On the Eigenvalues of Matrices for the Reconstruction of Missing Uniform Samples

Problem conditioning interpolation educational tool

Abstract: This paper presents an educational tool to be used in signal processing interpolation-related subjects. Besides the consolidation of acquired theoretical knowledge, the tool allows its users to apply three error patterns geometry to the signals and test minimum dimension and maximum dimension signal reconstruction algorithms. In the specific case of minimum dimension problems it can be solved using different solvers, iterative and direct linear equations methods. The developed tool allows the problem conditioning analysis through the spectral radius of the system matrix, the condition number and others parameters available in some specific methods. This feature gives the possibility to alter the problem definitions to the desired goal before the reconstruction begins and to choose the optimal method, depending on each problem constraints The time unit that measures the algorithms performance is expressed in terms of one Fourier Transform (FFT) calculation time. In this way the data is presented not in an absolute way but in a relative measure independent from the machine's architecture.

Signal Processing Interpolation and Problem Conditioning Educational Workbench

Abstract: This work presents a newer and more complete version of an educational tool to be used in signal processing interpolation-related subjects. Besides the consolidation of acquired theoretical knowledge, the tool allows now its users to apply three error geometry patterns, test minimum or maximum dimension signal reconstruction algorithms and problem conditioning through the analysis of the matrix spectral radius or the condition number: these new features gives the possibility to alter the problem definitions to the desired goal before the reconstruction begins. The time unit that measures the algorithms performance is ¶(nlogn) thus independent from the machine’s architecture. A video of the developed tool can be seen in: https://www.dropbox.com/s/t6yiiuy31ramxse/FILME_1.avi.

Matrix Analysis

Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

Prolate spheroidal wave functions, fourier analysis and uncertainty — II

Abstract: The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.

Toeplitz Forms and Their Applications.

Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

Toeplitz forms and their applications

Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

Super-resolution through Error Energy Reduction

Abstract: A new view of the problem of continuing a given segment of the spectrum of a finite object is presented. Based on this, the problem is restated in terms of reducing a defined ‘error energy’ which is implicit in the truncated spectrum. A computational procedure, which is readily implemented on general purpose computers, is devised which must reduce this error. It is demonstrated that by so doing, resolution well beyond the diffraction limit is attained. The procedure is shown to be very effective against noisy data.