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

Abstract

The invention discloses an FDA-MIMO sonar fast meshless joint distance-angle estimation method and product based on a two-dimensional generalized fast interior point method, and the method comprises the steps: s1, carrying out the frequency mixing and matched filtering of a received signal, and obtaining an expanded virtual received signal model; s2, an estimation problem is concluded as an NsCP problem, and related parameters of an interior point method are initialized; s3, updating an original variable and a dual variable by using a limited memory BFGS algorithm and an augmented KKT condition, performing 2D-FFT fast feasible region cone verification on the obtained variables, repeating the step until a result is in a feasible region, and turning to s4; s4, carrying out convergence check on the obtained variables, if convergence is carried out, obtaining an optimal solution of the NsCP problem, and turning to s5, otherwise, turning to step 3; and s5, according to the optimal solution, sparse atoms are retrieved through Vandermonde decomposition of the multi-stage Toeplitz matrix, and distance and angle estimation of the target is extracted by using a linear corresponding relation.

Description

Technical Field

[0001] The present invention belongs to the technical field of underwater detection, and particularly relates to a fast meshless joint range-angle estimation method and product for FDA-MIMO sonar based on two-dimensional generalized fast interior point method. Background Art

[0002] As an advanced underwater detection technology, the multiple-input multiple-output (MIMO) sonar system has broad application prospects in both civilian and military fields. Its core principle is to work with multiple transmitters and receivers simultaneously, transmitting signals using orthogonal waveforms, so as to obtain richer spatial information than traditional sonar systems. In recent years, the frequency diverse array (FDA) technology has been introduced into the MIMO architecture, forming an FDA-MIMO sonar system. By introducing small frequency increments in the transmit array, this system makes beamforming dependent on the distance and angle of the target, thereby realizing high-precision range-angle joint estimation of the target. The dense MIMO sonar with an FDA structure is an implementation form of high-resolution MIMO sonar, which closely arranges the transmit and receive antennas and constructs a large-aperture virtual array by cleverly utilizing the orthogonality of the transmitted signals. Compared with traditional phased array sonar systems, the FDA-MIMO sonar has significant advantages in terms of the accuracy, resolution, and degrees of freedom of target parameter estimation, can depict the characteristic information of the target more finely, and provides more powerful support for the identification and tracking of underwater targets.

[0003] In a sonar detection system, accurately estimating the position parameters of a target, including distance, angle, intensity, etc., is a key link for effective detection. For the detection of moving targets, due to the distance displacement between different pulses, this will interfere with the result of distance estimation, thereby affecting the accurate grasp of the target's motion state. To solve this problem, special technical means usually need to be applied under specific assumptions.

[0004] In the existing technical solutions, the parameter estimation methods based on the compressive sensing (CS) theory (such as the l1-svd, two-dimensional orthogonal matching pursuit (2D-OMP), and multi-dictionary sparse Bayesian learning multi-dimensional (MD–SBL) methods, etc.) are an implementation scheme relatively close to the present invention. The compressive sensing theory states that sparse data can be represented by a few elements in a predefined grid, and these elements encode the core information. The CS-based parameter estimation methods utilize the sparse representation characteristics of the data, rather than relying on statistical characteristics that only appear in large datasets, so as to be able to function under limited data or poor data statistical characteristics.

[0005] Meshless CS theory extends CS to the continuous domain, encoding effective information into atoms defined in a continuous-domain atom set rather than in predefined grid elements. Meshless CS methods in multi-dimensional parameter estimation mainly solve an atomic norm minimization (ANM) optimization problem by using some optimization tools such as the alternating direction method of multipliers (ADMM) or the CVX toolbox. Different applied optimization tools thus evolve into different numerical meshless CS parameter estimation methods. To achieve lightweight and real-time joint estimation of meshless multi-dimensional parameters, some new methods such as two-dimensional iterative Vandermonde decomposition and shrinkage thresholding (2D-IVDST) and decoupled atomic norm minimization for multi-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 restricted by the constraints of predefined grids and deviate from the actual target detection model to a certain extent. In practical applications, the distance and angle of the target are continuous values and it is difficult to exactly coincide with the points on the predefined grid, which limits the application effect of traditional CS methods.

[0007] The meshless CS optimization methods based on ADMM and CVX directly call the computationally expensive semidefinite programming (SDP), require accurate noise priors, and are slow and burdensome in calculation, making it difficult to meet the requirements of real-time and efficiency in the underwater detection environment. 2D-IVDST is essentially an accelerated proximal gradient technique. Although it has a fast convergence rate, it will destroy the original structure of the optimization variables in the threshold shrinkage step, resulting in the solution deviating from the optimal solution of ANM optimization; MMV-DANM depends on the covariance matrix reconstruction of the received data and the statistical assumptions of the data, and has poor robustness.

[0008] In summary, there are still many deficiencies in the application of existing technologies in FDA-MIMO sonar systems. Generally speaking, although the traditional CS parameter estimation method relying on prefabricated grids can play a role when the data in MIMO sonar detection is limited or the statistical characteristics are poor, the traditional CS method relies on predefined grids and deviates from the actual target detection model. And the current relatively advanced meshless CS technology relies on optimization tools such as ADMM or the CVX toolbox, which are high in computational cost, slow in speed, and burdensome, making it difficult to meet the real-time and efficiency requirements of underwater detection. Summary of the Invention

[0009] The object of the present invention is to overcome the defects of the prior art, and a fast meshless joint distance-angle estimation method for FDA-MIMO sonar based on the two-dimensional generalized fast interior point method is proposed to realize the fast joint estimation of the distance and angle of underwater targets. The present invention also discloses a computer program product, which realizes the fast meshless joint distance-angle estimation method for FDA-MIMO sonar based on the two-dimensional generalized fast interior point method.

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

[0011] Step 1: Mix and match-filter the signals received by the FDA-MIMO sonar system to obtain an extended virtual received signal model, and represent it in matrix form;

[0012] Step 2: Reduce the joint distance-angle estimation problem to an NsCP problem, and initialize the relevant parameters of the interior point method;

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

[0014] Step 4: Check the convergence of the obtained variables. If it converges, obtain the optimal solution of the NsCP problem, go to Step 5, otherwise, go to Step 3 to continue the iteration;

[0015] Step 5: According to the optimal solution, retrieve the sparse atoms through the Vandermonde decomposition of the multi-level Toeplitz matrix, and extract the distance and angle estimates of the target using the linear correspondence relationship.

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

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

[0018]

[0019] where k,l represents the signal sparse coefficient, E represents the noise term, the superscript T represents the transpose, a(f k) is the combined transmit - receive virtual steering vector, k , β k respectively represent the corresponding transmit and receive spatial normalized frequencies, k represents the k - th 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] In the formula, is the multi - level Toeplitz matrix formed by the optimization variables , sparse recovery signal, is the Hermitian matrix, τ is the hyperparameter associated with the soft - thresholding constraint , K is the sparsity level, is the 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: the barrier parameter update step size γ = 5, the regularization parameter τ, the relative convergence threshold ∈ relative and the absolute convergence threshold ∈ gap , i = 0 represents the current iteration number label, and the reference lower bound f LB = - ∞.

[0024] Preferably, in step 3, the limited - memory BFGS algorithm and the augmented KKT conditions are used to update the primal variables and dual variables, and the obtained variables are verified by 2D - FFT in the feasible region cone. If the verification result is outside the feasible region, this step is repeated, including:

[0025] When the duality gap η i ≥ ∈ gap , and the difference between the duality gaps of two consecutive iterations |η i-1 -η i |≥ ∈ relativ at this time e , construct the barrier optimization problem and solve for the updated 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] where z, S, V are the optimization variables in the dual domain corresponding to the optimization variables u, X, W respectively. In particular, S also represents the residual of sparse recovery;

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

[0032]

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

[0034] Calculate the dual function g(Λ i )

[0035]

[0036] Judge to satisfy Take the maximum value from f LB , g(Λ i ) as the reference lower bound f LB ; and update the barrier parameter t for the (i + 1)-th iteration according to the following formula i+1 :

[0037]

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

[0039] Preferably, step 5 includes:

[0040] Extract according to the Vandermonde decomposition of the following formula in

[0041]

[0042] Obtain the distance estimate of the k-th target according to the following formula and the angle estimate

[0043]

[0044] Among them, c is the speed of sound in water, and Δf is the frequency increment to ensure orthogonality between transmitted signals. represents the reference frequency.

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

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

[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 complexity of a single iteration is reduced from O(N 6 M 6 ) to O(N 3 (M 2 -M + 2)), and the number of iterations is reduced to less than 50 times, realizing 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 gridded compressive sensing method. At the same time, by combining the atomic norm soft threshold and multi-snapshot joint sparsity to suppress noise interference, the resolution ability under low signal-to-noise ratio is improved; in terms of algorithm stability, by using the logarithmic determinant barrier function to constrain the feasible region of the original variables, and innovatively proposing a cone verification method based on the 2D fast Fourier transform (FFT), the asymmetric feasible region constraint of high-dimensional dual variables is efficiently determined to ensure global convergence, solving the problem of insufficient stability caused by difficult verification of dual variables in the two-dimensional extension of the traditional FIPM. Thus, high-precision and high-efficiency joint distance-angle estimation is achieved in a complex underwater environment. Description of the Drawings

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

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

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

[0051] Figure 4 is the variation trend of the running time with the number N of receiving array elements;

[0052] Figure 5 shows the variation trend of RMSE with different signal-to-noise ratios SNR, where Figure 5(a) is RMSE r , and Figure 5(b) is RMSE θ . Detailed implementation manners

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

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

[0055] Embodiment 1

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

[0057] 1. Principle description

[0058] The algorithmic process of the FDA-MIMO sonar fast meshless joint distance-angle estimation method using the two-dimensional generalized fast interior point method (2D-GFIPM) is shown in the following figure. The specific steps are as follows. First, at the receiving array end, the received signal is processed through mixing and matched filtering to expand the virtual array received signal, and several snapshots are arranged as the received signal matrix according to the arrival wave timing. The second step is to reduce the problem of joint distance-angle estimation extracted from the above received signal matrix to the NsCP problem and initialize the relevant parameters of the interior point method. The third step is to update the primal variable and the dual variable using the limited memory BFGS (L-BFGS) algorithm and the augmented KKT conditions, and perform a 2D-FFT fast feasible region cone verification on the obtained variables. If the verification result falls outside the feasible region, repeat the variable update step just performed until the obtained variables pass the feasible region cone verification. Check the convergence of the above solution. If it does not converge, still return to the variable update step. If it converges, it means that we have obtained the optimal solution to the NsCP problem. Using this optimal solution, retrieve the sparse atoms through the Vandermonde decomposition of the multistage Toeplitz matrix, and extract the distance and angle estimates of the target using the linear correspondence relationship.

[0059] 2. Algorithmic Structure

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

[0061]

[0062] where denotes the reference frequency, and Δf is the frequency increment that ensures orthogonality between the transmitted signals. The signal transmitted by the m-th element 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, considering a distance of r kand an angle of θ k (k = 1, …, K) far-field targets. As shown in the figure, the propagation delay of the signal transmitted by the m-th transmitting array element, reflected by the k-th target, and received by the n-th receiving array element is

[0068]

[0069] where c is the speed of sound in water. The received signal at the n-th receiving array element associated with the l-th pulse echo is

[0070]

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

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

[0073] At the n-th receiving element of the n-th pulse, the matched filtering output of the m-th reference function is shown in Equation (6) below

[0074]

[0075] where The x n,m (l) in Equation (6) corresponds to the element in the n-th row and m-th column of the matrix, which can be written as:

[0076]

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

[0078] Rearranging each row of X(l) in Equation (7) by stacking can obtain an MN-dimensional virtual data vector:

[0079]

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

[0081] assuming that the zero - mean noise with variance \(\sigma\) 2 is identically distributed across array elements, pulses, and samples, the processed signal model for \(L\) snapshots is

[0082]

[0083] where \(E\) represents the noise term.

[0084] Furthermore, by exploiting the assumption that the number of signal sources in space is sparse, we can represent the range - angle estimation model of signal sources in the FDA - MIMO sonar system in the following form of atomic norm minimization, that is

[0085]

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

[0087]

[0088] where is the multi - level Toeplitz (MLT) matrix formed by the optimization variable , which encodes the parameters to be estimated (\(\alpha\) k , \(\beta\) k ). In addition, the Hermitian matrix and the sparse recovery signal are also part of the optimization variable \(\mu\). In Equation (11), \(\tau\) is the hyperparameter associated with the soft - threshold constraint , which controls the trade - off between the fidelity and sparsity inherent in any estimator involving the sparsity level \(K\). Choosing a larger \(\tau\) usually results in a smaller estimated value of \(K\). Following the simple extension discussed in the one - dimensional spectral frequency estimation problem, we set Numerical simulations have shown that this setting is usually effective and enables the proposed algorithm to be applied in practice.

[0089] The core idea of solving Equation (11) in the present invention is to approximate the hard constraint \(\kappa\) with a smooth barrier function, transforming the constrained optimization problem (11) into an unconstrained problem with a soft threshold. The appropriate logarithmic barrier function \(F(\mu)\) of \(\kappa\) is given by

[0090]

[0091] The barrier function F(μ) is a smooth and strictly convex function, which satisfies F(μ k ) → ∞ as the sequence of points μ k → μ bd , where μ bd represents a point on the κ boundary. This actually approximates the constraint By introducing the logarithmic barrier, we transform the constrained optimization problem into the following unconstrained problem (also known as 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, the t -1 F(μ) term gets closer and closer to the role of the indicator function, resulting in the constraints imposed by the original conditions becoming gradually more important. Eventually, this process leads to the solution of the primal problem with the barrier function approaching the optimal solution of the optimization problem (10).

[0093] In each step, μ* is solved from the solution of the previous step regarding the optimization problem (13), called the central 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 primal point on the central path of the problem (11).

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

[0098]

[0099] where represents the Lagrangian function, is called the dual cone, representing the feasible region of the dual variables.

[0100] From equation (16), we can solve for the solution of the dual variables:

[0101]

[0102] Regarding we hope to focus more on the essential discussion, that is, to give the results

[0103] Note that the MLT matrix T ML (u) is in the form as follows

[0104]

[0105] where

[0106]

[0107] (d = 1, 2, in the present invention N1 = M, N2 = N) represents an elementary Toeplitz matrix, which is 1 on the n d th diagonal line and 0 elsewhere

[0108] Therefore, for any Hermitian matrix there is

[0109]

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

[0111] In addition, we can introduce the adjoint operator T * to obtain a further expression. The operation acting on a certain Hermitian matrix is defined as q = T * (Q), and its n = (n1, n2)th element is given by the following formula

[0112]

[0113] and also satisfies Therefore, a more complete analytical expression of the dual solution related to the augmented KKT condition can be calculated

[0114]

[0115] where ξ = T *(I). If we ignore the matrix structure of the optimization variables μ and Λ and instead focus on the basic numerical variables, μ and Λ are only constrained by a finite number of variables. This greatly reduces the number of variables 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 primal problem (11) is

[0117]

[0118] Substitute the dual solution into the second equation (complementary condition) of the extended KKT condition (16) to obtain the primal solution:

[0119]

[0120] where 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 all other primal-dual variables have explicit solutions except for the variable u which can only provide implicit relationships.

[0121] To solve the variable u in (25), consider constructing a barrier optimization problem as follows:

[0122] min ψ(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)). The gradient of h(u) + t -1 H(u) vanishes when the optimization problem (26) reaches the optimal solution, and H(u) serves as a barrier function for the following constraint (MLT positive semi-definite cone):

[0124]

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

[0126] However, since the dual solution needs to be calculated from the primal solution μ (t) obtained by numerical methods, the dual solution cannot be guaranteed to always lie 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 for verifying the dual feasible region, that is:

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

[0128]

[0129] where represents the dual cone of the MLT positive semi-definite cone . And the judgment of the dual cone can be obtained by fast calculation, that is, for any if the result of its Fourier transform

[0130]

[0131] can always be satisfied, then where q n is the element in the sequence q at .

[0132] Obviously, the method provided in (29) adopts the form of 2D Fourier transform. The fast Fourier transform allows the effective calculation of (29) at a large number of points on . The above steps provide a low-complexity method to determine This method is approximate because (29) is sampled at a finite number of points on , but the approximation can be made arbitrarily accurate by increasing the number of evaluation points. In our simulation experiments, it is proved that the approximation is accurate enough for practical applications.

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

[0134] According to the optimality condition (14), the primal variables at μ = μ ＊ can satisfy This leads to the Lagrangian function for within the feasible region to reach the lower bound. Therefore, the dual function g can be calculated at the dual point (15):

[0135]

[0136] where the ideal dual gap is calculated as η ＊ = f(μ＊ ) - g(Λ) = t -1 (MN + L). Thus, for any 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 Equation (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 shown in the following equation:

[0139]

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

[0141] So far, we have completed the entire description of the principle of the present invention. The summarized method flow chart is as follows.

[0142]

[0143]

[0144] That is, a fast meshless joint distance - angle estimation method for FDA - MIMO sonar using a two - dimensional generalized fast interior point method of the present invention includes:

[0145] Step 1: Mix and match - filter the signals received by the FDA - MIMO sonar system to obtain an extended virtual received signal model and represent it in matrix form;

[0146] Step 2: Convert the joint distance - angle estimation problem into an NsCP problem and initialize the relevant parameters of the interior point method;

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

[0148] Step 4: Check the convergence of the obtained variables. If it converges, obtain the optimal solution of the NsCP problem and go to Step 5; otherwise, go back to Step 3 to continue the iteration;

[0149] Step 5: According to the optimal solution, retrieve the sparse atoms through the Vandermonde decomposition of the multi-level Toeplitz matrix, and use the linear correspondence relationship to extract the distance and angle estimates of the target.

[0150] 3. Simulation Experiments

[0151] Here, we conducted simulation experiments to evaluate the performance of the proposed algorithm under different numbers of receiving array elements (N). The distance-angle position parameters of the experimental setup targets were randomly set to 7 targets, and the number of receiving array elements N was varied. Figures 3 and Figure 4 respectively show the changes in RMSE and running time of several advanced 2D parameter estimation methods with the increase of N in the FDA-MIMO sonar distance-angle joint estimation.

[0152] At a signal-to-noise ratio of 10 dB, with an element spacing of 0.75 m, a linear array of 30 elements was set up to estimate four signal sources with signal source positions of θ = (-1°, 1°, 30°, 40°), and the regularization parameter τ = 0.5 was set. The array shown in Figure 3(a) has no failed elements, that is, all elements are in the active state. The array shown in Figure 3(b) randomly selects 1 / 3 of the elements not to receive signals.

[0153] As Figure 4 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 distance resolution limitation specified by the system bandwidth. It is worth noting that among all the listed algorithms, the proposed 2D-GFIPM achieves the largest magnitude of RMSE θ . And it performs excellently in both aspects of RMSE r and RMSE θ . For grid-based methods (such as 2D-OMP, MD-SBL), when N exceeds the critical threshold due to the influence of grid mismatch, the performance will be in a steady state. At the same time, since MMV-DANM relies on covariance matrix reconstruction and statistical assumptions about data, it suffers from unstable distance-angle estimation under limited snapshots.

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

[0155] In addition, we evaluated the performance of the proposed algorithm at different SNR levels. As in the previous experiment, 7 targets were randomly selected. The SNR ranged from -10 to 25 dB with a step size of 5 dB. The results presented in the figure show that the proposed 2D-GFIPM demonstrated robust performance, 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 various steps in Embodiment 1 above can be implemented.

[0158] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the embodiments, those of ordinary skill in the art should understand that any modification or equivalent replacement of the technical solutions of the present invention does not depart from the spirit and scope of the technical solutions of the present invention, and they should all be covered by the scope of the claims of the present invention.

Claims

1. A two-dimensional generalized fast interior point method for FDA-MIMO sonar fast gridless joint distance-angle estimation method, comprising: Step 1: Mix and match filter the signal received by the FDA-MIMO sonar system to obtain an extended virtual receiving signal model and express it in matrix form; Step 2: The joint distance-angle estimation problem is summarized as the NsCP problem, and the parameters related to the interior point method are initialized; Step 3: Use the restricted memory BFGS algorithm and augmented KKT conditions to update the original variables 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, pass the feasible domain cone verification, 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 to 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 estimation of the target are extracted using linear correspondence.

2. The FDA-MIMO sonar fast gridless joint distance-angle estimation method of 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 spacing between every 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 of 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, k,l represents the signal sparse coefficient, E represents the noise term, the superscript T represents the transposition, a(f k ) is the joint transmit-receive virtual steering vector, k ,β k They represent the corresponding normalized frequencies of the transmitting and receiving spaces, k represents the kth target, K represents the total number of targets, and L represents the total number of transmitted pulses.

4. The FDA-MIMO sonar fast gridless joint distance-angle estimation method of the two-dimensional generalized fast interior point method according to claim 3 is characterized in that: The NsCP problem of step 2 is: In the formula, The optimization variable 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 constraint, and the superscript H represents conjugate transpose.

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

6. The FDA-MIMO sonar fast gridless joint distance-angle estimation method of the two-dimensional generalized fast interior point method according to claim 5, characterized in that: 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 verified by 2D-FFT fast feasible domain cone. If the verification result is outside the feasible domain, the step is repeated, including: When the dual gap η of the i-th iteration i ≥ gap , and the dual gap difference between the two iterations |η i-1 -η i |≥∈ relat When , construct and solve the following obstacle optimization problem to 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 : L i : Among them, z, S, V are the optimization variables in the dual domain corresponding to the optimization variables u, X, W respectively, and S also represents the residual of sparse recovery; Update the original solution μ according to the following formula: i : m i :

7. The FDA-MIMO sonar fast gridless joint distance-angle estimation method of the two-dimensional generalized fast interior point method according to claim 6 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 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.

8. The FDA-MIMO sonar fast gridless joint distance-angle estimation method of the two-dimensional generalized fast interior point method according to claim 7, characterized in that: The step 5 comprises: Extraction according to Vandermonde decomposition In The distance estimate 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, Indicates the reference frequency.

9. 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 of the two-dimensional generalized fast interior point method described in claim 1 are implemented.

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