Neural network-aided sparse convex optimization algorithm for fast DOA estimation

In this paper, a fast sparse convex optimization algorithm based on a neural network is proposed to improve the direction of arrival estimation. First, a fast [Formula: see text]-sparse representation of the array covariance vector model based on the Hermitian Toeplitz structure of array covariance is established to reduce computational complexity in data dimension and variable number. Then, the estimation error upper bound problem is investigated, and a neural network-aided coefficient selection method is developed. The direction of arrival estimation problem is solved through spectral peak search. Finally, the algorithm is extended to the case of off-grid error. The algorithm’s advantages in accuracy, calculation speed and robustness is verified by the simulations.

Korovkin-type theorems on $$B({\mathcal {H}})$$ and their applications to function spaces

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, $$C^{*}$$ C ∗ -algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on $$B({\mathcal {H}})$$ B ( H ) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space $$A^{2}({\mathbb {D}})$$ A 2 ( D ) , Fock space $$F^{2}({\mathbb {C}})$$ F 2 ( C ) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk $${\mathbb {D}}$$ D and on the whole complex plane $${\mathbb {C}}$$ C . It is worthwhile to notice that so far such results are available for Toeplitz operators on circle only. This also establishes the role of Korovkin-type approximation techniques on function spaces with certain oscillation property. To address the function theoretic questions using these operator theory tools will be an interesting area of further research.

Gridless DOA Estimation for Minimum-Redundancy Linear Array in Nonuniform Noise

A gridless direction-of-arrival (DOA) estimation method to improve the estimation accuracy and resolution in nonuniform noise is proposed in this paper. This algorithm adopts the structure of minimum-redundancy linear array (MRA) and can be composed of two stages. In the first stage, by minimizing the rank of the covariance matrix of the true signal, the covariance matrix that filters out nonuniform noise is obtained, and then a gridless residual energy constraint scheme is designed to reconstruct the signal covariance matrix of the Hermitian Toeplitz structure. Finally, the unknown DOAs can be determined from the recovered covariance matrix, and the number of sources can be acquired as a byproduct. The proposed algorithm can be regarded as a gridless version method based on sparsity. Simulation results indicate that the proposed method has higher estimation accuracy and resolution compared with existing algorithms.

A fourth-order cumulant orthonormal propagator rooting method based on Toeplitz approximation

A novel Toeplitz fourth-order cumulant (FOC) orthonormal propagator rooting method (TFOC ‐ OPRM) of direction-of-arrival (DOA) estimation for uniform linear array (ULA) is proposed in this paper. Specifically, the modified (i.e., reduced-dimension) FOC (MFOC) matrix is achieved at first via removing the redundant information encompassed in the primary FOC matrix; then, the TFOC matrix which possesses Toeplitz structure can be recovered by utilizing the Toeplitz approximation method. To reduce the computational complexity, an effective method based on the polynomial rooting technology is adopted. Finally, the DOAs of incident signals can be estimated by exploiting orthonormal propagator rooting method. The theoretical analysis coupled with simulation results show that the proposed resultant algorithm can reduce the computational complexity significantly, as well as improve the estimation performance in both spatially white noise environment and spatially color noise environment.

Efficient Implementation and Numerical Analysis of Finite Element Method for Fractional Allen-Cahn Equation

We embed the fractional Allen-Cahn equation into a Galerkin variational framework and thus develop its corresponding finite element procedure and then prove rigorously its mathematical and physical properties for the finite element solution. Combining the merits of the conjugate gradient (CG) algorithm and the Toeplitz structure of the coefficient matrix, we design a fast CG for the linearized finite element scheme to reduce the computation cost and the storage to O(M log⁡ M ) and O(M), respectively. Numerical experiments confirm that the proposed fast CG algorithm recognizes accurately the mass and energy dissipation, the phase separation through a very clear coarse graining process, and the influences of different indices r of fractional Laplacian and different coefficients K,η on the width of the interfaces.

Knowledge-Aided Structured Covariance Matrix Estimator Applied for Radar Sensor Signal Detection

This study deals with the problem of covariance matrix estimation for radar sensor signal detection applications with insufficient secondary data in non-Gaussian clutter. According to the Euclidean mean, the authors combined an available prior covariance matrix with the persymmetric structure covariance estimator, symmetric structure covariance estimator, and Toeplitz structure covariance estimator, respectively, to derive three knowledge-aided structured covariance estimators. At the analysis stage, the authors assess the performance of the proposed estimators in estimation accuracy and detection probability. The analysis is conducted both on the simulated data and real sea clutter data collected by the IPIX radar sensor system. The results show that the knowledge-aided Toeplitz structure covariance estimator (KA-T) has the best performance both in estimation and detection, and the knowledge-aided persymmetric structure covariance estimator (KA-P) has similar performance with the knowledge-aided symmetric structure covariance estimator (KA-S). Moreover, compared with existing knowledge-aided estimator, the proposed estimators can obtain better performance when secondary data are insufficient.