A method for estimating multi-measurement vector line spectra, along with a computer device and storage medium.

Abstract

This invention provides a method for estimating the line spectrum of a multi-measurement vector model, along with a computer device and storage medium. The method involves sampling discrete signal data from a multi-measurement vector model, iteratively estimating the rank-minimum Toplitz matrix using a fast interior-point method framework, and then estimating the signal frequency using the Root-MUSIC algorithm. The fast interior-point method framework addresses the convex optimization problem of minimizing the atomic norm, deriving an iterative optimization process based on the necessary conditions for the optimal solution of a nonlinear programming problem, and introducing the BFGS algorithm to optimize the covariance matrix during the iteration process. The algorithm proposed in this invention represents the multi-measurement vector line spectrum estimation problem as a convex optimization model, thus exhibiting performance advantages in frequency resolution. Furthermore, at low signal-to-noise ratios, the estimated frequency accuracy is superior to the MUSIC and L1-SVD algorithms. Compared to the SDPT3-MMV-ANM algorithm, the algorithm proposed in this invention uses derived fast iterative expressions for atomic and dual variables, resulting in significant advantages in computational efficiency.

Description

Technical Field

[0001] This invention belongs to the field of spectrum analysis technology, specifically relating to a multi-measurement vector line spectrum estimation method, computer equipment, and storage medium. Background Technology

[0002] In recent years, with the widespread application of sonar and radar equipment in communication, detection, and guidance, spectrum analysis technology has also developed rapidly. Among spectrum analysis techniques, line spectrum estimation is a crucial research topic. This technique aims to estimate the line spectrum from noise-contaminated signals. For example, in passive sonar systems, line spectrum estimation can detect target ship information using radiated noise data. Since the power spectrum of ship radiated noise consists of a continuous broadband spectrum and a discrete narrowband line spectrum, and its narrowband line spectrum has characteristics such as high energy, long propagation distance, and good stability, utilizing line spectrum features for ship target identification is of great significance for passive underwater target detection. Furthermore, in the field of non-destructive testing, line spectrum features reflect the operating status of mechanical equipment; therefore, line spectrum estimation technology also plays an important role in mechanical fault diagnosis systems.

[0003] Line spectrum estimation techniques consider signal models that can be categorized into single measurement vectors (SMVs) and multiple measurement vectors (MMVs) signal models. Compared to SMV models, MMV models estimate signal frequencies by collecting multiple snapshots of data, thus improving the resolution performance and robustness of line spectrum estimation. Traditional line spectrum estimation methods based on MMV models estimate frequencies by exploring the properties of the signal subspace, such as Multiple Signal Classification (MUSIC) and Matrix Pencil Method (MPM). These algorithms are often significantly affected by noise; when the signal-to-noise ratio is low, the estimation accuracy is low and the robustness is poor. Furthermore, in practical applications, the resolution performance of traditional algorithms is not ideal. Because the line spectrum of a signal is sparse across the entire frequency range, sparse recovery-based line spectrum estimation methods have made significant progress in recent years. These methods transform the line spectrum estimation problem into an L1 norm optimization problem, leveraging the sparsity of the L1 norm to optimize the frequency components of the signal. Compared to traditional line spectrum estimation methods, they offer higher resolution. These algorithms include L1-SVD and Sparse Iterative Covariance-based Estimation (SPICE). However, both L1-SVD and SPICE suffer from mesh mismatch, meaning the preset mesh size affects the accuracy of frequency estimation. To address this mesh mismatch issue, meshless sparse recovery algorithms have been extensively studied. These algorithms introduce the concept of atomic norms, transforming the parameter estimation problem over continuous intervals into a convex optimization problem based on atomic norms. The signal frequency is then calculated using convex optimization algorithms. However, these algorithms require the SDPT3 optimization tool in the CVX toolbox to solve for the signal frequency. This optimization tool has high computational complexity and cannot be used in systems with high real-time requirements.

[0004] In sparse recovery algorithms, although L1-SVD and SPICE can be used to estimate signal frequencies, these algorithms require the construction of a discrete dictionary, and the estimated frequency must fall on the parameter grid corresponding to the discrete dictionary. Therefore, the grid spacing directly affects the algorithm's performance. A large grid spacing leads to a significant error between the estimated and actual frequencies; a small grid spacing increases computational complexity, increases the correlation between the basis vectors of the discrete dictionary, thus causing the algorithm's finite isometric property (RIP) to fail and reducing the sparsity of the obtained solution. Therefore, the performance of L1-SVD and SPICE algorithms is affected by the preset grid spacing.

[0005] To address the mesh mismatch problem in the aforementioned discrete sparse recovery algorithms, a meshless sparse recovery algorithm based on the atomic norm has been proposed. This algorithm transforms the parameter estimation problem over continuous intervals into a convex optimization problem based on atomic norm minimization (ANM), and calculates the signal frequency using this convex optimization algorithm. However, the meshless sparse recovery algorithm based on the atomic norm requires the SDPT3 optimization tool from the CVX toolbox to solve for the signal frequency. The SDPT3 optimization tool has low computational efficiency, making it difficult to use in scenarios with high real-time requirements. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of meshless sparse recovery algorithms, such as high computational complexity and inability to quickly estimate line spectra. Based on a multi-measurement vector signal model, a fast convex optimization algorithm framework is proposed. The computational efficiency of this framework is much higher than that of the SDPT3 optimization tool, which makes the meshless sparse recovery algorithm applicable to engineering projects with real-time requirements, thus increasing its practicality.

[0007] To achieve the above objectives, this invention proposes a multi-measurement vector line spectrum estimation method. This method samples discrete signal data from a multi-measurement vector model, iteratively estimates the rank-minimum Toplitz matrix using a fast interior-point method framework, and then estimates the signal frequency using the Root-MUSIC algorithm. The fast interior-point method framework addresses the convex optimization problem of minimizing the atomic norm by deriving an iterative optimization process based on the necessary conditions for the optimal solution of nonlinear programming, and introduces the BFGS algorithm to optimize the covariance matrix during the iteration process.

[0008] As an improvement to the above method, the method includes:

[0009] Step 1: Sampling to obtain observation data Y o Set initial values ​​for each parameter, including: hyperparameter γ; hyperparameter τ; maximum number of iterations J; stopping threshold β; Armijo line search condition parameter c; target lower bound function f. LB ;variable The t required for the first optimization;

[0010] in, The symbol for a real number is... Represents a 2N-1 dimensional real vector; N is the length of one of the measurement vectors in the multi-measurement vector model;

[0011] Step 2: Based on the objective function:

[0012] minh t (u)=g(u)+t -1 G(u)

[0013] The update direction Δu is calculated using the BFGS algorithm;

[0014] in,

[0015] t is the variable that needs to be updated in each iteration based on the difference between the original and dual variables. In the first iteration, the initial t from the input is used, and in subsequent iterations, the result calculated in step 9 is used; τ is the set hyperparameter; g is T. * (I), where T * (·) denotes the adjoint Toeplitz operator; I is an N×N identity matrix; Tr(·) denotes the trace of the computed matrix; T(·) denotes the Toeplitz mapping; u (t) u is obtained for the current t; H is the conjugate transpose symbol, and T is the transpose of the real matrix;

[0016] u is by The real variable consisting of the real and imaginary parts:

[0017]

[0018] Where Re(·) denotes taking the real part, and Im(·) denotes taking the imaginary part; The variable to be solved is initially unknown and represents the best solution that satisfies this optimization equation.

[0019] Step 3: Using formula u i =u i-1 The step size α is calculated using the Armijo line search condition for +α△u, i.e., α needs to satisfy:

[0020]

[0021] Where i is the number of iterations; for;

[0022]

[0023] Among them, t i Let t be the value of t in the i-th iteration. i The variables Y, W, and u are determined before each update. Where (u i ) 1:N Represents vector u i The first to Nth elements, and so on; Representation function Calculate the gradient of variable u; i Let u represent the result of the i-th iteration. i-1 This represents the result of u in the (i-1)th iteration;

[0024] Step 4: Update variable u i =u i-1 +α△u;

[0025] Step 5: Based on the current iteration result u i ,calculate:

[0026]

[0027] Where (u i ) 2:N Represents vector u i The 2nd to Nth elements, and so on;

[0028] Step 6: Update parameter W i ,Y i

[0029] Y i =T(u i )(T(u i )+I) -1 Y o

[0030] W i =(τt) -1 +(Y o ) H T -1 (u i )Y o

[0031] Among them, W i Y is the optimal solution for variable W in the function mino(μ) that satisfies the condition during the i-th iteration; i W is the optimal solution for variable Y in the function mino(μ) that satisfies the condition during the i-th iteration; that is, W i Y i u i Enables the function

[0032]

[0033] The value is the smallest; where, ||·|| F Denotes the Frobenius norm;

[0034] Step 7: Update parameter V i ,S i ,z i

[0035] V i =τI,z i =τg,S i =2(Y i -Y o)

[0036] Among them, V i S represents the value of the dual variable V during the i-th iteration; i z is the value of the dual variable S in the i-th iteration; i Let z be the value of the dual variable z in the i-th iteration; where S is the dual variable. i z i and V i It can make functions The variable with the largest value;

[0037] Step 8: Determine if the condition is met. Right now

[0038]

[0039] Where ω is any value in the range of 0 to 2π; S i The result S obtained in step 7 i V i V obtained in step 7 i ;z i z obtained in step 7 i ;q is through The calculated result is a vector consisting of 2N-1 dimensional real numbers; Represents the imaginary unit. It is an N-dimensional complex vector; represent The kth element;

[0040] If satisfied, update the target lower bound function:

[0041]

[0042] If not satisfied, then f LB constant;

[0043] Step 9: Calculate the original objective function And update parameter t:

[0044]

[0045] Where L indicates that the multi-measurement vector data matrix has L measurement vectors;

[0046] Step 10: Determine if the stopping condition is met:

[0047]

[0048] If the condition is met, proceed to step 11; otherwise, proceed to step 2.

[0049] Step 11: Obtain the optimal Construct the corresponding Toeplitz matrix

[0050] Step 12: Use the Root-MUSIC algorithm from Estimating the frequency vector f * .

[0051] As an improvement to the above method, the hyperparameter γ = 1.25; the hyperparameter τ = 1.5; the maximum number of iterations J is 2000; and the stopping threshold condition β = 10. -4 Armijo line search condition parameter c = 0.125; target lower bound function f LB =-∞, initialize t to 10 for the first time.

[0052] The present invention also provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method as described in any of the preceding claims.

[0053] The present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, causes the processor to perform the method described in any of the preceding claims.

[0054] Compared with the prior art, the advantages of the present invention are:

[0055] 1. The FastIPM-MMV-ANM algorithm proposed in this invention represents the multi-measurement vector line spectrum estimation problem as a convex optimization model, thus having a performance advantage in frequency resolution. Moreover, at low signal-to-noise ratios, the frequency estimation accuracy is better than that of the MUSIC algorithm and the L1-SVD algorithm.

[0056] 2. Compared with the SDPT3-MMV-ANM algorithm based on the SDPT3 optimization tool, the FastIPM-MMV-ANM algorithm proposed in this invention uses a fast iterative expression of derived atomic and dual variables, which has a significant advantage in computational efficiency. Attached Figure Description

[0057] Figure 1 The diagram shown is a flowchart of the sparse array design based on the meshless compressed sensing algorithm.

[0058] Figure 2 The figure shows the performance results of line spectrum estimation using the MUSIC algorithm (N=32, L=50, SNR=10dB).

[0059] Figure 3The figure shows the performance results of line spectrum estimation using the SDPT3-MMV-ANM algorithm (N=32, L=50, SNR=10dB).

[0060] Figure 4 The figure shows the performance results of line spectrum estimation using the FastIPM-MMV-ANM algorithm (N=32, L=50, SNR=10dB).

[0061] Figure 5 The figure shows the line spectrum estimation performance (N=32, L=50) under different signal-to-noise ratios;

[0062] Figure 6 The diagram shows the computation time of the algorithm (N=32, L=50) under different signal-to-noise ratios. Detailed Implementation

[0063] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0064] This invention proposes a Fast InteriorPoint Method (FastIPM) framework based on the perspective of meshless sparse recovery. This framework addresses the convex optimization problem of minimizing the atomic norm by deriving an iterative optimization process based on the necessary conditions (Karush-Kuhn-Tucker, KKT conditions) for the optimal solution of nonlinear programming. In the iterative process, the BFGS algorithm (a method of symmetric positive definite iterative matrix in quasi-Newton methods) is introduced to optimize the covariance matrix. Compared with the SDPT3 optimization tool, this framework greatly improves computational efficiency and increases the application value of meshless sparse recovery theory.

[0065] For convenience, the multi-measurement vector line spectrum estimation method based on the fast interior-point method proposed in this invention is referred to as FastIPM-MMV-ANM. This algorithm requires three steps to estimate the signal frequency: first, sampling the discrete signal data of the multi-measurement vector model; second, iteratively estimating the rank-minimum Toeplitz matrix using the FastIPM framework; and finally, estimating the signal frequency using the Root-MUSIC algorithm. The algorithm flow is as follows: Figure 1 As shown.

[0066] 1. Sparse Optimization Representation of Multi-Measurement Vector Signal Models

[0067] For a multi-measurement vector signal model, assume that a data matrix consisting of L discrete sinusoidal signals is observed. The (j,t)th element y jt It can be represented as

[0068]

[0069] in The imaginary unit, For the frequency of the signal, And [N] = {1, 2, ..., N}. Formula (1) means that each column of Y consists of K frequency sets {f k The amplitude set is {s} kt The discrete sinusoidal signal is composed of}. Formula (1) can be expressed in matrix form as:

[0070]

[0071] in c k =||s k ||2>0 and In the meshless sparse recovery framework, for the multi-measurement vector signal model corresponding to equations (1) and (2), the atom set is defined as:

[0072]

[0073] To measure the sparsity of frequencies, the atomic norm is introduced. The concept is defined as follows:

[0074]

[0075] Since the frequencies constituting the observed signal are sparse across the entire frequency domain, the atomic norm can be used to reconstruct the target signal Y and estimate its frequency f. This problem can then be expressed as an optimization problem in the following form:

[0076]

[0077] Where Y o This represents the observation data obtained through sampling; ||·|| F Let Frobenius norm be denoted; η is the hyperparameter of the noise estimator. When η = 0, Equation (5) considers the noise-free case. The convex optimization problem represented by Equation (5) is equivalent to the following positive semidefinite programming form:

[0078]

[0079] Where Tr(·) represents the calculation of the trace of the matrix. These are all variables that need optimization. T(·) represents the Toeplitz mapping, i.e.:

[0080]

[0081] From a statistical perspective, the optimized formula (6) yields... The corresponding Toeplitz matrix The covariance matrix of the target signal model (2) can be used to estimate the sparse frequencies f that constitute Y using the subspace method. * ,Right now It can be decomposed into the following form:

[0082]

[0083] in P = diag(p1,…,p) K And p k >0, k=1,…,K. k} is a frequency range Different points on the same surface. Therefore, the subspace class method can be used to optimize the resulting... Estimate the frequency f * In this invention, we use the Root-MUSIC algorithm to estimate the frequency f. * .

[0084] 2. Fast Interior Point Method Framework

[0085] For the optimization equation (6), the SDPT3 optimization tool from the CVX toolkit is often used to solve it; however, SDPT3 has low computational efficiency. This invention proposes a fast interior-point method framework based on the primordial-dual optimization problem to solve for the variables. Y * and W * To facilitate the introduction of the subsequent fast interior-point method framework, let's first discuss the variables. The properties of the Toeplitz operator will be discussed. Furthermore, due to subsequent... The update requires the use of a quasi-Newton algorithm like L-BFGS. This type of algorithm can only be used to update real variables and cannot be directly used to update complex variables. Therefore, define a with The associated new optimization variables are of length 2N-1. This variable is determined by The real variable formed by the real and imaginary parts of , i.e., u and The relationship between them can be represented as:

[0086]

[0087] Where Re(·) represents taking the real part and Im(·) represents taking the imaginary part. From another perspective, formula (9) represents... The relationship between u and u can be expressed in the following form:

[0088]

[0089] Where u 1:N This represents the first to Nth elements of vector u, and so on.

[0090] Furthermore, we also discuss the fundamental properties of the Toeplitz operator and its adjoint operator. For the Toeplitz operator T(·) of equation (9), its adjoint operator T * (·) is defined as:

[0091] T * (B)=(2β0,2Re(β1),…,2Re(β N-1 ),2Im(β1),…,2Im(β N-1 )) T (10)

[0092] in and Furthermore, based on the definition of the adjoint operator in formula (10), it is not difficult to derive T. * (·) has the following property: for any matrix B, we have This property will be used in the subsequent derivation of the fast interior point method framework.

[0093] In the fast interior-point method framework, it is necessary to start from the primal-dual optimization form and establish the iterative update equation of the optimization variables. Using the property of the adjoint Toepltz operator corresponding to formula (10), it is not difficult to see that the primal optimization problem (6) is equivalent to the following form:

[0094]

[0095] According to the properties of the Toeplitz operator, i.e. There is g = T * (I). For convenience, we define the set consisting of variables Y, W, and u as μ, i.e., μ = (Y, W, u). The objective function of formula (11) is expressed as f(μ), i.e. According to the constraint conditions of formula (11), μ needs to be on the constraint K of the convex cone surface, that is, μ needs to satisfy the following conditions:

[0096]

[0097] In order to ensure that the optimization variable μ satisfies the condition of falling on the convex cone surface K, we introduce a penalty function of logarithmic determinant to constrain the original variable μ, so that it satisfies the constraint K of falling on the convex cone surface, that is:

[0098]

[0099] Where μ∈intK indicates that the variable μ is inside the convex cone K, it can be seen that the feasible region condition of the function F(μ) is μ∈intK, thus making the solved μ satisfy the constraint condition corresponding to formula (12). By introducing the corresponding logarithmic determinant penalty function of formula (13), the original problem (11) can be rewritten as follows:

[0100] minf(μ)+t -1 F(μ) (14)

[0101] The parameter t>0 is an adjustable parameter, and as the parameter t increases, t -1 The closer F(μ) gets to the characteristic function, the better the optimized performance, and the closer it is to the effect of the constraint term in formula (11). In this invention, the value of parameter t updated in each iteration is related to the difference between the original objective function and the dual objective function.

[0102] The above is an analysis of the original optimization equation. By definition, the form of the dual problem needs to be obtained through the Lagrangian function L(μ,λ). The Lagrangian function of the original problem (11) is defined as follows:

[0103] L(μ,λ)=f(μ)+<μ,λ> (15)

[0104] Where λ is the dual variable, from It consists of three variables, namely λ=(S,V,z). Furthermore, <μ,λ> represents the inner product of corresponding elements of the original variable μ and the dual variable λ, i.e., <λ,μ>=Tr(V H W)+Re(Tr((S H Y)))+z T u. By definition, the dual optimization problem can be expressed in the following form:

[0105]

[0106] For convenience, the objective function of the dual optimization equation (16) is defined as g(λ), i.e. Furthermore, it can be seen from the constraint terms of formula (16) that the prerequisite for the validity of the result of the dual objective function g(λ) is that λ must fall on the dual cone K. * Above, that is, λ must satisfy However, directly applying this condition to determine whether the dual variable λ falls within the dual cone K... * The above is complex and difficult, so we propose a relatively simple judgment condition, namely K. * It can be represented as:

[0107]

[0108] Where C *Let represent the dual cone corresponding to the positive semi-definite Hermitian-Toeplitz matrix. In summary, λ lies on the dual cone K. * Only need to meet This condition is sufficient, and it is easy to determine. Equivalent to The following conditions must be met:

[0109]

[0110] It can be seen that formula (18) is actually the form of the Fourier transform, therefore The equivalent Fourier transform of q is non-negative, which in turn allows us to transform the dual cone K. * Make a quick judgment.

[0111] In summary, the primal and dual problems based on the atomic norm were established. Next, to enable iterative optimization and updates of the parameters, the KKT conditions for the primal and dual problems are listed as follows:

[0112]

[0113] By solving the first equation of the KKT conditions We can obtain the dual variable V. (t) ,S (t) ,z (t) The iterative update formula is:

[0114] V (t) =τI,z (t) =τg,S (t) =2(Y (t) -Y o (20)

[0115] By solving the last equation of the KKT conditions The original variable W can be obtained. (t) ,Y (t) ,u (t) The iterative update formula is:

[0116]

[0117] Where I represents an identity matrix of dimension N. (Through...) get u (t) It needs to meet the following form:

[0118] τw-τT * (φφ H )-t -1 T * (T -1 (u (t) ))=0 (22)

[0119] Where, φ=(T(u) (t) )+I) -1 Y o From formula (22), we can see that u is obtained. (t) The analytical solution is quite difficult. From another perspective, formula (22) can also be regarded as a convex optimization model, u (t) The solution is exactly the optimal point of the convex optimization model, that is, formula (22) can be regarded as minh t (u)=g(u)+t -1 The case of G(u), where:

[0120]

[0121] Among them, h t (u) is a convex function, and It is exactly equivalent to the form of formula (22), therefore minh t (u) is equivalent to nesting an inner optimization problem within the overall optimization problem (11), which is only used to solve for the optimal value of variable u. Furthermore, it can be observed that in the formula g(u)+t -1 In G(u), G(u) is equivalent to the logarithmic determinant penalty function of the variable T(u), such that u falls within the cone corresponding to the semi-definite Hermitian-Toeplitz matrix. Above. For the optimization equation minh t The optimization variable u in (u) is updated using the following iterative optimization form:

[0122] u i =u i-1 +α△u (24)

[0123] Where u i Minh t (u) represents the variable u corresponding to the i-th iteration of this internal optimization problem, where Δu represents the update direction and α represents the step size. In the gradient update method, Δu can be represented by the negative gradient. However, gradient methods have low iterative efficiency and are slow. While using the Hessian matrix instead of Δu improves iterative efficiency, the Hessian matrix is ​​a dense matrix, requiring significant storage space. Therefore, this invention introduces the BFGS method to calculate the update direction Δu, i.e. Here, matrix H is a matrix that approximates the inverse of the Hessian matrix. The BFGS method can calculate the sparse matrix H that best approximates the Hessian matrix by storing the key coefficients of the first n iterations, greatly saving storage space. After H is determined, the step size α also needs to be determined in order to ensure that the objective function g(u) + t -1To minimize G(u), this invention uses the Armijo line search condition to find the optimal step size α, which requires α to satisfy the following condition:

[0124]

[0125] Where c>0 is a hyperparameter that needs to be set. In this invention, c=0.125 is set.

[0126] Furthermore, in the primal-dual optimization framework, the parameter t of the logarithmic determinant penalty function needs to be updated. In this invention, the parameter t is updated by calculating the gap between the primal-dual objective functions, i.e.:

[0127]

[0128] Where γ>1 is a hyperparameter that needs to be set, in this invention, γ=1.5 is set. In summary, the iterative formulas for all variables that need to be updated in the original dual optimization framework are summarized, and the iterative optimization process of the FastIPM-MMV-ANM algorithm is expressed in the form of an algorithm flowchart, as shown in Table 1.

[0129] Table 1 Algorithm Flowchart

[0130]

[0131]

[0132]

[0133] The length N of the observed signal was set to 32, the number of snapshots L to 50, the inherent signal frequency f = [0.1, 0.11, 0.15, 0.2, 0.5], and the signal-to-noise ratio (SNR) was set to 10 dB. The performance of the MUSIC algorithm, the SDPT3-MMV-ANM algorithm, and the FastIPM-MMV-ANM algorithm proposed in this paper were compared. The results of 10 Monte Carlo experiments are as follows: Figure 2 , 3 As shown in Figure 4.

[0134] from Figure 2 , 3 As shown in section 4, the MUSIC algorithm failed to distinguish the frequency components of 0.1 and 0.11, while the multi-block ANM algorithm could. Therefore, the ANM framework has a certain advantage in super-resolution parameter estimation. Regarding the frequency estimation error, the average MSE error of FastIPM-MMV-ANM is 3.689 × 10⁻⁴. -6 The average MSE error of SDPT3-MMV-ANM is 3.303 × 10⁻⁶. -6It is evident that for the same ANM framework, the estimation errors are not significantly different. Regarding computational speed, in 10 Monte Carlo experiments, FastIPM-MMV-ANM's average execution time was 0.7145s, while SDPT3-MMV-ANM's average execution time was 30.7093s, clearly demonstrating FastIPM-MMV-ANM's advantage in computational efficiency.

[0135] The performance of the FastIPM-MMV-ANM algorithm under different signal-to-noise ratios (SNRs) was tested, and the proposed algorithm was compared with ROOT-MUSIC and SDPT3-MMV-ANM algorithms. The target frequency, observation length, and snapshot length were all set according to the conditions in Experiment 1. The SNR of the observed signal was set to SNR = [-5, 0, 5, 10, 15] dB. 100 Monte Carlo experiments were performed for each SNR. The experimental results are shown below. Figure 5 As shown.

[0136] from Figure 5 As can be seen, the performance of the FastIPM-MMV-ANM algorithm is similar to that of the SDPT3-MMV-ANM algorithm. Compared with the ROOT-MUSIC algorithm, the estimation accuracy of FastIPM-MMV-ANM is significantly better than that of the ROOT-MUSIC algorithm when the signal-to-noise ratio is between 0 and 5, indicating that the FastIPM-MMV-ANM algorithm has better robustness. At a high signal-to-noise ratio, i.e., SNR = 15dB, the errors of both the FastIPM-MMV-ANM and ROOT-MUSIC algorithms are on the order of 10. -7 .

[0137] Furthermore, comparing the average computation time of the FastIPM-MMV-ANM and SDPT3-MMV-ANM algorithms at each signal-to-noise ratio, the results are as follows: Figure 6 As shown. From Figure 6 As can be seen, the computation time of the FastIPM-MMV-ANM algorithm is within 10. -1 In terms of order of magnitude, while the SDPT3-MMV-ANM algorithm has a computation time of 10... 2 The order of magnitude difference clearly shows that the FastIPM-MMV-ANM algorithm is far more efficient than the SDPT3-MMV-ANM algorithm. Therefore, the experimental results confirm that the fast interior-point method framework greatly improves computational efficiency.

[0138] Compared to the MUSIC and L1-SVD algorithms, the FastIPM-MMV-ANM algorithm proposed in this invention represents the multi-measurement vector line spectrum estimation problem as a convex optimization model, thus exhibiting performance advantages in frequency resolution. Furthermore, at low signal-to-noise ratios, its frequency estimation accuracy surpasses that of the MUSIC and L1-SVD algorithms. Compared to the SDPT3-MMV-ANM algorithm based on the SDPT3 optimization tool, the FastIPM-MMV-ANM algorithm proposed in this invention uses derived fast iterative expressions for atomic and dual variables, resulting in significant advantages in computational efficiency.

[0139] The present invention also provides a computer device comprising: at least one processor, a memory, at least one network interface, and a user interface. The various components of this device are coupled together via a bus system. It is understood that the bus system is used to enable communication between these components. In addition to a data bus, the bus system also includes a power bus, a control bus, and a status signal bus.

[0140] The user interface may include a display, keyboard, or clicking device (e.g., mouse, trackball, touchpad, or touchscreen).

[0141] It is understood that the memory in the embodiments disclosed in this application may be volatile memory or non-volatile memory, or may include both volatile and non-volatile memory. The non-volatile memory may be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory may be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as Static Random Access Memory (SRAM), Dynamic Random Access Memory (DRAM), Synchronous DRAM (SDRAM), Double Data Rate Synchronous DRAM (DDRSDRAM), Enhanced Synchronous DRAM (ESDRAM), Synchlink DRAM (SLDRAM), and Direct Rambus RAM (DRRAM). The memories described herein are intended to include, but are not limited to, these and any other suitable types of memory.

[0142] In some implementations, the memory stores elements such as executable modules or data structures, or subsets thereof, or extended sets thereof: operating systems and applications.

[0143] The operating system includes various system programs, such as the framework layer, core library layer, and driver layer, used to implement various basic business functions and handle hardware-based tasks. The application programs include various applications, such as media players and browsers, used to implement various application functions. Programs implementing the methods of the embodiments of this disclosure can be included in the application programs.

[0144] In the above embodiments, the processor can also invoke programs or instructions stored in memory, specifically programs or instructions stored in an application program, for the following purposes:

[0145] Follow the steps described above.

[0146] The above methods can be applied to or implemented by a processor. The processor may be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above methods can be completed by integrated logic circuits in the processor's hardware or by software instructions. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the disclosed methods, steps, and logic block diagrams. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the disclosed methods can be directly implemented by a hardware decoding processor, or by a combination of hardware and software modules in the decoding processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory; the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above methods.

[0147] It is understood that the embodiments described in this invention can be implemented in hardware, software, firmware, middleware, microcode, or a combination thereof. For hardware implementation, the processing unit can be implemented in one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), general-purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in this application, or combinations thereof.

[0148] For software implementation, the technology of this invention can be achieved by executing the functional modules (e.g., procedures, functions, etc.) of this invention. The software code can be stored in memory and executed by a processor. The memory can be implemented in the processor or externally.

[0149] The present invention may also provide a non-volatile storage medium for storing a computer program. When the computer program is executed by a processor, it can implement the steps in the above method embodiments.

[0150] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for estimating the line spectrum of a multi-measurement vector model, wherein after sampling discrete signal data of a multi-measurement vector model, the method iteratively estimates the rank-minimum Toplitz matrix using a fast interior-point method framework, and then estimates the frequency of the signal using the Root-MUSIC algorithm; in, The fast interior point method framework addresses the convex optimization problem of minimizing the atomic norm by deriving an iterative optimization process based on the necessary conditions for the optimal solution of nonlinear programming, and introduces the BFGS algorithm to optimize the calculation of the covariance matrix during the iteration process. The method includes: Step 1: Sample and obtain observation data Y°; set initial values ​​for each parameter; Step 2: Based on the objective function: bright t (u)=g(u)+t -1 G(u); The update direction Δu is calculated using the BFGS algorithm; in, ; t is a variable that needs to be updated in each iteration based on the difference between the original and the dual. In the first iteration, the initial t of the input is used, and in subsequent iterations, the result calculated in step 9 is used. The hyperparameter is set; g is T. * (I), where T * (·) denotes the adjoint Toeplitz operator; I is an N×N identity matrix; Tr(·) denotes the trace of the computed matrix; T(·) denotes the Toeplitz mapping; u (t) u is obtained for the current t; H is the conjugate transpose symbol, and T is the transpose of the real matrix; u is by The real variable consisting of the real and imaginary parts: ； Where Re(·) denotes taking the real part, and Im(·) denotes taking the imaginary part; The variable to be solved is initially unknown and represents the best solution that satisfies this optimization equation. ; Step 3: Using formula u i =u i-1 The step size α is calculated using the Armijo line search condition +αΔu; Step 4: Update variable u i =u i-1 +αΔu; Step 5: Based on the current iteration result u i ,calculate: ； Among them, (u i ) 2:N Represents vector u i The 2nd to Nth elements, and so on; Step 6: Update parameter W i ,Y i ; ； ； Among them, W i Y is the optimal solution for variable W in the function mino(μ) that satisfies the condition during the i-th iteration; i W is the optimal solution for variable Y in the function min(o(μ)) at the i-th iteration; that is, W i Y i u i Enables the function The value is the smallest; where, ||·|| F Denotes the Frobenius norm; Step 7: Update parameter V i ,S i ,z i ; ； Among them, V i S represents the value of the dual variable V during the i-th iteration; i z is the value of the dual variable S in the i-th iteration; i Let S be the value of the dual variable z in the i-th iteration; where S i z i and V i It can make functions The variable with the largest value; Step 8: Determine if the condition is met. ; Step 9: Calculate the original objective function And update parameter t: ； Where L indicates that the multi-measurement vector data matrix has L measurement vectors; Step 10: Determine if the stopping condition is met; Step 11: Obtain the optimal Construct the corresponding Toeplitz matrix ; Step 12: Use the Root-MUSIC algorithm from Estimating the frequency vector f * .

2. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method as described in claim 1.

3. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, causes the processor to perform the method as described in claim 1.

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