matrix using the sampling data at T time instants: where p k is the power of the kth source, σ m 2 is the noise power at the mth array element, and σ m 2 are different for different array elements for non-uniform noise, and the superscript [·] H denotes the conjugate transpose operation, and diag{·} denotes a diagonal matrix constructed with the vector inside the brackets as the diagonal elements. (4) Vectorize the data covariance matrix to get: where vec(·) denotes the vectorization operation, ⊙ and are the Kronecker product and the Hadamard product, respectively, [·] * denotes the conjugate operation, e m denotes the m-th column of the M-dimensional identity matrix; The corresponding virtual array element position is Integrate the redundant elements in to get the virtual differential array receiving data: where is the set of all (m, n) that make true, is the set of unique elements in , denotes the cardinality of the set , and denote the n-th, n-th element of the set and the i-th element of the set , Γ 1(i,M(m-1)+n) denotes the i-th row, M(m-1)+n-th column element of Γ1 (5) with denotes the smallest uniform linear array comprising an atomic set based on the noiseless received data y to be recovered: define the atomic norm of y where inf{·} denotes the infimum and conv(·) denotes the convex hull of the atomic set; (6) Constructing the correspondence between the received data and the data to be recovered y: where Γ2 has dimension and its mth row, nth column element is denotes the cardinality of the set ; C is obtained by removing the first row and the row corresponding to the hole element in M +1 and dimension d +1. (7) simultaneously constraining the sparsity of the atomic composition of y and the degree of fitting of y and an optimization problem model based on atomic norm minimization is constructed: where ||·||2 represents the matrix two-norm, and ε is a pre-set fitting error threshold; (8) Derivation The distribution of the compliance: where, Γ'1 is obtained by removing the first row from Γ1, denotes the asymptotic chi-square distribution with degrees of freedom; from the chi-square distribution table, η can be selected to ensure that holds with a certain probability; (9) Since the atomic norm constrains sparsity in infinite dimensions, in combination with step (8), a solvable optimization problem form is derived: where T(y) represents a Hermitian Toeplitz matrix constructed with y as the first column, tr(·) represents the trace of a matrix, and T(y) > 0 is used to constrain T(y) to be a semi-definite matrix; (10) Deriving the SDP form of the dual problem of the atomic norm minimization problem: where Re(·) denotes the real part operation, denotes the inverse operation. (11) Target bearing is achieved using the MUSIC algorithm or polynomial root finding Estimation: where V is the noise subspace, which is given by the d M eigenvectors corresponding to the d is the optimal solution obtained by solving the optimization problem in step (9); or where, is the optimal solution obtained by solving the optimization problem in step (10).

2. The meshless parameter estimation method based on sparse array in non-uniform noise background according to claim 1, characterized in that, The role of the matrix Γ1 in step (4) is to average the elements in the virtual difference matrix corresponding to the same position in the real difference matrix to achieve the purpose of de-redundancy.

3. The non-uniform noise background based on sparse array meshless parameter estimation method according to claim 1, characterized in that, The method described in step (4) Corresponding to the virtual differential common array element position, the method does not require Is a non-porous continuous differential array, the array pattern of the application is not limited to the common coprime array or nested array.

4. The non-uniform noise background based on sparse array's meshless parameter estimation method according to claim 1, characterized in that, The atomic norm described in step (5) is a convex approximation of the atomic lo norm ​ 5. The non-uniform noise background based on sparse array's meshless parameter estimation method according to claim 1, characterized in that, The role of the matrix Γ2 in step (6) is to apply a virtual differential co-array to the received data The role of the matrix C is to extract the common data in x and y by complementing 0 at the hole positions The first element of x and y is also removed since the first element of x is contaminated by noise, and the first row elements of x and y.

6. The meshless parameter estimation method based on sparse matrix in non-uniform noise background according to claim 1, characterized in that, The objective function of the optimization problem described in step (9) is derived from the following relation:

7. The method of claim 1, wherein, The dual problem of the atomic norm minimization problem in step (10) is derived by Lagrangian analysis, and since strong duality holds, solving the dual problem is equivalent to solving the primal optimization problem in step (7).

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