The Fractional Fourier Transform: with Applications in Optics and Signal Processing
The discovery of the Fractional Fourier Transform and its role in optics and data management provides an elegant mathematical framework within which to discuss diffraction and other fundamental aspects of optical systems. This book explains how the fractional Fourier transform has allowed the generalization of the Fourier transform and the notion of the frequency transform. It will serve as the standard reference on Fourier transforms for many years to come.
Why Read This Book
You should read this book if you want a rigorous, single-source treatment of the Fractional Fourier Transform that covers its mathematical foundations, discrete algorithms, and concrete applications in optics and signal processing. It gives you the tools to apply FRFT-based transforms and filters in time-frequency problems, chirp processing, and optical propagation models.
Who Will Benefit
Graduate students, researchers, and DSP engineers working on time-frequency analysis, chirp/linear FM processing, optical signal processing, or advanced filter/transform design.
Level: Advanced — Prerequisites: Solid background in continuous and discrete Fourier transforms, linear systems and linear algebra; familiarity with complex exponentials and basic optics or signal-processing concepts is helpful.
Key Takeaways
- Understand the mathematical definition and key properties of the continuous Fractional Fourier Transform (FRFT), including additivity and rotation interpretation in time-frequency space.
- Implement and analyze discrete FRFT algorithms and practical numerical methods for efficient computation.
- Apply FRFT to time-frequency analysis tasks such as chirp signal detection, compression, and matched filtering.
- Relate the FRFT to the linear canonical transform and classical optical propagation/diffraction theory for signal and image processing uses.
- Design and reason about sampling, interpolation, and reconstruction in fractional domains for processing and communications applications.
- Employ FRFT-based techniques in pattern recognition, phase retrieval, and other applied optics/signal-processing problems.
Topics Covered
- Introduction and motivation: why fractionalize the Fourier transform?
- Mathematical preliminaries and notation
- Definition and properties of the continuous FRFT
- Integral representations and time-frequency (rotation) interpretation
- Relationship to the linear canonical transform and optics
- Discrete FRFT: definitions and challenges
- Fast and numerically stable algorithms for discrete FRFT
- Sampling, interpolation and reconstruction in fractional domains
- Applications in optics: propagation, diffraction, and imaging
- Applications in signal processing: chirps, filtering, and detection
- Applications in pattern recognition and phase retrieval
- Numerical examples, experiments and implementation notes
- Conclusions, open problems and future directions
- Appendices (mathematical proofs and background material)
Languages, Platforms & Tools
How It Compares
More focused and in-depth on FRFT (and optics) than broad time-frequency texts like Cohen or Boashash; complements Almeida's earlier FRFT introductions by providing fuller algorithms and optical applications.
