Fast gridless joint distance-angle estimation method and product for FDA-MIMO sonar based on two-dimensional generalized fast interior point method

Abstract

The present invention discloses a FDA-MIMO sonar fast gridless joint distance-angle estimation method and product based on a two-dimensional generalized fast interior point method, comprising: s1: performing frequency mixing and matched filtering on a received signal to obtain an extended virtual received signal model; s2: summarizing the estimation problem as an NsCP problem and initializing parameters related to the interior point method; s3: using a restricted memory BFGS algorithm and an augmented KKT condition to update the primal variable and the dual variable, performing a 2D-FFT fast feasible domain cone verification on the obtained variables, repeating this step until the result is within the feasible domain, and then going to s4; s4: performing a convergence check on the obtained variables, if converged, obtaining the optimal solution to the NsCP problem, and going to s5, otherwise going to step 3; s5: based on the optimal solution, retrieving sparse atoms through Vandermonde decomposition of a multi-level Toeplitz matrix, and extracting the distance and angle estimation of the target using linear correspondence.

Description

Technical Field

[0001] The present invention belongs to the field of underwater detection technology, and in particular relates to a FDA-MIMO sonar fast gridless joint distance-angle estimation method and product based on a two-dimensional generalized fast interior point method. Background Art

[0002] As an advanced underwater detection technology, multiple-input, multiple-output (MIMO) sonar systems have broad application prospects in both civilian and military fields. Their core principle is to utilize multiple transmitters and receivers operating simultaneously, transmitting signals using orthogonal waveforms, thereby obtaining richer spatial information than traditional sonar systems. In recent years, frequency diversity array (FDA) technology has been incorporated into the MIMO architecture, resulting in FDA-MIMO sonar systems. This system introduces small frequency increments into the transmit array, making beamforming dependent on the target's range and angle, thereby achieving high-precision joint range-angle estimation of the target. Dense MIMO sonar with an FDA structure is a high-resolution MIMO sonar arrangement. It closely arranges transmit and receive antennas, cleverly exploiting the orthogonality of the transmitted signals to construct a large-aperture virtual array. Compared to traditional phased array sonar systems, FDA-MIMO sonar offers significant advantages in target parameter estimation accuracy, resolution, and degrees of freedom. It can more precisely characterize target characteristics, providing stronger support for underwater target identification and tracking.

[0003] In sonar detection systems, accurately estimating target position parameters, including distance, angle, and intensity, is crucial for effective detection. When detecting moving targets, the presence of distance shifts between pulses can interfere with distance estimation, hindering accurate understanding of the target's motion. Addressing this issue often requires applying specialized techniques under specific assumptions.

[0004] Among existing technical solutions, parameter estimation methods based on compressed sensing (CS) theory (for example, l1-svd, two-dimensional orthogonal matching pursuit (2D-OMP), and multi-dictionary sparse Bayesian learning multidimensional (MD-SBL) methods) are an implementation scheme that is relatively close to the present invention. Compressed sensing theory points out that sparse data can be represented by a small number of elements in a predefined grid, which encode the core information. CS-based parameter estimation methods utilize the sparse representation characteristics of data rather than relying on statistical properties that only appear in large data sets. This allows them to work when data is limited or the data statistical properties are poor.

[0005] Meshless CS theory extends CS to the continuous domain, encoding effective information into atoms defined in the continuous domain atomic set rather than predefined grid elements. In multidimensional parameter estimation, meshless CS methods are mainly implemented by solving an atomic norm minimization (ANM) optimization problem using optimization tools such as the alternating direction method of multipliers (ADMM) or the CVX toolbox. Different numerical meshless CS parameter estimation methods have evolved due to the different optimization tools used. In order to achieve lightweight and real-time meshless multidimensional joint parameter estimation, some new methods such as two-dimensional iterative Vandermonde decomposition and shrinkage thresholding (2D-IVDST) and decoupled atomic norm minimization for multiple measurement vectors (DANM-MMV) have made certain progress. These methods avoid the high computational complexity of the SDP corresponding to ANM and develop new optimization problems and numerical solution methods.

[0006] However, traditional CS methods have some limitations. They are constrained by the predefined grid and deviate from actual object detection models. In real applications, the distance and angle of the target are continuous values, which are difficult to completely coincide with the points on the predefined grid. This limits the effectiveness of traditional CS methods.

[0007] Gridless CS optimization methods based on ADMM and CVX directly invoke computationally expensive semidefinite programming (SDP), requiring accurate noise priors. These methods are computationally slow and burdensome, making them difficult to meet the real-time and efficiency requirements of underwater detection environments. 2D-IVDST, essentially an accelerated proximal gradient technique, converges quickly, but the threshold shrinkage step destroys the original structure of the optimization variables, causing the solution to deviate from the optimal solution of ANM optimization. MMV-DANM, on the other hand, relies on the reconstruction of the covariance matrix of the received data and statistical assumptions about the data, resulting in poor robustness.

[0008] In summary, the application of existing technologies in FDA-MIMO sonar systems still has many shortcomings. While traditional CS parameter estimation methods based on prefabricated grids can be effective in MIMO sonar detection when data is limited or statistically poor, their reliance on predefined grids deviates from actual target detection models. Optimization tools derived from currently advanced gridless CS techniques, such as the ADMM or CVX toolbox, are computationally expensive, slow, and burdensome, making them difficult to meet the real-time and efficiency requirements of underwater detection. Summary of the Invention

[0009] The present invention aims to overcome the shortcomings of the prior art by proposing a fast gridless joint range-angle estimation method for FDA-MIMO sonar based on a two-dimensional generalized fast interior point method, thereby achieving fast joint range-angle estimation for underwater targets. The present invention also discloses a computer program product that implements the fast gridless joint range-angle estimation method for FDA-MIMO sonar based on a two-dimensional generalized fast interior point method.

[0010] In view of this, the present invention proposes a two-dimensional generalized fast interior point method FDA-MIMO sonar fast gridless joint distance-angle estimation method, comprising:

[0011] Step 1: Perform frequency mixing and matched filtering on the signal received by the FDA-MIMO sonar system to obtain an extended virtual received signal model and express it in matrix form;

[0012] Step 2: The joint distance-angle estimation problem is formulated as the NsCP problem, and the parameters related to the interior point method are initialized.

[0013] Step 3: Use the restricted memory BFGS algorithm and the augmented KKT condition to update the primal and dual variables, and perform 2D-FFT fast feasible domain cone verification on the obtained variables. If the verification result is outside the feasible domain, repeat this step until the verification result is within the feasible domain, that is, the feasible domain cone verification is passed, and go to step 4;

[0014] Step 4: Check the convergence of the obtained variables. If converged, the optimal solution of the NsCP problem is obtained and go to step 5. Otherwise, go to step 3 and continue iterating.

[0015] Step 5: Based on the optimal solution, sparse atoms are retrieved through Vandermonde decomposition of the multi-level Toeplitz matrix, and the distance and angle estimates of the target are extracted using linear correspondences.

[0016] Preferably, the transmitting array and receiving array of the FDA-MIMO sonar system are both uniform linear arrays, wherein the transmitting array includes M units, the receiving array includes N units, and the distance between every two transmitting array units is d t , the distance between each two receiving array elements is d r .

[0017] Preferably, the extended virtual received signal model Y obtained in step 1 is:

[0018]

[0019] in, k,l represents the signal sparse coefficient, E represents the noise term, the superscript T represents the transpose, a(f k) is the joint transmit-receive virtual steering vector, k ,β k represent the corresponding normalized frequencies of transmit and receive space, respectively, k represents the kth target, K represents the total number of targets, and L represents the total number of transmit pulses.

[0020] Preferably, the NsCP problem in step 2 is:

[0021]

[0022] Where, The optimization variables The multi-level Toeplitz matrix formed, Sparse recovery signal, is the Hermitian matrix, τ is the soft threshold constraint The associated hyperparameters are, K is the sparsity level, is a hard constraint, μ represents the constraint, and the superscript H represents the conjugate transpose.

[0023] Preferably, the initialization of the relevant parameters in step 2 includes: barrier parameter update step size γ = 5, regularization parameter τ, relative convergence threshold ∈ relative and the absolute convergence threshold ∈ gap , i=0, represents the number of current iterations, refer to the lower bound f LB =-∞.

[0024] Preferably, in step 3, the primal variable and the dual variable are updated using the restricted memory BFGS algorithm and the augmented KKT condition, and the obtained variables are subjected to 2D-FFT fast feasible domain cone verification. If the verification result is outside the feasible domain, the step is repeated, including:

[0025] When the duality gap η of the i-th iteration i ≥∈ gap , and the duality gap difference between the two iterations |η i-1 -η i |≥∈ relativ hour e , construct an obstacle optimization problem, solve and update u i :

[0026]

[0027] Where h(u) = τξ T u+τtr(Y H T ML -1 (u+τe0)Y), H(u)=-logdet(T ML (u)), is a unit vector whose first element is equal to 1 and all other elements are equal to 0, φ = T ML -1 (u+τe0)Y,ξ=T * (I);

[0028] Update the dual solution Λ according to the following formula i :

[0029]

[0030] Among them, z, S, V are the optimization variables in the dual domain corresponding to the optimization variables u, X, W. In particular, S also represents the residual of sparse recovery;

[0031] Update the original solution μ according to the following formula: i :

[0032]

[0033] Preferably, the convergence of the variables obtained in step 4 is checked. If converged, it is the optimal solution to the NsCP problem, including:

[0034] The dual function g(Λ i )

[0035]

[0036] Judgment Satisfaction From f LB ,g(Λ i ) takes the maximum value as the reference lower bound f LB ; and update the barrier parameter t for the i+1th iteration according to the following formula i+1 :

[0037]

[0038] Where γ>1 is the update step size.

[0039] Preferably, the step 5 comprises:

[0040] Extraction according to Vandermonde decomposition in

[0041]

[0042] The distance estimation of the kth target is obtained according to the following formula and angle estimation

[0043]

[0044] Where c is the speed of sound in water, Δf is the frequency increment that ensures orthogonality between the transmitted signals, Indicates the reference frequency.

[0045] On the other hand, the present invention provides a computer program product, comprising a computer program, which, when executed by a processor, implements the steps of the above-mentioned two-dimensional generalized fast interior point method FDA-MIMO sonar fast gridless joint distance-angle estimation method.

[0046] Compared with the prior art, the advantages of the present invention are:

[0047] The present invention significantly improves the performance of the FDA-MIMO sonar system by introducing 2D-GFIPM: in terms of computational efficiency, by using the L-BFGS algorithm for gradient calculation, the single iteration complexity is reduced from O(N) of the traditional SDP. 6 M 6 ) is reduced to O(N 3 (M 2 -M+2)) and reduces the number of iterations to less than 50, achieving lightweight real-time processing; in terms of parameter estimation accuracy, based on the ANM principle, the target parameters are directly recovered from the continuous domain, avoiding the mismatch problem of the grid-based compressed sensing method, and combining the atomic norm soft threshold and multi-snapshot joint sparsity to suppress noise interference, thereby improving the resolution capability under low signal-to-noise ratio; in terms of algorithm stability, the feasible domain of the original variables is constrained by the logarithmic determinant barrier function, and an innovative cone verification method based on 2D fast Fourier transform (FFT) is proposed to efficiently determine the asymmetric feasible domain constraints of high-dimensional dual variables, ensuring global convergence, and solving the problem of insufficient stability of traditional FIPM in two-dimensional extension due to the difficulty of dual variable verification, thereby achieving high-precision and high-efficiency joint distance-angle estimation in complex underwater environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 This is a flow chart of the FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method of the present invention;

[0049] Figure 2 It is the echo signal model and processing flow chart of the FDA-MIMO sonar system;

[0050] Figure 3 shows the variation trend of RMSE with the number of receiving array elements N, where Figure 3(a) is the RMSE r ,Figure 3(b) is RMSEθ;

[0051] Figure 4 is the trend of running time as a function of the number N of received array elements;

[0052] Figure 5 shows the trend of RMSE with different signal-to-noise ratios (SNRs), where Figure 5(a) shows the RMSE r , Figure 5(b) is the RMSE θ . DETAILED DESCRIPTION

[0053] The present invention realizes gridless super-resolution joint distance-angle estimation by constructing an asymmetric cone programming (NsCP) optimization problem related to atomic norm minimization (ANM) and utilizing the mathematical model of the processing results of the FDA-MIMO sonar system. At the same time, the present invention develops an efficient numerical solver 2D-GFIPM, which avoids calling the computationally expensive SDP problem, reduces computational complexity, and improves computational speed and robustness. In addition, the present invention proposes an effective numerical method to determine the feasible domain of dual variables, solves the core challenge of extending the fast interior point method to high-dimensional parameter estimation problems, and improves the versatility and extensibility of the method. Through these improvements, the present invention not only improves the accuracy and robustness of parameter estimation, but also meets the needs of real-time underwater detection, overcomes the limitations of the existing technology, and improves the overall performance of the underwater detection system.

[0054] The technical solution of the present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0055] Example 1

[0056] Embodiment 1 of the present invention proposes a fast gridless joint distance-angle estimation method for FDA-MIMO sonar based on a two-dimensional generalized fast interior point method, which is specifically described as follows:

[0057] 1. Principle Description

[0058] The algorithmic flow of the two-dimensional generalized fast interior point method (2D-GFIPM) for FDA-MIMO sonar fast gridless joint range-angle estimation is shown in the figure below. Specifically, the received signal is first processed at the receiving array through mixing and matched filtering to expand the virtual array received signal. Several snapshots are then arranged into a received signal matrix according to the arrival time sequence. The second step is to formulate the joint range-angle estimation problem from the received signal matrix as an NsCP problem, and the relevant parameters of the interior point method are initialized. The third step is to update the primal and dual variables using the restricted memory BFGS (L-BFGS) algorithm and augmented KKT conditions, and the resulting variables are then subjected to a 2D-FFT fast feasible cone verification. If the verification result indicates that the variable falls outside the feasible region, the variable update step is repeated until the resulting variable passes the feasible cone verification. The solution is then checked for convergence. If it does not converge, the variable update step is repeated. If convergence occurs, the optimal solution to the NsCP problem has been obtained. Using this optimal solution, sparse atoms are retrieved via Vandermonde decomposition of a multi-level Toeplitz matrix, and the distance and angle estimates of the target are extracted using linear correspondences.

[0059] 2. Algorithm Structure

[0060] Consider an FDA-MIMO sonar system as shown in the figure. Both the transmit and receive arrays are configured as uniform linear arrays. The transmit array consists of M elements with a spacing of d. t , and the receiving array consists of N units with a spacing of d r The center frequency of the signal transmitted by the M-th unit, m = 0, ..., M-1, is given by

[0061]

[0062] in represents the reference frequency, and Δf is the frequency increment that ensures orthogonality between the transmitted signals. The signal transmitted by the mth unit of the transmitter is expressed as

[0063]

[0064] Where T p is the pulse duration, and m=0, 1, ..., M-1 represents the narrowband envelope of the transmitted signal that satisfies orthogonality:

[0065]

[0066] where τ represents the time delay.

[0067] Using the signal received at the first element of the receiving array as a reference, consider the distance r kand the angle θ k As shown in the figure, the propagation delay of the signal transmitted by the mth transmitting element, reflected by the kth target and received by the nth receiving element is

[0068]

[0069] where c is the speed of sound in water. The received signal at the nth receive array element associated with the lth pulse echo is

[0070]

[0071] where n = 0, ..., N-1 and σ k，l is the scattering coefficient of the kth target for the lth pulse.

[0072] At the receiver, the signal is down-converted to baseband and processed by matched filtering, e.g. Figure 2 Assuming orthogonal waveforms in the co-located FDA-MIMO radar and negligible frequency increment relative to the carrier frequency, we adopt the following narrowband approximation: 1. The wavelength of the Mth carrier signal is m=0,…,M-1is approximately 2. The approximation of the narrowband envelope, i.e.

[0073] At the nth receiving element of the nth pulse, the mth reference function The matched filter output is shown in the following formula (6):

[0074]

[0075] in x in formula (6) n,m (l) Corresponding to the elements in the nth row and mth column of the matrix, the matrix can be written as:

[0076]

[0077] in and b tr (β k )=[1,…,exp{j2π(N-1)β k}] T Denote the receive steering vector and transmit-receive steering vector, respectively. Here, and denote the corresponding transmit and receive spatial normalized frequencies, respectively.

[0078] By rearranging each row of X(l) in equation (7), we can obtain a virtual data vector of MN dimension:

[0079]

[0080] where a(f k ) is the joint transmit-receive virtual steering vector, and

[0081] Assume that the variance σ 2 The zero-mean noise of is uniformly distributed across array elements, pulses, and samples, so the processing signal model of L snapshots is

[0082]

[0083] in E represents the noise term.

[0084] Furthermore, by taking advantage of the assumption that the number of sources in space is sparse, we can express the distance-angle estimation model of the source under the FDA-MIMO sonar system as the following atomic norm minimization form, that is,

[0085]

[0086] However, the above formula is extremely difficult to solve directly by definition, but it can be equivalent to the following optimization form, that is,

[0087]

[0088] In the formula The optimization variables The multi-level Toeplitz (MLT) matrix is ​​formed, which encodes the parameters to be estimated (α k ,β k ). In addition, the Hermitian matrix and sparse recovery signal They are also part of the optimization variable μ. In formula (11), τ is the soft threshold constraint is a hyperparameter associated with τ that controls the trade-off between fidelity and sparsity inherent in any estimator involving the sparsity level K. Choosing a larger τ generally leads to smaller estimates of K. Following the simple extension discussed in the one-dimensional spectral frequency estimation problem, we set Numerical simulations have demonstrated that this setup is generally effective and enables the application of the proposed algorithm in practice.

[0089] The core idea of ​​the solution to equation (11) is to approximate the hard constraint κ with a smooth barrier function, transforming the constrained optimization problem (11) into a more tractable unconstrained problem with a soft threshold. The appropriate logarithmic barrier function F(μ) of κ is given by

[0090]

[0091] The barrier function F(μ) is a smooth strictly convex function that satisfies F(μ k )→∞ as a sequence of points μ k →μ bd , where μ bd represents a point on the boundary of κ. This actually approximates the constraint By introducing a logarithmic barrier, we transform the constrained optimization problem into the following unconstrained problem (also called the primal problem):

[0092] min μ f(μ)+t -1 F(μ)(13) where t>0 is the barrier parameter. As t increases in each iteration, t -1 The term F(μ) increasingly approaches the action of the indicator function, causing the constraints imposed by the original condition to gradually become more important. Ultimately, this process leads to a solution to the original problem with the barrier function that approaches the optimal solution to the optimization problem (10).

[0093] In each step, μ* is obtained from the solution of the optimization problem (13) in the previous step, which is called the center point. The set of these points is called the central path of the problem. Due to the convexity of the optimization problem (13), each central point should satisfy the optimality condition:

[0094] ▽ μ {f(μ*)+t -1 F(μ★)}=0 (14)For a given t>0,

[0095] Λ=-t -1 ▽ μ F(μ*)=▽ μ f(μ*) (15)

[0096] is defined as the dual point corresponding to the original point on the central path of problem (11).

[0097] Points on the center path {(μ (t) ,Λ (t) ), t>0}, conditions (14) and (15) that need to be satisfied can be summarized as the central path condition, also known as the augmented KKT condition:

[0098]

[0099] in represents the Lagrangian function, It is called the dual cone and represents the feasible region of the dual variables.

[0100] From formula (16), we can solve the solution of the dual variable:

[0101]

[0102] about We hope to focus more on the discussion of the essence, that is, to give results.

[0103] Note that the MLT matrix T ML (u) is composed of the following formula

[0104]

[0105] in

[0106]

[0107] (d = 1, 2, in the present invention, N1 = M, N2 = N) represents an elementary Toeplitz matrix, in the nth d There are 1s along the diagonal and 0s everywhere else.

[0108] So for any Hermitian matrix Both

[0109]

[0110] The element in q with label n=(n1,n2) is represented by q n =tr{Θ n Q} is given.

[0111] In addition, we can introduce the adjoint operator T * To obtain Further expression of . Acting on a Hermitian matrix Operation Defined as q = T * (Q), whose n = (n1, n2)th element is given by:

[0112]

[0113] And also satisfied Therefore, a more complete analytical expression for the dual solution associated with the augmented KKT condition can be calculated as:

[0114]

[0115] where ξ=T *(I). If we ignore the matrix structure of the optimization variables μ and Λ and focus instead on the basic numerical variables, μ and Λ are only constrained by a finite number of variables. This greatly reduces the number of variables that need to be considered in the optimization. Taking advantage of this, the inner product of the primal and dual variables can be simplified to:

[0116] Substitute (22) and (23) into the dual function The dual problem corresponding to the original problem (11) is

[0117]

[0118] Substituting the dual solution into the second equation (complementarity condition) of the expanded KKT condition (16) yields the original solution:

[0119]

[0120] in is a unit vector whose first element is equal to 1 and all other elements are equal to 0, and φ = T ML -1 (u+τe0)Y. It is observed that except for the variable u, which can only provide implicit relations, all other primal-dual variables have explicit solutions.

[0121] In order to solve the variable u in (25), consider constructing an obstacle optimization problem as follows:

[0122] minψ(u)=h(u)+t -1 H(u) (26)

[0123] where h(u) = τξ T u+τtr(Y H T ML -1 (u+τe0)Y), H(u)=-logdet(T ML (u)). h(u)+t -1 The gradient of H(u) vanishes when the optimization problem (26) reaches the optimal solution, and H(u) serves as the barrier function for the following constraints (MLT semidefinite cone):

[0124]

[0125] Therefore, the constraint Solving (25) is equivalent to solving (26). In addition, (26) can be solved efficiently using the mature L-BFGS algorithm or Newton's method.

[0126] However, since the dual solution requires the original solution μ obtained by numerical methods (t)Therefore, the dual solution cannot be guaranteed to always be within the cone. Therefore, the fourth condition of the augmented KKT condition cannot always be guaranteed. To solve this problem, the present invention proposes a numerical method to verify the dual feasible region, namely:

[0127] The dual cone shown in equation (16) is equivalent to the following equation

[0128]

[0129] in Represents the MLT positive semidefinite cone The dual cone of . And for the dual cone The judgment can be obtained by quick calculation, that is, for an arbitrary If the result of Fourier transformation

[0130]

[0131] Can always be satisfied, then where q n is the number in sequence q The elements at .

[0132] Obviously, the method provided in (29) takes the form of a 2D Fourier transform. The fast Fourier transform allows (29) can be efficiently calculated on a large number of points on . The above steps provide a low-complexity method to determine This method is approximate because (29) is performed in The approximation is sampled on a finite number of points, but can be made arbitrarily accurate by increasing the number of evaluation points. In our simulations, we demonstrate that the approximation is sufficiently accurate for practical applications.

[0133] After obtaining a feasible solution (22)(25) in one iteration through the above steps including the feasible region determination process, we need to check the duality gap to determine whether the numerical iteration process has reached the optimal solution. Specifically, this requires further discussion on the update of the barrier parameter t.

[0134] According to the optimality condition (14), μ = μ ＊ The original variable at can satisfy This leads to the Lagrangian function For the feasible region The lower bound is reached. Therefore, the dual function g can be calculated at the dual point (15):

[0135]

[0136] where the ideal duality gap is calculated as η＊ =f(μ ＊ )-g(Λ)=t -1 (MN+L). Therefore, for any The dual gap η should satisfy

[0137] η=f(μ)-g(Λ)≥η ＊ =t -1 (MN+L) (31)

[0138] By calculating the dual function g(Λ) according to formula (24) in each iteration and recording the reference lower bound function f LB =max{f LB ,g(Λ)}, we can update the barrier parameter for the next iteration as follows:

[0139]

[0140] Here, γ>1 is artificially set to control the update step size.

[0141] We have now completed the entire description of the principles of the present invention, and the summarized method flow chart is shown below.

[0142]

[0143]

[0144] That is, the present invention provides a two-dimensional generalized fast interior point method FDA-MIMO sonar fast gridless joint distance-angle estimation method, comprising:

[0145] Step 1: Perform frequency mixing and matched filtering on the signal received by the FDA-MIMO sonar system to obtain an extended virtual received signal model and express it in matrix form;

[0146] Step 2: The joint distance-angle estimation problem is formulated as the NsCP problem, and the parameters related to the interior point method are initialized.

[0147] Step 3: Use the restricted memory BFGS algorithm and the augmented KKT condition to update the primal and dual variables, and perform 2D-FFT fast feasible domain cone verification on the obtained variables. If the verification result is outside the feasible domain, repeat this step until the verification result is within the feasible domain, that is, the feasible domain cone verification is passed, and go to step 4;

[0148] Step 4: Check the convergence of the obtained variables. If converged, the optimal solution of the NsCP problem is obtained and go to step 5. Otherwise, go to step 3 and continue iterating.

[0149] Step 5: Based on the optimal solution, sparse atoms are retrieved through Vandermonde decomposition of the multi-level Toeplitz matrix, and the distance and angle estimates of the target are extracted using linear correspondences.

[0150] 3. Simulation Experiment

[0151] Here, we conduct simulation experiments to evaluate the performance of the proposed algorithm under different numbers of receiving array elements (N). The experimental setting is that the range-angle position parameters of the target are randomly set to 7 targets, and the number of receiving array elements N is varied. Figure 4 The changes in RMSE and running time of several advanced 2D parameter estimation methods in FDA-MIMO sonar range-angle joint estimation as N increases are shown respectively.

[0152] With a signal-to-noise ratio of 10 dB and an element spacing of 0.75 m, a 30-element linear array was set up to estimate four signal sources at positions θ = (-1°, 1°, 30°, 40°). The regularization parameter τ was set to 0.5. Figure 3(a) shows an array with no failed elements, meaning all elements are active. Figure 3(b) shows an array with one-third of the elements randomly selected to not receive the signal.

[0153] like Figure 4 As shown, for most algorithms, the angle estimation error (RMSEθ) decreases with the increase of N, while the distance estimation error (RMSE r ) does not improve significantly due to the inherent range resolution limitation imposed by the system bandwidth. It is worth noting that among all the listed algorithms, the proposed 2D-GFIPM achieves the largest RMSE θ . And in RMSE r and RMSE θ It performs well in both aspects. For grid-based methods (e.g., 2D-OMP, MD-SBL), the performance will plateau when N exceeds a critical threshold due to the influence of grid mismatch. Meanwhile, since MMV-DANM relies on covariance matrix reconstruction and statistical assumptions about the data, it suffers from unstable distance-angle estimation under limited snapshots.

[0154] Regarding running time, such as Figure 4 As shown, the proposed algorithm maintains an actual computation time of less than 1 second among sparse methods, although low-complexity methods such as 2D-ESPRIT and 2D-OMP consume relatively short computation time. This demonstrates good scalability for real-time applications.

[0155] In addition, we evaluate the performance of the proposed algorithm at different SNR levels. As in the previous experiments, 7 targets were randomly selected. The SNR range was from -10 to 25dB with a step size of 5dB. The results presented in the figure show that the proposed 2D-GFIPM demonstrates robust performance compared to other methods, achieving lower RMSE at all SNR levels. r and RMSE θ This highlights the effectiveness and reliability of the method in the presence of varying noise interference.

[0156] Example 2

[0157] Embodiment 2 of the present invention provides a computer program product, including a computer program. When the computer program is executed by a processor, the steps in the above embodiment 1 can be implemented.

[0158] 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 two-dimensional generalized fast interior point method for FDA-MIMO sonar fast gridless joint range-angle estimation, including: Step 1: Perform frequency mixing and matched filtering on the signal received by the FDA-MIMO sonar system to obtain an extended virtual received signal model and express it in matrix form; Step 2: The joint distance-angle estimation problem is formulated as the NsCP problem, and the parameters related to the interior point method are initialized. Step 3: Use the restricted memory BFGS algorithm and the augmented KKT condition to update the primal and dual variables, and perform 2D-FFT fast feasible domain cone verification on the obtained variables. If the verification result is outside the feasible domain, repeat this step until the verification result is within the feasible domain, that is, the feasible domain cone verification is passed, and go to step 4; Step 4: Check the convergence of the obtained variables. If converged, the optimal solution of the NsCP problem is obtained and go to step 5. Otherwise, go to step 3 and continue iterating. Step 5: Based on the optimal solution, sparse atoms are retrieved through Vandermonde decomposition of the multi-level Toeplitz matrix, and the distance and angle estimates of the target are extracted using linear correspondences; The NsCP problem in step 2 is: Where, The optimization variables The multi-level Toeplitz matrix formed, is the sparse recovery signal, is the Hermitian matrix, τ is the soft threshold constraint The associated hyperparameters, K is the total number of targets, is a hard constraint, μ represents the constraint, and the superscript H represents the conjugate transpose The step 5 comprises: Extraction according to Vandermonde decomposition in Among them, a(f k ) is the joint transmit-receive virtual steering vector; The distance estimation of the kth target is obtained according to the following formula and angle estimation Where c is the speed of sound in water, Δf is the frequency increment that ensures orthogonality between the transmitted signals, represents the reference frequency, d t is the distance between every two transmitting array elements, d r is the distance between every two receiving array elements.

2. The FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method according to claim 1 is characterized in that: The transmitting array and receiving array of the FDA-MIMO sonar system are both uniform linear arrays, wherein the transmitting array includes M units, the receiving array includes N units, and the distance between each two transmitting array units is d t , the distance between each two receiving array elements is d r .

3. The FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method according to claim 1 is characterized in that: The extended virtual received signal model Y obtained in step 1 is: in, δ k,l represents the signal sparse coefficient, E represents the noise term, the superscript T represents the transpose, a(f k ) is the joint transmit-receive virtual steering vector, α k ,β k represent the corresponding normalized frequencies of transmit and receive space, respectively, k represents the kth target, K represents the total number of targets, and L represents the total number of transmit pulses.

4. The FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method according to claim 3 is characterized in that: The relevant parameters of step 2 are initialized, including: barrier parameter update step size γ = 5, regularization parameter τ, relative convergence threshold ∈ relative and the absolute convergence threshold ∈ gap , i=0, represents the number of current iterations, refer to the lower bound f LB =-∞.

5. The FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method according to claim 4 is characterized in that: In step 3, the primal and dual variables are updated using the restricted memory BFGS algorithm and the augmented KKT condition, and the obtained variables are verified by 2D-FFT fast feasible domain cone. If the verification result is outside the feasible domain, the step is repeated, including: When the duality gap η of the i-th iteration i ≥∈ gap , and the duality gap difference between the two iterations |η i-1 -ηi|≥∈ relative When , construct and solve the following obstacle optimization problem, solve and update u i : Where h(u) = τξ T u+τtr(Y H T ML -1 (u+τe0)Y), H(u)=-logdet(T ML (u)), is a unit vector whose first element is equal to 1 and all other elements are equal to 0, φ = T ML -1 (u+τe0)Y,ξ=T * (I); Update the dual solution Λ according to the following formula i : Among them, z, S, and V are the optimization variables in the dual domain corresponding to the optimization variables u, X, and W respectively; Update the original solution μ according to the following formula: i :

6. The FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method according to claim 5 is characterized in that: In step 4, the obtained variables are checked for convergence. If converged, it is the optimal solution to the NsCP problem, including: The dual function g(Λ i ) Judgment Satisfaction From f LB ,g(Λ i ) takes the maximum value as the reference lower bound f LB ; and update the barrier parameter t for the i+1th iteration according to the following formula i+1 : Where γ>1 is the update step size.

7. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the FDA-MIMO sonar fast gridless joint distance-angle estimation method based on the two-dimensional generalized fast interior point method described in claim 1 are implemented.

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