Time‒frequency and Time‒scale Methods - (Applied and Numerical Harmonic Analysis) by Jeffrey A Hogan (Hardcover)

Book Synopsis

Developed in this book are several deep connections between time-frequency (Fourier/Gabor) analysis and time-scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of these areas are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time-frequency and time-scale methods as a rich area of work.

Foundations of Time-Frequency and Time-Scale Methods will be suitable for applied mathematicians and engineers in signal/image processing and communication theory, as well as researchers and students in mathematical analysis, signal analysis, and mathematical physics.

From the Back Cover

Developed in this book are several deep connections between time--frequency (Fourier/Gabor) analysis and time--scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of developments in time--frequency and time--scale analysis are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time--frequency and time--scale methods as a rich area of work.

Topics and Features:

* Inclusion of historical, background material such as the pioneering ideas of von Neumann in quantum mechanics and Landau, Slepian, and Pollak in signal analysis

* Presentation of self-contained core material on wavelets, sampling reconstruction of bandlimited signals, and local trigonometric and wavelet packet bases

* Development of connections based on perspectives that emerged after the wavelet revolution of the 1980s

* Integrated approach to the use of Fourier/Gabor methods and wavelet methods

* Comprehensive treatment of Fourier uncertainty principles

* Explanations at the end of each chapter addressing other major developments and new directions for research

Applied mathematicians and engineers in signal/image processing and communication theory will find in the first half of the book an accessible presentation of principal developments in this active field of modern analysis, as well as the mathematical methods underlying real-world applications. Researchers and students in mathematical analysis, signal analysis, and mathematical physics will benefit from the coverage of deep mathematical advances featured in the second part of the work.

Review Quotes

"[The] book under review is written from a relatively novel perspective: its goal is to carefully and deeply study the similarities and differences between time-frequency and time-scale methods. Thus, one may view the present book as a continuation of the famous textbook of Ingrid Daubechies [Ten Lectures on Wavelets].... In my opinion, the two books nicely complement each other and both are a 'must have' for everyone who is interested in a deeper understanding of time-frequency and time-scale methods." --Zentralblatt MATH

"[This book] covers an enormous range in time-frequency analysis and time-scale analysis.... [It] is about methods and ideas. The main goal is to motivate, discuss, and describe important results and new developments in time-frequency and time-scale analysis. Full proofs are rarely given, but insightful sketches of proofs or at least descriptions of the underlying ideas are given for most results. This style makes it possible to cover an enormous amount of material, and indeed the book reviews the main results of (an estimated) 200 articles in some detail.

...The resulting book is like a gigantic mosaic. Each facet describes an interesting result. While neighboring facets are not necessarily related in any formal way, the entire mosaic, when viewed from a distance, displays a coherent and magnificent picture, the picture of time-frequency and time-scale analysis. The book is a great resource for experts to obtain an overview of the field and its literature. It is also recommended for advanced graduate students who want to escape their specialization and expand their horizons to a broader view of their field." --Bulletin of the AMS

"This book is about time-frequency and time-scale analyses, two topics that are historically and conceptually related.... Unlike many similar research monographs, which focus on the authors' own research, this book includesother researchers' work and covers a wide spectrum of topics, some in meticulous detail.... Because of the wide array of results presented in the book, I would recommend it to experts who work in the area; some parts of it might also be useful for advanced graduate students." --Mathematical Reviews