Positive trigonometric polynomials and signal processing applications

• Main Author: Dumitrescu, Bogdan (Author)
• Format: Electronic eBook
• Language: English
• Published: Dordrecht : Springer, 2017.
• Edition: Second edition.
• Series: Signals and communication technology.
• Subjects: Polynomials.; Signal processing > Mathematical models.; Toeplitz matrices.; elektronické knihy; electronic books
• ISBN: 9783319536880; 9783319536873
• ISSN: 1860-4862
• Physical Description: 1 online resource (xvi, 276 pages) : illustrations

This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily base...

Table of Contents:

• Preface; 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; 1.5.1 Chebyshev Polynomials; 1.5.2 Positive Polynomials in mathbbR[t] as Sum-of-Squares; 1.5.3 Proof of Theorem 1.11; 1.5.4 Proof of Theorem 1.13; 1.5.5 Proof of Theorem 1.15; 1.5.6 Proof of Theorem 1.17; 1.5.7 Proof of Theorem 1.18; 1.6 Bibliographical and Historical Notes; References; 2 Gram Matrix Representation; 2.1 Parameterization of Trigonometric Polynomials.
• 2.2 Optimization Using the Trace Parameterization2.3 Toeplitz Quadratic Optimization; 2.4 Duality; 2.5 Kalman
• Yakubovich
• Popov Lemma; 2.6 Spectral Factorization from a Gram Matrix; 2.6.1 SDP Computation of a Rank-1 Gram Matrix; 2.6.2 Spectral Factorization Using a Riccati Equation; 2.7 Parameterization of Real Polynomials; 2.8 Choosing the Right Basis; 2.8.1 Basis of Trigonometric Polynomials; 2.8.2 Transformation to Real Polynomials; 2.8.3 Gram-Pair Matrix Parameterization; 2.9 Interpolation Representations; 2.10 Mixed Representations; 2.10.1 Complex Polynomials and the DFT.
• 2.10.2 Cosine Polynomials and the DCT2.11 Fast Algorithms; 2.12 Details and Other Facts; 2.12.1 Writing Programs with Positive Trigonometric Polynomials; 2.12.2 Proof of Theorem 2.16; 2.12.3 Proof of Theorem 2.19; 2.12.4 Proof of Theorem 2.21; 2.13 Bibliographical and Historical Notes; References; 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 Matrix Parameterization of Multivariate Trigonometric Polynomials; 3.5 Sum-of-Squares Relaxations; 3.5.1 Relaxation Principle; 3.5.2 A Case Study.
• 3.5.3 Optimality Certificate3.6 Gram Matrices from Partial Bases; 3.6.1 Sparse Polynomials and Gram Representation; 3.6.2 Relaxations; 3.7 Gram Matrices of Real Multivariate Polynomials; 3.7.1 Gram Parameterization; 3.7.2 Sum-of-Squares Relaxations; 3.7.3 Sparseness Treatment; 3.8 Pairs of Relaxations; 3.9 The Gram-Pair Parameterization; 3.9.1 Basic Gram-Pair Parameterization; 3.9.2 Parity Discussion; 3.9.3 LMI Form; 3.10 Polynomials with Matrix Coefficients; 3.11 Details and Other Facts; 3.11.1 Transformation Between Trigonometric and Real Nonnegative Polynomials.
• 3.11.2 Pos3Poly Program with Multivariate Polynomials3.11.3 A CVX Program Using the Gram-Pair Parameterization; 3.12 Bibliographical and Historical Notes; References; 4 Polynomials Positive on Domains; 4.1 Real Polynomials Positive on Compact Domains; 4.2 Trigonometric Polynomials Positive on Frequency Domains; 4.2.1 Gram Set Parameterization; 4.2.2 Gram-Pair Set Parameterization; 4.3 Bounded Real Lemma; 4.3.1 Gram Set BRL; 4.3.2 BRL for Polynomials with Matrix Coefficients; 4.3.3 Gram-Pair Set BRL; 4.4 Positivstellensatz for Trigonometric Polynomials; 4.5 Proof of Theorem 4.11.