Toeplitz Matrices Research Articles

Reduced-complexity multiple parameters estimation via toeplitz matrix triple iteration reconstruction with bistatic MIMO radar

In this advanced exploration, we focus on multiple parameters estimation in bistatic Multiple-Input Multiple-Output (MIMO) radar systems, a crucial technique for target localization and imaging. Our research innovatively addresses the joint estimation of the Direction of Departure (DOD), Direction of Arrival (DOA), and Doppler frequency for incoherent targets. We propose a novel approach that significantly reduces computational complexity by utilizing the Temporal-Spatial Nested Sampling Model (TSNSM). Our methodology begins with a multi-linear mapping mechanism to efficiently eliminate unnecessary virtual Degrees of Freedom (DOFs) and reorganize the remaining ones. We then employ the Toeplitz matrix triple iteration reconstruction method, surpassing the traditional Temporal-Spatial Smoothing Window (TSSW) approach, to mitigate the single snapshot effect and reduce computational demands. We further refine the high-dimensional ESPRIT algorithm for joint estimation of DOD, DOA, and Doppler frequency, eliminating the need for additional parameter pairing. Moreover, we meticulously derive the Cramér-Rao Bound (CRB) for the TSNSM. This signal model allows for a second expansion of DOFs in time and space domains, achieving high precision in target angle and Doppler frequency estimation with low computational complexity. Our adaptable algorithm is validated through simulations and is suitable for sparse array MIMO radars with various structures, ensuring higher precision in parameter estimation with less complexity burden.

On galaxies of sequences of toeplitz matrix solutions of the Diophantine Equation Xn + Y n = Z n , n ≥ 3

<p>We construct the galaxies of sequences of Toeplitz matrix solutions of the Diophantine equation Xn + Y n = Z n , n ≥ 3, linked to Pythagorean triples.</p>

Isotropy groups of the action of orthogonal *congruence on Hermitian matrices

We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation.

Constructing Pavement Roughness of a Three-Dimensional Bridge in Abaqus for Vehicle Scanning Method

In this study, a procedure for constructing pavement roughness in a three-dimensional bridge subjected to a vehicle using Abaqus is proposed for numerical simulation of vehicle–bridge interaction problems. As the plug-in RufGen, a graphical user interface in Abaqus/CAE for surface interaction, creates a hexahedron with a rough surface and attaches it to the bridge deck, the resulting weight of a bridge model can be much larger than the original one. In the vehicle scanning method, it is essential to keep a finite element bridge model with an unchanged weight after imposing a rough surface on it in consideration of the roughness effect for bridge health monitoring. As such, the proposed procedure is aimed at resolving this issue, with the following two highlights: First, the Toeplitz matrix is introduced to generate a rough surface in three dimensions to ensure the randomness of roughness in the two perpendicular directions under a given roughness profile. Second, a shell-type rough pavement is established for directly using as the bridge pavement or attaching to the bridge deck, which minimizes weight increment in the bridge.

Computing a compact local Smith–McMillan form

We define a compact local Smith–McMillan form of a rational matrix R ( λ ) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of R ( λ ) . We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of R ( λ ) around an arbitrary complex point λ 0 , allows us to compute a compact local Smith-McMillan form of that rational matrix R ( λ ) at the point λ 0 , provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point λ 0 . Numerical tests illustrate the promising performance of the resulting algorithm.

Mathematical Foundations of the Non-Hermitian Skin Effect

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

Eigenvalue superposition for Toeplitz matrix-sequences with matrix order dependent symbols

The eigenvalues of Toeplitz matrices Tn(f) with a real-valued generating function f, satisfying some conditions and tracing out a simple loop over the interval [−π,π], are known to admit an asymptotic expansion with the formλj(Tn(f))=f(σj,n)+c1(σj,n)h+c2(σj,n)h2+O(h3), where h=1/(n+1), σj,n=πjh, and ck are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [−π,π].In this article we investigate the superposition caused over this expansion, when considering the following linear combinationλj(Tn(f0)+βn,1Tn(f1)+βn,2Tn(f2)), where βn,1,βn,2 are certain constants depending on n and the generating functions f0,f1,f2 are either simple loop or satisfy the weaker conditions mentioned before.We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting.The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis.

A note on the paper “A note on inversion of Toeplitz matrices” by X.-G. Lv and T.-Z. Huang

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (2007), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have given, especially when the size of the Toeplitz matrix is very large. Numerical experiments illustrate the effectiveness of our theoretical results.

Improved multiple-Toeplitz matrices reconstruction method using quadratic spatial smoothing for coherent signals DOA estimation

PurposeThe purpose of this paper is to exploit the multiple-Toeplitz matrices reconstruction method combined with quadratic spatial smoothing processing to improve the direction-of-arrival (DOA) estimation performance of coherent signals at low signal-to-noise ratio (SNRs).Design/methodology/approachAn improved multiple-Toeplitz matrices reconstruction method is proposed via quadratic spatial smoothing processing. Our proposed method takes advantage of the available information contained in the auto-covariance matrices of individual Toeplitz matrices and the cross-covariance matrices of different Toeplitz matrices, which results in a higher noise suppression ability.FindingsTheoretical analysis and simulation results show that, compared with the existing Toeplitz matrix processing methods, the proposed method improves the DOA estimation performance in cases with a low SNR. Especially for the cases with a low SNR and small snapshot number as well as with closely spaced sources, the proposed method can achieve much better performance on estimation accuracy and resolution probability.Research limitations/implicationsThe study investigates the possibility of reusing pre-existing designs for the DOA estimation of the coherent signals. The proposed technique enables achieve good estimation performance at low SNRs.Practical implicationsThe paper includes implications for the DOA problem at low SNRs in communication systems.Originality/valueThe proposed method proved to be useful for the DOA estimation at low SNR.

Fixed-point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices

UDC 517.9 In [V. M. Abramov, <em>Bull. Austral. Math. Soc.,</em> <strong>104</strong>, 108–117 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present paper, we provide an alternative proof  for the existence of a positive solution in the general case. The presented proof is based on the application of a version of the  M. A. Krasnosel'ski fixed-point theorem. The results are then extended to the equations with infinite matrices of a general type.

Direction of Arrival Estimation Method Based on Eigenvalues and Eigenvectors for Coherent Signals in Impulsive Noise

In this paper, a Toeplitz construction method based on eigenvalues and eigenvectors is proposed to combine with traditional denoising algorithms, including fractional low-order moment (FLOM), phased fractional low-order moment (PFLOM), and correntropy-based correlation (CRCO) methods. It can improve the direction of arrival (DOA) estimation of signals in impulsive noise. Firstly, the algorithm performs eigenvalue decomposition on the received covariance matrix to obtain eigenvectors and eigenvalues, and then the Toeplitz matrix is created according to the eigenvectors corresponding to its eigenvalues. Secondly, the spatial averaging method is used to obtain an unbiased estimate of the Toeplitz matrix, which is then weighted and added based on the corresponding eigenvalues. Next, the noise subspace of the Toeplitz matrix is reconstructed to obtain the one that has less angle information. Finally, the DOA of the coherent signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. The improved method based on the Toeplitz matrix can not only suppress the effect of impulsive noise but can also solve the problem of aperture loss due to its decoherence. A series of simulations have shown that they have better performances than other algorithms.

An Efficient Method for Solving Problems of Acoustic Scattering on Three-Dimensional Transparent Structures

The article contains a study of methods for solving integral equations in the context of acoustic problems. The methodology considered is applied to describe acoustic wave propagation and scattering. Efficient discretization methods are used together with iterative methods to solve the operator equations, including an apparatus for fast multiplication of the resulting post-discretization Toeplitz matrices by a vector using the fast Fourier transform. The theoretical analysis of the proposed numerical algorithm demonstrates its efficiency in terms of the required number of arithmetic operations and the memory footprint of the computing system. The presented numerical simulation demonstrates the possibility of solving the problem of acoustic wave propagation in transparent media using the proposed methods. A visualization of the obtained solutions for a practical problem with a high level of discretization of the solution volume domain is also presented.

A 3-D spatial–temporal near-field parameter estimation method with two correlation operations

This paper proposes a narrowband near-field (NF) source localization algorithm based on two correlation operations. By utilizing the correlation of incident signal in the time and spatial domains, the algorithm achieves direction-of-arrival (DOA) with high-precision and low-complexity and range estimation under the virtual single snapshot. To do so, in the processing of NF source received signals, this paper first utilizes the symmetry of the cross-cross array and obtains a covariance matrix containing only azimuth and elevation information through two correlation operations. Since the covariance matrix has a standard two-level Toeplitz matrix form, the estimated parameters can be directly extracted from the noise-free covariance matrix without pairing using the two-level Vandermonde decomposition technique. Finally, the estimated value of the range is obtained using a one-dimensional (1-D) spectral peak search. Computer simulation results verify the superior performance of the proposed algorithm in terms of complexity and estimation accuracy.

Efficient 2D DOA Estimation via Decoupled Projected Atomic Norm Minimization

This paper presents an efficient two-dimensional (2D) direction of arrival (DOA) estimation method, termed as decoupled projected atomic norm minimization (D-PANM), to solve the angle-ambiguity problem. It first introduces a novel atomic metric via projecting the original atom set onto a smoothing space, based on which we formulate an equivalent semi-definite programming (SDP) problem. Then, two relatively low-complexity decoupled Toeplitz matrices can be obtained to estimate the DOAs. We further exploit the structural information hidden in the newly constructed data to avoid pair matching for the azimuth and elevation angles when the number of sensors is odd, and then propose a fast and feasible decoupled alternating projections (D-AP) algorithm, reducing computational complexity to a great extent. Numerical simulations are performed to demonstrate that the proposed algorithm is no longer restricted by angle ambiguity scenarios, but instead provides a more stable estimation performance, even when multiple signals share the same angles in both azimuth and elevation dimensions. Additionally, it greatly improves the resolution, with control of the computation load compared with the existing atomic norm minimization (ANM) algorithm.

Modeling Excitable Cells with the EMI Equations: Spectral Analysis and Iterative Solution Strategy

In this work, we are interested in solving large linear systems stemming from the extra–membrane–intra model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations equipped with proper boundary conditions, we provide its finite element discretization and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate preconditioned Krylov solvers. Through numerical experiments, we show that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.

Fast Coupling Iterative Algorithm for Media Rough Surface with Multiple Targets Below

This paper proposes a fast mutual coupling iteration algorithm for analyzing the composite electromagnetic scattering characteristics of a Gaussian rough surface with multiple targets below it. Firstly, an electromagnetic scattering model of Gaussian rough surface with multiple targets underneath is established. To improve the efficiency of electromagnetic scattering calculations, the (banded matrix iterative approach canonical grid - conjugate gradient method (BMIA/CAG-CMG) algorithm is used to calculate the surface scattering of the rough surface, while for the scattering of the targets, the traditional numerical algorithm moment of methods (MoM) is used for the calculation. Its main acceleration principle is to decompose the matrix into banded and Toeplitz matrices during the solution process, and then use fast Fourier transform (FFT) for fast solution based on their characteristics. By comparing the calculation results of this algorithm with those of the MoM algorithm, the correctness of this algorithm is verified. Several examples are studied, and the impact of different parameters on the surface current and composite scattering characteristics is discussed, which have important significance for deepening the understanding of scattering characteristics in complex electromagnetic environments.

Efficient nonparametric estimation of Toeplitz covariance matrices

Abstract A new efficient nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the discrete cosine transform is proposed.

Block-encoding structured matrices for data input in quantum computing

The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related algorithms. We demonstrate how to construct block encoding circuits based on an arithmetic description of the sparsity and pattern of repeated values of a matrix. We present schemes yielding different subnormalisations of the block encoding; a comparison shows that the best choice depends on the specific matrix. The resulting circuits reduce flag qubit number according to sparsity, and data loading cost according to repeated values, leading to an exponential improvement for certain matrices. We give examples of applying our block encoding schemes to a few families of matrices, including Toeplitz and tridiagonal matrices.

A fast NTRU software implementation based on 5-way TMVP

The fast and efficient operation of post-quantum cryptographic algorithms has become an important research topic, especially with the recent developments and the process of the PQC Standardization Project by NIST. Various algorithms based on lattices are chosen to be finalists in this project. Number Theoretic Transform (NTT), Toom–Cook, Karatsuba, and Toeplitz matrix–vector product (TMVP) are considered in order to efficiently perform multiplications in quotient polynomial rings required by the cryptographic schemes based on lattices. In this paper, we propose a 5-way split TMVP-based algorithm for multiplication in polynomial quotient rings and determine its computational complexity. The new algorithm is derived from a 5-term polynomial multiplication algorithm using 13 multiplications with coefficients of only 1 or -1 that provide efficiency. We also implement it for all parameter sets of NTRU. The results show that we have up to 34%, 35%, and 157% speed-up against Toom4-Karatsuba implementation in key generation, encapsulation, and decapsulation, respectively.

Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study

AbstractIt is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.