A fast gridless method for sonar target range-angle estimation based on 2D-FIPM

Abstract

The present invention proposes a fast gridless method for sonar target range-angle estimation based on 2D-FIPM, comprising the following steps: performing frequency mixing and pulse compression on a received signal to obtain a discrete-time model, and expressing the model in matrix form; formulating the gridless range-angle estimation problem in an active sonar system as a semidefinite programming problem, and using Fast Fourier Transformation (FFT) to determine the dual cone corresponding to the Toeplitz semidefinite cone; constraining the original variables within a feasible region through a logarithmic determinant barrier function, and simultaneously updating the original variables and the dual variables through an augmented KKT condition to determine the feasible domain of the dual variables; establishing a relationship between the dual gap and the barrier function, obtaining an optimal solution through iteration, and estimating the unpaired frequency based on the optimal solution; constructing a manifold matrix, searching for the maximum element in each row or column of the matrix, completing the pairing process, realizing range-angle pairing, and thus completing target range and angle estimation.

Description

Technical Field

[0001] The present invention belongs to the technical field of target detection and positioning, and in particular relates to a fast gridless estimation method for sonar target distance-angle based on 2D-FIPM. Background Art

[0002] In active sonar systems, achieving accurate range and angle estimation is crucial for target detection and localization. These systems rely on emitting sound waves and capturing the echo signals reflected from the target to perform their functions. In recent years, with the rise of compressed sensing (CS) theory, numerous methods based on this theory have been widely applied to the field of position parameter estimation, demonstrating excellent performance. These CS-based methods have demonstrated significant advantages in position parameter estimation, particularly in challenging environments, such as coherent sources and targets, and limited echo data. This advantage lies in the fundamental principle of CS theory: if a signal is sparse within a domain, it can be recovered from measurements, even if the number of measurements is far fewer than required by traditional methods. From the perspective of CS, spatial signals are represented solely as presence or absence, unconstrained by coherence or incoherence, and therefore unconstrained by the quality of the recovered covariance matrix, providing more relaxed conditions for target signal recognition. In the context of sonar systems, this means being able to recover target range and angle information from even limited sonar echo data without relying on a complete dataset and making assumptions about noise statistics.

[0003] Existing sonar signal processing methods, particularly those based on sparse representation for position parameter estimation, typically rely on discretizing the signal parameter domain so that processing and analysis can be performed on a predefined grid. However, a major limitation of this approach lies in the so-called "grid mismatch problem," whereby the actual target parameters may not correspond exactly to the predefined grid points, thus reducing the accuracy of the estimate. This problem limits the effectiveness of conventional methods when dealing with non-grid-aligned target parameters, highlighting the need for novel solutions.

[0004] Specifically, the development of distance-angle estimation technology currently covers a variety of solutions, including the 2D Estimation of Signal Parameters via Rotational Invariance Techniques (2D-ESPRIT) method developed based on the subspace method and its accelerated FFT-ESPRIT method, the 2D Orthogonal Matching Pursuit (2D-OMP) method based on grid sparseness, and the Decoupled Atomic Norm Minimization (DANM) method based on gridless sparseness and its accelerated Alternating Direction Method of Multipliers (ADMM) developed by DANM-ADMM.

[0005] 2D-ESPRIT is a classic subspace method that estimates the angle of arrival and time of arrival (or distance) of a signal by leveraging the rotation-invariant properties of the signal subspace. This method is favored for its ability to estimate signal parameters in two dimensions, but it still suffers from high computational complexity, especially in large-scale computations. 2D-OMP is a grid-sparse method based on compressed sensing theory that estimates target parameters by iteratively selecting grid points that best match the observed data. This method's advantage lies in its ability to perform fast searches on a predefined grid, thus reducing computational complexity. However, 2D-OMP is limited by the grid setting, and its performance suffers when the target parameters do not match the grid points, limiting its application in complex environments. The DANM method is a gridless sparse method that estimates signal parameters by minimizing the atomic norm. This method does not rely on a predefined grid and can be optimized directly in the continuous domain, avoiding the grid mismatch problem. DANM demonstrates excellent estimation accuracy, especially when the target parameters are located at non-grid points, but it suffers from high computational complexity, especially when processing large datasets. ADMM is an iterative algorithm for solving large optimization problems. It simplifies the optimization process by breaking the problem down into smaller subproblems. In the context of the DANM problem, ADMM is used to accelerate the problem by iteratively updating variables to reach a solution. Although ADMM has low computational complexity per iteration, due to its proximal nature, it typically requires more iterations to reach the optimal solution, which limits its performance in real-time or near-real-time applications.

[0006] The limitations of these solutions mainly focus on the inability to strike a good balance between accuracy and computing time, and the inability to achieve fast, real-time and effective position parameter estimation under the technical requirements of high-precision detection. Summary of the Invention

[0007] The purpose of the present invention is to overcome the defects of the prior art and propose a fast gridless estimation method for sonar target range-angle based on 2D-FIPM.

[0008] In view of this, the present invention proposes a fast gridless sonar target range-angle estimation method based on 2D-FIPM, which is used for active sonar systems to estimate target range and angle based on received signals, including:

[0009] Step 1) performing frequency mixing and pulse compression on the received signal to obtain a discrete time model and express it in matrix form;

[0010] Step 2) The gridless distance-angle estimation problem in the active sonar system is summarized as a semidefinite programming problem, and the dual cone corresponding to the Toeplitz semidefinite cone is determined by using FFT;

[0011] Step 3) constraining the original variable to be within the feasible region through the logarithmic determinant barrier function, and simultaneously updating the original variable and the dual variable through the augmented KKT condition to determine the feasible region of the dual variable;

[0012] Step 4) Establish the relationship between the dual gap and the barrier function, and obtain the optimal solution through iteration.

[0013] Step 5) estimating the unpaired frequency based on the optimal solution;

[0014] Step 6) Construct a manifold matrix, search for the maximum element in each row or column of the matrix, complete the pairing process, realize distance-angle pairing, and thus complete the target distance and angle estimation.

[0015] Preferably, the active sonar is a uniform linear array having M identical elements with an element spacing of d=λ / 2, where λ is the operating wavelength.

[0016] Preferably, the discrete time model of step 1) is:

[0017]

[0018] Among them, y m (n) is the signal received by the mth antenna element, T s represents the sampling interval, a k represents the backscatter coefficient of the kth target, K represents the total number of targets, exp(-j2πmβ k) represents the phase offset generated when the echo of the kth target reaches the mth array element, f c is the carrier frequency, Noise term e m represents the additive white Gaussian noise on the mth array element, τ k is the time delay of the kth target, c is the speed of sound wave propagation underwater, and μ is the frequency change rate of the LFM signal.

[0019] Preferably, the matrix form Y of step 1) is:

[0020] Y=X+E,

[0021] where the matrix X is the ideal sparse signal to be recovered, is the additive white Gaussian noise matrix.

[0022] Preferably, the semidefinite programming problem in step 2) is:

[0023]

[0024] Among them, η is the weight regularization parameter, σ represents the variance of independent and identically distributed Gaussian noise, represents the noise control term, X,u x and u y represents the optimization variable of the optimization problem, and is the Toeplitz operator The first-level Toeplitz matrix of the mapping, N and M represent the sampling length and the number of array elements respectively.

[0025] Preferably, the step 2) uses FFT to determine the dual cone corresponding to the Toeplitz semi-definite cone, including: when the following formula is satisfied, the dual variable z is located in the dual cone Inside:

[0026]

[0027] in is the complex-valued representation of the real vector z, f represents the Fourier transform frequency, and exp(-j2πnf) represents the Fourier transform rotation factor.

[0028] Preferably, the logarithmic determinant barrier function F(μ) in step 3) is:

[0029]

[0030] Where R(μ)=T(u y )-X H T -1(u x )X;

[0031] The degree of the barrier function d f for:

[0032] d f =N+M

[0033] By introducing the barrier function as a penalty term into the optimization problem, the semidefinite programming problem in step 2) is solved by iteratively solving the following problem:

[0034] minf(μ)+t -1 F(μ)

[0035] Where t>0 is the barrier parameter;

[0036] By using the augmented KKT condition, the original variables μ=[u y T ,vec(X) T ,u x T ] T and dual variables Determine the feasible region of the dual variables; including:

[0037] Set the current number of iterations i and the barrier parameter t i , T groups of difference vectors Δu x,i ,Δψ i ,ΔΨ i , the gradient vector And the Hessian matrix approximation H i ,

[0038] Calculate search direction

[0039] Solve the damping factor by traversing from k=i-1,i-2,...,max(iT,1) Newton's approximate direction

[0040] use Update search direction: Traverse from k=max(iT,1) and solve the correction factor Update search direction d = d + Δu x,k (χ k -δ k );

[0041] Update the original solution μ=[u y T ,vec(X) T ,u x T ]T In: u x,i =u x,i-1 +σd

[0042] The central path (μ) consisting of the primal solution and the dual solution is updated according to the following steps: i ,λ i ):

[0043] The dual solution in:

[0044]

[0045] Original solution

[0046]

[0047] in, is a unit vector, its first element is equal to 1, and the rest of the elements are equal to 0;

[0048] Determine the feasible region f of the dual variable LB =max(g(λ i ),f LB ).

[0049] Preferably, the step 4) comprises:

[0050] Define the lower bound f LB =max(g(λ i ),f LB ), update the dual gap η according to the following formula i :

[0051] η i =f(μ i )-f LB .

[0052] Duality gap η i and the barrier parameter t i The relationship between them is:

[0053]

[0054] By introducing a multiplier γ>1 to adjust the change of the duality gap, the barrier parameter t of the next iteration is updated i+1 for:

[0055]

[0056] By iteratively increasing the barrier parameter t i , duality gap η i is effectively reduced, so that (μ i ,λ i) gradually approaches the optimal solution (μ*,λ).

[0057] Preferably, the step 5) comprises:

[0058] Based on the optimal u x * and u y *, extract the optimal normalized frequency estimates α* and β* from the corresponding Toeplitz matrix by Vandermonde decomposition:

[0059]

[0060] Among them, D α ,D β ≥0 is a diagonal matrix.

[0061] Preferably, the step 6) comprises:

[0062] Construct the manifold matrix A(α ＊ ) and B(β ＊ ), the estimate D of the matrix D is obtained by the following formula:

[0063]

[0064] Searches for the maximum element s in each row or column of the matrix |D| ij ,in is the element in the i-th row and j-th column of the matrix |D|, if but and Pairing, complete the pairing process, get the corresponding pairing The target distance estimation is obtained according to the following formula and angle estimation

[0065]

[0066] Where c is the speed of sound waves in water and λ is the wavelength.

[0067] Compared with the prior art, the advantages of the present invention are:

[0068] 1. The present invention utilizes the sparse structure of the sonar received signal to rearrange the effective parameter information and constructs a DANM optimization problem that can be solved by the gridless sparse theory, thereby making the present invention have a higher resolution. In addition, the application of the 2D-FIPM algorithm significantly reduces the computational complexity of each iteration from O(N 3 ) is reduced to O(N 2 ), which enables the algorithm to process large-scale data sets more quickly and improves computational efficiency while maintaining high accuracy compared to previous similar algorithms.

[0069] 2. The algorithm of the present invention has a clear structure, is simple to implement, and is easy to integrate with other sonar system components, which is beneficial to the upgrade of existing systems and the development of new systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 This is a schematic diagram of the target range-angle estimation scenario in active sonar;

[0071] Figure 2 shows the distance-angle estimation performance of various methods under different array sizes (M), where Figure 2(a) is the RMSE comparison and Figure 2(b) is the running time comparison;

[0072] Figure 3 It is the RMSE comparison of distance-angle estimation of different methods under different signal-to-noise ratios;

[0073] Figure 4 It is a flow chart of the fast method for gridless estimation of sonar target range-angle based on 2D-FIPM of the present invention. DETAILED DESCRIPTION

[0074] The present invention aims to propose a new solution, especially focusing on how to help existing technologies get rid of their dependence on preset grids, restore a more realistic range-angle parameter estimation problem scenario, and further reduce the system's calculation time on this basis to achieve gridless rapid estimation of sonar target range-angle.

[0075] This paper proposes a gridless and fast two-dimensional fast interior point method (2D-FIPM) for range-angle estimation in active sonar systems. Traditional range-angle estimation relies on the discretization of the area of ​​interest, and the use of predefined grids inevitably leads to grid mismatch problems. To address this drawback, we regard the range-angle estimation problem as a two-dimensional gridless compressed sensing problem and formulate it as a DANM optimization problem. The proposed 2D-FIPM method can quickly implement DANM and overcome grid dependence. The algorithm introduces an efficient method to determine the feasible region of the dual variables, constrains the original variables within the feasible region through the logarithmic determinant barrier function, and simultaneously updates the original variables and dual variables through the augmented Karush-Kuhn-Tucker (KKT) conditions, thereby reducing the number of iterations. In addition, we suggest using classical fast computing techniques to handle specific matrix operations in the algorithm implementation. The computational complexity of this method for an N+M order semidefinite matrix in each iteration is Extensive numerical simulation results demonstrate the computational advantages of the proposed algorithm, which provides excellent resolution and accuracy.

[0076] Principle description:

[0077] Consider K far-field, narrowband sources incident on a uniform linear array (ULA) with M identical elements and element spacing d = λ / 2, where λ is the operating wavelength.

[0078] Figure 1 Describes the kth target located at (r k ,θ k ) generates an echo. Therefore, the echo signal received by the mth array element can be expressed as

[0079]

[0080] where a k represents the backscatter coefficient of the kth target. k ) represents the phase offset generated when the echo of the kth target reaches the mth array element, where Noise term e m (t) represents the additive white Gaussian noise on the mth array element. The time delay τ of the kth target k Expressed as Where c is the speed of sound waves propagating underwater. Here, it is assumed that the linear frequency modulation (LFM) signal transmitted by the sonar system is as follows:

[0081]

[0082] Where rect(a) represents the rectangular function, which takes the value 1 when a∈[0,1] and 0 otherwise. c is the carrier frequency, T sym is the pulse length. Parameter μ=B / T sym It represents the rate of change of instantaneous frequency, where B is the bandwidth.

[0083] At the receiving end, after mixing and pulse compression, the signal received by the mth antenna element presents a two-dimensional sinusoidal waveform.

[0084]

[0085] With sampling frequency f s =1 / T s Sampling is performed, where T s Represents the sampling interval, and the discrete time model y can be obtained m (n), further organized into

[0086]

[0087] in α k =μτ k Ts .

[0088] We represent the received signal y in matrix form m (n) is

[0089] Y=X+E, (5)

[0090] where Y n,m =y m (n) represents the matrix The element at the nth row and mth column of . is the additive white Gaussian noise matrix.

[0091]

[0092] in Steering vector and With a Vandermonde structure, as follows:

[0093]

[0094] and They are defined as the corresponding manifold matrices. This produces a new matrix form atomic set It is defined on a continuous field and has the following form:

[0095]

[0096] where f = (α, β) is the combination of α and β. The atomic norm in matrix form is naturally defined as

[0097]

[0098] Atomic norm in matrix form represents the sparsity of the recovered signal x. Since the signal components that make up X are far less than the number of atoms in the continuous atomic set, the problem of estimating the values ​​of k components in the continuous set can be regarded as a sparse recovery problem, that is, minimizing the atomic norm As shown in (9), this newly formulated minimization problem can be equivalently expressed as the following semidefinite programming (SDP):

[0099]

[0100] in and is the Toeplitz operator The first-order (Hermitian) Toeplitz matrix of the mapping, L = M or N, acts on the real-valued vector and Using real-valued vectors The complex-valued form of provides a more intuitive understanding. The real-valued vector u corresponds to a unique complex-valued vector, namely The expression of T(u) is

[0101]

[0102] In order to ensure accurate reconstruction within a certain error range compared to the received signal Y, we introduce the error term By minimizing this error and regularizing the problem, we can adopt the DANM formulation, which leads to a regularized denoising problem to recover X. The formula is as follows:

[0103]

[0104] in represents the sparsity enforcement term, represents the noise control term. The weight regularization parameter η plays a crucial role in balancing the above two terms. Here we follow the classic setting method of most research methods, that is, to achieve stable sparse recovery, where σ represents the variance of independent and identically distributed Gaussian noise.

[0105] According to formulas (10) and (12), the gridless range-angle estimation problem in active sonar systems can be summarized as the following semidefinite programming (SDP) problem:

[0106]

[0107] It is crucial to emphasize the fundamental role of the positive semidefinite Toeplitz matrix cone in the constraints of the optimization problem (13). The properties of this cone and its dual cone are the key to developing 2D-FIPM to efficiently solve the problem (13). This section takes the N-dimensional positive semidefinite Toeplitz matrix cone as an example to derive the relevant theorem. The definition is as follows:

[0108]

[0109] At the same time, its corresponding dual cone can be expressed as:

[0110]

[0111] It should be pointed out that only by defining (15) To this end, we introduce a numerically feasible method for efficiently judging

[0112] First, we introduce the adjoint operator T for the Toeplitz operator T defined above. * For any Hermitian matrix Adjoint operator The role is:

[0113]

[0114] in Due to the Hermitian Toeplitz structure of the matrix T(u), for any Hermitian matrix B, the inner product of T(u) and B is a real number, and the relationship is as follows:

[0115]

[0116] In addition, regarding The gradient formula is:

[0117]

[0118] For a function F and its complex vector x=a+jb, the gradient is defined as

[0119] Using the adjoint operator T * , the inner product analysis in (17) can be further simplified. Based on Vandermonde decomposition, we have:

[0120]

[0121] where a(f i ) has the same Vandermonde structure as a(α) in (19), i.e., a(f) = [1, exp(-j2πf),…, exp(-j2π(N-1)f)] H ,f∈[0,1]. Therefore, the inner product (15) can be expressed as:

[0122]

[0123] Where Z is the Hermitian matrix, T * (Z)=z=[z0,…,z 2N-2 ] T According to the dual cone Definition (15), if Then the inner product (20) for all Neither is negative.

[0124] Because p i >0, we have obtained a fast method to determine the given Specifically, If and only if:

[0125]

[0126] in is the complex value representation of the real vector z. Obviously, Z(f) in formula (21) presents a similar The only difference is that f needs to be approximated by taking enough discrete points in the interval [0,1]. FFT technology makes it possible to efficiently evaluate Z(f)≥0, so as to quickly determine whether a given z is within the cone. Inside.

[0127] This insight is a key conclusion for the implementation of 2D-FIPM, as it allows us to quickly verify whether the non-analytical solution for the dual variable z exceeds the bound in an asymmetric cone programming problem, as discussed in the next section.

[0128] We continue to focus on the fast solution of the DANM optimization problem and reformulate (13) into the original problem:

[0129]

[0130] where g x =T * (I N ) and g y =T * (I M ). The optimization variables in (13) are reorganized into new optimization variables μ = [u y T ,vec(X) T ,u x T ] T . Proper cone represents the feasible region of μ, defined as

[0131]

[0132] The Lagrangian function is defined as The Lagrangian function induced by (22) is

[0133]

[0134] where λ = [z y T ,vec(S) T ,z x T ] T ,and is the dual variable.

[0135] The dual function is the lower bound of the Lagrangian function and is defined as Only when λ satisfies the following conditions, all The lower bound of :

[0136]

[0137] Substituting this type of λ into the Lagrangian function, we get the form of the dual problem:

[0138]

[0139] in yes The dual cone of

[0140]

[0141] The definition of the original dual cone, as shown in Equation (27), does not effectively help us judge Through analysis, we established the cone With cone and use the criterion in (21) to efficiently verify whether the dual solution in the current iteration satisfies So it is used in the next iteration. Specifically, if λ=[z y T ,vec(S) T ,z x T ] T satisfy

[0142]

[0143] but Which verify The method is shown in Equation (21) and can be quickly verified by using FFT technology.

[0144] We use the homogeneous logarithmic determinant barrier function to express

[0145]

[0146] Where R(μ)=T(u y )-X H T -1 (u x )X. The degree of the barrier function is given by

[0147] d f =N+M (30)

[0148] By introducing the barrier function as a penalty term into the optimization problem, the original problem (22) can be solved by iteratively solving the following problem:

[0149] minf(μ)+t -1 F(μ)(31)

[0150] Where t>0 is an increasing sequence, called the barrier parameter. By gradually increasing the parameter t according to the value of the duality gap in each iteration, the solution gradually converges to the optimal solution. In each step, the solution of μ is obtained by the solution of the previous step. As the parameter t increases, the term t -1 F(μ) gradually approaches the role of the indicator function. Therefore, the original constraint The constraints imposed become progressively more important.

[0151] The enhanced KKT conditions can be interpreted as continuous variations of the standard KKT conditions. They provide a numerical method to iteratively approximate the optimal solution as a series of suboptimal solutions. The main difference is the replacement of the complementarity condition. In the i-th iteration, the barrier parameter t i >0, the enhanced KKT condition of problem (31) can be expressed as:

[0152]

[0153] For a sequence of increasing t i >0 value, the set of solutions {(μ i ,λ i ):t i >0} forms the primal-dual central path. As t i As increases, this path will converge to the desired solution, i.e.

[0154] By solving (32), we obtain the dual solution in:

[0155]

[0156] Original solution It can be obtained in the following ways:

[0157]

[0158] as well as

[0159] u x,i :τg x -t -1 T * (T -1 (u x,i ))-τT * (ΛΛ H)=0 (35)

[0160] In formulas (34) and (35), Λ = T x -1 (u x,i +0.5τe0)Y, where is a unit vector whose first element is 1 and the rest are 0. The problem is that Equation (34) may not have an explicit solution, but we can think of its left side as a function The gradient of ψ(u x )=τg x T u x +τtr(Y H T -1 (u x,i +0.5τe0)Y),

[0161] and Ψ(u x )=-logdet(T(u x )). Therefore, we consider the following unconstrained optimization problem with a barrier function:

[0162]

[0163] function is convex, so solving the optimal solution of the optimization problem (36) is equivalent to solving This is consistent with the form in (35). Since Ψ(u x )yes Logarithmic barrier function, solve equation (35), and make It can be regarded as solving the optimization problem (36). In order to solve the optimization problem in (36), we use the line search method. In this method, the variable u x Update via:

[0164] u x,i =u x,i-1 +σd, (37)

[0165] where u x,i Indicates u in the i-th iteration x The solution is σ, d is the search direction. The search direction d of the i-th iteration is generally obtained as follows:

[0166]

[0167] in and Respectively expressed in u x,i-1The Hessian matrix and gradient at []. We use the L - BFGS algorithm (quasi - Newton method) to approximate the search direction d in (38) for fast solution. In the i - th iteration, the difference vectors are stored as:

[0168]

[0169] In the algorithm implementation, only the most recent T < N sets of difference vectors are retained, thus reducing memory consumption (note that when i < T, only i sets are stored). In addition, we use a heuristic diagonal matrix to approximate the Hessian matrix:

[0170]

[0171] where The analytical expression of this term can be obtained by the chain rule:

[0172]

[0173] After obtaining the search direction d using the L - BFGS algorithm, the step size σ must satisfy the Armijo rule:

[0174]

[0175] where the hyperparameter c ∈ (0, 1) is set to 0.05 in this invention. Once the step size σ and direction d are determined, u x The update in the i - th iteration is carried out according to (37).

[0176] Through these improvements, we can efficiently solve (35) with low complexity. In addition, by substituting u x,i into (33) and (34), we can obtain the primal - dual solution pair (μ i , λ i ), which is organized as shown in Algorithm 1 of Table 1.

[0177] Table 1

[0178]

[0179]

[0180] Algorithm structure

[0181] In the numerical process of 2D - FIPM, the log - homogeneous barrier function and line search method in (36) ensure that If Therefore, based on equation (34), in each iteration, the primal variable μ i satisfies

[0182]

[0183] This ensures However, it should be noted that the numerical solution u of (36) x,i It is not guaranteed that the dual solution (33) always lies in the cone To ensure the reliability and convergence of the solution, each update (μ i ,λ i ), use the fast method described in (28) to verify This ensures that the iteration process always occurs within the valid domain.

[0184] Obstacle parameter t i is updated based on the duality gap, which is equal to the difference between the optimal values ​​of the original problem and the dual problem. According to the duality theorem, the lower bound of the original problem is greater than the optimal value of the dual problem. In the i-th iteration, when When the barrier parameter t is updated by the dual gap η i We define the lower bound f LB =max(g(λ i ),f LB ), then update η i as follows:

[0185] η i =f(μ i )-f LB (44)

[0186] Duality gap η i It also provides the solution of μ i The suboptimality upper bound of f(μ i )-f ＊ ≤ i , where f ＊ is the optimal value. As the iteration proceeds, f LB will be replaced by g(λ i ) is updated. According to (32) and (30), the duality gap η i The relationship between and the barrier parameter t is roughly as follows:

[0187]

[0188] Calculate η i Finally, considering the inequality in (45), a multiplier γ>1 is introduced to adjust the change of the duality gap, and then the barrier parameter t of the next iteration can be updated. i+1 for:

[0189]

[0190] By iteratively increasing the barrier parameter t i, duality gap η i is effectively reduced, so that (μ i ,λ i ) gradually approaches the optimal solution (μ ＊ ,λ ＊ ).

[0191] Based on the optimal u x ＊ and u y ＊ , the optimal normalized frequency estimate α can be extracted from the corresponding Toeplitz matrix by Vandermonde decomposition ＊ and β ＊ This can be achieved efficiently using the root-MUSIC algorithm:

[0192]

[0193] Among them, D α ,D β ≥0 is a diagonal matrix. In addition, considering the extreme case where some frequency components overlap in one dimension but separate in another, the number of sources is determined to be K ＊ =max{rank(T(u x ＊ )),rank(T(u y ＊ ))}. The pairing step is important for identifying K pairs of frequencies It is crucial to improve the accuracy and robustness of the estimation. ＊ The fact that the bootstrap matrices can be generated by correctly pairing them leads to a simple pairing technique as shown in (6). Therefore, we try to estimate D as follows:

[0194]

[0195] The matrix |D| may have multiple nonzero elements in its rows or columns, but not in both. In either case, if but Should Pairing, where is the element in the i-th row and j-th column of the matrix |D|. This will give and does not introduce ambiguity. Accordingly, and

[0196] In summary, the distance-angle estimation method based on 2D-FIMP can be summarized by Algorithm 2, see Table 2:

[0197] Table 2

[0198]

[0199]

[0200] The technical solution of the present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0201] Example

[0202] like Figure 4 As shown, an embodiment of the present invention proposes a fast gridless sonar target range-angle estimation method based on 2D-FIPM, which is used for an active sonar system to estimate target range and angle based on received signals, including:

[0203] Step 1) The received signal is mixed and pulse compressed to obtain a discrete time model, which is expressed in matrix form as shown in formula (5).

[0204] Step 2) The gridless range-angle estimation problem in the active sonar system is summarized as a semidefinite programming problem, see formula (13), and the dual cone corresponding to the Toeplitz semidefinite cone is determined by FFT, see formula (21);

[0205] Step 3) Constrain the original variable within the feasible region through the logarithmic determinant barrier function, and simultaneously update the original variable and the dual variable through the augmented KKT condition to determine the feasible region of the dual variable, see formula (28);

[0206] Step 4) Establish the relationship between the dual gap and the barrier function, see formulas (45) and (46), and obtain the optimal solution through iteration, see Algorithm 1 in Table 1;

[0207] Step 5) estimating the unpaired frequency based on the optimal solution;

[0208] Step 6) Construct a manifold matrix, search for the maximum element in each row or column of the matrix, complete the pairing process according to formula (48), realize distance-angle pairing, and thus complete the target distance and angle estimation.

[0209] Simulation experiment

[0210] To validate the performance of the proposed algorithms, this section uses numerical simulations. We analyze the performance of different algorithms, including 2D-ESPRIT, FFT-ESPRIT, 2D-OMP, and DANM-ADMM, under different array configurations, where the number of array elements is represented by M. Simulations are performed with a signal-to-noise ratio (SNR) of 10 dB and a sampling length of N = 128. Five randomly located targets are considered in the comparison. Figure 2 shows the root mean square error (RMSE) and execution time as the value of M varies from 10 to 50 (in steps of 5).

[0211] Figure 2(a) shows that all distance-angle estimation methods exhibit a decreasing RMSE trend under different values ​​of M. The RMSE of several sparse methods is consistently lower than that of the traditional methods 2D-ESPRIT and FFT-ESPRIT. On the other hand, the accuracy of the gridded compressed sensing method (2D-OMP) does not improve significantly as M increases, due to the influence of the preset grid. Notably, in Figure 2(b), the proposed method runs significantly faster and is almost insensitive to changes in the array size M compared to all other methods except FFT-ESPRIT.

[0212] In addition, in the performance exploration under different signal-to-noise ratios (SNR), we randomly selected 5 targets and set N = 128 and M = 20. The SNR range is from -15dB to 30dB with a step size of 5dB. The results are shown in Figure 2. Figure 3 The results show that the proposed algorithm can maintain consistent accuracy at different SNR levels. In addition, the RMSE gradually decreases with the increase of SNR and the improvement of signal strength.

[0213] Finally, it should be noted that the above embodiments are intended only to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the embodiments, it should be understood by those skilled in the art 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 are intended to be encompassed by the claims of the present invention.

Claims

1. A fast gridless method for sonar target range-angle estimation based on 2D-FIPM, used in active sonar systems to estimate target range and angle based on received signals, including: Step 1) performing frequency mixing and pulse compression on the received signal to obtain a discrete time model and express it in matrix form; Step 2) The gridless distance-angle estimation problem in the active sonar system is summarized as a semidefinite programming problem, and the dual cone corresponding to the Toeplitz semidefinite cone is determined by using FFT; Step 3) constraining the original variable to be within the feasible region through the logarithmic determinant barrier function, and simultaneously updating the original variable and the dual variable through the augmented KKT condition to determine the feasible region of the dual variable; Step 4) Establish the relationship between the dual gap and the barrier function, and obtain the optimal solution through iteration; Step 5) estimating the unpaired frequency based on the optimal solution; Step 6) construct a manifold matrix, search for the maximum element in each row or column of the matrix, complete the pairing process, realize distance and angle pairing, and thus complete the target distance and angle estimation; The step 5) comprises: Based on the optimal u x ＊ and u y ＊ , extract the optimal normalized frequency estimate α from the corresponding Toeplitz matrix through Vandermonde decomposition ＊ and β ＊ : Among them, D α , is a diagonal matrix, u x and u y The optimization variables representing the optimization problem; The step 6) comprises: Construct the manifold matrix A(α ＊ ) and B(β ＊ ), the estimate D of the matrix D is obtained by the following formula: Searches for the maximum element s in each row or column of the matrix |D| ij ,in is the element in the i-th row and j-th column of the matrix |D|, if but and Pairing, complete the pairing process, get the corresponding pairing The target distance estimation is obtained according to the following formula and angle estimation Where c is the speed of sound waves in water and λ is the wavelength.

2. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 1, characterized in that: The active sonar is a uniform linear array with M identical elements and an element spacing d=λ / 2, where λ is the operating wavelength.

3. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 2, characterized in that: The discrete time model of step 1) is: Among them, y m (n) is the signal received by the mth antenna element, T s represents the sampling interval, a k represents the backscatter coefficient of the kth target, K represents the total number of targets, exp(-j2πmβ k ) represents the phase offset generated when the echo of the kth target reaches the mth array element, f c is the carrier frequency, Noise term e m represents the additive white Gaussian noise on the mth array element, τ k is the time delay of the kth target, c is the speed of sound wave propagation underwater, and μ is the frequency change rate of the LFM signal.

4. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 3, characterized in that: The matrix form Y of step 1) is: Y=X+E, where the matrix X is the ideal sparse signal to be recovered, is the additive white Gaussian noise matrix.

5. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 4, characterized in that: The semidefinite programming problem of step 2) is: Among them, η is the weight regularization parameter, σ represents the variance of independent and identically distributed Gaussian noise, represents the noise control term, X,u x and u y represents the optimization variable of the optimization problem, and is the Toeplitz operator The first-level Toeplitz matrix of the mapping, N and M represent the sampling length and the number of array elements respectively.

6. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 5, characterized in that: The step 2) uses FFT to determine the dual cone corresponding to the Toeplitz semi-definite cone, including: when the following formula is satisfied, the dual variable z is located in the dual cone Inside: in is the complex-valued representation of the real vector z, f represents the Fourier transform frequency, and exp(-j2πnf) represents the Fourier transform rotation factor.

7. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 6, characterized in that: The logarithmic determinant barrier function F(μ) in step 3) is: where R(μ)=T(u y )-X H T -1 (u x )X; The degree of the barrier function d f for: d f =N+M By introducing the barrier function as a penalty term into the optimization problem, the semidefinite programming problem in step 2) is solved by iteratively solving the following problem: minf(μ)+t -1 F(μ) Where t>0 is the barrier parameter; By using the augmented KKT condition, the original variables μ=[u y T ,vec(X) T ,u x T ] T and dual variables Determine the feasible region of the dual variables; including: Set the current number of iterations i and the barrier parameter t i , T groups of difference vectors Δu x,i ,Δψ i ,ΔΨ i , the gradient vector And the Hessian matrix approximation H i , Calculate search direction Solve the damping factor by traversing from k=i-1,i-2,…,max(iT,1) Newton's approximate direction use Update search direction: Traverse from k=max(iT,1) and solve the correction factor Update search direction d = d + Δu x,k (χ k -δ k ); Update the original solution μ=[u y T ,vec(X) T ,u x T ] T In: u x,i =u x,i-1 +σd, The central path (μ) consisting of the primal solution and the dual solution is updated according to the following steps: i ,λ i ): Among them, the dual solution Where: Original solution Where: X i =T(u x,i )T -1 (u x,i +0.5τe0)Y Where, is a unit vector, its first element is equal to 1, and the rest of the elements are equal to 0; Determine the feasible region f of the dual variable LB =max(g(λ i ),f LB ).

8. The fast gridless method for sonar target range-angle estimation based on 2D-FIPM according to claim 7, characterized in that: The step 4) comprises: Define the lower bound f LB =max(g(λ i ),f LB ), update the dual gap η according to the following formula i : or i =f(μ i )-f LB . Duality gap η i and the barrier parameter t i The relationship between them is: By introducing a multiplier γ>1 to adjust the change of the duality gap, the barrier parameter t of the next iteration is updated i+1 for: By iteratively increasing the barrier parameter t i , duality gap η i is effectively reduced, so that (μ i ,λ i ) gradually approaches the optimal solution (μ ＊ ,λ ＊ ).

Images