The Fractional Fourier Transform with Applications in Optics and Signal Processing

by Ozaktas, Haldun M.; Zalevsky, Zeev; Kutay, M. Alper

• ISBN13: 9780471963462
• ISBN10: 0471963461
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-02-08
• Publisher: WILEY

Summary

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.

Author Biography

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

Table of Contents

Preface	xv
Acknowledgments	xvii
Introduction	1	(6)
Fractional operations and the fractional Fourier transform	1	(3)
Applications of the fractional Fourier transform	4	(1)
Overview of the book	5	(2)
Signals, Systems, and Transformations	7	(56)
Signals	7	(4)
Signals	7	(1)
Notation	7	(1)
Some commonly used functions	8	(1)
Analytic signals and the Hilbert transform	9	(1)
Signal spaces	10	(1)
Systems	11	(4)
Systems	11	(1)
Linearity and superposition integrals	12	(1)
Some special linear systems	12	(1)
Shift invariance and convolution	13	(2)
Representations and transformations	15	(9)
Systems versus transformations	15	(1)
Basis sets and representations	16	(4)
Impulse and harmonic bases	20	(1)
Transformations between representations	21	(3)
Operators	24	(10)
Operators	24	(4)
Eigenvalue equations	28	(1)
Diagonalization and spectral expansion	29	(3)
Functions of operators	32	(2)
The Fourier transform	34	(8)
Definition and properties	34	(4)
Eigenfunctions of the Fourier transform	38	(4)
Some important operators	42	(8)
Coordinate multiplication and differentiation operators	42	(4)
Phase shift, translation, chirp multiplication, and chirp convolution operators	46	(3)
Annihilation and creation operators	49	(1)
Uncertainty relations	50	(2)
Random processes	52	(2)
Fundamental definitions	52	(1)
Power spectral density	53	(1)
Linear systems with random inputs	54	(1)
Generalization to two dimensions	54	(2)
Some additional definitions and results	56	(2)
The Radon transform and projection-slice theorem	56	(1)
Complex exponential integrals	57	(1)
Stationary-phase integral	58	(1)
Schwarz's inequality	58	(1)
Further reading	58	(1)
Appendix: Vector spaces and function spaces	58	(5)
Vector spaces	58	(1)
Inner products and norms	59	(4)
Wigner Distributions and Linear Canonical Transforms	63	(54)
Time-frequency and space-frequency representations	63	(6)
Short-time or windowed Fourier transform	63	(2)
Gabor expansion	65	(2)
Wavelet transforms	67	(1)
Remarks	68	(1)
The Wigner distribution and the ambiguity function	69	(17)
The Wigner distribution	69	(4)
The ambiguity function	73	(3)
Cohen's class of shift-invariant distributions	76	(2)
Smoothing of the Wigner distribution	78	(3)
Effect of linear systems on the Wigner distribution	81	(1)
Time-frequency filtering	82	(3)
Wigner distribution of random signals	85	(1)
Wigner distribution of analytic signals	86	(1)
Other properties	86	(1)
Sampling and the number of degrees of freedom	86	(7)
Linear canonical transforms	93	(22)
Definition and properties	93	(3)
Effect on Wigner distributions	96	(3)
Special linear canonical transforms	99	(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