Discrete Fourier And Wavelet Transforms: An Introduction Through Linear Algebra With Applications To Signal Processing
"This book is suitable as a textbook for an introductory undergraduate mathematics course on discrete Fourier and wavelet transforms for students with background in calculus and linear algebra. The particular strength of this book is its accessibility to students with no background in analysis. The exercises and computer explorations provide the reader with many opportunities for active learning. Studying from this text will also help students strengthen their background in linear algebra." Mathematical Association of America This textbook for undergraduate mathematics, science, and engineering students introduces the theory and applications of discrete Fourier and wavelet transforms using elementary linear algebra, without assuming prior knowledge of signal processing or advanced analysis. It explains how to use the Fourier matrix to extract frequency information from a digital signal and how to use circulant matrices to emphasize selected frequency ranges. It introduces discrete wavelet transforms for digital signals through the lifting method and illustrates through examples and computer explorations how these transforms are used in signal and image processing. Then the general theory of discrete wavelet transforms is developed via the matrix algebra of two-channel filter banks. Finally, wavelet transforms for analog signals are constructed based on filter bank results already presented, and the mathematical framework of multiresolution analysis is examined.
Why Read This Book
You will learn the discrete Fourier and wavelet transforms from a clean linear-algebra perspective that makes the subject intuitive and computationally practical. The book emphasizes accessibility and active learning through exercises and computer explorations, so you can quickly apply transforms to real problems in audio, radar, and communications without needing advanced analysis background.
Who Will Benefit
Undergraduate math, science, or engineering students (and practicing engineers retraining themselves) who know calculus and linear algebra and want a practical, concept-driven introduction to DFTs, FFTs, and wavelets for signal-processing applications.
Level: Intermediate — Prerequisites: Single-variable calculus and an introductory course in linear algebra (vectors, inner products, eigenvectors); basic programming experience (MATLAB/Python) is helpful but not strictly required.
Key Takeaways
- Compute and interpret the Discrete Fourier Transform (DFT) and its inverse as linear change-of-basis operations
- Apply and analyze Fast Fourier Transform (FFT) algorithms to speed up spectral computations
- Design and implement discrete wavelet transforms and multiresolution filter banks using linear-algebra tools
- Perform spectral analysis (windowing, leakage, power spectral density estimation) for practical signals
- Relate core linear-algebra concepts (orthogonality, bases, projections, eigenstructure) to digital filter and transform design
- Apply transforms to concrete problems in audio/speech, radar, and communications through exercises and computer explorations
Topics Covered
- 1. Introduction and Motivation: Transforms in Signal Processing
- 2. Linear-Algebra Foundations: Vectors, Inner Products, and Orthogonality
- 3. Linear Operators and Matrices: Change of Basis Viewpoint
- 4. The Discrete Fourier Transform: Definitions and Properties
- 5. Fourier Matrices, Convolution, and Circulant Matrices
- 6. Fast Fourier Transform Algorithms and Computational Issues
- 7. Spectral Analysis: Windowing, Leakage, and Power Spectra
- 8. Digital Filters and Basic Filter Design Concepts
- 9. Introduction to Wavelets and Multiresolution Analysis
- 10. Discrete Wavelet Transforms, Filter Banks, and Orthogonal Bases
- 11. Applications: Audio and Speech, Radar, and Communications Examples
- 12. Adaptive and Statistical Signal-Processing Connections
- 13. Computer Explorations, Projects, and Further Reading
- Appendices: MATLAB/Octave and Python Notes; Mathematical Background
Languages, Platforms & Tools
How It Compares
Compared with Oppenheim & Schafer's Discrete-Time Signal Processing (which is engineering- and systems-focused), Goodman's book emphasizes an undergraduate-friendly linear-algebra viewpoint; compared with Mallat's A Wavelet Tour (which is comprehensive and research-oriented), it is far more elementary and application-driven.
