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Fractional Fourier Transform With Applications in Optics and Signal Processing

ISBN-10: 0471963461

ISBN-13: 9780471963462

Edition: 2000

Authors: Haldun M. Ozaktas, M. Alper Kutay, Zeev Zalevsky

List price: $266.00
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Description:

The discovery of the Fractional Fourier Transform by the editors provides an elegant mathematical framework within which to discuss diffraction and other fundamental aspects of optical systems.
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Book details

List price: $266.00
Copyright year: 2000
Publisher: John Wiley & Sons, Incorporated
Publication date: 2/8/2001
Binding: Hardcover
Pages: 532
Size: 6.75" wide x 9.75" long x 1.25" tall
Weight: 2.288
Language: English

Preface
Acknowledgments
Introduction
Fractional operations and the fractional Fourier transform
Applications of the fractional Fourier transform
Overview of the book
Signals, Systems, and Transformations
Signals
Signals
Notation
Some commonly used functions
Analytic signals and the Hilbert transform
Signal spaces
Systems
Systems
Linearity and superposition integrals
Some special linear systems
Shift invariance and convolution
Representations and transformations
Systems versus transformations
Basis sets and representations
Impulse and harmonic bases
Transformations between representations
Operators
Operators
Eigenvalue equations
Diagonalization and spectral expansion
Functions of operators
The Fourier transform
Definition and properties
Eigenfunctions of the Fourier transform
Some important operators
Coordinate multiplication and differentiation operators
Phase shift, translation, chirp multiplication, and chirp convolution operators
Annihilation and creation operators
Uncertainty relations
Random processes
Fundamental definitions
Power spectral density
Linear systems with random inputs
Generalization to two dimensions
Some additional definitions and results
The Radon transform and projection-slice theorem
Complex exponential integrals
Stationary-phase integral
Schwarz's inequality
Further reading
Appendix: Vector spaces and function spaces
Vector spaces
Inner products and norms
Wigner Distributions and Linear Canonical Transforms
Time-frequency and space-frequency representations
Short-time or windowed Fourier transform
Gabor expansion
Wavelet transforms
Remarks
The Wigner distribution and the ambiguity function
The Wigner distribution
The ambiguity function
Cohen's class of shift-invariant distributions
Smoothing of the Wigner distribution
Effect of linear systems on the Wigner distribution
Time-frequency filtering
Wigner distribution of random signals
Wigner distribution of analytic signals
Other properties
Sampling and the number of degrees of freedom
Linear canonical transforms
Definition and properties
Effect on Wigner distributions
Special linear canonical transforms
Decompositions
Transformation of moments
Linear fractional transformations
Coordinate multiplication and differentiation operators
Uncertainty relation
Invariants and hyperdifferential forms
Differential equations
Symplectic systems
Connections to group theory
Generalization to two and higher dimensions
Further reading
The Fractional Fourier Transform
Definitions of the fractional Fourier transform
Definition A: Linear integral transform
Definition B: Fractional powers of the Fourier transform
Definition C: Rotation in the time-frequency plane
Definition D: Transformation of coordinate multiplication and differentiation operators
Definition E: Differential equation
Definition F: Hyperdifferential operator
Eigenvalues and eigenfunctions
Distinct definitions of the fractional Fourier transform
Transforms of some common functions
Properties
Rotations and projections in the time-frequency plane
Rotation of the Wigner distribution
Projections of the Wigner distribution
Other time-frequency representations
Coordinate multiplication and differentiation operators
Phase shift and translation operators
Fractional Fourier domains
Chirp bases and chirp transforms
Two-dimensional fractional Fourier transforms
Extensions and applications
Fractional Fourier transforms in braket notation
Complex-ordered fractional Fourier transforms
Relation to wavelet transforms
Application to neural networks
Chirplets and other approaches
Other fractional operations and transforms
Historical and bibliographical notes
Time-Order and Space-Order Representations
Introduction
The rectangular time-order representation
Optical implementation
The polar time-order representation
Relationships with the Wigner distribution and the ambiguity function
Applications of time-order representations
Other applications of the fractional Fourier transform in time- and space-frequency analysis
Historical and bibliographical notes
The Discrete Fractional Fourier Transform
Introduction
Discrete Hermite-Gaussian functions
The discrete fractional Fourier transform
Definition in hyperdifference form
Higher-order discrete analogs
Discussion
Discrete computation of the fractional Fourier transform
Historical and bibliographical notes
Optical Signals and Systems
Introduction
Notation and conventions
Wave optics
The wave equation
Plane wave decomposition
The paraxial wave equation
Hermite-Gaussian beams
Wave-optical characterization of optical components
Sections of free space
Thin lenses
Quadratic graded-index media
Extensions
Spatial filters
Fourier-domain spatial filters
General linear systems
Spherical reference surfaces
Remarks
Geometrical optics
The ray equation
Fermat's principle and the eikonal equation
Hamilton's equations
Geometrical-optical characterization of optical components
Sections of free space
Thin lenses
Quadratic graded-index media
Extensions
Spatial filters
Fourier-domain spatial filters
General linear systems
Spherical reference surfaces
Remarks
Partially coherent light
Fourier optical systems
Further reading
Phase-Space Optics
Wave-optical and geometrical-optical phase spaces
Quadratic-phase systems and linear canonical transforms
Optical components
Sections of free space
Thin lenses
Quadratic graded-index media
Extensions
Spatial filters
Fourier-domain spatial filters
General linear systems
Spherical reference surfaces
Discussion
Imaging and Fourier transformation
Imaging systems
Fourier transforming systems
General theorems for image and Fourier transform planes
Decompositions and duality in optics
Relations between wave and geometrical optics
Phase of the system kernel and Hamilton's point characteristic
Transport equations for the Wigner distribution
Discussion
Quadratic-exponential signals
Ray-like signals
Complex Gaussian signals
Optical invariants
Invariance of density and area in phase space
The symplectic condition and canonical transformations
The Lagrange invariant
The Smith-Helmholtz invariant and Abbe's sine condition
The constant brightness theorem
The unit-determinant condition for inhomogeneous media
Poisson brackets
The number of degrees of freedom
Partially coherent light
Further reading
The Fractional Fourier Transform in Optics
Applications of the transform to wave and beam propagation
Overview
Quadratic-phase systems as fractional Fourier transforms
Quadratic graded-index media
Fresnel diffraction
Multi-lens systems
Optical implementation of the fractional Fourier transform
Hermite-Gaussian expansion approach
General fractional Fourier transform relations in free space
The fractional Fourier transform and Fresnel's integral
Analysis
Synthesis
Propagation
Discussion
Illustrative applications
Fresnel diffraction as fractional Fourier transformation
The symmetric case
Fractional Fourier transform between planar surfaces
Classical single-lens imaging
Multi-lens systems as consecutive fractional Fourier transforms
General fractional Fourier transform relations for quadratic-phase systems
Fractional Fourier transformation in quadratic graded-index media
Propagation in quadratic-index media as fractional Fourier transformation
Analogy with the simple harmonic oscillator
Quadratic graded-index media as the limit of multi-lens systems
Gaussian beams through quadratic graded-index media
Hermite-Gaussian expansion approach
The fractional Fourier order and the Gouy phase shift
Spherical mirror resonators and stability
First-order optical systems
Quadratic-phase systems as fractional Fourier transforms
Geometrical-optical determination of fractional Fourier transform parameters
Differential equations for the fractional Fourier transform parameters
Fractional Fourier transform parameters and Gaussian beam parameters
Discussion
Fourier optical systems
Locations of fractional Fourier transform planes
Wave field reconstruction, phase retrieval, and phase-space tomography
Extensions and applications
Temporal optical implementation of the transform
Digital optical implementation of the transform
Optical implementation of two-dimensional transforms
Optical interpretation and implementation of complex-ordered transforms
Incoherent optical implementation of the transform
Applications to systems with partially coherent light
Other applications of the transform in optics
Practical considerations for implementing the transform
Other fractional operations and effects in optics
Historical and bibliographical notes
Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery
Introduction
Optimal Wiener filtering in fractional Fourier domains
Multistage, multichannel, and generalized filtering configurations
Introduction
Cost-performance trade-off
Extensions and generalizations
Applications of fractional Fourier domain filtering
Elementary signal separation examples
Optical signal separation
System and transform synthesis
Signal recovery and restoration
Signal synthesis
Free-space optical interconnection architectures
Convolution and filtering in fractional Fourier domains
Convolution and multiplication in fractional Fourier domains
Compaction in fractional Fourier domains
Filtering in fractional Fourier domains
Derivation of the optimal fractional Fourier domain filter
Continuous time
Discrete time
Optimization and cost analysis of multistage and multichannel filtering configurations
Determination of the optimal filters
Rectangular system matrices
Cost analysis
The fractional Fourier domain decomposition
Introduction and definition
Construction of the fractional Fourier domain decomposition
Pruning and sparsening
Repeated filtering in the ordinary time and frequency domains
Multiplexing in fractional Fourier domains
Historical and bibliographical notes
Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition
Introduction
Fractional correlation
Controllable shift invariance
Performance measures for fractional correlation
Fractional power filters
Performance measures and optimal filters
Optimal filters for fractional correlation
Fractional joint-transform correlators
Adaptive windowed fractional Fourier transforms
Time- or space-dependent windowed transforms
Applications
Applications with different orders in the two dimensions
Historical and bibliographical notes
Bibliography on the Fractional Fourier Transform
Other Cited Works
Credits
Index