Transforms and Fast Algorithms for Signal Analysis and Representations
This book is a comprehensive presentation of recent results and developments on several widely used transforms and their fast algorithms. In many cases, new options are provided for improved or new fast algorithms, some of which are not well known in the digital signal processing community. The book is suitable as a textbook for senior undergraduate and graduate courses in digital signal processing. It may also serve as an excellent self-study reference for electrical engineers and applied mathematicians whose work is related to the fields of electronics, signal processing, image and speech processing, or digital design and communication.
Why Read This Book
You should read this book if you want a broad, mathematically grounded survey of transform families and practical fast algorithms so you can choose or derive the most efficient transform for your DSP problem. It highlights both common (FFT, DCT) and less-familiar fast procedures, giving you tools to reduce computation and adapt transforms to signals in audio, speech, image, and communications work.
Who Will Benefit
Senior undergraduates, graduate students, and practicing DSP engineers who need a deeper, algorithmic understanding of transforms and fast implementations for analysis, filtering, and coding.
Level: Intermediate — Prerequisites: Familiarity with linear algebra, discrete-time signals and systems, complex arithmetic, basic Fourier/DFT concepts, and calculus.
Key Takeaways
- Implement and analyze a variety of fast DFT/FFT algorithms (including split-radix and common variants).
- Derive and apply fast algorithms for DCT/DST, Hadamard, and related orthogonal transforms.
- Evaluate computational complexity and memory trade-offs for different fast-transform strategies.
- Apply transform-based techniques to practical signal-analysis tasks such as filtering, spectral estimation, and transform coding.
- Adapt and select less-common fast algorithms (e.g., prime-factor, Winograd, chirp-z style methods) for specialized problems.
- Understand the role of multirate and multi-resolution (wavelet) transforms and their fast implementations.
Topics Covered
- 1. Introduction and Motivation for Transform-Based Representations
- 2. Mathematical Preliminaries (periodicity, orthogonality, matrix views)
- 3. The Discrete Fourier Transform and Basic Properties
- 4. FFT Algorithms: Cooley-Tukey, Split-Radix and Radix Variants
- 5. Prime-Factor and Rader-Type FFT Algorithms
- 6. Winograd Small-Factor and Minimal-Multiplication Methods
- 7. DCT, DST, and Real-Signal Fast Transforms
- 8. Hadamard, Walsh, and Related Transforms
- 9. Chirp-Z and Fractional Fourier-Type Algorithms
- 10. Fast Algorithms for Convolution and Correlation
- 11. Multi-resolution and Wavelet Transforms: Fast Filter Bank Implementations
- 12. Applications: Signal, Image and Speech Processing Examples
- 13. Implementation Issues: Complexity, Numerical Accuracy and Memory
- Appendices: Tables, Proofs and Algorithmic Pseudocode
How It Compares
Compared with Brigham's The Fast Fourier Transform and Its Applications, Bi's book spans a broader set of transforms (DCT/DST, Hadamard, wavelets) and emphasizes a larger collection of lesser-known fast algorithms; compared with general DSP texts (e.g., Oppenheim & Schafer) it is more algorithmic and transform-focused rather than a broad signals-and-systems treatment.
