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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 | |