A First Course in Wavelets with Fourier Analysis
A comprehensive, self-contained treatment of Fourier analysis and wavelets-now in a new edition Through expansive coverage and easy-to-follow explanations, A First Course in Wavelets with Fourier Analysis , Second Edition provides a self-contained mathematical treatment of Fourier analysis and wavelets, while uniquely presenting signal analysis applications and problems. Essential and fundamental ideas are presented in an effort to make the book accessible to a broad audience, and, in addition, their applications to signal processing are kept at an elementary level. The book begins with an introduction to vector spaces, inner product spaces, and other preliminary topics in analysis. Subsequent chapters feature: The development of a Fourier series, Fourier transform, and discrete Fourier analysis Improved sections devoted to continuous wavelets and two-dimensional wavelets The analysis of Haar, Shannon, and linear spline wavelets The general theory of multi-resolution analysis Updated MATLAB code and expanded applications to signal processing The construction, smoothness, and computation of Daubechies' wavelets Advanced topics such as wavelets in higher dimensions, decomposition and reconstruction, and wavelet transform Applications to signal processing are provided throughout the book, most involving the filtering and compression of signals from audio or video. Some of these applications are presented first in the context of Fourier analysis and are later explored in the chapters on wavelets. New exercises introduce additional applications, and complete proofs accompany the discussion of each presented theory. Extensive appendices outline more advanced proofs and partial solutions to exercises as well as updated MATLAB routines that supplement the presented examples. A First Course in Wavelets with Fourier Analysis , Second Edition is an excellent book for courses in mathematics and engineering at the upper-undergraduate and graduate levels. It is also a valuable resource for mathematicians, signal processing engineers, and scientists who wish to learn about wavelet theory and Fourier analysis on an elementary level.
Why Read This Book
You should read this book if you want a clear, mathematically sound introduction to both Fourier analysis and wavelets that connects theory to practical signal-analysis ideas. It builds core tools (Fourier series/transform, DFT, multiresolution analysis) so you can reason about and implement wavelet-based processing with confidence.
Who Will Benefit
Engineers and graduate students with basic calculus and linear algebra who need a solid mathematical grounding in Fourier and wavelet methods for DSP, spectral analysis, or signal-processing research.
Level: Intermediate — Prerequisites: Single-variable calculus, basic linear algebra (vector spaces, inner products) and elementary familiarity with signals and systems or basic complex analysis will help.
Key Takeaways
- Explain and derive Fourier series and Fourier transform properties relevant to signal analysis.
- Apply the discrete Fourier transform and understand sampling/periodization relationships.
- Formulate multiresolution analysis and construct orthonormal wavelet bases (including compactly supported examples).
- Implement and analyze the discrete wavelet transform and its connection to filter banks.
- Use continuous wavelet transforms for time-frequency analysis and understand reconstruction formulas.
- Relate Fourier and wavelet viewpoints to practical signal-processing tasks such as spectral estimation and localized analysis.
Topics Covered
- Preliminaries: Vector spaces, inner product spaces, and basics of functional analysis
- Fourier Series: Convergence, orthogonality, and examples
- Fourier Transform: L2 theory, basic properties, and transforms of common signals
- Discrete Fourier Transform and Sampling: Periodization, DFT properties and sampling theorems
- L2 Spaces and Approximation Theory: Orthogonal expansions and projection ideas
- Introduction to Wavelets: Motivation, time-frequency localization
- Multiresolution Analysis (MRA): Scaling functions and MRA axioms
- Construction of Orthonormal Wavelet Bases: Filter equations and examples (e.g., Daubechies)
- Discrete Wavelet Transform and Filter Banks: Implementation and perfect reconstruction
- Continuous Wavelet Transform: Analysis, synthesis, and time-scale representations
- Applications to Signal Analysis: Examples and exercises in spectral and localized analysis
- Appendices and Problems: Mathematical tools and worked problems
Languages, Platforms & Tools
How It Compares
More rigorous and mathematically structured than Mallat's 'A Wavelet Tour of Signal Processing' (which is more applied), and more accessible and less terse than Daubechies' 'Ten Lectures on Wavelets' (which is a classic but denser).
