A meshless parameter estimation method based on sparse matrix phase error correction

Abstract

The application discloses a kind of sparse array amplitude-phase error correction-based meshless parameter estimation method, and its implementation steps are: receiving end is laid out sparse array;The received signal of sparse array with array element gain phase error is modeled;Sample covariance matrix is constructed;The output signal of virtual differential common array is calculated by vectorization;Define the atom norm of introducing amplitude-phase error parameter, and construct the optimization model for target direction estimation;By dual atom norm, the original optimization problem is converted into its dual problem;The semi-positive definite programming form of dual problem is deduced;Dual polynomial is constructed, and the wave direction estimation result is obtained by spectral peak search.In addition, the application fully utilizes the virtual differential common array of sparse array to provide extended degrees of freedom and all information, and suppresses the influence of array element amplitude-phase error by defining new atom norm, solves the mesh mismatch problem caused by parameter domain discretization, and obtains high-precision meshless 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 incident signals when there is gain and phase error in a sparse array. Specifically, it is a meshless parameter estimation method based on sparse array amplitude and phase error correction. Background Technology

[0002] Array signal processing has significant application value in military and civilian fields such as radar, sonar, and wireless communication. Traditional direction-of-arrival estimation methods based on uniform linear arrays are limited by the number of physical array elements and cannot handle situations where the number of signal sources exceeds the number of sensors (underdetermined). To address this issue, sparse arrays have been proposed. By using virtual differential arrays, greater degrees of freedom can be obtained with a smaller number of sensors, thereby improving the accuracy of parameter estimation.

[0003] Traditional sparse array-based direction-of-arrival (DOA) methods, such as SS-MUSIC (CL Liu and PPVaidyanathan, “Remarks on the spatial smoothing step incoarray music,” IEEE Signal Processing Letters, vol. 22, no. 9, pp. 1438–1442, 2015.), largely rely on the exact unbiased condition of the array steering vector. However, in practical engineering applications, due to differences in manufacturing quality and aging rates, the gain and phase of each sensor channel can be affected by random errors, which may lead to a decrease in DOA estimation performance or even failure.

[0004] In recent years, some DOA estimation algorithms suitable for amplitude and phase correction have been proposed, mainly divided into two categories: active calibration and self-calibration. Active calibration typically requires additional auxiliary sensors or auxiliary signal sources at known locations, placing high demands on practical engineering environments and making it difficult to apply in many situations. Self-calibration, on the other hand, does not require auxiliary sensors or signal sources, making it more meaningful. For example, eigenvalue-based algorithms utilize the orthogonality of the signal and noise subspaces to simultaneously estimate DOA and gain phase deviation parameters. The drawbacks of this type of method are that it cannot be applied to correlated signal sources, has high requirements for signal-to-noise ratio and snapshot number, and cannot be applied to sparse array structures. Mesh-based sparse reconstruction calibration algorithms include STLS-based methods (K. Han and P. Yang, “Calibrating nested sensor arrays with model errors,” Conference Record-Asilomar Conference on Signals, Systems and Computers, vol. 2015, pp. 368–372, Apr. 2015.) and SBL-based methods (R. Lu, M. Zhang, X. Liu, X. Chen, and A. Zhang, “Direction-of-arrival estimation via coarray with model errors,” IEEE Access, vol. PP, no. 99, pp. 1–1, 2018.) overcomes the above problems, but these methods usually restrict the estimated parameters to a pre-defined grid, which may lead to a mismatch between the grid and the target. If the mismatch problem is overcome by setting a denser grid, on the one hand, it will bring high computational cost, and on the other hand, the correlation between adjacent angular steering vectors will lead to a decrease in estimation performance.

[0005] To address the challenges posed by discretizing continuous parameter domains, the Ori-ANM method (T. Chen, L. Shi, and L. Guo, “Gridless direction of arrival estimation exploiting sparse linear array,” IEEE Signal Processing Letters, vol. PP, no. 99, pp. 1–1, 2020) based on the original atomic norm was proposed for sparse arrays. It achieves more accurate DOA estimation performance by utilizing an atomic norm minimization method based on the received signal from a differential virtual array. However, existing atomic norm-based algorithms for sparse arrays do not consider the potential amplitude and phase errors of physical array elements, thus leading to a degraded parameter estimation performance. The atomic norm minimization method GP-ANM based on uniform linear arrays (P.Chen,Z.Chen,Z.Cao,X.Wang,A new atomic norm for DOAestimation with gain-phase errors,IEEE Trans.Signal Process.68(2020)4293-4306.) is limited to one-dimensional linear arrays, even though it considers the influence of amplitude and phase errors when defining the atomic norm.

[0006] Therefore, how to improve the underdetermined DOA estimation performance using a meshless method when there are gain and phase errors in sparse array antenna elements is a key problem to be solved. Summary of the Invention

[0007] 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 in the case of gain-phase errors in sparse arrays. This method, for signals received by a virtual differential array, defines an atomic norm structure that considers gain-phase errors, fully utilizing all the information provided by the virtual array and suppressing the influence of gain-phase errors, thereby improving DOA estimation performance.

[0008] This invention is achieved through the following technical solution, which includes the following steps:

[0009] (1) Construct a sparse array using M array elements, with the element positions as follows: Where λ is the wavelength of the incident signal. Let d be the set of positive integers. i Arranged in ascending order;

[0010] (2) K narrowband far-field signals at angles The incident signal is projected onto the sparse array; neglecting gain and phase error, the received signal of the sparse array in a single snapshot is:

[0011]

[0012] Where x(t)=[x1(t),x2(t),…,x M (t)] T The received signal vector; A = [α(θ1), ..., α(θ)] k ),…,α(θ K [)] is the manifold matrix. The guiding vector; where j represents the unit imaginary number, j 2 =-1, [·] T Indicates transpose; s(t) = [s1(t), ..., s K (t)] T Let n(t) be the incident signal vector; n(t) be zero-mean additive Gaussian noise.

[0013] Considering gain and phase error, the received signal under a single snapshot is:

[0014] x(t) = GAs(t) + n(t),

[0015] Where G = (I M +diag{g})diag{e jφ} = I M +diag{e},g=[g0,g1,…,g M-1 ] T Let φ be the gain error vector, φ = [φ0, φ1, ..., φ M-1 ] T Let φ be the phase error vector. n ∈[0,2π), e=[e0,e1,…,e M-1 ] T The gain phase of the relative corrected array elements is uncertain. I M Let be an M×M identity matrix, and diag{·} denote a diagonal matrix constructed with the vectors inside the brackets as diagonal elements;

[0016] (3) When the number of sampling snapshots is T, construct the covariance matrix of the array received signal:

[0017]

[0018] in, For the new guiding vector, p k Let σ be the power of the k-th source. 2 For noise power, superscript [·]. H This represents the conjugate transpose operation;

[0019] (4) Vectorize the covariance matrix of the received array signal to obtain the output signal of the virtual differential array:

[0020] r = vec(R) = y + σ 2 vecI M ,

[0021] Among them, the signal section p = [p1, p2, ..., p K ] T , ⊙ and These are the KR product and the Kronecker product, respectively, with [·]* representing the conjugate operation; the corresponding virtual array element positions are...

[0022] (5) Considering the sensor gain phase error, a special set of atoms is constructed for the virtual differential co-array received signal of the sparse array:

[0023]

[0024] Where C e Used to constrain the strength of gain phase error. and These are the variances of the gain and phase error, respectively.

[0025] (6) Define the atomic norm of the received signal in the virtual differential array as the atomic norm in the atomic set A. e The smallest number of atoms that can represent y is:

[0026]

[0027] Where inf{·} denotes the infimum, and conv(·) denotes the convex hull of the atomic set;

[0028] (7) Construct an optimization model for target orientation estimation by minimizing the atomic norm:

[0029]

[0030] Where ε is a pre-set fitting error threshold;

[0031] (8) Transform the optimization model in step (7) into its dual problem:

[0032]

[0033] in, Let sup{·} represent the dual atomic norm of the atomic norm described in step (6), and sup{·} represent the supremum.<u,y> =Re(y H u) denotes taking the real part of the inner product of vectors;

[0034] (9) The dual problem of the optimization model in step (8) can be expressed as an easily solvable equivalent semidefinite program (SDP) form:

[0035]

[0036]

[0037] Where ≥ indicates generalized inequality, Q i i = 0, 1, ..., 2d M This represents the sum of elements at different positions in matrix Q; specifically, it defines a matrix H(θ) = b * 9θ)b T (θ), The (m,n)th element of H(θ) is Where d m ={d j -d i |i, j=1,..M; m=(i-1)M+j}, d n ={d j -d i |i,j=1,..M; n=(i-1)M+j}; Q i It is the sum of a subset of elements in matrix Q, whose positions satisfy the condition that the exponential parts of elements at the same position in matrix H(θ) are equal in length. n -d m =i; The above SDP-type optimization problem can be solved using MATLAB's CVX toolbox;

[0038] (10) Construct a dual polynomial and obtain the target orientation estimate through spectral peak search.

[0039]

[0040] Among them, The optimal solution is obtained by solving the dual problem in the form of SDP.

[0041] Furthermore, the sparse array structure described in step (1) does not require a non-porous differential continuous segment, and the array types applicable to this invention are not limited to common coprime arrays or nested arrays.

[0042] Furthermore, the atomic norm described in step (6) yes The convex approximation.

[0043] Furthermore, the dual problem described in step (8) is obtained through Lagrange analysis.

[0044] Furthermore, the SDP-form constraint described in step (9) is a sufficient but not necessary condition for the dual atomic norm-form constraint in the dual problem of step (8).

[0045] Compared with the prior art, the present invention has the following advantages:

[0046] 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 high degree of freedom and large aperture provided by the virtual differential array.

[0047] Second, this invention defines a new atomic norm for the received signal of the virtual differential array that takes into account the amplitude and phase errors of the physical array elements, constructs a DOA estimation optimization problem based on minimizing the atomic norm, and overcomes the problems of grid mismatch caused by the discretization of the parameter domain by deriving the SDP form of its dual problem and searching the spectral peaks of the dual polynomial, thus realizing high-resolution gridless parameter estimation of narrowband far-field signals under the condition that the physical array elements have amplitude and phase errors. Attached Figure Description

[0048] Figure 1 This is the overall flowchart of a meshless parameter estimation method based on sparse array amplitude and phase error correction.

[0049] Figure 2(a) shows the azimuth estimation spectrum of the Ori-ANM algorithm without the influence of amplitude and phase errors.

[0050] Figure 2(b) shows the azimuth estimation spectrum of the method proposed in this invention without the influence of amplitude and phase errors.

[0051] Figure 2(c) shows the azimuth estimation spectrum of the method proposed in this invention under the influence of amplitude and phase errors.

[0052] Figure 3 This is a comparison chart showing the mean square error of DOA estimation as a function of signal-to-noise ratio between the proposed method and other parameter estimation methods.

[0053] Figure 4 A comparison of the mean square error of DOA estimation between the proposed method and other parameter estimation methods with the number of snapshots.

[0054] Figure 5 A comparison of the mean square error of DOA estimation with gain error between the proposed method and other parameter estimation methods.

[0055] Figure 6A comparison of the mean square error of DOA estimation with phase error between the method proposed in this invention and other parameter estimation methods. Detailed Implementation

[0056] 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 sparse array amplitude phase error correction, referring to... Figure 1 The implementation steps of this invention are as follows:

[0057] Step 1: Construct a sparse array using M=7 array elements, and arrange the array elements in... At the location, where λ is the wavelength of the incident signal.

[0058] Step 2: Inject K = 2 narrowband far-field signals onto the sparse array described above. The antenna receiver in the array samples the spatial target signal, and the array output signal is x(t) = [x1(t), x2(t), ..., x7(t)]. T x i (t), i = 1, 2, ... 7, represents the received signal of the i-th array element.

[0059] Step 3: Construct the covariance matrix of the array received signal using T = 15 sampling snapshots:

[0060]

[0061] Step 4: Vectorize the covariance matrix of the received array signal to obtain the output signal of the virtual differential array:

[0062] r = vec(R).

[0063] Step 5: Solve the following SDP-type optimization problem using MATLAB's CVX toolbox:

[0064]

[0065]

[0066] Where ε = 0.4 is the preset fitting error threshold. Used to constrain the strength of gain phase error. and The variances of gain and phase error, respectively, are Q. i i = 0, 1, ..., 2d M This represents the sum of elements at different positions in matrix Q; specifically, it defines a matrix H(θ) = b * (θ)b T (θ), d iThe i-th element of the corresponding set [0 3 5 6 9 10 12], the (m,n)-th element of H(θ) is... Where d m ={d j -d i |i, j=1,..M; m=(i-1)M+j}, d n ={d j -d i |i,j=1,..M; n=(i-1)M+j}, d i ,d j These correspond to the i-th and j-th elements in the set [0 3 5 6 9 1012], respectively. Q i It is the sum of a subset of elements in matrix Q, whose positions satisfy the condition that the exponential parts of elements at the same position in matrix H(θ) are equal in length. n -d m =i.

[0067] Step 6: Construct the dual polynomial and obtain the DOA estimate through spectral peak search.

[0068]

[0069] in, The optimal solution is obtained by solving the dual problem in the form of SDP.

[0070] The effects of the present invention will be further described below with reference to simulation examples.

[0071] Simulation Example 1: The antenna array is a sparse array with M = 7 elements, and the element positions are [0 3 5 6 9 10 12]λ / 2, where λ is the signal wavelength, and the noise is additive Gaussian noise. Assume two equal-power far-field narrowband sources are incident on the sparse array at -30° and 10°, respectively, with the snapshot number and signal-to-noise ratio (SNR) set to 15 and 10 dB, respectively. The element gain error g... m Follows a Gaussian distribution Where the standard deviation σ g =0.2, array element phase error φ m Follows a Gaussian distribution Where the standard deviation σ φ=5. Figure 2(a) shows the DOA estimation spectrum obtained by the parameter estimation method based on the original atomic norm when there is no amplitude and phase error in the sparse array; Figure 2(b) shows the DOA estimation spectrum of the method when there is amplitude and phase error in the array elements; and Figure 2(c) shows the DOA estimation spectrum of the method proposed in this invention when there is amplitude and phase error. Compared with the original atomic parameter minimization method based on the sparse array, the method proposed in this invention, while fully utilizing the information and degrees of freedom provided by the virtual differential co-array to provide meshless DOA estimation results, also suppresses the influence of amplitude and phase error in the array elements by defining a new atomic norm. Figure 3 As shown, the method proposed in this invention can perform well even in the presence of array element amplitude and phase errors, while the parameter estimation method based on the original atomic norm will have spurious peaks in the DOA spectrum under such circumstances, which will lead to serious estimation errors.

[0072] Simulation Example 2: 200 Monte Carlo experiments were conducted. The root-mean-square error (RMSE) of the estimated signal as a function of the incident signal-to-noise ratio was plotted when the signal snapshot number was 15. Figure 3 In the figure, SNR varies from 0dB to 30dB in 5dB increments. The curve of RMSE as a function of snapshot number is plotted on [the graph]. Figure 4 In the figure, the number of snapshots varies from 5 to 20 with a step size of 3, and the signal-to-noise ratio is set to 5 dB. As shown in the figure, the method proposed in this invention has a lower RMSE than the other four methods. This is because, for Ori-ANM and GP-ANM, although both overcome the grid mismatch problem based on the atomic norm, Ori-ANM does not consider the influence of element amplitude and phase errors, and GP-ANM can only be applied to uniform linear arrays; for subspace methods like SS-MUSIC, not only is the influence of element errors not considered, but also higher requirements are placed on the signal-to-noise ratio and the number of snapshots; for the SBL method of sparse array correction, it suffers from the grid mismatch problem, even if the incident source is located in a preset grid, the similarity of the steering vectors corresponding to adjacent grids in the dictionary matrix will reduce the DOA recovery probability. The sparse array correction method proposed in this invention not only overcomes the grid mismatch problem by suppressing element amplitude and phase errors, but also fully utilizes the degrees of freedom of sparse array expansion, achieving the effect of improving estimation accuracy.

[0073] Simulation Example 3: Studying the variation of mean square error (MSE) with element amplitude and phase error. SNR and snapshot number are set to 10dB and 15, respectively. The gain phase error σ... p For case 5, RMSE varies with amplitude and phase error σ g The changing curve is plotted on Figure 5 The curve of RMSE as a function of amplitude and phase error is plotted in [the image / plot]. Figure 6At this point, the gain phase error is fixed at 0.2. As shown in the figure, the method proposed in this invention has a better effect on suppressing gain phase error compared with other parameter estimation methods. Therefore, it has higher accuracy in estimating the target wave direction of arrival even in the presence of amplitude and phase errors.

[0074] In summary, this invention presents a meshless DOA estimation method for sparse arrays with gain and phase errors. By introducing an additional amplitude and phase error parameter into the received signal of the virtual differential co-array, a new atomic norm is designed, and a DOA parameter estimation optimization problem based on minimizing the new atomic norm is constructed. A new SDP form that is easy to solve is derived. This method fully utilizes the information and degrees of freedom provided by the virtual differential co-array of the sparse array, suppresses the influence of its gain and phase errors, and achieves DOA estimation in the continuous parameter domain through the atomic norm, overcoming the problems caused by grid discretization. This method exhibits superior performance in direction-of-arrival estimation and target detection.

Claims

1. A meshless parameter estimation method based on sparse array amplitude and phase error correction, characterized in that, Includes the following steps: (1) Construct a sparse array using M array elements, with the element positions as follows: Where λ is the wavelength of the incident signal. Let d be the set of positive integers. i Arranged in ascending order; (2) K narrowband far-field signals at angles The incident signal is projected onto the sparse array; neglecting gain and phase error, the received signal of the sparse array in a single snapshot is: Where, x(t) = [x1(t), x2(t), ..., x M (t)] T The received signal vector; A = [α(θ1), ..., α(θ)] k ), …, α(θ) K [)] is the manifold matrix. The guiding vector; where j represents the unit imaginary number, j 2 =-1, [·] T Indicates transpose; s(t) = [s1(t), ..., s K (t)] T Let n(t) be the incident signal vector; n(t) be zero-mean additive Gaussian noise. Considering gain and phase error, the received signal under a single snapshot is: x(t) = GAs(t) + n(t), Where G = (I M +diag{g})diag{e jφ } = I M +diag{e},g=[g0,g1,...,g M-1 ] T Let φ be the gain error vector, φ = [φ0, φ1, ..., φ2]. M-1 ] T Let φ be the phase error vector. n ∈[0, 2π), e=[e0, e1,...,e M-1 ] T The gain phase of the relative corrected array elements is uncertain. I M Let be an M×M identity matrix, and diag{·} denote a diagonal matrix constructed with the vectors inside the brackets as diagonal elements; (3) When the number of sampling snapshots is T, construct the covariance matrix of the array received signal: in, For the new guiding vector, p k Let σ be the power of the k-th source. 2 For noise power, superscript [·]. H This represents the conjugate transpose operation; (4) Vectorize the covariance matrix of the received array signal to obtain the output signal of the virtual differential array: r=vec(R)=y+σ 2 vecI M , Among them, the signal section p = [p1, p2, ..., p] K ] T , ⊙ and These are the KR product and the Kronecker product, respectively. * Indicates the conjugate operation; the corresponding virtual element position is (5) Considering the sensor gain phase error, a special set of atoms is constructed for the virtual differential co-array received signal of the sparse array: Where C e Used to constrain the strength of gain phase error. and These are the variances of the gain and phase error, respectively. (6) Define the atomic norm of the received signal in the virtual differential array as the atomic norm in the atomic set A. e The smallest number of atoms that can represent y is: Where inf{·} denotes the infimum, and conv(·) denotes the convex hull of the atomic set; (7) Construct an optimization model for target orientation estimation by minimizing the atomic norm: st||y+σ 2 vecI M -r||2≤ε, Where ε is a pre-set fitting error threshold; (8) Transform the optimization model in step (7) into its dual problem: in, Let sup{·} represent the dual atomic norm of the atomic norm described in step (6), and sup{·} represent the supremum.<u,y> =Re(y H u) represents the real part of the inner product of vectors; (9) The dual problem of the optimization model in step (8) can be expressed as an equivalent positive definite programming problem that is easy to solve: in, Q represents generalized inequality. i i = 0, 1, ..., 2d M This represents the sum of elements at different positions in matrix Q; specifically, it defines a matrix H(θ) = b * (θ)b T (θ), The (m, n)th element of H(θ) is Where d m ={d j -d i |i, j=1,..M; m=(i-1)M+j}, d n ={d j -d i |i, j=1,..M; n=(i-1)M+j}; Q i It is the sum of a subset of elements in matrix Q, whose positions satisfy the condition that the exponential parts of elements at the same position in matrix H(θ) are equal in length. n -d m =i; The above semidefinite programming optimization problem can be solved using MATLAB's CVX toolbox; (10) Construct a dual polynomial and obtain the target orientation estimate through spectral peak search. in, The optimal solution is obtained by solving the semidefinite programming form of the dual problem.

2. The meshless parameter estimation method based on sparse array amplitude and phase error correction according to claim 1, characterized in that, The sparse array structure described in step (1) does not require a non-porous differential continuous segment. The array types applicable to this invention are not limited to common coprime arrays or nested arrays.

3. The meshless parameter estimation method based on sparse array amplitude and phase error correction according to claim 1, characterized in that, The atomic norm mentioned in step (6) yes The convex approximation.

4. The meshless parameter estimation method based on sparse array amplitude and phase error correction according to claim 1, characterized in that, The dual problem described in step (8) is obtained through Lagrange analysis.

5. The meshless parameter estimation method based on sparse array amplitude and phase error correction according to claim 1, characterized in that, The semidefinite programming form constraint described in step (9) is a sufficient but not necessary condition for the dual atomic norm form constraint in the dual problem of step (8).

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