First Course in Wavelets with Fourier Analysis
This book fills the gap between volumes on wavelets that are either too advanced (in terms of mathematical background required) or that contain too little mathematical theory underlying wavelets. It presents most of the theory underlying Fourier analysis and wavelets in a clear and comprehensive fashion-- without requiring advanced background in real analysis. Provides a careful balance between theory and practical algorithms, and features a clear presentation of applications to digital signal processing--e.g., data compression, digital filtering and singularity detection. Includes illustrations and MATLAB code used in many of the algorithms. Inner Product Spaces. Fourier Series. The Fourier Transform. Discrete Fourier Analysis. Wavelet Analysis. Multiresolution Analysis. The Daubechies Wavelets. For anyone interested in Wavelets and Fourier Analysis.
Why Read This Book
You will learn a clear, approachable development of Fourier analysis and wavelets that balances rigorous theory with practical DSP algorithms; the book shows you how to move from inner-product spaces to implementable wavelet filter banks and MATLAB code for compression, denoising, and singularity detection. It’s ideal if you want the mathematical foundations without heavy real-analysis prerequisites and concrete examples you can reuse in engineering work.
Who Will Benefit
Senior undergraduates, graduate students, and practicing engineers in signal processing, communications, audio/speech, or radar who need a practical yet theoretical grounding in Fourier and wavelet methods to build algorithms and implementations.
Level: Intermediate — Prerequisites: Single-variable calculus, linear algebra (inner products, orthogonality), basic signals and systems or familiarity with Fourier series/transforms; familiarity with MATLAB is helpful but not strictly required.
Key Takeaways
- Understand inner product spaces and how they form the foundation for Fourier and wavelet analysis
- Apply Fourier series, Fourier transform, DFT, and FFT methods to analyze and process signals
- Design and implement digital filters and multirate/wavelet filter banks for signal decomposition
- Use discrete and continuous wavelet transforms for compression, denoising, and singularity detection
- Perform spectral analysis and time–frequency analysis using wavelets and Fourier tools
- Translate theory into practice by adapting the provided MATLAB examples and algorithms to real DSP problems
Topics Covered
- Preface and Motivation
- Inner Product Spaces and Orthogonality
- Fourier Series
- The Fourier Transform
- Discrete Fourier Analysis and the DFT
- Sampling, Reconstruction, and Aliasing
- Digital Filters and Multirate Signal Processing
- The Fast Fourier Transform (FFT)
- Introduction to Wavelets
- Multiresolution Analysis and Scaling Functions
- Discrete Wavelet Transform and Filter Banks
- Continuous Wavelet Transform and Time–Frequency Analysis
- Applications: Compression, Denoising, and Singularity Detection
- MATLAB Examples and Algorithm Implementations
- Appendices (mathematical background and tables)
Languages, Platforms & Tools
How It Compares
Less abstract and more application-focused than Daubechies' Ten Lectures, and more accessible than Mallat's A Wavelet Tour of Signal Processing—Boggess fills the gap between introductory texts and mathematically rigorous monographs.
