The Fractional Fourier Transform

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.Die Fraktionale Fourier-Transformation bietet ein elegantes mathematisches Fundament zur Diskussion der Beugung und anderer grundlegender Eigenschaften optischer Systeme. In diesem Buch erläutern die Entdecker ihre Methode einem größeren Publikum mit dem Ziel, die Entwicklung neuer Anwendungen zu fördern. Beschrieben werden auch Studien zur Fresnelschen Beugung, zur Ausbreitung optischer Strahlen, zu Resonatoren aus sphärischen Spiegeln und zur Signaldetektion. (02/00) Preface.

Acknowledgments.

Introduction.

Signals, Systems, and Transformations.

Wigner Distributions and Linear Canonical Transforms.

The Fractional Fourier Transform.

Time-Order and Space-Order Representations.

The Discrete Fractional Fourier Transform.

Optical Signals and Systems.

Phase-Space Optics.

The Fractional Fourier Transform in Optics.

Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery.

Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition.

Bibliography on the Fractional Fourier Transform.

Other Cited Works.

Credits.

Index. "...[the authors] explain the basic concepts from various perspectives and survey its application in two areas where it is widely used." (SciTech Book News, Vol. 25, No. 4, December 2001) Haldun M. Ozaktas Bilkent University, Ankara, Turkey
Zeev Zalensky Tel Aviv University, Tel Aviv, Israel
M. Alper Kutay TÜBTAK-UEKAE, Ankara, Turkey The Fractional Fourier Transform provide a comprehensive and widely accessible account of the subject covering both theory and applications. As a generalisation of the Fourier transform, the fractional Fourier transform is richer in theory and more flexible in applications but not more costly in implementation. This text consolidates knowledge on the transform and illustrates its application in diverse contexts. Applications studied so far fall mostly in the areas in optics and wave propagation and signal processing, including optical information processing, beam synthesis, phase retrieval, perspective projections, shift-variant filtering, image restoration, pattern recognition, tomography, data compression and time-frequency representations.
* Background material introduces time-frequency analysis emphasizing the Wigner distribution, ambiguity function and canonical transforms.
* Chapter on phase-space optics employs matrix algebra in a unified manner for both wave and geometrical optics, leading to many important results such as those on general Fourier transform planes and optical invariants.
* Separate discussion of optics for readers with no interest in optics.
Unifying knowledge from the mathematics, optics and signal processing literature in a manner accessible to a broad audience, this book is of interest to researchers, engineers, and senior undergraduate and graduate students in electrical engineering, physics, and mathematics.