The Fractional Fourier Transform

with Applications in Optics and Signal Processing

Haldun M. Ozaktas

Zeev Zalevsky

M. Alper Kutay

Little need be said of the importance and ubiquity of the ordinary Fourier transform in many areas of science and engineering. As a generalization of the ordinary Fourier transform, the fractional Fourier transform is only richer in theory and more flexible in applications--but not more costly in implementation. This book provides a comprehensive and widely accessible account of the transform covering both theory and applications.

Properties and applications of the ordinary Fourier transform are special cases of those of the fractional Fourier transform. The ordinary frequency domain is a special case of the continuum of fractional Fourier domains, which are intimately related to time-frequency representations such as the Wigner distribution. In every area in which Fourier transforms and frequency-domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.

So far applications of the transform have been studied mostly in the areas of optics and wave propagation, and signal analysis and processing. They include Fourier optics and optical information processing, beam propagation and spherical mirror resonators (lasers), optical systems and lens design, quantum optics, statistical optics, beam synthesis and shaping, optical and quantum wavefield reconstruction and phase retrieval, perspective projections, flexible and cost-effective time- or space-variant filtering with applications in signal and image restoration and reconstruction, signal extraction, signal and system synthesis, correlation, detection, pattern recognition, phase retrieval, radar, tomography, multiplexing, data compression, linear FM detection. study of time-frequency representations, differential equations, and harmonic motion. Many of these applications are discussed at length in this book. By consolidating knowledge on the transform and illustrating how it has been applied in diverse contexts, the book aims to make possible the discovery of new applications in other areas as well.

The considerable amount of background material is an important feature of the book which is of interest in its own right, as self-contained expositions or by presenting certain less encountered perspectives and results. This material includes an introduction to time-frequency analysis emphasizing the Wigner distribution and ambiguity function, and canonical transforms. The chapters on optics complement introductory texts on Fourier optics, dealing with optical systems in phase space (the space-frequency plane) in terms of canonical transforms. Matrix algebra is employed in a unified manner for both wave and geometrical optical perspectives, leading to many important and fundamental results, such as those on general Fourier transform planes and optical invariants.

- Of interest to graduate and senior undergraduate students, academicians, researchers, and professionals in branches of mathematics, science, and engineering where Fourier transforms and related concepts are used. A partial list of these areas is operator theory, harmonic analysis and integral transforms, linear algebra, group representation theory, phase-space methods, time- and space-frequency representations, transform theory and techniques, signal analysis and processing, wave propagation, and many areas of optics.

- Unifies knowledge from the mathematics, optics, and signal processing literature in a manner accessible to a broad audience.

- Includes a comprehensive bibliography.

- Discussion of optics completely segregated for readers with no interest or background in optics.

- May be used for self-study or in courses on the fractional Fourier transform and time-frequency analysis and their applications in optics and/or signal processing, advanced signal processing, advanced Fourier optics or information optics emphasizing phase-space concepts and the Wigner distribution.

Ozaktas is the recipient of the 1998 ICO International Prize in Optics (jointly with D. Mendlovic) and the 1999 Scientific and Technical Research Council of Turkey Science Award for his contributions to the subject.

Table of Contents (short):

1 Introduction

2 Signals, Systems, and Transformations

3 Wigner Distributions and Linear Canonical Transforms

4 The Fractional Fourier Transform

5 Time-Order and Space-Order Representations

6 The Discrete Fractional Fourier Transform

7 Optical Signals and Systems

8 Phase-Space Optics

9 The Fractional Fourier Transform in Optics

10 Applications to Filtering, Estimation, and Signal Recovery

11 Applications to Matched Filtering, Detection, and Pattern Recognition

Bibliography on the Fractional Fourier Transform

Other Cited Works

Table of Contents (long):

1 Introduction

1.1 Fractional operations and the fractional Fourier transform

1.2 Applications of the fractional Fourier transform

1.3 Overview of the book

2 Signals, Systems, and Transformations

2.1 Signals

2.2 Systems

2.3 Representations and transformations

2.4 Operators

2.5 The Fourier transform

2.6 Some important operators

2.7 Uncertainty relations

2.8 Random processes

2.9 Generalization to two dimensions

2.10 Some additional definitions and results

2.11 Further reading

2.12 Appendix: Vector spaces and function spaces

3 Wigner Distributions and Linear Canonical Transforms

3.1 Time-frequency and space-frequency representations

3.2 The Wigner distribution and the ambiguity function

3.3 Sampling and the number of degrees of freedom

3.4 Linear canonical transforms

3.5 Generalization to two and higher dimensions

3.6 Further reading

4 The Fractional Fourier Transform

4.1 Definitions of the fractional Fourier transform

4.2 Eigenvalues and eigenfunctions

4.3 Distinct definitions of the fractional Fourier transform

4.4 Transforms of some common functions

4.5 Properties

4.6 Rotations and projections in the time-frequency plane

4.7 Coordinate multiplication and differentiation operators

4.8 Phase shift and translation operators

4.9 Fractional Fourier domains

4.10 Chirp bases and chirp transforms

4.11 Two-dimensional fractional Fourier transforms

4.12 Extensions and applications

4.13 Historical and bibliographical notes

5 Time-Order and Space-Order Representations

5.1 Introduction

5.2 The rectangular time-order representation

5.3 Optical implementation

5.4 The polar time-order representation

5.5 Relationships with the Wigner distribution and the ambiguity function

5.6 Applications of time-order representations

5.7 Other applications of the fractional Fourier transform in time- and space-frequency analysis

5.8 Historical and bibliographical notes

6 The Discrete Fractional Fourier Transform

6.1 Introduction

6.2 Discrete Hermite-Gaussian functions

6.3 The discrete fractional Fourier transform

6.4 Definition in hyperdifference form

6.5 Higher-order discrete analogs

6.6 Discussion

6.7 Discrete computation of the fractional Fourier transform

6.8 Historical and bibliographical notes

7 Optical Signals and Systems

7.1 Introduction

7.2 Notation and conventions

7.3 Wave optics

7.4 Wave optical characterization of optical components

7.5 Geometrical optics

7.6 Geometrical optical characterization of optical components

7.7 Partially coherent light

7.8 Fourier optical systems

7.9 Further reading

8 Phase-Space Optics

8.1 Wave-optical and geometrical-optical phase spaces

8.2 Quadratic-phase systems and linear canonical transforms

8.3 Optical components

8.4 Imaging and Fourier transformation

8.5 Decompositions and duality in optics

8.6 Relations between wave and geometrical optics

8.7 Quadratic-exponential signals

8.8 Optical invariants

8.9 Partially coherent light

8.10 Further reading

9 The Fractional Fourier Transform in Optics

9.1 Applications of the transform to wave and beam propagation

9.2 Overview

9.3 General fractional Fourier transform relations in free space

9.4 Illustrative applications

9.5 Fractional Fourier transformation in quadratic graded-index media

9.6 Hermite-Gaussian expansion approach

9.7 First-order optical systems

9.8 Fourier optical systems

9.9 Locations of fractional Fourier transform planes

9.10 Wavefield reconstruction, phase retrieval, and phase-space tomography

9.11 Extensions and applications

9.12 Historical and bibliographical notes

10 Applications to Filtering, Estimation, and Signal Recovery

10.1 Introduction

10.2 Optimal Wiener filtering in fractional Fourier domains

10.3 Multi-stage, multi-channel, and generalized filtering configurations

10.4 Applications of fractional Fourier domain filtering

10.5 Convolution and filtering in fractional Fourier domains

10.6 Derivation of the optimal fractional Fourier domain filter

10.7 Optimization and cost analysis of multi-stage and multi-channel filtering configurations

10.8 The fractional Fourier domain decomposition (FFDD)

10.9 Repeated filtering in the ordinary time and frequency domains

10.10 Multiplexing in fractional Fourier domains

10.11 Historical and bibliographical notes

11 Applications to Matched Filtering, Detection, and Pattern Recognition

11.1 Introduction

11.2 Fractional correlation

11.3 Controllable shift-invariance

11.4 Performance measures for fractional correlation

11.5 Fractional joint-transform correlators

11.6 Adaptive windowed fractional Fourier transforms

11.7 Applications with different orders in the two dimensions

11.8 Historical and bibliographical notes

Bibliography on the Fractional Fourier Transform

Other Cited Works