The Fractional Fourier Transform: with Applications in Optics and Signal Processing: Haldun M. Ozaktas (Bilkent Univ., Ankara, Turkey); Zeev Zalevsky (Tel Aviv Univ., Tel Aviv, Israel); M. Alper Kutay (TÜBİTAK-UEKAE, Ankara, Turkey): 9780471963462: Book: Technology: eCampus.com

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.

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.

Preface

Acknowledgments

xvii

Introduction

(6)

Fractional operations and the fractional Fourier transform

(3)

Applications of the fractional Fourier transform

(1)

Overview of the book

(2)

Signals, Systems, and Transformations

(56)

Signals

(4)

Signals

(1)

Notation

(1)

Some commonly used functions

(1)

Analytic signals and the Hilbert transform

(1)

Signal spaces

(1)

Systems

(4)

Systems

(1)

Linearity and superposition integrals

(1)

Some special linear systems

(1)

Shift invariance and convolution

(2)

Representations and transformations

(9)

Systems versus transformations

(1)

Basis sets and representations

(4)

Impulse and harmonic bases

(1)

Transformations between representations

(3)

Operators

(10)

Operators

(4)

Eigenvalue equations

(1)

Diagonalization and spectral expansion

(3)

Functions of operators

(2)

The Fourier transform

(8)

Definition and properties

(4)

Eigenfunctions of the Fourier transform

(4)

Some important operators

(8)

Coordinate multiplication and differentiation operators

(4)

Phase shift, translation, chirp multiplication, and chirp convolution operators

(3)

Annihilation and creation operators

(1)

Uncertainty relations

(2)

Random processes

(2)

Fundamental definitions

(1)

Power spectral density

(1)

Linear systems with random inputs

(1)

Generalization to two dimensions

(2)

Some additional definitions and results

(2)

The Radon transform and projection-slice theorem

(1)

Complex exponential integrals

(1)

Stationary-phase integral

(1)

Schwarz's inequality

(1)

Further reading

(1)

Appendix: Vector spaces and function spaces

(5)

Vector spaces

(1)

Inner products and norms

(4)

Wigner Distributions and Linear Canonical Transforms

(54)

Time-frequency and space-frequency representations

(6)

Short-time or windowed Fourier transform

(2)

Gabor expansion

(2)

Wavelet transforms

(1)

Remarks

(1)

The Wigner distribution and the ambiguity function

(17)

The Wigner distribution

(4)

The ambiguity function

(3)

Cohen's class of shift-invariant distributions

(2)

Smoothing of the Wigner distribution

(3)

Effect of linear systems on the Wigner distribution

(1)

Time-frequency filtering

(3)

Wigner distribution of random signals

(1)

Wigner distribution of analytic signals

(1)

Other properties

(1)

Sampling and the number of degrees of freedom

(7)

Linear canonical transforms

(22)

Definition and properties

(3)

Effect on Wigner distributions

(3)

Special linear canonical transforms

(5)

Decompositions

104

(3)

Transformation of moments

107

(1)

Linear fractional transformations

108

(2)

Coordinate multiplication and differentiation operators

110

(1)

Uncertainty relation

111

(1)

Invariants and hyperdifferential forms

111

(1)

Differential equations

112

(1)

Symplectic systems

113

(1)

Connections to group theory

114

(1)

Generalization to two and higher dimensions

115

(1)

Further reading

116

(1)

The Fractional Fourier Transform

117

(70)

Definitions of the fractional Fourier transform

117

(20)

Linear integral transform

118

(4)

Fractional powers of the Fourier transform

122

(2)

Rotation in the time-frequency plane

124

(2)

Transformation of coordinate multiplication and differentiation operators

126

(3)

Differential equation

129

(3)

Hyperdifferential operator

132

(5)

Eigenvalues and eigenfunctions

137

(2)

Distinct definitions of the fractional Fourier transform

139

(4)

Transforms of some common functions

143

(9)

Properties

152

(7)

Rotations and projections in the time-frequency plane

159

(7)

Rotation of the Wigner distribution

159

(1)

Projections of the Wigner distribution

160

(2)

Other time-frequency representations

162

(4)

Coordinate multiplication and differentiation operators

166

(3)

Phase shift and translation operators

169

(2)

Fractional Fourier domains

171

(2)

Chirp bases and chirp transforms

173

(2)

Two-dimensional fractional Fourier transforms

175

(5)

Extensions and applications

180

(3)

Fractional Fourier transforms in braket notation

180

(1)

Complex-ordered fractional Fourier transforms

181

(1)

Relation to wavelet transforms

181

(1)

Application to neural networks

181

(1)

Chirplets and other approaches

182

(1)

Other fractional operations and transforms

182

(1)

Historical and bibliographical notes

183

(4)

Time-Order and Space-Order Representations

187

(14)

Introduction

187

(1)

The rectangular time-order representation

187

(2)

Optical implementation

189

(1)

The polar time-order representation

190

(4)

Relationships with the Wigner distribution and the ambiguity function

194

(3)

Applications of time-order representations

197

(2)

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

199

(1)

Historical and bibliographical notes

200

(1)

The Discrete Fractional Fourier Transform

201

(22)

Introduction

201

(1)

Discrete Hermite-Gaussian functions

202

(8)

The discrete fractional Fourier transform

210

(3)

Definition in hyperdifference form

213

(2)

Higher-order discrete analogs

215

(1)

Discussion

216

(2)

Discrete computation of the fractional Fourier transform

218

(2)

Historical and bibliographical notes

220

(3)

Optical Signals and Systems

223

(42)

Introduction

223

(1)

Notation and conventions

224

(3)

Wave optics

227

(11)

The wave equation

228

(3)

Plane wave decomposition

231

(2)

The paraxial wave equation

233

(2)

Hermite-Gaussian beams

235

(3)

Wave-optical characterization of optical components

238

(10)

Sections of free space

238

(1)

Thin lenses

239

(1)

Quadratic graded-index media

240

(4)

Extensions

244

(1)

Spatial filters

245

(1)

Fourier-domain spatial filters

246

(1)

General linear systems

247

(1)

Spherical reference surfaces

247

(1)

Remarks

248

(1)

Geometrical optics

248

(6)

The ray equation

249

(1)

Fermat's principle and the eikonal equation

250

(2)

Hamilton's equations

252

(2)

Geometrical-optical characterization of optical components

254

(5)

Sections of free space

254

(1)

Thin lenses

255

(1)

Quadratic graded-index media

256

(1)

Extensions

256

(1)

Spatial filters

257

(1)

Fourier-domain spatial filters

257

(2)

General linear systems

259

(1)

Spherical reference surfaces

259

(1)

Remarks

259

(1)

Partially coherent light

259

(1)

Fourier optical systems

260

(3)

Further reading

263

(2)

Phase-Space Optics

265

(54)

Wave-optical and geometrical-optical phase spaces

265

(4)

Quadratic-phase systems and linear canonical transforms

269

(1)

Optical components

270

(12)

Sections of free space

272

(2)

Thin lenses

274

(1)

Quadratic graded-index media

275

(2)

Extensions

277

(1)

Spatial filters

277

(1)

Fourier-domain spatial filters

278

(2)

General linear systems

280

(1)

Spherical reference surfaces

280

(1)

Discussion

281

(1)

Imaging and Fourier transformation

282

(12)

Imaging systems

282

(2)

Fourier transforming systems

284

(3)

General theorems for image and Fourier transform planes

287

(7)

Decompositions and duality in optics

294

(3)

Relations between wave and geometrical optics

297

(5)

Phase of the system kernel and Hamilton's point characteristic

297

(2)

Transport equations for the Wigner distribution

299

(3)

Discussion

302

(1)

Quadratic-exponential signals

302

(3)

Ray-like signals

302

(2)

Complex Gaussian signals

304

(1)

Optical invariants

305

(12)

Invariance of density and area in phase space

306

(2)

The symplectic condition and canonical transformations

308

(2)

The Lagrange invariant

310

(1)

The Smith-Helmholtz invariant and Abbe's sine condition

311

(2)

The constant brightness theorem

313

(1)

The unit-determinant condition for inhomogeneous media

313

(1)

Poisson brackets

314

(2)

The number of degrees of freedom

316

(1)

Partially coherent light

317

(1)

Further reading

318

(1)

The Fractional Fourier Transform in Optics

319

(68)

Applications of the transform to wave and beam propagation

319

(2)

Overview

321

(12)

Quadratic-phase systems as fractional Fourier transforms

323

(1)

Quadratic graded-index media

324

(1)

Fresnel diffraction

325

(1)

Multi-lens systems

326

(4)

Optical implementation of the fractional Fourier transform

330

(1)

Hermite-Gaussian expansion approach

331

(2)

General fractional Fourier transform relations in free space

333

(4)

The fractional Fourier transform and Fresnel's integral

333

(2)

Analysis

335

(1)

Synthesis

336

(1)

Propagation

336

(1)

Discussion

337

(1)

Illustrative applications

337

(10)

Fresnel diffraction as fractional Fourier transformation

338

(1)

The symmetric case

338

(1)

Fractional Fourier transform between planar surfaces

339

(2)

Classical single-lens imaging

341

(2)

Multi-lens systems as consecutive fractional Fourier transforms

343

(1)

General fractional Fourier transform relations for quadratic phase systems

344

(3)

Fractional Fourier transformation in quadratic graded-index media

347

(7)

Propagation in quadratic-index media as fractional Fourier transformation

347

(1)

Analogy with the simple harmonic oscillator

348

(2)

Quadratic graded-index media as the limit of multi-lens systems

350

(3)

Gaussian beams through quadratic graded-index media

353

(1)

Hermite-Gaussian expansion approach

354

(5)

The fractional Fourier order and the Gouy phase shift

354

(2)

Spherical mirror resonators and stability

356

(3)

First-order optical systems

359

(13)

Quadratic-phase systems as fractional Fourier transforms

359

(2)

Geometrical-optical determination of fractional Fourier transform parameters

361

(4)

Differential equations for the fractional Fourier transform parameters

365

(4)

Fractional Fourier transform parameters and Gaussian beam parameters

369

(2)

Discussion

371

(1)

Fourier optical systems

372

(5)

Locations of fractional Fourier transform planes

377

(1)

Wave field reconstruction, phase retrieval, and phase-space tomography

378

(3)

Extensions and applications

381

(4)

Temporal optical implementation of the transform

381

(1)

Digital optical implementation of the transform

381

(1)

Optical implementation of two-dimensional transforms

381

(1)

Optical interpretation and implementation of complex-ordered transforms

382

(1)

Incoherent optical implementation of the transform

382

(1)

Applications to systems with partially coherent light

382

(1)

Other applications of the transform in optics

383

(1)

Practical considerations for implementing the transform

383

(1)

Other fractional operations and effects in optics

384

(1)

Historical and bibliographical notes

385

(2)

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

387

(56)

Introduction

387

(1)

Optimal Wiener filtering in fractional Fourier domains

388

(4)

Multistage, multichannel, and generalized filtering configurations

392

(6)

Introduction

392

(2)

Cost-performance trade-off

394

(3)

Extensions and generalizations

397

(1)

Applications of fractional Fourier domain filtering

398

(21)

Elementary signal separation examples

399

(2)

Optical signal separation

401

(2)

System and transform synthesis

403

(1)

Signal recovery and restoration

403

(15)

Signal synthesis

418

(1)

Free-space optical interconnection architectures

418

(1)

Convolution and filtering in fractional Fourier domains

419

(5)

Convolution and multiplication in fractional Fourier domains

419

(2)

Compaction in fractional Fourier domains

421

(1)

Filtering in fractional Fourier domains

422

(2)

Derivation of the optimal fractional Fourier domain filter

424

(4)

Continuous time

425

(2)

Discrete time

427

(1)

Optimization and cost analysis of multistage and multichannel filtering configurations

428

(4)

Determination of the optimal filters

429

(1)

Rectangular system matrices

430

(1)

Cost analysis

431

(1)

The fractional Fourier domain decomposition

432

(5)

Introduction and definition

432

(2)

Construction of the fractional Fourier domain decomposition

434

(1)

Pruning and sparsening

435

(2)

Repeated filtering in the ordinary time and frequency domains

437

(2)

Multiplexing in fractional Fourier domains

439

(2)

Historical and bibliographical notes

441

(2)

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

443

(24)

Introduction

443

(1)

Fractional correlation

443

(4)

Controllable shift invariance

447

(3)

Performance measures for fractional correlation

450

(4)

Fractional power filters

450

(1)

Performance measures and optimal filters

451

(1)

Optimal filters for fractional correlation

452

(2)

Fractional joint-transform correlators

454

(1)

Adaptive windowed fractional Fourier transforms

455

(5)

Time- or space-dependent windowed transforms

455

(2)

Applications

457

(3)

Applications with different orders in the two dimensions

460

(4)

Historical and bibliographical notes

464

(3)

Bibliography on the Fractional Fourier Transform

467

(18)

Other Cited Works

485

(12)

Credits

497

(2)

Index

499

Haldun M. Ozaktas Bilkent University, Ankara, Turkey
Zeev Zalensky Tel Aviv University, Tel Aviv, Israel
M. Alper Kutay T+�BTAK-UEKAE, Ankara, Turkey


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