Computational Signal Processing with Wavelets (Applied and Numerical Harmonic Analysis)
Overview For over a decade now, wavelets have been and continue to be an evolving subject of intense interest. Their allure in signal processing is due to many factors, not the least of which is that they offer an intuitively satisfying view of signals as being composed of little pieces of wa'ues. Making this concept mathematically precise has resulted in a deep and sophisticated wavelet theory that has seemingly limitless applications. This book and its supplementary hands-on electronic: component are meant to appeal to both students and professionals. Mathematics and en gineering students at the undergraduate and graduate levels will benefit greatly from the introductory treatment of the subject. Professionals and advanced students will find the overcomplete approach to signal represen tation and processing of great value. In all cases the electronic component of the proposed work greatly enhances its appeal by providing interactive numerical illustrations. A main goal is to provide a bridge between the theory and practice of wavelet-based signal processing. Intended to give the reader a balanced look at the subject, this book emphasizes both theoretical and practical issues of wavelet processing. A great deal of exposition is given in the beginning chapters and is meant to give the reader a firm understanding of the basics of the discrete and continuous wavelet transforms and their relationship. Later chapters promote the idea that overcomplete systems of wavelets are a rich and largely unexplored area that have demonstrable benefits to offer in many applications.
Why Read This Book
You will learn how to turn wavelet theory into working signal-processing tools — from multiresolution analysis to practical discrete wavelet transforms and filter-bank implementations. The book balances mathematical clarity with computational recipes and application examples so you can apply wavelets to audio, speech, radar, and communications problems.
Who Will Benefit
Ideal for engineering students and practicing signal-processing engineers who want a computationally oriented introduction to wavelets and hands-on methods for DSP applications.
Level: Intermediate — Prerequisites: Familiarity with basic digital signal processing (discrete-time signals and systems), the Fourier transform/FFT, linear algebra, and elementary probability/statistics.
Key Takeaways
- Understand the foundations of multiresolution analysis and the theory behind wavelet bases and frames.
- Implement discrete and continuous wavelet transforms and their inverse for practical signal analysis.
- Design and analyze finite impulse response filter banks and perfect-reconstruction wavelet filter sets.
- Apply wavelet methods to real DSP tasks such as denoising, compression, feature extraction, and spectral analysis for audio, speech, radar, and communications.
- Use overcomplete/frame-based representations and wavelet packets to build flexible, application-specific signal representations.
- Combine wavelet techniques with statistical and adaptive filtering approaches for detection and estimation tasks.
Topics Covered
- Introduction and motivation: signals, time-frequency intuition, and wavelet history
- Mathematical preliminaries: Fourier analysis, linear algebra, and sampling basics
- Time–frequency representations and the short-time Fourier transform
- Multiresolution analysis: scaling functions and construction of wavelet bases
- Discrete Wavelet Transform (DWT) and algorithmic implementation
- Filter-bank theory: analysis/synthesis filters and perfect reconstruction
- Continuous wavelet transform and frame/overcomplete representations
- Wavelet packets, best-basis selection, and adaptive decompositions
- Numerical algorithms: FFTs, fast wavelet transforms, and computational considerations
- Applications in audio and speech processing: denoising, coding, and feature extraction
- Applications in radar and communications: detection, spectral analysis, and modulation-domain processing
- Adaptive and statistical signal processing with wavelets
- Implementation notes, examples, and hands-on computational experiments
- Appendices: reference transforms, tables, and supplementary code/materials
Languages, Platforms & Tools
How It Compares
Compared with Mallat's 'A Wavelet Tour of Signal Processing', Teolis is more computational and engineering-focused and presents a hands-on, overcomplete approach that is more approachable than the highly mathematical treatment in Daubechies' 'Ten Lectures on Wavelets'.
