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.

Table of Contents

1 – Normed vector spaces

2 – Analytic tools

3 – Fourier series

4 – The Fourier transform

5 – Compressive sampling

6 – Discrete transforms

7 – linear filters

8 -Windowed Fourier and continuous wavelet transforms. Frames

9 -Multiresolution analysis

1 0 – Discrete wavelet theory

11 – Biorthogonal filters and wavelets

12 – Parsimonious representation of data

Review

“Damelin and Miller provide a very detailed and thorough treatment of all the important mathematics related to signal processing. This includes the required background information found in elementary mathematics courses, so their book is really self-contained. The style of writing is suitable not only for mathematicians, but also for practitioners from other areas. Indeed, Damelin and Miller managed to write their text in a form that is accessible to nonspecialists, without giving up mathematical rigor.”

Kai Diethelm, Computing Reviews

“The authors’ aim is to use ideas from signal processing to motivate the mathematics behind the applications, and heir target readers are mathematics students who are interested in potential applications of wavelets and Fourier analysis, and engineering students who want to understand the mathematical foundation of their subject.”

Ahmed I. Zayed, Mathematical Reviews

Book Description

Develops mathematical and probabilistic tools needed to give rigorous derivations and applications of fundamental results in signal processing theory.

About the Author

Steven B. Damelin is currently Full Professor at Georgia Southern University and Visiting Full Professor at the University of the Witwatersrand in Johannesburg.

Willard Miller, Jr is Professor Emeritus in the School of Mathematics at the University of Minnesota.