Wavelet Transforms & Time-Frequency Signal Analysis
The last fifteen years have produced major advances in the mathematical theory of wavelet transforms and their applications to science and engineering. In an effort to inform researchers in mathematics, physics, statistics, computer science, and engineering and to stimulate furtherresearch, an NSF-CBMS Research Conference on Wavelet Analysis was organized at the University of Central Florida in May 1998. Many distinguished mathematicians and scientists from allover the world participated in the conference and provided a digest of recent developments, open questions, and unsolved problems in this rapidly growing and important field. As a follow-up project, this monograph was developed from manuscripts sub mitted by renowned mathematicians and scientists who have made important contributions to the subject of wavelets, wavelet transforms, and time-frequency signal analysis. This publication brings together current developments in the theory and applications of wavelet transforms and in the field of time-frequency signal analysis that are likely to determine fruitful directions for future advanced study and research.
Why Read This Book
You should read this book if you want a rigorous, research-oriented survey of wavelet theory and time–frequency methods that connects mathematical foundations to signal-processing applications. It collects advanced treatments and open problems from leading researchers, so you’ll gain depth and pointers to primary literature rather than just recipes.
Who Will Benefit
Advanced graduate students, researchers, and engineer-researchers with a solid math background who need a rigorous, theory-forward reference on wavelets and time-frequency analysis.
Level: Advanced — Prerequisites: Real analysis and linear algebra, familiarity with Fourier transforms and basic functional analysis; comfort with proofs and advanced calculus is recommended.
Key Takeaways
- Understand the mathematical foundations of the continuous and discrete wavelet transforms and multiresolution analysis.
- Analyze properties of wavelet bases, frames, biorthogonal constructions, and wavelet packets for signal representation.
- Apply time–frequency techniques (e.g., short-time Fourier, Wigner–Ville and wavelet-based spectra) to characterize nonstationary signals.
- Evaluate theoretical trade-offs: time vs. frequency localization, orthogonality vs. redundancy, and sampling/aliasing issues in wavelet expansions.
- Identify current research directions and open problems in wavelet analysis and advanced time–frequency methods.
Topics Covered
- Introduction and overview of time–frequency analysis
- Mathematical preliminaries: function spaces and transforms
- Short-time Fourier transform and classical time–frequency distributions
- Continuous wavelet transform: theory and properties
- Multiresolution analysis and discrete wavelet transform
- Construction of orthogonal and biorthogonal wavelet bases
- Frames, Riesz bases, and redundant representations
- Wavelet packets and best-basis methods
- Time–frequency distributions: Wigner–Ville and Cohen class
- Numerical aspects and implementation considerations
- Applications to signal and image processing (case studies)
- Recent advances, open questions, and research directions
Languages, Platforms & Tools
How It Compares
More mathematically oriented and research-focused than Mallat's 'A Wavelet Tour of Signal Processing' and broader/collection-style compared with Daubechies' concise 'Ten Lectures on Wavelets'.
