Foundations of Signal Processing
This comprehensive and engaging textbook introduces the basic principles and techniques of signal processing, from the fundamental ideas of signals and systems theory to real-world applications. Students are introduced to the powerful foundations of modern signal processing, including the basic geometry of Hilbert space, the mathematics of Fourier transforms, and essentials of sampling, interpolation, approximation and compression The authors discuss real-world issues and hurdles to using these tools, and ways of adapting them to overcome problems of finiteness and localization, the limitations of uncertainty, and computational costs. It includes over 160 homework problems and over 220 worked examples, specifically designed to test and expand students' understanding of the fundamentals of signal processing, and is accompanied by extensive online materials designed to aid learning, including Mathematica® resources and interactive demonstrations.
Why Read This Book
You should read this book if you want a rigorous, unifying view of signal processing that connects linear algebra and functional analysis to practical DSP tools like sampling, FFTs, multirate systems, wavelets and compression. It combines clear mathematical exposition with many worked examples and problems so you will both understand why algorithms work and how to adapt them to real-world constraints.
Who Will Benefit
Advanced undergraduates, graduate students, and practicing engineers who need a firm mathematical foundation for designing and analyzing DSP algorithms and systems.
Level: Advanced — Prerequisites: Familiarity with calculus, linear algebra (inner products, orthogonality), basic signals-and-systems concepts and Fourier transforms; basic familiarity with complex numbers and elementary probability is helpful.
Key Takeaways
- Formulate signals and systems in the language of vector spaces and Hilbert spaces to reason about projections, orthogonality, and bases.
- Apply Fourier analysis and transform properties to analyze and manipulate continuous- and discrete-time signals.
- Derive and use sampling and interpolation results, including nonuniform sampling and practical discretization issues.
- Design and analyze multirate systems, filter banks, and the construction of wavelet bases (multi-resolution analysis).
- Understand approximation, compression, and sparse representations and apply principles behind transform coding.
- Evaluate time-frequency localization and uncertainty principles to guide the choice of representations for specific tasks.
Topics Covered
- 1. Signals, Systems and Mathematical Background
- 2. Vector Spaces and Hilbert Spaces for Signals
- 3. Inner Products, Bases, and Orthogonal Projections
- 4. Fourier Transforms and Their Properties
- 5. Sampling, Interpolation and Discretization
- 6. Discrete-Time Signals and the DFT/FFT
- 7. Approximation Theory and Least-Squares Methods
- 8. Filter Banks and Multirate Signal Processing
- 9. Wavelets and Multi-Resolution Analysis
- 10. Sparse Representations and Compression
- 11. Time-Frequency Analysis and Uncertainty
- 12. Computational Considerations and Applications
Languages, Platforms & Tools
How It Compares
More mathematically foundational than Oppenheim & Willsky's Signals and Systems (which is more engineering-oriented), and broader in scope than Mallat's A Wavelet Tour (which focuses deeply on wavelets).
