The Mathematics of Signal Processing

Arising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Φx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.

Spis treści:
1. Introduction
2. Normed vector spaces
3. Analytic tools
4. Fourier series
5. Fourier transforms
6. Compressive sensing
7. Discrete transforms
8. Linear filters
9. Windowed Fourier transforms, continuous wavelets, frames
10. Multiresolution analysis
11. Discrete wavelet theory
12. Biorthogonal filters and wavelets
13. Parsimonious representation of data
Bibliography
Index.