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The Fractional Fourier Transform With Applications in Optics and Signal Processing

Haldun M. Ozaktas, Zeev Zalevsky, M. Alper Kutay
Format
Book
Published
Chichester [West Sussex, England] ; New York : J. Wiley, c2001.
Language
English
Series
Wiley Series in Pure and Applied Optics
ISBN
0471963461 (acid-free paper)
Contents
  • 1.1 Fractional operations and the fractional Fourier transform 1
  • 1.2 Applications of the fractional Fourier transform 4
  • 2 Signals, Systems, and Transformations 7
  • 2.1 Signals 7
  • 2.1.3 Some commonly used functions 8
  • 2.1.4 Analytic signals and the Hilbert transform 9
  • 2.1.5 Signal spaces 10
  • 2.2 Systems 11
  • 2.2.2 Linearity and superposition integrals 12
  • 2.2.3 Some special linear systems 12
  • 2.2.4 Shift invariance and convolution 13
  • 2.3 Representations and transformations 15
  • 2.3.1 Systems versus transformations 15
  • 2.3.2 Basis sets and representations 16
  • 2.3.3 Impulse and harmonic bases 20
  • 2.3.4 Transformations between representations 21
  • 2.4 Operators 24
  • 2.4.2 Eigenvalue equations 28
  • 2.4.3 Diagonalization and spectral expansion 29
  • 2.4.4 Functions of operators 32
  • 2.5 Fourier transform 34
  • 2.5.1 Definition and properties 34
  • 2.5.2 Eigenfunctions of the Fourier transform 38
  • 2.6 Some important operators 42
  • 2.6.1 Coordinate multiplication and differentiation operators 42
  • 2.6.2 Phase shift, translation, chirp multiplication, and chirp convolution operators 46
  • 2.6.3 Annihilation and creation operators 49
  • 2.7 Uncertainty relations 50
  • 2.8 Random processes 52
  • 2.8.1 Fundamental definitions 52
  • 2.8.2 Power spectral density 53
  • 2.8.3 Linear systems with random inputs 54
  • 2.9 Generalization to two dimensions 54
  • 2.10 Some additional definitions and results 56
  • 2.10.1 Radon transform and projection-slice theorem 56
  • 2.10.2 Complex exponential integrals 57
  • 2.10.3 Stationary-phase integral 58
  • 2.10.4 Schwarz's inequality 58
  • 2.12 Appendix: Vector spaces and function spaces 58
  • 2.12.2 Inner products and norms 59
  • 3 Wigner Distributions and Linear Canonical Transforms 63
  • 3.1 Time-frequency and space-frequency representations 63
  • 3.1.1 Short-time or windowed Fourier transform 63
  • 3.1.2 Gabor expansion 65
  • 3.1.3 Wavelet transforms 67
  • 3.2 Wigner distribution and the ambiguity function 69
  • 3.2.1 Wigner distribution 69
  • 3.2.2 Ambiguity function 73
  • 3.2.3 Cohen's class of shift-invariant distributions 76
  • 3.2.4 Smoothing of the Wigner distribution 78
  • 3.2.5 Effect of linear systems on the Wigner distribution 81
  • 3.2.6 Time-frequency filtering 82
  • 3.2.7 Wigner distribution of random signals 85
  • 3.2.8 Wigner distribution of analytic signals 86
  • 3.3 Sampling and the number of degrees of freedom 86
  • 3.4 Linear canonical transforms 93
  • 3.4.1 Definition and properties 93
  • 3.4.2 Effect on Wigner distributions 96
  • 3.4.3 Special linear canonical transforms 99
  • 3.4.4 Decompositions 104
  • 3.4.5 Transformation of moments 107
  • 3.4.6 Linear fractional transformations 108
  • 3.4.7 Coordinate multiplication and differentiation operators 110
  • 3.4.8 Uncertainty relation 111
  • 3.4.9 Invariants and hyperdifferential forms 111
  • 3.4.10 Differential equations 112
  • 3.4.11 Symplectic systems 113
  • 3.4.12 Connections to group theory 114
  • 3.5 Generalization to two and higher dimensions 115
  • 4 Fractional Fourier Transform 117
  • 4.1 Definitions of the fractional Fourier transform 117
  • 4.1.1 Definition A: Linear integral transform 118
  • 4.1.2 Definition B: Fractional powers of the Fourier transform 122
  • 4.1.3 Definition C: Rotation in the time-frequency plane 124
  • 4.1.4 Definition D: Transformation of coordinate multiplication and differentiation operators 126
  • 4.1.5 Definition E: Differential equation 129
  • 4.1.6 Definition F: Hyperdifferential operator 132
  • 4.2 Eigenvalues and eigenfunctions 137
  • 4.3 Distinct definitions of the fractional Fourier transform 139
  • 4.4 Transforms of some common functions 143
  • 4.5 Properties 152
  • 4.6 Rotations and projections in thez time-frequency plane 159
  • 4.6.1 Rotation of the Wigner distribution 159
  • 4.6.2 Projections of the Wigner distribution 160
  • 4.6.3 Other time-frequency representations 162
  • 4.7 Coordinate multiplication and differentiation operators 166
  • 4.8 Phase shift and translation operators 169
  • 4.9 Fractional Fourier domains 171
  • 4.10 Chirp bases and chirp transforms 173
  • 4.11 Two-dimensional fractional Fourier transforms 175
  • 4.12 Extensions and applications 180
  • 4.12.1 Fractional Fourier transforms in braket notation 180
  • 4.12.2 Complex-ordered fractional Fourier transforms 181
  • 4.12.3 Relation to wavelet transforms 181
  • 4.12.4 Application to neural networks 181
  • 4.12.5 Chirplets and other approaches 182
  • 4.12.6 Other fractional operations and transforms 182
  • 4.13 Historical and bibliographical notes 183
  • 5 Time-Order and Space-Order Representations 187
  • 5.2 Rectangular time-order representation 187
  • 5.3 Optical implementation 189
  • 5.4 Polar time-order representation 190
  • 5.5 Relationships with the Wigner distribution and the ambiguity function 194
  • 5.6 Applications of time-order representations 197
  • 5.7 Other applications of the fractional Fourier transform in time- and space-frequency analysis 199
  • 5.8 Historical and bibliographical notes 200
  • 6 Discrete Fractional Fourier Transform 201
  • 6.2 Discrete Hermite-Gaussian functions 202
  • 6.3 Discrete fractional Fourier transform 210
  • 6.4 Definition in hyperdifference form 213
  • 6.5 Higher-order discrete analogs 215
  • 6.7 Discrete computation of the fractional Fourier transform 218
  • 6.8 Historical and bibliographical notes 220
  • 7 Optical Signals and Systems 223
  • 7.3 Wave optics 227
  • 7.3.1 Wave equation 228
  • 7.3.2 Plane wave decomposition 231
  • 7.3.3 Paraxial wave equation 233
  • 7.3.4 Hermite-Gaussian beams 235
  • 7.4 Wave-optical characterization of optical components 238
  • 7.4.1 Sections of free space 238
  • 7.4.2 Thin lenses 239
  • 7.4.3 Quadratic graded-index media 240
  • 7.4.4 Extensions 244
  • 7.4.5 Spatial filters 245
  • 7.4.6 Fourier-domain spatial filters 246
  • 7.4.7 General linear systems 247
  • 7.4.8 Spherical reference surfaces 247
  • 7.4.9 Remarks 248
  • 7.5 Geometrical optics 248
  • 7.5.1 Ray equation 249
  • 7.5.2 Fermat's principle and the eikonal equation 250
  • 7.5.3 Hamilton's equations 252
  • 7.6 Geometrical-optical characterization of optical components 254
  • 7.6.1 Sections of free space 254
  • 7.6.2 Thin lenses 255
  • 7.6.3 Quadratic graded-index media 256
  • 7.6.4 Extensions 256
  • 7.6.5 Spatial filters 257
  • 7.6.6 Fourier-domain spatial filters 257
  • 7.6.7 General linear systems 259
  • 7.6.8 Spherical reference surfaces 259
  • 7.6.9 Remarks 259
  • 7.7 Partially coherent light 259
  • 7.8 Fourier optical systems 260
  • 8 Phase-Space Optics 265
  • 8.1 Wave-optical and geometrical-optical phase spaces 265
  • 8.2 Quadratic-phase systems and linear canonical transforms 269
  • 8.3 Optical components 270
  • 8.3.1 Sections of free space 272
  • 8.3.2 Thin lenses 274
  • 8.3.3 Quadratic graded-index media 275
  • 8.3.4 Extensions 277
  • 8.3.5 Spatial filters 277
  • 8.3.6 Fourier-domain spatial filters 278
  • 8.3.7 General linear systems 280
  • 8.3.8 Spherical reference surfaces 280
  • 8.4 Imaging and Fourier transformation 282
  • 8.4.1 Imaging systems 282
  • 8.4.2 Fourier transforming systems 284
  • 8.4.3 General theorems for image and Fourier transform planes 287
  • 8.5 Decompositions and duality in optics 294
  • 8.6 Relations between wave and geometrical optics 297
  • 8.6.1 Phase of the system kernel and Hamilton's point characteristic 297
  • 8.6.2 Transport equations for the Wigner distribution 299
  • 8.7 Quadratic-exponential signals 302
  • 8.7.1 Ray-like signals 302
  • 8.7.2 Complex Gaussian signals 304
  • 8.8 Optical invariants 305
  • 8.8.1 Invariance of density and area in phase space 306
  • 8.8.2 Symplectic condition and canonical transformations 308
  • 8.8.3 Lagrange invariant 310
  • 8.8.4 Smith-Helmholtz invariant and Abbe's sine condition 311
  • 8.8.5 Constant brightness theorem 313
  • 8.8.6 Unit-determinant condition for inhomogeneous media 313
  • 8.8.7 Poisson brackets 314
  • 8.8.8 Number of degrees of freedom 316
  • 8.9 Partially coherent light 317
  • 9 Fractional Fourier Transform in Optics 319
  • 9.1 Applications of the transform to wave and beam propagation 319
  • 9.2.1 Quadratic-phase systems as fractional Fourier transforms 323
  • 9.2.2 Quadratic graded-index media 324
  • 9.2.3 Fresnel diffraction 325
  • 9.2.4 Multi-lens systems 326
  • 9.2.5 Optical implementation of the fractional Fourier transform 330
  • 9.2.6 Hermite-Gaussian expansion approach 331
  • 9.3 General fractional Fourier transform relations in free space 333
  • 9.3.1 Fractional Fourier transform and Fresnel's integral 333
  • 9.3.2 Analysis 335
  • 9.3.3 Synthesis 336
  • 9.3.4 Propagation 336
  • 9.4 Illustrative applications 337
  • 9.4.1 Fresnel diffraction as fractional Fourier transformation 338
  • 9.4.2 Symmetric case 338
  • 9.4.3 Fractional Fourier transform between planar surfaces 339
  • 9.4.4 Classical single-lens imaging 341
  • 9.4.5 Multi-lens systems as consecutive fractional Fourier transforms 343
  • 9.4.6 General fractional Fourier transform relations for quadratic-phase systems 344
  • 9.5 Fractional Fourier transformation in quadratic graded-index media 347
  • 9.5.1 Propagation in quadratic-index media as fractional Fourier transformation 347
  • 9.5.2 Analogy with the simple harmonic oscillator 348
  • -- 9.5.3 Quadratic graded-index media as the limit of multi-lens systems 350
  • 9.5.4 Gaussian beams through quadratic graded-index media 353
  • 9.6 Hermite-Gaussian expansion approach 354
  • 9.6.1 Fractional Fourier order and the Gouy phase shift 354
  • 9.6.2 Spherical mirror resonators and stability 356
  • 9.7 First-order optical systems 359
  • 9.7.1 Quadratic-phase systems as fractional Fourier transforms 359
  • 9.7.2 Geometrical-optical determination of fractional Fourier transform parameters 361
  • 9.7.3 Differential equations for the fractional Fourier transform parameters 365
  • 9.7.4 Fractional Fourier transform parameters and Gaussian beam parameters 369
  • 9.8 Fourier optical systems 372
  • 9.9 Locations of fractional Fourier transform planes 377
  • 9.10 Wave field reconstruction, phase retrieval, and phase-space tomography 378
  • 9.11 Extensions and applications 381
  • 9.11.1 Temporal optical implementation of the transform 381
  • 9.11.2 Digital optical implementation of the transform 381
  • 9.11.3 Optical implementation of two-dimensional transforms 381
  • 9.11.4 Optical interpretation and implementation of complex-ordered transforms 382
  • 9.11.5 Incoherent optical implementation of the transform 382
  • 9.11.6 Applications to systems with partially coherent light 382
  • 9.11.7 Other applications of the transform in optics 383
  • 9.11.8 Practical considerations for implementing the transform 383
  • 9.11.9 Other fractional operations and effects in optics 384
  • 9.12 Historical and bibliographical notes 385
  • 10 Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery 387
  • 10.2 Optimal Wiener filtering in fractional Fourier domains 388
  • 10.3 Multistage, multichannel, and generalized filtering configurations 392
  • 10.3.2 Cost-performance trade-off 394
  • 10.3.3 Extensions and generalizations 397
  • 10.4 Applications of fractional Fourier domain filtering 398
  • 10.4.1 Elementary signal separation examples 399
  • 10.4.2 Optical signal separation 401
  • 10.4.3 System and transform synthesis 403
  • 10.4.4 Signal recovery and restoration 403
  • 10.4.5 Signal synthesis 418
  • 10.4.6 Free-space optical interconnection architectures 418
  • 10.5 Convolution and filtering in fractional Fourier domains 419
  • 10.5.1 Convolution and multiplication in fractional Fourier domains 419
  • 10.5.2 Compaction in fractional Fourier domains 421
  • 10.5.3 Filtering in fractional Fourier domains 422
  • 10.6 Derivation of the optimal fractional Fourier domain filter 424
  • 10.6.1 Continuous time 425
  • 10.6.2 Discrete time 427
  • 10.7 Optimization and cost analysis of multistage and multichannel filtering configurations 428
  • 10.7.1 Determination of the optimal filters 429
  • 10.7.2 Rectangular system matrices 430
  • 10.7.3 Cost analysis 431
  • 10.8 Fractional Fourier domain decomposition 432
  • 10.8.1 Introduction and definition 432
  • 10.8.2 Construction of the fractional Fourier domain decomposition 434
  • 10.8.3 Pruning and sparsening 435
  • 10.9 Repeated filtering in the ordinary time and frequency domains 437
  • 10.10 Multiplexing in fractional Fourier domains 439
  • 10.11 Historical and bibliographical notes 441
  • 11 Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition 443
  • 11.2 Fractional correlation 443
  • 11.3 Controllable shift invariance 447
  • 11.4 Performance measures for fractional correlation 450
  • 11.4.1 Fractional power filters 450
  • 11.4.2 Performance measures and optimal filters 451
  • 11.4.3 Optimal filters for fractional correlation 452
  • 11.5 Fractional joint-transform correlators 454
  • 11.6 Adaptive windowed fractional Fourier transforms 455
  • 11.6.1 Time- or space-dependent windowed transforms 455
  • 11.6.2 Applicationsl 457
  • 11.7 Applications with different orders in the two dimensions 460
  • 11.8 Historical and bibliographical notes 464
  • Bibliography on the Fractional Fourier Transform 467.
Description
xviii, 513 p. : ill. ; 25 cm.
Notes
Includes bibliographical references (p. [467]-496) and index.
Technical Details
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    g| 1.1 t| Fractional operations and the fractional Fourier transform g| 1 -- g| 1.2 t| Applications of the fractional Fourier transform g| 4 -- g| 2 t| Signals, Systems, and Transformations g| 7 -- g| 2.1 t| Signals g| 7 -- g| 2.1.3 t| Some commonly used functions g| 8 -- g| 2.1.4 t| Analytic signals and the Hilbert transform g| 9 -- g| 2.1.5 t| Signal spaces g| 10 -- g| 2.2 t| Systems g| 11 -- g| 2.2.2 t| Linearity and superposition integrals g| 12 -- g| 2.2.3 t| Some special linear systems g| 12 -- g| 2.2.4 t| Shift invariance and convolution g| 13 -- g| 2.3 t| Representations and transformations g| 15 -- g| 2.3.1 t| Systems versus transformations g| 15 -- g| 2.3.2 t| Basis sets and representations g| 16 -- g| 2.3.3 t| Impulse and harmonic bases g| 20 -- g| 2.3.4 t| Transformations between representations g| 21 -- g| 2.4 t| Operators g| 24 -- g| 2.4.2 t| Eigenvalue equations g| 28 -- g| 2.4.3 t| Diagonalization and spectral expansion g| 29 -- g| 2.4.4 t| Functions of operators g| 32 -- g| 2.5 t| Fourier transform g| 34 -- g| 2.5.1 t| Definition and properties g| 34 -- g| 2.5.2 t| Eigenfunctions of the Fourier transform g| 38 -- g| 2.6 t| Some important operators g| 42 -- g| 2.6.1 t| Coordinate multiplication and differentiation operators g| 42 -- g| 2.6.2 t| Phase shift, translation, chirp multiplication, and chirp convolution operators g| 46 -- g| 2.6.3 t| Annihilation and creation operators g| 49 -- g| 2.7 t| Uncertainty relations g| 50 -- g| 2.8 t| Random processes g| 52 -- g| 2.8.1 t| Fundamental definitions g| 52 -- g| 2.8.2 t| Power spectral density g| 53 -- g| 2.8.3 t| Linear systems with random inputs g| 54 -- g| 2.9 t| Generalization to two dimensions g| 54 -- g| 2.10 t| Some additional definitions and results g| 56 -- g| 2.10.1 t| Radon transform and projection-slice theorem g| 56 -- g| 2.10.2 t| Complex exponential integrals g| 57 -- g| 2.10.3 t| Stationary-phase integral g| 58 -- g| 2.10.4 t| Schwarz's inequality g| 58 -- g| 2.12 t| Appendix: Vector spaces and function spaces g| 58 -- g| 2.12.2 t| Inner products and norms g| 59 -- g| 3 t| Wigner Distributions and Linear Canonical Transforms g| 63 -- g| 3.1 t| Time-frequency and space-frequency representations g| 63 -- g| 3.1.1 t| Short-time or windowed Fourier transform g| 63 -- g| 3.1.2 t| Gabor expansion g| 65 -- g| 3.1.3 t| Wavelet transforms g| 67 -- g| 3.2 t| Wigner distribution and the ambiguity function g| 69 -- g| 3.2.1 t| Wigner distribution g| 69 -- g| 3.2.2 t| Ambiguity function g| 73 -- g| 3.2.3 t| Cohen's class of shift-invariant distributions g| 76 -- g| 3.2.4 t| Smoothing of the Wigner distribution g| 78 -- g| 3.2.5 t| Effect of linear systems on the Wigner distribution g| 81 -- g| 3.2.6 t| Time-frequency filtering g| 82 -- g| 3.2.7 t| Wigner distribution of random signals g| 85 -- g| 3.2.8 t| Wigner distribution of analytic signals g| 86 -- g| 3.3 t| Sampling and the number of degrees of freedom g| 86 -- g| 3.4 t| Linear canonical transforms g| 93 -- g| 3.4.1 t| Definition and properties g| 93 -- g| 3.4.2 t| Effect on Wigner distributions g| 96 -- g| 3.4.3 t| Special linear canonical transforms g| 99 -- g| 3.4.4 t| Decompositions g| 104 -- g| 3.4.5 t| Transformation of moments g| 107 -- g| 3.4.6 t| Linear fractional transformations g| 108 -- g| 3.4.7 t| Coordinate multiplication and differentiation operators g| 110 -- g| 3.4.8 t| Uncertainty relation g| 111 -- g| 3.4.9 t| Invariants and hyperdifferential forms g| 111 -- g| 3.4.10 t| Differential equations g| 112 -- g| 3.4.11 t| Symplectic systems g| 113 -- g| 3.4.12 t| Connections to group theory g| 114 -- g| 3.5 t| Generalization to two and higher dimensions g| 115 -- g| 4 t| Fractional Fourier Transform g| 117 -- g| 4.1 t| Definitions of the fractional Fourier transform g| 117 -- g| 4.1.1 t| Definition A: Linear integral transform g| 118 -- g| 4.1.2 t| Definition B: Fractional powers of the Fourier transform g| 122 -- g| 4.1.3 t| Definition C: Rotation in the time-frequency plane g| 124 -- g| 4.1.4 t| Definition D: Transformation of coordinate multiplication and differentiation operators g| 126 -- g| 4.1.5 t| Definition E: Differential equation g| 129 -- g| 4.1.6 t| Definition F: Hyperdifferential operator g| 132 -- g| 4.2 t| Eigenvalues and eigenfunctions g| 137 -- g| 4.3 t| Distinct definitions of the fractional Fourier transform g| 139 -- g| 4.4 t| Transforms of some common functions g| 143 -- g| 4.5 t| Properties g| 152 -- g| 4.6 t| Rotations and projections in thez time-frequency plane g| 159 -- g| 4.6.1 t| Rotation of the Wigner distribution g| 159 -- g| 4.6.2 t| Projections of the Wigner distribution g| 160 -- g| 4.6.3 t| Other time-frequency representations g| 162 -- g| 4.7 t| Coordinate multiplication and differentiation operators g| 166 -- g| 4.8 t| Phase shift and translation operators g| 169 -- g| 4.9 t| Fractional Fourier domains g| 171 -- g| 4.10 t| Chirp bases and chirp transforms g| 173 -- g| 4.11 t| Two-dimensional fractional Fourier transforms g| 175 -- g| 4.12 t| Extensions and applications g| 180 -- g| 4.12.1 t| Fractional Fourier transforms in braket notation g| 180 -- g| 4.12.2 t| Complex-ordered fractional Fourier transforms g| 181 -- g| 4.12.3 t| Relation to wavelet transforms g| 181 -- g| 4.12.4 t| Application to neural networks g| 181 -- g| 4.12.5 t| Chirplets and other approaches g| 182 -- g| 4.12.6 t| Other fractional operations and transforms g| 182 -- g| 4.13 t| Historical and bibliographical notes g| 183 -- g| 5 t| Time-Order and Space-Order Representations g| 187 -- g| 5.2 t| Rectangular time-order representation g| 187 -- g| 5.3 t| Optical implementation g| 189 -- g| 5.4 t| Polar time-order representation g| 190 -- g| 5.5 t| Relationships with the Wigner distribution and the ambiguity function g| 194 -- g| 5.6 t| Applications of time-order representations g| 197 -- g| 5.7 t| Other applications of the fractional Fourier transform in time- and space-frequency analysis g| 199 -- g| 5.8 t| Historical and bibliographical notes g| 200 -- g| 6 t| Discrete Fractional Fourier Transform g| 201 -- g| 6.2 t| Discrete Hermite-Gaussian functions g| 202 -- g| 6.3 t| Discrete fractional Fourier transform g| 210 -- g| 6.4 t| Definition in hyperdifference form g| 213 -- g| 6.5 t| Higher-order discrete analogs g| 215 -- g| 6.7 t| Discrete computation of the fractional Fourier transform g| 218 -- g| 6.8 t| Historical and bibliographical notes g| 220 -- g| 7 t| Optical Signals and Systems g| 223 -- g| 7.3 t| Wave optics g| 227 -- g| 7.3.1 t| Wave equation g| 228 -- g| 7.3.2 t| Plane wave decomposition g| 231 -- g| 7.3.3 t| Paraxial wave equation g| 233 -- g| 7.3.4 t| Hermite-Gaussian beams g| 235 -- g| 7.4 t| Wave-optical characterization of optical components g| 238 -- g| 7.4.1 t| Sections of free space g| 238 -- g| 7.4.2 t| Thin lenses g| 239 -- g| 7.4.3 t| Quadratic graded-index media g| 240 -- g| 7.4.4 t| Extensions g| 244 -- g| 7.4.5 t| Spatial filters g| 245 -- g| 7.4.6 t| Fourier-domain spatial filters g| 246 -- g| 7.4.7 t| General linear systems g| 247 -- g| 7.4.8 t| Spherical reference surfaces g| 247 -- g| 7.4.9 t| Remarks g| 248 -- g| 7.5 t| Geometrical optics g| 248 -- g| 7.5.1 t| Ray equation g| 249 -- g| 7.5.2 t| Fermat's principle and the eikonal equation g| 250 -- g| 7.5.3 t| Hamilton's equations g| 252 -- g| 7.6 t| Geometrical-optical characterization of optical components g| 254 -- g| 7.6.1 t| Sections of free space g| 254 -- g| 7.6.2 t| Thin lenses g| 255 -- g| 7.6.3 t| Quadratic graded-index media g| 256 -- g| 7.6.4 t| Extensions g| 256 -- g| 7.6.5 t| Spatial filters g| 257 -- g| 7.6.6 t| Fourier-domain spatial filters g| 257 -- g| 7.6.7 t| General linear systems g| 259 -- g| 7.6.8 t| Spherical reference surfaces g| 259 -- g| 7.6.9 t| Remarks g| 259 -- g| 7.7 t| Partially coherent light g| 259 -- g| 7.8 t| Fourier optical systems g| 260 -- g| 8 t| Phase-Space Optics g| 265 -- g| 8.1 t| Wave-optical and geometrical-optical phase spaces g| 265 -- g| 8.2 t| Quadratic-phase systems and linear canonical transforms g| 269 -- g| 8.3 t| Optical components g| 270 -- g| 8.3.1 t| Sections of free space g| 272 -- g| 8.3.2 t| Thin lenses g| 274 -- g| 8.3.3 t| Quadratic graded-index media g| 275 -- g| 8.3.4 t| Extensions g| 277 -- g| 8.3.5 t| Spatial filters g| 277 -- g| 8.3.6 t| Fourier-domain spatial filters g| 278 -- g| 8.3.7 t| General linear systems g| 280 -- g| 8.3.8 t| Spherical reference surfaces g| 280 -- g| 8.4 t| Imaging and Fourier transformation g| 282 -- g| 8.4.1 t| Imaging systems g| 282 -- g| 8.4.2 t| Fourier transforming systems g| 284 -- g| 8.4.3 t| General theorems for image and Fourier transform planes g| 287 -- g| 8.5 t| Decompositions and duality in optics g| 294 -- g| 8.6 t| Relations between wave and geometrical optics g| 297 -- g| 8.6.1 t| Phase of the system kernel and Hamilton's point characteristic g| 297 -- g| 8.6.2 t| Transport equations for the Wigner distribution g| 299 -- g| 8.7 t| Quadratic-exponential signals g| 302 -- g| 8.7.1 t| Ray-like signals g| 302 -- g| 8.7.2 t| Complex Gaussian signals g| 304 -- g| 8.8 t| Optical invariants g| 305 -- g| 8.8.1 t| Invariance of density and area in phase space g| 306 -- g| 8.8.2 t| Symplectic condition and canonical transformations g| 308 -- g| 8.8.3 t| Lagrange invariant g| 310 -- g| 8.8.4 t| Smith-Helmholtz invariant and Abbe's sine condition g| 311 -- g| 8.8.5 t| Constant brightness theorem g| 313 -- g| 8.8.6 t| Unit-determinant condition for inhomogeneous media g| 313 -- g| 8.8.7 t| Poisson brackets g| 314 -- g| 8.8.8 t| Number of degrees of freedom g| 316 -- g| 8.9 t| Partially coherent light g| 317 -- g| 9 t| Fractional Fourier Transform in Optics g| 319 -- g| 9.1 t| Applications of the transform to wave and beam propagation g| 319 -- g| 9.2.1 t| Quadratic-phase systems as fractional Fourier transforms g| 323 -- g| 9.2.2 t| Quadratic graded-index media g| 324 -- g| 9.2.3 t| Fresnel diffraction g| 325 -- g| 9.2.4 t| Multi-lens systems g| 326 -- g| 9.2.5 t| Optical implementation of the fractional Fourier transform g| 330 -- g| 9.2.6 t| Hermite-Gaussian expansion approach g| 331 -- g| 9.3 t| General fractional Fourier transform relations in free space g| 333 -- g| 9.3.1 t| Fractional Fourier transform and Fresnel's integral g| 333 -- g| 9.3.2 t| Analysis g| 335 -- g| 9.3.3 t| Synthesis g| 336 -- g| 9.3.4 t| Propagation g| 336 -- g| 9.4 t| Illustrative applications g| 337 -- g| 9.4.1 t| Fresnel diffraction as fractional Fourier transformation g| 338 -- g| 9.4.2 t| Symmetric case g| 338 -- g| 9.4.3 t| Fractional Fourier transform between planar surfaces g| 339 -- g| 9.4.4 t| Classical single-lens imaging g| 341 -- g| 9.4.5 t| Multi-lens systems as consecutive fractional Fourier transforms g| 343 -- g| 9.4.6 t| General fractional Fourier transform relations for quadratic-phase systems g| 344 -- g| 9.5 t| Fractional Fourier transformation in quadratic graded-index media g| 347 -- g| 9.5.1 t| Propagation in quadratic-index media as fractional Fourier transformation g| 347 -- g| 9.5.2 t| Analogy with the simple harmonic oscillator g| 348 --
    505
    8
    0
    g| 9.5.3 t| Quadratic graded-index media as the limit of multi-lens systems g| 350 -- g| 9.5.4 t| Gaussian beams through quadratic graded-index media g| 353 -- g| 9.6 t| Hermite-Gaussian expansion approach g| 354 -- g| 9.6.1 t| Fractional Fourier order and the Gouy phase shift g| 354 -- g| 9.6.2 t| Spherical mirror resonators and stability g| 356 -- g| 9.7 t| First-order optical systems g| 359 -- g| 9.7.1 t| Quadratic-phase systems as fractional Fourier transforms g| 359 -- g| 9.7.2 t| Geometrical-optical determination of fractional Fourier transform parameters g| 361 -- g| 9.7.3 t| Differential equations for the fractional Fourier transform parameters g| 365 -- g| 9.7.4 t| Fractional Fourier transform parameters and Gaussian beam parameters g| 369 -- g| 9.8 t| Fourier optical systems g| 372 -- g| 9.9 t| Locations of fractional Fourier transform planes g| 377 -- g| 9.10 t| Wave field reconstruction, phase retrieval, and phase-space tomography g| 378 -- g| 9.11 t| Extensions and applications g| 381 -- g| 9.11.1 t| Temporal optical implementation of the transform g| 381 -- g| 9.11.2 t| Digital optical implementation of the transform g| 381 -- g| 9.11.3 t| Optical implementation of two-dimensional transforms g| 381 -- g| 9.11.4 t| Optical interpretation and implementation of complex-ordered transforms g| 382 -- g| 9.11.5 t| Incoherent optical implementation of the transform g| 382 -- g| 9.11.6 t| Applications to systems with partially coherent light g| 382 -- g| 9.11.7 t| Other applications of the transform in optics g| 383 -- g| 9.11.8 t| Practical considerations for implementing the transform g| 383 -- g| 9.11.9 t| Other fractional operations and effects in optics g| 384 -- g| 9.12 t| Historical and bibliographical notes g| 385 -- g| 10 t| Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery g| 387 -- g| 10.2 t| Optimal Wiener filtering in fractional Fourier domains g| 388 -- g| 10.3 t| Multistage, multichannel, and generalized filtering configurations g| 392 -- g| 10.3.2 t| Cost-performance trade-off g| 394 -- g| 10.3.3 t| Extensions and generalizations g| 397 -- g| 10.4 t| Applications of fractional Fourier domain filtering g| 398 -- g| 10.4.1 t| Elementary signal separation examples g| 399 -- g| 10.4.2 t| Optical signal separation g| 401 -- g| 10.4.3 t| System and transform synthesis g| 403 -- g| 10.4.4 t| Signal recovery and restoration g| 403 -- g| 10.4.5 t| Signal synthesis g| 418 -- g| 10.4.6 t| Free-space optical interconnection architectures g| 418 -- g| 10.5 t| Convolution and filtering in fractional Fourier domains g| 419 -- g| 10.5.1 t| Convolution and multiplication in fractional Fourier domains g| 419 -- g| 10.5.2 t| Compaction in fractional Fourier domains g| 421 -- g| 10.5.3 t| Filtering in fractional Fourier domains g| 422 -- g| 10.6 t| Derivation of the optimal fractional Fourier domain filter g| 424 -- g| 10.6.1 t| Continuous time g| 425 -- g| 10.6.2 t| Discrete time g| 427 -- g| 10.7 t| Optimization and cost analysis of multistage and multichannel filtering configurations g| 428 -- g| 10.7.1 t| Determination of the optimal filters g| 429 -- g| 10.7.2 t| Rectangular system matrices g| 430 -- g| 10.7.3 t| Cost analysis g| 431 -- g| 10.8 t| Fractional Fourier domain decomposition g| 432 -- g| 10.8.1 t| Introduction and definition g| 432 -- g| 10.8.2 t| Construction of the fractional Fourier domain decomposition g| 434 -- g| 10.8.3 t| Pruning and sparsening g| 435 -- g| 10.9 t| Repeated filtering in the ordinary time and frequency domains g| 437 -- g| 10.10 t| Multiplexing in fractional Fourier domains g| 439 -- g| 10.11 t| Historical and bibliographical notes g| 441 -- g| 11 t| Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition g| 443 -- g| 11.2 t| Fractional correlation g| 443 -- g| 11.3 t| Controllable shift invariance g| 447 -- g| 11.4 t| Performance measures for fractional correlation g| 450 -- g| 11.4.1 t| Fractional power filters g| 450 -- g| 11.4.2 t| Performance measures and optimal filters g| 451 -- g| 11.4.3 t| Optimal filters for fractional correlation g| 452 -- g| 11.5 t| Fractional joint-transform correlators g| 454 -- g| 11.6 t| Adaptive windowed fractional Fourier transforms g| 455 -- g| 11.6.1 t| Time- or space-dependent windowed transforms g| 455 -- g| 11.6.2 t| Applicationsl g| 457 -- g| 11.7 t| Applications with different orders in the two dimensions g| 460 -- g| 11.8 t| Historical and bibliographical notes g| 464 -- t| Bibliography on the Fractional Fourier Transform g| 467.
    596
      
      
    a| 5
    650
      
    0
    a| Fourier transformations.
    650
      
    0
    a| Fourier transform optics.
    650
      
    0
    a| Control theory.
    700
    1
      
    a| Kutay, M. Alper.
    700
    1
      
    a| Zalevsky, Zeev.
    999
      
      
    a| QC20.7 .F67 O93 2001 w| LC i| X004474071 k| CHECKEDOUT l| STACKS m| SCI-ENG t| BOOK
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