Array signal direction-of-arrival phase recovery method and system based on meshless compressed sensing, computer and storage medium

Abstract

The invention discloses an array signal direction-of-arrival phase recovery method and system based on meshless compressed sensing, a computer and a storage medium, and the method comprises the steps: collecting the amplitude observation data of a target signal through a linear sampler, and forming an observation vector; a matrix variable is introduced, a linear relation between the observation vector and the matrix variable is established, and x is an array signal to be recovered; constructing a joint optimization model, wherein the model is a positive semidefinite programming model containing an atomic norm minimization constraint and a phase lifting constraint; performing numerical solution on the positive semidefinite programming model to obtain an optimal solution; and constructing a Toeplitz matrix according to the optimal solution to perform frequency estimation, and obtaining direction-of-arrival information of the target signal. According to the invention, under the condition of only signal amplitude measurement, phase information loss or serious distortion, high-precision recovery and direction-of-arrival estimation of array signals can be realized.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and in particular to a method, system, computer, and storage medium for recovering the direction-of-arrival phase of an array signal based on meshless compressed sensing in the event of missing phase information. Background Technology

[0002] Direction of arrival (DOA) estimation is a core problem in array signal processing and is widely used in radar, sonar, wireless communication and other fields. In actual measurements, due to motion errors, hardware limitations or noise interference, the phase information of the signal received by the array is often unreliable or completely missing, and only the amplitude (or power) information of the signal can be obtained. This constitutes the classic phase recovery problem.

[0003] Existing phase retrieval techniques, such as the PhaseLift method, convexify the problem by raising the signal vector to matrix space, but they do not utilize the inherent Vandermonde sparse structure of array signals in the frequency domain, resulting in low recovery accuracy when sampling is insufficient. Another class of sparse phase retrieval methods (such as CPRL), while introducing a sparsity prior, rely on discretizing the continuous frequency domain onto a predefined grid. When the actual signal frequency deviates from the grid points, a "grid error" occurs, leading to a significant performance degradation.

[0004] Therefore, there is an urgent need for a high-precision, robust DOA estimation method that can utilize the sparse structure of the signal and avoid grid errors when only amplitude observations are available. Summary of the Invention

[0005] The purpose of this invention is to provide a method, system, computer, and storage medium for array signal direction-of-arrival phase recovery based on meshless compressed sensing.

[0006] The technical solution to achieve the objective of this invention is: a method for recovering the direction-of-arrival phase of array signals based on meshless compressed sensing, comprising the following steps:

[0007] Step S1: Acquire amplitude observation data of the target signal using a linear sampler to form an observation vector;

[0008] Step S2: Introduce matrix variables Establish a linear relationship between the observation vector and the matrix variables, where x is the array signal to be recovered;

[0009] Step S3: Construct a joint optimization model, which is a positive semidefinite programming model that includes atomic norm minimization constraints and phase lifting constraints;

[0010] Step S4: Numerically solve the semidefinite programming model to obtain the optimal solution;

[0011] Step S5: Construct the Toeplitz matrix based on the optimal solution to estimate the frequency and obtain the direction of arrival information of the target signal.

[0012] Further, step S1: Acquire amplitude observation data of the target signal using a linear sampler to form an observation vector. ,in:

[0013]

[0014] In the formula, Let m be the m-th linear sampler, where m = 1, ..., M.

[0015] Further, step S2: Introduce matrix variables Establish a linear relationship between the observation vector and the matrix variables, expressed as:

[0016]

[0017] The above The stacked equations can be written in a concise form.

[0018]

[0019] In the formula, For sampling system Determined linear operators.

[0020] Further, step S3: Construct a joint optimization model, which is a positive semidefinite programming model containing atomic norm minimization constraints and phase lifting constraints, expressed as:

[0021]

[0022] In the formula, express The Toeplitz matrix corresponding to the vector is calculated as follows:

[0023]

[0024] at the same time , , Let N be a positive definite diagonal matrix, and N be the number of arrays. The trace of a matrix is ​​represented by the matrix trace. and This is an adjustable regularization parameter.

[0025] Further, step S4: Numerically solve the semidefinite programming model, specifically as follows:

[0026] The optimal solution is obtained by numerically solving a semidefinite programming problem using the CVX convex optimization toolkit and calling the semidefinite programming solver. .

[0027] Further, step S5: Construct a Toeplitz matrix based on the optimal solution to perform frequency estimation and obtain the direction of arrival information of the target signal. The specific method is as follows:

[0028] According to the optimal solution Constructing the Toeplitz matrix The ESPRIT algorithm is used to... Vandermonde decomposition is performed to estimate the frequency components of the target signal. That is, the direction of arrival information of the signal is obtained.

[0029] A gridless compressed sensing-based array signal direction-of-arrival phase recovery system includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the gridless compressed sensing-based array signal direction-of-arrival phase recovery method.

[0030] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the aforementioned array signal direction-of-arrival phase recovery method based on meshless compressed sensing.

[0031] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the aforementioned array signal direction-of-arrival phase recovery method based on meshless compressed sensing.

[0032] Compared with the prior art, the significant advantages of this invention are:

[0033] 1) Using the atomic norm minimization method to process array signal structures

[0034] This application employs the atomic norm minimization method from meshless sparse signal processing to leverage the special structure of the signal. The atomic norm can directly handle frequency components with continuous indices without discretization. Utilizing the Toeplitz matrix property, atomic norm minimization forms a positive semi-definite problem solvable in polynomial time, avoiding the infinite-dimensional programming problem under continuous indices.

[0035] 2) Construct a unified framework combining atomic norm minimization and phase lifting methods.

[0036] This application proposes a unified optimization framework that combines atomic norm minimization with phase lifting. This framework enables accurate recovery of sparse signals with Vandermonde structures even under conditions where only amplitude observations are available, phase information is missing, or the signal is severely affected by noise. This unified framework not only retains the advantages of phase lifting in problem transformation but also introduces atomic norm sparsity constraints, effectively utilizing the structural characteristics of the signal in direction-of-arrival estimation.

[0037] 3) Introducing matrix variables And based on matrix atomic norm optimization solution.

[0038] This application introduces matrix variables.

[0039]

[0040] The nonlinear phase retrieval problem in the original signal domain is transformed into a convex optimization problem in the matrix domain. Furthermore, atomic norm constraints are imposed on the matrix domain as an effective characterization of sparse priors.

[0041] 4) Add a noise regularization term to the objective function to improve robustness.

[0042] This application adds a noise regularization term to the objective function of the unified optimization framework. The least squares approach is used to constrain the observation residuals. This design effectively suppresses error accumulation when the observation signal is contaminated by noise, improves the robustness and stability of the algorithm, and ensures reliable direction-of-arrival estimation results even under low signal-to-noise ratio conditions. Attached Figure Description

[0043] Figure 1 This is a comparison chart of the recovery success rates of the method of this invention with those of PhaseLift and CPRL methods at discrete frequencies.

[0044] Figure 2 This is a comparison chart of the recovery success rates of the method of this invention with those of PhaseLift and CPRL methods at continuous frequencies.

[0045] Figure 3 This is a flowchart of the method of the present invention. Detailed Implementation

[0046] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0047] This invention discloses a method for direction-of-arrival (DOA) phase retrieval of array signals based on meshless compressed sensing. This method combines atomic norm minimization with phase boosting techniques to achieve sparse signal recovery in the continuous frequency domain, thereby enabling high-precision DOA estimation in low-sampling, high-noise environments. It includes:

[0048] Step 1, Data Acquisition: Obtain the amplitude observation vector of the signal;

[0049] Amplitude observation data of the target signal is acquired through a linear sampler to form an observation vector. ;

[0050] In many practical DOA applications, the signal exhibits frequency domain sparsity, and the signal to be recovered... This can be represented as a superposition of a small number of Vandermonde components:

[0051]

[0052] in Indicates the first The normalized frequency of each component, Indicates the first The complex amplitude of each component. Represents a Vandermonde vector, specifically as

[0053]

[0054] in This indicates the number of arrays, in typical direction-of-arrival (DOA) estimation applications. signal It satisfies the sparsity constraint. Assume the signal... Given a known linear sampler If sampling is performed, and only samples of squared amplitude are collected, then

[0055]

[0056] in, This represents the number of observed samples.

[0057] Step 2, Phase Boosting: Introducing matrix variables to linearize the nonlinear phase recovery problem;

[0058] PhaseLift is a classic phase recovery method that recovers the signal vector. Promoted to matrix variable , established and A linear mapping between them. Define a new matrix variable.

[0059]

[0060] Transform the observation equation into a relation Linear constraints:

[0061]

[0062] The above The equations can be stacked and written in a concise form. ,in For sampling system Determined linear operators.

[0063] When implementing PhaseLift The sparse pattern will be directly passed to Therefore, it is only necessary to... Sparsity optimization is performed on the above. This application uses the matrix atomic norm minimization method in meshless sparse signal processing to process continuous frequencies, and simultaneously optimizes the atomic norm minimization and phase lifting together.

[0064] Step 3, Optimize modeling steps: Construct a semidefinite programming model that integrates atomic norm sparsity constraints, phase boosting constraints, and noise regularization terms;

[0065] Phase lifting optimization modeling is performed in the matrix domain, while atomic norm constraints and noise constraints are applied, resulting in a positive semidefinite programming (SDP) form.

[0066]

[0067] in, express The Toeplitz matrix corresponding to the vector is calculated as follows:

[0068]

[0069] at the same time , , For a positive definite diagonal matrix, The frequency information of the signal can be obtained by performing Vandermonde decomposition. Represents the trace of a matrix. This comes from the phase lifting method, scalar This is an adjustable parameter. This factor takes into account signal noise and scalars. This is an adjustable parameter.

[0070] Step 4, Numerical Solution: Solve the semidefinite programming model to obtain the optimal variables;

[0071] Using the convex optimization toolkit CVX, the SDP solver is invoked to obtain the optimal solution.

[0072]

[0073] In addition to using the SDP solver in the CVX toolkit, corresponding fast algorithms can be developed, such as Projection Gradient Descent (PGD) and Alternating Direction Method of Multipliers (ADMM).

[0074] Step 5, Parameter Recovery: Directly estimate the direction of arrival of the signal from the optimal variables.

[0075] In the direction-of-arrival (DOA) estimation scenario, the target is often to estimate the frequency. , rather than It itself. In this case, it can be directly achieved through... Vandermonde decomposition is performed to obtain frequency information without the need for self-decomposition. recover Therefore, according to Calculate its corresponding Toeplitz matrix Using the ESPRIT algorithm from Extracting the frequency components of the target signal That is, the direction of arrival information of the signal is obtained.

[0076] This invention proposes an array signal direction-of-arrival estimation system based on meshless compressed sensing, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the array signal direction-of-arrival estimation method based on meshless compressed sensing.

[0077] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the aforementioned array signal direction-of-arrival estimation method based on meshless compressed sensing.

[0078] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the aforementioned array signal direction-of-arrival estimation method based on meshless compressed sensing.

[0079] In summary, compared to the traditional PhaseLift technique, this application introduces an atomic norm minimization constraint in the optimization modeling, fully utilizing the Vandermonde sparse structure of the signal. Traditional PhaseLift does not consider signal structural characteristics, requiring a large number of amplitude observation samples to achieve stable recovery, and its recovery accuracy and success rate decrease significantly when sampling is insufficient. This application, by combining PhaseLift with atomic norm constraints, achieves high recovery rate and accurate direction-of-arrival estimation under finite sampling conditions, significantly reducing the requirement for a large number of observation samples. Furthermore, due to the introduction of sparsity priors during optimization, the method in this application maintains stable recovery performance and high resolution even in low signal-to-noise ratio scenarios, exhibiting superior robustness compared to traditional PhaseLift.

[0080] Compared to sparse phase retrieval (CPRL) technology, this application employs a gridless atomic norm minimization method, enabling direct signal recovery in the continuous frequency domain. CPRL relies on frequency domain discretization, but in practical applications, frequencies are continuous values, often resulting in mismatch with the preset grid, leading to severe degradation in recovery performance. This application, by establishing an optimization model in the continuous frequency domain, fundamentally avoids estimation errors caused by grid mismatch, ensuring stability and accuracy under practical application conditions. Therefore, compared to CPRL, this application not only avoids the performance bottleneck caused by discretization but also maintains high recovery accuracy and stability under different array sizes and source conditions, exhibiting stronger universality and engineering applicability.

[0081] Example

[0082] To verify the effectiveness of the present invention, the following simulation experiment was conducted.

[0083] In this embodiment, a simulation experiment is conducted where the number of signal sources L=1, the number of arrays N=8, and the number of samples... The frequency ranges from 1 to 32. We designed two sets of experiments to compare PhaseLift, CPRL, and the method of this application at different frequency values, for a total of three methods. In the first set of experiments, we assumed the frequency... The second set of experiments assumes a frequency that falls on a given grid point. exist Random value is selected from within.

[0084] The results of the first set of experiments are shown in Figure 1. It is evident that in this scenario, the present application has a significant advantage over PhaseLift, but CPRL performs best. This result is reasonable because both CPRL and the present application utilize the structural information of the array signal, while PhaseLift does not. Therefore, PhaseLift is used for recovery. It requires the most samples. The reason why CPRL performs so well is that the frequencies fall exactly on the pre-defined grid points, satisfying the sparsity assumption of CPRL under the frequency discretization model.

[0085] The results of the second set of experiments are shown in Figure 2. As can be seen from the results, the performance of CPRL dropped from the best among the three to the worst, even worse than PhaseLift, which completely does not utilize the signal structure. The reason is as mentioned before: if... exist With random values, regardless of the mesh size (20 mesh points and 0.05 mesh spacing in this experiment), there will almost always be grid errors, leading to spectral leakage. Forcibly imposing sparsity constraints when the mesh is not actually sparse will cause the recovery process to deviate. In contrast, this application remains robust and maintains its advantage over PhaseLift. Therefore, we can conclude that CPRL is only effective under a very few specific conditions, which are rarely met in practical applications.

[0086] In summary, this technology enables high-precision recovery and direction-of-arrival estimation of array signals even when only signal amplitude measurement is available, phase information is missing, or there is severe distortion. This technology can be widely applied to scenarios involving sparse spectrum signal recovery and target orientation localization, such as synthetic aperture radar (SAR) 3D imaging, vehicle-mounted millimeter-wave radar imaging, target detection and tracking, large-scale MIMO communication channel estimation, sonar localization, and medical imaging.

[0087] The technical features of the above embodiments can be combined in any way. For the sake of brevity, 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.

[0088] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. A method for retrieval of the direction-of-arrival phase of an array signal based on meshless compressed sensing, characterized in that, Includes the following steps: Step S1: Acquire amplitude observation data of the target signal using a linear sampler to form an observation vector; Step S2: Introduce matrix variables Establish a linear relationship between the observation vector and the matrix variables, where x is the array signal to be recovered; Step S3: Construct a joint optimization model, which is a positive semidefinite programming model that includes atomic norm minimization constraints and phase lifting constraints; Step S4: Numerically solve the semidefinite programming model to obtain the optimal solution; Step S5: Construct the Toeplitz matrix based on the optimal solution to estimate the frequency and obtain the direction of arrival information of the target signal.

2. The array signal direction-of-arrival phase recovery method based on meshless compressed sensing according to claim 1, characterized in that, Step S1: Acquire amplitude observation data of the target signal using a linear sampler to form an observation vector. ,in: ； In the formula, Let m be the m-th linear sampler, where m = 1, ..., M.

3. The array signal direction-of-arrival phase recovery method based on meshless compressed sensing according to claim 2, characterized in that, Step S2: Introduce matrix variables Establish a linear relationship between the observation vector and the matrix variables, expressed as: ； The above The stacked equations can be written in a concise form. ； In the formula, For sampling system Determined linear operators.

4. The array signal direction-of-arrival phase recovery method based on meshless compressed sensing according to claim 3, characterized in that, Step S3: Construct a joint optimization model, which is a positive semidefinite programming model containing atomic norm minimization constraints and phase lifting constraints, expressed as: ； In the formula, express The Toeplitz matrix corresponding to the vector is calculated as follows: ； at the same time , , Let N be a positive definite diagonal matrix, and N be the number of arrays. The trace of a matrix is ​​represented by the matrix trace. and This is an adjustable regularization parameter.

5. The array signal direction-of-arrival phase recovery method based on meshless compressed sensing according to claim 4, characterized in that, Step S4: Numerically solve the semidefinite programming model. The specific method is as follows: The optimal solution is obtained by numerically solving a semidefinite programming problem using the CVX convex optimization toolkit and calling the semidefinite programming solver. .

6. The array signal direction-of-arrival phase recovery method based on meshless compressed sensing according to claim 1, characterized in that, Step S5: Construct the Toeplitz matrix based on the optimal solution to estimate the frequency and obtain the direction of arrival information of the target signal. The specific method is as follows: According to the optimal solution Constructing the Toeplitz matrix Using the ESPRIT algorithm to Vandermonde decomposition is performed to estimate the frequency components of the target signal. That is, the direction of arrival information of the signal is obtained.

7. A gridless compressed sensing-based array signal direction-of-arrival phase recovery system, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the gridless compressed sensing-based array signal direction-of-arrival phase recovery method according to any one of claims 1-6.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the array signal direction-of-arrival phase recovery method based on meshless compressed sensing as described in any one of claims 1-6.

9. A computer-readable storage medium having a computer program stored thereon, wherein when executed by a processor, the computer program implements the array signal direction-of-arrival phase recovery method based on meshless compressed sensing as described in any one of claims 1-6.

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