Book: Positive Trigonometric Polynomials and Signal Processing Applications (Signals and Communication Technology)
"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)
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.