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Among the directional angle calculation models of Bluetooth 5.1, the rectangular hollow array offers the advantage of shorter sampling time due to its fewer elements compared to traditional planar arrays. However, the antimultipath algorithms suitable for traditional planar arrays cannot be applied to rectangular hollow arrays. Therefore, this study proposes a virtual array filling algorithm, wherein four virtual matrices are inserted into the hollow matrix to transform the array into a uniform rectangular array. This algorithm ensures translation invariance of the rectangular array, enabling the application of the antimultipath coherent source algorithm to a rectangular hollow array. An algorithm for reconstructing Toeplitz matrices in two‐dimensional uniform planar arrays is also proposed. Through the analysis of the spatial spectrum and angle estimation results of various algorithms, the effectiveness of the signal angle of arrival estimation theory is verified. [ABSTRACT FROM AUTHOR]

A directed Toeplitz graph T n s 1 , ⋯ , s k ; t 1 , ⋯ , t l with vertices 1 , 2 , ⋯ , n is a directed graph whose adjacency matrix is a Toeplitz matrix. In this paper, we investigate the Hamiltonicity in directed Toeplitz graphs T n s 1 , ⋯ , s k ; t 1 , ⋯ , t l with s 1 = 1 and s 3 = 4. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the nonlinear Fredholm integro-differential equation of the second kind with singular kernel in two-dimensional NT-DFIDE. Furthermore, we study this new equation numerically. The existence of a unique solution of the equation is proved. The numerical results of NT-DFIDE are obtained by the following methods: Toeplitz matrix method (TMM) and product Nystrom method (PNM). The given applications showed the efficiency of these methods. [ABSTRACT FROM AUTHOR]

A principal singular vector based on multiple Toeplitz matrices is proposed to solve the accuracy problem of direction-of-arrival (DOA) estimation for coherent signals. First, the data matrix received by uniform linear array (ULA) is transformed into a Toeplitz matrix. An equivalent covariance matrix is obtained by square weighted summation method using the Toeplitz matrix. Then, a polynomial containing DOA information is constructed because the signal space and the steering matrix have the same column space; the Toeplitz matrix is built using polynomial coefficients. The problem is transformed into solving linear equations by establishing the relationship between the Toeplitz matrix and the signal subspace. Furthermore, the weighted least square method is used to obtain multiple candidates for linear equations. Finally, the maximum likelihood (ML) rule is used to select source signal candidates from multiple candidates. In comparison with currently known algorithm, the proposed algorithm has the characteristics of high estimation accuracy, low-complexity, and strong anti-interference ability and resolution. Even when the signal-to-noise ratio (SNR) is low, the snapshot number is small, and multiple signals exist; this method can still provide good estimation performance and resolution, which is more than 90% in most cases. Simulation experiments verify the superiority of the algorithm. [ABSTRACT FROM AUTHOR]

This paper adopts the matrix reconstruction method to analyze the digital image of urban landscape information in-depth and uses it to study and design the visual communication of its construction. Based on the special structure and properties of Toeplitz matrices, a stepwise structured Augmented Lagrange Multiplier (SALM) algorithm for Toeplitz matrix filling is proposed by introducing structured operators. The main idea of the algorithm is to structure the iteration matrix at each step, that is, to reassign the elements on each diagonal of the matrix by the operator. In this way, the approximation matrix always maintains the Toeplitz structure during the iterative process, and the fast singular value decomposition of the Toeplitz. A matrix can be exploited, thus saving time. The convergence theory of the new algorithm is further discussed. A lightweight progressive feature fusion module is designed to improve network learning efficiency, which consists of two components: progressive local connectivity and feature attention. Specifically, progressive local connectivity extracts multilevel features for fusion by layer-by-layer separation and local splicing. Feature attention evaluates the importance of features using both channel and spatial dual attention modules. The Oculus Rift virtual reality device system is used, and the OSG graphics rendering engine is used as the basis for the scene data transfer of the overall 3D cityscape towards the virtual reality headset display device system. With the support of the Oculus SDK, the secondary rendering of the overall 3D cityscape in the immersive virtual reality module is carried out, and the corresponding OSG camera browsing interface is constructed to realize the two-eye immersive virtual display of the 3D cityscape in a virtual reality headset display device, assisting in connecting the virtual cityscape to reality. [ABSTRACT FROM AUTHOR]

This paper proposes a method to address the problem of the joint direction of departure (DOD) and direction of arrival (DOA) estimation with augmented coprime array (CPA) bistatic multiple input multiple output (MIMO) radar using interpolating sensors. At first, we deduce the regular pattern of hole positions in the virtual array and interpolate a small number of sensors to augmented CPA to form a partially contiguous virtual array. Then, we use the diversity smoothing algorithm to reconstruct the Toeplitz matrix to obtain a spatial smoothing matrix. Finally, we combine the RD-MUSIC algorithm with the spatial smoothing matrix to estimate the spatial spectrum and achieve automatic matching of DODs and DOAs for the targets. Simulation results clearly illustrate the superiority of the method. [ABSTRACT FROM AUTHOR]

The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple. [ABSTRACT FROM AUTHOR]

In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability. [ABSTRACT FROM AUTHOR]

In order to realize the high-precision direction of arrival (DOA) estimation of the coherent source of two-dimensional multiple-input and multiple-output (MIMO) radar, a solution is given by combining Toeplitz matrix set reconstruction. MIMO radar obtains a larger aperture with fewer arrays. Traditional two-dimensional reconstruction Toeplitz-like algorithms use part of the information in the construction of two correlation matrices or covariance matrices to construct the Toeplitz matrix when performing two-dimensional coherent source DOA estimation, which makes the information utilization incomplete and requires additional denoising processing. To solve the above problems, this paper proposes an improved Toeplitz matrix set reconstruction algorithm based on the two-dimensional reconstruction Toeplitz class algorithm. The complete array element receiving signal vector is used to construct two Toeplitz matrix sets containing complete information, and then their conjugate transposes. Multiply and sum to correct the matrix to obtain a full-rank matrix, so as to achieve the purpose of decoherence and combine the traditional ESPRIT algorithm to perform two one-dimensional reconstruction processing through rotation invariance and then perform angle matching to achieve two-dimensional coherent signal angle estimation, while avoiding additional denoising processing. Finally, the simulation results of the cross array and the L-shaped array verify the effectiveness of the algorithm in this paper and further extend it to the two-dimensional MIMO radar array model and compare it with the traditional ESPRIT-like algorithm and the REC-FBSS-ESPRIT algorithm. In comparison, the algorithm in this paper has better performance under the conditions of successful resolution, DOA estimation accuracy, and low signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

A novel generalized nested multiple-input multiple-output (MIMO) radar for direction of arrival (DOA) estimation is proposed in this paper. The proposed structure utilizes the extended two-level nested array (ENA) as transmitter and receiver and adjusts the interelement spacing of the receiver with an expanding factor. By optimizing the array element configuration, we can obtain the best number of elements of the transmitter and receiver and the attainable degrees of freedom (DOF). Compared with the existing nested MIMO radar, the proposed MIMO array configuration not only has closed-form expressions for sensors' positions and the number of maximum DOF, but also significantly improves the array aperture. It is verified that the sum-difference coarray (SDCA) of the proposed nested MIMO radar can get higher DOF without holes. MUSIC algorithm based on Toeplitz matrix reconstruction is employed to prove the rationality and superiority of the proposed MIMO structure. [ABSTRACT FROM AUTHOR]

We consider the problem of clutter covariance matrix (CCM) estimation for space-time adaptive processing (STAP) radar in the small sample. In this paper, a fast efficient algorithm for CCM reconstruction is proposed to overcome this shortcoming for the linear structure. Particularly, we present a low-rank matrix recovery (LRMR) question about CCM estimation based on the Toeplitz structure of CCM and the prior knowledge of the noise. The closed-form solution is obtained by relaxing the nonconvex LRMR problem that the trace norm replaces the rank norm. The target can then be efficiently detected by using the recovered CCM according to the STAP theorem. We also analyze the algorithm model under the linear structure in the presence of unknown mutual coupling. It is shown that our method can obtain accurate CCM in the small sample, with even higher accuracy than the traditional algorithms in the same number of samples. It also can reduce the coupling effect and obtain more degrees of freedom (DOF) with limited sensors and pulses by utilizing sparse linear structure (SLS) and improve angle and Doppler resolutions. Finally, numerical simulations have verified the effectiveness of the proposed method in comparison with some of the existing methods. [ABSTRACT FROM AUTHOR]

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software. [ABSTRACT FROM AUTHOR]

We extend the concept of Pascal matrix to all matrices with the property that entries are the sum of left and above entries. We have obtained an algebraic relation that allows us to obtain a Toeplitz matrix from a Pascal matrix, and vice versa. This procedure can be applied to any Pascal matrices and any Toeplitz matrices, Hermitian matrices or not, moment matrices or not. The relationship with the difference operator and with the shift operator is also studied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The problem of parameter estimation of coherent signals impinging on an array with vector sensors is considered from a new perspective by means of the decomposition of tensors. Signal parameters to be estimated include the direction of arrival (DOA) and the state of polarization. In this paper, mild deterministic conditions are used for canonical polyadic decomposition (CPD) of the tensor-based signal model; i.e., the factor matrices can be recovered, as long as the matrices satisfy the requirement that at least one is full column rank. In conjoint with the estimation of signal parameters via the algebraic method, the DOAs and polarization parameters of coherent signals can be resolved by virtue of the first and second factor matrices. Hereinto, the key innovation of the proposed approach is that the proposed approach can effectively estimate the coherent signal parameters without sacrificing the array aperture. The superiority of the proposed algorithm is shown by comparing with the algorithms based on higher order singular value decomposition (HOSVD) and Toeplitz matrix. Theoretical and numerical simulations demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper proposes a superresolution two-dimensional (2D) direction of arrival (DOA) estimation algorithm for a rectangular array based on the optimization of the atomic l0 norm and a series of relaxation formulations. The atomic l0 norm of the array response describes the minimum number of sources, which is derived from the atomic norm minimization (ANM) problem. However, the resolution is restricted and high computational complexity is incurred by using ANM for 2D angle estimation. Although an improved algorithm named decoupled atomic norm minimization (DAM) has a reduced computational burden, the resolution is still relatively low in terms of angle estimation. To overcome these limitations, we propose the direct minimization of the atomic l0 norm, which is demonstrated to be equivalent to a decoupled rank optimization problem in the positive semidefinite (PSD) form. Our goal is to solve this rank minimization problem and recover two decoupled Toeplitz matrices in which the azimuth-elevation angles of interest are encoded. Since rank minimization is an NP-hard problem, a novel sparse surrogate function is further proposed to effectively approximate the two decoupled rank functions. Then, the new optimization problem obtained through the above relaxation can be implemented via the majorization-minimization (MM) method. The proposed algorithm offers greatly improved resolution while maintaining the same computational complexity as the DAM algorithm. Moreover, it is possible to use a single snapshot for angle estimation without prior information on the number of sources, and the algorithm is robust to noise due to its iterative nature. In addition, the proposed surrogate function can achieve local convergence faster than existing functions. [ABSTRACT FROM AUTHOR]

We first define Cesàro type classes of sequences of fuzzy numbers and equip the set with a complete metric. Then we compute the Köthe-Toeplitz dual and characterize some related matrix classes involving such classes of sequences of fuzzy numbers. [ABSTRACT FROM AUTHOR]

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices. [ABSTRACT FROM AUTHOR]

On the harmonic Dirichlet space of the unit disk, the commutativity of Toeplitz and Hankel operators is studied. We obtain characterizations of commuting Toeplitz and Hankel operators and essentially commuting (semicommuting) Toeplitz and Hankel operators with general symbols. [ABSTRACT FROM AUTHOR]

Let R denote the family of functions f(z) = z + Σn=2∞ anzn of bounded boundary rotation so that Re(f' (z)) > 0 in the open unit disk U = {z : |z| < 1}. We obtain sharp bounds for Toeplitz determinants whose elements are the coefficients of functions f ∈ R. [ABSTRACT FROM AUTHOR]

Application of the pure rank-one update algorithm as well as a combination of rank-one updates and the Sherman-Morrison formula in computing the Moore-Penrose inverse of the particular Toeplitz matrix is investigated in the present paper. Such Toeplitz matrices appear in the image restoration process and in many scientific areas that use the convolution. Four different approaches are developed, implemented, and tested on a number of numerical experiments. [ABSTRACT FROM AUTHOR]

We found that in the polydisk Dn there exist (n+1)(n+2)/2 different classes of commutative C⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that the C⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutative C⁎-algebra of Toeplitz operators. [ABSTRACT FROM AUTHOR]

In this paper, because of the acquisition principle and big data of remote sensing image, we suggest an encryption scheme using chaotic system to build up the compressed sensing framework. A two-dimensional generalized Arnold map is adopted here to generate the entries of the Toeplitz matrix used as measurement matrix. Then, the selecting row of the Toeplitz matrix is also decided by the chaotic sequence generated randomly by the Arnold map. With a further function of permutation operation, we can also encrypt remote sensing image after compression. The key space is constructed by the initial conditions in chaotic system. Experiment results show that the proposed method can be suitable to compressed sensing when the random measurement matrix is generated using chaotic system. Additionally, although the permutation operation is adopted, it is mainly used to enhance the security of cipher-image. The reconstruction result is only dependent on the standard recovery algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we essentially deal with Köthe-Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions. In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Finally, we obtain the Köthe-Toeplitz duals of fuzzy level sets with respect to the partial metric based on a partial ordering. [ABSTRACT FROM AUTHOR]

Compressive sensing (CS) takes advantage of the signal's sparseness in some domain, allowing the entire signal to be efficiently acquired and reconstructed from relatively few measurements. A proper measurement matrix for compressive sensing is significance in above processions. In most compressive sensing frameworks, random measurement matrix is employed. However, the random measurement matrix is hard to implement by hardware. So the randomness of the measurement matrix leads to the poor performance of signal reconstruction. In this paper, Toeplitz matrix is employed and optimized as a deterministic measurement matrix. A hardware platform for signal efficient acquisition and reconstruction is built by field programmable gate arrays (FPGA). Experimental results demonstrate the proposed approach, compare with the existing state-of-the-art method, and have the highest technical feasibility, lowest computational complexity, and least amount of time consumption in the same reconstruction quality. [ABSTRACT FROM AUTHOR]

It is time consuming to numerically solve fractional differential equations. The fractional ordinary differential equations may produce Toeplitz-plus-band triangular systems. An efficient iterationmethod for Toeplitz-plus-band triangular systems is presented with O(Mlog (M)) computational complexity and O(M) memory complexity in this paper, compared with the regular solution with O(M2) computational complexity and O(M2) memory complexity. M is the discrete grid points. Some methods such as matrix splitting, FFT, compress memory storage and adjustable matrix bandwidth are used in the presented solution. The experimental results show that the presented method compares well with the exact solution and is 4.25 times faster than the regular solution. [ABSTRACT FROM AUTHOR]

In this note it is shown that a fairly recent result on the asymptotic distribution of the zeros of generalized polynomials can be deduced from an old theorem of G. Polya, using a minimum of orthogonal polynomial theory.

The Toeplitz Procrustes problems are the least squares problems for the matrix equation AX= B over some Toeplitz matrix sets. In this paper the necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions of the Toeplitz Procrustes problemswhen the unknown matrices are constrained to the general, the triangular, and the symmetric Toeplitz matrices, respectively. The algorithms are designed and the numerical examples show that these algorithms are feasible. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a complete and practical method for numerical application of Padé approximation with the help of the -table analysis. We present an exhaustive list of useful formulas to compute a c-table related to a formal power series C(z) = ... cnzn . Some of these formulas are not widely known, because they were presented in publications of limited circulation. Some others were never published, as three symmetric Paszkowski-like formulas to overcome the blocks in a c-table or an extension of local error formula for Padé approximants in the blocks. All formulas are given in two versions: in terms of Toeplitz determinants (c-table) and in the version of Hankel determinants (c-table). We compare the theory with numerical observations by reproducing different computational aspects of software producing the c-tables with the presence of blocks and their evolution following the evolution of computer environment. [ABSTRACT FROM AUTHOR]

The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra of n-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, "On the fine spectra of triangular toeplitz operators" (2011) and Altun, "Fine spectra of tridiagonal symmetric matrices" (2011). Here, we generalize those results to the (2n +1)banded symmetric Toeplitz matrix operators for arbitrary positive integer n. [ABSTRACT FROM AUTHOR]

We focus on the behavior of solutions of the difference equation xn b1xn-1+b2xn-2+…+bnx0+yn, n = 1, 2,..., where (bk) is a fixed sequence of complex numbers, and (yk) is a fixed sequence in a complex Banach space. We give the general solution of this difference equation. To examine the asymptotic behavior of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz matrices. [ABSTRACT FROM AUTHOR]

We give a sufficient condition the solvability of two standard equations of Sylvester matrix by using the displacement structure of the Sylvester matrix, and, according to the sufficient condition, we derive a new fast algorithm for the inversion of a Sylvester matrix, which can be denoted as a sum of products of two triangular Toeplitz matrices. The stability of the inversion formula for a Sylvester matrix is also considered. The Sylvester matrix is numerically forward stable if it is nonsingular and well conditioned. [ABSTRACT FROM AUTHOR]

Let f be a nonnegative integrable function on [−π,π], Tn(f) the (n+1)×(n+1) Toeplitz matrix associated with f and λ1,n its smallest eigenvalue. It is shown that the convergence of λ1,n to minf(0) can be exponentially fast even when f does not satisfy the smoothness condition of Kac, Murdoch and Szegö (1953). Also a lower bound for λ1,n corresponding to a large class of functions which do not satisfy this smoothness condition is provided.