A sparse low-rank decomposition DOA estimation method based on virtual array interpolation

Abstract

The present application relates to the technical field of signal processing, in particular to a kind of sparse low-rank decomposition DOA estimation method based on virtual array interpolation, establish receiving signal model;The sampling covariance matrix of receiving signal model is calculated, and vectorization operation is carried out to sampling covariance matrix, since there is hole for a non-uniform virtual array in virtual array, virtual sensor is inserted at the hole position of virtual array, form virtual uniform array, reconstruct virtual uniform array receiving signal model as Toeplitz matrix, and carry out low-rank and sparse matrix decomposition into signal covariance matrix and noise covariance matrix;Signal covariance matrix is reconstructed using low-rank matrix recovery theory, and the expected signal covariance matrix is recovered, it is carried out eigenvalue decomposition using MUSIC algorithm, and the target DOA is estimated.This application makes full use of all virtual array element information, avoids the inaccuracy of DOA estimation caused by information loss.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and in particular to a sparse low-rank decomposition DOA estimation method based on virtual array interpolation. Background Technology

[0002] Direction of Arrival (DOA) estimation, a major research technique in array signal processing, is primarily used to obtain the spatial locations of multiple desired signals in the airspace, and has broad application prospects in both military and civilian fields. Currently, methods for DOA estimation on coprime arrays can be mainly divided into two categories: sparse reconstruction methods based on compressed sensing and subspace methods. Sparse reconstruction methods mainly utilize the sparsity of the signal source relative to the entire space to design and solve an optimization problem. One approach involves overcompletely representing the spatial power spectrum using predefined grid points, constructing a data fitting model based on sparsity theory, and using the Least Absolute Shrinkage and Selection Operator (Lasso) method for sparse signal recovery. However, this can lead to basis mismatch problems, resulting in estimation errors when estimating the direction of arrival. One of the main methods in the subspace class is the virtual domain-based DOA estimation method. The core idea of ​​this method is to perform vectorization operation on the virtual array data model, obtain a full-rank covariance matrix through spatial smoothing, and then introduce the MUSIC algorithm for DOA estimation. This method can effectively increase the degrees of freedom by taking advantage of the structural advantages of sparse arrays, but it only uses the continuous part of the virtual array, resulting in the loss of some information. Summary of the Invention

[0003] The purpose of this invention is to solve the basis mismatch problem caused by sparse reconstruction-based DOA estimation methods and the problem of insufficient utilization of virtual array element data by subspace-based methods. It proposes a sparse low-rank decomposition DOA estimation method based on virtual array interpolation, which makes full use of all virtual array element information and avoids inaccurate DOA estimation caused by missing information.

[0004] To achieve the above objectives, the technical solution adopted is:

[0005] A sparse low-rank decomposition DOA estimation method based on virtual array interpolation, comprising:

[0006] Establish a received signal model;

[0007] Calculate the sampling covariance matrix of the received signal model and perform vectorization on the sampling covariance matrix, due to the virtual array. The existence of holes is a non-uniform virtual array, in the virtual array Virtual sensors are inserted at the locations of the holes to form a virtual uniform array.

[0008] For virtual uniform array The received signal model is reconstructed into a Toeplitz matrix, and then decomposed into a low-rank and sparse matrix into a signal covariance matrix and a noise covariance matrix.

[0009] The signal covariance matrix is ​​reconstructed using low-rank matrix recovery theory, and the desired signal covariance matrix is ​​obtained. The MUSIC algorithm is then used to perform eigenvalue decomposition on the recovered covariance matrix, and the target DOA is estimated.

[0010] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, it is further assumed that K independent far-field narrowband signals come from different directions θ. k If (k = 1, 2, ..., K) are incident on a coprime array, then the first... The received signal of each array element at time t is represented as:

[0011]

[0012] Among them, s k (t) represents the original reference signal. The direction of arrival is θ k The array manifold of the signal. For the l i Zero-mean Gaussian white noise on each element;

[0013] The expression for the received signal model is:

[0014]

[0015] in, Let s(t) be the array manifold matrix, and s(t) be the original reference signal. It is zero-mean Gaussian white noise.

[0016] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, the formula for calculating the sampling covariance matrix of the received signal model is as follows:

[0017]

[0018] Where T represents the number of snapshots, (·) H Represents the conjugate transpose of a matrix;

[0019] Vectorizing the above equation yields:

[0020]

[0021] Among them, the virtual array manifold matrix As the guide vector, For the conjugate of the guide vector, Let be the incident signal power vector. Let I be the noise power, and I be the identity matrix.

[0022] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, further, the virtual uniform array... The received signal model is represented as follows:

[0023]

[0024] Among them, <·> l The received signal at the l-th virtual array element. This indicates that the l-th virtual array element is in the set But not in the collection middle.

[0025] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, the reconstructed Toeplitz matrix is ​​further as follows:

[0026]

[0027] in,

[0028] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, the Toeplitz matrix is ​​further decomposed into a signal covariance matrix and a noise covariance matrix:

[0029]

[0030] Where I is L sub ×L sub For an identity matrix, typically the number of sources is much smaller than the number of elements, then rank(R) can be obtained. s )=K≤L sub That is, the signal covariance matrix R s It has low-rank characteristics; the noise covariance matrix R n R is a full-rank matrix whose elements are all zero except for the diagonal elements, i.e., the noise covariance matrix. n It has sparse properties, and based on this, the signal covariance matrix is ​​reconstructed using low-rank matrix recovery theory.

[0031] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, the signal covariance matrix reconstructed using low-rank matrix recovery theory further includes:

[0032] Based on the covariance matrix T of the expected signals and the expected noise covariance matrix T n The low-rank and sparse properties are used to construct the following optimization problem:

[0033]

[0034] subject to R T =C○T s +T n

[0035] T s ≥0,T n ∈D +

[0036] Where rank(·) is the rank of the matrix, ||·||0 is the l0 norm, λ is the regularization parameter balancing the l0 norm and the error, and C is a matrix R. T Binary matrices of the same dimension have the same value in R. T The value at non-zero positions is 1, and the value at all other positions is 0. + T is a positive semi-definite diagonal matrix; s It is a matrix that satisfies the Hermitian Toplitz structure;

[0037] Since both the rank minimization problem and the l0 norm are difficult NP problems, the above equation is relaxed to the following form:

[0038]

[0039] subject to T s ≥0,T n ∈D +

[0040] Where tr(·) is the trace of the matrix, the covariance matrix T of the desired signal can be recovered. s .

[0041] According to the sparse low-rank decomposition DOA estimation method based on virtual array interpolation of the present invention, the MUSIC algorithm is further used to estimate the covariance matrix T of the desired signal. s The eigenvalues ​​are further divided into a signal subspace and a noise subspace, as expressed by:

[0042]

[0043] Among them, U s U n These are the signal subspace and the noise subspace, respectively, Σ s , Σ n K large eigenvalues ​​and L respectively subA diagonal matrix consisting of -K small eigenvalues; utilizing the orthogonality of the noise eigenvectors and the signal eigenvectors, the spatial spectral function is obtained:

[0044]

[0045] Where a(θ) is the steering vector, and as the angle θ changes, the value of the spatial spectral peak is the estimated value of the direction of arrival.

[0046] The beneficial effects achieved by adopting the above technical solution are:

[0047] The proposed sparse low-rank decomposition DOA estimation method based on virtual array interpolation employs virtual element interpolation technology. Virtual sensors are inserted at the locations of holes in the virtual array to form a virtual uniform array. This approach avoids discarding virtual elements containing holes, thus fully utilizing all virtual element information. A Toeplitz matrix is ​​constructed from the received signal model of the interpolated virtual uniform array, and a sparse and low-rank decomposition optimization problem is used to recover the missing data in the Toeplitz matrix. This achieves DOA estimation of the signal on a coprime array. The invention achieves higher estimation accuracy, and even higher estimation accuracy and resolution at low signal-to-noise ratios. Attached Figure Description

[0048] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings of the embodiments of the present invention will be briefly described below. The drawings are merely illustrative of some embodiments of the present invention and are not intended to limit the scope of the present invention to all embodiments.

[0049] Figure 1 This is a flowchart illustrating the sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to an embodiment of the present invention.

[0050] Figure 2 This is a spatial spectral distribution diagram of the method proposed in this embodiment of the invention;

[0051] Figure 3 This refers to the resolution of the method proposed in this embodiment of the invention;

[0052] Figure 4 The root mean square error of each method in the embodiments of the present invention varies with the signal-to-noise ratio;

[0053] Figure 5 The root mean square error of each method in the embodiments of the present invention varies with the number of snapshots. Detailed Implementation

[0054] The exemplary solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art.

[0055] like Figure 1 As shown, this embodiment discloses a sparse low-rank decomposition DOA estimation method based on virtual array interpolation, which includes the following steps:

[0056] Step S101: Establish the received signal model.

[0057] Assume K independent far-field narrowband signals originate from different directions θ k If (k = 1, 2, ..., K) are incident on a coprime array, then the first... The received signal of each array element at time t is represented as:

[0058]

[0059] Among them, s k (t) represents the original reference signal. The direction of arrival is θ k The array manifold of the signal. For the l i Zero-mean Gaussian white noise on each array element. Therefore, as shown in equation (1), the received signal of a single array element is expanded to the received signal on a coprime array, expressed as:

[0060]

[0061] The received signal can be represented as:

[0062]

[0063] The original reference signal can be represented as:

[0064] s(t)=[s1(t),s2(t),…,s K (t)] T (4)

[0065] Noise data can be represented as:

[0066]

[0067] The array manifold matrix can be represented as:

[0068]

[0069] Where the guiding vector It can be represented as:

[0070]

[0071] Step S102: Calculate the sampling covariance matrix of the received signal model, and perform a vectorization operation on the sampling covariance matrix. (Due to the virtual array...) The existence of holes is a non-uniform virtual array, in the virtual array Virtual sensors are inserted at the locations of the holes to form a virtual uniform array.

[0072] The formula for calculating the covariance matrix of the received signal model using formula (2) is as follows:

[0073]

[0074] in, Let I be the noise power, and let R be the (M+N-1)×(M+N-1) dimensional identity matrix. s Let be the signal covariance matrix.

[0075] In practical applications, the received signal model consists of T snapshots of sampled data. Since the number of snapshots is finite, formula (8) can be written as:

[0076]

[0077] Where T represents the number of snapshots, (·) H This represents the conjugate transpose of a matrix.

[0078] Performing a vector operation on formula (9) yields:

[0079]

[0080] Among them, the virtual array manifold matrix As the guide vector, For the conjugate of the guide vector, Let be the incident signal power vector. Let I be the noise power and I be the identity matrix. Steering vector. The values ​​of each element in the middle can be regarded as Then the set of virtual array element positions That is, the virtual array element positions are composed of the differences between the positions of the elements in the coprime array. Because there are gaps after subtracting the element positions, the virtual array at this point... It is a non-uniform virtual array. The common method for processing virtual arrays is to extract the continuous part of the non-uniform array as the processing object and discard the part containing holes. However, this actually discards some virtual array elements, resulting in missing information, which in turn affects the performance of subsequent DOA estimation.

[0081] This embodiment employs virtual array element interpolation technology in the virtual array. Virtual sensors are inserted at the locations of the holes to form a virtual uniform array. It's important to note that the virtual sensor inserted into the virtual array at this point is not a real device; therefore, it can be considered an inactive antenna element, and its received signal can be initialized to zero. (The virtual uniform array...) The remaining array elements are all virtual arrays. The array elements in the array receive signals, which is a virtual array. The received signal. Then the virtual uniform array The received signal model can be represented as:

[0082]

[0083] Among them, <·> l The received signal at the l-th virtual array element. This indicates that the l-th virtual array element is in the set But not in the collection middle.

[0084] Step S103: Apply the virtual uniform array The received signal model is reconstructed into a Toeplitz matrix, and then decomposed into a low-rank and sparse matrix into a signal covariance matrix and a noise covariance matrix.

[0085] Formula (11) can be viewed as a virtual uniform array. The rank deficiency problem caused by single-snapshot sampling data of the received signal can be solved by constructing a Toeplitz matrix. The matrix form is as follows:

[0086]

[0087] in,

[0088] The Toeplitz matrix at this point can also be considered as having a length of L. sub The covariance matrix constructed from the received data of the uniform array can be decomposed into a signal covariance matrix and a noise covariance matrix, i.e.:

[0089]

[0090] Where I is L sub ×L sub For an identity matrix, typically the number of sources is much smaller than the number of elements, then rank(R) can be obtained. s )=K≤L sub That is, the signal covariance matrix R sIt has low-rank characteristics; the noise covariance matrix R n R is a full-rank matrix whose elements are all zero except for the diagonal elements, i.e., the noise covariance matrix. n It exhibits sparse properties. Based on this, the signal covariance matrix can be reconstructed using low-rank matrix recovery theory.

[0091] Step S104: Reconstruct the signal covariance matrix using low-rank matrix recovery theory, recover the desired signal covariance matrix, perform eigenvalue decomposition on it using the MUSIC algorithm, and estimate the target DOA.

[0092] Based on the covariance matrix T of the expected signal s and the expected noise covariance matrix T n The low-rank and sparse properties are used to construct the following optimization problem:

[0093]

[0094] Where rank(·) is the rank of the matrix, ||·||0 is the l0 norm, λ is the regularization parameter balancing the l0 norm and the error, and C is a matrix R. T Binary matrices of the same dimension have the same value in R. T The value at non-zero positions is 1, and the value at all other positions is 0. + T is a positive semi-definite diagonal matrix; s It is a matrix that satisfies the Hermitian Toplitz structure.

[0095] Because the number of signal samples is limited in practical applications, matrix R T There is usually an estimation error, i.e., R T ≠C○T s +T n Based on the Frobenius norm, the following convex error model is constructed:

[0096]

[0097] Where δ is the covariance matrix T of the constrained desired signal during the optimization process. s Error parameters.

[0098] Based on the above convex error model, formula (14) can be expressed as:

[0099]

[0100] Since the rank minimization problem and the l0 norm in formula (16) are both difficult-to-solve NP problems, the above problem can be relaxed by the following convex relaxation:

[0101]

[0102] Due to matrix Rs Since it is a positive semi-definite matrix, formula (17) can be further expressed as:

[0103]

[0104] Where tr(·) is the trace of the matrix.

[0105] By solving the above optimization problem, the covariance matrix T of the desired signal can be recovered. s In practical applications, due to the influence of errors, the recovered desired signal covariance matrix T s Since the influence of noise cannot be completely eliminated, the MUSIC algorithm can be used to perform eigenvalue decomposition and further divide it into signal subspace and noise subspace, i.e.:

[0106]

[0107] Among them, U s U n These are the signal subspace and the noise subspace, respectively, Σ s , Σ n K large eigenvalues ​​and L respectively sub A diagonal matrix consisting of -K small eigenvalues; utilizing the orthogonality of the noise eigenvectors and the signal eigenvectors, the spatial spectral function is obtained:

[0108]

[0109] Where a(θ) is the steering vector, and as the angle θ changes, the value of the spatial spectrum peak can be obtained, which is the estimated value of the direction of arrival.

[0110] This invention performs simulation analysis on the proposed method from three aspects: spatial spectrum distribution, resolution, and direction-of-arrival estimation accuracy. A coprime array with M=5 and N=6 is set up, where the physical array element number is considered to be 10, and the virtual hole positions of the coprime array are {-23,-22,-21,-16,-11,11,16,21,22,23}. The noise environment is Gaussian white noise with zero mean and unit variance. It should be noted that since this invention decomposes the covariance matrix into a signal covariance matrix and a noise covariance matrix, the regularization parameter λ related to the noise covariance matrix should not be too large. In the actual simulation, the parameter λ in equation (18) is set to 0.1, and the grid precision in the peak search is set to 0.1°.

[0111] This invention first simulates the spatial spectrum of the DOA estimation method. The simulation uses 11 signals with incident angles uniformly distributed in the range [-57.33°, 55.44°], a signal-to-noise ratio of 5dB, and 500 snapshots. The spatial spectrum of this invention is shown below. Figure 2 As shown, from Figure 2 As can be seen from this, the present invention can accurately estimate 11 incident sources, and the spectral peaks are sharp.

[0112] This invention simulates the resolution of the DOA estimation method. The simulation uses two incident signals from different directions [31.5°, 34.6°], with a signal-to-noise ratio of 5dB and a snapshot count of 500. The resolution of this invention is as follows: Figure 3 As shown, this is verified by the spatial spectral distribution map. Figure 3 As can be seen, the present invention can accurately estimate two signal sources that are close to each other in terms of incident direction, with high resolution.

[0113] Finally, simulations were performed to verify the impact of different snapshot numbers and signal-to-noise ratios on the estimation accuracy of each method. The root mean square error (RMSE) was used to verify the direction-of-arrival estimation performance of all methods. The RMSE results were obtained by averaging L = 500 Monte Carlo experiments, as shown in the following formula:

[0114]

[0115] in For the j-th source estimated in the l-th Monte Carlo experiment, θ j Let j be the real source.

[0116] Suppose that 10 signals uniformly distributed in the interval [-57.33°, 55.44°] are incident on a coprime array. Figure 4 To investigate the impact of different signal-to-noise ratios (SNRs) on estimation accuracy at the same number of snapshots, the number of snapshots was set to 500, the SNR range was -10 to 20 dB, and the step size was 5 dB. Overall, the proposed method has a lower RMSE value than other methods, demonstrating superior estimation performance. Furthermore, at high SNRs, the proposed method's performance is similar to that of the McLeod algorithm. This is because the proposed method decomposes the matrix into a signal covariance matrix and a noise covariance matrix during low-rank matrix recovery, thereby reducing the impact of noise and thus outperforming other methods.

[0117] Figure 5 To investigate the impact of different snapshot numbers on estimation accuracy at the same signal-to-noise ratio, the signal-to-noise ratio was set to 5 dB, the snapshot number range was 10–3010, and the step size was 300. Figure 5 As can be seen, the RMSE curves of the proposed method and the comparison method gradually flatten out with the increase of the number of snapshots. Among them, due to the ability of the proposed method to suppress noise, its estimation performance is better than other methods at low number of snapshots.

[0118] Those skilled in the art will understand that all or part of the steps in the above methods can be implemented by a program instructing related hardware, and the program can be stored in a computer-readable storage medium, such as a read-only memory, a disk, or an optical disk. Optionally, all or part of the steps in the above embodiments can also be implemented using one or more integrated circuits. Accordingly, each module / unit in the above embodiments can be implemented in hardware or as a software functional module. This invention is not limited to any particular combination of hardware and software.

[0119] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit it. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A sparse low-rank decomposition DOA estimation method based on virtual array interpolation, characterized in that, Include: Step 1: Establish the received signal model; Step 2: Calculate the sampling covariance matrix of the received signal model and perform a vectorization operation on the sampling covariance matrix, since the virtual array The existence of holes is a non-uniform virtual array, in the virtual array Virtual sensors are inserted at the locations of the holes to form a virtual uniform array. Step 3: Apply the virtual uniform array The received signal model is reconstructed into a Toeplitz matrix, and the Toeplitz matrix is ​​decomposed into a signal covariance matrix and a noise covariance matrix: Where I is L sub ×L sub For an identity matrix, typically the number of sources is much smaller than the number of elements, then rank(R) can be obtained. s )=K≤L sub That is, the signal covariance matrix R s It has low-rank characteristics; the noise covariance matrix R n R is a full-rank matrix whose elements are all zero except for the diagonal elements, i.e., the noise covariance matrix. n It has sparse properties, and based on this, the signal covariance matrix is ​​reconstructed using low-rank matrix recovery theory; Step 4: Reconstruct the signal covariance matrix using low-rank matrix recovery theory to obtain the desired signal covariance matrix. Perform eigenvalue decomposition on this matrix using the MUSIC algorithm and estimate the target DOA. The reconstruction of the signal covariance matrix using low-rank matrix recovery theory includes: Based on the covariance matrix T of the expected signal s and the expected noise covariance matrix T n The low-rank and sparse properties are used to construct the following optimization problem: subject to R T =CoT s +T n T s ≥0,T n ∈D + Where rank(·) is the rank of the matrix, ||·||0 is the l0 norm, λ is the regularization parameter balancing the l0 norm and the error, and C is a matrix R. T Binary matrices of the same dimension have the same value in R. T The value at non-zero positions is 1, and the value at all other positions is 0. + T is a positive semi-definite diagonal matrix; s It is a matrix that satisfies the Hermitian Toplitz structure; Since both the rank minimization problem and the l0 norm are difficult NP problems, the above equation is relaxed to the following form: subject to T s ≥0,T n ∈D + Where tr(·) is the trace of the matrix, the covariance matrix T of the desired signal can be recovered. s .

2. The sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to claim 1, characterized in that, Assume K independent far-field narrowband signals originate from different directions θ k If (k = 1, 2, ..., K) are incident on a coprime array, then the first... The received signal of each array element at time t is represented as: Among them, s k (t) represents the original reference signal. The direction of arrival is θ k The array manifold of the signal. For the l i Zero-mean Gaussian white noise on each element; The expression for the received signal model is: in, Let s(t) be the array manifold matrix, and s(t) be the original reference signal. It is zero-mean Gaussian white noise.

3. The sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to claim 2, characterized in that, The formula for calculating the sampling covariance matrix of the received signal model is as follows: Where T represents the number of snapshots, (·) H Represents the conjugate transpose of a matrix; Vectorizing the above equation yields: Among them, the virtual array manifold matrix As the guide vector, For the conjugate of the guide vector, Let be the incident signal power vector. Let I be the noise power, and I be the identity matrix.

4. The sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to claim 3, characterized in that, Virtual uniform array The received signal model is represented as follows: Among them, <·> l The received signal at the l-th virtual array element. This indicates that the l-th virtual array element is in the set But not in the collection middle.

5. The sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to claim 4, characterized in that, The reconstructed Toeplitz matrix is ​​as follows: in, 6. The sparse low-rank decomposition DOA estimation method based on virtual array interpolation according to claim 5, characterized in that, The MUSIC algorithm is used to calculate the covariance matrix T of the desired signal. s The eigenvalues ​​are further divided into a signal subspace and a noise subspace, as expressed by: Among them, U s U n These are the signal subspace and the noise subspace, respectively, Σ s , Σ n K large eigenvalues ​​and L respectively sub A diagonal matrix consisting of -K small eigenvalues; utilizing the orthogonality of the noise eigenvectors and the signal eigenvectors, the spatial spectral function is obtained: Where a(θ) is the steering vector, and as the angle θ changes, the value of the spatial spectral peak is the estimated value of the direction of arrival.

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