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