Overview

This book gathers the main recent results on positive trigonometric polynomials within a unitary framework. The book has two parts: theory and applications. The theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The applications part is organized as a collection of related problems that use systematically the theoretical results.

Full Product Details

Author:   Bogdan Alexandru Dumitrescu
Publisher:   Springer
Imprint:   Springer
Edition:   1st ed. Softcover of orig. ed. 2007
Dimensions:   Width: 15.60cm , Height: 1.30cm , Length: 23.40cm
Weight:   0.397kg
ISBN:  

9789048172887

ISBN 10:   9048172888
Pages:   242
Publication Date:   15 December 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Replaced By:   9783319536873
Format:   Paperback
Publisher's Status:   Active
Availability:   In stock  
We have confirmation that this item is in stock with the supplier. It will be ordered in for you and dispatched immediately.

Table of Contents

1. Positive polynomials. 1.1 Types of polynomials. 1.2 Positive polynomials. 1.3 Toeplitz positivity conditions. 1.4 Positivity on an interval. 1.5 Details and other facts. 1.6 Bibliographical and historical notes. 2. Gram matrix representation. 2.1 Parameterization of trigonometric polynomials. 2.2 Optimization using the trace parameterization. 2.3 Toeplitz quadratic optimization. 2.4 Duality. 2.5 Kalman-Yakubovich-Popov lemma. 2.6 Spectral factorization from a Gram matrix. 2.7 Parameterization of real polynomials. 2.8 Choosing the right basis. 2.9 Interpolation representations. 2.10 Mixed representations. 2.11 Fast algorithms. 2.12 Details and other facts. 2.13 Bibliographical and historical notes. 3. Multivariate polynomials. 3.1 Multivariate polynomials. 3.2 Sum-of-squares multivariate polynomials. 3.3 Sum-of-squares of real polynomials. 3.4 Gram matrices of trigonometric polynomials. 3.5 Sum-of-squares relaxations. 3.6 Gram matrices from partial bases. 3.7 Gram matrices of real multivariate polynomials. 3.8 Pairs of relaxations. 3.9 The Gram pair parameterization. 3.10 Polynomials with matrix coefficients. 3.11 Details and other facts. 3.12 Bibliographical and historical notes. 4. Polynomials positive on domains. 4.1 Real polynomials positive on compact domains. 4.2 Polynomials positive on frequency domains. 4.3 Bounded Real Lemma. 4.4 Positivstellensatz. 4.5 Details and other facts. 4.6 Bibliographical and historical notes. 5. Design of FIR filters. 5.1 Design of FIR filters. 5.2 Design of 2-D FIR filters. 5.3 FIR deconvolution. 5.4 Bibliographical and historical notes. 6. Orthogonal filterbanks. 6.1 Two-channel filterbanks. 6.2 Signal-adapted wavelets. 6.3 GDFT modulated filterbanks. 6.4 Bibliographical and historical notes. 7. Stability. 7.1 Multidimensional stability tests. 7.2 Robust stability. 7.3 Convex stability domains. 7.4 Bibliographical and historical notes. 8. Design of IIR filters. 8.1 Magnitude design of IIR filters. 8.2 Approximate linear-phase designs. 8.3 2D IIR filter design. 8.4 Bibliographical and historical notes Appendix A: semidefinite programming. Appendix B: spectral factorization. References.

Reviews

From the reviews: The book under review is a new contribution on the topic, with a focus on signal processing applications. ! this is the first self-contained manuscript on this emerging research area, and hence it is a welcome and timely contribution to the technical literature. ! use of illustrative numerical examples, accompanied by Matlab scripts, allows the inexperienced reader to grasp the essential ideas without having to understand all the mathematical subtelties. In particular, signal processing engineers should benefit a lot from reading the book ! . (Didier Henrion, Zentralblatt MATH, Vol. 1126 (3), 2008) Trigonometric polynomials that are positive on the unit circle play an essential role in a number of digital filtering problems. ! The text would be quite suitable for use as the basis of lectures, for it has proofs written out, a survey of the literature and many exercises. (A. Bultheel, Mathematical Reviews, Issue 2007 m)

From the reviews: The book under review is a new contribution on the topic, with a focus on signal processing applications. ... this is the first self-contained manuscript on this emerging research area, and hence it is a welcome and timely contribution to the technical literature. ... use of illustrative numerical examples, accompanied by Matlab scripts, allows the inexperienced reader to grasp the essential ideas without having to understand all the mathematical subtelties. In particular, signal processing engineers should benefit a lot from reading the book ... . (Didier Henrion, Zentralblatt MATH, Vol. 1126 (3), 2008) Trigonometric polynomials that are positive on the unit circle play an essential role in a number of digital filtering problems. ... The text would be quite suitable for use as the basis of lectures, for it has proofs written out, a survey of the literature and many exercises. (A. Bultheel, Mathematical Reviews, Issue 2007 m)

Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions