Rates of convergence for the approximation of dual shift-invariant systems in l2(Z)l_2(Z)

A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the form $g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system $\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each function $f$ can be written as $f = \sum g_{m,n}$. The mathematical theory usually addresses this problem in infinite dimensions (typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in $l_2(Z)$. For compactly supported $g_{m,n}$ (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper

Fast Algorithms for Displacement and Low-Rank Structured Matrices

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures

New Methods for MLE of Toeplitz Structured Covariance Matrices with Applications to RADAR Problems

This work considers Maximum Likelihood Estimation (MLE) of a Toeplitz structured covariance matrix. In this regard, an equivalent reformulation of the MLE problem is introduced and two iterative algorithms are proposed for the optimization of the equivalent statistical learning framework. Both the strategies are based on the Majorization Minimization (MM) paradigm and hence enjoy nice properties such as monotonicity and ensured convergence to a stationary point of the equivalent MLE problem. The proposed framework is also extended to deal with MLE of other practically relevant covariance structures, namely, the banded Toeplitz, block Toeplitz, and Toeplitz-block-Toeplitz. Through numerical simulations, it is shown that the new methods provide excellent performance levels in terms of both mean square estimation error (which is very close to the benchmark Cram\'er-Rao Bound (CRB)) and signal-to-interference-plus-noise ratio, especially in comparison with state of the art strategies.Comment: submitted to IEEE Transactions on Signal Processing. arXiv admin note: substantial text overlap with arXiv:2110.1217

On the decay of the off-diagonal singular values in cyclic reduction

It was recently observed in [10] that the singular values of the off-diagonal blocks of the matrix sequences generated by the Cyclic Reduction algorithm decay exponentially. This property was used to solve, with a higher efficiency, certain quadratic matrix equations encountered in the analysis of queuing models. In this paper, we provide a theoretical bound to the basis of this exponential decay together with a tool for its estimation based on a rational interpolation problem. Numerical experiments show that the bound is often accurate in practice. Applications to solving n × n block tridiagonal block Toeplitz systems with n × n quasiseparable blocks and certain generalized Sylvester equations in O(n 2 log n) arithmetic operations are shown

A TBLMI Framework for Harmonic Robust Control

The primary objective of this paper is to demonstrate that problems related to stability and robust control in the harmonic context can be effectively addressed by formulating them as semidefinite optimization problems, invoking the concept of infinite-dimensional Toeplitz Block LMIs (TBLMIs). One of the central challenges tackled in this study pertains to the efficient resolution of these infinite-dimensional TBLMIs. Exploiting the structured nature of such problems, we introduce a consistent truncation method that effectively reduces the problem to a finite-dimensional convex optimization problem. By consistent we mean that the solution to this finite-dimensional problem allows to closely approximate the infinite-dimensional solution with arbitrary precision. Furthermore, we establish a link between the harmonic framework and the time domain setting, emphasizing the advantages over Periodic Differential LMIs (PDLMIs). We illustrate that our proposed framework is not only theoretically sound but also practically applicable to solving H 2 and H$\infty$ harmonic control design problems. To enable this, we extend the definitions of H 2 and H$\infty$ norms into the harmonic space, leveraging the concepts of the harmonic transfer function and the average trace operator for Toeplitz Block operators. Throughout this paper, we support our theoretical contributions with a range of illustrative examples that demonstrate the effectiveness of our approach

A polynomial fit preconditioner for band Toeplitz matrices in image reconstruction

The Preconditioned Conjugate Gradient is often applied in image reconstruction as a regularizing method. When the blurring matrix has Toeplitz structure, the modified circulant preconditioner and the inverse Toeplitz preconditioner have been shown to be effective. We introduce here a preconditioner for symmetric positive definite Toeplitz matrices based on a trigonometric polynomial fit which has the same effectiveness of the previous ones but has a lower cost when applied to band matrices. The case of band block Toeplitz matrices with band Toeplitz blocks (BTTB) corresponding to separable point spread functions is also considered