Sparse array direction of arrival estimation method and system based on Vandermonde decomposition and reconstruction

Abstract

The present application discloses a sparse array DOA estimation method and system based on Vandermonde decomposition and reconstruction, which relates to the field of array signal processing technology. The method includes constructing a covariance matrix completion optimization model based on sparse and uniform linear array signals, and performing Vandermonde decomposition using the characteristics of uniform linear arrays; updating the covariance matrix completion model by introducing the rank function in the nuclear norm optimization model; introducing auxiliary variables, converting the model into a solvable optimization problem, and solving it using the alternating direction multiplier method to obtain the optimal estimate. Finally, the DOA is estimated using a root-finding multiple signal classification algorithm. The present application uses a sparse array antenna to estimate the DOA of an incident signal, achieving better spatial resolution and maximizing the use of the additional degrees of freedom provided by the sparse array.

Description

Technical Field

[0001] The present application relates to the technical field of array signal processing, and in particular to a sparse array direction of arrival estimation method and system based on Vandermonde decomposition and reconstruction. Background Art

[0002] At present, direction of arrival (DOA) estimation using sensor arrays plays a vital role in many fields such as radar, wireless communication, acoustics and speech. According to the Nyquist sampling theorem, uniform linear arrays are the most commonly used configuration for direction of arrival estimation. In recent years, sparse arrays, including coprime arrays and nested arrays, have received widespread attention. These arrays provide larger apertures than uniform linear arrays with the same number of sensors, thereby enhancing spatial resolution. In addition, sparse arrays can break through the limitations of degrees of freedom. In order to utilize the additional degrees of freedom provided by sparse arrays, the difference coarray of sparse arrays is used for direction of arrival estimation, which is also called a virtual array. Considering the entire virtual array, the sparse signal reconstruction (SSR) algorithm realizes the direction of arrival estimation by constructing a signal sparse reconstruction problem.

[0003] However, the sparse signal reconstruction (SSR) algorithm suffers from basis mismatch problem and cannot maximize the use of the extra degrees of freedom provided by sparse arrays. The virtual array interpolation (VAI) algorithm and structured Nyquist correlation reconstruction (SNCR) algorithm may not give satisfactory estimation performance for signal sources with small angular spacing. Summary of the invention

[0004] The purpose of this application is to provide a sparse array direction of arrival estimation method and system based on Vandermonde decomposition and reconstruction, which can use sparse array antennas to estimate the direction of arrival of incident signals, achieve better spatial resolution and maximize the use of additional degrees of freedom provided by the sparse array.

[0005] To achieve the above objectives, this application provides the following solutions:

[0006] In a first aspect, the present application provides a sparse array direction of arrival estimation method based on Vandermonde decomposition and reconstruction, comprising:

[0007] Acquire array received signals; the array received signals include sparse array signals and uniform linear array signals; the sparse array signals are composed of a number of far-field narrowband and unrelated signals incident on the sparse array from any direction; the uniform linear array signals are signals received by an assumed uniform linear array; the assumed uniform linear array is obtained by transforming the sparse array through an interpolation method.

[0008] A covariance matrix completion optimization model is constructed according to the sparse array signal and the uniform linear array signal in the array received signal; the covariance matrix completion optimization model is a matrix completion model established by utilizing the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array.

[0009] With respect to the covariance matrix completion optimization model, a nuclear norm is introduced, and the rank function existing in the covariance matrix completion optimization model is replaced to obtain an updated covariance matrix completion optimization model.

[0010] For the updated covariance matrix completion optimization model, auxiliary variables are introduced, and the updated covariance matrix completion optimization model is transformed into an equivalent form to obtain a solvable optimization problem.

[0011] The solvable optimization problem is solved by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal.

[0012] According to the optimal estimated value of the covariance matrix of the sparse array signal, a root-finding multiple signal classification algorithm is used to perform direction of arrival estimation.

[0013] In a second aspect, the present application provides a sparse array direction of arrival estimation system based on Vandermonde decomposition and reconstruction, comprising:

[0014] A signal acquisition module is used to acquire array reception signals; the array reception signals include sparse array signals and uniform linear array signals; the sparse array signals are composed of a number of far-field narrowband and unrelated signals incident on the sparse array from any direction; the uniform linear array signals are signals received by an assumed uniform linear array; the assumed uniform linear array is obtained by converting the sparse array through an interpolation method.

[0015] A model building module is used to build a covariance matrix completion optimization model according to the sparse array signal and the uniform linear array signal in the array received signal; the covariance matrix completion optimization model is a matrix completion model established by utilizing the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array.

[0016] The first model optimization module is used to introduce a nuclear norm for the covariance matrix completion optimization model, replace the rank function existing in the covariance matrix completion optimization model, and obtain an updated covariance matrix completion optimization model.

[0017] The second model optimization module is used to introduce auxiliary variables for the updated covariance matrix completion optimization model, transform the updated covariance matrix completion optimization model into an equivalent form, and obtain a solvable optimization problem.

[0018] The calculation module is used to solve the solvable optimization problem by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal.

[0019] The direction of arrival estimation module is used to perform direction of arrival estimation using a root-finding multiple signal classification algorithm according to the optimal estimate of the covariance matrix of the sparse array signal.

[0020] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0021] The present application provides a sparse array direction of arrival estimation method and system based on Vandermonde decomposition reconstruction, the method comprising the following steps: firstly, an array receiving signal is obtained, the array receiving signal is composed of a sparse array signal and a uniform linear array signal; wherein the sparse array signal is formed by a number of far-field narrowband and unrelated signals incident on the sparse array from any direction; the uniform linear array signal is an assumed uniform linear array receiving signal; the assumed uniform linear array is obtained by converting the sparse array using an interpolation method. Based on the sparse array signal and the uniform linear array signal in the array receiving signal, a covariance matrix completion optimization model is constructed; the covariance matrix completion optimization model is a matrix completion model established by using the Vandermonde decomposition characteristics of the covariance matrix of the uniform linear array. For the covariance matrix completion optimization model, a nuclear norm is introduced, and the rank function in the model is replaced to obtain an updated covariance matrix completion optimization model. Subsequently, for the updated covariance matrix completion optimization model, auxiliary variables are introduced and converted into an equivalent form, thereby obtaining a solvable optimization problem. The solvable optimization problem is solved by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal. Finally, a root-finding multiple signal classification algorithm is used to estimate the direction of arrival of the incident signal using a sparse array antenna, thereby achieving a better spatial resolution and maximizing the use of the additional degrees of freedom provided by the sparse array. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0023] Figure 1 A schematic flow chart of a sparse array direction of arrival estimation method based on Vandermonde decomposition and reconstruction provided in one embodiment of the present application.

[0024] Figure 2 The RMSE provided in an embodiment of the present application varies with the parameter 's change curve graph.

[0025] Figure 3 The RMSE provided in an embodiment of the present application varies with the parameter 's change curve graph.

[0026] Figure 4 The incident angle provided in one embodiment of the present application is and Schematic diagram of the spatial spectrum of the test algorithm.

[0027] Figure 5 Schematic diagram of the spatial spectrum of the test algorithm when the number of signal sources provided in an embodiment of the present application is 9. (a) Schematic diagram of the spatial spectrum of the SSR algorithm; (b) Schematic diagram of the spatial spectrum of the VAI algorithm; (c) Schematic diagram of the spatial spectrum of the SNCR algorithm; (d) Schematic diagram of the spatial spectrum of the algorithm of the present application.

[0028] Figure 6 Schematic diagram of the spatial spectrum of the test algorithm when the number of signal sources provided in an embodiment of the present application is 12. (a) Schematic diagram of the spatial spectrum of the SSR algorithm; (b) Schematic diagram of the spatial spectrum of the VAI algorithm; (c) Schematic diagram of the spatial spectrum of the SNCR algorithm; (d) Schematic diagram of the spatial spectrum of the algorithm of the present application.

[0029] Figure 7 A schematic diagram of the structure of a sparse array direction of arrival estimation system based on Vandermonde decomposition and reconstruction is provided in one embodiment of the present application. DETAILED DESCRIPTION

[0030] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0031] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0032] Embodiment 1

[0033] like Figure 1 As shown, this embodiment provides a sparse array direction of arrival estimation method based on Vandermonde decomposition and reconstruction, including:

[0034] Step 101: Acquire array received signals; the array received signals include sparse array signals and uniform linear array signals; the sparse array signals are composed of a number of far-field narrowband and unrelated signals incident on the sparse array from any direction; the uniform linear array signals are signals received by an assumed uniform linear array; the assumed uniform linear array is obtained by converting the sparse array through an interpolation method.

[0035] Step 102: constructing a covariance matrix completion optimization model according to the sparse array signal and the uniform linear array signal in the array received signal; the covariance matrix completion optimization model is a matrix completion model established by utilizing the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array.

[0036] Step 103: For the covariance matrix completion optimization model, a nuclear norm is introduced to replace the rank function existing in the covariance matrix completion optimization model to obtain an updated covariance matrix completion optimization model.

[0037] Step 104: For the updated covariance matrix completion optimization model, an auxiliary variable is introduced, and the updated covariance matrix completion optimization model is converted into an equivalent form to obtain a solvable optimization problem.

[0038] Step 105: using an alternating direction multiplier method to solve the solvable optimization problem, and obtaining an optimal estimate of the covariance matrix of the sparse array signal.

[0039] Step 106: According to the optimal estimate of the covariance matrix of the sparse array signal, a root-finding multiple signal classification algorithm is used to perform direction of arrival estimation.

[0040] In order to deeply analyze and understand the structural characteristics of the array covariance matrix, an optimization model needs to be established. This model will better understand the relationship between the various sensors in the array and the propagation characteristics of the signal. By optimizing the model, the performance of the array can be further optimized and the accuracy and efficiency of signal processing can be improved.

[0041] In the layout design of array sensors, the Nyquist sampling position is a key concept. These positions refer to a set of sensor positions with fixed intervals, and their intervals are usually determined by the half-wavelength of the incident signal. This sampling method ensures uniform sampling of the signal in space, thereby avoiding the occurrence of aliasing. By accurately setting the Nyquist sampling position, it can be ensured that the array can effectively capture the details of the signal, thereby improving the accuracy and reliability of signal processing.

[0042] Specifically, the Nyquist sampling position Indicates a fixed interval A set of sensor locations, where is half the wavelength of the incident signal. It can be expressed as:

[0043] (1).

[0044] In the formula, yes A subset of .

[0045] In some embodiments, when performing steps 101-102, the specific steps may be as follows:

[0046] Acquire array received signals; the array received signals include sparse array signals and uniform linear array signals; the sparse array signals are composed of a number of far-field narrowband and unrelated signals incident on the sparse array from any direction; the uniform linear array signals are signals received by an assumed uniform linear array; the assumed uniform linear array is obtained by transforming the sparse array through an interpolation method.

[0047] According to the sparse array signal and the uniform linear array signal in the array received signal, the covariance matrix of the sparse array signal and the uniform linear array signal is calculated.

[0048] By utilizing the characteristics of Vandermonde decomposition, the covariance matrix of the uniform linear array is decomposed into a Vandermonde matrix and the corresponding coefficient vector.

[0049] The covariance matrix completion optimization model is constructed according to the Vandermonde matrix and the coefficient vector.

[0050] Specifically, when acquiring the signal, it is assumed that A far-field narrowband and uncorrelated signal from the direction When incident on a sparse array, the sparse array signal in the array received signal can be modeled as:

[0051] (2).

[0052] in, for the summation symbol; ,express dimensional signal vector, l ∈(1, L ); t is the sampling time; Represents a transpose operation; is a dimensional independent and identically distributed complex-valued additive Gaussian noise vector with zero mean and covariance matrix ; is the noise power; express dimensional array manifold matrix; The angle is of dimensional steering vector, expressed as:

[0053] (3).

[0054] in, represents the wavelength of the incident signal; j is an imaginary unit; e is the base of natural logarithms.

[0055] Based on the obtained sparse array , its covariance matrix is ​​expressed as:

[0056] (4).

[0057] in, It means to find the mathematical expectation; represents the signal covariance matrix; It means to generate a diagonal matrix with the elements of a vector as diagonal elements; Indicates l The power of a signal, is the noise term, H represents the conjugate transpose.

[0058] But in practice, the sparse array covariance matrix It can only be estimated through the sample covariance matrix, expressed as:

[0059] (5).

[0060] in, Indicates the number of time domain samples (snapshots).

[0061] In order to satisfy the Nyquist spatial sampling criterion (i.e., the sampling frequency must be at least twice the highest frequency component of the signal) while maintaining the aperture of the sparse array unchanged, a Nyquist spatial filling process is implemented to generate a hypothetical uniform linear array. Interpolation into a continuous uniform array , which solves the problem at position The problem of lack of actual sampling data is Include In theory, assuming a uniform linear array The received signal can be modeled as:

[0062] (6).

[0063] in, yes dimensional noise vector; express dimensional array manifold matrix; The angle is of dimensional steering vector, expressed as:

[0064] (7).

[0065] Among them, the sparse array The received signal Can be It is expressed as:

[0066] (8).

[0067] in, is a A dimensional compressed matrix containing only 0s and 1s is No. Elements and No. When elements overlap, ,express No. Line The first column is 1, and all other elements are 0. For example, for a sparse array and a uniform linear array , we can get:

[0068] (9).

[0069] Among them, the assumed uniform linear array The covariance matrix of is expressed as:

[0070] (10).

[0071] in, express The noise-free covariance matrix of .

[0072] dimensional sparse array covariance matrix Can be expanded to The assumed uniform array covariance matrix of as follows:

[0073] (11).

[0074] in, is a dimensional diagonal matrix. It can be noted that for all of , the augmented covariance matrix No. Row and The columns are zero vectors. Therefore, the problem of filling in the hypothetical sensors at the Nyquist sampling locations can be transformed into completing the covariance matrix The problem is to reconstruct the covariance matrix .

[0075] Hankel matrix transformation operator , indicating that Dimensional vector Mapping dimensional Hankel matrix ,Right now ,express No. elements equal to No. Line Elements of a column. The adjoint operator , indicating that Dimensional matrix Mapping Dimensional vector ,Right now ,express No. elements equal to The sum of the row and column subscripts is equal to The sum of the corresponding elements. , which means that the matrix Columns are extracted, i.e. ,in Representation Matrix No. List. The adjoint operator Indicates that Dimensional vector Mapping The dimension matrix is ​​defined as:

[0076] (12).

[0077] in, express The values ​​of are .

[0078] make ,in In theory, the noise-free array covariance matrix is It can be broken down into:

[0079] (13).

[0080] in, ,for dimensional matrix. The above decomposition is called the Vandermonde decomposition of a semi-positive definite Toeplitz matrix. is a Vandermonde matrix, and No. The columns correspond to No. Multiply a column by a factor Therefore, in this embodiment, It is called a Vandermonde structure.

[0081] Based on the above matrix decomposition, the noise term is excluded , for the matrix reconstruction, which involves finding a Vandermonde structure Dimensional Matrix , so that , but it is difficult to directly add constraints to the Vandermonde structure. Represents a zero column dimensional matrix, obviously ,in represents the rank of the matrix. When it has a Vandermonde structure, Therefore, minimizing Can encourage Vandermonde structures. In addition, is a Hermitian, Toeplitz matrix, and the available information in the augmented covariance is used as reference information for the reconstruction process. Therefore, the covariance matrix of the assumed uniform linear array can be reconstructed by solving the following optimization problem:

[0082] (14).

[0083] in, ,for dimensional matrix, Used to restrict the feasible domain of optimization variables, is the covariance matrix, which is a Hermitian and Toeplitz matrix with the first column being ; represents the Frobenius norm; the covariance fitting error is used to tolerate the noise term, and the regularization parameter is used to trade off the covariance fitting error and the rank function, is the Hankel matrix transformation operator, represents the wavelength of the incident signal, R represents the total number of columns, is the column operator, Indicates a A compressed matrix of dimension containing only 0 and 1, As shown in formula (5).

[0084] When executing steps 103-104, the specific steps may be as follows:

[0085] Since the rank function makes the minimization problem difficult to solve, the nuclear norm is introduced as a convex relaxation of the rank. The optimization problem (14) can be restated as:

[0086] (15).

[0087] in, represents the nuclear norm.

[0088] In practice, the number of signal sources is usually unknown. Therefore, the algorithm of this embodiment requires preset parameters To determine the matrix In order to solve the optimization problem conveniently, auxiliary variables are introduced and (15) is rewritten into the following equivalent form: (16).

[0089] In some embodiments, when executing step 105, the specific steps may be as follows:

[0090] The alternating direction multiplier method is used to solve the optimization problem, where the augmented Lagrangian function of the optimization problem (16) is:

[0091] (17).

[0092] In formula (17), represents the Lagrangian function; , and is the Lagrange multiplier; , represents the trace of a matrix; is the penalty parameter.

[0093] Next, this embodiment initializes the variables in the following manner .

[0094] First of all Perform eigenvalue decomposition, that is . Then, initialize the variable to ,in Indicates taking No. List to List.

[0095] The alternating direction multiplier method uses the following iterative scheme to solve the optimization problem (16):

[0096] (18).

[0097] (19).

[0098] (20).

[0099] (twenty one).

[0100] (twenty two).

[0101] (twenty three).

[0102] (twenty four).

[0103] 1) Yes Update as follows:

[0104] For update , we need to solve subproblem (18), which can be expressed as:

[0105] (25).

[0106] Since the objective function of the optimization problem (25) is a convex function, we define The conjugate matrix of , let the objective function be about The partial derivative of is 0, which gives the following linear equations to solve the problem: (26).

[0107] According to the operator and The definition of can be obtained:

[0108] (27).

[0109] Formula (27), is a vector whose The elements are the Hankel matrices No. The number of elements on the anti-diagonal, represents the Hadamard product. According to the operator and The definition of can be obtained:

[0110] (28).

[0111] in, is a matrix whose columns are Therefore, formula (26) can be expressed as:

[0112] (29).

[0113] in, is the right side of equation (29). Therefore, we can get The closed-form solution for each row of is:

[0114] (30).

[0115] in, express dimensional identity matrix, Represents the matrix inversion operation.

[0116] 2) Yes To update, you can do as follows:

[0117] For update , sub-problem (19) can be expressed as:

[0118] (31).

[0119] Update Similarly, The closed-form solution of is expressed as:

[0120] (32).

[0121] 3) Yes To update, the specific steps are as follows:

[0122] For update , sub-problem (20) can be expressed as:

[0123] (33).

[0124] Since the objective function of the optimization problem (33) is a convex function, let it be The partial derivative of is 0, so the solution to the subproblem is as follows:

[0125] (34).

[0126] Specifically, the variables involved in formula (34) are defined as follows:

[0127] (35).

[0128] (36).

[0129] In the formula, , and for dimensional vector; , and Respectively , and No. elements; ; Indicates that all elements are 1 Dimensional matrix.

[0130] 4) Yes To update, you can do as follows:

[0131] Finally, to update , we can get by solving the following problem Solution:

[0132] (37).

[0133] The solution of problem (37) is expressed as follows through the soft threshold operator:

[0134] (38).

[0135] In this iterative algorithm, in order to accelerate convergence, this embodiment adopts The strategy gradually increases the parameters , after completing the iteration, the reconstructed covariance matrix is ​​obtained .

[0136] In some embodiments, when executing step 106, the specific steps may be as follows:

[0137] Multiple signal classification algorithm is used for direction of arrival estimation.

[0138] Specifically, after obtaining the reconstructed covariance matrix After that, the Root-MUSIC algorithm is used to estimate the direction of arrival. The spatial spectrum of the MUSIC is as follows:

[0139] (39).

[0140] In the formula express The noise subspace of represents the steering vector. By searching The largest peaks, the direction of arrival estimation result can be obtained, where the number of signal sources is It can be estimated using the Akaike Information Criterion (AIC), the Minimum Description Length (MDL), or the Gail Circle Method (GDE).

[0141] In summary, the entire technical solution of this application is summarized as follows:

[0142] The input is a sparse array receiving signal ; Output is the direction of arrival estimation result ; variables are initialized to .

[0143] when When , execute: Update according to formula (30) , Update according to formula (32) , Update according to formula (34) ,for , updated according to formula (38) ,for , updated according to formula (24) , Update according to formula (22) and formula (23) , as well as ; then Perform a root-finding multiple signal classification algorithm for direction-of-arrival estimation.

[0144] Specifically, the method involved in this application provides an embodiment, which is described in detail as follows:

[0145] Consider a coprime array of 7 physical sensors with the sensor positions In the following simulation experiments, for the algorithm of this application, the parameters are set to , , these parameters are selected based on practical experience. Since the number of signal sources is unknown, the parameters Also set to the maximum value The number of snapshots is set to .

[0146] The estimation accuracy of the tested algorithm is evaluated by the root mean square error (RMSE), which is defined as:

[0147] (40).

[0148] in, represents the number of Monte Carlo trials, represents the number of incident signals, Indicates The first experiment An estimated angle, Indicates The true angle of the incident signal.

[0149] First, the parameters and parameters The impact of the estimation accuracy of the technical method of the present application is tested. The angle of the incident signal is set to , the signal-to-noise ratio (SNR) is , perform 5000 Monte Carlo experiments, such as Figure 2 The RMSE shown varies with the parameters It can be observed that even if The number of signal sources is set larger than the actual number, and the estimation accuracy of the technical method of the present application is only slightly reduced. Then, the signal-to-noise ratio is set to , , and , perform 500 Monte Carlo experiments, such as Figure 3 As shown in the figure, under different signal-to-noise ratios, RMSE increases with the parameter Obviously, for different SNRs, there is an optimal parameter Minimize RMSE. In addition, the higher the signal-to-noise ratio, the more parameters are set to minimize RMSE. The bigger.

[0150] Next, the technical method of this application is compared with the sparse signal reconstruction (SSR) algorithm, the virtual array interpolation (VAI) algorithm, and the structured Nyquist correlation reconstruction (SNCR) algorithm for estimation performance. First, the resolution performance of the test algorithm is compared. Assume that two signal sources with a small angle interval are respectively and incident, their signal-to-noise ratio is .like Figure 4 , showing the spatial spectrum of the tested algorithm. Other algorithms cannot distinguish between these two closely spaced signal sources, while the technical method of the present application can distinguish them. The comparison results show that the technical method of the present application has better resolution performance.

[0151] Finally, the performance of the algorithm in estimating multiple signal sources is compared. Consider 9 signal sources evenly distributed in In the range, the signal-to-noise ratio is .like Figure 5 As shown, Figure 5 (a), (b), (c), and (d) show the spatial spectra of the tested algorithms. SSR produces several pseudo peaks in the spatial spectrum, while the other three algorithms are able to accurately estimate the DOA of the 9 signal sources. To evaluate the maximum achievable degree of freedom of the technical method of this application. Figure 6 As shown, Figure 6 (a), (b), (c), and (d) show the spatial spectra of the tested algorithms. Obviously, except for the SSR algorithm, all algorithms successfully identified all 12 signal sources. However, the VAI algorithm produced several estimation results that were obviously offset from the true direction, while the SNCR algorithm and the technical method of the present application achieved higher estimation accuracy.

[0152] Embodiment 2

[0153] like Figure 7 As shown, this embodiment provides a sparse array direction of arrival estimation system based on Vandermonde decomposition and reconstruction, including:

[0154] The signal acquisition module 701 is used to acquire array received signals; the array received signals include sparse array signals and uniform linear array signals; the sparse array signals are composed of a number of far-field narrow-band and unrelated signals incident on the sparse array from any direction; the uniform linear array signals are signals received by an assumed uniform linear array; the assumed uniform linear array is obtained by converting the sparse array through an interpolation method.

[0155] The model building module 702 is used to build a covariance matrix completion optimization model according to the sparse array signal and the uniform linear array signal in the array received signal; the covariance matrix completion optimization model is a matrix completion model established by utilizing the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array.

[0156] The first model optimization module 703 is used to introduce a nuclear norm into the covariance matrix completion optimization model, replace the rank function existing in the covariance matrix completion optimization model, and obtain an updated covariance matrix completion optimization model.

[0157] The second model optimization module 704 is used to introduce auxiliary variables into the updated covariance matrix completion optimization model, and convert the updated covariance matrix completion optimization model into an equivalent form to obtain a solvable optimization problem.

[0158] The calculation module 705 is used to solve the solvable optimization problem by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal.

[0159] The direction of arrival estimation module 706 is used to perform direction of arrival estimation using a root-finding multiple signal classification algorithm according to the optimal estimate of the covariance matrix of the sparse array signal.

[0160] In summary, this application has the following technical effects:

[0161] The sparse signal reconstruction (SSR) algorithm in the prior art has basis mismatch, poor estimation performance when the number of signal sources exceeds the number of sensors in the sparse array, and the virtual array interpolation (VAI) algorithm and structured Nyquist correlation reconstruction (SNCR) algorithm may not be able to distinguish different incident signals for signal sources with small angular intervals. The present application can achieve better spatial resolution and maximize the use of the additional degrees of freedom provided by the sparse array when the angular interval of the incident signal is small and the number of signal sources exceeds the number of sensors in the sparse array.

[0162] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0163] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A sparse array direction of arrival estimation method based on Vandermonde decomposition and reconstruction, characterized in that: include: Acquire array receiving signals; the array receiving signals include sparse array signals and uniform linear array signals; The sparse array signal is composed of a number of far-field narrow-band and unrelated signals incident on the sparse array from any direction; the uniform linear array signal is a signal received by an assumed uniform linear array; the assumed uniform linear array is obtained by converting the sparse array through an interpolation method; According to the sparse array signal and the uniform linear array signal in the array received signal, a covariance matrix completion optimization model is constructed; the covariance matrix completion optimization model is a matrix completion model established by utilizing the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array; The steps of constructing the covariance matrix completion optimization model include: Calculating the covariance matrix of the sparse array signal and the uniform linear array signal; By using the characteristics of Vandermonde decomposition, the covariance matrix of the uniform linear array is decomposed into a Vandermonde matrix and the corresponding coefficient vector; Constructing the covariance matrix completion optimization model according to the Vandermonde matrix and the coefficient vector; The formula expression of the covariance matrix completion optimization model is: ； in, ,for dimensional matrix, Used to restrict the feasible domain of optimization variables, is the covariance matrix, which is a Hermitian and Toeplitz matrix with the first column being ; represents the Frobenius norm, is the Hankel matrix transformation operator, represents the wavelength of the incident signal, Indicates the total number of columns, r For the r List, is the column operator, Indicates a A compressed matrix of dimension containing only 0 and 1, is the sample covariance matrix; For the covariance matrix completion optimization model, a nuclear norm is introduced to replace the rank function existing in the covariance matrix completion optimization model to obtain an updated covariance matrix completion optimization model; For the updated covariance matrix completion optimization model, auxiliary variables are introduced to transform the updated covariance matrix completion optimization model into an equivalent form, and a solvable optimization problem is obtained; For the updated covariance matrix completion optimization model, auxiliary variables are introduced to transform the updated covariance matrix completion optimization model into an equivalent form to obtain a solvable optimization problem, including: The formula expression of the solvable optimization problem is: ； in, ; The solvable optimization problem is solved by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal; According to the optimal estimated value of the covariance matrix of the sparse array signal, a root-finding multiple signal classification algorithm is used to perform direction of arrival estimation.

2. The method for sparse array direction of arrival estimation based on Vandermonde decomposition and reconstruction according to claim 1, characterized in that: The formula expression of the sparse array signal is: ； in, represents the summation symbol; ,express dimensional signal vector, l ∈(1, L ); Indicates sampling time; Represents a transpose operation; Indicates a dimensional independent and identically distributed complex-valued additive Gaussian noise vector; express dimensional array manifold matrix; The angle is of Dimensional guide vector.

3. The method for sparse array direction of arrival estimation based on Vandermonde decomposition and reconstruction according to claim 2, characterized in that: The formula expression of the uniform linear array signal is: ； in, yes dimensional noise vector; express dimensional array manifold matrix; The angle is of Dimensional guide vector.

4. The method for sparse array direction of arrival estimation based on Vandermonde decomposition and reconstruction according to claim 3, characterized in that: Calculating the covariance matrix of the sparse array signal specifically includes: According to the formula ; Calculate the covariance matrix of the sparse array signal; in, It means to find the mathematical expectation; represents the signal covariance matrix; It means to generate a diagonal matrix with the elements of a vector as diagonal elements; Indicates l The power of a signal, is the noise term, H represents the conjugate transpose.

5. The method for sparse array direction of arrival estimation based on Vandermonde decomposition and reconstruction according to claim 4, characterized in that: Calculating the covariance matrix of the uniform linear array signal specifically includes: According to the formula Calculating the covariance matrix of the uniform linear array signal; in, express The noise-free covariance matrix of .

6. The method for sparse array direction of arrival estimation based on Vandermonde decomposition and reconstruction according to claim 5, characterized in that: The root-finding multiple signal classification algorithm is specifically: ； in, express The noise subspace of Represents the steering vector.

7. A sparse array direction of arrival estimation system according to the sparse array direction of arrival estimation method based on Vandermonde decomposition and reconstruction according to any one of claims 1 to 6, characterized in that: include: A signal acquisition module, used to acquire array received signals; the array received signals include sparse array signals and uniform linear array signals; The sparse array signal is composed of a number of far-field narrow-band and unrelated signals incident on the sparse array from any direction; the uniform linear array signal is a signal received by an assumed uniform linear array; the assumed uniform linear array is obtained by converting the sparse array through an interpolation method; A model building module, used to build a covariance matrix completion optimization model according to the sparse array signal and the uniform linear array signal in the array received signal; the covariance matrix completion optimization model is a matrix completion model established by using the characteristics of the Vandermonde decomposition of the covariance matrix of the uniform linear array; A first model optimization module is used to introduce a nuclear norm for the covariance matrix completion optimization model, replace the rank function existing in the covariance matrix completion optimization model, and obtain an updated covariance matrix completion optimization model; The second model optimization module is used to introduce auxiliary variables for the updated covariance matrix completion optimization model, convert the updated covariance matrix completion optimization model into an equivalent form, and obtain a solvable optimization problem; A calculation module, used for solving the solvable optimization problem by using an alternating direction multiplier method to obtain an optimal estimate of the covariance matrix of the sparse array signal; The direction of arrival estimation module is used to perform direction of arrival estimation using a root-finding multiple signal classification algorithm according to the optimal estimate of the covariance matrix of the sparse array signal.

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