Toeplitz matrix

In this paper two fast algorithms that use orthogonal simila rity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form a re developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost e fficiency by exploiting the rank structure of the input Toeplitz matrix. The proposed algorithms differ in the choi ce of the generator set for the rank structure of the input Toeplitz matrix.

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result

In this paper, a new efficient adaptive filtering algorithm belonging to the Quasi-Newton (QN) family is proposed. In the new algorithm, the autocorrelation matrix is assumed to be Toeplitz. Due to this assumption, the algorithm can be implemented in the frequency domain using the fast Fourier transform (FFT). The proposed algorithm turns out to be particularly suitable for adaptive channel equalization in wireless burst transmission systems. The algorithm exhibits a faster convergence rate and less computational ...

Let a, b and c be fixed complex numbers. Let M n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M n (a, b, c). We also show that all complex polynomials in M n (a, b, c) are Toeplitz matrices. In particular, the inverse of M n (a, b, c) is a Toeplitz matrix when it exists.

In certain imaging applications, conventional lens technology is constrained by the lack of materials which can effectively focus the radiation within a reasonable weight and volume. One solution is to use coded apertures-opaque plates perforated with multiple pinhole-like openings. If the openings are arranged in an appropriate pattern, then the images can be decoded and a clear image computed. Recently, computational imaging and the search for a means of producing programmable software-defined optics have revived interest in coded apertures. The former state-of-the-art masks, modified uniformly redundant arrays (MURAs), are effective for compact objects against uniform backgrounds, but have substantial drawbacks for extended scenes: (1) MURAs present an inherently ill-posed inversion problem that is unmanageable for large images, and (2) they are susceptible to diffraction: a diffracted MURA is no longer a MURA. We present a new class of coded apertures, separable Doubly-Toeplitz masks, which are efficiently decodable even for very large images-orders of magnitude faster than MURAs, and which remain decodable when diffracted. We implemented the masks using programmable spatial-light-modulators. Imaging experiments confirmed the effectiveness of separable Doubly-Toeplitz masks-images collected in natural light of extended outdoor scenes are rendered clearly. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

An adaptive disturbance rejection algorithm is developed for the standard control problem. The MIMO system and controller are represented as ARMARKOV/Toeplitz models, and the parameter matrix of the compensator is updated online by means of a gradient algorithm. The algorithm requires minimal knowledge of the plant, specifically, the numerator of the ARMARKOV transfer function from control to performance is required. No knowledge about the spectrum of the disturbance is needed. Experimental results demonstrating tonal and broadband disturbance rejection in an acoustic duct are presented

Popular preconditioners for conjugate gradient methods often reveal poor regularization properties that make them useless for very ill-conditioned linear systems arising in inverse problems. Recent results have awakened the interest towards the Tyrtyshnikov superoptimal preconditioners since it has been demonstrated that they exhibit good filtering capabilities. Here, in order to improve the regularizing behaviour, we generalize the definition of superoptimal

In a recent paper, we established asymptotic formulas for the eigenvalues of the $n\times n$ truncations of certain infinite Hessenberg Toeplitz matrices as $n$ goes to infinity. The symbol of the Toeplitz matrices was of the form $a(t)=t^{-1}(1-t)^{\alpha}f(t)$ ($t\in{\mathbb T}$), where $\alpha$ is a positive real number but not an integer and $f$ is a smooth function in $H^\infty$. Thus,