Mathematical Principles of Signal Processing: Fourier and Wavelet Analysis
From the reviews: "[…] the interested reader will find in Bremaud’s book an invaluable reference because of its coverage, scope and style, as well as of the unified treatment it offers of (signal processing oriented) Fourier and wavelet basics." Mathematical Reviews
Why Read This Book
You should read this book if you want a rigorous, unified mathematical foundation for signal processing that ties Fourier analysis, wavelets, and stochastic methods into a single framework. It’s especially valuable when you need precise theorems and proofs to support advanced algorithm design or research.
Who Will Benefit
Mathematically inclined engineers, researchers, and graduate students working on theoretical DSP, wavelets, or stochastic signal analysis who need rigorous tools and proofs.
Level: Advanced — Prerequisites: Undergraduate-level signals & systems, linear algebra, real analysis (or familiarity with Lebesgue integration), and basic probability theory.
Key Takeaways
- Formulate signal-processing problems using measure-theoretic and functional-analytic tools.
- Derive and justify Fourier-transform results in L1/L2 and distributional frameworks.
- Understand multiresolution analysis and the mathematical construction of wavelet bases.
- Analyze stochastic processes and their spectral representations for statistical DSP.
- Apply convergence, stability, and approximation theorems relevant to transforms and filters.
Topics Covered
- 1. Introduction and signal-processing motivations
- 2. Measure theory and distributions (generalized functions)
- 3. Hilbert spaces and L2 theory for signals
- 4. Fourier transform: definitions, properties, and inversion
- 5. Spectral representations and stationary stochastic processes
- 6. Linear systems and frequency-domain analysis
- 7. Multiresolution analysis and wavelet constructions
- 8. Wavelet transforms: bases, frames, and decomposition algorithms
- 9. Time-frequency methods and localization
- 10. Approximation, convergence, and stability results
- 11. Selected applications and examples
- Appendices: mathematical background and proofs
How It Compares
More mathematically rigorous and probability-oriented than Mallat's A Wavelet Tour of Signal Processing and somewhat complementary to Vetterli/Kovacevic's Foundations of Signal Processing, which is more applied and algorithm-focused.
