Positive Trigonometric Polynomials and Signal Processing Applications

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. Positive and sum-of-squares polynomials have received a special interest in the latest decade, due to their connections with semidefinite programming. Thus, efficient optimization methods can be employed to solve diverse problems involving polynomials. This book gathers the main recent results on positive trigonometric polynomials within a unitary framework; the theoretical results are obtained partly from the general theory of real polynomials, partly from self-sustained developments. The optimization applications cover a field different from that of real polynomials, mainly in signal processing problems: design of 1-D and 2-D FIR or IIR filters, design of orthogonal filterbanks and wavelets, stability of multidimensional discrete-time systems.
Positive Trigonometric Polynomials and Signal Processing Applications 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 presentation starts by giving the main results for univariate polynomials, which are later extended and generalized for multivariate polynomials. The applications part is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semidefinite programming form, ready to be solved with algorithms freely available, like those from the library SeDuMi.

Résumé
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)

Contenu

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.