Block Toeplitz Research Articles

Extremal solutions of matricial Carathéodory problem and Nevanlinna–Pick-type interpolation for Carathéodory matrix functions in the nondegenerate case

This article is devoted to the investigation of remarkable properties of extremal solutions F [U] and G [U] with a q × q unitary parameter U and their distinguished one-parameter families and with |w| = 1 within the solution sets of Problem (C) (the matricial Carathéodory problem) and Problem (NP) (a certain Nevanlinna–Pick-type interpolation for Carathéodory matrix functions), respectively, in the nondegenerate case. We show connections between the poles of the functions F [U] and at the unit circle of the complex plane and their Riesz–Herglotz measures, and discuss closely related matters for the extremal solutions F [U] and . These results further serve as a starting point to look for corresponding properties of the functions G [U] and based on the so-called modified block Toeplitz vector approach.

New preconditioners for systems of linear equations with Toeplitz structure

In this paper, we consider applying the preconditioned conjugate gradient (PCG) method to solve system of linear equations $$T x = \mathbf b $$ where $$T$$ is a block Toeplitz matrix with Toeplitz blocks (BTTB). We first consider Level-2 circulant preconditioners based on generalized Jackson kernels. Then, BTTB preconditioners based on a splitting of BTTB matrices are proposed. We show that the BTTB preconditioners based on splitting are special cases of embedding-based BTTB preconditioners, which are also good BTTB preconditioners. As an application, we apply the proposed preconditioners to solve BTTB least squares problems. Our preconditioners work for BTTB systems with nonnegative generating functions. The implementations of the construction of the preconditioners and the relevant matrix-vector multiplications are also presented. Finally, Numerical examples, including image restoration problems, are presented to demonstrate the efficiency of our proposed preconditioners.

Block circulant and block Toeplitz approximants of a class of spatially distributed systems—An LQR perspective

In this paper block circulant and block Toeplitz long strings of MIMO systems with finite length are compared with their corresponding infinite-dimensional spatially invariant systems. The focus is on the convergence of the sequence of solutions to the control Riccati equations and the convergence of the sequence of growth bounds of the closed-loop generators. We show that block circulant approximants of infinite-dimensional spatially invariant systems reflect well the LQR behavior of the infinite-dimensional systems. An example of an exponentially stabilizable and exponentially detectable infinite-dimensional system is also considered in this paper. This example shows that the growth bounds of the Toeplitz approximants, as the length of the string increases, and the growth bound of the corresponding infinite-dimensional model, can be significantly different. On the positive side, we give sufficient conditions for the convergence of the growth bounds.

A multicore solution to Block–Toeplitz linear systems of equations

There exist algorithms, also called "fast" algorithms, which exploit the special structure of Toeplitz matrices so that, e.g., allow to solve a linear system of equations in O(n 2) flops. However, ...

Estimates for the minimum eigenvalue and the condition number of Hermitian (block) Toeplitz matrices

We give a lower bound for the minimum eigenvalue of the Hermitian Toeplitz matrix Tn(|θ|α) and a corresponding upper bound for the spectral condition number κ2(Tn(|θ|α)). Our main theorem concerns more general cases and establishes a lower bound for the minimum eigenvalue of Tn(f) and a corresponding upper bound for κ2(Tn(f)), provided the non-negative real-valued symbol f satisfies certain conditions. We discuss some examples of symbols for which these estimates work and we see how the minimax principle can be applied together with our main result in order to obtain estimates of λmin(Tn(f)) and κ2(Tn(f)) even in some cases in which the symbol f does not satisfy the conditions of our main theorem. Finally, we provide an extension of the main result to the block Toeplitz case.

Blind Identification of Multi-Channel ARMA Models Based on Second-Order Statistics

This correspondence presents a new second-order statistical approach to blind identification of single-input multiple-output (SIMO) autoregressive and moving average (ARMA) system models. The proposed approach exploits the dynamical autoregressive information of the model contained in the autocorrelation matrices of the system outputs but does not require the block Toeplitz structure of the channel convolution matrix used by classical subspace methods. For the multi-channel model with the same autoregressive (AR) polynomial, sufficient conditions and an efficient identification algorithm are given such that the multi-channel model can be uniquely identified up to a constant scaling factor. Furthermore, an extension of the result to blind identification of multi-channel models with different AR polynomials is presented. Simulation results are given to show the effectiveness of the proposed approach.

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmosʼs Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamseʼs theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.

Statistical covariance-matching based blind channel estimation for zero-padding MIMO–OFDM systems

We propose a statistical covariance-matching based blind channel estimation scheme for zero-padding (ZP) multiple-input multiple-output (MIMO)–orthogonal frequency division multiplexing (OFDM) systems. By exploiting the block Toeplitz channel matrix structure, it is shown that the linear equations relating the entries of the received covariance matrix and the outer product of the MIMO channel matrix taps can be rearranged into a set of decoupled groups. The decoupled nature reduces computations, and more importantly guarantees unique recovery of the channel matrix outer product under a quite mild condition. Then the channel impulse response matrix is identified, up to a Hermitian matrix ambiguity, through an eigen-decomposition of the outer product matrix. Simulation results are used to evidence the advantages of the proposed method over a recently reported subspace algorithm applicable to the ZP-based MIMO–OFDM scheme.

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

AbstractWe consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix.For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix.

Hyponormality and subnormality of block Toeplitz operators

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit circle.First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic.Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[U∗??U∗] so that A becomes subnormal, where U is the unilateral shift on H2.

A Fast Algorithm for Errors-in-Variables Filtering

This note concerns the optimal estimation of the input and output sequences of linear time-invariant errors-in-variables (EIV) processes. An efficient recursive filtering algorithm is proposed. It is an innovation-based approach that relies on the triangular decomposition of block Toeplitz matrices. Unlike the other algorithms described in the literature, the proposed one is characterized by a computational complexity which increases only linearly with the order of the process. Both the SISO and MIMO cases are analyzed. An extension of the described algorithm to EIV models with colored input and output noises is considered as well.

Limiting spectral distribution of block matrices with Toeplitz block structure

We study two specific symmetric random block Toeplitz (of dimension k×k) matrices, where the blocks (of size n×n) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by nk) when (a) k is fixed and n→∞ (b) n is fixed and k→∞ (c) n and k go to ∞ simultaneously. Further, the LSDs obtained in (a) and (b) coincide with those in (c) when n or respectively k tends to infinity. This limit in (c) is the semicircle law in Case (i). In Case (ii), the limit is related to the limit of the random symmetric Toeplitz matrix as obtained by Bryc et al. (2006) and Hammond and Miller (2005).

Lax pairs and Fourier analysis: The case of sine-Gordon and Klein-Gordon equations

In this paper we construct a new Lax pair for the Klein-Gordon equation. The structure algebra of this Lax pair is the algebra \U0001d4af \U0001d49c2 of upper triangular Toeplitz block matrices with blocks. For the suitable choice of the values of the spectral parameter, the integrals of motion, obtained from the holonomy of the spatial part of the Lax pair, have simple expressions in terms of the Fourier data. We compare these integrals to the corresponding integrals of the sine-Gordon system.

Fast implementation of sparse iterative covariance-based estimation for source localization

Fast implementations of the sparse iterative covariance-based estimation (SPICE) algorithm are presented for source localization with a uniform linear array (ULA). SPICE is a robust, user parameter-free, high-resolution, iterative, and globally convergent estimation algorithm for array processing. SPICE offers superior resolution and lower sidelobe levels for source localization compared to the conventional delay-and-sum beamforming method; however, a traditional SPICE implementation has a higher computational complexity (which is exacerbated in higher dimensional data). It is shown that the computational complexity of the SPICE algorithm can be mitigated by exploiting the Toeplitz structure of the array output covariance matrix using Gohberg-Semencul factorization. The SPICE algorithm is also extended to the acoustic vector-sensor ULA scenario with a specific nonuniform white noise assumption, and the fast implementation is developed based on the block Toeplitz properties of the array output covariance matrix. Finally, numerical simulations illustrate the computational gains of the proposed methods.

A Novel Subspace Channel Estimation with Fast Convergence for ZP-OFDM Systems

Novel subspace channel estimation with fast convergence is proposed for single-input multiple-output (SIMO) zero-padded orthogonal frequency-division multiplexing (ZP-OFDM) systems. Stacking all the received OFDM symbols turns the channel matrix to be block Toeplitz. The block matrix scheme is then presented so a group of sub-vectors is formed from the stacked OFDM signal and the number of samples is increased. With the sub-vector samples, the perturbations of the sample correlation matrix and the noise subspace are reduced considerably and therefore, the channel estimation error is lowered. Computer simulations demonstrate that the proposed channel estimation is effective and robust compared with existing methods.

A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of Li et al. (2010), where the authors studied the eigenvalue distribution μ X N of random block Toeplitz band matrices with given block order m . In this paper, we will give explicit density functions of lim N → ∞ μ X N when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of m × m Gaussian unitary ensemble (GUE for short). The series { lim N → ∞ μ X N ∣ m ∈ N } can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest. The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.

Using the Sherman–Morrison–Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman–Morrison–Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman–Morrison–Woodbury inversion. Numerical experiments are provided to illustrate the theory.

A gap between hyponormality and subnormality for block Toeplitz operators

This paper concerns a gap between hyponormality and subnormality for block Toeplitz operators. We show that there is no gap between 2-hyponormality and subnormality for a certain class of trigonometric block Toeplitz operators (e.g., its co-analytic outer coefficient is invertible). In addition we consider the extremal cases for the hyponormality of trigonometric block Toeplitz operators: in this case, hyponormality and normality coincide.

The maximization of determinants of block Toeplitz matrices with applications in optimal design theory

We consider the maximization of the determinant of the trigonometric moment matrix of a matrix measure on the unit circle. The results are analog to those obtained by Dette and Studden (2005) for moment matrices of matrix measures on a compact interval. A statistical application in optimal design theory is considered and we present several optimal designs for Fourier regression models.